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[
"Holographic Renormalization of general dilaton-axion gravity",
"Holographic Renormalization of general dilaton-axion gravity"
] |
[
"Ioannis Papadimitriou [email protected] \nDepartment of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland\n"
] |
[
"Department of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland"
] |
[] |
We consider a very general dilaton-axion system coupled to Einstein-Hilbert gravity in arbitrary dimension and we carry out holographic renormalization for any dimension up to and including five dimensions. This is achieved by developing a new systematic algorithm for iteratively solving the radial Hamilton-Jacobi equation in a derivative expansion. The boundary term derived is valid not only for asymptotically AdS backgrounds, but also for more general asymptotics, including non-conformal branes and Improved Holographic QCD. In the second half of the paper, we apply the general result to Improved Holographic QCD with arbitrary dilaton potential. In particular, we derive the generalized Fefferman-Graham asymptotic expansions and provide a proof of the holographic Ward identities.
|
10.1007/jhep08(2011)119
|
[
"https://arxiv.org/pdf/1106.4826v3.pdf"
] | 53,572,374 |
1106.4826
|
a9ba90ae7e5134d1c01dc7db8bfc848b013d611d
|
Holographic Renormalization of general dilaton-axion gravity
23 Aug 2011
Ioannis Papadimitriou [email protected]
Department of Physics
CERN -Theory Division
CH-1211Geneva 23Switzerland
Holographic Renormalization of general dilaton-axion gravity
23 Aug 2011
We consider a very general dilaton-axion system coupled to Einstein-Hilbert gravity in arbitrary dimension and we carry out holographic renormalization for any dimension up to and including five dimensions. This is achieved by developing a new systematic algorithm for iteratively solving the radial Hamilton-Jacobi equation in a derivative expansion. The boundary term derived is valid not only for asymptotically AdS backgrounds, but also for more general asymptotics, including non-conformal branes and Improved Holographic QCD. In the second half of the paper, we apply the general result to Improved Holographic QCD with arbitrary dilaton potential. In particular, we derive the generalized Fefferman-Graham asymptotic expansions and provide a proof of the holographic Ward identities.
Introduction
Understanding the holographic dictionary for holographic models in non asymptotically AdS spaces has been a long standing problem. It has been a pressing question ever since physically promising holographic dualities involving non asymptotically AdS backgrounds, such as the Klebanov-Strassler [1] and Maldacena-Núñez [2] backgrounds, were found, but it has become even more relevant with the recent interest in the phenomenological application of holography to condensed matter physics and models of QCD. Even though numerous attempts have been made to understand aspects of the dictionary of some of these systems, there are very few cases where a systematic and extensive understanding has been achieved for non asymptotically AdS backgrounds. These include the analyses of the dictionary for non-conformal branes [3,4] and for Schrödinger backgrounds [5].
However, it cannot be overemphasized that the process of understanding the holographic dictionary, in any holographic model and any background, even beyond the supergravity approximation, can be split into two conceptually distinct steps. The first step is intrinsically related with the "bulk" holographic model. Namely, one must identify a suitable boundary in the bulk theory and construct a reduced phase space of the theory in terms of data on that boundary. This step is exactly analogous to the Fefferman-Graham reconstruction of the bulk geometry in asymptotically hyperbolic manifolds from boundary data [6]. A systematic way of addressing this question and its connection to a certain variational problem at infinity in the most general setting was discussed in [7]. The approach developed in [7] in principle allows one to algorithmically construct this reduced phase space for any bulk model. Having completed this step, one not only has achieved a reformulation of the bulk dynamics in terms of a symplectic space of boundary data that can unambiguously be identified with the symplectic space of renormalized observables in any holographically dual theory, but also has automatically made the variational problem of the bulk theory well defined, which implies that the on-shell action is finite [8,7].
Only once this first step has been completed, one can directly compare the symplectic space of boundary data with the symplectic space of gauge-invariant observables in any candidate holographic dual. This mapping is simply the classical version of the Hilbert space isomorphism one expects in a fully quantum mechanical holographic duality. On the bulk side one has the symplectic space of a classical system, being that classical strings or classical gravity, while on the field theory side the Hilbert space reduces to a classical symplectic space in some limit where the number of degrees of freedom becomes infinite [9].
In this paper we consider a generic dilaton-axion system coupled to Einstein-Hilbert gravity in arbitrary dimension with the action (2.1). This system contains the standard dilaton-axion system in AdS 5 dual to the complexified coupling of N = 4 super Yang-Mills in four dimensions, nonconformal branes [3,4], as well as Improved Holographic QCD [10] as special cases. The last two examples admit non asymptotically AdS vacua, and so the standard dictionary for asymptotically AdS gravity is not applicable. Our aim here will be to carry out this two-step procedure outlined above for this general dilaton-axion system and to explore in more detail the consequences for the model of Improved Holographic QCD.
The semi-phenomenological holographic model dual to large N c Yang-Mills theory put forward in [10] (see [11] for an extensive review) is based on the five dimensional two-derivative (Euclidean) bosonic supergravity action
S = −M 3 pl N 2 cˆd 5 x √ g R[g] − ξ 2 λ −2 ∂ µ λ∂ µ λ − Z(λ)∂ µ χ∂ µ χ + V (λ) ,(1.1)
where the Planck mass M 3 pl = 1/g 2 s ℓ 3 s is related to the five dimensional Newton's constant by (16πG 5 ) −1 = M 3 pl N 2 c . The field content of this action consists of the five dimensional metric g µν , a dilaton λ, and an axion χ. These are respectively designed to describe the dynamics of the lowest dimension gauge-invariant operators of pure Yang-Mills theory, namely the stress tensor, T ij , Tr (F 2 ) and Tr (F F ). In particular, λ is proportional to the 't Hooft coupling, N c g 2 Y M , while χ is related to the instanton angle θ Y M . The constants of proportionality are not known a priori, but can be determined by comparing the perturbative UV expansion of the beta functions for the 't Hooft coupling and θ Y M with the corresponding holographic beta functions for the bulk fields λ and χ respectively. These proportionality constants are nevertheless scheme dependent and so they do not affect the value of any physical observable [11].
The holographic model is defined by the potential V (λ) and the function Z(λ), as well as the constant ξ = 0, corresponding to the normalization of the kinetic term of the dilaton. 1 Anticipating that such a model could possibly originate in a non-critical string theory in five dimensions [10], the metric and the dilaton are expected to come from the NSNS sector while the axion comes from the RR sector. As was argued in [10], this implies that the kinetic term of the axion should be O(1/N 2 c ) relative to the Einstein-Hilbert term, the dilaton kinetic term, and the scalar potential. Hence, Z(λ) = O(1/N 2 c ), while ξ and V (λ) are O(N 0 c ). The form of the functions Z(λ) and V (λ) can be constrained by physical input. Firstly, asymptotic freedom means that at the UV the theory is conformal and so the dual string vacuum should be asymptotically AdS 5 . Of course, it also means that the theory is weakly coupled at the far UV, and so a two-derivative gravity approximation of the form (1.1) cannot a priori be trusted in that limit. However, the assumption in [10] is that the effect of the higher derivative terms affecting the dynamics of the lowest lying fields can somehow be incorporated into a an effective cosmological constant in the potential V (λ), thus allowing the two-derivative action (1.1) to admit asymptotically AdS solutions. Although this argument is admittedly difficult, if not impossible, to defend from a string theoretic point of view, the attitude in [10] is that this assumption can be a posteriori justified by the success of the model in reasonably describing various qualitative properties of Yang-Mills theory. Accepting this assumption, means that the dilaton λ must vanish in the far UV and the functions V (λ) and Z(λ) must admit Taylor expansions of the form 2 V (λ) = 12
ℓ 2 1 + ∞ n=1 V n λ n , Z(λ) = (M 3 pl N 2 c ) −1 ∞ n=0 Z n λ n ,(1.2)
where ℓ is the radius of the AdS corresponding to the UV fixed point, and V n and Z n are O(N 0 c ). The original motivation for these expansions was that they should be the holographic analogue of the perturbative expansions of the beta functions β λ and β χ of the 't Hooft coupling and instanton angle respectively [10]. In particular, the coefficients V 1 and V 2 were argued to be related respectively to the one-and two-loop beta function of the 't Hooft coupling. In order to account for the logarithmic running of the coupling, therefore, one demands that V 1 = 0. As we shall see below, the requirement that V 1 = 0 is indeed crucial for the asymptotic solutions to logarithmically deviate from strictly AdS asymptotics. However, the beta function is a scheme-dependent quantity which does not correspond to any physical observable. As we shall see below, the holographic Ward identities only involve physical, renormalization group invariant quantities.
Further constraints on the functions V (λ) and Z(λ) are imposed by the IR properties of the model, i.e. as λ → ∞. Specifically, confinement and the absence of certain 'bad' singularities in the IR require that [10] V (λ) ∼ λ 2Q (log λ) P , as λ → ∞, with 2/3 < Q < 2 √ 2/3, P arbitrary,
Q = 2/3, P ≥ 0. (1.3)
Moreover, requiring an asymptotically linear glueball spectrum, m 2 n ∼ n, uniquely picks out Q = 2/3, P = 1/2. Certain conditions for the strongly coupled limit of Z(λ) are also necessary in some cases [10]. Here we will keep the functions V (λ) and Z(λ) completely general, however, assuming only that 3
V (λ) = d(d − 1) ℓ 2 1 + V 1 λ + V 2 λ 2 + V (λ), V 1 = 0, V (λ) = o(λ 2 ), as λ → 0. (1.4)
In particular, we allow V (λ) to contain non analytic terms provided they are subleading compared to λ 2 as λ → 0. Given the functions V (λ) and Z(λ), the Yang-Mills vacuum is described by extrema of (1.1) with four-dimensional Poincaré invariance. Such solutions are domain walls of the form
ds 2 = dr 2 + e 2A(r) dx i dx i , λ = λ(r), χ = χ(r).
(1.5)
Such backgrounds are extrema of (1.1) provided they satisfẏ
A 2 − 1 d(d − 1) ξ 2 λ −2λ2 + Z(λ)χ 2 + V (λ) = 0, A + dȦ 2 − 1 d − 1 V (λ) = 0, 2ξ 2 λ −2 λ + dȦλ − λ −1λ2 − Z(λ)χ 2 + V ′ (λ) = 0, Z(λ)χ + Z ′ (λ)λχ + dZ(λ)Ȧχ = 0,(1.6)
where˙denotes differentiation w.r.t. the radial coordinate r. These equations are automatically solved provided A, λ and χ satisfy the first order flow equationṡ
A = − 1 d − 1 W (λ, χ),λ = ξ −2 λ 2 ∂W (λ, χ) ∂λ ,χ = Z −1 (λ) ∂W (λ, χ) ∂χ , (1.7)
where the 'superpotential' W (λ, χ) is determined by the scalar potential V (λ) and the function Z(λ) via the equation
V (λ) = −ξ −2 λ 2 ∂W ∂λ 2 − Z −1 (λ) ∂W ∂χ 2 + d d − 1 W 2 . (1.8)
In particular, any solution W (λ, χ) of (1.8) uniquely specifies a Yang-Mills vacuum. Notice that the last equation in (1.6) can be integrated exactly to obtaiṅ
χ = cZ −1 (λ)e −dA , (1.9)
where c is an integration constant. From the last equation in (1.7) then follows that
∂W (λ, χ) ∂χ = ce −dA .
(1.10)
As we will see later, this result has very significant consequences for the form of the counterterms required to make the variational problem for the action (1.1) well defined. As far as possible vacuum solutions are concerned, this relation means that there are two broad classes of vacua depending on whether c is zero or not. Namely, if c = 0, then W is independent of the axion and so the axion is just a constant in the background, corresponding to the value of the instanton angle θ Y M . Vacua with c = 0 allow for a non-trivial axion profile, corresponding to giving a VEV to the operator Tr (F F ). In particular, using the result (3.44) below, and assuming O χ = Tr (FF ), we obtain Tr (F F ) = c/κ 2 . Additionally, generic vacua are classified according to whether the operator dual to the dilaton acquires a vacuum expectation value. As follows from the exact onepoint function (3.40), this happens iff the UV expansion of the superpotential W (λ, χ) contains a non-zero term of the form
e −dA ∼ λ dν exp(−d/b 0 λ), where ν = 2ξ 2 d − 1 − 1 d − (d − 1)V 2 ξ 2 b 2 0 . (1.11)
Poincaré invariant vacua are, therefore, parameterized by three constants: the dilaton and axion VEVs, as well as the instanton angle θ Y M . As we shall see, the source of the dilaton is a gauge freedom and its value can be thought of as the energy scale. In particular, it's value is not a physical observable and does not correspond to a coordinate in the moduli space of vacua. This is in good agreement with what one expects from a holographic model describing pure Yang-Mills theory. The rest of the paper is organized as follows. In Section 2 we define the general dilaton-axion gravity theory we will consider and we formulate its dynamics in terms of a radial Hamiltonian. We then develop a systematic iterative procedure for solving the Hamilton-Jacobi equation in a derivative expansion for this general class of theories. Using this procedure the full boundary term that makes the variational problem at infinity well defined is derived for spacetime dimension up to and including five dimensions. This result, summarized in Tables 2 and 3, is the main result of the paper. In Section 3 we apply this general result to IHQCD. After writing down explicitly the boundary counterterms for IHQCD, we systematically derive the generalized Fefferman-Graham asymptotic expansions and give explicit expressions for the exact renormalized one-point functions in the presence of sources for the stress tensor, dilaton and axion operators. These are given in equations (3.41), (3.40) and (3.44) respectively. The section ends with a detailed discussion of the asymptotic bulk diffeomorphisms that preserve the form of the asymptotic expansions, leading to a proof of the holographic Ward identities. Some concluding remarks follow in Section 4, while some technical details of the main calculation are presented in Appendix A. Finally, in Appendix B we apply the main result to the case of a constant dilaton potential, and derive general expressions for the exact one-point functions in the presence of sources for the fully coupled stress tensor-dilatonaxion sector of N = 4 super Yang Mills in four dimensions. To our knowledge this is the first time that the general form of these one-point functions has been derived.
Boundary term for generic dilaton-axion gravity
Since the action (1.1) already contains essentially arbitrary functions of the dilaton, deriving the appropriate boundary term that makes its variational problem well defined is not much easier than considering instead the slightly more general action
S = − 1 2κ 2 ˆM d d+1 x √ g (R[g] − ∂ µ ϕ∂ µ ϕ − Z(ϕ)∂ µ χ∂ µ χ + V (ϕ)) +ˆ∂ M d d x √ γ2K , (2.1)
where the spacetime dimension is taken to be arbitrary and we have introduced a canonical dilaton field, ϕ, related to the dilaton λ in (1.1) by ϕ = ξ log λ. Moreover, we have added the standard Gibbons-Hawking term [12] and the constant κ is related to Newton's constant in d + 1 dimensions by κ 2 = 8πG d+1 . Note that this action contains as special cases a very large class of theories considered in the literature. Apart, from IHQCD, other special cases include the standard dilaton-axion of N = 4 super Yang-Mills 4 , as well as non-conformal branes [3,4]. By deriving the appropriate boundary term for the action (2.1), therefore, we automatically carry out the holographic renormalization of all these theories, whether asymptotically AdS or not, and without the need of any field redefinition. Explicit results for the case of non-conformal branes will be presented elsewhere [15].
Hamiltonian formulation of the variational problem
The variational problem at infinity for the action (2.1) is not well defined as it stands [8,7]. For the case of asymptotically AdS gravity it was first shown in [16] that a certain asymptotic solution of the radial Hamilton-Jacobi equation renders the on-shell action finite. It was later shown [17] that the boundary term obtained by solving the radial Hamilton-Jacobi solution is the same as the one obtained via the standard method of holographic renormalization [18,19,20,21,16,22,23,24]. In [8] it was pointed out that the boundary term required to make the on-shell action finite is in fact the same boundary term required to render the variational problem at infinity well defined, while in [7] it was pointed out that this conclusion holds more generally, not just for non-asymptotically AdS gravity, but also for non-gravitational variational problems with a boundary at infinity, provided the variations are confined within a space of asymptotic solutions carrying a well defined symplectic form. Moreover, this boundary term always corresponds to a solution of the radial Hamilton-Jacobi equation. In practical terms this means that the leading solution of the Hamilton-Jacobi equation determined from the leading asymptotic form of the solutions must not contain transverse derivatives [7]. If this is not the case, it simply means that the space of asymptotic solutions corresponding to the chosen leading asymptotics is not well defined and it does not carry a suitable symplectic form. In such cases, one must first perform some kind of 'Kaluza-Klein reduction' to trivialize the transverse derivatives appearing in the leading radial asymptotics, derive an effective action for the KK fields, and then solve the radial Hamilton-Jacobi equation for this effective action of the KK fields [7]. The condition that the leading asymptotic form of the solutions should give rise to a solution of the Hamilton-Jacobi equation whose leading asymptotic form does not contain any transverse derivatives is automatically satisfied in the case of IHQCD, asymptotically AdS gravity, or non-conformal branes and we will assume that it is the case in the analysis below.
To formulate the variational problem we start by writing the metric in an ADM decomposition [25], but with Hamiltonian time being replaced by the radial coordinate, r, emanating from the boundary at infinity. Namely, we write
ds 2 = (N 2 + N i N i )dr 2 + 2N i drdx i + γ ij dx i dx j ,(2.2)
where N and N i are respectively the lapse and shift functions, and γ ij is the induced metric on the hypersurfaces Σ r of constant radial coordinate r. The metric g µν is therefore replaced in the Hamiltonian description by the three fields {N, N i , γ ij } on Σ r . In terms of these variables the Ricci scalar takes the form
R[g] = R[γ] + K 2 − K ij K ij + ∇ µ (−2Kn µ + n ρ ∇ ρ n µ ),(2.3)
where R[γ] is the Ricci scalar of the induced metric γ ij , the extrinsic curvature, K ij , of the hypersurface Σ r is given by
K ij = 1 2N (γ ij − D i N j − D j N i ) ,(2.
4)
and D i is the covariant derivative w.r.t. the induced metric γ ij . Moreover, K = γ ij K ij and n µ = 1/N, −N i /N , is the outward unit normal vector to Σ r . The total derivative term in this decomposition of the bulk Ricci scalar is an indication of the need for the Gibbons-Hawking term. Evaluating this term on Σ r we see that it gives a contribution which is precisely canceled by the Gibbons-Hawking term. We therefore arrive at a Lagrangian description of the dynamics of the induced fields {N, N i , γ ij } on Σ r , namely
2κ 2 L = −ˆΣ r d d x √ γN R[γ] + K 2 − K i j K j i +ˆΣ r d d x √ γN 1 N 2 (φ 2 + Z(ϕ)χ 2 ) − 2N i N 2 (φ∂ i ϕ + Z(ϕ)χ∂ i χ) + γ ij + N i N j N 2 (∂ i ϕ∂ j ϕ + Z(ϕ)∂ i χ∂ j χ) − V (ϕ) . (2.5)
Note that this Lagrangian involves no kinetic terms for the fields N and N i , which are therefore Lagrange multipliers, leading to constraints. The canonical momenta conjugate to γ ij , ϕ and χ are now obtained respectively as
π ij ≡ δL δγ ij = − 1 2κ 2 √ γ Kγ ij − K ij , π ϕ ≡ δL δφ = 1 κ 2 N √ γ φ − N i ∂ i ϕ , π χ ≡ δL δχ = 1 κ 2 N √ γZ(ϕ) χ − N i ∂ i χ ,(2.6)
while the momenta conjugate to N and N i vanish identically. The Hamiltonian is given by
H =ˆΣ r d d x π ijγ ij + π ϕφ + π χχ − L =ˆΣ r d d x N H + N i H i ,(2.7)
where
H = 2κ 2 γ − 1 2 π i j π j i − 1 d − 1 π 2 + 1 4 π 2 ϕ + 1 4 Z −1 (ϕ)π 2 χ + 1 2κ 2 √ γ R[γ] − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ + V (ϕ) , H i = −2D j π ij + π ϕ ∂ i ϕ + π χ ∂ i χ. (2.8)
Hamilton's equations for the fields N and N i lead respectively to the Hamiltonian and momentum constraints
H = 0, H i = 0. (2.9)
Moreover, the symplectic form is given by
Ω =ˆΣ r d d x δπ ij ∧ δγ ij + δπ ϕ ∧ δϕ + δπ χ ∧ δχ ,(2.10)
and it is independent of the value of the radial coordinate r [26]. The variational problem for the action (2.1) can now be formulated in a regularized space, M ro , whose boundary is defined as the surface Σ ro for some fixed r o . Provided r o is sufficiently large, Σ ro is diffeomorphic to the boundary ∂M at infinity. Adding then a generic boundary term, S b , to the action (2.1) defined on M ro and considering a generic variation we obtain [7]
δ(S + S b ) =ˆM ro d d+1 x(EOM s) + (L +Ṡ b ) ro δr o +ˆΣ ro d d x π ij + δS b δγ ij δγ ij + π ϕ + δS b δϕ δϕ + π χ + δS b δχ δχ , (2.11)
where the integrand of the integral over the bulk of M ro is proportional to the equations of motion.
In order for the variational problem at r o → ∞ to be well defined with boundary conditions imposed on the induced fields γ ij , ϕ and χ, i.e. for a generic variation of the action with such boundary conditions to imply the equations of motion, one must demand that [7] (
L +Ṡ b ) ro ro→∞ − −−− → 0. (2.12)
The variational problem is then defined by considering variations of γ ij , ϕ and χ within the space of generic asymptotic solutions of the equations of motion so that
S b | ro = − S| ro , (2.13)
where S is Hamilton's principal functional, i.e. a solution of the Hamilton-Jacobi equation where the values of the induced fields γ ij , ϕ and χ on Σ ro are totally arbitrary. It follows from (2.11) that
π ij ro = δS δγ ij ro , π ϕ | ro = δS δϕ ro , π χ | ro = δS δχ ro , (2.14)
where S is identified with the on-shell value of the action S on solutions with arbitrary boundary values for γ ij , ϕ and χ on Σ ro . Including the boundary term S b in these relations leads to the canonically transformed momenta [7] Π ij ro =
δ(S + S b ) δγ ij ro , Π ϕ | ro = δ(S + S b ) δϕ ro , Π χ | ro = δ(S + S b ) δχ ro . (2.15)
In order to determine the boundary term S b we need to determine the asymptotic form of Hamilton's principal functional. In other words we need to solve the Hamilton-Jacobi equation in a certain asymptotic sense, which we will specify below. The Hamilton-Jacobi equation can be derived from the following simple argument. Let S ro ≡ S| ro denote the on-shell action S with arbitrary boundary values for γ ij , ϕ and χ on Σ ro . Then,
S ro = ∂S ro ∂r o +ˆΣ ro d d x γ ij [γ, ϕ, χ] δ δγ ij +φ[γ, ϕ, χ] δ δϕ +χ[γ, ϕ, χ] δ δχ S ro (2.14) = ∂S ro ∂r o +ˆΣ ro d d x π ijγ ij + π ϕφ + π χχ = ∂S ro ∂r o + H + L,(2.16)
where ∂/∂r o denotes the partial derivative with respect to r o . However, since S ro is the on-shell action, we must haveṠ ro = L and so we conclude that
∂S r ∂r + H = 0. (2.17)
For a generally covariant theory, like supergravity, the Hamiltonian vanishes identically since it is proportional to the constraints (2.9). It follows that, as a consequence of the general covariance of the theory, the on-shell action does not depend explicitly on the radial coordinate, r, but only through the induced fields on the hypersurface Σ r . Moreover, we see that the Hamilton-Jacobi equation in a generally covariant theory is equivalent to the vanishing of the constraints, i.e.
2κ 2 γ − 1 2 γ ik γ jl − 1 d − 1 γ ij γ kl δS r δγ ij δS r δγ kl + 1 4 δS r δϕ 2 + 1 4 Z −1 (ϕ) δS r δχ 2 = − 1 2κ 2 √ γ R[γ] − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ + V (ϕ) , −2D j δS r δγ ij + δS r δϕ ∂ i ϕ + δS r δχ ∂ i χ = 0. (2.18)
The task of the next subsection will be to systematically solve these equations is a certain asymptotic sense.
Recursive solution of the Hamilton-Jacobi equation
We now turn to the task of solving the Hamilton-Jacobi equations (2.18) in order to determine the boundary term S b that makes the variational problem for the action (2.1) at r o = ∞ well defined.
As was argued in [7] and reiterated above, provided the variational problem is formulated within a well defined space of asymptotic solutions, the leading form of the boundary term, obtained as a solution of the Hamilton-Jacobi equation, will contain no transverse derivatives. It follows that the full solution of the Hamilton-Jacobi equation that corresponds to the boundary term we seek to determine admits an expansion in transverse derivatives. One can, therefore, try to solve the Hamilton-Jacobi equation by writing down an ansatz containing all possible terms allowed by general covariance at each order in derivatives. Although this approach, which was used in, for example, [16] in the case of asymptotically AdS gravity, generically suffices for simple cases in low spacetime dimension and for a limited number of fields, it quickly becomes prohibitively inefficient and cumbersome. In particular, this approach not only unnecessarily includes terms in the ansatz that may happen to be absent in the particular theory, but also the number of equations one obtains for the undetermined functions in the ansatz is greater than the number of functions to be determined, and so many equations are redundant. Instead, the approach we will develop here is a systematic recursive procedure for solving the Hamilton-Jacobi equation, closer in spirit to the recursive method developed in [27]. In particular, the algorithm we will present computes systematically the nth term in the derivative expansion from lower terms, thus producing only the terms that do appear in the actual solution of the Hamilton-Jacobi equation. The basis of our algorithm is the following observation. Let us begin by writing Hamilton's principal function, S r , as
S r =ˆΣ r d d xL(γ, ϕ, χ). (2.19)
From the relations (2.14) then we have
π ij δγ ij + π ϕ δϕ + π χ δχ = δL + ∂ i v i (δγ, δϕ, δχ),(2.20)
for some vector field v i (δγ, δϕ, δχ). Since by construction the solution S r we are seeking admits a derivative expansion as r → ∞, S r increasingly approaches a solution of the form
S (0) = 1 κ 2ˆΣ r d d x √ γU (ϕ, χ),(2.21)
for some function U (ϕ, χ). Given the zero order solution (2.21), we can compute corrections to this action in a systematic expansion in eigenfunctions of the operator 5
δ γ =ˆd d x2γ ij δ δγ ij ,(2.22)
namely (dropping the subscript r from now on)
S = S (0) + S (2) + S (4) + · · · ,(2.23)
where δ γ S (2n) = (d−2n)S (2n) . It is easy to see that the resulting expansion is a derivative expansion. Note that the operator (2.22) agrees with the dilatation operator introduced in [17] in the case of a constant potential, i.e. for the usual dilaton-axion system in conformal theories. However, the would-be dilatation operator for an arbitrary dilaton potential would lead to an operator whose eigenfunctions are highly non-trivial and so, from a practical point of view, would not serve as a good basis for expanding Hamilton's principal function. Moreover, an expansion in eigenfunctions of the operator (2.22) has the advantage of leading to algebraic in the induced metric γ ij equations for determining the terms S (2n) , which is the fundamental ingredient in our algorithm.
Let us see how this works. Applying the general identity (2.20) to the variation δ γ we obtain
2π (2n) = (d − 2n)L (2n) + ∂ i v i (2n) . (2.24)
Since L is defined up to a total derivative, we can absorb the last term in this identity into L (2n) such that
2π (2n) = (d − 2n)L (2n) . (2.25)
The significance of this relation will become clear shortly, when we write down the equation determining L (2n) . Since (2.25) holds only for a certain choice of total derivative terms in L (2n) , in principle we should keep track of total derivative terms as well. However, as we shall see, this will not be necessary in our recursive procedure. In particular, we will determine L (2n) at each order up to total derivative terms. The canonical momenta at this order can then be obtained by differentiating S (2n) =´d d xL (2n) . Since total derivative terms do not influence the canonical momenta, we will still get the correct expressions for the canonical momenta. Inserting the leading term (2.21) of the above expansion in to the Hamilton-Jacobi equation (2.18) we find that the function U (ϕ, χ) satisfies the equation
(∂ ϕ U ) 2 + Z −1 (ϕ)(∂ χ U ) 2 − d d − 1 U 2 + V (ϕ) = 0. (2.26)
However, using the leading term (2.21) in the relations (2.14) and identifying the momenta from the expressions (2.6) we find that to leading order the induced metric takes the form 6
γ ij = e 2Aḡ (0)ij (x), (2.27) whereḡ (0)ij (x)
is an arbitrary metric on the boundary and the fields A, ϕ and χ satisfy the flow equations (1.7), but with W replaced with U . But then, these first order equations in combination with (2.26) imply the second order equations (1.6). As we have seen above, the second order equation for the axion is integrable, giving in this case
∂U (ϕ, χ) ∂χ =ce −dA ,(2.
28)
Sr and the corresponding canonical momenta in eigenfunctions of the dilatation operator, δD, obtained from the
relation ∂r =´d d x γij δ δγ ij + Σ fḟ δ δf r→∞ −−−→ 1 ℓ δD,
where f denotes generic matter fields. In that case expanding in eigenfunctions of the dilatation operator is the most efficient recursive method to solve the Hamilton-Jacobi equation, because it amounts to solving the zero order problem (2.26) and the one determining the higher derivative terms simultaneously. However, in more general cases the operator corresponding to the leading asymptotic behavior of ∂r is not very useful in practice since its eigenfunctions are not simple functions of curvature invariants and matter fields. In most cases it is easier in practice to first solve the zero order problem (2.26) to determine U (ϕ, χ) and then to expand in eigenfunctions of δγ . Of course in the case of asymptotically AdS or dS gravity the two approaches produce identical results, even though the two expansions differ order by order, the difference being that the latter expansion resums all zero derivative terms into the leading term. 6 We gauge-fix the lapse and shift functions to N = 1 and Ni = 0.
which is the analogue of (1.10). We therefore see that any χ dependence in U leads to a finite contribution in Hamilton's principal function S and hence we can set the integration constant c = 0. We conclude that the function U (ϕ) required to make the variational problem well defined can be taken, without loss of generality, to be independent of the axion χ. As we will see shortly, this leads to a significant simplification of the analysis to determine the required higher derivative terms in S. In order to determine the leading solution (2.21) of the Hamilton-Jacobi equation, therefore, one needs to solve the reduced equation
(∂ ϕ U ) 2 − d d − 1 U 2 + V (ϕ) = 0. (2.29)
For an arbitrary potential this equation can be transformed into an Abel's equation of the first kind [17], which is generically non-integrable. However, we need not find the general solution of this equation. Any solution of this equation that ensures that ϕ has the desired general asymptotics via the relationφ
= ± d d − 1 U 2 (ϕ) − V (ϕ), (2.30)
suffices. In particular, if U is such a solution and it is not isolated in the space of solutions 7 , then it is easy to see that any solution in the vicinity of U is of the form U + ǫ∆U , where ǫ is an infinitesimal parameter and ∆U = O(exp(−dA)). Hence, the difference between two such solutions only contributes a finite term in S (0) , and it is therefore irrelevant. The reason for restricting this argument to the vicinity of the original solution U is that there exist solutions at infinite parametric distance from U that change the leading asymptotic behavior of ϕ. Such solutions are excluded by the requirement that ϕ has the correct asymptotics. Now that we have determined that U (ϕ) is independent of the axion, inserting the above expansion of Hamilton's principal function in the Hamiltonian constraint and matching terms of equal δ γ eigenvalue we obtain for the higher order terms
U ′ (ϕ) δ δϕˆd d xL (2n) − d − 2n d − 1 U (ϕ)L (2n) = R (2n) , n > 0, (2.31) where R (2) = − 1 2κ 2 √ γ R[γ] − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ , R (2n) = −2κ 2 γ − 1 2 n−1 m=1 π (2m) i j π (2(n−m)) j i − 1 d − 1 π (2m) π (2(n−m)) + 1 4 π ϕ(2m) π ϕ(2(n−m)) + 1 4 Z −1 (ϕ)π χ(2m) π χ(2(n−m)) , n > 1. (2.32)
Importantly, these are linear equations and only involve a derivative w.r.t. the dilaton, ϕ, and not the induced metric γ ij or the axion. The absence of a derivative w.r.t. the induced metric is due to the relation (2.25), while the absence of a derivative w.r.t. the axion is because we have shown that U only depends on the dilaton. We will now solve these equations up to the order required in order to determine the boundary term S b for d = 4. In the analysis below we will take U ′ (ϕ) = 0, which is guaranteed by the requirement of a logarithmic running for the dilaton. However, the case of a constant potential V and, consequently, constant U arises in conformal theories such as N = 4 super Yang-Mills and so it is interesting on its own right. For completeness we discuss this case in Appendix B.
7 See Appendix B for an example where this happens.
Turning now to the case U ′ (ϕ) = 0, we notice that the linear equation (2.31) for L (2n) , n > 0, admits the homogeneous solution
L hom (2n) = F (2n) [γ, χ] exp d − 2n d − 1 ˆϕ dφ U ′ (φ) U (φ) , (2.33)
where F (2n) [γ, χ] is a covariant function of the induced metric and the axion of weight d − 2n. However, this solution contributes only to finite local counterterms and can be ignored. To see this notice that the leading order solution (2.21) implies via the Hamilton-Jacobi relations (2.14) that the induced metric takes to leading order the form (2.27) with
A = − 1 d − 1ˆϕ dφ U ′ (φ) U (φ). (2.34) Now, by construction, F (2n) [γ, χ] has weight d − 2n and so L hom (2n) ∼ e (d−2n)A × e −(d−2n)A = f inite. (2.35)
We are, therefore, only interested in the inhomogeneous solution of (2.31). In other words, formally,
L (2n) = e −(d−2n)A(ϕ)ˆϕ dφ U ′ (φ) e (d−2n)A(φ) R (2n) (φ). (2.36)
This integral is well defined if R (2n) does not involve derivatives of the dilaton, ϕ, but it requires some caution when it does. In general, we can write more precisely
L (2n) = e −(d−2n)A(ϕ) F (2n) , (2.37) where F (2n) satisfies δϕ U ′ (ϕ) e (d−2n)A(ϕ) R (2n) (ϕ) = δ ϕ F (2n) + e (d−2n)A(ϕ) ∂ i v (2n) i (ϕ, δϕ), (2.38)
for some vector field v (2n) i (ϕ, δϕ). In Table 1 we have listed the local functional F (2n) (ϕ) for a number of generic source terms R (2n) (ϕ) that we will need for our calculation. A detailed derivation of the formulas given in Table 1 is provided for the reader's convenience in Appendix A. Both in Table 1 and in Appendix A we make extensive use of the short-hand notation
ϕ n,m ≡ A ′ m e −(d−2n)Aˆϕ dφ U ′ e (d−2n)A A ′ −m , (2.39)
where A(ϕ) is given by (2.34). The integration formula (2.38) allows us to develop an algorithmic procedure for evaluating Hamilton's principal function, S, iteratively. Namely, given the source term of equation (2.31) at order n, (2.38) is used to obtain L (2n) up to an irrelevant total derivative term. Differentiating this with respect to the various induced fields gives the corresponding momenta at that order. Finally, using these momenta, as well as those of lower orders, one determines the source term at order n + 1 via (2.32). The procedure is then repeated to the desired order. This algorithm is schematically outlined in Fig. 1. Let us now carry out this general algorithm to order n = 2, which is sufficient for evaluating the boundary term for the action (2.1) in d = 4. In general, at each order n the source term, R (2n) and the corresponding inhomogeneous solution, L (2n) , of the functional equation (2.31) can be written respectively in the form Here, t i 1 i 2 ...im and t ij are arbitrary totally symmetric tensors independent of ϕ, while t ijkl (2.37), U i corresponds to a total derivative term in L (2n) and so we need not determine this term explicitly. Note that the source term r 1 2 2 only contributes to U i and so it can be ignored. A detailed derivation of these results is given in Appendix A.
R (2n) = − 1 2κ 2 √ γ Nn I=1 c I n (ϕ)T I n , (2.40) R (2n) e −(d−2n)A F (2n) r 1 m (ϕ)t i 1 i 2 ...im ∂ i 1 ϕ∂ i 2 ϕ . . . ∂ im ϕ ffl ϕ n,m r 1 m (φ)t i 1 i 2 ...im ∂ i 1 ϕ∂ i 2 ϕ . . . ∂ im ϕ r 2 (ϕ)t ij D i D j ϕ ffl ϕ n,1 r 2 (φ)t ij D i D j ϕ − ffl ϕ n,2 U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ n,1 r 2 (φ)t ij ∂ i ϕ∂ j ϕ r 1 2 2 (ϕ)t ijkl 1 + s 1 2 2 (ϕ)t ijkl 2 ∂ i ϕ∂ j ϕD k D l ϕ ffl ϕ n,3 s 1 2 2 (φ)t ijkl 2 ∂ i ϕ∂ j ϕD k D l ϕ r 2 2 (ϕ)t ijkl 1 + s 2 2 (ϕ)t ijkl 2 D i D j ϕD k D l ϕ ffl ϕ n,2 r 2 2 (φ)t ijkl 1 + ffl ϕ n,2 s 2 2 (φ)t ijkl 2 D i D j ϕD k D l ϕ −2 ffl ϕ n,3 U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ n,2 s 2 2 (φ)t ijkl 2 ∂ i ϕ∂ j ϕD k D l ϕ1 = 1 3 γ ik γ jl + γ il γ jk + γ ij γ kl , t ijkl 2 = 1 3 γ ik γ jl + γ il γ jk − 2γ ij γ kl . Moreover, F (2n) is given up to terms of the form e (d−2n)A D i U i , for some vector field U i . Since L (2n) is related to F (2n) as inR (2n) ffl − −−−−−−− → L (2n) δ − −−−−−− → {π (2n) } {π (2n+2) } δ ← −−−−−− − L (2n+2) ffl ← −−−−−−− − R (2n+2) R (2n+4) ffl − −−−−−−− → L (2n+4)
. . . Here, the operators ffl and δ stand respectively for the functional integration defined by the formula in (2.38) and for functional differentiation with respect to the induced fields. and
I T I 1 c I 1 (ϕ) P I 1 (ϕ) 1 R 1 ffl ϕ 1,0 1 ≡ −2Ξ(ϕ) 2 ∂ i ϕ∂ i ϕ −1 ffl ϕ 1,2 (−1) ≡ −M (ϕ) 3 ∂ i χ∂ i χ −Z(ϕ) ffl ϕ 1,0 (−Z(φ)) ≡ −Θ(ϕ)L (2n) = − 1 2κ 2 √ γ Nn I=1 P I n (ϕ)T I n , (2.41)
where c I n (ϕ) and P I n (ϕ) are scalar functions of ϕ and T I n are quantities involving fields other than ϕ, as well as derivatives of ϕ. The number N n is the number of such quantities at each order n. For n = 1 the source R (2) is given in (2.32). Applying the results in Table 1 to this sources we obtain L (2) . The result is summarized in Table 2.
In order to move to the next order we first need to evaluate the canonical momenta at order n = 1 by differentiating L (2) with respect ot the induced fields. After a little algebra we obtain:
π (2) ij = − 1 κ 2 √ γ ΞR ij − Ξ ′ D i D j ϕ + 1 2 (M − 2Ξ ′′ )∂ i ϕ∂ j ϕ + 1 2 Θ∂ i χ∂ j χ − 1 2 γ ij ΞR − 2Ξ ′ γ ϕ + 1 2 (M − 4Ξ ′′ )∂ k ϕ∂ k ϕ + 1 2 Θ∂ k χ∂ k χ , π ϕ(2) = − 1 κ 2 √ γ 1 2 M ′ ∂ i ϕ∂ i ϕ + M γ ϕ − 1 2 Θ ′ ∂ i χ∂ i χ − Ξ ′ R , π χ(2) = − 1 κ 2 √ γ Θ γ χ + Θ ′ ∂ i ϕ∂ i χ . (2.42)
Inserting these in the expression for R (4) in (2.32) results in an explicit expression for the source term at order n = 2. This is a rather complicated source, involving 20 different terms, and can be read out from the first three columns of Table 3. Using the results of Table 1 we then obtain the fourth column in Table 3, giving L (4) . This table is the main result of this paper providing together with Table 2, the boundary term necessary to make the variational problem of a generic action of the form (2.1) well defined in any dimension up to and including d + 1 = 5. Before we apply this general result to IHQCD, a technical comment is due. The boundary terms listed in Tables 2 and 3 generically have poles at d = 2 or d = 4. This happens whenever there is a conformal anomaly, or, in terms of the bulk language, whenever the boundary term required to make the variational problem well defined breaks the bulk diffeomorphisms corresponding to translations in the radial coordinate. The way to handle these poles is to relate the radial cut-off, r 0 , to the parameter d, as
8 r 0 = 1/(d − d * ),
where d * is the dimension of the boundary. An example of this issue arises in the case of a constant dilaton potential, discussed in Appendix B.
I T I 2 c I 2 (ϕ) P I 2 (ϕ) 1 R ij R ij 4Ξ 2 ffl ϕ 2,0 4Ξ 2 2 R 2 Ξ ′2 − d d−1 Ξ 2 ffl ϕ 2,0 (Ξ ′2 − d d−1 Ξ 2 ) 3 D i D j ϕD i D j ϕ 4Ξ ′2 4 ffl ϕ 2,2 Ξ ′2 4 D i D j ϕ∂ i ϕ∂ j ϕ −4Ξ ′ (M − 2Ξ ′′ ) 2 3 ffl ϕ 2,3 (4Ξ ′ (3Ξ ′′ − M ) − M M ′ − 2U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ 2,2 (6Ξ ′2 − M 2 )) 5 γ ϕ∂ i ϕ∂ i ϕ M M ′ + 2Ξ ′ (M − 4Ξ ′′ ) − 2 3 ffl ϕ 2,3 (4Ξ ′ (3Ξ ′′ − M ) − M M ′ − 2U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ 2,2 (6Ξ ′2 − M 2 )) 6 ( γ ϕ) 2 M 2 − 4Ξ ′2 ffl ϕ 2,2 M 2 − 4Ξ ′2 7 ∂ i ϕ∂ i ϕ 2 1 4 M ′2 + (3d−4) 4(d−1) M 2 − 2M Ξ ′′ ffl ϕ 2,4 ( 1 4 M ′2 + (3d−4) 4(d−1) M 2 − 2M Ξ ′′ ) 8 ∂ i χ∂ i χ 2 1 4 Θ ′2 + (3d−4) 4(d−1) Θ 2 1 4 ffl ϕ 2,0 (Θ ′2 + (3d−4) (d−1) Θ 2 ) 9 R ij D i D j ϕ −8ΞΞ ′ −8 ffl ϕ 2,1 ΞΞ ′ 10 R ij ∂ i ϕ∂ j ϕ 4Ξ (M − 2Ξ ′′ ) 4 ffl ϕ 2,2 Ξ(M − 2Ξ ′′ ) + 2U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ 2,1 ΞΞ ′ −2Ξ ′ (M − 2Ξ) −2 ffl ϕ 2,1 Ξ ′ (M − 2Ξ) 12 R∂ i ϕ∂ i ϕ −Ξ ′ M ′ − d d−1 ΞM + 4ΞΞ ′′ ffl ϕ 2,2 ( − Ξ ′ M ′ − d d−1 ΞM + 4ΞΞ ′′ + 2U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ 2,1 Ξ ′ (M − 2Ξ)) 13 R ij ∂ i χ∂ j χ 4ΞΘ ffl ϕ 2,0 4ΞΘ 14 D i D j ϕ∂ i χ∂ j χ −4Ξ ′ Θ −4 ffl ϕ 2,1 Ξ ′ Θ 15 ∂ i ϕ∂ i χ 2 Z −1 Θ ′2 + 2 (M − 2Ξ ′′ ) Θ ffl ϕ 2,2 (Z −1 Θ ′2 + 2 (M − 2Ξ ′′ ) Θ + 4U ′ A ′ ∂ 2 ϕ 1 A ′ fflφ 2,1 Ξ ′ Θ) 16 γ ϕ∂ i χ∂ i χ 2Ξ ′ Θ − M Θ ′ ffl ϕ 2,1 (2Ξ ′ Θ − M Θ ′ ) 17 R∂ i χ∂ i χ Ξ ′ Θ ′ − d d−1 ΞΘ ffl ϕ 2,0 (Ξ ′ Θ ′ − d d−1 ΞΘ) 18 ∂ i ϕ∂ i ϕ∂ j χ∂ j χ 1 2 Θ 4Ξ ′′ − d d−1 M − 1 2 M ′ Θ ′ 1 2 ffl ϕ 2,2 (Θ(4Ξ ′′ − d d−1 M ) − M ′ Θ ′ − 2A ′ U ′ ∂ 2 ϕ 1 A ′ fflφ 2,1 (2Ξ ′ Θ − M Θ ′ )) 19 ( γ χ) 2 Z −1 Θ 2 ffl ϕ 2,0 Z −1 Θ 2 20 γ χ∂ i ϕ∂ i χ 2Z −1 ΘΘ ′ ffl ϕ 2,1 2Z −1 ΘΘ ′
Application to IHQCD
The results presented in the previous section are applicable to any action of the form (2.1). In particular, Tables 2 and 3 provide the general boundary term that renders the variational problem of any action of the form (2.1) well posed for boundary dimension up to and including d = 4. However, additional assumptions that enter into specific models often lead to significant simplification of this boundary term. In this section we will apply these general results to the specific problem of IHQCD [10] described in the Introduction. The significance of this boundary term will be demonstrated by deriving asymptotic expansions analogous to the Fefferman-Graham expansion for asymptotically AdS gravity [6], providing general expressions for the one-point functions of the dual operators in terms of the coefficients of these expansions, and finally giving the correct holographic Ward identities for IHQCD.
In deriving the boundary term in Tables 2 and 3 we have assumed that the leading asymptotics of the induced fields follows from a leading solution (2.21) of the Hamilton-Jacobi equation which does not involve transverse derivatives. However, no specific asymptotics was assumed. Taking into account the leading asymptotic form of the induced fields of the IHQCD model allows for various simplifications of the result of the previous section. To see what simplifications occur let us start by writing ϕ = ξ log λ, for some non-zero constant ξ. Then, by abuse of notation, we will correspondingly write U (λ), A(λ), etc. as functions of λ. As we have argued above, U is a function of the dilaton λ only and not of the axion. Moreover, it is a solution of equation (2.26), which now becomes
d d − 1 U 2 − ξ −2 λ 2 ∂U ∂λ 2 = V (λ). (3.1)
At this point we need to invoke some information about the asymptotics. First, recall that we want to consider potentials of the form (1.4) as λ → 0. Moreover, U (λ) defines the leading form of the metric and dilaton asymptotics via the flow equations (these are the same as (1.7) but with W replaced by U , and follow from combining the relations (2.6) and (2.14) for the canonical momenta)
A = − 1 d − 1 U (λ),λ = ξ −2 λ 2 ∂U (λ) ∂λ . (3.2)
The leading metric and dilaton asymptotics adopted in the model of IHQCD is [10]
A ∼ ℓ −1 ,λ ∼ −b 0 ℓ −1 λ 2 + b 1 ℓ −1 λ 3 , as λ → 0 (3.3)
where the constants b 0 and b 1 were argued in [10] to be related respectively to the one and twoloop perturbative beta function coefficients. According to that argument, subleading corrections to the asymptotic form of the dilaton correspond to higher loop coefficients in the perturbative beta function, which are scheme dependent. In that sense, the terms in (3.3) are universal in the model of IHQCD, and any dilaton potential V (λ) chosen must be such that it is compatible with this leading asymptotics in the UV, while subleading terms in the UV expansion of the potential can differ for different choices of the dilaton potential. One should keep in mind, however, that the beta function of any operator whose coupling transforms inhomogeneously under scale transformations is scheme dependent and, therefore, not a physical observable. In other words, the beta function of any operator that acquires anomalous dimension is scheme dependent. For such operators the scaling dimension γ O is not constant along the RG flow. Only the value of this scaling dimension at the fixed points is a physical observable. Nevertheless, the combination β O · O(x) is a physical observable 9 , since this combination appears in the trace of the stress tensor, whose scaling dimension does not renormalize. As we shall see below from the trace Ward identity, the bulk dilaton λ should be thought of as the holographic dual to the operator β(λ Y M )Tr F 2 . Isolating the beta function factor in this operator is a schemedependent procedure, but the product transforms homogeneously under the renormalization group flow. It is precisely this combination that appears in the trace Ward identity. In fact, this is a generic mechanism of how one can accommodate operators with running scaling dimensions in a supergravity setting, thus going beyond the realm of operators with protected dimensions.
In order to ensure that the leading asymptotic behavior of the metric and dilaton is of the form (3.3) we take the function U (λ) to be of the form (3.4) where b 1 = 2ξ −2 U 2 , and we have introduced a term of order λ α , α > 2, in order to allow for potentials with non-integer powers of λ, as were considered e.g. in [11,28]. However, it should be emphasized that the full closed form of the function U (λ) is required in order to construct the necessary boundary term that makes the variational problem well defined. In particular, to write down explicitly the boundary term one necessarily needs an exact solution of (3.1). Such an exact solution is of course dependent on the particular choice of dilaton potential. Since we want to keep the discussion general here, we will not fix the dilaton potential or the function U (λ), beyond the requirement that as λ → 0 it behaves as in (3.4). Once a dilaton potential is specified, one then will simply need to find an exact solution of (3.1) that behaves asymptotically as in (3.4) and use it in the expressions below, all of which are expressed in terms of a generic U (λ). Of course, such a solution will not be unique since it can be shown [29] that if U 0 (λ) is such a solution, then there is a continuous family of deformations of this solution, with the deformation behaving as λ dν exp(−d/b 0 λ) as λ → 0, where the value of the exponent ν is given below. However, any member of this continuous family of solutions is equally good for the purposes of evaluating the boundary term since any difference arising from this deformation only contributes finite terms to the boundary term.
U (λ) = − d − 1 ℓ − ξ 2 b 0 ℓ λ + 1 ℓ U 2 λ 2 + 1 ℓ U α λ α + o(λ α ), α > 2,
Boundary term for IHQCD
In this section we will make use of the asymptotic form (3.4) of the function U (λ) in order to simplify the general boundary term derived in the previous section. To this end we start by noting that, depending on the value of the exponent α, the function A(λ) defined in (2.34) has the following asymptotic behavior as λ → 0
e A(λ) = λ −ν e 1 b 0 λ 1 − αUα (α−2)ξ 2 b 2 0 λ α−2 + o λ α−2 , α < 3, λ −ν e 1 b 0 λ 1 − 1 b 0 2U 2 ξ 2 b 0 2 − U 2 d−1 + 3U 3 ξ 2 b 0 λ + o (λ) , α = 3, λ −ν e 1 b 0 λ 1 − 1 b 0 2U 2 ξ 2 b 0 2 − U 2 d−1 λ + o (λ) , α > 3, (3.5) where ν = ξ 2 d−1 + 2U 2 ξ 2 b 2 0
. Using these asymptotic expansions and the integral identitŷ
λ dλ ′ λ ′µ−2 e ω λ ′ = − 1 ω e ω λ λ µ no n=0 Γ(n+µ) Γ(µ) 1 ω n λ n + O λ no+1 , ω > 0, ∀n o ∈ N, 1 µ−1 λ µ−1 + const., ω = 0, µ = 1, log λ + const., ω = 0, µ = 1,(3.6)
we can determine the asymptotic form of all functions listed in Tables 2 and 3. These functions involve integrals of the form λ n,m
λ ∆ ≡ ξ 2 λA ′ (λ) m e −(d−2n)A(λ)ˆλ dλ λ 2 U ′ (λ) e (d−2n)A(λ) λ A ′ (λ) −mλ ∆ ,(3.7)
whose asymptotic form we tabulate in Table 4. With these results one can now determine the asymptotic form of the functions in Tables 2 and 3, which we present respectively in Tables 5 Table 2. and 6. In deriving these asymptotic forms we have made repeated use of the crucial fact that ∂ i λ = O(λ 2 ). This follows from the general asymptotic form of the dilaton λ and will be derived in the next section. Note that even though we have listed the leading asymptotic behavior of the order n = 1 terms in Table 5, we actually need the exact closed form expressions of the functions M (λ), Ξ(λ) and Θ(λ) in the boundary term. The reason is quite obvious. Namely, the expansion of the solution of the Hamilton-Jacobi equation in eigenfunctions of the operator δ γ , i.e. the derivative expansion, is an expansion in powers of e −2A ∼ λ 2ν exp(−2/b 0 λ). This expansion, therefore, is non-perturbative in λ. However, at each order in the derivative expansion the boundary term contains an infinite expansion in powers of λ. In other words, the expansion of the boundary term is a double expansion. 10 Clearly, at orders n = 0 and n = 1 in the derivative expansion we must keep the entire perturbative expansions in λ, since there are divergences coming from any power of λ no matter how large. This is the reason why the full closed form expressions for U (λ) at order n = 0 and of M (λ), Ξ(λ) and Θ(λ) at order n = 1 must be kept. At order n = 2 in the derivative expansion, however, this is no longer necessary. Indeed, for d = 4 the factor e 4A coming from the volume element √ γ exactly cancels the factor e −4A coming from the four-derivative terms. Hence, the divergences at order n = 2 are only power-like or logarithmic in λ. The first few terms of the asymptotic form of the boundary terms listed in Table 6, therefore, suffice. Table 3.
d − 2n ∆ + m ffl λ n,m λ ∆ > 0 any ℓλ ∆ d−2n 1 + ∆+m d−2n − ξ 2 d−1 b 0 λ + O(λ 2 ) 0 = 0, 1 − ℓλ ∆−1 b 0 (∆−1+m) 1 + 1 ∆+m (∆ − 1 + 2m) 2U 2 ξ 2 b 0 + m ξ 2 b 0 d−1 λ + O(λ 2 ) 0 1 − ℓλ ∆−1 b 0 log λ + O(λ 0 ) 0 0 ℓλ ∆−1 b 0 1 − 2U 2 ξ 2 b 0 + ∆b 0 ν λ log λ + O(λ)I T I 1 P I 1 (λ) 1 R −2Ξ(λ) = ℓ d−2 1 − ξ 2 b 0 d−1 λ + O(λ 2 ) 2 ξ 2 λ −2 ∂ i λ∂ i λ −M (λ) = − ℓ d−2 1 + 2 d−2 − ξ 2 d−1 b 0 λ + O(λ 2 ) 3 ∂ i χ∂ i χ −Θ(λ) = − ℓ d−2 M 3 pl N 2 c −1 Z 0 + Z 1 − ξ 2 b 0 d−1 Z 0 λ + O(λ 2 )I T I 2 P I 2 (λ) 1 R ij R ij ℓ 3 (d−2) 2 b 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 log λ + O(λ 0 ) 2 R 2 − d 4(d−1) ℓ 3 (d−2) 2 b 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 log λ + O(λ 0 ) 3 ξ 2 λ −2 D i D j λD i D j λ − 2λ −3 ∂ i λ∂ j λD i D j λ + λ −4 (∂ i λ∂ i λ) 2 − ℓ 3 ξ 2 b 0 3(d−1) 2 (d−1) 2 λ + O(λ 2 ) 4 ξ 3 λ −3 ∂ i λ∂ j λD i D j λ − λ −4 (∂ i λ∂ i λ) 2 − 2ξ −1 ℓ 3 3(d−2) 2 b 0 λ −1 + O(λ 0 ) 5 ξ 3 λ −3 ∂ i λ∂ i λ λ − λ −4 (∂ i λ∂ i λ) 2 2ξ −1 ℓ 3 3(d−2) 2 b 0 λ −1 + O(λ 0 ) 6 ξ 2 λ −2 ( λ) 2 − 2λ −3 ∂ i λ∂ i λ λ + λ −4 (∂ i λ∂ i λ) 2 − ℓ 3 (d−2) 2 b 0 λ −1 + O(λ 0 ) 7 ξ 4 λ −4 ∂ i λ∂ i λ 2 − (3d−4)ℓ 3 12(d−2) 2 (d−1)b 0 λ −1 + O(λ 0 ) 8 ∂ i χ∂ i χ 2 (3d−4)ℓ 3 (M pl l 3 N 2 c ) −2 4(d−1)(d−2) 2 b 0 Z 2 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 Z 0 − 2Z 1 Z 0 log λ + O(λ 0 ) 9 ξR ij (λ −1 D i D j λ − λ −2 ∂ i λ∂ j λ) − 2ξℓ 3 (d−1)(d−2) 2 + O(λ) 10 ξ 2 λ −2 R ij ∂ i λ∂ j λ 2ℓ 3 (d−2) 2 b 0 λ −1 + O(λ 0 ) 2ξℓ 3 (d−2) 2 (d−1) + O(λ) 12 ξ 2 λ −2 R∂ i λ∂ i λ − dℓ 3 2(d−2) 2 (d−1)b 0 λ −1 + O(λ 0 ) 13 R ij ∂ i χ∂ j χ − 2ℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 b 0 Z 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 Z 0 − Z 1 log λ + O(λ 0 ) 14 ξ(λ −1 D i D j λ∂ i χ∂ j χ − λ −2 (∂ i λ∂ i χ) 2 ) 2ξℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 (d−1) Z 0 + O(λ) 15 ξ 2 λ −2 ∂ i λ∂ i χ 2 − 2ℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 b 0 Z 0 λ −1 + O(λ 0 ) 16 ξ(λ −1 λ − λ −2 ∂ j λ∂ j λ)∂ i χ∂ i χ − ξ −1 ℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 2 d−2 − 2ξ 2 d−1 Z 0 + O(λ) 17 R∂ i χ∂ i χ dℓ 3 (M 3 pl N 2 c ) −1 2(d−2) 2 (d−1)b 0 Z 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 Z 0 − Z 1 log λ + O(λ 0 ) 18 ξ 2 λ −2 ∂ i λ∂ i λ∂ j χ∂ j χ − dℓ 3 (M 3 pl N 2 c ) −1 2(d−2) 2 (d−1)b 0 Z 0 λ −1 + O(λ 0 ) 19 ( γ χ) 2 ℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 b 0 Z 0 λ −1 + 2ξ 2 b 0 d−1 − 2U 2 ξ 2 b 0 Z 0 − Z 1 log λ + O(λ 0 ) 20 ξλ −1 γ χ∂ i λ∂ i χ − ξ −1 ℓ 3 (M 3 pl N 2 c ) −1 (d−2) 2 b 0 Z 1 − ξ 2 b 0 d−1 Z 0 + O(λ)
The entire boundary term that renders the variational problem for IHQCD well defined, therefore, takes the form
S IHQCD b = − S (0) + S (2) + S (4) = − 1 κ 2ˆd 4 x √ γ U (λ) + 1 2 M (λ)ξ 2 λ −2 ∂ i λ∂ i λ + Ξ(λ)R + 1 2 Θ(λ)∂ i χ∂ i χ + 1 4 a(λ) R ij R ij − 1 3 R 2 + 1 4 b(λ) R ij − 1 3 Rγ ij ∂ i χ∂ j χ − 1 2 ( γ χ) 2 + 1 4 c(λ) ∂ i χ∂ i χ 2 ,(3.8)
where we have set d = 4 and introduced functions a(λ), b(λ) and c(λ), which are given by
a(λ) = − ℓ 3 2 b −1 0 λ −1 + 2ζ log λ + c 1 , b(λ) = ℓ 3 (M 3 pl N 2 c ) −1 Z 0 b −1 0 λ −1 + 2ζ − Z 1 b 0 Z 0 log λ + c 2 , c(λ) = − ℓ 3 3 (M 3 pl N 2 c ) −2 Z 2 0 b −1 0 λ −1 + 2ζ − 2Z 1 b 0 Z 0 log λ + c 3 . (3.9) Here ζ = ξ 2 3 − U 2 ξ 2 b 2 0 . (3.10)
and c 1 , c 2 and c 3 are arbitrary constants. The terms they multiply correspond to finite local counterterms and they reflect the usual scheme dependence. The functional derivative of each of these three terms w.r.t the induced metric gives a local transverse and traceless tensor. These three tensors are given in (3.33)-(3.35) below. Moreover, the tilde in the n = 2 solution, S (4) , of the Hamilton-Jacobi equation is there to remind us that this is not the full solution of the Hamilton-Jacobi equation at order n = 2. Namely, S (4) contains only the local divergent part (plus scheme dependence) of the n = 2 solution. In addition, there is a finite part, S (4) , that corresponds to the renormalized action and which we have not determined. Recall that all the homogeneous solutions of the equations in the previous section were ignored precisely because they contribute only to this finite part, which cannot be determined from the derivative expansion of the Hamilton-Jacobi equation alone. As usual, to determine this part one must impose some regularity or boundary condition in the deep interior of the spacetime. The form of this boundary term deserves some close inspection. By far the most remarkable feature is that it is totally covariant. In particular, contrary to the boundary term (B.12) for the strictly asymptotically locally AdS dilaton-axion system, it does not break the bulk diffeomorphisms corresponding to shifts of the radial coordinate. A direct consequence of this fact is that, as we shall show below, the trace of the stress tensor does not depend explicitly on the sources, but only implicitly through the dilaton one-point function. In other words, there is no conformal anomaly in IHQCD, although conformal invariance is broken via the dilaton one-point function. A related observation is that there are terms proportional to λ −1 ∼ r at order n = 2 in the derivative expansion. These terms are in fact nothing but the conformal anomaly for a gravity-axion system in asymptotically AdS space, as can be seen from (B.12) in Appendix B. This picture suggests that the field λ effectively acts as a compensator for scale transformations or shifts of the radial coordinate 11 . Later on we will confirm this by showing that the source of the dilaton λ can be removed by a bulk diffeomorphism that induces a Weyl rescaling of the boundary metric, while the one-point function of the dilaton contains exactly the would-be conformal anomaly of the gravityaxion system.
One-point functions and Fefferman-Graham expansions
Having solved the Hamilton-Jacobi equation for IHQCD asymptotically, we are in a position to derive the full asymptotic behavior of the bulk fields, i.e. the generalized Fefferman-Graham expansions, without solving asymptotically the second order equations of motion. Not only is this approach considerably more efficient than directly solving the second order equations of motion, but it also avoids the necessity of making an ansatz for the form of the asymptotic expansions. As we shall see, the structure of these asymptotic expansions can be highly involved and practically impossible to guess a priori. However, the asymptotic solution of the Hamilton-Jacobi equation that we have obtained above already contains all information about the form of the asymptotic expansions. Additionally, it will automatically tell us how the one-point functions, i.e. the renormalized momenta, are related to the coefficients of the asymptotic expansions.
The key ingredient in deriving the asymptotic expansions from the asymptotic solution of the Hamilton-Jacobi equation is the first order flow equationṡ
γ ij = 4κ 2 γ ik γ jl − 1 3 γ kl γ ij 1 √ γ δS δγ kl , λ = κ 2 ξ −2 λ 2 1 √ γ δS δλ , χ = κ 2 Z −1 (λ) 1 √ γ δS δχ , (3.11)
which follow from combining the two expressions for the canonical momenta given in (2.6) and (2.14). Now, the solution of the Hamilton-Jacobi equation we have determined above takes the form S = S (0) + S (2) + S (4) + S (4) , (3.12) where S (4) is undetermined and remains finite when the radial cut-off is removed, while the other terms are given in (3.8). Inserting this form of the solution in (3.11) we obtain the following flow equations.
λ = ξ −2 λ 2 ∂ λ U + 1 2 λ −1 (2M − λ∂ λ M ) ∂ i λ∂ i λ − M γ λ + 1 2 ξ −2 λ 2 ∂ λ Θ∂ i χ∂ i χ + ξ −2 λ 2 ∂ λ ΞR + 1 4ξ 2 λ 2 ∂ λ a(R ij R ij − 1 3 R 2 ) + ∂ λ b (R ij − 1 3 Rγ ij )∂ i χ∂ j χ − 1 2 ( γ χ) 2 + ∂ λ c(∂ i χ∂ i χ) 2 +κ 2 ξ −2 λ 2 1 √ γ π λ(4) , (3.13) χ = −Z −1 Θ γ χ + ∂ λ Θ∂ i λ∂ i χ − 1 2 Z −1 D i b(R ij − 1 3 Rγ ij )∂ j χ + 1 2 D i (b γ χ) + 2c∂ k χ∂ k χ∂ i χ +κ 2 Z −1 1 √ γ π χ(4) ,(3.
14)
γ ij = − 2 3 U γ ij −4ΞR ij + 4∂ λ ΞD i D j λ + 4 ∂ 2 λ Ξ − 1 2 ξ 2 λ −2 M ∂ i λ∂ j λ − 2Θ∂ i χ∂ j χ + 2 3 γ ij ΞR + 1 2 M λ −2 ξ 2 ∂ k λ∂ k λ + 1 2 Θ∂ k χ∂ k χ −2aR ikjl R kl + 2 3 aRR ij + 2D (i D k (aR k j) ) − γ (aR ij ) − 2 3 D i D j (aR) +γ ij 1 2 a(R kl R kl − 1 3 R 2 ) − 1 3 D k D l (aR kl ) + 1 3 γ (aR) + 1 2 bγ ij (R kl − 1 3 Rγ kl )∂ k χ∂ l χ − bR (ik ∂ k χ∂ j) χ + 1 3 bR∂ i χ∂ j χ − bR ikjl ∂ k χ∂ l χ + 1 3 bR ij ∂ k χ∂ k χ + D (i D k (b∂ k χ∂ j) χ) − 1 2 γ (b∂ i χ∂ j χ) − 1 6 γ ij D k D l (b∂ k χ∂ l χ) − 1 3 D i D j (b∂ k χ∂ k χ) + 1 6 γ ij γ (b∂ k χ∂ k χ) + bD i D j χ γ χ − D (i (b γ χD j) χ) + 1 6 γ ij D k (b γ χD k χ) − 1 4 γ ij b( γ χ) 2 + c 1 2 γ ij ∂ k χ∂ k χ − 2∂ i χ∂ j χ ∂ l χ∂ l χ +4κ 2 γ ik γ jl − 1 3 γ kl γ ij 1 √ γ π (4) kl . (3.15)
Here, π λ(4) , π χ(4) and π (4) kl denote respectively the functional derivatives of S (4) w.r.t. the dilaton, the axion and the induced metric. Since S (4) is undetermined, so are these renormalized momenta, which define the one-point functions of the dual operators [17]. The presence of these terms in the flow equation will lead directly to the identification of the normalizable modes in the asymptotic expansions.
Although these flow equations look rather complicated, they can actually be solved in a fairly straightforward way by noticing that the asymptotic solutions will in fact be two-scale expansions. Namely, there is an expansion in exponentials of r coming from the derivative expansion. Moreover, at each order in this expansion there is an expansion in powers and possibly logarithms of r. Since the structure of the expansion in exponentials of r is pretty clear from the derivative expansion, it is useful to separate the two expansions by writing explicitly γ ij (r, x) = e 2r/ℓ γ (0)ij (r, x) + e −2r/ℓ γ (2)ij (r, x) + e −4r/ℓ γ (4)ij (r, x) + · · · ,
λ(r, x) = λ (0) (r, x) + e −2r/ℓ λ (2) (r, x) + e −4r/ℓ λ (4) (r, x) + · · · , χ(r, x) = χ (0) (r, x) + e −2r/ℓ χ (2) (r, x) + e −4r/ℓ χ (4) (r, x) + · · · , (3.16)
where the coefficients of the exponentials here are undetermined functions of r, only constrained by the requirement that their asymptotic expansions only contain powers and logarithms of r, but not exponentials. Inserting these expansions into the flow equations leads to non-linear first order equations for the order zero coefficients and linear first order equations for the higher order coefficients. Namely, for the induced metric we getγ
(0)ij + 2 1 ℓ + 1 3 U (λ (0) ) γ (0)ij = 0, (3.17) γ (2)ij + 2 3 U (λ (0) )γ (2)ij = −4Ξ(λ (0) )R ij [γ (0) ] + 4Ξ ′ (λ (0) )D (0)i D (0)j λ (0) − 2Θ(λ (0) )∂ i χ (0) ∂ j χ (0) +4 Ξ ′′ (λ (0) ) − 1 2 ξ 2 λ (0) −2 M (λ (0) ) ∂ i λ (0) ∂ j λ (0) + 2 3 γ (0)ij −U ′ (λ (0) )λ (2) + Ξ(λ (0) )R[γ (0) ] (3.18) + 1 2 Θ(λ (0) )D (0)k χ (0) D (0) k χ (0) + ξ 2 2 λ (0) −2 M (λ (0) )D (0) k λ (0) D (0)k λ (0) , γ (4)ij + 2 3 U − 2 ℓ γ (4)ij = − 2 3 λ (4) U ′ + 1 2 λ (2) 2 U ′′ γ (0)ij − 2 3 λ (2) U ′ γ (2)ij −4λ (2) Ξ ′ R ij [γ (0) ] − 4Ξ D (0)k D (0)(i γ (2) k j) − 1 2 (0) γ (2)ij − 1 2 D (0)i D (0)j γ (2) +4λ (2) Ξ ′′ D (0)i D (0)j λ (0) + 4Ξ ′ D (0)i D (0)j λ (2) + 1 2 D (0)k γ (2)ij − D (0)(i γ (2)kj) D (0) k λ (0) +2 Ξ ′′ − ξ 2 2 λ (0) −2 M ∂ (i λ (0) ∂ j) λ (2) + 4λ (2) Ξ ′′′ − ξ 2 2 λ (0) −2 M ′ − 2λ (0) −1 M ∂ i λ (0) ∂ j λ (0) −2λ (2) Θ ′ ∂ i χ (0) ∂ j χ (0) − Θ∂ (i χ (0) ∂ j) χ (2) + 2 3 γ (2)ij ΞR[γ (0) ] + ξ 2 2 λ (0) −2 M D (0)k λ (0) D (0) k λ (0) + 1 2 ΘD (0)k χ (0) D (0) k χ (0) + 2 3 γ (0)ij λ (2) Ξ ′ R[γ (0) ] + Ξ D (0) k D (0) l γ (2)kl − (0) γ (2) − γ (2) kl R[γ (0) ] kl + ξ 2 2 λ (0) −2 × λ (2) (M ′ − 2λ (0) −1 M )D (0)k λ (0) D (0) k λ (0) + 2M D (0)k λ (0) D (0) k λ (2) − M γ (2) kl ∂ k λ (0) ∂ l λ (0) + 1 2 λ (2) Θ ′ D (0)k χ (0) D (0) k χ (0) + ΘD (0)k χ (0) D (0) k χ (2) − 1 2 Θγ (2) kl ∂ k χ (0) ∂ l χ (0) −2aR ikjl [γ (0) ]R kl [γ (0) ] + 2 3 aR[γ (0) ]R ij [γ (0) ] + 2D (0)(i D (0)k (aR k j) [γ (0) ]) − (0) (aR ij [γ (0) ]) − 2 3 D (0)i D (0)j (aR[γ (0) ]) + γ (0)ij 1 2 a(R kl [γ (0) ]R kl [γ (0) ] − 1 3 R 2 [γ (0) ]) − 1 3 D (0)k D (0)l (aR kl [γ (0) ]) + 1 3 (0) (aR[γ (0) ]) + 1 2 bγ (0)ij R kl [γ (0) ] − 1 3 R[γ (0) ]γ (0)kl D (0) k χ (0) D (0) l χ (0) −bR (ik [γ (0) ]D (0) k χ (0) D (0)j) χ (0) + 1 3 bR[γ (0) ]∂ i χ (0) ∂ j χ (0) − bR ikjl [γ (0) ]D (0) k χ (0) D (0) l χ (0) + 1 3 bR ij [γ (0) ]D (0)k χ (0) D (0) k χ (0) − 1 6 γ (0)ij D (0)k D (0)l (bD (0) k χ (0) D (0) l χ (0) ) − 1 2 (0) (b∂ i χ (0) ∂ j χ (0) ) + D (0)(i D (0) k (b∂ k χ (0) ∂ j) χ (0) ) − 1 3 D (0)i D (0)j (bD (0)k χ (0) D (0) k χ (0) ) + 1 6 γ (0)ij (0) (bD (0)k χ (0) D (0) k χ (0) ) + bD (0)i D (0)j χ (0) (0) χ (0) − D (0)(i (b (0) χ (0) D (0)j) χ (0) ) + 1 6 γ (0)ij D (0)k (b (0) χ (0) D (0) k χ (0) ) − 1 4 γ (0)ij b( (0) χ (0) ) 2 +c 1 2 γ (0)ij D (0)k χ (0) D (0) k χ (0) − 2∂ i χ (0) ∂ j χ (0) D (0)l χ (0) D (0) l χ (0) +4κ 2 γ (0)ik γ (0)jl − 1 3 γ (0)kl γ (0)ij 1 √ γ (0) π (4) kl . (3.19)
Similarly, the first order equations for the coefficients in the dilaton expansion arė
λ (0) − ξ −2 λ (0) U ′ (λ (0) ) = 0, (3.20) λ (2) − 2 ℓ + 2ξ −2 λ (0) U ′ (λ (0) ) + ξ −2 λ (0) 2 U ′′ (λ (0) ) λ (2) = ξ −2 λ (0) 2 Ξ ′ (λ (0) )R[γ (0) ] −M (λ (0) ) (0) λ (0) + 1 2 λ (0) −1 2M (λ (0) ) − λ (0) M ′ (λ (0) ) D (0)i λ (0) D (0) i λ (0) + 1 2 ξ −2 λ (0) 2 Θ ′ (λ (0) )D (0)i χ (0) D (0) i χ (0) , (3.21) λ (4) − 4 ℓ + 2ξ −2 λ (0) U ′ + ξ −2 λ (0) 2 U ′′ λ (4) = U ′ + 2λ (0) U ′′ + 1 2 λ (0) 2 U ′′′ ξ −2 λ (2) 2 −λ (2) M ′ (0) λ (0) + M γ (2) ij D (0)i D (0)j λ (0) − (0) λ (2) + 1 2 (2D (0)i γ (2) i j − D (0)j γ (2) )D (0) j λ (0) + ξ −2 2 λ (0) λ (2) 2Θ ′ + λ (0) Θ ′′ D (0)i χ (0) D (0) i χ (0) + ξ −2 λ (0) λ (2) 2Ξ ′ + λ (0) Ξ ′′ R[γ (0) ] + ξ −2 2 λ (0) 2 Θ ′ 2D (0) i χ (0) D (0)i χ (2) − γ (2) ij ∂ i χ (0) ∂ j χ (0) +ξ −2 λ (0) 2 Ξ ′ D (0) i D (0) j γ (2)ij − (0) γ (2) − γ (2) ij R[γ (0) ] ij + 1 2 λ (2) λ (0) −1 M ′ − λ (0) M ′′ D (0)i λ (0) D (0) i λ (0) + 1 2 λ (0) −1 M ′ − 2λ (0) −1 M × λ (2) D (0)i λ (0) D (0) i λ (0) + λ (0) γ (2) ij ∂ i λ (0) ∂ j λ (0) − 2λ (0) D (0)i λ (0) D (0) i λ (2) + 1 4ξ 2 λ (0) 2 a ′ R ij [γ (0) ]R ij [γ (0) ] − 1 3 R 2 [γ (0) ] + c ′ D (0)i χ (0) D (0) i χ (0) 2 +b ′ R ij [γ (0) ] − 1 3 R[γ (0) ]γ (0)ij D (0) i χ (0) D (0) j χ (0) − 1 2 ( (0) χ (0) ) 2 +κ 2 ξ −2 λ (0) 2 1 √ γ (0) π λ(4) . (3.22)
Finally, for the axion we getχ (0) = 0, (3.23)
χ (2) − 2 ℓ χ (2) = −Z −1 (λ (0) ) Θ(λ (0) ) (0) χ (0) + Θ ′ (λ (0) )D (0)i λ (0) D (0) i χ (0) ,(3.24)χ (4) − 4 ℓ χ (4) = −λ (2) Z −1 Θ ′ − Z −1 Z ′ Θ (0) χ (0) + Z −1 Θγ (2) ij D (0)i D (0)j χ (0) −Z −1 Θ (0) χ (2) − 1 2 2D (0)i γ (2) i j − D (0)j γ (2) D (0) j χ (0) −Z −1 Θ ′ D (0)i χ (0) D (0) i λ (2) + D (0)i χ (2) D (0) i λ (0) − γ (2) ij ∂ i λ (0) ∂ j χ (0) −Z −1 λ (2) Θ ′′ − Z −1 Z ′ Θ ′ D (0)i λ (0) D (0) i χ (0) − 1 2 Z −1 D (0) i b R ij [γ (0) ] − 1 3 R[γ (0) ]γ (0)ij D (0) j χ (0) + 1 2 D (0)i (b (0) χ (0) ) +2cD (0)k χ (0) D (0) k χ (0) ∂ i χ (0) + κ 2 Z −1 1 √ γ (0) π χ(4) .(3.25)
In all these first order equations all functions are functions of λ (0) (r, x) and primes denote derivative w.r.t. λ (0) . From these equations we can now easily construct the asymptotic expansions. The non-linear equations for the zero order coefficients can be integrated exactly in terms of the function U (λ (0) ). Using the asymptotic form of U (λ) in (3.4) one obtains the asymptotic expansions
γ (0)ij (r, x) = r ℓ 2ξ 2 /3 1 − 4U 2 3b 2 0 ℓ r log(r/ℓ) − 2 3 U 2 b 2 0 + b 0 ξ 2λ (0) (x) ℓ r + 4U α 3(α − 2)b α 0 ℓ r α−1 + O log 2 (r/ℓ)/r 2 ḡ (0)ij (x), (3.26) λ (0) (r, x) = ℓ b 0 r + 2U 2 ξ 2 b 3 0 ℓ r 2 log(r/ℓ) + ℓ r 2λ (0) (x) − αU α (α − 2)ξ 2 b α+1 0 ℓ r α +O log 2 (r/ℓ)/r 3 ,χ (0) (r, x) =χ (0) (x),(3.27)
whereḡ (0)ij (x),λ (0) (x) andχ (0) (x) are arbitrary and we identify them with the sources of the dual operators. A striking feature of the zero order dilaton expansion is that the source,λ (0) (x), that couples to the dual operator appears not in the leading order, but at O(1/r 2 ). This is related to the fact thatλ (0) (x) can be removed by a bulk diffeomorphism corresponding to shifts of the radial coordinate, which we will demonstrate below. However, we have already made extensive use of the identity ∂ i λ = O(λ 2 ) in deriving the results in Tables 5 and 6, which follows precisely from the observation that the source of the dilaton appears at subleading order. At the second order we get
γ (2)ij (r, x) =ḡ (2)ij (x) + O 1 r log(r/ℓ) , λ (2) (r, x) = ℓ r 2+2ξ 2 /3 λ (2) (x) + O 1 r log(r/ℓ) , χ (2) (r, x) = ℓ r 2ξ 2 /3 χ (2) (x) + O 1 r log(r/ℓ) , (3.28) whereḡ (2)ij (x) = − ℓ 2 2 R ij [ḡ (0) ] − 1 6ḡ (0)ij R[ḡ (0) ] (3.29) − M 3 pl N 2 c −1 Z 0 ∂ iχ(0) ∂ jχ(0) − 1 6ḡ (0)ijD(0)kχ(0)D(0) kχ (0) , λ (2) (x) = − ℓ 2 4 1 6b 0 R[ḡ (0) ] − (0)λ(0) (3.30) + 1 2ξ 2 b 2 0 M 3 pl N 2 c −1 Z 1 − ξ 2 b 0 3 Z 0 D (0)kχ(0)D(0) kχ (0) , χ (2) (x) = ℓ 2 4 M 3 pl N 2 c −1 (0)χ(0) . (3.31)
Finally, at fourth order the asymptotic expansions take the form
γ (4)ij (r, x) = ℓ r 2ξ 2 /3 r ℓ ḡ (4)ij (x) + log(r/ℓ)g (4)ij (x) +ĝ (4)ij (x) + O log(r/ℓ) r , λ (4) (r, x) = ℓ r 4ξ 2 /3 λ (4) (x) + O log(r/ℓ) r , (3.32) χ (4) (r, x) = ℓ r 4ξ 2 /3 r ℓ χ (4) (x) + log(r/ℓ)χ (4) (x) +χ (4) (x) + O log(r/ℓ) r ,
where the termsĝ (4)ij (x),λ (4) (x) andχ (4) (x) are undetermined and are therefore identified with the normalizable modes. The flow equations relate these to the undetermined renormalized momenta, thus leading to the expressions for the one-point functions in terms of the coefficients of the asymptotic expansions, which we present below. In order to write down the explicit expressions for the coefficients in these fourth order expansions, it is useful to define the traceless tensors
H 1ij = −2 R ikjl − 1 4 R klḡ(0)ij R kl + 2 3 R R ij − 1 4 Rḡ (0)ij + 1 3D (0)iD(0)j R − (0) R ij + 1 6ḡ (0)ij (0) R, (3.33) H 2ij = −R ikjl + 1 2ḡ (0)ij R kl − R (ikḡ(0)j)l − 1 6 Rḡ (0)ijḡ(0)kl + 1 3 Rḡ (0)ikḡ(0)jl + 1 3 R ijḡ(0)kl D (0) kχ (0)D(0) lχ (0) +D (0)(iD(0) k (∂ kχ(0) ∂ j)χ(0) ) − 1 2 (0) (∂ iχ(0) ∂ jχ(0) ) − 1 6ḡ (0)ijD(0)kD(0)l (D (0) kχ (0)D(0) lχ (0) ) − 1 3D (0)iD(0)j (D (0)kχ(0)D(0) kχ (0) ) + 1 6ḡ (0)ij (0) (D (0)kχ(0)D(0) kχ (0) ) +D (0)iD(0)jχ(0) (0)χ(0) −D (0)(i ( (0)χ(0)D(0)j)χ(0) ) + 1 6ḡ (0)ijD(0)k ( (0)χ(0)D(0) kχ (0) ) − 1 4ḡ (0)ij ( (0) χ (0) ) 2 , (3.34) H 3ij = 1 2ḡ (0)ijD(0)kχ(0)D(0) kχ (0) − 2∂ iχ(0) ∂ jχ(0) D (0)lχ(0)D(0) lχ (0) . (3.35)
As mentioned in the previous section, these correspond respectively to the derivative of the three terms proportional to c 1 , c 2 and c 3 in the boundary term (3.8) w.r.t. the induced metric. All curvatures here are curvatures of the boundary metricḡ (0)ij . The first two coefficients in the fourth order expansion of the metric are just linear combinations of these three traceless tensors. Namely,
g (4)ij (x) = ℓ 4 8 H 1ij − ℓ 4 4 Z 0 M 3 pl N 2 c −1 H 2ij + ℓ 4 12 Z 2 0 M 3 pl N 2 c −2 H 3ij , (3.36) g (4)ij (x) = ℓ 4 12 2U 2 b 2 0 − ξ 2 H 1ij − ℓ 4 4 Z 0 M 3 pl N 2 c −1 4U 2 3b 2 0 − 2ξ 2 3 + Z 1 b 0 Z 0 H 2ij + ℓ 4 12 Z 2 0 M 3 pl N 2 c −2 4U 2 3b 2 0 − 2ξ 2 3 + 2Z 1 b 0 Z 0 H 3ij . (3.37)
Moreover, for the coefficients of the axion expansion we havē
χ (4) (x) = ℓ 4 8D (0) i M 3 pl N 2 c −1 R ij [ḡ (0) ] − 1 3 R[ḡ (0) ]ḡ (0)ij D (0) jχ (0) + 1 2 D (0)i (0) χ (0) − 2 3 Z 0 M 3 pl N 2 c −2D (0)kχ(0)D(0) kχ (0) ∂ iχ(0) , (3.38) andχ (4) (x) = ℓ 4 8D (0) i M 3 pl N 2 c −1 8U 2 3b 2 0 − 2ξ 2 3 + Z 1 b 0 Z 0 × R ij [ḡ (0) ] − 1 3 R[ḡ (0) ]ḡ (0)ij D (0) jχ (0) + 1 2 D (0)i (0) χ (0) − 2 3 Z 0 M 3 pl N 2 c −2 8U 2 3b 2 0 − 2ξ 2 3 + 2Z 1 b 0 Z 0 D (0)kχ(0)D(0) kχ (0) ∂ iχ(0) . (3.39)
Finally, we can write down the general expressions for the one-point functions, i.e. the renormalized momenta, in terms of the coefficients of the asymptotic expansions. Starting from the dilaton, the exact one-point function in given by
O λ ren = − b 0 ℓ 3 8κ 2 32b 0 ξ 2 ℓ 4λ (4) + R ij [ḡ (0) ]R ij [ḡ (0) ] − 1 3 R 2 [ḡ (0) ] −2Z 0 M 3 pl N 2 c −1 R ij [ḡ (0) ] − 1 3 R[ḡ (0) ]ḡ (0)ij D (0) iχ (0)D(0) jχ (0) − 1 2 ( (0)χ(0) ) 2 + 2 3 Z 2 0 M 3 pl N 2 c −2 D (0) iχ (0)D(0)iχ(0) 2 . (3.40)
Notice that this gets contributions from the dilaton normalizable mode,λ (4) , plus a combination of the metric and axion sources which is nothing but the conformal anomaly of the axion-gravity system in strictly asymptotically AdS space (cf. (B.12)). All results quoted so far are valid for arbitrary dilaton sourceλ (0) (x). However, in order to simplify the expressions for the stress tensor and axion one-point functions, we will only give the expressions for constantλ (0) , independent of the transverse coordinates. This is not a big disadvantage since, as we shall see, the dilaton source can be removed or restored by a bulk diffeomorphism corresponding to a Weyl transformation of the boundary metric. The full dependence on a generic dilaton source can therefore be restored starting from the expressions given below for constant λ (0) (x), by a suitable boundary Weyl transformation.
The one-point function of the stress tensor can be written in the form
T ij ren = 2 κ 2 ℓ Ω ij − Tr Ωḡ (0)ij − 1 4b 0 O λ renḡ(0) ij ,(3.41)
where the tensor Ω ij is given by
Ω ij =ĝ (4)ij + ξ 2 b 0 3λ (4)ḡ(0)ij − ℓ 4 8 c ′ 1 H 1ij + ℓ 4 4 Z 0 M 3 pl N 2 c −1 c ′ 2 H 2ij − ℓ 4 12 Z 2 0 M 3 pl N 2 c −2 c ′ 3 H 3ij + ℓ 2 4 D (0)kD(0)(iḡ(2) k j) − 1 2 (0)ḡ(2)ij − 1 2D (0)iD(0)j Trḡ (2) − ℓ 2 8 Z 0 M 3 pl N 2 c −1 ∂ (iχ(0) ∂ j)χ(2) − ℓ 2 24ḡ (2)ij R + Z 0 M 3 pl N 2 c −1D (0)kχ(0)D(0) kχ (0) − ℓ 2 24ḡ (0)ij D (0) kD (0) lḡ (2)kl − (0) Trḡ (2) −ḡ (2) kl R kl − ℓ 2 12 Z 0 M 3 pl N 2 c −1ḡ (0)ij D (0) kχ (0)D(0)kχ(2) − 1 2ḡ (2) kl ∂ kχ(0) ∂ lχ(0) + ℓ 4 192ḡ (0)ij R kl R kl − 1 3 R 2 −2Z 0 M 3 pl N 2 c −1 R kl − 1 3 Rḡ (0)kl D (0) kχ (0)D(0) lχ (0) − 1 2 ( (0)χ(0) ) 2 + 2 3 Z 2 0 M 3 pl N 2 c −2 D (0) iχ (0)D(0)iχ(0) 2 . (3.42)
Here we have defined the shifted constants
c ′ 1 = c 1 + 2U 2 3b 2 0 + 2ξ 2 3 − 1 b 0λ(0) − 2ζ log b 0 + 1 4 4ξ 2 3 − 1 , c ′ 2 = c 2 + 2U 2 3b 2 0 + 2ξ 2 3 − 1 b 0λ(0) − 2ζ − Z 1 b 0 Z 0 log b 0 + 1 4 4ξ 2 3 − 1 , c ′ 3 = c 3 + 2U 2 3b 2 0 + 2ξ 2 3 − 1 b 0λ(0) − 2ζ − 2Z 1 b 0 Z 0 log b 0 + 1 4 4ξ 2 3 − 1 .
(3.43)
In principle, one can set these constants to zero by suitable choice of scheme, i.e. by a suitable choice of c 1 , c 2 and c 3 , but we give the full expressions so that one knows exactly what scheme needs to be chosen to achieve this. Finally, the one-point function of the operator dual to the axion is
O χ ren = − 4Z 0 κ 2 ℓχ (4) + ℓ 3 2κ 2 Z 0 M 3 pl N 2 c −1 c ′′ 2D(0) i R ij − 1 3 Rḡ (0)ij D (0) jχ (0) + 1 2D (0)i (0)χ(0) − ℓ 3 3κ 2 Z 2 0 M 3 pl N 2 c −2 c ′′ 3D(0) i D (0)kχ(0)D(0) kχ (0) ∂ iχ(0) + ℓ 2κ 2 Z 0 M 3 pl N 2 c −1 × (3.44) (0)χ(2) − 1 2 2D (0)iḡ(2) i j −D (0)j Trḡ (2) D (0) jχ (0) −ḡ (2) ijD (0)iD(0)jχ(0) ,
where again we have introduced the constants
c ′′ 2 = c ′ 2 + 2U 2 3b 2 0 + 2ξ 2 3 b 0λ(0) − Z 1 b 0 Z 0 , c ′′ 3 = c ′ 3 + 2U 2 3b 2 0 + 2ξ 2 3 b 0λ(0) − Z 1 b 0 Z 0 , (3.45)
to abbreviate the above expression.
Asymptotic diffeomorphisms and Ward identities
Now that we have determined the general form of the asymptotic expansions for IHQCD and we have identified the exact one-point functions, we can proceed with the derivation of the holographic Ward identities. These follow as a consequence of the existence of a class of asymptotic bulk diffeomorphisms that preserve the structure of the asymptotic expansions. Let us consider a generic infinitesimal bulk diffeomorphism, δx µ = −ξ µ , and demand that it preserves the gauge fixed form of the metric, namely that it does not modify the lapse and shift functions. This requirement leads to a pair of equations for the vector field ξ µ , namely L ξ g rr =ξ r = 0,
L ξ g ri = γ ij (ξ j + ∂ j ξ r ) = 0, (3.46)
where L ξ is the Lie derivative w.r.t. the bulk vector ξ µ . Solving these conditions gives
ξ r = δσ(x), ξ i = ξ i o (x) + ∂ j δσ(x)ˆ∞ r dr ′ γ ji (r ′ , x),(3.47)
where σ(x) is an arbitrary function of the transverse coordinates and ξ i o (x) is an arbitrary transverse vector field. Inserting now the asymptotic form of the induced metric we obtain
ξ i = ξ i o (x) + ℓ 2 e −2r/ℓ ℓ r 2ξ 2 3ḡ (0) ij ∂ j δσ(x) + O e −2r/ℓ r − 2ξ 2 3 −1 log(r/ℓ) . (3.48)
Under this bulk diffeomorphism then the induced fields transform as
δ ξ γ ij = L ξ g ij = L ξ γ ij + 2K ij ξ r = L ξo γ ij − 2 d − 1 U (λ)γ ij δσ(x) + O(r −2ξ 2 /3 ), δ ξ λ = L ξ λ = L ξ λ + ξ rλ = ξ i o ∂ i λ + ξ −2 λ 2 ∂U ∂λ δσ(x) + O(r −2−2ξ 2 /3 e −2r/ℓ ), δ ξ χ = L ξ χ = L ξ χ + ξ rχ = ξ i o ∂ i χ + O(r −2ξ 2 /3 e −2r/ℓ ),(3.49)
where L ξ denotes the Lie derivative w.r.t. the transverse components ξ i of the bulk vector field ξ. It follows that the sources,ḡ (0)ij (x),λ (0) (x) andχ (0) (x) transform under such a diffeomorphism as
δ ξḡ(0)ij = L ξoḡ(0)ij + 2 ℓ δσ(x)ḡ (0)ij , δ ξλ(0) = ξ i o (x)∂ iλ(0) − 1 b 0 ℓ δσ(x), δ ξχ(0) = ξ i o (x)∂ iχ(0) .
(3.50)
There is nothing surprising about the transformation of the metric and axion sources. They are exactly as they would be in the case of strictly asymptotically AdS space. Namely, the bulk diffeomorphisms that preserve the form of the asymptotic expansions correspond to arbitrary boundary diffeomorphisms parameterized my ξ i o (x), as well as boundary Weyl transformations, parameterized by the function σ(x). What is rather unusual, is the transformation of the dilaton source, λ (0) under the Weyl transformation δσ. Contrary to the usual multiplicative transformation of the sources, the transformation ofλ (0) is additive under boundary Weyl rescalings. This means that one can in fact remove the dilaton source completely by means of a boundary Weyl rescaling.
We can now also determine the transformation of the one-point functions under boundary Weyl rescalings. The above transformations of the sources imply that the corresponding functional derivatives transform as
δ σ δ δḡ (0)ij = − 2 ℓ δσ(x) δ δḡ (0)ij , δ σ δ δλ (0) = 0, δ σ δ δχ (0) = 0. (3.51)
Moreover, the renormalized action S ren = lim r→∞ S (4) , (3.52)
is invariant under any bulk diffeomorphism since the boundary term (3.8) does not break the bulk diffeomorphisms. Hence, Inserting the transformation of the sources under the bulk diffeomorphisms considered above and using the fact that δσ(x) and ξ i o (x) are arbitrary leads respectively to
δ σ T i j ren = δ σ 1 ḡ (0)ḡ (0) ik δS ren δḡ (0)kj = − 4 ℓ δσ T i j ren , δ σ O λ ren = δ σ 1 ḡ (0) δS ren δλ (0) = − 4 ℓ δσ O λ ren , δ σ O χ ren = δ σ 1 ḡ (0) δS ren δχ (0) = − 4 ℓ δσ O χ ren ,(3.T i i ren = − 1 b 0 O λ ren , D (0) i T i j ren + O λ ren ∂ jλ(0) + O χ ren ∂ jχ(0) = 0. (3.55)
An immediate consequence of the trace Ward identity is that the tensor Ω ij introduced above is traceless. Finally, let us examine a bit closer the relation between the dilaton sourceλ (0) and boundary Weyl transformations. Note that the assignment of sources in the leading order asymptotic solutions (3.26) is rather arbitrary. In particular, we could have included a factor of the dilaton source in the definition of the boundary metric. Suppose, in particular, that we definē
g (0)ij (x) = e −2b 0λ(0) (x)ǧ (0)ij (x),(3.56)
Then, inserting this boundary metric back in (3.26) 12 and evaluating the variation of the induced fields with respect to variations of the dilaton sourceλ (0) , one immediately sees from (3.49) that transformations of the dilaton source correspond precisely to boundary Weyl transformations upon the identification σ(x) = −b 0 ℓλ (0) (x). (3.57) This observation proves that the source of the dilaton is gauge freedom that can be removed by a bulk diffeomorphism corresponding to a boundary Weyl transformation. This property can also be used to restore the full dependence on the dilaton source of the one-point functions of the stress tensor and the axion.
Concluding remarks
We considered a generic dilaton-axion system coupled to Einstein-Hilbert gravity in arbitrary spacetime dimension and we carried out the procedure of holographic renormalization of this action for dimension up to and including five dimensions. The general boundary term that renders the variational problem for this action well defined is summarized in Tables 2 and 3. This result is applicable to a very wide range of holographic models in the literature, including N = 4 super Yang-Mills in four dimensions, Improved Holographic QCD and non-conformal branes. We explicitly evaluated this general boundary term for a constant dilaton potential, corresponding to the standard dilatonaxion system dual to the complexified coupling of N = 4 super Yang-Mills in four dimensions, in Appendix B, and for IHQCD in Section 3. In particular, we systematically derived the generalized Fefferman-Graham asymptotic expansions, provided exact expressions for the one-point functions in the presence of sources, and proved the holographic holographic Ward identities by studying the asymptotic bulk diffeomorphisms that preserve the form of the asymptotic expansions.
In the case of IHQCD, an important lesson from the analysis is that the source of the dilaton is not a physical coupling, but its value can be thought of as an energy scale. In particular, changes in the dilaton source can be absorbed by a Weyl rescaling of the boundary metric. Moreover, the operator dual to the dilaton field λ is the combination
O λ = β(λ Y M )Tr F 2 ,(4.1)
which has fixed scaling dimension 4 under renormalization group flow. This is the combination that appears in the trace Ward identity 3.55, since the coefficient relating the trace of the stress tensor to the operator O λ is a constant, independent of the renormalization group scaleλ (0) . Our calculation for IHQCD, independently of the legitimacy of the model as a physically sound holographic model dual to pure Yang-Mills theory in four dimensions, provides us with an explicit example of a gravity model that can accommodate operators with running scaling dimensions. This is particularly interesting since it allows us, in principle, to develop supergravity holographic models capturing the dynamics of operators with non-protected scaling dimensions.
A Functional integration
In this appendix we outline the derivation of the functional integration formulas in Table 1. In particular, the question we want to address is the following: given a local functional R (2n) (ϕ) of the scalar field, ϕ, what is the local functional F (2n) (ϕ) such that (cf. (2.38))
δϕ U ′ (ϕ) e an(ϕ) R (2n) (ϕ) = δ ϕ F (2n) (ϕ) + e an(ϕ) ∂ i v (2n) i (ϕ, δϕ), (A.1)
where a n (ϕ) is a prescribed function of ϕ and v (2n) i (ϕ, δϕ) is some vector field? If the source R (2n) (ϕ) does not involve spacetime derivatives of the scalar field the answer to this question is given by simple integration. However, if R (2n) (ϕ) involves derivatives of ϕ, then determining F (2n) (ϕ) becomes less trivial. It turns out that one can still find a general formula if R (2n) (ϕ) involves first derivatives of the scalar field, but once R (2n) (ϕ) contains second and higher order derivatives of the scalar field finding a general formula becomes much harder. What we will do instead here is to consider only the sources R (2n) (ϕ) which are relevant to our computation of the solution of the Hamilton-Jacobi equation in the main body of the paper.
• R (2n) (ϕ) = r 1 m (ϕ)t i 1 i 2 ...im ∂ i 1 ϕ∂ i 2 ϕ . . . ∂ im ϕ
The first example is a generic source that is polynomial in first derivatives. Here, t i 1 i 2 ...im is an arbitrary totally symmetric tensor that does depend on ϕ. In this case we can write F (2n) as
F (2n) = e an α(ϕ)t i 1 i 2 ...im ∂ i 1 ϕ∂ i 2 ϕ . . . ∂ im ϕ + D i β(ϕ)t ii 2 ...im ϕ∂ i 2 ϕ . . . ∂ im ϕ , (A.2)
where the functions α(ϕ) and β(ϕ) are to be determined. Evaluating the variation of this expression and inserting the result in (A.1) we obtain the two equations
β = m a ′ n α α ′ − m a ′′ n a ′ n α = r 1 m U ′ , (A.3)
which can be solved to determine α(ϕ) and β(ϕ). Since β(ϕ) contributes a total derivative to Hamilton's principal function, i.e. to e −an(ϕ) F (2n) , we are only interested in α(ϕ), which is given by
α(ϕ) = a ′m n e −anˆϕ dφ U ′ e an a ′−m n r 1 m (φ) = ϕ n,m r 1 m (φ). (A.4) • r 2 (ϕ)t ij D i D j ϕ
Similarly, for a source with a single second derivative of the scalar field we can write
F (2n) = e an α(ϕ)t ij D i D j ϕ + β(ϕ)t ij ∂ i ϕ∂ j ϕ + D i γ(ϕ)t ij ∂ j ϕ + δ(ϕ)D j t ij . (A.5)
Here, t ij is again a symmetric tensor that does not depend on ϕ. Inserting the variation of this expression in (A.1) leads to the following equations for the functions α(ϕ), β(ϕ), γ(ϕ) and δ(ϕ)
a ′ n (β + γ ′ ) + α ′′ − β ′ = 0, a ′ n (γ + δ ′ ) + 2(α ′ − β) = 0, a ′ n δ + α = 0, 2α ′ + a ′ n (α + γ) − 2β = r 2 U ′ . (A.6)
These can be immediately solved to obtain α(ϕ) = a ′ n e −anˆϕ dφ U ′ e an a ′−1 n r(φ) = ϕ n,1 r(φ), (A.7)
β(ϕ) = −a ′2 n e −anˆϕ dφ U ′ e an a ′−2 n U ′ a ′ n ∂ 2 ϕ 1 a ′ n α(φ) = − ϕ n,2 U ′ a ′ n ∂ 2 ϕ 1 a ′ n α(φ). • r 1 2 2 (ϕ)t ijkl 1 + s 1 2 2 (ϕ)t ijkl 2 ∂ i ϕ∂ j ϕD k D l ϕ + r 2 2 (ϕ)t ijkl 1 + s 2 2 (ϕ)t ijkl 2 D i D j ϕD k D l ϕ
As a final example we consider a generic term with four derivatives, but we restrict to covariantly constant tensors t ijkl 1 and t ijkl 2 . In particular, for our purposes it suffices to take these two tensors to be the two linearly independent tensors constructed out of the metric. Namely, we will take
t ijkl 1 = 1 3 γ ik γ jl + γ il γ jk + γ ij γ kl , t ijkl 2 = 1 3 γ ik γ jl + γ il γ jk − 2γ ij γ kl . (A.8)
Moreover, we need not consider a source for four first derivatives since we have already computed the result for an arbitrary number of first derivatives above. Writing then
F (2n) = e an A ijkl D i D j ϕD k D l ϕ + B ijkl ∂ i ϕ∂ j ϕD k D l ϕ + C ijkl ∂ i ϕ∂ j ϕ∂ k ϕ∂ l ϕ +D i E ijkl D j D k D l ϕ + H ijkl ∂ j ϕD k D l ϕ + G ijkl ∂ j ϕ∂ k ϕ∂ l ϕ , (A.9)
and inserting the variation of this expression in (A.1) we obtain the set of coupled equations
3A ′ijkl + a ′ n A ijkl + B ilkj + B ikjl − 2B ijkl + a ′ n H ijkl = 1 U ′ r 2 2 (ϕ)t ijkl 1 + s 2 2 (ϕ)t ijkl 2
2A ′ijkl + 2 B ′kjil + B ′ljik − 12C ijkl + a ′ n B ijkl + H ′ijkl + G klij + G ljik + G kjil = 1 U ′ r 1 2 2 (ϕ)t ijkl 1 + s 1 2 2 (ϕ)t ijkl 2 , −3C ′ijkl + B ′′(ijkl) + a ′ n C ijkl + G ′(ijkl) = 0, 4A ′ijkl + B kijl + B lijk − 2B ijkl + a ′ n E ′ijkl + H ijkl = 0, 2A ijkl + a ′ n E ijkl = 0. (A.10)
Here, the parentheses in the indices mean total symmetrization. Note that these are five equations for six undetermined tensors. They can be solved as follows. First we can use the last two equations to eliminate E ijkl and H ijkl . This leads to the decoupled equation
A ′ijkl + a ′ n − 2a ′′ n a ′ n A ijkl = 1 U ′ r 2 2 (ϕ)t ijkl 1 + s 2 2 (ϕ)t ijkl 2 , (A.11)
whose solution is A ijkl = a ′2 n e −anˆϕ dφ U ′ e an a ′−2 n r 2 2 (ϕ)t ijkl 1 + s 2 2 (ϕ)t ijkl 2 .
(A.12)
Next, we can set a ′ n G ijkl = 4C ijkl , (A.13) so that we obtain the following two equations for B ijkl and C ijkl : These results can be simplified considerably by noticing that the terms involving α combine into a total derivative, up to a homogeneous term that is irrelevant since it contributes a finite piece. To see this, integrate by parts the first term in αt ijkl 1 ∂ i ϕ∂ j ϕD k D l ϕ + ct ijkl 1 ∂ i ϕ∂ j ϕ∂ k ϕ∂ l ϕ,
2B ′ijkl + B ′kijl + B ′lijk + a ′ n −2a(A.22) which gives − 2αt ijkl 1 ∂ i ϕ∂ j ϕD k D l ϕ + (c − α ′ )t ijkl 1 ∂ i ϕ∂ j ϕ∂ k ϕ∂ l ϕ. (A.23)
We can now replace the first of these expressions in F (2n) by 2/3 of the first expression plus 1/3 of the second, such that the coefficient of t ijkl 1 ∂ i ϕ∂ j ϕD k D l ϕ vanishes. This replaces c bỹ c = c − α ′ /3. However, we have seen above that the only source of the equation satisfied by c, orc now, is the coefficient of t ijkl 1 ∂ i ϕ∂ j ϕD k D l ϕ. Since we have now set this coefficient to zero,c satisfies a homogeneous equation and hence we can setc = 0. We therefore conclude that, without loss of generality, we have B ijkl = β(ϕ)t ijkl 2 , C ijkl = 0, (A. 24) for any σ(ϕ). Notice, in particular, that the source r 1 2 2 (ϕ) gives no contribution at all.
B Dilaton-axion system with constant dilaton potential
In this appendix 13 we focus on the special case of a constant dilaton potential (ℓ = 1)
V (ϕ) = d(d − 1) = 12. (B.1)
This special case drastically simplifies the solution of both the zero order problem (2.26) and the linear recursion equations (2.31). In particular, the first order differential equations are replaced by algebraic equations. Namely, we have 14
− d d − 1 U 2 + d(d − 1) = 0, (B.3) − d − 2n d − 1 U L (2n) = R (2n) , n > 0. (B.4)
From these we immediately obtain U = −(d − 1),
L (2) = − 1 2(d − 2)κ 2 √ γ R[γ] − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ , (B.5)
where we have fixed the sign of U by demanding that the solution leads, via the relatioṅ
γ ij ∼ − 2 d − 1 U γ ij , (B.6)
to the correct asymptotics for the induced metric γ ij . In order to compute L (4) we need to compute the momenta from L (2) . We easily get
π (2) ij = 1 2(d − 2)κ 2 √ γ R ij − ∂ i ϕ∂ j ϕ − Z(ϕ)∂ i χ∂ j χ − 1 2 γ ij R − ∂ k ϕ∂ k ϕ − Z(ϕ)∂ k χ∂ k χ , π ϕ(2) = − 1 (d − 2)κ 2 √ γ γ ϕ − 1 2 Z ′ (ϕ)∂ i χ∂ i χ , π χ(2) = − 1 (d − 2)κ 2 √ γ Z(ϕ) γ χ + Z ′ (ϕ)∂ i ϕ∂ i χ . (B.7)
13 The author is grateful to David Mateos and Diego Trancanelli for checking the results of this appendix and for pointing out typos in a preliminary version. Of course, the author is solely responsible for any remaining typos.
14 In addition to the constant solution U (ϕ) = −(d − 1) of (2.29) there is additionally a one-parameter family of non-constant solutions given by [30,31]
U (ϕ) = −(d − 1) cosh d d − 1 (ϕ − ϕo) , (B.2)
for some arbitrary constant ϕo and AdS asymptotics requires that ϕ → ϕo asymptotically. However, this solution only allows for a constant non-normalizable mode for the dilaton, i.e. ϕo cannot be a function of the transverse coordinates x and hence it does not correspond to the most general asymptotics. In fact, a domain wall with such a 'fake superpotential' describes a vacuum where the operator dual to the dilaton field has acquired a VEV [31]. Finally note that a special feature of equation (2.26) with constant scalar potential is that the solution describing the most general asymptotics, i.e. U (ϕ) = −(d − 1), is isolated in the space of solutions, while the one-parameter family of solutions describes special asymptotics. This is the reverse of what happens for non-constant scalar potentials.
Hence, from (2.32), we evaluate
R (4) = −2κ 2 γ − 1 2 π (2) i j π (2) j i − 1 d − 1 π (2) 2 + 1 4 π ϕ(2) 2 + 1 4 Z −1 (ϕ)π χ(2) 2 = − 1 2κ 2 1 (d − 2) 2 √ γ R ij R ij + (∂ i ϕ∂ i ϕ) 2 + Z 2 (ϕ)(∂ i χ∂ i χ) 2 − 2R ij ∂ i ϕ∂ j ϕ −2Z(ϕ)R ij ∂ i χ∂ j χ + 2Z(ϕ)(∂ i ϕ∂ i ϕ)(∂ j χ∂ j χ) − d 4(d − 1) (R − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ) 2 + γ ϕ − 1 2 Z ′ (ϕ)∂ i χ∂ i χ 2 + Z(ϕ) γ χ + ∂ ϕ log Z(ϕ)∂ i ϕ∂ i χ 2 .
(B.8)
Now, the recursion relations tell us that L (4) is given by
(d − 4)L (4) = R (4) , (B.9)
which is ill defined in d = 4. This problem is of course well known and it is related to the breakdown of full diffeomorphism invariance of the Hamilton-Jacobi functional, which in turn leads to the conformal anomaly of the dual conformal field theory [18]. It was shown in [17] that the Hamilton-Jacobi approach can be still applied in this case and reproduces the results of the Fefferman-Graham expansion [6] provided the radial cut-off is related to the deviation of d from the desired value of 4. In particular, we set
r 0 = 1 d − 4 , (B.10)
and define L (4) | r 0 = −2r 0L(4) | r 0 , so thatL This then leads to the full counterterm action for d = 4. Namely, dropping again the subscript 0 from the radial cut-off,
S ct = 1 κ 2ˆd 4 x √ γ 3 + 1 4 R − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ − log e −2r 1 16 R ij R ij + (∂ i ϕ∂ i ϕ) 2 + Z 2 (ϕ)(∂ i χ∂ i χ) 2 − 2R ij ∂ i ϕ∂ j ϕ − 2Z(ϕ)R ij ∂ i χ∂ j χ +2Z(ϕ)(∂ i ϕ∂ i ϕ)(∂ j χ∂ j χ) − 1 3 (R − ∂ i ϕ∂ i ϕ − Z(ϕ)∂ i χ∂ i χ) 2 + γ ϕ − 1 2 Z ′ (ϕ)∂ i χ∂ i χ 2 + Z(ϕ) γ χ + ∂ ϕ log Z(ϕ)∂ i ϕ∂ i χ 2 . (B.12)
This expression does not quite agree with the one given in (26) of [32], which we believe is incorrect.
B.1 Asymptotic expansions
The above recursive solution of the Hamilton-Jacobi equation, implies that the canonical momenta take the form π ij = π (0) ij + π (2) ij + (−2r)π (4) ij + π (4) ij + . . . , (B.13) and similarly for the dilaton and the axion, obtained by differentiating Hamilton's principal function w.r.t. the corresponding induced field. The form of these expansions implies in turn that the induced fields admit the asymptotic expansions
Figure 1 :
1A schematic illustration of the algorithm used to determine Hamilton's principal function iteratively.
5 :
5The leading form of the solution (2.41) of the Hamilton-Jacobi equation at order n = 1 for IHQCD. The third column gives the asymptotic form of the terms in the fourth column of
53) that is all one-point functions transform homogeneously under boundary Weyl rescalings. The Ward identities now follow from the identity δ ξ S ren =ˆd d x − 1 2 δ ξḡ(0)ij T ij ren + δ ξλ(0) O λ ren + δ ξχ(0) O χ ren = 0. (3.54)
for B ijkl we first decompose it asB ijkl = α(ϕ)t ijkl 1 + β(ϕ)t ijkl 2 . (A.17)Inserting this into the above equation for B ijkl leads to the two decoupled equations4α ′ + a ′ n e −anˆϕ dφ U ′ e an a ′−2 n r 2 2 (φ), ω = s 1 2 2 (ϕ) − 2U ′ a ′an/4ˆϕ dφ U ′ e an/4 σ, β = a ′3 n e −anˆϕ dφ U ′ e an a ′−3 n ω. (A.20) Moreover, since B (ijkl) = αt ijkl 1 , we have C ijkl = −t ijkl 1 a ′4 n e −anˆϕ dφ U ′ e an a ′−4 n α ′′ . (A.21)
Table 1 :
1The result of the functional integration described by (2.38) for various source terms.
Table 2 :
2Summary of the source terms (2.40) and the corresponding solution (2.41) of the Hamilton-Jacobi equation at order n = 1. The source term is determined iteratively as in (2.32), while L (2n) are determined by solving the linear equations (2.31) via the integration procedure described in the text.
Table 3 :
3This table summarizes the source terms (2.40) and the corresponding solution (2.41) of the Hamilton-Jacobi equation at order n = 2. The
solution L (4) of the Hamilton-Jacobi equation at order n = 2 can simply be read off the fourth column via the relation (2.41). This table is the
main result of this paper providing, together with Table 2, the boundary term necessary to make the variational problem of a generic action of the
form (2.1) well defined in any dimension up to and including d + 1 = 5.
Table 4 :
4The asymptotic form of the integral (3.7) for a function U (λ) that behaves as in (3.4) as
λ → 0.
Table
Table 6 :
6The leading form of the solution (2.41) of the Hamilton-Jacobi equation at order n = 2 and d = 4 for IHQCD. The third column gives the asymptotic form of the terms in the fourth column of
In[10] ξ 2 = 4/3, but here we prefer to keep it arbitrary.2 Potentials that contain non-analytic terms at order higher than O(λ 2 ) have been considered in the literature. The analysis below applies to such more general potentials as well.
We will consider an arbitrary boundary dimension d from now on, in order to maintain a wider applicability of our results.
See Appendix B for the explicit results in this case and[13,14], where these results have been recently used.
In the case of asymptotically AdS or dS gravity coupled to matter, it was proposed in[17] that one expands
The radial cut-off r0 and the boundary dimension d are arbitrary parameters in the radial Hamiltonian formalism. However, the fact that this formalism, without relying on any additional input from the asymptotic expansions, can handle conformal anomalies in d * dimensions by relating the radial cut-off to the parameter d suggests that a radial cut-off regularization corresponds to dimensional regularization in the dual field theory. Taken at face value, this identification has profound consequences for the physical significance of the holographic direction.
We are grateful to Elias Kiritsis for pointing this out to us.
Generically, there are logarithmic in λ terms as well, but these can be included with the powers in λ since they do not affect the counting of the derivative expansion.
We are grateful to Kostas Skenderis for pointing out this interpretation to us.
The same can be done with the higher order terms in the asymptotic expansions but we will not do this explicitly here.
AcknowledgmentsWe thank Elias Kiritsis, Umut Gursoy, Kostas Skenderis, George Papadopoulos, David Mateos and Diego Trancanelli for useful discussions and comments. We are especially grateful to David Mateos and Diego Trancanelli for carefully checking the results in Appendix B and pointing out typos. We also thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work.Appendix γ ij = e 2r g (0)ij + e −2r g(2)ij + e −4r −2rh (4)ij + g (4)ij + . . . , ϕ = ϕ (0) + e −2r ϕ (2) + e −4r −2rφ (4) + ϕ (4) + . . . , χ = χ (0) + e −2r χ (2) + e −4r −2rχ (4) + χ (4) + . . . ,(B.14)where the dots denote subleading terms that are unambiguously determined in terms of the terms shown. These are precisely the well known Fefferman-Graham expansions[6]. The coefficients in these expansions can be easily deduced from the expressions for the canonical momenta we obtained in the previous section. In particular, inserting the asymptotic expansions for the induced fields in the expressions (2.6) for the canonical momenta on the one hand, and in the above expansions of the momenta on the other and comparing the two determines at second orderComparing the logarithmic terms giveswhich can be easily evaluated explicitly using the above expression forL (4) but we will not need these expressions here. Finally, the O(e −4r ) terms lead to the expressions for the renormalized one-point functions in the presence of sources, namelyNote that in some places we have symmetrized indices with weight one, i.e. (ij) ≡ 1 2 (ij + ji). It can be easily checked that the expression for the one-point function of the stress tensor evaluated at zero dilaton and axion sources completely agrees with the expression in (3.12) of[33].15The holographic Ward identities can now be derived as in the main body of the text in the case of IHQCD. Namely, the second of the equations in (2.18) holds order by order in the expansion of the Hamilton-Jacobi functional in eigenfunctions of the dilatation operator. In particular, we haveMoreover, considering the transformation of the renormalized action under the asymptotic diffeomorphisms that leave the Fefferman-Graham expansions form-invariant leads to the trace Ward identityis the conformal anomaly. As a final comment, we observe that sincẽthe terms involving h (4)ij ,φ(4)andχ(4)in the above expressions for the renormalized one-point functions can be ignored since they are scheme dependent and they can be removed by adding the finite counterterm − 1 4ˆd 4 x √ γA. (B.24)15In making the comparison one should keep in mind that R there ijkl = −R here ijkl . Moreover, we believe that there are two typos in (3.12) of[33]. Namely, in the third line of (3.12), −Tr g(2)g (2)ij should read Tr g(2)g(2)ij , while in the beginning of the fourth line R ik R k j should read 2R ik R k j .
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|
[] |
[
"Similarity of Objects and the Meaning of Words ⋆",
"Similarity of Objects and the Meaning of Words ⋆"
] |
[
"Rudi Cilibrasi [email protected] \nCWI\nKruislaan 4131098 SJAmsterdamThe Netherlands\n",
"Paul Vitanyi ⋆⋆ \nCWI\nKruislaan 4131098 SJAmsterdamThe Netherlands\n"
] |
[
"CWI\nKruislaan 4131098 SJAmsterdamThe Netherlands",
"CWI\nKruislaan 4131098 SJAmsterdamThe Netherlands"
] |
[] |
We survey the emerging area of compression-based, parameter-free, similarity distance measures useful in data-mining, pattern recognition, learning and automatic semantics extraction. Given a family of distances on a set of objects, a distance is universal up to a certain precision for that family if it minorizes every distance in the family between every two objects in the set, up to the stated precision (we do not require the universal distance to be an element of the family). We consider similarity distances for two types of objects: literal objects that as such contain all of their meaning, like genomes or books, and names for objects. The latter may have literal embodyments like the first type, but may also be abstract like "red" or "christianity." For the first type we consider a family of computable distance measures corresponding to parameters expressing similarity according to particular features between pairs of literal objects. For the second type we consider similarity distances generated by web users corresponding to particular semantic relations between the (names for) the designated objects. For both families we give universal similarity distance measures, incorporating all particular distance measures in the family. In the first case the universal distance is based on compression and in the second case it is based on Google page counts related to search terms. In both cases experiments on a massive scale give evidence of the viability of the approaches.All data are created equal but some data are more alike than others. We and others have recently proposed very general methods expressing this alikeness, using a new similarity metric based on compression. It is parameter-free in that it doesn't use any features or background knowledge about the data, and can without changes be applied to different areas and across area boundaries. Put differently: just like 'parameter-free' statistical methods, the new method uses essentially unboundedly many parameters, the ones that are appropriate. It is universal in that it approximates the parameter expressing similarity of the dominant feature in all pairwise comparisons. It is robust in the sense that its success appears independent from the type of compressor used. The clustering we use is hierarchical clustering in dendrograms based on a new fast heuristic for the quartet method. The method is available as an open-source software tool,[7].Feature-Based Similarities: We are presented with unknown data and the question is to determine the similarities among them and group like with like together. Commonly, the data are of a certain type: music files, transaction records of ATM machines, credit card applications, genomic data. In these data there are hidden relations that we would like to get out in the open. For example, from genomic data one can extract letteror block frequencies (the blocks are over the four-letter alphabet); from music files one can extract various specific numerical features, related to pitch, rhythm, harmony etc. One can extract such features using for instance Fourier transforms[39]or wavelet transforms [18], to quantify parameters expressing similarity. The resulting vectors corresponding to the various files are then classified or clustered using existing classification software, based on various standard statistical pattern recognition classifiers [39], Bayesian classifiers [15], hidden Markov models [9], ensembles of nearest-neighbor classifiers [18] or neural networks[15,34]. For example, in music one feature would be to look for rhythm in the sense of beats per minute. One can make a histogram where each histogram bin corresponds to a particular tempo in beats-per-minute and the associated peak shows how frequent and strong that particular periodicity was over the entire piece. In [39] we see a gradual change from a few high peaks to many low and spread-out ones going from hip-hip, rock, jazz, to classical. One can use this similarity type to try to cluster pieces in these categories. However, such a method requires specific and detailed knowledge of the problem area, since one needs to know what features to look for.Non-Feature Similarities: Our aim is to capture, in a single similarity metric, every effective distance: effective versions of Hamming distance, Euclidean distance, edit distances, alignment distance, Lempel-Ziv distance, and so on. This metric should be so general that it works in every domain: music, text, literature, programs, genomes, executables, natural language determination, equally and simultaneously. It would be able to simultaneously detect all similarities between pieces that other effective distances can detect seperately.The normalized version of the "information metric" of [32,3] fills the requirements for such a "universal" metric. Roughly speaking, two objects are deemed close if we can significantly "compress" one given the information in the other, the idea being that if two pieces are more similar, then we can more succinctly describe one given the other. The mathematics used is based on Kolmogorov complexity theory[32].
|
10.1007/11750321_2
|
[
"https://arxiv.org/pdf/cs/0602065v1.pdf"
] | 5,370,625 |
cs/0602065
|
3ad7befd3617b01075354af0d5ffb9af31e4e548
|
Similarity of Objects and the Meaning of Words ⋆
17 Feb 2006
Rudi Cilibrasi [email protected]
CWI
Kruislaan 4131098 SJAmsterdamThe Netherlands
Paul Vitanyi ⋆⋆
CWI
Kruislaan 4131098 SJAmsterdamThe Netherlands
Similarity of Objects and the Meaning of Words ⋆
17 Feb 2006
We survey the emerging area of compression-based, parameter-free, similarity distance measures useful in data-mining, pattern recognition, learning and automatic semantics extraction. Given a family of distances on a set of objects, a distance is universal up to a certain precision for that family if it minorizes every distance in the family between every two objects in the set, up to the stated precision (we do not require the universal distance to be an element of the family). We consider similarity distances for two types of objects: literal objects that as such contain all of their meaning, like genomes or books, and names for objects. The latter may have literal embodyments like the first type, but may also be abstract like "red" or "christianity." For the first type we consider a family of computable distance measures corresponding to parameters expressing similarity according to particular features between pairs of literal objects. For the second type we consider similarity distances generated by web users corresponding to particular semantic relations between the (names for) the designated objects. For both families we give universal similarity distance measures, incorporating all particular distance measures in the family. In the first case the universal distance is based on compression and in the second case it is based on Google page counts related to search terms. In both cases experiments on a massive scale give evidence of the viability of the approaches.All data are created equal but some data are more alike than others. We and others have recently proposed very general methods expressing this alikeness, using a new similarity metric based on compression. It is parameter-free in that it doesn't use any features or background knowledge about the data, and can without changes be applied to different areas and across area boundaries. Put differently: just like 'parameter-free' statistical methods, the new method uses essentially unboundedly many parameters, the ones that are appropriate. It is universal in that it approximates the parameter expressing similarity of the dominant feature in all pairwise comparisons. It is robust in the sense that its success appears independent from the type of compressor used. The clustering we use is hierarchical clustering in dendrograms based on a new fast heuristic for the quartet method. The method is available as an open-source software tool,[7].Feature-Based Similarities: We are presented with unknown data and the question is to determine the similarities among them and group like with like together. Commonly, the data are of a certain type: music files, transaction records of ATM machines, credit card applications, genomic data. In these data there are hidden relations that we would like to get out in the open. For example, from genomic data one can extract letteror block frequencies (the blocks are over the four-letter alphabet); from music files one can extract various specific numerical features, related to pitch, rhythm, harmony etc. One can extract such features using for instance Fourier transforms[39]or wavelet transforms [18], to quantify parameters expressing similarity. The resulting vectors corresponding to the various files are then classified or clustered using existing classification software, based on various standard statistical pattern recognition classifiers [39], Bayesian classifiers [15], hidden Markov models [9], ensembles of nearest-neighbor classifiers [18] or neural networks[15,34]. For example, in music one feature would be to look for rhythm in the sense of beats per minute. One can make a histogram where each histogram bin corresponds to a particular tempo in beats-per-minute and the associated peak shows how frequent and strong that particular periodicity was over the entire piece. In [39] we see a gradual change from a few high peaks to many low and spread-out ones going from hip-hip, rock, jazz, to classical. One can use this similarity type to try to cluster pieces in these categories. However, such a method requires specific and detailed knowledge of the problem area, since one needs to know what features to look for.Non-Feature Similarities: Our aim is to capture, in a single similarity metric, every effective distance: effective versions of Hamming distance, Euclidean distance, edit distances, alignment distance, Lempel-Ziv distance, and so on. This metric should be so general that it works in every domain: music, text, literature, programs, genomes, executables, natural language determination, equally and simultaneously. It would be able to simultaneously detect all similarities between pieces that other effective distances can detect seperately.The normalized version of the "information metric" of [32,3] fills the requirements for such a "universal" metric. Roughly speaking, two objects are deemed close if we can significantly "compress" one given the information in the other, the idea being that if two pieces are more similar, then we can more succinctly describe one given the other. The mathematics used is based on Kolmogorov complexity theory[32].
Introduction
Objects can be given literally, like the literal four-letter genome of a mouse, or the literal text of War and Peace by Tolstoy. For simplicity we take it that all meaning of the object is represented by the literal object itself. Objects can also be given by name, like "the four-letter genome of a mouse," or "the text of War and Peace by Tolstoy." There are also objects that cannot be given literally, but only by name and acquire their meaning from their contexts in background common knowledge in humankind, like "home" or "red." In the literal setting, objective similarity of objects can be established by feature analysis, one type of similarity per feature. In the abstract "name" setting, all similarity must depend on background knowledge and common semantics relations, which is inherently subjective and "in the mind of the beholder." [2], and follow-up work, a closely related notion of compression-based distances is proposed. There the purpose was initially to infer a language tree from different-language text corpora, as well as do authorship attribution on basis of text corpora. The distances determined between objects are justified by ad-hoc plausibility arguments and represent a partially independent development (although they refer to the information distance approach of [27,3]). Altogether, it appears that the notion of compression-based similarity metric is so powerful that its performance is robust under considerable variations.
Similarity Distance
We briefly outline an improved version of the main theoretical contents of [12] and its relation to [31]. For details and proofs see these references. First, we give a precise formal meaning to the loose distance notion of "degree of similarity" used in the pattern recognition literature.
Distance and Metric
Let Ω be a nonempty set and R + be the set of nonnegative real numbers. A distance function on Ω is a function D : Ω × Ω → R + . It is a metric if it satisfies the metric The value D(x, y) is called the distance between x, y ∈ Ω. A familiar example of a distance that is also metric is the Euclidean metric, the everyday distance e(a, b) between two geographical objects a, b expressed in, say, meters. Clearly, this distance satisfies the properties e(a, a) = 0, e(a, b) = e(b, a), and e(a, b) ≤ e(a, c) + e(c, b) (for instance, a = Amsterdam, b = Brussels, and c = Chicago.) We are interested in a particular type of distance, the "similarity distance", which we formally define in Definition 4. For example, if the objects are classical music pieces then the function D defined by D(a, b) = 0 if a and b are by the same composer and D(a, b) = 1 otherwise, is a similarity distance that is also a metric. This metric captures only one similarity aspect (feature) of music pieces, presumably an important one that subsumes a conglomerate of more elementary features.
Admissible Distance
In defining a class of admissible distances (not necessarily metric distances) we want to exclude unrealistic ones like f (x, y) = 1 2 for every pair x = y. We do this by restricting the number of objects within a given distance of an object. As in [3] we do this by only considering effective distances, as follows.
Definition 1.
Let Ω = Σ * , with Σ a finite nonempty alphabet and Σ * the set of finite strings over that alphabet. Since every finite alphabet can be recoded in binary, we choose Σ = {0, 1}. In particular, "files" in computer memory are finite binary strings. A function D : Ω × Ω → R + is an admissible distance if for every pair of objects x, y ∈ Ω the distance D(x, y) satisfies the density condition
∑ y 2 −D(x,y) ≤ 1,(1)
is computable, and is symmetric, D(x, y) = D(y, x).
If D is an admissible distance, then for every x the set {D(x, y) : y ∈ {0, 1} * } is the length set of a prefix code, since it satisfies (1), the Kraft inequality. Conversely, if a distance is the length set of a prefix code, then it satisfies (1), see for example [27].
Normalized Admissible Distance
Large objects (in the sense of long strings) that differ by a tiny part are intuitively closer than tiny objects that differ by the same amount. For example, two whole mitochondrial genomes of 18,000 bases that differ by 9,000 are very different, while two whole nuclear genomes of 3 × 10 9 bases that differ by only 9,000 bases are very similar. Thus, absolute difference between two objects doesn't govern similarity, but relative difference appears to do so.
Definition 2.
A compressor is a lossless encoder mapping Ω into {0, 1} * such that the resulting code is a prefix code. "Lossless" means that there is a decompressor that reconstructs the source message from the code message. For convenience of notation we identify "compressor" with a "code word length function" C : Ω → N , where N is the set of nonnegative integers. That is, the compressed version of a file x has length C(x). We only consider compressors such that C(x) ≤ |x| + O(log |x|). (The additive logarithmic term is due to our requirement that the compressed file be a prefix code word.) We fix a compressor C, and call the fixed compressor the reference compressor. Definition 3. Let D be an admissible distance. Then D + (x) is defined by D + (x) = max{D(x, z) : C(z) ≤ C(x)}, and D + (x, y) is defined by D + (x, y) = max{D + (x), D + (y)}. Note that since D(x, y) = D(y, x), also D + (x, y) = D + (y, x). Definition 4. Let D be an admissible distance. The normalized admissible distance, also called a similarity distance, d(x, y), based on D relative to a reference compressor C, is defined by
d(x, y) = D(x, y) D + (x, y) .
It follows from the definitions that a normalized admissible distance is a function d : Ω × Ω → [0, 1] that is symmetric: d(x, y) = d(y, x).
Lemma 1.
For every x ∈ Ω, and constant e ∈ [0, 1], a normalized admissible distance satisfies the density constraint
|{y : d(x, y) ≤ e, C(y) ≤ C(x)}| < 2 eD + (x)+1 .(2)
We call a normalized distance a "similarity" distance, because it gives a relative similarity according to the distance (with distance 0 when objects are maximally similar and distance 1 when they are maximally dissimilar) and, conversely, for every welldefined computable notion of similarity we can express it as a metric distance according to our definition. In the literature a distance that expresses lack of similarity (like ours) is often called a "dissimilarity" distance or a "disparity" distance.
Normal Compressor
We give axioms determining a large family of compressors that both include most (if not all) real-world compressors and ensure the desired properties of the NCD to be defined later.
Definition 5.
A compressor C is normal if it satisfies, up to an additive O(log n) term, with n the maximal binary length of an element of Ω involved in the (in)equality concerned, the following: Remark 1. These axioms are of course an idealization. The reader can insert, say O( √ n), for the O(log n) fudge term, and modify the subsequent discussion accordingly. Many compressors, like gzip or bzip2, have a bounded window size. Since compression of objects exceeding the window size is not meaningful, we assume 2n is less than the window size. In such cases the O(log n) term, or its equivalent, relates to the fictitious version of the compressor where the window size can grow indefinitely. Alternatively, we bound the value of n to half te window size, and replace the fudge term O(log n) by some small fraction of n. Other compressors, like PPMZ, have unlimited window size, and hence are more suitable for direct interpretation of the axioms.
Idempotency:
A reasonable compressor will see exact repetitions and obey idempotency up to the required precision. It will also compress the empty string to the empty string.
Monotonicity: A real compressor must have the monotonicity property, at least up to the required precision. The property is evident for stream-based compressors, and only slightly less evident for block-coding compressors.
Symmetry: Stream-based compressors of the Lempel-Ziv family, like gzip and pkzip, and the predictive PPM family, like PPMZ, are possibly not precisely symmetric. This is related to the stream-based property: the initial file x may have regularities to which the compressor adapts; after crossing the border to y it must unlearn those regularities and adapt to the ones of x. This process may cause some imprecision in symmetry that vanishes asymptotically with the length of x, y. A compressor must be poor indeed (and will certainly not be used to any extent) if it doesn't satisfy symmetry up to the required precision. Apart from stream-based, the other major family of compressors is block-coding based, like bzip2. They essentially analyze the full input block by considering all rotations in obtaining the compressed version. It is to a great extent symmetrical, and real experiments show no departure from symmetry.
Distributivity: The distributivity property is not immediately intuitive. In Kolmogorov complexity theory the stronger distributivity property
C(xyz) + C(z) ≤ C(xz) + C(yz)(3)
holds (with K = C). However, to prove the desired properties of NCD below, only the weaker distributivity property
C(xy) + C(z) ≤ C(xz) + C(yz)(4)
above is required, also for the boundary case were C = K. In practice, real-world compressors appear to satisfy this weaker distributivity property up to the required precision.
Definition 6. Define C(y|x) = C(xy) − C(x).(5)
This number C(y|x) of bits of information in y, relative to x, can be viewed as the excess number of bits in the compressed version of xy compared to the compressed version of x, and is called the amount of conditional compressed information.
In the definition of compressor the decompression algorithm is not included (unlike the case of Kolmorogov complexity, where the decompressing algorithm is given by definition), but it is easy to construct one: Given the compressed version of x in C(x) bits, we can run the compressor on all candidate strings z-for example, in length-increasing lexicographical order, until we find the compressed string z 0 = x. Since this string decompresses to x we have found x = z 0 . Given the compressed version of xy in C(xy) bits, we repeat this process using strings xz until we find the string xz 1 of which the compressed version equals the compressed version of xy. Since the former compressed version decompresses to xy, we have found y = z 1 . By the unique decompression property we find that C(y|x) is the extra number of bits we require to describe y apart from describing x. It is intuitively acceptable that the conditional compressed information C(x|y) satisfies the triangle inequality
C(x|y) ≤ C(x|z) + C(z|y).(6)
Lemma 2. Both (3) and (6) imply (4).
Lemma 3.
A normal compressor satisfies additionally subadditivity: C(xy) ≤ C(x) + C(y).
Subadditivity:
The subadditivity property is clearly also required for every viable compressor, since a compressor may use information acquired from x to compress y. Minor imprecision may arise from the unlearning effect of crossing the border between x and y, mentioned in relation to symmetry, but again this must vanish asymptotically with increasing length of x, y.
Normalized Information Distance
Technically, the Kolmogorov complexity of x given y is the length of the shortest binary program, for the reference universal prefix Turing machine, that on input y outputs x; it is denoted as K(x|y). For precise definitions, theory and applications, see [27]. The Kolmogorov complexity of x is the length of the shortest binary program with no input that outputs x; it is denoted as K(x) = K(x|λ) where λ denotes the empty input. Essentially, the Kolmogorov complexity of a file is the length of the ultimate compressed version of the file. In [3] the information distance E(x, y) was introduced, defined as the length of the shortest binary program for the reference universal prefix Turing machine that, with input x computes y, and with input y computes x. It was shown there that, up to an additive logarithmic term, E(x, y) = max{K(x|y), K(y|x)}. It was shown also that E(x, y) is a metric, up to negligible violations of the metric inequalties. Moreover, it is universal in the sense that for every admissible distance D(x, y) as in Definition 1, E(x, y) ≤ D(x, y) up to an additive constant depending on D but not on x and y. In [31], the normalized version of E(x, y), called the normalized information distance, is defined as
NID(x, y) = max{K(x|y), K(y|x)} max{K(x), K(y)} .(7)
It too is a metric, and it is universal in the sense that this single metric minorizes up to an negligible additive error term all normalized admissible distances in the class considered in [31]. Thus, if two files (of whatever type) are similar (that is, close) according to the particular feature described by a particular normalized admissible distance (not necessarily metric), then they are also similar (that is, close) in the sense of the normalized information metric. This justifies calling the latter the similarity metric. We stress once more that different pairs of objects may have different dominating features. Yet every such dominant similarity is detected by the NID . However, this metric is based on the notion of Kolmogorov complexity. Unfortunately, the Kolmogorov complexity is non-computable in the Turing sense. Approximation of the denominator of (7) by a given compressor C is straightforward: it is max{C(x),C(y)}. The numerator is more tricky. It can be rewritten as
max{K(x, y) − K(x), K(x, y) − K(y)},(8)
within logarithmic additive precision, by the additive property of Kolmogorov complexity [27]. The term K(x, y) represents the length of the shortest program for the pair (x, y).
In compression practice it is easier to deal with the concatenation xy or yx. Again, within logarithmic precision K(x, y) = K(xy) = K(yx). Following a suggestion by Steven de Rooij, one can approximate (8) best by min{C(xy),C(yx)} − min{C(x),C(y)}. Here, and in the later experiments using the CompLearn Toolkit [7], we simply use C(xy) rather than min{C(xy),C(yx)}. This is justified by the observation that block-coding based compressors are symmetric almost by definition, and experiments with various stream-based compressors (gzip, PPMZ) show only small deviations from symmetry. The result of approximating the NID using a real compressor C is called the normalized compression distance ( NCD ), formally defined in (10). The theory as developed for the Kolmogorov-complexity based NID in [31], may not hold for the (possibly poorly) approximating NCD . It is nonetheless the case that experiments show that the NCD apparently has (some) properties that make the NID so appealing. To fill this gap between theory and practice, we develop the theory of NCD from first principles, based on the axiomatics of Section 2.4. We show that the NCD is a quasi-universal similarity metric relative to a normal reference compressor C. The theory developed in [31] is the boundary case C = K, where the "quasi-universality" below has become full "universality".
Compression Distance
We define a compression distance based on a normal compressor and show it is an admissible distance. In applying the approach, we have to make do with an approximation based on a far less powerful real-world reference compressor C. A compressor C approximates the information distance E(x, y), based on Kolmogorov complexity, by the compression distance E C (x, y) defined as
E C (x, y) = C(xy) − min{C(x),C(y)}.(9)
Here, C(xy) denotes the compressed size of the concatenation of x and y, C(x) denotes the compressed size of x, and C(y) denotes the compressed size of y.
Lemma 4. If C is a normal compressor, then E C (x, y) + O(1) is an admissible distance.
Lemma 5. If C is a normal compressor, then E C (x, y) satisfies the metric (in)equalities
up to logarithmic additive precision.
Lemma 6.
If C is a normal compressor, then E + C (x, y) = max{C(x),C(y)}.
Normalized Compression Distance
The normalized version of the admissible distance E C (x, y), the compressor C based approximation of the normalized information distance (7), is called the normalized compression distance or NCD:
NCD(x, y) = C(xy) − min{C(x),C(y)} max{C(x),C(y)} .(10)
This NCD is the main concept of this work. It is the real-world version of the ideal notion of normalized information distance NID in (7). Actually, the NCD is a family of compression functions parameterized by the given data compressor C.
Remark 2. In practice, the NCD is a non-negative number 0 ≤ r ≤ 1 + ε representing how different the two files are. Smaller numbers represent more similar files. The ε in the upper bound is due to imperfections in our compression techniques, but for most standard compression algorithms one is unlikely to see an ε above 0.1 (in our experiments gzip and bzip2 achieved NCD 's above 1, but PPMZ always had NCD at most 1).
There is a natural interpretation to NCD(x, y): If, say, C(y) ≥ C(x) then we can rewrite
NCD(x, y) = C(xy) − C(x) C(y) .
That is, the distance NCD(x, y) between x and y is the improvement due to compressing y using x as previously compressed "data base," and compressing y from scratch, expressed as the ratio between the bit-wise length of the two compressed versions.
Relative to the reference compressor we can define the information in x about y as C(y) − C(y|x). Then, using (5),
NCD(x, y) = 1 − C(y) − C(y|x) C(y) .
That is, the NCD between x and y is 1 minus the ratio of the information x about y and the information in y.
Theorem 1. If the compressor is normal, then the NCD is a normalized admissible distance satsifying the metric (in)equalities, that is, a similarity metric.
Quasi-Universality: We now digress to the theory developed in [31], which formed the motivation for developing the NCD . If, instead of the result of some real compressor, we substitute the Kolmogorov complexity for the lengths of the compressed files in the NCD formula, the result is the NID as in (7). It is universal in the following sense: Every admissible distance expressing similarity according to some feature, that can be computed from the objects concerned, is comprised (in the sense of minorized) by the NID . Note that every feature of the data gives rise to a similarity, and, conversely, every similarity can be thought of as expressing some feature: being similar in that sense. Our actual practice in using the NCD falls short of this ideal theory in at least three respects:
(i) The claimed universality of the NID holds only for indefinitely long sequences x, y. Once we consider strings x, y of definite length n, it is only universal with respect to "simple" computable normalized admissible distances, where "simple" means that they are computable by programs of length, say, logarithmic in n. This reflects the fact that, technically speaking, the universality is achieved by summing the weighted contribution of all similarity distances in the class considered with respect to the objects considered. Only similarity distances of which the complexity is small (which means that the weight is large), with respect to the size of the data concerned, kick in.
(ii) The Kolmogorov complexity is not computable, and it is in principle impossible to compute how far off the NCD is from the NID . So we cannot in general know how well we are doing using the NCD of a given compressor. Rather than all "simple" distances (features, properties), like the NID , the NCD captures a subset of these based on the features (or combination of features) analyzed by the compressor. For natural data sets, however, these may well cover the features and regularities present in the data anyway. Complex features, expressions of simple or intricate computations, like the initial segment of π = 3.1415 . . ., seem unlikely to be hidden in natural data. This fact may account for the practical success of the NCD , especially when using good compressors.
(iii) To approximate the NCD we use standard compression programs like gzip, PPMZ, and bzip2. While better compression of a string will always approximate the Kolmogorov complexity better, this may not be true for the NCD . Due to its arithmetic form, subtraction and division, it is theoretically possible that while all items in the formula get better compressed, the improvement is not the same for all items, and the NCD value moves away from the NID value. In our experiments we have not observed this behavior in a noticable fashion. Formally, we can state the following: Theorem 2. Let d be a computable normalized admissible distance and C be a normal compressor. Then,
NCD(x, y) ≤ αd(x, y) + ε, where for C(x) ≥ C(y), we have α = D + (x)/C(x) and ε = (C(x|y) − K(x|y))/C(x)
, with C(x|y) according to (5).
Remark 3. Clustering according to NCD will group sequences together that are similar according to features that are not explicitly known to us. Analysis of what the compressor actually does, still may not tell us which features that make sense to us can be expressed by conglomerates of features analyzed by the compressor. This can be exploited to track down unknown features implicitly in classification: forming automatically clusters of data and see in which cluster (if any) a new candidate is placed.
Another aspect that can be exploited is exploratory: Given that the NCD is small for a pair x, y of specific sequences, what does this really say about the sense in which these two sequences are similar? The above analysis suggests that close similarity will be due to a dominating feature (that perhaps expresses a conglomerate of subfeatures). Looking into these deeper causes may give feedback about the appropriateness of the realized NCD distances and may help extract more intrinsic information about the objects, than the oblivious division into clusters, by looking for the common features in the data clusters.
Hierarchical Clustering
Given a set of objects, the pairwise NCD 's form the entries of a distance matrix. This distance matrix contains the pairwise relations in raw form. But in this format that information is not easily usable. Just as the distance matrix is a reduced form of information representing the original data set, we now need to reduce the information even further in order to achieve a cognitively acceptable format like data clusters. The distance matrix contains all the information in a form that is not easily usable, since for n > 3 our cognitive capabilities rapidly fail. In our situation we do not know the number of clusters a-priori, and we let the data decide the clusters. The most natural way to do so is hierarchical clustering [16]. Such methods have been extensively investigated in Computational Biology in the context of producing phylogenies of species. One the most sensitive ways is the so-called 'quartet method. This method is sensitive, but time consuming, running in quartic time. Other hierarchical clustering methods, like parsimony, may be much faster, quadratic time, but they are less sensitive. In view of the fact that current compressors are good but limited, we want to exploit the smallest differences in distances, and therefore use the most sensitive method to get greatest accuracy. Here, we use a new quartet-method (actually a new version [12] of the quartet puzzling variant [35]), which is a heuristic based on randomized parallel hill-climbing genetic programming. In this paper we do not describe this method in any detail, the reader is referred to [12], or the full description in [14]. It is implemented in the CompLearn package [7].
We describe the idea of the algorithm, and the interpretation of the accuracy of the resulting tree representation of the data clustering. To cluster n data items, the algorithm generates a random ternary tree with n − 2 internal nodes and n leaves. The algorithm tries to improve the solution at each step by interchanging sub-trees rooted at internal nodes (possibly leaves). It switches if the total tree cost is improved. To find the optimal tree is NP-hard, that is, it is infeasible in general. To avoid getting stuck in a local optimum, the method executes sequences of elementary mutations in a single step. The length of the sequence is drawn from a fat tail distribution, to ensure that the probability of drawing a longer sequence is still significant. In contrast to other methods, this guarantees that, at least theoretically, in the long run a global optimum is achieved. Because the problem is NP-hard, we can not expect the global optimum to be reached in a feasible time in general. Yet for natural data, like in this work, experience shows that the method usually reaches an apparently global optimum. One way to make this more likely is to run several optimization computations in parallel, and terminate only when they all agree on the solutions (the probability that this would arises by chance is very low as for a similar technique in Markov chains). The method is so much improved against previous quartet-tree methods, that it can cluster larger groups of objects (around 70) than was previously possible (around 15). If the latter methods need to cluster groups larger than 15, they first cluster sub-groups into small trees and then combine these trees by a super-tree reconstruction method. This has the drawback that optimizing the local subtrees determines relations that cannot be undone in the supertree construction, and it is almost guaranteed that such methods cannot reach a global optimum. Our clustering heuristic generates a tree with a certain fidelity with respect to the underlying distance matrix (or alternative data from which the quartet tree is constructed) called standardized benefit score or S(T ) value in the sequel. This value measures the quality of the tree representation of the overall oder relations between the distances in the matrix. It measures in how far the tree can represent the quantitative distance relations in a topological qualitative manner without violating relative order. The S(T ) value ranges from 0 (worst) to 1 (best). A random tree is likely to have S(T ) ≈ 1/3, while S(T ) = 1 means that the relations in the distance matrix are perfectly represented by the tree. Since we deal with n natural data objects, living in a space of unknown metric, we know a priori only that the pairwise distances between them can be truthfully represented in n − 1-dimensional Euclidian space. Multidimensional scaling, representing the data by points in 2-dimensional space, most likely necessarily distorts the pairwise distances. This is akin to the distortion arising when we map spherical earth geography on a flat map. A similar thing happens if we represent the n-dimensional distance matrix by a ternary tree. It can be shown that some 5-dimensional distance matrices can only be mapped in a ternary tree with S(T ) < 0.8. Practice shows, however, that up to 12-dimensional distance matrices, arising from natural data, can be mapped into a such tree with very little distortion (S(T ) > 0.95). In general the S(T ) value deteriorates for large sets. The reason is that, with increasing size of natural data set, the projection of the information in the distance matrix into a ternary tree gets necessarily increasingly distorted. If for a large data set like 30 objects, the S(T ) value is large, say S(T ) ≥ 0.95, then this gives evidence that the tree faithfully represents the distance matrix, but also that the natural relations between this large set of data were such that they could be represented by such a tree.
Applications of NCD
The compression-based NCD method to establish a universal similarity metric (10) among objects given as finite binary strings, and, apart from what was mentioned in the Introduction, has been applied to objects like music pieces in MIDI format, [11], computer programs, genomics, virology, language tree of non-indo-european languages, literature in Russian Cyrillic and English translation, optical character recognition of handwrittern digits in simple bitmap formats, or astronimical time sequences, and combinations of objects from heterogenous domains, using statistical, dictionary, and block sorting compressors, [12]. In [19], the authors compared the performance of the method on all major time sequence data bases used in all major data-mining conferences in the last decade, against all major methods. It turned out that the NCD method was far superior to any other method in heterogenous data clustering and anomaly detection and performed comparable to the other methods in the simpler tasks. We developed the CompLearn Toolkit, [7], and performed experiments in vastly different application fields to test the quality and universality of the method. In [40], the method is used to analyze network traffic and cluster computer worms and virusses. Currently, a plethora of new applications of the method arise around the world, in many areas, as the reader can verify by searching for the papers 'the similarity metric' or 'clustering by compression,' and look at the papers that refer to these, in Google Scholar.
Heterogenous Natural Data
The success of the method as reported depends strongly on the judicious use of encoding of the objects compared. Here one should use common sense on what a real world compressor can do. There are situations where our approach fails if applied in a straightforward way. For example: comparing text files by the same authors in different encodings (say, Unicode and 8-bit version) is bound to fail. For the ideal similarity metric based on Kolmogorov complexity as defined in [31] this does not matter at all, but for practical compressors used in the experiments it will be fatal. Similarly, in the music experiments we use symbolic MIDI music file format rather than wave-forms. We test gross classification of files based on heterogenous data of markedly different file types: (i) Four mitochondrial gene sequences, from a black bear, polar bear, fox, and rat obtained from the GenBank Database on the world-wide web; (ii) Four excerpts from the novel The Zeppelin's Passenger by E. Phillips Oppenheim, obtained from the Project Gutenberg Edition on the World-Wide web; (iii) Four MIDI files without further processing; two from Jimi Hendrix and two movements from Debussy's Suite Bergamasque, downloaded from various repositories on the world-wide web; (iv) Two Linux x86 ELF executables (the cp and rm commands), copied directly from the RedHat 9.0 Linux distribution; and (v) Two compiled Java class files, generated by ourselves. The compressor used to compute the NCD matrix was bzip2. As expected, the program correctly classifies each of the different types of files together with like near like. The result is reported in Figure 1 with S(T ) equal to the very high confidence value 0.984. This experiment shows the power and universality of the method: no features of any specific domain of application are used. We believe that there is no other method known that can cluster data that is so heterogenous this reliably. This is borne out by the massive experiments with the method in [19].
Literature
The texts used in this experiment were down-loaded from the world-wide web in original Cyrillic-lettered Russian and in Latin-lettered English by L. Avanasiev. The compressor used to compute the NCD matrix was bzip2. We clustered Russian literature in the original (Cyrillic) by Gogol, Dostojevski, Tolstoy, Bulgakov,Tsjechov, with three or four different texts per author. Our purpose was to see whether the clustering is sensitive enough, and the authors distinctive enough, to result in clustering by author. In Figure 2 we see an almost perfect clustering according to author. Considering the English translations of the same texts, we saw errors in the clustering (not shown). Inspection showed that the clustering was now partially based on the translator. It appears that the translator superimposes his characteristics on the texts, partially suppressing the characteristics of the original authors. In other experiments, not reported here, we separated authors by gender and by period.
Music
The amount of digitized music available on the internet has grown dramatically in recent years, both in the public domain and on commercial sites. Napster and its clones are prime examples. Websites offering musical content in some form or other (MP3, MIDI, . . . ) need a way to organize their wealth of material; they need to somehow classify their files according to musical genres and subgenres, putting similar pieces together. The purpose of such organization is to enable users to navigate to pieces of music they already know and like, but also to give them advice and recommendations ("If you like this, you might also like. . . "). Currently, such organization is mostly done manually by humans, but some recent research has been looking into the possibilities of automating music classification. For details about the music experiments see [11,12]. In Figure 4 we display classification of bird-flu virii of the type H5N1 that have been found in different geographic locations in chicken. Data downloaded from the National Center for Biotechnology Information (NCBI), National Library of Medicine, National Institutes of Health (NIH).
Bird-Flu Virii-H5N1
Google-Based Similarity
To make computers more intelligent one would like to represent meaning in computerdigestable form. Long-term and labor-intensive efforts like the Cyc project [23] and the WordNet project [36] try to establish semantic relations between common objects, or, more precisely, names for those objects. The idea is to create a semantic web of such vast proportions that rudimentary intelligence and knowledge about the real world spontaneously emerges. This comes at the great cost of designing structures capable of manipulating knowledge, and entering high quality contents in these structures by knowledgeable human experts. While the efforts are long-running and large scale, the overall information entered is minute compared to what is available on the world-wideweb.
The rise of the world-wide-web has enticed millions of users to type in trillions of characters to create billions of web pages of on average low quality contents. The sheer mass of the information available about almost every conceivable topic makes it likely that extremes will cancel and the majority or average is meaningful in a lowquality approximate sense. We devise a general method to tap the amorphous low-grade knowledge available for free on the world-wide-web, typed in by local users aiming at personal gratification of diverse objectives, and yet globally achieving what is effectively the largest semantic electronic database in the world. Moreover, this database is available for all by using any search engine that can return aggregate page-count estimates like Google for a large range of search-queries.
The crucial point about the NCD method above is that the method analyzes the objects themselves. This precludes comparison of abstract notions or other objects that don't lend themselves to direct analysis, like emotions, colors, Socrates, Plato, Mike Bonanno and Albert Einstein. While the previous NCD method that compares the objects themselves using (10) is particularly suited to obtain knowledge about the similarity of objects themselves, irrespective of common beliefs about such similarities, we now develop a method that uses only the name of an object and obtains knowledge about the similarity of objects by tapping available information generated by multitudes of web users. The new method is useful to extract knowledge from a given corpus of knowledge, in this case the Google database, but not to obtain true facts that are not common knowledge in that database. For example, common viewpoints on the creation myths in different religions may be extracted by the Googling method, but contentious questions of fact concerning the phylogeny of species can be better approached by using the genomes of these species, rather than by opinion.
Googling for Knowledge: Let us start with simple intuitive justification (not to be mistaken for a substitute of the underlying mathematics) of the approach we propose in [13]. While the theory we propose is rather intricate, the resulting method is simple enough. We give an example: At the time of doing the experiment, a Google search for "horse", returned 46,700,000 hits. The number of hits for the search term "rider" was 12,200,000. Searching for the pages where both "horse" and "rider" occur gave 2,630,000 hits, and Google indexed 8,058,044,651 web pages. Using these numbers in the main formula (13) we derive below, with N = 8, 058, 044, 651, this yields a Normal-ized Google Distance between the terms "horse" and "rider" as follows:
NGD(horse, rider) ≈ 0.443.
In the sequel of the paper we argue that the NGD is a normed semantic distance between the terms in question, usually in between 0 (identical) and 1 (unrelated), in the cognitive space invoked by the usage of the terms on the world-wide-web as filtered by Google. Because of the vastness and diversity of the web this may be taken as related to the current objective meaning of the terms in society. We did the same calculation when Google indexed only one-half of the current number of pages: 4,285,199,774. It is instructive that the probabilities of the used search terms didn't change significantly over this doubling of pages, with number of hits for "horse" equal 23,700,000, for "rider" equal 6,270,000, and for "horse, rider" equal to 1,180,000. The NGD(horse, rider) we computed in that situation was ≈ 0.460. This is in line with our contention that the relative frequencies of web pages containing search terms gives objective information about the semantic relations between the search terms. If this is the case, then the Google probabilities of search terms and the computed NGD 's should stabilize (become scale invariant) with a growing Google database.
Related Work: There is a great deal of work in both cognitive psychology [22], linguistics, and computer science, about using word (phrases) frequencies in text corpora to develop measures for word similarity or word association, partially surveyed in [37,38], going back to at least [24]. One of the most successful is Latent Semantic Analysis (LSA) [22] that has been applied in various forms in a great number of applications. As with LSA, many other previous approaches of extracting meaning from text documents are based on text corpora that are many order of magnitudes smaller, using complex mathematical techniques like singular value decomposition and dimensionality reduction, and that are in local storage, and on assumptions that are more restricted, than what we propose. In contrast, [41, 8,1] and the many references cited there, use the web and Google counts to identify lexico-syntactic patterns or other data. Again, the theory, aim, feature analysis, and execution are different from ours, and cannot meaningfully be compared. Essentially, our method below automatically extracts meaning relations between arbitrary objects from the web in a manner that is feature-free, up to the search-engine used, and computationally feasible. This seems to be a new direction altogether.
The Google Distribution
Let the set of singleton Google search terms be denoted by S . In the sequel we use both singleton search terms and doubleton search terms {{x, y} : x, y ∈ S }. Let the set of web pages indexed (possible of being returned) by Google be Ω. The cardinality of Ω is denoted by M = |Ω|, and at the time of this writing 8 · 10 9 ≤ M ≤ 9 · 10 9 (and presumably greater by the time of reading this). Assume that a priori all web pages are equi-probable, with the probability of being returned by Google being 1/M. A subset of Ω is called an event. Every search term x usable by Google defines a singleton Google event x ⊆ Ω of web pages that contain an occurrence of The event x consists of all possible direct knowledge on the web regarding x. Therefore, it is natural to consider code words for those events as coding this background knowledge. However, we cannot use the probability of the events directly to determine a prefix code, or, rather the underlying information content implied by the probability. The reason is that the events overlap and hence the summed probability exceeds 1. By the Kraft inequality, see for example [27], this prevents a corresponding set of code-word lengths. The solution is to normalize: We use the probability of the Google events to define a probability mass function over the set {{x, y} : x, y ∈ S } of Google search terms, both singleton and doubleton terms. There are |S | singleton terms, and counting each singleton set and each doubleton set (by definition unordered) once in the summation. Note that this means that for every pair {x, y} ⊆ S , with x = y, the web pages z ∈ x y are counted three times: once in x = x x, once in y = y y, and once in x y. Since every web page that is indexed by Google contains at least one occurrence of a search term, we have N ≥ M. On the other hand, web pages contain on average not more than a certain constant α search terms. Therefore, N ≤ αM. Define
g(x) = g(x, x), g(x, y) = L(x y)M/N = |x y|/N.(11)
Then, ∑ {x,y}⊆S g(x, y) = 1. This g-distribution changes over time, and between different samplings from the distribution. But let us imagine that g holds in the sense of an instantaneous snapshot. The real situation will be an approximation of this. Given the Google machinery, these are absolute probabilities which allow us to define the associated prefix code-word lengths (information contents) for both the singletons and the doubletons. The Google code G is defined by
G(x) = G(x, x), G(x, y) = log 1/g(x, y).(12)
In contrast to strings x where the complexity C(x) represents the length of the compressed version of x using compressor C, for a search term x (just the name for an object rather than the object itself), the Google code of length G(x) represents the shortest expected prefix-code word length of the associated Google event x. The expectation is taken over the Google distribution p. In this sense we can use the Google distribution as a compressor for Google "meaning" associated with the search terms. The associated NCD , now called the normalized Google distance ( NGD ) is then defined by (13), and can be rewritten as the right-hand expression:
NGD(x, y) = G(x, y) − min(G(x), G(y)) max(G(x), G(y)) = max{log f (x), log f (y)} − log f (x, y) log N − min{log f (x), log f (y)} ,(13)
where f (x) denotes the number of pages containing x, and f (x, y) denotes the number of pages containing both x and y, as reported by Google. This NGD is an approximation to the NID of (7) using the prefix code-word lengths (Google code) generated by the Google distribution as defining a compressor approximating the length of the Kolmogorov code, using the background knowledge on the web as viewed by Google as conditional information. In practice, use the page counts returned by Google for the frequencies, and we have to choose N. From the right-hand side term in (13) it is apparent that by increasing N we decrease the NGD , everything gets closer together, and by decreasing N we increase the NGD , everything gets further apart. Our experiments suggest that every reasonable (M or a value greater than any f (x)) value can be used as normalizing factor N, and our results seem in general insensitive to this choice. In our software, this parameter N can be adjusted as appropriate, and we often use M for N.
Universality of NGD: In the full paper [13] we analyze the mathematical properties of NGD , and prove the universality of the Google distribution among web author based distributions, as well as the universality of the NGD with respect to the family of the individual web author's NGD 's, that is, their individual semantics relations, (with high probability)-not included here for space reasons.
Applications
Colors and Numbers
The objects to be clustered are search terms consisting of the names of colors, numbers, and some tricky words. The program automatically organized the colors towards one side of the tree and the numbers towards the other, Figure 5. It arranges the terms which have as only meaning a color or a number, and nothing else, on the farthest reach of the color side and the number side, respectively. It puts the more general terms black and white, and zero, one, and two, towards the center, thus indicating their more ambiguous interpretation. Also, things which were not exactly colors or numbers are also put towards the center, like the word "small". We may consider this an example of automatic ontology creation. As far as the authors know there do not exist other experiments that create this type of semantic meaning from nothing (that is, automatically from the web using Google). Thus, there is no baseline to compare against; rather the current experiment can be a baseline to evaluate the behavior of future systems. As search terms we used only the names of texts, without the authors. The clustering is given in Figure 6; it automatically has put the books by the same authors together. The S(T ) value in Figure 6 gives the fidelity of the tree as a representation of the pairwise distances in the NGD matrix (1 is perfect and 0 is as bad as possible. For details see [7,12]). The question arises why we should expect this. Are names of artistic objects so distinct? (Yes. The point also being that the distances from every single object to all other objects are involved. The tree takes this global aspect into account and therefore disambiguates other meanings of the objects to retain the meaning that is relevant for this collection.) Is the distinguishing feature subject matter or title style? (In these experiments with objects belonging to the cultural heritage it is clearly a subject matter. To stress the point we used "Julius Caesar" of Shakespeare. This term occurs on the web overwhelmingly in other contexts and styles. Yet the collection of the other objects used, and the semantic distance towards those objects, determined the meaning of "Julius Caesar" in this experiment.) Does the system gets confused if we add more artists? (Representing the NGD matrix in bifurcating trees without distortion becomes more difficult for, say, more than 25 objects. See [12].) What about other subjects, like music, sculpture? (Presumably, the system will be more trustworthy if the subjects are more common on the web.) These experiments are representative for those we have performed with the current software. For a plethora of other examples, or to test your own, see the Demo page of [7].
Names of Literature
Systematic Comparison with WordNet Semantics
WordNet [36] is a semantic concordance of English. It focusses on the meaning of words by dividing them into categories. We use this as follows. A category we want to learn, the concept, is termed, say, "electrical", and represents anything that may pertain to electronics. The negative examples are constituted by simply everything else. This category represents a typical expansion of a node in the WordNet hierarchy. In an experiment we ran, the accuracy on the test set is 100%: It turns out that "electrical terms" are unambiguous and easy to learn and classify by our method. The information in the WordNet database is entered over the decades by human experts and is precise. The database is an academic venture and is publicly accessible. Hence it is a good baseline against which to judge the accuracy of our method in an indirect manner. While we cannot directly compare the semantic distance, the NGD , between objects, we can indirectly judge how accurate it is by using it as basis for a learning algorithm. In particular, we investigated how well semantic categories as learned using the NGD -SVM approach agree with the corresponding WordNet categories. For details about the structure of WordNet we refer to the official WordNet documentation available online. We considered 100 randomly selected semantic categories from the WordNet database.
For each category we executed the following sequence. First, the SVM is trained on 50 labeled training samples. The positive examples are randomly drawn from the WordNet database in the category in question. The negative examples are randomly drwan from a dictionary. While the latter examples may be false negatives, we consider the probability negligible. Per experiment we used a total of six anchors, three of which are randomly drawn from the WordNet database category in question, and three of which are drawn from the dictionary. Subsequently, every example is converted to 6-dimensional vectors using NGD . The ith entry of the vector is the NGD between the ith anchor and the example concerned (1 ≤ i ≤ 6). The SVM is trained on the resulting labeled vectors. The kernel-width and error-cost parameters are automatically determined using five-fold cross validation. Finally, testing of how well the SVM has learned the classifier is performed using 20 new examples in a balanced ensemble of positive and negative examples obtained in the same way, and converted to 6-dimensional vectors in the same manner, as the training examples. This results in an accuracy score of correctly classified test examples. We ran 100 experiments. The actual data are available at [10]. A histogram of agreement accuracies is shown in Figure 7. On average, our method turns out to agree well with the WordNet semantic concordance made by human experts. The mean of the accuracies of agreements is 0.8725. The variance is ≈ 0.01367, which gives a standard deviation of ≈ 0.1169. Thus, it is rare to find agreement less than 75%. The total number of Google searches involved in this randomized automatic trial is upper bounded by 100 × 70 × 6 × 3 = 126, 000. A considerable savings resulted from the fact that we can re-use certain google counts. For every new term, in computing its 6-dimensional vector, the NGD computed with respect to the six anchors requires the counts for the anchors which needs to be computed only once for each experiment, the count of the new term which can be computed once, and the count of the joint occurrence of the new term and each of the six anchors, which has to be computed in each case. Altogether, this gives a total of 6 + 70 + 70 × 6 = 496 for every experiment, so 49, 600 google searches for the entire trial.
-
D(x, y) = 0 iff x = y, -D(x, y) = D(y, x) (symmetry), and -D(x, y) ≤ D(x, z) + D(z, y) (triangle inequality).
Fig. 1 .
1Classification of different file types. Tree agrees exceptionally well with NCD distance matrix: S(T ) = 0.984.
Fig. 2 .Fig. 3 .
23Clustering of Russian writers. Legend: I.S. Turgenev, 1818-1883 [Father and Sons, Rudin, On the Eve, A House of Gentlefolk]; F. Dostoyevsky 1821-1881 [Crime and Punishment, The Gambler, The Idiot; Poor Folk]; L.N. Tolstoy 1828-1910 [Anna Karenina, The Cossacks, Youth, War and Piece]; N.V. Gogol 1809-1852 [Dead Souls, Taras Bulba, The Mysterious Portrait, How the Two Ivans Quarrelled]; M. Bulgakov 1891-1940 [The Master and Margarita, The Fatefull Eggs, The Heart of a Dog]. S(T Output for the 12-piece set. Legend: J.S. Bach [Wohltemperierte Klavier II: Preludes and Fugues 1,2-BachWTK2{F,P}{1,2}]; Chopin [Préludes op. 28: 1, 15, 22, 24 -ChopPrel{1,15,22,24}]; Debussy [Suite Bergamasque, 4 movements-DebusBerg{1,2,3,4}]. S(T ) = 0.968.
Fig. 4 .
4Set of 24 Chicken Examples of H5N1 Virii. S(T ) = 0.967.
x and are returned by Google if we do a search for x. Let L : Ω → [0, 1] be the uniform mass probability function. The probability of such an event x is L(x) = |x|/M. Similarly, the doubleton Google event x y ⊆ Ω is the set of web pages returned by Google if we do a search for pages containing both search term x and search term y. The probability of this event is L(x y) = |x y|/M. We can also define the other Boolean combinations: ¬x = Ω\x and x y = ¬(¬x ¬y), each such event having a probability equal to its cardinality divided by M. If e is an event obtained from the basic events x, y, . . ., corresponding to basic search terms x, y, . . ., by finitely many applications of the Boolean operations, then the probability L(e) = |e|/M. Google events capture in a particular sense all background knowledge about the search terms concerned available (to Google) on the web. The Google event x, consisting of the set of all web pages containing one or more occurrences of the search term x, thus embodies, in every possible sense, all direct context in which x occurs on the web.
Remark 4 .
4It is of course possible that parts of this direct contextual material link to other web pages in which x does not occur and thereby supply additional context. In our approach this indirect context is ignored. Nonetheless, indirect context may be important and future refinements of the method may take it into account.
Fig. 5 .
5Colors and numbers arranged into a tree using NGD .
Fig. 6 .
6Hierarchical clustering of authors. S(T ) = 0.940.
Fig. 7 .
7Histogram of accuracies over 100 trials of WordNet experiment.
Another example is English novelists. The authors and texts used are: William Shakespeare: A Midsummer Night's Dream; Julius Caesar; Love's Labours Lost; Romeo and Juliet . Jonathan Swift: The Battle of the Books; Gulliver's Travels; Tale of a Tub; A Modest Proposal; Oscar Wilde: Lady Windermere's Fan; A Woman of No Importance; Salome; The Picture of Dorian Gray.
Idempotency: C(xx) = C(x), and C(λ) = 0, where λ is the empty string. Idempotency: C(xx) = C(x), and C(λ) = 0, where λ is the empty string.
Monotonicity: C(xy) ≥ C(x). Monotonicity: C(xy) ≥ C(x).
Symmetry: C(xy) = C(yx). Symmetry: C(xy) = C(yx).
Distributivity: C(xy) + C(z) ≤ C(xz) + C(yz). Distributivity: C(xy) + C(z) ≤ C(xz) + C(yz).
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[
"ON SOME EQUIVALENT NORMS IN SOBOLEV SPACES ON BOUNDED DOMAINS AND ON THE BOUNDARIES",
"ON SOME EQUIVALENT NORMS IN SOBOLEV SPACES ON BOUNDED DOMAINS AND ON THE BOUNDARIES"
] |
[
"Bienvenido Barraza Martínez ",
"AND JAIRO HERNÁNDEZ MONZÓNJonathan González Ospino "
] |
[] |
[] |
We consider the equivalence of some norms in Sobolev spaces on bounded domains of R d and also in Sobolev spaces on the boundaries of those domains.
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[
"https://arxiv.org/pdf/2202.10856v1.pdf"
] | 247,025,923 |
2202.10856
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91bb7045f73b32b83c6a8b1dddaef17d97d6066c
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ON SOME EQUIVALENT NORMS IN SOBOLEV SPACES ON BOUNDED DOMAINS AND ON THE BOUNDARIES
22 Feb 2022
Bienvenido Barraza Martínez
AND JAIRO HERNÁNDEZ MONZÓNJonathan González Ospino
ON SOME EQUIVALENT NORMS IN SOBOLEV SPACES ON BOUNDED DOMAINS AND ON THE BOUNDARIES
22 Feb 2022
We consider the equivalence of some norms in Sobolev spaces on bounded domains of R d and also in Sobolev spaces on the boundaries of those domains.
Introduction
In this notes we will consider a bounded domain (i.e., a bounded open and connected subset) Ω of R d , d ∈ N, with enough regular boundary ∂Ω (this regularity will be made precise later). We will present a relativ general result about the equivalence of norms in the scalar Sobolev space W k,p (Ω) with k ∈ N and 1 ≤ p < ∞. The main result follows strongly the proof of Theorem 7.1 in [4]. For a domain Ω ⊂ R d (bounded or unbounded) the usual Sobolev space W m,p (Ω) for m ∈ N and 1 ≤ p ≤ ∞, is the subspace of L p (Ω) consisting of all complex fuctions u ∈ L p (Ω) such that its distributional (weak) derivatives ∂ α u, with α ∈ N d 0 and |α| ≤ m, belong to L p (Ω). A standard norm in W m,p (Ω) for 1 ≤ p < ∞ is given by for u ∈ W m,p (Ω). The Sobolev space W m,p 0 (Ω), for m ≥ 0 and p ≥ 1, is defined as the clousure of C ∞ c (Ω) in W m,p (Ω). Finally, for m < 0 and p ≥ 1, W m,p (Ω) is defined as the topological dual of W −m,q 0 (Ω), where q := p p−1 .
About the regularity of the boundaries of domains
In this section we will make precise the concept of continuous and smooth boundary for a bounded domain in R d . Let Ω a bounded domain in R d with boundary ∂Ω. Fig. 1) Usually it is said also that the domain Ω belongs to the class N 0,1 .
V ∩ ∂Ω = (x ′ r , a r (x ′ r )) : x ′ r ∈ ∆ r , V ∩ Ω = (x ′ r , x d r ) : x ′ r ∈ ∆ r , a r (x ′ r ) < x d r < a r (x ′ r ) + b , V ∩ (R d Ω) = (x ′ r , x d r ) : x ′ r ∈ ∆ r , a r (x ′ r ) − b < x d r < a r (x ′ r ) .Remark
Remark 2.4.
If Ω ∈ N 0,1 , then a normal vector exists almost everywhere on ∂Ω (cf. [4], Lemma 4.2 on pag. 83). In fact, since the functions a r in Definition 2.1 are Lipschitz continuous, they are differentiable almost everywhere in their domains. Therefore, for each part ∂Ω r of the ∂Ω, represented locally as the graph of the functions a r for some r ∈ {1, . . . , m}, the gradient ∇a r exists almost everywhere in ∆ r and since ∂Ω r is a level set of the function
∆ r ∋ x ′ r → a r (x ′ r ) − x d r , the vector (∇a r (x ′ r ), −1) = ∂ ∂x 1 r a r (x ′ r ), . . . , ∂ ∂x d−1 r a r (x ′ r )
, −1 defined almost everywhere in ∂Ω r , is normal to ∂Ω r pointing to the exterior of Ω. Then the outer unit normal vector to ∂Ω r is given by
ν := ν(x ′ r ) := 1 + |∇a r (x ′ r )| 2 −1/2 ∇a r (x ′ r ), −1 = 1 + ∂ ∂x 1 r a r (x ′ r ) 2 + · · · + ∂ ∂x d−1 r a r (x ′ r ) 2 −1/2 · ∂ ∂x 1 r a r (x ′ r ), . . . , ∂ ∂x d−1 r a r (x ′ r ), −1 almost everywhere on ∂Ω r .
Definition 2.5 (Domains of class N k,µ ). Let k ∈ N 0 and 0 ≤ µ ≤ 1. It is said that the domain Ω in Definition 2.1 belong to the class N k,µ if the functions a r , r = 1, . . . , m, given in that definition are of class C k,µ (∆ r ), i.e., if a r together with its derivatives of order ≤ k are Hölder continuous with exponent µ in ∆ r , which means that for each α ∈ N d−1 0 there is a constant c such that for all x ′ r , y ′ r ∈ ∆ r the estimate
|∂ α a r (x ′ r ) − ∂ α a r (y ′ r )| ≤ c|x ′ r − y ′ r | µ holds.
Remark 2.6. Note that the case k = 0 and µ = 1 is the Lipschitz case mentioned previously above. In case that µ = 0 we also say that Ω is a domain of class C k . The notion of continuous boundary given in the first definition of this section corresponds to the case k = 0 and µ = 0.
Lebesgue and Sobolev spaces on the boundary
Let Ω a bounded domain in R d with continuous boundary ∂Ω. The notations in this section refer to those given in Definition 2.1 1 .
Definition 3.1. Let 1 ≤ p ≤ ∞.
It is said that a complex function f defined almost everywhere on ∂Ω (which means that x ′ r → f (x ′ r , a r (x ′ r )) is defined almost everywhere in ∆ r , r = 1, . . . , m) belongs to the space L p (∂Ω) if the function
x ′ r → f r (x ′ r ) := f (x ′ r , a r (x ′ r )) belongs to L p (∆ r ) for each r ∈ {1, .
. . , m}. The space L p (∂Ω) is a Banach space with the norm given by
(3.1) f p,∂Ω := f L p (∂Ω) := m r=1 f r p L p (∆r) 1/p , if 1 ≤ p < ∞ max r=1,...,m f r L ∞ (∆r) , if p = ∞.
If Ω ∈ N 0,1 , the space L p (∂Ω), with 1 ≤ p < ∞, can be endowed with another useful norm, equivalent to the norm in (3.1), which is given in terms of a boundary integral. Next we define the boundary integral for a function in L 1 (∂Ω).
Definition 3.2.
Let Ω ∈ N 0,1 . With the notations of Definition 2.1, let
V r := (x ′ r , x d r ) ∈ R d : x ′ r ∈ ∆ r , a r (x ′ r ) − b < x d r < a r (x ′ r ) + b , r = 1, . . . , m. Now, let {ϕ r } m r=1 a partition of the unity on ∂Ω subordinate to the cover {V r } m r=1 , i.e., for each r = 1, . . . , m, ϕ r ∈ C ∞ c (V r ), 0 ≤ ϕ r ≤ 1, and it holds m r=1 ϕ r (x) = 1 for all x ∈ ∂Ω. For a function f ∈ L 1 (∂Ω) we have f = m r=1 ϕ r f and we define ∂Ω f dσ := m r=1 ∆r f (x ′ r , a r (x ′ r ))ϕ r (x ′ r , a r (x ′ r )) 1 + |∇a r (x ′ r )| 2 dx ′ r . Proposition 3.3.
Let Ω ∈ N 0,1 and 1 ≤ p < ∞. The functional
(3.2) f → ∂Ω |f | p dσ 1/p
is a norm in L p (∂Ω), equivalent to the norm given in (3.1).
Proof. See [4], Lemma 1.2., pag. 116.
The following is a standard definiton for the Sobolev spaces on the boundary ∂Ω.
Definition 3.4 (Sobolev spaces on the boundary). Let k ≥ 0, 1 ≤ p ≤ ∞ and Ω ∈ N ⌈k⌉−1,1 , where ⌈k⌉ is the smallest integer greater than or equal to k.
The Sobolev space W k,p (∂Ω) is the subspace of L p (∂Ω) consisting of all functions f ∈ L p (∂Ω) such that f r ∈ W k,p (∆ r ) for r = 1, . . . , m. The space W k,p (∂Ω) is endowed with the norm (3.3) f k,p,∂Ω := f W k,p (∂Ω) := m r=1 f r p k,p,∆r 1/p , if 1 ≤ p < ∞ max r=1,...,m f r k,∞,∆r , if p = ∞.
With this norm W k,p (∂Ω) is a Banach space.
Imbeddings and traces
Theorem 4.1 (Sobolev imbedding theorem).
Let Ω ∈ N 0,1 , p ≥ 1 and kp > d. [4], pag. 66.
Then W k,p (Ω) ֒→ C 0,µ (Ω), where µ = k − d p , if k − d p < 1, < 1, if k − d p = 1, = 1, if k − d p > 1. Proof. See Theorem 3.8 inTheorem 4.2 (A first trace theorem). Let Ω ∈ N 0,1 . For p ≤ q ≤ (d−1)p d−p if 1 ≤ p < d, or for q ≥ 1 if p = d, there exists a continuos linear mapping Z : W 1,p (Ω) → L q (∂Ω) such that Zu = u| ∂Ω if u ∈ C ∞ (Ω).
Proof. Since L q 2 (∂Ω) ֒→ L q 1 (∂Ω) for 1 ≤ q 1 ≤ q 2 , the result follows from Theorem 4.2 on page 79 and Theorem 4.6 on page 81 of [4]. Cf. also with Theorem 5.36 in [1].
From Theorems 4.1 and 4.2 above, we have
Theorem 4.3 (A second trace theorem). Let Ω ∈ N 0,1 and p ≥ 1. There is a bounded linear mapping γ 0 : W 1,p (Ω) → L p (∂Ω) such that γ 0 u = u| ∂Ω if u ∈ C ∞ (Ω).
Usually the mapping γ 0 is called trace map of order zero. The map γ 0 : W 1,p (Ω) → L p (∂Ω) is not surjective, but it holds that γ 0 (W 1,p (Ω)) is dense in L p (∂Ω) , whenever p ≥ 1 and Ω ∈ N 0,1 (see Theorem 4.9 on page 82 in [4]).
A further useful result about traces is the following.
Theorem 4.4 (A third trace theorem). Let k ∈ N, p > 1, Ω ∈ N k−1,1 and u ∈ W k,p (Ω). If l ∈ N 0 is such that l ≤ k − 1, then γ 0 ∂ α u ∈ W k−l− 1 p ,p (∂Ω) for all α ∈ N d 0 with |α| = l, and (4.1) ∂ l ν u k−l− 1 p ,p,∂Ω ≤ const. u k,p,Ω , where (4.2) ∂ l ν u = |α|=l l! α! ν α γ 0 ∂ α u,
where ν is the outer normal on ∂Ω.
Proof. See [4], Th. 5.5, pag. 95.
From now on we will omit sometimes the notation γ 0 u and write simply u.
Main results about equivalence of norms in Sobolev spaces on bounded domains
The following is the main result of this notes.
Theorem 5.1. Let d, k, l ∈ N, 1 ≤ p < ∞, Ω ⊂ R d a bounded domain with Lipschitz boundary ∂Ω, {f i } l i=1 a set of seminorms in W k,p (Ω) such that i) For each i = 1, . . . , l, there exists C i ≥ 0 with f i (v) ≤ C i v k,p for all v ∈ W k,p (Ω). ii) For v ∈ P k−1 := |α|≤k−1 C α x α : C α ∈ R, x ∈ R d it holds that l i=1 f p i (v) = 0 implies v = 0. Then, (5.1) u → u ′ k,p := l i=1 f p i (u) + |u| p k,p 1/p , with (5.2) |u| k,p := |α|=k Ω |∂ α u(x)| p dx 1/p
, is a norm in W k,p (Ω), equivalent to the standard one given in (1.1).
Proof. It is clear, due to i), that there exists b > 0 such that
u ′ k,p ≤ b u k,p (u ∈ W k,p (Ω)).
Now suppose that there does not exist a constant a > 0 such that
a u k,p ≤ u ′ k,p
for all u ∈ W k,p (Ω). Then, for each n ∈ N, there exists u n ∈ W k,p (Ω) with u n k,p = 1 and such that
(5.3) 1 n > l i=1 f p i (u n ) + |u n | p k,p 1/p . Therefore, for each multiindex α ∈ N d 0 with |α| = k, we have (5.4) ∂ α u n → 0 in L p (Ω), when n → ∞.
Theorem 6.3 in [4] ensures that the identity mapping Id : W 1,p (Ω) → L p (Ω) is compact, which implies that the identity mapping Id : W k,p (Ω) → W k−1,p (Ω) is also compact. Since (u n ) n∈N is a bounded sequence in (W k,p (Ω), · k,p ), there exists a subsequence (u nm ) m∈N of (u n ) n∈N , which converges in the space (W k−1,p (Ω), · k−1,p ). Let u := lim m→∞ u nm , where the limit is taked in W k−1,p (Ω).
We assert that u ∈ W k,p (Ω), ∂ α u = 0 for all multiindex α with |α| = k and that u nm → u in W k,p (Ω) when m → ∞. In fact, let ϕ ∈ C ∞ c (Ω) and α ∈ N d 0 with |α| = k. Then, there are β ∈ N d 0 with |β| = k − 1 and j ∈ {1, . . . , d} such that
α = β + e j , where e j = (δ ij ) d i=1 ∈ N d 0 with δ ij = 0 if i = j and δ ij = 1 if i = j. Due to (5.4) it holds that Ω u ∂ α ϕ dx = Ω u ∂ β ∂ e j ϕ dx = (−1) k−1 Ω ∂ β u ∂ e j ϕ dx = (−1) k−1 lim m→∞ Ω ∂ β u nm ∂ e j ϕ dx = (−1) k lim m→∞ Ω ∂ α u nm ϕ dx = 0.
Then, in weak sense, ∂ α u = 0 ∈ L p (Ω). So we have u ∈ W k,p (Ω) and, by virtue of (5.4), ∂ α u nm → ∂ α u in L p (Ω) when m → ∞, for all α ∈ N d 0 with |α| = k. Therefore, u nm → u in W k,p (Ω) when m → ∞. Since ∂ α u = 0 for all α ∈ N d 0 with |α| = k, we have that u ∈ P k−1 (see [3], Theorem 3.2). Now, since the f i , i = 1, . . . , l are seminorms in W k,p (Ω) and due to the assumption i), for each i = 1, . . . , l, we have
|f i (u nm ) − f i (u)| ≤ f i (u nm − u) ≤ C i u nm − u k,p −−−→ m→∞ 0. By virtue of (5.3) we obtain l i=1 f p i (u nm ) 1/p < 1 n m .
Taking limit when m → ∞ in the last inequality it follows l i=1 f p i (u) = 0, which implies u = 0 because of the assumption ii). But this is a contradiction with the fact that u k,p = lim m→∞ u nm k,p =1 = 1.
We came to this contradiction due to the assumption that there does not exist a constant a > 0 such that a u k,p ≤ u ′ k,p for all u ∈ W k,p (Ω). In consequence, this assumption is false and therefore, there exists a > 0 such that the inequality a u k,p ≤ u ′ k,p holds for all u ∈ W k,p (Ω). With this we end the proof of Theorem 5.1.
Remark 5.2. It is clear that the functional · ′
k,p given in Theorem 5.1 is a seminorm in W k,p (Ω). Theorem 3.2 in [3] implies that it is in fact a norm in W k,p (Ω), because if u ′ k,p = 0 for u ∈ W k,p (Ω), then ∂ α u = 0 for all α ∈ N d 0 with |α| = k and therefore u ∈ P k−1 . We would have also l i=1 f p i (u) = 0 and then u = 0 by assumption ii). Corollary 5.3. Let d, k ∈ N, 1 ≤ p < ∞, Ω ⊂ R d a bounded domain with sufficiently regular boundary ∂Ω (at least Ω ∈ N ⌈k− 1 p ⌉−1,1 ), ν the outer normal on ∂Ω, Γ ⊆ ∂Ω with σ(Γ) = 0, where σ is the (d − 1)-dimensional Lebesgue surface measure. Furthermore, suppose that Γ is not contained in a hyperplane of R d . Then, the functional
(5.5) u → k−1 i=0 Γ |∂ i ν u| p dσ + |α|=k Ω |∂ α u| p dx 1/p
is a norm in W k,p (Ω), equivalent to the standard one · k,p .
Proof. For i = 0, ..., k − 1, let f i be defined by
f i (u) := Γ |∂ i ν u| p dσ 1/p (u ∈ W k,p (Ω)).
It is easy to see that the functional f i , i = 0, . . . , k − 1, is a seminorm in W k,p (Ω). Furthermore, due to equation (4.1) in Trace theorem 4.4, we have that there exists C i ≥ 0 such that
f i (v) ≤ ∂Ω |∂ i ν v| p dσ 1/p ≤ const. ∂ i ν v k−i− 1 p , p,∂Ω ≤ C i v k,p,Ω for all v ∈ W k,p (Ω).
On the other side, [5], pag. 18). Now, if |α| = |β| and α = β, then there exists j ∈ {1, . . . , d} such that α j > β j , otherwise α i ≤ β i for all i ∈ {1, . . . , d} and since α = β, we would have α l < β l for some l ∈ {1, . . . , d} and then |α| < |β| which contradicts |α| = |β|. Therefore, if |α| = |β| we have ∂ α x β = 0 if α = β and ∂ α x β = α! if α = β. We recall also from (4.2) that
let v ∈ P k−1 , v(x) = |β|≤k−1 c β x β such that k−1 i=0 f p i (v) = 0. This implies ∂ i ν v = 0 on Γ for each i = 0, . . . , k − 1. We recall that ∂ α x β = α! β α x β−α with β α := β! α!(β−α)! if α ≤ β, β α = 0 otherwise (see∂ i ν v = |α|=i i! α! ∂ α v ν α . Then 0 = ∂ k−1 ν v = |α|=k−1 (k − 1)! α! ∂ α v ν α = |α|=k−1 (k − 1)! α! ν α |β|≤k−1 c β ∂ α x β = |α|=k−1 (k − 1)! α! ν α |β|=k−1 c β ∂ α x β = |α|=k−1 (k − 1)! α! ν α c α α! = |α|=k−1 (k − 1)!c α ν α .
Since Γ is not contained in a hiperplane, the powers ν α are linear independent and then c α = 0 for all α ∈ N d
0 with |α| = k − 1. Therefore v(x) = |β|≤k−2 c β x β . Similarly ∂ k−2 ν v = 0 implies c β = 0 for all β ∈ N d 0 with |β| = k − 2.
In this form we obtain that c β = 0 for all β ∈ N d 0 with |β| ≤ k − 1, i.e., v = 0. In consequence, we have proved that the functionals f i , i = 0, . . . , k − 1, satisfy the assumptions of Theorem 5.1 and we conclude that
u → k−1 i=0 f p i (u) + |u| p k,p 1/p = k−1 i=0 Γ |∂ i ν u| p dσ + |α|=k Ω |∂ α u| p dx 1/p
is a norm in W k,p (Ω), equivalent to the norm · k,p .
Theorem 5.4 (Generalized Poincaré inequality).
Let Ω ⊂ R d be open, bounded and connected with Lipschitz boundary ∂Ω (i.e. Ω ∈ N 0,1 ). Moreover, let 1 < p < ∞ and let M ⊂ W 1,p (Ω) be nonempty, closed and convex. Then the following assertions are equivalent for every u 0 ∈ M:
(1) There exists a constant C 0 < ∞ such that for all ξ ∈ R,
u 0 + ξ ∈ M =⇒ |ξ| ≤ C 0 .
(2) There exists a constant C < ∞ with
u L p (Ω) ≤ C ∇u L p (Ω) + 1 (u ∈ M).
If M in addition, is a cone with apex 0, i.e. if u ∈ M, r ≥ 0 =⇒ ru ∈ M, then the inequality in the assertion (2) can be replaced with
u L p (Ω) ≤ C ∇u L p (Ω) (u ∈ M).
Proof. See Then M ⊂ W 1,p (Ω) is nonempty because 0 ∈ M, closed because u → Γ u dσ : W 1,p (Ω) → C is continuous, and convex because of the linearity of this functional. Now let u 0 ∈ M and take C 0 := 0. For all ξ ∈ R we have
u 0 + ξ ∈ M =⇒ 0 = Γ (u 0 + ξ) dσ = Γ u 0 dσ + ξσ(Γ) = ξσ(Γ).
Then ξ = 0 which implies |ξ| ≤ C 0 . Since M is a cone with appex 0, we have in virtue of Theorem 5.4 that
u L p (Ω) ≤ C ∇u L p (Ω) (u ∈ M).
Let now u ∈ W 1,p (Ω) and define u := u − 1 σ(Γ) Γ u dσ. Then u ∈ M and we have
u L p (Ω) ≤ C ∇ u L p (Ω) .
Then
u L p (Ω) − µ(Ω) 1/p σ(Γ) Γ u dσ ≤ u − 1 σ(Γ) Γ u dσ L p (Ω) ≤ C ∇u L p (Ω) ,
where µ(Ω) is the d-dimensional Lebesgue measure of Ω. Therefore
u L p (Ω) ≤ C ∇u L p (Ω) + µ(Ω) 1/p σ(Γ) Γ u dσ
for all u ∈ W 1,p (Ω). In particular if u ∈ W 1,p Γ (Ω) we have u L p (Ω) ≤ C ∇u L p (Ω) because in this case Γ u dσ = 0.
Remark 5.6. In virtue of Corollary 5.5 we have that the functional u → ∇u L p (Ω) is a norm in W 1,p Γ (Ω), which is equivalent to the norm · k,p,Ω (compare with Corollary 5.3).
6. About equivalence of norms in Sobolev spaces on the boundary Theorem 6.1. Let 1 ≤ p < ∞ and Ω a bounded domain in R 2 with sufficiently regular boundary ∂Ω ( at least Ω ∈ N 2,1 ). Then, the functional
(6.1) u → u p 1,p,∂Ω + ∂ 2 τ u p p,∂Ω 1/p ,
where τ is the unit tangential vector on ∂Ω, is a norm in W 2,p (∂Ω), equivalent to the standard norm · 2,p,∂Ω given in (3.3).
Proof. let u ∈ W 2,p (∂Ω). With the notations of Sections 2 and 3 we have Since a r ∈ C 2,1 (∆ r ), there are constants c 1 r , c 2 r and c 3 r such that On the other side, again from (6.3), we have
θ ′ r (x 1 r ) θ r (x 1 r ) ≤ c 1 r , 0 < c 2 r ≤ θ r (x 1 r ) 2 ≤ c 3 r . (6.4) Then, due to (6.3) we have with [∂ 2 τ u] r (x 1 r ) := ∂ 2 τ u(x 1 r , a r (x 1 r )) that d dx 1 r 2 u r L p (∆r) ≤ θ ′ r θ r d dx 1 r u r L p (∆r) + θ 2 r [∂ 2 τ u] r L p (∆r) ≤ c 1 r d dx 1 r u r L p (∆r) + c 3 r [∂ 2 τ u] r L p (∆r) . Therefore d dx 1 r 2 u r p L p (∆r) ≤ 2 p (c 1 r ) p d dx 1 r u r p L p (∆r) + 2 p (c 3 r ) p [∂ 2 τ u] r p L p (∆r) .c 2 r [∂ 2 τ u] r L p (∆r) ≤ θ 2 r [∂ 2 τ u] r L p (∆r) ≤ d dx 1 r 2 u r L p (∆r) + c 1 r d dx 1 r u r L p (∆r)
, which implies
(c 2 r ) p [∂ 2 τ u] r p L p (∆r) ≤ 2 p d dx 1 r 2 u r p L p (∆r) + 2 p (c 1 r ) p d dx 1 r u r p L p (∆r) ,
i.e.,
[∂ 2 τ u] r p L p (∆r) ≤ max 2 p (c 2 r ) p , 2 p (c 1 r ) p (c 2 r ) p d dx 1 r 2 u r p L p (∆r) + d dx 1 r u r p L p (∆r) ≤ max 2 p (c 2 r ) p , 2 p (c 1 r ) p (c 2 r ) p u r p 2,p,∆r .
Therefore, with c 2 := max r=1,...,m
max 2 p (c 2 r ) p , 2 p (c 1 r ) p (c 2 r ) p ,≤ (1 + c 2 ) u p 2,p,∂Ω . With c 2 := (1 + c 2 ) 1/p it holds (6.6) u p 1,p,∂Ω + ∂ 2 τ u p p,∂Ω 1/p ≤ c 2 u 2,p,∂Ω .
From (6.5) and (6.6) follows the result.
Some useful interpolation type estimates for traces on Sobolev spaces
In this section we use as norm in L p (∂Ω), for a domain Ω ⊂ R d , the norm given in (3.2).
Proposition 7.1. Let R d + := x = (x ′ , x d ) ∈ R d−1 ×R : x d > 0 and u ∈ C 1 c (R d + ) with 1 ≤ p < ∞. The following estimate holds Proof. Let δ > 0 be such that supp u ⊂ B(0; δ) ∩ R d + and ν = (ν 1 , . . . , ν d ) the outer normal on ∂[B(0; δ) ∩ R d + ] (see Fig. 2). Then, by virtue of Gauß theorem of divergence and Hölder inequality it holds
∂R d + |u| p dσ = R d−1 |u| p dx ′ = − R d−1 |u| p (−1) dx ′ = − ∂[B(0;δ)∩R d + ] |u| p ν d dσ = − B(0;δ)∩R d + ∂ x d (|u| p ) dx = − R d + ∂ x d (uu) p/2 dx = − p 2 R d + (uu) p−2= − p R d + |u| p−2 Re[(∂ x d u)u] dx = − p Re R d + |u| p−2 (∂ x d u)u dx ≤ p R d + |u| p−2 |∂ x d u||u| dx = p R d + |u| p−1 |∂ x d u| dx ≤ p R d + |u| p dx p−1 p R d + |∂ x d u| p dx 1 p
≤ p u p−1 p,R d + u 1,p,R d + , which implies (7.1).
Proposition 7.2.
Let Ω a bounded domain in R d , Ω ∈ N 1,1 , 1 ≤ p < ∞ and u ∈ C 1 (Ω). Then, it holds (7.2) u p,∂Ω ≤ c p u p−1 p p,Ω u 1 p 1,p,Ω , with c p a constant, which depends on p, but not on u.
Proof. With the notations of Section 2, let
V r := (x ′ r , x d r ) ∈ R d : x ′ r ∈ ∆ r , a r (x ′ r ) − b < x d r < a r (x ′ r ) + b , r = 1, . . . , m.
Furthermore let V 0 ⊂⊂ Ω, V 0 abierto, such that Ω ⊂ m r=0 V r . Choose a C ∞ partition of the unity {ϕ r } m r=0 on Ω subordinate to the cover {V r } m r=0 , i.e., ϕ r ∈ C ∞ c (V r ), 0 ≤ ϕ r ≤ 1 and supp ϕ r ⊂ V r for r = 0, 1, . . . , m. Moreover uϕ r on Ω with uϕ r ∈ C 1 c (V + r ) for
Fig. 1 .
1∈ W m,p (Ω)).The Sobolev space W m,∞ (Ω) is usually endowed with the norm(1.2) u m,∞ := u m,∞,Ω := max |α|≤m ∂ α u L ∞ (Ω) (u ∈ W m,∞ (Ω)). Now, for m ∈ R, m > 0, m / ∈ Z, and 1 ≤ p < ∞, the Sobolev space W m,p (Ω) (also called Sobolev-Slobodetskii spaces) is the subspace of W [m],p (Ω), where [m] denotes the integer part of m, of functions u such that for α ∈ N d 0 with |α| = [|∂ α u(x) − ∂ α u(y)| p |x − y| d+p(m−[m]) dx dy < ∞.In this case the usual norm in W m,p (Ω) is given by(1.4) u m,p := u m,p,Ω := u p [m],p + |α|=[m] Ω Ω |∂ α u(x) − ∂ α u(y)| p |x − y| d+p(m−[m]) dx dy Domain Ω with continuous boundary ∂Ω.
. 5 .
5Let Ω ⊂ R d be open, bounded and connected with Lipschitz boundary ∂Ω. Moreover let 1 < p < ∞, Γ ⊆ ∂Ω with σ(Γ) = 0, where σ is the (d − 1)dimensional Lebesgue surface measure, and W 1,p Γ (Ω) := {u ∈ W 1,p (Ω) : u = 0 on Γ}. Then there exists a constant C < ∞ such that u L p (Ω) ≤ C ∇u L p (Ω) (u ∈ W 1,p Γ (Ω)). Proof. Let M be defined by M := u ∈ W 1,p (Ω) : Γ u dσ = 0 .
≤
max{1 + c 1 , c 3 } u p 1,p,∂Ω + ∂ 2 τ u p p,∂Ω . With c 1 := (max{1 + c 1 , c 3 })
Fig. 2 .
2Domain R d + := {x = (x ′ , x d ) ∈ R d−1 × R : x d > 0} and supp u.
(x) = 1 for all x ∈ Ω. Then, u = m r=0
Definition 2.1. The boundary ∂Ω is called continuous if there exist real numbers a > 0, b > 0, a system of local coordinates (x 1 , . . . , m and continuous functions a r : ∆ r → R, r = 1, . . . , m, where ∆ r are the open cubes in R d−1 defined by ∆ r := {(x 1 r , . . . , x d−1 r ) : |x j r | < a, j = 1, . . . , d − 1}, such that for each point on the boundary ∂Ω there is an open neighborhood V , such that for some r ∈ {1, . . . , m} the following holds (seer , . . . , x d−1
r , x d
r ) =: (x ′
r , x d
r ) : r =
1
2.2.Note that ∂Ω is continuos if locally it is the graph of a continuos function defined in a subset of R d−1 . Definition 2.3. If the functions a r in the Definition 2.1 are Lipschitz continuous it is said that Ω has a Lipschitz boundary or that the boundary ∂Ω is lipschitzian.
The spaces defined in this section are independent on the local system of coordinates choosen in Definition 2.1. The corresponding norms related to each local system of coordinates are all equivalents.
[(∂ x d u)u + u∂ x d u] dx
,(6.2)where [x 1 r → u r (x 1 r ) := u(x 1 r , a r (x 1 r ))] ∈ W 2,p (∆ r ) = W 2,p ((−a, a)), with a r ∈ C 1,1 (∆ r ) = C 1,1 ([−a, a]), r = 1, . . . , m. Note that ∆ r = (−a, a) for all r = 1, . . . , m. Now, fix r ∈ {1, . . . , m}. Taking in account that ∆ r ∋ x 1 r → (x 1 r , a r (x 1 r )) is a parametrization of a part of ∂Ω, we have that (1, a ′ r (x 1 r )) is a tangent vector to ∂Ω on that part. Set θ r (x 1 r ) := |(1, a ′ r (x 1 r ))| = 1 + (a ′ r (x 1 r )) 2 . Furthermore, the weak (or distributional) derivative d dx 1 r 2 u r is almost everywhere equal to the corresponding usual classical derivative in ∆ r (see Theorem 2.2. in[4], pag. 55). Then, it holds almost everywhere in ∆ r thatNow, for r = 1, . . . , m, we will consider the transformation of coordinatesgiven by) and therefore, extending by zero. Then, it follows (with several constants c r , c 1 , c 2 p , etc., which can depend on p, but not on u) Proof. Let (u n ) n∈N a sequence of functions of C ∞ (Ω) such that u n → u in W 1,p (Ω) whenever n → ∞. Due to trace theorem 4.3 we have that there exists a constant c, such that u n − u p,∂Ω ≤ c u n − u 1,p,Ω (n ∈ N).Then, u n − u p,∂Ω → 0 when n → ∞.Now, from Proposition 7.2 it follows thatMaking n → ∞ we obtain (7.3).From Proposition 7.3 follow also the following estimates.Proposition 7.4. Let Ω a bounded domain in R d , Ω ∈ N 2,1 , 1 ≤ p < ∞ and u ∈ W 2,p (Ω). The following estimate holds: Proof. For u ∈ C 2 (Ω), due to Proposition 7.3, the following estimates hold: also true for u ∈ W 2,p (Ω) due to the density of C 2 (Ω) in W 2,p (Ω).Proposition 7.5.Let Ω a bounded domain in R d , Ω ∈ N 3,1 , 1 ≤ p < ∞ and u ∈ W 3,p (Ω). The following estimate holds: Proof. Let u ∈ C 3 (Ω). Then, similarly as the proof of Proposition 7.4 and using again Propostion 7.3 (or Proposition 7.2), we haveDue to the density of C 3 (Ω) in W 3,p (Ω) we obtain that the estimate (7.5) holds also for u ∈ W 3,p (Ω), withĉ p := d c p , where c p is the constant of Proposition 7.3.Proposition 7.6.Let Ω a bounded domain in R d , Ω ∈ N 4,1 , 1 ≤ p < ∞ and u ∈ W 4,p (Ω). The following estimate holds: Proof. Similarly as the proof of Proposition 7.4 we obtain that for u ∈ C 4 (Ω) the estimate (7.6) holds. Then, due to the density of C 4 (Ω) in W 4,p (Ω), the estimate (7.6) holds also for u ∈ W 4,p (Ω), with the constant c p being the same of Proposition 7.4.
R A Adams, J F Fournier, Sobolev spaces. AmsterdamAcademic presssecond editionR. A. Adams and J. F. Fournier, Sobolev spaces, second edition, Academic press, Amsterdam (2003).
Linear Functional Analysis: An Application-Oriented Introduction. H W Alt, SpringerBerlinH. W. Alt, Linear Functional Analysis: An Application-Oriented Introduction., Springer, Berlin (2012).
Polynomial approximation of functions in Sobolev spaces, Mathematics of computation. T Dupont, R Scott, 34T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Mathe- matics of computation, volume 34, number 150, april 1980, 441-463.
Direct methods in the theory of elliptic equations. J Nečas, Springer VerlagBerlinJ. Nečas, Direct methods in the theory of elliptic equations, Springer Verlag, Berlin (2012).
Introduction to pseudodifferential operators, Posgrado en Matemáticas, Universidad Nacional de Colombia. G Schleinkofer, Medellín -ColombiaG. Schleinkofer, Introduction to pseudodifferential operators, Posgrado en Matemáticas, Uni- versidad Nacional de Colombia, Medellín -Colombia (1996).
J Wloka, Partial differential equations. LondonCambridge university pressJ. Wloka, Partial differential equations, Cambridge university press, London (1987).
|
[] |
[] |
[
"1★H F Stevance \nDepartment of Physics\nThe University of Auckland\nPrivate Bag 92019AucklandNew Zealand\n",
"S G Parsons \nDepartment of Physics and Astronomy\nThe University of Sheffield\nHicks BuildingS3 7RHSheffieldUnited Kingdom\n",
"J J Eldridge \nDepartment of Physics\nThe University of Auckland\nPrivate Bag 92019AucklandNew Zealand\n"
] |
[
"Department of Physics\nThe University of Auckland\nPrivate Bag 92019AucklandNew Zealand",
"Department of Physics and Astronomy\nThe University of Sheffield\nHicks BuildingS3 7RHSheffieldUnited Kingdom",
"Department of Physics\nThe University of Auckland\nPrivate Bag 92019AucklandNew Zealand"
] |
[
"MNRAS"
] |
We use the self-consistent stellar populations in the Binary Population A Spectral Synthesis (BPASS) models to assess whether NGC1850-BH1 is a black hole. Using search criteria based on reported physical properties in the literature we purposefully search for suitable systems with a black hole (or compact object) companion: we do not find any. Good matches to the observations are found in models where the bright component is a stripped star and the companion is natively (meaning we did not impose this in our search) 1 to 2.3 magnitudes fainter than the primary in the optical bands. This alone can explain the lack of detection of the companion in the MUSE spectra without the need to invoke rapid rotation, although the conservative mass transfer exhibited by these particular models is likely to lead to rapidly rotating companions which could further smear their spectroscopic signatures. We advise that future claims of unseen black holes in binary systems would benefit from exploring detailed binary evolution models of stellar populations as a sanity check.
|
10.1093/mnrasl/slac001
|
[
"https://arxiv.org/pdf/2112.00015v3.pdf"
] | 246,241,175 |
2112.00015
|
ec7ba76b096556de10cdc9e1a5fdbef5a36628b9
|
2021
1★H F Stevance
Department of Physics
The University of Auckland
Private Bag 92019AucklandNew Zealand
S G Parsons
Department of Physics and Astronomy
The University of Sheffield
Hicks BuildingS3 7RHSheffieldUnited Kingdom
J J Eldridge
Department of Physics
The University of Auckland
Private Bag 92019AucklandNew Zealand
MNRAS
0002021Accepted XXX. Received YYY; in original form ZZZPreprint 25 January 2022 Compiled using MNRAS L A T E X style file v3.0 To be or not to be a black hole: detailed binary population models as a sanity checkstars: evolution -stars: black holes -(stars:) binaries: close -software: simulations
We use the self-consistent stellar populations in the Binary Population A Spectral Synthesis (BPASS) models to assess whether NGC1850-BH1 is a black hole. Using search criteria based on reported physical properties in the literature we purposefully search for suitable systems with a black hole (or compact object) companion: we do not find any. Good matches to the observations are found in models where the bright component is a stripped star and the companion is natively (meaning we did not impose this in our search) 1 to 2.3 magnitudes fainter than the primary in the optical bands. This alone can explain the lack of detection of the companion in the MUSE spectra without the need to invoke rapid rotation, although the conservative mass transfer exhibited by these particular models is likely to lead to rapidly rotating companions which could further smear their spectroscopic signatures. We advise that future claims of unseen black holes in binary systems would benefit from exploring detailed binary evolution models of stellar populations as a sanity check.
INTRODUCTION
Increasing the sample of stellar mass black holes (BH) with well constrained physical characteristics is essential to improving our collective understanding of stellar evolution and compact remnants. Gravitational wave detectors have considerably boosted our sample of BHs in binaries (e.g. The LIGO Scientific Collaboration et al. 2021), but these systems are at the very end of their evolution; finding black holes in binaries prior to the formation of a second compact remnant provides further constraints on stellar evolution in intermediary stages of the life of these systems. Recently, several studies with detailed orbital motion analysis have identified multiple systems with a BH component, such as LB1 and HR 6819 (Liu et al. 2019;Rivinius et al. 2020). But the complexity of these analyses has led to controversy surrounding the discoveries, and further work by independent groups found evidence that BH components where not responsible for the observational properties (LB1 - Irrgang et al. 2020;Shenar et al. 2020;Abdul-Masih et al. 2020;El-Badry & Quataert 2020;HR6819 -Bodensteiner et al. 2020) A new study by Saracino et al. (2021) suggested the presence of a BH in the massive cluster NGC 1850 located in the Large Magellanic Cloud. Dubbed NGC 1850 BH1, it is reported to have a mass of 11M and be in a 5.04 day orbit around a main sequence turn-off donor star with M donor = 4.9 M . A few days later El-Badry & Burdge (2021) showed that the period-density relationship reduced the estimated donor mass to ∼1M (with an upper limit ∼2.5M ). They also used MESA (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019 to propose an alternative binary system to explain the observations, ★ E-mail: [email protected] but they emphasise that the model they present only serves as an example to show that "more banal" alternatives are possible.
In this letter we will use the Binary Population And Spectral Synthesis (BPASS) datasets to address two questions: 1) Is the existence of NGC 1850BH1 supported by stellar evolution? 2) What type of system is most likely to explain the physical and observational properties described in Saracino et al. (2021) and El-Badry & Burdge (2021)? BPASS includes an extensive pre-computed grid of detailed stellar evolution models which includes binary interactions -one of the key differences between MESA and BPASS is that the former is most often used to flexibly create individual systems as needed, whereas the grid of stellar models in the later was created and tested to self-consistently reproduce a wide range of observables (see Eldridge et al. 2017;Stanway & Eldridge 2018). In the context of studies such as this one it is very important to emphasise that we are not using the BPASS code to create a system on the fly to recreate the observations. Instead we are searching through existing data products to see if our populations can predict observed systems. This gives us confidence that the models we find are consistent with stellar evolution at large, and allows us to quantify how prevalent a matched model would be in a given population.
All the models used in this work use the BPASS fiducial initial mass function which is a Kroupa (2001) prescription with a maximum initial mass of 300 M . Amongst the 13 metallicities available, we use Z=0.010 which is the closest match to NGC 1850. Finally, to facilitate searches through the vast stellar library (∼250,000 models) we are using the freely available BPASS stellar library dataframes created with hoki (Stevance et al. 2020a;Stevance et al. 2020b).
In Section 2 we perform several searches to see if systems containing potential NGC1850-BH1 candidates are predicted, before removing the constrain of a compact companion and performing a search to identify the models that best match the observational properties of the system. In Section 3 we discuss the implications of our model search and conclude on the most likely nature of the "dark" companion.
MODEL SEARCH
Looking for NGC 1850 BH1
First let us consider the system as described by Saracino et al. (2021): a ∼5 M star with a ∼11 M black hole primary in a 5 day orbit. We perform a search on a wide window of parameters, selecting any secondary model with M=3-7M , ages between 80 and 300 Myrs, and periods ranging from 3 to 7 days. At this stage we do not attempt to match the observed photometry to our synthetic photometry. The closest match obtained are 5 M stars with black hole masses 6.3M , and we find no models with larger black hole masses. The originally described system is therefore not predicted in our simulations. We then perform a more specific black hole (or compact remnant) search taking into account the analysis of El-Badry & Burdge (2021), who placed limits on the mass of the binary components. We also include constraints on the synthetic photometry (± 1 mag) of f336w and f814w (the two filters present in our models with reported observational values). The search criteria are as follows (i) Must contain a compact remnant (ii) Age = [80, 120] Myrs (iii) f336w = -2.537 ± 1 mags (iv) f814w = -1.68 ± 1 mags
(v) P=5.04 ± 2 days (vi) M bright = [0.5, 2] M (vii) M dark = [2, 6] M
The mass limits are based on the best estimates in figure 2 of El-Badry & Burdge 2021. We do not use temperature as a search criteria to allow for discussion but we come back to it at the end of this section. Using these constraints we find two matching models (see Table 1), both of which have a dark companion with a mass around the upper mass limit for a neutron star.
The upper mass limit of neutron stars remains debated but estimates based on the binary neutron star merger GW 170817 (see figure 3 in Abbott et al. 2018) indicate that Model 175736 (2.55 M ) is very likely a black hole whereas Model 176986 (2.26M ) may be a black hole or a neutron star depending on the equation of state considered. Note that the ages are lower than that of NGC 1850 by 20 Myrs, which is actually necessary for these models to be viable matches: In BPASS to save computing time the primary and secondary stars are evolved in detail successively, not simultaneously, and the age of a secondary star model is its age since the last rejuvination episode. A multitude of primary models could provide such systems with compact companions within 20 to 30 Myrs, and it is not within the scope of this letter to assess the most likely primary route as it does not inform the question of the nature of NGC 1850-BH1 further.
Even without searching for the primary evolution of these systems, BPASS outputs provide an estimate of the occurrence rate of each model within a given simple stellar population. Models 175736 and 176986 have occurrence rates of 0.047 and 0.048 per 10 6 M , respectively, meaning that in NGC1850 (with M∼10 5 M ) there is roughly a 0.5 percent chance of the parent system occurring. Additionally, the phases at which the criteria are met are very short lived for each model (40 and 15,200 years, respectively), making their observation even more unlikely. Finally, we need to address the temperature discrepancy between the reported value in Saracino et al. (2021) -14,500 K -and the predicted values in our two models -35,000 to 63,000K (see Table 1). Not only are our best black hole system matches extremely rare, the predicted surface temperature of the bright component actually excludes them quite confidently as viable matches to the observations. Consequently our synthetic stellar population suggests that there is no compact remnant in NGC 1850.
But the "dark" companion does not have to be a compact remnant and we explore these possibilities in the next section.
Looking for the best match
We now search for binary models without compact remnants that match the physical properties of the system. We slightly tighten the constraint on the photometry to highlight the very best matches in our analysis. We find three models (166726, 163570 and 168485) matching these criteria; their properties are summarised in Tables 2 and 3, and their evolution is shown in Figure 1. All three models are very similar: The bright star is heavily stripped with mass ∼1.27 to 1.85M and the unseen component is a 3 to 5.5 M secondary which gained mass through Roche Lobe Overflow (RLOF) starting during the Main Sequence of the primary star. In this scenario the system is observed at the very end of the semi-detached phase when the primary undergoes a second episode of rapid mass loss as the remainder of the the hydrogen envelope expands due to the onset of hydrogen shell burning. BPASS models do not record spectra of individual stars but the absolute magnitudes of both stars are predicted. We find that in the B, V and R band (covering the wavelength range of the MUSE data obtained by Saracino et al. 2021), the stripped primary is 1 to 2.3 magnitudes brighter than its counterpart. For completeness we note that in the current models the evolution of the second star is treated approximately and is not calculated in detail, and the synthetic photometry values quoted here are for a ZAMS star of the same mass (since these models have just been rejuinated through binary interaction). In the raw BPASS data the second star is 3 to 4 magnitudes fainter than the bright component, therefore that values quoted here are conservative estimates. Ultimately the study of systems such as these could reveal more about the impact of mass transfer on the secondary stars of binary systems and allow us to further refine our models.
On the whole these models correspond to rather garden variety binary stars with primary Zero Age Main Sequence masses 6 M and secondary masses ∼2 to 3 M (see Table 3). As a result their occurrence rate is much higher than any model involving a compact remnant, with each individual system expected to occur ∼1.5 times in a massive cluster such as NGC 1850 (M∼10 5 M ). Collectively over 4 systems (on average) are expected to have the potential to match the observed properties in such a cluster, and the time span over which the models presented above do match our criteria ranges from 250,000 years for Model 168485 to over half a million years for Model 166726 -which means that these stars are evolving on the thermal timescale and are not unlikely to be observed in a 100 Myrs cluster such as NGC 1850.
DISCUSSION AND CONCLUSIONS
In their letter El-Badry & Burdge (2021) suggested that NGC1850-BH1 could be a stripped star, but could not exclude the presence of a black hole as their upper mass limit on the unseen companion extended to 6M . Using the stellar populations modeled with BPASS we confirm that NGC 1850-BH1 is a stripped star, and exclude the possibility that it would be a black hole (or neutron star). The original interpretation that the visible star was a main sequence turn off star is understandable when using isochrones, but evolutionary tracks including binary interactions can lead stars at the end of their RLOF to circle back towards their MS turn off, which can be missed in a single-star paradigm. El-Badry & Burdge (2021) suggested that the unseen companion could be hidden in the spectrum due to rapid rotation smearing the spectral lines. In our best three matching models the stripped star is brighter than its companion in the optical bands by 1 to 2.3 magnitudes, or a factor of roughly 2.5 to >8. It is therefore not surprising that the companion would appear "dark" and be left undetected in the spectrum; for example, advanced disentanglement techniques were required to identify the components of the supposed LB-1 system and the final results revealed a flux contribution of 55% and 45% for the primary and the secondary respectively (see table 1 in Shenar et al. 2020), whcih is a much smaller disparity than observed here. Overall rapid rotation likely does not need to be invoked to result in a non-detection, as suggested by El-Badry & Burdge (2021), but it is highly probable that the systems shown here would have rapidly rotating companions as the mass transfer episodes they underwent was rather conservative. The systems in Models 168485, 163570 and 166726 only lost 0.18 percent, 4.7 percent and 16 percent of their pre-RLOF mass, respectively. Conservative mass transfer leads to the spin-up of the accretor and as a result we could expect the unseen companion to be rapidly rotating at the time the system is observed, making it harder to detect in the spectra even with a very high signalto-noise. Since BPASS does not directly include rotation we are not able at this stage to quantify the rotation of the unseen companion. Stellar evolution models are not perfect, and population synthesis is itself limited by the fact that we can only simulate in detail a finite and discrete number of systems. That is why the range of values applied for our search criteria must be wider than the observational errors, and although all three of our best models are the result of Case A mass transfer, it would be incorrect to conclude at this stage that the system reported in Saracino et al. (2021) is necessarily a result of Case A mass transfer. Additionally, the use of a discrete grid of models means that matching radial velocities directly is not a realistic endeavour, that is why we set independent mass ranges for M bright and M dark -although they are rooted in the radial velocity analysis of El-Badry & Burdge 2021. Comparison to their figure 2 indicates that Model 166726 would have very high inclination approaching 90 degrees (resulting in an eclipse, which would have been seen), whereas Models 163570 and 168485 would have inclinations around or below 60 degrees consistent with a non-eclipsing system.
Overall, there is overwhelming evidence that our stellar evolution models cannot predict NGC 1850-BH1 and that frequently occurring intermediate-mass stripped binaries are the best explanation. Future observational studies suggesting the presence of an unseen black hole in a binary system would benefit from using BPASS and hoki to search for their systems in our stellar populations as a sanity check (e.g. Eldridge et al. 2020) 1 . If more precise matches to observations are desired, the information obtained from matching the stellar population synthesis in BPASS can then be used as a starting point to create a range of MESA models to iterate upon, thereby taking advantage of the strengths of the respective codes. It is also worth keeping in mind that triple systems can offer a natural explanation to quiescent BH candidates (e.g. HR6819 Romagnolo et al. 2021) and that dynamical ineractions play a role in the formation of BH-Main Sequence star systems (e.g. Banerjee 2018).
If further observations of the NGC 1850-BH1 system confirm the existence of NGC 1850-BH1, it would pose a direct challenge to our simulations and offer a fantastic constraint to our stellar evolution prescription, directly impacting our understanding of compact remnant pathways. We also plot the He core mass as it shows that the RLOF occurs on the main sequence before a He core has formed (Case A). Right: Evolution of the separation and of the stellar radii.The shaded regions show RLOF, and the dark blue doted line shows the time at which the models match the observations. support of the Marsden Fund Council managed through Royal Society Te Aparangi. SGP acknowledges the support of a Science and Technology Facilities Council (STFC) Ernest Rutherford Fellowship.
Third party software
This work made extensive use matplotlib Hunter (2007), numpy (Hunter 2007;Harris et al. 2020) and pandas pandas development team (2020); Wes McKinney (2010).
DATA AVAILABILITY
The code to reproduce all the analysis and figures can be found on GitHub (https://github.com/UoA-Stars-And-Supernovae/ ngc1850bh1) with large dataset stored on Zenodo https:// zenodo.org/record/5813956#.YdJK33VBzmE The BPASSv2.2 stellar library is available as a repository of individual text (Download bpass-v2.2-newmodels.tar.gz files 2 ) or as individual tables split by metallicity 3
Figure 1 .
1Left: Evolutionary track of the matching models in Hertzsprung-Russell Diagrams. The orange markers shows the time steps where Roche Lobe Overflow (RLOF) is occurring in the model. The dark blue markers show the steps at which the models match the search criteria. Middle: Evolution of the masses of both binary components.
Table 1 .
1Properties of the nearest matching models with compact remnantsMODEL Age
M bright
M "dark
log( bright / )
ID
(Myrs) (days)
M
M
175736
81.10
4.89
0.90
2.55
4.81
176986
80.31
5.42
1.16
2.26
4.55
Table 2 .
2Properties of the stripped star models that best match observationsMODEL Age
M bright
M "dark
log(T/K)
log(g)
ID
(Myrs) (days)
M
M
cm.s −2
166726
84.42
4.82917
1.52
3.24
4.03
2.84
166726
84.61
5.24026
1.41
3.28
4.02
2.77
166726
84.76
5.7367
1.27
3.32
4.01
2.68
163570
85.98
4.70969
1.70
4.33
4.02
2.89
163570
86.26
5.30375
1.60
4.43
4.01
2.82
168485
85.79
4.54248
1.85
5.47
4.01
2.91
168485
86.04
5.17962
1.74
5.58
4.01
2.85
Table 3. Zero Age Main Sequence masses and number of such systems
expected to occur per 10 6 M at Z=0.010
MODEL ID M bright
M "dark"
N/10 6 M
166726
6
1.8
2.88
163570
6
2.4
2.78
168485
6
3
2.72
MNRAS 000, 1-5 (2021)
https://drive.google.com/drive/folders/ 1BS2w9hpdaJeul6-YtZum--F4gxWIPYXl 3 https://zenodo.org/record/3905388#.YZ603JFBzDQ
ACKNOWLEDGEMENTSThe authors are grateful to the anonymous referee for their insightful suggestions. We are also thankful to Chris Usher for sharing the original observational data of the system with us so quickly, aswell as Sebastian Kamann and Nate Bastian for thought provoking questions in private correspondence. HFS and JJE acknowledge the1We have tutorials and jupyter notebooks freely available to make searching through our models as accessible as possible. We are also always happy to answer your questions on GitHub or via email. Tutorial: https://heloises.github.io/hoki/ModelSearch.html Further Example -based on(Stevance & Eldridge 2021): https://github.com/UoA-Stars-And-Supernovae/Binary_ pathways_to_SLSNe_I_17gci
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Wes Mckinney, 10.5281/zenodo.3509134doi:10.5281/zenodo.3509134Proceedings of the 9th Python in Science Conference. Stéfan van der Walt Jarrod Millman edsthe 9th Python in Science Conference6100a pandas development team T., 2020, pandas-dev/pandas: PandasWes McKinney 2010, in Stéfan van der Walt Jarrod Millman eds, Pro- ceedings of the 9th Python in Science Conference. pp 56 -61, doi:10.25080/Majora-92bf1922-00a pandas development team T., 2020, pandas-dev/pandas: Pandas, doi:10.5281/zenodo.3509134, https://doi.org/10.5281/zenodo. 3509134
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[
"Towards an explanation for the 30 Dor (LMC) Honeycomb nebula -the impact of recent observations and spectral analysis",
"Towards an explanation for the 30 Dor (LMC) Honeycomb nebula -the impact of recent observations and spectral analysis"
] |
[
"J Meaburn \nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nM13 9PLManchesterUK\n",
"M P Redman \nCentre for Astronomy\nSchool of Physics\nNational University of Ireland Galway\nUniversity RoadGalwayIreland\n",
"P Boumis \nInstitute of Astronomy & Astrophysics\nNational Observatory of Athens\nI. Metaxa & V. Pavlou\nGR-152 36 P. PenteliAthensGreece\n",
"E Harvey \nCentre for Astronomy\nSchool of Physics\nNational University of Ireland Galway\nUniversity RoadGalwayIreland\n"
] |
[
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nM13 9PLManchesterUK",
"Centre for Astronomy\nSchool of Physics\nNational University of Ireland Galway\nUniversity RoadGalwayIreland",
"Institute of Astronomy & Astrophysics\nNational Observatory of Athens\nI. Metaxa & V. Pavlou\nGR-152 36 P. PenteliAthensGreece",
"Centre for Astronomy\nSchool of Physics\nNational University of Ireland Galway\nUniversity RoadGalwayIreland"
] |
[
"Mon. Not. R. Astron. Soc"
] |
The unique Honeycomb nebula, most likely a peculiar supernova remnant, lies in 30 Doradus in the Large Magellanic Cloud. Due to its proximity to SN1987A, it has been serendipitously and intentionally observed at many wavelengths. Here, an optical spectral analysis of forbidden line ratios is performed in order to compare the Honeycomb high-speed gas with supernova remnants in the Galaxy and the LMC, with galactic Wolf-Rayet nebulae and with the optical line emission from the interaction zone of the SS433 microquasar and W50 supernova remnant system. An empirical spatiokinematic model of the images and spectra for the Honeycomb reveals that its striking appearance is most likely due to a fortuitous viewing angle. The Honeycomb nebula is more extended in soft X-ray emission and could in fact be a small part of the edge of a giant LMC shell revealed for the first time in this short wavelength domain. It is also suggested that a previously unnoticed region of optical emission may in fact be an extension of the Honeycomb around the edge of this giant shell. A secondary supernova explosion in the edge of a giant shell is considered for the creation of the Honeycomb nebula. A microquasar origin of the Honeycomb nebula as opposed to a simple supernova origin is also evaluated.
|
10.1111/j.1365-2966.2010.17204.x
|
[
"https://arxiv.org/pdf/1006.2692v1.pdf"
] | 118,733,099 |
1006.2692
|
0296fe7b00c63ec736d820d1dd30af92f24ffc4d
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Towards an explanation for the 30 Dor (LMC) Honeycomb nebula -the impact of recent observations and spectral analysis
2010
J Meaburn
Jodrell Bank Centre for Astrophysics
University of Manchester
M13 9PLManchesterUK
M P Redman
Centre for Astronomy
School of Physics
National University of Ireland Galway
University RoadGalwayIreland
P Boumis
Institute of Astronomy & Astrophysics
National Observatory of Athens
I. Metaxa & V. Pavlou
GR-152 36 P. PenteliAthensGreece
E Harvey
Centre for Astronomy
School of Physics
National University of Ireland Galway
University RoadGalwayIreland
Towards an explanation for the 30 Dor (LMC) Honeycomb nebula -the impact of recent observations and spectral analysis
Mon. Not. R. Astron. Soc
0002010Received; AcceptedPrinted (MN L A T E X style file v2.2)ISM: individual: Honeycomb nebula -ISM: supernova remnants -Mag- ellanic clouds
The unique Honeycomb nebula, most likely a peculiar supernova remnant, lies in 30 Doradus in the Large Magellanic Cloud. Due to its proximity to SN1987A, it has been serendipitously and intentionally observed at many wavelengths. Here, an optical spectral analysis of forbidden line ratios is performed in order to compare the Honeycomb high-speed gas with supernova remnants in the Galaxy and the LMC, with galactic Wolf-Rayet nebulae and with the optical line emission from the interaction zone of the SS433 microquasar and W50 supernova remnant system. An empirical spatiokinematic model of the images and spectra for the Honeycomb reveals that its striking appearance is most likely due to a fortuitous viewing angle. The Honeycomb nebula is more extended in soft X-ray emission and could in fact be a small part of the edge of a giant LMC shell revealed for the first time in this short wavelength domain. It is also suggested that a previously unnoticed region of optical emission may in fact be an extension of the Honeycomb around the edge of this giant shell. A secondary supernova explosion in the edge of a giant shell is considered for the creation of the Honeycomb nebula. A microquasar origin of the Honeycomb nebula as opposed to a simple supernova origin is also evaluated.
INTRODUCTION
The 30 Doradus nebula in the Large Magellanic Cloud (LMC) is the most massive and largest H ii complex in the Local Group of galaxies. Its 300 pc diameter core and adjacent halo exists between two 1000 pc diameter supergiant filamentary shells (Meaburn 1979;1980). The multitude of young massive stars scattered throughout this region identify it as a nursary of recent star formation consequently, the supernova (SN) rate is of particular interest.The explosion of SN1987A emphasised that these events are ongoing.
The SN rate in this nebular complex has been considered by several authors recently. In particular Lazendic, Dickel & Jones (2003) using high resolution optical, radio and X-ray imagery searched for young supernova remnants (SNRs) in its central 80 pc diameter region but came up with only four regions with high radio/Hα ratios. However, two ⋆ E-mail: [email protected] of these were associated with young HII regions with young embedded stellar objects. Chu et al. (2004) then showed the remaining two SNR candidates to be dust clouds or obscured star forming regions. This leaves N157 along with the Honeycomb nebula (Wang 1992) as the only firm SNR candidates but many more young (1000 yr old) remnants are likely to be present but as yet undetected. This probability lead Meaburn (1984; to suggest that such remnants could be identified by the high-speed (200-300 km s −1 ) velocity 'spikes', with predominantly approaching radial velocities, that are present in the longslit position-velocity (pv) arrays of optical line profiles (e.g. see fig.3 of Meaburn 1988). Chu et al. (1994) suggested quite correctly that these are not necessarily diagnostic of SN activity because wind-blown shells, particularly around Wolf-Rayet (WR) stars can have similar extents in radial velocity. Maybe individual SNRs just become merged and combine with particle winds to drive the giant (50-100 pc diameter) shells found around the many OB-associations in the 300 pc diameter halo of 30 Doradus.
However, it seemed worthwhile to investigate if optical line brightness ratios of only the high-speed component of the 30 Doradus radial velocity spikes could be used as a diagnostic of individual SN origin. This opportunity became viable for it was realised that in previous, separate observations of the Honeycomb nebula (Wang 1992), itself most likely of SN origin for it is a non-thermal radio source (Chu et al. 1995 though see Sect. 4.4 for a more radical alternative possibility), that one slit position for Hα and [N ii] 6548 & 6584Å longslit line profiles matched, within the 'seeing' disk, the position for the [S ii] 6717 & 6731Å profiles (Redman et al. 1999). Sound diagnostic optical line ratios of the high-speed ionized gas therefore became accessible.
In the present paper these line ratios of the Honeycomb nebula's high-speed gas are evaluated and they resemble those of LMC but not Galactic SNRs. The creation and structure of the strange, and possibly unique, Honeycomb nebula itself, is re-considered also aided by the most recent Chandra and other X-ray imagery plus Hubble Space Telescope (HST) and ESO New Technology Telescope (NTT) imagery. The impact of these results on an evaluation of the 30 Doradus SNe rate is also considered.
OBSERVATIONS
Due to it's location in the vicinity of SN1987A, the Honeycomb nebula has been observed by XMM-Newton and Chandra in the X-ray regime and in narrow band optical imaging with the NTT. Figure 1(a) shows narrow band images of the Honeycomb in Hα plus [N ii] 6548 & 6584Å. Figure 1
(b) is an [O iii] 5007Å
image, in which the Honeycomb is seen more extensively. Figure 1(c) is a Chandra X-ray image of the source in which it can be seen that the spatial extent of the X-ray emission closely matches the optical emission. Part of the Honeycomb was also serendipitously observed at the very edge of the field by the Hubble Space Telescope (HST) as part of a project to detect light echos from SN1987A (Crotts 1988). These broad-band observations are of much lower sensitivity than the narrow-band ones in Figs (Redman et al. 1999) line profiles were obtained using the Manchester Echelle Spectrometer (MES; Meaburn et al. 1984) on the Anglo-Australian telescope along the same (within the one arcsec wide seeing disk) slit position (Fig. 2a). The position-velocity (pv) array of [S ii] 6717 & 6731Å profiles along this slit is shown in Fig. 2b. The atmospheric conditions were photometric in both cases. The details of these observations and their analyses are fully described in the respective papers and will not be repeated here.
OPTICAL LINE BRIGHTNESS RATIOS
The two MES long slit pv arrays of line profiles had been converted into absolute surface brightnesses (B erg s −1 cm −2 sr −1Å−1 ) using the spectra of standard stars. The brightnesses along the slit length of each of the five emission lines were obtained in the heliocentric radial velocity ranges of V hel = 150-230 km s −1 and 240-300 km s −1 using YSTRACT in the STARLINK FIGARO suite of data reduction programmes. The example pv array in Fig. 2b reveals that the former velocity range contains only the high-speed Honeycomb features whereas the latter range contains the emission from predominantly the 30 Doradus host nebula. The peaks of the B values for all five lines in the 150-230 km s −1 range occurred at the positions marked 1-6 in Fig. 2b and each is coincident with a high-speed velocity 'spike' from the Honeycomb nebula. and Goudis, respectively. One black dot for RCW 104 is of particular interest for it is for a high speed knot. This is shown in Fig. 3 and I(λ 6717)/I(λ 6731) brightness ratios are ± 0.04, ± 0.04 and ± 0.05 respectively. Those for the brighter emission from the host 30 Doradus nebula are significantly smaller.
DISCUSSION
Line ratio diagnostics
The Honeycomb nebula is a non-thermal radio source and therefore most likely of SN origin (Chu et al. 1994;1995) though see Sect. 4.4 for a more radical, but unlikely, possibility. However, it is striking that the positions of the line ratios of the Honeycomb high-speed gas, on the diagnostic diagrams in Figs. 3-5 match most closely those occupied by LMC but not Galactic SNRs. These positions are significantly away from those for Galactic SNRs before considering that the nitrogen abundance of the LMC is lower by a factor of 2 compared with that of the Galaxy (Russell & Dopita 1992). This could proportionally lower the [N ii] 6584Å brightness from the Honeycomb nebulosity and move line ratios closer to the Galactic SNR zones in Figs. 3 & 5.
Furthermore, ratios for the high speed Honeycomb gas bear no resemblance to those of the two WR nebulae considered here (Fig. 3) even when Galactic and LMC abundance differences are taken into account. Even the high-speed component of the Galactic WR nebula, RCW 104, is far removed from the Honeycomb zone. When determining the SN rate for the halo of the 30 Doradus nebula from the number of high-speed velocity spikes over the region (Meaburn 1984; any confusion with those of WR origin is easily clarified by the relative positions of their line ratios on diagnostic diagrams.
Modelling kinematics and morphology
The SHAPE code of Steffen (see Steffen, Holloway & Pedlar 1996 for its initial use and Steffen & López 2006) permits the actual structure and kinematics of a nebula to be deduced from the imagery (Fig. 1 a-b) and long-slit pv arrays (such as that shown in Fig. 2b). No consideration of emission mechanisms or of the dynamics is involved here but the spatiokinematic modelling is useful to investigate how critical the viewing angle is to the unique Honeycomb morphology.
The Honeycomb was modelled as a set of cylinders with equal length and diameter (since the Rayleigh-Taylor instability, for example, tends to grow most rapidly for those modes of order of the thickness of the disturbed layer). The edges of some of the Honeycomb cells are visible in archive HST images yet are unresolved. The edge widths to cell diameter ratios are 0.01.
The Chandra images (eg. Fig. 1c) are of comparable resolution to the size of the cells but do seem more indicative of emission from the centre of each cell rather than poorly resolved boundary layers. A simple model would then be of hot X-ray emitting gas venting through gaps in a preexisting shell or layer of gas and accelerating the prexisting material in a boundary layer at the cell edges. Therefore, the Honeycomb cells were modelled with SHAPE as a series of very thin, nested optically emitting cylinders, with each cylinder inwards having an increasing characteristic velocity. The blueshifted velocity spikes (eg. Fig. 2b) then result from this range of velocities being simultaneously present within the unresolved edges of the Honeycomb cells. While the Honeycomb morphology and kinematics can be reproduced in such an empirical model, it should be stressed that it does not describe the radiation emission or hydrodynamics of the system.
More interesting perhaps is to examine what happens when the model system described above is viewed from a different angle. Figs. 6a-c show that the circular cells, when viewed from increasing viewing angle, rapidly overlap and become blended into arcs and knots, somewhat reminiscent of the tangled filamentary structure that is commonly seen in optical emission lines toward the edges of SNRs. Similarly, the striking velocity spikes characteristic of the Honeycomb blend into more confused and diffuse emission on the pv array. If the material forming the cell edges is being continually entrained and eroded then the extreme thinness of the cell edges suggests that the uniqueness of the Honeycomb could simply be due to it being a short-lived structure viewed from a fortuitous angle. The low cell edge width to diameter ratio of the most prominent cell is clear in the HST image in Fig. 7 and should be compared to its synthetic image in Fig. 6a.
However, the Honeycomb may not be being viewed exactly in the plane of the sky. The individual cells are almost all distinctly brighter on their western sides which is easily reproduced by the SHAPE code if they are viewed at a small angle to the line of sight. Furthermore, there are redshifted velocity spikes present in the Honeycomb that are not explained by the empirical model above though see possible explanation in Sect. 4.3.
A SN explosion in an expanding LMC giant shell
It was thought that the Honeycomb appearance and the pv arrays across the nebula could only be produced as a secondary SN blast wave encountered clumps of material in the nearside of a preceding expanding shell (e.g. see fig. 3 of Redman et al. 1999). This interpretation can certainly explain the nearly circular features in the imagery and the predominance of approaching velocity 'spikes' in the pv arrays of line profiles which are coincident with the edges of the sub-shells in the nebular image. However, it fails to explain the elongated, three ridge structure that is particularly evident in the Chandra X-ray image in Fig. 1c and the fact that positive velocity spikes occur over the most westerly ridge of Honeycomb shells which are most apparent in the [O iii] 5007Å image in Fig. 1b. Another possibility therefore is that such a secondary SN explosion has occurred in the edge of a preceding giant LMC shell but now viewed tangentially to this previous feature (see Fig. 8). The three ridges of secondary Honeycomb shells could then be in folds of the surface of the original shell. Direct, but partial, evidence for the existence of this preceding (2004). The Honeycomb nebula forms a small part of its western edge but only a northern ridge is clearly detected at X-ray wavelengths. The X-ray image of Dennerl et al. (2001) shows a bright feature at the Honeycomb location that also clearly extends in an arc to the northeast. This extended X-ray feature is compared in Fig. 9 to the area of the images in Figs. 1a-c. Strangely, fainter Honeycomb-like features are also present in optical images along parts of this ridge mostly outside the area of Figs 1a-c (but see top-left of Fig.1a) but as yet no line profiles have been obtained from them to confirm their Honeycomb nature.
A microquasar origin
The unusual, and possibly unique, morphology and kinematics of the Honeycomb nebula warrants the consideration of a more radical possibility than given in Sect. 4.3 for its origin. Could it be the manifestation of the collision of a precessing relativistic jet emitted by a binary microquasar similar to SS 433 (for a review see Fabrika 2004), Cygnus X-1 (Russel et al. 2007), IC 342 X-1 (Feng & Kaaret 2008) and that in the galaxy NGC 7793 (Soria et al. 2009)? Non-thermal radio emission as seen emanating from the Honeycomb nebula would also be expected in this interaction zone between the jet and the ambient gas.
The eight hard X-ray point sources found by Haberl et al. (2001) in the vicinity of the Honeycomb nebula, could be candidates for the origin of such a jet. Although Haberl et al. (2001) strongly favour background active galactic nuclei for the origin of these point, hard X-ray sources, they do not completely rule out the microquasar possibility in all cases.
The faintest of these eight point sources (Source 1 of Haberl et al. 2001) has no measured spectral index and can be seen in their fig. 5 to be well placed with respect to the Honeycomb nebula if such a microquasar mechanism is occurring and if this nebula is on the western edge of an elongated structure with this source central. A much fainter candidate is the marginally detected point source arrowed in Fig. 9 but seen more clearly in fig. 5 of Haberl et al. (2001) though not listed by these authors. This is towards the centre of the more spherical giant shell proposed here to have the Honeycomb nebula at its edge.
However, it is interesting that the optical line ratios for the Galactic SS 433/W50 nebulosity (Boumis et al. 2007) when placed into the diagnostic diagrams shown in Figs. 3-5 bear no resemblance to the Honeycomb ratios or any other phenomena except Galactic planetary nebulae. Perhaps, the X-rays from the SS 433 jet are radiatively ionizing the processed material of the W50 SNR envelope. Furthermore, the different X-ray properties of the Honeycomb Nebula and W50 do not support a microquasar origin for the former. Even though these points argue against a microquasar origin for the Honeycomb nebula it will still be worthwhile to see if any stars within the error boxes of the two point X-ray sources in Fig. 9 have microquasar characteristics.
CONCLUSIONS
The optical line ratios of the high-speed Honeycomb nebula confirm its most likely SNR origin. The velocity spikes on pv arrays of longslit line profiles found over the rest of the halo of 30 Doradus can be identified as of young SNR origin if they also occupy the LMC SNR zone on such diagrams. In this way they can be distinguished from high-speed WR shells. The circular structures of the Honeycomb gas and their corresponding approaching and receding, highly collimated, flows can be modelled as a young SNR in the edge of a larger and preceding giant shell. The appearance and kinematics of the separate Honeycomb cells is strongly dependent on viewing angle i.e.they are only apparent if the cell walls are viewed along their cylindrical axes. An alternative view that this unique nebula could be the consequence of a microquasar jet (similar to that from SS 433) is shown from comparative optical line ratios, and by other arguments, to be highly unlikely but not completely dismissed as there are two point X-ray sources in its vicinity, and because it is such a strange object. The positions of the line ratios of the SS 433/W 50 interaction nebulosity on the diagnostic diagrams are remarkable in their own right. . . Similar diagnostic diagram to that in Fig. 3 but again (as for Fig. 4) no values for the WR nebulae were available. 1a-c is shown as a dashed box against a subset of the XMM X-ray image from Dennerl et al. (2001). Softer X-rays become orange in this presentation and hard blue. SN1987A is the bright source towards the top right and the point X-ray source number 1 of Haberl et al. (2001) is towards the top left (J2000 coords).An arrow points to a marginally detected point source which is more apparent in the original image of Dennerl et al. (2001)
The log10 [Hα / ([N ii] 6548 & 6584Å) ] versus log10 [Hα /([S ii] 6716 & 6731Å)] brightness ratios for the high speed 'spikes' are shown in Fig. 3 for positions 1-6 as square dots. Similarly the [S ii] 6716 & 6731Å ratios I(λ 6717)/I(λ 6731) versus log10 [Hα /([N ii] 6548 & 6584Å)] and log10[Hα /([S ii] 6716 & 6731Å)] ratios are shown in Figs 4 and 5 respectively. The mean ratios for the 30 Dor host nebula from the same data are marked in each of Figs 3-5 by a cross. All of these values are compared with the regions occupied by line ratios of Galactic SNRs, H ii regions and planetary nebulae (PNe) as given in the diagnostic diagrams ofSabbadin, Minello & Bianchini (1977). The values for LMC SNRs(Payne, White & Filipovic 2008) are shown as diamonds. The large open circles and large black dots only inFig. 3are for the Galactic filamentary WR nebulae NGC 3199 and RCW 104 as given by
≈ 8 ′
8(≡ 130 pc) diameter giant shell centred on RA 05h 36m 30s DEC −69 • 19 ′ (J 2000) can be seen in the X-ray imagery in fig. 5 of Dennerl et al. (2001), fig 1l of Dunne, Points & Chu (2001) and fig. 2 of Smith & Wang
Figure 1 .Figure 2 .
12The a) Hα+[N ii] 6548 & 6584Å and b) [O iii] 5007Å images of the Honeycomb nebula are compared with c) the Chandra X-ray image. All are for the same area of the sky. (J2000 coords). a) The position of part of the longslit is shown against the Hα+[N ii] 6548 & 6584Å image of the Honeycomb nebula and compared in b) with the pv array of [S ii] 6717 & 6731Å profiles along this same length.The positions 1-6 coincident with negative velocity 'spikes' where line ratios were measured are also indicated.
Figure 3 .Figure 4 .Figure 5
345The six line intensity ratios (square dots) measured for positions 1-6 (Fig. 2b)for the high speed components in the Honeycomb line profiles are compared with the areas of this diagnostic diagram occupied by galactic SNRs, H ii regions and planetary nebulae. The mean of the line ratios for 30 Doradus in the same vicinity is marked as a cross. Also shown are line ratios for two galactic Wolf-Rayet nebulae RCW 104 and NGC 3199. The large dot with an arrow is for the high-speed gas in RCW 104 and the arrow indicates that the [S ii] 6717 & 6731Å lines were not detected. All other WR values are for the filamentary features. The values for LMC SNRs are shown as diamonds. The ratios for the W 50 nebulosity generated by the microquasar, SS 433, are shown by plus signs. Similar diagnostic diagram to that inFig. 3but no values for the WR nebulae were available.
Figure 6 .Figure 7 .Figure 8 .Figure 9 .
6789a) SHAPE code model of the Honeycomb as a set of cylinders viewed end-on. The image is on the left and the synthetic pv array on the right for the slit position marked in Fig. 2a. Sharp 'velocity spikes' are generated. The synthetic images and pv arrays in (b) and (c) are for the same model as in Figure 6a but viewed at angles of 45 and 60 degrees respectively. The Honeycomb structure loses coherence and the velocity spikes become shortened and blended. The radial velocity scale is with respect to systemic value for this region of the 30 Doradus halo.N The broadband HST (675 nm) image of part of the field shown inFig.1a-c. The low edge width to diameter ratio of the most prominant Honeycomb shell (top right) should be compared to its synthetic image inFig. 6a. The positive and negative high-speed velocity spikes in pv arrays can be generated by a young SNR in the edge of an LMC giant shell. A more realistic depiction would have clumpy material in the giant shell and it to have a more irregular structure than sketched here. Positive and negative flows in the cylindrical cells around clumps would be seen along the sight-line marked here. The area of Figs.
. 1a-b but do show that the edges of the Honeycomb cells are not resolved even at the resolution of the HST.The edge widths
to cell diameter ratios are remarkably
0.01. (see Sect.4.2
and Fig. 7).
Hα and [N ii] 6548 & 6584Å (Meaburn et al. 1993)
and [S ii] 6717 & 6731Å
with an arrow attached to show its limiting ratio as [S ii] 6717 & 6731Å was not detected. The uncertainties in the Honeycomb log10 [Hα /([N ii] 6548 & 6584Å)], log10[Hα /([S ii] 6716 & 6731Å)]
This paper has been typeset from a T E X/ L A T E X file prepared by the author.
ACKNOWLEDGMENTSWe thank Wolfgang Steffen for help using the SHAPE coded and Ravi Sankrit for past useful discussions and advice on Chandra data.
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|
[] |
[
"In-Vitro MPI-guided IVOCT catheter tracking in real time for motion artifact compensation",
"In-Vitro MPI-guided IVOCT catheter tracking in real time for motion artifact compensation"
] |
[
"Florian Grieseid \nInstitute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany\n\nSection for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany\n",
"Sarah Latus \nInstitute of Medical Technology\nHamburg University of Technology\nHamburg, Germany\n",
"Matthias Schlü Terid \nInstitute of Medical Technology\nHamburg University of Technology\nHamburg, Germany\n",
"Matthias Graeser \nInstitute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany\n\nSection for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany\n",
"Matthias Lutz \nDepartment of Internal Medicine\nUniversity Medical Center Schleswig-Holstein\nKielGermany\n\nUniversitatsklinikum Wurzburg\nGERMANY\n",
"Alexander Schlaefer \nInstitute of Medical Technology\nHamburg University of Technology\nHamburg, Germany\n",
"Tobias Knopp \nInstitute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany\n\nSection for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany\n",
"Wolfgang Rudolf Bauer "
] |
[
"Institute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany",
"Section for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany",
"Institute of Medical Technology\nHamburg University of Technology\nHamburg, Germany",
"Institute of Medical Technology\nHamburg University of Technology\nHamburg, Germany",
"Institute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany",
"Section for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany",
"Department of Internal Medicine\nUniversity Medical Center Schleswig-Holstein\nKielGermany",
"Universitatsklinikum Wurzburg\nGERMANY",
"Institute of Medical Technology\nHamburg University of Technology\nHamburg, Germany",
"Institute for Biomedical Imaging\nHamburg University of Technology\nHamburg, Germany",
"Section for Biomedical Imaging\nUniversity Medical Center Hamburg-Eppendorf\nHamburg, Germany"
] |
[] |
PurposeUsing 4D magnetic particle imaging (MPI), intravascular optical coherence tomography (IVOCT) catheters are tracked in real time in order to compensate for image artifacts related to relative motion. Our approach demonstrates the feasibility for bimodal IVOCT and MPI invitro experiments.Material and methodsDuring IVOCT imaging of a stenosis phantom the catheter is tracked using MPI. A 4D trajectory of the catheter tip is determined from the MPI data using center of mass sub-voxel strategies. A custom built IVOCT imaging adapter is used to perform different catheter motion profiles: no motion artifacts, motion artifacts due to catheter bending, and heart beat motion artifacts. Two IVOCT volume reconstruction methods are compared qualitatively and quantitatively using the DICE metric and the known stenosis length.
|
10.1371/journal.pone.0230821
| null | 208,309,998 |
1911.12226
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8206ced3598b41ffa1bfb1824d5c5a73c2555c2a
|
In-Vitro MPI-guided IVOCT catheter tracking in real time for motion artifact compensation
Published: March 31, 2020
Florian Grieseid
Institute for Biomedical Imaging
Hamburg University of Technology
Hamburg, Germany
Section for Biomedical Imaging
University Medical Center Hamburg-Eppendorf
Hamburg, Germany
Sarah Latus
Institute of Medical Technology
Hamburg University of Technology
Hamburg, Germany
Matthias Schlü Terid
Institute of Medical Technology
Hamburg University of Technology
Hamburg, Germany
Matthias Graeser
Institute for Biomedical Imaging
Hamburg University of Technology
Hamburg, Germany
Section for Biomedical Imaging
University Medical Center Hamburg-Eppendorf
Hamburg, Germany
Matthias Lutz
Department of Internal Medicine
University Medical Center Schleswig-Holstein
KielGermany
Universitatsklinikum Wurzburg
GERMANY
Alexander Schlaefer
Institute of Medical Technology
Hamburg University of Technology
Hamburg, Germany
Tobias Knopp
Institute for Biomedical Imaging
Hamburg University of Technology
Hamburg, Germany
Section for Biomedical Imaging
University Medical Center Hamburg-Eppendorf
Hamburg, Germany
Wolfgang Rudolf Bauer
In-Vitro MPI-guided IVOCT catheter tracking in real time for motion artifact compensation
Published: March 31, 2020Received: November 28, 2019 Accepted: March 9, 2020RESEARCH ARTICLE ☯ These authors contributed equally to this work. * [email protected] OPEN ACCESS Citation: Griese F, Latus S, Schlüter M, Graeser M, Lutz M, Schlaefer A, et al. (2020) In-Vitro MPI-guided IVOCT catheter tracking in real time for motion artifact compensation. PLoS ONE 15(3): e0230821. https://doi.org/10.1371/journal. pone.0230821 Editor: Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https:// Data Availability Statement: The raw data of the IVOCT and MPI measurements are uploaded at https://doi.org/10.5281/zenodo.3554935.
PurposeUsing 4D magnetic particle imaging (MPI), intravascular optical coherence tomography (IVOCT) catheters are tracked in real time in order to compensate for image artifacts related to relative motion. Our approach demonstrates the feasibility for bimodal IVOCT and MPI invitro experiments.Material and methodsDuring IVOCT imaging of a stenosis phantom the catheter is tracked using MPI. A 4D trajectory of the catheter tip is determined from the MPI data using center of mass sub-voxel strategies. A custom built IVOCT imaging adapter is used to perform different catheter motion profiles: no motion artifacts, motion artifacts due to catheter bending, and heart beat motion artifacts. Two IVOCT volume reconstruction methods are compared qualitatively and quantitatively using the DICE metric and the known stenosis length.
Results
The MPI-tracked trajectory of the IVOCT catheter is validated in multiple repeated measurements calculating the absolute mean error and standard deviation. Both volume reconstruction methods are compared and analyzed whether they are capable of compensating the motion artifacts. The novel approach of MPI-guided catheter tracking corrects motion artifacts leading to a DICE coefficient with a minimum of 86% in comparison to 58% for a standard reconstruction approach.
Conclusions
IVOCT catheter tracking with MPI in real time is an auspicious method for radiation free MPIguided IVOCT interventions. The combination of MPI and IVOCT can help to reduce motion artifacts due to catheter bending and heart beat for optimized IVOCT volume reconstructions.
Introduction
Optical coherence tomography (OCT) enables a high-resolution imaging of tissue structures [1][2][3]. In the field of cardiovascular diseases intravascular OCT (IVOCT) imaging is applied to assess the vascular wall structures and observe plaque formations and related stenosis lengths [4,5]. IVOCT highly benefits from a second imaging modality in order to align its catheter tip position within the global coordinate system of the patient. Using digital subtraction angiography (DSA), ionizing radiation is introduced and only 2D projections of the catheter tip positions are observed. Different methods have been presented to determine the 3D vascular shape using a combination of IVOCT and angiographic images. For example, a co-registration of both imaging modalities is applied to align the images to each other [6][7][8][9]. An improved 3D volume reconstruction method uses the information of both the vessel center line as well as the 3D catheter trajectory determined in bi-plane angiographic frames [10]. Most of the recent volume reconstruction methods assume a static imaging scenario neglecting heart beat motion, arterial vasomotion, and catheter bending leading to motion artifacts. Nevertheless, several publications depict a relevant influence of motion artifacts on the IVOCT volume reconstructions. For example, an irregular formation of stent struts are related to heart beat motion [11,12]. In a pre-clinical scenario a setup for ECG triggered IVOCT imaging with a duration of less than one second is proposed [13], hence heart beat motion artifacts can be minimized. Micro-motor catheters are proposed in order to deal with the problem of imaging artifacts due to bending of proximally rotated catheters [13,14]. However, the miniaturization of high-speed motors is a challenging and expensive task. Thus, a medically approved IVOCT catheter with micro motor has not been presented yet. Consequently, motion artifacts due to catheter bending and arterial vasomotion still arise in clinical scenarios and have an influence on the quantification of plaque formations. In addition, a contrast agent (iodine) is necessary for DSA imaging, which can be problematic in some patients with kidney diseases [15][16][17].
As an alternative, magnetic particle imaging (MPI) spatially resolves the distribution of superparamagnetic iron oxide nanoparticles (SPION) in 4D at high temporal resolution by using the particle's non-linear magnetization characteristics [18,19]. MPI applies static and oscillating magnetic fields to visualize the SPIONs. Thus in contrast to DSA, no ionizing radiation is induced to the patient. The changing magnetic fields are operated within the safety constraints of the peripheral nerve stimulation [20] and specific absorption rate (SAR) [21][22][23]. The SPIONs are biodegradable and decomposed within the liver [24,25]. Furthermore, MPI provides 3D information over time, while DSA only provides 2D projections over time. An advanced biplane DSA measurements can be post-processed to gain a 3D information over time, which however leads to a doubled radiation exposure. Further, MPI has demonstrated its beneficial usage in several interventional applications such as catheter tracking, stenosis identification and stenosis clearing [26][27][28]. Catheters and guide wires are coated with magnetic markers to track their position with MPI in real time [29][30][31][32][33]. The first bimodal experiments combining IVOCT and MPI are presented in [34,35]. The 3D vessel center line can be estimated from static MPI images. With the help of the estimated vessel center line the IVOCT images are oriented in 3D space to reconstruct the vessel volume.
In this work, we track the IVOCT catheter by labeling its tip with an MPI visible marker. Using real time MPI imaging we get the catheter position over time allowing for motion compensation. Both imaging modalities are registered to each other using a time synchronization. An experimental setup, including a custom built IVOCT adapter, enables the generation of different catheter motion profiles. In all experiments the MPI-tracked catheter trajectory is used to reconstruct a 4D IVOCT volume. A straight 3D printed vessel phantom with integrated stenosis is imaged. In a first experiment the plausibility and statistical error of the MPI catheter tracking is analyzed using a constant catheter velocity. In the following experiments motion artifacts due to catheter bending and heart beat are simulated. The reduction of motion artifacts and their statistical deviation are analyzed. The reduced artifacts in the reconstructed volumes are shown by quantitative measurements using the Dice similarity coefficient (DICE) factor and the estimated stenosis lengths in comparison to the ground-truth shape of the phantom.
Materials and methods
Experimental setup
The experimental setup is composed of a pre-clinical MPI scanner [36], a custom built IVOCT imaging adapter, a spectral domain OCT system (Telesto I, Thorlabs), and a control unit as shown in Fig 1. A straight vessel phantom with an inner diameter of 2.5 mm and total length of 20 mm is positioned within the MPI field of view (FoV). A stenosis with a length of 1.5 mm and an inner diameter of 1.5 mm is integrated in the phantom (see Fig 1, CAD sketch). A 3D printer (Form 2, Formlabs) based on stereolithography is used to build the phantom out of gray resin. An IVOCT catheter (Dragonfly Duo Kit, Abbott) with an outer diameter of 0.9 mm is used. The catheter consists of an optical fiber covered by a tight and flexible protection, which can rotate freely within a hollow plastic catheter. Within the catheter tip a prism directs the infrared light to the surface. To enable a MPI-based catheter tracking, the catheter tip is coated with a thin layer of magnetic lacquer (1 μL Magneto Magnetic Lacquer, Hand & Nail Harmony), as seen in Fig 1. The lacquer dries quickly on the catheter tip. The OCT images acquired with the marked catheter provide suitable image quality, whereas the phantom structures are still apparent for later segmentation algorithms.
MPI acquisition parameters.
For the MPI measurements a pre-clinical MPI scanner is used together with a custom-built receive coil [37]. The scanner excites the particles with three orthogonal sinusoidal excitation fields with frequencies f x = 2.5/102 MHz, f y = 2.5/96 MHz, and f z = 2.5/99 MHz. The magnetic field strength is set to 12 mT in all three directions while the gradient strength is set to 2.0 T m -1 in z-direction and 1.0 T m -1 in the x-and y-directions. The imaging period is 21.54 ms which equals a frame rate of 46.43 Hz. The FoV has a size of 24 mm × 24 mm × 12 mm and the MPI data acquisition is conducted with the system software Paravision (Bruker).
In order to reconstruct an MPI image using the frequency space approach [38], a calibration scan is required. This scan moves a small delta sample filled with SPIONs through the FoV while the system response at all attended positions is measured. The acquired data is used to set up the MPI system matrix, which characterizes the relation between the induced voltage signal and the particle distribution. In this work, the system matrix is acquired at 35 × 25 × 13 positions which cover a total volume of 35 mm × 25 mm × 13 mm. To prevent artifacts at the FoV boundaries, the calibration volume is chosen to be larger than the system FoV in all directions [39]. The delta sample has a size of 1 mm × 1 mm × 1 mm and is filled with 1 μL undiluted magnetic lacquer.
2.1.2 IVOCT acqusition parameters. The OCT system with an A-scan rate of f OCT = 91 kHz uses a central wavelength of 1315 nm. The axial OCT FoV is about 2.66 mm in air, whereas each A-scan consists of 512 pixels. The phantom is filled with distilled water yielding a pixel spacing of 4.5 μm between catheter and inner phantom wall, assuming a refractive index of 1.33. The custom-built catheter adapter enables a simultaneous rotation and translation of the IVOCT catheter. A rotational frequency of f rot = 6.25 Hz is used for all experiments. The center pullback velocity v 0 = −1.25 mm s −1 is varied during the experiments to simulate different motion artifacts. In all experiments, the catheter is pulled back over a total distance of s = 25 mm.
MPI image reconstruction and image processing
In frequency space MPI, the inverse problem to reconstruct an MPI image is treated with a first-order Tikhonov-regularized least-squares approach argmin c kSc À uk
2 2 þ lkck 2 2 ;ð1Þ
where S 2 C M�N is the MPI system matrix, u 2 C M is the measurement vector and c 2 Rþ N is the particle-concentration vector. This least-squares problem is iteratively solved by using the Kaczmarz method. The Kaczmarz method converges quickly for nearly orthogonal matrices, which is the case for MPI [40,41]. For the MPI reconstruction and data processing the Julia packages MPIFiles.jl [42] and MPIReco.jl [43] are used. The number of Kaczmarz iterations is set to 3 whereas the regularization parameter λ is set to λ = λ 0 � 10 −3 , where λ 0 = trace(S H S) N −1 . These reconstruction parameters have been optimized regarding the visual impression of the reconstructed MPI images.
MPI-Guided catheter tracking.
The 4D MPI images are block averaged with a factor of two over time prior to reconstruction, which leads to a temporal resolution of f MPI = 23.2 Hz. The set of MPI images is denoted by
I : O s � R ! R (O s � R 3 ) with I(x, t)
where x is the position and t is the time. The catheter localization is performed in three steps as generally described for more than one marker in [44]. At first, a threshold filter is applied to each image in order to separate the marker from the background. This results in the data set I seg :
O s � R ! R with I seg ðx; tÞ ¼ ( 1 if Iðx; tÞ � Y � max x Iðx; tÞ 0 otherwise;ð2Þ
where Θ 2 [0, 1] denotes the relative threshold. In our case the relative threshold is chosen to be Θ = 0.35. In a second step, the connected region O t 1 � O s , with the highest maximal intensity value max IðO t 1 ; tÞ is identified by connected-component labeling of I seg (O s , t), t 2 R. Finally, the position of the catheter marker is obtained by calculating the center of mass
cðtÞ ¼ R O t 1 x � I seg ðx; tÞdx R O t 1 I seg ðx; tÞdxð3Þ
of the voxel intensities of the corresponding connected region in the MPI image I. The accuracy for this sub-voxel approach is within the sub-millimeter range and the catheter position is determined only within cropped MPI FoV robustly. The positions M 1 to M 2 denote the positions when the catheter enters and leaves this cropped MPI FoV. Hence, we crop the MPI FoV for later 4D reconstruction methods. In x-direction the cropped MPI FoV has a length of approximately 10 mm. Outside this cropped MPI FoV the catheter position could not be determined as robust since more image artifacts are introduced by the rotation of the catheter. These outer positions are not considered for the later reconstruction methods. Additionally, outliers are removed with a Ransac algorithm and extreme outliers are excluded via thresholding. In these extreme cases, the images are affected by noise and the localization algorithm falsely detects a high intensity noise voxel as a marker position. Further, the trajectories are smoothed to ensure a continuous trajectory.
Volume reconstruction methods
We refer to two different methods as IVOCT catheter marker tracking (MT) and input parameter (IP) based volume reconstruction, respectively. For both reconstruction methods, the inner phantom wall is segmented in the IVOCT data using a semi-automatic algorithm [34,35,45]. Especially in the narrowed phantom parts, some manual corrections are applied. As a result the distance r between catheter and phantom wall is given for each A-scan. 3D point clouds are generated based on the MT and IP method, whereas their envelopes are used to quantify the volume reconstructions.
Input parameter (IP) method.
On the basis of the known input parameters of the custom built adapter (pullback and rotational speed) we take the distances r for each OCT Ascans and place a respective point in a 3D coordinate system. We assume a constant pullback and rotational velocity and align the 3D phantom boundary points on a helix with constant pitch p 0 = v 0 /f OCT and angle θ 0 = 360˚� f rot /f OCT .
Marker tracking (MT) method.
Using the MPI-guided IVOCT catheter tracking we can arrange the OCT A-scans along the actual catheter trajectory. A temporal synchronization of both imaging systems allows for image registration (Fig 2). The 4D volume reconstruction method is separated in two parts. First, the OCT and MPI data sets are registered via temporal correlation. The measurements are synchronized via a trigger signal sent from MPI to the IVOCT system. The related time events can be seen on the time line in Fig 3. One second after the MPI trigger arises (t trigger ), the catheter motion profile and OCT A-scan acquisition starts (t OCT,0 ). The time stamps t M1 and t M2 are related to the MPI volumes, wherein the catheter tip enters and leaves the cropped MPI FoV. Once the catheter motion profile is finished (t OCT,end ) the MPI measurement is stopped subsequently (t MPI,end ).
Then, we place points at distance r in 3D space considering both the spatial and temporal dependencies of MPI and OCT data. Due to substantial noise of the y-and z-component of the estimated 4D catheter trajectory, we only consider the x-coordinate (in pullback direction) as catheter position over time. For two successive catheter positions we determine the distance in space Δx and time Δt MT and distribute the meanwhile acquired A-scans equidistantly. Based on the given catheter rotation f rot , OCT frequency f OCT , and MPI volume rate f MPI up to four catheter positions are observed per catheter rotation. Assuming a constant catheter rotation
Experiments
We perform three experiments with the stenosis phantom repeating each experiment three times. As a first experiment, we conduct a standard pullback profile (SP) with a constant pullback velocity v 0 and a pullback distance s.
As a second experiment, a bending artifact profile (BA) is used to simulate the non-linear pullback of the catheter when the catheter is decelerated due to a bending and is then suddenly accelerated due to its elastic material. At first, the catheter is pulled back with velocity v 0 for the first 10 mm. Afterwards the catheter is simulated to be stuck and its velocity is set to v 1 = −0.625 mm s −1 for the next 5 mm. Finally, the velocity is set back to the initial velocity v 0 for the last 10 mm to simulate the elastic contraction of the catheter. The distances over time of the BA profile are shown in Fig 4a).
As a third experiment, we perform a measurement with a heart beat motion artifact (HBA) profile. A heart beat artifact is related to the heart contraction and the relative vessel motion w. r.t. the IVOCT catheter. This artifact results in multiple acquisitions of the same blood vessel part due to a back and forth movement of the vessel (Fig 5). We use a catheter motion profile (HBA) that simulates this relative motion. The velocity is set to v 0 for the first 15 mm. Exemplary sketch of heart beat motion artifact. Due to heart contraction the imaged artery is deformed for time stamp t 2 . Meanwhile, the catheter tip (blue dot) moves continuously backwards. After heart contraction (t 3 ) the artery gets back to its original shape (t 1 ). Again, the catheter motion is continued in between. This relative motion between catheter and artery leads to multiple IVOCT imaging of the sketched stenosis (black).
https://doi.org/10.1371/journal.pone.0230821.g005
PLOS ONE
In-Vitro MPI-guided IVOCT catheter tracking in real time Subsequently, the velocity is inverted to −v 0 for the following 5 mm to imitate the heart beat movement. Afterwards the velocity is adjusted back to v 0 for the last 15 mm. The distance over time of the HBA profile is shown in Fig 4b).
The described phantom experiments are performed in-vitro and do not involve human subjects. The datasets are acquired with the described imaging devices (pre-clinical MPI scanner, spectral-domain OCT system). The MPI data processing is implemented in Julia, while the OCT data processing is written in Matlab.
Results
The results are divided into three parts. First, the positions and the resulting velocities determined by 4D MPI catheter tracking are validated for three motion profiles SP, BA, and HBA. The mean absolute error (MAE) is calculated for the distance traveled only in x and for the distance traveled in x, y, z. The same is done for the velocities of the profiles. Second, we compare the IP and the MT volume reconstruction using the IVOCT and MPI data from the standard profile. The influence on both reconstruction methods in terms of bending artifacts is analyzed for the BA profile. Additionally, the HBA profile is used to investigate heart beat artifacts on both reconstruction methods. Third, the DICE factor is calculated for both reconstruction methods. In addition, the stenosis length is quantified for all reconstruction methods/profiles and compared to its ground-truth value.
Statistical validation of 4D MPI catheter tracking
For the standard profile the distance in x over time between M 1 = 18 mm and M 2 = 6 mm is shown in Fig 6a). From 18 mm to 11 mm the tracked x positions (black) are in good agreement with the expected x positions (red).
Between 11 mm to 6 mm the tracked x positions (black) seem to diverge slightly from the expected values (red). The mean values are used to fit a regression line (blue). The MAE for the distance in x is 0.44 mm ± 0.44 mm. The absolute error (AE) of the velocity using the regression line is 0.21 mm s -1 with a relative error (RE) of 16.8% as given in Table 1. For the BA profile the distance in x over time between 18 mm and 6 mm is presented in Fig 6b). Overall the tracked x positions (black) are in good agreement with the expected x positions (red). Only in the first segment a small deviation is visible. Again, all three measurements are shown The MAE for the distance in x is 0.26 mm ± 0.16 mm for the first segment, 0.35 mm ± 0.11 mm for the second segment and 0.20 mm ± 0.22 mm for the third segment. The AE and RE regarding the velocity of the regression in all three segments for the BA profile are given in Table 2.
For the HBA profile the distances in x over time between 18 mm and 6 mm are shown in Table 3.
Volume reconstructions
In Fig 8 the 4D volume reconstructions are compared for all motion profiles and both reconstruction methods. The volumes are shown with x cropped to the MPI FoV. A ground-truth volume with boundary information created by the parameters from the CAD sketch is depicted as a reference. The 4D boundary points are colored related to the underlying time, whereas the color map is shifted with respect to the time values of the positions x M 1 ¼ 18mm.
Table 1. The mean absolute error (MAE) is given for the distance in x-direction with its standard deviation (SD).
Additionally, the absolute error (AE) along with the relative error (RE) of the velocity using the 3D regression line between t M1 and t M2 is also reported.
Errors
Standard profile
PLOS ONE
The envelopes of all volume reconstructions show deviations compared to the ground-truth volume. The stenosis lengths are highlighted with red arrows. The MT reconstruction method leads to stenosis lengths and relative positions that are almost equal to the ground-truth volume for all motion profiles. The IP reconstruction method shows a larger deviation of the stenosis relative position. Furthermore, an obvious deviation of the depicted stenosis length using the IP volume reconstruction method are depicted for the BA and HBA profiles. Especially, for the BA profile with underlying deceleration of the catheter, the length is obviously increased.
In order to consider the complete pullback time for the BA and HBA profile, the related 4D volumes without cropping the x-axis are shown in Figs 9 and 10, respectively. Considering the BA profile (Fig 9), the overall IP volume results in an increased length with constant helical pitch p 0 . In contrast, the MT volume does not overestimate the total volume and especially the Fig 7. a) The measured distances in x agree with the set catheter movement. The turning points are clearly visible. The velocity, however, is underestimated in the first two segments. In the third segment the measured distances in x agree with the expected positions in x. b) The measure distance in y and c) z shows that the stenosis phantom is inserted slightly diagonal and the back and forth movement is also noticeable in the y and z dimension. d) The inversion of the velocity is visible and the mean velocity value are within the range of the expected velocities. However, the spread of the velocity is quite high.
https://doi.org/10.1371/journal.pone.0230821.g007 Table 3. The mean absolute error for the distance in x-direction with its standard deviation (SD) is presented for the HBA profile. The absolute and relative error of the velocity using the 3D regression line between t M1 and t M2 is also reported. PLOS ONE stenosis length. The varying catheter velocity, as depicted in Fig 6b), is apparent for the MT method by different densities of boundary points between x M 1 and x 2 = 10.2 mm compared to the points between x 2 and x 3 = 8 mm. In case of the HBA profile (Fig 10), the IP volume also shows a relevant overestimation of the total volume. Furthermore, in the volume reconstruction beyond x = 8 mm a second stenosis appears. The MT volume again presents an improved reconstruction method. Considering the tracked catheter motion, the 3D boundary points are arranged over time such that several boundary points overlay each other between x 1 and x 4 = 14 mm. Hence, the colored 4D volume (bottom) represents a catheter trajectory with a turning point within the stenosis.
Errors
Quantitative volume results
We determine the envelopes of the 3D boundary points of the IP and MT methods and quantify the volume reconstruction results using the DICE metric
DICE ¼ 1 N X N i¼1 2 � j U i \ V i j j U i j þ j V i j ;ð4Þ
whereas U i are the 2D projected shapes of the reconstructed envelopes for method IP and MT, respectively, compared to the ground-truth 2D projected shapes V i for all angles from 1 to N = 180˚. The mean DICE for all repetitions are listed in Table 4. The stenosis length is determined in x-direction as full at half width of the envelope decay of the volume shapes for all reconstructions, profiles and experimental repetitions. In case of the HBA profile, the stenosis length is determined as summation of the two stenosis lengths.
Discussion
The 4D catheter trajectory is tracked by the MPI for three different catheter motion profiles.
The statistical validation of all motion profiles and repetitions reveal small MAEs in x-direction of around 0.5mm, which is in good accordance with the estimated determination accuracy [44]. The ground truth for the trajectory in terms of the position in x, y and z directions is not known, since it is hardly possible to track the catheter's position within the MPI scanner with a second instance, e.g., an optical system. The ground truth for the trajectory is only known in terms of the pullback velocity and distance in 3D over a defined period. Therefore, the calculated MAEs in x-direction contain a small uncertainty because the y and z ground truth positions are assumed to be constant zero. The absolute and relative errors of the velocities determined by a regression line in 3D are comparable to the ground truth velocity of the IVOCT adapter. They show varying relative errors in the range of 3.2%-39% for all motion profiles and their segments. For the BA profile the first change of velocity could not be captured by the measured values. One reason could be that the velocity change is close to the border of the FoV. The voxel intensities representing the marked catheter tip have circle shaped form in the MPI images. If this circle shape has not entered the FoV completely, the center of mass localization algorithm might misinterpret the position. In addition, in case of the HBA profile deviations occur around the turning points seen in Fig 7a), which lead to an underestimation of the velocity. These deviations can be linked to the catheter setup with a proximal actuator such that the pullback is increased by the shrinkage and stretching of the flexible catheter. It is also worth noting that the back and forth motion of the catheter is also visible in the y-and z-positions seen in Fig 7b and 7c) as the vessel phantom is not placed perfectly in accordance with the x-axis. The novel MT volume reconstruction method based on MPI catheter tracking demonstrates a qualitative improvement in comparison to the IP method (Fig 8). Even in case of the SP profile without additional motion artifacts, the IP method shows worse results by means of the DICE metric. The illustrated results of the motion artifact profiles underline the need of a catheter tracking over time. In addition to the catheter tracking in 3D [7,10,34,35], the time synchronization of the IVOCT and MPI data leads to an optimized arrangement of OCT Ascans in 3D space. The DICE metrics and stenosis lengths in Table 4 emphasize the relevant errors in case of the IP method. Nevertheless, inaccuracies in volume shapes occur for all methods and profiles (DICE< 0.9), as other imaging artifacts have an influence on the IVOCT and MPI data. For example, non-uniform rotational distortions (NURD) of the catheter might appear due to the catheter setup. Furthermore, the boundary segmentation in the IVOCT data as well as the catheter tip segmentation in the MPI data contain inaccuracies. In future work, the results can be further improved by a correction of additional artifacts and image enhancements. On the one hand, a rotation tracking with MPI may be possible with an asymmetric marking of the catheter tip. On the other hand, the phantom centerline and catheter trajectory can be tracked using a multi-contrast MPI imaging approach [29][30][31][32]46, 47] visualizing the marker and the blood pool tracer inside the phantom. Both approaches can be used to minimize the effect of NURD artifacts.
Conclusion
A novel approach for MPI-guided IVOCT catheter tracking is presented considering both the 3D catheter trajectory and the time synchronization of IVOCT and MPI data in order to reconstruct volumes of a known vessel phantom shape in 4D. A DICE coefficient of up to 89% is achieved for different IVOCT motion artifact studies. The presented approach estimates the stenosis length for simulated artifacts more precisely with a relative error of up to 0.6% in comparison to 160% of the standard method.
Fig 1 .
1Experimental setup. A vessel phantom with a stenosis (a) is positioned within the MPI FoV. In the CAD sketch of the phantom (bottom right), the phantom and entire stenosis dimensions are depicted. The stenosis has a diameter and length of 1.5 mm. Triggered by MPI, an IVOCT catheter (b) is rotated and pulled backwards through the phantom using a custom built adapter. The catheter tip (blue dot) is coated with magnetic lacquer (c) without covering the OCT prism. The cropped MPI FoV is highlighted with a red box. https://doi.org/10.1371/journal.pone.0230821.g001
Fig 2 .
2Exemplary IVOCT and MPI data. The OCT A-scans are arranged over time (top). The segmented phantom boundary is highlighted in red. For three time stamps t i , the related MPI signals from catheter tip are shown within the CAD sketch (bottom). https://doi.org/10.1371/journal.pone.0230821.g002
Fig 3 .
3Time axis for synchronizing OCT device and pullback device with the help of the MPI trigger signal. https://doi.org/10.1371/journal.pone.0230821.g003PLOS ONEf rot , the A-scans are oriented with a fixed angle difference θ 0 around the actual catheter trajectory.
Fig 4 .
4a) In case of the BA profile, the catheter is pulled backwards with a velocity v 0 = −1.25 mm s −1 over the first 10 mm, then the velocity is reduced to v 1 = −0.625 mm s −1 for the next 5 mm, afterwards the velocity is increased back to v 0 for the last 10 mm. b) In case of HBA profile, the IVOCT catheter is pulled backwards over the first 15 mm, then the catheter is moved forward for 5 mm. Last, the catheter is pulled backwards in the original direction again for 15 mm. The catheter moves with the initial velocity v 0 for all motion directions.https://doi.org/10.1371/journal.pone.0230821.g004
Fig 5 .
5Fig 5. Exemplary sketch of heart beat motion artifact. Due to heart contraction the imaged artery is deformed for time stamp t 2 . Meanwhile, the catheter tip (blue dot) moves continuously backwards. After heart contraction (t 3 ) the artery gets back to its original shape (t 1 ). Again, the catheter motion is continued in between. This relative motion between catheter and artery leads to multiple IVOCT imaging of the sketched stenosis (black).
Fig 6 .
6a) MPI measurements for standard profile: The measured distance in x over time between 18 mm and 11 mm is in good agreement with the expected values. Expect in the last part the positions in x marginally deviate. b) MPI measurements for BA profile: The first change of velocity is not captured by the measurement. After the first change the measured distances in x over time are in agreement with the different velocity v 0 and v 1 with only slight deviations. https://doi.org/10.1371/journal.pone.0230821.g006 PLOS ONE In-Vitro MPI-guided IVOCT catheter tracking in real time as a box plot and illustrate the distribution of the tracked positions. The mean values are used to determine a regression line (blue).
Fig 7a .
7aOverall, the tracked x positions resemble the movement of the catheter being pulled back and forth. The turning points between velocities −v 0 , v 0 and again −v 0 can be clearly identified. However, in the first two segments the velocity is underestimated as the tracked x positions are not in full accordance with the expected x positions (red). In the third segments the tracked x positions agree with the expected values. The distances in y and z over time are presented in Fig 7b and 7c) and shows that the stenosis phantom has been inserted slightly diagonal as the y-values increase and the z-values decrease depending on the x-position. For a straight insertion we would expect a straight line in both dimensions. Only at the time points when the velocities change the tracked positions in x differ from the expected positions in x. The three measurements are depicted as box plots to show the distribution of the measurements. The regression lines for each segment are plotted in blue. The mean absolute error for the distance in x is 0.64 mm ± 36 mm for the first segment, 0.51 mm ± 55 mm for the second segment and 0.38 mm ± 45 mm for the third segment. In Fig 7d) the velocity in x over time is shown and the inversion of the velocity is visible. The absolute and relative error of the velocity using the regression line in x are 0.38 mm/s (30.0%) for the first segment, 0.49 mm/s (39.4%) for the second segment and 0.04 mm/s (3.2%) for the third segment. The errors regrading the HBA profile are given in
Fig 8 .
8Reconstructed volumes for all motion profiles w.r.t. the ground-truth volume (top) for the cropped MPI FoV. The distances x = 18 and x = 8 mm correspond to the time points t M1 and t M2 . The IP and MT volume reconstructions are labeled (left). The phantom boundary points are colored w.r.t. the time color map (right). The stenosis lengths are depicted with red arrows. https://doi.org/10.1371/journal.pone.0230821.g008
Fig 9 .
9Complete IP volume reconstruction compared to MT volume reconstruction for the bending profile. https://doi.org/10.1371/journal.pone.0230821.g009 Fig 10. Complete IP volume reconstruction compared to MT volume reconstruction for the heart beat profile. https://doi.org/10.1371/journal.pone.0230821.g010 PLOS ONE
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Table 4 .
4DICE quantification and related stenosis lengths in mm for 3D and 4D reconstruction methods for motion profiles SP, BA, and HBA, respectively.https://doi.org/10.1371/journal.pone.0230821.t004SP profile
BA profile
HBA profile
DICE IP
0.86
0.66
0.58
DICE MT
0.88
0.89
0.86
Stenosis IP (RE)
1.52 (1.3%)
3.91 (160%)
3.82 (154%)
Stenosis MT (RE)
1.15 (23%)
1.49 (0.6%)
1.19 (21%)
42 .
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[] |
[
"The study of influence of the Teslar technology on aqueous solution of some biomolecules",
"The study of influence of the Teslar technology on aqueous solution of some biomolecules"
] |
[
"E Andreev \nDepartment of Physics of Biological Systems\nInstitute of Physics\nNational Academy of Sciences\nProspekt Nauky 46UA-03028KyivUkraine\n",
"G Dovbeshko \nDepartment of Physics of Biological Systems\nInstitute of Physics\nNational Academy of Sciences\nProspekt Nauky 46UA-03028KyivUkraine\n",
"V Krasnoholovets \nDepartment of Physics\nInstitute for Basic Research\n90 East Winds Court34683Palm HarborFLU.S.A\n",
"Leif A Eriksson "
] |
[
"Department of Physics of Biological Systems\nInstitute of Physics\nNational Academy of Sciences\nProspekt Nauky 46UA-03028KyivUkraine",
"Department of Physics of Biological Systems\nInstitute of Physics\nNational Academy of Sciences\nProspekt Nauky 46UA-03028KyivUkraine",
"Department of Physics\nInstitute for Basic Research\n90 East Winds Court34683Palm HarborFLU.S.A"
] |
[
"Research Letters in Physical Chemistry"
] |
The possibility of recording physical changes in aqueos solutions caused by a unique field generated by the Teslar chip (TC) inside a quartz wristwatch has been studied using holographic interferometry. We show that the refraction index of degassed pure distilled water and aqueous solutions of L-tyrosine and b-alanine affected by the TC does not change during the first 10 minutes of influence. In contrast, a 1% aqueous solution of plasma extracted from the blood of a patient with heart-vascular disease changes the refractive index when affected by the TC. The characteristic time of reaction is about 10 2 s. Based on our prior research experience we state that the response of the system studied to the TC's field is similar to that stipulated by the action of a constant magnetic field with the intensity of 1.1 × 10 −3 T. Nevertheless, our team could unambiguously prove that the TC generates the inerton field, which is associated with a substructure of the matter waves (and, therefore, it does not relate to the electromagnetic nature).
|
10.1155/2007/94286
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[
"https://arxiv.org/pdf/1204.6062v1.pdf"
] | 55,088,654 |
1204.6062
|
aff8e437944d1953d2d6e1efe5434a80d47daa38
|
The study of influence of the Teslar technology on aqueous solution of some biomolecules
Hindawi Publishing CorporationCopyright Hindawi Publishing Corporation2007
E Andreev
Department of Physics of Biological Systems
Institute of Physics
National Academy of Sciences
Prospekt Nauky 46UA-03028KyivUkraine
G Dovbeshko
Department of Physics of Biological Systems
Institute of Physics
National Academy of Sciences
Prospekt Nauky 46UA-03028KyivUkraine
V Krasnoholovets
Department of Physics
Institute for Basic Research
90 East Winds Court34683Palm HarborFLU.S.A
Leif A Eriksson
The study of influence of the Teslar technology on aqueous solution of some biomolecules
Research Letters in Physical Chemistry
Hindawi Publishing Corporation94286200710.1155/2007/94286Received 5 June 2007; Accepted 25 July 2007 Recommended byarXiv:1204.6062v1 [physics.gen-ph] Correspondence should be addressed to V. Krasnoholovets,
The possibility of recording physical changes in aqueos solutions caused by a unique field generated by the Teslar chip (TC) inside a quartz wristwatch has been studied using holographic interferometry. We show that the refraction index of degassed pure distilled water and aqueous solutions of L-tyrosine and b-alanine affected by the TC does not change during the first 10 minutes of influence. In contrast, a 1% aqueous solution of plasma extracted from the blood of a patient with heart-vascular disease changes the refractive index when affected by the TC. The characteristic time of reaction is about 10 2 s. Based on our prior research experience we state that the response of the system studied to the TC's field is similar to that stipulated by the action of a constant magnetic field with the intensity of 1.1 × 10 −3 T. Nevertheless, our team could unambiguously prove that the TC generates the inerton field, which is associated with a substructure of the matter waves (and, therefore, it does not relate to the electromagnetic nature).
Abstract
The possibility of recording physical changes in aqueos solutions caused by a unique field generated by the Teslar chip (TC) inside a quartz wristwatch has been studied using holographic interferometry. We show that the refraction index of degassed pure distilled water and aqueous solutions of L-tyrosine and b-alanine affected by the TC does not change during the first 10 minutes of influence. In contrast, a 1% aqueous solution of plasma extracted from the blood of a patient with heart-vascular disease changes the refractive index when affected by the TC. The characteristic time of reaction is about 10 2 s. Based on our prior research experience we state that the response of the system studied to the TC's field is similar to that stipulated by the action of a constant magnetic field with the intensity of 1.1 × 10 −3 T. Nevertheless, our team could unambiguously prove that the TC generates the inerton field, which is associated with a substructure of the matter waves (and, therefore, it does not relate to the electromagnetic nature).
INTRODUCTION
Numerous experiments fix the influence of electromagnetic field of certain frequency-amplitude ranges on living organisms. For instance, the magnetic field with frequencies in the range 0.3 to 30 Hz and with the intensity that is comparable with the Earth magnetic field can effectively influence the living organism function. It is supposed that the mechanism of influence should be connected with the parametric, or Schumann Resonance. The first four harmonics of the Schumann resonance are known: 7.8 Hz ± 1.5 Hz, 14.5, 20, 26 Hz (± 0.3 Hz) [1][2][3]. Well-known are two main mechanisms of the resonance reaction of the organism to a weak electromagnetic field. The first one is the Alfa-rhythm concerned with the thought process; the second one, the parametric resonance of organs, or organ systems, could be responsible for primary human reception [4][5][6]. A number of physiological processes, such as the reductive-oxidative process in living cells, responsible for the oxygen input, oxygen transport, etc. could be taken into account in this case. The parametric resonance of biological tissue and surrounding medium could be also responsible for the medical action of the TC.
The aim of the present study is: 1) the influence of the TC on a biological model system and 2) registration of this influence in those cases when it is possible. The inventors [7] of this device state that the chip produces a longitudinal scalar wave/field (the notation was introduced by Nicola Tesla [8]). In this case a part of the energy is radiated in the form of a scalar longitudinal wave (also known as a Tesla free-standing wave). In more detail, this wave has been studied in Refs. [9,10].
A model object must be sensitive; it has to have a large gain factor and the method must be reproducible and stable, simultaneously. Since our goal is to account for biophysical aspects of the influence of the TC on living organisms, the model system should include components available in hypodermic tissues of the wrist. These conditions allow us to choose, as the model of primary reception, the following:
(i) saturated aqueous solution of amino acids (tyrosine, tryptophane and alanine);
(ii) diluted aqueous solution of human blood plasma. (Although a solution of human serum albumin is a dominating in blood protein, the preparation of its solution by conventional method will mean that we obtain an equilibrium system, and it will be very difficult to move such system from its deep potential minimum. In the case of biomolecules of plasma of blood, which we study in this work, we deal with a non-equilibrium system and even very small stimuli applied to it could be effective.)
The parameter under study has become the refraction index n of an aqueous solution.
MATERIALS AND METHOD
Holographic experiments have been carried out with the use of the holographic interferometer IGD-3, developed and produced in the Institute of Physics of Semiconductors of Nat. Acad. Sci. of Ukraine [11][12][13], whose optical scheme is given and described in Fig. 1. The He-Ne laser (1) radiation (power output equals 1 mW at λ= 632.8 nm) is divided by the beam splitter cube (2) into two beams: the object beam and the reference beam. In the object beam shoulder there is the mirror (3) and the collimator (4) consisting of negative and positive lenses, which forms a parallel beam, 5 cm in diameter. The beam passes through the object under study (6), and then arrives at the finely dispersed diffuse scatterer (9). According to Lambert's law, its every point is scattering the light in all directions. The light from the whole surface of the scatterer arrives at every point of the light sensitive thermoplastic (10) in which plane the object can be selected. At one inclination of plate (5) the increase of n has to result in the increase of the interference period, i.e. in the decrease of the number of bands. TC (7) has been put onto the top of a quartz cuvette (1 × 1 × 4.3 cm 3 ) filled with the solution studied. If the dielectric characteristics of the object studied are the same before and after the TC influence, the fringe pattern remains unaltered and interference bands inside and outside the object's profile continue each other. On the contrary, if an external factor caused changes of n, the fringe pattern within the limits of the object's profile will change.
The amino acids used in our experiments were produced by Sigma, Inc.
Aqueous solutions were prepared on the basis of pure bidistilled water. Prepared solutions, before the experiment, were maintained for 24 hours under 25 • C. The plasma blood solution was extracted from the blood of a heart vascular disorder patient just after the blood was drawn at hospital by conventional methods. We dilute the solution by distilled water as 1: 50 and 1: 100. The time between the blood extraction and the holographic measurement was 4 hours. The procedure of dynamic measurement consisted of a sequence of records of interference patterns on a special thermoplastic plate, which then was fixed by a digital video camera. Afterward, the images were input into the computer and evaluated. For determination of the interference band centre, the 10 points along the horizontal line of cuvette have been chosen.
The studies were conducted at temperature 20 ± 1.5 • C controlled by the thermocouple accurate to 0.2 • C. We suppose that absolute meaning of the temperature does not influence the process under study due to the fact that we have been recording a dynamics of redistribution of the optical density. The most important point in the experiment was to protect the cuvette from the temperature gradient and the airflow. The last two disturbed factors have determined an inner non-stability of the system.
In Fig. 2a, typical interference patterns of the aerial ambient space ("Air") and the aqueous solution ("Solution") are presented. Vertical black lines show the image of the cuvette corner (its size 1 × 1 × 4 cm 3 ). Thus in our experiments we have been able to observe an alteration of the reflective index in the surface zone of the cuvette equal to 1× 2 cm 2 that is determined by the cuvette size and the aperture of laser beam.
In Fig. 2a, typical interference patterns of the aerial ambient space ("Air") and the aqueous solution ("Solution") are presented. Vertical black lines show the image of the cuvette corner (its size 1× 1 × 4 cm 3 ). We have been able to observe an alteration of n in the surface zone 1 × 2 cm 2 of the cuvette (the cuvette size) and the aperture of laser beam. Deformations of the interference pattern in different points of the solution have been caused by changes in n in these points. The resolution is defined by a location of the optical wedge, namely, by a sum of horizontal interference lines. The space resolution is about 2 mm.
The method described gives a possibility to follow the response of the solution with the time factor of minimal discontinuous ability equal to 10 s. A sequence of pictures of the fringe pattern characterizes the space dynamics of the system studied in any place of the cuvette. As an example, Fig. Figure 1: Experimental holographic set. 1 He-Ne laser; 2 beam splitter cube; 3 mirror; 4 collimator; 5 plane-parallel plate; 6 quartz flask (cuvette) with the solution; 7 Teslar bracelet; 8 filter that divides two flasks; 9 scattering layer; 10 thermoplastic recording plate; 11 reference beam mirror; 12 reference beam lens; 13 TV camera.
2b shows the fringe pattern formed in about 4 minutes starting from the moment of action of the TC that has been spaced at 2 mm from the cuvette. Changes of interference bands occurred during this time are associated with an internal stimulus.
RESULTS
A primary series of the experiments was conducted with distilled water, the saturated aqueous solution of L-tyrosine and β-alanine at 25 • C . The experiments were conducted both in the morning and afternoon. The results showed typical slight changes of the fringe pattern in 400 s or larger time interval. These changes should be associated with the inner drift of liquid parameters. The curve of long-time dynamics does not show any influence on the side of the TC approximate to the cuvette.
The other behavior and picture have been observed in the case of the blood plasma solution. Without the TC action this solution has shown stable and reproducible characteristics during more than 4 hours.
During more than one hour the system of recording and the objects of study (the solution of plasma and water) were stable and reproducible. of human without the influence of the TC. The value of the effect is estimated by difference between the shift of the interference band in the cuvette with the solution and the position of same band in the air; we evaluated the shift of interference bands before and after the TC application. It is seen that during 4 minutes the bands in the cuvette have not been deformed and they have essentially not moved relative to those in the air.
In Fig. 3 we present the image of the cuvette with plasma blood solution affected by the TC. Black dots indicate the center position of one of the interference bands on the image plane. The value of the shift relating to zero line characterizes the degree of influence of the TC. Black rectangles (Fig. 3, right) show the positions of two Teslar chips relating to cuvette; the fringe pattern of the solution relating to the chip is deformed in different ways in different zones (short, mid, and far-distance). A physical mechanism of the change is associated with the increase of n in the short-distance zone; n remains unchanged in the mid-distance zone and decreases in the far-distance zone.
Moreover, it seems that slow laminar flows have been induced by the TC near the front wall and directed to it.
If the refraction index of the solution changes in one place under the influence of an external factor, the length of optical path will also change.
With the purpose of the registration of the changes, the device is designed in such a way that the "starting interferogram" constitutes a family of horizontal bands, bands of equal thickness. Depending on the character of changes of the optical density in the cuvette volume, the bands can be distorted (local changes of n), gaps between bands can expand without deformations (volumetric decreases of n). Thus arbitrary deformations of the fringe pattern are caused by a combination of local and global changes of the optical density.
Changes of n are produced by changes in the structure of the network of hydrogen bonds of water, which being under the influence of oxygen, biomolecules and the inerton field generated by the TC, forms long-lived structures. In the mentioned network those new structures try to minimize the total energy relative to the volume occupied by the water system. Such kinds of changes (structuring of the aqueous solution) occur sufficiently slowly and therefore allows the recording by optical methods.
The strongest changes in effects associated with the TC have been detected during the first 5 to 15 minutes starting from the moment of influence. The changes in the fringe pattern have been irregular in time. The saturation effect reaches at 10 minutes. Without the TC the number of interference bands remains the same in the field of the object and around it and is equal to five. Changes in the refractive index n of the sample affected by the TC estimated from equation where L is the thickness of the sample (the aqueous solution studied), ∆n is the change in refractive index, λ is the wavelength of the source of light (laser), ∆k is the change in the number of interference bands as a result of an external effect. Influence of the TC on water leads only to minor changes of the fringe pattern (∆n = 2 × 10 −5 ). The behavior of proteins is mainly determined by the influence of the TC. Effects associated with the TC and heating effects have shown the opposite trend/tendency. The temperature rise in the flask detected by the thermo sensor with 1 mW/cm 2 power density ranged between 0.2 to 0.5 • C. Therefore, heating caused by the laser radiation allows an evaluation of the role of temperature. The estimation of the temperature effect by using the thermal conductivity equation and the thermal balance equations show the following. The maximum heating of the aqueous solution without account of the thermal exchange, i.e. under the condition unfavorable for the thermal effect estimation, may amount to 1 • C. The calculation shows that even without the heat exchange between the aqueous solution and the environment the radiation effect with the power density of 10 mW/cm 2 may produce an increase of 1 • C of the temperature of the 1.5 cm 3 volume of aqueous solution within 6 minutes. The temperature coefficient of changes in n of water makes up ∆n = 2 × 10 −5 .
L∆n = λ∆k,(1)
The maximum change of n of the protein solution affected by the TC reached the value of ∆n = 2 × 10 −4 , which is an order of magnitude larger than the temperature changes of n. Thus the numerical estimates and the experimental data show that changes of n caused by the influence of the TC have been conditioned by non-thermal changes of the solution dielectric constant, which may be described as the total contribution of electronic, vibration and orientational components.
In Fig. 4, experimental dots show changes of n of the solution [the vertical axis] at the cuvette's back wall against the upper TC (rectangles) and the lower TC (triangles); recall the two TC are located near the front wall (see Fig. 3). Current time, in seconds, is plotted along the horizontal axis. Legends "H 2 O" (the blue background) and "H 2 O + Plasma" (the orange background) indicate different solutions in the cuvette. Time intervals are singled out for: 1) the magnet ("Magnet" on the green background) is applied to the front wall of the cuvette and 2) two links of the bracelet with two TC are set into the cuvette (the violet background). Moments of intermix of the solution ("Destruction") are shown by means of arrows; the intermix was made by using a medical syringe with 0.2 mm-needle: the solution was absorbed from the cuvette by the syringe and then poured back.
CONCLUSION
The TC does not affect distilled water. However, biomolecules of plasma of blood or an ensemble of such biomolecules in a micromolar concentration in water lead to the changes in reological characteristics, which allows the observation by optical methods, in particular, by the holographic interferometer. In the aqueous solution of blood plasma, biomolecules play a role of primary receptors of the TC radiation.
Changes in the aqueous solution affected by the TC cover all the macroscopic volume of the sample studied. This behavior can be associated with both inner convective flows (like in the case of Benar cells) and structural changes of water. The latter may bring about changes in the reflective index of the solution and the fringe pattern. Comparative responses of the aqueous solution to the mechanical, magnetic and TC influences point to a very specific action of the latter. A microscopic physical consideration of the phenomenon of the TC has already been performed in some detail [9,10,14]. We could prove [9,10] that the Teslars phenomenon belongs to the inerton field effects and hence does not relate to the electromagnetic nature. The inerton field (the field of inertia) appears as a basic field in the submicroscopic mechanics of canonical particles developed in the real physical space and ac-counts for the availability of the wave ψ−function in conventional quantum mechanics (see, e.g. Refs. [15,16]). This field transfers local deformations of space, which appear in physical terms as mass. Therefore, the inerton field transfers mass changing potential properties of the environment.
Consequently, the defect of mass ∆m becomes an inherent property not only of atomic nuclei but any physical and physical chemical systems [14] (including biophysical ones). A more extensive study and repeated examinations should be completed to shed more light upon the mechanism of action of the Teslar chip and the inerton field in general upon living organisms.
Figure 2 :
2Dynamics of the fringe pattern of the aqueous solution of plasma blood
Figure 3 :
3Dynamics of the fringe pattern of the aqueous solution of plasma of human blood after the insertion of 2 TC. The strong disturbance of the optical density of the solution is emerged already in 72 s, right figure. (The back covers of two sections of the bracelet are found at 4 mm from the right wall of the cuvette).
Figure 4 :
4Dynamics of changes of the fringe pattern of the cuvette's volume at different external factors. The horizontal axis shows current time in seconds.
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[
"Self-propulsion against a moving membrane: enhanced accumulation and drag force",
"Self-propulsion against a moving membrane: enhanced accumulation and drag force"
] |
[
"U Marini ",
"Bettolo Marconi \nScuola di Scienze e Tecnologie\nUniversità di Camerino\nVia Madonna delle Carceri62032Camerino\n\nINFN Perugia\nItaly\n",
"A Sarracino \nDipartimento di Fisica\nCNR-ISC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly\n",
"C Maggi \nDipartimento di Fisica\nCNR-NANOTEC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly\n",
"A Puglisi \nDipartimento di Fisica\nCNR-ISC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly\n"
] |
[
"Scuola di Scienze e Tecnologie\nUniversità di Camerino\nVia Madonna delle Carceri62032Camerino",
"INFN Perugia\nItaly",
"Dipartimento di Fisica\nCNR-ISC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly",
"Dipartimento di Fisica\nCNR-NANOTEC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly",
"Dipartimento di Fisica\nCNR-ISC\nSapienza Università di Roma, p.le A. Moro 200185RomaItaly"
] |
[] |
Self-propulsion (SP) is a main feature of active particles (AP), such as bacteria or biological micromotors, distinguishing them from passive colloids. A renowned consequence of SP is accumulation at static interfaces, even in the absence of hydrodynamic interactions. Here we address the role of SP in the interaction between AP and a moving semipermeable membrane. In particular, we implement a model of noninteracting AP in a channel crossed by a partially penetrable wall, moving at a constant velocity c. With respect to both the cases of passive colloids with c > 0 and AP with c = 0, the AP with finite c show enhancement of accumulation in front of the obstacle and experience a largely increased drag force. This effect is understood in terms of an effective potential localised at the interface between particles and membrane, of height proportional to cτ /ξ, where τ is the AP's re-orientation time and ξ the width characterising the surface's smoothness (ξ → 0 for hard core obstacles). An approximate analytical scheme is able to reproduce the observed density profiles and the measured drag force, in very good agreement with numerical simulations. The effects discussed here can be exploited for automatic selection and filtering of AP with desired parameters.I. INTRODUCTIONActive particles (AP) represent a large class of systems characterized by a conversion of internal energy into selfpropulsion [1]. The behavior of AP deeply differs from that of passive colloids in a thermal bath and shows typical features of nonequilibrium dynamics[2,3]. At the level of single trajectories, AP are characterized by persistent random walks and correlated motion. Instances of such systems can be found in the realm of bacteria and microorganisms [4], or in the context of man-made nano-devices[5].Several models have been proposed to study the physical properties of active matter systems, which show intriguing phenomena, such as nonequilibrium phase transitions, self-organization and collective behaviors. Let us mention the "run and tumble" model [6], characterized by directed motion interrupted by random reorientations, the "active Brownian" model[7,8], where particles are pushed by a constant force, whose direction changes stochastically, and the Vicsek model [9, 10], where the particle speed is fixed and the orientation depends on the average velocity of the neighbors. More recently, the Gaussian colored-noise (GCN) model has been proposed to account for the correlated motion (over a typical time τ ) characterizing AP systems[11], which allows for an analytical treatment within a specific scheme, known as Unified Colored Noise Approximation (UCNA)[12].Among the several nonequilibrium phenomena observed in AP systems, a surprising result reproduced also by the GCN model, is that, in the presence of a static repulsive potential, AP do accumulate around the obstacle, producing a nontrivial density profile[13,14]. This observation raises the question of what effects are produced when the obstacle is not static and moves with constant velocity, inducing a stationary current.The study of the density profiles in (passive) colloidal systems under the action of a moving obstacle, indeed, takes on great importance in several contexts and has been addressed from different perspectives. For instance, it is the central issue in active microrheology, where a tracer is (magnetically or optically) driven through a medium to probe its structural properties[15,16]. A moving potential barrier can also be realized by means of optical fields, with travelling waves or inverted traps[17][18][19]. Moreover, soft potential barriers with a finite height and width are also used to model the finite thickness of a semipermeable membrane in contact with fluids [20-23], or the translocation properties of polymer chains through nanopores[24,25]. Similar problems related to the study of the stationary currents and density profiles of colloids under the effect of moving potentials have been addressed with the formalism of the density functional theory, with applications to the motion of colloidal particles in narrow channels[26], or in polymer solutions[27,28].In this paper, we study a simple model for a semipermeable membrane moving at constant velocity c in a fluid of noninteracting GCN active particles of persistence time τ , see the sketch inFig. 1. Our analytical theory demonstrates the appearance of an effective dynamical potential arising from the coupling of self-propulsion with the nonequilibrium current induced by the moving obstacle: indeed it vanishes in both the limits of c → 0, and τ → 0 (passive colloids with thermal noise). Our approach, which generalizes the UCNA to non-vanishing steady currents, gives accurate predictions -when compared to numerical simulations -for the density profiles of AP and the effective drag force, in a wide range of parameters. The most striking consequence of the current-induced effective potential is an enhanced accumulation of AP at the interface, with respect to the static case or with respect to the behavior of passive colloids.
|
10.1103/physreve.96.032601
|
[
"https://arxiv.org/pdf/1709.03381v1.pdf"
] | 44,827,928 |
1709.03381
|
8d8284f7cf0777776229dfae6832f1353b546a7c
|
Self-propulsion against a moving membrane: enhanced accumulation and drag force
11 Sep 2017
U Marini
Bettolo Marconi
Scuola di Scienze e Tecnologie
Università di Camerino
Via Madonna delle Carceri62032Camerino
INFN Perugia
Italy
A Sarracino
Dipartimento di Fisica
CNR-ISC
Sapienza Università di Roma, p.le A. Moro 200185RomaItaly
C Maggi
Dipartimento di Fisica
CNR-NANOTEC
Sapienza Università di Roma, p.le A. Moro 200185RomaItaly
A Puglisi
Dipartimento di Fisica
CNR-ISC
Sapienza Università di Roma, p.le A. Moro 200185RomaItaly
Self-propulsion against a moving membrane: enhanced accumulation and drag force
11 Sep 2017
Self-propulsion (SP) is a main feature of active particles (AP), such as bacteria or biological micromotors, distinguishing them from passive colloids. A renowned consequence of SP is accumulation at static interfaces, even in the absence of hydrodynamic interactions. Here we address the role of SP in the interaction between AP and a moving semipermeable membrane. In particular, we implement a model of noninteracting AP in a channel crossed by a partially penetrable wall, moving at a constant velocity c. With respect to both the cases of passive colloids with c > 0 and AP with c = 0, the AP with finite c show enhancement of accumulation in front of the obstacle and experience a largely increased drag force. This effect is understood in terms of an effective potential localised at the interface between particles and membrane, of height proportional to cτ /ξ, where τ is the AP's re-orientation time and ξ the width characterising the surface's smoothness (ξ → 0 for hard core obstacles). An approximate analytical scheme is able to reproduce the observed density profiles and the measured drag force, in very good agreement with numerical simulations. The effects discussed here can be exploited for automatic selection and filtering of AP with desired parameters.I. INTRODUCTIONActive particles (AP) represent a large class of systems characterized by a conversion of internal energy into selfpropulsion [1]. The behavior of AP deeply differs from that of passive colloids in a thermal bath and shows typical features of nonequilibrium dynamics[2,3]. At the level of single trajectories, AP are characterized by persistent random walks and correlated motion. Instances of such systems can be found in the realm of bacteria and microorganisms [4], or in the context of man-made nano-devices[5].Several models have been proposed to study the physical properties of active matter systems, which show intriguing phenomena, such as nonequilibrium phase transitions, self-organization and collective behaviors. Let us mention the "run and tumble" model [6], characterized by directed motion interrupted by random reorientations, the "active Brownian" model[7,8], where particles are pushed by a constant force, whose direction changes stochastically, and the Vicsek model [9, 10], where the particle speed is fixed and the orientation depends on the average velocity of the neighbors. More recently, the Gaussian colored-noise (GCN) model has been proposed to account for the correlated motion (over a typical time τ ) characterizing AP systems[11], which allows for an analytical treatment within a specific scheme, known as Unified Colored Noise Approximation (UCNA)[12].Among the several nonequilibrium phenomena observed in AP systems, a surprising result reproduced also by the GCN model, is that, in the presence of a static repulsive potential, AP do accumulate around the obstacle, producing a nontrivial density profile[13,14]. This observation raises the question of what effects are produced when the obstacle is not static and moves with constant velocity, inducing a stationary current.The study of the density profiles in (passive) colloidal systems under the action of a moving obstacle, indeed, takes on great importance in several contexts and has been addressed from different perspectives. For instance, it is the central issue in active microrheology, where a tracer is (magnetically or optically) driven through a medium to probe its structural properties[15,16]. A moving potential barrier can also be realized by means of optical fields, with travelling waves or inverted traps[17][18][19]. Moreover, soft potential barriers with a finite height and width are also used to model the finite thickness of a semipermeable membrane in contact with fluids [20-23], or the translocation properties of polymer chains through nanopores[24,25]. Similar problems related to the study of the stationary currents and density profiles of colloids under the effect of moving potentials have been addressed with the formalism of the density functional theory, with applications to the motion of colloidal particles in narrow channels[26], or in polymer solutions[27,28].In this paper, we study a simple model for a semipermeable membrane moving at constant velocity c in a fluid of noninteracting GCN active particles of persistence time τ , see the sketch inFig. 1. Our analytical theory demonstrates the appearance of an effective dynamical potential arising from the coupling of self-propulsion with the nonequilibrium current induced by the moving obstacle: indeed it vanishes in both the limits of c → 0, and τ → 0 (passive colloids with thermal noise). Our approach, which generalizes the UCNA to non-vanishing steady currents, gives accurate predictions -when compared to numerical simulations -for the density profiles of AP and the effective drag force, in a wide range of parameters. The most striking consequence of the current-induced effective potential is an enhanced accumulation of AP at the interface, with respect to the static case or with respect to the behavior of passive colloids.
FIG. 1: A semipermeable membrane, modeled as a potential barrier U (x) (color-bar), moves at velocity c (denoted by the arrow) in a fluid of noninteracting active particles.
This effect yields a drag force whose intensity can be made large at will by tuning the model parameters. In the nonlinear regime of large c, we also observe a nonmonotonic behavior of the experienced drag force [29][30][31], which is well described within our analytical approach. Our results have practical applications, e.g. in sweeping up AP from a mixture of inert/active particles, or in selecting and filtering AP with specific parameters, by tuning the properties of the moving membrane.
II. MODEL
A channel, in generic dimension, contains suspended (active or passive) particles. A membrane separates the channel in two parts and moves with constant velocity c along the direction x perpendicular to itself, see Fig. 1. Since the particles are noninteracting, the only relevant direction is that parallel to the membrane movement. We assume the channel to be periodic and very large in the x direction. The dynamics of each particle is described by the overdamped Langevin equationẋ
(t) = F (x − ct) ζ + η(t),(1)F (x) = −d x U (x),(2)
where the potential U (x) represents the moving penetrable membrane. The width of the membrane is used as unit of length (see below, Eq. (5)), while the mass of the particle is 1. The quantity η(t) stands for a noise term, which is white (thermal) for passive colloids, or coloured, with correlation time τ , for active particles: in both cases η(t) = 0. When Eq. (1) models passive particles, we take η(t)η(t ′ ) = 2 ζ δ(t − t ′ ) and the host fluid has unitary temperature: therefore ζ is the viscosity of the host fluid in these particular units. When Eq. (1) models active particles, η(t) is GCN ("active noise"), i.e.η (t) = −ζη(t) + 2ζχ(t)
(3) χ(t)χ(t ′ ) = δ(t − t ′ ). (4) In this case ζ = 1/τ and the active effective temperature is set to 1 (or, equivalently, the active speed is set to 1). We notice that in both cases (passive and active), with chosen units, the bare diffusion coefficient of the particles (i.e. when U (x) ≡ 0) is 1/ζ. In the following, we use a smooth potential of the form
U (x) = U 0 {tanh[(x + 1)/ξ] − tanh[(x − 1)/ξ]},(5)
which is characterised by a steepness 1/ξ. In order to understand the main effects induced by self-propulsion in the presence of a stationary current, we focus on two quantities: i) the density profile around the moving obstacle and ii) the experienced drag force.
A. Effective potential
To proceed with our analysis, it is useful to notice that, when η(t) is GCN, we can time-derive Eq. (1), obtaininġ
x(t) = v(t),(6)v(t) = −ζg(x − ct)v(t) + F * (x − ct) + 2ζχ(t),(7)F * (x) = F (x) − c ζ dF (x) dx = − dU (x) dx + c ζ d 2 U (x) dx 2 ,(8)g(x) =1 + 1 ζ 2 d 2 U (x) dx 2 .(9)
In the above equations two terms deserve discussion: an effective force F * (x), which reduces to −dU/dx when c = 0, and an effective viscosity g(x). The latter -which is the only effect of self-propulsion when c = 0 -has been thoroughly discussed in [11,32,33]: it can be treated within an approximate equilibrium-like solution (known as UCNA), based upon an effective static potential U stat (x) = U (x) + 1
2ζ 2 dU(x) dx 2 − ln |g(x)|.
In the present case, the finite velocity of the obstacle c > 0 produces an additional contribution in the force term, which is responsible for new dynamical effects. These effects can be accounted for by a new approximate treatment (see Appendix A).
III. DYNAMICAL UCNA
In the case of a shifting barrier, one rewrites the stochastic differential equations (6)-(7) into the equivalent Fokker-Planck equation for the probability distribution of position and velocity P (y, v):
∂P (y, v) ∂t + v ∂ ∂y P (y, v) + F * (y) ∂ ∂v P (y, v) = ζ ∂ ∂v ∂ ∂v + g(y)v P (y, v),(10)
with y = x − ct. In order to proceed further, we consider the steady state solution of Eq. (10) and set ∂P (y,v) ∂t = −c ∂P (y,v) ∂y . By multiplying by powers of v and integrating w.r.t. v, one obtains a hierarchy of coupled first order ordinary differential equations for the velocity moments of P (y, v), whose first two members are the continuity equation for the density ρ(y) = dvP (y, v)
− c dρ(y) dy + d dy J(y) = 0,(11)
and the momentum balance equation for the current J(y) = dvvP (y, v):
− c dJ(y) dy + dΠ(y) dy − F * (y)ρ(y) − ζg(y)J(y) = 0,(12)
where Π(y) = dvv 2 P (y, v). According to Eq. (11) the current must be proportional to the density
J(y) = c[ρ(y) −ρ],(13)
whereρ is a constant such that the solution is periodic, ρ(L) = ρ(−L). The following distribution represents the exact solution of Eq. (10) in the regions where the force vanishes and contains adjustable parameters to obtain an approximate solution in the wall region:
P (y, v) = β(y) 2π [ρ(y) −ρ] exp − 1 2 β(y)(v − c) 2 +ρ exp − 1 2 β(y)v 2 ,(14)
where β(y) is a positive definite function. Remarkably, expression (14) also represents an (approximate) closure of the infinite hierarchy of equations (of which Eqs. (11) and (12) are the first two members) generated by the transformation of the partial differential equation (10) into a set of coupled ordinary differential equations for the velocity moments of P . Hence, according to the information contained in Eq. (14) the momentum flux reads Π(y) = ρ(y) β(y) + c 2 [ρ(y) −ρ], so that Eq. (12) becomes:
d dy ρ(y) β(y) = [F (y) − ζc]ρ(y) + ζcg(y)ρ,(15)
which has the following interpretation: the "active pressure" gradient d dy ρ(y) β(y) is balanced by the force due to the moving wall and by the friction force −ζg(y)J(y) (the second term in the r.h.s.). In the case of a very weak potential, ρ(y) ≈ρ and the current vanishes, whereas for high barriers ρ(y) ≫ρ and J(y) ≈ cρ(y). The static UCNA approximation is recovered by setting c = 0, i.e. J = 0 and β(y) = g(y). The density profile is given by:
ρ(y) β(y) = ρ(L) β(L) e −w(y)+w(−L)−cζ(y+L) + ζcρe −w(y)−ζcy y −L dse w(s)+cζs g(s),(16)
where w(y) is an effective potential defined by:
w(y) = y −L dsβ(s) dU (s) ds + ζc y −L ds[β(s) − 1],(17)
and ρ(L) is fixed by the normalization of the number of particles. The explicit expression of the constantρ is given in Appendix A. Interestingly, the second term in the r.h.s. of Eq. (17) can be identified with a dynamical potential U dyn (y) vanishing when either c = 0 (static barrier) or ζ → ∞ (passive particles). Therefore it is a peculiar feature of our model, arising from the coupling of self-propulsion with the nonequilibrium current. This term gives an effective trap -at the front of the moving potential -of height ∼ U 0 c/(ζξ), and a specular effective barrier at its tail. As one can see, the solution for c = 0 is not Boltzmann-like since the system is in a truly nonequilibrium state and therefore the density profile is not symmetric with respect to the transformation y → −y which characterizes the bare potential U (y).
Since the UCNA breaks down in regions with negative curvature of the potential [12], for the purpose of obtaining quantitative predictions for ρ(y), we empirically set β(y) = g(y) where g(x) ≥ 0, and β(y) = 0 otherwise. From the density profile ρ(y) we also obtain the average drag force acting on the moving barrier F = L −L dyF (y)ρ(y), which obeys the sum rule
F = ζc L −L dy[ρ(y) − g(y)ρ].(18)
IV. NUMERICAL RESULTS
The approximations underlying our theory have been fairly verified by comparison with numerical simulations of the model in Eq. (1), for both passive and active particles. The simulations implement a time-discretized scheme for Eq. (1) through a fourth-order Runge-Kutta algorithm [34], with a time step dt = 10 −4 . Averages are done on a single trajectory of length 5 × 10 8 in the used units. In the figures, error bars fall within the symbols.
In Fig. 2 we show the density profiles for two passive cases and two active cases, with static or moving potential. A first important information is the good match between simulations and theory. In the passive case (two top frames), switching on the external velocity from c = 0 to c > 0 leads to an imbalance of the density distribution with an accumulation at the front of the membrane (at x − ct = 1, see expression of the moving potential, Eq. (5)), and a depletion at its tail (at x − ct = −1) [41]. The two frames on the left (c = 0) demonstrate that switching from passive to active particles induces an accumulation of particles near both borders of the membrane potential, with a depletion inside the energetically unfavoured region. The novel effect discussed here appears, strikingly, in the active case with c > 0 (bottom-right frame): the accumulation of particles on the moving front of the membrane becomes much more important than the passive case with c > 0 or the active case with c = 0 (notice the log scale on the y axis).
To understand the behavior of the system in full generality, exploring the effects of all parameters, we focus on the global observable F (y) , which is the average drag force experienced by the moving membrane. Several results are shown in Fig. 3A and 3B, where a comparison is presented between passive and active cases for several values of ζ and ξ in a relevant range of velocities c. Again, we observe a fair superposition of numerical results with theoretical predictions, Eq. (18): this is expected for the passive cases, where the theory is exact, while it is not trivial at all in the active case. Surprisingly, even at low ξ, a reduction of ζ (longer activity persistence time τ ) may improve the agreement with the simulations. In all the cases considered (excluding the ξ → 0 limit for the passive case), at constant ξ and ζ, the average drag reaches a maximum at some value c * and then decreases for c > c * . This can be understood in terms of competition between "kinetic energy" ∼ c 2 and the potential barrier. In the passive case this leads to a value of c * which is roughly independent of ξ or ζ, and a saturation of F when ξ → 0, as seen in Fig. 3A. In the active case at large 1/(ξζ) the dynamic potential ∼ U 0 c/(ζξ) dominates, so that the energetic argument leads to c * ∼ U 0 /(ζξ). When the effective barrier is high and c < c * , very few particles cross it and the majority goes atẋ ∼ c, so that Eq. (1) on average gives the linear behavior F ≈ ζc, well visible in simulations at large values of 1/(ζξ). Estimating the maximum value of the drag force to be F max ≈ ζc * , we get for the active case F max ≈ U 0 /ξ, expected to hold at large (ζξ) −1 . The active case with a moving membrane, therefore, is qualitatively different from the passive case -or from any case at c = 0 -since the average drag force can increase indefinitely by reducing ξ. In Fig. 3C we have shown F max versus (ζξ) −1 for the active and passive cases: at intermediate values of (ζξ) −1 an interesting data collapse is found, together with a sharp increase with (ζξ) −1 for the active case. Such an increase eventually saturates if ζ is further decreased at constant ξ, or continues if ξ is reduced at constant ζ, demonstrating the qualitative difference between the active and the passive cases.
V. CONCLUSIONS
We have shown the existence of a dynamical enhancement of clustering and drag when a travelling barrier sweeps active particles. The synergy of two dynamical effects (active noise and non-zero current) leads to a scenario qualitatively new, as shown in Fig. 3C: indeed the average drag is sensitive to the persistence time 1/ζ and to the steepness of the membrane potential 1/ξ, and can be made indefinitely strong. We have discussed a theoretical treatment of this effect, fairly compared with numerical simulations. This is remarkable if one considers that predictive theoretical schemes are scarce in the framework of active particles, particularly in the non-linear regime with strong spatial currents as in our case. It is interesting to note that our theory truncates the Fokker-Planck hierarchy at the same order of the static UCNA scheme: however, unlike the static UCNA, it leads to a genuine non-equilibrium behavior [3,35].
The parameter values used in our simulations are in the range of realistic systems of AP, therefore they are within reach for experimental verification, e.g. in setups with optical travelling waves or inverted traps [17][18][19], taking care to avoid competing effects such as diffusiophoretic torques or hydrodynamic-induced wall-attachment [36,37]. For instance, taking as unit of length 10µm (order of magnitude of the width of lithographed micro-membrane), typical biological swimmers with speed ∼ 10µm/s and reorientation time τ ∼ 1s correspond to ζ ≈ 1. A straightforward application of our study is the possibility to separate a mixture of AP, filtering out those with given parameters (e.g. a certain value of τ ) by sweeping a membrane with well-tuned values of c and ξ.
Appendix A: Model equations
We consider a dilute solution of active particles dragged along the x-direction under the action of a travelling potential barrier with velocity c, modelled by a time dependent external potential, U (x, t) = U (x − ct), which acts on the colloidal particles but has negligible effects on the solvent [38][39][40]. For the sake of simplicity we neglect the interactions among the particles and any hydrodynamic effect and include only the friction, through a drag coefficien γ. The active forces are modelled by a coloured noise, i.e. Gaussian noise with exponential memory of characteristic time τ . Note that, in this Appendix, we introduce the model with all dimensional parameters and explicitly show the change of variables necessary to obtain the Equations studied in the paper.
a. Langevin description
The following stochastic dynamics is assumeḋ
x(t) = 1 γ F (x, t) + η(t) ,(A1)
where F = −∂U/∂x and η mimics the self-propulsion mechanism and is assimilated to an Ornstein-Uhlenbeck procesṡ
η(t) = − 1 τ η(t) + D 1/2 τ ξ(t). (A2)
The underlying stochastic force ξ(t) is a Gaussian and Markovian process distributed with zero mean and moments ξ(t)ξ(t ′ ) = 2δ(t − t ′ ). The coefficient D due to the activity is related to the correlation of the Ornstein-Uhlenbeck process η(t) via
η(t)η(t ′ ) = D τ exp − |t − t ′ | τ . (A3) b
. Fokker-Planck description
After differentiating with respect to time eq. (A1) and introducing a velocity v =ẋ, we may write the following system of equations:ẋ
= v v = − 1 τ 1 − τ γ ∂F ∂x v + 1 τ γ F + τ ∂F ∂t + D 1/2 τ η . (A4)
The latter equation in the case of the shifting potential becomes:
v = − 1 τ 1 − τ γ ∂F ∂x v + 1 τ γ F − τ c ∂F ∂x + D 1/2 τ η (A5)
and the associated Fokker-Planck (FP) equation for the "phase-space" distribution P (x, v, t) reads
∂P ∂t + v ∂P ∂x + F (x, t) + τ ∂F (x,t) ∂t γτ ∂P ∂v = 1 τ ∂ ∂v D τ ∂ ∂v + g(x, t)v P ,(A6)
where g(x, t) = 1 − τ γ ∂F ∂x . Now, defining F * = (F − τ c ∂F ∂x ) and considering the steady state regime of the system, where P (x, v, t) must have the travelling wave form P (x − ct, v), we can write:
− c ∂P ∂x + v ∂P ∂x + F * (x − ct) γτ ∂P ∂v = 1 τ ∂ ∂v D τ ∂ ∂v + g(x − ct)v P.(A7)
In the problem at hand, the shifting external potential U is localized within a finite region around the origin of the comoving reference frame and vanishes for x → ±∞.
c. Non dimensional variables
In order to proceed further, it is time saving to adopt non dimensional variables for positions, velocities, and time, and rescale forces accordingly. We define v T = D/τ , measure lenghts using the characteristic lenght, ℓ, of the potential and introduce the following non dimensional variables:
t ≡ t v T l ,v ≡ v v T , X ≡ x ℓ ,F (x,t) ≡ ℓF (x, t) Dγ , ζ = ℓ τ v T ,P = v T ℓ P,c = c v T ,(A8)
where ζ plays the role of a non dimensional friction. To lighten the notation we shall drop the bar over the non dimensional variables without incurring in ambiguities.
In the case of a shifting barrier, one can write the following Fokker-Planck equation in terms of the coordinate y = x − ct relative to the comoving reference frame:
−c ∂P (y, v) ∂y + v ∂ ∂y P (y, v) + F * (y) ∂ ∂v P (y, v) = ζ ∂ ∂v ∂ ∂v + g(y)v P (y, v) (A9) d
. Hydrodynamic theory
In order to proceed further, it is convenient to eliminate the v dependence of the phase-space distribution P (y, v), by multiplying by powers of v and integrating w.r.t. v. One obtains a set of coupled first order ordinary differential equations, the so-called Brinkman hierarchy, whose first two members are the continuity equation and the momentum balance equation, respectively:
−c dρ(y) dy + d dy J(y) = 0 , (A10) −c dJ(y) dy + dΠ(y) dy − F * (y)ρ(y) + ζg(y)J(y) = 0 (A11)
where we have introduced the density ρ(y), the current J(y) and the momentum current Π(y), respectively, via:
ρ(y) = dvP (y, v) ,(A12)J(y) = dvvP (y, v) ,(A13)Π(y) = dvv 2 P (y, v) .(A14)
According to the continuity equation (A10) the current must be proportional to the density
J(y) = c[ρ(y) −ρ] ,(A15)
whereρ is a constant such that the solution is periodic at ρ(L) = ρ(−L), where 2L is the box size. As we shall see later, for large systems L ≫ ℓ,ρ ≈ ρ(±L) and the current is almost vanishing at the boundaries. It can be easily verified that the following distribution is a solution of the eq. (A9) in regions where F * (y) = 0 and g(y) = 1:
P (y, v) = ρ(y) −ρ H 0 (v − c) +ρ H 0 (v) ,(A16)
where
H 0 (v) = 1 √ 2π exp − 1 2 v 2 (A17)
is a Hermite function of zeroth order. By substituting the ansatz (A16) in eq. (A9) (with F = 0) we obtain a solution provided ρ(y) satisfies the following condition:
dρ(y) dy = −ζc(ρ(y) −ρ) .(A18)
e. Solution in the presence of a force field Now, we insist in looking for a solution of eq. (A9) of the form:
P (y, v) = ρ(y) −ρ H 0 (y, v − c) +ρ H 0 (y, v) ,(A19)
even in the region where F (y) = 0. We have introduced the following (non uniform) Hermite functions, which are position dependent through β(y), an adjustable function:
H 0 (y, v) = β(y) 2π exp − β(y) 2 v 2 ,(A20)H 1 (y, v) = β(y) 2π β 1/2 (y) v exp − β(y) 2 v 2 .(A21)
If we do that, i.e. if we apply the full FP operator to the trial distribution (A19) we get:
H 1 (y, v − c) 1 β 1/2 dρ(y) dy − β(F * − ζgc)(ρ −ρ) − β ′ β (ρ −ρ) − H 1 (y, v)) 1 β 1/2 βF * ρ + β ′ βρ +ζ(g − β) (ρ −ρ)H 2 (y, v − c) +ρH 2 (y, v) − β ′ 2β 3/2 (ρ −ρ)H 3 (y, v − c) +ρH 3 (y, v) − cβ 1/2 H 2 (y, v) = 0 ,(A22)
where H 2 (y, v) and H 3 (y, v) are the Hermite functions of order 2 and 3, respectively, and given by the recursion relation:
H ν+1 (y, v) = − 1 β 1/2 ∂H ν (y, v) ∂v .
The trial solution fails to solve eq. (A9). However, if we limit ourselves to consider only the two lowest moments of the probability distribution, i.e. if after multiplying by (v − c), we integrate (A22) over v we obtain the following condition which gives the profile equation:
1 β dρ(y) dy − (F − ζc)ρ(y) − β ′ β 2 ρ(y) − ζcgρ = 0 .(A23)
If we continue the projection procedure beyond the first order in (v − c) there will be an error in the equation for the second moment, which becomes inconsistent with the value of the second moment imposed by the trial distribution (which, in fact, is already fixed by the trial form and therefore does not contain enough parameters to satisfy the extra conditions.).
f. Construction of the solution
Eq. (A23) can be rearranged as follows:
d dy ρ(y) β(y) = (F (y) − ζc)ρ(y) + ζcg(y)ρ.(A24)
Notice that the ansatz for the phase-space distribution, gives the following expression for the momentum flux:
Π(y) = ρ(y) β(y) + c 2 (ρ(y) −ρ) .(A25)
Notice that eq. (A23) is perfectly equivalent to eq. (A11) when the latter is endowed with a closure, indeed represented by eq. (A25). The static UCNA approximation is recovered by setting the arbitrary function β(y) = g(y) and c = 0, ,
where ρ(L) is fixed by the normalization and the effective potential w(y) is defined by
w(y) = y −L dsβ(s) dU (s) ds + ζc y −L ds[β(s) − 1] ,(A27)
The function β(y) is given by g(y) when g(y) > 0 and β(y) = 0 otherwise.
g. Average Force and sum rule
The average drag force is given by
< F >= L −L dyF (y)ρ(y) = ζc L −L dy[ρ(y) −ρg(y)].(A28)
The constantρ isρ and we can rewrite the solution as:
ρ(y) β(y) = ρ(L) β(L) e −w(y)+w(−L)−cζ(y+L) + ζcρe −w(y)−ζcy y −L dse w(s)+cζs g(s).(A29)
Finally, in order to regularize the problem we have chosen β(y) = g(y) when g(y) > 0 and β(y) = 0 otherwise.
Appendix B: The Dual picture
The same mathematical problem can describe a different physical set up. Consider a one dimensional system and a non uniform potential U (y) acting in a central region only, where F (y) = − dU dy = 0. The particles are subject to colored noise and to a uniform force E. There will be a constant current, say J 0 .
The obstacle is fixed in space, represented by the force F (y). There is a constant external field Ė
y = F (y) + E γ + η(t) (B1)
where η(t) is the standard colored noise as before. Time-differentiating Eq. (B1) we geṫ
y = v (B2) v = F ′ (y) γ v − v(t) τ + F + E τ γ + D 1/2 τ ξ .(B3)
Equivalently, we write the associated FP equation:
∂P ∂t + v ∂P ∂y + F (y) + E γτ ∂P ∂v = 1 τ ∂ ∂v D τ ∂ ∂v + g(y)v P ,(B4)
If one integrates over v and defines J(y) = dvvP (y, v), one finds:
d dy J(y) = 0 . (B7)
The current is, now, constant: J(y) = J 0 . Let us multiply by v and integrate eq. (B5):
d dy Π(y) − (F (y) + E)ρ(y) = −ζg(y)J 0 ,(B8)
with Π(y) = v 2 P (y, v)dv. Let us invert the relation and make the ansatz:
P (y, v) =ρH 0 (y, v − u) + (ρ(y) −ρ)H 0 (y, v)(B9)
Substituting (B9) in eq. (B6) when F = 0 and g = 1 we obtain
H 1 (v) dρ(y) dy − E(ρ(y) −ρ) − Eρ − ζcρ H 1 (v − u) = 0 ,(B10)
whose solution is:
dρ(y) dy − E(ρ(y) −ρ) = 0 (B11) Eρ = ζuρ (B12)
Now, we go back to eq. (B8) and use the following closure (contained already in the parametric form of the solution for P (y, v)):
J(y) = J 0 =ρu ,(B13)
Π(y) =ρu 2 + ρ(y) β(y) .
(B14)
So that the equation for ρ(y) reads:
d dy ρ(y) β(y) − (F (y) + E)ρ(y) = −ζug(y)ρ .
Now, such an equation is identical to the equation (A24) , provided we identify:
E = −ζc , (B16) u = −c , (B17) u = E ζ .(B18)
Thus, we have shown that the equation for ρ(y) is of the same type as the Nernst-Planck (NP) equation: The NP equation assumes that the constant current J 0 results from the combined effects of a diffusive current due to the random fluctuations ( the "thermal agitation" in other words) and a deterministic migration current due to the coupling to an external field E, which can be also modified by the presence of some localized potential U = − y F (s)ds:
J 0 = − 1 ζ 1 g(y)
d dy ρ(y) β(y) + 1 ζg(y) (F (y) + E)ρ(y) ,
with a space-dependent diffusion coefficient D(y) ≡ 1 ζ 1 β(y)g(y)
(B20) and a space-dependent mobility µ(y) = 1 ζ 1 g(y)
.
Notice that this is exactly the UCNA equation for the current, which can be derived without phase-space considerations. Finally, let us rewrite J 0 = − d dy D(y)ρ(y) + µ(y) F (y) + E − 1 β(y) d dy ln g(y) ρ(y)
There is an extra contribution from the drift stemming from the colored noise. Note that the mathematics is the same as for the original problem, but the interpretation of each term is now different. If we look at the profiles, we observe a crowding of active particles at the front of the potential (where the derivative of U is largest) and a depletion inside. Particle near the entrance loose mobility and therefore crowd there. With strong activity and sharp entrances (ζ → 0 and ξ → 0, respectively) the current should go to zero.
FIG. 2 :
2Density profiles with different kinds of noise (thermal or active) and different values of the barrier's velocity c: (a) thermal noise, c = 0: (b) thermal noise, c = 0.2; (c) active noise, c = 0; (d) active noise, c = 0.2.
FIG. 3 :
3(a) and (b): average drag force versus velocity c for thermal noise (a) and active noise (b). Symbols represent numerical simulations (legends are valid for both graphs), while dotted lines are the theoretical predictions; (c): maximum average drag force versus (ξζ) −1 . Symbols are numerical simulations, while lines are from theory.
(i.e. J = 0). The solution of the inhomogeneous equation in the case of c = 0 is ρ(y) = ρ(L) β(L) β(y)e −(w(y)−w(−L))−cζ(y+L) 1 + (e 2ζcL e w(L)−w(−L) − 1) y −L dse w(s)+cζs g(s) L −L dse w(s)+cζs g ( s)
dye w(y)+cζy g(y)
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Our theory predicts an asymptotic exponential decay of the density profiles both in front and past the moving wall, in agreement with what found for analogous problems in lattice systems. 31Our theory predicts an asymptotic exponential decay of the density profiles both in front and past the moving wall, in agreement with what found for analogous problems in lattice systems [31].
|
[] |
[
"An abstract proximal point algorithm",
"An abstract proximal point algorithm"
] |
[
"Laurenţiu Leuştean s:[email protected] \nThe Research Institute\nUniversity of Bucharest (ICUB)\nUniversity of Bucharest\nM. Kogȃlniceanu 36-46050107BucharestRomania\n\nFaculty of Mathematics and Computer Science\nUniversity of Bucharest\nAcademiei 14010014BucharestRomania\n\nSimion Stoilow Institute of Mathematics of the Romanian Academy\nCalea Griviţei 21010702BucharestRomania\n",
"Adriana Nicolae [email protected] \nDepartment of Mathematical Analysis -IMUS\nUniversity of Seville\nC/ Tarfia s/n41012SevillaSpain\n\nDepartment of Mathematics\nBabeş-Bolyai University\nKogȃlniceanu 1400084Cluj-NapocaRomania\n",
"Andrei Sipoş [email protected] \nSimion Stoilow Institute of Mathematics of the Romanian Academy\nCalea Griviţei 21010702BucharestRomania\n\nDepartment of Mathematics\nTechnische Universität Darmstadt\nSchlossgartenstrasse 764289DarmstadtGermany\n"
] |
[
"The Research Institute\nUniversity of Bucharest (ICUB)\nUniversity of Bucharest\nM. Kogȃlniceanu 36-46050107BucharestRomania",
"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nAcademiei 14010014BucharestRomania",
"Simion Stoilow Institute of Mathematics of the Romanian Academy\nCalea Griviţei 21010702BucharestRomania",
"Department of Mathematical Analysis -IMUS\nUniversity of Seville\nC/ Tarfia s/n41012SevillaSpain",
"Department of Mathematics\nBabeş-Bolyai University\nKogȃlniceanu 1400084Cluj-NapocaRomania",
"Simion Stoilow Institute of Mathematics of the Romanian Academy\nCalea Griviţei 21010702BucharestRomania",
"Department of Mathematics\nTechnische Universität Darmstadt\nSchlossgartenstrasse 764289DarmstadtGermany"
] |
[] |
The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.
|
10.1007/s10898-018-0655-9
|
[
"https://arxiv.org/pdf/1711.09455v2.pdf"
] | 53,026,512 |
1711.09455
|
0188d1ca05e6abee52c4f6401fd3b03bfbb80b36
|
An abstract proximal point algorithm
17 Apr 2018
Laurenţiu Leuştean s:[email protected]
The Research Institute
University of Bucharest (ICUB)
University of Bucharest
M. Kogȃlniceanu 36-46050107BucharestRomania
Faculty of Mathematics and Computer Science
University of Bucharest
Academiei 14010014BucharestRomania
Simion Stoilow Institute of Mathematics of the Romanian Academy
Calea Griviţei 21010702BucharestRomania
Adriana Nicolae [email protected]
Department of Mathematical Analysis -IMUS
University of Seville
C/ Tarfia s/n41012SevillaSpain
Department of Mathematics
Babeş-Bolyai University
Kogȃlniceanu 1400084Cluj-NapocaRomania
Andrei Sipoş [email protected]
Simion Stoilow Institute of Mathematics of the Romanian Academy
Calea Griviţei 21010702BucharestRomania
Department of Mathematics
Technische Universität Darmstadt
Schlossgartenstrasse 764289DarmstadtGermany
An abstract proximal point algorithm
17 Apr 2018Convex optimizationProximal point algorithmCAT(0) spacesJointly firmly nonexpansive familiesUniformly firmly nonexpansive mappingsProof miningRates of con- vergence Mathematics Subject Classification 2010: 90C25, 46N10, 47J25, 47H09, 03F10
The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.
Introduction
The first instance of what came later to be known as the proximal point algorithm can be found in a short communication from 1970 of Martinet [44]. He considered (among others) the issue of solving the minimization problem argmin x∈C f (x),
where C is a closed convex subset of a Hilbert space H and f is a (real-valued) lower semicontinuous convex function defined on C, that further has the property that for all a ∈ R, the set
{x ∈ C | f (x) ≤ a}
is bounded. One then starts from an arbitrary point x 0 ∈ C and afterwards iteratively builds a sequence (x n ) by the implicit (though uniquely determining) recurrence relation
f (x n+1 ) = min y∈C (f (y) + x n − y 2 ).
Martinet's Théorème 3 then asserts that any weak cluster point of this sequence is a solution to the given minimization problem.
In 1976, Rockafellar [47] took up the more general problem of finding a zero of a maximally monotone multi-valued operator A : H → 2 H , i.e. a point x such that 0 ∈ A(x) (single-valued monotone operators had already been considered by Martinet). This contains the previous case since the subdifferential ∂f of a function f having the properties considered above is a maximally monotone operator whose zeros coincide with the minimizers of f . The method used in this case in order to approach the desired solution was called the "proximal point algorithm" and generates starting from a point x 0 ∈ C a sequence using another implicit recurrence, namely
x n ∈ (id H + γ n A)(x n+1 ),
where (γ n ) is a sequence of positive real numbers. When A = ∂f , the relation reduces to the previous one if (γ n ) is the sequence constantly equal to 1/2. Theorem 1 of [47] shows that if inf n∈N γ n > 0, then (x n ) weakly converges to a zero of A. Strong convergence is proved under some additional uniformity assumptions (such as A −1 being Lipschitz continuous at 0), but it does not generally hold, as Güler [21,Corollary 5.1] later put forward a counterexample in this sense. Two years after Rockafellar's paper, Brézis and Lions [12] studied more general conditions one could impose on (γ n ) that still yield weak convergence of the sequence (x n ), there regarded as the "infinite product" of the resolvent operators J γnA := (id H + γ n A) −1 .
Those conditions continue to be the state of the art -e.g. for an arbitrary maximally monotone operator one may assume ( [12,Proposition 8]
) ∞ n=0 γ 2 n = ∞.
The proximal point algorithm has grown to become a versatile tool of convex optimization, being used, in addition to the applications already expounded upon, to solve a plethora of problems such as variational inequalities, minimax or equilibrium problems (some of these may already be found in the papers cited above). The book of Bauschke and Combettes [9] may serve as an introduction to the field in the context of Hilbert spaces.
Outside Hilbert spaces, a natural generalization of the resolvent operator was given in the 1990s by Jost [25] and Mayer [45] in the context of complete CAT(0) spaces, which can be regarded as the proper nonlinear analogue of Hilbert spaces. Using this definition and an appropriate notion of weak convergence introduced by Lim [43], called ∆-convergence, Bačák [3] extended in 2013 Théorème 9 of Brézis and Lions [12] in this context. More precisely, he proved the ∆-convergence of the sequence generated by the proximal point algorithm when f is a proper, convex and lower semicontinuous function that attains its minimum.
A separate strand of development came from fixed point theory. In the 1960s, Browder [15] and Halpern [22] studied the existence and computation of fixed points of nonexpansive mappings T : C → C (where C is a closed convex bounded subset of a Hilbert space). They considered the notion of a resolvent of order γ of T -that is, a mapping satisfying, for all x ∈ C,
R γ x = 1 1 + γ x + γ 1 + γ T R γ x.
In the above, we reparametrized their construction in order to obtain a better fit with the objects considered here. Their main result states that by letting γ → ∞, R γ x tends to the fixed point of T which is the closest to x. Halpern's particularly simple argument was later generalized to the Hilbert ball in [20] and to complete CAT(0) spaces in [27]. A proof in the latter setting that starts from minimal boundedness assumptions may be found in a 2014 paper of Bačák and Reich [6]. Note that [6] also contains a variant of the proximal point algorithm which constructs, by iterating the resolvents of T , a sequence that ∆-converges to a fixed point of T . As one may notice, every iterative sequence that was considered above under the name of "proximal point algorithm" follows a pattern: we have a mathematical object that we seek to optimize in some way, we construct associated "resolvent" operators, we take an initial arbitrary point x and finally we iterate those operators starting from x. One may ask whether there are some very general hypotheses which yield the convergence of the resulting sequence without explicitly considering the particular details of the optimization problem at hand. Our first main goal is to answer this question in the affirmative in the framework of CAT(0) spaces (and therefore also for Hilbert spaces) by deriving some conditions related to firm nonexpansivity that are satisfied by all the above types of families of resolvents. We give these conditions in weaker and stronger forms, and show that while the strongest one is generally satisfied, the weakest one suffices to obtain appropriate convergence results. We should mention here that it was known for a long time that individual resolvents are in particular firmly nonexpansive, and some abstract results in the same spirit were previously obtained by Ariza-Ruiz, the first author and López-Acedo in [1]. The present paper may be regarded as a natural continuation of the study initiated there (see also [2]). Section 2 introduces general notions and properties regarding geodesic metric spaces and mappings that are used in the sequel. Section 3 starts with some very general hypotheses for a sequence generated by a family of firmly nonexpansive mappings that yield its weak or ∆-convergence. In the process, we derive some lemmas that characterize various asymptotic aspects of the proximal point algorithm. Then we define two conditions that one may impose on a family (T n ) with respect to a sequence (γ n ). These conditions generalize the property of a mapping being firmly nonexpansive to a relation between two possibly different mappings which is then applied to each possible pair from the countable family. We claim that these definitions capture the residual property used in convergence proofs that corresponds to the way a family of resolvents (J γn ) behaves with respect to the sequence of step-sizes (γ n ). We consider then "jointly firmly nonexpansive families" and a somewhat weaker notion, "jointly (P 2 ) families" from which the general conditions can be obtained. In particular, the families of mappings involved in the problems discussed before (i.e. minimization of convex functions, finding fixed points of nonexpansive mappings and finding zeros of maximally monotone operators) satisfy these conditions.
The second main goal of this paper is to find quantitative variants of some convergence results for the proximal point algorithm. This falls within the purview of proof mining, an applied subfield of logic. Proof mining primarily concerns itself with the application of tools from proof theory to obtain computational content for theorems in ordinary mathematics with proofs that are not necessarily fully constructive. The project was first suggested in the 1950s by Kreisel under the name of "unwinding of proofs", but it gained considerable momentum after its extensive development in the 1990s and 2000s by Kohlenbach and his collaborators, culminating with the publication of general logical metatheorems, developed by Kohlenbach [30] and by Gerhardy and Kohlenbach [19], that tell us when a proof of a theorem proven in classical logic may be analyzed in order to obtain ("extract") its hidden quantitative information. A comprehensive reference for the major developments of the field up to 2008 is the monograph of Kohlenbach [31], while surveys of recent results are [32,33]. So far, proof mining has been successfully applied to obtain quantitative versions of celebrated results in various areas of mathematics such as approximation theory, nonlinear analysis, metric fixed point theory, ergodic theory, or topological dynamics. Recently, its methods have begun to be applied to convex optimization, for more details see [5,34,35,36,38,41,42].
Let us discuss the sort of quantitative results that we obtain. If (x n ) is a sequence in a metric space X and x ∈ X, then lim n→∞ x n = x if and only if
∀k ∈ N ∃N ∈ N ∀n ≥ N d(x n , x) ≤ 1 k + 1 .
A quantitative version of the above would be a rate of convergence for the sequence: a formula showing how to compute the N in terms of the k. However, very simple real-valued sequences have been shown by methods of mathematical logic to lack a computable rate of convergence. We recall, though, from the discussion above, that strong convergence of the proximal point algorithm could only be obtained under some extra uniformity assumptions. Fortunately, some of these conditions yield the uniqueness of the needed optimizing point (minimizer, fixed point or zero). This uniqueness was shown by the work of Kohlenbach [29], Kohlenbach and Oliva [37, Section 4.1] and Briseid [14] to guarantee the extraction of a rate of convergence, relative to some other piece of quantitative information. In Section 4, therefore, we define a general notion of uniformity applicable to our families of mappings (extending the similar notion given in [8] in the context of Hilbert spaces). One then shows that the concrete algorithms have corresponding "uniform" cases that are subsumed into this definition, e.g. finding zeros of uniformly monotone mappings or minimizing uniformly convex functions. Section 5 then shows that this definition suffices: a quantitative variant of the asymptotic lemmas from Section 3 fits in as the relevant piece of information that is then used, as per the above discussion, to obtain a highly uniform rate of convergence for this special case of the proximal point algorithm. As a byproduct, we obtain an alternate proof for the classical qualitative results of strong convergence.
Proximal methods are not limited to the classical problems of convex optimization. Therefore, a question that arises is to what extent a natural and abstract approach of the type provided here could be employed to capture other such variants, which are used, for example, in global (nonconvex) optimization [26,23], where the necessity of the existence of iterates requires one to assume weak forms of monotonicity. Another direction consists in considering multi-valued resolvent-type operators instead of single-valued ones. In this case the algorithm becomes nondeterministic (i.e. given a current iterate, the following one is not uniquely determined). Such a development would allow one to cover e.g. vector-valued optimization problems [11,16].
Preliminaries
We start by briefly recalling some notions and properties about geodesic spaces needed in the sequel. More details on geodesic spaces can be found, for example, in [46,13,4]. Let (X, d) be a metric space. A geodesic in X is a mapping γ :
[a, b] → X (where a, b ∈ R) such that for all s, t ∈ [a, b], d(γ(s), γ(t)) = |s − t|.
We say that X is a geodesic space if for all x, y ∈ X, there is a geodesic γ : [a, b] → X satisfying γ(a) = x and γ(b) = y. A geodesic space (X, d) is called a CAT(0) space if for all z ∈ X, all geodesics γ : [a, b] → X and all t ∈ [0, 1] we have that
d 2 (z, γ((1 − t)a + tb)) ≤ (1 − t)d 2 (z, γ(a)) + td 2 (z, γ(b)) − t(1 − t)d 2 (γ(a), γ(b)).(1)
It easily follows that every CAT(0) space is uniquely geodesic -that is, for any x, y in such a space X there is a unique geodesic γ : [0, d(x, y)] → X such that γ(0) = x and γ(d(x, y)) = y -and in this framework we shall denote, for any t ∈ [0, 1], the point γ(td(x, y)) by (1 − t)x + ty. Note that every CAT(0) space X is Busemann convex -i.e., for any x, y, u, v ∈ X and t ∈ [0, 1],
d((1 − t)x + ty, (1 − t)u + tv) ≤ (1 − t)d(x, u) + td(y, v).(2)
We will also make use of the quasi-linearization function ·, · : X 2 × X 2 → R introduced by Berg and Nikolaev in [10], which is defined, for any x, y, u, v ∈ X, by the following (where an ordered pair of points (w,
w ′ ) ∈ X 2 is denoted by − − → ww ′ ): − → xy, − → uv := 1 2 (d 2 (x, v) + d 2 (y, u) − d 2 (x, u) − d 2 (y, v)).(3)
Proposition 2.1 ([10, Proposition 14]). In any metric space (X, d), the mapping ·, · is the unique one that satisfies, for any x, y, u, v, w ∈ X, the following properties:
(i) − → xy, − → xy = d 2 (x, y); (ii) − → xy, − → uv = − → uv, − → xy ; (iii) − → yx, − → uv = − − → xy, − → uv ; (iv) − → xy, − → uv + − → xy, − → vw = − → xy, − → uw .
In particular, if X is a real Hilbert space with inner product ·, · , then
− → xy, − → uv = x − y, u − v = y − x, v − u ,(4)
for all x, y, u, v ∈ X. This justifies the notation.
The main result of [10], Theorem 1, gives a characterization of CAT(0) spaces in terms of the "Cauchy-Schwarz" inequality for ·, · . More precisely, a geodesic space (
X, d) is CAT(0) if and only if − → xy, − → uv ≤ d(x, y)d(u, v),(5)
for all x, y, u, v ∈ X. Furthermore, by [10, Theorem 6], a related condition for a geodesic space (X, d) to be CAT(0) is the following inequality
d 2 (x, v) + d 2 (y, u) ≤ d 2 (x, u) + d 2 (y, v) + d 2 (x, y) + d 2 (u, v),(6)
which is to be satisfied for all x, y, u, v ∈ X.
For the rest of the section, (X, d) is a geodesic space, unless stated otherwise. If T : X → X is a mapping, we denote by F ix(T ) the set of its fixed points.
The following generalization of firmly nonexpansive mappings to geodesic spaces was introduced in [1].
Definition 2.2. A mapping T : X → X is called firmly nonexpansive if for any x, y ∈ X and any t ∈ [0, 1] we have that d(T x, T y) ≤ d((1 − t)x + tT x, (1 − t)y + tT y).
As mentioned in [2] (see also [36]), if X is a CAT(0) space, every firmly nonexpansive mapping T : X → X satisfies the so-called property (P 2 ). Namely,
2d 2 (T x, T y) ≤ d 2 (x, T y) + d 2 (y, T x) − d 2 (x, T x) − d 2 (y, T y),
for all x, y ∈ X. In other words,
d 2 (T x, T y) ≤ −−−→ T xT y, − → xy ,(7)
for all x, y ∈ X. If X is a Hilbert space, property (P 2 ) is sufficient for firm nonexpansivity as (7) and (4) yield T x − T y 2 ≤ T x − T y, x − y , which is, in turn, equivalent to Definition 2.2 (see, e.g., [9, Proposition 4.2] for a proof). Moreover, from (7) and (5) one immediately obtains the following result.
Lemma 2.3. If X is a CAT(0) space and T : X → X satisfies property (P 2 ), then T is nonexpansive.
Let (x n ) be a bounded sequence in X and F ⊆ X be nonempty. For any y ∈ X, define
r(y, (x n )) := lim sup n→∞ d(y, x n ), r(F, (x n )) := inf{r(y, (x n )) | y ∈ F }.
Furthermore, A(F, (x n )) := {y ∈ F | r(y, (x n )) = r(F, (x n ))} and elements of A(F, (x n )) are called asymptotic centers of (x n ) with respect to F . We shall denote A(X, (x n )) by A((x n )) and call its elements asymptotic centers of (x n ). The next results will be used in the subsequent sections.
Lemma 2.4 ([40, Lemma 3.2]). Let (x n ) be a bounded sequence in X with A((x n )) = {c}
and (α n ), (β n ) be real sequences such that α n ≥ 0 for all n ∈ N, lim sup n→∞ α n ≤ 1 and lim sup n→∞ β n ≤ 0.
Assume that y ∈ X is such that there exist p, N ∈ N satisfying, for all n ≥ N ,
d(y, x n+p ) ≤ α n d(c, x n ) + β n .
Then y = c.
Proposition 2.5 ([17, Proposition 7])
. Every bounded sequence (x n ) in a complete CAT(0) space X has a unique asymptotic center with respect to any nonempty closed convex subset of X.
In order to state our main results, we need to introduce the notion of ∆-convergence which was defined by Lim [43] in metric spaces. We refer to [39,24,18] for equivalent notions in the setting of complete CAT(0) spaces, where ∆-convergence can be seen as a generalization of the weak convergence in Banach spaces (see [28]). In fact, in Hilbert spaces, ∆-convergence coincides with weak convergence (see [4, Exercise 3.1]). Definition 2.6. A bounded sequence (x n ) ∆-converges to a point x ∈ X if for any subsequence (u n ) of (x n ) we have that A((u n )) = {x}.
The notion of Fejér monotonicity will also play an important role in this work. Let (x n ) be a sequence in X and F ⊆ X be nonempty.
Definition 2.7. We say that (x n ) is Fejér monotone with respect to F if for all p ∈ F and all n ∈ N, we have that
d(x n+1 , p) ≤ d(x n , p).
It is obvious that if (x n ) is Fejér monotone with respect to F , then (d(x n , p)) converges for every p ∈ F and, furthermore, (x n ) is bounded.
Finally, we recall the following well-known result (see, for example, [7, Proposition 3.3.(iii)]), which turns out to be very useful in obtaining ∆-convergence results.
Proposition 2.8. Let X be a complete CAT(0) space and (x n ) be Fejér monotone with respect to F . Assume that the asymptotic center of every subsequence of (x n ) is in F . Then (x n ) ∆-converges to some x ∈ F .
An abstract Proximal Point Algorithm
We now begin the process of modularizing the proof(s) that guarantee the weak convergence of common instances of the proximal point algorithm. Theorem 3.5 is the first stage in this sense and provides some highly general conditions under which the iteration constructed by applying countably many mappings converges weakly. In proving it, we shall also show some fundamental properties of that iterative sequence, such as Fejér monotonicity and a form of asymptotic regularity.
In the following, X is a complete CAT(0) space and (T n ) n∈N is a family of self-mappings of X satisfying property (P 2 ) and having common fixed points. Set
F := n∈N F ix(T n ) = ∅.
For x ∈ X, we define the following iteration starting with x:
x 0 := x, x n+1 := T n x n for all n ∈ N.(8)
Let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞. The following conditions will also be considered in the sequel:
(C1) for all n, m ∈ N and w ∈ X, d(T n w, T m w) ≤ |γn−γm| γn d(w, T n w); (C2) the sequence d(xn,xn+1)
γn n∈N is nonincreasing.
We include below a series of preliminary results.
Lemma 3.1. Suppose that (C1) holds. Then F ix(T n ) = F for every n ∈ N.
Proof. It follows immediately.
Lemma 3.2. For all p ∈ F and all n ∈ N, we have that
d 2 (x n+1 , p) ≤ d 2 (x n , p) − d 2 (x n , x n+1 ).
In particular, (x n ) is Fejér monotone with respect to F .
Proof. Let p ∈ F and n ∈ N. Since T n satisfies property (P 2 ) and p ∈ F ix(T n ), we have that
2d 2 (T n x n , p) ≤ d 2 (x n , p) + d 2 (T n x n , p) − d 2 (x n , T n x n ). It follows that d 2 (T n x n , p) ≤ d 2 (x n , p) − d 2 (x n , T n x n ), hence the conclusion. Lemma 3.3. Assume that (C2) is satisfied. Then lim n→∞ d(x n , x n+1 ) = 0 and lim n→∞ d(x n , x n+1 ) γ n = 0. Proof. Since F = ∅, there exists p ∈ F . Let b > 0 be such that d(x, p) ≤ b. For every n ∈ N, we have, by Lemma 3.2, that n k=0 d 2 (x k , x k+1 ) ≤ n k=0 (d 2 (x k , p) − d 2 (x k+1 , p)) = d 2 (x, p) − d 2 (x n+1 , p) ≤ b 2 .
It follows that the series
∞ n=0 d 2 (x n , x n+1 ) converges, so lim n→∞ d(x n , x n+1 ) = 0.
We prove now that lim n→∞
d(xn,xn+1) γn = 0. Let ε > 0. Since ∞ n=0 γ 2 n = ∞, there exists N ∈ N such that N k=0 γ 2 k ≥ b 2 /ε 2 . If for all k ∈ {0, . . . , N } one has that d(x k ,x k+1 ) γ k > ε, we get that N k=0 d 2 (x k , x k+1 ) > N k=0 γ 2 k ε 2 ≥ b 2 ,alim n→∞ d(x n , T m x n ) = 0.
Proof. Let m ∈ N. We get that for all n ∈ N,
d(x n , T m x n ) ≤ d(x n , x n+1 ) + d(x n+1 , T m x n ) = d(x n , x n+1 ) + d(T n x n , T m x n ) ≤ d(x n , x n+1 ) + |γ n − γ m | γ n d(x n , T n x n ) by (C1) ≤ 2d(x n , x n+1 ) + γ m · d(x n , x n+1 ) γ n .
Our conclusion follows by applying Lemma 3.3.
We can prove now the main result of this section.
Theorem 3.5 (Abstract Proximal Point Algorithm). Let X be a complete CAT(0) space and (T n ) be a family of self-mappings of X satisfying property (P 2 ) and having common fixed points. Set F := n∈N F ix(T n ) = ∅. For x ∈ X, let (x n ) be defined by (8). Let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞. Assume that (C1) and (C2) hold. Then (x n ) ∆-converges to a point in F .
Proof. Note first that by Lemma 3.2, (x n ) is Fejér monotone with respect to F , hence bounded. Let (u n ) be an arbitrary subsequence of (x n ). By Proposition 2.5, (u n ) has a unique asymptotic center u. We shall prove that u ∈ F , so let m ∈ N be arbitrary. Note that
d(T m u, u n ) ≤ d(T m u, T m u n ) + d(u n , T m u n ) ≤ d(u, u n ) + d(u n , T m u n ).
Applying Lemma 2.4 with α n = 1, β n = d(u n , T m u n ), p = N = 0 and using the fact that lim n→∞ d(u n , T m u n ) = 0 (by Proposition 3.4), we get that T m u = u.
Finally, Proposition 2.8 yields that (x n ) ∆-converges to a point in F .
Jointly firmly nonexpansive families of mappings
We shall now proceed to the second stage of our abstraction -that is, giving a natural condition for a family (T n ) and a sequence (γ n ) of positive numbers such that the previous general conditions are satisfied. This can be regarded as an extension of the project initiated in [1] with the definition and the asymptotic behaviour of a firmly nonexpansive mapping to the case of a countable family of mappings. Recall that the notion of a firmly nonexpansive mapping in a Hilbert space has two analogues when considered within the more general setting of CAT(0) spaces. In the same spirit, we shall present here two definitions that apply to families of mappings which coincide when restricted to Hilbert spaces. In the sequel, X is a CAT(0) space, T n : X → X for every n ∈ N and (γ n ) is a sequence of positive real numbers.
Definition 3.6. The family (T n ) is said to be jointly firmly nonexpansive with respect to (γ n ) if for all n, m ∈ N, x, y ∈ X and all α, β ∈ [0, 1] such that
(1 − α)γ n = (1 − β)γ m , d(T n x, T m y) ≤ d((1 − α)x + αT n x, (1 − β)y + βT m y).(9)
Definition 3.7. We say that the family (T n ) is jointly (P 2 ) with respect to (γ n ) if for all n, m ∈ N and all x, y ∈ X,
1 γ m (d 2 (T n x, T m y) + d 2 (y, T m y) − d 2 (y, T n x)) ≤ 1 γ n (d 2 (x, T m y) − d 2 (x, T n x) − d 2 (T n x, T m y)).(10)
Lemma 3.8. If (T n ) is jointly firmly nonexpansive (resp. jointly (P 2 )) with respect to (γ n ), then each T n is firmly nonexpansive (resp. satisfies property (P 2 )).
Proof. Apply (9) (resp. (10)) for m = n. In the first case, remark that given t ∈ [0, 1], we take α = β = t.
Proposition 3.9. If (T n ) is jointly firmly nonexpansive with respect to (γ n ), then (T n ) is jointly (P 2 ) with respect to (γ n ).
Proof. Let m, n ∈ N and x, y ∈ X. We choose arbitrarily α ∈ (1 − min {γ m /γ n , 1} , 1) and set
β := 1 − (1 − α) γ n γ m .
Then β ∈ (0, 1) and (1 − α)γ n = (1 − β)γ m . Hence, applying the fact that (T n ) is jointly firmly nonexpansive and the inequality (1) twice, we get that
d 2 (T n x, T m y) ≤ d 2 ((1 − α)x + αT n x, (1 − β)y + βT m y) ≤ (1 − α)d 2 (x, (1 − β)y + βT m y) + αd 2 (T n x, (1 − β)y + βT m y) − α(1 − α)d 2 (x, T n x) ≤ (1 − α)(1 − β)d 2 (x, y) + (1 − α)βd 2 (x, T m y) − (1 − α)β(1 − β)d 2 (y, T m y)+ + α(1 − β)d 2 (T n x, y) + αβd 2 (T n x, T m y) − αβ(1 − β)d 2 (y, T m y) − α(1 − α)d 2 (x, T n x) = (1 − α)(1 − β)d 2 (x, y) + (1 − β)αd 2 (T n x, y) + (1 − α)βd 2 (x, T m y)+ + αβd 2 (T n x, T m y) − α(1 − α)d 2 (x, T n x) − β(1 − β)d 2 (y, T m y), so (1 − αβ)d 2 (T n x, T m y) ≤ (1 − α)(1 − β)d 2 (x, y) + (1 − β)αd 2 (T n x, y) + (1 − α)βd 2 (x, T m y)− − α(1 − α)d 2 (x, T n x) − β(1 − β)d 2 (y, T m y).
Dividing now the above inequality by 1 − α > 0, we obtain that
1 − αβ 1 − α d 2 (T n x, T m y) ≤ (1 − β)d 2 (x, y) + (1 − β)α 1 − α d 2 (T n x, y) + βd 2 (x, T m y)− − αd 2 (x, T n x) − β(1 − β) 1 − α d 2 (y, T m y).
By easy computations, one can see that
1 − αβ 1 − α = 1 + α γ n γ m , (1 − β)α 1 − α = α γ n γ m and β(1 − β) 1 − α = 1 − (1 − α) γ n γ m γ n γ m .
Therefore, we have that
1 + α γ n γ m d 2 (T n x, T m y) ≤ (1 − α) γ n γ m d 2 (x, y) + α γ n γ m d 2 (T n x, y) + 1 − (1 − α) γ n γ m d 2 (x, T m y)− − αd 2 (x, T n x) − 1 − (1 − α) γ n γ m γ n γ m d 2 (y, T m y).
Letting α → 1, we get that Divide by γ n to obtain (10), our required inequality.
1 + γ n γ m d 2 (T n x, T m y) ≤ γ n γ m d 2 (T n x, y) + d 2 (x, T m y) − d 2 (x, T n x) − γ n γ m d 2 (y, T m y),
Using the quasi-linearization function defined by (3), the joint (P 2 ) condition can equivalently be expressed as:
1 γ m − −−−− → T n xT m y, − −− → yT m y ≤ 1 γ n − −−−− → T n xT m y, − −− → xT n x ,(11)
for all n, m ∈ N.
Proposition 3.10. Suppose that (T n ) is jointly (P 2 ) with respect to (γ n ). Then for all m, n ∈ N and all w ∈ X,
d(T n w, T m w) ≤ |γ n − γ m | γ n d(w, T n w).
Proof. Let m, n ∈ N. We shall denote, for simplicity, T := T n , U := T m , λ := γ n , µ := γ m . We want to show that for all w ∈ X,
d(T w, U w) ≤ |λ − µ| λ d(w, T w).
If T w = U w, the statement is trivially true. Let w ∈ X be such that T w = U w.
Claim: (λ + µ)d 2 (T w, U w) ≤ (λ − µ)(d 2 (w, T w) − d 2 (w, U w)). Proof of claim: We have that 1 µ − −−− → T wU w, − −− → wU w ≤ 1 λ − −−− → T wU w, − −− → wT w ,
and, by multiplying with (−λ), we get that
− −−− → T wU w, − −− → T ww ≤ λ µ − −−− → T wU w, − −− → U ww .(12)
A simple expansion of ·, · shows that
d 2 (T w, U w) = d 2 (w, U w) − d 2 (w, T w) + 2 − −−− → T wU w, − −− → T ww .(13)
By exchanging the roles of T and U in the above equation, we obtain that
d 2 (U w, T w) = d 2 (w, T w) − d 2 (w, U w) + 2 − −−− → U wT w, − −− → U ww .(14)
Applying (12) and (13) and multiplying (14) by λ µ , we get that
d 2 (T w, U w) ≤ d 2 (w, U w) − d 2 (w, T w) + 2λ µ − −−− → T wU w, − −− → U ww , λ µ d 2 (U w, T w) = λ µ d 2 (w, T w) − λ µ d 2 (w, U w) + 2λ µ − −−− → U wT w, − −− → U ww .
As a consequence, it follows that
1 + λ µ d 2 (T w, U w) ≤ λ µ − 1 (d 2 (w, T w) − d 2 (w, U w)).
Multiply by µ to get the claim.
We distinguish now two cases, according to the sign of λ − µ.
When λ − µ is negative, we obtain, using the claim, that
(λ + µ)d 2 (T w, U w) ≤ (λ − µ)(d 2 (w, T w) − d 2 (w, U w)) = (µ − λ)(d 2 (w, U w) − d 2 (w, T w)) ≤ (µ − λ)((d(w, T w) + d(T w, U w)) 2 − d 2 (w, T w)) = (µ − λ)d(T w, U w)(d(T w, U w) + 2d(w, T w)).
Dividing by d(T w, U w) = 0, we have that
(λ + µ)d(T w, U w) ≤ 2(µ − λ)d(w, T w) + (µ − λ)d(T w, U w), so 2λd(T w, U w) ≤ 2(µ − λ)d(w, T w).
Thus,
d(T w, U w) ≤ µ − λ λ d(w, T w) = |λ − µ| λ d(w, T w),
as required. Now, when λ − µ is positive, we proceed as follows. By the reverse triangle inequality for metric spaces, we have that
d 2 (w, U w) ≥ |d(T w, U w) − d(w, T w)| 2 = d 2 (T w, U w) − 2d(w, T w)d(T w, U w) + d 2 (w, T w).
Applying the claim, we obtain that
(λ + µ)d 2 (T w, U w) ≤ (λ − µ)(d 2 (w, T w) − d 2 (w, U w)) ≤ (λ − µ)(2d(w, T w)d(T w, U w) − d 2 (T w, U w)) = (λ − µ)d(T w, U w)(2d(w, T w) − d(T w, U w)).
As above, one gets that
d(T w, U w) ≤ λ − µ λ d(w, T w) = |λ − µ| λ d(w, T w).
Corollary 3.11. Suppose that (T n ) is jointly (P 2 ) with respect to (γ n ). Then any two mappings of the family have the same set of fixed points.
Proof. It follows from Proposition 3.10 and Lemma 3.1.
Proposition 3.12. Assume that (T n ) is jointly (P 2 ) with respect to (γ n ). Let x ∈ X and (x n ) be given by (8). Then the sequence d(xn,xn+1) γn is nonincreasing.
Proof. Let n ∈ N. By (11),
1 γ n+1 −−−−−−−−−−→ T n x n T n+1 x n+1 , −−−−−−−−−− → x n+1 T n+1 x n+1 ≤ 1 γ n −−−−−−−−−−→ T n x n T n+1 x n+1 , − −−−− → x n T n x n , that is 1 γ n+1 −−−−−−→ x n+1 x n+2 , −−−−−−→ x n+1 x n+2 ≤ 1 γ n −−−−−−→ x n+1 x n+2 , − −−−− → x n x n+1 . Thus, 0 ≤ 1 γ n −−−−−−→ x n+1 x n+2 , − −−−− → x n x n+1 − d 2 (x n+1 , x n+2 ) γ n+1 = γ n+1 1 γ n γ n+1 −−−−−−→ x n+1 x n+2 , − −−−− → x n x n+1 − d 2 (x n+1 , x n+2 ) γ 2 n+1 ≤ γ n+1 d(x n , x n+1 ) γ n · d(x n+1 , x n+2 ) γ n+1 − d 2 (x n+1 , x n+2 ) γ 2 n+1 by (5) = γ n+1 · d(x n+1 , x n+2 ) γ n+1 d(x n , x n+1 ) γ n − d(x n+1 , x n+2 ) γ n+1 . It follows that d(xn+1,xn+2) γn+1 ≤ d(xn,xn+1) γn .
We give now another abstract version of the Proximal Point Algorithm.
Theorem 3.13. Let X be a complete CAT(0) space, T n : X → X for every n ∈ N and (γ n ) be a sequence of positive real numbers satisfying ∞ n=0 γ 2 n = ∞. Assume that the family (T n ) is jointly (P 2 ) with respect to (γ n ) (in particular, (T n ) may be jointly firmly nonexpansive) and that F := n∈N F ix(T n ) = ∅. Let x ∈ X and (x n ) be given by (8).
Then (x n ) ∆-converges to a point in F .
Proof. By Lemma 3.8, each T n satisfies property (P 2 ). We can now apply Theorem 3.5, as conditions (C1) and (C2) follow from Propositions 3.10 and 3.12, respectively.
The case of Hilbert spaces
Assume now that H is a Hilbert space with inner product ·, · . We show next that joint firm nonexpansivity coincides with the joint (P 2 ) condition.
Proposition 3.14. Let (T n ) be a family of self-mappings of H and (γ n ) be a sequence of positive real numbers. Then (T n ) is jointly (P 2 ) with respect to (γ n ) if and only if (T n ) is jointly firmly nonexpansive with respect to (γ n ).
Proof. "⇐" By Proposition 3.9. "⇒" Let m, n ∈ N, x, y ∈ H and α, β ∈ [0, 1] be such that (1 − α)γ n = (1 − β)γ m =: δ. A simple computation yields the following two identities
(1 − α)x + αT n x = T n x + δ γ n (x − T n x) and (1 − β)y + βT m y = T m y + δ γ m (y − T m y).
It follows that
((1 − α)x + αT n x) − ((1 − β)y + βT m y) 2 = (T n x − T m y) + δ γ n (x − T n x) − δ γ m (y − T m y) 2 = T n x − T m y 2 + δ 2 1 γ n (x − T n x) − 1 γ m (y − T m y) 2 + 2δ T n x − T m y, 1 γ n (x − T n x) − 1 γ m (y − T m y) .
In order to show that the right-hand side is greater than or equal to T n x − T m y 2 , which is what we are aiming to prove here, it is sufficient to show that
D := T n x − T m y, 1 γ n (x − T n x) − 1 γ m (y − T m y) ≥ 0. Remark that D = 1 γ n T n x − T m y, x − T n x − 1 γ m T n x − T m y, y − T m y = 1 γ n − −−−− → T n xT m y, − −− → xT n x − 1 γ m − −−−− → T n xT m y, − −− → yT m y by (4)
≥ 0 by (11).
Thus, (T n ) is jointly firmly nonexpansive with respect to (γ n ).
Since ∆-convergence coincides with weak convergence in Hilbert spaces, we get, as an immediate consequence of Theorem 3.13, the following abstract version of the Proximal Point Algorithm. Assume that the family (T n ) is jointly firmly nonexpansive with respect to (γ n ) and that F := n∈N F ix(T n ) = ∅. Let x ∈ H and (x n ) be given by (8).
Then (x n ) converges weakly to a point in F .
We are now in a position to prove that specific instances of the proximal point algorithm satisfy the stronger requirement that their associated families of resolvents are jointly firmly nonexpansive, thus justifying our choice of definitions. Three concrete problems -minimizing convex functions, finding fixed points of nonexpansive mappings and finding zeros of maximally monotone operators -are used to illustrate this fact. We may then apply Theorems 3.13 and 3.15 in order to obtain classical weak convergence results for these iterations.
Minimizers of convex proper lsc functions
In the sequel, X is a complete CAT(0) space and f : X → (−∞, ∞] is a convex, proper, lower semicontinuous (lsc) function. A point x ∈ X is said to be a minimizer of f if f (x) = inf y∈X f (y). The set of minimizers of f is denoted by Argmin(f ).
For any γ > 0, let us denote, following [3],
J γ : X → X, J γ (x) := argmin y∈X f (y) + 1 2γ d 2 (x, y) .
The mapping J γ , defined in the context of CAT(0) spaces by Jost [25], is called the (Moreau-Yosida) resolvent or the proximal mapping of f of order γ. We recall in the following proposition some well-known properties proven in [25].
Proposition 3.16. Let γ > 0. Then (i) F ix(J γ ) = Argmin(f ).
(ii) J γ is nonexpansive.
(iii) For all x ∈ X and all t ∈ [0, 1], the following holds:
J (1−t)γ ((1 − t)x + tJ γ (x)) = J γ (x).
Proposition 3.17. Let (γ n ) be a sequence of positive real numbers. Then the family (J γn ) is jointly firmly nonexpansive with respect to (γ n ).
Proof. Let m, n ∈ N, x, y ∈ X and α, β ∈ [0, 1] be such that (1 − α)γ n = (1 − β)γ m =: δ. Applying Proposition 3.16, we get that
d(J γn x, J γm y) = d(J (1−α)γn ((1 − α)x + αJ γn x), J (1−β)γm ((1 − β)y + βJ γm y)) = d(J δ ((1 − α)x + αJ γn x), J δ ((1 − β)y + βJ γm y)) ≤ d((1 − α)x + αJ γn x, (1 − β)y + βJ γm y).
As a consequence of Theorem 3.13, we get the following ∆-convergence result.
Theorem 3.18. Assume that Argmin(f ) = ∅ and let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞. For any x ∈ X, define the sequence (x n ), starting with x, by x 0 := x, x n+1 := J γn x n for all n ∈ N.
Then (x n ) ∆-converges to a minimizer of f .
Proof. For all n ∈ N, put T n := J γn . By Proposition 3.16.(i), F ix(T n ) = Argmin(f ) for all n ∈ N. Furthermore, by Proposition 3.17, the family (T n ) is jointly firmly nonexpansive with respect to (γ n ). Hence, we may apply Theorem 3.13 to derive our conclusion.
The above theorem is a slightly weaker variant (with a completely different proof) of a result due to Bačák [3, Theorem 1.4], since one uses here the stronger assumption ∞ n=0 γ 2 n = ∞ instead of ∞ n=0 γ n = ∞. We point out that an analysis of Bačák's original statement from the point of view of proof mining was previously carried out in [41,42].
Fixed points of nonexpansive mappings
We proceed now to give another application. Let X be a complete CAT(0) space and T : X → X be a nonexpansive mapping.
For x ∈ X and γ > 0 we define
G T,x,γ : X → X, G T,x,γ (y) := 1 1 + γ x + γ 1 + γ T y.
It is easy to see that this mapping is Lipschitz with constant γ 1+γ ∈ (0, 1). Therefore it admits a unique fixed point, which we shall denote by R T,γ x. We have thus defined a mapping R T,γ : X → X, called the resolvent of order γ of T , which satisfies, for any x ∈ X,
R T,γ x = 1 1 + γ x + γ 1 + γ T R T,γ x.(16)
We immediately obtain that F ix(R T,γ ) = F ix(T ) for all γ > 0.
Proposition 3.19. Let (γ n ) be a sequence of positive real numbers. Then the family (R T,γn ) is jointly firmly nonexpansive with respect to (γ n ).
Proof. Let m, n ∈ N, x, y ∈ X and α, β ∈ [0, 1] be such that (1 − α)γ n = (1 − β)γ m =: δ. Denote
u := (1 − α)x + αR T,γn x, v := (1 − β)y + βR T,γm y.
Then we have to show that
d(R T,γn x, R T,γm y) ≤ d(u, v).(17)
Using (16) and the definition of u, we may apply [1, Lemma 2.4.(iii)] to obtain that
R T,γn x = (1 − ν)u + νT R T,γn x, where ν := (1 − α) γn 1+γn 1 − α · γn 1+γn = δ 1 + δ .
We remark that ν = 1. Also note that, while the statement of [1, Lemma 2.4.(iii)] requires the four points to be pairwise distinct, its conclusion is trivial to show in the case of some of them are equal. We show similarly that
R T,γm y = (1 − ν)v + νT R T,γm y.
Applying (2) and the nonexpansivity of T , we get that
d(R T,γn x, R T,γm y) = d((1 − ν)u + νT R T,γn x, (1 − ν)v + νT R T,γm y) ≤ (1 − ν)d(u, v) + νd(T R T,γn x, T R T,γm y) ≤ (1 − ν)d(u, v) + νd(R T,γn x, R T,γm y).
It follows immediately that (17) holds.
As an immediate application of Theorem 3.13, we get Theorem 3.20. Assume that F ix(T ) = ∅ and let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞. For any x ∈ X, define the sequence (x n ) by
x 0 := x, x n+1 := R T,γn x n for all n ∈ N.
Then (x n ) ∆-converges to a fixed point of T .
We have therefore obtained a new proof of [6, Proposition 1.5].
Zeros of maximally monotone operators
In the following, H is a Hilbert space with inner product ·, · and A : H → 2 H is a maximally monotone operator. We denote by zer(A) the set of zeros of A. Given γ > 0, the resolvent J γA of order γ of A is defined by
J γA = (id H + γA) −1 .
It is well-known (see, e.g., [9]) that, for every γ > 0, J γA : H → H is a single-valued firmly nonexpansive mapping satisfying F ix(J γA ) = zer(A).
Proposition 3.21. Let (γ n ) be a sequence of positive real numbers. Then the family (J γnA ) is jointly firmly nonexpansive with respect to (γ n ).
Proof. By Proposition 3.14, we can prove, equivalently, that the family (J γnA ) is jointly (P 2 ) with respect to (γ n ). Let n, m ∈ N and x, y ∈ H. It is easy to see that
1 γ n (x − J γnA x) ∈ A(J γnA x) and 1 γ m (y − J γmA y) ∈ A(J γmA y).
By the monotonicity of A we obtain that
J γnA x − J γmA y, 1 γ n (x − J γnA x) − 1 γ m (y − J γmA y) ≥ 0, therefore 1 γ m J γnA x − J γmA y, y − J γmA y ≤ 1 γ n J γnA x − J γmA y, x − J γnA x .
As a consequence of Theorem 3.15 we derive the following well-known weak convergence result (see, e.g., [9,Theorem 23.41.(i)]).
Then (x n ) converges weakly to a zero of A.
4 Uniformly firmly nonexpansive and uniformly (P 2 ) mappings
As mentioned in the Introduction, if one wants to obtain strong convergence for the proximal point algorithm, one usually imposes a uniformity condition on the object that is being optimized. The aim of this section is to give such a condition in the abstract setting from the previous section. For a single mapping defined on a Hilbert space, this condition was also considered in [8,Section 3.4], under the name of uniform firm nonexpansivity with a given modulus. We will now generalize this notion to CAT(0) spaces and show how it may be applied for the families of mappings that arise from two of the concrete problems just discussed. Let X be a CAT(0) space, T : X → X, C ⊆ X be a nonempty subset of X and ϕ : [0, ∞) → [0, ∞) be an increasing function which vanishes only at 0. Definition 4.1. We say that T is (i) uniformly firmly nonexpansive on C with modulus ϕ if T (C) ⊆ C and, for all x, y ∈ C and all t ∈ [0, 1], the following holds:
d 2 (T x, T y) ≤ d 2 ((1 − t)x + tT x, (1 − t)y + tT y) − 2(1 − t)ϕ(d(T x, T y)).(19)
(ii) uniformly (P 2 ) on C with modulus ϕ if T (C) ⊆ C and, for any x, y ∈ C,
2d 2 (T x, T y) ≤ d 2 (x, T y) + d 2 (y, T x) − d 2 (x, T x) − d 2 (y, T y) − 2ϕ(d(T x, T y)).(20)
Obviously, if T is uniformly firmly nonexpansive (resp. (P 2 )) on C, then its restriction T | C : C → C is firmly nonexpansive (resp. (P 2 )). We remark also that the uniform (P 2 ) condition may be expressed using the quasi-linearization function as follows:
−−−→ T xT y, − − → yT y ≤ −−−→ T xT y, − − → xT x − ϕ(d(T x, T y)).(21)
Proposition 4.2. Suppose that T is uniformly firmly nonexpansive on C with modulus ϕ. Then T is uniformly (P 2 ) on C with the same modulus ϕ.
Proof. Let x, y ∈ C and t ∈ (0, 1). As in the proof of Proposition 3.9, we apply the uniform firm nonexpansivity condition and (1) twice to get that
d 2 (T x, T y) ≤ (1 − t) 2 d 2 (x, y) + t(1 − t)d 2 (T x, y) + t(1 − t)d 2 (x, T y) + t 2 d 2 (T x, T y) − t(1 − t)d 2 (x, T x) − t(1 − t)d 2 (y, T y) − 2(1 − t)ϕ(d(T x, T y)).
Divide now by 1 − t = 0 to obtain that
(1 + t)d 2 (T x, T y) ≤ (1 − t)d 2 (x, y) + td 2 (T x, y) + td 2 (x, T y) − td 2 (x, T x) − td 2 (y, T y) − 2ϕ(d(T x, T y)),
By taking t → 1 we get what is needed.
As in the non-uniform case, for Hilbert spaces, the two notions coincide.
Proposition 4.3.
Assume that X is a Hilbert space and that T is uniformly (P 2 ) on C with modulus ϕ. Then T is uniformly firmly nonexpansive on C with the same modulus ϕ.
Proof. Let x, y ∈ C and t ∈ [0, 1]. By the hypothesis, (21) and (4), we immediately get that
T x − T y, (x − T x) − (y − T y) ≥ ϕ( T x − T y ).
Consequently,
((1 − t)x + tT x) − ((1 − t)y + tT y) 2 = (T x − T y) + (1 − t)((x − T x) − (y − T y)) 2 = T x − T y 2 + (1 − t) 2 (x − T x) − (y − T y) 2 + 2(1 − t) T x − T y, (x − T x) − (y − T y) ≥ T x − T y 2 + 2(1 − t)ϕ( T x − T y ).
The following properties will be useful in the proof of our main quantitative result, Theorem 5.1.
Lemma 4.4. Let T be uniformly (P 2 ) on C with modulus ϕ. Then
ϕ(d(T x, z)) ≤ d(x, T x)d(T x, z),
for all x ∈ C and all z ∈ C ∩ F ix(T ).
Proof. Applying (20) for y := z, we get that T x, z)).
d 2 (T x, z) ≤ d 2 (x, z) − d 2 (x, T x) − 2ϕ(d(
It follows that
2ϕ(d(T x, z)) ≤ d 2 (x, z) − d 2 (x, T x) − d 2 (T x, z) ≤ (d(x, T x) + d(T x, z)) 2 − d 2 (x, T x) − d 2 (T x, z) = 2d(x, T x)d(T x, z).
As an immediate consequence, we obtain (ii) Applying (23) with u := x, v := J γ y and then with u := y, v := J γ x, we get that
d 2 (J γ x, J γ y) ≤ d 2 (x, J γ y) − d 2 (x, J γ x) − 2γ(f (J γ x) − f (J γ y)) − 2γψ(d(J γ x, J γ y)), d 2 (J γ y, J γ x) ≤ d 2 (y, J γ x) − d 2 (y, J γ y) − 2γ(f (J γ y) − f (J γ x)) − 2γψ(d(J γ y, J γ x)).
Summing up, we obtain
2d 2 (J γ x, J γ y) + d 2 (x, J γ x) + d 2 (y, J γ y) ≤ d 2 (x, J γ y) + d 2 (y, J γ x) − 4γψ(d(J γ y, J γ x)).
By (6), we have that
d 2 (x, J γ y) + d 2 (y, J γ x) ≤ d 2 (x, y) + d 2 (J γ x, J γ y) + d 2 (x, J γ x) + d 2 (y, J γ y),
from where we get our conclusion.
Proposition 4.7. Suppose that f is uniformly convex on C with modulus ψ. Let γ > 0 be such that J γ (C) ⊆ C. Then J γ is uniformly firmly nonexpansive on C with modulus 2γψ.
Proof. Let x, y ∈ C and t ∈ [0, 1]. Denote u := (1 − t)x + tJ γ x and v := (1 − t)y + tJ γ y.
By Proposition 3.16.(iii), we have that J (1−t)γ (u) = J γ x and J (1−t)γ (v) = J γ y. We get that
d 2 (J γ x, J γ y) = d 2 (J (1−t)γ (u), J (1−t)γ (v)) ≤ d 2 (u, v) − 4(1 − t)γψ(d(J (1−t)γ (u), J (1−t)γ (v)) by (24) = d 2 (u, v) − 4(1 − t)γψ(d(J γ x, J γ y).
Uniformly monotone operators
Fix now a Hilbert space H and C ⊆ H a nonempty subset. Let A : H → 2 H be a multi-valued operator and ϕ : [0, ∞) → [0, ∞) be an increasing function which vanishes only at 0. Then A is said to be uniformly monotone on C with modulus ϕ (see, e.g. [9, Definition 22.1]) if for all x, y ∈ C and u, v ∈ H with u ∈ A(x) and v ∈ A(y) we have that
x − y, u − v ≥ ϕ( x − y ).
Proposition 4.8. Assume that A is a maximally monotone operator which is uniformly monotone on C with modulus ϕ. Let γ > 0 be such that J γA (C) ⊆ C. Then J γA is uniformly firmly nonexpansive on C with modulus γϕ.
Proof. Let x, y ∈ C. As in the proof of Proposition 3.21, we get that
J γA x − J γA y, x − J γA x ≥ J γA x − J γA y, y − J γA y + γϕ( J γA x − J γA y ).
Thus, J γA is uniformly (P 2 ) on C with modulus γϕ. Apply now Proposition 4.3.
A rate of convergence for the uniform case
We shall now show that in the presence of the uniformity constraint described in the previous section, one indeed gets strong convergence of the proximal point algorithm in its most abstract form, given by Theorem 3.5. Moreover, as announced in the Introduction, we use the tools of proof mining to derive that result from a stronger one which is highly uniform and also quantitativei.e. also yields a rate of convergence for the sequence.
Let us recall that if (a n ) n∈N is a convergent sequence in a metric space (X, d) with lim n→∞ a n = a, then a rate of convergence of (a n ) is a function Φ : N → N such that for all k ∈ N and all n ≥ Φ(k), d(a n , a) ≤ 1 k + 1 .
Another needed quantitative notion will be that of a rate of divergence for a given diverging series ∞ n=0 b n = ∞, which is a function θ : N → N such that for all K ∈ N we have that
θ(K) n=0 b n ≥ K.
In this section, X is a complete CAT(0) space and T n : X → X for every n ∈ N. We assume that the family (T n ) has common fixed points and set
F := n∈N F ix(T n ) = ∅. Furthermore, ϕ : [0, ∞) → [0, ∞)
is an increasing function which vanishes only at 0 and (γ n ) is a a sequence in (0, ∞) such that ∞ n=0 γ 2 n = ∞ with rate of divergence θ. We can state now the main result of this section.
Theorem 5.1. Let b ∈ N, p ∈ F and C be the closed ball of center p and radius b. Assume that, for all n ∈ N, T n is uniformly (P 2 ) on C with modulus γ n ϕ.
For every x ∈ C, let (x n ) be defined by
x 0 := x, x n+1 := T n x n for all n ∈ N.(26)
Suppose that (C2) holds, that is, the sequence d(xn,xn+1) γn is nonincreasing. Then C ∩ F = {p} and (x n ) converges strongly to p with rate of convergence Ψ b,θ,ϕ , given by
with Σ b,θ (k) := θ(b 2 (k + 1) 2 ).
Before proving the theorem, let us give some consequences.
Proposition 5.2. Assume that H is a Hilbert space and A : H → 2 H is a maximally monotone operator with zer(A) = ∅. Let b ∈ N, p ∈ zer(A) and C be the closed ball of center p and radius b. Suppose that A is uniformly monotone on C with modulus ϕ. For any x ∈ C, let (x n ) be defined by (18). Then p is the unique zero of A in C and (x n ) converges strongly to p with rate of convergence Ψ b,θ,ϕ , given by (27).
Proof. We use the notation from Subsection 3.4. Since, for every n ∈ N, F ix(J γnA ) = zer(A) = ∅ and J γnA is nonexpansive, it is obvious that J γnA (C) ⊆ C. Thus, by Proposition 4.8, every J γnA is uniformly firmly nonexpansive on C with modulus γ n ϕ. Furthermore, (C2) is satisfied, by Propositions 3.21 and 3.12. An application of Theorem 5.1 for the family (J γnA ) yields the result.
The above proposition is a quantitative uniform version of Theorem 3.22. If we forget about the quantitative features, we get immediately the following well-known result (see, e.g., [9,Theorem 23.41.(ii)]). Corollary 5.3. Assume that H is a Hilbert space and A : H → 2 H is a maximally monotone operator with zer(A) = ∅. Let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞, x ∈ X and (x n ) be defined by (18). Suppose that A is uniformly monotone on every bounded subset of H.
Then (x n ) converges strongly to the unique zero of A.
The following result is a quantitative uniform version of Theorem 3.18.
Proposition 5.4. Assume that X is a complete CAT(0) space and f : X → (−∞, ∞] is a convex, proper, lsc function that attains its minimum. Let b ∈ N, p ∈ Argmin(f ) and C be the closed ball of center p and radius b. Suppose that f is uniformly convex on C with modulus ψ. For any x ∈ C, let (x n ) be defined by (15). Then p is the unique minimizer of f in C and (x n ) converges strongly to p with rate of convergence Ω b,θ,ψ := Ψ b,θ,2ψ .
Proof. By Proposition 3.16, F ix(J γn ) = Argmin(f ) = ∅ and J γn is nonexpansive, hence J γn (C) ⊆ C for all n. Use now Proposition 4.7 to get that every J γn is uniformly firmly nonexpansive on C with modulus 2γ n ψ. Since (C2) is satisfied (by Propositions 3.17 and 3.12), we can apply Theorem 5.1 for the family (J γn ) to get the result.
Proof of Theorem 5.1
Apply the fact that C ∩ F = ∅ and Corollary 4.5 to conclude that C ∩ F = {p}. Since, by Lemma 3.2, (x n ) is Fejér monotone with respect to F , we have that d(x n , p) ≤ b for all n ∈ N.
Assume that for all n ∈ {0, . . . , Σ b,θ (k)} we have that d(xn,xn+1) γn > 1 k+1 . It follows that We get a contradiction with (28). Thus, there exists N ≤ Σ b,θ (k) such that d(xN ,xN+1) γn ≤ 1 k+1 . By (C2), the claim follows.
Let k ∈ N and n ≥ Ψ b,θ,ϕ (k). Set n ′ := n − 1. Then n ′ ≥ Σ b,θ 2b ϕ( 1 k+1 )
, hence, by the claim,
d(x n ′ , x n ′ +1 ) γ n ′ ≤ 1 2b ϕ( 1 k+1 ) + 1 ≤ 1 2b ϕ( 1 k+1 ) = 1 2b · ϕ 1 k + 1 .
Applying Lemma 4.4 for x := x n ′ , z := p, T := T n ′ (and hence ϕ becomes γ n ′ ϕ), we get that γ n ′ ϕ(d(T n ′ x n ′ , p)) ≤ d(x n ′ , T n ′ x n ′ )d(T n ′ x n ′ , p).
Since x n ′ +1 = T n ′ x n ′ , it follows that ϕ(d(x n ′ +1 , p)) ≤ d(x n ′ , x n ′ +1 ) γ n ′ · d(x n ′ +1 , p) ≤ 1 2b · ϕ 1 k + 1 · b = 1 2 ϕ 1 k + 1 .
If d(x n ′ +1 , p) > 1 k+1 , then ϕ(d(x n ′ +1 , p)) ≥ ϕ 1 k+1 > 1 2 ϕ 1 k+1 , since ϕ is increasing and ϕ 1 k+1 = 0. We have got a contradiction. Thus, we must have
d(x n ′ +1 , p) ≤ 1 k + 1 ,
which is what we wanted to show, since n = n ′ + 1.
so γ n γ m d 2 (T n x, T m y) + d 2 (y, T m y) − d 2 (T n x, y) ≤d 2 (x, T m y) − d 2 (x, T n x) − d 2 (T n x, T m y).
Theorem 3 . 15 .
315Let H be a Hilbert space, T n : H → H for every n ∈ N and (γ n ) be a sequence of positive real numbers satisfying ∞ n=0 γ 2 n = ∞.
Theorem 3 . 22 .
322Assume that zer(A) = ∅ and let (γ n ) be a sequence of positive real numbers such that ∞ n=0 γ 2 n = ∞. For any x ∈ H, define the sequence (x n ) byx 0 := x, x n+1 := J γnA x n for all n ∈ N.
Claim: Σ b,θ is a rate of convergence of the sequence d(xn,xn+1) γn towards 0. Proof of claim: We reason as in the proof of [34, Lemma 8.3.(ii)]. Let k ∈ N. By the proof of Lemma 3.3, ∞ n=0 d 2 (x n , x n+1 ) ≤ b 2 .
contradiction. Hence, there exists M ∈ {0, . . . , N } such that d(xM ,xM+1) ≤ ε for all n ≥ M .Proposition 3.4. Assume that (C1) and (C2) are satisfied. Then for all m ∈ N,γM
≤ ε. Since, by (C2),
the sequence d(xn,xn+1)
γn
is nonincreasing, we get that d(xn,xn+1)
γn
Acknowledgements:Adriana Nicolae was partially supported by DGES (MTM2015-65242-C2-1-P). She would also like to acknowledge the Juan de la Cierva -Incorporación Fellowship Program of the Spanish Ministry of Economy and Competitiveness. Laurenţiu Leuştean and Andrei Sipoş were partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS -UEFISCDI, project number PN-II-ID-PCE-2011-3-0383.Corollary 4.5. If T is uniformly (P 2 ) on C with modulus ϕ, the set C ∩ F ix(T ) is at most a singleton.Proof. Let x, z ∈ C ∩ F ix(T ). Applying Lemma 4.4, we obtain that ϕ(d(x, z)) = 0. Since ϕ vanishes only at 0, we must have that x = z.We shall now check that the conditions introduced above are indeed satisfied by nontrivial particular cases in the concrete instances that we have presented.Uniformly convex functionsLet X be a complete CAT(0) space and f : X → (−∞, ∞] be a proper, convex, lsc function. We use the notation from Subsection 3.2.Let ψ : [0, ∞) → [0, ∞) be an increasing function which vanishes only at 0 and C ⊆ X be nonempty. Recall that f is said to be uniformly convex on C with modulus ψ if for all x, y ∈ C and all t ∈ [0, 1], the following holds:Lemma 4.6. Assume that f is uniformly convex on C with modulus ψ. Let γ > 0 be such that J γ (C) ⊆ C. Then:(ii) for all x, y ∈ C, d 2 (J γ x, J γ y) ≤ d 2 (x, y) − 4γψ(d(J γ x, J γ y)).Proof. (i) By the definition of J γ , we have that for all p ∈ X,Let t ∈ (0, 1) be arbitrary. Note that, by(1),Applying the first inequality (multiplied by γ) for p := (1 − t)v + tJ γ u, (25) and the uniform convexity of f on C (since v, J γ u ∈ C), we get thatDivide by 1 − t = 0 and let t → 1 to obtain that γ(f (J γ u) − f (v)) ≤ 1 2 (d 2 (u, v) − d 2 (u, J γ u) − d 2 (v, J γ u) − 2γψ(d(v, J γ u))),hence(23).
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|
[] |
[
"PRIMITIVE TRANSFORMATION SHIFT REGISTERS OVER FINITE FIELDS",
"PRIMITIVE TRANSFORMATION SHIFT REGISTERS OVER FINITE FIELDS"
] |
[
"Ambrish Awasthi ",
"Rajendra K Sharma "
] |
[] |
[] |
We consider the problem of existence and enumeration of primitive TSRs of order n over any finite field. Here we prove the existence of primitive TSRs of order two over finite fields of characteristic 2 and establish an equivalence between primitive TSRs and primitive polynomials of special form. A conjecture regarding the existence of these special type of primitive polynomials is submitted by us along with some experimental verification. Further we have attempted to enumerate primitive TSRs of order 2 over finite fields of characteristic 2. Finally we give a general search algorithm for primitive TSRs of odd order over any finite field and in particular of order two over fields of characteristic 2.
|
10.1142/s0219498819501718
|
[
"https://arxiv.org/pdf/1612.05367v1.pdf"
] | 119,592,330 |
1612.05367
|
6dd7a5fbd7a2a1b9b4b7197c925af52151dc7c8b
|
PRIMITIVE TRANSFORMATION SHIFT REGISTERS OVER FINITE FIELDS
16 Dec 2016
Ambrish Awasthi
Rajendra K Sharma
PRIMITIVE TRANSFORMATION SHIFT REGISTERS OVER FINITE FIELDS
16 Dec 2016
We consider the problem of existence and enumeration of primitive TSRs of order n over any finite field. Here we prove the existence of primitive TSRs of order two over finite fields of characteristic 2 and establish an equivalence between primitive TSRs and primitive polynomials of special form. A conjecture regarding the existence of these special type of primitive polynomials is submitted by us along with some experimental verification. Further we have attempted to enumerate primitive TSRs of order 2 over finite fields of characteristic 2. Finally we give a general search algorithm for primitive TSRs of odd order over any finite field and in particular of order two over fields of characteristic 2.
Introduction
Linear feedback shift registers (LFSRs) are systems consisting of a homogeneous linear recurrence relation over F q . They have wide applications in cryptography and are particularly useful for generating pseudorandom sequences in stream ciphers refer [15,9]. Sequences with maximum period are a necessary prerequisite for cryptographic applications. LFSRs which generate such sequences are known as primitive LFSRs. The characteristic polynomial of such LFSRs are primitive in nature. The cardinality of primitive LFSRs of order n over F q is given by (1) φ(q n − 1) n ,
where φ is Euler's totient function. Similarly the number of irreducible LFSRs (whose characteristic polynomials are irreducible) of order n over a finite field F q is given by
(2) 1 n d|n µ (d) q n d ,
where µ is the Möbius function. Zeng et. al [4] considered a generalization of LFSR which they called as σ-LFSR. These are word-oriented linear feedback shift registers, involving linear recurrence relation over F 2 m , with matrix coefficients coming from M m (F 2 ). They also gave a conjectural formula for the number of primitive σ-LFSRs of order n over F 2 m [4]. A further generalisation of σ-LFSR to F q m was done by Ghorpade and Hasan [12] who extended the conjecture formula over F q m . [12] states this number as
(3) φ(q mn − 1) mn q m(m−1)(n−1) m−1 i=1 (q m − q i ).
Refer to [12,13,3] for progress as well as complete proof of the conjecture. Further refer to [13,3,11] for the cardinality of irreducible σ-LFSRs which is given by
(4) 1 mn q m(m−1)(n−1) m−1 i=1 (q m − q i ) d|mn µ (d) q mn d .
Here we focus on transformation shift registers (TSRs) which are an extremely important and useful subclass of σ-LFSRs. TSRs find their origin in a problem posed by Bart Preneel [1] as a challenge to design fast and secure LFSRs which use the parallelism offered by the word operations of modern processors. The problem was addressed by the introduction of TSRs. Tsaban and Vishne [2] proved them to be faster and more efficient in software implementation than σ LFSRs. Refer to Dewar and Panario [6,7] for further developments on the theory of TSRs. Like σ LFSRs, TSRs are also classified as irreducible and primitive based on their characteristic polynomial. A study of irreducible TSRs was carried by Ram [11] who considered the problem of enumerating TSRs over a finite field and gave an explicit formula for the number of irreducible TSRs of order two. The problem was further investigated by Sartaj and Cohen [10] who gave an asymptotic formula for the number of irreducible TSRs in some special cases.So we see that some significant progress has been made on irreducible TSRs but the same cannot be said about primitive TSRs. In the context of stream ciphers, we are again basically interested in TSR sequences with maximum period i.e primitive TSRs. Answers to questions regarding cardinality, existence, construction etc of primitive TSRs are challenging and still remain elusive. In our submission we concentrate on these aspects. We have made here an attempt to address the problem of existence and generation of primitive TSRs of order n over F q m by establishing an equivalence between primitive TSRs and primitive polynomials of special type. These primitive polynomials serve as building blocks for primitive TSRs. We give a focussed search algorithm for generating primitive TSRs. A conjecture regarding the existence of these special type of primitive polynomials has been proposed by us along with some experimental results in its support. We give an explicit proof for the existence of primitive TSRs of order 2 over F 2 m . Finally an attempt has been made by us to give the cardinality of primitive TSRs of order 2 over F 2 m along with some bounds for primitive TSRs in general.
Preliminaries
We will be using the following notations throughout the paper. Let F q denote the finite field with q elements. F q [X] is the ring of polynomials with coefficients in F q . For every set C let |C| denote the cardinality of C. The set of all d × d matrices with entries in F q is denoted by M d (F q ).
Throughout the paper we fix positive integers m and n, and a vector space basis {α 0 , . . . , α m−1 } of F q m over F q . There exists a vector space isomorphism from F q m ←→ F m q such that s −→ s. here s denotes the corresponding co-ordinate vector (s 0 , . . . , s m−1 ) of s. Elements of F m q may be thought of as row vectors and so sC is a well-defined element of F m q for any s ∈ F m q and C ∈ M m (F q ). We now recall from [14] and [11] some definitions and results concerning transformation shift registers.
Definition 2.1. A polynomial f (X) ∈ F q [X]
of degree n is said to be a primitive polynomial if its root α generates the cyclic group F * q n , consisting of non zero elements of F q n . Definition 2.2. Let c 0 , c 1 , . . . , c n−1 ∈ F q and A ∈ M m (F q ). Given any n-tuple (s 0 , . . . , s n−1 ) of elements of F q m , let (s i ) ∞ i=0 denote the infinite sequence of elements of F q m determined by the following linear recurrence relation:
s i+n = s i (c 0 A) + s i+1 (c 1 A) + · · · + s i+n−1 (c n−1 A) i = 0, 1, . . . .(5)
The system (5) is a transformation shift register (TSR) of order n over F q m , while the sequence (s i ) ∞ i=0 is the sequence generated by the TSR (5). • The n-tuple (s 0 , s 1 , . . . , s n−1 ) is the initial state of the TSR.
• The polynomial I m X n − (c n−1 A)X n−1 − · · · − (c 1 A)X − (c 0 A) with matrix
coefficients is the tsr-polynomial of the TSR (5). Here I m denotes the m × m identity matrix over F q . • The sequence (s i ) ∞ i=0 is ultimately periodic if there are integers r, n 0 with r ≥ 1 and n 0 ≥ 0 such that s j+r = s j for all j ≥ n 0 .
• The least positive integer r with this property is the period of (s i ) ∞ i=0 and the corresponding least nonnegative integer n 0 is the preperiod of (s i ) ∞ i=0 . The sequence (s i ) ∞ i=0 is periodic if its preperiod is 0. We can associate a block companion matrix T with a TSR definition given in (5) as follows
(6) T = 0 0 0 . . 0 0 c 0 A I m 0 0 . . 0 0 c 1 A . . . . . . . . . . . . . . . . 0 0 0 . . I m 0 c n−2 A 0 0 0 . . 0 I m c n−1 A ,
where c 0 , c 1 , . . . , c n−1 ∈ F q , A ∈ M m (F q ) and 0 indicates the zero matrix in M m (F q ). The set of all such (m, n)-block companion matrices T over F q is denoted by TSR(m, n, q). The block companion matrix (6) is the state transition matrix for the TSR (5). Indeed, the k-th state S k := (s k , s k+1 , . . . , s k+n−1 ) ∈ F n q m of the TSR (5) is obtained from the initial state S 0 := (s 0 , s 1 , . . . , s n−1 ) ∈ F n q m by S k = S 0 T k , for any k ≥ 0.
Using a Laplace expansion or a suitable sequence of elementary column operations, we conclude that if T ∈ TSR(m, n, q) is given by (6), then det T = ± det(c 0 A). Consequently,
(7) T ∈ GL mn (F q ) ⇐⇒ c 0 = 0 and A ∈ GL m (F q ).
where GL m (F q ) is the general linear group of all m × m nonsingular matrices over F q . We denote here the intersection TSR(m, n, q) ∩ GL mn (F q ) by TSR * (m, n, q). Elements of TSR * (m, n, q) are exactly the state transition matrices of periodic TSRs of order n over F q m [14,Prop. 4]. It follows from (6) that T ∈ TSR * (m, n, q) iff T is of the form
(8) 0 0 0 . . 0 0 B I m 0 0 . . 0 0 c 1 B . . . . . . . . . . . . . . . . 0 0 0 . . I m 0 c n−2 B 0 0 0 . . 0 I m c n−1 B ,
where c 1 , . . . , c n−1 ∈ F q and B ∈ GL m (F q ). Henceforth we deal with periodic TSRs only, that is, a TSR of the form (8). The map
(9) Ψ : M mn (F q ) −→ F q [x]
defined by Ψ(T ) := det(XI mn − T ) will be referred to as the characteristic map. The characteristic polynomial of T is given by [14, Lemma 1]
(10) Ψ(T ) = det(X n I m − g T (X)B) where g T (X) = 1 + c 1 X + c 2 X + ...c n−1 X n−1 ∈ F q [X]
. We see that T is uniquely determined by g T (X) and B. For every matrix A we denote by Ψ A (X) the characteristic polynomial of A. It follows from (10) that for T ∈ TSR * (m, n, q)
(11) Ψ T (X) = g T (X) m Ψ B ( X n g T (X) )
thus f (X) ∈ Ψ(TSR * (m, n, q)) iff f (X) can be expressed in the form
(12) g(X) m h( x n g(X) )
for some monic polynomial h(X) ∈ F q [X] of degree m with h(0) = 0 and g(X) ∈ F q [X] of degree at most n − 1 with g(0) = 1. When f (X) ∈ Ψ(TSR * (m, n, q)) is a primitive polynomial then the representation (12) is unique and is said to be (m, n) decomposition of f (X) [11].
Primitive TSRs
A TSR is primitive if its characteristic polynomial is primitive.The set of primitive TSRs is denoted by TSRP(m, n, q) and the set of primitive polynomials in F q [X] of degree d is denoted by P(d, q).
Then the characteristic map
Ψ : M mn (F q ) −→ F q [X] defined by Ψ(T ) := det(XI mn − T ),
if restricted to the set TSRP(m, n, q) yields the map Ψ P : TSRP(m, n, q) −→ P(mn, q).
It was noted in [11] that the map Ψ P is not surjective in general. Denote the characteristic polynomial of A ∈ M mn (F q ) by Ψ A (X)
Lemma 3.1. Let η : M m (F q ) −→ F q [X] be defined by η(A) := det(XI m − A).
Then, for every p(X) ∈ P(m, q), we have,
η −1 (p(X)) = m−1 i=1 (q m − q i ). Proof: [5, Theorem 2].
Primitive TSRs of odd order n over F q m , q ≥ 3 and m ≥ 2.
Theorem 3.2. The number of primitive TSRs of odd order n over F q m where q ≥ 3 and m ≥ 2 is given by
|TSRP(m, n, q)| = |Ψ P (TSRP(m, n, q))| m−1 i=1 (q m − q i ).
Proof: Let us assume that f (X) ∈ Ψ P (TSRP(m, n, q)) is the characteristic polynomial of T ∈ TSRP(m, n, q) i.e Ψ T (X) = f (X) then f (X) can be uniquely expressed in the form (equation 12)
(13) g(X) m h X n g(X) T ∈ TSRP(m, n, q) =⇒ f (X) is primitive =⇒ h(X) is primitive, (refer [11]), where h(X) ∈ F q [X] ofm−1 i=1 (q m − q i ) hence proved.
Let P q (m, n) denote the set of primitive polynomials of the form X n − µg(X)
∈ F q m [X] where µ is a primitive element of F q m , g(X) ∈ F q [X]
such that g(0) = 1 and deg g(X) ≤ (n − 1).
Theorem 3.3. | TSRP(m, n, q)| = |Pq(m,n)| m |GLm(Fq)| q m −1
where n is odd, q ≥ 3 and m ≥ 2
Proof: We will prove the above results along the lines of the proof of [11,Theorem 6] Define Ω q (m, n) := Ψ P (TSRP(m, n, q)).
By theorem 3.2 | TSRP(m, n, q)| = |Ω q (m, n)| GL m (F q ) q m − 1 .
Define a map
Φ : P q (m, n) −→ F q m [X] by Φ(X n − µg(X)) := m−1 i=0 (X n − µ q i g(X))).
The product on the right is (m, n) decomposable. Let β be a root of X n − µg(X) in the extension field F q mn then the minimal polynomial of β over F q is Φ(X n −µg(X)).
Thus Φ(X n − µg(X)) is primitive in F q [X]. Since Ω q (m, n) is precisely the set of primitive (m, n) decomposable polynomials in F q [X], it follows that Φ(P q (m, n)) ⊆ Ω q (m, n).
Claim is Φ(P q (m, n)) = Ω q (m, n).
Let f (x) ∈ Ω q (m, n). since f is primitive, f has a unique (m, n) decomposition [11, theorem 3] say f (X) = g(X) m h( X n g(X) ). f (X) is primitive =⇒ h(X) is primitive in F q [X] and if µ is a root of h(X) in F q m , then Φ(X n − µg(X)) = f (X). Now |Φ −1 (f )| = m for each f ∈ Ω q (m, n) and therefore. |Ω q (m, n)| = |P q (m, n)| m .
Primitive TSRs of order n over F 2 m , m ≥ 2.
Existence of primitive TSRs
We denote by n odd whenever n is taken to be odd positive integer.
Let f (X) ∈ P q (m, n odd ) =⇒ f (X) ∈ F q m [X] and f (X) = X n −µg(X) where µ is a primitive element of F q m , g(X) ∈ F q [X] such that g(0) = 1 and deg g(X) ≤ (n − 1). Now Consider the reciprocal polynomial of f (X) which is of the form h(X) + µ −1 where h(X) ∈ F q [X], h(0) = 0 and µ −1 ∈ F q m is a primitive element. Denote the reciprocal polynomials of P q (m, n odd ) by P (m, n odd , q).
The existence of primitive TSRs of:
• odd order n over F q m , q ≥ 3 and m ≥ 2 denoted by TSRP(m, n odd , q)
• any order n ≥ 2 over F 2 m , m ≥ 2 denoted by TSRP(m, n, 2)
is directly connected to the problem of existence of primitive polynomials of the form P q (m, n odd ), q ≥ 3 and m ≥ 2 whereas when q = 2 it depends on primitive polynomials of the form P 2 (m, n) for any positive integer m ≥ 2, n ≥ 2.
Finally we have the following existence relation.
TSRP(m, n odd , q) ⇐⇒ P q (m, n odd ) ⇐⇒ P (m, n odd , q)
TSRP(m, n, 2) ⇐⇒ P 2 (m, n) ⇐⇒ P (m, n, 2)
Based on our experimental results (9) we propose a conjecture regarding the existence of primitive polynomials P (m, n, q) for any prime q and m, n ≥ 2. Denote the Galois group of automorphisms of F q m over F q by Gal(F q m /F q ) then a useful and alternate form of the conjecture 4.1 is as follows:
Conjecture 4.2.
For all m, n there exist polynomials f (X), g(X) ∈ F q [X] of degrees m and n respectively with f (X) primitive and g(0) = 0 such that f (g(x)) ∈ F q [X] is primitive of degree mn.
Proof: Suppose f (X) = g(X) + λ, as described in conjecture (4.1).
=⇒ σ∈Gal(F q m /Fq) σ(f (X)) ∈ F q [X] is primitive of degree mn Now h(X) = m−1 i=0 (X + λ q i ) ∈ F q [X] is primitive of degree m but h(g(X)) = σ∈Gal(F q m /Fq) σ(f (X)) is primitive of degree mn.
Therefore we have h(X) ∈ F q [X] primitive polynomial of degree m and g(X) ∈ F q [X] of degree n such that g(0) = 0. Conversely let f (X), g(X) ∈ F q [X] be as given in conjecture (4.2) such that f (g(X)) ∈ F q [X] is a primitive polynomial of degree mn.
Now f (X) = m−1 i=0 (X + λ q i ), λ q i are primitive roots of f (X) in F q m for i ∈ {0, . . . m − 1}. Therefore f (g(X)) = m−1 i=0 (g(X) + λ q i ) is primitive =⇒ g(X) + λ q i is primitive ∀i ∈ {0, . . . m − 1}. Hence h(X) = g(X) + λ is a primitive polynomial in F q m .
We now give a search algorithm for generating primitive TSRs of odd order n over F q m .
5.
Search algorithm for primitive TSRs of odd order n over F q m , q ≥ 3 step 1. Pick a primitive polynomial f (X) of degree m over F q . step 2. Pick a polynomial g(X) of odd degree n in F q such that g(0) = 0. step 3. Check if f (g(X)) primitive over F q . step 4. If primitive, proceed to step 5 else repeat step 1. step 5. Take k(X) = g(X) + α such that f (α) = 0. Compute the reciprocal polynomial of k(X) given by X n + λ(X n g( 1 X )) where λ = α −1 . Therefore reciprocal(k(X)) = X n +λ(c n−1 X n−1 +c n−2 X n−2 ....c 1 X+1) = X n +λL(X)
Step 6. Compute the minimal polynomial, say h(X), of λ in F q [X]. It is primitive.
Step 7. Compute matrix A in GL m (F q ) whose characteristic polynomial is h(X).
Step 8. The characteristic polynomial of TSR T is σ∈Gal(F q m /Fq) σ(X n + λL(X)).
It is primitive in
F q [X].
Step 9. T is given by
(14) T = 0 0 0 . . 0 0 A I m 0 0 . . 0 0 c 1 A . . . . . . . . . . . . . . . . 0 0 0 . . I m 0 c n−2 A 0 0 0 . . 0 I m c n−1 A .
The above algorithm can be exactly used for generating primitive TSRs of any order n over F 2 m . However for clarity we restate the algorithm. 6. Search algorithm for primitive TSRs of order n over F 2 m step 1. Pick a primitive polynomial f (X) of degree m over F 2 . step 2. Pick a polynomial g(X) of degree n in F 2 such that g(0) = 0. step 3. Check if f (g(X)) primitive over F 2 . step 4. If primitive, proceed to step 5 else repeat step 1. step 5. Take k(X) = g(X) + α such that f (α) = 0. Compute the reciprocal polynomial of k(X) given by X n + λ(X n g( 1 X )) where λ = α −1 . Therefore reciprocal(k(X)) = X n +λ(c n−1 X n−1 +c n−2 X n−2 ....c 1 X+1) = X n +λL(X)
Step 6. Compute the minimal polynomial, say h(X), of λ in F 2 [X]. It is primitive.
Step 7. Compute matrix A in GL m (F 2 ) whose characteristic polynomial is h(X).
Step 8. The characteristic polynomial of TSR T is
σ∈Gal(F 2 m /F2) σ(X n + λL(X)). It is primitive in F 2 [X].
Step 9. T is given by
(15) T = 0 0 0 . . 0 0 A I m 0 0 . . 0 0 c 1 A . . . . . . . . . . . . . . . . 0 0 0 . . I m 0 c n−2 A 0 0 0 . . 0 I m c n−1 A .
We now give an explicit proof for the existence of primitive TSRs of order 2 over F q m . Theorem 6.1. [8] There exists a primitive quadratic polynomial of trace 1 over F 2 m . Corollary 6.2. There exists a primitive quadratic polynomial of the form
X 2 + λX + λ in F 2 m [X], F * 2 m =< λ > ∀m ≥ 1. Theorem 6.3.
There exists a primitive TSRs of order 2 over F 2 m for all m.
Proof:-Using corollary (6.2) consider a primitive polynomial of the form X 2 + λX + λ. Now consider a map
F 2 m [X] ←− F 2 [X] f (X) −→ σ∈Gal(F 2 m /F2) σ(f (X)) X 2 + λX + λ −→ σ∈Gal(F 2 m /F2) σ(X 2 + λ(X + 1)) but σ(X 2 + λ(X + 1)) = (X + 1){ X 2 (X + 1) + σ(λ)} σ∈Gal(F 2 m /F2) σ(X 2 + λ(X + 1)) = g(X)h( X 2 g(X) )
where g(X) = (X + 1) and h(X) =
σ∈Gal(F 2 m /F2) (X − σ(λ))
Now g(X)h( X 2 g(X) ) is a primitive polynomial of degree 2m which gives a primitive TSR of order 2 over field F 2 m . 7. Cardinality of P 2 (m, 2)
We now consider the cardinality of primitive TSRs of order n over F q m for trivial values of m and n. The case n = 1 follows immediately from [12,Theorem 7.1]. In this case, the number of primitive TSRs of order one over F q m is given by
|GL m (F q )| (q m − 1) φ(q m − 1) m .
The case m = 1 is trivial and in this case, the number of primitive TSRs of order n is given by φ(q n − 1) n .
However, for general values of m and n, the enumeration of primitive TSRs does not seem to be an easy problem and remains open. We attempt to derive the cardinality of primitive TSRs of order 2 over F 2 m . Consider
P 2 (m, 2) := {f (X) = X 2 −α(X+1) : f (X) primitive, f (X) ∈ F 2 m [X], F * 2 m = < α >}. |P 2 (m, 2)| = |{X 2 + X + α : F * 2 m = < α >}|.
Consider the field F 2 2m generated by a primitive polynomial f (X) ∈ F 2 [X] of degree 2m and having a primitive root α ∈ F 2 2m . Consider the set
A = {i : (i, 2 2m − 1) = 1, 1 ≤ i ≤ 2 2m − 1}.
Then the set B = {α i : i ∈ A}, gives the primitive elements in F * 2 2m . Definition 7.1. [15] A cyclotomic coset C is of an element i s ∈ D modulo 2 n − 1 with respect to 2 is defined to be
C is = {i s , i s .2, i s .2 2 , i s .2 3 ....i s .2 ns−2 , i s .2 ns−1 }
where n s is the smallest positive integer such that i s ≡ i s .2 ns (mod 2 n − 1). The subscript i s is chosen as the smallest integer in C s , and i s is called the coset leader of C s .
Decompose A in cyclotomic cosets with respect to 2 modulo 2 2m − 1. Let C i1 denote the cyclotomic coset containing i 1 ∈ A. Corresponding to C i1 , α Ci 1 is the set containing α i1 ∈ B and all its conjugates over F 2 . We call this as conjugate
class of C i1 C i1 = {i 1 , i 1 .2, i 1 .2 2 , i 1 .2 3 ....i 1 .2 2m−2 , i 1 .2 2m−1 }, α Ci 1 = {α i1 , α i1.2 , α i1.2 3 ...α i1.2 2m−2 , α i1.2 2m−1 }.
where (i 1 , 2 2m − 1) = 1, α 2 2m −1 = 1 and |C i1 | = 2m. Let C i1 , C i2 , ...C in be the cyclotomic coset classes of A then A = ∪ j=n j=1 C ij , B = ∪ j=n j=1 α Ci j . For each conjugate class consisting of elements of B form polynomials
X 2 − (α j + α j.2 m )X + α j .α j.2 m , X 2 − (α j.2 + α j.2 m+1 )X + α j.2 .α j.2 m+1 , X 2 − (α j.2 2 + α j.2 m+2 )X + α j.2 2 .α j.2 m+2 , .
. .
X 2 − (α j.2 m−1 + α j.2 2m−1 )X + α j.2 m−1 .α j.2 2m−1 .
This gives the m polynomials given by the conjugate class α Cj corresponding to cyclotomic class C j . Here we have trace and norm of polynomials as
N = {α j+j.2 m , α (j+j.2 m ).2 , α (j+j.2 m ).2 2 ...α (j+j.2 m ).2 (m−1) }, T = {(α j + α j.2 m ), (α j + α j.2 m ) 2 , (α j + α j.2 m ) 2 2 ...(α j + α j.2 m ) 2 (m−1) }.
In one of the conjugate class of elements of B we will get primitive elements with trace one (6.1). Consequently trace of all the quadratic polynomials formed from that class will be 1. Counting such conjugate classes will give us the number of primitive quadratic polynomials with trace 1. Since each conjugate class gives, m degree 2, polynomials. Therefore total number of primitive quadratic polynomials over F 2 m with trace 1 will be multiple of m
|P 2 (m, 2)| = rm,
where r is the number of conjugate classes with trace 1. Hence the number of primitive TSRs of order 2 over F 2 m is
| TSRP(m, 2, 2)| = |P 2 (m, 2)| m |GL m (F 2 )| 2 m − 1 , | TSRP(m, 2, 2)| = r * |GL m (F 2 )| 2 m − 1 , r ≤ φ(2 m −1) m .
Here we give some values of r obtained experimentally using sage.
| TSRP(m, n, q)| ≤ (q n−1 − 1) φ(q m − 1) m GL m (F q ) q m − 1 ,
where n is odd. In particular, TSRs of order 2 over F 2 m are bounded by,
| TSRP(m, 2, 2)| ≤ (2 n−1 − 1) φ(2 m − 1) m GL m (F 2 ) 2 m − 1 .
Proof: Consider n to be odd then from (3.3) we have,
| TSRP(m, n, q)| = |P q (m, n)| m |GL m (F q )| q m − 1 .
Consider all primitive polynomials of deg n in F q m [x] of the form
U = {g(X) + λ : g(X) ∈ F q [X], g(0) = 0, λ primitive in F q m }, |P q (m, n)| = |U|, but |U| ≤ (q n−1 − 1)φ(q m − 1).
Similarly for the case when q = 2.
Some experimental verification for the proposed conjecture
The conjecture (4.1) is always true for n = 2 and q = 2 since we always have primitive elements in F 2 2m with trace 1 over F 2 m for any m ≥ 2. Therefore in F 2 m [X] we always have primitive polynomials f (X) of degree 2 of the form
f (X) = g(X) + λ,
where λ is a primitive element in F 2 m and g(X) = X 2 + X. However based on our experimental results we feel that the conjecture is always true. We give below some results in support of our claim. We recall from the conjecture (4.1) that P (m, n, q) denotes the primitive polynomials of degree n over F q m of the form g(X) + λ where g(X) ∈ F q [X] is of degree n such that g(0) = 0 and λ is a primitive element of F q m . Table 1. Primitive polynomials of the type P (2, 3, q) i.e of degree 3 over F q 2 where F * q 2 =< a > q primitive polynomials 2
x 3 + x 2 + x + a x 3 + x 2 + x + a + 1 3 x 3 + x 2 + x + a x 3 + x 2 + x + 2a + 1 5 x 3 + x 2 + x + 3a x 3 + x 2 + x + 2a + 3 7 x 3 + x 2 + x + 3a + 1 x 3 + x 2 + x + 3a + 3 x 3 + x 2 + x + 4a + 4 x 3 + x 2 + x + 4a + 6 11 x 3 + x 2 + 7a + 1 x 3 + x 2 + a + 10 x 3 + x 2 + a + 7 x 3 + x 2 + 4a + 7 x 3 + x 2 + 3a x 3 + x 2 + 8a + 13 x 3 + x 2 + a x 3 + x 2 + a 2 x 3 + x 2 + a 2 + a 4 x 3 + x 2 + a 3 + a + 1 x 3 + x 2 + a 3 + a 2 + a x 3 + x 2 + a 3 + a 2 + 1 x 3 + x 2 + a 3 + 1 5 x 3 + x 2 + a 4 + a 2 x 3 + x 2 + a 2 + a + 1 x 3 + x 2 + a 4 + a 3 + a 2 x 3 + x 2 + a 4 + a 3 + a 2 + 1 x 3 + x 2 + a 2 + a x 3 + x 2 + a 4 + a 3 x 3 + x 2 + a 4 + a 2 + 1 x 3 + x 2 + a 4 + a 3 + 1 x 3 + x 2 + a 4 + a 2 + a + 1 x 3 + x 2 + a 4 + a 2 + a 6
x 3 + x 2 + a 4 + a 3 + 1 x 3 + x 2 + a 5 + a 4 + a 3 + a x 3 + x 2 + a 5 + a 3 + a 2 x 3 + x 2 + a 4 + a 3 x 3 + x 2 + a 5 + a 4 + a 3 + a + 1 x 3 + x 2 + a 5 + a 3 + a 2 + 1 Table 3. Primitive polynomials of the type P (2, n, 2) i.e. of degree n over F 2 2 where F * 2 2 =< a > i.e n primitive polynomials Based on some specific observations we propose the existence of trace 1 trinomials of the special form over F q 2 for all prime q.
Conjecture 9.1. There always exist primitive polynomials of the type P (2, 3, q) which have the form x 3 + x 2 + x + α over F q 2 ∀ q where F * q 2 =< a >. Table 5. Primitive polynomials of degree 3 over F q 2 where F * q 2 =< a > q primitive polynomials 2
x 3 + x 2 + x + a x 3 + x 2 + x + a + 1 3 x 3 + x 2 + x + a x 3 + x 2 + x + 2a + 1 5 x 3 + x 2 + x + 3a x 3 + x 2 + x + 2a + 3 7
x 3 + x 2 + x + 3a + 1 x 3 + x 2 + x + 3a + 3 x 3 + x 2 + x + 4a + 4 x 3 + x 2 + x + 4a + 6 11
x 3 + x 2 + x + 9a + 2 x 3 + x 2 + x + 9a + 6 x 3 + x 2 + x + 6a + 5 x 3 + x 2 + x + 5a x 3 + x 2 + x + 6a + 4 x 3 + x 2 + x + 6a + 9 x 3 + x 2 + x + 2a + 9 x 3 + x 2 + x + 2a + 5 x 3 + x 2 + x + 5a + 6 x 3 + x 2 + x + 6a x 3 + x 2 + x + 5a + 7 x 3 + x 2 + x + 5a + 2 13
x 3 + x 2 + x + a x 3 + x 2 + x + 12a + 6 x 3 + x 2 + x + 10a + 9 x 3 + x 2 + x + 12a + 1 x 3 + x 2 + x + 11a + 9 x 3 + x 2 + x + 7a + 5 x 3 + x 2 + x + 9a + 5 x 3 + x 2 + x + 10a + 11 x 3 + x 2 + x + 2a + 7 x 3 + x 2 + x + a + 5 x 3 + x 2 + x + 4a + 1 x 3 + x 2 + x + 9a
Discussions
Primitive TSRs are very important for generating efficient word oriented stream ciphers. We have here dealt with the question of existence of primitive TSRs of order 2 over F 2 m and attempted to derive a formula for their cardinality. However, we saw that the problem of computing the number of primitive TSRs of order 2 is related to computing the number of primitive elements in F 2 2m with trace 1 over F 2 m . Also a general construction algorithm for finding primitive TSRs of order 2 over F 2 m is related to the construction algorithm for finding primitive elements in F 2 2m with trace 1 over F 2 m . As far as the question of existence of primitive TSRs of odd order n over F q m , where q ≥ 2, is concerned, we have proposed a conjecture (4.1) regarding the existence of primitive polynomials of special type P (m, n, q). We propose the following questions for further study.
1. Computing the number of conjugate classes of primitive elements in F 2 2m with trace 1 over F 2 m . 2. Construction algorithm for finding primitive elements in F 2 2m with trace 1 over F 2 m . 3. Existence of primitive polynomials of the form P (m, n, q) and P (m, n, 2).
degree m with h(0) = 0 and g(X) ∈ F q [X] of degree at most n − 1 with g(0) = 1. Clearly g T (X) = g(X) and Ψ B (X) = h(X) refer (equation 11). The number of such T is equal to the number of possible values of B with Ψ B (X) = h(X). Since h(X) is primitive, by Lemma 3.1, the number of such B is
Theorem 3. 4 .
4The number of primitive TSRs of order n over F 2 m where m ≥ 2 is given by |TSRP(m, n, 2)| = |Ψ P (TSRP(m, n, m − 2 i ).Proof: Exactly along the line of the proof (Theorem 3.2) with q = 2.
Theorem 3. 5 .
5| TSRP(m, n, 2)| = |P2(m,n)| m |GLm(F2)| 2 m −1 where n ≥ 1 and m ≥ 2 Proof: Exactly along the line of the proof (Theorem 3.3) with q = 2.
Conjecture 4. 1 .
1There exists a primitive polynomial f (X) of degree n over F q m of the following form f (X) = g(X) + λ, ∀ m, n ≥ 2 and ∀q, where g(X) ∈ F q [X] such that g(0) = 0 and λ is a primitive element in F q m .
Table 2 .
2Primitive polynomials of the type P (m,3,2) i.e of degree 3 over F 2 m where F * 2 m =< a >m
primitive polynomials
Table 4 .
4Primitive polynomials of the type P (m,3,3) i.e.of degree 3 over F 3 m where F * 3 m =< a >. i.em
primitive polynomials
x 4 + x 3 + x 2 + x + a x 4 + x 3 + x 2 + x + a + 1 6x 6 + x 5 + x + a x 6 + x 5 + x + a + 1 7
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[] |
[
"Bell test with time-delayed two-particle correlations",
"Bell test with time-delayed two-particle correlations"
] |
[
"A V Lebedev \nTheoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia\n",
"G Blatter \nTheoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia\n",
"L D Landau \nTheoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia\n"
] |
[
"Theoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia",
"Theoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia",
"Theoretische Physik\nInstitute for Theoretical Physics, RAS\nETH-Zurich\nSchafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia"
] |
[] |
Adopting the frame of mesoscopic physics, we describe a Bell type experiment involving timedelayed two-particle correlation measurements. The indistinguishability of quantum particles results in a specific interference between different trajectories; the non-locality in the time-delayed correlations manifests itself in the violation of a Bell inequality, with the degree of violation related to the accuracy of the measurement. In addition, we demonstrate how the interrelation between the orbital-and the spin-exchange symmetry can by exploited to infer knowledge on spin entanglement from a measurement of orbital entanglement.
|
10.1103/physrevb.77.035301
|
[
"https://arxiv.org/pdf/0706.3437v2.pdf"
] | 119,170,186 |
0706.3437
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305c5b492e0f04d1906ac0337ffed8e52fb6629a
|
Bell test with time-delayed two-particle correlations
10 Jan 2008 (Dated: February 1, 2008)
A V Lebedev
Theoretische Physik
Institute for Theoretical Physics, RAS
ETH-Zurich
Schafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia
G Blatter
Theoretische Physik
Institute for Theoretical Physics, RAS
ETH-Zurich
Schafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia
L D Landau
Theoretische Physik
Institute for Theoretical Physics, RAS
ETH-Zurich
Schafmattstrasse 32CH-8093, 119334Zürich, MoscowSwitzerland and, Russia
Bell test with time-delayed two-particle correlations
10 Jan 2008 (Dated: February 1, 2008)numbers: 0365Ud7323-b0560Gg
Adopting the frame of mesoscopic physics, we describe a Bell type experiment involving timedelayed two-particle correlation measurements. The indistinguishability of quantum particles results in a specific interference between different trajectories; the non-locality in the time-delayed correlations manifests itself in the violation of a Bell inequality, with the degree of violation related to the accuracy of the measurement. In addition, we demonstrate how the interrelation between the orbital-and the spin-exchange symmetry can by exploited to infer knowledge on spin entanglement from a measurement of orbital entanglement.
I. INTRODUCTION
Fundamental quantum phenomena, such as nonlocality and entanglement of quantum degrees of freedom, have regained a lot of interest recently, mainly due to their potential usefulness as a computational resource. Mesoscopic physics provides a new platform for the investigation of these phenomena, important issues being the creation, quantification, and verification of nonlocality/entanglement. In this paper, we describe an experiment where two electrons with different orbital wave functions are superposed in an interferometer and analyzed in a Bell type experiment involving two-particle correlation measurements, see Fig. 1. The particular feature of this Bell test is the replacement of the four different settings of local detectors in the original setup by four different time-delays in the measured correlators. The main physical property we want to exploit is the indistinguishability of quantum particles, which results in a specific interference between different trajectories. We wish to convey three messages: first, the non-locality in the time-delayed correlations due to indistinguishability manifests itself in the violation of a Bell inequality. Second, the degree of violation is related to the accuracy of the measurement and is reduced, once the local measurement can distinguish between the different orbital wave functions of the particles. The above two items refer to spinless objects (or particles with equal spin). Third, adding the spin degree of freedom, we show how the symmetry relation between spin-and orbital components allows to extract information on spin-entanglement from an orbital measurement.
By now, numerous proposals have been made how to create entangled states in mesoscopic setups, both for orbital-and spin degrees of freedom 1 . The verification and quantification of entanglement can be carried out using Bell inequality checks 2 or state tomography 3,4,5 . The indistinguishability of (spinless) particles producing a two-particle Aharonov-Bohm effect and entanglement has been exploited in a Hanbury-Brown Twiss interferometer 6,7 ; here, we offer an alternative implementation which makes use of an electron beam split- ter. Using the same setup as discussed here, Burkhard et al. 8 have demonstrated how to distinguish between singlet and triplet spin-states by a measurement of zerofrequency cross-correlations. Below, we exploit that the orbital measurement of the Bell parameter preserves the spin-entanglement; this feature allows us to find a lower bound on the concurrence of the spin wave function.
In the following, we consider a setup with two incoming leads, denoted asū andd, connected to two outgoing leads u and d through a reflectionless four-terminal beam splitter, see Fig. 1. At time t = 0 two electrons with normalized orbital wave functions f (x) and g(x) and common spin state χ(σ 1 , σ 2 ) are injected into the leadsū and d. The state, factorizable in orbital and spin parts (with factorized orbital part and general spin part), is conveniently written within a second quantized formalism,
|Ψ in = dx 1 dx 2 f (x 1 )g(x 2 ) (1) × σ1σ2 χ(σ 1 , σ 2 )ψ † uσ1 (x 1 )ψ † dσ2 (x 2 )|0 ;
here,ψ † ασ (x) creates electrons at the position x in lead α and |0 is the vacuum state with no electrons. After mixing in a four-terminal splitter (with mixing angle θ), we will analyze correlations in the system through detection of particles in time (see Fig. 1) or space separated intervals A and B (see Fig. 3). Our focus then is on the entanglement of the lead indices u and d with respect to bipartitioning of the system between the time or space intervals A and B.
II. BELL TEST
The particles in the outgoing leads u and d are subjected to a Bell test expressed through time-resolved current-current correlators in the leads α 1 and α 2 , α 1 , α 2 ∈ {u, d} (both auto-α 1 = α 2 and crossed-α 1 = α 2 correlators are considered),
C α1α2 (AB) = 1 δt 2 A dt 1 B dt 2 Î α1 (x 1 , t 1 )Î α2 (x 2 , t 2 ) ,(2)
whereÎ α (x, t) is the total current operator (summed over spin degrees of freedom) in lead α at position x and time t. The time integration is taken over a finite time interval A = [t A − δt/2, t A + δt/2] (same for B) with the width δt accounting for the finite time-resolution of the current measurement, cf. Fig. 1; the limit δt → 0 corresponds to a measurement of the instantaneous current. In the following, we will assume that all correlators are measured at some fixed symmetric position x 1 = x 2 and omit the coordinate variable.
With only two electrons present in the system and for non-overlapping time-intervals A∩B = 0, the correlation function C α1α2 (AB) is proportional to the joint probability P α1α2 (AB) for the detection of two particles during the time intervals A and B in the leads α 1 and α 2 , see Ref. 9. There are four distinct possibilities to distribute two electrons between the outgoing leads and we can define the properly normalized ( α1α2 P α1α2 (AB) = 1) probabilities as
P α1α2 (AB) = C α1α2 (AB) α1α2 C α1α2 (AB) .(3)
Out of these, we define the two-particle Bell inequality in the Clauser-Horne 10 form in the same way as it is done in the usual optics context 11 : we introduce the Bell correlation functions
E AB = [P uu − P ud − P du + P dd ] AB(4)
and obtain the Bell inequality
E AB − E AB ′ + E A ′ B + E A ′ B ′ ≤ 2.(5)
Here, the polarizations ± in the optics context are replaced by the lead indices u and d and the role of the four different polarization settings of the detectors is played by four different time intervals A, B and A ′ , B ′ . The violation of this inequality for a particular choice of time intervals shows that non-local correlations are present in the system, i.e., the result of the measurement cannot be simulated by any local-variable theory. Let us demonstrate that the above Bell inequality indeed can be violated by the incoming state (1) after proper projection. We then have to calculate the four current-current auto-and cross-correlators C α1α2 with α 1 = α 2 and α 1 = α 2 , respectively. This is done within the scattering matrix approach to quantum noise 12 : we assume that the Fourier components f (k) and g(k) of the single-particle wave functions are concentrated near the wave vector k 0 > 0, allowing us to linearize the energymomentum dispersion near k 0 . The time evolution of the incoming state (1) then is described by the propagation of the single-particle wave packets f (x) and g(x) with constant velocity v 0 =hk 0 /m to the right, f (x, t) = f (ξ) and g(x, t) = g(ξ), where ξ = x − v 0 t is a retarded variable. With the scattering matrix of the beam splitter (parametrized by the angle θ),
u d = cos θ − sin θ sin θ cos θ ū d ,(6)
we can express the current operatorsÎ α (x, t) in the outgoing leads α ∈ {u, d} through the electronic scattering states. Averaging the product of current operators in Eq.
(2) over the incoming state |Ψ in one arrives at the results Î u (ξ 1 )Î u (ξ 2 ) = (ev 0 ) 2 cos 2 θ |f (ξ 1 )| 2 (7)
+ sin 2 θ |g(ξ 2 )| 2 δ(ξ 1 − ξ 2 ) + sin 2 θ cos 2 θ |f (ξ 1 )| 2 |g(ξ 2 )| 2 + |g(ξ 1 )| 2 |f (ξ 2 )| 2 −Q f (ξ 1 )g * (ξ 1 )g(ξ 2 )f * (ξ 2 ) + c.c. , Î u (ξ 1 )Î d (ξ 2 ) = (ev 0 ) 2 cos 4 θ |f (ξ 1 )| 2 |g(ξ 2 )| 2 (8) + sin 4 θ |g(ξ 1 )| 2 |f (ξ 2 )| 2 + sin 2 θ cos 2 θ Q f (ξ 1 )g * (ξ 1 )g(ξ 2 )f * (ξ 2 ) + c.c. },
where Q = σ1σ2 χ(σ 1 , σ 2 )χ * (σ 2 , σ 1 ) describes the overlap between the spin states of the two electrons in the incoming state (1). The two other correlation functions Î d (ξ 1 )Î d (ξ 2 ) and Î d (ξ 1 )Î u (ξ 2 ) are obtained by exchanging cos θ and sin θ in Eqs. (7) and (8). Substituting these expressions for the current correlators into Eq. (2) and integrating over (non-overlapping) time intervals A and B, one arrives at the Bell correlation function Eq. (4)
E AB = − cos 2 (2θ) − sin 2 (2θ) Q S A S * B + S * A S B F A G B + G A F B ,(9)
where we have introduced the particle densities F A,B and G A,B averaged over the time intervals A and B,
F A,B = 1 δt t∈A,B dt |f (ξ)| 2(10)
(and similarly for G A,B with f replaced by g). The overlap S A,B between different single particle wave functions reads
S A,B = 1 δt t∈A,B dt f (ξ)g * (ξ).(11)
In the following, we apply the result (9) first to spinless fermions and plane wave states f and g and confirm the violation of the Bell inequality in this simple situation. We then proceed with a rederivation of the expression (9) with space-like separated measurement intervals A and B in order to make the origin of the entanglement more transparent. A formulation in terms of reduced density matrices leading to an expression of the Bell correlator in terms of concurrences completes the discussion.
A. Spinless fermions
We first concentrate on spinless particles; this situation can be realized by preparing the two electrons in equal spin-states, χ(σ 1 , σ 2 ) = δ σ1↑ δ σ2↑ with corresponding overlap Q = 1. To begin with, we choose a plane wave form for the wave packets with different momenta k 1 and k 2 close to k 0 (in order to allow for the linearized spectrum), f (x) = exp(ik 1 x) and g(x) = exp(ik 2 x). The correlation function E AB takes the form
E AB = − cos 2 (2θ) − V sin 2 (2θ) cos ϕ AB ,(12)
where ϕ AB = δω (t A − t B ) is the relative phase shift accumulated by the two waves between the two measurement intervals and δω = v 0 (k 1 − k 2 ) is the frequency mismatch between the two plane waves. The phase shift ϕ AB replaces the angle between the two polarizers in the conventional Bell setup. The visibility factor 0 ≤ V ≤ 1 accounts for the width δt of the time interval,
V = sin 2 (δωδt/2) (δωδt/2) 2 .(13)
The other correlation functions involving intervals A ′ and B ′ are obtained in the same way; their combination into the Bell inequality Eq. (5) produces a maximal violation for the angles
ϕ AB = ϕ A ′ B = ϕ A ′ B ′ = π/4 and ϕ AB ′ = 3π/4, corresponding to measurement intervals with relative distance t B = t A + τ /8, t A ′ = t A + τ /4, and τ B ′ = t A + 3τ /8, where τ =E = V √ 2 sin 2 (2θ) + cos 2 (2θ) ≤ 1.(14)
For given V , the maximal violation E max = V √ 2 is reached for a symmetric beam splitter with θ = π/4, cf. Fig. 2. Furthermore, the maximally allowed degree of violation E = √ 2 can be attained only for V ≈ 1, corresponding to a short time measurement of the current value with δt < 1/v 0 δk =h/δε: hence, the maximal violation of the Bell inequality can be obtained for indistinguishable particles, while a time interval with length beyond Heisenberg's uncertainty bound δt >h/δε allows for a distinction between the two particles and the Bell inequality cannot be violated in this classical situation. For V ≤ 1/ √ 2 the Bell inequality is always satisfied. Within the region 1/ √ 2 < V ≤ 1, the Bell inequality (14) is always violated for any mixing angle 0 < θ < π/2, although to a lesser degree then in the symmetric point θ = π/4. Let us discuss the physical origin of the violation. The correlator Eq. (2) measured in the Bell test is finite, provided that both electrons are detected within the time windows A and B; in this case, it is proportional to the probability P α1α2 (AB). Although, formally, the electrons have different energies ε and thus are distinguishable in principle, given a small time resolution δt <h/δε of the local current measurements one cannot distinguish between the energies ε 1 =hv 0 k 1 and ε 2 =hv 0 k 2 . Under this circumstances the electrons indeed can be considered as indistinguishable particles. Then, according to the rules of quantum mechanics, there are two quantum alternatives contributing to a coincident detection of the electrons in A and B: either the electrons with energies ε 1 and ε 2 are detected in the time windows A and B, respectively, or vice a versa. These two alternatives contribute to the measurement outcome with different phases: in the first case, the phase factor acquired by the two-particle wave function after the first measurement at t A due to the propagation of the second particle until t B is given by exp[−iε 2 (t B −t A )], while in the second case this phase assumes the value exp[−iε 1 (t B − t A )]. The phase difference between the two alternatives leads to quantum interference and a corresponding oscillatory dependence (with frequency δω = (ε 2 −ε 1 )/h) of the probability P α1α2 (AB) as a function of time, with an amplitude proportional to the visibility factor V . The precise bound on δt allowing for a violation of the Bell inequality is given by 1/ √ 2 < V ≤ 1 or (ε 2 − ε 1 )δt ≤ 2h, corresponding to a measurement where the Heisenberg uncertainty principle for energy-time variables is violated.
B. Space-separated domains
In order to understand better the nature of the entanglement observed in (14), we consider a slightly different experiment, where instead of using time-separated detection intervals, the two observers Alice and Bob are measuring the simultaneous appearance of particles in spatially separated regions A and B of the setup, see Fig. 3; for particles with a linear dispersion, these two experiments are equivalent since a time delayed measurement with δt = t 2 − t 1 at the point x corresponds to a coincident measurement at time t with δx = (t 2 − t 1 )v 0 . We first concentrate on plane-wave incoming states, where the present setup with spatially separated detectors provides additional insights. In particular, we will see that it is the projection of the non-entangled incoming state onto the two domains A and B that defines a bipartition of the system with respect to which the lead index becomes entangled. On the other hand, in order to perform a Bell inequality check, we need a set of local 'rotations' of the measurement apparatus: in our setup, the parameters generating a suitable set of local 'rotations' are determined by the distance between the measurement domains A and B and by the mixing angle θ.
A central element in our discussion below is the interchangeability of mixing U ⊗ U and projection P AB onto the domains A and B, where U denotes the oneparticle scattering matrix of the beam splitter and the tensor product U ⊗U acts on our two-particle state. This interchangeability is a trivial consequence of these two operations affecting different degrees of freedom, coordinates x 1 and x 2 and lead indices u and d. In terms of these operators, we can relate the incoming and outgoing states via
|Ψ out AB = P AB U ⊗ U |Ψ in .(15)
Assuming that the projections onto A, B in the outgoing leads and ontoĀ,B in the incoming leads (see Fig. 3) are ballistically separated (i.e., the measurement in A, B involves the appropriate ballistic delay time) we can write
|Ψ out AB = U ⊗ U PĀB|Ψ in .(16)
Hence, in our discussion we are free to interchange the two operations of mixing and projection. Consider then an incoming state (before mixing) with single-particle wave functions f (x) = e ik1x and g(x) = e ik2x with shifted momenta. The state incident from leadsū andd can be written as a simple Slater determinant, and thus is non-entangled. The lead index x ∈ {ū,d} of the electron field operatorψ x is conveniently regarded as a pseudo-spin.
|Ψ in = dx 1 dx 2 f (x 1 )g(x 2 )ψ † u (x 1 )ψ † d (x 2 )|0 ,(17)
To start with, we analyze the coincident detection of two particles within the non-overlapping regionsĀ and B of the incoming leads, see Fig. 3, and select only those events, where each of the observers (Alice inĀ and Bob inB) finds only one particle. For two particles, this projection can be described by the operator PĀB =N (Ā)N (B), with the particle number operator N (X) = X dx (ψ † u (x)ψū(x) +ψ † d (x)ψd(x)) counting particles in the region X of the incoming leads. Projecting the incoming state (17) one arrives at the state
|Ψ in AB = dx 1 dx 2 [fĀ(x 1 )gB(x 2 ) (18) +fB(x 1 )gĀ(x 2 )]ψ † u (x 1 )ψ † d (x 2 )|0
, where f X (x) and g X (x) are equal to f (x) and g(x) for x ∈ X and vanishing outside. This projected state is no longer a simple Slater determinant and describes a twoparticle state entangled in the lead indices and shared between the regionsĀ andB of the incoming leads. It is instructive to rewrite the state (18) in a pseudo-spin notation: Assuming for simplicity that the intervalsĀ andB are reduced to individual points xĀ and xB we have
|Ψ in AB ∝ e iϕĀB /2 | ↑ Ā| ↓ B + e −iϕĀB /2 | ↓ Ā| ↑ B ,(19)
where | ↑ X and | ↓ X denote states of particles localized in X and residing in leadū andd, respectively; the orbital part of the wave function contributes the phase factors exp(±iϕĀB/2) with ϕĀB = δk(xĀ−xB), where δk = k 1 − k 2 is a momentum mismatch. The projected state (19) is in fact maximally entangled in the lead-or pseudo-spin index with respect to bipartitioning the system between the regionsĀ andB. In the following, we wish to detect this entanglement in a Bell test.
The implementation of a Bell test relies on the ability to locally change the pseudo-spin basis of the particles. To do so, we transmit the original incoming state Eq. (17) through a beam splitter before measuring the presence of particles in the intervals A and B, now located in the leads u and d to the right of the mixer, see Fig. 3; the mixing then acts as an equal rotation of the (pseudo-spin) basisū,d for both particles. However, such a global rotation of the original basis alone is not sufficient to perform the Bell test, as locally distinct rotations are required as well; the latter are implemented through different choices in the separation δx = x 2 − x 1 between the regions A and B. Exploiting the interchangeability of projection and mixing, cf. Eqs. (15) and (16), we see that this change in distance results in a relative rotation with the angle ϕĀB = ϕ AB around the original (ū,d)-polarization axis of the pseudo spins, see Eq. (19). Writing the outgoing state (16) in pseudo-spin notation, we obtain the expression
|Ψ out AB = − cos(ϕ AB /2) sin(2θ) | ↑ A | ↑ B − | ↓ A | ↓ B √ 2 + cos(ϕ AB /2) cos(2θ) | ↑ A | ↓ B + | ↓ A | ↑ B √ 2 +i sin(ϕ AB /2) | ↑ A | ↓ B − | ↓ A | ↑ B √ 2 .(20)
This projected state describes two spatially separated localized particles with entangled pseudo spin indices. Choosing different space separations between the regions A and B allows one to change the phase ϕ AB and mixing by U ⊗ U generates a second rotation parametrized by the angle θ. Calculating the joint probabilities P α1α2 (AB) ∝ | α 1 α 2 |Ψ out AB | 2 for the four settings α 1 α 2 ∈ {uu, ud, du, dd} we find the Bell correlation functions E AB as given by (9) and choosing appropriate angles ϕ AB , ϕ AB ′ , ϕ A ′ B , ϕ A ′ B ′ and θ one finds the Bell inequalities violated.
C. Density matrix formulation
In a last step, we reformulate our analysis in terms of density matrices and express the Bell inequality in terms of concurrences of density matrices reduced after projection to the intervals A and B. We rewrite the projected state (18) incident from leadsū andd in pseudo spin representation,
|ΨĀB = dx 1 dx 2 |ΨĀB(x 1 , x 2 ) , where |ΨĀB(x 1 , x 2 ) = fĀ(x 1 )gB(x 2 ) | ↑ Ā| ↓ B +fB(x 1 )gĀ(x 2 ) | ↓ Ā| ↑ B .(21)
The joint measurement of the pseudo-spin index in the regionsĀ andB is described by the coordinate-reduced two-particle density operator
ρĀB ∝ dx 1 dx 2 |ΨĀB(x 1 , x 2 ) ΨĀB(x 1 , x 2 )|.
We introduce the two-particle pseudo-spin basis {| ↑↑ , |↑↓ , |↓↑ , |↓↓ }, where the first (second) arrow refers to the particle localized inĀ (B); the normalized density matrix then assumes the form
ρĀB = 1 FĀGB +GĀFB 0 0 0 0 0 FĀGB −SĀS * B 0 0 −S * A SB GĀFB 0 0 0 0 0 , (22) where F X = X |f (x)| 2 dx, G X = X |g(x)| 2 dx and S X = X f (x)g * (x)dx with X ∈ {Ā,B}.
Although initially the two particles have been in a pure state, the reduced density matrix (22) corresponds to a mixed state with ρ 2ĀB =ρĀB. The calculation of the entanglement in the mixed two-particle state Eq. (22) corresponds to finding the concurrence C(ρĀB) of a two-qubit problem, and thus can be calculated following the scheme introduced by Wootters 13
, C(ρĀB) = max{0, √ λ 1 − √ λ 2 − √ λ 3 − √ λ 4 } where λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥
0 are the eigenvalues of the matrixρĀBqĀB withqĀB = (σ y ⊗ σ y )ρ * ĀB (σ y ⊗ σ y ), σ y is a Pauli matrix, and ⊗ denotes the tensor product. The result of this calculation provides us with the expression
C(ρĀB) = 2|SĀ||SB| FĀGB + GĀFB .(23)
This quantity is indeed restricted to the interval [0, 1], as follows from the Cauchy-Schwartz inequality and the inequality ( FĀGB − FĀGB) 2 > 0, 2|SĀ||SB| ≤ 2 FĀGĀFBGB ≤ FĀGB + GĀFB. The state described by Eq. (22) is trivial (i.e., not entangled or classical) only for zero overlap SĀ = 0 and/or SB = 0. Physically, the vanishing of the overlap between the wave functions f (x) and g(x) in either of the two regionsĀ andB implies, that these orbital states are perfectly distinguishable via a local measurement. In this situation the corresponding density matrixρĀB can be written in a convex form
ρĀB = i p iρ (i) A ⊗ρ (i) B
, with probabilities p i ≥ 0 and i p i = 1, and thus is separable. On the other hand, for SĀ ,B = 0 one cannot perfectly distinguish between the different orbital states via local measurements, resulting in an interference between the different terms of the anti-symmetric wave function Eq. (21), a non-separable density matrix, and a finite concurrence.
Next, we analyze the reduced density matrix in the outgoing leads, i.e., in the new basis {uu, ud, du, dd}. Exploiting the possibility of exchanging real space projection and mixing, we can simply rotate the projected density matrix according to
ρ AB = (U ⊗ U )ρĀB(U † ⊗ U † ).(24)
The diagonal elements of the density matrix ρ AB directly provide the detection probabilities P α1α2 (AB), which then can be used in the calculation of the Bell correlation function E AB , Eq. (4),
E AB = − cos 2 (2θ) − sin 2 (2θ) C(ρ AB ) cos ϕ AB ,(25)
where the angle ϕ AB is given by the overlap integrals, ϕ AB = arg(S A S * B ); combining Eqs. (23) and (25) we immediately recover the original expression (9). Choosing four different intervals A, B, A ′ , and B ′ (note that the selection of these intervals is non-trivial in the general situation discussed here, as ϕ AB now involves overlap integrals), we can set up the Bell inequality (5) and find the result expressed in terms of concurrences
C A ′ B = C(ρ A ′ B ), sin 2 (2θ) C AB cos ϕ AB − C AB ′ cos ϕ AB ′ (26) +C A ′ B cos ϕ A ′ B + C A ′ B ′ cos ϕ A ′ B ′ + 2 cos 2 (2θ) ≤ 2.
Choosing plane waves for f (x) and g(x), the concurrences take the value C AB = C AB ′ = C A ′ B = C A ′ B ′ = V and the Bell inequality reduces to the simpler form found earlier, see Eq. (14); for the general case, the degree of violation depends separately on the shapes f (x) and g(x) of the orbital wave functions in each region A, A ′ , B, and B ′ via the corresponding concurrences.
D. Particles with spin
So far, we have considered only spinless particles or, more exactly, two electrons in a spin-triplet state with the same spin polarization of the electrons, χ tr +1 (σ 1 , σ 2 ) = δ σ1↑ δ σ2↑ and χ tr −1 (σ 1 , σ 2 ) = δ σ1↓ δ σ2↓ . Since all spindependence of the Bell inequality is encoded in the overlap Q of the spin wave-functions, see Eq. (9), one concludes that the above results are valid as well for the third maximally entangled triplet state, χ tr 0 (σ 1 , σ 2 ) = (δ σ1↑ δ σ2↓ + δ σ1↓ δ σ2↑ )/ √ 2 with Q = 1. On the other hand, the character of violation is modified for the spinsinglet state χ sg (σ 1 , σ 2 ) = (δ σ1↑ δ σ2↓ − δ σ1↓ δ σ2↑ )/ √ 2 with Q = −1. Choosing a set of optimal time intervals A, B, A ′ , and B ′ , the resulting Bell inequality takes the form
E = V √ 2 sin 2 (2θ) − cos 2 (2θ) ≤ 1.(27)
The main difference to the previous result for spin-triplet states is that this inequality can be violated only for a sufficiently large visibility factor 1/ √ 2 < V ≤ 1 and a beam splitter with a mixing angle θ sufficiently close to optimal, θ ∈ [π/4 − θ c , π/4 + θ c ], where the critical angle θ c is given by sin 2 (2θ c ) = 1/(1 + V √ 2), cf. Fig. 2. This result allows one to distinguish between triplet and singlet incoming states by measuring a Bell inequality involving only orbital degrees of freedom, see also Ref. 8. Moreover, assuming that the incident electrons have opposite spin polarization, i.e., their spin state can be written as a superposition χ in = αχ tr 0 + βχ sg , the degree of violation of the orbital Bell inequality gives a lower bound on the value of the concurrence in the spin part of the wave function. Indeed, in this case Q(χ in ) = |α| 2 − |β| 2 and the maximal violation of the orbital Bell inequality for V = 1 and symmetric scattering is given by E max = |Q| The voltage pulse V (t) injects a singlet-pair of electrons into the lead s; the π/4 fourterminal splitter distributes the particles with equal probabilities among the two leadsū ′ andd ′ . The resonances in the quantum dots select the desired energies ε1 and ε2; dots residing in the Coulomb blockade regime inhibit the propagation of two electrons into the same lead, such that the two-particle incident state involves one particle in each of the leadsū and d.
the concurrence C(χ in ) = |α 2 − β 2 | ≥ |Q|, with equality established for real α and β (note that C(χ in ) gives the degree of (useful) spin entanglement in the outgoing leads u and d). Hence, measuring the entanglement E max of the orbital part of the wave function (which leaves the spin component untouched), provides a (lower) estimate of the degree of spin entanglement of the incoming state.
If the spin wave function of the incoming electrons factorizes, χ in (σ 1 , σ 2 ) = δ σ1↑ δ σ2↓ , the orbital Bell inequality never can be violated since in this situation the electrons are distinguishable and thus the detection of an electron with given spin in one of the outgoing leads always allows to determine its origin.
III. SPIN-SINGLET/TRIPLET SOURCES
Finally, we discuss the potential experimental realization of the proposed Bell test. The source of spinentangled incoming particles can be realized with the help of a beam splitter, followed by leads with dots serving as energy filters defined through resonance levels at energies ε 1 and ε 2 , see Fig. 4. We assume that both dots reside in the strong Coulomb blockade regime; applying a single-electron voltage pulse to the source lead s, two electrons in a singlet state are detached from the Fermi see 14,15 . There is only one scattering process, involving trajectories where the electrons tunnel through different quantum dots, for which the two electrons reach the second beam splitter. The incoming state in the leadsū and d then is of a spin-singlet type with different energies ε 1 and ε 2 as defined through the dot resonances. All other scattering processes with only one or no electrons propagating towards the second beam splitter are irrelevant as they do not contribute to the correlation measurement.
A spin entangled triplet state can be generated with the help of spin-polarized reservoirs with polarizations ↑ and ↓ attached to the leads s ands, re-spectively. Applying a single-electron voltage-pulse to each reservoir, two electrons with opposite spins are injected into the leads s ands, see Fig. 4. The state |Ψ ss = dx 1 dx 2 f (x 1 )g(x 2 )ψ † s↑ (x 1 )ψ † s↓ (x 2 )|0 incident on the symmetric beam splitter emerges with a component
|Ψū′d′ ∝ dx 1 dx 2 g(x 1 )f (x 2 )ψ † u ′ ↓ (x 1 )ψ † d ′ ↑ (x 2 ) (28) +f (x 1 )g(x 2 )ψ † u ′ ↑ (x 1 )ψ † d ′ ↓ (x 2 ) |0
describing electrons scattered into different leadsū ′ and d ′ ; it is this component which can propagate through the subsequent energy filter and contribute to the current correlators. The propagation of this component through the quantum dots results in an entangled spin-triplet state of the form given by Eq.
(1) with f (x) = exp(ik 1 x 1 ) and g(x) = exp(ik 2 x). Above, we have considered an idealized situation where only two electrons are present in the system, while in a realistic situation one deals with electronic reservoirs at finite temperature. The associated equilibrium fluctuations then generate noise signals which are of the same order as the correlations associated with the injection of the two electrons. We note, however, that the corresponding equilibrium current correlators Î α1 (x, t 1 )Î α2 (x, t 2 ) eq assume significant values only for instantaneous or ballistically retarded variables, i.e., at times t 2 = t 1 and t 2 − t 1 = 2ℓ/v F in the same leads and t 2 − t 1 = 2ℓ/v F in opposite leads (here, ℓ denotes the distance between the position of measurement and the reflecting dots). The Bell test involves correlations at time differences of the order of τ = 2π/δω and a proper choice of the frequency mismatch δω always allows one to render the contribution from equilibrium fluctuations negligible. Another restriction on τ is due to dephasing and electron-electron interactions; we then have to assume that the characteristic times associated with these processes are larger then τ .
IV. CONCLUSION
We have discussed how to make use of quantum indistinguishability as a resource to generate non-classical correlations: the indistinguishability of particles enforces proper symmetrization of their wave function and results in non-factorizable states. We have demonstrated how to generate such states with the help of quantum dots residing in the Coulomb blockade regime and have determined their degree of entanglement as measured in a Bell inequality test based on auto-and cross-current correlators. In a real experiment, the latter are measured over a finite time or space domain. As a result, we obtain an interesting interplay between the measurement accuracy (time or space resolution) and the degree of non-locality as measured in the Bell inequality test: the more information is gained that locally distinguishes between the particles, the smaller is the degree of violation. Once the uncertainty principle allows for the identification of the particle, the Bell inequality cannot be violated any longer. This feature can be exploited in the design of experiments testing the above predictions: choosing a small energy difference δε = ε 2 − ε 1 allows for a slow measurement with a less stringent time resolution, while the violation of the Bell test remains observable. On the other hand, the energy difference δε has to be chosen sufficiently large in order to avoid the influence of decoherence or interactions.
The above setup for spinless particles provides an alternative for the observation of the two-particle interference as proposed by Samuelsson et al. 6 and recently observed by Neder et al. 7 ; here, the role of the magnetic flux Φ penetrating the Hanbury-Brown Twiss interferometer is replaced by the time-delay of subsequent measurements in the correlator. Adding the spin degree of freedom, we are confronted with two distict situations: if the spin degree of freedom allows to distinguish between the particles (this is the case for the spin-state χ(σ 1 , σ 2 ) = δ σ1↑ δ σ2↓ ) the Bell inequality is never violated. On the other hand, entangled spin states in the singlet or triplet sector (these are the states χ sg and χ tr 0 ) can generate maximal violation of the Bell inequality; finally, the superposition of these states reduces the spin-entanglement and the degree of violation in the orbital Bell inequality gives a lower bound on the spin-concurrence, with an ideal measurement providing the best bound.
We thank Gordey Lesovik for discussions and acknowledge the financial support from the Swiss National Foundation, the RFBR grant No. 06-02-17086-a and the Programm "Quantum Macrophysics" of RAS.
FIG. 1 :
1Particles incident in leadsū andd with wave functions f (x) and g(x) are mixed in a four-terminal splitter (characterized by the mixing angle θ) and analyzed at x1 = x2 through measurement of time-correlations during the time intervals t1 ∈ A and t2 ∈ B (time axis drawn perspectively into the plane). We are interested in the entanglement of the lead indices u and d with respect to bipartitioning of the system between the time intervals A and B. For details on the implementation of the source (shaded area) seeFig. 4.
FIG. 2 :
2Bell inequality violation for maximal visibility V = 1 versus mixing angle θ (solid line: spin-triplet states; dashed line: spin-singlet state).
FIG. 3 :
3Particles incident in leadsū andd with wave functions f (x) and g(x) are mixed in a four-terminal splitter and analyzed through measurement of equal-time correlations within the space intervals A and B centered around xA and xB. The interchangeability of projection (to the intervals A and B) and mixing allows to shift the measurement intervals to the positionsĀ andB in the incoming leads; provided that the measurements inĀ andB and in A and B are ballistically delayed in time, the measurement outcome is the same.
√ 2 .FIG. 4 :
24At the same time, Spin-singlet source:
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|
[] |
[
"Topological aspects of nonlinear excitonic processes in noncentrosymmetric crystals",
"Topological aspects of nonlinear excitonic processes in noncentrosymmetric crystals"
] |
[
"Takahiro Morimoto \nDepartment of Physics\nUniversity of California\n94720BerkeleyCA\n",
"Naoto Nagaosa \nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan\n\nDepartment of Applied Physics\nThe University of Tokyo\n113-8656TokyoJapan\n"
] |
[
"Department of Physics\nUniversity of California\n94720BerkeleyCA",
"RIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan",
"Department of Applied Physics\nThe University of Tokyo\n113-8656TokyoJapan"
] |
[] |
We study excitonic processes second order in the electric fields in noncentrosymmetric crystals. We derive formulas for shift current and second harmonic generation produced by exciton creation, by using the Floquet formalism combined with the Keldysh Green's function method. It is shown that (i) the steady dc shift current flows by exciton creation without dissociation into free carriers and (ii) second harmonic generation is enhanced at the exciton resonance. The obtained formulas clarify topological aspects of these second order excitonic processes which are described by Berry connections of the relevant valence and conduction bands.
|
10.1103/physrevb.94.035117
|
[
"https://arxiv.org/pdf/1512.00549v3.pdf"
] | 119,210,357 |
1512.00549
|
6cba102bc16ba54cf99f27b3d0164419750d686a
|
Topological aspects of nonlinear excitonic processes in noncentrosymmetric crystals
Takahiro Morimoto
Department of Physics
University of California
94720BerkeleyCA
Naoto Nagaosa
RIKEN Center for Emergent Matter Science (CEMS)
351-0198WakoSaitamaJapan
Department of Applied Physics
The University of Tokyo
113-8656TokyoJapan
Topological aspects of nonlinear excitonic processes in noncentrosymmetric crystals
(Dated: July 12, 2016)
We study excitonic processes second order in the electric fields in noncentrosymmetric crystals. We derive formulas for shift current and second harmonic generation produced by exciton creation, by using the Floquet formalism combined with the Keldysh Green's function method. It is shown that (i) the steady dc shift current flows by exciton creation without dissociation into free carriers and (ii) second harmonic generation is enhanced at the exciton resonance. The obtained formulas clarify topological aspects of these second order excitonic processes which are described by Berry connections of the relevant valence and conduction bands.
I. INTRODUCTION
Nonlinear optical processes in solids are the important subject in condensed matter physics, which are also of crucial importance for applications [1][2][3]. In particular, noncentrosymmetric crystals host the second order processes in electric fields such as shift current, optical rectification and second harmonic generation (SHG). We have recently revealed that these second order optical processes are topological in nature and are described by the Berry connections of Bloch wavefunctions of conduction and valence bands for non-interacting electrons [4]. However, the role of Coulomb interaction is often essential in the electronic processes in insulating solids. In particular, excitons, bound pairs of electron and hole via Coulomb interaction, are relevant to these processes by enhancing these nonlinear effects in many cases [3] Therefore, it is an important issue to study the interaction effect on the topological nonlinear optical effects, which we address in this paper.
One example of the second order optical effects is the photocurrent, which is also relevant to the solar cell action. Light irradiation creates the electrons and holes in the crystal, which are often bound to form neutral excitons. It is believed that the excitons cannot contribute to the steady dc current. If this is the case, the dissociation of the excitons into free electrons and holes is essential to produce the dc photocurrent. This process is usually achieved by the potential gradient at p-n junction or by the applied electric field. The efficiency of the solar cell action is largely determined by the probability of the thermally activated dissociation process which is competing with the annihilation of the excitons [5]. However, we show below that the excitons can support the dc current under the steady light irradiation due to the geometrical nature of the Bloch wavefunctions.
In the past decades, it has been recognized that the free carriers are not necessarily needed for the current. The representative example is the polarization current in ferroelectrics [6]. In an insulator with a band gap, electrons occupying the valence band can support current corresponding to the time derivative of the polarization. This current is characterized by the Berry phase of the valence electrons. Specifically, the Berry phase is related to the "intra-cell" coordinates, i.e., the band dependent shift of the wavepacket made from the Bloch wavefunctions. This shift of the electrons described by the Berry phase is the origin of the electric polarization that leads to the polarization current. However, the polarization current cannot be a steady dc current. Since the polarization current usually appears in the process of the polarization reversal in ferroelectrics, it inevitably vanishes when the polarization reversal is completed. Quantum pumping current proposed by Thouless [7], on the other hand, can support dc current, but it requires a nontrivial topological (winding) number defined in the parameter space of the Hamiltonian which is only achieved with a large deformation of the Hamiltonian in the parameter space. The Hall current in the quantum Hall effect is also such current characterized by a nontrivial Berry phase [8]. While the quantum Hall current is steady dc current, it is carried by the edge channels (not through the bulk) and is realized under the nontrivial topological (Chern) number which usually requires an application of a large external magnetic field. Therefore, it has been considered to be difficult to realize dc current in the presence of a band gap when the Hamiltonian does not possess any nontrivial topological number.
The restriction on obtaining dc current is relaxed in the non-equilibrium state under the light irradiation. Of particular interest is the shift current as a mechanism of the photocurrent in noncentrosymmetric crystal [4,[9][10][11][12][13]. This might be relevant to the recent experiments showing the high efficiency solar cell action [14][15][16][17][18]. Shift current is induced by the change in the intra-cell coordinates associated with the interband transitions. Namely, the difference of the Berry phases between the conduction and valence bands induces the steady dc current in noncentrosymmetric crystals even in the absence of an external dc electric field. However, it is assumed here that electrons and holes are independent free particles, i.e., the single particle approximation is employed.
The other example of the second order optical effects is the second harmonic generation (SHG) [1][2][3]. SHG is a common optical process which is used to detect the in-
ψ v ψ 1 ψ c = ψ 2 hΩ − FIG.
1. Schematic picture of the Floquet two band model. Bands labeled by ψ1 and ψ2 denote the valence band with Floquet index −1 and the conduction band with Floquet index 0, respectively. Quasienergy for the Floquet band ψ1 is smaller than that for ψ2. The exciton formation is described by an effective mixing of these two bands in the mean field approximation of the electron-electron interaction.
version symmetry breaking both in the bulk crystal and at interfaces or surfaces. When the incident light has the frequency Ω, the second order nonlinear responses can have two output frequencies, i.e., 0 and 2Ω. The first one corresponds to the shift current discussed above or optical rectification if it is detected optically in sufficiently low frequencies. The optical rectification is important in generating terahertz (THz) light. The latter 2Ω response corresponds to the SHG. It is experimentally shown that the excitons can contribute to the optical rectification [19,20] and the SHG [21][22][23]. While the optical rectification and the SHG are well-known nonlinear optical effects, their enhancement due to the exciton resonance has not been fully explored from the viewpoint of topology and geometry of the Bloch electrons.
In the present paper, we study the role of exciton formation on the second order optical processes and demonstrate that they are topological in nature. We show that (i) the shift current originating from the Berry phase remains nonvanishing even when the excitons are formed due to the attractive interaction between an electron and a hole created by the light irradiation, and (ii) the SHG is expressed by the similar expression to the shift current in terms of the Berry connection and is enhanced at the exciton resonance. This is achieved by using the Floquet two band model developed in Ref. [4] by incorporating the attractive interaction. This formalism enables us to concisely describe nonequilibrium steady states with exciton formation and study various nonlinear current responses produced by exciton creation.
II. FLOQUET TWO BAND MODEL FOR EXCITONS
We study a two band model that describes the exciton formation in a system driven by an external electric field of light. We consider a d-dimensional system in which the electric field is applied along the ith direction and the repulsive interaction is present between the valence electron and the conduction electron. Then the Hamiltonian of the two band model is given (with the convention e = = 1) by
H = α=c,v k 0 i [k + A(t)e i ]ψ † α,k ψ α,k + k A(t)[v 12 (k)ψ † v,k ψ c,k + h.c.] − k,k V kk ψ † c,k ψ v,k ψ † v,k ψ c,k ,(1)
where ψ v and ψ c are annihilation operators for valence and conduction bands with the energy dispersions 0 v (k) and 0 c (k), respectively. Electrons are driven by an electric field E(t)e i (with the ith unit vector e i ) which is periodic in time as
E(t) = Ee −iΩt + E * e iΩt .(2)
This electric field is introduced to the Hamiltonian by the substitution k → k + A(t)e i with the gauge potential is given by A(t) = iAe −iΩt − iA * e iΩt with A = E/Ω. In addition to the first term in Eq. (1) that corresponds to this substitution in the energy dispersion, the electric field leads to an interband effect described by the second term in Eq. (1), i.e., the coupling to the current matrix element
v 12 (k) = ψ v,k |v|ψ c,k(3)
between the valence and conduction bands where v is the velocity operator in the ith direction. The attractive interaction between electron and hole is described by V k,k which leads to the exciton formation. We have picked up only the interaction terms which are relevant to the formation of excitons with zero center-of-mass momentum corresponding to the uniform electric field of light. This is analogous to the BCS Hamiltonian of superconductivity, but we do not discuss the condensate of the excitons here.
The nonequilibrium steady state under the light irradiation is concisely described by using the Floquet formalism combined with the Keldysh Green's function method [4,[24][25][26][27][28][29]. The Floquet formalism offers a description of periodically driven systems in terms of Floquet bands. Specifically, we define a Floquet Hamiltonian H F with Fourier transformation of the time dependent Hamiltonian H 0 (t) of the period T as
(H F ) mn = 1 T T 0 dte i(m−n)Ωt H 0 (t) − nΩδ mn ,(4)
where m and n are Floquet indices and Ω = 2π/T . While Floquet bands obtained from H F determines eigenstates in periodically driven systems, they lack the information of how they are occupied in the steady state. The occupation of the Floquet bands in the steady state can be fixed by coupling the system to a heat bath having the Fermi energy and the temperature that we want to impose onto the system. This is concisely described by using the Keldysh Green's function method and include the effect of the heat bath as a self energy. The Keldysh Green's functions in the Floquet formalism are written by the Dyson equation as
G R G K 0 G A −1 = ω − H F + Σ.(5)
Here we consider two contributions to the self energy as Σ = Σ bath + Σ ex , where Σ bath is the self energy arising from a coupling to the bath and Σ ex is the self energy arising from the exciton formation due to the electronelectron interaction. The self energy Σ bath for the heat bath is given by
(Σ bath ) mn = iΓδ mn 1 2 −1 + f (ω + mΩ) 0 − 1 2 ,(6)
for Floquet indices m and n. This form of Σ bath assumes that each site is coupled to a heat bath which has a wide spectrum and the distribution function f ( ), and Γ measures the strength of the coupling between the system and the bath [24]. The inclusion of Σ bath fixes the occupation of the Floquet bands properly through the Keldysh component of the Dyson equation. The self energy Σ ex describes the exciton formation in the driven system which we incorporate in the mean-field approximation for the interaction term by keeping the Fock term in the Keldysh Green's function. Now we apply this formalism to the two band model in Eq. (1). When the electric field is weak, we can focus on two Floquet bands, i.e., the valence band dressed with one photon and the conduction band dressed with zero photon, which are denoted by annihilation operators ψ 1 and ψ 2 , respectively, as schematically illustrated in Fig. 1. Here, subscripts 1 and 2 are shorthands for the valence band with Floquet index −1 and the conduction band with Floquet index 0, respectively. In this case, the Floquet Hamiltonian is given by [4]
H F = ψ † 1,k ψ † 2,k H F ψ 1,k ψ 2,k(7)H F = 1 (k) −iA * [v 12 (k) + v * (k)] iA[v 21 (k) + v (k)] 2 (k) ≡ + d · σ,(8)where 1 = 0 v + Ω, 2 = 0 c , and v 21 = v * 12 . The de- tuning d z (k) = − 1 2 [ 0 c (k) − 0 v (k)
− Ω] is negative for any value of k, because we are interested in the excitonic bound state where the photon energy is smaller than the band gap. The nonzero expectation value of excitons effectively modifies the dipole matrix element by
iAv (k) = − dk V kk ∆(k ),(9)∆(k) = ψ † 1,k ψ 2,k .(10)
This mean field treatment of excitons in H F is equivalent to including the retarded component of the exciton self energy Σ R ex into the mean field Floquet Hamiltonian H F in the Dyson equation [Eq. (5)].
The exciton formation is captured by the selfconsistency equation for ∆(k) which we solve by employing the Keldysh Green's function in the following. First, the lesser Green's function for the Floquet two band model is given by [4]
G < = G R Σ < G A = (ω − − iΓ + d · σ)Σ < (ω − + iΓ + d · σ) [(ω − − iΓ) 2 − d 2 ][(ω − + iΓ) 2 − d 2 ] ,(11)
with
Σ < = Σ R + Σ K − Σ A 2 = iΓ 1 + σ z 2 .(12)
Here the lesser self-energy Σ < describes the occupation of Floquet bands and is determined by the heat bath as Σ < ∼ = Σ < bath . The above form of Σ < assumes that the Fermi energy is located within the energy gap of the original band structure [24]. Specifically, the final equation in Eq. (12) follows from f ( 0 v ) = 1 and f ( 0 c ) = 0 since (Σ < bath ) mn = iΓδ mn f (ω + mΩ). Next, in the case of two band model, general expectation values ψ † (b · σ) T ψ for any b = (b x , b y , b z ) can be evaluated by using the above lesser Green's function as [4]
ψ † (b · σ)ψ = −iTr[G < (b · σ)] = dk 1 d 2 + Γ 2 4 Γ 2 (−d x b y + d y b x ) + (d x b x + d y b y )d z + (d 2 z + Γ 2 4 )b z ,(13)
where Tr denotes the trace of a matrix and integration over k and ω. The self-consistency condition for ∆(k) is written by using the above equation with b · σ = (σ x + iσ y )/2 as
∆(k) = ψ † 1,k ψ 2,k = −i dωG < 21 = iA(v 21 + v )(d z − i Γ 2 ) 2(d 2 + Γ 2 4 ) ,(14)
which is essentially equivalent to the Dyson equation for the retarded component of the self energy Σ R [29]. This leads to the integral equation,
∆(k) = iAv 21 d z − i Γ 2 2(d 2 + Γ 2 4 ) − d z − i Γ 2 2(d 2 + Γ 2 4 ) dk V kk ∆(k ).(15)
If we assume that the attractive interaction has the separable form
V kk = w * (k)w(k ),(16)
we can solve the integral equation as
v (k) = −w * (k)B, B = 1 iA dk w(k )∆(k ),(17)
where the integral equation for ∆(k) reduces to the linear equation for B given by
B = dkw(k)v 21 d z − i Γ 2 2(d 2 + Γ 2 4 ) − dk|w(k)| 2 d z − i Γ 2 2(d 2 + Γ 2 4 ) B.(18)
When A|v 21 + v | is much smaller than |d z | and Γ (i.e., the external electric field is not too strong), v is written as
v (k) = −w * (k) C 1 1 + C 2 ,(19)
with
C 1 = dk w(k)v 21 2(d z + i Γ 2 ) , C 2 = dk |w(k)| 2 2(d z + i Γ 2 )
.
Intuitively, 1/(1 + C 2 ) corresponds to the propagator of the exciton, and the resonance to the exciton state takes place when Re(1+C 2 ) = 0 is satisfied by the incident light frequency Ω. In particular, when the detuning d z (k) is constant as a function of k (as in flat bands) and the interaction is of a contact type [i.e., w(k) is a constant satisfying dk|w(k)| 2 = V ], the exciton resonance takes place at the frequency Ω = 0
c − 0 v − V .
In the following, we show that the shift current is nonvanishing in the presence of the exciton formation where no free electrons and holes are created.
Now we study the current J ≡ ψ † ∂ kj H F ψ in the jth direction in the presence of exciton formation. The current expectation value is obtained by setting b x −ib y = −iA * (∂ kj v) 12 and b z = (∂ kj 1 − ∂ kj 2 )/2 in Eq. (13), which gives
J = dk(j 1 + j 2 ),(21)
with
j 1 = dk Re[(d z − i Γ 2 )(d x + id y )(b x − ib y )] d 2 + Γ 2 4 = |A| 2 Re{(d z − i Γ 2 )[(∂ kj v) 12 (v 21 + v )]} d 2 + Γ 2 4 ,(22)j 2 = (d 2 z + Γ 2 4 )(∂ kj 1 − ∂ kj 2 ) 2(d 2 + Γ 2 4 )
.
When |A(v 21 + v )| and Γ are much smaller than |d z |, we can replace d 2 with d 2 z in the denominators. Then j 2 vanishes after the integration over k because dk∂ kj α = 0; we focus on the contribution from j 1 hereafter. In the two band model, the derivative of the velocity operator in Eq. (22) is written as [4] ∂v
∂k j 12 = ∂v 12 ∂k j − ∂ kj u 1 |v|u 2 − u 1 |v|∂ kj u 2 = v 12 (R 1 + iR 2 ),(24)
with
R 1 = ∂ kj log |v 12 | + v 11 − v 22 1 − 2 ,(25)R 2 = ∂ kj Im[log v 12 ] + a 1 − a 2 ,(26)
Here, u α is the periodic part of the Bloch wave function and a α = −i u α |∂ kj u α is the Berry connection of the band α. We note that R 1 and R 2 have dimensions of the length. In particular, R 2 is known as the shift vector and describes the shift of the wavepackets in the valence and conduction bands. Intuitively, the shift vector R 2 is k-resolved version of electric polarization and originates from the difference of intra-cell coordinates for the valence and conduction bands which is expressed by the Berry connections. Indeed, the k-integral of R 2 is the difference of electric polarizations of the valence and conduction bands as can be seen from
dkR 2 = dka 1 − dka 2 ,(27)
where the contribution of ∂ kj Im[log v 12 ] vanishes because it is a total derivative with respect to k j . In the presence of the time reversal symmetry (TRS), R 1 and R 2 are odd and even in k, respectively, and |v 12 | 2 is even in k. Thus, the photocurrent from the excitons in Eq. (21) reduces in the presence of TRS to
J = J con + J ex ,(28a)J con = |A| 2 dk Γ 2 d 2 z + Γ 2 4 |v 12 | 2 R 2 ,(28b)J ex = |A| 2 dk 1 d z [Re(v 12 v )R 1 − Im(v 12 v )R 2 ]. (28c)
The first term J con describes the conventional shift current that involves creation of a pair of free electron and hole which is present for Ω > E g with the band gap E g . The second term J ex describes the shift current carried by excitons and is nonvanishing even when Ω < E g . We note that we dropped Γ in the second term because we can assume Γ |d z | in describing excitons. We study properties of the exciton photocurrent J ex in the following. First, the photocurrent J ex is generated through the real transitions to create excitons, because only the imaginary part of the exciton propagator 1/(1+C 2 ) contributes to the photocurrent J ex in Eq. (28). This is reasonable from the viewpoint of the energy conservation. It is easy to explicitly show this fact in the presence of the TRS by assuming that the time reversal operation is represented by a complex conjugation (T = K). In this case, the velocity operator obeys the equation v ij (−k) = −(v ij (k)) * and the separable interaction term w(k) can be chosen to satisfy w(−k) = w * (k) without loss of generality. Since the real part of w(k)v 21 (k) is odd in k, C 1 is pure imaginary at the exciton resonance where we can drop Γ in the denominator. By noticing that w * (−k)v 12 (−k) = −[w * (k)v 12 (k)] * , we can write the photocurrent as
J ex = |A| 2 dk Im(C 1 ) d z Im 1 1 + C 2 × {Re[w * (k)v 12 (k)]R 1 − Im[w * (k)v 12 (k)]R 2 } ,(29)
which is proportional to Im [1/(1 + C 2 )] and manifests that the photocurrent is generated by real transition to the exciton state. Second, the expression for J ex can be further simplified for shallow excitons. Shallow excitons are those with small binding energy that are formed by Bloch states near the band gap. Specifically, in this case of shallow excitons, the k-integrals are contributed only from the small region (δk) d around k = 0 where the band gap is the smallest; we replace dk with dk(δk) d δ(k) in Eq. (29). By doing so, the mean field solution of the exciton in Eq. (20) leads to
Im[C 1 ] = (δk) d w(0) 2d z (0) Im[v 21 (0)],(30)Im 1 1 + C 2 = −2d z (0) Γ [2d z (0) + V ] 2 + Γ 2 ,(31)
with V = |w(0)| 2 (δk) d . Thus the photocurrent J ex for shallow excitons is given by
J ex ∼ = |A| 2 V (δk) d |d z (0)| Γ [2d z (0) + V ] 2 + Γ 2 |v 12 (0)| 2 R 2 (0),(32)
where we used the relations Im[v 12 (0)]Im[v 21 (0)] = −|v 12 (0)| 2 and d z < 0. In this expression, the factor Γ/{[2d z (0) + V ] 2 + Γ 2 } describes the real transition into the exciton state at the resonance frequency
Ω = 0 c − 0 v − V .
In addition, the shift current for shallow excitons is proportional to the weight of the exciton V (δk) d /[2|d z (0)|]. This clearly shows that the nonzero shift current flows by creating excitons below the band gap. Furthermore, we notice that J ex is proportional to the contribution to the conventional shift current J con at k = 0 that has a factor |v 12 (0)| 2 R 2 (0) as seen in Eq.(28b). This indicates that the conventional shift current in the noninteracting system is partly transfered to the exciton resonance below the band gap due to the exciton formation with the attractive interaction. In fact, by recovering the energy broadening Γ in Eq. (28c), one obtains the term |A| 2 Γ/2 12 v ] which gives negative shift current contribution for the electron-hole continuum as follows. For simplicity, we focus on the case where the resonance condition is satisfied at the band gap at k = 0 (i.e., d z (0) = 0 and d z (k) = 0 for k = 0). In this case, Eq. (20) gives C 1 = −iπw(0)v 21 (0)D(0) and C 2 = −iπ|w(0)| 2 D(0) where D(0) is the joint density of states at k = 0, and the above term is expressed as |A| 2 Γ/2
d 2 z +Γ 2 /4 Im[(∂ kj v)d 2 z +Γ 2 /4 |v 12 (0)| 2 R 2 (0)[− π|w(0)| 2 D(0) 1+(π|w(0)| 2 D(0)) 2 ]
. This should be compared with J con and clearly describes the partial suppression of the conventional shift current above the band gap due to the exciton formation.
An optical process that is closely related to the shift current is optical rectification. The optical rectification is the second order nonlinear optical effect that optically measures emission of low frequency light, typically in the THz regime. Namely, the optical rectification is a low frequency optical analog of the shift current and is important for application for THz generation. Since the shift current is enhanced at the exciton resonance below the band gap, the optical rectification is also enhanced at the exciton resonance. Thus strong THz generation is expected by shining the light to noncentrosymmetric crystals at the exciton resonance. Indeed, there are experimental reports on enhanced THz emissions for GaAs when the laser frequency is resonant to the excitons [19,20].
III. SECOND HARMONIC GENERATION
Exciton formation also enhances the SHG for the photon energy below the band gap in a similar manner to the case of shift current. The SHG is the current response of the frequency 2Ω when the incident light has the frequency Ω. In our formalism, the SHG can be studied by using the formula for time-dependent current,
J(t) = −i m Tr[v(t)G < mn ]e −i(m−n)Ωt ,(33)
where subscripts m and n denote the Floquet indices, and Tr denotes a trace over the band indices and ω and k integration. Here the time-dependent current operator is given by
v(t) = v + (iAe −iΩt ∂ k v + h.c.) + O(A 2 ).(34)
The frequency 2Ω component of the current J 2Ω is decomposed into two contributions as
J 2Ω = J 1ph + J 2ph ,(35)
where the first term and the second term represent onephoton contribution and the two-photon contributions, respectively. In the standard perturbation theory [10], the one-photon contribution corresponds to the bubble diagram where the diamagnetic current is induced by the external electric field coupling to the usual current operator, while the two-photon contribution corresponds to the diagram where the usual current response is induced by the external electric field coupling to the diamagnetic current. In the following, we compute the one-photon contribution and the two-photon contribution separately. The one-photon contribution J 1ph is obtained by setting m = n + 1 and v(t) → iAe −iΩt ∂ k v in Eq. (33). Since this is the same Floquet two band model as in Eq. (8), we can compute J 1ph in a similar way to the shift current. In particular, the same self-consistent equation Eq. (15) holds for the exciton formation. By using the formula for the lesser Green's function [4],
(G < ) 21 = (d x + id y )( Γ 2 + id z ) 2(d 2 + Γ 2 4 ) ,(36)
the one-photon contribution is written as
J 1ph = −A 2 dk 1 2d z Re[(∂ k v) 12 v ] = −A 2 dk Im(C 1 ) 2d z Im 1 1 + C 2 × {Re[w * (k)v 12 (k)]R 1 − Im[w * (k)v 12 (k)]R 2 } ,(37)
where we only kept terms relevant to the exciton resonance. Thus the one-photon contribution is the same as the shift current J ex except that it has the factor −A 2 /2 instead of |A| 2 in Eq. (29). The two-photon contribution J 2ph is obtained by setting m = n + 2 and v(t) → v in Eq. (33). Therefore, the two photon contribution arises from another two band model in which valence and conduction bands are separated by two Floquet indices. This is given bỹ
H F = ψ † 1ψ † 2 H F ψ 1 ψ 2 ,(38)H F = 0 v + 2 Ω − 1 2 A 2 [(∂ k v) 12 + (∂ kṽ ) * ] − 1 2 A 2 [(∂ k v) 21 + ∂ kṽ ] 0 c ,(39)
whereψ 1 ,ψ 2 are annihilation operators for the valence band with Floquet index −2 and the conduction band with Floquet index 0, respectively, and (∂ k v) 12 = ψ v,k |∂ k v|ψ c,k . The off-diagonal term originate from the time-dependent Hamiltonian expanded up to the order of A 2 . Specifically, when we keep terms up to the order of A 2 , the time-dependent Hamiltonian reads
H(t) = H 0 + A(t)v + 1 2 A(t) 2 ∂ k v,(40)
and the Fourier components of e ±i2Ωt produces the offdiagonal terms in Eq. (39). In this case, the self consis-tent equation is given by
− 1 2 A 2 ∂ kṽ = − dk V kk ∆ (k ),(41)∆(k) = ψ † 1,kψ 2,k = −(1/2)A 2 [(∂ k v) 21 + ∂ kṽ ](d z − i Γ 2 ) 2(d 2 + Γ 2 4 )
. (42) These equations describe excitons formed by two photon absorption and are different from Eq. (15) for the excitons formed by one photon absorption. The self consistent equation is solved in a similar manner by assuming the separable form for the interaction V kk = w * (k)w(k ) as
∂ kṽ (k) = −w * (k)C 1 1 +C 2 ,(43)
with
C 1 = dk w(k)(∂ k v) 21 2(d z + i Γ 2 ) = dk w(k)v 21 2(d z + i Γ 2 ) (R 1 − iR 2 ),(44)C 2 = dk |w(k)| 2 2(d z + i Γ 2 )
.
Here we used the identity (
∂ k v) 21 = [(∂ k v) 12 ] * = [v 12 (R 1 + iR 2 )] * = v 21 (R 1 − iR 2 ).
Then we obtain the two photon contribution as
J 2ph = − A 2 2 dk 1 2d z Re[v 12 (∂ kṽ ) 21 ] = −A 2 dk Re(C 1 ) 4d z Im 1 1 +C 2 Im[w * (k)v 12 (k)],(46)
where we only kept the term relevant to the exciton resonance. When we equate the first line and the second line, we used constraints from the TRS. Specifically, the TRS
(T = K) requires w * (k)v 12 (k) = −[w * (−k)v 12 (−k)] * , w(k)∂ k v(k) = [w(−k)∂ k v(−k)] * .
ThusC 1 is real when we neglect i Γ 2 in the denominator, and Re[w * (k)v 12 (k)] is odd in k and vanishes after k-integration. Since this expression shows J 2ph ∝ Im[1/(1 +C 2 )], we again find that the two photon contribution to SHG is generated by the real transition to the exciton state.
Next we study SHG in the case of shallow excitons. We assume that the integral is contributed near k = 0 due to the factor 1/d z , and replace dk with dk(δk) d δ(k). The one-photon contribution J 1ph reduces to −1/2 times Eq. (32). In the case of the two-photon contribution, the self consistent solution reduces in the shallow exciton limit to
Re[C 1 ] = (δk) d w(0) 2d z (0) Im[v 21 (0)]R 2 (0),(47)Im 1 1 +C 2 = −2πd z (0)δ[2d z (0) + V ],(48)
Here we used Re[v 21 (0)] = R 1 (0) = 0 under the TRS in the first line and took Γ → 0 limit in the second line. By using these equations in Eq. (46), the two photon contribution is written as
J 2ph ∼ = −πA 2 V (δk) d 4d z (0) |v 12 (0)| 2 R 2 (0)δ[2d z (0) + V ].(49)
Combining these two contributions, we obtain the SHG from shallow excitons as
J 2Ω ∼ = πA 2 V (δk) d |v 12 (0)| 2 R 2 (0) × δ( 0 v − 0 c + V + Ω) 2( 0 v − 0 c + Ω) − δ( 0 v − 0 c + V + 2 Ω) 4( 0 v − 0 c + Ω)
.
(50)
This clearly shows that the SHG is enhanced with exciton formation when the photon energy Ω is the same as or the half of the exciton creation energy ( 0
c − 0 v − V )
. Finally we comment on the relationship between the SHG and the shift current. Let us define nonlinear conductivities for SHG and shift current as
J 2Ω = σ (2) (Ω)E(Ω) 2 ,(51)J = σ (0) (Ω)|E(Ω)| 2 .(52)
In the case of noninteracting systems, the real part of the nonlinear conductivity for SHG is related to that for shift current as [4] Re[σ (2) (Ω)] = − 1 2
σ (0) (Ω) + 1 4 σ (0) (2Ω).(53)
This is obtained by replacing v with v in expressions for SHG [Eq. (37) and Eq. (46)] and comparing it with J con . While the above relation still holds for the one-photon contribution with exciton formation, i.e., J 1ph = − 1 2 J ex , the two photon contribution does not satisfy this relation because the mean-field solution for the two-photon contribution involves the k-integral of ∂ k v inC 1 in contrast to v in C 1 . However, this relationship recovers in the case of shallow excitons as is noticed by comparing Eq. (32) and Eq. (50). Thus the SHG and the shift current are closely related with each other even in the presence of exciton formation, and both are governed by the shift vector R k (k) which is essentially a topological quantity described by Berry connections. Since the k-integral of R k (k) over the Brillouin zone coincides with the difference of polarizations of valence and conduction bands, both SHG and shift current are considered to be topological phenomena akin to electric polarization phenomena in ferroelectric materials.
IV. DISCUSSIONS
We have shown that the excitons can produce shift current under the steady light irradiation. The absence of the inversion symmetry, i.e., the noncentrosymmetric crystal structure, is essential for this effect, since otherwise the two contributions from k and −k cancel each other as discussed in Ref. [4]. In addition, the experimental test of the prediction in the present paper requires (i) well-defined exciton absorption peak separated from the electron-hole continuum, (ii) low enough temperature to suppress the thermal dissociation of excitons into electrons and holes, (iii) well-separated electrodes from the light irradiation spot to eliminate the contribution from the exciton dissociation at electrodes. It is also mentioned here that the shift current of excitons can be generalized to that of spin waves in noncentrosymmetric magnets, e.g., the electromagnons in chiral magnets.
Shift current of excitons can be also detected in optical measurements. When the incident light has two frequencies Ω 1 and Ω 2 , the second order nonlinear effect allows that two harmonics Ω 1 −Ω 2 and Ω 1 +Ω 2 are generated. In particular, the former one is used to generate the THz light [19,20], and corresponds to the shift current when Ω 1 = Ω 2 = Ω. Therefore, by tuning Ω 1 , Ω 2 , one can see whether the current remains finite in the limit of Ω 1 − Ω 2 → 0. This offers an experimental test of the dc shift current without the complications related to the contact to the leads. Furthermore, this indicates that the THz generation and the SHG are enhanced when the incident light is resonant to the exciton state below the band gap. One example of noncentrosymmetric materials to study such nonlinear optical effects of excitons would be transition metal dichalcogenide monolayers such as MoS 2 , because MoS 2 monolayers are noncentrosymmetric and known to show strong exciton binding [30][31][32].
A comment is in order for the mechanism of relaxations. In our model, the relaxation originates from the fact that each site is coupled to a heat bath with a fixed distribution function. This is introduced by the self energy Σ and realizes the nonequilibrium steady state with finite shift current. Here, it is assumed that the exchange of electrons between the system and the heat bath does not lead to a change in polarization. In contrast, the recombination of electron-hole pairs (which is also a source of relaxation) results in a decrease in polarization and reduces the shift current. Therefore, the shift current from the exciton formation requires a relaxation process which involves no change in polarization and whose efficiency is larger than the recombination process. For example, this requirement is satisfied by an isotropic heat bath such as a partially filled band that can exchange charge degrees of freedom with the two band system involved in the exciton formation.
Finally, the physical picture of the exciton shift current is sketched. The exciton formation results in the polarization due to the shift between a hole in the valence band and an electron in the conduction band which is quantified by the Berry phase. When excitons are constantly created in the nonequilibrium situation, the continuous increase of the polarization in time produces the steady dc current. This mechanism is analogous to the quantum Ratchet motion in the presence of the asymmetry, and in sharp contrast to the charge pumping in the ground state. In the latter case, large amplitude deformation of the Hamiltonian is required to achieve a nontrivial winding number; in the quantum Rachet motion, only a small amplitude oscillation of a parameter in the Hamiltonian is sufficient to support the constant dc current and energy supply. Therefore, the nonequilibrium states will offer a new avenue for the physics of Berry phase.
arXiv:1512.00549v3 [cond-mat.mes-hall] 8 Jul 2016k
E
valence band
conduction band
valence band
+1 photon
ACKNOWLEDGMENTSWe thank enlightening discussions with Y. Tokura, Y. Nakamura, M. Kawasaki, J. E. Moore, J. Orenstein, and B. M. Fregoso. This work was supported by the EPiQS initiative of the Gordon and Betty Moore Foundation (TM), and by JSPS Grant-in-Aid for Scientific Research (Grants No. 24224009 and No. 26103006) from MEXT, Japan, and ImPACT Program of Council for Science, Technology and Innovation (Cabinet office, Government of Japan) (NN).
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] |
[] |
In experiments with superconducting quantum circuits, characterizing the photon statistics of propagating microwave fields is a fundamental task. We quantify the n 2 + n photon number variance of thermal microwave photons emitted from a black-body radiator for mean photon numbers 0.05 n 1.5. We probe the fields using either correlation measurements or a transmon qubit coupled to a microwave resonator. Our experiments provide a precise quantitative characterization of weak microwave states and information on the noise emitted by a Josephson parametric amplifier. PACS numbers: 42.50.Pq,02.50.-r,85.25.HvAs propagating electromagnetic fields in general [1-3], propagating microwaves with photon numbers on the order of unity are essential for quantum computation[4,5], communication[6], and illumination [7-10] protocols. Because of their omnipresence in experimental setups, the investigation of propagating thermal states is a fundamental task. Specifically in the microwave regime, sophisticated experimental techniques for their generation, manipulation, and detection have been developed in recent years. In this context, an important aspect is the generation of propagating thermal microwaves using thermal emitters[11][12][13]. These emitters can be spatially separated from the setup components used for manipulation and detection[14,15], which allows one to individually control the emitter and the setup temperature. Due to the low energy of microwave photons, the detection of these fields typically requires the use of nearquantum-limited amplifiers[16][17][18], cross-correlation detectors [13, 14, 19], or superconducting qubits[20][21][22].The unique nature of propagating fields is reflected in their photon statistics, which is described by a probability distribution either in terms of the number states or in terms of moments. The former were studied by coupling the field to an atom or qubit and measuring the coherent dynamics[23][24][25]or by spectroscopic analysis[26]. The moment-based approach, in practice, requires knowledge on the average photon number n and its variance Var(n) to distinguish many states of interest. To this end, the second-order correlation function g (2) (τ ) has been measured to analyze the photon statistics of thermal[27][28][29]or quantum [30-32] emitters ever since the ground-breaking experiments of Hanbury Brown and Twiss[33,34]. While these experiments use the time delay τ as control parameter, at microwave frequencies the photon number n can be controlled conveniently[11,26,[35][36][37]. In the specific case of a thermal field at frequency ω, the Bose-Einstein distribution yields n(T ) = [exp( ω/k B T ) − 1] −1 and Var(n) = n 2 + n, which can be controlled by the temperature T of the emitter. In practice, one wants to distinguish this relation from both the classical limit Var(n) = n 2 and the Poissonian behavior Var(n) = n characteristic for coherent states[35]or shot noise[20,38]. Hence, as shown inFig.
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Photon Statistics of Propagating Thermal Microwaves
September 27, 2016)
J Goetz
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
S Pogorzalek
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
F Deppe
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
K G Fedorov
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
P Eder
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
Nanosystems Initiative Munich (NIM)
Schellingstraße 480799MünchenGermany
M Fischer
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
Nanosystems Initiative Munich (NIM)
Schellingstraße 480799MünchenGermany
F Wulschner
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
E Xie
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
Nanosystems Initiative Munich (NIM)
Schellingstraße 480799MünchenGermany
A Marx
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
R Gross
Walther-Meißner-Institut
85748Bayerische Akademie der Wissenschaften, GarchingGermany
Physik-Department
Technische Universität München
85748GarchingGermany
Nanosystems Initiative Munich (NIM)
Schellingstraße 480799MünchenGermany
Photon Statistics of Propagating Thermal Microwaves
September 27, 2016)
In experiments with superconducting quantum circuits, characterizing the photon statistics of propagating microwave fields is a fundamental task. We quantify the n 2 + n photon number variance of thermal microwave photons emitted from a black-body radiator for mean photon numbers 0.05 n 1.5. We probe the fields using either correlation measurements or a transmon qubit coupled to a microwave resonator. Our experiments provide a precise quantitative characterization of weak microwave states and information on the noise emitted by a Josephson parametric amplifier. PACS numbers: 42.50.Pq,02.50.-r,85.25.HvAs propagating electromagnetic fields in general [1-3], propagating microwaves with photon numbers on the order of unity are essential for quantum computation[4,5], communication[6], and illumination [7-10] protocols. Because of their omnipresence in experimental setups, the investigation of propagating thermal states is a fundamental task. Specifically in the microwave regime, sophisticated experimental techniques for their generation, manipulation, and detection have been developed in recent years. In this context, an important aspect is the generation of propagating thermal microwaves using thermal emitters[11][12][13]. These emitters can be spatially separated from the setup components used for manipulation and detection[14,15], which allows one to individually control the emitter and the setup temperature. Due to the low energy of microwave photons, the detection of these fields typically requires the use of nearquantum-limited amplifiers[16][17][18], cross-correlation detectors [13, 14, 19], or superconducting qubits[20][21][22].The unique nature of propagating fields is reflected in their photon statistics, which is described by a probability distribution either in terms of the number states or in terms of moments. The former were studied by coupling the field to an atom or qubit and measuring the coherent dynamics[23][24][25]or by spectroscopic analysis[26]. The moment-based approach, in practice, requires knowledge on the average photon number n and its variance Var(n) to distinguish many states of interest. To this end, the second-order correlation function g (2) (τ ) has been measured to analyze the photon statistics of thermal[27][28][29]or quantum [30-32] emitters ever since the ground-breaking experiments of Hanbury Brown and Twiss[33,34]. While these experiments use the time delay τ as control parameter, at microwave frequencies the photon number n can be controlled conveniently[11,26,[35][36][37]. In the specific case of a thermal field at frequency ω, the Bose-Einstein distribution yields n(T ) = [exp( ω/k B T ) − 1] −1 and Var(n) = n 2 + n, which can be controlled by the temperature T of the emitter. In practice, one wants to distinguish this relation from both the classical limit Var(n) = n 2 and the Poissonian behavior Var(n) = n characteristic for coherent states[35]or shot noise[20,38]. Hence, as shown inFig.
In experiments with superconducting quantum circuits, characterizing the photon statistics of propagating microwave fields is a fundamental task. We quantify the n 2 + n photon number variance of thermal microwave photons emitted from a black-body radiator for mean photon numbers 0.05 n 1.5. We probe the fields using either correlation measurements or a transmon qubit coupled to a microwave resonator. Our experiments provide a precise quantitative characterization of weak microwave states and information on the noise emitted by a Josephson parametric amplifier. As propagating electromagnetic fields in general [1][2][3], propagating microwaves with photon numbers on the order of unity are essential for quantum computation [4,5], communication [6], and illumination [7][8][9][10] protocols. Because of their omnipresence in experimental setups, the investigation of propagating thermal states is a fundamental task. Specifically in the microwave regime, sophisticated experimental techniques for their generation, manipulation, and detection have been developed in recent years. In this context, an important aspect is the generation of propagating thermal microwaves using thermal emitters [11][12][13]. These emitters can be spatially separated from the setup components used for manipulation and detection [14,15], which allows one to individually control the emitter and the setup temperature. Due to the low energy of microwave photons, the detection of these fields typically requires the use of nearquantum-limited amplifiers [16][17][18], cross-correlation detectors [13,14,19], or superconducting qubits [20][21][22].
The unique nature of propagating fields is reflected in their photon statistics, which is described by a probability distribution either in terms of the number states or in terms of moments. The former were studied by coupling the field to an atom or qubit and measuring the coherent dynamics [23][24][25] or by spectroscopic analysis [26]. The moment-based approach, in practice, requires knowledge on the average photon number n and its variance Var(n) to distinguish many states of interest. To this end, the second-order correlation function g (2) (τ ) has been measured to analyze the photon statistics of thermal [27][28][29] or quantum [30][31][32] emitters ever since the ground-breaking experiments of Hanbury Brown and Twiss [33,34]. While these experiments use the time delay τ as control parameter, at microwave frequencies the photon number n can be controlled conveniently [11,26,[35][36][37]. In the specific case of a thermal field at frequency ω, the Bose-Einstein distribution yields n(T ) = [exp( ω/k B T ) − 1] −1 and Var(n) = n 2 + n, which can be controlled by the temperature T of the emitter. In practice, one wants to distinguish this relation from both the classical limit Var(n) = n 2 and the Poissonian behavior Var(n) = n characteristic for coherent states [35] or shot noise [20,38]. Hence, as shown in Fig. 1, the most relevant regime for experiments is 0.05 n 1, which translates into temperatures between 100 mK and 1 K at approximately 6 GHz for the thermal emitter [22].
In this Letter, we experimentally confirm the theoretically expected photon number variance Var(n) of thermal microwave fields for 0.05 n 1.5 using two fundamentally distinct experimental setups. On the one hand, we use a superconducting transmon qubit [39] interacting with the propagating fields via a dispersively coupled microwave resonator. Differently to approaches relying on the coherent dynamics [23][24][25], in our experiments the change of the qubit dephasing rate induced by the field directly reflects the photon number variance. We furthermore can distinguish between the Poissonian statistics of coherent states and shot noise because the resonator has a different decay constant for these two input fields. On the other hand, we extract the super-Poissonian photon statistics of propagating thermal microwaves from direct correlation measurements and from measurements using a near-quantum-limited Josephson parametric am- plifier (JPA) [17,40] as preamplifier. The results show that the noise added by the JPA inevitably alters the photon statistics of the amplified field. Our results provide a quantitative picture of propagating thermal microwaves, which is especially relevant for the characterization of more advanced quantum states in the presence of unavoidable thermal background fields. With respect to superconducting qubits, we gain systematic insight into a dephasing mechanism which may become relevant for state-of-the-art devices with long coherence times [41,42].
In our experiments, we generate the thermal fields using a temperature-controllable, 50 Ω-matched attenuator acting as a black-body emitter. This emitter is thermally only weakly coupled to the 35 mK base temperature stage of a dilution refrigerator. Heating the attenuator up to 1.5 K results in the emission of thermal microwave radiation, which we guide to our detection setups. Experimentally, we achieve a high photon number stability δn/n 0.01 due to the precise temperature stabilization of the emitter. In addition, we generate coherent states using a microwave source and shot noise with 200 MHz bandwidth using an arbitrary function generator (AFG) at room temperature.
We measure the photon number variance of propagating fields with the qubit setup depicted in Fig. 2(a). To this end, we operate a frequency-tunable transmon qubit at its maximum transition frequency ω q /2π = 6.92 GHz (see Ref. 22 for experimental details).
The qubit is coupled with strength g/2π 67 MHz to a quarterwavelength coplanar waveguide resonator with resonance frequency ω r /2π = 6.07 GHz and external coupling rate κ x /2π = 8.5 MHz. Hence, the system is in the dispersive regime, where the detuning δ ≡ ω q − ω r fulfills χ g. Here, [39] χ ≡ [g 2 /δ][α/(δ + α)] −2π × 3.11 MHz and α/2π −315 MHz is the transmon anharmonicity. In the dispersive regime, the qubit couples to the photon number n r in the resonator via the interaction Hamiltonian H int = χ[n r + 1/2]σ z . Because the coupling is mediated by the Pauli operatorσ z , temporal fluctuations n r (τ ) introduce qubit dephasing [43]. More precisely, the fluctuations are characterized by the correlator C(τ ) = n r (0)n r (τ ) and generate a shift δϕ(τ ) of the qubit phase. While the first moment of this phase shift has a vanishing arithmetic mean, the second moment [44]
δϕ 2 = 4χ 2 τ 0 dτ C(τ ) enters into the Ram- sey decay envelope exp[−γ 1 (n r )τ /2 − γ ϕ0 τ − δϕ 2 /2].
Here, γ 1 (n r ) is the total qubit relaxation rate and γ ϕ0 is the qubit dephasing rate due to all other noise sources except for those described by C(τ ). Thermal fields exhibit a super-Poissonian correlator [45,46] C th (τ ) =(n 2 r + n r ) exp(−κ x τ ). The Poissonian nature of coherent states and shot noise inside a resonator follows C coh (τ ) = n r exp(−κ x τ /2) and C sh (τ ) = n r exp(−κ x τ ), respectively. For all three states one obtains δϕ 2 /2 = γ ϕn (n r )τ , i.e., an exponential de-cay envelope. The photon-number-dependent dephasing rates are then defined by [35,41,[44][45][46] γ th ϕn (n r ) =κ x θ 2 0 (n 2 r + n r )≡ s th 0 (n 2 r + n r ) ,
γ coh ϕn (n r ) = 2κ x θ 2 0 n r ≡ s coh 0 n r ,(1)γ sh ϕn (n r ) = κ x θ 2 0 n r ≡ s sh 0 n r .(2)
Here, [44] θ 0 = tan −1 (2χ/κ x ) is the accumulated phase of the resonator photons due to the interaction with the qubit. Remarkably, the factor two between γ coh ϕn and γ sh ϕn is due to the fact that the impact of the fluctuations onto the qubit is larger if the resonator decays slower [43,45]. For thermal states, we control n r via the temperature of the emitter, n r ∝ n(T ). For coherent states and shot noise, we vary the output power P of the microwave generator, n r ∝ P . The exact calibra- tion procedure is presented in Ref. 22. As a consequence of Eqs. (1) -(3), measurements of the Ramsey decay rate γ 2 (n r ) = γ 1 (n r )/2 + γ ϕ0 + γ ϕn (n r ) allow us to extract the photon number variance as a function of n r . We obtain γ 1 (n r ) from an independent measurement [22]. Furthermore, we emphasize that during our sweeps of the attenuator temperature, the sample box is stabilized at 35 mK. Also, all low-frequency components of the thermal field, usually responsible for dephasing, are strongly suppressed by the filter function of the resonator. Therefore, γ ϕ0 can be taken as a constant and we can extract γ ϕn from a numerical fit to the decay envelope of a Ramsey time trace.
In the absence of external microwave fields, the transmon qubit is relaxation-limited with the rates γ 1 (n r 0)/2π 4 MHz and γ 2 (n r 0)/2π 2 MHz. In Fig. 2(b), we show the Ramsey time traces for the attenuator temperatures T = 50 mK and T = 1 K. As expected, the latter shows a significantly increased Ramsey decay rate. A systematic temperature sweep reveals γ th ϕn (n r ) ∝ n 2 r + n r as displayed in Fig. 2(c). For small photon numbers n r 0.5, we observe that the dephasing rate approaches a linear trend with slope s th 0 ≡ ∂γ th ϕn /∂n r | nr = 0 . This finite slope clearly allows us to rule out the validity of the classical limit Var(n r ) = n 2 r in this regime. From a fit of Eq. (1) to the data, we find s th 0 /2π = 3.9 MHz, which is marginally enhanced compared to the expected value κ x θ 2 0 /2π = 3.4 MHz. We attribute this slight deviation to a constant thermal background field emitted from attenuators at higher temperature stages and from the sample environment. Applying a beam splitter model [47], we extract the reasonable contribution of n th c = 0.15 photons. This value corresponds to an effective mode temperature of approximately 140 mK.
As a cross-check for our setup, we confirm the wellexplored [20,35,44] linear variance of fields with Poissonian photon statistics. To this end, we first expose the resonator to shot noise emitted at room temperature by the AFG [48]. As shown in Fig. 2(c), we indeed find a constant slope s sh ≡ ∂γ sh ϕn /∂n r 2π × 4.6 MHz, which is in reasonable agreement with s th 0 . In terms of additional thermal population and effective mode temperature [47], we obtain n sh c 0.19 ≈ n th c and 150 mK, respectively. In the next step, we investigate measurementinduced dephasing caused by coherent states as displayed in Fig. 2(c).
We again find a linear slope s coh ≡ ∂γ coh ϕn /∂n r 2π × 9.3 MHz ≈ 2s sh . The excellent quantitative agreement with the shot noise result is also reflected in n coh c = n sh c . Due to the enhancement of s coh by a factor of two, the qubit can discriminate between a coherent field and thermal field as well as shot noise already for n r 0.3.
In order to complement our studies of thermal microwaves with the qubit setup, we directly probe field correlations with the dual-path state reconstruction method [12][13][14]49]. We use the setup depicted in Fig. 3(a), where a cryogenic beam splitter equally divides the signal along two paths, which are subsequently amplified independently. From the averaged auto-and crosscorrelations, we retrieve all signal moments (â † ) nâm up to fourth order (0 ≤ n + m ≤ 4 with n, m ∈ N 0 ) in terms of the annihilation and creation operators,â andâ † . To calibrate the average photon number n bs = â †â ∝ n(T ) at the input of the beam splitter, we perform a Planck spectroscopy experiment [12]. More details on the setup can be found in Ref. 15. Notwithstanding the very different experimental requirements in the microwave regime, direct correlation measurements on propagating light fields are inspired from quantum optics. For this reason, we characterize the photon number variance of the thermal microwave fields via the unnormalized correlation functiong (2) (0) ≡ n 2 bs g (2) (0) = Var(n bs ) − n bs + n 2 bs .
As shown in Fig. 3(b), the correlation functiong (2) (0) of the thermal source follows the expected quadratic behavior. A numerical fit of the polynomial functioñ g (2) (0) = ρ n 2 bs using ρ as a free parameter yields ρ = 2.07. This result coincides nicely withg (2) (0) = 2n 2 bs predicted for thermal states by Eq. (4). In the same way as with the qubit setup, we are therefore able to reliably map out the n 2 + n dependence and not only the classical n 2 limit experimentally found in earlier work [11].
To lower the statistical scatter of the data points in Fig. 3(b), we repeat the correlation measurement using a JPA operated in the phase insensitive mode. In this mode, the JPA works as a near-quantum-limited, phase-preserving amplifier [17] with power gain G 1. At the input of the beam splitter, one then obtains n bs ≈ G(n jpa + n n + 1). Here, n jpa ∝ n(T ) are the signal photons and n n are the noise photons added by the JPA, which we again obtain from a Planck spectroscopy experiment [12]. We compare measurements using two different JPAs (JPA 1 and JPA 2) based on frequency-tunable quarter wavelength resonators with operating frequencies ω jpa /2π 5.35 GHz and typical gains G 14 dB (specific parameters are summarized in Ref. 47). After calibrating for G, we record the modified correlation functioñ g (2) (0) = 2(n jpa + n n + 1) 2 ,
which can be derived from an input-output model for the JPA [47]. In Eq. (5), there is an n jpa -independent offset g (2) n (0) = 2n 2 n + 4n n + 2 due to the JPA gain and noise. In our model, we assume that the JPA noise is thermal, i.e., Var(n n ) = n 2 n + n n . In Fig. 3(c) we plot the experimentally obtained correlationsg (2) (0) −g (2) n (0) versus the photon number n jpa at the JPA input. From fits to the formula ρn 2 jpa + ξn jpa , we find ρ 2.2 in all three data sets in agreement with the expected value of ρ = 2. Therefore, also the JPA assisted measurements confirm a super-Poissonian statistics of the thermal fields. From the fits, we also find that the values ξ is reduced by a factor of approximately 2 compared to the expected value 4 + 4n n . This observation is confirmed by the values extracted forg (2) n (0), which deviate to a similar extent. The reduced experimental values suggest that the JPA noise appears to contain a significant contribution with non-thermal statistics.
Finally, we compare the performance of the qubit and the dual-path setup. Although we operate on and below the single photon level, the qubit and the dual-path setup (without JPA) systematically reproduce the n 2 + n law with a high accuracy. Currently, the statistical spread for the qubit setup is one order of magnitude lower than the one for the dual-path setup. The accuracy of the qubit setup is limited by the low-frequency variations of the qubit relaxation rate described in Ref. 22. Their standard deviation of 5 % well explains the spread of the experimental data points in Fig. 2(c). Assuming that these variations decrease proportionally to the qubit decoherence rate, we estimate that for the best performing superconducting qubits [41], the accuracy can be improved by at least two orders of magnitude. The dual-path setup (without the JPA) is limited by the data processing rate of our digitizer card and by the noise temperature T n 3 K of the cryogenic amplifiers. When the JPA is on, the noise temperature of these amplifiers is insignificant. Concerning adaptability, the dual-path setup in principle gives access to all signal moments, whereas the qubit is limited to amplitude and power correlations. While our measurements including a JPA decrease the statistical spread by two orders of magnitude, they also indicate that the statistics of the JPA noise can influence the statistics of the amplified field.
In conclusion, we have quantitatively characterized the photon number variance of propagating thermal microwaves using two fundamentally different approaches: indirect measurements with a superconducting qubitresonator system and direct ones, with a dual-path detector. With both setups, we are able to quantitatively recover the n 2 + n photon number variance of thermal fields in the single photon regime with a high resolution in comparison with existing experimental achievements [11]. In particular, we analyze the resolution limits and find that they can be improved by several orders of magnitude in both setups. For our current dual-path setup, we make the remarkable observation that noise added by the JPAs has a significant non-thermal contribution. Our results demonstrate that the three types of propagating microwave states we investigate can be reliably distinguished below the single photon level in an experiment by their photon statistics. Therefore, both setups are promising candidates to explore decoherence mechanisms possibly limiting high-performance superconducting qubits [41,42] and the properties of more advanced quantum microwave states.
The JPAs used in this work are kindly provided by K. Inomata (RIKEN Center for Emergent Matter Science), T. Yamamoto (NEC IoT Device Research Laboratories), and Y. Nakamura (RIKEN, RCAST at the University of Tokyo). We thank E. Solano, R. Di Candia, M. Sanz (Department of Physical Chemistry, University of the Basque Country) for fruitful discussions on the correlation measurement setup. We acknowledge financial support from the German Research Foundation through SFB 631 and FE 1564/1-1, EU projects CCQED, PROMISCE, the doctorate programs ExQM of the Elite Network of Bavaria, and the International Max Planck Research School "Quantum Science and Technology". Similar to the calculations of attenuated propagating microwaves, we calculate the variance of an amplified thermal field using input-output relations. Following Ref. 49 and Ref. 50, we describe the amplified field by the operator a(t) = √ Gb + √ G − 1ĉ † , where n bs = â †â . The quantity n n = ĉ †ĉ describes the noise photons added by the JPA, which we assume to be thermal. Based on these assumptions, we obtain the photon number variance Var(n bs ) = (â †â − â †â ) 2 = G 2 n 2 jpa + G 2 n jpa + G(G − 1)n jpa + 2G(G − 1)n jpa n n + (G − 1) 2 n 2 n + (G − 1) 2 n n + G(G − 1)n n + G(G − 1) ,
which approaches the variance n 2 jpa + n jpa of a thermal state for G → 1 (no amplification). For strong amplification (G 1), we obtain Var(n bs ) ≈ G 2 (n jpa + n n + 1) 2 = n 2 bs . As a consequence, the unnormalized g (2) function of the amplified field becomesg (2) (0) ≡ n 2 bs g (2) (0) = Var(n bs ) − n bs + n 2 bs ≈ 2G 2 (n jpa + n n + 1) 2 .
We fit this relation to our data as discussed in the main text.
PACS numbers: 42.50.Pq,02.50.-r,85.25.Hv
Figure 1 .
1Photon number correlations. [Var(n)] 1/2 plotted versus photon number for thermal fields (black), their classical limit (red), and coherent states (blue). The inset shows the regime that we capture in our experiments. arXiv:1609.07353v2 [quant-ph] 26 Sep 2016
Figure 2 .
2(a) Sketch of the qubit setup. We measure the photon number variance Var(n) of microwave fields encoded in the photon correlator C(τ ) by detecting the dephasing rate γϕn of a superconducting qubit. (b) Qubit excited state probability pe for a Ramsey experiment plotted versus waiting time τ between two π/2 pulses. The solid lines are exponentially decaying sinusoidal fits. The inset shows the Ramsey pulse sequence followed by a readout (RO) pulse. (c) Qubit dephasing rates γϕn of prototypical input fields plotted versus the average resonator population nr. The super-Poissonian n 2 r + nr statistics of thermal fields is fitted using Eq. (1). The blue and the grey line are fits of Eq. (2) and Eq. (3) to the data for coherent states and shot noise, respectively.
Figure 3 .
3(a) Sketch of the dual-path setup, which we use to directly probe field correlations between two amplification chains behind a cryogenic microwave beam splitter. We can switch on and off the JPA. (b) Unnormalized second-order correlation functiong(2) (0) plotted versus photon number n bs at the beam splitter input without using the JPA. The solid line is a fit to the data using the functiong (2) (0) = ρ n 2 bs . (c) Unnormalized second-order correlation functiong(2) (0) corrected for the constant offsetg
and plotted versus the photon number njpa at the JPA input. For the measurements of JPA 2a and JPA 2b we use slightly different operating points described in detail in Ref.47.
PHOTON NUMBER VARIANCE OF ATTENUATED AND AMPLIFIED SIGNALSIn this section, we discuss how attenuation (loss) and amplification processes modify the photon statistics of a signal. We describe the attenuation process with a beam splitter model. The attenuation factor η translates into the splitting ratio of the device and a thermal state with population nc is incident of the second input port. We define the thermal state emitted by the heatable attenuator with the bosonic operatorsb † andb and n = b †b . Similarly, the background thermal photons are defined byĉ † andĉ leading to n c = ĉ †ĉ . That way, we obtainâ(t) = √ ηb(t) + √ 1 − ηĉ(t) for the attenuated input state leading to the total photon number n r = â †â = ηn + (1 − η)n c . Then, following Ref.43, the correlatordescribes the total field entering the resonator. From Eq. (S1) we see that the beam splitter model predicts the thermal photon statistics of the emitted field for η → 1 (no background field) and the thermal photon statistics of the background field for η → 0 (strong background field). In a similar way, we calculate the correlator of an attenuated coherent fieldwhich approaches the variance of a coherent state for η → 1 and the thermal photon statistics of the cold attenuator for η → 0.
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See Supplemental Material for experimental techniques and theoretical methods. See Supplemental Material for experimental techniques and theoretical methods.
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Supplemental Materials: Photon Statistics of Propagating Thermal Microwaves SAMPLE DETAILS AND EXPERIMENTAL SETUP. Supplemental Materials: Photon Statistics of Propagating Thermal Microwaves SAMPLE DETAILS AND EXPERIMENTAL SETUP
For the measurements based on the dual-path setup, we use the experimental setup presented in Ref. 15. This setup comprises a flux-driven Josephson parametric amplifier (JPA) with gain G consisting of a quarter-wavelength transmission line resonator, which is short-circuited to the ground by a DC SQUID. We couple an on-chip antenna inductively to the DC SQUID loop to apply a strong coherent pump tone ω p at approximately twice the resonant frequency ω jpa of the JPA. For the measurements based on the qubit setup, we use the experimental setup presented in Ref. 22.. This pump scheme amplifies the incoming signal with a typical gain G as summarized in Tab. IFor the measurements based on the qubit setup, we use the experimental setup presented in Ref. 22. For the measurements based on the dual-path setup, we use the experimental setup presented in Ref. 15. This setup comprises a flux-driven Josephson parametric amplifier (JPA) with gain G consisting of a quarter-wavelength transmission line resonator, which is short-circuited to the ground by a DC SQUID. We couple an on-chip antenna inductively to the DC SQUID loop to apply a strong coherent pump tone ω p at approximately twice the resonant frequency ω jpa of the JPA. This pump scheme amplifies the incoming signal with a typical gain G as summarized in Tab. I.
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[] |
[
"Balance functions in coalescence model",
"Balance functions in coalescence model"
] |
[
"A Bialas [email protected] ",
"M Smoluchowski ",
"\nInstitute of Physics Jagellonian University\nCracow\n",
"\nReymonta 430-059KrakowPoland\n"
] |
[
"Institute of Physics Jagellonian University\nCracow",
"Reymonta 430-059KrakowPoland"
] |
[] |
It is shown that the quark-antiquark coalescence mechanism for pion production allows to explain the small pseudorapidity width of the balance function observed for central collisions of heavy ions, provided effects of the finite acceptance region and of the transverse flow are taken into account. In contrast, the standard hadronic cluster model is not compatible with this data. * ; 1 Note that in this model the quark-gluon structure of the system is entirely ignored.
|
10.1016/j.physletb.2003.10.106
|
[
"https://arxiv.org/pdf/hep-ph/0308245v2.pdf"
] | 119,468,616 |
hep-ph/0308245
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21162c9acf599925357a8a542a8c4c8586253df4
|
Balance functions in coalescence model
21 Oct 2003 November 1, 2018
A Bialas [email protected]
M Smoluchowski
Institute of Physics Jagellonian University
Cracow
Reymonta 430-059KrakowPoland
Balance functions in coalescence model
21 Oct 2003 November 1, 2018
It is shown that the quark-antiquark coalescence mechanism for pion production allows to explain the small pseudorapidity width of the balance function observed for central collisions of heavy ions, provided effects of the finite acceptance region and of the transverse flow are taken into account. In contrast, the standard hadronic cluster model is not compatible with this data. * ; 1 Note that in this model the quark-gluon structure of the system is entirely ignored.
1.
Recently, measurements of balance functions [1] in central collisions of heavy ions were reported by the STAR collaboration [2]. The striking feature observed in the data is the small width of the balance functions (in rapidity and in pseudorapidity), as compared to the expectations from the expanding thermally equilibrated quark-gluon plasma [3]. This indicates that hadronization occurs only at the very late stage of the development of the system [4].
One may then ask if the hadronization properties of the system produced in central collisions of heavy ions (as reflected in the balance function) are similar to those observed in nucleon-nucleon collisions. To investigate this problem we have evaluated the expected width of the balance function in pseudorapidity, using the pion cluster model which some time ago was successfully applied to nucleon-nucleon data [5] 1 . Corrections due to the finite acceptance region of [2] and the effects of the transverse flow were included.
Our estimates show that to obtain consistency with the data of [2] for central collisions, the decay width of the pion cluster (in its rest frame) must be substantially narrower than that corresponding to isotropic decay. Thus one must conclude that the hadronization of the system produced in heavy ion collisions is rather different from that produced in hadronic collisions, where isotropic clusters can approximately account for the data [5]
2 .
Looking for a more adequate description, we considered the coalescence model [6], which we generalized to include correlations. We thus assume that the hadronization proceeds in three steps. First, partons form neutral clusters. Subsequently, each cluster decays into some number of gluons and one 3 qq pair (either uū or dd) 4 . Quarks and antiquarks then recombine into positive, negative and neutral pions. The remaining gluons form new clusters and the process is continuing until all partons are transformed into hadrons.
We show that this generalized coalescence model gives a good description of the data from [2], provided the decay of clusters into quarks and antiquarks is isotropic, which seems a rather natural assumption. The obtained reduction of the width of the balance function (essential to account for the data) is a natural consequence of the coalescence process. It follows simply from the fact that the dispersion of the average of two independent random variables is smaller than the dispersion of each of them by factor √ 2. It should be emphasized that we are discussing here only the angular distributions (expressed in terms of pseudorapidity). The natural assumption of approximately isotropic and uncorrelated cluster decay is then sufficient to describe the width of the balance function. This is not the case for rapidity distribution where more detailed information on cluster decay is needed.
Our conclusion is that the generalized coalescence model provides a natural explanation of the very narrow width of the balance function observed in [2]. This result is a consequence of a very general fenomenon, characteristic for coalescence mechanism and thus, in our opinion, it does not depend on details of the specific structure of correlations between partons proposed in this note. Since, as we have seen, the model based solely on hadronic degrees of freedom is not adequate, we feel that our result provides a rather strong argument in favour of the coalescence scenario.
2. The balance functions can be expressed in terms of the single and double particle densities [3,7]. Assuming, for simplicity, the (+−) symmetry we have
B(∆ 2 |∆ 1 ) = D(−, ∆ 2 |+, ∆ 1 ) − D(+, ∆ 2 |+, ∆ 1 )(1)
with
D(+, ∆ 2 |+, ∆ 1 ) = D(−, ∆ 2 |−, ∆ 1 ) = ∆ 2 dη 2 ∆ 1 dη 1 d 2 n ++ dη 1 dη 2 ∆ 1 dη + dn + dη + (2) D(−, ∆ 2 |+, ∆ 1 ) = D(+, ∆ 2 |−, ∆ 1 ) = ∆ 2 dη 2 ∆ 1 dη 1 d 2 n +− dη 1 dη 2 ∆ 1 dη + dn + dη +(3)
where dn/dη and dn/dη 1 dη 2 are the corresponding particle densities in pseudorapidity. The measurements of STAR require both particles to be in an interval −∆ ≤ η 1 , η 2 ≤ ∆ (acceptance) while the difference of (pseudo)rapidities is kept fixed. This suggests a change of variables:
η 1 − η 2 = δ; (η 1 + η 2 )/2 = z.(4)
The integrations must be performed over dz with δ being fixed. This implies that z must be kept inside the interval −∆ ≤ z ≤∆ where∆ = ∆ − |δ|/2. Thus the balance function measured in [2] is given by
B s (δ; ∆) = −∆ −∆ dz [d 2 n +− /dzdδ − d 2 n ++ /dzdδ] +∆ −∆ dη + dn + /dη +(5)
3. Consider first a model in which pions are produced in neutral, isotropic clusters. It is well known that such models can account for the gross features of the nucleon-nucleon data [5]. The distribution of clusters is denoted by ρ c (Y ) where Y is the cluster pseudorapidity.
To simplify the problem we assume that clusters decay into two charged particles and any number of neutrals.
The distribution in cluster decay is
dN +− dη + dη − = h(z − Y )f (η + − Y )f (η − − Y )(6)
where h(z − Y ) is responsible for correlations: h ≡ 1 if the decay products are uncorrelated. The single particle distribution in cluster decay ρ(η + − Y ) is obtained by integration of (6) over rapidity of one particle. Both distributions are normalized to unity.
The distribution of all particles is the convolution
dn + dη + = dY ρ c (Y )ρ(η + − Y )(7)
with identical formula for negative particles.
To evaluate the two-particle distributions one has to take into account that some pairs may come from different clusters and some others from one cluster. As is well-known (and can be easily confirmed by explicit calculation) the contribution from different clusters cancels in the balance function and thus only (+−) pairs from one cluster do contribute. Their distribution is given by
d 2 n +− dη + dη − = dY ρ c (Y )h(z − Y )f (η + − Y )f (η − − Y ).(8)
Introducing (8) into (5) we have
B(δ; ∆) = +∆ −∆ dz dY ρ c (Y )h(z − Y )f (η + − Y )f (η − − Y ) ∆ −∆ dη dY ρ(η − Y ) .(9)
4. To continue, we perform a simple exercise, assuming that all functions are Gaussians. We take 5
ρ c (Y ) = < N c > A √ π exp(−Y 2 /A 2 ); f (u) = 1 f √ π exp(−u 2 /f 2 ); h(u) = 1 + f 2 /2g 2 exp(−u 2 /g 2 )(10)
Using (10) one can evaluate the single particle distribution in cluster decay:
ρ(η) = 1 a √ π exp(−η 2 /a 2 ); a = f 1 + f 2 /4g 2 1 + f 2 /2g 2 .(11)
This gives
dn + dη + = < N c > π(A 2 + a 2 ) exp[−η 2 + /(A 2 + a 2 )]; dY ρ c (Y )h(z − Y )f (η 1 − Y )f (η 2 − Y ) = =< N c > 1 f √ 2π exp(−δ 2 /2f 2 ) 1 F √ π exp(−z 2 /F 2 )(12)
with
F 2 = f 2 + 2A 2 [1 + f 2 /2g 2 ] 2[1 + f 2 /2g 2 ](13)
This result introduced into (9) allows to express the integrals over z and y in terms of the error function. The result is
B(δ; ∆) = 1 f √ 2π exp(−δ 2 /2f 2 ) erf [(∆ − |δ|/2)/F ] erf [∆/ √ A 2 + a 2 ](14)
which completes the calculation. This exercise shows that -in the limit of large acceptance-the width of the balance function is determined by the parameter f which describes the cluster decay.
5. These results can be used to evaluate expectations from isotropically decaying clusters which were found roughly compatible with the data on hadron-hadron collisions [5]. In this case the cluster decay distribution is
ρ(η) = 1 2(cosh η) 2(15)
giving the cluster decay width < |η| >= log 2. This can be approximated by a Gaussian of the form (11) with a = log 2 √ π ≈ 1.23. Ignoring for the time beeing the effect of finite acceptance, we thus conclude from (11) that the expected width of the balance function < |δ| > must be larger than a √ 2/ √ π ≈ .98 and thus by far exceeds the one measured in [2].
Finite acceptance [∆ = 1.3] of the STAR measurements [2] reduces the observed width of the balance function, as seen from (14). This is not sufficient, however, to bring the data in agreement with the model of isotropic pion clusters. The width < |δ| > calculated from (11) for isotropic clusters with f /g = 0 (uncorrelated dcay) equals 0.67 at A = 3.5 (this value of A is roughly consistent with data for central collisions [8]), and does not change significantly when A varies around this value. Since, furthermore, < |δ| > increases with increasing f /g, there is no chance to meet the experimental value of 0.55.
Another effect which may be responsible for the small width of the balance function is the transverse flow. Indeed, the clusters which are isotropic in their rest frame will not appear isotropic when moving with a transverse velocity. As shown in [9], the distribution (15) is then -to a good approximationreplaced by
ρ(η) = 1 2 cosh 2 η cosh Y ⊥ [1 + sinh 2 Y ⊥ tanh 2 η] 3/2 (16) where Y ⊥ is the transverse rapidity, Y ⊥ = 0.5 log[(1 + v)/(1 − v)],
and v is the tranverse velocity of the cluster. We have calculated the width of the balance function with (16) approximated by a Gaussian 6 of the same width as that of (16). The results are shown in Fig.1 where < |δ| > is plotted versus v, for A = 3.5. The measured values for most central events (as reported in [2]) are also indicated. One sees that the calculated width decreases with increasing transverse velocity of clusters. One also sees that to obtain quantitative agreement with data the transverse velocity must approach 0.8c. Such a large value seems difficult to reconcile with other estimates of the transverse velocity [10].
6. The pion cluster model discussed so far ignores entirely the parton structure of the final system of hadrons. One may therefore not be surprized that it fails to describe the data from central heavy ion collisions. This argument suggests to try a model in which the parton structure is built in from the beginning. To this end we investigated coalescence model [6] which we generalized to include the correlations inside the system.
To introduce correlations we assume that -just before hadronization-the QGP forms the weakly correlated neutral clusters. The clusters decay into quarks 4 , antiquarks and gluons. One cluster provides one 3 qq pair (either uū or dd) and any number of gluons. In the final step quarks and antiquarks coalesce into observed hadrons. The remaining gluons form again neutral clusters and the process continues.
Thus the model we consider is basically the well-known coalescence model [6] supplemented by a prescription for correlations. Since the coalescence model was rather succesful in description of single particle spectra in central collisions of heavy ions [6,11], it seems worthwhile to investigate its extention to correlation phenomena (see also [12]). Admittedly, the proposed extension is very simple -perhaps even simplistic. It contains, however, all ingredients necessary to formulate and study the width of balance functions which is of interest in this paper. Therefore we do not find necessary at the moment to formulate and discuss a more general and/or detailed approach.
7. To evaluate the balance function we need the distribution of pairs of charged pions, same charge as well as opposite charge. The pairs of same charge can be constructed by coalescence of the decay products of four clusters (two U-clusters and two D-clusters) 7 . The distribution of the pairs of opposite charge consists of two terms: one identical to the distribution of same charge pairs and another one, arizing from coalescence of the decay products of two clusters (one U-cluster and one D-cluster). Thus the contributions involving four clusters exactly cancel and we only have to consider the distribution of pions of opposite charge which result from coalescence of decay products of one U-cluster and one D-cluster.
This distribution can be expressed as
ρ(η + , η − ) = dY U dY D ρ G (Y C , ∆ Y ) dη u dηūf q (η u − Y U )f q (ηū − Y U )h q ([η u + ηū]/2 − Y U ) dη d dηdf q (η d − Y D )f q (ηd − Y D )h q ([η d + ηd]/2 − Y D ) δ[η + − (η u + ηd)/2]δ[η − − (η d + ηū)/2]Φ[η u − ηd]Φ[η d − ηū](17)
where f q and h q are responsible for the distribution of quarks in decay of either U or D cluster, while Φ summarizes the properties of the coalescence process. Finally, ρ G (Y C , ∆ Y ) denotes the joint distribution of U and D clusters with average rapidity Y C , where
Y C = Y U + Y D 2 ; ∆ Y = Y U − Y D(18)
To simplify the discussion, in the following we shall assume that ρ G factorizes:
ρ G (Y C , ∆ Y ) = ρ C (Y C )ρ(∆ Y )(19)
Taking advantage of the delta functions we can rewrite (17) as
ρ(η + , η − ) = dY C d∆ Y ρ G (Y C , ∆ Y ) du + du − f q η + + u + 2 − Y C f q η − + u − 2 − Y C h q z + u + + u − 4 − Y C f q η − − u − 2 − Y C f q η + − u + 2 − Y C h q z − u + + u − 4 − Y C Φ (u + + ∆ Y ) Φ (u − + ∆ Y )(20)
where z = (η + + η − )/2. To proceed, we again consider Gaussians
f q (x) = 1 c √ π e −x 2 /c 2 ; h q (x) = 1 + a 2 /2h 2 exp[−x 2 /h 2 ] Φ(x) = 1 p √ π e −x 2 /p 2 .(21)
With this Ansatz, the integrals over du + du − can be performed. The result is
ρ(η + , η − ) = Ce −δ 2 /c 2 dY C ρ C (Y C ) exp − 4 [z − Y C ] 2 c 2 (1 + c 2 /2h 2 )(22)
where δ = η + − η − and C is a constant, irreleveant for further discussion. The formula (22) can be now introduced into (5) and thus the balance function can be calculated. In the limit of very large acceptance we obtain
B s (δ; ∆) ∆ → ∞ = 1 c √ π e −δ 2 /c 2(23)
One sees that -in this limit-the width of the balance function depends on one parameter which -to a large degree-determines also the distribution in decay of a cluster into free quark and antiquark. Indeed, using (17) and (21), one can show that the decay distribution in the rest frame of the cluster is given by
ρ p (η u ) = 1 d √ π exp − η 2 u d 2(24)
where
d 2 = c 2 1 + c 2 /4h 2 1 + c 2 /2h 2 .(25)
8. It seems natural to assume that -in their rest frame-clusters decay isotropically. This means that their decay distribution is given by (15), the same as for the clusters of pions considered before. It follows that -for an uncorrelated decay (c/h ≈ 0)-the parameters f in (10) and c in (21) are identical. Comparing (14) and (23) we thus conclude that in the coalescence model the width of the balance function is expected to be by factor √ 2 smaller than that obtained for pion clusters. The reason is clear: the dispersion of the pion rapidity is reduced by precisely this factor when the pion is formed by random coalescence of a quark and an antiquark.
Repeating the argument of the section 5, we thus conclude that -ignoring for the moment the corrections for finite acceptance and effects of transverse flow-the width of the balance function is expected to lie between .69 and .98 (the lower limit is obtained for α = (c/h) 2 = 0, i.e. when the decay products of a cluster are uncorrelated).
To compare this result with the data we have to estimate the corrections. To this end we take the Gaussian Ansatz for ρ C :
ρ C (Y C ) = 1 A √ π e −Y 2 C /A 2(26)
which allows to evaluate explicitely the integrals in (22). We obtain
B s (δ; ∆) = 1 c √ π e −δ 2 /c 2 erf [2(∆ − |δ|/2)/b] erf [2∆/ √ b 2 + c 2 ](27)
where
b 2 = c 2 + 4A 2 (1 + c 2 /2h 2 ) 1 + c 2 /2h 2(28)
Using (27) one can now follow the argument of section 5 and evaluate the width of the balance function, taking into account the finite acceptance and the transverse flow. In Figure 1 the width of the balance function evaluated from (27) is plotted versus v for A = 3.5 and two values of the parameter α = c 2 /h 2 . One sees that these effects reduce substantially the calculated width. The value found in [2] for central collisions is reproduced with transverse velocity below 0.5, consistent with other estimates of the transverse flow [10].
One also sees from the Fig. 1 that in the coalescence model the calculated width is smaller than the value 0.65 found in [2] for peripheral collisions. This is not surprizing: in peripheral collisions a substantial part of the particle production should resemble the elementary nucleon-nucleon collisions which are not expected to follow the coalescence mechanism [11] and are characterized by a significantly larger width of the balance function [1,5]. As seen from Fig.1, the width of the balance function calculated from the pion cluster model (adequate for nucleon-nucleon collisions) is indeed close to 0.65.
9. In conclusion, we have shown that the coalescence mechanism implies a substantial reduction of the pseudorapidity width of the balance function. This allows to explain the small width observed for central collisions of heavy ions [2], provided the corrections due to the finite acceptance region and to the transverse flow are taken into account. This result supports the coalescence mechanism as the final stage of the process of hadronization.
Figure 1 :
1Width of balance function versus velocity of transverse flow.
This conclusion is not surprizing since the measured rapidity width of the balance function in nucleon-nucleon collisions[1] is about twice as large as that in central heavy ion collisions[2]. Our calculation shows that neither finite acceptance nor transverse flow effects can account for this difference.3 We consider just one pair for simplicity. This is not essential for the conclusions.4 Throughout this paper by quarks and antiquarks we always mean -in the spirit of the coalescence model-the constituent quarks and antiquarks.
The normalization of the expression for h(u) guarantees the correct normalization in(6).
It was shown in[9] that this is a good approximation.
To shorten the wording, we call by U-cluster the one decaying into uū and by D-cluster the one decaying into dd. Their distributions and decay properties are identical.
AcknowledgementsThanks are due to W. Broniowski
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Biro Heavy Ion Phys. 17 (2003) 205 and references quoted there. J Zimanyi, P Levai, T S , J.Pisut, N.Pisutova, Acta Phys. Pol. 282817J.Zimanyi, P.Levai and T.S.Biro Heavy Ion Phys. 17 (2003) 205 and references quoted there; J.Pisut, N.Pisutova, Acta Phys. Pol. B28 (1997) 2817;
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. E G See, G Van Buren, Nucl. Phys. 715129See, e.g., G.Van Buren, Nucl. Phys. A715 (2003) 129c;
. T Chujo, Nucl.Phys. 715151T.Chujo, Nucl.Phys. 715 (2003) 151c;
. W Broniowski, A Baran, W Florkowski, Acta Phys.Pol. 33and references quoted thereW.Broniowski, A.Baran and W.Florkowski, Acta Phys.Pol. B33 (2002) 4235 and references quoted there.
. E G See, A Bialas, Phys. Lett. 442449See, e.g., A.Bialas, Phys. Lett. B442 (1998) 449;
. J Zimanyi, Phys. Lett. 472243J.Zimanyi et al., Phys. Lett. B472 (2000) 243;
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|
[] |
[
"Gapless Triplet Superconductivity in Magnetically Polarized Media",
"Gapless Triplet Superconductivity in Magnetically Polarized Media"
] |
[
"Marios Georgiou \nDepartment of Physics\nNational Technical University of Athens\nGR-15780AthensGreece\n",
"Georgios Varelogiannis \nDepartment of Physics\nNational Technical University of Athens\nGR-15780AthensGreece\n"
] |
[
"Department of Physics\nNational Technical University of Athens\nGR-15780AthensGreece",
"Department of Physics\nNational Technical University of Athens\nGR-15780AthensGreece"
] |
[] |
We reveal that in a magnetically polarized medium, a specific triplet commensurate pair density wave superconducting (SC) state, the staggered d-wave Π-triplet state, may coexist with homogeneous triplet SC states and even dominate eliminating them under generic conditions. When only this TPDW SC state is present, we have the remarkable phenomenon of gapless superconductivity. This may explain part of the difficulties in the realization of the engineered localized Majorana fermion modes for topological quantum computation. We point out qualitative characteristics of the tunneling density of states, specific heat and charge susceptibility that identify the accessible triplet SC regimes in a spinless medium.
| null |
[
"https://arxiv.org/pdf/1809.02879v1.pdf"
] | 119,518,108 |
1809.02879
|
dad9cbd2ab7860a26b9fbaa0794e94024a0e0c25
|
Gapless Triplet Superconductivity in Magnetically Polarized Media
8 Sep 2018
Marios Georgiou
Department of Physics
National Technical University of Athens
GR-15780AthensGreece
Georgios Varelogiannis
Department of Physics
National Technical University of Athens
GR-15780AthensGreece
Gapless Triplet Superconductivity in Magnetically Polarized Media
8 Sep 2018
We reveal that in a magnetically polarized medium, a specific triplet commensurate pair density wave superconducting (SC) state, the staggered d-wave Π-triplet state, may coexist with homogeneous triplet SC states and even dominate eliminating them under generic conditions. When only this TPDW SC state is present, we have the remarkable phenomenon of gapless superconductivity. This may explain part of the difficulties in the realization of the engineered localized Majorana fermion modes for topological quantum computation. We point out qualitative characteristics of the tunneling density of states, specific heat and charge susceptibility that identify the accessible triplet SC regimes in a spinless medium.
Singlet superconductivity (SC) and ferromagnetism (FM) are directly competing phenomena. The discovery of SC coexisting with FM in UGe 2 [1], and other bulk FM-SC [2,3], in heterostructures where proximity of SC and FM is enforced [4][5][6] necessarily involves exotic spin-triplet SC states. Numerous theoretical models with homogeneous triplet SC states possibly odd in frequency have been proposed [7][8][9][10][11]. For the 2-D SC state that develops at the interfaces of some oxide insulators like LaAlO 3 /SrTiO 3 [12] in the presence of FM [13] a modulated or Pair Density Wave (PDW) triplet state of FFLO type has also been suggested [14].
The observation of proximity induced SC in the half metallic (fully polarized) FM CrO 2 in contact with SC NbTiN [4] demonstrates that effectively spinless systems may exhibit SC as well. Triplet SC in spinless systems is of enormous interest because a spinless triplet SC wire can exhibit at its two edges localized Majorana fermion modes [15,16] Such localized Majorana fermions [15] would allow for non-local quantum information storage avoiding local decoherence as well as for logical manipulations through braiding because of their non-abelian character [16]. The localization of these modes is a crucial requirement for quantum-bit realizations and braiding manipulations, and it can occur only if a finite SC gap is present [17].
In the present Letter, based on a systematic study of the interplay of all SC condensates allowed by symmetry in a fully spin-polarized medium, we show that under realistic generic conditions a triplet SC state exhibiting a commensurate density wave modulation of the superfluid density may coexist with, or even dominate eliminating it, the homogeneous (zero momentum) triplet SC. When this triplet commensurate pair density wave SC (TCPDWSC) state dominates we have robust gapless SC, a situation that would be catastrophic for the engineered topological quantum bits. We report phase transitions between the various types of accessible triplet SC states including transitions between gapped and gapless SC states, as well as qualitative physical characteristics in the density of states, specific heat and charge susceptibility that would allow to identify the type of triplet SC in which the system of interest is in.
A similar TCPDWSC state channel has been suggested to occur in the high field SC state of CeCoIn 5 coexisting with singlet SC and spin density waves [18,19] explaining fascinating neutron scattering results [20]. There have also been studies of TCPDWSC in the singlet channel, also called η-pairing, [21][22][23][24][25]27] mainly motivated by the extraordinary physics in the pseudogap and other stripe regimes of cuprates [28,29] where extended FS in the SC state has been reported as well [22].
Our starting point is a BCS-type Hamiltonian with frozen spin:
H = k ξ k c † k c k − k (∆ 0 k c † k c † −k + h.c) − k (Π Q k c † k c † −(k+Q) + h.c).
The first term in describes a tight binding dispersion which generically can be written as a sum of particle-hole symmetric terms and particlehole asymmetric terms: ξ k = γ k + δ k . When δ k = 0 there is particle-hole symmetry or perfect nesting while finite values of δ k destroy the nesting conditions. The second term ∆ 0 k = k V 0 k,k c −k c k represents unconventional SC with zero pair momentum, and the last term Π Q k = k V Q k,k c −(k +Q) c k is the TCPDWSC or modulated SC state. Although our TCPDWSC bear some resemblance with the FFLO state [30] because Cooper pairs have finite total pair momentum and the the superfluid density is inhomogeneous, they are fundamentally different. In fact, our TCPDWSC is a spin-triplet state whereas the FFLO is a spin-singlet trying to survive the Zeeman field. The wavevector of the superfluid modulation in our TCPDWSC is the commensurate nesting vector Q. In the FFLO state instead, the wavevector of the superfluid modulation is variable scaling with the magnitude of the magnetic field.
The effective interactions of the itinerant quasiparticles V 0 k,k , V Q k,k may have a purely electronic origin in the case of FM superconductors. However, our approach is generic irrespective of the microscopic origin of the effective interactions, and the validity of our findings is generic as well. In the case of heterostructures, we assume within our approach that the effective potentials arXiv:1809.02879v1 [cond-mat.supr-con] 8 Sep 2018 incorporate the proximity effects as well. Naturally, we would expect in that case a real space dependence of the potentials, that we neglect here. We only focus on qualitative symmetry questions that would not be affected by a smooth space dependence. In fact, the modulation of the superfluid density in our TCPDWSC state has a wavelength negligible compared to the coherence length and the characteristic lengths of the heterostructure. We therefore expect our qualitative findings to hold for bulk materials and for nanostructures as well.
To treat both types of SC order parameters (OPs) in a compact manner we introduce a Nambu-type representation using the spinors Ψ † k = c † k , c −k , c † k+Q , c −k−Q . and we use the basis provided by the tensor products ρ i = σ i ⊗1 2 ) and σ i = 1 2 ⊗ σ i ), where σ i with i = 1, 2, 3 are the usual 2x2 Pauli matrices and1 2 the unit 2x2 matrix. The absence of spin index in the hamiltonian affects the symmetry classification of the acceptable triplet SC states. for which we produced a systematic phase map. In fact, the OPs are normally classified by their behavior under inversion (Î) k → −k, translation (t Q ) k → k + Q and time reversal (T ).
Instead of the latter we may use complex conjugation (K) which is related to time reversal via the re-
lationsT ≡ −K(∆ 0 k ) andT ≡ÎK(∆ Q k ).
Since the spins are frozen, the homogeneous (q = 0) SC pair states may only have odd parity:
∆ 0 −k = −∆ 0 k . Un- der translation we have both signs ∆ 0 k+Q = ±∆ 0 k and underT we getT ∆ 0 k = −∆ 0 * k . TCPDWSC states may have both parities Π Q −k = ±Π Q k and both signs under translation since Π Q k+Q = −Π Q −k = ∓Π Q k .
Time reversal demands thatT Π Q k = Π Q * −k implying the rela-tionT =ÎK for the TCPDWSC states. The break of time reversal allows finally four possible SC OPs, two homogeneous SC states and two TCPDWSC states:
∆ 0I−− k , ∆ 0I−+ k , Π QI−+ k , Π QR+− k .
Here the first index q = 0 or q = Q indicates the total momentum of the pair (or the characteristic wavevector of the superfluid density), the second index R or I indicates whether the OP is real or imaginary, the third index ± indicates parity under inversionÎ and the last index denotes gap symmetry undert Q . The symmetry properties of the OPs under inversionÎ and translationt Q imply a specific structure in k-space. Every OP M k is written in the form M k = M f k where the form factors f k belong to the different irreducible representations of the point group.
According to the above symmetry classification there exist four possible pairs of competing homogeneous and modulated SC states. Using our formalism we calculate Greens functions and from them self-consistent systems of coupled gap equations for each case.
The pairs ∆ 0I−− k with Π QR+− k and ∆ 0I−− k with Π QI−+ k obey the system of coupled equations:
∆ k = k V ∆ k,k ∆ k ± 1 4E±(k ) tanh( E±(k ) 2T ) and Π k = k V Π k,k Π k ± A k ±γ k 4E±(k )A k tanh( E±(k ) 2T )where A k ≡ [δ 2 k + Π 2 k ] 1/2 and the quasiparticle energies are E ± (k) = [( δ 2 k + Π 2 k ± γ k ) 2 + ∆ 2 k ] 1/2 . The remain- ing two cases, competition of ∆ 0I−+ k with Π QR+− k and ∆ 0I−+ k with Π QI−+ k
, obey the following equa-
tions: ∆ k = k V ∆ k,k ∆ k ± B k ±Π 2 k 4E±(k )B k tanh( E±(k ) 2T ) and Π k = k V Π k,k Π k ± B k ±γ 2 k ±∆ 2 k 4E±(k )B k tanh( E±(k ) 2T ) where B k ≡ [γ 2 k A 2 k + ∆ 2 k Π 2 k ] 1/2 and the dispersions are E ± (k) = ∆ 2 k δ 2 k A −2 k + (A k ± γ 2 k + ∆ 2 k Π 2 k A −2 k ) 2 1/2 . The effective potentials V ∆ k,k , V Π k,k have the form V k,k = V f k f k (separable potentials).
We have solved self consistently the above systems of equations on a square lattice with γ k = −t 1 (cos k x + cos k y ) and δ k = −t 2 cos k x cos k y and Q = (π, π). The choice of a tetragonal dispersion is motivated by the fact that CrO 2 as well as strongly FM superconductors like UGe 2 and URhGe exhibit all a tetragonal structure, however, our qualitative findings are generic. The corresponding form factors belong to irreducible representations of the tetragonal group D 4h . Specifically:
∆ 0I−− k ∼ sin k x + sin k y (s-wave), ∆ 0I−+ k , Π QI−+ k ∼ sin(k x + k y ) (p-wave) and Π QR+− k ∼ cos k x − cos k y (d-wave)
. For every competing pair we have performed a large number of self-consistent calculations varying pairing potentials in the two channels, temperatures and ratios t 2 /t 1 .
The first important result is that the TCPDWSC Π QI−+ k OP can never survive. Specifically, the Π QI−+ k gap is zero regardless of the values of the pairing potentials and the particle-hole asymmetry t 2 /t 1 term. We conclude that although the state Π QI−+ k is allowed by symmetry, it is never realized. Therefore, we only report results about the relevant competition of the remaining TCPDWSC OP Π QR+− k with both zero momentum SC states.
The phase sequences as t 2 /t 1 grows starting from zero and for various values of the pairing potentials for the competition Π QR+− k with ∆ 0I−− k and Π QR+− k with ∆ 0I−+ k are shown in the respective panels of Fig. 1. Arrows in Fig. 1 indicate the cascade of phases observed when the ratio t 2 /t 1 grows starting from zero at each region of the map. The variation of t 2 /t 1 may simulate various effects such that chemical doping, or stress effects as well as proximity effects. Since we consider a spin-polarized background, all states reported coexist with FM, and the transitions to the FM state reported at high values of t 2 /t 1 has the meaning of a transition to a state that is only ferromagnetic with no SC OP present.
Both cases share the characteristic feature that the TCPDWSC state Π QR+− k is finite in the largest part of the maps of the pairing potentials. Thus, since we do not limit to a specific microscopic model that could correspond to a specific value for the pairing potentials, the existence of the modulated TCPDWSC phase can be considered generically plausible. The interplay of Π QR+− k with ∆ 0I−− k favors the coexistence of both (q = 0 and q = Q) SC states at low-T over a wide range of values of the pairing potentials (Fig. 1a). The transition from a coexistence state to a homogeneous (q = 0) SC state as t 2 /t 1 grows is always continuous (second order ) and dominates the V ∆ , V Π parameter space.
V Δ V Π F M Δ Δ+Π F M Δ Π F M Π F M Δ Δ+Π ( a ) V Δ V Π F M Δ F M Π F M Δ Π Π F M Δ Δ+Π ( b )
The low temperature regime is different in the interplay of Π QR+− k with ∆ 0I−+ k . Coexistence of the two SC states is allowed again but now is restricted to a small portion of the V ∆ , V Π map (Fig. 1b). The most interesting feature is now the the domination of the TCPDWSC (modulated SC) state for the smaller values of t 2 /t 1 . Thus, in this case the formation of the Π QR+− TCPDWSC state is favored. As particle hole asymmetry grows (t 2 /t 1 grows) we may have transitions from TCPDWSC to a state of coexistence or to a homogeneous SC state.
The stability of the solutions of the self consistentequations has been verified by free-energy calculations as well.
The free-energy difference ∆F between the normal and the condensed state is given by:
∆F = ∆ 2 V ∆ + Π 2 V Π − 1 2β k j=±,i=± ln( 1+e −jβE i (k) 1+e −jβ i (k)
) where E ± (k) the energy dispersions for each competing pair and ± (k) the energy dispersions obtained when both gaps are zero. The safest way to ensure that the solutions of the coupled gap equations correspond to the minimum of the free-energy difference is to vary ∆F with respect to the magnitudes of the gaps and verify that ∆F attains its minimum for these values. We report in Fig. 2 the variations of the free-energy difference with ∆ 0I−+ k and Π QR+− k at low-T for t 2 /t 1 = 0 and for V ∆ = V Π = 3 to illustrate the dominance of the TCPDWSC state. These values of the pairing potentials correspond to the cascade of transitions Π → ∆ → F M when t 2 /t 1 grows (cf. Fig. 1b). It is clear that ∆F attains its minimum value for (∆, Π) = (0, 0.94t 1 ), thus the ground state consists solely of theTCPDWSC phase.
We report in Fig. 3 the dependence of the OPs on t 2 /t 1 at low-T (Fig. 3a) and the phase diagram (Fig. 3b) obtained by the coupled-gap equations. We stress that at low-T for t 2 /t 1 = 0 the values of the gaps are in full agreement with the ∆F minimum requirement, i.e (∆, Π) = (0, 0.94t 1 ). The t 2 /t 1 transition from the modulated to the homogeneous SC state is first order, and we note that the TCPDWSC gap is significantly larger than the homogeneous SC gap despite the fact that the pairing potentials have the same magnitude (Fig. 3a). The phase diagram shows that the transition Π → ∆ with t 2 /t 1 is not limited to low-T. The modulated SC phase extends to higher temperatures (Fig. 3b) than the homogeneous SC phase. The boundary separating the two SC states remains first order and ends at a tricritical point. Decreasing the temperature moves the boundary to lower t 2 /t 1 -values. This allows a first order transition with respect to temperature within the superconducting phase from the TCPDWSC to the homogeneous SC state. An example of such a transition realized for t 2 /t 1 = 2 is shown in the inset of Fig. 3b. A question that naturally arises is how the exotic TCPDWSC state Π QR+− k can be identified experimentally. Quite remarkably, specific heat measurements at low-T may be sufficient. Specifically, isolated TCPDWSC states exhibit an extended FS whereas the FS is limited to two Fermi points in the coexistence phase ∆ + Π . Therefore a polynomial behavior of the specific heat at low-T is a signature of the coexistence phase. As particle-hole asymmetry t 2 /t 1 grows for example with gate voltage, only the modulated SC state Π QR+− k continues to exhibit extended FS whereas the zero momentum SC states as well as the coexistence phase present limited FS consisting of isolated Fermi points. We note that the extended FS is also a feature of the spin-singlet η-pairing [22].
Consequently the TCPDWSC state Π QR+− k is the sole SC state that exhibits a linear low-T behavior of the specific heat and this is robust since it holds even for finite values of t 2 /t 1 . This is illustrated in Fig. 4 where the Fermi surface and the specific heat for t 2 /t 1 = 1.0 in the TCPDWSC phase (red) and t 2 /t 1 = 2.5 in the homogeneous SC phase (blue) of Fig. 3 are reported. We observe that in the Π QR+− k phase the Fermi surface is extended imposing the linear behavior of the specific heat at low-T. On the other hand, in the ∆ 0I−+ k phase we only have two Fermi points and the specific heat at low-T exhibits a polynomial behavior. This is also the case for the other homogeneous SC state ∆ 0I−− k as well as for the coexistence phase ∆ + Π. Therefore linear low-T specific heat in the SC state identifies the triplet TCPDWSC state. The difference in the FS is reflected in the behavior of the electronic density of states (DOS) N(ω) accessible by tunneling. In our spinor formalism: N(ω)= − 1 π Im k Tr{G(k, iω n → ω + in)}. Performing the analytical continuation it can be shown to take the form:
N (ω) = k {δ ω + E ± (k) + δ ω − E ± (k) }.
As an example we present in Fig. 5 the DOS in the Π QR+− k , the ∆ 0I−− k state and the coexistence phase ∆ + Π for t 2 /t 1 = 0 (left) and t 2 /t 1 = 1 (right). In each case the pairing potential is 3t 1 . The vanishing DOS at the Fermi level identifies the coexistence phase ∆ + Π in the case of perfect nesting t 2 /t 1 = 0, whereas the finite DOS for particle-hole asymmetry t 2 /t 1 = 0 is a direct signature of the TCPDWSC state. We note that finite DOS at the Fermi level has also been reported in spin-singlet PDW states [21]. 10 -π -π/2 0 π/2 π P(q, iω n=0 ) diagonal (q x =q y ) Finally, measurements related with the charge-charge correlation function P (q, iω n ) for the lowest Matsubara frequency ω n=0 = πT along the diagonal and the antidiagonal of the first Brillouin zone (FBZ) may provide the ultimate experimental strategy to distinguish the different accessible triplet SC states.
Π QR+-(d-wave) ∆ 0I--(s-wave) ∆ 0I-+ (p-wave) ∆ 0I--+Π QR+- -4 -2 0 2 4 -π -π/2 0 π/2 π P(q, iω n=0 ) antidiagonal (q x =-q y ) Π QR+-(d-wave) ∆ 0I--(s-wave) ∆ 0I-+ (p-wave) ∆
We illustrate this in Fig. 6 where P (q, iω n=0 ) is reported along the diagonal (left) and the antidiagonal (right) of the FBZ in the Π QR+− k (red), the ∆ 0I−− k (green), the ∆ 0I−+ k (blue) and the coexistence phase ∆ + Π (magenta). The unique feature of the TCPDWSC state Π QR+− k is that it is the sole state for which the charge-charge correlation function is the same in both directions of the FBZ. In case of the homogeneous ∆ 0I−− k state measuring the correlation function along the diagonal direction reveals a double-peak structure around the points ±(π/2, π/2) and a decrease around the center of the FBZ whereas for all the other states it exhibits only one peak at ±(π/2, π/2) and remains practically constant around the center of the FBZ. Quite remarkably, the double-peak structure around the points ±(π/2, π/2) as well as the decrease around the center of the FBZ become characteristic features of the ∆ 0I−+ k state along the antidiagonal direction. For the coexistence ∆ + Π state, in the diagonal direction it is the sole state that exhibits peaks only at the edges of the BZ, while in the antidiagonal direction there are again peaks at the edges of the BZ but they are on the negative side and two new smaller peaks at ±(π/2, π/2) that are on the positive side.
In summary, within a generic microscopic mean field theory we explored systematically the interplay of all possible triplet SC states in an effectively spinless system. We find that the inhomogeneous TCPDWSC state Π QI−+ k having the p-wave symmetry can never survive the two allowed by symmetry homogeneous triplet SC states. However, the other TCPDWSC Π QR+− k having d-wave symmetry may either appear alone or coexist with the homogeneous SC OPs driving the phenomenon of gapless SC over a wide parameter range. Our findings are universally applicable to any strongly ferromagnetic system that develops superconductivity including devices designed to host localized Majorana modes for topological quantum computation. Geometry and the presence of one-spin triplet SC does not guarantee the relevance of a device designed to host Majorana qubits, it should be tested against the eventual emergence of catastrophic gapless triplet SC and we have identified some experimental paths for such tests.
We are grateful to Alexandros Aperis, Panagiotis Kotetes and Georgios Livanas for illuminating discussions.
PACS numbers: 74.81.-g, 74.20.Rp, 74.25.Dw
FIG. 1 .
1Maps of the dependence of phase sequences on the effective interactions V ∆ and V Π for low temperature. Arrows indicate the cascade of phases obtained when t2/t1 grows starting from zero. The black dots separate regions of different phase sequences under growing t2/t1. All phases coexist with ferromagnetism (FM). The phases indicated as FM only FM is present. Panel (a) depicts the interplay of Π QR+− with ∆ 0I−− whereas panel (b) that of Π QR+− with ∆ 0I−+ . The potentials are in units of t1.
FIG. 2 .
2(Color online) Contour plot of the condensation free energy ∆F as a function of the OPs Π t 2 /t 1 = 0 and V ∆ = V Π = 3t 1 . The lowest free energy is situated at the point (∆, Π) = (0, 0.94t 1 ) where only Π QR+− k is finite despite the fact that it exhibits gapless SC.
on t 2 /t 1 at low-T. (b) t 2 /t 1 -temperature phase diagram. Closed symbols mark 2nd order and open symbols 1st order transitions. A first order transition, for t 2 /t 1 = 2, within the SC phase from the TCPDWSC to homogeneous SC, is possible with decreasing temperature (inset). The values of the pairing potentials are V ∆ = V Π = 3t 1 .
FIG. 4 .
4(Color online) Fermi surface (left) and specific heat at low-T (right) in the Π QR+− k state (red) for t 2 /t 1 = 1.0 and the ∆ 0I−+ k state (blue) for t 2 /t 1 = 2.5. The extended FS in the TCPDWSC state causes the linear behavior of the specific heat, whereas the polynomial dependence in the ∆ 0I−+ k state is a direct consequence of the presence of Fermi points instead of FS. The pairing potentials are V ∆ = V Π = 3t 1 .
FIG. 5 .
5(Color online) DOS for t 2 /t 1 = 0 (left) and t 2 /t 1 = 1 (right) at low-T in the ∆ 0I−− k (blue), the Π QR+− k (red) and the coexistence phase ∆ + Π. The pairing potentials are equal V ∆,Π = 3t 1 .
(
Color online) Charge-charge correlation function P (q, iω n=0 ) along the diagonal (left) and the antidiagonal (right) of the FBZ in the Π ) and the coexistence phase ∆ + Π (magenta) at low-T for t 2 /t 1 = 0. The pairing potentials are V ∆,Π = 3t 1 .
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|
[] |
[
"All-Electron APW+lo calculation of magnetic molecules with the SIRIUS domain-specific package",
"All-Electron APW+lo calculation of magnetic molecules with the SIRIUS domain-specific package"
] |
[
"Long Zhang \nQuantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA\n\nCenter for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA\n",
"Anton Kozhevnikov \nSwiss National Supercomputing Centre\nZurichSwitzerland\n",
"Thomas Schulthess \nSwiss National Supercomputing Centre\nZurichSwitzerland\n",
"S B Trickey \nQuantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA\n\nCenter for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA\n",
"Hai-Ping Cheng \nQuantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA\n\nCenter for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA\n",
"\nDepartment of Physics\nUniversity of Florida\n32611GainesvilleFLUSA\n"
] |
[
"Quantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA",
"Center for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA",
"Swiss National Supercomputing Centre\nZurichSwitzerland",
"Swiss National Supercomputing Centre\nZurichSwitzerland",
"Quantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA",
"Center for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA",
"Quantum Theory Project\nUniversity of Florida\n32611GainesvilleFLUSA",
"Center for Molecular Magnetic Quantum Materials\nUniversity of Florida\n32611GainesvilleFLUSA",
"Department of Physics\nUniversity of Florida\n32611GainesvilleFLUSA"
] |
[] |
We report APW+lo (augmented plane wave plus local orbital) density functional theory (DFT) calculations of molecule systems using the domain specific SIRIUS multi-functional DFT package. Compared to other packages the additional APW and FLAPW task and data parallelism and the additional eigensystem solver provided by the SIRIUS package can be exploited for performance gains in in the ground state Kohn-Sham calculation. This is in contrast with the use of SIRIUS as a library backend to some other APW+lo or FLAPW (full-potential linearized AWP) code.We benchmark the code and demonstrate performance on several magnetic molecule and metal organic framework systems. We show that the SIRIUS package in itself is capable of handling systems as large as a few hundreds of atoms in the unit cell without losing the accuracy needed for magnetic systems.
| null |
[
"https://arxiv.org/pdf/2105.07363v2.pdf"
] | 248,240,186 |
2105.07363
|
8c7326bef255b75204d1e88c430de1a4a6825693
|
All-Electron APW+lo calculation of magnetic molecules with the SIRIUS domain-specific package
Long Zhang
Quantum Theory Project
University of Florida
32611GainesvilleFLUSA
Center for Molecular Magnetic Quantum Materials
University of Florida
32611GainesvilleFLUSA
Anton Kozhevnikov
Swiss National Supercomputing Centre
ZurichSwitzerland
Thomas Schulthess
Swiss National Supercomputing Centre
ZurichSwitzerland
S B Trickey
Quantum Theory Project
University of Florida
32611GainesvilleFLUSA
Center for Molecular Magnetic Quantum Materials
University of Florida
32611GainesvilleFLUSA
Hai-Ping Cheng
Quantum Theory Project
University of Florida
32611GainesvilleFLUSA
Center for Molecular Magnetic Quantum Materials
University of Florida
32611GainesvilleFLUSA
Department of Physics
University of Florida
32611GainesvilleFLUSA
All-Electron APW+lo calculation of magnetic molecules with the SIRIUS domain-specific package
1 arXiv:2105.07363v2 [cond-mat.mtrl-sci] 19 Apr 2022
We report APW+lo (augmented plane wave plus local orbital) density functional theory (DFT) calculations of molecule systems using the domain specific SIRIUS multi-functional DFT package. Compared to other packages the additional APW and FLAPW task and data parallelism and the additional eigensystem solver provided by the SIRIUS package can be exploited for performance gains in in the ground state Kohn-Sham calculation. This is in contrast with the use of SIRIUS as a library backend to some other APW+lo or FLAPW (full-potential linearized AWP) code.We benchmark the code and demonstrate performance on several magnetic molecule and metal organic framework systems. We show that the SIRIUS package in itself is capable of handling systems as large as a few hundreds of atoms in the unit cell without losing the accuracy needed for magnetic systems.
I. INTRODUCTION
Molecular magnetism, including molecular magnetic materials, is a very active interdisciplinary research area that has significant implications for computational investigations.
Motivated in no small measure by the promise of high impact on quantum computing 1,2 , molecular magnetism deals with design, synthesis, and physical and chemical characterization of both single-molecule magnets (SMMs) 3,4 and condensed aggregates thereof. Predictive and interpretive first principles calculations are valuable both for experimental progress and for formulation and parameterization of models of the molecular spin systems. We begin therefore with a brief overview of the physical problem class, then turn to attendant computational challenges and some progress in meeting them.
A. Motivation from physical systems
Molecular magnetism involves a large range of dimensionality, from isolated SMMs 5,6 and 1D chain magnets 7,8 through 2D molecular layers 9 to 3D polymers and metal organic framework 10,11 materials that exhibit collective ordering of magnetic moments.
Research activity, both theoretical and experimental, has grown in recent decades with focus on (1) low dimensional materials (motivated by their potential application in high-density magnetic storage and nano-scale devices) 12,13 and (2) so-called functional materials 5,12,14,15 (because of their strong response to changing external conditions). Because molecular magnets have well-localized magnetic moments, they are a nearly perfect arena for investigation of intriguing phenomena and testing models. Unlike bulk magnetic materials, their quantum size effects suggest applications beyond conventional high-density information storage. Such applications include spintronics and qubits for quantum computing.
The molecules involved are large and rather complicated. An example category is Mn 12 complexes 16 . Their investigation dates to synthesis by Weinland and 4 ] complex was studied beginning in 1988, 19 and its ground state eventually determined 20,21 to have S = 10. For computational context, note that this last system has 176 atoms and 1210 electrons.
A major class of experimental effort has focused on design and synthesis of multi-nuclear clusters containing 4 Mn 3+ , because the axially Jahn-Teller distorted Mn 3+ ions usually possess large magnetic anisotropy due to spin-orbital coupling. A large number of Mn(III)based SMMs has been reported, including those with Mn 6 22 , Mn 19 23-25 , Mn 25 26,27 , Mn 31 28 and Mn 84 29 . In addition, inorganic chemists have also synthesized many transition metal The all electron full potential linearized augmented plane wave (FLAPW) method is one flavor of implementation of the Kohn-Sham DFT. The method is based on a partitioning of the material's unit cell into non-overlapping muffin-tin (MT) spheres, centered at the atomic nuclei, and an interstitial (IS) region between the MT spheres, as illustrated in Fig. 1. Bloch wave-vector k and plane-wave vector G is
ϕ G k (r) = ,m ν A α,k mν (G)u α ν (r)Y m (r), r ∈ α (1/ √ Ω)e i(G+k)·r , r / ∈ α
Here u α lν (r) is the solution of the (energy dependent) radial Schrodinger equation in the MT sphere labeled α; Y m (r) are spherical harmonics; A α,k mν (G) are the matching coefficients for connection with the interstitial plane wave; and m are the azimuthal and magnetic quantum numbers in a particular sphere; and ν is the order of energy derivative of the radial function (the APW basis set does not have continuous radial first derivatives at the sphere boundaries.) The dependence of the radial functions upon the energy for which they are solved is the difficulty with the APW basis set. Continuity at sphere boundaries requires those solution energies to correspond to KS eigenvalues. That correspondence makes the KS secular equation highly non-linear in the one-electron energies. The problem is remedied by introducing the energy derivative of the radial function in the APW basis,u α ν = [∂/∂ ]u α ν , that makes the basis a linearized APW (LAPW) basis. The basis can be enhanced with additional local orbitals (lo), which are radial functions and energy derivatives, that vanish at the MT boundaries.
The electron density and the Kohn-Sham effective potential are expanded in correspondence to the space division in the unit cell. In the interstitial region they are expanded in plane waves and inside MT spheres in real spherical harmonics R m (r):
n(r) = m n α m (r)R m (r), r ∈ α Gñ (G)e iG·r , r / ∈ α(1)
and v KS (r) =
m v α m (r)R m (r), r ∈ α Gṽ (G)e iG·r , r / ∈ α .(2)
Here n α m (r),ñ(G), v α m (r), andṽ(G) are expansion coefficients determined through the self-consistent solution of the KS equation.
Due to the fact that it contains no shape approximation to the atomic potential and includes all electrons into consideration, FLAPW DFT is often considered as a highly precise realization of density functional theory and can be use as a gold standard to measure the precision of other approaches.
The present work focuses on the domain specific SIRIUS library package. Our goal is to make fast and efficient all-electron, full-potential DFT calculations routinely feasible for large and complicated systems such as magnetic molecules and their aggregates. In the following sections, we first describe the package and its intended usage for accelerating plane-wave based DFT codes. We then explain its value when used as a stand-alone FLAPW package.
After describing the library, we demonstrate its capabilities by calculations on a selection of magnetic molecules. From a development perspective, these commonalities offer an opportunity. Rather than have the developers of each individual code devise -and in many cases essentially reinvent -various schemes for computational speed and scalability, we can proceed via separation of concerns, identify the general calculation elements within the plane wave Kohn-Sham DFT formalism and create a library with explicit support for FLAPW/APW+lo and PW-PP calculations. By abstracting and encapsulating the common objects such as the examples just given, the focus then can be on optimizing computational performance. The SIRIUS library was created as such an optimized collection. It was designed from the outset with both task parallelization and data parallelization. It has been optimized for multiple MPI levels as well as OpenMP parallelization and also optimized for GPU utilization.
II. THE SIRIUS PACKAGE
SIRIUS can be interfaced directly with both existing FLAPW/APW+lo codes and with PW-PP codes as a DFT library. We discuss that use of SIRIUS elsewhere 59 . Though that was the originally intended primary usage SIRIUS does have basic FLAPW/LAPW+lo stand-alone capability. The present study investigates its benefits to large-scale magnetic system calculations in that form.
The SIRIUS package is written in C++ in combination with the CUDA 60 backend to provide the following features: (1) low-level support such as pointer arithmetic and type casting as well as high-level abstractions such as classes and template meta-programming;
(2) easy interoperability between C++ and widely used Fortran90; (3) full support from the standard template library (STL) 61 ; (4) easy integration with the CUDA nvcc compiler 62 .
The SIRIUS code provides dedicated API functions to interface to QuantumEspresso and exciting.
Importantly for large-system calculations, SIRIUS is designed and implemented with both task distribution and data (large array) distribution in mind. Virtually all KS electronic structure calculations rely at minimum on two basic functionalities: distributed complex matrix-matrix multiplication (pzgemm in LAPACK) and a distributed generalized eigenvalue solver (pzhegvx also in LAPACK). SIRIUS handles these two major tasks with data distribution and multiple levels of task distribution.
The eigenvalue solver deserves scrutiny. Development of exciting revealed that considerable changes in code were required to scale the calculation to a larger number of distributed tasks, for example, by making the code switchable from LAPACK to ScaLAPACK. This can be seen by comparing the task distribution and data distribution of the base ground state subroutine in recent versions of exciting (version Nitrogen for example) and ELK (version 5.2.14 or earlier version for example). Evidently the ELK developer team had emphasized physics features and functionalities rather than adding support for ScaLAPACK. The situation is similar with Exciting-Plus 63 ; it has only LAPACK support and does not have data distribution of large arrays.
Eigenvalue solver performance depends upon intrinsic limitations of the underlying algorithms. Widely used linear algebra libraries such as LAPACK and ScaLAPACK implement robust full diagonalization algorithms that can handle system size up to about 10 6 . But, unfortunately, because of the very large plane wave cutoff needed for FLAPW/APW+lo calculations on systems as large as 100+ atoms, the eigensystem often is several times larger than that. A Davidson-type 64 iterative diagonalization algorithm is often suitable for such large systems because it suffices to solve for the lowest 10-20 percent of all eigenvalues and the associated eigenvectors to obtain the valence band structure around the Fermi energy.
But Davidson-type diagonalization algorithms are not offered in standard linear algebra libraries. That omission is at least in part because such algorithms involve repeated application of the Hamiltonian to a sub-space of the system. Thus the algorithm depends on construction details of the Hamiltonian matrix, i.e. it depends on the specific DFT formalism. The SIRIUS library takes this into consideration and offers an efficient implementation of Davidson-type diagonalization 65 for FLAPW/APW+lo and PW-PP codes.
For computational capability, switching from LAPACK to ScaLAPACK gives the benefit of data parallelism but does not remove the diagonalization algorithm limitation. Switching from LAPACK (or ScaLAPACK) to a Davidson-type diagonalization addresses that. Doing so requires taking into consideration both the diagonalization algorithm and the handling of large data sets as the system size grows. This is especially so in the case of iterative Estimates are about 50 ± 30 meV but as high as a few hundred meV. Note that usual XC density functional approximations, suffer from a self-interaction error and hence tend to favor the LS state to reduce spurious self-repulsion with the result of overestimated ∆E HL values ? . That is not of concern here since what we are testing is algorithmic efficiency.
Several factors can affect a DFT calculation of ∆E HL of the molecule significantly.
For consistency with condensed phase calculations, it is appropriate to study the isolated molecule in a large, periodically bounded box. To ensure accuracy, the plane wave cutoff needs to be large. To treat the isolated molecule, the vacuum volume in the computational unit cell requires even larger cutoffs. This makes the APW+lo calculation an intensive job because of the large Hamiltonian to be diagonalized and the large G-vector related arrays.
The quality of APW+lo calculation is governed by the dimensionless quantity R M T min · |G k | max , where R M T min is the minimum muffin-tin radius of all atomic species and |G k | max is the maximum length of the G + k vectors. The standard value of R M T min · |G k | max is 7 in most FLAPW codes, and can be 10 if one wants to push the accuracy of total energy to be close to µHa. For organic molecules, the smallest R M T is normally around 1.4 a 0 of the hydrogen atom, which makes |G k | max ∈ [5, 7] a −1 0 . The size of the Hamiltonian in the first variational step is determined by |G k | max , the total number of atoms and the local orbitals added to each atom.
For the Mn-taa molecule, if one sets R M T min · |G k | max = 7 and other parameters as shown in Table.II, the size of the first variational Hamiltonian is about 330,000. This is actually within the theoretical limit of the LAPACK's full diagonalization algorithm. However, in order to run the DFT calculation, not only the G k vectors but also the G vectors for expanding density and potential have to be handled efficiently. For molecules in a large cube unit cell of the size of 20Å with irregular vacuum space, we found (from SIRIUS calculation) that the |G| for density and potential needs to be as large as 25-30 a −1 0 to make the total energy when |G| = 30 a −1 0 or larger, the calculation crashed with out-of-memory (OOM) error, even if one node was assigned with only one MPI task. To have a more specific measure of the memory issue we ran it on the UF HPC large memory node and observed the peak memory consumption values as shown in Table-I. For the Sirius calculation of Mn-taa, we used the experimentally determined low spin [Mn(taa)] structures and the PBE 67 XC generalized gradient approximation (GGA). We set R M T min · |G k | max = 5, |G| max = 30 a −1 0 , and other input parameter values listed in Table-II. There was no Hubbard U . The same structures and XC were used in a VASP 68 calculation for comparing the results, also in Table-II The two triangular Mn 3 units are parallel, with each having the magnetic S = 6 ground state from intra-Mn 3 ferromagnetic couplings. The inter-Mn 3 interaction also has been determined to be ferromagnetic. There is no experimental evidence for noticeable interactions between the two dimers. That is consistent with the absence of significant inter-dimer contacts in the structure and the relatively large distance between Mn ions in adjacent dimers. For this size of the system (436 atoms, 2435 electrons), we used R M T min · G max = 5 and similar other cutoffs as used for previous systems. The large number of atoms does not make the Hamiltonian significantly larger but costs much more time in each iteration. With the cutoff parameters listed in VI we managed to converge the system to a non-magnetic ground state.
VII. SUMMARY AND CONCLUSION
The main outcome of this work is to demonstrate, by several concrete examples, the value of the SIRIUS architecture and implementation for all-electron DFT calculations on fairly large molecules with complicated spin manifolds. By inference, this capacity extends to aggregates of such systems. SIRIUS as a stand-alone package provides performance gains through refined diagonalization methods and task and data parallelization improvements.
The result is an advance in capability of the FLAPW-APW+lo DFT treatment of complex molecular aggregates.
In detail, the eigenvalue solver and the distribution of G-vector arrays in community We showed results from molecules using the APW+lo basis. The resulting total energy and magnetization are in agreement with experimental measurements and corresponding VASP calculations. In these test calculations, good scaling in band parallelization is observed, which is particularly crucial for a single k-point calculations ([Mn(taa)], Mn 3 dimer).
The results also indicate that SIRIUS parallelization works well on contemporary high performance systems and the computational time is drastically reduced compared to ELK and Exciting-Plus for example. For a system of the size of DTN supercell studied in this work, the main physical feature is captured by extraction of exchange constants J from the total energies. The results are qualitatively in agreement with experiment and VASP calculations.
Looking ahead, all-elctron FLAPW and APW+lo calculations of medium to large molecule and MOF systems is a relatively little-explored area. It is plausible thatf important core effects from, for example, spin-orbit coupling will be uncovered by such all-electron investigations. At the least, the use of plane-wave-PAW codes to drive ab initio BOMD
Fischer in 1921 17 . The crystal structure was not determined until 1980 18 . That particular Mn 12 molecule was built from four Mn 4+ (S = 3/2) and eight Mn 3+ (S = 2) ions coupled by oxygen atoms. The [Mn 12 O 12 (O 2 CPh) 16 (H 2 O)
FIG. 1 .
1Muffin-tin partitioning of a unit cell. This method originated from the APW method proposed by Slater 55,56 . It brought great progress with the idea of linear methods 57,58 introduced by Andersen and applied by Koelling and Arbman using the model of muffin-tin shaped approximation to the potentials. The basis set is constructed according to the same space partition. One piece of the basis function for
Irrespective of the particular code, LAPW/APW+LO/lo calculations obviously have the same underlying formalism. Because the basic conceptualization starts with plane waves, those codes also share significant formal and procedural elements with plane-wave pseudopotential (PW-PP) codes. Specifically the following tasks are shared: unit cell setup, atomic configurations, definition and generation of reciprocal lattice vectors G and combinations with Bloch wave vectors G + k, definition of basis functions on regular grids as Fourier expansion coefficients; construction of the plane wave contributions to the KS Hamiltonian matrix, generation of the charge density, effective potential, and magnetization on a regular grid; symmetrization operations on the charge density, potential and occupation matrix; iteration-to-iteration mixing schemes for density and potential; diagonalization of the secular equation. Compared to PW-PP codes, FLAPW/APW+lo additionally have everything expanded in radial functions and spherical harmonics inside the MT spheres, and enforcement of matching conditions on sphere surfaces.
diagonalization implemented in FLAPW/LAPW+lo basis codes. When the unit cell is as large as tenÅ 3 and contains a hundred or more multi-electron atoms, the plane-wave cutoff needed is normally about 25-30 a −1 0 to reach a properly converged ground state.This causes the reciprocal-lattice-vector (G-vector) related arrays to become very large.Again, the problem is illustrated by the ELK and exciting codes. Each has several multidimensional arrays that have one dimension for the global G-vector indices. Because those arrays are not handled in a distributed way, they become very memory consuming in a single MPI task. Eventually they become the real bottleneck, once the diagonalization algorithm limitation is removed via a Davidson-type method. Though designed originally for use as a library, SIRIUS, as written, treats the G-vector related multi-dimensional arrays in distributed manner from the beginning. And, it has basic FLAPW/LAPW+lo DFT ground state capability when running stand-alone. That makes SIRIUS stand-alone calculation much more suitable for large molecule systems. We turn to some examples.III. SIRIUS: [MN(TAA)] MOLECULE The so-called [Mn(taa)] molecule ([M n 3+ (pyrol) 3 (tren)]) is a meridional pseudo-octahedral chelate complex of a single Mn as the magnetic center and the hexadentate tris[(E)-1-(2-azolyl)-2-azabut-1-en-4-yl]amine ligand. It is a spin-crossover system of long-standing interest. Originally studied by Sim and Sinn 66 , it was the first known example of a manganese(III) d 4 spin-crossover system. Experimentally it is found that the Mn 3+ cation goes from a low-spin state to a high-spin state (HS) at a transition temperature of about 45 K.The [Mn(taa)] structure is sufficiently large that it has non-negligible intra-molecular dispersion interactions with a critical HS-LS difference. The HS state involves anti-bonding molecular orbital occupation, hence the octahedral HS complex tends to have weaker and longer metal-ligand bonds than the LS bonds. Therefore, the dispersion contribution to the metal-ligand bonding is lower for the HS case than for the more compact LS state. Mn(taa) poses challenges to the computational determination of the ground state because the purely molecular (non-thermal) ∆E HL = E HS − E LS is very small compared to the total energies.
.FIG. 2. Mn-taa molecule
The interaction quantum mechanically couples the two Mn 3 units. The structure is robust and resists any significant deformation or distortion that might affect the weak inter-Mn 3 coupling.With 137 atoms and 748 electrons the [Mn 3 ] 2 SMM dimer is the largest isolated molecule system we have treated with all-electron DFT calculations. Like the Mn-taa, this calculation also benefits from the band parallelization within a single k-point and the distributed storage of the G-vector related arrays. The molecule is also placed in a 20Å periodically bounded box with single k-point Brillouin Zone sampling. The G max cutoff and LO/lo configuration for the Mn atoms were the same as for [Mn(taa)]. Also as before, we used the experimental geometry 73 and the PBE exchange-correlation functional 67 (again, no Hubbard U ). We are able to converge the system to the anti-ferromagnetic ground state, where there are two spin-up Mn and one spin-down Mn in each [Mn 3 ], and the total moment is zero.
FIG. 3 .
3Mn3 · |G k | max 5 |G max | (a −1 0 ) for ρ and V eff 30 l max for APW 8 l max for ρ and V LOMO gap (eV) 0.23 (VASP: 0.27) µ tot (µ B ) each Mn atom (Sirius): ±1.88 each Mn atom (Vasp): ±1.95 V. SIRIUS: DTN MOLECULE The insulating organic compound XCl 2 -[SC(NH 2 ) 2 ] 4 where X=Ni or Co (DTN and DTC molecules, respectively) is a molecule-based framework structure in which magnetic and electric order can couple with each other. It has been experimentally studied for its quantum magnetism 74-77 . (NiCl 2 -[SC(NH 2 ) 2 ] 4 ) has a tetragonal molecular crystal structure with two Ni atoms as magnetic centers in one unit cell. Each Ni has four S atoms and two Cl atoms as nearest neighbors, as shown in Fig. 4. Thus it forms an octahedral structure similar to the BO 6 octahedra in ABO 3 perovskites. The base octahedral structures pack in a body-centered tetragonal structure, with Ni-Cl-Cl-Ni bonding along the c-axis and hydrogen bonding between theiourea molecules in the a-b-plane. The four thiourea [SC(NH 2 ) 2 ] molecules around each Ni ion are electrically polar. The a-b-plane components of the thiourea electric polarization cancel, while the c-axis components are in the same direction, thereby creating a net c-axis electric polarization that could be responsible for magnetic field-modified ferroelectricity. At temperatures below 1.2 K and below a critical magnetic field, DTN is a quantum paramagnet. As the magnetic field perpendicular to the a-b-plane reaches the first critical value, DTN experiences a quantum phase transition into an XY-antiferromagnetic state in which all Ni spins lie within the a-b-plane. When the field is increased further the spins begin to be more aligned with a corresponding increase in magnetization. When a second critical field is reached, the magnetization saturates and the material enters a spin-polarized state with all spins aligned along the c-axis and parallel to the applied magnetic field. There are three non-equivalent Ni-Ni spin couplings in the tetragonal DTN bulk: J c along the c-axis (along the Ni-Cl-Cl-Ni bonding), J ab in the a-b-plane (nearest neighbors in that plane), and J diag between the corner Ni at corner and the one at the tetagonal body center.Experiments suggest that the exchange coupling along the c-axis via the Ni-Cl-Cl-Ni chain is strong 78 , about about an order of magnitude larger than the other two Ni-Ni couplings.The system can behave as a quasi one-dimensional AFM with S = 1 chains of Ni 2+ ions.Recent DFT+U study 79 has shown that rather large Hubbard U correction (about 5-6 eV, relative to GGA-PBE XC approximation) is needed to match experimentally fit coupling constant values.However the main qualitative physical feature, that J c is about an order of magnitude
FIG. 5
5. the Fe4 molecule
FLAPW
codes, for example ELK, exciting and Exciting-Plus are major bottleneck in calculations of large systems. The LAPACK/ScaLAPACK full diagonalization algorithm cannot handle Hamiltonian matrices larger than ∼ 10 6 , and the G-vector arrays cannot be efficiently handled without distributed storage. The SIRIUS package provides better data distribution and options to the use of the various (LAPACK, ScaLAPACK, Davidson) diagonalization algorithms. One can easily perform Davidson-type diagonalization of the Hamiltonian in the self-consistent loop and benefit from multiple MPI and thread-level parallelization within k-points and within bands. Together with the proper distribution of G-vector related arrays, the SIRIUS package can do plane wave based LAPW and APW+lo calculation of systems larger than many community LAPW/APW+lo codes.
( TM )
TMSMMs based upon anisotropic V 3+ , Fe 2+/3+ , Ni 2+ , and Co 2+ ions; see Refs. 3, 30-35. of Mn 12 complexes. Other significant synthesis effort has been focused on enhancing the magnetic anisotropy of the molecule 37-40 . That has stimulated the development of several different groups of SMMs, such as cyano-bridged SMMs 41-43 , Ln-based SMMs 44-48 , 3d-4f electron elements based SMMs 46,49 , actinide-based SMMs 47 , radical-bridged SMMs 50 and organo-metallic SMMs 51 . The essential context for insightful computational studies illustrated by this brief overview is that SMMs are large, structurally and electronically intricate molecules with complicated spin manifolds. Materials comprised of them are more complicated and demanding. Many of the chemical details of SMMs are largely irrelevant at this stage. However, the presence of heavy nuclei and the importance of anisotropy both implicate the significance of relativistic effects, including spin-orbit coupling. Predictive, materials-specific simulations of systems composed from SMMs thus are extremely challenging.At present, density functional theory (DFT) 52 is the most widely used first-principles calculation method in material science. General DFT is based on the Hohenberg-Kohn theorems stating that every observable of an interacting electron system is a functional of its ground-state charge density, that minimizes the total energy of the system 53 . The theorems do not answer the question how to obtain such a ground-state density. A recipe for obtaining the ground state charge density is given by Walter Kohn and Lu Jeu Sham who introduce an auxiliary system of non-interacting particles54 . The non-interacting particles, described by the Schrodinger-like Kohn-Sham equations, share the same ground state charge density with the interacting particles. With a numerical basis, one can expand the wavefunctions, charge density and potential using the basis and convert the Kohn-Sham equations to a set of linear equations that can be solved using developed eigen system methods. Since the effective potential of the Kohn-Sham Hamiltonian depends on the charge density, which is not known for the true ground state, the Kohn-Sham equations are solved in an iterative procedure which requires the calculation of the density and the potential based on the updated wavefunctions in each iteration.The octa-nuclear cluster [Fe 8 O 2 (OH) 12 ](tacn) 6 ] 8+ (Fe 8 ) reported by Sangregorio et al. 31 is an
example. It has 174 atoms and 756 electrons with an S = 10 ground state that arises from
competing anti-ferromagnetic interactions among eight Fe 3+ (S = 5/2) ions 36 . Magnetic
measurements revealed that it has an anisotropy barrier of 17 cm −1 , much smaller than that
B. Predictive computational approaches
TABLE I .
IPeak memory consumption of single k-point Mn-taa calculation in a cube unit cell of 20Å. Other input parameters are same asTable.II.converge. If the arrays containing G vector indices are not stored in a distributed way during the calculation, they become the major memory consuming quantities and make the calculation very slow. We have tried to use community codes like ELK and Exciting-Plus to calculate the ground state of Mn-taa and found it to be practically difficult. If one performs single k-point calculation for an isolated system, the lack of band parallelism within a k-point makes it run with one MPI task only. The desired large |G| cutoff for density and potential requires a significant amount of memory that makes the calculation slow. A test to run the Mn-taa system using Exciting-Plus on NERSC Cori (128GB memory per node) found,
TABLE II .
IIInput parameters for the [Mn(taa)] LS stateOne highly desirable property for potential application to SMMs would be to demonstrate quantum mechanical coupling of two or more SMMs to one other or to a surfacestructure
Mn-taa, LS state structure
unit cell
20 × 20 × 20Å box
number of atoms
55
R mt (a 0 )
Mn: 2.2; O: 1.6; C: 1.4; H: 1.0;
R M T
min · |G k | max
5
|G max | (a −1
0 ) for ρ and V eff
30
l max for APW
8
l max for ρ and V eff
8
size of 1st variational Hamiltonian
≈ 155,000
k-point grid
1 × 1 × 1
(L)APW configuration
l = −0.15 eV; ∂ E = 0;
lo configuration
Mn: s, p, d; O/C: s, p; H: s;
treated as core state
Mn: 1s, 2s, 2p, 3s; O/C: 1s
total energy tolerance
10 −6 Har
potential tolerance
10 −7 Har
run job setup:
16 MPI tasks
16 OMP threads per task
number of SCF iterations
35
average time per SCF iteration
55 s
HUMO-LOMO gap (eV)
0.68 (VASP: 0.66)
µ tot (µ B )
total: 2.00 (VASP: 2.00)
Mn atom: 1.68 (VASP: 1.77)
IV. SIRIUS: [Mn 3 ] 2 MOLECULE
or other device component, all the while retaining their isolated-molecule magnetic prop-
erties to a useful degree. For this, an SMM-SMM coupled structure was identified for
hydrogen-bonded supramolecular pairs of S = 9/2 SMMs [Mn 4 O 3 Cl 4 (O 2 CEt) 3 (py) 3 ] 69-71 .
Since hydrogen-bonded inter-SMM interactions do not provide easy control of oligomeriza-
tion nor guarantee retention of the oligomeric structure in solution, covalently organic linked
SMM-SMM structures were developed.
First along that line apparently was the [Mn 3 ] 4 SMM tetramer 72 , which is covalently
linked with dioximate linker groups. A recent further development is the [Mn 3 ] 2 dimer
molecule 73 . It is comprised of two Mn 3 units covalently joined via dpd2-dioximate linkers.
TABLE IV .
IVExchange coupling constants from Sirius calculation compared to VASP results.79 for J ab . The J diag value can be obtained with the primitive unit cell. The supercells contain a number of atoms similar to that for the Mn 3 -dimer molecule in the preceding example, namely 140 atoms and 888 electrons Though the supercell is smaller by half (approximately 10 × 10 × 20Å 3 ) than that needed for the Mn-taa, we found proper convergence in total energy still requires the PW cutoff magnitude for density and potential to be as large aslarger than J ab , is already captured without Hubbard U correction in the DFT calcula-
tion. Therefore, we calculated the J constants without U using APW+lo as implemented
in SIRIUS. To estimate the J values, one calculates the energy of the ferromagnetically
and anti-ferromagnetically ordered state energies E F M and E AF M respectively. 80 . Then, as-
suming a Heisenberg Hamiltonian of the form H = J[S 1 · S 2 ], one can determine J from
E F M -E AF M . The calculation involves creating two supercells, 1 × 1 × 2 for J c and 2 × 1 × 1
|G max |=30 (a −1
0 ).
FIG. 4. base octahedral unit of DTN molecule 79
TABLE V .
VVI. SIRIUS: [FE(tBu 2 qsal) 2 ] MOLECULEThe [Fe(tBu 2 qsal) 2 ] molecule is a newly created spin crossover system 81 . Here (tBu 2 qsal) stands for 2,4-diterbutyl-6-((quinoline-8-ylimino)methyl)phenolate. The crystallized structure was determined by X-ray diffraction to be monoclinic, space group P 2 1 /c. It is potentially a functional molecule magnets material because [Fe(tBu 2 qsal) 2 ] can be sublimed at 473-573 K and 10 −3 -10 −4 mbar, hence its thin-film deposition on a substrate is possible.The system undergoes a hysteretic spin transition. The average Fe-N and Fe-O bond lengths elongate from 1.949(2)Å at 100 K to 2.167(2)Å at 230 K and from 1.945(1) at 100 K to 1.997(1)Å at 230 K, respectively. These changes indicate that a conversion from the LS (S = 0) to HS (S = 2) structure takes place as the temperature is increased. In addition, pseudo-potential based DFT calculations 82 found the electron transfer is minimal for the molecule on a monolayer of Au(111) that suggests very small changes in the electronic structure and magnetic properties.Input parameters and outputs of DTN
DTN 1 × 1 × 2 supercell
unit cell
≈ 10 × 10 × 20Å box
number of atoms in unit cell
140
R mt (a 0 )
Na/Co: 2.2; Cl/S/N: 1.4; C: 1.2; H: 1.2;
R M T
min · |G k | max
6
|G max | (a −1
0 ) for ρ and V eff
30
l max for APW
8
l max for ρ and V eff
8
size of 1st variational Hamiltonian
≈ 225,000
k-point grid
2 × 2 × 2
(L)APW configuration
l = −0.15 eV; ∂ E = 0;
for l ≤ l APW
max
lo configuration
Na/Co: s, p, d
Cl/S/C/N: s, p; H: s
treated as core state
Ni/Co: 1s,2s,2p,3s
Cl/S/C/N: 1s
total energy tolerance
10 −7 Har
potential tolerance
10 −7 Har
run job setup:
64 MPI tasks
16 OMP threads per task
number of SCF iterations
90
average time per SCF iteration
≈ 450 s
HUMO-LOMO gap (eV)
0.16 (VASP: 0.18)
magnetic moment
Ni atom: ±0.75 (VASP: ±0.82)
TABLE VI .
VIInput parameters of Fe 4 molecule, S=0 Fe 4 MOF unit cell 15 × 16 × 17.5Å number of atoms in unit cell 430 Cl/S/C/N: s, p; H: s treated as core state Ni/Co: 1s, 2s, 2p, 3sCl/S/C/N: 1sR mt (a 0 )
Fe: 2.2; O: 1.4; C: 1.2; H: 1.0;
R M T
min · |G k | max
5
|G max | (a −1
0 ) for ρ and V eff
30
l max for APW
8
l max for ρ and V eff
8
size of 1st variational Hamiltonian
≈ 325,000
k-point grid
1 × 1 × 1
(L)APW configuration
l = −0.15 eV; ∂ E = 0;
lo configuration
Fe: s, p, d
total energy tolerance
10 −6 Har
potential tolerance
10 −7 Har
run job setup:
64 MPI tasks
16 OMP threads per task
number of SCF iterations
135
average time per SCF iteration
≈ 1400 s
HUMO-LOMO gap (eV)
0.0 (VASP: 0.0)
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|
[] |
[
"Flavorful Z ′ signatures at LHC and ILC",
"Flavorful Z ′ signatures at LHC and ILC"
] |
[
"Shao-Long Chen \nDepartment of Physics and Center for Theoretical Sciences\nNational Taiwan University\nTaipeiTaiwan\n",
"Nobuchika Okada \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDUSA\n\nTheory Group\nKEK\n305-0801TsukubaJapan\n"
] |
[
"Department of Physics and Center for Theoretical Sciences\nNational Taiwan University\nTaipeiTaiwan",
"Department of Physics\nUniversity of Maryland\n20742College ParkMDUSA",
"Theory Group\nKEK\n305-0801TsukubaJapan"
] |
[] |
There are lots of new physics models which predict an extra neutral gauge boson, referred as Z ′ -boson. In a certain class of these new physics models, the Z ′ -boson has flavor-dependent couplings with the fermions in the Standard Model (SM). Based on a simple model in which couplings of the SM fermions in the third generation with the Z ′ -boson are different from those of the corresponding fermions in the first two generations, we study the signatures of Z ′ -boson at the Large Hadron Collider (LHC) and the International Linear Collider (ILC). We show that at the LHC, the Z ′ -boson with mass around 1 TeV can be produced through the Drell-Yan processes and its dilepton decay modes provide us clean signatures not only for the resonant production of Z ′ -boson but also for flavor-dependences of the production cross sections. We also study fermion pair productions at the ILC involving the virtual Z ′ -boson exchange. Even though the center-of-energy of the ILC is much lower than a Z ′ -boson mass, the angular distributions and the forward-backward asymmetries of fermion pair productions show not only sizable deviations from the SM predictions but also significant flavor-dependences.
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10.1016/j.physletb.2008.09.029
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[
"https://arxiv.org/pdf/0808.0331v3.pdf"
] | 14,701,337 |
0808.0331
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95c46a585f35dd5e432c20d79f3cd8e9a77c7235
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Flavorful Z ′ signatures at LHC and ILC
27 Sep 2008
Shao-Long Chen
Department of Physics and Center for Theoretical Sciences
National Taiwan University
TaipeiTaiwan
Nobuchika Okada
Department of Physics
University of Maryland
20742College ParkMDUSA
Theory Group
KEK
305-0801TsukubaJapan
Flavorful Z ′ signatures at LHC and ILC
27 Sep 2008
There are lots of new physics models which predict an extra neutral gauge boson, referred as Z ′ -boson. In a certain class of these new physics models, the Z ′ -boson has flavor-dependent couplings with the fermions in the Standard Model (SM). Based on a simple model in which couplings of the SM fermions in the third generation with the Z ′ -boson are different from those of the corresponding fermions in the first two generations, we study the signatures of Z ′ -boson at the Large Hadron Collider (LHC) and the International Linear Collider (ILC). We show that at the LHC, the Z ′ -boson with mass around 1 TeV can be produced through the Drell-Yan processes and its dilepton decay modes provide us clean signatures not only for the resonant production of Z ′ -boson but also for flavor-dependences of the production cross sections. We also study fermion pair productions at the ILC involving the virtual Z ′ -boson exchange. Even though the center-of-energy of the ILC is much lower than a Z ′ -boson mass, the angular distributions and the forward-backward asymmetries of fermion pair productions show not only sizable deviations from the SM predictions but also significant flavor-dependences.
The search for new physics beyond the Standard Model (SM) is one of the most important issues of particle physics today. In a class of new physics models, the SM gauge group is embedded in a larger gauge group and such a model often predicts an electrically neutral massive gauge boson, referred as Z ′ -boson, associated with the original gauge symmetry breaking into the SM one. There are many example models such as the left-right symmetric model [1], Grand Unified Theories based on the gauge groups SO(10) [2] and E 6 [3], and string inspired models [4] (for a review, see, for example, [5]).
It will be very interesting if a Z ′ -boson is discovered at future collider experiments such as the LHC and ILC. Current limits for the direct production at the Tevatron and indirect effects from LEP experiments imply that the Z ′ -boson is rather heavy and has a very small mixing with the SM Z-boson. No evidence of a signal has been found, and the lower limits on Z ′ mass at 95% confidence level are set to be in the range from 650 to 900 GeV, depending on the considered theoretical models [6].
Recently studies about measurement of the Z ′ -boson at the LHC have been performed [7]. Through the Drell-Yan process, pp → Z ′ X → ℓ + ℓ − X, a Z ′ -boson could be discovered at the LHC if its mass lies around TeV scale with typical electroweak scale couplings to the SM fermions. Once a Z ′ -boson resonance is observed at the LHC, the Z ′ -boson mass can be precisely measured. The next task is to precisely measure other properties of the Z ′ -boson, such as couplings to the SM particles, its spin, etc. Future e + e − linear colliders, such as the ILC, will be capable of such a task, even if the collider energy is not sufficiently high to produce the Z ′ -boson. For example, the precision goal of the ILC can allow us to indicate the existence of Z ′ -boson with mass up to 6 times of center-of-mass energies of the collider [8].
In general, the coupling of Z ′ -boson with the SM fermions can be flavor-dependent. In fact, such a class of models has been proposed by many authors [9,10,11,12]. If this is the case, the signature of Z ′ -boson should show flavor-dependences and the collider phenomenology of Z ′ -boson would be more interesting. In this Letter, we take a simple model recently proposed [12], where the SM fermions in the third generation have couplings with the Z ′ -boson different from those of the corresponding fermions in the first two generations, and study (flavor-dependent) Z ′ -boson signatures at the LHC and ILC.
Let us first give a brief review on a recently proposed "top hypercharge" model [12]. This model is based on the gauge group SU(3) C × SU(2) L × U(1) 1 × U(1) 2 and the SM fermions in the first two generations have hypercharges only under U(1) 1 while the third generation fermions have charges only under U(1) 2 . A complex scalar field, Σ, in the representation (1, 1, 1, 1/2, −1/2) is introduced, by whose vacuum expectation value (VEV) ( Σ = u/ √ 2) the gauge symmetry U(1) 1 × U(1) 2 is broken down to the SM U(1) Y . Associated with this gauge symmetry breaking, the mass eigenstates of two gauge bosons are described as
B μ B µ = cos φ sin φ − sin φ cos φ B 1 µ B 2 µ ,(1)
where B 1.2 µ are the gauge boson of U(1) 1,2 , and the mixing angle φ is defined by tan φ = g ′ 1 /g ′ 2 , the ratio between the coupling constants of U(1) 1,2 . The corresponding masses are m 2 Bµ = 0 and m 2B µ = (g ′2 1 + g ′2 2 )u 2 /4, and the massless state B µ is nothing but the SM U(1) Y gauge field.
In terms of B µ andB µ , the covariant derivative with respect to SU(2) L × U(1) 1 × U(1) 2 for a fermion with a charge (Y 1 , Y 2 ) under U(1) 1 × U(1) 2 is given as
D f µ = ∂ µ − igW a µ T a − ig ′ Y B µ − ig ′ (−Y 1 tan φ + Y 2 cot φ)B µ ,(2)
where Y = Y 1 + Y 2 is a hypercharge under the U(1) Y , and the U(1) Y gauge coupling g ′ is defined as
1 g ′2 = 1 g ′2 1 + 1 g ′2 2 .(3)
We can also express the coupling constants in terms of the electron charge e and the analog of the weak mixing angle θ W in the SM as
g = e sin θ W , g ′ = e cos θ W .(4)
To break the SU(2) L × U(1) Y gauge symmetry down to the U(1) EM , two scalar Higgs doublets Φ 1,2 are introduced, which transform under SU(2) L × U(1) 1 × U(1) 2 as (2, 1/2, 0) and (2, 0, 1/2), respectively, and develop VEVs as
Φ 1,2 = 1 √ 2 0 v 1,2 .(5)
Associated with this symmetry breaking, the W -boson gets mass M 2 W = g 2 (v 2 1 + v 2 2 )/4 while the mass-squared matrix of the neutral gauge bosons is given by
M 2 neutral = u 2 4 0 0 0 0 (g 2 + g ′2 )ǫ g ′2 ǫ sin θ W cot φ cos 2 β (tan 2 φ − tan 2 β) 0 ... 4g ′2 sin 2 2φ + g ′2 ǫ cos 2 β tan 2 φ + sin 2 β cot 2 φ ,(6)
under the basis (A µ, Z µ,Bµ ) with
A µ Z µ = cos θ W sin θ W − sin θ W cos θ W B µ W 3 µ ,(7)
and
ǫ = v 2 1 + v 2 2 u 2 , tan β ≡ v 2 v 1 .(8)
There is a mixing between Z µ andB µ which is constrained to be very small 10 −3 by the electroweak precision measurements [13]. Thus we fix model-parameters to realize such a small mixing, so that the fieldB µ is identified as the Z ′ -boson. Note that from Eq. (6), the mixing between Z µ andB µ vanishes for tan 2 φ = tan 2 β. In the following, we take tan 2 φ = tan 2 β, for simplicity.
Assigning Y 2 = 0 for fermions in the first two generations, while Y 1 = 0 for fermions in the third generation, we obtain the interactions between the Z ′ -boson and the SM fermions such as
− L Z ′ =ψ f γ µ (g f L P L + g f R P R )ψ f Z ′ µ ,(9)
where g L,R for each SM fermion are given as
g u,d,c,s L = − 1 6 e cos θ W tan φ , g t,b L = 1 6 e cos θ W cot φ ; g νe,νµ,e,µ L = 1 2 e cos θ W tan φ , g ντ ,τ L = − 1 2 e cos θ W cot φ ; g u,c R = − 2 3 e cos θ W tan φ , g t R = 2 3 e cos θ W cot φ ;(10)g d,s R = 1 3 e cos θ W tan φ , g b R = − 1 3 e cos θ W cot φ ; g e,µ R = e cos θ W tan φ , g τ R = − e cos θ W cot φ ; g νe,νµ,ντ R = 0 .
As a result, the Z ′ -boson couples differently to the first two and the third generations.
In general, the family non-universal couplings generate tree-level flavor changing neutral currents (FCNCs) [12,14,15,16] and therefore, they are severely constrained by current experimental data. The constraints on the model parameters due to the FCNC processes have been examined in Ref. [12], and in our analysis on the Z ′ -boson phenomenology, a parameter set is chosen to be consistent with the current experiments. The branching ratios of the decay Z ′ →ff as a function of | cos φ| are shown in Fig. 1. As can be expected, the branching ratios into the fermions in the first two generations are different from those into the corresponding fermions in the third generation (except for tan φ = 1). In the limit, cot φ = 0, the couplings between the Z ′ -boson and the third generation fermions are switched off, while the other limit, tan φ = 0, the couplings with the first two generation fermions vanish. For tan φ = 1, the couplings of the third generation fermions becomes the same (up to sign) as those of the corresponding fermions in the first two generations. Since the sign difference does not appear in the Z ′ decay width, no flavordependence can be seen in Fig. 1 for tan φ = 1. However, as will be shown below, this sign difference causes significant differences at collider phenomenologies. Now we investigate the Z ′ -boson production at the LHC. We calculate the dilepton production cross sections through the Z ′ exchange together with the SM processes mediated by the Z-boson and photon 1 . The significance of the Z ′ -boson discovery through the process pp → ℓ + ℓ − X (ℓ + ℓ − = e + e − , µ + µ − ) has been investigated in Ref. [12]. Here we show the 1 The quark pair production channel, in particular, top-quark pair production via the Z ′ -boson exchange is also worth investigating [17], since top quark, which electroweakly decays before hadronization, can be used as an ideal tool to probe new physics beyond the Standard Model [18]. dependence of the cross section on the final state invariant mass M ll described as 2
dσ(pp → ℓ + ℓ − X) dM ll = a,b 1 −1 d cos θ 1 M 2 ll E 2 CMS dx 1 2M ll x 1 E 2 CMS × f a (x 1 , Q 2 )f b M 2 ll x 1 E 2 CMS , Q 2 dσ(qq → ℓ + ℓ − ) d cos θ ,(11)
where E CMS = 14 TeV is the center-of-mass energy of the LHC. In our numerical analysis, we employ CTEQ5M [19] for the parton distribution functions with the factorization scale Q = M Z ′ . 2 shows the differential cross section for pp → e + e − (µ + µ − ) and τ + τ − for M Z ′ = 1.5 TeV and tan φ = 1.0 (left) and 1.5 (right), together with the SM cross section mediated by the Z-boson and photon. Although for tan φ = 1 the peak cross sections are the same, the dependence of dilepton invariant mass shows a remarkable flavor-dependence. This is because for the cross sections away from the peak, the interference between the Z ′ -boson mediated process and the SM processes dominates and the sign difference of the couplings in Eq. (11) between the first two and third generation fermions causes the strong flavordependence of the cross sections. For tan φ = 1.5, the flavor-dependence appears even in the peak cross sections. When we choose a kinematical region for the invariant mass in the range, M Z ′ −Γ Z ′ = 1.35 TeV ≤ M ll ≤ M Z ′ = 1.5 TeV, for example, 7.8 × 10 3 and 8.8 × 10 3 signal events would be observed for e + e − (µ + µ − ) and τ + τ − channels, respectively, with an integrated luminosity of 100 fb −1 , while the number of evens for the SM background would be about 100. In the case tan φ = 1.5, we would expect 4.7 × 10 4 and 9.4 × 10 3 signal events for e + e − (µ + µ − ) and τ + τ − channels, respectively, for the kinematical range around the peak, M Z ′ − Γ Z ′ = 1.35 TeV ≤ M ll ≤ M Z ′ + Γ Z ′ = 1.65 TeV, with an integrated luminosity of 100 fb −1 .
Once a resonance of the Z ′ -boson has been discovered at the LHC, the Z ′ -boson mass can be determined from the peak energy of the dilepton invariant mass. The difference between the cross sections of different dilepton channels at the LHC could be a good distinction between flavor-dependent Z ′ models and the flavor-universal ones. The ILC can provide more precise measurement of the Z ′ -boson properties such as couplings with each (chiral) SM fermion, spin and etc., even if its center-of-mass energy is far below the Z ′ -boson mass [8]. Then, we next study ILC phenomenologies on Z ′ -boson. GeV fixed, we show the differential cross sections for the process e + e − →ff for tan φ = 1 (upper figures) and 1.5 (lower figures) in Fig. 4. Since the collider energy is lower than M Z ′ , the interference between the SM processes and the virtual Z ′ -boson exchange causes the deviations of the cross sections from the SM results. We can see that the deviations from the SM results are flavor-dependent. With high integrated luminosity, the deviations could be used to determine the Z ′ couplings and therefore identify the Z ′ -boson properties. Furthermore, the forward-backward asymmetries A ℓ F B are very distinct for different generation dileptons 3 as shown in Fig. 5. Here we have defined the forward-backward asymmetry for e + e − → ℓ + ℓ − (ℓ = µ(e) or τ ) as
A F B ≡ 1 0 dσ d cos θ d cos θ − 0 −1 dσ d cos θ d cos θ 1 0 dσ d cos θ d cos θ + 0 −1 dσ d cos θ d cos θ(12)
with θ being the scattering angle between the final state ℓ − and the initial e − beam directions. The precise measurements of A ℓ F B at the ILC would reveal not only the existence of the Z ′boson at very high energy but also its flavor-dependent couplings.
In summary, we have studied the signatures of a new charge neutral gauge boson, Z ′ , at the LHC and ILC. Such a gauge boson has been predicted in many new physics models beyond the SM. In particular, we have concentrated on a class of these models where Z ′boson has flavor-dependent couplings with the SM fermions. As a concrete example of such models, we have taken the top hypercharge model proposed in [12], where the SM fermions in the third generation have couplings with the Z ′ -boson different from those of the corresponding SM fermions in the first two generations. For a Z ′ -boson mass around 1 TeV, the dilepton production cross sections via the Z ′ -boson in the s-channel well-exceed the SM background, so that the discovery of the Z ′ -boson resonance would be promising in this case. In addition, the dependence of the cross sections on the dilepton invariant mass shows a clear flavor-dependence around the resonance peak and therefore, the flavornon-universality of the Z ′ -boson resonance could be also observed at the LHC. We have also analyzed the Z ′ -boson effects at the ILC. Even if the energy at the ILC is far below a Z ′ -boson mass, the differential cross sections and the forward-backward asymmetries for the fermion pair production processes show not only sizable deviations from the SM results but also significant flavor-dependences, through which the ILC with a high integrated luminosity could precisely measure the flavor-dependent couplings of the Z ′ -boson with the SM fermions in different generations. Finally, although the analysis in this Letter was based on the model proposed in [12], our strategy is applicable to general models of Z ′ -boson with flavordependent couplings.
Figure 1 :
1The branching ratios of the decay Z ′ →f f as a function of | cos φ| for M Z ′ = 1.5 TeV .
Figure 2 :
2The differential cross section for pp → ℓ + ℓ − X at the LHC for M Z ′ = 1.5 TeV and tan φ = 1.0 (left) and1.5 (right), together with the SM cross section mediated by the Z-boson and photon (dotted line).
Figure 3 :
3The cross sections for e + e − →ff at the ILC as a function of the center-of-mass energy √ s.
Fig.
Fig. 2 shows the differential cross section for pp → e + e − (µ + µ − ) and τ + τ − for M Z ′ = 1.5 TeV and tan φ = 1.0 (left) and 1.5 (right), together with the SM cross section mediated by the Z-boson and photon. Although for tan φ = 1 the peak cross sections are the same, the dependence of dilepton invariant mass shows a remarkable flavor-dependence. This is because for the cross sections away from the peak, the interference between the Z ′ -boson mediated process and the SM processes dominates and the sign difference of the couplings
Figure 4 :
4The differential cross section dσ/d cos θ for the process e + e − →f f at ILC with √ s = 500 GeV.
Figure 5 :
5The forward-backward asymmetry A ℓ F B as a function of the center-of-mass energy √ s.We begin with calculating the cross sections of the process e + e − →ff at the ILC with different energies and the results are depicted inFig. 3for M Z ′ = 1.5 TeV and tan φ = 1 (upper figures) and 1.5 (lower figures), together with the SM cross sections. The results show a similar behavior as inFig. 2and we can see sizable deviations from the SM results and also clear flavor-dependences of the cross sections, even for s ≪ M 2 Z ′ . For √ s = 500
For explicit formulas for the production cross section etc, see, for example, Appendix in[17].
Similar analysis on the model proposed in[11] was performed in[20].
AcknowledgmentsThe work of S.-L.C. was supported in part by the NSC, Taiwan, R.O.C. He would also like to thank Theory Group, KEK for supports and hospitality during his visit. N.O. would like to thank the Maryland Center for Fundamental Physics, and especially Rabindra N. Mohapatra for their hospitality and financial support during his stay. The work of N.O. is supported in part by the National Science Foundation Grant No. PHY-0652363, and the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan (No. 18740170).
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. W M Yao, J. Phys. G. 331Particle Data GroupW. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
. P Langacker, M Plumacher, Phys. Rev. D. 6213006P. Langacker and M. Plumacher, Phys. Rev. D 62, 013006 (2000).
. K R Lynch, E H Simmons, M Narain, S Mrenna, Phys. Rev. D. 6335006K. R. Lynch, E. H. Simmons, M. Narain and S. Mrenna, Phys. Rev. D 63, 035006 (2001).
. V Barger, C W Chiang, P Langacker, H S Lee, Phys. Lett. B. 580186V. Barger, C. W. Chiang, P. Langacker and H. S. Lee, Phys. Lett. B 580, 186 (2004);
. X G He, G Valencia, Phys. Rev. D. 7413011X. G. He and G. Valencia, Phys. Rev. D 74, 013011 (2006);
. K Cheung, C W Chiang, N G Deshpande, J Jiang, Phys. Lett. B. 652285K. Cheung, C. W. Chiang, N. G. Deshpande and J. Jiang, Phys. Lett. B 652, 285 (2007).
. M See, N Arai, K Okada, V Smolek, Simak, arXiv:0804.3740hepph. references thereinSee, for example, M. Arai, N. Okada, K. Smolek and V. Simak, arXiv:0804.3740 [hep- ph], references therein.
. C T Hill, S J Parke, Phys. Rev. D. 494454C. T. Hill and S. J. Parke, Phys. Rev. D 49, 4454 (1994).
. J Pumplin, D R Stump, J Huston, H L Lai, P Nadolsky, W K Tung, JHEP. 0712J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. Nadolsky and W. K. Tung, JHEP 07 (2002) 012.
. E Nardi, Phys. Rev. D. 494394E. Nardi, Phys. Rev. D 49, 4394 (1994);
. E Nardi, T G Rizzo, Phys. Rev. D. 50203E. Nardi and T. G. Rizzo, Phys. Rev. D 50, 203 (1994).
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[
"A RECURSION FOR DIVISOR FUNCTION OVER DIVISORS BELONGING TO A PRESCRIBED FINITE SEQUENCE OF POSITIVE INTEGERS AND A SOLUTION OF THE LAHIRI PROBLEM FOR DIVISOR FUNCTION σ x (n)",
"A RECURSION FOR DIVISOR FUNCTION OVER DIVISORS BELONGING TO A PRESCRIBED FINITE SEQUENCE OF POSITIVE INTEGERS AND A SOLUTION OF THE LAHIRI PROBLEM FOR DIVISOR FUNCTION σ x (n)"
] |
[
"Vladimir Shevelev "
] |
[] |
[] |
For a finite sequence of positive integersAs a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function σ x (n) similar to the classic pentagonal identity in case of x = 1.
| null |
[
"https://arxiv.org/pdf/0903.1743v4.pdf"
] | 14,703,128 |
0903.1743
|
27532fa50a23323035fd41e97e1452cfbe5ef374
|
A RECURSION FOR DIVISOR FUNCTION OVER DIVISORS BELONGING TO A PRESCRIBED FINITE SEQUENCE OF POSITIVE INTEGERS AND A SOLUTION OF THE LAHIRI PROBLEM FOR DIVISOR FUNCTION σ x (n)
23 Mar 2009
Vladimir Shevelev
A RECURSION FOR DIVISOR FUNCTION OVER DIVISORS BELONGING TO A PRESCRIBED FINITE SEQUENCE OF POSITIVE INTEGERS AND A SOLUTION OF THE LAHIRI PROBLEM FOR DIVISOR FUNCTION σ x (n)
23 Mar 2009
For a finite sequence of positive integersAs a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function σ x (n) similar to the classic pentagonal identity in case of x = 1.
Introduction and main results
We start with the two well known beautiful classical recursions. Let p(n) be the number of all partitions of positive integer n and σ(n) be the sum of its divisors. Then (sf [1], [5]) we have (1) p(n) = p(n−1)+p(n−2)−p(n−5)−p(n−7)+p(n−12)+p(n−15)−...
(2) σ(n) = σ(n−1)+σ(n−2)−σ(n−5)−σ(n−7)+σ(n−12)+σ(n−15)−...
where the numbers 1,2,5,7,12,15,... appearing in the successive terms in (1)- (2) are the positive pentagonal numbers {v m } given by (3) v m = m(3m ∓ 1)/2, m = 1, 2, ...
In identities (1)- (2) we accept that p(m) = 0, σ(m) = 0 when m < 0. The only formal difference is that (1) is true with the understanding that (4) p(0) = 1, while (2) is valid with the understanding that (5) σ(0) = n.
Note that, formulas (1)- (2) are proved with help of the famous Euler pentagonal identity
(6) ∞ n=1 (1 − q n ) = ∞ m=−∞ (−1) m q m(3m−1)/2 .
In its turn, a combinatorial proof of (6) is based on the following statement (sf [1]). Let p e (n) (p o (n)) denote the number of partitions of n into even (odd) number of distinct parts. Then (7) p e (n) − p o (n) = (−1) m , if n = m(3m ∓ 1)/2, 0, otherwise .
Let σ x (n) denote the sum of the xth powers of the divisors of n. In 1969, Lahiri [3] noticed that every definition of σ k (0) = f (n), k = 1 is irrelevant in order to keep the classical identity (2) and posed the following problem: "Whether analogous identities exist for divisor function σ k (n) of higher degree?" Formally, for every not necessarily integer value of x, −∞ < x < ∞, for σ x (n) we could consider an identity of type (2) of the form
σ(n) = g x (n) + σ(n − 1) + σ(n − 2) − σ(n − 5) − σ(n − 7) + ...,
where {g x (n)} is some "compensating sequence," and a solution of the Lahiri problem consists of a description of the compensating sequence for every n without a reference to its divisors. In particular, by the definition of σ x (n), and accepting as in (5) σ(0) = n, we find
g x (1) = 0, g x (2) = 2 x − 2, g x (3) = 3 x − 2 x − 1, g x (4) = 4 x − 3 x − 1, g x (5) = 5 x − 4 x − 3 x − 2 x + 4, g x (6) = 6 x − 5 x − 4 x , ...
At first sight, this sequence is even more complicated than σ x (n), and it seems hardly probable to find a required description of it. Our paper, in particular, is devoted to this aim. For a simplification of our transformations, below we accept the unique convention
(8) σ x (n) = 0, if n ≤ 0
It is easy to see that in this case we have only a little change of the compensating sequence in the identity of the same form
σ x (n) = h x (n) + σ x (n − 1) + σ x (n − 2) − σ x (n − 5) − σ x (n − 7) + ..., such that h x (n) = g x (n) + (−1) m−1 , if n = m(3m ∓ 1)/2, 0, otherwise .
Note that this relation is so simple only due to Euler pentagonal identity (6); in more general case (see below Theorem 1) the corresponding relations could be very complicated and the convention (8) plays the unique role for the obtaining of general result. In particular, we write (1)-(2) in just a little another form. Namely, according to (8), instead of conventions (4)-(5), we accept the unique convention
p(0) = 0, σ(0) = 0.
Then with help of (7) it is easy to see that, instead of (1)-(2), we have
p(n) = h (p) (n)+ (9) p(n − 1) + p(n − 2) − p(n − 5) − p(n − 7) + p(n − 12) + p(n − 15) − ...,
where the compensating sequence has the form
(10) h (p) (n) = (−1) m−1 , if n = m(3m ∓ 1)/2, 0, otherwise .
and, in view of the same structure of (1) and (2) and taking into account (4)-(5), we see that
σ(n) = h (σ) (n)+ (11) σ(n − 1) + σ(n − 2) − σ(n − 5) − σ(n − 7) + σ(n − 12) + σ(n − 15) − ..., where (12) h (σ) (n) = (−1) m−1 n, if n = m(3m ∓ 1)/2, 0, otherwise .
Before formulating a generalization of (9) and (11), we study the divisor function over divisors belonging to a prescribed finite sequence A of positive integers. In the trivial case of a one-element sequence A = {a} we put
(13) σ ({a}) x (n) = a x , if a|n, n > 0, 0, otherwise , x ∈ (−∞, +∞).
According to (13), we accept
(14) σ ({a}) x (n) = 0, n ≤ 0, such that (15) σ ({a}) x (n) = σ ({a}) x (n − a) + a x , if n = a, 0, otherwise .
Consider now, for a fixed k ≥ 1, an arbitrary sequence
(16) A = {a j } k j=1
of positive integers. For a fixed x, let us consider an associated sequence
(17) B(A; x) = {b i (x)} 2 k i=1 , where (18) b i (x) = a x j 1 + a x j 2 + a x j 3 + ... + a x jr , if the binary expansion of i − 1 is (19) i − 1 = 2 j 1 −1 + 2 j 2 −1 + ... + 2 jr−1 , 1 ≤ j 1 < j 2 < ... < j r , 1 ≤ r ≤ k. In particular, since 2 k − 1 = 2 1−1 + 2 2−1 + ... + 2 k−1 , then (20) b 2 k (x) = a x 1 + a x 2 + ... + a x k ,
while, since to i = 1 corresponds the empty set of terms in (19), then
(21) b 1 (x) = 0.
Furthermore,
(22) b 2 (x) = a x 1 , b 3 (x) = a x 2 , b 4 (x) = a x 1 + a x 2 , etc. Moreover, denote (23) b i (1) = b i , 1 ≤ i ≤ 2 k .
For n ≥ 1, consider divisor function over sequence A
(24) σ (A) x (n) = d|n, d∈A d x
in the understanding that every term d x repeats correspondingly to the multiplicity of d in sequence A. Besides, we accept the convention [2] which is defined as
(25) σ (A) x (n) = 0, if n ≤ 0. Denote by {t n } the Thue-Morse sequence [4],(26) t n = (−1) s(n) ,
where s(n) denotes the number of ones in the binary expansion of n.
Theorem 1. In convention σ(n ≤ 0) = 0, we have the following recursion
(27) σ (A) x (n) = h (A) x (n) + 2 k i=2 t 2i−1 σ (A) x (n − b i )
where the compensating sequence h
(A) x (n) is defined as (28) h (A) x (n) = i≥2: b i =n t 2i−1 b i (x).
Remark 1. Taking into account that
1 + s(i − 1) = s(2(i − 1) + 1) = s(2i − 1), we prefer to write t 2i−1 instead of −t i−1 .
Note that, as follows from (28), for n > b 2 k , h
(A) x (n) = 0 such that (29) σ (A) x (n) = 2 k i=2 t 2i−1 σ (A) x (n − b i ), n > b 2 k .
Consider now the divisor function
(30) σ x (n) = d|n d x .
Putting here (1)) if the binary expansion of i − 1 is defined by (19), we obtain the following result.
(31) b i (x) = j x 1 + j x 2 + j x 3 + ... + j x r (and b i = b iTheorem 2. We have σ x (n) = h x (n)+ (32) σ x (n−1)+σ x (n−2)−σ x (n−5)−σ x (n−7)+σ x (n−12)+σ x (n−15)−...,
where the compensating sequence {h x (n)} is defined as
(33) h x (n) = i≥2: b i =n t 2i−1 b i (x), n ≥ 1.
Theorem 2 gives a solution of the Lahiri problem for divisor function σ x (n).
Proof of Theorem 1
We use the induction over the number of elements of sequence A, the base of which is given by (15). Note that if, instead of A = {a 1 , ..., a k }, to consider the sequence (34)
A ′ = {a 1 , ..., a k , a k+1 }, then we have
(35) σ (A ′ ) x (n) = σ (A) x (n) + σ ({a k+1 }) x (n).
Furthermore, in the case of A ′ , to every i, 1 ≤ i ≤ 2 k , with the binary expansion (19) of i − 1 corresponds bijectively the number 2 k + i from [2 k + 1, 2 k+1 ] with the expansion
2 k + i − 1 = 2 j 1 −1 + 2 j 2 −1 + ... + 2 jr−1 + 2 k
such that the associated sequence has the form
(36) b i (x) = a x j 1 + a x j 2 + a x j 3 + ... + a x jr , if 1 ≤ i ≤ 2 k , a x j 1 + a x j 2 + a x j 3 + ... + a x jr + a x k+1 , if 2 k + 1 ≤ i ≤ 2 k+1
. This means that, for 1 ≤ l ≤ 2 k , we have
(37) b l+2 k (x) = b l (x) + a x k+1 (in particular, b 1+2 k (x) = a x k+1 ). Notice also, that (38) t 2 k+1 +1 = 1; t 2(l+2 k )−1 = t 2l+2 k+1 −1 = −t 2l−1 and (39) 1≤l≤2 k t 2l−1 = − 1≤l≤2 k t l−1 = 0.
Suppose now that the theorem is true up to k. Then, using (37)-(38), we have
2 k+1 i=2 t 2i−1 σ (A ′ ) x (n − b i ) = 2 k i=2 t 2i−1 σ (A ′ ) x (n − b i ) + 2 k+1 i=2 k +1 t 2i−1 σ (A ′ ) x (n − b i ) = 2 k i=2 t 2i−1 σ (A ′ ) x (n − b i ) + 2 k l=1 t 2(l+2 k )−1 σ (A ′ ) x (n − b l+2 k ) = (40) 2 k i=2 t 2i−1 σ (A ′ ) x (n − b i ) − 2 k l=1 t 2l−1 σ (A ′ ) x ((n − b l ) − a k+1 ).
Furthermore, by (40) and (35), we have
2 k+1 i=2 t 2i−1 σ (A ′ ) x (n − b i ) = 2 k i=2 t 2i−1 σ (A) x (n − b i ) + 2 k i=2 t 2i−1 σ ({a k+1 }) x (n − b i ) (41) − 2 k i=1 t 2i−1 σ (A) x ((n − b i ) − a k+1 ) − 2 k i=1 t 2i−1 σ ({a k+1 }) x ((n − b i ) − a k+1 ).
Note that, according to (15),
2 k i=2 t 2i−1 σ ({a k+1 }) x (n − b i ) − 2 k i=1 t 2i−1 σ ({a k+1 }) x ((n − b i ) − a k+1 ) (42) = σ ({a k+1 }) x (n) + a x k+1 1≤i≤2 k : n−b i =a k+1 t 2i−1 .
Therefore, from (41) we find
2 k+1 i=2 t 2i−1 σ (A ′ ) x (n − b i ) = σ ({a k+1 }) x (n) + a x k+1 1≤i≤2 k : n−b i =a k+1 t 2i−1 + (43) 2 k i=2 t 2i−1 σ (A) x (n − b i ) − 2 k i=1 t 2i−1 σ (A) x ((n − a k+1 ) − b i ),
or, using the inductive hypothesis, we have
2 k+1 i=2 t 2i−1 σ (A ′ ) x (n − b i ) = σ ({a k+1 }) x (n) + a x k+1 1≤i≤2 k : n−b i =a k+1 t 2i−1 − (44) (σ (A) x ((n − a k+1 ) − h (A) (n − a k+1 )) + σ (A) x (n) − h (A) (n). Furthermore, 2≤i≤2 k+1 : b i =n t 2i−1 b i (x) = 2≤i≤2 k : b i =n t 2i−1 b i (x)+ 2 k +1≤i≤2 k+1 : b i =n t 2i−1 b i (x) = h (A) x − 1≤l≤2 k : b l+2 k =n t 2l−1 b l+2 k (x) = (45) h (A) x (n) − h (A) x (n − a k+1 ) − a x k+1 1≤l≤2 k : b l =n−a k+1 t 2l−1 .
Finally, summing the results of (44) and (45), we complete our proof:
2 k+1 i=2 t 2i−1 σ (A ′ ) x (n − b i ) + 2≤i≤2 k+1 : b i =n t 2i−1 b i (x) = σ ({a k+1 }) x (n) + σ (A) x (n) = σ (A ′ ) x (n).
Proof of Theorem 2
If to consider as a finite sequence A the sequence A = A k = {1, 2, ..., k}, then, for n ≤ k, we have
(46) σ (A k ) x (n) = σ x (n)
and, by Theorem 1, the (±)-structure of σ
(A k ) x
(n) is the same as in the case of x = 1 (see (11)). Therefore, independently from the summands (either σ 1 (n) or σ x (n)) we have the same reductions, i.e.
σ x (n) = σ (A k ) x (n) = h (A k ) x (n)+ σ (A k ) x (n − 1) + σ (A k ) x (n − 2) − σ (A k ) x (n − 5) − σ (A k ) x (n − 7)+ (47) σ (A k ) x (n − 12) + σ (A k ) x (n − 15) − ..., (n ≤ k),
with the compensating sequence
(48) h (A k ) x (n) = i≥2: b i =n t 2i−1 b i (x)
where b i (x) are defined by (31). If, instead of A k , to consider N, then for every n we actually consider a finite part of (47) which corresponds to A n = {1, 2, ..., n}. Thus (47) is true for A = N, and (32)-(33) follow. Example 1. Consider the case of x = 1, i.e. the case of sum-of-divisors function.
Then we have
h (N ) 1 (n) = n i≥2: b i =n t 2i−1 = −n i≥2: b i =n (−1) s(i−1) = n(p o (n) − p e (n))
and, in view of (7), we obtain (11) as a special case of Theorem 2.
Expression of compensating sequence {h
x (n)} = {h x (n) (N ) } via known sequences
Note that from the definition of sequence b n (x) (see (31) and (19)) it follows that if
(49) n − 1 = i≥1 β(i)2 i−1
is the binary expansion of n − 1, then
(50) b n (x) = i≥1 β(i)i x , such that (51) b n = i≥1 β(i)i.
Notice that, (51) is Sequence A029931(n-1) in [6]). Denoting Thus, according to (59), we find
h 0 (7) = 1 − 2 − 2 − 2 + 3 = −2.
Some another identities
In case of the finite set
A k = {2 j−1 } k j=1 , according to (18), we have (60) b i (x) = 2 (j 1 −1)x + 2 (j 2 −1)x + ... + 2 (jr−1)x , if (61) i − 1 = 2 j 1 −1 + 2 j 2 −1 + ... + 2 jr−1 , 1 ≤ j 1 < j 2 < ... < j r , 1 ≤ r ≤ k.
Thus b i has an especially simple form:
(62) b i = i − 1.
Let n = 2 α 1 −1 + ... + 2 αm−1 .
Then, according to Theorem 1 and (62), we find Considering now the infinite sequence of powers of 2: The sequence {σ (A) 0 (n)) − 1} n≥1 is well-known so-called "the binary carry sequence" (A007814 in [6]). In the case of x = 1, we obtain the identity (67) n j=1 (−1) s(n−j) σ (A) 1 (j)) = (−1) s(n)−1 n.
h (A k ) x (n) = i≥2: b i =n t 2i−1 b i (x) =(A = {2 j−1 } j≥1 ,
The sequence {σ (A) 1 (n))} n≥1 is also well-known (see A038712 in [6]).
Example 2 .
2Consider the case of x = 0, i.e. the case of the number of divisors of n. Then, by (50) and (53), the compensating sequence has the form (54) h 0 (n) = j≥1: η(j)=n (−1) s(j)−1 s(j), where s(n), as in the above, is the number of ones in the binary expansion of n. The first terms of compensating sequence {h Note that, in view of (49), the expression (51) gives the number of all partitions with distinct parts of a fixed values of b n . This means that if to denote by e n the set of the terms of A029931 for which η(j) = n : e 1 = {1}, e 2 = {2}, e 3 = {3, 4}, e 4 = {5, 8}, e 5 = {6, 9, 16}, e 6 = {7, 10, 17, 32}, e 7 = {11, 12, 18.33.64} ..., then the concatenation of this sets leads to the ordering of all partitions of n with distinct parts : n = i≥1 β(i)i respectively to the values of i≥1 β(i)2 i−1 . Thus this way leads us to the Adams-Watters sequence "Decimal equivalent of binary encoding of partitions into distinct parts" (see A118462 in [10, 17, 32, 11, 12, 18, 33, 64, ... Denote Sequence (56) via W (n) and put |e n | = R(n), where, for n ≥ 1, {R(n)} is Sequence A000009[6] that is the number of partitions of n into distinct parts. s(W (T (n)−m))−1 b W (T (n)−m)+1 (x).
Example 3 .
3Let us calculate the seventh term h 0 (7) of sequence (55). s(W (T (n)−m))−1 s(W (T (n) − m)). If n = 7, then we have from the corresponding tables of [6]: R(n) = 5, T (n) = 18, W (18) = 64, W (17) = 33, W (16) = 18, W (15) = 12, W (14) = 11.
63) t 2n+1 b n+1 (x) = (−1) s(2n+1) (2 (α 1 −1)x + ... + 2 (αm−1)x )andσ (A k )x (n) = (−1) s(2n+1) (2 (α 1 −1)x + ...
( 0
0x (n) = (−1) s(2n+1) (2 (α 1 −1)x + ... −1) s(n−j) σ (A) x (j)), where n = 2 α 1 −1 + ... + 2 αm−1 .In particular, in the case of x =
The theory of partitions. G E Andrews, Addison-WesleyG. E. Andrews, The theory of partitions, Addison-Wesley, 1976.
The fractal structure of rarefied sums of the Thue-Morse sequence. S Goldstein, K A Kelly, E R Speer, J. of Number Theory. 42S. Goldstein, K. A. Kelly and E. R. Speer, The fractal structure of rarefied sums of the Thue-Morse sequence, J. of Number Theory 42 (1992), 1-19.
Identities connecting elementary divisor function of different degrees, and allied congruences. D B Lahiri, Math. Scand. 24D. B. Lahiri, Identities connecting elementary divisor function of different degrees, and allied congruences, Math. Scand., 24 (1969), 102-110.
Reccurent geodesics on a surface of negative curvature. M Morse, Trans. Amer. Math. Soc. 22M. Morse, Reccurent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84-100.
Zuckerman An introduction to the theory of numbers. I Niven, H , John WileyNew YorkI. Niven and H. S. Zuckerman An introduction to the theory of numbers, John Wiley, New York, 1960.
N J A Sloane, .com) Departments of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105. The On-Line Encyclopedia of Integer Sequences. Israel. e-mail:[email protected]. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (http: //www.research.att.com) Departments of Mathematics, Ben-Gurion University of the Negev, Beer- Sheva 84105, Israel. e-mail:[email protected]
|
[] |
[
"Learning Class-Level Bayes Nets for Relational Data",
"Learning Class-Level Bayes Nets for Relational Data"
] |
[
"Oliver Schulte [email protected] \nSchool of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada\n",
"Hassan Khosravi [email protected] \nSchool of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada\n",
"Bahareh Bina \nSchool of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada\n",
"Flavia Moser [email protected] \nSchool of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada\n",
"Martin Ester [email protected] \nSchool of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada\n"
] |
[
"School of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada",
"School of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada",
"School of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada",
"School of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada",
"School of Computing Science\nSimon Fraser University Vancouver-Burnaby\nB.CCanada"
] |
[] |
Many databases store data in relational format, with different types of entities and information about links between the entities. The field of statistical-relational learning (SRL) has developed a number of new statistical models for such data. In this paper we focus on learning class-level or first-order dependencies, which model the general database statistics over attributes of linked objects and links (e.g., the percentage of A grades given in computer science classes). Classlevel statistical relationships are important in themselves, and they support applications like policy making, strategic planning, and query optimization. Most current SRL methods find class-level dependencies, but their main task is to support instance-level predictions about the attributes or links of specific entities. We focus only on class-level prediction, and describe algorithms for learning class-level models that are orders of magnitude faster for this task. Our algorithms learn Bayes nets with relational structure, leveraging the efficiency of single-table nonrelational Bayes net learners. An evaluation of our methods on three data sets shows that they are computationally feasible for realistic table sizes, and that the learned structures represent the statistical information in the databases well. After learning compiles the database statistics into a Bayes net, querying these statistics via Bayes net inference is faster than with SQL queries, and does not depend on the size of the database.
| null |
[
"https://arxiv.org/pdf/0811.4458v2.pdf"
] | 14,708,437 |
0811.4458
|
a8cf61a6d80af00ad804139b10a7352154fe3f2e
|
Learning Class-Level Bayes Nets for Relational Data
Oliver Schulte [email protected]
School of Computing Science
Simon Fraser University Vancouver-Burnaby
B.CCanada
Hassan Khosravi [email protected]
School of Computing Science
Simon Fraser University Vancouver-Burnaby
B.CCanada
Bahareh Bina
School of Computing Science
Simon Fraser University Vancouver-Burnaby
B.CCanada
Flavia Moser [email protected]
School of Computing Science
Simon Fraser University Vancouver-Burnaby
B.CCanada
Martin Ester [email protected]
School of Computing Science
Simon Fraser University Vancouver-Burnaby
B.CCanada
Learning Class-Level Bayes Nets for Relational Data
Many databases store data in relational format, with different types of entities and information about links between the entities. The field of statistical-relational learning (SRL) has developed a number of new statistical models for such data. In this paper we focus on learning class-level or first-order dependencies, which model the general database statistics over attributes of linked objects and links (e.g., the percentage of A grades given in computer science classes). Classlevel statistical relationships are important in themselves, and they support applications like policy making, strategic planning, and query optimization. Most current SRL methods find class-level dependencies, but their main task is to support instance-level predictions about the attributes or links of specific entities. We focus only on class-level prediction, and describe algorithms for learning class-level models that are orders of magnitude faster for this task. Our algorithms learn Bayes nets with relational structure, leveraging the efficiency of single-table nonrelational Bayes net learners. An evaluation of our methods on three data sets shows that they are computationally feasible for realistic table sizes, and that the learned structures represent the statistical information in the databases well. After learning compiles the database statistics into a Bayes net, querying these statistics via Bayes net inference is faster than with SQL queries, and does not depend on the size of the database.
Introduction
Many real-world applications store data in relational format, with different tables for entities and their links. Standard machine learning techniques are applied to data stored in a single table, that is, in nonrelational, propositional or "flat" format [18]. The field of statistical-relational learning (SRL) aims to extend machine learning algorithms to relational data [8,3]. One of the major machine learning tasks is to use data to build a generative statistical model for the variables in an application domain [8]. In the single-table learning setting, the goal is often to represent predictive dependencies between the attributes of a single individual (e.g., between the intelligence and ranking of a student). In the SRL setting, the goal is often to represent, in addition, dependencies between attributes of different individuals that are related or linked to each other (e.g., between the intelligence of a student and the difficulty of a course given that the student is registered in the course).
Task Description: Modelling Class-Level Dependencies Many SRL models distinguish two different levels, a class or type dependency model G M and an instance dependency model G I [7,14,19]. In a graphical SRL model, the nodes in the instance dependency model represent attributes of individuals or relationships. The nodes in the class dependency model correspond to attributes of the tables. The use of the term "class" here is unrelated to classification; [16] views it as analogous to the concept of class in object-oriented programming. An example of a class-level probabilistic dependency is "among students with high intelligence, the rate of GPA = 4.0 is 80%". An instance-level prediction would be "given that Jack is highly intelligent, the probability that his GPA is 4.0 is 80%". Thus class-level dependencies are concerned with the rates at which events occur, or at which properties hold within a class, whereas instance-level dependencies are concerned with specific events or the properties of specific entities [11,1]. In terms of predicate logic, the classlevel model features terms that involve first-order variables (e.g., intelligence(S ), where S is a variable ranging over a domain like Students), whereas the instance-level graph features terms that involve constants (e.g., intelligence(jack ) where jack is a constant that denotes a particular student) [3,14,5].
Typically, SRL systems instantiate a class-level model G M with the specific entities, their attributes and their relationships in a given database to obtain an instance dependency graph G I . An important purpose of the instance graph is to support predictions about the attributes of individual entities. A key issue for making such predictions is the combining problem: how to combine information from different related entities to predict a property of a target entity. Current SRL model construction algorithms learn class-level dependencies and instance-level predictions at the same time.
What is new about our approach is that we focus on classlevel variables only rather than making predictions about individual entities. We apply Bayes net technology to design new algorithms especially for learning class-level dependencies. In experiments our algorithms run at two orders of magnitude faster than a benchmark SRL method. Our models thus trade-off tractability of learning with the ability to answer queries about individual entities.
Applications. Examples of applications that provide motivation for the class-level statistical models include the following.
1. Policy making and strategic planning. A university administrator may wish to know which program characteristics attract high-ranking students in general, rather than predict the rank of a specific student in a specific program.
2. Query optimization is one of the applications of SRL where a statistical model predicts a probability for given table join conditions that can be used to infer the size of the join result [9]. Estimating join sizes is a key problem for database query optimization. In queries that involve several tables being joined together, the ideal scenario is to have smaller intermediate joins [17]. A class-level statistical model may be used for estimating frequency counts in the database to select smaller joins, and so to optimize the speed of answering queries by taking more efficient intermediate steps.
For example, suppose we wish to predict the size of the join of a Student table with a Registered table that records which students are registered in which courses, where the selection condition is that the student have high intelligence. In a logical query language like the domain relational calculus [26], this join would be expressed by the conjunctive formula Registered (S , C ), intelligence(S ) = high. A query to a Join Bayes Net can be used to estimate the frequency with which this conjunction is true in the database, which immediately translates into an estimate of the size of the join that corresponds to the conjunction. The join conditions often do not involve specific individuals.
3. Private Data. In some domains, information about individuals is protected due to confidentiality concerns. For example, [6] analyzes a database of police crime records. The database is anonymized and it would be unethical for the data mining researcher to try and predict which crimes have been committed by which individuals. However, it is appropriate and important to look for general risk factors associated with crimes, for example spatial patterns [6]. Under the heading of privacy-preserving data mining, researchers have devoted much attention to the problem of discovering class-level patterns without compromising sensitive information about individuals [27].
Approach. We apply Bayes nets (BNs) to model classlevel dependencies between attributes that appear in separate tables. Bayes nets [20] have been one of the most widely studied and applied generative model classes. A BN is a directed acyclic graph whose nodes represent random variables and whose edges represent direct statistical associations. Our class-level Bayes nets contain nodes that correspond to the descriptive attributes of the database tables, plus Boolean nodes that indicate the presence of a relationship; we refer to these as Join Bayes nets (JBNs). To apply machine learning algorithms to learn the structure of a Join Bayes Net from a database, we need to define an empirical database distribution over values for the class-level nodes that is based on the frequencies of events in the database. In a logical setting, the question is how to assign database frequencies for a conjunction of atomic statements such as intelligence(S ) = high, Registered (S , C ), difficulty(C ) = high. We follow the definition that was established in fundamental AI research concerning the combination of logic and statistics, especially the classic work of Halpern and Bacchus [11,1]. This research generalized the concept of single-table frequencies to the relational domain: the frequency of a first-order formula in a relational database is the number of instantiations of the variables in the formula that satisfy the formula in the database, divided by the number of all possible instantiations. In the example above, the instantiation frequency would be the number of student-course pairs where the student is highly intelligent, the course is highly difficult, and the student is registered in the course, divided by all possible student-course pairs. In terms of table joins, the instantiation frequency is the number of tuples in the join that corresponds to the conjunction, divided by the maximum size of the join.
Our learn-and-join algorithm aims to find a model of the database distribution. It upgrades a single table BN learner, which can be chosen by the user, to carry out relational learning by decomposing the learning problem for the entire database into learning problems for smaller tables. The basic idea is to apply the BN learner repeatedly to tables and join tables from the database, and merge the resulting graphs into a single graphical model for the entire database. This algorithm treats the single-table BN learner as a module within the relational structure learning system. This means that only minimal work is required to build a statisticalrelational JBN learner from a single-table BN learner.
The main algorithmic problem in parameter estimation (CP-table estimation) for Join Bayes nets is counting the number of satisfying instantiations (groundings) of a firstorder formula in a database. We show how the recent virtual join algorithm of [15] can be applied to solve this problem efficiently. The virtual join algorithm is an efficient algorithm designed to compute database instantiation frequencies that involve non-existent links.
Our experiments provide evidence that the learn and join algorithm leverages the efficiency, scalability and reliability of single-table BN learning into efficiency, scalability and reliability for statistical-relational learning. We benchmark the computational performance of our algorithm against structure learning for Markov Logic Networks (MLNs), one of the most prominent statistical-relational formalisms [5]. In our experiments on small datasets, the run-time of the learn-andjoin algorithm is about 20 times faster than the state-of-the art Alchemy program [5] for learning MLN structure. On medium-size datasets, such as the Financial database from the PKDD 1999 cup, Alchemy does not return a result given our system resources, whereas the learn-and-join algorithm produces a Join Bayes net model within less than 10 min. To evaluate the learned structures, we apply BN inference algorithms to predict relational frequencies and compare them with gold-standard frequencies computed via SQL queries. The learned Join Bayes nets predict the database frequencies very well. Our experiments on prediction take advantage of the fact that because JBNs use the standard Bayes net format, class-level queries can be answered by standard BN inference algorithms that are used "as is". Other SRL formalisms focus on instance-level inference, and do not support class-level inference, at least in their current implementations. Our datasets and code are available for ftp download from ftp://ftp.fas.sfu.ca/pub/cs/oschulte.
Paper Organization. As statistical-relational learning is a complex subject and a variety of approaches have been proposed, we review related work in some detail. A preliminary section introduces Bayes nets and predicate logic. We formally define Join Bayes nets and the instantiation frequency distribution that they model. The main part of the paper describes structure and parameter learning algorithms for Join Bayes nets. We evaluate the algorithms on one synthetic dataset and two public domain datasets (MovieLens and Financial), examining the learning runtimes and the predictive performance of the learned models.
Contributions. The main contributions of our work are as follows.
(1) A new type of first-order Bayes net for modelling the database distribution over attributes and relationships, which is defined by applying classic AI work on probability and logic.
(2) New efficient algorithms for learning the structure and parameters for the first-order Bayes net models.
(3) An evaluation of the algorithms on three relational databases, and an evaluation of the predictive performance of the learned structures.
Related Work
A preliminary version of our results appeared in the proceedings of the STRUCK and GKR workshops at IJCAI 2009.
There is much AI research on the theoretical foundations of the database distribution, and on representing and reasoning with instantiation frequencies in an axiomatic framework based on theorem proving [11,1]. Our approach utilizes graphical models for learning and inference with the database distribution rather than a logical calculus. For inference, our approach appears to be the first that utilizes standard BN inference algorithm to carry out probabilistic reasoning about database frequencies. For learning, our work appears to be the first to use Bayes nets to learn a statistical model of the database distribution.
Other SRL formalisms. Various formalisms have been proposed for combining logic with graphical models, such as Bayes Logic Networks (BLNs) [14], parametrized belief networks [21], object-oriented Bayes nets [16], Probabilistic Relational Model (PRMs) [7], and Markov Logic Networks [5]. We summarize the main points of comparison. General SRL overviews are provided in [14,8,3]. The main difference between JBNs and other SRL formalisms is their semantics. The semantics of JBNs is specified at the class level without reference to an instance-level model; a JBN models the database distribution defined by the instantiation frequencies.
Syntactically, Join Bayes Nets are very similar to parametrized belief networks and to BLNs. BLNs require the specification of combining rules, a standard concept in Bayes net design, to solve the combining problem. For instance, the class-level model may specify the probability that a student is highly intelligent given that she obtained an A in a difficult course. A set of grades thus translates into a multiset of probabilities, which the combining rule translates into a single probability. Essentially, a JBN is a BLN without combining rules.
The key difference between PRMs and BLNs is that PRMs use aggregation functions (like average), not combining rules [3]. [14,Sec.10.4.3] shows how aggregate functions can be added to BLNs; the same construction works for adding them to JBNs. Object-oriented Bayes nets use aggregation like PRMs, and have special constructs for capturing class hierarchies; otherwise their expressive power is similar to BLNs and JBNs.
Markov Logic Networks combine ideas from undirected graphical models with a logical representation. An MLN is a set of formulas, with a weight assigned to each. MLNs are like JBNs, and unlike BLNs and PRMs, in that an MLN is complete without specifying a combining rule or aggregation function. They are like BLNs and PRMs, and unlike JBNs, in that their semantics is specified in terms of an instance-level network whose nodes contain ground formulas (e.g., age(jack ) = 40 ). Instance-level prediction is defined by using the log-linear form of Markov random fields [5,Sec.12.4].
Inference and Learning Because JBNs use the Bayes net format, class-level probabilistic queries that involve firstorder variables and no constants can be answered by standard efficient Bayes nets inference algorithms, which are used "as is". This can be seen as an instance of lifted or first-order probabilistic inference, a topic that has received a good deal of attention recently [21,4]. Other SRL formalisms focus on instance-level queries that involve constants only, and do not support class-level inference, at least in their current implementations.
The most common approach to SRL structure learning is to use the instance-level structure G I to assign a likelihood to a given database D. This likelihood is the counterpart to the likelihood of a sample given a statistical model in single table learning. Learning searches for a parametrized model G M whose instance-level graph G I assigns a maximum likelihood to the given database D, where typically the likelihood is balanced with a complexity penalty to prevent overfitting [7,14]. This approach is a conceptually elegant way to learn class-level dependencies and instance-level predictions at the same time. JBNs separate the task of finding generic class-level dependencies from the task of predicting the attributes of individuals. Two key advantages of learning directed graphical models such as Bayes nets at the class-level only include the following.
(1) Class-level learning avoids the problem that the instantiated model may contain cycles, even if the classlevel model does not. For example, suppose that the classlevel model indicates that if a student S 1 is friends with another student S 2 , then their smoking habits are likely to be similar, so Smokes(S 1 ) predicts Smokes(S 2 ). Now if in the database we have a situation where a is friends with b, and b with c, and c with a, the instance level model would contain [7,19]. If learning is defined in terms of the instance model, cycles cause difficulties because concepts and algorithms for directed acyclic models no longer apply. In particular, the likelihood of a database that measures data fit is no longer defined. These difficulties have led researchers to conclude that "the acyclicity constraints of directed models severely limit their applicability to relational data" [19] (see also [5,24]).
a cycle Smokes(a) → Smokes(b) → Smokes(c) → Smokes(a)
(2) Other directed SRL formalisms require extra structure to solve the combining problem for instance-level predictions, such as aggregation functions or combining rules. While the extra structure increases the representational power of the model, it also leads to considerable complications in learning.
Preliminaries
We combine the concepts of graphical models from machine learning, relational schemas from database theory, and conjunctions of literals from first-order logic. This section in-troduces notations and definitions for these background subjects.
Bayes nets
A random variable is a pair X = dom(X), P X rangle where dom(X) is a set of possible values for X called the domain of X and P X : dom(X) → [0, 1] is a probability distribution over these values. For simplicity we assume in this paper that all random variables have finite domains (i.e., discrete or categorical variables). Generic values in the domain are denoted by lowercase letters like x and a. An atomic assignment assigns a value X = x to random variable x, where x ∈ dom(X). A joint distribution P assigns a probability to each conjunction of atomic assignments; we write P (X 1 = x 1 , ..., X n = x n ) = p, sometimes abbreviated as P (x 1 , ..., x n ) = p. To compactly refer to a set of variables like {X 1 , ..X n } and an assignment of values x 1 , .., x n , we use boldface X and x. If X and Y are sets of variables with P (Y = y) > 0, the conditional probability P (X = x|Y = y) is defined as P (X = x, Y = y)/P (Y = y).
We employ notation and terminology from [20,23] for a Bayesian Network. A Bayes net structure is a directed acyclic graph (DAG) G, whose nodes comprise a set of random variables denoted by V . When discussing a Bayes net, we refer interchangeably to its nodes or its variables. The parents of a node X in graph G are denoted by PA G X , and an assignment of values to the parents is denoted as pa G X . When there is no risk of confusion, we often simply write pa X . A Bayes net is a pair G, θ G rangle where θ G is a set of parameter values that specify the probability distributions of children conditional on instantiations of their parents, i.e. all conditional probabilities of the form P (X = x|pa G X ). These conditional probabilities are specified in a conditional probability table (CP-table) for variable X. A BN G, θ G rangle defines a joint probability distribution over V = {v 1 , .., v n } according to the formula
P (v = a) = n i=1 P (v i = a i |pa i = a pa i )
where a i is the value of node v i specified in assignment a, the term a pa i denotes the assignment of a value to each parent of v i specified in assignment a, and P (v i = a i |pa i = a pa i ) is the corresponding CP-table entry. Thus the joint probability of an assignment is obtained by multiplying the conditional probabilities of each node value assignment given its parent value assignments.
Relational Schemas and First-Order Formulas
We begin with a standard relational schema containing a set of tables, each with key fields, descriptive attributes, and foreign key pointers. A database instance specifies the tuples contained in the tables of a given database schema. Table reftable:university-schema shows a relational schema Student(student id, intelligence, ranking) Course(course id , difficulty, rating) Professor (prof essor id, teaching ability, popularity) reg (student id, course id , grade, satisf action) ra (student id, prof id, salary, capability) for a database related to a university (cf. [7]), and Figure reffig:university-tables displays a small database instance for this schema. A table join of two or more tables contains the rows in the Cartesian products of the tables whose values match on common fields. Table reftable:university-schema.
If the schema is derived from an entity-relationship model (ER model) [26,Ch.2.2], the tables in the relational schema can be divided into entity tables and relationship tables. Our algorithms generalize to any data model that can be translated into logical vocabulary based on first-order logic, which is the case for an ER model. The entity types of Schema reftable:university-schema are students, courses and professors. There are two relationship tables: Registration records courses taken by each student and the grade and satisfaction achieved, while ra records research assistantship contracts between students and professors.
In our university example, there are two entity tables: a Student table and a C table. There is one relationship table reg with foreign key pointers to the Student and C tables whose tuples indicate which students have registered in which courses. Intuitively, an entity table corresponds to a type of entity, and a relationship table represents a relation between entity types.
It is well known that a relational schema can be translated into [26]. We follow logic-based approaches to SRL that use logic as a rigorous and expressive formalism for representing regularities found in the data. Specifically, we use first-order logic with typed variables and function symbols, as in [5,21]. This formalism is rich enough to represent the constraints of an ER schema via the following transla-tion: Entity sets correspond to types, descriptive attributes to functions, relationship tables to predicates, and foreign key constraints to type constraints on the arguments of relationship predicates.
Formulas in our syntax are constructed using three types of symbols: constants, variables, and functions. Sometimes we refer to logical variables as first-order variables, to distinguish them from random variables. A type or domain is a set of constants. In the case of an entity type, the constants correspond to primary keys in an entity table. Each constant and variable is assigned to a type, and so are the arguments (inputs) and the values (outputs) of functions. A predicate is a function whose values are the special truth values constants T , F . [5] discusses the appropriate use of first-order logic for SRL. While our logical syntax is standard, we give the definition in detail, to clarify the space of statistical patterns that our learning algorithms can be applied to, and to give a rigorous definition of the database frequency, which is key to our learning method. Table reftable:vocab defines the logical vocabulary. and illustrates the correspondence to an ER schema. A term θ is any expression that denotes a single object; the notation θ denotes a vector or list of terms. If P is a predicate, we sometimes write P (θ) for P (θ) = T and ¬P (θ) for P (θ) = F . Terms are recursively constructed as follows.
A constant or variable is a term.
2. Let f be a function term with argument type τ (f ) = (τ 1 , . . . , τ n ). A list of terms θ 1 , . . . , θ n matches τ (f ) if each θ i is of type τ i . If θ matches the argument type of f , then the expression f (θ) is a function term whose type is dom(f ), the value type of f .
An atom is an equation of the form θ = θ where the types of θ and θ match. A negative literal is an atom of the form P (θ) = F ; all other atoms are positive literals. The formulas we consider are conjunctions of literals, or for short just conjunctions. We use the Prolog-style notation Ł 1 , . . . , Ł n for Ł 1 ∧ · · · ∧ Ł n , and vector notation Ł, C for conjunctions of literals. A term (literal) is ground if it contains no variables; otherwise the term (literal) is open.
If F = {f 1 , . . . , f n } is a finite set of open function terms F , an assignment to F is a conjunction of the form A = (f 1 = a 1 , .
. . , f n = a n ), where each a i is a constant. A relationship literal is a literal with two or more variables.
A database instance D (possible world, Herbrand interpretation) assigns a denotation constant to each ground function term f (a) which we denote by Function Domain In this case, we assign the descriptive attribute the special value ⊥ for "undefined". The general constraint is that for any descriptive attribute f P of a predicate P ,
[f (a)] D .dom(f ) = τ i The value domain of f dom(intelligence) = {1 , 2 , 3 } dom(Student) = {T , F } Predicate P, E, R, etc. A function with domain {T, F } Student, Course, Registration Relationship R, R ,[f P (a)] D = ⊥ ⇐⇒ [P (a)] D = F .
For a ground literal Ł, we write D |= Ł if Ł evaluates as true in D, and D |= Ł otherwise. If P is a predicate, we sometimes write D |= P (a) for D |= (P (a) = T ) and D |= ¬P (a) for D |= (P (a) = F ). A ground conjunction C evaluates to true just in case each of its literals evaluate to true. We make the unique names assumption that distinct constants denote different objects, so
D |= (θ = θ ) ⇐⇒ [θ] D = [θ ] D .
If E is an entity type, the domain of E is the set of constants satisfying E (primary keys in the E table). The domain of a variable X T of type T is the same as the domain of the type, so dom D (X T ) = dom D (T ). A grounding γ for a set of variables X 1 , . . . , X k assigns a constant of the right type to each variable X i (i.e., γ(X i ) ∈ dom D (X i )). If γ is a grounding for all variables that occur in a conjunction C, we write γC for the result of replacing each occurrence of a variable X in C by the constant γ(X). The number of groundings that satisfy a conjunction C in D is defined as
# D = |{γ : D |= γC}|
where γ is any grounding for the variables in C, and |S| denotes the cardinality of a set S.
The ratio of the number of groundings that satisfy C, over the number of possible groundings is called the instantiation frequency or the database frequency of C. Formally we define
(3.1) P D (C) = # D (C) |dom D (X 1 )| × · · · × |dom D (X k )|
where X 1 , . . . , X k , k > 0, is the set of variables that occur in C.
Discussion. When all function terms contain the same single variable X, Equation (3.1) reduces to the standard definition of the single-table frequency of events. For example, P D (intelligence(S ) = 3 ) is the ratio of students with an intelligence level of 3, over the number of all students. Halpern gave a definition of the frequency with which a first-order formula holds in a given database [11,Sec.2], which assumes a distribution µ over the domain of each type. The instantiation frequency (3.1) is a special case of his with a uniform distribution µ over the elements of each domain [11, fn.1]. As Halpern explains, an intuitive interpretation of this definition is that it corresponds to generic regularities or random individuals, such as "the probability that a randomly chosen bird will fly is greater than .9". The conditional instantiation frequency plays an important role for discriminative learning in Inductive Logic Programming (ILP). For instance, the classic FOIL system generalizes entropy-based decision tree learning to first-order rules, where the goal is to predict a class label l. FOIL uses the conditional instantiation frequency P D (l|C) to define the entropy of the empirical class distribution conditional on the body C of a rule.
Literal(s) L # D (Ł) P D (Ł) intelligence(S ) = 1 1 1/3 ¬intelligence(S ) = 1 2 2/3 difficulty(C ) = 2 1 1/2 Registered (S , C ) 4 4/6 ¬Registered (S , C ) 2 2/6 grade(S , C ) = B 2 2/6 grade(S , C ) = B , salary(S , P ) = hi 1 1/12
Structure Learning for Join Bayes Nets
We define the class of Bayes net models that we use to model the database distribution and present a structure learning algorithm. . . , f n (θ n ) = a n . We use the term "Join" because such conjunctions correspond to table joins in databases. A key step in model learning is to define an empirical distribution over the random variables in the model. Since the random variables in a JBN are function terms, this amounts to associating a probability with a conjunction C of literals given a database. We can use the instantiation frequency P D (C) as the empirical distribution over the nodes in a JBN. The goal of JBN structure learning is then to construct a model of P D given a database D as input.
The Learn-and-Join Algorithm
We describe our structure learning algorithm, then discuss its scope and limitations. The algorithm is based on a general schema for Table reftable:university-schema.
upgrading a propositional BN learner to a statistical relational learner. By "upgrading" we mean that the propositional learner is used as a function call or module in the body of our algorithm. We require that the propositional learner takes as input, in addition to a single table of cases, also a set of edge constraints that specify required and forbidden directed edges. The output of the algorithm is a DAG G for a database D with variables as specified in Definition refdef:jbn. Our approach is to "learn and join": we apply the BN learner to single tables and combine the results successively into larger graphs corresponding to larger table joins.
In principle, a JBN may contain any set of open function terms, depending on the attributes and relationships of interest. To keep the description of the structure learning algorithm simple, we assume that a JBN contains a default set of nodes as follows: (1) one node for each descriptive attribute, of both entities and relationships, (2) one Boolean indicator node for each relationship, (3) the nodes contain no constants. For each type of entity, we introduce one firstorder variable. The algorithm has four phases (pseudocode shown as Algorithm refalg:structure).
(1) Analyze single tables. Learn a BN structure for the descriptive attributes of each entity table E of the database separately (with primary keys removed). The aim of this phase is to find within-table dependencies among descriptive attributes (e.g., intelligence(S ) and ranking(S )).
(2) Analyze single join tables. Each relationship table R is considered. The input table for each relationship R is the join of that table with the entity tables linked by a foreign key constraint (with primary keys removed). Edges between attributes from the same entity table E are constrained to agree with the structure learned for E in phase (1). Additional edges from variables corresponding to attributes from different tables may be added. The aim of this phase is to find dependencies between descriptive attributes conditional on the existence of a relationship. This phase also finds dependencies between descriptive attributes of the relationship table R.
(3) Analyze double join tables. The extended input relationship tables from the second phase (joined with entity tables) are joined in pairs to form the input tables for the BN learner. Edges between variables considered in phases (1) and (2) are constrained to agree with the structures previously learned. The graphs learned for each join pair are merged to produce a DAG G. The aim of this phase is to find dependencies between descriptive attributes conditional on the existence of a relationship chain of length 2.
(4) Satisfy slot chain constraints. For each link A → B in G, where A and B are attributes from different tables, arrows from Boolean relationship variables into B are added if required to satisfy the following constraints: (1) A and B share variable among their arguments, or (2) the parents of B contain a chain of foreign key links connecting A and B. Figure reffig:structure-learn illustrates the increasingly Figure 3: To illustrate the learn-and-join algorithm. A given BN learner is applied to each table and join tables (nodes in the graph shown). The presence or absence of edges at lower levels is inherited at higher levels.
large joins built up in successive phases. Algorithm refalg:structure gives pseudocode.
Discussion
We discuss the patterns that can be represented in our current JBN implementation and the dependencies that can be found by the learn-and-join algorithm.
Number of Variables for each Entity Type. There are data patterns that require more than one variable per type to express. For a simple example, suppose we have a social network of friends represented by a Friend (X , Y ) relationship both of whose arguments are of type Person. There may be a correlation between the smoking of a person and that of her friends, so a JBN may place a link between nodes smokes(X ) and smokes(Y ), which requires two variables of type Person. In principle, the case of several variables for a given entity type can be translated to the case of one variable per entity type as follows [22]. For each variable of a given type, make a copy of the entity table. In our example, we would obtain two Person tables, Person X and Person Y . Each copy can be treated as its own type with just one associated variable. We leave for future work an efficient implementation of the learn-and-join algorithm with several variables for a given entity type. Econstraints += Get-Constraints(PBN(Em , ∅)) 5: end for 6: for m=1 to r do 7:
Nm := join of Rm and entity tables linked to Rm 8:
Econstraints += Get-Constraints(PBN(Nm, Econstraints)) 9: end for 10: for all N i and N j with an entity table foreign key in common do 11:
K ij := join of N i and N j 12:
Econstraints += Get-Constraints(PBN(K ij , Econstraints)) 13: end for 14: Construct DAG G from Econstraints 15: Add edges from Boolean relationship variables to satisfy slot chain constraints 16: Return G Correlation Coverage. The join-and-learn algorithm finds correlations between descriptive attributes, within a single table and between attributes from different linked tables.
It does not, however, find dependencies between relationship variables (e.g., Married (X , Y ) predicts Friend (X , Y )). A JBN search for such dependencies could use local search methods such as those described in [14,7]. In sum, the learn-and-join algorithm is suitable when the goal is to find correlations between descriptive attributes conditional on a given link structure.
Parameter Learning in Join Bayes Nets
This section treats the problem of computing conditional frequencies in the database distribution, which corresponds to computing sample frequencies in the single table case. The main problem is computing probabilities conditional on the absence of a relationship. This problem arises because a JBN includes relationship indicator variables such as reg(S, C), and building a JBN therefore requires modelling the case where a relationship does not hold. We apply the recent virtual join (VJ) algorithm of [15] to address this computational bottleneck. The key constraint that the VJ algorithm seeks to satisfy is to avoid enumerating the number of tuples that satisfy a negative relationship literal. A numerical example illustrates why this is necessary. Consider a university database with 20,000 Students, 1,000 Courses and 2,000 TAs. If each student is registered in 10 courses, the size of a Registered table is 200,000, or in our notation # D (Registered (S , C )) D ) = 2 × 10 5 . So the number of complementary student-course pairs is # D (¬Registered (S , C ))) = 2 × 10 7 − 2 × 10 5 , which is a much larger number that is too big for most database systems. If we consider joins, complemented tables are even more difficult to deal with: suppose that each course has at most 3 TAs. Then # D (Registered (S , C ), TA(T , C )) < 6 × 10 5 , whereas # D (¬Registered (S , C ), ¬TA(T , C )) is on the order of 4 × 10 10 . After explaining the basic idea behind the VJ algorithm, we give pseudocode for utilizing it in to estimate CP-table entries via joint probabilities. We conclude with a run-time analysis.
The Virtual Join algorithm for CP-tables. Instead of computing conditional frequencies, the VJ algorithm computes joint probabilities of the form P D (child , parent values), which correspond to conjunctions of literals. Conditional probabilities are easily obtained from the joint probabilities by summation. While generally it is easier to find conditional rather than joint probabilities, there is an efficient dynamic programming scheme (informally a "1 minus" trick) that relies on the simpler structure of conjunctions. The enumeration of groundings for negative literals can be avoided by recursively applying a probabilistic principle (a "1-minus" scheme). Consider the equation P (C) = P (C, Ł) + P (C, ¬Ł), which entails P (C, ¬Ł) = P (C) − P (C, Ł). (5.2) where C is a conjunction of literals and Ł a literal. This equation shows that the computation of a probability involving a negative relationship literal ¬Ł can be recursively reduced to two computations, one with the positive literal Ł and one with neither Ł nor ¬Ł, each of which contains one less negative relationship literal. In the base case, all literals are positive, so the problem is to find P D (C) for database instance D where C contains positive relationship literals only. This can be done with a standard database table join. Figure reffig:example illustrates the recursion.
The VJ algorithm computes the database frequency for just a single input conjunction of literals. We adapt it for CP-table estimation with two changes: (1) We compute all frequencies that are defined by the same join table at once when the join has been built, and (2) we use the CP-table itself as our data structure for storing the results of intermediate computations from which other database frequencies are derived. The algorithm can be visualized as a dynamic program that successively fills in rows in a joint probability table, or JP-table, where we first fill in rows with 0 nonexistent relationships, then rows with 1 nonexistent relationship, etc. A JP-table is just like a CPtable whose rows correspond to joint probabilities rather than conditional probabilities. Algorithm refalg:adapted shows the pseudocode for the VJ algorithm.
Implementation and Complexity Analysis. The intermediate results of the computation are stored in an extended JP-table structure that features a third value * for relationship predicates (in addition to T , F ). This is used to represent the frequencies where a relationship predicate and its attributes are unspecified.
The algorithm satisfies the key constraint that it never enumerates the groundings that satisfy a negative relationship literal. A detailed complexity analysis of the VJ algorithm is given in [15]; we summarize the main points that are relevant for CP-table estimation. Essentially, each computation step fills in one entry in the extended JP-table. Compared to the CP-tables for a given Join Bayes net structure, our algorithm adds an extra auxilliary value * to the domain of each relationship indicator variable. Thus the increase in the size of the data structure is manageable compared to the CP-table in the original JBN. An important point is that the recursive update in Line refline:update does not require a database access. Data access occurs only in Lines refline:start-join-refline:end-join. Therefore the cost in terms of database accesses is essentially the cost of counting frequencies in a join table comprising all m relationships that occur in the child or parent nodes (Line refline:join with i = m). This computation is necessary for any algorithm that estimates the CP-table entries. It can be optimized using the tuple ID propagation method of [28,29]. The crucial parameter for the complexity of this join is the number m of relationship predicates. In our learn-and-join algorithm, this parameter is bounded at m = 2. [15] provide an analysis whose upshot is that for typical SRL applications this parameter can be treated as a small constant. A brief summary of the reasons is as follows. (1) The space of models or rules that need to be searched becomes infeasible if too many relationships are considered at once. (2) Patterns that involve many relationships, for instance relationship chains of length 3 or more, are hard to understand for users. (3) Objects related by long relationship chains tend to carry less statistical information about a target object. We next examine empirically the performance of our our structure and parameter learning algorithms.
Empirical Evaluation of JBN Learning Algorithms
The learn-and-join algorithm trades off learning complexity with the coverage of data correlations. In this section we evaluate both sides of this trade-off on three datasets: the run-time of the algorithm, especially as the dataset grows larger, and its performance in predicting database probabilities at the class level, which is the main task that motivates our algorithm. The more important dependencies the algorithm misses, the worse its predictive performance, so evaluating the predictions provides information about the quality of the JBN structure learned.
Implementation and Datasets
All experiments were done on a QUAD CPU Q6700 with a 2.66GHz CPU and 8GB of RAM. The implementation used many of the procedures in version 4.3.9-0 of CMU's Tetrad package [25]. For single table BN search we used the Tetrad implementation of GES search [2] with the BDeu score (structure prior uniform, ESS=8). Edge constrains were implemented using Tetrad's "knowledge" functionality. JBN inference was carried out with Tetrad's Rowsum Exact Updater algorithm. Our datasets and code are available for ftp download from ftp://ftp.fas.sfu.ca/pub/cs/oschulte.
Datasets University Database. In order to check the correctness of our algorithms directly, we manually created a small dataset, based on the schema given in reftable:university-schema. The entity tables contain 38 students, 10 courses, and 6 Professors. The reg table has 92 rows and the RA table has 25 rows.
MovieLens Database. The second dataset is the Movie-Lens dataset from the UC Irvine machine learning repository. It contains two entity tables: User with 941 tuples and Item with 1,682 tuples, and one relationship table Rated with 80,000 ratings. The User table has 3 descriptive attributes age, gender , occupation. We discretized the attribute age into three bins with equal frequency. The table Item represents information about the movies. It has 17 Boolean attributes that indicate the genres of a given movie; a movie may belong to several genres at the same time. For example, a movie may have the value T for both the war and the action attributes. We performed a preliminary data analysis and omitted genres that have only weak correlations with the rating or user attributes, leaving a total of three genres.
Financial Database. The third dataset is a modified version of the financial dataset from the PKDD 1999 cup. There are two entity tables, Client with 5,369 tuples and Account with 4,500 tuples. Two relationship tables, CreditCard with 2,676 tuples and Disposition with 5,369 tuples relate a client with an account. The Client table has 10 descriptive attributes: the client's age, gender and 8 attributes on demographic data of the client. The Account table has 3 descriptive attributes: information on loan amount associated with an account, account opening date, and how frequently the account is used.
Learning: Experimental Design
To benchmark the runtimes of our learning algorithm, we applied the structure learning routine learnstruct (default options) of the Alchemy package for MLNs [5]. We chose MLNs for the following reasons: (1) MLNs are currently one of the most active areas of SRL research (e.g., [5,12]). Part of the reason for this is that undirected graphical models avoid the computational and representational problems caused by cycles in instance-level directed models (Section refsec:related). Discriminative MLNs can be viewed as logic-based templates for conditional Markov random fields, a prominent formalism for relational classification [24]. (2) Alchemy provides open-source, state-of-the-art learning software for MLNs. Structure learning software for alternative systems like BLNs and PRMs is not easily available. 1 (3) BLNs and PRMs require the specification of additional structure like aggregation functions or combining rules. This confounds our experiments with more parameters to specify. Also, incorporating the extra structure complicates learning for these formalisms, so arguably a direct comparison with JBN learning is not fair. In contrast, the MLN formalism, like the JBN, does not re-Algorithm 2 A dynamic program for estimating JP-table entries in a Join Bayes Net.
Notation: A row r corresponds to a partial or complete assignment for function terms. The value assigned to function term f (θ) in row r is denoted by f r . The probability for row r stored in JP-table τ is denoted by τ (r). Input: database D; child variable and parent variables divided into a set R 1 , . . . , R m of relationship predicates and a set C of function terms that are not relationship predicates. Calls: initialization function JOIN-FREQUENCIES(C, T = R 1 = · · · = R k ). Computes join frequencies conditional on relationships R 1 , . . . , R k being true. Output: Joint Probability table τ such that the entry τ (r) for row r ≡ (C = C r , R = R r ) is the frequencies P D (C = C r , R = R r ) in the database distribution D. if r has exactly i true relationship literals R 1 , .., R i then {r has m−i unspecified relationship literals} 5: find P D (C = C r , R = R r ) using JOIN-FREQUENCIES(C = C r , R = R r ). Store the result in τ (r). if r has exactly i false relationship literals R 1 , .., R i then {find conditional probabilities when R 1 is true and when unspecified} 13: Let r T be the row such that (1)
R 1 rT = T , (2) f R 1
rT is unspecified for all attributes f R 1 of R 1 , and (3) r T matches r on all other variables. 14: Let r * be the row that matches r on all variables f that are not R 1 or an attribute of R 1 and has R 1 r * unspecified. 15: {The rows r * and r T have one less false relationship variable than r.} In the MovieLens dataset, the algorithm finds a number of cross-entity table links involving the age of a user. Because genres have a high negative correlation, the algorithm produces a dense graph among the genre attributes. The richer relational structure of the Financial dataset is reflected in a more complex graph with several cross-table links. The birthday of a customer (translated into discrete age levels) has especially many links with other variables. The CP-tables for the learned graph structures were filled in using the dynamic programming algorithm refalg:adapted. Table reftable:runtime presents a summary of the run times for the datasets.
The databases translate into ground atoms for Alchemy input as follows: University 390, MovieLens 39,394, and Financial 16,129. On our system, Alchemy was able to process the University database, but did not have sufficient computational resources to return a result for the MovieLens and Financial data. We therefore subsampled the datasets to obtain small databases on which we can compare Alchemy's runtime with that of the join-and-learn algorithm. Because Alchemy returned no result on the complete datasets, we formed three subdatabases by randomly selecting entities for each dataset. We restricted the relationship tuples in each subdatabase to those that involve only the selected entities. The resulting subdatabase sizes are as follows. For Movie-Lens, (i) 100 users, 100 movies = 1,477 atoms (ii) 300 users, 300 movies = 9698 atoms (iii) 500 users, 400 movies = 19,053 atoms. For Financial, (i) 100 Clients, 100 Accounts = 3,228 atoms, (ii) 300 Clients, 300 Accounts = 9,466 atoms, (iii) 500 Clients, 500 Accounts = 15,592 atoms. For the Financial dataset, which contains numerous descriptive attributes, Alchemy returned a result only for the smallest subdatabase (i). Table reftable:runtime shows that the runtime of the JBN learning algorithm applied to the entire dataset is 600 times faster than Alchemy's learning time on a dataset about half the size. We emphasize that this is not a criticism of Alchemy structure learning, which aims to find a structure that is optimal for instance-level predictions (cf. Section refsec:related). Rather, it illustrates that the task of finding class-level dependencies is much less computationally complex taken as an independent task than when it is taken in conjunction with optimizing instance-level predictions. The next section compares the probabilities estimated by the JBN with the database frequencies computed directly from SQL queries. is difficult as the available implementations support only instance-level queries. 2 This reflects the fact that current SRL systems are designed for the task of predicting attributes of individual entities, rather than class-level prediction. We therefore evaluated the class-level predictions of our system against gold standard frequency counts obtained directly from the data using SQL queries. We follow the approach of [9] in our experiments and use the sample frequencies for 2 For example, Alchemy and the Balios BLN engine [14] support only queries with ground atoms as evidence. We could not obtain source code for PRM inference. parameter estimation. This is appropriate when the goal is to evaluate whether the statistical model adequately summarizes the data distribution. If the data frequencies are smoothed to support generalizations beyond the data, the first step is to compute the data frequencies, so our experiments are relevant to this case too. We randomly generated 10 queries for each data set according to the following procedure. First, choose randomly a target node V and a value a such that P (V = a) is the probability to be predicted. Then choose randomly the number k of conditioning variables, ranging from 1 to 3. Make a random selection of k variables V 1 , . . . , V k and corresponding values a 1 , . . . , a k . The query to be answered is then P (V = a|V 1 = a 1 , . . . , V k = a k ). Table reftable:inference shows representative test queries. It compares the probabilities predicted by the JBN with the frequencies in the database as computed by an SQL query, as well as the runtimes for computing the probability using the JBN vs. the SQL.
Inference: Results
The averages reported are taken over 10 random queries for each dataset. We see in Table reftable:inference and Figure reffig:results that the predicted probabilities are very close to the data frequencies: The average difference is less than 3% for MovieLens, and less than 8% for Financial. The measurements for Financial are taken on queries with positive relationship literals only, because the SQL queries that involve negated relationships did not terminate with a result for the Financial dataset. The graph shows the nontermination as corresponding to a high time-out number.
Observations about processing speed that hold for all datasets include the following. (1) For queries involving negated relationships, JBN inference was much faster on the MovieLens dataset. SQL queries with negated relationships were infeasible on the Financial dataset, whereas the JBN returns an answer in around 10 seconds. (2) Both JBN and SQL queries speed up as the number of conditioning variables increases. (3) Higher probability queries are slower with SQL, because they correspond to larger joins, whereas the size of the probability does not affect JBN query processing.
Other observations depend on the difference between the datasets: MovieLens has relatively many tuples and few attributes, whereas Financial has relatively few tuples and many attributes. These factors differentially affected the performance of SQL vs. JBN inference as follows (for queries involving positive relationships only). (1) A larger number of attributes and/or a larger number of categories in the attributes in the schema decreases the speed of both JBN inference and SQL queries. But the slowdown is greater for JBN queries because the required marginalization steps are expensive (summing over possible values of nodes). (2) The number of tuples in the database table is a very significant Table 5: The table shows representative randomly generated queries. We compare the probability estimated by our learned JBN model (P(Model)) with the database frequency computed from direct SQL queries (P(SQL)), and the execution times (in seconds) for each inference method. The SQL query in the last line exceeded our system resources. For simplicity we omitted first-order variables. factor for the speed of SQL queries but does not affect JBN inference. This last point is an important observation about the data scalability of JBN inference: While the cost of the learning algorithms depends on the size of the database, once the learning is completed, query processing is independent of database size. So for applications like query optimization that involve many calls to the statistical inference procedure, the investment in learning a JBN model is quickly amortized in the fast inference time.
Conclusion
Class-level generic dependencies between attributes of linked objects and of links are important in themselves, and they support applications like policy making, strategic planning, and query optimization. The focus on class-level dependencies brings gains in tractability of learning and inference. The theoretical foundation of our approach is classic AI research which established a definition of the frequency of a first-order formula in a given relational database. We described efficient and scalable algorithms for structure learning and parameter estimation in Join Bayes nets, which model the database frequencies over attributes of linked ob-jects and links. Our algorithms upgrade a single-table Bayes net learner as a self-contained module to perform relational learning. JBN inference can be carried out with standard algorithms "as is" to answer class-level probabilistic queries. An evaluation of our methods on three data sets shows that they are computationally feasible for realistic table sizes, and that the learned structures represented the statistical information in the databases well. Querying database statistics via the net is often faster than directly with SQL queries, and does not depend on the size of the database. A fundamental limitation of our approach is that Join Bayes nets do not directly support instance-level queries (our model does not include a solution to the combining problem). Limitations that can be addressed in future research include restrictions on the types of correlation that our structure learning algorithm can discover, such as dependencies between relationships, and dependencies that require more than one first-order variable per entity type to represent.
Figure 1 :
1A database instance for the schema in
2 :
2Definition of Logical Vocabulary with typed variables and function symbols. The examples refer to the database instance of Figure reffig:university-tables. database instance of Figure reffig:university-tables since Registered ((kim, 101 ) is false in D.)
Examples. The examples refer to the database instance D of Figure reffig:university-tables. The entity types are Student, Course, Professor . For each entity type we introduce one variable S, C, P . The constants of type Student are {jack , kim, paul }. True ground literals include intelligence(jack ) = 3 and ¬Registered (kim, 101 ). Table reftable:literals shows various open literals and their frequency in database D as derived from the number of true groundings.
3 :
3Examples of open literals and their frequency in the database instance of Figure reffig:university-tables. The bottom four lines show relationship literals.
DEFINITION 1 .
1Let D be a database with associated logical vocabulary vocab. A Join Bayes Net (JBN) structure for D is a directed acyclic graph whose nodes are a finite set {f 1 (θ 1 ), . . . , f n (θ n )} of open function terms. The domain of a node v i = f i (θ i ) is the range of f i (the set of possible output values for f i ).
Figure reffig :
reffiguniversity-JBN shows an example of a Join Bayes net. We also refer to relationship terms that appear in a JBN as relationship indicator variables or simply relationship variables. A JBN assigns probabilities to conjunctions of literals of the form f 1 (θ 1 ) = a 1 , .
Figure 2 :
2A Join Bayes Net for the relational schema shown in
Figure 4 :
4A frequency with negated relationships (nonexistent links) can be computed from frequencies that involve positive relationships (existing links) only. The leaves in the computation tree involves existing database tables only. The subtractions involve looking up results of previous computations. To reduce clutter, we abbreviated some of the predicates.
1: {fill in rows with no false relationships using table joins} 2: for all valid rows r with no assignments of F to relationship predicates do 3:for i = 0 to m do 4:
(r) := τ (r * ) − τ (r T ).19: end for quire extra structure beyond the relational schema.6.3 Learning: ResultsWe present results of applying our learning algorithms to the three relational datasets. The resulting JBNs are shown in Figures reffig:university-JBN, reffig:structmovie, and reffig:structfinancial.
6. 4
4Inference: Experimental Design A direct comparison of class-level inference with other SRL formalisms
Figure 5 :
5The JBN structure learned by our merge learning algorithm refalg:structure for the MovieLens Dataset.
Figure 6 :
6The JBN structure learned by our merge learning algorithm refalg:structure for the Financial Dataset.
Gender = M |date = 1, status = A, Disp = F ) 0.5187/10.65No result
Figure 7 :
7Comparing the probability estimates and times (in sec) for obtaining them from the learned JBN models versus SQL queries.
Table 1 :
1A relational schema for a university domain. Key fields are underlined.
The value of descriptive relationship attributes is not defined for tuples that are not linked by the relationship.(For example, grade(kim, 101 ) is not well defined in the Constants T, F, ⊥, a 1 , a 2 , . . . ⊥ for "undefined" jack , 101 , A, B , 1 , 2 , 3 Types τ 1 , τ 2 , . . . list of types Student, A, B , C , DDescription
Notation
Comment/Definition
Example
Functions
f 1 , f 2 , . . .
number of arguments =
arity of f
intelligence, difficulty, grade
Student, Registration
Table
Table
Algorithm 1 Pseudocode for structure learning Input: Database D with E 1 , ..Ee entity tables, R 1 , ...Rr Relationship tables, Output: A JBN graph for D. Calls: PBN: Any propositional Bayes net learner that accepts edge constraints and a single table of cases as input. Notation: PBN(T, Econstraints) denotes the output DAG of PBN. Get-Constraints(G) specifies a new set of edge constraints, namely that all edges in G are required, and edges missing between variables in G are forbidden. 1: Add descriptive attributes of all entity and relationship tables as variables to G.Add a boolean indicator for each relationship table
to G.
2: Econstraints = ∅ [Required and Forbidden edges]
3: for m=1 to e do
4:
{Recursively extend the table to JP-table entrieswith false relationships.} 10: for all rows r with at least one assignment of F to a relationship predicate do6:
end if
7:
end for
8: end for
9: 11:
for i = 1 to m − 1 do
12:
Table 4 :
4The runtimes-in seconds-for structure learning (SL) and parameter learning (PL) on our three datasets.
The webpage[13] lists different SRL systems and which software is available for them. The Balios BLN engine[14] supports only parameter learning, not structure learning; see[13]. We could not obtain source code for PRM structure learning.
AcknowledgmentsThis research was supported by Discovery Grants to the first and fifth author from the Natural Sciences and Engineering Research Council of Canada. A preliminary version of our results was presented at the STRUCK and GKR workshops at IJCAI 2009, and at the Computational Intelligence Forum at the University of British Columbia. We thank the audiences and Ke Wang for helpful comments.
. P( Int(s)=1 Reg, (s , C =f, P Ra(s, =f) =, P( Int(S)=1,Reg(S,C) =F, RA(S,P)=F) =
S)=1) =. P( Int, P(Int(S)=1) =
. Reg (s Int=1, C)= T Ra=f) =, p(Int=1, Reg(S,C)= T, RA=F) =
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Exploring the power of heuristics and links in multi-relational data mining. X Yin, J Han, Springer-Verlag. Berlin HeidelbergX. Yin and J. Han. Exploring the power of heuristics and links in multi-relational data mining. In Springer-Verlag Berlin Heidelberg 2008, 2008.
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[
"A Unified Definition and Computation of Laplacian Spectral Distances",
"A Unified Definition and Computation of Laplacian Spectral Distances"
] |
[
"Giuseppe Patané \nConsiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche Genova\nItaly\n"
] |
[
"Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche Genova\nItaly"
] |
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Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting[38,39]to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.
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10.1016/j.patcog.2019.04.004
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[
"https://arxiv.org/pdf/1906.03900v2.pdf"
] | 145,998,084 |
1906.03900
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b98d190f15b9545dba0979b50b717bb763d0d41d
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A Unified Definition and Computation of Laplacian Spectral Distances
Giuseppe Patané
Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche Genova
Italy
A Unified Definition and Computation of Laplacian Spectral Distances
Laplacian spectrumspectral distancesspectral kernelsheat kerneldiffusion distances and geometryshape and graph analysis
Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting[38,39]to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.
can be computed by approximating the filter with a rational function and converting their evaluation to the solution of a set of differential equations that involve only the Laplace-Beltrami operator.
The proposed discrete approximation of the spectral kernels (Sect. 3) is equivalent to solving r sparse, symmetric, and well-conditioned linear systems, where r is the degree of the rational polynomial approximation of the filter. Applying an iterative linear solver, the computation of the spectral kernel takes O(rτ(n)) time, where τ(n) typically varies from τ(n) = n to τ(n) = n log n and depends on the number n of shape samples and on the sparsity of the coefficient matrix. In a similar way, the computation of the spectral distance between two points has the same order of complexity of the evaluation of the corresponding spectral kernel at the same input points.
Through the spectrum-free approximation, the computed spectral kernel is not affected by the Gibbs phenomenon, as a consequence of the high accuracy of the rational approximation with respect to the low-pass filter associated with the truncated spectral approximation. Furthermore, the spectrum-free approximation is free of user-defined parameters (e.g., the number of Laplacian eigenpairs).
Finally (Sects. 4,5), we experimentally verify the accuracy, efficiency, and numerical robustness of the definition and spectrum-free computation of the spectral kernels and distances.
Novelties with respect to previous work
With respect to previous work and our recent results on the definition and computation of discrete spectral distances [39,38], the main novelties of this paper are
• the discussion of the optimality of the Laplacian spectrum for the approximation of the spectral kernel (Sect. 2.1);
• the study of the relation between the spectral and Green kernels (Sect. 2.4), associated with differential operators (Sect. 2.2);
• the analysis of the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues (Sect. 2.3).
In fact, the evaluation of the Laplacian spectrum is generally affected by small perturbations of the Laplace-Beltrami operator;
• a novel spectrum-free computation of the spectral distances (Sect. 2.5), which is achieved by properly approximating the associated filter with a rational polynomial and is expressed in terms of the canonical basis;
• the computation of the discrete spectral distances (Sect. 3.2), which reduces to the solution of a set of sparse, symmetric, and well-conditioned linear systems.
This analogy between the continuous and discrete cases shows the generality of the proposed approach and it has not been addressed by previous work. Furthermore, these results are complementary and more general than the spectrum-free approaches presented in [39,38];
• a new upper bound to the conditioning number of the linear systems associated with the spectrum-free approximation, which confirms the numerical stability and well-conditioned computation of the class of the spectrum-free approaches (Sect. 3.1);
• new experiments (Sect. 4), which enrich the tests initially presented in [39,38] and address the robustness of the computation of the Laplacian spectral distances and kernels with respect to the data resolution, partial sampling, geometric or topological noise, and deformations.
Laplacian spectral distances
Firstly, we present the Laplace-Beltrami operator, its spectrum, and the optimality of the Laplacian eigenbasis (Sect. 2.1). Recalling the definition of the spectral operator and kernel introduced in [39,37,38], we review equivalent representations of the spectral distances in terms of the spectral norm of the δ-functions, the Laplacian spectrum, the spectral operator, and the spectral kernel (Sect. 2.2). As novel contribution, we discuss the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues, which are generally affected by small perturbations of the Laplace-Beltrami operator (Sect. 2.3). Finally, we introduce a novel relation between the spectral and Green kernels (Sect. 2.4) and
propose a novel spectrum-free computation of the spectral distances in the continuous case, which is presented in a way analogous to the discrete case [38,39] and shows the generality of the proposed approach (Sect. 2.5).
Laplace-Beltrami operator
Let N be a smooth surface, possibly with boundary, equipped with a Riemannian metrics and let us consider the inner product f , g 2 := N f gdµ defined on the space L 2 (N ) of square integrable functions on N and the corresponding norm · 2 .
The Laplace-Beltrami operator ∆ is self-adjoint ∆ f , g 2 = f , ∆g 2 , and positive semi- [44].
definite ∆ f , f 2 ≥ 0, ∀ f , g,
Laplacian eigenbasis and its optimality. Recalling that the Laplace-Beltrami operator is self-adjoint and positive semi-definite, it has an orthonormal eigensystem (λ n , φ n ) +∞ n=0 , ∆φ n = λ n φ n , λ 0 = 0, λ n ≤ λ n+1 , in L 2 (N ). Expressing a function f in L 2 (N ) in terms of the Laplacian eigenfunctions, we have that
f = ∑ +∞ n=0 α n φ n , α n := f , φ n 2 ; f 2 2 = ∑ +∞ n=0 α 2 n , ∇ f 2 2 = ∑ +∞ n=0 α 2 n λ n .
According to [1], the Laplacian eigenfunctions are an optimal basis for the representation of signals with bounded gradient magnitude. In fact, the spectral decomposition f n = ∑ n i=0 f , φ i 2 φ i is optimal for the approximation of functions with L 2 bounded gradient magnitude; i.e., the residual error r n := f − f n is bounded as
r n 2 2 ≤ ∇ f 2 2 λ n+1 .(1)
The spectral decomposition is also optimal for the approximation of functions with respect to the error estimation in (1). In fact, for any 0 ≤ α < 1 there is no integer n and no sequence (ψ i ) n i=0 of linearly independent functions in L 2 such that
f − n ∑ i=1 f , ψ i 2 ψ i 2 ≤ α ∇ f 2 λ n+1 , ∀ f .
Theoretical background: spectral kernels and distances
We briefly review the equivalent definitions of the spectral kernels (Sect. 2.2.1) and distances (Sect. 2.2.2), which will be useful to introduce their spectrum-free computation and main properties.
Spectral operator and kernel
According to [38,39], let ρ : R + → R be a positive filter map that is square integrable and admits the power series ρ(s) = ∑ +∞ n=0 α n s n . Considering the orthonormal Laplacian eigensystem (λ n , φ n ) +∞ n=0 , the spectral representation of the functions ∆ i f and
(∆ † ) i f is ∆ i f = +∞ ∑ n=0 λ i n f , φ n 2 φ n , (∆ † ) i f = +∞ ∑ n=1 1 λ i n f , φ n 2 φ n .(2)
If λ k = 0, then we neglect the corresponding entry in (∆ † ) i f . In L 2 (N ), we define the spectral operator
Φ ρ f = +∞ ∑ n=0 α n ∆ n f = +∞ ∑ n=0 ρ(λ n ) f , φ n 2 φ n ,(3)
which is linear, continuous, self-adjoint, and Φ ρ f = K ρ , f 2 , with
K ρ (p, q) = +∞ ∑ n=0 ρ(λ n )φ n (p)φ n (q)(4)
spectral kernel. The spectral kernel operator and kernel are well-posed if the filter is bounded or square-integrable. The spectral kernel is also symmetric and self-adjoint, as a consequence of its definition. From these relations, it follows that
• the spectral kernel K ρ (p, ·) = Φ ρ δ p is achieved as the action of the spectral operator on the δ-function at p;
• the spectral kernel is the solution to the differential equation Φ 1/ρ K ρ (p, ·) = δ p .
In Sect. 2.4, this property is further investigated in terms of the relation between the spectral and Green kernels.
To express the spectral distances in terms of the spectral operator (Sect. 2.5), we show that the pseudo-inverse of Φ ρ is induced by the filter function 1/ρ; i.e., Φ † ρ = Φ 1/ρ . In fact, from the spectral representation (3) of Φ 1/ρ and Φ ρ , we get that
Φ ρ Φ 1/ρ Φ ρ = Φ ρ , Φ 1/ρ Φ ρ Φ 1/ρ = Φ 1/ρ , Φ 1/ρ Φ ρ f , g 2 = f , Φ 1/ρ Φ ρ g 2 .(5)
In the following, we assume that both Φ ρ and Φ 1/ρ are well-defined; for instance, this hypothesis is satisfied if ρ is not null only on a compact interval of R + and it is valid in the discrete case (Sect. 3), where the Laplacian spectrum belongs to the interval
I := [0, λ max (L)], with λ max (L) maximum Laplacian eigenvalue.
Spectral distances
Through the spectral operator, we introduce the scalar product and the corresponding distance as [38,39]
f , g := Φ 1/ρ f , Φ 1/ρ g 2 = ∑ +∞ n=0 f ,φn 2 g,φn 2 ρ 2 (λn) , (a) d 2 ( f , g) := f − g 2 = ∑ +∞ n=0 | f −g,φn 2 | 2 ρ 2 (λn) . (b)(6)
Indicating with δ p the map that takes value 1 at p and 0 otherwise, and selecting f := δ p , g := δ q in Eq. (6b), the spectral distance ( Fig. 1) on N is defined as
d 2 (p, q) := δ p − δ q 2 = Eq. (6b) +∞ ∑ n=0 |φ n (p) − φ n (q)| 2 ρ 2 (λ n ) = Eq. (6a) Φ 1/ρ (δ p ) − Φ 1/ρ (δ q ) 2 2 = K 1/ρ (p, ·) − K 1/ρ (q, ·) 2 2 = K 1/ρ (p, p) − 2K 1/ρ (p, q) + K 1/ρ (q, q);
i.e., these equivalent formulations involve the Laplacian spectrum, the spectral operator and kernel. The third equality follows from the identity Φ ρ (δ p ) = K ρ (p, ·) and will be applied to the computation of the spectral distances (Sect. 3.2); in fact, it is independent of the evaluation of the Laplacian spectrum. The last identity is achieved by applying the relation
d 2 (p, q) = K 1/ρ (p, ·) − K 1/ρ (q, ·) 2 2 = +∞ ∑ n=0 ρ −2 (λ n )φ n (p)φ n (p) − 2 +∞ ∑ n=0 ρ −2 (λ n )φ n (p)φ n (q) + +∞ ∑ n=0 ρ −2 (λ n )φ n (q)φ n (q).
Stability of the eigenpairs of the spectral operator
Generalising the results in [38,39], we show that the computation of a single eigenvalue of the spectral operator is numerically stable and instabilities are generally due to repeated or close eigenvalues. Firstly, we notice that if λ is a Laplacian eigenvalue of multiplicity m then µ := ρ(λ) is an eigenvalue of Φ ρ and its multiplicity is equal to or greater than m. The corresponding eigenfunction is the Laplacian eigenfunction associated with the eigenvalue λ; i.e., ∆φ = λφ and Φ ρ φ = ρ(λ)φ.
We perturb the spectral operator by δE, δ → 0, and compute the eigenpair (µ(δ), φ(δ))
of the corresponding operator Φ ρ + δE; i.e.,
Φ ρ + δE φ(δ) = µ(δ)φ(δ), φ(0) = φ, µ(0) = µ.(7)
Recalling that the derivative of µ(δ) measures the variation of the eigenvalue, deriving (7) with respect to δ, and evaluating the resulting relation at 0, we have that
Eφ + Φ ρ φ (0) = µ (0)φ + µφ (0).(8)
From (8) and the self-adjointness of Φ ρ , it follows that φ,
Φ ρ φ (0) 2 = µ φ, φ (0) 2 .
Multiplying both sides of (8) by φ and applying the previous identity we get that
|µ (0)| = | φ, Eφ 2 | ≤ E 2 φ 2 2 = E 2 ;
i.e., the computation of the eigenvalue of Φ ρ with multiplicity one is stable.
Assuming that µ := ρ(λ) is an eigenvalue of Φ ρ with multiplicity m and rewriting the characteristic polynomial as p(s) = (s − µ) m q(s), where q(·) is a polynomial of degree n − m and q(µ) = 0, we get that This unstable computation of multiple eigenpairs of the spectral operator generally affects the accuracy of the truncated spectral approximation
(s − µ) m = p(s) q(s) ≈ O(δ) q(s) , δ → 0, as p(s) → 0, when s → µ; ρ t (s) = s ρ t (s) = s 2 ρ(s) = s 3 ρ(s) = s 2 log(1 + s)d k (p, q) := k ∑ n=0 |φ n (p) − φ n (q)| 2 ρ 2 (λ n )(9)
of the corresponding distances. In fact, each filtered eigenvalue ρ(λ n ), n = 1, . . . , k, which appears at the denominator of d k , can further accentuate the numerical error of λ n in the distance computation. Furthermore, the computation of the Laplacian spectrum is time-consuming and it is difficult to properly select the number of eigenpairs that is necessary to accurately approximate the spectral distance. Indeed, we propose an evaluation of the spectral distance that is independent of the computation of the Laplacian spectrum and is equivalent to a set of differential equations involving only the Laplace-Beltrami operator and its pseudo-inverse. This novel spectrum-free computation is a generalisation of the discrete approach recently presented in [38,39].
Relation between spectral and Green kernels
To study the relation between the Green and the spectral kernels, let us consider a linear differential operator L and the corresponding Green kernel K : N × N → R such that LK(p, q) = δ(p − q). Then, the solution to the differential equation Lu = f is expressed in terms of the Green kernel as u(p) = K(p, ·), f 2 .
Noting that the eigensystem (µ n , ψ n ) +∞ n=0 of the integral operator A K f := K(·, ·), f 2 induced by the Green kernel satisfies the relation Lψ n = µ −1 n ψ n , µ n = 0, we have that the eigensystem of L is (µ −1 n , ψ n ) +∞ n=1 . Indeed, L and A K have the same eigenfunctions and reciprocal eigenvalues. In particular, the spectral representation of the Green kernel K(p, q) = ∑ +∞ n=0 µ n ψ n (p)ψ n (q) is uniquely defined by the spectrum of L. Combining the previous results with the properties of the spectral operator and kernel (Sect. 2.2.2), we get that
• for the harmonic operator L = ∆, the Green kernel is the commute-time kernel
K ∆ (p, q) = +∞ ∑ n=1 1 λ n φ n (p)φ n (q);
• for the bi-Laplacian operator L := ∆ 2 , the Green kernel is the bi-harmonic ker- • for the diffusion operator L = exp(t∆), the Green kernel is the diffusion kernel
(Fig. 2) K t (p, q) = +∞ ∑ n=0 exp(−tλ n )φ n (p)φ n (q);
• for the spectral operator L = Φ 1/ρ , the Green kernel is the spectral kernel K ρ defined in Eq. (4).
Spectrum-free approximation of kernels and distances
Recalling the relation Φ † ρ = Φ 1/ρ in Eq. (5) and noting that and equipped with the L 2 (N × N ) scalar product. Since
d( f , g) = u 2 , u = Φ 1/ρ ( f − g) ⇐⇒ Φ ρ u = f − g,K ρ − K ϕ 2 2 = +∞ ∑ n=0 |ρ(λ n ) − ϕ(λ n )| 2 ≤ ρ − ϕ 2 2 ,
the approximation of a given spectral kernel K ρ with a new kernel K ϕ in K(N ) is reduced to the approximation of ρ by ϕ on a proper subspace of functions (e.g., the space of polynomials or rational polynomials). The class of functions used for the approximation of the input filter ρ is selected in such a way that K ϕ provides a good approximation of the input filter K ρ and is easily computable.
L 2 e iL 3 e iL 10 e i (L † ) 2 e i (L † ) 3 e i (L † ) 10 e i
Rational approximation based on the canonical basis
We apply the Padé-Chebyshev rational approximation to the filter map 1/ρ. Let R l r be the space of all rational functions
c rl (s) := p l (s) q r (s) = β 0 + β 1 s + . . . + β l s l α 0 + α 1 s + . . . + α r s r , s ∈ [a, b].
We briefly recall that the representation of the rational approximation is not unique, unless we impose that it is irreducible; for example, by choosing q r (a) = 1. Given a filter ρ : [a, b] → R, there exists a unique best approximation of 1/ρ in R l r with respect to the ∞ norm [20] (Ch. 9), which is represented in terms of the canonic basis
B := {s i } l i=0 ∪ {s −i } r i=1 .
Truncated spectral approx. P.C. approx. Recalling that the dimension of the space R l r of rational polynomial of degree (r, l) is l + r + 1, let us express the best rational approximation c rl of ρ in terms of these basis functions. To this end, we can either apply algebraic rules or impose interpolating constraints at (l + r + 1) points in order to compute the new coefficients through the
identity l ∑ i=0 a i s i + r ∑ i=1 b i s −i = c rl (s).
The canonical basis used to represent the rational approximation of the input filter simplifies the computation of the spectral kernel; in fact, we need to evaluate only the functions ∆ i f and (∆ † ) i f . Instead of applying the spectral representation (2), we define a recursive procedure as follows. To compute g 1 = ∆ † f , let us multiply both sides by ∆ spectrum-free and (b) truncated spectral approximation (k = 500). This last method produces some artefacts that become evident as we move far from the seed point. and notice that
∆g 1 = ∆∆ † f = f − f , φ 0 2 φ 0 ,(10)
where φ 0 = is the constant eigenfunction equal to 1. By definition of pseudoinverse, g 1 is the least-squares solution to the equation ∆g 1 =f , wheref is equal to f minus its mean f , φ 0 2 φ 0 . For the general case, we apply the recursive relation
g i := (∆ † ) i f = ∆ † (∆ † ) i−1 f = ∆ † g i−1 , i ≥ 2,
which reduces to the previous case. In a similar way, g i = ∆ i f is calculated as h i = ∆h i−1 , i = 1, . . . , r, with h 0 := f . Indeed, the proposed approach requires the solution of (l + r)
Laplace equation with a different right-hand side.
Convergence and accuracy. To verify that the sequence
(Φ (r) 1/ρ f ) +∞ r=0 , Φ (r) 1/ρ f := +∞ ∑ n=0 c rl (λ n ) f , φ n 2 φ n ,
induced by the rational polynomial approximation c rl of 1/ρ, converges to Φ 1/ρ f , we apply the upper bound
Φ (r) 1/ρ f − Φ 1/ρ f 2 2 ≤ c rl − 1/ρ 2 ∞ +∞ ∑ n=0 | f , φ n 2 | 2 = σ 2 rl f 2 2 , σ rl ≈ O(s r+l+1 ), s → 0;
where σ rl is the approximation error between 1/ρ and c rl . For the diffusion operator [52], σ rr = 10 −5 .
Spectrum-free approximation of kernels and distances
We briefly summarise the discretisation of the spectral kernel and distances [38,39], which is then used to establish a simple relation between the spectral kernel and its pseudo-inverse (Sect. 3.1). Then, we introduce the spectrum-free approximation of the spectral kernels and distances (Sect. 3.2) and discuss the computational cost of the main steps of the proposed approach (Sect. 3.3). and mass matrix respectively (e.g., cotangent [40], Voronoi-cotg [15], linear FEM [42] weights), and the corresponding spectral decomposition is LX = BXΛ, X BX = I, where X := [x 1 , . . . , x n ] is the eigenvectors' matrix and Λ is the diagonal matrix of the eigenvalues (λ i ) n i=1 . Analogous discretisations apply to polygonal [3,22] and tetrahedral [27,51] meshes, or point sets [29].
Discretisation of spectral kernels and distances
Under these assumptions, we introduce the spectral kernels and distances (Sect. 3.1.1), together with their spectrum-free approximation (Sect. 3.1.2), which is based on a rational approximation of the input filter.
Discrete spectral kernels and distances
The spectral operator Φ 1/ρ is discretised by the spectral kernel matrix
K 1/ρ such that ρ −1 (λ i ) = Eq. (3) Φ 1/ρ φ i , φ j 2 = x i K 1/ρ Bx j , ∀i = 1, . . . , n. Indeed, K 1/ρ = Xρ † (Λ)X B is the pseudo-inverse of the spectral kernel K ρ = ρ(L),
which is a filtered version of the Laplacian matrix. Here,
ρ † (Λ) = diag(1/ρ(λ i )) n i=1
and its entry is null if ρ(λ i ) = 0. Then, the discrete spectral distances [38,39] are
d 2 (p i , p j ) = K 1/ρ (e i − e j ) 2 B = n ∑ l=1 | x l , e i − e j B | 2 ρ 2 (λ l ) ,
where e i is the vector of the canonical basis of R n , f, g B := f Bg and f 2 B := f Bf are the scalar product and the norm induced by the mass matrix, respectively. In this case, we have derived the spectral distances by applying the continuous expression of the spectral operator instead of the spectral kernel, as done in [38,39].
Analogously to the definitions in Sect. 2.2, the spectral distance is equal to the norm d(f, g) := u B of the solution to the linear system K ρ u = f − g or to the norm of the vector u = K 1/ρ (f − g). In a similar way, the spectral distance between two points reduces to
d(p i , p j ) = K 1/ϕ (e i − e j ) B .
Indeed, this approximation of the spectral distances involves the spectral kernel only and allows us to bypass numerical inaccuracies due to repeated or close Laplacian eigenvalues [20] ( § 7).
Spectral kernel approximation and conditioning
Since the eigenvalues of the spectral kernel matrix K ρ are (ρ(λ i )) n i=1 , the approximation of K ρ with a new kernel K ϕ reduces to the approximation of the corresponding filters with respect to the ∞ norm; in fact,
K ρ − K ϕ 2 = K ρ−ϕ 2 = max i=1,...,n {|ρ(λ i ) − ϕ(λ i )|} ≤ ρ − ϕ ∞ .
The approximation ϕ of ρ is computed on the interval [0, λ max (L)], where the maximum Laplacian eigenvalue is evaluated by the Arnoldi method [20], or is set equal to the [26,48].
upper bound λ max (L) ≤ min{max i {∑ jL (i, j)}, max j {∑ iL (i, j)}}
Conditioning of the spectral kernel. We now analyse the conditioning of the spectral kernel. Assuming that ρ is an increasing function (i.e., 1/ρ is a low pass filter), the conditioning number of the filtered Laplacian matrix is bounded as
κ 2 (K ρ ) = κ 2 (ρ(L)) = max i=1,...,n {ρ(λ i )} min i=1,...,n {ρ(λ i )} = ρ ∞ ρ(0) ,
and it is ill-conditioned when ρ(0) is close to zero or ρ is unbounded. If ρ is bounded and ρ(0) is not too close to 0, then the filtered Laplacian matrix is well-conditioned. If ρ(0) is null, then we consider the smallest and not null filtered Laplacian eigenvalue at the denominator of the previous relation.
Spectrum-free approximation of distances: canonical basis
Analogously to the discussion in Sect. 2.5.2, the basis B used to represent the rational approximation of the input filter simplifies the computation of the spectral kernel;
in fact, we need to compute only the following vectorsL i f and( L i ) † f. Applying the spectral representation of the Laplacian matrix
L = XΛX B, Λ = diag(λ i ) n i=1 , λ 1 = 0, λ i ≤ λ i+1 , i = 2, . . . , n − 1,
we can represent its powers and the corresponding pseudo-inverse matrices as i.e., g 1 is the least-squares solution to the sparse and symmetric linear system
L i = XΛ i X B. (L i ) † = X(Λ i ) † X B, Λ † = diag(0, λ −1 i ) n i=2 . t = 10 −1 t = 10 −2 (a) (b) (c) (d)Lg 1 = Bg 0 , g 0 := f − (1 Bf)1,
where g 0 is achieved by subtracting to f its mean value f, 1 B (c.f., Eq. (10)). For the general case, we apply the recursive relation which reduces to the previous case (Fig. 3). In a similar way, g i =L i f is calculated by recursively solving the sparse, symmetric, and positive-definite linear systems
g i := (L † ) i f =L † (L † ) i−1 f =L † g i−1 , i ≥ 2,Bg i+1 = Lg i , i = 1, . . . , r − 1, with g 0 := f.
Computational cost
The truncated spectral approximation, whose computational cost depends on the sparsity degree of the Laplacian matrix, takes from O(kn log n) to O(kn 2 ) time, where k is the number of selected eigenpairs. Selecting a rational approximation of the input filter of degree (r, l), the evaluation of the corresponding spectral kernel is reduced to solve l linear systems whose coefficient matrix is B and r linear systems whose coefficient matrix is L. Through iterative solvers, the computational cost is O((r + l)τ(n)),
where τ(n) is the cost for the solution of a sparse linear system, which varies from O(n) to O(n 2 ), according to the sparsity of the coefficient matrix, and it is O(n log n)
in the average case.
In a similar way, the computation of the spectral distance between two points has the same order of complexity of the evaluation of the corresponding spectral kernel at the same input points. For the computation of the one-to-all spectral distance, we lump or pre-factorise B (if not already diagonal) and pre-factorise L. Then, the overall computational cost varies from O(rn) (B diagonal) to O(n log n + rn) (B not diagonal),
where O(n log n) is the time for the factorisation of L.
In spite of this different computational cost, the truncated spectral approximation
Discussion
We now discuss the selection of the filter map and main examples on the computation of the Laplacian spectral distances.
The spectral kernels and distances depend only on the behaviour of the filter in the spectral domain and on the Laplacian spectrum: increasing or decreasing the decay of the filter to zero encodes global or local shape details, respectively. Recalling that an arbitrary filter 1/ρ can be approximated in R l r with an accuracy of order O(s l+r+1 ) with respect to the ∞ -norm (Sect. 3.2), we can use the space R l r to define any filter and the corresponding spectral distances. In this way, we reduce the degree of freedom in the definition of the filter to the selection of (l + r + 1) coefficients and without losing the richness of the resulting spectral distances. For the selection of the coefficients of the rational filter and the filter frequencies, we can apply the rules proposed in [24] for Laplacian spectral smoothing. In particular, the filter frequencies are derived from the dimension of a bounding box placed around the chosen feature F and whose axis are aligned with the eigenvectors of the covariance matrix of F.
The truncated spectral approximation (c.f., Eq. (9)) of the diffusion, and more generally of the spectral distances, is affected by (i) small undulations (especially at small scales, Fig. 6), (ii) heuristics for the selection of the number of Laplacian eigenpairs with respect to the target approximation accuracy and the scale of the shape features, Figure 12: Behaviour of the diffusion kernel on (almost) isometric 3D shapes. and (iii) the overall computational cost and storage overhead for the computation of a subpart of the Laplacian spectrum. The comparison of these results with the ones induced by the spectrum-free computation (Fig. 7) shows the improvement of the proposed approach in terms of smoothness, regularity, and accuracy of the computed distances.
At small scales ( Fig. 8(e,f)), the truncated spectral approximation of the diffusion kernel generally have small negative values as a matter of the slow decay of the exponential filter to zero for small eigenvalues. For the Padé-Chebyshev approximation ( Fig. 8(a-d)), the kernel values are positive at all the scales, as we approximate the filter with a higher accuracy with respect to the truncated spectral approximation, which applies a low-pass filter.
Finally, the analogous distribution and shape of the level-sets of the heat kernel and diffusion distance confirm the robustness of the Padé-Chebyshev of the approximation with respect to surface resolution (Fig. 7), partial sampling ( Fig. 9), geometric (Fig. 10) and topological (Fig. 11) noise, almost isometric deformations (Fig. 12).
Conclusions and future work
We have presented a unified approach to the definition, discretisation, and computation of the spectral kernels and distances, which are defined by filtering the Laplacian spectrum and generalise the commute-time, bi-harmonic, diffusion, and wave kernel and distances. Even though analytic filters are easily defined and encode global/local shape properties, the corresponding spectral kernels and distances are not capable of characterising a specific shape class (e.g., humans with respect to four-legs animals, chairs with respect to tables). Indeed, the main topic for future research is a deeper analysis of the constraints on the filter in order to define "optimal" spectral kernels and distances for shape comparison. To achieve this goal, we plan to apply and specialise learning methods, as initially investigated in [2,8]. Another interesting aspect is the study of spectral kernels for the adaptive hash retrieval [5] and the identification of those PDEs, whose Green kernels are useful for shape analysis, as already demonstrated by the bi-harmonic and diffusion kernels.
Indeed, the Laplacian eigenfunctions are an optimal basis for the definition of scalar functions on a given domain, such as the Laplacian spectral kernels and distances on 3D shapes (Sect. 3). From the computational point of view, they have two main drawbacks: a high computational cost and storage overhead, which prevent the evaluation of a large number of Laplacian eigenpairs, and numerical instabilities with respect to the surface discretisation[38,39].
Figure 1 :
1Different filters ρ induce spectral distances from a seed point with a different behaviour, in terms of the locality, shape, and distribution of the level-sets and according to the decay of 1/ρ to zero. These distances have been computed with the Padé-Chebyshev approximation (r = 5). i.e., s ≈ µ + O(δ 1 m ). Indeed, modifying the Laplacian matrix in such a way that the filtered eigenvalues are perturbed by δ := 10 −m corresponds to a change of order 0.1 in µ (i.e., s ≈ µ + 0.1) and this amplification becomes larger as the multiplicity of the eigenvalue increases.While multiple eigenvalues are typically associated with symmetric shapes, numerically close or switched eigenvalues are present regardless of the surface regularity.
Figure 2 :
2(a,b) Level-sets of the diffusion kernel at two different scales and at different seed points, which confirm the locality, smoothness, and shape-awareness of the heat kernel. the spectral distance is equivalent to the norm of (i) thefunction u = Φ 1/ρ ( f − g)and (ii) the solution of the differential equation Φ ρ u = f − g. Through these relations, the spectral kernels will be computed by approximating the filter with a rational function (Sect. 2.5.1) and by converting the evaluation of the corresponding distances (Sect. 2.5.2) to the solution of a set of differential equations that involve the Laplace-Beltrami operator. The choice of one of these two equivalent representations depends on the selected filter and the simplicity of evaluating either Φ ρ or Φ 1/ρ .2.5.1. Kernel approximation, convergence, and accuracyLet us introduce the space of the Laplacian spectral kernels as K(N ) := {K ρ : N × N → R, K ρ spectral kernel in Eq. (4)},
Figure 3 :
3Level-sets of the basis functions at p i associated with the rational polynomial approximation of the spectral distances.
Figure 4 :
4a) k = 100 (b) k = 1000 (c) k = 2K (d) ε ∞ = 10 Smoothness and locality of the diffusion kernel at a seed point (red dot) and at different scales, which have been computed with (a-c) the truncated spectral approximation (i.e., k Laplacian eigenpairs) and(d) the Padè-Chebyshev approximation. Since the input shape has 2K vertices, the spectral approximation (c) provides the ground-truth.
ρ, trunc. spect. approx.
Figure 5 :
5Comparison of the spectral distance induced by the same filter and approximated with the (a)
Figure 6 :
6Reducing the scale t, the corresponding diffusion distances become locally unstable, as we move far from the seed point (on the feet). See alsoFig. 7.
Let us consider a (triangular, polygonal, volumetric) mesh M := (P, T ), which discretises a domain N , where P := {p i } n i=1 is the set of n vertices and T is the connectivity graph. On M, a scalar function f : M → R is identified with the vector f := ( f (p i )) n i=1 of f -values at P. Let us introduce the Laplacian matrixL := B −1 L, where L and B are the stiffness
Figure 7 :
7Analogous behaviour of the heat kernel with respect to a different resolution (n vertices) of the input shape, in terms of shape and distribution of the level-sets.
Figure 8 :
8(c,d) Small undulations and negative values of the truncated spectral approximation at small scales. These local undulations are not present in the Padé-Chebyshev approximation (a,b), whose values are always positive.To compute g 1 =L † f, let us multiply both sides byL and notice thatLg 1 =LL † f = XΛΛ † X Bf, X BX = I, = X(I − e 1 e 1 )X Bf, e 1 := [1, 0, . . . , 0] , = f − (1 Bf)1, 1 := [1, 1, . . . , 1] ;
Figure 9 :
9Stability of the spectrum-free computation of the diffusion kernel with respect to partially-sampled surfaces.
(Figure 10 :
10Fig. 4)is affected by small geometric undulations (especially at small scales), the use of heuristics for the selection of the number of Laplacian eigenpairs with respect Behaviour and robustness of the spectrum-free approximation of the diffusion kernel on (b-d) the noisy shapes (bottom part, red surfaces), which have been plotted on (a) the initial shape.to the target approximation accuracy, and the scale of features of the input shape.The spectrum-free computation(Fig. 5) generally provides better results in terms of smoothness, regularity, and accuracy of the computed spectral basis. For further comparison examples, we refer the reader to Sect. 4.
Figure 11 :
11Robustness of the diffusion kernel at different scales and on self-intersecting surfaces.
Acknowledgements. We thank the Reviewers for their thorough review and constructive comments, which helped us to improve the technical part and presentation of the revised paper. This work is supported by the H2020 ERC Advanced Grant CHANGE, grant agreement No. 694515. Shapes are courtesy of the AIM@SHAPE Repository, the SHREC 2010 and 2016 data sets; tet-meshes have been generated by the TETGEN software http://wias-berlin.de/software/tetgen/.
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|
[] |
[
"Turbulence Accelerating Cosmology from an Inhomogeneous Dark Fluid",
"Turbulence Accelerating Cosmology from an Inhomogeneous Dark Fluid"
] |
[
"I Brevik ",
"A V Timoshkin ",
"Ye Rabochaya ",
"S Zerbini ",
"\nDepartment of Energy and Process Engineering\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n",
"\nTomsk State Pedagogical University\n634061TomskRussia\n",
"\nDepartment of Physics\nEurasian National University\nAstanaKazakhstan\n",
"\nand Gruppo Collegato di Trento, Sezione INFN di Padova\nUniversity of Trento\nItaly\n"
] |
[
"Department of Energy and Process Engineering\nNorwegian University of Science and Technology\nN-7491TrondheimNorway",
"Tomsk State Pedagogical University\n634061TomskRussia",
"Department of Physics\nEurasian National University\nAstanaKazakhstan",
"and Gruppo Collegato di Trento, Sezione INFN di Padova\nUniversity of Trento\nItaly"
] |
[] |
Specific dark energy models with a linear inhomogeneous timedependent equation of state, within the framework of 4d Friedman-Robertson-Walker (FRW) cosmology, are investigated. It is demonstrated that such 4d inhomogeneous fluid models may lead to a turbulence FRW cosmology. Both one-component and two-component models from 4d inhomogeneous dark fluid models are considered. In the one-component model the universe may develop from a viscous era with, for instance, a constant bulk viscosity, into a turbulent era. In the two-component model the fluid can be decomposed into two components, one non-turbulent (ideal) and another turbulent part, obeying two different equations of state. Conditions for the appearance of the turbulent dark energy universe in terms of the parameters in the equation of state (EoS) without introducing the turbulence concept explicitly are are obtained. An equivalent description in terms of an inhomogeneous fluid for the viscous Little Rip (LR) cosmology is also developed.
|
10.1007/s10509-013-1506-2
|
[
"https://arxiv.org/pdf/1307.6006v1.pdf"
] | 118,582,048 |
1307.6006
|
6551ea6c5e0e692a9b675b3da36ced2763479016
|
Turbulence Accelerating Cosmology from an Inhomogeneous Dark Fluid
23 Jul 2013 May 11, 2014
I Brevik
A V Timoshkin
Ye Rabochaya
S Zerbini
Department of Energy and Process Engineering
Norwegian University of Science and Technology
N-7491TrondheimNorway
Tomsk State Pedagogical University
634061TomskRussia
Department of Physics
Eurasian National University
AstanaKazakhstan
and Gruppo Collegato di Trento, Sezione INFN di Padova
University of Trento
Italy
Turbulence Accelerating Cosmology from an Inhomogeneous Dark Fluid
23 Jul 2013 May 11, 2014arXiv:1307.6006v1 [gr-qc]
Specific dark energy models with a linear inhomogeneous timedependent equation of state, within the framework of 4d Friedman-Robertson-Walker (FRW) cosmology, are investigated. It is demonstrated that such 4d inhomogeneous fluid models may lead to a turbulence FRW cosmology. Both one-component and two-component models from 4d inhomogeneous dark fluid models are considered. In the one-component model the universe may develop from a viscous era with, for instance, a constant bulk viscosity, into a turbulent era. In the two-component model the fluid can be decomposed into two components, one non-turbulent (ideal) and another turbulent part, obeying two different equations of state. Conditions for the appearance of the turbulent dark energy universe in terms of the parameters in the equation of state (EoS) without introducing the turbulence concept explicitly are are obtained. An equivalent description in terms of an inhomogeneous fluid for the viscous Little Rip (LR) cosmology is also developed.
Introduction
A variety of complicated problems in cosmology can be explained by the discovery of the accelerated expansion of the universe [1,2] in terms of dark energy [3,4,5]. According to recent observations the dark energy currently accounts for about 69% of the total mass/energy of the universe [6]. It possesses a negative pressure and/or negative entropy. The EoS parameter w is still determined up to some uncertainty: it is not clear if w is less than −1, equal to −1, or larger than −1. According to present observations, w = −1 +0.09 −0.10 [7,8]. The most interesting case is when the thermodynamic parameter w = p/ρ < −1 (phantom dark energy). An essential property of this kind of energy is the Big Rip future singularity [9] (see also [10,11]), where the scale factor becomes infinite at a finite time in the future. In the mild phantom models where w asymptotically tends to −1, the singularity occurs in the infinite future [12,13,14,15]. Such Rip phenomena take place for mild phantom scenarios like Little Rip or Pseudo Rip.
In a series of previous works [16,17,18,19] we considered the non-viscous models of the cosmic fluid. The case of such a fluid (also called an ideal fluid) is quite an idealized model; it will often be useless in practical situations, especially when fluid motion near boundaries is involved. Also under boundary-free conditions (isotropic turbulence, for instance), the influence from viscosity can be most important.
When working to the first order in deviations from thermodynamic equilibrium one has in principle to introduce two viscosity coefficients, namely the shear viscosity η and the bulk viscosity ζ. We shall assume, in conformity with usual practice, that spatial anisotropies (present in in the Kasner universe, for example), become smoothed out. Thus only the coefficient ζ will be included.
In the present article we point out the equivalence between 1) expansion of the universe described in terms of time-dependent parameters of the inhomogeneous dark fluid model, and 2) viscous Little Rip (LR) cosmology for the dark fluid in the late universe.
Our work is based upon, and extends, prior work of Ref. [21] (see also Refs. [22,23]).
Dark fluid with bulk viscosity
A theory of viscous LR cosmology was recently given in Ref. [19]. We consider now viscous LR cosmology in an isotropic cosmic fluid in the later stages of the development of the universe.
We shall assume viscosity-dependent governing equations. We suppose that the viscosity function ξ(H), defined as 3ζH, is a constant:
3ζH ≡ ξ 0 = const,(1)
with H =ȧ/a the Hubble parameter. Then the expression for the time dependent energy density becomes [19]
ρ(t) = ξ 0 A + √ ρ 0 exp( √ 6πG At) − ξ 0 A 2 ,(2)
with A a positive constant. This is a characteristic property of LR cosmology, now met under viscous conditions. Next, let us consider LR cosmology from the point of view of 4d FRW non-viscous cosmology. Here it is natural to associate t = 0 with the present time, so that ρ 0 becomes the present time energy density. The Friedman equation for a spatially flat universe is
ρ = 3 k 2 H 2 ,(3)
where ρ is the energy density, and k 2 = 8πG. Assume that our universe is filled with an ideal fluid (dark energy) obeying an inhomogeneous equation of state [20,17]
p = w(t)ρ + Λ(t),(4)
with w(t) and Λ(t) as time-dependent parameters and p the pressure. The energy conservation law iṡ
ρ + 3H(p + ρ) = 0,(5)
and the derivative of ρ with respect to cosmic time iṡ
ρ = 6 √ 2πG(ξ 0 + A √ ρ 0 ) exp( √ 6πG At)H.(6)
Taking into account Eqs. (2), (4)-(6) we obtain
2 √ 2πG( √ 3 AH + ξ 0 ) + [1 + w(t)]3AH 2 + Λ(t) = 0.(7)
Solving with respect to Λ(t), we have
Λ(t) = −3A[1 + w(t)]H 2 − 2 √ 2πG( √ 3 AH + ξ 0 ).(8)
If the parameter w(t) is chosen as
w(t) = −1 − δ 3AH 2 ,(9)
with δ is a positive constant, the "cosmological constant" becomes
Λ(t) = δ − 2 √ 2πG( √ 3 AH + ξ 0 ).(10)
Consequently, we have achieved the solution (10) which is in conformity with the LR expression (2), valid when the condition (1) is satisfied. The noticeable point is that such a LR behavior is now induced purely via the Λ− sector.
The turbulent approach
Let us consider the dark energy universe in its later stages, where it approaches the future singularity. The fluid system may then be regarded as quasi-stationary, and it becomes natural to take into account a transition into turbulence motion. We write the effective energy density as a sum of two terms [21]:
ρ eff = ρ + ρ turb ,(11)
where ρ denotes the laminar ordinary energy density and ρ turb its turbulent part. We assume that ρ turb is proportional to the scalar expansion θ = U µ ;µ = 3H and write the effective energy density as
ρ eff = ρ(1 + 3τ H),(12)
with τ a proportionality factor.
Analogously we split the effective pressure p eff into two terms,
p eff = p + p turb .(13)
The non-turbulent quantities p and ρ are connected by the standard relationship
p = wρ,(14)
where −1 < w < −1/3 in the quintessence region and w < −1 in the phantom region.
We take the dependence of p turb on ρ turb to be as simple as possible,
p turb = w turb ρ turb ,(15)
with w turb a constant. We shall consider two different possibilities for the value of w turb . First, we put w turb equal to w in Eq. (14), meaning that turbulent matter behaves in the same way non-turbulent matter as far as the equation of state is concerned. As second option, we shall assume w turb to be different from w.
1)
Assume that w turb = w < −1.
Even in this case the time development of ρ will be different from that of ρ turb . The ratio between turbulent and non-turbulent energy density becomes [21] ρ
ρ turb = 3τ H = 3τ H 0 Z .(16)
The Hubble parameter becomes
H = H 0 Z ,(17)
where we have defined Z as
Z = 1 + 3 2 γH 0 t,(18)with γ = 1 + w.(19)
Here H 0 is the initial value of H at the present time t = t 0 .
From the first Friedman equation we get a Big Rip behavior for the turbulent energy density [21],
ρ = 3H 2 0 k 2 1 Z 1 Z + 3τ H 0 .(20)
We now findρ
= − 9γH 3 0 2k 2 2Z + 3τ H 0 [Z(Z + 3τ H 0 )] 2 .(21)
using Eqs. (4), (20) and (21) the energy conservation can be rewritten as
3γH 2 0 Z 2 1 − 1 + w(t) γk 2 (1 + 3τ H 0 /Z) − Λ(t) = 0.(22)
Thus, if the parameter w(t) is chosen as
w(t) = −1 − k 2 3H 2 0 Z(Z + 3τ H 0 ),(23)
the "cosmological constant" becomes equal to
Λ(t) = 1 + 3γH 2 0 Z 2 .(24)
If t → +∞, then w → −∞, Λ → −∞, and the universe lies in the phantom region.
2) A milder variant: the Little Rip scenario.
Option 1) above was concerned with the Big Rip, meaning that the future singularity is encountered in a finite time. The Little Rip is a milder variant, as the time needed to obtain the singularity is infinite. Taking the equation of state in the form p = −ρ − A √ ρ with A the same positive constant as in Eq.
(2), we get [21]
ρ = ξ 2 0 9A 2 1 + 3A √ ρ 0 ξ 0 − 1 exp 1 2 √ 3 At 2 ,(25)
which shows that the increase of ρ towards infinity occurs only exponentially. We now findρ
= (3A √ ρ 0 − ξ 0 ) exp √ 3 2 kAt H.(26)
The energy conservation law takes takes the form
√ 3AH + k [1 + w(t)] 3 k 2 H 2 − ξ 0 3 + Λ(t) = 0.(27)
We solve this equation with respect to Λ(t) and insert for the parameter w(t) the expression
w(t) = −1 − δk 2 3H 2 ,(28)
with δ a positive constant. Then we obtain
Λ(t) = ξ 0 3 + δ − √ 3 k AH.(29)
In this case the LR is caused by the quantity w. When t → ∞, w(t) → −1, Λ(t) → −∞. The future behavior of this universe will depend on the choice of the model parameters ξ 0 , A and δ.
Consequently, if we start from a perfect fluid whose equation of state is given in the form (4), within the framework of 4d FRW cosmology, we realize the viscous Little Rip via the choice (28) for the parameter w(t), corresponding to the expression (29) for Λ(t).
A one-component dark fluid
There is an alternative way of approaching the problem, namely to consider the cosmic fluid as a one-component fluid. The universe can be assumed to start from the present time t = 0 as an ordinary viscous fluid with bulk viscosity ζ, developing with time according to the Friedman equations in the viscous era towards a future singularity. We assume that the EoS parameter w < −1, so that the future singularity should on the basis of these conditions be unavoidable. Before the singularity is encountered we assume, however, that at some instant t = t * there occurs a sudden transition of the whole fluid into a turbulent state after which the EoS parameter is w turb > −1 and the pressure is equal to p turb = w turb ρ turb . On the laminar side of the transition point, p * = wρ * < 0, while on the turbulent side, p * = w turb ρ * will even be positive if w turb > 0. The density is continuous at t = t * whereas the pressure is not. For simplicity we now take ζ, w and w turb to be constants.
In the viscous era 0 < t < t * the energy density is [21]
ρ = ρ 0 e 2t/tc [1 − 3 2 |γ|H 0 t c (e t/tc − 1)] 2 ,(30)
where t c is the "viscosity time"
t c = 3 2 k 2 ζ −1 .(31)
From this we can calculate the energy density at t = t * . Take now the derivative of the energy density with respect to cosmic time,
ρ = 2ρ 0 e 2t/tc t c 1 + 3 2 |γ|H 0 t c [1 − 3 2 |γ|H 0 t c (e t/tc − 1)] 3 ,(32)
and use Eqs. (4), (5), (30) and (32) to obtain the energy conservation law
2 t c 1 + 3 2 |γ|H 0 t c + 3[1 + w(t)] + k 2 e t tc Λ(t) H 2 = 0.(33)
When solving this with respect to Λ(t),
Λ(t) = − H 2 k 2 e t/tc 3[1 + w(t)] + 2 t c 1 + 3 2 |γ|H 0 t c(34)
we obtain, by choosing the parameter w(t) in the form
w(t) = −1 − δ 3H 2 e t tc(35)
with δ a positive constant, the following expression for Λ(t):
Λ(t) = δ k 2 − 2 1 + 3 2 |γ|H 0 t c k 2 t c H 2 e − t tc .(36)
If t → +∞, then Λ(t) → δ/k 2 .
In the turbulent era we make in the expression (30) the substitutions t c → +∞, t → t − t * , γ → γ turb (γ turb > 0) and ρ 0 → ρ * . Then [21]
ρ = ρ * 1 + 3 2 γ turb H * (t − t * )
and we now findρ
= − 9 k 2 γ turb H 3 .(38)
The energy conservation law is
3 k 2 γ turb H 2 − [1 + w(t)] 3 k 2 H 2 − Λ(t) = 0.(39)
Solving (39) with respect to Λ(t),
Λ(t) = 3 k 2 H 2 [γ turb − w(t) − 1],(40)
and choosing the parameter w(t) as
w(t) = −1 − δk 2 3H 2(41)
with δ a positive constant, we find the "cosmological constant" to be
Λ(t) = δ + 3γ turb k 2 H 2 .(42)
As both ρ and H go to zero when t → ∞ in the turbulent era, it follows from the expression (42) that Λ(t) → δ when t → ∞. Thus, we have shown how the transition of a one-component cosmic fluid from the viscous era into the turbulent era can alternatively be looked upon as a 4d FRW cosmology situation in which the EoS equation takes the general form (4) above.
Conclusion
This investigation can be taken as a demonstration of the diversity of cosmological fluid mechanical theory. Our starting point was the inclusion of turbulence in the cosmic fluid; this is natural approach in view of the fundamental property of classical fluids in general. As is known, the dark energy is often considered to be some kind of a classical fluid with unusual properties. It would seem physically reasonable to think that turbulence phenomena may be important for the dark energy, especially in the very late violent universe. What we have essentially shown, is that an equivalent description of viscous Little Rip cosmology for the dark fluid in the late universe can be obtained in terms of an inhomogeneous fluid within the framework of 4d FRW cosmology. The central form of the EoS equation is that of Eq. (4) above.
Thus a two-component model, in which the fluid system was assumed to be quasi-stationary with turbulent properties, was treated as an inhomogeneous fluid within 4d FRW cosmology.
A one-component model was also treated as an inhomogeneous fluid, in the viscous epoch as well as in the turbulent epoch, also in that case in terms of an inhomogeneous fluid in the 4d FRW theory.
The effect of turbulence and/or viscosity may thus always be tracked back to an appropriate effective equation of state.
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|
[] |
[
"VARIABLE TRANSFORMATION FOR EXPLICIT WALL-SHEAR STRESS FORMULA A PREPRINT",
"VARIABLE TRANSFORMATION FOR EXPLICIT WALL-SHEAR STRESS FORMULA A PREPRINT"
] |
[
"Aleksandar Jemcov [email protected] \nDepartment of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN\n",
"Joseph P Maruszewski [email protected] \nDepartment of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN\n",
"Ryan T Kelly [email protected] \nDepartment of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN\n"
] |
[
"Department of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN",
"Department of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN",
"Department of Aerospace and Mechanical Engineering\nNotre Dame Turbomachinery Laboratory University of Notre Dame South Bend\nUniversity of Notre Dame Notre Dame\nANSYS Inc. Evanston\n46556, 60201, 46602IN, IL, IN"
] |
[] |
We formulate a general variable transformation for existing wall functions that allows for an explicit wall-shear stress term. The proposed transformation aims to enable an explicit expression of wallshear stress and simplify the implementation of existing wall functions in simulation codes through a simple transformation of variables. The transformation is defined by introducing a new velocity scale allowing the definition of the new wall unit variable (r + ) and the corresponding normalized viscosity (η + t ). We also demonstrate that the law of the wall in new variables is equivalent to the one expressed in wall-normal variables (y + τ ) and (ν + t ). The new form of the law is particularly suitable for implementations in computational fluid dynamics codes as it does not require an iterative procedure to evaluate the wall shear stress. We show how to transform several known and often used expressions of wall functions with and without pressure gradients to allow for an explicit wall-shear stress expression. Finally, we illustrate the new form of wall functions by performing illustrative numerical simulations.
| null |
[
"https://arxiv.org/pdf/2012.14928v1.pdf"
] | 229,923,134 |
2012.14928
|
573c14b6f0ac019a28c6d03ffb5340b2d2e7d9d0
|
VARIABLE TRANSFORMATION FOR EXPLICIT WALL-SHEAR STRESS FORMULA A PREPRINT
January 1, 2021
Aleksandar Jemcov [email protected]
Department of Aerospace and Mechanical Engineering
Notre Dame Turbomachinery Laboratory University of Notre Dame South Bend
University of Notre Dame Notre Dame
ANSYS Inc. Evanston
46556, 60201, 46602IN, IL, IN
Joseph P Maruszewski [email protected]
Department of Aerospace and Mechanical Engineering
Notre Dame Turbomachinery Laboratory University of Notre Dame South Bend
University of Notre Dame Notre Dame
ANSYS Inc. Evanston
46556, 60201, 46602IN, IL, IN
Ryan T Kelly [email protected]
Department of Aerospace and Mechanical Engineering
Notre Dame Turbomachinery Laboratory University of Notre Dame South Bend
University of Notre Dame Notre Dame
ANSYS Inc. Evanston
46556, 60201, 46602IN, IL, IN
VARIABLE TRANSFORMATION FOR EXPLICIT WALL-SHEAR STRESS FORMULA A PREPRINT
January 1, 2021Variable Transformation · Incompressible Turbulent Boundary Layer · Wall-Shear Stress · Near-Wall Units
We formulate a general variable transformation for existing wall functions that allows for an explicit wall-shear stress term. The proposed transformation aims to enable an explicit expression of wallshear stress and simplify the implementation of existing wall functions in simulation codes through a simple transformation of variables. The transformation is defined by introducing a new velocity scale allowing the definition of the new wall unit variable (r + ) and the corresponding normalized viscosity (η + t ). We also demonstrate that the law of the wall in new variables is equivalent to the one expressed in wall-normal variables (y + τ ) and (ν + t ). The new form of the law is particularly suitable for implementations in computational fluid dynamics codes as it does not require an iterative procedure to evaluate the wall shear stress. We show how to transform several known and often used expressions of wall functions with and without pressure gradients to allow for an explicit wall-shear stress expression. Finally, we illustrate the new form of wall functions by performing illustrative numerical simulations.
Introduction
The practice of using boundary conditions based on the law of the wall is commonly used in computational fluid dynamics to reduce mesh requirements near no-slip walls. This approach for handling wall boundary conditions is commonly known by the name of wall functions [1,2]. Wall functions rely on the law of the wall to set the correct wall shear stress τ w , to compute local turbulent viscosity µ t , and set boundary conditions for the velocity and turbulence quantities. The implementation of wall functions requires knowledge of the friction velocity u τ = |τ w |/ρ to determine the wall shear stress which is obtained from the law of the wall for the turbulent boundary layer [1,2,3]. In the case of incompressible equilibrium turbulent boundary layers under zero pressure gradient conditions, the law of the wall is given as a functional relationship between non-dimensional quantities u + and y + τ [3] u + = u u τ (1)
y + τ = u τ y ν(2)
arXiv:2012.14928v1 [physics.flu-dyn] 29 Dec 2020
The symbol u in Eq. (1) denotes a streamwise component of the time averaged velocity vector, ν denotes molecular kinematic viscosity, and y denotes distance in the wall-normal direction. In the case of zero gradient of pressure incompressible turbulent boundary layers, the relationship between u + and y + τ is obtained by either solving the simplified incompressible Reynolds-averaged Navier-Stokes (RANS) equations
1 + ν + t du + dy + τ = 1, ν + t = ν t ν(3)
or by using analytic expressions that are defined from theoretical considerations [3,4,5,6,7,8]. The local value of the wall shear stress is readily obtained by computing the local friction velocity from the known local value of u + [2]. The procedure of obtaining the wall shear stress in the case of turbulent boundary layers with a pressure gradient is similar with the addition of the normalized pressure p + to the expression of the wall of the law [9,10,11].
For the purpose of discussion in this work, we distinguish between various expressions of the law of the wall that do not require and the ones that require an iterative procedure to compute the wall shear stress. The wall function expressions that do not require an iterative procedure to evaluate the wall shear stress are referred to as the explicit wall function.
Similarly, wall functions that need an iterative algorithm for the computation of the wall shear stress are called implicit wall functions. The need for an iterative procedure arises because the friction velocity appears on both sides of the law of the wall u + (u τ ) = f (y + τ (u τ )). The earliest example of implicitly defined wall function in terms of wall units y + τ and u + was given by Prandtl [12] that was based on the following set of equations:
u + = y + τ , y + τ ≤ 5.0(4)
u + = 1 κ ln(y + τ ) + B, y + τ > 10.8, κ = 0.41, B = 5.0
Eq. (4) is obtained by integrating Eq. (3) under the assumption ν + t → 0 for the values of y + τ between 0 and 5. The second expression is obtained by letting ν + t 1 and formally integrating Eq. (3) from y + τ = 0 to y + τ → ∞. Ignoring the buffer layer, two expressions are matched for the value of y + τ = 10.8 to produce the law of the wall for all values of y + τ expressed in Eq. (4) and Eq. (5). The constant of integration B was obtained by fitting the expression in Eq. (5) to experimental data. To set the correct boundary condition, wall shear stress must be obtained from Eq. (4) and Eq. (5) by computing the friction velocity. Given the definition of u + and y + τ , it is evident that an iterative procedure is required to obtain the friction velocity from Eq. (5). This requirement becomes obvious if we cast Eq. (5) in the following form
u u τ = 1 κ ln u τ y ν + B(6)
Since the right-hand side has u τ under the logarithmic function, it is not possible to solve for u τ without using an iterative procedure. In the context of computational codes using the finite volume formulation, the iterative procedure has to be performed for each discrete face of the discretized wall boundary surface, thus increasing the overall computational operations count. A similar iterative procedure is performed for the computation of wall shear stress in finite difference and finite element codes. While the wall function defined by Eq. (4) and Eq. (5) is relatively simple and the iterative procedure of computing u τ converges well, the wall function is not smooth in the buffer region. Therefore, the wall function based on Eq. (4) and Eq. (5) is not accurate for computations for wall meshes that fall into the buffer region of the boundary layer.
Spalding [4] proposed another form of the implicit wall function for zero pressure gradient boundary layers. Spalding's law of the wall consists of a single analytical expression that is valid in the laminar sublayer, the buffer zone, and the logarithmic region of the boundary layer. Spalding's function is given in the form y + τ = f (u + ) in contrast to usual dependence of u + on y + τ . Spalding's function is one of the earliest examples of an analytical expression suitable for a wide range of y + τ values, and it has found wide acceptance and implementation in computational fluid dynamics codes. One advantage of the Spalding's formulation is that only one expression is needed throughout the boundary layer instead of piece-wise formulation as in the case of Prandtl's formulation. Spalding's function was derived to satisfy the requirement that the ratio of the total (turbulent and molecular) and molecular kinetic viscosity + = (ν + ν t )/ν grows with at least third power of y + τ away from the wall in the wall-normal direction. To satisfy this requirement, Spalding derived the following function
y + τ = u + + 0.1108 e 0.4u + − 1 − 0.4u + − (0.4u + ) 2 2! − (0.4u + ) 3 3! − (0.4u + ) 4 4!(7)
It is clear from expression in Eq. (7) that Spalding's wall function requires an iterative method to evaluate the wall shear stress. Newton's iterative procedure is typically used to evaluate the wall shear stress. Iterations are performed until the set convergence criterion is achieved, or the maximum number of iterations is exceeded. The quadratic rate of convergence characterizes newton iterations, and the iterative procedure typically converges within several iterations. However, Newton's method is known to be sensitive to initial guess, and this property may lead to erroneous results in the computation of the wall shear stress if the unsuitable initial guess is used in the iterative procedure. The computational complexity of the implementation of the Spalding's wall function is high since the iterative procedure is required to be performed for each discrete face of the wall surface. Given the lack of pressure gradient influence in the formulation of Spalding's wall function and its computational complexity makes the formulation to be computationally expensive in complex three-dimensional flows where adverse and favorable pressure gradient plays a significant role.
Musker [11,5] proposed an analytical expression for u + as a function of y + τ by rejecting the Spalding's function in favor of a simpler expression. Musker postulates the following expression for the variation of ν + t ∼ (y + τ ) 3 in the boundary layer 1
ν + t = 1 C(y + τ ) 3 + 1 κy + τ , C = 0.001093, κ = 0.41(8)
Musker's formulation uses inverse weighting to blend two different functions, one with the correct scaling of turbulent viscosity in laminar sublayer f 2 (y + τ ) = κy + τ , and the other one suitable in the inertial region of the turbulent boundary layer f 1 (y + τ ) = C(y + τ ) 3 . The blending is performed using the reciprocal function thus allowing the continuous law of the wall formulation throughout laminar and inertial layers of the turbulent boundary layer. Using the Reynolds averaged equations for the boundary layer without pressure gradient, the dimensionless velocity gradient
du + dy + τ = κ + C(y + τ ) 2 κ + C(y + τ ) 2 + C(y + τ ) 3(9)
is integrated to yield the following closed-term expression
u + = 5.424 tan −1 2y + τ − 8.15 16.7 + ln (y + τ − 10.6) 4.6 ((y + τ ) 2 − 8.15y + τ + 86.0) 2 − 3.52(10)
Musker's approach suffers from the same shortcoming of needing to use an iterative procedure to compute the friction velocity since both u + and y + τ depend on friction velocity u τ . Therefore, Newton's method is used to evaluate friction velocity. Similarly to Spalding's wall function, the computational complexity of evaluation of the wall shear stress from the Musker's wall function is high. The iterative procedure must be applied to each facet of the discrete wall boundary surface. Musker's wall function suffers from similar computational complexity as Spalding's wall function since the pressure gradient is not included in the formulation. Therefore, the application of the Musker's wall function is limited to zero pressure gradient boundary layers.
In contrast, Werner and Wengle [6,13] defined an explicit function for the wall shear stress. The Werner and Wengle formulation is based on the following functions y + τ = u + , y + τ ≤ 11.81 (11)
u + = A(y + τ ) B , A = 8.3, B = 1 7 , y + τ > 11.81(12)
Werner and Wengle model, Eq. (11) and Eq. (12), is integrated to yield the following expression for the wall shear stress
|τ w | ρ = ν |u| y , u ≤ ν 4y A 2 1−B (13) τ w ρ = 1 − B 2 A 1+B 1−B ν y 1+B + 1 + B A ν y B |u| 1 1+B , u > ν 4y A 2 1−B(14)
While the expression in Eq. (14) of Werner-Wengle does not require an iterative procedure to compute the wall shear stress, it does not take into account pressure gradient in the formulation and therefore, it is only applicable to flows without pressure gradients. Werner-Wengle law has found application in many computational codes due to its simplicity and ease of implementation, it i not accurate for large values of y + τ . Additionally, the expression for the wall shear stress in Eq. (14) is not directly extensible to account for the pressure gradient effects. Also, the transition from the viscous sublayer to inertial region is not smooth due to nature of expressions in Eq. (11) and Eq. (12). The lack of smoothness is not desirable as it leads to less accurate results in the buffer region of the turbulent boundary layer.
Another approach to avoiding the iterative procedure in computing the wall shear stress that is commonly in use in conjunction with k − turbulence model [1,14,15,16] is to use the following expression [17] u τ = C 1 4 µ k 1 2 , C µ = 0.09 (15) However, the expression in Eq. (15) cannot be used in turbulence models that do not use the transport equation for turbulent kinetic energy. Furthermore, asymptotic behavior of k as a function of y + τ has to be known to make the value u τ consistent with the turbulence model. Given that the transport equation for k is heuristically obtained, the asymptotic behavior of k near the wall carries all assumptions that are built in the turbulence transport model thus making the value of u τ dependent on those assumptions.
To simplify the computation of the wall shear stress, Kalitzin at al. [7] proposed the use of the new variable denoted here by symbol r + = u + y + τ to define the law of the wall. A significant advantage of the new variable is that it is not defined in terms of friction velocity, thus bypassing the need for the iterative procedure to determine wall shear stress. In their work, the new variable r + was labeled erroneously as the local Reynolds number, and in this work, we show that the new variable is defined as the normalized velocity where the normalization is performed with respect to local velocity scaleũ = ν/y. In the approach proposed by Kalitzin at al., a numerical method is used to solve the simplified RANS equations displayed in Eq. (3), and the obtained values are used to create a table of wall shear stress as a function of local Reynolds number. The resulting tables utilize an interpolation procedure whenever the value of the local variable r + falls between two tabulated values. Introduction of the velocity scaling simplified the computation of the wall shear stress significantly as it allowed the use of the lookup tables without the need to use an iterative procedure. From the classification point of view adopted in this work, the approach of Kalitzin at al. constitutes an explicit wall function, albeit in tabular form. However, the main drawback of the wall function by Kalitzin at al. is the absence of the pressure gradient from the formulation. This drawback severely limits the use of the defined wall function in practical computations to flows without pressure gradients. A further drawback of Kalitzin at al. approach is that the law of the wall is formulated in the tabular form. Therefore, an interpolation procedure is required to compute the values of the wall shear stress for the values of r + falling between two values in the table. While using the interpolation procedure does not significantly add to the computational complexity of implementation of the wall function in computational codes, it is more convenient to use analytic expressions whenever possible.
Many other wall functions proposed by von Karman [18], Deissler [19], Rannie [20], and Reichardt [21] fall in the same category of implicitly defined wall functions. This is not a complete list of proposed wall functions, but all listed ones share the property of excluding the pressure gradient effects from their definition in addition to requiring an iterative method to compute the wall shear stress.
Several attempts to generalize the law of the wall in the presence of pressure gradient is evidenced by numerous references [9,10,11,17]. In Kim and Choudhury [17], the introduction of the pressure gradient to the law of the wall was obtained by the generalization of the two-layer approach by Launder [22]. Kim and Choudhury proposed a logarithmic velocity profile that is based on the kinetic energy budget in the boundary layer. The resulting expression for the wall shear stress requires the knowledge of the turbulent kinetic energy that is assumed to be constant in the turbulent part of the boundary layer. The wall function expression proposed by Kim and Choudhury is generalized here to be applicable to turbulence models that do not compute turbulent kinetic energy transport as follows
u + = y + τ + 1 2 (y + τ ) 2 p + if y + τ < 10.8 1 κ ln(y + τ ) + B + p + 2κ [y + τ + 10.8 ln(y + τ + 11.35] otherwise(16)
where p + is defined as follows
p + = ν ρu 3 τ dp dx(17)
The expression in Eq. (16) falls in the implicitly defined wall functions as an iterative procedure is required to compute the wall shear stress. Furthermore, the resulting velocity profile deviates significantly from experimental results for large y + τ values for moderate to strong pressure gradients. Furthermore, Röber [23] proposed a continuous formulation of the wall function with pressure gradient by blending the analytic solution for the viscous sublayer and a van Driest damping function. The resulting law of the wall requires an iterative procedure to evaluate the wall shear stress, and it deviates from experimental results for large values of the wall distance y + τ . The common behavior of Kim and Choudhury and Röber wall functions is that they are not bounded functions with increasing y + τ values. Therefore, the utility of the proposed wall functions is limited to lower values of y + τ . Duprat at al. [24] proposed another approach to a continuous wall functions with pressure gradient. The approach is based on velocity scaling and van Driest damping to produce a continuous wall function [25]. The resulting expression for the wall shear stress required an iterative procedure to compute the friction velocity. Furthermore, for large values of the wall distance, the proposed wall function have shown a significant discrepancy between experimental and computed results similarly to models of Kim and Choudhury and of Röber.
Popovac and Hanjalić [26] propose the pressure sensitized wall function that includes acceleration terms as follows
u + = 1 κψ ln(Ey + τ ) (18) ψ = 1 − C + U y + τ u + κ (19) C + U = ν ρu 3 τ ρ ∂u ∂t + ρu ∂u ∂x + ρv ∂u ∂y + ∂p ∂x(20)
where E = e Bκ nd C + U is the acceleration term. The proposed wall function is simplified here to read
u + = 1 κ ln(y + τ ) + B + C + U y + τ κ(21)
The wall function of Popovac and Hanjalić also belongs to the class of implicitly defined functions. Therefore, and iterative procedure is also required to compute the wall shear stress. Furthermore, the proposed wall functions tends to produce the negative velocities for large values of y + τ in adverse pressure gradient environment. in favorable pressure gradient environment, the wall function in Eq. (21) tends to produce large values of the velocity in contrast to experimental data. Similarly to all wall functions with pressure gradients discussed thus far, the proposal of Popovac and Hanjalić also produce the unbounded behavior of the wall function for large values of y + τ The proposal of Shih at al. [9] is based on the new velocity scale that includes both viscous and pressure gradient effects have demonstrated consistent velocity profiles for the extended range of the values of wall distance under moderate to severe pressure gradients. The model also included the blending of the buffer region to produce continuous wall functions for all y + τ values. Shih's model is based on the Lumley's analytic solution for the boundary layers with pressure gradient [3]. The agreement between Shih's model and experimental results is satisfactory for the broad range of y + τ values under moderate to severe pressure gradients. However, the model requires an iterative procedure, and its implementation can be difficult. However, Shih's wall function does not exhibit unbounded behavior for large values of y + τ and it can be used for the computation of the wall shear stress over the broad range of y + τ values. Irrespective of the presence of the pressure gradient, it is apparent that an iterative procedure is required to determine the value of the wall shear stress in all formulations except for Werner and Wengle formulation [6,13]. The goal of this work is to define a wall function that does not require an iterative procedure, and that is universally applicable to flows with and without pressure gradient. One of the main obstacles in achieving this goal is the use wall distance y + τ and normalized velocity u + since both are defined with reference to friction velocity u τ . Therefore, a different choice of variables can remove the need for an iterative procedure. The focus of this work is on defining the law of the wall in such variables. We generalize Shih's at al. model to be expressed in the new velocity scaling variable r + and we define the explicit expression for the wall shear stress for boundary layers with and without gradient of pressure.
The paper is organized as follows: in section (2), we introduce the variable r + in Shih's at al. model. We also introduce the new viscosity variable η + t related to r + and we show the the relationship between η + t and ν + t . We then state the boundary layer equations in terms of η + t , r + , u + , and p + . We show the equivalence of boundary layer equations and consequently, the law of the wall in the new (η + , r + ) and old variables (ν + t , y + τ ). We show the equivalence between the new and old formulations by integrating the boundary layer equations for zero pressure gradient flow conditions. We then modify the Shih at al. model to be formulated in terms of new variables to compute the wall shear stress explicitly. In Section 3, we apply the new definition of Shih at al. wall function to k − ω-SST model of turbulence [27]. In Section (4) we perform example calculations for the backward facing step [28] and planar diffuser problem [29] at various mesh resolutions and compare results to experiments. In Section (5) we provide the summary and conclusions
Formulation of the Boundary Layer Equation in the New Wall Units
Following Shih at al. [9], we introduce the incompressible boundary layer equation in wall units
1 + ν + t du + dy + τ = 1 + p + y + τ(22)
The quantities in Eq. (22) are defined as follows:
u + = u 1 u τ , y + τ = u τ y ν , ν + t = ν t ν , p + = ν ρu 3 τ dp dx , u 2 τ = |τ w | ρ (23)
where τ w is the wall shear stress and u τ is the friction velocity. For simplicity, curvature effects are neglected in Eq. (22) and it is assumed that the flow is mainly in the x-direction while y-direction is normal to the wall. Furthermore, symbol u represents the streamwise component of the time-averaged velocity vector, p denotes the time-averaged pressure, and ν represents the molecular viscosity of the fluid. The Boussinesq hypothesis is used to relate turbulent shear stress to the gradient of the mean velocity u
−u v = ν t du dy(24)
where u and v represent components of the turbulent velocity fluctuations in the streamwise and wall-normal directions respectively..
The solution of Eq. (22) is called the law of the wall, and it symbolically was written as the following expression
u + = f (y + τ )(25)
Given the definitions in Eq. (23), it is clear that if the objective of the computation is to determine the value of the wall shear stress, Eq. (25) requires an iterative procedure to compute the friction velocity u τ . Therefore, if some other velocity scale is used, it is in principle possible to avoid the implicit character of the law of the wall. One such scale is introduced in this work, and it is defined as follows:û
= ν y(26)
The local averaged velocity is normalized to introduce a new variable r +
r + = û u(27)
We seek to express the law of the wall in terms of new variable i.e., u + = g(r + ). To be able to achieve that goal, the boundary layer equation must be written in terms of the new variable. A transformation between old and new variables is needed to cast the law of the wall in terms of new wall units.
To define the transformation, we observe that the new variable r + can be written in terms of u + and y + as follows
r + = û u = u + y + τ(28)
With the help of the following identity
d(r + ) = d(u + y + τ ) = y + τ du + + u + dy + τ(29)
Eq (22) is transformed into the following form
u + (1 + ν + t ) + y + τ du + = dr + + p + r +(30)
The new variable η + t that represent the transformed normalized turbulent viscosity is defined as
η + t = u + 1 + ν + t + y + τ − 1(31)
so that the boundary layer equation takes the following form
1 + η + t du + = dr + + p + r +(32)
Eq. (22) and Eq. (32) are equivalent and can be transformed to each other by using the transformation defined in Eq. (28) and Eq. (31).
The main advantage of the new form of the boundary layer equation in new variables is that the evaluation of the wall shear stress can now be obtained without the recourse to an iterative procedure. Therefore, the law of the wall u + = g(r + ) becomes explicit, and the direct computation of u + is possible by performing a simple function evaluation. Moreover, y + τ is readily available once the value of u + is computed by using the relationship
y + τ = r + u +(33)
Solution of Eq. (32) represent the desired law of the wall in new variables that is symbolically given as u + = g(r + ).
Since the definition of r + does not depend on friction velocity, normalized velocity u + is evaluated directly. Moreover, all that is required for this computation is the local distance from the wall y, local mean velocity u, and local value of molecular viscosity ν.
Asymptotic Solutions of Turbulent Boundary Layer without Pressure Gradient
Boundary layer equation for the case of zero pressure gradient (p + = 0) reduces to the following equation in wall units
1 + ν + t du + = dy + τ(34)
Similarly, boundary layer equation in new variables is
1 + η + t du + = dr +(35)
With no slip boundary condition, Eq. (36) is integrated to yield to following expression
u + = y + τ , y + τ < 5(37)
The extent of the laminar sublayer is the range of y + τ values between 0 and 5. In order to demonstrate that Eq. (35) can be integrated in a similar way, we use the condition ν + t 1 to evaluate the viscosity η +
t η + t = u + + y + τ − 1(38)
Substituting expression from Eq. (38) in Eq. (35), we obtain the following expression
(u + + y + τ )du + = dr +(39)
Since in viscous sublayer according to Eq. (37) u + = y + τ , the solution to Eq. (35) in new variables is
u + = √ r + , r + < 5(40)
The inertial region of the boundary layer is recovered with the help of the Prandtl's assumption about the normalized turbulent viscosity
ν + t = κy + τ(41)
where κ = 0.41 is an experimentally determined constant. In the inertial region the nomralized turbulent viscosity is large and it is assumed that ν + t 1. With this assumption, Eq. (34) becomes
ν + t du + = dy + τ(42)
With the help of Eq. (41), the integration of Eq. (42) yields the well known logarithmic law
u + = 1 κ ln(y + τ ) + B(43)
where B = 5.0 is experimentally fitted constant.
Solution for the inertial region of the boundary layer can be expressed in new variables by enforcing the condition ν + t 1 in expression for modified turbulent viscosity
η + t = u + ν + t + y + τ − 1(44)
Substituting Eq. (44) in Eq. (35), we obtain
u + ν + t + y + τ du + = dr +(45)
To be able to solve Eq. (45) in closed form in new variables, we introduce and equivalent Prandtl's assumption as follows
ν + t = kr + − y + τ u +(46)k = κ + 1 u +(48)
Clearly, function k is not a constant and it is related to the value of κ. The contribution of the normalized velocity to values of k is negligible for large r + values. For smaller values of r + the contribution of the normalized velocity u + to values of k function is finite since the validity of the modified Prandtl's hypothesis extends only to the lower limit of inertial region bounded by r + >= 10.8 2 . We approximate k by a constant value determined by the curve fit to minimize error for the range of r + values. Estimated values values of k and C constants are as follows:
k = 1 2.1108 , C = 0.7576(49)
The comparison between the law of the wall defined by Eq. (43) in the traditional wall distance variables and the new one defined according to Eq. (47) is shown in Fig. 1. It is clear from the graph in Fig. 1 that the two expressions are equivalent. The viscous sublayer solution is extended beyond the buffer region (y τ ≥ 5) to intersect the inertial region at the point y τ = 10.8 as is often done in practical implementations of the law of the wall in computational fluid dynamics codes. The practiec of ignoring the buffer region leads to errors in the buffer region and it is done here only to compare two asymptotic solutions. Fig. 1 indicates that the law of the wall in the new scaling and old wall distance variables produce physically the same values of u + . Transformation of various profiles that appear in the literature [4,6,8] is in principle possible with some effort. The process becomes difficult for more complex expressions of the normalized turbulent viscosity. One such example is the Musker's law [5,11]. Musker's profile represents the continuous function that is applicable to both laminar sublayer, buffer, and inertial region. Moreover, the profile extends to the wake region. We are interested in representing Musker's function in new variables and instead of integrating the boundary layer equation, a best fit optimization is used to define the new function. The Musker's function in new variables takes the following form
u + = 1 k ln √ r + + a a + α a + 4α (a − 4α) ln a ( √ r + − α) 2 + β 2 2α( √ r + + a) 2 (50) + 2α(5a − 4α) β tan −1 √ r + − α β + tan −1 α β
The coefficient definitions of Eq. (50) are as follows:
a = l + 1 9k 2 l + 1 3k (51) l = 1 2 4s+27k 3 27k ks + 2s + 27k 3 54k 3 s 1 3 , k = 0.23, s = 8.347 × 10 −4 (52) 2α = a − 1 k , β 2 = 2aα − α 2
The new form os the Musker's function is structurally the same as the original function [5,11] with the modified coefficients. Furthermore, the independent variable that keeps the same form of the original Musker's function is √ r + instead of r + . The utility of the new variables becomes apparent when we consider the Musker's function. Since the variable r + is readily available, computation of u + requires only one evaluation of Eq. (50). If the traditional wall units y + τ are used, an assumed value of friction velocity must be iteratively improved in order to compute y + τ and u + . This process requires several evaluations of Musker's function in the course of iterations.
The comparison between Musker's function in wall distance and new scaling variables is shown in Fig. 2. It is evident that the agreement between the old and new variables is close with small differences in the buffer region.
Asymptotic Solutions of Turbulent Boundary Layer with Pressure Gradient
Asymptotic solutions for boundary layers without pressure gradients have a limited utility in practical computations as virtually all flows involve pressure gradients. Therefore, we are interested in expanding the use of new variables to wall functions that include pressure gradient. We focus on the generalized wall functions introduced by Shih at al. [9,10]. Shih's proposal is built on previous work by Lumley [3] in which a new velocity scale is introduced to encompass both viscous and pressure gradient effects
u c = u τ + u p(53)
Friction velocity is defined as before
u τ = |τ w | ρ (54)
The new velocity scale related to pressure effects was defined earlier and we repeat its definition for clarity u p = ν ρ dp w dx The asymptotic solution of Eq. (56) was obtained by Shih at al. [9] in the following form:
u u c = τ w ρu 2 τ u τ u c f 1 y + c u τ u c + dp dx | dp dx | u p u c f 2 y + c u p u c(57)
The non-dimensional wall distance y + c was defined by using the velocity scale u c
y + c = u c y ν(58)
It can be observed that unlike the previous definition of the wall distance y + c that was defined with the help of the friction velocity only, the new definition involves both friction and pressure related components. The solution in Eq. (57) was obtained by observing that the expression of Eq. (56) is linear thus allowing to separate the expression in two parts [3] u = u 1 + u 2 (59)
The velocity u 1 is related to the wall shear stress while the velocity u 2 is related to the pressure gradients. The decomposition of Eq (56) in two parts yields the following two equations [3,9] ν ∂u 1 ∂y
− u v 1 = τ w ρ (60) ν ∂u 2 ∂y − u v 2 = y ρ dp w dy(61)
The asymptotic solutions to Eq. (60) and Eq. (61) are
u 1 u τ = τ w ρu 2 τ f 1 (y + τ ) (62) u 2 u p = ν ρ dpw dx u 3 p f 2 (y + p )(63)
Asymptotic solution in Eq. (63) uses the non-dimensional wall distances y + p defined as follows
y + p = u p y ν(64)
whereas y + τ was previously defined in Eq. (2). The non-dimensional wall unit y + τ takes into account viscous effects, whereas y + p incorporates pressure gradient effects in its definition. Functions f 1 is obtained from the solution of Eq. (60) so that satisfies boundary conditions and represent velocity profile in viscous sublayer, buffer region, and inertial sublayer [9]
f 1 (y + τ ) = y + τ + 0.01(y + τ ) 2 − 2.9 × 10 −3 (y + τ ) 3 if y + τ ≤ 5 −0.872 + 1.465y + τ − 0.0702(y + τ ) 2 + 0.00166(y + τ ) 3 −1.495 × 10 −5 (y + τ ) 4 if 5 ≤ y + τ ≤ 30 8.6 + 0.1864y + τ − 0.002(y + τ ) 2 + 1.144 × 10 −5 (y + τ ) 3 −2.551 × 10 −8 (y + τ ) 4 if 30 ≤ y + τ ≤ 140 2.439 ln(y + τ ) + 5 if 140 ≤ y + τ(65)
Similarly, f 2 function is given by the following expression [10]
f 2 (y + p ) = 0.5(y + p ) 2 − 7.31 × 10 −3 (y + p ) 3 if y + p ≤
The sgn(arg) function which evaluates to +1 if arg > 0 and −1 if arg < 0 is introduced so that Eq. (57) is written in the following form
u u c = sgn(τ w ) u τ u c f 1 y + τ + sgn dp w dx u p u c f 2 y + p(67)
Algebraic manipulations lead to the following form of Eq. (66)
u u τ = sgn(τ w )f 1 y + τ + sgn dp w dx u p u τ f 2 y + p(68)
Eq. (68) is solved for the friction velocity
u τ = u − u p sgn dpw dx )f 2 (y + p sgn(τ w )f 1 (y + τ )(69)
Combining Eq. (63), Eq.(68), and Eq. (59), the definition of friction velocity becomes
u τ = sgn(τ w ) u 1 f 1 (y + τ )(70)
Since the friction velocity u τ is positive by definition, and f 1 (y + τ ) is a positive function, we have sgn(τ w ) = sgn(u 1 ) (71) Therefore, Eq. (69) becomes
u τ = u − u p sgn dpw dx )f 2 (y + p sgn(u 1 )f 1 (y + τ )(72)
Evaluation of Eq. (72) requires an iterative procedure due to the presence of u τ on both sides of equality. As before, we seek to define the the generalized wall function in terms of variable r + The new velocity scaling variable r + has a significant advantage that it does not involve either u + or y + τ in its definition directly. In other words, the new scaling variable is defined completely in terms of physical quantities. Therefore, if the expression for the friction velocity is defined in terms of r + , computation of the wall shear stress becomes straightforward evaluation of the expression similar to Eq. (72).
To define Eq. (72) as a function of the new scaling variable, we perform the curve fitting procedure for the function g 1 (r + ) defined in Eq. (65) with respect to the new scaling variable r + to obtain the following expression
g 1 r + = √ r + if 0 ≤ r + < 24.τ w = ρ u − u p sgn( dpw dx )f 2 (y + p ) sgn(u 1 )g 1 (r + ) 2(74)
Therefore, the boundary conditions for the momentum equation is easily specified by computing the wall shear stress from Eq. (74). The boundary conditions for transported turbulence quantities are also easily specified by providing the asymptotic behavior of the turbulence model in boundary layer [7,17].
The explicit expression for the wall shear stress in Eq. (68) is easily adapted to work with other formulations of the gradient-free wall functions. All that is required is to define the function g 1 (r + ) for each law of the wall by transforming from y + to r + units. A simple example of analytic transformation was shown in the case of the Prandt's hypothesis [12] and the log-law, as shown in Eq. (40) and Eq. (50). In case of more complicated expressions, a curve fit procedure can be used to obtain the function g 1 (r + ) as demonstrated in case of Musker's [5]) and Shih's [9]) law of the wall. Curve fit, or a transformation of variables can also be applied to the Werner and Wengle law of the wall, Eq. (11) and Eq. (12). Expression of Eq. (74) represents a unifying framework for several proposed laws of the wall with pressure gradient effect in the new wall units r + . As such, the proposed expression can be used in numerous numerical codes to provide the pressure effects on the wall shear stress. The essential advantage of the proposed law of the wall is that it is obtained in the new wall units that do not require an iterative procedure to evaluate the wall shear stress.
Sample Applications of Wall Functions in Transformed Variables
In this section we illustrate the application of the generalized wall function in new set of variables using widely used k − ω SST model [27]. The governing system of equations for incompressible turbulent flow consists of the transport equation for the momentum and turbulence quantities. The following expression gives the momentum equation in Cartesian tensor notation with Einstein summation rule over the repeated indexes:
∂ (ρu i ) ∂t + ∂ (ρu i u j ) ∂x j = − ∂p ∂x i + ∂σ ij ∂x j(75)
Indexes i and j take the values from the set (x, y, z) that labels three Cartesian directions. As before, all quantities in momentum and turbulence transport equations are time averaged. The symbol σ ij represents the viscous stress tensor consisting of molecular, and Reynolds averaged stress tensor
σ ij = (µ + µ t ) 2S ij (76) S ij = 1 2 ∂u i ∂x j + ∂u j ∂x i(77)
Since we are considering incompressible turbulent flows, the density ρ and molecular viscosity µ are assumed to be constant. All quantities in Eq. (75) are time averaged to produce the Reynolds averaged Navier-Stokes equations. In addition, all quantities in subsequent expressions are time averaged and no special notation is used to denote time averaging.
To close Eq. (75), turbulent viscosity field µ t must be known. In this work, we compute the turbulent viscosity field by solving the k − ω SST transport equations [27]. All time-averaged quantities must be specified on the boundary of the domain to define the well-posed problem. Boundary conditions away from solid walls pose no difficulties, and they are prescribed as in [27]. At the solid wall where no-slip boundary condition is applied, special care must be taken to specify boundary values of k and ω fields. Boundary conditions for k and ω fields are obtained from the asymptotic near-wall behavior of the corresponding transport equations for k and ω [7].
We distinguish between computations that are wall-resolved and not wall-resolved. We consider the computation to be wall-resolved if the first cell of the computational mesh of the wall produces a y + value that less or equal to one.
In this work, we are concerned with computational meshes that are not wall-resolved. The values of y + τ in the first computational point off the wall take much larger values, i.e., y + τ 1. The wall shear stress for such computational meshes is evaluated from Eq. (72). Additional specification of k and ω fields near the wall must also be specified. The specific dissipation field ω in the laminar sublayer of the turbulent boundary layer is specified as follows:
ω + vis = 6 β 1 (y + ) 2(78)
where ω + is normalized specific dissipation defined as
ω + = ων u 2 τ(79)
Eq. (78) contains a singularity as y + τ → 0. The singularity is removed by evaluating the value of ω + at the finite distance from the wall. In cell-centered finite volume methods, the boundary condition is obtained by using the the value of y + τ that corresponds to the cell center of the first cell next to the wall. The following expression gives the value of ω + in the logarithmic region of the turbulent boundary value [7]:
ω + log = 1 κ C µ y + , C µ = 0.09(80)
The intermediate region between laminar sublayer and logarithmic layer is interpolated by the use of the following blending equation:
ω + = ω + vis 2 + ω + log 2(81)
The functional dependence of k + in the boundary layer is obtained from the expression
k + = ν + t ω +(82)
The normalized turbulent kinetic energy k + is defined per following expression
k + = k u 2 τ(83)
It is clear from Eq. (79) and Eq. (83) that the knowledge of the friction velocity is required for the boundary conditions for k and ω transport equations. The value of the friction velocity is computed from Eq. (72) making use of the wall functions with pressure gradients.
Computational Examples
To illustrate the application of the new generalized wall functions, two computational studies were performed. The first study is the flow over backward-facing step [28], while the second study was concerned with the flow in a diffuser [29]. Both flows exhibit adverse pressure gradient and the flow separation thus making them suitable for the illustration of the application of wall function boundary condition as defined in Eq. (72), Eq. (81), and Eq. (82).
The second-order finite volume solver library Caelus [30,31] was used in all computations. Second-order upwind discretization for the convective and central discretization of diffusive fluxes was used for momentum and turbulence equations [32]. The computational approach consisted of the successive uniform mesh refinement, and all computations were performed on the progression of four meshes. The main characteristic of the uniform mesh refinement is that it allows for the constant aspect ratio of cells across all computational meshes while at the same time, different values of y + τ were obtained. In this way, the performance of the proposed generalized wall functions is assessed in a controlled numerical experiment. Additionally, k − ω SST transport model [27] in all computations. However, the proposed generalized wall functions are equally applicable to other equilibrium and non-equilibrium turbulence transport models.
Backward Facing Step Study
The experimental setup of Driver and Seegmiller [28] produces the variable pressure gradient across the channel that affects the reattachment point of the shear layer emanating from the backward-facing step. The experiment was performed with various deflections of the top wall, demonstrating the influence of the pressure gradient on the reattachment point of the jet. In this work, we focus on a zero deflection angle for in numerical illustrations. In addition to reattachment point measurements, Driver and Seegmiller provide the detailed measurements on the pressure and friction coefficients along the top and bottom walls as well as velocity profiles downstream of the backward-facing step.
The computational domain was constructed using a hexahedral mesh extending 40 step heights, h, upstream of the step and 50h downstream of the step to ensure flow development is not influenced by the boundary conditions. Fig. 3 shows a magnified section, centered on the backward step, of the 65,000 cell mesh and relative domain dimensions. Boundary conditions used in computations correspond to atmospheric pressure and temperature while the inlet velocity is 44.2 m/s and turbulence intensity is 0.061%. As indicated, four the meshes, with varying grid resolution, were generated so that the effect of y + τ variations can be examined. The increasing grid refinement levels are shown in Table 1. ,000 30 2 65,000 15 3 261,000 6 4 1,046,000 3 Table 1: Range of mesh element count (N ), wall spacing (y + τ ), and aspect ratio (AR) used for the backward facing step simulations. The grids were refined requiring AR consistency over the desired y + τ range. Figure 4 shows the comparison between experimental and computed results for the range of values of y + τ at the lower wall for the normalized axial distance ranging from x/h = −5 to x/h = 35. The friction coefficient for values of y + τ = 30 and y + τ = 15 show the discrepancy between computed and measured results in the region of reattachment (5 ≤ x/h ≤ 15). The error is larger for the mesh with y + τ = 30, and the error becomes progressively smaller as the mesh is refined. All meshes underpredict the values of friction coefficient for 0 ≤ x/h ≤ 5. Overall, the shape of computed friction coefficient curves is similar to the experimental one for all meshes. Similarly, the comparison of computed and measured pressure coefficient in Fig. 4 is consistent for all computational meshes with underprediction of the pressure coefficient for −5 ≤ x/h ≤ 6. The pressure coefficient is overpredicted for 7 ≤ x/h ≤ 15. [28] and simulation results are represented by lines with symbols: square -y + τ = 30, triangle -y + τ = 15, diamond -y + τ = 6, and x -y + τ = 3.
Grid Refinement Level N y + τ 1 16
Velocity profiles u/U ref are in a good agreement with experimental findings for x/h = −4, x/h = 1, and x/h = 4 as shown in Fig. 6 to Fig. 9. The velocity of the first point of the lower wall for the coarsest mesh y + τ = 30 displays the discrepancy of the axial location where the velocity achieves the value of zero (no slip condition) compared to other meshes and experimental results. All other meshes predict the velocity profiles in close agreement with experiments. The difference in velocity profiles between experiments and computations become progressively larger for streamwise measurement locations of x/h = 6 and larger.
Asymmetric Plane Diffuser Study
The asymmetric flow in a plane diffuser experiment [29,33] is the recreation of the original experiment performed by Obi and Matsui [34]. The fully turbulent flow on the inlet to the diffuser causes separation and reattachment of the flow at the lower wall. The complex flow features in the plane diffuser pose a challenge for many turbulence models. The presence of the separation region is particularly attractive for testing of wall functions.
The computational domain consists of a two-dimensional hexahedral grid, shown in Fig. 10, with an upstream channel height, h, and expansion ratio of 4.7. The flow enters the domain 110 channel heights upstream and exits 55 channel heights downstream of the plane diffuser.
The boundary conditions are prescribed according to those measured experimentally [29]. The inlet flow velocity is u = 0.3 m/s with turbulent kinetic energy and specific dissipation of k = 0.2945755 m 2 /s 2 , and ω = 97.37245 1/s, Figure 5: Comparison lower wall coefficient of pressure, C p , measured experimentally -filled circles [28] and simulated -unfilled symbols detailed in figure legend.
respectively. The upper and lower walls satisfy the no-slip condition, and the outlet boundary was modeled using the zero Neumann condition.
The computational approach consisted of the simulation of a sequence of uniformly refined meshes with decreasing y + τ , as shown in Table 2. Again, the uniform grid refinement was performed to obtain a sequence of meshes with consistent cell aspect ratio and varying y + τ .
Grid Refinement Level N y + τ 1 5,000 40 2 20,000 20 3 82,000 10 4 332,000 5 Table 2: Range of mesh element count (N ), wall spacing (y + τ ), and aspect ratio (AR) used for the asymetric plane diffuser simulations. The grids were refined requiring AR consistency over the desired y + τ range. Figure 11 displays the distribution of y + τ along the lower wall for four meshes. The range of y + τ values are between 5 and 40. The comparison between computed and measured velocity profiles in the diffuser are shown in Fig. 13 where a good agreement between computed and measured results can be seen. The mesh that corresponds to the y + τ value shows the larges discrepancy between computed and measured velocity profile. Also, Fig. 13 shows that with the uniform mesh refinement the discrepancy becomes smaller, indicating the mesh convergence.
Summary and Conclusions
A new formulation of wall functions is proposed in the new set of variables that are more convenient for implementation in computational codes. The main advantage of the new formulation is that it allows the computation of the wall shear stress without the need for an iterative procedure. Moreover, the new formulation is written in an explicit form so that it can be used to generalize several proposed laws of the wall that do not include pressure gradient effects. To achieve this goal, the new near-wall variable r + was introduced through defining the normalization using the local velocity. The introduction of the normalization with respect to the local velocity is the key to recasting the existing laws of the wall in the form that yields an explicit expression for the wall shear stress. Also, it was demonstrated that the law of the wall in the new and old variables leads to the same values of the wall shear stress. Therefore, the newly proposed formulation is consistent with the classic one with the ability to produce explicit expressions for the wall shear stress. Two computational studies involving the flow over backward-facing step and the flow in the diffuser demonstrate that the new form of generalized wall functions performs well while including the gradient of pressure effects. It is also shown that the new formulation converges to physical results obtained by measurements in the grid refinement study.
Eq.(34) and Eq. (35) have the same form and they are linked through transformation of variables given by Eq.(28) and Eq.(31). Both equations can be integrated if the boundary conditions and the form of normalized turbulent viscosity are known.In the case of the laminar sublayer, turbulent fluctuations are negligible as viscous forces dominate the flow. Since ν + t 1, Eq. (34) takes the following form du + = dy + τ
Figure 1 :
1Comparison of wall functions in old and new variables: solid line labeled f (y + τ ) corresponds to wall function defined with respect to y + τ and given by Eq. (37) and Eq. (40) while dashed line labeled g(r + ) corresponds to wall function defined in terms of r + and given by Eq. (40) and Eq. (47). The viscous sublayer solution is extended to values y τ ≤ 10.8 for comparison purposes.
Figure 2 :
2of the velocity scale u c is well posed as it never becomes zero i.e., in the absence of the pressure gradient it takes the value of the friction velocity. The opposite is valid as in the absence of wall shear stress, u c takes the value of the velocity induced by the pressure gradients u p . The velocity scale u c is used to scale the linearized x-momentum equation for boundary layers ν ∂u ∂y − u v = τ w ρ + y ρ dp w dx (56) Comparison of theoretical law of the wall and Musker function. Left vertical axis and solid lines represent u + values. Solid line without symbols -u + functions, solid line with circles -Musker's function, solid line with diamonds -Eq. (30).
44 2
44.8957 ln(r + ) − 4.4958 if 24.44 ≤ r + < 378.3 2.3513 ln(r + ) − 1.2777 if 378.3 ≤ r + < 2275 2.1973 ln(r + ) − 0.0873 otherwise (73) Eq. (73) and Eq. (66) together with the definition of the wall shear stress, friction velocity, and the knowledge of the physical distance and the local velocity magnitude is sufficient to evaluate the wall shear stress. The expression for the wall shear stress in the new variable becomes
Figure 3 :
3Magnified section view of Backward facing step 65,000 cell mesh. Domain dimensions are normalized by the step height, h.
Figure 4 :
4Channel lower wall coefficient of friction (C f ) between x/h = −5 and x/h = 35. Solid circles correspond to the experimental measurements of Driver and Seegmiller
Figure 6 :
6Stream-wise velocity profiles at the location x/h = −4 with respect to wall distance, y, normalized by step height, h. The velocity profile is normalized the reference velocity, U ref , defined in Ref[28] for each simulation.
Figure 9 :
9Stream-wise velocity profiles at the location x/h = 6 with respect to wall distance, y, normalized by step height, h. The velocity profile is normalized the reference velocity, U ref , defined in Ref[28] for each simulation.
Figure 10 :
10Plane diffuser 20,000 cell computational grid structure.[29] C U Buice and J K Eaton. Experimental investigation of flow through and asymmetric plane diffuser. Journal of Fluids Engineering, 122:433-435, 2000.
Figure 11 :
11Lower wall y + τ variations for each grid resolution.
Figure 12 :
12Coefficient of pressure along the asymmetric plane diffuser lower wall.
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Verification and validation of the caelus library: Incompressible turbulence models. D W Stephens, C Jemcov, Sideroff, Proceeding of the ASME Fluids Engineering Division Summer Meeting. eeding of the ASME Fluids Engineering Division Summer Meeting1D W Stephens, A Jemcov, and C Sideroff. Verification and validation of the caelus library: Incompressible turbulence models. In Proceeding of the ASME Fluids Engineering Division Summer Meeting, volume 1. ASME, 2017.
Verification and validation of the caelus library: Incompressible flow solvers. A Jemcov, C Stephens, Sideroff, Proceeding of the ASME Fluids Engineering Division Summer Meeting. eeding of the ASME Fluids Engineering Division Summer Meeting1A Jemcov, D W Stephens, and C Sideroff. Verification and validation of the caelus library: Incompressible flow solvers. In Proceeding of the ASME Fluids Engineering Division Summer Meeting, volume 1. ASME, 2017.
. J H Ferziger, M Perićc, Computational Methods or Fluid Dynamics. Springer-Verlag3 editionJ H Ferziger and M Perićc. Computational Methods or Fluid Dynamics, chapter 8. Springer-Verlag, Springer- Verlag Berlin Heidelberg, 3 edition, 2002.
M A D D Apsley, Leschziner, Advanced separated flow modelling in a diffuser. Flow, Turbulence and Combustion. 63D D Apsley and M A Leschziner. Advanced separated flow modelling in a diffuser. Flow, Turbulence and Combustion, 63:81-112, 1999.
Experimental and computational study of turbulent separating flow in an unsymmetric diffuser. S S A Obi, Masuda, Ninth Symposium on Turbulent Shear Flows. Kyoto, Japan305S A Obi and S Masuda. Experimental and computational study of turbulent separating flow in an unsymmetric diffuser. In Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, August 16 -19, page 305, 1993.
|
[] |
[
"SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES",
"SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES",
"SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES",
"SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES"
] |
[
"Ruben A Hidalgo ",
"Ruben A Hidalgo "
] |
[] |
[] |
An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the sum of the cardinalities of the connected components of the locus of fixed points of two or three different symmetries of a handlebody.2010 Mathematics Subject Classification. 30F10, 30F40.
| null |
[
"https://arxiv.org/pdf/1710.07520v3.pdf"
] | 227,240,567 |
1710.07520
|
0a97fa18a227379d2aefb35ff3b77be91ac34456
|
SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES
30 Nov 2020
Ruben A Hidalgo
SHARP UPPER BOUNDS FOR THE NUMBER OF FIXED POINTS COMPONENTS OF TWO AND THREE SYMMETRIES OF HANDLEBODIES
30 Nov 2020arXiv:1710.07520v3 [math.GT]
An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the sum of the cardinalities of the connected components of the locus of fixed points of two or three different symmetries of a handlebody.2010 Mathematics Subject Classification. 30F10, 30F40.
Introduction
A uniformization of a closed Riemann surface S is a triple (∆, Γ, P : ∆ → S ) where Γ is a Kleinian group, ∆ is a Γ-invariant connected component of its region of discontinuity and P : ∆ → S is a regular covering map with Γ = Deck(P) (in the classical terminology of Kleinian groups the pair (Γ, ∆) is called a function group; if ∆ is simply connected, it is a B-group). The collection of uniformizations of S is partially ordered where the highest ones are produced when ∆ is simply-connected (for instance, if S has genus at least two, ∆ can be assumed to be the hyperbolic plane and these are the Fuchsian uniformizations). Koebe's retrosection theorem [1,10] asserts that S can be uniformized by a Schottky group and, in this case, these provide the Schottky uniformizations. It is known that Schottky uniformizations provide the lowest ones and have not been exploited in this context as for the case of Fuchsian ones.
The Schottky space S g , consisting of the PSL 2 (C)-conjugacy classes of Schottky groups of rank g ≥ 1, is a complex manifold. If g = 1 it is isomorphic to the unit punctured disc and, for g ≥ 2, it has dimension 3(g − 1) [1,22]. On S g there is a natural real structure, this induced by the complex conjugation map. The fixed points of such a real structure can be identified with the PSL 2 (C)-conjugacy classes of those extended Kleinian groups (discrete groups of conformal and anticonformal automorphisms of the Riemann sphere C, necessarily containing anticonformal ones) admitting a Schottky group of rank g as a normal subgroup of finite index. Examples of these real points are given by the extended Schottky groups (extended Kleinian group whose orientation-preserving half is a Schottky group). The structural description of these groups, in terms of the Klein-Maskit combination theorems [13,14], was obtained in [3] (we recall it in Section 3.5).
It is well known that a closed Riemann surface S can be described by irreducible complex projective algebraic curves (this is a consequence of Riemann-Roch's theorem in one direction and the implicit function theorem in the other one). Weil's descent theorem [25] asserts that such a curve can be chosen to be defined over the reals if and only if S admits a symmetry (an anticonformal involution); in this case S is called symmetric. As a consequence of the known topological actions of symmetries, if τ : S → S is a symmetry, then there is a Schottky uniformization (∆, Γ, P : ∆ → S ) so that τ lifts, that is, there is an anticonformal automorphism σ : ∆ → ∆ so that τP = P σ. The automorphism σ is the restriction of an extended Möbius transformation satisfying σ 2 ∈ Γ and σΓσ −1 = Γ. The group K = Γ, σ is an example of an extended Schottky group whose orientationpreserving half is the Schottky group Γ. As a consequence, each symmetry on a closed Riemann surface can be realized by an extended Schottky group.
An extended Kleinian group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. In the above context of symmetries, the dihedral extended Schottky group induces two different symmetries on the surface uniformized by the Schottky group. In [7] there is provided an example of a Riemann surface admitting two different symmetries both of which cannot be realized by a dihedral extended Schottky group.
The structural description of those dihedral extended Schottky groups containing no reflections was obtained in [7]. In this paper we complete such a description to include the possibility of reflections in the dihedral extended Schottky groups (Theorem 4.3). As an application, we obtain sharp upper bounds on the sum of connected components of fixed points of orientation reversiong involutions on handlebodies of genus at least two (Theorem 2.1). In a forthcomming article we plan to use this structural description to study the connectivity of the real locus of Schottky space.
Symmetries on handlebodies
Let Γ be a Schottky group of rank g, with region of discontinuity Ω, and set S Γ = Ω/Γ (a closed Riemann surface of genus g). If H 3 denotes the hyperbolic 3-space, then M Γ = (H 3 ∪ Ω)/Γ is a handlebody of genus g (we say that Γ induces a Schottky structure on the handlebody). Its interior M 0 Γ = H 3 /Γ carries a natural complete hyperbolic structure and S Γ is its conformal boundary.
Let τ : M Γ → M Γ be a symmetry, that is, an order two orientation-reversing homeomorphism whose restriction to M 0 Γ is a hyperbolic isometry. The symmetry τ induces by restriction a symmetry of S Γ . By lifting τ to the universal cover, we obtain an extended Schottky group K τ whose orientation-preserving half is Γ. As a consequence of the geometrical structure of extended Schottky groups [3], the locus of fixed points of τ has at most g + 1 connected components (τ is called maximal if it has g + 1 connected components of fixed points) and each of such connected components is either (i) an isolated point in M 0 Γ or (ii) a 2-dimensional bordered compact surface (which may or not be orientable) whose border is contained in its conformal boundary S Γ (see also [9]). The projection of an isolated fixed point of τ produces a point in the orbifold M 0 Γ / τ = H 3 /K τ admitting a neighborhood which locally looks like a cone over the projective plane.
If τ is another symmetry of M Γ , then (again by the lifting process as above) we obtain another extended Schottky group K τ with Γ as its orientation-preserving half. In this way, the group K, generated by K τ and K τ , is a dihedral extended Schottky group. As a consequence of our structural description we obtain sharp upper bounds. In the following D r denotes the dihedral group of order 2r.
Theorem 2.1. Let M be a handlebody of genus g ≥ 2, with a given Schottky structure.
(1) If τ 1 and τ 2 are two different symmetries of M, q ≥ 2 is the order of τ 1 τ 2 and m j is the number of connected components of fixed points of τ j , then
m 1 + m 2 ≤ 2 g − 1 q + 4.
Moreover, for every integer q ≥ 2, the above upper bound is sharp for infinitely many values of g. (2) If τ 1 , τ 2 and τ 3 are three different symmetries, H = τ 1 , τ 2 , τ 3 and m j is the number of connected components of fixed points of τ j , then
m 1 + m 2 + m 3 ≤ 5 if g = 2. 8 if g = 3. g + 5
if g ≥ 4 and H Z 2 × D r for any r. (r + 1)g + 5r − 1 r if g ≥ 4 and H Z 2 × D r for some r.
Moreover, the above upper bounds are sharp for g = 2, 3 and for infinite many values of g ≥ 4.
The above upper bounds are obtained in Section 5. In Section 6, we construct explicit examples to see that they are sharp. Theorem 2.1 asserts the following fact (already observed in [7] if both symmetries only have isolated fixed points).
Corollary 2.2. Let τ 1 and τ 2 be two different symmetries of a handlebody of genus g ≥ 2, with a Schottky structure, such that τ j has m j connected components of fixed points. Then m 1 + m 2 ≤ g + 3. In particular, (i) a handlebody of genus g ≥ 2 admits at most one maximal symmetry and (ii) the upper bound m 1 + m 2 = g + 3 only occurs if q = 2, that is, when
τ 1 , τ 2 = Z 2 2 .
Remark 2.3 (Connection to symmetries of Riemann surfaces). Let S be a closed Riemann surface of genus g. (1) If τ is a symmetry of S , then each connected component of fixed points of τ is a simple loop, called an oval and, by Harnack's theorem [4], the total number of ovals is at most (g + 1). (2) If τ 1 , τ 2 are two different symmetries of S , then in [2] it was observed that the total number of ovals of these two symmetries is bounded above by 2(g − 1)/q + 4 (if q is odd) and 4g/q + 2 (if q is even), where q is the order of the product of them. If moreover, q ≥ 3 and q does not divides g − 1, then the sharp upper bound is [2(g − 1)/q] + 3 (where [ ] stands for the integer part) [11]. We may observe that the upper bounds given in Theorem 2.1 are different from the ones obtained for the case of Riemann surfaces. This difference comes from the fact that there Riemann surfaces S admitting two different symmetries which cannot be realized by a common Schottky group uniformizing S [7]. The number of ovals of two symmetries is at most 2g + 2, in particular, there must be the possibility for S to admit two different maximal symmetries.
In [23], Natanzon proved that if a Riemann surface admits two maximal symmetries, then it is necessarily hyperelliptic. This is again a different situation as that for handlebodies (in a handlebody there is at most one maximal symmetry). (3) In [24] there were obtained a sharp upper bound for the number of ovals for three different symmetries of S , this being 2(g + 2). The above bounds are mainly a consequence of the well known structure of non-Euclidian crystallographic (NEC) groups [12]. This upper bound is again different from that of Theorem 2.1.
Preliminaries and previous results
In this section we briefly review several definitions and basic facts we will need in the rest of the paper. More details on these topics may be found, for instance, in [15,20].
3.1. Extended Kleinian groups. We denote by M the group of Möbius and extended Möbius transformations (the composition of a Möbius transformation with the complex conjugation) and by M its index two subgroup of Möbius transformations. The group M can also be viewed, by the Poincaré extension theorem, as the group of hyperbolic isometries of the hyperbolic space H 3 ; in this case, M is the group of orientation-preserving ones. Möbius transformations are classified into parabolic, loxodromic (including hyperbolic) and elliptic. Similarly, extended Möbius transformations are classified into pseudoparabolic (the square is parabolic), glide-reflections (the square is hyperbolic), pseudoelliptic (the square is elliptic), reflections (of order two admitting a circle of fixed points on C) and imaginary reflections (of order two and having no fixed points on C). Each imaginary reflection has exactly one fixed point in H 3 and this point determines such reflection uniquely. If K is a subgroup of M not contained in M, then K + = K ∩ M is its canonical orientation-preserving subgroup. A Kleinian group is a discrete subgroup of M and an extended Kleinian group is a discrete subgroup of M necessarily containing extended Möbius transformations. If K is a (extended) Kleinian group, then its region of discontinuity is the subset Ω of C composed by the points on which it acts discontinuously. Note that K is an extended Kleinian groups if and only if K + is a Kleinian group; both of them with the same region of discontinuity.
3.2.
Klein-Maskit's combination theorems. Theorem 3.1 (Klein-Maskit's combination theorem [13,14]).
(1) (Free products) Let K j be a (extended) Kleinian group with region of discontinuity Ω j , for j = 1, 2. Let F j be a fundamental domain for K j and assume that there is simple closed loop Σ, contained in the interior of F 1 ∩ F 2 , bounding two discs ∆ 1 and ∆ 2 , so that, for j ∈ {1, 2}, Σ ∪ ∆ j ⊂ Ω 3− j is precisely invariant under the identity in K 3− j . Then (i) K = K 1 , K 2 is a (extended) Kleinian group with fundamental domain F 1 ∩ F 2 and K is the free product of K 1 and K 2 (ii) every finite order element in K is conjugated in K to a finite order element of either K 1 or K 2 and (iii) if both K 1 and K 2 are geometrically finite, then K is so.
(2) (HNN-extensions) Let K 0 be a (extended) Kleinian group with region of discontinuity Ω, and let F be a fundamental domain for K 0 . Assume that there are two pairwise disjoint simple closed loops Σ 1 and Σ 2 , both of them contained in the interior of F 0 , so that Σ j bounds a disc ∆ j such that (Σ 1 ∪ ∆ 1 ) ∩ (Σ 2 ∪ ∆ 2 ) = ∅ and that Σ j ∪ ∆ j ⊂ Ω is precisely invariant under the identity in K 0 . Let T be either a loxodromic transformation or a glidereflection so that T (Σ 1 ) = Σ 2 and T (∆ 1 ) ∩ ∆ 2 = ∅. Then (i) K = K 0 , f is a (extended) Kleinian group with fundamental domain F 1 ∩ (∆ 1 ∪ ∆ 2 ) c and K is the HNN-extension of K 0 by the cyclic group T , (ii) every finite order element of K is conjugated in K to a finite order element of K 0 and (iii) if K 0 is geometrically finite, then K is so.
3.3. Kleinian 3-manifolds and their automorphisms. If K is a Kleinian group and Ω is its region of discontinuity, then assocated to K is a 3-dimensional orientable orbifold M K = (H 3 ∪ Ω)/K; its interior M 0 K = H 3 /K has a hyperbolic structure and its conformal boundary S K = Ω/K has a natural conformal structure. If K is torsion free, then M K and M 0 K are orientable 3-manifolds and S K is a Riemann surface; we say that M K is a Kleinian 3-manifold and that M K and S K are uniformized by K. Now, if K is an extended Kleinian group, then the 3-orbifold M K + admits the orientation-reversing homeomorphism τ : M K + → M K + of order two induced by K − K + and M K + / τ = (H 3 ∪ Ω)/K. Let Γ be a torsion free Kleinian group, so M Γ = (H 3 ∪Ω)/Γ is a Kleinian 3-manifold. An automorphism of M Γ is a self-homeomorphism whose restriction to its interior M 0 Γ is a hyperbolic isometry. An orientation-preserving automorphism is called a conformal automorphim and an orientation-reversing one an anticonformal automorphism. A symmetry of M Γ is an anticonformal involution. We denote by Aut(M Γ ) the group of automorphisms of M Γ and by Aut + (M Γ ) the subgroup of conformal automorphisms. Let π 0 : H 3 → M 0 Γ be the universal covering induced by Γ. Clearly, π 0 extends to a universal covering π : H 3 ∪ Ω → M Γ with Γ as the group of Deck transformations. If H ⊂ Aut(M Γ ) is a finite group and we lift it to the universal covering space H 3 under π 0 , then we obtain an (extended) Kleinian group K containing Γ as a normal subgroup of finite index such that H = K/Γ. The group H contains orientation-reversing automorphisms if and only if K is an extended Kleinian group.
3.4. Schottky groups. The Schottky group of rank 0 is just the trivial group. A Schottky group of rank g ≥ 1 is a Kleinian group Γ generated by loxodromic transformations A 1 , . . . , A g , so that there are 2g disjoint simple loops,
C 1 , C ′ 1 , . . . , C g , C ′ g , with a 2g- connected outside D ⊂ C, where A i (C i ) = C ′ i , and A i (D) ∩ D = ∅, for i = 1, . . . , g.
The region of discontinuity Ω of Γ is known to be connected and dense in C, S Γ is a closed Riemann surface of genus g and M Γ is a handlebody of genus g. In this case, M 0 Γ carries a geometrically finite complete hyperbolic Riemannian metric with injectivity radius bounded away from zero. If g ≥ 2, then Aut(M Γ ) has order at most 24(g − 1) and Aut + (M Γ ) has order at most 12(g − 1) [26,27]. Each conformal (respectively anticonformal) automorphism of M Γ induces a conformal (respectively anticonformal) automorphism of the conformal boundary S Γ and the later determines the former due to the Poincare extension theorem. As a consequence of Koebe's retrosection theorem [1,10], every closed Riemann surface is isomorphic to S Γ for a suitable Schottky group Γ.
A Schottky group of rank g can be defined as a purely loxodromic Kleinian group of the second kind which is isomorphic to a free of rank g [16] and it can also be defined as a purely loxodromic geometrically finite Kleinian group which is isomorphic to a free of rank g (essentially a consequence of the fact that a free group cannot be the fundamental group of a closed hyperbolic 3-manifold). It follows that every Kleinian structure on a handlebody is provided by a Schottky group, we called it a Schottky structure. The geometrically finite hyperbolic structures on the interior of a handlebody, with injectivity radius bounded away from zero, are provided by Schottky groups.
Let M be a topological handlebody of genus g and let H be a finite group of homeomorphisms of M. It is known that there are: (i) a (extended) Kleinian group K, containing as a finite index normal subgroup a Schottky group Γ of rank g, and (ii) an orientationpreserving homeomorphism f : M → M Γ , with f H f −1 = K/Γ. This is consequence of the fact that a handlebody is a compression body (see also [27]).
3.5. Extended Schottky groups. An extended Schottky group of rank g is an extended Kleinian group whose orientation-preserving half is a Schottky group of rank g. In the case that it does not contains reflections (Klein-Schottky groups) a geometrical structure description was provided in [7]. A geometric structural description of all extended Schottky groups, in terms of the Klein-Maskit combination thorems, is as follows.
Theorem 3.2 ([3]). An extended Schottky group is the free product (in the Klein-Maskit combination theorem sense) of the following kind of groups:
(i) cyclic groups generated by reflections, (ii) cyclic groups generated by imaginary reflections, (iii) cyclic groups generated by glide-reflections, (iv) cyclic groups generated by loxodromic transformations, and (v) real Schottky groups (that is groups generated by a reflection and a Schottky group keeping invariant the corresponding circle of its fixed points).
Conversely, a subgroup of M constructed using α groups of type (i), β groups of type (ii), γ groups of type (iii), δ groups of type (iv) and ε groups of type (v), is an extended Schottky group if and only if α + β + γ + ε > 0. If, in addition, the ǫ real Schottky groups above have the ranks r 1 , . . . , r ε ≥ 1, then it has rank g = α+β+2(γ+δ)+ε−1+r 1 +. . .+r ε . Corollary 3.3. Let K be an extended Schottky group constructed, as in Theorem 3.2, using α groups of type (i), β groups of type (ii), γ groups of type (iii), δ groups of type (iv) and ε groups of type (v). If Γ is its orientation-preserving half, then K induces a symmetry of M Γ whose connected components of fixed points consist of α two dimensional closed discs, β isolated points, and ε two dimensional non-simply connected compact surfaces. In particular, If τ is a symmetry of a Kleinian manifold homeomorphic to a handlebody of genus g, and n 0 is the number of isolated fixed points of τ, n 1 is the number of total ovals in the conformal boundary and n 2 is the number of two-dimensional connected components of the set of fixed points of τ, then n 0 + n 1 , n 0 + n 2 ∈ {0, 1, . . . , g + 1}.
3.6. A lifting criteria. We recall a simple criterion for lifting loops which we will need in Section 4. This is a direct consequence of the Equivariant Loop Theorem [21], whose proof is based on minimal surfaces, that is, surfaces that minimize locally the area. In [6] there is provided a proof whose arguments is proper to (planar) Kleinian groups. A function group is a pair (K, ∆), where K is a finitely generated Kleinian group and ∆ is a K-invariant connected component of its region of discontinuity.
Theorem 3.4. [6,21] Let (K, ∆) be a torsion free function group such that S = ∆/K is a closed Riemann surface of genus g ≥ 2. Let P : ∆ → S a regular covering with K as its deck group. If H is a group of automorphism of S , then it lifts to the above regular planar covering if and only if there is a collection F of pairwise disjoint simple loops on S such that:
(i) F defines the regular planar covering P : ∆ → S ; and (ii) F is invariant under the action of H.
3.7.
A counting formula. Let K be an extended Kleinian group containing a Schottky group Γ of rank g as a normal subgroup of finite index (the last trivially holds if g ≥ 1). Let us denote by θ : K → H = K/Γ, the canonical projection. If τ ∈ H is an involution which is the θ-image of an extended Möbius transformation, then Γ = θ −1 ( τ ) is an extended Schottky group whose orientation-preserving hals is Γ. By Theorem 3.2, Γ is constructed using α reflections, β imaginary reflections and ε real Schottky groups. These values can be computed from Θ. Before, we provide some necessary definitions.
A complete set of symmetries of K is a maximal collection C = {c i ∈ K : i ∈ I} of anticonformal involutions (i.e. reflections and imaginary reflections) which are non-conjugate in K( we shall refer to its elements as to canonical symmetries). As K is geometrically finite (as it is a finite extension of a Schottky group), C is finite.
For each i ∈ I we set I(i) ⊂ I defined by those j ∈ I so that θ(c i ) and θ(c j ) are conjugate in H (in particular, i ∈ I(i)). Note that it may happen that for j ∈ I(i), c j can be imaginary reflection even if c i is a reflection and viceversa (this occurs when θ(c i ) is a symmetry of M Γ whose locus of fixed points has isolated points and also two-dimensional components). We set by J(i) the subset of I(i) defined by those j for which c j is an imaginary reflection. We also set by F(i) ⊂ I(i) \ J(i) for those j for which c j has finite centralizer in K and set
E(i) = I(i) \ (J(i) ∪ F(i)).
As Γ has finite index in K, a reflection c ∈ K has an infinite centralizer C(K, c) in K if and only if it has an infinite centralizer in Γ.
2, where α = j∈F(i) [C(H, θ(c j )) : θ(C(K, c j ))], β = j∈J(i) [C(H, θ(c j )) : θ(C(K, c j ))] and ε = j∈E(i) [C(H, θ(c j )) : θ(C(K, c j ))].
Structural picture of dihedral extended Schottky groups
In this section we provide an structural picture of the dihedral extended Schottky groups, in terms of the Klein-Maskit combination theorems, extending the one obtained in [7]. We first start with a simple criteria to determine when an extended Kleinian group is a dihedral extended Schottky group and then we describe some basic examples of these type of groups which will be the main actors in the structural description.
4.1.
A simple criteria. Let K be a dihedral extended Schottky group, say generated by the extended Schottky groups K 1 and K 2 with the same orientation-preserving half given by the Schottky group Γ. If Ω is the region of discontinuity of K (which is the same for K j and Γ), then K j induces a symmetry τ j on the Riemann surface S = Ω/Γ. If p > 1 is the order of τ 1 τ 2 , then there exists a surjective homomorphism Φ : K → D p = τ 1 , τ 2 such that K j = Φ −1 ( τ j ), ker(Φ) = Γ and Φ(K + ) = τ 1 τ 2 . The converse is provided by the following.
Proposition 4.1. An extended Kleinian group K is a dihedral extended Schottky group if and only there is a surjective homomorphism ϕ : K → D p = a, b : a 2 = b 2 = (ab) p = 1 (for some positive integer p > 1) whose kernel is a Schottky group Γ and ϕ(K + ) = ab .
Proof. One direction was already noted above. For the other, as ϕ(K + ) = ab , then K 1 = ϕ −1 ( a ) and K 2 = ϕ −1 ( b ) are extended Kleinian groups. As K j contains Γ as an index two subgroup, its is an extended Schottky group (where Γ is its index two preserving half). Since K is generated by K 1 and K 2 , it is a dihedral extended Schottky group.
Remark 4.2. If K is a dihedral extended Schottky group, then its limit set is totally disconnected (as K contains a Schottky group of finite index) and contains no parabolic transformations. As a consequence of the classification of funtion groups [17,18,19], every finitely generated Kleinian group, without parabolic nor elliptic elements and with totally disconnected limit set, is a Schottky group.
4.2.
Examples: Basic dihedral extended Schottky groups.
4.2.1. Let K be constructed (see Figure 1), by the Klein-Maskit combination theorem, as a free product of α cyclic groups generated by a reflection, β cyclic groups generated by an imaginary reflections, γ cyclic groups generated by a loxodromic transformation, δ cyclic groups generated by a glide-reflection, ρ cyclic groups generated by an elliptic transformation of finite order and η groups generated by an elliptic transformation of finite order and a loxodromic transformation commuting with the elliptic one and κ groups generated by an elliptic transformation of finite order and a glide-reflection transformation both sharing their fixed points (so the glide reflection conjugates the elliptic into its inverse).
Let Ω be the region of discontinuity of K. Then (i) Ω/K is the connected sum of β + 2γ + 2δ + 2κ + 2η real projective planes (if β + δ + κ > 0) or a genus γ + η orientable surface (if β = δ = κ = 0), with exactly α boundary components and with 2ρ cone points in the interior (in pairs of the same cone order); and (ii) Ω/K + is a closed Riemann surface of genus g = α + β + 2(γ + δ + κ + η) − 1 with 4ρ cone points (in cuadruples with the same cone order). As a consequence of Proposition 4.1, K will be a dihedral extended Schottky group if and only if either (i) α + β + δ + κ ≥ 2 or (ii) α + β + δ + κ = 1 and γ + η ≥ 1. In this case, that proposition, one may take p as the lest common multiple of the orders of the ρ + η + κ elliptic generators. The γ +η loxodromic transformations are send to the identity or powers of ab, the reflections, imaginary reflections and glide-reflections are sent to either a or b and the elliptic elements are sent to powers of ab (preserving the order).
Basic extended dihedral groups.
Let C ⊂ C be a circle and let σ be the reflection on it. Consider a (finite) non-empty collection of parwise disjoint closed discs ∆ j , each one with boundary circle Γ j being orthogonal to C. We denote by Int(∆ j ) and Ext(∆ j ) the interior and exterior, respectively, of ∆ j .
(1) Take 2r of these circles (it could be r = 0), say Γ 1 , · · · , Γ 2r , and loxodromic transformation L 1 , . . . , L r such that L j (Σ j ) = Σ r+ j , L j (Int(∆ j )) ∩ Int(∆ j+r ) = ∅ and L j commuting with σ. (The group generated by σ and L 1 , . . . , L r is a real Schottky group). (2) Now, for some each others Γ i , we consider an elliptic transformation
E i , with both fixed point on C, such that E i (Ext(∆ i )) ⊂ Int(∆ i ) and σE i σ = E −1 i (each E i σ is a reflection).
(3) For others Γ k , we consider an elliptic transformation F k , with both fixed point on C, such that F k (Ext(∆ k )) ⊂ Int(∆ k ) and a loxodromic transformation M k , such that
M k F k = F k M k , M k σ = σM k and σF k σ = F −1 k (both fixed points of F k are also the fixed points of M k ).
(4) For others Γ s we take an elliptic transformation of order two D s (whose fixed points are not on C) keeping invariant Γ s (so permuting both discs bounded by it) and commuting with σ. (5) Finally, for each of the rest of the circles Σ l we consider the reflection τ l on it (so it commutes with σ). The group K generated by σ and all of the above transformations is an extended Kleinian group (by the Klein-Maskit combination theorem) called a basic extended dihedral group (see Figure 2).
If Ω is the region of discontinuity of K, then S = Ω/K is a bordered (orbifold) Klein surface, which is orientable if and only if the loxodromic transformations L k keep invariant each of the two discs bounded by C. Each elliptic element E i produces two cone points on the border, both with cone order the order of E i . Each D s produces a cone point of order two in the interior. Each reflection in (5) produces two cone points of order two in the boundary. In particular, Ω/K + provides of a compact orientable orbifold with some even number of cone points admitting a symmetry permuting them.
As a consequence of Proposition 4.1, K is a dihedral extended Schottky group. In this case, we take p as the least common multiple of the orders of all elliptic transformations E i , F k and D s . We send the loxodromic transformations L j and M k to the identity, the reflection σ to a and the reflections τ l to b (for instance). (iv) δ cyclic groups generated by glide-reflections;
(v) ρ cyclic groups generated by an elliptic transformation of finite order; (vi) η groups generated by an elliptic transformation of finite order and a loxodromic transformation commuting with the elliptic one; (vii) κ groups generated by an elliptic transformation of finite order and a glide-reflection transformation both sharing their fixed points (so the glide reflection conjugates the elliptic into its inverse); (viii) ε basic extended dihedral groups K 1 , . . . , K ε ; such that α + β + δ + κ + ε > 0.
(2) An extended function group constructed as in (1), is a dihedral extended Schottky group if and only if either: (i) α + β + δ + κ + ε ≥ 2 or (ii) α + β + δ + κ + ε = 1 and γ + ρ + η > 0. (1) The condition α + β + δ + κ + ε > 0 in part (1) of Theorem 4.3 is needed for the constructed group to have orientatation-reversing elements.
(2) Let K be an extended Kleinian group, constructed as a free product (in the sense of the Klein-Maskit combination theorem) using α groups of type (i), β groups of type (ii) γ groups of type (iii), δ groups of type (iv), ρ groups of type (say of orders l 1 , . . . , l ρ ) (v), η groups of type (vi), κ groups of type (vii) and ε basic extended dihedral groups K 1 , . . . , K ε . Assume that the corresponding orbifold Ω i /K + i has genus g i and 2r i cone points of orders t i1 , t 1i , . . . , t ir i , t ir i ≥ 3 and 2n i cone points of order two, where Ω i is the region of discontinuity of K i . If |t| denotes the order of the transformation t, then (a) the region of discontinuity Ω of K is connected and the orbifold Ω/K + has genus g = α + β + 2(γ + δ + κ + η) + ε + g 1 + · · · + g ε − 1, has 4ρ conical points of orders l 1 , l 1 , l 1 , l 1 , . . . , l ρ , l ρ , l ρ , l ρ , 2(r 1 + · · · + r ε ) conical points, of orders |t 11 |, |t 11 |, |t 12 |, |t 12 |, . . . , |t εr ε |, |t εr ε | and 2(n 1 + · · · + n ε ) conical points of order 2; and (b) if K is a dihedral extended Schottky group containing a Schottky group Γ of rank g as a normal subgroup such that K/Γ D p , then (by the Riemann-Hurwitz formula)
g = p( g − 1) + 1 + 2p ρ i=1 1 − 1 l i + p ε i=1 r i k=1 1 − 1 |t ik | + p 2 ε i=1 n i .
4.4.
Proof of Theorem 4.3. Part (2) follows from Proposition 4.1. We proceed to prove part (1).
4.4.1. Let K be a dihedral extended Schottky group, generated by two different extended Schottky groups G 1 and G 2 , such that G + 1 = G + 2 = Γ (a Schottky group of rank g) and let G + be its index two orientation-preserving half. All the groups Γ, G 1 , G 2 , K and K + have the same region of discontinuity Ω. Set S + = Ω/Γ, S 1 = Ω/G 1 , S 2 = Ω/G 2 , S + = Ω/K + and S = Ω/K. Proof. As [K + : Γ] < ∞ and Γ contains no parabolic elements, the same holds for K + . As K + is a geometrically finite function group, the result follows from [5].
As K is a finite extension of Γ, and S + is a closed Riemann surface, (i) S is a compact not necessarily orientable orbifold, with a finite number of orbifold points and possible non-empty boundary, and (ii) S + is a closed Riemann surface with some finite number of orbifold points.
Since
Γ has index 2 in G i , for i = 1, 2, if η ∈ G i \ Γ, then η 2
i ∈ G and either η is: (i) a reflection or (ii) an imaginary reflection or (iii) glide-reflection. In particular, G i induces a symmetry τ i on S + with S i = S + / τ i and every orientation-reversing element of K either acts without fixed points on Ω or they are reflections.
The group J = τ 1 , τ 2 is isomorphic to the dihedral group D p , where p ≥ 2 is the order of τ 1 τ 2 . It follows that S + = S + / τ 1 τ 2 , S = S + /J and that every orientation-reversing element of J is conjugate in J to either τ 1 or τ 2 .
On S + we have an anticonformal involution induced by J, preserving the finite set of orbifold points, so that quotient of S + by it is S .
Let us denote by Φ : K → J = τ 1 , τ 2 the canonical surjective homomorphism with kernel Γ which is defined by sending the elements of G i \ Γ to the symmetry τ i . It follows that Φ −1 ( τ i ) = G i . Remark 4.6. If p is odd, then τ 1 and τ 2 are conjugated in J and, in particular, G 1 and G 2 are conjugated in K.
Proposition 4.7. Every orientation-reversing element of K is either glide-reflection or an imaginary reflection or a reflection.
Proof. An orientation-reversing element a ∈ K is a lift of a conjugate of either τ 1 or τ 2 , each one a symmetry of S + , so a 2 ∈ Γ. In this way, either (i) a 2 = 1, in which case a is either a reflection or an imaginary reflection, or (ii) a 2 is a loxodromic transformation, in which case it is a glide-reflection.
Structural loops and regions.
Let us consider the regular planar Schottky covering P : Ω → S + . As the group J lifts under P (such a lifting is the group K), Theorem 3.4 asserts the existence of a J-invariant collection of pairwise disjoint loops F in S + which divides S + into genus zero surfaces (since we are dealing with a Schottky covering) and each loop lifting to a simple loop on Ω. If A is a connected component of S \ F and J A is its J-stabilizer, then (as J is a dihedral group) the subgroup J A is either: (i) trivial; or (ii) a cyclic conformal group generated by a power of τ 1 τ 2 ; or (iii) a cyclic group of order two generated by a symmetry (which is conjugated to either τ 1 or τ 2 ); or (iv) a dihedral subgroup of J.
In either case (iii) or (iv) we have that A is invariant under symmetry τ ∈ J. If τ is either (a) a reflection containing a loop of fixed points in A or (b) an imaginary reflection, then we may find a simple loop β ⊂ A which is invariant under τ; moreover, if τ is reflection, then β is formed of only fixed points of it. We may add such a loop and its J-translated to F without destroying the conditions of Theorem 3.4. In this way, we may also assume that F satisfies the following extra property:
(v) Every symmetry in J A is necessarily a reflection whose circle of fixed points is not completely contained in A (it intersects some boundary loops).
Let us observe that we may assume F to be minimal in the sense that there is no proper subcollection of it satisfying (i)-(v) above. The loops in F are called the base structure loops and the connected components of S + \ F are called the base structure regions. Both collections are J-invariants.
Let G be the collection of loops on Ω obtained by the lifting of those in F ; these are called the structure loops. The connected components of Ω \ G will be called the structure regions. Both collections, G and the set of structure regions, are K-invariant.
Stabilizers of structure regions.
Let R be a structure region and K R be its K-stabilizer. As P : R → P(R) = A is a homeomorphism, K R is isomorphic to J A . As a consequence we obtain the following fact.
Proposition 4.8. If R is a structure region, then K R is either:
(i) trivial;
(ii) a finite cyclic group generated by an elliptic transformation; (iii) a cyclic group of order two generated by a reflection whose circle of fixed points is not completely contained on R; (iv) a dihedral group generated two reflections, each one with its circle of fixed points intersecting some boundary loop of R. Proposition 4.9. If R is a structure region with K R = {I}, then the restriction to R of the projection map from Ω to S = S + /J = Ω/K is a homeomorphism onto its image.
Proof. If k ∈ K, then either k ∈ K R = {I}, in which case, k(R) = R, or k K R , in which case, k(R) ∩ R = ∅.
4.4.5.
Stabilizers of structure loops. If β ∈ G, then we denote its K-stabilizer as K β . As J + = τ 1 τ 2 (a cyclic group), the orientation-preserving half of K β is either trivial or a finite cyclic group generated by some elliptic element.
By Proposition 4.7, an orientation-reversing transformation inside K is either an imaginary reflection or a reflection or a glide-reflection transformation. As the structure loop β is contained in Ω, a glide-reflection cannot belong to K β . In particular, the only orientationreversing transformations in K that keep invariant some structure loop can be either an imaginary reflection or a reflection. Also, K β cannot contain two different imaginary reflections, for the product of two distinct imaginary reflections is always hyperbolic.
The structure loop β can be stabilized by a reflection in two different manners. One is that it fixes it point-wise, that is, β is the circle of fixed points of the reflection. The second one is that β is not point-wise fixed by the reflection, in which case, there are exactly two fixed points of the reflection on it. These two points divide the loop into two arcs which are permuted by the reflection.
Summarizing all the the above is the following.
Proposition 4.10. If β ∈ G, then K β is either:
(i) trivial;
(ii) a cyclic group generated by an elliptic element of finite order whose fixed points are separated by β; (iii) a cyclic group generated by an elliptic element of order two with its fixed points on β; (iv) a cyclic group generated by an imaginary reflection;
(v) a cyclic group generated by an reflection with β as its circle of fixed points; (vi) a cyclic group generated by a reflection and β containing exactly two fixed points of it; (vii) a group generated by a reflection, with exactly two fixed points on β, and an elliptic involution with these two points as its fixed points. In this case, the composition of these two is a reflection with β as circle of fixed points and K β is isomorphic to Z 2 2 ; (viii) a group generated by an elliptic involution, whose fixed points are separated by β, and a reflection with β as its circle of fixed points. In this case, the composition of these two is an imaginary reflection and K β is isomorphic to Z 2 2 ; (ix) a group generated by an elliptic involution, with both fixed points on β, and a reflection whose circle of fixed points intersects β at two points and separating the fixed points of the elliptic involution. In this case, K β is isomorphic to Z 2 2 ; and (x) a group generated by a reflection, with exactly two fixed points on β, and an elliptic involution with both fixed points on the circle of fixed points of the reflection. In this case, K β is isomorphic to Z 2 2 ; 4.4.6. Later, we will construct a connected compact domain by gluing a finite set of structure regions an loops such that no two of these structure regions are K-equivalent. If two structure regions share a boundary structural loop, then they are K-equivalent if there is some element of K sending a boundary loop of one to a boundary loop of the other. This permits to obtain the following simple fact. Proposition 4.11. Let R and R ′ be any two different structure regions with a common boundary loop β. Then, they are K-equivalent if and only if either: (i) K β contains an element (necessarily of order two) which does not belong to K R or (ii) there is another boundary loop β ′ of R and an element k ∈ K\K R so that k(β ′ ) = β (necessarily a loxodromic or glide-reflection).
4.4.7. Next result is related to those structure regions with K-stabilizer being either trivial or a cyclic group generated by a reflection.
Proposition 4.12. Let R be a structure region with K R being either trivial or a cyclic group of order two generated by a reflection. If β is a boundary loop of R such that K R ∩ K β = {I}, then there is a non-trivial element k ∈ K \ K R so that k(β) still a boundary loop of R.
Proof. The hypothesis on K R asserts that the only possibilities in Proposition 4.10 for K β are (i), (iii), (iv) and (v). In all of these cases, with the exception of (i), K β contains an element outside K R . In case (i) K β is trivial. The projection of β on S + is a simple loop β * which has trivial J-stabilizer. We have that β is free homotopic to the product of the other boundary loops of R. If none of the other boundary loops of R is equivalent to β under K, then we may delete β * and its J-translates from F , contradicting the minimality of F .
Structure loops with non-trivial conformal stabilizer.
Let R be a structure region with K R neither trivial or a cyclic group generated by a reflection. Proposition 4.8 asserts that K R is either a cyclic group generated by an elliptic transformation or it is a dihedral group generated by two reflections. In either situation, H = K + R is a non-trivial elliptic cyclic group. The structure region R has either 0, 1 or 2 boundary structure loops being stabilized by H. The other boundary structure loops, if any, have trivial H-stabilizers.
If no structure loop on the boundary of R is stabilized by H, then both fixed points of H lie in R. Also, a boundary structure loop is stabilized by H if and only if it separates the fixed points of H.
If K R is a dihedral group and R contains only one of the fixed points of H, then such a fixed point is fixed by the reflections in K R . This, in particular, asserts that each reflection in K R will commute with the elements of H; this obligates to have K R Z 2 2 . Proposition 4.13. Let R be a structure region with K +
R = H being non-trivial. If there is a fixed point of H in R, then both fixed points of H lie in R.
Proof. Suppose there is only one fixed point in R of the cyclic group H. Then there is a unique structure loop β on the boundary of R stabilized by H. Every other structure loop, on the boundary of R, has H-stabilizer the identity. If K R = H, then it follows that if were to fill in the discs bounded by the other structure loops on the boundary of R, then β would be contractible; that is, if we delete the projection of β and their J-translates from our list of base structure loops, this would leave unchanged the smallest normal subgroup containing the base structure loops raised to appropriate powers. Since we have chosen our base structure loops to be minimal, this cannot be. If K R H, then we have a reflection τ ∈ K R whose circle of fixed points is not completely contained in R, and K R = H, τ is a dihedral group. In this case, the fixed point of H contained in R is also fixed by τ (so both fixed points of H are fixed by τ as τHτ = H), K R Z 2 2 and H Z 2 . In this case, we may also delete the projection of β and its J-translates from F in order to get a contradiction to the minimality of F .
The previous result asserts that a non-trivial elliptic transformation in K + R either has both fixed points on the structure region R or none of them belong to it. In the last case, there are (exactly) two boundary structure loops of R, each one invariant under such an elliptic transformation.
Proposition 4.14. Let R be a structure region with non-trivial K + R = H. If β 1 , β 2 are two different boundary loop of R which are invariant under H, then there is a (non-trivial) element k ∈ K so that k(β 1 ) = β 2 (such an element is either loxodromic or glide-reflection).
Proof. Let us assume that there is no such element of K as desired and let R * be the other structure region sharing β 2 in its boundary. On the region R * there is nother boundary loop β 3 which is invariant under H. All other boundary loops of R ∪ R * (with the exception of β 1 , β 2 and β 3 ) have trivial K-stabilizers. In particular, they are not K-equivalents to β 1 , β 2 and β 3 . Also, β 2 is neither K-equivalent to β 1 and β 3 (by our assumption). If we project the region R ∪ R * ∪ β 2 on S + , we obtain an homeomorphic copy and the projected loop from β 2 is not J-equivalent to none of its boundary loops. In particular, we may delete it (and its J-translates) obtaining a contradiction to the minimality of F .
4.4.9.
Structure regions with trivial stabilizers. Let R be a structure region with trivial stabilizer K R = {I}. As consequence of Proposition 4.12, every other structure region is necessarily K-equivalent to R. It follows that R is a fundamental domain for K and the boundary loops are paired by either reflections, imaginary reflections, loxodromic transformations or glide-reflections. In this case we obtain that K is the free product, by the Klein-Maskit combinatioin theorem, of groups of types (i)-(iv) as described in the theorem.
Structure regions with non-trivial stabilizers.
Let us now assume there is no structural region with trivial K-stabilizer.
Proposition 4.11, and the fact that S is compact and connected, permits us to construct a connected set R obtained as the union of a finite collection of K-non-equivalent structure regions (each of them has non-trivial K-stabilizer) together their boundary structure loops.
(1) Let R ⊂ R be a structure region with K R = τ , where τ is a reflection. The circle of fixed points C τ of τ intersects some structure loops in the boundary of R. If γ = C τ ∩ R, then R − γ consists of two domains R 0 1 and R 0 2 . Set R * j = R 0 j ∪ γ ⊂ R. By Proposition 4.12, the structural loops contained in the interior of R * 1 ∪ R * 2 are paired by either reflections, imaginary reflections, loxodromic transformations or glide-reflection transformations. This process provides free products (in the sense of the Klein-Maskit combination theorem) of groups of types (i)-(iv) as described in the theorem.
The other boundary loops of R intersect C τ . Let β be any structure loop in the boundary of R which intersects C τ , necessarily at two points.
(1.1) If β belongs to the boundary of R, then we already noted that either (a) there is an involution k ∈ K (conformal or anticonformal) with k(β) = β and k( R) ∩ R = β; or (b) there is another boundary loop β ′ of R and an element σ ∈ K (which is either loxodromic or a glide-reflection) so that σ(β) = β ′ and σ( R) ∩ R = β ′ . If σ is a glide-reflection, then τσ is a loxodromic with the same property. In any of the two situations above, we perform the HNN-extension (in the sense of Kleini-Maksti combination theorem) to produce factors of type (2), (4) or (5) in the descriotion of basic extended dihedral groups as in Section 4.2.2.
(1.2) If β is in the interior of R, then we have another structure region R ′ ⊂ R with β as one of its border loop. In this case, as consequence of Proposition 4.11, we should have K β = τ and τ ∈ K R ′ .
The above process produces basic extended dihedral groups using the circle C τ and its reflection τ.
(2) Let R ⊂ R be a structure region with K R = φ , where φ is an elliptic transformation. In this case we have two possibilities: either (i) both fixed points of φ belong to R or (ii) there are two boundary loops β 1 and β 2 of R, each one invariant under φ, and there is some k ∈ K with k(β 1 ) = β 2 , k( R) ∩ R = β 2 . We have that each other boundary loop β of R is either (i) the boundary of another structure region inside R (so it has trivial stabilizer in K) or (ii) it belongs to the boundary of R (in which case we have either (a) or (b) above). This process produces the free product of groups of types (v), (vi) and (vii) as in the theorem.
(3) Assume one of the structure regions R ⊂ R has stabilizer a dihedral group. In this case, K + R is a non-trivial cyclic group generated by an elliptic transformation. We may proceed similarly as above and we again produce basic extended dihedral groups. 4.4.11. All the above together permits to see that K can be constructed, by the Klein-Maskit combination theorem, by using the groups as stated in the theorem.
Proof of Theorem 2.1
Let Γ be a Schottky group of rank g ≥ 2 with M = (H 3 ∪ Ω)/Γ, where Ω is the region of discontinuity of Γ.
5.1.
Proof of part (1). By lifting both τ 1 and τ 2 to H 3 we obtain an extended Kleinian group K containing Γ as a normal subgroup so that G = K/Γ = τ 1 , τ 2 D q , that is, K is a dihedral extended Schottky group. Let θ : K → G be the canonical surjection and by c 1 , . . . , c r ∈ K a complete set of symmetries. By Theorem 4.3, K is constructed using reflections ζ 1 ,. . . , ζ α , imaginary reflections η 1 ,. . . , η β , γ cyclic groups generated by loxodromic transformations, δ cyclic groups generated by glide-reflections, ρ cyclic groups generated by an elliptic transformation of finite order, η groups generated by an elliptic transformation of finite order and a loxodromic transformation commuting with it, κ groups generated by an elliptic transformation of finite order and a glide-reflection (conjugating the elliptic into its inverse) and ε basic extended dihedral groups K 1 ,. . . , K ε .
Each K i is generated by a reflection σ i and some other loxodromic and elliptic transformations, where some of them do not commute with loxodromic ones, say t i1 ,. . . , t im i (all of them of order at least 3 and not commuting with σ i ) and some f i imaginary reflections and reflections (each of them commuting with σ i ); let us denote these involutions by σ i1 ,. . . , σ i f i . By the Riemann-Hurwitz formula (see Remark 4.4), g ≥ q( g − 1)
+ 1 + q 2 ε i=1 f i , where g = α + β + 2(γ + δ + κ + η) + ε + g 1 + · · · + g ε − 1 ≥ α + β + ε − 1. So, it follows that g − 1 q + 2 ≥ g + 1 + 1 2 ( f 1 + · · · + f ε ) ≥ α + β + ε + 1 2 ( f 1 + · · · + f ε ).
Also, note that a complete set of symmetries for K is given by ζ 1 ,. . . , ζ α , η 1 ,. . . , η β , σ i , σ ik , where k = 1, . . . , f i and i = 1, . . . , ε. If c denotes any of the above symmetries, then c < C( K, c) and so θ(c) ⊆ θ(C( K, c)) ⊆ C( G, θ(c)) Z 2 2 , q even Z 2 , q odd Now, it is easy to see the following θ(C( K; ζ j )) = θ(ζ j ) = Z 2 , θ(C( K; η j )) = θ(η j ) = Z 2 , θ(C( K; σ jk )) = θ( σ j , σ jk ) = Z 2 2 , θ(C( K; σ j )) = θ( σ j , σ j1 , . . . , σ j f j , ǫ j1 , . . . , ǫ jm j ) = Z 2 2 .
Finally, it follows from Theorem 3.5 that
m 1 + m 2 ≤ 2(α + β) + ε + ( f 1 + · · · + f ε ) ≤ 2(α + β + ε) + ( f 1 + · · · + f ε ) ≤ 2 g − 1 q + 4.
5.2.
Proof of part (2). Let q i j be to denote the order of τ i τ j , where i < j. By a permutation of the indices, we may assume that 2 ≤ q 12 ≤ q 13 ≤ q 23 . As consequence of Part (1) one has the inequality
(1) m 1 + m 2 + m 3 ≤ g − 1 q 12 + g − 1 q 13 + g − 1 q 23 + 6. 5.2.1. Case g = 2.
In this case, m 1 + m 2 + m 3 ≤ 6. As m i + m j ≤ 4, either m 1 + m 2 + m 3 ≤ 5 or else m 1 = m 2 = m 3 = 2. Claim 5.1 ends with the proof for g = 2.
Claim 5.1. The case m 1 = m 2 = m 3 = 2 is not possible for g = 2.
Proof. Assume there is a Schottky group Γ of rank two, with region of discontinuity Ω, such that the handlebody M = (H 3 ∪ Ω)/Γ admits three symmetries τ 1 , τ 2 and τ 3 , each of them having exactly two connected components of fixed points. Set H = τ 1 , τ 2 , τ 3 . Let π : H 3 ∪ Ω → M be the universal covering induced by Γ, and K be the extended Kleinian group obtained by lifting H under π. There is a surjective homomorphism θ : K → H with Γ as its kernel. Let K j = θ −1 ( τ r , τ s ), where r s and r, s ∈ {1, 2, 3} − { j}, and p j be the order of τ r τ s . The extended Kleinian group K j is a dihedral extended Schottky group (of finite index in K) with region of discontinuity Ω. By Theorem 4.3, K j is constructed by using α cyclic groups generated by reflections, β cyclic groups generated by imaginary reflections, γ cyclic groups generated by loxodromic transformations, δ cyclic groups generated by glide-reflections, ρ cyclic groups generated by an elliptic transformation of finite order, η groups generated by an elliptic transformation of finite order and a loxodromic transformation commuting with it, κ groups generated by an elliptic transformation of finite order and a glide-reflection (conjugating the elliptic into its inverse) and ε basic extended dihedral groups Γ 1 ,..., Γ ε .
If Ω i is the region of discontinuity of Γ i , then Ω i /Γ + i has genus g i , has 2r i cone points with cone orders |t i1 |, |t i1 |, . . . , |t ir i |, |t ir i | (each of them bigger tan two) and 2n i cone points of order two.
Then, Ω/K + j has genus g = α+β+2(γ+δ+κ+η)+ε+g 1 +· · ·+g ε −1, with 2(r 1 +· · ·+r ε ) conical points of orders |t 11 |, |t 11 |, |t 12 |, |t 12 |. . . , |t εr ε |, |t εr ε |, 2(n 1 + · · · + n ε ) conical points of order 2, and 4ρ conical points of orders l 1 , l 1 , l 1 , l 1 , . . . , l ρ , l ρ , l ρ , l ρ , and
2 = p j ( g − 1) + 1 + 2p j ρ j=1 1 − 1 l j + p j ε i=1 r i k=1 1 − 1 |t ik | + p j 2 ε i=1 n i ≥ p j ( g − 1) + 1.
It follows that g ∈ {0, 1}. If α = β = ε = 0 (so δ > 0), then g = 2(γ + δ + κ + η) − 1, so g = 1 = δ + κ and η = γ = 0. It follows that K j is a elementary group (either (i) generated by a glide-reflection or (ii) a glide-reflection and an elliptic element with the same fixed points), a contradiction (as it must contains a Schottky group of rank two).
If α+β+ε > 0, it must be that γ = δ = κ = η = 0. If ε = 0, then 2 = α+β and K j will be elementary group, again a contradiction. So ε ≥ 1. As α+β+ε+g 1 +· · ·+g ε = g+1 ∈ {1, 2}, we must have ε = 1. In fact, if ε ≥ 2, then ε = 2, α = β = g 1 = g 2 = 0, g = 1 and
1 = p j 2 i=1 r i k=1 1 − 1 |t ik | + p j 2 2 i=1 n i .
If, for instance, n 1 ≥ 1, then (as p j ≥ 2) r 1 = r 2 = 0 = n 2 and n 1 = 1, from which we obtain that K j is elementary, a contradiction. In particular, n 1 = n 2 = 0. The other case can be worked similarly to obtain a contradiction. Assume now that ε = 1. Then α + β + g 1 ∈ {0, 1}. If g 1 = 1, then α = β = 0 and K j is again elementary. So, g 1 = 0 and g = α + β ∈ {0, 1}, that is (α, β) ∈ {(0, 0), (1, 0), (0, 1)}.
In the case (α, β) = (0, 0), we have g = 0 and
2 ≥ 1 − n + n r 1 k=1 1 − 1 |t ik | + nn 1 2 .
As each |t i j | ≥ 2, the above ensures that 2 ≥ n(n 1 + r 1 − 2). Now, if n 1 + r 1 ≥ 3, then n = 2 (as required) and, necessarily, n 1 + r 1 = 3. If n 1 + r 1 ≤ 2, then K j will be elementary, a contradiction. In the case (α, β) ∈ {(1, 0), (0, 1)}, we have g = 1 and
2 ≥ 1 + n r 1 k=1 1 − 1 |t ik | + nn 1 2 .
Again, as |t 1k | ≥ 2, one obtains that 2 ≥ n(n 1 + r 1 ). Then n = 2 and n 1 + r 1 = 1, but again this makes K j to be elementary, a contradiction.
As a consequence of the above, we see that K j is generated by a reflection ρ j1 and three other involutions ρ j2 , ρ j3 , ρ j4 , each one commuting with ρ j1 (some of the involutions may be reflections and others may be imaginary reflections). Note also that K j keeps invariant the circle of fixed points of ρ j1 , that is, the limit set of K j = Z 2 ⊕(Z 2 * Z 2 * Z 2 ) is contained in the circle of fixed points of ρ j1 . As K j has finite index in K, the limit set of K is contained in such a circle, and as the limit set of K is infinite, such a circle is uniquely determined by it. As a consequence, ρ 11 = ρ 21 = ρ 31 = ρ. This asserts that θ(ρ) ∈ θ(K 1 ) ∩ θ(K 2 ) ∩ θ(K 3 ).
As, by the definition of the groups K j 's, (i) τ j θ(K 1 ) ∩ θ(K 2 ) ∩ θ(K 3 ), (ii) θ(ρ) is a symmetry and (iii) the symmetries in H are exactly τ 1 , τ 2 , τ 3 and τ 1 τ 2 τ 3 , the only possibility is to have θ(ρ) = τ 1 τ 2 τ 3 , from which, for instance, τ 1 τ 2 τ 3 ∈ τ 1 , τ 2 ; obligating to have that τ 3 ∈ τ 1 , τ 2 , a contradiction. 5.2.2. Case g = 3. If q 23 ≥ 3, then m 2 + m 3 ≤ 4. Since m 1 ≤ g + 1 = 4, we get in this case that m 1 + m 2 + m 3 ≤ 8. Let us now assume q 12 = q 13 = q 23 = 2, in which case τ 1 , τ 2 , τ 3 Z 3 2 . We may reorder again the indices to assume that m 1 ≤ m 2 ≤ m 3 ≤ 4. If m 3 = 4, then inequality (1) asserts that m 1 + m 2 ≤ 2, so m 1 + m 2 + m 3 ≤ 8. Now, the only case with m 3 ≤ 3 for which we do not have m 1 + m 2 + m 3 ≤ 8 is when m 1 = m 2 = m 3 = 3. Claim 5.2 ends the proof for g = 3.
Claim 5.2. The case m 1 = m 2 = m 3 = 3 is not possible for g = 3.
Proof. The Proof follows the same ideas as for the Claim 5.1. Let Γ be a Schottky group of rank g = 3, with region of discontinuity Ω, so that the handlebody M = (H 3 ∪ Ω)/Γ admits three symmetries τ 1 , τ 2 and τ 3 , each of them having exactly three connected components of fixed points, and with H = τ 1 , τ 2 , τ 3 = Z 3 2 . Let π : H 3 ∪ Ω → M be the universal covering induced by Γ and let K be the extended Kleinian group obtained by lifting H under π. There is a surjective homomorphism θ : K → H with Γ as its kernel. Let K j = θ −1 ( τ r , τ s ), where r s and r, s ∈ {1, 2, 3} − { j}; so θ(K j ) = Z 2 2 . The extended Kleinian group K j is a dihedral extended Schottky group and has index two in K, so it has region of discontinuity Ω. As consequence of Theorem 4.3, K j is constructed by using α cyclic groups generated by reflections, β cyclic groups generated by imaginary reflections, γ cyclic groups generated by loxodromic transformations, δ cyclic groups generated by glide-reflections, ρ cyclic groups generated by an elliptic transformation of order two, η groups generated by an elliptic transformation of order two, κ groups generated by an elliptic transformation of order two and a glide-reflection and ε basic extended dihedral groups Γ 1 ,..., Γ ε , so that α + β + δ + ε > 0. As before, Ω/K + j will have genus g = α + β + 2(γ + δ + κ + η) + ε + g 1 + · · · + g ε − 1, with 2(2ρ + r 1 + · · · + r ε + n 1 + · · · + n ε ) conical points of order 2, and so that
2 ≥ 2( g − 1) + ε i=1 (r i + n i ).
In particular, g ∈ {0, 1}. If α = β = ε = 0 (so δ + κ > 0), then g = 2(γ + δ + κ + η) − 1, so γ = η = 0 and δ + κ = 1, from which we obtain that K j is elementary, a contradiction. As α + β + ε > 0, then γ = δ = κ = η = 0. If ε = 0, the same asserts that K j will be elementary group containing the non-elementary group Γ, a contradiction, so ε ≥ 1. If ε ≥ 3, then α + β < 0, a contradiction. If ε = 2, then α + β = g − 1, so the only possible case is to have g = 1 and α = β = 0. In this way, (α, β, ε; g) ∈ {(0, 0, 1; 0), (1, 0, 1; 1), (0, 1, 1; 1), (0, 0, 2; 1)}.
Let us consider the case (α, β, ε; g) = (0, 0, 2; 1). In this case K j is free product of two groups Γ 1 = ρ 1 , η 1 = Z 2 2 and Γ 2 = ρ 2 , η 2 = Z 2 2 , where ρ k is reflection and η k is either a reflection or an imaginary reflection. In this case a complete set of symmetries of K j is given by {ρ 1 , ρ 2 , η 1 , η 2 }. Let us note that θ(ρ k ) θ(η k ) since otherwise the elliptic element of order two ρ k η k ∈ Γ, a contradiction to the fact that Γ is torsion free. As C(K j , ρ k ) = C(K j , η k ) = ρ k , η k = Z 2 2 and that θ(ρ k ) θ(η k ), we may see from Theorem 3.5 that the number of fixed points of any of the symmetries θ(ρ 1 ) and θ(η 1 ) is at most 2, a contradiction to the fact we are assuming the symmetries τ 1 , τ 2 and τ 3 each one has exactly 3 components.
Let us consider the case (α, β, ε; g) = (0, 0, 1; 0). In this case K j = ρ, η 1 , η 2 , η 3 , η 4 , where ρ is a reflection and each of the η k is either a reflection or an imaginary reflection commuting with ρ (each element of K j keeps invariant the circle of fixed points of ρ). As before, θ(ρ) θ(η k ), for each k = 1, 2, 3, 4. So, in particular, θ(η 1 ) = θ(η 2 ) = θ(η 3 ) = θ(η 4 ). In this case, a complete set of symmetries of K j is given by {ρ, η 1 , η 2 , η 3 , η 4 } and
C(K j , ρ) = K j , C(K j , η k ) = ρ, η k = Z 2 2
and it follows, from Theorem 3.5, that θ(ρ) has at most 1 connected components of fixed points, a contradiction to the assumption the symmetries have 3 conected components.
Let now (α, β, ε; g) = (1, 0, 1; 0) (similar arguments for the case (α, β, ε; g) = (0, 1, 1; 0)). In this case K j is a free product of a cyclic group generated by a reflection ζ and a group K 0 j = ρ, η 1 , η 2 , η 3 , η 4 , where ρ is a reflection and each of the η k is either a reflection or an imaginary reflection commuting with ρ (each element of K 0 j keeps invariant the circle of fixed points of ρ). Again, as before, θ(ρ) θ(η k ), for each k = 1, 2, 3, 4. So, in particular, θ(η 1 ) = θ(η 2 ) = θ(η 3 ) = θ(η 4 ). In this case, a complete set of symmetries of K j is given by {ζ, ρ, η 1 , η 2 , η 3 , η 4 } and C(K j , ζ) = ζ = Z 2 , C(K j , ρ) = K 0 j , C(K j , η k ) = ρ, η k = Z 2 2 . + 6 = (r + 1)g + 5r − 1 r .
Examples
Example 6.1 (Sharp upper bound in part (1) of Theorem 2.1). Let q ≥ 2 and let us consider an extended Schottky group K, constructed by Theorem 3.2 using exactly r + 1 reflections E 1 ,. . . , E r+1 . The orbifold uniformized by K is a planar surface bounded by exactly r + 1 boundary loops. Let us consider the surjective homomorphism θ : K → D q = x, y : x 2 = y 2 = (yx) q = 1 : θ(E 1 ) = x, θ(E 2 ) = · · · = θ(E r+1 ) = y.
Let Γ = ker(θ). If we set L = E 2 E 1 and C j = E r+1 E j+1 , for j = 1, . . . , r − 1, then it is not difficult to see that Γ = L q , C 1 , . . . , C r−1 , LC 1 L −1 , . . . , LC r−1 L −1 , . . . , L q−1 C 1 L −q+1 , . . . , L q−1 C r−1 L −q+1 is a Schottky group of rank g = (r − 1)q + 1. Let us consider the extended Schottky groups
Γ 1 = θ −1 ( x ) = E 1 , Γ , Γ 2 = θ −1 ( y ) = E 2 , Γ .
As the group K contains no imaginary reflections nor real Schottky groups, it follows that Γ j is constructed, by Theorem 3.2, using α i reflections and β i loxodromic and glidereflection transformations. The handlebody M Γ admits two symmetries, say τ 1 and τ 2 induced by Γ 1 and Γ 1 respectively. The number of connected components of fixed points of τ i is exactly α i . As consequence of Theorem 2.1, we should have α 1 + α 2 ≤ 2(r + 1). Next, we proceed to see that in fact we have an equality, showing that the upper bound in Theorem 2.1 is sharp. In order to achieve the above, we use Theorem 3.5. A complete set of symmetries in K is provided by E 1 ,. . . , E r+1 . We also should note that C(K, E i ) = E i , for every j, and that J(i) = ∅ and I(i) = F(i).
Case q odd. In this case, I(1) = {1, 2, . . . , r + 1} and C(D q , x) = x . It follows, from Theorem 3.5, that α 1 = α 2 = r + 1 and we are done.
Case q even. In this case, I(1) = {1}, I(2) = {2, . . . , r + 1}, C(D q , x) = x, (yx) q/2 Z 2 2 and C(D q , y) = y, (yx) q/2 Z 2 2 . It follows, from Theorem 3.5, that α 1 = 2, α 2 = 2r and we are done. Example 6.2 (Sharp upper bounds in part (2) of Theorem 2.1 for g = 2). Let K be the extended Kleinian group generated by four reflections, say η 1 ,. . . , η 4 , where η 1 (z) = z, η 2 (z) = −z, η 4 (z) = 1/z and η 3 is the reflection on a circle Σ which is orthogonal to the unit circle and disjoint from the real and imaginary axis. One may see that K = η 4 × ( η 1 , η 2 * η 3 ) Z 2 × (D 2 * Z 2 ).
If Γ = A = η 1 η 3 , B = η 1 η 2 η 3 η 2 , then it is a Schottky group of rank 2 with a fundamental domain bounded by the 4 circles C 1 = Σ, C ′ 1 = η 1 (Σ), C 2 = η 2 (Σ) and C ′ 2 = η 2 η 1 (Σ) such that A(C 1 ) = C ′ 1 and B(C 2 ) = C ′ 2 . As η 1 Aη 1 = η 3 Aη 3 = A −1 , η 2 Aη 2 = η 4 Bη 4 = B, η 4 Aη 4 = η 2 Bη 2 = A, η 1 Bη 1 = B −1 and η 3 Bη 3 = A −1 B −1 A, it follows that Γ is a normal subgroup of K. Moreover, K/Γ Z 3 2 = Z 2 × D 2 . On the handlebody M Γ both η 1 and η 3 induce the same symmetry τ 1 with exactly 3 connected components of fixed points (each of them a disc), η 2 induces a symmetry τ 2 with exactly one connected component of fixed points (a dividing disc) and η 4 induces a symmetry τ 3 with exactly one connected component of fixed points (this being an sphere with three borders). It follows that the three induced symmetries are non-conjugated. . Let K be the extended Kleinian group generated by three reflections, η 1 (z) = z, η 2 (z) = −z and η 3 the reflection on a circle Σ disjoint from the real and imaginary lines. One has that K = η 1 , η 2 * η 3 D 2 * Z 2 .
The group Γ = A 1 = (η 3 η 1 ) 2 , A 2 = (η 3 η 2 ) 2 , A 3 = η 3 η 2 η 1 η 3 η 1 η 2 is a Schottky group of rank 3 with a fundamental domain bounded by the 6 circles C 1 = η 1 (Σ), C ′ 1 = η 3 (C 1 ), C 2 = η 2 (Σ), C ′ 2 = η 3 (C 2 ), C 3 = η 2 (C 1 ) and C ′ 3 = η 3 (C 3 ), such that A 1 (C 1 ) = C ′ 1 , A 2 (C 2 ) = C ′ 2 and A 3 (C 3 ) = C ′ 3 . Similarly to the previous case, one may check that Γ is a normal subgroup of K and that K/Γ Z 3 2 . The reflection η j induces a symmetry τ j (for each j = 1, 2, 3) on the handlebody M Γ . In this case (either by direct inspection or by using Theorem 3.5) each of τ 1 and τ 2 has exactly 2 connected components of fixed points, and τ 3 has 4 connect components of fixed points. In some cases the handlebody M Γ will have extra automorphisms conjugating τ 1 with τ 2 (for instance, when Σ is orthogonal to the line L = {Re(z) = Im(z)}); but in the generic case this will not happen (that is, the three of them will be non-conjugated).
Example 6.4 (Sharp upper bounds in part (2) of Theorem 2.1 for g ≥ 4 and H Z 2 × D r ). Let K be an extended Schottky group constructed by using 2n + 3 reflections (or imaginary reflections or combination of them), say η 1 ,. . . , η 2n+3 . Consider the surjective homomorphism θ : K → D 3 = a, b : a 2 = b 2 = (ab) 3 = 1 , defined by θ(η 1 ) = · · · = θ(η 2n+2 ) = a, θ(η 2n+3 ) = b. In this case Γ = ker(θ) is a Schottky group of rank g = 6n + 4. The handlebody M Γ admits the symmetries τ 1 = a, τ 2 = b and τ 3 = bab. By direct inspection (or by using Theorem 3.5) it can be checked that m 1 = m 2 = m 3 = 2n + 3.
Example 6.5 (Sharp upper bounds in part (2) of Theorem 2.1 for g ≥ 4 and H Z 2 × D r ). Let K be an extended Schottky group constructed by using 3 reflections (or imaginary reflections or combination of them), say η 1 , η 2 and η 3 . Consider the surjective homomorphism θ : K → Z 2 × D 3 = c × a, b : a 2 = b 2 = (ab) 3 = 1 , defined by θ(η 1 ) = c, θ(η 2 ) = a and θ(η 3 ) = b. In this case Γ = ker(θ) is a Schottky group of rank g = 2r + 1. The handlebody M Γ admits the symmetries τ 1 = c, τ 2 = a and τ 3 = b. In this case, m 1 = 2r and m 2 = m 3 = 4.
Theorem 3.5 ([3]). Let K be an extended Kleinian group containing a Schottky group Γ as a finite index normal subgroup. Let θ : K → H = K/Γ be the canonical projection and C = {c i : i ∈ I} be a complete set of symmetries of K. Then Γ i = θ −1 ( θ(c i ) ) = Γ, c i is an extended Schottky group, constructed using α reflections, β imaginary reflections and ε real Schottky groups as in Theorem 3.
Figure 1 .
1Examples of dihedral extended Schottky groups: τ i reflections, η i imaginary reflections, L i loxodromics, N i glide-reflections, E, F elliptics, M either loxodromic or glide-reflection.
Figure 2 .
2Examples of a basic extended dihedral group. 4.3. The structural description.Theorem 4.3.(1) A dihedral extended Schottky group is the free product (in the sense of the Klein-Maskit combination theorems) of the following groups (seeFigures 1 and 2)(i) α cyclic groups generated by reflections;(ii) β cyclic groups generated by imaginary reflections; (iii) γ cyclic groups generated by loxodromic transformations;
Proposition 4. 5 .
5If k ∈ K + is an elliptic transformation with one fixed point in Ω, then both fixed points of k belong to Ω.
Example 6 . 3 (
63Sharp upper bounds in part (2) of Theorem 2.1 for g = 3)
The only way for both symmetries in θ(K j ) to have exactly 3 connected components of fixed points is to have that θ(ζ) = θ(ρ). But in this case we should have that Γ is the Schottky group generated by the loxodromic transformations ρζ, η 4 η 1 , η 4 η 2 , η 4 η 3 , which is of rank 4, a contradiction. 5.2.3. Case g ≥ 4. In this case, inequality (1) assertsIf q −1 12 +q −1 13 +q −1 23 ≤ 1, then the above ensures that m 1 +m 2 +m 3 ≤ g+5. If q −1 12 +q −1 13 +q −1 23 > 1, then we have the following cases:where Z 2 = τ 1 , r = τ 2 , τ 3 and A 4 , S 4 and A 5 are generated by τ 1 τ 2 , τ 1 τ 3 in each case. In the cases q 13 = 3 one has that τ 1 , τ 3 D 3 , so τ 1 and τ 3 are conjugated, that is, m 1 = m 3 . It follows thatNow, for the case q 12 = q 13 = 2 and q 23 = r ≥ 2, inequality (1) asserts
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[] |
[
"Design and performance evaluation of a state-space based AQM",
"Design and performance evaluation of a state-space based AQM"
] |
[
"Yassine Ariba ",
"Yann Labit ",
"Frédéric Gouaisbaut "
] |
[] |
[] |
Recent research has shown the link between congestion control in communication networks and feedback control system. In this paper, the design of an active queue management (AQM) which can be viewed as a controller, is considered. Based on a state space representation of a linearized fluid flow model of TCP, the AQM design is converted to a state feedback synthesis problem for time delay systems. Finally, an example extracted from the literature and simulations via a network simulator NS (under cross traffic conditions) support our study.
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10.1109/ctrq.2008.15
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[
"https://arxiv.org/pdf/0902.0920v1.pdf"
] | 190,947 |
0902.0920
|
1d9e3867829d41252b91933deecb853d752745fe
|
Design and performance evaluation of a state-space based AQM
5 Feb 2009 February 2008
Yassine Ariba
Yann Labit
Frédéric Gouaisbaut
Design and performance evaluation of a state-space based AQM
5 Feb 2009 February 2008
Recent research has shown the link between congestion control in communication networks and feedback control system. In this paper, the design of an active queue management (AQM) which can be viewed as a controller, is considered. Based on a state space representation of a linearized fluid flow model of TCP, the AQM design is converted to a state feedback synthesis problem for time delay systems. Finally, an example extracted from the literature and simulations via a network simulator NS (under cross traffic conditions) support our study.
Introduction
Congestion control is a very active research area in network community. In order to supply the well known transmission control protocol (TCP), active queue management mechanisms have been developed. AQM regulates the queue length of a router by actively dropping packets. Various mechanisms have been proposed in the literature such as Random Early Detection (RED) [4], Random Early Marking (REM) [1], Adaptive Virtual Queue (AVQ) [10] and many others [19]. Their performances have been evaluated in [19] and empirical studies have shown their effectiveness (see [12]). Recently, significant studies proposed by [7] have redesigned the AQMs using control theory and P , P I have been developed in order to cope with the packet dropping problem. Then, using dynamical model developed by [16], many research have been devoted to deal with congestion problem in a control theory framework (for example see [22]). Nevertheless, most of these papers do not take into account the delay and ensure the stability in closed-loop for all possible delays which could be conservative in practice. Modeling the congestion control using time delay is not new and global stability analysis has been studied by [15] and [17] via Lyapunov-Krasovskii theory. Also, in [9], a delay dependent state feedback controller is provided by compensation of the delay with a memory feedback control. This latter methodology is interesting in theory but hardly suitable in practice. Based on a recently developed Lyapunov-Krasovskii functional, an AQM stabilizing the TCP model is designed. This synthesis problem is carried out as state feedback synthesis for time delay systems. Then, this method is applied on an augmented system in order to vanish the steady state error in spite of disturbance. The paper is organized as follows. The second part presents the model of a network supporting TCP and the time delay system representation. Section 3 is dedicated to the Figure 1: Network configuration design of the AQM ensuring the stabilization of TCP. Section 4 presents application of the exposed theory and simulation results using NS-2 (see [3]) before concluding this work.
Problem statement
The linearized TCP fluid-flow model
In this paper, we consider the network topology consisting of N homogeneous TCP sources (i.e with the same propagation delay) connected to a destination node through a router (see figure 1). The bottleneck link is shared by N flows and TCP applies the well known congestion avoidance algorithm to cope with the phenomenon of congestion collapse [8]. Many studies have been dedicated to the modeling of TCP and its AIMD (additive-increase multiplicative-deacrease) behavior [13], [21], [22] and references therein. We consider in this note the model (1) developed by [16]. This latter may not capture with high accuracy the dynamic behavior of TCP but its simplicity allows us to apply our methodology. Let consider the following model
Ẇ (t) = 1 R(t) − W (t)W (t−R(t)) 2R(t−R(t)) p(t − R(t)) q(t) = W (t) R(t) N − C + d(t)(1)
where W is the TCP window size, q is the queue length of the router buffer, R is the round trip time (RTT) and can be expressed as R = q/C + T p . C, T p and N are parameters related to the network configuration and represent the transmission capacity of the router, the propagation delay and the number of TCP sessions respectively. The variable p is the marking/dropping probability of a packet (that depends whether the ECN option, explicit congestion notification, is enabled, see [18]). In the mathematical model (1), we have introduced an additional signal d(t) which models cross traffics through the router and filling the buffer. These traffics are not TCP based flows (not modeled in TCP dynamic) and can be viewed as perturbations since they are not reactive to packets dropping (for example, UDP based traffic). A linearization and some simplifications of (1) was carried out in [7] to allow the use of traditional control theory approach. The linearized fluid-flow model of TCP is as follow,
δẆ (t) = − N R 2 0 C δW (t) + δW (t − h(t)) − 1 R 2 0 C δq(t) − δq(t − h(t)) − R0C 2 2N 2 δp(t − h(t)) δq(t) = N R0 δW (t) − 1 R0 δq(t) + d(t)(2)
where δW . = W − W 0 , δq . = q − q 0 and δp . = p − p 0 are the perturbated variables about the operating point. The operating point (W 0 , q 0 , p 0 ) is defined by The input of the model (2) corresponds to the drop probability of a packet. This probability is fixed by the AQM. This latter has for objective to regulate the queue size of the router buffer. In this paper, this regulation problem is addressed in Section 3 with the design of a stabilizing state feedback for time delay systems. Indeed, an AQM acts as a controller (see figure 2) and in order to design it, we have to solve a synthesis problem. Considering a state feedback, the queue management strategy of the drop probability will be expressed as
Ẇ = 0 ⇒ W 2 0 p 0 = 2 q = 0 ⇒ W 0 = R0C N , R 0 = q0 C + T pp(t) = p 0 + k 1 δW (t) + k 2 δq(t).
(3) where k 1 and k 2 are the components of the matrix gain K which we have to design. Note that the input p(t) = u(t) + p 0 of the system (4) is delayed.
Time delay system approach
In this paper, we choose to model the dynamics of the queue and the congestion window as a time delay system. Indeed, the delay is an intrinsic phenomenon in networks and taking into account its characteristic should improve the precision of our model with respect to the TCP behavior. The linearized TCP fluid model (2) can be rewritten as the following time delay system:
ẋ(t) = Ax(t) + A d x(t − h) + Bu(t − h) + B d d(t) x 0 (θ) = φ(θ), with θ ∈ [−h, 0] (4) with A = − N R 2 0 C − 1 CR 2 0 N R 0 − 1 R 0 , A d = − N R 2 0 C 1 R 2 0 C 0 0 ,B = − C 2 R 0 2N 2 0 (5) B d = [0 1] T , x(t) = [δW (t) δq(t)] T is the state vector and u(t) = δp(t) the input. φ(θ) is the initial condition.
There are mainly three methods to study time delay system stability: analysis of the characteristic roots, robust approach and Lyapunov theory. The latter will be considered because it is an effective and practical method which provides LMI and BMI (Linear/Bilinear Matrix Inequalities, [2]) criteria. To analyze and control the system (4), the Lyapunov-Krasovskii approach (see [6]) is used which is an extension of the traditional Lyapunov theory.
Stability analysis of time delay systems
In this subsection, our goal is to derived a condition which takes into account an upperbound of the delay. The delay dependent case starts from a system stable without delays and looks for the maximal delay that preserves stability. Usually, all methods involve a Lyapunov functional, and more or less tight techniques to bound some cross terms and to transform system [6]. These choices of specific Lyapunov functionals and overbounding techniques are the origin of conservatism. In the present paper, we choose a recently developed Lyapunov-Krasovskii functional (6) [5]:
V (x t ) = x T (t)P x(t) + t t− h r t θẋ T (s)Rẋ(s)dsdθ + t t− h r x(s) x(s − 1 r h) . . . x(s − r−1 r h) T Q x(s) x(s − 1 r h) . . . x(s − r−1 r h) ds(6)
where P ∈ S n is a positive definite matrix, Q ∈ S rn and R ∈ S n are two positive definite matrices. r ≥ 1 is an integer corresponding to the discretization step. Using this functional, Let us introduce the following proposition.
Proposition 1
If there exist symmetric positive definite matrices P , R ∈ R n×n , Q ∈ R rn×rn , a scalar h m > 0 and an integer r ≥ 1 such that
S ⊥ T ΓS ⊥ < 0 (7) where Γ = hm r R P 0 . . . 0 P − r hm R r hm R . . . 0 r hm R − r hm R . . . . . . . . . . . . 0 . . . . . . . . . 0 + 0 . . . 0 . . . Q . . . 0 . . . 0 + 0 . . . 0 . . . . . . Q(8)
and
S = −1 A 0 n×(r−1)n A d (9) then, system (4) (with u(t) = 0 and d(t) = 0) is stable for all h ≤ h m .
Proof: It is always possible to rewrite (4) as
Sξ = 0 where ξ = 2 6 6 6 6 6 6 4ẋ (t) x(t) x(t − 1 r h) . . . x(t − r−1 r h) x(t − h) 3 7 7 7 7 7 7 5 ∈ R (r+2)n(10)
and S is defined as (9). Using the extended variable ξ(t) (10), the derivative of V along the trajectories of system (4) leads to:
V (x t ) = ξ T h r R P 0 . . . 0 P − r h R r h R . . . 0 r h R − r h R . . . . . . . . . . . . 0 . . . . . . . . . 0 ξ + ξ T 0 . . . 0 . . . Q . . . 0 . . . 0 ξ − ξ T 0 . . . 0 . . . . . . Q ξ < 0 s.t. [ −1 A 0 · · · 0 A d ] ξ = 0 (11) ⇔ V (x t ) = ξ T Γξ < 0 s.t. [ −1 A 0 · · · 0 A d ] ξ = 0(12)
where Γ ∈ S (r+2)n depends on P , R, Q and the delay h. Using projection lemma [20], expression (12) is equivalent to (7). Remark 1:
• There exists another equivalent form of this LMI provided in [5] and based on quadratic separation.
• In the same paper, it is shown that for r = 1, this proposed function (6) is equivalent to the main classical results of the literature. Moreover, it is also proved that for r > 1 conservatism is reduced.
A first result on synthesis
Given the analysis condition (7) and applying the delayed state feedback (3) on system (4) (in this subsection, the disturbance is not taken into account), the following proposition is obtained.
Proposition 2
If there exist symmetric positive definite matrices P , R ∈ R n×n , Q ∈ R rn×rn , a matrix X ∈ R (r+2)n×n , a scalar h m > 0, an integer r ≥ 1 and a matrix K ∈ R m×n such that Γ + XS + S T X T < 0
where Γ is defined as (8) and
S = −1 A 0 n×(r−1)n A d + BK(14)
then, system (4) can be stabilized for all h ≤ h m for the control law u(t) = Kx(t) (and for d(t) = 0).
Proof: Considering the system (4) with the state feedback (3), the following interconnected system is deducedẋ
(t) = Ax(t) +Ā d x(t − h),(15)
whereĀ d = A d + BK and A, A d and B are defined as (5). Then, we can apply the analysis condition (7) on (15). Using Finsler lemma [20], there exists a matrix X ∈ R (r+2)n×n such that if (13) is satisfied then (7) Figure 3: Design of an AQM as a dynamic state feedback variables"which can reduced conservatism and may be interesting for synthesis purpose as well as robust control purpose. Remark 2:
• To solve the synthesis criterion (13), one has to use a BMI solver.
• In [11], a relaxation algorithm is provided with LMI condition to find a stabilizing state feedback.
Delay dependent state feedback with an integral action
In the previous section, the design of a state feedback control for time delay systems has been exposed. The use of a such controller has been carried out in [11]. However, it appears that in some cases, the queue size is no longer regulated at the desired level (this phenomenon is only observed on the network simulator NS). It thus appears a slight steady state error which can be explain by an inaccuracy of the model. Futhermore, the introduction of non responsive flows like UDP (user datagram protocol) traffics which appear as a disturbance affects the queue size equilibrium and changes the steady state. In order to overcome these problems, the AQM is supplemented with an integral action. The idea is to apply the previously exposed synthesis method over an augmented time delay system composed of the original system (4) and an integrator (see figure 3). The augmented system has the following forṁ
z = A 0 0 0 1 0 z(t) + A d 0 0 0 0 0 z(t − h)(16)+ B 0 δp(t − h)
with z T = [δW δq u] T is the extended state variable. Then, the global control which correspond to our AQM, is a dynamic state feedback
δp(t) = K δW (t) δq(t) u(t) = k 1 δW (t) + k 2 δq(t) + k 3 t 0 δq(t)dt.(17)
In our problem, non modelled crossing traffics d(t) such as UDP based applications are introduced as exogenous signals (see figures 2 and 3). The queue dynamic is modified asq
(t) = W (t) R(t) N − C + d(t)(18)
Considering equations (17), the first equation of (2) and (18), we obtain the transfer function T (s) from the disturbance D(s) to the queue size (about the operating point) ∆Q(s):
T (s) = b(s)s (s + 1 R 0 )sb(s) + N R 0 h s R 2 0 C (1 − e −hs ) + a(s)sk2 + a(s)k3 i ,(19)
with a(s) = − R0C 2 2N 2 e −hs and b(s) = s + N R 2 0 C (1 + e −hs ) + a(s)k 1 . It can be easily shown that for a step type disturbance, the queue size still converges to its equilibrium.
Estimation of the congestion window
In these last two parts, a state feedback synthesis has been performed for the congestion control of TCP flows and the management of the router buffer. So far we have considered that the whole state was accessible. However, although the congestion window can be measured in NS (few lines have to be added in the TCP code), it is not the case in reality. That's why, in this paper it is proposed to estimate this latter variable using the aggregate flow incoming to the router buffer. The sending rate of single TCP source can be approximated by
x i (t) = W (t) R(t) .(20)
The above approximation is valid as long as the model does not describe the communication at a finer time scale than few round trip time (see [13]). Consequently, the whole incoming rate observed by the router is x(t) = N W (t)/R(t). The measure of the aggregate flow has already been proposed and successfully exploited in [10] and [9] for the realization of the AVQ and a PID type AQM respectively. It is worth noting that queue-based AQMs like RED or PI can be assimilated as output feedbacks according to the queue length. Conversely, AVQ can be viewed as an output feedback with respect to the aggregate flow, belonging thus to the rate-based AQM class.
NS-2 simulations
As a widely adopted numerical illustration extracted from [7] (see figure 1 for the network topology), consider the case where q 0 = 175 packets, T p = 0.2 second and C = 3750 packets/s (corresponds to a 15 Mb/s link with average packet size 500 bytes). Then, for a load of N = 60 TCP sessions, we have W 0 = 15 packets, p 0 = 0.008, R 0 = 0.246 seconds. According to the synthesis criteria presented in Section 3, the state feedback matrices
are calculated for the construction of the control laws (3) and (17) respectively. We aim at proving the effectiveness of our method using NS-2 [3], a network simulator widely used in the communication networks community. Taking values from the previous numerical example, we apply the new AQM based on a state feedback. The target queue length q 0 is 175 packets while buffer size is 800. The average packet length is 500 bytes. The default transport protocol is TCP-New Reno without ECN marking. For the convenience of comparison, we adopt the same values and network configuration than [7] who design a PI controller (Proportional-Integral). This PI is configured as follow, the coefficients a and b are fixed at 1.822e − 5 and 1.816e − 5 respectively, the sampling frequency is 160Hz. The RED has been also tested using the parametric configuration recommended in [7]. In figure 4, simulations are performed under an external perturbation. This latter is composed of 7 additional sources (CBR applications over UDP protocol) sending 1000 bytes packet length with a 1Mbytes/s throughput between t = 40s and t = 100s. The two DSF (see figure 4 for the DSF based on the congestion window and the aggregate flow) regulate faster than others and are able to reject the disturbance swiftly. Conversely, figure 4 shows the time response of the queue length with a simple state feedback K SF (3) as an AQM. One can note that the queue is stabilized slightly above the desired level (around 200 pkts). Futhermore, the non reponsive cross traffic affects the steady state. The table 1 summarizes the benefits of the two K SF I AQMs (according to simulations with UDP cross traffics). Classical statistical characteristics are calculated during the whole simulation, then only during the UDP cross traffic and finally after the UDP cross traffic (come back to steady state). These characteristics are mean, standard deviation (Sdt) and the square of the variation coefficient (CV 2 = (Std/mean) 2 ). This latter calculation assess the relative dispersion of the queue length around its mean. The mean points out the control precision and the standard deviation shows the ability of the AQM to keep the queue size close to its equilibrium. In
Conclusion
In this preliminary work, we have proposed the design of an AQM for the congestion control in communications networks. The developed AQM has been constructed using a dynamic state feedback control law. An integral action has been added to reject the steady state error in spite of disturbance, d(t) (cross traffic). Finally, the AQM has been validated using NS simulator. Future work consist in the improvement about control laws (theoritical part) extended to a greater network using a decentralized approach to reduce the weakness of this method on one side and validation on emulation platform (experimental part) on the other side.
Figure 2 :
2Design of an AQM as a state feedback
K
SF
Figure 4 :
4Time evolution of the queue length, AQM = {RED, P I, K SF , K SF I(cwnd) , K SF I(Aggf l) } under UDP crossing traffic
Figure 5 :
5Time evolution of the queue length, AQM = {RED, AV Q, REM, P I, K SF I(cwnd) , K SF I(Aggf l) } under cross traffic [5] F. Gouaisbaut and D. Peaucelle. Delay-dependent stability analysis of linear time delay systems. In IFAC Workshop on Time Delay System (TDS'06), Aquila, Italy, July 2006.
is true. Matrix X is called "slack+
+
TCP
Dynamic
Dynamic
Queue
W(t)
Delay
K
1/s
u(t)
traffics
crossing
p(t)
+
+
Delay
AQM
p 0
W 0
δq(t)
table 1, we can observe that K SF I (cwnd) maintains a very good control on the buffer queue during the whole simulation. Even though K SF I (aggf l) is slightly slower than the previous one, statistics (Std and CV2) show again a good regulation. Although PI reject the perturbation quite fast, extensive fluctuations appear during the steady state. To conclude, the two DSF are efficient AQMs which provide the best precision and are able to regulate faster and closer to the mean compared to others AQMs. To complete our simulation, we propose another NS-2 simulation betweenTable 1: Statistical characteristics for different AQMs (units are pkts) at different periods (B, D and A: before, during and after CBR applications) different AQMs: REM, AVQ, RED, PI, KSFI(cwnd) and KSFI(aggfl). We consider different levels of CBR cross traffics (13 sources, 1Mb). RED, REM, PI and AVQ are fixed to same values as in[14], which is a performance analysis of AQM under DoS attacks. The additional sources are sending 1000 bytes packet length with a 1Mbytes/s throughput between t = 60s and t = 180s. The simulation is illustrated in the figure 5. In these last two cases, one can imagine that AQMs could detect cross traffics or traffic anomalies. Moreover, AQM = {K SF I(cwnd) , K SF I(Aggf l) } still have well behaviours under cross traffic conditions.AQMs
RED
PI
KSF
KSF I
(cwnd)
KSF I
(aggfl)
Mean
235.7
176.7 263.9
175.9
175.5 B
Sdt
112.40 71.19 78.59
54.57
63.64 B
CV2
0.227
0.162 0.088
0.096
0.131 B
Mean
270.3
178.3 338.0
173.4
175.6 D
Sdt
57.39
40.42 41.21
35.08
28.32 D
CV2
0.045
0.051 0.014
0.040
0.026 D
Mean
201.4
177.8 236.0
176.2
174.8
A
Sdt
22.24
36.64 30.48
30.22
32.64 A
CV2
0.012
0.042 0.016
0.029
0.034 A
Stabilization: design of an AQMIn Section 2, the model of TCP/AQM has been addressed as time delay system. The congestion problem needs the construction of a controller which regulates the buffer queue length. In this section, we are first going to present a delay dependent stability analysis condition for time delay systems. Then, based on this criterion, a synthesis method to derive a stabilizing state feedback is deduced.
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Analysis and design of an adaptive virtual queue (avq) algorithm for active queue management. S Kunniyur, R Srikant, SIGCOMM'01. San Diego, CA, USAS. Kunniyur and R. Srikant. Analysis and design of an adaptive virtual queue (avq) algorithm for active queue management. In SIGCOMM'01, pages 123-134, San Diego, CA, USA, aug 2001.
Design of lyapunov based controllers as tcp aqm. Y Labit, F Ariba, Gouaisbaut, 2nd IEEE Workshop on Feedback control implementation and design in computing systems and networks (FeBID'07). Munich, GermanyY. Labit, Y Ariba, and F. Gouaisbaut. Design of lyapunov based controllers as tcp aqm. In 2nd IEEE Workshop on Feedback control implementation and design in computing systems and networks (FeBID'07), pages 45-50, Munich, Germany, May 2007.
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|
[] |
[
"MODEL OF COMMUNITIES ISOLATION AT HIERARCHICAL MODULAR NETWORKS",
"MODEL OF COMMUNITIES ISOLATION AT HIERARCHICAL MODULAR NETWORKS"
] |
[
"Pawe L Kondratiuk \nFaculty of Physics\nCenter of Excellence for Complex Systems Research\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland\n",
"Janusz A Ho \nFaculty of Physics\nCenter of Excellence for Complex Systems Research\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland\n"
] |
[
"Faculty of Physics\nCenter of Excellence for Complex Systems Research\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland",
"Faculty of Physics\nCenter of Excellence for Complex Systems Research\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland"
] |
[] |
The model of community isolation was extended to the case when individuals are randomly placed at nodes of hierarchical modular networks. It was shown that the average number of blocked nodes (individuals) increases in time as a power function, with the exponent depending on network parameters. The distribution of time when the first isolated cluster appears is unimodal, non-gaussian. The developed analytical approach is in a good agreement with the simulation data.PACS numbers: 89.75. Hc, 89.75.Da, 89.75.Fb (1)
|
10.12693/aphyspola.121.b-67
|
[
"https://arxiv.org/pdf/1106.0439v1.pdf"
] | 14,218,026 |
1106.0439
|
c9cc5ad1b2883595fb6864642dabe067cc00b52a
|
MODEL OF COMMUNITIES ISOLATION AT HIERARCHICAL MODULAR NETWORKS
2 Jun 2011
Pawe L Kondratiuk
Faculty of Physics
Center of Excellence for Complex Systems Research
Warsaw University of Technology
Koszykowa 75PL-00-662WarsawPoland
Janusz A Ho
Faculty of Physics
Center of Excellence for Complex Systems Research
Warsaw University of Technology
Koszykowa 75PL-00-662WarsawPoland
MODEL OF COMMUNITIES ISOLATION AT HIERARCHICAL MODULAR NETWORKS
2 Jun 2011numbers: 8975Hc0250-r8975-k8975Da8975Fb
The model of community isolation was extended to the case when individuals are randomly placed at nodes of hierarchical modular networks. It was shown that the average number of blocked nodes (individuals) increases in time as a power function, with the exponent depending on network parameters. The distribution of time when the first isolated cluster appears is unimodal, non-gaussian. The developed analytical approach is in a good agreement with the simulation data.PACS numbers: 89.75. Hc, 89.75.Da, 89.75.Fb (1)
INTRODUCTION
Recently, hierarchical systems have been attracting attention of scientists working on complex networks [1,2,3,4,5]. In fact many real networks are hierarchically organized, e.g. WWW network, actor network, or the semantic web [1]. Dynamics at such networks can be qualitatively and quantitatively different from that at regular lattices (see [2,3,4]).
The Ising model at a network with a hierarchical topology was studied by Komosa and Ho lyst [2]. The analyzed parameters were, among others, magnetization, magnetic susceptibility, critical temperature and correlations of magnetization between different hierarchies. It was shown that the critical temperature is a power function of the network size and of the ratio k 2 k , where k stands for a node degree.
Opinion formation in hierarchical organizations was studied by Laguna et al. [3]. Agents, belonging to various authority strata, try to influence others opinions. The probability that an opinion of an agent of a certain authority prevails in the community depends on the size distribution of the authority strata. Phase diagrams can be obtained, where each phase corresponds to a distinct dominant stratum (or a sequence of the strata, with the decreasing probability of prevailing).
Fashion phenomena at hierarchical networks were studied by Galam and Vignes [4]. Interactions were imposed between social groups at different levels of hierarchy. A renormalization group approach was used to find the optimal investment level of the producer and to assess the influence of counterfeits on the probability of a new product success.
One of fundamental topics in social dynamics are conflict situations and many different sociophysics approaches [6,7,8] or prisoner's dilemma-type games [9] have been proposed. Recently a simple model of communities isolation has been introduced by Sienkiewicz and Ho lyst [10]. The model can describe such various issues as strategy at battlefields or formation of cultures. The idea behind this model is similar to the game of Go and it takes into account a natural leaning of people to avoid being surrounded by members of another (potentially hostile) community [11].
In this paper we extend the model of communities isolation studied for chains, hypercubic, random and scale-free networks [10,12] to hierarchical networks proposed by [1].
HIERARCHICAL NETWORKS
The model of hierarchical networks was proposed by Ravasz and Barabási [1] and modified by Suchecki and Ho lyst [5]. Such networks possess 3 parameters determining their structure:
• The degree of hierarchy h ∈ N ∪ {0} • The distribution P M (m), where m ∈ N, determining number of nodes at each level of hierarchy (in particular, the size of the cliques at the lowest level of hierarchy is m + 1) • The parameter determining the density of edges p ∈ [0, 1] Two models (referred to as P1 and PD models) were analyzed, which differ in the density of edges. Each network has a central node, referred to as a center of hierarchy. A network of hierarchy h = 0 is a complete graph of size m + 1 (m is a random number, chosen with probability P M (m)). The center of hierarchy, due to the symmetry, is an arbitrary node. In order to construct a network of hierarchy h > 0, one has to construct m + 1 subnetworks of hierarchy h − 1 and choose one of them -its center of hierarchy becomes a center of hierarchy v of the whole network. Afterwards, new connections (edges) are created: for each node w of m remaining subnetworks a connection (edge) (v, w) is created with probability p (in case of the P1 model) or p h (in case of the PD model). Sample networks created this way Periodic oscillations in degree distribution of such networks can be observed in log-log scale. The period, the amplitude and the shape of the peaks depend on the parameters of the network [5].
In this paper only the case with P M (m ′ ) = δ m ′ ,m (m = const) was considered, which corresponds to the original Ravasz and Barabási model [1].
BASIC ISOLATION MODEL
The model of communities isolation was proposed by Sienkiewicz and Ho lyst [10]. The rules are similar to those of the game of Go. A number of communities compete with each other, settling nodes of a network. In each step a random empty node is chosen. It is then settled by a member of randomly chosen community. A cluster of nodes occupied by one community becomes blocked when it gets surrounded by another community. The surrounded nodes are no more active in the game, i.e. they can not take part in surrounding other communities.
The case of communities competing at a chain was analyzed in [10]. Two functions describing the evolution were studied: the average number of blocked nodes over time and a mean critical time, i.e. the moment, when the first blocked cluster appears. In [12] the influence of external bias was considered when settling rates of competing communities are different.
In this paper the case of two competing communities at P1 and PD hierarchical networks is considered. Two parameters are analyzed: the average number of blocked nodes Z(t) and the critical time distribution P r(t c ).
NUMBER OF BLOCKED NODES OVER TIME
Case p = 0
For p = 0 models P1 are PD equivalent. The network consists of N m+1 isolated cliques of size m + 1. In such case
Z(0) = 0 Z(t + 1) = Z(t) + m i=1 ip i ,(1)
where p i -probability that in the (t + 1)th step i nodes will be blocked,
p i = t 2N m m i .(2)
After short algebra we obtain
Z(0) = 0 Z(t + 1) = Z(t) + m 2 t N m .(3)
The solution of this recursive equation is a (m + 1)th degree polynomial, which can be approximated by substituting the sum with the integral:
Z(t) = t−1 i=0 m 2 i N m ≈ t 0 m 2 x N m dx = m m + 1 t m+1 2N m .(4)
As one can see, Z(t) is a power function. The exponent β depends only on the m parameter, β = m + 1.
Case p = 1
In this case models P1 and PD are also equivalent. For networks of hierarchy h = 1:
Z (1) (t) = ρ 0 ρ m 1 m 2 + mρ m+1 1 m + 1 2 = 1 2 mρ 0 ρ m 1 (1 + (m + 1)ρ 1 ),(5)
where ρ i is a reduced density:
ρ i = ρ i (t) ≡ 0 for t < i t−i N −i for t ≥ i(6)
For networks of higher hierarchies, h ≥ 1, a recursive equation well approximating Z (h) (t) can be derived. The idea behind the formulas is that a clique can only be blocked if all the nodes of higher hierarchies neighboring with it are filled. Therefore Z (h) (t) = 0 if the center of hierarchy of the network (which neighbors with all the other nodes) is empty. In the opposite case,
Z (h) (t) depends on Z (h−1) (t), which describes each of m + 1 subnetworks. Z (1) i (t) = 1 2 mρ i ρ m i+1 (1 + (m + 1)ρ i+1 ) Z (h) i (t) = Z (h−1) i (t) + 1 2 mρ i Z (h−1) i+1 (t) + 1 4 ρ i ρ i+1 ρ (m+1) h −1 i+2 (m + 1) h + 1 Z (h) (t) ≡ Z (h) 0 (t)(7)
This equation can only be solved numerically. The solutions are presented in fig. 2 and 3. It can be noticed, that within a wide range of time t, Z (h) can be with reasonable accuracy approximated with a power function
Z (h) (t) ∝ t β ,(8)
where the exponents β are higher than in the case of p = 0 and they are close to m + 4.
General case
An analytical approximation of Z (h) (t) for networks with higher hierarchies (h > 1) when the parameter p is different from zero and one is far more difficult. Instead of searching for such a formula, an alternative approach was chosen. It was assumed that Z (h) (t) can be estimated from the proportion log Z
p=0 (t) − log Z (h) (t) log Z (h) (t) − log Z (h) p=1 (t) ≈ f (p, h) 1 − f (p, h) ,(9)
where f (p, h) ∈ [0, 1] should be an increasing function of p which, while not being too complicated, would give a reasonable approximation for the widest possible ranges of p and h. It turned out that in the case of the P1 model, choosing f (p, h) = p results in a good agreement of the Z (h) (t) function with simulation data. For the PD model, f (p, h) = p h 2 is a good choice.
CRITICAL TIME DISTRIBUTION
Case p = 0
As it was previously mentioned, in the case of p = 0 the network consists of N m+1 = (m + 1) h isolated cliques of m + 1 nodes. In order to find the distribution of critical time (time, when the first blocked cluster appears), one has to consider the probability that at time t there are no blocked nodes yet. It means that at time t the only completely filled cliques are those filled with members of one community, which leads to the formula
P r(t c > t) = 1 − α t N m+1 N m+1 ,(10)
where α ≡ 1 − 2 −m . The cumulative critical time distribution can be immediately obtained
P r(t c ≤ t) = 1 − 1 − α t N m+1 N m+1 ,(11)
as well as the critical time distribution in the approximation of continuous time: The mean critical time can be also calculated analytically:
P r(t c = t) = P r(t c ≤ t) − P r(t c ≤ t − 1) ≈ d dt P r(t c ≤ t) = α 1 − α t N m+1 N m+1 −1 t N m .(12)t c = N 0 tP r(t c = t)dt ≈ N 0 t d dt P r(t c ≤ t)dt = N m + 1 α − 1 m+1 B N m + 1 + 1, 1 m + 1 − B 2 −m ; N m + 1 + 1, 1 m + 1 ,(13)
where
B(a, b) ≡ 1 0 t a−1 (1 − t) b−1 dt (Euler beta function) and B(x; a, b) ≡ x 0 t a−1 (1 − t) b−1 dt (incomplete Euler beta function).
Case p = 1
For networks of hierarchy h = 0
P r(t c > t) = 1 − α t N m+1 = 1 − αρ m+1 0 .(14)
For networks with hierarchy h = 1
P r(t c > t) = 1 − ρ 0 + ρ 0 (1 − ρ m 1 ) 1 − αρ m+1 1 m .(15)
For networks with any degree of hierarchy, h ≥ 0, a recursive formula for the cumulative critical time distribution can be expressed as
F (0) i (t) = αρ m+1 i F (h) i (t) = ρ i − ρ i (1 − ρ m i+1 ) h−1 d=0 (1 − F (d) i+1 (t)) m P r (h) (t c ≤ t) ≡ F (h) 0 (t).(16)
The mean critical time can be obtained by numerical integration of P r (h) (t c ≤ t):
t c = N − N 0 P r (h) (t c ≤ t)dt.(17)
6. DISCUSSION AND CONCLUSIONS
Number of blocked nodes Z(t)
In all cases the function Z(t), defined as the average number of blocked nodes at time t, can be approximated with a high accuracy by a power function 13) and (17)).
Z(t) ∝ t β ,(18)
rescaling, therefore
Z (h+1) (ρ) ≈ (m + 1)Z (h) (ρ) = (m + 1)Cρ β = C ′ t β .(19)
For p = 0, the parameter β can be analytically found: β = m + 1. The result is in agreement with the simulated data. Increasing the density of connections (the p parameter) leads to the increase of β -up to approximately m + 4 for p = 1.
There is an important distinction in the way Z(t) was approximated for hypercubic and hierarchical networks. For hypercubic networks, the number of isolated nodes was calculated using the following approximation: all blocked nodes were blocked alone, i.e. they do not neighbor with other blocked nodes (of the same community). Although this approximation might seem coarse, resulting analytical predictions turned out to be in quite good agreement with simulated data [10,12]. For modular hierarchical networks, such an approximation would not be reasonable. Because of the fact that at the lowest level of hierarchy such networks consist of cliques of m + 1 nodes, the most probable are situations when m+1 2 nodes are simultaneously blocked.
Critical time t c
The second analyzed parameter was critical time t c , i.e. the moment, when the first isolated cluster appears. It is a random variable. The critical time distribution P r(t c ) was studied, as well as mean critical time t c . More precisely, a critical density (or a critical relative time)
ρ c ≡ t c N(20)
was often shown so networks with different parameters could be easily compared.
The P r(ρ c ) distribution is always unimodal. The mode (arg max P r(ρ c )) decreases with the increase of h and the standard deviation σ(ρ c ) decreases with m.
For p = 0 it was possible to find the analytical formula for both P r(t c ) and t c . The distribution P r(t c ) is a polynomial of degree m(m + 1)((m + 1) h − 1) (see eq. 12) and the average t c is a scaled difference of two Euler beta functions (see eq. 13). The average ρ c decreases with h and for a fixed h it reaches a minimum for m ≈ 2 (see fig. 5).
For p = 1 the distribution P r(ρ c ) reaches a constant, non-zero value for ρ c ∈ [1 − ǫ, 1] (for h ≤ 3, ǫ ≈ 0.1), which means that processes when blocked clusters firstly appear at the very end of the evolution are not unlikely.
The values of ρ c can be compared with those obtained for hypercubic networks. Similar trends can be observed in hypercubic and hierarchical networks: ρ c decreases with the network size N and increases with the average degree. However, for modular hierarchical networks the dependence of ρ c on the average degree (which equals m for p = 0 and rises with p) is very weak in comparison to hypercubic networks. Typical values of ρ c for hierarchical networks correspond to the ones obtained for two-or threedimensional networks, even for m ≫ 3.
Fig. 1 .
1Sample P1 networks with parameters P M (m) = Unif(2, 4), p = 0.5, with different degrees of hierarchy: (a) h = 0, (b) h = 1, (c) h = 2, (d) h = 3 are presented in fig. 1. Let us stress that the subnetworks do not have to be connected, especially if p is small. Some basic properties of such networks can be concluded from the construction algorithm: • For h ∈ {0, 1}, as well as for p ∈ {0, 1}, models P1 i PD are equivalent. • For p = 0 the network consists of isolated cliques of size m + 1 (mrandom variable). • For P M (m ′ ) = δ m ′ ,m the number of nodes (vertices) of the network equals N = (m + 1) h+1 .
Fig. 2 .Fig. 3 .
23(color online) Average number of blocked nodes, Z(t), for various networks of P1 model. Symbols correspond to data from computer simulations. Lines show analytical approximations (eq. (4),(7) and(9)). Left side -linear scale, right side -log-log scale. (color online) Average number of blocked nodes, Z(t), for various networks of PD model. Symbols correspond to data from computer simulations. Lines show analytical approximations (eq. (4),(7) and(9)). Left side -linear scale, right side -log-log scale.
Fig. 4 .Fig. 5 .
45(color online) Critical time distribution for networks of hierarchy h = 2 (left) and h = 3 (right), with p = 0 (top) and p = 1 (bottom). Symbols -simulated data, smooth lines -analytical approximation (eq. (12) and (16)).The β exponent depends on the parameters of the network. For d-dimensional hypercubic networks (including the 1-dimensional ones, i.e. chains) β = 2d − 1. For modular hierarchical networks β depends mainly on m and p parameters, i.e. on the sizes of basic cliques at the lowest hierarchy and on the density of inter-clique connections. The dependence on the degree of hierarchy h (and on the network size) is weak, what can be explained by the fact, that increasing the degree of hierarchy h is a process similar to system (color online) Mean critical time for various networks. Symbols correspond to data from the simulations, lines -to the analytical approximations (eq. (
AcknowledgmentsThe authors acknowledge support from the European COST Action MP0801 Physics of Competition and Conflicts and from the Polish Ministry of Science Grant No. 578/N-COST/2009/0.
. E Ravasz, A.-L Barabási, Phys. Rev. E. 6726112E. Ravasz, and A.-L. Barabási, Phys. Rev. E, 67, 026112 (2003).
Ising model at hierarchical network. S Komosa, J A Ho, to be publishedS. Komosa and J. A. Ho lyst, Ising model at hierarchical network, to be pub- lished.
. M F Laguna, S Risau Gusman, G Abramson, S Gonalves, J R Iglesias, Physica A. 351580M.F. Laguna, S. Risau Gusman, G. Abramson, S. Gonalves and J.R. Iglesias, Physica A 351, 580 (2005).
. S Galam, A Vignes, Physica A. 351605S. Galam and A. Vignes, Physica A 351, 605 (2005).
. K Suchecki, J A Ho, Acta Phys. Pol. B. 362499K. Suchecki and J. A. Ho lyst, Acta Phys. Pol. B 36, 2499 (2005).
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. G Deffuant, F Amblard, G Weisbuch, T , Faure JASSS. 5G. Deffuant, F. Amblard, G. Weisbuch, and T. Faure JASSS 5 (2002).
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. S Lee, P Holme, Z.-X Wu, Phys. Rev. Lett. 10628702S. Lee, P. Holme and Z.-X. Wu Phys. Rev. Lett 106, 028702 (2011).
. J Sienkiewicz, J A Ho, Phys. Rev. E. 8036103J. Sienkiewicz and J. A. Ho lyst, Phys. Rev. E 80, 036103 (2009).
. T C Schelling, J. of Math. Soc. 1143T.C. Schelling, J. of Math. Soc. 1, 143 (1971).
. J Sienkiewicz, G Siudem, J A Ho, Phys. Rev. E. 8257101J. Sienkiewicz, G. Siudem, J. A. Ho lyst, Phys. Rev. E 82, 057101 (2010).
|
[] |
[
"Strong-Field Polarizability-Enhanced Dissociative Ionization",
"Strong-Field Polarizability-Enhanced Dissociative Ionization"
] |
[
"Lun Yue \nInstitute for Physical Chemistry and Abbe Center for Photonics\nFriedrich-Schiller University\n07743JenaGermany\n",
"Philipp Wustelt \nInstitute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany\n",
"A Max Sayler \nInstitute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany\n",
"Gerhard G Paulus \nInstitute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany\n",
"Stefanie Gräfe \nInstitute for Physical Chemistry and Abbe Center for Photonics\nFriedrich-Schiller University\n07743JenaGermany\n"
] |
[
"Institute for Physical Chemistry and Abbe Center for Photonics\nFriedrich-Schiller University\n07743JenaGermany",
"Institute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany",
"Institute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany",
"Institute of Optics and Quantum Electronics\nFriedrich-Schiller University\n07743JenaGermany",
"Institute for Physical Chemistry and Abbe Center for Photonics\nFriedrich-Schiller University\n07743JenaGermany"
] |
[] |
We investigate dissociative single and double ionization of HeH + induced by intense femtosecond laser pulses. By employing a semi-classical model with nuclear trajectories moving on field-dressed surfaces and ionization events treated as stochastical jumps, we identify a strong-field mechanism wherein the molecules dynamically align along the laser polarization axis and stretch towards a critical internuclear distance before getting dissociative ionized. As the tunnel-ionization rate is greater for larger internuclear distance and for aligned samples, ionization is enhanced. The strong dynamical rotation is traced back to a maximum in the parallel component of the internucleardistance-dependent polarizability tensor. Qualitative agreement with our experimental observations is found. Finally the criteria for observing the isotope effect for the ion angular distribution is discussed.PACS numbers: 33.80. Wz, 33.80.Eh, 42.50.Hz The ionization and dissociation of small molecules in intense laser fields is of fundamental interest and has captured the attention of physicists for many years[1][2][3]. When the ratio of the laser frequency to the peak electric field is sufficiently small, the ionization process can be considered as an electron tunneling through the instantaneous barrier formed by the field and the Coulomb potential of the system [4]. In molecules, such tunnelionization rates depend on the spatial separation between the nuclei [5-9], as well as the molecular orientation with respect to the laser polarization axis, where the ionization rate follows the shape of the highest-occupied molecular orbital[10][11][12]. In addition to these fixed-nuclei properties, the molecules will dynamically rotate and stretch in the field, potentially leading to fragmentation[8]. Here, we theoretically identify a new fragmentation pathway that involves the combination of the aforementioned strong-field dynamics. Namely, due to the force resulting from the internuclear-distance-dependent polarizability tensor, the molecule is simultaneously aligned and stretched towards a specific internuclear distance in the field-dressed ground state before being ionized. We denote it as polarizability-enhanced dissociative ionization (PEDI) and support our findings with experimental data.Polarizability effects have been explored extensively in strong-field physics, e.g. it has regularly appeared in the interpretation of strong-field ionization experiments[13,14], and the anisotropic polarizability is often exploited in molecular alignment experiments[15][16][17]. In dissociative ionization studies involving short laser pulses (tens of femtoseconds (fs) duration), often only the vibrational degrees of freedom are considered, while the rotational dynamics are disregarded. This is based on the intuition that the field-free rotational timescale of picoseconds is much greater than the vibrational timescale of fs and thus rotational motion can be safely neglected.However, as several works employing semiclassical methods[7,18,19]have shown, rotational dynamics are crucial for the understanding of the angular distribution of the final ion fragments. Even at lower intensities where ionization is negligible and pure dissociation is the dominating fragmentation process, it was shown that molecular rotations play a role[20][21][22][23][24]for pulses as short as ∼ 5 fs.For our studies, we choose to focus on the simplest stable polar molecule, HeH + , sketched inFig. 1(d). Aside from HeH + being a reference system, our choice is motivated by the fact that the two lowest electronic states, X 1 Σ + and A 1 Σ + [seeFig. 1(a)], have a large energy separation of 13 − 39 eV in the Franck-Condon region and a vanishing dipole coupling between them at large R. This highly limits the effect of the excited states and leads to the essential physics occuring in the electronic ground state. This is in stark contrast to H + 2 , where the two lowest charge-resonance states[25]are energetically separated by a few IR photons at intermediate R, and degenerate and strongly coupled at large R. Indeed, most of the prominent breakup processes first discovered in H + 2 depend on the efficient population of the first excited state. These processes include above-threshold dissociation[26][27][28][29], bond-hardening and bond-softening[30][31][32][33], electron localization [6, 34-37], above-threshold Coulomb explosion [38] and charge-resonance enhanced ionization[5]. Although PEDI should be present in H + 2 as well as in other more complicated molecules, more prominent population and coupling of excited states would lead to many processes being intertwined, making an identification of PEDI difficult. It should be mentioned that the first excited state in HeH + has received attention in terms of the enhanced ionization (EI) phenomenon in polar molecules[9,39]: at a critical internuclear distance, due to the crossing of the two lowest Stark-shifted energy levels, enhanced population and ionization of the arXiv:1807.00779v1 [physics.atom-ph]
|
10.1103/physreva.98.043418
|
[
"https://arxiv.org/pdf/1807.00779v1.pdf"
] | 53,692,434 |
1807.00779
|
decc8569abb213121a09453fa2927278cc91144d
|
Strong-Field Polarizability-Enhanced Dissociative Ionization
2 Jul 2018
Lun Yue
Institute for Physical Chemistry and Abbe Center for Photonics
Friedrich-Schiller University
07743JenaGermany
Philipp Wustelt
Institute of Optics and Quantum Electronics
Friedrich-Schiller University
07743JenaGermany
A Max Sayler
Institute of Optics and Quantum Electronics
Friedrich-Schiller University
07743JenaGermany
Gerhard G Paulus
Institute of Optics and Quantum Electronics
Friedrich-Schiller University
07743JenaGermany
Stefanie Gräfe
Institute for Physical Chemistry and Abbe Center for Photonics
Friedrich-Schiller University
07743JenaGermany
Strong-Field Polarizability-Enhanced Dissociative Ionization
2 Jul 2018(Dated: July 3, 2018)
We investigate dissociative single and double ionization of HeH + induced by intense femtosecond laser pulses. By employing a semi-classical model with nuclear trajectories moving on field-dressed surfaces and ionization events treated as stochastical jumps, we identify a strong-field mechanism wherein the molecules dynamically align along the laser polarization axis and stretch towards a critical internuclear distance before getting dissociative ionized. As the tunnel-ionization rate is greater for larger internuclear distance and for aligned samples, ionization is enhanced. The strong dynamical rotation is traced back to a maximum in the parallel component of the internucleardistance-dependent polarizability tensor. Qualitative agreement with our experimental observations is found. Finally the criteria for observing the isotope effect for the ion angular distribution is discussed.PACS numbers: 33.80. Wz, 33.80.Eh, 42.50.Hz The ionization and dissociation of small molecules in intense laser fields is of fundamental interest and has captured the attention of physicists for many years[1][2][3]. When the ratio of the laser frequency to the peak electric field is sufficiently small, the ionization process can be considered as an electron tunneling through the instantaneous barrier formed by the field and the Coulomb potential of the system [4]. In molecules, such tunnelionization rates depend on the spatial separation between the nuclei [5-9], as well as the molecular orientation with respect to the laser polarization axis, where the ionization rate follows the shape of the highest-occupied molecular orbital[10][11][12]. In addition to these fixed-nuclei properties, the molecules will dynamically rotate and stretch in the field, potentially leading to fragmentation[8]. Here, we theoretically identify a new fragmentation pathway that involves the combination of the aforementioned strong-field dynamics. Namely, due to the force resulting from the internuclear-distance-dependent polarizability tensor, the molecule is simultaneously aligned and stretched towards a specific internuclear distance in the field-dressed ground state before being ionized. We denote it as polarizability-enhanced dissociative ionization (PEDI) and support our findings with experimental data.Polarizability effects have been explored extensively in strong-field physics, e.g. it has regularly appeared in the interpretation of strong-field ionization experiments[13,14], and the anisotropic polarizability is often exploited in molecular alignment experiments[15][16][17]. In dissociative ionization studies involving short laser pulses (tens of femtoseconds (fs) duration), often only the vibrational degrees of freedom are considered, while the rotational dynamics are disregarded. This is based on the intuition that the field-free rotational timescale of picoseconds is much greater than the vibrational timescale of fs and thus rotational motion can be safely neglected.However, as several works employing semiclassical methods[7,18,19]have shown, rotational dynamics are crucial for the understanding of the angular distribution of the final ion fragments. Even at lower intensities where ionization is negligible and pure dissociation is the dominating fragmentation process, it was shown that molecular rotations play a role[20][21][22][23][24]for pulses as short as ∼ 5 fs.For our studies, we choose to focus on the simplest stable polar molecule, HeH + , sketched inFig. 1(d). Aside from HeH + being a reference system, our choice is motivated by the fact that the two lowest electronic states, X 1 Σ + and A 1 Σ + [seeFig. 1(a)], have a large energy separation of 13 − 39 eV in the Franck-Condon region and a vanishing dipole coupling between them at large R. This highly limits the effect of the excited states and leads to the essential physics occuring in the electronic ground state. This is in stark contrast to H + 2 , where the two lowest charge-resonance states[25]are energetically separated by a few IR photons at intermediate R, and degenerate and strongly coupled at large R. Indeed, most of the prominent breakup processes first discovered in H + 2 depend on the efficient population of the first excited state. These processes include above-threshold dissociation[26][27][28][29], bond-hardening and bond-softening[30][31][32][33], electron localization [6, 34-37], above-threshold Coulomb explosion [38] and charge-resonance enhanced ionization[5]. Although PEDI should be present in H + 2 as well as in other more complicated molecules, more prominent population and coupling of excited states would lead to many processes being intertwined, making an identification of PEDI difficult. It should be mentioned that the first excited state in HeH + has received attention in terms of the enhanced ionization (EI) phenomenon in polar molecules[9,39]: at a critical internuclear distance, due to the crossing of the two lowest Stark-shifted energy levels, enhanced population and ionization of the arXiv:1807.00779v1 [physics.atom-ph]
We investigate dissociative single and double ionization of HeH + induced by intense femtosecond laser pulses. By employing a semi-classical model with nuclear trajectories moving on field-dressed surfaces and ionization events treated as stochastical jumps, we identify a strong-field mechanism wherein the molecules dynamically align along the laser polarization axis and stretch towards a critical internuclear distance before getting dissociative ionized. As the tunnel-ionization rate is greater for larger internuclear distance and for aligned samples, ionization is enhanced. The strong dynamical rotation is traced back to a maximum in the parallel component of the internucleardistance-dependent polarizability tensor. Qualitative agreement with our experimental observations is found. Finally the criteria for observing the isotope effect for the ion angular distribution is discussed. The ionization and dissociation of small molecules in intense laser fields is of fundamental interest and has captured the attention of physicists for many years [1][2][3]. When the ratio of the laser frequency to the peak electric field is sufficiently small, the ionization process can be considered as an electron tunneling through the instantaneous barrier formed by the field and the Coulomb potential of the system [4]. In molecules, such tunnelionization rates depend on the spatial separation between the nuclei [5][6][7][8][9], as well as the molecular orientation with respect to the laser polarization axis, where the ionization rate follows the shape of the highest-occupied molecular orbital [10][11][12]. In addition to these fixed-nuclei properties, the molecules will dynamically rotate and stretch in the field, potentially leading to fragmentation [8]. Here, we theoretically identify a new fragmentation pathway that involves the combination of the aforementioned strong-field dynamics. Namely, due to the force resulting from the internuclear-distance-dependent polarizability tensor, the molecule is simultaneously aligned and stretched towards a specific internuclear distance in the field-dressed ground state before being ionized. We denote it as polarizability-enhanced dissociative ionization (PEDI) and support our findings with experimental data.
Polarizability effects have been explored extensively in strong-field physics, e.g. it has regularly appeared in the interpretation of strong-field ionization experiments [13,14], and the anisotropic polarizability is often exploited in molecular alignment experiments [15][16][17]. In dissociative ionization studies involving short laser pulses (tens of femtoseconds (fs) duration), often only the vibrational degrees of freedom are considered, while the rotational dynamics are disregarded. This is based on the intuition that the field-free rotational timescale of picoseconds is much greater than the vibrational timescale of fs and thus rotational motion can be safely neglected.
However, as several works employing semiclassical methods [7,18,19] have shown, rotational dynamics are crucial for the understanding of the angular distribution of the final ion fragments. Even at lower intensities where ionization is negligible and pure dissociation is the dominating fragmentation process, it was shown that molecular rotations play a role [20][21][22][23][24] for pulses as short as ∼ 5 fs.
For our studies, we choose to focus on the simplest stable polar molecule, HeH + , sketched in Fig. 1(d). Aside from HeH + being a reference system, our choice is motivated by the fact that the two lowest electronic states, X 1 Σ + and A 1 Σ + [see Fig. 1(a)], have a large energy separation of 13 − 39 eV in the Franck-Condon region and a vanishing dipole coupling between them at large R. This highly limits the effect of the excited states and leads to the essential physics occuring in the electronic ground state. This is in stark contrast to H + 2 , where the two lowest charge-resonance states [25] are energetically separated by a few IR photons at intermediate R, and degenerate and strongly coupled at large R. Indeed, most of the prominent breakup processes first discovered in H + 2 depend on the efficient population of the first excited state. These processes include above-threshold dissociation [26][27][28][29], bond-hardening and bond-softening [30][31][32][33], electron localization [6,[34][35][36][37], above-threshold Coulomb explosion [38] and charge-resonance enhanced ionization [5]. Although PEDI should be present in H + 2 as well as in other more complicated molecules, more prominent population and coupling of excited states would lead to many processes being intertwined, making an identification of PEDI difficult. It should be mentioned that the first excited state in HeH + has received attention in terms of the enhanced ionization (EI) phenomenon in polar molecules [9,39]: at a critical internuclear distance, due to the crossing of the two lowest Stark-shifted energy levels, enhanced population and ionization of the first excited state occurs. However, the EI discovery was based on a fixed-nuclei picture, and we find for a more realistic scenario, i.e. with moving nuclei and a molecule which is initially prepared close to the equilibrium R, the dominating effect is that ionization is enhanced due to the joined dynamics of rotation and stretching of the molecule. Recent quantum calculations with vibrating HeH + also confirm the dominance of ionization from the ground state [40].
In this letter, strong-field dissociative single and double ionization (SI and DI) of HeH + is simulated employing a semiclassical approach with classical nuclear trajectories moving on field-dressed surfaces and ionization treated as stochastical jumps. Such an approach allows us to treat rotation, stretching, single and double ionization, while at the same time keep the computational effort manageable to perform scans over an extensive laser-parameter space with inclusion of the initial rotational temperature, vibrational-and intensity-focal-volume-averaging effects. A full quantum treatment at the laser intensities considered in this work ( 10 16 W/cm 2 ) is computationally prohibitive.
In the HeH + sketch in Fig. 1(d), R denotes the internuclear distance and θ the angle between the field polarization and the molecular axes. The field-dressed energy surfaces read (atomic units are used unless stated otherwise) the Born-Oppenheimer curves, µ (s) the permanent dipole moments and α (s) ⊥ (α (s) ) the molecular-frame perpendicular (parallel) components of the polarizability tensors (obtained on the CASSCF(15/2)/aug-cc-pVQZ [41] level of theory calculated with MOLCAS [42]). For HeH Fig. 1(b)], which leads to distinct wells at R c and cos θ = ±1 in the cycleaveraged field-dressed energy surface [ Fig. 1(c)]. The peak in α (s) is understood in terms of the plotted density isosurfaces in Fig. 1
E (s) (R, θ, t) = E (s) 0 (R) − µ (s) (R)F (t) cos θ − 1 2 F 2 (t) α (s) ⊥ (R) + α (s) (R) − α (s) ⊥ (R) cos 2 θ ,(1)+ (X 1 Σ + ), the anisotropic polarizability α (s) −α (s) ⊥ exhibits a distinct maximum at R c = 2.6 [(b): For R → 0 (R → ∞)
, the electron cloud is spatially confined and resides almost completely on the united-atom 5 Li nucleus ( 4 He nucleus), resulting in small polarizabilities, while at intermediate R's the density is spatially extended, resulting in large polarizabilities. Recent quantum chemistry calculations on alkali dimers have also noticed the polarizability maximum and its possible implications for alignment experiments [43]. Each trajectory moves classically on the instantaneous HeH field-dressed surface [Eq. (1)], with ionization treated as stochastical jumps [18,44]. For the pulses considered in this work, the Keldysh parameter is γ < 1, indicating the tunneling regime of strong-field ionization [4]. We employ lowest-order many-electron weak-fieldasymptotic theory (ME-WFAT) [45], where the ioniza-
+ (X 1 Σ + )→ HeH 2+ (1sσ)+e − 0.0 0.2 0.4 0.6 0.8 HeH 2+ (1sσ)→ HeH 3+ +e − |G 000 (θ)| 2 (a.u.) θ (π) θ (π) R = 1.tion rate Γ (s) (F, θ; R) = G (s) (θ; R) 2 W (s) (F ; R)
is given in terms of the structure factor G (s) (θ; R) 2 and the field factor
W (s) = κ (s) 2 4κ (s) 2 F 2Z (s) /κ (s) −1 exp − 2κ (s) 3 3F ,(2)
with κ (s) = 2I the Rdependent ionization potentials. An additional emperical factor exp[−14Z (s) 2 F/κ (s) 5 ] is applied to counteract the overestimation of the rates at large F [11]. The structure factors for R = 1 − 4 are obtained using the method in Ref. [46] and extrapolated to the values of the relevant atoms for R → ∞. As shown in Fig. 2, the electron favourably tunnels from the hydrogen side, in agreement with earlier works [9,12,47]. The initial conditions of R and p R for each trajectory are chosen randomly according to a Husimi distribution of a given vibrational state, and the molecules are chosen to be randomly oriented. The initial angular momenta p θ are distributed according to a Boltzmann-distribution to account for the high experimental rotational temperature in the ion source, T = 3400 ± 300 K [48,49]. The laser wavelength is 800 nm and the pulse has a sin 2 envelope with the full width at half maximum τ = 34 fs. For each intensity and initial vibrational state, 10 3 trajectories are released. All the results presented in this work are averaged over the initial vibrational-state distribution [48], the laser beam focal volume [50], and the carrier-envelope phase, unless states otherwise.
The experimental setup is identical to that described in Ref. [51]. Briefly, the HeH + ions are synthesized in a duoplasmatron ion source, accelerated to 10 keV kinetic energy, and focused towards the laser interaction region, where a tabletop Ti:Sapphire laser system provides 800nm-pulses with peak intensities of up to 10 17 W/cm 2 and intensity full width at half maximum duration τ ∼ 34 fs. Figure 3 shows the kinetic energy release (KER) and angular distributions of the nuclear fragments from SI The SI experimental result in Fig. 3(e) shows that the fragments have KERs of 7-17 eV with the angular distribution aligned along the laser polarization direction (cos θ = 1). Due to the Faraday cup used in the experimental setup to block the non-fragmented HeH + beam, the yields at cos θ ≈ 1 are not fully detected. Also, due to the finite size of the detector, the measureable range of cos θ is limited, resulting in zero counts in the grayed region in Figs. 3(e). Our simulation results with T = 3500 K in Fig. 3(d) agrees with the experiment, with the fragments aligned along the laser polarization axis and having KER 7-18 eV, corresponding to ionization at R ∼ 1.5 − 3.5 (see Fig. 4). Compared with Fig. 3(c), a nonzero temperature is seen to broaden the angular distribution. Three intertwined effects lead to the final aligned ion distribution: dynamic alignment during the laser pulse, geometric alignment [8,22] denoting the orientation-dependent ionization (Fig. 2), and postionization alignment [19]. By artificially disabling the molecular rotation in the simulations, we see in Fig. 3(b) that the angular distribution is quasi-flat and cannot mimick the experimental results, thus geometric alignment plays a lesser role for the final ion angular distribution. Simulation results where we disabled the molecular stretching shown in Fig. 3(a) have too high KER (peaked at 17 eV) compared to the full result in Fig. 3(d), corresponding to ionization events occuring at smaller R's.
The measured DI yields in Fig. 3(j) are also aligned, with KER 8-25 eV, corresponding to the second ionization event occuring between R ∼ 5 − 25 [51]. The simulation with T = 3500 K in Fig. 3(i) agrees with these observations. The angular distributions for SI and DI are similar, because after a molecule is singly ionized, it will dissociate and its moment of inertia M R 2 (M the reduced mass) increases, effectively freezing θ before DI. For both SI and DI, the ionization probabilities P SI = 0.192 [ Fig. 3(d)] and P DI = 0.071 [ Fig. 3(i)] are larger when the molecules are allowed to strech and rotate, indicating increased ionization due to dynamical alignment and stretching towards R = R c .
We estimate the rotational timescale as the time it takes a trajectory to reach from half of the well depth in Fig. 1(c) to the potential minima, i.e. τ rot = √ M h(R)/F , with h(R) ≈ 2.62R/ α − α ⊥ . At R c , with M HeH + = 1469 and F = 0.37, we have τ rot = 12.8 fs which is comparable to the field-free vibrational period τ vib = 11.5 fs and 37 times shorter than that of the fieldfree rotational period of 478 fs. Dynamical rotation is expected to be prominent when the rotational timescale τ rot is comparable to or shorter than the pulse duration τ , which is the case for τ = 34 fs used in the experiment. Fig. 4 shows the (R j1 , θ j1 )-distribution of the singlyionized molecules at the instant of ionization. For the results in Fig. 4(c) with rotation artificially disabled, the ionization occur over a broad range of cos θ j1 . With increasing cos θ j1 , the ionization are seen to be shifted towards larger R j1 , e.g. we have R j1 ≈ 1.8 for cos θ j1 = 0, and R j1 ≈ 2.2 for cos θ j1 = 1. Also, more ionization events occur for the aligned molecules (cos θ ≈ 1). The reason is clear: without rotation, the aligned molecules need to stretch towards larger R [see Fig. 1(c)] where the lower I p and the favourable structure factor in Fig. 2 leads to more ionization. The projected R j1 -distribution in the upper panel of Fig. 4(c) peaks at R j1 ≈ 1.9, corresponding to E N ≈ 14 eV by reflection. This is in disagreement with the experimental KER maximum in Fig. 3(e) at E N ≈ 11 eV. For the simulation results in Fig. 4(b) where rotational motion is included, most molecules ionize after they are dynamically aligned, with the projected R j1 -distribution peaked at R ≈ 2.1, corresponding to a reflected KER E N ≈ 12.5 eV, in better agreement with the experimental value. For the short pulse τ = 4.9 fs τ rot in Fig. 4(a), the molecules do not have much time to rotate and stretch, resulting in a smaller R j1 's and thus much higher KERs. We mention that we have not taken the EI effect [9,39,47] into account in our ionization rates, which will take place at larger R (horizontal dashed lines in Fig. 4). Since most ionization events occur before this region, we expect its effect to be small.
To understand more in detail the bound-state dynamics before ionization, we have considered the artificial case with ionization switched off. The time-evolution of the bound nuclear densities are shown in Fig. 5. Since the low-intensity regions in the focal-volume dominate, we have only considered a single intensity I = 5×10 15 W/cm 2 without performing the focal-volume averaging. For the short pulse in Fig. 5(a), τ < τ rot ≈ τ vib , and the trajectories do not have time to rotate and vi- brate, which results in a minimal change of the density during the pulse. This is different for a long pulse shown in Fig. 5(b). The molecules have time to align and stretch towards R c , and at the field maximum t = 46.7 fs a substantial part of the density has reached R ∈ [2,3]. During the second half of the pulse, the already aligned molecules contract back towards the equilibrium R 0 , with density oscillations due to the vibrational motion distinctly observed in Fig. 5(b). When the molecule is not allowed to rotate [ Fig. 5(c)], similar to the case of a short pulse, the stretching of the nuclei follows the dressed potential, but only molecules initially aligned along the laser polarization direction (cos θ = 1) stretch towards R c .
The degree of alignment can be characterized by the alignment parameter cos 2 θ (equal to 1 for completely aligned samples and 1 3 for randomly oriented samples) and is given as a function of I for HeH + in Fig. 6(a). For SI, higher temperature and shorter pulses result in broader distributions. With increasing intensities, the detected fragments are seen to be less aligned, despite the rotational timescale τ rot ∝ 1/F being shorter. This is indeed a characteristic of dynamical rotation: with increasing I, ionization will occur earlier during the pulse, leaving the molecule with less time to rotate [8]. The experimental values for cos 2 θ are shown in Fig. 6(a) as a circle and square for SI and DI, respectively. To avoid inaccuracies resulting from the properties of the experimental setup we consider only the counts in the interval E N ∈ [0, E max ] in Figs. 3(e) and 3(j), with E max = 7 eV, and fitted the missing yields at cos θ ≈ 1 with a function f (cos θ) = a cos 2 θ + b cos c θ. The bars belonging to the data points are for the extremal values of cos 2 θ for E max ∈ [4,9] eV and are seen to be consistent with the simulation results.
Since τ rot ∝ √ M , the dynamical rotation is influenced by the specific isotope. Indeed, recently the isotope effect in tunneling ionization of molecules has attracted considerable attention [52,53]. We finally investigate whether an isotope effect can be observed in the ion angular distributions of HeH + and HeD + . Fig. 6(b) presents the simulated and experimental results for HeD + . With the employed experimental laser parameters, however, this isotope effect cannot be conclusively resolved. To experimentally identify this would require shorter pulses, as depicted in Fig. 6(c) showing the normalized difference between the curves in Figs. 6(a) and 6(b), S ≡ cos 2 θ HeH + − cos 2 θ HeD + / cos 2 θ HeH + + cos 2 θ HeD + . For the 34 fs, 3500 K case, S < 0.014, while for the 9.7 fs pulse S has more than doubled to 0.033. In Figs. 6(d)-(f) we show cos 2 θ for HeH + , HeD + and their relative differences over an extensive pulse regime. It is seen that the isotope effect for dynamical rotation is pronounced for pulse durations of less than 10 fs, potentially allowing for an experimental detection.
In conclusion, we have identified polarizabilityenhanced dissociative ionization, a strong-field molecular breakup pathway where the molecules dynamically align and stretch towards a specific internuclear distance before ionization, with geometric alignment playing a lesser role. For our studies we have focused on the fundamental polar molecules HeH + and HeD + , and we believe that the effect is quite general and more pronounced for polar diatomics. Indeed, the maximum in the anisotropy polarizability is present in all diatomics except for oddcharged molecular ions where the parallel polarizability will monotonically increase with R [25].
The authors acknowledge support from the German Research Foundation (DFG-SPP-1840 "Quantum Dynamics in Tailored Intense Fields"). L. Y. thanks Johannes Steinmetzer for aiding in the calculation of the molecular polarizabilities.
PACS numbers: 33.80.Wz, 33.80.Eh, 42.50.Hz
with the superscript (s) indicating either the ground states of HeH + and HeH 2+ , or the Coulomb potential of HeH 3+ [Fig. 1(a)], F the instantaneous electric field, E (s) 0
FIG. 1 .
1(a) Field-free BO curves considered in this work for HeH + and its daughter ions. The Franck-Condon region from the initial vibrational distribution in HeH + is indicated by the shaded region. (b) Components of the polarizability tensors as a function of internuclear distance. The densities (isosurface value 0.04) of HeH + (X 1 Σ + ) are plotted for R =1.0, 2.6 and 6.0. (c) Cycle-averaged field-dressed potential for F = 0.53 [see Eq. (1) and the following text]. (d) Sketch of HeH + .
FIG. 2 .
2Calculated structure factors for single and (sequential) double ionization G (s) (θ; R) 2 included in the ME-WFAT ionization rates.
PFIG. 3 .
3, Z (s) the nuclear charge seen by KER-and orientation-dependent fragmentation yields from single (upper panels) and double dissociative ionization (lower panels) of HeH + for λ = 800 nm, I = 9 × 10 15 W/cm 2 and τ = 34 fs. Panels (e) and (j): experimental results at laser peak intensity (∼ 10 17 W/cm 2 ), with the shaded area denoting absolute zero counts due to the experimental setup (see text). Colorscale in arbitrary linear units.the outgoing electron asymptotically and I (s) P
FIG. 4 .
4Simulated distribution of R and cos θ for the singlyionized molecules at the instant of ionization for I = 9 × 10 15 W/cm 2 and T = 0 K. (a) τ = 4.9 fs; (b) τ = 34 fs; (c) fixed θ (disabled rotation) with τ = 34 fs. The vertical dashed line indicates Rc, the horizontal dotted line approximately traces the enhanced ionization region from [39] (not included in our simulations), and the and the upper (blue) axis shows the kinetic energy by the reflection principle. and DI. Due to the high number of optical cycles, the directional ionization asymmetry for θ = 0 and θ = π [Fig. 2] are averaged out, resulting in symmetric yields in the intervals cos θ ∈ [0, 1] and [−1, 0], of which we only consider the former.
FIG. 5 .
5Simulated time-evolution of the nuclear densities in the artificial case where ionization is turned off, for I = 5 × 10 15 W/cm 2 and T = 0 K. (a) τ = 4.9 fs; (b) τ = 34 fs; (c) fixed θ (disabled rotation) with τ = 34 fs. The results are vibrational averaged, but not intensity-focal-volume averaged.
FIG. 6 .
6Alignment parameter cos 2 θ for (a,d) HeH + , (b,e) HeD + , and (c,f) their normalized difference. In (a,b), the fitted experimental results at peak intensity, 800 nm and 34 fs for SI and DI are given by the (black) circle and (red) square, with the bars denoting the uncertainty in the fit (see text).
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[
"Probabilistic approach to Appell polynomials",
"Probabilistic approach to Appell polynomials"
] |
[
"Bao Quoc \nDepartment of Natural Sciences/Mathematics and Statistics\nÅbo Akademi University\nFIN-20500ÅboFinland\n",
"Ta \nDepartment of Natural Sciences/Mathematics and Statistics\nÅbo Akademi University\nFIN-20500ÅboFinland\n"
] |
[
"Department of Natural Sciences/Mathematics and Statistics\nÅbo Akademi University\nFIN-20500ÅboFinland",
"Department of Natural Sciences/Mathematics and Statistics\nÅbo Akademi University\nFIN-20500ÅboFinland"
] |
[] |
In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also discuss moment representations of Appell polynomials. In the latter part of the paper the presented general theory is applied to study some classical explicitly given polynomials.
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10.1016/j.exmath.2014.07.003
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[
"https://arxiv.org/pdf/1311.4999v1.pdf"
] | 119,715,392 |
1311.4999
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fee1ccd6d6acfbce73fe4571738e79034b6e6662
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Probabilistic approach to Appell polynomials
20 Nov 2013
Bao Quoc
Department of Natural Sciences/Mathematics and Statistics
Åbo Akademi University
FIN-20500ÅboFinland
Ta
Department of Natural Sciences/Mathematics and Statistics
Åbo Akademi University
FIN-20500ÅboFinland
Probabilistic approach to Appell polynomials
20 Nov 2013arXiv:1311.4999v1 [math.PR]Appell polynomialsgamma distributionBernoulliEu- lerHermiteLaguerre polynomials 2010 mathematics subject classification: 11B6833C4560E05
In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also discuss moment representations of Appell polynomials. In the latter part of the paper the presented general theory is applied to study some classical explicitly given polynomials.
Introduction
Paul Appell introduced and studied in his paper [2] published 1880, a system of polynomials {Q n , n = 0, 1, . . . }, where Q n is of order n, satisfying the recursive differential equation
d dx Q n (x) = nQ n−1 (x), n ≥ 1.
Such polynomials, now called Appell polynomials, have since Appell's pioneering work appeared and applied in a variety of mathematical fields, e.g., number theory, numerical analysis and probability theory. Some much studied classical polynomials like Bernoulli, Euler, Hermite polynomials are, in fact, Appell polynomials. Our main interest in Appell polynomials arises from fairly recent observation due to Novikov and Shiryaev [13], see also Kyprianou and Surya [10] and Salminen [16], that Appell polynomials are of key importance when solving optimal stopping problems with power reward functions and a Lévy process as the underlying. Another application of Appell polynomials is in constructing time-space martingales for Lévy processes (see, e.g., [17,19]). Picard and Lefèvre [14] apply Appell polynomials for the problem of ruin in insurance models. We refer to [8,11,15] for more applications in insurance mathematics. With the applications above in mind we find it motivated to present in detail in this survey paper basic properties of Appell polynomials. We focus on the probabilistic approach in which to every random variable with finite moments or some exponential moments is associated a family of Appell polynomials, see Definition 2.1 and 2.3 in the next section. Connecting Appell polynomials to random variables provides us, in particular, when the random variable has some exponential moments, with powerful tools to deeper understanding of the behaviour and properties of Appell polynomials. Our general discussion is completed with examples of classical Appell polynomials: Bernoulli, Euler, Hermite and Laguerre polynomials. It is demonstrated that the probabilistic approach offers us new proofs of some old results for these polynomials but yields also interesting new developments, e.g., the moment representations.
The paper is organized as follows. In the next section Appell polynomials are defined, and many general properties are scrutinized. In particular, we give a simple proofs of the mean value property, find conditions for the moment representation of Appell polynomials and discuss the property of the unique positive root (which is important in optimal stopping). In section 3 the results in section 2 are illustrated and further analysed for Bernoulli, Euler, Hermite and Laguerre polynomials.
On Appell polynomials 2.1 Definition
Consider a random variable ξ having some exponential moments, i.e., for some λ > 0, M ξ (u) := E(e uξ ) < ∞ for |u| < λ. Then, as is well known, E(ξ n ) < ∞ for all n = 1, 2 . . . , and
E(e uξ ) = ∞ n=0 u n n! E(ξ n ), |u| < λ.(1)
Taking on the RHS of (1) u to be a complex number then the RHS defines a complex analytic function z → Ψ(z), |z| < λ, z ∈ C. Since M ξ (0) = 1 it follows from the continuity of Ψ that |Ψ(z)| > 0 for |z| < λ. Consequently, z → 1 Ψ(z) is a well defined analytic function and representable via power
series 1 Ψ(z) = ∞ n=0 c n z n , |z| < ρ,
where ρ > 0 is the radius of convergence. In particular, for z = u ∈ R such that |u| < ρ we obtain
1 E(e uξ ) = ∞ n=0 c n u n = ∞ n=0ĉ n u n n! , whereĉ n = c n n!. For x ∈ R and |u| < ρ it holds e ux E(e uξ ) = ∞ l=0 u l l! x l ∞ k=0 u k k!ĉ k = ∞ l=0 ∞ k=0 u l+k l!k!ĉ k x l = ∞ n=0 u n n! n k=0 n k ĉ k x n−k ,(2)
where we have used the fact that the value of the sum does not depend on the order of summation since the both series are absolutely convergent in a neighbourhood of the origin. Guided by (2), we give now the definition of the Appell polynomials associated with a random variable.
u n n! Q (ξ) n (x) = e ux E(e uξ ) ,(3)
where Q (ξ) n is of order n, are called the Appell polynomials associated with ξ.
Remark 2.2.
Recall from, e.g., [17] that given two analytic functions f and g such that g(0) = 0, g ′ (0) = 0, f (0) = 0 then the polynomials S n , n = 1, 2, . . . satisfying (u) are called the Sheffer polynomials. The discussion above leading to Definition 2.1 reveals that Appell polynomials are, in fact, special Sheffer polynomials.
∞ n=0 u n n! S n (x) = f (u)e xg
In the next definition we introduce the Appell polynomials in case ξ has some moments (but not necessarily exponential moments). Definition 2.3. Let ξ be a random variable which has the moments up to N , i.e., E(|ξ n |) < ∞, n = 0, 1, . . . , N.
The Appell polynomials {Q (ξ) n , n = 0, 1, 2, . . . , N } associated with ξ are de- fined via Q (ξ) 0 (x) = 1 for all x,(4)Q (ξ) n (x) := n i=0 n i Q (ξ) n−i (0)x i , n = 1, . . . , N(5)
where Q (ξ) j (0), j = 1, 2, . . . , n are generated by the recurrence formula
Q (ξ) j (0) = − j−1 i=0 j i Q (ξ) j−i (0)E(ξ i ).(6)
It is proved below (see Proposition 2.9) that in case ξ has the moment generating function, i.e., some exponential moments, then the polynomials in Definition 2.1 and Definition 2.3 are the same.
Remark 2.4. The Appell polynomials associated with a random variable ξ can be defined also via the differential equation (9) together with the normalisation (see, e.g., Kyprianou [9,p 250])
E(Q (ξ)
n (ξ)) = 0.
General properties
We now state and prove several general properties of the Appell polynomials, many of them are elementary and can be found in [9,13,16] but we wish to give a fairly complete discussion to make the paper self-contained.
Basic formulas
Putting
u = 0 in (3) yields Q (ξ) 0 (x) ≡ 1. Letting ξ ≡ 0 then clearly Q (0)
n (x) = x n , n = 0, 1, 2, . . . .
Proposition 2.5. Series expansion:
Q (ξ) n (x + y) = n k=0 n k Q (ξ) k (x)y n−k .(7)
Letting y = (m − 1)x, m = 1, 2, . . . yields the multiplication formula
Q (ξ) n (mx) = n k=0 n k Q (ξ) k (x)(m − 1) n−k x n−k .
Putting x = 0 in (7) gives
Q (ξ) n (y) = n k=0 n k Q (ξ) k (0)y n−k . (8) Proof. We have ∞ n=0 u n n! Q (ξ) n (x + y) = e u(x+y) E(e uξ ) = e ux E(e uξ ) e uy = ∞ k=0 u k k! Q (ξ) k (x) ∞ l=0 u l l! y l = ∞ k=0 ∞ l=0 u k+l k!l! Q (ξ) k (x)y l = ∞ n=0 u n n! n l=0 n l Q (ξ) n−l (0)y l ,
where we have substituted n = k + l. Notice that the two last equalities follow since the order of summation can be changed due to the absolute convergence of the series.
Proposition 2.6. Recursive differential equation:
d dx Q (ξ) n (x) = nQ (ξ) n−1 (x),(9)
or, equivalently,
Q (ξ) n (x) = Q (ξ) n (0) + n x 0 Q (ξ) n−1 (z)dz.
Proof. This follows directly from (8) by differentiating.
Proposition 2.7. Mean value property:
E(Q (ξ) n (ξ + x)) = x n , n = 1, 2 . . . .(10)
Proof. Case 1: ξ has some exponential moments. Putting x = 0 in (3) yields
1 = E(e uξ ) ∞ n=0 u n n! Q (ξ) n (0) = ∞ m=0 u m m! E(ξ m ) ∞ n=0 u n n! Q (ξ) n (0).
Since the series are absolute convergence we have for all |u| < λ
1 = ∞ k=0 u k k n=0 1 n!(k − n)! Q (ξ) n (0)E(ξ k−n ),
and, hence,
k n=0 1 n!(k − n)! Q (ξ) n (0)E(ξ k−n ) = 0, ∀k ≥ 1.(11)
From (8) we have
E(Q (ξ) n (x + ξ)) = n i=0 n i Q (ξ) n−i (0)E((x + ξ) i ) = n i=0 n i Q (ξ) n−i (0) i j=0 i j x j E(ξ i−j ) = n j=0 x j n i=j n i i j Q (ξ) n−i (0)E(ξ i−j ).(12)
Consider
A := n i=j n i i j Q (ξ) n−i (0)E(ξ i−j ), j < n. Substituting m = i − j yields for k ≥ 1 A = n−j m=0 n m + j m + j j Q (ξ) n−j−m (0)E(ξ m ) = n! j! n−j m=0 1 (n − j − m)!m! Q (ξ) n−j−m (0)E(ξ m ) = n! (n − k)! k m=0 1 (k − m)!m! Q (ξ) k−m (0)E(ξ m ) = 0,
by (11). Consequently, from (12), we have now E(Q (ξ)
n (x + ξ))) = x n . Case 2: ξ has the moments up to N , N ≥ 1. In this case the Appell polynomials Q (ξ) n , n = 1 . . . , N are defined by Definition 2.3. It is seen that the mean value property holds for n = 1. Assume that the mean value property holds for n = m ≤ N . From (5) and the fact E(|ξ| m ) < ∞, it follows that
d dx E(Q (ξ) m+1 (x + ξ)) = d dx m+1 i=0 m + 1 i Q (ξ) m+1−i (0) i j=0 i j E(ξ i−j )x j = m+1 i=1 m + 1 i Q (ξ) m+1−i (0) i j=1 i j E(ξ i−j )jx j−1 = m+1 i=1 m + 1 i Q (ξ) m+1−i (0) i i j=1 (i − 1)! (i − j)!(j − 1)! E(ξ i−j )x j−1 = m+1 i=1 m + 1 i Q (ξ) m+1−i (0) i E(x + ξ) i−1 = (m + 1) m+1 i=1 m! (m + 1 − i)!(i − 1)! Q (ξ) m+1−i (0)E(x + ξ) i−1 = (m + 1)E Q (ξ) m (x + ξ) = (m + 1)x m .
Hence,
E(Q (ξ) m+1 (x + ξ)) = c + x m+1 , where c is some constant. Since from (5) and (6) E(Q (ξ) m+1 (ξ)) = m+1 j=0 m + 1 j Q (ξ) m+1−j (0)E(ξ j ) = 0
it is seen that c = 0 and we obtain
E(Q (ξ) m+1 (x + ξ)) = x m+1 ,
as claimed.
Remark 2.8.
(i) Using the definition of the Appell polynomials as given in Remark 2.4, the mean value property is proved in [9] by exploiting the dominated convergence theorem.
(ii) Notice that by the mean value property and Definition 2.
1 it holds ∞ n=0 u n n! E(Q (ξ) n (x + ξ)) = ∞ n=0 u n n! x n = E e u(x+ξ) E(e uξ ) = E ∞ n=0 u n n! Q (ξ) n (x + ξ).x ∈ R Q (ξ) n (x) = Q (ξ) n (x), n = 0, 1, 2 . . . .(13)
Proof. Obviously equality (13) holds for n = 0, 1. Assume (13) holds for n = k, we will show that (13) also holds for n = k + 1. We have
d dx Q (ξ) k+1 (x) = (k + 1)Q (ξ) k (x),
and from (9) d dxQ
(ξ) k+1 (x) = (k + 1)Q (ξ) k (x). So we obtain d dx (Q (ξ) k+1 (x) − Q (ξ) k+1 (x)) = 0, and, hence,Q (ξ) k+1 (x) − Q (ξ) k+1 (x) = c, for all x.
Consequently, since bothQ
(ξ) k+1 (x) and Q (ξ) k+1 (x) satisfy the mean value prop- erty, E(Q (ξ) k+1 (x + ξ)) − E(Q (ξ) k+1 (x + ξ)) = c, which yields c = 0.
The next result is also an application of the mean value property. Proposition 2.10. Representation of powers:
x n = n k=0 n k Q (ξ) k (x)E(ξ n−k ),(14)
in other words
Q (ξ) n (x) = x n − n−1 k=0 n k Q (ξ) k (x)E(ξ n−k ).(15)
Proof. From (7) we have
Q (ξ) n (x + ξ) = n k=0 n k Q (ξ) k (x)ξ n−k .
Taking expectation and using the mean value property yields (14).
Example 2.11. Let ξ be a standard log-normally distributed random vari- able , i.e., ξ = e η , where η ∼ N (0, 1). Then µ k = E(ξ k ) = e k 2 /2 . Hence, the infinite series ∞ n=0 u n n! E(ξ n ) = ∞ n=0 u n n! e n 2 /2
diverges and ξ has no exponential moments. The Appell polynomials Q (ξ) n , n = 1, 2 . . . can be defined as in Definition 2.3. Using the recurrence formula (15) we can calculate
Q (ξ) 1 (x) = x − e 1/2 ; Q (ξ) 2 (x) = x 2 − 2x + 2e 1/2 − e 2 , Q (ξ) 3 (x) = x 3 − 3e 1/2 x 2 − 3(e 2 − 2e 1/2 )x − 6e − e 9/2 , . . . From (8) it is seen that the coefficients of Q (ξ) n are determined by Q (ξ) k (0), k = 1, 2, . . . , n.
To calculate these we may use the triangular system (6) to obtain for k = 1, 2, . . .
Proposition 2.12. It holds
Q (ξ) k (0) = −det µ k k k−1 µ 1 . . . k 2 µ k−2 k 1 µ k−1 µ k−1 1 . . . k−1 2 µ k−3 k−1 1 µ k−2 . . . . . . . . . . . . . . . µ 2 0 . . . 1 2 1 µ 1 µ 1 0 . . . 0 1 ,(16)
where µ k := E(ξ k ), k = 1, . . . , n are the moments of ξ.
Proof. From (6) solving the equation with variables Q (ξ)
k (0), k = 1, 2, . . . , n k j=1 k j Q (ξ) j (0)E(ξ k−j ) = −µ k ,
yields (16).
We have also the following relation (see [16, formula 2.9]) between the cumulants of ξ. Recall that the cummulants κ n of ξ are defined by
∞ n=0 u n n! κ n = log(E(e uξ )).(17)Proposition 2.13. For n = 1, 2 . . . it holds κ n+1 = n j=0 n j µ j+1 Q (ξ) n−j (0).(18)
Proof. Taking derivative in u in (17) we obtain
∞ n=0 u n n! κ n+1 = E(ξe uξ ) E(e uξ ) = ∞ n=0 u n n! µ n+1 ∞ n=0 u n n! Q (ξ) n (0),
which yields (18).
Probabilistic properties
In this section we focus on properties of Appell polynomials induced by the transforms of the underlying random variable.
Proposition 2.14. Let η := −ξ then for all n and x
Q (η) n (x) = (−1) n Q (ξ) n (−x).(19)
In particular,
Q (η) n (0) = (−1) n Q (ξ) n (0). Proof. We have e ux E(e uη ) = ∞ n=0 u n n! Q (η) n (x), and e ux E(e uη ) = e −u(−x) E(e −uξ ) = ∞ n=0 u n n! (−1) n Q (ξ) n (−x).
Hence, (19) holds.
Corollary 2.15. If ξ is a symmetric random variable then
Q (ξ) n (x) = (−1) n Q (ξ) n (−x), i.e., Q(ξ)
2n+1 is an odd polynomial and Q (ξ) 2n is even, n = 1, 2, . . . In particular, Q
(ξ) 2n+1 (0) = 0.
Proposition 2.16. Let ξ 1 and ξ 2 be two independent random variables and put ξ := ξ 1 + ξ 2 then
Q (ξ) n (x + y) = n k=0 n k Q (ξ 1 ) k (x)Q (ξ 2 ) n−k (y).(20)
In particular,
n k=0 n k Q (ξ 1 ) k (x)Q (ξ 2 ) n−k (−x) = n k=0 n k Q (ξ 1 ) k (0)Q (ξ 2 ) n−k (0).(21)
Proof. (see also [16]) By the independence of ξ 1 and ξ 2 , we have
e u(x+y) E(e uξ ) = e ux E(e uξ 1 ) e uy E(e uξ 2 )
.
Hence ∞ n=0 u n n! Q (ξ) n (x + y) = ∞ k=0 u k k! Q (ξ 1 ) k (x) ∞ l=0 u l l! Q (ξ 2 ) l (y) = ∞ k=0 ∞ l=0 u k+l k!l! Q (ξ 1 ) k (x)Q (ξ 2 ) l (y).
Substituting n = k + l yields (20). Notice that by choosing ξ 2 ≡ 0 then Q
k (x) = x k the series expansion (7) can be obtained from (20).
Q (ξ 1 ) n (x) = ∞ −∞ Q (ξ) n (x + y)P(ξ 2 ∈ dy).(22)
Remark 2.18. If ξ 1 , ξ 2 are i.i.d symmetric random variables then ξ = ξ 1 +ξ 2 is also symmetric. Therefore, from (21) and Corollary 2.14 , we have
0 = Q (ξ) 2n+1 (0) = 2n+1 k=0 2n + 1 k (−1) k Q (ξ 1 ) k (x)Q (ξ 1 ) 2n+1−k (−x).
Moment representation
Let η be a complex or real random variable with finite moments of all orders. Clearly, the polynomialQ n (x) := E(x + η) n satisfies Appell differential equation, see (9):
d dxQ n (x) = nQ n−1 (x).
A natural question is whether there exists a random variable θ such that
Q (θ) n (x) =Q n (x), i.e., Q (θ) n (x) = E(x + η) n .(23)
This equality is then called the moment representation of the Appell polynomial. Hence, from (23)
E θ×η ((θ + η) n ) = 0,(24)
where E θ×η denotes the product measure induced by θ and η. Consequently, taking n = 2, it is seen that (24) holds if and only if
θ + η = 0, P θ×η − a.s.(25)
But θ and η are independent under P θ×η and, hence, it follows from (25) that there exists a constant c such that θ = −η = c.
Although there does not exists non-constant real random variables satisfying (23) it is possible in some interesting cases to find for a given real θ a complex random variable η such that (23) holds. To develop this, consider the following, Proposition 2.20. Let θ be a random variable with the moment generating function
M θ (u) := E(e uθ ),
which is assumed to be well defined and positive for all u ∈ R. Then if
ψ(u) := 1 M θ (u)
is the characteristic function of a real random variable ζ having finite moments of all orders then
Q (θ) n (x) = E(x + iζ) n ,(26)
where i is the imaginary unit. In particular,
Q (θ) 2n (0) = (−1) n E(ζ 2n ). Proof. We have e ux M θ (u) = e ux E(e uθ ) = ∞ n=0 u n n! Q (θ) n (x),(27)
and
e ux M θ (u) = e ux E(e iuζ ) = E(e u(x+iζ) ) = ∞ n=0 E(x + iζ) n .(28)
The claim follows now from (27) and (28).
Remark 2.21. Notice that ζ in Proposition 2.20 is symmetric (since its characteristic function is real valued). Hence, we have the relationship
1 E(e uθ )
= E(e iuζ ) = E(cos(uζ)),
and, consequently, also θ must be symmetric. In fact, for symmetric θ having positive moment generating function for all u ∈ R it holds
(i) d du M θ (u)| u=0 = 0, (ii) u → M θ (u)
is convex, increasing for u ≥ 0 and decreasing for u ≤ 0.
(iii) M θ (u) ≥ 1 for all u and lim u→∞ M θ (u) = +∞.
In the next section we give several examples of random variables such that their Appell polynomials have the moment representations.
Appell polynomials with unique positive root
In this section we will give conditions for a random variable which guarantee that its Appell polynomials have a unique positive root. This property plays an important role in studying optimal stopping problems of power reward function for Lévy processes, for more details we refer to [10,13,16].
Let us first recall Descartes' rule of signs (see, e.g., [24]): Let p n (x) = n k=0 λ k x σ k be a polynomial with non-zero real coefficients λ k and powers σ k which are integers satisfying 0 ≤ σ 0 < σ 1 < · · · < σ n . Then the number of positive zeros of p n (x) (counted with multiplicities) is either equal to the number of variations in sign in the sequence {λ k } or less than that by an even integer number. Proof. We show that from condition (ii) it follows that Q k (z) > ε for all z ∈ (−δ(ε), δ(ε)). Choosing a ∈ (0, δ(ε)) we have by (i)
E(Q (ξ) k (ξ)1 {ξ<a} ) > ε P(ξ < a) > 0.(29)
Since E(Q (ξ) k (ξ)) = 0 it follows from (ii)
E(Q (ξ) k (ξ)1 {ξ<a} ) = −E(Q (ξ) k (ξ)1 {ξ≥a} ) < 0,
which contradicts (29). Hence Q
k (x) = k i=0 k i Q (ξ) i (0)x k−i . Conse-
quently, the coefficients of Q
Examples of Appell polynomials
In this section we consider some well known Appell polynomials, e.g., Bernoulli, Euler, Hermite and Laguerre polynomials and connect them with random variables. We use the characterization via random variables to review some properties of these polynomials. From (6) we obtain a recursive formula for the Bernoulli numbers:
Bernoulli polynomials and uniform distribution
B n = − n−1 k=0 n k B k 1 n − k + 1 ,(30)
which givesB
0 = 1,B 1 = − 1 2 ,B 2 = 1 6 , B 3 = 0, B 4 = −1/30 . . . .
Notice also that
u e u − 1 + u 2 − 1 = ∞ n=2 u n n! B n (0).(31)
Since the left hand side of (31) defines an even function it follows that B 2k+1 = 0 for k = 1, 2, . . . . Furthermore, from (8) we have
B n (x) = n m=0 n m B n−m x m .
We now exploit the properties of the Appell polynomials under the probabilistic approach to provide new proofs of many properties of the Bernoulli polynomials; some of these can be found in [1, p 804]). First we prove the relationship between the cumulants κ n of ξ and the Bernoulli numbersB n . Clearly, the first cumulant
κ 1 = E(ξ) = 1/2 = −B 1 .
Property (i). Cumulant relationship:
κ n+1 =B n+1 n + 1 , n = 1, 2 . . . .(32)
Proof. From (30)
B n+1 = − n j=0 n + 1 j B j 1 n − j + 2 = −(n + 1) n j=0 n j B j 1 n − j + 1 1 n − j + 2 = −(n + 1) n j=0 n j B j 1 n − j + 1 + (n + 1) n j=0 n j B j 1 n − j + 2 .
In the last equality, the first sum is 0 by (30), and using (18) the second sum equals κ n+1 and we therefore obtain formula (32).
Property (ii). First difference property:
B n (x + 1) − B n (x) = nx n−1 ,(33)
which yields
m k=1 k n = B n+1 (m + 1) − B n (m) n + 1 .
Proof. Using the mean value property of the Appell polynomials we get
x n−1 = E(Q (ξ) n−1 (x + ξ)) = n−1 m=0 n − 1 m B m E(x + ξ) n−1−m = n−1 m=0 n − 1 m B m 1 n − m (x + 1) n−m − x n−m = 1 n n−1 m=0 n m B m (x + 1) n−m − x n−m = 1 n [B n (x + 1) − B n (x)],
implying (33).
Property (iii). Symmetry:
B n (1 − x) = (−1) n B n (x).(34)
Proof. Set η := 1 − ξ. Then also η ∼ U(0, 1). Therefore, the Appell polynomials associated with η are also the Bernoulli polynomials B n (x).
On the other hand,
e ux E(e uη ) = e u(x−1) E(e u(−ξ) ) ,
and combining this with (19) gives
Q (η) n (x) = Q (−ξ) n (x − 1) = (−1) n Q (ξ) n (1 − x),
proving (34).
Property (iv). Second difference property:
(−1) n B n (−x) − B n (x) = nx n−1 .
Proof. This can be obtained from (33) and (34).
Property (v). Representation of powers:
x n = 1 n + 1 n k=0 n + 1 k B k (x).(35)
Proof. From (14) and the fact E(ξ n−k ) = 1 n − k + 1
we have
x n = n k=0 n k 1 n − k + 1 B k (x),
and this implies (35).
Next we give a new proof of the moment representation for Bernoulli polynomials first presented by Sun [21] (see also Srivastava and Vignat [20]).
Property (vi). Moment representation:
B n (x) = E(x − 1 2 + iζ) n ,(36)
where ζ has the logistic distribution with the density
f (ζ) (x) = π 2 sech 2 (πx), x ∈ R.
Proof. We make a symmetric random variable from the uniform random variable ξ by putting η := ξ − 1/2. Then η has the uniform distribution on [−1/2, 1/2]. We have
E(e uη ) = e −u/2 E(e uξ ) = e u/2 − e −u/2 u ,
and, hence,
Q (η) n (x) = B n (x + 1/2). Consider the cosine Fourier transform E(cos(uζ)) = 1 E(e uη ) = u e u/2 − e −u/2 = u/2 sinh(u/2) ,
The inverse of this transform is given by (see [6, (2)p 30])
f (ζ) (x) = π 2 1 cosh 2 (πx) = π 2 sech 2 (πx).
Therefore, we have the moment representation
Q (η) n (x) = E(x + iζ) n ,
and, hence,
B n (x) = Q (η) n (x − 1 2 ) = E(x − 1 2 + iζ) n .
The following result follows from the formula of the Appell polynomials associated to the sum of two independent variables.
Property (vii). For n = 0, 1, 2, . . .
B n (x) = 2 n−1 B n ( x 2 ) + B n ( x + 1 2 ) .(37)
Proof. Let η ∼ U (0, 1) and ξ ∼ Ber(1/2), i.e., P(ξ = 0) = P(ξ = 1) = 1/2. Assume that η and ξ are independent. Then (see [22,Proposition 1])
Q (η+ξ) n (x) = 2 n B n (x/2).
From which and (22) we obtain
Q (η) n (x) = E(Q (η+ξ) n (x + ξ)) = 2 n−1 B n ( x 2 ) + B n ( x + 1 2 ) . But B n (x) ≡ Q (η)
n (x). So the proof is complete.
Property (viii). For n = 0, 1, 2 . . .
B n (1/2) = −(1 − 2 1−n )B n (0) (38) Proof. Formula (38) is equivalent with B n (0) = 2 n−1 B n (1/2) + B n (0) ,
which is obtained directly by putting x = 0 in (37).
We next give a simple proof for the sign of the Bernoulli numbers (see [4])
Property (ix). The sign of the Bernoulli numbers:
(−1) n−1 B 2n (0) > 0, for all n ≥ 1.(39)
Proof. From the moment representation (36) we have B 2n (1/2) = (−1) n E(ζ 2n ).
Combining this fact with (38) yields
(−1) n−1 B 2n (0) = E(ζ 2n ) (1 − 2 1−2n ) > 0,
where ζ is as in Property (vi), from which readily follows (39).
Euler polynomials and Bernoulli distribution
Let ξ ∼ Ber(1/2), i.e.,
P(ξ = 0) = P(ξ = 1) = 1 2 .(40)
Then E(e uξ ) = e u + 1 2 , and, hence,
e ux E(e uξ ) = 2e ux e u + 1 = ∞ n=0 u n n! E n (x),(41)
where E n (x) are the so called the Euler polynomials (see [1, p 804]). Consequently, the Appell polynomials associated with ξ coincide with the Euler polynomials and we have from (7)
E n (x) = n m=0 n m Ẽ n−m x m ,
where the numbersẼ k = E k (0), k = 0, 1 . . . n can be generated from the recurrence equation (6) which -since E(ξ k ) = 1/2 -takes the form
E n = − 1 2 n−1 k=0 n k Ẽ k .(42)
Calculating from (42) we obtaiñ
E 0 = 1,Ẽ 1 = − 1 2 ,Ẽ 2 = 0,Ẽ 3 = 1 4 ,Ẽ 4 = 0,Ẽ 5 = − 1 2 , . . . .
Notice from (41) that
2 e u + 1 + u 2 − 1 = ∞ n=2 u n n!Ẽ n .
Consequently, since the function on the LHS is odd it follows thatẼ 2k = 0 for k ≥ 1.
Similarly as for the Bernoulli polynomials, we also discuss the following properties (see also [1, p 804]) of the Euler polynomials via the probabilistic approach:
Property (i). Difference property:
E n (x + 1) + E n (x) = 2x n ,(43)
which yields
m k=0 (−1) k k n = (−1) m E n (m + 1) + E n (0) 2 .(44)
Proof. Using again the mean value property, we get
x n = E(Q (ξ) n (x + ξ)) = n m=0 n m Ẽ m 1 2 (x + 1) n−m + x n−m ,
which yields (43). The proof of (44) is obtained from (43) by observing
m k=0 (−1) k k n = m k=0 (−1) k E n (k) − (−1) k+1 E n (k + 1) 2 .
Property (ii). Symmetry:
E n (1 − x) = (−1) n E n (x).
Property (iii). Representation of powers:
x n = E n (x) + 1 2 n−1 k=0 n k E k (x).
Property (iv). Euler numbers:
E n := 2 n E n ( 1 2 ), n = 0, 1, 2, . . . .
the so-called Euler numbers, are integers.
Proof. From (iii), putting x = 1/2 we have the recurrence for the Euler numbersÊ
n = 1 − n−1 k=0 n k 2 n−k−1Ê k .
Using induction here it easily seen that the Euler numbers are integers.
Property (v). Relationship between the Bernoulli and Euler polynomials:.
E n (x) = 2 n+1 n + 1 B n+1 x + 1 2 − B n+1 x 2 .(45)
Proof. Let η and ξ be two independent random variables with η ∼ U (0, 1) and ξ ∼ Ber (1/2). Then as in property (vi) for Bernoulli polynomials,
Q (η+ξ) n (x) = 2 n B n (x/2).(46)
Using (46) and (22) we obtain
E n (x) = Q (ξ) n (x) = 1 0 Q (η+ξ) n (x + y)dy = 1 0 2 n B n x + y 2 dy.
Setting z = (x + y)/2 we now have
E n (x) = 2 n+1 x+1 2 x 2 B n (z)dz = 2 n+1 B n+1 x+1 2 − B n+1 x 2 n + 1 .(47)
Notice that the last equality of (47) is obtained from recursive differential equation (9).
Property (vi). Moment representation:
E n (x) = E(x + iζ) n ,
where ζ has the hyperbolic secant distribution with the density f (ζ) (y) = sech(πy).
In particular, the Euler numbers satisfŷ
E 2n = (−1) n 2 2n E(ζ 2n ) andÊ 2n+1 = 0, n ≥ 1,
from which it follows the sign property of the Euler numbers (see also [4])
(−1) nÊ 2n > 0.
Proof. Let θ := ξ − 1/2. Then θ is symmetric and we get E(e uθ ) = e −u/2 E(e uξ ) = e u/2 + e −u/2 2 .
Hence,
Q (θ) n (x) = E n (x + 1 2 ).
We have the cosine transform E(cos(uζ)) = 1 E(e uη ) = 2 e u/2 + e −u/2 = 1 cosh(u/2) , and, hence, inverting this transform gives (see [6, (1)p 30])
f (ζ) (y) = sech(πy),
which results to the moment representation for Q
(θ) n Q (θ) n (x) = E(x + iζ) n .
Consequently (see also [21])
E n (x) = E(x − 1 2 + iζ) n .
Remark 3.1. Talacko [23] considers a class of symmetric distributions which is called the family of Perks' distributions with the density function
f (x) = c e x + k + e −x ,
where c is the normalizing constant and k > −2, k ∈ R . For k = 0 we have the so-called hyperbolic secant distribution and for k = 2 the logistic distribution. In [23] the following characterizations are also proved: (i) Let ξ 2 be a random variable having the logistic distribution, then
ξ 2 (d) = ∞ j=1 L j 2jπ , where L i are i.i.ξ 0 (d) = ∞ j=1 L j (2j − 1)π .
Sun [21] uses these characterizations to give the moment representations for Bernoulli polynomials and Euler polynomials. Such representations are also used in Srivastava and Vignat [20].
Hermite polynomials and normal distribution
Let ξ ∼ N (0, 1). Then E(e uξ ) = e u 2 /2 , and, hence, e ux E(e uξ ) = e ux−u 2 /2 .
So the Appell polynomials Q (ξ) n associated with ξ are the Hermite polynomials He n (see also [16]):
Q (ξ) n (x) = He n (x) = n! [n/2] k=0 (−1) k x n−2k (n − 2k)!k!2 k . Remark 3.2. Letξ ∼ N (µ, σ 2 ) thenξ (d)
= µ + σξ. Denote by H n the Appell polynomials associated withξ, from (3) we have
H n (x) = σ n He n x − µ σ .
We discuss now some basis properties of He n .
Property (i). Symmetry:
He n (x) = (−1) n He n (−x).
Proof. Since the random variable η := −ξ has also the standard normal distribution, the property follows from (19).
Property (ii). Moment representation:
He n (x) = E(x + iξ) n ,
In particular,
He 2n (0) = (−1) n E(ξ 2n ) = (−1) n (2n)! 2 n n! .
Proof. It is seen that ξ is symmetric and the function = ξ ∼ N (0, 1) and E(ζ 2n ) = (2n)! 2 n n! .
ψ(u) = (E(e uξ )) −1 = e u 2 /
Remark 3.3.
Withers [25] derives the moment representation for multiple Hermite polynomials by using Taylor expansion of E(e u(x+iZ) ), where Z is a multidimensional normally distributed random variable.
Property (iii). Representation of powers:
x 2n = n i=1
2n 2k (2(n − k))! 2 n−k (n − k)! He 2k (x).
Next we consider the orthogonality of Hermite polynomials. As proved in [3] (see also [18]) the Hermite polynomials are the only Appell polynomials which are orthogonal. Here we give a new proof of this fact via the moment generating function. Recall that by Favard's theorem (see [5, p 21], [17]) the family of polynomials {P n , P 0 ≡ 1, n = 1, 2, . . . } is orthogonal if and only if there exist sequences {a n } ∞ n=1 and {b n } ∞ n=1 such that P n+1 (x) = (x − a n )P n (x) − b n P n−1 (x), n = 1, 2, . . . .
Property (iv). Let ξ be a random variable having some exponential moments. The Appell polynomials {Q
Laguerre polynomials and gamma distribution
Let ξ ∼ Γ(β, α), α, β > 0, i.e., ξ is a gamma distributed random variable with parameters α and β. Then As explained in subsection 2.2.4 the property that an Appell polynomial has a unique positive root is crucial in the theory of optimal stopping of Lévy processes. In the next proposition it is seen that if the parameter β ≤ 1 then the gamma distribution has this property.
Corollary 2. 17 .
17Let ξ 1 , ξ 2 and ξ be as in Proposition 2.16. Then
Proposition 2. 19 .
19The only pairs of real random variables satisfying(23) are the deterministic ones, i.e., θ = −η = c for some constant c ∈ R.Proof. The Appell polynomial Q (θ) n associated with θ satisfies E(Q (θ) n (θ)) = 0.
Proposition 2 . 22 .
222Let ξ be a non-negative random variable having the moments up to n. If the following conditions hold for all a > 0(i) P(ξ < a) > 0, (ii) E(Q (ξ) k (ξ)1 {ξ≥a} ) > 0, k = 1, 2, . . . , n,then the Appell polynomials Q (ξ) k , k = 1, 2, . . . , n have a unique positive root.
k
(0) ≤ 0, k = 1, 2 . . . , n. Indeed, assume that Q(ξ) k (0) > 0, by the continuity of Q (ξ) k there exists ε > 0 and δ(ε) > 0 such that Q (ξ)
k ( 0 )
0≤ 0 for all k = 1, 2 . . . , n. Furthermore, we have Q
k
have only one change of sign. By Descartes' rule of signs we conclude that Q (ξ) k , k = 1, 2 . . . , n have a unique positive zero.
the Appell polynomials associated with the uniformly distributed random variable coincide with the Bernoulli polynomials. The numbers B k := B k (0), k = 0, 1, . . . are called the Bernoulli numbers. We have E(ξ k ) = 1 0 y k dy = 1/(k + 1), k = 1, 2, . . . .
n
} n≥0 associated ξ are orthogonal if and only if ξ is normally distributed, i.e., Q (ξ) n are Hermite polynomials He n , n = 0, 1 . . . .Proof. Since it is assumed that ξ has some exponential moments, the cumulants κ n , n = 1, 2, . . . are obtained via formula E(e uξ )
n−m x m , where β n − m = β(β − 1) . . . (β − (n − m)
Definition 2.1. Let ξ be a random variable having some exponential moments. The polynomials Q(ξ)
n , n = 0, 1, 2, . . . , satisfying
∞
n=0
d of sequence of Laplace distributed random variables with Let ξ 0 have the hyperbolic secant distribution, thenthe density g(x) =
1
2
e −|x| .
(ii)
2
is well defined for all u. Consider the cosine transformE(cos(uζ)) = 2
∞
0
f (ζ) (y) cos(uy)dy = e −u 2 /2 .
From [6, (11) p 15]
∞
0
e −y 2 /2
√
2π
cos(uy)dy =
1
2
e u 2 /2 ,
and, hence, ζ
(d)
Acknowledgements. The author would like to thank Professor Paavo Salminen for helpful discussions and valuable comments which improved this paper. The financial support from the Finnish Doctoral Programme in Stochastics and Statistics is gratefully acknowledged.which is equivalent withConsequently,and, hence, we obtain the representationIt follows from Favard's theorem that Q n can be represented in terms of the Bell polynomials Be n as follows (see Madan and Yor[12])and then by using formula(8)in[19]we also obtain formula (48).for all 0 ≤ m ≤ n − 1 and q n = 1. Since 0 < β ≤ 1 thenObviously, q m ≤ 0 for all 0 ≤ m ≤ n − 1.By the Descartes' rule of signs the proof is complete.Remark 3.6. For arbitrary α > 0 it holdsIn particular, for β = 1, i.e., ξ ∼ Exp(α) we have (see also[16])We conclude by presenting the following representation of powers via Laguerre polynomialsTo prove this, use(14)and the fact that
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[] |
[
"Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures",
"Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures"
] |
[
"A Erpenbeck \nInstitute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany\n",
"C Hertlein \nInstitute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany\n",
"C Schinabeck \nInstitute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany\n\nInstitute of Physics\nUniversity of Freiburg\nHermann-Herder-Str. 3D-79104FreiburgGermany\n",
"M Thoss \nInstitute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany\n\nInstitute of Physics\nUniversity of Freiburg\nHermann-Herder-Str. 3D-79104FreiburgGermany\n"
] |
[
"Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany",
"Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany",
"Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany",
"Institute of Physics\nUniversity of Freiburg\nHermann-Herder-Str. 3D-79104FreiburgGermany",
"Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B2D-91058Germany",
"Institute of Physics\nUniversity of Freiburg\nHermann-Herder-Str. 3D-79104FreiburgGermany"
] |
[] |
The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline an extension of the HQME approach, which uses a re-summation over poles and can be applied to calculate transient currents at a numerical cost that is independent of temperature and band structure of the baths. We demonstrate the performance of the extended HQME approach for noninteracting tight-binding model systems of increasing complexity as well as for the spinless Anderson-Holstein model. arXiv:1805.10161v1 [cond-mat.mes-hall]
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10.1063/1.5041716
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[
"https://arxiv.org/pdf/1805.10161v1.pdf"
] | 52,013,285 |
1805.10161
|
b1e8d886270cb41018a16eb1971ab1e8817a6e6f
|
Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures
A Erpenbeck
Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials
Friedrich-Alexander-Universität Erlangen-Nürnberg
Staudtstr. 7/B2D-91058Germany
C Hertlein
Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials
Friedrich-Alexander-Universität Erlangen-Nürnberg
Staudtstr. 7/B2D-91058Germany
C Schinabeck
Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials
Friedrich-Alexander-Universität Erlangen-Nürnberg
Staudtstr. 7/B2D-91058Germany
Institute of Physics
University of Freiburg
Hermann-Herder-Str. 3D-79104FreiburgGermany
M Thoss
Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials
Friedrich-Alexander-Universität Erlangen-Nürnberg
Staudtstr. 7/B2D-91058Germany
Institute of Physics
University of Freiburg
Hermann-Herder-Str. 3D-79104FreiburgGermany
Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures
(Dated: May 28, 2018)
The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline an extension of the HQME approach, which uses a re-summation over poles and can be applied to calculate transient currents at a numerical cost that is independent of temperature and band structure of the baths. We demonstrate the performance of the extended HQME approach for noninteracting tight-binding model systems of increasing complexity as well as for the spinless Anderson-Holstein model. arXiv:1805.10161v1 [cond-mat.mes-hall]
I. INTRODUCTION
Quantum transport through nanostructures, such as, e.g., molecular junctions, is an active field of research, which combines the possibility to study fundamental aspects of non-equilibrium many-body quantum physics at the nanoscale with the perspective for applications in nanoelectronic devices. [1][2][3][4][5] From the theoretical side, there are several approaches capable of describing quantum transport in nanosystems. 6 Approximate methods include quantum master equations, [7][8][9][10][11][12][13][14][15][16][17][18][19] scattering theory, [20][21][22][23][24] and the non-equilibrium Green's function approach. 18,[25][26][27][28][29][30][31][32][33][34][35][36][37] A numerically exact treatment can, for example, be provided by means of path integrals, [38][39][40][41][42][43][44] multiconfigurational wave-function methods, 45,46 numerical renormalization-group theories, 47-50 a combination of reduced density matrix techniques and impurity solvers, 51,52 and the hierarchical quantum master equations (HQME) approach (also called hierarchical equation of motion (HEOM) method). The latter method, which will be the focus of this paper, was originally developed by Tanimura et al. to describe relaxation dynamics in quantum systems, 53,54 and later extended by Yan et al. [55][56][57][58][59][60][61] and Härtle et al. [62][63][64][65][66] to study charge transport.
In its original formulation, the HQME method employs a decomposition of the bath correlation function in terms of exponentials, which are also referred to as poles. 55,57,62 Although mathematically exact, only a finite number of exponentials can be taken into account in numerical calculations, thus effectively restricting the efficient application of the HQME method to systems at higher temperatures coupled to simple baths. In order to circumvent this restriction, more efficient decomposition schemes for the bath correlation function allowing for a systematic construction of a hierarchy were proposed, such as the Pade decomposition, 67,68 the Chebyshev decomposition, [69][70][71] an expansion in terms of a com-plete set of orthogonal functions, 72 or hybrid approaches combining different decomposition schemes. 73 Although these extensions improved the applicability of the HQME approach profoundly, they still rely on a decomposition that needs truncation for numerical applications, thus introducing an approximation, which limits the applicability to certain parameter regimes. In order to lift this restriction, we propose an extension of the HQME approach, which employs an analytic re-summation of the decomposition thus avoiding the truncation in numerical applications. This extends the applicability of the HQME method to quantum transport in systems at low temperatures coupled to structured baths.
The outline of this paper is as follows: In Sec. II, we establish the theory. To this end, we introduce the model system in Sec. II A and review the HQME method in Sec. II B. In Sec. II C, we outline the extension of the HQME method, that uses a re-summation over poles. In Sec. III we demonstrate the performance of this extended HQME method by calculating transient currents for model systems of increasing complexity and comparing the results to traditional HQME calculations. A conclusion is given in Sec. IV.
II. THEORY
A. Model
In order to study quantum transport through nanosystems, we consider a typical model for a molecular junction, comprising the molecule (in the following referred to as 'system'), which is coupled to two macroscopic leads (representing the 'bath'). The corresponding Hamiltonian is given by
H = H S + H L + H R + H SL + H SR ,(1)
where H S describes the system, H L/R the leads and H SL/R the coupling between the system and the leads, which enables transport. The left and right lead is modeled as a continuum of noninteracting electronic states of energy k ,
H L/R = k∈L/R k c † k c k ,(2)
where c † k /c k are the corresponding creation and annihilation operators. The coupling between the system and the leads is described by the Hamiltonian
H SL/R = ν∈S k∈L/R V νk c † k d ν + h.c. ,(3)
with d † ν /d ν being the electronic creation and annihilation operators of the system. This form of the coupling between the system and the leads gives rise to the spectral density of the leads
Γ L/Rνν ( ) = 2π k∈L/R V νk V * ν k δ( − k ),(4)
which depends on the electronic energies k and hence incorporates the band structure of the lead.
B. HQME approach
The HQME theory provides an equation of motion for the reduced density matrix ρ(t) of a system coupled to one or several baths, which are in our case the leads. For a system coupled to the leads via the Hamiltonian (3), all information about the influence of the leads on the system is encoded in the two-time bath correlation function
C ± lνν (t − t ) = 1 2π d e ± i (t−t ) Γ lνν ( )f (± , ±µ l ),(5)
with the spectral density Γ lνν ( ) of lead l and the Fermi distribution function f ( , µ) = (1 + exp(β( − µ))) −1 . Here, β = 1 kBT where k B is the Boltzmann constant, T the temperature and µ the chemical potential. In order to obtain a closed set of equations within the HQME approach, it is expedient to represent the bath correlation function as a sum over exponentials. 55 To this end, the Fermi distribution and the spectral density of states Γ lνν ( ) are separately represented by a sum-over-poles scheme,
C ± lνν (t − t ) ≡ ∞ q=1 η lνν q± e −γ lνν q± (t−t )(6)= ∞ p =1η lνν p ± e −γ lνν p ± (t−t ) + ∞ p=1η
lνν p e −γ lp± (t−t ) .
In the notation used here, the parametersη lνν p ± and γ lνν p ± correspond to the decomposition of Γ lνν ( ), whereasη lνν p andγ lp± stem from the decomposition of the Fermi function. Common approaches for obtaining these representation are the Matsubara 54,55,74 and the Pade decomposition. 67,68 For details on the derivation of the HQME for a system-bath coupling of the form Eq. (3) we refer to Refs. 55, 57, and 62. The equation of motion for the n th -tier auxiliary density operator is given by
∂ ∂t ρ (n) j1...jn (t) = − i L S − n m=1 γ jm ρ (n) j1...jn (t) −i n m=1 (−1) n−m C jm ρ (n−1) j1...jm−1jm+1...jn (t) − i 2 j A σj νj ρ (n+1) j1...jnj (t),(7)
with the multi-index j i = (l i , ν i , ν i , q i , σ i ), where l i ∈ {L, R}, ν i , ν i are electronic indices of the system, σ i = ±1 and q i being the pole-index stemming from the decomposition of the bath correlation function in terms of exponentials. ρ (0) (t) is the reduced density operator of the system, the higher tier auxiliary density operators encode the influence of the leads on the system dynamics.
Further, σ = −σ and L S O = [H S , O].
The objects A σ ν and C j couple the n th -tier to the (n + 1) th -and (n − 1) thtier, respectively, and act upon the auxiliary density operators as
A σ ν ρ (n) (t) = d σ ν ρ (n) (t) + (−1) n ρ (n) (t)d σ ν ,(8a)C j ρ (n) (t) = η lνν qσ d σ ν ρ (n) (t) − (−1) n η * lνν qσ ρ (n) (t)d σ ν . (8b)
In these equations, we have used the shorthand notation d − ν ≡ d ν and d + ν ≡ d † ν . This leads to an infinite set of coupled equations of motion. To this point, the HQME approach is exact for the Hamiltonian of the above given form and does not include any approximation. For applications, however, the hierarchy needs to be truncated in a suitable manner. [75][76][77][78] Further, only a finite number of poles q i can be used to represent the bath correlation function. The HQME approach is therefore particularly efficient for the description of systems, where a manageable number of poles represents a good approximation, which is the case for simple spectral densities at higher temperatures.
Within the HQME approach, observables are represented via the reduced density matrix and the auxiliary density operators. For the study of transport through nanosystems, the electronic population of the system and the current are important observables of interest. While the population is given by the diagonal elements of the density matrix ρ(t), the current for lead l is given by
I l (t) = ie 2 νν q Tr S d ν ρ (1) lνν q+ (t) − d † ν ρ (1) lνν q− (t) ,(9)
where Tr S denotes the trace over the system degrees of freedom.
C. Re-summation over poles
In order to extend the HQME formalism, we introduce the following weighted sum over poles
R (n) a1...an (t, t 1 , . . . , t n ) = q1...qn n m=1 e γj m (t−tm) ρ (n) j1...jn (t),(10)
such that an infinite number of poles is treated within one object.
Here, we use the multi-index a i = (l i , ν i , ν i , σ i ), with l i ∈ {L, R}, ν i , ν i electronic indices of the system and σ i = ±1, which does not have a pole-index. As this extension of the HQME approach relies on a re-summation over poles, we will abbreviate it as RSHQME in the following. Taking the derivative of Eq. (10) with respect to time t and using Eq. (7), we arrive at the equation of motion
∂ ∂t R (n) a1...an (t, t 1 , . . . , t n ) = − i L S R (n) a1...an (t, t 1 , . . . , t n ) − i 2 a A σa νa R (n+1) a1...ana (t, t 1 . . . , t n , t) −i n m=1 C am (t, t m )R (n−1) a1...am−1am+1...an (t, t 1 , . . . , t m−1 , t m+1 , . . . , t n ),(11)
where we have defined
C am (t, t m )R (n) = (−1) n+1−m ξ am (t, t m )d σ ν m R (n) − (−1) nξ am (t, t m )R (n) d σ ν m ,(12)
with
ξ a (t, t m ) = ∞ q=1 η lνν q± e −γ lνν q± (t−tm) , (13a) ξ a (t, t m ) = ∞ q=1 η * lνν q∓ e −γ lνν q± (t−tm) . (13b)
The only remainder of the decomposition of the bath correlation function in terms of exponentials are the sums ξ a andξ a . Notice that ξ a is indeed the bath correlation function as becomes apparent upon comparing Eqs. (6) and (13a). For exact results, these sums need to be evaluated analytically. Approximate results, can be obtained by numerically evaluating the sums, where several thousand poles can easily be included. Considering the R (n) where all time arguments are equal, the previous auxiliary density operators, summed over all poles q1...qn , are recovered, which are important for the representation of observables. Of particular interest for this work is the electronic current, which is given by
I l (t) = ie 2 νν Tr S d ν R (1) lνν + (t, t) − d † ν R (1) lνν − (t, t) .(14)
Although the RSHQME method is an extension of the HQME approach that can deal with the decomposition of the bath correlation function in an exact way, it comes with a numerical drawback. As the objects R (n) depend on (n + 1) time-arguments, the approach scales as O(t n+1 ) given that n is the maximal tier considered. Thus, the RSHQME method is best suited to simulate the dynamics of a system for short to intermediate times.
The HQME method, on the other hand, scales as O(t·q n ), where q is the total number of poles taken into account. This restricts the HQME approach to systems that are well approximated by a small number of poles. Some numerical details of the RSHQME method are given in Appendix A.
III. ILLUSTRATIVE APPLICATIONS
In this section, we apply the RSHQME method to representative model systems of increasing complexity. To this end, we explicitly consider the simple but generic model system consisting of a single electronic state of energy 0 coupled to one vibrational mode of frequency Ω as described by the system Hamiltonian
H S = 0 d † d + Ωa † a + λ(a † + a)d † d.(15)
Here, d/d † denote the electronic and a/a † the vibrational creation and annihilation operators, respectively. λ is the electronic-vibrational coupling strength. For λ = 0, we recover a purely electronic transport problem, to which we refer as "noninteracting". In the following, we consider the model system with the parameters 0 = 0.2eV, Γ L = Γ R = Γ = 0.06eV, which are representative for molecular junctions. 23,37,66,[79][80][81][82][83][84] The bias, defined as the difference between the chemical potentials of the left and the right lead, is assumed to drop symmetrically,
µ L = −µ R .
The temperature of the system is T = 0.1K. For the numerical simulations, we assumed that the total density matrix factorizes at time t = 0. Further, at t = 0 the molecular electronic state is unpopulated, the vibrational mode is in its ground state, and the leads are in their thermal state. We start with a noninteracting system (λ = 0) within the wide-band approximation. In order to establish the applicability of the RSHQME method, we compare its results to the outcome of the traditional HQME approach. These results provide the possibility to benchmark HQME calculations including different numbers of poles. Subsequently, we consider a model for the leads beyond the wide-band limit, demonstrating that the RSHQME method can describe the influence of band structure effects on electronic transport. Finally, we show that the RSHQME is applicable to interacting systems, thus being a numerically affordable extension of the exact HQME method to low temperatures and complex band structures. To this end, we consider the spinless Anderson-Holstein model (λ = 0), which serves as an example for an interacting model.
A. Noninteracting system attached to leads in the wide-band limit
We first consider a noninteracting model described by the system Hamiltonian Eq. (15) with λ = 0, coupled to leads modeled in the wide-band limit, where the energydependence of the spectral density is neglected.
In the wide-band limit, it is not feasible to perform a decomposition of the constant Γ L/R ( ) in terms of exponentials. It is more efficient to drop the first sum in the second line of Eq. (6) and instead include one additional auxiliary object in every tier of the hierarchy, which is calculated as
ρ (n+1) (l,σ)j1...jn = − i Γ 4 · d σ , ρ (n) j1...jn (−1) n+1 , (16a) R (n+1) (l,σ)a1...an = − i Γ 4 · d σ , R (n) a1...an (−1) n+1 ,(16b)
and not by forward propagation. As this is a technical aspect which is identical for the HQME and the RSHQME method, we refrain from giving details here and refer the reader to Refs. 85-89 for more information.
For the RSHQME method, we represent the Fermidistribution employing the Matsubara decomposition, yielding
γ lp± = π β (2p − 1) ∓ i µ l ,(17a)η lp = − iΓ β .(17b)
With these expressions for γ lp± and η lp , we obtain
ξ am (t, t m ) = − iΓ β · e ( 3π β −σm i µ l )(t−tm) 1 − e 2π β (t−tm) = −ξ am (t, t m ).(18)
Using Eq. (18) in Eq. (11), we arrive at a closed set of equations for the re-summed auxiliary density operators R (n) . For the HQME method, we apply the Pade spectral decomposition considering 100, 150, 200 and 500 poles in the numerical calculation. Notice that the number of poles considered here is at the limit of numerical accessibility. Usually, HQME calculations take tenths of poles into account. 62,64,66,68,73,90,91 We refrain from discussing results obtained using the Matsubara decomposition for a finite number of poles, as even taking into account 500 Matsubara poles fails to provide a reasonable result for any bias voltage.
It is noted that for a noninteracting system, the hierarchy terminates at the 2nd-tier. 55,92 In the wide-band limit, it suffices to only include the 1st-tier auxiliary density matrices for exact results for single-particle observables such as the current. 85,86,88 value of the current depend on the numerical details of the different methods. The transient behavior has already been studied in detail elsewhere. 45,52,[93][94][95] We compare the current obtained by the different numerical methods. Generally, the quality of the spectral decomposition increases with an increasing number of poles, 67,68 thus the results calculated by the HQME method converge towards the RSHQME results with increasing number of poles. For 0.2V, which is representative for the non-resonant transport regime, the HQME calculation accounting for 200 Pade poles performs reasonably well, whereas the results for the resonant transport regime at 0.8V still exhibit deviations from the converged current. This is due to the fact that for the resonant transport regime, a larger part of the leads' structure in energy space has to be represented adequately, leading to an increased requirement in the number of poles taken into account with increased bias. 67 The HQME approach taking into account 500 Pade poles gives the same results as the RSHQME method. Only for very short times below ∼ 0.3fs, we find a small deviation between the HQME and the RSHQME result in the resonant transport regime. This can be explained by the fact that the short-time dynamics depends on a large energy span of the leads' structure. 85
B. Noninteracting system attached to leads with a finite bandwidth
Next, we go beyond the wide-band description of the leads, demonstrating the capability to include bandeffects in the RSHQME approach. Thereby, the advantage of using the RSHQME method is that the numerical cost does not depend on the band structure. Using the HQME method, on the other hand, the numerical cost depends on the model for the leads. Representing leads with a band structure increases the number of poles that need to be taken into account for the calculation, thus increasing the numerical effort.
We consider the noninteracting model described by the Hamiltonian Eq. (15) with λ = 0, the spectral density of the leads is modeled by a box function with smooth edges as described by the function
Γ l,µ l ( ) = Γ 1 + e α( − C −µ l · 1 + e −α( + C −µ l(19)
with α = 25eV −1 and C = 0.95eV. This form of Γ l,µ l ( ) is also referred to as wide-band limit with soft cut-off and was already used by Schmidt et al. 93 who investigated the influence of the band structure on the transient current or, in a different context, by Mühlbacher and Rabani 38 . The parameters α and C where chosen such that for low bias voltages, the system behaves like in the wide-band case, however for higher bias voltages, band edge effects become important. The bias voltage is assumed to shift the energy levels of the leads. Γ l,µ l ( ) as a function of applied bias voltage is depicted in Fig. 2 (top).
µ L µ R 0 0.2V µ L µ R 0 0.4V µ L µ R 0 0.8V µ L µ R 0 1.5V µ L µ R 0 2.0V
In order to describe this non-trivial spectral density within the RSHQME methodology, we represent Eq. (19) by a modified Pade decomposition
Γ l,µ l ( ) = Q l,µ l ( ) P l,µ l ( ) ≈ N −1 n=0 q n n 1 + N n=1 p n n .(20)
Sampling Γ l,µ l ( ) by 2N -values leads to a set of coupled linear equations, which can be solved for q n and p n . Applying the Bairstow algorithm, 96 we determine the roots of P l,µ l ( ) which are used to perform the energy integral in Eq. (5) using Cauchy's residual theorem. In this way, we obtain the values forη lp ± andγ lp ± . However, the details of the decomposition of Γ l,µ l ( ) are not of importance, because the RSHQME approach works with any decomposition scheme. As only the sum over poles enters the equation of motion, any number of poles N can be included in the calculation. Thus the RSHQME method opens up the possibility to accurately describe leads of arbitrary complexity. The approach of fitting complex band structures in order to represent band effects was already discussed in Refs. 97 and 98 and was for example used in explicit calculations by Xie et al. 90 , who expanded the density of states of 1D tight-binding chains in terms of Lorentzians and therefrom performed HQME calculations. Fig. 2 (bottom) compares the time-dependent current for a system attached to leads with different band structures. The solid lines represent the current for different bias voltages, where the leads are modeled by Eq. (19). The associated band structure as a function of bias voltage is given in Fig. 2 (top). The dashed lines in Fig. 2 (bottom) correspond to the results obtained in the wideband limit already discussed in Sec. III A.
The most striking difference between the results for the two lead models is observed for the current at short times. The fact that the wide-band description of the leads results in an instantaneous finite current, which is unphysical, was already studied by Schmidt et al. 93 . Furthermore, the precise form of the transient current also depends on the details of the lead band structure. For bias voltages 0.2V, 0.4V and 0.8V, the current for the system attached to leads with a finite band approaches after a certain time a value similar to that obtained in the wide-band limit. For the larger bias voltages 1.5V and 2V, band edge effects become important also in the steady state. The long-time current is smaller than in the wide-band limit because the density of states in the leads in resonance with the molecular electronic states is decreased with bias.
C. Interacting model system
To demonstrate the applicability of the RSHQME approach to interacting systems, we consider the spinless Anderson-Holstein model given by the Hamiltonian Eq. (15), where we set the energy of the vibrational mode to Ω = 0.2eV. We consider two different coupling strengths, λ = 0.05eV and λ = 0.1eV, and compare the results to the noninteracting system λ = 0. These model parameters are in the typical range for molecular junctions, similar parameters have been used in our recent work on the spinless Anderson-Holstein model. 66 Notice that we consider a system in the anti-adiabatic regime, Ω > Γ, as the influence of the electronic-vibrational interaction on the short-time dynamics is most pronounced under these conditions. The leads are modeled in the wide-band limit. Fig. 3 compares the current for different coupling strengths, λ = 0eV, 0.05eV, 0.1eV, obtained by the RSHQME method. In the numerical calculations, we used three vibrational basis states for λ = 0.05eV and six vibrational basis states for λ = 0.1eV. The finding that a relatively small vibrational basis set is sufficient to obtain converged results for the current is due to the comparably weak coupling strength λ, leading to a small non-equilibrium vibrational excitation, and the fact that the current is not very sensitive with respect to the vibrational basis for the parameters studied here. Note also that typically a smaller number of vibrational basis states is necessary to obtain converged results for short to intermediate times. For the chosen parameters, the current was converged employing the hierarchy up to the 2nd-tier. Converged HQME calculations would require about 500 Pade poles, which is numerically very demanding and demonstrates the improvement obtained with the RSHQME method.
The results in Fig. 3 have been obtained for a range of bias voltages, 0.2V, 0.4V, and 0.8V, which span a variety of different transport processes. The current at 0.2V is predominantly composed of non-resonant elastic and inelastic co-tunneling processes. For the bias voltages 0.4V and 0.8V, elastic and inelastic resonant transport processes are of importance. Thereby, at 0.8V, more resonant inelastic transport processes are energetically possible than for 0.4V. A detailed analysis of transient currents through a system with electronic-vibrational interaction has been given in Ref. 99.
IV. CONCLUSION
In this paper, we have proposed an extension of the HQME method, which avoids the limitations imposed by the decomposition of the bath correlation function inherent to the construction of the hierarchy. In contrast to other extensions, the methodology proposed here relies on an analytic re-summation over poles rather than a more efficient parametrization of the bath correlation function, thus circumventing the shortcomings of the traditional HQME approach with respect to bath parametrization.
In order to demonstrate the performance of the novel RSHQME method, we have applied the approach to model systems of increasing complexity including both interacting and noninteracting systems. The results show that the RSHQME method is able to reproduce the outcome of traditional HQME calculations for parameters where the latter can be converged with respect to the number of poles. Furthermore, we applied the RSHQME method to systems, where traditional HQME calculations are numerically prohibitively expensive.
Despite of the appeal of the newly introduced RSHQME method, numerical calculations become increasingly expensive with simulation time, which confines the applications of the RSHQME method in its current formulation to simulations for short or intermediate times. Similar as in other time-dependent density-matrix schemes, this limitation may be overcome by exploiting the fact that for realistic systems the bath correlation function decays in time. ACKNOWLEDGEMENT We thank R. Härtle for helpful discussions. This work was supported by the German Research Foundation (DFG) through SFB 953 and a research grant.
Appendix A: Propagation scheme of the RSHQME method
In this appendix, we give some details on the propagation of the coupled differential equations (11), which are the basis of the RSHQME method. To this end, we exemplify how the lowest order auxiliary density matrices are calculated. A generalization to the calculation of higher tier auxiliary density operators is straightforward.
In this work, the density matrix ρ(t) = R (0) (t) for a time t is calculated via propagation, which corresponds to integrating its differential equation
ρ(t) = t 0 ∂ τ ρ(τ ) dτ (A1) = t 0 F 0 ρ(τ ), R (1) a1 (τ, τ ) dτ.
Here, F 0 indicates that according to Eq. (11), the timederivative ∂ τ ρ(τ ) is a function of ρ(τ ) and R (1) a1 (τ, τ ). Consequently, in order to calculate ρ(t), the density matrix ρ(τ ) needs to be known at all previous times τ , which is trivial when using propagation, and also R (1) a1 (τ, τ ) need to be calculated for every τ < t. The latter is done by integrating the corresponding differential equation (11) as R (1) a1 (τ, τ ) = τ 0 ∂ τ1 R (1) a1 (τ 1 , τ ) dτ 1 (A2) = τ 0 F 1 ρ(τ 1 ), R (1) a1 (τ 1 , τ ), R (2) a1a2 (τ 1 , τ, τ 1 ) dτ 1 .
F 1 indicates that ∂ τ1 R(1)
a1 (τ 1 , τ ) is a function of ρ(τ 1 ), R (1) a1 (τ 1 , τ ) and R (2) a1a2 (τ 1 , τ, τ 1 ) for τ 1 < τ < t. Notice, that only the first time argument, which corresponds to the actual physical time, is propagated. The later time argument is a parameter that is kept constant during this propagation. The necessary 2nd-tier auxiliary density operators are then calculated as R (2) a1a2 (τ 1 , τ, τ 1 ) = τ1 0 ∂ τ2 R (2) a1a2 (τ 2 , τ, τ 1 ) dτ 2 (A3) = τ1 0 F 2 R (1) a1 (τ 2 , τ 1 ), R (1) a1 (τ 2 , τ ), R (2) a1a2 (τ 2 , τ, τ 1 ), R (3) a1a2a3 (τ 2 , τ, τ 1 , τ 2 ) dτ 2 with τ 2 < τ 1 < τ < t. Again, only the first time argument is propagated, whereas the others are kept constant throughout the propagation. This scheme straightforwardly generalizes to higher tiers.
ρ(τ)
R (1) a 1 (τ 1 ,τ 2 ) In order to provide some more insight, we explicitly consider the simplistic case of a noninteracting system attached to leads described in the wide-band limit, which can be calculated using only the density matrix and the 1st-tier auxiliary density matrices. The corresponding propagation scheme is visualized in Fig. 4 for two infinitesimal time steps centered around time t. In order to propagate ρ from t − dt to t, which is highlighted by a red arrow, we need the 1st-tier auxiliary operators R (1) a1 (t − dt, t − dt). These objects are calculated by setting the second time argument to t − dt and propagating the first time argument from 0 to t − dt, as indicated by a red arrow. Thereby, we implicitly assumed that the density matrix of previous times is known. In the next propagation step of the density matrix from t to t + dt, the object R (1) a1 (t, t) needs to be known which is calcu-lated by setting the second time argument to t and then propagating the first time argument from 0 to t.
Fig. 1
1depicts the current as a function of time for different bias voltages and different numerical routines. For 0.2V, the system is in the non-resonant transport regime, whereas for 0.8V it is in the resonant regime. For 0.4V, the chemical potential of the left lead µ L is in resonance with the molecular electronic state 0 . All currents show a transient behavior before approaching the steady-state value. Both the transient behavior and the steady state
FIG. 1 .
1Current as a function of time for an applied bias voltage of 0.2V, 0.4V and 0.8V. The different colors corresponds to the different bias voltages, the line style represents the numerical method used to calculate the current.
FIG. 2 .
2Top: Energy level scheme of the molecule and the leads as a function of applied bias voltage. Bottom: Current as a function of time for an applied bias voltage of 0.2V-2.0V. The dashed lines depict the current in the wide-band limit, the solid lines are obtained for a system coupled to leads with a finite bandwidth. The colors represent different applied bias voltages.
FIG. 3 .
3Current as a function of time for an applied bias voltage of 0.2V, 0.4V and 0.8V. The different colors correspond to the different bias voltages, the line style represents the electronic-vibrational interaction strength λ = 0eV, 0.05eV, 0.1eV.
FIG. 4 .
4Propagation scheme for the RSHQME method including the 0th and the 1st-tier only. Red arrows visualize propagation of the first time argument, blue dashed arrows indicate the dependency of the propagation step of ρ on the 1st-tier auxiliary density operator.
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|
[] |
[
"Median Area for Broken Sticks",
"Median Area for Broken Sticks"
] |
[
"Steven Finch "
] |
[] |
[] |
Breaking a line segment L in two places at random, the three pieces can be configured as a triangle T with probability 1/4. We determine both the PDF and CDF for area(T ) in terms of elliptic integrals. In particular, if L has length 1, then the median area 0.031458... can be calculated to arbitrary precision. We also mention the analog involving cyclic quadrilaterals -with corresponding probability 1/2 -and ask some unanswered questions.
| null |
[
"https://arxiv.org/pdf/1804.09602v1.pdf"
] | 119,313,081 |
1804.09602
|
73c3ab30393c5a96a9aaf04bf83bc37c0168b187
|
Median Area for Broken Sticks
25 Apr 2018 April 25, 2018
Steven Finch
Median Area for Broken Sticks
25 Apr 2018 April 25, 2018arXiv:1804.09602v1 [math.HO]
Breaking a line segment L in two places at random, the three pieces can be configured as a triangle T with probability 1/4. We determine both the PDF and CDF for area(T ) in terms of elliptic integrals. In particular, if L has length 1, then the median area 0.031458... can be calculated to arbitrary precision. We also mention the analog involving cyclic quadrilaterals -with corresponding probability 1/2 -and ask some unanswered questions.
For simplicity's sake, we start with a stick of length 2 (not 1). A triangle, formed by two independent uniform breaks in the stick, has sides a + b + c = 2 satisfying
0 < a < b + c, 0 < b < a + c, 0 < c < a + b hence 0 < a < 1, 0 < b < 1, 1 < a + b < 2.
The joint density function for (a, b) is thus 2 (constant) over the shaded triangular region in Figure 1; the marginal density for a is 2x if 0 < x < 1; the cross-correlation between a and b is −1/2. By Heron's formula, the mean area of the triangle is [1] E (area) = In contrast, a cyclic quadrilateral [2,3], formed by three independent uniform breaks in the stick, has sides a + b + c + d = 2 in this order satisfying 0 < a < b + c + d, 0 < b < a + c + d, 0 < c < a + b + d, 0 < d < a + b + c hence 0 < a < 1, 0 < b < 1, 0 < c < 1, 1 < a + b + c < 2. The joint density for (a, b, c) is thus 3/2 (constant) over the shaded hexahedral region in Figure 2; note the additional complexity of two missing corners, not just one. The marginal density for a is 3 4 (1 + 2x − 2x 2 ) for 0 < x < 1; the cross-correlation between a and b is −1/3. By Brahmagupta's formula, the mean area of the cyclic quadrilateral is [4]
E (area) = 1 0 1−x 0 1 1−x−y 3 2 (1 − x)(1 − y)(1 − z)(x + y + z − 1) dz dy dx + 1 0 1 1−x 2−x−y 0 3 2 (1 − x)(1 − y)(1 − z)(x + y + z − 1) dz dy dx = 4 17π 525 − π 2 160
and the mean square area is similarly 4 2 /560 = 1/35. As far as we know, no one has previously determined the exact density for the area of a triangle or a cyclic quadrilateral created via broken sticks. From such an expression would come a numerical estimate of the median area (50%-tile), obtained via a single integration. We succeed in finding the density for triangles, but unfortunately not for quadrilaterals. Even better would be an exact cumulative distribution function -allowing us to avoid the integration -and, surprisingly, this too is possible.
PDF for Triangle Area
We work with z = area 2 for now, returning to √ z = area at the conclusion. The system of equations
z = (1 − x)(1 − y)(x + y − 1), w = y
has two solutions:
x = 1 − w 2 ± (1 − w)w 2 − 4z 2 √ 1 − w , y = w
and the map (x, y) → (z, w) has absolute Jacobian determinant
−(1 − y)(2x + y − 2) −(1 − x)(x + 2y − 2) 0 1 = √ 1 − w (1 − w)w 2 − 4z.
Since the joint density for (x, y) is 2 and the map to (z, w) is two-to-one, the joint density for (z, w) is [5] 4
√ 1 − w (1 − w)w 2 − 4z = 4 (w − c)(w − a)(b − w)(1 − w)
where c(z) < 0 < a(z) < w < b(z) < 1 are the three zeroes of the cubic polynomial (1 − w)w 2 − 4z:
a(z) = 1 3 + 1 − i √ 3 6 θ(z) −1/3 + 1 + i √ 3 6 θ(z) 1/3 , b(z) = 1 3 + 1 + i √ 3 6 θ(z) −1/3 + 1 − i √ 3 6 θ(z) 1/3 , c(z) = 1 3 − 1 3 θ(z) −1/3 − 1 3 θ(z) 1/3 and θ(z) = −1 + 54z + 6 √ 3 √ −z + 27z 2 .
It follows that the marginal density for z is [6] g
(z) = b a 4 dw √ 1 − w (1 − w)w 2 − 4z = 8 (1 − a)(b − c) K (b − a)(1 − c) (1 − a)(b − c)
where K is the complete elliptic integral of the first kind:
K[m] = 1 0 dτ √ 1 − τ 2 √ 1 − mτ 2 (consistent with Mathematica). The marginal density for √ z is therefore f (ζ) = d dζ P √ z < ζ = d dζ P z < ζ 2 = 2ζ · g ζ 2 where 0 < ζ < 1/ 3 √ 3 ; see. Let u = 2 − x − y and v = y − x, so that 0 < u < 1, −1 < v < 1 and (u+v)(u−v)(1−u) = (2−2x)(2−2y) [1 − (2 − x − y)] = 4(1−x)(1−y)(x+y−1) = 4z hence u 2 − v 2 = 4z 1 − u hence |v| = u 2 − 4z 1 − u = (1 − u)u 2 − 4z 1 − u = q(z, u).
The pair (u, v) is uniform on the domain {(u, v) : 0 < u < 1 and |v| < u}, a triangle of unit area; thus the probability that area 2 exceeds z is
b(z) a(z) q(z,u) −q(z,u) dv du = 2 b(z) a(z) q(z, u)du
where the zeros c(z) < a(z) < u < b(z) < 1 are exactly as before. If t = √ 1 − u, then u = 1 − t 2 and du = −2t dt; it follows that
q(z, u)du = t 2 (1 − t 2 ) 2 − 4z t 2 (−2t dt) = −2 (1 − t 2 ) 2 t 2 − 4z dt = −2 (1 − t 2 ) t − 2 √ z (1 − t 2 ) t + 2 √ z dt = −2 (t 2 − α 2 ) (β 2 − t 2 ) (γ 2 − t 2 ) dt where β = √ 1 − a, α = √ 1 − b and γ = √ 1 − c. From 1 − c > 1 − a > 1 − u > 1 − b > 0, we have 0 < α(z) < t < β(z) < γ(z).
The preceding argument leading to the formula
P {area > ζ} = 4 β(ζ 2 ) α(ζ 2 ) (1 − t 2 ) 2 t 2 − 4ζ 2 dt = 4J (ζ)
is due to an anonymous student [7]. Our only contribution is to link this with Dieckmann's [8] integral evaluation:
8J = β γ 2 − α 2 α 2 + β 2 + γ 2 E (β 2 − α 2 ) γ 2 (γ 2 − α 2 ) β 2 + α 2 β γ 2 − α 2 α 2 + β 2 − 5γ 2 K (β 2 − α 2 ) γ 2 (γ 2 − α 2 ) β 2 − α 2 β γ 2 − α 2 (α + β − γ) (α − β − γ) (α − β + γ) (α + β + γ) Π β 2 − α 2 β 2 , (β 2 − α 2 ) γ 2 (γ 2 − α 2 ) β 2
where E and Π are complete elliptic integrals of the second and third kind:
E[m] = 1 0 √ 1 − mτ 2 √ 1 − τ 2 dτ, Π[n, m] = 1 0 dτ (1 − nτ 2 ) √ 1 − τ 2 √ 1 − mτ 2 .
Solving numerically the equation 8J (µ) = 1 gives the median to essentially infinite precision.
Cyclic Quadrilaterals
On the one hand, arbitrary angles α and β in a cyclic quadrilateral are distributed according to what we call a bivariate tent density:
ϕ(π − y, x)
if π − y < x < y and π/2 < y < π, ϕ(x, y) if x < y < π − x and 0 < x < π/2, ϕ(π − x, y)
if π − x < y < x and π/2 < x < π, ϕ(y, x)
if y < x < π − y and 0 < y < π/2 A sketch of the proof is given in Appendix I; a less complicated example appears in [9]. Clearly α and β are uncorrelated yet dependent. The univariate density for α is ψ 1 (x) − 16ψ 2 (x) ln(sin(x)/2) + ψ 3 (x) ln(tan(x/2)) 16 cos(x) 3 sin(x) 5 where trigonometric polynomials ψ 1 , ψ 2 , ψ 3 are given by ψ 1 (x) = −25 cos(x) + 7 cos(3x) + 17 cos(5x) + cos(7x), ψ 2 (x) = 42 cos(x) + 19 cos(3x) + 3 cos(5x), ψ 3 (x) = 378 + 489 cos(2x) + 150 cos(4x) + 7 cos(6x) (Figures 4 and 5).
It follows that E(α) = π/2, but a closed-form expression for On the other hand, finding the density for area has eluded us -witness Appendix II -and computer simulation suggests that it is approximately linear (Figure 6). Of all quadrilaterals with sides a, b, c, d in this order, there is a unique one with maximal area, the cyclic quadrilateral [10,11]. The natural analog of this theorem to n-gons for n ≥ 5 is true [12,13].
When breaking a stick in n − 1 places at random, the n pieces can be configured as an n-gon with probability 1 − n/2 n−1 [14].
A concise formula for the area of a cyclic pentagon, generalizing those of Heron and Brahmagupta, apparently does not exist. It is known that (4 · area) 2 satisfies a 7 th degree polynomial equation with coefficients involving elementary symmetric functions σ k of squares of sides [15,16,17,18,19,20,21]. One of two 7 th degree polynomials is satisfied for cyclic hexagons. For n ≥ 7, the equations become inconceivably lengthy, possessing degree 38 for cyclic heptagons and octagons, and almost a million terms when expanding with regard to σ k .
Unless a theoretical breakthrough occurs, broken sticks will never be fully understood for large n. A numerical approach is perhaps mandatory. We wonder if even the mean area (let alone the median area) of a cyclic pentagon is too much for which to ask.
Appendix I
The bivariate density for two angles of a triangle is easily obtained in [22]; the corresponding work for a cyclic quadrilateral is harder. Let adjacent angles α 1 , α 2 be opposite angles α 3 = π − α 1 , α 4 = π − α 2 . Let sides s 1 , s 2 determine α 3 and sides s 3 , s 4 determine α 1 (see the picture in [9]). By the Law of Cosines, s 2 1 + s 2 2 − 2s 1 s 2 cos(α 3 ) = s 2 3 + s 2 4 − 2s 3 s 4 cos(α 1 ), s 2 2 + s 2 3 − 2s 2 s 3 cos(α 4 ) = s 2 1 + s 2 4 − 2s 1 s 4 cos(α 2 ) hence s 2 1 + s 2 2 − s 2 3 − s 2 4 = −2(s 1 s 2 + s 3 s 4 ) cos(α 1 ), .
s 2 2 + s 2 3 − s 2 1 − s 2 4 = −2(s 2 s 3 + s 1 s 4 ) cos(α 2 ) hence α 1 = arccos s 2 3 + (2 − s 1 − s 2 − s 3 ) 2 − s 2 1 − s 2 2 s 3 (2 − s 1 − s 2 − s 3 ) + s 1 s 2 , α 2 = arccos s 2 1 + (2 − s 1 − s 2 − s 3 ) 2 − s 2 2 − s 2 3 s 1 (2 − s 1 − s 2 − s 3 ) +
We first rewrite this in terms of α 1 , α 2 , s 3 , remembering not only s 1 > 0, s 2 > 0 but also s 1 + s 2 + s 3 < 2. To do this, perform the substitutions
s 1 = 1 − (1 − s 3 ) tan α 2 2 tan α 1 2 , s 2 = −s 3 sin α 1 + α 2 2 + (2 − s 3 ) sin α 2 2 sin(α 1 − α 2 2 ) + sin α 2 2 .
The reciprocal of the determinant is then integrated with respect to s 3 , with lower limit Denote the integral by I(α 1 , α 2 ). The tent-like appearance of the surface plot of I suggests necessary simplifications leading to our formula for the joint density. Finally, the details of further integrating out α 2 are elaborate and thus omitted.
Appendix II
We work with r 1 = area 2 . The system of equations
r 1 = (1 − s 1 )(1 − s 2 )(1 − s 3 )(s 1 + s 2 + s 3 − 1), r 2 = s 2 , r 3 = s 3
has two solutions:
s 1 = 1 − r 2 + r 3 2 ± (1 − r 2 )(1 − r 3 )(r 2 + r 3 ) 2 − 4r 1 2 √ 1 − r 2 √ 1 − r 3 , s 2 = r 2 , s 3 = r 3
and the map (s 1 , s 2 , s 3 ) → (r 1 , r 2 , r 3 ) has absolute Jacobian determinant
√ 1 − r 2 √ 1 − r 3 (1 − r 2 )(1 − r 3 )(r 2 + r 3 ) 2 − 4r 1 .
Since the joint density for (s 1 , s 2 , s 3 ) is 3/2 and the map to (r 1 , r 2 , r 3 ) is two-to-one, the joint density for (r 1 ,
r 2 , r 3 ) is [5] 3 √ 1 − r 2 √ 1 − r 3 (1 − r 2 )(1 − r 3 )(r 2 + r 3 ) 2 − 4r 1 = 3 (r 3 − c)(r 3 − a)(b − r 3 )(1 − r 3 )(1 − r 2 )
where c(r 1 , r 2 ) < a(r 1 , r 2 ) < r 3 < b(r 1 , r 2 ) < 1 are the three zeroes of the cubic polynomial (1 − r 2 )(1 − r 3 )(r 2 + r 3 ) 2 − 4r 1 (regarded as a function of r 3 only). What troubles us is that, given sufficiently small r 1 > 0, there is a nonempty interval Ω ⊆ [0, 1] for which r 2 ∈ Ω implies a(r 1 , r 2 ) < 0. (As an example, if r 1 = 0.03, then Ω = [0.4807..., 0.8227...].) This implies that an integral with respect to r 3 must possess lower limit max {0, a(r 1 , r 2 )}. While this should not present an obstacle numerically, it does create havoc symbolically. To find exactly the endpoints of Ω, that is, to solve the equation a(r 1 , r 2 ) = 0 for two values 0 < r ′ 2 < r ′′ 2 < 1 via computer algebra, introduces a complexity roadblock in our stochastic analysis. Such difficulties did not arise in Section 1 because a(z) was always positive. Our hope is that someone else will see a workaround.
Acknowledgements
I am grateful to Andreas Dieckmann [8], who promptly evaluated the integral containing three quadratic factors at my request. It is impressive to see the recent work of students in [4,7,23,24] on variations of the broken stick; I appreciate efforts of their teachers in keeping the flame of original research alive.
Figure 1 :
1Triangular support for bivariate side density
Figure 2 :
2Hexahedral support for trivariate side density (two views)
Figure 3 .µ
3Numerically = 0.1258338431386510592028005... = (4)(0.0314584607846627648007001...) to precision limited only by the accuracy of the integration routine. A symbolic antiderivative of f (ζ) would seem infeasible, at least at first glance.
Figure 3 :
3Area density for triangles2. CDF for Triangle AreaAgain, we denote sides by x, y and area 2 by z
Figure 4 :
4Bivariate tent densitiy on [0, π] × [0, π] where ϕ(x, y) = [4 cos(x) − 3 cos(2y) − 1] tan (x/2) 2 2 [sin(x) + sin(y)] 2 sin(y) 2 .
Figure 5 :
5Angle density for cyclic quadrilaterals
Figure 6 :
6Area density for cyclic quadrilateralsWe nonrigorously estimate the median area to be 0.1696... via such experimentation. Can this be calculated to high precision?
s 2 s 3
3because the stick has length 2. The map (s 1 , s 2 , s 3 ) → (α 1 , α 2 , s 3 ) has absolute Jacobian determinant 2(s 1 + s 3 )(1 − s 3 ) [s 1 s 2 + s 3 (2 − s 1 − s 2 − s 3 )] [s 2 s 3 + s 1 (2 − s 1 − s 2 − s 3 )]
((α 1 ) − 3 cos(2α 2 ) − 1 (α 1 ) − 3 cos(2α 2 ) − 1
121121from s 2 = 0 and s 1 + s 2 + s 3 = 2). As an example, s 2 sin(α 1 ) + sin(α 2 )] 2 sin(α 2 ) 2 by a double angle formula and since, by the product-to-sum identity, (α 1 )+sin(α 2 ).
Copyright c 2018 by Steven R. Finch. All rights reserved.
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. Steven Finch, Mit Sloan School Of Management Cambridge, M A , Steven Finch MIT Sloan School of Management Cambridge, MA, USA steven [email protected]
|
[] |
[
"Covariance-free Partial Least Squares: An Incremental Dimensionality Reduction Method",
"Covariance-free Partial Least Squares: An Incremental Dimensionality Reduction Method"
] |
[
"Artur Jordao [email protected] \nComputer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil\n",
"Maiko Lie [email protected] \nComputer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil\n",
"Victor Hugo Cunha De Melo [email protected] \nComputer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil\n",
"William Robson Schwartz [email protected] \nComputer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil\n"
] |
[
"Computer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil",
"Computer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil",
"Computer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil",
"Computer Science Department\nSmart Sense Laboratory\nUniversidade Federal de Minas Gerais\nBrazil"
] |
[] |
Dimensionality reduction plays an important role in computer vision problems since it reduces computational cost and is often capable of yielding more discriminative data representation. In this context, Partial Least Squares (PLS) has presented notable results in tasks such as image classification and neural network optimization. However, PLS is infeasible on large datasets (e.g., ImageNet) because it requires all the data to be in memory in advance, which is often impractical due to hardware limitations. Additionally, this requirement prevents us from employing PLS on streaming applications where the data are being continuously generated. Motivated by this, we propose a novel incremental PLS, named Covariance-free Incremental Partial Least Squares (CIPLS), which learns a low-dimensional representation of the data using a single sample at a time. In contrast to other state-of-the-art approaches, instead of adopting a partially-discriminative or SGD-based model, we extend Nonlinear Iterative Partial Least Squares (NI-PALS) -the standard algorithm used to compute PLSfor incremental processing. Among the advantages of this approach are the preservation of discriminative information across all components, the possibility of employing its score matrices for feature selection, and its computational efficiency. We validate CIPLS on face verification and image classification tasks, where it outperforms several other incremental dimensionality reduction methods. In the context of feature selection, CIPLS achieves comparable results when compared to state-of-the-art techniques.
|
10.1109/wacv48630.2021.00146
|
[
"https://arxiv.org/pdf/1910.02319v1.pdf"
] | 203,837,582 |
1910.02319
|
9a4a9d25784bb73209d6349b11bded9803d1a11d
|
Covariance-free Partial Least Squares: An Incremental Dimensionality Reduction Method
5 Oct 2019
Artur Jordao [email protected]
Computer Science Department
Smart Sense Laboratory
Universidade Federal de Minas Gerais
Brazil
Maiko Lie [email protected]
Computer Science Department
Smart Sense Laboratory
Universidade Federal de Minas Gerais
Brazil
Victor Hugo Cunha De Melo [email protected]
Computer Science Department
Smart Sense Laboratory
Universidade Federal de Minas Gerais
Brazil
William Robson Schwartz [email protected]
Computer Science Department
Smart Sense Laboratory
Universidade Federal de Minas Gerais
Brazil
Covariance-free Partial Least Squares: An Incremental Dimensionality Reduction Method
5 Oct 2019
Dimensionality reduction plays an important role in computer vision problems since it reduces computational cost and is often capable of yielding more discriminative data representation. In this context, Partial Least Squares (PLS) has presented notable results in tasks such as image classification and neural network optimization. However, PLS is infeasible on large datasets (e.g., ImageNet) because it requires all the data to be in memory in advance, which is often impractical due to hardware limitations. Additionally, this requirement prevents us from employing PLS on streaming applications where the data are being continuously generated. Motivated by this, we propose a novel incremental PLS, named Covariance-free Incremental Partial Least Squares (CIPLS), which learns a low-dimensional representation of the data using a single sample at a time. In contrast to other state-of-the-art approaches, instead of adopting a partially-discriminative or SGD-based model, we extend Nonlinear Iterative Partial Least Squares (NI-PALS) -the standard algorithm used to compute PLSfor incremental processing. Among the advantages of this approach are the preservation of discriminative information across all components, the possibility of employing its score matrices for feature selection, and its computational efficiency. We validate CIPLS on face verification and image classification tasks, where it outperforms several other incremental dimensionality reduction methods. In the context of feature selection, CIPLS achieves comparable results when compared to state-of-the-art techniques.
Introduction
Dimensionality reduction is widely used in computer vision applications from image classification [11] [2] to neural network optimization [8]. The idea behind this technique is to estimate a transformation matrix that projects the high-dimensional feature space onto a low-dimensional latent space [20] [7]. Previous works have demonstrated that dimensionality reduction can improve not only computational cost but also the effectiveness of the data representation [18] [31] [29]. In this context, Partial Least Squares (PLS) has presented remarkable results when compared to other dimensionality reduction methods [29]. This is mainly due to the criterion through which PLS finds the low dimensional space, which is by capturing the relationship between independent and dependent variables. Another interesting aspect of PLS is that it can operate as a feature selection method, for instance, by employing Variable Importance in Projection (VIP) [21]. The VIP technique employs score matrices yielded by NIPALS (the standard algorithm used for traditional PLS) to compute the importance of each feature based on its contribution in the generation of the latent space.
Despite achieving notable results, PLS is not suitable for large datasets (e.g., ImageNet [5]) since it requires all the data to be in memory in advance, which is often impractical due to hardware limitations. Additionally, this requirement prevents us from employing PLS on streaming applications, where the data are being generated continuously. Such limitation is not particular to PLS, many dimensionality reduction methods, such as Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA), also suffer from this problem [32] [2].
To handle the aforementioned problem, many works have proposed incremental versions of traditional dimensionality reduction methods. The idea behind these methods is to estimate the projection matrix using a single data sample (or a subset) at a time while keeping some properties of the traditional dimensionality reduction methods. A well-known class of incremental methods is the one based on Stochastic Gradient Descent (SGD) [3] [2]. These methods interpret dimensionality reduction as a stochastic optimization problem of an unknown distribution. As shown by Weng et al. [32], incremental methods based on SGD are computationally expensive, present convergence problems and require many parameters that depend on the nature of (a) IPLS projection.
(b) SGDPLS projection.
(c) CIPLS (Ours) projection. Figure 1. Projection on the first (x-axis) and second (y-axis) components using different dimensionality reduction techniques. Our method (CIPLS) separates the feature space better than IPLS and SGDPLS, which are state-of-the-art incremental PLS-based methods. For IPLS and SGDPLS, the class separability is effective only on a single dimension of the latent space, while for CIPLS it is retained on both dimensions. Blue and red points denote positive and negative samples, respectively.
the data. To address this problem, Zeng et al. [34] proposed an efficient and low-cost incremental PLS (IPLS). In their work, the first dimension (component) of the latent space is found incrementally, while the other dimensions are estimated by projecting the first component onto the reconstructed covariance matrix, which is employed to address the issue of impractical memory requirements of a full covariance matrix. Even though IPLS achieves better performance than SGD-based and other state-of-the-art incremental methods, the discriminability of its higher-order components (i.e., all except the first) is not preserved, as shown in Figure 1 (a). This behavior appears because the higher-order components are estimated using only the independent variables, that is, it is based on an approximation of the covariance matrix X ⊤ X (similar to PCA) instead of X ⊤ Y employed in PLS. This can degrade the discriminability of the latent model since preserving the relationship between independent and dependent variables is an important property of the original PLS [7]. It is important to emphasize that, for highdimensional data, employing several components often provides better results [29][9][10] [17], hence, IPLS might not be suitable on these cases.
Motivated by limitations and drawbacks in incremental PLS-based approaches, we propose a novel incremental method 1 . Our method relies on the hypothesis that the estimation of higher-order components using the covariance matrix, as proposed by Zeng et al. [34], is inadequate since the relationship between independent and dependent variables is lost. Therefore, to preserve this characteristic, we extend NIPALS [1] to avoid the computation of X ⊤ Y and, consequently, enable it for incremental operation. Since our proposed extension is based on a simple algebraic decomposition, we preserve the simplicity and efficiency that makes NIPALS popular, and we ensure that the relationship between independent and dependent variables is propagated to all components, differently from other methods.
As shown in Figure 1, our method is capable of separating data classes better than IPLS, mainly on the second component (i.e., y-axis). Since the proposed method does not use the covariance matrix (X ⊤ X) to estimate higherorder components, we refer to it as Covariance-free Incremental Partial Least Squares (CIPLS). Besides providing superior performance, our method can easily be extended as a feature selection technique since it provides all the requirements to execute VIP. Existing incremental PLS methods, on the other hand, require more complex techniques to operate as feature selection [21].
We compare the proposed method on the tasks of face verification and image classification, where it outperforms several other incremental methods in terms of accuracy and efficiency. In addition, in the context of feature selection, we evaluate and compare the proposed method to state-ofthe-art methods, where it achieves competitive results.
Related Work
To enable PCA to operate in an incremental scheme, Weng et al. [32] proposed to compute the principal components without estimating the covariance matrix, which is unknown and impossible to be calculated in incremental methods. For this purpose, their method (CCIPCA) updates the projection matrix for each sample x, replacing the unknown covariance matrix by the sample covariance matrix (xx ⊤ ). While CCIPCA provides a minimum reconstruction error of the data, it might not improve or even preserve the discriminability of the resulting subspace since label information is ignored (similarly to traditional PCA) [20].
To achieve discriminability, incremental methods based on Linear Discriminant Analysis (LDA) have been proposed [12] [19]. In particular, this class of methods is less explored since they present issues such as the sample size problem [13], which makes them infeasible for some tasks. Different from incremental LDA methods, incremental PLS methods are more flexible and present better results [34]. Motivated by this, Arora et al. [3] proposed an incremental PLS based on stochastic optimization (SGDPLS), where the idea is to optimize an objective function using a single sample at a time. Similarly to Arora et al. [3], Stott et al. [30] proposed applying stochastic gradient maximization on NIPALS, extending it for incremental processing. Even though they present promising results on synthetic data, their approach presented convergence problems when evaluated on real-world datasets. Thus, in this work, we consider only the approach by Arora et al. [3], which was the one that converged for several of the datasets evaluated and presented better results.
While SGDPLS is effective, as demonstrated by Weng et al. [32] and Zeng et al. [34], SGD-based methods applied to dimensionality reduction are computationally expensive and present convergence problems. In addition, this class of approaches requires careful parameter tuning and their results are often sensitive to the type of dataset [32].
To address convergence problems in SGD-based PLS, Zeng et al. [34] proposed to decompose the relationship between independent and dependent matrices (variables) into a sample relationship (i.e., a single sample with its label). This process is performed only to compute the first component, the higher-order components are estimated by projecting the first component onto an approximated covariance matrix using a few PCA components. As we mentioned earlier, since traditional PCA cannot be employed in incremental methods, Zeng et al. [34] used CCIPCA to reconstruct the principal components of the covariance matrix.
In contrast to existing incremental PLS methods, our method presents superior performance in both accuracy and execution time for estimation of the projection matrix, which is an important requirement for time-sensitive and resource-constrained tasks. In particular, our method outperforms IPLS and SGDPLS in 32.48 and 24.83 percentage points, respectively, when using only higher-order components. The reason for these results is the quality of our higher-order components, which keeps the properties of traditional PLS.
Besides dimensionality reduction, another group of techniques widely employed to reduce computational cost are feature selection methods. One of the most recent and successful feature selection methods is the work by Roffo et al. [28], which proposed to interpret feature selection as a graph problem. In their method, named infinity feature selection (infFS), each feature represents a node in an undirected fully-connected graph and the paths in this graph represent the combinations of features. Following this model, the goal is to find the best path taking into account all the possible paths (in this sense, all the subsets of features) on the graph, by exploring the convergence property of the geometric power series of a matrix. Improving upon this model, Roffo et al. [27] suggested quantizing the raw features into a small set of tokens before applying the process of Roffo et al. [28]. By using this pre-processing, their method (referred to as infinity latent feature selection -ilFS) achieved even better results than Roffo et al. [28]. Even though Roffo et al. [28] [27] achieved state-of-the-art results on the context of neural network optimization, Jordao et al. [16] showed that PLS+VIP attains superior performance. We show that CIPLS+VIP achieves comparable results when compared to PLS+VIP and other state-of-the-art feature selection techniques.
Proposed Approach
In this section, we start by describing the traditional Partial Least Squares (PLS). Then, we present the proposed Covariance-free Incremental Partial Least Squares (CIPLS) and the Variable Importance in Projection (VIP) technique, which enables PLS and CIPLS to be employed for feature selection. Unless stated otherwise, let X ⊂ R n×m be the matrix of independent variables denoting n training samples in a m-dimensional space. Furthermore, let Y ⊂ R n×1 be the matrix of dependent variables representing the binary class label. Finally, let x n ⊂ R 1×m and y n ⊂ R 1×1 be a single sample of X and Y , respectively. We highlight that, in the context of streaming data, x n is a data sample acquired at time n.
Partial Least Squares
Given a high m-dimensional space, PLS finds a projection matrix W (w 1 , w 2 , ..., w c ), which projects this space onto a low c-dimensional space, where c ≪ m. For this purpose, PLS aims at maximizing the covariance between the independent and dependent variables. Formally, PLS constructs W such that
w i = maximize(cov(Xw, Y )), s.t w = 1,(1)
where w i denotes the ith component of the c-dimensional space. The exact solution to Equation 1 is given by
w i = X ⊤ Y X ⊤ Y .(2)
From Equation 2, we can compute all the c components using either Nonlinear Iterative Partial Least Squares (NI-PALS) [1] or Singular Value Decomposition (SVD). Most works employ NIPALS since it is capable of finding only the c first components, while SVD always finds all the m components, being computationally inefficient compared to NIPALS [1].
Covariance-free Incremental PLS
The core idea in our method is to ensure that, as in traditional PLS, the relationship between independent and dependent variables (Equation 2) is kept on all the c components. To achieve this goal, our method works as follows. First, we need to center the data to the mean of the training samples X. However, different from traditional methods, in incremental approaches the mean is unknown since we cannot assume that all the data are known a priori [32] [34]. To face this problem, we center the current data sample using an approximate centralization process [32], which consists of estimating an incremental mean using the nth sample. According to Weng et al. [32], we can compute the incremental mean µ n w.r.t. the nth data sample as
µ n = n − 1 n µ (n−1) + 1 n x n .(3)
Once we have centralized the sample, the next step in our method is to compute the component w i following Equation 2. As we mentioned, X and its respective Y are unknown or are not in memory in advance, which prohibits us to apply Equation 2 directly. However, as suggested by Zeng et al. [34], we employ the following decomposition:
X T Y = n−1 k=1 x k y k + x n y n .(4)
By replacing X ⊤ Y in Equation 2 by Equation 4, it is possible to calculate the ith component of PLS considering a single sample at a time. In other words, Equation 4 enables to compute w i incrementally.
To compute the higher-order components (w i , i > 1), we employ a deflation process, which consists of subtracting the contribution of the current component on the sample before estimating the next component. Following the NIPALS algorithm, the deflation process works as follows
t = Xw i ,(5)p = X ⊤ t, q = Y ⊤ t,(6)X = X − tp ⊤ , Y = Y − tq ⊤ ,(7)
where t denotes the projected samples onto the current component w i , and p and q represent the loadings of this projection. It should be noted that while t works in an incremental scheme (since we can project one sample at a time), p and q cannot be computed since X and Y are neither known nor are in memory in advance. However, in light of Equation 4, we can decompose p and q as p = n−1 k=1
x k t k + x n t n , q = n−1 k=1 y k t k + y n t n .
Algorithm 1: CIPLS Algorithm. Input : nth data sample x n and its label y n Number of components c Projection matrix W (n−1) ⊂ R c×m Loading matrix P (n−1) ⊂ R c×m Loading matrix Q (n−1) ⊂ R c×1 Output: Updated W , P and Q 1 Update µ n using Equation 3
2x n = x n − µ n
3 for i = 1 to c do 4 w i =x n y n + w i(n−1) , where w i ∈ W 5 t n =x nwi xnwi 6 p i =x n t n + p i(n−1) , where p i ∈ P 7 q i = y n t n + q i(n−1) , where q i ∈ Q 8x n =x n − t n p ⊤ i 9 y n = y n − t n q ⊤ i 10 end
By embedding Equation 8 on the deflation process, we can remove the contribution of the current component and repeat the process to compute a single component w i . Observe that Equation 7 deflates each sample by its reconstructed value, therefore, Equation 7 can be computed sample-by-sample, working in an incremental scheme. With this formulation, we are now capable of computing the c components incrementally. Algorithm 1 summarizes the steps of the proposed method. It should be mentioned that the matrices W , P and Q are initialized with zeros.
According to Algorithm 1, the proposed method maintains the property of capturing the relationship between X and Y for all components (step 4 in Algorithm 1). In addition, since we compute all components at once, our method has a time complexity of O(ncm), where n, c and m denote the number of samples, number of components, and dimensionality of the data, respectively.
CIPLS for Feature Selection
An advantage of PLS is that, after estimating the projection matrix W , it is possible to estimate the importance of each feature, enabling PLS to operate as a feature selection method. For this purpose, it is possible to employ Variable Importance in Projection (VIP), which estimates the importance of each feature f j w.r.t its contribution to yield the low dimensional space. According to [21], VIP is defined as
f j = m c i=1 q 2 i t ⊤ i t i (w ij / w i 2 )/ c i=1 q 2 i t ⊤ i t i .(9)
Once we have estimated the score of each feature, we can remove a percentage of features based on their scores. As can be verified in Algorithm 1, CIPLS preserves the ability of traditional PLS to be employed as a feature selection method via VIP (Equation 9). On the other hand, it is important to emphasize that IPLS and SGDPLS cannot be used to compute VIP since they do not provide the loading matrix Q (q 1 , q 2 , ..., q c ).
Experimental Results
In this section, we first introduce the experimental setup and the tasks employed to validate the proposed method. Then, we present the procedure conducted to calibrate the parameters of the methods. Next, we compare the proposed method with other incremental partial least squares methods as well as with the traditional PLS. Afterwards, we present the influence of higher-order components on the classification performance. Finally, we discuss the time complexity of the methods, their performance on a streaming scenario and compare our method on the feature selection context. Experimental Setup. Throughout the experiments, we use a linear SVM for binary classification (face verification) because, according to Zeng et al. [34], a linear SVM coupled with dimensionality reduction is able to achieve remarkable results while being computationally efficient. In addition, it has been shown that simple classifiers when feed by features from convolutional networks are able to achieve results comparable to more sophisticated classifiers [6] [25]. For multi-class problems (image classification), on the other hand, we prefer to use a multilayer perceptron because it handles the multi-class problem naturally, avoiding the need for employing a binary classifier on a one-versus-rest fashion, which would be computationally expensive. All experiments and methods were executed on an Intel Core i5-8400, 2.4 GHz processor with 16 GB of RAM.
To assess the differences in efficacy and efficiency among the compared methods, throughout the experiments we follow the approach by Jain et al. [15] and perform statistical tests based on a paired t-test using 95% of confidence. We highlight that the statistical tests were conducted only for face verification due to the computational cost of retraining (i.e., fine-tuning) the convolutional neural network for image classification, which is considerably high since we employ large-scale datasets in our assessment. Face Verification. Given a pair of face images, face verification determines whether this pair belongs to the same person. For this purpose, we use a three-stage pipeline [24] [4], which works as follows. First, we extract a feature vector of each face using a deep learning model. In this work, we use the feature maps from the last convolutional layer of the VGG16 model, learned on the VGGFaces dataset [23], as feature vector. Then, we compute the distance between the We conduct our evaluation on two face verification datasets, namely Labeled Faces in the Wild (LFW) [14] and Youtube Faces (YTF) [33]. Image Classification. Image classification consists of deciding to which one of a given set of categories an image belongs. Traditionally, this is done by extracting features from the samples and feeding these features to a classifier, which determines the category to which each image belongs. For this purpose, we use the feature maps from the last convolutional layer of the VGG16 model as features.
For the image classification task, we consider two versions of the ImageNet dataset, with images of size 224×224 and 32×32 pixels. The former is used since it is the original version of the dataset, while the latter is used because it has been demonstrated to be more challenging than the original version [26] [22]. Number of Components. One of the most important aspects in dimensionality reduction methods is the number of components c of the resulting latent space. Therefore, to choose the best number of components for each method, we vary c from 1 to 10 and select the value for which the method achieved the highest accuracy on the validation set (10% of the training set). Once the best c is chosen, we use the training and validation set to learn the projection method and the classifier. We repeat this process for each dataset. Comparison with Incremental Methods. This experiment compares our CIPLS with other incremental dimensionality reduction methods. Table 1 summarizes the results and shows that, on LFW, our method outperformed SGDPLS and IPLS by 1.08 and 1.38 percentage points (p.p.), respectively. Similarly, on the YTF dataset, CIPLS outperformed SGDPLS and IPLS by 0.88 and 1.88 p.p., in this order.
Finally, on the ImageNet dataset, the difference in accuracy compared to IPLS was of 0.07 and 1.35 p.p., for the 32 × 32 and 224 × 224 versions, respectively. It is important to mention that we do not consider SGDPLS on these datasets due to convergence problems and high computa- Table 2. Accuracy of existing incremental methods when using only higher-order components. Values computed considering the average accuracy across all tasks in our assessment.
Average Accuracy CCIPCA [32] 63.48 SGDPLS [3] 58.41 IPLS [34] 50.76 CIPLS (Ours) 83.24
tional cost. Moreover, due to memory constraints, it was not possible to run the traditional PLS on the ImageNet dataset.
Comparison with Partial Least Squares. As suggested by Weng et al. [32], we compare the incremental methods with the traditional approach (in our case, traditional PLS), in which the closer to the accuracy of the baseline, the better. According to Table 1, besides providing better results than IPLS and SGDPLS, CIPLS achieved the closest results to traditional PLS. For instance, on LFW, the difference in accuracy between PLS and CIPLS was 0.79 p.p. while on the YTF dataset it was 1.86 p.p. In contrast, the difference in accuracy between PLS and SGDPLS is higher -1.87 p.p. on LFW and 2.74 p.p. on the YTF dataset. In addition, the difference in accuracy between PLS and IPLS is among the highest, 2.17 and 3.74 p.p. for the LFW and YTF, respectively. In particular, the results for PLS and CIPLS are statistically equivalent, while IPLS and SGDPLS present results statistically inferior compared to PLS.
It should be noted that the results of IPLS are closer to CCIPCA than PLS since only the first component of IPLS maintains the relationship between independent and dependent variables. On the other hand, the proposed method preserves this relation along higher-order components, which provides better discriminability, as seen in our results. Higher-order Components. This experiment assesses the discriminability of the higher-order components of CIPLS compared to each of the other incremental methods. For this purpose, we follow a process suggested by Martinez [20], which consists of removing the first component of the latent space before presenting the projected data to the classifier. This evaluates the performance of the remaining components, not only the first one which tends to be better. Table 2 shows the results. According to Table 2, CIPLS outperformed IPLS by 32.48 p.p. Observe that when all the components are used, CIPLS outperformed IPLS by 1.17 p.p. This larger difference when removing the first component is an effect of the better discriminability achieved by the components extracted by CIPLS. As we have argued, CIPLS preserves the relationship between dependent and independent variables across higher-order components, yielding more accurate results. Compared to SGDPLS, CIPLS outperforms it by 24.83 p.p. Table 3. Comparison of incremental dimensionality reduction methods in terms of time complexity and execution time (in seconds) for estimating the projection matrix. m, n denote dimensionality of the original data and number of samples, while c, L and T denote number of PLS components, number of PCA components and convergence steps, respectively. Time Complexity CCIPCA [32] O(nLm) SGDPLS [3] O(T cm) IPLS [34] O(nLm + c 2 m) CIPLS (Ours) O(ncm)
Time Issues. To demonstrate the efficiency of CIPLS, in this experiment, we compare its time complexity to compute the projection matrix with the incremental methods evaluated. Following Weng et al. [32] and Zeng et al. [34], we report this complexity w.r.t. dimensionality of the original data (m), number of samples (n), number of components (c) and number of PCA components (L -required only by IPLS and CCIPCA). Table 3 shows the time complexity of the methods. According to Table 3, CIPLS presents a low time complexity for estimating the projection matrix. The complexity of CIPLS is not only on the same class as CCIPCA, which is the fastest among the compared methods, but it also has a very small constant factor. This constant factor is the number of components, c for CIPLS and L for CCIPCA. Experimentally, we found that the optimal constant factor for PLS is negligible, c = 2 resulted in the highest accuracies. While, for fairness, the same number of components was adopted for all methods in Table 3, typically c < L on practical applications. This is a known advantage of PLS, it has been shown to require substantially less components to achieve its optimal accuracy than PCA [29]. Finally, we report the average computation time (considering 30 executions) of the methods for estimating the projection matrix for one new sample. To make a fair comparison, we set c = 4 for all methods and for the other parameters we use the values where the methods achieved the best results in validation. As shown in Figure 2, SGDPLS is the slowest incremental PLS method, which is a consequence of its strategy for estimating the projection matrix, where for each sample the convergence step is run T times. Our experiments showed that T ≥ 100 is required for good results. The computation time for estimating the projection matrix of our method was statistically equivalent (according to a paired t-test) to that of CCIPCA, which is the fastest among the incremental dimensionality reduction methods assessed. Moreover, CIPLS was statistically faster than IPLS and SGDPLS, demonstrating that it is the fastest among the compared incremental PLS methods. Incremental Methods on the Streaming Scenario. As we argued before, incremental methods can be employed on streaming applications, where the training data are continuously generated. To demonstrate the robustness of our method on these scenarios, in this experiment, we evaluate the methods on a synthetic streaming context, as proposed by Zeng et al. [34]. The procedure works as follows. First, the training data is divided into k blocks, where k = 20. The idea behind this process is to interpret each block as a new instance of arriving data. Then, we create a new training set and insert each kth block at a time. Each time we insert a new block, we learn the projection method and evaluate its accuracy on the testing set. For instance, when adding the tenth block, all the 1, 2, ..., 10 blocks are being used as training. It is important to mention that a block contains more than one sample, however, this does not modify the strategy of the incremental methods, which is to estimate the projection matrix by using a single sample at a time. Figure 3 (a) and (b) show the results on the LFW and YTF datasets, respectively. On LFW, until the fifth block, it is not possible to determine the best method since the accuracy presents high variance, however, from the sixth block onwards, CIPLS outperforms all other methods. On YTF, our method achieves the highest accuracy for all blocks. These results show that the proposed method is more adequate for streaming applications than existing incremental PLS methods. Comparison with Feature Selection Methods. Our last experiment evaluates the performance of CIPLS as a feature selection method. Table 4 shows the results for different percentages of kept features on LFW and YTF. According to Table 4, CIPLS achieves comparable results when compared to state-of-the-art feature selection techniques. For example, on LFW the difference in accuracy, on average, from CIPLS to infFS and ilFS is of 0.15 and 0.25 p.p., respectively. In contrast, on YTF for some percentages of kept features (e.g., 15% and 20%), CIPLS outperforms infFS and ilFS. We highlight that these methods were designed specifically for feature selection. Additionally, the difference, on average, between CIPLS and PLS is of 0.27, 0.14 and 0.50 p.p. on the LFW and YTF datasets, respectively. Moreover, the largest accuracy difference between PLS and CIPLS is of 0.4 p.p., on LFW dataset with 10% of features kept. This result reinforces that the proposed decompositions to extend the NIPALS and enable the employment of VIP are a good approximation of the original method.
Based on the results shown, it is possible to conclude that, besides dimensionality reduction, CIPLS achieves state-of-the-art results in the context of feature selection.
Conclusions
This work presented a novel incremental partial least squares method, named Covariance-free Incremental Partial Least Squares (CIPLS). The method extends the NI-PALS algorithm for incremental operation and enables computation of the projection matrix using one sample at a time while still presenting the main property of traditional PLS, namely preserving the relation between dependent and independent variables. Compared to existing incremental partial least squares methods, CIPLS attains superior performance besides being computationally efficient. In addition, different from previous incremental partial least squares, CIPLS can easily to operate as a feature selection method. In this context, the proposed method is able to achieve comparable results to the state-of-the-art.
Figure 2 .
2Average prediction time (in seconds) for estimating the projection matrix, lower values are better. Black bars denote the confidence interval.
Faces in the Wild (LFW).
Figure 3 .
3Comparison of incremental methods on a streaming scenario. The x-axis denotes the data arriving sequentially.
Table 1 .
1Comparison of existing incremental methods in terms of accuracy. The symbol '-' denotes that it was not possible to execute the method on the respective dataset due to memory constraints or convergence problems (see the text). PLS denotes the use of the traditional PLS.LFW
YTF
ImageNet
32x32
ImageNet
224x224
CCIPCA [32] 89.87 81.48
40.30
52.58
SGDPLS [3] 90.60 83.22
-
-
IPLS [34]
90.30 82.22
43.24
65.74
CIPLS
91.68 84.10
43.31
67.09
PLS
92.47 85.96
-
-
two feature vectors employing the ℓ 1 -distance metric and
present the result of the distance metric to a classifier.
Table 4 .
4Comparison of feature selection methods using different percentages of kept features. CIPLS (Ours)+VIP 91.63 91.55 91.80 92.18 86.48 86.92 87.02 87.40LFW
YTF
Percentage of Kept Features
Percentage of Kept Features
10
15
20
50
10
15
20
50
infFS [28]
91.92 91.58 92.03 92.23
86.64 86.68 87.14 87.30
ilFS [27]
92.03 91.67 92.25 92.23
86.60 86.94 86.84 87.54
PLS+VIP
92.05 91.67 92.13 92.38
86.70 86.82 87.18 87.68
The code is available at: https://github.com/arturjordao/IncrementalDimensionalityReduction
AcknowledgmentsThe authors would like to thank the Brazilian National
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"Signatures of the light gluino in the top quark production",
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"Chong Sheng Li \nDepartment of Physics\nPeking University\n100871BeijingChina\n",
"P Nadolsky \nDepartment of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA\n",
"C.-P Yuan \nDepartment of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA\n",
"Hong-Yi Zhou \nInstitute of Modern Physics\nDepartment of Physics\nTsinghua University\n100084BeijingChina\n"
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"Department of Physics\nPeking University\n100871BeijingChina",
"Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA",
"Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA",
"Institute of Modern Physics\nDepartment of Physics\nTsinghua University\n100084BeijingChina"
] |
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If a light gluino, with a mass of the order of GeV, exists in the minimal supersymmetric extension of the Standard Model, then it can contribute to the production rate of the top quark pairs at hadron colliders viagg → tt. Because the top quark is heavy, the masses of the superpartners of the lefthanded and right-handed top quarks can be very different such that a parityviolating observable can be induced in the tree level production process. We discuss the phenomenology of this parity violatiing asymmetry at the CERN Large Hadron Collider.
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10.1103/physrevd.58.095004
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"https://arxiv.org/pdf/hep-ph/9804258v1.pdf"
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hep-ph/9804258
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7bbbcea52080b879ec60bd22daa43b2228f2a058
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Signatures of the light gluino in the top quark production
Apr 1998 (February 1, 2018)
Chong Sheng Li
Department of Physics
Peking University
100871BeijingChina
P Nadolsky
Department of Physics and Astronomy
Michigan State University
48824East LansingMIUSA
C.-P Yuan
Department of Physics and Astronomy
Michigan State University
48824East LansingMIUSA
Hong-Yi Zhou
Institute of Modern Physics
Department of Physics
Tsinghua University
100084BeijingChina
Signatures of the light gluino in the top quark production
Apr 1998 (February 1, 2018)arXiv:hep-ph/9804258v1 8
If a light gluino, with a mass of the order of GeV, exists in the minimal supersymmetric extension of the Standard Model, then it can contribute to the production rate of the top quark pairs at hadron colliders viagg → tt. Because the top quark is heavy, the masses of the superpartners of the lefthanded and right-handed top quarks can be very different such that a parityviolating observable can be induced in the tree level production process. We discuss the phenomenology of this parity violatiing asymmetry at the CERN Large Hadron Collider.
I. INTRODUCTION
In despite of the success of the Standard Model (SM) in explaining and predicting experimental data, it is widely believed that new physics has to set in at some high energy scale. One of such new physics models is the the minimal supersymmetric extension of the Standard Model (MSSM). Various supersymmetry (SUSY) models, such as gravity-mediated and gauge-mediated supersymmetry breaking models [2], have been extensively considered in the literature to explain why the masses of the superparticles are not the same as those of the SM particles. In general, the masses of the superparticles are predicted to be around a few hundred GeV or at the TeV region. There are ample studies in the literature to examine the detection of these non-standard particles in the current and future experiments, including those at the CERN Large Hadron Collider (LHC), and at the future Linear Colliders.
Among the superparticles of the MSSM, some models of SUSY breaking predict the existence of a light gluino with the masses around 1 GeV or less [6]. If this scenario is true, then there is rich phenomenology predicted for the current experimental data which can be used to either confirm or constrain models. In Ref. [3], ALEPH Collaboration used the data on the cross-sections of dijet production and the angular distributions in 4-jet production to derive the ratios of the color factors C A /C F and T F /C F . Based on the obtained values, ALEPH excluded the existence of the gluinos with the mass lighter than 6.3 GeV at 95% confidence level. The result was criticized by Farrar [5], who argued that ALEPH's analysis underestimated the theoretical uncertainties in the knowledge of hadronization and resummation of large logarithms arising in the separation of jets from soft radiation. If these uncertainties are taken into account, the light gluino is excluded only at 1σ level. This problem was further examined by Csikor and Fodor in Ref. [4], where they determined the color factors of underlying gauge theory by studying the behavior of the ratios R γ = σ(e + e − → jets)/σ(e + e − → µ + µ − ), R τ = Γ(τ − → ν τ + jets)/Γ(τ − → ν τ e −ν e ), R Z = Γ(Z → hadrons)/Γ(Z → µ + µ − ) in the region of 5 GeV to M Z scale. They concluded that the O(α 3 s ) analysis of these quantities allows to exclude the light gluino with the mass between 3 and 5 GeV at 93% confidence level, and with the mass less than 1.5 GeV at 70.8% confidence level. If their results are combined with the χ 2 -distribution from ALEPH analysis, the exclusion confidence level is improved to 99.97% and 99.89%, respectively. This conclusion is quite insensitive to the overall error of ALEPH's results; for instance, the exclusion limits of the combined analysis are still above 95% if ALEPH's systematic error is increased by a factor of 3. However, in order to extract the number of active fermions from the experimental data, both methods [3] and [4] rely on the state-of-art usage of perturbative theory. None of the separate analyses can exclude the light gluino at the confidence level ≥ 70%, and combined and complicated analysis is needed to overcome the flaws of each separate method. Another significant limitation on the possible parameter space of the models with light gluinos was recently imposed by the negative results of the search for the production of charginos with the mass less than m W at LEP2 [21]. This result disfavors the models with the masses of all gauginos vanishing at tree level at GUT scale [6], in which gluino has the mass of the order GeV, and at the electroweak scale at least one of the charginos is necessarily lighter than W -boson. However, the LEP2 data can not rule out the models with the other spectra of gaugino masses, for instance, the models of gauge-mediated symmetry breaking where the gluino can be the only light gaugino [7]. As mentioned before, the analysis of [3,4] already puts strong constraints on the possibility of the light mass of the gluino, however, due to the aforementioned theoretical difficulties it seems that more study is needed.
There are a few other methods discussed in the literature to look for light gluino. If gluino is light and hadronizes before reaching the detector, it should be possible to observe its bound states, for example, R 0 -mesons, created by binding of gluon and gluino [8]. Although the region of R 0 masses is significantly restricted by KTeV measurements [9], R 0 can still exist in the mass region 1.4 − 2.2 GeV [10].
If the squark masses are of the order of several hundreds GeV, the light mass of gluino can lead to the noticeable peaks in the dijet invariant mass and angular distributions at TEVATRON or LHC, arising due to the resonant production of massive squarks in the quark-gluino fusion [11]. The already existing TEVATRON data allows to exclude the light gluino models with the masses of the lighter squarks lying between 150 and 650 GeV [11]; it would be desirable to continue the search for the resonant peaks at TEVATRON, as well as at LHC, where the increased dijet production cross-section would allow to cover a larger region of squark masses.
In the pQCD theory, the existence of a light gluino would change the running of the strong coupling, as well as the form of the Dokshitser-Gribov-Lipatov-Altarelli-Parizi (DGLAP) equations. Therefore, to describe the existing DIS data, it is necessary to account not only for the quark and gluon distribution functions inside the initial hadron(s), but also for the gluino distribution which has different renormalization group properties. An obvious question is whether the currently available hadronic data is consistent with the existence of light gluino. The last analysis of this type was done in 1994 publications [17,18], which showed that the existence of the light gluino didn't contradict DIS data available at that time. However, those analyses did not include the more recent data from H1, ZEUS and NMC experimental groups [12][13][14] covering the region of lower x and Q 2 . These new data can be crucial for testing the scenario of having a light gluino in the supersymmetry models, because the existence of a light gluino would imply a slower running of the parton distribution functions from the low to high Q 2 .
The existence of new types of particle interactions can be proved if one observes the violation of the symmetries of the Standard Model, for instance, the significant violation of the discrete symmetry with respect to space reflections (P -parity) in strong interactions. Experimental search for parity-violating effects could be performed relatively easy in the processes with t-quarks in the final state, due to the possibility to trace the polarization of the tops decaying through the channel t → W + + b. Therefore, in this work we would like to concentrate on the production of top quark pairs. For the top quark pairs produced at hadron colliders, the SM allows the production processes qq, GG → G → tt to violate Pparity in the next-to-leading orders due to the presence of W and Z bosons in loop diagrams. However, this effect is shown to be negligible [15]. On the other hand, for certain choices of SUSY parameters in the MSSM, it is possible to obtain a large difference between the masses of right-stop and left-stop, which in principle can lead to some noticeable asymmetries in the production of right-and left-handed top quarks. These asymmetries arise either in the next-to-leading order of the SUSY QCD process qq, GG → G → tt, or at the tree level of the SUSY QCD processgg → tt. The asymmetries of the first type were studied earlier in [16]. It was shown that at TEVATRON the difference in the cross-sections of right-and left-handed t-quark production can be of the order 2 − 3% provided the right-stop is light. This conclusion holds for a wide range of gluino masses. As it will be shown below, the asymmetries of the second type can only be noticeable if gluinos are light, and the parton density of gluinos in the nucleon is comparable with that of the sea quarks.
The primary goal of this article is to present the leading order (LO) study of the second scenario, and to evaluate the impact of the small mass of gluino on the production of t-quarks at the CERN Large Hadron Collider (LHC). In the first part of this study we obtained the LO distributions of the partons in the nucleon with the account for the possible non-zero contents of light gluinos. For this purpose we modified the fitting program used previously to obtain CTEQ4L parton distributions [19]. Since our study is the leading order calculation, we considered it sufficient not to perform the complete NLO analysis of parton distributions, contrary to what was done in [17,18].
In the course of the study, it was a surprise for us to find that the account for the new hadronic data from H1, ZEUS and NMC groups [12][13][14], which was not available at the time of the previous studies [17,18], tends to increase the overall χ 2 of the fit after the inclusion of a light gluino. The reason for this is that these new data cover the region of lower x and Q 2 , thus making the analysis more sensitive to the slower running of the parton distributions in the SUSY QCD theory with a light gluino. Nonetheless, we would like to be extremely cautious about this observation and refrain from any final conclusions about the consistency of the current experimental data and the SUSY QCD theory with a light gluino before more thorough next-to-leading order global analysis of hadronic data is made. Instead, we would like to concentrate on the primary goal of this paper, namely, on the calculations of the top quark production asymmetries at the LHC. For this process, the Bjorken x of the initial state partons are allowed to lie in the range
4m 2 t s = 6.25 · 10 −4 ≤ x 1,2 ≤ 1,(1)
where the parton distributions are less dependent on the low x data. We therefore expect the results of this work to be stable with respect to the possible changes in the parton distributions, and that these changes will not introduce an uncertainty more important than those coming from the other sources (e.g. next-to-leading order corrections). After obtaining the parton distribution functions (PDFs) in the SUSY QCD theory with a light gluino, we calculate the degree of parity violation in the tt pairs produced via the LOgg → tt. Thus, the paper consists of 3 main sections: the description of the parton distribution functions for the SUSY QCD theory with a light gluino, the calculation of the cross-sections for the process pp(gg) → t L,Rt , and the numeric analysis of the asymmetries in the left-and right-handed top quark production. Finally, the conclusion summarizes the obtained results.
II. PARTON DISTRIBUTIONS
We start the construction of parton distributions by assuming that the only superparticle, actively present in the nucleons at the energies of the supercolliders, is gluino, with its mass much smaller than the typical scales of tt production ( less than 1.5 GeV compared to m t ≈ 175 GeV). For the purpose of our calculation, we incorporate the gluino sector into the PDF evolution package, used recently to build the set of CTEQ4 unpolarized parton distributions [19]. In order to simplify the modifications in the fitting program, we used the approach close to the one adopted by the authors of GRV distributions [18]. The input scale Q 0 for the parton distributions was chosen to be lower than in CTEQ4L and equal to the mass of gluino (assumed to be mg = 0.5 GeV in this study, unless stated otherwise). At this scale, the only input distributions are of gluons and lighter (u, d, s) quarks, while the non-zero PDFs of gluinos and heavy quarks are radiatively generated at scales above their mass thresholds.
In the presence of light gluino, two aspects of the leading-order evolution of parton distributions are different from that in the standard QCD. First, the one-loop β-function, determining the running of the strong coupling α s ,
α s (Q 2 ) = 4π β 0 ln(Q 2 /Λ 2 ) ,(2)
now has the form
β 0 = 11 − 2 3 n f − 2ng,(3)
where n f and ng are the number of active quark flavors and gluinos, respectively. In our analysis, we use α s (M Z ) = 0.118 and Λ = 7.65 MeV for 5 flavors. The matching of α s between 4 and 5, and 5 and 6 flavors takes place at Q = 5.0 GeV and Q = 175 GeV respectively, which are defined as the bottom and top quark masses. Second, the leading order DGLAP equations now should account for the splittingsg → qq, g → gg and g →gg, so that the singlet equation takes the form of
d dt q S (x, Q 2 ) G(x, Q 2 )) g(x, Q 2 )) = α S (Q 2 ) 2π 1 x P qq (x/y) P qG (x/y) P qg (x/y) P Gq (x/y) P GG (x/y) P Gg (x/y) Pg q (x/y) Pg G (x/y) Pgg(x/y) × q S (y, Q 2 ) G(y, Q 2 ) g(y, Q 2 ) dy y .(4)
The splitting functions used in (4) can be found, e.g., in [20].
With the help of the upgraded evolution package we performed the fit of the experimental data, closely following the procedure of the construction of CTEQ4L PDF set, as described in [19]. However, for simplicity, the fit didn't use the jet data, and the value of the strong coupling was fixed to be equal to the world-average value α s (M Z ) = 0.118. As a result, we obtained the set of parton distributions SUSYL, which was used throughout the rest of the paper.
III. CALCULATION OF THE MATRIX ELEMENTS
At the LHC, the tt pairs will be dominantly produced from the standard QCD processes of qq and GG interactions, as shown in Figs. 1a-d. All of these diagrams preserve P -parity. In the SUSY QCD theory, if the mass of gluino is small (mg ∼ 1 GeV), we will also expect a noticeable contribution due to the annihilation of gluinos, described by the three diagrams of Figs. 1e-g. The s-channel diagram of Fig. 1e is equivalent, up to a color factor, to the analogous qq diagram of Fig. 1a and doesn't break parity; however, the parity symmetry is broken in the t-and u-channels due to the mechanism of squark mass mixing which is briefly described below.
In MSSM the left-squark and the right-squark, superpartners of the the left-and righthanded quarks, do not have definite mass but instead are a mixture of two mass eigenstates. These mass eigenstatesq 1 andq 2 are related to the current eigenstatesq L andq R bỹ
q 1 =q L cos θ q +q R sin θ q ,q 2 = −q L sin θ q +q R cos θ q .(5)
Due to this, MSSM in general allows to have nonzero asymmetries in qq-pair production, defined by
A q = σ(pp → q Lq ) − σ(pp → q Rq ) σ(pp → q Lq ) + σ(pp → q Rq ) ,(6)
where σ denotes the cross-section of the tt-pair production, integrated over the relevant part of the phase space to be discussed below. The best chance to observe a non-zero A q is provided by the tt production process, where the mixing between the squarks is the largest due to the large mass of the top quark. In the following, we ignore the mass mixing for the lighter 5 quarks.
In MSSM, the squark-quark-gluino interaction Lagrangian is given by
Lgqq = −g s T a jkq k [(a 1 − b 1 γ 5 )q 1j + (a 2 − b 2 γ 5 )q 2j ]g a + h.c. ,(7)
where g s is the strong coupling constant, T a are SU(3) C generators and a 1 , b 1 , a 2 , b 2 are given by
a 1 = 1 √ 2 (cos θ q − sin θ q ) = −b 2 , b 1 = − 1 √ 2 (cos θ q + sin θ q ) = a 2 .(8)
The mixing angle θ t and the masses mt 1 , mt 2 can be calculated by diagonalizing the following mass matrix:
M 2 t L m t m LR m t m LR M 2 t R ,(9)
where M 2 t L,R and m LR are the parameters of the soft-breaking terms in the MSSM. From Eq.(9), we can derive the expressions for m 2 t 1,2 and θ t :
m 2 t 1,2 = 1 2 M 2 t L + M 2 t R ∓ (M 2 t L − M 2 t R ) 2 + 4m 2 t m 2 LR ,(10)tan θ t = m 2 t 1 − M 2 t L m t m LR .(11)
Inversely,
M 2 t R,L = 1 2 m 2 t 1 + m 2 t 2 ∓ (m 2 t 2 − m 2 t 1 ) 2 − 4m 2 t m 2 LR .(12)
The asymmetry A t depends on the angle of mixing θ t in the following manner. The denominator of A t is dominated by the large contributions from the quark and gluon channels (Figs.1a-d) and therefore shows little dependence on the masses of stops. The numerator of the asymmetry depends both on the splitting and the mixing of the squark masses. Since left/right-handed quarks couple only to left/right squarks, in the case of no mass mixing (m LR = 0) the asymmetry is completely determined by the difference of masses Mq L − Mq R . In this limit θ t ≈ −π/2, provided that Mt R < Mt L .
For fixed mass eigenvalues mt 1,2 , the relationship (12) for the mass parameters Mt L,R puts the upper bound on m LR :
m LR ≤ m 2 t 2 − m 2 t 1 2m t .(13)
For largest m LR ,
θ t = − π 4(14)
and
Mt L = Mt R .(15)
In this limit, stop mass eigenstates have the maximal mixing between the left-and right-stops, so that the asymmetry A t becomes zero. Thus, for fixed mass eigenstates, the asymmetries are expected to decrease with the growth of m LR . In gravity-mediated supersymmetry breaking models (mSUGRA), the masses of left-and right-stops satisfy the relations
M 2 t L = m 2 t L + m 2 t + ( 1 2 − 2 3 sin 2 θ W ) cos(2β)m 2 Z , M 2 t R = m 2 t R + m 2 t + 2 3 sin 2 θ W cos(2β)m 2 Z , m LR = −µ cot β + λ t ,(16)
where m 2 t L , m 2 t R are the soft SUSY-breaking mass terms of left-and right-stops, µ is the coefficient of the H 1 -H 2 mixing term in the superpotential, λ t is the parameter describing the strength of soft SUSY-breaking trilinear scalar interactiont LtR H 2 , tan β = v u /v d is the ratio of the vacuum expectation values of the two Higgs doublets. In the minimal supergravity models, the soft SUSY breaking parameters m 2 q L and m 2 q R are equal to each other, so that the mass splitting M 2 q L − M 2 q R is small, and of the same order of magnitude for all quark flavors. In this case, it is hard to expect observable asymmetries. On the other hand, in the general MSSM right-and left-squark masses Mt L,R are considered to be independent parameters, in which case there is no theoretical limitations on the splitting of stop masses. In the following, the second point of view is accepted, so that Mt R is assumed to be of the order 90-175 GeV, while Mt L is varied between 150 and 1000 GeV.
The cross-sections entering the asymmetry (6) are calculated in a usual way by convolution of the squared and spin-and color-averaged hard scattering matrix elements |M k 1 k 2 | 2 L,R with the appropriate parton distributions f i (x):
σ(pp → t L,Rt ) = β 32πŝ 1 −1 d cos θ dx 1 dx 2 i 1 ,i 2 f i 1 (x 1 )f i 2 (x 2 )|M i 1 i 2 (ŝ,t,û)| 2 L,R ,(17)
whereŝ,t,û are the parton Mandelstam variables, β ≡ 1 − 4m 2 t /ŝ and the particle momenta for the partons q i 1,2 in the initial state are defined as q i 1 (p 1 ) + q i 2 (p 2 ) → t(p 3 ) +t(p 4 ). The numerator of the asymmetry (6) is determined solely by the diagrams Figs. 1f,g (containing stops), which give the following matrix elements for the production of left-handed top:
M t M † u + M u M † t L = i,j a 2 i a 2 j (t − m 2 t i )(û − m 2 t j ) × 4m 2 tŝ (1 − C i C j ) (1 − C i C j ) + (C i − C j ) cos θ −2(m 2 t −t)(m 2 t −û)(1 − C 2 i )(1 − C 2 j ) ,(18)|M t | 2 L = i,j a 2 i a 2 j 2(t − m 2 t i )(t − m 2 t j ) (m 2 t −t)ŝ(1 + C i C j ) (A + B cos θ) ,(19)|M u | 2 L = i,j a 2 i a 2 j 2(û − m 2 t i )(û − m 2 t j ) (m 2 t −û)ŝ(1 + C i C j ) (A − B cos θ) .(20)
In these formulas
C i ≡ b i a i , i = 1, 2,(21)A ≡ (1 + C i )(1 + C j )(1 − β) + (1 − C i )(1 − C j )(1 + β),(22)B ≡ (1 + C i )(1 + C j )(1 − β) − (1 − C i )(1 − C j )(1 + β),(23)
the summation (i, j) goes over the two stop masses. The squared matrix element |Mgg| 2 L entering (17) can be written in terms of (18)(19)(20) as
|Mgg| 2 L = 1 256 16 3 |M t | 2 + |M u | 2 + 2 3 M t M † u + M u M † t L .(24)
The matrix elements for the production of right-handed top are obtained by the substitution
C i,j → −C i,j .(25)
If (18)(19)(20) are combined with the explicit formulas (8) for a i , b i , it is possible to get the following expression for the difference of the matrix elements for producing the left-and right-handed top quarks in the tt pairs:
|Mgg| 2 L − |Mgg| 2 R = 4 cos 2θ t (X 11 − X 22 )(β − cos θ) +(Y 21 − Y 12 ) cos θ + (Z 11 − Z 22 )(β + cos θ) ,(26)
where
X ij ≡ (m 2 t −t)ŝ 96(t − m 2 t i )(t − m 2 t j ) ,(27)Y ij ≡ m 2 tŝ 192(t − m 2 t i )(û − m 2 t j ) ,(28)Z ij ≡ (m 2 t −û)ŝ 96(û − m 2 t i )(û − m 2 t j ) .(29)
Equation (26) depends on the mass mixing angle θ t only through the common factor cos 2θ t . This proves the argument given before that for fixed mt 1,2 the asymmetry should be the largest at m LR = 0 and θ t = −π/2. The diagrams in Figs. 1a-e do not violate the parity and need to be included only in the denominator of the asymmetry (6). The matrix elements for the pure QCD processes (in Figs. 1a-d) are well-known, while the s-channelgg diagram (in Fig. 1e) differs from the analogous qq one only by a color factor:
|M qq | 2 = 4 9 (m 2 t −t) 2 + (m 2 t −û) 2 + 2m 2 tŝ s 2 ,(30)|M GG | 2 = 1 16 (m 2 t −t)(m 2 t −û) 12ŝ 2 + 8 3 (m 2 t −t)(m 2 t −û) − 2m 2 t (m 2 t +t) (m 2 t −t) 2 + 8 3 (m 2 t −t)(m 2 t −û) − 2m 2 t (m 2 t +û) (m 2 t −û) 2 − 2 3 m 2 t (ŝ − 4m 2 t ) (m 2 t −t)(m 2 t −û) −6 (m 2 t −û)(m 2 t −t) + m 2 t (û −t) s(m 2 t −t) −6 (m 2 t −û)(m 2 t −t) − m 2 t (û −t) s(m 2 t −û) ,(31)|Mgg s | 2 = 27 32 |M qq | 2 .(32)
In the above, the spin and color factors in both the final and the initial states are all properly summed and averaged. One can also obtain the total parton cross-sections by the integration of (18-20,30-32) over the scattering angle θ. For the t and u-channels we define
C ≡ 2(1 − β 2 )(1 − C i C j )(C i − C j ) (33) D ≡ β 2 (1 − C 2 i )(1 − C 2 j ) (34) E ≡ 2(1 − β 2 )(1 − C i C j ) 2 − (1 − C 2 i )(1 − C 2 j ),(35)v i (ŝ, β) ≡ 2m 2 t i +ŝ + βŝ − 2m 2 t 2m 2 t i +ŝ − βŝ − 2m 2 t .(36)
Then
[σgg tu ] L = i,j g 4 s a 2 i a 2 j 24576π [8(1 + C i C j )f 1 (m 2 1 β 2 − 8D s β + 2Eβ 2ŝ2 + (4m 2 t i + 2ŝ − 4m 2 t ) ×(2Dm 2 t i + Dŝ + Cβŝ − 2Dm 2 t ) ln v i (ŝ, β) + 2Eβ 2ŝ2 + (4m 2 t j + 2ŝ − 4m 2 t ) (39) ×(2Dm 2 t j + Dŝ − Cβŝ − 2Dm 2 t ) ln v j (ŝ, β) × 1 s 2 (m 2 t i + m 2 t j +ŝ − 2m 2 t ) .(40)
When mt i = mt j , we have
f 1 (m 2 t i = m 2 t j ) = 1 β − 8B s β + 4(4Bm 2
The cross-sections of the other sub-processes, corresponding to (30-32), are given by
σ qq = g 4 s 108πŝ β(2 + ρ) (43) σ GG = g 4 s 48πŝ (1 + ρ + ρ 2 16 ) ln 1 + β 1 − β − β( 7 4 + 31 16 ρ) ,(44)σgg s = g 4 s 128πŝ β(2 + ρ),(45)
where ρ ≡ 4m 2 t /ŝ.
IV. NUMERIC RESULTS
To estimate the largest possible asymmetries, we varied the squark mass eigenvalues mt 1,2 with m LR set to be zero (see the discussion in the previous Section). No assumption was made about any model-specific relationships between the values of the mass parameters Mt L,R and m LR (c.f. eq. (16)).
If m LR = 0, the left/right-handed quarks couple independently to the left/right-stops. Correspondingly, for mt 1 = mt 2 , the production rates of the left-and right-handed tops will be different. The asymmetry A t is expected to grow when the mass splitting mt 1 = mt 2 increases. In this work, the asymmetries were calculated for mt 1 = 90 GeV (which is consistent with the current LEP2 data [21]), mt 1 = m t = 175 GeV and various values of mt 2 . Two values of factorization scale µ = m t and 2m t were used. Various sets of masses will be further denoted as (mt 1 , mt 2 , m LR ), with numerical values in GeV. As before, the gluino mass is assumed to be equal to 0.5 GeV.
In the SM, both t andt decay into b (b) and W ± with an almost unit probability, with a subsequent decay of the W -bosons into 2 jets or 2 leptons. In the MSSM, when both the gluino and the stop are light, the top quark can also decay via t →gt 1 , so that the branching ratio of t → W + + b decreases. Assuming that all the other supersymmetric particles are heavier than the top quark, and θ t = −π/2, the branching ratio for t → W + + b is equal to 0.29 and 1 for mt 1 = 90 GeV and 175 GeV, respectively. The CDF collaboration has measured the branching ratio of t → W + + b to be 0.87 +0.13 −0.30 +0.13 −0.11 [23]. Hence, the chosen sets of the values for mt 1 and mg are still allowed by data within 95% c.l.
It is convenient to study the asymmetry A t using the semileptonic modes of decay, with t → bl + ν l (l = e, µ) andt →bqq (or vice versa), which have a branching ratio of about 0.086 and 24/81 for mt 1 = 90 GeV and 175 GeV, respectively. In the following we assume that it will be possible to reconstruct the kinematics of tt-pair from the momenta of the decay products by requiring the transverse momenta p jets T ≥ 30 GeV, p leptons T ≥ 20 GeV, E T ≥ 20 GeV, the rapidities of the jets and leptons |y| < 2.0, and the jet cone separation ∆R > 0.4 [22]. We also assume that it will be necessary to tag one b-quark with an efficiency C b = 50%. We estimate the statistical error in the measurement of the asymmetry by
δA t = 1 LT C b (σ tot L + σ tot R ) ,(46)
where we assume the observation time T = 1 year and the luminosity L = 100 f b −1 /year, corresponding to the second run of the LHC. The imposed selection cuts and branching ratio significantly reduce the total cross-section of tt production, typically from around 340 pb down to 3.5 and 12 pb for mt 1 = 90 GeV and 175 GeV, respectively. As an example, Fig. 2 shows various differential cross-sections including the GG, qq andgg subprocesses for the squark masses (90, 1000, 0) and µ = 2m t , obtained with the kinematic cuts and branching ratios applied. It can be readily seen that the dominant part of the tt pairs is produced due to the gluon-gluon subprocess, which contributes around 71% of the total rate. The quark-antiquark and gluino-gluino shares are 22% and 7%, respectively. The gluino contribution is comparable with the conventional QCD uncertainties in the knowledge of the total rate (about 5% to 10%), however, the presence of the light gluino will change the shape of the cross-section distributions. Thus, in principle there is a possibility to detect the light gluino by carefully fitting the event rate distributions and comparing them with the predictions of perturbative QCD. Fig. 3 shows the sum of the differential cross-sections of left-and right-handed top quarks production dσ L /dM tt + dσ R /dM tt , and their difference dσ L /dM tt − dσ R /dM tt (scaled by a factor of 100), as functions of the invariant mass of the tt pair M tt . One can see that the asymmetry is most noticeable in the region of small and intermediate values of M tt . This is different from the behavior of the asymmetry produced due to the presence of superpartners in the loop corrections [16]. In that case, the asymmetry becomes significant in the region of large M tt , where in the case of the light gluino it can have the value of 2−3%. In this respect, we expect the minimal interference between the tree-level and loop-generated asymmetries, since the main contributions to them come from different kinematic regions.
The dependence of the cross-sections on two other kinematic parameters, the transverse momentum of the t-quark and the cosine of the scattering angle in the tt rest frame, is shown in Figs. 4 and 5. As can be seen from Fig. 5, the difference dσ L /d cos θ − dσ R /d cos θ changes its sign around cos θ ≈ ±0.8, so that one can enhance the asymmetry by separately considering the cross-sections integrated over either large or small angles. It can also be shown that the asymmetries at small angles can be further enlarged by rejecting the events with transverse momenta larger than 100 GeV/c. For the other combinations of stop masses, dσ L / cos θ −dσ R / cos θ changes its sign at slightly lower | cos θ|, approximately 0.75 −0.8. We therefore present the asymmetries of the cross-sections integrated separately over the region | cos θ| ≤ 0.8, or the region | cos θ| > 0.8 with p T ≤ 100 GeV/c. Table 1 shows the values of the asymmetry A t obtained after the integration of the rate with the aforementioned cuts in | cos θ| and p T . As can be seen from the Table, for various stop masses the asymmetry ranges from 0.3% to 1.1%. The behavior of the asymmetries with the growth of mt 2 is different at large and small angles. At | cos θ| ≤ 0.8 the asymmetry monotonously increases with the growth of mt 2 , while at | cos θ| > 0.8 the asymmetry has a maximum around mt 2 = 200 GeV and then starts to decrease. At small angles (| cos θ| ≤ 0.8) the asymmetry quickly decreases with the growth of the mass of the lighter squark and becomes practically unnoticeable for mt 1 ≥ m t .
For the comparison, we also give in the same Table the statistical errors δA t from Eq. (46). These errors are mostly determined by GG and qq cross-sections, so that they hardly depend on the choice of the squark masses. For most combinations of the stop masses, the obtained values of A t can in principle be distinguished from the statistical error δA t at a 2σ level or better. However, what can be more important are the experimental systematic uncertainties related to the measurement of the asymmetries of the order 1%. In particular, it can be challenging to reach the necessary accuracy in the reconstruction of the kinematics of the tt-pair, and the determination of the top quark polarization. Nevertheless, the predictive power of this analysis can be increased if it is combined with the search for the signature of the light gluinos in the other kinematic regions, for instance, for the loop-generated asymmetries in the production of the top-antitop pairs with large invariant masses.
V. CONCLUSION
In this work we propose a new method, based on the search of the possible violations of the discrete symmetries of the Standard Model, to test the existence of a light gluino in the MSSM. This is in contrast to many other methods presented in the literature (see the Introduction section), in which one has to assume how a light gluino hadronizes into hadron states to be compared with the experimental measurement.
We study the consequences the small mass of the gluino would have for the production of top quarks at the LHC via the tree level processgg → tt. We show that with a large mass splitting in the masses of superpartners (top-squarks) of the top quark, the gluinogluino fusion process can generate the parity-violating asymmetry in the production of leftand right-handed t-quarks. Since the SM QCD theory preserves the discrete symmetry of P -parity, a small violation of such a symmetry may be observed from a large tt data sample at the LHC.
For mg ≈ 0 the largest values of the parity-violating asymmetry discussed in the previous sections is around 0.3 − 1.1% for various choices of SUSY parameters. Hence, it can in principle be observed, taking into the account the high rate of the top production at the LHC. In order to measure the asymmetry with a small statistical error, the experiment should be preferably done during the second run of LHC with an integrated luminosity of 100 f b −1 /year. The rate of the top quark production does not seem to be the major obstacle for the measurement of the parity-violating asymmetry. However, it demands a good understanding of the systematic errors, better than 1%, to reach the precision of the measurement sufficient to test the existence of a light gluino in tt pair production. The solid line, stars, circles and dashed line correspond to the full differential cross-section and the contributions of gluon, quark and gluino subprocesses, respectively. The factorization scale µ = 2m t , the squark masses are (90,1000,0). Fig. 3. Dependence of the sum dσ L /dM tt + dσ R /dM tt (solid line) and the difference dσ L /dM tt − dσ R /dM tt (dashed line, magnified by 100) of the differential cross-sections of the production of the left-and right-handed tops on the invariant mass of the tt pairs M tt . The factorization scale µ = 2m t , the squark masses are (90,1000,0). Fig. 4. Dependence of the sum dσ L /dp T + dσ R /dp T (solid line) and the difference dσ L /dp T − dσ R /dp T (dashed line, magnified by 100) of the differential cross-sections of the production of the left-and right-handed tops on the transverse momentum of the t-quark p T . The factorization scale µ = 2m t , the squark masses are (90,1000,0).
Fig. 5.
Dependence of the sum dσ L /d cos θ + dσ R /d cos θ(solid line) and the difference dσ L /d cos θ − dσ R /d cos θ (dashed line, magnified by 100) of the differential cross-sections of the production of the left-and right-handed tops on the cosine of the scattering angle in the tt pair rest frame. The factorization scale µ = 2m t , the squark masses are (90,1000,0).
Fig. 1 .
1Leading order diagrams contributing to the production of top quarks in SUSY QCD theory.
Fig. 2 .
2The dependence of the cross-section of tt pair production on various kinematic parameters: tt pair invariant mass M tt , t-quark transverse momentum p T and rapidity y.
. The asymmetries A t (in % ) predicted by SUSY QCD, and the estimated statistical errors of their measurement δA t for the LHC luminosity L = 100 f b −1 /year. The entries with the hyphen correspond to the asymmetries which are too small to be observed.
Fig. 1 f) g)
×100
µ = m t µ = 2m t Masses (GeV)| cos θ| ≤ 0.8 | cos θ| > 0.8 | cos θ| ≤ 0.8| cos θ| > 0.8
(mt
1 , mt
2 , m LR )
A t
δA t
A t
δA t
A t
δA t
A t
δA t
(90,150,0)
-0.31
0.23
1.14
0.54
-0.35
0.25
1.20
0.60
(90,200,0)
-0.57
0.23
1.30
0.54
-0.62
0.25
1.42
0.60
(90,500,0)
-1.31
0.23
1.20
0.54
-1.39
0.25
1.33
0.60
(90,1000,0)
-1.50
0.23
1.05
0.54
-1.61
0.25
1.
t i , m 2 t j ) + f 2 (m 2 t i , m 2 t j )],(37)f 1 (m 2 t i , m 2 t j ) = 1 β(m 2 t i − m 2 t j ) − 8B s β(m 2 t i − m 2 t j ) +4(m 2 t i − m 2 t )(2Bm 2 t i + Bŝ + Aβŝ − 2Bm 2 t )/ŝ 2 ln v i (ŝ, β) −4(m 2 t j − m 2 t )(2Bm 2 t j + Bŝ + Aβŝ − 2Bm 2 t )/ŝ 2 ln v j (ŝ, β) ,(38)f 2 (m 2 t i , m 2 t j ) =
t i + Bŝ + Aβŝ − 4Bm 2 t )/ŝ 2 ln v i (ŝ, β) +8(m 2 t i − m 2 t )(2Bm 2 t i + Bŝ + Aβŝ − 2Bm 2 t ) ×( 1 2m 2 t i +ŝ + βŝ − 2m 2 t − 1 2m 2 t i +ŝ − βŝ − 2m 2 t )/ŝ 2 .(41)Again, [σgg tu ] R can be obtained by the substitutionC i → −C i , C j → −C j .(42)
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|
[] |
[
"LONG-RANGE INTERACTIONS AND NETWORK SYNCHRONIZATION",
"LONG-RANGE INTERACTIONS AND NETWORK SYNCHRONIZATION"
] |
[
"Ernesto Estrada ",
"Lucia Valentina Gambuzza ",
"\nDepartment of Mathematics and Statistics\nDepartment of Electrical, Electronic and Computer Engineering\nUniversity of Strathclyde\n26 Richmond StreetG1 1HXGlasgowUnited Kingdom\n",
"\nUniversity of Catania\nViale A. Doria 6CataniaItaly\n",
"\nDepartment of Electrical, Electronic and Computer Engineering\nMATTIA FRASCA\nUniversity of Catania\nViale A. Doria 6CataniaItaly\n"
] |
[
"Department of Mathematics and Statistics\nDepartment of Electrical, Electronic and Computer Engineering\nUniversity of Strathclyde\n26 Richmond StreetG1 1HXGlasgowUnited Kingdom",
"University of Catania\nViale A. Doria 6CataniaItaly",
"Department of Electrical, Electronic and Computer Engineering\nMATTIA FRASCA\nUniversity of Catania\nViale A. Doria 6CataniaItaly"
] |
[] |
The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of d−path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for realworld networks and artificial models. Our analysis reveals that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.
| null |
[
"https://arxiv.org/pdf/1704.01349v2.pdf"
] | 119,255,796 |
1704.01349
|
3db69b2cec999e8a63222599ad3a04b2f47a1f31
|
LONG-RANGE INTERACTIONS AND NETWORK SYNCHRONIZATION
31 Jul 2017
Ernesto Estrada
Lucia Valentina Gambuzza
Department of Mathematics and Statistics
Department of Electrical, Electronic and Computer Engineering
University of Strathclyde
26 Richmond StreetG1 1HXGlasgowUnited Kingdom
University of Catania
Viale A. Doria 6CataniaItaly
Department of Electrical, Electronic and Computer Engineering
MATTIA FRASCA
University of Catania
Viale A. Doria 6CataniaItaly
LONG-RANGE INTERACTIONS AND NETWORK SYNCHRONIZATION
31 Jul 2017
The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of d−path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for realworld networks and artificial models. Our analysis reveals that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.
Introduction
The fact that most of complex systems are networked has made complex networks an important paradigm for studying such systems [51,9,20]. A complex network is a graph that represents the skeleton of such complex systems, ranging from biological and ecological to social and infrastructural ones. Representing such a variety of systems by a single mathematical object can be considered as a drastic simplification. However, complex networks have been very useful in explaining many properties of complex systems, which has been empirically validated their use for this purpose. In order to fill the gaps left by this simplified representation of complex systems a few extensions beyond the simple graph have been proposed. They include the use of hypergraphs, multiplexes and multilayer networks [16,32], temporal graphs [29] and more recently the use of simplicial complexes [14,22].
The previously mentioned extensions of complex systems representations try to ameliorate the emphasis paid by graphs on binary relations only. Then, in either of the previously developed representations of complex systems-hypergraphs, multiplexes, simplicial complexes-the binary relation is replaced by a unified k-ary one, in which the individual is replaced by the group. Just to illustrate one example, in the hypergraph the binary relation between individuals is replaced by the k-ary hyperedges in which nodes are assumed to be identical in their connectivity inside this group. Then, an important missing aspect of these representations is how to capture the influence of nodes in a network in a way that gradually decays with the separation of these nodes in the graph.
Recently, such approach has been proposed to account for long-range influences in a network by using extensions of the graph-theoretic concepts of adjacency and connectivity [23,21]. In these works, a new paradigm is developed in which a node in a network is not only influenced by its nearest neighbours but by any other node in the graph. However, such influence is gradually tuned by the shortest path distance at which the influencers are from the influencee. It is obvious that this type of representation is not general for any kind of complex systems, as there are cases where such long-range influence does not exists. However, in the case of social systems such kind of long-range influence is certainly manifested in the so-called indirect peers pressure. Individuals in a social group are not only directly influenced by those connected to them but also by those socially close to them.
The approach proposed in [23,21] is here applied to study how the long-range influences affect synchronization of the network nodes. In fact, when the units of a network are dynamical systems (for instance, periodic or chaotic oscillators), a collective phenomenon, characterized by the emergence of a common rhythm in all the units and observed in many natural and artificial systems, may emerge as the result of the interactions [61]. In this context it is known that the topology of the connections plays a fundamental role in determining the characteristics of the synchronous motion, its onset and stability [6]. Most of the works on the subject, however, consider interactions as only dictated by direct links, whereas, in this paper, we take into account influences through paths of length grater than one. Other studies [59,42,5,36] have considered the case of oscillators embedded on a geometrical space and coupled with an intensity decaying with the geometric distance between them. In our paper, instead, the oscillators are viewed as the nodes of a network and the intensity of the interactions is weighted by considering the distance of the oscillators as measured by the length of the shortest path connecting them. This scenario has been modelled through the use of d−path Laplacian matrices, whose spectra are shown to determine the synchronizability of the system. In particular, we prove how increasingly weighting the long-range interactions always leads to the best scenario possible for synchronization and verify the result on network models and real-world structures. The quantitative impact of the long-range interactions depends on the network topology and on its specific properties such as the diameter, the density and the degree distribution. Finally, we study the effect of edge removal in networks with and without long-range interactions and, as a major finding, observe that it is more critical in the presence of such interactions, still exhibiting a higher synchronizability in this case than when the long-range interactions are not present.
Intuition and Mathematical formulation
In this section we formulate the general mathematical equations for considering long-range interactions (LRIs) in a system of coupled dynamical units. Let G = (V, E) be a simple, undirected graph without self-loops having N nodes and m edges. Let us now write the equations of N coupled dynamical units on a graph. Let us consider N identical oscillators coupled with a coupling constant σ, where the oscillator i has state variables x i ∈ R n . Then,
(2.1)ẋ i = f (x i ) + σ (i,j)∈E (H(x j ) − H(x i )) ,
where f (x i ) : R n → R n represents the uncoupled dynamics, and H(x j ) : R n → R n the coupling function.
In matrix form Eqs. (2.1) can be written as follow
(2.2)˙ x = f ( x) − σ ∇ · ∇ T ⊗ I n · H( x), where x = [x 1 x 2 . . . x N ] T , f ( x) = [f (x 1 ) f (x 2 ) . . . f (x N )]
T , I n indicates the identity matrix of order n, and
H( x) = [H(x 1 ) H(x 2 ) . . . H(x N )] T .
The entries of the node-to-edges incidence matrix ∇ ∈ R N ×m of the graph are defined as
(2.3) ∇ ij = 1 −1 0
if node i is the head of the edge j, if node i is the tail of edge j, otherwise.
We recall that the matrix L = ∇∇ T is known as the Laplacian matrix of the graph and has entries
(2.4) L ij = k i −1 0 if i = j , if (i, j) ∈ E, otherwise,
where k i is the degree of the node i, the number of nodes adjacent to it. It is worth noting that the Laplacian matrix is related to the adjacency matrix of the graph via: L = K − A, where K is the diagonal matrix of degrees and the adjacency matrix has entries Schematic representation of long-range influences in a social group. It is assumed that σ dmax < · · · < σ 3 < σ 2 < σ 1 , where d max is the diameter of the graph.
(2.5) A ij = 1 0 if (i, j) ∈ E, otherwise.
Now, let us consider the following scenario of N oscillators which are coupled according to a graph G with a coupling strength σ 1 . Consider that the oscillators can also couple in a weaker way if they are separated at a shortest path distance two. Let σ 2 be the strength of the coupling between those oscillators, such that σ 2 < σ 1 . In a similar way we can consider that oscillators at a given shortest path distance can couple together with a coupling strength which depends on their separation in the network. For instance, pairs of oscillators at distance three can couple with strength σ 3 < σ 2 < σ 1 . This situation may well represents social scenarios as illustrated in Figure 2.1. In a social network, individuals are connected by certain social ties, such as friendship, collaboration, etc. Then, two individual directly connected to each other can influence each other in a relatively strong way. However, an individual in this network can also receive certain influence from others which are not directly connected to her. This influence is supposed to be smaller than the ones received from direct acquaintances, but not one that can be discarded at all. Let us consider a simple example of a scientific collaboration network. Two individuals are connected in this network if they have collaborated on a certain topic, e.g., they have published a paper together. It is clear that they have influenced each other in terms of their scientific styles. However, these two scientists are also influenced by individuals which are close to their topic of research although they have not collaborated together. This closeness is reflected in the fact that they are relatively close in this collaboration network in terms of shortest path distance. An individual from a completely different field is expected to be far from them in terms of the shortest path distance, and so to have a lower influence in their scientific styles.
In mathematical terms this intuition can be formulated as follows. Let σ 1 , σ 2 , . . . , σ dmax account for the strength of the cupling between pairs of oscillators separated by shortest path distances of one, two, and so forth up to the diameter of the graph, d max . Then, we can write
(2.6)ẋ i = f (x i ) + σ 1 (i,j)∈E (H(x j ) − H(x i )) + σ 2 d(i,k)=2 (H(x k ) − H(x i )) + · · · · · · + σ dmax d(i,r)=dmax (H(x r ) − H(x i )) .
The simplest way to account for these long-range influences is by considering that the coupling strength between individuals decays as certain law of the shortest path distance. That is, if we consider a power-law decay of the strength of coupling with the distance we get, assuming that σ 1 = σ and that the rest are just a fraction of it,
(2.7)ẋ i = f (x i ) + σ (i,j)∈E (H(x j ) − H(x i )) + σ2 −s d(i,k)=2 (H(x k ) − H(x i )) + · · · · · · + σd −s max d(i,r)=dmax (H(x r ) − H(x i )) .
Similarly, if we assume an exponential decay we obtain
(2.8)ẋ i = f (x i ) + σ (i,j)∈E (H(x j ) − H(x i )) + σe −2λ d(i,k)=2 (H(x k ) − H(x i )) + · · · · · · + σe −dmaxλ d(i,r)=dmax (H(x r ) − H(x i )) .
We call these equations the Mellin and Laplace transforms, respectively, of the N coupled dynamical units on a graph. Let us now define the d-path incidence matrices which account for these coupling of non-nearest-neighbours in the graph. Let P l,ij denote a shortest path of length l between i and j. The nodes i and j are called the endpoints of the path P l,ij . Because there could be more than one shortest path of length l between the nodes i and j we introduce the following concept. The irreducible set of shortest paths of length l in the graph is the set P l = {P l,ij , P l,ir , ..., P l,st } in which the endpoints of every shortest path in the set are different. Every shortest path in this set is called an irreducible shortest path. Let d max be the graph diameter, i.e., the maximum shortest path distance in the graph.
Definition 1. Let d ≤ d max .
The d-path incidence matrix, denoted by ∇ d ∈ R n×p , of a connected graph of N nodes and p irreducible shortest paths of length d is defined as:
(2.9) ∇ d,ij = 1 −1 0
if node i is the head of the irreducible shortest path j, if node i is the tail of the irreducible shortest path j, otherwise.
Obviously ∇ 1 = ∇. Let us now rewrite our Mellin and Laplace transformed equations, respectively, in matrix-vector form using the d-path incidence matrix as follow
(2.10)˙ x = f ( x) − σ dmax d=1 d −s ∇ d · ∇ T d ⊗ I n · H( x) , (2.11)˙ x = f ( x) − σ ∇ · ∇ T ⊗ I n · H( x) − σ dmax d=2 e −λd ∇ d · ∇ T d ⊗ I n · H( x) .
The parameters s and λ account for the strength of the coupling of the oscillators at a given distance. The smallest the values of these parameters the stronger the coupling between the oscillators at a given distance. For instance when s → ∞ (λ → ∞) there is a very weak influence of non-nearest neighbours and we recover the classical model in which there is no long-range coupling. When s → 0 (λ → 0) the strength of the coupling between oscillators at any distance is the same, which corresponds to the situation of the classical model of coupled oscillators on a complete graph K n . Thus, in every case we always recover the original model of coupled oscillators on graphs for large values of the parameters in the transforms of the d-path incidence matrices and we approach the coupling on a complete graph when these parameters tend to zero. Note that in our approach the coupling strength depends on the shortest path distance. This differs from the notion of accessability matrix [35], which accounts for the existence of paths of arbitrary length between the nodes with unitary weights.
2.1.
Example. Long-range interactions in the Kuramoto model. In this section we particularize the equations for the coupled system to the case of phase oscillators, so that to consider LRIs in the Kuramoto model [58]. Let us consider N phase oscillators on graph G = (V, E) coupled with an identical coupling constant σ, where the oscillator i has phase θ i and intrinsic frequency ω i . Then,
(2.12)θ i = ω i + σ (i,j)∈E sin (θ j − θ i ) , or in matrix form (2.13)˙ θ = ω − σ∇ · sin ∇ T θ .
The consideration of LRIs in the way we have described previously will give rise to the following transforms of the Kuramoto model:
(2.14)θ i = ω i + σ (i,j)∈E sin (θ j − θ i ) + σ2 −s d(i,k)=2 sin (θ k − θ i ) + · · · + σd −s max d(i,r)=dmax sin (θ r − θ i ) ,
for the Mellin transform, and
(2.15)θ i = ω i + σ (i,j)∈E sin (θ j − θ i ) + σe −2λ d(i,k)=2 sin (θ k − θ i ) + · · · · · · + σe −dmaxλ d(i,r)=dmax sin (θ r − θ i ) ,
for the Laplace transform. In matrix-vector form they are given by
(2.16)˙ θ = ω − σ dmax d=1 d −s ∇ d · sin ∇ T d θ ,
for the Mellin transform, and
(2.17)˙ θ = ω − σ ∇ · sin ∇ T θ − σ dmax d=2 e −λd ∇ d · sin ∇ T d θ .
for the Laplace transform. Here again when s → ∞ (λ → ∞) the coupling between oscillators at distance larger than one is almost zero and we recover the classical Kuramoto model where there is no LRIs. On the other hand, when s → 0 (λ → 0) the strength of the coupling between oscillators at any distance is the same and we obtain the Kuramoto model on a complete graph K N .
Synchronizability and Laplacian Spectra
The Laplacian matrix of the graph is positive semi-definite with eigenvalues denoted by: 0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ N . If the network is connected, the multiplicity of the zero eigenvalue is equal to one, i.e., 0 = λ 1 < λ 2 ≤ · · · ≤ λ N , and the smallest nontrivial eigenvalue λ 2 is known as the algebraic connectivity of the network. It is now well known that there are two types of networks with bounded and unbounded synchronization regions in the parameter space. One large class of dynamic networks have an unbounded synchronized region specified by
(3.1) σλ 2 > α 1 > 0,
where constant α 1 depends only on the node dynamics, a bigger spectral gap λ 2 implies a better network synchronizability, namely a smaller coupling strength σ > 0 is needed [12,17,18,39]. Another large class of dynamic networks have a bounded synchronized region specified by
(3.2) σλ 2 , . . . , σλ N ∈ (α 2 , α 3 ) ⊂ (0, ∞),
where constants α 2 , α 3 depend only on the node dynamics as well, and a bigger eigenratio λ 2 /λ n implies a better network synchronizability, which likewise means a smaller coupling strength is needed [54,30].
Here we only consider the latter criterion while the former can be discussed similarly. As introduced, in this scenario the synchronizability of the graph depends on the eigenratio of the second smallest and the largest eigenvalues of the Laplacian matrix:
(3.3) Q = λ 2 λ N .
Now, we have generalized the Laplacian matrix to the so-called d-path Laplacian matrices which are defined as follow.
Definition 2. Let d ≤ d max .
The d-path Laplacian matrix, denoted by L d ∈ R n×n , of a connected graph of n nodes is defined as:
(3.4) L d,ij = δ k (i) −1 0 if i = j , if d ij = d otherwise,
where δ k (i) is the number of irreducible shortest paths of length d that are originated at node i, i.e., its d-path degree. It is straightforward to realize that L d = ∇ d ∇ T d in a similar way as it happens for the graph Laplacian. We now extend the notion of a connected component of a graph to the d-connected component.
Definition 3. Let G = (V, E) be a simple graph and let d ≤ d max . A d-connected component of G is a subgraph G ′ = (V ′ , E ′ ), V ′ ⊆ V , E ′ ⊆ E, such that for all p, q ∈ V ′ there is a shortest path of length d (p, q) = d.
If the d-connected component includes all the vertices and edges of G, the graph is said to be d-connected.
The following is an important property of the d-path Laplacian matrix proved by Estrada in 2012 [19]. The previous result has an important implication for the study of synchronization on graphs using the d-path Laplacian matrix. Although a graph can be connected in the usual way-here it should be said 1-connected-it not necessarily should be d-connected. Then, a synchronization process taking place between agents separated at distance d in that graph may not give rise to a unique synchronized state. For instance, a synchronization process taking place on the pairs of nodes separated at distance 2 in the pentagon C 5 gives rise to a unique synchronized state because the graph is 2-connected, but a similar process on the hexagon C 6 conduces to two synchronized states because there are two 2-connected components in this graph. Then, the use of the Mellin and Laplace transformed dynamical systems, like the ones introduced in this work, require a transformation of the d-path Laplacian matrices such that they account for the decay of the influence of oscillators separated at different distances in the graph. Then, we have the following.
Definition 5. Let G = (V, E) be a simple connected graph and let d ≤ d max . The Mellin and Laplace transformed d-path Laplacian matrices of G are given by
(3.5)L τ = dmax d=1 d −s L d , L + dmax d=2 e −λd L d , for τ = Mell, s > 0 for τ = Lapl, λ > 0.
Now, we can interpret the LRI-model in the following way. Consider a simple connected graph G = (V, E) and the following transformation: f :
G (V, E) → G ′ (V, E ′ , φ, W ), such that E ′ = {E ∪ (p, q) |p, q ∈ V, (p, q) /
∈ E } and φ : W → E ′ is a surjective mapping that assigns a weight to each of the elements of E ′ . The weights w ij ∈ W are given by the Mellin or Laplace transforms and they are specific for each graph, that is w ij = d −s ij or w ij = e −λdij for d ij > 1, and w ij = 1 for connected pairs of nodes. The main consequence of this is that we can generalize the synchronizability definition (3.3) to:
(3.6) Q τ = λ 2 L τ λ n L τ ,
where λ n L τ and λ 2 L τ are the largest and second smallest eigenvalues of the transformed d-path Laplacian matrices.
The most important consequence of this new formulation is that when s, λ → ∞ the weighted graph G ′ (V, E ′ , φ, W ) tends to the original graph G = (V, E). In addition, when s, λ → 0 the weighted graph G ′ (V, E ′ , φ, W ) tends to the complete graph with N nodes, K N . It is known that the Laplacian eigenvalues of K N are all equal to N but one which is equal to zero. Then, the following result holds.
Lemma 6. Let G = (V, E) and letL τ be the Mellin or Laplace transformed d-path Laplacian of the graph. Then, we have the following behavior of the generalized eigenratio
(3.7) Q τ → Q 1 if s, λ → ∞, if s, λ → 0.
Similarly, for networks of dynamical units with unbounded synchronized region, if one considers synchronizability as measured by the smallest non-zero eigenvalue of the d-path Laplacian normalized by the number of nodes, i.e., λ 2 /N , the following lemma holds. Lemma 7. Let G = (V, E) and letL τ be the Mellin or Laplace transformed d-path Laplacian of the graph. Then, we have the following behavior of the normalized smallest eigenvalue
(3.8) λ 2 N τ → λ 2 N if s, λ → ∞ 1 if s, λ →0.
In addition, the following Lemma characterizes the behavior of the smallest non-zero eigenvalue and of the largest eigenvalue of the transformed d-path Laplacian with respect to the transform parameter (s for the Mellin transform or λ for the Laplace transform).
Lemma 8. Let G = (V, E) be a simple graph and letL τ (G, w) be the transformed d-path Laplacian of G with parameter w ij where w ij = d −s ij (τ =Mellin) and w ij = e −λdij for d ij > 1 ( τ =Laplace). Let λ n (G, w) be the largest eigenvalue of L τ (G, w). Then, if s ′ < s (Mellin) or λ ′ < λ (Laplace) we have that λ 2 (G, w ′ ) ≥ λ 2 (G, w) and λ n (G, w ′ ) ≥ λ n (G, w). That is, λ 2 (G, w) and λ n (G, w) are non-increasing with s or λ.
Proof. Let x be a nontrivial vector. Using the Rayleigh-Ritz theorem we have that
λ 2 (G, w) = min x =0, x⊥ 1 x TL τ (G, w) x x T x and λ n (G, w) = max x =0 x TL τ (G, w) x x T x .
Let us select a normalized vector x for the sake of simplicity. Then,
(3.9) λ 2 (G, w) = min x =0, x⊥ 1 i,j w ij (x i − x j ) 2 , (3.10) λ n (G, w) = max x =0 i,j w ij (x i − x j ) 2 .
Then, for a fixed vector x, a decrease of the parameter w ij , i.e., increase of the parameters s or λ, makes that λ 2 (G, s) and λ n (G, s) decay or remain at their previous value, so in general they are non-increasing with s or λ.
Based on these considerations and on the fact that λ 2 ≥ λ n , one could expect that Q τ is also increasing on average when s or λ are decreasing. Numerical results, which are shown in the next section, along with the fitting functions, show that, in addition, the decay is monotonic. This implies that for any network of coupled oscillators, independently of its topology and the unit dynamics, the increase of the long-range coupling between oscillators, i.e., s, λ → 0, produces the best possible synchronizability.
We illustrate this result by calculating λ 2,τ /N and Q τ for Erdős-Renyi (ER) random graphs with N = 100 and k = 4 for increasing values of the parameters s and λ of the d-path Laplacian matrices. Fig. 3.1 clearly shows that for s, λ → 0 the best possible synchronizability is obtained, whereas for s, λ → ∞ the synchronizability of the original graph is recovered.
A note of caution should be written here. The fact that for s, λ → 0 the network behaves like a complete graph does not mean that the current results are trivial in the sense that we replace the graph under study by a complete graph. If we consider values of s, λ close but not exactly equal to zero, the synchronizability of the networks approach the best possible value, but (and this is an important but) it still depends on the topology of the network. In order to illustrate this fact we investigate next how the values of λ 2,τ /N and Q τ change with s or λ for a few different types of graphs. In particular, we have done the calculations for a series of graphs: the star topology, the triangular and square lattices, the ring, the Barbell graph and the path graph. For each of these graphs we have applied the Mellin and the Laplace transform and computed the synchronizability measures for different values of s or λ. The results, shown in Fig. 3.2, include as reference the synchronizability of the complete graph, which clearly does not depend on the transform parameters. The curves clearly show that the synchronizability tends to one for all the graphs as LRIs are increasingly weighted (s, λ → 0), but the impact of LRIs is a function of the graph topology.
To further corroborate our finding, we considered a dataset of 54 real-world networks (see Appendix for descriptions of the networks) and calculated λ 2,τ /N and Q τ without and with LRIs. Fig. 3.3(a) shows the results of the change of Q τ -the results for λ 2,τ /N are very similar-with the parameter s of the Mellin transform. This confirms our previous result that the rate of change of the spectral ratio with the parameters of the transforms used in the synchronization models Table 1 in Appendix) for which the relative value of Q τ is approximately 30% of the best possible value while for others (such as Skipwith, network 29 of Table 1 in Appendix) it approaches 60%. This demonstrates that the topology of the network continues to play a fundamental role in the synchronizability of the system even when the long-range influence of the nodes is very high. Knowing how the network topology influences the rate of change of the spectral parameters and consequently of their synchronizability is an important open question that should be further investigated. Some insights on this is provided by the following considerations on how to fit the data of Fig. 3.3(a). To illustrate this, let us first consider the star topology. For this network we have been able to find a closed formula for the eigenvalues of the d-path transformed Laplacian matrix. Let λ i (L), λ i (L d ), and λ i (K n ) indicate the i-th eigenvalue of the original Laplacian matrix, of the d-path transformed Laplacian matrix (here, for simplicity we consider only the Mellin transform, but similar results hold for the Laplace one) and of the Laplacian matrix of the complete graph.
For the star topology, we have:
(3.11) λ i (L d ) = λ i (K n )p −s + λ i (L)(1 − p −s )
with p = 2 (this expression has been analytically checked up to n = 6 and numerically verified for larger n).
We have then generalized this formula for an arbitrary network as follows: The different curves correspond to these graphs: complete graph (black, filled circles); star topology (cyan, stars); triangular lattice (yellow, triangles); square lattice (magenta, squares); ring (green, open circles); Barbell graph (blue, dots); path graph (red, asterisks).
(3.12) λ i (L d ) = λ i (K n )p −s i + λ i (L)(1 − p −s i )
where now p i is a fitting parameter (one for each eigenvalue to fit) that depends on the topology we are investigating. Note that for s = 0 we have λ i (L d ) = λ i (K n ) = n and for s → ∞ we have λ i (L d ) = λ i (L) (the eigenvalues of the original network).
Finally we propose to fit Q Mell as:
(3.13) Q Mell = λ 2 (K n )p −s 2 + λ 2 (L)(1 − p −s 2 ) λ n (K n )p −s n + λ n (L)(1 − p −s n )
An example for three real-world networks is shown in Fig. 3.3(b). This fitting provides an approximate formula describing how Q Mell changes with s. Noticeably, the formula depends on the characteristics of the topology through the eigenvalues λ 2 and λ n of the original network Laplacian and some fitting parameters p 2 and p n .
In the case of the Kuramoto model we show some further results confirming the beneficial effect of LRIs. For this model, in fact, it is known that λ 2 plays a fundamental role. In particular, for identical oscillators, it is known that the synchronization time scales with the inverse of λ 2 . Based on this and on the result of Lemma 7, we thus expect that LRIs in the Kuramoto model promote synchronization. For the numerical simulations, we considered two different network topologies, ER and scale-free (SF) random graphs [4], and monitored synchronization by measuring the order parameter r, defined as
(3.14) r = 1 N N i=1 e jθi T ,
where T is a sufficiently large averaging window. Values of r close to one indicates a high degree of synchronization, while low values (close to zero) the absence of coherence among the oscillators. Fig. 3.4(a) shows the behavior of the order parameter r vs. the coupling σ for an ER network with N = 100 and different values of s. As expected, decreasing s favors synchronization. Similar results are obtained for SF networks (Fig. 3.4(b)). The results are illustrated for Mellin transformed d-path Laplacian matrices, but also hold when the Laplace transform is considered.
How does network topology influence the role of LRIs?
The analysis of the effects of LRIs on the synchronizability of real-world networks has revealed that the quantitative impact of LRIs varies from network to network. In this section we study how the effects of LRIs on synchronizability depend on some network characteristics by considering a series of artificial networks with controlled attributes.
To begin, we start investigating how the effects of LRIs depend on network diameter. To this aim, we have considered the Watts-Strogatz model generating small-world networks through a rewiring process applied to a pristine regular graph [65]. More specifically, we started from a ring with 4-nearest neighbors and then rewired the links with a progressively increasing rewiring probability, labeled as p r . This produces networks with the same number of nodes and links, but with a different diameter (large for p r = 0, small for p r = 1). Figs. 4.1(a) and (d) illustrate the results. LRIs always lead to enhancement of synchronizability, but the larger is the diameter the larger are the (beneficial) effects of LRIs.
We have then studied how the effects of LRIs are influenced by the average degree k . To this aim, we have considered networks with the same number of nodes and a growing number of links. The ER model is adopted. Figs. 4.1(b) and (e) shows that the smaller is the average degree the larger are the effects of LRIs on synchronizability, so that we conclude that LRIs are more important in networks with smaller degree.
Finally, we have investigated the effect of degree heterogeneity by simulating networks with the same number of nodes, number of links and diameter, but different degree distributions. These are obtained by using the network model described in [28], which parameterizes with α the tuning from one network type to the other (α = 0 corresponds to a SF network, while α = 1 to an ER network). Fig. 4.1(c) and (f) shows that the effects on λ 2,τ /N poorly depend on the topology, but those on the ratio Q τ are much more important on SF networks rather than in ER structures. So, SF networks receive more benefits from the inclusion of LRIs.
Critical edges for synchronization
To further study the interplay between synchronizablity and structure, in this section we investigate the effect of the edge removal on synchronization in the absence and in the presence of LRIs. More specifically, edge removal is performed according to different strategies; we consider either the removal of links chosen at random or according to some criterion ranking the edges. Since synchronization occurs through the exchange, among the network nodes, of the information on their dynamical state, those links in which information traffic is larger should be considered as the most critical. To account for this, we considered different measures of edge centrality.
The first one is edge degree calculated as k i + k j − 2, where k i and k j are the node degrees of i and j. The larger is the edge degree the more critical is the edge, so first we remove edges with larger edge degree.
Edge degree, however, only considers one-hop information exchanges, whereas information in a network is transmitted through the many existing paths from node to node. If one limits to consider shortest paths, edge centrality can be measured by edge betweeness centrality (EBC) defined as
(5.1) EBC(e) = vi∈V vj ∈V ρ(v i , e, v j ) ρ(v i , v j ) ,
where ρ(v i , e, v j ) is the number of shortest paths from node v i to v j passing through e and ρ(v i , v j ) the total number of shortest paths between nodes v i and v j . The larger is the value of the EBC, the more critical is the edge and so has to be removed first. If information is assumed to flow not only through shortest paths, then one can takes into account two other measures of edge centrality as recently introduced in [24]: the communicability function and the communicability angle. The first is defined as
(5.2) G ij = ∞ k=0 (A k ) ij k! = (e A ) ij .
The communicability function is calculated for the network edges, that is,G ij where (i, j) ∈ E, providing a measure to rank them: the smaller isG ij the poorer is the communicability between i and j, so the more critical is the edge. Therefore, if one wants to remove the critical edges according to this measure, those edges with small values ofG ij should be removed first.
Finally, the communicability angle is defined as
(5.3) θ ij = cos −1 G ij G ii G jj .
Restricting the analysis to the network edges, one hasθ ij where (i, j) ∈ E. The larger isθ ij the poorer is the communicability between i and j, so the more critical is the edge. Edges with high values ofθ ij should be then removed first.
Given a graph G, for each of the edge centralities considered, we have ranked the edges in decreasing order and removed a percentage of those that do not disconnect the graph, creating a graph G ′ with the same nodes of G and edges E ′ = E\{e}. We have then compared the synchronization measures (λ 2 /N and Q) for network G ′ and that of the original network G. We have then considered LRIs in both G and G ′ and again compared the synchronization measures for these graphs. The analysis has been performed for each of the 54 real-world networks of the dataset. Fig. 5.1 shows how the synchronizability of real-world networks, measured by the normalized parameters λ * 2 /λ 2 ( Fig. 5.1(a)-(c)) and Q * /Q (Fig. 5.1(d)-(f)), is affected by LRIs (here, λ * 2 and Q * indicate the values of λ 2 and Q for the network after removing the edges). To facilitate the visualization, the networks have been ordered according to the descending values of λ * 2 /λ 2 , where the removal of the links has been done according to decreasing values of the communicability angle. We observe that the edge removal affects more significantly synchronization in graphs with LRI than in the no-LRI scheme due to the fact that, when a "physical" link is removed, then also its long-range influence is removed. Synchronizability of the network without the removed edges is still larger if LRIs are allowed than if it is not, which means that the In these results G ′ has been obtained by removing the 20% of the original links from G. system can work better after the edge removal if such long-range interactions are present than if not. Finally, comparing the different edge removal methods, we notice that edge degree does not identify important edges, as the removal by this index leaves the networks almost unaffected in terms of the synchronizability. On the contrary, it seems that the best identifiers of critical edges are those accounting for effects going beyond nearest neighbors interactions, that is, the edge betweenness (shortest paths) and the communicability angle (all walks), because the removal by them affected the most the synchronizability.
Conclusions
In this work we have studied the impact of long-range interactions (LRIs) on synchronization. To account for LRIs, oscillators are coupled through d−path Laplacian matrices into a model that generalizes the traditional one, based on the classical Laplacian of a graph, and includes it as a special case for d = 1. As network synchronizability is essentially determined by the spectra of the Laplacian, and in particular by the smallest non-zero eigenvalue or by the ratio between the smallest non-zero eigenvalue and the largest one, we have thus studied these quantities for the graphs with LRIs.
Our theoretical considerations led to the conclusion that, increasing the weight of LRIs, any network of coupled oscillators, independently of the topology and the dynamics of the units, approaches the best possible scenario for synchronization. The specific path towards this limit condition depends on the network properties. We have performed numerical simulations both on real-world examples and on network models, and found results in perfect agreement with the theoretical expectations.
A significant result is the dependence of the impact of LRIs on some topological properties of interest, such as the diameter, the density and the degree distribution. We have found that a larger diameter, a smaller average degree or a higher heterogeneity in the degree distribution are all factors contributing to increase the positive impact of LRIs on synchronizability with respect to the scenario without LRIs.
Finally, we have studied the effects of the removal of critical edges in the absence and in the presence of LRIs, where criticality has been measured according to different topological measures of edge centrality. This analysis, carried out on real-world networks of different sizes and characteristics, points out scenarios common to all the examples considered. First of all, we have found that edge removal has a larger impact on the synchronizability of networks with LRIs rather than on the structures without LRIs. This is due to the fact that, when a physical link is removed, then also the long-range interactions it allowed are affected. However, synchronizability of the networks without the removed edges is still larger if LRIs are allowed than if not, confirming the general result that the presence of long-range interactions always favors synchronizability. Finally, we have observed that the most critical edges for synchronization are those having high values of the edge betweenness or of the communicability angles, i.e., are ranked according to measures taking into account effects going beyond nearest neighbors interactions.
The current approach can be extended to directed graphs. The first step in doing so is to generalize the d-path Laplacian matrices for such graphs. In this case there are two kinds of d-path Laplacians, namely the in-and the out-degree d-path Laplacians. That is, we should first generalize the in-and out-degrees to account for the indirect influence of nodes. The generalization is, however, straightforward. We only need to consider the number of directed shortest paths of length d from the node i to any other node in the graph to account for the out-d-degree of the node i. Similarly, we can define the in-d-degree of the node i by taking the number of directed shortest paths of length d from any node in the graph to the node i. For the synchronization dynamics we should consider the out-degree d-path Laplacians in a similar way as we have done in the current work for the undirected graphs. For directed graphs synchronizability has to be checked in the complex plane as the Laplacian eigenvalues are in general complex. However, provided that the graph is strongly connected, the generalization of d-path Laplacian matrix accounting for oriented paths still yields for s → 0 (λ → 0) a complete graph, thus favoring synchronizability. The mathematical properties of these in-and the out-degree d-path Laplacians are not trivial and deserve to be considered in a separate work.
Figure 2.1. Schematic representation of long-range influences in a social group. It is assumed that σ dmax < · · · < σ 3 < σ 2 < σ 1 , where d max is the diameter of the graph.
Theorem 4 .
4Let G be a simple graph and let d ≤ d max . The matrix L d is positive semidefinite. Moreover, the multiplicity of the zero eigenvalue of L d is equal to the number of d-connected components of the graph.
Figure 3 . 1 .
31Synchronizability of LRI ER networks. The d-path Laplacian matrices are Mellin transformed in (a)-(b) and Laplace transformed in (c)-(d). Panels (a) and (c) illustrates λ 2 /N , whereas panels (b) and (d) illustrate λ 2 /λ N . Results are averaged on 50 networks with N = 100 and k = 4. The solid lines represent mean value of the measures for synchronizability, while the shadow regions indicate the range of variability (minimum and maximum values). depends on the topology of the network. Just to give a clear example of the meaning of this finding, when s = 1 there are networks (such as SoftwareMysql, network 17 of
Figure 3 . 2 .
32Synchronizability of a few graphs with LRIs. The d-path Laplacian matrices are Mellin transformed in (a)-(b) and Laplace transformed in (c)-(d). Panels (a) and (c) illustrates λ 2,τ /N , whereas panels (b) and (d) illustrate Q τ .
Figure 3 . 3 .
33(a) Change of the synchronizability parameter (spectral ratio) for graphs with LRIs as a function of the Mellin transform parameter s for 54 real-world networks studied here. (b) Fitting of the synchronizability parameter (spectral ratio) for three real-world graphs (network 33 in blue, network 49 in red, network 51 in green). Numerical data are reported with symbols, while the continuous line is the fitting through Eq. (3.13) with p 2 = 3.16 and p n = 2.08 for network 33, p 2 = 2.07 and p n = 2.00 for network 49, and p 2 = 10.45 and p n = 4.05 for network 51.
Figure 3 . 4 .
34Synchronization order parameter r vs. the coupling strength σ for an ER (a) and SF (b) network of N = 100 Kuramoto oscillators with LRIs. Mellin transform is used to weight the LRIs.
Figure 4 . 1 .
41Values of λ 2,Mell /λ 2 (a)-(c) and of Q Mell /Q (d)-(f) for artificial networks with LRIs. LRIs are weighted with the Mellin transform. Panels (a) and (d) refer to networks with different diameter (see text for the details on the generation of the networks). Panels (b) and (e) refer to artificial networks with different average degree k . Panels (c) and (f) refer to artificial networks with different heterogeneity levels and the same average degree, k = 8. Here α tunes the heterogeneity of the network (α = 0 corresponds to a SF network, while α = 1 to an ER network). The curves refer to different values of s (s = 0.1 blue filled triangles, s = 0.5 red open triangles, s = 1 magenta filled squares, s = 2 cyan open circles, s = 5 green open squares, s = 10 black filled cyrcles). All networks have N = 500 nodes.
Figure 5 . 1 .
51Effect of edge removal on λ * 2 /λ 2 (a)-(c) and on Q/Q * (d)-(f) for real-world networks without LRIs (a) and (d) and with LRIs (b) and (e) (Mellin transform) and (c) and (f) (Laplace transform). The different bars refer to removal according to these edge centralities: from the bottom of each panel to the top, communicability angle (dark blue bars); edge degree (light blue bars); edge betweenness (light green bars); communicability function (orange bars); and random (red bars). The path Laplacians are obtained by applying the Mellin transform or the Laplace transform with s = 2 or λ = 2.
Table 1 .
1Dataset of real-world networks: network name, domain, N number of nodes, m number of links, and reference. The networks have been ordered according to the descending values of λ 2 /N . [15] Gerald F Davis, Mina Yoo, and Wayne E Baker. The small world of the american corporate elite, 1982-2001. Strategic organization, 1(3):301-326, 2003. [16] Manlio De Domenico, Albert Solé-Ribalta, Emanuele Cozzo, Mikko Kivelä, Yamir Moreno, Mason A Porter, Sergio Gómez, and Alex Arenas. Mathematical formulation of multilayer networks. Physical Review X, 3(4):041022, 2013.No. Dataset
Domain
N
m
λ *
2 /λ 2
Ref.
1
ReefSmall
ecological
50
524
0.9904
[53]
2
StMartin
ecological
44
218
0.6589
[43]
3
Internet-1998
technological 3522
6324
0.6503
[25]
4
Trans-urchin
biological
45
80
0.5974
[47]
5
KSHV
biological
50
122
0.5143
[63]
6
ODLIS
informational 2898 16381
0.4696
[2]
7
Software Abi
technological 1035
1736
0.4036
[50]
8
Internet-1997
technological 3015
5156
0.3488
[25]
9
USA Air 97
technological
332
2126
0.3095
[2]
10
Sawmill
social
36
62
0.2802
[46]
11
Grassland
ecological
75
113
0.2789
[44]
12
World-trade
informational
80
875
0.2726
[8]
13
Trans-Ecoli
biological
328
456
0.2715
[47]
14
Neurons
biological
280
1973
0.2122
[66]
15
Ythan1
ecological
134
597
0.2102
[31]
16
Hpyroli
biological
710
1396
0.1978
[37]
17
Software Mysql
technological 1480
4221
0.1850
[50]
18
YeastS
biological
2224
7049
0.1736
[10]
19
Geom
social
3621
9461
0.1666
[8]
20
Chesapeake
ecological
33
72
0.1664
[13]
21
Benguela
ecological
29
191
0.1490
[67]
22
Software Digital
technological
150
198
0.1470
[50]
23
Hi-tech
social
33
91
0.1420
[33]
24
Zackar
social
34
78
0.1375
[69]
25
PRISON-Sym
social
67
142
0.1309
[41]
26
ScotchBroom
ecological
154
370
0.1269
[45]
27
PIN Ecoli
biological
230
695
0.1231
[11]
28
Roget
informational
994
3641
0.1171
[3]
29
Skipwith
ecological
35
364
0.1098
[68]
30
StMarks
ecological
48
221
0.1011
[27]
31
social3
social
32
80
0.1002
[70]
32
Malaria PIN
biological
229
604
0.0999
[34]
33
Canton
ecological
108
708
0.0943
[62]
34
Drugs
social
616
2012
0.0917
[1]
35
BridgeBrook
ecological
75
547
0.0896
[55]
36
Stony
ecological
112
832
0.0844
[7]
37
Dolphins
social
62
159
0.0826
[40]
38
ElVerde
ecological
156
1441
0.0813
[57]
39
PIN Human
biological
2783
6438
0.0733
[60]
40
electronic2
technological
252
399
0.0730
[48]
41
Software-VTK
technological
771
1369
0.0717
[50]
42
Transc-yeast
biological
662
1062
0.0668
[47]
43
CorporatePeople
social
1586 13126
0.0654
[15]
44
electronic3
technological
512
819
0.0595
[48]
45
Software-XMMS technological
971
1809
0.0471
[50]
46
electronic1
technological
122
189
0.0463
[48]
47
SmallW
informational
233
994
0.0422
[2]
48
Shelf
ecological
81
1476
0.0356
[38]
49
Coachella
ecological
30
261
0.0300
[64]
50
Power grid
technological 4941
6594
0.0067
[65]
51
ColoSPG
social
324
347
1.8 · 10 −13 [56]
52
Drosophila PIN
biological
3039
3715
3 · 10 −14
[26]
53
Pin-Bsubtilis
biological
84
98
1.7 · 10 −15 [52]
54
PIN-Afulgidus
biological
32
38
1.4 · 10 −15 [49]
The real-world networks used in this paper belong to different domains: ecological (includes food webs and ecosystems), social (networks of friendships, communication networks, corporate relationships), technological (internet, transport, software development networks), informational (vocabulary networks, citations) and biological (protein-protein interaction networks, transcriptional regulation networks). The dataset comprises networks of different sizes, ranging from N = 29 to N = 4941 nodes. The networks are listed inTable 1.
Data for this project were provided, in part, by NIH grants DA12831 and HD41877. Data for this project were provided, in part, by NIH grants DA12831 and HD41877, 2001.
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|
[] |
[
"SUNGLASS: A new weak lensing simulation pipeline",
"SUNGLASS: A new weak lensing simulation pipeline"
] |
[
"A Kiessling \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K\n",
"A F Heavens \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K\n",
"A N Taylor \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K\n"
] |
[
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEdinburghU.K"
] |
[
"Mon. Not. R. Astron. Soc"
] |
A new cosmic shear analysis pipeline SUNGLASS (Simulated UNiverses for Gravitational Lensing Analysis and Shear Surveys) is introduced. SUNGLASS is a pipeline that rapidly generates simulated universes for weak lensing and cosmic shear analysis. The pipeline forms suites of cosmological N-body simulations and performs tomographic cosmic shear analysis using line-of-sight integration through these simulations while saving the particle lightcone information. Galaxy shear and convergence catalogues with realistic 3D galaxy redshift distributions are produced for the purposes of testing weak lensing analysis techniques and generating covariance matrices for data analysis and cosmological parameter estimation. We present a suite of fast medium resolution simulations with shear and convergence maps for a generic 100 square degree survey out to a redshift of z = 1.5, with angular power spectra agreeing with the theory to better than a few percent accuracy up to ℓ = 10 3 for all source redshifts up to z = 1.5 and wavenumbers up to ℓ = 2000 for the source redshifts z 1.1. At higher wavenumbers, there is a failure of the theoretical lensing power spectrum reflecting the known discrepancy of theSmith et al. (2003)fitting formula at high physical wavenumbers. A two-parameter Gaussian likelihood analysis of σ 8 and Ω m is also performed on the suite of simulations, demonstrating that the cosmological parameters are recovered from the simulations and the covariance matrices are stable for data analysis. We find no significant bias in the parameter estimation at the level of ∼ 0.02. The SUNGLASS pipeline should be an invaluable tool in weak lensing analysis.
|
10.1111/j.1365-2966.2011.18540.x
|
[
"https://arxiv.org/pdf/1011.1476v1.pdf"
] | 118,651,313 |
1011.1476
|
6f85822475a97b26d4e3247592f3c4745e6bcc88
|
SUNGLASS: A new weak lensing simulation pipeline
5 Nov 2010 Printed 8 November 2010
A Kiessling
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEdinburghU.K
A F Heavens
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEdinburghU.K
A N Taylor
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEdinburghU.K
SUNGLASS: A new weak lensing simulation pipeline
Mon. Not. R. Astron. Soc
0005 Nov 2010 Printed 8 November 2010Accepted -. Received -; in original form -.(MN L A T E X style file v2.2)Gravitational lensing -Cosmology: large scale structure of Universe - Methods: N -Body simulations
A new cosmic shear analysis pipeline SUNGLASS (Simulated UNiverses for Gravitational Lensing Analysis and Shear Surveys) is introduced. SUNGLASS is a pipeline that rapidly generates simulated universes for weak lensing and cosmic shear analysis. The pipeline forms suites of cosmological N-body simulations and performs tomographic cosmic shear analysis using line-of-sight integration through these simulations while saving the particle lightcone information. Galaxy shear and convergence catalogues with realistic 3D galaxy redshift distributions are produced for the purposes of testing weak lensing analysis techniques and generating covariance matrices for data analysis and cosmological parameter estimation. We present a suite of fast medium resolution simulations with shear and convergence maps for a generic 100 square degree survey out to a redshift of z = 1.5, with angular power spectra agreeing with the theory to better than a few percent accuracy up to ℓ = 10 3 for all source redshifts up to z = 1.5 and wavenumbers up to ℓ = 2000 for the source redshifts z 1.1. At higher wavenumbers, there is a failure of the theoretical lensing power spectrum reflecting the known discrepancy of theSmith et al. (2003)fitting formula at high physical wavenumbers. A two-parameter Gaussian likelihood analysis of σ 8 and Ω m is also performed on the suite of simulations, demonstrating that the cosmological parameters are recovered from the simulations and the covariance matrices are stable for data analysis. We find no significant bias in the parameter estimation at the level of ∼ 0.02. The SUNGLASS pipeline should be an invaluable tool in weak lensing analysis.
INTRODUCTION
Cosmic shear analysis is an excellent method for probing the dark Universe (for reviews, see Mellier 1999; Bartelmann & Schneider 2001;Refregier 2003;Schneider 2006;Munshi et al. 2008;Massey et al. 2010, and references therein). It is also a reasonably new field of research with cosmic shear first being observed just ten years ago (Bacon et al. 2000;Kaiser et al. 2000;Van Waerbeke et al. 2000;Wittman et al. 2000). Weak gravitational lensing effects on a cosmic scale are a mere 1% change in shape and observational systematics makes the measurement of these changes challenging. However, the combination of the wellunderstood underlying physics and the expected precision of cosmological parameter estimation make the effort worthwhile.
Next generation telescope surveys will observe more of the sky than ever before and the volume of data they will produce is unprecedented. Future surveys promise to determine the equation of state of dark energy to 1% as well as ⋆ E-mail: [email protected] probing the possibilities of extra dimensional gravity models and alternative cosmologies. The first Pan-STARRS 1 telescope is currently undertaking a cosmic shear survey of the entire visible sky from its location in Hawaii and new projects such as VST-KIDS 2 , DES 3 , HALO 4 , Euclid 5 and LSST 6 and are planned to perform wide field cosmic shear surveys, measuring both large, linear scales, and small, nonlinear scales.
Due to the relative youth of this field, techniques are still being developed to exploit the weak lensing data from these surveys to provide further understanding on the nature of the Universe. To realise the potential of these new telescope surveys and to test new weak lensing analysis techniques, challenges must be met. To achieve the small statistical errors required, experiments require full end-to-end simulations of huge volumes which also probe the non-linear regime to assist in understanding the limitations of the analysis techniques. Simulations offer data sets with known parameters which are essential when testing analysis pipelines. Simulations can also include effects which may be difficult to model theoretically, such as source clustering and galaxy alignments, as well as other systematics and real-world effects. An additional role for simulations is in accurate estimation of the covariance of observable quantities. This is needed for the analysis of surveys and analytic approximations can be wholly inadequate (e.g. Semboloni et al. 2007). Monte Carlo analysis can be performed with simulations to provide covariance matrices that are required for data analysis and cosmological parameter estimation. Simulations are also required for rigorous testing and development so all analysis methods can be analysed blindly before the same techniques are applied to real data. To address these challenges, the SUNGLASS, Simulated UNiverses for Gravitational Lensing Analysis and Shear Surveys, pipeline has been developed to produce simulations and mock shear and convergence catalogues rapidly for weak lensing and cosmic shear analysis. The purpose of this paper is to introduce SUNGLASS and show rigorous testing of its outputs.
Many weak lensing studies use simulations with very high resolution to run their analysis (e.g. Fosalba et al. 2008;Hilbert et al. 2009;Teyssier et al. 2009;Schrabback et al. 2010). The computational cost of running these simulations is high and consequently there is often only a single realisation available. However, it is very important to ensure that covariance matrices calculated from these simulations are not contaminated by correlations in the simulations (Hartlap et al. 2007). In order to ensure uncorrelated data, a Monte Carlo suite of simulations should be used to determine the covariance matrix (Sato et al. 2009). In this work, 100 independent simulations were constructed using SUNGLASS.
To date, there are still reasonably few weak lensing simulations available. Of the few that are available, many implement a ray-tracing technique where light rays are propagated from an observer to a lensing source plane (e.g. Jain et al. 2000;Vale & White 2003;Hilbert et al. 2009;Teyssier et al. 2009;Sato et al. 2009;Dietrich & Hartlap 2010;Vafaei et al. 2010). Ray-tracing is computationally intensive and time consuming when solving the full ray-tracing equations. If the Born approximation is used in the raytracing, the time to run the analysis is reduced but the process is still computationally intensive and the simulation data still needs to be binned in three dimensions to perform the calculations. An alternative to ray-tracing is lineof-sight integration, which uses the Born approximation to calculate rapidly the weak lensing signal through a lightcone (e.g. White & Hu 2000;Fosalba et al. 2008). This method is not suitable in the strong lensing regime but in the weak lensing regime, it is rapid and requires fewer computational resources than ray-tracing techniques. In this paper, a new line-of-sight integration technique, implemented in the SUN-GLASS pipeline, for measuring convergences in an N-body simulation is introduced. This new method is rapid and can be run on a single processor of a desktop computer. In contrast to ray-tracing, the method does not bin in the radial direction, using all of the redshift information available. Although the catalogues are suitable for real-space analysis, SUNGLASS analyses and tests our mock weak lensing surveys in Fourier space, using power spectra, as it is possible to cleanly distinguish between linear and nonlinear regimes in Fourier space. We are also able to easily identify scales where the simulations are reliable by determining the region of the power spectrum in Fourier space that lies between the size of the simulated volume at low wavenumbers and shotnoise due to particle discreteness and pixelization effects at high wavenumbers.
The outline of this paper is as follows. Section 2 introduces the SUNGLASS pipeline. Details of the simulations are in section 2.1 and the line-of-sight integration method for determining shear and convergence without radial binning is described in section 2.2. Section 2.3 presents the shear and convergence power spectrum analysis and section 2.4 deals with the generation of the mock galaxy shear catalogues. An application of the mock catalogues is discussed in section 3 where Gaussian likelihood estimates of Ωm and σ8 are performed. A summary of the pipeline and methods concludes the paper in section 4.
DETAILS OF THE SUNGLASS PIPELINE
SUNGLASS is a pipeline that generates cosmic shear and convergence catalogues using N-body simulations. The pipeline creates mock galaxy shear catalogues that can be used to test the cosmic shear analysis software used on telescope survey data sets. The nature of the pipeline also allows many simulation realisations to be generated rapidly to produce covariance matrices for data analysis and cosmological parameter estimation. The pipeline begins by creating a suite of cosmological N-body simulations. Lightcones are generated through the simulations and tomographic shear and convergence maps are determined using line-of-sight integrations at multiple lensing source redshifts. Finally, mock galaxy catalogues with fully 3D shear and convergence information and galaxy redshift distributions are assembled from the lightcones and the tomographic shear and convergence planes. The following sections detail each step of the SUNGLASS pipeline.
The N-body Simulations
All of the simulations presented in this work were run on a modest Xeon cluster, using 4 nodes with dual Xeon E5520 2.27 GHz quad-core processors per node and 24Gb shared memory per node. The simulations were run using the cosmological structure formation software package GADGET2 (Springel 2005). GADGET2 represents bodies by a large number, N (in this work we use 512 3 ), particles. Each particle is 'tagged' with its own unique kinematic and physical properties that evolve with the particle over time. GADGET2 models the dynamics of dark matter particles using a Tree-PM scheme and for the purposes of this work, only dark matter particles were considered. The pre-initial particle distribution for the simulations used in this work is a glass which has sub-Poissonian noise properties (White 1994). This distribution has no preferred direction with forces on each particle being close to zero. If a glass is used as the initial condition in a standard integrator, structures do not evolve. Particle displacements are imposed manually as an initial step to enable structure formation. The initial power spectrum was imposed on the particles using the parallel initial conditions generator N-GenIC that was provided by Volker Springel. The initial particle displacement field is formed by using the Zel'dovich approximation (Zel'dovich 1970) to perturb the particles, imposing an Eisenstein & Hu (1998) matter power spectrum on the particles, and giving each particle an initial velocity.
Multiple medium-resolution simulations were run with 512 3 dark-matter particles, in a box of L = 512h −1 Mpc comoving side length with periodic boundary conditions. The following cosmological parameters were used for a flat concordance ΛCDM model consistent with the WMAP 7year results (Jarosik et al. 2010): ΩΛ = 0.73, Ωm = 0.27, Ω b = 0.045, ns = 0.96, σ8 = 0.8 and h = 0.71 in units of 100 km s −1 Mpc −1 . The particle mass is 7.5 × 10 10 M⊙and the softening length is 33h −1 kpc. The simulations were all started from a redshift of z = 60 and allowed to evolve to the present.
The simulation data were stored at 26 output times corresponding to a 128h −1 Mpc comoving separation, between z = 1.5 and the present. These snapshots were chosen to fall within the photometric redshift error of σz < 0.05(1 + z) corresponding to a displacement of ≃ 147h −1 Mpc at z = 1. In a 512 3 particle simulation, this amounts to 100GB data per simulation and takes approximately 21hrs to run on the Xeon cluster's 32 processors.
Shear and Convergence Map Generation
We begin by determining the shear and convergence for a source plane at fixed comoving distance rs. We consider a distribution of sources in Section 2.4.
The effects of weak gravitational lensing on a source can be described by two fields, the spin-2 shear, γ, which describes the stretching or compression of an image, and a scalar convergence, κ, which describes its change in angular size. These can be related to a lensing potential field, φ, by
κ = 1 2 ∂ 2 φ,(1)γ = γ1 + iγ2 = 1 2 ∂∂φ,(2)
where γ1 and γ2 are the orthogonal components of the shear distortion, and ∂ = ∂x + i∂y is a complex derivative on the sky. We want to generate shear and convergence maps along a lightcone through the simulation. Instead of using ray tracing to determine the lightcone (e.g. Wambsganss et al. 1998;Jain et al. 2000;Teyssier et al. 2009;Hilbert et al. 2009), a line-of-sight integration was implemented using the Born approximation where one integrates along an unperturbed path (e.g. Cooray & Hu 2002;Vale & White 2003). Fosalba et al. (2008) build their convergence maps by adding slices from their simulation with the appropriate lensing weight and averaging over a pixel; κ(θi, rs) = rs 0 dr K(r, rs)δ(θi, r) Figure 1. Lightcone geometry through a simulation box volume. The lightcone travels through the first 128h −1 Mpc of the first simulation and then the next 128h −1 Mpc of the next simulation etc. At the end of the simulation volume, the next volume snapshots have their centroids shifted and are randomly rotated to avoid repeated structures along the lightcone.
where θi is the position if the i th pixel on the sky and j is a bin in the radial direction which is at a distance of rj and has a width of ∆rj. An overline denotes an average over a pixel on the sky. The expansion factor at each radial bin j is given by aj and the comoving radial distance of the lensing source plane is given by rs. In order to make these calculations, the 3D matter overdensityδ(θ, r) must be calculated by binning the simulation data in three dimensions. A limitation of this approach is memory, speed and accuracy. Here we propose, in the SUNGLASS pipeline, a new method for the line-of-sight integration so that no radial binning is required to determine the convergence. The particles are binned in a fine angular grid while allowing them to keep their radial co-ordinate.
Rewriting equation (4) we find the average convergence in an angular pixel, with no radial binning, is given bȳ
κp(rs) = j K(rj , rs) ∆Ωpn(rj)r 2 j − rs 0 dr K(r, rs),(5)
where ∆Ωp = ∆θx∆θy is the pixel area and K(r, rs) is the scaled lensing kernel:
K(r, rs) = 3H 2 0 Ωm 2c 2 (rs − r)r rsa(r) .(6)
Hereafter we drop the overline and assume all fields are averaged over an angular pixel. A derivation of equation (5) is given in Appendix A. In practice equation (5) can be calculated by a running summation so that it is not necessary to re-calculate the convergence from scratch for each source redshift.
The convergence maps are generated by adding the particles that fall within the lightcone to the line-of-sight integration. To show evolution through the lightcone, the simulation volumes are split into 128h −1 Mpc sections. The first 128h −1 Mpc of the first (z = 0) snapshot is used, the second 128h −1 Mpc of the second (z > 0) snapshot and so on until the end of the simulation box volume is reached at snapshot 4 as shown in Figure 1. The centroid of the next simulation box is then shifted and the simulation box is rotated randomly to try to avoid repeated structures along the lineof-sight (e.g. White & Hu 2000;Vale & White 2003). The boxes are always periodic in the transverse direction. This continues through all of the snapshots out to a redshift of z = 1.5. The source redshifts have been placed at ∆z = 0.1 intervals because the change in convergence between these redshifts is small enough that desired redshift values in between can be accurately determined by interpolation.
Once the convergences have been calculated at each of the source redshifts, the shear values can be determined on a flat-sky. The flat-sky shear and convergence Fourier coefficients are related by
γ1(ℓ) = κ(ℓ) (ℓ 2 x − ℓ 2 y ) (ℓ 2 x + ℓ 2 y ) ,(7)γ2(ℓ) = κ(ℓ) 2ℓxℓy l 2 x + l 2 y ,(8)
where κ(ℓ) is the Fourier transform of the convergence and ℓx and ℓy are the Fourier variables. The Fast Fourier transform used throughout this paper is FFTW 7 . The periodic nature of FFTW requires that the field is buffered with a small number of bins that are trimmed off after the shear has been calculated. To test the algorithm we also estimated B-modes by calculating the unphysical imaginary part of the convergence β = imag(κ), from the shear, Figure 2 is an example of a convergence and shear map for a field that is 100 square degrees at a source redshift of 7 The Fastest Fourier Transform in the West http://www.fftw.org/ z = 0.8. There are 2048 bins in each transverse direction and no binning in the radial direction. The background of the map shows the integrated convergence along the lightcone up to z = 0.8 and the white ticks show the shear at this source redshift. The length of the ticks has been multiplied by an arbitrary constant to make them visible as the magnitude of the shear is at the percent level. The red patches show areas of the highest convergence and the shear ticks clearly trace these regions tangentially. These maps can be generated for the standard simulations at multiple source redshifts quite rapidly once the simulations have been run. The most time consuming module in this code is reading in the snapshots due to their reasonably large size of 100GB. This module can be optimised by using the fastest available data transfer rates on the drive where the snapshot data is stored.
β(ℓ) = γ1(ℓ) 2ℓxℓy l 2 x + l 2 y + γ2(ℓ) ℓ 2 x − ℓ 2 y ℓ 2 x + ℓ 2 y .(9)
Shear and Convergence Power Spectra
In order to verify the accuracy of the shear and convergence maps, the shear and convergence power spectra are determined for each source redshift. From equation (4), the theoretical prediction for the shear and convergence power spectrum for sources at redshift z is given by (Munshi et al. 2008) where P (ℓ/r; r) is the 3D matter density power spectrum at a redshift z. From the simulations it is possible to determine an angle-averaged power spectrum from the convergence and shear calculated in the lightcones. When taking in to consideration the conventions used in FFTW, the discretised convergence power spectrum for a slice in redshift is given as the sum over logarithmic shells in ℓ-space as
C γγ ℓ (z) = C κκ ℓ (z) = 9H 4 0 Ω 2 m 4c 4 rs 0 dr P ℓ r ; r [rs(z) − r] 2 r 2 s (z)a 2 (r) ,(10)ℓ(ℓ + 1)Ĉ κκ ℓ (z) 2π = ℓ in shell |κ(ℓ, z)| 2 n 2 ∆ ln ℓ ,(11)
where n is the total number of bins in the Fourier transform and ∆ ln ℓ represents the thickness of the shell in log ℓ-space, andĈ κκ ℓ is the estimated power. Similarly the shear power is estimated by
ℓ(ℓ + 1)Ĉ γγ ℓ (z) 2π = ℓ in shell |γ1(ℓ, z)| 2 + |γ2(ℓ, z)| 2 n 2 ∆ ln ℓ .(12)
The B-mode power is estimated in the same way as the convergence.
The modes in this power spectrum are arranged on a square grid, which causes discreteness errors when binned in annuli at small ℓ. To correct for this, the power is scaled by the ratio of the measured number of modes to the expected number of modes,
Nexp = gπ(ℓ 2 max − ℓ 2 min ),(13)
where g = (L/2π) 2 is the density of states, L is the size of the field in radians, ℓmax and ℓmin are the minimum and maximum wave numbers in this shell. The effect of this normalisation correction is about 10% at the lower wave numbers while the higher wavenumbers remain largely unaffected. The discreteness correction is not perfect which is why the same slight zig-zag of the power spectrum is evident in all of the source redshift planes at wavenumbers ℓ < 100. We can compare our simulated shear and convergence power spectra with the theoretical expectation. The theoretical power spectrum we use is determined using a code kindly provided by Benjamin Joachimi (as demonstrated in Joachimi & Schneider 2008, and extensively tested against iCosmo 8 (Refregier et al. 2008)). This code uses the method of Smith et al. (2003) for the non-linear power spectrum, the matter transfer function of Eisenstein & Hu (1998) and the analytical expression for the linear growth factor as given in Heath (1977).
Due to the discrete number of particles in an N-body simulation, the measured power spectrum measured will be the combined real shear and convergence power plus a shotnoise power contribution,
C κκ ℓ = C κκ ℓ + C SN ℓ ,(14)
whereĈ κκ ℓ is the power estimated from the simulation. The shot-noise power can be derived from equation (10) using a white-noise power spectrum, PSN (k, r) = 1/n3(r), wherē n3(r) is the 3-D mean comoving number density of particles in the simulation. The shot-noise power for the shear and convergence is then given by
C SN ℓ = 9H 4 0 Ω 2 m 4c 4 rs 0 dr (rs − r) 2 n(r)r 2 s a(r) 2 .(15)
Usually, for simulated particles,n will be a constant in comoving coordinates. Figure 3 shows the mean, normalised 2-D shear power spectra estimated from 100 independent simulations (black points and line), with the error bars showing the scatter on the estimated mean. The figures show the shear power for sources at redshifts of z = 0.3, 0.6, 0.8, 1.0, 1.3 and 1.5. The smooth (red) line shows the theoretical prediction for the ensemble-averaged shear power spectrum, while the diagonal (blue) lines show the shot-noise power for each source redshift. The (light blue) curve between the simulated data and the theory curve shows the mean power spectrum with the expected shot-noise subtracted and the lower (magenta) curve shows the estimated B-mode power spectrum.
The bottom panel of each figure shows the percentage difference between the measured shear power spectrum and the ensemble-average theory prediction (black), while the lower (light blue) points show the shot-noise subtracted shear power spectrum. Overall the mean shear power agrees well, to within a few percent, with the ensemble-averaged theoretical model over the ℓ-range ℓ < 1000 for all source redshifts. The difference of a few percent is due to the fact that the theory 3D matter density power spectrum is a few percent lower than the measured data power spectrum. Calculating the highly non-linear power spectrum is currently not accurate to a few percent and many calculations of this theory curve do not agree with each other to within a few percent. The Joachimi theory curve was the closest fit to the simulations and was used for all subsequent calculations. At low ℓ the measured signal drops as we reach the size of the simulation box, while at high ℓ, the estimated mean shear power becomes shot-noise dominated before reaching the highest mode allowed by the resolution of the angular pixels beyond ℓ = 1/θpix ≃ 10 4 . Before reaching pixel-resolution, the measured shear power at high-ℓ agrees well with the predicted shot-noise. This agreement suggests that the shot-noise model works well in this regime, even though the initial particle distribution is a glass (see Baugh et al. 1995, for a discussion). This suggests an improved estimate of the mean power can be found by subtracting off the shot-noise contribution. However, the shot-noise subtracted shear power does not follow the ensemble-averaged theoretical power estimated from the theory code. It is likely this is a failure of the theoretical model of lensing -on small-scales the Smith et al. (2003) nonlinear correction formula is known to underestimate the matter-density power spectrum, P (k), by up to 10% at wavenumbers of k < 1 and as great as 50% at k = 10 Mpc −1 (Giocoli, private communication) and hence has been shown to underestimate the shear and convergence power spectrum by up to 30% on scales of ℓ < 10 4 (Hilbert et al. 2009). In the absence of accurate fitting formulae, simulations like those presented in this paper may be used to improve theoretical predictions. However, this needs to be explored in more detail before it is fully understood so in subsequent analysis in this paper we will restrict our analysis to the region of the measured power spectrum that agrees with the theoretical prediction. Figure 3 also shows the estimated B-mode power spectra. When galaxies trace the shear signal, we expect the B-mode power to pick up a shot-noise dependence. But here the shear signal is a pixelized field which would be continuous in the limit of infinite pixels. Therefore we do not expect there to be a noise-generated B-mode. However, Bmodes can still be generated due to leakage of power from the convergence field caused by the finite window function when we generate the shear field from equation (8). As a consequence the induced B-mode has the shape of the shear power, but suppressed by around three orders of magnitude.
In this section we have shown that the SUNGLASS algorithm for calculating the shear and convergence maps and the power spectra in redshift slices is accurate to a few percent over a wide range of scales and redshifts. Wavenumbers up to 1500 can be recovered for the source redshifts z 1.1 with this simulation resolution. For shot-noise subtracted power spectra, the recovered modes increase before the angular pixel resolution cuts off the power.
Mock 3-D Weak Lensing Galaxy Catalogues
Real, 3-D weak lensing data analysis is applied to a galaxy catalogue where galaxy angular positions and redshift are added to estimated shears for each galaxy. For a 2-D analysis, individual redshifts are ignored and the theory uses only the redshift distribution. It is straightforward to generate a simple 3-D mock weak lensing galaxy catalogue with the information in the lightcones we have generated from the simulations. Shear and convergence maps are generated for each lensing source redshift and then each particle in the simulation is assigned a shear and convergence by interpolating between adjacent planes. The error introduced by linearly interpolating the shear and convergence between source redshift planes separated by ∆z = 0.1 was estimated by comparing with much higher redshift-sampled planes and found to be substantially below the theoretical prediction (∆C γγ ℓ < 10 −7 ) except at angular wavenumbers where shotnoise becomes dominant. With the interpolated shear and convergence assigned to each particle, we now have a fullysampled 3-D mock weak lensing galaxy catalogue, which can be down-sampled to generate realistic weak lensing surveys.
To down-sample the full 3-D weak lensing simulated lightcone to construct a realistic 3-D weak lensing galaxy catalogue, we use a galaxy redshift distribution (Refregier et al. 2004)
n(z) ∝ z α exp − z z0 β ,(16)
where z0, α and β set the depth, low-redshift slope and highredshift cut-off for a given galaxy survey. We take α = 2, β = 2 and z0 = 0.78, yielding a median redshift of zm = 0.82, similar to the CFHTLens Survey. As the particles in our simulation are in comoving coordinates, we transform this redshift distribution to a probability distribution for the particle to enter our catalogue given its comoving radial distance,
p(r) ∝ r α dr dz exp − z(r) z0 β ,(17)
where
dr dz = c H(z) ,(18)
and Figure 5. 2D shear power spectrum for the lightcone suite with the n(z) particle distribution. In the upper panel, the smooth (red) line is the theory prediction and the diagonal (blue) line is the shot noise prediction. The (black) points and line is the mean measured power spectrum for the suite of mock catalogues with the errors representing the error on the mean and the curve between the theory prediction and the measured simulation data is the shot-noise subtracted power spectrum. The diagonal (magenta) line shows the mean of the B-modes for the suite of mock catalogues with errors on the mean. The bottom panel shows the percentage difference of the data from the theory curve with errors on the mean (black) and the lower (light blue) points represent the shot-noise subtracted data.
H(z) = H0 [Ωm(1 + z) 3 + ΩK (1 + z) 2 + ΩΛ] 1/2 ,(19)
where H0 is the current Hubble value, Ωm is the current matter density, ΩΛ is the current dark energy density and ΩK is the curvature parameter. Throughout we have assumed a flat, ΩK = 0, cosmology for our simulations. We sample the particle distribution so our final galaxy catalogue has a surface density of around 15 galaxies per square arcmin, with a maximum redshift cut-off at z = 1.5. The left panel of Figure 4 is an example of a redshift distribution taken from the full particle lightcone. The red line shows the theoretical distribution from equation 17, normalised to the number of particles selected, that the simulation particles were drawn from. The clustered nature of the particles in the distribution is apparent as the peaks and troughs around the theoretical curve can be seen.
Our 3-D weak lensing catalogue currently assumes that the redshift to each galaxy is accurately known. This would be appropriate for a spectroscopic redshift survey, but with such large surveys we can expect most weak lensing catalogues will contain photometric redshift estimates for each galaxy. To account for photometric redshift errors, we randomly sample the measured redshift from the true redshift using a Gaussian distribution with uncertainty
σz = σ0(zg)(1 + zg),(20)
where zg is the true redshift of the particle. For the purposes of this work we assume a fixed σ0 = 0.05. The right-hand panel of Figure 4 shows what the distribution on the left looks like with photometric redshift errors. The structures are smoothed out and the distribution becomes featureless. The photometric redshift errors were implemented by specifying a Gaussian error. Figure 5 shows the ensemble-averaged 2-D shear power spectrum estimated from 100 mock weak lensing surveys in the top panel (black dots) with errors on the mean, compared the theoretical prediction in red, and the ensembleaveraged B-mode power in magenta. The (blue) diagonal line shows the shot-noise prediction for these galaxy redshift distributed lightcones. The shot-noise was determined by running the SUNGLASS analysis on a number of simulation box volumes filled with randomly distributed particles. The power spectrum of these lightcones represents shot-noise estimate for the simulations and is a remarkably straight power law. The (light blue) curve between the shot noise and the measured power spectrum is the shot-noise subtracted power spectrum. The bottom panel shows the fractional difference between the average of the mock surveys and the theory curve, with the error on the mean (black) and the shot-noise subtracted points below (light blue). This shows that the mock weak lensing survey agrees with the theoretical expectation from wavenumbers from ℓ = 200 to ℓ = 2000, where the disagreement with theory can be ascribed to the uncertainty on the theory curve, and the rise of shot-noise. The shot-noise subtraction in this case is a few percent lower than the theoretical prediction and the reason for this is not well understood and is the subject of ongoing investigation. The analyses in this paper will use the measured simulation power spectrum only. The B-mode power appears to follow a shot-noise profile which is consistent with the effect of sampling from the full particle lightcone. A secondary source for B-modes is source clustering, which appears to be sub-dominant.
We found a dependence for the recovered shear and convergence power on the number of pixels used to estimate the 2-D lensing power. With too many bins, there were a number of empty pixels and this reduced the amplitude of the power spectrum. The amplitude of the power spectrum increased with fewer empty bins before converging at the true amplitude. However, by using too few bins, the number of ℓ modes recovered was reduced due to pixelization effects. It was found that for this work, 768 2 bins provided a stable amplitude for the power spectrum with the largest number of modes possible without causing this amplitude to fall. In this case, 0.03% of the bins are empty. If this number is increased to 5% empty, the amplitude of the power spectrum drops by up to 10%. This effect will also be important for observational studies and should be considered when binning survey data to determine 2D lensing power spectra.
PARAMETER ESTIMATION
As described in the previous Section 100 simulations have been generated using the SUNGLASS pipeline. The mock survey parameters are given in Table 1.
For each of these mock lensing surveys the shear and convergence power spectra has been estimated, and the ensemble average power and its scatter measured. Here we want to use the mock surveys to test a maximum likelihood cosmological parameter estimation analysis, typically used to extract parameters from weak lensing surveys. Here we try and recover the amplitude of the matter clustering, σ8, and density parameter, Ωm, from a 2-D weak lensing survey.
In Section 2.4 we showed that our simulations could produce unbiased estimates of the shear power from a mock survey over a range of ℓ-modes from 200 to 2000. For parameter estimation we need to know the conditional probability distribution of shear power, p(Ĉ γγ ℓ |σ8, Ωm), for the likelihood function, where we have fixed all other parameter at their fiducial values. This is usually assumed to be Gaussian (although, see Hartlap et al. 2009, who study non-Gaussian likelihoods). Here we test this assumption on our mock catalogues. Figure 6 shows the distribution of variations about the mean of theĈ ℓ 's, ∆Ĉ γγ ℓ , divided by the ensemble-averaged scatter in the power, σ(Ĉ γγ ℓ ). If the distribution is Gaussian, these distributions should all lie on the unit-variance Gaussian. The left panel shows a histogram of the distribution of points for modes of ℓ < 400 which is close to the linear region of the power spectrum. The middle panel shows the distribution of C γγ ℓ for modes of 400 < ℓ < 1300 which represents the non-linear region of the power spectrum. The final panel shows the distribution for modes ℓ > 1300 which is the shot-noise dominated regime. The smooth (red) line in each of the panels is a normalised unit-Gaussian curve. In each of the panels, the histogram of points is peaked slightly to the left of the Gaussian peak which indicates a slight non-Gaussianity of the distribution of points. This slight non-Gaussianity may bias the Gaussian likelihood analysis but the dominant effect is currently the inaccurate fitting of the matter power spectrum by the Smith et al. (2003) formula at high k (Giocoli et al. 2010).
The cosmological parameters of the simulations were estimated using Gaussian likelihood analysis where the likelihood is given by
L(Ĉ γγ ℓ |σ8, Ωm) = 1 (2π) N/2 (det M ℓℓ ′ ) 1/2 exp −χ 2 2 ,(21)
where
χ 2 = ℓℓ ′ (Ĉ γγ ℓ − C γγ ℓ )M −1 ℓℓ ′ (Ĉ γγ ℓ ′ − C γγ ℓ ′ ),(22)
and M ℓℓ ′ is the covariance matrix of the shear power spectra given by
M ℓℓ ′ = ∆C γγ ℓ ∆C γγ ℓ ′ .(23)
The inverse covariance matrix was determined by performing a singular value decomposition (SVD) on the covariance matrix (Press et al. 1992). The resulting inverse covariance matrix is, however, biased due to noise in the covariance matrix. Hartlap et al. (2007) propose a correction for this bias by multiplying the inverse covariance matrix by a factor:
M −1 ℓℓ ′ = NS − Np − 2 NS − 1 M −1 ℓℓ ′ ,(24)
where NS is the number of simulations used to determine the covariance matrix, Np is the number of bins in the power spectrum andM −1 ℓℓ ′ is the unbiased covariance matrix. The likelihood analysis relies on accurate estimation of the covariance matrix to show the degree of correlations. Figure 6. Histogram of the distribution of power spectra for the suite of lightcones with the n(z) particle distribution. The left panel shows the distribution of the C γγ ℓ s less than ℓ = 400, the middle panel shows the C γγ ℓ distribution from 400 < ℓ < 1300 and the right panel shows the distribution from at ℓ > 1300.
The correlation coefficients are
r ℓℓ ′ = M ℓℓ ′ √ M ℓℓ M ℓ ′ ℓ ′ .(25)
The correlation coefficient matrix is equal to 1 along the diagonal and the off diagonal components will show how correlated the ℓ modes are, with numbers close to zero indicating low correlation and numbers close to (minus) one indicating high (anti-)correlation. Figure 7 shows the correlation coefficient matrix for the ℓ modes being considered between 100 < ℓ < 2500. The modes with a low correlation are represented in black and dark blues and the modes with a high correlation shown in yellows and reds. This shows the the bandpowers at low ℓ have very little correlation between them, as we would expect, since for an all-sky survey the linear power is uncorrelated. At higher ℓ bandpower, the modes become more correlated, due to cross-talk between different scales due to nonlinear clustering in the matter power spectrum. The variations in this coefficient matrix indicate an error of around 10% which is suitable for the studies in this paper. This error can be reduced by introducing more realisations into the calculations. In our analysis we shall consider modes up to ℓ = 1500, where the correlation coefficient is around r ℓℓ ′ ≈ 0.6 . Figure 8 shows the χ 2 -distribution in the σ8-Ωm plane for our ensemble of simulations. The black lines represent the χ 2 two-parameter, 1, 2 and 3σ (which should contain 68.3%, 95.4% and 99.7% of the points assuming a bivariate Gaussian distribution), contours of parameter space for the cosmological parameters. However, this clearly is not a bivariate Gaussian distribution. The contours shown are representative and come from the simulation that had the best fit parameters that were closest to the true input parameters (the point shown by the red polygon). The blue triangles represent the best fit points for each of the 100 realisations. With this distribution, 68% of the points lie within the 1σ contour, 93% within the 2σ contour and 97% within the 3σ. The black diamond represents the best fit for the combined χ 2 estimate as discussed below.
The results from this analysis give us very encouraging results for the parameter estimation. Figure 9 shows the results of combining the likelihoods for all 100 realisations, as if we have one hundred independent 100 square degree surveys. Even for this test we see the maximum likelihood recovered parameter values lie within the 1 − σ confidence contour. The marginalised error on the measured parameters for the combined 100 surveys is ∆Ωm = 0.012 and ∆σ8 = 0.022, within expected errors. There is no significant bias in this result at the level of ∼ 0.02.
DISCUSSION AND CONCLUSIONS
This work introduces SUNGLASS -Simulated UNiverses for Gravitational Lensing Analysis and Shear Surveys. SUN-GLASS is a new, rapid pipeline that generates cosmological Figure 8. Gaussian likelihood estimate. The black contours come from the simulation with the closest fit to the true cosmological parameters. The blue triangles show the best fit cosmological parameters for the suite of lightcones. The true cosmological parameters are shown at the red polygon and the combined χ 2 best fit parameter is shown at the black diamond. Figure 9. Combined χ 2 likelihood. The black lines show the combined χ 2 1, 2 and 3σ contours. The blue triangle shows the best fit parameters for the combined χ 2 and the red star shows the true cosmological parameters.
N-body simulations with GADGET2. It computes weak lensing effects along a lightcone using line-of-sight integrations with no radial binning and the Born approximation to determine the convergence and shear at multiple source redshifts. This information is interpolated back on to the particles in the lightcone to generate mock shear catalogues in 3D for testing weak lensing observational analysis techniques.
In this work, SUNGLASS was used to generate 100 simulations with 512 3 particles, a box length of 512h −1 Mpc and a WMAP7 concordance cosmology. The corresponding mock shear catalogues were 100 sq degrees with a source redshift distribution with median zm = 0.82 and 15 galaxies per square arcminute. The parameters are easily changed within the SUNGLASS pipeline so that the mock shear catalogues matches the survey of interest.
To show the reliability of the lightcones generated with SUNGLASS, E-and B-mode power spectra were shown at multiple source redshifts. The results show that at low redshifts, the signal becomes dominated by shot-noise at reasonably low ℓ. With increasing source redshift, the power spectrum recovers the theoretical prediction over a wider range of modes, ℓ < 2500.
Given that the measured power spectrum of the simulations appears to follow the predicted shot noise at higher modes, the shot noise was subtracted from the power spectra to increase the recovered range. The theoretical prediction is expected to under predict the power spectrum around the turn over and consequently, the simulations could be recovering the power spectrum up to around ℓ = 5 × 10 4 at the highest redshift planes.
The multiple source redshift plane shear and convergence was interpolated onto the particles in the lightcone to generate a mock shear catalogue. A redshift sampling was also imposed on the lightcone to mimic an observed shear catalogue. Binning this distribution too finely resulted in empty bins which had the effect of suppressing the power spectrum. This has implications for observations where the number of objects per square arcminute should be taken into account, as well as the density of the binning, when determining the accuracy of the power spectrum.
The mock shear catalogues were used to determine a covariance matrix which is essential for both parameter estimation and data analysis. A strength of SUNGLASS is the ability to rapidly produce Monte Carlo realisations of these catalogues, ensuring independent mock data sets for the generation of the covariance matrices.
The mock catalogues were also used to perform a simple parameter estimation using Gaussian likelihood analysis. The distribution of power spectra were shown to be reasonably Gaussian and the resulting parameter estimation contours for a single realisation showed a good agreement with the input parameters within the 2-parameter 1,2 and 3σ error contours.
The combined likelihood from the 100 simulations shows narrow likelihood contours and accurate parameter recovery within the expected errors, with no evidence of significant bias at the level of ∼ 0.02. Current and future telescope surveys promise to provide an enormous amount of data for weak lensing analysis. Weak lensing is still a young field and analysis techniques are still being developed. It is essential that the strengths and weaknesses of these techniques are fully understood before using them on real data with unknown parameters. Using the simulations, lightcones and mock shear catalogues provided by the SUNGLASS pipeline, and demonstrated in this paper, is an excellent way to test these observational weak lensing analysis techniques. The outputs of this pipeline have been rigorously tested and are well understood, making them ideal for generating covariance matrices that are critical to many observational analysis techniques.
where part are the particles in the pixel with ri rs. Substituting this sum of delta-functions into equation (A1) yields the average convergence per pixel on the sky, p, with no radial binning; κp = 1 ∆Ω p d 2 θ κ = i k(ri, rs) ∆Ωpn(ri)r 2 i − rs 0 dr k(r, rs), (A5) where ∆Ωp = ∆θx∆θy.
Figure 2 .
2Convergence and shear map for a simulated survey of 100 square degrees with a single source redshift of z = 0.8. The colour-scale background shows the convergence while the white ticks show the shear signal.
Figure 3 .
3Simulated slices of the shear power spectra for N-body particle data at source redshifts of z = 0.3, 0.6, 0.8, 1.0, 1.3 and 1.5. The smooth (red) line shows the theoretical predictions, the straight diagonal (blue) line is the predicted shot-noise at each source redshift. The black points are the mean power spectrum of the simulated data for the 100 realisations with errors on the mean shown and the (light blue) curve under these points is the simulation data with the shot-noise subtracted. The sub-shot-noise (magenta) curve is the estimated induced B-mode. The lower panel shows the fractional percentage difference between the simulated shear power and the theoretical prediction with black points representing the simulated data and light blue points representing the shot-noise subtracted simulation data. c 0000 RAS, MNRAS 000, 1-??
Figure 4 .
4Left: The galaxy distribution, n(z), in the mock galaxy catalogue. The smooth (red) line shows the theoretical n(z) and the black histogram shows the distribution from a single simulation lightcone. The histogram shows the clustered nature of the lightcone. Right: The galaxy distribution in the mock galaxy catalogue with photometric redshift errors assigned to each galaxy. The structures visible in the true redshift lightcone have been smoothed out with the addition of the photo-z errors.
Figure 7 .
7Correlation coefficient matrix. This figure shows the correlation between the bandpower ℓ-modes in the covariance matrix. The higher ℓ bandpowers are strongly correlated (shown in reds), while the lower bandpowers are only weakly correlated (shown in blues).
8 http://www.icosmo.orgN Surveys
Area
n
z median
σ 0
zmax
100
100
15
0.82
0.05
1.5
Table 1. Table of mock weak lensing survey parameters used in
this paper.
c 0000 RAS, MNRAS 000, 1-??
ACKNOWLEDGMENTSAK acknowledges the support of the European DUEL RTN, project MRTN-CT-2006-036133. AK would also like to thank Benjamin Joachimi for the use of his power spectrum theory code as well as John Peacock, Richard Massey, Tom Kitching and Martin Kilbinger for their very helpful discussions on this work.APPENDIX A: DERIVATION OF THE LINE-OF-SIGHT CONVERGENCE WITH NO RADIAL BINNINGThis appendix shows how the line-of-sight convergence shown in equation 5 was derived.Start with the general equation for the convergence,where rs is the lensing source redshift, δ(r) is the fractional matter overdensity and K(r, rs) is the kernel K(r, rs) = (rs − r)r rsa(r)The overdensity δ(r) is given bywheren(r) is the average density at the comoving radial distance r and is constant in comoving co-ordinates.The particle number density, n(r), is given by a sum of 3D delta functions n(r) = i=part δ 3D (r−ri) = i δ 1D (r − ri) r 2 δ 2D (θ−θi), (A4)
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|
[] |
[
"Cosmic Ray Transport with Magnetic Focusing and the \"Telegraph\" model",
"Cosmic Ray Transport with Magnetic Focusing and the \"Telegraph\" model"
] |
[
"M A Malkov \nCASS and Department of Physics\nUniversity of California\nSan Diego, La Jolla92093CA\n",
"R Z Sagdeev \nUniversity of Maryland\n20742-3280College ParkMD\n"
] |
[
"CASS and Department of Physics\nUniversity of California\nSan Diego, La Jolla92093CA",
"University of Maryland\n20742-3280College ParkMD"
] |
[] |
Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation, whereby not all the particles propagate at realistic speeds, causes attempts to justify an alternative approach based on the "telegraph" equation. However, its derivations often lack rigor and transparency leading to incorrect results.The classic Chapman-Enskog method is applied to the pitch-angle averaged spatial CR transport. We show that the convective term arises only from the magnetic focusing effect and no "telegraph" (second order time derivative) term emerges in any order of the proper asymptotic expansion with systematically eliminated short time scales. However, this term may formally be converted from the fourth order hyper-diffusive term in the Chapman-Enskog expansion. But, within the method's validity range, it may only be important for a short relaxation period associated with either strong pitchangle anisotropy or spatial inhomogeneity of the initial CR distribution. However, in neither of these two cases a treatment based on merely the isotropic CR component is sufficient, regardless of the correction type (telegraph or hyper-diffusive). Moreover, in a long-time asymptotic regime these corrections are generally insignificant, as the angular anisotropy decays rapidly.
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10.1088/0004-637x/808/2/157
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[
"https://arxiv.org/pdf/1502.01799v2.pdf"
] | 531,781 |
1502.01799
|
1043b070bad789ae5d8fc00cbd22870a0647af4e
|
Cosmic Ray Transport with Magnetic Focusing and the "Telegraph" model
6 Feb 2015
M A Malkov
CASS and Department of Physics
University of California
San Diego, La Jolla92093CA
R Z Sagdeev
University of Maryland
20742-3280College ParkMD
Cosmic Ray Transport with Magnetic Focusing and the "Telegraph" model
6 Feb 2015Received ; accepted
Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation, whereby not all the particles propagate at realistic speeds, causes attempts to justify an alternative approach based on the "telegraph" equation. However, its derivations often lack rigor and transparency leading to incorrect results.The classic Chapman-Enskog method is applied to the pitch-angle averaged spatial CR transport. We show that the convective term arises only from the magnetic focusing effect and no "telegraph" (second order time derivative) term emerges in any order of the proper asymptotic expansion with systematically eliminated short time scales. However, this term may formally be converted from the fourth order hyper-diffusive term in the Chapman-Enskog expansion. But, within the method's validity range, it may only be important for a short relaxation period associated with either strong pitchangle anisotropy or spatial inhomogeneity of the initial CR distribution. However, in neither of these two cases a treatment based on merely the isotropic CR component is sufficient, regardless of the correction type (telegraph or hyper-diffusive). Moreover, in a long-time asymptotic regime these corrections are generally insignificant, as the angular anisotropy decays rapidly.
Preliminary Considerations
The problem addressed here is fundamental but not new to the cosmic ray (CR) transport studies. It can be formulated very plainly: How to describe CR transport by only their isotropic component, after the anisotropic one has been suppressed by scattering on magnetic irregularities? Suppose the angular distribution of CRs is given by the function f (µ,t, z) obeying an equation from which the rapid gyro-phase rotation is already removed (drift approximation, e.g., Vedenov et al. 1962;Kulsrud 2005)
∂ f ∂t + vµ ∂ f ∂ z = ∂ ∂ µ 1 − µ 2 D (µ) ∂ f ∂ µ .
(1)
Here z is the local coordinate along the ambient magnetic field, µ is the cosine of the particle pitch angle, and D is the pitch angle diffusion coefficient. Now, we make the next step in simplifying the transport description and seek an equation for the pitch-angle averaged distribution
f 0 (t, z) ≡ 1 2 1 −1 f (µ,t, z) dµ ≡ f .
The basic solution to this problem has been known for at least half a century (e.g., Jokipii 1966 and references therein). To the leading order in 1/D (assuming the characteristic scale and time of the problem being longer than particle mean free path and collision time) it can be obtained straightforwardly by averaging eq.(1)
∂ f 0 ∂t = − v 2 ∂ ∂ z 1 − µ 2 ∂ f ∂ µ ,
and substituting ∂ f /∂ µ ≪ f 0 from eq.(1) as:
∂ f ∂ µ ≈ − v 2D ∂ f 0 ∂ z .(2)
Thus, the following diffusion equation for f 0 results
∂ f 0 ∂t = v 2 4 ∂ ∂ z 1 − µ 2 D ∂ f 0 ∂ z .(3)
The only questionable step in this derivation was our neglect of ∂ f /∂t compared to v∂ f /∂ z in eq.(2). It is somewhat justified a posteriori by the small parameter1/D ≪ 1 in the final result given by eq.(3), making ∂ f /∂t hopefully small. On the other hand this is true for ∂ f 0 /∂t but not necessarily for ∂ f /∂t since it may contain also the rapidly decaying anisotropic partf = f − f 0 in the initial distribution. For Dt 1, however,f must die out and neglecting ∂ f /∂t appears plausible for the long-time evolution of CR distribution.
However convincing the justification, the CR diffusion model encounters the problem of a superluminal, or simply too fast particle propagation. Although rather common for diffusive models, the problem is not important if the number of such particles is small. There are cases, however, such as that of the propagation of ultra high-energy cosmic rays, where this problem must be addressed (Aloisio et al. 2009). Various attempts, starting as early as in 60s, e.g., (Axford 1965), have been made to devise a better transport equation for CRs. Unfortunately, in our view, they lack mathematical rigor and clarity and often lead to incorrect results.
In the most recent such treatment, due to Litvinenko & Schlickeiser (2013), a higher order in 1/D ≪ 1 term was included by retaining ∂ f /∂t, dropped in the simplest derivation above. This strategy gave rise to an additional ∂ 2 f 0 /∂t 2 -term in the "master" equation. This additional term transforms eq.(3) into a "telegraph" equation:
∂ f 0 ∂t − v 2 4 ∂ ∂ z κ ∂ f 0 ∂ z + τ ∂ 2 f 0 ∂t 2 = 0 (4) with κ = 1 − µ 2 D , τ = μ 0 dµ/D 2 /κ(5)
From a mathematical perspective, asymptotic reduction schemes should be continuable to higher orders. Proceeding to higher orders using the above approach will apparently generate an infinite series of higher time derivatives, thus introducing a series of shorter time scales into the master equation. Note that the goal should be the opposite, i.e., to eliminate short-time and transient phenomena in favor of the long-time evolution. At a minimum, the treatment of Litvinenko & Schlickeiser (2013) appears to encounter a truncation problem which is also present in the earlier work due to Earl (1973). The approach there was based on the decomposition of eq.(3) using the eigenfunctions of the angular operator on its r.h.s. This approach has been extended and described in detail by Schwadron & Gombosi (1994).
Leaving aside the validity of its derivation for a moment, the telegraph equation has both advantages and disadvantages compared to the plain diffusion equation introduced, e.g., by Jokipii (1966). One obvious advantage is that if the initial conditions make the second and the third term dominate over the first one (at least in the early phase of evolution), CRs must propagate nearly ballistically and if τ has a proper value, their speed may also be realistic. For example, this speed was derived in (Earl 1973) to be v/ √ 3. This is just the rms velocity projection of an isotropic CR distribution on z, which appears to be a potentially plausible solution to the problem of superluminal propagation.
The disadvantage of eq.(4) is that it is no longer an evolution equation and requires the time derivative ∂ t f 0 as an initial condition. Although this can be inferred from the angular distribution at t = 0 (if available) with recourse to the primary equation (1), the description of CR transport using eq.(4), by contrast with eq.(3), is not self-contained. We will also show below, that the transport coefficient τ in this equation, that was iteratively obtained from eq.(1) in (Litvinenko & Schlickeiser 2013) and by truncation of eigenfunction expansion in (Earl 1973), is not consistent with the regular asymptotic treatment and incorrect. The difference is due to the ∂ 4 f 0 /∂ z 4 -term, neglected in eq.(4) but contributing to the same order as the τ-term. Being converted into ∂ 2 f 0 /∂t 2 -form by using the leading order approximation given by eq.(3), it comes out with a markedly different coefficient τ. In addition, the τ-term in eq.(4) is sub-dominant compared to the other two terms for Dt 1, thus representing a transient process in the CR transport. Strictly speaking, it should be omitted in the asymptotic transport description along with the small hyper-diffusion term ∼ ∂ 4 f 0 /∂ z 4 , particularly if the term ∼ ∂ 3 f 0 /∂ z 3 does not vanish or when the magnetic focusing is present. These effects contribute to the CR transport at a lower order of approximation.
There are at least two reasons why we undertake a derivation of master equation to higher (fourth) order approximation using the Chapman-Enskog method. First, as we explained above, it is necessary to clarify the role of the telegraph τ-term entertained in the literature as an allegedly viable alternative to the standard diffusion model. Second, it is necessary to obtain the transport coefficients valid for arbitrary D (µ), that is for an arbitrary spectrum of the scattering magnetic fluctuations. In particular, the diffusion equation in eq.(3) supplemented by a convective term u (z) ∂ f 0 /∂ z for the case of the bulk fluid motion with velocity u, has been the main tool in studies of diffusive shock acceleration (DSA). In most DSA applications, it is crucial to allow for an arbitrary fluctuation spectrum and its dependence upon f 0 . This dependence directly affects the particle spectrum and acceleration time.
In the next section, the basic transport equation with magnetic focusing is introduced and the shortcomings of the reduction scheme based on direct iteration are demonstrated. The appropriate asymptotic method is elaborated in Sec.3. Apart from what we discussed above regarding the telegraph equation, the objective of Sec.3 is to create a framework suitable also for nonlinear (e.g., Ptuskin et al. 2008;Malkov et al. 2010b) and quasi-linear (Fujita et al. 2011;Malkov et al. 2013) versions of CR transport which are important for both the DSA and subsequent escape of CR. In these settings, the CR pressure is high enough to strongly modify at least the pitch-angle diffusion coefficient D and possibly the shock structure itself (Malkov et al. 2010b). In Sec.3.1 the implications of our results for the telegraph model are discussed and Sec.4 concludes the paper.
CR Transport Equation and its Asymptotic Reduction
Energetic particles (e.g., CRs) in a magnetic field, slowly varying on the particle gyro-scale, are transported according to the following gyro-phase averaged equation, e.g. (Vedenov et al. 1962;Jokipii 1966;Kulsrud 2005)
∂ f ∂t + vµ ∂ f ∂ z + v σ 2 1 − µ 2 ∂ f ∂ µ = ∂ ∂ µ νD (µ) 1 − µ 2 ∂ f ∂ µ(6)
Here v and µ are the particle velocity and pitch angle, z points in the local field direction, σ = −B −1 ∂ B/∂ z is the magnetic mirror inverse scale and ν is the (constant) pitch angle scattering rate, while D (µ) ∼ 1 depends on the spectrum of magnetic fluctuations. As the fastest transport is assumed to be in µ, we introduce the following small parameter
ε = v lν ≪ 1,
where l is the characteristic scale of the problem. It may be determined by B (z) variation, CR source, or propagation distance. By measuring time in ν −1 , z in l, and simply replacing σ l → σ ∼ 1, the above equation is transformed as follows
∂ f ∂t − ∂ ∂ µ D (µ) 1 − µ 2 ∂ f ∂ µ = −ε µ ∂ f ∂ z + σ 2 1 − µ 2 ∂ f ∂ µ(7)
A suitable scheme for asymptotic reduction of the above equation using ε ≪ 1 is due to Chapman and Enskog, suggested in development of the earlier ideas by Hilbert (a good discussion of the history of this method with mathematical details is given by Cercignani 1988). Originally, it was applied to Boltzmann equation in a strongly collisional regime. Similar approaches have been used in plasma physics, e.g., in regards to the hydrodynamic description of collisional magnetized plasmas (Braginskii 1965) and the problem of run-away electrons (Gurevich 1961;Kruskal & Bernstein 1964).
Regardless of the asymptotic scheme, eq. (7) suggests to seek f as a series in ε
f = f 0 + ε f 1 + ε 2 f 2 + . . . ≡ f 0 +f (8) where f = f 0 , with · = 1 2 1 −1 (·) dµ,(9)
so that˜ f = f n>0 = 0. The equation for f 0 , which is the main ("master") equation of the method, takes the following form
∂ f 0 ∂t = −ε ∂ ∂ z + σ µ f = − ε 2 2 ∂ ∂ z + σ ∞ ∑ n=1 ε n−1 1 − µ 2 ∂ f n ∂ µ (10)
We see from this equation that, similarly to the case of Lorentz's gas in an electric field (Gurevich 1961;Kruskal & Bernstein 1964), f 0 depends on the "slow time" t 2 = ε 2 t rather than on t. Indeed, the two problems are similar in that they describe a secular expansion of particles in phase space. The expansion occurs in z-direction for CR diffusion and in energy for runaway electrons. It is driven by a rapid isotropization in pitch angle plus the convection in z-direction, or acceleration in the electric field direction, for the CR transport and electron runaway, respectively.
The slow dependence of f 0 on time in eq.(10) may suggest to attribute the time derivative term in eq.(7) to a higher order approximation (thus moving it to the r.h.s.). Such ordering has been employed by Litvinenko & Schlickeiser (2013) and the term ∝ ∂ 2 f 0 /∂t 2 has been produced in eq.(10). Note, that a series of terms with progressively higher time derivatives, multiplied by small parameters, would have been generated on the r.h.s of eq.(10), had this scheme been continued to higher orders. These terms would be responsible for the initial transport relaxation occurring in progressively shorter times. They would be associated with small scales in the initial angular distribution. These transient phenomena will be removed using the asymptotic reduction scheme in the next section.
Unlike f 0 ,f in eq.(8) does depend on t as on a "fast" time. Therefore, it is illegitimate to attribute the first term on the l.h.s of eq.(7) to any order of approximation different from that of the second term, notwithstanding its fast decay for t 1. Thus, using eq.(7) we must apply the following ordering
∂ f n ∂t − ∂ ∂ µ D (µ) 1 − µ 2 ∂ f n ∂ µ = −µ ∂ f n−1 ∂ z − σ 2 1 − µ 2 ∂ f n−1 ∂ µ(11)
The above expansion scheme is sufficient to recover the leading order of f 0 evolution from eq.(10) by substituting there ∂ f 1 /∂ µ ≈ − (2D) −1 ∂ f 0 /∂ z, obtained from the last equation for t 1. However, this scheme is not suitable for determining f n for n ≥ 2 to submit to eq.(10). Indeed, as it may be seen from eq.(11), the solubility condition for f 2 at t ≫ 1 is
(∂ /∂ z + σ ) 1 − µ 2 ∂ f 1 /∂ µ ≈ − (∂ /∂ z + σ ) 1 − µ 2 /2D ∂ f 0 /∂ z = 0
. This is clearly too strong a restriction. The reason for this inconsistency of the direct asymptotic expansion is that f 0 depends on the slow time t 2 = ε 2 t, so that the multiple time scales are involved in the problem. The Chapman-Enskog method is known to be the right way to treat this situation, on which we elaborate in the next section.
Chapman-Enskog Expansion
As we have seen, the reduction of the original CR propagation problem given by eq.(7) to its isotropic part requires a multi-time asymptotic expansion. In the classic Chapman-Enskog method the operator ∂ /∂t is expanded instead. Perhaps more customary today and equivalently is to introduce a hierarchy of formally independent time variables (e.g., Nayfeh 1981) t → t 0 , t 1 , . . ., so that
∂ ∂t = ∂ ∂t 0 + ε ∂ ∂t 1 + ε 2 ∂ ∂t 2 . . .(12)
Instead of eq.(11), from eq. (7) we have
∂ f n ∂t 0 − ∂ ∂ µ D (µ) 1 − µ 2 ∂ f n ∂ µ = −µ ∂ f n−1 ∂ z − σ 2 1 − µ 2 ∂ f n−1 ∂ µ − n ∑ k=1 ∂ f n−k ∂t k (13) ≡ L n−1 [ f ] (t 0 , . . .,t n ; µ, z)
where the conditions f n<0 = 0 are implied. The solution of this equation should be sought in the following form
f n = f n (t 2 ,t 3 , . . . ; µ) +f n (t 0 ,t 1 , . . .; µ)
wheref n and f n satisfy, respectively, the following two equations:
∂f n ∂t 0 − ∂ ∂ µ D (µ) 1 − µ 2 ∂f n ∂ µ = L n−1 f (t 0 , . . . ,t n ; µ, z)(15)
and
− ∂ ∂ µ D (µ) 1 − µ 2 ∂f n ∂ µ = L n−1 f (t 2 , . . . ,t n ; µ, z)(16)
The solution forf n takes the formf
n = ∞ ∑ k=1 C (n) k (t)e −λ k t 0 ψ k (µ)(17)
and it can be easily found for an arbitrary n by expanding both sides of eq.(15) in a series of eigenfunctions of the diffusion operator on its l.h.s.:
− ∂ ∂ µ D (µ) 1 − µ 2 ∂ ψ k ∂ µ = λ k ψ k ,
For D = 1, for example, ψ k are the Legendre polynomials with λ k = k (k + 1), k = 0, 1, . . .. The time dependent coefficients C (n) k are determined by the initial values off n (anisotropic part of the initial CR distribution) and the r.h.s. of eq.(15), that depends onf n−1 , obtained at the preceding step. It is seen, however, that allf n exponentially decay in time for t 1 and we may ignore them 1 as we are interested in evolving the system over times t ε −2 ≫ 1 and longer. Starting from n = 0 and using eq.(13), for the slowly varying part of f we have
∂ f 0 ∂t 0 = 0.(18)
The solubility condition for f 1 (obtained by integrating both sides of eq.[13] in µ) also gives a trivial result
∂ f 0 ∂t 1 = 0,(19)
so the last two conditions make the decomposition in eq.(14) consistent with eqs. (18-19), since from eq.(16) with n = 1 we havef
1 = − 1 2 W ∂ f 0 ∂ z(20)
and, thus bothf 0 andf 1 are, indeed, independent of t 0 and t 1 . We have introduced the function W (µ) here by the following two relations
∂W ∂ µ = 1 D , W = 0.(21)
The solubility condition for f 2 yields the nontrivial and well-known (e.g., Jokipii 1966) result, which is actually the leading term of the ∂ f 0 /∂t expansion in ε ≪ 1
∂ f 0 ∂t 2 = 1 4 ∂ ∂ z + σ κ ∂ f 0 ∂ z ,(22)where κ = 1 − µ 2 D .
The solubility conditions for f 3 , f 4 , ... will generate the higher order terms of our expansion which, after some algebra, can be manipulated into the following expressions for the third and fourth orders of approximation
∂ f 0 ∂t 3 = − 1 4 ∂ ∂ z + σ ∂ ∂ z + σ 2 µW 2 ∂ f 0 ∂ z (23) ∂ f 0 ∂t 4 = 1 8 ∂ ∂ z + σ × ∂ ∂ z + σ 2 2 W 2 U ′ − κ + 1 2 ∂ ∂ z + σ ∂ ∂ z [κ (1 − µ) +U ] 2 D (1 − µ 2 ) ∂ f 0 ∂ z .(24)
We have denoted
U ≡ μ 1 1 − µ 2 D dµ,
and U ′ = ∂U/∂ µ. The pitch-angle diffusion coefficient D (µ) and magnetic focusing σ are considered z-independent for simplicity, a limitation that can easily be relaxed by re-arranging the operators containing ∂ /∂ z in eq.(24). We can proceed to the higher orders of approximation ad infinitum since terms containing 1 − µ 2 ∂ f n /∂ µ can be expressed through f n−1 , f n−2 , .... According to eqs. (18-19), of interest is the evolution of f 0 on the time scales t 2 1 or t ε −2 so, as we already mentioned, the contributions off n (µ) to all the solubility conditions, similar to those given by eqs. (22-24), have to be dropped (as they become exponentially small) and onlyf n (µ)contributions should be retained. Using eqs. (18)(19)(22)(23)(24) to form the combinations ε n ∂ n f 0 /∂t n and summing up both sides, on the l.h.s. of the resulting equation we simply obtain ∂ f 0 /∂t (see eq.
[12]). Therefore, the evolution of f 0 up to the fourth order in ε takes the following form
∂ f 0 ∂t = ε 2 4 ∂ ′ z κ − ε∂ ′′ z µW 2 − ε 2 2 ∂ ′′ z 2 W 2 κ −U ′ − 1 2 ∂ ′ z ∂ z [κ (1 − µ) +U ] 2 D (1 − µ 2 ) ∂ f 0 ∂ z (25)
where ∂ ′ z = ∂ z + σ and ∂ ′′ z = ∂ z + σ /2. The above algorithm allows one to obtain the master equation to an arbitrary order in ε. By construction, in no order of approximation will higher time derivatives emerge, as has been intentionally devised in the Chapman-Enskog method. We have truncated this process at the fourth order, ε 4 . As we show in the next subsection, this is the lowest order required to clarify the origin of the telegraph equation. Higher order terms can in principle be calculated at the expense of a more involved algebra. We argue below that such calculations would be of no avail.
Telegraph Term and Arguments against it
By contrast to the telegraph equation given by eqs.(4-5), that has been derived with no order control, eq.(25) is derived to the fourth order in ε. Yet, it has no second order time derivative and this challenges the derivation of eq.(4). Various versions of the telegraph equation have been obtained either without clear ordering, e.g., (Litvinenko & Schlickeiser 2013), or using a specific and simple D (µ), e.g. (Schwadron & Gombosi 1994). In an earlier treatment by (Earl 1973), an eigenfunction expansion was truncated, also with no order control. In most of these treatments care has not been exercised to systematically eliminate the short time scales which are irrelevant to the long-time evolution of the isotropic part of the CR distribution. In principle, this is acceptable if the reduction scheme is based on an exact solution of the original equation, to include all required orders of approximation into the master equation. Such approach appears to be pursued in (Schwadron & Gombosi 1994). The drawback here is a strongly limited choice of possible D (µ) and its f 0 -independence (see a brief discussion in Sec.4).
To understand the differences between the telegraph eq.(4) and eq.(25) obtained using the Chapman-Enskog multi-time decomposition, we simplify eq.(25) to remove noncritical terms. First, the telegraph coefficient τ does not contain magnetic focusing effect, so we may set σ = 0 in eq.(25) for the purpose of this section. Second, τ does not vanish in the case of symmetric pitch-angle scattering coefficient, that is D (−µ) = D (µ). This particular choice removes the term ∼ ε 3 containing ∂ 3 z in eq.(25). It is not included in the telegraph equation derived for magnetic focusing by Litvinenko & Schlickeiser (2013), anyway. Using these simplifications and the slow time T = ε 2 t/4, eq.(25) rewrites:
∂ f 0 ∂ T = κ ∂ 2 f 0 ∂ z 2 − ε 2 K ∂ 4 f 0 ∂ z 4(26)
where K is the hyper-diffusion coefficient
K = 1 2 W 2 κ −U ′ − 1 2 [κ (1 − µ) +U ] 2 D (1 − µ 2 )(27)
To the same order in ε ≪ 1, the last equation can indeed be rewritten in form of a telegraph equation, cf. eq.(4):
∂ f 0 ∂ T = κ ∂ 2 f 0 ∂ z 2 − ε 2 K κ 2 ∂ 2 f 0 ∂ T 2 .(28)
However, the comparison of eqs.(26-27) with eqs.
(4-5) shows that, apart from the ε 2 factor, originating from normalization, the telegraph coefficient in eq.(28) differs from the respective coefficient τ given in eq.(5). The reason is that eq.(4) has been obtained in, e.g., (Litvinenko & Schlickeiser 2013) by iterations not including all the fourth order terms which contribute to the telegraph term. A technically rigorous treatment by Schwadron & Gombosi (1994) gives an expression for τ, likely to be consistent with our result here but, as it was obtained for a power-law D (µ), a side by side comparison would require further steps, not worth taking here.
More importantly, the telegraph term in eq.(28) belongs to the type of terms with small parameter at highest derivative. The role of such terms is well known from other multi-scale problems, most notably the boundary layer problems. These terms correspond to corrections that are insignificant in a smooth part of the solution but become crucial near and inside boundary layers, thus determining their structure and scale. In the context of the telegraph equation, the boundary layer translates into the initial relaxation phase of the CR distribution 2 . However, as we have seen in Sec.3, this relaxation is primarily associated with the CR anisotropy, that is with f n , which we have shown to decay rapidly during the initial phase at t 1. This is exactly the characteristic time scale (T ∼ ε 2 ) when the telegraph term (second) on the r.h.s. of eq.(28) may come into play. To realize such scenario, one must start with an initial condition that either leads to a subsequent rapid relaxation in time or is strongly inhomogeneous. These two options become obvious after returning to the original time variable in eq.(28):
1 + 4 K κ 2 ∂ ∂t ∂ f 0 ∂t = ε 2 4 κ ∂ 2 f 0 ∂ z 2 .(29)
Namely, in the limit ε → 0 there are two modes, of which the first being f 0 = f 0 (z). This is the main diffusion mode that slowly evolves in time when 0 < ε ≪ 1, and, as we are interested in the evolution over the time scales t ε −2 , the telegraph term becomes sub-dominant and can be discarded altogether. The second mode corresponds to a rapid decay of the initial distribution ∼ exp (−γt), at the rate γ = κ 2 /4K ∼ 1. It is associated with the decay of a strong initial anisotropy. The latter process cannot be adequately described by the equation for f 0 alone, and the telegraph term must be understood as a "ghost" term reflecting a rapid decay of an anisotropic part of the initial distribution. It follows that the rapidly changing partf in the decomposition in eq.(14) needs to be retained in the short-time analysis on an equal basis with f 0 .
The next possibility to make the telegraph term work is to start with a highly inhomogeneous initial distribution, so that the r.h.s. of eq.(29) is O (1) despite of small ε. The telegraph term becomes temporarily (t K/κ 2 ) dominant and the initial distribution indeed propagates ballistically rather than diffusively away from the CR source with the telegraph mode speeds, C = ±εκ 3/2 /4 √ K. However, the scale of initial profile must be shorter than the CR m.f.p. which, again, invalidates the equation in use. Besides, this case is not much different from the previous one as the sharp spatial inhomogeneity almost automatically results in strong angular anisotropy of the CR distribution. Therefore, the telegraph regime has virtually no validity range.
We conclude this section by adding yet another argument in disfavor of the telegraph equation. A consistent asymptotic reduction method must be continuable to infinity in powers of small ε. The Chapman-Enskog scheme clearly is. The outcome will be a series of terms ∼ ε n ∂ n z f 0 on the r.h.s. of eq.(25). To solve the resulting equation, only the initial distribution f 0 (0, z) is needed, as the equation remains evolutionary. Turning to the "telegraph" version of this equation we realize that progressively higher time derivatives, with higher powers of ε, will comprise the asymptotic series. The resulting equation will not be evolutionary and an arbitrarily large set of initial time derivatives of f 0 would then be needed to solve the initial value problem. These data can be extracted only from the full anisotropic distribution with recourse to the full (anisotropic) equation. Therefore, the telegraph equation is also not self-contained and unrealistic for practical use.
Summary and Conclusions
Using the Chapman-Enskog method, we have extended the CR diffusion equation with magnetic focusing to the fourth order in a small parameter ε = λ /l (CR mean free path to the problem scale). This clarifies the nature of the so called telegraph transport regime, widely publicized in the literature as a promising alternative to the diffusive CR propagation models. We have shown that the telegraph-term extension of the diffusion equation can be mapped from the hyper-diffusive term of the Chapman-Enskog expansion, although with a significantly different coefficient τ (cf. eqs.[4-5] with eqs. [27][28][29]). In addition, the telegraph equation (even with corrected coefficient τ), by contrast to the original Chapman-Enskog equation, is not self-contained and requires an initial condition for also the anisotropic part of the CR distribution. Furthermore, an attempt to proceed to higher orders in ε introduces progressively shorter time scales associated with the "ghost" terms reflecting a quick relaxation of the initial anisotropy or strong spatial inhomogeneity. Conversely, the classic Chapman-Enskog method is devised to eliminate the short time scales irrelevant to the long-term asymptotic evolution of the CR distribution. In particular, should an unacceptably short scale be present in the initial condition, the hyper-diffusive Chapman-Enskog term will quickly erase it, thus making the further evolution consistent with the asymptotic method used.
Apart from resolving the telegraph equation controversy, we have derived the CR transport equation for an arbitrary D (µ) in combination with magnetic focusing effect. This form of the transport equation, (25), is suitable for describing CR transport and acceleration problems where the phenomenon of self-confinement (D is a functional of f , D = D [ f ; µ,t]) is critical, e.g. (Ptuskin et al. 2008;Malkov et al. 2010bMalkov et al. , 2013Fujita et al. 2011). An accounting for the magnetic focusing effect is required for describing particle acceleration in CR-modified shocks with oblique magnetic fields, since the field increases towards such shocks due to the pressure exerted by the accelerated CRs (Malkov & Drury 2001), thus producing a mirror effect.
In conclusion, by comparison even with corrected telegraph equation, the Chapman-Enskog expansion is a considerably more useful and flexible tool to describe the long-time CR propagation. Efforts on improving the CR diffusion models, where their drawbacks are important, need to address the lower level transport, including anisotropic component of the CR distribution, directly. Recent treatments of this kind can be found in, e.g., (Aloisio et al. 2009;Malkov et al. 2010a). When the diffusive treatment is well within the method's validity range (weakly anisotropic spatially smooth CR distributions) neither the telegraph nor the hyper-diffusive term (∼ ε 4 ) is essential to the CR transport and can be omitted. Of interest are the magnetic mirror and ε 3 terms as they capture convective transport of the CRs.
The work of MM was supported by the NASA ATP-program under the Grant NNX14AH36G. He is also indebted to the University of Maryland for hospitality and partial support during this work. Partial support from US DoE is also appreciated.
In fact we must do so, as our asymptotic method has a power accuracy in ε ≪ 1, but not the exponential accuracy.
The possibility of internal boundary layer (front propagation) can be disregarded as long as the problem remains linear, but becomes important when D = D ( f 0 ), e.g.,(Malkov et al. 2010b).
. R Aloisio, V Berezinsky, A Gazizov, Astrophys. J. 6931275Aloisio, R., Berezinsky, V., & Gazizov, A. 2009, Astrophys. J. , 693, 1275
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S I Braginskii, Transport Processes in a Plasma. 1205Braginskii, S. I. 1965, Transport Processes in a Plasma, Vol. 1, 205
. C Cercignani, BoltzmannEquation, Applied Mathematical Sciences Series. Springer-VerlagCercignani, C. 1988, BoltzmannEquation, Applied Mathematical Sciences Series (Springer- Verlag)
. J A Earl, Astrophys. J. 180227Earl, J. A. 1973, Astrophys. J. , 180, 227
. Y Fujita, F Takahara, Y Ohira, K Iwasaki, Mon. Not. R. Astron. Soc. 4153434Fujita, Y., Takahara, F., Ohira, Y., & Iwasaki, K. 2011, Mon. Not. R. Astron. Soc. , 415, 3434
. A V Gurevich, Soviet Journal of Experimental and Theoretical Physics. 12904Gurevich, A. V. 1961, Soviet Journal of Experimental and Theoretical Physics, 12, 904
. J R Jokipii, Astrophys. J. 146480Jokipii, J. R. 1966, Astrophys. J. , 146, 480
. M D Kruskal, I B Bernstein, Physics of Fluids. 7407Kruskal, M. D., & Bernstein, I. B. 1964, Physics of Fluids, 7, 407
R M Kulsrud, Plasma physics for astrophysics. Kulsrud, R. M. 2005, Plasma physics for astrophysics
. Y E Litvinenko, R Schlickeiser, Astronomy and Astrophys. 55459Litvinenko, Y. E., & Schlickeiser, R. 2013, Astronomy and Astrophys., 554, A59
. M A Malkov, P H Diamond, L O'c. Drury, R Z Sagdeev, Astrophys. J. 721750Malkov, M. A., Diamond, P. H., O'C. Drury, L., & Sagdeev, R. Z. 2010a, Astrophys. J. , 721, 750
. M A Malkov, P H Diamond, R Z Sagdeev, Plasma Physics and Controlled Fusion. 52124006Malkov, M. A., Diamond, P. H., & Sagdeev, R. Z. 2010b, Plasma Physics and Controlled Fusion, 52, 124006
. M A Malkov, P H Diamond, R Z Sagdeev, F A Aharonian, I V Moskalenko, Astrophys. J. 76873Malkov, M. A., Diamond, P. H., Sagdeev, R. Z., Aharonian, F. A., & Moskalenko, I. V. 2013, Astrophys. J. , 768, 73
. M A Malkov, L O Drury, Reports on Progress in Physics. 64429Malkov, M. A., & Drury, L. O. 2001, Reports on Progress in Physics, 64, 429
Introduction to perturbation techniques. A H Nayfeh, Nayfeh, A. H. 1981, Introduction to perturbation techniques
. V S Ptuskin, V N Zirakashvili, A A Plesser, Advances in Space Research. 42486Ptuskin, V. S., Zirakashvili, V. N., & Plesser, A. A. 2008, Advances in Space Research, 42, 486
. N A Schwadron, T I Gombosi, J. Geophys. Res. 9919301Schwadron, N. A., & Gombosi, T. I. 1994, J. Geophys. Res. , 99, 19301
. A A Vedenov, E P Velikhov, R Z Sagdeev, NUCLEAR FUSION. 465Vedenov, A. A., Velikhov, E. P., & Sagdeev, R. Z. 1962, NUCLEAR FUSION, 465
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[] |
[
"IDEALS IN THE TEMPERLEY-LIEB CATEGORY",
"IDEALS IN THE TEMPERLEY-LIEB CATEGORY"
] |
[
"Frederick M Goodman ",
"Hans Wenzl "
] |
[] |
[] |
This note will appear as an appendix to the paper of Michael Freedman, A magnetic model with a possible Chern-Simons phase [F]; it may, however, be read independently of [F]. The purpose of this paper is to prove the following result.Theorem 0.1. For a generic value of the parameter, the Temperley-Lieb category has no non-zero, proper tensor ideal. When the parameter d is equal to 2 cos(π/n) for some n ≥ 3, then the Temperley-Lieb category has exactly one non-zero, proper ideal, namely the ideal of negligible morphisms.Our notation in the appendix differs slightly from that in [F]. We write t instead of −A 2 , T n for the Temperley-Lieb algebra with n strands, and T L for the Temperley-Lieb category. We trust that this notational variance will not cause the reader any difficulty.1. The Temperley-Lieb Category 1.1. The Generic Temperley Lieb Category. Let t be an indeterminant over C, and let d = (t + t −1 ). The generic Temperley Lieb category TL is a strict tensor categor whose objects are elements of N 0 = {0, 1, 2, . . . }. The set of morphisms Hom(m, n) from m to n is a C(t) vector space described as follows:If n − m is odd, then Hom(m, n) is the zero vector space. For n − m even, we first define (m, n)-TL diagrams, consisting of:1. a closed rectangle R in the plane with two opposite edges designated as top and bottom. 2. m marked points (vertices) on the top edge and n marked points on the bottom edges. 3. (n + m)/2 smooth curves (or "strands") in R such that for each curve γ, ∂γ = γ ∩ ∂R consists of two of the n + m marked points, and such that the curves are pairwise non-intersecting. Two such diagrams are equivalent if they induce the same pairing of the n + m marked points. Hom(m, n) is defined to be the C(t) vector space with Date: written April, 2001. 2000 Mathematics Subject Classification. 17B37, 05E10. We are grateful to Michael Freedman for bringing the question of tensor ideals in the Temperley-Lieb category to our attention and for allowing us to present the proof as an appendix to his paper.
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"https://export.arxiv.org/pdf/math/0206301v1.pdf"
] | 14,902,191 |
math/0206301
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4f453e28c2699f575f851d6d05c5d9481aa4bde2
|
IDEALS IN THE TEMPERLEY-LIEB CATEGORY
27 Jun 2002
Frederick M Goodman
Hans Wenzl
IDEALS IN THE TEMPERLEY-LIEB CATEGORY
27 Jun 2002
This note will appear as an appendix to the paper of Michael Freedman, A magnetic model with a possible Chern-Simons phase [F]; it may, however, be read independently of [F]. The purpose of this paper is to prove the following result.Theorem 0.1. For a generic value of the parameter, the Temperley-Lieb category has no non-zero, proper tensor ideal. When the parameter d is equal to 2 cos(π/n) for some n ≥ 3, then the Temperley-Lieb category has exactly one non-zero, proper ideal, namely the ideal of negligible morphisms.Our notation in the appendix differs slightly from that in [F]. We write t instead of −A 2 , T n for the Temperley-Lieb algebra with n strands, and T L for the Temperley-Lieb category. We trust that this notational variance will not cause the reader any difficulty.1. The Temperley-Lieb Category 1.1. The Generic Temperley Lieb Category. Let t be an indeterminant over C, and let d = (t + t −1 ). The generic Temperley Lieb category TL is a strict tensor categor whose objects are elements of N 0 = {0, 1, 2, . . . }. The set of morphisms Hom(m, n) from m to n is a C(t) vector space described as follows:If n − m is odd, then Hom(m, n) is the zero vector space. For n − m even, we first define (m, n)-TL diagrams, consisting of:1. a closed rectangle R in the plane with two opposite edges designated as top and bottom. 2. m marked points (vertices) on the top edge and n marked points on the bottom edges. 3. (n + m)/2 smooth curves (or "strands") in R such that for each curve γ, ∂γ = γ ∩ ∂R consists of two of the n + m marked points, and such that the curves are pairwise non-intersecting. Two such diagrams are equivalent if they induce the same pairing of the n + m marked points. Hom(m, n) is defined to be the C(t) vector space with Date: written April, 2001. 2000 Mathematics Subject Classification. 17B37, 05E10. We are grateful to Michael Freedman for bringing the question of tensor ideals in the Temperley-Lieb category to our attention and for allowing us to present the proof as an appendix to his paper.
The Temperley-Lieb Category
1.1. The Generic Temperley Lieb Category. Let t be an indeterminant over C, and let d = (t + t −1 ). The generic Temperley Lieb category TL is a strict tensor categor whose objects are elements of N 0 = {0, 1, 2, . . . }. The set of morphisms Hom(m, n) from m to n is a C(t) vector space described as follows:
If n − m is odd, then Hom(m, n) is the zero vector space. For n − m even, we first define (m, n)-TL diagrams, consisting of:
1. a closed rectangle R in the plane with two opposite edges designated as top and bottom. 2. m marked points (vertices) on the top edge and n marked points on the bottom edges. 3. (n + m)/2 smooth curves (or "strands") in R such that for each curve γ, ∂γ = γ ∩ ∂R consists of two of the n + m marked points, and such that the curves are pairwise non-intersecting. Two such diagrams are equivalent if they induce the same pairing of the n + m marked points. Hom(m, n) is defined to be the C(t) vector space with Date: written April, 2001. 2000 Mathematics Subject Classification. 17B37, 05E10. We are grateful to Michael Freedman for bringing the question of tensor ideals in the Temperley-Lieb category to our attention and for allowing us to present the proof as an appendix to his paper. basis the set of equivalence classes of (m, n)-TL diagrams; we will refer to equivalence classes of diagrams simply as diagrams.
The composition of morphisms is defined first on the level of diagrams. The composition ba of an (m, n)-diagram b and an (ℓ, m)-diagram a is defined by the following steps:
1. Juxtapose the rectangles of a and b, identifying the bottom edge of a (with its m marked points) with the top edge of b (with its m marked points). 2. Remove from the resulting rectangle any closed loops in its interior.
The result is a (n, ℓ)-diagram c. 3. The product ba is d r c, where r is the number of closed loops removed. The composition product evidently respects equivalence of diagrams, and extends uniquely to a bilinear product Hom(m, n) × Hom(ℓ, m) −→ Hom(ℓ, n), hence to a linear map Hom(m, n) ⊗ Hom(ℓ, m) −→ Hom(ℓ, n).
The tensor product of objects in TL is given by n⊗n ′ = n+n ′ . The tensor product of morphisms is defined by horizontal juxtposition. More exactly, The tensor a ⊗ b of an (n, m)-TL diagram a and an (n ′ , m ′ )-diagram b is defined by horizontal juxtposition of the diagrams, the result being an (n + n ′ , m + m ′ )-TL diagram.
The tensor product extends uniquely to a bilinear product
Hom(m, n) × Hom(m ′ , n ′ ) −→ Hom(m + m ′ , n + n ′ ),
hence to a linear map Hom(m, n) ⊗ Hom(m ′ , n ′ ) −→ Hom(m + m ′ , n + n ′ ).
For each n ∈ N 0 , T n := End(n) is a C(t)-algebra, with the composition product. The identity 1 n of T (n) is the diagram with n vertical (noncrossing) strands. We have canonical embeddings of T n into T n+m given by
x → x ⊗ 1 m . If m > n with m − n even, there also exist obvious embeddings of Hom(n, m) and Hom(m, n) into T m as follows: If ∩ and ∪ denote the morphisms in Hom(0, 2) and Hom(2, 0), then we have linear embeddings
a ∈ Hom(n, m) → a ⊗ ∪ ⊗(m−n)/2 ∈ T m and b ∈ Hom(m, n) → b ⊗ ∩ ⊗(m−n)/2 ∈ T m .
Note that these maps have left inverses which are given by premultiplication by an element of Hom(n, m) in the first case, and postmultiplication by an element of Hom(m, n) in the second. Namely,
a = d −(m−n)/2 (a ⊗ ∪ ⊗(m−n)/2 ) • (1 n ⊗ ∩ ⊗(m−n)/2 ) and b = d −(m−n)/2 (1 n ⊗ ∪ ⊗(m−n)/2 ) • (b ⊗ ∩ ⊗(m−n)
/2 ) By an ideal J in TL we shall mean a vector subspace of n,m Hom(n, m) which is closed under composition and tensor product with arbitrary morphisms. That is, if a, b are composible morphisms, and one of them is in J, then the composition ab is in J; and if a, b are any morphisms, and one of them is in J, then the tensor product a ⊗ b is in J.
Note that any ideal is closed under the embeddings described just above, and under their left inverses.
1.2. Specializations and evaluable morphisms. For any τ ∈ C, we define the specialization TL(τ ) of the Temperley Lieb category at τ , which is obtained by replacing the indeterminant t by τ . More exactly, the objects of TL(τ ) are again elements of N 0 , the set of morphisms Hom(m, n)(τ ) is the Cvector space with basis the set of (m, n)-TL diagrams, and the composition rule is as before, except that d is replaced by d(τ ) = (τ + τ −1 ). Tensor products are defined as before. T n (τ ) := End(n) is a complex algebra, and x → x ⊗ 1 m defines a canonical embedding of T n (τ ) into T n+m (τ ). One also has embeddings Hom(m, n) → T n and Hom(n, m) → T n , when m < n, as before. An ideal in TL(τ ) again means a subspace of n,m Hom(n, m) which is closed under composition and tensor product with arbitrary morphisms.
Let C(t) τ be the ring of rational functions without pole at τ . The set of evaluable morphisms in Hom(m, n) is the C(t) τ -span of the basis of (n, m)-TL diagrams. Note that the composition and tensor product of evaluable morphisms are evaluable. We have an evaluation map from the set of evaluable morphisms to morphisms of TL(τ ) defined by
a = s j (t)a j → a(τ ) = s j (τ )a j ,
where the s j are in C(t) τ , and the a j are TL-diagrams. We write x → x(τ ) for the evaluation map. The evaluation map is a homomorphism for the composition and tensor products. In particular, one has a C-algebra homomorphism from the algebra T τ n of evaluable endomorphisms of n to the algebra T n (τ ) of endomorphisms of n in TL(τ ).
The principle of constancy of dimension is an important tool for analyzing the specialized categories TL(τ ). We state it in the form which we need here:
Proposition 1.1. Let e ∈ T n and f ∈ T m be evaluable idempotents in the generic Temperley Lieb category. Let A be the C(t)-span in Hom(m, n) of a certain set of (m, n)-TL diagrams, and let A(τ ) be the C-span in Hom(m, n)(τ ) of the same set of diagrams. Then
dim C(t) eAf = dim C e(τ )A(τ )f (τ ). Proof. Let X denote the set of TL diagrams spanning A. Clearly dim C(t) A = dim C A(τ ) = |X|. Choose a basis of e(τ )A(τ )f (τ ) of the form {e(τ )xf (τ ) : x ∈ X 0 }, where X 0 is some subset of X. If the set {exf :
x ∈ X 0 }, were linearly dependent over C(t), then it would be linearly dependent over C [t], and evaluating at τ would give a linear dependence of
{e(τ )xf (τ ) : x ∈ X 0 } over C. It follows that dim C(t) eAf ≥ dim C e(τ )A(τ )f (τ )
. But one has similar inequalities with e replaced by 1 − e and/or f replaced by 1 − f . If any of the inequalities were strict, then adding them would give dim C(t) A > dim C A(τ ), a contradiction.
1.3. The Markov trace. The Markov trace Tr = Tr n is defined on T n (or on T n (τ )) by the following picture, which represents an element in End 0 ∼ = C(t) (resp. End(0) ∼ = C). a = Tr(a) ∈ End(0) On an (n, n)-TL diagram a ∈ T n , the trace is evaluated geometrically by closing up the diagram as in the figure, and counting the number c(a) of components (closed loops); then Tr(a) = d c(a) .
It will be useful to give the following inductive description of closing up a diagram. We define a map ε n : T n+1 → T n (known as a conditional expectation in operator algebras) by only closing up the last strand; algebraically it can be defined by
a ∈ T n+1 → (1 n ⊗ ∪) • (a ⊗ 1) • (1 n ⊗ ∩).
If k > n, the map ε n,k is defined by ε n,k = ε n • ε n+1 ... • ε k−1 . It follows from the definitions that Tr(a) = ε 0,n for a ∈ T n .
It is well-known that Tr is indeed a functional satisfying Tr(ab) = Tr(ba); one easily checks that this equality is even true if a ∈ Hom(n, m) and b ∈ Hom(m, n). We need the following well-known fact:
Lemma 1.2. Let f ∈ T n+m and let p ∈ T n such that (p ⊗ 1 m )f (p ⊗ 1 m ) = f , where p is a minimal idempotent in T n . Then ε n,n+m (f ) = γp, where γ = T r n+m (f )/T r n (p)
Proof. It follows from the definitions that
pε n,n+m (f )p = ε n,n+m ((p ⊗ 1 m )f (p ⊗ 1 m )) = ε n,n+m (f ).
As p is a minimal idempotent in T n , ε n,n+m (f ) = γp, for some scalar γ. Moreover, by our definition of trace, we have T r n+m (f ) = T r n (ε n,n+m (f )) = γT r n (p). This determines the value of γ.
The negligible morphisms Neg(n, m) are defined to be all elements a ∈ Hom(n, m) for which T r(ab) = 0 for all b ∈ Hom(m, n). It is well-known that the negligible morphisms form an ideal in TL. All Young diagrams in this paper will have at most two rows. For λ a Young diagram with n boxes, a Young tableau of shape λ is a filling of λ with the numbers 1 through n so that the numbers increase in each row and column. The number of Young tableax of shape λ is denoted by f λ . The generic Temperley Lieb algebras T n are known ( [J]) to decompose as direct sums of full matrix algebras over the field C(t), T n = λ T λ , where the sum is over all Young diagrams λ with n boxes (and with no more than two rows), and T λ is isomorphic to an f λ -by-f λ matrix algebra.
When λ and µ are Young diagrams of size n and n + 1, one has a (nonunital) homomorphism of T λ into T µ given by x → (x ⊗ 1)z µ , where z µ denotes the minimal central idempotent in T n+1 such that T µ = T n+1 z µ . Let g λ,µ denote the rank of (e ⊗ 1)z µ , where e is any minimal idempotent in T λ . It is known that g λ,µ = 1 in case µ is obtained from λ by adding one box, and g λ,µ = 0 otherwise.
One can describe the embedding of T n into T n+1 by a Bratteli diagram (or induction-restriction diagram), which is a bipartite graph with vertices labelled by two-row Young diagrams of size n and n + 1 (corresponding to Figure 3. Bratteli diagram for the sequence (T n ) the simple components of T n and T n+1 ) and with g λ,µ edges joining the vertices labelled by λ and µ. That is λ and µ are joined by an edge precisely when µ is obtained from λ by adding one box. The sequence of embeddings T 0 → T 1 → T 2 → . . . is described by a multilevel Bratteli diagram, as shown in Figure 3.
❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ∅
A tableau of shape λ may be identified with an increasing sequence of Young diagrams beginning with the empty diagram and ending at λ; namely the j-th diagram in the sequence is the subdiagram of λ containing the numbers 1, 2, . . . , j. Such a sequence may also be interpreted as a path on the Bratteli diagram of Figure 3, beginning at the empty diagram and ending at λ.
Path idempotents.
One can define a familiy of minimal idempotents p t in T n , labelled by paths t of length n on the Bratteli diagram (or equivalently, by Young tableaux of size n), with the following properties:
1. p t p s = 0 if t, s are different paths both of length n. 2. z λ = {p t : t ends at λ}. 3. p t ⊗ 1 = {p s : s has length n + 1 and extends t} Let t be a path of length n and shape λ and let µ be a Young diagram of size n + m. It follows that (p t ⊗ 1 m )z µ = 0 precisely when there is a path on the Bratteli diagram from λ to µ. It has been shown in [J] that (in our notations) Tr(p t ) = [λ 1 − λ 2 + 1], where [m] = (t m − t −m )/(t − t −1 ) for any integer m, and where λ is the endpoint of the path t. Observe that we get the same value for diagrams λ and µ which are in the same column in the Bratteli diagram.
The idempotents p t were defined by recursive formulas in [W2], generalizing the formulas for the Jones-Wenzl idempotents in [W1].
2.3. Specializations at non-roots of unity. When τ is not a proper root of unity, the Temperley Lieb algebras T n (τ ) are semi-simple complex algebras with the "same" structure as generic Temperley Lieb algebras. That is, T n (τ ) = λ T λ (τ ), where T λ (τ ) is isomorphic to an f λ -by-f λ matrix algebra over C. The embeddings T n (τ ) → T n+1 (τ ) are described by the Bratteli diagram as before. The idempotents p t , and the minimal central idempotents z λ , in the generic algebras T n , are evaluable at τ , and the evaluations p t (τ ), resp. z λ (τ ), satisfy analogous properties.
Specializations at roots of unity and evaluable idempotents.
We require some terminology for discussing the case where τ is a root of unity. Let q = τ 2 , and suppose that q is a primitive ℓ-th root of unity. We say that a Young diagram λ is critical if w(λ) := λ 1 − λ 2 + 1 is divisible by ℓ. For τ a proper root of unity, the formulas for path idempotents in [W1] and [W2] generally contain poles at τ , i.e. the idempotents are not evaluable. However, suitable sums of path idempotents are evaluable.
Suppose w(λ) ≤ ℓ and t is a path of shape λ which stays strictly to the left of the first critical line (in case w(λ) < ℓ), or hits the first critical line for the first time at λ (in case w(λ) = ℓ); then p t is evaluable at τ , and furthermore Tr(
p t ) = [w(λ)] τ = (τ w(λ) − τ −w(λ) )/(τ − τ −1 ).
For each critical diagram λ of size n, the minimal central idempotent z λ in T n is evaluable at τ . Furthermore, for each non-critical diagram λ of size n, an evaluable idempotent z L λ = p t ∈ T n was defined in [GW] as follows: The summation goes over all paths t ending in λ for which the last critical line hit by t is the one nearest to λ to the left and over the paths obtained from such t by reflecting its part after the last critical line in the critical line (see Figure 5).
These idempotents have the following properties (which were shown in [GW]):
1. {z λ (τ ) : λ critical } ∪ {z L µ (τ ) : µ non-critical } is a partition of unity in T n (τ ); that is, the idempotents are mutually orthogonal and sum to the identity. Figure 4. Critical lines 2. z λ (τ ) is a minimal central idempotent in T n (τ ) if λ is critical, and z L λ (τ ) is minimal central modulo the nilradical of T n if λ is not critical (see [GW], Theorem 2.2 and Theorem 2.3). 3. For λ and µ non-critical, z L λ (τ )T n (τ )z L µ (τ ) = 0 only if λ = µ, or if there is exactly one critical line between λ and µ which reflects λ to µ. If in this case µ is to the left of λ, z L λ T n z L λ ′ ⊆ T µ (in the generic Temperley Lieb algebra). 4. Let z reg n = p t , where the summation goes over all paths t which stay strictly to the left of the first critical line, and let z nil n = 1 − z reg n . Then both z reg n and z nil n are evaluable; this is a direct consequence of the fact that z reg n = λ z L λ , where the summation goes over diagrams λ with n boxes with width w(λ) < ℓ.
❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅
Proposition 2.1. The ideal of negligible morphisms in TL(τ ) is generated by the idempotent p [ℓ−1] (τ ) ∈ T ℓ−1 (τ ). Figure 5. A path and its reflected path.
❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅
Proof. Let us first show that z nil n (τ ) is in the ideal generated by p [ℓ−1] (τ ) for all n. This is clear for n < ℓ, as z nil ℓ−1 = p [ℓ−1] and z nil n = 0 for n < ℓ − 1. Moreover, z nil n is a central idempotent in the maximum semisimple quotient of T n , whose minimal central idempotents are the z L λ with w(λ) ≥ ℓ. One checks pictorially that p [ℓ−1] z L λ = 0 for any such λ (i.e. the path to [ℓ−1] can be extended to a path t for which p t is a summand of z L λ ). This proves our assertion in the maximum semisimple quotient of T n ; it is well-known that in this case also the idempotent itself must be in the ideal generated by p [ℓ−1] . In particular, Hom(n, m)z nil m (τ ) + z nil n (τ )Hom(n, m) is also contained in this ideal.
By [GW], Theorem 2.2 (c), for λ a Young diagram of size n, with w(λ) < ℓ, z L λ T n z L λ (τ ) is a full matrix algebra, which moreover contains a minimal idempotent p t of trace Tr(p t ) = [w(λ)] τ = 0. Therefore
z L λ T n z L λ (τ ) ∩ Neg(n, n) = (0).
Furthermore, z reg n T n z reg n (τ ) = z L λ T n z L λ (τ ), by Fact 4 above, so z reg n T n z reg n (τ ) ∩ Neg(n, n) = (0) as well. Now for x ∈ Neg(n, n), one has z reg n (τ )xz reg n (τ ) = 0, so
x ∈ T n (τ )z nil n (τ ) + z nil n (τ )T n (τ ).
We have shown that Neg(n, n) is contained in the ideal of TL(τ ) generated by p [l−1] , for all n. That the same is true for Neg(m, n) with n = m follows from using the embeddings, and their left inverses, described at the end of Section 1.1.
Ideals
Proposition 3.1. Any proper ideal in TL (or in TL(τ )) is contained in the ideal of negligible morphisms.
Proof. Let a ∈ Hom(m, n). For all b ∈ Hom(n, m), tr(ba) is in the intersection of the ideal generated by a with the scalars End(0). If a is not negligible, then the ideal generated by a contains an non-zero scalar, and therefore contains all morphisms.
Corollary 3.2. The categories TL and TL(τ ) for τ not a proper root of unity have no non-zero proper ideals.
Proof. There are no non-zero negligible morphisms in TL and in TL(τ ) for τ not a proper root of unity.
Theorem 3.3. Suppose that τ is a proper root of unity. Then the negligible morphisms form the unique non-zero proper ideal in TL(τ ).
Proof. Let J be a non-zero proper ideal in TL(τ ). By the embeddings discussed at the end of Section 1.1, we can assume J ∩ T n = 0 for some n.
Now let a be a non-zero element of J ∩ T n (τ ). Since {z λ (τ )} ∪ {z L µ (τ )} is a partition of unity in T n (τ ), one of the following conditions hold:
(a) b = az µ (τ ) = 0 for some critical diagram µ. (b) b = z L µ (τ )az L µ (τ ) = 0 for some non-critical diagram µ. (c) b = z L λ (τ )az L λ ′ (τ )
= 0 for some pair λ, λ ′ of non-critical diagrams which are reflections of one another in a critical line. In this case, let µ denote the leftmost of the two diagrams λ, λ ′ .
In each of the three cases, one has b ∈ e(τ )T n (τ )f (τ ), where e, f are evaluable idempotents in T n such that eT n f ⊆ T µ . Let α be a Young diagram on the first critical line of size n + m, such that there exists a path on the generic Bratteli diagram connecting µ and α. Then one has dim C z α (τ )(e(τ ) ⊗ 1 m )(T n (τ ) ⊗ C 1 m )(f (τ ) ⊗ 1 m ) = dim C(t) z α (e ⊗ id m )(T n ⊗ C(t)1 m )(f ⊗ 1 m ) = dim C(t) eT n f = dim C e(τ )T n (τ )f (τ )
where the first and last equalities result from the principle of constancy of dimension, and the second equality is because x → z α (x ⊗ 1 m ) is injective from T µ to T α . But then it follows that x → z α (τ )(x ⊗ 1 m ) is injective on e(τ )T n (τ )f (τ ). In particular (b ⊗ 1 m )z α is a non-zero element of J ∩ T α . Hence there exists c ∈ T α such that f = c(b ⊗ 1 m )z α is an idempotent. After conjugating (and multiplying with p [ℓ−1] ⊗ 1 m , if necessary), we can assume f to be a subidempotent of p [ℓ−1] ⊗ 1 m . But then ε ℓ−1+m,ℓ−1 (f ) is a multiple of p [ℓ−1] , by Lemma 1.2, with the multiple equal to the rank of f in T α . This, together with Prop. 2.1, finishes the proof.
It is easily seen that TL has a subcategory TL ev whose objects consist of even numbers of points, and with the same morphisms between sets of even points as for TL. The evaluation TL ev (τ ) is defined in complete analogy to TL(τ ).
Corollary 3.4. If τ 2 is a proper root of unity of degree ℓ with ℓ odd, the negligible morphisms form the unique non-zero proper ideal in TL ev .
Proof. If ℓ is odd, p [ℓ−1] is a morphism in TL ev . The proof of the last theorem goes through word for work (one only needs to make sure that one stays within TL ev , which is easy to check).
Figure 1 .
1A
Figure 2 .
2The categorical trace of an element a ∈ T n .
2 .
2The structure of the Temperley Lieb algebras 2.1. The generic Temperley Lieb algebras. Recall that a Young diagram λ = [λ 1 , λ 2 , ... λ k ] is a left justified array of boxes with λ i boxes in the i-th row and λ i ≥ λ i+1 for all i. For example, [5, 3] = .
The m-th critical line on the Bratteli diagram for the generic Temperly Lieb algebra is the line containing the diagrams λ with w(λ) = ml. See Figure 4. Say that two non-critical diagrams λ and µ with the same number of boxes are reflections of one another in the m-th critical line if λ = µ and |w(λ) − mℓ| = |w(µ) − mℓ| < ℓ. (For example, with ℓ = 3, [2, 2] and [4] are reflections in the first critical line w(λ) = 3.)
A magnetic model with a possible Chern-Simons phase. quant- ph/0110060preprint, A magnetic model with a possible Chern-Simons phase, preprint 2001, quant- ph/0110060.
The Temperley-Lieb Algebra at Roots of Unity. F Goodman, H Wenzl, Pac. J. Math. 161F. Goodman and H. Wenzl, The Temperley-Lieb Algebra at Roots of Unity, Pac. J. Math., 161 (1993) 307-334.
Index for subfactors. V F R Jones, Invent. Math. 721Jones, V. F. R. Index for subfactors. Invent. Math. 72 (1983), no. 1, 1-25.
On sequences of projections. H , Math. Rep. C.R. Acad. Sc. Canada. 9H. Wenzl, On sequences of projections, Math. Rep. C.R. Acad. Sc. Canada 9 (1987) 5-9.
Hecke algebras of type An and subfactors. H , Invent. math. 92H. Wenzl, Hecke algebras of type An and subfactors, Invent. math. 92 (1988) 349- 383.
|
[] |
[
"Input Fast-Forwarding for Better Deep Learning",
"Input Fast-Forwarding for Better Deep Learning"
] |
[
"Ahmed Ibrahim \nVirginia Polytechnic Institute and State University\nUSA\n\nBenha University\nEgypt\n",
"A Lynn Abbott [email protected] \nVirginia Polytechnic Institute and State University\nUSA\n",
"Mohamed E Hussein [email protected] \nUniversity of Science and Technology\nEgypt-Japan, Egypt\n\nAlexandria University\nEgypt\n"
] |
[
"Virginia Polytechnic Institute and State University\nUSA",
"Benha University\nEgypt",
"Virginia Polytechnic Institute and State University\nUSA",
"University of Science and Technology\nEgypt-Japan, Egypt",
"Alexandria University\nEgypt"
] |
[] |
This paper introduces a new architectural framework, known as input fast-forwarding, that can enhance the performance of deep networks. The main idea is to incorporate a parallel path that sends representations of input values forward to deeper network layers. This scheme is substantially different from "deep supervision," in which the loss layer is re-introduced to earlier layers. The parallel path provided by fast-forwarding enhances the training process in two ways. First, it enables the individual layers to combine higher-level information (from the standard processing path) with lower-level information (from the fast-forward path). Second, this new architecture reduces the problem of vanishing gradients substantially because the fast-forwarding path provides a shorter route for gradient backpropagation. In order to evaluate the utility of the proposed technique, a Fast-Forward Network (FFNet), with 20 convolutional layers along with parallel fast-forward paths, has been created and tested. The paper presents empirical results that demonstrate improved learning capacity of FFNet due to fast-forwarding, as compared to GoogLeNet (with deep supervision) and CaffeNet, which are 4× and 18× larger in size, respectively. All of the source code and deep learning models described in this paper will be made available to the entire research community 5 . 5 https://github.com/aicentral/FFNet
|
10.1007/978-3-319-59876-5_40
|
[
"https://arxiv.org/pdf/1705.08479v1.pdf"
] | 9,464,934 |
1705.08479
|
b4720674dcd92d28978e24727d5b40edb363dfe9
|
Input Fast-Forwarding for Better Deep Learning
Ahmed Ibrahim
Virginia Polytechnic Institute and State University
USA
Benha University
Egypt
A Lynn Abbott [email protected]
Virginia Polytechnic Institute and State University
USA
Mohamed E Hussein [email protected]
University of Science and Technology
Egypt-Japan, Egypt
Alexandria University
Egypt
Input Fast-Forwarding for Better Deep Learning
This paper introduces a new architectural framework, known as input fast-forwarding, that can enhance the performance of deep networks. The main idea is to incorporate a parallel path that sends representations of input values forward to deeper network layers. This scheme is substantially different from "deep supervision," in which the loss layer is re-introduced to earlier layers. The parallel path provided by fast-forwarding enhances the training process in two ways. First, it enables the individual layers to combine higher-level information (from the standard processing path) with lower-level information (from the fast-forward path). Second, this new architecture reduces the problem of vanishing gradients substantially because the fast-forwarding path provides a shorter route for gradient backpropagation. In order to evaluate the utility of the proposed technique, a Fast-Forward Network (FFNet), with 20 convolutional layers along with parallel fast-forward paths, has been created and tested. The paper presents empirical results that demonstrate improved learning capacity of FFNet due to fast-forwarding, as compared to GoogLeNet (with deep supervision) and CaffeNet, which are 4× and 18× larger in size, respectively. All of the source code and deep learning models described in this paper will be made available to the entire research community 5 . 5 https://github.com/aicentral/FFNet
Introduction
Developments in deep learning have led to networks that have grown from 5 layers in LeNet [10], introduced in 1998, to 152 layers in the latest version of ResNet [5]. One consequence of deeper and deeper networks is the problem of vanishing gradients during training. This problem occurs as error values, which depend on the computed gradient values, are propagated backward through the network to update the weights at each layer. With each additional layer, a smaller fraction of the error gradient is available to guide the adjustment of network weights. As a result, the weights in early layers are updated very slowly; hence, the performance of the entire training process is degraded.
Many models have been proposed to overcome the vanishing-gradient problem. One approach is to provide alternative paths for signals to travel, as compared to traditional layer-to-layer pathways. An example of this approach is the Deeply-Supervised Network (DSN) [11], where a companion objective function is added to each hidden layer in the network, providing gradient values directly to the hidden layers. DSN uses Support Vector Machines (SVM) [3] in its companion objective function, which means that end-to-end training of the network is not supported. Another example is relaxed deep supervision [12], where an improvement over a holistic edge detection model [19] is made by providing relaxed versions of the target edge map to the earlier layers of the network. This approach provides a version of the gradient directly to the early layers. However, relaxed deep supervision is suitable only for problems where relaxed versions of the labels can be created, such as for maps of intensity edges. GoogLeNet [13] is another model that uses a mechanism to address the problem of vanishing gradients. More relevant details about GoogLeNet will be given in section 2 because it serves as a baseline for comparison with our proposed model.
The novel approach that is proposed here provides parallel signal paths that carry simple representations of the input to deeper layers through what we call a fast-forwarding branch. This approach allows for a novel integration of "shallower information" with "deeper information" by the network. During training the fast-forwarding branch provides an effective means for back-propagating errors so that the vanishing-gradient problem is reduced.
To demonstrate the efficacy of the model, we created a 20 layer network with fast-forwarding branches, which we call FFNet. To study the effect of the fastforwarding concept, the network layers are made of simple convolutional layers followed by fully connected layers with no additional complexities. The results that we have obtained using the the relatively small and simple FFNet model have been surprisingly good, especially when compared with the performance of bigger and more complex models.
The rest of this paper is organized as follows. Section 2 presents a brief survey of related work, including a discussion of the models that will be used as a baseline to be compared with FFNet. Section 3 provides details concerning the proposed model. In order to gauge the performance of this approach, experimental results from FFNet were compared with results from several well-known deep models. These experiments are described in Section 4. Finally, concluding remarks are given in Section 5.
Related Work
Deep Learning
Deep learning is a machine-learning technique that has become increasingly popular in computer vision research. The main difference between classical machine learning (ML) and deep learning is the way that features are extracted. For classical ML techniques such as support vector machines (SVM) [3], feature extraction is performed in advance using techniques crafted by the researchers. Then, the training procedure develops weights or rules that map any given feature vector to an output class label. In contrast, the typical deep-learning procedure is to directly feed signal values as inputs to the training procedure, without any preliminary efforts at feature extraction. The network takes the input signal (pixel values, in our case), and assigns a class label based on those signal values directly. Because the deep-learning approach implicitly must derive its own features, many more training samples are required than for traditional ML approaches.
Several deep-learning packages are available for researchers. The popular package that we have used to evaluate the proposed model is Caffe [7], which was created with computer vision tasks in mind. Caffe is relatively easy to use, flexible, and powerful. It was developed in C++ using GPU optimization libraries, such as CuDNN [2], BLAS [18], and ATLAS [17]. In the next sections, we will discuss briefly two well-known deep models, AlexNet and GoogLeNet. These two models will be used as a baseline for comparison with the proposed FFNet model.
AlexNet and CaffeNet
AlexNet [9] was the first deep model to win the ILSVRC [4] challenge. For the ILSVRC-2012 competition, AlexNet won with a top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry. This model consists of five convolutional layers followed by three fully-connected layers. The creators of Caffe [7] introduced a slightly modified version of AlexNet by switching the order of pooling and normalization layers. They named the modified version CaffeNet [1]. As the only modification done to the network is switching the order of pooling and normalization layers, the size of the network is exactly the same as AlexNet.
AlexNet and CaffeNet will be used to provide baseline cases of simple architectures that rely on huge numbers of parameters. The number of filters in the convolutional layers range from 96 to 384 in AlexNet, while the proposed FFNet model uses only 64 filters in each convolutional layer. AlexNet uses a 4069-node fully-connected layer followed by another layer of the same size, whereas FFNet uses only a 400-node fully connected layer followed by a 100-node layer. The total size of AlexNet is therefore approximately 18 times bigger than FFNet.
GoogleNet
GoogLeNet [13] is another winner of the ILSVRC challenge. This model won the ILSVRC-2014 competition with a top-5 test error rate of 6.6%. The network consists of 22 layers with a relatively complex design called "inception." The inception module, which is used to implement the layers of GoogLeNet, consists of parallel paths of convolutional layers of different sizes concatenated together. The number of filters in the convolutional layers inside the inception modules ranges from 16 to 384. (By comparison, in FFNet the number of filters in each convolutional layer is fixed.) In addition to using the inception design, GoogLeNet uses three auxiliary classifiers connected to the intermediate layers during training. GoogLeNet is of interest to us as a baseline for comparison because of its depth, because of its complex architecture, and especially because of the auxiliary classifiers. GoogLeNet is 4 times bigger and far more complex than the proposed FFNet model.
Benchmarking Datasets
Many datasets have been created to aid in machine learning for computer vision. To evaluate the proposed FFNet model, we selected two publicly available datasets, COCO-Text-Patch and CIFAR-10.
COCO-Text-Patch [6], contains approximately 354, 000 images of size 32×32 that are each labeled as "text" or "non-text." This dataset was created to address the problem of text verification, which is an essential stage in the end-to-end text detection and recognition pipeline. The dataset is derived from COCO-Text [15], which contains 63, 686 images of real-world scenes with 173, 589 instances of text.
CIFAR-10 [8] is a labeled subset of the "80 million tiny images" dataset [14]. They were collected by the creator of AlexNet. The CIFAR-10 dataset consists of 60, 000 color images of size 32 × 32 in 10 classes, with 6, 000 images per class.
Proposed Model: FFNet
The new FFNet model consists of convolutional units that are organized into a sequence of stages. Within each stage, as illustrated in figure 1, computations are performed in 2 parallel paths. The left branch in the figure represents a standard convolutional path, whereas the right branch represents an extra parallel data path. It is this parallel, "fast-forwarding", path that delivers the improved performance of the network.
The input to the stage, S1, arrives from the previous layer, and the output to the next layer is shown as S2. The standard (deep) branch consists of three consecutive 3 × 3 × 64 convolutional layers. Each layer is followed by an in-place Rectified Linear Unit (ReLU). The last layer of the deep branch is padded with zeros, for reasons that are described below.
Let the input S1 be of size N × N × C. The value of C is the number of channels, which is typically 128 except for the first stage where C = 3 to match the input data. Refer to a stage's deep convolutional layers as S2C1, S2C2, and S2C3, as shown in the figure. The deep branch's output S2C3 can be represented as follows, where CON V is the convolutional operation, s is the stride, and p is the padding: S2C3 = CON V 3×3,s=1,p=1 (CON V 3×3,s=1,p=0 (CON V 3×3,s=1,p=0 (S1))) (1)
The size of S2C3 will be (N − 2) × (N − 2).
The fast-forwarding branch consists of a single 5 × 5 × 64 convolutional layer followed by a ReLU. This branch takes S1 as input, and generates the output B2C1 that can be represented as follows: B2C1 = CON V 5×5,s=1,p=0 (S1) (2) No padding is used for the fast-forwarding branch, so that the resulting output size is also (N − 2) × (N − 2). This branch will provide a "shallower" representation of the input S1 to the next stage. The outputs of the deep branch and of the fast-forwarding branch are concatenated to create the single stage output S2. The size of S2 will be (N − 2) × (N − 2) × 128. Because the last layer of the deep branch is padded with zeros, both branches provide data of the same size to the output.
To evaluate the fast-forwarding concept, we built a Fast-Forwarding Network (FFNet) that consists of 6 consecutive fast-forwarding stages followed by two fully connected layers plus an output layer, as shown in figure 2. The 6 fastforwarding stages consist of a total of 18 convolutional layers, each of size 3 × 3 × 64. The first layer of the two fully-connected layers consists of 400 nodes, while the second layer consists of 100 nodes. Fig. 1. A single fast-forwarding stage. Node S1 represents the input, and S2 is the output. The left pathway contains common convolutational blocks. At the right is the fast-forward path.
Evaluation
To evaluate the performance of the proposed model, a number of experiments were conducted that compare FFNet to AlexNet, CaffeNet, and GoogLeNet.
The publicly available datasets CIFAR-10 [8] and COCO-Text-Patch [6] were used in the evaluation, as described previously. FFNet was implemented using Caffe [7]. Standard 10-crop augmentation was applied to the datasets. All the training and testing were performed on a GPU with batch size 32. The training was stopped after 150, 000 iterations as the validation accuracy and loss started to plateau.
A summary of results is provided in table 1. Despite its relatively small size, the performance of the proposed FFNet model exceeded the performance of CaffeNet and GoogleNet in these experiments. The accuracy and validation loss graphs shown in figure 3 demonstrate how the proposed model converges with the same speed as CaffeNet and GoogLeNet. These trends provide evidence of the effectiveness of the fast-forwarding approach in fighting the vanishing-gradient problem.
Conclusion
This paper has presented a new concept, called input fast-forwarding, which results in improved performance for deep-learning systems. The approach utilizes parallel data paths that provide two advantages over previous approaches. One Table 1. Performance comparison of the proposed FFNet model with several common alternatives. Although FFNet is much smaller than the other models, its error rate was lower than the others (with one exception), using publicly available test sets.
Model
Error Rate (%) Description
Layers Size (MB) Time*(ms) CIFAR-10 CTP** AlexNet with dropout [ 9.0 * Average forward path time per image on a K80 GPU ** CTP: COCO-Text-Patch dataset [6] advantage is the explicit merging of higher-level representations of data with lower-level representations. A second advantage is a substantial reduction to the effects of the vanishing gradients problem.
To evaluate the model, we built a 20-layer network (FFNet) that implements the fast-forwarding concept. The network consists of simple convolutional layers, with no added complexities, to prove that the outstanding performance of the model is primarily the result of the fast-forwarding approach. Empirical results also showed convergence during training at virtually the same rate as the bigger and more complex models. FFNet achieved an error rate of 13.6% on the CIFAR-10 dataset, which is on par with one variation of AlexNet. When tested on COCO-Text-Patch, FFNet's performance surpassed that of CaffeNet and GoogLeNet, which are all significantly larger in size.
These results suggest that similar advantages may be obtained through the application of fast-forwarding to other models, and with different benchmark datasets.
Fig. 2 .
2Proposed FFNet model. Because of fast-forwarding, this relatively small network has yielded empirical results that are better than much larger deep networks.
Fig. 3 .
3COCO-Text-Patch validation accuracy and loss for the proposed FFNet model (red), CaffeNet (blue), and GoogLeNet (green).
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"Quantum boomerang effect for interacting particles"
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"Jakub Janarek \nInstitute of Theoretical Physics\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland\n\nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance\n",
"Dominique Delande \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance\n",
"Nicolas Cherroret \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance\n",
"Jakub Zakrzewski \nInstitute of Theoretical Physics\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland\n\nMark Kac Complex Systems Centre\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland\n"
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"Institute of Theoretical Physics\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland",
"Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance",
"Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance",
"Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75004ParisFrance",
"Institute of Theoretical Physics\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland",
"Mark Kac Complex Systems Centre\nJagiellonian University in Krakow\nLojasiewicza 11, 30-348KrakówPoland"
] |
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When a quantum particle is launched with a finite velocity in a disordered potential, it may surprisingly come back to its initial position at long times and remain there forever. This phenomenon, dubbed "quantum boomerang effect", was introduced in [Phys. Rev. A 99, 023629(2019)]. Interactions between particles, treated within the mean-field approximation, are shown to partially destroy the boomerang effect: the center of mass of the wave packet makes a U-turn, but does not completely come back to its initial position. We show that this phenomenon can be quantitatively interpreted using a single parameter, the average interaction energy.
|
10.1103/physreva.102.013303
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[
"https://arxiv.org/pdf/2003.09903v1.pdf"
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2003.09903
|
80a70dbb4db8d892e9d8654de6cf21e4d4dea58d
|
Quantum boomerang effect for interacting particles
Jakub Janarek
Institute of Theoretical Physics
Jagiellonian University in Krakow
Lojasiewicza 11, 30-348KrakówPoland
Laboratoire Kastler Brossel
Sorbonne Université
CNRS
ENS-Université PSL, Collège de France
4 Place Jussieu75004ParisFrance
Dominique Delande
Laboratoire Kastler Brossel
Sorbonne Université
CNRS
ENS-Université PSL, Collège de France
4 Place Jussieu75004ParisFrance
Nicolas Cherroret
Laboratoire Kastler Brossel
Sorbonne Université
CNRS
ENS-Université PSL, Collège de France
4 Place Jussieu75004ParisFrance
Jakub Zakrzewski
Institute of Theoretical Physics
Jagiellonian University in Krakow
Lojasiewicza 11, 30-348KrakówPoland
Mark Kac Complex Systems Centre
Jagiellonian University in Krakow
Lojasiewicza 11, 30-348KrakówPoland
Quantum boomerang effect for interacting particles
(Dated: March 24, 2020)
When a quantum particle is launched with a finite velocity in a disordered potential, it may surprisingly come back to its initial position at long times and remain there forever. This phenomenon, dubbed "quantum boomerang effect", was introduced in [Phys. Rev. A 99, 023629(2019)]. Interactions between particles, treated within the mean-field approximation, are shown to partially destroy the boomerang effect: the center of mass of the wave packet makes a U-turn, but does not completely come back to its initial position. We show that this phenomenon can be quantitatively interpreted using a single parameter, the average interaction energy.
I. INTRODUCTION
Anderson Localization (AL) [1], i.e. inhibition of transport in disordered media, has been the source of various, often counter-intuitive phenomena discovered over the last 60 years [2]. Already in one dimension (1D) the fact that even the tiniest random disorder generically leads to a full localization of eigenstates is totally against a classical way of thinking. This effect is a clear manifestation of the inherently interferometric nature of AL, typically explained as the effect of quantum wave interference. The phenomenon was observed in many experimental setups including light [3,4], sound waves [5] as well as matter waves [6][7][8][9][10]. The closely related phenomenon of Aubry-André localization in quasi-periodic potentials [11] was also observed in a cold atomic setting [12]. Last years have also led to a number of studies of the many-body counterpart of AL [13,14], the so called many-body localization (MBL) (for recent reviews see [15,16]). It has already been observed in cold atomic experiments in quasi-periodic potentials [17,18]. While studies of MBL have been very extensive, even its very existence has been questioned recently [19], provoking a vivid debate [20][21][22].
The physics of a single particle in a random potential, particularly in 1D, has much stronger foundations, while still bringing novel features such as studies of random (or quasi-random) potentials revealing the presence of mobility edge [23,24], i.e. situations where localized and extended states appear at different energies. Even for a pure random standard case, one may find new counter-intuitive phenomena as exemplified by the quantum boomerang effect [25]. As a classical boomerang returns to the initial position from which it was launched, the center of mass of a wave packet with a nonzero initial velocity returns to its origin due to AL. The effect * [email protected] † [email protected] ‡ [email protected] § [email protected] is quite general and occurs in Anderson localized multidimensional systems [25] whose Hamiltonians preserve time-reversal invariance. The aim of this work is to investigate how interactions between particles affect the boomerang effect. Our study focuses on the limit of weak interactions, a regime where it was previously shown that AL of wave packets is replaced by a subdiffusive evolution at very long time [26][27][28][29][30]. Computing the temporal evolution of a many-body wave packet in a disordered potential is in general a formidable task, even in 1D. When interactions are sufficiently weak however, one may use a mean-field approximation and describe the dynamics with a onedimensional Gross-Pitaevskii equation (GPE). Within this formalism, we provide an in depth numerical analysis of the quantum boomerang effect. For simplicity, we study the weak-disorder case, for which an analytic description of the boomerang effect in the absence of interaction is available. However, the conclusions are expected to be qualitatively identical when the disorder strength is increased. At short times, of the order of a few disorder scattering times, we observe that the dynamics of the wave packet is essentially not affected by interactions, with an initial ballistic flight followed by a U-turn of the center of mass, slowly returning towards its initial position. The main effect in this regime is a small modification of the scattering time and scattering mean free path due to interactions. At longer time, we observe that the center of mass, instead of returning to its initial position, stops at a finite distance from it, which increases with the interaction strength. We show that this phenomenon can be interpreted in terms of a break time, the time scale beyond which interference effects are typically destroyed by interactions. We finally show that in the presence of interactions, the boomerang effect can be quantitatively described in terms of a single scaling parameter, a variant of the interaction energy, computed at the break time.
The paper is organized as follows. After formulating the problem and introducing the main parameters in Sec. II, we numerically analyze the influence of interactions on the boomerang effect in Sec. III. In Sec. IV, we then perform several numerical studies which establish that arXiv:2003.09903v1 [cond-mat.dis-nn] 22 Mar 2020 the dynamics can be described by a single parameter. The appendix shows that the short-time dynamics can also be understood using the same parameter. We finally conclude and briefly discuss open questions.
II. THE MODEL
To study the boomerang effect in the presence of interactions, we use the one-dimensional, time dependent Gross-Pitaevskii equation (GPE) [31]:
i ∂ψ(x, t) ∂t = p 2 2m + V (x) + g|ψ(x, t)| 2 ψ(x, t),(1)
where V (x) is a disordered potential, while g represents the interaction strength. Wave functions are normalized to unity, |ψ(x, t)| 2 dx = 1. Throughout this work we assume that the disordered potential is a Gaussian uncorrelated random variable, i.e.
V (x) = 0, V (x)V (x ) = γδ(x − x ),(2)
where the overbar denotes the average over disorder realizations. The parameter γ measures the disorder strength and determines the characteristic time and length scales for scattering: the mean scattering time τ 0 (the typical time for a particle to be scattered by the disorder) and the corresponding mean scattering length 0 , usually called mean free path. The uncorrelated disorder model is sufficient to capture the main features of the boomerang effect, which depends only on a small set of well defined parameters, τ 0 , 0 and time t itself. In the Born approximation, τ 0 and 0 are given by [32]:
τ 0 = 3 k 0 2mγ , 0 = k 0 m τ 0 = 4 k 2 0 2m 2 γ .(3)
Non-interacting 1D systems remain always strongly localized, i.e. their eigenstates decay exponentially over the localization length ξ loc = 2 0 [33].
Following [25], as the initial state for time evolution we take a Gaussian wave packet with mean velocity k 0 /m:
ψ(x, t = 0) = 1 πσ 2 1/4 exp(−x 2 /2σ 2 + ik 0 x),(4)
where σ and k 0 are chosen such that the initial wave function is sharply peaked around k 0 in momentum space, i.e. k 0 σ 1. Moreover, we focus on the weak-disorder regime where the mean free path is much longer than the de Broglie wavelength, that is k 0 0 1 [32]. In this regime, a full analytic theory, based on the Berezinskii diagrammatic technique [34] exists in the non-interacting limit. It makes it possible to express the average centerof-mass position (CMP):
x(t) = x|ψ(x, t)| 2 dx ,(5)
as:
x(t) 0 = f t τ 0 ,(6)
where f is a universal function whose Taylor expansion at short time and asymptotic behavior at long time are exactly known [25]. In particular x(t) ≈ t 0 /τ 0 at short time t τ 0 , meaning that the initial motion is ballistic, and
x(t) ≈ 64 0 ln(t/4τ 0 )τ 2 0 t 2(7)
at long time t τ 0 . The two assumptions of weak disorder, k 0 0 1, and narrow wave packet in momentum space, k 0 σ 1, make it possible to have a well controlled non-interacting limit [35]. If they are not valid, the wave packet will contain a broad energy distribution, and consequently a distribution of scattering time (which depends on energy), making the analysis more tedious. However, the phenomena described below are expected to be very similar.
Another important ingredient of the model are interactions. Our interest is only in small values of g, which corresponds to a weak interaction regime [36][37][38].
In the following, we study numerically the propagation of the initial wave function ψ(x, t = 0) for different disorder realizations. It yields the averaged density profile |ψ(x, t)| 2 , from which we compute the CMP, Eq. (5). The numerical technique is as follows. The one-dimensional configuration space is discretized on a regular grid, over which the wave function is computed. The temporal propagation is performed using a splitstep algorithm of step ∆t, alternating propagations of the linear part of the GPE, exp −i(p 2 /2m + V )∆t/ , and of the nonlinear part, which is simply a phase factor exp(−ig|ψ(x, t)| 2 ∆t/ ). The linear part of the evolution operator is expanded in a series of Chebyshev polynomials, as described in [39][40][41][42].
Throughout our work, we express lengths in units of 1/k 0 , times in m/ k 2 0 , and energies in 2 k 2 0 /m. The interaction strength is expressed in units of 2 k 0 /m, and the disorder strength γ in units of 4 k 3 0 /m 2 . Numerical results have been obtained on a discretized grid of size 4000/k 0 (sufficiently large for the wave function to be vanishingly small at the edges; open boundary conditions have been used) divided into 20000 points, so that discretization effects are negligible. We have used a disorder strength γ = 0.1 4 k 3 0 /m 2 , so that, in the Born approximation (3), one has k 0 0 = 5, i.e. the disorder is weak. Note that, while the true mean free path and scattering time may slightly differ from their expressions (3) at the lowest order Born approximation, the dynamics of the non-interacting quantum boomerang effect is still given by Eq. (6), provided corrected values of τ 0 and 0 are used. We have performed calculations for various widths σ of the initial wave packet ψ(x, t = 0), as well as various values of the interaction strength g. Here, x is in units of 1/k0 and time t in units of m/ k 2 0 . All curves have been averaged over more than 5·10 5 disorder realizations. The short-time behavior remains almost unchanged, whereas the long-time evolution clearly depends on the interaction strength. The error bars represent statistical average errors. Center-of-mass trajectories among different disorder realizations are normally distributed, such that we use the standard error of the mean as estimator of the errors.
III. THE BOOMERANG EFFECT WITH INTERACTIONS
A. Role of the interaction strength
In Fig. 1 we present results obtained for a wave packet with initial width σ = 10/k 0 for various interaction strengths g. Similarly to the non-interacting case, af-ter the initial ballistic motion of the center of mass, we observe a subsequent retro-reflection towards the origin, that is a boomerang effect. However, the long-time behavior is affected by interactions. For all non-zero studied values of g, the center of mass does not return to the origin but saturates at some finite value. In other words, the boomerang effect is only partial [43]. This can be understood as follows. For g = 0, the disorder is static and full Anderson localization sets in; the complicated interference between multiply scattered paths leads to full localization at infinite time, and also to full return of the center of mass to its initial position. For non-zero but small g, the nonlinear term in Eq. (1), g|ψ(r, t)| 2 , plays the role of a small additional effective potential which is time-dependent, thereby adding a fluctuating phase along each scattering path. This breaks the interference between multiple scattering paths and thus destroys both Anderson localization [26][27][28][29][30] and the full boomerang effect at long time. For all studied widths of the wave packet, namely σ = 5/k 0 , σ = 10/k 0 , σ = 20/k 0 , and σ = 40/k 0 we observe a similar saturation effect. Of course, the phase scrambling progressively develops over time. The characteristic break time over which it kills coherent transport and boomerang effect is discussed in detail in the sequel of this paper. Note that the interpretation of the effect of interactions in terms of a decoherence mechanism may be questioned at very long time, where thermalization comes into play and affects the momentum distribution as well as the dynamics of the system [44,45]. In our case, both disorder and interactions are small, so that thermalization takes place at times significantly longer than the ones considered here. Neglecting thermalization nevertheless restricts our analysis to the regime where the long-time CMP x 0 (see Eq. (8) below), which constrains the maximum value of g.
To study in detail the long-time evolution, we run numerical simulations up to time t max ≈ 2500τ 0 . From these simulations we calculate the long-time average of the CMP, x , defined in the following way:
x = 1 t 2 − t 1 t2 t2 x(t) dt ,(8)
where we choose t 1 ≈ 1200τ 0 , t 2 ≈ 2500τ 0 . The results are essentially independent of these bounds, provided they are much longer than τ 0 . This definition provides us with a very good estimate of the infinite-time CMP. Figure 2 shows the dependence of x on the interaction strength g. For small values of g, the CMP dependence is quadratic in g, x ∝ g 2 , and becomes approximately linear for larger g. In the following, we will mostly concentrate on the quadratic regime of small interactions, leaving the more difficult case of larger g for future studies.
B. Role of the wave-packet width
Another important parameter is the width σ of the initial wave packet. In Fig. 3, we show x(t) for different σ values. While, for g = 0, it follows the analytic prediction (7) independently of σ, for interacting particles the long-time behavior strongly depends on σ. A simple qualitative explanation is that the destruction of the boomerang effect is controlled by the nonlinear term g|ψ(x)| 2 in Eq. (1). This term is larger for a spatially narrow wave packet, so that interference between scattered waves are suppressed at shorter time, giving a higher x value.
Although the boomerang effect is affected by a change of either g or σ in the interacting case, one may guess that the CMP is not an independent function of these two parameters. Indeed, closer investigation reveals that similar "trajectories" of x(t) can be achieved for different combinations of g and σ. In particular, the same values of x can be obtained from different initial states, provided g is properly adjusted. This property is illustrated in Fig. 4, where we have computed x(t) for various values of σ and have adjusted g so to that the curves fall on top of each other.
IV. UNIVERSAL SCALING OF THE INTERACTING BOOMERANG EFFECT
From the results of the previous section, it is natural to ask whether or not the interacting boomerang effect, and more specifically its long-time average x -Eq. (8) -can be described in terms of a single parameter, in the spirit of scaling approaches well-known in the context of Anderson localization of non-interacting particles [46]. We now address this question.
A. Break time
Before attempting to rescale the CMP, let us introduce an important parameter that will turn useful in the following. We recall that in the non-interacting limit, the CMP at long time is given by [25]:
x(t) ≈ 64 0 ln(t/4τ 0 )τ 2 0 t 2 .(9)
If we neglect the logarithmic part, x(t) decays as t −2 . It suggests that one may identify a characteristic time scale connected with weak interactions which is inversely proportional to g. In our analysis we dub this time scale the break time t b and define it by the relation:
t b (g) : x(t b ) g=0 = x (g),(10)
where for the left-hand-side we use the analytical prediction of the non-interacting theory, Eq. (9). A given time scale in quantum mechanics corresponds to some energy scale, here the break energy: which will turn out to be a key parameter in our rescaling of the CMP.
E b = 2π t b ,(11)
B. Rescaling of the CMP: first attempt
A closer inspection of Fig. 4 reveals that, when trying to superimpose the CMP curves, broader wave packets require to adjust the interaction strength to larger values. A first natural candidate to represent the CMP is thus the interaction energy,
E int (t) = g 2 |ψ(x, t)| 4 dx .(12)
We recall that the total energy, conserved by the GPE, is the sum of the non-interacting part and the interaction energy: E tot = p 2 /2m + V + E int . A related important quantity is the interaction energy at initial time, E int (t = 0) = g/(2 √ 2πσ). Notice that while the CMP curves in Fig. 4 are obtained from quite different values of g and σ, they are all associated with comparable interaction energies at time t = 0. This is a very clear hint that E int (t = 0) is a crucial parameter for describing the impact of interactions on the boomerang effect. We can thus try to rescale the results by plotting them vs. the initial interaction energy. This is done for x in Fig. 5, and for the break energy -see Eq. (11) -in Fig. 6. In these plots we also show, for comparison, the formal limit σ → ∞ of infinitely large wave packets, where ψ 0 (x) → √ ρ 0 exp(ik 0 x) reduces to a plane wave with E int (t = 0) = gρ 0 /2. In this limit, the sole parameter gρ 0 controls the boomerang effect. Note that despite the flatness of the density profile in the plane-wave limit, in practice it is still possible to study the boomerang effect by measuring an effective CMP defined as:
x(t) σ=∞ ≡ 1 m t 0 p(t ) dt .(13)
We have verified numerically, in particular, that in the non-interacting case the definition (13) agrees with results for wave packets of finite width, thereupon with the theoretical prediction (7). The curves in Figs. 5 and 6, which correspond to different values of σ, are qualitatively similar, despite the fact that the g values are widely different. This suggests that the interaction energy is indeed an important parameter. Moreover, note that the break energy E b is comparable (within a factor 4) to the initial interaction energy. In particular, because E int (t = 0) is proportional to the interaction strength g for all initial states, at small values of E int (t = 0) we see the expected linear dependence of E b with E int (t = 0). Nevertheless, it is clear that plots based on E int (t = 0) do no fall on the same universal curve: x and E b deviate from each other as σ is changed, approaching the upper limit curve σ = ∞ as σ increases (this limit is represented by green square symbols in Figs. 5 and 6).
C. Nonlinear energy at break time
The reason why E int (t = 0) does not allow for a universal rescaling of the CMP stems from the fact that the interaction energy E int (t) varies significantly from t = 0 onwards. This evolution is shown in Fig. 7 for two finite values of σ and for the plane-wave limit σ = ∞. The figure reveals that the time evolution of the interaction . Initial states are chosen so to clearly emphasize the specklelization phenomenon. It is, however, generically present for any interaction strength. In the plane-wave limit, specklelization doubles Eint, which then remains stationary. At finite σ, specklelization is not full and, due to the wave-packet dilution, Eint decays in time. From t = t b onwards, however, the decay is very slow (the location of t b is indicated by arrows). energy generally consists of two stages. In a first stage, which takes place on a time scale of a few scattering times, the interaction energy roughly doubles. This can be understood by noticing that E int (t) depends on the fourth moment of the field, see Eq. (12), which obeys:
|ψ(x, t)| 4 = |ψ(x, t)| 2 2 + Var |ψ(x, t)| 2 .(14)
In the plane-wave limit where the profile is flat, the initial density variance is zero. During the temporal evolution however, the density develops fluctuations depending on the realization of the disorder, which makes the interaction energy increase. The factor 2 enhancement is obtained by assuming that, after a few scattering times, ψ(x, t) is a complex Gaussian random variable. The variance in Eq. (14) is then |ψ(x, t)| 2 2 , so that |ψ(x, t)| 4 is doubled. Due to similarities of this phenomenon with optical speckles in scattering media [47] we dub it specklelization of the interaction energy. It implies that E int (t τ 0 ) = 2E int (t = 0) in the limit σ = ∞. For wave packets of finite size σ, the effect is also present, albeit slightly smaller [48].
In the plane-wave limit σ = ∞, the interaction energy remains constant once the specklelization process has ended. For finite σ on the contrary, a second stage occurs, where E int decreases in time, see Fig. 7. This decrease is due to the spreading of the wave packet, which becomes more and more dilute. The spreading is initially fast, and then slows down very quickly.
The time evolution of E int makes a detailed rescaling analysis of the boomerang effect very complex. The curves in Fig. 7, however, suggest the simple idea of using as scaling parameter the interaction energy at the break time t b , instead of the initial interaction energy. Although at finite σ such a rescaling can only be approximate, since wave packets keep evolving slowly in time beyond t b (indicated as arrows in Fig. 7), we show below that it provides satisfactory results.
Before applying this strategy, a final adjustment must be performed. The quantum boomerang being a dynamical effect governed by the GPE (1), its evolution is not strictly governed by the interaction energy, g|ψ(r, t)| 4 /2, but rather by the nonlinear energy, defined as
E NL (t) = 2E int (t) = g |ψ(x, t)| 4 dx .(15)
In the appendix, we discuss the dynamical behavior of the system at short time and show analytically and numerically that it is indeed E NL , rather than E int , which governs the evolution.
D. Rescaling of the boomerang effect
We can now re-analyze the boomerang effect using the nonlinear energy at the break time, E NL (t = t b ), as a control parameter of interactions. We show in Fig. 8 the long-time average x of the CMP as a function of the nonlinear energy calculated at the break time. In contrast with Fig. 5, now all points collapse on a single universal curve. As expected, in the regime of small nonlinear energy, x shows a quadratic dependence.
In Fig. 9 we also compare the break energy E b = 2π /t b with the nonlinear energy at the break time t b for wave packets of size σ = 10/k 0 , 40/k 0 and σ = ∞, for various values of g. There is a compelling evidence that these quantities are very similar. A small difference is observed in the plane-wave limit, which we attribute to the slow residual decay of x due to prethermalization processes, which lead to an underestimation of the break energy. Such a good agreement shows that a simple model involving a single parameter, the nonlinear energy at the break time, captures quantitatively the essential features of a complex dynamical process like the quantum boomerang effect for interacting particles. As discussed in the appendix, the same parameter turns out to also control the short-time dynamics and the correction of the scattering time due to interactions.
V. CONCLUSION
We have analyzed the quantum boomerang phenomenon in the presence of interactions on the basis of the one-dimensional Gross-Pitaevskii equation. We have found that weak interactions do not destroy the quantum boomerang effect, in the sense that the center of mass of a wave packet launched with a finite velocity is still retro-reflected after a few scattering times, slowly returning towards its initial position. The boomerang effect is only partial though, as the quantum particle does not return to its initial position but stops on the way back. We have interpreted this phenomenon as a consequence of the destruction of interference between multiple scattering paths induced by a time-dependent nonlinear phase acquired along a path. To characterize this phenomenon, we have introduced a break time -and the corresponding break energy -the characteristic time beyond which the destruction of interference prevents the wave packet to further move back to its initial position. We have finally shown that different initial states and interaction strengths can all be described by means of a single parameter, the nonlinear energy estimated at the break time.
Our analysis is limited to the regime of weak disorder and weak interactions. For stronger interactions, the break time is likely to decrease until it becomes comparable to the scattering time. Whether a single parameter also controls this regime is an interesting question left for future studies. Another important question -especially if one envisions experiments with ultra-cold atoms -is to know whether the observed softening of the boomerang effect due to interactions remains valid beyond the mean-field description. Studying the full manybody boomerang effect is a challenging task.
Self energy in interacting systems
The self energy is a key concept in the description of disordered quantum systems. It is a complex valued function of the state energy E and its momentum k 0 describing the disorder-induced energy shift and the exponential decay of average Green's functions in configuration space. It is noted Σ E (k 0 ) and defined by:
G R E (k 0 ) = 1 E − E 0 − Σ E (k 0 ) ,(A1)
where G R denotes the averaged retarded Green's function and E 0 = 2 k 2 0 /2m the disorder-free energy. The selfenergy vanishes in the absence of disorder and is much smaller than E 0 in the weak-disorder limit (by a factor 1/k 0 0 ) [32].
The evolution operator is the temporal Fourier transform of the Green's function. If the self-energy is a smooth function of E, one obtains for the evolution of a plane wave:
ψ 0 |ψ(t) = e −i(E k 0 +Σ E (k0))t/ = e −i(E k 0 +Re Σ E 0 (k0))t/ e Im Σ E 0 t/ , (A2)
so that Re Σ is an energy shift and − Im Σ the decay rate induced by the disorder. At the Born approximation, we have in 1D:
Σ (0) E0 (k 0 ) = − i 2τ 0 ,(A3)
where the superscript (0) refers to zero interactions.
In the presence of interactions, the situation is in general much more complicated. Because of the nonlinearity of the GPE, the notion of evolution operator no longer exists and the overlap Eq. (A2) has no reason to be an exponential function of time. However, it is possible to define an effective self-energy using Eq. (A2), the lefthand-side of the equation being computed numerically from the solution |ψ(t) of the GPE. The obtained selfenergy Σ (g) E0 (k 0 ) depends on time.
To analyze the impact of interactions on the self energy, we introduce its nonlinear part Σ (g) , defined as:
Σ (g) = Σ (g) E0 (k 0 ) − Σ (0) E0 (k 0 ),(A4)
where both Σ (g) E0 (k 0 ) and Σ (0) E0 (k 0 ) are calculated numerically. The real part of this quantity is plotted in Fig. 10 in the plane-wave limit σ = ∞. Σ (g) increases over a few mean scattering times and then saturates at roughly twice its initial value.
It is easy to compute Σ (g) at t = 0 from the GPE. The result is:
Σ (g) (t = 0) = E NL (t = 0) = gρ 0 ,(A5)
where the first equality is valid for any initial state, while the second holds only for a plane wave. At time longer than the scattering time, the specklelization phenomenon described in Sec. IV C is responsible for a doubling of the nonlinear energy. It is thus very natural, and fully confirmed by the numerical results in Fig. (10) as well as by a theoretical approach [38] to have:
Σ (g) (t τ 0 ) = E NL (t τ 0 ) = 2gρ 0 .(A6)
This close connection between the nonlinear energy and the nonlinear part of the self-energy also exists at intermediate times, as shown in Fig. 10, where we also plot numerically computed nonlinear energies. After an initial growth, both Σ (g) (t) and E NL (t) saturate around 2gρ 0 and follow a close evolution (even though the growth rate of Re Σ (g) is slightly lower than for the nonlinear energy). Altogether, it suggests that Re Σ (g) and E NL may have a similar status for the problem of interacting disordered systems. This corroborates the conclusion of Sec. IV C, since Re Σ is typically involved in the calculation of any observable, in particular of the CMP.
Modification of the scattering time
We now show that the nonlinear energy, Eq. (15), which governs the long-time behavior of the quantum boomerang effect, also controls the change in the mean scattering time.
As described in Sec. III, during the first part of the time evolution, precisely in the range t < 20 − 30τ 0 , see Fig. 1, the CMP is essentially not modified by interactions. The only difference between the interacting and non-interacting cases is that the mean scattering time and mean free path are increased. We have used the theoretical prediction for x(t) [25] in the non-interacting limit, which depends on τ and , to fit the data in the interacting case in the short-time region, and thus access the dependence of τ and on the interaction strength g. Fits have been performed including weights inversely proportional to the square of the statistical errors in the time window t ∈ [0, t fit ]. Our choice is t fit = 20τ 0 . This value is chosen such that the fits enclose the whole ballistic motion and the beginning of the retro-reflection. The value of t fit slightly influences the fitted parameters, but the changes are smaller than the error originating from the fitting procedure.
The fits return both τ and for a given interaction strength g. This allows us to calculate the average velocity v = /τ . From our data, we observe that the average velocity remains almost unaffected across all studied interaction values, although it is a little higher than the predicted value k 0 /m at the Born approximation. This apparent discrepancy is caused by higher order corrections to the Born approximation and is of the order of 1/k 0 0 1. We can thus restrict the analysis to the mean scattering time τ only.
The fitted values of τ are shown in Fig. 11 as a function of the nonlinear energy averaged over the fit time window, E short NL . To explain these curves, we expand the mean scattering time τ (E E 0 + E NL ) to leading order in E NL E 0 , using the Born approximation, Eq. (3). This yields:
τ τ 0 + 2k 0 γ E NL .(A7)
The linear increase of τ is well visible in Fig. 11. The fact that curves at different σ are slightly shifted upwards is due to the (small) dependence of the g−independent part, τ 0 , on σ, which shows up beyond the Born approximation. Eq. (A7) is shown in Fig. 11, where for τ 0 we use the numerical value of the scattering time for σ = 40/k 0 and g = 0. The agreement between Eq. (A7) and the data is very good. At larger values of E short NL , the curves start to deviate from the linear behavior, bending upwards. This effect is smaller for decreasing wave-packet widths σ. The observed change corresponds to a relatively strong boomerang breakdown, so that the use of the non-interacting theoretical prediction for the fit becomes less reliable.
FIG. 1 .
1CMP time evolution for an initial wave packet with σ = 10/k0, for different values of the interaction strength g.
FIG. 2 .
2Long-time average x vs. interaction strength g for a wave packet with initial width σ = 40/k0. x is in units of 1/k0, and g is in units of 2 k0/m. Data have been averaged over more than 5 · 10 5 disorder realizations.
FIG
. 3. a) Comparison of non-interacting CMP x(t) for wave packets of widths σ = 10/k0, 20/k0, 40/k0. All three curves overlap, indicating that x(t) is σ-independent in the non-interacting limit. b) Same as a) but for non-zero interaction strength g = 2.0 2 k0/m. Here, the saturation point of x(t) is higher for initially narrower wave packets. Thecenter of mass x(t) is in units of 1/k0 and time t in units of m/ k 2 0 . Error bars are only shown in panel a) for σ = 10/k0 to indicate their order of magnitude.
FIG. 4 .
4CMP x(t) vs. time for different initial states chosen such that all of them saturate around the value x = 0.4/k0. Results have been averaged over 16000 disorder realizations. The CMP is shown in units of 1/k0 and time is in units of m/ k 2 0 . Error bars are shown only for σ = 5/k0.
FIG. 5 .FIG. 6 .
56Dependence of the long-time average x , Eq. (8), on the initial interaction energy. For wave packets Eint(t = 0) = g/(2 √ 2πσ), while Eint(t = 0) = gρ0/2 in the plane-wave limit σ = ∞. The interaction energy is expressed in units of 2 k 2 0 /m, and the CMP in units of 1/k0. Error bars represent standard deviation of the averaged points. Break energy E b vs. initial interaction energy Eint(t = 0). The break energy is defined by E b = 2π /t b , where the break time t b , calculated according to Eq. (10), is the characteristic time beyond which the boomerang effect disappears. Energies are in units of 2 k 2 0 /m.
FIG. 7 .
7Interaction energy vs. time for different initial states and interaction strengths. The plot shows data for wave packets with [σ = 10/k0, g = 0.25 2 k0/m], [σ = 40/k0, g = 1.0 2 k0/m], and [σ = ∞, g = 45.0 2 k0/m, ρ0 = 0.00025k0]. Eint is in units of 2 k 2 0 /m and time t in units of m/ k 2 0
FIG. 8 .FIG. 9 .
89Long-time averages x of the CMP for different initial states vs. the nonlinear energy computed at the break time,ENL(t = t b ).x is expressed in units of 1/k0, while energy is in units of 2 k 2 0 /m. All results have been averaged over more than 5 · 10 5 disorder realizations. In this representation, data fall on the same single curve. Comparison of break energy, E b , and nonlinear energy at break time, ENL(t = t b ), for increasing values of the interaction strength. Results are shown for wave packets of size a) σ = 10/k0, b) σ = 40/k0 and c) σ = ∞. Energy is in units of 2 k 2 0 /m, interaction strength g in units of 2 k0/m. All results have been averaged over 5 · 10 5 disorder realizations.
FIG. 10 .
10Numerically calculated real parts of self energy Σ (g) (solid lines) for plane waves vs. time for several values of the interaction strength. In the plot we additionally show nonlinear energies ENL(t) (dashed lines), and indicate as dotted lines the value 2gρ0. Energies are expressed in units of 2 k 2 0 /m and time in units of m/ k 2 0 .
FIG. 11 .
11Fitted values of the mean scattering time τ for wave packets of different initial widths vs. the nonlinear energy averaged over the fit time window, E short NL . Time is in units of m/ k 2 0 and energy in units of 2 k 2 0 /m. The black dashed line shows the prediction of Eq. (A7), where the wave packet with σ = 40/k0 is used for τ0.
ACKNOWLEDGMENTSWe kindly thank PL-Grid Infrastructure for providing computational resources. JJ and JZ acknowledge support of National Science Centre (Poland) under the project OPUS11 2016/21/B/ST2/01086. We also acknowledge support of Polish-French bilateral project Polonium 40490ZE.Appendix A: Short time behaviorWe have seen that interactions modify the long-time dynamics of the center of mass position, and that a change of either parameters σ or g can be encompassed in the use of the nonlinear energy. In this appendix, we show that the very same concept -the nonlinear timedependent energy -also accurately describes the shorttime dynamics of the system, through a change of the real part of the self-energy and of the scattering time.
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|
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"Beating the channel capacity limit for linear photonic superdense coding",
"Beating the channel capacity limit for linear photonic superdense coding",
"Beating the channel capacity limit for linear photonic superdense coding",
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Quantum frequency combs from chip-scale integrated sources are promising candidates for scalable and robust quantum information processing (QIP). However, to use these quantum combs for frequency domain QIP, demonstration of entanglement in the frequency basis, showing that the entangled photons are in a coherent superposition of multiple frequency bins, is required. We present a verification of qubit and qutrit frequency-bin entanglement using an on-chip quantum frequency comb with 40 mode pairs, through a two-photon interference measurement that is based on electro-optic phase modulation. Our demonstrations provide an important contribution in establishing integrated optical microresonators as a source for high-dimensional frequency-bin encoded quantum computing, as well as dense quantum key distribution.
|
10.1364/oe.26.001825
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"https://arxiv.org/pdf/1707.02276v2.pdf"
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1707.02276
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28b9f0a64959820fbffe0b3d0bd5101c52f2475e
|
Beating the channel capacity limit for linear photonic superdense coding
1998. 2005. 2000. 2008. 2007. 2015. 2006. 2001. 2009. 2009. 2003. 2010
. A W Steane ; P
J Shor
J T Preskill ; 5
T.-C Barreiro
P G Wei
R Kwiat ; Gisin
S Thew ; V. Giovannetti
L Lloyd
; E Maccone
R Knill
G J Laflamme
Y J Milburn
R L Lu
Z Y Campbell
Ou
Beating the channel capacity limit for linear photonic superdense coding
Z. Xie, T. Phys. Rev. Lett
619138041998. 2005. 2000. 2008. 2007. 2015. 2006. 2001. 2009. 2009. 2003. 20106 These authors contributed equally to this work * [email protected] References and linksLarge-alphabet quantum key distribution using energy-time entangled bipartite states," Phys. Rev. Lett. 98, 060503 (2007). 8. T. 11. J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum controlled-NOT gate," Nature 426, 264-267 (2003). 12. A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, "High-dimensional single-photon quantum gates: Concepts and experiments," Phys. Rev. Lett. 119, 180510 (2017). 16. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. Langford, and A. Zeilinger, "Discrete tunable color entanglement," 19. C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, "Shaping frequency-entangled qudits," Phys. Rev. A 88, 032322 (2013). 20. C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, "Generation of multiphoton entangled quantum states by means of integrated frequency combs," Science 351, 1176-1180 (2016). 21. F. MazeasOCIS codes: (2700270) Quantum Optics(2705585) Quantum information and processing(1904410) Nonlinear optics, parametric processes
Quantum frequency combs from chip-scale integrated sources are promising candidates for scalable and robust quantum information processing (QIP). However, to use these quantum combs for frequency domain QIP, demonstration of entanglement in the frequency basis, showing that the entangled photons are in a coherent superposition of multiple frequency bins, is required. We present a verification of qubit and qutrit frequency-bin entanglement using an on-chip quantum frequency comb with 40 mode pairs, through a two-photon interference measurement that is based on electro-optic phase modulation. Our demonstrations provide an important contribution in establishing integrated optical microresonators as a source for high-dimensional frequency-bin encoded quantum computing, as well as dense quantum key distribution.
Introduction
Quantum information processing (QIP) has gained massive attention in recent years as it promises to solve some exponentially hard problems in polynomial time through quantum computation [1][2][3], as well as having other unique capabilities such as fully secure communications through quantum key distribution [4][5][6][7][8], and enhanced sensing through quantum metrology [9]. Typical QIP systems are based on two-level quantum states, also called qubits. To simplify the complexity of quantum circuits [10,11] and increase the practicality of quantum computation, high-dimensional entangled states (entangled qudits) are strong candidates as a result of their robustness and stronger immunity to noise, compared to two-dimensional systems [12][13][14][15].
In photonics, amongst different degrees of freedom capable of high-dimensionality, the frequency domain-using single or entangled photons in a coherent superposition of multiple frequency bins-offers both compatibility with fiber transmission and more robust and scalable systems because it does not require stabilization of interferometers or complex beam shaping [14,16]. But while frequency-bin entangled photons (also referred to as biphoton frequency combs or BFCs) have been explored through spontaneous parametric down-conversion (SPDC) together with cavity and programmable spectral filtering [17][18][19], the bulk platform is faced with the drawback of low scalability and high cost. To overcome these disadvantages, integrated optical microresonators offer a solution that is highly compatible with semiconductor foundries. Such chipscale devices have been used to generate entangled photons with a comb-like spectrum [20][21][22]. Time-bin entanglement for a single comb line pair from microresonators has been verified in [20,21,23], and in [22] time-bin entanglement was demonstrated for multiple comb line pairs simultaneously. Yet these studies did not show the ensuing photon states to be in a coherent superposition of multiple frequency-bins. The difficulty of this measurement stems from the large Free Spectral Range (FSR) of conventional microring resonators (typically a few hundred GHz) which results in temporal correlation trains with periods of order several picoseconds, much faster than the timing resolution of standard single-photon detectors (∼100 ps). As a result, direct detection of the comb-like photon pairs is incapable of showing spectral phase sensitivity, a condition required to prove frequency-bin entanglement.
Phase modulation has been used to mix frequency states of biphotons generated by SPDC [18]; frequency-bin entanglement was tested through analysis of two-photon interference as a function of the phase modulation amplitude. Here we use phase modulation to overlap sidebands from either two or three different comb line pairs which are preselected and adjusted for equal amplitude by a programmable pulse shaper. This provides an indistinguishable superposition of frequency states for two-photon interference measurements which prove phase coherence and frequency-bin entanglement in two dimensions (qubits) and three dimensions (qutrits) for a silicon nitride on-chip BFC. In contrast to [18], our approach provides a close analog with Franson interferometry methods that have been widely used for characterization of time-bin entanglement [20][21][22][23]. We presented a subset of these results in [24]; earlier we demonstrated the feasibility of this approach in measurements involving entangled photons generated via SPDC in [25]. A similar technique was developed independently and presented in [26], exploring frequency-bin entanglement for a Hydex microring resonator with 200 GHz FSR. Our experiments explore a larger microresonator with ∼50 GHz FSR. Due to the denser resonance spacing, we are able to identify up to 40 frequency modes from the Joint Spectral Intensity (JSI), a factor of 4 higher than in [26], which suggests substantial potential to push towards higher dimensionality. From a practical perspective, the more closely spaced resonances should provide a better match to the capabilities of practical phase modulator technology, allowing a greater number of frequency modes to be superimposed for future studies of higher dimensionality entanglement. Demonstration of frequency-bin entanglement is a major step in qualifying integrated biphoton frequency comb sources for applications in scalable high capacity quantum computation [27] and dense quantum communications [28].
5 ! p ! S k S k+1 I k+1 I k c. b.
Experiments
For our experiments, we use a silicon nitride microring resonator [ Fig. 1(a)] with a loaded quality factor of ∼ 2 × 10 6 to generate entangled photons. The field possible within the microring corresponds to resonant modes with linewidths of ∼100 MHz and frequency separations just under 50 GHz. Hence when we pump the ring with a tunable continuous-wave laser (operating in the C-band), the spontaneous four wave mixing (SFWM) process leads to the generation of a quantum frequency comb with a frequency spacing and linewidth that mirror the resonance structure of the ring. Further details on the microring and experimental procedures are provided in Appendix A. Generally, the BFC state can be written as:
|Ψ = N k=1 α k |k, k SI (1) |k, k SI = ∫ dΩ Φ (Ω − k∆ω) |ω P + Ω, ω P − Ω SI (2)
where |k, k SI represents the signal and idler photons from the k th comb line pair, α k is a complex number describing the amplitude and phase of the k th comb line pair and N is the total number of mode pairs, Φ(Ω) is the lineshape function, ∆ω is the FSR and ω P is the pump frequency. The coherent superposition of |k, k SI states implied by Eq. (1) requires phase coherence between the frequency mode pairs, i.e., the different |k, k SI must be able to interfere.
Joint spectral intensity
We characterized the spectro-temporal correlations between combinations of frequency modes spanning a 38×38 space (signal and idler lines 3-40). Using a programmable pulse shaper [29] as a tunable frequency filter, we route different signal and idler photons to a pair of single-photon detectors (SPDs) and record the relative arrival time of each photon pair with a Time Interval Analyzer (TIA). As expected, we observe tight temporal correlations only between energy matched comb lines spanning up to the 40 th mode, as presented in the form of the JSI in Fig. 1(b); the high diagonal coincidences reflect the energy matching in the SFWM process. The calculated lower bound of the Schmidt number for this JSI is k min = 20, which is a figure of merit for the degree of frequency correlations [30]. Here we note that the JSI, unlike the joint spectral amplitude, lacks any phase information and cannot show phase coherence between different frequency mode pairs.
Two-dimensional frequency-bin entanglement
To show phase coherence between different comb line pairs, we implement the setup depicted in Fig. 1(d). The output of the microring is coupled into pulse shaper 1, where in the first experiment we select only comb line pairs 6 (S 6 I 6 ) and 7 (S 7 I 7 ). Subsequently, we will use this pulse shaper to apply optical spectral phase to the comb lines. We also note that we use the first pulse shaper to equalize the contribution of the modes to coincidence counts. By doing so, we are making sure that |α k | = |α k+1 | for the rest of the experiments, which optimizes contrast in quantum interference. The selected lines are then coupled into an electro-optic phase modulator, which creates optical sidebands at frequency offsets equal to multiples of the radio frequency (rf) of the driving sinusoidal waveform, which we set to yield sidebands at half the spacing of the BFC [ Fig. 1(c)]. Then with pulse shaper 2, we pick out the sidebands which overlap midway between S 6 -S 7 and I 6 -I 7 [solid blue curves in Fig. 1(c)], and route them to the SPDs and the TIA. Our frequency-bin entanglement verification scheme is a frequency domain analog of the Franson interferometry approach [31] widely used to verify time-bin entanglement [see Fig. 2]. In Franson interferometry the two time-bin input state passes through an imbalanced interferometer with time delay τ equal to the time difference between the time bins. This produces 3 different states projections {|1 , |2 , |S } at the output, where |S is the superposition state defined as
|S = 1 √ 2 |1 + e iφ |2(3)
and φ is a relative phase varied in one of the interferometer arms. In our scheme, we pass a two frequency-bin input state with frequency spacing ∆ f through the phase modulator, which produces upper and lower sidebands at frequency offsets ±∆ f /2 from each of the parent signals and idlers. In Fig. 2 we label this operation as a "frequency splitter". The upper sideband from one parent signal (idler) frequency overlaps with the lower sideband from the other parent signal (idler) frequency. Accordingly, at the output of the frequency splitter, we will have 3 different state projections {|1 , |2 , |S } where |S is the superposition state again defined as in Eq. (3), but now with φ corresponding to a phase imposed onto the biphoton by the first pulse shaper prior to the phase modulator. We can apply different relative phases between the parent frequency bins, and therefore the superposition state |S can have different representations according to Eq.
(3). We note that unlike Franson interferometry, where phase stabilization is needed, here the phases in our frequency interferometry approach are intrinsically stable.
To be able to measure the optimum frequency overlap and maximize the indistinguishability between different phase modulation sidebands, first we apply a relative phase shift of π between S 6 I 6 and S 7 I 7 using pulse shaper 1-inducing a π/2 phase on both S 6 and I 6 -to create a destructive interference between these two modes. We proceed to measure the coincidences as we sweep the rf frequency to yield a sideband separation from 24.54 to 25.14 GHz. We observe a dip with a maximum visibility of 89% at 24.84 GHz, as shown in Fig. 3(a). The full width at half maximum of this dip is measured to be ∼100 MHz, similar to the resonance linewidth of the microring. We note that background accidentals were subtracted from the plot in Fig. 3(a) and subsequent results in the rest of the paper, where the coincidence to accidental ratio was about 2:1. This reduction in coincidence to accidental ratio in the phase measurement experiments compared to the JSI measurement is due to the additional loss that the extra pulse shaper and phase modulator introduce to our biphotons; as a consequence we are forced to use higher pump power and biphoton flux, which reduces the ratio. Now that we have superposition of the sidebands, we should be able to observe an interference pattern by changing the relative phases of the comb line pairs. Using the first pulse shaper to vary the phases of S 7 and I 7 simultaneously, we obtain a sinusoidal interference pattern in the measured coincidences [ Fig. 3(b)]. The resulting visibility of 93% ± 13% shows strong phase coherence between the comb line pairs S 6 I 6 and S 7 I 7 . Following the same procedure but selecting comb line pairs S 5 I 5 and S 6 I 6 and sweeping the phases of S 6 and I 6 simultaneously, we obtain a visibility of 86% ± 11% [ Fig. 3(c)]. Our results establish a two-photon interferometry approach for frequency-bin entangled photons that is in close analogy with the Franson (time-imbalanced) interferometer approach widely used for characterization of time-bin entangled photons [31].
0 1 2 3 ! p ! |1i I |2i I |+i I |Li I |Li S |+i S |1i S |2i S I 6 I 7 S 7 S 6
Quantum state tomography
We perform quantum state tomography by measuring a complete set of 16 projections of the two-qubit entangled state [32,33] which allows us to estimate the density matrix. We performed coincidence measurements between signal and idler photons in the 16 possible combinations of the states {|1 , |2 , |L , |+ }. Here, |L and |+ are the superposition states in Eq. (3) when φ is equal to π/2 and 0, respectively, as shown in Fig. 4. Because we can make an exact analogy between our approach for projecting frequency-bin qubits and the Franson interferometry approach for projecting time-bin qubits, we can perform quantum state tomography of two-photon frequency-bin qubit states using an exact transcription of the measurement protocol for two-photon time-bin qubit states detailed in [33]. The estimated real and imaginary parts of the density matrix are shown in Figs. 5(a) and 5(b), respectively. (See Appendix B for more details). To evaluate the amount of entanglement in the measured two-qubit state, we use the Peres-Horodecki criterion [34,35] and calculate an entanglement monotone called negativity. The negativity of a density matrixρ is defined as:
0 1 2 3 ! p ! I 6 I 7 S 7 S 6 S 5 I 5N(ρ) = 3 i=0 |λ i |−λ i 2 ,
where λ i are the eigenvalues of the partial positive transposed version ofρ. A two-qubit density matrix is separable iff N(ρ) = 0, and N(ρ) > 0 signifies entanglement. For a maximally entangled state N(ρ) = 0.5, and for the experimentally recovered state given in Appendix B we find N(ρ) = 0.34, strongly indicating inseparability.
Three-dimensional frequency-bin entanglement
The results presented so far have been for two-dimensional quantum states. Our observation of strong interference contrast involving comb line pairs S 5 I 5 -S 6 I 6 and S 6 I 6 -S 7 I 7 individually suggests phase coherence across lines 5, 6 and 7 jointly. For a proof of such high-dimensionality, however, we must examine phase coherence across the selected comb line pairs simultaneously.
Here we consider a biphoton state initially made up of three comb line pairs (two entangled qutrits). We use the first pulse shaper to select the comb line pairs S 5 I 5 , S 6 I 6 and S 7 I 7 ; after the phase modulator, we overlap the first sidebands for the 5 th and 6 th comb line pairs together with the third sideband from the 7 th comb line pair. In order to ensure equal mixing weights for all three sidebands, we send a continuous-wave test laser through the modulator and adjust the electrical drive power such that the first and third phase modulation sidebands are equalized, as verified with an optical spectrum analyzer. We also use the first pulse shaper to balance the intensities of the biphoton sideband pairs such that individually they each contribute equal coincidence counts (so the three diagonal terms in the JSI are equal), thereby maximizing the potential Bell inequality violation. Additionally, we compensate for the relative phases on the comb line pairs induced by fiber dispersion. Now we use the second pulse shaper to select the overlapping sidebands from the signal and idler triplets [blue curves in Fig. 6], which arise from an indistinguishable superposition of contributions from S 5 , S 6 , S 7 and I 5 , I 6 , I 7 , respectively. Pulse shaper 1 places spectral phases on the signal and idler lines such that the ideal state after the second pulse shaper can be written in the form |ψ ∝ |5, 5 SI + e i(φ S +φ I ) |6, 6 SI + e i2(φ S +φ I ) |7, 7 SI . Extensive numerical searches [36] suggest that the largest violation of the 3-dimensional Bell inequality is realized by measurement bases with the property that the phase applied to the 7 th signal and idler should be twice the phase put on the 6 th comb line pair [37]. Now, by setting the phase parameters φ S and φ I to appropriate specific values, we construct a three-dimensional CGLMP inequality (I 3 ≤ 2) adapted from [36] and described in detail for time-bin and frequency-bin entangled photons in [37,38], respectively (see Appendix C). We calculate I 3 = 2.63 ± 0.2 which surpasses the classical limit of 2 by more than three standard deviations. This shows a phase coherence spanning three comb line modes and validates high dimensional frequency-bin entanglement for our BFC.
Discussion
While we have demonstrated frequency-bin entanglement for up to 3 dimensions, it is of great interest to extend this scheme to investigate entanglement and reconstruct density matrices for even higher dimensions. Significant improvements to the experimental apparatus that would help in this endeavor can be readily foreseen. The most obvious would be to replace the InGaAs single-photon detectors with superconducting nanowire single-photon detectors (SNSPDs) [26]. Such detectors can provide quantum efficiencies >80% and dead times on the order of 100 ns. Therefore, with an upgrade to SNSPD detectors, the count rate in our experiments would be increased by an impressive factor of ∼100 (a factor of 10 from the improved efficiency of two detectors and another factor of 10 from reduced dead time). Furthermore, since the SNSPDs have only <100 dark counts/sec, three orders of magnitude better than our current detectors, the background counts should be strongly reduced. These factors would allow us to reach significantly higher visibilities even without background subtraction. Additional enhancement is possible using parallel detection. Commercial pulse shaper technology supports programmable routing of different frequency channels to more than one dozen different output fibers. As an example, in our two qubit frequency-bin quantum state tomography scheme, if all six output frequency channels (three for signal, three for idler) were routed to six different output fibers connected to parallel SNSPDs and multi-channel timing electronics, it should be possible to reduce the number of measurement cycles from 36 to 4, providing a further factor of 9 improvement. This is similar to the measurement speed-up reported in two qubit time-bin quantum state tomography [33] where |1 , |2 and |S photons are time resolved and recorded in the same measurement cycle. Parallel detection schemes would also be beneficial for higher dimensional experiments. Although the phase modulator spreads energy from individual signal or idlers into multiple sidebands, only one sideband per signal or idler is used in the current experiments; energy spread to the unused sidebands is lost. For example, our frequency conversion efficiency using the PM is currently about 30% when we optimize photon transfer to the first sideband and about 15% when we transfer to both first and third sidebands. With parallel detection with a sufficient number of detectors, multiple sidebands lying between original biphoton comb lines could be used, substantially mitigating unnecessary loss and opening the door to stronger phase modulation to construct superpositions of a larger number of frequency bins. Tailoring the rf waveform driving the phase modulator could also contribute to improving efficiency [27].
Algorithmic innovations may also aid in quantifying frequency-bin entanglements over larger subspaces. The number of measurements required to fully reconstruct the density matrix through quantum state tomography grows rapidly with increased dimensionality. New methods which provide bounds on high dimensional entanglement based on measurements that only partially sample the density matrix [39] should provide a more favorable scaling.
Finally, we note that while the on-chip biphoton source is fairly efficient, we incur a large loss of about 15 dB simply due to the insertion loss of the discrete off-chip components (phase modulator and two pulse shapers). It will be interesting to investigate the potential for reducing this loss through photonic integration. Quantum photonic chips based on arrays of interferometers are now an active area of research [40]. For studies of frequency bin entanglement, a more appropriate architecture could include the microresonator biphoton source and on-chip phase modulators and pulse shapers. The pulse shapers, for example, could be constructed from thermally tunable arrays of microring resonators, which have been demonstrated with both spectral amplitude [41] and spectral phase shaping functionalities [42] for applications in rf photonic and optical signal processing.
Conclusion
In summary, our research offers a scalable integrated platform to generate high dimensional photonic states in a superposition of different frequency bins. Due to its robustness and weak interaction with the environment, the frequency degree of freedom in photonic states is a potential candidate to move this research towards experimental realization of high dimensional quantum computing protocols. The use of these high dimensional entangled states offers a clear path to having more complex quantum circuits within reach, as well as denser information encoding [5,13].
Appendices:
A. Experimental details
Our scheme for characterizing the frequency bin entanglement is based on commercial instrumentation such as pulse shapers, phase modulators, and single photon detectors, all of which are fiber pigtailed and compatible with operation in the lightwave C band. The microring resonator is formed from waveguides with dimensions 1.6 µm wide by 870 nm thick fabricated in a SiN film. In-and out-coupling to the SiN chips are performed with lensed fibers. U-grooves etched into the chip [see Fig. 1(a)] provide support points that enhance the stability of the coupling [43]. Interference filters [not depicted in Fig. 1(d)] follow the microring and strongly attenuate the pump line; sideband pairs S 1 I 1 and S 2 I 2 are also attenuated in the process. Pulse shapers 1 and 2 (Finisar WaveShaper models 1000S and 4000S, respectively) allow us to perform programmable filtering with 10 GHz resolution and 1 GHz addressability over the wavelength ranges 1527.4-1567.5 nm and 1527.4-1600.8 nm, respectively. Pulse shaper 2 also supports programmable frequency selective routing to four fiber output ports (only two are used in the current experiments). Based on the availability of phase modulators (lithium niobate integrated optic modulators from EOSpace), we used a 20-GHz bandwidth modulator for the frequency qubit measurements of Fig. 3. We modulated with an rf frequency of 12.4 GHz and used the ±2 sidebands corresponding to ±24.8 GHz frequency offset to get the frequency overlapped superposition. For the frequency qutrit measurements of Fig. 6, a higher (40 GHz) bandwidth modulator was available, allowing us to modulate directly at 24.8 GHz. An advantage of using a relatively large microresonator with correspondingly small (49.6 GHz) free spectral range is its relatively good match with practical rf modulation frequencies; by working with low-order modulation sidebands, we are able to shift a relatively large fraction of the signal and idler power into the sidebands used for superposition. Coincidences are measured using a pair of InGaAs single-photon avalanche diodes (Aurea Technology) connected to a two-channel time-to-digital converter module (PicoQuant HydraHarp). The detectors have specified 25% quantum efficiency, 1000 ns dead time, and 10 5 dark counts/sec with a gate frequency of 1.25 MHz.
Using this experimental setup, we first find the best rf drive frequency for maximum indistinguishability between the frequency bins S 6 I 6 and S 7 I 7 . In this process, we program pulse shaper 1 (taking into account the estimated dispersion of fiber leads) for a phase difference of π between S 6 I 6 and S 7 I 7 ; this condition is expected to yield destructive interference and the minimum number of coincidences after the rf frequency is optimized [see Fig. 3(a)]. To obtain a complete interference pattern, we sweep the phase of S 7 I 7 over a range of 2π [ Fig. 3(b)], recording coincidences for ten minutes at each phase point. To extract the visibilities, we use the expression V = (C max − C min ) /(C max + C min ), where C max and C min correspond to the phase settings where the maximum and minimum coincidences are expected. This procedure for estimating the visibility is repeated three times to yield an average and standard deviation for the visibility estimate.
The effect of dispersion due to fiber leads can be seen in Figs. 3(b) and 3(c) as a shift in the sinusoidal interference patterns. Without dispersion, the maxima of the interference patterns should be at zero phase, but we can see a shift of ∼ π/4 in the measured interference patterns. From this shift, the amount of standard single mode fiber in our setup can be estimated (∼35 m). We use this calculated fiber length to compensate for dispersion (by programming the pulse shaper for additional phase to offset the frequency dependent phase from the dispersion) in our measurement of the three-dimensional CGLMP inequality described in section 2.4.
B. Density matrix reconstruction
The measurement protocol and coincidence count data for the quantum state tomography (Section 2.3) are given in Table1. Table 1 may be understood as follows. Since the two-qubit density matrix is 4 × 4, we require a complete set of 16 projections |Ψ ν (ν = 1 : 16) , written in terms of its basis coefficients ( 11| Ψ ν , 12| Ψ ν , 21| Ψ ν , 22| Ψ ν ) . We perform these projections by acquiring data in four different phase configurations (φ S , φ I ) = {(0, 0), (0, π/2), (π/2, 0), (π/2, π/2)} , columns 5-8. Here, in performing tomography on the S 6 I 6 -S 7 I 7 qubit pair, φ S and φ I are the signal and idler phases applied to the 7 th comb line pair via pulse shaper 1 in the experimental setup shown in Fig. 1(d). For each projection columns 2 and 3 specify which signal and idler frequency channel are routed to the respective single photon detectors. Referring to Fig. 4, |1 and |2 in columns 2 and 3 correspond to unique frequency channels, whereas |+ and |L are both sideband superpositions measured when the same physical frequency channel is routed for detection. Therefore, an entry in column 2 of |+ or |L signifies both routing of the signal superposition frequency channel for detection and application of the appropriate phase to the 7 th signal line (0 phase for |+ , data reported in column 5 or 6; π/2 phase for |L , data reported in column 7 or 8). An entry in column 3 of |+ or |L has similar meaning, but refers to the idler superposition frequency channel (data in column 5 or 7 for |+ , column 6 or 8 for |L ). As an example, for |Ψ 8 we have (φ S , φ I ) = (π/2, 0), column 7, and we obtain:
|Ψ 8 = 1 √ 2 |1 S + e iφ S |2 S . 1 √ 2 |1 I + e iφ I |2 I = 1 2 |1, 1 SI + 1 2 |1, 2 SI + i 2 |2, 1 SI + i 2 |2, 2 SI = 1 2 , 1 2 , i 2 , i 2(4)
In this notation |x, y SI = |x S |y I , in which signal and idler photons are in frequency bins x and y, respectively. Also, as explained in [33], for each of the signal and idler photons, measurement in a nonsuperposition basis (|1 or |2 ) involves a factor of two loss relative to measurement in the superposition channel. This is understood in the time-bin case as the loss incurred at the output beam splitter of the interferometer, since for nonsuperposition bases, half of the photons go to the unused output port. For the superposition cases, with constructive interference such loss is avoided. The same argument holds in our frequency-bin approach. These factors of two that arise for each of signal and idler are accounted for by noting for projections such as |Ψ 1 = |11 , which incur a factor of four loss, coincidences may be measured for each of the four phase configurations. The corresponding coincidence counts are listed in columns 5 to 8 and are added to give a total coincidence count (column 9). Likewise, projections such as |Ψ 6 = |1+ incur a factor of two loss but may be measured in two phase configurations, and projections such as |Ψ 7 = |++ incur no extra loss but are measured in only a single-phase configuration. Overall, 36 independent measurements are performed, and the total number of coincidence counts obtained by adding the entries in columns 5-8 (column 9, n ν ) provides the correct normalization across the different projections.
As in [32,33], we perform a maximum likelihood estimate to obtain the density matrix that best fits our projection measurement data (the n ν ) while satisfying the requirement for a physical density matrix that the eigenvalues lie in the interval [0,1]. This estimation is calculated using the minimization of the following likelihood function: Table 1. Projection measurements for frequency-bin density matrix estimation. For each measurement coincidences were acquired over a 10-minute period. A dash (-) indicates that the phase setting indicated by the respective column is not involved in the projection measurement indicated by the respective row; hence coincidence counts were not obtained.
L = 16 ν=1 C Ψ ν |ρ |Ψ ν − n ν 2 2C Ψ ν |ρ |Ψ ν(5)
Signal where C is the normalization constant defined by:
Idler (φ S , φ I ) ν Photon Photon |Ψ ν (0, 0) 0, π 2 π 2 , 0 π 2 , π 2 n ν 1 |1 |1 (1,C = 4 ν=1 n ν(6)
As a result of this optimization, we found the following physical density matrix:
ρ = 0.4388 + 0.0000i −0.0115 − 0.0699i −0.0721 − 0.0193i 0.3745 + 0.0166i −0.0115 + 0.0699i 0.0574 + 0.0000i 0.0279 − 0.0244i 0.0084 − 0.0227i −0.0721 + 0.0193i 0.0279 + 0.0244i 0.0281 + 0.0000i −0.0280 − 0.0211i 0.3745 − 0.0166i 0.0084 + 0.0227i −0.0280 + 0.0211i 0.4757 + 0.0000i (7)
C. CGLMP inequality for two qutrits
In this section, following [37,38], we describe how we evaluate the CGLMP inequality for two entangled frequency-bin qutrits. As described in the main text, we measure coincidences between signal and idler frequency channels selected to represent superpositions from three parent signal and idler frequencies, respectively. Reference [37] evaluated the three-dimensional Bell's inequality for time-bin entangled qutrit states using three-arm interferometers coupled to three different output ports via a 3 × 3 splitter. They constructed a CGLMP inequality expressed in a form equivalent to the following:
I 3 = 3 P 11 (0, 0) + P 21 (0, 1) + P 22 (0, 0) + P 12 (0, 0) −3 P 11 (0, 1) + P 21 (0, 0) + P 22 (0, 1) + P 12 (1, 0) ≤ 2(8)
where P xy (a, b) is the probability of getting a coincidence count between detector a on the signal and detector b on the idler side, using the measurement basis A x and B y for signal and idler photons, respectively. We note that the original form of the 3-dimensional Bell inequality consists of 24 total measurement probabilities [36]; the reduction to 8 terms [Eq. (8)], however, is valid under the assumption of an unbiased 3 × 3 splitter and an input quantum state containing sufficient symmetries. In particular, as we show below, the above reduction holds for a density matrixρ taken to be the incoherent mixture of a maximally entangled state and white noise [37]. Such an assumption is physically reasonable and common in visibility-based Bell-violation tests. For the time-bin case, the measurement bases correspond to the sets of phases applied to short, medium, and long interferometer arms. The particular sets of phases used are [A 1 = (0, 0, 0), A 2 = (0, π/3, 2π/3)] for the signal and [B 1 = (0, π/6, π/3), B 2 = (0, −π/6, −π/3)] for the idler. These choices of phases have been shown to give the largest violation of the CGLMP inequality for a maximally entangled state [37]. In the classical picture, if the signal and idler are two independent systems, meaning a measurement on the signal does not affect the idler, and vice versa, then I 3 ≤ 2. However, for an entangled state this classical limit may be violated, and with the set of phases specified, a maximum violation I 3 max = 2.872 is predicted for a maximally entangled state. In our frequency bin case, the different measurement bases are constructed by putting different sets of phases on different comb line triplets. For example, for a triplet consisting of comb lines 5-7 as in our experiment, for signal measurement basis A 2 we would place phases (0, π/3, 2π/3) on signal lines 5, 6, and 7, respectively. However, unlike the 3 × 3 beam splitter case, we have only a single detector each for signal and idler. This can be accounted for by imposing additional phases on the comb lines according to the equivalent transfer function of the 3 × 3 beam splitter [44,45]. Equalizing the power in the ±1 and ±3 phase modulation sidebands gives us the ability to perform an unbiased beam splitter in frequency, thereby satisfying one of the key assumptions behind the reduced form [Eq. (8)]. The phases are chosen from {0, −2π/3, 2π/3} according to which "beam splitter output" is involved in the projection that we are mapping from the three-output time-bin case to our one-output frequency-bin case [37,38]. In this way, we adapt the CGLMP inequality for time-bin entangled photons to our frequency bins by applying different sets of phases to our comb lines [38]. For our experiment involving comb lines 5, 6, and 7, the phases applied to signal and idler lines k are given by:
Φ x S k (a) = (k − 5)φ x S (a) = 2π 3 (k − 5) (a + α x ) (9) Φ y I k (b) = (k − 5)φ y I (b) = 2π 3 (k − 5) −b + β y(10)
Here, Φ x S k (a) and Φ y I k (b) are the phases applied to the k th signal and idler frequency bin, respectively, expressed in terms of fundamental phases φ x S (a) and φ y I (b) for each basis choice x for signal and y for idler; the a, b = {0, 1, 2} correspond to the output channel used in the 3 × 3 splitter version of the projection. The α x and β y parameters relate to the measurement bases and are chosen as α 1 = 0, α 2 = 1/2, β 1 = 1/4 and β 2 = −1/4 . These correspond to the measurement bases A x and B y discussed above and yield phase triplets A x = (0, (2π/3) α x , (4π/3) α x ) and B y = (0, (2π/3) β y , (4π/3) β y ). These are modified by the addition of phase triplets (0, (2π/3) a, (4π/3) a) and (0, (−2π/3) b, (−4π/3) b) to signal and idler, respectively, in accord with the a and b parameters.
As our quantum state, we assume a density matrix of the form
ρ = λ |ψ ψ| + (1 − λ)ρ N(11)
with 0 ≤ λ ≤ 1 , where |ψ is the maximally entangled state represented as:
|ψ = 1 √ 3 |5, 5 SI + |6, 6 SI + |7, 7 SI(12)
andρ N is our particular noise model, taken to be symmetric, or white: ρ N = 1 9 |5, 5 5, 5| SI + |5, 6 5, 6| SI + |5, 7 5, 7| SI + |6, 5 6, 5| SI + |6, 6 6, 6| SI + |6, 7 6, 7| SI + |7, 5 7, 5| SI + |7, 6 7, 6| SI + |7, 7 7, 7| SI
Following the discussion surrounding Eqs. (9) and (10), the projective measurements done on each photon are:
Therefore, the probabilities measured are given by:
Combined, Eqs. (17) and (18) justify the simplification from a full 24-term Bell parameter to the 8-term I 3 in Eq. (8), which is based on symmetries in the combinations of outcomes a and b.
The noise terms show no dependence on a and b [Eq. (17)], while the pure state contribution [Eq. (18)] varies only via the difference a − b, modulo 3. Thus, under our particular noise model, we only need to obtain 8 probability estimates. This model is consistent with the measured JSI, which shows a roughly constant background on the off-diagonal terms within the two-qutrit subspace. (We note that a Bell test with no such symmetry assumptions would be possible by testing all 24 projections separately.) Table 2. Parameters for evaluations of the CGLMP inequality. The coincidences were measured in 10-minute spans; measurements were done three times to obtain standard deviations. To achieve the maximum and minimum number of coincidences, the phases of φ x S (a) = φ y I (b) = 0 and φ x S (a) = φ y I (b) = π/3 were put on the biphotons, respectively. To calculate each of the probabilities that appear in Eq. (8), the corresponding coincidence counts have to be divided by the maximum number of coincidences P max (0, 0).
Term
x y a b φ x S (a) φ y I (b) Coincidences P 11 (0, 0) 1 1 0 0 0 π/6 150±10 P 21 (0, 1) 2 1 0 1 π/3 −π/2 141±23 P 22 (0, 0) 2 2 0 0 π/3 −π/6 152±21 P 12 (0, 0) 1 2 0 0 0 −π/6 146±16 P 11 (0, 1) 1 1 0 1 0 −π/2 54±4 P 21 (0, 0) 2 1 0 0 π/3 π/6 33±6 P 22 (0, 1) 2 2 0 1 π/3 −5π/6 49±12 P 12 (1, 0) 1 2 1 0 2π/3 −π/6 32±10 P max (0, 0) --0 0 0 0 160±18 P min (0, 0) --0 0 π/3 π/3 15±13 In the Table 2, the first column corresponds to the individual terms in Eq. (8). Columns 6 and 7 evaluate Eqs. (9) and (10) to obtain the signal and idler phase parameters φ x S (a) and φ y I (b). Our coincidence data are given in column 8. We calculate I 3 = 2.63 ± 0.2 which is more than three standard deviations away from the classical limit and indicates three-dimensional frequency-bin entanglement.
Funding
National Science Foundation (NSF) (ECCS-1407620), DARPA PULSE Program (W31P40-13-1-0018), Oak Ridge National Laboratory (ORNL).
Fig. 1 .
1(a) Microscope picture of the microring and U-grooves to support fiber coupling. (b) Joint spectral intensity for comb line pairs from 3 to 40. The background accidentals are not subtracted in this measurement and the coincidence to accidental ratio is about 10:1. (c) Illustration of biphoton spectrum after phase modulation. (d) Experimental setup.
Fig. 2 .Fig. 3 .
23Analogy between a 1-bit delay interferometer for forming projections of time-bin qubits and a frequency splitter for forming projections of a frequency-bin qubit. The green frequency bins after the frequency splitter are phase modulation sidebands from |1 and |2 . (a) Coincidence dip as a function of sideband frequency to maximize the indistinguishability. (b) Coincidences of the S 6 I 6 and S 7 I 7 superposition versus phase applied on S 7 I 7 . (c) Coincidences of the S 5 I 5 and S 6 I 6 superposition versus phase applied on S 6 I 6 . The coincidences reported are in (a) 20 minutes. and (b), (c) 10 minutes. and after background subtraction. Each data point was measured three times to obtain the standard deviation indicated by the error bars.
Fig. 4 .Fig. 5 .
45Phase modulation scheme for quantum state tomography. Red peaks represent the input signal and idler, each of which is in one of two frequency bins. Blue curves represent projections of signal and idler after the phase modulator (frequency splitter) into three new frequency positions. Solid blue is a projection of the superposition state; dashed blue peaks represent a projection from a single signal or idler frequency bin. (a) Real and (b) imaginary parts of the estimated density matrix for comb line pairs S 6 I 6 and S 7 I 7 .
Fig. 6 .
6Illustration of overlapped phase modulation sidebands for comb line pairs S
|5 S + e iφ x S (a) |6 S + e i2φ x S (a) |7 S 5| S + e −iφ x S (a) 6| S + e −i2φ x S (a) 7| S I + e iφ y I (b) |6 I + e i2φ y I (b) |7 I 5| I + e −iφ y I (b) 6| I + e −i2φ y I (b) 7| I
AcknowledgmentWe acknowledge Allison L. Rice for designing the graphics of the experimental setup inFig. 1. J.A.J. acknowledges support by Colciencias and Fulbright Colombia. A portion of this work was performed at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under contract no. DE-AC05-00OR22725.
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[] |
[
"BIFURCATIONS OF MULTI-VORTEX CONFIGURATIONS IN ROTATING BOSE-EINSTEIN CONDENSATES",
"BIFURCATIONS OF MULTI-VORTEX CONFIGURATIONS IN ROTATING BOSE-EINSTEIN CONDENSATES"
] |
[
"C García-Azpeitia ",
"And D E Pelinovsky "
] |
[] |
[] |
We analyze global bifurcations along the family of radially symmetric vortices in the Gross-Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency Ω. The families are constructed in the smallamplitude limit when the chemical potential µ is close to an eigenvalue of the Schrödinger operator for a quantum harmonic oscillator. We show that for Ω near 0, the Hessian operator at the radially symmetric vortex of charge m0 ∈ N has m0(m0 +1)/2 pairs of negative eigenvalues. When the parameter Ω is increased, 1 + m0(m0 − 1)/2 global bifurcations happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross-Pitaevskii equation and the zeros of Hermite-Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex (m0 = 1), the asymmetric vortex pair (m0 = 2), and the vortex polygons (m0 ≥ 2). 1 The threshold value of the rotation frequency for the bifurcation of local minimizers in[5,29]is smaller than the first critical value in[13,14], at which the charge-one vortex becomes the global minimizer of energy.
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10.1007/s00032-017-0275-8
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[
"https://arxiv.org/pdf/1701.01494v2.pdf"
] | 119,590,442 |
1701.01494
|
0407027b6771d0eab6fcdd80cd9bc8d130339f28
|
BIFURCATIONS OF MULTI-VORTEX CONFIGURATIONS IN ROTATING BOSE-EINSTEIN CONDENSATES
C García-Azpeitia
And D E Pelinovsky
BIFURCATIONS OF MULTI-VORTEX CONFIGURATIONS IN ROTATING BOSE-EINSTEIN CONDENSATES
Gross-Pitaevskii equationrotating vorticesharmonic potentialsLyapunov- Schmidt reductionsbifurcations and symmetries
We analyze global bifurcations along the family of radially symmetric vortices in the Gross-Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency Ω. The families are constructed in the smallamplitude limit when the chemical potential µ is close to an eigenvalue of the Schrödinger operator for a quantum harmonic oscillator. We show that for Ω near 0, the Hessian operator at the radially symmetric vortex of charge m0 ∈ N has m0(m0 +1)/2 pairs of negative eigenvalues. When the parameter Ω is increased, 1 + m0(m0 − 1)/2 global bifurcations happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross-Pitaevskii equation and the zeros of Hermite-Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex (m0 = 1), the asymmetric vortex pair (m0 = 2), and the vortex polygons (m0 ≥ 2). 1 The threshold value of the rotation frequency for the bifurcation of local minimizers in[5,29]is smaller than the first critical value in[13,14], at which the charge-one vortex becomes the global minimizer of energy.
Introduction
This work addresses the Gross-Pitaevskii equation describing rotating Bose-Einstein condensates (BEC) placed in a symmetric harmonic trap. It is now well established from the energy minimization methods that vortex configurations become energetically favorable for larger rotating frequencies (see review [7] for physics arguments). Ignat and Millot [13,14] confirmed that the vortex of charge one near the center of symmetry is a global minimizer of energy for a frequency above the first critical value. Seiringer [30] proved that a vortex configuration with charge m 0 becomes energetically favorable to a vortex configuration with charge (m 0 − 1) for a frequency above the m 0 -th critical value and that radially symmetrically vortices of charge m 0 ≥ 2 cannot be global minimizers of energy. The questions on how the m 0 individual vortices of charge one are placed near the center of symmetry to form an energy minimizer remain open since the time of [13,14,30].
For the vortex of charge one, it is shown by using variational approximations [5] and bifurcation methods [29] that the construction of energy minimizers is not trivial past the threshold value for the rotation frequency, where the radially symmetric vortex becomes a local minimizer of energy 1 . Namely, in addition to the radially symmetric vortex, which exists for all rotation frequencies, there exists another branch of the asymmetric vortex solutions above the threshold value, which are represented by a vortex of charge one displaced from the center of rotating symmetric trap. The distance from the center of the harmonic trap increases with respect to the detuning rotation frequency above the threshold value, whereas the angle is a free parameter of the asymmetric vortex solutions. Although the asymmetric vortex is not a local energy minimizer, it is nevertheless a constrained energy minimizer, for which the constraint eliminates the rotational degree of freedom and defines the angle of the solution family uniquely. Consequently, both radially symmetric and asymmetric vortices are orbitally stable in the time evolution of the Gross-Pitaevskii equation for the rotating frequency slightly above the threshold value [29].
Further results on the stability of equilibrium configurations of several vortices of charge one in rotating harmonic traps were found numerically, from the predictions given by the finitedimensional system for dynamics of individual vortices [2,21,26]. The two-vortex equilibrium configuration arises again above the threshold value for the rotation frequency with the two vortices of charge one being located symmetrically with respect to the center of the harmonic trap. However, the symmetric vortex pair is stable only for small distances from the center and it losses stability for larger distances. Once it becomes unstable, another asymmetric pair of two vortices bifurcate, where one vortex has a smaller-than-critical distance from the center and the other vortex has a larger-than-critical distance from the center. The asymmetric pair is stable in numerical simulations and coexist for rotating frequencies above the threshold value with the stable symmetric vortex pair located at the smaller-than-critical distances [26]. The symmetric pair is a local minimizer of energy above the threshold value, whereas the asymmetric pair is a local constrained minimizer of energy, where the constraint again eliminates the rotational degree of freedom [18].
This work continues analysis of local bifurcations of vortex configurations in the Gross-Pitaevskii (GP) equation with a cubic repulsive interaction and a symmetric harmonic trap. In a steadily rotating frame with the rotation frequency Ω, the main model can be written in the normalized form (1.1) iu t = −(∂ 2 x + ∂ 2 y )u + (x 2 + y 2 )u + |u| 2 u + iΩ(x∂ y − y∂ x )u, (x, y) ∈ R 2 . The associated energy of the GP equation is given by (1.2) E(u) = R 2 |∇u| 2 + |x| 2 |u| 2 + 1 2 |u| 4 + i 2 Ωū(x∂ y − y∂ x )u − i 2 Ωu(x∂ y − y∂ x )ū dxdy.
Compared to work in [29], we do not use the scaling for the semi-classical limit of the GP equation and parameterize the vortex solutions in terms of the chemical potential µ arising in the separation of variables u(t, x, y) = e −iµt U (x, y). The profile U satisfies the stationary GP equation in the form
(1.3) µU = −(∂ 2 x + ∂ 2 y )U + (x 2 + y 2 )U + |U | 2 U + iΩ(x∂ y − y∂ x )U, (x, y) ∈ R 2 .
Local bifurcations of small-amplitude vortex solutions in the GP equation (1.1) have been addressed recently in many publications. We refer to these small-amplitude vortex solutions as the primary branches. Classification of localized (soliton and vortex) solutions from the triple eigenvalue was constructed by Kapitula et al. [16] with the Lyapunov-Schmidt reduction method. Existence, stability, and bifurcations of radially symmetric vortices with charge m 0 ∈ N were studied by Kollar and Pego [20] with shooting methods and Evans function computations. Symmetries of nonlinear terms were used to continue families of general vortex and dipole solutions from the linear limit by Contreras and García-Azpeitia [4] by using equivariant degree theory [15] and bifurcation methods [8]. Existence and stability of stationary states were analyzed in [9,10] with the amplitude equations for the Hermite function decompositions and their truncation at the continuous resonant equation. Vortex dipoles were studied with normal form equations and numerical approximations in [12]. Numerical evidences of existence, bifurcations, and stability of such vortex and dipole solutions can be found in a vast literature [22,23,25,28,31].
Compared to the previous literature, our results will explore the recent discovery of [29] of how bifurcations of unconstrained and constrained minimizers of energy are related to the spectral stability problem of radially symmetric vortices in the small-amplitude limit, in particular, with the eigenvalues of negative Krein signature which are known to destabilize dynamics of vortices [20]. Therefore, we consider bifurcations of secondary branches of multi-vortex solutions from the primary branch of the radially symmetric vortex of charge m 0 ∈ N. The primary branch is parameterized by only one parameter ω := µ + m 0 Ω in the small-amplitude limit, whereas the secondary branches of multi-vortex configurations are parameterized by two parameters ω and Ω.
As a particular example with m 0 = 2, we show that the asymmetric pair of two vortices of charge one bifurcates from the radially symmetric vortex of charge two for Ω below but near Ω 0 = 2. Similarly to the symmetric charge-two vortex [16,20], the asymmetric pair of two charge-one vortices is born unstable but it is more energetically favorable near the bifurcation threshold compared to the charge-two vortex in the case of no rotation (Ω = 0). If the chargetwo vortex is a saddle point of the energy E in (1.2) with three pairs of negative eigenvalues for Ω = 0, it has only one pair of negative eigenvalues for Ω below but near Ω 0 = 2.
We note that the bifurcation technique developed here is not feasible by the methods developed in [16] because of the infinitely many resonances at Ω 0 = 2. Nevertheless, we show that these resonances are avoided for Ω below but near Ω 0 = 2.
For a charge-one vortex, a similar bifurcation happens for Ω below but near Ω 0 = 2, which has been already described in [29] in other notations and with somewhat formal analysis. The results developed here allows us to give a full justification of the results of [29] for a chargeone vortex, but also to extend the analysis to the charge-two vortex, as well as to a radially symmetric vortex of a general charge m 0 ∈ N.
We also consider all other secondary bifurcations of the radially symmetric vortices of charge m 0 ∈ N when the frequency parameter Ω is increased from zero in the interval (0, 2). We show that each bifurcation results in the disappearance of a single pair of negative eigenvalues in the characterization of radially symmetric vortices as saddle points of the energy E in (1.2).
As a particular example, we show that the symmetric charge-two vortex has a bifurcation at Ω near Ω * = 2/3, where another secondary branch bifurcates. The new branch contains three charge-one vortices at the vertices of an equilateral triangle and a vortex of anti-charge one at the center of symmetry. Again, the secondary branch inherits instability of the radially symmetric vortex along the primary branch in the small-amplitude limit. Past the bifurcation point, the radially symmetric vortex of charge two has two pairs of negative eigenvalues. The bifurcation result near Ω * = 2/3 was not obtained in the previous work [16].
In the case of the multi-vortex configurations of the total charge two, we can conjecture that the local minimizers of energy given by the symmetric pair of two charge-one vortices as in [26] can be found from a tertiary bifurcation along the secondary branch given by the asymmetric pair of charge-one vortices. However, it becomes technically involved to approximate the secondary branch near the bifurcation point and to find the tertiary bifurcation point.
The following theorem represents the main result of our paper. A schematic illustration is given on Figure 1. (iii) There exist C m 0 > 0 and D m 0 ≥ 0 such that for small a, 1 + B(m 0 ) global bifurcations occur when the parameter Ω is increased in the interval
[a 2 D m 0 , 2 − a 2 C m 0 ], where B(m 0 ) = 1 2 m 0 (m 0 − 1).
For Ω a 2 D m 0 , the family of radially symmetric vortices has only 2N (m 0 ) negative eigenvalues and a simple zero eigenvalue, and it losses two of these negative eigenvalues past each non-resonant bifurcation point. If 1 ≤ m 0 ≤ 16, the family has 2(m 0 − 1) negative eigenvalues for Ω 2 − a 2 C m 0 . (iv) A new smooth family of multi-vortex configurations is connected to the family of radially symmetric vortices on one side of each non-resonant bifurcation point (of the pitchfork type). Furthermore, on the right (respectively, left) side of the bifurcation point, the new family has one more (respectively, one less) negative eigenvalue compared to the family of radially symmetric vortices.
(v) For a non-resonant bifurcation point Ω m,n ∈ (0, 2) with m > m 0 and n ≥ 0, the new family has a polygon configuration of (m−m 0 ) charge-one vortices surrounding a center with total charge 2m 0 − m. For the "last" bifurcation point Ω m 0 +1,0 = 2 + O(a 2 ), the new family consists of the charge-one asymmetric vortex (m 0 = 1), the asymmetric pair of charge-one vortices (m 0 = 2), and a configuration of vortices near the center of total charge m 0 (m 0 ≥ 3).
Remark 1. By global bifurcation, we mean that the bifurcating family that originates from the family of radially symmetric vortices of charge m 0 either reaches the boundaries Ω = 0 or Ω = 2, diverges to infinity for a value of Ω ∈ (0, 2), or returns to another bifurcation point along the family of radially symmetric vortices of charge m 0 .
Remark 2. For 1 ≤ m 0 ≤ 3, we have D m 0 = 0 in item (iii), therefore, the 1 + B(m 0 ) global bifurcations arise when Ω is increased from Ω = 0 to Ω = 2 − a 2 C m 0 . However, we do not know if D m 0 = 0 in a general case. If D m 0 = 0, up to Z(m 0 ) additional bifurcations may appear if Ω is increased from Ω = 0 to Ω = a 2 D m 0 .
Remark 3. For 1 ≤ m 0 ≤ 3, all bifurcation points are non-resonant in items (iii)-(v). Resonant bifurcation points may exist in a general case for m 0 ≥ 4. In this case, the statements (i)-(iii) remain valid, but for each bifurcation point of multiplicity k, the family of radially symmetric vortices losses 2k negative eigenvalues past the bifurcation point. In the resonant case, the statements (iv)-(v) require further estimates. However, these resonances are unlikely to be present as the more likely scenario is that the multiple eigenvalues at a = 0 split into simple nonzero eigenvalues of order O(a 2 ). From a technical point of view, the proof of Theorem 1 is developed by using the equivariance of the bifurcation problem under the action of the group O(2) × O (2). The global bifurcation result is proven by using the restriction of the bifurcation problem to the fixed-point space of a dihedral group. This restriction leads to a simple eigenvalue in the fixed-point space, which allows us to apply the global Crandall-Rabinowitz result, see Theorem 3.4.1 of [27]. This method is also helpful to get additional information on the symmetries of the bifurcating solutions which is essential to localize the distributions of zeros for the individual vortices in the multi-vortex configurations.
The paper is structured as follows. In Section 2, we review eigenvalues of the Schrödinger operator for quantum harmonic oscillator and give definitions for the primary and secondary branches of multi-vortex solutions. In Section 3, we analyze distribution of eigenvalues of the Hessian operators along the primary branches at the secondary bifurcation points. In Section 4, we justify bifurcations of the secondary branches at the non-resonant bifurcation points by using bifurcation theorems. In Section 5, we study distribution of individual vortices in the multi-vortex configurations along the secondary branches.
Preliminaries
We denote the space of square integrable functions on the plane by L 2 (R 2 ) and the space of radially symmetric squared integrable functions integrated with the weight r by L 2 r (R + ). We also use the same notations for the L 2 -based Sobolev spaces such as H 2 (R 2 ) and H 2 r (R + ). The weighted subspace of L 2 with | · | 2 u L 2 < ∞ are denoted by L 2,2 (R 2 ) and L 2,2 r (R + ). We distinguish notations for the two sets: N = {1, 2, 3, . . .} and N 0 = {0, 1, 2, 3, . . .}. Notation b a means that there is an a-independent constant C such that b ≤ Ca for all a > 0 sufficiently small. If X is a Banach space, notation u = O X (a) means that u X a for all a > 0 sufficiently small. Similarly, ω = O(a) means that |ω| a for all a > 0 sufficiently small.
2.1. Schrödinger operator for quantum harmonic oscillator. Recall the quantum harmonic oscillator with equal frequencies in the space of two dimensions [3,24]. In polar coordinates on R 2 , the energy levels of the quantum harmonic oscillator are given by eigenvalues of the Schrödinger operator L written as
(2.1) L := −∆ (r,θ) + r 2 : H 2 (R 2 ) ∩ L 2,2 (R 2 ) → L 2 (R 2 ),
where ∆ (r,θ) = ∂ 2 r +r −1 ∂ r +r −2 ∂ 2 θ . As is well-known [3,24], the eigenvalues of L are distributed equidistantly and can be enumerated by two indices m ∈ Z for the angular dependence and n ∈ N 0 for the number of zeros of the eigenfunctions in the radial direction. To be more precise, the eigenfunction f m,n for the eigenvalue λ m,n can be written in the form f m,n (r, θ) = e m,n (r)e imθ , m ∈ Z, n ∈ N 0 , where e m,n is an L 2 r (R + )-normalized solution of the differential equation (2.2) −∆ m + r 2 e m,n (r) = λ m,n e m,n (r), ∆ m := ∂ 2 r + r −1 ∂ r − r −2 m 2 with n zeros on R + and the eigenvalue λ m,n is given explicitly as (2.3) λ m,n = 2(|m| + 2n + 1), m ∈ Z, n ∈ N 0 .
In particular, λ 0,0 = 2 is simple, λ 1,0 = λ −1,0 = 4 is double, λ 2,0 = λ −2,0 = λ 0,1 = 6 is triple, and so on. For fixed m ∈ Z, the spacing between the eigenvalues is 4. Multiplicity of an eigenvalue λ = 2 for ∈ N is .
2.2.
Primary branches of radially symmetric vortices. Stationary solutions of the GP equation (1.1) are given in the form u(t, x, y) = e −iµt U (x, y), where U satisfies (1.3) and µ ∈ R is a free parameter which has the physical meaning of the chemical potential. In polar coordinates (r, θ), U satisfies the stationary GP equation in the form
(2.4) µU = −∆ (r,θ) U + r 2 U + |U | 2 U + iΩ∂ θ U.
Radially symmetric vortices of a fixed charge m 0 ∈ N are given in the form
(2.5) U (r, θ) = e im 0 θ ψ m 0 (r), ω = µ + m 0 Ω,
where (ψ m 0 , ω) is a root of the nonlinear operator
(2.6) f (u, ω) : H 2 r (R + ) ∩ L 2,2 r (R + ) × R → L 2 r (R + ), given by f (u, ω) := −∆ m 0 u + r 2 u + u 3 − ωu.
By Theorem 1 in [4], for every m 0 ∈ N, there exists a unique smooth family of radially symmetric vortices of charge m 0 parameterized locally by amplitude a such that , and the normalization e m 0 ,0 L 2 r = 1 has been used. By using the explicit expression for the L 2 r (R + )-normalized Hermite-Gauss solutions of the Schrödinger equation (2.2) with λ m 0 ,0 = 2(m 0 + 1) given by (2.9) e m 0 ,0 (r) =
√ 2 √ m 0 ! r m 0 e − r 2 2 , m 0 ∈ N 0 ,
we compute explicitly
(2.10) ω m 0 ,0 = e m 0 ,0 4 L 4 r = (2m 0 )! 4 m 0 (m 0 !) 2 .
Since e m 0 ,0 (r) > 0 for all r > 0, the property ψ m 0 (r; a) > 0, r > 0 holds 2 at least for sufficiently small a. The family of radially symmetric vortices approximated by (2.7) and (2.8) in the small-amplitude limit is referred to as the primary branch.
(2.11) E µ (u) = E(u) − µQ(u),
where E(u) is given by (1.2) and Q(u) = u 2 L 2 . Expanding E µ (u) near the critical point U given by (2.5)
with u = U + v, where v is a perturbation term in H 1 (R 2 ) ∩ L 2,1 (R 2 )
, we obtain the quadratic form at the leading order
E µ (U + v) − E µ (U ) = Hv, v L 2 + O( v 3
H 1 ∩L 2,1 ), where the bold notation v is used for an augmented vector with components v andv and the Hessian operator H can be defined in the stronger sense as the linear operator (2.12) H :
H 2 (R 2 ) ∩ L 2,2 (R 2 ) → L 2 (R 2 ), with (2.13) H = −∆ (r,θ) + r 2 + iΩ∂ θ − µ + 2ψ 2 m 0 ψ 2 m 0 e 2im 0 θ ψ 2 m 0 e −2im 0 θ −∆ (r,θ) + r 2 − iΩ∂ θ − µ + 2ψ 2 m 0 .
By using the Fourier series
(2.14) v = m∈Z V m e imθ ,v = m∈Z W m e imθ ,
the operator H is block diagonalized into blocks H m that acts on V m and W m−2m 0 for m ∈ Z.
We recall that ψ m 0 = ψ m 0 (·; a), ω = µ + m 0 Ω = ω m 0 (a), and write the blocks H m as linear operators
(2.15) H m : H 2 r (R + ) ∩ L 2,2 r (R + ) → L 2 r (R + )
, with explicit dependence on the parameters (a, Ω) as follows:
(2.16) H m (a, Ω) = K m (a) − Ω(m − m 0 )R, where K m (a) = −∆ m + r 2 − ω m 0 (a) + 2ψ 2 m 0 (r; a) ψ 2 m 0 (r; a) ψ 2 m 0 (r; a)
−∆ m−2m 0 + r 2 − ω m 0 (a) + 2ψ 2 m 0 (r; a) and R = 1 0 0 −1 .
A secondary bifurcation along the primary branch of radially symmetric vortices given by (2.7) corresponds to the nonzero solutions in H 2 r (R + ) ∩ L 2,2 r (R + ) of the spectral problem
(2.17) K m (a) V m W m−2m 0 = Ω(m − m 0 )R V m W m−2m 0 .
This spectral problem (2.17) coincides with the stability problem for the primary branch (2.7) in the absence of rotation. The spectral parameter λ of the stability problem 3 is given for each m ∈ Z by λ := Ω(m − m 0 ). The parameter m for the angular mode satisfying the eigenvalue problem (2.17) corresponds to the bifurcating mode superposed on the primary branch of vortex solutions.
Secondary branches of multi-vortex solutions.
We can look for the secondary branches bifurcating along the primary branch of radially symmetric vortices given by (2.7) and (2.8).
Consequently, we write
(2.18) U (r, θ) = e im 0 θ ψ m 0 (r; a) + v(r, θ),
where v is a root of the nonlinear operator
(2.19) g(v; a, Ω) : H 2 (R 2 ) ∩ L 2,2 (R 2 ) × R × R → L 2 (R 2 ),
given by
g(v; a, Ω) = −∆ (r,θ) v + r 2 v + iΩ (∂ θ v − im 0 v) + 2ψ 2 m 0 (r; a)v + e 2im 0 θ ψ 2 m 0 (r; a)v + e −im 0 θ ψ m 0 (r; a)v 2 + 2e im 0 θ ψ m 0 (r; a)|v| 2 + |v| 2 v − ω m 0 (a)v. (2.20)
The Jacobian operator of g(v; a, Ω) at v = 0 is given by the Hessian operator (2.13), which is block-diagonalized by the Fourier series (2.14) into blocks (2.15)-(2.16).
In the next two lemmas, we analyze symmetries of the individual blocks of the spectral problem (2.17).
Lemma 1.
There exists a 0 such that for every 0 < a < a 0 , the spectrum of H m 0 (a, Ω) = K m 0 (a) is strictly positive except for a simple zero eigenvalue, which is related to the gauge symmetry spanned by the eigenvector
(2.21) K m 0 (a) ψ m 0 −ψ m 0 = 0 0 .
Consequently, no bifurcations arise in the Ω continuation from the block H m 0 (a, Ω) = K m 0 (a).
Proof. For m = m 0 , H m 0 (a, Ω) = K m 0 (a) is independent of the rotation frequency Ω. If the primary branch (2.7) describes vortices with ψ m 0 (r; a) > 0 for all r > 0, that is, if n 0 = 0, then the assertion on the spectrum of K m 0 (a) for small a follows from the previous works [6,20].
[V m 0 −k , W −m 0 −k ] = [W −m 0 +k , V m 0 +k ].
It follows from Lemmas 1 and 2 that it is sufficient to consider the spectrum of H m (a, Ω) for m > m 0 and to count negative and zero eigenvalues of H m (a, Ω) in pairs. If a = 0 and Ω = 0,
we have H m (0, 0) = K m (0), where (2.22) K m (0) = −∆ m + r 2 − λ m 0 ,0 0 0 −∆ m−2m 0 + r 2 − λ m 0 ,0 .
The spectrum of K m (0) is obtained from eigenvalues of the Schrödinger equation (2.2). The first diagonal entry of K m (0) has strictly positive eigenvalues
µ + m,n (0) := 2(m + 2n − m 0 ) > 0, m > m 0 , n ∈ N 0 .
The second diagonal entry of K m (0) has eigenvalues
µ − m,n (0) := 2(|m − 2m 0 | + 2n − m 0 ), m > m 0 , n ∈ N 0 .
Let N (m 0 ) and Z(m 0 ) be the cardinality of the sets
N (m 0 ) = m > m 0 , n ∈ N 0 : µ − m,n (0) < 0 and Z(m 0 ) = m > m 0 , n ∈ N 0 : µ − m,n (0) = 0 .
The following lemma gives the count of N (m 0 ) and Z(m 0 ).
Lemma 3. For every m 0 ∈ N, we have (2.23) N (m 0 ) = m 0 (m 0 + 1) 2 , Z(m 0 ) = m 0 . Proof. To count Z(m 0 ), we note that µ − m,n (0) = 0 if and only if |m − 2m 0 | + 2n = m 0 . The cardinality of the set {( , n) ∈ Z × N 0 : | | + 2n = m 0 } coincides with the multiplicity of the eigenvalue λ m 0 ,0 = 2(m 0 + 1) of the Schrödinger equation (2.2), which is m 0 + 1. Since | | ≤ m 0 translates to m 0 ≤ m ≤ 3m 0 and since m = m 0 contains one zero eigenvalue with n = 0, we obtain Z(m 0 ) = m 0 + 1 − 1 = m 0 .
To count N (m 0 ), we follow the same idea. The largest negative eigenvalue µ − m,n (0) = −2 corresponds to |m − 2m 0 | + 2n = m 0 − 1, which coincides with the multiplicity of the eigenvalue λ m 0 −1,0 = 2m 0 , which is m 0 . The next negative eigenvalue µ − m,n (0) = −4 corresponds to |m − 2m 0 | + 2n = m 0 − 2, which coincides with the multiplicity of the eigenvalue λ m 0 −2,0 = 2(m 0 − 1), which is m 0 − 1. The count continues until we reach the smallest negative eigenvalue µ − m,n (0) = −2m 0 , which corresponds to |m−2m 0 |+2n = 0 and which is simple for m = 2m 0 and n = 0. Summing integers from 1 to m 0 , we obtain N (m 0 ) = 1+2+· · ·+m 0 = m 0 (m 0 +1)/2.
σ(K 2 ) = {−2, 2, 2, 6, 6, · · · }, σ(K 3 ) = {0, 4, 4, 8, 8, · · · }, σ(K 4 ) = {2
, 6, 6, 10, 10, · · · }, . . . (2.24) so that N (1) = 1 and Z(1) = 1.
If m 0 = 2, then λ 2,0 = 6 and 4,8,8,12,12, · · · }, σ(K 7 ) = {2, 6, 10, 10, 14, 14, · · · }, . . . (2.25) so that N (2) = 3 and Z(2) = 2. If m 0 = 3, then λ 3,0 = 8 and In what follows, we fix a > 0 small enough and consider a continuation of eigenvalues of H m (a, Ω) given by (2.16) with respect to the parameter Ω in the interval (0, 2). When one of the eigenvalues of H m (a, Ω) reaches zero, we say that a secondary bifurcation occurs along the primary branch of radially symmetric vortices given by (2.7) and (2.8).
σ(K 3 ) = {−2, 2, 2, 6, 6, · · · }, σ(K 4 ) = {−4, 0, 4, 4, 8, 8, · · · }, σ(K 5 ) = {−2, 2, 6, 6, 10, 10, · · · }, σ(K 6 ) = {0, σ(K 4 ) = {−2, 2, 2, 6, 6, · · · }, σ(K 5 ) = {−4, 0, 4, 4, 8, 8, · · · }, σ(K 6 ) = {−6, −2, 2, 6, 6, 10, 10, · · · }, σ(K 7 ) = {−4, 0, 4, 8, 8, 12, 12, · · · }, σ(K 8 ) = {−2,
We will show that for every m = m 0 + 2 , 1 ≤ ≤ m 0 , there is an a-independent constant D m,m 0 ≥ 0 such that the zero eigenvalue of K m (0) becomes a positive eigenvalue of H m (a, Ω) for small a and for Ω D m,m 0 a 2 . The maximum of
D m 0 +2 ,m 0 for 1 ≤ ≤ m 0 is denoted by D m 0 .
We further show that there is another a-independent constant C m 0 > 0 such that when Ω is increased in the interval (D m 0 a 2 , 2 − C m 0 a 2 ), then 1 + B(m 0 ) secondary bifurcations occur, where B(m 0 ) = m 0 (m 0 − 1)/2, at which a negative eigenvalue of H m (a, Ω) for some m and for Ω below the bifurcation point becomes a positive eigenvalue of H m (a, Ω) for the same m and for Ω above the bifurcation point. The first B(m 0 ) secondary bifurcations occur for values of Ω sufficiently distant from the value Ω 0 = 2, whereas the last secondary bifurcation occurs for the value of Ω near but below the value Ω 0 = 2. The latter case has to be handled in the presence of infinitely many resonances in the limit a → 0. The aforementioned claims proved in Section 3 will provide proofs of item (iii) in Theorem 1.
At each non-resonant bifurcation point, a new secondary branch of vortex solutions is born for Ω on one side of the bifurcation point among the roots of the nonlinear operator g given by (2.19) and (2.20). The secondary branch represents a multi-vortex configuration near the origin of the total charge m 0 , where the radial symmetry is now broken. The aforementioned claims proved in Section 4 and 5 will provide respectively proofs of items (iv) and (v) in Theorem 1.
Secondary bifurcations as Ω increases
Let the primary branch of radially symmetric vortices be defined by (2.7) and (2.8) in the small-amplitude limit. Expanding the family of operators K m (a) in powers of a, we obtain
K m (a) = −∆ m + r 2 − λ m 0 ,0 0 0 −∆ m−2m 0 + r 2 − λ m 0 ,0 + a 2 −ω m 0 ,0 + 2e 2 m 0 ,0 (r) e 2 m 0 ,0 (r) e 2 m 0 ,0 (r) −ω m 0 ,0 + 2e 2 m 0 ,0 (r) + O(a 4 ),
where the correction term is given by a bounded potential on R + . Also recall from (2.16) that the operator H m (a, Ω) is expanded as a → 0 with the leadingorder term given by the diagonal operator H m (0, Ω) with the entries given by two linear operators:
(3.1) L + := −∆ m + r 2 − λ m 0 ,0 − Ω(m − m 0 ), L − := −∆ m−2m 0 + r 2 − λ m 0 ,0 + Ω(m − m 0 ).
We shall now analyze how eigenvalues of H m (a, Ω) cross zero when Ω is increased in the interval (0, 2). Lemma 4. For every m 0 ∈ N, there exists a 0 > 0 and D m 0 ≥ 0 such that for every 0 < a < a 0 , Ω > D m 0 a 2 , and 1 ≤ ≤ m 0 , there is a small positive eigenvalue of H m 0 +2 (a, Ω) which is continuous in (a, Ω) and converges to the zero eigenvalue of K m 0 +2 (0) as a → 0 and Ω → 0.
Proof. The zero eigenvalue of K m (0) for m = m 0 + 2 , 1 ≤ ≤ m 0 corresponds to the second diagonal operator in K m (0). Let us show by the perturbation theory argument that the zero eigenvalue is continued as a small O(a 2 ) eigenvalue of K m (a) for all a sufficiently small.
The eigenfunction of K m (0) with m = m 0 + 2 for the zero eigenvalue is obtained from the balance
λ m−2m 0 ,n = λ m 0 ,0 ⇒ n( ) = m 0 − |2 − m 0 | 2 .
Since the zero eigenvalue of K m 0 +2 (0) is simple, the regular perturbation theory in [17] implies the existence of a small eigenvalue µ (a) of the linear operator K m 0 +2 (a) and the corresponding eigenvector (V m 0 +2 , W −m 0 +2 ), which are analytic functions of a. Their Taylor expansions are given by
V m 0 +2 = a 2Ṽ m 0 +2 + O L 2 r (a 4 ), W −m 0 +2 = c −m 0 +2 e |m 0 −2 |,n( ) + a 2W −m 0 +2 + O L 2 r (a 4 ), µ = a 2μ + O(a 4 ),(3.2)
where c −m 0 +2 = 0 is arbitrary,Ṽ m 0 +2 ,W −m 0 +2 , andμ are obtained by the standard projection algorithm, and the correction terms are defined uniquely by the method of Lyapunov-Schmidt reductions. In particular,μ is obtained from
(3.3)μ = −ω m 0 ,0 + 2 e 2 m 0 ,0 , e 2 |m 0 −2 |,n( ) L 2 r . Ifμ = 0, the eigenvalue µ (a) is generally nonzero but O(a 2 ) small.
It follows from (3.1) that the Ω-term in L − is a positive perturbation to K m (0) for m > m 0 . Therefore, there exists an a-independent constant D ,m 0 ≥ 0 such that the eigenvalue µ (a) continued with respect to the parameter Ω is strictly positive for Ω > D ,m 0 a 2 . The assertion of the lemma is proved by taking the largest of D ,m 0 for all admissible 1 ≤ ≤ m 0 as D m 0 .
(3.4) e m,1 (r) = √ 2 (m + 1)! r m (m + 1 − r 2 )e − r 2 2 .
Then, we obtain from (3.3) for m 0 ≥ 2:
µ m 0 −1 = 2 e 2 m 0 ,0 , e 2 m 0 −2,1 L 2 r − e m 0 ,0 4 L 4 r = (2m 0 )!(m 2 0 + m 0 − 1) 4 m 0 (m 0 !) 2 > 0.
By the symmetry, we also have µ 1 = µ m 0 −1 > 0. Thus, for 1 ≤ m 0 ≤ 3, we haveμ > 0 for all admissible 1 ≤ ≤ m 0 . Since H m (a, Ω) is self-adjoint, by regular perturbation theory in [17], the operator H m (a, Ω) has a set of eigenvalues counted by n ∈ N 0 :
(3.6) µ + m,n (a, Ω) := λ m,n − λ m 0 ,0 − Ω(m − m 0 ) + O(a 2 ), µ − m,n (a, Ω) := λ m−2m 0 ,n − λ m 0 ,0 + Ω(m − m 0 ) + O(a 2 )
. For m > m 0 , n ∈ N 0 , and Ω < 2, we have
λ m,n − λ m 0 ,0 − Ω(m − m 0 ) = (2 − Ω)(m − m 0 ) + 4n > 0.
Therefore, the eigenvalues µ + m,n (a, Ω) never become zero for small a and Ω < 2. On the other hand, the eigenvalues µ − m,n (a, Ω) become zero when Ω = Ω m,n (a) given by
(3.7) Ω m,n (a) = 2 m 0 − |m − 2m 0 | m − m 0 − 4n m − m 0 + O(a 2 ).
Let B(m 0 ) denote the number of eigenvalues µ − m,n crossing zero at Ω = Ω m,n (a) with Ω m,n (0) ∈ (0, 2). The following lemma gives the count of B(m 0 ).
Lemma 5.
For every m 0 ∈ N, we have
(3.8) B(m 0 ) = m 0 (m 0 − 1) 2 .
Proof. To count B(m 0 ), we count the values of m > m 0 for the first values of n ∈ N 0 , when Ω m,n (0) ∈ (0, 2):
• For n = 0, the inequality 0 < m 0 −|m−2m 0 | < m−m 0 is true for 2m 0 +1 ≤ m ≤ 3m 0 −1; • For n = 1, the inequality 0 < m 0 − |m − 2m 0 | − 2 < m − m 0 is true for m 0 + 3 ≤ m ≤ 3m 0 − 3; • For n = 2, the inequality 0 < m 0 − |m − 2m 0 | − 4 < m − m 0 is true for m 0 + 5 ≤ m ≤ 3m 0 − 5. For a general n ∈ N, we have Ω m,n (0) ∈ (0, 2) for m 0 + 2n + 1 ≤ m ≤ 3m 0 − 2n − 1 provided that the range for m is nonempty. Summing up all cases, we have B(m 0 ) = m 0 − 1 + ∞ n=1 [2m 0 − 4n − 1] +
where [a] + is a when a ≥ 0 and 0 if a < 0. The sum is finite as n terminates at the last entry for which 2m 0 − 4n − 1 > 0. If m 0 is odd, then the last entry corresponds to N = (m 0 − 1)/2 and we obtain
∞ n=1 [2m 0 − 4n − 1] + = N n=1 (2m 0 − 4n − 1) = m 2 0 − 3m 0 + 2 2 .
If m 0 is even, then the last entry corresponds to N = m 0 /2 − 1 and we obtain
∞ n=1 [2m 0 − 4n − 1] + = N n=1 (2m 0 − 4n − 1) = m 2 0 − 3m 0 + 2 2 .
Adding m 0 − 1 to this number, we obtain (3.8) in both cases.
In
S m := V m 2 L 2 r − W m−2m 0 2 L 2 r . Since V m → 0 as a → 0, we have S m < 0 for each m ∈ {m 1 , . . . , m k }, provided a is small enough.
Remark 11. The quantity S m defined by (3.11) is referred to as the Krein quantity. The sign of S m gives the Krein signature of the neutrally stable eigenvalues of the spectral stability problem associated with the radially symmetric vortices in the case of no rotation [20]. We will show that if Ω is defined at a particular value denoted by Ω m 0 +1,0 (a) = 2 + O(a 2 ), for which the spectral stability problem (2.17) with m = m 0 + 1 admits a nontrivial solution, then the blocks H m (a, Ω) of the Hessian operator for every m ≥ m 0 + 2 are invertible in L 2 r (R + ) near Ω = Ω m 0 +1,0 (a) and the smallest eigenvalue of H m (a, Ω m 0 +1,0 (a)) is proportional to O(a 2 ). At the same time, the block H m 0 +1 (a, Ω m 0 +1,0 (a)) has a simple zero eigenvalue and a simple positive eigenvalue proportional to O(a 2 ). We also show for 1 ≤ m 0 ≤ 16 that the blocks H m (a, Ω m 0 +1,0 (a)) for m 0 + 2 ≤ m ≤ 2m 0 have exactly one small negative eigenvalue proportional to O(a 2 ), whereas all other eigenvalues are strictly positive.
The following lemma gives the precise location of Ω m 0 +1,0 (a) = 2 + O(a 2 ).
Lemma 6.
There exists a 0 > 0 such that for every 0 < a < a 0 , there exists Ω m 0 +1,0 (a) < 2 given asymptotically by such that H m 0 +1 (a, Ω m 0 +1,0 (a)) has a simple zero eigenvalue.
Proof. We solve the bifurcation equation (2.17) for m = m 0 + 1 near Ω = 2 in powers of a.
Since Ω = 2 is a double (semi-simple) eigenvalue of the bifurcation equation (2.17) at a = 0, we use the two-parameter perturbation theory with the Taylor expansion
V m 0 +1 = c m 0 +1 e m 0 +1,0 + a 2Ṽ m 0 +1 + O L 2 r (a 4 ), W −m 0 +1 = c −m 0 +1 e m 0 −1,0 + a 2W −m 0 +1 + O L 2 r (a 4 ), Ω = 2 + a 2Ω + O(a 4 ),(3.13)
where (c m 0 +1 , c −m 0 +1 ) = (0, 0) are to be determined, the correction termsṼ m 0 +1 ,W −m 0 +1 , and Ω are a-independent, and the reminder terms are uniquely defined by the Lyapunov-Schmidt reductions. The admissible values of (c m 0 +1 , c −m 0 +1 ) = (0, 0) andΩ are found from the matrix eigenvalue problemÃ
c m 0 +1 c −m 0 +1 =Ω c m 0 +1 c −m 0 +1 , wherẽ A = (−ω m 0 ,0 + 2e 2 m 0 ,0 )e m 0 +1,0 , e m 0 +1,0 L 2 e 2 m 0 ,0 e m 0 −1,0 , e m 0 +1,0 L 2 − e 2 m 0 ,0 e m 0 +1,0 , e m 0 −1,0 L 2 − (−ω m 0 ,0 + 2e 2 m 0 ,0 )e m 0 −1,0 , e m 0 −1,0 L 2 = (2m 0 )! 4 m 0 (m 0 −1)!(m 0 +1)! (2m 0 )! 4 m 0 m 0 ! √ (m 0 −1)!(m 0 +1)! − (2m 0 )! 4 m 0 m 0 ! √ (m 0 −1)!(m 0 +1)! − (2m 0 )! 4 m 0 (m 0 !) 2 ,
and we have used the explicit formula (2.9). Eigenvalues ofà and their normalized eigenvectors are given byΩ = 0 :
c m 0 +1 c −m 0 +1 = 1 √ 2m 0 + 1 √ m 0 + 1 − √ m 0 (3.14) andΩ =Ω m 0 +1,0 := − (2m 0 )! 4 m 0 m 0 !(m 0 + 1)! : c m 0 +1 c −m 0 +1 = 1 √ 2m 0 + 1 √ m 0 − √ m 0 + 1 . (3.15)
SubstitutingΩ =Ω m 0 +1,0 from (3.15) to (3.13), we obtain the asymptotic expansion (3.12). SinceΩ m 0 +1,0 < 0 in (3.15), we have Ω m 0 +1,0 (a) < 2 for small a.
Remark 12. Lemma 6 yields the existence of constant C m > 0 in item (iii) of Theorem 1.
In order to compute eigenvalues of the blocks H m (a, Ω m 0 +1,0 (a)) for small a, we write explicitly the following expansion in powers of a:
H m (a, Ω m 0 +1,0 (a)) = −∆ m + r 2 − 2(m + 1) 0 0 −∆ m−2m 0 + r 2 + 2(m − 2m 0 − 1) + a 2 −ω m 0 ,0 − (m − m 0 )Ω m 0 +1,0 + 2e 2 m 0 ,0 (r) e 2 m 0 ,0 (r) e 2 m 0 ,0 (r) −ω m 0 ,0 + (m − m 0 )Ω m 0 +1,0 + 2e 2 m 0 ,0 (r) + O(a 4 ).
We consider now eigenvalues of H m (a, Ω m 0 +1,0 (a)) denoted by λ near zero as a → 0. The following three lemmas summarize the results of computations of the perturbation theory. Lemma 7. There exists a 0 > 0 such that for every 0 < a < a 0 , the block H m 0 +1 (a, Ω m 0 +1,0 (a)) has a simple zero eigenvalue and a simple positive eigenvalue of the order O(a 2 ), whereas all other eigenvalues are strictly positive.
Proof. For m = m 0 + 1, computations of the perturbation theory similar to the expansion (3.13) are repeated as follows:
V m 0 +1 = c m 0 +1 e m 0 +1,0 + a 2Ṽ m 0 +1 + O L 2 r (a 4 ), W −m 0 +1 = c −m 0 +1 e m 0 −1,0 + a 2W −m 0 +1 + O L 2 r (a 4 ), λ = a 2λ + O(a 4 ).
(3.16)
The Lyapunov-Schmidt reduction method results now in the matrix eigenvalue problem
A c m 0 +1 c −m 0 +1 =λ c m 0 +1 c −m 0 +1 , whereà = (2m 0 )! 4 m 0 (m 0 !) 2 (2m 0 )! 4 m 0 m 0 ! √ (m 0 −1)!(m 0 +1)! (2m 0 )! 4 m 0 m 0 ! √ (m 0 −1)!(m 0 +1)! (2m 0 )! 4 m 0 (m 0 −1)!(m 0 +1)! .
Eigenvalues of A and their normalized eigenvectors are given bỹ Proof. For m ≥ 2m 0 + 1, the zero eigenvalue of H m (0, 2) is simple and all other eigenvalues are strictly positive. The one-parameter perturbation expansion for the small eigenvalue is developed as follows: It remains to consider the blocks H m (a, Ω m 0 +1,0 (a)) for m 0 +2 ≤ m ≤ 2m 0 . Before continuing with the technical details, we note the example of m 0 = 2. The results of [16,20] imply that no real eigenvalues exist in the neighborhood of Ω = 2 and a = 0 among eigenvalues of the bifurcation equation (2.17) for m = 4 = 2m 0 . This is due to oscillatory instability of the radially symmetric vortex of charge two (m 0 = 2), which arises in the small-amplitude limit of the primary branch. See Remark 6.9 in [16]. More general results were obtained in [10], see Proposition 8.3, where all vortices with m 0 ≥ 2 were found unstable but the number of unstable modes is smaller than m 0 − 1 if m 0 is sufficiently large. The following result is in agreement with the outcomes of the stability computations in [10,16].
λ = 0 : c m 0 +1 c −m 0 +1 = 1 √ 2m 0 + 1 √ m 0 − √ m 0 + 1 (3.17) andλ = (2m 0 + 1)! 4 m 0 m 0 !(m 0 + 1)! : c m 0 +1 c −m 0 +1 = 1 √ 2m 0 + 1 √ m 0 + 1 √ m 0 . V m = c m e m,0 + a 2Ṽ m + O L 2 r (a 4 ), W m−2m 0 = a 2W m−2m 0 + O L 2 r (a 4 ), λ = a 2λ + O(a 4 ).
Lemma 9. Let 2 ≤ m 0 ≤ 16. There exists a 0 > 0 such that for every 0 < a < a 0 , the block H m (a, Ω m 0 +1,0 (a)) with m 0 + 2 ≤ m ≤ 2m 0 has two small eigenvalues of the order O(a 2 ) (one is positive and the other one is negative), whereas all other eigenvalues are strictly positive.
Proof. For m 0 +2 ≤ m ≤ 2m 0 , the zero eigenvalue of H m (0, 2) is double and all other eigenvalues are strictly positive. The two-parameter perturbation expansion for the small eigenvalue is developed as follows: For m 0 = 2 (with m = 4) and m 0 = 3 (with m = 5, 6), the entries ofà are computed explicitly. Since the first diagonal entry is positive and the second diagonal entry is negative,à has one positive and one negative eigenvalueλ. We have checked numerically that this property remains true for every 2 ≤ m 0 ≤ 16. The expansion (3.21) yields one positive and one negative eigenvalue of the order O(a 2 ) in the block H m (a, Ω m 0 +1,0 (a)) for small a. It remains to prove that the zero eigenvalue of H m 0 +1 (a, Ω m 0 +1,0 (a)) becomes a small positive eigenvalue of H m 0 +1 (a, Ω) for Ω Ω m 0 +1,0 (a) and a small negative eigenvalue of H m 0 +1 (a, Ω) for Ω Ω m 0 +1,0 (a). This follows from the derivative (3.10) and the Krein signature of the zero eigenvalue of H m 0 +1 (a, Ω m 0 +1,0 (a)) defined by (3.11). We obtain from the expansion (3.16)
V n = c m e m,0 + a 2Ṽ m + O L 2 r (a 4 ), W m−2m 0 = c m−2m 0 e 2m 0 −m,0 + a 2W m−2m 0 + O L 2 r (a 4 ), λ = a 2λ + O(a 4 ).(3.24) S m 0 +1 = V m 0 +1 2 L 2 r − W −m 0 +1 2 L 2 r = c 2 m 0 +1 − c 2 −m 0 +1 + O(a 2 ),
where (c m 0 +1 , c −m 0 +1 ) is given by the eigenvector ofà that corresponds toλ = 0. From (3.17), we obtain S m 0 +1 < 0, hence the corresponding eigenvalue of H m 0 +1 (a, Ω) is an increasing 4 function of Ω. and so on. This finding corresponds to the result of Proposition 8.3 in [10] where the number of complex eigenvalues is found to be smaller than m 0 −1 if m 0 is sufficiently large. If R(m 0 ) = 0, then there exist R(m 0 ) bifurcation curves connected to the point Ω 0 = 2 from below. As follows from the count on negative eigenvalues in
N (m 0 ) − B(m 0 ) − 1 = m 0 − 1,
which coincides with the number of negative eigenvalues in Lemma 9, these additional bifurcation curves for 4 ≤ m 0 ≤ 16 are located above the curve Ω m 0 +1,0 . However, for m 0 ≥ 17, thanks to the computations in Remark 13, some of the positive eigenvalues of H m (a, Ω) for m 0 + 2 ≤ m ≤ 2m 0 become negative eigenvalues for Ω Ω m 0 +1,0 (a) and the total number of negative eigenvalues at Ω Ω m 0 +1,0 (a) exceeds m 0 − 1. Therefore, some of the R(m 0 ) bifurcation curves are located below the curve Ω m 0 +1,0 for m 0 ≥ 17.
Secondary branches of multi-vortex solutions
Recall that the solution U to the stationary GP equation (2.4) is a critical point of the energy functional E µ (u) in (2.11), therefore, the bifurcation problem for g(v; a, Ω) in (2.19)-(2.20) has a variational structure. The number of negative eigenvalues of the Jacobian operator H(a, Ω) in (2.12)-(2.13) (which is known as the Morse index ) changes at every bifurcation curve as Ω crosses Ω m,n (a), according to Propositions 1 and 2, where the values of Ω m,n (a) are given by Lemmas 5 and 6, see equations (3.7) and (3.12).
Here we prove that for each fixed a and for each non-resonant bifurcation point, there is a continuous branch of solutions of g(v; a, Ω) bifurcating from (0; a, Ω m,n (a)) on one side of the bifurcation point Ω = Ω m,n (a). The new family of multi-vortex solutions is parameterized by two parameters (a, Ω).
Besides proving the local bifurcation result, we discuss symmetries of the bifurcating branches and their global continuation with respect to parameter Ω. For definitions and methods used to prove the equivariant bifurcation we refer to [1,11,15].
In section 4.1, symmetries of g(v; a, Ω), in particular, its equivariant properties are analyzed. In section 4.2, we prove the local bifurcation result for a non-resonant bifurcation point Ω m,n (a), with a simple zero eigenvalue of H(a, Ω). We also discuss symmetries and asymptotic estimates of the bifurcating branches, which are needed to study the location of the individual vortices in the multi-vortex configurations. In section 4.3, we prove the global continuation of the solution branches. Similarly, we have g(ρ(κ)v) = ρ(κ)g(v).
As is explained in Section 2.2, the component v is extended to the vector v = (v, w) with the constraint w =v, so that the root finding problem is formulated for the analytic nonlinear operator g(v) = (g(v, w),ḡ(v, w)). The natural extension of the action of the group O(2) to the second component of
v = (v, w) is (4.2) ρ(ϕ)w(r, θ) = e im 0 ϕ w(r, θ + ϕ), ρ(κ)w(r, θ) =w(r, −θ).
In the Fourier basis
v = m∈Z V m (r)e imθ , w = m∈Z W m (r)e imθ ,
the action of the group O(2) = S 1 ∪ κS 1 is given by
ρ(ϕ)V m = e i(m−m 0 )ϕ V m , ρ(κ)V m =V m , ρ(ϕ)W m = e i(m+m 0 )ϕ W m , ρ(κ)W m =W m . so that ρ(ϕ)(V m , W m−2m 0 ) = e i(m+m 0 )ϕ (V m , W m−2m 0 ), (4.3) ρ(κ)(V m , W m−2m 0 ) = (V m ,W m−2m 0 ).
Therefore, the subspaces of functions (V m , W m−2m 0 ) are composed of similar irreducible representations under the action of the group O(2).
The subspace (V m , W m−2m 0 ) has as isotropy group, the dihedral group D m−m 0 generated by the elements κ and ζ = 2π/(m − m 0 ). The dihedral group D m−m 0 will be used to find the symmetry-breaking bifurcations of the primary branch into the multi-vortex solutions along the secondary branches. Due to the symmetries of D m−m 0 , the multi-vortex solution is represented by a (m − m 0 )-polygon of individual vortices.
For a fixed value of m ∈ Z, the action of ρ(ζ) is given by
ρ(ζ)(V j , W j−2m 0 ) = exp 2πi j − m 0 m − m 0 (V j , W j−2m 0 ), j ∈ Z.
The fixed point space
Fix(D m−m 0 ) = {(v, w) ∈ L 2 (R 2 ) : ρ(γ)(v, w) = (v, w) for γ ∈ D m−m 0 } is composed of functions with real components (V j , W j−2m 0 ) such that j − m 0 is a multiple of m − m 0 . If (v,v) ∈ Fix(D m−m 0 ), then v can be characterized by v(r, θ) = j∈m 0 +(m−m 0 )Z V j (r)e ijθ = e im 0 θ j∈(m−m 0 )Z V m 0 +j (r)e ijθ ,
where all functions {V j (r)} j∈m 0 +(m−m 0 )Z are real-valued. Writing v(r, θ) = e im 0 θ φ(r, θ), we deduce that φ satisfies the symmetry constraints:
(4.4) φ(r, θ) =φ(r, −θ) = φ(r, θ + ζ).
Since g is O(2)-equivariant, the operator g(v) restricted to Fix(D m−m 0 ) is well defined. Therefore, we can consider the bifurcation problem
(4.5) g D m−m 0 (v; a, Ω) : X ∩ Fix(D m−m 0 ) × R × R → Fix(D m−m 0 ),
where X := H 2 (R 2 ) ∩ L 2,2 (R 2 ) is the graph norm of the Jacobian operator H. A schematic illustration of the local bifurcations of the primary and secondary branches is given on Figure 2. By Schur's lemma, the Jacobian operator H for g has a diagonal decomposition in the subspaces of similar irreducible representations given by the components (V m , W m−2m 0 ). Indeed, this has been done in (2.13) and (2.16), where the operator H in the subspace (V m , W m−2m 0 ) is represented by the block H m . Consequently, for a fixed m ∈ Z, the Jacobian operator of g D m−m 0 consists of the blocks H j corresponding to j − m 0 ∈ (m − m 0 )Z. Moreover, in the subspace Fix(D m−m 0 ) we have w =v, so that W j−m 0 = V −(j−m 0 ) and the blocks H j with negative j − m 0 are determined by those with j − m 0 ∈ N. Hence, we denote
H D m−m 0 = diag{H j } j∈m 0 +(m−m 0 )N .
The operator H has a zero eigenvalue in the block j = m 0 due to the gauge invariance of the original problem. This zero eigenvalue is not present for the operator H The local bifurcation results are obtained for the non-resonance bifurcation points, according to the following definition. This definition extends Definition 1.
Definition 2. For a fixed a > 0, we say that Ω m,n (a) ∈ (0, 2) is a non-resonant bifurcation point if the kernel of H D m−m 0 (Ω m,n (a)) has dimension one. We say that Ω m,n ∈ (0, 2) is a non-resonant curve if this condition holds for each small a.
For each curve Ω m,n , the non-resonant condition is given by the following equivalent conditions:
(i) H j (a, Ω m,n (a)) is invertible; (ii) Ω m,n (a) = Ω j,k (a); (iii) µ − j,k (a, Ω m,n (a)) = 0; where j takes values in m 0 + (m − m 0 ) , ∈ N\{1}, k ∈ N 0 , and a > 0 is arbitrary but sufficiently small.
As we discussed in Remark 10, the bifurcation curves are all non-resonant for 1 ≤ m 0 ≤ 3 and the first resonance happens for m 0 = 4 because Ω 10,0 (0) = Ω 7,1 (0) = 2/3. In view of the restriction on the range of j in the space Fix(D m−m 0 ), however, the bifurcation curve Ω 10,0 is non-resonant because H D 6 is composed of blocks H j with j = 4, 10, 16, ... and the zero eigenvalue µ − 7,1 (0, Ω 10,0 (0)) is not included in the spectrum of H D 6 . On the other hand, the bifurcation curve Ω 7,1 may be resonant because H D 3 is composed of blocks H j with j = 4, 7, 10, ... and the zero eigenvalue µ − 10,0 (0, Ω 7,1 (0)) is included in the spectrum of H D 3 . To know exactly if Ω 7,1 (a) is resonant with Ω 10,0 (a) one needs to compute the normal form in a, which is out of the scope of our presentation.
Ω m,0 (0) = Ω j,k (0) for j − m 0 = (m − m 0 ) with ∈ N\{1}, then f (m) = f (j) − 2k j − m 0 , where f (j) = m 0 − |j − 2m 0 | j − m 0 = 1 m 0 < j ≤ 2m 0 2m 0 j−m 0 − 1 2m 0 < j Note that f is a strictly decreasing function on [2m 0 , ∞). If k = 0, then j > 2m 0 , hence f (m) = f (j) is true only if j = m ( = 1), which is excluded. If k ≥ 1, then f (m) < f (j), which implies that m > j or j − m 0 < m − m 0 . Therefore, the possible resonant block H j with m 0 < j < m is not in H D m−m 0 = diag(H m 0 , H m , H 2m−m 0 , . . .).
Remark 17. The curve Ω m 0 +1,0 (a) = 2 + O(a 2 ) is non-resonant as long as the matricesà arising in the matrix eigenvalue problem (3.22) are invertible. We have checked this condition numerically for 1 ≤ m 0 ≤ 100.
The following proposition follows from the Crandall-Rabinowitz theorem, see Theorem I.5.1 in [19]. It covers the non-resonant bifurcation curve Ω m,n , for which Proposition 1 applies. It does not cover the curve Ω m 0 +1,0 in Remark 17. is the eigenvector of H m (a, Ω m,n (a)) associated with the zero eigenvalue µ − m,n (a, Ω m,n (a)).
Proof. The local bifurcation problem (4.5) is well-defined for the operator g D m−m 0 . The operator g D m−m 0 has a linearization given by H D m−m 0 and its kernel is spanned by the eigenvector f m,n associated to the simple zero eigenvalue µ − m,n (a; Ω m,n (a)) under the assumption of the proposition. Since H D m−m 0 has a uniformly bounded inverse operator in the complement of the kernel, according to Proposition 1, we are in the position to define the bifurcation equation as in Theorem I.5.1 in [19].
The only condition to be verify is that
∂ Ω H D m−m 0 f m,n = i∂ θ f m,n
is not in the range of H D m−m 0 . Thanks to the basis in (3.5) and the fact that the zero eigenvalue corresponds to µ − m,n , the leading-order approximation of the eigenvector f m,n is given by (4.8) for a > 0 sufficiently small. Then, i∂ θ f m,n / ∈ Ran(H D m−m 0 ) because
i∂ θ f m,n , f m,n L 2 = m − 2m 0 + O(a 2 ) = 0.
The existence of the new root of g D m−m 0 (v; a, Ω) and the estimate (4.7) for a > 0 sufficiently small follow from the Crandall-Rabinowitz theorem, where the scaling O X (ab 2 ) is due to the cubic terms in the expressions for g in (2.20). This theorem gives also the estimate Ω(a, b) = Ω m,n (a) + O(b). Furthermore, the S 1 -action (4.3) of the element ϕ = π/(m + m 0 ) in the kernel generated by f m,n is given by ρ(ϕ) = −1. Therefore, the bifurcation equation is odd and ∂ vv g D m−m 0 (0)(f m,n , f m,n ) = 0. The estimate (4.6) is obtained from formula (I. 6.3) in [19].
Remark 18. The new family (4.6) and (4.7) exists on one side of the bifurcation curve Ω m,n , that is,
Ω(a, b) = Ω m,n (a) + cb 2 + O(b 4 )
, where c can be computed from (I.6.11) in [19]. If c > 0 the bifurcation is supercritical pitchfork (to the right of the bifurcation curve) and the Jacobian operator at the new (secondary) branch of solutions has one more negative eigenvalue compared to that at the primary branch. If c < 0 the bifurcation is subcritical pitchfork (to the left of the bifurcation curve) and the Jacobian operator at the new branch of solutions has one less negative eigenvalue compared to that at the primary branch. Because the new family can be rotated in the (x, y) plane, the Jacobian operator at the new branch has an additional zero eigenvalue related to this rotation symmetry.
The following proposition covers the non-resonant bifurcation curve Ω m+1,0 , for which Proposition 2 applies. The following result holds because the Jacobian operator H given by (2.12) and (2.13) is bounded and has closed range for |Ω| < 2.
Lemma 10. Let X = H 2 (R 2 ) ∩ L 2,2 (R 2 ) be the domain space for the Jacobian operator H. For every |Ω| < 2 there is a positive constant c such that the operator (H + cI) : X → L 2 (R 2 ) is positive definite and (H + cI) −1 : X → X is compact.
Proof. The eigenvalues of H are given by µ ± m,n (a, Ω) expanded as in (3.6). For |Ω| < 2, the eigenvalues µ ± m,n are bounded from below and do not accumulate at a finite value. Therefore, there is a positive constant c such that the bounded operator H+cI : X → L 2 (R 2 ) is positive definite and invertible. Since X is compactly included in L 2 (R 2 ), then the inverse operator (H + cI) −1 : X → L 2 → X is compact.
Remark 21. Observe in (3.6) that for |Ω| = 2 the eigenvalues µ ± m,n (a, Ω) can accumulate at a finite value as a → 0, while for |Ω| > 2 the eigenvalues µ ± m,n (a, Ω) are unbounded both from above and from below. As a result, the operator H + cI : X → L 2 (R 2 ) does not have a closed range for |Ω| = 2, its inverse (H + cI) −1 : L 2 (R 2 ) → X is not bounded, and the inverse operator (H + cI) −1 : X → X is not compact. If Ω m,n is a non-resonant bifurcation curve, then it is obvious that η Dm,n (Ω m,n ) = 1. The following proposition gives the global bifurcation result for each non-resonant bifurcation curve. Proof. Since X is a Banach algebra with respect to pointwise multiplication and g(0; a, Ω) = 0, we obtain the expansion g(v; a, Ω) = H(Ω)v + O X (v 2 ). We can apply the global Rabinowtz theorem to the nonlinear operator where B 2ε and B 2ρ are ball of radius 2ε and 2ρ around 0 ∈ X ∩ Fix(D m−m 0 ) and Ω m,n ∈ [0, 2), respectively.
f (v, Ω) = (H + cI) −1 g(v; a, Ω)v = Iv − c (H + cI) −1 v + O X (v 2 ),
Remark 22. If the branch from (0, Ω m,n ) returns to another bifurcation point (0, Ω m ,n ), then the sum of all the bifurcation indices (4.13) at the bifurcation points has to be equal zero. Therefore, the knowledge of the exact factor ± in (4.13) is helpful to obtain information of where the branches can return. The exact factor ± in (4.13) can be computed for all the bifurcation curves using the fact that
deg(f D (x, Ω); B 2ε ) = (−1) n D m−m 0 (Ω) ,
since H+cI is positive definite. For example, for the last bifurcation from Ω m 0 +1,0 with 1 ≤ m 0 ≤ 16, the exact index is
deg x − ε, f D 1 (x, Ω); B 2ε × B 2ρ = (−1) m 0 − (−1) m 0 −1 = (−1) m 0 2.
Therefore, this branch can return to a single bifurcation point Ω 0 only if the latter point has index −2(−1) m 0 . Remark 23. Proposition 5 provides a proof of the claim in item (iii) of Theorem 1 that the bifurcations in the interval [a 2 D m 0 , 2 − a 2 C m 0 ] are global.
Individual vortices in the multi-vortex configurations
We can assume a > 0 in the expansions (2.7) and (2.8) for the primary branch after a change of phase. Also, we can choose the sign of b by a shift of θ, i.e. we can assume b > 0 in the expansions (4.7) and (4.10) for the secondary branch. Here we analyze the location of individual vortices in the multi-vortex configurations bifurcating along the secondary branch.
First, we prove that the total vortex charge is preserved near the origin when the secondary branch bifurcates off from the primary branch.
Lemma 11. Fix R > 0. There exists b 0 > 0 such that the degree of the bifurcating solution U along the secondary branch on the circle of the radius R is m 0 for every b ∈ [0, b 0 ).
Proof. We recall that a > 0 and ψ m 0 (r) > 0 for every r ∈ (0, ∞). For every fixed R > 0, there exists a sufficiently small b 0 > 0 such that the bifurcating solution U (r, θ) in Propositions 1 and 2 is nonzero at r = R for every b ∈ [0, b 0 ). This follows from the smallness of the error terms in the expansions (4.7) and (4.10) in the norm of X = H 2 (R 2 ) ∩ L 2,2 (R 2 ), which is embedded in C 0 (R 2 ). Since U (r, θ) is nonzero at r = R, the degree of U on the disk B R of radius R is well defined and does not change for every b ∈ [0, b 0 ). Since the degree is m 0 at b = 0, it remains m 0 for every b ∈ [0, b 0 ).
Remark 24.
Because ψ m 0 (r) → 0 as r → ∞, we are not able to claim that additional zeros of U (r, θ) cannot come from infinity as b = 0. If such zeros exist, additional individual vortices come from infinity on a very small background U (r, θ).
Next, we rewrite the eigenfunctions e m,n (r) of the linear eigenvalue problem (2.2) in the form The following proposition deals with the secondary bifurcations described in Proposition 3.
Proposition 6. Let 0 < b a and consider the bifurcating solution to the stationary GP equation (2.4) in the form (2.18) given by the expansions (2.7) and (4.7). Let r 0 be the first positive zero of the function To determine the small radius ρ in the limit b → 0, we factorize the factor e −r 2 /2 in the eigenfunctions (5.1) and truncate the error term φ. Since we assume that p m,n (r) is positive for r near zero, then the right-hand side of (5.4) is strictly positive for θ = kζ and r 0. For θ = (k + 1/2)ζ, we have e −i(m−m 0 )θ = −1, hence the right-hand side of (5.4) has a zero only if z(r) in (5.3) has a positive root.
Let r 0 be the first positive root of z in (5.3) and assume that it is simple. Since the function φ(r, θ) is small in the norm of X = H 2 (R 2 ) ∩ L 2,2 (R 2 ) by Proposition 3, an application of the implicit function theorem proves that the representation (5.4) with θ = ζ/2 = π/(m − m 0 ) has the (m − m 0 ) polygon of simple zeros at the circle of radius ρ, where ρ = r 0 + O(a 2 ).
Remark 25. For the last bifurcation with m = m 0 + 1 described in Proposition 4, a similar result cannot be proven because the small parameter a is scaled out from the expansion (4.10). The remainder term φ = O X (ab 2 ) in the representation for U (r, θ) may give a contribution to the distribution of individual vortices, which is comparable with the leading-order term aψ m 0 (r)e im 0 θ and the bifurcating mode abf m 0 +1,0 (r, θ; a).
In the rest of this section, we study individual vortices in the bifurcating multi-vortex configurations. 5.1. (m − m 0 )-polygons of vortices. Polygons made of vortices rotating at a constant speed have been studied for many models: fluids, BECs and superconductors. It has been found that these relative equilibria of m vortices are stable for m ≤ 7, see, e.g., [8,21] and references therein. We have found that similar multi-vortex configurations appear along the secondary branches bifurcating from the primary branch of the radially symmetric vortex of charge m 0 ≥ 2. As an example, we give precise information about the vortex polygons in the particular cases n = 0 and n = 1. For n = 1, the bifurcation is similar to the bifurcation of complex multi-vortex solutions described in Lemma 3.3 of [16] for m 0 = 6. 5.1.1. Case n = 0: Vortex polygons with a central vortex. Bifurcation occurs at the bifurcation curve Ω m,n ∈ (0, 2) with n = 0 and 2m 0 + 1 ≤ m ≤ 3m 0 − 1 (when m 0 ≥ 2) in accordance with Lemma 5, Propositions 1, 3, and 6. By using (5.2) and (5.3), we write explicitly
z(r) = ap m 0 ,0 (r) − bp m−2m 0 ,0 (r) = √ 2 √ m 0 ! r m−2m 0 ar 3m 0 −m − b √ m 0 ! (m − 2m 0 )! ,
where we recall that 2m 0 < m < 3m 0 . If 0 < b a, the first positive zero of z(r) is located at
(5.5) r 0 = b a √ m 0 ! (m − 2m 0 )! 1/(3m 0 −m)
. We claim that the degree of each simple zero of U (r, θ) is +1, which means that each zero of U on the (m − m 0 )-polygon represents a vortex of charge one. By symmetry of D m−m 0 , each zero in the (m − m 0 )-polygon has equal degree, hence it is sufficient to compute the degree at the simple zero (r, θ) = (ρ, ζ/2). Using Taylor expansion of U in (5.6) for b a, we obtain cU (r, θ) = z (ρ)(r − ρ) + ib(m − m 0 )p m−2m 0 ,0 (ρ)(θ − ζ/2) + O(2),
By
where c ∈ C is constant and O(2) denotes quadratic remainder terms of the Taylor expansion. Because m − 2m 0 < m 0 , we have z(r) < 0 for r > 0 sufficiently small, therefore, z (ρ) > 0 for b > 0 sufficiently small. On the other hand, m > 2m 0 and p m−2m 0 ,0 (ρ) > 0 in the same limit. Therefore, the degree of U (r, θ) at (r, θ) = (ρ, ζ/2) is +1.
In addition, U (r, θ) in (5.6) has a zero at r = 0 if the remainder term e im 0 θ φ(r, θ) is truncated. Let d be the degree of U in a neighborhood of r = 0. The degree in the disk B R of a sufficiently large radius R is equal to sum of the local degrees in the disk. By Lemma 11, we have d + m − m 0 = m 0 , hence d = 2m 0 − m < 0.
When the remainder term e im 0 θ φ(r, θ) is taken into account in (5.6), the multiple zero of U at r = 0 may split from the origin. However Each vortex has charge one by using the same arguments as in the case n = 0.
Remark 28. If m = 2m 0 , the polygon of m 0 charge-one vortices surrounds the origin with no central vortex. For m 0 = 3, the bifurcation point is Ω 6,1 = 2/3 + O(a 2 ) and the secondary branch has three charge-one vortices located at the equilateral triangle. For m 0 = 6 studied in [16], the bifurcation point is Ω 12,1 = 4/3 + O(a 2 ) and the secondary branch has six charge-one vortices at a hexagon (top panel of Figure 4).
5.2.
Asymmetric vortex and asymmetric vortex pair. Bifurcation occurs at the bifurcation curve Ω m 0 +1,0 = 2 + O(a 2 ) (when m 0 ≥ 1) in accordance with Lemma 6, Propositions 2 and 4. By using (4.10) and (4.11), we write explicitly U (r, θ) = a p m 0 ,0 (r) + bc m 0 +1 p m 0 +1,0 (r)e iθ + bc −m 0 +1 p m 0 −1,0 (r)e −iθ e −r 2 /2 e im 0 θ +φ(r, θ)e im 0 θ , where (c m 0 +1 , c −m 0 +1 ) is obtained from the eigenvector in (3.17) and φ = O X (ab 2 ), see Remark 25. In particular, we have c m 0 +1 > 0 and c −m 0 +1 < 0.
Remark 30. If m 0 = 1 and 0 < b a, the simple zero of U (r, θ) near the origin is located at ρ = b|c 0 | + O(b 2 ). The degree of U near the simple zero at (r, θ) = (ρ, π) is again +1, so that the corresponding vortex has charge one. Since no other zeros of U (r, θ) are located near the origin, the bifurcating solution at the secondary branch corresponds to the asymmetric vortex obtained in [29] (center panel of Figure 4).
Remark 31.
If m 0 = 2 and 0 < b a, the double zero of U (r, θ) at the origin for b = 0 split to the distances ρ ± = O(b) according to the roots of the quadratic equation
(5.7) r 2 ± b|c −1 | √ 2 + b 2 β = 0,
where β is a numerical constant obtained from the remainder term φ(r, θ), whereas the plus and minus signs correspond to the choice θ = 0 and θ = π respectively. Only positive roots of the quadratic equations (5.7) are counted, and according to Lemma 11, we should have the total of two positive roots at both sign combinations. Indeed, if β > 0, the two positive roots ρ ± = O(b) exist for θ = π and no positive roots for θ = 0, while if β < 0, one positive root ρ + exists for θ = 0 and one positive root exists for θ = π. In both cases, ρ + = ρ − , so that the bifurcating solution at the secondary branch corresponds to the asymmetric pair of two charge-one vortices obtained in [26] (bottom panel of Figure 4 in the case β < 0).
Remark 32. If m 0 ≥ 3 and 0 < b a, the multiple root of U (r, θ) at the origin for b = 0 split to the distances ρ ± = O(b) according to the roots of the n-th order polynomial equation, which is obtained from computations of the remainder term φ(r, θ) up to the order of b n . By Lemma 11, there must exist exactly n roots to the two polynomial equations for θ = 0 and θ = π but the precise characterization of these roots depend on the coefficients of the polynomial equation.
Remark 33. Proposition 6 and computations in Sections 5.1 and 5.2 yield the proof of item (v) of Theorem 1. All items of Theorem 1 have been proved.
Theorem 1 .
1Fix an integer m 0 ∈ N and denote ω := µ + m 0 Ω.
Figure 1 .
1A schematic illustration of the bifurcation curves in the parameter plane (Ω, a), where a defines ω. The bifurcating solutions form surfaces parameterized by (Ω, a) close to the curves Ω m,n . (i) There exists a smooth family of radially symmetric vortices of charge m 0 with a positive profile U satisfying (1.3) with ω = ω(a) given by ω(a) = 2(m 0 + 1) + (2m 0 )! 4 m 0 (m 0 !) 2 a 2 + O(a 4 ), where the "amplitude" a parameterizes the family. (ii) For Ω = 0 and small a, the vortices are degenerate saddle points of the energy E in (1.2) with 2N (m 0 ) negative eigenvalues, a simple zero eigenvalue, and 2Z(m 0 ) small eigenvalues of order O(a 2 ), where N (m 0 ) = 1 2 m 0 (m 0 + 1) and Z(m 0 ) = m 0 .
Remark 4 .
4For m 0 ≥ 4, there are R(m 0 ) additional bifurcations near Ω 0 = 2. For 4 ≤ m 0 ≤ 16, the additional bifurcation arise past the last bifurcation point at Ω m 0 +1,0 . For m 0 ≥ 17, some of the R(m 0 ) bifurcations arise before the "last" bifurcation point. We have found numerically that R(4) = R(5) = 1, R(6) = 2, R(7) = R(8) = 3, etc.
m 0 (r) ≡ ψ m 0 (r; a) = ae m 0 ,0 (r) + O H 1 r (a 3 ) and (2.8) ω ≡ ω m 0 (a) = λ m 0 ,0 + a 2 ω m 0 ,0 + O(a 4 ),
Remark 5 .
5Item (i) in Theorem 1 is just a reformulation of the result of Theorem 1 in [4]. Every solution U of the stationary GP equation (2.4) is a critical point of the energy functional
Lemma 2 .
2Eigenvalues of the spectral problem (2.17) with m < m 0 are identical to eigenvalues of the spectral problem (2.17) for m > m 0 . Proof. We observe the symmetry ∆ m = ∆ m 0 +(m−m 0 ) and ∆ m−2m 0 = ∆ m 0 −(m−m 0 ) with respect to the symmetry point at m = m 0 . As a result, for each k ∈ N, if λ = Ωk is an eigenvalue of the spectral problem (2.17) with m = m 0 + k for the eigenvector [V m 0 +k , W −m 0 +k ], then λ = Ωk is the same eigenvalue of the spectral problem (2.17) with m = m 0 − k for the eigenvector
Remark 6 .
6Lemma 3 yields the proof of item (ii) of Theorem 1. Let us give some explicit examples. If m 0 = 1, then λ 1,0 = 4 and
2, 6, 10, 10, 14, 14, · · · }, σ(K 9 ) = {0, 4, 8, 12, 12, 16, 16, · · · }, σ(K 10 ) = {2, 6, 10, 14, 14, 18, 18, · · · }, . . .(2.26) so that N (3) = 6 and Z(3) = 3.
3. 1 .
1Zero eigenvalues of K m (0). When a = 0 and Ω = 0, each operator block H m (0, 0) = K m (0) has a simple zero eigenvalue for m = m 0 + 2 , 1 ≤ ≤ m 0 . See examples in (2.24), (2.25), and (2.26). The following lemma tells us that the zero eigenvalue of such H m (0, 0) becomes a positive eigenvalue of H m (a, Ω) for every sufficiently small a, provided the values of Ω are sufficiently large and positive.
Remark 7 .
7Lemma 4 yields the existence of constant D m ≥ 0 in item (iii) of Theorem 1.
Remark 8 .
8Ifμ > 0 in the perturbation result (3.3), then µ > 0 for every small a > 0 and Ω > 0. If this is true for every 1 ≤ ≤ m 0 , then D m 0 = 0 in Lemma 4. In particular, this is true for 1 ≤ m 0 ≤ 3. Indeed, for = m 0 and n( ) = 0, we obtain from (2.10) and (For = m 0 −1 and n( ) = 1, we use the following formula for the L 2 r (R + )-normalized Hermite-Gauss solutions of the Schrödinger equation (2.2) with λ m,1 = 2(m + 3):
Remark 9 .
9It remains unclear ifμ > 0 for the other values in 2 ≤ ≤ m 0 − 2 for m 0 ≥ 4.3.2.Zero eigenvalues of H m (a, Ω) for Ω ∈ (0, 2). For m > m 0 , the leading-order diagonal operator H m (0, Ω) given by the operators L + and L − in (3.1) has an eigenbasis(3.5) {(e m,n , 0); (0, e |m−2m 0 |,n )} n∈N 0 .
,0 ( 0 ) = 2/3; • For m 0 = 3, three bifurcations occur at Ω 7,0 ( 0 ) = 1 , Ω 8,0 ( 0 ) = 2/5, and Ω 6,1 ( 0 ) = 2/3; • For m 0 = 4, six bifurcations occur at Ω 9,0 ( 0 Remark 10 .
00100010particular, we have B(1) = 0, B(2) = 1, B(3) = 3, and B(4) = 6. See examples in (2.24), (2.25), and (2.26). Let us list the bifurcation values of Ω for these examples: • For m 0 = 1, no bifurcations occur; • For m 0 = 2, the only bifurcation occurs at Ω 5Lemma 5 yields the number B(m 0 ) in item (iii) of Theorem 1. Note that the bifurcation points of Ω are simple for 1 ≤ m 0 ≤ 3. Multiple bifurcation points exist in a general case for m 0 ≥ 4, e.g. Ω 10,0 (0) = Ω 7,1 (0) = 2/3 for m 0 = 4. The following proposition summarizes properties of H m (a, Ω) near each bifurcation point. These properties are needed for the bifurcation analysis in Section 4. Proposition 1. For every m 0 ∈ N, let Ω * (a) be one of the bifurcation points defined by (3.7). Assume it has multiplicity k and corresponds to m 1 , . . . , m k > m 0 . There exists a 0 > 0, C m 0 > 0, and E m 0 > 0 such that for every 0 < a < a 0 , |Ω − Ω * (a)| < C m 0 a 2 , and every m > m 0 such that m / ∈ {m 1 , . . . , m k }, the operator H m (a, Ω) is invertible in L 2 r (R + ) 1 , . . . , m k }. Moreover, the number of negative eigenvalues of H m (a, Ω), m / ∈ {m 1 , . . . , m k } remains the same for every Ω in |Ω − Ω * (a)| < C m 0 a 2 . On the other hand, the number of negative eigenvalues for H m (a, Ω), m ∈ {m 1 , . . . , m k } is reduced by one when Ω crosses Ω * (a) in |Ω − Ω * (a)| < C m 0 a 2 . Proof. First, we note that for each m > m 0 , there may be at most one eigenvalue of H m (a, Ω) which becomes zero at Ω = Ω * (a). Bound (3.9) follows from the fact that H m (a, Ω * (a)) with m / ∈ {m 1 , . . . , m k } has no eigenvalues in the neighborhood of zero. On the other hand, each simple eigenvalue of H m (a, Ω * (a)) with m ∈ {m 1 , . . . , m k } is continued in Ω according to the derivative (3.10) ∂H m ∂Ω (a, Ω) = −(m − m 0 )R. Let (V m , W m−2m 0 ) be the corresponding eigenvector for the zero eigenvalue of H m (a, Ω * (a)). Since m > m 0 , the eigenvalue is positive for Ω Ω * (a) and negative for Ω Ω * (a) if S m < 0, where (3.11)
Definition 1 .
1If k = 1 in Proposition 1, we say that the bifurcation point Ω * (a) is non-resonant. 3.3. Zero eigenvalues of H m (a, Ω) for Ω near 2. Consider the rotation frequency Ω = 2 + O(a 2 ). According to (2.17) and (2.22), see examples in (2.24), (2.25), and (2.26), there are infinitely many resonances for a = 0.
( 3 .
312)Ω m 0 +1,0 (a) := 2 − (2m 0 )! 4 m 0 m 0 !(m 0 + 1)! a 2 + O(a 4 ),
eigenvalue in (3.17) corresponds to the choice Ω = Ω m 0 +1,0 (a) at the bifurcation point. The positive eigenvalue in (3.18) gives the positive eigenvalue of the order O(a 2 ) in (3.16). The other eigenvalues of H m 0 (0, 2) are strictly positive and they remain so in H m 0 +1 (a, Ω m 0 +1,0 (a)) for small a.
Lemma 8 .
8There exists a 0 > 0 such that for every 0 < a < a 0 , the block H m (a, Ω m 0 +1,0 (a)) with m ≥ 2m 0 + 1 has a simple positive eigenvalue of the order O(a 2 ), whereas all other eigenvalues are strictly positive.
condition yields the only eigenvalue given bỹλ = 2 e 2 m 0 ,0 e m,0 , e m,0 L 2 r − ω m 0 ,0 − (m − m 0 )Ω m 0 ,0 = 2(m 0 + m)! 2 m 0 +m m 0 !m! + (2m 0 )!(m − 2m 0 − 1) 4 m 0 m 0 !(m 0 + 1)! > 0. (3.20)Sinceλ > 0, the expansion (3.19) yields the positive eigenvalue of the order O(a 2 ) in the block H m (a, Ω m 0 +1,0 (a)) for small a.
Remark 13 .
13For m 0 ≥ 17, the matrixà for some m in the range m 0 + 2 ≤ m ≤ 2m 0 has two negative eigenvalues and the number of such m-values grows with the number m 0 . No zero eigenvalues ofà are found numerically for at least m 0 ≤ 100.The following proposition summarizes the previous computations of the perturbation theory. The corresponding result is needed for the bifurcation analysis in Section 4.
Proposition 2 .
2For every integer 1 ≤ m 0 ≤ 16, there exists a 0 > 0, C m 0 ∈ (0, |Ω m 0+1 ,0 |), and E m 0 > 0 such that for every 0 < a < a 0 , |Ω − Ω m 0 +1,0 (a)| < C m 0 a 2 , and every m ≥ m 0 + 2, the operator H m (a, Ω) is invertible in L 2 r (R + ) with the bound(3.23) H m (a, Ω) −1 L 2 r →H 2 r ∩L 2,2 r ≤ E m 0 a −2 , m ≥ m 0 + 2.Moreover, all eigenvalues of H m (a, Ω) are strictly positive, except for m 0 − 1 simple negative eigenvalues, which correspond to m 0 + 2 ≤ m ≤ 2m 0 . On the other hand, all eigenvalues of H m 0 +1 (a, Ω) are strictly positive except one simple eigenvalue, which is negative for Ω Ω m 0 +1,0 (a) and positive for Ω Ω m 0 +1,0 (a).Proof. Eigenvalues and invertibility of H m (a, Ω m 0 +1,0 (a)) for m ≥ m 0 + 2 with the bound (3.23) follows from the outcomes of the perturbation theory in Lemmas 7, 8, and 9, where the m-independent constant E m 0 exists thanks to the fact that the O(a 2 ) positive eigenvalue in(3.20) is bounded away from zero.
Remark 14 .
14Propositions 1 and 2 complete the proof of item (iii) in Theorem 1. Remark 15. Eigenvalues λ := Ω(m − m 0 ) of the bifurcation problem (2.17) near λ m,m 0 = 2(m − m 0 ) are either complex or real for m 0 + 2 ≤ m ≤ 2m 0 , depending on whether the mth mode of the m 0 -th vortex is spectrally unstable or stable. When all such eigenvalues are complex, which happens for 1 ≤ m 0 ≤ 3, no other bifurcation curve is connected to the point Ω 0 = 2 from below, besides the curve Ω m 0 +1,0 . When m 0 ≥ 4, we have found that there are R(m 0 ) pairs of real eigenvalues λ of the bifurcation problem (2.17) near λ m,m 0 , e.g. • R(4) = 1 with m = 8; • R(5) = 1 with m = 10; • R(6) = 2 with m = 11, 12; • R(7) = 3 with m = 12, 13, 14; • R(8) = 3 with m = 14, 15, 16;
4. 1 .
1Symmetries and equivariance of g(v; a, Ω). We define the action of the group O(2) = S 1 ∪ κS 1 by(4.1) ρ(ϕ)v(r, θ) = e −im 0 ϕ v(r, θ + ϕ), ρ(κ)v(r, θ) =v(r, −θ).The operator g(v; a, Ω) given by(2.19)-(2.20) is O(2)-equivariant by the action of the group given by (4.1). That is, we have g(ρ(ϕ)v) = ρ(ϕ)g(v) sincee im 0 ϕ g(ρ(ϕ)v)(r, θ − ϕ) = −ω m (a) − ∆ (r,θ) + r 2 + Ωi(∂ θ − im 0 ) v(r, θ) − e im 0 θ ψ m 0 (r; a) 3+ e im 0 θ ψ m 0 (r; a) + v(r, θ) 2 e im 0 θ ψ m 0 (r; a) + v(r, θ) .
Figure 2 .
2The isotropy lattice for the symmetry-breaking bifurcations.
D m−m 0 in Fix(D m−m 0 ) because the reflection κ ∈ D m−m 0 excludes the gauge invariance. Furthermore, the double eigenvalues of H in the blocks with positive and negative j − m 0 become the simple eigenvalues of H D m−m 0 in Fix(D m−m 0 ) again due to the reflection κ.
4. 2 .
2Local bifurcation results. Here we prove a local bifurcation from a simple eigenvalue of H D m−m 0 that exists at Ω = Ω m,n (a) for small a, according to (3.7) and (3.12) in Lemmas 5 and 6. The restriction of the space X to the fixed-point space Fix(D m−m 0 ) is useful in two aspects. First, it allows us to prove the local bifurcation from a simple eigenvalue by avoiding resonances from the components that are not contained in Fix(D m−m 0 ). Second, it gives additional information on symmetries of the bifurcating solutions v. The symmetries are useful to understand the distributions of individual vortices in the (m − m 0 )-polygons.
Remark 16 .
16The curves Ω m,0 for 2m 0 + 1 ≤ m ≤ 3m 0 − 1 are non-resonant. Even if the resonance occurs in H, e.g. for m 0 = 4, it does not show up in H D m−m 0 . Indeed, if
Proposition 3 .
3For each non-resonant curve Ω m,n ∈ (0, 2) parameterized by a > 0 sufficiently small, the operator g D m−m 0 (v; a, Ω) in (4.5) admits a new family of roots v ∈ Fix(D m−m 0 ) and Ω ∈ (0, 2) parameterized by real b such that
r, θ; a, Ω(a, b)) = bf m,n (r, θ; a) + O X (ab 2 , b 3 ), where (4.8) f m,n (r, θ; a) = e |m−2m 0 |,n (r)e i(2m 0 −m)θ + O X (a 2 )
Proposition 4 .
4For the non-resonant curve Ω m 0 +1,0 parameterized by a > 0 sufficiently small, the operator g D 1 (v; a, Ω) in (4.5) admits a new family of roots v ∈ Fix(D 1 ) and Ω ∈ (0, 2) parameterized by real b such that
a, b) = Ω m 0 +1,0 (a) + O(a 2 b 2 ) and (4.10) v(r, θ; a, Ω(a, b)) = a bf m 0 +1,0 (r, θ; a) + O X (b 2 ) , where f m 0 +1,0 is the eigenvector of H m 0 +1 (a, Ω m 0 +1,0 (a)) associated with the zero eigenvalue. Proof. The scaling of a in (4.10) is needed due to the loss of O(a −2 ) in the bound (3.23) on the inverse operator (H D 1 ) −1 , according to Proposition 2. Since ψ m 0 = O(a), the nonlinear terms in the operator g D 1 (v; a, Ω) are now scaled by O(a 3 ), hence the loss of O(a −2 ) produces the terms of the expansion (4.10) at the order O(a) and higher. Hence the bifurcation problem is closed at the order O(a) and the proof follows the one in Proposition 3 with a new parameter b, which is scaled independently of a. Remark 19. The local bifurcation in Proposition 4 is also of the pitchfork type. Thanks to the computations in Lemma 7, the leading-order approximation of the eigenvector f m 0 +1,0 is given by(4.11) f m 0 +1,0 (r, θ; a) = c m 0 +1 e m 0 +1,0 (r)e i(m 0 +1)θ + c −m 0 +1 e m 0 −1,0 (r)e i(m 0 −1)θ + O X (a 2 ), for a > 0 sufficiently small, where (c m 0 +1 , c 1−m 0 ) is an eigenvector of the matrixà computed in (3.17). Remark 20. Propositions 3 and 4 yield the proof of item (iv) in Theorem 1.4.3. Global bifurcation. We obtain the global bifurcation result in the fixed-point space Fix(D m−m 0 ) by using the topological degree theory in the case of simple eigenvalues. It is usually referred to as the global Rabinowitz result, see Theorem 3.4.1 of [27]. The global bifurcation result means that the solution branch (v, Ω) that originates at the non-resonant bifurcation curve (0, Ω m,n ) either reaches the boundaries Ω = 0 and Ω = 2 or return to another bifurcation point (0, Ω * ) or diverges to infinite values of v for a finite value of Ω ∈ [0, 2).
By
Lemma 10, the Jacobian operator H is Fredholm of the degree zero for Ω ∈ [0, 2). Also the restricted operator H D m−m 0 (Ω) is a self-adjoint Fredholm operator for every Ω ∈ [0, 2). Since H D m−m 0 (Ω) is invertible for Ω close but different from the non-resonant bifurcation curve Ω m,n , then the Morse index n D m−m 0 (Ω) of H D m−m 0 (Ω) restricted to ker H D m−m 0 (Ω m,n ) for Ω close to Ω m,n is well defined. Let η D m−m 0 (Ω m,n ) be the net crossing number of eigenvalues of H D m−m 0 (Ω) defined by (4.12) η D m−m 0 (Ω m,n ) := lim ε→0 n D m−m 0 (Ω m,n + ε) − n D m−m 0 (Ω m,n − ε) .
Proposition 5 .
5Fix a > 0 sufficiently small, if η D m−m 0 (Ω m,n ) is odd for Ω m,n ∈ [0, 2), the nonlinear operator g(v; a, Ω) has a global bifurcation of solutions (v, Ω) in Fix(D m − m 0 ) × [0, 2) arising from (v, Ω) = (0, Ω m,n ).
where c > 0 is defined inLemma 10. The operator f is also equivariant and can be restrictedto Fix(D m−m 0 ) denoted by f D m−m 0 . The index for bifurcation of f in Fix(D m−m 0 ) is up to an orientation factor, the jump on the local indices as Ω crosses Ω m,n . That is, since η D m−m 0 (Ω m,n ) is odd, then deg x − ε, f D m−m 0 (x, Ω); B 2ε × B 2ρ = deg(f D m−m 0 (x, Ω − ρ); B 2ε ) − deg(f D m−m 0 (x, Ω + ρ); B 2ε ) = ± 1 − (−1) η D m−m 0 = ±2, (4.13)
m,n (r) = p m,n (r)e −r 2 /2 , where p m,n (r) is a polynomial of degree |m| + 2n, which is chosen to be positive for r near zero. The first eigenfunctions e m,0 (r) and e m,1 (r) in (2.9) and(3.4) are given by (5.1) with (5.2) p m,0 (r) = √ 2 √ m! r m , p m,1 (r) = √ 2 (m + 1)! r m (m + 1 − r 2 ).
r) := ap m 0 ,0 (r) − bp |m−2m 0 |,n (r) and assume that it is a simple zero 5 . Then, the bifurcating solution has simple zeros arranged in the (m − m 0 )-polygon on a circle of radius ρ with ρ = r 0 + O(a 2 ).Proof. By combining (2.7), (2.18), (4.7), and (4.8), we obtain an asymptotic representation of the bifurcating solutions U in the formU (r, θ) = ae m 0 ,0 (r)e im 0 θ + be |m−2m 0 |,n (r)e i(2m 0 −m)θ + O X (a 3 , a 2 b, ab 2 , b 3 ).Zeros of U (r, θ) are equivalent to the zeros of(5.4) e −im 0 θ U (r, θ) = ae m 0 ,0 (r) + be |m−2m 0 |,n (r)e −i(m−m 0 )θ + φ(r, θ), where φ = O X (a 3 ,a 2 b, ab 2 , b 3 ) satisfies the symmetry constraints (4.4). The function e −i(m−m 0 )θ is real only if θ = kζ and θ = (k + 1/2)ζ, where ζ = 2π/ (m − m 0 ) and k ∈ Z. For these angles, the function φ(θ, r) is real by the symmetries (4.4). Therefore, the function (5.4) is real if and only if θ = kζ and θ = (k + 1/2)ζ. These two choices of angles give two choices of the (m − m 0 )-polygons of zeros along a circle of radius ρ.
Figure 3 .
3The left and right columns illustrate the norm and phase of the truncated solution U in (5.6). Top: m 0 = 2 near Ω 5,0 . Middle: m 0 = 3 near Ω 7,0 . Bottom: m 0 = 3 near Ω 8,0 .
Proposition 6, we have a (m − m 0 )-polygon of simple zeros of the function (5.6) U (r, θ) = ap m 0 ,0 (r) + bp m−2m 0 ,0 (r)e −i(m−m 0 )θ e −r 2 /2 e im 0 θ + φ(r, θ)e im 0 θ , at points (r, θ) = (ρ, ζ/2+kζ), where ζ = 2π/(m−m 0 ), k ∈ {0, 1, ..., m−m 0 −1}, ρ = r 0 +O(a 2 ), and r 0 is given by (5.5).
3
3, by the symmetry in D m−m 0 , if the central vortex splits, then it breaks into m − m 0 vortices of equal charge |d|/(m − m 0 ). Since m − 2m 0 ≤ m 0 − 1 < m 0 + 1 ≤ m − m 0 , then |d|/(m − m 0 ) < 1 and the central vortex never splits. Remark 26. For the case m 0 = 2, we have the bifurcation point Ω 5,0 = 2/3 + O(a 2 ). Since m = 5 and n = 0, we have a configuration of three vortices of charge one that form an equilateral triangle and a central vortex of charge d = 2m 0 − m = −1 (top panel of Figure 3). Remark 27. For the case m 0 = 3, we have two bifurcation points Ω 7,0 = 1 + O(a 2 ) and Ω 8,0 = 2/5 + O(a 2 ). At the former bifurcation, the bifurcating branch has four vortices of charge one that form a square and the central vortex of charge d = 2m 0 − m = −1 (middle panel of Figure 3). At the latter bifurcation, the bifurcating branch has five vortices of charge one that form an equilateral pentagon and the central vortex of charge d = 2m 0 − m = −2 (bottom panel of Figure 3). 5.1.2. Case n = 1: Vortex polygons without a central vortex. Bifurcation occurs at the bifurcation curve Ω m,n with n = 1 and m 0 + 3 ≤ m ≤ 3m 0 − 3 (when m 0 ≥ r) = ap m 0 ,0 (r) − bp |m−2m 0 |,1 |m−2m 0 | ar m 0 −|m−2m 0 | − bC m,m 0 (|m − 2m 0 | + 1 − r 2 ) , − 2m 0 | + 1)! and we recall that |m − 2m 0 | < m 0 . If 0 < b a, the first positive zero of z(r) 6, we have the (m−m 0 ) polygon of vortices on the circle of radius ρ = r 0 +O(a 2 ).
Remark 29 .
29If m = 2m 0 , U (r, θ) has zero at r = 0 if the remainder term φ(r, θ) is truncated. By Lemma 11, the central zero of U corresponds to the vortex of charge d = 2m 0 − m, where −m 0 < d < m 0 . The central vortex may split into m − m 0 vortices of equal charge only if |d| is divisible by m − m 0 .
Figure 4 .
4The left and right columns illustrate the norm and phase of the truncated solution U . Top: m 0 = 6 near Ω 12,1 . Center: m 0 = 1 near Ω 2,0 . Bottom: m 0 = 2 near Ω 3,0 .
More general vortex families with n0 zeros on R + have also been constructed in[4], but our work will focus on the case n0 = 0.
When the vortex is unstable, a complex eigenvalue λ of the stability problem does not correspond to the secondary bifurcation associated with the eigenvalue problem (2.17).
If (cm 0 +1, c−m 0 +1) in(3.24) is given by the other eigenvector ofà that corresponds toλ > 0, then it follows from (3.18) that Sm 0 +1 > 0. Hence, the corresponding small positive eigenvalue of Hm 0 +1(a, Ω) is a decreasing function of Ω. Nevertheless, for Ω Ωm 0 +1,0(a), these two small eigenvalues of Hm 0 +1(a, Ω) are ranged in the same order of O(a 2 ) as at Ω = Ωm 0 +1,0(a).
The assumption is always satisfied if 0 < b a since p |m−2m 0 |,n (r) is positive for small r and |m−2m0| < m0.
Acknowledgement. The authors are indebted to A. Contreras, P. Kevrekidis and M. Tejada-Wriedt for discussions related to this project and collaboration.
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Rotating matter waves in Bose-Einstein condensates. T Kapitula, P G Kevrekidis, R Carretero-González, Physica D. 233T. Kapitula, P.G. Kevrekidis, and R. Carretero-González, "Rotating matter waves in Bose-Einstein con- densates", Physica D 233 (2007), 112-137.
T Kato, Perturbation theory for linear operators. BerlinSpringer-VerlagT. Kato, Perturbation theory for linear operators (Springer-Verlag, Berlin, 1995).
On the characterization of vortex configurations in the steady rotating Bose-Einstein condensates. P G Kevrekidis, D E Pelinovsky, arXiv:1708.03683P.G. Kevrekidis and D.E. Pelinovsky, "On the characterization of vortex configurations in the steady rotating Bose-Einstein condensates", arXiv:1708.03683 (2017)
Bifurcation Theory, An Introduction with Applications to Partial Differential Equations. H Kielhöfer, Applied Mathematical Sciences. SpringerH. Kielhöfer. Bifurcation Theory, An Introduction with Applications to Partial Differential Equations. Ap- plied Mathematical Sciences, Springer, 2012.
Spectral stability of vortices in two-dimensional Bose-Einstein condensates via the Evans function and Krein signature. R Kollar, R L Pego, Appl. Math. Res. eXpress. 2012R. Kollar and R.L. Pego, "Spectral stability of vortices in two-dimensional Bose-Einstein condensates via the Evans function and Krein signature", Appl. Math. Res. eXpress 2012 (2012), 1-46.
A tale of two distributions: from few to many vortices in quasi-two-dimensional Bose-Einstein condensates. T Kolokolnikov, P G Kevrekidis, R Carretero-González, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 47018T. Kolokolnikov, P.G. Kevrekidis, and R. Carretero-González, "A tale of two distributions: from few to many vortices in quasi-two-dimensional Bose-Einstein condensates", Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), 20140048 (18 pp).
Size and dynamics of vortex dipoles in dilute Bose-Einstein condensates. P Kuopanportti, J A M Huhtamäki, M Möttönen, Phys. Rev. A. 8311603P. Kuopanportti, J. A. M. Huhtamäki, and M. Möttönen, "Size and dynamics of vortex dipoles in dilute Bose-Einstein condensates", Phys. Rev. A 83 (2011), 011603.
Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates. S Middelkamp, P J Torres, P G Kevrekidis, D J Frantzeskakis, R Carretero-Gonzalez, P Schmelcher, D V Freilich, D S Hall, Phys. Rev. A. 8411605S. Middelkamp, P. J. Torres, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher, D. V. Freilich, and D. S. Hall, "Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates", Phys. Rev. A 84 (2011), 011605.
M Morrison, The Joy of Quantum Physics. Oxford University PressM. Morrison, The Joy of Quantum Physics, Oxford University Press 2013.
Stationary vortex clusters in nonrotating Bose-Einstein condensates. M Möttönen, S M M Virtanen, T Isoshima, M M Salomaa, Phys. Rev. A. 7133626M. Möttönen, S. M. M. Virtanen, T. Isoshima, and M. M. Salomaa, "Stationary vortex clusters in nonro- tating Bose-Einstein condensates", Phys. Rev. A 71 (2005), 033626.
Dynamics of a few corotating vortices in Bose-Einstein condensates. R Navarro, R Carretero-González, P J Torres, P G Kevrekidis, D J Frantzeskakis, M W Ray, E Altuntas, D S Hall, Phys. Rev. Lett. 110225301R. Navarro, R. Carretero-González, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altun- tas, and D.S. Hall, "Dynamics of a few corotating vortices in Bose-Einstein condensates", Phys. Rev. Lett. 110 (2013), 225301.
Topics in Nonlinear Functional Analysis. Courant Lecture Notes. L Nirenberg, L. Nirenberg. Topics in Nonlinear Functional Analysis. Courant Lecture Notes. 2001
Variational approximations of trapped vortices in the large-density limit. D Pelinovsky, P G Kevrekidis, Nonlinearity. 24D. Pelinovsky, P.G. Kevrekidis, "Variational approximations of trapped vortices in the large-density limit", Nonlinearity 24 (2011), 1271-1289.
D Pelinovsky, P G Kevrekidis, Bifurcations of Asymmetric Vortices in Symmetric Harmonic Traps. 2013D. Pelinovsky, P.G. Kevrekidis, "Bifurcations of Asymmetric Vortices in Symmetric Harmonic Traps", Applied Mathematics Research eXpress 2013 (2013), 127-164.
Gross-Pitaevskii theory of the rotating Bose gas. R Seiringer, Commun. Math. Phys. 229R. Seiringer, "Gross-Pitaevskii theory of the rotating Bose gas", Commun. Math. Phys. 229 (2002), 491- 509.
Dynamics of vortex dipoles in confined Bose-Einstein condensates. P J Torres, P G Kevrekidis, D J Frantzeskakis, R Carretero-Gonzalez, P Schmelcher, D S Hall, Phys. Lett. A. 375P. J. Torres, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher, and D. S. Hall, "Dynamics of vortex dipoles in confined Bose-Einstein condensates", Phys. Lett. A 375 (2011), 3044-3050.
. Matemáticas Departamento De, Universidad Nacional Autónoma deFacultad de CienciasDepartamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de
|
[] |
[
"ON THE SUMS OF MANY BIQUADRATES IN TWO DIFFERENT WAYS",
"ON THE SUMS OF MANY BIQUADRATES IN TWO DIFFERENT WAYS"
] |
[
"Izadi Farzali ",
"Mehdi And ",
"Baghalaghdam "
] |
[] |
[] |
The beautiful quartic Diophantine equation A 4 + hB 4 = C 4 + hD 4 , where h is a fixed arbitrary positive integer, has been studied by some mathematicians for many years. Although Choudhry, Gerardin and Piezas presented solutions of this equation for many values of h, the solutions were not known for arbitrary positive integer values of h. In a separate paper (see the arxiv), the authors completely solved the equation for arbitrary values of h, and worked out many examples for different values of h, in particular for the values which has not already been given a solution. Our method, give rise to infinitely many solutions and also infinitely many parametric solutions for the equation for arbitrary rational values of h. In the present paper, we use the above solutions as well as a simple idea to show that how some numbers can be written as the sums of two, three, four, five, or more biquadrates in two different ways. In particular we give examples for the sums of 2, 3, · · · , and 10, biquadrates expressed in two different ways.
| null |
[
"https://arxiv.org/pdf/1701.02687v1.pdf"
] | 119,585,946 |
1701.02687
|
ab21f8fb4aabc8330a308ddd0d2fede1a7572347
|
ON THE SUMS OF MANY BIQUADRATES IN TWO DIFFERENT WAYS
9 Jan 2017
Izadi Farzali
Mehdi And
Baghalaghdam
ON THE SUMS OF MANY BIQUADRATES IN TWO DIFFERENT WAYS
9 Jan 2017
The beautiful quartic Diophantine equation A 4 + hB 4 = C 4 + hD 4 , where h is a fixed arbitrary positive integer, has been studied by some mathematicians for many years. Although Choudhry, Gerardin and Piezas presented solutions of this equation for many values of h, the solutions were not known for arbitrary positive integer values of h. In a separate paper (see the arxiv), the authors completely solved the equation for arbitrary values of h, and worked out many examples for different values of h, in particular for the values which has not already been given a solution. Our method, give rise to infinitely many solutions and also infinitely many parametric solutions for the equation for arbitrary rational values of h. In the present paper, we use the above solutions as well as a simple idea to show that how some numbers can be written as the sums of two, three, four, five, or more biquadrates in two different ways. In particular we give examples for the sums of 2, 3, · · · , and 10, biquadrates expressed in two different ways.
3086, 3193, 3307, 3319, 3334, 3347, 3571, 3622, 3623, 3628, 3644, 3646, 3742, 3814, 3818, 3851, 3868, 3907, 3943, 3980, 4003, 4006, 4007, 4051, 4054, 4099, 4231, 4252, 4358, 4406, 4414, 4418, 4478, 4519, 4574, 4583, 4630, 4643, 4684, 4870, 4955, 4999,
see [S.T].
Gerardin and Piezas found solutions of this equation when h is given by polynomials of degrees 5, and 2, respectively, (See [T.P], [S.T]). Also Choudhry presented several new solutions of this equation when h is given by polynomials of degrees 2, 3, and 4. (See [A.C2])
In a seprate paper, by using the elliptic curves theory, we completely solved the above Diophantine equation for every arbitrary rational values of h, in particular, for every arbitrary positive integer values of h, (See [I.B]). In this paper by using the results of the previous paper as well as a simple idea, we easily show that how some numbers can be written as the sums of two, three, four, five, or more biquadrates in two different ways.
Our main result is the following:
Main Theorem 1.1. The Diophantine equation n i=1 a 4 i = n i=1 b 4 i , where n ≥ 2, is a fixed arbitrary natural number, has infinitely many nontrivial positive integer solutions. Furthermore, we may obtain infinitely many parametric solutions for the aforementioned Diophantine equation. This shows that how some numbers can be written as the sums of many biquadrates in two different ways.
Firstly, we prove the following theorem, which is our main theorem in the pervious paper, then this theorem easily implies the above main result.
The Diophantine equation
A 4 + hB 4 = C 4 + hD 4
Theorem 2.1. The Diophantine equation :
(2.1) A 4 + hB 4 = C 4 + hD 4 ,
where h is an arbitrary rational number, has infinitely many integer positive solutions. Furthermore, we may obtain infinitely many parametric solutions for the above Diophantine equation as well. Moreover, we have A + C = B + D. By letting h = v u , this also solves the equation of the form uA 4 + vB 4 = uC 4 + vD 4 , for every arbitrary integer values of u, and v.
We use some auxiliary variables for transforming the above quartic equation to a cubic elliptic curve of the positive rank in the form
Y 2 = X 3 + F X 2 + GX + H,
where the coefficients F , G, and H, are all functions of h. Since the elliptic curve has positive rank, it has infinitely many rational points which give rise to infinitely many integer solutions for the above equation too.
Proof. Let: A = m − q, B = m + p, C = m + q, and, D = m − p, where all variables are rational numbers. By substituting these variables in the above equation we get −8m 3 q − 8mq 3 + 8hm 3 p + 8hmp 3 = 0.
Then after some simplifications and clearing the case m = 0, we get:
(2.2) m 2 (hp − q) = −hp 3 + q 3 .
We may assume that hp − q = 1, and m 2 = −hp 3 + q 3 . By plugging q = hp − 1, into the equation (2.2), and some simplifications, we obtain the elliptic curve:
(2.3) m 2 = (h 3 − h)p 3 − (3h 2 )p 2 + (3h)p − 1.
Multiplying both sides of this elliptic curve by (h 3 − h) 2 , and letting
(2.4) Y = (h 3 − h)m, and (2.5) X = (h 3 − h)p,
we get the new elliptic curve
(2.6) E(h) : Y 2 = X 3 − (3h 2 )X 2 + (3h(h 3 − h))X − (h 3 − h) 2 .
Then if for given h, the above elliptic curve has positive rank, by calculating m, p, q, A, B, C, D, from the relations (2.4), (2.5), q = hp − 1, A = m − q, B = m + p, C = m + q, D = m − p, after some simplifications and canceling the denominators of A, B, C, D, we obtain infinitely many integer solutions for the Diophantine equation.
If the rank of the elliptic curve (2.6) is zero, we may replace h by ht 4 , for an appropriate arbitrary rational number t such that the rank of the elliptic curve (2.6) becomes positive. Then, we obtain infinitely many integer solutions for the Diophantine equation A 4 + (ht 4 )B 4 = C 4 + (ht 4 )D 4 . Then by multiplying t 4 , to the numbers B 4 , D 4 ,(written as A 4 + h(tB) 4 = C 4 + h(tD) 4 ), we get infinitely many positive integer solutions for the main Diophantine equation A 4 + hB 4 = C 4 + hD 4 (see the examples.).
Finally, for getting infinitely parametric solutions, we mention that each point on the elliptic curve can be represented in the form ( r s 2 , t s 3 ), where r,
s, t ∈ Z. So if we put nP = ( rn s 2 n , tn s 3 n ),
where the point P is one of the elliptic curve generators, we may obtain a parametric solution for each case of the Diophantine equations by using the new point P ′ = nP = ( rn s 2 n , tn s 3 n ). Also by using the new points of infinite order and repeating the above process, we may obtain infinitely many nontrivial parametric solutions for each case of the Diophantine equations. (see [L.W], page 83, for more information about the computations of r n , s n , t n .) Now the proof of the above theorem is complete. It is interesting to see that
A + C = B + D, too.
Remark 1. Note that by putting h = v u , we may solve the Diophantine equation of the form uA 4 + vB 4 = uC 4 + vD 4 , for every arbitrary integer values of u, and v.
Proof of Theorem 1.1. From the theorem 2.1, we know that the Diophantine equation A 4 + hB 4 = C 4 + hD 4 , where h is an arbitrary fixed rational number, has infinitely many positive integer solutions and we may obtain infinitely many nontrivial parametric solutions for the aforementioned Diophantine equation too. Now, in the equation A 4 + hB 4 = C 4 + hD 4 , let us take h = ±h 4 1 ± h 4 2 ± +h 4 3 ± · · · ± h 4 n−1 , where h i are arbitrary fixed rational numbers, then we get
(2.7) A 4 +(±h 4 1 ±h 4 2 ±+h 4 3 ±· · ·±h 4 n−1 )B 4 = C 4 +(±h 4 1 ±h 4 2 ±+h 4 3 ±· · ·±h 4 n−1 )D 4 .
Now by multiplying h 4 i , to the numbers B 4 , D 4 , and by writing the positive terms in the one side and the negative terms in the other side, we get n positive terms of fourth powers in the both sides, and then obtain infinitely many nontrivial solutions and infinitely many nontrivial parametric solutions for the Diophantine equation n i=1 a 4 i = n i=1 b 4 i . Now the proof of the main theorem is complete.
Remark 2. Surprisingly, we may solve the general Diophantine equation Example 2. Sums of 3 biquadrates in two different ways: Example 3. Sums of 4 biquadrates in two different ways: Example 4. Sums of 5 biquadrates in two different ways: Example 5. Sums of 6 biquadrates in two different ways: Solution:
X 4 1 + X 4 2 + X 4 3 = Y 4 1 + Y 4 2 + YX 4 1 + X 4 2 + X 4 3 + X 4 4 = Y 4 1 + Y 4 2 + Y 4 3 + Y 4 4 . h = 23 = 5 4 −1 4 −4 4 2 4 . E(23) : Y 2 = X 3 − 1587X 2 + 837936X − 147476736.X 4 1 + X 4 2 + X 4 3 + X 4 4 + X 4 5 = Y 4 1 + Y 4 2 + Y 4 3 + Y 4 4 + Y 4 5 . h = 3 17 = 1 4 +2 4 +3 4 +11 4X 4 1 + X 4 2 + X 4 3 + X 4 4 + X 4 5 + X 4 6 = Y 4 1 + Y 4 2 + Y 4 3 + Y 4 4 + Y 4 5 + Y 4 6 . h = 66 25 = 1 4 +2 4 +3 4 +4 4 +6 4
103059413079145 4 +31484981684799 4 +62969963369598 4 +94454945054397 4 + 125939926739196 4 +188909890108794 4 = 203229351055175 4 +11450994089593 4 + 22901988179186 4 +34352982268779 4 +45803976358372 4 +68705964537558 4 .
Example 6. Sums of 7 biquadrates in two different ways: Example 7. Sums of 8 biquadrates in two different ways: Example 8. Sums of 9 biquadrates in two different ways: Example 9. Sums of 10 biquadrates in two different ways:
X 4 1 + X 4 2 + X 4 3 + X 4 4 + X 4 5 + X 4 6 + X 4 7 = Y 4 1 + Y 4 2 + Y 4 3 + Y 4 4 + Y 4 5 + Y 4 6 + Y 4 7 .X 4 1 + X 4 2 + · · · + X 4 8 = Y 4 1 + Y 4 2 + · · · + Y 4 8 . h = 10 = 7 4 +1 4 +2 4 −3 4 −4 4 −5 4 −6 4 2 4 . E(10) : Y 2 = X 3 − 300X 2 + 29700X − 980100.X 4 1 + X 4 2 + · · · + X 4 9 = Y 4 1 + Y 4 2 + · · · + Y 4 9 . h = 21 8 = 8 4 +1 4 −2 4 −3 4 −4 4 +5 4 −6 4 −7 4 4 4 . E( 21 8 ) : Y 2 = X 3 −
X 4 1 + X 4 2 + · · · + X 4 10 = Y 4 1 + Y 4 2 + · · · + Y 4 10 .
h = −63 = 14 4 +1 4 +2 4 +3 4 +4 4 +5 4 −6 4 −11 4 −13 4 By choosing the other points on the above elliptic curves such as 3P , 4P , · · · , (or changing the value of h, and getting new elliptic curves) we obtain infinitely many solutions for the above Diophantine equation.
The Sage software was used for calculating the rank of the elliptic curves, (see [S.A]).
a j y 4 j , by taking h = ±a 1 h 4 1 ± · · · ± a m h 4 m in the equationA 4 + hB 4 = C 4 + hD 4 . Now we are going to work out many examples. Example 1. Sums of 2 biquadrates in two different ways (h = 1): A 4 + B 4 = C 4 + D 4 . h = 16 = 2 4 . E(16) : Y 2 = X 3 − 768X 2 + 195840X − 16646400. ′ , m ′ , q ′ ) = ( 2929 8748 , −1070183 118098 , 9529 2187 ), (p ′′ , m ′′ , q ′′ ) = (
105 4 + 16 4 + 64 4 = 83 4 + 80 4 + 21 4 + 84 4 , 1264542442 4 + 2646847655 4 + 22479447 4 + 89917788 4 = 2368240398 4 + 112397235 4 + 529369531 4 + 2117478124 4 .
) : Y 2 = X 3 − 27 289 X 2 − 7560 83521 X − 705600 24137569 . rank=1; Generator: P = (X, Y ) , m, q) = ( −170 21 , −187 21 , −51 21 ), (p ′ , m ′ , q ′ ) = ( −1112429 406560 , −5325029 8944320 , −200957 135520 ). Solutions: 2312 4 + 357 4 + 714 4 + 1071 4 + 3927 4 = 4046 4 + 17 4 + 34 4 + 51 4 + 187 4 , 134948261 4 + 29798467 4 + 59596934 4 + 89395401 4 + 327783137 4 = 315999247 4 + 19148409 4 + 38296818 4 + 57445227 4 + 210632499 4 .
+ 34572 4 + 23048 4 + 28810 4 + 4637 4 + 9274 4 + 13911 4 = 33963 4 + 27822 4 + 18548 4 + 23185 4 + 5762 4 + 11524 4 + 17286 4 ,X 1 = 653165044877947269, X 2 = 1215694385212406862, X 3 = 810462923474937908, X 4 = 1013078654343672385, X 5 = 41335991086309694, X 6 = 82671982172619388, X 7 = 124007973258929082, Y 1 = 1385020210743079782, Y 2 = 248015946517858164, Y 3 = 165343964345238776, Y 4 = 206679955431548470, Y 5 = 202615730868734477, Y 6 = 405231461737468954, Y 7 = 607847192606203431.
rank = 1 ;
1Generator: P = (X, Y ) = (165, 495). Points: P , 2P = ( 505 4 , −85 8 ), 3P = ( 172029 961 , −20192733 29791 ). (p, m, q) = ( 1 6 , 1 2 , 2 3 ), (p ′ , m ′ , q ′ ) = ( 101 792 , −17 1584 , 109 396 ), (p ′′ , m ′′ , q ′′ ) = ( 5213 28830 , −203967 297910 , 2330 2883 ).
) : Y 2 = X 3 − 11907X 2 + 47246976X − 62492000256. rank = 1; Generator: P = (X, Y ) = (4960, 30752). Points: P , 2P = ( 4096948 961 , 74223316 29791 ). (p, m, q) = ( −5 252 , −31 252 , 1 4 ), (p ′ , m ′ , q ′ ) = ( −1024237 60058656 , −18555829 1861818336 , 70925 953312 ).
312344 4 + 2040328 4 + 255041 4 + 1275205 4 + 16446 4 + 24669 4 + 32892 4 + 49338 4 + 57561 4 = 1299616 4 + 65784 4 + 8223 4 + 41115 4 + 510082 4 + 765123 4 + 1020164 4 + 1530246 4 + 1785287 4 .By choosing the other points on the elliptic curve such as 2P , 3P , · · · , (or changing the value of h, and getting new elliptic curve) we obtain infinitely many solutions for the above Diophantine equation as well. + 456 4 + 57 4 + 114 4 + 171 4 + 316 4 + 395 4 + 474 4 + 553 4 .1323
64 X 2 + 498771
4096 X − 62678889
262144 .
rank = 1; Generator: P = (X, Y ) = ( 163241
11552 , 46525193
3511808 ).
Point: P .
(p, m, q) = ( 6928
7581 , 123409
144039 , 505
361 ).
Solution:
h = −3
2 = 8 4 +1 4 +2 4 +3 4 −4 4 −5 4 −6 4 −7 4
4 4
.
rank = 1; Generator: P = (X, Y ) = ( 85
16 , 55
64 ).
Point: P .
(p, m, q) = ( −17
6 , −11
24 , 13
4 ).
Solution:
356 4 + 632 4 + 79 4 + 158 4 + 237 4 + 228 4 + 285 4 + 342 4 + 399 4 =
268 4
Solutions :
Solutions141 4 + 252 4 + 18 4 + 36 4 + 54 4 + 72 4 + 90 4 + 78 4 + 143 4 + 169 4 = 48 4 + 182 4 + 13 4 + 26 4 + 39 4 + 52 4 + 65 4 + 108 4 + 198 4 + 234 4 , 235608531 4 +352150232 4 +25153588 4 +50307176 4 +75460764 4 +100614352 4 + 125767940 4 + 39586554 4 + 72575349 4 + 85770867 4 = 179941044 4 + 92368626 4 + 6597759 4 + 13195518 4 + 19793277 4 + 26391036 4 + 32988795 4 + 150921528 4 + 276689468 4 + 326996644 4 .
On The Diophantine Equation A 4 + hB 4 = C 4 + hD 4. C] A Choudhry, Indian J. Pure Appl. Math. 2611C] A. Choudhry, : On The Diophantine Equation A 4 + hB 4 = C 4 + hD 4 , Indian J. Pure Appl. Math, 26(11), pp.1057 − 1061, (1995).
. C2] A, Choudhry, A Note on the Quartic Diophantine Equation A 4 +hB 4 = C 4 +hD 4 , available at arXiv.C2] A. Choudhry, A Note on the Quartic Diophantine Equation A 4 +hB 4 = C 4 +hD 4 , available at arXiv. (2016)
Is the quaratic Diophantine equation A 4 +hB 4 = C 4 + hD 4 , solvable for any integer h?. B] F Izadi, M Baghalaghdam, submittedB] F. Izadi, and M. Baghalaghdam, "Is the quaratic Diophantine equation A 4 +hB 4 = C 4 + hD 4 , solvable for any integer h?", submitted, (2017).
A collection of algebraic identities. P] T Piezas, P] T. Piezas, A collection of algebraic identities, available at https://sites.google.com/site/tpiezas/0021e, (accessed on 7 April (2016)).
. T] S Tomita, T] S. Tomita, https://www.maroon.dti.ne.jp/fermat/dioph121e.html, ( accessed on 7 April (2016)).
. A] Sage Software, A] Sage software, available from http://sagemath.org.
W] L C Washington, Elliptic Curves: Number Theory and Cryptography. Chapman-HallW] L. C. Washington, Elliptic Curves: Number Theory and Cryptography, Chapman- Hall, (2008)
|
[] |
[
"R-DIAGONAL DILATION SEMIGROUPS",
"R-DIAGONAL DILATION SEMIGROUPS"
] |
[
"Todd Kemp "
] |
[] |
[] |
This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , a k are * -free R-diagonal operators in a II1 factor, then Dt(ai 1 · · · ai n ) = e −nt ai 1 · · · ai n defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1, . . . , a k . We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1, . . . , a k . Moreover, we show that Dt satisfies an optimal ultracontractive property: Dt : L 2 → L ∞ ∼ t −1 for small t > 0.above: if a 1 , . . . , a d are R-diagonal and * -free, then D t is defined by D t (a i 1 · · · a in ) = e −nt a i 1 · · · a in . Phillippe Biane asked the author if this dilation semigroup generally has a completely positive extension to the von Neumann algebra W * (a 1 , . . . , a d ), as it does in the special case that each operator a j is circular. The first main theorem (proved in Section 2) of this paper answers that question in the affirmative.Theorem 1.1. Let d ∈ {1, 2, . . . , ∞}, and let a 1 , . . . , a d be * -free R-diagonal operators. Then the dilation semigroup D t , defined on the algebra generated by a 1 , . . . , a d (and not a * 1 , . . . , a * d ) by D t (a i 1 · · · a in ) = e −nt a i 1 · · · a in , has a completely positive extension to W * (a 1 , . . . , a d ), given by D t (a ǫ 1 i 1 · · · a ǫn in ) = e −|ǫ 1 +···+ǫn|t a ǫ 1 i 1 · · · a ǫn in where ǫ j ∈ {1, −1} and a −1 j is interpreted as a * j . This extension is precisely the kind of semigroup considered in the case of Haar unitary generators in[15]and[14]. (For example, if d = 1 and the single generator is unitary u, then D t is simply the homomorphism generated by u → e −t u.) However, in the circular case, this extension is not the natural one (it is much simpler than Biane's free O-U semigroup which is diagonalized by free products of Techebyshev polynomials). In fact, the above completely positive extension is generally non-unique: in the circular case, the dilation semigroup also has the free O-U semigroup as a CP extension. In Section 2, we will provide a framework for more natural CP extensions (including the O-U semigroup), depending on a Markovian character of the distribution of the absolute value of the R-diagonal generators.The second half of this paper concerns L p -bounds of the non-selfadjoint semigroup D t of Theorem 1.1. In the classical context of the O-U semigroup U t acting in L 2 (γ) or restricted to L 2 hol (γ), for any finite p > 2 the map U t is bounded into L p for sufficiently large t. However, it is never bounded into L ∞ . This is not the case in the free analogue. Let us fix some notation. Notation 1.2. Let A = {a 1 , . . . , a d } be * -free R-diagonal operators in a II 1 -factor with trace ϕ. Denote by L 2 hol (a 1 , . . . , a d ) the Hilbert subspace of L 2 (W * (A), ϕ) generated by the (non- * ) algebra generated by A.In[18]we proved the following.Theorem (Theorem 5.4 in [18]). Suppose that a 1 , . . . , a d are * -free R-diagonal operators satisfying C = sup 1≤j≤d a j / a j 2 < ∞ (e.g. if d is finite). Then for t > 0,
| null |
[
"https://arxiv.org/pdf/0708.2562v2.pdf"
] | 12,956,255 |
0708.2562
|
f090006d84eec3596b09adfec51ecac0c3cf720b
|
R-DIAGONAL DILATION SEMIGROUPS
11 Feb 2008
Todd Kemp
R-DIAGONAL DILATION SEMIGROUPS
11 Feb 2008arXiv:0708.2562v2 [math.FA]
This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , a k are * -free R-diagonal operators in a II1 factor, then Dt(ai 1 · · · ai n ) = e −nt ai 1 · · · ai n defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1, . . . , a k . We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1, . . . , a k . Moreover, we show that Dt satisfies an optimal ultracontractive property: Dt : L 2 → L ∞ ∼ t −1 for small t > 0.above: if a 1 , . . . , a d are R-diagonal and * -free, then D t is defined by D t (a i 1 · · · a in ) = e −nt a i 1 · · · a in . Phillippe Biane asked the author if this dilation semigroup generally has a completely positive extension to the von Neumann algebra W * (a 1 , . . . , a d ), as it does in the special case that each operator a j is circular. The first main theorem (proved in Section 2) of this paper answers that question in the affirmative.Theorem 1.1. Let d ∈ {1, 2, . . . , ∞}, and let a 1 , . . . , a d be * -free R-diagonal operators. Then the dilation semigroup D t , defined on the algebra generated by a 1 , . . . , a d (and not a * 1 , . . . , a * d ) by D t (a i 1 · · · a in ) = e −nt a i 1 · · · a in , has a completely positive extension to W * (a 1 , . . . , a d ), given by D t (a ǫ 1 i 1 · · · a ǫn in ) = e −|ǫ 1 +···+ǫn|t a ǫ 1 i 1 · · · a ǫn in where ǫ j ∈ {1, −1} and a −1 j is interpreted as a * j . This extension is precisely the kind of semigroup considered in the case of Haar unitary generators in[15]and[14]. (For example, if d = 1 and the single generator is unitary u, then D t is simply the homomorphism generated by u → e −t u.) However, in the circular case, this extension is not the natural one (it is much simpler than Biane's free O-U semigroup which is diagonalized by free products of Techebyshev polynomials). In fact, the above completely positive extension is generally non-unique: in the circular case, the dilation semigroup also has the free O-U semigroup as a CP extension. In Section 2, we will provide a framework for more natural CP extensions (including the O-U semigroup), depending on a Markovian character of the distribution of the absolute value of the R-diagonal generators.The second half of this paper concerns L p -bounds of the non-selfadjoint semigroup D t of Theorem 1.1. In the classical context of the O-U semigroup U t acting in L 2 (γ) or restricted to L 2 hol (γ), for any finite p > 2 the map U t is bounded into L p for sufficiently large t. However, it is never bounded into L ∞ . This is not the case in the free analogue. Let us fix some notation. Notation 1.2. Let A = {a 1 , . . . , a d } be * -free R-diagonal operators in a II 1 -factor with trace ϕ. Denote by L 2 hol (a 1 , . . . , a d ) the Hilbert subspace of L 2 (W * (A), ϕ) generated by the (non- * ) algebra generated by A.In[18]we proved the following.Theorem (Theorem 5.4 in [18]). Suppose that a 1 , . . . , a d are * -free R-diagonal operators satisfying C = sup 1≤j≤d a j / a j 2 < ∞ (e.g. if d is finite). Then for t > 0,
INTRODUCTION AND BACKGROUND
This paper is a sequel to [18], in which the authors discuss an important norm inequality (the Haagerup inequality) in the context of certain non-normal operators (R-diagonal operators) in free probability. The motivation for these papers comes from the classical Ornstein-Uhlenbeck semigroup in Gaussian spaces, which we will briefly recall now. Let γ d denote Gauss measure on R d (the standard n-dimensional normal law). The Ornstein-Uhlenbeck semigroup U t is the C 0 Markov semigroup on L 2 (R d , γ d ) associated to the Dirichlet form of the measure (f, f ) → |∇f | 2 dγ. Its infinitesimal generator N , called the Ornstein-Uhlenbeck operator or number operator, is given by N f (x) = −∆f (x) + x · ∇f (x). The O-U semigroup can be expressed as a multiplier semigroup (with integer eigenvalues) in terms of tensor products of Hermite polynomials.
The space L 2 (R 2d , γ 2d ) contains many holomorphic functions; for example, all monomials z n = z n 1 1 · · · z n d d with n = (n 1 , . . . , n d ). The space of holomorphic L 2 -functions, L 2 hol (C d , γ 2d ), is a Hilbert space that reduces the O-U semigroup. Since ∆h = 0 for h ∈ L 2 hol (C d , γ 2d ), the restriction of the number operator is N h(z) = z · ∇h(z) which is sometimes called the Euler operator, the infinitesimal generator of dilations. As a result, for holomorphic h it follows that U t h(z) = h(e −t z). In terms of monomials, U t (z n ) = e −|n|t z n , where |n| = n 1 + · · · + n d . This simpler action has many important consequences for norm estimates (in particular hypercontractivity) in such spaces; see [13,9,8].
There is a natural analogue of the complex variable z in free probability. Let s, s ′ be free semicircular operators in a II 1 factor; these are natural analogues of independent normal random variables. (For basics on free probability, see the book [23].) Then c = (s + is ′ )/ √ 2 is Voiculescu's circular operator. Aside from its obvious analogous appearance to a complex standard normal random variable in L 2 hol (C, γ 2 ), it can also be thought of as a limit N → ∞ of the N × N Ginibre ensemble of matrices with all independent complex normal entries (of variance 1/2N ). To mimic the random vector z = (z 1 , . . . , z d ) one can take * -free circular operators c 1 , . . . , c d , and d can even be infinite. The analogue of the O-U semigroup is then simply D t (c i 1 · · · c in ) = e −nt c i 1 · · · c in . In this context, the same kinds of strong norm estimates referred to above are discussed in the author's paper [16]. This dilation semigroup is actually a restriction of a semigroup of completely positive maps, the free O-U semigroup, on the full von Neumann algebra W * (c 1 , . . . , c d ), as considered in [3,4].
Circular operators are the prime examples of R-diagonal operators. Introduced in [21], R-diagonal operators form a large class of non-self-adjoint operators that all have rotationally-invariant distributions in a strong sense. They have played important roles in a number of different problems in free probability; see [10,20,26]). In [18], the authors proved a strong form of a norm inequality (the Haagerup inequality) for an L 2 hol -space in the context of R-diagonal operators. A corollary to the estimates therein is a norm inequality (ultracontractivity) for a dilation semigroup akin to the one D t : L 2 hol (a 1 , . . . , a d ) → W * (a 1 , . . . , a d ) ≤ 515 √ e C 2 t −1 .
(In [18] this Theorem is stated only in the case that a 1 , . . . , a d are identically-distributed, but a glance at the proof of Theorem 1.3 in [18] shows that the the theorem was actually proved in the generality stated above.) The following theorem shows that this ultracontractive bound is, in fact, sharp. Theorem 1.3. Let a 1 , . . . , a d be * -free R-diagonal operators, at least one of which is not a scalar multiple of a Haar unitary. Then there are constants α, β > 0 such that, for 0 < t < 1, α t −1 ≤ D t : L 2 hol (a 1 , . . . , a d ) → W * (a 1 , . . . , a d ) ≤ β t −1 . Moreover, this bound is achieved on the algebra generated by a single non-Haar-unitary a j .
In fact, in Section 3 we will give sharp bounds for the action of D t from L 2 hol to L p for any even integer p ≥ 2, at least in the case that the generator a j has non-negative cumulants. This is not quite enough to yield the bound of Theorem 1.3, but only the infinitesimally smaller lower-bound t −1+ǫ for any ǫ > 0. The full theorem is proved instead using a clever L 4 estimate suggested by Haagerup. In what follows, we will provide a minimum of technical background on the free probabilistic tools used when needed. We suggest that readers consult the "Free Probability Primer" (Section 2) in [18], and the excellent book [23] for further details. Remark 1.4. A note on constants. The D t : L 2 → L p -estimates considered in this paper are of interest for the order of magnitude blow-up as t → 0. Multiplicative constants will be largely ignored. As such, the symbols α, β will sometimes be used to represent arbitrary positive constants, and so some equations may seemingly imply false relations like 2α = α 2 = α/α = √ α = α. The author hopes this will not cause the reader any undue stress.
COMPLETELY POSITIVE EXTENSIONS
2.1. Preliminaries. We begin with a few basic facts about R-diagonal operators. Fix a II 1 -factor A with trace ϕ. N C(n) denotes the lattice of non-crossing partitions of the set {1, . . . , n}. The free cumulants {κ π ; π ∈ n N C(n)} relative to ϕ are multilinear functionals A n → C, given by the Möbius inversion formula κ π [a 1 , . . . , a n ] = σ≤π ϕ σ [a 1 , . . . , a n ] Moeb(σ, ϕ),
(2.1)
where Moeb is the Möbius function of the lattice N C(n), and ϕ σ [a 1 , . . . , a n ] is the product of moments of the arguments corresponding to the partition π: if the blocks of σ are {V 1 , . . . , V r }, then
ϕ σ = ϕ V 1 · · · ϕ Vr , where, if V = {i 1 < · · · < i k }, ϕ V [a 1 , . . . , a n ] = ϕ(a i 1 · · · a i k ). For example, if π = {{1, 4}, {2, 5}, {3}} then ϕ π [a 1 , . . . , a 5 ] = ϕ(a 1 a 4 ) ϕ(a 2 a 5 ) ϕ(a 3 )
. Let κ n stand for κ 1n where 1 n is the one block partition {1, . . . , n}. Then, for example,
κ 1 [a] = ϕ(a) is the mean, while κ 2 [a, b] = ϕ(ab) − ϕ(a)ϕ(b)
is the covariance. It is important to note that the functionals κ π also share the same factorization property as the functionals
ϕ π : if π = {V 1 , . . . , V r } then κ π = κ V 1 · · · κ Vr , where, if V = {i 1 < · · · < i k }, κ V [a 1 , . . . , a n ] = κ k [a i 1 , . . . , a i k ].
In this way, any free cumulant can be factored as a product of block cumulants κ k . Equation 2.1 is designed so that the following moment-cumulant formula holds true:
ϕ(a 1 · · · a n ) = π∈N C(n) κ π [a 1 , . . . , a n ].
(2.2) Equations 2.1 and 2.2 show that there is a bijection between the mixed-moments and free cumulants of a collection of random variables a 1 , . . . , a n ∈ A . The benefit of using the free cumulants in this context is their relation to freeness. The following can be taken as the definition: a 1 , . . . , a d ∈ A are free if and only if their mixed cumulants vanish. That is, for any n ∈ N and collection i 1 , . . . , i n ∈ {1, . . . , d} not all equal, κ n [a i 1 , . . . , a in ] = 0.
Two important examples of operators with particularly nice free cumulants are circular operators and Haar unitaries. If c is circular, then among all free cumulants in c and c * , only κ 2 [c, c * ] = κ 2 [c * , c] = 1 are non-zero. On the other hand, for Haar unitary u, there are non-zero free cumulants of all even orders; the non-zero ones are
κ 2n [u, u * , . . . , u, u * ] = κ 2n [u * , u, . . . , u * , u] = (−1) n C n−1 ,
where C n is the Catalan number 1 n+1 2n n . In both cases (c and u), the non-vanishing free cumulants must alternate between the operator and its adjoint. This is the definition of R-diagonality. Definition 2.1. An operator a in a II 1 -factor is called R-diagonal if the only non-zero free cumulants of {a, a * } are of the form κ 2n [a, a * , . . . , a, a * ] or κ 2n [a * , a, . . . , a * , a].
The terminology "R-diagonal" relates to the multi-dimensional R-transform in [23]; the joint R-transform of a, a * , for R-diagonal a, has the form (z, w) → n≥0 α n (zw) n + β n (wz) n , and so is supported "on the diagonal". In Section 3, we will use Definition 2.1 directly. Here, it is more convenient to have the following alternate characterization of R-diagonality.
Theorem (Theorem 15.10 in [23]). An operator a is R-diagonal if and only if, for any Haar unitary u * -free from a, ua has the same distribution as a.
Remark 2.2. To be clear, the above equi-distribution statement means that if P is a non-commutative polynomial in two variables then ϕ(P (a, a * )) = ϕ(P (ua, a * u * )). Corollary 15.14 in [23] re-interprets this equi-distribution property in terms of the polar decomposition of a: at least in the case that ker a = {0}, a is R-diagonal iff its polar decomposition is of the form a = ur where r ≥ 0, u is Haar unitary, and r, u are * -free. In the case that a is R-diagonal but ker a = {0}, it is still possible to write a = ur with u Haar unitary and r ≥ 0, but this is not the polar decomposition of a and here u, r are not * -free; cf. Proposition 15.13 in [23].
Let a be R-diagonal. For exponents ǫ 1 , . . . , ǫ n ∈ {1, * }, our immediate aim is to appropriately bound general moments of the form ϕ(a ǫ 1 · · · a ǫn ). Denote the string (ǫ 1 , . . . , ǫ n ) as S, and denote a ǫ 1 · · · a ǫn by a S . The following specialization of Equation 2.2 is vital to the combinatorial understanding of R-diagonal moments. Proposition 2.3. Let S = (ǫ 1 , . . . , ǫ n ) be a string of 1s and * s. Let N C(S) denote the set of all partitions π ∈ N C(n) such that each block of π is of even size and alternates between 1 and * in S. Let a be R-
diagonal. Then ϕ(a S ) = π∈N C(S) κ π [a ǫ 1 , . . . , a ǫn ]. (2.3)
For example, consider the word a 3 a * 2 aa * 2 with exponent string S = (1, 1, 1, * , * , 1, * , * ). The set N C(S) consists of the three partitions in Figure 1. Proof. In Equation 2.2, consider the general term κ π [a ǫ 1 , . . . , a ǫn ] in the summation. This factors into terms κ V [a ǫ 1 , . . . , a ǫn ] over the blocks V of π. Since a is R-diagonal, the only such non-zero terms are of the form κ 2m [a, a * , . . . , a, a * ] or κ 2m [a * , a, . . . , a * , a]. Thus, each V must be even in size, and must alternate between a and a * for the term to contribute. It follows that the non-zero terms in the sum 2.2 are all indexed by π ∈ N C(S).
1 1 1 * * 1 * * 1 1 1 * * 1 * * 1 1 1 * * 1 * *
Corollary 2.4.
Let a be R-diagonal, and let S be a string. Then ϕ(a S ) is 0 unless S is balanced: it must have equal numbers of 1s and * s. In particular, S must have even length.
Proof. In each term κ π in 2.3, each block of the partition π ∈ N C(S) is of even length and alternates between 1 and * . Hence, each block has equal numbers of 1s and * s, and thus only balanced S contribute to the sum.
Corollary 2.5. R-diagonal operators are rotationally-invariant: let a be R-diagonal, and let θ ∈ R. Then e iθ a has the same distribution as a.
Remark 2.6. If a were a normal operator, then the above equi-distribution statement is precisely the same as requiring the spectral measure of a to be a rotationally-invariant measure on C.
Proof. Let P be a non-commutative monomial in two variables, P (x, y) = x n 1 y m 1 · · · x nr y mr . Set n 1 + · · · + n r = n and m 1 + · · · + m r = m. Then P (e iθ a, e −iθ a * ) = e i(n−m)θ P (a, a * ). By Corollary 2.4, if n = m then ϕ(P (a, a * )) = ϕ(e i(n−m)θ P (a, a * )) = 0. Otherwise, the two elements are equal and so have the same trace. 4 The following orthogonality relation will be important in the sequel, and is an immediate consequence of Corollary 2.4. Corollary 2.7. Let a 1 , a 2 , . . . , a d be * -free R-diagonal operators in (A , ϕ). Then (a j ) n and (a k ) m are orthogonal in L 2 (A , ϕ) whenever j = k or n = m.
Proof. The inner product is ϕ(a n j (a m k ) * ). Applying Equation 2.2, this is a sum of terms of the form κ π [a j , . . . , a j , a * k , . . . , a * k ]. If j = k, this is a mixed cumulant of * -free random variables, and so vanishes. If j = k and m = n, the inner product is ϕ(a S ) for an imbalanced string, and so it also vanishes by Corollary 2.4.
Completely positive extensions for rotationally-invariant generators.
Here we prove Theorem 1.1, which actually holds in the wider context of * -free rotationally-invariant generators.
Proof of Theorem 1.1. Let a 1 , . . . , a d be * -free rotationally-invariant operators. Since the law of the generators determines the von Neumann algebra they generate, for any θ ∈ R and any index j ∈ {1, . . . , d} there is a * -automorphism
α (j) θ : W * (a j ) → W * (a j ) determined by α (j)
θ (a j ) = e iθ a j . Since the generators are * -free, W * (a 1 , . . . , a d ) is naturally isomorphic to W * (a 1 ) * · · · * W * (a d ), and so we have a * -automorphism
α θ = α (1) θ * · · · * α (d) θ : W * (a 1 , . . . , a d ) → W * (a 1 , . . . , a d ).
(2.4)
Note, any * -automorphism is automatically completely-positive.
Let P (r, θ) denote the Poisson kernel for the unit disc in C; that is, for r ∈ [0, 1] and θ ∈ [0, 2π),
P (r, θ) = Re 1 + re iθ 1 − re iθ = 1 − r 2 1 − 2r cos θ + r 2 = ∞ k=−∞ r |k| e ikθ .
(2.5)
For any fixed r < 1, this kernel is strictly positive and bounded on the circle θ ∈ [0, 2π). For any operator x ∈ W * (a 1 , . . . , a d ), define
D r x = 1 2π 2π 0 P (r, θ) α θ (x) dθ.
Because the Poisson kernel is continuous and bounded on a compact set, this integral converges in operator norm. Each uniformly convergent Riemann sum is therefore of the form p j α θ j , where the p j (samples of the Poisson kernel) are positive numbers. Each such sum is thus completely positive, and so the uniform limit D r is a completely positive operator as well.
Now we need only check the action of D r on monomials a ǫ 1 i 1 · · · a ǫn in , where ǫ j ∈ {1, * }. Note that
α (i k ) θ (a ǫ 1 i 1 · · · a ǫ k i k · · · a ǫn in ) = e ǫ k iθ (a ǫ 1 i 1 · · · a ǫ k i k · · · a ǫn in )
, where ǫ = * is interpreted as ǫ = −1 on the right-hand-side. Hence, α θ (a ǫ 1 i 1 · · · a ǫn in ) = e i(ǫ 1 +···+ǫn)θ (a ǫ 1 i 1 · · · a ǫn in ). Hence, from the third equality in Equation 2.5,
P (r, θ) α θ (a ǫ 1 i 1 · · · a ǫn in ) = ∞ k=−∞
r |k| e i(k+ǫ 1 +···+ǫn)θ (a ǫ 1 i 1 · · · a ǫn in ).
Integrating term-by-term around the circle, the only term that survives is k = −(ǫ 1 + · · · + ǫ n ), and the integral there is just 1. Hence, D r (a ǫ 1 i 1 · · · a ǫn in ) = r |ǫ 1 +···+ǫn| a ǫ 1 i 1 · · · a ǫn in . Setting D t = D e −t yields the formula in Theorem 1.1. Note that this restricts to the dilation semigroup when all ǫ j = 1. Hence, D t has a completely-positive extension.
Indeed, every dilation semigroup associated to * -free rotationally-invariant generators (for example * -free R-diagonal generators) has a completely positive extension. In the case of a single unitary generator u, the action of D t on Laurent polynomial in u is simply D t (u n ) = e −|n|t u n , which "counts unitaries". However, this action is not particularly natural in the general setting: it has no connection with the distribution of the generators. In the circular setting, this is not the free O-U semigroup, which is also completely positive (as proved in [4]). Therefore, this extension may not be unique.
2.3. Completely positive extensions for Markov kernels. A different CP extension is possible for some generating distributions. Let µ be a (compactly-supported) probability measure on R. Then the monomials {1, x, x 2 , . . .} are dense in L 2 (R, µ), and Gram-Schmidt orthogonalization produces the orthogonal polynomials {p 0 , p 1 , p 2 , . . .} associated to µ. (If µ has infinite support, all monomials are linearly independent; if µ has support of size n then p k = 0 for k > n.) The polynomial p n has degree n, and p 0 (x) = 1 while p 1 (x) = x. If µ is the semicircle law, the associated polynomials are the Tchebyshev polynomials of type II, usually denoted u n .
Given µ with associated orthogonal polynomials {p n }, letp n denote the normalized polynomials (in L 2 (µ), so that {p n } forms an o.n. basis). Consider the following integral kernel (which may take infinite values):
m µ (r; x, y) = n≥0 r np n (x)p n (y). (2.6)
Here r ∈ [0, 1) and x, y range over supp µ. The formula converges at least when r|xy| < 1. We refer to the kernel in Equation 2.6 as a Mehler kernel. If the measure is chosen as the standard normal law (which is not compactly-supported but has sufficient tail decay to ensure the L 2 -density of polynomials), then the polynomials p n are the Hermite polynomials and the kernel is the Mehler kernel. On the other hand, if µ is the semicircle law, setting r = e −t yields the kernel of the (one-dimensional) free O-U semigroup in [4].
Let M µ (r) denote the integral operator associated to the kernel m µ (r, ·, ·); that is, for f ∈ L ∞ (µ) at least, let
(M µ (r)f )(x) = R m µ (r; x, y) f (y) dµ(y).
Since µ is a probability measure, L ∞ (µ) ⊂ L 2 (µ) and so any such f has an L 2 -expansion f = n≥0 f npn for an ℓ 2 (N)-sequence (f n ) ∞ n=0 . From the orthonormality of the polynomialsp n in L 2 (µ) it is then easy to see that the action of M µ (r) is
M µ (r)f = n≥0 r n f npn . (2.7)
That is, M µ (r) is a polynomial multiplier semigroup (in multiplicative form): M µ (r) p n = r n p n . Contingent on convergence in L ∞ (µ), we may then ask the question of whether M µ (r) is completely positive. In this case, as a bounded operator on the commutative von Neumann algebra L ∞ (µ), complete positivity is equivalent to positivity of the kernel: M µ (r) is CP if and only if m µ (r; x, y) ≥ 0 for x, y ∈ supp µ. In other words, M µ is CP if and only if m µ is a Markov kernel.
Example 2.8. For the point mass δ 0 , p 0 = 1 and all other p n are 0, so the kernel m δ 0 is trivially Markovian. Let λ > 0 and set µ = 1 2 (δ λ + δ −λ ). Then we may easily calculate thatp 0 (x) = 1 and p 1 (x) = x/λ, while all higher polynomials are 0. Thence m µ (r; x, y) = 1 + rxy/λ 2 , and on the support of µ x, y ∈ {±λ} we have m µ = 1 + r ≥ 0. Hence m µ is Markovian. Example 2.9. On the other hand, consider an arbitrary symmetric measure with 3-point support,
ν = a(δ λ + δ −λ ) + (1 − 2a)δ 0 where 0 ≤ a ≤ 1
2 and λ > 0. A simple calculation shows that in this case
m ν (r; λx, λy) = 1 + r xy 2a + r 2 (x 2 − 2a)(y 2 − 2a) 2a(1 − 2a) . 6
The arguments λx, λy are in supp ν if and only if x, y ∈ {0, ±1}; note that
m ν (r; 1, −1) = 1 − r 1 2a + r 2 1 − 2a 2a = (1 − r) 1 + r − r 2a .
If a < 1 4 , this is < 0 for some r ∈ [0, 1), and so m ν is not Markovian for some choices of a. It is a historically challenging problem to determine, for a given measure µ, whether the associated Mehler kernel m µ is a Markov kernel. (See, for example, [1,19,24,27].) The motivating example (the Mehler kernel) is Markovian: with µ = γ 1 (Gauss measure on R),
m γ 1 (r; x, y) = (1 − r 2 ) −1/2 exp y 2 /2 + (1 − r 2 ) −1/2 (rx − y) ,
which is strictly positive on R = supp γ 1 for 0 ≤ r < 1. Writing a formula for a general Mehler kernel is a hopeless task. Nevertheless, the following positivity condition affords many examples of Markovian Mehler kernels. (1) m µ (r; x, y) ≥ 0 a.s. on supp µ.
(2) For f ≥ 0 a.s. on supp µ, and in L 1 (µ), M µ (r)f ≥ 0 a.s. on supp µ. (3) For f ∈ L 1 (µ), M µ (r)f L 1 (µ) ≤ f L 1 (µ) . (4) For f ∈ L ∞ (µ), M µ (r)f L ∞ (µ) ≤ f L ∞ (µ) . (5) For all p ∈ [1, ∞], M µ (r)f L p (µ) ≤ f L p (µ) .
Remark 2.11. The statement is that m µ is Markovian if and only if M µ is a contraction on L 1 , or on L ∞ , or on L p for all p between 1 and ∞. This proposition and the following proof are borrowed from [13], but the results really go back to Beurling and Deny [2].
Proof. The equivalence of (1) and (2) is elementary. Condition (4) follows from (3) by duality, since M µ (r) is self-adjoint on L 2 (µ). Condition (5) follows from (3) and (4) by the Riesz-Thorin interpolation theorem, and evidently (5) implies (3).
Suppose condition (1) holds. Then
|M µ (r)f (x)| = m µ (r; x, y)f (y) dµ(y) ≤ |m µ (r; x, y)| |f (y)| dµ(y) = m µ (r; x, y) |f (y)| dµ(y) = M µ (r)|f |(x).
The reader can easily check that M µ (r) is trace-preserving:
M µ (r)g dµ = g dµ. Hence M µ (r)|f | dµ = |f | dµ = f L 1 (µ) , and so M µ (r)f L 1 (µ) = |M µ (r)f | dµ ≤ M µ (r)|f | dµ = f L 1 (µ) ,
verifying property (3).
On the other hand, suppose condition (3) holds. Take f ∈ L 1 (µ) with f ≥ 0. Then by assumption (3),
|M µ (r)f (x)| dµ(x) = M µ (r)f L 1 (µ) ≤ f L 1 (µ) = f dµ, and as above we have f dµ = M µ (r)f dµ. Hence, |M µ (r)f | dµ ≤ M µ (r)f dµ. Since µ is a positive measure, this means that M µ (r)f (x) ≥ 0 for x ∈ supp µ, verifying property (2).
Example 2.12. In [6] the authors introduced the q-Gaussian factors, with their associated q-Gaussian measures σ q . When q = 1, σ q = γ 1 is the standard normal law; when q = 0, σ 0 is the semicircle law, and q = −1 yields a two-point Bernoulli measure as in Example 2.8. All the measures σ q with −1 ≤ q < 1 are compactly supported and symmetric. The associated orthogonal polynomials are the q-Hermite polynomials H (q) n given by the following tri-diagonal recursion:
H (q) 0 (x) = 1, H (q) 1 (x) = x, H (q) n+1 (x) = x H (q) n (x) + [n] q H (q) n−1 (x),
where [n] q = 1 + q + · · · + q n−1 . When q = 1 these are the Hermite polynomials, associated to Gauss measure; for q = 0 the recurrence produces the Tchebyshev II polynomials, orthogonal for the semicircle law. The L 2 (σ q ) normalization factor is 1/
[n] q ! where [n] q ! = [n] q ·[n − 1] q · · · [2] q ·[1] q .
The Mehler kernel m σq is the kernel of the q-O-U semigroup considered in [4]. There, Biane proved Nelson's hypercontractivity inequalities for the associated semigroup, which include as a special case condition 5 in Proposition 2.10. Hence, there is a continuous family of Mehler kernels that are both symmetric and Markovian.
We now come to the question of completely positive extensions for R-diagonal dilation semigroups. Let a 1 , . . . , a d be * -free R-diagonal operators in a II 1 -factor (traciality is necessary here). From [22], there are self-adjoint even elements x j (that is, the distribution µ x j is symmetric on R) with the same free cumulants as those of a j . Hence,
κ 2n [x j , x j , . . . , x j , x j ] = κ 2n [a j , a * j , . . . , a j , a * j ] = κ 2n [a * j , a j , . . . , a * j , a j ]. (2.8) (
The odd cumulants of a j , a * j are 0 by definition, and so are those of x j since it is even; all odd moments are 0, and so too are all odd cumulants.) This means that |a j | is equal in distribution to |x j |. In [12] (Corollary 3.2), the authors show that if s is self-adjoint, even, free from x j and s 2 = 1, then sx j is R-diagonal and indeed has the same distribution as a j . What's more, the construction of x j from a j takes place within the W * -algebra generated by a j , and so x 1 , . . . , x d are free. In other words, we may represent the generators a j in the form a 1 = sx 1 , . . . , a d = sx d where x 1 , . . . , x d , s are all free, self-adjoint, even, and s 2 = 1. Note, then, that
W * (a 1 , . . . , a d ) ∼ = W * (x 1 , . . . , x d , s).
Let µ j denote the distribution of x j on R. Since the x j are free, we have
W * (a 1 , . . . , a d ) ∼ = L ∞ (µ 1 ) * · · · * L ∞ (µ d ) * W * (s).
We may then define, for 0 ≤ r < 1, the operator T r on Of course the Id map on W * (s) is also CP and trace preserving. Then by Theorem 3.8 in [5], the operator T r is completely positive and trace preserving on
L ∞ (µ 1 ) * · · · * L ∞ (µ d ) * W * (s) by T r = M µ 1 (r) * · · · * M µ d (r) * Id.L ∞ (µ 1 ) * · · · * L ∞ (µ d ) * W * (s) ∼ = W * (a 1 , . . . , a d ).
Now, by definition the orthogonal polynomial p µ 1 (x) for any measure µ is a scalar multiple of x, and hence M µ j (r) x j = r x j . Then the action of T r on words in the generators and not their adjoints is:
T r (a i 1 a i 2 · · · a in ) = T r (sx i 1 sx i 2 · · · sx in ) = s(r x i 1 )s(r x i 2 ) · · · s(r x in ) = r n a i 1 a i 2 · · · a in .
Setting T t = D e −t , this means that T t , restricted to the (non- * ) algebra generated by a 1 , . . . , a d , is the associated R-diagonal dilation semigroup D t . We have therefore proved the following. Remark 2.14. The notion of correspondence in the final statement of Theorem 2.13 is as follows.
The free O-U semigroup U t acts on W * (σ 1 , . . . , σ d ) where σ j = (c j + c * j )/ √ 2 are free semicircular operators -the c j are * -free circular operators. In this circular case, the symmetrization of the distribution of |c j | is also the semicircle law σ: this is easy to check from Equation 2.8 and the fact that κ 2 [c, c * ] = κ 2 [c * , c] = κ 2 [σ, σ] = 1 and all other free cumulants are 0. Hence, the construction above W * (c 1 , . . . , c d ) ∼ = W * (x 1 , . . . , x d , s) yields free semicircular x j , and so we can view U t acting in W * (x 1 , . . . , x d ). Its action is
U t (u n 1 (x i 1 ) · · · u n k (x i k )) = e −(n 1 +···+n k )t u n 1 (x i 1 ) · · · u n k (x i k ),
where u n are the Tchebyshev II polynomials, the orthogonal polynomials for the semicircle law, and the indices i ℓ are consecutively distinct. From Equation 2.7, this is precisely the action of M σ (e −t ) * · · · * M σ (e −t ) on W * (x 1 , . . . , x d ) ∼ = L ∞ (σ) * · · · * L ∞ (σ), and so U t is the restriction from W * (x 1 , . . . , x d , s) to W * (x 1 , . . . , x d ) of the Markov extension T t of the circular R-diagonal dilation semigroup. Note that T t acts on the full von Neumann algebra W * (x 1 , . . . , x d , s) through a very similar formula:
T t s ǫ 0 u n 1 (x i 1 )s ǫ 1 u n 2 (x i 2 ) · · ·s ǫ n−1 u n k (x i k )s ǫn = e −(n 1 +···+n k )t s ǫ 0 u n 1 (x i 1 )s ǫ 1 u n 2 (x i 2 ) · · · s ǫ k−1 u n k (x i k )s ǫ k ,
where ǫ j is either 0 or 1 (i.e. either the s is included or not). In this case, if an s separates u n (x i ) from u m (x j ), it is not required that i = j; the s stands in for free product. . , x n }. As a result, and monomial x n can be expanded x n = n k=0 α n,k p k (x) as a finite sum. Since µ is symmetric, p n is even if n is even and odd if n is odd, and so it follows that only every second α n,k is non-zero. What's more, the leading term of p n is x n , and so α n,n = 1. So, for example, x 3 = p 3 (x) + α 3,1 x. (The other coefficients are certain combinations of the moments of the measure µ; for example, α 3,1 = x 4 dµ/ x 2 dµ.) This allows for the easy determination of the action of T t on arbitrary words in the generators a 1 , . . . , a d and their adjoints.
Example 2.16. Consider the word a * 1 a 1 a * 1 a 2 2 a * 1 . We rewrite this as a * 1 a 1 a * 1 a 2 2 a * 1 = (x 1 s)(sx 1 )(x 1 s)(sx 2 ) 2 (x 1 s) = x 3 1 x 2 s x 2 x 1 s.
(2.10)
Let p n denote the orthogonal polynomials of the distribution of x 1 , and let α n,k be the relevant coefficients, as explained in Remark 2.15. Then x 3 1 = p 3 (x 1 ) + α 3,1 x 1 , and so T t (a * 1 a 1 a * 1 a 2 2 a * 1 ) = e −3t p 3 (
x 1 ) + e −t α 3,1 x 1 (e −t x 2 )s(e −t x 2 )(e −t x 1 )s = e −6t p 3 (x 1 )x 2 s x 2 x 1 s + e −4t α 3,1 x 1 x 2 s x 2 x 1 s. Rewriting p 3 as p 3 (x 1 ) = x 3 1 − α 3,1 x 1 yields T t (a * 1 a 1 a * 1 a 2 2 a * 1 ) = e −6t x 3 1 x 2 s x 2 x 1 s + (e −4t − e −6t )α 3,1 x 1 x 2 s x 2 x 1 s. From Equation 2.10 we have x 3 1 x 2 s x 2 x 1 s = a * 1 a 1 a * 1 a 2 2 a * 1 ;
for the second term, we introduce 1 = s 2 between x 1 x 2 yielding,
x 1 x 2 s x 2 x 1 s = (x 1 s)(sx 2 )(sx 2 )(x 1 s) = a * 1 a 2 2 a * 1 .
In this way, any monomial in x 1 , . . . , x d , s may be converted into a unique monomial in a 1 , , . . . , a d and their adjoints. Hence, for this example, we have T t (a * 1 a 1 a * 1 a 2 2 a * 1 ) = e −6t a * 1 a 1 a * 1 a 2 2 a * 1 + (e −4t − e −6t )α 3,1 a * 1 a 2 2 a * 1 . The leading term is multiplication by e −nt on any word of length n, mimicking the action of D t on words in the generators and not their inverses; this is to be contrasted with the alternating degree count in the generic extension of Theorem 1.1. There are then correction terms involving lower-degree words, with coefficients that are polynomial in e −t that vanish at t = 0. All such correction terms vanish in the case of words in the generators without inverses, thus resulting in the R-diagonal dilation semigroup as per Theorem 2.13.
OPTIMAL ULTRACONTRACTIVITY
We now wish to consider the action of an R-diagonal dilation semigroup D t , relative to generators a 1 , . . . , a d , on L 2 hol (a 1 , . . . , a d ) taking values in L p (W * (a 1 , . . . , a d ), ϕ) for p = 2, 4, . . . , ∞ (where the L ∞ is just W * (a 1 , . . . , a d ) equipped with its operator norm). The initial idea is to approximate the operator norm through the L p -norms as p → ∞ to prove Theorem 1.3. In fact, the following approximations are slightly too weak to make this approach work; the resultant lower-bound involves a constant which tends to 0 as p → ∞, missing the target at p = ∞ by an infinitesimal exponent. A separate argument based on similar combinatorics is given to prove Theorem 1.3. The L p -estimates are included below for independent interest; in particular, they are in line with the conjecture that the spaces L p hol (a 1 , . . . , a d ) are complex interpolation scale. 3.1. Bounding |N C(S)|. A generic string may be written in the form S = (1 n 1 , * m 1 , . . . , 1 nr , * mr ) where r ≥ 1 and n 1 , m 1 , . . . , n r , m r ≥ 1. (There are implied commas: 1 3 = (1, 1, 1).) The number r of alternations between 1 and * is an important statistic; refer to it as the number of runs in S. We will shortly provide a lower-bound on the size of N C(S), which depends fundamentally on this number r. First, we specialize N C(S) to pairings. For example, for the string in Figure 1, the first two partitions are in N C 2 (S) (the last is not). Of course N C 2 (S) is a subset of N C(S), so we may estimate its size to find a lower-bound on |N C(S)|. It follows from Equation 2.3 and the form of the cumulants of a circular operator c that ϕ(c S ) = |N C 2 (S)| for any S, and so it is no surprise that this set plays an important role in the following estimates. For the regular strings S = (1 n , * n ) r , |N C 2 (S)| was calculated exactly in our paper [18]. Our proof was very topological, but a simpler recursive proof is given in [7]. The result is as follows:
|N C 2 ((1 n , * n ) r ) | = ϕ ((c n c * n ) r ) = C (n) r = 1 nr + 1 (n + 1)r r . (3.1)
The numbers C is on the order of (n + 1) r−1 . This structure is reflected in the following estimate. Proof. The case r = 1 is simple: this means that S has the form (1 n 1 , * m 1 ), and since S is balanced, this means n 1 = m 1 = i. It is therefore a regular string of the form mentioned above, and so |N C 2 (S)| = C Proceeding by induction on r ≥ 2, suppose that for any string S with precisely r−1 runs, it holds that |N C 2 ( S)| ≥ (1 +ĩ) r−2 , whereĩ is the minimal block size in S. Let S = (1 n 1 , * m 1 , . . . , 1 nr , * mr ) be any balanced string with r runs, and minimum block size i. We may cyclically permute the entire string without affecting the size of |N C 2 (S)|, and so without loss of generality we may assume that n 1 = i. (Note: if the minimum occurs on a * block, we can rotate and then reverse the roles of 1 and * .) Now, since n 1 = i is the minimum, m r ≥ i, and so one possible way to pair each of the initial 1s in S is with the last i * s in the final block; the resulting leftover string S is ( * m 1 , 1 n 2 , . . . , * m r−1 , 1 nr , 0 mr −i ), which can be rotated to the string (1 n 2 , * m 2 , . . . , 1 nr , * m 1 +mr−i ) which is, by construction, still balanced, and has r−1 runs. Therefore, by the induction hypothesis, there are at least (1 +ĩ) r−2 pairings of this internal string, and since it is a substring of S,ĩ ≥ i; hence, with the initial block of 1s all paired at the end, there are at least (1 + i) r−2 pairings in N C 2 (S).
More generally, let 1 ≤ ℓ ≤ i = n 1 . Since m 1 ≥ i ≥ ℓ, the last ℓ 1s in this first block can be paired to the first ℓ * s, with the remaining i − ℓ 1s pairing to the final i − ℓ ≤ m r * s in the final block, as above. The remaining internal string is then 1 m 1 −ℓ , 1 n 2 , . . . , * m r−1 , 1 nr , * mr −(i−ℓ) which can be rotated to 1 n 2 , * m 2 , . . . , 1 nr , * m 1 +mr−i once again. Thus, as above, for each choice of ℓ between 1 1 1 1 * * * * 1 1 1 1 1 * * * 1 1 1 1 * * * * * S = S FIGURE 2. One of the i + 1 configurations for the first block of 1s, yielding all the pairings ofS; in this example, i = 3, and ℓ = 2.
and i, we have at least (1 + i) r−2 distinct pairings of S, and the different pairings for different ℓ are distinct. Adding these i(1 + i) r−2 pairings to the (1 + i) r−2 in the case above, we see that N C 2 (S) indeed contains at least (1 + i) r−1 pairings. Remark 3.3. This proof actually yields a somewhat larger lower-bound, given as a product of iterated minima (1 + i 1 )(1 + i 2 ) · · · (1 + i r−1 ) where i 1 = i is the global minimum and each i k+1 is the minimum of the leftover string after the inductive step has been applied at stage k (i.e. i 2 =ĩ from the proof). It is possible to construct examples where this iterated minimum product is much larger than the stated lower bound; it is also easy to construct strings with arbitrary length that achieve the bound. Regardless, the result of Proposition 3.2 is sufficient for our purposes.
We will also require an upper-bound for the size of |N C 2 (S)|. To achieve it, we need a convenient way to understand the restrictions a string S enforces over pairings. Consider the string (1, 1, 1, 1, * , * , 1, 1, * , * , * , * , * , 1, 1, * ) for example; if the second 1 were paired to the second * , the substring so-contained would be (1, 1, * ), which is not balanced and so has no internal pairings. Therefore, in order for the whole string to be paired off, it is not possible for the second 1 to pair to the second * . To understand which pairings may be made, associate to any string S a lattice path P(S): start at the origin in R 2 , and for each 1 in the string, draw a line segment of direction vector (1, 1); for each * draw a line segment of direction vector (1, −1). If the balanced string S has length 2n (n 1s and n * s), then the associated lattice path P(S) is a ±1-slope piecewise-linear curve joining (0, 0) to (2n, 0) (and each such curve, with slope-breaks at integer points, corresponds to a balanced 1- * string). Figure 3 shows the lattice path corresponding to the string S considered above. The lattice path P(S) gives an easy geometric condition on allowed non-crossing pairings of S. Any 1 must be paired with a * in such a way that the substring between them is balanced; since the line-segments in P(S) slope up for 1 and down for * , this means that any pairing must be from an up slope to a down slope at the same vertical level. In Figure 3, these levels are marked with dotted lines, and labeled along the vertical axis.
The statistic of a string which is important for the upper bound is the lattice path height h(S), which is simply the total height (total number of vertical increments) in P(S); that is, h(S) is the number of distinct labels needed on the vertical axis of the lattice path. In Figure 3, h = 5.
|N C 2 (S)| ≤ C (h) r ≤ r r−1 r! (1 + h) r−1 . (3.2)
Proof. This is a purely combinatorial fact, owing to a nice inclusion N C 2 (S) ⊆ N C 2 (1 h , * h ) r , as follows. In the lattice path P (1 h , * h ) r , locally pair those peaks and troughs whose height-labels do not occur at the corresponding levels in the lattice path P(S). (If the lattice path dips below the horizontal axis, "locally" may mean matching the first block to the last one; one could take care of this by first rotating the string so its minimal block is first.) The remaining unpaired entries in (1 h * h ) r form a copy of S, and since the labels correspond, there is a bijection between pairings of S and pairings of this inclusion of substrings. This gives the inclusion, and the result follows from Equation 3.1. Figure 4 demonstrates the inclusion. Remark 3.5. Note that the lattice path height is the smallest h which can be used in the proof of Lemma 3.4, since all labels appearing in P(S) must be present in P (1 h , * h ) r . Unfortunately, h(S) can be quite large in comparison to the average (or even maximum) block size in S: consider the string (1 ,k , * ) ,ℓ , (1, * ,k ) ,ℓ . The maximum block size is k, while the lattice path height is (k − 1)ℓ + 1. Indeed, this string has length 2(k + 1)ℓ, and so the height is about half the total length. In general, this is about the best that can be said, and so the only generally useful corollary is the following.
Corollary 3.6. Let S be a balanced string of length 2n (n 1s and n * s), with r runs. Then
|N C 2 (S)| ≤ r r−1 r! (1 + n) r−1 . (3.3)
Proof. From Lemma 3.4, it suffices to show that if S has n 1s then h(S) ≤ n. Let us rotate S so that its minimum block is first. This means that the lattice path P(S) never drops below the horizontal axis, and so each vertical label corresponds to at least one up-slope (the first one where it appears, for example). This means that the lattice path height h(S) is bounded above by the number of up slopes, which is n. This bound is achieved only when r = 1.
Remark 3.7. The bound in Corollary 3.6 is quite large and essentially never achieved. A better bound, proved in our paper [17], replaces h in Equation 3.2 with the average block size; in view of Equation 3.1, this is the best possible general bound. Its proof is very involved, and the very rough estimate of Equation 3.3 is sufficient for our purposes in what follows.
For small r, it is possible to give explicit formulas for the sizes |N C(S)| and |N C 2 (S)| for all strings S. Following are such formulas in the case r = 2, which we will use in Section 3.3 below. Proof. If π ∈ N C(S), then each of its blocks alternates between 1s and * s; from the form of S this means blocks must have length at most 4, so π consists of 2-blocks and 4-blocks. If π has a 4-block, all remaining blocks of π are nested between its consecutive pairings, and since a 4-block must connect one element from each of the four segments in S, it follows from the non-crossing condition that all remaining blocks of π are 2-blocks, and are determined by the position of the 4block. This is demonstrated in Figure 5. Hence, the number of π ∈ N C(S) with a 4-block is equal 1 1 * * * * * * 1 1 1 1 1 1 1 1 * * * * 1 1 * * FIGURE 5. The 4-block (dark) determines all other blocks of the partition in this 2-run string. The high and low peaks must be paired locally since there are no other choices at the same level; the 4-block can be placed at any height within the first (minimal) string of 1s, and once in place it determines all other pairings: local inside, and far outside. These pairings are dotted in the partition diagram.
to the number of positions the 4-block can occupy. If we rotate S so that n 1 = min{n 1 , m 1 , n 2 , m 2 } as in Figure 5 then it is clear this number is precisely n 1 : the height of the 4-block must be within the smallest block n 1 , and this height determines the rest of the partition.
If, on the other hand, π contains no 4-blocks, then it is in N C 2 (S). Rotate again so n 1 is the minimum block size. Since pairings must be made at the same vertical level in the lattice diagram of S, all the points below the axis or above n 1 in height must be paired locally. The unpaired entries of S form a regular string (1 n 1 , * n 1 ) 2 whose pairings number C (n 1 ) 2 = 1 + n 1 . This proves the result. Remark 3.9. A formula in the case of 3 runs can also be written down with a little more difficulty. In fact, there is a general formula for |N C 2 (S)| expressed as a sum of binomial coefficients, index but rooted trees determined by the string S; see [17]. 13 3.2. L p -bounds of D t . Let p be a positive even integer, p = 2r, and let a 1 , . . . , a d be * -free Rdiagonal operators in (A , ϕ). Our goal in this section is to give optimal bounds on the norm of an R-diagonal dilation semigroup D t : L 2 hol (a 1 , . . . , a d ) → L p (A , ϕ). In [18], we proved such a bound in the case p = ∞, through an application of our strong Haagerup inequality. In the circular case (generators c 1 , . . . , c d free * -free circular operators), the same techniques we developed in that paper demonstrate the following estimate holds for small t > 0:
D t : L 2 hol (c 1 , . . . , c d ) → L p (A , ϕ) ≤ α p t −1+ 1 p .
One can check from the asymptotics of the Fuss-Catalan numbers that c n p ∼ n 1 2 − 1 p , and as a result the above estimate cannot be improved by means of a Haagerup inequality approach similar to that in Section 5 of [18]. Nevertheless, this estimate is not optimal. In the case of a single circular generator c, the correct bound is
D t : L 2 hol (c) → L p (W * (c), ϕ) ∼ t −1+ 2 p ,(3.4)
for even p and t > 0. As a lower-bound, this holds more generally for any R-diagonal generator with non-negative free cumulants, or any infinite * -free R-diagonal generating set (through a straightforward application of the free central limit theorem). We proceed to prove these bounds in the following; first, we begin with a well-known estimate which will be key to the proofs.
Lemma 3.10.
Let 0 ≤ q < ∞. There are constants α q , β q > 0 such that, for 0 < t < 1,
α q t −q−1 ≤ n≥0 n q e −nt ≤ β q t −q−1 .
Proof. For the lower bound, for any integer k ≥ 1 look at just those terms n that lie strictly between 1 (k+1)t and 1 kt ; since the difference is 1 t 1 k(k+1) , there are at least 1 t 1 (k+1) 2 such terms. For each one, n ≤ 1 kt and so e −nt ≥ e −1/k , while n q ≥ 1 ((k+1)t) q . So in total, we have 1 (k+1)t ≤n≤ 1 kt n q e −nt ≥ e −1/k 1 t
1 (k + 1) 2 1 ((k + 1)t) q = e −1/k 1 (k + 1) q+2 t −q−1 .
The largest such term is achieved with k = 1, but we can add them up; since e −1/k ≥ e −1 , this yields that the sum over all n ≥ 0 is at least e −1 k≥1 1 (k+1) q+2 t −q−1 , which yields the results.
As to the upper bound, note that the function x → x p e −x/2 is bounded for x ≥ 0, with maximum value (2q) q e −q achieved at x = 2q. Therefore x q e −x ≤ (2q) q e −q e −x/2 , and plugging in x = nt we have
n≥0 (nt) q e −nt ≤ (2q) q e −q n≥0 e −nt/2 = (2q) q e −q 1 − e −t/2 .
The function t/(1 − e −t/2 ) is bounded near 0 and increasing on (0, 1); at 1 its value is < 3, and so (1 − e −t/2 ) −1 ≤ 3/t on the interval (0, 1), yielding the result.
Following is the upper-bound half of Equation 3.4.
Theorem 3.11. Let T ∈ L 2 hol (c), and let D t denote the associated R-diagonal dilation semigroup (in this case, D t is the free O-U semigroup). For each even p ≥ 4, there is a constant α p so that, for 0 < t < 1,
D t T p ≤ α p t −1+ 2 p T 2 .
Remark 3.12. A version of the following proof works for any finite circular generating set as well, but is significantly more complicated. We do not include it here since the p = ∞ case, which is our main interest, is proved independently in Section 3.3.
Proof. Let T = n λ n c n . By Corollary 2.7, this is an orthogonal sum. Also, Now, let p = 2r for r ∈ N. Note that D t T = n e −nt λ n c n , and so
[(D t T )(D t T ) * ] r = n,m e −(|n|+|m|)t λ n λ m c n 1 c * m 1 · · · c nr c * mr ,
where n = (n 1 , . . . , n r ) and m = (m 1 , . . . , m r ) range independently over N r , and λ n = λ n 1 · · · λ nr . From Corollary 2.4, ϕ(c n 1 c * m 1 · · · c nr c * mr ) = 0 unless |n| = n 1 + · · · + n r = m 1 + · · · + m r = |m|, and so
D t T p p = ∞ n=0 e −2nt
|n|=|m|=n λ n λ m ϕ(c n 1 c * m 1 · · · c nr c * mr )). We now employ the estimate of Equation 3.3:
|N C 2 (S n,m )| ≤ r r−1 r! (1 + n) r−1 .
Thus,
D t T p p ≤ r r−1 r! ∞ n=0
(1 + n) r−1 e −2nt |n|=|m|=n λ n λ m . and using the optimal ℓ 1 -ℓ 2 estimate | k∈S a k | 2 ≤ |S| k∈S |a k | 2 and the fact that the set of ordered integer partitions {n ∈ N r ; |n| = n} is counted by the binomial coefficient n+r−1 r−1 , we have |n|=n λ n 2 ≤ n + r − 1 r − 1 |n|=n |λ n | 2 .
15
Combining with Equation 3.8 yields
D t T p p ≤ r r−1 r! ∞ n=0 (1 + n) r−1 n + r − 1 r − 1 e −2nt |n|=n |λ n | 2 . (3.9)
Let S(t) = sup n≥0 (1 + n) r−1 n+r−1 r−1 e −2nt ; then we can roughly estimate Equation 3.9 by
D t T p p ≤ r r−1 r! S(t) ∞ n=0 |n|=n |λ n | 2 ,
and this latter sum n |λ n | 2 = ( n |λ n | 2 ) r is simply T 2r 2 = T p 2 from Equation 3.5. We are left, therefore, to estimate S(t). Note that
n + r − 1 r − 1 = (n + r − 1) · · · (n + 1) (r − 1)! ≤ (n + r − 1) r−1 (r − 1)! ,
and so estimating n + 1 ≤ n + r − 1,
S(t) ≤ 1 (r − 1)! sup n≥0 (n + r − 1) 2(r−1) e −2nt = 1 (r − 1)! sup n≥0 (n + r − 1) e − n r−1 t 2(r−1)
.
Elementary calculus yields that the supremum over all real n ≥ 0 of (n + r − 1)e − n r−1 t is r−1 t e t−1 , and so
S(t) ≤ (r − 1) 2(r−1) (r − 1)! e 2(r−1)(t−1) t −2r+2 .
Altogether, then, we have
D t T p p ≤ (r(r − 1) 2 ) r−1 r!(r − 1)! e 2(r−1)(t−1) t −p+2 T p 2 .
(3.10)
For 0 < t < 1, e 2(r−1)(t−1) ≤ 1, and so taking pth roots and letting α p represent the ratio following the ≤ in Equation 3.10 yields the desired result.
Remark 3.13. The constant α p developed through Equation 3.10 tends to ∞ as p → ∞. However, the p = ∞ case of Theorem 3.11 is actually a special case of Theorem 5.4 in [18], following a different technique.
We now turn to the lower-bound in Equation 3.4, and show that it is optimal in considerable generality.
Theorem 3.14. Let a 1 , . . . , a d be * -free R-diagonal operators, and suppose that a 1 has non-negative free cumulants: for each n, κ 2n [a 1 , a * 1 , . . . , a 1 , a * 1 ] ≥ 0 and κ 2n [a * 1 , a 1 , . . . , a * 1 , a 1 ] ≥ 0. Then for each even integer p ≥ 2, there is a constant α p > 0 such that, for 0 < t < 1,
D t : L 2 hol (a 1 , . . . , a d ) → L p (W * (a 1 , . . . , a d ), ϕ) ≥ α p t −1+ 2 p .
Moreover, this bound is achieved on the subspace L 2 hol (a 1 ) ⊂ L 2 hol (a 1 , . . . , a d ).
Proof. Set p = 2r, and denote a 1 by a. We will show that t 1−1/r · D t : L 2 hol (a) → L 2r (W * (a), ϕ) is bounded above 0 for small t. In fact, we will show that for each small t > 0, there is an element ψ t ∈ L 2 hol (a) so that t 2r−2 D t ψ t 2r 2r / ψ t 2r 2 ≥ α r for a t-independent constant α r > 0. Indeed, for fixed t > 0 define ψ t = n≥0 e −nt a n , 16 where we rescale a so that a 2 = 1. (Formally ψ t is D t ψ where ψ = n≥0 a n = (1 − a) −1 if 1 − a is invertible in L 2 hol (a).) Thus D t ψ t = ψ 2t , and so we wish to consider the ratio ψ 2t 2r 2r / ψ t 2r 2 . We begin by expanding the numerator. e −2n 1 t a n 1 e −2m 1 t a * m 1 · · · e −2nrt a nr e −2mrt a * mr . (3.11) It is convenient to add two more summation indices: n = n 1 + · · · + n r and m = m 1 + · · · + m r . Equation 3.11 then becomes ψ 2t 2r 2r = n,m≥0 e −2(n+m)t n 1 ,...,nr ≥0 n 1 +···+nr =n m 1 ,...,mr ≥0 m 1 +···+mr =m ϕ(a n 1 a * m 1 · · · a nr a * mr ). (3.12) Referring back to Corollary 2.4, the only non-zero terms in Equation 3.12 are those for which the word a n 1 a * m 1 · · · a nr a * mr is balanced: there must be as many as as a * s, and so we must have n 1 + · · · + n r = m 1 + · · · + m r ; i.e. n = m. Thus ϕ(a n 1 a * m 1 · · · a nr a * mr ). κ π [a ,n 1 , a * ,m 1 , . . . , a ,nr , a * ,mr ].
(3.14)
(The formerly implied commas are now explicit in the exponents; this is to make clear that the free cumulants have 2n arguments and are not being evaluated at products of arguments.) By assumption, all cumulants in a, a * are ≥ 0, and so we may restrict the summation in Equation 3.14 to those π that are pairings.
ϕ(a n 1 a * m 1 · · · a nr a * mr ) ≥ Now, for each pairing π ∈ N C 2 (S n 1 ,...,nr m 1 ,...,mr ), each block π matches an a with an a * ; there are n such blocks in total, and each is κ 2 [a, a * ] or κ 2 [a * , a]. Since a is R-diagonal, ϕ(a) = κ 1 [a] = 0 and ϕ(a * ) = κ 1 [a * ] = 0; thus κ 2 [a, a * ] = ϕ(aa * ) − ϕ(a)ϕ(a * ) = a 2 2 = 1, and κ 2 [a * , a] = ϕ(a * a) − ϕ(a * )ϕ(a) = ϕ(a * a) = a 2 2 = 1 by the traciality assumption. In general, then, we have κ π [a ,n 1 , a * ,m 1 , . . . , a ,nr , a * ,mr ] = a 2n 2 = 1. We will now throw away all of the terms where any index n j or m j is 0; in this case, since each of the r indices n j (or m j ) is at least 1, their sum n must be at least r, and so we have We can break up the set Ω n r according to the size of the minimal block in each element. Let Ω n,i r denote the subset of Ω n r of all balanced, length 2n, r run strings with minimal block size i. (Note: this set is empty unless n ≥ ir.) Evidently the sets Ω n,i r are disjoint for different i, and indeed Ω n r = n/r i=1 Ω n,i r . Combining this with Equations 3.18 and 3.17, and reordering the sum, this yields
ψ 2t 2r 2r ≥ ∞ i=1 ∞ n=ir e −4nt S∈Ω n,i r |N C 2 (S)|. (3.19)
Now employing Proposition 3.2, we have that for each S ∈ Ω n,i r , |N C 2 (S)| ≥ (1 + i) r−1 . Hence
ψ 2t 2r 2r ≥ ∞ i=1 (1 + i) r−1 ∞ n=ir e −4nt |Ω n,i r |. (3.20)
For fixed i, n it is relatively straightforward to enumerate the set Ω n,i r . Let S ∈ Ω n,i r ; then S can be written as S n 1 ,...,nr m 1 ,...,mr for indices n j , m j satisfying n 1 + · · · + n r = m 1 + · · · + m r = n and n j , m j ≥ i, and where at least one of n 1 , . . . , m r is equal to i. We will consider here only those terms for which n 1 = i. Let n ′ j = n j − i and m ′ j = m j − i; then we can rewrite S as the string (1 0+i , * m ′ 1 +i , . . . , 1 n ′ r +i , * m ′ r +i ), where the n ′ j , m ′ j are ≥ 0 and sum to n − ri. That is, there is an injection
Ω n,i r ←֓ {(0, m ′ 1 , . . . , n ′ r , m ′ r ) ; ∀j n ′ j , m ′ j ≥ 0 & n ′ 2 + · · · + n ′ r = m ′ 1 + m ′ 2 + · · · + m ′ r = n − ir}. (3.21)
The set on the right-hand-side of Equation 3.21 is a Cartesian product of the (ordered) integer partition sets {(n ′ 2 , . . . , n ′ r ) ; ∀j n ′
j ≥ 0 & n ′ 2 + · · · + n ′ r = n − ir} and {(m ′ 1 , . . . , n ′ r ) ; ∀j n ′ j ≥ 0 & m ′ 1 + · · · + m ′ r = n − ir}∞ i=1 (1 + i) r−1 ∞ n=ir e −4nt n − ir + r − 2 r − 2 n − ir + r − 1 r − 1 . (3.22)
We can lower bound the binomial coefficient as follows.
n − ir + r − 1 r − 1 = (n − ir + r − 1)(n − ir + r − 2) · · · (n − ir + 1) (r − 1)! ≥ (n − ir) r−1 (r − 1)! ,1 (r − 2)!(r − 1)! ∞ i=1 (1 + i) r−1 ∞ n=ir (n − ir) 2r−3 e −4nt .
(3.23)
Reindexing the internal sum, we have and an analysis completely analogous to that leading up to Equation 3.5 (coupled with the normalization a 2 = 1) yields a n 2 2 = 1 as well. Thus, ψ t 2 2 = n≥0 e −2nt = (1 − e −2t ) −1 , and so from Equation 3.26 we have t 2r−2 · ψ 2t 2r 2r ψ t 2r 2 ≥ t 2r−2 · α r t −3r+2 · (1 − e −2t ) r = α r (1 − e −2t ) r t −r .
(3.28)
The function t → (1 − e −2t )/t is bounded and decreasing on (0, 1), and so is ≥ (1 − e −2 ) on this interval. We therefore have that t 2r−2 · ψ 2t 2r 2r / ψ t 2r 2 ≥ α r for 0 < t < 1. Taking 2rth roots, this means that for each t we have D t ψ t 2r ψ t 2 = ψ 2t 2r ψ t 2 ≥ α r t −1+ 1 r .
Since ψ t ∈ L 2 hol (a) for each t > 0, this proves the theorem. Remark 3.15. While the constant α p in the above calculation is strictly positive, it decreases exponentially fast to 0 as r → ∞. Hence, Theorem 3.14 does not yield the optimal ultracontractive bound in [18], but rather a slightly weaker statement: it follows that D t : L 2 hol (a 1 , . . . , a d ) → W * (a 1 , . . . , a d ) ≥ α ǫ t −1+ǫ for any ǫ > 0. The fully optimal sharp bound does hold true, however, and also does not require the stringent non-negative cumulant condition of Theorem 3.14. For this purpose, we require an alternate technique which s is the subject of Section 3.3.
Remark 3.16. The condition of non-negative free cumulants is not entirely superfluous, as can easily be seen in the example of a single Haar unitary generator (some of whose free cumulants are negative). In this case, an analysis similar to the proof of Theorem 3.11 yields an L 2 → L p bound of order t − 1 2 + 1 p , which is optimal.
Theorem 3.17. Let a 1 , a 2 , . . . be an infinite set of * -free R-diagonal operators, and let A = W * (a 1 , a 2 , . . .). Let p ≥ 2 be an even integer. Then there is a constant α p > 0 so that, for 0 < t < 1, D t : L 2 hol (a 1 , a 2 , . . .) → L p (A , ϕ) ≥ α p t −1+ 2 p .
Proof. This is an application of the free central limit theorem due to R. Speicher, [25]. Let A = {a 1 , a 2 , . . .}, where the generators are renormalized so that a j 2 = 1 for all j. Since a j is Rdiagonal, we then have ϕ(a j ) = ϕ(a * j ) = 0, and ϕ(a * j a j ) = ϕ(a j a * j ) = a j 2 2 = 1 for each j, while ϕ(a 2 j ) = ϕ(a * 2 j ) = 0 (thanks to Corollary 2.4). Hence, by Theorem 3 (and more specifically the remark following Theorem 6) in [25], the sequence of elements a (N ) = 1 √ N (a 1 + · · · + a N ) ∈ L 2 hol (A)
converges in * -distribution to a standard circular element c. Then Define, for each t > 0, Following the proof of Theorem 3.14, we have ψ and ψ t , using the above limit-in-distribution, and then using the normality of ϕ.) Appealing to Theorem 3.14, since ψ t ∈ L 2 hol (c) for each t > 0 and c is R-diagonal with non-negative cumulants, as N → ∞ we have ψ (N ) 2t 2r 2r converges to a limit which is ≥ α r t −3r+2 for some α r > 0. Now following the conclusion of the proof of Theorem 3.14, we have
lim N →∞ ψ (N ) 2t 2r ψ (N ) t 2 ≥ α r t −1+ 1 r .
Since ψ (N ) t ∈ L 2 hol (A) for each N and each t > 0, the result now follows.
3.3. Optimal Ultracontractivity. We now prove Theorem 1.3. It is restated here as Theorem 3.18, with conditions that appear slightly different from those state in Theorem 1.3; however, the two are equivalent, as is explained in Remark 3.21.
Theorem 3.18. Let a 1 , . . . , a d be * -free R-diagonal operators such that sup a j / a j 2 < ∞, and suppose that one of the generators a = a j satisfies a 2 = 1 while a 4 > 1. Then there are constants α, β > 0 so that, for 0 < t < 1, α t −1 ≤ D t : L 2 hol (a 1 , . . . , a d ) → W * (a 1 , . . . , a d ) ≤ β t −1 . Moreover, this optimal lower bound is achieved on the subspace L 2 hol (a) ⊂ L 2 hol (a 1 , . . . , a d ). Remark 3.19. The upper bound of Theorem 3.18 was proved as Theorem 5.4 in [18], stated in the special case that all the generators are identically-distributed. In fact, the proof therein yields the above-stated theorem without modification, and so we only prove the lower bound below.
FIGURE 1 .
1The three partitions in N C(1, 1, 1, * , * , 1, * , * ).
Proposition 2 . 10 .
210The following conditions are equivalent.
suppose that the kernels m µ 1 , . . . , m µ d are in fact Markovian (for example, satisfying the conditions of Proposition 2.10). Then the operators M µ 1 (r), . . . , M µ d (r) are all completely positive (they are positive operators on commutative von Neumann algebras). Moreover, as was stated in the proof of Proposition 2.10, they are trace-preserving ( M µ f dµ = f dµ).
Remark 2. 15 .
15The orthogonal polynomials {p 0 , p 1 , p 2 , . . .} for µ are constructed from {1, x, x 2 , . . .} by Gram-Schmidt orthogonalization, which means that the span (in L ∞ ) of {p 0 , . . . , p n } is the same as the span of {1, x, . .
( 2 . 11 )
211Remark 2.17. Equation 2.11 is typical of the action of the Markov extension T t of D t (when it exists).
Definition 3. 1 .
1Given a string S, let N C 2 (S) denote the set of all pairings in N C(S).
Fuss-Catalan numbers. As a function of n, C (n) r
Proposition 3. 2 .
2Let S = (1 n 1 , * m 1 , . . . , 1 nr , * mr ) be a balanced string, and let i be the minimum block size, i = min{n 1 , m 1 , . . . , n r , m r } ≥ 1. Then |N C 2 (S)| ≥ (1 + i) r−1 .
i) 1−1 , proving this base case correct.
FIGURE 3 .
3The lattice path P(1, 1, 1, 1, * , * , 1, 1, * , * , * , * , * , 1, 1, * ).
Lemma 3. 4 .
4Let S be a balanced 1- * string with lattice path height h = h(S) and r runs. Then
FIGURE 4 .
4S is injected into S h r , with extraneous labels (dark lines) paired locally.
Theorem 3. 8 .
8Let S = (1 n 1 , * m 1 , 1 n 2 , * m 2 ) be any balanced string with 2 runs (so n 1 + n 2 = m 1 + m 2 ). Then |N C 2 (S)| = 1 + min{n 1 , m 1 , n 2 , m 2 }, |N C(S)| = 1 + 2 min{n 1 , m 1 , n 2 , m 2 }.
= 1 .
1[c, . . . , c, c * , . . . , c * ] by Equation 2.3 and the fact that only κ 2 = 0 for c, where S = (1 n , * n ). This is a regular pattern treated by Equation 3.1, and so the number of such π is C Indeed, the only element of N C(S) is the fully nested pairing ̟: (All partitions in N C(S) are pairing, since each block of ̟ = FIGURE 6. The only partition in N C(1, . . . , 1, * , . . . , * ). such a partition must alternate between 1 and * , and all * s in S follow all 1s.) Thus, c n 2 2 = κ ̟ [c, . . . , c, c * , . . . , c * ] = (κ 2 [c, c * ]) n = 1,
From
Equation 2.3, ϕ(c n 1 c * m 1 · · · c nr c * mr )) = π∈N C(Sn,m) κ π [c n 1 , c * m 1 , . . . , c nr , c * mr ] where S n,m = (1 n 1 , * m 1 , . . . , 1 nr , * mr ). Since only κ 2 = 0 for circular variables, the sum is really over N C 2 (S n,m , and all such terms are equal to 1 since κ 2 [c, c * ] = 1. λ m |N C 2 (S n,m )|.(3.7)
,...,nr ≥0 m 1 ,...,mr ≥0
,...,nr ≥0 n 1 +···+nr =n m 1 ,...,mr ≥0 m 1 +···+mr =n
let S n 1 ,...,nr m 1 ,...,mr denote the string (1 n 1 , * m 1 , . . . , 1 nr , * mr ). Equation 2.3 then yields that ϕ(a n 1 a * m 1 · · · a nr a * mr ) = π∈N C(S n 1 ,...,nr m 1 ,...,mr )
,...,nr m 1 ,...,mr )κ π [a ,n 1 , a * ,m 1 , . . . , a ,nr , a * ,mr ].(3.15)
,...,nr ≥0 n 1 +···+nr =n m 1 ,...,mr ≥0 m 1 +···+mr =n |N C 2 (S n 1 ,...,nr m 1 ,...,mr )|. (3.16)
,...,nr ≥1 n 1 +···+nr =n m 1 ,...,mr ≥1 m 1 +···+mr =n |N C 2 (S n 1 ,...,nr m 1 ,...,mr )|.(3.17) Now, fix n and let us reorganize the internal summation. As the indices n 1 , . . . , n r and m 1 , . . . , n r range over their summation sets, the string S n 1 ,...,nr m 1 ,...,mr ranges over all possible balanced 1- * strings (beginning with 1 and ending with * ) with length 2n and r runs. Let us denote this set of strings by Ω n r . Then the internal sum in Equation 3.17 can be rewritten as n 1 ,...,nr ≥0 n 1 +···+nr =n m 1 ,...,mr ≥0 m 1 +···+mr =n |N C 2 (S n 1 ,...,nr m 1 ,...,mr )| = S∈Ω n r |N C 2 (S)|. (3.18)
≥
(n − ir) r−2 /(r − 2)!. Combining with Equation 3.
(
− ir) 2(r−1) e −4nt = n≥0 n 2r−3 e −4(n+ir)t = e −4irt n≥0 n 2r−3 e −4nt .Appealing to Lemma 3.10, the sum n≥0 n 2(r−1) e −4nt ≥ α r (4t) −(2r−3)−1 for small t. As before, α r is used for an arbitrary r-dependent constant.) Applying Lemma 3.10 again, the remaining summation may be estimated∞ i=i (1 + i) r−1 e −4irt ≥ ∞ i=1 (i − 1) r−1 e −4irt = i≥0 i r−1 e −4(i+1)rt ≥ e −4rt · α r t −(r−1)−1 .
e
−nt [a (N ) ] n ∈ L 2 hol (A).
2
= (1 − e −2t ) 1/2 (since a (N ) 2 = 1 for each N ). As before, D t ψ ) * r converges, asN → ∞, to ϕ[(ψ 2t ψ * 2t ) r ] where ψ t = n≥0 e −nt c n . (This follows by truncating the infinite sums in the definitions of ψ (N ) t
Theorem 2.13. Let a 1 , . . . , a d be * -free R-diagonal operators, and suppose that the Mehler kernels associated to the symmetrizations µ j of the distributions of |a j | are Markovian. Then Equation 2.9 with T t = D e −t defines a CP trace preserving extension of the R-diagonal dilation semigroup D t of {a 1 , . . . , a d } which is different from the extension of Theorem 1.1. In particular, in the case that each a j is circular, T t corresponds to the free O-U semigroup.
; they have sizes n−ir+r−2r−2
and n−ir+r−1
r−1
respectively. Thus,
Equation 3.20 yields
ψ 2t
2r
2r ≥
0 < t < 1, we have e −4rt > e −4r . Collecting all constants and combining Equations 3.Now turning to the denominator, since different powers of a are orthogonal (by Corollary 2.24 and
3.25, we have
ψ 2t
2r
2r ≥ α r t −3r+2 .
(3.26)
7)
we have
ψ t
2
2 =
n≥0
e −2nt a n 2
2 ,
(3.27)
Acknowledgments.The author wishes to thank Uffe Haagerup, Karl Mahlburg, Mark Meckes, and Luke Rogers for useful conversations.Proof. The main technique we use here is the following simple estimate. If x is a bounded operator and ϕ is a state on C * (x), then since xx * is a positive semidefinite operator it is less than or equal to xx * 1; that is, xx * 1 − xx * is positive semidefinite. A product of commuting positive semidefinite operators is positive semidefinite, and so xx * xx * − xx * xx * ≥ 0. Since ϕ is a state, it follows that ϕ(xx * xx * ) ≤ ϕ( xx * xx * ) = x 2 ϕ(xx * ).In other words, we have the estimate x 2 ≥ x 4 4 / x 2 2 . Now, let a be R-diagonal with a 2 = 1 and a 4 > 1, and for fixed t > 0 let x = D t ψ t from the proof of Theorem 3.14:Then x = D t ψ t = ψ 2t . We need to estimate ψ 2t 4 4 from below, but the estimate of Equation 3.26 will not suffice because it holds valid only for a with non-negative free cumulants; this is because we ignored non-pairing cumulants to develop it. Here, instead, we will estimate this norm more accurately with all relevant partitions, using Theorem 3.8. ϕ(a n 1 a * m 1 a n 2 a * m 2 ),(3.29)where ϕ(a n 1 a * m 1 a n 2 a * m 2 ) = π∈N C(1 n 1 , * m 1 ,1 n 2 , * m 2 ) κ π [a ,n 1 , a * ,m 1 , a ,n 2 , a * ,m 2 ]. Break up this sum into those π that contain a 4-block, and those that are pairings. From the proof of Theorem 3.8, if π contains a 4-block then it is the only one, and all other blocks are pairings. We normalized a 2 = 1, so that κ 2 [a, a * ] = κ 2 [a * , a] = 1. The four block yields a term κ 4 [a, a * , a, a * ] or κ 4 [a * , a, a * , a]. From page 177 in[23]we calculate these asLet v(a) = a 4 4 − 1. Thus, if π contains a 4-block, then κ π [a ,n 1 , a * ,m 1 , a ,n 2 , a * ,m 2 ] = v(a) − 1, while if π ∈ N C 2 then κ π [a ,n 1 , a * ,m 1 , a ,n 2 , a * ,m 2 ] = 1. From the enumeration of N C(S) and N C 2 (S) in Theorem 3.8, we then have ϕ(a n 1 a * m 1 a n 2 a * m 2 ) = µ (v(a) − 1) + µ + 1 = µ · v(a) + 1, where µ = min{n 1 , m 1 , n 2 , m 2 }. Note that the assumption a 4 > 1 is precisely to ensure that v(a) > 0 in this expression.From Equation 3.29, we have This sum can be evaluated exactly, but for this estimate it is sufficient to look only at the terms 0≤i,j≤n min{i, j, n − i, n − j} ≥ 0≤i≤j≤n/2 min{i, j, n − i, n − j}.Since i ≤ j, n − j ≤ n − i, and so the summation is over min{i, n − j}. We can then write this as n/2 j=0 j i=0 min{i, n − j}, and since j ≤ n/2, n − j ≥ n/2 ≥ i so we have Since v(a) > 0, this proves the result.21Remark 3.22. Letting e −t = 1/λ, the proofs of Theorems 3.14 and 3.18 are estimates for various norms of the resolvent of a: ψ t = n≥0 e −nt a n = λ/(λ − a). We have provided here only rough estimates of constants involved. In fact, there is completely universal behaviour for this resolvent function over all R-diagonal a; the only dependence in the blow-up is on v(a). These sharp estimates will be discussed in[11].
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A note on the characterization of conditional expectation operators. D Wulbert, Pacific J. Math. 34MA 02139 E-mail address: [email protected], D.: A note on the characterization of conditional expectation operators. Pacific J. Math. 34, 285-288 (1970) DEPARTMENT OF MATHEMATICS, MIT, 2-172, 77 MASSACHUSETTS AVENUE, CAMBRIDGE, MA 02139 E-mail address: [email protected]
|
[] |
[
"Resistively-detected NMR lineshapes in a quasi-one dimensional electron system",
"Resistively-detected NMR lineshapes in a quasi-one dimensional electron system"
] |
[
"M H Fauzi \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"A Singha \nDepartment of Electrical Engineering\nIndian Institute of Technology Bombay\n400076MumbaiIndia\n",
"M F Sahdan \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"M Takahashi \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"K Sato \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"K Nagase \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"B Muralidharan \nDepartment of Electrical Engineering\nIndian Institute of Technology Bombay\n400076MumbaiIndia\n",
"Y Hirayama \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n\nCenter for Spintronics Research Network\nTohoku University\n980-8577SendaiJapan\n"
] |
[
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Electrical Engineering\nIndian Institute of Technology Bombay\n400076MumbaiIndia",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Electrical Engineering\nIndian Institute of Technology Bombay\n400076MumbaiIndia",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Center for Spintronics Research Network\nTohoku University\n980-8577SendaiJapan"
] |
[] |
We observe variation in the resistively-detected nuclear magnetic resonance (RDNMR) lineshapes in quantum Hall breakdown. The breakdown is locally occurred in a gate-defined quantum point contact (QPC) region. Of particular interest is the observation of a dispersive lineshape occured when the bulk 2D electron gas (2DEG) is set to ν b = 2 and the QPC filling factor to the vicinity of νQPC = 1, strikingly resemble the dispersive lineshape observed on a 2D quantum Hall state. This previously unobserved lineshape in a QPC points to simultaneous occurrence of two hyperfinemediated spin flip-flop processes within the QPC. Those events give rise to two different sets of nuclei polarized in the opposite direction and positioned at a separate region with different degree of electronic spin polarization.Recent advent in NMR technique through a resistive detection (RDNMR) has made it possible to study various spin physics in a 2D quantum Hall system[1][2][3][4][5][6][7], and a quasi-1D channel[8,9]. Despite the success achieved, a certain aspect related to the origin of the RDNMR lineshape variations noted experimentally in continuous wave (cw) mode is still poorly understood. One of them involved the puzzling observation of a dispersive lineshape in the quantum Hall state, a resistance dip followed by a resistance peak resonance line with increasing radio frequency[10]. It is first reported by Desrat et al[11]in the vicinity of ν b = 1 and has been confirmed in a number of follow-up papers[7,[12][13][14][15][16][17]. Similar dispersive like lineshape has been observed as well in the vicinity of ν b = 2/9[18], ν b = 2/3, ν b = 1/3[19], and at ν b = 2 Landau level crossing[20]. A number of appealing explanations has been put forward, but none of them provides a comprehensive explanation. Part of the reason why it still is difficult to unravel its physical origin is that we do not have a mature level of understanding about manybody 2D electronic states at the first Landau level yet, let alone their coupling to the nuclear spin. Thus, it would be highly desirable to study the lineshape variations in a platform where one can avoid such complexity.In this Rapid Communication, we resort to a quasione dimensional system in a gate-defined quantum point contact (QPC) to study various possible lineshapes including the dispersive lineshape noted experimentally in cw mode. Unlike on the 2D system, the mechanism for generation and resistive detection of nuclear spin polarization is tractable, allowing conveniently a direct interpretation of the observed lineshapes.Generation and detection of nuclear spin polarization are achieved by setting the filling factor in the bulk 2DEG to ν b = 2 and ν QPC = 1 in the QPC[21-30].Fig. 1(a)-(b) schematically displays how the nuclear polarization affects the transmission probability through the potential barrier of the QPC. For ν QPC < 1 (the down-spin channel T ↓ does not affect the transport), the up-spin channel T ↑ sees an increase(decrease) in the barrier potential in the presence of positive(negative) nuclear polarization, where positive (negative) means nuclear polarization is parallel (opposite) to the external magnetic field. Consequently, the transmission probability of the up-spin channel is reduced(enhanced). Therefore, the transmission is modified by a dynamic nuclear polarization (DNP) under a steady state where nuclear spins diffuse from the polarized regions to the center of the QPC. At sufficiently high current densities, there are two possible tractable DNPs by hyperfine-mediated inter edge spinflip scattering within the lowest Landau level, namely forward and backward spin-flip scatterings[21,22,31]. The first (second) one involves a spin-flip scattering from the forward propagating up-spin (down-spin) channel to the forward (backward) propagating down-spin (up-spin) channel, which in turn produces the positive (negative) nuclear polarization through the spin flip-flop process. On sweeping the rf field after the polarization reaches a steady state, those two different sets of nuclear polarization would leave a different trace in the RDNMR signal; with the positive (negative) one resulting in a resistance dip (peak). Here we demonstrate that under certain electronic state in the QPC, those two sets of nuclei can be generated simultaneously in a separate region within the QPC. Since they experience different degree of electron spin polarization, one can observe a combination of a resistance dip and peak resonance line in the RDNMR spectrum, namely dispersive lineshape.Our studies are carried out on a 20-nm-wide doped GaAs quantum well with the 2DEG located 165 nm beneath the surface. The wafer is photo-lithographically carved into a 30-µm-wide and 100-µm-long Hall bar geometry. The low temperature electron mobility is 84.5 m 2 /Vs at an electron density of 1.0 × 10 15 m −2 . A single QPC defined by triple Schottky gates is patterned on top of the Hall bar by Ti/Au evaporation. The bulk 2DEG density n can be tuned by applying back gate voltage (V BG ) to Si-doped GaAs substrate. It enables us to control the filling factor of interest in the bulk 2DEG ν = h eB n with back gate V BG and magnetic field B. The samples are mounted inside a single-shot cryogenicfree 3 He refrigerator with a temperature of 300 mK. A arXiv:1703.03520v2 [cond-mat.mes-hall]
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10.1103/physrevb.95.241404
|
[
"https://arxiv.org/pdf/1703.03520v2.pdf"
] | 119,064,342 |
1703.03520
|
cbe301281e5293b3cedb2dc205f74a5f61ef6a8d
|
Resistively-detected NMR lineshapes in a quasi-one dimensional electron system
(Dated: October 2, 2018) 22 May 2017
M H Fauzi
Department of Physics
Tohoku University
980-8578SendaiJapan
A Singha
Department of Electrical Engineering
Indian Institute of Technology Bombay
400076MumbaiIndia
M F Sahdan
Department of Physics
Tohoku University
980-8578SendaiJapan
M Takahashi
Department of Physics
Tohoku University
980-8578SendaiJapan
K Sato
Department of Physics
Tohoku University
980-8578SendaiJapan
K Nagase
Department of Physics
Tohoku University
980-8578SendaiJapan
B Muralidharan
Department of Electrical Engineering
Indian Institute of Technology Bombay
400076MumbaiIndia
Y Hirayama
Department of Physics
Tohoku University
980-8578SendaiJapan
Center for Spintronics Research Network
Tohoku University
980-8577SendaiJapan
Resistively-detected NMR lineshapes in a quasi-one dimensional electron system
(Dated: October 2, 2018) 22 May 2017
We observe variation in the resistively-detected nuclear magnetic resonance (RDNMR) lineshapes in quantum Hall breakdown. The breakdown is locally occurred in a gate-defined quantum point contact (QPC) region. Of particular interest is the observation of a dispersive lineshape occured when the bulk 2D electron gas (2DEG) is set to ν b = 2 and the QPC filling factor to the vicinity of νQPC = 1, strikingly resemble the dispersive lineshape observed on a 2D quantum Hall state. This previously unobserved lineshape in a QPC points to simultaneous occurrence of two hyperfinemediated spin flip-flop processes within the QPC. Those events give rise to two different sets of nuclei polarized in the opposite direction and positioned at a separate region with different degree of electronic spin polarization.Recent advent in NMR technique through a resistive detection (RDNMR) has made it possible to study various spin physics in a 2D quantum Hall system[1][2][3][4][5][6][7], and a quasi-1D channel[8,9]. Despite the success achieved, a certain aspect related to the origin of the RDNMR lineshape variations noted experimentally in continuous wave (cw) mode is still poorly understood. One of them involved the puzzling observation of a dispersive lineshape in the quantum Hall state, a resistance dip followed by a resistance peak resonance line with increasing radio frequency[10]. It is first reported by Desrat et al[11]in the vicinity of ν b = 1 and has been confirmed in a number of follow-up papers[7,[12][13][14][15][16][17]. Similar dispersive like lineshape has been observed as well in the vicinity of ν b = 2/9[18], ν b = 2/3, ν b = 1/3[19], and at ν b = 2 Landau level crossing[20]. A number of appealing explanations has been put forward, but none of them provides a comprehensive explanation. Part of the reason why it still is difficult to unravel its physical origin is that we do not have a mature level of understanding about manybody 2D electronic states at the first Landau level yet, let alone their coupling to the nuclear spin. Thus, it would be highly desirable to study the lineshape variations in a platform where one can avoid such complexity.In this Rapid Communication, we resort to a quasione dimensional system in a gate-defined quantum point contact (QPC) to study various possible lineshapes including the dispersive lineshape noted experimentally in cw mode. Unlike on the 2D system, the mechanism for generation and resistive detection of nuclear spin polarization is tractable, allowing conveniently a direct interpretation of the observed lineshapes.Generation and detection of nuclear spin polarization are achieved by setting the filling factor in the bulk 2DEG to ν b = 2 and ν QPC = 1 in the QPC[21-30].Fig. 1(a)-(b) schematically displays how the nuclear polarization affects the transmission probability through the potential barrier of the QPC. For ν QPC < 1 (the down-spin channel T ↓ does not affect the transport), the up-spin channel T ↑ sees an increase(decrease) in the barrier potential in the presence of positive(negative) nuclear polarization, where positive (negative) means nuclear polarization is parallel (opposite) to the external magnetic field. Consequently, the transmission probability of the up-spin channel is reduced(enhanced). Therefore, the transmission is modified by a dynamic nuclear polarization (DNP) under a steady state where nuclear spins diffuse from the polarized regions to the center of the QPC. At sufficiently high current densities, there are two possible tractable DNPs by hyperfine-mediated inter edge spinflip scattering within the lowest Landau level, namely forward and backward spin-flip scatterings[21,22,31]. The first (second) one involves a spin-flip scattering from the forward propagating up-spin (down-spin) channel to the forward (backward) propagating down-spin (up-spin) channel, which in turn produces the positive (negative) nuclear polarization through the spin flip-flop process. On sweeping the rf field after the polarization reaches a steady state, those two different sets of nuclear polarization would leave a different trace in the RDNMR signal; with the positive (negative) one resulting in a resistance dip (peak). Here we demonstrate that under certain electronic state in the QPC, those two sets of nuclei can be generated simultaneously in a separate region within the QPC. Since they experience different degree of electron spin polarization, one can observe a combination of a resistance dip and peak resonance line in the RDNMR spectrum, namely dispersive lineshape.Our studies are carried out on a 20-nm-wide doped GaAs quantum well with the 2DEG located 165 nm beneath the surface. The wafer is photo-lithographically carved into a 30-µm-wide and 100-µm-long Hall bar geometry. The low temperature electron mobility is 84.5 m 2 /Vs at an electron density of 1.0 × 10 15 m −2 . A single QPC defined by triple Schottky gates is patterned on top of the Hall bar by Ti/Au evaporation. The bulk 2DEG density n can be tuned by applying back gate voltage (V BG ) to Si-doped GaAs substrate. It enables us to control the filling factor of interest in the bulk 2DEG ν = h eB n with back gate V BG and magnetic field B. The samples are mounted inside a single-shot cryogenicfree 3 He refrigerator with a temperature of 300 mK. A arXiv:1703.03520v2 [cond-mat.mes-hall]
We observe variation in the resistively-detected nuclear magnetic resonance (RDNMR) lineshapes in quantum Hall breakdown. The breakdown is locally occurred in a gate-defined quantum point contact (QPC) region. Of particular interest is the observation of a dispersive lineshape occured when the bulk 2D electron gas (2DEG) is set to ν b = 2 and the QPC filling factor to the vicinity of νQPC = 1, strikingly resemble the dispersive lineshape observed on a 2D quantum Hall state. This previously unobserved lineshape in a QPC points to simultaneous occurrence of two hyperfinemediated spin flip-flop processes within the QPC. Those events give rise to two different sets of nuclei polarized in the opposite direction and positioned at a separate region with different degree of electronic spin polarization.
Recent advent in NMR technique through a resistive detection (RDNMR) has made it possible to study various spin physics in a 2D quantum Hall system [1][2][3][4][5][6][7], and a quasi-1D channel [8,9]. Despite the success achieved, a certain aspect related to the origin of the RDNMR lineshape variations noted experimentally in continuous wave (cw) mode is still poorly understood. One of them involved the puzzling observation of a dispersive lineshape in the quantum Hall state, a resistance dip followed by a resistance peak resonance line with increasing radio frequency [10]. It is first reported by Desrat et al [11] in the vicinity of ν b = 1 and has been confirmed in a number of follow-up papers [7,[12][13][14][15][16][17]. Similar dispersive like lineshape has been observed as well in the vicinity of ν b = 2/9 [18], ν b = 2/3, ν b = 1/3 [19], and at ν b = 2 Landau level crossing [20]. A number of appealing explanations has been put forward, but none of them provides a comprehensive explanation. Part of the reason why it still is difficult to unravel its physical origin is that we do not have a mature level of understanding about manybody 2D electronic states at the first Landau level yet, let alone their coupling to the nuclear spin. Thus, it would be highly desirable to study the lineshape variations in a platform where one can avoid such complexity.
In this Rapid Communication, we resort to a quasione dimensional system in a gate-defined quantum point contact (QPC) to study various possible lineshapes including the dispersive lineshape noted experimentally in cw mode. Unlike on the 2D system, the mechanism for generation and resistive detection of nuclear spin polarization is tractable, allowing conveniently a direct interpretation of the observed lineshapes.
Generation and detection of nuclear spin polarization are achieved by setting the filling factor in the bulk 2DEG to ν b = 2 and ν QPC = 1 in the QPC [21][22][23][24][25][26][27][28][29][30]. Fig. 1(a)-(b) schematically displays how the nuclear polarization affects the transmission probability through the potential barrier of the QPC. For ν QPC < 1 (the down-spin channel T ↓ does not affect the transport), the up-spin channel T ↑ sees an increase(decrease) in the barrier potential in the presence of positive(negative) nuclear polarization, where positive (negative) means nuclear polarization is parallel (opposite) to the external magnetic field. Consequently, the transmission probability of the up-spin channel is reduced(enhanced). Therefore, the transmission is modified by a dynamic nuclear polarization (DNP) under a steady state where nuclear spins diffuse from the polarized regions to the center of the QPC. At sufficiently high current densities, there are two possible tractable DNPs by hyperfine-mediated inter edge spinflip scattering within the lowest Landau level, namely forward and backward spin-flip scatterings [21,22,31]. The first (second) one involves a spin-flip scattering from the forward propagating up-spin (down-spin) channel to the forward (backward) propagating down-spin (up-spin) channel, which in turn produces the positive (negative) nuclear polarization through the spin flip-flop process. On sweeping the rf field after the polarization reaches a steady state, those two different sets of nuclear polarization would leave a different trace in the RDNMR signal; with the positive (negative) one resulting in a resistance dip (peak). Here we demonstrate that under certain electronic state in the QPC, those two sets of nuclei can be generated simultaneously in a separate region within the QPC. Since they experience different degree of electron spin polarization, one can observe a combination of a resistance dip and peak resonance line in the RDNMR spectrum, namely dispersive lineshape.
Our studies are carried out on a 20-nm-wide doped GaAs quantum well with the 2DEG located 165 nm beneath the surface. The wafer is photo-lithographically carved into a 30-µm-wide and 100-µm-long Hall bar geometry. The low temperature electron mobility is 84.5 m 2 /Vs at an electron density of 1.0 × 10 15 m −2 . A single QPC defined by triple Schottky gates is patterned on top of the Hall bar by Ti/Au evaporation. The bulk 2DEG density n can be tuned by applying back gate voltage (V BG ) to Si-doped GaAs substrate. It enables us to control the filling factor of interest in the bulk 2DEG ν = h eB n with back gate V BG and magnetic field B. six-turn rf coil wrapped the sample to be able to apply an oscillating magnetic field in the plane of the 2DEG. Throughout this study, the amplitude of rf power delivered to the top of the cryostat is fixed to −30 dBm (unless specified otherwise). Fig. 1(c) displays two sets of diagonal resistance traces as a function of split gate bias voltage across the QPC measured at a field of 4.5 (black line) and 4.25 (red line) T. The center gate is fixed to V CG = −0.425 V and V CG = −0.4 V, respectively [32]. We start with fully filled first Landau level in the bulk 2DEG (ν b = 2), where both the up-spin and down-spin electrons are available for transmission. Applying negative voltage on the split gates allows us to selectively transmit the up-spin channel through the constriction and reflect the down-spin channel. The nuclear spins is dynamically polarized by applying I AC = 10 nA at a selected operating point along the diagonal resistance trace on both sides of the ν QPC = 1 plateau. Typically, the resistance increases exponentially and reaches a point of saturation on the time scale of a few hundred seconds with the characteristic exponential rise time of about 150 seconds (see the lower inset of Fig. 1), similar time scale characteristic is reported previously on other QPC structures [28]. Once the resistance saturated, the rf is swept across the Larmor frequency of 75 As nuclei while measuring its resistance. The rf sweep rate is set to 100 Hz/s [33].
We observe variation in the RDNMR lineshape spectra on both flank of the ν QPC = 1 plateau as displayed in Fig. 2 and 3. Let us start with the RDNMR spectra for ν QPC < 1 case observed at a field of 4.5 T shown in Fig. 2
(a), measured from V SG = −0.41 up to V SG = −0.7 V.
For ease of comparison, we plot the resistance variation ∆R d with respect to the off-resonance resistance at f = 33 MHz. The salient feature appears in a narrow portion of the split gate bias voltage region, −0.50 ≤ V SG ≤ −0.41 V, very close to the ν QPC = 1 plateau. The spectra have a curious dispersive lineshape, strikingly resemble the dispersive lineshape previously observed in a number of reports on a 2D quantum Hall system in the vicinity of ν b = 1 [7,[11][12][13][14][15][16][17]. The lineshape we observe in our system is found to be highly sensitive to the rf power such that the resistance peak resonance line vanishes at a relatively high rf power of -15 dBm [34].
The corresponding signal amplitude normalized to the off resonance resistance |∆R d | /R d is displayed in Fig. 2(b). All the signal amplitude observed here falls below 1%, similar to the previous reports in Ref. [28,29]. Starting from the observable signal closest to the plateau V SG = −0.41 V, the dip amplitude shows a sharp upturn and reaches a maximum value at V SG = −0.44 V. It is then followed by a downturn and takes on a minimum value at V SG = −0.50 V, precisely at the transition between dispersive-to-single lineshape. The peak amplitude has a smaller amplitude than the dip amplitude and shows a monotonically decrease from V SG = −0.42 V and eventually vanishes at V SG = −0.51 V. The spectrum evolves into an expected single dip lineshape for V SG ≤ −0.51 V with the signal amplitude gradually increases. It can be partially explained by an increase in the current density locally in the constriction. Altogether, the facts that the lineshapes, signal amplitudes, as well as resonance point variations with the split gate bias voltage constitute firm evidence that the nuclei is polarized locally in the QPC.
We plot in Fig. 2(c)-(d) the raw RDNMR spectra at the two most extreme cases V SG = −0.70 and V SG = −0.41, respectively. In order to extract the Knight shift for each spectrum, here we plot in Fig. 2(d) (red dots) the reference signal taken close to ν b = 2 with nearly zero Knight shift. The spectrum is fitted with a Gaussian function [27], centered at 33.057 MHz and FWHM of 8.8 kHz (red line). Note that the long tail in the higher radio frequency side in the reference spectrum is nothing but reflects a long T1 time [35]. Fig. 2(f). The ∆f value continuously drops down to 12 kHz in an obviously linear fashion up until V SG = −0.46 V from its initial value of 18.3 kHz at V SG = −0.41 V, bearing a similarity to ∆f − B plot around ν b = 1 observed on the 2D system [16]. The value remains constant at about 12 kHz throughout the remaining split gate values, an indication that the electronic state in the QPC does not change sig-nificantly. Similar trend is observed as well for a field of 4.25 T[37].
We now move on to discuss the RDNMR taken at the opposite side of the plateau (ν QPC > 1) as shown in Fig. 3. The data show similar lineshape trend, but with inverted signal and much smaller amplitude than its counterpart. At a field of 4.5 T displayed in Fig. 3(a), the RDNMR signal is visible only in a confined split gate bias range, −0.32 ≤ V SG ≤ −0.30 V. The spectra measured very close to the plateau are hindered by a large resistance fluctuation in particular at the point where the diagonal resistance abruptly changes. Nevertheless, one can verify the existence of the inverted dispersive lineshape for ν QPC > 1 (see the line-cuts at V SG = −0.313 and V SG = −0.302 V in Fig. 3(b) for better visual). The RDNMR signal measured at a field of 4.25 T displayed in Fig. 3(c) has less resistance fluctuation and hence offers better signal to noise ratio. The inverted dispersive lineshape appears at V SG = −0.29 V (upper Fig. 3d) and turns into a resistance peak lineshape at V SG = −0.285 V (lower Fig. 3d). In contrast to the case for ν QPC < 1 where the RDNMR signal is observed in a wide range of split gate bias voltages, the signal observed here vanishes very quickly far from the ν QPC = 1 plateau region. Recall that the hyperfine-mediated spin flip-flop process relies on the spatial overlap between the up-spin and down-spin channels [21]. Thus, the absence of RD-NMR signal indicates the critical current for breakdown is higher than for ν QPC < 1 since the channel is opened wider [38]. The results presented in Fig. 2−3 provide important insights onto mechanisms leading to the dispersive lineshape observed in the vicinity of ν QPC = 1 plateau. Fig. 4 displays all possible hyperfine-mediated spin-flip scattering events where the QPC filling factor is tuned slightly less than 1 for two different alternating current cycles. The forward and backward spin-flip scattering could occur simultaneously within the QPC. The forward scattering occurs at the central region of the QPC where the degree of electron spin polarization is finite, not zero. On the other hand, the backward spin-flip scattering occurs slightly outside the central region where the electron spin polarization is zero. Those scattering events polarize the nuclei in opposite direction and spatially separated. On sweeping of rf with increasing frequency, the positive nuclear polarization is destroyed first due to Knight shift. It results in an increase in the transmissivity of the upspin channel. On further sweeping the rf, the positive nuclear polarization starts to build up and negative nuclear polarization is destroyed. This results in a decrease in the transmissivity of the up-spin channel. The backward spin-flip scattering is highly suppressed when the QPC filling factor is further tuned to ν QPC < 1, leaving only positive nuclear polarization build-up at the central region of the QPC. The RDNMR spectrum switches from dispersive-like to dip resonance lineshape. In this sce- nario, the Knight shift at the central region is determined by K S ∝ n ↑ − n ↓ ∝ T ↑ − T ↓ , where n ↑ (n ↓ ) and T ↑ (T ↓ ) are up(down)-spin electron density and up(down)spin transmission probability, respectively. The Knight shift reaches a maximum value when the up-spin channel is completely transmitted (T ↑ = 1) while the down spin channel is completely reflected (T ↓ = 0). It decreases with reduction of T ↑ , agreeing well with the experimental data shown in Fig. 2(e).
For ν QPC > 1 case, similar scenario happens. However, the Overhauser field from the polarized nuclei now affects the transmission of the down-spin channel while the fully transmitted up-spin channel is left unaffected. The nuclear polarization influences the transmissivity of the down-spin channel in an opposite way than that of the up-spin channel. This is the reason why the RDNMR spectrum gets inverted as experimentally confirmed in Fig. 3 and noted in Ref. [29].
To summarize, here we observe four variation of the RDNMR lineshapes in a gate-defined QPC. Of particular interest is the emergence of the dispersive lineshape in the RDNMR signal when the bulk filling factor is set to ν b = 2 and the QPC filling factor to the vicinity of the ν QPC = 1 plateau. It can be accounted by considering simultaneous occurrence of two hyperfine-mediated spinflip scattering events due to current-induced dynamic nuclear polarization. These phenomena give rise to localized regions with opposite nuclear polarization in the QPC. Although both of them are in contact with electrons in the QPC, they polarize in a region with different degree of electron spin polarization. Our experimental results further cemented the idea that the observation of the dispersive lineshapes on the 2D system, in particular around ν b = 1, should reflect the nuclear spin interaction with two electronic sub-systems as suggested by the authors in Ref. [7,16].
FIG. 1 .
1The samples are mounted inside a single-shot cryogenicfree 3 He refrigerator with a temperature of 300 mK. A arXiv:1703.03520v2 [cond-mat.mes-hall] (a)-(b) Schematic of potential barrier seen by upspin and down-spin electrons without (solid line) and with (dashed line) the presence of positive and negative nuclear polarization, respectively. The chemical potential window sits at νQPC < 1, so that only the up-spin channel affects the transport. (c) Differential diagonal resistance R d ≡ dV d /dIAC curve versus split gate bias voltage (VSG) at a field of 4.5 T (black) and 4.25 T (red). The left and right split gate are biased equally. The center gate voltage VCG is fixed to −0.425 V and −0.4 V, respectively. Upper inset displays a schematic drawing of device. Cross marks represent Ohmic contact pads. Triple Schottky gates deposited on top of the Hall bar defined a quantum point contact (see SEM image). The lithographic gap(width) between(of) a pair of split gate is 600(500) nm. An extra gate (center gate) with lithographic width of 200 nm is deposited in between the split gates. An excitation current IAC = 1 nA with f = 13.7 Hz is applied to the device for transport measurement. Lower inset shows typical R d time trace during current-induced dynamic nuclear polarization with IAC = 10 nA.
FIG. 2 .
2(a) Lower plot shows a 2D color map of 75 As RDNMR traces at the upper flank of the νQPC = 1 plateau, −0.70 ≤ VSG ≤ −0.41 V, measured at 4.5 T. The background resistance has been subtracted from the spectrum. Upper plot shows the blown-up spectra in between −0.46 ≤ VSG ≤ −0.41 V to accentuate the dispersive structure. (b) The RDNMR amplitude percentage vs split gate normalized to the off-resonance resistance, |∆Rd| /Rd, extracted from panel (a). (c)-(d) Raw RDNMR data sliced at the VSG = −0.7 and VSG = −0.41 V, respectively. RDNMR in red dots superimposed in panel (d) measured very close to the bulk 2DEG ν = 2 plateau, served as a reference signal with almost zero Knight shift. The signal is obtained by applying IAC = 100 nA. The red line is a Gaussian fit to the spectrum with the FWHM of 8.8 kHz. (e) The position of peak resonance frequency (black dots) and dip resonance frequency (red dots) extracted from panel (a) for −0.50 ≤ VSG ≤ −0.41 V. (f) The peak-to-dip resonance frequency separation ∆f extracted from panel (e). All the spectra measured with IAC = 10 nA (except the ref. signal) and RF power is −30 dBm.
Comparing with the reference signal, the observed spectrum at V SG = −0.70 V is only Knight shifted by about 8 kHz, reasonable value for the spectrum very far from the plateau at a field of 4.5 T. The dip frequency in the dispersive lineshape at V SG = −0.41 V gives the largest observable shift by about 18 kHz. Interestingly, its peak frequency appears to be substantially unshifted as it is aligned reasonably well with the reference resonance point. RDNMR measurement performed at a smaller field of 4.25 T reveals similar lineshape patterns[36].
Fig. 2 (
2e) displays the dip and peak resonance line points extracted from the split gate bias voltage segment between −0.41 to −0.50 V, where the dispersive lineshape is observed. The peak resonance line lies at the resonance reference point with very small variation throughout the range, substantially not Knight shifted. On the other hand, the dip resonance line is upshifted in a linear fashion up to V SG = −0.46 V and then followed by a slight downshift. The resulting ∆f values extracted from panel (e) is plotted in
FIG. 3 .
3(a) 2D color map of 75 As RDNMR traces at the lower flank of the νQPC = 1 plateau (νQPC > 1) measured at a field of 4.5 T. (b) Raw RDNMR traces sliced at VSG = −0.313 (upper) and VSG = −0.302 (lower) V, respectively. (c) 2D color map of 75 As RDNMR traces at the lower flank of the νQPC = 1 plateau (νQPC > 1) measured at a field of 4.25 T. (d) Raw RDNMR traces sliced at VSG = −0.29 (upper) and VSG = −0.285 (lower) V, respectively.
FIG. 4 .
4(a) Schematics of Landauer-Büttiker edge channel with forward (up-to-down spin flip) and backward (down-toup spin flip) hyperfine-mediated spin-flip scatterings occurred at filling factor slightly smaller than νQPC ≈ 1 during the first half-clock alternating current cycle (µS > µD) and (b) during the second half-clock alternating current cycle (µS < µD).Lighter edges indicate an empty channel while darker edges indicate a filled channel. The drain is held at ground (µD = 0) while the source chemical potential µS = 0 oscillates at a frequency of 13.7 Hz.
We would like to thank K. Muraki of NTT Basic Research Laboratories for supplying high quality wafers for this study. We thank K. Hashimoto
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25) T, we effectively make the channel wide so that the resistance increases gradually for νQPC < 1. This gradual changes in the diagonal resistance make systematic RDNMR measurement possible for νQPC < 1. field of 4.5(4.field of 4.5(4.25) T, we effectively make the channel wide so that the resistance increases gradually for νQPC < 1. This gradual changes in the diagonal resistance make system- atic RDNMR measurement possible for νQPC < 1.
See Supplemental Material at [URL will be inserted by publisher] for RF sweep rate dependence characteristics. See Supplemental Material at [URL will be inserted by publisher] for RF sweep rate dependence characteristics.
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"Impedances of Anisotropic Round and Rectangular Chambers",
"Impedances of Anisotropic Round and Rectangular Chambers"
] |
[
"Igor Zagorodnov \nDeutsches Elektronen-Synchrotron\nNotkestrasse 8522603HamburgGermany\n"
] |
[
"Deutsches Elektronen-Synchrotron\nNotkestrasse 8522603HamburgGermany"
] |
[] |
We consider the calculation of electromagnetic fields generated by an electron bunch passing through an anisotropic transversally non-homogeneous vacuum chamber of round or rectangular cross-section with translational symmetry in the beam direction. The described algorithms are implemented in a numerical code and cross-checked on several examples.
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10.1103/physrevaccelbeams.21.064601
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[
"https://arxiv.org/pdf/1805.11263v1.pdf"
] | 119,064,020 |
1805.11263
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a553f902107d82b635666bc29a1e1b7bb4752ca0
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Impedances of Anisotropic Round and Rectangular Chambers
April 2018 29 May 2018
Igor Zagorodnov
Deutsches Elektronen-Synchrotron
Notkestrasse 8522603HamburgGermany
Impedances of Anisotropic Round and Rectangular Chambers
April 2018 29 May 2018(Dated: April 6, 2018) Submitted to Physical Review Accelerators and Beams2numbers: 4160-m2927Bd0260Cb0270Bf
We consider the calculation of electromagnetic fields generated by an electron bunch passing through an anisotropic transversally non-homogeneous vacuum chamber of round or rectangular cross-section with translational symmetry in the beam direction. The described algorithms are implemented in a numerical code and cross-checked on several examples.
The anisotropic dielectrics may be incorporated either intentionally or unintentionally (processing-induced anisotropy) [4]. Dielectric permittivity and conductivity depend on the direction of wave propagation and polarization in many materials. The anisotropy can have a significant effect on modal coupling and must be accounted for in the design and analysis of such structures.
There are many papers which describe impedance calculations of steady-state impedance for isotropic round and flat layered chambers [5][6][7][8][9][10] with translational symmetry in the beam direction. The solutions for isotropic structures are obtained in analytical form or a field-matching approach can be used to reduce the problem to a simple matrix equation. In this paper we consider anisotropic transversally nonhomogeneous round and rectangular chambers where the field-matching technique does not work.
We start in Section II from formulation of the problem. Then we review the general form of the impedance for round and rectangular waveguides in non-relativistic case. For a special case of uniaxial anisotropy in the beam direction the same field matching approach as for the isotropic case [5] can be used. We consider shortly the required modifications in Section III. For transversally non-homogeneous materials the structure can be approximated by many layers. However such approach could be computationally expensive as it requires calculation of modified Bessel or exponential functions (of complex argument) for each layer.
A fully anisotropic case is treated in Section IV. The field-matching technique does not work in this case. We consider several possible analytical formulations and choose one with non-singular differential equations. For this choice we describe a simple finite-difference scheme. With a proper permutation of mesh indexes we reduce a sparse matrix with 7 bands to a pentadiagonal one. It allows a fast algorithm of complexity O(N) (N-number of mesh steps) for calculation of impedances for non-homogeneous anisotropic materials. Open boundary conditions are formulated for the case when the last layer has an uniaxial anisotropy in the beam direction.
The finite difference method allows treating of full anisotropy but could be timeconsuming. For the case when the anisotropic layers are thin we suggest in Section V a combination of the field matching and the finite-difference approaches.
Finally in Section VI the described methods are cross-checked on several numerical examples. The algorithms are implemented in numerical code ECHO [11].
II. PROBLEM FORMULATION
FIG. 1: Examples of "round" and "rectangular" geometry.
We consider a point-charge q moving with constant velocity v through a structure with round or rectangular cross-section. In the following we call the structure "round" if it is axially symmetric. If the structure has a constant width between two perfectly conducting planes and has rectangular cross-sections then we call such structure "rectangular". Fig. 1 shows examples of round and rectangular structures.
In the following we consider only an anisotropic materials with diagonal material permittivity and permeability tensors, where the optical axes coincide with coordinate ones. Hence their diagonals are given by complex vectors , µ.
We assume that the charge is moving along a straight line parallel to the longitudinal axis of the system, and we neglect the influence of the wakefields on the charge motion. For round structures we will use cylindrical coordinates r, ϕ, z. The charge density in the frequency domain can be expanded in Fourier series
ρ(r, ϕ, z, k) = e −ikz/β ∞ m=0 ρ m (r) cos(m(ϕ − ϕ 0 )), ρ m (r) = qδ(r − r 0 ) πvr 0 (1 + δ m0 ) ,(1)
where r 0 , ϕ 0 are coordinates of the point charge q, β = v/c, c is velocity of light in vacuum, and δ m0 = 1 if m = 1, 0 otherwise.
From the linearity of Maxwell's equations the components of the electromagnetic field can be represented by infinite sums:
H ϕ (r, ϕ, z, k) E r (r, ϕ, z, k) E z (r, ϕ, z, k) = e −ikz/β ∞ m=0 H ϕ,m (r, k) E r,m (r, k) E z,m (r, k) sin(mϕ), E ϕ (r, ϕ, z, k) H r (r, ϕ, z, k) H z (r, ϕ, z, k) = e −ikz/β ∞ m=0 E ϕ,m (r, k) H r,m (r, k) H z,m (r, k) cos(mϕ).(2)
The electric displacement D and the magnetic induction B are defined using com-plex permittivity and permeability diagonal tensors
D = r (r, k) 0 0 0 ϕ (r, k) 0 0 0 z (r, k) E, B = µ r (r, k) 0 0 0 µ ϕ (r, k) 0 0 0 µ z (r, k) H.
We do not have to assume any particular frequency dependence. In order to include conductivity and other losses in our code ECHO1D we use the following expressions (here we consider as example r-component):
r (r, k) = r (r)(1 + iδ r (r)) + i κ r (r) ω(1 + iωτ r (r)) , µ r (r, k) = µ r (r)(1 + iδ µ r (r)), ω = kc,
where r is the real part of the complex permettivity, µ r is the real part of the complex permeability, and the loss can be introduced with the help of dielectric loss tangent δ r , magnetic loss tangent δ µ r or/and with AC conductivity following the Drude model [12], where κ r is the DC conductivity of the material and τ r its relaxation time. We use similar expressions for ϕand z-components of the permittivity and the permeability tensors.
For each mode number m we can write an independent system of equations
m r H z,m + i k β H ϕ,m = iω r E r,m , − i k β H r,m − ∂ ∂r H z,m = iω ϕ E ϕ,m , 1 r ∂ ∂r (rH ϕ,m ) − m r H r,m = iω z E z,m + vρ m , − m r E z,m + i k β E ϕ,m = −iωµ r H r,m , − i k β E r,m − ∂ ∂r E z,m = −iωµ ϕ H ϕ,m , 1 r ∂ ∂r (rE ϕ,m ) + m r E r,m = −iωµ z H z,m , 1 r ∂ ∂r (rH r,m µ r ) − m r H ϕ,m µ ϕ − ikH z,m µ z = 0, 1 r ∂ ∂r (rE r,m r ) + m r E ϕ,m ϕ − ikE z,m z = ρ m .(3)
We have reduced the initial three-dimensional problem to an infinite set of independent dimensional problems, Eqs.
(3), for the Fourier componets of the field.
In rectangular case we choose a coordinate system with y in the vertical and x in the horizontal directions; the z coordinate is directed along the beam direction. The structures considered in this paper have constant width 2w in x-direction between two perfectly conducting side walls.
The charge density can be expanded in Fourier series
ρ(x, y, z, k) = e −ikz/β w ∞ m=1 ρ m (y) sin(k x,m x 0 ) sin(k x,m x), k x,m = πm 2w , ρ m (y) = qδ(y − y 0 ) v ,
where x 0 , y 0 are coordinates of the point charge. Again it follows from the linearity of Maxwell's equations that the components of electromagnetic field can be represented by infinite sums:
H x (x, y, z, k) E y (x, y, z, k) E z (x, y, z, k) = e −ikz/β w ∞ m=1 H x,m (y, k) E y,m (y, k) E z,m (y, k) sin(k x,m x), E x (x, y, z, k) H y (x, y, z, k) H z (x, y, z, k) = e −ikz/β w ∞ m=1 E x,m (y, k) H y,m (y, k) H z,m (y, k) cos(k x,m x).
For each mode number m we can write an independent system of equations
− k x,m H z,m + i k β H x,m = iω y E y,m , − i k β H y,m − ∂ ∂y H z,m = iω x E x,m , ∂ ∂y H x,m + k x,m H y,m = iω z E z,m + vρ m , k x,m E z,m + i k β E x,m = −iωµ y H y,m , − i k β E y,m − ∂ ∂y E z,m = −iωµ x H x,m , ∂ ∂y (E x,m ) − k x,m E y,m = −iωµ z H z,m , ∂ ∂y (H y,m µ y ) + k x,m H x,m µ x − ikH z,m µ z = 0, ∂ ∂y (E y,m y ) − k x,m E x,m x − ikE z,m z = ρ m .(4)
We are interested in coupling impedances as defined in [5,14]. For round pipe the coupling impedance can be written as Z (r 0 , ϕ 0 , r, ϕ, k, γ) = ∞ m=0 Z m (k, γ)I m kr 0 γβ I m kr γβ cos(m(ϕ − ϕ 0 )) + Z sc (r 0 , ϕ 0 , r, ϕ, k, γ),
Z sc (r 0 , ϕ 0 , r, ϕ, k, γ) = − kZ 0 2π(γ 2 − 1) K 0 k r 2 0 + r 2 − 2r 0 rcos(ϕ − ϕ 0 ) γβ ,(5)
where γ is the relative relativistic energy and we have written explicitly the space
charge contribution Z sc .
For a rectangular pipe the impedance reads
Z (x 0 , y 0 , x, y, k) = 1 w ∞ m=1 Z m (y 0 , y, k, γ) sin(k x,m x 0 ) sin(k x,m x) + Z sc (x 0 , y 0 , x, y, k, γ), Z sc (x 0 , y 0 , x, y, k, γ) = − kZ 0 2π(γ 2 − 1) K 0 k (x − x 0 ) 2 + (y − y 0 ) 2 γβ ,(6)
where
Z m (y 0 , y, k, γ) = Z cc m (k, γ) cosh(k y,m y 0 ) + Z sc m (k, γ) sinh(k y,m y 0 ) cosh(k y,m y) + Z cs m (k, γ) cosh(k y,m y 0 ) + Z ss m (k, γ) sinh(k y,m y 0 ) sinh(k y,m y), k y,m = k 2 x,m + k 2 γ 2 β 2 .
In Eqs.(5, 6) the infinite sum defines a so-called wall impedance. The longitudinal and the transverse wall impedances are connected by Panofsky-Wentzel theorem (see [5] for a detailed discussion).
The wake field effect in time domain is described by a longitudinal wake function which can be obtained by the Fourier transform of the longitudinal impedance
w || (s) = c 2π ∞ −∞ Z || (k)e iks/β dk,
where s is the distance between the source and the test particles [14].
III. FIELD MATCHING FOR UNIAXIAL ANISOTROPY
In the general anisotropic case from system of first-order Eqs.
(2) we obtain the second-order coupled equations for z-components of the electric and the magnetic fields:
1 r ∂ ∂r
r r ν 2 rϕ ∂ ∂r E z,m − m 2 ϕ r 2 ν 2 ϕr + z E z,m + m rv ∂ ∂r 1 ν 2 rϕ − 1 ν 2 ϕr ∂ ∂r H z,m = iqδ(r − r 0 ) πr 0 (1 + δ m0 )ω , 1 r ∂ ∂r rµ r ν 2 ϕr ∂ ∂r H z,m − m 2 µ ϕ r 2 ν 2 rϕ + µ z H z,m − m rv ∂ ∂r 1 ν 2 ϕr − 1 ν 2 rϕ ∂ ∂r E z,m = 0, ν 2 rϕ = k 2 β −2 − ω 2 2 r µ 2 ϕ , ν 2 ϕr = k 2 β −2 − ω 2 2 ϕ µ 2 r .(7)
The field matching technique for round and flat isotropic pipes was considered, for example, in [5][6][7][8][9][10]. For the case of uniaxial anisotropy along z-axis we use the same technique, which we describe shortly in this Section.
FIG. 2: Examples of "round" and "rectangular" layered geometry.
We consider the uniaxial anisotropy when the permittivity and the permeability tensors are diagonal and for their elements the following relations hold
r (r) = ϕ (r), µ r (r) = µ ϕ (r).
Inside of each layer where the complex permeability and permittivity are constants (independent from r) Eqs.(7) reduce to the decoupled equations
1 r ∂ ∂r r ∂ ∂r E z,m − m 2 r 2 + ν 2 r z r E z,m = iqδ(r − r 0 )ν 2 r πr 0 (1 + δ m0 )ω r , 1 r ∂ ∂r r ∂ ∂r H z,m − m 2 r 2 + ν 2 r µ z µ r H z,m = 0, ν 2 r = k 2 β −2 − ω 2 2 r µ 2 r .(8)
A general solution of homogeneous hyperbolic Eqs. (8) can be written in form
E z,m (r) = C m I I m (ν r r) + C m K K m (ν r r), H z,m (r) = D m I I m (ν µ r r) + D m K K m (ν µ r r),(9)ν r = ν r z / r , ν µ r = ν r µ z /µ r ,
where I m , K m are modified Bessel functions of complex argument.
In the following we will numerate the layers by index j and r = a j defines interface between the layers with numbers j and j + 1. In order to find the constants C m, j I , C m, j K , D m, j I , D m, j K in Eqs. (9) we can use 4 conditions at the interfaces between the layers:
E j z,m (a j ) = E j+1 z,m (a j ), H j z,m (a j ) = H j+1 z,m (a j ), j r E j r,m (a j ) = j+1 r E j+1 r,m (a j ), µ j r H j r,m (a j ) = µ j+1 r H j+1 r,m (a j ),(10)
where the radial field components are defined through the longitudinal ones as
E j r,m (r) = ik ν 2 r 1 β ∂ ∂r E j m,z + mcµ r r H z,m , H j r,m (r) = ik ν 2 r 1 β ∂ ∂r H j m,z + mc r r E z,m .(11)
From Eqs. (9)-(11) at each interface r = a j we obtain the relations
(C m, j+1 I , C m, j+1 K , D m, j+1 I , D m, j+1 K ) T = M j (C m, j I , C m, j K , D m, j I , D m, j K ) T ,
where M j is a complex matrix of order 4. We do not write the explicit form of the elements of the matrix M j . They can be written as a combination of modified Bessel functions and the expressions are similar to those obtained in [5] for an isotropic case.
The matrix connecting the coefficients from vacuum layer to the coefficients of the last layer can be found as a matrix product
M = M N−1 M N−2 ...M 1 M 0 . M 11 M 13 0 0 M 21 M 23 −1 0 M 31 M 33 0 0 M 41 M 43 0 −1 C m,0 I /C m,0 K D m,0 I /C m,0 K C m,N K /C m,0 K D m,N K /C m,0 K = −M 12 −M 22 −M 32 −M 42 .(12)
After numerically solving of Eq. (12) the modal longitudinal impedance in Eq. (5) can be found as
Z m (k, γ) = − ikZ 0 2π(γ 2 − 1) C m,0 I C m,0 K .
If the last layer, j = N, is closed with perfectly electric conducting (PEC) material at r = a N , then we use a modified matrix
M = M C2F N M N−1 M N−2 ...M 1 M 0 , where M C2F
N is a matrix converting the field coefficients in the field components H r , H ϕ and their derivatives:
H r,m (a N ) H ϕ,m (a N ) ∂ ∂r [H ϕ,m r]| r=a N ∂ ∂r [µ r H r,m r]| r=a N = M C2F N C m,N I C m,N K D m,N I D m,N K .
Again , k µ y,m = k 2 x,m + ν 2 y µ z µ y , ν 2 y = k 2 β −2 − ω 2 2 y µ 2 y .
In the following we consider only the case where the rectangular structure is symmetric in the y-direction (up-bottom symmetry). In this case Eq. (6) has a simpler form Z m (y 0 , y, k, γ) = Z cc m (k, γ) cosh(k y,m y 0 ) cosh(k y,m y) + Z ss m (k, γ) sinh(k y,m y 0 ) sinh(k y,m y). Hence we are looking for the solution of the following system
M 11 + M 12 M 13 − M 14 0 0 M 21 + M 22 M 23 − M 24 −1 0 M 31 + M 32 M 33 − M 34 0 0 M 41 + M 42 M 43 − M 44 0 −1 C m,0 + /(C m,0 − − C m,0 + ) D m,0 + /(C m,0 − − C m,0 + ) C m,N − /(C m,0 − − C m,0 + ) D m,N − /(C m,0 − − C m,0 + ) = −M 12 −M 22 −M 32 −M 42 .(13)
After numerical solution of Eq. (13) the item Z cc m (k, γ) can be found as
Z cc m (k, γ) = − 2ikZ 0 π(γ 2 − 1)k 0 y,m C m,0 + (C m,0 − − C m,0 + ) , k 0 y,m = k 2 x,m + k 2 γ 2 β 2 .
The item Z ss m (k, γ) can be found from the solution of another problem in the half of the domain with electric boundary condition at the symmetry plane E z,m (0) = 0. We are looking for the solution of the following system
M 11 − M 12 M 13 + M 14 0 0 M 21 − M 22 M 23 + M 24 −1 0 M 31 − M 32 M 33 + M 34 0 0 M 41 − M 42 M 43 + M 44 0 −1 C m,0 + /(C m,0 − + C m,0 + ) D m,0 + /(C m,0 − + C m,0 + ) C m,N − /(C m,0 − + C m,0 + ) D m,N − /(C m,0 − + C m,0 + ) = −M 12 −M 22 −M 32 −M 42 .(14)
After numerical solution of Eq.(14) the item Z ss m (k, γ) can be found as
Z ss m (k, γ) = − 2ikZ 0 π(γ 2 − 1)k 0 y,m C m,0 + (C m,0 − + C m,0 + )
.
If the last layer, j = N, is closed with perfectly conducting material at y = a N then we use a modified matrix in the same way as described above for the round geometry. We will not consider here a rectangular structure without symmetry. In general, matrix M is a composition of matrices for all layers. It can be found and treated in the same way as described in [5] for an isotropic case.
IV. FINITE-DIFFERENCE METHOD FOR FULL ANISOTROPY
In this section we describe a finite-difference method to treat the round and the rectangular structures with arbitrary anisotropic materials. We start with the round case. At the beginning we have to decide which equations to use. The system
mµ ϕ rµ z H 0 ϕ,m ,(16)b ϕ (r) = ω 2 µ ϕ r − k 2 r r β 2 − m 2 µ ϕ r 3 r µ z , b r (r) = ω 2 ϕ r − k 2 rµ r β 2 − m 2 ϕ r 3 µ r z .
In order to remove the discontinuity of the azimuthal component in the charge location r 0 we present the azimuthal component of the magnetic field in the form
H ϕ,m = H s ϕ,m + H 0 ϕ,m , H 0 ϕ,m = (1 + δ m0 )H 0 ϕ , H 0 ϕ = θ(r − r 0 ) 2πr ,
where θ(r) is Heaviside function and H 0 ϕ presents a monopole harmonic of the self field of relativistic charge in free space. Let us note that H s ϕ,m has the meaning of the scattered field only for the lowest monopole mode, m = 0, and the relativistic charge. Another choice could be to take H 0 ϕ,m as a true m-harmonic of the selffield but this introduces additional terms into the right-hand side of Eqs. (15), (16) without any clear improvement of the accuracy of the numerical solution. We introduce one dimensional mesh with shifted positions of the transverse magnetic filed components as shown in Fig. 3. The mesh in material is not equidistant in general. It is chosen to sample the wave length in the material properly and depends on the wavenumber k = ω/c. We use the standard second order approximations of the derivatives [15] and the finite-difference scheme reads 1 (17) where we have introduced the discrete field components h ϕ,i = H s ϕ,m (r i )r i , h r,i+0.5 = µ r (r i+0.5 )H r,m (r i+0.5 )r i+0.5 and the following notation
r i+0.5 − r i−0.5 a ϕ (r i+0.5 ) h ϕ,i+1 − h ϕ,i r i+1 − r i − a ϕ (r i−0.5 ) h ϕ,i − h ϕ,i−1 r i − r i−1 + b ϕ (r i )h ϕ,i + c ϕ (r i ) h r,i+0.5 − h r,i−0.5 r i+0.5 − r i−0.5 − d ϕ (r i+0.5 )h r,i+0.5 − d ϕ (r i−0.5 )h r,i−0.5 r i+0.5 − r i−0.5 = f ϕ (r i ), 1 r i − r i−1 a r (r i ) h r,i+0.5 − h r,i−0.5 r i+0.5 − r i−0.5 − a r (r i−1 ) h r,i−0.5 − h r,i−1.5 r i−0.5 − r i−1.5 + b r (r i−0.5 )h r,i−0.5 + c r (r i−0.5 ) h ϕ,i − h ϕ,i−1 r i − r i−1 − d r (r i )h ϕ,i − d r (r i−1 )h ϕ,i−1 r i − r i−1 = f r (r i−0.5 ),a ϕ (r i+0.5 ) = 1 r i+0.5 z (r i+0.5 ) , c ϕ (r i ) = m r 2 i r (r i )µ z (r i ) , d ϕ (r i+0.5 ) = m r 2 i+0.5 z (r i+0.5 )µ r (r i+0.5 ) , f ϕ (r i ) = −b r (r i )[H 0 ϕ,m (r i )r i ] a r (r i ) = 1 r i µ z (r i ) , c r (r i−0.5 ) = m r 2 i−0.5 ϕ (r i−0.5 ) z (r i−0.5 ) , d r (r i ) = m r 2 i µ ϕ (r i )µ z (r i ) , f r (r i−0.5 ) = d r (r i )[H 0 ϕ,m (r i )r i ] − d r (r i−1 )[H 0 ϕ,m (r i−1 )r i−1 ] r i − r i−1 .
At the axis of axially symmetric geometry we have magnetic boundary condition [H ϕ,r r]| r=0 = 0, ∂ ∂r [µ r H r,m r]| r=0 = 0, and the equations for i = 1 can be written in the form 1
r 1.5 − r 0.5 a ϕ (r 1.5 ) h ϕ,2 − h ϕ,1 r 2 − r 1 − a ϕ (r 0.5 ) h ϕ,1 r 1 + b ϕ (r 1 )h ϕ,1 + c ϕ (r 1 ) h r,1.5 − h r,0.5 r 1.5 − r 0.5 − d ϕ (r 1.5 )h r,1.5 − d ϕ (r 0.5 )h r,0.5 r 1.5 − r 0.5 = f ϕ (r 1 ), 1 r 1 a r (r 1 ) h r,1.5 − h r,0.5 r 1.5 − r 0.5 + b r (r 0.5 )h r,0.5 + c r (r 0.5 ) h ϕ,1 r 1 − d r (r 1 )h ϕ,1 r 1 = f r (r 0.5 ),
If the exterior boundary is perfectly conducting at r N+0.5 = b then we have electric boundary condition for the magnetic field
where the matrix M has dimensions 2N × 2N and the seven band structure shown in Fig. 4 on the left side. In order to use a direct method of solution of linear system (18) we introduce the permutation matrix P σ defined by permutation of indexis
σ i = 2N + 1 − i 2 , i even, N + 1 − i−1 2 , i odd.(19)
It converts the sparse seven band matrix M in pentadiagonal form P σ MP T σ shown in Fig. 4 on the right side. , G = ikZ 0 2π(γ 2 − 1) .
In the case of rectangular geometry we again consider only the case with symmetry plane at y = 0. In this case we have to solve two problems in half of the computational domain. The first problem for Z cc has a magnetic boudary condition at the symmetry plane (H z,m (0) = 0) and we approximate it in the same way as it was done at the axis for round geometry. The second problem for Z ss has an electric boundary condition (E z,m = 0) at the symmetry plane and we approximate it in the same way as it was done for round geometry at PEC boundary.
y z k y,m − k 2 x,m k µ y,m ∂ ∂y H x + ν 2 y H x − k x,m ν 2 y k µ y,m − y z k y,m − k 2 x,m k µ y,m H y,m = 0, µ y µ z k µ y,m − k 2 x,m k y,m ∂ ∂y H y + ν 2 y H y − k x,m ν 2 y k y,m − µ y µ z k µ y,m − k 2 x,m k y,m H x,m = 0.
The longitudinal electric field component and the impedance in the rectangular case can be found as in analytical form, Eq.(9), the middle layer is anisotropic and could be treated only with finite-difference method. Let us denote the coefficients in the first layer as C m,1 I , C m,1 K , D m,1 I , D m,1 K and the coefficients in the third layer as C m,3 I , C m,3 K , D m,3 I , D m,3 K . In order to use the matrix approach of Section III we need to find matrix M FD 13 , converting the first set of coefficients in the second one:
E z,m (y 0 ) = − i ω z (y 0 ) ∂ ∂y H x,m | y=y 0 + k x,m H y,m (y 0 ) , Z cc m (k) = C m,3 I C m,3 K D m,3 I D m,3 K = M FD 13 C m,1 I C m,1 K D m,1 I D m,1 K .
The matrix M 13 can be found as a product of several simple complex matrices of size 4 × 4:
M 13 = M F2C 2 M F2F 2 M FD 12 M F2F 1 M C2F 1 ,(21)
where M CF
M r h r = f r ,(24)
h r = (h ϕ,2 , h ϕ,3 , ..., h ϕ,N+1 , h r,1.5 , h r,2.5 , ..., h r,N+0.5 ) t , The matrix M r of system (24) has the form shown in Fig. 6 The field components Analogously we will find the third and the fourth columns of this matrix.
f r = ( f r 1 , f r 2 , f 3 , f N , f r 0.5 , f r 1.5 , f r 2.5 ..., f N−0.5 ) t , M r i, j = M i, j+2 , i = 1,H r,m (a − 2 ) = (h r,N+0.5 + h r,N−0.5 )/2, H ϕ,m (a − 2 ) = h ϕ,N , ∂ ∂r [H ϕ,m r]| r=a − 2 = h r,N+1 − h r,N−1 r N+1 − r N−1 , ∂ ∂r [µ r H r,m r]| r=a − 2 = h r,
As can be seen from the above description we need to solve the problem 4 times in the anisotropic layer only. If the layer is thin then the suggested method is faster than the finite-difference method of the previous section where the whole domain has to be discretized to sample the electromagnetic field everywhere. At the rectangular geometry the algorithm is exactly the same with corresponding equations for the rectangular case.
VI. NUMERICAL EXAMPLES
Recently, experimental demonstration of energy modulations in dielectric pipes was observed at the PITZ facility [17]. The experiment was performed with a dielectric pipe with an isotropic dielectric layer of permittivity = 4.41 0 . The layer starts at radius a 0 = 0.45 mm and is closed with PEC at a 1 = 0.55 mm. We take this dielectric pipe as our first example and calculate the steady-state wake of a relativistic Gaussian bunch with rms length σ z = 25 µm. In Fig. 7 we show the longitudinal and the transverse wake potentials near the pipe axis. The longitudinal wake potential for the charge distribution λ(s) is defined as
W (s) = s −∞ w (s )λ(s − s )ds .
The transverse wake potential is defined analogously and W ⊥ (s) means here the dipole component of the transverse wake normalized by offset [14]. ECHO2D [11]. In order to obtain the steady-state wake in time-domain we have subtracted the wake for pipe of length 10 cm from the wake of pipe of length 11 cm.
The agreement of the curves from two different methods confirms the correctness of the results. In Fig. 8 Table I. Let us note that in this example we have used a small conductivity κ = 1 S/m to resolve the real part of the impedance.
For the same aperture size the cylindrical geometry allows to obtain the highest accelerating gradients. Due to technological difficulties in preparing cylindrical structures with stringent requirements to tolerances the rectangular structures are
Method Round Rectangular
Field Matching (Section III) 31 5
Finite-Difference (Section IV) MeV, charge 100 nC and bunch length σ z = 1.5mm is considered. The dependence of the longitudinal electric field component E z at the symmetry axis produced by the bunch on the distance s = vt − z behind it is shown in Fig. 9. The solid line corresponds to anisotropic sapphire, the dashed line corresponds to isotropic filling. The wave number k was sampled from 1 m −1 to 20e4 m −1 with step 0.2 and we have calculated 5 the lowest odd Fourier harmonics in Eq. (6). At this example we used a small conductivity κ = 0.05 S/m to resolve the real part of the impedance.
The data in Fig. 9 agree with the results published in [3]. A frequency shift with a little influence on the wake field amplitudes can be seen.
The execution times of different methods discussed in this paper for the rectangular example are shown in Table I. It can be seen again that for the same accuracy the combined method requires less computational time as compared to a fully finitedifference one.
waveguides are under extensive study as accelerating structures excited by charged beams [1]. Quartz and cordierite structures have been beam tested, and accelerating gradient exceeding 100 MV/m has been demonstrated [2]. Several materials used for accelerating structures (sapphire, ceramic films etc) possess significant anisotropic properties. It is shown, for example, in [3] that the dielectric anisotropy causes a frequency shift in comparison to dielectric-lined waveguides with isotropic dielectric loadings.
From the boundary condition at the axis we have D m,0 K = 0. If the last layer, j = N, is infinite with finite conductivity then we have open boundary condition.The field should decay at infinity, giving C msolution of the following simple system
we do not write the explicit form of the elements of the matrix M C2F N . They can be written as a combination of modified Bessel functions and the expressions are easy to obtain from Eqs. (3), (9) in any computer program supporting symbolic calculations.The boundary conditions for perfectly conducting material at a N can be written as H r,m (a N ) = 0, ∂ ∂r [H ϕ,m r]| r=a N = 0. Hence in order to find the impedance we again use Eqs.(12) where the right hand side has the same form but the vector of un-knowns is different: (C m,0 I /C m,0 K , D m,0 I /C m,0 K , H ϕ,m (a N )/C m,0 K , ∂ ∂r [µ r H r,m r]| r=a N /C m,0 K ) T .For rectangular geometries we follow the same approach. The field in the homogeneous uniaxially anisotropic layer can be presented as sum of complex exponents
E z,m (r) = C m + e k y,m y + C m − e −k y,m y , H z,m (r) = D m + e k µ y,m y + D m − e −k µ y,m y , k y,m = k 2 x,m + ν 2 y z y
The item Z cc m (k, γ) can be found from the solution of the problem in the half of the domain with magnetic boundary condition at the symmetry plane H z,m (0) = 0. If the last layer, j = N, is infinite with finite conductivity then we have open boundary condition. The field should decay at infinity and it results in C m
( 3 )
3contains 8 first-order equations for 6 unknown field components. It can be reduced only to 2 second-order equations. For example we can use Eqs. (7) for longitudinal components of electric and magnetic fields. However for relativistic beam in vacuum these equations degenerate: the coefficients in highest derivatives go to infinity. We would like to have a pair of equations which are non-singular and give the field components even in a perfectly conducting vacuum pipe. The relativistic charge in the limit v = c in perfectly conducting pipe does not have the longitudinal filed components. Hence the equations should be ones for the transverse field components. A possible choice could be to write equations for the radial components of electric and magnetic fields. However for higher order modes, m > 0, these equations have singular coefficients as well.We suggest to solve the well-posed problem for transverse components of mag-
FIG. 3 :
3One dimensional mesh and positions of the transverse magnetic field components.
[µ r H r,m r]| r=b = 0, ∂ ∂r [H ϕ,m r]| r=b = 0, and the equations for i = N can be written in form(17)with h ϕ,N+1 = h ϕ,N , h r,N+0.5 = 0. Hence we have to solve a linear system Mh = f , h = (h ϕ,1 , h ϕ,2 , ..., h ϕ,N , h r,0.5 , h r,1.5 , ..., h r,N+0.5 ) t ,
FIG. 4 :
4Reduction of seven band matrix to pentadiagonal form.
The new system allows for a direct solution with complexity O(N) [16], meaning that the solution time is proportional to the number of mesh points. If the last layer is infinite then we need an open boundary condition to truncate the matrix at r = b. If the last material has only uniaxial anisotropy like one considered in Section III then we can easily write such a condition. Indeed from the definition of the modified Bessel functions of the second type, the open boundary
If the last layer of the rectangular geometry is infinite and has only uniaxial anisotropy then the open boundary condition for the longitudinal field components read ∂ ∂y E z,m + k y,m E z,m = 0, ∂ ∂y H z,m + k µ y,m H z,m = 0, Combining them with Maxwell's equations (4) we can derive the open boundary conditions for the transverse components of the magnetic field in the rectangular case:
x,m , H y,m are solutions of the corresponding problem with magnetic or electric boundary condition at the symmetry plane. V. COMBINATION OF FIELD MATCHING AND FINITE-DIFFERENCE METHODS FOR ANISOTROPIC WAVEGUIDES The finite-difference method of the previous section allows treating the full anisotropy but it could also be time-consuming as it requires a mesh in the whole domain. In this Section we suggest a combination of the field matching technique and of the finite-difference method. Again we will start with a round geometry. In order to describe the method, let us consider example shown in Fig.5: the first and the third layers allow solutions FIG. 5: One dimensional mesh of combined method and positions of the transverse magnetic field components.
the magnetic field components (and their derivatives) H r,m (a − 1 ), H ϕ,m (a − 1 ), ∂ ∂r [H ϕ,m r]| r=a − 1 , ∂ ∂r [µ r H r,m r]| r=a − 1 . Here theIn order to find h r,−0.5 we use the second order approximation of the fourth boundary condition and Eq. (23) for i = 1. After a simple algebra we obtain: h r,−0.5 = − M N+1,1 h ϕ,1 + M N+1,2 h ϕ,2 + M N+1,N+4 h r,0.5 + M N+1,N+5 D r (r 1.5 − r −0.5 ) M N+1,N+3 + M N+1,N+5 , where M i, j are elements of matrix M in Eq. (22). Through excluding of h ϕ,0 , h ϕ,1 , h r,−0.5 , h r,0.5 from system Eq.(22) we obtain a matrix equation with reduced matrix M r of size 2N × 2N:
..., 2N, j = 1, ..., N, M r i, j = M i, j+4 , i = 1, ..., 2N, j = N + 1, ..., 2N, where f r 1 = f 1 − (M 1,1 h ϕ,0 + M 1,2 h ϕ,1 + M 1,N+4 h r,0.5 ), f r 2 = f 2 − M 2,2 h ϕ,1 , f r 0.5 = f 0.5 − (M N+1,1 h ϕ,0 + M N+1,2 h ϕ,1 + M N+1,N+3 h r,−0.5 + M N+1,N+4 h r,0.5 ), f r 1.5 = f 1.5 − (M N+2,2 h ϕ,1 + M N+2,N+4 h r,0.5 ).
FIG. 6 :
6Reduction of seven band matrix of combined method to upper triangular form.
the elements of the first column of matrix M FD 12 . The second column can be found from the solution of the same equations but with another boundary condition at r = a 1 : H r,m (a + 1 ) = 0, H ϕ,m (a + 1 ) = 1, ∂ ∂r [H ϕ,m r]| r=a + 1 = 0, ∂ ∂r [µ r H r,m r]| r=a + 1 = 0.
FIG. 7 :
7The longitudinal and the transverse wake potentials near the pipe axis as obtained by time-domain code ECHO2D (solid black line) and by frequency-domain code ECHO1D (grey dashed line). The gray dashed line shows the results obtained with field matching method as described in Section III. The solid line is obtained with time-domain code FIG. 8: The longitudinal wake and the real part of the longitudinal impedance for dielectric pipe at PITZ. The solid black lines show the results for isotropic case and the dashed grey line presents the result for anysotropic case.
the longitudinal wake potential and the real part of the longitudinal impedance are shown. The solid black lines show the results for the isotropic case and the dashed grey line presents the result for the anisotropic case when we have changed only the permittivity in radial direction, r = 6 0 . We see a clear shift in the modal frequencies for the anisotropic case. It cannot be treated with the field matching only. Here we have used methods described in Sections IV, V. The wave number k was sampled from 1 m −1 to 10 5 m −1 with step 0.2. The execution times for all methods are shown in
FIG. 9 :
9The longitudinal electric field component and the real part of the longitudinal impedance for anisotropic (solid black line) and isotropic (dashed gray line) rectangular structures. considered as well. As a next example we consider a Gaussian relativistic electron bunch with parameters of the Argonne wakefield accelerator in the sapphirebased rectangular accelerating structure [1, 3]. The rectangular structure has width 2w = 11mm, the anisotropic layer starts at a 0 = 1.5 mm and is closed by PEC at a 1 = 2.39 mm. The permittivities along main axes are: x = z = 9.4 0 , y = 11.5 0 . It corresponds to a frequency of 25.0 GHz of the accelerating mode of the structure. For comparison a waveguide with isotropic dielectric filling with = 10.45 0 corresponds to the base frequency of 24.23 GHz. The electron bunch with energy 15
and it can be reduced with the permutations (19) to the upper triangular matrix shown on the right. Hence the system requires only O(N) operations to solve it.In order to find matrix M FD 12 we need to solve the same equations but with 4 different sets of boundary conditions at r = a 1 . The boundary conditions at r = a 1 for the first problem readH r,m (a +
1 ) = 1, H ϕ,m (a +
1 ) = 0,
∂
∂r
[H ϕ,m r]| r=a +
1 = 0,
∂
∂r
[µ r H r,m r]| r=a +
1 = 0.
TABLE I :
IExecution time in seconds for different methods.
AcknowledgementsThe author thanks K.L.F. Bane, M. Dohlus, F. Lemery and G. Stupakov for helpful discussions. [3] I.L. Sheinman, Y.S. Sheinman, Wake fields in a rectangular dielectric-lined accelerating structure with transversal isotropic loading, arXiv:1703.04037, (2017).condition in the round geometry reads (see Eq. (9)) ∂ ∂r E z,m + K m r E z,m = 0, K m = m + rν r K m−1 (ν r r) K m (ν r r) , ∂ ∂r H z,m + K µ m r H z,m = 0, K µ m = m + rν µ r K m−1 (ν µ r r) K m (ν µ r r).We approximate these boundary condition on the one dimensional mesh with second order by finite differences[15]. The final matrix will have the same structure as in previous situation with the perfectly conducting boundary (seeFig.4).After numerical solution of the linear system (18) the longitudinal electric field component and the impedance can be found aswhere M is a non-square matrix of size 2N ×(2N +4). In order to reduce the number of the unknowns to 2N we will use the boundary conditions at r = a 1 and exclude It is easy to write the second order approximation of the first three equations (23) and obtain the expressions for h ϕ,0 , h ϕ,1 , h r,0.5 :h r,0.5 = B r , h ϕ,0 = B ϕ − (r 0.5 − r 0 )D ϕ , h ϕ,1 = B ϕ + (r 1 − r 0.5 )D ϕ .
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Calculation of wakefields in 2D rectangular structures. I Zagorodnov, K L F Bane, G Stupakov, Phys. Rev. ST Accel. Beams. 18104401I. Zagorodnov, K.L.F. Bane, G. Stupakov, Calculation of wakefields in 2D rectangular structures, Phys. Rev. ST Accel. Beams 18, 104401 (2015).
A W Chao, Physics of Collective Beam Instabilities in High Energy Accelerators. New YorkWileyA.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wi- ley, New York, 1993).
A A Samarskii, The Theory of Difference Schemes. New YorkMarcel Dekker, IncA.A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, Inc., New York, 2001).
T A Davis, Direct Methods for Sparse Linear Systems. SIAM, Philadelphia, Pa, USAT.A. Davis, Direct Methods for Sparse Linear Systems (SIAM, Philadelphia, Pa, USA, 2006).
Experimental demonstration of ballistic bunching with dielectric-lined waveguides at PITZ. F Lemery, Proceedings of International Particle Accelerator Conference. International Particle Accelerator ConferenceCopenhagen, Denmark122F. Lemery et al, Experimental demonstration of ballistic bunching with dielectric-lined waveguides at PITZ, in Proceedings of International Particle Accelerator Conference 2017, (Copenhagen, Denmark,2017) p. WEPAB122.
|
[] |
[
"On the emergence of Lorentzian signature and scalar gravity",
"On the emergence of Lorentzian signature and scalar gravity"
] |
[
"F Girelli \nSISSA\nVia Beirut 2-434014TriesteItaly\n",
"S Liberati \nSISSA\nVia Beirut 2-434014TriesteItaly\n",
"L Sindoni \nSISSA\nVia Beirut 2-434014TriesteItaly\n",
"Infn Sezione \nSISSA\nVia Beirut 2-434014TriesteItaly\n",
"Di Trieste \nSISSA\nVia Beirut 2-434014TriesteItaly\n"
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[
"SISSA\nVia Beirut 2-434014TriesteItaly",
"SISSA\nVia Beirut 2-434014TriesteItaly",
"SISSA\nVia Beirut 2-434014TriesteItaly",
"SISSA\nVia Beirut 2-434014TriesteItaly",
"SISSA\nVia Beirut 2-434014TriesteItaly"
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[] |
In recent years, a growing momentum has been gained by the emergent gravity framework. Within the latter, the very concepts of geometry and gravitational interaction are not seen as elementary aspects of Nature but rather as collective phenomena associated to the dynamics of more fundamental objects. In this paper we want to further explore this possibility by proposing a model of emergent Lorentzian signature and scalar gravity. Assuming that the dynamics of the fundamental objects can give rise in first place to a Riemannian manifold and a set of scalar fields we show how time (in the sense of hyperbolic equations) can emerge as a property of perturbations dynamics around some specific class of solutions of the field equations. Moreover, we show that these perturbations can give rise to a spin-0 gravity via a suitable redefinition of the fields that identifies the relevant degrees of freedom. In particular, we find that our model gives rise to Nordström gravity. Since this theory is invariant under general coordinate transformations, this also shows how diffeomorphism invariance (albeit of a weaker form than the one of general relativity) can emerge from much simpler systems.
|
10.1103/physrevd.79.044019
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[
"https://arxiv.org/pdf/0806.4239v2.pdf"
] | 118,624,257 |
0806.4239
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50ef014eb1a425371798fb0bbe74a1b226a7e264
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On the emergence of Lorentzian signature and scalar gravity
19 Feb 2009 (Dated: February 19, 2009)
F Girelli
SISSA
Via Beirut 2-434014TriesteItaly
S Liberati
SISSA
Via Beirut 2-434014TriesteItaly
L Sindoni
SISSA
Via Beirut 2-434014TriesteItaly
Infn Sezione
SISSA
Via Beirut 2-434014TriesteItaly
Di Trieste
SISSA
Via Beirut 2-434014TriesteItaly
On the emergence of Lorentzian signature and scalar gravity
19 Feb 2009 (Dated: February 19, 2009)PACS numbers: 0460-m; 0450Kd Keywords: emergent gravity, scalar gravity
In recent years, a growing momentum has been gained by the emergent gravity framework. Within the latter, the very concepts of geometry and gravitational interaction are not seen as elementary aspects of Nature but rather as collective phenomena associated to the dynamics of more fundamental objects. In this paper we want to further explore this possibility by proposing a model of emergent Lorentzian signature and scalar gravity. Assuming that the dynamics of the fundamental objects can give rise in first place to a Riemannian manifold and a set of scalar fields we show how time (in the sense of hyperbolic equations) can emerge as a property of perturbations dynamics around some specific class of solutions of the field equations. Moreover, we show that these perturbations can give rise to a spin-0 gravity via a suitable redefinition of the fields that identifies the relevant degrees of freedom. In particular, we find that our model gives rise to Nordström gravity. Since this theory is invariant under general coordinate transformations, this also shows how diffeomorphism invariance (albeit of a weaker form than the one of general relativity) can emerge from much simpler systems.
I. INTRODUCTION
In spite of being the first force of Nature to be understood in physical terms, gravity is somehow still a riddle for physicists. Not only it keeps evading a full quantum description as well as any form of unification with the other interactions, it also puzzles us with profound questions and unexpected features. We will not attempt here to present a complete list of these startling aspects of gravitation theory, but we can recall, for example, the surprising connection between gravity and thermodynamics associated to black hole physics [1,2,3] as well as the deep questions associated to the nature of inertia and time [4,5,6].
In recent years, a new approach to these old problems has been gaining momentum and many authors have been advancing the idea that gravity could all in all be an intrinsically classic/large scale phenomenon similar to a condensed matter state made of many atoms [7]. In this sense gravity would not be a fundamental interaction but rather a large scale/numbers effect, something emergent from a quite different dynamics of some elementary quantum objects. In this sense, many examples can be brought up, starting from the causal set proposal [8], passing to group field theory [9] or the recent quantum graphity models [10] and other approaches (see e.g. [11]).
All these models and many others share a common scheme: they consider a fundamental theory which is not General Relativity and examine, using different techniques often borrowed from condensed matter physics, how space, time and their dynamics could emerge in some regime. In this sense a leading inspirational role also been played by another stream of research which goes under the name of "analogue models of gravity" [12]. These are condensed matter systems which have provided toy models showing how at least the concept of a pseudo-Riemannian metric and Lorentz invariance of matter equations of motion can be emergent. For example, non-relativistic systems which admit some hydrodynamics description can be shown to have perturbations (phonons) whose propagation is described, at low energies, by hyperbolic wave equations on an effective Lorentzian geometry [12]. While these models have not provided so far also an analogy of emergent gravitational dynamics equations they do have provided a new stream of ideas about many other pressing problems in gravitation theory (see for example recent works on the origin of the cosmological constant in emergent gravity [13]).
In this paper we shall try to further advance in the direction of a consistent emergent gravity picture by proposing a toy model that will show some of the salient features that would be desirable to see in an eventual full theory of emergent gravity. In particular we shall focus here on the possible dynamical origin of time 1 (in the sense of pseudo-Riemannian signature of the metric) as well as of another crucial aspect of gravitation theory, i.e. diffeomorphism invariance (see however the discussion in the conclusion). It is perhaps important to stress here, that a Lorentzian signature has been considered by some authors as a necessary, but not sufficient, condition for the emergence of time. This is because these authors associate to the latter not just a special signature, but also to a dimension with a univocal direction (arrow of time), which is of course much less easy to rigorously characterize [15]. Missing a general agreement on this issue, we will stick here to the more technical language of Lorentzian signature, although we think that this feature is the very essence of the nature of time.
In building up our model we shall make the conjecture that the dynamics of some elementary quantum objects (what one could call the "atoms of spacetime") is such to produce, in some semi-classical or large number limit (hydrodynamic limit, or condensation...), a Riemannian manifold (essentially R 4 equipped with the trivial metric δ µν ), a set of scalar fields Ψ i , and their Lagrangian L. Hence we do not assume a priori any notion of time. Instead we shall consider, largely following the intuition gained from condensed matter analogues of gravity, the possibility that Lorentzian dynamics can emerge as a property of the equations associated to perturbations around some solutions of the equations of motion. These perturbations will be associated to the emergent gravity and matter degrees of freedom which will characterize our gravitational dynamics.
The plan of the paper will be the following. In the first section, we shall look for a specific Lagrangian so that the perturbations around some solutions of the equations of motion will see an effective pseudo-Riemannian metric, even though the fundamental theory is Euclidean. In other words, we shall be looking for specific properties of the Lagrangian so that time can emerge.
In the second section, we shall consider the perturbations around a specific solution of the equations of motion (e.g. such that these perturbations propagate on Minkowski spacetime), and identify among them the degrees of freedom corresponding to matter and to gravity. More precisely, we shall show how the equations of motion for the perturbations can be reinterpreted as the Einstein-Fokker equations of motion. These latter are historically the first equations of motion for a relativistic theory of gravity, written in a diffeomorphism invariant way. They describe a relativistic scalar gravity theory known as Nordström gravity. Our construction identifies therefore a possible mechanism to obtain diffeomorphisms invariant equations of motion for gravity, in an emergent framework.
II. TIME EMERGENCE
As explained in the Introduction, we consider that the fundamental unknown theory gives rise in some large number limit to simple structures such as R 4 equipped with the Euclidean metric δ µν , and a set of scalar fields Ψ i (x µ ), i = 1, ..., N (x µ ∈ R 4 ) with their Euclidean Lagrangian L. Since we do not know this fundamental theory, we choose such Lagrangian to be of the simple shape 2 L = F (X 1 , ..., X N ).
(1)
with X i = δ µν ∂ µ Ψ i ∂ ν Ψ i .
It is easy to see that this Lagrangian is invariant under the Euclidean group ISO(4). The equations of motion are then simply for a given field Ψ i
∂ µ ∂F ∂X i ∂ µ Ψ i = 0 = Σ j ∂ 2 F ∂X i ∂X j ∂ µ X j ∂ µ Ψ i + ∂F ∂X i ∂ µ ∂ µ Ψ i .(2)
Let us now consider a specific solution of the above equations of motion, ψ i and perturbations ϕ i around it. For Ψ i = ψ i + ϕ i , the kinematic term X i becomes then
X i → X i + δX i , with X i = δ µν ∂ µ ψ i ∂ ν ψ i and δX i = 2∂ µ ψ i ∂ µ ϕ i + ∂ µ ϕ i ∂ µ ϕ i .(3)
We intend now to identify some specific F such that the Lagrangian for the perturbations ϕ i is invariant under the Poincaré group ISO(3, 1). To determine the Lagrangian for the perturbations ϕ i , we expand (1) using (3).
F (X 1 , .., X N ) → F (X 1 , .., X N ) + j ∂F ∂X j X δX j + 1 2 jk ∂ 2 F ∂X j ∂X k X δX j δX k + 1 6 jkl ∂ 3 F ∂X j ∂X k ∂X l X δX j δX k δX l + ...(4)
The first term F (X 1 , .., X N ) is the Lagrangian for the classical solution ψ i . The second term, the one linear in δX j , contains a term linear in ∂ µ ϕ i , which is zero on shell. We can also identify the quadratic contribution for ∂ µ ϕ k ∂ ν ϕ k :
for k = l, ∂ µ ϕ k ∂ ν ϕ l 2 ∂ 2 F ∂X k ∂X l X ∂ µ ψ k ∂ ν ψ l ,(5)for k = l, ∂ µ ϕ k ∂ ν ϕ k ∂F ∂X k X δ µν + 1 2 ∂ 2 F (∂X k ) 2 X ∂ µ ψ k ∂ ν ψ k .(6)
The contribution (5) introduces some mixing between fields in the kinematic term. To simplify the analysis, we demand that they cancel, which puts a constraint on the choice of F , i.e.
∂ 2 F ∂X k ∂X l X = 0, if k = l. A specific solution is then F (X 1 , .., X N ) = f 1 (X 1 ) + ... + f N (X N ).(7)
We can identify in (6) the effective or emergent metrics 3 for each field ϕ k , (taking into account (7))
g µν k ≡ df k dX k X k δ µν + 1 2 d 2 f k (dX k ) 2 X k ∂ µ ψ k ∂ ν ψ k .(8)
Since a priori f i = f j and ψ i = ψ j if i = j, we are dealing with a multi-metric structure: each field sees its own metric. However, we can enforce a mono-metric structure by constraining the solution ψ k and the derivatives of f k at X k to be independent of k
f k = f, ψ k = ψ, ∀k.(9)
So far we have just shown that the perturbations around a solution of the field equations on a Riemannian manifold can propagate, for suitably chosen Lagrangians, on an effective geometry which is not the fundamental one, δ µν , but rather a rank 2 tensor constructed from it and partial derivatives of the chosen background solution. Note that, in order for this to be possible, it was crucial to have a starting Lagrangian with non-canonical kinetic terms as it can be clearly evinced by the second contribution to the metrics in equation (8). As a next step, we show now how for some solutions of the equations of motion, such effective metric can be of pseudo-Riemannian form. In fact, we can even ask that the metric (8) is the Minkowski metric η µν . This will put some constraints on the derivative of f , evaluated at X = ∂ µ ψ∂ µ ψ.
In order to do so, we shall need to specify a particular solution,ψ, of the equations of motion. Let us take it to be an affine function of the coordinates,ψ = α µ x µ + β. It is easy to check that this is indeed a solution of our field equations (2). Moreover, thanks to the SO(4) symmetry, we can always make a rotation such that
ψ = αx 0 + β.(10)
The choice of the coordinate x 0 is completely arbitrary, what only matters is that there is one coordinate which is pinpointed. Finally, we ask for the metric to have the signature (−, +, +, +). This puts some constraint on the value of the derivatives of f
df dX X + 1 2 d 2 f (dX) 2 X ∂ 0ψ ∂ 0ψ < 0, df dX X + 1 2 d 2 f (dX) 2 X ∂ aψ ∂ aψ > 0, a = 1, 2, 3(11)
which using (10) imply
df dX X + α 2 2 d 2 f (dX) 2 X < 0, df dX X > 0.(12)
Note that, due to the choice of a solution of the form (10), the conditions (11) are not only implying a pseudo-Riemannian signature but also the constancy of the metric components, which hence can be easily rescaled so to take the familiar Minkowskian form diag(−1, +1, +1, +1). Of course, there are many possible choices of f (X) and α which can fulfill the above requirements. For example, we can pick up the specific combination
f (X) = −X 2 + X, 1 3 < α 2 < 1 2 .(13)
However, in what follows we should not make use of any particular form of f (X) and α and simply assume that they are such that (12) are satisfied.
To summarize, since g µν k ≡ η µν , ∀k, the (free) perturbations ϕ i are propagating on a Minkowski space, even though the fundamental theory is Euclidean (c.f. (1)). At this point few remarks are in order.
So far, our theory does not posses any fundamental speed scale. This is natural since the fundamental theory is Euclidean. At this level, there is no coordinate with time dimension and therefore one cannot define a constant with speed dimension. The invariant speed c, which will relate the length x 0 to an actual time parameter t, could be determined experimentally by first introducing a coordinate with time dimension (as it would be natural to do given the hyperbolic form of the equations of motion for the perturbations) and then by defining c as the signal speed associated to light cones in the effective spacetime 4 .
Second, a comment is due about our choice of the background solution around which we have considered the dynamics for perturbations. It is obvious that within our model this choice is arbitrary. It simply shows that there are some background solutionsψ for which a pseudo-Riemannian metric can emerge. Obviously, different background solutions could lead to alternative metrics, e.g. one could also obtain the Euclidean metric δ µν (for example if ψ is constant), a degenerate metric or more complicated structures according to the possible solutions ψ. While it is conceivable that in a more complicate model we could have some mechanism for selecting the specific background solution that leads to an emergent Lorentzian signature, it is not obvious at all that such a feature should be built in the emergent theory. In fact, one generally minimizes an energy functional to select the ground state of the theory. However, when looking at Lorentzian signature emergence starting from an Euclidean set up as in our model, there is no initial notion of time and hence no energy functional to minimize. It is therefore unclear how a ground state could be selected from within the emergent system.
On the other hand, it is also conceivable that the actual background solution in which the initial system of fields (1) emerges from the fundamental (pre-manifold) theory, can be depending on the conditions for which the "condensation" of the fundamental objects takes place. In this sense, the right ground state or background solution would be selected from minimizing some functional defined at the level of the atoms of space-time. To use an analogy, the same fundamental constituents, e.g. carbon atoms, can form very different materials, diamond or graphite, depending on the external conditions during the process of formation. Similarly, in a Bose-Einstein condensation the characteristics of the background solution (the classical wave function of the condensate), such as density and phase, are determined by physical elements (like the shape of the EM trap or the number and kind of atoms involved) which pre-exist the formation of the condensate.
In conclusion, we have identified the fundamental Lagrangian so that the perturbations ϕ i have a kinematic term determined by the Minkowski metric.
L eff (ϕ 1 , ...ϕ N ) = i η µν ∂ µ ϕ i ∂ ν ϕ i .(14)
In this sense, we have a toy-model for the emergence of the Poincaré symmetries. This construction can be seen as a generalization of the typical situation in analogue models of gravity [12] where one has Poincaré symmetries emerging from fundamental Galilean symmetries [12]. However, let us stress that in our case no preferred system of reference is present in the underling field theory given that the fundamental Lagrangian is endowed with a full Euclidean group ISO(4). Moreover, the emergence of a pseudo-Riemannian metric is in our model free of the usual problems encountered in the context of continuous signature change (e.g. degenerate metrics) 5 given that the former arises as a feature of the dynamics of perturbations around some solution of the equations of motion. Similarly one can see that the invariance under Lorentz transformations is only an approximate property of the field equations (as usual for emergent systems), valid up to some order in perturbation theory. In particular, if we analyze the third order contribution in (4) we get 6
∂ α ϕ k ∂ β ϕ k ∂ γ ϕ k d 2 f (dX k ) 2 X ∂ α ψδ βγ + 1 6 d 3 f (dX k ) 3 X (∂ α ψ∂ β ψ∂ γ ψ) .(15)
This contribution is clearly not Lorentz invariant if the solution ψ pinpoints a specific direction, as for example when the Minkowski metric is emergent. As a matter of fact our theory will show aether like effects beyond second order. So far, we have hence generalized and extended results familiar to the analogue gravity community. However, as said, a typical drawback of analogue gravity models is related to the fact that they show only the emergence of a background Lorentzian geometry while they are unable to reproduce a geometrodynamics of any sort. In what follows, we shall show that our model overcomes this drawback and indeed is able to describe the emergence of a theory for scalar gravity. This theory will come out to be the only known other theory of gravitation, apart from General Relativity, which satisfy the strong equivalence principle [18], i.e. Nordström gravity (for details see the Appendix).
III. EMERGENCE OF NORDSTRÖM GRAVITY
In this section, we describe how we can recover a relativistic scalar gravity theory from a Lagrangian of the type (1), when ground state is such that the perturbations are living (at the lowest order in perturbation theory) in a Minkowski spacetime. So, let us start from the truncated Lagrangian for the perturbations (14) that we obtained in the previous section. This Lagrangian can simply be rewritten in terms of the (real) multiplet ϕ = (ϕ 1 , ..., ϕ N ) as
L eff (ϕ) = η µν (∂ µ ϕ) T (∂ ν ϕ).(16)
This system has a global O(N ) symmetry which has emerged as well from the initial Lagrangian (1). It is hence quite natural to rewrite the multiplet ϕ by introducing an amplitude characterized by a scalar field Φ(x) and a multiplet φ(x) with N components such that 7
ϕ 1 . . . ϕ N = Φ φ 1 . . . φ N , with |φ| 2 ≡ i φ 2 i = ℓ 2 .(17)
ℓ is an arbitrary length parameter to keep the dimension right. In particular, Φ is dimensionless and φ has the dimension of a length. Φ is the field invariant under O(N ) transformations, whereas φ does transform under O(N ). As we shall see, this field redefinition will provide us the means to identify gravity and matter degrees of freedom. The Lagrangian for the perturbations (16) reads now as 8
L eff (ϕ 1 , ...ϕ N ) → L eff (Φ, φ 1 , ...φ N ) = ℓ 2 η µν ∂ µ Φ∂ ν Φ + i Φ 2 η µν ∂ µ φ i ∂ ν φ i + λ(|φ| 2 − ℓ 2 ),(18)
where λ is a Lagrange multiplier. We recognize in particular the action for a non-linear sigma model given in terms of the fields φ i . The associated equations of motion are
η µν (ℓ 2 ∂ µ ∂ ν Φ − Φ i ∂ µ φ i ∂ ν φ i ) = 0,(19)η µν (2∂ µ Φ∂ ν φ i + Φ 2 ∂ µ ∂ ν φ i + 1 ℓ 2 ∂ µ φ j ∂ ν φ k δ jk φ i ) = 0,(20)|φ| 2 − ℓ 2 = 0.(21)
If we introduce the (conformally flat) metric
g µν (x) = Φ 2 (x)η µν ,(22)
the equations of motion (20) can be simply rewritten as
( √ −g) −1 ∂ µ ( √ −gg µν ∂ ν φ i ) + 1 ℓ 2 g µν ∂ µ φ j ∂ ν φ k δ jk φ i = g φ i + 1 ℓ 2 g µν ∂ µ φ j ∂ ν φ k δ jk φ i = 0,(23)
where we have introduced the d'Alembertian g for the metric g and used that √ −g = Φ 4 and g µν = Φ −2 η µν . (Incidentally, let us note that equation (20) can be rewritten in the form (23) using the metric redefinition (22) only in four dimensions.) To be consistent, the change of variable Φ → g µν should be completed with the constraint that g µν is conformally flat, that is
C αβγδ (g) = 0,(24)
where C αβγδ is the Weyl tensor. Eq. (23) suggests that the gravitational degree of freedom should be encoded in the scalar field Φ, whereas matter should be encoded in the φ i . We are therefore aiming at a scalar theory of gravity with actions:
S eff = dx 4 √ −η L eff = S grav + S matter ,(25)S grav = ℓ 2 dx 4 √ −η η µν ∂ µ Φ∂ ν Φ,(26)S matter = dx 4 √ −η i Φ 2 η µν ∂ µ φ i ∂ ν φ i + λ(|φ| 2 − ℓ 2 ) ,(27)
where we have explicitly written the volume element √ −η = 1 so to make clear that these actions are given in flat spacetime.
It is easy to see that the very same actions can be recast in the form of actions in a curved spacetime endowed with the metric (22). In particular for the matter action in (27) one has
S matter = dx 4 i Φ 2 η µν ∂ µ φ i ∂ ν φ i + λ(|φ| 2 − ℓ 2 ) = √ −gdx 4 i g µν ∂ µ φ i ∂ ν φ i + λ ′ (|φ| 2 − ℓ 2 ) ,
where we have suitably rescaled the Lagrange multiplier to λ ′ . This allows to construct the stress-energy tensor T µν for the non-linear sigma model, and its trace T with respect to the metric g:
T µν = 2 √ −g δS matter δg µν = i ∂ µ φ i ∂ ν φ i − 1 2 g µν (g αβ ∂ α φ i ∂ β φ i ) , T = g µν T µν = −Φ −2 i η µν ∂ µ φ i ∂ ν φ i .
Finally, the above result, together with the recognition that the Ricci scalar R, associated to the metric g µν , can be written as R = −6 η Φ/Φ 3 , allows us to rewrite Eq. (19) as the Einstein-Fokker equation 9
η Φ = 1 ℓ 2 η µν Φ i ∂ µ φ i ∂ ν φ i ⇔ R = 6 ℓ 2 T.(28)
In summary, we can gather together the equations of motion (23), (24), (28), obtained by introducing the metric (22), we have
R = 6 ℓ 2 T, C αβγδ = 0. (29) g φ i + 1 ℓ 2 g µν ∂ µ φ j ∂ ν φ k δ jk φ i = 0, |φ| 2 − ℓ 2 = 0.(30)
We recognize the equations of motion as those for Nordström gravity 9 The very same equation could be derived from the action (25) if one also notices that the gravitational action (26) can be rewritten as well in a curved spacetime form by a simple integration by parts and the addition of a set of Lagrange multipliers implementing the vanishing of the Weyl tensor.
R = 24πG N T, C αβγδ = 0,(31)
coupled to a non-linear sigma model. Indeed, the rewriting of (19)-(21) into the form (29)-(30), is a special case of the procedure suggested by Einstein and Fokker so to cast Nordström gravity in a geometrical form [27]. We see from the above equation that the Newton constant G N in our model has to be proportional to ℓ −2 . However, in identifying the exact relation between the two quantities, some care has to be given to the fact that the stress-energy tensors appearing respectively in equation (29) and equation (31) do not share the same dimensions. This is due to the fact that the fields φ i have the dimension of a length rather than the usual one of an energy. This implies that in order to really compare the expressions one has to suitably rescale our fields with a dimensional factor, Ξ, which in the end would combine with ℓ so to produce an energy, dim[ℓ Ξ] = energy. In particular, is easy to check that one has to assume 4πℓ 2 Ξ 2 ≡ E 2 Planck in order to recover the standard value of G N (assuming c as the observed speed of signals and as the quantum of action). As a final remark, we should stress that the scale ℓ is completely arbitrary within the emergent system and in principle should be derived from the physics of the "atoms of spacetime" whose large N limit gives rise to (1).
Accidentally, the above discussion also shows that, once the fields are suitably rescaled so to have the right dimensions, the constraint appearing in Eq.(30) is fixing the norm of the multiplet to be equal to the square of the Planck energy. This implies that the interaction terms in the aforementioned equation are indeed Planck-suppressed and hence negligible at low energy. This should not be a surprise, given that in the end ℓΞ is the only energy scale present in our model. It is conceivable that more complicate frameworks, possibly endowed with many dimensional constants, will introduce a hierarchy of energy scales and hence break the degeneracy between the scale of gravity and the scale of matter interactions.
IV. REMARKS AND CONCLUSIONS
In the first section, we have considered fields that live in a Euclidean space, and showed that there exists a class of Lagrangians such that the perturbations around some classical solutionsψ propagate in a Minkowski spacetime. In this caseψ is essentially picking up a preferred direction, so that we have a spontaneous symmetry breaking of the Euclidean symmetry. The apparent change of signature is free of the problems usually met in signature change frameworks since the theory is fundamentally Euclidean. Lorentz symmetry is only approximate, and in this sense it is emergent.
The main lesson we want to emphasize here is that Lorentzian signature can emerge from a fundamental Euclidean theory and this process can in principle be reconstructed by observers living in the emergent system. In fact, while from the perturbations point of view it is a priori difficult to see the fundamental Euclidean nature of the world, this could be guessed from the fact that some Lorentz symmetry breaking would appear at high energy (in our case in the form of a non-dynamical ether field).
In the second part, using a natural field redefinition, we have identified from the perturbations ϕ i , a scalar field Φ encoding gravitational degrees of freedom and a set of scalar fields φ i (a non-linear sigma model) encoding matter. In this sense, gravity and matter are both emergent at the same level 10 . This approach is then rather different from the one of analogue models of gravity where one usually identifies the analogue of the gravitational degrees of freedom with the "background" fields, i.e. the condensate or the solution ψ of the equations of motion. Indeed, following this line of thought in looking for a theory of gravitational dynamics, we would be led to require that the fundamental field theory (1) must be endowed with diffeomorphisms invariance from the very start -the symmetries of the background are identical by construction to the ones of the fundamental theory. This would imply that one would have to obtain gravity from a theory which is already diffeomorphisms invariant and hence most probably with a form very close to some known theory of gravitation. For these reasons, we do expect that if an emergent picture is indeed appropriate for gravitation, then it should be of the sort presented here, with both matter and gravity emerging at the same level 11 .
In particular, this allows not only for an emergent local Lorentz invariance for the perturbations dynamics but it leads as well to an emergent diffeomorphisms invariance. In fact, we saw how the equations of motion (19) and (20) could be rewritten in a completely equivalent way using a conformally flat metric (22). Most noticeably, they can be rewritten in an evidently diffeomorphisms invariant form, from the point of view of "matter fields observers" 12 . In agreement with the fact that diffeomorphisms invariance is emergent in our system, it can be noted that the cubic contribution (15) ends up breaking it at the same level it breaks Lorentz invariance. Moreover, our derivation obviously holds for small perturbations ϕ i , and hence small Φ, implying that in our framework one would predict strong deviations from the weak field limit of the theory whenever the gravitational field becomes very large. We think this is an intriguing aspect of this proposal which might deserve further investigation.
Coming back to the emergence of diffeomerphism invariance, we note that Nordström gravity is also a nice framework for discussing the subtle distinction between background independence and diffeomorphisms invariance [21]. We call background some geometrical degrees of freedom that are not dynamical. For example, in General Relativity the topology of the manifold and its dimension, or the signature of the metric, can be considered as (trivial) background quantities. We can therefore have some specific background structures while still having diffeomorphisms invariance. Nordström gravity is encoded in conformally flat metrics. If one considers fields which are conformally coupled to the metric (such as the electromagnetic field), these fields only see the metric η µν which is of course not dynamical. The Minkowski metric can be see then as a background structure, this is what one may call a "prior geometry" (e.g. see [19]). One may hence say that diffeomorphism invariance is somewhat of a weaker form in Nordström gravity with respect the one present in general relativity.
In particular, while the essence of diffeomorphism invariance in GR is encoded in the associated Hamiltonian constraints, these are not defined in the present formulation of Nordström gravity. Furthermore, in the most general implementations of Norström theory, quantities can be built which manifestly include the background structure η µν and hence are not diffeomorphism invariant. However, within our model, the prior geometry cannot be detected 13 . Indeed, in order to detect the Minkowski background, one should be able to propose a method to pinpoint the conformal factor Φ 2 in the relation g µν = Φ 2 η µν . However, a careful analysis shows that this is actually impossible. Let us elaborate on this point. If we perform a conformal transformation, x µ →x µ (x), the equations of motions associated to (16) are transforming like
η ϕ i = 0 → η ϕ i = 0,(32)
where η andη are two different Minkowski metrics related by some conformal factor λ(x). Therefore, η andη are indistinguishable, due to conformal invariance the equations of motion for ϕ i . Hence, what appears to be a background structure, namely η µν , is ambiguously defined, and the coordinates x µ in which the equations of motion for the fields ϕ i are written have no operational meaning, they are mere labels. Furthermore, this ambiguity in the definition of what would be called a background structure implies an ambiguity on the definition of the conformal factor relating the physical metric to the would-be background structure. In this sense, within this very specific implementation of the model which has conformal invariance, there is no Minkowski geometry as a background. There is a background structure, which is the conformal structure of Minkowski spacetime. This is a mild limitation of our simple toy model as a diffeomorphism invariant, background independent system. Of course, the above discussion holds only at the lowest order in the fields ϕ i . As previously discussed, higher orders in perturbation theory will produce terms like (15) producing a breaking of the conformal symmetry and hence the appearance of the background structures, i.e. the Euclidean space and the ∂ µψ which have selected the timelike direction.
Finally, Nordström gravity is only a scalar gravity theory, which has been falsified by experiments (e.g. the theory does not predict the bending of light). In order to obtain a more physical theory, in particular General Relativity, one should surely look for more complicated emergent Lagrangians than (1). Of course, one would in this case aim to obtain the emergence of a theory characterized by spin-2 gravitons (while in Nordström theory the graviton is just a scalar). This would open a door to a possible conflict with the so called Weinberg-Witten theorem [22]. However, there are many ways in which such a theorem can be evaded (see e.g. [23]) and in particular one may guess that analogue models inspired mechanisms like the one discussed here will generically lead to Lagrangian which show Lorentz and diffeomorphism invariance only as approximate symmetries for the lowest order in the perturbative expansion.
It is unclear which sort of generalization may still lead to some viable gravitational theory from the perturbations dynamics. For example, the simple addition of a potential will in general prevent the selection of a preferred direction, except in regions where the potential is almost flat. Moreover, it would also spoil the metric interpretation of the theory. For example, the terms |ϕ| n for n ≥ 1 and = 4 cannot be rewritten as an interaction between the matter field fields φ living on the conformal metric Φ 2 η µν , when using the change of variables (17) (although it is interesting to note that a |ϕ| 4 term would give Nordström gravity with a cosmological constant). However, this "rigidity" of the model is most probably due to its simplicity: considering a more complex emergent field theory with fields such as spinors or tensors could possibly allow to have a preferred direction pinpointed while giving rise to more physical Lagrangians for the perturbations. We see this as a possible development of the research presented in this work.
APPENDIX A: NORDSTRÖM GRAVITY
Nordström gravity was a key step in the search for a relativistic theory of gravity [24,25,26]. It is a scalar gravity theory which is a natural generalization of Newtonian gravity. Indeed after the inception of Special Relativity in 1905, it was quickly realized that a relativistic theory for gravity was needed. The naive extension consisted into generalizing the Poisson equation to a relativistic equation, ∆Φ = ρ → Φ = ρ, but some ambiguities both for the Newton equation (the dynamics of matter in a gravitational field) and for the (relativistic) source for gravity, arose. In particular, this theory was not accounting for the characteristic non-linear behaviour of gravity which should be expected by the special relativistic equivalence of energy and mass.
After several attempts 14 , Nordström managed to propose an improved theory using the newly introduced stressenergy tensor as a source for gravity 15 . Einstein and Fokker then showed that Nordström's theory could be encoded in purely geometry manner [27] using the metric formalism. In the modern language, Nordström gravity (minimally coupled to a scalar field) can be summarized as follows
C αβγδ = 0, (A1) R = κT, (A2) ( g + m 2 )φ = 0,(A3)
where T is the trace of the stress energy tensor, defined with respect to g µν , for the matter field φ, R is the Ricci scalar for g µν and κ is proportional to the Newton constant G N (indeed κ = 24πG N ). Equation (A1) encodes the fact that the Weyl tensor is zero, i.e. that the metric g µν is conformally flat and there is some coordinate system in which it takes the form
g µν = Φ 2 η µν ,(A4)
where η µν is the Minkowski metric. Equation (A2) is not the trace of the Einstein equations restricted to the conformal metric, due to the absence of a minus sign in front of the Ricci scalar. Equation (A3) is simply the equation of motion of the scalar field φ propagating in the metric g µν . This theory was introduced before Eddington's observation showing the bending of light. It is clear that since the metric is conformally flat, no bending of light can occur there, so that Nordström gravity is experimentally ruled out. Note however that a massless minimally coupled scalar field would see non trivial gravitational effects. Nordström theory predicts also an anomalous perihelion precession of Mercury, which disagrees in both sign and magnitude with the observed anomalous precession.
Nordström gravity is interesting as it shares many aspects of General Relativity, such as diffeomorphisms invariance and the strong equivalence principle [18]. It is also an example of the subtle distinction between diffeomorphisms invariant theories and background invariant theories as we discussed in the Discussion section. In general, scalar gravities have attracted much attention due to their simplicity and similitude with General Relativity (see e.g. [28] which also contains a complete list of references for scalar gravity theories).
For early, alternative work in this direction see also[14] and references therein.2 We could also consider a dependence on crossed terms of the kind h µν ∂µΨ i ∂ν Ψ j , however this is not changing the final result.
Actually, we show here the inverse metrics from which the actual metrics can be derived once invertibily conditions are imposed. In our case of interest, this will always be true.
Noticeably, a similar situation is encountered in the von Ignatowsky derivation of Special Relativity[16] where, given a list of simple axioms, one derives the existence of a universal speed, observer independent, which is not fixed a priori to be the speed of light but has to be identified via actual experiments.5 Bose-Einstein condensate analogue models of signature change events have indeed been considered in the literature together with the associated particle production (see e.g.[17] and references therein). These works are rather different from the one presented here as in that context the fundamental Lagrangian is non-relativistic and simply the emergent metric for the perturbations can have Lorentzian or Riemannian signature depending on the experimental possibility of changing at will the sign of the atomic interaction.
We are in the mono-metric case, so that F (X 1 , ..., X N ) = f (X 1 ) + ... + f (X N ), and ψ k = ψ, ∀k.7 Our field redefinition is the generalization of the so-called Madelung representation[12].8 We use the normalization condition |φ| 2 = ℓ 2 , which implies in particular P i φ i ∂µφ i = 0.
Note that while the purely gravitational sector of Nordström gravity could be reproduced by a single free massless scalar field, the emergence of gravity and matter requires of course the introduction of several independent degrees of freedom, i.e. many fields.11 Of course, we cannot exclude that a full fledged theory of gravity could emerge, together with the notion of manifold, in a single step from the eventual semiclassical/large number limit of the fundamental objects. In this case, however, we would still have a very different picture from the one envisaged in analogue models of gravity.
Following the standard hole argument (see e.g.[20]), this also implies that the coordinates xµ, used to parameterized our theory, do not have any physical meaning from the point of view of the φ i "matter observers". They are merely parameters.13 We want to thank S. Sonego for discussions on this point.
See[25] for all references and some very interesting historical summary of the different steps that prepared the advent of General Relativity.15 In Nordström's (second) formulation, the relativistic alter-ego of the Poisson equation is Φ η Φ = Tη, where Tη = Tµν η µν and Tµν is the matter stress-energy tensor defined with respect to the Minkowski metric. This formulation is not in accordance to the modern formulation encoding gravitational degrees of freedom in the metric.
ACKNOWLEDGMENTSThe authors wish to thank S. Sonego, T. Sotirou, M. Visser and S. Weinfurtner for useful remarks and critical
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In this paper we discuss open problems concerning L 2 -invariants focusing on approximation by towers of finite coverings. Date: January, 2015 Comment 1: Has to be updated. 2000 Mathematics Subject Classification. 59Q99,22D25,46L99, 58J52.
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29 Jan 2015
29 Jan 2015SURVEY ON APPROXIMATING L 2 -INVARIANTS BY THEIR CLASSICAL COUNTERPARTS: BETTI NUMBERS, TORSION INVARIANTS AND HOMOLOGICAL GROWTH LÜCK, W.and phrases L 2 -invariantsapproximationhomological growth 1
In this paper we discuss open problems concerning L 2 -invariants focusing on approximation by towers of finite coverings. Date: January, 2015 Comment 1: Has to be updated. 2000 Mathematics Subject Classification. 59Q99,22D25,46L99, 58J52.
Introduction
We want to study in this paper the following general situation:
Setup 0.1. Let G be a (discrete) group together with a descending chain of subgroups
G = G 0 ⊇ G 1 ⊇ G 2 ⊇ · · · (0.2) such that G i is normal in G, the index [G : G i ] is finite and i≥0 G i = {1}.
Let p : X → X be a G-covering. Put X[i] := G i \X.
We obtain a [G : G i ]-sheeted covering p[i] : X[i] → X. Its total space X[i] inherits the structure of a finite CW -complex, a closed manifold or a closed Riemannian manifold respectively if X has the structure of a finite CW -complex, a closed manifold or a closed Riemannian manifold respectively.
Let α be a classical topological invariant such as the Euler characteristic, the signature, the nth Betti number with coefficients in the field Q or F p , torsion in the sense of Reidemeister or Ray-Singer, the minimal number of generators of the fundamental group, the minimal number of generators of the nth homology group with integral coefficients, or the logarithm of the cardinality of the torsion subgroup of the nth homology group with integral coefficients. We want to study the sequence
α(X[i]) [G : G i ] i≥0 .
Problem 0.3 (Approximation Problem).
(1) Does the sequence converge?
(2) If yes, is the limit independent of the chain? (3) If yes, what is the limit?
The hope is that the answer to the first two questions is yes and the limit turns out to be an L 2 -analogue α (2) of α applied to the G-space X, i.e., one can prove an equation of the type Here N (G) stands for the group von Neumann algebra and is a reminiscence of the fact that the G-action on X plays a role. Equation (0.4) is often used to compute the L 2 -invariant α (2) (X; N (G)) by its finite-dimensional analogues α(X[i]). On the other hand, it implies the existence of finite coverings with large α(X[i]), if α (2) (X; N (G)) is known to be positive.
For some important invariants α one can prove (0.4), for instance for α the Euler characteristic, the signature or the n-th Betti number with rational coefficients. In other very interesting cases Problem 0.3 and the equality (0.4) are open, and hence there is the intriguing and hard challenge to find a proof. Here we are thinking of α to be one of the following invariants:
• the n-th Betti number b n (X[i]; F p ) of X[i] with coefficients in the field F p for a prime p; • the minimal number of generators d(G i ) or the deficiency def(G i ) of G i = π 1 (X[i]), if X is contractible; • Reidemeister or Ray-Singer torsion ρ an (X[i]) if X is a closed Riemannian manifold; • the logarithm of the cardinality of the torsion in the n-th singular homology with integer coefficients ln tors H n (X[i]) , if X is an aspherical closed manifold and X its universal covering.
Here are two highlights of open problems which will be treated in more detail later in the manuscript. (4) If d = 2n + 1 is odd, we get for the L 2 -torsion (−1) n · ρ (2) an M ≥ 0; If d = 2n + 1 is odd and M carries a Riemannian metric with negative sectional curvature, we have (−1) n · ρ (2) an M > 0; The earliest reference, where a version of Problem 0.3 appears, is to our knowledge Kazhdan [48], where the inequality lim sup i→∞ bn(X[i];F ) [G:Gi] ≤ b (2) n (X; N (G)) for X a closed manifold is discussed, see also Gromov [46,pages 13,153].
Commencing with Section 12 we will drop the condition that [G : G i ] is finite. We assume that the reader is familiar with basic concepts concerning L 2 -Betti numbers and L 2 -torsion. More information about these notions can be found for instance in [66,67].
Most of the article consists of surveys of open problems and known results. There are a few new aspects in this manuscript:
• In Section 4.1 we introduce the truncated Euler characteristic which leads to a high-dimensional version of the rank gradient and to Question 4.2 about asymptotic Morse inequalities; • In Theorem 16. 16 we discuss a strategy to prove the Approximation Conjecture for Fuglede-Kadison determinants under a uniform logarithmic estimate; • The vanishing of the regulators on the homology comparing the inner product coming from a Riemannian metric with the one coming from a triangulation in the L 2 -acyclic case, see Theorem 7.7, or more generally Theorem 14.10. • The question whether L 2 -torsion can be approximated by integral torsion depends only on the QG-chain homotopy type of a finite based free L 2acyclic ZG-chain complex, see Lemma It is known that it is multiplicative under finite coverings, but the proof is however more involved than the one for the Euler characteristic. It follows for instance from Hirzebruch's Signature Theorem, see [47], or Atiyah's L 2 -index theorem [6, (1.1)] in the smooth case, for closed topological manifolds see Schafer [88,Theorem 8].
Since this implies sign(X) = sign(X[i]) [G: Gi] , the answer in this case is yes for all three questions appearing in Problem 0. 3 1.3. Signature of finite Poincaré complexes. The next level of generality is to pass from a topological manifold to a finite Poincaré complex whose definition is due to Wall [97]. For them the signature is still defined if the dimension is divisible by 4. There are Poincaré complexes X for which the signature is not multiplicative under finite coverings, see [85,Example 22.28], [97,Corollary 5.4.1]. Hence the situation is more complicated here. Nevertheless, it turns out that the answer in this case is yes for all three questions appearing in Problem 0.3 and the limit is where sign (2) (X; N (G)) denotes the L 2 -signature which is in general different from sign(X) for a finite Poincaré complex X.
Actually, for any closed oriented 4k-dimensional topological manifold M one has sign(M ) = sign (2) (M ; N (G)). Equation (1.3) for finite Poincaré complexes extends to finite Poincaré pairs. For a detailed discussion of these notions and results we refer to [72,73].
Betti numbers
2.1. Characteristic zero. Fix a field F of characteristic zero. We consider the n-th Betti number with F -coefficients b n (X; F ) := dim F (H n (X; F )). Notice that b n (X; F ) = b n (X; Q) = rk Z (H n (X; Z)) holds, where rk Z denotes the rank of a finitely generated abelian group. In this case the answer is yes for all three questions appearing in Problem 0.3 by the main result of Lück [62].
Theorem 2.1. Let F be a field of characteristic zero and let X be a finite CWcomplex. Then we have
lim i→∞ b n (X[i]; F ) [G : G i ] = b (2) n (X; N (G)),
where b (2) (X; N (G)) denotes the n-th L 2 -Betti number.
2.2. Prime characteristic. Fix a prime p. Let F be a field of characteristic p. We consider the n-th Betti number with F -coefficients b n (X; F ) := dim F (H n (X; F )). Notice that b n (X; F ) = b n (X; F p ) holds, where F p is the field of p-elements. In this setting a general answer to Problem 0.3 is only known in special cases. The main problem is that one does not have an analogue of the von Neumann algebra in characteristic p and the construction of an appropriate extended dimension function, see [65], is not known in general.
If G is torsionfree elementary amenable, one gets the full positive answer by Linnell-Lück-Sauer [58,Theorem 0.2], where more explanations, e.g., about Ore localizations are given and actually virtually torsionfree elementary amenable groups are considered.
Theorem 2.2. Let F be a field (of arbitrary characteristic) and X be a connected finite CW -complex. Let G be a torsionfree elementary amenable group. Then:
dim Ore F G H n (X; F ) = lim n→∞ b n (X[i]; F ) [G : G n ] .
For a brief survey on elementary amenable groups we refer for instance to [ Here is another special case taken from Bergeron-Lück-Linnell-Sauer [9], see also Calegari-Emerton [17,18], where we know the answer only for special chains. Let p be a prime, let n be a positive integer, and let φ : G → GL n (Z p ) be a homomorphism, where Z p denotes the p-adic integers. The closure of the image of φ, which is denoted by Λ, is a p-adic analytic group admitting an exhausting filtration by open normal subgroups
Λ i = ker Λ → GL n (Z/p i Z) . Put G i = φ −1 (Λ i ).
Theorem 2.3. Let F be a field (of arbitrary characteristic). Put d = dim(Λ). Let X be a finite CW -complex. Then for any integer n and as i tends to infinity, we have:
b n (X[i]; F ) = b (2) n (X; F ) · [G : G i ] + O [G : G i ] 1−1/d . where b(2)
n (X; F ) is the nth mod p L 2 -Betti numbers occurring in [9, Definition 1.3]. In particular
lim i→∞ b n (X[i]; F ) [G : G i ] = b (2) n (X; F ).
By the universal coefficient theorem we have b n (X[i]; Q) ≤ b n (X[i]; F ) for any field F and hence by Theorem 2.1 the inequality
lim inf i→∞ b n (X[i]; F ) [G : G i ] ≥ b (2) n (X; N (G)).
If p is a prime and we additionally assume that each index [G : exists, see [9, Theorem 1.6].
G i ]
Conjecture 2.4 (Approximation in zero and prime characteristic). We get
lim i→∞ b n (X[i]; F ) [G : G i ] = b (2) n (X; N (G))
for all fields F and n ≥ 0, provided that X is contractible, or, equivalently, that X is aspherical, G = π 1 (X) and X is the universal covering X.
The assumption that X is contractible is necessary in Conjecture 2.4, see [58,Example 6.2]. An obvious modification of [58, Example 6.2] applied to G = Z and X = S 1 ∨ Y for a finite aspherical CW -complex Y with H n (Y ; Q) = 0 and H n (Y ; F p ) = 0 yields a counterexample, where X is aspherical, (but X is not the universal covering).
Estimates of the growth of Betti-numbers in terms of the volume of the underlying manifold and examples of aspherical manifolds, where this growth is sublinear, are given in [87].
2.3.
Minimal number of the generators of the homology. Recall the standard notation that d(G) denotes the minimal number of generators of a finitely generated group G. The Universal Coefficient Theorem implies d H n (X[i]; Z) ≥ b n (X[i]; F ) if F has characteristic zero, but this inequality is not necessarily true in prime characteristic One can make the following version of Conjecture 2.4. which is in some sense stronger, see the discussion in [68, Remark 1.3 and Lemma 2.13].
Conjecture 2.5 (Growth of number of generators of the homology). We get
lim i→∞ d H n (X[i]; Z) [G : G i ] = b (2) n (X; N (G)),
provided that X is contractible.
Rank gradient and cost
Let G be a finitely generated group. Let (G i ) i≥0 be a descending chain of subgroups of finite index of G. The rank gradient of G (with respect to (G i )) is defined by
RG(G; (G i ) i≥0 ) = lim i→∞ d(G i ) − 1 [G : G i ] . (3.1)
The above limit always exists since for any finite index subgroup H of G one has is a monotone decreasing sequence of non-negative rational numbers.
The rank gradient was originally introduced by Lackenby [53] as a tool for studying 3-manifold groups, but is also interesting from a purely group-theoretic point of view, see, e.g., [3,4,81,90].
In the sequel let (G i ) i≥0 be a descending chain of normal subgroups of finite index of G with i≥0 G i = {1}. The following inequalities are known to hold: Question 3.3 (Rank gradient, cost, and first L 2 Betti number). Let G be an infinite finitely generated residually finite group. Let (G i ) i≥0 be a descending chain of normal subgroups of finite index of G with i≥0 G i = {1}.
(3.2) b(2)1 (G) − b(2)0 (G) ≤ cost(G) − 1 ≤ RG(G; (G i ) i≥0 ).
Do we have b
1 (G) = cost(G) − 1 = RG(G; (G i ) i≥0 )(2)
? Question 3.4 (Rank gradient, cost, first L 2 -Betti number and approximation). Let G be a finitely presented residually finite group. Let (G i ) be a descending chain of normal subgroups of finite index of G with i≥0 G i = {1}. Let F be any field.
Do we have
lim i→∞ b 1 (G i ; F ) − 1 [G : G i ] = b (2) 1 (G) − b (2) 0 (G) = cost(G) − 1 = RG(G; (G i ) i≥0 )?
Notice that a positive answer to the questions above also includes the statement, that lim i→∞
bn(X[i];F ) [G:Gi]
and RG(G; (G i ) i≥0 ) are independent of the chain. One can ask for any finitely generated group G (without assuming that it is residually finite) whether b
(2) 1 (G) = cost(G) − 1 is true, and whether the Fixed Prize Conjecture is true which predicts that the cost of every standard action of G, i.e., an essentially free G-action on a standard Borel space with G-invariant probability measure, is equal to the cost of G.
lim i→∞ d(G i ) − rk Z (H 1 (G)) [G : G i ] = 0.
This is surprising since in general one would not expect that for a finitely generated group H the minimal number of generators d(H) agrees with the rank of its abelianization rk Z (H 1 (G)). So a positive answer to Question 3.4 would imply that this equality is true asymptotically. This raises the question whether this equality holds for a "random group" in the sense of Gromov. Remark 3.7 (All conditions are necessary). One cannot drop in Question 3.4 the assumption that the intersection i≥0 G i is trivial. Namely, there exists a finitely presented group G together with a descending chain (G i ) i≥0 of normal subgroups G i of finite index of G, but with non-trivial intersection i≥0 G i such that
lim i→∞ b 1 (G i ; Q) [G : G i ] < lim i→∞ b 1 (G i ; F p ) [G : G i ] < lim i→∞ d(H 1 (G i ; Z)) [G : G i ] < RG(G; (G i ))
holds, see Ershof-Lück [33,Section 4].
The condition that each subgroup G i is normal in G cannot be discarded in Question 3.4. Namely one can conclude from Abért and Nikolov [4,Proposition 14], see [36,Proposition 3.14] for details, that there exists for every prime p a finitely presented group together with a descending chain (G i ) i≥0 of (not normal) subgroups of finite index of G with i≥0 G i satisfying
lim i→∞ b 1 (G i ; Q) [G : G i ] < b (2) 1 (G) < lim i→∞ b 1 (G i ; F p ) [G : G i ] < lim i→∞ d(H 1 (G i ; Z)) [G : G i ] < RG(G; (G i )).
One can find also examples where G is the fundamental group of an oriented hyperbolic 3-manifold of finite volume and the rank gradient is positive for a descending chain (G i ) i≥0 of (not normal) subgroups of finite index of G with i≥0 G i , whereas the first L 2 -Betti number of G is zero, see Girão [40,41]. Also the condition that G is finitely presented has to appear in Question 3.4. (Notice that in Question 3.3 we demand G only to be finitely generated.) For finitely generated G and F of characteristic zero one knows at least lim sup i→∞ b1(Gi;F ))
[G:Gi] ≤ b (2) 1 (G), see Lück-Osin [71, Theorem 1.1].
However, for every prime p there exists an infinite finitely generated residually p-finite p-torsion group G such that for any descending chain of normal subgroups (G i ) i≥0 , for which [G : G i ] is finite and a power of p and i≥0 G i is trivial,
0 = lim i→∞ b 1 (G i ; Q) [G : G i ] < b (2) 1 (G) ≤ lim i→∞ b 1 (G i ; F p ) [G : G i ]
holds. This follows from Ershof-Lück [
χ trun d (X) := min{χ(Y d ) | Y ∈ CW d (X)} if d is even; max{χ(Y d ) | Y ∈ CW d (X)} if d is odd, where χ(Y d ) is the Euler characteristic of the d-skeleton Y d of Y . If X is a finite CW -complex, then χ trun d (X) = χ(X) if d ≥ dim(X)
. Fix a G-covering X → X. Consider Y ∈ CW d (X). Choose a homotopy equivalence h : Y → X. We obtain a G-covering Y → Y by applying the pullback construction to X → X and h : Y → X. We get using [66, Theorem 6.80 (1) on
page 277] χ(Y d ) = χ (2) (Y d ; N (G)) = d n=0 (−1) n · b (2) n (Y d ; N (G)) = (−1) d · b (2) d (Y d ; N (G)) + d−1 n=0 (−1) n · b (2) n (Y d ; N (G)) = (−1) d · b (2) d (Y d ; N (G)) + d−1 n=0 (−1) n · b (2) n (Y ; N (G)) = (−1) d · b (2) d (Y d ; N (G)) − b (2) d (Y ; N (G)) + d n=0 (−1) n · b (2) n (Y ; N (G)) = (−1) d · b (2) d (Y d ; N (G)) − b (2) d (Y ; N (G)) + d n=0 (−1) n · b (2) n (X, N (G)). Since b (2) d (Y d ; N (G)) ≥ b (2) d (Y ; N (G)
) holds, we always have the inequality
χ(Y d ) ≥ d n=0 (−1) n · b (2) n (X, N (G)) if d is even; ≤ d n=0 (−1) n · b (2) n (X, N (G)) if d is odd. This implies that χ trun d (X) is a well-defined integer satisfying χ trun d (X) ≥ d n=0 (−1) n · b (2) n (X, N (G)) if d is even; ≤ d n=0 (−1) n · b (2) n (X, N (G)) if d is[i] ∈ CW d (X[i]) such that χ trun d (X[i]) = χ(Y [i] d ) holds. We can find a [G i : G i+1 ]-sheeted covering Y [i + 1] → Y [i] such that Y [i + 1] belongs to CW d [X[i + 1]). Obviously χ(Y [i + 1] d ) = χ(Y [i] d ) · [G i : G i+1 ]. Suppose that d is even. We conclude χ trun d (X[i + 1]) [G : G i+1 ] = χ trun d (X[i + 1]) [G : G i ] · [G i : G i+1 ] ≤ χ(Y [i + 1] d ) [G : G i ] · [G i : G i+1 ] = χ(Y [i] d ) [G : G i ] = χ trun d (X[i]) [G : G i ] .
Hence the sequence
χ trun d (X[i]) [G:Gi] i≥0
is monotone decreasing. Since we get by an argument similar to the one above
χ trun d (X[i]) [G : G i ] ≥ d n=0 (−1) n · b (2) n (X[i]; N (G/G i )) = d n=0 (−1) n · b n (X[i]; Q) [G : G i ]
for all i, we conclude from Theorem 2.1 that the sequence
χ trun d (X[i]) [G:Gi] i≥0 is bounded from below by d n=0 (−1) n · b(2)
n (X; N (G)). Hence its limit exists and satisfies
lim i→∞ χ trun d (X[i]) [G : G i ] ≥ d n=0 (−1) n · b (2) n (X; N (G)),
if d is even. Provided that d is odd, one analogously shows that the sequence
χ trun d (X[i]) [G:Gi] i≥0
is monotone increasing, bounded from above by d n=0 (−1) n ·b (2) n (X; N (G)) and hence converges with
lim i→∞ χ trun d (X[i]) [G : G i ] ≤ d n=0 (−1) n · b (2) n (X; N (G)).
This leads to
Question 4.2 (Asymptotic Morse equality). Let X → X be a G-covering and let d be a natural number such that CW d (X) is not empty. When do we have
lim i→∞ χ trun d (X[i]) [G : G i ] = d n=0 (−1) n · b (2) n (X; N (G))?
In this generality, the answer to Question 4.2 is not positive in general. For instance if G is trivial and d = 1, a positive answer to Question 4.2 would mean for a connected CW -complex X with non-empty CW 1 (X) that π 1 (X) is finitely generated and satisfies d(π 1 (X)) = b 1 (X) which is not true in general.
Of particular interest is the case, where X is contractible, or, equivalently, X = BG and X = EG. Since χ trun d (X) depends only on the homotopy type of X, we will abbreviate χ trun d (G i ) := χ trun d (BG i ), provided that CW d (BG) for some (and hence all) model for BG is not empty. Then Question 4.2 reduces to the following question, for which we do not know an example, where the answer is negative.
1 − def(G i ) [G : G i ] = b (2) 2 (G) − b(2)1 (G) + b (2) 0 (G),
where def(H) denotes for a finitely presented group H its deficiency, i.e, the maximum over the numbers g − r for all finite presentations H = s 1 , s 2 , . . . , s g | R 1 , R 2 , . . . R r .
Remark 4.5 (Asymptotic Morse inequalities imply Approximation for Betti numbers over any field). Let G be a group with a finite model for BG. It is not hard to show that Conjecture 2.4 is true for G, provided that the answer to Question 4.3 is positive for all d ≥ 0. The main idea of the proof is to show for every field F and CW -complex Z with non-empty CW d (Z) More generally, Conjecture 2.5 is true for G, provided that the answer to Ques- (2) k (G) = 0 for k = 0, 1, 2 . . . , d. If k ∈ {0, 1, 2, . . . , d} is even, we get
χ trun d (Z) ≥ d n=0 (−1) n · b n (Z; F ) if d is even; ≤ d n=0 (−1) n · b n (Z; F ) if dtion 4.3 is positive for all d ≥ 0, since for Y ∈ CW d (Z) we get b d (Y d ; Q) = d(H d (Y d ; Z)) ≥ d(H d (Y ; Z)) = d(H d (Z; Z)0 = k n=0 (−1) n · b (2) n (G) ≤ lim i→∞ χ trun k (G i ) [G : G i ] ≤ lim i→∞ χ (X[i]) k [G : G i ] = lim i→∞ k n=0 c n (X[i]) [G : G i ] = k n=0 lim i→∞ c n (X[i]) [G : G i ] = 0, and hence lim i→∞ χ trun k (G i ) [G : G i ] = 0.
The proof in the case where k is odd is analogous.
Lemma 4.8. Let 1 → K j − → G q − → Q → 1
be an extensions of groups. Suppose that K has slow growth in dimensions ≤ d. Suppose that there is a model for BQ with finite d-skeleton or that there is a model for BG with finite d-skeleton.
Then G has slow growth in dimensions ≤ d.
Proof. If BG has a model with finite d-skeleton, then also BQ has a model with finite d-skeleton by [64,Lemma 7.2 (2)]. Hence it suffices to treat the case, where BQ has a model with finite d-skeleton.
Consider any chain (G i ) i≥0 of normal subgroups of finite index with trivial intersection. Put K i = j −1 (G i ) and Q i = q(G i ). We obtain an exact sequence of
groups 1 → K i ji − → G i qi − → Q i → 1,
where j i and q i are obtained from j and q by restriction. The subgroups K i ⊆ K, G i ⊆ G and Q i ⊆ Q are normal subgroups of finite index and [G :
G i ] = [K : K i ] · [Q : Q i ]. We have i≥0 K i = {1}.c n (Z i ) = [Q : Q i ] · c n (Z).
There is a fibration X[i] → BG i → Z i such that after taking fundamental groups we obtain the exact sequence 1
→ K i ji − → G i qi − → Q i → 1.
Then one can find a CW -complex Y i which is a model for BG i such that we get for the number of k-cells for k ∈ {0, 1, 2, . . . , d}
c k (Y i ) = m+n=k c m (X[i]) · c n (Z i ),
see for instance [34,Section 3]. This implies for k ∈ {0, 1, 2, . . . , d} Obviously Z has slow growth in dimensions ≤ d for all natural numbers d since any non-trivial subgroup K of Z is isomorphic to Z again and has S 1 as model for BK.
lim i→∞ c k (Y i ) [G : G i ] = lim i→∞ m+n=k c m (X[i]) · c n (Z i ) [G : G i ] = lim i→∞ m+n=k c m (X[i]) [K : K i ] · c n (Z i ) [Q : Q i ] = m+n=k c n (Z) · lim i→∞ c m (X[i]) [K : K i ] = 0.
We conclude from Lemma 4.8 that any infinite virtually poly-Z-group has slow growth in dimensions ≤ d.
Moreover, if G is any group which possesses a finite sequence K 0 ⊆ K 1 ⊆ · · · ⊆ K n = G of subgroups such that K 0 ∼ = Z, K i is normal in K i+1 and B(K i+1 /K i ) has a model with finite d-skeleton for i = 0, . . . , (n − 1), then G has slow growth in dimensions ≤ d by Lemma 4.8.
Speed of convergence
The speed of convergence of
lim i→∞ b n (X[i]; F ) [G : G i ] = b (2) n ( X) (if it converges) and of lim i→∞ d(G i ) − 1) [G : G i ] = RG(G; (G i ) i≥0 )
can be arbitrary slow for one chain and very fast for another chain in the following sense. Fix a prime p and two functions F s ,
F f : {i ∈ Z | i ≥ 1} → (0, ∞) such that lim i→∞ F s (i) = 0; lim i→∞ F f (i) = 0; lim i→∞ i · F f (i) = ∞.
Betti numbers.
Theorem 5.1. For every integer n ≥ 1, there is a (2n + 1)-dimensional Riemannian manifold X with non-positive sectional curvature and two chains (G s i ) i≥0 and (G f i ) i≥0 for G = π 1 (X) such that G s i and G f i are normal subgroups of G of finite p-power index, the intersections i≥0 G s i and i≥0 G f i are trivial, and we have for
every field F lim i→∞ b n (X s [i]; F ) [G : G s i ] = b (2) n ( X) = 0; b n (X s [i]; F ) [G s : G s i ] ≥ F s ([G : G s i ]); lim i→∞ b n (X f [i]; F ) [G : G f i ] = b (2) n ( X) = 0; b n (X f [i]; F ) [G : G f i ] ≤ F f ([G : G f i ]).
Proof. Consider any finite connected CW -complex Y with universal covering Y → Y and b (2) n ( Y )+b (2) n−1 ( Y ) > 0 such that K = π 1 (Y ) is infinite and residually p-finite. exists and is greater than 0. Hence there exist real numbers C 1 and C 2 (independent of i) with 0 < C 1 ≤ C 2 such that for each i ≥ 1
C 1 ≤ b n (Y [i]; F ) + b n−1 (Y [i]; F ) [K : K i ] ≤ C 2 .
Let k i be the natural number for which [K : K i ] = p ki holds. Then (k i ) i≥0 is a monotone increasing sequence of natural numbers with lim i→∞ k i = ∞. Since lim k→∞ F s p k · p m = 0 holds for any integer m ≥ 0, we can construct a strictly monotone increasing sequence of natural numbers (j i ) i≥0 such that we get for all i ≥ 0
F s p kj i · p i ≤ C 1 · p −i .
Since lim n→∞ p n · F f p k · p n = ∞ for any natural number k, we can construct a strictly monotone increasing sequence of natural numbers (n f i ) i≥0 satisfying
C 2 · p −n f i ≤ F f p kj i · p n f i . Put X = Y × S 1 . Then G = π 1 (X) can be identified with K × Z. We conclude bG s i = K ji × (p i · Z); G f i = K ji × (p n f i · Z). Then (G s i ) i≥0
is a chain of normal subgroups of G with finite index [G : G s i ] = [K : K ji ] · p i which is a p-power, namely p kj i +i , and trivial intersection i≥0 G s i = {1}, and analogously for (G f i ) i≥0 . We estimate using then Künneth formula
b n (X s [i]; F ) [G : G s i ] = b n (Y [j i ]; F ) + b n−1 (Y [j i ]; F ) [K : K ji ] · p i ≥ C 1 · p −i ≥ F s p kj i · p i = F s ([G : G s i ]), and b n (X s [i]; F ) [G : G s i ] = b n (Y [j i ]; F ) + b n−1 (Y [j i ]; F ) [K : K ji ] · p i ≤ C 2 · p −i .
The latter implies lim i→∞
bn(X s [i];F ) [G:G s i ] = 0. We estimate b n (X f [i]; F ) [G : G f i ] = b n (Y [j i ]; F ) + b n−1 (Y [j i ]; F ) [K : K ji ] · p n f i ≤ C 2 · p −n f i ≤ F f p kj i · p n f i = F f ([G : G f i ]). Since lim i→∞ C 2 · p −n f i = 0, we also get lim i→∞ bn(X[i];F ) [G:G s i ]
= 0. It remains to construct the desired finite CW -complex Y . The fundamental group of an oriented closed surface of genus ≥ 2 is residually free and hence residually p-finite for any prime p by [8]. The L 2 -Betti numbers of its universal covering are all zero except in dimension 1, where it is non-zero, see [66, Example 1.36 on page 40] We conclude from the Künneth formula for L 2 -Betti numbers [66, Theorem 6.54 (5) on page 266] that an example for Y is the direct product of n closed oriented surfaces of genus ≥ 2. So we can arrange that X is an aspherical closed (2n + 1)-dimensional Riemannian manifold with non-positive sectional curvature.
Theorem 5.1 implies that one can find for any ǫ > 0 two such chains (G s
i ) i≥0 and (G f i ) i≥0 satisfying lim i→∞ b n (X[i]; F ) [G : G s i ] 1−ǫ = ∞; lim i→∞ b n (X[i]; F ) [G : G f i ] ǫ = 0, since we can take F s (i) = i −ǫ/2 and F f (i) = i ǫ/2−1 .
The condition lim i→∞ i · F f (i) = ∞ is reasonable. Namely, in most cases one would expect lim i→∞ b n (X[i]; F ) = ∞ and if this is true, we get
lim i→∞ [G : G i ] · b n (X[i]; F ) [G : G i ] = ∞. 5.2. Rank gradient.
Theorem 5.2. Let G be the product of Z with a finitely generated free group of rank ≥ 2 or the product of Z with the fundamental group of a closed surface of genus ≥ 2.
Then there are two chains (G s i ) i≥0 and (G f i ) i≥0 such that G s i and G f i are normal subgroups of G of finite p-power index, the intersections i≥0 G s i and i≥0 G f i are trivial, and we have
RG(G, (G s i ) i≥0 ) = 0; d(G s i ) − 1 [G : G s i ] ≥ F s ([G : G s i ]); RG(G, (G f i ) i≥0 ) = 0; d(G f i ) − 1 [G : G f i ] ≤ F f ([G : G f i ]).
Proof. Essentially the same argument as in the proof of Theorem 5.1 also applies to the rank gradient. Let K be a finitely presented group with b
(2)
1 (K) > 0. Choose any chain (K i ) i≥0 of normal subgroups of K of finite index [K : K i ] which is a power of p and with trivial intersection i≥0 K i = {1}. Then d(Ki)−1 [K:Ki] i≥0
is a monotone decreasing sequence. Its limit RG(K,
(K i ) i≥0 ) := lim i→∞ d(Ki)−1 [K:
Ki] exists and is greater than b
(2) 1 (K). Hence we can choose constants C 1 > 0 and C 2 > 0 such that such that for each i ≥ 1 we get
C 1 ≤ d(K i ) − 1 [K : K i ] ≤ d(K i ) [K : K i ] ≤ C 2 .
Put G = K × Z. Now construct the sequences (j i ) i≥0 and (n s i ) i≥0 and define G s i and G f i as in the proof of Theorem 5.1.
Then d(K ji ) ≤ d(G s i ) ≤ d(K ji ) + 1 and d(K ji ) ≤ d(G f i ) ≤ d(K ji ) + 1
holds. Now a calculation analogous to the one in the proof of Theorem 5.1 shows
d(G s i ) − 1 [G : G s i ] ≥ F s ([G s : G s i ]); d(G f i ) − 1 [G : G f i ] ≤ F f ([G f : G f i ]).
If K is a finitely generated free group of rank ≥ 2 or the fundamental group of a closed surface of genus ≥ 2, then K is finitely presented, residually p-finite and b
(2) 1 (K) > 0.
The Approximation Conjecture for Fuglede-Kadison determinants
Let A ∈ M r,s (QG) be a matrix. It induces by right multiplication a G-equivariant bounded operator r (2)
A : L 2 (G) r → L 2 (G) s . We denote by det (2) N (G) r (2) A : L 2 (G) r → L 2 (G) s its Fuglede-Kadison determinant. Denote by A[i] ∈ M r,s (Q[G/G i ])
the matrix obtained from A by applying elementwise the ring homomorphisms QG → Q[G/G i ] induced by the projection
G → G/G i . It induces a C-homomorphism of finite-dimensional complex Hilbert spaces r (2) A[i] : C[G/G i ] r → C[G/G i ] s . Consider a C-homomorphism of finite-dimensional Hilbert spaces f : V → W . It induces an endomorphism f * f : V → V . We have ker(f ) = ker(f * f ). Denote by ker(f ) ⊥ the orthogonal complement of ker(f ). Then f * f induces an automorphism (f * f ) ⊥ : ker(f ) ⊥ → ker(f ) ⊥ . Define det ′ (f ) := det (f * f ) ⊥ . (6.1) If 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · are the non-zero eigenvalues (listed with multiplicity) of the positive operator f * f : V → V , then det ′ (f ) = j≥1 λ j .
If f is an isomorphism, then det ′ (f ) reduces to det(f * f ).
Conjecture 6.2 (Approximation Conjecture for Fuglede-Kadison determinants).
Consider a matrix A ∈ M r,s (QG). Then we get ln det
(2) N (G) (r (2) A ) = lim i→∞ ln det ′ (r (2) A[i] ) [G : G i ] .
Remark 6.3. If r = s and A ∈ M r,r (QG) is invertible, then the following equality is always true ln det
(2) N (G) (r (2) A ) = lim i→∞ ln det(r (2) A[i] ) [G : G i ] .
However, for applications to L 2 -torsion we have to consider the case, where r and s may be different and the maps (r
(2) A ) * r(2)
A and (r
(2) A[i] ) * r (2)
A[i] may not be injective. There exists a sequence of integers 2 ≤ n 1 < n 2 < n 3 < · · · and a real number s such that for G = Z and G i = n i · Z and the (1, 1)-matrix A given by the element
z − exp(2πis) in C[Z] = C[z, z −1 ] we get for all i ≥ 1 ln det (2) N (G) (r (2) A ) = 0; ln det(r (2) A[i] ) [G : G i ] ≤ −1/2.
Remark 6.5 (Status of Conjecture 6.2). Conjecture 6.2 has been proved for G = Z by Schmidt [91], see also [
(2) N (G) (r (2) A ) ≥ lim sup i→∞ ln det ′ (r (2) A[i] ) [G : G i ] . (6.6)
For G = Z n the last inequality for the limit superior is known to be an equality by Lê [55]. But nothing seems to be known beyond virtually finitely generated free abelian groups.
Torsion invariants
7.1. L 2 -torsion. Let D * be a finite based free Z-chain complex, for instance the cellular chain complex C * (Y ) of a finite CW -complex Y . The C-chain complex D * ⊗ Z C inherits from the Z-basis on D * and the standard Hilbert space structure on C the structure of a Hilbert space and the resulting L 2 -chain complex is denoted by D
(2) * with differentials d (2) p := d p ⊗ Z id C . Define its L 2 -torsion by ρ (2) (D (2) * ; N ({1})) := − n≥1 (−1) n · ln det (2) N ({1}) (d (2) n ) ∈ R. (7.1) Notice that det (2) N ({1}) (c (2) n ) is the same as det ′ (c (2) n ) which we have introduced in (6.1).
More generally, if C * is a finite based free ZG-chain complex, we obtain a finite Hilbert N (G)-chain complex C (2) * := C * ⊗ ZG L 2 (G) and we define its L 2 -torsion
ρ (2) (C (2) * ; N (G)) := − n≥1 (−1) n · ln(det (2) N (G) c (2) n ) ∈ R. (7.2) Conjecture 7.3 (Approximation Conjecture for Fuglede-Kadison determinants). Let C * be a finite based free ZG-chain complex. Denote by C[i] * the finite free Z-chain complex given by C[i] * = C * ⊗ Z[Gi] Z = C * ⊗ ZG Z[G/G i ].
Then we get
ρ (2) (C (2) * ; N (G)) = lim i→∞ ρ (2) (C[i] (2) * ; N ({1})) [G : G i ] .
Obviously Conjecture 7.3 is just the chain complex version of Conjecture 6.2 and these two conjectures are equivalent for a given group G.
7.2.
Analytic and topological L 2 -torsion. Let X be a closed Riemannian manifold. Let ρ an (X[i]) be the analytic torsion in the sense of Ray and Singer of the closed Riemannian manifold X[i]. Denote by ρ (2) an (X) the analytic L 2 -torsion of the Riemannian manifold X with isometric free cocompact G-action. Conjecture 7.4 (Approximation Conjecture for analytic torsion). Let X be a closed Riemannian manifold. Then
ρ (2) an (X; N (G)) = lim i→∞ ln(ρ an (X[i])) [G : G i ] .
There are topological counterparts which we will denote by ρ top (X[i]) and ρ (2) top (X) which agree with their analytic versions by results Cheeger [21] and Müller [77] and Burghelea-Friedlander-Kappeler-McDonald [16]. So the conjecture above is equivalent to its topological counterpart. Conjecture 7.5 (Approximation Conjecture for topological torsion). Let X be a closed Riemannian manifold. Then
ρ (2) top (X; N (G)) = lim i→∞ ln(ρ top (X[i])) [G : G i ] .
Remark 7.6 (Dependency on the triangulation and the Riemannian metric). Let X be a closed smooth manifold. Fix a smooth triangulation. Since this induces a structure of a free finite G-CW -complex on X, we get a ZG-basis for C * (X) and hence can consider ρ (2) (C (2) * (X); N (G)). The cellular ZG-basis for C * (X) is not unique, only up to permutation of the basis elements and multiplying base elements with trivial units, i.e., elements of the shape ±g for g ∈ G, but it turns out that ρ (2) (C (2) * (X); N (G)) is independent of these choices after we have fixed a smooth triangulation of X. However, if we pass to a subdivision of the smooth triangulation of C, then ρ (2) (C (2) * (X); N (G)) changes in general. Let X be a closed smooth Riemannian manifold. Then ρ (2) an (X; N (G)) and ρ (2) top (X; N (G)) are independent of the choice of smooth triangulation and hence depend only on the isometric diffeomorphism type of X. However, changing the Riemannian metric does in general change ρ (2) an (X; N (G)) and ρ (2) top (X; N (G)). If we have b (2) n (X; N (G)) = 0 for all n ≥ 0, then ρ (2) an (X; N (G)) and ρ (2) top (X; N (G)) are independent of the Riemannian metric and depend only on the diffeomorphism type of X, actually, they depend only on the simple homotopy type of X. There is a lot of evidence that in this situation only the homotopy type of X matters.
The next result is a special case of Theorem 14.10.
Theorem 7.7 (Relating the Approximation Conjecture for Fuglede-Kadison determinant and torsion invariants). Suppose that X is a closed Riemannian manifold such that b (2) n (X, N (G)) vanishes for all n ≥ 0. If G satisfies Conjecture 6.2 for all matrices A ∈ M r,s (QG) and all natural numbers r, s, then Conjecture 7.4 and Conjecture 7.5 hold for X.
Remark 7.8 (On the L 2 -acyclicity assumption). Recall that in Theorem 7.7 we require that b (2) n (M ; N (G) = 0 holds for n ≥ 0. This assumption is satisfied in many interesting cases. It is possible that this assumption is not needed for Theorem 7.7 to be true, but our proof does not work without it.
Integral torsion.
Definition 7.9 (Integral torsion). Define for a finite Z-chain complex D * its integral torsion
ρ Z (D * ) := n≥0 (−1) n · ln tors(H n (D * )) ∈ R,
where tors(H n (D * )) is the order of the torsion subgroup of the finitely generated abelian group H n (D * ). Given a finite CW -complex X, define its integral torsion ρ Z (X) by ρ Z (C * (X)), where C * (X) is its cellular Z-complex. Remark 7.10 (Integral torsion and Milnor's torsion). Let C * be a finite free Zchain complex. Fix for each n ≥ 0 a Z-basis for C n and for H n (C)/ tors(H n (C)). They induce Q-bases for Q ⊗ Z C n and H n Q ⊗ Z C * ) ∼ = Q ⊗ Z H n (C)/ tors(H n (C) . Then the torsion in the sense of Milnor [76, page 365] is ρ Z (C * ).
The following two conjectures are motivated by [ n (X; N (G)) vanishes for all n ≥ 0 ensures that the definition of the topological L 2 -torsion ρ (2) top (X; N (G)) makes sense for X also in the case of a connected finite CW -complex.
Conjecture 7.11 (Approximation Conjecture for integral torsion). Let X be a finite connected CW -complex. Suppose that b (2) n (X; N (G)) vanishes for all n ≥ 0. Then ρ (2) top (X; N (G)) = lim i→∞ ρ Z (X[i]) [G : G i ] .
The chain complex version of Conjectures 7.11 is (1) Let f : ZG r → ZG r be a ZG-homomorphism such that f (2) :
L 2 (G) r → L 2 (G) r is a weak isomorphism of Hilbert N (G)-modules. Let f [i] := f ⊗ ZGi Z = f ⊗ ZG Z[G/G i ] : Z[G/G i ] r → Z[G/G i ] r be the induced Z-homomorphism. Then det (2) N (G) (f (2) ) = lim i→∞ tors coker(f [i]) 1/[G:Gi] ;
(2) Let C * be a finite based free ZG-chain complex. Suppose that C
(2) * is L 2 - acyclic, i.e., b (2) p C (2) * = 0 for all p ≥ 0. Let C[i] * := C * ⊗ ZGi Z = C * ⊗ ZG Z[G/G i ] be the induced finite based free Z-chain complex. Then ρ (2) C (2) * = lim i→∞ ρ Z (C[i] * ) [G : G i ] ;
In Conjecture 7.11 and Conjecture 7.12 it is necessary to demand that f is a weak isomorphism and that C * and X are L 2 -acyclic, otherwise there are counterexamples, see Remark 9.2.
Here are some results about the conjecture above which will be proved in Section 18.
Theorem 7.13.
(1) Let f : ZG r → ZG s be a ZG-homomorphism. Then ln det
(2) N (G) (f (2) ) ≥ lim sup i→∞ tors coker(f [i]) [G : G i ] ;
(2) Suppose in the situation of assertion ]. An extension of the results in this paper is given by Raimbault [84].
(1) of Conjecture 7.12 that f ⊗ Z id Q : Q[G] r → Q[G] r is
On the relation of L 2 -torsion and integral torsion
Let C * be a finite based free Z-chain complex, for instance the cellular chain complex C * (X) of a finite CW -complex. We have introduced in Subsection 7.1 the L 2 -chain complex C
(2) * = C * ⊗ Z C with differentials c (2) n := c n ⊗ Z id C and its L 2 -torsion ρ (2) (C (2) * ; N ({1})). Let H (2) n (C (2) * ) be the L 2 -homology of C (2) *
with respect to the von Neumann algebra N ({1}) = C. The underlying complex vector space is the homology H n (C * ⊗ I C) of C * ⊗ Z C, but it comes now with the structure of a Hilbert space. For the reader's convenience we recall this Hilbert space structure. Let
∆ (2) n = c (2) n * • c (2) n + c (2) n+1 • c (2) n+1 * : C (2) n → C (2)
n be the associated Laplacian. Equip ker(∆
(2) n ) ⊆ C (2) n
with the induced Hilbert space structure. Equip H (2) n (C (2) * ) with the Hilbert space structure for which the obvious C-isomorphism ker(∆ (2) n ) → H (2) n (C (2) * ) becomes an isometric isomorphism. This is the same as the Hilbert subquotient structure with respect to the inclusion ker c (2) n → C (2) n and the projection ker c We have the canonical C-isomorphism
α n : H n (C * ) f (2) := H n (C * )/ tors(H n (C * ) ⊗ Z C ∼ = − → H (2) n (C (2) * ). (8.2)
Choose a Z-basis on H n (C * ) f . This and the standard Hilbert space structure on C induces a Hilbert space structure on H n (C * ) f (2) . Now we can consider the logarithm of the Fuglede-Kadison determinant
R n (C * ) = ln det N ({1}) α n : H n (C * ) f ) (2) → H (2) n (C (2) * ) , (8.3)
which is some times called the nth regulator. It is independent of the choice of the Z-basis of H n (C * ) f , since the absolute value of the determinant of an invertible matrix over Z is always 1. If {b 1 , b 2 , . . . , b r } is an integral basis of H n (C * ) f and we equip H n (C * ) f ⊗ Z C with an inner product −, − for which the map α n of (8.2) becomes an isometry, then
R n (C * ) = ln det C (B) 2 , where B is the Gram-Schmidt matrix b i , b j i,j . The next result is proved for instance in [68, Lemma 2.3].
Lemma 8.4. Let C * be a finite based free Z-chain complex. Then
ρ Z (C * ) − ρ (2) C (2) * ; N ({1}) = n≥0 (−1) n · R n (C * ).
Remark 8.5 (Comparing conjectures for L 2 -torsion and integral torsion). Consider the following three statements:
(1) Every finite based free ZG-chain complex C * with b
(2) n C
(2) * = 0 for all n ≥ 0 satisfies
ρ (2) C (2) * = lim i→∞ ρ (2) C[i] (2) * [G : G i ] ;
(2) Every finite based free ZG-chain complex C * with b
(2) n C
(2) * = 0 for all n ≥ 0 satisfies assertion (2) of Conjecture 7.12, i.e.,
ρ (2) C (2) * = lim i→∞ ρ Z (C[i] * ) [G : G i ] ;
(3) Every finite based free ZG-chain complex C * with b
(2) n C (2) * = 0 for all n ≥ 0 satisfies lim i→∞ n≥0 (−1) n · R n (C[i] * ) [G : G i ] = 0,
By Lemma 8.4 they are all true if two of them hold.
Lemma 8.6. Let X be an oriented closed smooth manifold of dimension d. Fix a smooth triangulation. Let s n be the number of n-simplices of the triangulation of X. Then we get (2) d and has norm [G :
R d (C * (X[i])) = ln([G : G i ] · s d ) 2 ; R 0 (C * (X[i])) = − ln([G : G i ] · s 0 ) 2 ; lim i→∞ R n (C * (X[i])) [G : G i ] = 0 for n = 0, d.G i ] · s d . Hence R d (C * (X[i])), which is the logarithm of the norm of σ[i] d considered as an element in C (2) d (X[i]), is ln([G:Gi]·s d ) 2 . Consider the element σ[i] 0 ∈ C 0 (X[i]) given by the sum of the 0-simplices of X[i]. The number of 0-simplices in X[i] is [G : G i ] · s 0 . The element σ[i] 0 considered as element in C
0 since it is orthogonal to any element of the shape e 1 -e 0 for 0-simplices e 0 and e 1 and hence to the image of c
(2) 1 : C (2) 1 (X[i]) → C (2) 0 (X[i]). The augmentation map C 0 (X[i]) → Z sending a 0-simplex to 1 induces an isomorphism H 0 (C * (X[i])) ∼ = − → Z. This shows that σ[i] 0 represents [G : G i ] · s 0 times the generator of H 0 (X[i]; Z). Hence R 0 (C * (X[i])) which is the logarithm of the norm of σ[i]0 [G:Gi]·s0 is − ln([G:Gi]·s0) 2 .
In particular we get lim i→∞
Rn(C * (X[i])) [G:Gi]
= 0 for n = 0, d.
Elementary example about L 2 -torsion and integral torsion
Consider integers a, b, k, l, and g ≥ 1, such that (a, b) = (1) and (k, l) = (1). Consider the following finite based free Z-chain complex C * which is concentrated in dimensions 0, 1, 2 and 3 and given there by
0 · · · → 0 → Z c3= −l k − −−−−−− → Z 2 c2= gka gla gkb glb − −−−−−−−−−−− → Z 2 c1= −b a − −−−−−−−− → Z → 0 → · · · .
Notice that any matrix homomorphism c 2 : Z 2 → Z 2 whose kernel has rank one is of the shape above. One easily checks that
ker(c 3 ) = {0}; im(c 3 ) = {n · −l k | n ∈ Z}; ker(c 2 ) = {n · −l k | n ∈ Z}; im(c 2 ) = {n · ga gb | n ∈ Z}; ker(c 1 ) = {n · a b | n ∈ Z}; im(c 1 ) = Z.
This implies
H i (C * ) = Z/g i = 1; {0} i = 1.
We conclude from [66, Lemma 3.15 (4) on page 129] ln det
(2) N ({1}) c (2) 3 = 1 2 · ln det (2) N ({1}) (c(2)
3 ) * • c
(2) 3 = 1 2 · ln det (2) N ({1}) (k 2 + l 2 ) : C → C ) = ln(k 2 + l 2 ) 2 .
Analogously one shows ln det
(2) N ({1}) c (2) 1 = ln(a 2 + b 2 ) 2 .
The kernel of c
2 is the subvector space of C 2 generated by
1 √ k 2 +l 2 · −l k and the image of c (2) 2 is the subvector space of C 2 generated by 1 √ a 2 +b 2 · a b
. Hence the orthogonal complement ker c
(2) 2 ⊥ of the kernel of c (2) 2 is the subvector space of C 2 generated by 1 √ k 2 +l 2 · k l . Since 1 √ a 2 +b 2 · a b and 1 √ k 2 +l 2 · k l
have norm 1 and
c (2) 2 1 √ k 2 + l 2 · k l = g · k 2 + l 2 · a 2 + b 2 · 1 √ a 2 + b 2 · a b , we conclude from [66, Lemma 3.15 (3) on page 129] ln det (2) N ({1}) c (2) 2 = ln(g) + ln(a 2 + b 2 ) + ln(k 2 + l 2 ) 2 .
Notice that Lemma 8.4 predicts ρ (2) C
(2) * = ρ Z (C * ) which is consistent with the direct computation ρ (2) C (2) * = − ln det (2) N ({1}) c (2) 3 + ln det (2) N ({1}) c (2) 2 − ln det (2) N ({1}) c (2) 1 = ln(g) = ln tors H 1 (C * ) = ρ Z (C * ).
We also compute the combinatorial Laplace operators of C * . We get for their matrices
∆ 3 = (k 2 + l 2 ); ∆ 2 = g 2 k 2 a 2 + g 2 k 2 b 2 + l 2 g 2 kla 2 + g 2 klb 2 − kl g 2 kla 2 + g 2 klb 2 − kl g 2 l 2 a 2 + g 2 l 2 b 2 + k 2 ; ∆ 1 = g 2 k 2 a 2 + g 2 l 2 a 2 + b 2 g 2 k 2 ab + g 2 l 2 ab − ab g 2 k 2 ab + g 2 l 2 ab − ab g 2 k 2 b 2 + g 2 l 2 b 2 + a 2 ; ∆ 0 = (a 2 + b 2 ).
This implies
det Z (∆ i ) = det (2) N ({1}) ∆ (2) i = k 2 + l 2 i = 3; (a 2 + b 2 ) · g 2 · (k 2 + l 2 ) 2 i = 2; (a 2 + b 2 ) 2 · g 2 · (k 2 + l 2 ) i = 1; (a 2 + b 2 ) i = 0.
This is consistent with the formula
ρ (2) C (2) * ; N ({1}) = − 1 2 · i≥0 (−1) i · i · ln det (2) N ({1}) ∆ (2) i .
Remark 9.1 (No relationship between the differentials and homology in each degree). We see that there is no relationship between ln det N ({1}) ∆ This shows that a potential proof of Conjecture 7.12 will require more input than one would expect for a potential proof of Conjecture 7.3, Conjecture 7.4, or Conjecture 7.5.
Remark 9.2 (L 2 -acyclicity is necessary for the homological version). This example can also be used to show that the condition of L 2 -acyclicity appearing in Conjecture 7.12 is necessary. This is not a surprise since ρ Z (C[n] * ) depends only on the Z[Z]-chain homotopy type of C * which is not true for ρ (2)
(C (2) * ) unless C (2) * is L 2 -acyclic.
Namely, consider the 1-dimensional Z-chain complex chain complex D * whose only non-trivial differential d 1 is the differential c 2 in the chain complex C * above.
Let E * := D * ⊗ Z Z[Z]. Put E * [n] = E * ⊗ Z[Z] Z[Z/n]. Then E * [n] = D * ⊗ Z Z[Z/n].
We conclude from the computations above and [66, Theorem 3.14 (5) and (6)
ρ (2) E (2) * ; N (Z) = ln(g) + ln(a 2 + b 2 ) + ln(k 2 + l 2 ) 2 ; ρ (2) E[n] (2) * ; N ({1}) n = ln(g) + ln(a 2 + b 2 ) + ln(k 2 + l 2 ) 2 ; ρ Z (E[n] * ) n = ln(g).
Hence we have
ρ (2) E (2) * ; N (Z) = lim n→∞ ρ (2) E[n] (2) * ; N ({1}) n but ρ (2) E (2) * ; N (Z) = lim n→∞ ρ Z (E[n] * )
n .
Notice that the condition of L 2 -acyclicity is not demanded in Conjecture 7.3, Conjecture 7.4, Conjecture 7.5, Conjecture 14.5, and Conjecture 14.8.
Aspherical manifolds
The following conjecture is in our view the most advanced and interesting one. It combines Conjecture 7.11 that one can approximate L 2 -torsion by integral torsion in the L 2 -acyclic case with the conjecture that for closed aspherical manifolds X the L 2 -cohomology of X and asymptotically the homology of X[i] are concentrated in the middle dimension. Notice that in assertion (1) and (4) we are not demanding that G = π 1 (M ) is residually finite. This assumption only enters in assertions (2) and (5) , where the chain (G i ) i≥0 occurs. (1), where negative sectional curvature is required, one has to add an assumption on π 1 (X), for instance that π 1 (X) is a CAT(-1)-group.
1 ( M ) = lim i→∞ b 1 (M [i]; F ) [G : G i ] = lim i→∞ d G i /[G i , G i ] [G : G i ] = RG(G, (G i ) i≥0 ) = 0 d = 2; −χ(M ) d = 2;(2)
Assertion (1) of Conjecture 10.1 in the case that M carries a Riemannian metric with non-positive sectional curvature is the Singer Conjecture The Singer Conjecture and also the related Hopf Conjecture are discussed in detail in [66,Section 11].
Assertion (2) is closely related to Conjecture 2.4, Conjecture 2.5 and Question 3.4.
The parity condition about the L 2 -torsion appearing in assertion (4) Here is a concrete and already very interesting special case. Bergeron-Segun-Venkatesh [11] consider the equality above for arithmetic hyperbolic 3-manifolds and relate it to a conjecture about classes in the second integral homology.
Some numerical evidence for the equality above is given in Sengun [93].
H 1 (M [i]; Z)) ∼ = Z r[i,1] ⊕ Z/p m[i,1]
Mapping tori
A very interesting case is the example of a mapping torus T f of a selfmap f : Z → Z of a connected finite CW -complex Z (which is not necessarily a homotopy equivalence). The canonical projection q : T f → S 1 induces an epimorphism pr : G = π 1 (T f ) → Z. Let K be its kernel which can be identified with the colimit of the direct system of groups indexed by Z.
· · · π1(f ) −−−→ π 1 (Z) π1(f ) −−−→ π 1 (Z) π1(f ) −−−→ · · ·
In particular the inclusion Z → T f induces a homomorphism j : π 1 (Z) → K which corresponds to the structure map in the description of K as a colimit at 0 ∈ Z. We obtain a short exact sequence
1 → K j − → G pr − → Z → 1. We use the Setup (0.1) with X = T f and X = T f . Put K i = j −1 (G i ). Let d i ∈ Z be the integer for which pr(G i ) = d i ·Z. We obtain an induced exact sequence 1 → K i ji − → G i pr i − − → d i ·Z → 1. We have [G : G i ] = [K : K i ] · d i .
If π 1 (f ) is an isomorphism, then j : π 1 (Z) → K is an isomorphism.
Let p i : S 1 → S 1 be the d i -sheeted covering given by z → z di . It is the covering associated to d i · Z ⊆ Z. Let p i : T f [i] ′ → T f be the d i -sheeted covering given by the pullback
T f [i] ′ q[i] ′ / / pi S 1 pi T f q / / S 1 It is the covering associated to pr −1 (d i · Z) ⊆ G = π 1 (T f ). Let q i : T f [i] → T f [i] ′ be the [K : K i ]-sheeted covering which is associated to G i ⊆ pr −1 (d i · Z) = π 1 (T f [i] ′ ). The composite T f [i] qi − → T ′ f [i] pi − → T f is the [G : G i ]-sheeted covering associated to G i ⊆ G = π 1 (T f ). Let q i : Z[i] → Z be the [K :
K i ]-sheeted covering given by the pullback
Z[i] qi / / T f [i] qi Z i / / T f [i] ′
It is the covering given by
K/K i × π1(Z)/j −1 (Ki) Z.
Since T f [i] ′ is obtained from the d i -fold mapping telescope of f by identifying the left and the right end by the identity, there is an obvious map T f [i] ′ → T f d i which turns out to be a homotopy equivalence. Hence we can choose a homotopy equivalence
u : T f d i ≃ − → T f [i] ′ .
Define the homotopy equivalence u :
T f d i ≃ − → T f [i] by the pullback T f d i u / / qi T f [i] qi T f d i u / / T f [i] ′
There is a finite CW -structure on T f d i such that the number of n-cells c n (T f d i ) is c n (Z)+c n−1 (Z), where c n (Z) is the number of n-cells in Z. Since q i :
T f d i → T f d i is a [K : K i ]-sheeted covering, there is a finite CW -structure on T f d i such that the number of n-cells c n (T f d i ) is [K : K i ] · (c n (Z) + c n−1 (Z)). This implies c n (T f d i ) [G : G i ] = c n (Z) + c n−1 (Z) d i .
Hence we get for any field
F b n (T f [i]; F ) [G : G i ] ≤ c n (Z) + c n−1 (Z) d i ; d(π 1 (T f [i])) ≤ c 1 (Z) + c 0 (Z) d i ; def(π 1 (T f [i])) [G : G i ] ≤ c 2 (Z) + c 1 (Z) + c 0 (Z) d i ; χ trun n (T f [i]) [G : G i ] ≤ n i=0 c i (Z) d i .
11.1. The case i≥0 d i · Z = {1}. Suppose that i≥0 d i · Z = {1}, or, equivalently, lim i→∞ d i = ∞ holds. Then we conclude for any field F
lim i→∞ b n (T f [i]; F ) [G : G i ] = 0; lim i→∞ d(π 1 (T f [i])) [G : G i ] = 0; lim i→∞ def(π 1 (T f [i])) [G : G i ] = 0; lim i→∞ χ trun n (T f [i]) [G : G i ] = 0. Since b (2) n ( T f ) =i ≥ 0, otherwise replace T f by T f [i 0 ], G by G i0 , Z by n i0 · Z, and (G i ) i≥0 by (G i ) i≥i0 .
We conclude from Theorem 2.1, Theorem 2.2 and (11.1)
lim i→∞ b n (T f [i]; F ) [G : G i ] = 0. (11.2)
provided that F has characteristic zero. We get the same conclusion (11.2) for any field F provided that G is torsionfree elementary amenable and residually finite by Theorem 2. = 0. We do not know whether (11.2) holds for arbitrary fields and arbitrary residually finite groups G, as predicted by Conjecture 2.4.
We do not know whether the rank gradient RG(G, (G i ) i ≥ 0 ) is zero for any chain (G i ) i≥0 as predicted by Conjecture 3.3 in view of (11.2), but at least for chains with i≥0 d i · Z = {1} this follows from Subsection 11.1. This illustrates why it would be very interesting to know whether the rank gradient RG(G, (G i ) i≥0 ) is independent of the chain (G i ) i≥0 . The same remark applies to the more general Question 4.3.
One can express b n (T f [i]; F ) in terms of f as follows. Obviously K i is a normal subgroup of G, the automorphisms π 1 (f ) : G = π 1 (Z) → G = π 1 (Z) sends K i to K i and we have [G :
G i ] = [K : K i ] for all i ≥ 0. Put f [0] = f : Z = Z[0] → Z = Z[0].f [i] / / Z[i] Z[i − 1] f [i−1] / / Z[i − 1] commutes. Then T f [i] is T f [i] .
We have the Wang sequence of R-modules for any commutative ring R
(11.3) · · · → H n (Z[i]; R) id −Hn(f [i];R) −−−−−−−−−→ H n (Z[i]; R) → H n (T f [i]) → H n−1 (Z[i]; R) id −Hn−1(f [i];R) − −−−−−−−−−− → H n−1 (Z[i]; R) → · · · This implies (11.4) b n (T f [i]; F ) = dim F coker id −H n (f [i]; F ) + dim F ker id −H n−1 (f [i]; F ) .
11.3. Selfhomeomorphism of a surface. Now assume that Z is a closed orientable surface of genus g and f : Z → Z be an orientation preserving selfhomeomorphism.
If g = 0, we get T f = S 1 × S 2 and in this case everything can be computed directly.
If g = 1, then π 1 (T f ) is poly-Z and T f is aspherical, and hence we know already that Conjecture 2.4 and Conjecture 2.5 are true, the answers to Questions 3.
b n (T f [i]; F ) = dim F coker id −H 1 (f [i]; F ) + 1 n = 1; dim F ker id −H 1 (f [i]; F ) + 1 n = 2; 1 n = 0, 3; 0 n ≥ 4.
We know for all n ≥ 0 (11.5) provided that F has characteristic zero. Notice that (11.5) for a field of characteristic zero is equivalent to
lim i→∞ b n (T f [i]; F ) [G : G i ] = 0,lim i→∞ dim Q coker(id −H 1 (f [i]; Z)) ⊗ Z Q [G : G i ] = 0.
Next we consider the case that F is a field of prime characteristic p. Then we do know (11.5) in the situation of Subsection 11.1 but not in the situation of Subsection 11.2. Recall that Conjecture 2.4 predicts (11.5) in view of (11.1) also in this case. In order to prove (11.2) also for a field F of prime characteristic p for all n ≥ 0 in the situation of Subsection 11.1, it suffices to show
lim i→∞ dim Fp tors(H 1 (T f [i]; Z)) ⊗ Z F p [G : G i ] = 0(T f [i]; Z)) [G : G i ] > 0, lim i→∞ dim Fp tors(H 1 (T f [i]; Z)) ⊗ Z F p [G : G i ] = 0, lim i→∞ d tors(H 1 (T f [i]; Z)) [G : G i ] = 0,
or, because of the Wang sequence (11.3) equivalently,
lim i→∞ ln tors coker(id −H 1 (f [i]; Z)) [G : G i ] > 0, lim i→∞ dim Fp tors coker(id −H 1 (f [i]; Z)) ⊗ Z F p [G : G i ] = 0, lim i→∞ d tors coker(id −H 1 (f [i]; Z)) [G : G i ] = 0.
Dropping the finite index condition
From now on we want to drop the condition that the index of the subgroups G i in G is finite and that the index set for the chain is given by the natural numbers. So we will consider for the remainder of this paper the following more general situation: Setup 12.1 (Inverse system). Let G be a group together with an inverse system {G i | i ∈ I} of normal subgroups of G directed by inclusion over the directed set I such that i∈I G i = {1}.
If I is given by the natural numbers, this boils down to a nested sequence of normal subgroups G = G 0 ⊃ G 1 ⊇ G 2 ⊇ · · · satisfying n≥1 G n = {1}. If we additionally assume that [G : G i ] is finite, we are back in the previous special situation (0.2). Some of the following conjectures reduce to previous conjectures in this special case. The reason is that for a finite group H and a based free finite ZH-chain complex D * we have If G = colim i∈I G i is the colimit of the directed system {G i | i ∈ I} of groups indexed by the directed set I (with not necessarily injective structure maps) and each G i belongs to F , then G belongs to F ; (3) Inverse limits If G = lim i∈I G i is the limit of the inverse system {G i | i ∈ I} of groups indexed by the directed set I and each G i belongs to F , then G belongs to F ; (4) Subgroups
b (2) p (D (2) * ; N (H)) = b (2) p (D (2) * ; N ({1})) |H| ; ρ (2) (D (2) * ; N (H)) = ρ (2) (D (2) * ; N ({1})) |H| .
If H is isomorphic to a subgroup of a group G with G ∈ F , then H ∈ F ; (5) Quotients with finite kernel Let 1 → K → G → Q → 1 be an exact sequence of groups. If K is finite and G belongs to F , then Q belongs to F ; (6) Sofic groups belong to F . The class of sofic groups is very large. It is closed under direct and free products, taking subgroups, taking inverse and direct limits over directed index sets, and is closed under extensions with amenable groups as quotients and a sofic group as kernel. In particular it contains all residually amenable groups. One expects that there exists non-sofic groups but no example is known. More information about sofic groups can be found for instance in [32] and [82].
A : L 2 (G) r → L 2 (G) s = lim i∈I dim N (G/Gi) ker r (2) A[i] : L 2 (G/G i ) r → L 2 (G/G i ) s ;(2)
(2) CW -complex version Let X be a G-CW -complex of finite type. Then X[i] := G i \X is a G/G i -CW -complex of finite type and b (2) p (X; N (G)) = lim
i∈I b (2) p (X[i]; N (G/G i )).
The two conditions appearing in Conjecture 13.4 are equivalent by [66, Lemma 13.4 on page 455].
We will frequently make the following assumption:
Assumptiondet N (G) r (2) A : L 2 (G) r → L 2 (G) s = lim i∈I det N (G/Gi) r (2) A[i] : L 2 (G/G i ) r → L 2 (G/G i ) s
, where the existence of the limit above is part of the claim. (2) * using the standard Hilbert structure on L 2 (G) and L 2 (G/G i ). We emphasize that in the sequel after fixing a QG-basis for C * the Q[G/G i ]-basis for C * [i] and the Hilbert structures on C (2) * has to be chosen in this particular way.
Denote by We have the following chain complex version which is obviously equivalent.
ρ (2) C (2) * := − p≥0 (−1) p · ln det (2) N (G) c (2) p ; (14.3) ρ (2) C[i]
Conjecture 14.5 (Approximation Conjecture for L 2 -torsion of chain complexes).
A group G together with an inverse system {G i | i ∈ I} as in Setup 12.1 satisfies the Approximation Conjecture for L 2 -torsion of chain complexes if for any finite based free QG-chain complex C * we have It is conceivable that Theorem 14.10 remains to true if we drop the assumption that b 14.
ρ (2) C (2) * = lim i∈I ρ (2) C[i]
3. An inequality. We always have the following inequality.
Theorem 14.11 (Inequality). Consider a group G together with an inverse system {G i | i ∈ I} as in Setup 12.1. Suppose that Assumption 13.5 holds. Consider a matrix A ∈ M r,s (QG) with coefficients in QG.
Then we get the inequality
det (2) N (G) r (2) A : L 2 (G) r → L 2 (G) s ≥ lim sup i∈I det (2) N (G/Gi) r (2) A[i] : L 2 (G/G i ) r → L 2 (G/G i ) s .
The proof of Theorem 14.11 will be given in Section 17.
14.4. Matrices invertible in L 1 (G).
Theorem 14.12 (Invertible matrices over L 1 (G)). Consider a group G together with an inverse system {G i | i ∈ I} as in Setup 12.1. Consider an invertible matrix
A ∈ GL d (L 1 (G)) with coefficients in L 1 (G). The projection G → G/G i induces a ring homomorphism L 1 (G) → L 1 (G/G i ). Thus we obtain for each i ∈ I an invertible matrix A[i] ∈ GL d (L 1 (G/G i )).
Then the Approximation Conjecture for Fuglede-Kadison determinants 14.1 holds for A, i.e.,
det (2) N (G) r (2) A : L 2 (G) d → L 2 (G) d = lim i∈I det (2) N (G/Gi) r (2) A[i] : L 2 (G/G i ) d → L 2 (G/G i ) d .
Theorem 14.12 has already been proved by Deninger [24,Theorem 17] (1) Let C * be a finite based free L 1 (G)-chain complex which is acyclic. Then
ρ (2) C (2) * = lim i∈I ρ (2) C * [i] (2) ;
(2) Let C * and D * be finite based free L 1 (G)-chain complexes. Suppose that they are L 1 (G)-chain homotopy equivalent. Then (2) .
ρ (2) C (2) * − ρ (2) D (2) * = lim i∈I ρ (2) C * [i] (2) − ρ (2) D * [i]
The proofs of Theorem 14.12 and Corollary 14.13 will be given in Section 17.
15. The L 2 -de Rham isomorphism and the proof of Theorem 14.10
In this section we investigate the L 2 -de Rham isomorphism in order to give the proof of Theorem 14.10.
Let M be a closed Riemannian manifold. Fix a smooth triangulation K of M . Consider a (discrete) group G and a G-covering pr : M → M . The smooth triangulation K of M lifts to G-equivariant smooth triangulation K of M . Denote by pr : K → K the associated G-covering. Equip M with the Riemannian metric for which pr : M → M becomes a local isometry.
In the sequel we will consider the de Rham isomorphism
Int p : H p (2) (M ) ∼ = − → H p (2) (K). (15.1)
from the space of harmonic L 2 -integrable p-forms on M to the L 2 -cohomology of the free simplicial G-complex K. It is essentially given by integrating a p-form over a p-simplex and is an isomorphism of finitely generated Hilbert N (G)-modules. For more details we refer to [26] or [66, Theorem 1.59 on page 52].
There is the de Rham cochain map (for large enough fixed k) (see [26] or [66, (1.77)
on page 61]
A * : H k− * Ω p (M ) → C * (2) (K) (15.2)
where H k− * Ω p (M ) denotes the Sobolev space of p-forms on M . Lemma 15.3. Assume that for every simplex σ of K we can find a neighborhood V σ together with a diffeomorphism η σ : R m → V σ . (This can be arranged by possibly passing to a d-fold barycentric subdivision of K.) Fix an integer p with 0 ≤ p ≤ dim(M ).
Then there exist constants L 1 , L 2 > 0 which depend on data coming from M and K, but do not depend on G and pr : M → M such that for every ω ∈ H k−p Ω p (M ) we have L 1 · ||ω|| H k−p ≤ ||A p (ω)|| L 2 ≤ L 2 · ||ω|| H k−p , and we get for the operator norm of the operator Int p of (15.1)
L 1 ≤ || Int p || ≤ L 2 .
Proof. Dodziuk [26,Lemma 3.2] proves for a given G-covering pr : M → M and p ≥ 0 that the map A p :
H k−p Ω p (M ) ∼ = − → C p (2) (K) is bounded, i.e., there exists a constant L 2 such that for ω ∈ H k−p Ω p (M ) we have ||A p (ω)|| L 2 ≤ L 2 · ||ω|| H k−p .
Next we will analyze Dodziuk's proof and explain why the constant L 2 depends only on data coming from M and K and does not depend on G and pr : M → M .
For every p-simplex σ in K we choose a relatively compact neighborhood U σ of σ with U σ ⊆ V σ . Choose N to be an integer such that every point x ∈ M belongs to at most N of the sets U σ , where σ runs through the p-simplices of K, e.g., take N to be the number of p-simplices of K. We can apply [26, Lemma 3.1] to σ ⊆ V σ and obtain a constant C σ > 0 such that for any p-form ω in H k−p Ω p (M ) and any
p-simplex σ of K sup x∈σ |ω(x)| ≤ C σ · ||ω|| Uσ H k−p + ||ω|| Uσ L 2
holds. The real number |ω(x)| is the norm of ω(x) as an element in Λ p T * x M , and ||ω|| Uσ H k−p and ||ω|| Uσ L 2 are the Sobolev norm and L 2 -norm of ω restricted to U σ . Let C be the maximum of the numbers C σ , where σ runs through all p-simplices of K. Let E be the maximum over the volumes of the p-simplices of K. Obviously the numbers C, E and N depend only on data coming from M and K, but do not depend on G and pr : M → M .
Since V σ is contractible, the restriction of pr : M → M to pr −1 (V σ ) is trivial and hence pr −1 (V σ ) is G-diffeomorphic to G × V σ . Hence there are for every p-simplex σ ∈ M open neighborhoods U σ and V σ which are uniquely determined by the property that they are mapped diffeomorphically under pr : M → M onto U pr(σ) and V pr(σ) . Notice for the sequel that pr | V σ : V σ → V pr(σ) and pr | U σ : U σ → U pr(σ) are isometric diffeomorphisms. Hence we get for every p-form ω in H k−p Ω p (M ) and any p-simplex σ of K
sup x∈σ |ω(x)| ≤ C · ||ω|| U σ H k−p + ||ω|| U σ L 2 .
Here the real number |ω(x)| is the norm of ω(x) as an element in Λ p T * x M , and ||ω|| V σ H k−p and ||ω|| V σ L 2 are the Sobolev norm and L 2 -norm of the restriction of ω to V σ . One easily checks that every point x ∈ M belongs to at most N of the sets U σ , where σ runs through the p-simplices of K and that the volume of every p-simplex of K is bounded by E. Put
L 2 := √ 4 · C 2 · E 2 · N .
Then L 2 depends only on data coming from M and K, but does not depend on G and pr : M → M .
Next we perform essentially the same calculation as in [26,Lemma 3.2] We estimate for a p-simplex σ of K and an element ω ∈ H k−p Ω p (M )
σ ω 2 ≤ sup x∈σ |ω(x)| · vol(σ) 2 ≤ C · ||ω|| Uσ H k−p + ||ω|| Uσ L 2 · vol(σ) 2 ≤ C 2 · 2 · ||ω|| Uσ H k−p 2 + |ω|| Uσ L 2 2 · E 2 .
This implies for ω ∈ H k−p Ω p (M ), where σ runs through the p-simplices of K.
σ σ ω 2 ≤ σ C 2 · 2 · ||ω|| U σ H k−p 2 + |ω|| U σ L 2 2 · E 2 ≤ 2 · C 2 · E 2 · σ ||ω|| Uσ H k−p 2 + ||ω|| Uσ L 2 2 ≤ 2 · C 2 · E 2 · N · ||ω|| H k−p 2 + |ω|| L 2 2 ≤ 2 · C 2 · E 2 · N · 2 · ||ω|| 2 H k−p = L 2 2 · ||ω|| 2 H k−p .
We conclude that the de Rham map
A p : H k−p Ω p (M ) → C p
(2) (K) is a bounded operator whose norm is less or equal to L 2 .
Dodziuk [26,Lemma 3.7] (see also [66, (1.78) on page 61]) constructs a bounded G-equivariant operator (15.4) and gives an upper bound for its operator norm by a number p-simplices of K · max ||W p σ|| H k−p | σ p-simplex of K .
W p : C p (2) (K) → H k−p Ω p (M ),
Define
L 1 := 1 p-simplices of K · max ||W σ|| H k−p | σ p-simplex of K .
Obviously L 1 depends only on data coming from M and K, but not on G and pr : M → M . The maps A p of (15.2) and W p of (15.4) induce bounded G-operators (see [
H p (2) (A * ) : H p (2) H k− * Ω * (M ) → H p (2) C * (2) (K) ; H p (2) (W * ) : H p (2) C * (2) (K) → H p (2) H k− * Ω * (M )
, such that we obtain for their operator norms
H p (2) (A * ) ≤ L 2 ; H p (2) (W * ) ≤ 1 L 1 .
Since W p • A p = id and H p (2) (A p ) is an isomorphism (see [26, (3.6), Lemma 3.8 and Lemma 3.10] and [66, (1.79) and (1.80) on page 61], the map H p (2) (W * ) is the inverse of H p (2) (A * ). This implies
L 1 ≤ H p (2) (A * ) ≤ L 2 .
Since the canonical inclusion We assume that the Approximation Conjecture for L 2 -torsion of chain complexes 14.5 is true. Hence we get
i p : H p (2) ∼ = − → H p (2) H k− * Ω * (M ) is an isometric G-isomorphism (ρ (2) K; N (G) = lim i∈I ρ (2) (K[i]; N (G/G i ) . (15.5)
In the sequel we will use the theorem of Burghelea, Friedlander, Kappeler and McDonald [16] that the topological and the analytic L 2 -torsion agree. Since M and hence K is L 2 -acyclic, we get from the definitions
ρ (2) an M ; N (G) = ρ (2) K; N (G) ; ρ (2) an M [i]; N (G/G i ) = ρ (2) K[i]; N (G/G i ) − p≥0 (−1) p · det N (G/Gi) (Int[i] p ), where Int[i] p : H p (2) (M [i]) ∼ = − → H p (2) (K[i])
is the L 2 -de Rham isomorphism of (15.1). Hence it suffices to show for p ≥ 0 lim i∈I ln det This assumption is satisfied in many interesting cases. It is possible that this assumption is not needed for Theorem 14.10 to be true, but our proof does not work without it. We can drop this assumption, if we can generalize (15.6) to lim i∈I ln det N (G) (Int[i] p ) = ln det N (G) (Int p ) .
b (2) p M [i]; N (G/G i ) = dim N (G/Gi) H p (2) (M [i]) = dim N (G/Gi) H p (2) (K[i]) , we conclude ln(L 1 ) · b (2) p M [i]; N (G/G i ) ≤ ln det (2) N (G/Gi) (Int[i] p ) ≤ ln(L 2 ) · b (2) p (M [i]; N (G/G i ) .
A strategy to prove the Approximation Conjecture for
Fuglede-Kadison determinants 14.1
16.1. The general setup. Throughout this section we will consider the following data:
• G is a group, B is a matrix in M d (N (G)) and tr : M d (N (G)) → C is a faithful finite normal trace.
B[i] : L 2 (Q i ) d → L 2 (Q i ) d with respect to the trace tr i . If r (2)
B is positive, we get F (λ) = tr(E λ ) for {E λ | λ ∈ R} the family of spectral projections of r Recall that for a directed set I and a net (x i ) i∈I of real numbers one defines lim inf i∈I x i := sup{inf{x j | j ∈ I, j ≥ i} | i ∈ I}; (16.1) lim sup i∈I x i := inf{sup{x j | j ∈ I, j ≥ i} | i ∈ I}.
B[i] : L 2 (Q i ) d → L 2 (Q i ) d ≥ κ for each i ∈ I, (iv) Suppose r (2) B : L 2 (G) d → L 2 (G) d and r (2) B[i] : L 2 (Q i ) d → L 2 (Q i ) d for i ∈ I are positive;
(v) The uniform integrability condition is satisfied, i.e., there exists ǫ > 0 such that
ǫ 0+ sup F [i](λ) − F [i](0) λ i ∈ I dλ < ∞.
Then:
(1) If conditions (i), (ii) (iii), and (iv) are satisfied, then det r
(2) B : L 2 (G) d → L 2 (G) d ≥ lim sup i∈I det i r (2) B[i] : L 2 (Q i ) d → L 2 (Q i ) d ;
(2) If conditions (i), (ii) (iii), (iv) and (v) are satisfied, then det r
(2) B : L 2 (G) d → L 2 (G) d = lim i∈I det i r (2) B[i] : L 2 (Q i ) d → L 2 (Q i ) d .
Proof. Completely analogously to the proof of [66, Theorem 13.19 on page 461] we prove
F (λ) = F + (λ) = F + (λ) for λ ∈ R; (16.4) F (0) = lim i∈I F [i](0); (16.5) κ ≤ ln det (2) r (2) B : L 2 (G) d → L 2 (G) d . (16.6)
The proof of [66, (13.22) and (13.23) on page 462] carries directly over and yields ln(det(r Since F and F are monotone increasing bounded functions, there are only countably many elements λ ∈ [0, ∞) such that F (λ) = F + (λ) or F (λ) = F + (λ) hold.
(2) B )) = ln(K) · (F (K) − F (0)) − K 0+ F (λ) − F (0) λ dλ; (16.7) ln(det i (r (2) B[i] )) = ln(K) · (F [i](K) − F [i](0)) − K 0+ F [i](λ) − F [i](0) λ dλ.
We conclude from (16.4) that there is a countable set S ⊆ [0, ∞) such that for all λ ∈ [0, ∞) \ S the limit lim i→∞ F [i](λ) exists and is equal to F (λ). Since S has measure zero and we have (16.5), we get almost everywhere for λ ∈ [0, ∞)
lim i∈I F [i](λ) − F [i](0) λ = F (λ) − F (0) λ (16.10)
Analogously to the proof of [66, (13.28) on page 463] one shows
K 0+ lim i∈I F [i](λ) − F [i](0) λ dλ ≤ lim inf i∈I K 0+ F [i](λ) − F [i](0) λ dλ.
This implies
K 0+ F (λ) − F (0) λ dλ ≤ lim inf i∈I K 0+ F [i](λ) − F [i](0) λ dλ. (16.11)
Now assertion (1) follows from (16.7), (16.8), and (16.11). Next we prove assertion (2). We can apply Lebesgue's Dominated Convergence Theorem to (16.10) because of the assumption (v) and obtain
K 0+ lim i∈I F [i](λ) − F [i](0) λ dλ = lim i∈I K 0+ F [i](λ) − F [i](0) λ dλ.
Now assertion (2) follows from (16.7), and (16.8). This finishes the proof of Theorem 16.3.
16.3. The uniform integrability condition is not automatically satisfied.
The main difficulty to apply Theorem 16.3 to the situations of interest is the verification of the uniform integrability condition (v) appearing in Theorem 16.3. We will illustrate by an example that one needs extra input to ensure this condition since there are examples where this condition is violated but all properties of the spectral density functions which are known so far are satisfied. Define the following sequence of functions f n :
[0, 1] → [0, 1] f n (λ) = λ 0 ≤ λ ≤ e −3n ; (e −2n −λ)·e −3n +(λ−e −3n )· 1 − ln(e −2n ) +e −2n e −2n −e −3n e −3n ≤ λ ≤ e −2n ; 1 − ln(λ) + λ e −2n ≤ λ ≤ e −n ; 1 − ln(e −n ) + e −n e −n ≤ λ ≤ 1 − ln(e −n ) + e −n ; λ 1 − ln(e −n ) + e −n ≤ λ ≤ 1.
Lemma 16.12.
(1) The function f n (λ) is monotone non-decreasing and continuous for n ≥ 1;
(2) f n (0) = 0 and f n (1) = 1 for all n ≥ 1;
(3) lim n→∞ f n (λ) = λ for λ ∈ [0, 1]; (4) We have for all n ≥ 1 and λ ∈ [0, 1)
λ ≤ f n (λ) ≤ 1 − ln(λ) + λ ≤ 2 − ln(λ) ; (5) We have for λ ∈ [0, e −1 ]
sup f n (λ) | n ≥ 0} = 1 − ln(λ) + λ;
(6) We have 1 0+ sup{f n (λ) | n ≥ 0} λ dλ = ∞;(7)
We get for all n ≥ 1 1 0+ f n (λ) λ dλ ≥ ln(2) + 1;
(8) We have 1 0+ lim n→∞ f n (λ) λ dλ < lim inf n→∞ 1 0+ f n (λ) λ dλ ≤ lim sup n→∞ 1 0+ f n (λ) λ dλ;
(9) We get for all n ≥ 1
1 0+ f n (λ) λ dλ ≤ 4.
Proof.
(1) One easily checks that the definition of f n makes sense, in particular at the values λ = e −3n , e −2n , e −n .
The first derivative of f n (λ) exists with the exception of λ = e −3n , e −2n , e −n , 1 − ln(e −n ) + e −n and is given by − ln(λ) + λ is monotone non-decreasing. We have λ ≤ 1 − ln(λ) for λ ∈ (0, 1). (5) From assertion (4) we conclude sup f n (λ) | n ≥ 0} ≤ 1 − ln(λ) + λ for λ ∈ [0, 1). Since for λ with 0 < λ ≤ e −1 we can find n ≥ 1 with e −2n ≤ λ ≤ e −n and hence f n (λ) = 1 − ln(λ) + λ holds for that n, we conclude sup f n (λ) | n ≥ 0} = 1 − ln(λ) + λ for λ ∈ [0, e −1 ]. (6) We compute using assertion (5) for every ǫ ∈ (0, e −1 )
f ′ n (λ) = 1 0 ≤ λ < e −3n ; 1 + 1 2n·(e −2n −e −3n ) e −3n < λ < e −2n ; 1 λ·ln(λ) 2 + 1 e −2n < λ < e −n ; 0 e −n ≤ λ ≤ 1 − ln(e −n ) + e −n ;1 0+ sup f n (λ) | n ≥ 0} λ dλ ≥ e −1 ǫ sup f n (λ) | n ≥ 0} λ dλ = e −1 ǫ 1 + 1 λ · (− ln(λ)) dλ = e −1 − ǫ + [− ln(− ln(λ))] e −1 ǫ = e −1 − ǫ − ln(− ln(e −1 ) + ln(− ln(ǫ))
= e −1 − ǫ + ln(− ln(ǫ)) ≥ ln(− ln(ǫ)).
Since lim ǫ→0+ ln(− ln(ǫ)) = ∞, assertion (6) follows.
We estimate for given n ≥ 1 using the conclusion f n (λ) − λ ≥ 0 for λ ∈ [0, 1] from assertion (4)
1 0+ f n (λ) λ dλ = 1 0+ f n (λ) − λ λ dλ + 1 ≥ e −n e −2n f n (λ) − λ λ dλ + 1 = e −n e −2n(9) We estimate 1 0+ f n (λ) λ dλ = e −3n 0+ f n (λ) λ dλ + e −2n e −3n f n (λ) λ dλ + e −n e −2n f n (λ) λ dλ + 1 − ln(e −n ) +e −n e −n f n (λ) λ dλ + 1 1 − ln(e −n ) +e −n f n (λ) λ dλ = e −3n 0+ λ λ dλ + e −2n e −3n (e −2n − λ) · e −3n + (λ − e −3n ) · 1 − ln(e −2n ) + e −2n e −2n − e −3n · 1 λ dλ + e −n e −2n 1 − ln(λ) + λ λ dλ + 1 n +e −n e −n 1 − ln(e −n ) + e −n λ dλ + 1 1 n +e −n λ λ dλ = e −3n 0+ 1 dλ + e −2n e −3n 1 + 1 2n · (e −2n − e −3n ) + e −3n 2n · (e −2n − e −3n ) · 1 λ dλ + e −n e −2n 1 − 1 ln(λ) · λ dλ + 1 n +e −n e −n 1 n + e −n λ dλ + 1 1 n +e −n 1 dλ = e −3n + e −2n e −3n 1 + 1 2n · (e −2n − e −3n ) dλ + e −2n e −3n e −3n 2n · (e −2n − e −3n ) · 1 λ dλ + e −n e −2n 1 dλ − e −n e −2n 1 ln(λ) · λ dλ + 1 n +e −n e −n 1 n + e −n λ dλ + 1 − 1 n − e −n = e −3n + e −2n − e −3n · 1 + 1 2n · (e −2n − e −3n ) + e −3n · ln(λ) 2n · (e −2n − e −3n )+ 1 − 1 n − e −n = 1 − 3 2n + e −3n · (−2n + 3n) 2n · (e −2n − e −3n )
− ln(n) − ln(2n) + 1 n + e −n · ln 1 n + e −n − ln(e −n ) = 1 − 3 2n + 1 2 · (e n − 1) + ln(2) + 1 n + e −n · ln e n n + 1
≤ 1 + 1 2 · (e − 1) + ln(2) + 2 n · ln(2e n ) = 1 + 1 2 · (e − 1) + ln(2) + 2 · ln(2) ≤ 4.
This finishes the proof of Lemma 16.12.
Remark 16.13 (Exotic behavior at zero). The sequence of functions (f n ) n≥0 has an exotic behavior close to zero in a small range depending on n. There are no C > 0 and ǫ > 0 such that f ′ n (λ) ≤ C holds for all n ≥ 1 and all λ ∈ (0, ǫ) for which the derivative exists.
This exotic behavior is responsible for the violation of the the uniform integrability condition, see Lemma 16.12 (6). It is very unlikely that such a sequence (f n ) n≥0 actually occurs as the sequence of spectral density functions of the manifolds G i \M for some smooth manifold M with proper free cocompact G-action and G-invariant Riemannian metric. The example above shows that we need to have more information on such sequences of spectral density functions. 16.4. Uniform estimate on spectral density functions. The crudest way to ensure the uniform integrability condition (v) appearing in Theorem 16.3 is to assume a uniform gap in the spectrum, namely we have the obvious Lemma 16.14. Suppose that the uniform gap in the spectrum at zero condition is satisfied, i.e., there exists ǫ > 0 such that for all i ∈ I and λ ∈ [0, ǫ] we have
F [i](λ) = F [i](0).
Then the uniform integrability condition (v) appearing in Theorem 16.3 is satisfied.
However, this is an unrealistic condition in our situation for closed aspherical manifolds because of the following remark.
Remark 16.15 (The Zero-in-the-Spectrum Conjecture). Let M be an aspherical closed manifold. If one wants to use Lemma 16.14 in connection with Theorem 16.3 to prove Conjecture 6.2 or more generally Conjecture 14.8 for M , one has to face the fact that the assumption that one has in each dimension a uniform gap in the spectrum at zero implies that b (2) p ( M ) vanishes and the p-th Novikov-Shubin invariant satisfies α p ( M ) = ∞ * for all p ≥ 0. In other words, M must be a counterexample to the Zero-in-the-Spectrum Conjecture which is discussed in detail in [66, Chapter 12 on pages 437 ff]. Such counterexample is not known to exist and it is evident that it is hard to find one. Therefore the the uniform gap in the spectrum at zero condition is not useful in this setting.
There are examples where Lemma 16.14 does apply when one allows to twist with representations in favorable case, see for instance [12,74,78,79].
Here is a more promising version.
Theorem 16.16 (The uniform logarithmic estimate). Suppose that there exists constants C > 0, 0 < ǫ < 1 and δ > 0 independent of i such that
F [i](λ) − F [i](0) ≤ C (− ln(λ)) 1+δ ,
Then the uniform integrability condition (v) appearing in Theorem 16.3 is satisfied.
Proof. This follows from the following calculation.
ǫ +0 C λ · (− ln(λ)) 1+δ dλ λ=exp(µ) = ln(ǫ) −∞ C exp(µ) · (− ln(exp(µ))) 1+δ · exp(µ) dµ = ln(ǫ) −∞ C (−µ) 1+δ dµ µ=−ν = ∞ − ln(ǫ) C ν 1+δ dν = lim x→∞ x − ln(ǫ) C ν 1+δ dν = lim x→∞ −δ · x −δ − (− ln(ǫ)) −δ = δ · (− ln(ǫ)) −δ < ∞,
The p-th spectral density function F p (X[i]) of X[i] is defined as the spectral density function F c (2) p (X[i]) ) of the p-th differential of C
F p (X[i])(λ) − F p (X[i])(0) [G : G i ] ≤ C − ln(λ)
. (16.17) But this is not enough, since one cannot take δ = 0 in Theorem 16.16, namely, we have for every C > 0 and 0 < ǫ < 1
ǫ +0 C λ · (− ln(λ)) dλ = lim x→0+ ǫ x C λ · (− ln(λ)) dλ = lim x→0+
C · (− ln(− ln(ǫ)) + ln(− ln(x))) = ∞.
Condition (16.17) is implied by the stronger condition that there exist for each p ≥ 0 constants C p > 0, ǫ p > 0 and α p > 0 independent of i such that we have for all λ ∈ [0, ǫ p ) and all i = 1, 2, . . . [44], where they show that there exists an explicit element a in the integral group ring of the wreath product Z 3 ≀ Z such that the spectral density function of the associated N (Z 3 ≀ Z)-map r a : N (Z 3 ≀Z) → N (Z 3 ≀Z) does not satisfy condition (16.18). This implies that there exists a closed Riemannian manifold M with fundamental group G = π 1 (X) = Z 3 ≀Z such that for some p condition (16. There is no counterexample known to the condition appearing in Theorem 16.16 but the constant δ has to be depend on the group G, see Grabowski [43]. 17. Proof of Theorem 14.11,Theorem 14.12,and Corollary 14.13 In this section we derive the proofs of Theorem 14.11, 14.12 from Theorem 16.3. This needs some preparation.
F p (X[i])(λ) − F p (X[i])(0) [G : G i ] ≤ C · λ αp .(
Define for a matrix B ∈ M r,s (L 1 (G)) the real number
K G (B) := rs · max{||b i,j || L 1 | 1 ≤ i ≤ r, 1 ≤ j ≤ s}, (17.1)
where for a = g∈G λ g · g its L 1 -norm ||a|| L 1 is defined by g∈G |λ g |. Consider ǫ > 0. We can choose a finite subset S ⊆ G with e ∈ S such that g∈G,g / ∈S |λ g (r, r)| < ǫ/d holds for all r ∈ {1, 2, . . . , d}. Since i∈I G i = {1}, there exists an index i S such that ψ i (g) = e ⇒ g = e holds for all g ∈ S and i ≥ i S . This
implies for i ≥ i S tr N (G) (B) − tr N (G/Gi) (B[i]) = d r=1 λ e (r, r) − d r=1 g∈G,ψi(g)=e λ g (r, r) ≤ d r=1 λ e (r, r) − g∈G,ψi(g)=e λ g (r, r) = d r=1 g∈G,g / ∈S,ψi(g)=e λ g (r, r) ≤ d r=1 g∈G,g / ∈S,ψi(g)=e λ g (r, r) ≤ d r=1 g∈G,g / ∈S λ g (r, r) ≤ d r=1 ǫ/d = ǫ.
Next we give the proof of Theorem 14.11.
Proof of Theorem 14.11. We have to show for A ∈ M r,s (QG).
(17.4) det N (G) r (2) A : L 2 (G) r → L 2 (G) s ≥ lim sup i∈I det N (G/Gi) r (2) A[i] : L 2 (G/G i ) r → L 2 (G/G i ) s .
We first deal with the special case that A ∈ M r,s (ZG). We will apply Theorem 16.3 to the following special situation:
• B = A * A; • Q i = G/G i ; • B[i] = A[i] * A[i];
• tr is the von Neumann trace tr N (G) : N (G) → C;
• tr i is the von Neumann trace tr N (G/Gi) : N (G/G i ) → C.
We have to check that the conditions of Theorem 16.3 (1) are satisfied. We obtain Condition (i) appearing in Theorem 16.3 from Lemma 17.2 since the projection L 1 (G) → L 1 (G/G i ) has operator norm at most 1 and hence we get for the number K G (B) defined in (17.1)
K Qi (B[i]) ≤ K G (B).det N (G) r (2) B : L 2 (G) r → L 2 (G) r ≥ lim sup i∈I det N (G) r (2) B[i] : L 2 (G/G i ) r → L 2 (G/G i ) r .
Since we get from [66, Lemma 3.15 (4) on page 129]
det N (G) r (2) A = det N (G) r (2) B ; det N (G(Gi ) r (2) A[i] = det N (G(Gi r (2) B[i] ,det N (H) f • m id L 2 (H) r 2 = det N (H) f • m id L 2 (H) r * • f • m id L 2 (H) r = det N (H) f * • f • m 2 id L 2 (H) r = det N (H) f * • f • m 2 id L 2 (H) r ker f * •f •m 2 id L 2 (H) r ⊥ = det N (H) f * • f ker(f * f ) ⊥ • m 2 id ker(f * f ) ⊥ = det N (H) f * • f ker(f * f ) ⊥ · det N (H) m 2 id ker(f * f ) ⊥ = det N (H) f * • f · m 2·dim N (H) ker(f * f ) ⊥ = det N (H) (f ) 2 · m 2r−2 dim N (H) (ker(f * f )) = det N (H) (f ) · m r−dim N (H) (ker(f )) 2 .
Thus we have shown
det N (H) f • m id L 2 (H) r = det N (H) f ) · m r−dim N (H) (ker(f )) . (17.5)
Let m ≥ 1 be an integer such that mI r ·A belongs to M r,s (ZG), where mI r is obtained from the identity matrix by multiplying all entries with m. If we apply (17.5) in the case H = G and H = G/G i to f = r (2) A and f = r
(2) A[i] , we obtain det N (G) r (2) mIr ·A = det N (G) r (2) A · m r−dim N (G) (ker(r (2) A )) ; det N (G/Gi) r (2) mIr ·A[i] = det N (G/Gi) r(2)A[i] · m r−dim N (G/G i ) (ker(r(2)
A[i] )) .
Since det N (G) r (2) mIr ·A ≥ 1 follows from (17.4) and the assumption that we have det N (G/Gi) r (2) mIr ·A[i] ≥ 1 for i ∈ I, we get det N (G) r (2) A > 0. We conclude (17.6) det N (G/Gi) r
(2) A[i] det N (G) r (2) A = det N (G/Gi) r (2) mIr A[i] det N (G) r (2) mIr A · m | dim N (G) (ker(r (2) A ))−dim N (G/G i ) (ker(r (2) A[i] ))| .
We derive
lim i∈I dim N (G) ker r (2) A − dim N (G/Gi) ker r (2) A[i] = 0 (17.7)
from Theorem 13.6. Since (17.4) holds for mI r A, it holds also for A by (17.6) and (17.7). This finishes the proof of Theorem 14.11.
Next we give the proof of Theorem 14.12.
Proof of Theorem 14.12. We will apply Theorem 16.3 to the following special situation:
• B = A * A;
• Q i = G/G i ; • B[i] = A[i] * A[i]; •K G/Gi (B[i] −1 ) ≤ K.
hold. We conclude ||r Hence condition (iii) is satisfied if we take κ := d · ln(K). We conclude from (17.8) that also condition (v) is satisfied. We conclude from Theorem 16.3 (2)
det (2) N (G) r (2) B : L 2 (G) d → L 2 (G) d = lim i∈I det 2 N (G/Gi) r (2) B[i] : L 2 (G/G i ) d → L 2 (G/G i ) d .
Since we get from [
N (G) r (2) A = det (2) N (G) r (2) B ; det (2) N (G/Gi) r (2) A[i] = det(2)
N (G/Gi) r
B[i] , Theorem 14.12 follows.
Next we prove Corollary 14.13 Proof of Corollary 14.13. (1) Since C * is acyclic over L 1 (G) and finitely generated free, we can choose an L 1 (G)-chain contraction γ : C * → C * +1 . Then (c + γ) odd : C odd ∼ = − → C ev is an isomorphism of finitely generated based free L 1 (G)modules. It induces an isomorphism of finitely generated Hilbert N (G)-modules (c + γ) (2) odd : C
(2) odd ∼ = − → C(2)ρ (2) C (2) * := ln det (2) N (G) (c + γ) (2) odd : C (2) odd ∼ = − → C (2) ev .
Analogously we prove for each i ∈ I
ρ (2) C[i]
(2) * := ln det
(2) N (G/Gi) (c[i] + γ[i])(2)odd : C[i] (2) odd ∼ = − → C[i] (2) ev .
Now assertion (1) follows from Theorem 14.12.
(2) We begin with the case of an isomorphism f * : C * ∼ = − → D * of finitely generated based free L 1 (G)-chain complexes. We conclude from [66, Lemma 3.41 on page 146] for all i ∈ I
ρ (2) D (2) * − ρ (2) C (2) * = p≥0 (−1) p · ln det det (2) N (G) (f (2) p ) ; ρ (2) D[i] (2) * − ρ (2) C[i] (2) * = p≥0 (−1) p · ln det det (2) N (G/Gi) (f [i] (2) p ) .
Now the claim follows in this special case from Theorem 14.12.
Finally we consider an L 1 (G)-chain homotopy equivalence f * : C * ≃ − → D * . Let cyl(f * ) be its mapping cylinder and cone(f * ) be its mapping cone. Let cone(C * ) be the mapping cone of C * . We obtain based exact sequences of L 1 (G)-chain complexes
0 → C * → cyl(f * ) → cone(f * ) → 0 and 0 → D * → cyl(f * ) → cone(C * ) → 0.
Since f * is a L 1 (G)-chain homotopy equivalence, cone(f * ) is contractible. Since cone(C * ) is contractible, we can find isomorphisms of L 1 (G)-chain complexes (cf. [
u * : C * ⊕ cone(f * ) ∼ = − → cyl(f * ); v * : D * ⊕ cone(C * ) ∼ = − → cyl(f * ).
Since we have already treated the case of a chain isomorphism, we conclude
ρ (2) C * ⊕ cone(f * ) (2) − ρ (2) cyl(f * ) (2) = lim i∈I ρ (2) C[i] * ⊕ cone(f [i] * ) (2) − ρ (2) cyl(f [i] * ) (2) ; and ρ (2) D * ⊕ cone(C * ) (2) − ρ (2) cyl(f * ) (2) = lim i∈I ρ (2) D[i] * ⊕ cone(C[i] * ) (2) − ρ (2) cyl(f [i] * ) (2) .
This implies
ρ (2) C (2) * + ρ (2) cone(f (2) * ) − ρ (2) D (2) * − ρ (2) cone(C * ) (2) = lim i∈I ρ (2) C[i] (2) * + ρ (2) cone(f [i] * ) (2) − ρ (2) D[i]
(2) * We conclude from assertion (1)
ρ (2) cone(f (2) * ) = lim i∈I ρ (2) cone(f [i] * ) (2) ; ρ (2) cone(C (2) * ) = lim i∈I ρ (2) cone(C[i] * ) (2) .
This implies
ρ (2) C (2) * − ρ (2) D (2) * = lim i∈I ρ (2) C[i] (2) * − ρ (2) D[i] (2) * .
This finishes the proof of Corollary 14.13.
18. Proof of Theorem 7.13
Next we want to prove Theorem 7.13. First we deal with homotopy invariance and with the relationship between L 2 -torsion and integral torsion. Lemma 18.3. Let u : Z r → Z s be a homomorphism of abelian groups. Let j : ker(u) → Z r be the inclusion and pr : Z s → coker(u) f be the canonical projection. Choose Z-basis for ker(u) and coker(u) f .
Then det N ({1}) j (2) and det N ({1}) pr (2) are independent of the choice of the Z-basis for ker(u) and coker(u) f , and we have (2) ;
det N ({1}) (u (2) ) = det N ({1}) j (2) · tors(coker(u)) · det N ({1}) prand 1 ≤ det N ({1}) (j (2) ) ≤ det N ({1}) (u (2) ); 1 ≤ det N ({1}) pr (2) ≤ det N ({1}) (u (2) ); 1 ≤ tors(coker(u)) ≤ det N ({1}) (u (2) ).
The point of the next lemma is that the chain complexes live over ZG but the chain homotopy equivalence has only to exist over QG. Lemma 18.4. Let C * and D * be two finite free ZG-chain complexes. Suppose that C * ⊗ Z Q and D * ⊗ Z Q are QG-chain homotopy equivalent and that C
ρ (2) D (2) * − ρ (2) C (2) * = lim i∈I ρ Z D[i] * − ρ Z C[i] * [G : G i ] . (18.7) ker tors H p (C ′′ [i] * ) → tors H p−1 (C[i] * ) im tors H p (C ′ [i] * ) → tors H p (C ′′ [i] * ) ∼ = coker (H p (C[i] * ) f → H p (C ′ [i] * ) f ) , and (18.8) ker tors H p (C[i] * ) → tors H p (C ′ [i] * ) im tors H p+1 (C[i] ′′ * ) → tors H p (C[i] * ) ∼ = 0. Obviously (H p (C[i] * ) f and hence ker (H p (C[i] * ) f → H p (C ′ [i] * ) f ) are torsionfree.
On the other hand ker (
H p (C[i] * ) f → H p (C ′ [i] * ) f ) is finite, since tors(H p (C ′ [i] * )) is finite and ker (H p (C[i] * ) → H p (C ′ [i] * )) is a quotient of H p+1 (C ′′ [i] * ) and hence finite. Hence ker (H p (C[i] * ) f → H p (C ′ [i] * ) f ) is trivial. We conclude from (18.6) (18.9) ker tors H p (C ′ [i] * ) → tors H p (C ′′ [i] * ) = im tors H p (C[i] * ) → tors H p (C ′ [i] * ) .
The cokernel of the map
H p (C[i] * ) → H p (C ′ [i] * ) is a submodule of H p−1 (C ′′ [i] * )
and hence annihilated by multiplication with m. The cokernel of
H p (C[i] * ) f → (H p (C ′ [i] * ) f is a quotient of the cokernel of H p (C[i] * ) → H p (C ′ [i] * ). Hence coker H p (C[i] * ) f → H p (C ′ [i] * ) f is annihilated by multiplication with m.
Therefore we obtain an epimorphism
H p (C ′ [i] * ) f /m · H p (C ′ [i] * ) f → coker H p (C[i] * ) f → H p (C ′ [i] * ) f .
This implies
coker H p (C[i] * ) f → H p (C ′ [i] * ) f ≤ m rkZ(Hp(C ′ [i] * )) .
Since C
(2) * is L 2 -acyclic, and C * ⊗ Z Q and C ′ * ⊗ Z Q are QG-chain homotopy equivalent, (C ′ )
(2) * is L 2 -acyclic. We conclude from [62, Theorem 0.1] for all p ≥ 0
lim i∈I rk Z H p (C ′ * [i]) [G : G i ] = 0.
Since m is independent of p, we conclude
lim i∈I ln coker H p (C[i] * ) f → H p (C ′ [i] * ) f [G : G i ] = 0. (18.10)
Taking the logarithm of the order of a finite abelian group is additive under short exact sequences of finite abelian groups. Hence we get for any finite-dimensional chain complex E * of finite abelian groups p≥0 (−1) p · |E p | = p≥0 (−1) p · |H p (E * )|.
If we apply this to the left column in the diagram (18.5), we conclude from (18.7), (18.8), and (18.9)
p≥0 (−1) p · ln tors H p (C[i] * )) − p≥0 (−1) p · ln tors H p (C ′ [i] * ) + p≥0 (−1) p · ln tors H p (C ′′ [i] * ) = p≥0 (−1) p · ln (|coker (H p (C[i] * ) f → H p (C ′ [i] * ) f )|) ≤ p≥0 ln (|coker (H p (C[i] * ) f → H p (C ′ [i] * ) f )|) .
This together with (18.10) implies
lim i∈I ρ Z (C[i] * ) [G : G i ] − ρ Z (C ′ [i] * ) [G : G i ] + ρ Z (C ′′ [i] * ) [G : G i ] = 0. (18.11)
We conclude from [66, Lemma 3.68 on page 153]
ρ (2) C (2) * − ρ (2) (C ′ ) (2) * + ρ (2) (C ′′ ) (2) * = 0. (18.12)
Hence it suffices to show
ρ (2) (C ′′ ) (2) * = lim i∈I ρ Z (C ′′ [i] * ) [G : G i ] . (18.13)
We conclude from Corollary 14.13 (1)
ρ (2) (C ′′ ) (2) * = lim i∈I ρ (2) C ′′ [i](2)ln det N (G) (f (2) ) ≥ lim sup i∈I ln det N ({1}) (f [i] (2) ) [G : G i ] .
Now apply Lemma 18.3.
(2) Obviously it suffices to prove the claim for chain complexes. Notice that
ρ (2) C[i] (2) * [G : G i ] = ρ (2) C[i](2)ρ (2) C (2) * = lim i∈I ρ (2) C[i] (2) * [G : G i ] .
Since H p C[i] * ⊗ Z Q vanishes for all p ≥ 0 and i ∈ I, assertion (2)
H k (C * ) ⊗ Z Q ∼ = H k (C * ) ⊗ Z[Z] Q[Z] ∼ = t k j=1 Q[Z]/(p m k,j k,j ).
By multiplying the elements p k,j with some natural number and a power of the generator t ∈ Z, we can arrange that the elements p k,1 , p k,2 , . . . , p k,t k belong to
Z[t]. Since Q[Z]/(p m k,j k,j ) ∼ = Z[Z]/(p m k,j k,j ) ⊗ Z Q, there is a map of Z[Z]-modules ξ k : t k j=1 Z[Z]/(p m k,j k,j ) → H k (C * )
which becomes an isomorphism of Q[Z]-modules after applying −⊗ Z Q. By possibly enumerating the polynomials p k,j we can arrange, that for some integer s k with 0 ≤ s k ≤ t k + 1 a polynomial p k,j has some root of unity as a root if and only if j ≤ s k . Consider j ∈ {1, 2, . . . , s k }. Let d k,j ≥ 2 be the natural number for which p k,j has a primitive d k,j -th root of unity as zero. Recall the d-th cyclotomic polynomial Φ d is a polynomial over Z[t] with Φ d k,j (0) = ±1 and is irreducible over
Q[t]. Hence we can find a unit in u ∈ Q[Z] such that u · Φ d k,j = p k,j . Every unit in Q[Z] = Q[t, t −1 ]
is of the shape rt l for some r ∈ Q, r = 0 and l ∈ Z. Since p k,j is a polynomial in Z[t], we can arrange p k,j = Φ d k,j . To summarize, we have achieved that p k,j is Φ d k,j for j ∈ {1, 2, . . . , s k } and that no root of unity is a root of p k,j for j ∈ {s k + 1, s k + 2, . . . , t k }. Let F k,j * for j ∈ {1, 2, . . . , t k } be the Z[Z]-chain complex which is concentrated in dimensions (k+1) and k and whose (k+1)-th differential is the Z[Z]-homomorphism Z[Z] By construction H k (f * ) ⊗ Z Q is bijective for all k ≥ 0. We conclude from Lemma 18.4 that we can assume without loss of generality
C * = k≥0 t k j=1 F k,j * .
Obviously assertion (3) is satisfied for a direct sum D * ⊕ E * of two based free L 2acyclic Z[Z]-chain complexes if both D * and E * satisfy assertion (3). Hence we only have to treat the case, where C * is concentrated in dimension 0 and 1 and its first differential is given by p · id : Z[Z] → Z[Z] for some non-trivial polynomial p such that either p is of the shape φ m d for some natural numbers d and m or no root of unity is a root of p.
We begin with the case where p is of the shape φ m d for some natural numbers d and m. Then all roots of p have norm 1 and hence ln ρ (2) (C * ) = ln det N (Z) p · id : L 2 (Z) → L 2 (Z) = 0 by [66, (3.23) on page 136]. Now the claim follows from assertion (1).
Finally we treat the case, where no root of unity is a root of p. Fix i ∈ I. Put n = [Z : Z i ]. Then Z/Z i = Z/n. For l ∈ Z/n let C l be the unitary Z/nrepresentation whose underlying Hilbert space is C and on which the generator in Z/n acts by multiplication with ζ l n , where we put ζ n := exp(2πi/n). We obtain a unitary Z/n-isomorphism ω :
Miscellaneous
We briefly mention some variations of the problems considered here or some other prominent open conjectures about L 2 -invariants. 19.1. Approximation for lattices. In our setting we approximate the universal covering of a closed manifold or compact CW-complex by a tower of finite coverings corresponding to the normal chain (G i ) i≥0 of normal subgroups of G with finite index and trivial intersection.
One can also look at a uniformly discrete sequence of lattices (G i ) i≥0 in a connected center-free semisimple Lie group L without compact factors and study the quotients M [i] = X/G i , where X is the associated symmetric space L/K for K ⊆ L a maximal compact subgroup. There is a notion of BS-convergence for lattices which generalizes our setting. One can ask whether for such a convergence sequence of cocompact lattices the sequence bn(M[i];Q) vol(M[i]) converges to the L 2 -Betti number of X. This setup and various convergence questions are systematically examined in the papers by Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet [1,2].
Another paper containing interesting information about these questions is Bergeron-Lipnowski [10].
19.2. Twisting with representations. We have already mentioned that one can twist the analytic torsion with special representations. This has in favorite situations the effect that one obtains a uniform gap for the spectrum of the Laplace operators and can prove the desired approximations results, see Remark 16.15. For more information we refer for instance to [12,74,78,79].
In [70] twisted L 2 -torsion for finite CW -complex X with b
n ( X) = 0 for all n ≥ 0 will be introduced for finite-dimensional representations which are given by restricting finite-dimensional Z d -representations with any homomorphism π 1 (M ) → Z d . In particular one can twist the L 2 -torsion for a given element φ ∈ H 1 (X; Z) with the 1-dimensional representation whose underlying complex vector space is C and on which g ∈ π 1 (X) acts by multiplication with t φ(g) . This yields the L 2torsion function (0, ∞) → R whose value at 1 is the L 2 -torsion itself. The proof that this function is well-defined is based on approximation techniques. This function seem to contain very interesting information, in particular for 3-manifolds, see for instance [27,28,29,30]. In particular there is the conjecture that one can read off the Thurston norm of φ from the asymptotic behavior at 0 and ∞ if X is a connected compact orientable 3-manifold with infinite fundamental group and empty or toroidal boundary which is not S 1 × D 2 .
19.3. Atiyah's Question. Atiyah [6, page 72] asked the question, whether the L 2 -Betti numbers b (2) p ( M ) for a closed Riemannian manifold M are always rational numbers. Meanwhile it is known that the answer can be negative, see for instance [7,42,83]. However, the following problem, often referred to as the strong Atiyah Conjecture, remains open.
A ) ∈ Z; (2) For every closed manifold M with G ∼ = π 1 (M ) and n ≥ 0 we have d · b (2) n ( M ) ∈ Z. Notice that we can choose d = 1 if G is torsionfree. For a discussion, a survey on the literature and the status of this Question 19.1, we refer for instance to [66,Chapter 10].
The Approximation Conjecture 13.4, which is known by Remark 13.2 and Theorem 13.6 for a large class of groups, can be used to enlarge the class of groups for which the answer to part (1) of Question 19.1 is positive. Namely, if G is torsionfree and possesses a chain of normal subgroups (G i ) i≥0 with trivial intersection i≥0 G i = {1} such that the answer to part (1) of Question 19.1 is positive for each quotient G/G i , then the answer to part (1) of Question 19.1 is positive for each quotient G/G i . Here it becomes important that we could drop the condition that each G/G i is finite. An example for G is a finitely generated free group whose descending central series gives such a chain (G i ) i≥0 with torsionfree nilpotent quotients G/G i .
Notice that Conjecture 10.1 implies a positive answer to part (2) of Question 19.1 if M is an aspherical closed manifold.
One can ask an analogous question in the mod p case as soon as one has a replacement for the L 2 -Betti number in the mod p case. In some special cases this replacement exists and the answer is positive, see for instance Theorem 2.2 for torsionfree elementary amenable groups, and Theorem 2.3 for torsionfree G taking into account that is the nth mod p L 2 -Betti numbers b If the closed orientable manifold M has a selfmap f : M → M of degree different from −1, 0, 1, then one easily checks that its simplicial volume ||M || vanishes. If its minimal volume is zero, i.e., for every ǫ > 0 one can find a Riemannian metric on M whose sectional curvature is pinched between −1 and 1 and for which the volume of M is less or equal to ǫ, then its simplicial volume ||M || vanishes. This follows from [45, page 37].
If one replaces in Conjecture 19.2 the simplicial volume by the minimal volume, whose vanishing implies the vanishing of the simplicial volume, then the claim for the L 2 -Betti numbers in Conjecture 19.2 has been proved by Sauer [86, Second Corollary of Theorem A].
There are a versions of the simplicial volume such as the integral foliated simplicial volume and stable integral simplicial volume which are related to Conjecture 19.2 and may be helpful for a possible proof, and reflect a kind of approximation conjecture for the simplicial volume, see for instance [35,60,92].
More information about the simplicial volume and the literature can be found for instance in [45], [59], [66, Section 14.1].
Entropy, Fuglede-Kadison determinants and amenable exhaustions.
In recent years the connection between entropy and Fuglede-Kadison determinant has been investigated in detail, see for instance [23,25,56,57]. In particular the amenable exhaustion approximation result for Fuglede-Kadison determinants of Li-Thom [57, Theorem 0.7] for amenable groups G is very interesting, where the Fuglede-Kadison determinant of a matrix over ZG is approximated by finitedimensional analogues of its "restrictions" to finite Fölner subsets of the group G. if we write p(z) as a product
p(z) = c · z k · r i=1 (z − a i )
for an integer r ≥ 0, non-zero complex numbers c, a 1 , . . ., a r and an integer k. This implies M (p) ≥ 1. Remark 19.6 (Lehmer's polynomial). There is even a candidate for which the minimal Mahler measure is attained, namely, Lehmer's polynomial L(z) := z 10 + z 9 − z 7 − z 6 − z 5 − z 4 − z 3 + z + 1.
It is conceivable that for any non-trivial element p ∈ Z[Z] with M (p) = 1 M (p) ≥ M (L) = 1.17628 . . .
holds.
For a survey on Lehmer's problem were refer for instance to [13,14,19,95]. Consider an element p = p(z) ∈ C[Z] = C[z, z −1 ]. It defines a bounded Zoperator r (2) p : L 2 (Z) → L 2 (Z) by multiplication with p. Suppose that p is not zero. Then the Fuglede-Kadison determinant of r N (G) r (2) A : N (G) r → N (G) s , where A runs through all (r, s)-matrices A ∈ M r,s (ZG) for all r, s ∈ Z with r, s ≥ 1 for which det (2) N (G) (r (2) A ) > 1 holds. If we only allow square matrices A such that r (2) A : N (G) r → N (G) r is injective and det (2) N (G) (r (2) A ) > 1, then we denote the corresponding infimum by Λ w (G) ∈ [1, ∞)
Obviously we have Λ(G) ≤ Λ w (G). We suggest the following generalization of Lehmer's problem to arbitrary groups. Problem 19.8 (Lehmer's problem for arbitrary groups). For which groups G is Λ(G) > 1 or Λ w (G) > 1?
For a discussion and results on this problems see [22,Question 4.7] and [69].
Question 3 . 4 (
34Rank gradient, cost, first L 2 -Betti number and approximation) Let G be a finitely presented residually finite group. Let (G i ) be a descending chain of normal subgroups of finite index of G with i≥0 G i = {1}. Let F be any field.When do we havelim i→∞ b 1 (G i ; F ) G) = cost(G) − 1 = RG(G; (G i ) i≥0 ),where cost(G) denotes the cost and RG(G; (G i ) i≥0 ) the rank gradient?Conjecture 10.1 (Homological growth and L 2 -torsion for aspherical closed manifolds) Let M be an aspherical closed manifold of dimension d ≥ 1 and fundamental group G = π 1 (M ). Let M be its universal covering. Then (1) For any natural number n with 2n = d we get b (2) n ( M ) = 0. If d = 2n, we have (−1) n · χ(M ) = b (2) n ( M ) ≥ 0. If d = 2n and M carries a Riemannian metric of negative sectional curvature, then (−1) n · χ(M ) = b (2) n ( M ) > 0; (2) Let (G i ) i≥0 be any chain of normal subgroups G i ⊆ G of finite index [G : G i ] and trivial intersection i≥0 G i = {1}. Put M [i] = G i \ M . Then we get for any natural number n and any field F b (2) n ( M ) = lim i→∞ b n (M [i]; F ) [G : G i ] = lim i→∞ d H n (M [i]; Z) [G : G i ] , where d H n (M [i]; Z) is the minimal numbers of generators of H n (M [i]; M ) if d is even and 2m ≥ d; 0 otherwise;
( 5 )
5Let (G i ) i≥0 be any chain of normal subgroups G i ⊆ G of finite index [G : G i ] and trivial intersection i≥0 G i = {1}. Put M [i] = G i \ M .Then we get for any natural number n with 2n + 1 = dlim i→∞ ln tors H n (M [i]) [G : G i ] = 0, and we get in the case d = 2n + 1 lim i→∞ ln tors H n (M [i]) [G : G i ] = (−1) n · ρ (2) an M ≥ 0.
1. 1 .
1Euler characteristic. Let us begin with one of the oldest and most famous invariants, the Euler characteristic χ(X) for a finite CW -complex. It is defined as n≥0 (−1) n · c n , where c n is the number of n-cells. It is easy to see that it is multiplicative under finite coverings. Since this implies χ(X) = χ(X[i]) [G:Gi] , the answer in this case is yes for all three questions appearing in Problem 0.Signature of closed oriented manifolds. Next we consider the signature of a closed oriented topological 4k-dimensional manifold M . It is defined as the signature of the non-degenerate symmetric bilinear R-pairing given by the intersection form H 2k (M ; R) × H 2k (M ; R) → R, (x, y) → x ∪ y, [M ] R .
:H] ≤ d(G) − 1 by the Schreier index formula and hence the sequence d(Gi)−1 [G:Gi]
Remark 3 . 5 (
35Minimal number of generators versus rank of the abelianization). A positive answer to Question 3.4 is equivalent to the assertion that
Question 4 . 3 (
43Asymptotic Morse equality for groups). Let G be a group and let d be a natural number such that CW d (BG) is not empty. When do we have
Example 4. 4 (
4Morse relation in degree d = 1, 2). Question 4.3 is in the case d = 1 precisely Question 3.4, since a group H is finitely generated if and only if there is a model for BH with finite 1-skeleton and in this case χ trun 1 (H) = 1 − d(H). In the case d = 2 Question 4.3 can be rephrased as the question when for a finitely presented group G we have lim i→∞
is odd, and then show in the situation of Conjecture 2.4 for n = 0, 1, 2, . . . by induction using Theorem 2.1 the equality lim i→∞ b n (X[i]; F ) [G : G i ] = b (2) n ( X).
Definition 4 . 6 (
46Slow growth in dimensions ≤ d). We say that a residually finite group has slow growth in dimension ≤ d if for any chain (G i ) i≥0 of normal subgroups of finite index with trivial intersection there is a choice of CW -complexes (X[i]) i≥0 such that X[i] has a finite d-skeleton and is a model for BG i for each i ≥ 0, and lim i→∞ c k (X[i]) [G:Gi] = 0 holds for every k = 0, 1, 2 . . . , d, where c k (X[i]) is the number of k-cells in X[i].
Lemma 4 . 7 .
47Suppose that G has slow growth in dimension ≤ d. Then we get for k = 0, 1, 2, . . . Proof. By assumption there is a choice of CW -complexes (X[i]) i≥0 such that X[i] has a finite d-skeleton and is a model for BG i for each i ≥ 0, and lim i→∞ cn(X[i]) [G:Gi] = 0 holds for every n = 0, 1, 2 . . . , d. Since b n (G i ; Q) ≤ c n (X[i]) holds, we conclude lim i→∞ bn(Gi;Q) [G:Gi] = 0 for every n = 0, 1, 2 . . . , d. Theorem 2.1 implies b
By assumption there is a choice of CW -complexes (X[i]) i≥0 such that X[i] has a finite d-skeleton and is a model for BK i for each i ≥ 0, and lim i→∞ cm(X[i]) [K:Ki] = 0 holds for every m = 0, 1, 2 . . . , d, where c m (X[i]) is the number of m-cells in X[i]. Choose a CW -model Z for BQ with finite d-skeleton. Let Z i → BQ be the [Q : Q i ]-sheeted finite covering associated to Q i ⊆ Q. Equip Z i with the CW -structure induced by the one of Z. Then Z i is a model for BQ i , has a finite d-skeleton, and we get for the number of n-cells for n ∈ {0, 1, 2, . . . , d}
Example 4. 9 (
9Examples of groups with slow growth in dimensions ≤ d). A residually finite group has slow growth in dimensions ≤ 0 if and only if it is infinite.
Choose any chain (K i ) i≥0 of normal subgroups of K of finite index [K : K i ] which is a power of p and with trivial intersection i≥0 K i = {1}. Because of Theorem 2.1 and [9, Theorem 1.6] the limit bn(Y [i];F )+bn−1(Y [i];F ) [K:Ki]
Remark 6. 4
4(Q coefficients are necessary). Conjecture 6.2 does not hold if one replaces Q by C by the following result appearing in [66, Example 13.69 on page 481].
12 ,
12Conjecture 1.3] and [66, Conjecture 11.3 on page 418 and Question 13.52 on page 478]. They are true in special cases by Theorem 10.4. The assumption that b
Conjecture 7 .
712 (Approximating Fuglede-Kadison determinants and L 2 -torsion by homology).
bijective. Then the conclusion appearing there is true. Suppose in the situation of assertion (2) of Conjecture 7.12 that H p (C * )⊗ Z Q = 0 for all p ≥ 0. Then the conclusion appearing there is true; (3) If G is the infinite cyclic group Z, then Conjecture 7.12 is true; Assertion (2) of Theorem 7.13 is generalized in Lemma 18.4. Assertion (3) of Theorem 7.13 has already been proved by Bergeron-Venkatesh [12, Theorem 7.3]. Applied to cyclic coverings of a knot complement this reduces to a theorem of Silver-Williams [94, Theorem 2.1
) * ).
Notation 8 . 1 .
81If M is a finitely generated abelian group, define M f := M/ tors(M ).
X[i]) has norm [G : G i ] · s 0 and lies in the kernel of c[i]
tors H i (C * ) or between ln det
and ln tors H i (C * ) for each individual i ∈ Z, there is only a relationship after taking the alternating sum over i ≥ 0. LÜCK, W.
Conjecture 10. 1 (
1Homological growth and L 2 -torsion for aspherical closed manifolds). Let M be an aspherical closed manifold of dimension d and fundamental group G = π 1 (M ). Let M be its universal covering. Then
( 1 )
1For any natural number n with 2n = d we getb (2) n ( M ) = 0. If d = 2n, we have (−1) n · χ(M ) = b (2)n ( M ) ≥ 0. If d = 2n and M carries a Riemannian metric of negative sectional curvature, then (−1) n · χ(M ) = b(2) n ( M ) > 0;
( 2 )
2Let (G i ) i≥0 be any chain of normal subgroups G i ⊆ G of finite index [G : G i ] and trivial intersection i≥0 G i = {1}. Put M [i] = G i \ M .Then we get for any natural number n and any field Fb (2) n ( M ) = lim i→∞ b n (M [i]; F ) [G : G i ] = lim i→∞ d H n (M [i]; Z) [G : G i ] ;and for n = 1 b
M ) if d is even and 2m ≥ d; 0 otherwise;(4) If d = 2n + 1 is odd, we have (−1) n · ρ (2) an M ≥ 0; If d = 2n + 1 is odd and M carries a Riemannian metric with negative sectional curvature, we have(−1) n · ρ (2) an M > 0; (5) Let (G i ) i≥0be any chain of normal subgroups G i ⊆ G of finite index [G : G i ] and trivial intersection i≥0 G i = {1}. Put M [i] = G i \ M . Then we get for any natural number n with 2n + 1 = d lim i→∞ ln tors H n (M [i]) [G : G i ] = 0, and we get in the case d = 2n + 1 lim i→∞ ln tors H n (M [i]) [G : G i ] = (−1) n · ρ (2) an M ≥ 0.
Remark 10. 2 (
2Rank growth versus torsion growth). Let us summarize what Conjecture 10.1 means for an aspherical closed manifold M of dimension d. It predicts that the rank of the singular homology grows in dimension m sublinear if 2m = d, and grows linearly if d = 2m and M carries a Riemannian metric of negative sectional curvature. It also predicts that the cardinality of the torsion of the singular homology grows in dimension m subexponentially if 2m + 1 = d and grows exponentially if d = 2m + 1 and M carries a Riemannian metric of negative sectional curvature. Roughly speaking, the free part of the singular homology is asymptotically concentrated in dimension m if d = 2m and the torsion part is asymptotically concentrated in dimension m if d = 2m + 1. A vague explanation for this phenomenon could be that Poincaré duality links the rank in dimensions m and d − m, whereas the torsion is linked in dimensions m and d − 1 − m and there must be some reason that except in the middle dimension the growth of the rank and the growth of the cardinality of the torsion block one another in dual dimensions.
Remark 10. 3 (
3Finite Poincaré complexes). One may replace in the formulation of Conjecture 10.1 the aspherical closed manifold M by an aspherical finite Poincaré complex. In the formulation of the part of assertion
Theorem 10. 4 .
4Let M be an aspherical closed manifold with fundamental group G = π 1 (M ). Suppose that M carries a non-trivial S 1 -action or suppose that G contains a non-trivial elementary amenable normal subgroup. Then we get for all n ≥ 0 and fields Flim i→∞ b n (M [i]; F ) [G : G i ] ( M ) = 0. In particular Conjecture 2.4, Conjecture 2.5, Conjecture 7.4, Conjecture 7.5, Conjecture 7.11 and Conjecture 10.1 with the exception of assertion (3) are known to be true for G = π 1 (M ) and X = M . Estimates of the growth of the torsion in the homology in terms of the volume of the underlying manifold and examples of aspherical manifolds, where this growth is subexponential, are given in [87]. Sometimes one can express for certain classes of closed Riemannian manifolds M the L 2 -torsion of the universal covering M by the volume ρ (2) an ( M ) = C dim(M) · vol(M ), where C dim(M) ∈ R is a dimension constant depending only on the class but not on the specific M . This follows from the Proportionality Principle due to Gromov, see for instance [66, Theorem 1.183 on page 201]. Typical examples are locally symmetric spaces of non-compact type, for instance hyperbolic manifolds, see [66, Theorem 5.12 on page 228]. Since ρ (2) an ( M ) vanishes for even-dimensional closedRiemannian manifolds, only the case of odd dimension is interesting. For locally symmetric spaces of non-compact type with odd dimension d one can show that (−1) (d−1)/2 · C d ≥ 0 holds. Thus one obtains some computational evidence for Conjecture 10.1.
Example 10. 5 (
5. Suppose that M is a closed hyperbolic 3-manifold. Then ρ an ( M ) is known to be − 1 6π · vol(M ), see[66, Theorem 4.3 on page 216], and hence Conjecture 10.volume is always positive, the equation above implies that | tors H 1 (G i | growth exponentially in [G : G i ]. This is in contrast to the question appearing in the survey paper by Aschenbrenner-Friedl-Wilton [5, Question 9.13] whether a hyperbolic 3-manifold of finite volume admits a finite covering N → M such that tors H 1 (N ) is non-trivial. However, a positive answer to this question and evidence for Conjecture 10.1 for closed hyperbolic 3-manifolds is given in Sun[96, Corollary 1,6], where it is shown that for any finitely generated abelian group A, and any closed hyperbolic 3-manifold M , there exists a finite cover N of M , such that A is a direct summand of H 1 (N ; Z).
Gi] = 0 holds. Next we discuss the case, where M possesses a Riemannian metric with negative sectional curvature and dim(M ) = 2n. Then Conjecture 10.
2 and (11.1), since Theorem 2.2 implies for torsionfree elementary G that lim i→∞ bn(T f [i];F ) [G:Gi]exists and is independent of the chain (G i ) i≥0 and because of Subsection 11.1 there exists an appropriate chain, for instance (G i ∩pr −1 (2 i ·Z)) i≥0 , with lim i→∞ bn(T f [i];F ) [G:Gi]
interesting (and open) case is g ≥ 2. In this situation equality(11.4) becomes
13 .
13Review of the Determinant Conjecture and the Approximation Conjecture for L 2 -Betti numbers We begin with the Determinant Conjecture (see [66, Conjecture 13.2 on page 454]).
Conjecture 13. 1 (
1Determinant Conjecture for a group G). For any matrix A ∈ M r,s (ZG), the Fuglede-Kadison determinant of the morphism of Hilbert modules r (2) A : L 2 (G) r → L 2 (G) s given by right multiplication with A satisfies det (2) N (G) r
Notation 13 . 3 (
133Inverse systems and matrices). Let R be a ring with Z ⊆ R ⊆ C. Given a matrix A ∈ M r,s (RG), let A[i] ∈ M r,s (R[G/G i ]) be the matrix obtained from A by applying elementwise the ring homomorphismRG → R[G/G i ] induced by the projection G → G/G i . Let r A : RG r → RG s and r A[i] : R[G/G i ] r → R[G/G i ] s bethe RG-and R[G/G i ]-homomorphism given by right multiplication with A and A[i]. Let r (2) A : L 2 (G) r → L 2 (G) s and r (2) A[i] : L 2 (G/G i ) r → L 2 (G/G i ) s be the morphisms of Hilbert N (G)-and Hilbert N (G/G i )-modules given by right multiplication with A and A[i]. Next we deal with the Approximation Conjecture for L 2 -Betti numbers (see [89, Conjecture 1.10], [66, Conjecture 13.1 on page 453]).
Conjecture 13. 4 (
4Approximation Conjecture for L 2 -Betti numbers). A group G together with an inverse system {G i | i ∈ I} as in Setup 12.1 satisfies the Approximation Conjecture for L 2 -Betti numbers if one of the following equivalent conditions hold:(1) Matrix version Let A ∈ M r,s (QG) be a matrix. Then dim N (G) ker r
Notation 14.2 (Inverse systems and chain complexes). Let C * be a finite based free QG-chain complex. In the sequel we denote byC[i] * the Q[G/G i ]-chain complex Q[G/G i ] ⊗ QG C * , by C (2) * the finite Hilbert N (G)-chain complex L 2 (G) ⊗ QG C * , and by C[i] (2) * the finite Hilbert N (G/G i )-chain complex L 2 (G/G i ) ⊗ Q[G/Gi] C[i] * .The QG-basis for C * induces a Q[G/G i ]-basis for C[i] * and Hilbert space structures on C
2 -torsion over N (G) and N (G/G i ) respectively.
2. L 2 -torsion. Let M be a Riemannian manifold without boundary that comes with a proper free cocompact isometric G-action. Denote by M [i] the Riemannian manifold obtained from M by dividing out the G i -action. The Riemannian metric on M [i] is induced by the one on M . There is an obvious proper free cocompact isometric G/G i -action on M [i] induced by the given G-action on M . Notice that M = M /G is a closed Riemannian manifold and we get a G-covering M → M and a G/G i -covering M [i] → M which are compatible with the Riemannian metrics. Denote by ρ (2) an M ; N (G) ∈ R; (14.6) ρ (2) an M [i]; N (G/G i ) ∈ R, (14.7)their analytic L 2 -torsion over N (G) and N (G/G i ) respectively.
Conjecture 14.8 (Approximation Conjecture for analytic L 2 -torsion). Consider a group G together with an inverse system {G i | i ∈ I} as in Setup 12.1. Let M be a Riemannian manifold without boundary that comes with a proper free cocompact isometric G-action. Thenρ (2) an M ; N (G) = lim i∈I ρ (2) an M [i]; N (G/G i ) .Remark 14.9. The conjectures above imply a positive answer to[24, Question 21] and [66, Question 13.52 on page 478 and Question 13.73 on page 483]. They also would settle [50, Problem 4.4 and Problem 6.4] and [51, Conjecture 3.5]. One may wonder whether it is related to the Volume Conjecture due to Kashaev [49] and H. and J. Murakami [80, Conjecture 5.1 on page 102].We will prove in Section 15 the following result which in the weakly acyclic case reduces Conjecture 14.8 to Conjecture 14.1.Theorem 14.10. Consider a group G together with an inverse system {G i | i ∈ I} as in Setup 12.1. Let M be a Riemannian manifold without boundary that comes with a proper free cocompact isometric G-action. Suppose that b
M ; N (G)) = 0 for all p ≥ 0. Assume that the Approximation Conjecture for L 2 -torsion of chain complexes 14.5 (or, equivalently, Conjecture 14.1) holds for G. Then Conjecture 14.8 holds for M , i.e., ρ (2) an M ; N (G) = lim i∈I ρ (2) an M [i]; N (G/G i ) .
M ; N (G)) vanishes for all p ≥ 0, but our present proof works only under this assumption (seeRemark 15.7).
see [ 66 ,
66Lemma 1.75 on page 59]) and the mapInt p : H p (2) → H p (2) (K) is the composite H p (2) (A * ) • i p ,Lemma 15.3 follows. Now we are ready to give Proof of Theorem 14.10. Let M be a Riemannian manifold without boundary that comes with a proper free cocompact isometric G-action such that b
all p ≥ 0. Fix a smooth triangulation K of M = G\M . By possibly subdividing it we can arrange that Lemma 15.3 will apply to M and K. The triangulation K lifts to a G-equivariant smooth triangulation K of M and to a G/G i -equivariant smooth triangulation K[i] of M [i] := G i \M .
from Lemma 15.3 constants L 1 > 0 and L 2 > 0 which are independent of i ∈ I such that for every i ∈ I L 1 ≤ || Int[i] p || ≤ L 2 holds for the operator norm of Int[i] p . Since
Since the Approximation Conjecture 13.4 holds for M by Theorem 13.6 and we have b
M
[i]; N (G/G i ) = 0. Now (15.6) follows. This finishes the proof of Theorem 14.10.Remark 15.7 (On the L 2 -acyclicity assumption). Recall that in Theorem 14.10 we require that b
p
(M ; N (G) = 0 holds for p ≥ 0.
• I is a directed set. For each i ∈ I we have a group Q i , a matrix B[i] ∈ M d (N (Q i )) and a faithful finite normal trace tr i :M d (N (Q i )) → C such that tr i (I d ) = d holds for the unit matrix I d ∈ M d (N (Q i )).Faithful finite normal trace tr means that tr is C-linear, satisfies tr(B 1 B 2 ) = tr i (B 2 B 1 ), sends B * Bto a real number tr(B * B) ≥ 0 such that tr(B * B) = 0 ⇔ B = 0, and for f ∈ N (G), which is the supremum with respect to the usual ordering ≤ of positive elements of some monotone increasing net {f j | j ∈ J} of positive elements in N (G), we get tr(f ) = sup{tr(f j ) | j ∈ J}. The trace tr or tr i respectively may or may not be the von Neumann trace (see [66, Definition 1.2 on page 15]) tr N (G) or tr N (Qi) respectively. Let F : [0, ∞) → [0, ∞) be the spectral density function of r (2) B : L 2 (G) d → L 2 (G) d with respect to tr as defined in [66, Definition 2.1 on page 73]. Let F [i] : [0, ∞) → [0, ∞) be the spectral density function of r (2)
B
(see [66, Lemma 2.3 on page 74 and Lemma 2.11 (11) on page 77]). The analogous statement holds for F [i].
.F
The general strategy. To any monotone non-decreasing function f : [0, ∞) → [0, ∞) we can assign a density function, i.e., a monotone non-decreasing rightcontinuous function,f + : [0, ∞) → [0, ∞), [i](λ).Let det and det i be the Fuglede-Kadison determinant with respect to tr and tr i (compare [66, Definition 3.11 on page 127]). If tr is the von Neumann trace tr N (G) , then det is the Fuglede-Kadison determinant det N (G) as defined in [66, Definition 3.11 on page 127]. We want to prove Theorem 16.3. Consider the following conditions, where K > 0 and κ > 0 are some fixed real numbers: (i) The operator norms satisfy ||r
B
|| ≤ K and ||r
i] || ≤ K for all i ∈ I; (ii) For every polynomial p with real coefficients we have tr(p(B)) = lim i∈I tr i (p(B[i]));(iii) We have det i r(2)
1 1 −
1ln(e −n ) + e −n < λ ≤ 1. Hence for n ≥ 1 the function f n is smooth with non-negative derivative on the open intervals (0, e −3n ), (e −3n , e −2n ), (e −2n , e −n ), and (e −n , 1), and is continuous on the closed intervals [0, e −3n ], [e −3n , e −2n ], [e −2n , e −n ], and [e −n , 1]. Hence each f n is continuous and monotone non-decreasing. (2) This is obvious. (3) This follows since lim n→∞ 1 − ln(e −n ) + e −n = 0, f n (λ) = λ for 1 − ln(e −n ) + e −n ≤ λ ≤ 1 and f n (0) = 0 for all n ≥ 1. (4) We conclude λ ≤ f n (λ) ≤ 1 − ln(λ) + λ for λ ∈ [0, 1) by inspecting the definitions since λ ≤ 1 − ln(λ) + λ holds and1
This follows from assertion (3),(7).
−n − e −2n − [ln(− ln(λ))]
( 2 )
2* (X[i]). If we now consider the Setup 0.1, we get from [62, Theorem 0.3] for all p ≥ 0 at least the inequality
18) is not satisfied for X = M and the p-th Novikov-Shubin invariant of M is zero, disproving a conjecture of Lott and Lück, see [61, Conjecture 7.1 and 7.2]. It may still be possible that Condition 16.18 and the conjecture of Lott and Lück hold for an aspherical closed manifold M .
The proof of the following result is analogous to the proof of [66, Lemma 13.33 on page 466].Lemma 17.2. We get for B ∈ M r,s (L 1 (G)) ||r (2)B : L 2 (G) r → L 2 (G) s || ≤ K G (B).Consider the setup 12.1.Lemma 17.3. Consider B ∈ M d (L 1 (G)). Let B[i] ∈ M d (L 1 (G/G i ))be obtained from B by applying the map L 1 (G) → L 1 (G/G i ) induced by the projection ψ i : G → G/G i . Then tr N (G) (B) = lim i∈I tr N (G/Gi) (B[i]).Proof. Suppose that B looks like g∈G λ g (r, s)
tr is the von Neumann trace tr N (G) : N (G) → C; • tr i is the von Neumann trace tr N (G/Gi) : N (G/G i ) → C. We have to check that the conditions of Theorem 16.3 (2) are satisfied. Condition (ii) appearing in Theorem 16.3 follows from Lemma 17.3. Condition (iv) appearing in Theorem 16.3 is satisfied because of B = A * A and B[i] = A[i] * A[i]. Let B −1 be the inverse of B in GL d (L 1 (G)). Put K := max{K G (B), K G/Gi (B[i])}. Since the projection L 1 (G) → L 1 (G/G i ) has operator norm at most 1, we get for the numbers K G (B) and K G (B −1 ) defined in (17.1) and all i ∈ I K G/Gi (B[i]) ≤ K;
i] −1 || ≤ K for all i ∈ I from Lemma 17.2. In particular condition (i) appearing in Theorem 16.3 is satisfied. Since r i] , we conclude from [66, Lemma 213 (2) on page 78, Theorem 3.14 (1) on page 128 and Lemma 3.15 (6) on page 129] F [i](λ) = 0 for all λ < K −1 and i ∈ I; · ln(K) for all i ∈ I. (17.9)
Lemma 18 . 1 ..
181Let G be a group for which the Determinant Conjecture 13.1 is true. Let f * : D * → E * be a ZG-chain homotopy equivalence equivalence of finite based free ZG-chain complexes. Suppose that D Proof. This follows from [66, Theorem 3.93 (1) on page 161 and Lemma 13.6 on page 456]. Notation 18.2. Let A be a finitely generated free abelian group and let B ⊆ A be a subgroup. Define the closure of B in A to be the subgroup B = {x ∈ A | n · x ∈ B for some non-zero integer n}. Notice that A/B and M f := M tors(M ) are finitely generated free and we have ker(f ) = ker(f ) for a homomorphism f : A 0 → A 1 of finitely generated free abelian groups. The proof of the next result can be found in [68, Lemma 2.11].
2 -acyclic. Then D (2) * is L 2 -acyclic and
→ Z[Z] given by multiplication with p k,j . There is an obvious identification of Z[Z]-modules H k F k,j * ∼ = Z[Z]/(p k,j )andH i F k,j * = 0 for i = k. Since F k,j *has projective chain modules and is concentrated in dimensions (k + 1) and k and we have the exact sequence of Z[Z]modules C k+1 c k+1 − −− → ker(c k ) → H k (C * ), we can construct a Z[Z|-chain map f k,j * : F k,j * → C * such that H k f k,j *agrees with the restriction of ξ k to the j-th summand. Define a Z[Z]-chain map
Hence p[i](2) :Z[Z/n] (2) → Z[Z/n] (2) is an isomorphism. Therefore p[i] : Z[Z/n] → Z[Z/n]is rationally an isomorphism. Now the claim follows from assertion(2). This finishes the proof of Theorem 7.13.
Question 19 . 1 .
191Let G be a group for which there exists natural number d such that the order of any finite subgroup divides d. Then:(1) For any A ∈ M m,n (ZG) we get for the von Neumann dimension of the kernel of the induced G-equivariant bounded operator r(2) A : L 2 (G) m → L 2 (G) n d · dim N (G) ker(r
an integer for torsionfree G.
19. 4 .
4Simplicial volume. The following conjecture is discussed in[66,.
Conjecture 19.2 (Simplicial volume and L 2 -invariants). Let M be an aspherical closed orientable manifold of dimension ≥ 1. Suppose that its simplicial volume ||M || vanishes.
19. 6 .
6Lehmer's problem. Let p(z) ∈ C[Z] = C[z, z −1 ] be a non-trivial element. Its Mahler measure is defined by
Problem 19.5 (Lehmer's Problem). Does there exist a constant Λ > 1 such that for all non-trivial elements p(z) ∈ Z[Z] = Z[z, z −1 ] with M (p) = 1 we have M (p) ≥ Λ?
with the Mahler measure of p by [66, (3.23) on page 136].
Definition 19.7 (Lehmer's constant of a group). Define Lehmer's constant Λ(G) of a group G Λ(G) ∈ [1, ∞) to be the infimum of the set of Fuglede-
The case i≥0 d i · Z = {1} 30 11.2. The case i≥0 d i · Z = {1}18.4.
5. Speed of convergence
14
5.1. Betti numbers
14
5.2. Rank gradient
16
6. The Approximation Conjecture for Fuglede-Kadison determinants
16
7. Torsion invariants
18
7.1. L 2 -torsion
18
7.2. Analytic and topological L 2 -torsion
18
7.3. Integral torsion
19
8. On the relation of L 2 -torsion and integral torsion
20
9. Elementary example about L 2 -torsion and integral torsion
22
10. Aspherical manifolds
24
11. Mapping tori
28
11.1. 30
11.3. Selfhomeomorphism of a surface
31
12. Dropping the finite index condition
32
13. Review of the Determinant Conjecture and the Approximation
Conjecture for L 2 -Betti numbers
32
14. The Approximation Conjecture for Fuglede-Kadison determinants and
L 2 -torsion
34
14.1. The chain complex version
34
14.2. L 2 -torsion
35
14.3. An inequality
35
14.4. Matrices invertible in L 1 (G)
36
15. The L 2 -de Rham isomorphism and the proof of Theorem 14.10
36
16. A strategy to prove the Approximation Conjecture for Fuglede-Kadison
determinants 14.1
40
16.1. The general setup
40
16.2. The general strategy
40
16.3. The uniform integrability condition is not automatically satisfied
42
16.4. Uniform estimate on spectral density functions
46
17. Proof of Theorem 14.11, Theorem 14.12, and Corollary 14.13
47
18. Proof of Theorem 7.13
52
19. Miscellaneous
57
19.1. Approximation for lattices
57
19.2. Twisting with representations
57
19.3. Atiyah's Question
58
19.4. Simplicial volume
58
19.5. Entropy, Fuglede-Kadison determinants and amenable exhaustions
59
19.6. Lehmer's problem
59
References
60
1. Euler characteristic and signature
66, Section 6.4.1 on page 256ff]. Solvable groups are examples of elementary amenable groups. Every elementary amenable group is amenable, the converse is not true in general.Notice that Theorem 2.2 is consistent with Theorem 2.1 since for a field F of characteristic zero and a torsionfree elementary amenable group G we have b(2) n (X; N (G)) = dim Ore F G H n (X; F ) . The latter equality follows from [66, Theorem 6.37 on page 259, Theorem 8.29 on page 330, Lemma 10.16 on page 376, and Lemma 10.39 on page 388].
The first inequality is due toGaboriau [38, Corollaire 3.16, 3.23] and the second was proved by Abért and Nikolov[4, Theorem 1]. See[37,38,39] for the definition and some key results about the cost of a group.It is not known if either inequality in (3.2) can be strict. It is strict if G is finite since then all values are |G| −1 . The following questions remain open:
It is also positive for limit groups by Bridson-Kochlukova [15, Theorem A and Corollary C], where all values are −χ(G).Remark 3.6 (Known cases). The answer to Question 3.3 and 3.4 is known to be
positive if G contains a normal infinite amenable subgroup. Namely, in this case all
values are 0 since RG(G; (G i ) i≥0 ) = 0 for all descending chains (G i ) i≥0 of normal
subgroups of finite index of G with trivial intersection, as proved by Lackenby [53,
Theorem 1.2] when G is finitely presented, and by Abért and Nikolov [4, Theorem 3]
in general, where actually more general chains are considered.
One may speculate about the following higher dimensional analogue of Question 3.4.4.1. Truncated Euler characteristics.Let d be a natural number and let X be a space. Denote by CW d (X) the set of CW -complexes Y which have a finite d-skeleton Y d and are homotopy equivalent to X.33, Theorem 1.6] and Lück-Osin [71, Theo-
rem 1.2].
4. A high dimensional version of the rank gradient
Provided that CW d (X) is not empty, define the d-th truncated Euler character-
istic of X by
(4.1)
odd .
oddNext we show that the limit lim i→∞ always exists. Consider a natural number i. Choose an element Yχ trun
d
(X[i])
[G:Gi]
).4.2. Groups with slow growth. The answer to Question 4.3 is positive by Bridson-
Kochlukova [15, Theorem A and Corollary C] if G is a limit group. Their proofs
are based on various notions of groups with slow growth. It is interesting that limit
groups may have non-trivial first L 2 -Betti numbers.
Here is a another case, where the answer to Question 4.3 is positive. Following
Bridson-Kochlukova [15, page 4] we make
Proof. The fundamental class [X[i]] is a generator of the infinite cyclic group H d (X[i]; Z), and is represented by the cycle σ[i] d in C d (X[i]) given by the sum over all d-dimensional simplices. The number of d-simplices in X[i] is [G : G i ] · s d . If we consider σ[i] d as an element in C(2)
d (X[i]), it belongs to the kernel of ∆[i]
The number max{t[i, n, p] | p prime} grows sublinearly in comparison with [G : G i ]. There is a prime p such that the number t[i,n,p] j=1 n[i, n, p] j grows linearly with [G : G i ]. A concrete example is the case, whenThe inequality
lim sup
i→∞
ln tors H 1 (G i
[G : G i ]
≤
1
6π
· vol(M ).
is proved by Thang [54] for a compact connected orientable irreducible 3-manifold
M with infinite fundamental group and empty or toroidal boundary.
Remark 10.6 (Possible Scenarios). Consider the situation of Conjecture 10.1. We
can find for each i ≥ 0, n ≥ 0 and prime number p integers r[i, n] ≥ 0, t[i, n, p] ≥
0, and n[i, n, p] 1 , n[i, n, p] 2 , . . . , n[i, n, p] t[i,n,p] ≥ 1 such that the set {p prime |
t[i, n, p] > 0} is finite and
H n (M [i]; Z) ∼ = Z r[i,n] ⊕
p prime
t[i,n,p]
j=1
Z/p n[i,n,p]j .
Then
b n (M [i]; Q) = r[i, n];
b n (M [i]; F p ) = r[i, n] + t[i, n, p];
d H n (M [i]; Z) = r[i, n] + max{t[i, n, p] | p prime};
ln tors H n (M [i])
=
p
t[i,n,p]
j=1
n[i, n, p] j · ln(p).
Let us discuss first the case, where M possesses a Riemannian metric with negative
sectional curvature and dim(M ) = 2n + 1. Then Conjecture 10.1 predicts
lim
i→∞
r[i, n]
[G : G i ]
= 0;
lim
i→∞
max{t[i, n, p] | p prime}
[G : G i ]
= 0;
lim
i→∞
p
t[i,n,p]
j=1
n[i, n, p] j · ln(p)
[G : G i ]
> 0.
There are two scenarios which can explain these expected statements. Of course
there are other scenarios as well, but the two below illustrate nicely what may
happen.
• The number max{t[i, n, p] | p prime} grows sublinearly in comparison with
[G : G i ]. The number of primes p for which t[i, n, p] ≥ 1 grows linearly with
[G : G i ]. A concrete example is the case, where
H 1 (M [i]; Z)) ∼ = Z r[i,1] ⊕
p∈P[i]
Z/p,
where P[i] is a set of primes satisfying lim i→∞
|P[i]|
[G:Gi] > 0, and lim i→∞
r[i,1]
[G:Gi] =
0 holds;
•
holds for n ≥ 0 by[63, Theorem 2.1], this gives evidence for Conjecture 2.4 and Conjecture 2.5, and positive answers to Questions 3.3 and Question 4.3, provided that Z is aspherical. 11.2. The case i≥0 d i ·Z = {1}. Next we consider the hard case i≥0d i ·Z = {1}.Then there exists an integer i 0 such that d i = d i0 for all i ≥ i 0 . We can assume without loss of generality that d i = 1 holds for all0
(11.1)
We can choose for each i ≥ 1 selfhomotopy equivalences f [i] : Z[i] → Z[i] for which the following diagram with the obvious coverings as vertical maps Z[i]
is even more mysterious. Suppose that f : Z → Z is an orientation preserving irreducible selfhomeomorphism of a closed orientable surface Z of genus g ≥ 2. If f is periodic, T f is finitely covered by S 1 × Z and Conjecture 10.1 is known to be true. Therefore we consider from now on the case, where f is not periodic. Then f is pseudo-Anosov, see[20, Theorem 6.3] and T f carries the structure of a hyperbolic 3-manifold by[75, Theorem 3.6 on page 47, Theorem 3.9 on page 50]. Hence Conjecture 10.1 predicts, see Example 10.5,;
or equivalently
lim
i→∞
dim Fp tors coker(id −H 1 (f [i]; Z)) ⊗ Z F p
[G : G i ]
= 0.
So one needs to understand more about the maps id −H 1 (f [i]; Z) : H 1 (Z[i]; Z) →
H 1 (Z[i]; Z) for i ≥ 0.
The status of Conjecture 10.1 (5) lim
i→∞
ln tors(H 1
13.5 (Determinant Conjecture). For each i ∈ I the quotient G/G i satisfies the Determinant Conjecture 13.1. Suppose that each quotient G/G i is finite. Then Assumption 13.5 is fulfilled by Remark 13.2, and we recover Theorem 2.1 from Theorem 13.6. For more information about the Approximation Conjecture and the Determinant Conjecture we refer to [66, Chapter 13 on pages 453 ff] and [89]. 14. The Approximation Conjecture for Fuglede-Kadison determinants and L 2 -torsion Next we turn to Fuglede-Kadison determinants and L 2 -torsion. 14.1. The chain complex version. Conjecture 14.1 (Approximation Conjecture for Fuglede-Kadison determinants). A group G together with an inverse system {G i | i ∈ I} as in Setup 12.1 satisfies the Approximation Conjecture for Fuglede-Kadison determinants if for any matrix A ∈ M r,s (QG) we get for the Fuglede-Kadison determinantTheorem 13.6 (The Determinant Conjecture implies the Approximation Conjec-
ture for L 2 -Betti numbers). If Assumption 13.5 holds, then the conclusion of the
Approximation Conjecture 13.4 holds for {G i | i ∈ I}.
Proof. See [66, Theorem 13.3 (1) on page 454] and [89].
in the case d = 1. Notice that Deninger [24, page 46] uses a different definition of Fuglede-Kadison determinant which agrees with ours for injective operators by [66, Lemma 3.15 (5) on page 129]. Corollary 14.13. Consider a group G together with an inverse system {G i | i ∈ I} as in Setup 12.1.
26, Corollary on page 162 and Corollary on page 163])
16.18) Condition (16.18) is known for p = 0, see [52, Theorem 1.1]. However, there is an unpublished manuscript by Grabowski and Virag
equation (17.4) follows from Theorem 16.3 for A ∈ M r,s (ZG). Next we reduce the general case A ∈ M r,s (QG) to the case above. Consider any real number m > 0, any group H and any morphism f : L 2 (H) r → L 2 (H) s . We conclude from [66, Lemma 1.18 on page 24 and Theorem 3.14 (1) on page 128 and Lemma 3.15 (3), (4) and (7) on page 129]
ev . We conclude from [66, Lemma 3.41 on page 146]
* [G : G i ] . Since H p (C ′′ [i]) * ⊗ Z Q vanishes for all p ≥ 0 and i ∈ I, (18.13) follows from Lemma 8.4. This finishes the proof of Lemma 18.4. Now we are ready to prove Theorem 7.13. Proof of Theorem 7.13. (1) Notice that ln det N ({1}) (f [i] (2) ) [G : G i ] = ln det N (G/Gi) (f [i] (2) ) holds by [66, Theorem 3.14 (5) on page 128]. Theorem 14.11 implies
* ; N (G/G i )holds by [66, Theorem 3.35 (7) on page 143]. We conclude from Corollary 14.13(1)
Obviously it suffices to prove the chain complex version. Let C * be a finitebased free Z[Z]-chain complex that is L 2 -acyclic. If Q[Z] (0) is the quotient field of the integral domain Q[Z], then H k (C * ) ⊗ Z[Z] Q[Z] (0) is trivial for k ≥ 0 because of [66, Lemma 1.34 (1) on page 35]. Since Q[Z]is a principal ideal domain, we can find non-negative integers t k and pairwise prime irreducible elements p k,1 , p k,2 , . . . , p k,t k in Q[Z] and natural numbers m k,1 , m k,2 , . . . m k,t k such that we have isomorphisms of Q[Z]-modulesfollows
from Lemma 8.4
(3)
Acknowledgments. This paper is financially supported by the Leibniz-Preis of the author, granted by the Deutsche Forschungsgemeinschaft DFG. The author thanks Nicolas Bergeron, Florian Funke, Holger Kammeyer, Clara Löh, Roman Sauer, and Andreas Thom for their useful comments.Proof. Let g * : C * ⊗ Z Q → D * ⊗ Z Q be a QG-chain homotopy equivalence. Since C * and D * are finite based free ZG-chain complexes, we can find a ZG-chain map f * : C * → D * and an integer l such that f * ⊗ Z Q = l · g * . Obviously f * ⊗ Z Q is a QG-chain homotopy equivalence. In the sequel we abbreviate C ′ * := cyl(f * ) and C ′′ * := cone(f * ). By the chain homotopy invariance of integral torsion and of L 2 -torsion (see Lemma 18.1) it suffices to prove the claim for C * and C ′ * instead of C * and D * .We have the obvious exact sequence of finite based free ZG-chain complexesSince each C ′′ * is a finite free ZG-chain complex, we can find an integer m and ZG-maps δ p : We have the long exact homology sequenceThe group H p C ′′ [i] * is a finite abelian group for each p ≥ 0. We obtain the following commutative diagram with exact rows(18.5). . .. . .
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|
[] |
[
"The Luminosity Function of Quasars by the Principle of Maximum Entropy",
"The Luminosity Function of Quasars by the Principle of Maximum Entropy"
] |
[
"Alexandre Andrei \nUniversidade Federal do Rio de Janeiro\nObservatório do Valongo\nLadeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil\n\nObservatório Nacional\nMCTIC\nRua Gal. José Cristino 7720921-400Rio de JaneiroCEPBrazil\n\nSYRTE\nObservatoire de Paris\n61 Avenue de l'Observatoire75014ParisFrance\n",
"Bruno Coelho \nInstituto de Telecomunicações\nCampus Universitário de Santiago3810-193AveiroPortugal\n",
"† ",
"Leandro Guedes \nPlanetarium Foundation of the City of Rio de Janeiro\nRua Vice-Governador Rúbens Berardo 100 -Gávea22451-070Rio de Janeiro -RJCEPBrazil\n",
"‡ ",
"Alexandre Lyra \nUniversidade Federal do Rio de Janeiro\nObservatório do Valongo\nLadeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil\n"
] |
[
"Universidade Federal do Rio de Janeiro\nObservatório do Valongo\nLadeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil",
"Observatório Nacional\nMCTIC\nRua Gal. José Cristino 7720921-400Rio de JaneiroCEPBrazil",
"SYRTE\nObservatoire de Paris\n61 Avenue de l'Observatoire75014ParisFrance",
"Instituto de Telecomunicações\nCampus Universitário de Santiago3810-193AveiroPortugal",
"Planetarium Foundation of the City of Rio de Janeiro\nRua Vice-Governador Rúbens Berardo 100 -Gávea22451-070Rio de Janeiro -RJCEPBrazil",
"Universidade Federal do Rio de Janeiro\nObservatório do Valongo\nLadeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil"
] |
[
"MNRAS"
] |
We propose a different way to obtain the distribution of the luminosity function of quasars by using the Principle of Maximum Entropy. The input data comes from the SDSS-DR3 quasars counts, extending up to redshift 5 and limited from apparent magnitude i = 15 to 19.1 at z 3 to i = 20.2 for z 3. Using only few initial data points, the Principle allows us to estimate probabilities and hence that luminosity curve. We carry out statistical tests to evaluate our results. The resulting luminosity function compares well to earlier determinations. And our results remain consistent either when the amount or choice of sampled sources is unbiasedly altered. Besides this we estimate the distribution of the luminosity function for redshifts in which there is only observational data in the vicinity.
|
10.1093/mnras/stz162
|
[
"https://arxiv.org/pdf/1907.06984v1.pdf"
] | 196,831,358 |
1907.06984
|
3a4db94ab497eaba342e65e872bbae71700f1499
|
The Luminosity Function of Quasars by the Principle of Maximum Entropy
2018
Alexandre Andrei
Universidade Federal do Rio de Janeiro
Observatório do Valongo
Ladeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil
Observatório Nacional
MCTIC
Rua Gal. José Cristino 7720921-400Rio de JaneiroCEPBrazil
SYRTE
Observatoire de Paris
61 Avenue de l'Observatoire75014ParisFrance
Bruno Coelho
Instituto de Telecomunicações
Campus Universitário de Santiago3810-193AveiroPortugal
†
Leandro Guedes
Planetarium Foundation of the City of Rio de Janeiro
Rua Vice-Governador Rúbens Berardo 100 -Gávea22451-070Rio de Janeiro -RJCEPBrazil
‡
Alexandre Lyra
Universidade Federal do Rio de Janeiro
Observatório do Valongo
Ladeira Pedro Antonio, 4320080-090Rio de JaneiroRJ, CEPBrazil
The Luminosity Function of Quasars by the Principle of Maximum Entropy
MNRAS
0002018Accepted XXX. Received YYY; in original form ZZZPreprint 17 July 2019 Compiled using MNRAS L A T E X style file v3.0methods: data analysis -quasars: general -galaxies: luminosity function
We propose a different way to obtain the distribution of the luminosity function of quasars by using the Principle of Maximum Entropy. The input data comes from the SDSS-DR3 quasars counts, extending up to redshift 5 and limited from apparent magnitude i = 15 to 19.1 at z 3 to i = 20.2 for z 3. Using only few initial data points, the Principle allows us to estimate probabilities and hence that luminosity curve. We carry out statistical tests to evaluate our results. The resulting luminosity function compares well to earlier determinations. And our results remain consistent either when the amount or choice of sampled sources is unbiasedly altered. Besides this we estimate the distribution of the luminosity function for redshifts in which there is only observational data in the vicinity.
INTRODUCTION
The quasar luminosity function gives a measure for the bidimensional distribution of quasars in luminosity and redshift. Fundamentally it indicates that the universe is not in a stationary state. As consequence it requires the due interpretation before using quasars to determine cosmological parameters, but at the same time it informs about the evolution of quasars themselves, and the changing content of the space intervening between distant quasars and the observer. The function usually describing the quasar luminosity function, as a function of redshift and absolute luminosity, basically starts from the modulus distance formulae and incorporates several corrections, to accommodate line emission, the expanding universe scale of distance, the intrinsic dependency of quasar light emission on wavelength, terms of self and media absorption, etc. The result is an empirical description, which exponents and coefficients are adjusted to each sample examined. It is interesting thus to build an independent function, able to describe the quasar luminosity func-tion in a simpler form and from different physical principles. Although by necessity also incorporating the astrophysical and cosmological assumptions, an alternative, simpler form for the quasar luminosity function can be derived from the statistical mechanics methods.
The concept of entropy, since Clausius, became part of thermodynamics. In addition, it also became part of statistical mechanics. The study of systems in equilibrium and out of equilibrium is closely related to the notions of entropy as well as its production. There is a vast bibliography about it with warm discussions. We can cite three important related principles: Ziegler's maximum entropy production principle (e. g. Ziegler (1983), Ziegler (1987), see also Dewar (2005)); Prigogine's minimum entropy production principle (Prigogine (1967), Prigogine (1978), Kondepudi et al (1999)) 1 ; and the Maximum Entropy Principle (MaxEnt) (Jaynes (1957)). This paper employs the last one.
Derivations of the first two principles from MaxEnt can be found in literature, as seen in Martyushev et al (2006). In that review the authors make a very interesting description of the MaxEnt focusing on the production of entropy. Other authors emphasize that Jaynes's MaxEnt formulation of sta-tistical mechanics provides a theoretical basis for Maximum Entropy Production Principle (Dewar-Maritan (2014)). The applications of the MaxEnt are many. We'll see below related issues and discuss how they are connected to the focus of our treatment, that aims to find the distribution of the luminosity function of quasars.
Despite this vast reach there are authors who restrict the MaxEnt applications (see eg. Shimony (1985), and references therein).
Some of these critiques were addressed by Jaynes himself (Jaynes (1989)). In this paper Jaynes also provides a fairly complete description of MaxEnt from its roots to its implications. On the other hand, we can not fail to mention that there are also papers written exactly in defense of Jaynes's Principle as in Tikochinsky et al (1984), stating with: "The only consistent algorithm is one that leads to the distribution of Maximum entropy subject to constraints given.". There are other papers with very interesting critiques that bring out points for and against MaxEnt and provide quite compelling references on the subject, like in the Appendix A of Pontzen et al (2013), where the authors sketch Jaynes's reasoning, "that the maximization of entropy subject to certain constraints is equivalent to testing whether these constraints encapsulate later the physics of the situation...", and the use of the method to derive the phase space distribution of a virialized dark matter halo.
In addition, there are several other areas in physics and astrophysics where it can be applied. Some examples are, in spectral analysis (Ables (1974)), where "the method produces superior spectral representations when compared with more traditional methods...". as well as a powerful technique of image reconstruction (Skilling-Bryan (1984)), in the same paper other applications of MaxEnt in astronomy can be found. In Gull et al. (1978) MaxEnt is applied in radio and X-ray astronomy. It is interesting that the method is also applied in X-ray tomographic image reconstruction and restoration (Mohammad-Djafari (1989)). In the case of astrophysics and cosmology, we see papers where the dark energy equation of state w(z) is reconstructed using the MaxEnt (Zunckel et al (2007)).
In gravitation, with the confirmation in 2016 of the existence of the gravitational waves predicted by A. Einstein, the study of the black holes assumes still greater importance. The earliest detections were precisely on collisions of black-holes (Abbott et al. (2016)). The traditional second law of thermodynamics was modified into a generalized second law for the study of black holes (Bekenstein (1974)). The Jaynes's method of maximum entropy was also used by Bekenstein to determine the probability distribution for a system containing a Kerr black hole (Bekenstein (1975)).
This paper presents a new approach to find the distribution of probabilities of the luminosity function using the MaxEnt technique. Even with some criticisms like those cited above, we believe that the MaxEnt is extremely useful to be applied when we have partial information about a certain system. So this principle allows us to know accurate probabilities (see formula (4)) from a small data set. Although the number of known quasars is constantly increasing, to get perhaps to a million known objects in the next decade, small subsamples are useful and had not been yet designed by lack of elements. On one hand, the quasars zoo is also growing, different types of active galaxies conceivably exhibiting luminosity functions peculiar by a certain degree. On the other hand, the capability of mapping in detail particular thin slices of the universe in redshift is long sought, nonetheless to better define the complex form of the luminosity function. Finally, it is important to be able to drawn different samples of a large dataset for sanity check control. Is this paper we will explore such capability of the MaxEnt description of the luminosity distribution.
Since quasars discovery (Schmidt (1963), Matthews & Sandage (1963)) their energy output and magnitude have been object of much observation and increasingly complex theories. Conversely, that information became much used for studies as surrounding host galaxies, gravitational lenses, in situ and intergalactic absorption, up to the cosmological scale of distances in an expanding universe. The socalled luminosity function is all important to make sense of such extraordinary energy output and to those astrophysical quantities from it derived. The evolution of the quasar luminosity function with redshift is an important observational tool, that allows us to put constraints on the formation and growth history of supermassive black holes, and their coevolution with host galaxies. It also give us a measure of the contribution of quasars in the cosmological reionization of the Universe. For all these the study of the quasar luminosity function has received the attention of several works (eg. Richards et al (2006), Masters et al (2012), Ross et al (2013), Manti et al (2017)).
So, among some successful applications of MaxEnt in astrophysics, we are going now to explore a new one, in the study of the quasar luminosity function.
The Sloan Digital Sky Survey (SDSS) 2 provided observations of quasars in different redshifts, being responsible for the identification of the vast majority of the known Quasi-Stellar Objects (QSOs; Pâris et al. (2018)). However there are observational limitations, to the effect that one can ask: what would be the quasar distribution on each redshift slice if we could consider unobserved magnitudes? These observational limitations must be taken into account when computing the quasar luminosity function, however this does not constitute the aim of this work. So we are going to use already corrected counts of QSOs computed by Richards et al (2006) in their study of the quasar luminosity function. We show here that the MaxEnt can provide a good distribution of probabilities for the luminosity function from few values of a limited sample in each redshift.
The luminosity function provides the density distribution of classes of objects, per unit volume and assuming a statistically complete sample. In the case of quasars this indicates more or less probable scenarios for their formation and evolution, as well as their relationship with the host galaxy. Quasars have been found out by several projects, chiefly the SDSS, relying on different strategies to single them out from the more numerous contaminants of other celestial bodies. The ESA cornerstone mission Gaia combines the recognition of such known quasars, with micro arcsecond determination of proper motions over five years, therefore providing direct means to cleanse away the intruding false positives, as nearby red dwarfs. On top of it, Gaia will use a neural network strategy leading to autonomous recognition of quasars. Combined to the all sky repeated sweeping of objects up to near-red twentieth-second magnitude, it will produce an unprecedent complete sample of quasars. Therefore to establish an alternative, independent, and physics robust method of tracing the quasar luminosity function affords a strong way of checking upon and getting feedback from the usual Schechter based determination. In short, the motivations for these studies are threefold, an independent study of the luminosity function on the quasar population in the SDSS DR3, the development of an independent tool for determining the luminosity function based on maximum entropy physics, and it is a comparative assessment on a limited sample with views to application on the all sky, statistically complete sample of quasars in final Gaia catalogue.
Elseways the definition of the quasars luminosity function have so far been done using a modified template of the Schechter exponential for galaxies. Such a description although well adapted to the somehow simpler quasar case, since it is in practice free from the surface brightness issue, limits the reliability of the astrophysical and cosmological interpretation of the luminosity function. To mention a few, it is known that the shape and turnover of the luminosity function would favor either models for the growth of the super massive black hole from mergers or by inflow and host galaxy instabilities. The bright end of the luminosity function can favor intrinsic properties about which time black holes are increasing in mass rapidly whereas the faintest end would indicate about the length of time quasars spend at relatively low accretion rates.
The remainder of this work is organized as follows. In section 2 we will briefly review the MaxEnt method. With this we establish our main formula, the equation (4), which defines from MaxEnt the probability of the luminosity function. In section 3 we will summarize our technique to determine the luminosity function of quasars, and we show the details of how the Lagrange multipliers were calculated for the studied cases, in addition we will show the comparison between our result by MaxEnt and the Schechter's based Richards et al (2006) one.
In section 4 we will describe the statistical tests we use. In section 5 from few observational data in particular redshifts, we will make a prediction of the PDF (Probability Density Function) for in between redshifts, that is, we will estimate the distribution of the luminosity function. Finally, a summary discussion and conclusions are presented in the last section.
THE JAYNES APPROACH TO MAXIMUM ENTROPY PRINCIPLE
We can sum up the Maximum Entropy Principle as we shall see in the sequel 3 . As there is a vast bibliography regarding this principle, we will only make a brief account.
Initially we assume that a quantity x can have the discrete values xi(i = 1, 2, ..., n), but we do not know the corresponding probabilities pi. All we have is the expectation 3 Here we will follow Jaynes (1957).
value of the function f (x),
f (x) = n i=1 pif (xi)(1)
Based on this information, how can we obtain the expectation value of another function of the system g(x)? Jaynes responds to this apparently insoluble question. The given information is insufficient to determine the probabilities pi.
The equation (1) and the normalization condition
pi = 1(2)
would have to be supplemented by (n − 2) more conditions before g(x) could be found.
In order to find a solution to this problem, Jaynes's method uses the following expression for entropy
H(p1, p2, ..., pn) = −k i pi ln pi ,(3)
where k is a positive constant. Since H is just the expression for entropy as found in statistical mechanics, it will be called the "entropy of the probability distribution pi". The entropy H, given in (3) is maximized subject to the constraints (1) and (2). In order to achieve a final expression for the probability of xi, we use the method of Lagrangian multipliers, usually noted by and µ, where is associated with the normalization equation, i.e. the equation (2) and µ is associated with the equation of the expectation value (1). With this methodology we obtain the probability
pi = e −−µf (x i ) .(4)
This formula gives an important expression, which can be associated to the function of the luminosity distribution of the objects to which we wish to estimate the distribution, and the method used in its determination is called the Maximum Entropy Principle. See the complete development from data to Lagrange multipliers at Appendix B.
THE LUMINOSITY FUNCTION OF QSO(S) FROM MAXENT
To summarize what will be done next, from MaxEnt we will determine the distribution of the luminosity function of the quasars in a certain redshift z k , by using the probability distribution (4). Notice that the strong energy released by quasars make possible to observe them from the nearby Universe at least up to redshifts greater than 7 (eg. Bañados et al. (2018)). This large range of distance, hence an evolving luminosity function, allowed us to inspect how consistent are the predictions from MaxEnt, and compare the results against those originally derived from the same observed data, used here as control result.
In the MaxEnt methodology, for the consistency of the principle, the strongest symmetry that we could have "a priori" would be the uniform distribution, but this is not the case. We know that if we have a single constraint, that is associated with normalization n i=1 pi = 1 , we get exactly for n = N → pi = 1/N or the uniform distribution. The other constraint, associated with equation (1), breaks this symmetry. Let us also remember that as it is well placed in Caticha-Preuss (2004) : "The method of maximum entropy (ME) is designed for updating from a prior probability distribution to a posterior distribution when the information to be processed takes the form of a constraint ... ". Then, we assume that we can extract a certain expected value obtained through some luminosity values provided by the system observations, which obviously have the uniformity between all values broken. These values are randomly chosen, and under these conditions we will apply MaxEnt with their two constraints: (1) and (2). This is the central point of the methodology, namely that from just some values a strong estimate of the luminosity function of the distribution of all values in this redshift can be made 4 .
For the present quasar luminosity function derivation by MaxEnt, we have tested different sets from the whole of the initial data, seeing in every case great accordance between the Luminosity curve from MaxEnt and the control result. In order to analyze the most realistic scenario, the one for which the sample is small and, thus, not necessarily containing a perfect representation of the data population, we choose to analyze here the results from random initial data. We have picked up just three luminosities in each redshift as initial data.
The starting point of using MaxEnt is the calculation of µ and from the equations (1) and (2) (see details in Appendix B). From a certain redshift, the mean value to be used in the Lagrange multiplier method is calculated from three luminosities randomly chosen, to each of which is assigned the corrected number of quasars in that luminosity bin after applying the selection function of Richards et al (2006), Table 6, p. 2782. Those values will be used to calculate the weighted mean luminosity Lz , which is the value to be used in the equation (1). The other Lagrange multiplier comes from the normalization of the probability, or, n i=1 pi = 1.
Errors have been calculated using a bootstrap method. In each case, three random luminosities were drawn 200 times and the mean value used to find a different and µ that, applied to original data, gave us a different set of points. The extreme values stand as the upper and lower limits of the error bars to the results from the principle. Likewise the errors on the control result were calculated using probabilities from bootstrap draws.
Verifying our assumptions, the calculated probabilities by MaxEnt and the ones of the control results show similar behavior.
For each redshift the complete table leading to the control result is in Appendix A, Table A1., and the three ones randomly chosen in each redshift are on the lines indicated in bold at the first column.
The conversion from calibrated magnitudes to luminosities was done using the following relation
where L (in ergs s −1 Hz −1 ) is the luminosity and Mi the magnitude.
The curves obtained for each redshift are shown in Figure 1. We can see clearly that a correspondence is found at the sampled redshifts, within the error bars, between the MaxEnt results and those for the control.
STATISTICAL TESTS
As discussed in the previous paragraphs, and detailed in Appendix B, the MaxEnt approach, from robust yet simple physical principles and computational algorithms, delivers a statistical probability distribution of the luminosity function which is cosmologically plausible, vis a vis the literature on the subject. The magnitude and redshift data used for that is taken from the SDSS project. It is natural thus that the outcomes from the luminosity function here obtained shall be compared with those from the SDSS analysis.
At the start of the current application of the Max-Ent principle to derive the quasars luminosity function, several approaches were used. Choosing by hand representative data, choosing data from quartiles of the distribution, and picking up the extreme and mean values. The outcomes were always concurrent (they are available under request), what served as sanity check, as well as gave us ground to adopt the random draws finally used. The plots in Figure 1 are compelling to show the agreement between the two luminosity function statistical probability distributions. Such agreement can be quantified. Table 1 shows results of the statistical tests comparing the two distributions of probabilities, the one from MaxEnt and the one from the control results, at each redshift.
As indicated in section 3 a minimal number of points were randomly drawn from the data. Using only these few data points, MaxEnt can provide us an estimate luminosity function to be compared with the luminosity function obtained from the control results. The results are compared by verifying the mutual correlation. The Spearman's correlation test is used because of the small number of chosen points, as well to not assume their normal distribution. The ρ is quite close to unity. Notice however that although the luminosity function is best represented as an exponential progression, the pair of points of the two compared distributions are not necessarily so, thus we have also used the F-test and the Student's T-Test because these are nonparametric tests.
Since the shape of the curves is obviously similar but not the error bars, while the number of points is small, the F-test for variances is advisable. The table of the F-distribution indicates that the null hypothesis (no difference) must be accepted to a large degree of statistical certainty, with two exceptions, out of the limit redshift. Those exceptions lie at z = 0.87 and z = 1.25, for which the null hypothesis certainty is mediocre. In both cases, that befalls upon the large error bars seeing at the one brightest luminosities. The F-Test without those points give us results 1.64 and 0.92 respectively, that take us back to a null hypothesis scenario.
On views of the outturn of the correlation and variance tests pointing to the agreement of the MaxEnt and control results, the two-samples Student's T-test is next justified. On this one, as Table 1 shows, in all cases -even for the troublesome redsfifts as detected in the previous tests -the null-hypothesis on the means can not be rejected for usual statistical standards. Table 1 brings the three statistics for the distributions. On Table 2, instead, the same statistical tests are applied to compare their error budgets. Notice, at start, that the error bars are asymmetrical, and therefore up and down pairs are formed. The correlations are poor, though they undeniably exist. On the other hand, the F-test and T-test for the errors show the MaxEnt method and the control results faring quite alike also in this respect. We thus can further conclude for the independence of the methods, but similar efficiency.
ESTIMATION OF THE LUMINOSITY FUNCTION FOR OTHER REDSHIFTS
In this section we use the MaxEnt luminosity function presented in this paper to investigate the outcomes for a redshift in which we suppose that data exist only in its vicinity. For this simulation, the redshift z = 3.5 is chosen. As shown in Figure 2 at this redshift the luminosity L seems to increase again after a drop between z = 2.75 and z = 3.25, at the same time there are enough input data and good results for the neighbor redshifts. From those the mean value Lz=3.5 is interpolated, and next we will obtain by MaxEnt the distribution of L for the redshift 3.5. In practice, we start from the same set of data used before, from Richards et al (2006), plotting all the available redshifts with respective mean luminosities. Then the curve of best fitting to the observational data is obtained, and from this fitting curve we associate a mean luminosity to the redshift aimed at. Next, in order to procure the Lagrange multipliers µ and λ a set of observed luminosities is demanded. Those were picked up at random from the luminosities actually present for the neighbor redshifts.
The point now is to verify whether using this quite ar-bitrary choice the MaxEnt formulation is capable to issue a credible luminosity function. We thus compare the MaxEnt formulation results to a direct interpolation of the control results and of the MaxEnt results themselves (both depicted on Figure 1). Figure 3 shows these three results. It is seen that the MaxEnt formulation based on neighbor data gives a result comparable to the direct interpolation results, but at the same time it delivers a smoother curve.
This type of situation occurs frequently in astrophysics, and MaxEnt demonstrates here to be a very useful tool to estimate values, what later can be tested later as more data becomes available.
CONCLUSIONS
The quasar luminosity function is intended as a measure of the actual distribution of quasars in luminosity and redshift. For that observational, astrophysical, and cosmological restricting factors must be accounted for and often different surveys must be combined, before a complete population is inferred. That satisfied, most quasar luminosity functions available in the literature are represented either by a double power-law regimen or by a modified Schechter function. The adjustments are semi-empirical, having as usual parameters a normalization factor, a break magnitude, a reference redshift, and bright and faint ends slopes.
By contrast, the MaxEnt method, on top of being quite simple to handle, offers three strong features. First it represents a physically distinct approach, thus bringing the known benefits of different bias, limitations, and systematics. Secondly because it is purely statistical, it depends of less astrophysical and cosmological assumptions, in special the key ones break magnitude and reference redshift. Thirdly, a hallmark of MaxEnt is to deliver trustful conclusions from small samples. This last quality is particularly suited to deal with limited dedicated surveys, as well as to piece off portions of the luminosity function without further requirements to the mathematical representation of the function itself. By the same token it is suited to try out luminosity functions for putative new classes of quasars and their location, either within large clusters or relatively isolated.
In this pioneer derivation we took the SDSS DR3 quasar population, and the normalization made by Richards et al (2006) there in. The luminosity functions and corresponding curves were used here as control results. The comparisons hold very well, being practically immaterial whether the whole luminosity population or samples as small as three random elements were used.
As Jaynes has stated, that MaxEnt is the generalization of the Principle of Insufficient Reason. In our case, we show that little information of the system (quasars luminosities) gave us consistent results. In so it is an effective way of practical generalization. As a result, the Lagrange multipliers behaved in a stable manner, enabling to use bootstrapping for determination of errors. The aspect of updating the knowledge when of the outcome of a much larger data set, as expected from Gaia, is foreseen to be coherently accommodated, as well as to investigate piecemeal the luminosity function. Table A1 was obtained from Richards et al (2006), with the addition of the probability required to our objective and derived from their data, plus the probability we obtained for the comparison.
Object1 A1 Object2 A2 Object3 A3 ... ... Objectn An
where each Object i is a quasar and Ai its respective luminosity, with i = (1, 2, 3, ..., n). From now on we adapt Jaynes's notation to our work. Thus, we will call the luminosities by Ai, and their mean value by A . Each Ai has a probability pi to occur and we get from the data an average value A that can be obtained from arithmetic mean, weighted average, or from a more accurate form, using expression (1). This expression may be rewritten as
A = n i=1 Aipi ,(B1)
where at one redshift z k , the index n varies in the sum of i = 1, ..., n only on selected objects, that is, only in those three chosen luminosities in this redshift. Considering that the data set contains all possible values to occur, we have the bond condition that the summation of all probabilities must be equal to 1, see (2), or
n i=1 pi = 1 (B2)
The two Lagrange multipliers µ and are associated to these two equations respectively, (B1) and (B2). Then next they will be placed into a new form of the above equations. From B1 we have
µ n i=1 Aipi − A = 0 ,(B3)
and from B2
n i=1 pi − 1 = 0.(B4)
According to Jaynes, the method consists in the determination of the distribution function, pi, by maximizing the so-called informational entropy
H ≡ H(pi, p2, ...pn) = −K n i=1
pi ln pi , this can be done by the standard method using the additional conditions (B1) and (B2) and the Lagrange multipliers and µ. The maximization procedure leads to the following result pi = e −µA i − .
The two equations that we have to adjust to compute are obtained by taking (B5) into the equations of constraints (B1) and (B2), so we obtain the equations
e − n i=1 Aie −µA i = A (B6) e − n i=1 e −µA i = 1 .(B7)
The equation (B6) inform us that
e = n i=1 Aie −µA i A .(B8)
Taking (B8) into equation (B7) we obtain an equation in µ to be solved:
A n i=1 e −µA i n i=1 Aie −µA i = 1.(B9)
To find , the obtained values of µ are taken into (B8). That is, the sequence of procedures to find µ from equation (B9) and substitute it into the equation (B8) to find . With both Lagrange Multipliers found, we can get by MaxEnt the resulting probability (B5) and the average value (B1) for each redshift.
Figure 1 .
1Comparison between probabilities calculated from the MaxEnt method and from the control results. Plots show luminosities on the horizontal axis and probabilities on the vertical ones. Values at horizontal axis should be multiplied by 10 −31 (ergs s −1 Hz −1 ) at Z=0.49, 0.87, 1.25 and by 10 −32 (ergs s −1 Hz −1 ) at the others.
Figure 2 .
2Mean luminosity in bins of redshift from z = 0.5 to z = 4.25 derived from SDSS DR3 as inRichards et al (2006), not taking into account any corrections for the Malmquist bias.
Figure 3 .
3Luminosity function obtained from MaxEnt using the fitting curve to z=3.5.
ACKNOWLEDGMENTS A. L. thanks the colleagues at the Valongo Observatory, H. M. Boechat Roberty and M. Assafin, for suggestions in the beginning of this work. A. A. thanks CPNp Grant Bolsa de Produtividade em Pesquisa 306775/2018. B. C. acknowledges support from the Advanced EU Network of Einfrastructures for Astronomy with SKA (AENEAS), funded by the European Commission Framework Programme Horizon 2020 RIA under grant agreement n. 731016 and from the ENGAGE SKA RI, grant POCI-01-0145-FEDER-022217, funded by COMPETE 2020 and FCT, Portugal. We must also thank the anonymous referee for valuable suggestions and comments. APPENDIX A: TABLE A1
APPENDIX B: LAGRANGE MULTIPLIERS METHOD: DATA, CONSTRAINTS AND COMPUTATION To develop the fundamentals of MaxEnt, consider the following set of data
Table 1 .
1Statistical tests comparing the MaxEnt and control results luminosity curves.Spearman
F-Test
Student-t
ρ
P-Value
T-status
P-value
0.49 0.93 1.17 × 10 −5
1.32
10 −99
1
0.87 1
10 −99
4.40
8.88 ×10 −16
0.99
1.25 1
10 −99
3.26
10 −99
1
1.63 0.99 6.65 × 10 −64
1.10
10 −99
1
2.01 0.99 6.65 × 10 −64
2.75
10 −99
1
2.4
1
10 −99
2.53
1.99 × 10 −16
0.99
2.8
1
10 −99
1.12
2.21 × 10 −16
0.99
3.25 0.99 3.76 × 10 −9
1.48
3.35 × 10 −16
0.99
3.75 1
10 −99
1.50
10 −99
1
4.25 1
10 −99
1.87
10 −99
1
Table 2 .
2Statistical tests comparing the error budgets over the MaxEnt and control results luminosity curves.Spearman
F-Test
Student-t
ρ
P-Value
T-status P-value
0.49 0.77 1.02 × 10 −5
0.70
1.48
0.15
0.87 0.48 0.02
0.19
0.71
0.48
1.25 0.63 0.00
0.22
0.11
0.91
1.63 0.60 0.00
0.29
0.44
0.67
2.01 0.53 0.02
0.48
0.03
0.97
2.4
0.46 0.07
0.29
0.50
0.62
2.8
0.60 0.01
0.16
0.55
0.58
3.25 0.56 0.01
0.41
0.45
0.65
3.75 0.65 0.01
0.41
0.45
0.65
4.25 0.58 0.05
0.24
1.49
0.15
Table A1 .
A1The Redshift, Luminosities and ProbabilityZ
L (×10 30 )
Prob
Prob
(erg/s/Hz)
MaxEnt
0.49
13.70
3.32 × 10 −3
3.84 × 10 −4
0.49
10.42
6.336 × 10 −3
1.89 × 10 −3
0.49
7.90
1.00 × 10 −2
6.36 × 10 −3
0.49 5.60
1.55 × 10 −2
1.59 × 10 −2
0.49
4.55
2.83 × 10 −2
3.20 × 10 −2
0.49 3.45
4.70 × 10 −2
5.42 × 10 −2
0.49
2.62
6.95 × 10 −2
8.09 × 10 −2
0.49 1.99
1.08 × 10 −1
1.10 × 10 −1
0.49
1.51
1.52 × 10 −1
1.38 × 10 −1
0.49
1.14
1.96 × 10 −1
1.65 × 10 −1
0.49
0.87
2.83 × 10 −1
1.88 × 10 −1
0.49
0.66
8.16 × 10 −2
2.08 × 10 −1
0.87
41.50
9.74 × 10 −4
1.07 × 10 −2
0.87
31.50
3.70 × 10 −3
2.07 × 10 −2
0.87 23.90
5.61 × 10 −3
3.40 × 10 −2
0.87
18.10
1.23 × 10 −2
4.96 × 10 −2
0.87
13.70
2.62 × 10 −2
6.61 × 10 −2
0.87
10.40
3.46 × 10 −2
8.21 × 10 −2
0.87
7.91
5.74 × 10 −2
9.69 × 10 −2
0.87
5.60
9.09 × 10 −2
1.10 × 10 −1
0.87 4.55
1.2 × 10 −1
1.21 × 10 −1
0.87
3.45
1.419 × 10 −1
1.30 × 10 −1
0.87
2.62
1.77 × 10 −1
1.37 × 10 −1
0.87 1.99
3.30 × 10 −1
1.43 × 10 −1
1.25
72.10
1.69 × 10 −3
2.79 × 10 −3
1.25
54.70
3.29 × 10 −3
8.38 × 10 −3
1.25 41.50
4.14 × 10 −3
1.93 × 10 −2
1.25
31.50
1.04 × 10 −2
3.63 × 10 −2
1.25
23.90
1.98 × 10 −2
5.87 × 10 −2
1.25
18.10
3.52 × 10 −2
8.45 × 10 −2
1.25
13.70
6.27 × 10 −2
1.11 × 10 −1
1.25 10.40
1.11 × 10 −1
1.37 × 10 −1
1.25
7.90
1.37 × 10 −1
1.61 × 10 −1
1.25 5.60
1.807 × 10 −1
1.82 × 10 −1
1.25
4.55
4.33 × 10 −1
1.99 × 10 −1
1.63
125.00
2.57 × 10 −3
1.55 × 10 −4
1.63
95.00
3.79 × 10 −3
1.14 × 10 −3
1.63
72.10
7.57 × 10 −3
5.20 × 10 −3
1.63
54.70
1.86 × 10 −2
1.64 × 10 −2
1.63 41.50
3.46 × 10 −2
3.93 × 10 −2
1.63 31.50
6.30 × 10 −2
7.61 × 10 −2
1.63
23.90
1.07 × 10 −1
1.26 × 10 −1
1.63
18.10
1.82 × 10 −1
1.84 × 10 −1
1.63 13.70
2.51 × 10 −1
2.46 × 10 −1
1.63
10.40
3.31 × 10 −1
3.06 × 10 −1
2.01
165.00
1.65 × 10 −3
4.48 × 10 −3
2.01 125.00
4.87 × 10 −3
1.25 × 10 −2
2.01
95.00
7.54 × 10 −3
2.71 × 10 −2
2.01
72.00
1.81 × 10 −2
4.89 × 10 −2
2.01
54.70
3.00 × 10 −2
7.65 × 10 −2
2.01 41.50
5.72 × 10 −2
1.07 × 10 −1
2.01
31.50
9.06 × 10 −2
1.39 × 10 −1
2.01
23.90
1.50 × 10 −1
1.69 × 10 −1
2.01 18.10
2.44 × 10 −1
1.96 × 10 −1
2.01
13.70
3.96 × 10 −1
2.19 × 10 −1
2.4
165.00
6.57 × 10 −3
7.35 × 10 −3
2.4
125.00
8.08 × 10 −3
2.07 × 10 −2
2.4
95.00
2.22 × 10 −2
4.56 × 10 −2
2.4
72.00
4.44 × 10 −2
8.28 × 10 −2
2.4
54.70
7.71 × 10 −2
1.30 × 10 −1
2.4
41.50
1.31 × 10 −1
1.84 × 10 −1
2.4
31.50
2.12 × 10 −1
2.39 × 10 −1
2.4
23.90
4.98 × 10 −1
2.91 × 10 −1
Table A1 -continued
A148 × 10 −2 3.75 72.10 6.57 × 10 −2 9.16 × 10 −2 3.75 54.70 1.04 × 10 −1 1.35 × 10 −1 3.75 41.49 1.54 × 10 −1 1.82 × 10 −1 3.75 31.50 1.98 × 10 −1 2.28 × 10 −1Z
L (×10 30 )
Prob
Prob
(erg/s/Hz)
MaxEnt
2.8
274.00
1.04 × 10 −2
3.64 × 10 −3
2.8
218.00
1.77 × 10 −2
1.09 × 10 −2
2.8
165.00
3.21 × 10 −2
3.01 × 10 −2
2.8
125.00
4.34 × 10 −2
6.53 × 10 −2
2.8
95.00
8.65 × 10 −2
1.17 × 10 −1
2.8
72.10
1.81 × 10 −1
1.83 × 10 −1
2.8
54.70
3.00 × 10 −1
2.57 × 10 −1
2.8
41.50
3.29 × 10 −1
3.32 × 10 −1
3.25
287.00
4.09 × 10 −3
4.64 × 10 −4
3.25
218.00
7.11 × 10 −3
2.32 × 10 −3
3.25 165.00
6.63 × 10 −3
7.85 × 10 −3
3.25 125.00
1.63 × 10 −2
2.00 × 10 −2
3.25
95.00
3.25 × 10 −2
3.99 × 10 −2
3.25
72.10
4.80 × 10 −2
6.80 × 10 −2
3.25
54.70
6.79 × 10 −2
1.02 × 10 −1
3.25 41.50
1.15 × 10 −1
1.38 × 10 −1
3.25
31.50
1.45 × 10 −1
1.75 × 10 −1
3.25
23.90
2.25 × 10 −1
2.08 × 10 −1
3.25
18.10
3.32 × 10 −1
2.38 × 10 −1
3.75
165.00
1.65 × 10 −2
1.14 × 10 −2
3.75
125.00
2.74 × 10 −2
2.78 × 10 −2
3.75
95.00
6.42 × 10 −2
5.3.75
23.90
3.69 × 10 −1
2.70 × 10 −1
4.25 218.00
4.65 × 10 −2
4.22 × 10 −2
4.25
125.00
9.28 × 10 −2
1.11 × 10 −1
4.25 95.00
1.09 × 10 −1
1.53 × 10 −1
4.25
72.10
1.73 × 10 −1
1.94 × 10 −1
4.25
54.70
2.18 × 10 −1
2.33 × 10 −1
4.25 41.50
3.61 × 10 −1
2.67 × 10 −1
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|
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[
"A property of cyclotomic polynomials 1",
"A property of cyclotomic polynomials 1"
] |
[
"Giovanni Falcone [email protected] \nDipartimento di Metodi e Modelli Matematici\nViale delle Scienze Ed. 8I-90128PalermoItaly\n\nSupported by M.I.U.R\nUniversità di Palermo\n"
] |
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"Dipartimento di Metodi e Modelli Matematici\nViale delle Scienze Ed. 8I-90128PalermoItaly",
"Supported by M.I.U.R\nUniversità di Palermo"
] |
[] |
Given two cyclotomic polynomials Φn(x) and Φm(x), n = m, we determine the minimal natural number k such that we can writewith a(x) and b(x) integer polynomials.
| null |
[
"https://arxiv.org/pdf/0708.1423v1.pdf"
] | 14,329,725 |
0708.1423
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4e297ff3fd4c88d156124a60c42cf507e119500c
|
A property of cyclotomic polynomials 1
10 Aug 2007
Giovanni Falcone [email protected]
Dipartimento di Metodi e Modelli Matematici
Viale delle Scienze Ed. 8I-90128PalermoItaly
Supported by M.I.U.R
Università di Palermo
A property of cyclotomic polynomials 1
10 Aug 2007
Given two cyclotomic polynomials Φn(x) and Φm(x), n = m, we determine the minimal natural number k such that we can writewith a(x) and b(x) integer polynomials.
In June 2000 the author of this note gave a talk at the conference "Combinatorics 2000" in Gaeta (Italy), titled Dividing Cyclotomic Polynomials. The content of that talk was proposed to the American Mathematical Monthly as Problem 10914 in Volume 109, January 2002, and a composite solution by R. Stong and N. Komanda was published on Volume 110, October 2003. The problem was also solved by R. Chapman, C. P. Rupert, the GCHQ Problem Solving Group and the proposer, whose original solution is given in this note.
A property of cyclotomic polynomials. We denote, as usual, with Φ k (x) the cyclotomic polynomial (with integer coefficients) of index k, defined inductively through the identity
x m − 1 = Φ k (x),
where k runs among the divisors of m. The basic properties of cyclotomic polynomials are well known, as well as the role they play in several branches of Mathematics. Here we just recall that Φ m (1) = p, if m is a p-power (p prime), Φ m (1) = 1 otherwise.
Given two different cyclotomic polynomials Φ m (x), Φ n (x), we can find two polynomials s(x), t(x) ∈ Q[x] such that
1 = s(x)Φ m (x) + t(x)Φ n (x),
Φ m (x) and Φ n (x) being irreducible as element of the euclidean ring Q[x]. Since the ring Z[x] is a unique factorization domain, but not a euclidean ring, here this property fails. But the existence of an integer k and of two integer polynomials a(
x), b(x) ∈ Z[x], such that k = a(x)Φ m (x) + b(x)Φ n (x), is manifestly guaranteed.
This note is essentially an integration of [1], where we used cyclotomic polynomials to define minimal polynomials of finite automorphisms of groups. The knowledge of the integer k is useful in order to give a unique definition of the minimal polynomial for elements of finite order in a ring, as well as a decomposition of a group, on which a finite automorphism is acting, into the direct sum of invariant subgroups. We refer to [1] for more details.
According to [1], the following proposition holds good:
Proposition. Let m > n be two integers. If n does not divide m then two polynomial a(x), b(x) ∈ Z[x] exist, such that 1 = a(x)Φ m (x) + b(x)Φ n (x).
In order to evaluate k we can confine ourself then to the case in which n is a divisor of m:
Proposition. Let Φ m (x), Φ n (x) be two cyclotomic polynomials, and let n be a divisor of m. Then two polynomials a(
x), b(x) ∈ Z[x] exist, such that k = a(x)Φ m (x) + b(x)Φ n (x),
where k = 1, if m n is not a prime-power, or k = p, if m n = p t (p a prime). In the latter case, p is the smallest positive integer with that property.
Proof. Note that, since n divides m, then Φ m (x) is a divisor of the polynomial Φ m n (x n ). In fact, if ω ∈ C is a primitive m-th root of unity, then ω generates a cyclic group of order m and ω n generates a cyclic group of order m n , that is ω n is a primitive m n -th root of unity, hence Φ m n (ω n ) = 0. Each root of Φ m (x) is then a root of Φ m n (x n ). Since Φ m (x) is irreducible, this means that Φ m (x) is a divisor of Φ m n (x n ), even over Z[x] by Gauß' Lemma.
Write Φ m n (x n ) = a(x)Φ m (x) and let σ ∈ C be a primitive n-th root of unity, so we have
a(σ)Φ m (σ) = Φ m n (σ n ) = Φ m n (1) = k,
where k = 1, if m n is not a prime-power, or k = p, if m n = p t (p prime). Divide a(x)Φ m (x) by Φ n (x) and write
a(x)Φ m (x) = b(x)Φ n (x) + r(x),
where r(x) is an integer polynomial of lower degree than the one of Φ n (x). We get k = r(σ), but this is possible only if r(x) is a constant (equal to k), since the degree of r(x) − k is lower than the one of the irreducible polynomial Φ n (x).
In order to prove that, if m n = p t (p prime), no other natural number 0 < k < p is such that
k = a(x)Φ m (x) + b(x)Φ n (x),
we note that Φ m (x) and Φ n (x) have, in this case, the same roots on a field K of characteristic p (see the following Note), and for such a root α ∈ K we would have
k = a(α)Φ m (α) + b(α)Φ n (α) = 0, a contradiction. 2
Note. Let K be a field of characteristic p, α ∈ K and h = kp r , p not dividing k. Then Φ h (α) = 0 ⇐⇒ Φ k (α) = 0.
In fact, one can immediatily see that on Z[x] the following two properties of cyclotomic polynomials hold good: i) if p is a prime, not dividing n, then Φ n (x)Φ pn (x) and Φ n (x p ) have precisely the same roots, hence
Φ pn (x) = Φ n (x p ) Φ n (x) ;
ii) if n = p r 1 1 · · · p rs s is the prime factorization of n, then
Φ n (x) = Φ p 1 ···ps (x (p (r 1 −1) 1 ···p (rs−1) s ) ).
In fact ω is a primitive n-th root of unity if and only if ω (p (r 1 −1)
1 ···p (rs−1) s )
is a primitive (p 1 · · · p s )-th root of unity. Now let k = p r 1 1 · · · p rs s be the prime factorization of k, then h = kp r = p r 1 1 · · · p rs s p r is the prime factorization of h and we have
Φ h (x) = Φ p 1 ···psp (x (p (r 1 −1) 1 ···p (rs−1) s p (r−1) ) ).
Put for short y = x (p (r 1 −1) 1 ···p (rs−1) s ) , and rewrite Φ h (x) = Φ p 1 ···psp (y p (r−1) ) = Φ p 1 ···ps (y p r ) Φ p 1 ···ps (y p (r−1) ) .
Since we are evaluating on a field of characteristic p, we have now Φ h (x) = Φ p 1 ···ps (y) p r Φ p 1 ···ps (y) p (r−1) = Φ p 1 ···ps (y) p (r−1) (p−1) .
But we have, as well Φ k (x) = Φ p 1 ···ps (x (p (r 1 −1) 1 ···p (rs−1) s ) ) = Φ p 1 ···ps (y), so we are done.
AMS MCS 11C08. 2 Supported by M.I.U.R., Università di Palermo.
Cyclotomic polynomials and finite automorphisms of groups. G Falcone, Geom. Dedicata. 841-3G. Falcone "Cyclotomic polynomials and finite automorphisms of groups", Geom. Dedicata 84 (2001), Nos. 1-3, 235-244
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"JuxtaPiton: Enabling Heterogeneous-ISA Research with RISC-V and SPARC FPGA Soft-cores"
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"Jonathan Balkind [email protected] \nUniversity of Washington\nPrinceton University\nPrinceton University\n\n",
"David Wentzlaff [email protected] \nUniversity of Washington\nPrinceton University\nPrinceton University\n\n"
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"University of Washington\nPrinceton University\nPrinceton University\n",
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Energy efficiency has become an increasingly important concern in computer architecture due to the end of Dennard scaling. Heterogeneity has been explored as a way to achieve better energy efficiency and heterogeneous microarchitecture chips have become common in the mobile setting.Recent research has explored using heterogeneous-ISA, heterogeneous microarchitecture, general-purpose cores to achieve further energy efficiency gains. However, there is no open-source hardware implementation of a heterogeneous-ISA processor available for research, and effective research on heterogeneous-ISA processors necessitates the emulation speed provided by FPGA prototyping. This work describes our experiences creating JuxtaPiton by integrating a small RISC-V core into the OpenPiton framework, which uses a modified OpenSPARC T1 core. This is the first time a new core has been integrated with the OpenPiton framework, and JuxtaPiton is the first open-source, general-purpose, heterogeneous-ISA processor. JuxtaPiton inherits all the capabilities of OpenPiton, including vital FPGA emulation infrastructure which can boot full-stack Debian Linux. Using this infrastructure, we investigate area and timing effects of using the new RISC-V core on FPGA and the performance of the new core running microbenchmarks.
|
10.1145/3289602.3293958
|
[
"https://arxiv.org/pdf/1811.08091v1.pdf"
] | 53,762,010 |
1811.08091
|
2bcab8ba9a1fbaa52313a282b49c8ec24ed017d6
|
JuxtaPiton: Enabling Heterogeneous-ISA Research with RISC-V and SPARC FPGA Soft-cores
Katie Lim [email protected]
University of Washington
Princeton University
Princeton University
Jonathan Balkind [email protected]
University of Washington
Princeton University
Princeton University
David Wentzlaff [email protected]
University of Washington
Princeton University
Princeton University
JuxtaPiton: Enabling Heterogeneous-ISA Research with RISC-V and SPARC FPGA Soft-cores
Energy efficiency has become an increasingly important concern in computer architecture due to the end of Dennard scaling. Heterogeneity has been explored as a way to achieve better energy efficiency and heterogeneous microarchitecture chips have become common in the mobile setting.Recent research has explored using heterogeneous-ISA, heterogeneous microarchitecture, general-purpose cores to achieve further energy efficiency gains. However, there is no open-source hardware implementation of a heterogeneous-ISA processor available for research, and effective research on heterogeneous-ISA processors necessitates the emulation speed provided by FPGA prototyping. This work describes our experiences creating JuxtaPiton by integrating a small RISC-V core into the OpenPiton framework, which uses a modified OpenSPARC T1 core. This is the first time a new core has been integrated with the OpenPiton framework, and JuxtaPiton is the first open-source, general-purpose, heterogeneous-ISA processor. JuxtaPiton inherits all the capabilities of OpenPiton, including vital FPGA emulation infrastructure which can boot full-stack Debian Linux. Using this infrastructure, we investigate area and timing effects of using the new RISC-V core on FPGA and the performance of the new core running microbenchmarks.
INTRODUCTION
Energy efficiency has become an increasingly important concern for modern processors. The end of Dennard scaling means that power dissipation no longer decreases with feature length. This has resulted in new research problems such as dark silicon [22] that require a new emphasis on energy efficiency. Additionally, demands in both the datacenter and the mobile setting have made power and energy efficiency more important. Datacenters now account for several percent of global energy usage [21]. As we increasingly rely on datacenters for computation, energy efficiency becomes more important both for economic and environmental reasons [6]. In a mobile setting with limited cooling and demand for better battery life, energy efficiency cannot be avoided.
In response to the need for energy efficiency, research has explored introducing heterogeneity into processors. Heterogeneous processors seek to cater to the fact that not all applications have the same computational demands.
Processors with heterogeneous microarchitecture have become common in modern cellphones for their energy efficiency benefits. Examples include ARM's big.LITTLE architectures [3] and Apple's A11 processor [2]. These processors use smaller, lower power cores for applications that do not need the computational power of larger, more power hungry cores to achieve better energy efficiency without significantly impacting the user experience. * This work done at Princeton University Recent research by Venkat et al. explored using general-purpose cores with heterogeneous microarchitecture and heterogeneous ISA in simulation [23]. They found potential energy and performance benefits from this type of architecture. However, further work on this topic has been difficult due to a lack of suitable prototyping platforms which can explore the implications of heterogeneous-ISA processor designs in a full-stack system at speeds suitable for rapid prototyping
To assist heterogeneous-ISA research, we have created JuxtaPiton by integrating PicoRV32 [24], a small RISC-V core, into the OpenPiton framework [5], an open-source manycore research platform which uses a modified OpenSPARC T1 core [17]. This is the first time another core has been integrated into OpenPiton, and we believe the JuxtaPiton FPGA platorm is the first open-source implementation of a general-purpose, heterogeneous-ISA processor. Given that heterogeneous ISA architecture is an emerging area of research that requires investigation of issues in hardware and software, an FPGA implementation will prove the most helpful to heterogeneous-ISA researchers. Architects will be able to modify any aspect of the design and prototype their research ideas on FPGA. At the same time, OS researchers can run complex, fullstack software on an FPGA-speed hardware system to evaluate their designs.
Heterogeneous-ISA processors pose interesting challenges to software. Some prior research has been done in simulation [9]. However, OS research projects such as Popcorn Linux [7] or K2 [13] have relied on hardware platforms that do not support shared memory. While this is a common design point, shared memory systems are easier to program and common in homogeneous-ISA processors, like those used in mobile systems on chip (SoCs). Additionally, hardware shared memory is needed for efficient process migration between ISAs [9,14]. However, it is difficult to build a heterogeneous-ISA, shared memory processor using off-the-shelf parts. In JuxtaPiton, the PicoRV32 core has fully cache coherent shared memory with the SPARC core enabled by OpenPiton's P-Mesh cache system. Table 1: A summary of the major differences between the OpenSPARC T1 core and the PicoRV32 core This paper details our experience using OpenPiton and RISC-V to create JuxtaPiton. By leveraging the OpenPiton infrastructure, we were able to quickly construct a functional system and implement software infrastructure to run static C binaries on the PicoRV32 core hosted by the OpenSPARC T1 core, and the PicoRV32 core is able to proxy syscalls to the OpenSPARC T1 core. We also evaluate some of the trade-offs of the PicoRV32 core compared to the OpenSPARC T1 core. We look at potential area and timing improvements as well as the performance impact of using the simpler PicoRV32 core instead of the OpenSPARC T1 core.
ARCHITECTURE
To construct our framework, we leverage the PicoRV32 and the OpenPiton framework. We integrated these two open-source projects by connecting PicoRV32 to the OpenPiton cache hierarchy. These cores were chosen because they are very different. Table 1 summarizes differences between the two cores.
OpenPiton
OpenPiton has a tiled manycore architecture. Each tile has a core, an FPU, three P-Mesh NoC routers, and caches. The original OpenPiton core is a modified OpenSPARC T1 core implementing the SPARCv9 ISA, and has a 6-stage in-order pipeline. The core has an instruction cache and a data cache. Each tile also has two levels of cache: the L1.5 and the L2. The L1.5 cache is equivalent to a private L2 cache, and the L2 cache is equivalent to a shared, distributed Last Level Cache (LLC). An OpenSPARC tile is shown in Figure 1 as Tile 0. Cache coherence is maintained using OpenPiton's cache coherence protocol, P-Mesh. The OpenPiton framework supports running designs in simulation as well as implementing designs for FPGA.
PicoRV32
PicoRV32 is a multicycle implementation of RV32I, the 32-bit core RISC-V ISA. It has no caches, and it does not support virtual memory. The core also does not implement the RISC-V privileged specification, so it is hosted by the OpenSPARC T1 core.
We decided on the PicoRV32 core for several reasons. First, the Pi-coRV32 core is open-source and is written in synthesizable Verilog RTL. It also has been applied in a number of settings by the community and has been the subject of formal verification [25]. Second, its simpler microarchitecture meant we would have a heterogeneous-ISA, heterogeneous microarchitecture system. We chose to use a core with a vastly simpler microarchitecture compared to the OpenSPARC T1 in order to research differences in microarchitecture.
Integration
To integrate the PicoRV32 core into the OpenPiton infrastructure, we connected it behind OpenPiton's L1.5 cache by adding transducers that convert the memory requests from the PicoRV32 core to OpenPiton L1.5 cache operations. This creates a tile where the OpenSPARC T1 core is replaced by the PicoRV32 core. We also removed the FPU from the tile with the PicoRV32 core, since the core does not support the RISC-V floating point extension. A diagram of an OpenSPARC T1 tile and a PicoRV32 tile connected to each other and memory is shown in Figure 1.
Connecting the PicoRV32 core to the L1.5 cache enables it to use the P-Mesh cache coherence protocol without modification to any existing infrastructure, therefore OpenSPARC T1 cores and PicoRV32 cores can share memory. Because interrupts traverse the caches, connecting the PicoRV32 core to the L1.5 cache also enables the PicoRV32 core to receive interprocessor interrupts from the OpenSPARC T1 core. The PicoRV32 core is brought out of reset using an interrupt sent from the OpenSPARC T1 core.
When we integrated the PicoRV32 core, we had to consider byte endianness differences, because SPARC is a big endian ISA whereas RISC-V is a little endian ISA. We chose to flip the outgoing and incoming data buses for the PicoRV32 core, such that the data accessed by the PicoRV32 core is stored little endian in memory. SPARCv9 does support little endian data accesses with use of special assembly instructions, but in our higher-level C code, we choose to use endian-flipping macros when interacting with data for the PicoRV32 cores.
The PicoRV32 core interacts with the L1.5 cache slightly differently than the OpenSPARC T1 core does. This is because the PicoRV32 core does not have an L1 cache, so the L1.5 is the Pi-coRV32 core's first-level cache whereas the OpenSPARC T1 core has L1 caches, making the L1.5 cache the OpenSPARC T1 core's second-level cache.
To improve performance, we chose to have the L1.5 cache instructions and data for the PicoRV32 core. This is in contrast with the OpenSPARC T1 core where instructions are not cached in the L1.5 cache, but only in the L1 and the L2 caches.
Second, OpenPiton's L1 cache is write-through, so writes must always go to the L1.5 cache when using the OpenSPARC T1 core. However, the L1 cache is the same associativity and capacity as the L1.5 cache, so any read that would hit in the L1.5 cache also hits in the L1 cache. This means that reads do not go to the L1.5 cache. However, when using the PicoRV32 core, both reads and writes will go to the L1.5 cache.
We found that modifying the OpenPiton framework to fit our needs was quick and the changes were relatively minimal even though we were replacing a core which is a relatively major change.
The changes were isolated to the transducers and slight modifications to core and tile instantiations as well as instantiations within the tile. so we could select between OpenSPARC T1 and PicoRV32 cores. Once the core was integrated, we were able to instantiate multiple tiles with different cores without further modification to the infrastructure. JuxtaPiton maintains all the functionality of the original Open-Piton framework. For example, the PicoRV32 core is able to access all of the I/O devices from the original OpenPiton framework including the SD card and the UART. We were also able to have the PicoRV32 core read instructions straight from the SD card and write characters to the UART during testing and evaluation.
Furthermore, because the OpenPiton framework contains pushbutton scripts for implementation on FPGA, and the PicoRV32 core was synthesizable, we could immediately test our design on FPGA, and we used the FPGA to efficiently prototype JuxtaPiton. Being able to implement JuxtaPiton on FPGA was also crucial to our software prototyping, which we describe in more detail in Section 3.
SOFTWARE SUPPORT
We are able to run the RISC-V assembly test suite and staticallylinked C binaries on the PicoRV32 core. We augmented the Open-Piton simulation infrastructure to enable us to compile and run RISC-V assembly tests by adding a script to generate the proper memory image for the PicoRV32 core. The PicoRV32 core integrated in the OpenPiton framework is able to run and pass all of the RV32UI tests from the official riscv-tests distribution [20].
For evaluation, we also built software infrastructure to enable the PicoRV32 core to run statically-linked C binaries hosted by the OpenSPARC T1 core. The OpenSPARC T1 core is responsible for loading the binaries into memory and proxying any syscalls from the PicoRV32 core.
The OpenPiton software stack consists of full-stack Debian Linux and a lightly modified version of the OpenSPARC T1 hypervisor. To support the PicoRV32 core, we wrote a userspace proxy program to load binaries and proxy syscalls for the PicoRV32 core to OpenSPARC T1 core running full-stack Debia Linux. This enables the PicoRV32 core to access OS resources, such as the file system, transparently. We also added two new Linux syscalls, and added a new hypercall.
An overview of the process to run a binary on the PicoRV32 core is shown in Figure 2. The userspace proxy program on the OpenSPARC T1 core with the name of the binary to be run provided as a command line argument (Step 0). The added syscall pico_setup is then used to allocate a region of physical memory for the PicoRV32 core (Step 1). Once the syscall returns, the binary is loaded into the allocated memory (Step 2). After the binary is completely loaded, the OpenSPARC T1 core send the start interrupt to the PicoRV32 core using the new syscall pico_start (Step 3). The syscall then calls the new hypercall hycall_pico_start (Step 4). Finally, the hypervisor sends the start interrupt to the PicoRV32 core (Step 5).
The userspace proxy program continues running to proxy syscalls from the PicoRV32 core. We take advantage of the fact that the OpenSPARC T1 core is running full-stack Linux and have the OpenSPARC T1 core perform syscalls. Binaries that run on the PicoRV32 core are linked against a version of Newlib where the syscall stubs are modified to write the syscall number and arguments out to the shared piece of memory. The OpenPiton userspace program polls on this memory and when it sees the PicoRV32 core needs a syscall serviced, it reads the number and arguments out of memory and makes the syscall in Linux itself. To return the result, the OpenSPARC T1 core writes it back to the shared memory region. The PicoRV32 core can then read this result and make use of it.
We host the PicoRV32 core since it does not implement the RISC-V privileged specification, but a core with a privileged specification implementation could also be hosted using the same setup. Additionally, system software support for heterogeneous-ISA systems with self-hosting cores in a shared memory system is an active area of research.
When developing our software infrastructure, having JuxtaPiton running on FPGA was crucial. When running on FPGA, the SPARC core is able to run Linux, which is practically impossible in behavioral simulation due to the orders of magnitude difference in speed of simulation (tens of kilohertz) versus FPGA emulation speed (ten of megahertz). Linux provides a much more fully featured enviroment for developing software, which enabled us to develop a RISC-V ELF binary loader on the SPARC core. Additionally, Linux is required for PicoRV32 to be able to proxy syscalls.
EVALUATION
All evaluation was done using a Digilent Genesys2 FPGA board using Xilinx Vivado 2015.4 to implement designs for the boards.
Area Analysis
For our area analyses we used Xilinx Vivado to build bitfiles for the Genesys2. The Genesys2 uses the Xilinx Kintex-7 FPGA (XC7K325T-2FFG900C) [10]. We used Vivado's default synthesis strategy and the phys_opt_design implementation strategy.
We looked at building designs with 1 OpenSPARC T1 tile or 1 Pi-coRV32 tile at frequencies between 50 MHz and 100 Mhz. We saw no significant change in resource utilization for either an OpenSPARC T1 tile or a PicoRV32 tile when increasing frequency.
There is a slight gain in maximum frequency when using the PicoRV32 core over the OpenSPARC T1 core. The maximum frequency that meets timing is 109.091 Mhz for an OpenSPARC T1 tile and 114.286 MHz for a PicoRV32 tile. For the OpenSPARC T1 tile, the critical path is in the D-TLB. For the PicoRV32 tile, the critical path is in the L2 cache.
We did see a significant area improvement gained by using the PicoRV32 core over the OpenSPARC T1 core. Utilization is shown in Table 2. We found that a PicoRV32 core uses approximately 1 30 th the look-up tables (LUTs) of an OpenSPARC T1 core. The resulting
Memory Hierarchy Latency
We also investigated the latency in cycles to different parts of the memory hierarchy to better understand performance of the Pi-coRV32 core. Cycle counts for other instructions that are only dependent on the core itself are already available on the GitHub page for the core.
We measured latency of using a memory operation between two rdcycle instructions. The results are shown in Table 3. The raw measurements given are the difference between the cycle counts before and after a memory operation. However, the PicoRV32 core is unpipelined and takes multiple cycles to execute each instruction. As such, the raw measurements must be adjusted to gain more insight into how much of the latency is actually from operations within the memory hierarchy versus the time for the other portions of the instruction to execute.
We first looked at determining the true L1.5 cache hit time from the raw L1.5 cache hit time of 17 cycles. Every instruction requires at least one cache access to the L1.5 cache to fetch it, so a memory instruction is actually two accesses to the L1.5 cache. Some of the latency is also from fetching the second rdcycle instruction. Thus, the instructions used to measure memory latency account for 3 accesses to the L1.5 cache. In addition, load and store instructions take 5 cycles in the PicoRV32 core. This was given by the official documentation for the PicoRV32 core and verified in simulation. For accesses to the L1.5 cache, after subtracting off the 5 cycles for the memory instruction and dividing by the 3 memory hierarchy accesses, we get that an L1.5 cache hit for the PicoRV32 takes 4 cycles. The time for a DRAM is much higher at around 113 cycles, and there is some slight variance in the measurements. This is most likely due to the fact that requests that go all the way to DRAM cross clock domains and need to go through asynchronous FIFOs. Depending on when the request gets to the FIFO relative to the other clock domain, there may be variance in the number of cycles it is waiting. In this part of the test, both instruction reads go to the L1.5 cache and only the actual memory access goes all the way out to memory. With this in mind, one operation to memory for the PicoRV32 core is about 100 cycles. Latency from L1.5 cache to L2 cache is the same for PicoRV32 and OpenSPARC cores and varies with core count and the L2 cache homing policy as described in [5].
Microbenchmarks
We ran three microbenchmarks to compare the performance of the PicoRV32 core integrated into the OpenPiton framework to that of the OpenSPARC T1 core. The bitfile was running at a frequency of 66.667 MHz and had one OpenSPARC T1 tile and one PicoRV32 tile. The first microbenchmark was a program that simulated solving Table 3: Memory latency measurements for PicoRV32 as measured using a sequence of 3 instructions. The measured latency is the raw cycle count from the test whereas the true latency is adjusted for cycles spent in the cache hierarchy Figure 3: Slowdown from running the microbenchmarks on the PicoRV32 core versus the OpenSPARC T1 core. The slowdown values are also given over each bar the Towers of Hanoi puzzle recursively (hanoi). The second was a binary search program (binsearch), and the third was a quicksort program (quicksort). The Towers of Hanoi is run with a height of 7. The benchmark recursively calls the same function to simulate moving the disks. The binary search benchmark searches for 10 32-bit integer keys randomly chosen in an array of 10,000 32-bit integers. The quicksort benchmark sorts an array of 100 32-bit integers shuffled randomly.
The slowdown of running each of these benchmarks is shown in Figure 3. As expected, all microbenchmarks experienced a slowdown when running on the PicoRV32 core since it is a more simplistic core. hanoi and quicksort both saw about an 8x slowdown. binsearch experienced a smaller slowdown at 4x.
binsearch's performance was affected less by running on the PicoRV32 core, because its working set does not fit in the L1.5 cache, which is 8KB. The working set does fit within the L2 cache although there is still the possibility of conflict misses. As a result of the working set size, both cores are forced to access the L2 cache or memory often. Since operations that must go to the L1.5 cache or beyond take approximately the same amount of time for the PicoRV32 core and the OpenSPARC T1 core, binsearch is less impacted by running on the PicoRV32 core.
Although microbenchmarks running on the PicoRV32 suffer reduced performance, the PicoRV32 is designed to minimize area and maximize frequency, essentially trading performance for area and timing. We also expect the PicoRV32 core would consume less energy.
In our evaluation, the OpenSPARC T1 core and the PicoRV32 core were running at the same clock frequency. To take advantage of the PicoRV32 core's higher maximum frequency, the PicoRV32 core could be put in a different clock domain from the rest of the design and run at a higher frequency to lessen the performance difference.
It is worth noting that the OpenSPARC T1 core was designed for throughput and not single-threaded performance. For example, a thread will be descheduled until a branch is resolved. The core originally had 4 threads to overlap useful work from other threads with long latency instructions.
These trade-offs between performance and other metrics is an intended consequence of having a heterogeneous system architecture. An intelligent scheduler would optimize for these trade-offs and make use of the most appropriate core for its performance and energy-consumption goals.
RELATED WORKS
Kumar et al. explored using multicore processors where cores had heterogeneous microarchitectures but a common ISA [11,12]. Using a variety of simulated cores, they found performance and energy efficiency benefits by scheduling applications on the cores that best match the applications' demands. This work motivated research into heterogeneous architectures, but only looked at cores using one ISA.
Venkat et al. used simulation to explore cores with heterogeneous microarchitecture and heterogeneous ISAs [23]. They used combinations of ARM Thumb, x86, and Alpha cores and found further performance and energy efficiency benefits over just heterogeneous microarchitecture. However, they built their system in simulation, which is of limited usefulness for prototyping hardware and software infrastructure.
The PULP Platform HERO project does provide a heterogeneous-ISA platform [18] . They use an ARM core and RISC-V cores. Although the RISC-V cores are implemented on FPGA and can be modified, the ARM core is a hard core. This limits its use for prototyping since the ARM core cannot be modified.
Mantovani et al. implemented an FPGA-based framework for prototyping and analyzing heterogeneous SoCs [15,16]. However, their focus is on accelerators rather than general-purpose cores.
DeVuyst et al. [9] built a compiler and infrastructure for runtime migration in heterogeneous-ISA systems. Using ARM and MIPS cores, they were able to migrate binaries during runtime between cores. Taking advantage of shared memory, they were able to achieve a total performance loss of under 5% even when migrating every few hundred milliseconds. A key to achieving this performance, however, was the availability of hardware shared memory, so they performed their experiments in simulation. Additionally, they did not use an OS in their evaluation.
The researchers behind Popcorn Linux have also explored building a compiler that allows for runtime migration as well as OS support for heterogeneous-ISA systems [7,14]. They used their multikernel model to investigate a potential OS design for a heterogeneous-ISA system by compiling a copy of their kernel for each ISA. For their evaluations, they used a hardware x86-ARM system where the cores were connected over PCI, but the system did not have hardware shared memory, which meant that migration of binaries during execution was expensive due to the overhead of copying state. JuxtaPiton could provide better insight into the cost of migration of binaries in Popcorn Linux since it has shared memory available.
Lin et al. built K2 OS [13], an OS which assumes multiple coherence domains where cores in different domains do not have coherent memory. In their hardware model, they assume that cores in different domains can be of different ISAs. Using modified Linux kernels, they run a main kernel in one domain and a shadow kernel in another and replicate state between them. Although K2 is able to run without shared memory, their model supports heterogeneous-ISA cores and could be used in a shared memory system as well.
Enabled research
Although the PicoRV32 core we incorporated had no caches to simplify interfacing with the L1.5 caches, the same method of adding transducers between a different core's L1 cache and OpenPiton's L1.5 cache could be used to add a more complex core. Our initial experience investigating the integration of more complex cores indicates that this should be relatively straightforward. Other RISC-V cores that could be integrated include "medium" cores such as Ariane [19] or Rocket [4] or "large" cores such as Anycore [1] or BOOM [8].
JuxtaPiton could also be paired with another open-source FPGA framework, like that developed by Mantovani et al. which focuses on accelerators, to create a platform with numerous heterogeneous elements. This would enable researchers to explore heterogeneous architectures with the ability to modify any component of the system and prototype their design on FPGA.
JuxtaPiton can also help enable systems research by providing shared memory on an FPGA. Shared memory is a familiar programming model and allows for efficient migration between cores as found by previous work. At the same time, emulating the design on FPGA enables research into complex, full-stack software that would not be practical in simulation. We expect these unique benefits that JuxtaPiton provides will enable future OS and runtime migration work.
CONCLUSION
With an increasing emphasis on energy efficiency in computer systems, it is becoming common to see architectures with heterogeneous processing elements, creating a need for better frameworks for use in research and prototyping. We built JuxtaPiton to enable heterogeneous-ISA research by integrating two open-source projects: OpenPiton and PicoRV32. JuxtaPiton is the first time a new core has been integrated into the OpenPiton framework, and we belive it is the first open-source, general-purpose, heterogeneous-ISA processor. We evaluated trade-offs of using the PicoRV32 core or the OpenSPARC T1 core. We found that although the PicoRV32 core experienced a slowdown in the microbenchmarks we ran, it used much less area than the OpenSPARC T1 core and could improve timing in the design. We believe that this FPGA implementation of a heterogeneous-ISA, shared memory, multiprocessor will enable future research. Architects will be able to modify it and prototype their designs on FPGA. OS researchers will be able to evaluate more complex software designs on realistic hardware prototypes while also taking advantage of shared memory.
Figure 1 :
1Architecture of an OpenSPARC T1 tile and a Pi-coRV32 tile connected to memory. (Derived from[5])
Figure 2 :
2Flowchart of the process used to load and start running a binary for the PicoRV32 core
arXiv:1811.08091v1 [cs.AR] 20 Nov 2018OpenSPARC T1
PicoRV32
ISA
SPARC v9
RISC-V I
Word size
64 bits
32 bits
Endianness
Big endian
Little endian
Implementation
6-stage in-order
Multicycle
pipeline
L1 Cache
Yes
No
MMU/TLB
Yes
No
FPU
Yes
No
Privileged Mode
Yes
No
Tile Type
TileCore LUTs Tile LUTs Core BRAMsOpenSPARC T1
36756
64695
24
PicoRV32
1076
21862
0
Table 2 :
2Resource utilization for OpenSPARC T1 and Pi-coRV32 cores and tiles on the Xilinx Kintex-7 FPGA rd the LUTs of an OpenSPARC T1 core. We can fit 2 OpenSPARC T1 cores on the Genesys2 or 7 PicoRV32 cores. In both cases, the limiting resource is the LUTs.PicoRV32 tile uses approximately 1
3
ACKNOWLEDGMENTSThis work was partially supported by the NSF under Grants No. CNS-1823222, CCF-1217553, CCF-1453112, and CCF-1438980,
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|
[
"https://github.com/anycore/anycore-riscv",
"https://github.com/pulp-platform/ariane",
"https://github.com/cliffordwolf/picorv32."
] |
[
"A Practical Approach to Sizing Neural Networks",
"A Practical Approach to Sizing Neural Networks"
] |
[
"Gerald Friedland [email protected] ",
"Alfredo Metere ",
"Mario Michael Krell [email protected] "
] |
[] |
[] |
Memorization is worst-case generalization. Based on MacKay's information theoretic model of supervised machine learning[23], this article discusses how to practically estimate the maximum size of a neural network given a training data set. First, we present four easily applicable rules to analytically determine the capacity of neural network architectures. This allows the comparison of the efficiency of different network architectures independently of a task. Second, we introduce and experimentally validate a heuristic method to estimate the neural network capacity requirement for a given dataset and labeling. This allows an estimate of the required size of a neural network for a given problem. We conclude the article with a discussion on the consequences of sizing the network wrongly, which includes both increased computation effort for training as well as reduced generalization capability.4. For perceptrons in series (e.g., in subsequent layers),
|
10.2172/1476219
|
[
"https://arxiv.org/pdf/1810.02328v1.pdf"
] | 52,919,352 |
1810.02328
|
69a22bfabb3b215805cf53ce194e7d2112e6db52
|
A Practical Approach to Sizing Neural Networks
October 4th, 2018
Gerald Friedland [email protected]
Alfredo Metere
Mario Michael Krell [email protected]
A Practical Approach to Sizing Neural Networks
October 4th, 2018
Memorization is worst-case generalization. Based on MacKay's information theoretic model of supervised machine learning[23], this article discusses how to practically estimate the maximum size of a neural network given a training data set. First, we present four easily applicable rules to analytically determine the capacity of neural network architectures. This allows the comparison of the efficiency of different network architectures independently of a task. Second, we introduce and experimentally validate a heuristic method to estimate the neural network capacity requirement for a given dataset and labeling. This allows an estimate of the required size of a neural network for a given problem. We conclude the article with a discussion on the consequences of sizing the network wrongly, which includes both increased computation effort for training as well as reduced generalization capability.4. For perceptrons in series (e.g., in subsequent layers),
Introduction
Most approaches to machine learning experiments currently involve tedious hyperparameter tuning. As the use of machine learning methods becomes increasingly important in industrial and engineering applications, there is a growing demand for engineering laws similar to the ones existing for electronic circuit design. Today, circuits can be drawn on a piece of paper and their behavior can be predicted exclusively based of engineering laws. Fully predicting the behavior of machine learning, as opposed * UC Berkeley and Lawrence Livermore National Lab † International Computer Science Institute, Berkeley. ‡ International Computer Science Institute, Berkeley.
to relying on trial and error, requires insights into the training and testing data, the available hypothesis space of a chosen algorithm, the convergence and other properties of the optimization algorithm, and the effect of generalization and loss terms in the optimization problem formulation. As a result, we may never reach circuit-level predictability. One of the core questions that machine learning theory focuses on is the complexity of the hypothesis space and what functions can be modeled, especially in connection with real-world data. Practically speaking, the memory and computation requirements for a given learning tasks are very hard to budget. This is especially a problem for very large scale experiments, such as on multimedia or molecular dynamics data. Even though artificial neural networks have been popular for decades, the understanding of the processes underlying them is usually based solely on anecdotal evidence in a particular application domain or task (see for example [25]). This article presents general methods to both measure and also analytically predict the experimental design for neural networks based on the underlying assumption that memorization is worst-case generalization.
We present 4 engineering rules to determine the maximum capacity of contemporary neural networks:
1. The output of a single perceptron yields maximally one bit of information.
2. The capacity of a single perceptron is the number of its parameters (weights and bias) in bits.
3. The total capacity C tot of M perceptrons in parallel is C tot = M i=1 C i where C i is the capacity of each neuron.
the capacity of a subsequent layer cannot be larger than the output of the previous layer.
After presenting related work in Section 2 and summarizing MacKay's proof in Section 3, we derive the above principles in Sections 4 and 5. In Section 6, we then present and evaluate a heuristic approach for fast estimation of the required neural network capacity, in bits, for a given training data set. The heuristic method assumes a network with static, identical weights. Even if such a network will still be able to approximately learn any labeling for a data set, it would require too many neurons. We then assume that training is able to cut down the number of parameters exponentially, when compared to the untrained network. Section 7 discusses the practical implications of memory capacity for generalization. Finally, in Section 8, we conclude the article with future work directions.
Related Work
The perceptron was introduced in 1958 [29] and since then, it has been extended in many variants, including, but not limited to, the structures described in [10,11,21,22]. The perceptron uses a k-dimensional input and generates the output by applying a linear function to the input, followed by a gating function. The gating function is typically the identity function, the sign function, a sigmoid function, or the rectified linear unit (ReLU) [18,26]. Motivated by brain research [12], perceptrons are stacked together to networks and they are usually trained by a chain rule known as backpropagation [30,31].
Even though perceptrons have been utilized for a long time, its capacities have been rarely explored beyond discussion of linear separability. Moreover, catastrophic forgetting has so far not been explained satisfactorily. Catastrophic forgetting [24,27] is a phenomenon consisting in the very quick loss of the network's capability to classify the first set of labels, when the net is first trained on one set of labels and then on another set of labels. Our interpretation of the cause of this phenomenon is that it is simply a capacity overflow.
One of the largest contributions to machine learning theory comes from Vapnik and Chervonenkis [37], including the Vapnik-Chervonenkis (VC) dimension. The VC dimension has been well known for decades [38] and is defined as the largest natural number of samples in a dataset that can be shattered by a hypothesis space. This means that for a hypothesis space having VC dimension D V C , there exists a dataset with D V C samples such that for any binary labeling (2 D V C possibilities) there exists a perfect classifier f in the hypothesis space, that is, f maps the samples perfectly to the labels. Due to perfect memorizing, it holds that D V C = ∞ for 1-nearest neighbor. Tight bounds have so far been computed for linear classifiers (k + 1) as well as decision trees [3]. The definition of VC dimension comes with two major drawbacks, however. First, it only considers the potential hypothesis space but not other aspects, like the optimization algorithm, or loss and regularization function affecting the choice of the hypothesis [2]. Second, it is sufficient to provide only one example of a dataset to match the VC dimension. Hence, given a more complex structure of the hypothesis space, the chosen data can take advantage of this structure. As a result, shatterability can be increased by increasing the structure of the data. While these aspects do not matter much for simple algorithms, they constitute a major point of concern for deep neural networks. In [39], Vapnik et al. suggest to determine the VC dimension empirically, but state in their conclusion that the described approach does not apply to neural networks as they are "beyond theory". So far, the VC dimension has only been approximated for neural networks. For example, Mostafa argued loosely that the capacity must be bounded by N 2 with N being the number of perceptrons [1]. Recently, [33] determined in their book that for a sigmoid activation function and a limited amount of bits for the weights, the loose upper bound of the VC dimension is O(|E|) where E is the set of edges and consequently |E| the number of non-zero weights. Extensions of the boundaries have been derived for example for recurrent neural networks [20] and networks with piecewise polynomials [4] and piecewise linear [17] gating functions. Another article [19] describes a quadratic VC dimension for a very special case. The authors use a regular grid of n times n points in the two dimensional space and tailor their multilayer percep-tron directly to this structure to use only 3n gates and 8n weights. One measure that handles the properties of given data is the Rademacher complexity [5]. For understanding the properties of large neural networks, Zhang et al. [41] recently performed randomization tests. They show that their observed networks can memorize the data as well as the noise. This is proven by evaluating that their neural networks perfectly learn with random labels or with random data. This shows that the VC dimension of the analyzed networks is above the size of the used dataset. But it is not clear what the full capacity of the networks is. This observation also explains why smaller size networks can outperform larger networks. A more elaborate extension of this evaluation has been provided by Arpit et al. [2].
Summarizing the contribution by [9], MacKay is the first to interpret a perceptron as an encoder in a Shannon communication model ([23], Chapter 40). MacKay's use of the Shannon model allows the measurement of the memory capacity of the perceptron in bits. Furthermore, it allows for the discussion of a perceptron's capabilities, without taking into account the number of bits used to store the weights (64 bit doubles, real-valued, etc.). He Characteristic curve examples of the T (n, k) function for different input dimensions k and the two crucial points at n = k for the VC dimension and n = 2k for the MacKay capacity. Right: Measured characteristic curve example for different number of hidden layers for a configuration of scikit-learn [13]. The tools to measure and compare the characteristic curves of concrete neural network implementations are available in our public repository (see Section 8).
also points out that there are two distinct transition points in the error measurement. The first one is discontinuous and happens at the VC dimension. For a single perceptron with offset, that point is D V C = k + 1, when k is the dimensionality of the data. Below this point the error should be 0, given perfect training, because the perceptron is able to generate all possible shatterings of the hypothesis space. For clarification, we summarize this proof in Section 3 and present initial work on an extension in [13].
Another important contribution using information theory comes from Tishby [35]. They use the information bottleneck principle to analyze deep learning. For each layer, the previous layers are treated as an encoder that compresses the data X to some better representation T which is then decoded to the labels Y by the consecutive layers. By calculating the respective mutual information I(X, T ) and I(T, Y ) for each layer they analyze networks and their behavior during training or when changing the amount of training data. Our Principle 4 is a direct consequence of his work.
This questions of generalization and network architecture have recently become a heated academic discussion again as deep learning surprisingly seems to outperform shallow learning. For deep learning, single perceptrons with a nonlinear and continuous gating function are con-catenated in a layered fashion. Techniques like convolutional filters, drop out, early stopping, regularization, etc., are used to tune performance, leading to a variety of claims about the capabilities and limits of each of these algorithms (for example [41]). We are aware of recent questioning of the approach of discussing the memory capacity of neural networks [2,41]. Occam's razor [7] dictates that one should follow the path of least assumptions, as perceptrons were initially conceived as a "generalizing memory", as detailed for example, in the early works of Widrow [40]. This approach has also been suggested by [1] and, as mentioned earlier, later explained in depth by MacKay [23]. Also, the Ising model of ferromagnetism, which is a well-known model used to explain memory storage, has already been reported to have similarities to perceptrons [15,16] and also to the neurons in the retina [36].
Capacities of a Perceptron
Here we summarize the proof elaborated in [9] and [23], Chapter 40.
The functionality of a perceptron is typically explained by the XOR example (i. e., showing that a perceptron with 2 input variables k, which can have 2 k = 4 states, can only model 14 of the 2 2 k = 16 possible output functions). XOR and its negation cannot be linearly separated by a single threshold function of two variables and a bias. For an example of this explanation, see [28], section 3.2.2. Instead of computing binary functions of k variables, MacKay effectively changes the computability question to a labeling question: given n points in general position, how many of the 2 n possible labelings in {0, 1} n can be trained into a perceptron. Just as done by [9,28], MacKay uses the relationship between the input dimensionality of the data k and the number of inputs n to the perceptron, which is denoted by a function T (n, k) that indicates the number of "distinct threshold functions" (separating hyperplanes) of n points in general position in k dimensions. The original function was derived by [32]. It can be calculated as:
T (n, k) = 2 k−1 l=0 n − 1 l(1)
Most importantly, it holds that
T (n, k) = 2 n ∀k : k ≥ n.(2)
This allows to derive the VC dimension D of a neuron with k parameters shattering a set of n points in general position. The number of possible binary labelings for n points is 2 n and T (n, n = k) = 2 n . This is the D = k, since all possible labelings of the k = n points can be realized. When k < n, the T (n, k) function follows a calculation scheme based on the Pascal Triangle [8], which means that the loss due to incomplete shattering is still predictable. MacKay uses an error function based on the cumulative distribution of the standard Gaussian to perform that prediction and approximate the resulting distribution. More importantly, he defines a second point, at which only 50 % of all possible labelings can be separated by the binary classifier. He proofs this point to be at n = 2k for large k and illustrates that there is a sharp continuous drop in performance at this point. MacKay then follows Cover's conclusion that the information theoretic capacity of a perceptron is 2k. We call this point MacKay dimension in [13].
When comparing and visualizing the T (n, k) function, it is only natural to normalize function values by the number of possible labelings 2 n and to normalize the argu-ment by the number of inputs k which is equal to the capacity of the perceptron. Figure 3 displays these normalized functions for different input dimensions k. The functions follow a clear pattern like the characteristic curves of circuit components in electrical engineering.
Information Theoretic Model
To the best of our knowledge, MacKay is the first person to interpret a perceptron as an encoder in a Shannon communication model ( [23], Chapter 40). In our article, we use a slightly modified version of the model depicted in Fig. 4.
As explained in Section 3, the input of the encoder are n points in general position and a random labeling. The output of the encoder are the weights of a perceptron. The decoder receives the (perfectly learned) weights over a lossless channel. The question is: given the received set of weights and the knowledge of the data, can the decoder reconstruct the original labels of the points? In other words, the perceptron is interpreted as memory that stores a labeling of n points relative to the data: how much information can then be stored by training a perceptron? We address this question by interpreting MacKay's definition of neuron capacity as a memory capacity. The use of Shannon's model has an advantage: The mathematical framework of information theory can be applied to machine learning. Moreover, it allows to predict and measure neuron capacity in the unit of information: bits.
We are interested in an upper bound. Therefore, we are only interested in the cases where we can guarantee lossless reproduction of the trained function; in other words, we are interested in the lossless memory capacity of neurons and networks of neurons. The definition of general position used in the previous section is typically used in linear algebra and is the most general case needed for a perceptron that uses a hyperplane for linear separation (see also Table 1 in [9]). For neural networks, a stricter setting is required because they can implement arbitrary non-linear separations. We must therefore assume that the data points are in completely random positions. This is, the coordinates of the data points are equiprobable.
Networks of Perceptrons
For the remainder of this article, we will assume that the network is a feedforward network consisting of traditional perceptrons (threshold units with activation function) with real-valued weights. Each unit has a bias, which counts as an additional real-valued weight [28,23]. We will additionally assume that the perceptrons are part of a neural network embedded in the model depicted in Figure 4, thus solving a binary labeling task. Because our discussion concerns the upper bounds, it is agnostic about training algorithms.
We define perceptrons to be in parallel when they are connected to the same input. A layer is a set of perceptrons in parallel. We define perceptrons to be in series when they are connected in such a way that as the ones exclusively relying on the outputs of other perceptrons.
We note that Figure 1 b) shows a perceptron that is connected in parallel. Principle 1. The output of a single perceptron yields maximally one bit of information A perceptron uses a decision function f ( w, x, b) of the form
f ( w, x, b) = 1 if w · x > b 0 otherwise(3)
where x = {x 1 , x 2 , . . . , x N } and w = {w 1 , w 2 , . . . , w N } are real vectors and b is a real scalar. Therefore, w · x represents a dot product:
w · x = N i=1 w i x i(4)
Because the inequality describes a binary condition (it is either greater or not), it follows that each perceptron ultimately behaves as a binary classifier, thus outputting a symbol o = f ( w, x, b) ∈ {0, 1}. If each state of o is equiprobable, the information content encoded in the output of the perceptron is log 2 (2) = 1 bit, else, if each state of o is not equiprobable, the information content it is less than 1 bit. It is worth remarking that an analytic approximation of the step function f ( w, x, b), for example a sigmoid, a rectified linear unit, or any other space dividing function, does not affect the aforementioned analysis. This is guaranteed by the data processing inequality [23] (p. 144).
Principle 2. The lossless storage capacity of a single perceptron is the number of parameters in bits.
This follows intuitively from Section 3, because n bits of labels can be stored with k = n parameters. This is, each parameter models one bit of labeling. However, confusion often arises over the fact that the weights are assumed real-valued. We therefore introduce the following lemma showing that a perceptron behaves analogous to a memory cell. This is, given fixed random input, it can model 2 k different output states, where k is the number of parameters stored by the perceptron.
Assume a perceptron as defined in Principle 1 in the model defined in Section 4. Let C(k) be the number of bits of labeling storable by k parameters. We now rewrite Eq. 4 as:
N i=1 s i |w i |x i(5)
where |w i | is the absolute value of w i and s i is the sign of each w i , this is s i ∈ {−1, 1}.
It is now clear that, given an input x i , the choice of s i in training is the only determining factor for the outcome of f ( w, x, b). The values of |w i | merely serve as scaling factors 1 .
Since s i ∈ {−1, 1} and |{−1, 1}| = 2, it follows that each s i can be encoded using log 2 (2) = 1 bit. This is, the maximum number of encodable outcome changes for f ( w, x, b) is N . This inevitably results in the memory capacity of a perceptron being C(N ) = N .
Case 2: b = 0 Using the same approach as above, we begin by separating the bias and its sign: b = s b |b|, where |b| is the absolute value of b and s b is the sign of b, this is s b ∈ {−1, 1}. We can now reformulate the equation as:
N i=1 s i |w i |x i > |b|s b (6) 1 s b N i=1 s i |w i |x i > |b|(7)
Since s b is not dependent on i, s b can only be trained to correct all decisions at once. |b| is strictly positive. This is, the inequality can be decided just by comparing the sign of s b and the sign of the sum. Again, s b ∈ {−1, 1} and thus s b encodes log 2 (2) = 1 bit. As a result, b contributes 1 bit of memory capacity. In total, a perceptron with nonzero bias can therefore maximally memorize N +1 bits of changes to the outcomes of f ( w, x, b). Since k = N + 1, it inevitably follows that C(k) = k.
Principle 3. The total capacity C tot of M perceptrons in parallel is:
C tot = M i=1 C i (8)
where C i is the capacity of each neuron.
1 Such scaling maybe important for generalization and training but is not relevant for computing the decision changing capabilities. Consistent with MacKay's interpretation, connecting, for example, two perceptrons in parallel is analogous to using two memory cells with capacity C 1 and C 2 . The storage capacity of such a circuit is maximally C tot = C 1 + C 2 bits.
For the following lemma we assume two perceptrons connected to the same input, each with a number of parameters k 1 and k 2 . Due to the associativity of addition, We can do this Without loss of generality.
Lemma 5.2 (Perceptrons in parallel). C(k 1 + k 2 ) = k 1 + k 2
Proof. We know from Lemma 5.1 that C(k 1 ) = k 1 and C(k 2 ) = k 2 . Since we assume all points of the data to be in equiprobable positions, each perceptron i can now maximally label k i points independently. This is, the two perceptrons can maximally label k 1 + k 2 points. This is,
C(k 1 + k 2 ) = k 1 + k 2 .
Principle 4. For perceptrons in series, the capacity of a subsequent layer cannot be larger than the largest possible amount of information output of the previous layer.
As explained in Section 2, Tishby [35] treats each layer in a deep perceptron network as an encoder for a subsequent layer. The work analyzes the mutual information between layers and points out that the data processing inequality holds between them, both theoretically and empirically. We are able to confirm this result and note that channel capacity C in general is defined as C = sup p X (x) I(X; Y ), where the supremum is taken over all possible choices of p X (x)( [34]). The data processing inequality ( [23], p. 144) states that if X → Y → Z is a Markov chain then I(x; y) I(x; z), where I(x; y) is the mutual information. In our model (See Section 4), the channel is the identity channel and the label distribution is assumed as equiprobable. These two assumptions make the channel capacity identical to the memory capacity and to the mutual information. As a result, the capacity of a subsequent layer is upper bounded by the output of the previous layer.
Without loss of generality, we assume two layers of perceptrons. The output of perceptron layer 1 is the sole input for perceptron layer 2. We denote the total capacity of layer 2 with C L2 , the number of parameters in layer 2 with k L2 and the number of bits in the output of layer 1 with o L1 .
Lemma 5.3 (Perceptrons in Series
). C L2 = min(C(k L2 ), o L1 ) Proof. Let create the Markov chain X → Y → Z,
where X is the random variable representing the input to layer 1, Y is the random variable representing the output of layer 1 and Z is representing the output of layer 2. It is clear that the sup p X (x) I(Y ; Z) is bounded by I(X; Y ), which we know to be o L1 . If o L1 > C(k L2 ), then C(K L2 ) limits the number of bits that can be stored in layer 2. If o L1 ≤ C(k L2 ), then the data processing inequality does not allow for the creation of information and sup p X (x) I(Y ; Z) ≤ o L1 . As a consequence,
C L2 = min(C(k L2 ), o L1 )
When generalizing to more than two layers, it is important to keep in mind that any capacity constraint from an earlier layer will upper bound all subsequent layers. This is, capacity can never increase in subsequent layers. Note that the input layer counts as a layer as well. Figure 2 shows a screen shot of our neural network capacity web demo (link see Section 8) with an example of Principle 4 in action. Figure 1 discusses various architecture capacities practically applying the computation principles presented here.
There is a notable illusion that sometimes makes it seem that Principle 4 does not hold. In training, weights are initialized, for example at random. This initial configuration can create the illusion that a layer has more states available than dictated by the principle. For example, a layer that has only 1 bit of capacity using Principle 4 can be in more than 2 states before the weights have been updated in training based on the information passed by a previous layer.
Measuring Capacity
It is possible to practically measure the capacity of concrete neural networks implementations with varying architectures and learning strategies. This is done by generating n random data data points in d dimensions and training the network to memorize all possible 2 n binary labeling vectors. Once a network is not able to learn all labelings anymore, we reached capacity. While this effectiveness measurement is exponential in run time, it only needs to be performed on a small representative subnet as capacity scales linearly.
We found that the effectiveness of neural network implementations actually varies dramatically (always below the theoretical upper limit). Therefore capacity measurement alone allows for a task-independent comparison of neural network variations. Our experiments show that linear scaling holds practically and our theoretical bounds are actionable upper bounds for engineering purposes. All the tested threshold-like activation functions, including sigmoid and ReLU exhibited the predicted behavior -just as explained in theory by the data processing inequality. Our experimental methodology serves as a benchmarking tool for the evaluation of neural network implementations. Using points in random position, one can test any learning algorithm and network architecture against the theoretical limit both for performance and efficiency (convergence rate). Figure 3 (right) shows an example measurement curve. These results as well as all tools are available in our public repository (See Section 8).
Capacity Requirement Estimate
Algorithm 1 Calculating the maximum and approximated expected capacity requirement of a binary classifier neural network for given training data. Require: data: array of length i contains d-dimensional vectors x, labels: a column of 0 or 1 with length i procedure M axCapReq((data, labels)) thresholds ← 0 for all i do [1] thresholds ← thresholds + 1 end if end for maxcapreq ← thresholds * d + thresholds + 1 expcapreq ← log 2 (thresholds + 1) * d print "Max: "+maxcapreq+" bits" print "Exp: "+expcapreq+" bits" end procedure
The upper-bound estimation of the capacity allows the comparison of the efficiency of different architectures independent of a task. However, sizing a network properly to a task requires an estimate of the required capacity. We propose a heuristic method to estimate the neural network capacity requirement for a given dataset and labeling.
The exact memorization capacity requirement based on our model in Figure 4 is the minimum description length of the data/labels table that needs to be memorized. In practice, this value is almost never given. Furthermore, in a neural network, the table is recoded using weighted dotproduct threshold functions, which, as discussed in Section 3, has intrinsic compression capabilities. This is, often the labels of n points can be stored with less than n parameters. As we have done throughout the article, we will ignore the compression capabilities of neurons and work with the worst case.
Upper Limit Network Size
This section presents our proposed heuristic for a worst case sized network. Our idea for the heuristic method stems from the definition of the perceptron threshold function (see Principle 1). We observe that the dot product has d + 1 variables that need to be tuned, with d being the dimensionality of the input vector x. This makes perceptron learning and backpropagation NP-complete [6]. However, for an upper limit estimation, we chose to ignore the training of the weights w i by fixing them to 1: we only train the biases. This is done by calculating the dot products with w i := 1, essentially summing up the data rows of the table. The result is a two-column table with these sums and the classes. We now sort this twocolumn table by the sums before we iterate through it and record the need of a threshold every time a class change occurs. Note that we can safely ignore column sums with the same value (collisions): If they don't belong to the same class, they count as a threshold. If an actual network was built, training of the weights would potentially resolve this collision. As a last step, we take the number of thresholds needed and estimate the capacity requirement for a network by assuming that each threshold is implemented by a neuron in the hidden layer firing 0 or a 1. The number of inputs for these neurons is given by the dimensionality of the data. We then need to connect a neuron in the output layer that has the number of hidden layer neurons as input. The threshold of that output neuron is 0 and the input weights are +1 for class 1 and −1 for class 0. The reader is encouraged to check that such a network is able to label any table (ignoring collisions). Our algorithm is bounded by the runtime of the sorting, which is O(n log(n)) in the best case. Since we are able to effectively create a network that memorizes the labeling given the data without tuning the weights, we consider this the upper limit network. Any network that uses more parameters would therefore be wasting resources. Figure 1 shows pseudo code for this algorithm and the expected capacity presented in the next section.
Approximately Expected Capacity
We estimate the expected capacity by assuming that training the weights and biases is maximally effective. This is, it can cut down the number of threshold comparisons exponentially to log 2 (n) where n is the number of thresholds. The rationale for this choice is that a neural network effectively takes an input as a binary number and matches it against stored numbers in the network to determine the closest match. The output layer then determines the class for that match. That matching is effectively a search algorithm which in the best case can be implemented in logarithmic time. We call this the approximately expected capacity requirement as we need to take into account that real data is never random. Therefore, the network might be able to compress by a factor of 2 or even a much higher margin. Table 1 shows experimental results for various data sets. We show the maximum and the approximately expected capacity as generated by the heuristic method. We then show the achieved accuracy using an actual validation experiment using a neural network of the indicated capacity. The AND classifier requires one perceptron without bias. We implemented XOR using a shortcut network (see also [28]). The Gaussians and the circle, checker, and spiral patterns are available as part of the Tensorflow Playground. For the ImageNet experiment, we took 2000 random images from 2 classes ("hummingbird" and "snow leopard") and in lieu of a convolution layer, we compressed all images aggressively with JPEG quality 20 [14]. The image channels were combined from RGB into only the Y component (grayscale). We then trained . We note that image experiments like these are often anecdotal as many factors play into the actual achieved accuracy, including initial conditions of the initialization, learning rate, regularization, and others. We therefore made the scripts and data available for repetition in our public repository (Link see Section 8). The results show that our approximation of the expected capacity is very close to the actual capacity.
Experimental Results
From Memorization to Generalization
Training the network with random points makes the upper bound neural network size analytically accessible because no inference (generalization) is possible and the best possible thing any machine learner can do is to memorize. This methodology, which is not restricted to neural networks, therefore operates at the lower limit of generalization.
In reality, especially with a large set of samples, one is very unlikely to encounter data with equiprobable distribution. A network trained based on the principles presented here is therefore overfitting. A first consequence is that using more capacity than required for memorization wastes memory and computation resources. Secondly, it will complicate any attempt at explaining the inferences made by the network.
To avoid overfitting and to have a better chance of ex-plaining the data in a human comprehensible way, it is therefore advisable to reduce the number of parameters. This is, again, consistent with Occam's razor. For a given task, we therefore recommended to size the neural network for memorization at first and then successively retrain the network while reducing the number of parameters. It is expected that accuracy on the training data reduces with the network capacity reduction. Generalization capability, which should be quantified by measuring accuracy against a validation set, should increase, however.
In the best case, the network loses the ability to memorize the lowest significant digits of the training data. The lowest significant digits are likely insignificant with regard to the target function. This is, they are noise. Cutting the lowest-significant digits first, we expect the decay of training accuracy to follow a logarithmic curve (this was also observed in [14]). Ultimately, the network with the smallest capacity that is still able to represent the data is the one that maximizes generalization and the chances at explainability. The best possible scenario is a single neuron that can represent an infinite amount of points (above and below the threshold).
Conclusion and Future Work
We present an alternative understanding of neural networks using information theory. The main trick, that is not specific to neural networks, is to train the network with random points. This way, no inference (generalization) is possible and the best thing any machine learner can do is to memorize. We then present engineering principles to quantify the capabilities of a neural network given it's size and architecture as memory capacity. This allows the comparison of the efficiency of different architectures independently of a task. Second, we introduce and experi-mentally validate a heuristic method to estimate the neural network capacity requirement for a given dataset and labeling. We then relate this result to generalization and outline a process for reducing parameters. The overall result is a method to better predict and measure the capabilities of neural networks. Future work in continuation of this research will explore non-binary classification, recursive architectures, and self-looping layers. Moreover, further research into investigating convolutional networks, fuzzy networks and RBF kernel networks would help put these types of architectures into a comparative perspective. We will also revisit neural network training given the knowledge that we have gained doing this research. During the backpropagation step, the data processing inequality is reversed. This is, only one bit of information is actually transmitted backwards through the layers. All results and the tools for measuring capacity and estimating the required capacity are available in our public repository: https:// github.com/fractor/nntailoring. An interactive demo showing how capacity can be used is available at: http://tfmeter.icsi.berkeley.edu.
Figure 1 :
1The perceptron a) has 3 bits of capacity and can therefore memorize the 14 Boolean functions of two variables that can have a truth table of 8 or less states (removing redundant states). The shortcut network b) has 3 + 4 = 7 bits of capacity and can therefore implement all 16 Boolean functions. The 3-layer network in c) has 6 + min(3, 2) = 8 bits of capacity. Last but not least the deep network d) has 6 + min(6, 2) + min(3, 2) = 10 bits of capacity.
Figure 2 :
2Our web demo based on the Tensorflow Playground (link see Section 8) showing the Principle 4 in action. The third hidden layer is dependent on the second hidden layer. Therefore it only holds 1 bit of information (smoothed by the activation function) despite consisting of 6 neurons with a stand-alone capacity of 18 bits.
Figure 3 :
3Left:
Figure 4 :
4Shannon's communication model applied to labeling in machine learning. A dataset consisting of n sample points and the ground truth labeling of n bits are sent to the neural network. The learning method converts it into a parameterization (i. e., network weights). In the decoding step, the network then uses the weights together with the dataset to try to reproduce the original labeling.
Lemma 5 . 1 (
51Lossless Storage Capacity of a Perceptron). C(k) = k Proof. Let us consider a case distinction over b. Case 1: b = 0
DatasetMax Capacity Requirement Expected Capacity Requirement Validation (% accuracy)AND, 2 variables
4 bits
2 bits
2 bits (100%)
XOR, 2 variables
8 bits
4 bits
7 bits (100%)
Separated Gaussians (100 samples) 4 bits
2 bits
3 bits (100%)
2 Circles (100 samples)
224 bits
12 bits
12 bits (100%)
Checker pattern (100 samples)
144 bits
12 bits
12 bits (100%)
Spiral pattern (100 samples)
324 bits
14 bits
24 bits (98%)
ImageNet: 2000 images in 2 classes 906984 bits
10240 bits
10253 bits (98.2 %)
Table 1 :
1Experimental validation of the heuristic capacity estimation method using the structures available both in our public repository and in the online demo. a 3-layer neural network and increased the capacity successively. The best result was achieved at the capacity shown in the table; fewer parameters made the memorization result worse -all other parameters being the same (e.g. 94.6% accuracy at 5 kbit capacity, 97.3% accuracy at 9 kbit capacity and 97.9% accuracy at 11 kbit capacity)
AcknowledgementsThis work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. It was also partially supported by a Lawrence Livermore Laboratory Directed Research & Development grants (17-ERD-096 and 18-ERD-021). IM release number LLNL-TR-758456. Mario Michael Krell was supported by the Federal Ministry of Education and Research (BMBF, grant no. 01IM14006A) and by a fellowship within the FITweltweit program of the German Academic Exchange Service (DAAD). This research was partially supported by the U.S. National Science Foundation (NSF) grant CNS 1514509. 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 any sponsoring institution, the U.S. government or any other entity. We want to cordially thank Raúl Rojas for in depth discussion on the chaining of the T () function. We also want to thank Jerome Feldman for discussions on the cognitive backgrounds, especially the concept of actionability. Kannan Ramchandran's intuition of signal processing and information theory was invaluable. We'd like thank Sascha Hornauer for his advise on the imagenet experiments as well as Alexander Fabisch, Jan Hendrik Metzen, Bhiksha Raj, Naftali Tishby, Jaeyoung Choi, Friedrich Sommer, Alyosha Efros, Andrew Feit, and Barry Chen for their insightful advise. Special thanks go to Viviana Reverón for proofreading.
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|
[] |
[
"Post-Newtonian Celestial Dynamics in Cosmology: Field Equations",
"Post-Newtonian Celestial Dynamics in Cosmology: Field Equations"
] |
[
"Sergei M Kopeikin [email protected] ",
"Alexander N Petrov ",
"\nDepartment of Physics & Astronomy\nUniversity of Missouri\n322 Physics Bldg65211ColumbiaMOUSA\n",
"\nSternberg Astronomical Institute\nLomonosov State University\nUniversitetskii Prospect 13119992MoscowMoscow M. VRussia\n",
"\nI. NOTATIONS\n\n"
] |
[
"Department of Physics & Astronomy\nUniversity of Missouri\n322 Physics Bldg65211ColumbiaMOUSA",
"Sternberg Astronomical Institute\nLomonosov State University\nUniversitetskii Prospect 13119992MoscowMoscow M. VRussia",
"I. NOTATIONS\n"
] |
[] |
Post-Newtonian celestial dynamics is a relativistic theory of motion of massive bodies and test particles under the influence of relatively weak gravitational forces. Standard approach for development of this theory relies upon the key concept of the isolated astronomical system supplemented by the assumption that the background space-time is flat. The standard post-Newtonian theory of motion was instrumental in explanation of the existing experimental data on binary pulsars, satellite and lunar laser ranging, and in building precise ephemerides of planets in the solar system. Recent studies of the formation of large-scale structure in our universe indicate that the standard post-Newtonian mechanics fails to describe more subtle dynamical effects in motion of the bodies comprising the astronomical systems of larger size -galaxies and clusters of galaxies -where the Riemann curvature of the expanding FLRW universe interacts with the local gravitational field of the astronomical system and, as such, can not be ignored.The present paper outlines theoretical principles of the post-Newtonian mechanics in the expanding universe. It is based upon the gauge-invariant theory of the Lagrangian perturbations of cosmological manifold caused by an isolated astronomical N-body system (the solar system, a binary star, a galaxy, a cluster of galaxies). We postulate that the geometric properties of the background manifold are described by a homogeneous and isotropic Friedman-Lemaître-Robertson-Walker (FLRW) metric governed by two primary components -the dark matter and the dark energy. The dark matter is treated as an ideal fluid with the Lagrangian taken in the form of pressure along with the scalar Clebsch potential as a dynamic variable. The dark energy is associated with a single scalar field with a potential which is hold unspecified as long as the theory permits. Both the Lagrangians of the dark matter and the scalar field are formulated in terms of the field variables which play a role of generalized coordinates in the Lagrangian formalism. It allows us to implement the powerful methods of variational calculus to derive the gauge-invariant field equations of the post-Newtonian celestial mechanics of an isolated astronomical system in an expanding universe. These equations generalize the field equations of the post-Newtonian theory in asymptotically-flat spacetime by taking into account the cosmological effects explicitly and in a self-consistent manner without assuming the principle of liner superposition of the fields or a vacuole model of the isolated system, etc. The field equations for matter dynamic variables and gravitational field perturbations are coupled in the most general case of arbitrary equation of state of matter of the background universe.We introduce a new cosmological gauge which generalizes the de Donder (harmonic) gauge of the post-Newtonian theory in asymptotically flat spacetime. This gauge significantly simplifies the gravitational field equations and allows to find out the approximations where the field equations can be fully decoupled and solved analytically. The residual gauge freedom is explored and the residula gauge transformations are formulated in the form of the wave equations for the gauge functions. We demonstrate how the cosmological effects interfere with the local system and affect the local distribution of matter of the isolated system and its orbital dynamics. Finally, we worked out the precise mathematical definition of the Newtonian limit for an isolated system residing on the cosmological manifold. The results of the present paper can be useful in the solar system for calculating more precise ephemerides of the solar system bodies on extremely long time intervals, in galactic astronomy to study the dynamics of clusters of galaxies, and in gravitational wave astronomy for discussing the impact of cosmology on generation and propagation of gravitational waves emitted by coalescing binaries and/or merging galactic nuclei.
|
10.1103/physrevd.87.044029
|
[
"https://arxiv.org/pdf/1301.5706v1.pdf"
] | 51,836,197 |
1301.5706
|
c45bbf3d1d187d8bcddc57d04af6d1b79eb4678c
|
Post-Newtonian Celestial Dynamics in Cosmology: Field Equations
24 Jan 2013 (Dated: December 12, 2013)
Sergei M Kopeikin [email protected]
Alexander N Petrov
Department of Physics & Astronomy
University of Missouri
322 Physics Bldg65211ColumbiaMOUSA
Sternberg Astronomical Institute
Lomonosov State University
Universitetskii Prospect 13119992MoscowMoscow M. VRussia
I. NOTATIONS
Post-Newtonian Celestial Dynamics in Cosmology: Field Equations
24 Jan 2013 (Dated: December 12, 2013)PACS numbers: 04.20.-q, 04.80.Cc * Electronic address:
Post-Newtonian celestial dynamics is a relativistic theory of motion of massive bodies and test particles under the influence of relatively weak gravitational forces. Standard approach for development of this theory relies upon the key concept of the isolated astronomical system supplemented by the assumption that the background space-time is flat. The standard post-Newtonian theory of motion was instrumental in explanation of the existing experimental data on binary pulsars, satellite and lunar laser ranging, and in building precise ephemerides of planets in the solar system. Recent studies of the formation of large-scale structure in our universe indicate that the standard post-Newtonian mechanics fails to describe more subtle dynamical effects in motion of the bodies comprising the astronomical systems of larger size -galaxies and clusters of galaxies -where the Riemann curvature of the expanding FLRW universe interacts with the local gravitational field of the astronomical system and, as such, can not be ignored.The present paper outlines theoretical principles of the post-Newtonian mechanics in the expanding universe. It is based upon the gauge-invariant theory of the Lagrangian perturbations of cosmological manifold caused by an isolated astronomical N-body system (the solar system, a binary star, a galaxy, a cluster of galaxies). We postulate that the geometric properties of the background manifold are described by a homogeneous and isotropic Friedman-Lemaître-Robertson-Walker (FLRW) metric governed by two primary components -the dark matter and the dark energy. The dark matter is treated as an ideal fluid with the Lagrangian taken in the form of pressure along with the scalar Clebsch potential as a dynamic variable. The dark energy is associated with a single scalar field with a potential which is hold unspecified as long as the theory permits. Both the Lagrangians of the dark matter and the scalar field are formulated in terms of the field variables which play a role of generalized coordinates in the Lagrangian formalism. It allows us to implement the powerful methods of variational calculus to derive the gauge-invariant field equations of the post-Newtonian celestial mechanics of an isolated astronomical system in an expanding universe. These equations generalize the field equations of the post-Newtonian theory in asymptotically-flat spacetime by taking into account the cosmological effects explicitly and in a self-consistent manner without assuming the principle of liner superposition of the fields or a vacuole model of the isolated system, etc. The field equations for matter dynamic variables and gravitational field perturbations are coupled in the most general case of arbitrary equation of state of matter of the background universe.We introduce a new cosmological gauge which generalizes the de Donder (harmonic) gauge of the post-Newtonian theory in asymptotically flat spacetime. This gauge significantly simplifies the gravitational field equations and allows to find out the approximations where the field equations can be fully decoupled and solved analytically. The residual gauge freedom is explored and the residula gauge transformations are formulated in the form of the wave equations for the gauge functions. We demonstrate how the cosmological effects interfere with the local system and affect the local distribution of matter of the isolated system and its orbital dynamics. Finally, we worked out the precise mathematical definition of the Newtonian limit for an isolated system residing on the cosmological manifold. The results of the present paper can be useful in the solar system for calculating more precise ephemerides of the solar system bodies on extremely long time intervals, in galactic astronomy to study the dynamics of clusters of galaxies, and in gravitational wave astronomy for discussing the impact of cosmology on generation and propagation of gravitational waves emitted by coalescing binaries and/or merging galactic nuclei.
I. NOTATIONS
This section summarizes notations used in the present paper. We use G to denote the universal gravitational constant and c for the ultimate speed in Minkowski spacetime. Every time, when there is no confusion about the system of units, we shall choose a geometrized system of units such that G = c = 1. We put a bar over any function that belongs to the background manifold of the FLRW cosmological model. Any function without such a bar belongs to the perturbed manifold.
The notations used in the present paper are as follows:
• Greek indices α, β, γ, . . . run through values 0, 1, 2, 3, and Roman indices i, j, k, . . . take values 1, 2, 3,
• Einstein summation rule is applied for repeated (dummy) indices, for example, P α Q α ≡ P 0 Q 0 + P 1 Q 1 + P 2 Q 2 + P 3 Q 3 , and P i Q i ≡ P 1 Q 1 + P 2 Q 2 + P 3 Q 3 ,
• g αβ is a full metric on the cosmological spacetime manifold,
•ḡ αβ is the FLRW metric on the background spacetime manifold,
• f αβ is the metric on the conformal spacetime manifold,
• η αβ = diag{−1, +1, +1, +1} is the Minkowski metric,
• T and X i = {X, Y, Z} are the coordinate time and isotropic spatial coordinates on the background manifold,
• X α = {X 0 , X i } = {cη, X i } are the conformal coordinates with η being a conformal time,
• x α = {x 0 , x i } = {ct,
x i } is an arbitrary coordinate chart on the background manifold,
• a bar,F above a geometric object F , denotes the unperturbed value of F on the background manifold,
• a prime F ′ = dF/dη denotes a total derivative with respect to the conformal time η,
• a dotḞ = dF/dη denotes a total derivative with respect to the cosmic time T ,
• ∂ α = ∂/∂x α is a partial derivative with respect to the coordinate x α ,
• a comma with a following index F ,α = ∂ α F is another designation of a partial derivative with respect to a coordinate x α ,
• a vertical bar, F |α denotes a covariant derivative of a geometric object F (a scalar, a vector, a tensor) with respect to the background metricḡ αβ ,
• a semicolon, F ;α denotes a covariant derivative of a geometric object F (a scalar, a vector, a tensor) with respect to the conformal metric f αβ ,
• the tensor indices of geometric objects on the background manifold are raised and lowered with the background metricḡ αβ ,
• the tensor indices of geometric objects on the conformal spacetime are raised and lowered with the conformal metric f αβ ,
• the scale factor of the FLRW metric is denoted as R = R(T ), or as a = a(η) = R[T (η)],
• the Hubble parameter, H =Ṙ/R, and the conformal Hubble parameter, H = a ′ /a.
Other notations will be introduced and explained in the main text of the paper.
II. INTRODUCTION
Post-Newtonian celestial mechanics is a branch of fundamental gravitational physics [12,58,92] that deals with the theoretical concepts and experimental methods of measuring gravitational fields and testing general theory of relativity both in the solar system and beyond [97,104]. In particular, the relativistic celestial mechanics of binary pulsars (see [72], and references therein) was instrumental in providing conclusive evidence for the existence of gravitational radiation as predicted by Einstein's theory [93,101].
Over the last few decades, various groups within the International Astronomical Union (IAU) have been active in exploring the application of the general theory of relativity to the modeling and interpretation of high-accuracy astrometric observations in the solar system and beyond. A Working Group on Relativity in Celestial Mechanics and Astrometry was formed in 1994 to define and implement a relativistic theory of reference frames and time scales. This task was successfully completed with the adoption of a series of resolutions on astronomical reference systems, time scales, and Earth rotation models by 24th General Assembly of the IAU, held in Manchester, UK, in 2000. The IAU resolutions are based on the first post-Newtonian approximation of general relativity which is a conceptual basis of the fundamental astronomy in the solar system [91].
The mathematical formalism of the post-Newtonian approximations is getting progressively complicated as one goes from the Newtonian to higher orders [23,88]. For this reason the theory has been primarily developed under a basic assumption that the background spacetime is asymptotically flat. Mathematically, it means that the full spacetime metric, g αβ , is decomposed around the background Minkowskian metric, η αβ = diag(−1, 1, 1, 1), into a linear combination
g αβ = η αβ + h αβ ,(1)
where the perturbation,
h αβ = c −2 h [2]
αβ + c −3 h [3] αβ + c −4 h [4] αβ + . . . ,
is the post-Newtonian series with respect to the powers of 1/c, where c is the ultimate speed in general relativity (equal to the speed of light). Post-Newtonian approximations is the method to determine h αβ by solving Einstein's field equations with the tensor of energy-momentum of matter of a localized astronomical system, T αβ (Φ, h αβ ), taken as a source of the gravitational field h αβ , by iterations starting from h αβ = 0 in the expression for T αβ . The solution of the field equations and the equations of motion of the astronomical bodies are derived in some coordinates r α = {ct, r} where t is the coordinate time, and r = {x, y, z} are spatial coordinates. The post-Newtonian theory in asymptotically-flat spacetime has a well-defined Newtonian limit determined by:
1) equation for the Newtonian potential, U = h [2] 00 /2,
U (t, r) = V ρ(t, r ′ )d 3 r ′ |r − r ′ | ,(3)
where ρ = c −2 T 00 , is the density of matter producing the gravitational field, 2) equation of motion for massive particlesr
= ∇U ,(4)
where ∇ = {∂ x , ∂ y , ∂ z } is the operator of gradient, r = r(t) is time-dependent position of a particle (worldline of the particle), and the dot denotes a total derivative with respect to time t,
3) equations of motion for light (massless particles)r = 0 .
These equations are considered as fundamentals for creation of astronomical ephemerides of celestial bodies in the solar system [58] and in any other localized system of self-gravitating bodies like a binary pulsar [72]. In all practical cases they have to be extended to take into account the post-Newtonian corrections sometimes up to the 3-d post-Newtonian order of magnitude [105]. It is important to notice that in the Newtonian limit the coordinate time t of the gravitational equations of motion (4), (5) coincides with the proper time of observer τ that is practically measured with an atomic clock. The formalism of the present paper has been employed in [60] to check the theoretical consistency of (3)- (5) and to analyse the outcome of some experiments like the anomalous Doppler effect discovered by J. Anderson et al [3,4] in the orbital motion of Pioneer 10 and 11 space probes. So far, the post-Newtonian theory was mathematically successful ("unreasonably effective" as Clifford Will states [105]) and passed through numerous experimental tests with a flying color. Nevertheless, it hides several pitfalls. The first one is the problem of convergence of the post-Newtonian series and regularization of divergent integrals that appear in the post-Newtonian calculations at higher post-Newtonian orders [88]. The second problem is that the background manifold is not asymptotically-flat Minkowskian spacetime but the FLRW metric,ḡ αβ . We live in the expanding universe which rate of expansion is determined by the Hubble constant H 0 . Therefore, the right thing would be to replace the post-Newtonian decomposition (1) with a more adequate post-Friedmannian series [96] g αβ =ḡ αβ + κ αβ ,
where
κ αβ = κ {0} αβ + Hκ {1} αβ + H 2 κ {2} αβ + . . . ,(7)
is the metric perturbation around the cosmological background represented as a series with respect to the Hubble parameter, H. Each term of the series has its own expansion into post-Newtonian series like (2). Generalization of the theory of post-Newtonian approximations from the Minkowski spacetime to that of the expanding universe is important for extending the applicability of the post-Newtonian celestial mechanics to testing cosmological effects and for more deep understanding of the process of formation of the large-scale structure in the universe and gravitational interaction between galaxies and clusters of galaxies. Whether cosmological expansion affects gravitational dynamics of bodies inside a localized astronomical system was a matter of considerable efforts of many researchers [11,30,31,62,77,78,89]. A recent article [16] summarizes the previous results and provides the reader with a number of other valuable resources. Most of the previous works on celestial mechanics in cosmology were based on assumption of spherical symmetry of gravitational field and matching two (for example, Schwarzschild and Friedmann) exact solutions of Einstein's equations. The matching was achieved in many different ways. McVittie's solution [78] is perhaps the most successful mathematically but yet lacks a clear physical interpretation [16]. Moreover, its practical application is doubtful since it is valid only for sphericallysymmetric case.
We need a precise mathematical formulation of the post-Newtonian theory for a self-gravitating localized astronomical system not limited by the assumption of the spherical symmetry, embedded to the expanding universe and coupled through the gravitational interaction with the time-dependent background geometry. Theoretical description of the post-Newtonian theory for a localized astronomical system in expanding universe should correspond in the limit of vanishing H to the post-Newtonian theory obtained in the asymptotically-flat spacetime. Such a description will allow us to directly compare the equations of the standard post-Newtonian celestial mechanics with its cosmological counterpart. Therefore, the task is to derive a set of the post-Newtonian equations in cosmology in some coordinates introduced on the background manifold, and to map them onto the set of the Newtonian equations (3)- (5) in asymptotically-flat spacetime. Such a theory of the post-Newtonian celestial mechanics would be of a paramount importance for extending the tools of experimental gravitational physics to the field of cosmology, for example, to properly formulate the cosmological extension of the PPN formalism [103]. The present article discusses the main ideas and principal results of such a theoretical approach in the linearized approximation with respect to the gravitational perturbations of the cosmological background caused by the presence of a localized astronomical system. The paper is organized as follows. In the following section we describe a brief history of the development of the theory of cosmological perturbations. Section IV describes the Lagrangian of gravitational field, the matter of the background cosmological model, and an isolated astronomical system which perturbs the background cosmological manifold. Section V describes the geometric structure of the background spacetime manifold of the cosmological model and the corresponding equations of motion of the matter and field variables. Section VI introduces the reader to the theory of the Lagrangian perturbations of the cosmological manifold and the dynamic variables. Section VII makes use of the preceding sections in order to derive the field equations in the gauge-invariant form. Beginning from section VIII we focus on the spatially-flat universe in order to derive the post-Newtonian field equations that generalize the post-Newtonian equations in the asymptotically-flat spacetime. These equations are coupled in the scalar sector of the proposed theory. Therefore, we consider in section IX a few particular cases when the equations can be fully decoupled one from another, and solved in terms of the retarded potentials. Appendix provides a proof of the Lorentz-invariance of the retarded potentials for the wave equations describing propagation of weak gravitational and sound waves on the background cosmological manifold.
III. BRIEF HISTORY OF COSMOLOGICAL PERTURBATION THEORY
In order to solve the problem of the interaction of the gravitational field of an isolated astronomical system with expanding universe one has to resort to the theory of cosmological perturbations. The immediate goal of cosmological perturbation theory is to relate the physics of the early universe to CMB anisotropy and large-scale structure and to provide the initial conditions for numerical simulations of structure formations. The ultimate goal of this theory is to establish a mathematical link between the fundamental physical laws at the Planck epoch and the output of the gravitational wave detectors which are the only experimental devices being able to map parameters of the universe at that time [56].
Originally, two basic approximation schemes for calculation of cosmological perturbations have been proposed by Lifshitz with his collaborators [68,69] and, later on, by Bardeen [7]. Lifshitz [68] worked out a coordinate-dependent theory of cosmological perturbations while Bardeen [7] concentrated on finding the gauge-invariant combinations for perturbed quantities and derivation of a perturbation technique based on gauge-invariant field equations. At the same time, Lukash [73] had suggested an original approach for deriving the gauge-invariant scalar equations based on the thermodynamic theory of the Clebsch potential also known in cosmology as the scalar velocity potential [64,90] or the Taub potential [90,95]. It turns out that the variational principle with a Lagrangian of cosmological matter formulated in terms of the Clebsch potential, is the most useful mathematical device for developing the theory of relativistic celestial mechanics of localized astronomical systems embedded in expanding cosmological manifold. It is for this reason, we use the Clebsch potential in the present paper.
A few words of clarification regarding a comparison between Lifshitz's and Bardeen's approaches should be relayed to the reader. The approach established in the papers by Lifshitz [68,69], is fully correct. Lifshitz decided to work in synchronous gauge and realized that this fixing of coordinates allows for a residual gauge ambiguity, which can also be fixed by picking a synchronous, comoving coordinate system. Bardeen [7] wrote his paper, because there was some confusion in the 1970s about that issue (which could have beed avoided if people would have studied Lifshitz's papers carefully). He demonstrated, that any coordinate could be chosen and that there exist quantities which are independent of that choice, which he identified with the physical degrees of freedom. However, this is not where the story of the cosmological perturbation theory ends. Closer inspection shows that what is really relevant is not the choice of coordinates (which do not have a physical meaning), rather the choice of spacetime foliation is relevant, for example, it makes a physical difference if one defines the Harrison-Zel'dovich spectrum [46,107] of the primordial perturbations on a synchronous, comoving hypersurface or a shear free hypersurface. Pitfalls in understanding this issue are subtle and, sometimes, may be not easily recognized (see [25,42,76] and [40] for a detailed discussion of the role of foliations in cosmology and in general relativity).
In the years that followed, the gauge-invariant formalism was refined and improved by Durrer and Straumann [27,28], Ellis et al. [32][33][34] and, especially, by Mukhanov et al. [81,82]. Irrespectively of the approach a specific gauge must be fixed in order to solve equations for cosmological perturbations. Any gauge is allowed and its particular choice is simply a matter of convenience. Imposing a gauge condition eliminates four gauge degrees of freedom in the cosmological pertrubations and brings the differential equations for them to a solvable form. Nonetheless, the residual gauge freedom originated from the tensor nature of the gravitational field remains. This residual gauge freedom leads to appearance of unphysical perturbations which must be disentangled from the physical modes. Lifshitz theory of cosmological perturbations [68,69] is worked out in a synchronous gauge and contains the spurious modes but they are easily isolated from the physical perturbations [41]. Other gauges used in cosmology are described in Bardeen's paper [7] and used in cosmological perturbation theory. Among them, the longitudinal (conformal or Newtonian) gauge is one of the most common. This gauge is advocated by Mukhanov [81] because it removes spurious coordinate degrees of freedom out of scalar perturbations. Detailed comparison of the cosmological perturbation theory in the synchronous and conformal gauges was given by Ma and Bertschinger [74].
Unfortunately, none of the previously known cosmological gauges can be applied for analysis of the cosmological perturbations caused by localized matter distributions like an isolated astronomical system which can be a single star, a planetary system, a galaxy, or even a cluster of galaxies. The reason is that the synchronous gauge has no the Newtonian limit and is applicable only for freely falling test particles while the longitudinal gauge separates the scalar, vector and tensor modes present in the metric tensor perturbation in the way that is incompatible with the technique of the approximation schemes having been worked out in asymptotically flat space-time [58]. We also notice that standard cosmological perturbation technique often operates with harmonic (Fourier) decomposition of both the metric tensor and matter perturbations when one is interested in statistical statements based on the cosmological principle. This technique is unsuitable and must be avoided in sub-horizon approximation for working out the post-Newtonian celestial mechanics of self-gravitating isolated systems. Current paradigm is that the cosmological generalization of the Newtonian field equations of an isolated gravitating system like the solar system or a galaxy or a cluster of galaxies can be easily obtained by just making use of the linear principle of superposition with a simple algebraic addition of the local system to the tensor of energy momentum of the background matter. It is assumed that the superposition procedure is equivalent to operating with the Newtonian equations of motion derived in asymptotically-flat spacetime and adding to them ("by hands") the tidal force due to the presence of the external universe (see, for example, [77]). Though such a procedure may look pretty obvious it lacks a rigorous mathematical analysis of the perturbations induced on the background cosmological manifold by the local system. This analysis should be done in the way that embeds cosmological variables to the field equations of standard post-Newtonian approximations not by "hands" but by precise mathematical technique which is the goal of the present paper. The variational calculus on manifolds is the most convenient for joining the standard theory of cosmological perturbations with the post-Newtonian approximations in asymptotically-flat spacetime. It allows us to track down the rich interplay between the perturbations of the background manifold with the dynamic variables of the local system which cause these perturbations. The output is the system of the post-Newtonian field equations with the cosmological effects incorporated to them in a physically-transparent and mathematically-rigorous way. This system can be used to solve a variety of physical problems starting from celestial mechanics of localised systems in cosmology to gravitational wave astronomy in expanding universe that can be useful for deeper exploration on scientific capability of such missions as LISA and Big Bang Observer (BBO) [22] In fact, the problem of whether the cosmological expansion affects the long-term evolution of an isolated N-body system (galaxy, solar system, binary system, etc.) has a long controversial history. The reason is that there was no an adequate mathematical formalism for describing cosmological perturbations caused by the isolated system so that different authors have arrived to opposite opinions. It seems that McVittie [78] was first who had considered the influence of the expansion of the universe on the dynamics of test particles orbiting around a massive point-like body immersed to the cosmological background. He found an exact solution of the Einstein equations in his model which assumed that the mass of the central body is not constant but decreases as the universe expands. Einstein and Straus [30,31] suggested a different approach to discuss motion of particles in gravitationally self-interacting systems residing on the expanding background. They showed that a Schwarzschild solution could be smoothly matched to the Friedman universe on a spherical surface separating the two solutions. Inside the surace ("vacuole") the motion of the test particles is totally unaffected by the expansion. Thus, Einstein and Straus [30,31] concluded that the cosmic expansion is irrelevant for the Solar system. Bonnor [11] generalized the Einstein-Straus vacuole and matched the Schwarzschild region to the inhomogeneous Lemaître-Tolman-Bondi model thus, making the average energy density inside the vacuole be independent of the exterior energy density while in the Einstein-Straus model they must be equal. Bonnor [11] concluded that the local systems expand but at a rate which is negligible compared with the general cosmic expansion. Similar conclusion was reached by Mashhoon et al. [77] who analysed the tidal dynamics of test particles in the Fermi coordinates.
The vacuole solutions are not appropriate for adequate physical solution of the N-body problem in the expanding universe. There are several reasons for it. First, the vacuole is spherically-symmetric while majority of real astronomical systems are not. Second, the vacuole solution imposes physically unrealistic boundary conditions on the matching surface that relates the central mass to the size of the surface and to the cosmic energy density. Third, the vacuole is unstable against small perturbations. In order to overcome these difficulties a realistic approach based on the approximate analytic solution of the Einstein equations for the N-body problem immersed to the cosmological background, is required. In the case of a flat space-time there are two the most advanced techniques for finding approximate solution of the Einstein equations describing gravitational field of an isolated astronomical system. The first is called the post-Newtonian approximations and we have briefly discussed this technique in the introduction. The second technique is called post-Minkowskian approximations [23]. The post-Newtonian approximations is applicable to the systems with weak gravitational field and slow motion of matter. The post-Minkowskian approximations also assume that the field is weak but does not imply any limitation on the speed of matter. The post-Newtonian iterations are based on solving the elliptic-type Poisson equations while the post-Minkowskain approach operates with the hyperbolic-type Dalambert equations. The post-Minkowskian approximations naturally includes description of the gravitational radiation emitted by the isolated system while the post-Newtonian scheme has to use additional mathematical methods to describe generation of the gravitational waves [17]. In the present paper we concentrate on the development of a generic scheme for calculation of cosmological perturbations caused by a localized distribution of matter which preserves many advantages of the post-Minkowskian approximation scheme. The cosmological post-Newtonian approximations are derived from the general perturbation scheme by making use of the slow-motion expansion with respect to a small parameter v/c where v is the characteristic velocity of matter in the N-body system and c is the fundamental speed.
There were several attempts to work out a physically-adequate and mathematically-rigorous approximation schemes in general relativity in order to construct and to adequately describe small-scale self-gravitating structures in the universe. The most notable works in this direction have been done by Kurskov and Ozernoi [63], Futamase et al. [10,37,38], Buchert and Ehlers [14,29], Mukhanov et al. [1,[81][82][83], Zalaletdinov [106]. These approximation schemes have been designed to track the temporal evolution of the cosmological perturbations from a very large down to a small scale up to the epoch when the perturbation becomes isolated from the expanding cosmological background. These approaches looked hardly connected between each other until recent works by Clarkson et al [20,21], Li & Schwarz [66,67], Räsänen [87], Buchert & Räsänen [15] and Wiegand & Schwarz [102]. In particular, Wiegand & Schwarz [102] have shown that the idea of cosmic variance (that isa standard way of thinking) is closely related to the cosmic averages defined by Buchert and Ehlers [14,29]. All researchers agree that the second and higher-order non-linear approximations are important to understand the back-reaction of the cosmological perturbations propagating on the cosmological background used in the linearized theory (see, for example, [1, 37,50,51,82,106]).
Papers [24,61,86] attempted to construct an approximation scheme being compatible with both the post-Newtonian and post-Minkowskian approximations in asymptotically-flat space-time and the gauge-invariant theory of cosmological perturbations caused by a localized astronomical system. We have succeeded in solving this problem in the work [61] which assumes the dust-dominated background universe with spatial curvature k = 0. We remind that in standard Bardeen's approach [7] the metric tensor perturbations h αβ are decomposed in irreducible scalar, vector, and tensor parts which are combined with themselves and with matter perturbations in order to obtain some gauge-invariant quantities that do not contain spurious modes invoked by the freedom in doing coordinate transformations on cosmological background manifold. Bardeen [7] reformulates Einstein's field equations in terms of these gauge-invariant variables which are further decomposed in Fourier harmonics. The field equations become then of the Helmholtz type and are solved by constructing Green's function. This specific procedure of Bardeen's approach is incompatible with the post-Newtonian or post-Minkowskian approximations which do not decompose the metric tensor in scalar, vector, and tensor harmonics and do not expand them to the Fourier series. Therefore, we have used a different procedure based on introduction of auxiliary scalar, φ, and vector, ζ, fields which are used along with the metric tensor perturbation h αβ as basic elements for decomposing the perturbed stress-energy tensor of matter δT αβ and selecting from this decomposition that part of δT αβ which has the same transformation property as the perturbed Einstein tensor δG αβ . This process makes the rest of the perturbation of δT αβ gauge-invariant so that it can be identified with the bare (external) perturbation imposed on the cosmological background by the presence of a material system like a single star, solar system, galaxy, etc. The auxiliary scalar, φ, and vector, ζ, fields are also determined by this procedure. For example, it is proved out [61] that the vector field ζ is identically equal to zero while the scalar field φ is found from the equation following from the Bianchi identity of the perturbed Einstein equations. The entire approach is gauge-invariant but the equations for the scalar and metric perturbations are strongly coupled in general case. We have shown that there is a special cosmological gauge generalizing the harmonic gauge of general relativity in such a way that the reduced field equations are completely decoupled and significantly simplified. More specifically, the linearized Einstein equations for the metric tensor perturbations h 0i and h ij are decoupled both from each other and from h 00 component which couples only with the auxiliary scalar field φ. However, it turns out that our gauge [61] admits a simple linear combination, χ, of h 00 , h kk , and φ such that the equation for χ decouples from any other perturbation. The equations for χ, h 0i and h ij are of a wave-type and have the bare stress energy-tensor of matter as a source in their right-hand sides. These equations have simple Green functions given in terms of the retarded integrals with the Minkowskian null cone defined by the conformally-flat part of the FLRW metric.
In the work [86] this linearized approach has been extended to the background cosmological models governed by a perfect fluid with the barotropic equation of statep = qǭ, wherep andǭ are pressure and energy density of the background matter respectively, and q is a constant parameter taking value in the range from -1 to +1. We have shown [86] that the overall perturbative scheme for calculation of the cosmological perturbations in such model can be significantly streamlined and simplified if one formally replaces the stress-energy tensor of the perfect fluid with one of a classic scalar field minimally coupled with metric. A specific (exponential) form of the potential V (Φ) of the scalar field Φ is fixed by two conditions: Although a minimally coupled scalar field can be viewed to a certain extent as a perfect fluid one should keep in mind that its barotropic equation of state does not hold generally in the perturbed universe (see [13] and discussion in [42,76]). This may impose some technical difficulties in handling the mathematical analysis of cosmological perturbations caused by localized distributions of matter. We explain how to get around these difficulties by making use of the Lagrangian-based variational technique of the gauge-invariant perturbations of curved manifolds. This makes our perturbative approach [61,86] more efficient in developing the post-Newtonian celestial mechanics in cosmology as compared with the standard technique [7, 27, 28, 32-34, 68, 69, 74, 81, 82].
Development of observational cosmology and gravitational wave astronomy demands to extend the linearized theory of cosmological perturbations to second and higher orders of approximation. A fair number of works have been devoted to solving this problem. Non-linear perturbations of the metric tensor and matter affect evolution of the universe and this back-reaction of the perturbations should be taken into account. This requires derivation of the effective stressenergy tensor for cosmological perturbations like freely-propagating gravitational waves and scalar field [1, [81][82][83].
The laws of conservation for the effective stress-energy tensor are important for better understanding of physics of the expanding universe [6,43].
In the present paper we construct a non-linear theory of cosmological perturbations for isolated systems which generalizes the post-Minkowskian approximation scheme for calculation perturbations of gravitational field in asymptotically flat space-time. We rely upon the basic results of the linearized gauge-invariant theory from our previous works [61,86] in order to derive a decoupled system of equations for quadratic cosmological perturbations of a spatially flat FLRW universe. We implement the Lagrangian-based theory of dynamical perturbations of gravitational field on a curved background manifold which has been worked out in [44,85] (see also [6]). This theory has a number of specific advantages over other perturbation methods among which the most important are:
• Lagrangian-based approach is covariant and can be implemented for any curved background spacetime that is an exact solution of the Einstein gravity field equations;
• the system of the partial differential equations describing dynamics of the perturbations is determined by a dynamic Lagrangian L D which is derived from the total Lagrangian L by making use of its Taylor expansion with respect to the perturbations and accounting for the background field equations. The dynamic Lagrangian L D defines the conserved quantities for the perturbations (energy, angular momentum, etc.) that depend on the symmetries of the background manifold;
• the dynamic Lagrangian L D and the corresponding field equations for the perturbations are gauge-invariant in any order of the perturbation theory. Gauge transformations map the background manifold onto itself and are associated with arbitrary (analytic) coordinate transformations on the background space-time;
• the entire perturbation theory is self-reproductive and is extended to the next perturbative order out of a previous iteration so that the linearized approximation is the basic starting point.
IV. LAGRANGIAN AND FIELD VARIABLES
We accept the Einstein's theory of general relativity and consider a universe filled up with matter consisting of three components. The first two components are: (1) an ideal fluid composed of particles of one type with transmutations excluded; (2) a scalar field; and (3) a matter of the localized astronomical system. The ideal fluid consists of baryons and cold dark matter, while the scalar field describes dark energy [2]. We assume that these two components do not interact with each other directly, and are the source of the Friedmann-Lemître-Robertson-Walker (FLRW) geometry. There is no dissipation in the ideal fluid and in the scalar field so that they can only interact through the gravitational field. It means that the equations of motion for the fluid and the scalar field are decoupled, and we can calculate their evolution separately. In other words, the Lagrangian of the ideal fluid and that of the scalar field depend only on their own field variables.
The tensor of energy-momentum of matter of the localized astronomical system is not specified in agreement with the approach adopted in the post-Newtonian approximation scheme developed in the asymptotically-flat spacetime [23,57]. This allows us to generate all possible types of cosmological perturbations: scalar, vector and tensor modes. We are the most interested in developing our formalism for application to the astronomical system of massive bodies bound together by intrinsic gravitational forces like the solar system, galaxy, or a cluster of galaxies. It means that our approach admits a large density contrast between the background matter and the matter of the localized system. The localized system perturbs the background matter and gravitational field of FLRW universe locally but it is not included to the matter source of the background geometry, at least, in the approximation being linearized with respect to the metric tensor perturbation. Our goal is to study how the perturbations of the background matter and gravitational field are incorporated to the gravitational field perturbations of the standard post-Newtonian theory of relativistic celestial mechanics.
Let us now consider the action functional and the Lagrangian of each component.
A. The Action Functional
We shall consider a theory with the action functional
S = M Ld 4 x ,(8)
where the integration is performed over the entire spacetime manifold M. The Lagrangian L is comprised of four terms
L = L g + L m + L q + L p ,(9)
where L g , L m , L q are the Lagrangians of gravitational field, the dark matter, the scalar field that governs the accelerated expansion of the universe [39], and L p is the Lagrangian describing the source of the cosmological perturbations. Gravitational field Lagrangian is
L g = − 1 16π √ −gR ,(10)
where R is the Ricci scalar built of the metric g αβ and its first and second derivatives [79]. Other Lagrangians depend on the metric and the matter variables. Correct choice of the matter variables is a key element in the development of the Lagrangian theory of the post-Newtonian perturbations of the cosmological manifold caused by a localized astronomical system.
B. The Lagrangian of the Ideal Fluid
The ideal fluid is characterized by the following thermodynamic parameters: the rest-mass density ρ m , the specific internal energy Π m (per unit of mass), pressure p m , and entropy s m where the sub-index 'm' stands for 'matter'. We shall assume that the entropy of the ideal fluid remains constant, that excludes it from further consideration. The standard approach to the theory of cosmological perturbations preassumes that the constant entropy excludes rotational (vector) perturbations of the fluid component from the start, and only scalar (adiabatic) perturbations are generated [2,81,99,100]. However, the present paper deals with the cosmological perturbations that are generated by a localized astronomical system which is described by its own Lagrangian (see section IV D) which is left as general as possible. This leads to the tensor of energy-momentum of the matter of the localized system that incorporates the rotational motion of matter which is the source of the rotational perturbations of the background ideal fluid. This extrapolates the concept of the gravitomagnetic field of the post-Newtonian dynamics of localized systems in the asymptotically-flat spacetime [12,19,58] to cosmology. Further details regarding the vector perturbations are given in section VII E of the present paper.
The total energy density of the fluid
ǫ m = ρ m (1 + Π m ) .(11)
One more thermodynamic parameter is the specific enthalpy of the fluid defined as
µ m = ǫ m + p m ρ m = 1 + Π m + p m ρ m .(12)
In the most general case, the thermodynamic equation of state of the fluid is given by equation
p m = p m (ρ m , Π m ),
where the specific internal energy Π m is related to pressure by the first law of thermodynamics.
Since the entropy has been assumed to be constant, the first law of thermodynamics reads
dΠ m + p m d 1 ρ m = 0 .(13)
It can be used to derive the following thermodynamic relationships
dp m = ρ m dµ m ,(14)dǫ m = µ m dρ m ,(15)
which means that all thermodynamic quantities are solely functions of the specific enthalpy µ m , for example, ρ m = ρ m (µ m ), Π m = Π m (µ m ), etc. The equation of state is also a function of the variable µ m , that is
p m = p m (µ m ) .(16)
Derivatives of the thermodynamic quantities with respect to µ m can be calculated by making use of equations (14) and (15), and the definition of the (adiabatic) speed of sound c s of the fluid
∂p m ∂ǫ m = c 2 s c 2 ,(17)
where the partial derivative is taken under a condition that the entropy, s m , of the fluid does not change. Then, the derivatives of the thermodynamic quantities take on the following form
∂p m ∂µ m = ρ m , ∂ǫ m ∂µ m = c 2 c 2 s ρ m , ∂ρ m ∂µ m = c 2 c 2 s ρ m µ m ,(18)
where all partial derivatives are performed under the same condition of constant entropy. The Lagrangian of the ideal fluid is usually taken in the form of the total energy density, L m = √ −gǫ m [79]. However, this form is less convenient for applying the variational calculus on manifolds. The above thermodynamic relationships and the integration by parts of the action (8) allows us to recast the Lagrangian L m = √ −gǫ m to the form of pressure, L m = − √ −gp m , so that the Lagrangian density becomes (see [58, pp. 334-335 ] for more detail)
L m = − √ −gp m = √ −g (ǫ m − ρ m µ m ) .(19)
Theoretical description of the ideal fluid as a dynamic system on space-time manifold is given the most conveniently in terms of the Clebsch potential, Φ which is also called the velocity potential [90]. In the case of a single-component ideal fluid the Clebsch potential is introduced by the following relationship
µ m w α = −Φ ,α .(20)
In fact, equation (20) is a solution of relativistic equations of motion of the ideal fluid [64]. The Clebsch potential is a primary field variable in the Lagrangian description of the isentropic ideal fluid. The four-velocity is normalized to w α w α = g αβ w α w β = −1, so that the specific enthalpy can be expressed in the following form
µ m = −g αβ Φ ,α Φ ,β .(21)
One may also notice that
µ m = w α Φ ,α .(22)
It is important to notice that the Clebsch potential Φ has no direct physical meaning as it can be changed to another value Φ → Φ ′ = Φ +Φ such that the gauge function,Φ, is constant along the worldlines of the fluid: w αΦ ,α = 0. In terms of the Clebsch potential the Lagrangian (19) of the ideal fluid is
L m = √ −g ǫ m − ρ m −g αβ Φ ,α Φ ,β .(23)
Metrical tensor of energy-momentum of the ideal fluid is obtained by taking a variational derivative of the Lagrangian (23) with respect to the metric tensor,
T m αβ = 2 √ −g δL m δg αβ .(24)
Calculation yields
T m αβ = (ǫ m + p m ) w α w β + p m g αβ ,(25)
where w α = dx α /dτ is the four-velocity of the fluid, and τ is the proper time of the fluid element taken along its worldline. This is a standard form of the tensor of energy-momentum of the ideal fluid [79]. Because the Lagrangian (23) is expressed in terms of the dynamical variable Φ, the Noether approach based on taking the variational derivative of the Lagrangian with respect to the field variable, can be applied to derive the canonical tensor of the energymomentum of the ideal fluid. This calculation has been done in [58, pp. 334-335 ] and it leads to the expression (25). It could be expected because we assumed that the the ideal fluid consists of bosons. The metrical and canonical tensors of energy-momentum for the liquid differ, if and only if, the liquid's particles are fermions (see [58, pp. 331-332] for more detail). We do not consider the fermionic liquids in the present paper.
C. The Lagrangian of the Scalar Field
The Lagrangian of the scalar field Ψ is given by
L q = √ −g 1 2 g αβ ∂ α Ψ∂ β Ψ + W ,(26)
where W ≡ W (Ψ) is a potential of the scalar field. We assume that there is no direct coupling between the scalar field and the matter of the ideal fluid. They can interact only through the gravitational field. Many different potentials of the scalar field are used in cosmology [2]. At this step, we do not chose a specific form of the potential which will be selected later.
Metrical tensor of energy-momentum of the scalar field is obtained by taking a variational derivative
T q αβ = 2 √ −g δL q δg αβ ,(27)
that yields
T q αβ = ∂ α Ψ∂ β Ψ − g αβ g µν ∂ µ Ψ∂ ν Ψ + W (Ψ) .(28)
The canonical tensor of energy-momentum of the scalar field is obtained by applying the Neother theorem and leads to the same expression (28). One can formally reduce the tensor (28) to the form similar to that of the ideal fluid by making use of the following procedure. First, we define the analogue of the specific enthalpy of the scalar field "fluid"
µ q = −g σν ∂ σ Ψ∂ ν Ψ ,(29)
and the effective four-velocity, v α , of the "fluid"
µ q v α = −∂ α Ψ .(30)
The four-velocity v α is normalized to v α v α = −1. Therefore, the scalar field enthalpy µ q can be expressed in terms of the partial derivative from the scalar field
µ q = v α ∂ α Ψ .(31)
Then, we introduce the analogue of the rest mass density ρ q of the scalar field "fluid" by defining,
ρ q ≡ µ q = v α ∂ α Ψ = −g σν ∂ σ Ψ∂ ν Ψ .(32)
As a consequence of the above definitions, the energy density, ǫ q and pressure p q of the scalar field "fluid" can be introduced as follows
ǫ q ≡ − 1 2 g σν ∂ σ Ψ∂ ν Ψ + W (Ψ) = 1 2 ρ q µ q + W (Ψ) ,(33)p q ≡ − 1 2 g σν ∂ σ Ψ∂ ν Ψ − W (Ψ) = 1 2 ρ q µ q − W (Ψ) .(34)
One notices that a relationship
µ q = ǫ q + p q ρ q ,(35)
between the specific enthalpy µ q , the density ρ q , the pressure p q and the energy density ǫ q , of the scalar field "fluid" formally holds on the same form (12) as in the case of the barotropic ideal fluid. After applying the above-given definitions in equation (28), it is formally reduced to the tensor of energy-momentum of an ideal fluid
T q αβ = (ǫ q + p q ) v α v β + p q g αβ .(36)
It is worth emphasizing that the analogy between the tensor of energy-momentum (36) of the scalar field "fluid" with that of the barotropic ideal fluid (25) is rather formal since the scalar field, in the most general case, does not satisfy all required thermodynamic equations because of the presence of the potential W = W (Ψ) in the energy density ǫ q , and pressure p q of the scalar field.
D. The Lagrangian of the Localized Astronomical System
The Lagrangian L p of matter of the localized astronomical system which perturbs the geometry of the background manifold of the FLRW universe, can be chosen arbitrary. We shall call the perturbation of the background manifold that is induced by L p , the bare perturbation. We assume that the matter of the bare perturbation is described by a set of scalar potentials θ which are analogues of the Clebsch potential of the matter supporting the background geometry. The Lagrangian density of the bare perturbation is given by L p = √ −gL p (θ, g αβ ). Tensor of energy-momentum of the matter of the bare perturbation, T αβ , is obtained by taking a variational derivative
T αβ = 2 √ −g δL p δg αβ .(37)
Tensor T αβ is a source of the bare gravitational perturbation of the background manifold that will be determined by solving Einstein's field equations derived in next sections.
V. BACKGROUND MANIFOLD
A. The Hubble Flow
We shall consider the background universe as described by the Friedmann-Lemître-Robertson-Walker (FLRW) metric. The functional form of the metric depends on the coordinates introduced on the manifold. Because the FLRW metric describes homogeneous and isotropic spacetime there is a preferred class of coordinates which clearly reveal these properties of the background manifold. These coordinates materialize a special set of freely falling observers, called comoving observers. These observers are following with the flow of the expanding universe and have constant values of spatial coordinates. The proper distance between the comoving observers increases in proportion to the scale factor R(T ). In the preferred cosmological coordinates, the time coordinate of the FLRW metric is just the proper time as measured by the comoving observers. A particle moving relative to the local comoving observers has a peculiar velocity with respect to the Hubble flow. An observer with a non-zero peculiar velocity does not see the universe as isotropic.
For example, the peculiar velocity of the solar system implies the dipole anisotropy of cosmic microwave background (CMBR) radiation corresponding to |v ⊙ | = 369.0 ± 0.9 km·s −1 , towards a point with the galactic coordinates (l, b) = (264 • , 48 • ) [47,53]. Such a solar system's velocity implies a velocity |v LG | = 627±22 km·s −1 toward (l, b) = (276 • , 30 • ) for our Galaxy and the Local Group of galaxies relative to the CMBR [35,55]. The existence of the preferred frame in cosmology should not be understood as a violation of the Einstein principle of relativity. Indeed, any coordinate chart can be used in order to describe the FLRW universe. A preferred frame exists merely because the FLRW metric admits only six-parametric group (3 spatial translations and 3 spatial rotations) as contrasted with the ten-parametric group of Minkowski (or De Sitter) spacetime which includes the time translation and three Lorentz boosts as well. The metric of FLRW does not remain invariant with respect to the time translation and the Lorentz transformations because its expansion makes different spacelike hypersurfaces non-equivalent. It may lead to some interesting observational predictions of cosmological effects within the solar system [60].
B. The Friedmann-Lemître-Robertson-Walker metric
In what follows, we shall consider the problem of calculation of the post-Newtonian perturbations in the expanding universe described by the FLRW class of models. The FLRW metric is an exact solution of Einstein's field equations of general relativity that describes a homogeneous, isotropically expanding or contracting universe. The general form of the metric follows from the geometric properties of homogeneity and isotropy of the manifold [99,100]. Einstein's equations are only needed to derive the scale factor of the universe as a function of time.
The most general form of the FLRW metric is given by
ds 2 = −dT 2 + R 2 dρ 2 1 − kρ 2 + ρ 2 d 2 ϑ + sin 2 ϑd 2 υ ,(38)
where T is the coordinate time, {ρ, ϑ, υ} are spherical coordinates, R = R(T ) is the scale factor depending on time and characterizing the size of the universe compared to the present value of R = 1. The time T has a physical meaning of the proper time of a comoving observer that is being at rest with respect to the cosmological frame of reference.
The present epoch corresponds to the value of the time T = T 0 . The constant k can take on three different values k = {−1, 0, +1}, where k = −1 corresponds to the spatial hyperbolic geometry, k = 0 does the spatially flat FLRW model, and k = +1 does the spatially closed world [79]. The Hubble parameter H characterizes the rate of the temporal evolution of the universe. It is defined by
H ≡Ṙ R = 1 R dR dT .(39)
For mathematical reasons, it is convenient to introduce a conformal time, η, via differential equation
dη = dT R(T ) .(40)
If the time dependence of the scale factor is known, the equation (40) can be solved, thus, yielding T = T (η). It allows us to re-express the scale factor R(T ) in terms of the conformal time, R (T (η)) ≡ a(η). The conformal Hubble parameter is, then, defined as
H ≡ a ′ a = 1 a da dη .(41)
The two expressions for the Hubble parameters are related by means of equation
H = H a ,(42)
that allows us to link their time derivatives
a 2Ḣ = H ′ − H 2 ,(43)a 3Ḧ = H ′′ − 4HH ′ + 2H 3 ,(44)
and so on. It is also convenient to introduce the isotropic Cartesian coordinates X i = {X, Y, Z}, by transforming the radial coordinate
ρ = r 1 + k 4 r 2 ,(45)
and defining r 2 = X 2 + Y 2 + Z 2 = δ ij X i X j . In the isotropic coordinates the interval (38) takes on the following form
ds 2 = G αβ dX α dX β ,(46)
where the coordinates X α = {X 0 , X 1 , X 2 , X 3 } = {η, X, Y, Z}, and the metric has a conformal form
G αβ = a 2 (η)g αβ(47)g 00 = −1 , g 0i = 0 , g ij = δ ij 1 + k 4 r 2 2 .(48)
Determinant of the metric G αβ is G = a 8 g, where g = − 1 + kr 2 /4 −6 . The spacetime interval in the isotropic Cartesian coordinates reads
ds 2 = a 2 (η) −dη 2 + δ ij dX i dX j 1 + k 4 r 2 2 .(49)
The distinctive property of the isotropic coordinates in the FLRW universe is that the radial coordinate r is defined in such a way that the three-dimensional space looks exactly Euclidean and null cones appear in it as round spheres irrespectively of the value of the space curvature k. The isotropic coordinates do not represent proper distances on the sphere, nor does the radial coordinate r represents a proper radial distance measured with the help of radar astronomy technique. The proper spatial distance in the isotropic coordinates is (1 + kr 2 /4) −1 ar [99]. The FLRW metric presented in the conformal form by equation (49) singles out a preferred cosmological reference frame defined by the congruence of worldlines of the fiducial test particles being at rest with respect to the spatial coordinates X i . Four-velocity of a fiducial particle is denoted asŪ α = dX α /dτ , where dτ = −ds is the proper time on the worldline of the particle. In the isotropic conformal coordinates,Ū α = (1/a, 0, 0, 0). The four-velocity is a unit vector,Ū αŪ α = G αβŪ αŪ β = −1. It implies that the covariant components of the four-velocity areŪ α = (−a, 0, 0, 0). In the preferred frame the universe looks homogeneous and isotropic. The choice of the isotropic Cartesian coordinates reflects these fundamental properties explicitly in the symmetric form of the metric (47). However, the set of the fiducial particles is a mathematical idealization. In reality, any isolated astronomical systems (galaxy, binary star, the solar system, etc.) have a peculiar velocity with respect to the preferred cosmological frame formed by the Hubble flow. We have to introduce a locally-inertial coordinate chart which is associated with the isolated system and moves along with it. Transformation from the preferred cosmological frame to the local chart must include the Lorentz boost and a geometric part due to the expansion and curvature of cosmological spacetime. It can take on multiple forms which originate from certain geometric and/or experimental requirements [16,18,48,54].
We do not impose specific limitations on the choice of coordinates on the background manifold and keep the overall formalism of the post-Newtonian approximations, covariant. The arbitrary coordinates are denoted as x α = (x 0 , x i ) and they are related to the preferred isotropic coordinates X α = (η, X i ) by the coordinate transformation x α = x α X β . This transformation has inverse X α = X α x β , at least in some local domain of the background manifold. In this domain, the matrices of the coordinate transformations
Λ α β = ∂x α ∂X β , M α β = ∂X α ∂x β ,(50)
and they satisfy to the apparent equalities Λ α γ M γ β = δ α β and M α γ Λ γ β = δ α β . Four-velocity of the Hubble observers written in the arbitrary coordinates has the following form
u α = Λ α βŪ β = a −1 Λ α 0 ,ū α = M β α U β = −aM 0 α .(51)
The background FLRW metric written down in the arbitrary coordinates, x α , takes on the following form
g αβ (x α ) = a 2f αβ (x α ) .(52)
Here the scalar function a(x α ) ≡ a [η(x α )], and the conformal metric
f αβ (x α ) = M µ α M ν β g µν (X i ) .(53)
Any metric admits 3+1 decomposition with respect to a congruence of a timelike vector field [79]. FLRW universe admits a privileged congruence formed by the four-velocityū α of the Hubble observers which is a physically privileged vector field. The 3+1 decomposition of the FLRW metric is applied in arbitrary coordinates and has the following formḡ
αβ = −ū αūβ +P αβ ,(54)
where the tensorP
αβ = a 2 M i α M j β g ij ,(55)
describes the metric on the spacelike hypersurface being everywhere orthogonal to the four-velocityū α of the Hubble flow. TensorP αβ is the operator of projection on this hypersuface. It can be also interpreted as a metric on the hypersurace of orthogonality to the Hubble vector flow. Equation (54) can be used in order to prove thatP αβ satisfies the following relationshipP
βµP β ν =P µν ,(56)
which can be confirmed by inspection. The traceP α α =ḡ αβP αβ =P αβP αβ = 3. Now, we consider how to express the partial derivatives of any scalar function F = F (η), which depends only on the conformal time η = η(x α ), in terms of the four-velocityū α of the Hubble flow. Taking into account that η = x 0 and applying equation (51), we obtain
F ,α = ∂F ∂x α = dF dη ∂η ∂x α = F ′ M 0 α = − F ′ aū α = −Ḟū α .(57)
In particular, the partial derivative from the scale factor, a ,α = −ȧū α = −Hū α , and the partial derivative from the Hubble parameter H ,α = −Ḣū α .
C. The Christoffel Symbols and Covariant Derivatives
In the following sections of the paper we will need to calculate the covariant derivatives from various geometric objects on the background cosmological manifold covered by an arbitrary coordinate chart x α = (x 0 , x i ). The calculation engages the affine connectionΓ α βγ of the background manifold which is decomposed into an algebraic sum of two connections (the Christoffel symbols) because of the conformal structure of the FLRW metric [98]. By definition,Γ
α βγ = 1 2ḡ αν (ḡ νβ,γ +ḡ νγ,β −ḡ βγ,ν ) ,(58)whereḡ αβ,γ = −2Hḡ αβūγ + a 2f αβ,γ .(59)
Separating terms in the right side of (58) yieldsΓ
α βγ =Ā α βγ +B α βγ ,(60)whereĀ α βγ = −H δ α βū γ + δ α γū β −ū αḡ βγ ,(61)andB α βγ = 1 2f αµ (f µβ,γ +f µγ,β −f βγ,µ ) .(62)
The non-vanishing components of the connections are given in the isotropic Cartesian coordinates X α bȳ
A α 0β = Hδ α β ,Ā 0 ij = Hg ij ,B i pq = − k 2 δ i p X q + δ i q X p − δ pq X i 1 + k 4 r 2 ,(63)
where X q ≡ δ qj X j , and all the other components of the connections vanish.
A covariant derivative of a geometric object (scalar, vector, etc.) on the background manifold is denoted in this paper with a vertical bar. For example, the covariant derivative of a vector field F α is
F α |β = F α ,β +Γ α βγ F γ ,(64)
where a comma in front of sub-index β denotes a partial derivative with respect to coordinate x β . Equation (64) can be brought to yet another form if we denote the covariant derivative of the affine connectionB α βγ with a semicolon. Making use of (60) in equation (64)transforms it to the following form
F α |β = F α ;β +Ā α βγ F γ .(65)
The covariant derivative of a covector F α is defined in a similar way,
F α|β = F α,β −Γ γ αβ F γ(66)
which is equivalent to
F α|β = F α;β −Ā γ αβ F γ ,(67a)F α;β = F α,β −B γ αβ F γ (67b)
Equations for tensors of higher rank can be presented in a similar way. Of course, the covariant derivative of a scalar field F always coincides with its covariant derivative by definition,
F |α = F ;α = F ,α .(68)
We also provide an equation for the covariant derivative of the four-velocity of the Hubble flow. Doing calculations in the isotropic coordinates X α for the four-velocityŪ α , and applying the tensor law of transformation to arbitrary coordinates x α , results inū
α|β = HP αβ ,ū α |β = H δ a β +ū αū β ,ū α|β = HP αβ ,(69)
where the tensor indices are raised and lowered with the metricḡ αβ .
D. The Riemann Tensor
The Riemann tensor is defined bȳ
R α βµν =Γ α βν,µ −Γ α βµ,ν +Γ α µγΓ γ βν −Γ α νγΓ γ βµ .(70)
and can be calculated directly from this equation. We prefer a slightly different way by making use of the algebraic decomposition of the Riemann tensor into the irreducible parts
R αβµν =C αβµν + 1 2 S αµḡβν +S βνḡαµ −S ανḡβµ −S βµḡαν +R 12 (ḡ αµḡβν −ḡ ανḡβµ ) ,(71)
whereC αβµν is the Weyl tensor,S
µν =R µν − 1 4Rḡ µν ,(72)
R µν =ḡ αβR αµβν is the Ricci tensor, and R =ḡ αβR αβ is the Ricci scalar. FLRW cosmological metric (38) has a remarkable property -it can be always brought up to the conformally-flat form by applying an appropriate coordinate transformation [49]. However, the Weyl tensor of any conformally-flat spacetime vanishes identically,
C αβµν ≡ 0 .(73)
Direct evaluation of other tensors entering (71) by making use of the FLRW metric (47), (48) yields
R µν = 1 a 2 H ′ (ḡ µν − 2ū µūν ) + 2 H 2 + k (ḡ µν +ū µūν ) ,(74)R = 6 a 2 H ′ + H 2 + k .(75)
Making use of equations (73) - (75) in the decomposition (71) of the Riemann tensor, yields the following result
R αβµν = 1 a 2 H ′ (ḡ αµḡβν −ḡ ανḡβµ ) − H ′ − H 2 − k P αµPβν −P ανPβµ ,(76)
whereP αβ =ḡ αβ +ū αūβ is the operator of projection that was introduced in (55).
E. The Friedmann Equations
The Einstein tensorĒ αβ =R αβ −ḡ αβR /2 of the FLRW cosmological model is derived from equations (74) and (75). It readsĒ
αβ = − 1 a 2 2 H ′ − H 2 − k P αβ + 3 H 2 + k ḡ αβ .(77)
Einstein's field equations on the background spacetime takes on the following form
E αβ = 8πT αβ ,(78)
where the tensor of energy-momentum of the background spacetime manifold includes the background matter and the scalar fieldT
αβ =T m αβ +T q αβ .(79)
Here, tensors of energy-momentum in the right side of Einstein's equations are derived from the Lagrangians (23) and (26), and represent an algebraic sum of tensors (25) and (29). Each tensor of energy-momentum,T m αβ andT q αβ , is Lie-invariant with respect to the group of symmetry of the background FLRW metric independently, and each of them have the form of the tensor of energy-momentum of the perfect fluid. Hence, the tensor of energy-momentum T αβ in the right side of (78) has the form of a perfect fluid as well,
T αβ = (ǭ +p)ū αūβ +pḡ αβ .(80)
It imposes a certain restriction on the effective energy densityǭ and pressurep which must obey Dalton's law for a partial energy density and pressure of the background matter and the scalar field components [9] ǫ =ǭ m +ǭ q ,
(81) p =p m +p q .(82)
Here,ǭ m andp m are the energy density and pressure of the ideal fluid, andǭ q andp q are the energy density and pressure of the scalar field which are related to the time derivativeΨ of the scalar field and its potentialW =W (Ψ) by equations (33), (34). On the background spacetime these equations takes on the following form
ǫ q = 1 2ρ qμq +W ,(83)p q = 1 2ρ qμq −W ,(84)
whereμ q is the background specific enthalpy of the scalar field defined by (29), andρ q =μ q is the background density of the scalar field "fluid". It is worthwhile to remind to the reader that due to the homogeneity and isotropy of the FLRW universe, all matter variables on the background manifold are functions of the conformal time η only when being expressed in the isotropic Cartesian coordinates. Einstein's equations (78) can be projected on the direction of the background four-velocity of matter and on the spatial hypersurface being orthogonal to it. It yields two Friedmann equations for the evolution of the scale factor a,
H 2 = 8π 3ǭ − k a 2 ,(85)2Ḣ + 3H 2 = −8πp − k a 2(86)
whereǭ andp are the effective energy density and pressure of the mixture of matter and scalar field as defined above. A consequence of the Friedmann equations (85), (86) is an equatioṅ
H − k a 2 = −4π (ǭ +p) ,(87)
relating the time derivative of the Hubble parameter with the sum of the overall energy density and pressure, which can be expressed in terms of the density and specific enthalpy of the background components of matter,
ǫ +p =ρ mμm +ρ qμq .(88)
In order to solve the Friedmann equations (85), (86) we have to employ the equation of state of matter. Customarily, it is assumed that each matter component obeys its own cosmological equation of state,
p m = w mǭm ,p q = w qǭq ,(89)
where w m and w q are parameters lying in the range from −1 to +1. In the most simple cosmological models, parameters w m and w q are fixed. More realistic models admit that the parameters of the equation of state may change in the course of the cosmological expansion, that is they may depend on time. The equation of state does not close the system of the Friedmann equations, which have to be complemented with the equations of motion of the scalar field and of the ideal fluid in order to make the system of differential equations for the gravitational and matter field variables complete.
F. The Hydrodynamic Equations of the Ideal Fluid
The background value of the Clebsch potential of the ideal fluid,Φ, depends only on the conformal time η of the FLRW metric. The partial derivative of the potential, taken in arbitrary coordinate chart on the background manifold, can be expressed in accordance with equation (57) in terms of the background four-velocityū α as follows
Φ |α = −Φū α .(90)
It allows us to write down the specific enthalpy of the ideal fluid in terms of the Clebsch potential. Taking background value of equation (22), we obtainμ
m ≡ū αΦ |α =Φ .(91)
The background equation of continuity for the rest mass densityρ m of the ideal fluid is
(ρ mū α ) |α = 0 ,(92)
that is equivalent toρ
m|α − 3Hρ mūα = 0 ,(93)
where we have used (69). The background equation of conservation of energy is
ǫ m|α − 3H (ǭ m +p m )ū α = 0 ,(94)
where we have employed definition of the energy (11), and equation (93) along with (13).
G. The Scalar Field Equations
Background equation for the scalar fieldΨ is derived from the action (8) by taking variational derivatives with respect toΨ. It yieldsḡ
αβΨ |αβ − ∂W ∂Ψ = 0 .(95)
In terms of the time derivatives with respect to the conformal time η, equation (95) reads
Ψ + 3HΨ + ∂W ∂Ψ = 0 .(96)
Here, we have taken into account that the background value of the scalar field,Ψ, depends only on time η, and its derivative (with respect to η) is proportional to the background four-velocitȳ
Ψ |α = −Ψū α .(97)
If we use definition of the background enthalpy of the scalar field
µ q ≡ū αΦ |α =Ψ ,(98)
and account for definition (33) of the specific energy ǫ q of the scalar field, the equation (96) will becomē
ǫ q|α − 3H (ǭ q +p q )ū α = 0 .(99)
that looks similar to the hydrodynamic equation (93) of conservation of energy of the ideal fluid. Because of this similarity, the second Friedmann equation (86) can be derived from the first Friedmann equation (85) by taking a time derivative and applying the energy conservation equations (94) and (99). The background densityρ q of the scalar filed "fluid" isρ q =μ q in accordance with (32). The equation of continuity for the densityρ q of the ideal fluid is obtained by differentiating definition ofρ q , and making use of (96). It yields
(ρ qū α ) |α = − ∂W ∂Ψ ,(100)
or, equivalently,ρ
q|α − 3Hρ qūα = ∂W ∂Ψū α ,(101)
which shows that the "density" of the scalar field "fluid" is not conserved. We emphasize that there is no any violation of physical laws, since (101) is simply another way of writing equation (95), and the scalar field is not thermodynamically equivalent to the ideal fluid. Equation (101) is convenient in the calculations that follows in next sections.
H. Equations of Motion of Matter of the Localized Astronomical System
Matter of the localized astronomical system is described by the tensor of energy-momentum T αβ defined in (37) in terms of the Lagrangian derivative. It can be given explicitly as a function of field variables after we chose a specific form of matter, for example, gas, liquid, solid, or something else. We do not restrict ourselves with a particular form of this tensor, and shall develop a more generic approach that is applicable to any kind of matter comprising the localized astronomical system..
Background equation of motion of matter of the astronomical system is given by the conservation law
T αβ |β = 0 .(102)
It can be also written down in terms of a covariant derivative of the conformal metric
√ −ḡT αβ ;β + √ −ḡĀ α βγ T βγ = 0 ,(103)
where the connectionĀ α βγ is defined in (61). It is natural to write down this equation in 3+1 form by projecting it on the direction of 4-velocity of the Hubble flow,ū α , and on the hypersurface being orthogonal to it. This is achieved by introducing the following projections
σ ≡ū µūν T µν , (104a) τ ≡P µν T µν , (104b) τ α ≡ −P α µūν T µν , (104c) τ αβ ≡P α µP β ν T µν ,(104d)
which corresponds to the kinemetric decomposition of T µν introduced by Zelmanov [108]. Quantity σ is the energy density of matter of the localized system, t α is a density of linear momentum of the matter, and t αβ is the stress tensor of the matter. Equations of motion (102) of the localized matter can be rewritten in terms of the kinemetric quantities as follows, In the present paper, FLRW background manifold is defined by the metricḡ αβ which dynamics is governed by background matter fields -the Clebsch potentialΦ of the ideal fluid and the scalar fieldΨ. We assume that the background metric and the background values of the fields are perturbed by a localized astronomical system which is considered as a bare perturbation associated with a field variable Θ. Perturbations of the metric and the matter fields caused by the bare perturbation are considered to be small so that the perturbed metric and the matter fields can be split in their backgrounds values and the corresponding perturbations,
(σū α + τ α ) |α = −Hτ , (105a) τ αβ +ū β τ α |β = −H (τ α −ū α τ ) .g αβ =ḡ αβ + κ αβ , Φ =Φ + φ , Ψ =Ψ + ψ .(106)
These equations are exact. We emphasize that all functions entering equation (106) are taken at one and the same point of the background manifold. The bare perturbation does not remain the same in the presence of the perturbations of the metric and the matter fields. Therefore, the field variable Θ corresponding to the bare perturbation, is also perturbed
Θ =Θ + θ .(107)
We consider the perturbations of the metricκ αβ , the Clebsch potentialφ, and the scalar fieldψ as being weak with respect to their corresponding background valuesḡ αβ ,Φ, andΨ, which dynamics is governed by the background equations that have been explained in section V. Because the field variable Θ is the source of the bare perturbation, we postulate that its background value is equal to zero:Θ = 0. The perturbations κ αβ , φ, and ψ have the same order of magnitude as θ.
Perturbation of the contravariant component of the metric is determined from the condition g αγ g γβ =ḡ αγḡ γβ = δ β α , and is given by
g αβ =ḡ αβ − κ αβ + κ α γ κ γβ + . . . ,(108)
where the ellipses denote terms of the higher order. It turns out [44,85] that a more convenient field variable of the gravitational field in the the theory of Lagrangian perturbations of curved manifolds, is a contravariant (Gothic) metric
g αβ = √ −gg αβ .(109)
The convenience of the Gothic metric stems from the fact that it enters the de Donder (harmonic) gauge conditions which significantly simplifies the Einstein equations [65,98]. The Gothic metric variable is also indispensable for concise and elegant formulation of dynamic field theories on curved manifolds [26]. Making use of the Gothic metric allows us to significantly reduce the amount of algebra in taking the first and second variational derivatives from the Hilbert Lagrangian and the Lagrangian of the background matter in FLRW universe as explains in the rest of this section.
The covariant Gothic metric g βγ is defined by means of equation
g αβ g βγ = δ α γ ,(110)
that yields g αβ = g αβ / √ −g. We accept that g αβ is expanded around its background value,ḡ αβ = √ −ḡḡ αβ , as follows
g αβ =ḡ αβ + h αβ ,(111)
which is an exact equation. Further calculations prompt that it is more suitable to single out √ −ḡ from h αβ , and operate with a variable
l αβ ≡ h αβ √ −ḡ .(112)
This variable splits the dynamic degrees of freedom of the gravitational perturbations from the background manifold which evolves in according with the unperturbed Friedmann equations. Tensor indices of l αβ are raised and lowered with the help of the background metric, for example, l αβ ≡ḡ αµḡβν l µν . The field variable l αβ relates to the perturbation κ αβ of the metric tensor. To establish this relationship, we start from (109), substitute equation (111) to its left side, and expand its right side in the Taylor series with respect to κ αβ . It results in
h αβ = ∂ḡ αβ ∂ḡ µν κ µν + 1 2 ∂ 2ḡαβ ∂ḡ µν ∂ḡ ρσ κ µν κ ρσ + . . . .(113)
where the partial derivatives are calculated by successive application of the following rules
∂ḡ αβ ∂ḡ µν = − 1 2 √ −ḡ ḡ αµḡβν +ḡ ανḡβµ −ḡ αβḡµν ,(114a)∂ḡ αβ ∂ḡ µν = − 1 2 ḡ αµḡβν +ḡ ανḡβµ ,(114b)∂ √ −ḡ ∂ḡ µν = + 1 2 √ −ḡḡ µν ,(114c)
which can be easily confirmed by inspection. Replacing the partial derivatives in (113) and making use of the definition (112), yields the relationship between l αβ and κ αβ as follows
l αβ = −κ αβ + 1 2ḡ αβ κ + κ µ(α κ β) µ − 1 2 κ αβ κ − 1 4ḡ αβ κ µν κ µν − 1 2 κ 2 + . . . ,(115)
where κ ≡ κ σ σ =ḡ ρσ κ ρσ , and ellipses denote terms of the cubic and higher order in κ αβ .
Perturbations of four-velocities, w α and v α , entering definitions of the energy-momentum tensors (25), (36), are fully determined by the perturbations of the metric and the potentials of the matter fields. Indeed, according to definitions (20) and (32) the four-velocities are defined by the following equations
w α = − Φ ,α µ m , v α = − Ψ ,α µ q .(116)
where µ m = −g αβ Φ ,α Φ ,β and µ q = −g αβ Ψ ,α Ψ ,β in accordance with (21) and (29) respectively. We define perturbation of the covariant components of the four-velocities as follows
w α =ū α + δw α , v α =ū α + δv α ,(117)
where the unperturbed values of the four-velocities coincide and are equal to the four-velocity of the Hubble flow due to the requirement of the homogeneity and isotropy of the background FLRW metric. Substituting these expansions to the left side of definitions (116), and expanding its right side by making use of the expansions (106) and (108) of the scalar fields and the metric, yields
δw α = − 1 µ mP β α φ |β − 1 2 qū α , δv α = − 1 µ qP β α ψ |β − 1 2 qū α ,(118)
where we have introduced a new notation
q ≡ −ū αūβ κ αβ ,(119)
for the gravitational perturbation of the metric tensor projected on the background four-velocity of the Hubble flow.
Making use of l αβ , the previous equation can be recast to
q ≡ū αūβ l αβ + l 2 ,(120)
where l ≡ l α α =ḡ αβ l αβ . Remembering thatḡ αβ =P αβ −ū αūβ , we can put equation (120) yet to another form
q ≡ 1 2 ū αūβ +P αβ l αβ ,(121)
which is useful in the calculations that follows.
B. The Perturbative Expansion of the Lagrangian
We have introduced the Lagrangian of the theory in section IV. The Hilbert Lagrangian of the gravitational field is L g = − √ −gR/16π, where R is the Ricci scalar. The Lagrangian density of matter is L m = √ −gL m (Φ, g αβ ), and the Lagrangian density of the scalar field L q = √ −gL q (Ψ, g αβ ). The matter, the scalar field as well as the spacetime manifold are perturbed by a matter of N-body system described by a set of field variables Θ with the Lagrangian density L p = √ −gL p (Θ, g αβ ). The action of the unperturbed FLRW universe is a functional
S = M d 4 xL ,(122)
depending on the unperturbed LagrangianL
=L g +L m +L q ,(123)
taken on the background values of the field variablesḡ αβ ,Φ, andΨ. The presence of a localized astronomical system perturbs the spacetime manifold and the background values of the field variables. The perturbed Lagrangian becomes an algebraic sum of four terms
L = L g + L m + L q + L p ,(124)
where the Lagrangian L p describes the bare perturbation, and L g , L m , L q are perturbed values of the Lagrangian of the FLRW universe.
The perturbed Lagrangian can be decomposed in a Taylor series with respect to the perturbed values of the field variables. It is achieved by substituting expansions (106) to the Lagrangian (124) and expanding it around the background values of the variables. It yields
L =L + L 1 + L 2 + L 3 + ... ,(125)
where L 1 , L 2 , L 3 , . . . , are the Lagrangian perturbations which are linear, quadratic, cubic, and so on, with respect to the perturbations of the field variables, h αβ , φ, ψ, and θ. More specifically [85],
L 1 = h µν δL δḡ µν + φ δL δΦ + ψ δL δΨ + L p ,(126a)L 2 = 1 2! h αβ δ δḡ αβ h µν δL δḡ µν + 1 2! φ δ δΦ φ δL δΦ + 1 2! ψ δ δΨ ψ δL δΨ (126b) + h αβ δ δḡ αβ φ δL δΦ + h αβ δ δḡ αβ ψ δL δΨ + h αβ δL p δḡ αβ ,
and so on. Here, the variational derivatives from the Lagrangian density,L, depending on the field variables and their derivatives, are defined as follows
δL δḡ µν ≡ ∂L ∂ḡ µν − ∂ ∂x α ∂L ∂ḡ µν ,α + ∂ 2 ∂x α ∂x β ∂L ∂ḡ µν ,αβ , (127a) δL δΦ ≡ ∂L ∂Φ − ∂ ∂x α ∂L ∂Φ ,α + ∂ 2 ∂x α ∂x β ∂L ∂Φ ,αβ , (127b) δL δΨ ≡ ∂L ∂Ψ − ∂ ∂x α ∂L ∂Ψ ,α + ∂ 2 ∂x α ∂x β ∂L ∂Ψ ,αβ ,(127c)
The variational derivative with respect to the metric densityḡ µν relates to the derivative with respect to the metric g µν by an algebraic operator
δ δḡ µν = ∂ḡ αβ ∂ḡ µν δ δḡ αβ = 1 2 √ −ḡ δ α µ δ β ν + δ α ν δ β µ −ḡ µνḡ αβ δ δḡ αβ .(128)
One has to notice that the expansion (125) is defined up to the terms which are represented as a total covariant derivative from a vector density (the, so-called, divergent terms). For example, the direct Taylor series expansion shows that the Lagrangian L 2 has a term with cross-coupling of φ and ψ. This term was eliminated from L 2 because it can be represented as a covariant divergence from a vector that vanishes identically after taking the Lagrangian derivative [85]. The divergency terms can be important in the discussion of the boundary conditions but they do not enter the equations of motion of fields which represent a system of differential equations in partial derivatives for the perturbations of the dynamic (field) variables. Furthermore, it is straightforward to prove that any of the Lagrangian derivatives (127a)-(127c), applied to a partial derivative of a geometric object F = F (ḡ αβ ;Φ;Ψ;ḡ αβ ,γ ;Φ ,γ ;Ψ ,γ ; . . .), vanishes [80] δ δḡ αβ
∂F ∂x α = 0 , δ δΦ ∂F ∂x α = 0 , δ δΨ ∂F ∂x α = 0 .(129)
Equations (129) does not hold for a covariant derivative [80]. We shall use equation (129) for bringing the Lagrangian derivatives to a simpler form. The field equations are obtained by taking the variational derivatives from the perturbed action with respect to various variables subject to the least action principle. In accordance with this principle, the variational derivatives from the perturbed Lagrangian must vanish,
δL δg µν = 0 , δL δΦ = 0 , δL δΨ = 0 .(130)
We substitute the Taylor decomposition (125) of the Lagrangian to equations (130) and separate the background value of the derivatives from their perturbed values. We assume that gravitational dynamics the unperturbed universe obeys the background field equations. Then, the perturbed part of the equations represent a series of equations of the first, second, third, etc. order, which can be solved by successive iterations. In this paper we restrict ourselves with the linearized approximation of the first order with respect to the perturbations. It generalizes the first post-Minkowskian approximation to the case of the expanding universe.
C. The Background Field Equations
The dynamics of the background universe is governed by the variational equations
δL g δḡ αβ + δL m δḡ αβ + δL q δḡ αβ = 0 , (131a) δL m δΦ = 0 ,(131b)δL q δΨ = 0 .(131c)
After performing the derivatives, equation (131a) becomes the Einstein equation (78), equation (131b) is reduced to equation of continuity (92) after taking into account the thermodynamic relationship (15), and equation (131c) is equivalent to (95). These equations have been thoroughly discussed in section V. Solution of these equations depend on equation of state of background matter. We assume that the solution exists and that the time dependence of the FLRW metricḡ αβ =ḡ αβ (η), the Clebsch potentialΦ =Φ(η), and the scalar fieldΨ =Ψ(η) is explicitly known.
D. The Gravitational Field Perturbations
The gravitational field perturbations satisfy the following (exact) differential equation
F µν = 8π (T µν + t µν ) ,(132)
which generalizes the Einstein field equations of the post-Minkowskian approximations in asymptotically flat spacetime to the case of the expanding universe. Tensor F µν is an algebraic superposition
F µν ≡ F g µν + F m µν + F q µν ,(133)
where the linear operators in the right side are defined through the Lagrangian derivatives as follows,
F g µν ≡ − 16π √ −ḡ δ δḡ µν h αβ δL g δḡ αβ ,(134a)F m µν ≡ − 16π √ −ḡ δ δḡ µν h αβ δL m δḡ αβ + φ δL m δΦ , (134b) F q µν ≡ − 16π √ −ḡ δ δḡ µν h αβ δL q δḡ αβ + ψ δL q δΨ .(134c)
The right side of equation (132) contains the tensor of energy-momentum T µν of the bare gravitational perturbation which is generated by the matter of the localized astronomical system and can be calculated as the Lagrangian derivative (37) . The right side of (132) also contains the non-linear corrections that are given by
t µν = 2 √ −ḡ δL 2 δḡ µν + δL 3 δḡ µν + . . . .(135)
In what follows, we shall neglect the contribution of t µν as it is of the higher order compared with other terms in (132). The differential operator, F g µν , represents a linearized perturbation of the Ricci tensor, and after calculation of (134a), is given by
F g µν = 1 2 l µν |α |α +ḡ µν l αβ |αβ − l αµ|ν |α − l αν|µ α ,(136)
where each vertical bar denotes a covariant derivative with respect to the background metricḡ µν . Operators F m µν and F q µν depend essentially on a particular choice of the Lagrangian of matter and scalar field, and take on different forms depending on the specific analytic dependence of L m and L q on the field variables. In the particular case of the ideal fluid, the term embraced in the round parentheses in the right side of equation (134b) is
h αβ δL m δḡ αβ + φ δL m δΦ = 1 2 h αβ T m αβ − 1 2ḡ αβT m + φ ∂ α ρ m √ −ḡū α ,(137)
whereū α ≡ −ḡ αβΦ ,β /μ m , andT m αβ is given in (25). We emphasize that though the ideal fluid satisfies the equation of continuity (92), it should not be immediately implemented in (137) because this expression is to be further differentiated with respect to the metric tensor according to (134b).
For the scalar field, the term enclosed to the round parentheses in the right side of (134c) is
h αβ δL q δḡ αβ + ψ δL q δΨ = 1 2 h αβ T q αβ − 1 2ḡ αβT q + ψ √ −ḡ ∂W ∂Ψ + ∂ α ρ q √ −ḡū α ,(138)
whereū α ≡ −ḡ αβΨ ,β /μ q ,ρ q =μ q ,T q αβ is given in (36), and the equation of continuity for the scalar field (100) should not be implemented until differentiation with respect to the metric tensor (134c) is completed.
Taking the variational derivatives with respect toḡ µν from the expressions (137) and (138), and applying thermodynamic equations (18), allows us to write down the right sides of equations (134b), (134c) as follows,
F m µν = −4π (p m −ǭ m )l µν + 1 − c 2 c 2 s (ǭ m +p m )qū µūν (139) +8πρ m ū µ φ ,ν +ū ν φ ,µ + 1 − c 2 c 2 s ū µūν −ḡ µν ū α φ ,α , F q µν = −4π (p q − ǫ q ) l µν − 2ḡ µν ∂W ∂Ψ ψ + 8πρ q (ū µ ψ ,ν +ū ν ψ ,µ −ḡ µνū α ψ ,α ) ,(140)
which is a consequence of thermodynamic relationships and definition of a partial derivative. The ratio of the partial derivatives in (142) is not reduced to w m in case when w m depends on some other thermodynamic parameters implicitly depending on the specific enthalpy. For example, in case of an ideal gas the equation of state p m = w m ǫ m , where w m = kT /mc 2 , k is the Boltzmann constant, m -mass of a particle of the ideal fluid, and T is the fluid temperature. The speed of sound c 2 s = c 2 (∂p m /∂ǫ m ) sm=const. = Γw m > w m = p m /ǫ m , where Γ > 1 is the ratio of the heat capacities of the gas taken for the constant pressure and the constant volume respectively.
The scalar field with the potential function W (Ψ) = 0 does not bear all thermodynamic properties of an ideal fluid. Nevertheless, we can formally define the speed of "sound"ĉ s propagating in the scalar field "fluid", by equation being similar to (142). More specifically,ĉ 2 s
c 2 = (∂p q /∂µ q ) Ψ=const. (∂ǫ q /∂µ q ) Ψ=const. .(143)
Simple calculation reveals that the speed of "sound" of the scalar field is always equal to the fundamental speed
c s = c ,(144)
irrespectively of the value of the potential function W (Ψ). It explains why the terms being proportional to the factor 1 − c 2 /ĉ 2 s , do not appear in the expression (140) as contrasted with (139).
E. The Ideal Fluid Perturbations
The perturbed field equations for the ideal fluid are obtained by taking the variational derivatives with respect to the field Φ from the Lagrangian (124) -it corresponds to the middle equation in (130). Taking into account the background equation (131b) yields the equation of sound waves propagating in the fluid as small perturbations,
F m = 8πΣ m ,(145)
where the linear differential operator
F m ≡ − 1 √ −ḡ δ δΦ h µν δL m δḡ µν + φ δL m δΦ ,(146)
and the source term
Σ m ≡ 1 8π √ −ḡ δL m 2 δΦ + δL m 3 δΦ + ... .(147)
In the case of a single-component ideal fluid, the Lagrangian (23) depends merely on the derivative of the Clebsch potential Φ and on the metric tensor. Therefore, the explicit form of the linear operator F m is reduced to a covariant divergence
F m = Y α |α ,(148)
where a vector field
Y α ≡ ∂ ∂Φ ,α l µν − 1 2 lḡ µν ∂L m ∂ḡ µν − 1 2 g µνL m + φ ,β ∂L m ∂Φ ,β ,(149)
where the partial derivatives are taken from the Lagrangian L m , but not from its density L m = √ −gL m . More specifically, calculations yield
Y α ≡ρ m µ m φ |α −ρ m l αβū β + 1 − c 2 c 2 s ρ m µ mū αūβ φ |β − 1 2ρ mū α q .(150)
Similar equations were derived by Lukash [73] who used the variational method to analyze production of sound waves in the early universe.
F. The Scalar Field Perturbations
Equations for the scalar field perturbations are derived by taking the variational derivative from the Lagrangian (124) with respect to the field variable Ψ -see the last equation in (130). Subtracting the background equation (131c) leads to
F q = 8πΣ q ,(151)
where the linear differential operator
F q ≡ − 1 √ −ḡ δ δΨ h µν δL q δḡ µν + ψ δL q δΨ ,(152)
and the source term
Σ q ≡ 1 8π √ −ḡ δL q 2 δΨ + δL q 3 δΨ + ... .(153)
According to equation (26), the Lagrangian density of the scalar field L q = √ −gL q depends on both the field Ψ and its first derivative, Ψ ,α . For this reason, the differential operator F q is not reduced to the covariant derivative from a vector field as the partial derivative of the Lagrangian with respect to Ψ does not vanish. We have
F q ≡ Z α |α − l 2 ∂W ∂Ψ − ψ ∂ 2W ∂Ψ 2(154)
where l ≡ḡ αβ l αβ , and vector field
Z α ≡ ∂ ∂Ψ ,α l µν − 1 2 lḡ µν ∂L q ∂ḡ µν − 1 2ḡ µνL q + ψ ,β ∂L q ∂Ψ ,β .(155)
Performing the partial derivatives in equation (155), yields a rather simple expression
Z α ≡ ψ |α −ρ q l αβū β ,(156)
where we have used equationΨ |α = −ū βΨ |βūα = −ρ qūα . The reader is invited to compare equation (156) with (150) to observe the differences between the Lagrangian perturbations of the ideal fluid and the scalar field. One may observe that (150) becomes identical with (156) in the limit c s → c, andρ m →μ m . This corresponds to the case of an extremely rigid equation of state w m = 1 in equation (89). According to the discussion following equations (143), (144) the speed of 'sound'ĉ s in the scalar field 'fluid' is always equal to c. However, it does not assume that the parameter w q of the equation of state of the scalar field,p q = w qǭq , in (89) is equal to unity. This is because the scalar field is not completely equivalent to the ideal fluid in the sense of thermodynamic [2]. .
G. The Lagrangian Equations for Field Variables
Equations for the metric tensor perturbations
Linearized equations for gravitational field variables, l µν , are obtained from (131a) after neglecting in its right side the non-linear source t µν , and rendering a series of transformations which re-arrange and sort out similar terms. First, let us make use of Einstein's equations (139) and (140) to find
F m µν + F q µν = 4π (ǭ −p) l µν (157) + 8πρ m ū µ φ ,ν +ū ν φ ,µ −ḡ µν u α φ ,α + 1 − c 2 c 2 s ū α φ ,α − 1 2μ m q ū µūν + 8πρ q ū µ ψ ,ν +ū ν ψ ,µ −ḡ µν u α ψ ,α +ḡ µν ∂W (Ψ) ∂Ψ ψ µ q .
Second step is to transform the linear differential operator F g µν in (136) to a more convenient form that will allow us to single out the gauge-dependent terms denoted by
A µ ≡ l µν |ν .(158)
Changing the order of the covariant derivatives in (136) and taking into account that the commutator of the second covariant derivatives is proportional to the Riemann tensor, we can recast (136) to the following form,
F g µν ≡ 1 2 l µν |α |α +ḡ µν A α |α − A µ|ν − A ν|µ −R α (µ l ν)α −R µαβν l αβ ,(159)
where the round brackets around indices denote symmetrization. The terms with the Ricci and Riemann tensors can be expressed in terms of the total background energy and pressure of the ideal fluid and scalar field by making use of equations (71), (73) and Einstein's equations (78). It yields
R α (µ l ν)α +R µαβν l αβ = 4π 5ǭ 3 −p l µν + l 2 p −ǭ 3 ḡ µν + (ǭ +p) (2ū αū µ l να + 2ū αū ν l µα −ū µūν l −ḡ µν q) . (160)
Finally, substituting equations (157), (159) and (160) to (132) results in
l µν |α |α +ḡ µν A α |α − A µ|ν − A ν|µ (161) −16π ǭ 3 l µν + l 4 p −ǭ 3 ḡ µν + (ǭ +p) ū αū µ l να +ū αū ν l µα − 1 2ū µūν l − 1 2ḡ µν q +16πρ m ū µ φ ,ν +ū ν φ ,µ −ḡ µν u α φ ,α + 1 − c 2 c 2 s ū α φ ,α − 1 2μ m q ū µūν +16πρ q ū µ ψ ,ν +ū ν ψ ,µ −ḡ µν u α ψ ,α +ḡ µν ∂W (Ψ) ∂Ψ ψ µ q = 16πT µν ,
where the non-linear term t µν was neglected. The first term in (161) is a covariant Laplace-Beltrami operator, l µν |α |α ≡ḡ αβ l µν|αβ , that is a rather complicated geometric object. Its explicit expression can be developed by making use of the Christoffel symbols given in (60). Tedious but straightforward calculation yields l µν |α |α = l µν ;α ;α + 2Hū α l µν;α − 2 (Hū α l αµ ) |ν − 2 (Hū α l αν ) |µ (162)
+ 2H (ū µ A ν +ū ν A µ ) + 2H ′ (l µν −ū αū µ l να −ū αū ν l µα ) + 2H 2 2l µν + 3ū µū α l αν + 3ū νū α l αµ −ḡ µνū αūβ l αβ −ū µūν l ,
where the semicolon denotes a covariant derivative that is calculated with the Christoffel symbols B α µν like in (67b), and the differential operator l µν ;α ;α ≡ḡ αβ l µν;αβ . Further derivation of the differential equations for linearized metric tensor perturbations can be significantly simplified if we re-define the gauge function, A α ≡ l µν |ν , in the following form
A α = −2Hl αβū β + 16π (ρ m φ +ρ q ψ)ū α + B α ,(163)
where B α is an arbitrary gauge function. This choice of the gauge function A α allows us to eliminate two terms in equation (162)
+ 2 H ′ + H 2 (l µν +ū µū α l αν +ū νū α l αµ − lū µūν ) (164) − 2k a 2 l µν + 2ū µū α l αν + 2ū νū α l αµ − lū µūν − q + l 2 ḡ µν +16πū µūν ρ m 1 − c 2 c 2 s ū α φ ,α − 1 2μ m q − 2 ∂W ∂Ψ ψ − 4H (ρ m φ +ρ q ψ) +ḡ µν B α |α − B µ|ν − B ν|µ + 2H (ū µ B ν +ū ν B µ −ḡ µνū α B α ) = 16πT µν .
This equation is fully covariant and is valid in any gauge and/or coordinate chart. It clarifies the advantage in the choice of the gauge function (163). Indeed, if one works in the isotropic coordinates associated with the Hubble flow, whereŪ α = (1/a, 0, 0, 0), it allows us to fully decouple the differential equations for l 00 , l 0i and l ij components of the metric tensor perturbations. Let us assume, for simplicity, B α = 0 that is an analogue of the harmonic gauge in asymptotically-flat spacetime. Then, different tensor components of equation (164)
become q + 2Hq ;0 + 4kq − 4π 1 − c 2 c 2 s ρ mμm q = 8π (T 00 + T kk ) − (165a) 8πa ρ m 1 − c 2 c 2 s φ ,0 − 2a ∂W ∂Ψ ψ − 4H (ρ m φ +ρ q ψ) , l 0i + 2Hl 0i;0 + 2kl 0i = 16πT 0i , (165b) l <ij> + 2Hl <ij>;0 + 2 (H ′ − k) l <ij> = 16πT <ij> , (165c) l kk + 2Hl kk;0 + 2 (H ′ + 2k) l kk = 16πT kk .(165d)
Here, we denoted l µν =f αβ l µν;αβ , q = (l 00 + l kk ) /2, l kk = l 11 +l 22 +l 33 , l <ij> = l ij −(1/3)δ ij l kk , and the same index notations are applied to the tensor of energy-momentum T ij of the localized astronomical system. These equations are clearly decoupled from one another, thus, demonstrating the advantage of the gauge condition (163) used along with B α = 0. Equations (165b)-(165d) can be solved independently if the initial and boundary conditions are known, and the tensor of energy-momentum of the localized astronomical system is well-defined. Equation (165a) for scalar q besides knowledge of T αβ , demands to know the scalar field perturbations, φ and ψ, that contribute to the source of q standing in the right side of (165a). Equations for these perturbations are obtained in the following text.
Equations for the ideal fluid perturbations
The ideal fluid perturbations, φ, evolve in accordance with the Lagrangian equation (145). In the linear approximation we can neglect the source term Σ m in its right side. The covariant derivative in the definition of the linear operator F m given by (148), can be explicitly performed that yields equation for the Clebsch potential
φ |α |α − 2μ m Hq + 16πμ m (ρ m φ +ρ q ψ) + 1 − c 2 c 2 s ū αūβ φ |αβ − 1 2μ mū α q ,α =μ mū α B α ,(166)
where the gauge function (163 has been used. The gauge B α remains yet unspecified so that equation (166) is covariant and is valid in any coordinate chart.
Equations for the scalar field perturbations
Linearized equation for the scalar field perturbations, ψ, is obtained from the Lagrangian equation (151) after neglecting the (non-linear) source term Σ q . After performing the covariant differentiation in equation (154), we get equation for the scalar field perturbation
ψ |α |α − 2μ q H + ∂W ∂Ψ q + 16πμ q (ρ m φ +ρ q ψ) − ∂ 2W ∂Ψ 2 ψ =μ qū α B α ,(167)
where equation (163) has been used along with the equalityρ q =μ q . The gauge function B α is kept unspecified so that equation (166) is covariant and is valid in any coordinates.
VII. GAUGE-INVARIANT FIELD EQUATIONS IN 3+1 FORMALISM
A. Algebraic Decomposition of the Metric Perturbations.
We have derived the system of coupled differential equations (164), (166), (167) for the field variables l αβ , φ and ψ, describing perturbations of the gravitational field, the ideal fluid, and the scalar field respectively. These system of equations can be split into a set of gauge-invariant differential equations for the scalar, vector, and tensor parts which is convenient for theoretical study of the evolution of the perturbations in arbitrary coordinates. This 3+1 split is achieved by making use of the operator of projectionP αβ onto a hypersurface being orthogonal to the congruence of worldlines of the Hubble flow.
The theory under development admits four, algebraically-independent scalar perturbations. Two of them are the Clebsch potential of the ideal fluid φ and the scalar field ψ. The two other scalars characterize the scalar perturbations of the gravitational field. They can be chosen, for example, as a projection of the metric tensor perturbation on the direction of the background four-velocity field,ū αūβ l αβ , and the trace of the metric tensor perturbation, l =ḡ αβ l αβ . However, it is more convenient to work with two other scalars, defined as their linear combinations,
q ≡ 1 2 ū αūβ +P αβ l αβ ,(168a)p ≡ ū αūβ +ḡ αβ l αβ ,(168b)
Notice that the scalar q has been introduced earlier in (140). The scalar p is, in fact, projection of l αβ onto the spacelike hypersurface being orthogonal everywhere to the worldlines of fiducial observers moving with the background four-velocityū α of the Hubble flow. Indeed, after accounting for definition (54), equation (168b) can be written as
p =P αβ l αβ .(169)
Vectorial gravitational perturbations are defined by a spacial-temporal projection
p α ≡ −P α βūγ l βγ ,(170)
where minus sign was taken for the sake of mathematical convenience. Due to its definition, vector p α =ḡ αβ p β is orthogonal to the four-velocityū α , that isū α p α = 0. Hence, it describes a space-like vector-like gravitational perturbations with three algebraically-independent components.
Tensorial gravitational perturbations are associated with the projection
p ⊺ αβ ≡ p αβ − 1 3P αβ p ,(171)
where
p αβ ≡P α µP β ν l µν .(172)
Here, the tensor p αβ is a double projection of l αβ onto space-like hypersurface being orthogonal to the worldlines of fiducial observers moving with the four-velocityū α of the Hubble flow. The trace of this tensor coincides with the scalar p. Indeed,ḡ
αβ p αβ =ḡ αβP α µP β ν l µν =P βµP β ν l µν =P µν l µν = p ,(173)
where the property of the projection tensorP βµP β ν =P µν has been used. Equation (173) makes it clear that tensor p ⊺ αβ is traceless, that isḡ αβ p ⊺ αβ = 0. Because of this property, and four orthogonality conditions,ū α p ⊺ αβ = 0, the symmetric tensor p ⊺ αβ has only five, algebraically-independent components. Gravitational perturbation l αβ can be decomposed into the algebraically-irreducible scalar, vector and tensor parts as follows
l αβ = p ⊺ αβ +ū α p β +ū β p α + ū αūβ + 1 3P αβ p + 2ū αūβ (q − p) .(174)
One should not confuse the pure algebraic decomposition of the metric tensor perturbation with its decomposition in a functional (Hilbert) space. This decomposition was pioneered by Lifshitz and Khalatnikov [70] and later on, structured by Arnowitt, Deser and Misner Arnowitt et al. [5] (see also [79]). It is commonly used in the research on the relativistic theory of formation of the large-scale structure in the universe. The functional ADM decomposition of the metric tensor perturbations is done with respect to the direction of propagation of weak gravitational waves and singles out the longitudinal (L), transversal (T) and transverse-traceless (TT) parts of the perturbations. In other words, the functional decomposition make sure that the vector p α and the tensor parts of the gravitational perturbation, p ⊺ αβ , are further decomposed in the functionally-irreducible components that are reduced to two more scalars, and two transverse vectors each of which has only two, functionally-independent components. The remaining part of the tensor perturbations, p ⊺ αβ , is transverse-trackless and has only two functionally-independent components denoted as p TT αβ . The ADM decomposition of the metric tensor is a powerful technique in the theory of gauge-invariant cosmological perturbations [8]. However, it is not convenient in the development of the post-Newtonian dynamics of celestial bodies in cosmology, and shall not be used in the present paper.
Our next step is derivation of the field equations for the algebraically-irreducible components of matter and gravitational field. Before doing this derivation, let us discuss the gauge transformations of the corresponding field variables.
B. The Gauge Transformation of the Field Variables
Gauge invariance is a cornerstone of the modern theoretical physics with a long and interesting history [52]. Gauge invariance should be distinguished from the coordinate invariance or the general covariance because, by definition, a gauge transformation changes only field variables of the theory under consideration but not coordinates. Discussion of gauge transformation and invariance requires introducing a gauge field and a new geometric object -an affine connection -on a fibre bundle manifold describing the intrinsic degrees of freedom of corresponding field variables of the gauge field theory.
The present paper discusses physical perturbations of the field variables l αβ , Φ, Ψ on the cosmological spacetime manifold in the framework of general relativity. The affine connection on the spacetime manifold of general relativity is represented by the Christoffel symbols while the gauge transformation is generated by a flow of an arbitrary vector (gauge) field ξ α that maps the manifold into itself. Gauge transformation of the fields on a curved manifold is associated with a Lie transport of the fields along the vector flow ξ α [65,99]. Infinitesimal gauge transformation is a Lie derivative of the field taken at the value of the parameter on the curves of the vector flow equal to 1 [58, chapter 3.6].
Let us consider a mapping of spacetime manifold into itself induced by a vector flow, ξ α = ξ α (x β ). It means that each point of the manifold with coordinates x α is mapped to another point with coordinateŝ
x α = x α − ξ α (x) .(175)
This mapping of the manifold into itself can be interpreted as a local diffeomorphism which transforms the field variables in accordance to their tensor properties. The transformed value of the field variable is pulled back to the point of the manifold having the original coordinates x α , and is compared with the value of the field at this point. The difference between the transformed and the original value of the field, generated by the diffeomorphism (175) is the gauge transformation of the field that is given by the Lie derivative taken along the vector field ξ α at the point of the manifold with coordinates x α . Let us denote the transformed values of the field variables with a hat. In the linearized perturbation theory of the cosmological manifold, the gauge transformations of the field variables are given by equationŝ To find out the gauge-invariant content of the theory one should search for the gauge-invariant field variables and to derive the gauge-invariant equations for them. This program has been completed by Bardeen [8] who used the functional 3+1 decomposition of the metric tensor perturbations and the vector field ξ α to build the gauge-invariant variables out of the various projections of the metric tensor components on space an time. Various modifications of Bardeen's approach can be found, for example, in [13,29,32,33,73,82] and in the book by Mukhanov [81]. We use algebraic 3+1 decomposition of the metric tensor perturbations (174) that allows us to build gauge-invariant scalars. Vector and tensor perturbations remain gauge-dependent in this approach. In order to suppress the gauge degrees of freedom in these variables we impose a particular gauge condition B α = 0 in equation (163). This limits the freedom of the gauge field ξ α by a particular set of differential equations which are discussed in section (VII G).
κ αβ = κ αβ + ξ α|β + ξ β|α ,(176a)l αβ = l αβ − ξ α|β − ξ β|α +ḡ αβ ξ γ |γ , (176b) φ = φ +Φ |α ξ α , (176c) ψ = ψ +Ψ |α ξ α ,(176d)
C. The Gauge-invariant Scalars
The existence of the preferred four-velocity,ū α , of the Hubble flow in the expanding universe provides a natural way of separating the perturbations of the field variables in scalar, vector, and tensor components. This section discusses how to build the gauge-invariant scalars. Vector and tensor perturbations are discussed afterwards.
The gauge-invariant scalar perturbations can be build from the perturbation of the Clebsch potential, φ, the perturbation of the scalar field ψ, and two scalars associated with the trace of the metric tensor, q, and its projection on the worldlines of the Hubble flow, q. To build the first gauge-invariant scalar, we introduce the scalar perturbations
χ m ≡ φ µ m , χ q ≡ ψ µ q ,(177)
that normalize perturbations of the Clebsch potential φ and that of the scalar field ψ to the corresponding background values of the specific enthalpy,μ m andμ q . The gauge transformations for the three scalars q, χ m , and χ q are obtained from (176b)-(176d), and readq
= q − 2ū αūβ ξ α|β , (178a) χ m = χ m −ū α ξ α , (178b) χ q = χ q −ū α ξ α ,(178c)
where we have used the definition of the background four-velocityū α = −Φ |α /μ m = −Ψ |α /μ q in terms of the partial derivatives of the background values of the scalar fields Φ and Ψ. Equations (178b), (178c) immediately reveal that the linear combination
χ ≡ χ m − χ q ,(179)
is gauge-invariant,χ = χ, that is the diffeomorphism (175) does not change the value of the scalar variable χ. Two other gauge-invariant scalars are defined by the following equations,
V m ≡ū α χ m|α − q 2 , (180a) V q ≡ū α χ q|α − q 2 ,(180b)
or, more explicitly,
V m = 1 µ mū α φ |α − q 2 + 3 c 2 s c 2 Hχ m , (181a) V q = 1 µ qū α ψ |α − q 2 + 3Hχ q + χ q µ q ∂W ∂Ψ ,(181b)
where the last terms in the right side of these equations were obtained by making use of thermodynamic relationships (18), the equalityρ q =μ q , and the equations of continuity (93) and (101) for the density of the ideal fluid,ρ m , and that of the scalar field,ρ q , respectively. One can easily check that both scalars, V m and V q remain unchanged after making the infinitesimal coordinate transformation (175). Indeed, the gauge transformation of the derivativeŝ
χ m|α = χ m|α − HP αβ ξ β −ū β ξ β |α ,(182a)χ q|α = χ q|α − HP αβ ξ β −ū β ξ β |α ,(182b)
whereP αβ =ḡ αβ +ū αūβ is the operator of projection on the hypersurface being orthogonal to the Hubble flow of four-velocityū α . Making the coordinate transformation (175), and substituting the transformations of functions q, χ m and χ q to the definitions of V m and V q we find
V m = V m ,V q = V q ,(183)
that proves the gauge-invariant property of the scalars V m and V q . Physical meaning of the gauge-invariant quantity V m can be understood as follows. We consider the perturbation of the specific enthalpy µ m defined in equation (21). Substituting the decomposition (106) of the field variables to equation (21) and expanding, we obtain
µ m =μ m + δµ m ,(184)
where the perturbation δµ m of the specific enthalpy is defined (in the linearized order) by
δµ m =ū α φ |α − 1 2μ m q .(185)
It helps us to recognize that
V m = δµ m µ m + 3 c 2 s c 2 Hχ m .(186)
Fractional perturbation of the specific enthalpy can be re-written with the help of thermodynamic equations (18) in terms of the perturbation δǫ m of the energy density of the ideal fluid,
δµ m µ m = c 2 s c 2 δǫ m ǫ m +p m ,(187)
or, by making use of equation (15), in terms of the perturbation δρ m of the density of the ideal fluid
δµ m µ m = c 2 s c 2 δρ m ρ m .(188)
This allows us to write down equation (186) as follows
V m = c 2 s c 2 δρ m ρ m + 3Hχ m ,(189)
which elucidates the relationship between the gauge-invariant variable V m and the perturbation δρ m of the rest mass density of the ideal fluid. More specifically, V m is an algebraic sum of two scalar functions, δρ m and χ m neither of each is gauge-invariant. The gauge transformation of the ideal-fluid density perturbation is
δρ m = δρ m −ρ m|α ξ α = δρ m + 3Hρ mūα ξ α ,(190)
and the gauge transformation of the variable χ m is given by (178b). Their algebraic sum in equation (189) does not change under the diffeomorphism (175) showing that V m is the gauge-invariant density fluctuation that does not depend on a particular choice of coordinates on spacetime manifold. Similar considerations, applied to function V q reveals that it can be represented as an algebraic sum of the perturbation, δρ q , of the density of the scalar field, and the function χ q ,
V q = δρ q ρ q + 3Hχ q .(191)
It is easy to check that each term in the right side of this equation is not gauge-invariant but their linear combination does. The reader should notice that standard textbooks on cosmological theory (see, for example, [71,84,99,100]) derive equations for the density perturbations δρ/ρ but those equations are not gauge-invariant and, hence, their solutions should be interpreted with care (see discussion in section IX A 2).
D. Field Equations for the Scalar Perturbations.
Equation for a scalar q.
Function q was defined in (168a). In order to derive a differential equation for q, we apply the covariant Laplace-Beltrami operator to q, and make use of the covariant equations (161) and (163). Straightforward but fairly long calculation yields
q |α |α − 2 Ḣ + H 2 − 2k a 2 q + 8πρ mμm 1 − c 2 c 2 s V m − 1 + 3 c 2 s c 2 Hχ m (192) −16πρ q ∂W ∂Ψ + 2Hμ q χ q − 2ū αūβ B α|β − 4Hū α B α = 8π (σ + τ ) ,
where the source density σ + τ for the field q is
σ + τ = ū αūβ +P αβ T αβ ,(193)
in accordance with the definitions introduced in (104a), (104b). The reader should notice that equation (192) depends on the gauge function B α which remains arbitrary so far.
Equation for a scalar p.
Function p was defined in (168b). In order to derive equation for p, we apply the covariant Laplace-Beltrami operator to the definition of p, and make use of the covariant equations (161) and (163). It results in a wave equation
p |α |α + 4k a 2 p + B α |α − 2ū αūβ B α|β − 6Hū α B α = 16πτ .(194)
where the source density τ has been defined in (104b). Equation (194) depends on the arbitrary gauge function B α .
Equation for a scalar χ.
Equation for the gauge invariant scalar, χ = χ m − χ q , is derived from the definitions (177) and the field equations (166), (167). Replacing φ and ψ in those equations with χ m and χ q , and making use of equations (87), (88) for reshuffling some terms, yields
χ |α m |α + 2Hū α χ m|α − Ḣ − 4k a 2 χ m + 4HV m + 1 − c 2 c 2 s ū α V m|α − 16πρ qμq χ =ū α B α ,(195a)χ |α q |α + 2Hū α χ q|α − Ḣ − 4k a 2 χ q + 4HV q + 2 µ q ∂W ∂Ψ V q + 16πρ mμm χ =ū α B α . (195b)
Subtracting (195b) from (195a) cancels the gauge-dependent term,ū α B α , and brings about the field equation for χ,
χ |α |α + 6Hū α χ |α + 3Ḣχ = 2 µ q ∂W ∂Ψ V q − 1 − c 2 c 2 s ū α V m|α .(196)
This equation is apparently gauge-invariant since any dependence on the arbitrary gauge function B α disappeared. It is also covariant that is valid in any coordinates. Equation (196) can be recast to the form of an inhomogeneous wave equation:
(ρ m χ) |α |α = 2ρ m ρ q ∂W ∂Ψ V q − 1 − c 2 c 2 s ρ mū α V m|α .(197)
Yet another form of equation (196) is obtained in terms of the variable ρ q χ. By simple inspection we can check that equation (196) is transformed to
(ρ q χ) |α |α − ∂ 2W ∂Ψ 2 (ρ q χ) = 2 ∂W ∂Ψ V m − 1 − c 2 c 2 s ρ qū α V m|α .(198)
This is an inhomogeneous Klein-Gordon equation for the field (ρ q χ) governed by V m . The 'mass' of the scalar field ρ q χ depends on the second derivative of the potential functionW which defines the 'coefficient of elasticity' of the background scalar fieldΨ. Inhomogeneous equations (196), (197), (198) have the source terms that is determined by variables V m and V q . We derive differential equations for these field variables in the next sections.
Equation for a scalar Vm.
Equation for the field variable V m is derived from the equations for functions χ m and q that enter its definition (180a). By applying the Laplace-Beltrami operator to function V m we get
V |α m |α =ū β χ |α m |α |β + 2Hχ |α m |α − 1 2 q |α |α +ū βRα β χ m|α + 2Hū α V m + 1 2 q |α + 3H 2 V m + 1 2 q .(199)
The Laplace-Beltrami operator for function χ m is given in equation (195a) which is not gauge-invariant. Taking the covariant derivative from this equation and contracting it withū α brings about the first term in the right side of equation (199),
u β χ |α m |α |β = − 1 − c 2 c 2 s ū αūβ V m|αβ − 6Hū α V m|α − 5Ḣ + 4k a 2 V m (200) −Hū α q |α − 1 2Ḣ + 2k a 2 q − 3H 1 + c 2 s c 2 Ḣ − 3 + c 2 s c 2 k a 2 χ m +8πρ q ∂W ∂Ψ (4χ q − 3χ m ) + 16πρ qμq ū α χ |α − 6χ + 3 4 H 1 − c 2 s c 2 χ m +ū αūβ B α|β .
The Laplace-Beltrami operator for function q has been derived in (192). Now, we make use of equations (192), (195a), (200) in calculating the right side of (199). After a significant amount of algebra, we find out that all terms explicitly depending on q and the gauge functions B α cancel out, so that equation for V m becomes
V |α m |α + 1 − c 2 c 2 s ū αūβ V m|αβ + 2 3 − c 2 c 2 s Hū α V m|α + (201) 2 Ḣ + 3H 2 + 2k a 2 − 4πρ mμm 1 − c 2 c 2 s V m − 16πρ qμq ū α χ |α − 3 H + 1 2μ q ∂W ∂Ψ χ = −4π (σ + τ ) .
Second-order covariant derivatives in this equation read
ḡ αβ + 1 − c 2 c 2 s ū αūβ V m|αβ ≡ − c 2 c 2 sū αūβ +P αβ V m|αβ ,(202)
and they form a hyperbolic-type operator describing propagation of sound waves in the expanding universe from the source of the sound waves towards the field point with the constant velocity c 2 s . Additional terms in the left side of equation (201) depend on the Hubble parameter H, and take into account the expansion of the universe. Equation (201) contains only gauge-invariant scalars, V m and χ. Moreover, it does not depend on the choice of coordinates on the background manifold. It also becomes clear that the field variables V m and χ are coupled through the differential equations (198) and (201) which should be solved simultaneously in order to determine these variables. Solution of the coupled system of differential equations is a very complicated task which cannot be rendered analytically in the most general case. Only in some simple cases, the equations can be decoupled. We discuss such cases in section IX.
Equation for a scalar Vq.
The field variable V q is not independent since it relates to V m and χ by a simple relationship
V q = V m −ū α χ |α ,(203)
which is obtained after subtraction of equation (180a) from (180b). Equation for V q is derived directly from (203) and equations (201) and (196) for V m and χ respectively. We obtain,
V |α q |α + 4 H + 1 2μ q ∂W ∂Ψ ū α V q|α (204) + 2 Ḣ + 3H 2 + 2k a 2 − 4πρ mμm 1 − c 2 c 2 s + 2 µ q 5H + 1 µ q ∂W ∂Ψ ∂W ∂Ψ + 2 ∂ 2W ∂Ψ 2 V q +4πρ mμm 3 + c 2 c 2 s ū α χ |α − 3 c 2 s c 2 Hχ = −4π (σ + τ ) .
This equation can be also derived by the procedure being similar to that used in the previous subsection in deriving equation for V m . We followed this procedure and confirm that it leads to (204) as expected. Equation (204) is clearly gauge-invariant. It couples with the variable χ and should be solved along with equation (196).
E. Field Equations for Vector Perturbations
Vector perturbations of the ideal fluid and scalar field are gradients, φ |α and ψ |α . However, they are insufficient to build a gauge-invariant vector perturbation out of the vector perturbation of the metric tensor p α . Field equations for vector p α can be derived by applying the covariant Laplace-Beltrami operator to both sides of definition (170) and making use of equation (164). After performing the covariant differentiation and a significant amount of algebra, we derive the field equation
p α |β |β − 2Hū α p β |β − 2Ḣ + 3H 2 − 2k a 2 p α +P α βūγ B β|γ + B γ|β + 2Hū γ B β = 16πτ α ,(205)
where the matter current σ α is defined in (104c). This equation is apparently gauge-dependent as shown by the appearance of the gauge function B α . This equation reduces to a much simpler form
p α |β |β − 2Hū α p β |β − 2Ḣ + 3H 2 − 2k a 2 p α = 16πτ α ,(206)
in a special gauge B α =0 which imposes a restriction on the divergence of the metric tensor perturbation in equation (163). Equation (205) points out that the vector perturbations are generated by the current of matter τ a existing in the localized astronomical system which physical origin may be a relict of the primordial perturbations. We do not discuss this interesting scenario in the present paper as it would require a non-conservation of entropy and non-isentropic background fluid -the case which we have intentionally excluded in order to focus on derivation of cosmological generalization of the post-Newtonian equations of relativistic celestial mechanics [58].
F. Field Equations for Tensor Perturbations
Field equations for traceless tensor p ⊺ αβ can be derived by applying the covariant Laplace-Beltrami operator to the definition (171) and making use of equation (164) along with a tedious algebraic transformations. This yields the following equation
p ⊺ αβ |γ |γ − 2H ū α p ⊺ βγ |γ +ū β p ⊺ αγ |γ − 2 H 2 + k a 2 p ⊺ αβ −P α µP β ν B µ|ν + B ν|µ + 2 3P αβP µν B µ|ν = 16πτ ⊺ αβ . (207)
Here the transverse and traceless tensor source of the tensor perturbations is
τ ⊺ αβ ≡ τ αβ − 1 3P αβ τ ,(208)
where τ αβ has been introduced in (104d), and τ =P αβ τ αβ in accordance with equation (104b). Tensor τ ⊺ αβ is traceless, that isḡ αβ τ ⊺ αβ =P αβ τ ⊺ αβ = 0. Equation (207) is gauge-dependent. The gauge freedom is significantly reduced by imposing the gauge condition B α = 0 which brings equation (207) to the following form,
p ⊺ αβ |γ |γ − 2H ū α p ⊺ βγ |γ +ū β p ⊺ αγ |γ − 2 H 2 + k a 2 p ⊺ αβ = 16πτ ⊺ αβ .(209)
G. The Residual Gauge Freedom
The gauge freedom of the theory under discussion is associated with the gauge function B α appearing in equation (163). The most favourable choice of the gauge condition is
B α = 0 ,(210)
which drastically simplifies the above equations for vector and tensor gravitational perturbations. The gauge (210) is a generalization of the harmonic (de Donder) gauge condition used in the gravitational wave astronomy and in the post-Newtonian dynamics of extended bodies. This choice of the gauge establishes differential relationships between the algebraically-independent metric tensor components introduced in section VII A. Indeed, substituting the algebraic decomposition (174) of the metric tensor perturbations to equation (163) and imposing the condition (210) yields p ⊺ αβ |β +ū α p β |β +ū β p α |β − ū αūβ − 1 3P αβ p |β + 2ū αūβ q |β + 2Hp α + 2Hqū α = 16π (ρ mμm χ m +ρ qμq χ q )ū α . (211) Projecting this relationship on the direction of the background 4-velocity,ū α , and on the hypersurface being orthogonal to it, we derive two algebraically-independent equations between the perturbations of metric tensor components and of the matter variables. They are
p β |β +ū β (2q − p) |β + 2Hq = 16π (ρ mμm χ m +ρ qμq χ q ) ,(212a)p ⊺ αβ |β +ū β p α |β + 1 3P αβ p |β + 2Hp α = 0 .(212b)
The gauge (210) does not fix the gauge function ξ α uniquely. The residual gauge freedom is described by the gauge transformations that preserve equations (212a), (212b). Substituting the gauge transformation (176b) of the gravitational field perturbation l αβ to equation (163) and holding on the gauge condition (210), yields the differential equation for the vector function ξ α
ξ α|β |β +ḡ αγ ξ β |γβ − ξ β |βγ + 2H ξ α|βū β + ξ β|αū β − ξ β |βū α − 16π (ρ mμm +ρ qμq ) ξ βū βū α = 0 ,(213)
which can be further recast to
ξ α|β |β + 2H ξ α|βū β + ξ β|αū β − ξ β |βū α + 2 Ḣ − k a 2 ξ βū βū α + Ḣ + 3H 2 + 2k a 2 ξ α = 0 .(214)
The gauge function ξ α can be decomposed in time-like, ξ ≡ −ξ βū β , and space-like, ζ α ≡P α β ξ β , components,
ξ α = ζ α +ū α ξ .(215)
Calculating covariant derivatives from ξ and ζ α and making use of equation (214), yield equations
ξ |β |β + 2Hū β ξ |β − Ḣ − 4k a 2 ξ = 0 , (216a) ζ α|β |β + 2H ū β ζ α |β −ū α ζ β |β + Ḣ + H 2 + 2k a 2 ζ α = 0 .(216b)
These equations have non-trivial solutions which describe the residual gauge freedom in choosing the coordinates on the background manifold subject to the gauge condition (210). It is remarkable that equations (216a), (216b) are decoupled and can be solved separately. It means that the residual gauge transformations along the worldlines of the Hubble flow are functionally independent of those performed on the hypersurface being orthogonal to the Hubble flow.
VIII. POST-NEWTONIAN FIELD EQUATIONS IN A SPATIALLY-FLAT UNIVERSE
A. Cosmological Parameters and Scalar Field Potential
Equations of the field perturbations given in the previous section are generic and valid for any model of the FLRW universe. They neither specify the equation of state of matter, nor that of the scalar field, nor the parameter of the space curvature k. By choosing a specific model of matter and picking up a value of k = −1, 0, +1, we can solve, at least, in principle the field equations governing the time evolution of the background cosmological manifold. Realistic models of the cosmological matter are rather sophisticated and, as a rule, include several components. It leads to the system of the coupled field equations which can be solved only numerically. However, the large scale structure of the universe is formed at rather late stages of the cosmological evolution being fairly close to the present epoch. Therefore, the study of the impact of cosmological expansion on the post-Newtonian dynamics of isolated astronomical systems is based on the recent and present states of the universe.
Radiometric observations of the relic CMB radiation and photometry of type Ia supernova explosions reveal that at the present epoch the space curvature of the universe, k = 0, and the evolution of the universe is primarily governed by the dark energy and dark matter, which make up to 74% and 24% of the total energy density of the universe respectively, while 4% of the energy density of the universe belongs to visible matter (baryons), and a tiny fraction of the energy density occupies by the CMBR radiation [35,47,53,55]. It means that we can neglect the effects of the baryonic matter and CMB radiation field in consideration of the post-Newtonian dynamics of astronomical systems in the expanding universe.
We shall assume that the dark matter is made of an ideal fluid and the dark energy is represented by a scalar field with a potential functionW which structure should be further specified. In doing this, we shall follow discussion in [2] assuming that the spatial curvature k = 0, and the potential,W , of the scalar field relates to its derivative by a simple equation
∂W ∂Ψ = − √ 8πλW ,(217)
where the time-dependent parameter, λ = λ(Ψ), characterizes the slope of the field potentialW . The time evolution of the background universe can be described in terms of the parameter λ and two other parameters, x 1 = x 1 (Ψ) and x 2 = x 2 (Ψ), which are functions of the density,ρ q =μ q =Ψ, of the background scalar field, and the potential,W , scaled to the Hubble parameter, H. These parameters are defined more specifically as follows,
ρ 2 q = 3H 2 4π x 1 ,(218)W = 3H 2 8π x 2 .(219)
The energy density of the scalar field,ǭ q , is expressed in terms of the parameters x 1 and x 2 and the parameter Ω q ≡ 8πǭ q /3H 2 , by a simple relationship
Ω q = x 1 + x 2 .(220)
The time evolution of the parameters x 1 and x 2 is given by the system of two ordinary differential equations which are obtained by differentiating the definitions (218), (219) and making use of the equations (101) along with the Friedmann equation (87) with k = 0. It yields
dx 1 dω = −6x 1 + λ √ 6x 1 x 2 + 3x 1 [(1 − w m ) x 1 + (1 + w m ) (1 − x 2 )] ,(221a)dx 2 dω = −λ √ 6x 1 x 2 + 3x 2 [(1 − w m ) x 1 + (1 + w m ) (1 − x 2 )] ,(221b)
where ω ≡ ln a is the logarithmic scale factor characterizing the number of e-folding of the universe, w m is the parameter entering the hydrodynamic equation of state (89), and the parameters x 1 and x 2 are restricted by the condition imposed by the Friedmann equation (85), that is
x 1 + x 2 = 1 − Ω m ,(222)
where
Ω m ≡ 8πǭ m /3H 2 .
The parameter λ obeys the following equation
dλ dω = − √ 6x 1 λ 2 (Γ q − 1) ,(223)
where
Γ q = ∂ 2W /∂Ψ 2 ∂W /∂Ψ 2W ,(224)
If Γ q = 1, the parameter λ is constant, and equation (217) can be integrated yielding an exponential potential
W (Ψ) =W 0 exp(− √ 8πλΨ) .(225)
In this case, and under assumption that, w m = const., the system of two differential equations (221a), (221b) is closed. If Γ q = 1, three equations (221a)-(223) must be solved in order to describe the evolution of the background cosmological manifold.
In the general case, derivatives of the potentialW are expressed in terms of the parameters under discussion. Namely,
∂W ∂Ψ = − 3λ √ 8π H 2 x 2 , ∂ 2W ∂Ψ 2 = 3Γ q λ 2 H 2 x 2 .(226)
It is also useful to express the productsρ qμq andρ mμm in terms of the parameters x 1 and x 2 . Forμ q =ρ q , one can use definition (218) to obtainρ
qμq = 3H 2 4π x 1 .(227)
The productρ mμm =ǭ m +p m , so that making use of the matter equation of state,p m = w mǭm , and equation (222), we deriveρ
mμm = 3H 2 8π (1 + w m )Ω m ,(228)
where Ω m = 1 − x 1 − x 2 . These equations allow us to recast equation (87) for the time derivative of the Hubble parameter to the following formḢ
= − 3 2 (1 + w eff ) H 2 ,(229)
where
w eff ≡ w m + (1 − w m )x 1 − (1 + w m )x 2 ,(230)
is the (time-dependent) parameter of the effective equation of state of the mixture of the ideal fluid and the scalar field.
B. Conformal Cosmological Perturbations
The FLRW metric (52) is a product of the scale factor a and a conformal metricf αβ . The conformal spacetime is comoving with the Hubble flow and is not globally expanding. In case of the flat spatial curvature, k = 0, the conformal spacetime becomes equivalent to the Minkowski spacetime which is used as a starting point in the standard theory of the post-Newtonian or post-Minkowskian approximations [23]. Therefore, it is instructive to formulate the field equations for cosmological perturbations in the conformal spacetime.
Let us define the cosmological perturbations, h αβ , of gravitational field in the conformal spacetime with the background metricf αβ as follows,
κ αβ = a 2 (η)h αβ ,(231)
where perturbations κ αβ has been defined in (106), and a(η) is the scale factor of the FLRW universe. Perturbation l αβ relates to κ αβ by equation (115), and can be also represented in the conformal form
l αβ = a 2 (η)γ αβ ,(232)
where
γ αβ = −h αβ + 1 2f αβ h ,(233)
with h ≡f αβ h αβ . In what follows, tensor indices of geometric objects in the conformal spacetime are raised and lowered with the help of the conformal metricf αβ . We assume that the scale factor a of the universe remains unperturbed. This assumption is justified since we can always include the perturbation of the scale factor to that of the conformal metric. Thus, the perturbed physical spacetime interval, ds, of the FLRW universe relates to the perturbed conformal spacetime interval, ds, by the conformal transformation
ds 2 = a 2 (η)ds 2 .(234)
The perturbed conformal spacetime interval reads
ds 2 = f αβ dx α dx β ,(235)
where
f αβ =f αβ + h αβ ,(236)
is the perturbed conformal metric. Here,f αβ is the unperturbed conformal metric defined in (53), h αβ is the perturbation of the conformal metric, and x α = (x 0 , x i ) are arbitrary coordinates which are the same as in the physical spacetime manifold. It is worth emphasizing that in case of the space curvature k = 0, the background conformal metric, g αβ (η, X i ), expressed in the isotropic Cartesian coordinates (η, X i ), is the diagonal Minkowski metric, g αβ (η, X i ) = η αβ = diag (−1, 1, 1, 1). Therefore, the background metricf αβ remains the Minkowski metric with the components expressed in arbitrary coordinates by means of tensor transformation
f αβ = M µ α M ν β η µν ,(237)
where the matrix of transformation has been defined in (50). If the matrix of transformation, M µ α , is the Lorentz boost, the conformal metric,f αβ , remains flat,f αβ = η αβ . It is worth noticing that, in general, the unperturbed conformal metric can be chosen flat even in case of k = −1, +1 [49]. It means that our formalism is applicable to FLRW universe with any space curvature. However, the conformal factor in this case is not merely the scale factor a(η) of the FLRW universe but a more complicated function of coordinates. Though it is not difficult to handle all three cases of k = −1, 0, +1 but it burdens equations for the field perturbations and we restrict ourselves only to the case of the spatially flat universe with k = 0.
Similarly to (174) the conformal metric perturbation, γ αβ , can be split in the algebraically-irreducible components
γ αβ = p ⊺ αβ +v α p β +v β p α + v αvβ + 1 3π αβ p + 2v αvβ (q − p) ,(238)
where the four-velocityv α = aū α ,v α =f αβv β = a −1ḡ αβū β = a −1ū α , and
π αβ =f αβ +v αvβ ,(239)
is the operator of projection on the conformal space which represents a hypersurface being everywhere orthogonal to the congruence of worldlines of four-velocityv α . Four-velocityv α is an analogue of the Hubble flow in the conformal spacetime. We also notice thatP αβ = a 2π αβ . Different pieces of the conformal metric perturbation, γ αβ , are related to those of the physical metric perturbation, l αβ , by the powers of the scale factor,
p ⊺ αβ = a 2 p ⊺ αβ , p α = ap α , p = p , q = q .(240)
More specifically,
q = 1 2 (v µvν +π µν ) γ µν ,(241a)p =π µν γ µν ,(241b)p α = −π α βvγ γ βγ ,(241c)p ⊺ αβ = p αβ − 1 3π αβ p ,(241d)
where
p αβ =π α µπ β ν γ µν .(242)
The trace of the gravitational perturbation, γ =f αβ γ αβ = 2(p − q). The components h αβ = −γ αβ +f αβ γ/2 are used in calculating dynamical behavior of particles and light in the conformal spacetime as well as in matching theory with observables. The components of h αβ are
h αβ = −p ⊺ αβ −v α p β −v β p α + 2 3π αβ p − (v αvβ +π αβ ) q ,(243)H ′ = − 1 2 (1 + 3w eff )H 2 .(244)
We shall use this expression in the calculations that follows.
C. The Post-Newtonian Field Equations in Conformal Spacetime
The set of the post-Newtonian field equations in cosmology consists of equations for perturbations of the background matter and gravitational field. Perturbations of matter are described by four scalars, V m , V q , χ m and χ q but only three of them are functionally-independent because of equality (203), that is
V m − V q =ū α (χ m − χ q ) |α .(245)
Depending on a particular situation, any of the three scalars can be taken as independent variables. The gravitational field perturbations are q, p, p α , p ⊺ αβ but among them the scalar q is not independent and can be expressed either in terms of χ m and V m in accordance with (180a),
q = −2 V m −ū α χ m,α ,(246)
where we have also used the equality q = q as follows from (240). The scalar q can be also expressed in terms of χ q and V q in accordance with (180b). Hence, as soon as the pairs, V m and χ m or V q and χ q are known, the scalar gravitational perturbation q can be easily calculated from (246). Functions p, p α , p ⊺ αβ are independent and decouple both from each other and from the other perturbations. Thus, the most difficult part of the theory is to find out solutions of the scalar perturbations which are coupled one to another.
The field equations in the conformal spacetime for variables χ m , χ q , V m and for p, p α , p ⊺ αβ are derived from the equations of the previous section by transforming all functions and operators from physical to conformal spacetime. The important part of the transformation technique is based on formulas converting the covariant Laplace-Beltrami wave operators, defined on the background spacetime manifold, to their conformal spacetime counterparts.
The Laplace-Beltrami operator
Let F be an arbitrary scalar, F α -an arbitrary covector, and F αβ -an arbitrary covariant tensor of the second rank. We have three Laplace-Beltrami operators on the curved background manifold: scalar -F |µ |µ , vector -F α |µ |µ , and tensor -F αβ |µ |µ types where the covariant derivatives are taken with respect to the affine connectionΓ α βγ being compatible with the metricḡ αβ (see equation 58). Covariant derivatives are the most convenient for the invariant description of differential equations of mathematical physics on curved manifolds. For practical purposes of finding solutions of the differential equations, the covariant operators must be expressed in terms of partial derivatives with respect to coordinates chosen for solving the equations.
Transformation of the covariant Laplace-Beltrami operators to the partial derivatives is achieved after writing down the covariant derivatives for scalar, vector and tensor in explicit form by making use of the Christoffel symbols given in (60)- (62). Tedious but straightforward calculations of the covariant derivatives yield the scalar, vector and tensor Laplace-Beltrami operators in the following form
F |µ |µ = 1 a 2 F − 2Hv µ F ;µ , (247a) F α |µ |µ = 1 a 2 F α − 2Hv µ F µ;α + 2Hv αf µν F µ;ν + H ′ + 2H 2 F α − 2H 2v αv µ F µ ,(247b)F αβ |µ |µ = 1 a 2 F αβ + 2Hv µ F αβ;µ − 2Hv µ F µα;β − 2Hv µ F µβ;α + 2Hf µν (v α F βµ;ν +v β F αµ;ν ) (247c) + 2 H ′ + H 2 F αβ − 4H 2 v µv α F βµ +v µv β F αµ − 1 2v αvβf µν F µν − 1 2f αβv µvν F µν ,
where we have introduced notations
F ≡f µν F ;µν , F α ≡f µν F α;µν , F αβ ≡f µν F αβ;µν ,(248)
of the wave operators for scalar, vector and tensor fields in the conformal spacetime and in arbitrary coordinates. Notice that although the conformal spacetime coincides, in case of k = 0, with the Minkowski spacetime, the metric f αβ is not the diagonal Minkowski metric η αβ unless the coordinates are Cartesian. Of course, the covariant derivative from a scalar must be understood as a partial derivative, that is F ;α = F ,α . We will need several other equations to complete the transformation of the Laplace-Beltrami operators to the conformal spacetime since the wave operator acts on functions like (240) which are made of a product of some power n of the scale factor, a = a(η), with a geometric object, ̥ = ̥(x α ), which can be a scalar, a vector or a tensor of the second rank (we have suppressed the tensor indices of ̥ since they do not interfere with the derivation of the equations which follow). These equations are (a n ̥) ;µ = a n (̥ ;µ − nHv µ ̥) ,
(a n ̥) ;µν = a n ̥ ;µν − nH (v µ ̥ ;ν +v ν ̥ ;µ ) + n H ′ + nH 2 v µvν ,
and they allow us to write down the wave operator from the product of a n and ̥ in the following form (a n ̥) = a n ̥ − 2nHv µ ̥ ;µ − n H ′ + nH 2 ̥ ,
It is easy to confirm that contraction of (249b) with the conformal four-velocity,v α , brings about another differential operatorv µvν (a n ̥) ;µν = a n v µvν ̥ ;µν + 2nHv µ ̥ ;µ + n H ′ + nH 2 ̥ .
We remind that if the object ̥ is a scalar, the covariant derivative is reduced to a partial derivative, ̥ ;α = ̥ ,α . In case, when ̥ is either a vector or a tensor, the covariant derivative must be calculated with taking into account the affine connectionB α βγ defined in (62). It is also interesting to notice that in the expanding universe the conformal Laplace operator, ∆̥ ≡π µν ̥ ;µν is the scale invariant in the sense that ∆ (a n ̥) = a n ∆̥ ,
where ̥ is a tensor of an arbitrary rank. Equation (252) can be proven by adding up (250) and (251), and accounting for definition (239) of the projection operator on the hypersurface being orthogonal tov α . Now, we are ready to formulate the field equations for cosmological perturbations in the conformal spacetime.
Equations for the matter perturbations
We accept the gauge condition imposed by equations (163), (210) and convert the covariant derivatives taken with respect to the background metric,ḡ αβ , to the partial derivatives of the conformally-flat metric,f αβ , in equation (201) for scalar V m . We use equation (247a) for the Laplace-Beltrami operator, and expressions for various cosmological parameters given in section VIII B. After arranging terms with respect to the powers of the Hubble parameter H, we obtain the sound-wave equation for function V m describing perturbations of the ideal fluid,
V m + 1 − c 2 c 2 s v αvβ V m;αβ + 3 − c 2 c 2 s Hv α V m,α (253) +3 1 − w eff − 1 2 (1 + w m ) 1 − c 2 c 2 s Ω m H 2 V m +12H 2 v α χ ,α − 3 1 − 3 8x 1 λx 2 Hχ x 1 a = −4πa 2 (σ + τ ) .
Similar procedure applied to equation (204) leads to a wave equation for function V q describing perturbations of the scalar field,
V q + 2 1 − 3 2x 1 λx 2 Hv µ V q,µ (254) +3 1 − w eff − 1 2 (1 + w m ) 1 − c 2 c 2 s Ω m H 2 V q +λx 2 3λ 2Γ q + x 2 x 1 − 5 6 x 1 H 2 V q + 3 2 H 2 (1 + w m ) 3 + c 2 c 2 s v µ χ ,µ − 3 c 2 s c 2 Hχ Ω m a = −4πa 2 (σ + τ ) .
Equations (253) and (254) contains function χ which obeys equation (196). Making use of the same transformations as above, we recast (196) to a wave equation for χ,
χ + 4H 1 − 3 8x 1 λx 2 v α χ ,α − 9 2 (1 + w eff ) H 2 χ = −a 6 x 1 λx 2 HV m + 1 − c 2 c 2 s v α V m,α .(255)
Equations (253)-(255) are closed with respect to the variables V m , V q and χ. The gauge-invariant scalar, V m describes propagation of sound waves in the ideal fluid filling up the expanding universe. It can be found from solving two equations (253) and (255) simultaneously after imposing a certain (cosmological) boundary conditions. As soon as the gauge-invariant scalar χ is known, the potential, V q , can be determined as a particular solution of the inhomogeneous equation (254) or, more simple, from equation (203).
We also need equations for the normalized Clebsch and scalar potentials, χ m and χ q . These potentials are required to determine the gravitational perturbation, q, with the help of (246) and/or to get the check on self-consistency of the solutions of equations in the matter sector of the perturbation theory. Conformal-spacetime equations for χ m and χ q are derived from their definition (177) and the field equations (166) and (167). They are
χ m + 3 2 (1 + w eff ) H 2 χ m = 12H 2 x 1 χ − a 4HV m + 1 − c 2 c 2 s v α V m,α ,(256)χ q + 3 2 (1 + w eff ) H 2 χ q = −6H 2 (1 + w m )Ω m χ − a 4 − 6 x 1 λx 2 HV q .(257)
By subtracting one of these equations from another, we get back to equation (255).
Equations for the metric perturbations
Post-Newtonian equations for gravitational perturbations in physical spacetime are (192), (194), (205) and (207). We remind to the reader that the gauge conditions (163), (210) has been imposed. In this gauge, equations for the conformal metric tensor perturbations become
q − 2Hv α q ,α + (1 + 3w eff ) H 2 q = 8πa 2 (σ + τ ) − 24H 2 3x 1 8 λx 2 − Hx 1 χ q a (258a) −3 (1 + w eff ) H 2 Ω m 1 − c 2 c 2 s V m − H 1 + 3 c 2 s c 2 χ m a , p − 2Hv α p ,α = 16πa 2 τ ,(258b)p α − 2Hv β p α;β + (1 + 3w eff ) H 2 p α = 16πaτ α , (258c) p ⊺ αβ − 2Hv γ p αβ;γ = 16πτ ⊺ αβ .(258d)
The reader can observe that equations (258a)-(258d) for linearized metric perturbations are decoupled from each other. Moreover, equations (258b-(258d) are decoupled from the matter perturbations V m , χ m , etc. Only equation (258a) for q is coupled with the matter perturbations governed by equations (253), (256), (257) so that these equations should be solved together. As we have mentioned above, function q is a linear combination of V m and χ m according to (246). Hence, in order to determine q it is, in fact, sufficient to solve (253) and (256). Nevertheless, it is convenient to present the differential equation (258a) for q explicitly for the sake of mathematical completeness and rigour. It can be used for independent validation of the solution of the system of equations (253), (256) and (246). Unfortunately, these equations are strongly mixed up and cannot be solved analytically in the most general situation of a multi-component background universe governed by the dark energy and dark matter. Solution of (253)-(257) and (258a)-(258d) would require an application of the methods of numerical integration. It would be instrumental to get better insight to the post-Newtonian theory of cosmological perturbations by making some simplifying assumptions about the background model of the expanding universe in order to decouple the system of the post-Newtonian equations and to find their analytic solution explicitly. We discuss these assumptions and the corresponding system of the decoupled post-Newtonian equations in the section IX.
D. The Residual Gauge Freedom in the Conformal Spacetime
The gauge conditions (163), (210) in the physical space are given by (212a), (212b). After transforming to the conformal spacetime the gauge conditions read
p β ;β +v β (2q − p) ,β + 2Hq = 16πa (ρ mμm χ m +ρ qμq χ q ) ,(259a)p ⊺αβ ;β +v β p α ;β + 1 3π αβ p ,β + 2Hp α = 0 .(259b)
The residual gauge freedom in the conformal spacetime is described by two functions, ζ ≡ ξ/a and ζ α , where ξ and ζ α have been defined in section VII G. Differential equations for ζ and ζ α are obtained by making transformation of equations (216a), (216b) to the conformal spacetime. The calculation is straightforward and results in
ζ − 2Hv β ζ ,β + (1 + 3w eff ) H 2 ζ = 0 ,(260a)ζ α − 2Hv β ζ α ;β = 0 .(260b)
Solutions of equations (258a)-(258d) are determined up to the gauge transformationŝ q = q + 2v α ζ ,α + 2Hζ , (261a) p = p + ζ α ;α + 3v α ζ ,α + 6Hζ , (261b)
p α = p α +π αβ v γ ζ β ;γ − ζ ,β + 2Hζ β , (261c) p αβ = p αβ − (π µαπβ ν +π µβπα ν ) ζ µ ;ν +π αβ (ζ α ;α +v α ζ ,α + 2Hζ) ,(261d)
where the gauge functions ζ, ζ α are solutions of the differential equations (260a), (260b).
IX. THE DECOUPLED SYSTEMS OF THE POST-NEWTONIAN FIELD EQUATIONS
A. The Universe Governed by the Ideal Fluid and Cosmological Constant
Case 1: Arbitrary equation of state of the ideal fluid
Let us consider a special case of the dark energy represented by the cosmological constant Λ. In this case, the equation of state of the scalar field is w q = 1, and we haveρ qμq =ǭ q +p q = 0. The parameter x 1 = 0, and x 2 = Λ/(3H 2 ). It yields the parameter Ω q = x 2 , and Ω m = 1 − x 2 . Since the cosmological constant corresponds to a constant potentialW of the scalar field, we get for its derivative ∂W /∂Ψ = 0, and equation (217) points out that the parameter λ = 0.
In the universe governed by the ideal fluid and the cosmological constant the parameter of the effective equation of state
w eff = w m − (1 + w m ) Λ 3H 2 .(262)
Hence, the time derivative of the Hubble parameter defined in (229), is reduced to a more simple expression,
H = 1 2 (1 + w m ) Λ − 3H 2 .(263)
On the other hand, equation (87) tells us that in this model of the universė
H = −4πρ mμm ,(264)
The field equation (253) for scalar V m is reduced to that describing the time evolution of the perturbation of the ideal fluid density, δρ m . Indeed, the scalar V m defined by equation (180a), can be recast to the form given by equation (189), that is
V m = c 2 s c 2 δ m ,(265)
where the gauge-invariant scalar perturbation Equation (255) for potential χ makes no sense since the normalized perturbation χ q = ψ/μ q of the scalar field diverges due to the condition µ q = ρ q = 0. Equation for the perturbation of the scalar field ψ itself is obtained from (167) and is reduced to a homogeneous wave equation
δ m ≡ δρ m ρ m + 3Hχ m ,(266)ψ − 2Hv µ ψ ,µ = 0 .(268)
Equation for the normalized Clebsch potential, χ m , is derived from equation (256). In the case of the universe under consideration this equation reads
χ m + 1 2 (1 + w m ) 3H 2 − a 2 Λ χ m = 1 − c 2 s c 2 av µ δ m,µ − 4aH c 2 s c 2 δ m .(269)
This is an inhomogeneous equation that can be solved as soon as one knows δ m from equation (267). Gravitational potential q can be determined directly from equation (246) after solving equations (267) and (269) or by solving equation (258a) which takes on the following form,
q − 2Hv µ q ,µ + (1 + 3w m ) H 2 − (1 + w m ) a 2 Λ q = 8πa 2 σ + τ +ρ mμm 1 − c 2 s c 2 δ m + H 1 + 3 c 2 s c 2 χ m a .
(270a) Equations for the remaining gravitational perturbations are found from (258b)-(258d) which read
p − 2Hv µ p ,µ = 16πa 2 τ ,(270b)p α − 2Hv µ p α;µ + (1 + 3w m ) H 2 − (1 + w m ) a 2 Λ p α = 16πaτ α , (270c) p ⊺ αβ − 2Hv µ p ⊺ αβ;µ = 16πτ ⊺ αβ .(270d)
Equations given in this section are valid for arbitrary cosmological equation of state of the ideal fluid,p m = w mǭm , that is physically reasonable. The parameter w m of the equation of state should not be replaced with the ratio of c 2 s /c 2 which characterizes the derivative of pressurep m with respect to the energy densityǭ. This is because the parameter w m can depend in the most general case on the other thermodynamic quantities (like enthropy, etc.) which may implicitly depend onǭ. Equations (267)-(270d) are decoupled in the sense that all of them can be solved one after another starting from solving equation (267) for δ m , which is a primary equation.
Case 2: Dust equation of state
Equations of the previous section can be further simplified for some particular equations of state of the ideal fluid. For example, in the case when the ideal fluid is made of dust, the background pressure of matter drops to zero making parameter of the equation of state w m = 0. Sound waves do not propagate in dust. Hence, the speed of sound c s = 0. For this reason all terms being proportional to c 2 s and w m vanish in equation (267). Moreover, dust has the specific enthalpy, µ m = 1 making the energy density of dust equal to its rest mass densityǭ m =ρ m , and the normalized perturbation χ m of the Clebsch potential of dust is equal to the perturbation φ of the Clebsch potential itself, χ m = φ. The Friedmann equation (85) tells us that
H 2 = a 2 3 (8πρ m + Λ) .(271)
Accounting for this result in equation (267), and neglecting all terms being proportional to the speed of sound, c s , we obtainv
αvβ δ m;αβ + Hv α δ m,α − 4πa 2ρ m δ m = 4πa 2 (σ + τ ) ,(272)
where the terms depending on the cosmological constant, Λ, have cancelled out. This equation is more familiar when is written down in the preferred FLRW frame, wherev α = (1, 0, 0, 0). Equation (272) assumes the "canonical" form
δ m + Hδ m − 4πa 2ρ m δ m = 4πa 2 (σ + τ ) ,(273)
which can be found in many textbooks on cosmology [71,81,84,99,100]. Equation (273) has been derived by previous researchers without resorting to the concept of the Clebsch potential of the ideal fluid. For this reason, the density contrast, δ m , was interpreted as the ratio of the perturbation of the dust density to its background value, δ = δρ m /ρ m , without taking into account the perturbation, φ, of the Clebsch potential. However, the quantity δ is not gauge-invariant which was considered as a drawback. The scrutiny analysis of the underlying principles of hydrodynamics in the expanding universe given in the present paper, reveals that equation (273) is, in fact, valid for the gauge-invariant density perturbation δ m defined above in (266). Another distinctive feature of equation (273) is the presence of the source of a bare perturbation in its right side. The bare perturbation is caused by the effective density σ + τ of the matter which comprises the isolated astronomical system and initiates the growth of instability in the cosmological matter that, in its own turn, induces formation of the large scale structure of the universe [84,100]. Standard approach to cosmological perturbation theory always set σ + τ = 0 and operates with the spectrum of the primordial perturbation of the density δρ m /ρ m (but not with the spectrum for δ m ).
Equation (269) in case of dust reads,
χ m + 1 2 3H 2 − a 2 Λ χ m = av α δ m,α ,(274)
where χ m is reduced to the perturbation of the Clebsch potential, χ m = φ, for the reason that has been mentioned above.
If equations (273) and (274) are solved, the gravitational perturbations can be found from equations (270a)-(270d), which take on the following form
q − 2Hv α q ,α + H 2 − a 2 Λ q = 8πa 2 σ + τ +ρ m δ m + H χ m a , (275a) p − 2Hv α p ,α = 16πa 2 τ ,(275b)p α − 2Hv β p α;β + H 2 − a 2 Λ p α = 16πaτ α , (275c) p ⊺ αβ − 2Hv γ p ⊺ αβ;γ = 16πτ ⊺ αβ . (275d)
It is interesting to notice that besides the bare density perturbation, σ + τ , the source for the scalar gravitational perturbation, q, contains the induced density perturbationρ m (δ m + Hχ m /a) = δρ m + Hρ m φ in the right side of equation (275a). This induced density perturbation changes the initial mass of the isolated astronomical system in the course of the evolution (expansion) of the universe. This explains the origin of the time-dependence of the central point-like mass in the cosmological solution found by McVittie [78] (see also discussion in [16]).
B. The Universe Governed by a Scalar Field
In this section we explore the case of the universe governed primarily by a scalar field with all other matter variables being unimportant. In this case, the time evolution of the background universe is defined exceptionally by equations (221a), (221b). The most general solution of (221a), (221b) is complicated and can not be achieved analytically. Numerical analysis shows that the solution evolves in the phase space of the two variables {x 1 , x 2 } from an unstable to a stable fixed point by passing through a saddle point [2]. The cosmic acceleration is realized by the stable point with the values of x 1 = λ 2 /6 and x 2 = 1 − λ 2 /6, which is equivalent to the equations of state (89) with the values of the parameters, w m = 0, and, w q = −1 + λ 2 /3. It also requires the energy density of the background matterǭ m = 0, that is Ω m = 0. In such a universe the derivatives of the potential of the scalar field are
1 µ q ∂W ∂Ψ = − 3 2 H (1 − w q ) , ∂ 2W ∂Ψ 2 = 9 2 H 2 1 − w 2 q .(276)
Moreover, becauseρ mμm =ǭ m +p m = 0, the time derivative of the Hubble parameter iṡ
H = −4πρ qμq = − 3 2 H 2 (1 + w q ) .(277)
In the point of the attractor of the scalar field, perturbations of the ideal fluid are fully suppressed that is the Clebsch potential of the fluid, χ m = 0. It makes the function V m = q/2, that is reduced to the perturbation of the scalar component of the gravitational field only. Perturbations of the scalar field are described by the scalar field variable, χ q . In particular, after substituting the derivatives (276) of the scalar field potential along with the derivative (277) of the Hubble parameter, in equation (204), one obtains the post-Newtonian equation for function V q ,
V q − (1 − 3w q ) Hv µ V q,µ + 3 2 H 2 (1 − w q ) (1 + 3w q ) V q = −4πa 2 (σ + τ ) .(278)
Field equation for the perturbation of the scalar field, χ q , is reduced to
χ q − 2Hv µχ q,µ + H 2 (1 + 3w q )χ q = − (1 + 3w q ) HV q ,(279)
where the variable,χ q ≡ χ q /a, has been used for the notational convenience. Post-Newtonian equations for gravitational perturbations are (258a)-(258d). After substituting the values of the parameters x 1 , x 2 , w eff , etc., corresponding to the model of the universe governed by the scalar field alone, the post-Newtonian equations for the metric perturbations become
q − 2Hv µ q ,µ + (1 + 3w q ) H 2 q = 8πa 2 (σ + τ ) + 3 (1 + w q ) (1 + 3w q ) H 3χ q , (280a) p − 2Hv µ p ,µ = 16πa 2 τ ,(280b)p α − 2Hv µ p α;µ + (1 + 3w q ) H 2 p α = 16πaτ α , (280c) p ⊺ αβ − 2Hv µ p ⊺ αβ;µ = 16πτ ⊺ αβ .(280d)
One can see that the field equations for the scalar field and metric perturbations are decoupled, and can be solved separately starting from the primary equation (278).
C. Post-Newtonian Potentials in the Linearized Hubble Approximation
The metric tensor perturbations
The post-Newtonian equations for cosmological perturbations of gravitational and matter field variables crucially depend on the equation of state of the matter fields in the background universe. It determines the time evolution of the scale factor a = a(η) and the Hubble parameter H = H(η) which are described by the wide range of elementary and special functions of mathematical physics (see, for example, the books by Amendola and Tsujikawa [2], Macías, A., Cervantes-Cota, J. L. & Lämmerzahl, C. [75], Stephani et al. [94] and references therein). It is not the goal of the present article to provide the reader with an exhaustive list of the mathematical solutions of the perturbed equations which requires a meticulous development of cosmological Green's function (see, for example, [45,68,69,86]).
In this section we shall focus on the observation that the post-Newtonian equations for the field perturbations have identical mathematical structure if all terms that are quadratic with respect to the Hubble parameter, H, are neglected. In such a linearized Hubble approximation the differential equations for cosmological perturbations are not only decoupled from one another, but their generic solution can be found irrespectively of the equation of state governing the background universe. Indeed, if we neglect all quadratic with respect to H terms, the field equations for the conformal metric perturbations are reduced to the following set, q − 2Hv α q ,α = 8πa 2 (σ + τ ) ,
p − 2Hv α p ,α = 16πa 2 τ , (281b)
p α − 2Hv β p α;β = 16πaτ α , (281c) p ⊺ αβ − 2Hv γ p ⊺ αβ;γ = 16πτ ⊺ αβ ,(281d)
where the wave operator has been defined in (248), and the source of the perturbation is the tensor of energymomentum of a localized astronomical system with the matter having a bounded support in space -see section (V H). The differential structure of the left side of equations (281a)-(281d) is the same for all functions. The equations differ from each other only in terms of the order of H 2 which have been omitted.
In order to bring equations (281a)-(281d) to a solvable form, we resort to relationship (250) which reveals that in the linearized Hubble approximation, the equations can be reduced to the form of a wave equation (aq) = 8πa 3 (σ + τ ) , (282a) (ap) = 16πa 3 τ , (282b) (ap α ) = 16πa 2 τ α , (282c)
ap ⊺ αβ = 16πaτ ⊺ αβ .(282d)
So far, we did not impose any limitations on the curvature of space that can take three values: k = {−1, 0, +1}. Solution of wave equations (282a)-(282d) can be given in terms of special functions in case of the Riemann (k = +1) or the Lobachevsky (k = −1) geometry [68,69]. The case of the spatial Euclidean geometry (k = 0) is more manageable, and will be discussed below.
If the FLRW universe is spatially-flat universe, k = 0, and we chose the Cartesian coordinates x α related to the isotropic coordinates X α of the FLRW universe by a Lorentz transformation, the operator becomes a wave operator in the Minkowski spacetime, = η µν ∂ µν .
(283)
In this case, equations (282a)-(282d) are reduced to the inhomogeneous wave equations which solution depends essentially on the boundary conditions imposed on the metric tensor perturbations at conformal past-null infinity J − of the cosmological manifold [79]. We shall assume a no-incoming radiation condition also known as Fock-Sommerfeld's condition [23,36] lim r→+∞ t+r=const.
n γ ∂ γ [a(η)rl αβ (x γ )] = 0 ,(284)
where x γ = (x 0 , x i ), η = η(x γ ), the null vector n α = {1, x i /r}, and r = δ ij x i x j is the radial distance. This condition ensures that there is no infalling gravitational radiation arriving to the localized astronomical system from the future null infinity J + . Effectively, it singles out the retarded solution of the wave equation.
A particular solution of the wave equations satisfying condition (284), is the retarded integral [65] aq(t,
x) = −2 V a 3 [η (s, x ′ )] [σ (s, x ′ ) + τ (s, x ′ )] d 3 x ′ |x − x ′ | ,(285a)ap(t, x) = −4 V a 3 [η(s, x ′ )] τ (s, x ′ ) d 3 x ′ |x − x ′ | , (285b) ap α (t, x) = −4 V a 2 [η(s, x ′ )] τ α (s, x ′ ) d 3 x ′ |x − x ′ | , (285c) ap ⊺ αβ (t, x) = −4 V a [η(s, x ′ )] τ ⊺ αβ (s, x ′ ) d 3 x ′ |x − x ′ | ,(285d)
where the scale factor a in the left side of all equations is a ≡ a [η(s, x)], and the argument s of the functions appearing in the integrands, is the retarded time
s = t − |x − x ′ | .(286)
The retarded time s is a characteristic of the null cone in the conformal Minkowski spacetime that determines the causal nature of the gravitational field of the localized astronomical system in the expanding universe with k = 0 [58]. Solutions (285a)-(285d) are Lorentz-invariant as shown by calculations in Appendix A. Integration in (285a)-(285d) is performed over the volume, V, occupied by the matter of the localized astronomical system. In case of the system comprised of N massive bodies that are separated by distances being much larger than their characteristic size, the matter occupies the volumes of the bodies. In this case the integration in equations (285a)-(285d) is practically performed over the volumes of the bodies. It means that each post-Newtonian potential q, p, p α , p ⊺ αβ is split in the algebraic sum of N pieces
q = N A=1 q A , p = N A=1 p A , p α = N A=1 p Aα , p ⊺ αβ = N A=1 p ⊺ Aαβ ,(287)
where each function with sub-index A has the same form as one of the corresponding equations (285a)-(285d) with the integration performed over the volume, V A , of the body A.
The gauge functions
The residual gauge freedom describe the arbitrariness in adding solution of homogeneous equations (285a)-(285d) with the right side being equal to zero. It is described by two functions, ζ ≡ ξ/a and ζ α . Since we neglected the terms being quadratic with respect to the Hubble parameter, the gauge functions satisfy the following equations ζ − 2Hv β ζ ,β = 0 ,
ζ α − 2Hv β ζ α ;β = 0 .
They are equivalent to the homogeneous wave equations in the conformal flat spacetime (aζ) = 0 , (aζ α ) = 0 ,
which point out that (in the approximation under consideration) the products, aζ and aζ α , are the harmonic functions. Potentials q, p, p α , p ⊺ αβ must satisfy the gauge conditions (259a), (259b). Neglecting terms being quadratic with respect to the Hubble parameter, the gauge conditions (259a), (259b) can be written down as follows (ap α ) ,α +v α (2aq − ap) ,α + Hap = 0 ,
ap ⊺αβ ,β +v β (ap α ) ,β + 1 3π αβ (ap) ,β + Hap α = 0 ,
where the potentials p α and p ⊺αβ are obtained from p α and p ⊺ αβ by rising the indices with the Minkowski metric and taking into account that the indices of τ α and τ ⊺ αβ in the integrands of (285c) and (285d) should be raised with the full background metricḡ αβ = a −2 η αβ taken at the point of integration. It yields
ap α (t, x) = −4 V a 4 [η(s, x ′ )] τ α (s, x ′ ) d 3 x ′ |x − x ′ | ,(291a)ap ⊺αβ (t, x) = −4 V a 5 [η(s, x ′ )] τ ⊺αβ (s, x ′ ) d 3 x ′ |x − x ′ | .(291b)
It is instrumental to write down solutions for the products of the potentials p and p α = η αβ p β with the Hubble parameter. Multiplying both sides of equations (282b), (282c) with the Hubble parameter H, and neglecting the quadratic with respect to H terms, we obtain (Hap) = 16πa 3 Hτ , (Hap α ) = 16πa 4 Hτ α ,
which solutions are the retarded potential
Hap(t, x) = −4 V a 3 [η(s, x ′ )] H [η(s, x ′ )] τ (s, x ′ ) d 3 x ′ |x − x ′ | ,(293a)Hap α (t, x) = −4 V a 4 [η(s, x ′ )] H [η(s, x ′ )] τ (s, x ′ ) d 3 x ′ |x − x ′ | .(293b)
Substituting functions q, p, p α , p ⊺αβ and Hap, Hap α to the gauge equations (290a), (290b), bring about the following integral equations V a 4 τ α +v α a 3 σ ,α + a 3 Hτ
d 3 x ′ |x − x ′ | = 0 , (294a) V a 5 τ ⊺αβ + a 4vβ τ α + 1 3π αβ a 3 τ ,β + a 4 Hτ α d 3 x ′ |x − x ′ | = 0 ,(294b)
where all functions in the integrands are taken at the retarded time s and at the point x ′ , for example, a = a[η(s, x ′ )], H = H[η(s, x ′ )], σ = σ[(s, x ′ )], and so on. These equations are satisfied by the equations of motion (105a), (105b) of the localized matter distribution. Indeed, divergences of any vector F α and a symmetric tensor F αβ obey the following equalities
F α |α = 1 √ −ḡ √ −ḡF α ,α ,(295)F αβ |β = 1 √ −ḡ √ −ḡF αβ ,β +Γ α βγ F βγ .(296)
Moreover, the root square of the determinant of the background metric tensor is expressed in terms of the scale factor, √ −ḡ = a 4 , while the four-velocityū α =v α /a. Applying these expressions along with equations (295), (296) in equations of motion (105a), (105b), transforms them to a 4 τ α +v α a 3 σ ,α + a 3 Hτ = 0 , (297a) a 4 τ αβ + a 3vβ τ α ,β + 2a 3 Hτ α = 0 .
Equation (297a) proves that the integral equation (294a) and, hence, the gauge condition (290a) are valid. In order to prove the second integral equation (294b), we multiply equation (297b) with the scale factor a, and reshuffle its terms. It brings (297b) to the following form a 5 τ αβ + a 4vβ τ α ,β + a 4 Hτ α = 0 .
Applying relationship (250) in (306) allows us to recast it to 1 a n (a n V q ) + 2 n + 1 − 3 2x 1 λx 2 Hv α V m,α = −4πa 2 (σ + τ ) .
Choosing, n = n q = −1 + 3/(2x 1 )λx 2 , eradicates the second term in the left side of (307) that results in (a nq V q ) = −4πa 2+nq (σ + τ ) .
This is the wave equation in flat spacetime. We pick up the retarded solution as the most physical one. It reads
a nq V q = V a 2+nq (s, x ′ ) [σ (s, x ′ ) + τ (s, x ′ )] d 3 x ′ |x − x ′ | ,(309)
where the retarded time s has been defined in (286). Perturbations χ m and χ q can be found by integrating equations (180a) and (180b) that can be written as v α χ m,α = a V m + q 2
,v α χ q,α = a V q + q 2 .
(310)
These are the ordinary differential equations of the first order. Their solutions are
χ m = t t0 a[t, x(t)]{V m [t, x(t)] + 1 2 q[t, x(t)]}dt ,(311a)χ q = t t0 a[t, x(t)]{V q [t, x(t)] + 1 2 q[t, x(t)]}dt , .(311b)
where t 0 is an initial epoch of integration, and the integration is performed along the characteristics of the unperturbed equations of motion of matter of the background universe
dx i dt =v i (t, x) .(312)
These characteristics make up the Hubble flow. Therefore, the most simple way to integrate equations (310) would be to work in the preferred coordinate frame X α = (η, X i ) where the velocityv i = 0, and the coordinates X i = const. After the calculations in the rest frame of the Hubble flow are finished, the transformation to a moving frame can be done with the Lorentz boost.
where the matrix of the Lorentz boost [79] Λ 0 0 = γ ,
Λ i 0 = Λ 0 i = −γβ i , Λ i j = δ ij + γ − 1 β 2 β i β j ,(A5)
the boost four-velocity u α = {u 0 , u i } = u 0 {1, β i } is constant, and
γ = u 0 = 1 1 − β 2 ,(A6)
is the constant Lorentz-factor. The inverse Lorentz transformation is given explicitly as follows
η = γ(t + β · x) ,(A7)X = r + γ 2 1 + γ (β · r)b ,(A8)
where
r = x + βt ,(A9)
and the boost three-velocity, β = {β i } = {u i /u 0 }. Let us reiterate (A2) by introducing a one-dimensional Dirac's delta function and integration with respect to time η,
V (η, X) = ∞ −∞ V σ X (η ′ , X ′ )δ(η ′ − ζ) dη ′ d 3 X ′ |X − X ′ | ,(A10)
where ζ is the retarded time given by (A3). Then, we transform coordinates X ′α = (η ′ , X ′ ) to x ′α = (t ′ , x ′ ) with the Lorentz boost (A4). The Lorentz transformation changes functions entering the integrand of (A10) as follows,
σ(η ′ , X ′ ) = σ x (t ′ , x ′ ) ,(A11)|X − X ′ | = |r − r ′ | 2 + γ 2 [β · (r − r ′ )] 2 ,(A12)
where the coordinate difference
r − r ′ = x − x ′ + β(t − t ′ ) .(A13)
The coordinate volume of integration remains Lorentz-invariant
dη ′ d 3 X ′ = dt ′ d 3 x ′ .(A14)
Let us denote F η (η ′ ) ≡ η ′ − ζ where ζ is given by (A3). After making the Lorentz transformation this function changes to
F η (η ′ ) = F t (t ′ ) = γ [t ′ − t − β · (x − x ′ )] + |x − x ′ | 2 − (t ′ − t) 2 + γ 2 [β · (x − x ′ ) − (t ′ − t)] 2 ,(A15)
where we have used equations (A7), (A8) and (A12) and relationship γ 2 β 2 = γ 2 − 1, to perform the transformation. Integral (A10) in coordinates x α becomes
V (t, x) = ∞ −∞ V σ x (t ′ , x ′ )δ(F t (t ′ )) dt ′ d 3 x ′ |r − r ′ | 2 + γ 2 [β · (r − r ′ )] 2 ,(A16)
The delta function has a complicated argument F t (t ′ ) in coordinates x α . It can be simplified with a well-known formula
δ [F t (t ′ )] = δ(t ′ − s) F t (s) ,(A17)
whereḞ t (s) ≡ [dF t (t ′ )/dt ′ ] t ′ =s , and s is one of the roots of equation F t (t ′ ) = 0 that is associated with the retarded interaction. It is straightforward to confirm by inspection that the root is given by formula,
s = t − |x − x ′ | .(A18)
The time derivative of function F t (t ′ ) iṡ
F t (t ′ ) = γ + γ 2 β 2 (t ′ − t) − β · (x − x ′ ) |x − x ′ | 2 − (t ′ − t) 2 + γ 2 [β · (x − x ′ ) − (t ′ − t)] 2 .(A19)
After substituting t ′ = s, with s taken from equation (A18), we obtain,
F t (s) = 1 γ |x − x ′ | |x − x ′ | + β · (x − x ′ ) .(A20)
Performing now integration with respect to t ′ in equation (A16) with the help of the delta-function, we arrive to
V (t, x) = V σ x (s, x ′ ) d 3 x ′ F t (s)|X − X ′ | t ′ =s ,(A21)
where |X − X ′ | t ′ =s must be calculated from (A12) with t ′ = s where s is taken from (A18). It yieldṡ
F t (s)|X − X ′ | t ′ =s = |x − x ′ | ,(A22)
and proves that the retarded potential (A2) is Lorentz-invariant
V σ X (ζ, X ′ ) d 3 X ′ |X − X ′ | = V σ x (s, x ′ ) d 3 x ′ |x − x ′ | .(A23)
We have verified the Lorentz invariance for the scalar retarded potential. However, it is not difficult to check that it is valid in case of a source σ α1α2...α l that is a tensor field of rank l. Indeed, the Lorentz transformation of the source leads to Λ β1 α1 Λ β2 α2 ...Λ β l α l σ β1β2...β l but the matrix Λ α β is constant, and can be taken out of the sign of the retarded integral. Because of this property, all mathematical operations given in this appendix for a scalar retarded potential, remain the same for the tensor of any rank. Hence, the Lorentz invariance of the retarded integral is a general property of the wave operator in Minkowski spacetime.
1. it reproduces all functional relationships of the background FLRW cosmological model; 2. it maintains the background equation of statep = qǭ.
105a) is equivalent to the law of conservation of energy of matter of the localized system. Equation (105b) is analogues to the Euler equation of motion of fluid or the equation of the force balance in case of solids. VI. LAGRANGIAN PERTURBATIONS OF FLRW MANIFOLD A. The Concept of Perturbations
whereρ q ≡Ψ/a in accordance with definition(32). The potential energy of the scalar field,W =W (Ψ), remains arbitrary as yet.It is important to emphasize that in the most general case the ratio c 2 s /c 2 of the speed of sound in fluid to the fundamental speed c, is not equal to the parameter w m of the equation of state(89), that is w m = (c s /c) 2 . Indeed, the speed of sound is defined as a partial derivative of pressure p m with respect to the energy density ǫ m taken under the condition of a constant entropy s m , m /∂µ m ) sm=const.(∂ǫ m /∂µ m ) sm=const. ,
which depend on the first covariant derivatives with respect to the background metricḡ αβ . Moreover, it allows to eliminate a number of terms depending on the first derivatives of the fields φ and ψ in equation (161). Since we keep the gauge function B α arbitrary, the equation (163) does not fix any gauge. The choice of the gauge is controlled by the gauge function B α . One substitutes the gauge function (163) to equations (162) and (161) and make use of the background Friedmann equations (85), (86) to replace the background values of the energy density,ǭ, and pressure,p, with the Hubble parameter H and its time derivative H ′ . It brings about equation (161) to the following form l µν ;α ;α + 2Hū α l µν;α
where the hat above each symbol denotes a new value of the field variable after applying the gauge transformation, and all functions are calculated at the same value of coordinates x α . The gauge transformations of the field variables are expressed in terms of the covariant derivatives on the manifold and, thus, are coordinate-independent. Equation (176b) is derived from the Lie transformation (176a) of the metric tensor perturbation, and the relationship (115) between κ αβ and l αβ .Gauge invariance of the Lagrangian perturbation theory means that the gauge transformations of the field variables do not change the content of the theory. In other words, the equations for the field variables must be invariant with respect to the gauge transformations (176a)-(176d). However, direct inspection of equations (164), (166), (167) shows that they do depend on the choice of the gauge in the form of the gauge function B α introduced in equation(163).
and h =f αβ h αβ = 2(p − q) = γ.It turns out that the conformal Hubble parameter, H = a ′ /a is more convenient in the conformal spacetime thanH =Ṙ/R = R −1 dR/dT ,where T is the cosmological time (see section V B). Relationships between H and H, and their derivatives are shown in equations (42)-(44). These relationships along with equations (43) and (229) are employed in order to express the time derivative, H ′ , of the conformal Hubble parameter in terms of H 2 and the parameter w eff of the effective equation of state
is a linear combination of the perturbation of the mass density of the fluid and the normalized Clebsch potential. Replacing expression (265) in equation (253), yields the exact equation for δ m that is 2 + (1 + w m ) H 2 δ m + 1 2 (1 + w m ) 1 − 3 c 2 s c 2 a 2 Λδ m = 4πa 2 (σ + τ ) . This equation describes propagation of the ideal fluid density perturbation δ m in the form of sound waves with velocity c s .
The authors thank the anonymous referees for a number of fruitful comments and constructive suggestions that helped us to improve the manuscript.Substituting, τ αβ = τ ⊺αβ + (1/3a 2 )π αβ τ , to (298) and comparing with the integrand of (295) makes it clear that (294b) is valid. It proves the second gauge condition (290b). We conclude that the retarded integrals (285a)-(285d) yield the complete solution of the linearised wave equations (282a)-(282d). Thus, we can chose the gauge functions ζ = ζ α = 0.The matter field perturbationsWhat remains is to find out solutions for the scalar functions V m and V q and χ m and χ q . In the linearized Hubble approximation equation for V m is obtained from (253) by discarding all terms of the order of H 2 . It yieldsApplying relationships (250), (251) in equation(299)allows us to recast it to 1 a n (a n V m ) + 1 − c 2 c 2 s v αvβ (a n V m ) ,αβ + 3 + (2n − 1)where n is yet undetermined real number. Choosing, n = n s , withannihilates the term being proportional to H in the left side of (300) and reduces it toThis equation describes propagation of perturbation V m with the speed of sound c s . Indeed, let us introduce the sound-wave Laplace-Beltrami operatorThen, equation (302) readsThis equation has a well-defined Green function with characteristics propagating with the speed of sound c s . We discard the advanced Green function because we assume that at infinity the function V m and its first derivatives vanish. Solution of (304) is explained in Appendix B, and has the following formwhere the retarded time ς is given by equation (B18), β = β i =v i /c, γ = 1/ 1 − β 2 is the Lorentz factor, and the unit vector n = (Linearized equation for V q is obtained from (254) after discarding all terms being proportional to H 2 . It yieldsAppendix A: Lorentz Invariance of the Retarded PotentialWe use a prime in the appendices exclusively as a notation for time and spatial coordinates which are used as variables of integration in volume integrals (see, for example, equations (A2), (A3), and so on). It should not be confused with the time derivative with respect to the conformal time used in the main text of the present paper.Let us consider an inhomogeneous wave equation for a scalar field, V = V (η, X), written down in a coordinate chart X α = (X 0 , X i ) = (η, X),where ≡ η αβ ∂ αβ , ∂ α = ∂/∂X α , and σ X = σ X (η, X) is the source (a scalar function) of the field V with a compact support (bounded by a finite volume in space). Equation (A1) has a solution given as a linear combination of advanced and retarded potentials. Let us focus only on the retarded potential which is more common in physical applications. Advanced potential can be treated similarly. We assume the field, V , and its first derivatives vanish at past null infinity. Then, the retarded solution (retarded potential) of (A1) is given by an integral,whereis the retarded time, and we assume the fundamental speed c = 1. Physical meaning of the retardation is that the field V propagates in spacetime with the fundamental speed c from the source σ X , to the point with coordinates X α = (η, X) where the field V is measured in correspondence with equation (A2). Left side of equation (A1) is Lorentz-invariant. Hence, we expect that solution (A3) must be Lorentz-invariant as well. As a rule, textbooks prove this statement for a particular case of the retarded (Liénard-Wiechert) potential of a moving point-like source but not for the retarded potential given in the form of the integral (A2). This appendix fulfils this gap. Lorentz transformation to coordinates, x α = (t, x) linearly transforms the isotropic coordinates X α = (η, X) of the FLRW universe as followsAppendix B: Retarded Solution of the Sound-Wave EquationLet us consider an inhomogeneous sound-wave equation for a scalar field U = U (η, X) written down in the isotropic coordinates X α = (η, X),where τ X = τ X (η, X) is the source of U having a compact support, and the sound-wave differential operator s was defined in (303). It is Lorentz-invariant and readswherev α is four-velocity of motion of the medium with respect to the coordinate chart, c s is the constant speed of sound in the medium, and we keep the fundamental speed c in the definition of the operator for dimensional purposes. We assume that c s < c. The case of c s = c is treated in section A, and the case of c s ≥ c makes a formal mathematical sense in discussion of the speed of propagation of gravity in alternative theories of gravity since the equation describing propagation of gravitational potential U has the same structure as (B1) after formal replacement of c s with the speed of gravity c g[59,104]. In particular, in the Newtonian theory the speed of gravity c g = ∞, and the operator (B2) is reduced to the Laplace operatorwhere the constant projection operator,π αβ , has been defined in (239). We are looking for the solution of (B1) in the Cartesian coordinates x α = (t, x) moving with respect to the isotropic coordinates X α with constant velocity β i . Transformation from X α to x α is given by the Lorentz transformation (A4). In coordinates X α the four-velocityv α = (1, 0, 0, 0). Therefore, in these coordinates, equation (B1) is just a wave equation for the field U propagating with speed c s . It has a well-known retarded solution,whereis the retarded time. Equation (B1) is Lorentz-invariant. Hence, its solution must be Lorentz-invariant as well. Our goal is to prove this statement. To this end, we take solution (B4) and perform the Lorentz transformation (A7), (A8). We recast the retarded integral (B4) to another form with the help of one-dimensional delta-functionIt looks similar to (A2) but one has to remember that the retarded time η s differs from ζ that was defined in (A3) on the characteristics of the null cone defined by the fundamental speed c. Transformation of functions entering integrand in(B6)is similar to what we did in section A but, because c s = c, calculations become more involved. It turns out more preferable to handle the calculations in tensor notations, making transition to the coordinate language only at the end of the transformation procedure. Let us consider two events with the isotropic coordinates X α = (η, X) and X ′α = (η ′ , X ′ ). We postulate that in the coordinate chart, x α , these two events have coordinates, x α = (t, x), and, x ′α = (t ′ , x ′ ), respectively. We define the components of a four-vector, r α = (t ′ − t, x − x ′ ) which is convenient for doing mathematical manipulations with the Lorentz transformations. For instance, the Lorentz transformation of the Euclidean distance between the spatial coordinates of the two events, is given by awhereπ αβ is the operator of projection on the hyperplane being orthogonal tov α (the same operator as in (B3)). Equation (B7) is a Lorentz-invariant analogue of expression (A12) and matches it exactly. Transformation of the source, τ X (X α ) = τ x (x α ) is fully equivalent to that of σ X as given by equation(A11). Coordinate volume of integration transforms in accordance with (A14). We need to transform the argument, η ′ − η s , of delta-function which we shall denote in coordinates X α as f η (η ′ ) ≡ η ′ −η s . The argument is a scalar function which is transformed asTransformation of the delta-function in the integrand of integral (B6) iswhereḟ t (ς) ≡ [df t (t ′ )/dt ′ ] t ′ =ς , and ς is one of the roots of equation f t (t ′ ) = 0 that is associated with the retarded interaction. Eventually, after accounting for transformation of all functions and performing integration with respect to time, integral (B6) assumes the following formwhere |X − X ′ | t ′ =ς denotes the expression (B7) taken at the value of t ′ = ς. What remains is to calculate the instant of time, ς, and the value of functions entering denominator of the integrand in (B10). Calculation of ς is performed by solving equation f t (ς) = 0, that defines the characteristic cone of the sound waves, and has the following explicit form,which is derived from (B8). This is a quadratic algebraic equation with respect to the time variable r 0 = ς − t. It readsCoefficient α s defines the speed of propagation of the sound waves, v s ≡ c s /α s , as measured by observer moving with speed β i with respect to the Hubble flow. Thus, the value of the speed of sound, v s , depends crucially on the motion of observer. Derivative of the function,ḟ t (ς), is given byḟwhere the partial derivative ∂r α /∂ς = δ α 0 = (1, 0, 0, 0). Making use of (B8), the partial derivativewhich has to be calculated at the instant of time, t ′ = ς, where ς is given by (B18). In order to calculate the denominator in the integrand in (B10), we account for (B7), (B11) and combine (B20), (B21) together. We getIt is straightforward to check that after using (B16) the above equation is reduced to |X −X ′ |ḟ x (ς) = (c/c s ) √ B 2 − AC, or more explicitly,Finally, the retarded Lorentz-invariant solution of (B1) iswith the retarded time ς calculated in accordance with (B18). This solution reduces to the retarded potential (A23) in the limit of c s → c.
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Inaugural Article: On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics. C M Will, 10.1073/pnas.1103127108Proceedings of the National Academy of Science. the National Academy of Science108C. M. Will. Inaugural Article: On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics. Proceedings of the National Academy of Science, 108:5938-5945, April 2011. doi: 10.1073/pnas.1103127108.
The Averaging Problem in Cosmology and Macroscopic Gravity. R Zalaletdinov, 10.1142/S0217751X08040032International Journal of Modern Physics A. 23R. Zalaletdinov. The Averaging Problem in Cosmology and Macroscopic Gravity. International Journal of Modern Physics A, 23:1173-1181, 2008. doi: 10.1142/S0217751X08040032.
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[] |
[
"A new analysis of the GJ581 extrasolar planetary system",
"A new analysis of the GJ581 extrasolar planetary system"
] |
[
"M Tadeu [email protected] \nInstituto de Astronomia\nGeofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo\nBrasil\n",
"Dos Santos \nInstituto de Astronomia\nGeofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo\nBrasil\n"
] |
[
"Instituto de Astronomia\nGeofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo\nBrasil",
"Instituto de Astronomia\nGeofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo\nBrasil"
] |
[] |
We have done a new analysis of the available observations for the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the ocurrence of aliases and the additional bodies -the planets f and gannounced in Vogt et al. 2010. Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the Habitable Zone (HZ) of the star GJ581. This region, for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period ≈ 450 days, but we are judicious about any affirmation concernig this body because its signal is in the threshold of detection and the high period is in a spectral region where the ocorruence of aliases is very common. Besides, we outline some dynamical features of the habitable zone with the dynamical map and point out the role played by some resonances laying there.
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10.1007/s10569-012-9407-1
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"https://arxiv.org/pdf/1203.3140v1.pdf"
] | 118,547,077 |
1203.3140
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de41a1b4585608baa44597eb864b6c929dd88750
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A new analysis of the GJ581 extrasolar planetary system
14 Mar 2012
M Tadeu [email protected]
Instituto de Astronomia
Geofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo
Brasil
Dos Santos
Instituto de Astronomia
Geofísica e Ciências Atmosféricas(IAG) Universidade de São Paulo São Paulo
Brasil
A new analysis of the GJ581 extrasolar planetary system
14 Mar 2012Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) M. Tadeu dos Santos · G. G. Silva · S. Ferraz-Mello · T.A. MichtchenkoGJ581 exoplanetary-system · Orbit Determination · Habitable Zone
We have done a new analysis of the available observations for the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the ocurrence of aliases and the additional bodies -the planets f and gannounced in Vogt et al. 2010. Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the Habitable Zone (HZ) of the star GJ581. This region, for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period ≈ 450 days, but we are judicious about any affirmation concernig this body because its signal is in the threshold of detection and the high period is in a spectral region where the ocorruence of aliases is very common. Besides, we outline some dynamical features of the habitable zone with the dynamical map and point out the role played by some resonances laying there.
Introduction
The diversity of the exoplanets known today is quite big: large and small masses, semi-major axes, eccentricities, etc. However there are only very few planets known in the regions where it is possible to find favourable conditions for the development of life, the habitable zones. The importance of the planetary system around the 0.31 M ⊙ , M3V star GJ581 is in this context: According to von Paris et al. (2011) and Kaltenegger et al.(2011), GJ581 d is a potentially habitable exoplanet. Besides, a fifth putative planet, GJ581 g, could exist in the central part of the habitable zone. This gives particular importance to the study of the dynamics and stability of this system. The possibility of a near-resonant scenario makes necessary a good knowledge of the masses and elements. Dynamical studies are often very time-consuming and near resonances critically depend on some elements. It is thus necessary to have an extended analysis of the existing observations so that we may know in advance the level of confidence that can be attributed to the masses and elements. This analysis was done using the 119 HARPS radial velocity measurements of this system available in the VizieR data base and the 109 Lick observations done between August, 1999 and May, 2010.
From the first set, Mayor et al (2009;hereafter M09), concluded the existence of four planets, while from the second one, Vogt et al (2010; hereafter V10), claimed the discovery of two additional bodies: GJ581 f, with period ≈ 433 days and GJ581 g with period ≈ 36 days. We have done a new analysis of these observations and we have investigated some of the problems pointed in earlier works. One of them, very common in the analysis of time series, is the aliasing indetermination of the period of GJ581 d. As shown by Fabrycky & , there are two almost equivalent solutions for planet d, one of them with period ≈ 67 days, as found in M09 and, the other, ≈ 1.0124 days. Unfortunately, this problem is intimately related with the spacing of the data in the observations time series and there is no mathematical way to distinguish which one is the correct period. However the joint analysis of the two time series allowed us to decide in favour of the longest period among the two possible ones. The other problem is the existence of the two planets claimed in V10. The planet GJ581 f is technically detectable and the second -GJ581 g -is also possible if the orbits of the other planets in the system (mainly GJ581 d) are assumed to be circular. The orbit determinations resulting from the two existing data series and the problems related to the period of GJ581 d, are discussed in Section 2. Section 3 is devoted to the joint analysis of the two data sets and to the tests concerning the detection of GJ581 g. It also includes an analysis of the dynamics inside the Habitable Zone. The detection of the planet GJ581 f is discussed in Section 4, and, at last, Section 5 presents a final discussion and the conclusions.
The previous orbit determination
This section presents independent analyses based on the two existing series. In all cases, the optimization procedure used was the genetic algorithm proposed by Charbonneau (1995Charbonneau ( , 2002 followed by a simplex optimization (Press et al. 1992), It was done using the same package used in the study of the CoRoT 7 system (Ferraz-Mello et al. 2011). A basic tool in the analysis of the data was the DCDFT (date-compensated discrete Fourier transform; Ferraz-Mello, 1981), which was used as a diagnostic tool to assess the periodicities in the data. However, it is worth stressing the fact that in all models studied in this paper, the given periods result from the joint best-fit of all elements. The DCDFT spectra were mainly used to check if a given periodicity on the system can be obtained by random data showing the same distribution. These random samples were obtained by shuffling the data (but keeping the dates unchanged). The spectra obtained from the randomly distributed samples allow us to determine in each case a minimum level of confidence for the peaks. In order to be accepted, the peaks of the analyzed data must appear significantly higher than this minimum confidence level. In general, to define the confidence level, we have used 1000 random samples. For the sake of comparing results obtained with the different models analyzed in this paper, we adopt as goodness-of-fit estimator the weighted root mean square of the residuals (see Beaugé et al. 2008):
(wrms) 2 = S N − 1 N i=1 (O i − C i ) 2 σ 2 i(1)
where
S −1 = 1 N N i=1 1 σ 2 i .(2)
O i and σ i are the radial velocities measured and the errors, respectively. N is the number of observations, M the total number of parameters determined and C i the radial velocities calculated with the model considered. We also give, to ease comparisons with results in other papers, the quantity Q given by
Q = S N − M N i=1 (O i − C i ) 2 σ 2 i(3)
often called normalized chi-square, whose usage as goodness-of-fit estimator is very common in the literature. In all solutions presented here we give the values of both parameters.
The M09 data set
The initial observations of this series were used by Bonfils et al. (2005) to announce the discovery of GJ581 b. In a next paper, Udry et al (2007) reported the discovery of the planets GJ581 c and GJ581 d. In that paper, the periods of the 3 planets found, b, c and d were, respectively, ≈ 5.36 d, ≈ 12.9 d and ≈ 83.4 d. The fourth planet, GJ581 e, was announced in M09, using the same 119 observations considered here. In addition, the period of the planet d was recalculated as being ≈ 66 days. Corrections of this kind are a common fact in the study of time series and are due to aliasing effects of the spacing of the data (all observations in one site being done near the same sidereal time), when a sampling can represent more than one periodic function. The alias appears as a peak at a spurious frequency in the power spectrum. There is no mathematical tool to solve this kind of problem; the only possible solution is to have observations done at different sidereal times (or, equivalently, in sites well separated in longitude). Figure 1 shows the normalized DCDFT (Ferraz-Mello 1981) of the M09 data, after having eliminated from them the part of the signal coming from the planets b and c, by means of a harmonic filtering. We see one peak in the period ≈ 3.14 d and also another one in ≈ 66 d, corresponding respectively to the planets e and d (M09). The vertical dashed line represent the Nyquist frequency ≈ 0.5d −1 where a mirror effect can be seen: the peaks in both sides of that line are nearly mirror images one from another. They are the result of the aliasing. This problem, for this system, was already detected by Fabrycky & . Figure 1 shows that the inspection of the power spectrum does not allow us to decide which alias corresponds to the actual period. The results of complete least-squares solutions ( Table 1) show that this procedure is also not sufficient to decide among the two solutions corresponding to planet GJ581 d (but it allows us to decide between the periods ≈ 3.15 and ≈ 1.47 d for the planet e). The dynamical analyses can be helpful in getting the correct period: numerical integrations of the conditions of the Table 1 are stable for a long period. However, we verify that the eccentricities of the planet d given by (1.1) and (1.2) are excited as soon as the integration begins. After that, the variation of this element stabilize in a value different of the initially proposed. This phenomenon does not occur with (1.3) and (1.4) and it allows us conclude that the solutions with period ≈ 1.0123d, in solutions (1.1) and (1.2), are not consistent from the dynamical point of view. We have also compared the model with circular orbits to one model with elliptic orbits. The improvement in the wrms when non-zero eccentricities are used is not statistically significant, given the increase in the number of degrees of freedom, as an F-test can show. Table 1 Complete least-square solutions using the data in M09 considering, separately, the two possible periods for GJ581 d and two models with all planets in circular or eccentric orbits. In all solutions presented in this paper, the orbital longitude and argument of pericenter corresponds to the epoch JD-2451409.76222 Table 2 Complete least-squares solutions using the data in V10 considering the planets GJ581 d in circular or eccentric motion, respectively.
Solution 1.1 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.5 ± 0.6 5.3687 ± 0.0005 0.001 ± 0.07 233.2 ± 7 253.4 ± 9 15.7 ± 0.3 GJ581 c 3.2 ± 0.6 12.931 ± 0.002 0.2 ± 0.2 267 ± 50 133. ± 50 5.3 ± 0.2 GJ581 d 2.7 ± 1. 1.0124 ± 0.0001 0.37 ± 0.3 158 ± 40 285. ± 40 1.8 ± 0.8 GJ581 e 1.67 ± 0.7 3.149 ± 0.002 0.18 0.5 −0.1 133 ± 35 4.1 ± 30 1.72 +0.7 −0.3 V o = (−0.20 ± 0.3)m/s Q = 2.53 wrms = 1.54m/s Solution 1.2 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.33 ± 0.3 5.368 ± 0.002 0.0(f ixed) 0.0(f ixed) 162. ± 6 15.5 ± 0.4 GJ581 c 2.8 ± 1.7 12.93 ± 0.03 0.0(f ixed) 0.0(f ixed) 20. ± 25 4.7 ± 2 GJ581 d 2.5 ± 1.5 1.0124 ± 0.002 0.0(f ixed) 0.0(f ixed) 311. ± 30 1.8 ± 0.2 GJ581 e 1.5 ± 1. 3.15 ± 0.001 0.0(f ixed) 0.0(f ixed) 204. ± 20 1.6 ± 0.4 V o = (0.0 ± 0.4)m/s Q = 3.07 wrms = 1.77m/s Solution 1.3 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.4 ± 1. 5.3688 ± 0.0007 0.02 −0.02 +0.3 20 ± 20 142. ± 30 15.5 ± 0.5 GJ581 c 3.2 ± 1.2 12.931 ± 0.3 0.16 −0.1 +0.3 231.025 154. ± 30 5.3 +0.7 −1.7 GJ581 d 2.7 ± 1.5 66.8 ± 0.5 0.38 ± 0.3 322 ± 50 107. ± 50 7.2 ± 0.8 GJ581 e 1.9 ± 1.6 3.157 ± 0.08 0.11 −0.1 +0.7 145 ± 50 2.5 ± 50 2.0 1.7 −1 V o = (−0.34 ± 0.3)m/s Q = 2.57 wrms = 1.55m/s Solution 1.4 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.6 ± 0.6 5.3688 ± 0.0004 0.0(f ixed) 0.0(f ixed) 162. ± 6 15.8 ± 0.2 GJ581 c 3.06 ± 0.5 12.92 ± 0.01 0.0(f ixed) 0.0(f ixed) 123. ± 10 5.13 ± 0.1 GJ581 d 2.21 ± 0.5 66.74 ± 0.4 0.0(f ixed) 0.0(f ixed) 61. ± 30 6.4 ± 0.2 GJ581 e 1.75 ± 0.6 3.15 ± 0.015 0.0(f ixed) 0.0(f ixed) 146. ± 35 1.8 ± 0.2 V o = (0.00 ± 0.3)m/s Q = 2.91 wrms = 1.72m/sSolution 2.1 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.25 ± 0.6 5.369 ± 0.001 0.0(f ixed) 0.0(f ixed) 286. ± 17 15.4 ± 0.6 V o = (1.11 ± 0.5)m/s Q = 4.68 wrms = 3.28m/s Solution 2.2 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b
12.38 ± 0.6 5.369 ± 0.001 0.059 ± 0.04 43.5 ± 10 245.6 ± 10 15.5 ± 0.5
V o == (1.01 ± 0.5)m/s Q = 4.67 wrms = 3.24m/s 2.2 The V10 data set
In V10, 109 measurements done at the Lick observatory over 11 years were published. With this data set alone we can infer only the presence of GJ581b, with the period of ≈ 5.4 d, as can be seen in Table 2 and Figure 2. It is also not possible to obtain a significantly better solution (F-test) using an elliptic orbit instead of a circular one. The right panel of figure 2 shows the DCDFT of the residuals obtained after eliminating from the data the contribution of GJ581 b in an elliptic orbit. It shows that no important peak appears in the interval 0 -0.5d −1 (It is enough to look at this interval because one possible peak with a higher frequency would produce an alias in this interval.) They are all below a minimum confidence level determined by taking randomly shuffled sets of the data and seeing the height of the peaks that can be randomly formed; 15% of the spectra have peaks higher than the level shown. Therefore, the V10 ′ s data set alone does not allow other bodies to be detected in the system.
3 The combined data set. GJ581 g
The discussions in the previous section have shown the limitation of the two existing sets of data. However, with the combined data set (V10+M09) it was possible to solve some of the difficulties discussed in previous section. It was possible to solve the alias problem thanks to the ≈ 3h difference in longitude of the two observatories. The joint V10+M09 data did also allowed Vogt et al. (2010) to claim the discovery of two new bodies: GJ581 f and GJ581 g, with periods ≈ 433 d and ≈ 36 d respectively. The claimed planet GJ581 g lays in the habitable zone, which, for the star GJ581, is in the range [0.11 − 0.21] AU (Von Braun et al. (2011)).
Fourier Analysis
Again, by harmonic filtering, we remove from the combined data the contribution of the planets b and c and Fourier analyze the residuals. The result is shown in Figure 3. The highest peak is now the one in the period ≈ 67 d, but we also see the peak corresponding to the period ≈ 3.14 d (planet e) and their aliases. As before, it is not yet possible to choose from the spectrum which one is the true period of planet d. A detailed analysis of all possible solutions is still necessary. In Table 3, we give the complete least-squares orbital solutions for the combined data set with 4 planets considering the two possible periods for planet d. Let us stress the fact that in these solutions the periods are not fixed a priori but result from the least-squares best-fit procedure used. We just bracket the interval where the solution is searched. In one case we searched for a solution in the interval [40 − 80] d while in the other we searched it in the interval [0.5 − 10] d. In Table 3 we show that the difference in the resulting wrms, in both cases, is significant allowing us to decide in favor of the longest of the two periods. One may note, however, that the differences between the wrms of the models with circular Table 3 Least-squares solutions considering four bodies and the joint data sets M09 and V10.
Solution 3.1 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b
12.53 ± 0.9 5.369 ± 0.002
0.0(f ixed) 0.0(f ixed) 287. ± 10 15.7 ± 0.3 GJ581 c 3.01 ± 0.7 12.902 ± 0.008 0.0(f ixed) 0.0(f ixed) 35. ± 27 3.8 ± 0.3 GJ581 d 1.9 ± 1. 66.81 ± 0.4 0.0(f ixed) 0.0(f ixed) 36. ± 12 5.5 ± 2 GJ581 e 1.47 ± 0.8 3.149 ± 0.8 0.0(f ixed) 0.0(f ixed) 300. ± 40 1.5 ± 0.5 V (M 09) = (−0.42 ± 0.5)m/s V (V 10) = (1.1 ± 1.5)m/s Q = 3.13 wrms = 2.11m/s Solution 3.2 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.5 ± 0V (M 09) = (−0.34 ± 0.9)m/s V (V 10) = (1. ± 1.)m/s Q = 3.01 wrms = 2.03m/s Solution 3.3 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b 12.4 ± 0.5 5.368 ± 0.004 0.0(f ixed) 0.0(f ixed) 284. ± 8 15.5 ± 0.5 GJ581 c 2.73 ± 0.8 12.93 ± 0.01 0.0(f ixed) 0.0(f ixed) 51. ± 40 4.6 ± 0.2 GJ581 d 1.64 ± 0.6 1.012 ± 0.001 0.0(f ixed) 0.0(f ixed) 72. ± 18 1.2 ± 0.2 GJ581 e 1.3 ± 1. 3.15 ± 0.02 0.0(f ixed) 0.0(f ixed) 42. ± 30 1.4 ± 0.6 V (M 09) = (−0.5 ± 1.)m/s V (V 10) = (0.8 ± 2.)m/s Q = 3.17 wrms = 2.18m/s Solution 3.4 K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581 b
12 . and elliptical orbits still does not make possible to decide in favor of one model or another. The spectra of the residuals of the solutions given in Table 3 are shown in Figure 4, where it is possible to devise the existence of GJ581 g, in the residuals of the circular model. It is also shown that such a period does not appear in the power spectrum of the residuals of the eccentric model.
The Biased Monte Carlo (BMC)
The BMC method consists in applying incomplete least square fits to intial guesses chosen at random in a given domain of the space of parameters. It was first used in Ferraz-Mello et al.(2005a) to map the wrms values in the neighborhood of the minimum and serves a double purpose: on one hand, the map thus obtained allow us to define the confidence interval of these results; on the other hand, Table 4 Best-fit solution of the model with 5 planets in circular orbits. The fifth planet included in the analysis is GJ 581 g. The longitude of planet g is not constrained, so we can get good fits in all longitudes, the given value corresponds to the best fit of the period. it allows us to check if the minimum found by the optimization procedure is not just one among others. Each random guess is propagated running the optimization algorithms for a pre-determined number of iterations. The results farther from the minimum than a given amount are discarded. This is the case for many of the runs. Given the large number of unknowns, one pure random process is not feasible as the probability of getting one point near the minimum by pure chance is almost zero. The parameters of the incomplete optimization are set to be such that the procedure gives a reasonable amount of sampled points near the least squares solution and allows us to map the wrms on a significantly wide domain of the phase space. The solutions thus obtained are called 'good-fit solutions'. They are not mathematically as good as the best-fit, but they are statistically equivalent to it, since they belong to the confidence domain of the best fit. We use the BMC technique here in order to verify if each one of the orbital elements are well or poorly determined. It makes possible a better comparison between models and also allow us to seek for secondary minima, thus improving the results of the previous section. Therefore, we take the V010+M09 data set and look for good fits of the model with four planets in circular and eccentric orbits. Afterwards we compare this results with those obtained after the inclusion of the planet g. We have thus applied the BMC technique to four different models: four planets with all orbits circular or with two of them eccentric (as described below) and including or not the fifth planet in circular orbit. These results are shown in Figure 5. In panel (A), black dots indicate the good fits obtained when we consider four planets, two of them, GJ581 d and e, with periods 66 d and 3.15 d, in eccentric orbits and the other two, GJ581 b and c, in circular orbits. We remind that in the solution 3.2, the orbits of these two planets are almost circular. The innermost planet (GJ581 e) could also be assumed in circular orbit (see Papaloizou, 2011) but was kept as elliptic because of the nonzero value found for its eccentricity in solution 3.2. The horizontal black dashed line represents the best solution found, with wrms = 2.08m/s. In the same panel, red dots indicate the results when a fifth planet, in circular orbit, is added to the model, in which case the best-fit solution is obtained at wrms = 2.06m/s. So, from (A), it is not possible to see any statistical difference between the two considered models and the inclusion of the planet GJ581 g in this case is unjustified, as it does not bring any improvement to the wrms of the best-fit solutions. The best-fit solutions with 5 circular planets and with 4 planets (2 in circular and 2 in ellipitic orbits) are given in Table 5.
In panel (B), we have a similar situation but considering all planets in circular orbits. Now the wrms of the best-fit solution in the model with 4 planets is 2.17m/s and that of the model with 5 planets is 2.03m/s. In this case, the inclusion of the fifth planet (GJ581 g) means a real improvement and would support a conclusion positive concerning the existence of the planet GJ581 g. The best-fit solutions with 5 circular planets and with 4 planets (2 in circular and 2 in ellipitic orbits) are given in Table 5. Figure 6 shows BMC-fits of some orbital elements of GJ581 g using the model with circular orbits. It allows us to estimate the quality of the solution given in Table 5.1. The best value of the amplitude is K g = 1.2m/s. The period P g , however, shows two minima, ≈ 33 d and ≈ 36 d, which appear represented with the labels P ′ and P . This difference is due to an aliasing indetermination due to observations concentrated near the oppositions. We indeed have 1/33d ≈ (1/36d) + (1/365d). At last, we add that the orbital phase at the epoch, ℓ g , is not constrained as equally good fits are found at all longitudes. This is a consequence of the large interval of the confidence for the period of GJ581g. Variations in the period inside this interval are enough to give values of the longitude at epoch over all possible values. Table 5 Best fit solution with 5 planets in circular orbits (5.1) and with 4 planets, two of which in eccentric orbits (5.2). In these solutions, the longitudes of the planets e and g are not constrained. The given value corresponds to the nominal period. Figure 7 is the analogous to Figure 5 obtained when only the data in the set M09 are considered. We see that, in this case, the improvement of the results when the fifth planet is included is not significant. We have found solutions with periods in the range [25][26][27][28][29][30][31][32][33][34][35][36] d, but in all cases it was not possible to get a robust inference about this additional body. The plot for amplitude and phase are similar to those shown in Figure 6. The smaller wrms of the solutions of Table 1 results from the homogeneity of the set M09. It is important to stress that the wrms is an evaluation of the statistical errors and does not include possible systematic differences between the two data sets.
Solutions initially with 5 planets in circular orbits, or with only 4 of them, but in elliptic orbits with GJ581 d initially in a very eccentric orbit (e = 0.43), were simulated using precise numerical integrations for times up to 300 Myr. The solutions found did not show any appreciable variation in the considered time span. They also appeared to be very robust with respect to small variations in the initial conditions.
The dynamics in the Habitable Zone
In this section, we give some dynamical constraints concerning the existence of one body between the planets GJ581 c and d, using for the 4 planets the values of the solution 5. the fifith planet and one of the neighboring planets, GJ581 b, c and d. They may be generally written as
k b n b + k c n c + k t n t + k d n d ≃ 0.(4)
We selected those corresponding to the main instabilities appearing in the maps. Their coefficients are indicated by the n-uples (k b , k c , k t , k d ). Among them, we point out the resonances (0, 0, 1, −2) (i.e. n t − 2n d ≃ 0) at a ≃ 0.138 AU, and (0, 1, −2, 0) (i.e. n c − 2n t ≃ 0) at a ≃ 0.116 AU, as the most important, affecting the solutions at all eccentricities. It is also worth noting that no low-order resonances were found near some importante features of the maps, which appear to be associated with three-body resonances. Some of them exhibit a highly chaotic dynamics (at moderate-to-loweccentricities) which may be explained by the formation of resonances multiplets (see Cachucho et al. 2010). One such multiplet was found around the resonance (0, −1, 3 − 1) (i.e. −n c + 3n t − n d ≃ 0 at a ≃ 0.135 AU) shown in Figure 8. The formation of such overlapping multiplets is responsible for diffusion across the resonances and the more intense chaotic behavior shown by the semi-majo axis ( Figure 8A). At last, we have to stress the fact that a similar study using the eccentric solutions 5.2 lead only to unstable solutions. This shows that the eccentricity of GJ581 d plays a crucial role in the stability of a planet placed in the Habitable Zone. The high values for its eccentricity found here and in other determinations (e ≃ 0.4 − 0.5) make dinamically impossible the existence of any other body in the HZ. In that case, the perihelic distance of GJ581 d lies in the middle of the HZ and is responsible for the rapid ejection of all test bodies placed there.
4
The planet f Figure 9 shows the normalized DCDFT of the residuals of the solution (5.1) given in Table 5. One may see a peak corresponding to the period ≈ 455 d. However, its confidence level, estimated by comparing shuffled data spectra is ≃ 4%. So, the planet f is in the limit of detection. For the sake of completeness, we give in Table 6 the orbital elements obtained with 6 circular orbits.
Conclusion
From a purely statistical point of view, it is not incorrect to state the existence of GJ581 g. However, this can only be done by assuming that all planets are in circular orbits. The explanation of this fact Table 6 Best-fit circular solution with 6 planets. The longitudes of planets e,g and f are not constrained. 2009), happens when the periods are close to a commensurability relation (the period of GJ581 d is almost twice the period of GJ581 g). In this circumstance, the eccentricity of the external body will be overestimated and can hide totally the signal of the internal body. In this case, the analysis of the radial velocity measurements does not allow us to distinguish the signal of the putative fifth planet (GJ581 g) from an overestimated eccentricity of the orbit of GJ581 d. Additional tests were done, but we have not been able to verify the existence of the planet g when we consider the d with a fixed and low (≈ 0.1) but not zero eccentricity. Of course, additional observations can help solving the dilemma about planet g. Forveille et al. 2011, using new observations, were not able to confirm the existence of the planets g and f. But, this statement could not be verified because the new data were not yet made available in public domain.
Solution 6.1 K(m/s) P (d) e ω(•) ℓ(•) M (M × ) GJ581 b 12.5 ± 0.6 5.3687 ± 0.001 0.0(f ixed) 0.0(f ixed) 288 ± 5 15.7 ± 0.3 GJ581 c 2.7 ± 1. 12.9 ± 0.4 0.0(f ixed) 0.0(f ixed) 71 ± 30 4.5 ± 1.5 GJ581 d 2.4 ± 0.7 66.8 ± 0.3 0.0(f ixed) 0.0(f ixed) 64 ± 27 6.9 ± 1 GJ581 e 1.6 ± 1. 3.15 ± 0.01 0.0(f ixed) 0.0(f ixed) 25 1.7 ± 0.3 GJ581 f 1.1 ± 0.
In adition, we have that the analysis done by them did not include the V10 data set. The signal with the frequency of the alleged sixth planet -GJ581 f -was found, but it is in the threshold of confidence level.We cannot discard the hypotesis that the period of the sixth planet is produced by some complicated beating with the observation window of 1yr, once both are comparables.
Summary
Our best-fit with 4 planets in eccentric orbits (Solution 3.2 in the text) is:
K(m/s) P (d) e ω(•) ℓ(•) M (M ⊗ ) GJ581
Fig. 1
1Normalized DCDFT of the M09 data after having eliminated from them the part of the signal coming from the planets b and c by means of a harmonic filtering, The vertical dashed line represent the Nyquist frequency 0.5d −1 .
Fig. 2 (
2A) Normalized DCDFT spectra of the V10 data showing the peak corresponding to GJ581b, (B) DCDFT of the residuals. The dashed horizontal line shows the confidence level defined by shuffled data.
Fig. 3
3Normalized DCDFT of the joint M09+V10 data after elimination of the contributions due to planets b and c. The vertical dashed line represent the Nyquist frequency ≈ 0.5d −1 .
0 ± 100 104 ± 100 1.45 ± 0.5 V (M 09) = (−0.4 ± 0.9)m/s V (V 10) = (0.8 ± 1.)m/s Q = 3.66 wrms = 2.23m/s Fig. 4 (A) Normalized DCDFT spectrum of the residuals of the circular solution (3.1), showing the peak corresponding to GJ581 g at 33.2 d. (B) Normalized DCDFT spectrum of the residuals of the eccentric solution(3.2). In this case, the peak corresponding to GJ581 g is not seen (its position is represented with a vertical dashed line). The horizontal dashed lines represent the minimum level of confidence of the peaks.
Fig. 5
5Comparision of BMC fits of differents models: (A) the planets b, c and g are in circular orbits and d and e in eccentric orbits and in (B) all planets are in circular orbits. Red and black points are the results obtained when the planet g is respectively included or not included in the model. The horizontal dashed lines show the minimal values of the wrms (best-fit) reached in each case.
Fig. 6
6BMC-fits of the orbital parameters of GJ581 g in the model of the 5 planets in circular orbits.
Fig. 7
7Same asFigure 5when considering only the M09 data set.
1 .
1The mass and the initial longitude of the fifth planet were taken from solution 5.1. Its semi-axis and eccentricity were taken on a grid in the interval: a g = [0.1 − 0.18], ∆ a = 0.002 AU and e g = [0.0 − 0.5], ∆ e = 0.0025. Each point of the grid was integrated for 10 5 yrs and we calculated Michtchenko ′ s spectral numbers associated with the evolutions of its eccentricity and semi-major axis (seeFerraz-Mello et al., 2005). Solutions are considered unstable (chaotic) when the spectral number is high (log N ≥ 1.8) or when the test planet crosses the orbit of one of the neighboring planets after leaving the Habitable Zone. These situations are represented in the map by the red painted areas. The blue and purple regions represent the domains of regular motion.The white/black mark shows the position of planet GJ581 g in solution 5.1 (the eccentricity was taken as 0.001 as it is expected to be due to the forcing by the other planets).Figure(8) shows that several mean-motion resonances (MMR) act as source of instability in the Habitable Zone. The location of some selected resonances coinciding with the chaotic regions appearing on the map is shown by vertical lines. They are two-and three-body resonances involving
Fig. 8
8Dynamical map in the region of the planet g, calculated with the initial conditions and masses of solution 5.1 . In (A) and (B) we have the log of spectral number of semimajor-axis and eccentricity, respectively.
Fig. 9
9Normalized DCDFT of the residuals of the solution 5.1. The dashed horizontal line is the minimum level of confidence of the peaks.
well-known phenomenon, described in detail in Anglada-Escudé et al. (2009, 2010) and Giuppone et al. (
ixed) 0.0(f ixed) 53. ± 30 1.7 ± 0.3 GJ581 b 12.96 ± 0.5 5.3687 ± 0.0002 0.0(f ixed) 0.0(f ixed) 288. ixed) 0.0(f ixed) 78. ± 20 6.1 ± 3 V (M 09) = (−0.3 ± 0.9)m/s V (V 10) = (0.9 ± 1.)m/s Q = 3.06 wrms = 2.15m/s
e 1.6 ± 1.3.15 ± 0.2
0.21 0.6
−0.2
132 ± 30 182 ± 30 1.64 0.6
−0.3
GJ581 b 12.5 ± 0.4 5.369 ± 0.005 0.002 +0.01
−0.0001
44 ± 30
243 ± 30 15.7 +0.3
−3
GJ581 c 2.8 ± 0.8
12.92 ± 0.05
0.01 +0.3
−0.01
321 ± 50 96 ± 30
4.7 +0.3
−1.3
GJ581 d 2.7 ± 1.2
66.9 ± 0.6
0.54 +0.2
−0.5
324 ± 30 117 ± 30 6.56 +0.4
Acknowledgements MTS whish to thank to São Paulo State Research Foundation (FAPESP). GGS thanks to Capes for the doctoral fellowship and to Raúl E. Puebla for his hints about the IDL language. We are also grateful for the comments of two anonymous referees.
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|
[] |
[
"Effective attraction induced by repulsive interaction in a spin-transfer system",
"Effective attraction induced by repulsive interaction in a spin-transfer system"
] |
[
"Ya B Bazaliy \nDepartment of Physics and Astronomy\nInstituut Lorentz\nLeiden University\nThe Netherlands\n\nInstitute of Magnetism\nUniversity of South Carolina\nColumbiaSC\n\nNational Academy of Science\nUkraine\n"
] |
[
"Department of Physics and Astronomy\nInstituut Lorentz\nLeiden University\nThe Netherlands",
"Institute of Magnetism\nUniversity of South Carolina\nColumbiaSC",
"National Academy of Science\nUkraine"
] |
[] |
In magnetic systems with dominating easy-plane anisotropy the magnetization can be described by an effective one dimensional equation for the in-plane angle. Re-deriving this equation in the presence of spin-transfer torques, we obtain a description that allows for a more intuitive understanding of spintronic devices' operation and can serve as a tool for finding new dynamic regimes. A surprising prediction is obtained for a planar "spin-flip transistor": an unstable equilibrium point can be stabilized by a current induced torque that further repels the system from that point. Stabilization by repulsion happens due to the presence of dissipative environment and requires a Gilbert damping constant that is large enough to ensure overdamped dynamics at zero current.
|
10.1063/1.2822407
|
[
"https://arxiv.org/pdf/0705.0508v1.pdf"
] | 119,159,625 |
0705.0508
|
6ce758b8757d853cc16833c2a3c233a30c3ebbe0
|
Effective attraction induced by repulsive interaction in a spin-transfer system
May 2007
Ya B Bazaliy
Department of Physics and Astronomy
Instituut Lorentz
Leiden University
The Netherlands
Institute of Magnetism
University of South Carolina
ColumbiaSC
National Academy of Science
Ukraine
Effective attraction induced by repulsive interaction in a spin-transfer system
May 2007(Dated: April, 2006)
In magnetic systems with dominating easy-plane anisotropy the magnetization can be described by an effective one dimensional equation for the in-plane angle. Re-deriving this equation in the presence of spin-transfer torques, we obtain a description that allows for a more intuitive understanding of spintronic devices' operation and can serve as a tool for finding new dynamic regimes. A surprising prediction is obtained for a planar "spin-flip transistor": an unstable equilibrium point can be stabilized by a current induced torque that further repels the system from that point. Stabilization by repulsion happens due to the presence of dissipative environment and requires a Gilbert damping constant that is large enough to ensure overdamped dynamics at zero current.
In physics, there are cases where due to the presence of complex environment a repulsive force can lead to actual attraction of the entities. A well known example is a superconductor, where the Cooper pairs are formed from electrons repelled by the Coulomb forces due to the dynamical elastic environment. Here we report a phenomena of effective attraction induced by the repulsive spin-transfer torque in the presence of highly dissipative environment. The spin-transfer effect producing the repulsive torque is a non-equilibrium interaction that arises when a current of electrons flows through a non-collinear magnetic texture [1,2,3]. This interaction can become significant in nanoscopic magnets and is nowadays studied experimentally in a variety of systems. Its manifestations -either current induced magnetic switching [4] or magnetic domain wall motion [5] -serve as an underlying mechanism for a number of suggested memory and logic applications.
Here we consider a conventional spin-transfer device consisting of a a magnetic polarizer (fixed layer) and a small magnet (free layer) with electric current flowing from one to another (Fig 1). Both layers can be described by a macro-spin model due to large exchange stiffness. The free layer is influenced by the spin transfer torque, while the polarizer is too large to feel it. Magnetic dynamics of the free layer is described by the Landau-Lifshitz-Gilbert (LLG) equation with the spin transfer torque term [2,6].
The solutions of LLG are easy to find for the simplest easy axis magnetic anisotropy of the free layer. There exists a critical current at which the free layer either switches between the two minima of magnetic energy, or goes into a state of permanent precession, powered by the current source [2,6,7]. The same basic processes happen in the case of realistic anisotropies, however the complexity of the calculations increases substantially. In a nanopillar device [8] one additionally finds that stabilization of magnetic energy maxima is possible ("canted states" [6]) and that multiple precession modes exist with transitions between them happening as the current is increased [7,9,10]. The anisotropy of a nanopillar device is a combination of a magnetic easy plane and magnetic easy axis directed in that plane. Experimentally, the easy plane anisotropy energy is usually much larger than the easy axis energy, i.e. the system is in the regime of a planar spintronic device [11] (Fig. 1). This limit of dominating easy plane energy is characterized by another simplification of the dynamic equations [12,13], which comes not from the high symmetry of the problem, but from the existence of a small parameter: the ratio of the energy modulation within the plane to the easy plane energy. The deviation of the magnetization from the plane becomes small, making the motion effectively one dimensional.
In this paper we present a general form of effective planar equation describing a macrospin free layer in the presence of spin transfer torques. Its relationship to the first order expansion in the current magnitude used in Ref. 13 is discussed at the end. We then use this equation to study the "spin-flip transistor": a planar device in which the spin polarizer is perpendicular to the direction favored by the magnetic anisotropy energy. It was predicted [14] that the competition between the anisotropy and spin transfer torques leads to a 90 degrees jump of the magnetization at the critical current. Whether the jump happens into the parallel or antiparallel state with respect to the polarizer is determined by the direction of the current.
Here it is shown that the behavior of the spin-flip transistor is more complicated than expected from the simple picture above. Namely, the current inducing a jump into the parallel direction can also stabilize the antiparallel direction. This conclusion is certainly counter-intuitive because the spin torque repels the magnetization from this already unstable saddle point of the energy. However, a combination of two destabilizing torques manages to result in a stable equilibrium. We will see that this happens due to the dissipation terms and a sufficiently large (but still small compared to unity) Gilbert damping constant is required to observe the phenomena.
The magnetization of the free layer M = M n has a constant absolute value M and a direction given by a unit vector n(t). The LLG equation [2,6] reads:
n = γ M − δE δn × n + u(n)[n × [s × n]] + α[n ×ṅ] . (1)
Here γ is the gyromagnetic ratio, E(n) is the magnetic energy of the free layer, and α is the Gilbert damping constant. The second term on the right is the spin transfer torque, where s is a unit vector along the direction of the polarizer, and the spin transfer strength u(n) is proportional to the electric current I [6,13]. In general, spin transfer strength is a function of the angle between the polarizer and the free layer u(n) = f [(n · s)] I, with the function f [(n · s)] being material and device specific.
Equation (1) can be written in polar angles (θ(t), φ(t)):
θ + αφ sin θ = − γ M sin θ ∂E ∂φ + u(s · e θ ) ≡ F θ , φ sin θ − αθ = γ M ∂E ∂θ + u(s · e φ ) ≡ F φ ,(2)with tangent vectors e φ = [ẑ × n]/ sin θ, e θ = [e φ × n].
The easy plane is chosen at θ = π/2, and the magnetic energy has the form
E = (K ⊥ /2) cos 2 θ + E r (θ, φ),
where E r is the "residual" energy. In the planar limit, K ⊥ → ∞, the energy minima are very close to the easy plane and the low energy solutions of LLG have the property θ(t) = π/2 + δθ with δθ → 0. Equations (2) can then be expanded in small parameters |E r |/K ⊥ ≪ 1, |u(n)|/K ⊥ ≪ 1. Assuming time-independent u and s we obtain an effective equation of the in-plane motion
1 ω ⊥φ + α ef fφ = − γ M ∂E ef f ∂φ ,(3)
which has has the form of the Newton's equation of motion for a particle in external potential E ef f (φ) with a variable viscous friction coefficient α ef f (φ). The expressions for the effective friction and energy are
α ef f (φ) = α − (Γ φ + Γ θ )/ω ⊥ ,(4)Γ φ = (∂F φ /∂φ) θ=π/2 , Γ θ = (∂F θ /∂θ) θ=π/2 , and E ef f (φ) = E r (π/2, φ) + ∆E(φ) ,(5)∆E = − M γ φ u(n)(s · e θ ) − Γ θ ω ⊥ F φ θ= π 2 dφ ′ .
Equation (3) with definitions (4,5) gives a general description of a planar device in the presence of spin transfer torque. At non-zero current the effective friction can become negative (see below), and the effective energy is not necessarily periodic in φ (e.g. in the case of "magnetic fan" [13,15]). Physically this reflects the possibility of extracting energy from the current source, and thus developing a "negative dissipation" in the system. In many planar devices the polarizer direction s lies in the easy plane, θ s = π/2, with a direction defined by the azimuthal angle φ s . At the same time the residual energy has a property (∂E r /∂θ) θ=π/2 = 0, i.e. does not shift the energy minima away from the plane. We will also use the simplest form f [(n · s)] = const for the spin transfer strength. A more realistic function will not change the result qualitatively and can be easily used if needed. With these restrictions the effective friction and the energy correction get the form:
α ef f = α + 2u cos(φ s − φ) ω ⊥(6)∆E = − M u 2 2γω ⊥ cos 2 (φ s − φ) .
In a spin-flip transistor the polarizer direction is given by φ s = π/2. Following Ref. 14, we consider in-plane anisotropy energy E r (π/2, φ) = −(K || /2) cos 2 φ corresponding to an easy axis. Then the effective friction is α ef f = α + (2u sin φ)/ω ⊥ and effective energy equals (γ/M )E ef f = −[(ω || − u 2 /ω ⊥ )/2] cos 2 φ + const with ω || = γK || /M . Equilibrium points φ = 0, ±π/2, π are the minima and maxima of the effective energy, and do not depend on u. Stability of any equilibrium in one dimension depends on whether it is a minimum or a maximum of E ef f and on the sign of α ef f at the equilibrium point. It is easy to check, that out of four possibilities only an energy minimum with α ef f > 0 is stable. In the case of a spin-flip transistor the energy landscape changes above a threshold |u| > √ ω || ω ⊥ : the energy minima at φ = 0, π become maxima, and, vice versa, the energy maxima at φ = ±π/2 switch to minima. Effective friction at φ = 0, π is positive independent of u, while at φ = ±π/2 it changes sign at u = ∓αω ⊥ /2.
The behavior of the spin-flip transistor is summarized in a switching diagram Fig. 2 plotted on the plane of the material characteristic α and the experimental parameter u ∼ I. For definiteness we will discuss a current with u > 0. The effect of the opposite current is completely symmetric. For small values of Gilbert damping one observes stabilization of the φ = π/2 (parallel) equilibrium to which the spin torque attracts the magnetization of the free layer, while the opposite (antiparallel) direction remains unstable. This is in accord with the results of Ref. 14. However, when the damping constant is larger than the critical value α * = 2 ω || /ω ⊥ , a window of stability of the antiparallel equilibrium opens on the diagram. Since α ≪ 1, a sufficiently large easy plane energy is required to achieve α * < α ≪ 1.
+π/2 −π/2 0 π α * u α α E eff 0 +π/2 π −π −π/2 eff α E eff 0 +π/2 π −π −π/2 eff α E eff 0 +π/2 π −π −π/
If one thinks about the stability of the (θ, φ) = (π/2, −π/2) equilibrium for u > 0 in terms of Eq. (1), this prediction seems completely unexpected. The anisotropy torques do not stabilize this equilibrium because it is a saddle point of the total magnetic energy E, and the added spin transfer torque repels n from this point as well. The whole phenomena may be called "stabilization by repulsion". To check the accuracy of the planar approximation (3), the result was verified using the LLG equations (2) with no approximations for the axis-andplane energy E = (K ⊥ /2) cos 2 θ − (K || /2) sin 2 θ cos 2 φ. Calculating the eigenvalues of the linearized dynamic matrices [6] at the equilibrium points (π/2, ±π/2) we obtained the same switching diagram and confirmed the stabilization of the antiparallel direction. Typical trajectories n(t) numerically calculated from the LLG equation with no approximations are shown in Fig. 3 trate the predictions. At u > √ ω || ω ⊥ the φ = −π/2 equilibrium is stabilized. In accord with the predictions of Eqs. (3), (6), the wedge of its stability consists of two regions (b) and (c) characterized by overdamped and underdamped dynamics during the approach to the equilibrium. The dividing dashed line is given by u = ω || /α + αω ⊥ /4. It was checked that small deviations of the polarizer s from the (π/2, π/2) direction do not change the behavior qualitatively. Larger deviations eventually destroy the effect, especially the out-of-plane deviation which produces the "magnetic fan" effect [15] leading to the full-circle rotation of φ in the plane.
As the current is further increased to u > αω ⊥ /2, the antiparallel state looses stability and the trajectory approaches a stable precession cycle (Fig. 3(d)). The existence of the precession state is easy to understand from (3) viewed as an equation for a particle in external potential. Just above the stability boundary the effective friction α ef f (φ) is negative in a small vicinity of φ = −π/2, and positive elsewhere. Within the α ef f < 0 region the dissipation is negative and any small deviation from the equilibrium initiates growing oscillations. As their amplitude exceeds the size of that region, part of the cycle starts to happen with positive dissipation. Eventually the amplitude reaches a value at which the energy gain during the motion in the α ef f < 0 region is exactly compensated by the energy loss in the α ef f > 0 region: thus a cycle solution emerges. The effective planar description allows for the analysis of the further evolution of the cycle with transitions into different precession modes, which will be a subject of another publication.
The fact that α > α * condition is required for the stabilization means that dissipation terms play a crucial role entangling two types of repulsion to produce a net attraction to the reversed direction. Note that an interplay of a strong easy plane anisotropy and dissipation terms produces unexpected effects already in conventional (u = 0) magnetic systems. The effective planar equation (3) at u = 0 was discussed in Ref. 12. It was found that the same threshold α * represents a boundary between the oscillatory and overdamped approaches the equilibrium. Above α * the familiar precession of a magnetic moment in the anisotropy field is replaced by the dissipative motion directed towards the energy minimum. When the easy plane anisotropy is strong enough to ensure α ≫ α * , one can drop the second order time derivative term in Eq. (3) and use the resulting first order dissipative equation. In the presence of spin transfer, α ef f (φ, u) depends on the current and can assume small values even for α ≫ α * , thus no general statement about theφ term can be made.
The simplest easy axis energy expression E r (π/2, φ) = −(K || /2) cos 2 φ happens to have the same angular dependence as ∆E(φ) given by Eq. (6). Due to this special property the energy profile flips upside down at u = √ ω || ω ⊥ . For a generic E r (π/2, φ) with minima at φ = 0, π and maxima at φ = ±π/2 the nature of equilibria will change at different current thresholds. This will make the switching diagram more complicated, but will not affect the stabilization by repulsion phenomena. Similar complications will be introduced by a generic f [(n·s)] angular dependence of the spin transfer strength.
In Ref. 13 the known switching diagram for the collinear (φ s = 0) devices [6,9,10] were reproduced by equation (3) with E ef f = E r (π/2, φ). The ∆E term (6) was dropped as being second order in small u. This approximation gives a correct result for the following reason. In a collinear device (γ/M )E ef f = −[(ω || + u 2 /ω ⊥ )/2] cos 2 φ + const and the current never changes the nature of the equilibrium from a maximum to a minimum. Consequently, dropping ∆E does not affect the results. As was already noted in Ref. 13, the first order expansion in u is insufficient for the description of a spin-flip transistor, where the full form (6) is required.
In summary, we derived a general form of the effective planar equation (3) for a macrospin free layer in the presence of spin transfer torque produced by a fixed spin-polarizer and time-independent current. Qualitative understanding of the solutions of planar equation is obtained by employing the analogy with a one-dimensional mechanical motion of a particle with variable friction coefficient in an external potential. The resulting predictive power is illustrated by the discovery of the stabilization by repulsion phenomena in the spin-flip device. Such stabilization relies on the form of the dissipative torques in the LLG equation and happens only for a large enough Gilbert damping constant. The new stable state and the corresponding precession cycle can be used to engineer novel memory or logic devices, and microwave nano-generators with tunable frequency.
To observe the phenomena experimentally, one has to fabricate a device with α > α * , and initially set it into a parallel or antiparallel state by external magnetic field. Then the current is turned on and the field is switched off. Both states should be stabilized by a moderate current √ ω || ω ⊥ < u < αω ⊥ /2, but cannot yet be distinguished by their magnetoresistive signals. The difference can be observed as the current is increased above the αω ⊥ /2 threshold: the parallel state will remain a stable equilibrium, while the antiparallel state will transform into a precession cycle and an oscillating component of magnetoresistance will appear.
The author wishes to thank C. W. J. Beenakker, G. E. W. Bauer, and Yu. V. Nazarov for illuminating discussions. Research at Leiden University was supported by the Dutch Science Foundation NWO/FOM. Part of this work was performed at KITP Santa Barbara supported by the NSF grant No. PHY99-07949, and at Aspen Physics Institute during the Winter program of 2007.
FIG. 1 :
1Planar spin-transfer devices. Hashed parts of the devices are ferromagnetic, white parts are made from a nonmagnetic metal.
FIG. 2 :
2Switching diagram of the spin-flip transistor. In each zone one or two arrows show the possible stable directions of the free layer magnetization. Directions of the easy axis and spin polarizer are defined in the right bottom corner. Angular dependencies of α ef f and E ef f are given in insets. Stable subregions "b" and "c" differ in overdamped vs. underdamped approach to the equilibrium.
FIG. 3 :
3Typical trajectories of n(t) for ω || /ω ⊥ = 0.01, α = 1.5 α * . The plot labels correspond to the regions inFig. 2, the current magnitude is given in the units of u/ p ω || /ω ⊥ and we look at the stability of the φ = −π/2 equilibrium: (a) 0.93, unstable (b): 1.08, stabilized with overdamped approach (c): 1.38, stable, but with oscillatory approach (d): 1.53, unstable; a stable cycle is formed around the equilibrium.
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[] |
[
"ON VERY STABLITY OF PRINCIPAL G−BUNDLES",
"ON VERY STABLITY OF PRINCIPAL G−BUNDLES"
] |
[
"Hacen Zelaci "
] |
[] |
[] |
Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.
|
10.1007/s10711-019-00447-z
|
[
"https://arxiv.org/pdf/1804.04881v2.pdf"
] | 119,167,890 |
1804.04881
|
a446acd9bc311dab374655b57e255c86a6897dda
|
ON VERY STABLITY OF PRINCIPAL G−BUNDLES
17 Apr 2018
Hacen Zelaci
ON VERY STABLITY OF PRINCIPAL G−BUNDLES
17 Apr 2018arXiv:1804.04881v2 [math.AG]
Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.
Introduction
Let X be a smooth irreducible projective curve over C of genus g 2. Denote its canonical bundle by K X . Let E be a stable vector bundle of degree 0 and rank r over X. Let H E be the Hitchin map:
H E : H 0 (X, E ⊗ E * ⊗ K X ) −→ W := r i=1 H 0 (X, K i X ).
defined by associating to a Higgs field φ : E → E ⊗ K X the coefficients of its characteristic polynomial ((−1) i Tr(Λ i φ)) i=1,...,r ∈ W.
Following [Lau88], the vector bundle E is called very stable if H −1 E (0) = {0}. In other words, the vector bundle E is very stable if and only if it has no nilpotent Higgs field other than 0. With these notations, the following theorem has been proven recently in [PPN18].
Theorem 1.1. The vector bundle E is very stable if and only if H E is finite.
We will give a general and more elementary proof for this. Let G be a semisimple connected linear algebraic group. In a natural way, a principal G−bundle E over X is called very stable if the fiber of the Hitchin map (see Section 2) over 0 is reduced to 0. We will show that this is equivalent to say that the bundle ad(E) ⊗ K X has non nilpotent section with respect to the adjoint action. Our main result is then the following Theorem 1.2. The G−bundle E is very stable if and only if H E is finite.
Our proof is purely algebraic and is independent of the geometry of the moduli space of semistable G−Higgs pairs. In particular it induces an elementary proof of Theorem 1.1. Actually we will show the following Theorem 1.3. Let f : A n → A n be a morphism given by homogeneous polynomials such that f −1 (0) = {0}, then f is finite.
I want to thank C. Pauly for suggesting this problem and for useful remarks that considerably improve this notes, and D. Huybrechts for useful discussions. I also want to thank I. Grosse-Brauckmann and A. Peón-Nieto.
Preliminaries
We recall here the definitions and the basic facts about principal Higgs bundles and the Hitchin fibration.
Let G be a semisimple linear algebraic group. A principal G−bundle E over X is a smooth variety E → X with a free (right) G−action which is equivariant with respect to the trivial action on X such that it is locally trivial in theétale topology, i.e. there exists a covering (U i ) of X such E| Ui ∼ = U i × G with the right standard G−action.
Let E be a G−bundle. Given a finite dimensional representation ρ : G → GL(V ), E induces a vector bundle
E(V ) = E × G V := E × V /G,
for the diagonal action of G on E × V . Using this, we say that E is semistable if there exists a (equivalently for any) faithful representation ρ : G → GL(V ) such that E(V ) is a semistable vector bundle over X.
Let P ⊂ G be a subgroup. A reduction of the structure group of the G−bundle E to P is a section σ : X → E(G/P ). The pullback σ * E of E via σ is a P −bundle over X which is also called abusively a reduction of the structure group of E to P . Remark that σ * E(G) ∼ = E.
Ramanathan in his thesis, gave the following definition of semistability:
Definition 2.1. The G−bundle E is semistable if for any maximal parabolic subgroup P ⊂ G and for any reduction of the structure group σ : X −→ E(G/P ), one has deg(σ * T E(G/P ) ) 0, where T E(G/P ) = E(g/p) is the tangent bundle. We say that E is stable if the above inequality is strict.
Actually the two definitions are equivalent Theorem 2.2 ( [Ram96a], [Ram96b]). The G−bundle E is semistable if and only if it is semistable in the sense of Ramanathan.
Note that this is not true if we replace semistable with stable.
Consider now the Lie algebra g := Lie(G). The adjoint bundle ad(E) is defined to be the vector bundle E(g) associated to the adjoint representation of G on g. Since G is semisimple, ad(E) is a self-dual vector bundle over X of Lie algebras isomorphic to g. Moreover, the adjoint representation is faithful, hence E is semistable if and only if ad(E) is semistable vector bundle. Following [HM04], we say that E is ad-stable if ad(E) is stable vector bundle. Note also that there are no ad-stable bundles for a non semisimple algebraic group G (see loc.cit).
Very stability and finiteness of the Hitchin map
Let Q 1 , . . . , Q m be a basis of the algebra of invariant polynomials on g such that Q i is homogeneous of d i . This basis defines a map called the Hitchin morphism
(1) H E : H 0 (X, ad(E) ⊗ K X ) −→ W G := ⊕ m i=1 H 0 (K di X ). A G−Higgs pair is a pair (E, φ), with φ ∈ H 0 (X, ad(E) ⊗ K X ). We call (E, φ) semistable if the Higgs bundle (ad(E), φ) is semistable.
Moreover, by [Hit87], we have Proposition 3.1. With the assumption of semi-simplicity of G we have
dim(H 0 (X, ad(E) ⊗ K X )) = dim(W G ).
Let M X (G) be the moduli space of semistable G−Higgs pairs (E, φ). The Hitchin map gives a fibration
H : M X (G) −→ W G .
By Faltings [Fal93], H is proper. However, in general H E is not necessarily proper (take E to be the trivial G−bundle for example), and this is due to the fact that the canonical map
H 0 (X, ad(E) ⊗ K X ) −→ M X (G) is not proper in general. Definition 3.2. A G−bundle E over X is called very stable if the fiber of H E over 0 is equal to {0}.
Let n ⊂ g be the nilpotent cone. Since n is G−invariant, it makes sense to consider sections contained in n. Such sections are called nilpotent ( [BR94]). So the above definition is equivalent to say that the only nilpotent section of ad(E) ⊗ K X is the zero one. Indeed, let S(g * ) be the symmetric algebra on g * . Then the ring S(g * ) G of invariant polynomials on g with respect to the adjoint action is generated by trace polynomials: g → C
x → Tr(ad(x) k ) for x ∈ g and k 0. So we see that sections in the fiber of H E over 0 are exactly those being in the nilpotent cone n. For a detailed proof of this fact see [Kos63,Proposition 16].
Proposition 3.3. A very stable G−bundle over X is stable. The locus of very stable G−bundles is a nonempty open subset of M X (G).
Proof. The proof is given in [BR94, Proposition 5.2]. However for the completeness, we give a detailed argument. Note firstly that this is not true over P 1 (see [Lau88]).
Let's treat the case of vector bundles firstly. Consider a non stable vector bundle E of degree 0. So there exist vector bundles F and H such that deg(F ) = −deg(H) 0, and an exact sequence
0 → F → E → H → 0.
Using Riemann-Roch theorem one deduces that H 0 (H * ⊗ F ⊗ K X ) = 0. Let φ ′ be a non zero section in that vector space, then φ ′ gives a non trivial map G → F ⊗ K X . Finally define φ to be the composition
E → H → F ⊗ K X → E ⊗ K X .
Clearly φ is a non trivial nilpotent Higgs field of E.
Assume now that the G−bundle E is not stable. Let P ⊂ G be a maximal parabolic subgroup, put p = Lie(P ). Denote by F = σ * E a reduction of the structure group of E with respect to some section σ. Note that F (G) = E and F (g) = ad(E). Let g/p be the isotropy representation of P . Then σ * T E(G/P ) = F (g/p) and by assumption deg(F (g/p)) 0. Now let p ⊥ be the orthogonal of p with respect to the Killing form. It is actually the nilpotent radical of p. The representation of P on p ⊥ is the dual to the isotropy representation, hence the vector bundle F (p ⊥ ) has degree 0. This implies by Riemann-Roch that H 0 (F (p ⊥ ) ⊗ K X ) is non zero. But we have an exact sequence 0 → p → g → g/p → 0.
Taking the dual we get the exact sequence
0 → p ⊥ → g * → p * → 0.
Since g * ∼ = g, we deduce that F (p ⊥ ) ֒→ ad(E). This implies that E is not very stable.
This proof implies that if E is a very stable G−bundle, then for any maximal parabolic subgroup P of G, and for any reduction of the structure group σ : X → E(G/P ), we have deg(σ * T E(G/P ) ) r(g − 1), where r = dim(g/p) = rk(σ * T E(G/P ) ).
Recall that the group C * acts on H 0 (ad(E) ⊗ K X ) by (λ, φ) → λφ. Consider the weighted action of C * on W G given by
λ · (s i ) i = (λ di s i ),
where s i ∈ H 0 (X, K di X ). Then the Hitchin map H E is equivariant with respect to these actions. Now we come to our main theorem.
Theorem 3.4. The G−bundle E is very stable if and only if H E is finite.
Note that when H E is finite then it is quasi-finite. Hence E is very stable because the fiber over 0 is stable under the C * −action.
It is also not difficult to show that H E is quasi-finite when E is assumed to be very stable. Indeed assume that E is very stable. Consider the C * −action on both spaces in (1), H E is equivariant with respect to these actions. Consider the map from W G to N ∪ {−∞} given by
s −→ dim(H −1 E (s))
. This map is upper semi-continuous ([DG67, Théorème 13.1.3]). In particular its restriction to each C * −orbit is again upper semi-continuous. From this and since 0 is in the closure of any C * −orbit, we deduce that the dimension of each fiber is smaller or equal to the dimension of the special fiber over 0. But E is very stable, thus by definition dim(H −1 E (0)) = 0. It follows that dim(H −1 E (s)) 0 for any s ∈ W . Hence H is quasi-finite. However, it is still not clear why H E should be finite.
The following Theorem is a slight generalization of our main result.
Theorem 3.5. Let n 1 and f : A n → A n be a morphism given by homogeneous polynomials such that f −1 (0) = {0}. Then f is finite. In particular, it is proper.
Proof. Denote f = (P 1 , · · · , P n ) and let X 1 , · · · , X n be the coordinates on A n . Let d i = deg(P i ) 1.
For k = 1, · · · , n, and for i = 1, . . . , n, consider the polynomials P k i = P i (T 1 , · · · , 1, · · · , T n ), where 1 is at the k th position and T i = X i /X k . Since f −1 (0) = {0}, the polynomials P k 1 , . . . , P k n have no common zero in A n−1 . Hence the ideal generated by them contains 1. Let U k i be some polynomials in T i 's such that
n i=1 U k i P k i = 1.
Take c to be a positive integer bigger than d i + deg(U k i ) for all i and all k. Then we see that
X c k = n i=1Ũ k i P i , whereŨ k i = X c−di k U k i . In particular, since c > d i + deg(U k i ), we have deg(Ũ k i ) = c − d i < c. Let A = C[P 1
, · · · , P n ] and M = C[X 1 · · · , X n ]. For α = (α 1 , · · · , α n ) ∈ N n , let X α = X α1 1 · · · X αn n be the corresponding monomial in M . We claim that the A−module M is generated by the finite set S := {X α | α ∈ N n such that α i < c for all i}.
Let N be the A−module generated by S. Clearly N ⊂ M . To prove the converse inclusion, we will show, by induction on the total degree |α| = i α i , that X α ∈ N for any α ∈ N n . For α ∈ N n such that |α| < c, we have by definition X α ∈ S. Hence the basis of the induction. Now, let m be an integer such that m c and assume that for all α ∈ N n such that |α| < m we have X α ∈ N . Let α ∈ N n such that |α| = m. If for all i, α i < c then X α ∈ S and we are done. Otherwise, there exists k ∈ {1, · · · , n} such that α k c. Let β = (α 1 , · · · , α k − c, · · · , α n ). Using the above decomposition of X c k we deduce that
X α = n i=1 X βŨ k i P i .
But the polynomials X βŨ k i are homogeneous of degrees |α| − c + deg(Ũ k i ) < |α| = m. It follows by the induction hypothesis that for all i, X βŨ k i ∈ N . Hence X α ∈ N . This ends the proof.
Remark 3.6.
(1) Using Hilbert zero theorem, we see that actually the condition f −1 (0) = {0} implies that m N ⊂ P i , where m = X 1 , . . . , X n . Hence X N k ∈ P i for each k. But using this, one can not control the degrees of the coefficientsŨ k i .
(2) The condition that the polynomials P i are homogeneous means that f is equivariant with respect to an appropriate C * −actions on the base and target.
Since the Hitchin morphism H E is of finite presentation, then H E is finite if and only if it is proper, hence we deduce the following Another consequence is the following Corollary 3.8. Let E be a very stable G−bundle, then H E is surjective. In particular, for any very stable vector bundles E, if q :X s → X is the spectral curve associated to a general spectral data s ∈ W GL r , then there exists a line bundle L overX s such that q * L ∼ = E.
Proof. For the second part, use [BNR89, Proposition 3.6].
Note that the Hitchin map is never injective. To see this let d be the degree of Q l one of the generators of the ring of invariants. Let s = (s 1 , · · · , s m ) ∈ W G be such that s l = 0 and s i = 0 for all i = l. Then assuming that E is very stable, by the surjectivity of the Hitchin map, there exists a Higgs field φ such that H E (φ) = s. But then H E (ξφ) = s for any d th root of unity ξ ∈ C.
Ad-stable vs very stable SL 2 −bundles
In this section, we study the relation between very stability and ad-stability of SL 2 −bundles over X. We will show that there is no implication between these two notions.
In the following, we denote by U X (2, 0) the moduli space of semistable vector bundle of rank 2 and degree 0, and by SU X (2) the moduli space of vector bundle with trivial determinant.
Let E be an SL 2 −bundle, we denote by E v the associated vector bundle which has trivial determinant. Note that ad(E) = End 0 (E v ) the Lie algebra bundle of traceless endomorphisms of E v . Hence E is ad-stable if and only if End 0 (E v ) is stable. We assume hereafter that E (equivalently E v ) is semistable.
Lemma 4.1. If the G−bundle E is ad-stable, then E v is a stable vector bundle.
Proof. We know that E v is semistable because E is semistable G−bundle (actually it is stable by [HM04]). Assume that E v is not stable. Let η ∈ Pic 0 (X) such that
0 → η → E v → η −1 → 0.
Let ν : η −1 → η ⊗ η −2 be the canonical map. Then the composition
E v → η −1 ν −→ η ⊗ η −2 → E v ⊗ η −2 defines a non zero map φ : E v → E v ⊗ η −2 which is nilpotent, i.e. φ 2 = 0. Hence Tr(φ) = 0. So we deduce that H 0 (End 0 (E v ) ⊗ η −2 ) = 0, or, in other words, η 2 ֒→ End 0 (E v ). Hence End 0 (E v ) is not stable. So E is not ad-stable.
Let η ∈ J X [2] be a non trivial 2−torsion line bundle over X. Denote q : X η → X the associatedétale double cover. Note that the map q * : Pic 0 (X η ) −→ U X (2, 0) is defined everywhere. We define S η ⊂ U X (2, 0) to be the image of q * . Then we have Proof. If E v is not stable then E is not ad-stable by Lemma 4.1. So assume E v is stable. Now, End 0 (E v ) is semistable and H 0 (End 0 (E v )) = 0. Since it has rank three, End 0 (E v ) is not stable if and only if there exists a line bundle η ∈ J X such that H 0 (End 0 (E v ) ⊗ η) = 0. But since η = O X by the above, we have
H 0 (End 0 (E v ) ⊗ η) = H 0 (End(E v ) ⊗ η).
So we deduce, from the stability of E v , that these last spaces are non zero if and
only if E v ∼ = E v ⊗ η.
Hence taking the determinant we get η 2 = O X . From [NR75], we deduce that E v ∼ = q * L for some line bundle L ∈ J Xη , hence E v ∈ S η . The converse is straightforward. Now let κ be a theta characteristic and denote by Θ κ ⊂ SU X (2, 0) the associated theta divisor given by
Θ κ := {E ∈ SU X (2) | h 0 (E ⊗ κ) = 0}.
Then we have Proposition 4.3. The divisor Θ κ is included in the complement of the locus of very stable vector bundles.
Proof. For E ∈ Θ κ , we have an exact sequence
0 → L −1 → E → L → 0.,
where L −1 is the image of κ −1 in E. Since H 0 (X, L −1 ⊗ κ) = 0, we deduce that H 0 (X, L −2 ⊗ K X ) = 0. Let s be a non zero global section of L −2 ⊗ K X . Then the composition E ։ κ s −→ κ −1 ⊗ K X ֒→ E ⊗ K X is a non trivial Higgs field which is clearly nilpotent. This shows that E is not very stable.
It was pointed to me by Pauly that this result has been already known, see [PP18, and references therein]. Now, consider the map Ψ d : Ξ 2d × Θ d −→ Pic 0 (X η ) that associates to (M, N ) the line bundle M ⊗ q * N −1 . These maps are never surjective. Indeed, if 2d g − 2, we have dim(Im(Ψ d )) dim(Ξ 2d × Θ d ) = 2d + g < 2g − 1.
Otherwise 2d g − 1, and in this case we also have dim(Im(Ψ d )) dim(Ξ 2d × Θ d ) = 2d + 2g − 2d − 2 = 2g − 2 < 2g − 1.
Moreover, the union of the images of Ψ d , for d = 0, . . . , g − 1, is certainly not the whole of Pic 0 (X η ). Let L ∈ Pic 0 (X η ) not in the union of the images of Ψ d and such that E ′ = q * L is stable (note that such line bundle exists by [Zel16, Proposition 6.2]). Now let E = E ′ ⊗ δ −1 = q * (L ⊗ q * δ −1 ) where δ is a line bundle such that δ 2 = det(E ′ ). Then E is not ad-stable because E ∈ S, and by the remark in the beginning of this proof, it is very stable because it has no line subbundle L of degree −0, . . . , −(g − 1) such that K X L −2 has non zero global section. This ends the proof.
Mathematical Institute of the university of Bonn. E-mail address: [email protected]
Date: April 18, 2018. 2010 Mathematics Subject Classification. Primary 14H60, 14H70.
Proposition 2.3 ([HM04]). If the G−bundle E is ad-stable, then it is stable (in the sense of Ramanathan).
Corollary 3 . 7 .
37The G−bundle E is very stable if and only if H E proper.
Proposition 4 . 2 .
42The SL 2 −bundle E is ad-stable if and only if E v is stable and does not belong to the set S := η∈JX [2] {OX } S η .
Corollary 4.4. There exists an ad-stable SL 2 −bundle which is not very stable.Proof. Since the locus Θ κ ⊂ SU X (2) is a divisor, so its dimension is 3g − 4, but we see that S ∩ Θ κ inside U X (2) has dimension at most g − 1. Indeed, consider the double cover q : X η → X associated to some η. Then the vector bundle q * L has trivial determinant if and only if Nm(L) = η, where Nm : Pic(X η ) → Pic(X) is the norm map. Hence S ∩ Θ κ is a finite union (over η) of the direct image by q * of the intersection of Nm −1 (η) and the Riemann theta divisor inΘ q * κ ⊂ Pic 0 (X η ). But dim(Nm −1 (η)) = g − 1, hence the claim. Moreover, since g 2, we have g − 1 < 3g − 4. In particular, there exists a vector bundle E ∈ Θ κ S. So E is ad-stable, but not very stable.Conversely, we show the existence of a very stable SL 2 −bundle which is not ad-stable.Proposition 4.5. There exists a very stable SL 2 −bundle which is not ad-stable.Proof. Let E be a non very stable vector bundle in SU X (2), and let φ be a non zero nilpotent Higgs field. If L −1 = ker(φ) then we get the following diagramwhere ν is a non zero global section of L −2 ⊗ K X . This implies that deg(L) g − 1 and deg(L) 0 since E is semistable.Let η ∈ J X [2] non trivial, and let q : X η → X the associated unramified double cover. For d = 0, . . . , g − 1, let Θ d ⊂ Pic d (X) (resp. Ξ 2d ⊂ Pic 2d (X η )) be the locus of line bundles L such that K X ⊗ L −2 (resp. L) has a non zero global section. The locus Ξ 2d is called the Brill-Noether locus in Pic(X η ) of degree 2d and its dimension is 2d (for d g − 1). While the quotient of the locus Θ d by Pic 0 (X)[2] is isomorphic to the Brill-Noether locus in Pic(X) of degree 2g − 2d − 2, hence its dimension is given by
Spectral curves and the generalised theta divisor. A Beauville, M S Narasimhan, S Ramanan, Journal fur die reine und angewandte Mathematik. 398A. Beauville, M.S. Narasimhan, and S. Ramanan. Spectral curves and the generalised theta divisor. Journal fur die reine und angewandte Mathematik, volume 398:169-179, 1989.
An infinitesimal study of the moduli of Hitchin pairs. I Biswas, S Ramanan, Journal of the London Mathematical Society. 492I. Biswas and S. Ramanan. An infinitesimal study of the moduli of Hitchin pairs. Journal of the London Mathematical Society, 49(2):219-231, 1994.
Eléments de géométrie algébrique IV, Etude locale des schémas et des morphismes de schémas. Publications mathématiques de l'I. J Dieudonné, .H.É.SA Grothendieck, .H.É.SJ. Dieudonné and A. Grothendieck. Eléments de géométrie algébrique IV, Etude locale des schémas et des morphismes de schémas. Publications mathématiques de l'I.H.É.S, 1967.
Stable G-bundles and projective connections. G Faltings, J. Algebraic Geom. 23G. Faltings. Stable G-bundles and projective connections. J. Algebraic Geom., 2(3):507- 568, 1993.
Stable bundles and integrable systems. N Hitchin, Duke Math. J. 541N. Hitchin. Stable bundles and integrable systems. Duke Math. J., 54(1):91-114, 1987.
Note on the stability of principal bundles. D Hyeon, D Murphy, Proceedings of the American Mathematical Society. 132D. Hyeon and D. Murphy. Note on the stability of principal bundles. Proceedings of the American Mathematical Society, 132:2205-2213, 2004.
Lie group representations on polynomial rings. Bertram Kostant, American Journal of Mathematics. 853Bertram Kostant. Lie group representations on polynomial rings. American Journal of Mathematics, 85(3):327-404, 1963.
Un analogue global du cône nilpotent. G Laumon, Duke Mathematical Journal. 572G. Laumon. Un analogue global du cône nilpotent. Duke Mathematical Journal, 57(2):647-671, 1988.
Generalised Prym varieties as fixed points. M S Narasimhan, S Ramanan, Journal of the Indian Mathematical Society. 391-4M. S. Narasimhan and S. Ramanan. Generalised Prym varieties as fixed points. Journal of the Indian Mathematical Society, 39(1-4), 1975.
The wobbly divisors of the moduli space of rank-2 vector bundles. S Pal, C Pauly, arXive:1803.11315PreprintS. Pal and C. Pauly. The wobbly divisors of the moduli space of rank-2 vector bundles. Preprint, arXive:1803.11315, 2018.
Very stable bundles and properness of the Hitchin map. C Pauly, A Peón-Nieto, Geometriae Dedicata. C. Pauly and A. Peón-Nieto. Very stable bundles and properness of the Hitchin map. Geometriae Dedicata, pages 1-6, 2018.
Moduli for principal bundles over algebraic curves: I. A Ramanathan, Proc. Indian Acad. Sci. (Math. Sci.). A. Ramanathan. Moduli for principal bundles over algebraic curves: I. Proc. Indian Acad. Sci. (Math. Sci.), page 106: 301, 1996.
Moduli for principal bundles over algebraic curves: II. A Ramanathan, Proc. Indian Acad. Sci. (Math. Sci.). 421A. Ramanathan. Moduli for principal bundles over algebraic curves: II. Proc. Indian Acad. Sci. (Math. Sci.), page 106: 421, 1996.
Hitchin systems for invariant and anti-invariant vector bundles. H Zelaci, arXiv:1612.06910PreprintH. Zelaci. Hitchin systems for invariant and anti-invariant vector bundles. Preprint, arXiv:1612.06910, 2016.
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[] |
[
"Learning from the Success of MPI",
"Learning from the Success of MPI"
] |
[
"William D Gropp \nMathematics and Computer Science Division\nArgonne National Laboratory\nArgonne\n"
] |
[
"Mathematics and Computer Science Division\nArgonne National Laboratory\nArgonne"
] |
[] |
The Message Passing Interface (MPI) has been extremely successful as a portable way to program high-performance parallel computers. This success has occurred in spite of the view of many that message passing is difficult and that other approaches, including automatic parallelization and directive-based parallelism, are easier to use. This paper argues that MPI has succeeded because it addresses all of the important issues in providing a parallel programming model.
|
10.1007/3-540-45307-5_8
|
[
"https://arxiv.org/pdf/cs/0109017v1.pdf"
] | 3,259,974 |
cs/0109017
|
c43c5c1e54b0834cb1a2f5031aa669f2237ae384
|
Learning from the Success of MPI
13 Sep 2001
William D Gropp
Mathematics and Computer Science Division
Argonne National Laboratory
Argonne
Learning from the Success of MPI
13 Sep 2001Illinois 60439gropp@mcsanlgovWWW home page: wwwmcsanlgov/~gropp
The Message Passing Interface (MPI) has been extremely successful as a portable way to program high-performance parallel computers. This success has occurred in spite of the view of many that message passing is difficult and that other approaches, including automatic parallelization and directive-based parallelism, are easier to use. This paper argues that MPI has succeeded because it addresses all of the important issues in providing a parallel programming model.
Introduction
The Message Passing Interface (MPI) is a very successful approach for writing parallel programs. Implementations of MPI exist for most parallel computers, and many applications are now using MPI as the way to express parallelism (see [1] for a list of papers describing applications that use MPI). The reasons for the success of MPI are not obvious. In fact, many users and researchers complain about the difficulty of using MPI. Commonly raised issues include the complexity of MPI (often as measured by the number of functions), performance issues (particularly the latency or cost of communicating short messages), and the lack of compile or runtime help (e.g., compiler transformations for performance; integration with the underlying language to simplify the handling of arrays, structures, and native datatypes; and debugging). More subtle issues, such as the complexity of nonblocking communication and the lack of elegance relative to a parallel programming language, are also raised [2]. With all of these criticisms, why has MPI enjoyed such success?
One might claim that MPI has succeeded simply because of its portability, that is, the ability to run an MPI program on most parallel platforms. But while portability was certainly a necessary condition, it was not sufficient. After all, there were other, equally portable programming models, including many message-passing and communication-based models. For example, the socket interface was (and remains) widely available and was used as an underlying communication layer by other parallel programming packages, such as PVM [3] and p4 [4]. An obvious second requirement is that of performance: the ability of the programming model to deliver the available performance of the underlying hardware. This clearly distinguishes MPI from interfaces such as sockets. However, even this is not enough. This paper argues that six requirements must all be satisfied for a parallel programming model to succeed, that is, to be widely adopted. Programming models that address a subset of these issues can be successfully applied to a subset of applications, but such models will not reach a wide audience in high-performance computing.
Necessary Properties
The MPI programming model describes how separate processes communicate. In MPI-1 [5], communication occurs either through point-to-point (two-party) message passing or through collective (multiparty) communication. Each MPI process executes a program in an address space that is private to that process.
Portability
Portability is the most important property of a programming model for highperformance parallel computing. The high-performance computing community is too small to dictate solutions and, in particular, to significantly influence the direction of commodity computing. Further, the lifetime of an application (often ten to twenty years, rarely less than five years) greatly exceeds the lifetime of any particularly parallel hardware. Hence, any application must be prepared to run effectively on many generations of parallel computer, and that goal is most easily achieved by using a portable programming model.
Portability, however, does not require taking a "lowest common denominator" approach. A good design allows the use of performance-enhancing features without mandating them. For example, the message-passing semantics of MPI allows for the direct copy of data from the user's send buffer to the receive buffer without any other copies. 1 However, systems that can't provide this direct copy (because of hardware limitations or operating system restrictions) are permitted, under the MPI model, to make one or more copies. Thus MPI programs remain portable while exploiting hardware capabilities.
Unfortuately, portability does not imply portability with performance, often called performance portability. Providing a way to achieve performance while maintaining portability is the second requirement.
Performance
MPI enables performance of applications in two ways. For small numbers of processors, MPI provides an effective way to manage the use of memory. To understand this, consider a typical parallel computer as shown in Figure 1.
The memory near the CPU, whether it is a large cache (symmetric multiprocessor) or cache and memory (cluster or NUMA), may be accessed more rapidly than far-away memory. Even for shared-memory computers, the ratio of the number of cycles needed to access memory in L1 cache and main memory is roughly a hundred; for large, more loosely connected systems the ratio can exceed ten to one hundred thousand. This large ratio, even between the cache and Fig. 1. A typical parallel computer local memory, means that applications must carefully manage memory locality if they are to achieve high performance.
The separate processes of the MPI programming model provide a natural and effective match to this property of the hardware. This is not a new approach. The C language provides register, originally intended to aid compilers in coping with a two-level memory hierarchy (registers and main memory). Some parallel languages, such as HPF [6], UPC [7], or CoArray Fortran [8], distiguish between local and shared data. Even programming models that do not recognize a distinction between local and remote memory, such as OpenMP, have implementations that often require techniques such as "first touch" to ensure that operations make effective use of cache. The MPI model, based on communicating processes, each with its own memory, is a good match to current hardware.
For large numbers of processors, MPI also provides effective means to develop scalable algorithms and programs. In particular, the collective communication and computing routines such as MPI Allreduce provide a way to express scalable operations without exposing system-specific features to the programmer. Also important for supporting scalability is the ability to express the most powerful scalable algorithms; this is discussed in Section 2.4.
Another contribution to MPI's performance comes from its ability to work with the best compilers; this is discussed in Section 2.5. Also discussed there is how MPI addresses the performance-tradeoffs in using threads with MPI programs.
Unfortunately, while MPI achieves both portability and performance, it does not achieve perfect performance portability, defined as providing a single source that runs at (near) acheivable peak performance on all platforms. This lack is sometimes given as a criticism of MPI, but it is a criticism that most other programming models also share. For example, Dongarra et al [9] describe six different ways to implement matrix-matrix multiply in Fortran for a single processor; not only is no one of the six optimal for all platforms but none of the six are optimal on modern cache-based systems. Another example is the very existence of vendor-optimized implementations of the Basic Linear Algebra Subroutines (BLAS). These are functionally simple and have implementations in Fortran and C; if compilers (good as they are) were capable of producing optimal code for these relatively simple routines, the hand-tuned (or machined-tuned [10]) versions would not be necessary. Thus, while performance portability is a desir-able goal, it is unreasonable to expect parallel programming models to provide it when uniprocessor models cannot. This difficulty also explains why relying on compiler-discovered parallelism has usually failed: the problem remains too difficult. Thus a successful programming model must allow the programmer to help.
Simplicity and Symmetry
The MPI model is often criticized as being large and complex, based on the number of routines (128 in MPI-1 with another 194 in MPI-2). The number of routines is not a relevant measure, however. Fortran, for example, has a large number of intrinsic functions; C and Java rely on a large suite of library routines to achieve external effects such as I/O and graphics; and common development frameworks have hundreds to thousands of methods.
A better measure of complexity is the number of concepts that the user must learn, along with the number of exceptions and special cases. Measured in these terms, MPI is actually very simple.
Using MPI requires learning only a few concepts. Many MPI programs can be written with only a few routines; several subsets of routines are commonly recommended, including ones with as few as six functions. Note the plural: for different purposes, different subsets of MPI are used. For example, some recommend using only collective communiation routines; others recommend only a few of the point-to-point routines. One key to the success of MPI is that these subsets can be used without learning the rest of MPI; in this sense, MPI is simple. Note that a smaller set of routines would not have provided this simplicity because, while some applications would find the routines that they needed, others would not.
Another sign of the effective design in MPI is the use of a single concept to solve multiple problems. This reduces both the number of items that a user must learn and the complexity of the implementation. For example, the MPI communicator both describes the group of communicating processes and provides a separate communication context that supports component-oriented software, described in more detail in Section 2.4. Another example is the MPI datatype; datatypes describe both the type (e.g., integer, real, or character) and layout (e.g., contiguous, strided, or indexed) of data. The MPI datatype solves the two problems of describing the types of data to allow for communication between systems with different data representations and of describing noncontiguous data layouts to allow an MPI implementation to implement zero-copy data transfers of noncontiguous data.
MPI also followed the principle of symmetry: wherever possible, routines were added to eliminate any exceptions. An example is the routine MPI Issend. MPI provides a number of different send modes that correspond to different, wellestablished communication approaches. Three of these modes are the regular send (MPI Send) and its nonblocking versions (MPI Isend), and the synchronous send (MPI Ssend). To maintain symmetry, MPI also provides the nonblocking synchronous send MPI Issend. This send mode is meaningful (see [11, Section 7.6.1]) but is rarely used. Eliminating it would have removed a routine, slightly simplifying the MPI documentation and implementation. It would have created an exception, however. Instead of each MPI send mode having a nonblocking version, only some send modes would have nonblocking versions. Each such exception adds to the burden on the user and adds complexity: it is easy to forget about a routine that you never use; it is harder to remember arbitrary decisions on what is and is not available.
A place where MPI may have followed the principle of symmetry too far is in the large collection of routines for manipulating groups of processes. Particularly in MPI-1, the single routine MPI Comm split is all that is needed; few users need to manipulate groups at all. Once a routine working with MPI groups was introduced, however, symmetry required completing the set. Another place is in canceling of sends, where significant implementation complexity is required for an operation of dubious use.
Of course, more can be done to simplify the use of MPI. Some possible approaches are discussed in Section 3.1.
Modularity
Component-oriented software is becoming increasingly important. In commecial software, software components implementing a particular function are used to implement a clean, maintainable service. In high-performance computing, components are less common, with many applications being built as a monolithic code. However, as computational algorithms become more complex, the need to exploit software components embodying these algorithms increases.
For example, many modern numerical algorithms for the solution of partial differential equations are hierarchical, exploiting the structure of the underlying solution to provide a superior and scalable solution algorithm. Each level in that hierarchy may require a different solution algorithm; it is not unusual to have each level require a different decomposition of processes. Other examples are intelligent design automation programs that run application components such as fluid solvers and structural analysis codes under the control of a optimization algorithm.
MPI supports component-oriented software. Both describe the subset of processes participating in a component and to ensure that all MPI communication is kept within the component, MPI introduced the communicator. 2 Without something like a communicator, it is possible for a message sent by one component and intended for that component to be received by another component or by user code. MPI made reliable libraries possible.
Supporting modularity also means that certain powerful variable layout tricks (such as assuming that the variable a in an SPMD program is at the same address on all processors) must be modified to handle the case where each process may have a different stack-use history and variables may be dynamically allocated with different base addresses. Some programming models have assumed that all processes have the same layout of local variables, making it difficult or impossible to use those programming models with modern adaptive algorithms.
Modularity also deals with the complexity of MPI. Many tools have been built using MPI to provide the communication substrate; these tools and libraries provide the kind of easy-to-use interface for domain-specific applications that some developers feel are important; for example, some of these tools eliminate all evidence of MPI from the user program. MPI makes those tools possible. Note that the user base of these domain-specific codes may be too small to justify vendor-support of a parallel programming model.
Composability
One of the reasons for the continued success of Unix is the ease with which new solutions can be built by composing existing applications.
MPI was designed to work with other tools. This capability is vital, because the complexity of programs and hardware continues to increase. For example, the MPI specification was designed from the beginning to be thread-safe, since threaded parallelism was seen by the MPI Forum as a likely approach to systems built from a collection of SMP nodes. MPI-2 took this feature even further, acknowledging that there are performance tradeoffs in different degrees of threadedness and providing a mechanism for the user to request a particular level of thread support from the MPI library. Specificically, MPI defines several degrees of thread support. The first, called MPI THREAD SINGLE, specifies that there is a single thread of execution. This allows an MPI implementation to avoid the use of thread-locks or other techniques necessary to ensure correct behavior with multithreaded codes. Another level of thread support, MPI THREAD FUNNELLED, specifies that the process may have multiple threads but all MPI calls are made by one thread. This matches the common use of threads for loop parallelism, such as the most common uses of OpenMP. A third level, MPI THREAD MULTIPLE, allows multiple threads to make MPI calls. While these levels of thread support do introduce a small degree of complexity, they reflect MPI's pragmatic approach to providing a workable tool for high-performance computing.
The design of MPI as a library means that MPI operations cannot be optimized by a compiler. However, it also means that any MPI library can exploit the newest and best compilers and that the compiler can be developed without worrying about the impact of MPI on the generated code-from the compiler's point of view, MPI calls are simply generic function calls. 3 The ability of MPI to exploit improvements in other tools is called composability. Another example is in debuggers; because MPI is simply a library, any debugger can be used with MPI programs. Debuggers that are capable of handling multiple processes, such as TotalView [14], can immediately be used to debug MPI programs. Additional refinements, such as an interface to an abstraction of message passing that is described in [15], allows users to use the debugger to discover information about pending communication and unreceived messages.
More integrated approaches, such as language extensions, have the obvious benefits, but they also have significant costs. A major cost is the difficulty of exploiting advances in other tools and of developing and maintaining a large, integrated system.
OpenMP is an example of a programming model that achieves the effect of composability with the compilers because OpenMP requires essentially orthogonal changes to the compiler; that is, most of the compiler development can ignore the addition of OpenMP in a way that more integrated languages cannot.
Completeness
MPI provides a complete programming model. Any parallel algorithm can be implemented with MPI. Some parallel programming models have sacrified completeness for simplicity. For example, a number of programming models have required that synchronization happens only collectively for all processes or tasks. This requirement significantly simplifies the programming model and allows the use of special hardware affecting all processes. Many existing programs also fit into this model; data-parallel programs are natural candidates for this model. But as discussed in Section 2.4, many programs are becoming more complex and are exploiting software components. Such applications are difficult, if not impossible, to build using restrictive programming models.
Another way to look at this is that while many programs may not be easy under MPI, no program is impossible. MPI is sometimes called the "assembly language" of parallel programming. Those making this statement forget that C and Fortran have also been described as portable assembly languages. The generality of the approach should not be mistaken for an unnecessary complexity.
Summary
Six different requirements have been discussed, along with how MPI addresses each. Each of these is necessary in a general-purpose parallel programming system.
Portability and performance are clearly required. Simplicity and symmetry cater to the user and make it easy to learn and use safely. Composibility is required to prevent the approach from being left behind by the advance of other tools such as compilers and debuggers.
Modularity, like completeness, is required to ensure that tools can be built on top of the programming model. Without modularity, a programming model is suitable only for turnkey applications. While those may be important and easy to identify as customers, they represent the past rather than the future.
Completeness, like modularity, is required to ensure that the model supports a large enough community. While this does not mean that everyone uses every function, it means that the functionality that a user may need is likely to be present. An early poll of MPI users [16] in fact found that while no one was using all of the MPI-1 routines, essentially all MPI-1 routines were in use by someone.
The open standards process (see [17] for a description of the process used to develop MPI) was an important component in its success. Similar processes are being adopted by others; see [18] for a description of the principles and advantages of an open standards process.
Where Next?
MPI is not perfect. But any replacement will need to improve on all that MPI offers, particularly with respect to performance and modularity, without sacrificing the ability to express any parallel program. Three directions are open to investigation: improvements in the MPI programming model, better MPI implementations, and fundamentally new approaches to parallel computing.
Improving MPI
Where can MPI be improved? A number of evolutionary enhancements are possible, many of which can be made by creating tools that make it easier to build and maintain MPI programs.
1. Simpler interfaces. A compiler (or a preprocessor) could provide a simpler, integrated syntax. For example, Fortran 90 array syntax could be supported without requiring the user to create special MPI datatypes. Similarly, the MPI datatype for a C structure could be created automatically. Some tools for the latter already exist. Note that support for array syntax is an example of support for a subset of the MPI community, many of whom use data structures that do not map easily onto Fortran 90 arrays. A precompiler approach would maintain the composability of the tools, particularly if debuggers understood preprocessed code. 2. Elimination of function calls. There is no reason why a sophisticated system cannot remove the MPI routine calls and replace them with inline operations, including handling message matching. Such optimizations have been performed for Linda programs [19] and for MPI subsets [20]. Many compilers already perform similar operations for simple numerical functions like abs and sin. This enhancement can be achieved by using preprocessors or precompilers and thus can maintain the composability of MPI with the best compilers. 3. Additional tools and support for correctness and performance debugging.
Such tools include editors that can connect send and receive operations so that both ends of the operation are presented to the programmer, or performance tools for massively parallel programs. (Tools such as Vampir and Jumpshot [21] are a good start, but much more can be done to integrate the performance tool with source-code editors and performance predictors.)
4. Changes to MPI itself, such as read-modify-write additions to the remote memory access operations in MPI-2. It turns out to be surprisingly difficult to implement an atomic fetch-and-increment operation [22,Section 6.5.4] in MPI-2 using remote memory operations (it is quite easy using threads, but that usually entails a performance penalty).
Improving MPI Implementations
Having an implementation of MPI is just the beginning. Just as the first compilers stimulated work in creating better compilers by finding better ways to produce quality code, MPI implementations are stimulating work on better approaches for implementing the features of MPI. Early work along this line looked at better ways to implement the MPI datatypes [23,24]. Other interesting work includes the use of threads to provide a lightweight MPI implementation [25,26]. This work is particularly interesting because it involves code transformations to ensure that the MPI process model is preserved within a single, multithreaded Unix process.
In fact, several implementations of MPI fail to achieve the available asymptotic bandwidth or latency. For example, at least two implementations from different vendors perform unnecessary copies (in one case because of layering MPI over a lower-level software that does not match MPI's message-passing semantics). These implementations can be significantly improved. They also underscore the risk in evaluating the design of a programming model based on a particular implementation.
1. Improvement of the implementation of collective routines for most platforms.
One reason, ironically, is that the MPI point-to-point communication routines on which most MPI implementations build their collective routines are too high level. An alternative approach is to build the collective routines on top of stream-oriented methods that understand MPI datatypes. 2. Optimization for new hardware, such as implementations of VIA or Infiniband. Work in this direction is already taking place, but more can be done, particularly for collective (as opposed to point-to-point) communication. 3. Wide area networks (1000 km and more). In this situation, the steps used to send a message can be tuned to this high-latency situation. In particular, approaches that implement speculative receives [27], strategies that make use of quality of service [28], or alternatives to IP/TCP may be able to achieve better performance. 4. Scaling to more than 10,000 processes. Among other things, this requires better handling of internal buffers; also, some of the routines for managing process mappings (e.g., MPI Graph create) do not have scalable definitions. 5. Parallel I/O, particularly for clusters. While parallel file systems such as PVFS [29] provide support for I/O on clusters, much more needs to be done, particularly in the areas of communication aggregation and in reliability in the presence of faults.
6. Fault tolerance. The MPI intercommunicator (providing for communication between two groups of processes) provides an elegant mechanism for generalizing the usual "two party" approach to fault tolerance. Few MPI implementations support fault tolerance in this situation, and little has been done to develop intercommunicator collective routines that provide a well-specified behavior in the presence of faults. 7. Thread-safe and efficient implementations for the support of "mixed model"
(message-passing plus threads) programming. The need to ensure threadsafety of an MPI implementation used with threads can significantly increase latency. Architecting an MPI implementation to avoid or reduce these penalties remains a challenge.
New Directions
In addition to improving MPI and enhancing MPI implementations, more revolutionary efforts should be explored. One major need is for a better match of programming models to the multilevel memory hierarchies that the speed of light imposes, without adding unmanageable complexity. Instead of denying the importance of hierarchical memory, we need a memory centric view of computing.
MPI's performance comes partly by accident; the two-level memory model is better than a one-level memory model at allowing the programmer to work with the system to achieve performance. But a better approach needs to be found.
Two branches seem promising. One is to develop programming models targeted at hardware similar in organization to what we have today (see Figure 1). The other is to codevelop both new hardware and new programming models. For example, hardware built from processor-in-memory, together with hardware support for rapid communication of functions might be combined with a programming model that assumed distributed control. The Tera MTA architecture may be a step in such a direction, by providing extensive hardware support for latency hiding by extensive use of hardware threads. In either case, better techniques must be provided for both data transfer and data synchronization.
Another major need is to make it harder to write incorrect programs. A strength of MPI is that incorrect programs are usually deterministic, simplifying the debugging process compared to the race conditions that plague sharedmemory programming. The synchronous send modes (e.g., MPI Ssend) may also be used to ensure that a program has no dependence on message buffering.
Conclusion
The lessons from MPI can be summed up as follows: It is more important to make the hard things possible than it is to make the easy things easy. Future programming models must concentrate on helping programmers with what is hard, including the realities of memory hierarchies and the difficulties in reasoning about concurrent threads of control.
This is sometimes called a zero-copy transfer.
The context part of the communicator was inspired by Zipcode[12].
There are some conflicts between the MPI model and the Fortran language; these are discussed in[13, Section 10.2.2]. The issues are also not unique to MPI; for example, any asynchronous I/O library faces the same issues with Fortran.
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Introduction to UPC and language specification. W W Carlson, J M Draper, D Culler, K Yelick, E Brooks, K Warren, CCS-TR-99-157IDA, Bowie, MDCenter for Computing SciencesTechnical ReportCarlson, W.W., Draper, J.M., Culler, D., Yelick, K., Brooks, E., Warren, K.: In- troduction to UPC and language specification. Technical Report CCS-TR-99-157, Center for Computing Sciences, IDA, Bowie, MD (1999)
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Using MPI: Portable Parallel Programming with the Message Passing Interface. W Gropp, E Lusk, A Skjellum, MIT PressCambridge, MA2nd editionGropp, W., Lusk, E., Skjellum, A.: Using MPI: Portable Parallel Programming with the Message Passing Interface, 2nd edition. MIT Press, Cambridge, MA (1999)
The design and evolution of Zipcode. A Skjellum, S G Smith, N E Doss, A P Leung, M Morari, Parallel Computing. 20Skjellum, A., Smith, S.G., Doss, N.E., Leung, A.P., Morari, M.: The design and evolution of Zipcode. Parallel Computing 20 (1994) 565-596
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|
[] |
[
"The atmospheric charged kaon/pion ratio using seasonal variation methods",
"The atmospheric charged kaon/pion ratio using seasonal variation methods"
] |
[
"E W Grashorn [email protected]. \nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n\nCenter for Cosmology and Astro-Particle Physics\nOhio State University\n43210ColumbusOHUSA\n",
"J K De Jong \nPhysics Division\nIllinois Institute of Technology\n60616ChicagoIllinoisUSA\n\nDepartment of Physics\nUniversity of Oxford\nDenys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom\n",
"M C Goodman \nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n",
"A Habig \nDepartment of Physics\nUniversity of Minnesota -Duluth\n55812DuluthMinnesotaUSA\n",
"M L Marshak \nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"S Mufson \nIndiana University\n47405BloomingtonIndianaUSA\n",
"S Osprey \nDepartment of Physics\nUniversity of Oxford\nDenys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom\n",
"P Schreiner \nPhysics Department\nBenedictine University\n60532LisleIllinoisUSA\n"
] |
[
"University of Minnesota\n55455MinneapolisMinnesotaUSA",
"Center for Cosmology and Astro-Particle Physics\nOhio State University\n43210ColumbusOHUSA",
"Physics Division\nIllinois Institute of Technology\n60616ChicagoIllinoisUSA",
"Department of Physics\nUniversity of Oxford\nDenys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom",
"Argonne National Laboratory\n60439ArgonneIllinoisUSA",
"Department of Physics\nUniversity of Minnesota -Duluth\n55812DuluthMinnesotaUSA",
"University of Minnesota\n55455MinneapolisMinnesotaUSA",
"Indiana University\n47405BloomingtonIndianaUSA",
"Department of Physics\nUniversity of Oxford\nDenys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom",
"Physics Department\nBenedictine University\n60532LisleIllinoisUSA"
] |
[] |
Observed since the 1950's, the seasonal effect on underground muons is a well studied phenomenon. The interaction height of incident cosmic rays changes as the temperature of the atmosphere changes, which affects the production height of mesons (mostly pions and kaons). The decay of these mesons produces muons that can be detected underground. The production of muons is dominated by pion decay, and previous work did not include the effect of kaons. In this work, the methods of Barrett and MACRO are extended to include the effect of kaons. These efforts give rise to a new method to measure the atmospheric K/π ratio at energies beyond the reach of current fixed target experiments. These methods were applied to data from the MINOS far detector. A method is developed for making these measurements at other underground detectors, including OPERA, Super-K, IceCube, Baksan and the MINOS near detector.
|
10.1016/j.astropartphys.2009.12.006
|
[
"https://arxiv.org/pdf/0909.5382v3.pdf"
] | 118,545,711 |
0909.5382
|
ec943663bcc4247f2fa099c80ed44c092e479086
|
The atmospheric charged kaon/pion ratio using seasonal variation methods
16 Aug 2010 17 August 2010
E W Grashorn [email protected].
University of Minnesota
55455MinneapolisMinnesotaUSA
Center for Cosmology and Astro-Particle Physics
Ohio State University
43210ColumbusOHUSA
J K De Jong
Physics Division
Illinois Institute of Technology
60616ChicagoIllinoisUSA
Department of Physics
University of Oxford
Denys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom
M C Goodman
Argonne National Laboratory
60439ArgonneIllinoisUSA
A Habig
Department of Physics
University of Minnesota -Duluth
55812DuluthMinnesotaUSA
M L Marshak
University of Minnesota
55455MinneapolisMinnesotaUSA
S Mufson
Indiana University
47405BloomingtonIndianaUSA
S Osprey
Department of Physics
University of Oxford
Denys Wilkinson Building, Keble RoadOX1 3RHOxfordUnited Kingdom
P Schreiner
Physics Department
Benedictine University
60532LisleIllinoisUSA
The atmospheric charged kaon/pion ratio using seasonal variation methods
16 Aug 2010 17 August 2010Preprint submitted to Astroparticle Physics* Corresponding author.
Observed since the 1950's, the seasonal effect on underground muons is a well studied phenomenon. The interaction height of incident cosmic rays changes as the temperature of the atmosphere changes, which affects the production height of mesons (mostly pions and kaons). The decay of these mesons produces muons that can be detected underground. The production of muons is dominated by pion decay, and previous work did not include the effect of kaons. In this work, the methods of Barrett and MACRO are extended to include the effect of kaons. These efforts give rise to a new method to measure the atmospheric K/π ratio at energies beyond the reach of current fixed target experiments. These methods were applied to data from the MINOS far detector. A method is developed for making these measurements at other underground detectors, including OPERA, Super-K, IceCube, Baksan and the MINOS near detector.
Introduction
When cosmic rays interact in the stratosphere, mesons are produced in the primary hadronic shower. These mesons either interact again and produce lower energy hadronic cascades, or decay into high energy muons which can penetrate to detectors deep underground. The temperature of the stratosphere remains nearly constant, only changing slowly over longer timescales such as seasons (with the exception of the occasional Sudden Stratospheric Warming events observed during wintertime at high latitudes [1]). An increase in temperature of the stratosphere causes a decrease in density, reducing the chance of meson interaction, resulting in a larger fraction decaying to produce muons. This results in a higher muon rate observed deep underground [2,3,4,5,6]. The effect increases as higher energy muons are sampled, because higher energy mesons with increased lifetimes (due to time dilation) are involved. This effect permits the measurement of the atmospheric charged kaon/pion production ratio. The rate of low energy muons at the surface of the earth is also affected by the temperature because the varying production altitude changes the chances of the muon decaying before reaching earth, but this effect is not relevant for detectors deeper than 50 mwe [3] (meters water equivalent).
Muon Intensity Underground
The intensity of muons underground is directly related to the production of mesons in the stratosphere by hadronic interactions between cosmic rays and the nuclei of air molecules. It is assumed that meson production falls off exponentially as e −X/Λ N where Λ N is the absorption mean free path of the cosmic rays and X is the slant depth of atmospheric material traversed. It is also assumed that the mesons retain the same direction as their progenitors, that the cosmic ray sky is isotropic in solid angle at the top of the atmosphere, and ionization is neglected. These assumptions are particularly valid for the large energies of the mesons that produce muons seen in deep underground detectors such as Baksan [7], Super-K [8], IceCube [9] and MINOS far detector (MINOS FD) and near detector (MINOS ND) [10]. In this approximation, Λ N is constant. Two meson absorption processes will be considered: further hadronic interactions, dX/Λ M , where dX is the amount of atmosphere traversed, and M → µν µ decay. The fractional loss of mesons by decay is given by
m M c p r dX ρcτ 0 ,(1)
where ρ = air density, τ 0 = mean M lifetime (at rest) [2], and p r is the meson rest frame momentum. M is either a π or K meson (charm and heavier meson production doesn't become important until ∼ 10 5 TeV). For an isothermal, exponentially vanishing atmosphere, the atmospheric scale height H(T) = RT/Mg. The density ρ is then related to X by ρ = X cos θ/H(T ). The critical energy ǫ M , the energy that separates the atmospheric interaction and decay regimes, is given by
ǫ M = m M c 2 H(T ) cτ M .(2)
Since most interactions take place in the first few interaction lengths [11], and to first order H(T ) ≈ H 0 = 6.5 km, ǫ π = 0.115 TeV, ǫ K = 0.850 TeV, the differential meson intensity M(E, X, cos θ) can be written as a function of X [2,11]:
dM dX = Z N M Λ N N 0 (E)e −X/Λ N − M(E, X, cos θ) 1 Λ M + ǫ M EX cos θ(3)
for relativistic M, where N 0 (E) is the differential M production spectrum which has the form E
−(γ+1) M
, Λ N is the nucleon interaction length, and Z N M is the spectrum-weighted inclusive cross section moment. This differential equation is straightforward to solve using an integrating factor and rewriting [2,11]:
M(E, X, θ) = Z N M Λ N N 0 (E)e −X/Λ M X −ǫ M /E cos θ X 0 X ′ǫ M /E cos θ e −X ′ /Λ ′ M dX ′ = Z N M Λ N N 0 (E)e −X/Λ M X × 1 ǫ M /E cos θ + 1 − X/Λ ′ M ǫ M /E cos θ + 2 + 1 2! (X/Λ ′ M ) 2 ǫ M /E cos θ + 3 − ... ,(4)where 1/Λ ′ M ≡ 1/Λ N − 1/Λ M .
Now that an expression for the production and propagation of mesons through the atmosphere has been found, a function describing the production of muons must be found. Muons are produced from mesons via the two body decay process M → µν. The rest frame momentum for this decay is p r = (1 − m 2 µ /m 2 M )m M /2 (since the neutrino has negligible mass). The differential flux per unit cross section is proportional to the differential flux per energy, which can be written
dn dE = Bm M 2p r P L ,(5)
where B is the branching ratio and P L is the momentum of the decaying particle in the lab frame. The muon production spectrum for meson M parents is given by Gaisser [11]:
P µ (E, X, cos θ) = mesons Emax E min dn(E, E ′ ) dE ǫ M EX cos θ M(E ′ , X, cos θ)dE ′ .(6)
Inserting Eq. 5 into the muon production spectrum (Eq. 6) gives
P µ (E, X, cos θ) = mesons ǫ M X cos θ(1 − r M ) Eµ/r M Eµ dE E M(E, X, cos θ) E ,(7)
where r M = m 2 µ /m 2 M . Muons are sampled by detectors at one particular depth, so the production spectrum must be integrated over the whole atmosphere to find the energy spectrum of interest.
The relevant energy spectrum is written:
dI µ dE µ = ∞ 0 P µ (E, X)dX ≃ C 0 × E −(γ+1) A π 1 + 1.1E µ cos θ/ǫ π + 0.635 A K 1 + 1.1E µ cos θ/ǫ K ,(8)
where γ = 1.7 is the muons spectral index [12], the branching ratio B(K → ν µ µ) = 0.635, and B(π → ν µ µ) ≃ = 1. The parameters A π,K are constants involving the amount of inclusive meson production in the forward fragmentation region, the masses of the mesons and muons, and the muon spectral index [11]:
A π(K) ≡ Z N,π(K) (1 − r π(K) ) 1 − (r π(K) ) γ+1 γ + 1 .(9)
The integral of the production spectrum can be written in the form [2]
I µ (E) = ∞ E th dE µ dI µ dE µ .(10)
This is the total number of muons with energy greater than the minimum required to reach an underground detector. The threshold surface energy required for a muon to survive to slant depth d(θ, φ) (mwe) increases exponentially as a function of d and parameters a(E) and b(E) [11]. Since a and b depend on energy, an iterative procedure can be used to find the threshold energy [12]:
E th = E n+1 th (θ, φ) = E n + a b e bd(θ,φ) − a b ,(11)
where the energy-dependent parameters a = 0.00195 + 1.09 × 10 −4 ln(E) GeV cm 2 /g and b = 1.381 × 10 −6 + 3.96 × 10 −6 ln(E) GeV cm 2 /g [12], at column depth d(θ, φ). The threshold energy at the minimum depth of the detectors considered in Sec. 1 are shown in Table 1. Eq. 10 is approximated [2] as
I µ ≃ C 1 × E −γ th 1 γ + (γ + 1)1.1E th cos θ/ǫ π + 0.054 γ + (γ + 1)1.1E th cos θ/ǫ K ,(12)
where 0.635·A K /A π ·r(K/π) = 0.054 [11], [5] and r(K/π) is the atmospheric K/π ratio.
Temperature Effect on Muon Intensity
The temperature changes that occur in the atmosphere are not uniform, instead occurring at multiple levels, and neither muon nor meson production occurs at one particular level (see Figs. 1, 2). The perturbations that variations in temperature cause in muon intensity are small, and as a result properly chosen atmospheric weights can be used to approximate the effective temperature of the atmosphere as a whole,
T eff . Define η(X) ≡ (T (X) − T eff )/T eff , and ǫ M = ǫ 0 M (1 + η), where ǫ 0 M
is the constant value of ǫ M when T = T eff . This is the temperature that would cause the observed Table 1 Threshold muon energy for current underground detectors.
Detector
Min. Depth (MWE) E th (GeV)
MINOS ND [13] 225 51
Baksan [3] 850 234
IceCube [6] 1450 466
MINOS FD [5] 2100 730
Super-K [8] 2700 1196
OPERA [3] 3400 1833 muon intensity if the atmosphere were isothermal. To quantify the temperature effect on intensity, the temperature dependence of Eq. 2 needs to be considered. The meson production term in Eq. 3 (which applies to any charged meson: K, π, etc) can then be expanded:
dM dX = Z N M Λ N N 0 e −X/Λ N − M(E, X, cos θ) 1 Λ M + ǫ 0 M (1 + η) EX cos θ .(13)
The analytic solution to this differential equation is difficult to find since η(X ′ ) is an arbitrary function of X'. A solution to first order in η(X ′ ) can be found by expanding the exponential in a power series, and then following the procedure outlined above, beginning with Eq. 5. This solution can be written as M(E, X, cos θ)
= M 0 + M 1 , where M 0 (E, X)
is the solution where ǫ M = ǫ 0 M , which occurs at temperature T = T eff and M 1 (E, X) is given by:
M 1 (E, X, θ) = Z N M Λ N N 0 (E)e −X/Λ M X Λ M −ǫ 0 M /E cos θ ǫ 0 M E cos θ X 0 dX ′ ηΛ M X ′ X ′ Λ M ǫ 0 M /E cos θ+1 × 1 ǫ 0 M /E cos θ + 1 − X ′ /Λ ′ M ǫ 0 M /E cos θ + 2 + 1 2! (X ′ /Λ ′ M ) 2 ǫ 0 M /E cos θ + 3 − ... .(14)
If E cos θ ≫ ǫ 0 M , then the integrand is very small and η(X ′ ) = η(X). This is the case when interactions dominate, as time dilation effects allow these very high energy mesons to travel great distances before decaying. If E cos θ ≪ ǫ 0 M , then the mesons will not travel very far before decaying and the integrand is large only when X' is near X, and again, η(X ′ ) can be taken out of the integral [2].
Writing the solution of M where T = T eff as M 0 and letting ǫ M = ǫ 0 M (1 + η), an expression for the change in muon production induced by temperature variations can be found. Define ∆M ≡ M − M 0 , then to first order in η
∆M = Z N M Λ N N 0 (E)e −X/Λ M ǫ 0 M ηX E cos θ × 1 (ǫ 0 M /E cos θ + 1) 2 − 2X/Λ ′ M (ǫ 0 M /E cos θ + 2) 2 (15) + 1 2! 3(X/Λ ′ M ) 2 (ǫ 0 M /E cos θ + 3) 2 − ... .
Using Eq. 7 and Eq. 8, an expression for the change in differential muon intensity can be found:
∆ dI µ dE µ = Z N M Λ N ǫ 0 M E µ cos θ 2 E −(γ+1) µ (1 − r M ) ∞ 0 dXe −X/Λ M ηI M ,(16)
where
I M = 1/r M 1 dz z −(γ+2) × 1 (ǫ 0 M /E µ cos θ + z) 2 − 2X/Λ ′ M (ǫ 0 M /E µ cos θ + 2z) 2 + 1 2! 3(X/Λ ′ M ) 2 (ǫ 0 M /E µ cos θ + 3z) 2 − ... .(17)
Now, a solution to this integral can be found for
E µ ≫ ǫ 0 M (I H ) and for E µ ≪ ǫ 0 M (I L )
:
I H M (E µ ) = 1 γ + 3 1 − (r M ) γ+3 (1 − e −X/Λ ′ M ) Λ ′ M X , I L M (E µ ) = 1 γ + 1 1 − (r M ) γ+1 E µ cos θ ǫ 0 M 2 (1 − X/Λ ′ M )e −X/Λ ′ M .(18)
These expressions can be combined in a form that is valid for all energies (Eq. 8):
∆ dI µ dE µ ≃ E −(γ+1) µ 1 − Z N N ∞ 0 dX(1 − X/Λ ′ M ) 2 e −X/Λ M η(X)A 1 M 1 + B 1 M K(X) (E µ cos θ/ǫ 0 M ) 2 ,(19)
where
A 1 K ≡ 0.635 Z N,K Z N,π 1 − (r K ) γ+1 1 − (r π ) γ+1 (1 − r π ) (1 − r K ) , A 1 π ≡ 1, B 1 M ≡ (γ + 3) (γ + 1) 1 − (r M ) γ+1 1 − (r M ) γ+3 , K(X) ≡ (1 − X/Λ ′ M ) 2 (1 − e −X/Λ ′ M )Λ ′ M /X .
The exact solution for I L M (E µ ) has been replaced with an approximation that preserves the physical behavior of the system at low energies. These low energy mesons are relatively insensitive to changes in temperature because they decay before they have a chance to interact. So, this equation describes the expected behavior that mesons at very low energies will decay fairly high in the atmosphere. These mesons will not contribute any muons to an underground detector, because the muons they produce will be below the threshold energy.
There is a slight dip in this distribution as X approaches Λ ′ M , which results from the approximation made to join the high and low energy solutions for the approximation to Eq. 18. The low energy solution will go to zero when X = Λ ′ M and below zero when E th ≪ ǫ 0 M . The reason for this is that these low energy muons have such little energy that they decay in flight, producing a deficit in muons anticorrelated to positive temperature changes (the "negative temperature coefficient" related in older literature [2,14,15]). This effect is not seen by detectors deeper than 50 mwe. The fact that there is a dip and subsequent rise in Fig. 1 for X > 480 g/cm 2 does not affect an analysis for a detector deeper than 50 mwe since the weight is integrated over the entire atmosphere in discrete steps of dX and properly normalized, so this atmospheric depth is unimportant for the production of relevant muons.
Remembering that η(X) ≡ (T (X) − T eff )/T eff , the relationship between atmospheric temperature fluctuations and intensity variations can be written:
∆I µ = ∞ E th ∆ dI µ dE µ dE µ = ∞ 0 dXα(X) ∆T (X) T eff ,(20)
where the temperature coefficient α(X) can be written:
α(X) = (1 − X/Λ ′ M ) 2 e −X/Λ M ∞ E th dE µ A 1 M E −(γ+1) µ 1 + B 1 M K(X) (E µ cos θ/ǫ 0 M ) 2 = W M (X)E −(γ+1) th ,(21)
with W M (X) given by:
W M (X) ≃ (1 − X/Λ ′ M ) 2 e −X/Λ M A 1 M γ + (γ + 1) B 1 M K(X) (E th cos θ/ǫ 0 M ) 2 .(22)
The approximation to the integral follows from arguments made by Barrett [2]. The derivative of this expression agrees with the integrand in the energy region of interest to within 2%. The weights as a function of X using the threshold energies of the detectors under consideration can be seen in Fig. 1. The fact that the lines are nearly on top of each other shows that the weight of the particular atmospheric depth does not depend very much on the threshold energy. Recalling that M applies equally to K and π mesons and that the total muon intensity is the sum of the contribution by K and π (Eq. 12), the temperature induced change in muon intensity can be written:
∆I µ = ∞ 0 dXα π (X) ∆T (X) T eff + ∞ 0 dXα K (X) ∆T (X) T eff .(23)
Letting T eff be defined such that when T (X) = T eff , ∆I µ = 0 gives
T eff = ∞ 0 dXT (X)α π (X) + ∞ 0 dXT (X)α K (X) ∞ 0 dXα π (X) + ∞ 0 dXα K (X) .(24)
Since the temperature is usually measured at discrete levels, the integral is calculated numerically over the atmospheric levels ∆X n :
T eff ≃ N n=0 ∆X n T (X n ) W π n + W K n N n=0 ∆X n (W π n + W K n )
.
(25) W π,K n is W π,K evaluated at X n , 1/Λ ′ π,K ≡ 1/Λ N − 1/Λ π,K , Λ N = 120 g/cm 2 , Λ π = 160 g/cm 2 and Λ K = 180 g/cm 2 [11]. With this definition of Effective Temperature, an Effective Temperature coefficient, α T , can be defined:
T (K)
α T = 1 I 0 µ ∞ 0 dXα π (X) + ∞ 0 dXα K (X) ,(26)
where I 0 µ is the intensity for a given temperature T. Now that the atmospheric temperature has been parametrized and α T defined, the relationship between atmospheric temperature fluctuations and intensity variations can be written:
∆I µ I 0 µ = ∞ 0 dXα(X) ∆T (X) T eff = α T ∆T eff T eff .(27)
Note that the expression to calculate T eff in the pion scaling limit, ignoring the kaon contribution is the same as the MACRO [3] calculation for effective temperature. This distribution reflects the dominant atmospheric phenomena that produce muons visible to a detector underground. High energy mesons produced at the top of the atmosphere have the greatest influence on the seasonal variation because they are created where the density is lower, so they have the highest probability to decay into muons. High energy mesons that are produced lower in the atmosphere have a greater probability of interacting a second time, and thus greater probability of producing muons that are not seen by an underground detector.
Theoretical Effective Temperature Coefficient
The theoretical prediction of α T for properly weighted atmospheric temperature distribution can be written (Eq. 27):
α T = T I 0 µ ∂I µ ∂T .(28)
Barrett [2] shows that for a muon spectrum such as Eq. 8, the theoretical α T can be written:
α T = − E th I 0 µ ∂I µ ∂E th − γ.(29)
The prediction for α T can be calculated using the intensity found in Eq. 12 and a little algebra:
α T = 1 D π 1/ǫ 0 K + A 1 K (D π /D K ) 2 /ǫ 0 π 1/ǫ 0 K + A 1 K (D π /D K )/ǫ 0 π ,(30)
where
D π(K) = γ γ + 1 ǫ 0 π(K) 1.1E th cos θ + 1.(31)
Note that this can be reduced to MACRO's previously published expression α T π [3], which was only valid for pion induced muons, by setting A 1 K = 0 (no kaon contribution). This approximation can be extended to a kaon-only temperature coefficient, (α T ) K by setting the pion term (first term in parentheses) in Eq. 12. The result is an independent model of the temperature coefficient for each of the meson species:
(α T ) π,K = 1 γ γ + 1 ǫ 0 π,K 1.1E th cos θ + 1 .(32)
To compare the experimental α T to the theoretical expectation, a simple numerical integration using a Monte Carlo method was performed. A muon energy and cos θ were chosen out of the differential muon intensity (Eq. 12). A random azimuthal angle, φ, was chosen and combined with cos θ. Column depth was calculated as d = h/ cos θ, where h is the detector depth in mwe for standard rock with flat overburden. The threshold surface energy required for a muon to survive this column depth is found from the expression for threshold energy (Eq. 11). If the chosen E µ was greater than E th , it was used in the calculation of the theoretical α T . This was repeated for 10,000 successful muons with E µ > E th , at depths from 0 to 4,000 mwe. The result of this calculation, along with the experimental result from the MINOS experiment [5] can be seen in Fig. 3 as the solid line. This curve includes the "negative temperature effect" (muon decay correction) term, δ ′ = (1/E cos θ)(m µ c 2 H/cτ µ )(γ/γ + 1) ln(1030/Λ N cos θ) [2], which goes to zero for E µ > 50 GeV.
The kaon component of air showers that can be observed at 1400 mwe is about 10%, but the energy is too low for kaon-induced muon production to be affected by changes in temperature. The result is the large gap between the pion only curve and the Kπ curve. As the depth increases, the energy of sampled muons also increases, which results in a greater contribution to α T by kaon induced Fig. 3. The theoretical α T (X) (solid curve), the α T (X) π (dashed curve), the α T (X) K (dotted curve) for slant depths up to 4,000 mwe. The MINOS data point is from [5]. The cross-hatched regions indicate the sensitivity (separation between the three models for a particular depth) that current underground detectors have to measurements of α T (X) .
muons. The asymptotic behavior of the theoretical α T approaching one as primary energy increases is expected from Eq.28. At very high primary energies, the intensity is proportional to the critical meson energy, which depends on temperature. Thus, for an isothermal atmosphere, intensity will be directly proportional to the temperature (the constant of proportionality, α T , will be one).
Method for Measurement of Atmospheric K/π Ratio
The theoretical uncertainty of the atmospheric K/π ratio is of order 40 % [16]. There was not a measurement of this ratio with cosmic rays until 2009, when it was made using MINOS FD data [5]. Previous measurements have been made at accelerators for p+p collisions [17], Au+Au collisions [18], Pb+Pb collisions [19,20].
The expression for the experimental α T is written in Eq. 27. The kaon influence causes an overall decrease in total α T , shown in Sec. 4. Because the left hand side of Eq. 27 depends only on counting rate, it can be broken into meson components:
∆R π µ + ∆R K µ R π µ + R K µ = α T ∆T eff T eff ,(33)
which can be rewritten:
T eff α T ∆T eff ∆R π µ R π µ + ∆R K µ R π µ − 1 = R K µ R π µ .(34)
Recall that in the pion scaling limit, only pions are assumed to contribute to the seasonal effect.
From that, a model for pion-only and kaon-only temperature coefficients were developed in Eq. 32. Such a seasonal effect can be written:
∆R π,K µ R π,K µ = (α T ) π,K ∆T eff T eff .(35)
The ratio of the muon counting rates R K µ /R π µ is equivalent to the ratio of muons from kaons to muons from pions N K µ /N π µ , which will be written r µ (K/π). Rearranging and inserting Eq. 35 for both kaons and pions into Eq. 33 gives:
r µ (K/π) = 1 α T (α T ) π + (α T ) K R K µ R π µ − 1 (36) = (α T ) π /α T − 1 1 − (α T ) K /α T .(37)
The value for r µ (K/π) can be predicted by integrating Eq. 12:
r µ (K/π) = I K µ I π µ = C 2 × A 1 K ,(38)
where
I K µ = ∞ E th cos θ A 1 K E −γ µ 1 + 1.1E µ cos θ/ǫ K dE µ cos θ,(39)I π µ = ∞ E th cos θ E −γ µ 1 + 1.1E µ cos θ/ǫ 0 π dE µ cos θ.(40)
The parameter A 1 K is defined as
A 1 K = 0.635 × r(K/π) (1 − r π ) (1 − r K ) 1 − (r K ) γ+1 1 − (r π ) γ+1 ,(41)
where r(K/π) = Z NK Z Nπ [11] is the ratio of kaons to pions produced in the primary cosmic ray interactions. Inserting Eq. 41 into Eq. 38 and rearranging gives an expression for r(K/π) in terms of r µ (K/π):
r(K/π) = 1 C 1 × r µ (K/π) × 1 0.635 (1 − r K ) (1 − r π ) 1 − (r π ) γ+1 1 − (r K ) γ+1 .(42)
The MINOS-FD [5] measurement is shown in Fig. 4, along with STARS [18], NA49 [19,20] and ISR [17] accelerator measurements. The cross-hatched regions show the energy regimes to which the underground detectors discussed in this paper are sensitive for K/π ratio measurements using the method described. An OPERA measurement would extend to the region an order of magnitude Fig. 4. A compilation of selected measurements of r(K/π) for various primary particle center of mass energies ( √ s). The STARS value was from Au+Au collisions at RHIC [18], the NA49 measurement was from Pb+Pb collisions at SPS [19,20], the ISR measurement was from p+p collisions [17], and the MINOS value was from cosmic ray primaries + atmospheric nuclei collisions [5].The thick horizontal bars near the bottom of the graph show the typical ranges of cosmic ray primary energies for the collisions that produce muons observed by the underground detectors indicated.
beyond the energy of current fixed target experiments. With a detector area roughly 1,000 times the area of the MINOS FD, IceCube should have a cosmic ray muon rate of upon completion of construction (Boreal Spring, 2011) of 1,700 Hz [21]. Assuming a temperature data set of comparable quality to the BADC ECMWF data [22] used by the MINOS-FD analysis [5], the statistical uncertainty in α T could be reduced to ±0.001. Since the absorber material surrounding IceCube is ice instead of rock containing iron veins, the column depth should be more well known. This could reduce the uncertainty in the depth map, the dominant source of uncertainty in (α T ) π,K , by half. These factors taken together could reduce the uncertainty in the measurement of r(K/π) by 16-30%.
Summary
A new method was developed to include the effect of kaons in measurement of seasonal variations in underground muon intensity. A temperature coefficient that accounts for the kaon contribution was described, and a kaon-inclusive model was defined. These methods were applied to MINOS-FD data [5], and the new model fit the data better than the pion only model [3]. A formula was described so that other underground experiments, OPERA, Super-K, IceCube, Baksan and the MI-NOS ND could quickly apply this method to their data. Pion and kaon decay are affected differently by temperature variations, and this difference suggested a method to measure the atmospheric K/π ratio. This method was developed and a formula was offered for other underground experiments to follow.
Fig. 1 .
1The average mid-latitude summer temperature at various atmospheric depths (dashed line). The vertical range is from 1000 hPa (1 hPa = 1.019 g/cm 2 ), near Earth's surface, to 1 hPa (nearly 50 km), near the top of the stratosphere. The solid lines are the weight as a function of atmospheric depth used to find T eff (Eq. 22). The blue lines used the OPERA threshold energy, the red lines used the Super-K threshold energy, the black lines used the MINOS FD threshold energy, the green lines used the IceCube threshold energy, the violet lines used the Baksan threshold energy and the magenta lines used the MINOS ND threshold energy .
Fig. 2 .
2The solid lines show the meson intensity as a function of atmospheric depth (Eq. 4) and the dotdash lines show the muon intensity as a function of atmospheric depth (Eq. 7). The normalization in Eq. 4 (N 0 Z N M /Λ N ) was set to 1 to show the dependence on X more clearly. The range of the expressions were adjusted so that the maximum value of the OPERA meson intensity was equal to 1, and the other equations were scaled appropriately. These figures were produced with particular energy values corresponding to the threshold energy of the detectors under consideration. The blue lines used the OPERA threshold energy, the red lines used the Super-K threshold energy, the black lines used the MINOS FD threshold energy, the green lines used the IceCube threshold energy, the violet lines used the Baksan threshold energy and the magenta lines used the MINOS ND threshold energy .
AcknowledgmentsWe thank our many colleagues who provided vital input as these methods were developed, especially Tom Kelley for providing comments on the presentation of the mathematics. This work was supported by the U.S. Department of Energy, the U.K. Science and Technologies Facilities Council, the U.S. National Science Foundation, the Center for Cosmology and AstroParticle Physics at Ohio State University and the University of Minnesota. We also acknowledge the BADC and the ECMWF for providing the environmental data for this project.
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[
"CONSTRUCTION OF SOME GENERALISED MODULAR SYMBOLS",
"CONSTRUCTION OF SOME GENERALISED MODULAR SYMBOLS"
] |
[
"B Speh ",
"T N Venkataramana "
] |
[] |
[] |
We give a criterion for the non-vanishing of certain modular symbols on a locally symmetric manifold. The criterion is in terms of the non-vanishing of some cohomology classes on the compact dual of the locally symmetric manifold. Using this, we construct nonzero modular symbols for SL 2n over an imaginary quadratic extension of Q, which represent ghost classes. We also construct nonzero modular symbols in certain non-compact Shimura varieties and give an example of a modular symbol that generates an infinite dimensional module under the action of the Hecke algebra.
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"https://export.arxiv.org/pdf/math/0409376v1.pdf"
] | 204,899,696 |
math/0409376
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fba4a52e6e3b3194c9218e0ea495e31fa7fb4cef
|
CONSTRUCTION OF SOME GENERALISED MODULAR SYMBOLS
21 Sep 2004
B Speh
T N Venkataramana
CONSTRUCTION OF SOME GENERALISED MODULAR SYMBOLS
21 Sep 2004
We give a criterion for the non-vanishing of certain modular symbols on a locally symmetric manifold. The criterion is in terms of the non-vanishing of some cohomology classes on the compact dual of the locally symmetric manifold. Using this, we construct nonzero modular symbols for SL 2n over an imaginary quadratic extension of Q, which represent ghost classes. We also construct nonzero modular symbols in certain non-compact Shimura varieties and give an example of a modular symbol that generates an infinite dimensional module under the action of the Hecke algebra.
Introduction
Let G and H be semi-simple algebraic groups over Q and f : H → G a morphism with finite kernel of Q-algebraic groups. Let K H be a maximal compact subgroup of the group H(R) of real points of H and K ⊃ f (K H ) a maximal compact subgroup of G(R). We then get a map Y → X of the associated symmetric spaces Y = H(R)/K H and X = G(R)/K. Suppose that X and Y are of dimension D and d respectively.
Assume that Γ ⊂ G(Q) is a torsion-free congruence subgroup. Then S(Γ) = Γ\X is a manifold with finite volume under the G(R) invariant metric (and volume form ) on X. Write Γ ∩ H for f −1 (Γ) and S H (Γ) = Γ∩H\Y . We have an immersion S H (Γ) → S(Γ) induced by the map f . This identifies S H (Γ) with a closed subspace of S(Γ). This follows as in [A] 2.7. For ease of notation, we will assume that the groups G(R) and H(R) are connected. Then, both the symmetric spaces X and Y are orientable with an orientation fixed by G(R) and H(R) respectively. This in turn makes S(Γ) and S H (Γ) orientable. Now, compactly supported cohomology classes in H d c (S(Γ)) may be pulled back to S H (Γ) and integrated on S H (Γ). Since H d c (S H (Γ)) = C integration on S H (Γ) defines a linear form on H d c (S(Γ)) which may be thought of -by Poincaré duality-as an element [S H (Γ)] of H D−d (S(Γ)). It is called the generalized modular symbol corresponding to H.
In the arithmetic of modular curves and classical automorphic forms modular symbols have served as an indispensable tool linking geometry and arithmetic. "Period integrals" of Eisenstein classes or cuspidal cohomology classes over compact modular symbols have been used by G.Harder to obtain information about special values of L-functions [H1], [H2]. In 1990 A. Ash and A.Borel showed that the Levi factors of parabolic subgroups define nonzero modular symbols [A-B], [R-S]. Later Ash, Ginzburg and Rallis give 6 families of pairs (G,H) where they can show that that any cuspidal cohomology class for Γ over a generalized modular symbol corresponding to H. [A-G-R]. One such pair is G = Sp n and H = Sp m × Sp k which we also consider in theorem 3.
In this paper we give a criterion for the non-vanishing of a modular symbol [S H (Γ)]. The criterion is in terms of the compact duals spaceŝ X andŶ of the symmetric spaces X and Y . We recall the construction ofX andŶ ; there is a Cartan decomposition of the Lie algebra g 0 of G(R) with respect to the maximal compact K, which leaves the Lie algebra h 0 of H(R) stable. Write g 0 = k 0 ⊕ p 0 and h 0 = k 0 ∩ h 0 ⊕ p 0 ∩ h 0 as the Cartan decompositions. Let G u be the compact subgroup of G(C) with Lie algebra k 0 ⊕ ip 0 . Then X = G u /K. The compact dual Y is defined analogously. One has an embedding of Y in X, and the fundamental class [ Y ] of Y in X is a cohomology class in H D−d ( X).
The Borel map j from the cohomology H * ( X) into the cohomology H * (S(Γ)) is defined as follows: we identify H * ( X) with H * (g, K, C) and H * (S(Γ)) with H * (g, K, C ∞ (Γ\G(R))) where g = g 0 ⊗C; then the map j is induced by the inclusion of the constant functions C in the space C ∞ (Γ\G(R)) of smooth functions on Γ\G(R).
With notation as above, we prove the following theorem in section 2.
Theorem 1. Suppose that G is a simply connected group which has no R-anisotropic factors defined over Q. Then, the class j([ Y ]) is a linear combination of Hecke translates of the generalised modular
symbol [S H (Γ ′ )] ∈ H D−d (S(Γ ′ )) for some congruence subgroup Γ ′ of Γ. In particular, if j([ Y ]) = 0, then the modular symbol [S H (Γ ′ )] does not vanish.
The proof of this result relies on the work of J. Franke [F].
In section 3 we deduce, from Theorem 1 and some computations in the cohomology of classical compact symmetric spaces, the following theorem.
Theorem 2. If E is a totally imaginary number field, G = R E/Q (SL 2n ) and H = R E/Q (Sp 2n ), then the modular symbol [S H (Γ)] does not vanish for some congruence subgroup Γ.
If E is a totally imaginary quadratic extension of a totally real number field F , G = R E/Q SL 2n+1 and H = R F/Q SL 2n+1 , then the modular symbol [S H (Γ)] does not vanish for some congruence subgroup Γ.
In the above, R denotes the (Weil) restriction of scalars.
A cohomology class was called a ghost class by A.Borel if it restricts trivially to each boundary component of the Borel Serre compactification,but its restriction to the full boundary is not zero. The first example of ghost class was constructed by G.Harder in the cohomology of GL 3 over totally imaginary fields using Eisenstein classes. Later a whole family of ghost classes was constructed by J. Franke using cohomology classes represented by invariant forms. We show Corollary 1. The G(A f )-span of the modular symbols [S H (Γ)] in theorem 2 contains ghost classes.
Theorem 1 is especially useful in the case when the symmetric spaces X and Y are of Hermitian type and the embedding Y → X is holomorphic. In particular, we prove Theorem 3. Let G = Sp 2g be the split symplectic group over Q and let H = Sp 2g i ⊂ Sp 2g with g i = g, be the natural inclusion. Then, the modular symbol [S H (Γ)] is non-zero, for a suitable congruence subgroup Γ.
Analogously, for the unitary group, we prove Theorem 4. Suppose q ≥ p ≥ 1 are integers, and let G = U(p, q) be the unitary group in p + q variables.
(1) If p i and q i are integers such that p i = p and q i ≤ q, and H = U(p i , q i ) then the modular symbol [S H (Γ)] is non-zero, for some congruence subgroup Γ of G.
(2) Under the natural embedding of H = Sp 2g in G = U(g, g), the modular symbol [S H (Γ)] is non-zero for some congruence subgroup Γ.
In some instances, one can even determine if the space of the span of G(A f )-translates of the modular symbol [S H (Γ)] is infinite dimensional.
Theorem 5. Suppose that G = U(1, q), and H = U(1, r) with r = q−2 or r = q − 1. Then there exists a congruence subgroup Γ such that the G(A f )-translates of the modular symbol [S H (Γ)] is infinite dimensional.
2. some results of Franke and proof of Theorem 1 2.1. Notation. Let Γ be as in Theorem 1. Let K f be the closure of Γ in G(A f ), where A f denotes the ring of finite adeles over Q. Put H * (S G ) for the direct limit limH * (S(Γ)) as Γ varies over congruence subgroups of G(Q) (if Γ ′ ⊂ Γ, then there is a natural inclusion H * (S(Γ)) in H * (S(Γ ′ )) which gives us a direct system of finite dimensional complex vector spaces, and the direct limit is with respect to these inclusions). On this direct limit, the group G(A f ) operates, and if K is the closure of Γ in G(A f ) then, the space of K f -invariants in H * (S G ) is exactly H * (S(Γ)) (we are using strong approximation here which is guaranteed under the assumptions of Theorem 1).
Let K 0 be a good maximal compact subgroup of G(A f ). Put Γ 0 = G(Q) ∩ K 0 . We fix a subgroup Γ ′ of finite index in Γ ∩ Γ 0 .
It is clear from the definition of the modular symbol ξ Γ = [S H (Γ)] that the sum of translates of ξ Γ ′ over a set of coset representatives of Γ/Γ ′ (resp. Γ 0 /Γ ′ ) is a non-zero multiple of ξ Γ (resp. ξ Γ 0 ). Therefore, if we show that j([ Y ]) is a linear combination of G(A f )-translates of S H (Γ 0 ) then we have proved Therem 1.
Set H 0 to be the space of complex valued compactly supported K 0 -biinvariant functions on the finite adelic group G(A f ). This is an algebra (Hecke algebra corresponding to K 0 ) under convolutions and acts on the cohomology group H * (S(Γ 0 )) = H * (S G ) K 0 . Let C denote the trivial one dimensional G(A f )-module. On this module, H 0 operates, and we get a homomorphism χ : H 0 → C. Let m = m χ denote the kernel of the map χ. This is a maximal ideal in H 0 . Denote by H * (S G ) K 0 m the space of vectors in H * (S G ) K 0 which are annihilated by some power of the ideal m. Clearly, this space is a direct summand as a Hecke module.
However, the main theorem (Theorem 1 of [F]) of [F] asserts that for the good maximal compact subgroup K 0 , this space coincides with the space of vectors annihilated by the first power of the maximal ideal m and that it is also isomorphic to the space of co-invariants for G(A f ) of the module H * (S G ) . Moreover, according to [F], the latter is naturally isomorphic to H * (U) K∩P (R) where P is a fixed minimal parabolic Qsubgroup of G and U is the following open set in the compact dual X: U = X − ∪ X L where the union is over Levi subgroups L of parabolic Q-subgroups of G containing the minimal one P ; X L denotes the compact dual of the symmetric space associated to the group L 0 ⊂ L(R). Here, L 0 is the subgroup of L, which is the intersection of the kernels of rational characters on L. The above isomorphism takes the restriction res(
[x]) to H * (U) of a class [x] ∈ H * ( X), into the vector j([x]).
Now the space V = H * (S G ) K 0 is a finite dimensional complex vector space which is a module over the ring H 0 . Let J be the kernel of the map H 0 → End(V ) and let R = H 0 /J be the quotient ring (which is a finite dimensional algebra over C). Then, m is the inverse image of a maximal ideal m R of R under the quotient map. Let I denote the annihilator in R of the modular symbol [S H (Γ 0 )]. We may write
I = m k R A where A is an ideal coprime to m R . The span W [S H (Γ 0 )] of H 0 -translates of the class [S H (Γ 0 )] is isomorphic as an H 0 module, to R/I = R/m k R ⊕ R/A. By Franke's Theorem, if k ≥ 1 then k = 1. Consequently, if the projection I([S H (Γ 0 )]) of the modular symbol [S H (Γ 0 )] to the space of "invariants" is non-zero, then the projection I([S H (Γ 0 )]) lies in W [S H (Γ 0 )] and hence it is a linear combination of Hecke translates of [S H (Γ 0 )].
We will complete the proof of Theorem 1 by showing that this pro-
jection I([S H (Γ 0 )]) is a nonzero multiple of j([ Y ]).
The space of m invariants is a direct summand of the H 0 -module H * (S(Γ 0 )). We identify it as a vector space using the map j with H * (U). Hence under Poincare duality we can identify H * c (U) with the space of m invariants in H * c (S(Γ 0 )). Note that since the ring H 0 is equipped with a nice involution induced from g → g −1 on G(A f ), we can consider these spaces as dual modules of H 0 -modules.
For a given class
[w] in j(H d c (U)) we consider the map W [S H (Γ 0 )] → H D c (S(Γ 0 )) defined [v] → [v] ∧ [w] as a linear form on W [S H (Γ 0 )] . Since [w] is H 0 -invariant this form is also H 0 -invariant and so it follows that for [w] ∈ H d c (U), we have [v] ∧ [w] = I([S H (Γ 0 )]) ∧ [w].
However the map
H d c (U) → H D c (S G ) K 0 = j(H D ( X))
given by wedging with the fundamental class [S H (Γ 0 )] is up to a nonzero multiple the integral of [w] over Y and is hence equal to )
[ Y ] ∧ j * ([w] where j * : H * c (U) → H * ( X)
is the dual of the restriction map from the cohomology of X to that of U. By definition, this means that
[S H (Γ 0 )] ∧ [w] = j([ Y ]) ∧ [w].
This holds for all
[w] ∈ H d c (U). Hence I([S H (Γ 0 )]) = j([ Y ]).
Here, under the isomorphism of Franke mentioned in an earlier paragraph, the image of j has been identified with the image of the restriction map from the cohomology of X into that of U.
This proves Theorem 1.
2.2.
Remark. The foregoing proof is an adaptation of [V] where the analogous result for the compact case is proved. In the compact case, the fact that the action by Hecke operators is completely reducible is used in a crucial way. The extension in the present paper to the noncompact case (i.e. when S(Γ) is not compact) is achievable only because of the result of Franke that the space of Hecke algebra invariants is a direct summand -as a Hecke module-of H * (S G ) K 0 for the good maximal compact K 0 .
Applications
In this section we apply the criterion of Theorem 1. Before doing so we note another result from [F] ( see [F], (7.1), equation (54)) on the kernel of the Borel map
j : H * ( X) → H * (S(Γ)).
We have a dual map
j * : H * c (S(Γ)) → H * ( X).
Franke's theorem says that a class [x]
∈ H * ( X) lies in the image of the compactly supported cohomology H * c (S(Γ)) if and only if the restriction of the class [x] to X L vanishes for all Levi subgroups L of all Q-parabolic subgroups of G. Dually, this means that the kernel of the Borel map j is the orthogonal complement of the space of vectors [v] ∈ H * ( X) whose restriction to X L vanishes for al the Levis L as above. Here orthogonal complement means the following: the cohomology algebra H * ( X) comes equipped with a non-degenerate bilinear form (Poincare duality) and the orthogonal complement is with respect to this bilinear form.
3.1. Example. Let G = R E/Q (SL n ) with E an imaginary quadratic extension of Q. Then, X = SU(n) and its cohomology is an exterior algebra ∧ = ∧(e 3 , e 5 , · · · , e 2n−1 ) on primitive generators e 2i−1 of degree 2i − 1. Fix k ≥ 1 and consider the inclusion SU(k) ⊂ SU(n). The cohomology algebra of SU(k) is the exterior algebra ∧(e 3 , e 5 , · · · , e 2k−1 ) and the restriction map in the cohomology from SU(n) to SU(k) is given by e i → e i if i ≤ 2k − 1 and e i → 0 otherwise (this is well known; e.g., see [M-T], Chapter (III), p.148, Theorem (6.5) (4)).
¿From the description of the Levi subgroups (they are of the form R E/Q (SL m 1 ×· · ·×SL m k )) it follows that their compact duals are products of lower dimensional unitary groups. Using this and the definition of the generators e 2i−1 , it can be proved that a class v ∈ ∧ restricts trivially to all these compact duals if and only if it is divisible by the class e 2n−1 . It is clear that the orthogonal complement of the ideal generated by e 2n−1 is itself. Hence by Franke's theorem quoted at the beginning of this section, j(v) = 0 if and only if v is divisible by e 2n−1 .
Replace n by 2n and consider the embedding H = R E/Q (Sp 2m ) ⊂ G = R E/Q (SL 2m ). The compact dual Y is simply the group Sp 2n whose cohomology is an exterior algebra ∧(e 3 , e 7 , · · · , e 4n−1 ), on odd degree generators e 2i−1 of degree 2i − 1 with i ≥ 2. The cohomology of X is (note that n is replaced by 2n) as we said before, ∧(e 3 , e 5 , · · · , e 4n−1 ).
Moreover, the restriction map from X to Y takes e i to e i if i = 4j − 1 and to 0 otherwise. This can easily be proved by looking more closely at the spectral sequences used to obtain the cohomology of X and Y (see [M-T], p.119, Ch. (III), Theorem (3.10) (1) and (2)).
Consequently, the fundamental class is
[ Y ] = e 5 ∧ e 9 ∧ · · · ∧ e 4n−3 , and is not divisible by e 4n−1 . This proves that j([ Y ]) = 0. Therefore, the modular symbol [S H (Γ)] is non-zero for some congruence subgroup Γ ⊂ G(Q).
Since [ Y ] is not divisible by e 4n−3 , Franke's result shows that j([ Y ])
is not in the image of the cohomology with compact support. On the other hand the restriction of
[ Y ] = e 5 ∧ e 9 ∧ · · · ∧ e 4n−3 to the Levi subgroup R E/Q (SL n−1 ) is in the kernel of the image of j for R E/Q (SL n−1 ) since it is divisible by e 4n−3 . Thus j([ Y ]) is a ghost class. 3.2. Example. Consider the embedding SL 2n+1 ⊂ R E/Q (SL 2n+1 ) of Q-groups,
where E is an imaginary quadratic extension of Q. The embedding Y = SU(2n + 1)/SO(2n + 1) ⊂ X = SU(2n + 1) is induced by g → gg t from SU(2n + 1) into itself. The cohomology of SU(2n + 1)/SO(2n + 1) is ( see [F], p. 35, Proposition 7) an exterior algebra ∧(e 5 , e 9 , · · · , e 4n+1 ) and the restriction map from H * ( X) = ∧(e 3 , e 5 , · · · , e 4n−1 , e 4n+1 ) to H * ( Y ) is given by sending e 2i+1 to e 2i+1 if i is odd and to 0 otherwise.
Consequently, the fundamental class is
[ Y ] = e 3 ∧ e 7 ∧ · · · ∧ e 4n−1 and is therefore not divisible by e 4n+1 . Hence it does not lie in the kernel of the Borel map, and by Theorem 1, the modular symbol [S H (Γ)] is non-zero for a suitable Γ.
The rest of Theorem 2 is proved in an entirely analogous way. The argument in the previous example shows that j([ Y ]) = j(e 3 ∧ e 7 ∧ · · · ∧ e 4n−1 ) is a ghost class.
3.3. Remark. Suppose that H is any linear algebraic group over a totally real number field F such that
H(F ⊗ R) = SL 2n+1 (R) m with m the degree of F over Q (that is, H is a F -form of SL 2n+1 (R) m )
. Let E/F be a totally imaginary quadratic extension and set G = R E/F (H). Then, [S H (Γ)] is non-zero for some congruence subgroup of G(E). This follows by the same method as in the case when H = SL n over F (i.e. when H is the standard F -form).
In the next two sections, we assume that both the symmetric spaces X and Y are Hermitian symmetric and that the embedding of Y in X is holomorphic. Under these assumptions, one can show in a large number of cases that the associated modular symbol [S H (Γ)] is nonzero for some Γ. We will discuss two important cases, when G is the symplectic group Sp g and the unitary group U(p, q).
The Symplectic Group
In this section, G = Sp 2g denotes the symplectic group defined and split over Q. The associated symmetric space is the Siegel upper half space X = Sp 2g (R)/U(g). Denote by X its compact dual. Let T be the group of diagonals in K = U(g); it is a maximal torus in K and in G(R). The cohomology of the classifying space BT is generated by Chern classes of line bundles arising from characters of T , and may be identified with the polynomial algebra H * (BT ) = C[Λ], where X * (T ) is the group of characters on T and Λ ⊂ X * (T ) is a suitable "positive subset" (in the case of U(g)), consisting of non-negative integral linear combination of the characters of T occurring in the standard representation of U(g) on C g . Clearly, the ring R = C[Λ] is C[x 1 , · · · , x g ] where x i is (by a mild abuse of notation) the Chern class of the character x i (the (i,i)-th entry) in the diagonal torus T . Note that the ring R is a module for the Weyl groups W G = (Z/2Z) g × S g and W K = S g of G and K respectively. Here S g is the symmetric group on the g letters x 1 , x 2 , · · · , x g and W G is a semi-direct product of (Z/2Z) g with S g ; the elements of W G act on the x i by ±x σ(i) , with σ ∈ S G . Let σ 1 , · · · , σ g denote the elementary symmetric functions in the variables x 1 , · · · , x g . Lemma 6. The cohomology of X is the ring C[σ 1 , · · · , σ g ] modulo the ideal generated by the graded relation i (1 − x 2 i ) = 1 (i runs from 1 to g).
Proof. The cohomology of the compact dual X may be identified ( [M-T]) with the ring R W K of W K -invariants in R modulo the ideal generated by positive degree elements in the ring R W G of W G -invariants. The above description of the Weyl groups in question then implies the Lemma.
Next, we determine the kernel of the Borel map, using the criterion of Franke. Let K = U(g) ⊂ Sp 2g be the natural inclusion, where G u = Sp 2g is the compact form of G = Sp 2g .
Lemma 7. If there is any embedding i of the unitary group U(g) in the compact form Sp 2g such that K ∩ i(SU(g)) = O(g) the orthogonal group and the restriction of the embedding i to O(g) is identity, then, the restriction map from the cohomology (with C-coefficients) of X into that of the subsymmetric space i(SU(g))/SO(g) is zero except in degree zero.
Proof. To see this, first note that the cohomology of X is generated by Chern classes of the homogeneous vector bundle arising from the standard representation of U(g) on C g . Thus it is enough to show that these Chern classes vanish on the subsymmetric space. Now the restriction E of this bundle to i(SU(g))/SO(g) is also homogeneous, and arises from the standard representation ρ of SO(g). Obviously, ρ extends to a representation of i(SU(g)), which shows that the vector bundle E admits a trivialisation. Hence all its higher degree Chern classes are zero. But these classes are the restriction of Chern classes of the vector bundle on X with which we started. Therefore, the restriction of the cohomology of X to i(SU(g))/SO(g) is trivial.
Let P ⊂ G be a standard maximal parabolic subgroup defined over Q, and L its Levi component. One may identify L with GL g−k × Sp 2k for some k ≤ g. At the level of compact duals, one then has X L = SU(g − k)/SO(g − k) × Sp 2k /U(k). Thus the restriction map from the cohomology of X to X L factors through the product Sp 2g−2k /U(g − k) × Sp 2k /U(k). From Lemma 7 (applied to Sp 2g−2k ) one sees that the kernel of the restriction map from H * ( X) to H * ( X L ) is the same as the kernel of the restriction map from H * ( X) to H * ( Sp 2k /U(k)).
Let J be the kernel of the restriction map H * ( X) → H * ( X L ) where the product runs over all the Levi subgroups L of standard maximal parabolic Qsubgroups P . Therefore, we have proved that the kernel of the foregoing restriction map is the same as the kernel of the map H * ( X) → H * ( Sp 2g−2 /U(g − 1)). Let Z = Sp 2g−2 /U(g − 1), and let τ 1 , · · · , τ g−1 denote the elementary symmetric functions in the g − 1-variables x 2 , · · · , x g . By Lemma 6, we have the identifications
H * ( X) = C[σ 1 , · · · , σ g ]/ i=g i=1 (1 − x 2 i ) = 1 and H * (Z) = C[τ 1 , · · · , τ g−1 ]/ i=g i=2 (1 − x 2 i ) = 1.
Lemma 8. The kernel of the Borel map H * ( X) → H * (S(Γ)) is orthogonal to the ideal generated by σ g .
Proof. By Franke, the kernel of the Borel map is precisely the orthogonal complement (with respect to Poincaré duality on H * X)) of the kernel of the map H * ( X) → H * ( X L ). By the discussion preceding the lemma, the latter is the kernel of the restriction map H * ( X) → H * ( Sp 2g−2 /U(g − 1)) = H * (Z).
Consider the map p from the ring C[σ 1 , · · · , σ g ]
/( i (1 − x 2 i ) = 1) into the ring C[τ 1 , · · · , τ g−1 ]/( j (1 − x 2 j ) − 1) induced by the map C[x , · · · , x g ] → C[x 2 , · · · , x g ], with x 1 → 0, x 2 → x 2 , · · · , x g → x g .
Under the identifications made just before Lemma 8, p is nothing but the restriction map of the previous paragraph. The kernel of p is easily seen to be generated by σ g .
In the proof of Theorem 3, we need the following lemmata.
Lemma 9. If k ≤ g, then in the ring C[σ 1 , · · · , σ g ]/( (1 − x 2 i ) − 1), the product σ 2 k σ k+1 · · · σ g vanishes. Proof. By induction on g. Denote by β the product σ 2 k σ k+1 · · · σ g . Consider the map p into C[τ 1 , · · · , τ g−1 ]/( (1 − x 2 j ) − 1). The element α = σ 2 k σ k+1 · · · σ g−1 maps, under p, to the element τ 2 k τ k+1 · · · τ g−1 which is zero by induction assumption. Therefore, by Lemma 8, the element α is divisible by σ g . Now, β = ασ g and is thus divisible by σ 2 g . However, σ 2 g is the highest degree term in the graded equation (1 − x 2 i ) = 1 and hence is zero. Therefore, β also vanishes.
Lemma 10. The element σ 1 σ 2 · · · σ g is non-zero, i.e. it generates the top degree (degree g(g + 1)) cohomology of X.
Proof. By induction on g. Since the only class in degree two is σ 1 , it is clear that it is the Kahler class of X. Hence, the top degree cohomology is generated by σ g(g+1) 2 1 . We will prove that σ g(g+1)/2 1 is a multiple of σ 1 · · · σ g . This will prove the lemma. Now, the cohomology of Z = Sp 2g−2 /U(g) is the quotient of that of X, by the ideal σ g . Since the dimension of Z is g(g − 1)/2, it follows that σ g(g+1)/2 1 vanishes on Z. Thus, in the ring H * ( X), we have σ (g+1)g/2 1 = σ g ψ where ψ is a degree g(g − 1)/2 -element. Now, by induction assumption, the restriction of the element ψ to Z is a multiple of σ 1 · · · σ g−1 . Consequently, ψ = λσ 1 · · · σ g−1 + σ g φ for some φ ∈ H * ( X). Multiplying this last equation by σ g and noting that σ 2 g = 0, we see that σ g(g+1)/2 1 = λσ 1 · · · σ g , proving the lemma.
We will now begin the proof of Theorem 3. By the criterion of Theorem 1, it is enough to show that under the Borel map, the image of the compact dual class [ Y ] is non-zero , where Y is the compact dual of the symmetric space of H. We will assume for simplicity that H is a product of two symplectic groups: put a + b = g with a ≥ b, and set H = Sp 2a × Sp 2b ⊂ Sp 2g . The proof in the general case is tedious, and we omit it. Now, the cohomology ring H * ( Sp 2a /U(a)) may be identified with
C[x 1 , · · · , x a ] Sa /( i (1 − x 2 i ) = 1),
where S a is the symmetric group on the a letters x 1 , · · · , x a and i runs from 1 to a. Let α 1 , · · · , α a be the elementary symmetric functions in the variables x 1 , · · · , x a .
Similarly, the cohomology ring H * ( Sp 2b /U(b)) may be identified with
C[x a+1 , · · · , x a+b = x g ] S b /( i (1 − x 2 i ) = 1),
where S b is the symmetric group on the b letters x a+1 , · · · , x a+b = x g and i runs from a + 1 to a + b = g. Let β 1 , · · · , β b be the elementary symmetric functions in the variables x a+1 , · · · , x g .
The top degree term (of degree 2( a(a+1)
2 + b(b+1)
2 )) in the cohomology group of the product H * ( Sp 2a /U(a) × Sp 2b /U(b)) is easily seen (by Lemma 10) to be generated by the element γ = α 1 · · · α a ⊗ β 1 · · · β b .
The above description of the cohomology of the compact duals of the spaces Sp 2g /U(g), Sp 2a /U(a) and Sp 2b /U(b) identifies them as the rings generated by certain elementary symmetric functions modulo the ideal of relations i (1 − x 2 i ) = 1 for certain integers i (where i runs respectively from 1 to g, 1 to a and a + 1 to g = a + b). From this it is easy to see that the image of σ k in the cohomology of the product Sp 2a × Sp 2b /U(a) × U(b) is the sum r α r ⊗ β k−r where r runs from 0 to k (if r ≥ a + 1 then α r = 0 by convention). In particular, the image of σ g is α a ⊗ β b . By induction on k, and by using Lemma 9 applied to the symplectic groups Sp 2a and Sp 2b , one can show that the image of σ g σ g−2 · · · σ g−2k is
α a α a−1 · · · α a−r ⊗ β b β b−1 · · · β b−r .
In particular, if k = b we get that the image of σ g σ g−2 · · · σ g−2b is the element α a α a−1 · · · α a−b ⊗ β b β b−1 · · · β 1 . Thus, the top degree term γ (of the cohomology of the product of Sp 2a /U(a) and Sp 2b /U(b)) of the last but one paragraph, lies in the image of the element θ = (σ g σ g−2 · · · σ g−2b )(σ 1 σ 2 · · · σ a−b−1 ) under the restriction map of the cohomology of X to the cohomology of the product Sp 2a /U(a) × Sp 2b /U(b).
By the definition of the class [ Y ] ( Y being the compact dual of the symmetric space associated to Sp 2a × Sp 2b ), this implies the equality θ ∧ [ Y ] = σ 1 · · · σ g , namely a generator of the top degree (of degree 2( g(g+1 2 )) cohomology of X. Since this wedge product is non-zero, and θ is divisible by σ g , it follows that the class [ Y ] is not orthogonal to the ideal generated by σ g . Now, Lemma 8 implies that [ Y ] is not in the kernel of the Borel map. This proves Theorem 3.
The Unitary Group
In this section, G = U(p, q) will denote the unitary group in n = p+q variables, with 1 ≤ p ≤ q. The Q structure is defined as follows. Let V be an n-dimensional vector space over an imaginary quadratic extension E over Q. Consider the E-valued Hermitian form in n variables given by
h(v, v) = i=p i=1 | z i | 2 − i=n i=p+1 | z i | 2
with v = (z 1 , · · · , z n ) ∈ V . The group preserving this Hermitian form is a Q-algebraic group and the group of its real points is the group U(p, q).
The group K = U(p) × U(q) is a maximal compact subgroup of U(p, q) and the group T of diagonals in K is a maximal torus of G and K. As in section (4.1), if X is the compact dual of G/K ( X is the Grassmanian of p planes in n-dimensional complex vector space), then the cohomology of X may be identified with the quotient ring (see [M-T])
C[x 1 , · · · , x p ; y i , · · · , y q ] Sp×Sq (1 + x i ) (1 + y j ) = 1
In this equality, S p (resp, S q ) is the permutation group of the p letters x 1 , · · · , x p (resp. the q letters y 1 , · · · , y q ). The superscript denotes the ring of invariants under the product group S p ×S q . The variable i (resp, j) runs from 1 to p (resp. 1 to q). The equation (1+x i ) (1+y j ) = 1 is a graded equation.
Let σ 1 , · · · , , σ p (resp. τ 1 , · · · , τ q ) denote the elementary symmetric functions in the variables x 1 , · · · , x p (resp. y 1 , · · · , y q ). Then, the cohomology of X is the ring C[σ 1 , · · · , σ p ; τ 1 , · · · , τ q ] (1 + σ 1 + · · · + σ p )(1 + τ 1 + · · · + τ q ) = 1 where as before, the equation involving σ's and τ 's is a graded one.
Lemma 11. With the foregoing notation, the kernel of the Borel map for G = U(p, q) is the orthogonal complement of the ideal generated by σ p and τ q in the cohomology ring H * ( X).
Proof. As p ≤ q, the Q-rank of the group G = U(p, q) is p. The Hermitian form h defining G is a direct sum of p "hyperbolic" (i.e. isotropic over Q) Hermitian forms h 1 ⊕ · · · ⊕ h p and an anisotropic hermitian form h 0 . It is then easy to see that a Levi subgroup is of the form GL k × U(p − k, q − k) for some k ≤ p. Thus, X L is the product of the symmetric spaces SU(k)/SO(k) and U(p+q−2k)/(U(p−k)×U(q−k)).
An argument involving Chern classes of homogeneous vector bundles similar to that in the proof of Lemma 7 shows that all the (positive degree) cohomology classes on U(2k)/U(k)×U(k) vanish on i(SU(k))/SO(k), for any embedding i of SU(k) in U(2k) such that the intersection i(SU(k)) ∩ (U(k) × U(k)) = SO(k) and i is the identity on SO(k). In particular, it shows (cf. the paragraph preceding Lemma 8) that the kernel of the map H * ( X) → H * ( X L ) (where the product is over all the standard Levi subgroups as before), is the kernel of the map H * ( X) → H * (U(p + q − 2)/(U(p − 1) × U(q − 1))).
The latter kernel may easily be shown to be the ideal generated by σ p and τ q .
By Franke, the kernel of the Borel map is precisely the orthogonal complement of this ideal. Hence the Lemma.
Lemma 12. With the previous notation, the element τ p q generates the top degree cohomology (in degree 2pq) of X.
Proof. By induction on p.
Consider the restriction map from H * ( X) to H * ( Y 1 ), where Y 1 is the compact dual of the symmetric space associated to U(p−1, q) ⊂ U(p, q). The kernel of this map is generated by σ p , as may be easily seen. In particular, if R pq−p−q be the graded piece of the cohomology ring R of X (since H (p−1)q ( Y 1 ) is one dimensional) it follows that σ p R pq−p−q is of codimension one in H (p−1)q ( X). But, wedging by τ q kills this subspace (since, by the graded relation, τ q σ p = 0). By definition, wedging by the cycle class [ Y 1 ] also kills this subspace. Therefore, [ Y 1 ] = λτ q for some non-zero scalar λ.
By induction, the restriction of τ p−1 q generates the top degree cohomology of Y 1 . By the definition of the cycle class [ Y 1 ], τ p−1 q [ Y 1 ] is non-zero, i.e. generates the top degree cohomology of X. By the previous paragraph, this element is, up to non-zero scalars, τ p q .
Remark. Note that since σ 1 is the Kahler class on X and restricts to a Kahler class on Y 1 , we have also proved that τ q (σ 1 ) (p−1)q = τ p q = 0. We now begin the proof of Theorem 4. 5.1. Part 1 of Theorem 4. The Hermitian space V may be split into a direct sum of Hermitian spaces V i and W , where, on each V i , the form h is of type (p i , q i ); on W , it is of type (0, q − q i ). Then p i = p and q i ≤ q. We thus get an embedding of H = U(p i , q i ) in U(p, q).
Let ξ 1 , · · · , ξ l (resp. ω 1 , · · · , ω l ) denote the analogues of τ q (resp. of σ 1 ) for the groups U(p 1 , q 1 ), · · · , U(p l , q l ) and Z = Z 1 × · · · Z l be the products of the compact symmetric spaces associted to the product group. The restriction of τ q to this product variety Z is clearly the tensor product ξ 1 ⊗ · · · ⊗ ξ l . Here we use the fact that p i = p. An easy computation shows that if d (resp d i ) is the dimension of Z (resp. Z i ) then the restriction of the element τ q σ d 1 to the product variety Z is the tensor product ξ 1 ω d 1 1 ⊗· · · ⊗ξ l ω d l l . By Remark 5 applied to U(p i , q i ) for each i, we get that this tensor product element is the top degree cohomology class of Z. Thus, τ q σ d 1 ∧ [Z] = 0. Therefore, the cycle class of the compact dual associated to U(p i , q i ) is not orthogonal to an element of τ q R. In particular (from Lemma 11), [Z] does not lie in the kernel of the Borel map. This proves , by Theorem 1, that the modular symbol [S H (Γ)] is non-zero. 5.2. Part (2) of Theorem 4. The cohomology of the compact dual Y associated to Sp 2g is (see Lemma 6)
C[σ 1 , · · · , σ g ] (1 − x 2 i ) = 1 .
The cohomology of X for the group U(g, g) is
C[σ 1 , · · · , σ g ; τ 1 , · · · , τ g ] (1 + x i )(1 + y i ) = 1
The restriction map from the cohomology of X to that of Y , is induced by x i → x i and y i → −x i . Therefore, the top degree class σ 1 · · · σ g of Y is in the image of the product τ 1 · · · τ g . Hence, [ Y ] ∧ τ 1 · · · τ g generates the top degree class of X. This means that the cycle class [ Y ] is not orthogonal to the ideal generated by τ g . Thus, by Lemma 11, [ Y ] is not in the kernel of the Borel map, whence, by Theorem 1, the modular symbol [S H (Γ)] is non-zero.
5.3.
Proof of Theorem 5. The Hermitian space (E, h) is such that for the associated group G, we have G(R) = SU(1, q). We first prove the following lemma.
Lemma 13. If α is a non-zero holomorphic 1-form on S(Γ), ω is the Kahler class on X and j the Borel map, then the cup-product α ∧ j(ω) is non-zero.
Proof. It is enough to prove (since ω is G(A f )-invariant), that for some g ∈ G(A f ), the cup-product g(α) ∧ ω is non-zero. Consider the three dimensional hermitian subspace (F, h | F ) of the Hermitian space (V, h), (a vector space over the field E) where the Hermitian form does not represent a zero. By weak approximation, this is possible (since one may locate a three dimensional subspace over p-adic field E v of the hermitian vector space V ⊗ E v , h ⊗ E v where the Hermitian form does not represent a zero).
This is an anisotropic subspace, whence the associated group H 0 = SU(F, h) is anisotropic over Q and is isomorphic to SU(1, 2) over R. Thus, the subvariety S H 0 (Γ) is compact. If α is a non-zero holomorphic 1-form on S(Γ) class on X, then there exists a g ∈ G(A f ) such that the restriction of the form g * (α) to S H 0 (Γ) is non-zero. By replacing α by g * (α), we may assume that g = 1. However, since S H 0 (Γ) is compact, this means that the restriction of α ∧ α ∧ j(ω) to S H 0 (Γ) is non-zero. In particular, α ∧ j(ω) = 0.
In the above, α denotes the anti-holomorphic 1-form which is the complex conjugate of the holomorphic form α. The complex conjugation is on the cohomology group H 1 (S(Γ), C) = H 1 (S(Γ), R) ⊗ C.
Proof. of Theorem 5. We will argue by contradiction. Suppose that for H = SU(1, q − 1) the cycle class [S H (Γ)] is always (i.e. for every Γ) G(A f )-invariant. By Theorem 1, this class is then equal to j([ Y ]). However, H 2 ( X) is one dimensional ( X = P q , the complex projective q-space). Hence [ Y ] = ω the Kahler class.
If α is a non-zero holomorphic form on S(Γ) (such forms exist by [K]), then α ∧ j([ Y ]) = 0 by Lemma 13. Thus, the restriction of α to S H (Γ) does not vanish, for every Γ. Thus, at the level of H 1 the restriction map from SU(1, q) to SU(1, q − 1) is injective for every Γ. In particular, the holomorphic cohomology classes of degree one associated to SU(n, 1) restrict injectively to those on SU(n − 1, 1).
We now recall the criterion of [V2], for the span of Hecke-translates of [S H (Γ] to be infinite dimensional. In the statement of the following theorem, the map Res refers to an "Oda style" restriction map.
Theorem 14. (see [V2], Theorem 1). Suppose that G is almost Qsimple and that 1. The centralizer Z G (H) ∩ K is not contained in the center Z(G) of G.
2. For some integer m ≤ d = dim(S H (Γ)) (dimension as a complex manifold), the restriction map is non-zero.
Then, there exists a congruence subgroup Γ ′ of Γ such that the cycle class [S H (Γ ′ )] is not G(A f ) invariant.
For the group G with G(R) = SU(n, 1) up to compact factors there exist elements in G(Q) which centralize SU(n − 1, 1) but do not lie in the center of G, since the centralizer of SU(n − 1, 1) is the group U(1) (all these viewed, by restriction of scalars, as groups over Q).
Thus, the conditions of Theorem 1 of [V2] are satisfied and so by Theorem 1 of [V2], there exists a Γ so that the cycle class [S H (Γ)] is not G(A f )-invariant, which contradicts our assumption.
Res : H m,0 (S(Γ)) →g∈G(Q)H m,0 (S H (gΓg −1 ))
Acknowledgement. This work was begun when both the authors were visiting MPI, Bonn in May-June of 2001; the hospitality of MPI is gratefully acknowledged. The authors also thank Arvind Nair for very helpful conversations. Birgit Speh was partially supported by the NSF grant DMS-007056.
Non square integrable cohomology of arithmetic groups. A Ash, Duke Math. Journal. 47A.Ash., Non square integrable cohomology of arithmetic groups, Duke Math. Journal 47 (1980) 435-449.
Generalized modular symbols. B , A Ash, A Borel, Lecture Notes in Mathematics. 1447Springer-VerlagB] A.Ash and A.Borel, Generalized modular symbols, Lecture Notes in Math- ematics 1447, Springer-Verlag (1990), 57 -75.
Vanishing periods of cusp forms over modular symbols. -G-R] A Ash, D Ginzburg, S Rallis, Math. Annalen. 296-G-R] A.Ash, D.Ginzburg and S.Rallis, Vanishing periods of cusp forms over modular symbols, Math. Annalen 296 (1993), 709 -723.
A Topological Model for Some Summand of the Eisenstein Cohomology of Congruence Subgroups. J Franke, Preprint, Bielefeld UniversityJ.Franke, A Topological Model for Some Summand of the Eisenstein Co- homology of Congruence Subgroups, Preprint, Bielefeld University, 1991.
Some applications of the Weil representation. D Kazhdan, Journal d'analyse. 32D.Kazhdan, Some applications of the Weil representation, Journal d'analyse, 32, 233-248 (1977).
Toda T] Mimura, Topology of Lie Groups (I) and (II. AMer. Math. Soc91T] Mimura and Toda, Topology of Lie Groups (I) and (II), Translations of Math. Monographs, Vol 91, AMer. Math. Soc. (1991).
Eisenstein cohomology of arithmetic groups. The case GL 2 . Invent. G Harder, Math. 89G.Harder, Eisenstein cohomology of arithmetic groups. The case GL 2 . In- vent. Math 89, 37-118, (1987)
Pseudo-eisenstein Forms and the cohomology of arithmetic groups. G Harder ; R-S] J.Rohlfs, B Speh, Tata InstituteModular symbols and special values of L-functions. to appear in the proceedings of the conference in honor of RagunathanG.Harder, Modular symbols and special values of L-functions, available at ftp://ftp.math.uni-bonn.de/people/harder/Eisenstein/Modsym.pdf [R-S] J.Rohlfs and B.Speh, Pseudo-eisenstein Forms and the cohomology of arith- metic groups, to appear in the proceedings of the conference in honor of Ragunathan, Tata Institute 2002.
Cohomology of compact locally symmetric spaces. T N Venkataramana, Compositio.Math. 125T.N.Venkataramana, Cohomology of compact locally symmetric spaces, Compositio.Math 125, 221-253, (2001).
T N Venkataramana, On Cycles on Compact Shimura varieties. Monatshefte Mathematik. 135T.N.Venkataramana, On Cycles on Compact Shimura varieties. Monat- shefte Mathematik 135, 221-244 (2002).
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"Nonequilibrium Phase Transitions",
"Nonequilibrium Phase Transitions"
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"Zoltán Rácz \nInstitute for Theoretical Physics\nEötvös University\nPázmány sétány 1/a1117BudapestHungary\n"
] |
[
"Institute for Theoretical Physics\nEötvös University\nPázmány sétány 1/a1117BudapestHungary"
] |
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Nonequilibrium phase transitions are discussed with emphasis on general features such as the role of detailed balance violation in generating effective (long-range) interactions, the importance of dynamical anisotropies, the connection between various mechanisms generating power-law correlations, and the emergence of universal distribution functions for macroscopic quantities. Quantum spin chains are also discussed in order to demonstrate how to construct steady-states carrying fluxes in quantum systems, and to explain how the fluxes may generate power-law correlations.
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"https://export.arxiv.org/pdf/cond-mat/0210435v1.pdf"
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cond-mat/0210435
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70604e34bb1f9aaed510e1e37c3bccc3b01e4192
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Nonequilibrium Phase Transitions
20 Oct 2002 July 2002
Zoltán Rácz
Institute for Theoretical Physics
Eötvös University
Pázmány sétány 1/a1117BudapestHungary
Nonequilibrium Phase Transitions
20 Oct 2002 July 2002Lecture Notes, Les Houches,
Nonequilibrium phase transitions are discussed with emphasis on general features such as the role of detailed balance violation in generating effective (long-range) interactions, the importance of dynamical anisotropies, the connection between various mechanisms generating power-law correlations, and the emergence of universal distribution functions for macroscopic quantities. Quantum spin chains are also discussed in order to demonstrate how to construct steady-states carrying fluxes in quantum systems, and to explain how the fluxes may generate power-law correlations.
There are many reasons for studying nonequilibrium phase transitions. Let us start be mentioning a few which carry some generality.
First and foremost, equilibrium in nature is more of an exception than the rule, and structural changes (which constitute a significant portion of interesting phenomena) usually take place in nonequilibrium conditions. Thus there is much to be learned about the complex ordering phenomena occurring far from equilibrium.
Second, while very little is understood about the general aspects of nonequilibrium systems, the equilibrium critical phenomena have been much studied and have been shown to display universal features. This universality emerges from large-scale fluctuations in such a robust way that one can expect that similar mechanisms will work in nonequilibrium situations as well. Thus, investigating the similarities and differences of equilibrium and nonequilibrium orderings may help to discover the distinguishing but still robust properties of nonequilibrium systems.
Third, power law correlations are present in many nonequilibrium phenomena and there have been many attempts to explain these correlations through general mechanisms. Closer examination, however, usually reveals a close connection to equilibrium or nonequilibrium critical phenomena.
My lectures were designed to revolve around problems related to the above points. The lectures are built on the theory of equilibrium phase transitions [1,2], thus I assume knowledge about both static and dynamic critical phenomena at least on the level of familiarity with the basic concepts (symmetry breaking, order parameter, diverging correlation length, order parameter, scaling, universality classes, critical slowing down, dynamical symmetries).
I discuss simple examples throughout so that enterprising students could try out their luck implementing their ideas in simple calculations. Due to space restrictions, however, not all of the details discussed in the lectures and afterwards are covered in the written notes. In particular, the picturesque parts of the explanations are often left out since they take up disproportionally large part of the allowed space. Nevertheless, these pictures may be important in both understanding and memorizing, and I strongly encourage the reader to go through the slides of the lectures, as well. They can be found through the homepage of the school, http://dpm.univ-lyon1.fr/houches ete/lectures/ or at http://poe.elte.hu/∼racz/.
A. Nonequilibrium steady states
A general feature that distinguishes a nonequilibrium steady state (NESS) from an equilibrium one is the presence of fluxes of physical quantities such as energy, mass, charge, etc.
Thus the study of NESS is, in a sense, a study of the effects of fluxes imposed on the system either by boundary conditions, or by bulk driving fields, or by some combination of them.
A nonequilibrium steady stateswell known example is shown on Fig.1. This is the Rayleigh-Bénard experiment [3] in which a horizontal layer of viscous fluid (the system) is heated from below i.e. it has two heat baths of temperatures T 1 and T 2 attached (boundary conditions generating an energy flux). The presence of gravity (the bulk drive)
is also important (it generates mass and momentum fluxes at large δT = T 1 − T 2 > 0).
For T 1 = T 2 this system relaxes into a quiescent equilibrium state while a small δT will also result in a quiescent state but it is already a NESS since energy flux is flowing through the system. Increasing δT , this steady state displays a nonequilibrium phase transition (Rayleigh-Bénard instability), first to a stationary convective pattern, and then to a series of more complicated structures which have fascinated researchers for the past century [3].
Starting from the Navier-Stokes equations, one can arrive at a mean-field level of understanding of the above phenomena. It is, however, not the level of sophistication one got used to in connection with equilibrium phase transitions. There, we have simple exactly solvable models such as e.g. the Ising model which give much insight into the mechanism of ordering and, furthermore, this insight can be used to develop theories which reveal the universal features of equilibrium orderings [1,2].
The trouble with the Rayleigh-Bénard system is that we do not have a theory even for the NESS. The reason for this is that the fluxes result in steady state distributions , P * n , which break the detailed balance condition w n→n ′ P * n = w n ′ →n P * n ′ , where n and n ′ are two "microstates" and w n→n ′ is the rate of the n → n ′ transition. As a consequence of the breaking of detailed balance, a NESS is characterized not only by the probability distribution, P * n , but also by the probability currents in the phase space. Unfortunately, we have not learned yet how to handle the presence of such loops of probability currents.
The main lesson we should learn from the Rayleigh-Bénard example is that, in order to have Ising type models for describing phase transitions in NESS, one should use models which relax to steady states with fluxes present. Such models have been developed during the last 20 years, and most of my lectures are about these stochastic models defined through "microscopic" elementary processes. The first level of description is in terms of master equations which are conceptually simple and allow one to make use of general results (uniqueness of stationary state, etc.) which in turn are helpful in defining dynamics that leads to NESS (Sec.II). The next level is to describe the same problem in terms of Langevin equations and develop field-theoretic techniques for the solution. Our discussions will include both levels of description and I hope that at the end an understanding will emerge about a few results which grew in importance in the last decade (generation of long-range interactions and effects of dynamical anisotropies (Sec.II), connection between mechanisms generating power-law correlations (Sec.III), and universality of distribution functions for macroscopic quantities (Sec.IV)).
Before starting, however, I would like to insert here a little essay about effective temperatures. This concept is being widely discussed in connection with slowly relaxing systems, the topic of this school. So it may be of interest to present here a view from the perspective of NESS.
B. Problems with usual thermodynamic concepts
Systems close to equilibrium may retain many properties of an equilibrium state with the slight complication that the intensive thermodynamic variables (temperature, chemical potential, etc.) become inhomogeneous on long lengthscales and they may slowly vary in time. This type of situations are successfully dealt with using the so called local equilibrium approximation [4], with the name giving away the essence of the approximation. The applicability of the concept of local equilibrium should diminish, however, as a system is driven far from equilibrium. Nevertheless, questions of "how large drive produces a farenough state" and "couldn't one try to find a new equilibrium state near-by" are regularly asked and have legitimacy. So I will try to illuminate the problems on the example of the fluctuation-dissipation theorem much discussed nowadays due to attempts of associating effective temperatures with the various stages of relaxation in glasses [5] or with steady states in granular materials [6].
Let us consider a simple system of Ising variables σ with Hamiltonian H 0 and in equilibrium at temperature β = 1/(k B T ). Assuming that there is an external field H coupling linearly to the macroscopic magnetization M = i σ i , one can write the equilibrium distribution function as
P eq (σ) = Z −1 e −βH 0 (σ)+βHM (σ)(1)
The average value of the magnetization is given by
M = Z −1 σ M(σ)e −βH 0 (σ)+βHM (σ) .(2)
and the static limit of the fluctuation-dissipation theorem is obtained as
χ M = ∂ M ∂H H→0 = β M 2(3)
where we assumed the system to be in the high-temperature phase ( M = 0). Note the simplicity and the accompanying generality of this derivation. It uses only the fact that the external field is linearly coupled to the quantity (M) we are considering.
The fluctuation-dissipation theorem is used in many ways. It helps simplify field-theoretic studies of fluctuations through diagrammatic expansions and it also gives a powerful checking procedure in both experiments and Monte Carlo simulations. Note that eq.(3) can also be used to define the temperature of the system through β = χ M / M 2 , and the temperature defined in this way would be the same when using different "M"-s and conjugate fields "H".
It is clear that a fluctuation-dissipation theorem generalized to NESS would be extremely useful. Let us now try to imagine how a similar relationship may arise when we drive the above system away from equilibrium (e.g. by attaching two heat baths of different temperatures). If the system relaxes to a NESS then there will be steady-state distribution function, (P ) but the effective Hamiltonian (ln P ) will contain all the interactions allowed by the symmetries of the system. Thus, assuming that the effective Hamiltonian can be expanded in H, one finds in the H → 0 limit
P ne (σ) ∼ e −aH 1 (σ)+bH[M (σ)+S 3 (σ)+...](4)
where S 3 is a notation for sums over all three-spin clusters with different couplings for different types of spatial arrangements of the three spins. Furthermore, a, b and all other newly generated couplings depend on the original couplings in H 0 and on the temperatures of the heat baths. Now a derivation of the fluctuation-dissipation theorem similar to the equilibrium case yields a more complicated equation
χ M = ∂ M ∂H H→0 = a M 2 + MS 3 + ... .(5)
There are two ways a simple form for the fluctuation-dissipation theorem may emerge from eq.(5). One is that a nonlinear field
Q = M + S 3 + ...(6)
that is conjugate to H can be introduced (and effectively worked with). Then one obtains
χ Q = a Q 2(7)
and thus a becomes the nonequilibrium β.
The other possibility is that a mean-field type decoupling scheme works well and then
MS 3 = M 2 f (C 2 )(8)
where f (C 2 ) is a functional of the two-point correlations (and similar expressions are obtained for MS 2n+1 ). Then equation (5) becomes
χ M = aF (C 2 ) M 2 .(9)
If the theory provides F (C 2 ) then the effective temperature can again be read of from the above generalized fluctuation-dissipation relationship.
There are problems with both lines of reasoning. Apart from the practical difficulties of nonlinear fields and the validity of mean-field type approaches, the main conceptual difficulty is the fact that changing from the magnetization to other fields (e.g. energy) the nonequilibrium version of the fluctuation-dissipation theorem leads to different values for the same "temperature" [10]. There are cases where the above schemes generalized to timedependent processes work both at the theoretical [5,7,8] and the experimental levels [9] but there are clear examples when the concept of effective temperature does not apply [10].
Thus the meaning and use of nonequilibrium temperature has not been clarified enough to make a verdict on it.
II. PHASE TRANSITIONS FAR FROM EQUILIBRIUM
As mentioned in the Introduction, nonequilibrium phase changes constitute a large part of interesting natural phenomena and they are studied without worries about wider contexts. From a general perspective, on the other hand, the investigations of nonequilibrium phase transitions [11,12] can be viewed as an attempt to understand the robust features of NESS. This view is based on the expectation that the universality displayed in critical phase transitions carries over to criticality in NESS as well. If this is true then studies of the similarities to and differences from equilibrium will lead to a better understanding of the role and general consequences of the dynamics generating NESS.
In the following subsections, we shall construct, describe, and discuss models which display nonequilibrium phase transitions. Apart from getting familiar with a few interesting phenomena, the main general conclusion of these discussions should be that dynamical anisotropies often yield dipole-like effective interactions [13,14,15] and, furthermore, competing non-local dynamics (anomalous diffusion) generates long-range, power-law effective interactions [16]. Along the way, we shall also understand that the detailed-balance violating aspects of local relaxational dynamics do not affect the universality class of the nonequilibrium phase transitions [15,17].
A. Differences from equilibrium -constructing models with NESS
The violation of detailed balance has the consequence that not only the interactions determine the properties of the NESS but the dynamics also plays an important role. In order to understand and characterize the role of dynamics, a series of simple examples will be discussed in the following subsections.
First, let us discuss how to construct a model which yields a NESS in the long-time limit.
A simple way is to attach two heat baths to a system, each generating a detailed-balance dynamics but at different temperatures. To see an actual implementation, let us consider how this is done for the one dimensional kinetic Ising model. This type of models have been much studied and a collection of mini-reviews about them can be found in [18].
The state of the system {σ} ≡ {. . . , σ i , σ i+1 , . . .} is specified by stochastic Ising variables σ i (t) = ±1 assigned to lattice sites i = 1, 2, . . . , N. The interaction is short ranged (nearest neighbor) −Jσ i σ i+1 and periodic boundary conditions (σ N +1 = σ 1 ) are usually assumed. The dynamics of the system is generated by two heat baths (labeled by α = 1, 2) at temperatures T α , meaning that the heat baths try to bring the system to equilibrium at temperature T α by e.g. spin flips and spin exchanges, respectively.
Let us denote the rate of the flip of i-th spin (σ i → −σ i ) by w (1) i ({σ}), and let the rate of the exchanges of spins at sites i and j (σ i ↔ σ j ) be w (2) ij ({σ}). Then the dynamics is defined by the following master equation for the probability distribution P ({σ}, t) :
∂ t P ({σ}, t) = i w (1) i ({σ} i ) P ({σ} i , t) − w (1) i ({σ}) P ({σ}, t) + ij w (2) ij ({σ} ij ) P ({σ} ij , t) − w (2) ij ({σ}) P ({σ}, t)(10)
where the states {σ} i and {σ} ij differ from {σ} by the flip of the i-th spin and by the exchange of the i-th and j-th spins, respectively.
The assumption that the dynamics is generated by heath baths means that the rates satisfy detailed balance at the appropriate temperatures:
w (α) i(j) ({σ}) P eq α ({σ}) = w (α) i(j) ({σ} i(j) ) P eq α ({σ} i(j) ) ,(11)
where P eq α ∼ exp [−J/T α i σ i σ i+1 ] is the equilibrium distribution of the Ising model at temperature T α . Eq.(11) leaves some freedom in the choice of w α -s, and one is usually guided by simplicity. The most general spin flip rate that depends only on neighboring spins has the following form [19] w (1)
i (σ) = 1 2τ 1 1 − γ 2 σ i (σ i+1 + σ i−1 ) 1 + δσ i+1 σ i−1 .(12)
Without any other heath baths, equations (10) and (12) define the Glauber model [19] which relaxes to the equilibrium state of the Ising model at temperature T 1 defined through γ = tanh(2J/k B T 1 ). The time-scale for flips is set by τ 1 and δ is restricted to the interval −1 < δ < 1.
The competing dynamical process is the generation of spin exchanges (Kawasaki dynamics [20]) by a second heath bath at a temperature T 2 = T 1 . In the simplest case, the exchanges are between nearest neighbor sites and the rate of exchange satisfying detailed balance (11) is given by
w (2) i (σ) = 1 2τ 2 1 − γ 2 2 (σ i−1 σ i + σ i+1 σ i+2 ) .(13)
where γ 2 = tanh(2J/k B T 2 ) and τ 2 sets the timescale of the exchanges. It is often assumed that the exchanges are random (T 2 = ∞) and thus w (2) i (σ) = 1/(2τ 2 ). Equations (10), (12) and (13) from a bulk driving field) should be obvious. Just as it should be obvious that there are not too many exactly solvable models in this field and most of the results are coming from simulations [11,12].
Before turning to results, let us also introduce a Langevin equation description of competing dynamics. The Langevin approach has been successful in dynamic critical phenomena [2] where the counterparts of the Glauber and Kawasaki models are called Model A and B, correspondingly. This correspondence makes the "two-heath-baths" generalization straightforward. The coarse grained magnetization of the Ising model is replaced by n-component order-parameter field, S i (x, t), (i = 1, ..., n) (n = 1 is the Ising model, and n → ∞ is the spherical limit that allows simple analytic calculations as shown below). The system evolves under the combined action of local relaxation (Model A) satisfying detailed balance at temperature T 1 and diffusive dynamics (Model B) at temperature T 2 which yields the following Langevin equation for the Fourier transform S i q (t)
∂ t S i q = −L (1) q S i q + η i 1 (q, t) − L (2) q S i q + η i 2 (q, t) .(14)
Here L (α)
q S i q = Γ (α)
q δF (α) /δS i −q with F (α) being the free energy at temperature T α , and the kinetic coefficient Γ
L (α) q S i q = Γ (α) q (r α 0 + q 2 )S i q + u n j=1 q ′ q ′′ S j q ′ S j q ′′ S i q−q ′ −q ′′ (15)
where r α 0 is linear in T α and u ∼ 1/n in the n → ∞ spherical limit. In order to ensure that in case of a single heat bath, the system relaxes to equilibrium satisfying detailed balance, the noise terms in eq.(14 are Gaussian-Markovian random forces with correlations of the form:
η i α (q, t)η j α ′ (q ′ , t ′ ) = 2Γ (α) q δ αα ′ δ ij δ(q + q ′ )δ(t − t ′ ) .(16)
Eqs. (14), (15) and (16) define the model for the particular competing dynamics chosen and we are now ready to deduce some features of the NESS generated. Of course, just as in case of kinetic Ising models, the number of possible competing dynamics is infinite and the question is whether conclusions of some generality could be reached.
B. Generation of long-range interactions -nonlocal dynamics
The remarkable consequences of competing dynamics can be seen already on the example of d = 1 flip-and-exchange model which may produce ordering even though the interactions are of short range. Indeed, if T 1 temperature spin flips are competing with T 2 = ∞ spin exchanges of randomly chosen pairs then the system orders below a certain T 1c [16].
It turns out that the transition is of mean-field type and this gives a clue to understanding.
Indeed, let us imagine that the rate of spin exchanges is large compared to the rate of flips.
Then the random exchanges mix the spins in between two flips and the flipping spin sees the "average spins" in its neighborhood -a condition for mean-field to apply.
The mean field result can also be interpreted as the generation of infinite-range effective interactions. This interpretation can be put on more solid base by studying the above model with T 2 = ∞ spin exchanges where the probability of exchange at a distance r is decaying with r as p(r) ∼ 1/r d+σ (the spins exchanges are σ dimensional Levy flights in dimension d).
The system orders again below a T 1c and the examination of the critical exponents reveals [16] that the transition is in the universality class of long-range interactions decaying with r as J(r) ∼ 1/r d+σ . It is important to note that the above results are nonequilibrium effects which would disappear if the spin exchanges would also be at T 1 .
Let us now see if the same results can be derived from the Langevin equation approach.
The spin flips are translated into the Model A part of the dynamics while the Levy flights can be represented [21] by anomalous diffusion with Γ
(2) q = D (2) q 2 replaced by Γ (2) q = D (2) q σ with 0 < σ < 2S i q = −Γ 0 (r 0 + q 2 )S i q −Γ 0 u n j=1 q ′ q ′′ S j q ′ S j q ′′ S i q−q ′ −q ′′ + η i q (t) −Dq σ S i q +η i q (t) .(17)
Note that due to the randomness of the Levy flights (T 2 = ∞), the interaction and the nonlinear terms are missing in the Levy flight part (second line) of the equation. As discussed in Sec.II A, the η-s are Gaussian-Markoffian random forces with correlations η i
q (t)η j q ′ (t ′ ) = 2Γ 0 δ ij δ(q + q ′ )δ(t − t ′ ) and η i q (t)η j q ′ (t ′ ) = 2Dq σ δ ij δ(q + q ′ )δ(t − t ′ )
. In order to see the generation of long range interaction in the above model, let us first make an exact calculation [16] in the spherical limit (n → ∞) where fluctuations in u S j q (t)S j q ′ (t) may be neglected and this quantity may be replaced by
u n j=1 S j q (t)S j q ′ (t) = unC(q, t)δ(q + q ′ ) ,(18)
where the brackets denote averaging over both the initial conditions and the noises η andη (we restrict ourselves to the study of the high-temperature phase where the dynamic structure factor C(q, t) = S j q (t)S j −q (t) is independent of j). The decoupling (18) leads to a linear equation of motion and so the self-consistency equation for C(q, t) can be easily derived
C(q, t) = 2(Γ 0 + Dq σ ) t 0 dt ′ e −2 t ′ 0 {Γ0[r0+q 2 +unC(q,s)]+Dq σ }ds .(19)
Here, the initial condition C(q, 0) = 0 was used for simplicity. The t → ∞ limit does not depend on the initial condition and the equation for the steady state structure factor
C(q) = C(q, t → ∞) becomes C(q) = Γ 0 + Dq σ Γ 0 (r 0 + q 2 + unS) + Dq σ ,(20)
where S = dqC(q).
The long-wavelength instabilities are determined by the q → 0 form of C(q) which for 0 < σ < 2 can be written as
C(q) ≈ (r 0 + λq σ + unS) −1 ,(7)
with λ = D/Γ 0 . This form coincides with the long-wavelength limit of the equilibrium structure factor of a spherical model in which the interactions decay with distance as r −d−σ .
Consequently, both the self-consistency equation for r = r 0 + unS and the critical behavior that follows from it are identical to that of the equilibrium long-range model. Thus we can conclude that the critical properties of the NESS are dominated by an effective long-range potential proportional to r −d−σ .
The above conclusion should be valid quite generally for finite n as well. Looking at eq. (17), one can see that the correlations in the effective noise (η ef f = η +η) have an amplitude 2(Γ 0 + Dq σ ). One expects that the Dq σ term can be neglected in the longwavelength limit and thus that the noiseη in the Levy-flight exchanges can be omitted.
Withoutη, however, the system described by equation (2) satisfies detailed balance and has an effective Hamiltonian which, apart from the usual short-range interaction pieces, contains the expected long-range part λ dqq σ i S i (q)S i (−q). We should note here that σ → 2 corresponds to usual diffusion and that the above arguments changes for σ = 2 since no long-range interactions are generated any more, and no change in critical behavior occurs. This result is another way of saying that Model A type dynamics is robust against diffusive perturbations which break detailed balance [17]. Note also that if both the competing dynamics are relaxational then, adding up the corresponding deterministic and noise parts in the Langevin equation (14), one can easily deduce that the breaking of the detailed balance does not change the universality class of the equilibrium phase transition.
C. Generation of long-range interactions -dynamical anisotropies
In order to understand the meaning of dynamical anisotropy, let us consider the twotemperature, diffusive kinetic Ising model [22] on a square lattice. Two heat baths are attached and both of them generates nearest neighbor spin exchanges. Exchanges along one of the axis (called 'parallel' direction) satisfy detailed balance at temperature T while exchanges in the 'perpendicular' direction are produced by a heat bath of temperature T ⊥ .
It is important to note that the interactions Jσ i σ j are the same along both axes. It is the dynamics that is anisotropic.
For T ⊥ = T = T , this is the Kawasaki model [20] which relaxes to the equilibrium Ising model at T and, consequently, it displays a continuous transition. Since the dynamics conserves the total magnetization, the ordering for T < T c appears as a phase separation.
For T ⊥ = T , on the other hand, there is a flow of energy between the and ⊥ heath baths and the system relaxes to a NESS. MC simulations [14,22] show that a critical phase transition is present for T ⊥ = T as well but the phase separation is distinct from that occurring in equilibrium. The interfaces between the domains of up and down spins align with normals along the directions of lower temperatures. Thus the symmetries of the ordered states are different from the symmetry of the equilibrium order where interfaces with normals along any of axes coexist (isotropic ordering). As a consequence, the universality classes of the and ⊥ orderings are found to be distinct from the Ising class [22,23]. Renormalization group calculations actually show that the universality class of the nonequilibrium ordering is that of a uniaxial ferromagnet with dipolar interactions [24].
A dramatic demonstration of the long-range nature of the interactions generated by anisotropic dynamics comes from the generalization of the above model to n = 2 component spins (2T − XY model). One finds that the NESS in this system displays an ordering transition [14], a fact that would be in contradiction with the Mermin-Wagner theorem [25] should the effective interaction be short-ranged.
Let us try now understand the generation of the dipole-like interactions using the Langevin equation approach and considering the spherical limit again. The two-temperature, diffusive Ising model corresponds to competition of two Model B type dynamics along and ⊥ axes. Thus the equation of motion isṠ i
q (t) = L (q)S i q + η i (q, t) + L ⊥ (q)S i q + η i ⊥ (q, t)
with the diffusion in the α = , ⊥ directions described by the corresponding L α terms:
L α S i q = D α q 2 α (r α 0 + q 2 )S i q + u n j=1 q ′ q ′′ S j q ′ S j q ′′ S i q−q ′ −q ′′ .(21)
where q = (q , q ⊥ ). Note the isotropy of the interaction (q 2 term) and the different temper-
atures (r α 0 ) for diffusion in different directions. The noise correlations follow from detailed balance requirements, η i α (q, t)η j α ′ (q ′ , t ′ ) = 2D α q 2 α δ αα ′ δ ij δ(q + q ′ )δ(t − t ′ )
. Just as in the kinetic Ising model for r 0 = r ⊥ 0 , we have Model B with anisotropic diffusion (not a dynamical anisotropy!) and with an equilibrium steady state, while a NESS is produced for r 0 = r ⊥ 0 . The nature of the phase transitions in the NESS becomes transparent in the spherical limit (n → ∞ and u ∼ 1/n) where the fluctuations in u S j q (t)S j q ′ (t) may be neglected. This linearizes the equation of motion and allows to write down a selfconsistency equation for C(q) = S j q (t)S j q ′ (t) . The t → ∞ limit then yields the steady state structure factor in the following form [14]
C(q) = q 2 + aq 2 ⊥ q 2 (r 0 + q 2 + S) + aq 2 ⊥ (r ⊥ 0 + q 2 + S) ,(22)
where a = D ⊥ /D and S = un dqC(q).
One can see now the origin of dipole-like effective interactions. Because of the dynamical anisotropy, the q → 0 limit of C(q) is different whether first q → 0 and then q ⊥ → 0 or vice versa. This singularity of the long-wavelength limit translates into such power law correlations in real space which are characteristic of dipole interactions in equilibrium systems. Hence the conclusion [14,24] that the dynamical anisotropy has generated dipolelike interactions. Note that this is a nonequilibrium effect. The long-wavelength singularity disappears as soon as the heat baths have equal temperatures (r 0 = r ⊥ 0 ). The dynamical anisotropy is a strong effect and its mechanism of action is rather simple as we have seen above. Accordingly, it is the most viable candidate to change the universality class of equilibrium phase transitions by breaking detailed balance [15].
D. Driven lattice gases, surface growth
Models of NESS have a long history but the first one that became the center of attention and was recognized as the "Ising model" of NESS was the driven lattice gas (see [11] and references therein). The model can be understood as a kinetic Ising model with the up-spins being the particles in the lattice gas. Spin exchange dynamics at temperature T represents the particle diffusion and an external bulk field E x drives the up-spins (particles) along one of the lattice axes (x). In order to have a NESS with particle current one must also use periodic boundary conditions in the field direction.
This model displays a critical phase transition in its NESS and the phase separation that follows at low T is characterized by strong anisotropy: the interfaces align parallel with E x .
Thus one can see some similarities with the two-temperature diffusive model of Sec.II C.
One can easily recognize that dynamical anisotropy is at work here. The driving field can be considered as a second heath bath which generates the essential part of the dynamics along the x direction. There is a difference, however, from the two-temperature model of Sec.II C in that the drive now has a directionality (the forward-backward symmetry of diffusion is broken). As a result the phase transition is expected to belong to a new (nonequilibrium) universality class distinct from that of the dipole class. This is indeed what has been obtained in renormalization group calculations [11]. Unfortunately, the structure of the long-wavelength singularities in the two systems are similar and thus there are difficulties in observing the differences in numerical work. This problem has generated some debate that is still going on [26]. We believe the debate will not modify the general picture summarized in [11], and it will not change the conclusion about the importance of dynamical anisotropy.
An interesting and important field where even the simplest systems show "effective" critical behavior due to the unbounded long-wavelenth fluctuations is the field of surface growth processes. Most of the roughening transitions and transitions between various rough phases are genuine nonequilibrium phase transitions and have been much studied [27,28].
Remarkably, however, these transitions have not provided new insight into the general features of nonequilibrium criticality, they merely confirmed that the dynamics and dynamical anisotropy play an important role in determining the universality classes of growth processes.
One of the unsolved problems that continues to fascinate researchers is the phase transition in the Kardar-Parisi-Zhang equation [29]. We shall discuss the problem of growing surfaces including KPZ equation in connection with the nonequilibrium distribution functions in Sec.IV.
E. Flocking behavior
Up to this point, we have considered usual physical systems driven out of equilibrium.
Here I would like to give a taste of what awaits one if the studies are extended to the living realm.
Living creatures can also be viewed as units attached to two heat baths. One of them is the internal energy source which on a short time-scale is an infinite bath from which energy can be drawn at a given rate. The other one is the surroundings to which energy is lost by dissipation (friction, heat loss, ...) due to the activity of the unit. This view suggests that a collection of such self-propelled units will show orderings (nonequilibrium phase transitions) depending on the interactions between the units, on their density, on their possible motions and on the dissipation mechanisms. Indeed, collective behavior is often observed in flocks of birds, in schools of fish, in swimming cells, etc. and, as shown below, some of these phenomena can be described in terms of a surprisingly simple model.
Model [30] was introduced to describe the collective motion of self-propelled particles with birds and bacteria being candidates for these particles. We shall use the language of "birds" below.
θ i (t + τ ) = θ(t) r + η i(23)
where η i is random noise with amplitude η.
Assumption (i) handles the energy in-and outflow by strictly equating them, while assumption (ii) handles the interactions by seemingly reducing them to interactions in the space of velocity directions. This is not quite so, however, since the motion of the birds
x i (t + τ ) = x i (t) + v i (t)
τ couples the directional and spatial motions.
The control parameters in the system are r, η and the density of particles ρ. Keeping r and ρ fixed while varying the "temperature" η, one finds that the birds are flying randomly for η > η c (r, ρ) while collective motion develops below η c where the birds tend to move in the same direction. An order parameter characterizing this spontaneously symmetry breaking can be chosen e.g. φ = | N i v i |/N. Fig.2 shows the time evolution deep in the ordered regime (φ = 0.8) starting from a random configuration. One can see that local orientational order develops in the initial stages of relaxation (the state here shows resemblance to the states in classical XY ferromagnet) while the stationary state with almost full orientational order shows large density fluctuations. The structure and the large density fluctuations observed in the ordered state and, furthermore, the measurements of the critical exponents of the transition [30] suggest that the ordering in this system is in a universality so far not encountered.
A remarkable field theory has also been constructed for flocking [31]. It is a generalized Navier-Stokes equation with additional Model A type terms which drive the velocity to | v i | = 1. This theory explains the large density fluctuations present in the ordered state.
The investigation of ordering transition is at a higher level of difficulty, however, and has not been completed yet.
Clearly, much remains to be done before we understand flocking and before the model can be compared with experiments quantitatively. Nevertheless, activity is expected in this direction since the model of flocking is not much more complicated than the more standard NESS models discussed above, and, at the same time, it has close connection with experimentally observable, truly "far-from-equilibrium" phenomena. Hopefully, by designing and understanding similar models, a kind of "universality map" of the collective dynamics of self-driven units can be found. the white-dwarf light emission [35], the flow of sand through an hourglass [36], the number of daily trades in the stock market [37], water flow fluctuations of rivers [38], the spike trains of nerve cells [39], the traffic flow on a highway [40], interface fluctuations [27], dissipation in the turbulent systems [41]].
Understanding the (possibly) common origins of scaling in the above phenomena appears to be a highly nontrivial task. Power laws, of course, arise naturally in critical phenomena and we understand them: their origins are in the diverging fluctuations at the critical point.
Thus the first question one may ask is the following.
• Can the power laws just be the result of nonequilibrium phase transitions and the associated critical behavior?
In equilibrium systems, however, one must tune a parameter to its critical value in order to observe scale-invariant behavior while nonequilibrium systems appear to be in scale-invariant states without any tuning. Thus the answer to the first question appears to be negative.
The wide variety of the phenomena in the above list suggests that the next question could be as follows.
• Can the scaling merely be a natural outcome of complex dynamics?
After all, we have seen in Sec.II C that competing dynamics may generate long-range (powerlaw) interactions which may be at the origin of scaling even away from a critical point. The answer to this question may be a yes but, unfortunately, many of the problems mentioned are not amenable to an analysis in terms of simple competing dynamics and then the following question remains unanswered:
• What are the ingredients of complex dynamics which determine the existence and the characteristics (e.g. exponents) of the power laws?
There is an attempt to answer all the above question in affirmative along a logic that begins with the notion of self-organized criticality (SOC) introduced by Bak, Tang and Wiesenfeld [42]. According to this notion, systems with complex dynamics tune themselves to a state with a kind of avalanche type dynamics that is underlying a large number of scale-invariant phenomena. The notion of SOC has now been understood in terms of an interplay between local and non-local dynamics [43] which indeed tunes the system [44] to a nonequilibrium (absorbing-state) critical point. Then the problem of SOC is reduced to investigating the absorbing state transitions [45] and the characteristics of power laws can be determined by studying the universality of absorbing state transition. This is an interesting and active field of research and it is worth understanding the main points. Accordingly, I will discuss SOC in the next subsection, and will explain the connection to absorbing state transitions in the following one.
A. Self-organized criticality (SOC)
The first model of SOC was introduced to describe sandpiles. Later developments, however, made energy packets from the grains of sand, so the balls in Fig.3 will be grains at the beginning but will be called energy packets later. The dynamics defining the model consists of local and non-local elements. The sites of a (usually) two-dimensional lattice are occupied by grains and the local aspect of their dynamics is in the redistribution of the grains. If a site contains more than z c grains (e.g. z c = 4 on a square lattice) then the site is active and z c grains are redistributed to the neighboring lattice sites. The redistribution of particles (avalanche) continues until active sites are found. Clearly, an avalanche stops after a while since the redistribution leads to loss of particles at the boundaries (or, in terms of the energy model, dissipation occurs at the boundaries). Once the avalanche stopped, an external supervisor notices it (this is the nonlocal part of the dynamics) and starts to add new particles (energy is injected into the system) until a new avalanche starts.
The above dynamics yields a stationary state in the long-time limit, and the steady state characteristics of avalanches can be measured. Such a characteristics is e.g. the number of sites s which become active during the process, and the remarkable feature of this model is that the distribution of s (and of other quantities such as the spatial size and the lifetime of the avalanches) is found to display a power law form Then the particles source is switched on again.
P (s) ∼ s −τ .(24)
Thus one discovers that although the model does not contain parameters to tune, nevertheless it shows critical behavior (z c can be changed without changing the criticality of the outcome). This observation generated a large amount of activity and criticality-withouttuning was seen in a number of similar models [46]. The resulting notion of self-organized criticality grew in importance [47] and, accordingly, new effort was put into understanding how SOC works.
An important feature that was recognized quite early [43] is the existence of a non-local supervisor who watches the activity of the avalanches and, upon ceasing of the activity, switches on the source of particles (or of energy). In principle, non-local dynamics can generate long-range correlations in both time and space so the emergence of criticality is not necessary a surprise. Viewing the problem from another angle, the non-local dynamics separates the timescales of the avalanches and of the particle injection. Thus, in practice, there is tuning. Namely, the system is considered in the limit of particle injection rate going to zero (actually, the dissipation is also tuned to zero since the particles disappear only at the boundaries of the system).
The zero injection rate, however, does not have to be a critical point, and the next important development was [44] the demonstration that it is indeed a nonequilibrium (absorbing state) critical point.
B. Absorbing state transitions and their connection to SOC
Absorbing state transitions appear in many contexts in nonequilibrium statistical physics [12,48], and they are studied intensively since they are thought to be one of the truly nonequilibrium phenomena without counterpart in equilibrium systems. In order to understand the basics of it, let us consider a fixed energy sandpile model [49] shown on Fig.4. Once the absorbing state transition is understood, it is easy to make the connection to SOC. Indeed, let us assume that we have the fixed energy sandpile in the active state (E > E c ) and let us switch on the dissipation at the boundaries. Then the energy decreases slowly (note that the dissipation is proportional to the surface of the sample while E is proportional to the volume). This lowering of energy will continue until E reaches just below E c when the system falls into the absorbing state and thus the dissipation stops.
Let us now return to the fixed energy sandpile but this time let us start from an absorbing state (E < E c ) and switch on the "external supervisor" who is injecting energy into the system. The supervisor is required to stop the injection if adding of the last energy packet started activity in the system. This process increases the energy E infinitesimally slowly and brings the system near and perhaps slightly past the threshold of activity E = E c . Now, if both the dissipation at the boundaries and the "external supervisor" are present then the fixed energy sandpile model is nothing else but the sandpile model generating SOC. And we see that SOC emerges because the combined action of the dissipation and the "supervisor" brings the system to the critical point of the absorbing state transition of the fixed energy sandpile model.
The mechanism unmasked above is rather general and present in many models of self organized criticality [44,50]. The value of recognizing this mechanism lies in making it possible to describe and calculate scaling properties of SOC by studying "usual" nonequilibrium phase transitions. In particular, one may hope that field-theoretic description of SOC may be obtained through studies of the appropriate absorbing state transitions.
Of course, absorbing state transitions are numerous and it is not obvious which one is in the same universality class as a given system displaying SOC. In general, continuous phase transitions to an absorbing state are in the universality class of directed percolation [51,52,53] that can be described by the following reaction diffusion process
A → A + A , A → 0 , A0 ↔ 0A .(25)
Directed percolation is rather robust to various changes in its rules but the presence of extra symmetries (conservation laws) may change the universality class of an absorbing state transition. A well known example is the parity conserving process [54,55,56] which has the following reaction-diffusion representation
A → A + A + A , A + A → 0 , A0 ↔ 0A .(26)
Both of the above processes have been much investigated and the scaling properties have been accurately determined. Furthermore, understanding (if not complete solution) has emerged even on field theoretic level [51,53,56]. Unfortunately, neither of the above processes have been directly related to models of SOC. Accordingly, the present day research is concentrated on absorbing state transitions which have more contact with SOC. An example is the critical point observed in the so called pair contact process [57] where particles diffuse only through the birth-death processes given by the reaction scheme
A + A → A + A + A , A + A → 0 ,(27)
where the first and the second reactions take place with probabilities p and 1−p, respectively.
A related problem is the epidemic model [58] where the reaction scheme
A + B → B + B , B → A , B0 ↔ 0B ,(28)
describes static healthy subjects (A) getting infected by diffusing infectious agents (B) who, in turn, recover with time.
The last two models are close to the fixed energy sandpile models (and thus to SOC)
in that both of them have an infinite number of absorbing states and their coarse-grained description involves an order parameter (the active particles) coupled to a static field (the temporarily immobile particles). It has recently been suggested that the similarity may go deeper, i.e. they all belong to the same universality class [59]. This conclusion is based on a field-theoretic calculation near dimension d = 6 [59] using Langevin equations which were suggested on phenomenological grounds for the processes (27) and (28) [49,60]. At this point there is still a debate about both the applicability of the Langevin equations and the validity of the results in lower dimensions. Nevertheless, it appears that the approach of SOC through absorbing state transitions may be coming to an interesting and satisfactory conclusion.
Of course, one should not forget that apart from the connection to SOC, absorbing state transitions in general constitute an important problem in the theory of NESS. The field is developing fast and there are many interesting details scattered across the papers. A guide to the models and to the extensive literature about them can be found in recent reviews [48,61].
IV. DISTRIBUTION FUNCTIONS IN NONEQUILIBRIUM STEADY STATES
The simplicity of the description of equilibrium system lies in the existence of the Gibbs distribution i.e. in the elimination of the dynamics from the calculation of averages. Although dynamics is clearly important in nonequilibrium steady states, it is not inconceivable that a prescription exist for a nonequilibrium equivalent of the Gibbs distribution which would include the essential features of the dynamics. Such a distribution function may have singularities as shown in simple examples [62,63] or it may have problems with the additivity of the associated entropies (which is not unexpected in systems with long-range correlations) [64,65]. Nevertheless, a prescription with well defined restrictions on its applicability would be valuable and the search for nonequilibrium distribution function(s) has been going on for some time.
A phenomenological approach to the above problem is the non-extensive statistical mechanics [64], an approach that takes its name from the nonextensive character of the postulated entropy. This approach has been much developed during the last decade, and not surprisingly, is has its success in connection with systems which have long-range interactions or display (multi)fractal behavior [64].
Below we shall present a alternative approach that is somewhat less general but it is based on the extention of our knowledge of universality of distribution functions in strongly fluctuating systems.
A. Power laws and universality of nonequilibrium distributions
Distribution functions of additive quantities such as e.g. the total magnetization in the Ising model are Gaussian in usual equilibrium systems. This Gaussianity follows from the central limit theorem that is applicable due to the correlations being short ranged away from critical points. At critical points, however, the power law correlations result in nongaussian distributions. The emerging distributions are quite restricted in their possible shapes, however, the reason being that the distribution functions at critical points are scaling functions and their shape is determined by the universality class associated with the given critical point.
The above observation can be used to develop a classification of nonequilibrium distribution functions. Namely, one knows that "effective" criticality (i.e. strong fluctuations and power-law correlations) is the norm for nonequilibrium steady states. Of course, the "effective" critical behavior is determined not only by the interactions but by the dynamics as well. Accordingly, one may expect that the scaling functions (and thus the distribution of macroscopic quantities) are determined by the nonequilibrium universality classes. Once we build a gallery of such scaling functions, we can use them in the same way as in the equilibrium case: We can identify symmetries and underlying mechanisms in experimental systems; we may find seemingly different systems belonging to the same universality class, and thus we can discover common underlying processes present in those systems. We can also use these distribution functions to find the critical dimension of a model and the applications are restricted only by imagination. Below we shall show how to calculate these distribution functions in simple systems and present a few applications.
B. Picture gallery of scaling functions
The simplest nonequilibrium systems displaying "effective" criticality are the growing surfaces [27,28]. They are rough quite generally which means that the mean-square fluctuations of the surface diverge with system size. The roughness is defined by
w 2 = 1 A L r h( r, t) − h 2 ∼ L χ ,(29)
where A L is the area of the substrate of characteristic linear dimension L, h = r h( r, t)/A L is the average height of the surface, and χ is a critical exponent characterizing the given universality class. We shall be interested in the steady-state distribution of P (w 2 )dw 2 and expect that due to criticality, the diverging scale w 2 will be the only relevant scale in P (w 2 ) and, consequently, it can be written in a scaling from
P L (w 2 ) ≈ 1 w 2 L Φ w 2 w 2 L ,(30)
where Φ(x) is a scaling function characteristic of the universality class the growth process belong to. Below, we show how to calculate Φ for a simple growth process (Edwards-Wilkinson equation [27]) and will demonstrate that the Φ-s are different for growth processes distinct in the sense of distinct universality classes. Let us begin by discussing the equations for growing surfaces. In general, deposition of particles on a substrate, under the assumption that the surface formed is a single valued function h(x, t), can be described by the equation
∂ t h = v(h) where v(h)
gives the velocity of advance of the interface perpendicular to the substrate (Fig.6). The velocity is usually A simple form for F (h) follows from the assumption that particles like to stick at points with large number of neighbors i.e. at large ∂ 2 x . Then, F (h) is approximated as F (h) = ν∆ 2
written as v(h) = v 0 + F (h) + η where v
x and, in the frame moving with v 0 , one has the Edwards-Wilkinson (EW) model [27] of surface growth
∂ t h(x, t) = ν∆h(x, t) + η(x, t) .(31)
This equation can be solved and one finds that the steady-state probability distribution is given by
P[h(x)] = Ae − σ 2 L 0 (∇h) 2 dx(32)
where σ is related to ν and to the amplitude of the white noise.
Once P[h(x)] is known, P (w 2 ) is formally obtained from
P (w 2 ) = δ(w 2 − [h 2 − h 2 ])(33)
where the average is over all h(x) with the distribution function P[h(x)] (note that h n is a spatial average and it is still a fluctuating quantity). In practice, it is more convenient to calculate the generating function
G(s) = ∞ 0 e −sw 2 P (w 2 )dw 2 = e −s(h 2 −h 2 )(34)
with the above expression demonstrating why P (w 2 ) can be calculated analytically in simple models. Namely, if the partition function with P can be found then the generating function
G(s) = ∞ n=1 1 + sL σπ 2 n 2 −1 .(35)
Now one can find the average width diverging w 2 = −∂G(s)/∂s| s=0 = L/(6σ) in the L → ∞ limit. Using w 2 to eliminate L from (35), one observes that G(s) is a function of the product s w 2 only and, consequently the inverse Laplace transform yields P (w 2 ) the scaling form (30). The calculation of the scaling function Φ(x) consist of collecting contributions from the poles in G(s) and one obtains Fig.7 shows the above function displaying a characteristic shape of exponential decay Φ(x) ∼ e −π 2 x/6 at large x and essential singularity Φ(x) ∼ x −5/2 e −3/(2x) for x → 0. On Fig.7 we have also included the results for the so called curvature driven growth process which is also called the Mullins-Herring model of surface growth [27]. This is a model where the rearrangement of deposited particles goes on by surface diffusion and the particle current j h is towards places where there are many neighboring particles i.e. ∆h is large. This means that j h ∼ ∇∆h and F (h) = −ζ∆ 2 h. The resulting equation is called the Mullins-Herring (MH) equation
Φ(x) = π 2 3 ∞ n=1 (−1) n−1 n 2 exp − π 2 6 n 2 x .(36)∂ t h(x, t) = −ζ∆ 2 h(x, t) + η(x, t) .(37)
The surfaces described by the MH equation belong to a universality class distinct from that of the EW growth. Indeed, the MH equation can be solved easily and one finds that
w 2 M H ∼ L 2 in contrast to the EW result w 2 EW ∼ L.
Accordingly, the scaling function should also be different. A calculation similar to that described above for the EW case verifies this expectation [67] as can be observed on Fig.7.
An important point to remark about the comparisons of the EW and MH curves is that they are well distinguishable. Their maximum, their small x cutoff, and their decay at large
x are all sufficiently different so that no ambiguity would arise when analyzing experimental data. Indeed, the d = 2 versions of the above Φ-s as well as a number of others characterizing various growth processes have been obtained in [68] and it did not appear to be difficult to pick the scaling function which was corresponding to a given set of experiments [68].
Finding out the universality class of a growth process is one possible application if one has a sufficiently developed gallery of scaling functions. Below we discuss other possibilities for application.
C. Upper critical dimension of the KPZ equation
The KPZ equation [29] is the simplest nonlinear model describing growth in terms of a moving interface. It differs from the EW model by taking into account that the surface grows in the direction of its normal provided the incoming particles have no anisotropy in their arrival direction. Then the z component of the velocity of the surface has a correction term proportional to (∇h) 2 as shown on Fig.6 and the equation in lowest order in the nonlinearities becomes the so called KPZ equation
∂ t h = ν ∇ 2 h + λ( ∇h) 2 + η .(38)
Here ν and λ are parameters, while η( r, t) is again a Gaussian white noise. The steady state surfaces generated by (38) Of course, our result is coming from numerical work with the same general conclusion as in previous studies. So, why is it more believable? Because one of the main criticism of numerical studies does not apply to it. Namely, no fitting parameters and fitting procedures are used in contrast to the usual determination of critical exponents. One just builds the histograms, calculates the averages to determine the scaling variable and plots the scaled histogram. This is clear and well understood but there is another remarkable feature in these scaling functions the origin of which is less obvious. Namely, the scaling functions in principle should depend on the size of the system,
Φ(x) = Φ L (w 2 / w 2 L ) .(39)
What is observed, however, is that the L dependence is practically all in w 2 L and L dependence of the shape of Φ (explicit dependence on L) disappears already at small L.
These are important points and the KPZ application of the scaling functions was mainly chosen to emphasize them.
It seems that these scaling functions are versatile tools which can be used in computer science [72] as well as in understanding the propagation of chemical fronts [73]. An interesting application was e.g. the establishment of a connection between the energy fluctuations in a turbulence experiment and the interface fluctuations in the d = 2 Edwards-Wilkinson model [74], and thus prompting a search for an interface interpretation of the dissipative structure in the turbulent system [75]. In another case, it helped to make a link between the much studied 1/f noise and the extreme value statistics [76]. Since the effective criticality is a real feature of many nonequilibrium system, we expect that many more use will be found for the scaling functions discussed in this section. The only problem is how to force a quantum system into a non-equilibrium steady state.
An obvious way is to attach two heat baths of different temperatures at the two ends of a spin chain. Unfortunately, this makes the problem unsolvable (even numerically) for any reasonable size system [79] and thus it is practically impossible e.g. to draw conclusions about the long-range correlations generated in the system. Below we show a way to avoid the problem of heat baths. The idea is that the nonequilibrium steady states always carry some flux (of energy, particle, momentum, etc.). Thus a steady state that is presumably not very far from the one generated by boundary conditions may be constructed by constraining the quantum system to have a flux equal to the one generated by the boundary conditions.
For example, in the case of the transverse Ising chain treated below, we shall constrain the system to carry an energy current and will investigate the correlations in this constrained state. 1
A. Spin chains with fluxes
As a simple model with critical phase transition, we consider the d = 1 Ising model in a transverse field h which has the following Hamiltonian:
H I = − N ℓ=1 σ x ℓ σ x ℓ+1 + h 2 σ z ℓ .(40)
Here the spins σ α ℓ (α = x, y, z) are represented by 1/2 times the Pauli matrices located at the sites ℓ = 1, 2, ..., N of a one-dimensional periodic chain (σ α N +1 = σ α 1 ). The transverse field, h, is measured in units of the Ising coupling, J, which is set to J = 1 in the following.
This model can be solved exactly [84,85] and it is known that a second order phase transition takes place in the system as h is decreased. The order parameter is the expectation value σ x i.e. σ x = 0 for h > 1, while σ x = 0 for h < 1 and h c = 1 is a critical point.
The scaling behavior at and near h c belongs to the d = 2 Ising universality class.
In order to constrain the above system to carry a given energy flux J E we shall use the Lagrange multiplier method. Namely, we add a term λĴ E to the Hamiltonian whereĴ E is the local energy flux operator summed over all sites, and find that value of λ which produces a ground state with the expectation value Ĵ E = J E .
The above scheme requires the knowledge of the local energy current,Ĵ ℓ . It can be obtained using the quantum mechanical equation of motion for the energy densityε ℓ = i/h[Ĥ I , ε ℓ ], and representing the result as a divergence of the energy currentε ℓ = J ℓ − J ℓ+1 .
The calculation yields (h = 1 is used in the following)
J ℓ = h 4 σ y ℓ (σ x ℓ−1 − σ x ℓ+1 )(41)
and this allows to construct the 'macroscopic' currentĴ E = ℓĴ ℓ . Adding it toĤ I with a
Lagrange multiplier, −λ,Ĥ
=Ĥ I − λĴ E .(42)
we obtain the Hamiltonian whose ground states with Ĵ E = J E = 0 will give us information about the current carrying states ofĤ I .
In order to avoid confusion, we emphasize that the energy current,Ĵ E , is associated witĥ H I and not with the new Hamiltonian,Ĥ. We also note thatĤ is just another equilibrium [80], and one arrives to a system of free fermions with a spectrum of excitation energies given by ω q = |Λ q | where
Λ q = 1 2 1 + h 2 + 2h cos q + λh 4 sin q .(43)
with the wave numbers restricted to −π ≤ q ≤ π in the thermodynamic limit (N → ∞). The ground-state properties do change when Λ q < 0 in an interval [q − , q + ] and these q states become occupied. Due to the resulting asymmetry in the occupation of the q and −q states, the energy current becomes nonzero. The line λ c (h) which separates the region of unchanged transverse Ising behavior from the J E = 0 region is obtained from the conditions Λ q = 0 and ∂Λ q /∂q = 0, and is displayed on the phase diagram (Fig.10) as a solid line.
Another phase boundary on Fig.10 is shown by dashed line. It separates the magnetically ordered (h < 1, λ < 2/h) and disordered (h ≥ 1, λ < 2) transverse Ising regions. Since the ground state is independent of λ for λ < λ c , one has the same second order transition across the dashed line as at h = 1 and λ = 0 i.e. it belongs to the d = 2 Ising universality class [85].
Clearly, one can view the region λ < λ c as an equilibrium phase while the λ > λ c region as a nonequilibrium one since there is a nonzero energy flux through the latter. This flux can actually be calculated easily with the simple result
j E = Ĵ E /N = (4π) −1 (1 − 4/λ 2 ) (h 2 − 4/λ 2 ) .(44)
Apart from the fact that J E = 0, the λ > λ c region should also be considered as a distinct phase since the long-range magnetic order existing for h < 1 breaks down when J E = 0 and the magnetic correlations become oscillatory with amplitudes decaying as a power of distance. Indeed, this can be seen by investigating the σ x ℓ σ x ℓ+n correlations which can be expressed through Pfaffians [81] and thus making possible numerical calculations for n ≤ 100. In the presence of long-range order one should have σ x ℓ σ x ℓ+n → σ x ℓ 2 = 0 for n → ∞ while we find that the correlations decay to zero at large distances as
σ x ℓ σ x ℓ+n ∼ Q(h, ζ) √ n cos(kn)(45)
where the wavenumber, k = arccos (2/λh). The above result (45) is coming from numerics and it is exact in the λ → ∞ limit where the correlations are those of the d = 1 XX model [81].
One can observe power-law correlations for λ > λ c in other physical quantities as well.
For example, the envelopes of both σ z ℓ σ z ℓ+n and Ĵ ℓĴℓ+n correlations behave as n −2 in the large n limit [80]. Thus we arrive to the main conclusion of this section, namely that a simple, exactly soluble quantum system shows power-law correlations in the current carrying state in agreement with the notion that power-law correlations are a ubiquitous feature of nonequilibrium steady states.
Actually, remembering that power-law correlations in quantum models are associated with a gapless excitation spectrum, we can reformulate the transverse Ising model result to see a general connection between the emergence of power-law correlations and the presence of a current. Indeed, let us assume that a system with HamiltonianĤ 0 has a spectrum with a gap between the ground-state and the lowest excited state. Furthermore, letĴ be a 'macroscopic' current of a conserved quantity such that [Ĥ 0 ,Ĵ] = 0. Generally, there is no current in the ground state and adding −λĴ toĤ 0 does not change the Ĵ = 0 result for small λ. Current can flow only if some excited states mix with the ground state and, consequently, a branch of the excitation spectrum must come down and intersect the groundstate energy in order to have Ĵ = 0. Once this happens, however, the gap disappears and one can expect power-law correlations in the current-carrying state. Admittedly, the above argument is not strict and is just a reformulation (in general terms) of what we learned from the transverse Ising model. We believe, however, that the above picture is robust and suggestive enough to try to find other soluble examples displaying the flux → power-law-correlations relationship.
B. Quantum effective interactions
When using the Lagrange multiplier method, one assumes that the flux generated by boundary conditions can be replaced by the effective interactions contained in an appropriately chosen global fluxĴ. These interactions are generally short ranged since the flux is usually a sum of local terms. The short-range nature of the effective interactions is actually not in contradiction with the power-law correlation being generated since the Lagrange multiplier gets tuned in order to achieve a given flux and, furthermore, the tuning is not quite trivial since λ must be increased past a critical value in order to have e.g. a nonzero flux energy. The question nevertheless arises whether adding a flux term inĤ was an adequate description for the nonequilibrium steady state which is expected to display power-law correlations.
In order to investigate the above question one would have to solve the problem with the boundary drive but, as discussed above, an exact solution does not seem to be feasible.
Instead, however, one can prepare initial conditions which will lead in the long-time limit to a steady flux. Then the steady state obtained in this natural way can be compared to the one found by the Lagrange multiplier method. This program has been carried out [83] for the XX chain defined by the following Hamiltonian
H XX = − N ℓ=1 σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + hσ z ℓ .(46)
In this model, the transverse magnetization M z = i σ z i is conserved and one can investigate the nonequilibrium states which carry a given magnetization flux by using the Lagrange multiplier method [82]. At the same time, the model is simple enough so that one can solve the time dependent problem where a steady magnetization flux is achieved by starting with an inhomogeneous initial state that is the ground state at fixed magnetization but with m = s z n reversed from m 0 for n ≤ 0 to −m 0 for n > 0. The time-evolution of this step-like initial state can be followed exactly and the magnetization profile emerging in the long-time limit is shown on Fig.11. The remarkable feature of this magnetization profile is the middle part which is an m = 0 homogeneous state carrying a magnetization flux j(m 0 ). Comparing this state to the one generated by adding the flux term to the Hamiltonian and fixing the It should be noted, however, that recent calculation with an inhomogeneous initial state of different temperatures (T 1 for x < 0 and T 2 for x > 0) has yielded a different result for the asymptotic state carrying an energy flux [86]. Namely, it was shown that, at least in the neighborhood of x = 0 and in the t → ∞ limit, the properties of the flux-carrying state can be interpreted in terms of the ground state of an effective Hamiltonian
H eff =Ĥ XX + N n=1 µ n N j=1Q (n) j ,(47)
where Q (n) j is a product of local operators at sites j and j + n, and the interaction is of long-range type since µ n ∼ 1/n [86] (remarkably, the first two operatorsQ Although the homogeneity of the asymptotic state was not shown and thus the comparison may be questioned, the above result indicates that the Lagrange multiplier approach may be only a first approximation in describing the flux-carrying states.
In summary, the studies of quantum systems described above strengthen the view that fluxes generate long-range correlations. Furthermore, the quantum systems also give a simple picture of how the emergence of these correlations is related to the closing a gap in the excitation spectrum above the ground state. Finally, I was asked to provide entertainment for readers by trying to guess the future developments in connection with nonequilibrium orderings. Well, one of the present problem of the field is the lack of simple experimental systems which can be be taken far enough from equilibrium and compared to elementary models of NESS. Search for such systems will intensify and I expect that there will be a shift towards biological problems. There the condition of being far from equilibrium is satisfied and there may be surprisingly simple phenomena under the guise of complicated pictures. This line of research may in the future meet up with game theories generalized to take into account spatial structures.
Search for better understanding of the emerging effective interactions will also continue, just as the sorting out of the absorbing-state transitions (surprising connections may be still found there, in addition the existing one to SOC). I also believe that theory of nonequilibrium distributions will be much developed, and limiting distributions such as the ones emerging in extreme statistics will have a much wider use in physics. These are more or less safe bets.
And then there is the unpredictable part of future.
FIG. 1 :
1Setup for Rayleigh-Bénard experiments.
define a model which can be shown to have a NESS and one can start to ask questions about the phase transitions in this steady state. The generalizations to higher dimensions, to various combinations competing dynamics (flip -flip, flip -exchange, exchange -exchange), and other types of dynamical steps (resulting e.g.
is enforcing the conservation laws. In particular, Γ without conservation laws, and Γ (α) q = D (α) q 2 for Model B with diffusive dynamics conserving the total magnetization. In case F (α) is the Landau-Ginsburg functional, we have
being the dimension of the Levy flight. Thus the Langevin equation becomeṡ
FIG. 2 :
2Flocking: Trajectories of 10000 self-propelled particles in the model described in the text. The parameters are chosen so that the stationary order parameter is φ = 0.8. Each particle is represented by a point marking its current position as well as a continuous line showing its recent (10 time step long) trajectory. (a) Initial stage of the relaxation, (b) the stationary regime. Pictures are curtesy of A. Czirók and T. Vicsek. The basic assumptions of the model are that (i) the birds fly with constant speed | v i | = 1 (i is the bird index), and (ii) the birds adjust their direction θ i in time intervals of τ = 1 to the average direction of other birds within a distance r
III. WHERE DO THE POWER-LAWS COME FROM?Systems displaying power law behavior in their various characteristics (correlation inspace or time, fluctuation power spectra, size-distributions, etc.) are abundant in nature. The most impressive examples are found in biology (e.g. the metabolic rate vs. mass rela-tionship for living creatures displays scaling over 28 decades [32]) but there are remarkable examples in solid state physics (power spectra of voltage fluctuations when a current is flowing through a resistor [33] -6 decades of scaling), in geology (the number of earthquakes vs. their magnitude [34] -5 decades) and scaling over 2-3 decades is seen everywhere [see e.g.
FIG. 3 :
3Sandpile model. Particles (energy packets) are deposited on a two-dimensional substrate (one-dimensional section is shown). Injection stops when a site becomes active, i.e. it is occupied by more than z c particles. Then redistribution to neighboring sites take place and particles disappear (dissipation of energy) at the edges. The process continues until all active sites are eliminated.
FIG. 4 :
4A one-dimensional fixed energy sandpile model. Dynamics is defined by the energy (particles) being redistributed if a site contains more than z c units of energy (the units which are redistributed in the next time-step are marked by large dots in their center). The total energy of the system is conserved since the boundaries prevent the loss of energy. This model differs from the sandpile model by calling the particles energy units, and by the absence of both the injection and the dissipation of energy (particles). Thus the total energy E is conserved and as one can easily see from Fig.4 the behavior of the system is essentially different at small and large values of E. At small E < E c , the activity (redistribution) ceases in the long-time limit and the system falls into a so called absorbing state. For large E > E c , on the other hand, there are always active sites and the redistribution continues forever. For t → ∞, the system settles into a steady state which is called active state and the activity can be quantified by measuring e.g. the number of active sites. One finds then that the absorbing state transition (i.e. the absorbing-active state transition) is a critical phase transition with the activity changing continously through the transition point E c (see Fig.5).
FIG. 5 :
5Activity as a function of the total energy for the fixed energy sandpile model described onFig.4. The evolution of the system as dissipation at the boundaries or the energy injection is switched given by the left and right arrows, respectively.
FIG. 6 :
6Surface growth. The height of the surface above the substrate is given by h(x, t) and the width of the surface w 2 is characterized by the mean-square fluctuation. The vertical velocity of the surface in general a function of the local properties of the surface, v(∂ x h, ∂ 2 x h, ...).
0 is the average velocity due to the average rate of deposition, F (h) is related both to the motion of the particles on the surface and to the dependence of the growth on the inclination of the surface. Finally the fluctuations in the above processes are collected in η which is assumed to be a Gaussian white noise in both space and time.
( 34 )
34is just the partition function of the model with a quadratic term added and such term usually does not spoil the solvability of the problem. Indeed, e.g. in case of the d=1 EW model with periodic boundary conditions, the problem is reduced to the evaluation of the partition function of a d=1 quantum oscillator thus obtaining[66]
FIG. 7 :
7Comparison of the scaling functions for the EW and MH models.
appear to be rough (critical) in any dimension if the nonequilibrium drive (λ) is large enough. Since eq.(38) gives account of a number of interesting phenomena (Burgers turbulence, directed polymers in random media, etc.) lots of efforts have been spent on finding and understanding the scaling properties of its solutions [27, 28]. Nevertheless, a number of unsolved issues remain, the question of upper critical dimension (d u ) being the most controversial one. On one hand mode-coupling and other phenomenological theories suggest that d u = 4 [69] while all the numerical work fail to find a finite d u and the indication is that d u = ∞ [70]. Below I would like to show how the scaling functions of the roughness can shed some light on this controversy [71]. Let us begin with the observation that scaling functions do not change above d u . Thus if we build Φ(x) in dimensions d = 1 − 5 and observe that they differ significantly in d = 4 and 5 then we can conclude that d u > 4. Since Φ(x) cannot be exactly calculated for d ≥ 2 we must evaluate it through simulations with the results displayed on Fig.8. As one can see on Fig.8, the scaling functions change smoothly with d. The Φ(x)-s get narrower and more centered on x = 1 with increasing d, and there is no break in this behavior at d = 4. The equality of the d=4 and 5 scaling functions appears to be excluded.
FIG. 8 :
8Roughness distribution for KPZ steady-state surfaces in dimensions d = 1 − 5.
V. QUANTUM PHASE TRANSITIONSQuantum critical points are associated with the change of the symmetry of the ground state of a quantum system as the interactions or an external field (control parameter) are varied. They have been much investigated in recent years with the motivation coming from solid state physics[78]. Namely, the strongly-correlated electron systems often produce power law correlations, and the origin of the observed scale-invariance is suggested to be the presence of a quantum critical point at T = 0 provided the effect of the quantum phase transition is felt at finite T as well.From our point of view, the quantum phase transitions are interesting because they are good candidates for studying the effects of a nonequilibrium drive on well established symmetry-breaking transitions. The advantage of these systems is that there is no arbitrariness in their dynamics (it is given by quantum mechanics), the one-dimensional systems are simple with examples of exactly solvable models displaying genuine critical phase transitions (see e.g. the transverse Ising chains discussed below) and, furthermore, there is much previous work to build on.
FIG. 9 :
9Spectrum of the transverse Ising model in the presence of a field (λ) which drives the current of energy. The excitation energies are given as ω q = |Λ q |. Increasing the drive makes the ground-state change at a critical λ = λ c (λ c = 3 for h = 2/3) when negative energy states start to appear and get occupied. The qualitative picture is the same at all transverse fields h.
Fig. 9 FIG. 10 :
910displays the spectrum for h = 2/3 and various λ and one can see that the q → −q symmetry of the spectrum is broken for λ = 0. Nevertheless, for small λ the ground-state remains that of the transverse Ising model (λ = 0) since Λ q ≥ 0 and the occupation number representation of the ground state does not change. Accordingly, no energy current flows (J E = 0) for λ < λ c . This rigidity of the ground state against the symmetry-breaking field which drives the energy current is a consequence of the facts that the fermionic spectrum of the transverse Ising model has a gap and that the operatorĴ E commutes withĤ I (similar rigidity is observed in the studies of energy flux through transverse XX chain [82Phase diagram of the driven transverse Ising model in the h − λ plane where h is the transverse field while λ is the effective field which drives the flux of energy. Power-law correlations are present in the nonequilibrium phase (J E = 0) and on the Ising critical line in the equilibrium phase (J E = 0, dashed line).
FIG. 11 :
11Time evolution of the magnetization profile starting from a step-like initial conditions shown as solid line. There are two fronts going out to ±∞. They diminish the magnetization and leave behind a homogeneous s z n = 0 state. In the scaling limitt → ∞, n/t → x, the magnetization m(n, t) ≈ Φ(n/t) is given by Φ(x) = m 0 for −1 < x, Φ(x) = m 0 − π −1 arccos(x) for −1 < x < − cos(πm 0 ), Φ(x) = 0 for − cos(πm 0 ) < x < 0, and Φ(x) = −Φ(−x) for x > 0 (dashed line).Lagrange multiplier to have the same j(m 0 ), we find that various expectation values such as the energy, the occupation number in fermionic representation are all equal in the two states. Thus the Lagrange multiplier yields a correct description of the states carrying a magnetization flux.
in the Lagrange multiplier treatment of the energy flux in the XX chain).
are topics which are important but were not discussed in these lectures. To mention a few, there is a large body of work on one-dimensional systems displaying nonequilibrium phase transitions, on orderings of granular gases under shear, on pattern formation with phase transitions described by the complex-coefficient Landau-Ginzburg equation, and the list could be continued. My choice of topics mainly reflects my past work and my attempts to develop simple starting points for making inroads into the beautiful but rather difficult field of far from equilibrium phenomena.
Hamiltonian, it differs fromĤ I by an extra term which breaks the left-right symmetry of H I . Finding the ground state ofĤ, however, gives us the minimum energy state ofĤ I which carries an energy current, J E = Ĵ E . Thus the ground-state properties ofĤ provide us with the properties of the nonequilibrium steady states of the transverse Ising model. It turns out that [Ĥ I ,Ĵ E ] = 0 andĤ can be diagonalized by the same transformations which diagonalizeĤ I
It should be noted that this section was not discussed during the main lectures of the course. It was described only in a seminar for interested students.
AcknowledgmentsMy research partially described in this lecture series has been supported by the Hungarian Academy of Sciences (Grant No. OTKA T029792). I thank the organizers for providing ideal surroundings for delivering these lectures. Thanks are also due to M. Droz and F. van Wijland for helpful discussions and comments on the manuscript.
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|
[] |
[
"Stochastic Impedance",
"Stochastic Impedance"
] |
[
"Bart Cleuren \nTheoretical Physics\nHasselt University\nB-3590DiepenbeekBelgium\n",
"Karel Proesmans \nTheoretical Physics\nHasselt University\nB-3590DiepenbeekBelgium\n"
] |
[
"Theoretical Physics\nHasselt University\nB-3590DiepenbeekBelgium",
"Theoretical Physics\nHasselt University\nB-3590DiepenbeekBelgium"
] |
[] |
The concept of impedance, which characterises the current response to a periodical driving, is introduced in the context of stochastic transport. In particular, we calculate the impedance for an exactly solvable model, namely the stochastic transport of particles through a single-level quantum dot.
|
10.1016/j.physa.2019.122789
|
[
"https://arxiv.org/pdf/1904.05854v1.pdf"
] | 119,071,180 |
1904.05854
|
2ca3b6290292b0a80c3d12851e7ef9fc8261f4de
|
Stochastic Impedance
Bart Cleuren
Theoretical Physics
Hasselt University
B-3590DiepenbeekBelgium
Karel Proesmans
Theoretical Physics
Hasselt University
B-3590DiepenbeekBelgium
Stochastic Impedance
impedance, stochastic transport
The concept of impedance, which characterises the current response to a periodical driving, is introduced in the context of stochastic transport. In particular, we calculate the impedance for an exactly solvable model, namely the stochastic transport of particles through a single-level quantum dot.
Introduction
Electrical impedance [1,2] is a well known concept from undergraduate courses on electromagnetism. Students learn about it when analysing simple electrical circuits composed out of resistors, capacitors and inductors. When driven by an alternating voltage, the current response of these circuits is far richer compared to any direct current measurement. Apart from the amplitude, now also the phase shift between the voltage and current signal comes into play. By combining amplitude and phase into a single complex quantity, the impedance, students learn to appreciate the effectiveness of a complex notation. In a typical experimental setup, a swipe of the driving frequency allows access to detailed information about the composition and topology of the underlying electrical network.
Impedance can also be found in many other areas such as acoustics [3] and mechanical response [4]. All it requires is a linear and time-invariant system, properties which are easily found in various branches of physics. In this work, we introduce the concept of impedance in the context of stochastic transport of particles. We do so by considering one of the simplest stochastic systems available, namely a single-level quantum dot connected to two electron reservoirs. Periodic driving is achieved by modulating the difference in chemical potential between the reservoirs. Such a driving brings the system out of equilibrium and induces a flow of particles between the reservoirs via the intermediate quantum dot. As we demonstrate, even for this simple setup the response behaviour is rich and intricate. And quite remarkable, for certain parameter values the impedance of this system can be mapped exactly to the one of an equivalent electrical circuit.
The inspiration for this work comes from a series of remarkable experiments performed at the X-LAB facilities of Hasselt University [5]. Subject of these investigations are cable bacteria, a recently discovered organism found in fresh and seawater sediments [6,7,8,9]. These multicellular organisms form centimeter long and unbranched filaments. Due to their specific living environment, they have developed a unique metabolism which requires them to transport electrons from one end of the filament to the other end. Recent experiments in which a voltage difference is applied across the filament, show that these cable bacteria are capable of conducting electrons over centimeter long distances, an organic tour de force. The mechanism by which they achieve this, however, remains elusive, and is the subject of ongoing research. One possibility is an incoherent hopping of electrons between discrete sites along the conductive pathway [10]. In this context, the use of impedance spectroscopy is a well tried technique to further characterise the electrical conduction. In the present work, we investigate if and how such a stochastic hopping motion would be reflected in a measurement of the impedance. This paper is organised as follow. In section 2 we set the scene and briefly discuss the electrical impedance. This section introduces the basic notions, and the impedance of a simple electrical circuit is calculated. With hindsight, this impedance will be of relevance in section 3. In that section, we introduce the concept of stochastic impedance by studying the stochastic transport through a quantum dot. Furthermore, we discuss the relation between stochastic and electrical impedance. In section 4 we conclude and discuss several possibilities for future research.
Electrical Impedance
Consider an electrical circuit composed of passive and linear components, and driven by a sinusoidal voltage V (t) = V 0 cos(ωt + ϕ) with frequency ω. The linear dependence between voltage and current implies that the current signal has the exact same frequency, albeit with a different phase. Combining amplitude V 0 and phase ϕ of the voltage signal into a single complex quantity V = V 0 e iϕ , the applied voltage can be written as the following real part
V (t) = Re Ve iωt ,(1)
and a similar expression holding for the induced current
I(t) = Re Ie iωt .(2)
The impedance is then defined as the ratio of these complex amplitudes For the basic electrical components this leads to:
Z = V/I.(3)Z R = R resistor R, Z C = 1/(iωC) capacitor C, Z L = iωL inductor L.(4)
A major advantage of the complex impedance is that the familiar rules apply for serial and parallel combination. As an example, consider the electric circuit given in Fig. 1. This elementary circuit (and variants) is commonly found in impedance experiments [11]. The resistor R and capacitance C can be seen as originating from the contact resistance and capacitance of the measurement setup, whereas Z D is the (unknown) impedance of the device of interest. The total impedance of this circuit follows immediately:
Z = R + iωC + 1 Z D −1 .(5)
The case of a purely resistive device, ie. for Z D ≡ R D , leads to
Z = R + R D 1 + C 2 R 2 D ω 2 − i ωCR 2 D 1 + C 2 R 2 D ω 2 .(6)
A visualization of the impedance can be done by a parametric plot of the real and complex parts. In case of Z D ≡ R D , this results in the well-known semicircle shown in the right panel of Fig. 1. A quick calculation indeed confirms that
Re(Z) − R − R D 2 2 + Im(Z) 2 = R 2 D 4 ,(7)
hence the real and imaginary parts lie on a circle of radius R D /2 with its centre on the real (horizontal) axis.
Stochastic Impedance: transport through a quantum dot
We now apply the concept of impedance to stochastic transport. We do this by considering one of the simplest stochastic systems that allows a flow T T μ r μ l Figure 2: Sketch of the system. A single level quantum dot is connected by two reservoirs, allowing the exchange of particles. Indicated in each reservoir is the Fermi distribution and the chemical potentials. Changing the chemical potential in the left reservoir corresponds to a raising/lowering of the distribution compared to the distribution in the right reservoir. The temperatures in both reservoirs are equal.
of particles: a single quantum dot in simultaneous contact with two electron reservoirs, see Fig. 2. A difference in chemical potential µ l − µ r = qV leads to a flow of electrons via the intermediate quantum dot. We restrict ourselves to a quantum dot with only one active level at energy E which is occupied by at most one electron. Hence the quantum dot is either empty (unoccupied), with probability p 0 (t), or occupied with probability p 1 (t). Changes in the state are due to the exchange of electrons with the two reservoirs, and are described by the following master equatioṅ
p(t) = W (l) (t) + W (r) (t) p(t),(8)
with p(t) = (p 0 (t), p 1 (t)), and W (l/r) (t) is the time dependent transition matrix associated with the left/right reservoir
W (l) (t) = −k l (t) l l (t) k l (t) −l l (t) ,(9)
and a similar expression for W (r) (t). Periodic driving is achieved by time dependent chemical potentials: µ r (t) for the right reservoir and µ l (t) for the left reservoir. The temperature is constant and for notational simplicity, we set k B T = 1. The transition rates satisfy the local detailed balance condition
k l (t) = e −(E−µ l (t)) l l (t) ; k r (t) = e −(E−µr(t)) l r (t).(10)
This ensures that when the quantum dot is connected to only one reservoir at a constant chemical potential, the time evolution eventually leads to thermal equilibrium. Unlike the electrical setup, in which there is a single unique electrical current, there are now two possible flows: j l describing the flow between the left reservoir and the quantum dot, and j r describing the flow between the quantum dot and the right reservoir:
j l = k l (t)p 0 (t) − l l (t)p 1 (t) ; j r = l r (t)p 1 (t) − k r (t)p 0 (t)(11)
In a stationary non equilibrium situation, with constant but different chemical potentials, these flows are identical, but in case of time dependent chemical potentials, this is no longer the case.
In the context of impedance, we consider a time periodic driving. In that situation, eventually the flows and probabilities will become time periodic. Eq. (8) can be solved exactly, and gives (after short-time initial-state corrections have vanished) [12]:
p(t) = p ad (t) − ∞ 0 dτ e − τ 0 ds[k(t−s)+l(t−s)]ṗ ad (t − τ ),(12)
with
p ad (t) = 1 k(t) + l(t) l(t) k(t) ,(13)
the steady-state distribution if the driving was fixed at time t, and k(t) = k l (t) + k r (t) and l(t) = l l (t) + l r (t).
The flow from the left reservoir into the quantum dot then reads:
j l = k l (t)l r (t) − l l (t)k r (t) k(t) + l(t) + ∞ 0 dτ e − τ 0 ds(k(t−s)+l(t−s)) k l (t) + l l (t) × k (t − τ )l(t − τ ) − k(t − τ )l(t − τ ) (k(t − τ ) + l(t − τ )) 2 ,(15)
and a similar expression for j r . As these expressions depend nonlinearly on the chemical potentials, we first need to linearise them. Without loss of generality, we set the chemical potential of the right reservoir constant, µ r (t) = µ 0 , and the driving ∆µ(t) of the left reservoir small enough µ l (t) = µ 0 + ∆µ(t). Expanding k l (t) and l l (t) to first order in ∆µ(t) l l (t) ≈ l l;0 + ∆µ(t)l l;1 ; k l (t) ≈ k l;0 + ∆µ(t)k l;1 ,
while the rates associated with right reservoir are constant, k r (t) = k r;0 and l r (t) = l r;0 . The local detailed balance condition, Eq. (10), leads to the following relation between the coefficients k l;0 l l;0 = k r;0 l r;0 ; k l;1 = k l;0 + k l;0 l l;1 l l;0 .
Linearising Eq. (15) in terms of the driving ∆µ(t) gives, j l = ∆µ(t)k l;0 l r;0 k 0 + l 0 + k l;0 l 0 (k l;0 + l l;0 )
(k 0 + l 0 ) 2 ∞ 0 dτ e −(k0+l0)τ ∆μ(t − τ ),(18)
with k 0 = k l;0 + k r;0 , l 0 = l l;0 + l r;0 . The corresponding result for j r reads:
j r = ∆µ(t)k l;0 l r;0 k 0 + l 0 − k l;0 l 0 (k r;0 + l r;0 ) (k 0 + l 0 ) 2 ∞ 0 dτ e −(k0+l0)τ ∆μ(t − τ ).(19)
Similar expressions for the flows can be obtained when considering other configurations, for example ∆µ l (t) = −∆µ r (t) = ∆µ/2, which allows us to postulate the following generic expression:
j = A 1 ∆µ(t) + A 2 ∞ 0 dτ e −(k0+l0)τ ∆μ(t − τ ).(20)
This flow is decomposed in two parts. The first term represents an 'adiabatic' contribution, which is the steady-state flux associated with the gradient ∆µ(t) fixed at time t. The flow associated with this term is directly proportional to the gradient, and can be considered as an Ohmic contribution. The second term is due to the finite speed of the driving. This term will lead to a phase difference between the current j and gradient ∆µ. Introducing ∆µ(t) = µe iωt allows to calculate the impedance Z = ∆µ(t)/j 1 :
Z = A 1 + iωA 2 k 0 + l 0 + iω −1 .(21)
Since the sign of A 1 can always be made positive by an appropriate choice for the direction of the current, it is clear from this expression that the qualitative dependence of Z on the parameters is fully determined by the ratio α ≡ A 2 /A 1 .
Introducing ω = ω/(k 0 + l 0 ) leads to
A 1 Z = 1 + ω 2 (1 + α) − iω α 1 + ω 2 (1 + α) 2 .(22)
The following result is immediate:
Re(A 1 Z) − 2 + α 2 + 2α 2 + Im(A 1 Z) 2 = α 2 + 2α 2 .(23)
That is, the real and imaginary part are always located on a circle with the centre on the real axis. In fact, when the frequency is varied from 0 to ∞, Re(Z) and Im(Z) always trace out a semicircle, since Im(Z) does not changes sign, and starts/ends in 0. The analogy with the electrical circuit shown in Fig. 1 with Z D ≡ R D follows immediate. A direct comparison with Eq. (6) gives the following identification
R = 1 A 1 (1 + α) ; R D = α A 1 (1 + α) ; C = A 1 (1 + α) 2 α(k 0 + l 0 ) .(24)
While the analogy is there, the interpretation of such an identification is not straightforward. In fact, depending on the value of α, the signs of R, R D and C can change. These changes in sign can be used to identify three different regions in the (Re(Z), Im(Z))-plane by varying α from −∞ to +∞. A graphical representation of Eq. (22) is given in Fig. 3. The first region, for positive α and shown as the blue region in Fig. 3, corresponds to positive values for R, R D and C. When α = 0 (or A 2 = 0) the current only contains the adiabatic contribution. Hence Im(Z) = 0 and the radius of the semi-circle reduces to zero. This result is equivalent with an electrical circuit containing a single resistor R. As α decreases further, the values for R, R D and C can become negative. For −1 < α < 0 (the red region in Fig. 3) we find R > 0 and both R D and C negative (see for example [13,14]). This region ends at α = −1. For that specific value the radius of the semi-circle diverges and becomes a straight vertical line. This impedance is equivalent to that of a series combination of a resistor and inductor, as Eq. (21) reduces to
Z = 1 A 1 + iω A 1 (k 0 + l 0 ) .(25)
Finally, the third region, indicated by the green color in Fig. 3, corresponds to α < −1. In this case R < 0, R D > 0 and C < 0 (the green region in Fig. 3). These results show that the impedance in a stochastic setup, even for a simple system as considered here, can be quite diverse. The qualitative behaviour strongly depends on the values of the various parameters. Unlike the electrical setup, the characteristics of the impedance are not due to the presence of different components. In contrast, here they have a dynamic origin and are due to the difference in time scales of the driving frequency ω and the stochastic events as determined by the transition rates.
So far the calculations were done for general transition rates satisfying the detailed balance condition, without further assumptions. A specific choice is made by considering the connected thermal reservoirs as metallic leads described by a Fermi-Dirac distribution (see for example [15]), leading to:
k l (t) = Γ l f (x l ) ; l l (t) = Γ l (1 − f (x l )) (26) k r (t) = Γ r f (x r ) ; l r (t) = Γ r (1 − f (x r )) (27) with f (x) = 1 e x + 1 ; x l = E − µ l (t) kT ; x r = E − µ r (t) kT .(28)
The presence of E is only visible in the end results via a prefactor in the currents, hence without loss of generality, we can set E = 0 and (as before) kT = 1. Further setting µ r (t) = µ 0 = 0 and µ l (t) = µe iωt we end up with the following (linearised) currents:
j l = µe iωt Γ l (Γ r + iω) 4 (Γ l + Γ r + iω)(29)
and j r = µe iωt Γ l Γ r 4 (Γ l + Γ r + iω)
. The corresponding impedances are
Z l = 4 Γ l + 4Γ r Γ 2 r + ω 2 − i 4ω Γ 2 r + ω 2(31)
and
Z r = 4 Γ l + 4 Γ r + i 4ω Γ l Γ r .(32)
These results show that it is not possible to assign a unique impedance to a stochastic system. Unlike the electrical counterpart, the current here depends on the location at which it is measured. A calculation of the impedance based upon either j l or j r yields quite a different result. Comparing Z l with Eq. (6) shows that this impedance is located in the blue region (α > 0) and hence is equivalent to the circuit to the electrical circuit shown in Fig. 1 with Z D ≡ R D . Z r on the other hand corresponds to the vertical dashed line in Fig. 3 with α = −1.
Conclusions
In conclusion, we have introduced and applied the concept of impedance in a stochastic setting, namely the stochastic flow of particles. For a single-level quantum dot connecting two electron reservoirs, the impedance can be calculated exactly. Even this seemingly simple setup displays a diverse response to an alternating driving. For certain parameter values, the impedance is equivalent to that of an actual electrical circuit. An interesting question for further research would be how general this result is, i.e., to what extend is it possible to map a stochastic system on an equivalent electrical circuit and vice versa and whether a general procedure exists to do this mapping. This might be done using general approaches such as macroscopic fluctuation theory [16,17].
Another open question, is whether an actual impedance measurement on an experimental system (such as nanowires or the cable bacteria mentioned in the introduction) shows any signs of the underlying transport mechanism. This is clearly of interest, as it leads to new insights concerning the internal structures of the experimental system at hand. Results obtained so far for cable bacteria only show a purely resistive behaviour [18] up to frequencies of 1MHz. Comparing this frequency with the typical hopping rates applicable in bacterial nanowires, which are of the order of 10 13 s −1 (see for example [19]), shows that these driving frequencies are rather low. And as a result, the imaginary part of the impedance, responsible for the capacitive signature, becomes insignificant.
Apart from the biological relevance, stochastic impedance also has a fundamental appeal. The field of thermodynamics has made tremendous progress in the last decades with the development of stochastic thermodynamics [20,21]. In particular, the analysis done in this work bears similarities with the framework for linear stochastic thermodynamics for periodically driven systems [22,23,24,25]. In these works, the thermodynamic quantities such as for example heat, (chemical) work and entropy have been defined as time-averages over one period of the driving signal. The use of impedance allows to describe these quantities in a fully time dependent setting, and might reveal new characteristics concerning the thermodynamics of small-scaled systems.
Figure 1 :
1(left panel) A standard electrical circuit pertaining to an impedance spectroscopy experiment, with the purpose to characterise the unknown impedance Z D . R and C represent the resistance and capacity originating from the measurement setup. (right panel) Plot of the imaginary and real part of the impedance Z of the circuit shown in the left panel, with Z D replaced by a (pure) resistor R D .
Figure 3 :
3Plot of the imaginary and real part of the impedance Z given by Eq. (22) as a function of the ratio α. The three regions (see text) are indicated by the boxed interval of α values and by the different colours. Solid arrows show the direction of variation in α, dashed arrows indicate specific values of α.
In a thermodynamic setting the driving force for a particle current is the difference in chemical potential divided by kT .
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[] |
[
"Hastings-Levitov Aggregation in the Small-Particle Limit",
"Hastings-Levitov Aggregation in the Small-Particle Limit"
] |
[
"James Norris ",
"Amanda Turner "
] |
[] |
[] |
We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Carathéodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.
|
10.1007/s00220-012-1552-6
|
[
"https://arxiv.org/pdf/1106.3546v2.pdf"
] | 8,069,611 |
1106.3546
|
26f03099d6ac5511501f92220590f6ac947832df
|
Hastings-Levitov Aggregation in the Small-Particle Limit
2 Nov 2011 November 3, 2011
James Norris
Amanda Turner
Hastings-Levitov Aggregation in the Small-Particle Limit
2 Nov 2011 November 3, 2011
We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Carathéodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.
Introduction
Consider an increasing sequence (K n : n 0) of compact subsets of the complex plane, starting from the closed unit disc K 0 centred at 0. Set D n = (C ∪ {∞}) \ K n and assume that D n is simply connected. Write K n as a disjoint union K 0 ∪ P 1 ∪ · · · ∪ P n . Think of K n as a cluster formed by attaching a sequence of particles P 1 , . . . , P n to K 0 . By the Riemann mapping theorem, there is a unique normalized conformal map Φ n : D 0 → D n . Here, by normalized we mean that Φ n (z) = e cn z + O(1) as |z| → ∞ for some c n ∈ R. By a conformal map D 0 → D n we always mean a conformal isomorphism, in particular a bijection. The constant c n is the logarithmic capacity cap(K n ) and the sequence (c n : n 0) is increasing. We can write Φ n = F 1 • · · · • F n , where each F n is a normalized conformal map from D 0 to a neighbourhood of ∞ in D 0 . Moreover, any sequence (F n : n ∈ N) of such conformal maps is associated to such a sequence of sets (K n : n 0) in this way.
Hasting and Levitov [8] introduced a family of models for random planar growth, indexed by a parameter α ∈ [0, 2]. We shall study a version of the case α = 0, which may be described as follows. Let P be a non-empty and connected subset of D 0 , having 1 as a limit point. Set K = K 0 ∪ P and D = (C ∪ {∞}) \ K. Assume that K is compact and that D is simply connected. We think of P as a particle attached to K 0 at 1. For example, P could be a disc of diameter δ tangent to K 0 at 1, or a line segment (1, 1 + δ]. We sometimes allow the case where P has other limit points in K 0 , for example P = {z ∈ D 0 : |z − 1| δ}, but always give 1 the preferred status of attachment point. Write F for the unique normalized conformal map D 0 → D and set c = cap(K). We assume throughout that F extends continuously to the closureD 0 . This is known to hold if and only if K is locally connected. Let (Θ n : n ∈ N) be a sequence of independent random variables, each uniformly distributed on [0, 2π). Define for n 1 F n (z) = e iΘn F (e −iΘn z), Φ n = F 1 • · · · • F n .
Write (K n : n ∈ N) and (P n : n ∈ N) for the associated sequences of random clusters and particles. Note that cap(K n ) = cn. Note also that P n+1 = Φ n (e iΘ n+1 P ). Since harmonic measure is conformally invariant, conditional on K n , the random point Φ n (e iΘ n+1 ) at which P n+1 is attached to K n is distributed on the boundary of K n according to the normalized harmonic measure from infinity. However P n+1 is not a simple copy of P , as would be natural in a model of diffusion limited aggregation, but is distorted 4 by the map Φ n .
We obtain results which describe the limiting behaviour of the growing cluster when the basic particle P has small diameter δ, identifying both its overall shape and the distribution of random structures of 'fingers' and 'gaps'. Some of these results are stated in Section 3. The results are accompanied by illustrations of typical clusters for certain cases of the model. We need some basic estimates for conformal maps, which are derived in Section 4. A simplifying feature of the case α = 0 is that fact that, for Γ n = Φ −1 n , the process (Γ n (z) : n 0) is Markov, for all z ∈ D 0 . This enables us to do a fluid limit analysis in Section 5 for the random flows Γ n as the particles become small, showing that after adding n particles, the cluster fills out a disc of radius e cn , with only small holes. In Section 6, we obtain some further estimates which show that the harmonic measure from infinity on the boundary of the cluster is concentrated near the circle of radius e cn and spread out evenly around the circle. We also bound the distortion of individual particles. Section 7 reviews some weak approximation theorems for the coalescing Brownian flow from [15]. These are then applied to the flow of harmonic measure on the cluster boundary in Section 8. In conjunction with the results of Section 6, this finally allows us to identify the weak limit of the fingers and gaps. 4 If we suppose (unrealistically) that Φ ′ n is nearly constant on the scale of P , then a rough compensation for the distortion would be achieved by replacing P in the definition of P n+1 by a scaled copy of diameter δ n+1 = |Φ ′ n (e iΘn+1 )| −1 δ. More generally, we could interpolate between these models by taking δ n+1 = |Φ ′ n (e iΘn+1 )| −α/2 δ for some fixed α ∈ [0, 2]. This is the family proposed by Hastings and Levitov.
Review of related work
There has been strong interest in models for the random growth of clusters over the last 50 years. Early models were often set up on a lattice, such as the Eden model [5], Witten and Sander's diffusion limited aggregation (DLA) [17], and the family of dielectric breakdown models of Niemeyer et al. [13]. The primary interest in these and other related processes has been in the asymptotic behaviour of large clusters. Computational investigations of these lattice based models have revealed structures, of fractal type, which in some cases resemble natural phenomena. However, such investigations have also shown sensitivity to details of implementation, in particular to the geometry of the underlying lattice. For example, in [1] and [12] different fractal dimensions are obtained for DLA constructed with different lattice dependencies. This suggests that lattice-based models may not be the most effective way to describe these physical structures. In addition, lattice based models have proved difficult to analyse. There are few notable mathematical results, with the exception of Kesten's 1987 growth estimate for DLA [11], and there is much that remains to be understood about the large-scale behaviour of these models and in particular about the structure of fingers which is characteristically observed.
In 1998, Hastings and Levitov [8] formulated a family of continuum growth models in terms of sequences of iterated conformal maps, indexed by a parameter α ∈ [0, 2]. They argue, by comparing local growth rates, that their models share features with lattice dielectric breakdown in the range α ∈ [1,2], so that α = 1 corresponds to the Eden model, and α = 2 to DLA. Further exploration of this relation is discussed in the survey paper by Bazant and Crowdy [2].
The Hastings-Levitov family of models has been discussed extensively in the physics literature from a numerical point of view. In their original paper, Hastings and Levitov found experimental evidence of a phase transition at α = 1, and further studies can be seen in, for example, [4] where estimates for the fractal dimensions of clusters are obtained, [9] where the multifractal properties of harmonic measure on the cluster are explored, and [7] where the dependence of the fractal dimension on α is investigated.
Although this conformal mapping approach to planar random growth processes has proved more tractable than the lattice approach, there have been few rigorous mathematical results, particularly in the case α > 0. Carleson and Makarov [3], in 2001, obtained a growth estimate for a deterministic analogue of the DLA model. In 2005, Rohde and Zinsmeister [16] considered the case α = 0 in the Hastings-Levitov family. They established a long-time scaling limit, for fixed particle size and showed that the limit law was supported on clusters of dimension 1. They also gave estimates for the dimension of the limit sets in the case of general α, and discussed limits of deterministic variants. Recently, Johansson Viklund, Sola and Turner [10] studied an anisotropic version of the Hastings-Levitov model in the α = 0 case, and established deterministic scaling limits for the macroscopic shape and evolution of harmonic measure on the cluster boundary.
In this paper, we also consider the case α = 0 but in the limiting regime where the particle diameter δ becomes small and where the size of the cluster is of order 1 or larger. We obtain a precise description of the macroscopic shape and growth dynamics of these clusters, as well as a fine scale description of the underlying branching structure. In the process of obtaining these results, we show that the evolution of harmonic measure on the cluster boundary converges to the coalescing Brownian flow, also known as the Brownian web [6]. An early version of some parts of the present paper, along with its companion paper [15], appeared in [14].
Statement of results
We state here our main results on the shape and structure of the Hastings-Levitov cluster. Our main result on the harmonic measure flow, which cannot be stated so directly, is Theorem 8. 1. For simplicity, we assume in this section that the basic particle P is either a slit (1, 1 + δ] or a disc {|z − 1 − δ/2| δ/2}, and that δ ∈ (0, 1/3]. We shall prove our results under some general conditions (2),(10),(12) on the basic particle P , which can be readiliy checked for the slit and disc models. We shall see that under one of these conditions (2) the logarithmic capacity c = cap(K) = log F ′ (∞) of K satisfies δ 2 /6 c 3δ 2 /4. Our first result expresses that the cluster K n is contained in a disc of approximate radius e cn and fills out that disc with only small holes. Moreover, there is a rough correspondence between the time at which a particle arrives and its distance from the origin. |z − e cn+iΘn | εe cn for all z ∈ P n and dist(w, K n ) εe cn whenever |w| e cn and |z| (1 − ε)e cm for all z ∈ P n ′ .
Assume that ε = δ 2/3 (log(1/δ)) 8 and m = ⌊δ −6 ⌋.
Then P(Ω[m, ε]) → 1 as δ → 0.
This result is a special case of Theorem 6.5 below. Note that Ω[m, ε] is decreasing in m and increasing in ε. We have made some effort to maximise the power 2/3 in this statement. It will be crucial later that 2/3 > 1/2. We shall take particular interest in the case where m is of order δ −2 and in the case where m is of order δ −3 , when the diameter of the cluster K m is of order 1 and δ −1 respectively. We have not attempted to optimise the power 8 in the logarithm.
In Figure 1, we present some realizations of the cluster when P is a slit 5 (1, 1+δ], for various values of δ. We observe in Figure 1(b), when δ = 1, that incoming particles are markedly distorted and that particles arriving later tend to be larger. This effect is diminished when we examine smaller values of δ. In Figure 1(e), the cluster is a rough disc, as predicted by Theorem 3.1 but with some sort of internal structure. The colours label arrivals in different epochs, showing that there is a close relationship between the time of arrival and the distance from the origin at which a particle sticks, as in Theorem 3. 1. Figure 1(f) focuses on the motion of points on the boundary of the unit circle, under the inverse map Γ n = Φ −1 n and over a longer timescale than for the other simulations. This motion suggests the behaviour of coalescing Brownian motions, which is confirmed in Theorem 8.1 below.
We now fix N ∈ N and state two results describing the internal geometry of the cluster K N in terms of coalescing Brownian motions, which will follow from Theorems 3.1 and 8. 1. DefineK
n = {z ∈ C : e z ∈ K n },D n = {z ∈ C : e z ∈ D n } and determine ρ = ρ(P ) ∈ (0, ∞) by ρ 2π 2π 0 (g(θ) − θ) 2 dθ = 1
where g is the unique continuous map (0, 2π) → (0, 2π) such that g(π) = π and G(e iθ ) = e ig(θ) for all θ. We shall show in Proposition 4.3 that δ −3 /C ρ Cδ −3 for an absolute constant C < ∞. Note that K N has a natural notion of ancestry for its constituent particles: we say that P k is the parent of P n+1 if Φ n (e iΘ n+1 ) ∈ P k . This notion is inherited by the covering clusterK N and will allow us to identify path-like structures within the cluster. For Re(z) 0, denote byP 0 (z) the closest particle to z inK N , and recursively denote byP m (z) the parent ofP m−1 (z) until m = m(z) whenP m(z) (z) is attached to the imaginary axis, at a(z) say. Consider the compact set
finger(z) = {a(z)} ∪ m(z) m=0P m (z).
We shall describe also the structure of the complementary setD N , using a choice of paths in this set. The notion of ancestry is not available, so we look instead for paths in the gaps which lead mainly outwards, that is to the right in the logarithmic picture. In order to enforce this outwards property, we impose a condition of minimal length, which requires a suitable completion of the set of paths. By a gap path we mean a rectifiable path (p τ ) τ 0 in 5 The normalized conformal map G = F −1 : D → D 0 can be obtained in this case as φ −1 •g 1 •φ, where φ takes D 0 to the upper half plane H 0 by φ(z) = i(z − 1)/(z + 1) and g 1 (z) = ( C, parametrized by arc length, such that Re(p τ ) → ∞ as τ → ∞ and such that, for some continuous map h : [0, ∞)×[0, 1] → C and for all τ 0, we have p τ = h(τ, 1) and h(τ, t) ∈D N for all t ∈ [0, 1). For R > 0, define L R (p) = inf{τ 0 : Re(p τ ) = R}. Write p 0 (z) for the closest point to z which is not in the interior ofK N . SinceD N is simply connected and K N is compact, there exists a unique gap path p(z) starting from p 0 (z) and minimizing L R (p) over all gap paths starting from p 0 (z), for all sufficiently large R. The path p(z) may be thought of as a long piece of thread outside the cluster, with one end attached to p 0 (z) and drawn tight by pulling from the right. Set
z 2 + t)/(1 − t) takes H = H 0 \ (0, i √ t] to H 0 , where t = δ 2 /(2+δ) 2 . A straightforward calculation gives c = c(δ) = − log G ′ (∞) = − log(1−t) ≍ δ 2 /4.gap(z) = {p τ (z) : τ 0}.
Note that, by minimality, for all τ 1 , τ 2 0 with τ 1 < τ 2 and such that the open line segment I = (p τ 1 (z), p τ 2 (z)) is contained inD N , we have p τ (z) ∈ I for all τ ∈ (τ 1 , τ 2 ). These definitions are illustrated in Figure 2. Both fingers and gaps depend implicitly on N, although we have suppressed this in the notation. . This is only a representation and in general the particles will be distorted both by the conformal mapping and by the logarithmic transformation.
In order to capture the limiting fluctuations of the fingers and gaps we have to rescale. We do this in two ways, defining horizontal and vertical scaling operators σ andσ by σ(r + iθ) = (δ * r, θ),σ(r + iθ) = (r, θ/ √ δ * ), r 0, θ ∈ R where δ * = (ρc) −1 . Note that δ/C δ * Cδ for an absolute constant C < ∞. Alsō
σ = σ δ * • σ where σ δ * is the diffusive scaling σ δ * (s, x) = (s/δ * , x/ √ δ * ), s 0, x ∈ R.
The horizontal scaling identifies global random behaviour in the fingers and gaps over very long time scales, whereas the vertical scaling identifies local fluctuations in the fingers and gaps while the size of the cluster is of order 1. Denote by S the space of closed subsets of [0, ∞) × R, equipped with a local Hausdorff
metric. Define F, G : [0, ∞) × R → S andF,Ḡ : [0, ∞) × R → S by F = σ • finger • σ −1 , G = σ • gap • σ −1 F =σ • finger •σ −1 ,Ḡ =σ • gap •σ −1 .
Thus, for e = (s(e), x(e)),
F(e) = {σ(w) : w ∈ finger(s(e)/δ * + ix(e))}, G(e) = {σ(w) : w ∈ gap(s(e)/δ * + ix(e))} F(e) = {σ(w) : w ∈ finger(s(e) + ix(e) √ δ * )},Ḡ(e) = {σ(w) : w ∈ gap(s(e) + ix(e) √ δ * )}.
We consider F(e),F(e), G(e),Ḡ(e) as random variables in S. We state first the long time result. Fix T > 0 and let E be a finite subset of [0, T ] × R. Take N = ⌊ρT ⌋ so that K N is approximately a disc of radius e T /δ * . Denote by ν P E and η P E the respective laws of (F(e) : e ∈ E) and (G(e) : e ∈ E) on S E . Let (B e : e ∈ E) be a family of 2π-coalescing Brownian motions, B e running backwards in time from x(e) at time s(e).
Thus B e = (B e t : 0 t s(e)) and for all e, e ′ ∈ E, B e and B e ′ are independent until (time running backwards) their difference is an integer multiple of 2π, at which point it freezes. Let (W e : e ∈ E) be a family of 2π-coalescing Brownian motions, with W e running forwards in time from x(e) at time s(e). Denote by ν E and η E the laws on S E of the families of random sets ({(t, B e t ) : 0 t s(e)} : e ∈ E) and ({(t, W e t∧T ) : t s(e)} : e ∈ E). Theorem 3.2. We have ν P E → ν E and η P E → η E weakly on S E as δ → 0. Thus, for small δ, we can construct on a common probability space, the cluster K N and backwards and forwards 2π-coalescing Brownian motions, such that the union of fingers inK N starting from points s(e)/δ * + ix(e), e ∈ E is, with probability close to 1, close in Hausdorff metric to the set e∈E {t/δ * +iB e t : 0 t s(e)}, and hence the union of fingers in K N , starting from points exp(s(e)/δ * +ix(e)), e ∈ E looks approximately like the set e∈E {exp(t/δ * +iB e t ) : 0 t s(e)}. Similarly, the union of gaps in K N , starting from points exp(s(e)/δ * + ix(e)), e ∈ E looks approximately like the set e∈E {exp(t/δ * + iW e t∧T ) : t ≥ s(e)}. A simulation of e∈E {exp(t/δ * + iB e t ) : 0 t s(e)} and e∈E {exp(t/δ * + iW e t ) : s(e) t T } is shown in Figure 3(a).
For the local result we take now N = ⌊c −1 T ⌋ so that K N is approximately a disc of radius e T . Denote byν P E andη P E the laws of (F(e) : e ∈ E) and (Ḡ(e) : e ∈ E) on S E . Let (B e : e ∈ E) be a family of coalescing Brownian motions,B e running backwards in time from x(e) at time s(e). ThusB e = (B e t : 0 t s(e)) and for all e, e ′ ∈ E,B e andB e ′ are independent until (time running backwards) they collide, at which time they coalesce. Let Theorem 3. 3. We haveν P E →ν E andη P E →η E weakly on S E as δ → 0.
Thus, for small δ, we can construct on a common probability space, the cluster K N and backwards and forwards coalescing Brownian motions, such that the union of fingers inK N starting from points s(e) + ix(e) √ δ * , e ∈ E is, with probability close to 1, close in Hausdorff metric to the set e∈E {t + iB e t √ δ * : 0 t s(e)}, and hence the union of fingers in K N , starting from points exp(s(e)+ix(e) Figure 3
√ δ * ), e ∈ E looks approximately like the set e∈E {exp(t+ iB e t √ δ * ) : 0 t s(e)}. Similarly, the union of gaps in K N , starting from points exp(s(e) + ix(e) √ δ * ), e ∈ E looks approximately like the set e∈E {exp(t + iW e t∧T √ δ * ) : t ≥ s(e)}. A simulation of e∈E {exp(t + iB e t √ δ * ) : 0 t s(e)} and e∈E {exp(t + iW e t √ δ * ) : s(e) t T } is shown in(b).
Theorems 3.2 and 3.3 are obvious corollaries of Theorem 8.2, which identifies also the limiting joint law of fingers and gaps.
Some basic estimates
We derive in this section some estimates for quantities associated to the basic particle P . In some special cases one could use instead an explicit calculation. By proving general estimates we are able to demonstrate some universality for the small-particle limit. Recall that K = K 0 ∪ P and D = (C ∪ {∞}) \ K, with K compact and locally connected and D simply connected in C ∪ {∞}. The following assumptions are in force throughout this section δ ∈ (0, 1/3] and P ⊆ {z ∈ C : |z − 1| δ} and 1 + δ ∈ P and P = {z : z ∈ P }. (2) Consider the map ψ(z) =z −1 on C ∪ {∞} by reflection in the unit circle S. SetP = ψ(P ) andD = ψ(D),D 0 = ψ(D 0 ). Define also P * = P ∪ I ∪P , where I is the set of limit points of P in S, and set D * = (C ∪ {∞}) \ P * . By the Riemann mapping theorem, there is a conformal mapĜ :D →D 0 and a constant c ∈ R such thatĜ(z) = e c z + O(|z| 2 ) as |z| → 0, andĜ and c are unique. MoreoverĜ extends to a conformal map G * : D * → (C ∪ {∞}) \ J for some
interval J ⊆ S, with G * • ψ = ψ • G * on D * . Write G for the restriction of G * to D. Then G is a conformal map D → D 0 and G(z) = e −c z + O(1) as |z| → ∞,
The constant c is the logarithmic capacity cap(K). The well known fact that c is positive will emerge in the course of the proof of Proposition 4. 1. Note that D * is simply connected and G * (z)/z = 0 for all z ∈ D * . So we may choose a branch of the logarithm so that log(G * (z)/z) is continuous on D * with limit c at 0 and then, for some constant C(K) < ∞, we have
log Ĝ (z) z − c C(K)|z|, z ∈D and so log G(z) z + c C(K) |z| , z ∈ D.
In fact the following stronger estimate holds.
Proposition 4.1. There is an absolute constant C < ∞ such that log G(z) z + cap(K) C cap(K) |z − 1| , |z − 1| > 2δ, z ∈ D. Proof. Set H(z) = u(z) + iv(z) = log(G * (z)/z).
Then H is bounded and holomorphic on D * and H(z) → −c as |z| → ∞. Fix z ∈ C and let B be a complex Brownian motion starting from z. Suppose that z ∈ D and consider the stopping time
T = inf{t 0 : B t ∈ D}.
Then T < ∞ and |B T | 1 almost surely, and |B T | > 1 with positive probability. Also u(B t ) → − log |B T | as t ↑ T almost surely. Hence, by optional stopping,
u(z) = −E(log |B T |) < 0. Set r = δ/(2 − δ) and define P * 1 = {z ∈ C : |z − 1| r|z + 1|}. Then set D * 1 = (C ∪ {∞}) \ P * 1 , P 1 = P * 1 ∩ D 0 , D 1 = D * 1 ∩ D 0 , K 1 = K 0 ∪ P 1 .
Then
P * ⊆ P * 1 ⊆ {z ∈ C : |z − 1| δ/(1 − δ)}.
The boundary of D 1 consists of two circular arcs, one contained in S, where u = 0, the other contained in P 1 , which we denote by A. The normalized conformal map G 1 :
D 1 → D 0 can be obtained as φ −1 • g 1 • φ, where φ takes D 0 to the upper half-plane by φ(z) = i(z − 1)/(z + 1) and g 1 (z) = (z + r 2 /z)/(1 − r 2 ). Hence we obtain G * 1 (z) = z(γz − 1)/(z − γ) for z ∈ D * 1 , where γ = (1 − r 2 )/(1 + r 2 ), and G 1 (A) = {e iθ : |θ| < θ 0 }, where θ 0 = cos −1 γ. Set F 1 = G −1 1 .
Then u • F 1 is bounded and harmonic on D 0 . Suppose now that z ∈ D 0 and consider the stopping time
T 0 = inf{t 0 : B t ∈ D 0 }.
Then T 0 < ∞ almost surely and, by optional stopping,
u(F 1 (z)) = E(u(F 1 (B T 0 ))) = 1 2π |θ| θ 0 u(F 1 (e iθ )) Re z + e iθ z − e iθ dθ.
On letting |z| → ∞ we obtain
c = − 1 2π |θ| θ 0 u(F 1 (e iθ ))dθ > 0 so u(F 1 (z)) + c = 1 2π |θ| θ 0 u(F 1 (e iθ )) Re 2e iθ z − e iθ dθ.
Hence, for z ∈ D 1 ,
|u(z) + c| 2c dist(G 1 (z), G 1 (A))
.
By an elementary calculation, we have
|(G * 1 ) ′ (z) − γ| 6γ/7 whenever |z − 1| 7δ/4 and δ ∈ (0, 1/3]. Set A ′ = {z ∈ C : |z−1| = 7δ/4}. Then dist(G 1 (z), G 1 (A)) dist(G 1 (z), G 1 (A ′ )) whenever |z − 1| 7δ/4. By the mean value theorem, there is an absolute constant C 1 < ∞ such that dist(G 1 (z), G 1 (A ′ )) |z − 1|/C 1 , |z − 1| 2δ.
Hence
|u(z) + c| 2C 1 c/|z − 1|, |z − 1| 2δ, z ∈ D
and the same estimate extends to D * by reflection. Then, by a standard estimate for harmonic functions (differentiate the Poisson kernel),
|∇v(z)| = |∇u(z)| 8C 1 c/|z − 1| 2 , |z − 1| 2δ, z ∈ D(3)
and so
|v(z)| ∞ 0 |∇v(z + s(z − 1)||z − 1|ds 8C 1 c/|z − 1|, |z − 1| 2δ, z ∈ D,
giving the required bound.
Corollary 4.2.
We have δ 2 /6 cap(K) 3δ 2 /4.
Proof.
We use notation from the preceding proof. By uniqueness, we have
G 1 = G † • G, where G † is the normalized conformal map G(D 1 ) → D 0 . Hence cap(K) cap(K) + cap(G(K 1 \ K)) = cap(K 1 ) = log 1 + r 2 1 − r 2 3δ 2 4 . Also, since 1 + δ ∈ P , G = G ‡ • G 2 , where G 2 is the normalized slit map D 2 = D 0 \ (1, 1 + δ] → D 0 referred to in Section 3 and G ‡ is the normalized conformal map G 2 (D) → D 0 . Let K 2 = K 0 ∪ (1, 1 + δ].
Then
cap(K) = cap(K 2 ) + cap(G 2 (K \ K 2 )) ≥ − log 1 − δ 2 (2 + δ) 2 ≥ δ 2 6 .
We shall do most of the analysis in logarithmic coordinates. SetD = {z ∈ C : e z ∈ D} and D 0 = {z ∈ C : Re(z) > 0}. There are unique conformal mapsG :D →D 0 andF : (3) provide the following estimates forG(z)
D 0 →D such thatG(z) − z + c → 0 andF (z) − z − c → 0 as Re(z) → ∞. ThenF andG are 2πi-periodic andF =G −1 . Also G • exp = exp •G and F • exp = exp •F . Proposition 4.1 and|G(z) − z + c| Cc |e z − 1| , |G ′ (z) − 1| Cc|e z | |e z − 1| 2 , |e z − 1| 2δ, z ∈D.(4)
We introduce some further functions associated toG andF . Recall the definitions of I and J from the start of this section. Since P is symmetric, we can write I = {e iθ : |θ| p} and J = {e iθ : |θ| q} for some p ∈ [0, π) and q ∈ (0, π). Then there exist unique nondecreasing right-continuous functions g + and f + on R such that the functions θ → g + (θ) − θ and θ → f + (θ) − θ are 2π-periodic and such that
g + (θ) = ±q, ±θ ∈ (0, p] Im(G(iθ)), |θ| ∈ (p, π] , f + (θ) = 0, |θ| ∈ [0, q) Im(F (iθ)), |θ| ∈ (q, π] .(5)
Here we have used the continuous extensions ofG andF to certain intervals of the imaginary axis. Define, for θ ∈ R g 0 (θ) = g + (θ) − θ and, for x ∈ (0, 1] such that x + iθ ∈D, define
g x (θ) = Im(G(x + iθ)) − θ.
Proposition 4. 3. There is an absolute constant C < ∞ such that, for α = Cδ and |θ| π, |g 0 (θ)| α 2 |θ| ∨ α and the same estimate holds for |g x (θ)| when x ∈ (0, 1] and x + iθ ∈D. Moreover C may be chosen so that
δ 3 /C 1 2π 2π 0 g 0 (θ) 2 dθ Cδ 3 , 1 2π 2π 0 |g 0 (θ)g 0 (θ + a)|dθ Cδ 4 a log 1 δ whenever a ∈ [δ, π].
Proof. The first estimate follows from the first estimate in (4), using the non-decreasing property of g + and the maximum principle to deal with the case where |e x+iθ − 1| < 2δ, and using cap(K) 3δ 2 /4. This leads directly to the upper bound in the second estimate and the third estimate.
For the lower bound, note that
1 2π 2π 0 g 0 (θ) 2 dθ 1 π q 0 (q − θ) 2 dθ = q 3 3π
and q = πP ∞ (B T ∈ P ). We give an argument which uses neither the symmetry assumption P = {z : z ∈ P } nor the assumption 1 + δ ∈ P and instead assumes only that |z − 1| = δ for some z ∈ P . This will be useful in Lemma 6.1. Denote by P (2) the union of P with its reflection in the line ℓ joining z and 0. Denote by w the image of 1 under this reflection and by A the shorter arc in the unit circle joining w and 1. Then, since P is connected, we have
2P ∞ (B T ∈ P ) P ∞ (B hits P (2) before K 0 ) P ∞ (B hits K 0 in A) ∨ P ∞ (B hits ℓ before K 0 ) (|w − 1| ∨ (|z| − 1))/(2π) δ/(4π)(6)
which gives the claimed lower bound.
Fluid limit analysis for random conformal maps
Define conformal mapsF n andΦ n onD 0 bỹ
F n (z) =F (z − iΘ n ) + iΘ n ,Φ n =F 1 • · · · •F n where (Θ n : n ∈ N)
is the sequence of independent uniformly distributed random variables specified in the Introduction. WriteΓ n for the inverse mapΦ −1 n :D n →D 0 . It will be convenient to use the filtration (F n : n 0) given by F n = σ(Θ 1 , . . . , Θ n ). Recall that we write c for the logarithmic capacity cap(K). Assumption (2) remains in force in this section.
For ε ∈ [2δ, 1] and m ∈ N, denote by Ω(m, ε) the event defined by the following conditions: for all z ∈D 0 and all n m, we have
|Φ n (z) − z − cn| < ε whenever Re(z) 5ε
and z ∈D n and |Γ n (z) − z + cn| < ε whenever Re(z) cn + 4ε.
Note the round brackets -this is not the same event as Ω[m, ε], defined above. We shall use the following estimate in the case where m = ⌊δ −6 ⌋ and ε = δ 2/3 log(1/δ) when, using the bound c 3δ 2 /4 from Corollary 4.2, it implies that Ω(m, ε) has high probability as δ → 0. The proof is based on a fluid limit approximation for each Markov process (Γ n (z) : n 0), optimized using explicit martingale estimates. Local uniformity in z is achieved by combining the estimates for individual starting points with an application of Kolmogorov's Hölder criterion.
Proposition 5.1. There is an absolute constant C < ∞ such that, for all ε ∈ [2δ, 1] and all m ∈ N, P(Ω \ Ω(m, ε)) C(m + ε −2 )e −ε 3 /(Cc) .
Proof. It will suffice to consider the case where ε 3
c. Set M = ⌈cm/(2ε)⌉. Fix k ∈ {1, . . . , M} and set R = 2(k + 1)ε. Consider the vertical line ℓ R = {z ∈ C : Re(z) = R}.
Write N for the largest integer such that cN R − 2ε. Consider the stopping time
T = T R = inf{n 0 : z ∈D n or Re(Γ n (z)) R − cn − ε for some z ∈ ℓ R } ∧ N. Note that Re(Γ T −1 (z)) > ε > δ > log(1 + δ) so z ∈D T R for all z ∈ ℓ R . Consider the events Ω R = sup n T R , z∈ℓ R |Γ n (z) − z + cn| < ε , Ω 0 (m, ε) = M k=1 Ω 2(k+1)ε .
We shall show that there is an absolute constant C < ∞ such that
P(Ω \ Ω R ) Cε −6/5 e −ε 3 /(Cc)(7)
from which it follows that
P(Ω \ Ω 0 (m, ε)) C(cm/ε + 1)ε −6/5 e −ε 3 /(Cc) C(m + ε −2 )e −ε 3 /(Cc) . Note that, on Ω R , we have |Γ T R (z) − z + cT R | < ε for all z ∈ ℓ R , which forces T R = N and so z ∈D n whenever Re(z) R and cn R − 2ε. Then, sinceΓ n (z) − z + cn is a bounded holomorphic function onD n , we have on Ω R sup cn R−2ε, Re(z) R |Γ n (z) − z + cn| = sup cn R−2ε, z∈ℓ R |Γ n (z) − z + cn| < ε.
For n m, we can choose k so that R − 4ε cn R − 2ε. Then, if Re(z) cn + 4ε, then Re(z) R, so on Ω 0 (m, ε) we have z ∈D n and |Γ n (z) − z + cn| < ε. Moreover, on Ω 0 (m, ε), the imageΓ n (ℓ R ) lies to the left of ℓ 5ε , and hence, if Re(w) 5ε, we have w =Γ n (z) for some
Re(z) R so that |Φ n (w) − w − cn| = |z −Γ n (z) − cn| < ε.
We have shown that Ω 0 (m, ε) ⊆ Ω(m, ε), which implies the claimed estimate. It remains to prove (7). The functionG 0 (z) =G(z) − z is holomorphic, bounded and 2πi-periodic onD withG 0 (z) → −c as Re(z) → ∞. Hence
1 2π 2π 0G 0 (z − iθ)dθ = −c, Re(z) > δ. Let q(r) = r ∧ r 2 . Then |G 0 (z) + c| C 1 c Re(z) − δ , |G ′ 0 (z)| 2C 1 c q(Re(z) − δ) , Re(z) 2δ, where C 1 is the absolute constant in (4). Set M n (z) =Γ n (z) − z + cn, z ∈D n .
Then
M n+1 (z) − M n (z) =G 0 (Γ n (z) − iΘ n+1 ) + c.
So (M n (z)) n T is a martingale for all z ∈ ℓ R . For z ∈ ℓ R and n T − 1,
|M n+1 (z) − M n (z)| C 1 c (Re(Γ n (z)) − δ) C 1 c (R − cn − ε − δ) and N −1 n=0 C 2 1 c 2 (R − cn − ε − δ) 2 R−2ε 0 C 2 1 cds (R − s − ε − δ) 2 = C 2 1 c ε − δ 2C 2 1 c ε .
So, by the Azuma-Hoeffding inequality, for all z ∈ ℓ R ,
P sup n T |M n (z)| ε/2 2e −ε 3 /(16C 2 1 c) .(8)
Fix z, z ′ ∈ ℓ R , defineM n = M n (z) − M n (z ′ ) and set
f (n) = E sup k T ∧n |M k | 2 .
Note that |Γ n (z) −Γ n (z ′ )| |z − z ′ | + |M n | so, for n T − 1,
|M n+1 −M n | = |G 0 (Γ n (z) − iΘ n+1 ) −G 0 (Γ n (z ′ ) − iΘ n+1 )| 2C 1 c(|z − z ′ | + |M n |) q(R − cn − ε − δ) .
Then, by Doob's L 2 -inequality,
f (n) 4E |M T ∧n | 2 4 n−1 k=0 E(|M k+1 −M k | 2 1 {k T } ) 32C 2 1 c 2 n−1 k=0 |z − z ′ | 2 + f (k) q(R − ck − ε − δ) 2 so, by a Gronwall-type argument, E sup n T |M n | 2 = f (N) |z − z ′ | 2 exp ∞ ε−δ 32C 2 1 cds q(s) 2 − 1 . Now, for r ∈ (0, 1], ∞ r ds q(s) 2 = 1 3 2 + 1 r 3
and ε/2 ε − δ 1 and ε 3 c. So we deduce the existence of an absolute constant
16C 2 1 < C 2 < ∞ such that f (N) C 2 c|z − z ′ | 2 /ε 3 .
Hence by Kolmogorov's lemma, C 2 may be chosen so that, for some random variable M, with E(M 2 ) C 2 c/ε 3 , we have
sup k T |M k (z) − M k (z ′ )| M|z − z ′ | 1/3
for all z, z ′ ∈ ℓ R . So, by Chebyshev's inequality, for any L ∈ N,
P sup k T |M k (z) − M k (z ′ )| ε/2 for some z, z ′ ∈ ℓ R with |z − z ′ | π/L (π/L) 2/3 4C 2 c/ε 5 .
On combining this with (8), we obtain P(Ω \ Ω R ) Le −ε 3 /(C 2 c) + (π/L) 2/3 4C 2 c/ε 5 from which (7) follows on optimizing over L.
We note two consequences of the event Ω(m, ε). First we deduce an estimate for the normalized conformal maps Φ n . On Ω(m, ε), for n m and |z| = e 5ε , we have | log(e −cn Φ n (z)) − log z| < ε and so |e −cn Φ n (z) − z| < εe 6ε .
The last estimate then holds whenever |z| e 5ε by the maximum principle. Second, we show that on the event Ω(m, ε), for n m and R cn, there is no disc of radius 56ε with centre on the line ℓ R = {z ∈ C : Re(z) = R} which is disjoint fromK n .
Since the setsK n are increasing in n, we may assume that R > c(n − 1). Fix y ∈ R and set w = 6ε + iy. Note that |Φ n (w) − (R + iy)| < ε + |6ε + cn − R| < 7ε + c < 8ε. Here we have used c 3δ 2 /4 δ/4 < ε. By Cauchy's integral formulã
Φ ′ n (w) = 1 + 1 2πi |z−w|=εΦ n (z) − z − cn (z − w) 2 dz so |Φ ′ n (w)| 2.
Then, by Koebe's 1/4 theorem,
d(Φ n (w), ∂D n ) 4|Φ ′ n (w)|d(w, ∂D 0 ) 48ε.
and so d(R + iy, ∂D n ) 56ε. (9) 6 Harmonic measure and the location of particles
In this section we obtain an estimate on the location of the particles P n+1 = Φ n (e iΘ n+1 P ) in the plane. From the preceding section, we know that Φ n ((1 + ε)e iθ ) is close to (1 + ε)e cn+iθ with high probability, when ε is suitably large in relation to the particle radius δ. This must break down as ε → 0, at least when particles are attached at a single point, since the map θ → Φ n (e iθ ) parametrizes the whole cluster boundary by harmonic measure. Nevertheless, we shall show that the approximation breaks down only on a set of very small harmonic measure, and in fact the whole of each particle P n+1 is close to e cn+iΘ n+1 , in a sense made precise below. Throughout this section, we assume that condition (2) holds and we make also the following non-degenerate contact condition
P ⊆ {z ∈ C : Re(z) > 1}.(10)
Lemma 6.1. There is an absolute constant C < ∞ with the following properties. Let D * be any simply connected neighbourhood of ∞ in D 0 and set K * = C \ D * . Denote by µ the harmonic measure from ∞ in D * of K * \ K 0 and by N the number of connected components of K * \ K 0 . Then
P(K * ∩ K 1 = K 0 ) CN √ µ.(11)
Assume further that 16πµ δ. Then
P(K * ∩ K ∞ = K 0 ) CN √ µ/δ.
Proof.
By the estimate (6), each of the N connected components of K * \ K 0 is contained in a disc of radius 8πµ with centre on the unit circle. The non-degenerate contact assumption then allows us to choose C 1 < ∞ such that P 1 intersects that component only if e iΘ 1 lies in a concentric disc of radius C 1 √ µ. The estimate (11) follows.
Consider a complex Brownian motion B with B 0 uniformly distributed on the circle of radius 2 centred at 0, and independent of Θ 1 . Set r = 1 + δ/2 and note that K * ⊆ rK 0 . Set
T (K) = inf{t 0 : B t ∈ K}.
Note that, since Θ 1 is uniformly distributed on [0, 2π), the events {T (rK 0 )
T (K 1 )} and {T (K * ) < T (K 0 )} are independent. We use the estimate (6) and our assumption that 1+δ ∈ P to obtain P(T (rK 0 ) T (K 1 )) 1 − δ/C 1 .
Note that, since T (rK 0 ) T (K * ), we have
{T (K * ) < T (K 1 )} ⊆ {T (rK 0 ) T (K 1 )} ∩ {T (K * ) < T (K 0 )}.
Hence
P(T (K * ) < T (K 1 )) (1 − δ/C)P(T (K * ) < T (K 0 )) = (1 − δ/C 1 )µ. Set H n = P(T (K * ) < T (K n )|F n )
and note that H 0 = µ. By conformal invariance of Brownian motion, we have
H n = P(T (K * n ) < T (K 0 )|F n ) where K * n = Γ n (K * \ K n ) ∪ K 0 , and moreover E(H n+1 |F n ) = P(T (K * n ) < T (K ′ 1 )|F n ) where K ′
1 is an independent copy of K 1 . Since K * n ⊆ rK 0 , the argument of the preceding paragraph applies to show that E(H n+1 |F n ) (1 − δ/C 1 )H n . Hence E(H n ) (1 − δ/C 1 ) n µ for all n.
On the event {K * ∩ K n = K 0 }, the set K * n \ K 0 has N connected components, and its harmonic measure from ∞ in C \ K * n is H n . Define P ′ 1 = Γ n (P n+1 ). Then P ′ 1 has the same distribution as P 1 and is independent of F n . So the argument leading to (11) applies to give
P(K * ∩ P n+1 = ∅|F n ) = P(K * n ∩ P ′ 1 = ∅|F n ) C 1 N H n . Hence P(K * ∩ K ∞ = K 0 ) ∞ n=0 P({K * ∩ K n = K 0 } ∩ {K * ∩ P n+1 = ∅}) ∞ n=0 C 1 NE( H n ) C 2 1 N √ µ/δ.
WriteP for the connected component ofK \K 0 near 0. Set
P n =Φ n−1 (P + iΘ n ),Ã n =Φ n−1 (iΘ n ).
ThenP n is a component of the 2πi-periodic setK n \K n−1 and it is attached toK n−1 at A n . For the next result, we shall use a further assumption on the particle P which allows us to prove that none of the setsP n contain a certain size of fjord, even though they have been distorted by the mapsΦ n−1 . The useful form of this assumption is expressed in terms of harmonic measure. After stating this, we will give a geometrically more obvious sufficient condition. We assume the following harmonic measure condition.
For all sequences (z 1 , w 1 , z 2 , w 2 ) of points in ∂P , listed anticlockwise, and for any interval I of ∂D 0 , if for i = 1 and i = 2 at least the 3/4 of the harmonic measure on ∂D 0 from w i is carried on I, then for either i = 1 or i = 2 at least 1/4 of the (12) harmonic measure on ∂D 0 from z i is carried on I.
This condition is implied by the following property of the image φ(P ), where φ is the conformal map from D 0 to the upper half-plane H 0 , as in footnote 4. For z = x + iy and z ′ = x ′ + iy ′ in H 0 , write S(z, z ′ ) for the smallest closed square in H 0 containing all the points x − y, x + y, x ′ − y ′ , x ′ + y ′ . Then the preceding harmonic measure condition is implied by the following square condition. For all z, z ′ ∈ ∂(φ(P )), at least one of the boundary arcs of ∂(φ(P )) from z to z ′ is contained in S(z, z ′ ). To see this, suppose I ⊆ R is an interval which carries at least 3/4 of the harmonic measure on R starting from z, then (x − y, x + y) ⊆ I. Hence, if the same is true for z ′ , then S(z, z ′ ) ∩ R ⊆ I. Then, for any point w ∈ S(z, z ′ ), I carries at least 1/4 of the harmonic measure on R starting from w. We have used here the fact that the harmonic measure on R starting from i places equal mass on the intervals (−∞, −1), (−1, 0), (0, 1), (1, ∞). It is easy to check the square condition for P = (1, 1 + δ] and P = {|z − 1 + δ/2| = δ/2}, when φ(P ) is also a slit or a disc.
Consider for ν ∈ [0, ∞) the event Ω(m, ε, ν) = {Re(z) > c(n ∧ m) − εν for all z ∈P n+1 and all n 0} ∩ Ω(m, ε).
In conjunction with Proposition 5.1, the following estimate implies that, when m = ⌊δ −6 ⌋ and ε = δ 2/3 log(1/δ) and ν = (log(1/δ)) 2 , the event Ω(m, ε, ν) has high probability as δ → 0.
Proposition 6.2. There exists an absolute constant C < ∞ such that, for all ε ∈ [2δ, 1] and ν ∈ [0, ∞), P(Ω(m, ε) \ Ω(m, ε, ν)) Cm(m + δ −1 )e −ν/C .
Proof.
We use the following Beurling estimate. There is an absolute constant A ∈ [1, ∞) with the following property. For any η ∈ (0, 1] and any connected set K in C joining the circles of radius η and 1 about 0, the probability that a complex Brownian motion, starting from 0, leaves the unit disc without hitting K is no greater than A √ η.
Fix n m with εν cn. Condition on F n and on Ω(n, ε). For all z ∈ C with 0 Re(z) cn, there exists w ∈K n such that |z − w| 56ε. Set β = 56A 2 e 2 and ν 0 = ⌊ν/(2β)⌋. We assume without loss that ν 0 6. Define R(k) = cn − βεk and note that R(2ν 0 ) 0. Fix k ∈ {0, 1, . . . , ν 0 − 1} and z ∈ ℓ R(k) and consider a complex Brownian motion B starting from z. By the Beurling estimate, B hits ℓ R(k+1) without hittingK n with probability no greater than A 56/β = e −1 . Then, by the strong Markov property, for all z ∈ ℓ R(0) , almost surely on Ω(n, ε),
P z (B hits ℓ R(ν 0 ) beforeK n |F n ) e −ν 0 e −ν/(2β)+1 .(13)
There exists a family of disjoint open intervals ((θ j , θ ′ j ) : j = 1, . . . , N n ) in R/(2πZ) such that, for w j =Φ n (iθ j ) and w ′ j =Φ n (iθ ′ j ), we have Re(w j ) = Re(w ′ j ) = R(ν 0 ) and j (w j , w ′ j ) + 2πiZ disconnectsD n ∩ ℓ R(2ν 0 ) from ∞ inD n . We choose the unique such family minimizing j |w j − w ′ j |. Then w j ∈P k(j) for some k(j) n for all j. We shall show that the integers k(1), . . . , k(N n ) must all be distinct, so N n n.
Suppose k(j) = k(j ′ ) = k 0 + 1 for some distinct j and j ′ . Then there exist α < β < α ′ < β ′ < α + 2π such that, for z =Φ k 0 +1 (iα), z ′ =Φ k 0 +1 (iα ′ ), w =Φ k 0 +1 (iβ) and w ′ =Φ k 0 +1 (iβ ′ ), we have z, z ′ , w, w ′ ∈ ∂P k 0 +1 and Re(z) = Re(z ′ ) = R(2ν 0 ) and Re(w) = Re(w ′ ) = R(ν 0 ). Then, since we are on Ω(m, ε), we must have c(k 0 + 1) + 4ε R(ν 0 ), so ck 0 R(ν 0 ) − 4ε − c R(ν 0 + 1). Hence there exists an interval I of ∂D k 0 with endpoints p, p ′ in ℓ R(3ν 0 /2) such that z, z ′ are separated from ∂D k 0 \ I by I ∪ [p, p ′ ]. By a variation of the Beurling and strong Markov argument above, all but e −ν 0 /2 of the harmonic measure on ∂D k 0 starting from z is carried on I, and the same is true for z ′ . Then, by conformal invariance of harmonic measure, all but e −ν 0 /2+1 < 1/4 of the harmonic measure on iR starting fromF k 0 +1 (iα) is carried oñ Γ k 0 (I), and the same is true for α ′ . So, by our harmonic measure condition, either more than 1/4 of the harmonic measure on iR starting fromF k 0 +1 (iβ) is carried onΓ k 0 (I), or the analogous statement holds for β ′ . But, by the Beurling and strong Markov argument again, no more than e −ν 0 /2+1 < 1/4 of the harmonic measure on ∂D k 0 starting from w is carried on I, and the same is true for w ′ . So, by conformal invariance, no more than 1/4 of the harmonic measure on iR starting fromF k 0 +1 (iβ) is carried onΓ k 0 (I), and the same is true for β ′ , a contradiction.
Each path (Φ n (iθ) : θ ∈ (θ j , θ ′ j )), together with the line segment [w j , w ′ j ], forms the boundary of a connected subset ofD n . Denote by S n the union of these subsets. Define K * n = {eΓ n(z) : z ∈ S n } ∪ K 0 and D * n = (C ∪ {∞}) \ K * n . Then D * n is a simply connected neighbourhood of ∞ in D 0 , the set K * n \ K 0 has N n connected components and, by (13), the harmonic measure from ∞ of K * n \ K 0 in D * n is no greater than e −ν/(2β)+1 . So, on Ω(n, ε),
P((e iΘ n+1 P ) ∩ K * n = ∅|F n ) C 1 N n e −ν/(4β)+1/2 ,
where C 1 is the absolute constant from Lemma 6.1. But, if e iΘ n+1 P does not meet K * n , then Re(z) > cn − νε for all z ∈P n+1 . Of course this inequality holds also in the case where cn < νε.
It remains to deal with the case where n m+1. We may assume that ν 2β log(16πe/δ) or the estimate is trivial. Then, for µ = e −ν/(2β)+1 , we have 16πµ δ. So we can apply Lemma 6.1 with K * = K * m to obtain, on Ω(m, ε), P(Re(z) cm − εν for some z ∈P n+1 and some n m|F m ) C 1 N m e −ν/(4β)+1/2 /δ.
The estimates we have obtained combine to prove the proposition. Remark 6. 3. An analogous result to Proposition 6.2 can be obtained by bounding the contribution to the length of the cluster boundary made by each particle. This extends the class of allowable basic particles beyond that specified by (12), but at the expense of a weaker bound on the probability.
Suppose that (2) and (10) hold and that, in addition, ∂P is rectifiable, with length L, and is given by β : [0, L] → ∂P , where the parametrization is by arc length. We assume further that β is piecewise differentiable in such a way that there exist C(δ) ∈ (0, ∞), k(δ) ∈ N and 0 = a 0 < a 1 < · · · < a k(δ) = L such that r i :
(a i , a i+1 ) → (1, 1 + δ) given by r i (t) = |β(t)| is differentiable with |r ′ i (t)| > C(δ) on (a i , a i+1 ) for all i = 0, . . . , k(δ) − 1. Set r(δ) = k(δ)/C(δ)
. Letl n+1 be the contribution to the length of the boundary of ∂K n+1 that comes from particlẽ P n+1 . Thenl
n+1 = L 0 |Φ ′ n (β(t)e iΘ n+1 )| |Φ n (β(t)e iΘ n+1 )| dt.
So, by a similar argument to that in the proof of Theorem 4 of [16],
E(l n+1 |K n ) = k(δ)−1 i=0 2π 0 a i+1 a i |Φ ′ n (r i (t)e iθ )| |Φ n (r i (t)e iθ )| dtdθ r(δ) 2π 0 1+δ 1 |Φ ′ n (re iθ )| |Φ n (re iθ )| drdθ C 1 r(δ)(cnδ) 1/2 ,
for some absolute constant C 1 < ∞. Therefore, if N n is defined as in the proof of Proposition 6.2, for all ζ > 0, P(N n > ζ) n j=1 P(l j > ζνεn −1 ) C 1 r(δ)δ 3/2 n 5/2 /(ζνε). Hence, there exists some absolute constant C < ∞ such that P(Ω(m, ε) \ Ω(m, ε, ν)) Cζ(m + δ −1 )e −ν/C + Cr(δ)δ 3/2 m 7/2 /(ζνε).
On optimizing over ζ, it can be shown that there exists another absolute constant C < ∞ such that P(Ω(m, ε) \ Ω(m, ε, ν)) Cr(δ) 1/2 m 7/4 δ 3/4 ν −1/2 ε −1/2 (m + δ −1 ) 1/2 e −ν/C .
Define for z ∈D 0 N(z) = inf{n 0 : z ∈D n }.
Denote by Ω(m, ε, ν, η) the subset of Ω(m, ε, ν) defined by the following condition: for all z ∈D 0 ∩K ∞ with N(z) m and all n N(z) − 1, we have | Im(Γ n (z) − z)| < ε + 2η.
In conjunction with Propositions 5.1 and 6.2, the following estimate implies that, when m = ⌊δ −6 ⌋ and ε = δ 2/3 log(1/δ) and ν = (log(1/δ)) 2 and η = δ 2/3 (log(1/δ)) 6 , the event Ω(m, ε, ν, η) has high probability as δ → 0. Proposition 6. 4. There is an absolute constant C < ∞ such that, for all ε ∈ [2δ, 1/6], ν ∈ [0, ∞) and η ∈ (0, ∞),
P (Ω(m, ε, ν) \ Ω(m, ε, ν, η)) Cm η exp − η Cδ + Cνεδ c (1 + log (1/δ)) . Proof. Fix z ∈D 0 ∩K ∞ with N(z) m.
Write N 0 (z) for the maximum of 0 and the largest integer such that cN 0 (z) Re(z) − 4ε. Write N 1 for the smallest integer such that cN 1 (ν + 4)ε. Then, on Ω(m, ε, ν), we have N(z) − 1 N 0 (z) + N 1 and, since Ω(m, ε, ν) ⊆ Ω(m, ε), we have also | Im(Γ k (z) − z)| < ε for all k N 0 (z).
We showed in Proposition 4.3 that, for some absolute constant C 1 < ∞, for α = C 1 δ and for all z ∈D 0 with Re(z) 1, Im(G(z)) g * (Im(z))
where g * (θ) = θ + g * 0 (θ) and g * 0 is the 2π-periodic function given by
g * 0 (θ) = α 2 |θ| ∨ α , θ ∈ (−π, π].
Then, for N 0 (z) n N(z) − 1, Im(Γ n (z)) Y (n 0 ,y 0 ) n where n 0 = N 0 (z), y 0 = Im(Γ n 0 (z)) and where, recursively for n n 0 , Y n = Y
(n 0 ,y 0 ) n is defined by Y n 0 = y 0 , Y n+1 = g * (Y n − Θ n+1 ) + Θ n+1 = g * 0 (Y n − Θ n+1 ) + Y n . Note that Re(Γ n (z))
Re(Γ n 0 (z)) | Re(Γ n 0 ) − Re(z) + cn 0 | + Re(z) − cn 0 < 5ε + c < 1. Hence, g * 0 is non-negative and g * is non-decreasing, so Y Note that |E| Cm/η, so
P(Ω \ Ω 0 ) CmP(Y N 1 > η)/η where Y = Y (0,0) . Now E(e Y 1 /α ) = 1 2π π −π e g * 0 (θ)/α dθ = 1 + α(e − 1) π + 1 π π α
(e α/θ − 1)dθ exp{(αe/π)(1 + log(π/α))} so P(Y N 1 > η) exp{−η/α + (N 1 αe/π)(1 + log(π/α))}.
Choose j ∈ {1, . . . , M} so that (j − 1)h y 0 jh. Then, for n N(z) − 1, on Ω 0 , Im(Γ n (z)) Y (n 0 ,jh) n jh + η Im(Γ n 0 (z)) + 2η Im(z) + ε + 2η.
A similar argument allows us to bound the downward variation of Im(Γ n (z)) up to N(z) − 1.
Hence P(Ω(m, ε, ν) \ Ω(m, ε, ν, η) 2P(Ω \ Ω 0 ) which gives the claimed estimate.
Theorem 6. 5. Assume that the basic particle P satisfies conditions (2), (10) and (12). Consider for ε 0 ∈ (0, 1] and m ∈ N the event Ω[m, ε 0 ] specified by the following conditions: for all n m and all n ′ m + 1,
|z − e cn+iΘn | ε 0 e cn for all z ∈ P n and dist(w, K n ) ε 0 e cn whenever |w| e cn and |z| (1 − ε 0 )e cm for all z ∈ P n ′ .
Assume that ε 0 = δ 2/3 (log(1/δ)) 8 and m = ⌊δ −6 ⌋. Then P(Ω[m, ε 0 ]) → 1 uniformly in P as δ → 0.
Proof. Set ε = δ 2/3 log(1/δ) and ν = (log(1/δ)) 2 and η = δ 2/3 (log(1/δ)) 6 . We have shown that the event Ω(m, ε, ν, η) has high probability as δ → 0. We complete the proof by showing that, for δ sufficiently small, the defining conditions for Ω[m, ε 0 ] are all satisfied on Ω(m, ε, ν, η). Fix n m and z ∈P n . On Ω(m, ε) we have Re(z) < cn + 4ε and, restricting to Ω(m, ε, ν), we have also Re(z) > c(n−1)−νε. Restricting further to Ω(m, ε, ν, η), we have | Im(Γ n−1 (z)− z)| < ε + 2η. ButΓ n−1 (z) ∈P + 2πiΘ n , so |Γ n−1 (z) − 2πiΘ n | δ. Hence, on Ω(m, ε, ν, η), we have (since ν 4) |e z − e cn+2πiΘn | e cn (e νε+c − 1) + e cn+νε+c (ε + 2η + δ).
We can choose δ sufficiently small that (e νε+c − 1) + e νε+c (ε + 2η + δ) ε 0 .
Then on Ω(m, ε, ν, η) we have, for all z ∈ P n , |z − e cn+2πiΘn | ε 0 e cn .
Next, using (9), for 0 Re(w) cn, on Ω(m, ε), there exists z ∈K n with |z − w| 56ε. Then e z ∈ K n and |e z − e w | 56εe cn+4ε .
We can choose δ sufficiently small that 56εe 4ε ε 0 .
Then dist(w, K n ) ε 0 e cn whenever |w| e cn and n m.
Finally, for n m + 1 and z ∈P n , on Ω(m, ε, ν), we have Re(z) > cm − νε. Hence |w| > e cm−νε (1 − ε 0 )e cm for all w ∈ P n . Here and below, we drop the ± where the quantity computed takes the same value for both versions. Fix f ∈ D * and define ρ = ρ(f ) ∈ (0, ∞) by ρ 2π Let (Θ n : n ∈ Z) be a sequence of independent random variables, all uniformly distributed on [0, 2π). Define for each non-empty bounded interval I ⊆ R a pair of random functions
Φ I = {Φ − I , Φ + I } by Φ ± I = f ± Θn • · · · • f ± Θm where f ± θ (x) = f ± (x − θ) + θ
and where m and n are, respectively, the smallest and largest integers in the rescaled interval ρI. If ρI ∩ Z = ∅ then Φ I = id. Write I = I 1 ⊕ I 2 if I 1 , I 2 are disjoint intervals with sup I 1 = inf I 2 and I = I 1 ∪ I 2 . Note that the family Φ = (Φ I : I ⊆ R) has the following flow property
Φ ± I 2 • Φ ± I 1 = Φ ± I , whenever I = I 1 ⊕ I 2 .(14)
Moreover (see [15]), almost surely, for all I, Φ − I is the left-continuous modification of Φ + I , so
Φ I = {Φ − I , Φ + I } ∈ D.
We call Φ the disturbance flow with disturbance f . For ε ∈ (0, 1], we make the diffusive rescaling
Φ ε,± I (x) = ε −1 Φ ± ε 2 I (εx), x ∈ R
and call (Φ ε I : I ⊆ R) the ε-scale disturbance flow with disturbance f . In order to formulate a weak convergence result about these disturbance flows, we introduce metrics on D andD and then we define certain metric spaces which will serve as state-spaces for Φ and Φ ε . First, define for f, g ∈ D d D (f, g) = inf{ε 0 : f + (x) g + (x + ε) + ε and g + (x) f + (x + ε) + ε for all x ∈ R}.
For f, g ∈D, define
dD(f, g) = ∞ n=1 2 −n (d n (f, g) ∧ 1) where d n (f, g) = inf{ε 0 : f + (x) g + (x+ε)+ε and g + (x) f + (x+ε)+ε for all x ∈ [−n, n − ε]}.
Then d D is a metric on D and the metric space (D, d D ) is complete. In fact (D, d D ) is isometric to the set of periodic contractions on R with period 2π, with supremum metric, by drawing new axes for the graph of f ∈ D at a rotation of π/4. Also, dD is a metric onD and the metric space (D, dD) is complete. See [15].
Consider now a family φ = (φ I : I ⊆ R), where φ I ∈ D and I ranges over all non-empty bounded intervals. Say that φ is a weak flow if,
φ − I 2 • φ − I 1 φ − I φ + I φ + I 2 • φ + I 1 , whenever I = I 1 ⊕ I 2 .(15)
Say that φ is cadlag if, for all t ∈ R, (14) is not preserved under limits in d D . We refer to [15] for the specification of d D .
d D (φ (s,t) , id) → 0 as s ↑ t and d D (φ (t,u) , id) → 0 as u ↓ t.
The disturbance flow Φ with disturbance f is then a D • (R, D)-valued random variable, and the law of Φ is a Borel probability measure on D • (R, D), which we denote by µ f A . The ε-scale disturbance flow Φ ε is a D • (R,D)-valued random variable, so the law of Φ ε is a Borel probability measure on D • (R,D), which we denote by µ f,ε A . For e = (s, x) ∈ R 2 and φ ∈ D • (R, D), the maps and, for e = (s, x), e ′ = (s ′ , x ′ ) ∈ R 2 , we write T ee ′ for the collision time
t → φ − (s,t] (x) : [s, ∞) → R, t → φ + (s,T ee ′ = inf{t s ∨ s ′ : Z e t − Z e ′ t ∈ 2πZ}.
We make the same definitions for φ ∈ D • (R,D), except to define as collision timē
T ee ′ = inf{t s ∨ s ′ : Z e t = Z e ′ t }.
The space C • (R, D) is a convenient state-space for the coalescing Brownian flow on the circle where it has the following characterization (see [15,Theorem 6.1]). There exists a unique Borel probability measure µ A on C • (R, D) such that, for all e = (s, x), e ′ = (s ′ , x ′ ) ∈ R 2 , the processes (Z e t ) t s and (Z e t Z e ′ t −(t−T ee ′ ) + ) t s∨s ′ are continuous local martingales in the filtration (F t ) t∈R . Moreover, for all e ∈ R 2 , we have, µ A -almost surely, Z e,+ = Z e,− .
Similarly, the space C • (R,D) is a state-space for the coalescing Brownian flow (on the line). There exists a unique Borel probability measureμ A on C • (R,D) such that, for all e = (s, x), e ′ = (s ′ , x ′ ) ∈ R 2 , the processes (Z e t ) t s and (Z e t Z e ′ t −(t−T ee ′ ) + ) t s∨s ′ are continuous local martingales in the filtration (F t ) t∈R . Moreover, for all e ∈ R 2 , we have,μ A -almost surely, Z e,+ = Z e,− .
We consider a limit where f becomes an increasingly well-localized perturbation of the identity map. We quantify this localization in terms of the smallest constant λ = λ(f, ε) ∈ (0, 1] such that ρ 2π We can now state Theorem 6.1 from [15]. We have
µ f A → µ A weakly on D • (R, D) uniformly in f ∈ D * as ρ(f ) → ∞ and λ(f, 1) → 0 (16) and µ f,ε A →μ A weakly on D • (R,D) uniformly in f ∈ D * as ε → 0 with ε 3 ρ(f ) → ∞ and λ(f, ε) → 0.(17)
The harmonic measure flow
We return to the aggregation model. We assume throughout this section that condition (2) holds. The boundary ∂K n of the cluster K n has a canonical parametrization by [0, 2π) given by θ → Φ n (e iθ ). For θ 1 < θ 2 , the normalized harmonic measure (from ∞) of the positively oriented boundary segment from Φ n (e iθ 1 ) to Φ n (e iθ 2 ) is then (θ 2 − θ 1 )/(2π). We consider the related parametrization θ →Φ n (iθ) : R → ∂K n . For m n, each point z ∈ ∂K n has a unique ancestor point A mn (z) ∈ ∂K m , which is either z itself or the point of ∂K m to which the particle containing z is attached, possibly through several generations. On the other hand, each point in z ∈ ∂K m , except those points where particles are attached, has a unique escape point E mn (z) ∈ ∂K n , which is either z itself or is connected to z by a minimal path inK n , subject to not crossing any particles nor passing through any attachment points. If P is attached at a single point, then E mn (z) = z for all z ∈ ∂K m . These definitions are illustrated in Figure 4. Figure 4: Diagram illustrating ancestor points A mn (z) ∈ ∂K m for z ∈ ∂K n and escape points E mn (z) ∈ ∂K n for z ∈ ∂K m , whereK m is shown in red,K n \K m is shown in white, and attachment points are shown in blue.
2 π i 0 z E (z) w mn A (w) mn x = E (x) mn = A (x) mn
We define the forwards and backwards harmonic measure flows on R, respectively, for 0 m < n by
Φ P nm (x) = −iΓ n • E nm •Φ m (ix), Φ P mn (x) = −iΓ m • A mn •Φ n (ix).(18)
We shall show that, when embedded suitably in continuous-time, these flows converge weakly to the coalescing Brownian flow, as the diameter δ of the basic particle P tends to 0. Then, in the same limiting regime, we shall deduce the behaviour of fingers and gaps in the aggregation model. First we give an alternative presentation of the flows. Recall the functions g + and f + defined at (5) and write g − and f − for their left-continuous versions. Then g = {g − , g + } ∈ D and f = {f − , f + } = g −1 . Since P is non-empty and is invariant under conjugation, g is not the identity function but is an odd function. Hence g ∈ D * . Recall that the sequence of clusters (K n : n 0) is constructed from a sequence of independent random variables (Θ n : n ∈ N), uniformly distributed on [0, 2π). Define f θ , g θ ∈ D for θ ∈ [0, 2π) as in Section 7. Then define for 0 m < n
Φ P,± nm = g ± Θn • · · · • g ± Θ m+1 , Φ P,± mn = f ± Θ m+1 • · · · • f ± Θn .
We can check (just as for the disturbance flow) that, almost surely, Φ P nm = {Φ P,− nm , Φ P,+ nm } ∈ D and Φ P mn = {Φ P,− mn , Φ P,+ mn } ∈ D, with (Φ P mn ) −1 = Φ P nm . Moreover, a straightforward induction shows that this definition agrees with the more geometric formulation in (18).
In formulating a limit statement, it is convenient to embed the harmonic measure flow in continuous time. We do this in two ways. Hence δ * = (ρc) −1 satisfies δ/C δ * Cδ for an absolute constant C < ∞. In particular ρ → ∞ and ε → 0 and ε 3 ρ δ −3/2 /C → ∞ as δ → 0. Also, from Proposition 4.3, for a ∈ [δ, π], we have ρ 2π 2π 0 |g 0 (θ)g 0 (θ + a)|dθ Cδ a log 1 δ so λ(g, 1) λ(g, ε) → 0 as δ → 0. The result thus follows from (16) and (17).
We can now deduce the limiting joint distribution of fingers and gaps. Recall that S denotes the space of locally compact subsets of [0, ∞) × R, equipped with a local Hausdorff metric. We have fixed T > 0 and a finite subset E of [0, T ]×R. Recall that we study the cluster K N and have introduced in Section 3 associated path-like random sets finger(z) and gap(z), along with rescaled sets F(e),F(e), G(e) andḠ(e). Write µ P E for the law of (F(e), G(e) : e ∈ E) when N = ⌊ρT ⌋, considered as a random variable in (S E ) 2 where Φ is a coalescing Brownian flow on the circle and where we set Φ st = Φ −1 ts for s t. Write alsoμ E for the corresponding law when we replace Φ by a coalescing Brownian flowΦ on the line.
Theorem 8.2. Assume that the basic particle P satisfies conditions (2), (10) and (12). Then µ P E → µ E andμ P E →μ E weakly on (S E ) 2 , uniformly in P as δ → 0.
Proof. We consider first the long time case. Given ε 0 > 0, there exist ε > 0 and ε ′ ∈ (0, ε/3] such that, for any coalescing Brownian flow Φ = (Φ ts : 0 s t T ) on the circle, with probability exceeding 1 − ε 0 /3, for all e ∈ E and all t ∈ [0, T ], we have Φ ts(e) (x(e)) − ε 0 Φ ts(e) (x(e) − 5ε) − 5ε, Φ ts(e) (x(e) + 5ε) + 5ε Φ ts(e) (x(e)) + ε 0 and, for all s, s ′ , t,
t ′ ∈ [0, T ] with |s − s ′ |, |t − t ′ | 3ε ′ and all x ∈ R Φ ts (x) Φ t ′ s ′ (x + ε) + ε.
Note that these conditions imply 5ε ε 0 . Here we have used some standard estimates for Brownian motion and the fact that the map (s, t) → Φ ts : [0, T ] 2 → D is uniformly continuous, almost surely. Here and below such inequalities are each to be understood as a pair of inequalities, one for left-continuous versions and the other for right-continuous versions.
Then, by Theorem 8.1, and using a standard result on weak convergence, there exists a δ 0 > 0 such that, for all δ ∈ (0, δ 0 ] and all basic particles P satisfying (2), for N = ⌊ρT ⌋, we can construct, on some probability space, an HL(0) process Φ P = (Φ P n : n N) with basic particle P and a coalescing Brownian flow Φ = (Φ ts : 0 s t T ) on the circle with the following property. With probability exceeding 1 − ε 0 /3, for all 0 m < n N, for t = m/ρ and s = n/ρ, and for all x ∈ R, we have
Φ ts (x − ε) − ε Φ P mn (x) Φ ts (x + ε) + ε.
Here (Φ P mn : 0 m < n N) is the backwards harmonic measure flow of Φ P (which determines (Θ n : 1 n N) and hence Φ P uniquely).
Moreover, by Theorem 6.5, we may choose δ 0 so that, with probability exceeding 1 − ε 0 /3, for all e ∈ E, writing z(e) = σ −1 (e) = s(e)/δ * + ix(e) and σ(p 0 (z(e))) = (s 0 , x 0 ), we have |s 0 − s(e)| ε ′ /3, |x 0 − x(e)| ε and, for all (s, x) ∈ [0, T ] × R, there exists w ∈K N such that σ(w) = (t, y) satisfies |s − t| ε ′ /3, |x − y| ε and, for all n N − 1 and all z ∈P n+1 , σ(z) = (s, x) satisfies
|s − n/ρ| ε ′ /3, |x − Θ n+1 | ε.
From this point on, we condition on the good event Ω 0 of probability exceeding 1−ε 0 where all of the properties discussed above hold. Suppose that we fix j, k ∈ Z and m, n N − 1 and w ∈P m+1 + 2πij and z ∈P n+1 + 2πik, withP m+1 + 2πij an ancestor particle ofP n+1 + 2πik. Write σ(w) = (t, y) and σ(z) = (s, x). Then we must have m = ρt ′ n = ρs ′ , with |s − s ′ |, |t − t ′ | ε ′ /3 and |y − (Θ m+1 + 2πj)|, |x − (Θ n+1 + 2πk)| ε. Now Φ P mn is continuous and Θ m+1 + 2πj = Φ P mn (Θ n+1 + 2πk) so
y Θ m+1 + 2πj + ε = Φ P mn (Θ n+1 + 2πk) + ε Φ t ′ s ′ (Θ n+1 + 2πk + ε) + 2ε Φ t ′ s ′ (x + 2ε) + 2ε Φ ts (x + 3ε) + 3ε.
and by a similar argument also y Φ ts (x − 3ε) − 3ε. Here we have extended Φ by setting Φ ts = Φ t∧T,s∧T .
Fix e ∈ E and (t, y) ∈ F(e). Write (t, y) = σ(w) andP 0 (z(e)) =P n+1 + 2πik. We can choose z ∈P 0 (z(e)) with σ(z) = (s, x) and |s−s(e)| ε ′ /3 and |x−x(e)| ε. Set u = t∧s(e). Then w and z are related as in the preceding paragraph and t t ′ + ε ′ /3 s ′ + ε ′ /3 s + 2ε ′ /3 s(e) + ε ′ s(e) + ε 0 so |t − u| ε ′ . Hence y Φ ts (x + 3ε) + 3ε Φ us(e) (x + 4ε) + 4ε Φ us(e) (x(e) + 5ε) + 5ε Φ us(e) (x(e)) + ε 0 and similarly y Φ us(e) (x(e)) − ε 0 .
Since (t, y) was arbitrary, we have shown that We complete the proof by obtaining an analogous estimate for G(e). Recall that G(e) = {σ(p τ ) : τ 0} where p = p(z(e)) is the minimal length gap path starting from p 0 (z(e)), the closest point to z(e) which is not in the interior ofK N . Write σ(p 0 (z(e))) = (s 0 , x 0 ).
First we show that minimal gap paths cannot backtrack too much. Suppose that t < s(e) − ε ′ and p makes an excursion left of the line {t/δ * + iy : y ∈ R}, with endpoints w − , w + , say. Then the open line segment (w − , w + ) must contain a point ofK N , say w ∈P m+1 + 2πij. Set σ(w) = (t, y). Then, since p cannot crossK N , there must exist z ∈P n+1 + 2πik, an ancestor particle ofP m+1 + 2πij, with σ(z) = (s, x), say, and s s 0 . But then s(e) s 0 + ε ′ /3 s + ε ′ /3 n/ρ + 2ε ′ /3 m/ρ + 2ε ′ /3 t + ε ′ < s(e) which is impossible. Hence there is no such excursion and so
G(e) ⊆ {(s, x) : s s(e) − ε ′ , x ∈ R}.
Consider (t, y) = σ(w) with w ∈P m+1 + 2πij and m N − 1 and t s(e) − 3ε ′ and y Φ vs(e) (x(e)) + ε 0 , where v = s(e) ∨ t ∧ T . Note that t T + ε ′ /3 and |v − t| 3ε ′ . Suppose (s, x) = σ(z) with z ∈P n+1 + 2πik and |s − s(e)| ε ′ and whereP n+1 + 2πik is an ancestor particle ofP m+1 + 2πij. Then There exists a continuous function (y(t) : t ∈ I) such that, for all t ∈ I, setting v = s(e) ∨ t ∧ T , we have y(t) > Φ vs(e) (x(e)), d((t, y(t)), Φ(e)) = ε 0 + ε + 5ε ′ .
x Φ st (y − 3ε) − 3ε Φ s(e)v (y − 4ε) − 4ε x(e) + ε.
Define recursively a sequence τ 0 , . . . , τ M by setting τ 0 = s(e) − 2ε ′ and then taking τ n+1 as the supremum of the set {τ ∈ [τ n , T ] : |(τ, y(τ )) − (τ n , y(τ n ))| = ε ′ } until n = M − 1 when this set is empty and we set τ M = T . For n = 0, 1, . . . , M, choose w n ∈K N with σ(w n ) = (t n , y n ) and |t n − τ n | ε ′ and |y n − y(τ n )| ε. Note that t 0 s(e) − ε ′ and t M T − ε ′ and t n ∈ [s(e) − 3ε ′ , T + ε ′ ] for all n. Set
B 0 = M −1 n=0 [w n , w n+1 ), B 1 = {t/δ * + iy M : t t M }, B = B 0 ∪ B 1 .
Then, for any w ∈ B 0 , for (t, y) = σ(w), we have |(t, y) − (τ n , y(τ n ))| ε + 2ε ′ for some n, so ε 0 + 3ε ′ d((t, y), Φ(e)) ε 0 + 2ε + 7ε ′ and so y Φ vs(e) (x(e)) + ε 0 , where v = s(e) ∨ t ∧ T . The final inequality obviously extends to B. Suppose p crosses B, and does so for the first time at τ (1). Consider first the case where p τ (1) ∈ [w n , w n+1 ). Then, since w n and w n+1 are both connected to the imaginary axis inK N and p cannot crossK N , it must eventually hit [w n , w n+1 ] again after τ (1), at time τ (2) say, except possibly if p τ (1) = w n . If the open line segment (p τ (1) , p τ (2) ) contains a point w ∈K N with σ(w) = (t, y), then for all z ∈ finger(w) with σ(z) = (s, x) and |s − s(e)| ε ′ we have x x(e) + ε ′ . But this is impossible because w is disconnected from the imaginary axis by {s 0 /δ * + ix : x x 0 } ∪ {p τ : τ 0}. Hence (p τ (1) , p τ (2) ) ⊆D N , so p τ ∈ [p τ (1) , p τ (2) ] for all τ ∈ (τ (1), τ (2)), contradicting our crossing assumption. In the case p τ (1) = w n , if p does not return to [w n , w n+1 ], then it must hit [w n−1 , w n ] instead and this also leads to a contradiction by a similar argument. The case where p τ (1) ∈ B 1 also leads to a contradiction of minimality by a similar argument. Hence p never crosses B. So, for all (t, y) ∈ G(e) with y Φ vs(e) (x(e)), we have d((t, y), Φ(e)) ε 0 + 2ε + 7ε ′ 2ε 0 . A similar argument establishes this estimate also in the case y Φ vs(e) (x(e)). Since G(e) is a connected set joining (s 0 , x 0 ) to {T } × R, this implies d H (G(e), Φ(e)) 2ε 0 .
We turn now to the local fluctuations. The argument is mainly similar. It becomes crucial that Theorem 6.5 provides approximation on a scale just larger than δ 2/3 , allowing us to transfer fluctuation results from Theorem 8.1 at scale δ 1/2 to the cluster. There is also some loss of compactness in the local limit which requires attention.
Given 0 < ε 0 < 1/3, there exist ε > 0 and R ∈ [1, ∞) and ε ′ ∈ (0, ε/3] such that, for any coalescing Brownian flowΦ = (Φ ts : 0 s t T ) on the line, with probability exceeding 1 − ε 0 /3, for all e ∈ E and all t ∈ [0, T ], we have |Φ ts(e) (x(e))| R andΦ ts(e) (x(e)) − ε 0 Φ ts(e) (x(e) − 5ε) − 5ε,Φ ts(e) (x(e) + 5ε) + 5ε Φ ts(e) (x(e)) + ε 0 and, for all s, s ′ , t, t ′ ∈ [0, T ] with |s − s ′ |, |t − t ′ | 3ε ′ and all |x| 2R Φ ts (x) Φ t ′ s ′ (x + ε) + ε.
Uniform continuity of the map (s, t) →Φ ts : [0, T ] 2 →D now provides only local estimates in x, hence the need for the cut-off R.
Then, by Theorem 8.1, there exists a δ 0 > 0 such that, for all δ ∈ (0, δ 0 ] and all basic particles P satisfying (2), for N = ⌊c −1 T ⌋, we can construct, on some probability space, an HL(0) process Φ P = (Φ P n : n N) with basic particle P and a coalescing Brownian flow Φ = (Φ ts : 0 s t T ) on the line with the following property. Write (Φ P mn : 0 m < n N) for the backwards harmonic measure flow of Φ P and setΦ P mn (x) = (δ * ) −1/2 Φ P mn ((δ * ) 1/2 x). With probability exceeding 1 − ε 0 /3, for all 0 m < n N, for t = cm and s = cn, and for all |x| 2R, we haveΦ
ts (x − ε) − ε Φ P mn (x) Φ ts (x + ε) + ε.
Moreover, by Theorem 6.5, we may choose δ 0 so that, with probability exceeding 1 − ε 0 /3, for all e ∈ E, writing z(e) =σ −1 (e) = s(e) + i(δ * ) 1/2 x(e) andσ(p 0 (z(e))) = (s 0 , x 0 ), we have |s 0 − s(e)| ε ′ /3, |x 0 − x(e)| ε and, for all s ∈ [0, T ] and all x ∈ R, there exists w ∈K N such thatσ(w) = (t, y) satisfies |s − t| ε ′ /3, |x − y| ε and, for all n N − 1 and all z ∈P n+1 ,σ(z) = (s, x) satisfies
|s − cn| ε ′ /3, |x − Θ n+1 / √ δ * | ε.
From this point on, we condition on the good event Ω 0 of probability exceeding 1−ε 0 where all of the properties discussed above hold. Suppose that we fix j, k ∈ Z and m, n N − 1 and w ∈P m+1 + 2πij and z ∈P n+1 + 2πik, withP m+1 + 2πij an ancestor particle ofP n+1 + 2πik. Writeσ(w) = (t, y) andσ(z) = (s, x) and suppose that |x| + 2ε 2R. Then we must have m = c −1 t ′ n = c −1 s ′ , with |s − s ′ |, |t − t ′ | ε ′ /3 and |y − (Θ m+1 + 2πj)/ √ δ * |, |x − (Θ n+1 + 2πk)/ √ δ * | ε, so y (Θ m+1 + 2πj)/ √ δ * + ε =Φ P mn ((Θ n+1 + 2πk)/ √ δ * ) + ε Φ t ′ s ′ ((Θ n+1 + 2πk)/ √ δ * + ε) + 2ε Φ t ′ s ′ (x + 2ε) + 2ε Φ ts (x + 3ε) + 3ε.
and by a similar argument also y Φ ts (x − 3ε) − 3ε. Here we have extendedΦ by settinḡ Φ ts =Φ t∧T,s∧T .
Fix e ∈ E and (t, y) ∈F(e). Write (t, y) =σ(w) andP 0 (z(e)) =P n+1 + 2πik. We can choose z ∈P 0 (z(e)) withσ(z) = (s, x) and |s − s(e)| ε ′ /3 and |x − x(e)| ε. In particular |x| + 2ε |x(e)| + 3ε 2R. Set u = t ∧ s(e). Then w and z are related as in the preceding paragraph and t t ′ + ε ′ /3 s ′ + ε ′ /3 s + 2ε ′ /3 s(e) + ε ′ s(e) + ε 0 so |t − u| ε ′ . Hence y Φ ts (x + 3ε) + 3ε Φ us(e) (x + 4ε) + 4ε Φ us(e) (x(e) + 5ε) + 5ε Φ us(e) (x(e)) + ε 0 Define recursively a sequence τ 0 , . . . , τ M by setting τ 0 = s(e) − 2ε ′ and then taking τ n+1 as the supremum of the set {τ ∈ [τ n , T ] : |(τ, y(τ )) − (τ n , y(τ n ))| = ε ′ } until n = M − 1 when this set is empty and we set τ M = T . For n = 0, 1, . . . , M, choose w n ∈K N withσ(w n ) = (t n , y n ) and |t n − τ n | ε ′ and |y n − y(τ n )| ε. Note that t 0 s(e) − ε ′ and t M T − ε ′ and t n ∈ [s(e) − 3ε ′ , T + ε ′ ] and |y n | + 3ε 2R for all n. Set
B 0 = M −1 n=0 [w n , w n+1 ), B 1 = {t + i √ δ * y M : t t M }, B = B 0 ∪ B 1 .
Then, for any w ∈ B 0 , for (t, y) =σ(w), we have |(t, y) − (τ n , y(τ n ))| ε + 2ε ′ for some n, so ε 0 + 3ε ′ d((t, y),Φ(e)) ε 0 + 2ε + 7ε ′ and so y Φ vs(e) (x(e)) + ε 0 , where v = s(e) ∨ t ∧ T . The final inequality obviously extends to B. Suppose p crosses B, and does so for the first time at τ (1). Consider first the case where p τ (1) ∈ [w n , w n+1 ). Then, since w n and w n+1 are both connected to the imaginary axis inK N and p cannot crossK N , it must eventually hit [w n , w n+1 ] again after τ (1), at time τ (2) say, except possibly if p τ (1) = w n . If the open line segment (p τ (1) , p τ (2) ) contains a point w ∈K N withσ(w) = (t, y), then for all z ∈ finger(w) withσ(z) = (s, x) and |s − s(e)| ε ′ we have x x(e) + ε ′ . But this is impossible because w is disconnected from the imaginary axis by {s 0 + i √ δ * x : x x 0 } ∪ {p τ : τ 0}. Hence (p τ (1) , p τ (2) ) ⊆D N , so p τ ∈ [p τ (1) , p τ (2) ] for all τ ∈ (τ (1), τ (2)), contradicting our crossing assumption. In the case p τ (1) = w n , if p does not return to [w n , w n+1 ], then it must hit [w n−1 , w n ] instead and this also leads to a contradiction by a similar argument. The case where p τ (1) ∈ B 1 also leads to a contradiction of minimality by a similar argument. Hence p never crosses B. So, for all (t, y) ∈Ḡ(e) with y Φ vs(e) (x(e)), we have d((t, y),Φ(e)) ε 0 + 2ε + 7ε ′ 2ε 0 . A similar argument establishes this estimate also in the case y Φ vs(e) (x(e)). SinceḠ(e) is a connected set joining (s 0 , x 0 ) to {T } × R, this implies d H (Ḡ(e),Φ(e)) 2ε 0 .
Theorem 3 . 1 .
31Consider for ε ∈ (0, 1] and m ∈ N the event Ω[m, ε] specified by the following conditions: for all n m and all n ′ m + 1,
ofΓ n (e 2πix )/(2πi) for δ = 0.02, with t = n/10 6 .
Figure 1 :
1The slit case of HL(0)
Figure 2 :
2Diagram illustrating fingers and gaps inK N (repeating periodically)
(W e : e ∈ E) be a family of coalescing Brownian motions, withW e running forwards in time from x(e) at time s(e). Denote byν E andη E the laws on S E of the families of random sets ({(t,B e t ) : 0 t s(e)} : e ∈ E) and ({(t,W e t∧T ) : t s(e)} : e ∈ E).
approximation of a finite set of fingers and gaps in K N , with N = ⌊ρT ⌋, when T = 1 and δ * = 0.05. (b) An approximation of a finite set of fingers and gaps in K N , with N = ⌊c −1 T ⌋, when T = 1 and δ * = 0.01.
Figure 3 :
3Geometric illustration of Theorems 3.2 and 3.3, where fingers are denoted in dark blue, and gaps in light blue.
(n 0
0,y 0 ) n is non-decreasing in n and y 0 . Set M = ⌈2π/η⌉ and h = 2π/M so that h η. Consider the set of time-space starting points E = {(n 0 , jh) : n 0 ∈ {0, 1, . . . , m}, j ∈ {0, 1, . . . , M − 1}} and the event Ω 0 = {Y (n 0 ,jh) n 0 +N 1 jh + η for all (n 0 , jh) ∈ E}.
7
Weak convergence of the localized disturbance flow to the coalescing Brownian flow We review in this section the main results of [15]. Denote byD the set of all pairs f = {f − , f + }, where f + is a right-continuous, non-decreasing function on R and where f − is the left-continuous modification of f + . Denote by D the subset of those f ∈D such that x → f + (x) − x is periodic of period 2π. Write id for the identity function id(x) = x and, for f ∈D, write f ± 0 for the periodic functions f ± − id. Denote by D * the subset of D where f 0 is not identically zero but has
We write D • (R, D) for the set of all cadlag weak flows. For the disturbance flow Φ, almost surely, for all t ∈ R, for all sufficiently small ε > 0, we haveΦ (t−ε,t) = Φ (t,t+ε) = id. So Φ takes values in D • (R, D). Define similarly D • (R,D) and note that Φ ε takes values in D • (R,D). Fix φ ∈ D • (R, D) and suppose that φ {t} = id for all t ∈ R. Then φ (s,t) = φ (s,t] = φ [s,t) = φ [s,t]for all s, t ∈ R with s < t. Denote all these functions by φ ts and set φ tt = id for all t ∈ R.The map (s, t) → φ ts : {(s, t) ∈ R 2 : s t} → D is then continuous. We write C • (R, D) for the set of such continuous weak flows φ and we write C • (R,D) for the analogous subset in D • (R,D). We can and do make D • (R, D) and D • (R,D) into complete separable metric spaces by the choice of Skorokhod-type metrics, both denoted d D . The metrics d D have the following two further properties. The associated Borel σ-algebras coincide with those generated by the evaluation maps φ → φ + I (x) as x ranges over R and I ranges over bounded intervals in R. Moreover, for any sequence (φ n : n ∈ N) in D • (R, D) and any φ ∈ C • (R, D), we have d D (φ n , φ) → 0 if and only if d D (φ n I , φ I ) → 0 uniformly over subintervals I of compact sets in R. In particular, C • (R, D) is closed in D • (R, D). Analogous statements hold in the non-periodic case. However, the flow property
t] (x) : [s, ∞) → R are cadlag. Hence we obtain a measurable maps Z e = Z e,+ and Z e,− on D • (R, D) with values in D e = D x ([s, ∞), R) by setting Z e,± (φ) = (φ ± (s,t] (x) : t s). The restrictions of Z e,± to C • (R, D) then take values in C e = C x ([s, ∞), R). We define a filtration (F t ) t 0 on D • (R, D) by F t = σ(Z e r : e = (s, x) ∈ R 2 , r ∈ (−∞, t] ∩ [s, ∞))
(x + a)f 0 (x)|dx λ, a ∈ [ελ, 2π − ελ].
For a bounded interval I ⊆ [0, ∞), set Φ P I = Φ P nm , where m + 1 and n are respectively the smallest and largest integers in ρI. We set Φ P I = id if there are no such integers. Then (Φ P I : I ⊆ [0, ∞)) takes values in D • ([0, ∞), D). Set δ * = (ρc) −1 and defineΦ P I (x) = (δ * ) −1/2 Φ P nm ((δ * ) 1/2 x), wherem + 1 andn are the smallest and largest integers in c −1 I. Then (Φ P I : I ⊆ [0, ∞)) takes values in D • ([0, ∞),D). Theorem 8.1. Assume that the basic particle P satisfies condition (2). Then the harmonic measure flow (Φ P I : I ⊆ [0, ∞)) converges weakly in D • ([0, ∞), D) to the coalescing Brownian flow on the circle, uniformly in P as δ → 0. Moreover, the rescaled harmonic measure flow (Φ P I : I ⊆ [0, ∞)) converges weakly in D • ([0, ∞),D) to the coalescing Brownian flow on the line. Proof. The flow (Φ P I : I ⊆ [0, ∞)) is a disturbance flow with disturbance g and (Φ P I : I ⊆ [0, ∞)) is an ε-scale disturbance flow with disturbance g, where ε = √ δ * . From Corollary 4.2 we know that δ 2 /6 c 3δ 2 /4 and from Proposition 4.3, we have δ −3 /C ρ Cδ −3 .
. Similarly, writeμ P E for the law of (F(e),Ḡ(e) : e ∈ E) when N = ⌊c −1 T ⌋. Write µ E for the law on (S E ) 2 of the family of random sets ({(t, Φ ts(e) (x(e))) : t ∈ [0, s(e)]}, {(t, Φ t∧T,s(e) (x(e))) : t s(e)} : e ∈ E)
F
(e) ⊆ {(t, y) : t ∈ [0, s(e)] and |y − Φ ts(e) (x(e))| ε 0 } ∪ {(t, y) : t ∈ [s(e), s(e) + ε 0 ] and |y − x(e)| ε 0 } and, since F(e) is a connected set joining (s, x) to the imaginary axis, this implies for the Hausdorff metric d H that d H (F(e), {(t, Φ ts(e) (x(e))) : 0 t s(e)}) 2ε 0 .
Hence F(t, y) does not meet the vertical half-line {(s 0 , x) : x x 0 }. Define Φ(e) = {(t, Φ t∧T,s(e) (x(e))) : t s(e)} and set I = [s(e) − 2ε ′ , T ].
Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK 2 Research supported by EPSRC grant EP/103372X/1 3 Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK
Since (t, y) was arbitrary, we have shown that F(e) ⊆ {(t, y) : t ∈ [0, s(e)] and |y −Φ ts(e) (x(e))| ε 0 } ∪ {(t, y) : t ∈ [s(e), s(e) + ε 0 ] and |y − x(e)| ε 0 } and. sinceF(e) is a connected set joining (s, x) to the imaginary axis, this implies for the Hausdorff metric d H that d H (F(e), {(t,Φ ts(e) (x(e))) : 0 t s(e)}) 2ε 0Since (t, y) was arbitrary, we have shown that F(e) ⊆ {(t, y) : t ∈ [0, s(e)] and |y −Φ ts(e) (x(e))| ε 0 } ∪ {(t, y) : t ∈ [s(e), s(e) + ε 0 ] and |y − x(e)| ε 0 } and, sinceF(e) is a connected set joining (s, x) to the imaginary axis, this implies for the Hausdorff metric d H that d H (F(e), {(t,Φ ts(e) (x(e))) : 0 t s(e)}) 2ε 0 .
We complete the proof by obtaining an analogous estimate forḠ(e). Recall thatḠ(e) = {σ(p τ ) : τ 0} where p = p(z(e)) is the minimal length gap path starting from. p 0 (z(e)), the closest point to z(e) which is not in the interior ofK N . Writeσ(p 0 (z(e)We complete the proof by obtaining an analogous estimate forḠ(e). Recall thatḠ(e) = {σ(p τ ) : τ 0} where p = p(z(e)) is the minimal length gap path starting from p 0 (z(e)), the closest point to z(e) which is not in the interior ofK N . Writeσ(p 0 (z(e)))
Suppose that t < s(e) − ε ′ and p makes an excursion left of the line {t + i √ δ * y : y ∈ R}, with endpoints w − , w + , say. Then the open line segment (w − , w + ) must contain a point of K N , say w ∈P m+1 + 2πij. Setσ(w) = (t, y). Then, since p cannot crossK N , there must exist z ∈P n+1 + 2πik, an ancestor particle ofP m+1 + 2πij, with σ(z) = (s, x), say, and s s 0 . But then s(e) s 0 + ε ′ /3 s + ε ′ /3 cn + 2ε ′ /3 cm + 2ε ′ /3 t + ε ′ < s(e) which is impossible. Hence there is no such excursion and sō G(e) ⊆ {(s, x) : s s(e) − ε ′ , x ∈ R}= (s 0 , x 0 ). Suppose that t < s(e) − ε ′ and p makes an excursion left of the line {t + i √ δ * y : y ∈ R}, with endpoints w − , w + , say. Then the open line segment (w − , w + ) must contain a point of K N , say w ∈P m+1 + 2πij. Setσ(w) = (t, y). Then, since p cannot crossK N , there must exist z ∈P n+1 + 2πik, an ancestor particle ofP m+1 + 2πij, with σ(z) = (s, x), say, and s s 0 . But then s(e) s 0 + ε ′ /3 s + ε ′ /3 cn + 2ε ′ /3 cm + 2ε ′ /3 t + ε ′ < s(e) which is impossible. Hence there is no such excursion and sō G(e) ⊆ {(s, x) : s s(e) − ε ′ , x ∈ R}.
Consider (t, y) =σ(w) with w ∈P m+1 + 2πij and m N − 1 and t s(e) − 3ε ′ and |y| + 3ε 2R and y Φ vs. e) (x(e)) + ε 0 , where v = s(e) ∨ t ∧ T . Note that t T + ε ′ /3Consider (t, y) =σ(w) with w ∈P m+1 + 2πij and m N − 1 and t s(e) − 3ε ′ and |y| + 3ε 2R and y Φ vs(e) (x(e)) + ε 0 , where v = s(e) ∨ t ∧ T . Note that t T + ε ′ /3
Suppose (s, x) =σ(z) with z ∈P n+1 + 2πik and |s − s(e)| ε ′ and wherẽ P n+1 + 2πik is an ancestor particle ofP m+1 + 2πij. ′ , Then x Φ st (y − 3ε) − 3ε Φ s(e)v (y − 4ε) − 4ε x(e) + εand |v − t| 3ε ′ . Suppose (s, x) =σ(z) with z ∈P n+1 + 2πik and |s − s(e)| ε ′ and wherẽ P n+1 + 2πik is an ancestor particle ofP m+1 + 2πij. Then x Φ st (y − 3ε) − 3ε Φ s(e)v (y − 4ε) − 4ε x(e) + ε.
There exists a continuous function y(t) : I → R such that, for all t ∈ I, setting v = s(e) ∨ t ∧ T. HenceF(t, y) does not meet the vertical half-line {(s 0 , x) : x x 0 }. DefineΦ (e) = {(t,Φ t∧T. s(e) (x(e))) : t s(e)} and set I = [s(e) − 2ε ′ ,. we have y(t) >Φ vs(e) (x(e)), d((t, y(t)),Φ(e)) = ε 0 + ε + 5ε ′HenceF(t, y) does not meet the vertical half-line {(s 0 , x) : x x 0 }. DefineΦ (e) = {(t,Φ t∧T,s(e) (x(e))) : t s(e)} and set I = [s(e) − 2ε ′ , T ]. There exists a continuous function y(t) : I → R such that, for all t ∈ I, setting v = s(e) ∨ t ∧ T , we have y(t) >Φ vs(e) (x(e)), d((t, y(t)),Φ(e)) = ε 0 + ε + 5ε ′ .
Anisotropy and cluster growth by diffusion-limited aggregation. Robin C Ball, Robert M Brady, Giuseppe Rossi, Bernard R Thompson, Phys. Rev. Lett. 55Robin C. Ball, Robert M. Brady, Giuseppe Rossi, and Bernard R. Thompson. Anisotropy and cluster growth by diffusion-limited aggregation. Phys. Rev. Lett., 55:1406-1409, Sep 1985.
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|
[] |
[
"Comparison-limited Vector Quantization",
"Comparison-limited Vector Quantization"
] |
[
"Joseph Chataignon [email protected] \nTélécom Saint-Étienne\nUniversité Jean Monnet\nFrance\n",
"Stefano Rini [email protected] \nDepartment of Electrical and Computer Engineering\nNational Chiao Tung University\nTaiwan\n"
] |
[
"Télécom Saint-Étienne\nUniversité Jean Monnet\nFrance",
"Department of Electrical and Computer Engineering\nNational Chiao Tung University\nTaiwan"
] |
[] |
In this paper a variation of the classic vector quantization problem is considered. In the standard formulation, a quantizer is designed to minimize the distortion between input and output when the number of reconstruction points is fixed. We consider, instead, the scenario in which the number of comparators used in quantization is fixed. More precisely, we study the case in which a vector quantizer of dimension d is comprised of k comparators, each receiving a linear combination of the inputs and producing the output value one/zero if this linear combination is above/below a certain threshold. In reconstruction, the comparators' output is mapped to a reconstruction point, chosen so as to minimize a chosen distortion measure between the quantizer input and its reconstruction. The Comparison-Limited Vector Quantization (CLVQ) problem is then defined as the problem of optimally designing the configuration of the compactors and the choice of reconstruction points so as to minimize the given distortion. In this paper, we design a numerical optimization algorithm for the CLVQ problem. This algorithm leverages combinatorial geometrical notions to describe the hyperplane arrangement induced by the configuration of the comparators. It also relies on a genetic genetic meta heuristic to improve the selection of the quantizer initialization and avoid local minima encountered during optimization. We numerically evaluate the performance of our algorithm in the case of input distributions following uniform and Gaussian i.i.d. sources to be compressed under quadratic distortion and compare it to the classic Linde-Buzo-Gray (LBG) algorithm.
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10.1109/ieeeconf44664.2019.9048997
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[
"https://arxiv.org/pdf/2105.14464v1.pdf"
] | 153,312,588 |
2105.14464
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875bc7c2144a614b5eb58a8d67bbd1ecf5c000bf
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Comparison-limited Vector Quantization
Joseph Chataignon [email protected]
Télécom Saint-Étienne
Université Jean Monnet
France
Stefano Rini [email protected]
Department of Electrical and Computer Engineering
National Chiao Tung University
Taiwan
Comparison-limited Vector Quantization
10.3390/xx010005Received: date; Accepted: date; Published: dateArticleConstrained quantizationSource compressionHyperplane arrangementLBG algorithmGaussian quantizerUniform quantizer
In this paper a variation of the classic vector quantization problem is considered. In the standard formulation, a quantizer is designed to minimize the distortion between input and output when the number of reconstruction points is fixed. We consider, instead, the scenario in which the number of comparators used in quantization is fixed. More precisely, we study the case in which a vector quantizer of dimension d is comprised of k comparators, each receiving a linear combination of the inputs and producing the output value one/zero if this linear combination is above/below a certain threshold. In reconstruction, the comparators' output is mapped to a reconstruction point, chosen so as to minimize a chosen distortion measure between the quantizer input and its reconstruction. The Comparison-Limited Vector Quantization (CLVQ) problem is then defined as the problem of optimally designing the configuration of the compactors and the choice of reconstruction points so as to minimize the given distortion. In this paper, we design a numerical optimization algorithm for the CLVQ problem. This algorithm leverages combinatorial geometrical notions to describe the hyperplane arrangement induced by the configuration of the comparators. It also relies on a genetic genetic meta heuristic to improve the selection of the quantizer initialization and avoid local minima encountered during optimization. We numerically evaluate the performance of our algorithm in the case of input distributions following uniform and Gaussian i.i.d. sources to be compressed under quadratic distortion and compare it to the classic Linde-Buzo-Gray (LBG) algorithm.
Introduction
Quantization represents one of the crucial operations in signal processing as it allows for the mapping of analog signals into discrete values, i.e. A2D conversion. Being such a fundamental processing operation, quantization has been an important subject of research and numerous results have been derived for problem. In this work we consider a variation of the classic quantization setting which appears to be so far neglected in the literature. Vector quantizers are typically produced using operational amplifier (op-amp) comparators that take a signal and a bias as inputs and produce a zero/one voltage as output, depending on whether the signal is larger/smaller than the bias. Generally speaking, op-amp comparators have high power consumption and manufacturing cost. Additionally, generating a precise set of bias values for each comparator gives rise to a rather complex circuit structure. For this reason, in scenarios in which complexity, fabrication cost, or energy consumption are severely restricted, it is desirable to evaluate the quantizer performance in terms of the number of comparators it requires to produce its output. This is in contrast with the classic approach, in which the quantizer performance is optimized under a constraint on the output cardinality, i.e. the number of bits used to represent the output in A2D conversion. As a concrete example of the difference between the classic setting and the setting we propose, let us consider the case of a two dimensional quantizer (d = 2) in which each dimension is quantized with a rate of 1.5 bits-per-sample (R = 1.5 bps), so that In the classic setting, the quantizer is designed under the constraint that each pair of inputs is represented as one of three possible reconstruction points. Under this constraint, a quantizer can be represented as in Fig. 1: here each of the three reconstruction points is separated by the others through a edge in the Voronoi region, so that quantizer design requires three comparisons to distinguish the three reconstruction points. Accordingly, k = 3 comparisons are required to implement this quantization rule. More generally, since every two reconstruction points are separated by an edge of the Voronoi region, a vector quantizer might require up to 2 R (2 R − 1)/2 comparators. This scaling of the number of comparators with the output cardinality is generally valid only for low-rate regime, since the number of neighbors of each reconstruction point is large. Given that this quantization regime is naturally associated with low-cost and low-complexity devices, this cost scaling is particularly disadvantageous. A question that naturally arises is whether a better scaling of the number of comparators with the output cardinality can be attained. To address this question, note that number of comparators, k, equals the total number of hyperplane segments in the Voronoi regions. In this view, the best scaling is attained when the k hyperplanes induce the largest number of partitions of the space of dimension d. After some rather straightforward geometrical consideration, one realizes that for the case of d = 2 and R = 1.5, the largest number of partition is 7. This corresponds to the quantizer in Fig. 1. It is now apparent that there exists a large gap in the optimal quantizer design whether one considers a constraint on the number of points used in reconstruction or on the number of comparators employed. It is indeed the aim of this paper to develop a first understanding of this problem, which we term the "Comparison-Limited Vector Quantization (CLVQ) problem".
Relevant Results
Vector Quantization (VQ) [1,2] is a topic of vast interest, so that we shall briefly review only the literature which is most closely related to the topic of this publication. Suffices to say that VQ is widely adopted in speech coding [3][4][5], image coding [6][7][8], and video coding [9,10]. VQ has been successfully used for speaker identification [11,12], digital watermarking [13,14], and clustering [15,16].
In the following, we follow the information theoretical study of the quantizer performance as described by rate-distortion theory [17], originally developed by Shannon in his seminal work [18]. As originally formulated, the rate distortion function is the function describing the minimum rate at which a source of information can be compressed so that the reconstruction is to within a prescribed distortion from the original value, in the limit of infinitely many source samples are compressed. This approach is concerned with the compression rate needed to attain a prescribed distortion level, that is the number of bits required on average to represent a source sample after quantization. The idea of accounting for the number of comparators required for A2D conversion emerges from the literature of mmWave MIMO communication [19], in which high-resolution quantization is no longer feasible, duo to the high energy consumption in high-speed A2D conversion [20]. For this reason, authors have investigated the transmission performance achievable when quantization only preserves the sign of the channel output [19,21]. Generalizing the one-bit quantization case, in [22] the authors investigate the capacity of a MIMO channel with output quantization constraint, i.e. the MIMO channel in which the channel output is processed at the receiver using a finite number of one-bit threshold quantizers. The channel model of [22] is relevant in mm-wave communications which allows for a large number of receiver antennas, while the number of A2D conversion modules remains small due to limitation in the energy and costs of radio-frequency modules. Building on an idea of [21], a connection between combinatorial geometry and the MIMO channel with output quantization constraints of [22] is drawn in [23]. In particular, in [23] it is shown that each quantizer can be interpreted as a hyperplane bisecting the transmitter signal space; for this reason the largest rates are attained by the configuration allowing for the largest number of partitions induced by the set of hyperplanes. To the best of our knowledge, the problem of hardware limited quantization has so far only been considered in [24]. The difference between our approach and that of [24] is our focus on vector quantization, and our consideration of comparator limitations on the vector, rather than scalar, quantizer input.
Deep learning techniques have been used successfully in the domain of quantization, for task-based quantization problems [25] as well as particular use cases (MIMO channel estimation in [26] or high-dimensional signal recovery from one-bit quantization in [27]). The idea of adapting the quantizer design to the subsequent task, or task-based quantization, is explored in [28] and studied in the context of hardware limitations in [24] [29]. Tree-Structured Vector Quantization uses a set of scalar quantizers to gain speed, but comes at a disadvantage on the shape of quantization regions.
Signal recovery from one-bit quantized signals is a topic previously investigated in the literature, such as in [30] and [31]. An approach to one-bit signal recovery using deep network is considered in [32]. These approaches can be interpreted as one-dimensional formulations of the CLVQ quantization problem we study here. Algorithm for the design of quantizers under quadratic distortion is studied by Max and Lloyd for scalar quantization [33,34], and extended to vector quantization by Linde,Buzo, and Gray [35].
Contribution
Our main contributions, in the following paper, consist in the definition of the Comparison-Limited Vector Quantization (CLVQ) problem in its full generality and identify the connections between this problem and the classic quantization problem. In particular, we highlight a connection between the CLVQ problem and combinatorial geometry and provide novel insights on the ultimate performance of low-resolution quantization.
We also provide a first class of algorithms for the CLVQ design: although not optimal, this first approach investigates the combinatorial geometric aspects of the CLVQ design. Numerical evaluations are provided for the case of uniform and Gaussian i.i.d. sources. The performance of the proposed quantizer is compared with the classic optimal vector quantizer design obtained by Linde-Buzo-Gray (LBG) algorithm [35].
Paper Organization
The paper is organized as follows: Sec. 2 presents related results in the literature and introduces some useful combinatorial notions. Sec. 3 presents the CLVQ model and the performance evaluation for this model. Sec. 4 introduces a class of algorithms for the CLVQ problem. Sec. 5 provides relevant numerical evaluations for the case of an iid Gaussian source to be reconstructed under quadratic distortion. Finally, Sec. 6 concludes the paper.
Notation
In the remainder of the paper, all logarithms are taken in base two. With x = [x 1 , . . . , x N ] ⊆ X N we indicate a sequence of elements from X with length N. The notation x j i indicates the substring [x i , . . . , x j ] of x. The function sign(x) returns a vector with values in {−1, +1} which equals the sign of each entry of the vector x. The set {1, . . . , N} is indicated as [N]. Finally, the 2-norm operator is denoted as · , expectation as E[·], and 1(·) is used for the indicator function.
Related Results
In this section let us review three topics of relevance in the remainder of the paper: (i) rate-distortion theory, that is the information theoretical study of the quantization, (ii) quantizer design algorithms, such as the LBG algorithms which we use as a point of comparison of our results, and (iii) some combinatorial geometrical notions that are useful in describing quantizer configuration in the CLVQ problem.
Rate-Distortion Theory
The theoretical framework we introduce for the study of the CLVQ problem-see Sec. 3-expands on the classic formulation of rate-distortion theory [36, Ch. 10 ]. Let us briefly review some relevant results in this section.
Broadly speaking, rate-distortion theory is the information theoretical study of A2D conversion and addresses the quantization performance in the limit of an infinite source sequence when the number of bit-per-sample is kept constant.
Consider the system model in Fig. 2: here the source sample X n = [X 1 , . . . , X n ] with support X n is represented through the index m n using the source encoder mapping
f enc : X n → [2 nR ],(1)
so that the cardinality of m n is 2 nR when the source sequence has length n. The index m n is communicated to a source decoder through a noiseless channel. In turn, the source encoder produces the reconstruction X n through the mapping
f dec : [2 nR ] → X n ,(2)
so as to minimize the distortion measure ρ n (X n , X n ) between the original source sequence X n and its reconstruction X n . In (2), ρ n for n ∈ N is a positive and bounded function which captures the distortion, or degradation, between X n and X n . The best attainable performance when reconstructing the original source encoder X n source decoder symbol for a given value of R is evaluated through the rate-distortion function at rate R and block-length n, defined as
m n ∈ [2 dR ] X nD n (R) = inf ρ n (X n , X n ),(3)
where the infimization is over all source encoder/decoder mappings as in (1)/(2) . One is generally interested in the performance when n goes to infinity, since it provides an upper bound to the ultimate compression performance. We refer to this limit as the rate distortion function at rate R:
D(R) = lim inf n→∞ D n (R).(4)
It is well known that, when (i) the source sample is i.i.d. from the distribution P X and (ii) the distortion function is an additive distortion function, i.e.
ρ n (X n ; X n ) = ∑ i∈[n] ρ(X i , X i ),(5)
for some positive and bounded ρ(
x i , x i ), we have D(R) = inf P X|X , R≥I(X, X) E ρ(X, X) ,(6)
where I(X, X) indicates the mutual information between the source X and the reconstruction X. As an example of the result in (6) consider the case in which X has a standard normal distribution to be reconstructed under quadratic distortion, then
D(R) = 2 −2R ,(7)
so that D ∈ [0, 1].
Quantization Design Algorithms
The approach in Sec. 2.1 provides the theoretical limit to the quantization performance. In order to approach such performance, various algorithms have been proposed in the literature for the design of high-dimensional quantizers.
In the following, we shall compare the performance of the CLVQ design with the standard LBG algorithm which we shall describe next. The LBG algorithm, also known as the Max-Lloyd algorithm or k-means clustering algorithm, is an iterative algorithm for the design of quantizers for the case of quadratic distortion [33][34][35]. It is known to approach the optimal solution although this problem is NP-hard in the Euclidean space. The pseudo-code for the algorithm is provided in Algo. 1: the algorithm performs an alternate optimization between the reconstruction regions and the position of the reconstruction points. In the first step, the sample space X is partitioned by a Voronoi tessellation with the reconstruction points as generators. This means that each point in the sample space is assigned to the closest reconstruction points. reconstruction point has a region, and every point in space belongs to the region of the nearest reconstruction point. This procedure results in a set of regions {V m } m∈ [M] so that each point in the region V m contains the points that have the minimum distortion when reconstructed as the point c m , instead of any other reconstruction point. In the second step, the position of the reconstruction points c m is optimized: each reconstruction point c m is chosen as the mean of the sample points in the region V m . The two steps are repeated in succession until convergence is achieved. Various procedures can be designed to initialize the initial positions of the set of reconstruction points { c m } m∈ [M] as it is known that the choice of initialization set its crucial to attain fast convergence.
In the literature, many variations of the LBG algorithm as been studied with applications ranging from unsupervised learning to low-ranking approximations. Let us mention two variations of interest as a generalization of the settings in this paper.
Entropy-constrained vector quantizers (ECVQs) are quantizers in which the quantization objective is the distortion minimization subject an additional a bound on the entropy of the quantizer output. ECVQ are useful when they are used in tandem with variable-rate noiseless coding systems to provide locally optimal variable-rate block source coding. Another class of quantizers of interest are lattice quantizers. In lattice quantizers are quantizers in which the set of reconstructions points form a lattice. Note that the problem of determining the closest reconstruction point to a given quantizer input is, in general, a nearest neighbor problem. This class of problems has a complexity linear in the number of neighbors: when a quantizer has high dimension, this complexity is not amenable to real-time applications. For this reason, one is often interested in introducing further structure in the set { c m } m∈ [M] with the aim of speeding up the quantization operation. One class of such quantizers are lattice vector quantizers, that is, quantizers in which the set of reconstruction points form a lattice. It has been shown that lattice quantizers have good performance while also allowing for a fast quantization [37,38].
Some combinatorial geometric notions
This section briefly introduces a few combinatorial concepts useful in the remainder of the paper to describing the geometrical properties of the proposed quantization scenario. In this section and through the paper we mostly use the nomenclature and notation of [39] A hyperplane arrangement A is a finite set of n affine hyperplanes in R m for some n, m ∈ N. where A is obtained by letting each row i equals a T i and
letting b = [b 1 . . . b n ] T . A region, R i of an arrangement A is a connected component of the complement A of the hyperplanes, defined as A = R m − A.(8)
Let r(A) be the number of regions in which the hyperplane arrangement A divides the space R m , so that
A = r(A) i=1 R i .(9)
A plane arrangement A is said to be in General Position (GP) if and only if:
{H 1 , ..., H p } ⊆ A, p ≤ n ⇒ dim(H 1 ∩ ... ∩ H n ) = n − p {H 1 , ..., H p } ⊆ A, p > n ⇒ H 1 ∩ ... ∩ H n = ∅{ c m (0)} m∈[M] , i.e. c m (0) ∼ P X n 4: for t in [T max ] do 5: update the Voronoi regions {V m (t)} m∈[M] as V m (t) = x ∈ X s.t. (c m (t − 1) − x) 2 ≤ (c j (t − 1) − x) 2 , ∀j ∈ [M] \ mc m (t) = E[X|X ∈ V m (t)] E[1 {X∈V m (t) }] 7: end for d-parser X i v 1 v 2 v 3 v 4 X n = X d(i+1) di+1 + + + + t 4 t 3 t 2 t 1 sign(·) sign(·) sign(·) sign(·) n-parser Y 1n Y 2n Y 3n Y 4n source encoder Y n = [Y 1n , . . . , Y kn ] source decoder m n ∈ [2 dR ] X nr(m, n) = m ∑ i=0 n i ≤ 2 n .(10)
Lemma 2. [39, Prop. 2.4].
A hyperplane arrangement of size n in R m where all the hyperplanes pass through the origin divides R m into at most r 0 regions for
r 0 (n, m) = 2 m−1 ∑ i=0 n − 1 i .(11)
Lemma 3. [39, Prop. 2.4]. Let
A be a hyperplane arrangement of size l in R m and consider a hyperplane arrangement B of size dl with d ∈ N hyperplanes parallel to each of the hyperplanes in A, then B divides R m into at most r p regions for
r p (m, n, d) = m ∑ i=0 l i d i ≤ (1 + d) l .(12)
A necessary condition to attain the equality in Lem. 1, 2, and Lem. 3 is for the hyperplane arrangement A to be in GP. As we shall see in the next section,
Comparison-Limited Vector Quantization (CLVQ) Problem
After the introductory notion in Sec. 2 let us introduce the Comparison-Limited Vector Quantization (CLVQ) problem, also conceptually depicted in Fig. 3.
A source sequence {X i } i∈N , where each X i has support X , is parsed in super-symbols {X n } n∈N of dimension d with X n = [X d(n−1)+1 , . . . , X dn ] for n ∈ N where d is referred to as the dimension of the vector quantizer. The j th comparator computes a linear combination of each super-symbol X n and outputs a signal Y jn as
Y jn = sign v j X n + t j ,(13)
for j ∈ [k]. The value k is called resolution of the vector quantizer; v j ∈ R d , t j ∈ R are fixed and known. For more convenience, we express the k outputs of the quantizers of (13) as
Y n = sign (VX n + t) ,(14)
where V =∈ R k×d is such that its i th row is equal to the vector v i from (13). Similarly, t ∈ R k has the i th entry equal to t i and Y n = [Y 1n , . . . , Y kn ]. The set [V, t] is referred to as the configuration of the quantizer. The super-symbol Y n is provided to a source encoder that produces a bit-restricted representation of the quantizers' output as m n ∈ [2 dR ] where R is the name given to the rate of the quantizer through the source encoding mapping
f enc : {−1, +1} k → [2 dR ].(15)
The message m n is sent into the source decoder which outputs a reconstruction of the source super-symbol
X n as X n = [ X dn+1 , . . . , X d(n+1) ] with X i ∈ X , through the source decoding mapping f dec : [2 dR ] → X d .(16)
We measure the effectiveness of a vector quantizer with a measure of distortion
ρ n (X n ; X n ) : X n × X n → R + ,(17)
for n ∈ N which is assumed positive and non-decreasing in n. For a given configuration of the linear combiners [V, t], source encoder/decoder mappings f enc / f dec , and given a distortion measure between input and reconstruction sequence ρ n (X n ; X n ), the performance of the quantizer is evaluated as
ρ = lim sup n→∞ ρ n (X n ; X n ).(18)
The optimal quantizer performance regarding the distortion ρ, dimension d, resolution k and rate R can be written as
D(d, k, R) = inf ρ,(19)
where the infimization is over all configuration of the linear combiners [V,t] and all the source encoder/decoder mappings. Similarly to (4), one is often interested in determining the limiting performance as the dimension of the quantizer grows to infinity. In this case, it is natural to let the number of quantizer grow exponentially with the dimension, so that the number of comparators per dimension is kept constant, with ration approximatively equal to α. Let k = αd , then the comparison-limited distortion-rate function is defined as
D α (R) = lim d→∞, k= αd D(d, k, R),(20)
for D(d, k, R) in (19), that is D α (R) is the minimum distortion attainable as the quantizer dimension grows to infinity while k/d ≈ α.
Underlying Assumptions
The underlying assumptions in our problem formulation are as follows: op-amp voltage comparators are employed in nearly all analog-to-digital converters to obtain multilevel quantization. A reconfigurable receiver front-end might be able to re-configure the comparators' inputs so as to perform more complex operations. Given the receiver's ability to partially reconfigure its circuitry depending on the channel realization, we wish to determine which configuration of the comparators yields the largest capacity. We restrict our results to the case of linear analog pre-coding, as the capacity of the model without output quantization constraints can be attained through linear pre-coding strategies. The extension to non-linear processing is possible but not pursued here.
Discussion
Let us connect the CLVQ problem as defined above with the results in Sec. 2. As it can be gathered by comparing Fig. 2 versus Fig. 3, the CLVQ problem explicitly embeds a model of the hardware architecture performing A2D. In particular, in this formulation, we assume that quantization is performed through a bank of op-amp amplifier. Each op-amp amplifier receives a linear combination of the quantizer input and a bias.
It is immediate to connect the problem formulation in Fig. 3 with the setting in Sec. 2.3. The set of the comparators output Y n in (14) represent the membership function of each quantizer input X n with respect to each of the hyperplanes [v i , t i ]. This output is then compressed as in the classic rate-distortion problem in Sec. 2.1. When the number of sign quantizers is sufficiently large, a hyperplane can be used to separate any pair of reconstruction points, as in the Voronoi region of Algo. 1. In this regime, the rate in (6) is determined by the number of regions induced by the hyperplane arrangement. Given the considerations above, it is now useful to imagine the CLVQ problem in the context of Fig. 1: each comparator induces a hyperplane partitioning the sample space. The set of comparators then can be imagined as a hyperplane arrangement partitioning the sample space. The set of comparator outputs is then compressed as in the classic rate-distortion problem, thus further reducing the number of bits required to represent Y n by exploiting its distribution.
Let us next provide some more specific remarks.
Remark 1. Extension to the vector version
In the following we consider the case in which the source sequence {X i } i∈N is scalar. The case of a vector input sequence is not considered here. For the vector case, the dependency among different dimensions can be addressed in different manner: two come readily to mind. One approach would be to choose the linear combination as the whitening transformation that make the covariance of the input symbol unitary. Alternatively, the different dimension can be processed in parallel by the comparator network while the source encoder is tasked with accounting for the correlation across dimensions.
Remark 2. Entropy coding:
Note that the CLVQ problem formulation also captures the digital processing following A2D conversion. After quantization, the quantized samples are further compressed using a digital source compressor, such as an entropy coder. This is captured by the source encoder/decoder in (15)/(16) and as in Fig. 3. This is similar to the ECVQ discussed in Sec. 2.2.
Remark 3. Limitations in the comparator configurations:
In (19) we assume that the CLVQ performance can be optimized over all possible configuration of the linear combiners [V, t]. As a configuration of a linear combiner corresponds to an analog circutry implementation, this assumption might be unfeasible in some scenarios, as generating precise linear combinations and voltage references might require a high circuit complexity. To accommodate for such further limitations, on can further restrict the optimization in (19) over a set of configurations which can be efficiently implemented. Such an approach is similar to that in [22] and further discussion on this approach can be found there.
Remark 4. Lattice quantization for CLVQ:
Note that one can extend the construction of lattice codes for quantization to the CLVQ problem by considering regular hyperplane arraignments. In this case, it is possible to similarly define computationally-efficient quantization algorithms as discussed in Sec. 2.2.
A CLVQ Quantizer Design Algorithm
In the following, we shall introduce an algorithm for the design of the CLVQ linear combiner configuration [V, t] for given value of d, k, and R in (19). In other words, the algorithm attempts to numerically determine the optimal solution to (19) by assuming that the optimal choice of source encoder/decoder function is performed in a successive optimization step.
For the design of this numerical optimization algorithm, we take inspiration from the quantizer design algorithms discussed in Sec. 2.2, although the CLVQ problem is more general than considered is Sec. 2.2. Indeed, we shall leverage the notion in Sec. 2.3 to more efficiently represent and manipulate the linear combiner configuration during the optimization. The theoretical analysis of the asymptotically optimal solution as in (20) is left for future research.
For simplicity, in the design of our algorithm, we simplify the general setting of Sec. 3 as follows:
(a) i.i.d. sources -with distribution unimodal distribution P X , (b) quadratic distortion -also known as Mean Squared Error (MSE) distortion, i.e. ρ n (X n ;
X n ) = 1 n n ∑ i=1 X i − X i 2 .(21)
(c) infinite compression rate -that is R = ∞ in (19), i.e. D(d, k, ∞).
Regarding assumption (b), this assumption is relaxed in Sec. 5.3.3, here we discuss the use of entropy as a measure of distortion measure in the numerical optimization. Assumption (c) is chosen so as to simply the design of the comparator configuration. An effective, but sub-optimal, approach to address this constraint would be choose the encoding and decoding function which implement an efficient lossy-compression algorithm for discrete data -see [41].
Note that a more efficient approach to address the case R < ∞ would be to consider an entropy constrain, as discussed in Rem. 2. The design of such an algorithm would be similar to the problem of entropy-constrained quantization as in [42]. This further design step is not considered at the present.
Given the three assumptions above, (19) simplifies to
D(d, k, ∞) = d ∑ i=1 E X i − X i 2 ,(22)
where X = [X i , . . . , X d ] and X = [ X 1 , . . . , X d ] for X i , X i in (22) are i.i.d. distributed, so that the subscript n can be dropped. 1 For the problem in (22), the linear combiner optimization problem is mapped to a combinatorial geometric optimization problem as in Sec. 2.3 in which one is interested in choosing the 1 That is, every super-symbol X i is quantized in the same manner, regardless of n.
hyperplane arrangement which divides the space in {R i } regions for which MSE distortion is minimized as in Algo. 1 for the case in which each source vector X n is mapped to the closest centroid in the set {c i } i∈[r(A)] . The set of centroids is obtained from the hyperplane arrangement A as
c i = E[X|X ∈ R i ] E[1 {X∈R i }] ,(23)
similarly to the LBG algorithm.
Let us further detail the proposed approach: our algorithm is divided in two steps: (i) an initialization step -discussed in Sec. 4.1 -and (ii) an optimization step -detailed in Sec. 4.2.
Generally speaking, the optimization step is computationally complex, so the role of the initialization step is to choose of a set of initial configurations of quantizer that can span the large set of choices for the combiner configuration. By considering multiple restarts of the optimization point with a different initialization, one hopes to converge to an optimal solution and avoid local minima.
Note that there is a large number of symmetries in the linear combiner configurations, since one can permute each quantizer configuration and input values and obtain an equivalent configuration. This induces a number of local minima which hinder the convergence of the optimization step to the globally optimal solution. For this reason, the choice of the set of combiner configuration used for initialization is crucial to span the space of possible solutions.
Initialization methods
In the following, we explore two possibilities for initialization: -random initialization: in which the set of initial linear combiner configurations selected using random samples from the source distribution. -genetic initialization: in which we utilize a genetic algorithm to generate the set of initial configurations.
Random initialization
This first approach to the generation of initialization steps is the simplest, but yet provides good results. To obtain random hyperplanes that cut through dense zones of the source distribution, it is sufficient to randomly generate d points following that same source distribution and the hyperplane passing through these points can be used as a linear combiner configuration. By repeating this operation k times, one obtains an initial configuration for every comparator. This random initialization has constant time complexity and allows us to quickly generate a large number of possible candidate arrangements.
Genetic Initialization
Given the poor performance of the random initialization in Sec. 4.1.1 and the computational complexity of the exhaustive listing of all possible configuration in App. C, we next consider a genetic algorithm that retains elements of the two previous approaches. A genetic algorithm is a meta-heuristic taking inspiration from genetic evolution to apply some processes such as selection, crossover and mutation to optimization problems. It consists of a few steps that are repeated over several iterations. An initial population of potential solutions has to be generated. At each iteration, a part of the current generation is selected to breed the next one, based on a fitness function measuring its efficiency regarding the problem. Then, another generation is formed by performing crossover (which corresponds to mixing some elements between pairs of solutions) and mutation (small alterations of solution) on the current generation.
In our case, after generating a few starting configurations according to the random approach in Sec. 4.1.1, we proceed to genetic selection in three distinct stages:
(a) select a set of configurations based on their fitness (the MSE in our case) (b) perform crossover on these configurations to obtain the next generation with improved fitness (c) apply random mutations to the resulting configurations • In step (a), the configurations are reordered by increasing distortion (we use the opposite of distortion as fitness function).The k first configurations are kept for the next step, k being a fixed parameter. In our simulations, we set k to 80% of the number of configurations. • In step (b), we cross pairs of arrangements following one of two policies. The first one consists in randomly selecting hyperplanes to form pairs. With the second policy, we use a dissimilarity value to form the pairs of hyperplanes. This dissimilarity value is given by θ × d(p 1 , p 2 ), where θ is the angle between the hyperplanes; d is the euclidean distance; and p 1 , p 2 are the nearest point of each hyperplane to the centroid of the source distribution. Measures of the dissimilarity of each pair of hyperplanes are placed in a matrix, and hyperplanes are associated two by two in a way that produces the lowest average dissimilarity. Once pairs are formed, a new configuration is formed by selecting one hyperplane from each pair.
• Step (c) consists in mutating the resulting configurations.To apply mutation, we first form vectors of random numbers following a normal probability law with mean µ = 1 and with a standard deviation σ = 0.2 that was empirically chosen. Mutation is only applied to the half of configurations that have the lowest fitness score. The intuition for applying mutation this way is that solutions with a good fitness value might be made worse by a mutation, while the ones which fitness is already not good have better chances of being improved by the mutation.
These steps are repeated for a pre-determined number of iterations. Each step has a constant complexity, so the complexity of each iteration is linear with the number of initial hyperplane arrangements. This process provides us a set of arrangements with low distortion By repeating the process multiple times, we can accordingly generate a set of good initialization points.
Pseudo-code of the proposed genetic algorithm is also described in Algo. 2.
Remark 5.
An idea that naturally arises would be to consider all possible hyperplane configurations and simply optimize each of them. In App. A, we explain some notions of matroid theory that provide a framework for the exhaustive generation of hyperplane configurations. In App. B, we define a way to describe the structure of hyperplane arrangements with graphs. Then in App. C, we expose how one can progressively add nodes to this graph representation in order to generate graph configurations, and the reasons for this approach not being applicable in practice.
Optimization step
Being inspired from LBG algorithm (described in 1), our quantizer design algorithm also features two steps that are being iterated until convergence: (i) the optimization of the set of reconstruction points X for a given combiner configuration [V t] and (ii) the optimization of the combiner configuration for a given set of reconstruction points.
Step (i) consists in choosing the reconstruction points as the centroids (with regards to the source distribution) of the regions defined by the hyperplane arrangement [V t] in the space R d [43]. The optimization step (ii) presents more complexity. In that step we update the hyperplane arrangement [V t] using two approaches: a global configuration update and a local one as described in Algorithm 3. The global configuration update consists in adding a random perturbation to every hyperplane, perturbation which variance decreases over iterations. With the local configuration update, one hyperplane is randomly selected among the hyperplanes of the configuration. Its position and orientation are then set so as to for each pair (m 1 , m 2 ) do 8: perform crossover between A t,m 1 and A t,m 2 9: end for 10 minimize the distortion of the resulting output. In order to do that, we calculate the resulting MSE for several values of the hyperplane coefficients and apply interpolation between these values to find the best ones. At each iteration, one of the two configuration update approaches is chosen at random. The probability of choosing the local configuration update augments with the number of iterations, at a speed depending of an empirically fixed parameter s.
The reasoning behind differentiating local and global update is the following. Consider a lower triangular matrix M of size r(d, k) × r(d, k) and let the element in position i × j in M be equal to one of the hyperplanes separating X i and X j if such reconstruction point exists or equal to zero otherwise. 2 The matrix M can be thought of as one among a finite number of ways in which hyperplanes separate the reconstruction points.
In this view, the local update maximizes the quantizer performance in a given value of M. The global update, instead, allows "hyperplanes to jump over centroids", resulting in a different matrix M. This is illustrated in Fig. 4, where passing from step a. to step b. requires a global update (that is changing the structure of the arrangement) but passing from step b. to c. and c. to d. only require local updates (optimizing the configuration).
Algorithm 3 Quantizer design algorithm
Input: s and [T max ] ∈ N, a hyperplane arrangement A 0 , d the dimension Output: a hyperplane arrangement for quantization
for t in [T max ] do if random(0, 1) < exp(−s.t) then let H i,t be the i th hyperplane of A t let V i,
Discussion
A fundamental element in step (ii) of the design algorithm is the evaluation of the distortion (the MSE in our case) for a given hyperplane arrangement and set of reconstruction points. Given numerical precision limitations, our MSE evaluation has to be an approximation method, that uses numerical integration methods and particle filters. More precisely, we generate random points following the source distribution, until a certain number of points is reached for every region of the hyperplane arrangement. 2 Note that some hyperplane arrangements induce less that r(d, k) regions: we assume that there exists a natural numbering of the possible r(d, k) regions. For each random point generated, the output of the quantizer is the reconstruction point of the region in which that random point is located. Finally, the average of the squared distance between the random points and their corresponding output point is taken as the estimation of the MSE.
We use a similar approach to estimate the centroids of the regions formed by a hyperplane arrangement, since that estimation requires numerical integration too. To estimate the centroid positions, random points are also generated following the source distribution. Each point is assigned to the region it is in. We can then obtain the position of each centroid by computing the average coordinates of all the points generated in each region.
Note that these two steps (estimation of the centroids and estimation of the MSE) are done separately. In particular, they use distinct samples of the input distribution.
With the alternance of steps (i) and (ii) of the algorithm, the hyperplane configuration converges to an optimum. Multiple random restarts of the algorithm occasionally lead to convergence to distinct local minima. These minimal values are due either to a limitation in the precision of the numerical integration or to a local minimum in the quantizer configuration. In the next section, we shall further comment on this and other aspects of the proposed optimization algorithm in view of numerical evaluation results.
Numerical evaluations
In this section, we numerically investigate the a few aspects of the proposed algorithm. In Sec. 5.1 we will investigate the performance of the genetic algorithm discussed in Sec. 4.1.2. In Sec. 5.2, we discuss the optimization step in Sec. 4.2. Finally, in Sec. 5.3, we provide a performance comparison of the proposed algorithm and the classic LBG algorithm in Sec. 2.2.
Genetic Initialization Performance
Let us return to the genetic initialization discussed in Sec. 4.1.2. The nature of this genetic algorithm is such that the performance depends on the number of comparator configurations in input to the algorithm (see line 1 of Algo 2). The more are the available configurations, the bigger is the pool of genes that can be used. On the other hand, adding too many configurations can be computationally expensive. Fig. 5 shows the result obtained by the genetic algorithm with an input of dimension 3 and configurations of 3 or 4 hyperplanes in Fig. 5a and Fig. 5b respectively. Every run of the algorithm used a pool of 10 configurations. These configurations are then mixed following the steps described in Sec. 4.1.2. The algorithm was run twice for 5a and twice again for 5b. Both (a) and (b) display these two runs of the algorithm in light blue, and their average in dark blue. Note that in these two particular executions, in both Fig. 5a and Fig. 5b, that the distortion decreases very irregularly. This curve is composed of flat sections and a few sudden drops: this suggests that improvements happen from time to time when the right combination of two configurations is formed. For this reason, added to the general propensity of genetic algorithms to converge to local minima, this algorithm cannot be used for the whole optimization process. However, its computational efficiency makes it a good pre-optimization tool. Indeed, the distortion function is the often the most computationally heavy part of the optimization. This implementation calls to the distortion function 2 × n times per iteration (with n the number of configurations), while the previous algorithm we described in Sec. 4 calls it 40 × H × d with H the number of hyperplanes and d the dimension. Thus, this genetic algorithm can be used to quickly obtain a rough estimate of the global minimum, before a more precise and more computation-costly algorithm can finish the optimization.
Optimization Step Performance
Next, we numerically investigate the optimization step discussed in Sec. 4.2. As discussed in this section, the proposed alternate optimization encounters a number of local minima: Fig. 6 (showing the quantizers obtained by 2 distinct random restarts of the optimization algorithm) demonstrates the possibility of convergence to distinct local minima.While the arrangement in Fig. 6.a has 6 centroids, the one in Fig 6.b has 7, though both of them score similar performance for a standard Gaussian source distribution. Regarding the structure of the arrangement, however, these two configurations are rather different, and the proposed algorithm is unable to go from the configuration in Fig.6.a to the slightly better configuration 6.b.
LBG performance comparison
Finally, we come to perhaps the most interesting numerical evaluations: the comparison between the proposed quantization approach and the classic LBG algorithm. Such comparison indeed addresses the observation in Fig. 1 which is the initial motivation behind the proposed approach. This section presents the performance of the quantizer described in Sec. 3 with a configuration resulting from the algorithm in Sec. 4 for the case of (i) standard Gaussian and (ii) unitary uniform distribution. We obtained similar performance results in both cases. In all cases, we compared the performance attained with the performance of the quantizer obtained by the classic LBG algorithm.
The code for this algorithm is available online: see [44].
Gaussian distribution
As one can observe on Fig. 8a, the quantizer designed by our algorithm performs better than the quantizer design using LBG algorithm. For a quantizer using 5 comparators and an input signal in R 2 , its distortion is 0.64 that of LBG quantizer. Fig. 7 presents the performance of the output of LBG algorithm, compared to that of the proposed algorithm. One can observe there that the quantizer designed by the proposed algorithm performs better than the quantizer designed by LBG algorithm in every configuration represented on the figure.
As could be expected, the quantizer obtained with the proposed algorithm performs slightly worse than the optimal quantizer obtained with LBG algorithm using the same number of reconstruction points (ie the same output bit rate). This shows that using the CLVQ framework allows for quantizer designs that cost little degradation in performance in the classic quantization (where the constraint is on the cardinality of the output), but generate important performance gains in the view of comparison-limited quantization.
A result that may be surprising and seems counter-intuitive is that the hyperplanes do not always form as many regions as possible. Curiously, the hyperplanes sometimes converge to positions in which they form less regions than the maximum possible, but with more regular shapes, which implies a lesser probability of values at a long distance from the representation point, thus a lower MSE. Fig. 6.b is an example of such configuration. These configurations turn out to perform relatively well and sometimes better than configurations with more regions.
Uniform distribution
Running the algorithm with a unitary uniform distribution as source distribution gives similar results as with a Gaussian distribution. In this case again, our algorithm performs better than LBG with the same number of comparators. For a quantizer using 5 comparators and an input signal in R 2 , its distortion is 0.72 that of LBG quantizer. It also performs slightly worse than LBG algorithm with the same number of reconstruction points.
Because of the square shape of the distribution support, the hyperplanes are even more likely to form rectangular regions with a uniform distribution than with the Gaussian distribution, as Fig. 4 illustrates.
Entropy maximization
As commented in Sec. 3.2, one is often interested in characterizing the entropy of the quantizer output, as this indicates whether it is necessary to perform a further compression of the quantizer output. Setting as the objective as maximization of the entropy of the output, instead of minimization of the MSE, does not require one to make any significant changes to the optimization.
In our simulations, the result of the entropy maximization is rather similar to that of the MSE minimization. The most notable difference in using entropy as a measure is that in cases like Fig. 6, it favors configurations of type Fig. 6.a rather than type Fig. 6.b.
Conclusion
In this paper, a novel paradigm for vector quantization is considered, in which the performance of the vector quantizer is constrained by the number of comparators needed to obtain the quantized signal, For comparison purpose, the light blue curve shows the performance of a quantizer obtained with LBG algorithm, but with as many regions as the quantizer obtained with the proposed algorithm (using more comparators than the proposed algorithm). For each plot, the values were averaged over numerous restarts of the algorithm. instead of being constrained by the output bit rate as in the classic vector quantization problem. We study the scenario where the vector quantizer is made of k comparators that each receive a linear combination of the quantizer input and a constant bias, and produce the sign of the received signal as output. We focus on the task of finding the linear combinations and constant biases, given k, the distribution of the quantizer input and a distortion measure, so as to minimize the distortion between input and output of the quantizer.
We present an algorithm to solve this optimization problem and show the performance it attains in the case of mean squared error distortion and Gaussian and uniform i.i.d. source distributions. The proposed algorithm is composed of an initialization step and a distinct optimization step. In the initialization step, a set of possible initial configurations are produced which span the set of possible solutions. Diverse approaches are presented for this step: in particular we propose a genetic algorithm which can produce a rather good initial configuration for a linear complexity. In the optimization step, the algorithm iteratively optimized the position of the reconstruction points and the comfiguration of linear combinations received by the comparators. We perform numerical simulations for this algorithm and compare the performance attained to that of the classic Linde-Buzo-Gray quantizer. We show that one can obtain similar performance by using a smaller number of comparators than the classic approach would require.
A number of research directions remain open from this new vector quantizer architecture. In particular, we are investigating the optimal performance attainable in the limit of infinitely long vector quantizer in which the number of available comparators k and bits available to represent the quantizer inputs both grow to infinity at a given constant ratio α. This limit should result in a rather interesting generalization of the classic distortion-rate function.
Appendix A -Some notion of matroid theory
Oriented matroids are defined as combinatorial objects that can contain properties of linear dependence in vector spaces. Matroids have a natural way of being mapped to some objects such as sets of points or sets of hyperplanes, while retaining relations of alignment and coplanarity between their elements. Oriented matroids also have these properties, but add order relations to it. In consequence, convexity relations inside vector or hyperplane arrangements are also contained in oriented matroids [45]. An oriented matroid is defined as follows: for a linear functional x T ∈ R m , define the covector of x
C A (x) = sign(a T 1 x − b 1 ), . . . , sign(a T n x − b n ) ,(A1)
and let us define the set of all possible covectors of A as
L A = {C A (x) : x ∈ R m } ,(A2)
where the sign takes value zero when a T i x = b i , so that the support of sign is {−1, 0, +1}. The set M = −([n], L A ) is defined as the oriented matroid associated to A. Using the oriented matroid framework, one is able to state questions such as the what is the set of covectors that can be generated through the arrangement of n hyperplanes in dimension m. Regrettably, Vakil showed [46] using Mnëv's universality theorem [47] that most of the stimulating problems that could be framed with oriented matroids have no computationally efficient way to be solved.
Appendix B -Graph representation
In order to efficiently describe various combinations of hyperplane arrangements, we have used graphs to represent them. The object we need to work with has to be able to represent a hyperplane arrangement's structure rather than one particular arrangement. Thus, the object has to be invariant to Figure A1. The left hand side is an arrangement of 3 hyperplanes that form 7 regions, with the grey circle representing infinity. The right hand side is the corresponding graph (7 vertices for the 7 regions, and one for the sphere). rotation, translation and scaling and to any hyperplane modification that does not change the overall structure of the arrangement. A satisfying representation is the one shown in Fig. A1.
The interpretation for Fig. A1 is the following: regions of the hyperplane arrangement (left side) are individually associated to nodes in the graph (right side). If two regions in the arrangement share an edge, the related nodes are connected on the graph (the color of the edge on the graph corresponds to that of the hyperplane in the arrangement). In consequence, the number of edges of a region is translated into the degree of the corresponding node. Projectivization is used in order to accurately represent unbounded regions. A finite hypersphere is drawn around the zone of interest, and the space R m is projected onto that hypersphere so that coordinates at infinity are mapped to its edge. Then, the edge of the sphere is seen as an edge of the infinite regions. This way, regions of infinite size can be fully characterized. Another way to describe this representation is to define it as the dual graph of the graph that has hyperplane intersections as vertices and the hyperplanes themselves as edges. The same reasoning for using projectivization to characterize regions can be applied in a similar way.
Note that graph representation in Fig. A1 is substantially a representation of the oriented matroid in App. A. Indeed, given a label for each region, these labels can be mapped to the set of all possible covectors L A of the arrangement. In order to obtain such region labels, one can proceed as follows. First, give an arbitrary orientation to the hyperplanes. Each hyperplane defines two half-spaces, one of which is attributed a positive label and the other one a negative label. Then, each region can be attributed a unique label by concatenating the positive or negative labels given by each hyperplane (depending on whether the region is on the positive or negative side of the hyperplane). Thus, each region receives a unique label composed of a number of binary variables equal to the number of hyperplanes in the arrangement. Fig. A2 shows an example of such labeling process for an arrangement of 3 hyperplanes. Figure A2. Illustration of the labeling process. In this arrangement, each of the 3 hyperplanes is oriented: it has a positive side (indicated by an arrow) and a negative side. Each region is identified in a unique manner by its location relative to each hyperplane. If it is on the positive side (resp. negative side) of the k th hyperplane, it has a "+" (resp. "-") in the k th position of its label.
Appendix C -Progressive arrangement growth
By numerical experimentation, it is easy to observe that the optimization step which relies on random initialization in Sec. 4.1.1 has a high risk of being stuck in a local minimum. In order to alleviate this problem, we have considered a second approach to linear comparator initialization that makes use of the notions described in App. A and of the graph representation described in App. B. More specifically, (i) we begin by considering a simple graph with two vertices and one edge, representing two regions separated by one hyperplane then, recursively, (ii) one generates a new set of graphs by considering all the possible ways in which a new configuration can be generated from the old one adding one more hyperplane.
The recursive repetition of step (ii) above results in an exponentially growing number of configurations. Moreover, this process is susceptible to produce duplicates or equivalent configurations. To detect and remove duplicate configurations, one can consider verifying if the graph representations of two configurations are isomorphic. Verifying graph isomorphisms is a well studied problem that can be solved in logarithmic time in the case of planar graphs (which is the case of representation graphs for arrangements in 2 dimensions). However it is in sub-exponential time for non-planar graphs (and thus for higher dimension arrangements). This presents an obstacle to the problem of determining all possible hyperplane configurations. More generally, there exist two other obstacles that prevent one from generating all possible initial configurations and optimize each of them. First, the task of generating all these configurations is computationally very heavy. This is even more true in higher dimensions, as complexity grows super-exponentially with the number of dimensions. Second, the task of converting a hyperplane arrangement structure (that is a graph representation in our case) into an actual hyperplane arrangement with numerical values is highly complex, as is implied by Mnëv's universality theorem [47]. The high computational complexity of these two steps cause the approach not to be applicable in practice.
Figure 1 .
1Schematic representation of the vector quantization in which two input symbol are mapped into reconstruction points. In (a), three reconstruction points are obtained using three comparators while, in (b) seven reconstruction points are obtained from three comparators
Figure 2 .
2The rate-distortion problem in Sec. 2.1.
Algorithm 1
1Max-Lloyd/k-means clustering/Linde-Buzo-Gray (LBG) Algorithm 1: Input: source distribution P X n , number of reconstruction points M = 2 nR , maximum number of iterations T max 2: Output: position of the reconstruction points { c m } m∈[M] , Voronoi region of each reconstruction point {V m } m∈[M] 3: randomly pick the initial reconstruction points
position of the reconstruction points {c m (t)} m∈[M] as
Figure 3 .
3The Comparison-Limited Vector Quantization (CLVQ) problem with k = 4 quantizers. Lemma 1. [40, Th. 1.2]. A hyperplane arrangement of size n in R m divides R m into at most r(m, n) region for
Input: T max , M and k < M ∈ N, a set {A 0,m } m∈[M] of random hyperplane arrangements, a fixed parameter 2: Output: one hyperplane arrangement with low distortion 3: for t in [T max ] do 4: order {A t,m } m∈[M] (A t,m 1 , A t,m 2 ) i (read below about the 2 existing policies to chose (m 1 , m 2 ) pairs) 7:
t be a vector of d realisations of N (0, σ t ) (with σ t a parameter decreasing over t)for all i, H i,t ← H i,t + V i,t else select one hyperplane H of A t for i in [d] dolet p i be the i th coefficient of H and compute distortion for alternate values of p i interpolate the function f : p i → distortion p i ← min( f ) p i ∈R end for end if end for
Figure 4 .
4Hyperplane arrangement at different steps of the optimization algorithm. The source distribution is unitary uniform and its support is represented by the grey square. The crosses show the centroids of each region. The regions are becoming more and more regular over iterations.
Two random restarts of the genetic algorithm in gray and their average in blue, on 30 iterations, Two random restarts of the genetic algorithm in gray and their average in blue on 30 iterations, 4 hyperplanes in 3 dimensions.
Figure 5 .
5Genetic algorithm evolution.
Figure 6 .
62 local minima obtained with the algorithm in Sec. 4. The center of the Gaussian source distribution is marked by the grey dot, and the centroid of each region is indicated by a cross.
Figure 7 .
7Diagram showing the average distortion (Mean Squared-Error here) values attained by the classic LBG algorithm (green) and the proposed algorithm (blue) on a Gaussian input signal. The number d of dimensions ranges from 2 to 6, and the number of regions (for LBG) or hyperplanes (for the proposed algorithm) ranges from d to 8. The results are averaged over several random restarts of the algorithms. For better readability, these results are spread on a X-axis representing the ratio between number of hyperplanes or regions, and dimension.
quantizer (for comparison) LBG, same number of comparators proposed algorithm (b) Quantizer performance for a uniform unitarian input distribution in R 2 .
Figure 8 .
8The red curve shows the distortion of the proposed algorithm and the dark blue curve shows the distortion from the quantizer of LBG algorithm, in function of the number of comparators available.
: order {A t,m } m∈[M] by decreasing fitness {A T max ,m } m∈[M] )11:
for m ∈ [M] do
12:
if m < M
2 then
13:
A t+1,m ← A t,m
14:
else
15:
A t+1,m ← mutate(A t,m )
16:
end if
17:
end for
18: end for
19: select max(
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Acknowledgments:We would like to give thanks to Arnau Padrol from Sorbonne Université for his help on understanding oriented matroids.Funding:The work of S. Rini is supported by the MOST grant number 109 -2927-I-009 -512.Conflicts of Interest:The authors declare no conflict of interest.AbbreviationsThe following abbreviations are used in this manuscript:MSEMean Squared Error LBG Linde-Buzo-Gray algorithm MIMO Massive Input Massive Output GP General Position A2DAnalog-to-Digital VQ Vector Quantization CLVQ Comparison-Limited Vector Quantization
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[] |
[
"Elaborating Transition Interface Sampling Methods",
"Elaborating Transition Interface Sampling Methods"
] |
[
"Titus S Van Erp \nLaboratoire de Physique / Centre Européen de Calcul Atomique et Moléculaire\nEcole Normale Supérieure de Lyon\n46 allée d'Italie69364, Cedex 07LyonFrance\n",
"Peter G Bolhuis \nDepartment of Chemical Engineering\nUniversiteit van Amsterdam\nNieuwe Achtergracht 1661018 WVAmsterdamThe Netherlands\n"
] |
[
"Laboratoire de Physique / Centre Européen de Calcul Atomique et Moléculaire\nEcole Normale Supérieure de Lyon\n46 allée d'Italie69364, Cedex 07LyonFrance",
"Department of Chemical Engineering\nUniversiteit van Amsterdam\nNieuwe Achtergracht 1661018 WVAmsterdamThe Netherlands"
] |
[] |
We review two recently developed efficient methods for calculating rate constants of processes dominated by rare events in high-dimensional complex systems. The first is transition interface sampling (TIS), based on the measurement of effective fluxes through hypersurfaces in phase space. TIS improves efficiency with respect to standard transition path sampling (TPS) rate constant techniques, because it allows a variable path length and is less sensitive to recrossings. The second method is the partial path version of TIS. Developed for diffusive processes, it exploits the loss of long time correlation. We discuss the relation between the new techniques and the standard reactive flux methods in detail. Path sampling algorithms can suffer from ergodicity problems, and we introduce several new techniques to alleviate these problems, notably path swapping, stochastic configurational bias Monte Carlo shooting moves and order-parameter free path sampling. In addition, we give algorithms to calculate other interesting properties from path ensembles besides rate constants, such as activation energies and reaction mechanisms.
|
10.1016/j.jcp.2004.11.003
|
[
"https://arxiv.org/pdf/cond-mat/0405116v1.pdf"
] | 119,008,878 |
cond-mat/0405116
|
6c029c0f9389b7b1d66aca2e0bb71852addc5de9
|
Elaborating Transition Interface Sampling Methods
6 May 2004
Titus S Van Erp
Laboratoire de Physique / Centre Européen de Calcul Atomique et Moléculaire
Ecole Normale Supérieure de Lyon
46 allée d'Italie69364, Cedex 07LyonFrance
Peter G Bolhuis
Department of Chemical Engineering
Universiteit van Amsterdam
Nieuwe Achtergracht 1661018 WVAmsterdamThe Netherlands
Elaborating Transition Interface Sampling Methods
6 May 2004Preprint submitted to Elsevier Science 25 October 2018(Titus S. van Erp ), [email protected] (Peter G. Bolhuis).rare eventsreaction ratetransition path samplingtransition interface sampling PACS: 8220-w3115Qg0510Ln * Corresponding author
We review two recently developed efficient methods for calculating rate constants of processes dominated by rare events in high-dimensional complex systems. The first is transition interface sampling (TIS), based on the measurement of effective fluxes through hypersurfaces in phase space. TIS improves efficiency with respect to standard transition path sampling (TPS) rate constant techniques, because it allows a variable path length and is less sensitive to recrossings. The second method is the partial path version of TIS. Developed for diffusive processes, it exploits the loss of long time correlation. We discuss the relation between the new techniques and the standard reactive flux methods in detail. Path sampling algorithms can suffer from ergodicity problems, and we introduce several new techniques to alleviate these problems, notably path swapping, stochastic configurational bias Monte Carlo shooting moves and order-parameter free path sampling. In addition, we give algorithms to calculate other interesting properties from path ensembles besides rate constants, such as activation energies and reaction mechanisms.
Introduction
Molecular simulation has become indispensable as a modern tool to gain insight in the kinetics of processes in complex environment by supplying detailed atomistic information that is not (easily) experimentally accessible. Using either classical or ab initio based atomistic force fields [1,2], techniques such as molecular dynamics (MD) [3,4] can model reactive events on a reasonable realistic level. In contrast to most experiments where kinetic properties such as the reaction rate are obtained by measuring the macroscopic population densities of reactant and product states over a long time (seconds), molecular dynamics simulations have to obtain good statistics with much smaller systems (usually ∼ 100 to 100000 molecules) in the accessible time range of nanoseconds-microseconds using a time step of a few femtoseconds, as dictated by the molecular vibrations. This small timescale and system size limits the application to activated processes with relatively low barriers between reactant and product states. The computation of rate constants with straightforward MD becomes inefficient when the process of interest has to overcome a high activation barrier because the probability to observe a reactive event on this time-and system-scale decreases exponentially with the barrier height. The system will spend a long time in one of the stable states and occasionally jump -in relatively short time-to the other state. This separation of time scales results in two state kinetics: the exponential relaxation of the population densities [5].
The time-scale problem is traditionally solved by a two-step reactive flux method [6,7,8,9]. One first calculates the free energy as a function of a reaction coordinate describing the process. The transition state theory (TST) rate constant is then related to the probability to be at the maximum of the free energy barrier. This rate is only an approximation and the second part of the reactive flux methods computes the correction, the transmission coefficient, by starting many fleeting trajectories from the top of the barrier [6,7,8,9]. However, the success of this method depends strongly on the choice of reaction coordinate. If the reaction coordinate fails to capture the molecular mechanism the corresponding transmission coefficient will be extremely low, making an accurate evaluation of the rate problematic if not impossible. For high dimensional complex systems, for instance chemical reactions in solution, or protein folding, a good reaction coordinate can be extremely difficult to find and usually requires detailed a priori knowledge of the transition mechanism. Hence, TST based reactive flux methods will be ineffective for complex processes for which no prior knowledge is available.
Chandler and collaborators [10,11,12,13,14] devised a method for which no reaction coordinate is needed, but only a definition of the reactant and product state. This method, called transition path sampling (TPS), gathers a collec-tion of trajectories connecting the reactant to the product stable region by employing a Monte Carlo (MC) procedure called shooting and shifting. The resulting path ensemble gives an unbiased insight in the mechanism of the reaction. TPS has been successfully applied to such diverse systems as cluster isomerization, auto-dissociation of water, ion pair dissociation and on isomerization of a dipeptide, as well a reactions in aqueous solution (see Ref. [13] for an overview). A drawback of TPS is that the calculation of rate constants is rather computer time consuming. We therefore developed the more efficient transition interface sampling (TIS) method [15]. TIS allows a variable path length, thereby limiting the required MD time steps to the strict necessary minimum. The TIS rate equation is based on an effective positive flux formalism and is less sensitive to recrossings. The shifting moves used in TPS to enhance statistics, are unnecessary in the TIS algorithm. Also, multidimensional or even discrete order parameters can easily be implemented in TIS. Recently, we showed that for diffusive processes one can exploit the loss of correlation along trajectories. This lead to the development of the partial path TIS (PPTIS) method, a variation of TIS that samples much shorter paths [16].
In this paper we re-derive the basic concepts of TIS and PPTIS in a more intuitive way and relate them to the calculation of the transmission coefficient. For the mathematical validation of the expressions we refer to Refs. [15,16]. The paper is organized as follows. In section II we discuss the relation between several different microscopic expressions for the phenomenological rate constant present in the literature, and derive the positive effective flux formalism on which both interface path sampling methods are based. In section III we present the TIS and PPTIS formalism and precise algorithm. In section IV, we introduce new algorithms for alleviating ergodicity problems that might occur in path sampling simulations. The last section V, is reserved for new ways of extracting interesting properties from path ensembles, such as the activation energy of a reaction. We end with concluding remarks.
Microscopic rate equations
The calculation of reaction rate constants by computer simulation requires an expression for the rate constant in terms of microscopic properties. Such a microscopic rate expression needs a proper characterization of the reactant state A and product state B for each separate reaction, but should not be too sensitive to these state definitions, otherwise an unrealistic ill-defined rate will result. Once we have a rate expression, there are several ways to compute the reaction rate. The standard reactive flux method measures the flux through a single hypersurface in phase-space dividing the reactant state A from the product state B. In TPS the rate constant is taken from a time derivative of a correlation function, which can be calculated by slowly confining a completely free path ensemble to an ensemble that connects reactant to product. The TIS approach measures a reactive flux through many interfaces between A and B. These three methods can be related to each other, as they ultimately compute the same properties. The TPS correlation function at t = 0 becomes equivalent to the TST approach when A and B are adjacent in phase space [14]. The TIS effective positive flux formalism for a single interface is equivalent to TST-based transmission coefficient calculations [15]. The TIS rate equation can also be recast in terms of a TPS-like correlation function but then based on the so-called overall states of the system. In the following subsections, we will explain the reactive flux, TPS, and TIS methods and their connections in detail.
Transition state theory
The first step in TST is to choose a reaction coordinate λ describing the transition from a stable reactant state A to a stable product state B. This reaction coordinate can be any function λ(x) of phase space point x ≡ {r, p}, with r the particle coordinates and p the momenta. Next, the free energy F (λ) = −k B T ln(P (λ)) is calculated by determining the probability P (λ) to be at λ using, for instance, biased sampling techniques [17,18,19]. Here, k B is the Boltzmann constant and T is the temperature. The maximum λ * in F (λ) defines the dividing surface {x|λ(x) = λ * } separating state A from state B. By convention, the system is in A if λ(x) < λ * and in B if λ(x) > λ * . For a phase point x in A, the probability to be at the top of the barrier is:
P (λ * ) x∈A ≡ δ(λ(x) − λ * ) θ λ * − λ(x) = e −βF (λ * ) λ * −∞ dλ e −βF (λ) ,(1)
where the brackets . . . denote the equilibrium ensemble averages, θ(x) and δ(x) are the Heaviside step-function and the Dirac delta function, respectively, and β = (k B T ) −1 . TST assumes that trajectories that cross λ * do not recross the dividing surface Hence, the TST expression is equivalent to the positive flux through the dividing surface λ * :
k T ST AB = λ (x)θ λ (x) λ * P (λ * ) x∈A ,(2)
where the dot denotes a time derivative and the subscript λ * to the ensemble brackets indicates that the ensemble is constrained to the top of the barrier on the dividing surface λ * . The TST rate constant is sensitive to the choice of reaction coordinate λ(x) and will only be correct if the surface {x|λ(x) = λ * } corresponds to the true transition state dividing surface: the so-called separatrix. For complex systems, it is impossible to know the location and shape of this curved multidimensional separatrix. It is possible, however, to correct the TST expression with a dynamical factor that is called the transmission coefficient.
Transmission coefficients
Traditionally, the dynamical corrected rate constant is derived by applying a small perturbation to the equilibrium state and invoking the fluctuationdissipation theorem [20,5,4]. This leads, for instance, to the well known Bennett-Chandler (BC) [8,9] expression for the reaction rate
k BC AB (t) = λ (x 0 )δ λ(x 0 ) − λ * θ λ(x t ) − λ * θ λ * − λ(x 0 )(3)
where x t specifies the coordinates and momenta of the system at time t as obtained from a short molecular dynamics (MD) trajectory starting at x 0 . The ensemble average is taken over all phase points x 0 . For exponentially relaxing two state kinetics with a well defined rate constant, there is a separation of timescales: the reaction time τ rxn (or expectation time for one single event) is much longer than the molecular time τ mol that the system spends on the barrier. In that case, Eq. (3) will reach a plateau value for τ mol ≪ t ≪ τ rxn , which is equal to the correct phenomenological rate constant k AB . The function k BC AB (t) will sensitively depend on the choice of the reaction coordinate λ, but the plateau value will not. In the limit t → 0 + , the BC rate reduces to the TST expression Eq. (2) The transmission coefficient is defined as the ratio between the real rate constant and the TST expression: κ ≡ k AB /k T ST AB :
κ BC (t) = λ (x 0 )θ λ(x t ) − λ * λ * λ (x 0 )θ λ (x 0 ) λ *(4)
The numerator in Eq. (4) counts trajectories with a positive but also with a negative weight. The latter trajectories leave the surface at t = 0 with a negative velocityλ(x 0 ), but are eventually found at the B side of the surface after a (few) recrossing(s). However, untrue B → B trajectories do not contribute to the rate because the positive and negative terms cancel 1 (See Fig. 1 counted only once [15]. Although Eq. (3) gives the correct rate constant, it is rather unsatisfactory to sample only trajectories forward in time not knowing which contribute to the rate and which do not. Therefore, alternative expressions for the rate constant have been proposed taking the past into account. Here, they are referred to as the BC2 [8,9] expression
κ BC2 (t) = λ (x 0 )θ λ * − λ(x −t ) θ λ(x t ) − λ * λ * λ (x 0 )θ λ (x 0 ) λ *(5)
and the positive flux PF [21] expression
κ pf (t) = λ (x 0 )θ λ (x 0 ) θ λ(x t ) − λ * λ * λ (x 0 )θ λ (x 0 ) λ * − λ (x 0 )θ λ (x 0 ) θ λ(x −t ) − λ * λ * λ (x 0 )θ λ (x 0 ) λ *(6)
In Eq. (5) the theta functions guarantee that only true A → B events are counted. Still, the numerator in Eq. (5) contains negative terms: those phase points x 0 with a negative velocityλ(x 0 ) and with corresponding backward surface but with a different velocity. Still, this is the case. The absolute value of the flux of a trajectory is at each intersecting surface the same. A lower crossing velocityλ is compensated by a higher probability to measure the crossing point as the trajectory spends more time at the surface. and forward trajectory that ends up in A and B, respectively. Eq. (6) counts only positive crossings, but cancellation with a negative term can occur when the backward trajectory also ends up at the B side of the dividing surface. At first sight, Eq. (6) seems to overcount A → B trajectories with multiple λ * crossings. However, if one realizes that each A → B trajectory has an equivalent trajectory B → A by reversing the time, an overall cancellation of positive and negative terms ensures a proper final outcome.
For completeness, we mention that there are also similar expressions by Berne [22,23] and a relation by Hummer [24] that counts both positive and negative crossings with a positive weight, but only if the corresponding trajectory ends at opposite sides of the surface and with a weight lower than |λ| if its trajectory has more than just one crossing. Ruiz-Montero et al. designed a transition zone method in which they measure the flux on many places at the top of the barrier and weight them to the inverse free energy [25].
The effective positive flux formalism
A more intuitive, yet sound, alternative to the above expressions is the effective flux formalism. We can illustrate this formalism with an analogy to the migration of people from country A to B. To determine the emigration rate we can simply count the number of persons that cross the border from A to B within a certain time interval. However, we should not count tourists. This group consist of people who have a nationality A and will only spend a short time in B, or have a nationality B and are actually on their way back. Moreover, we have to be aware that some emigrants might cross the frontier several times on their way. To prevent overcounting, we should only count one specified crossing for each person, for instance, the first or the last crossing of the emigration journeys from A to B. The same reasoning can be applied when calculating the rate constant of a reaction. In a molecular simulation we can check the 'nationality' of the system and the one-crossing condition by simply following the equations of motions backward and forward in time. This procedure, to count only true events and to avoid counting recrossings is what we call the effective positive flux formalism. In Sec. 2.5 we give the mathematical expression of the effective positive flux.
It is surprising that the effective positive flux counting strategy is not so common. To our knowledge only two slightly different expressions of a transmission coefficient based on the effective positive flux have been proposed in Refs. [26,27]. In all other expressions found in the literature the counting of recrossings is not avoided, but the final rate constant follows through cancellation of many negative and positive terms. The effective flux transmission coefficients formulation is most useful when applying a single dividing surface and when recrossings are apparent [23]. In general, we note that any averaging method counting only zero and positive values will show a faster convergence than one that is based on cancellation of positive en negative terms. Moreover, in the effective flux formalism many trajectories will be assigned as unreactive after just a few MD steps (See Fig. 1), thus reducing the number of required force evaluations. A comparative study of ion channel diffusion [23] showed that the algorithm based on effective positive flux expression of Anderson [26] was superior to the other transmission rate expressions. Moreover, it was found as efficient as an optimized version of the more complicated Ruiz-Montero method [25].
TPS correlation function
In TPS one also has to define an order parameter λ(x), but this does not have to be a properly chosen reaction coordinate capturing the essence of the dynamical mechanism. Instead, it is sufficient but necessary that this function is able to characterize the basins of attraction of the stable states [13]. By
definition the system is in A if λ(x) < λ A and in B if λ(x) > λ B with λ A < λ B .
Clearly, the two states are not connected and the intermediate barrier region, belongs neither to A nor to B. By introducing following characteristic functions
h A (x) = 1, if x ∈ A, else h A (x) = 0 h B (x) = 1, if x ∈ B, else h B (x) = 0.(7)
the TPS-correlation function is defined as:
C(t) = h A (x 0 )h B (x t ) h A (x 0 ) .(8)
If there is a separation of timescales, this correlation function grows linearly in time, C(t) ∼ k AB t, for times τ mol < t < τ rxn In that case, the time dependent reaction rate
k T P S AB (t) = d dt C(t)(9)
reaches a plateau for τ mol < t < τ rxn . C(t) can be calculated in a path sampling simulation employing the shooting and shifting Monte Carlo moves, in combination with an umbrella sampling algorithm in which the final region B is slowly shrunk from the entire phase space to the final stable state B [14].
The disadvantage of such a procedure is that it can take a relatively long time τ mol before C(t) reaches a plateau (longer than in a transmission coefficient calculation [14]).
All paths in the path sampling should have a minimal length T > τ mol and as a result unnecessarily long periods are spent inside the stable state basins of attraction. Moreover, inspection of Eqs. (8) and (9) shows that a necessary cancellation of positive and negative terms can slow down the convergence of the MC sampling procedure. In the case of adjacent A and B regions, the TPS formalism becomes equivalent to the TST approximation in the limit t → 0 [14].
The road to TIS
The TIS method is based on the measurement of the fluxes though multiple dividing surfaces. Consider a set of n + 1 non-intersecting multidimensional interfaces {0, 1 . . . n} described by an order parameter λ(x) that does not have to correspond to the real reaction coordinate. We choose λ i , i = 0 . . . n such that λ i−1 < λ i , and that the boundaries of state A and B are described by λ 0 and λ n , respectively. For each phase point x and each interface i, we define a backward time t b i (x) and forward time t f i (x):
t b i (x 0 ) ≡ − max [{t|λ(x t ) = λ i ∧ t ≤ 0}] t f i (x 0 ) ≡ + min [{t|λ(x t ) = λ i ∧ t ≥ 0}] ,(10)
which mark the points of first crossing with interface i on a backward (forward) trajectory starting in x 0 . Note that t b i and t f i defined in this way always have positive values. Following Ref. [15], we then introduce two-fold characteristic functions that depend on two interfaces i = j,
h b i,j (x) = 1 if t b i (x) < t b j (x), 0 otherwise ,h f i,j (x) = 1 if t f i (x) < t f j (x), 0 otherwise(11)
which measure whether the backward (forward) time evolution of x will reach interface i before j or not. However, as the interfaces do not intersect, the time evolution has to be evaluated only for those phase points x that are in between the two interfaces i and j. In case i < j, we know in advance that
t b,f i (x) < t b,f j (x) if λ(x) < λ i and t b,f i (x) > t b,f j (x) if λ(x) > λ j .
When the system is ergodic, both interfaces i and j will be crossed in finite time and
thush b i,j (x) +h b j,i (x) =h f i,j (x) +h f j,i (x) = 1.
The two backward characteristic functions define the TIS overall states A and B:
h A (x) =h b 0,n (x), h B (x) =h b n,0 (x).(12)
Together, the overall states cover the entire phase space and, within certain limits, do not sensitively depend on the precise boundaries of stable states A and B. With these new characteristic functions we can write down a correlation function similar to Eq. (8):
C(t) = h A (x 0 )h B (x t ) h A (x 0 ) ,(13)
This correlation function exhibits a linear regime ∼ k AB t for 0 < t < τ rxn . Therefore, we can simply take the time derivative at t = 0 yielding
k AB = h b 0,n (x 0 )λ(x 0 )δ(λ(x 0 ) − λ n ) h A (x 0 ) .(14)
One can easily verify that here only positive terms contribute to the rate. The connection to the transmission coefficient can be made by using following relation [15]:
h b i,kλ δ(λ(x) − λ k ) = h b i,jλ δ(λ(x) − λ j )h f k,i(15)
for λ i < λ j < λ k . Using this equality, we can write down a transmission coefficient similar to the ones in Sec. 2.2 but then based on the effective positive flux [27]:
κ T IS = h b 0,i (x 0 )λ(x 0 )θ λ (x 0 ) h f n,0 (x 0 ) λ i λ (x 0 )θ λ (x 0 ) λ i(16)for λ i = λ * . Although, in principle θ λ (x 0 ) is redundant in the numerator of Eq. (16) ash b 0,i (x 0 ) = 0 ifλ(x 0 ) < 0,
it is there to highlight that only positive crossings are counted. Trajectories started at x 0 on interface i are followed backward in time until they reach stable region A or recross interface i. Then, only the ones that reach stable region A are also followed forward in time until they reach one of the stable regions. The slightly different effective flux expression of Ref. [26] follows trajectories until reaching the plateau region time and counts for each A → B trajectory only the last crossing instead of the first.
(Partial path) transition interface sampling
Formalism
In a system for which the correct reaction coordinate λ is known in advance and that is not dominated by recrossings, the effective positive flux formalism of (Eq. (16) and Ref. [26]) is probably the best choice when using a single dividing surface [23]. However, for complex systems, for instance chemical reactions in solution, any intuitively chosen reaction coordinate can give arbitrary small transmission coefficients, making an accurate computation prohibitive. To improve reaction coordinates by e. g. taking solvent degrees into account is generally a difficult job. Some progress has been made by using the coordination number as reaction coordinate [28,29], but this ad hoc approach probably only works for specific systems. For instance, we showed that a proton transfer reaction in water depends very sensitively on the angular orientation of the surrounding water molecules [30]. Similarly, the degrees of freedom in a protein are so large that dynamical folding processes are at best only very qualitatively described by order parameters. Quantities such as radius of gyration or number of native contact do usually not correspond to reaction coordinates [31]. Subtle effects, e.g. the solvent structure, play also here a role. To incorporate all these subtleties in a single one-dimensional reaction coordinate is an immense task and can only be successful if the precise reaction mechanism is already known in advance. The TPS and TIS techniques do not rely on a reaction coordinate. The TIS hypersurfaces do not have to coincide with the transition state dividing surface.
At the end of this section we give TIS (and PPTIS) rate expressions that can be employed in a computer algorithm. First, as the derivation of the TIS and PPTIS formalism requires a proper notation, we introduce following flux function
φ ij (x) ≡h b j,i (x)|λ(x)|δ(λ(x) − λ i ) =h b j,i (x) lim ∆t→0 1 ∆t θ ∆t − t f i (x)(17)
The first equality has the same flux notation as Eq. (14), but the second equality is more useful in practice. An MD trajectory might cross interface λ i , but consists of discrete time slices that are never exactly on the surface (as opposed to a transmission coefficient calculation). However, φ ij (x) can still be defined for the discrete MD set of time-slices by taking ∆t equal to the molecular time-step. In words, φ ij (x) equals 1/∆t if the forward trajectory crosses λ i in one single ∆t time-step and the backward trajectory crosses λ j before λ i . Otherwise φ ij (x) vanishes. In addition, we introduce a flux function that incorporates also the forward trajectory
Φ lm ij (x) ≡ φ ij (x)h f l,m (x)(18)
By making use of Eq. (15) we can write for λ i < λ j < λ k :
φ ki (x) = Φ ki ji (x)(19)
and, thus, the rate constant (14) becomes
k AB = φ n,0 / h A = Φ n,0 i,0 / h A(20)
for each λ i with 0 ≤ i ≤ n.
The second step is to define a conditional crossing probability that depends on the location of four interfaces:
P ( l m | i j ) ≡ Φ lm ij / φ ij .(21)
In words, this is the probability for the system to reach interface l before m under the condition that it crosses at t = 0 interface i, while coming directly from interface j in the past (see Fig. 2). The probabilities in Eq. (21) are the building blocks for both TIS as PPTIS to construct expressions for the rate constant. The probabilities P ( l m | i j ) are defined on any set of four interfaces. The case, where m = j = 0 and m = j = n, is of special interest for TIS and will be annotated as follows
P A (λ j |λ i ) ≡ P ( j 0 | i 0 ), P B (λ j |λ i ) ≡ P ( j n | i n )(22)
For PPTIS, two types of crossing probabilities are required: the one interface crossing probabilities
λ m λ j λ i λ l Fig. 2.
The conditional crossing probability P ( l m | i j ) for a certain configuration of interfaces λ i , λ j , λ l , and λ m . The condition | i j ) is depicted by the arrow and the solid line for two phase points (the dots): from this phase point one should cross λ i in one single ∆t time-step in the forward direction, and, besides, its backward trajectory should cross λ j before λ i . Two possible forward trajectories are given by the dashed line. The upper crosses λ m before λ l , the lower crosses λ l as first. The fraction whose forward trajectories behave like the last case equals P ( l m | i j ).
p ± i ≡ P ( i+1 i−1 | i i−1 ), p ∓ i ≡ P ( i−1 i+1 | i i+1 ) p = i ≡ P ( i−1 i+1 | i i−1 ), p ‡ i ≡ P ( i+1 i−1 | i i+1 ),(23)
and the long distance crossing probabilities
P + i ≡ P ( i 0 | 1 0 ), P − i ≡ P ( 0 i | i−1 i ).(24)
Using these probabilities, the TIS rate constant can be written in terms that can be determined in a computer simulation [15]
k AB = φ 1,0 h A P A (λ n |λ 1 ) P A (λ n |λ 1 ) = n−1 i=1 P A (λ i+1 |λ i )(25)
The first factor φ 1,0 h A is a flux and can be calculated by straightforward MD as λ 1 will be close to A (see Sec. 3.2). The second factor, the crossing probability P A (λ n |λ 1 ), is calculated using the factorization in Eq. (25) into probabilities P A (λ i+1 |λ i ) that are much higher than the overall crossing probability. These can be calculated using the shooting algorithm as will be explained in Sec. 3.3.
For PPTIS the set of equations are as follows [16]:
k AB = φ 1,0 h A P + n , k BA = φ n−1,n h B P − n (26) P + j ≈ p ± j−1 P + j−1 p ± j−1 + p = j−1 P − j−1 , P − j ≈ p ∓ j−1 P − j−1 p ± j−1 + p = j−1 P − j−1(27)
The factor φ 1,0 h A is identical to the TIS flux factor, whereas to obtain the reverse rate k BA only a single extra factor φ n−1,n h B is needed. The P + n and P − n are obtained via the recursive relations (27) once all single crossing probabilities of Eq. (23) are known. Starting with P + 1 = P − 1 = 1, we can iteratively determine (P + j , P − j ) for j = 2, . . . until j = n. The one-hopping probabilities (23) can again be calculated using the shooting algorithm. The PPTIS formalism basically transforms the process of interest into a Markovian sequence of hopping events. Yet, if the dynamics is diffusive and the interfaces are sufficient far apart, the rate formalism (26) and (27) will be almost exact [16].
The flux algorithm
The flux factor φ 1,0 h A is the effective flux through λ 1 of the trajectories coming from λ 0 (from A). This factor is most conveniently computed with the first two interfaces identical. Although φ 1,0 h A is not well defined for λ 1 = λ 0 , we can think that λ 1 = λ 0 + ǫ in the limit ǫ → 0. In this way, the effective positive flux will be equal to the simple positive flux through λ 1 (trajectories cannot recross without re-entering A, hence, all crossings are counted.). Similarly, for the reverse rate k BA we can set λ n−1 = λ n − ǫ. If λ 1 is chosen close enough to A the flux factor can be obtained by straightforward MD initialized in A and counting the positive crossings through λ 1 = λ 0 during the simulation run:
φ 1,0 h A = 1 ∆t N + c N MD(28)
with ∆t the MD time step, N MD the number of MD steps, and N + c the number of counted positive crossings. To calculate the rate at constant temperature instead of constant energy, one can apply a Nosé-Hoover [32,33,34,35] or Andersen [36] thermostat. However, one should be aware that these thermostats do give the correct canonical distribution at a given temperature, but modify the dynamics in an unphysical way. Hence, static averages A(x) will be correct, but time correlation functions A(x 0 )B(x t ) most likely not. As
N + c ∼ θ λ 1 − λ(x 0 ) θ λ(x ∆t ) − λ 1
is actually a correlation function over a very short time, this effect will be small. However, if necessary one can easily correct for this by explicitly counting only phase points x that in absence of the thermostat will cross λ 1 in one ∆t time-step. Applying this correction is computationally cheap as it does not require any additional force calculations. In Appendix A we describe some possibilities for further optimization of the flux algorithm.
The path sampling algorithm
To calculate the conditional probabilities in TIS and PPTIS we use a path sampling algorithm [14]. However, there are some differences with the classic TPS algorithm. Most importantly, in (PP)TIS the path length is variable, which has a small implication for the acceptance criterion for the shooting move. In appendix B we derive this acceptance rule for arbitrary (stochastic or deterministic) dynamics. The main tools in the MC sampling of trajectory space are the shooting move and the time-reversal move [14]. In particular for PPTIS time-reversal moves can be quite effective. Shifting moves that enhanced statistics in TPS are not needed and even useless in (PP)TIS.
TIS algorithm:
The quantity of interest in TIS is the crossing probability P A (λ i+1 |λ i ) (or P B (λ i−1 |λ i ) for the reverse rate constant k BA ). To calculate this probability by sampling in the λ i interface ensemble one needs an initial path that starts in A (at λ 0 ), crosses the interface λ i at least once, and finally ends by either crossing λ 0 or λ i+1 . In general one can take simply a successful path from the previous λ i−1 interface ensemble that reached λ i , and complete its evolution till reaching either A or λ i+1 . For more details on initial path generation we refer to Ref. [14]). The phase space point x 0 is then defined as the first crossing point of this path with interface λ i . It is convenient to use a discrete time index τ = int(t/∆t), and let τ b ≡ int(t b 0 (x 0 )/∆t) and
τ f ≡ int(min[t f 0 (x 0 ), t f i+1 (x 0 )]/τ ′ , with −τ b ≤ τ ′ ≤ τ f .(2)
Change all momenta of the particles at time-slice τ by adding small randomized displacements δp = δw √ m with δw taken from a Gaussian distribution with width σ w and m the mass of the particle [14].
(3) In case of constant temperature (NVT) simulations: accept the new momenta with a probability [4]:
min 1, exp β E(x (o) τ ′ ∆t ) − E(x (n) τ ′ ∆t ) .
Here, E(x) is the total energy of the system at phase space point x. In case of constant energy (NVE) simulations in which possibly also total linearor angular momentum should be conserved: rescale all the momenta of the system according to the procedure described in Ref. [37] and accept the new rescaled momenta. If the new momenta are accepted continue with step 4, else reject the whole shooting move and return to the main loop. (4) Take a uniform random number α 2 in the interval [0 : 1] and determine a maximum allowed path length for the trial move by: Finally, the probability P A (λ i+1 |λ i ) follows from:
N (n) max = int(N (o) /α 2 ).(5)P A (λ i+1 |λ i ) = N p (0 → i + 1) N p (total)(29)
with N p (0 → i + 1) the number of sampled paths that connect λ 0 with λ i+1 and N p (total) the total number sampled paths in the ensemble of interface λ i .
Time reversal moves do not require any force calculations. On the other hand two subsequent time reversals will just result in the same path. Therefore, we usually take γ = 0.5 giving shooting and time reversal move an equal probability. Similar reasoning is applied to the choice of σ w . If σ w is large, many trial moves will fail to create a proper path. On the other hand a too small value of σ w will result in almost the same path. Practice has shown that an optimal value of σ w is established when approximately 40% of the paths is accepted [12]. This will usually imply that σ w will be larger for the interfaces λ i close to A than the ones closer to B. The mass weighted momenta change at step 2 of the shooting algorithm is chosen such that the velocity rescaling at step 3 maintains detailed balance [37]. In principle, NVT simulations do not require rescaling and δp can be taken from any symmetric distribution. The integration of the equations of motion at step 5 and 6 of the shooting move are normally performed by constant energy MD simulations without using a thermostat to describe the actual dynamics as realistic as possible.
The temperature only appears at the acceptance criterion at step 3. In this algorithm we go from one phase point x by means of many MD steps. Therefore, it has a strong similarity with hybrid MC [38]. Hence, the argument that the dynamics should be time reversible and area preserving [4] should also be applied here. For this reason, we strongly advice to use the velocity Verlet [39] algorithm rather than higher order schemes such as Runga-Kutta. The maximum allowed path length N (n) max in step 4 is introduced to maintain detailed balance when sampling paths of different length and to avoid having to reject very long trial paths afterward [15].
PPTIS algorithm: the four one-interface probabilities p ± i , p = i , p ∓ i , and p ‡ i for a single interface λ i can be calculated simultaneously [16] with paths that start at λ i−1 or λ i+1 and end by crossing either λ i−1 or λ i+1 . All paths should have at least one crossing with λ i . Hence,
τ b ≡ int(min[t b i−1 (x 0 ), t b i+1 (x 0 )]/∆t) and τ f ≡ int(min[t f i−1 (x 0 ), t f i+1 (x 0 )]/∆t).
The path sampling is then identical to the TIS algorithm except that λ i−1 is used instead of λ 0 , time reversal moves are always accepted and the backward integrating at step 5 is not rejected when reaching λ i+1 as paths may start from both sides. The one-interface crossing probabilities are then given by
p ± i = N p (i − 1 → i + 1) N p (i − 1 → i + 1) + N p (i − 1 → i − 1) p ∓ i = N p (i + 1 → i − 1) N p (i + 1 → i − 1) + N p (i + 1 → i + 1) p = i = 1 − p ± i , p ‡ i = 1 − p ∓ i(30)
Defining the interfaces
The order parameter λ in TPS and TIS does not have to correspond to a reaction coordinate that captures the essence of the reaction mechanism. The only requirement is that λ can distinguish between the two basins of attraction. In TIS this occurs via the two outer interfaces λ 0 and λ n that define state A and B. The definitions of A and B are more strict than in TPS [15]. The boundaries λ 0 and λ n should be defined such that each trajectory between the stable states is a rare event for the reaction we are interested in. In addition, the probability that after this event the reverse reaction occurs shortly thereafter must be as unlikely as an entirely new event. In other words, a trajectory that starts in A and ends in B is allowed to leave region B shortly thereafter, but the chance that it re-enters region A in a short time must be highly unlikely. Sometimes it is not sufficient for a proper definition of the boundaries λ 0 and λ n to only use configuration space. In the dimer study of Ref. [15] an additional kinetic energy constraint was introduced to ensure the stability of state A and B.
The intermediate interfaces can be chosen freely and should be placed to optimize the efficiency. This is, of course, system dependent, but reasonable estimates can be made a priori. Let us write down the total computation time as CPU ∼ N W i=1 N i L i with N W the number of windows (interface ensembles), N i the number of paths in the ensemble of interface i required to obtain a desired precision ǫ i , and L i the average path length. Here, we neglect the influence of rejections and the fact that two successive pathways in the MC sequence are not completely uncorrelated. We chose the interface separations and the number of paths such that P (λ i+1 |λ i ) = p and N i = n p , resulting in ǫ i = ǫ for all i. The total error ǫ tot , that we fix, is related by
ǫ 2 tot = N W ǫ 2 with ǫ 2 ∼ (1 − p)/(pn p ). Hence, n p ∼ N W (1 − p)/p yielding CPU ∼ N W i=1 L i N W (1 − p)/p.
The number of windows follows from p N W = P (λ n |λ 0 ) ⇒ N W ∼ −1/ ln(p). Except for diffusive barrier crossings [16], that are most conveniently treated by PPTIS, the average path length L i has a linear dependence ∼ i(λ n − λ 0 )/N W [15]. Taking this all into account, the final result gives CPU ∼ ln(p) −2 (1 −p)/p that has a minimum for p = 0.2. Although, we made several assumptions in this derivation, we believe that in general P (λ i+1 |λ i ) ≈ 0.2 for all i is close to an optimum efficiency.
Between the interface positions one can use of a finer grid of sub-interfaces to obtain the crossing probability function P A (λ|λ 1 ) [15] which is the path space analogy to a Landau free energy profile F (λ). For PPTIS different requirements exist for the position of interfaces. As the PPTIS formalism is based on a complete memory loss over distances larger than the interface separations, the PPTIS interfaces should be set sufficiently far apart. The calculation of memory loss functions can help to determine the minimum required distance to establish this [16].
We would like to stress that although PPTIS transforms the system into a (pseudo) Markovian hopping sequence based on local transition probabilities, it still maintains considerable history dependence. For example, the chance to go from interface i to interface i + 1 is assumed to be equal for the path that arrived at i via the sequence i − 2 → i − 1 → i or via the sequence i → i − 1 → i. However, this transition to i + 1 from i can still be different when its history had hopping sequence i + 1 → i.
Improving the sampling
Parallel path swapping
Biased sampling methods such as constrained dynamics [19], multicanonical [40] or umbrella sampling [17,18] can suffer from substantial ergodicity problems when the order parameters are not equal to the reaction coordinate. This lack of ergodicity usually shows up in hysteresis in the free energy curves (see e. g. [30,41]), and gives, besides a low transmission coefficient, rise to an additional error in the rate constant estimate.
i i−1 i+1 i+1 i i+2 i+1 ensemble i ensemble i+1 ensemble i+1 i i+2 i i−1 i+1 i ensemble λ λ λ λ
Transition path sampling was precisely devised to avoid this problem with reaction coordinates, and, in a way, also avoids ergodicity problems due to the non-local nature of the shooting move. This advantage showed up in the water trimer study [37] where the TPS algorithm was capable of finding two reaction mechanisms across different saddle points separated by a barrier higher than the total energy of the NVE simulation. We stress that this would have been much more difficult to achieve or even impossible in an umbrella sampling algorithm with several narrow windows. However, path sampling can also suffer from ergodicity problems if large barriers separate multiple reaction channels in a high dimensional rough energy landscape. In particular in the case of PPTIS, the short paths are much less likely to overcome such barriers.
Parallel tempering techniques (also known as Replica Exchange methods) can facilitate the sampling [42], but requires a rather large computational effort and cannot be applied at constant energy. Here, we propose a less expensive parallel method especially tailored for PPTIS to enhance ergodicity. This parallel path swapping (PPS) technique is based on the exchange of paths between two subsequent interface ensembles. Fig. 3 shows one path in the λ i ensemble, consisting of all possible paths crossing λ i while starting and ending at either λ i−1 or λ i+1 , and one in the λ i+1 ensemble consisting of all paths crossing λ i+1 at least once, while starting and ending at either λ i or λ i+2 . We introduce a new MC move that attempts swapping the current path of the λ i ensemble with that of the λ i+1 -ensemble, as depicted in Fig. 3. The swap move will be rejected if the λ i ensemble path does not end at λ i+1 or if the λ i+1 ensemble path does not start at λ i . Otherwise, the move is accepted and the two trajectories are swapped from one ensemble to the other. Integrating the equations of motion backward (for the λ i ensemble) and forward (for the λ i+1 ensemble) will result in two entirely new paths for both ensembles. The acceptance/rejection criterion appears before any expensive computation of MD trajectories. Moreover, once accepted we obtain a new path for both ensembles for price of effectively only one path. This makes the path swapping move useful even if for systems not suffering from ergodicity problems.
Another advantage of PPS is that it allows to go beyond the pseudo-Markovian description of PPTIS. Fig. 3 shows that the paths at the right hand side, if we include the dashed trajectory part, can connect four interfaces instead of only three. This extension allows for a long range verification of the memory loss assumption. Also, the development of new, smart algorithms based on PPS might be able to correct for memory effects or to search for ideal interface positions on the fly.
While PPS is very effective when the confinement of short paths in PPTIS can cause sampling problems, even TIS and TPS algorithms might benefit from path swapping when multiple reaction channels exist.
CBMC based shooting moves
Originally developed to sample polymers at high densities, the Configurational Bias Monte Carlo (CBMC) technique grows chain molecules in a biased fashion in order to avoid unfavorable overlap of the beads [43,44,45,46]. The similarity between growing polymers and generating dynamical trajectories was the inspiration for the development of TPS and has been exploited in the sampling of the stochastic path action [10,47]. However, this CBMC-like technique was found to be less effective than the shooting algorithm [14]. Here, we propose a combination of the shooting move with CBMC for diffusive systems that suffer from low acceptance due to a non flat rough free energy barrier. When shooting from one basin of attraction in such systems, the Lyapunov instability causes the paths to diverge and return to the same basin of attraction before crossing the barrier. The use of some stochastic noise allows shooting in only one time direction and alleviates this problem slightly [48,31], but at the price that independent pathways are generated only after a number of accepted shooting moves from the barrier region. This slow exploration of path space is even worse for processes proceeding via multiple dynamical bottlenecks, for instance reactions taking place though a short lived intermediate state forward and backward for a time τ L , resulting in N s trajectory segments Fig. 4-A). The time interval τ L should be large enough to decide whether a trajectory has a chance of being successful, but much smaller than the average path length of a complete trajectory. All path segments are given a weight w j w (n)
s j ≡ {x (j) (τ ′ −τ L )∆t , . . . , x (j) (τ ′ +τ L )∆t }, for j = 1, . . . , N s (Seej = Ψ δp (n) F (s j )(31)
where Ψ equals 1 (else 0) for accepted momenta changes δp at the shooting point τ ′ . The biasing function F should be chosen to give the highest weight w j to those segments that are most likely to produce a complete path of the corresponding interface ensemble. One possibility is to choose F = exp(α∆λ) with ∆λ = λ(x (τ ′ +τ L )∆t ) − λ(x (τ ′ −τ L )∆t ) and α a parameter optimized to the steepness of the barrier at x τ ′ ∆t . In that case, F is a function only of the backward and forward end points of the path segments s j . The Rosenbluth factor for the set of trajectory segments is
W (n) ≡ Ns j=1 w j(32)
One of the segments s i is selected with a probability w i /W (n) . To correct for this bias and to obey detailed balance, we also have to calculate the Rosenbluth factor W (o) for the old path. The procedure is the same as above, but now we apply N s − 1 new random momenta changes {δp (o) } to the momenta of s i at the same shooting point and again generate a set of segments of length 2τ L ∆t. This set is completed by adding segment s 0 of the same length from the old path. The Rosenbluth factor for the old path equals
W (o) ≡ Ns−1 j=0 w (o) j .(33)
where w By imposing super detailed balance [4] the acceptance probability of segment i becomes
P acc (s 0 → s i ) = min 1, w (o) 0 W (n) ρ(x (n) τ ′ ∆t ) w (n) i W (o) ρ(x (o) τ ′ ∆t ) .(34)
Here, the weight functions w i and the distributions ρ are still present, because they do not cancel as in the standard CBMC expression. The accepted segment is integrated to the complete path just as in the normal shooting move of Sec. 3.3. Of course, this procedure is computationally more expensive than the standard shooting move. However, the biasing function F allows to choose a segment with much higher probability to become a accepted path. We expect an increase in sampling efficiency when the gain in acceptance outweighs the cost of the construction of the trajectory segment sets.
In the above algorithm we only can bias the growth of the first segment of the trajectory (the analog of the polymer in standard CBMC) because the rest of the trajectory follows deterministically once the first segment has been chosen. In the standard polymer CBMC a bias is introduced at each segment, and we can make use of the full power of CBMC if we consider stochastic trajectories. Introducing a small amount of stochasticity by for instance the Andersen thermostat [48] or by making use of the periodic boundary condition [49] will hardly change the dynamical properties of the transition process.
Stochasticity allows us to create trajectory jets at several points along the paths (See Fig. 4-B). The first segment is created as in the deterministic procedure above. However, the chosen segment is not integrated to the full path length. Instead, we start with the end point of the forward trajectory and integrate a 'jet' of forward trajectory segments each evolving differently according to its own random noise. Each segment j of this 'jet' k has a weight w jk similar to Eq. (31) and each jet will have a total weight W k = j w jk . We select a segment i according to its relative weight w ik /W k , and continue with the next jet of forward segments. The same is done for the backward paths, until the path is completed. After generating the new path, we have to repeat the 'jet' procedure for the old path as depicted in Fig. 4 in order to calculate the Rosenbluth factor of the old path. The total Rosenbluth factors are now
W (n) RF = k W (n) k , W (o) RF = k W (o) k(35)
where k runs over all the jets including the one at the shooting point x τ ′ ∆t . The final acceptance criterion obeying super detailed balance is then
P acc (o → n) = min 1, W (n) RF ρ(x (n) τ ′ ∆t ) k w (o) 0k W (o) RF ρ(x (o) τ ′ ∆t ) k w (n) ik (36) where w (o)
0k is the segment weight at jet k on the old path and w (n)
ik is the weight of the selected segment of jet k on the new path. To take into account the change in path-length one should include a factor min[1, N (o) /N (n) ], but this is usually implemented by defining a maximum path length as explained at step 4 of the shooting algorithm in Sec. 3.3. The above algorithm could be useful when the standard shooting move suffers from extreme low acceptance ratios.
Time as transition parameter
In TIS the choice of the order parameter is not critical as λ does not have to correspond to the reaction coordinate. Yet, it is possible that the order parameter λ can bias the outcome of transition mechanism and rate constantsalthough much less than for the TST reactive flux method-, for instance, when the reaction mechanism leads in a direction that λ does not allow. In principle, an order parameter-free sampling method is, therefore, highly desirable when examining unexpected contra-intuitive reaction mechanisms. One possibility for such a bias-free method is by using the time on the path outside A as transition parameter (we use 'transition' instead of 'order' to indicate that it is not a traditional order parameter as it is not based on a phase point).
For a particular stable state A definition λ 0 , P A (T i+1 |T i ) is the probability that a path, starting from λ 0 and remaining outside A over a time T i , remains even longer outside A until at least T i+1 > T i . To calculate the probability P A (T i+1 |T i ) by a bias-free TIS simulation we generate an ensemble of trajectories that have path lengths between T i and T i+1 using the shooting algorithm of Sec. 3.3. At the shooting point, we integrate backward until reaching λ 0 or until the length of the trial trajectory exceeds T i+1 (or N (n) max ∆t as defined at step 4 of the shooting algorithm). If the backward trajectory exceeds either T i+1 or N (n) max ∆t the shooting move is rejected. The forward trajectory is continued until reaching λ 0 , or until a path length of T i+1 , or N (n) max ∆t. The trial path is rejected if N (n) max ∆t is exceeded or if the trajectory ends at λ 0 in a time shorter than T i . In the subsequent ensemble, the probability P A (T i+2 |T i+1 ) for T i+2 > T i+1 is calculated for all paths with at least a length T i+1 .
This method, as illustrated in Fig. 5, will thus explore automatically the re-
1)
2)
3) 4)
A At panel 1), P A (T i+1 |T i ) is the fraction of of trajectories that stay outside A longer than T i+1 (open arrows). All trajectories have at least a length T i . The solid arrows are the paths that return to A before T i+1 . At panel 2), P A (T i+2 |T i+1 ) is calculated for paths that remain outside A longer than T i+1 . The minimum length of the paths is further increased at panel 3). Incidentally, a path will end up in the yet unknown state B. At panel 4) the minimum path length constraint forces all the paths into the metastable state region B. From here, they will not return. Hence P A (T |0) will show a plateau.
gions further and further outside A. At some moment it will find the closest stable state region (state B). Trajectories reaching this region will not go back to A, hence, the overall crossing probability function P A (T |0) will show a plateau at some time T similar to standard TIS.
Two-ended path sampling methods, such as TIS, PPTIS and TPS can only treat processes in which both stable states A and B are known. They cannot find the final state starting from a single stable state, a fact already discussed by Dellago and Chandler [50]. The algorithm described here, might be a solution to this problem.
Extracting information from path ensembles
Reaction mechanism
The ensemble of paths collected by the TIS algorithm can be used to investigate the reaction mechanism. We believe that for this purpose the TIS path ensembles might even be more useful than the TPS path ensembles. The TPS method, first samples paths that all successfully reach B in the part to obtain the reactive flux function and then in the second step samples artificially short trajectories of fixed length to calculate the time correlation function C(t). Because of this constraint, the resulting ensembles do not give useful information about the reaction. The TIS λ i -ensembles, on the other hand, contain the correct distribution of paths that have crossed λ i and are either going on to λ i+1 or return to A. Some hidden order parameters can only be discovered by carefully comparing configurations along reactive and unreactive trajectories that are similar in terms of order parameters which at first sight were considered as being the (only) important ones. For instance, the comparison of reactive and unreactive geometries with an almost identical orientation of the reactants showed that precise tetrahedral ordering of the solvent water molecules was an important factor in the hydration reaction of ketones [30]. Although there is currently no systematic way to extract the reaction coordinates from a path ensemble, once a reaction coordinate is postulated based on physical insight it can be tested using committor distributions [13].
Activation energies
The activation energy E a is an important experimentally accessible quantity and is defined by the Arrhenius law
k = Ae −βEa ,(37)
where A is a system dependent prefactor. In fact, A and E a may also be temperature dependent. Such non Arrhenius behavior can be quite severe: sometimes reaction rates are even decreasing with increasing temperature, resulting in a 'negative activation energy' (see e.g. [51]). ¿From Eq. (37) it follows that
E a = − ∂ ln k AB (β) ∂β .(38)
An algorithm to calculate E a in a TPS simulation was given in Ref. [52]. Here, we use a similar approach to calculate E a in a canonical TIS simulation. Substitution of Eq. (25) in Eq. (38) results in
E a = − ∂ ∂β ln φ 1,0 − ln h A + n−1 i=1 ln Φ i+1,0 i,0 − ln φ i,0(39)
For any function A(x) we can write
− ∂ ln A(x) ∂β = E(x) A − E(x) (40) with E(x) A = A(x)E(x) / A(x) . Using E(x) Φ i+1,0 i,0 = E(x) φ i+1,0 ,(41)
most terms in Eq. (39) cancel, only leaving
E a = E(x) Φ n,0 n−1,0 − E(x) h A ,(42)
which is the difference between the average energy of state A and the energy of the transition pathways connecting A with B. Consequently, the calculation of the E a does not require all interface ensembles, but only the last ensemble λ n−1 . However, if all the path ensembles i = 1, . . . , n − 1 are available an activation energy function
E a (λ i ) = E(x) Φ i,0 i−1,0 − E(x) h A(43)
can be calculated that should converge to a plateau analogous to the crossing probability P (λ|λ 1 ). A finer grid of sub-interfaces can be applied to obtain a continuous smooth function E a (λ).
Again, there is a subtle difference between the TPS and TIS algorithms. For the reaction rate determination, TPS requires a plateau in the time correlation function of Eq. (8), while TIS should give a plateau in λ for the crossing probability P (λ|λ 1 ). Similarly, the TPS activation energy is expressed as a time dependent function that will converge to a plateau at times t = T [52], while the TIS activation energy reaches a plateau in terms of λ.
Not all reactions show Arrhenius behavior. Therefore, it would be interesting to determine k AB (β) for a range of temperatures. One can estimate the rate for a slightly different temperature by reweighting the crossing probabilities [17,18]. If A(x) is the average of an observable A(x) at inverse temperature β, then A(x)e −∆βE(x) / e −∆βE(x) should be the average at inverse temperature β + ∆β. This reweighting technique can also be applied to the crossing probabilities (21) and the flux (28). Of course, ∆β should be small to obtain good enough statistics. The calculation of the temperature dependence of individual crossing probabilities has the advantage that the origin of possible non-Arrhenius behavior might be located (in terms of λ) along the reaction path.
Summary and Conclusions
We reviewed the basic concepts of TIS and PPTIS and explained their relation to TST based methods and TPS. We believe that path sampling methods, TPS, TIS and PPTIS, are powerful when dealing with high dimensional complex process for which is a reaction coordinate is lacking. Among these methods, TIS can be considered as an improvement upon the original TPS giving a complete non-Markovian description of the reactive event, but more efficient. PPTIS improves the efficiency even more, but relies on the assumption of memory loss between interfaces. Hence, it should only be applied for diffusive barrier crossings. In addition to this review, we have introduced several new techniques in this paper. These novel methods comprise the CBMC based shooting moves, order parameter free methods, parallel path swapping and the calculation of activation energies. The efficiency of these methods should be tested by future simulations. We plan study this in the near future.
A The flux revisited
In some cases, we can improve the efficiency of the flux calculation by separating the flux into the probability to be on the λ 1 surface times a factor integrating over all possible velocities when leaving the surface. The flux term can then be calculated by combining straightforward MD with, as soon as we cross the λ 1 surface, the sampling of sets of randomized Gaussian distributed velocitiesλ, after which the MD trajectory is continued with the old original momenta. In this way, we make optimal use of the statistics of the crossing points. The velocity sampling does not require force calculations and is therefore cheap. In the following we assume that we always take λ 1 = λ 0 . Similar to Eq. (2) we can write:
φ 1,0 h A = θ(λ)λ λ 1 P (λ 1 ) x∈A (A.1) with P (λ 1 ) x∈A ≡ δ(λ(x) − λ 1 ) h A (A.2)
The two terms φ 1,0 h A and P (λ 1 ) x∈A can be obtained in the same MD simulation. As δ(λ(x) − λ 1 ) dλ is equal to the probability to find the system in the interval [λ 1 − 1 2 dλ : λ 1 + 1 2 dλ], it can be measured by defining a width dλ and performing a MD (or MC) simulation starting in A:
P (λ 1 ) x∈A = 1 dλ N λ 1 N MD (A.3)
with N λ 1 the number of counts in the specified interval and N MD the number of MD steps. However, this number can depend sensitively on the choice of bin width dλ. Ideally one would like dλ to be as small as possible, at the cost of having to perform a very long simulation run for a statistically accurate number N λ 1 . A better option is to weigh the crossings with a function depending on the velocity. Assume that we cross λ 1 in one MD step from x i∆t to x (i+1)∆t . If dλ is small neither of these points will lie inside the interval. However, assuming a linear dynamics between these points, the system traverses from x i∆t to x (i+1)∆t in N sub equidistant sub steps. The number of phase points N λ 1 that lie in the dλ interval of this short linear trajectory is approximately dλN sub /|λ(x (i+1)∆t )−λ(x i∆t )|. The total number of MD moves N MD , of course, also increases by a factor N sub . So Eq.( A.3) becomes
P (λ 1 ) x∈A = 1 N MD i * 1 |λ(x (i+1)∆t ) − λ(x i∆t )| (A.4)
where the * indicates that the summation has to be performed only for points i along the trajectory for which x i∆t → x (i+1)∆t showed a crossing (positive or negative) with interface λ 1 . Further optimization can be achieved by writing
|λ(x (i+1)∆t ) − λ(x i∆t )| = |λ x i∆t |∆t + O(∆t 2 ) = |λ x (i+1)∆t |∆t + O(∆t 2 )
, but the velocityλ at the interface would give the most exact result. If we also assume a linear change in time for the velocities between i and i + 1, our best estimate for P (λ 1 ) x∈A is:
P (λ 1 ) x∈A = 1 N MD ∆t i * 1 |λ(x i∆t ; λ 1 )| (A.5)
where we have introduced the notation g(x i∆t ; λ j ) to denote the function g(x) at the crossing point of interface λ j obtained by a linear interpolation of the function between two successive trajectory points x i∆t → x (i+1)∆t :
g(x i∆t ; λ j ) ≡ 1 λ(x (i+1)∆t ) − λ(x i∆t ) [λ(x (i+1)∆t ) − λ j ]g(x i∆t ) + [λ j − λ(x i∆t )]g(x (i+1)∆t ) (A.6)
The factor θ(λ)λ λ 1
in Eq. (A.1) can be calculated in the same MD simulation with an additional sampling procedure. In some simple cases, there is even an analytically expression. For instance, in case the x-coordinate of particle j is the order parameter, λ(x) = r jx in a constant temperature (NVT) simulation, we would obtain θ(λ)λ
λ 1 = 1 √ 2πβm j
with m j the mass of this particle. However, for more complex λ(x), such as the distance between two particles i and j, λ(x) = |r i − r j |, no simple analytic expression exists. The calculation of θ(λ)λ λ 1 can then be calculated by sampling a random set of N MC velocitiesλ as soon as a crossing is detected:
θ(λ)λ λ 1 = i * 1 |λ(x i∆t ;λ 1 )| N MC j θ(λ)λ N MC i * 1 |λ(x i∆t ;λ 1 )| (A.7)
where i runs over all MD crossings with interface λ 1 ,λ(x i∆t ; λ 1 ) is the MD crossing velocity through λ 1 , j runs over the N MC 'artificial' velocitiesλ that are taken from a proper distribution P (λ|x i∆t ). For NVT simulations without additional constraints this distribution P (λ|x i∆t ) does not depend on the phase point x i∆t and we can simply sampleλ({p}) where the momenta {p} definingλ are taken from a Gaussian distribution For NVE simulations, the distribution P (λ|x i∆t ) does depend x i∆t and we have to change all momenta and distribute them on the hypersphere defined by the kinetic energy K = E − V (x i∆t ; λ 1 ) with E the total energy and V (x i∆t ; λ 1 ) the total potential energy at the crossing point. The proper sampling of momenta distributions in the presence of linear constraints, such as linear and angular momentum is explained in Ref. [37]. Clearly, if P (λ|x i∆t ) = δ λ −λ(x i∆t ; λ 1 ) , Eq. (A.7) would be equal to N + c / i * |λ(x i∆t ; λ 1 )| −1 leaving Eq. (A.1) identical to Eq. (28) from which we started.
B TIS shooting acceptance criterion for stochastic dynamics
Although we assume throughout the paper that the equations of motion were deterministic, it is sometimes useful to implement some stochasticity into the dynamics, or consider completely stochastic equation of motion such as Brownian Dynamics [14,48,31]. Quantities likeh A (x 0 ) are, then, no longer just 1 or 0 but turn into probabilities with a fractional value. Moreover, for stochastic dynamics it is not trivial whether we are allowed to use the path that generated x 0 . In this appendix we derive the acceptance probability for the shooting algorithm for arbitrary dynamics along the same lines as in Ref. [14]. At start, we try to be as general as possible making the least possible assumptions on the type of dynamics or on whether the system is in equilibrium or not. For this purpose, it is most convenient to use the path space description, instead of phase space. The weight or probability density P[x] for a single path x ≡ {x −τ b ∆t , . . . ; x 0 ; . . . , x +τ f ∆t } is than not only determined by the distribution ρ(x 0 ) of x 0 , but also by the probabilities of arriving along this precise route from x −τ b ∆t in the past and continuing upto x +τ f ∆t in the future.
P[x] = ρ(x 0 ) −τ b i=−1 p(x (i+1)∆t ← x i∆t ) τ f i=1 p(x (i−1)∆t → x i∆t ) (B.1)
where p(x → y) is the forward transition probability (more accurate: probability density) to go from x to y and p(y ← x) is the probability that, if the system is at y, it came from x in the past. Here, ρ(x 0 ) is not necessarily the Boltzmann distribution or even a distribution in equilibrium. It is applicable to all systems that (at least to some approximation) are described by a steady state. We can express p(y ← x) in terms of forward transition probabilities as p(y ← x) = ρ(x)p(x → y) dx ′ ρ(x ′ )p(x ′ → y) = ρ(x)p(x → y) ρ(y) .
(B.2)
where dx ′ ρ(x ′ )p(x ′ → y) = ρ(y) results from the steady state behavior. Using this relation, one can show that Eq. (B.1) is exactly equal to the probability of the first point ρ(x −τ b ∆t ) times the forward evolution probabilities
P[x] = ρ(x −τ b ∆t ) τ f −1 i=−τ b p(x i∆t → x (i+1)∆t ) (B.3)
which is identical to the weight for a path in the TPS-ensemble [14] with x −τ b ∆t instead of x 0 . Note that so far, we have assumed nothing about the nature of dynamics (irreversible or reversible, stochastic or deterministic). When restricted to the TIS ensemble for interface i, the probability density of a path can be written as
P λ i [x] ≡ĥ i (x)P[x]/Z(λ i ) (B.4)
whereĥ i is unity if the path goes from λ 0 , crosses λ i and ends either at λ i+i or goes back to λ 0 . Otherwise it is zero. The normalizing factor Z(λ i ) equals
Z(λ i ) ≡ Dxĥ i (x)P[x] (B.5)
where the integral is taken over all possible paths x of all lengths, starting in all possible initial conditions x −τ b . Note that, contrary to TPS, Eq. (B.3) and Eq. (B.4) are not directly related to the relative probabilities of all paths in the TIS ensemble. This is a result of the path ensemble containing paths of different lengths. Eq. (B.3) turns into a true probability only when multiplied with the infinitesimal volume element in path space Dx ≡ τ f i=−τ b dx i∆t ∼ dx N . Hence, a long path has an infinitely smaller probability than a shorter one for stochastic dynamics. Therefore, the concept of path space may sound peculiar for TIS. Still, it is instrumental to derive proper acceptance rules for TIS obeying detailed balance.
When performing the random walk in the TIS path space using the shooting algorithm, the detailed balance condition is
P gen [x (o) → x (n) ] P gen [x (n) → x (o) ] P acc [x (o) → x (n) ] P acc [x (n) → x (o) ] = P λ i [x (n) ] P λ i [x (o) ] (B.6)
where o and n, denote the old and new path respectively. The usual Metropolis acceptance rule is then Note that this rule only applies at the trajectory space level, it has nothing to do with whether the underlying dynamics is stochastic or deterministic, or even reversible or irreversible.
The generation probability to create a new path from an old path using the shooting move is given by
P gen [x (o) → x (n) ])] = P (δp) N (o) P f gen [x (o) → x (n) ]P b gen [x (o) → x (n) ] (B.8)
where 1/N (o) is the chance to choose the shooting point τ ′ with −τ b(o) ≤ τ ′ ≤ τ f (o) ) at the old path, P (δp) the chance to select the randomized momenta displacements. As δp is normally taken from a symmetric distribution, hence P (δp) = P (−δp), this term will cancel in Eq. (B.7). The last two factors in Eq. (B.8) are the probabilities to generate trajectories from the shooting point point τ ′ with the new momenta. These generation probabilities are given by the underlying dynamics used to generate the trajectories. If one starts from a shooting point at τ ′ on the old existing path (with −τ
(o) b < τ ′ < τ (o)
f ) the generation probability for the forward segment is
P f gen [x (o) → x (n) ] = τ (n) f −1 i=τ ′ p x (n) i∆t → x (n) (i+1)∆t . (B.9)
This generation probability is exactly identical to the weight of the forward segment. The integration of the backward segment is not always trivial especially when dealing with irreversible processes. However, in general, when reversible dynamics is applied, the backward segment is obtained by reversing the momenta, integrating forward in time and reversing the momenta again [14]. Accordingly, the backward segment's generation probability equals
P b gen [x (o) → x (n) ] = τ ′ −1 i=−τ (n) b p x (n) (i+1)∆t →x (n)
i∆t . (B.10)
wherex ≡ {r, −p} for a phase point x ≡ {r, p}. Using these generation probabilities and the path weight Eq. (B.3) the factor within the min function of Eq (B.7) can be written as
P[x (n) ]P gen [x (n) → x (o) ] P[x (o) ]P gen [x (o) → x (n) ] = ρ[x −τ (n) b ]N (o) ρ[x −τ (o) b ]N (n) × (B.11) τ ′ −1 i=−τ (n) b p[x (n) i∆t → x (n) (i+1)∆t ] p[x (n) (i+1)∆t →x (n) i∆t ] τ ′ −1 i=−τ (o) b p[x (o) (i+1)∆t →x (o) i∆t ] p[x (o) i∆t → x (o) (i+1)∆t ] .
where the generation probability and the weight of the forward parts of the trajectories have canceled each other. Eq. (B.3) simplifies tremendously for dynamics that obey the microscopic reversibility condition [14] p(x → y) p(ȳ →x) = ρ(y) ρ(x) .
(B.12)
This condition can be seen as a special case of Eq. (B.2) for time-reversible dynamics with p(y ← x) = p(ȳ →x), but is very general and valid for a broad class of dynamics applying to both equilibrium and non-equilibrium systems. As result, if the microscopic reversibility condition (B.12) is satisfied and thus p(ȳ →x) = p(y ← x), almost all terms in Eq. (B.11) cancel, except for the steady state distributions ρ(x τ ′ ) of the shooting points.
P acc [x (o) → x (n) ] =ĥ i (x (n) ) min 1, ρ(x (n) τ ′ ) ρ(x (o) τ ′ ) N (o) N (n) . (B.13)
which is exactly the same as for deterministic dynamics.
This is an important result as it allows to perform the acceptance/rejection rule (step 3 of Sec
∆t) be the backward and forward terminal time slice indices, respectively. Including x 0 , the initial path then consists of N (o) = τ b + τ f + 1 time slices. Choosing a probability γ < 1 and a Gaussian width σ w we now start following loop:• Main loop (1) Take a uniform random number α 1 in the interval [0 : 1]. (2) If α 1 < γ perform a time-reversal move. Otherwise, perform a shooting move. (3) If the trial path generated by either the time-reversal or shooting move is a proper path in the λ i ensemble accept the move and replace the old path by the new one, otherwise keep the old path. Update averages and repeat from step 1. • Time-reversal move (1) If the current path ends at λ i+1 reject the time-reversal move and return to the main loop. (2) If the current path starts and ends at λ 0 , reverse the momenta and the order of time-slices. On this reverse path, x 0 is the new first crossing point with λ i . Return to the main loop. • Shooting move (1) On the current path with length N (o) choose a random time slice
Integrate equations of motion backward in time by reversing the momenta at time slice τ ′ , until reaching either λ 0 , λ i+1 or exceeding the maximum path length N (n) max . If the backward trajectory did not reach λ 0 reject and go back the main loop. Otherwise continue with step 6. (6) Integrate from time slice τ ′ forward until reaching either λ 0 , λ i+1 or exceeding the maximum path length N (n) max . Reject and go back to the main loop if the maximum path length is exceeded or if the entire trial path has no crossing with interface λ i . Otherwise continue with the next step. (7) Accept the new path, reassign x 0 to be the first crossing point with λ i and return to the main loop.
Fig. 3 .
3Path swapping move for PPTIS. The last half of the path in the λ i ensemble and the first half of the path in the λ i+1 are swapped to the λ i+1 and λ i ensembles, respectively.
Fig. 4 .
4Within the shooting algorithm, CBMC can be applied both at the shooting point (the random time slice for which we change the momenta) and along the path by introducing some stochastic noise. At the shooting point τ ′ we generate a set of N s momenta displacements {δp (n) }, and accept these displacements using step 3 in Sec.3.3. Each phase point is then integrated CBMC shooting move. (A) At the shooting point at the old trajectory (dashed line) four trial segments are released. In this example the momenta of segment 1 and 2 have been rejected and are not integrated further. Segment 0 is retracing the old path. Of the trial segments, segment 3 has come farthest in its forward (solid arrow) and backward (open arrow) time evolution and will consequently have the highest weight. (B) The use of stochasticity allows the creation of trajectory jets at several points along the path. At each junction the path will follow the most favorable direction (bold solid line). The creation of trajectory jets at the old path (bold dashed line) is required to maintain super-detailed balance.
the weight of segment s 0 , and w (o) j with j = 1, . . . , N s − 1 are the weights for the segments that follow from {δp (o) }.
Fig. 5 .
5Time as transition parameter. The square denotes the definition of the boundary for state A. The thin lines are free energy contour lines. The four panels show the representation of generated trajectories in successive time-interface ensembles.
our instrument to search for the new phase point x (n)
P
acc [x (o) → x (n) ] =ĥ i (x (n) ) min 1, P[x (n) ] P[x (o) ] P gen [x (n) → x (o) ] P gen [x (o) → x (n) ] , (B.7)
. 3.3) for the new momenta at the shooting point even if the energy along the path changes giving a different weight to ρ(x τ ′ ∆t ) as to ρ(x 0 ). The ratio N (o) /N (n) in Eq. (B.13) can, of course, not be known in advance at the shooting point. However, this is effectively circumvented by defining N max at step 4 of the shooting algorithm in Sec. 3.3 leaving only the ρ(x the acceptance rule (step 3).
). Similarly, the A → B trajectories with multiple λ * crossings are effectivelyA
B
Fig. 1. Illustration of the difference in counting in the transmission coefficient
Eqs. (4), (5), and (16). For simplicity, assume that the system consists of three
kind of possible trajectories, as shown by this figure, that cross the dividing
surface with the same speed v orthogonal to the surface. To the seven phase
points on the surface (from top to bottom) the numerator of Eq. (4) will assign
the values [−v, v, v, −v, v, 0, 0], while these are [0, 0, v, −v, v, 0, 0] for Eq. (5) and
[0, 0, v, 0, 0, 0, 0] for Eq. (16). The sum of these give the same result v. Evaluation
of Eq. (16) in an actual computer algorithm requires the fewest MD steps as only
phase points similar to the 3rd and 7th phase points would need the integration
until reaching stable state regions. For instance, the fifth crossing point can be as-
signed zero already as soon as one detects that its backward trajectory recrosses
the surface.
This cancellation might seem to be not apparent if a trajectory recrosses the same
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|
[] |
[
"WEIGHING BLACK HOLES USING TIDAL DISRUPTION EVENTS",
"WEIGHING BLACK HOLES USING TIDAL DISRUPTION EVENTS"
] |
[
"Brenna Mockler \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n\nDARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark\n",
"James Guillochon \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n",
"Enrico Ramirez-Ruiz \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n\nDARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark\n"
] |
[
"Department of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA",
"DARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Department of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA",
"DARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark"
] |
[] |
While once rare, observations of stars being tidally disrupted by supermassive black holes are quickly becoming commonplace. To continue to learn from these events it is necessary to robustly and systematically compare our growing number of observations with theory. We present a tidal disruption module for the Modular Open Source Fitter for Transients (MOSFiT) and the results from fitting 14 tidal disruption events (TDEs). Our model uses FLASH simulations of TDEs to generate bolometric luminosities and passes these luminosities through viscosity and reprocessing transformation functions to create multi-wavelength light curves. It then uses an MCMC fitting routine to compare these theoretical light curves with observations. We find that none of the events show evidence for viscous delays exceeding a few days, supporting the theory that our current observing strategies in the optical/UV are missing a significant number of viscously delayed flares. We find that the events have black hole masses of 10 6 − 10 8 M , and that the masses we predict are as reliable as those based on bulk galaxy properties. We also find that there is a preference for stars with mass < 1M , as expected when low-mass stars greatly outnumber high-mass stars.
|
10.3847/1538-4357/ab010f
|
[
"https://arxiv.org/pdf/1801.08221v3.pdf"
] | 119,451,471 |
1801.08221
|
598f7391f47801e6f0f7a148d2a8d6914e7491c6
|
WEIGHING BLACK HOLES USING TIDAL DISRUPTION EVENTS
January 26, 2018
Brenna Mockler
Department of Astronomy and Astrophysics
University of California
95064Santa CruzCAUSA
DARK
Niels Bohr Institute
University of Copenhagen
Blegdamsvej 172100CopenhagenDenmark
James Guillochon
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Enrico Ramirez-Ruiz
Department of Astronomy and Astrophysics
University of California
95064Santa CruzCAUSA
DARK
Niels Bohr Institute
University of Copenhagen
Blegdamsvej 172100CopenhagenDenmark
WEIGHING BLACK HOLES USING TIDAL DISRUPTION EVENTS
January 26, 2018Draft version Preprint typeset using L A T E X style AASTeX6 v. 1.0stars: black holes -galaxies: active -galaxies: supermassive black holes
While once rare, observations of stars being tidally disrupted by supermassive black holes are quickly becoming commonplace. To continue to learn from these events it is necessary to robustly and systematically compare our growing number of observations with theory. We present a tidal disruption module for the Modular Open Source Fitter for Transients (MOSFiT) and the results from fitting 14 tidal disruption events (TDEs). Our model uses FLASH simulations of TDEs to generate bolometric luminosities and passes these luminosities through viscosity and reprocessing transformation functions to create multi-wavelength light curves. It then uses an MCMC fitting routine to compare these theoretical light curves with observations. We find that none of the events show evidence for viscous delays exceeding a few days, supporting the theory that our current observing strategies in the optical/UV are missing a significant number of viscously delayed flares. We find that the events have black hole masses of 10 6 − 10 8 M , and that the masses we predict are as reliable as those based on bulk galaxy properties. We also find that there is a preference for stars with mass < 1M , as expected when low-mass stars greatly outnumber high-mass stars.
INTRODUCTION
One of the most promising avenues for studying black holes in quiescent galaxies is through tidal disruption events (TDEs). Unlucky stars that pass too near a black hole are torn apart, lighting up previously dormant black holes (Rees 1988) and encoding the resultant light curves with a wealth of information about the nature of disruptor and disruptee (Lodato et al. 2009;Guillochon et al. 2009;Kesden 2012;Law-Smith et al. 2017;Cheng & Bogdanović 2014;Evans & Kochanek 1989;Kobayashi et al. 2004;Rosswog et al. 2009;Haas et al. 2012;Ayal et al. 2000).
For a TDE to occur, the tidal disruption radius, R t ≡ (M h /M * ) 1/3 R * of a star of mass M * and radius R * by a black hole of mass M h must be outside the gravitational radius of the black hole (e.g. MacLeod et al. 2012), else the black hole will swallow the star whole. For most stars, black holes 10 8 M are the most likely disruptors. This makes TDEs all the more exciting, as they are probing lower mass black holes that are otherwise difficult to study, and whose mass determinations are uncertain.
[email protected]
The fallback rate and the peak timescale of TDEs are dependent on the mass of the disrupting black hole, the mass of the star, and the stellar structure of the star (Lodato et al. 2009;Guillochon & Ramirez-Ruiz 2013). Because the dependence on the mass and radius of the star largely cancel one another out on the main sequence, the peak timescale is sensitive to the mass of the black hole. Thus, if a TDE's luminosity follows the fallback rate (i.e. is "prompt" Guillochon & Ramirez-Ruiz 2015a), the light curve can be used to measure the black hole's mass and the properties of the disrupted star. In order for the luminosity to follow the fallback rate, the stellar debris that initially returns on highly eccentric orbits must circularize on a timescale that is shorter than the fallback timescale Bonnerot et al. 2015;Hayasaki et al. 2016). As we show here, the optical and UV events that we modeled all require prompt circularization, suggesting that we can use their light curves to acquire reliable black hole mass measurements.
New TDEs have been uncovered at a steady rate in recent years and the rate of discoveries will continue to increase. As such, it has become imperative to be able to systematically quantify the key variables responsible for shaping TDE light curves so that we can compare these variables across events and develop a statistical understanding of the physical ingredients at play. To facilitate this, it is important for TDE data to be accessible, and the Open TDE Catalog Guillochon et al. 2017b) is aiming to do this by collecting TDE data and hosting it online in a standardized format. To compare and contrast between different TDEs it is important to fit the events consistently, and to this end in this paper we introduce a theoretical model for fitting TDEs as part of MOSFiT, the modular Open-Source Fitter for Transients (Guillochon et al. 2017b). This model has been implemented in MOSFiT and is available immediately.
Along with the model we present fits to the optical and UV data of 14 TDEs from the Open TDE Catalog. Using MOSFiT we are able to extract posterior distributions for key parameters, most notably the black hole mass. We attempt to capture the broad features of a TDE while minimizing the number of free parameters in our model. Our model ingredients are outlined in Section 2.1 and our TDE sample is described in Section 3. Our black hole mass estimates are presented in Section 4 along with a detailed comparison with those derived using other methods. In Section 5 we discuss how the posteriors from our fits can help inform TDE emission models and presents a summary of our findings.
METHOD
The tidal disruption model in MOSFiT uses FLASH simulations of the mass fallback rate (Guillochon & Ramirez-Ruiz 2013) as inputs to fit data of TDEs. It is modeled similarly to TDEFit, a code for fitting tidal disruption events, originally described in Guillochon et al. (2014), but excludes a few features of that code that will be ported to future versions of the MOSFiT model (see Section 5.3). In the sections that follow we provide a detailed description of the model components along with a brief overview of the fitting procedure.
MOSFiT Modules
The MOSFiT platform sub-divides the components of a model into independent modules such that common operations for fitting transients can be utilized by various transient types. This means any new model implemented in MOSFiT re-uses many existing modules, reducing the chance of coding errors and improving overall performance. Below, we describe the new modules added to MOSFiT specifically created for modeling TDEs, which include new engine (source of radiant emission), transform (reprocessing of radiant emission), and photosphere (conversion of bolometric flux to a distribution of flux as a function of wavelength) modules. T viscous (days) Log 10 −3 10 5 a The parameter b is a proxy for β as the relationship between β and ∆M bound to the black hole differs for different γ. Minimum disruptions for both β 5/3 and β 4/3 correspond to b = 0 and full disruptions for both β correspond to b = 1. Disruptions with b = 2 correspond to β 5/3 = 2.5 and β 4/3 = 4.0 respectively. b For our fit of iPTF16fnl we narrowed the range of t disruption as MOSFiT was having difficulty isolating the relatively short peak for that event, it is clear from the photometry that t first fallback is 500 days before the first observation. c The parameter t first fallback is different from the time of disruption. For any combination of disruption parameters (β, γ) there exists a fixed time between t disruption and t first fallback . This delay can be affected by the precession of debris out of the original orbital plane, however it does not affect the determination of M h because the mass-energy distribution remains intact during this delay (see Section 5.1). The engine for the TDE model comes from converting the fallback rate of material onto the black hole postdisruption directly to a bolometric flux via a constant efficiency parameter . To model this process we used hydrodynamical simulations of polytropic stars tidally disrupted by supermassive black holes (SMBHs) (Guillochon & Ramirez-Ruiz 2013). As stars of different masses are better represented by different polytropes, we take stars with mass ≤ 0.3M and mass ≥ 22M to be represented by 5/3 polytropes while stars with masses between 1M and 15 M are modeled as 4/3 polytropes. For stars in the transition ranges (0.3M -1M , 15M -22M ), we use hybrid fallback functions that smoothly blend between the 4/3 and 5/3 polytopes, the details of which are described later in this section. The simulations were run for a wide range of impact parameters (β = R t /R p ), varying from interactions that barely disrupted the star to interactions with β values significantly larger than what is needed for full disruption. Stars are considered to be fully disrupted when no surviving core remains post-disruption, which for SMBH encounters yields a fallback mass ∆M = M * /2. Because both the mass of the black hole and the mass of the star enter into the rate of fallback as simple scaling parameters (Guillochon & Ramirez-Ruiz 2013, 2015b, all simulations were run with M h = 10 6 M and M * = 1M .
The hydrodynamical simulations provide us with the distribution of debris mass dm/de as a function of specific binding energy e after it is torn apart. This dis-tribution is dependent on the structure of the star, a feature that is particularly important when fitting the shape of the light curve and its power-law decline at late times. To obtain the fallback rate dm/dt =Ṁ , dm/de is converted into a mass distribution in time using the de/dt calculated from orbital dynamics.
After collectingṀ for various values of β and γ, values for β, M * and M h are input into the fallback module, which linearly interpolates in β-M * space (using the mapping between M * and γ described above) to obtain fallback curves as a functions of both parameters. In order to provide accurate description for the light curve with M * and M h , we make use of the following scalings.
M ∝ M −1/2 h M 2 * R −3/2 * ,(1)
and
t(Ṁ ) ∝ M 1/2 h M −1 * R 3/2 * ,(2)
where t(Ṁ ) is the time of a given rate of fallback. We use Tout et al. (1996) to get R * from M * for M * ≥ 0.1M . Below that mass we assume that the radius is constant and use R * ,Tout (M * = 0.1M ) ≈ 0.1R , roughly the radius of Jupiter.
We also assume the stars are zero-age main-sequence stars (ZAMS) and that they have solar metallicity. Both the ZAMS and composition assumption as well as the assumption that the stars are represented by blends of 4/3 and 5/3 polytropes are simplifying assumptions that allow us to build this minimal model without introducing excessive numbers of free parameters. In future work we plan to use simulations of realistic stars for a wide range of ages and compositions as inputs into our fallback module (Law-Smith et al. 2017).
At the end of the fallback module, we convertṀ to luminosity by assuming a constant efficiency , which we allow to vary as a free parameter in our fitting procedure, yielding L = Ṁ c 2 . This freedom allows us to remain agnostic about the physical mechanism driving this conversion, which can be sub-percent if originating from a stream-stream collision (Jiang et al. 2016) or up to 40% if the conversion occurs at the ISCO of a maximally-spinning black hole (Beloborodov 1999). We also introduce a soft cut at the Eddington limit L Edd ≡ 4πGM h m p c/σ T to prevent the radiated luminosity from exceeding this value; this is motivated by both the fact that the peak bolometric luminosities derived observationally for optical/UV TDEs appear to be sub-Eddington Wevers et al. 2017) and that other accreting black hole systems (such as AGN) rarely show evidence for large thermal Eddington luminosity excesses.
Viscous Delay
The assumption that the luminosity closely follows the fallback rate is a bold assertion that, if correct, gives us a deterministic way to relate how stellar debris circularizes and how it accretes onto the black hole. If the viscous time about the black hole were short as compared to the fallback time, the accretion rate onto the black hole from the forming diskṀ d should be equal to the fallback rateṀ fb . However, as has been found in several numerical works (Guillochon et al. 2014;Shiokawa et al. 2015;Bonnerot et al. 2015;Hayasaki et al. 2016), circularization about the black hole might be very ineffective, resulting in viscous times that are potentially hundreds of times longer than the orbital period of the most-bound debris (Cannizzo et al. 1990;Guillochon & Ramirez-Ruiz 2015a;Dai et al. 2015). This would result in a central accretion disk with R ≈ R t that is starved of mass, with much of the mass being held aloft for long periods of time in an elliptical superstructure Guillochon et al. 2014). While the exact details of how matter is received by the disk and then later accreted by the black hole remain elusive (Sadowski et al. 2016), the primary effect of the viscous slow-down is likely well-approximated as a "low-pass" filter on the fallback rate,
M d (t) =Ṁ fb (t) − M d (t)/T viscous ,(3)
where the elliptical disk that forms acts as a reservoir where a mass M d remains suspended outside of the black hole's horizon for roughly a viscous time. The solution to this expression iṡ
M d (t) = 1 T viscous e −t/Tviscous t 0 e t /TviscousṀ fb (t )dt ,(4)
which shows that the accretion rate exponentially approaches the fallback rate after a viscous time. We implement the above expression in our viscous module, inputting the luminosities from our fallback module through the transform, which yields viscously-delayed luminosities that are used to compute light curves.
Photosphere
Regardless of the process or combination of processes responsible for generating the emission, the kinetic energy of the returning debris must eventually be dissipated in order to be observed. Even if some energy is deposited by circularization at large distances, the energy will be primarily dissipated by processes that operate closest to the black hole simply because the velocities there are the greatest. However, this would imply most of the radiation would be emitted at very high energies (X-rays), and instead we observe many TDEs with significant (and sometimes dominant) optical/UV flux. A reprocessing layer, either static or outflowing, can help explain the observed emission by reprocessing the luminosity generated by the various dissipation processes at play (Guillochon et al. 2014;Roth et al. 2015;Piran et al. 2015;Metzger & Stone 2015;Jiang et al. 2016;Bogdanović et al. 2004;Coughlin & Begelman 2014;Gaskell & Rojas Lobos 2014;Loeb & Ulmer 1997;Miller 2015;Strubbe & Quataert 2009;Ulmer et al. 1998). In this work we assume a simple blackbody photosphere for the reprocessing layer, so that the observed flux becomes
F ν = 2πhν 3 c 2 1 exp(hν/kT eff ) − 1 R 2 phot D 2 ,(5)
with a temperature
T eff = L 4πσ SB R 2 phot 1/4 .(6)
Many observations of TDEs have thermal temperatures that don't exhibit much variation near peak, but tend to increase at late times. For blackbody emission, the radius must increase as the luminosity (andṀ fb ) increase, and decrease as the luminosity decreases, in order for the temperature not to change significantly as the luminosity evolves. This simple behavior also explains the rise in temperatures at late times as the photospheric radius decreases and the bulk of the observed radiation shifts to higher energies. To model this dependence we assume that the radius of the photosphere has a power law dependence on the luminosity and fit for both the power law exponent l and radius normalization R ph0 ,
R phot = R ph0 a p (L/L edd ) l .(7)
Here a p is the semi-major axis of the accreting mass at peakṀ fb . This provides a reasonable typical scaling for the radius of the photosphere, with a minimum photosphere size set by R isco and a maximum photosphere size set by the semi-major axis of the accreting mass. One of the appealing aspects of this photosphere model is that it remains agnostic towards the mechanism ultimately responsible for generating the luminosity, but does make a number of simplifying assumptions regarding the source function of the radiation. In particular, it assumes that all of the radiation is efficiently thermalized at the scale of the photosphere radius. The resultant spectrum is compatible with what one would expect from a "veiled" TDE , and, as such, this model cannot reproduce the x-ray emission that is observed in a small fraction of TDEs found in optical surveys (e.g. ASASSN-14li, Miller et al. 2015). In the future, we plan to include an accretion disk module which will be used to describe the x-ray emission that sometimes is observed to accompany optical/UV TDEs ).
LIGHT CURVE FITS
The characteristics of the population of TDEs as a whole can be derived by fitting a significant fraction of the existing TDE candidates to a shared model. In what follows we describe the data used in this study as well as the results from the fitting procedure.
Data Selection
The data from our fits is public and can be found on the Open TDE Catalog 1 . There does not exist a single agreed upon test for classifying a transient as a TDE, and therefore multiple clues must be taken together to determine the likelihood that a transient is in fact the result of a TDE. First of all, astrometry must place the transient near the center of its host galaxy. Next, unique light curve features (blue optical/UV colors, minimal color evolution, and a large brightening above the quiescent level) are used to separate TDEs from other transients occurring in the cores of galaxies such as AGN flares (e.g., Gezari et al. 2009). Spectra of the events, in particular transient broad features of hydrogen and helium (Arcavi et al. 2014), are also used to separate the events from other phenomena, particularly supernovae. Finally, we theoretically expect the bolometric light curves to have a power law decline at late times (Rees 1988;Lodato 2012;Guillochon & Ramirez-Ruiz 2013), as opposed to an exponential decline that might be better associated with nuclear decay and thus a supernova origin.
In selecting data we were limited by the confines of our current model. For example, we currently do not fit x-ray radiation, and therefore we required events in our sample to have bolometric luminosities dominated by emission in the optical/UV. In addition to this, we only fit light curves that could be decently matched by a single temporal component, and are thus unable to fit events such as ASASSN-15lh that have a significant late time re-brightening that might arise from an emerging accretion disk (Margutti et al. 2017). We additionally select those events whose light curves have either wellsampled peaks or near-peak early time upper limits, or alternatively if they had detailed data of the decline in at least three optical/UV bands, as we expected these datasets to be capable of yielding informative measurements of black hole mass.
Events that satisfy these broad requirements are PS1-10jh (Gezari et al. 2012;Gezari et al. 2015), PS1-11af (Chornock et al. 2014
Fitting Procedure
MOSFiT currently uses a variant of the emcee ensemblebased MCMC routine (Foreman-Mackey et al. 2013) to find the combinations of parameters that yield the highest likelihood matches for a given input model (Guillochon et al. 2017a), where model errors are fitted simultaneously with model parameters by the variance parameter σ. To quantify how well the various combinations of parameters in the model fit each light curve, MOSFiT uses the Watanabe-Akaike information criteria (Watanabe 2010) or widely applicable Bayesian criteria (WAIC). This is used in place of the total evidence of the model: for objective functions where the likelihood function is not analytic and separable (such as in this semi-analytic model), it is difficult to evaluate the evidence exactly. While the WAIC score does not directly scale with the evidence, it is correlated with it, and can be used to rank fits between models (see Section 7 of Gelman et al. 2014). The WAIC is evaluated as follows,
WAIC = log p n − var(log p n ),(8)
where p n is the mean log likelihood score and var(log p n ) its variance. In addition to measuring the goodness of fit, it is important to ascertain whether or not a fit has converged. To this end, we use the Gelman-Rubin metric, or Potential Scale Reduction Factor (PSRF, signified withR) to gauge convergence (Gelman & Rubin 1992). This metric measures how well mixed each individual chain is as well as the degree of mixture between the different chains (for the definition, see Guillochon et al. 2017a).
For this multi-parameter model we used the maximum of the PSRFs computed for each parameter, so that the convergence of each fit was determined by the parameter with the slowest convergence. We attempted to run all of our fits until they reached a PSRF ≤ 1.2 (ensuring that the walkers are well-mixed within the regions of convergence (Brooks & Gelman 1998), however this was not possible for every fit. The 4 events with PSRF > 1.2 were refit multiple times, and continued to converge to the solutions we present here. For the work presented in this paper a minimum of 200 walkers and 30,000 iterations were used to recover the distribution of model fits.
Results
We show the results of the light curve fits in Figure 1, and the posterior distributions of the walkers in Figure 2. In Figure 1, the ensemble of light curves from the final walker positions are plotted. Although the model priors allow for long viscous times, the light curves of highest likelihood continue to closely follow the fallback rates. The viscous timescales and t peak values are shown in Table 1. The preferred viscous delays are less than 1% of t peak for all events modeled in this work; this preference is visible in the first column of panel plots in Figure 2. The minimal viscous delay of these events allows us to obtain precise black hole mass measurements as the luminosity evolution is still best described using the fallback rate, and the primary dependence of t peak is upon M h (as shown in Equation (2)).
In the absence of good photometry around peak, early time upper limits can still help us constrain the peak timescale and therefore the corresponding black hole mass, as shown in the plots for events D1-9, D3-13, PTF09djl, ASASSN-14li, ASASSN-15oi and ASASSN-14ae. For events without early time data or upper limits we can sometimes still obtain decent fits. The mass fallback rate and bolometric luminosity do not decline with a constant power law, and this helps MOSFiT find fits to events with well-sampled photometry but without early time data. For example, our initial fit to ASASSN-15oi was completed before we realized there existed an early time upper limit, however the black hole mass we measure with the addition of that upper limit is the same as what we found without it. The other parameters similarly maintained their previously measured values, the upper limit simply reduced the uncertainty in the measurements.
Good band coverage is also important, as it allows MOSFiT to accurately pin down different sections of the SED. This breaks multiple parameter degeneracies. For example, having data in several bands makes it possible to constrain the photosphere parameters R ph0 and l (the power law constant and exponent, as defined in Equation (7)).
The majority of the events in this sample are very well described by our current single-component model. These include PS1-10jh, PS1-11af, PTF09ge, PTF09djl, ASASSN-14ae, OGLE16aaa, D3-13, iPTF16axa, and iPTF16fnl. The light curves for these events have one clear peak in the optical and/or UV and monotonically decrease afterwards. They resemble veiled TDEs, in which the accretion disk is likely obscured by an optically thick photosphere or wind . These events are also seen to radiate most of their bolometric luminosity at UV/optical wavelengths. However, there are a few TDEs in this sample (ASASSN-14li, ASASSN-15oi, D1-9) that are not as well described by our current single-component model and likely require a secondary component to explain their late-time behavior.
As can be seen in Figure 7, the radius of the reprocess-ing layer in our model decreases at late times. Once the photosphere has receded to the size of the accretion disk, we expect higher energy photons to start contributing and ultimately dominating the light curve. As the luminosity decreases, the radiation from the accretion disk is expected to soften, potentially shifting the peak of the emission back into the optical bands. At the same time, as the photosphere recedes, less x-rays from the accretion disk are expected to be reprocessed, allowing us to observe them. These additional late-time components can change the decline of the light curve. Of this sample, it is likely that for ASASSN-14li, D1-9 and ASASSN-15oi ) these additional components might play a more prominent role in their light curves.
Although we did not model the origin of x-ray emission in this work, ASASSN-14li shows significant energy emitted at these wavelengths, which could be explained by the presence of a partially obscured accretion disk. In addition to this, the late time optical and UV data shows that the transient is re-brightening (Holoien et al. 2016b) (we did not include this late time data in our fit). ASASSN-15oi also has recently observed late time data that is fairly flat at optical/UV bands but shows an increasing x-ray component (Holoien et al. in prep), and we similarly did not include it in our fit of the event. D1-9 has poor late time coverage, however it appears to exhibit a re-brightening in g-band around MJD 53620 while remaining unusually flat at other optical bands. Another potential example of a two-component TDE in the literature is ASASSN-15lh. If ASASSN-15lh is indeed a TDE, then it requires a secondary late time component to explain the behavior of its light curve.
BLACK HOLE MASS PREDICTIONS
As discussed in the previous section, events with wellobserved peaks and data in multiple bands have wellconstrained black hole masses. The distributions of black hole masses for each event are shown in the last column of Figure 2, and the 68% confidence intervals are listed in Table 1. Figure 2 shows 2D histograms of all parameters plotted against black hole mass in order to see correlations between the different variables. The most obvious and consistent correlation is between the black hole mass and the time of peak. Nevertheless, we might expect other parameters to be mildly correlated with black hole mass as well. For example, the efficiency ( ), β, and the star mass all enter into the peak luminosity scaling relation with M h . However, when looking at columns 2, 5 and 6 in Figure 2, we see that none of these variables have a clear correlation with black hole mass-perhaps their combined influence dilutes their individual correlations with M h .
The masses of the black holes we fit are all in the ex- pected range between 10 6 and 10 8 solar masses. In Figure 3 we compare our results to mass measurements of the central black holes in the corresponding host galaxies using standard methods, and we find consistent results within reasonable errors. In this mass range there exist few black hole measurements and both the M − σ and bulge mass relations suffer from significant uncertainty (Greene et al. 2010;McConnell & Ma 2013). Therefore it is not surprising that masses derived using different galaxy scalings do not exactly match, as measurements in this range are rare and the required galaxy properties are difficult to measure. This makes the construction of an independent method even more valuable. We do note that our method results in systematically higher black hole masses than the M − σ relation. As we argue in Section 5, this provides a consistent picture on the nature of TDEs in which prompt flares, those that circularized quickly, are expected to be more frequent for higher mass black holes. The error bars from MOSFiT's measurements of black hole masses in Figure 3 are quite small. Although MOSFiT marginalizes over the errors in all of our model's free parameters, it is likely that we are underestimating the total error because our model provides a simple Example of the effect of a viscous delay on a TDE light curve. The plot shows g-band light curves for PS1-10jh with all parameters but the viscous time set to the best fit values (g-band is shown because it had good coverage over most of the light curve -all other bands are similarly affected). The best fit light curves are those with very small viscous delays. The plot also shows that Tviscous/t peak 0.1 yields a light curve that is essentially identical to the case with no viscous delay. There were no viscous delays 10 days or 10% of the peak timescale derived in any of our presented fits.
approximation of a complicated physical phenomenon. For example, changing the models for the disrupted stars from ZAMS polytropes with solar composition to more realistic MESA models will prevent the stellar mass of the disrupted star from being uniquely determined without additional knowledge about its evolutionary stage (and through that its radius). This will in turn affect the determination of the peak luminosity and peak timescale, allowing for those parameters to vary more and increasing the uncertainty in the black hole mass.
Influence of stellar mass
To test how changing the mass of the star changes the resulting fit, we performed fits of PS1-10jh while keeping the parameter for the mass of the star constant. We performed these tests for three different star masses: 0.1, 1, and 10 M . We found that all three tests achieved comparably good scores, implying that the mass of the star is a degenerate parameter that is difficult to measure accurately with our current model. However, the mass of the black hole does not change dramatically when fixing the stellar mass to different values-despite the uncertainty in the mass of the star we are still able to measure the mass of the black hole, however the variation in the black hole mass between tests implies larger uncertainty than our fits in which we leave the stellar mass free. Although only slightly favored by the evidence from the light curve fits, lower mass stars are far more common (Kroupa et al. 1993) and thus it is likely that the lower stellar masses are closer to the true value. The results from these tests are shown in Table 2 and are described further in Section 5.3.
We note that we find a slight preference for stellar masses near 0.1M for some events, which is near the peak in the initial mass function. In addition to such stars being much more common, the preference is likely contributed to by the fact that below this mass the radius of the star no longer cancels out the effect of the mass of the star on the time of peak of the light curve (see Equation 2) -the mass continues to decrease while the radius remains relatively constant as the mass transitions into the brown dwarfs. For simplicity we assumed the radius was constant below 0.1M in our current model, although in reality it is likely the radius will actually slightly increase below this mass, see Burrows et al. (2011). This changing mass-radius relationship means that the shortest possible peak times are achieved at M * ∼ 0.1M , and thus masses near 0.1M are favored for events in which short peak times are desired.
DISCUSSION
Luminosity Follows Fallback Rate
In Section 3.3 we briefly discussed how the luminosity appears to closely follow the fallback rate and that none of the events necessitate a viscous delay. Figure 4 shows how varying the viscous timescale changes the light curve of PS1-10jh -it is clear that the data is best fit when T viscous is a very small fraction of t peak .
For the luminosity to follow the fallback rate, the debris from the disruption must circularize promptly (or more precisely, while maintaining its initial mass-energy distribution) upon its return to pericenter (Guillochon et al. 2014). General relativistic effects are expected to play an important role for disruptions in which R p is comparable to the gravitational radius R g ≡ GM h /c 2 . Rapid circularization might be achieved through the effects of general relativity, which can strongly influence the trajectories of infalling material. GR precession effects can, for example, cause the stream of infalling debris to intersect itself (e.g., Dai et al. 2013), enabling a dissipation of kinetic energy, as seen in several recent hydrodynamical simulations (Hayasaki et al. 2013). This will naturally lead to rapid circularization.
If spin is included in the calculation, the stream deflects not only within its own orbital plane, but also out of this plane. The result is that the stream does not initially collide with itself (Stone & Loeb 2012) and circularization does not immediately occur. Because little dissipation occurs, the stream is extremely thin (Kochanek 1994;Guillochon et al. 2014) and wraps around the black hole many times (Guillochon & Ramirez-Ruiz 2015b). After a critical number of orbits, stream-stream interactions finally begin to liber-ate small amounts of gas. This eventually leads to a catastrophic runaway in which all streams simultaneously collapse onto the black hole, circularizing rapidly. For these events, the luminosity should still follow the original fallback rate so long as the mass-energy distribution of the debris remains unchanged (similarly to if rapid circularization had occurred), albeit after a fixed delay time post-disruption. Additionally, once circularization occurs the infalling material is likely to collect around the SMBH into a quasi-spherical layer. This layer is expected to quickly engulf the forming accretion disk, potentially leading to significant reprocessing of the emanated radiation. , we expect that lines with slopes of −2/3 will map to stars of different masses. Here we have assumed the Tout et al. (1996) relations for R * (M * ). There is a dependence on the impact parameter as well, and here we have set β = 1 for the dashed lines, however most of the fits prefer β near 1 and, as the plot implies, they also prefer stars between 0.1 and 1 M .
In Figure 5 we see that the majority of the fits prefer highly relativistic encounters, which naturally leads to the luminosity following the fallback rate. As mentioned in the previous section, we also find slightly larger black hole masses than those derived using standard galaxy scalings. Larger black holes have larger R g and can thus more easily cause relativistic disruptions. In Figure 5 we show that once M h is a few times 10 7 M , R g ≈ R t for M * ≈ 0.1M (the peak of the IMF), meaning that all disruptions in that parameter space are highly relativistic. In general, most of the fits prefer R p /R g 10. If R p /R g is calculated using the black hole masses from the M − σ relation (the masses that are systematically smaller than what MOSFiT measures), R p /R g increases from an average value of ≈ 12 to ≈ 25 for those disruptions (not all events in this selection have M − σ measurements for their black holes).
It has previously been postulated that we should expect a large number of TDEs to be viscously delayed, around 75% for the black hole mass range probed by the TDEs in this paper (Guillochon & Ramirez-Ruiz 2015b). Our results imply that we are therefore missing a number of viscously delayed TDEs. It is natural to ask why we seem to be biased towards these prompt, relativistic events. The most obvious explanation is simply that events that fall into this category tend to be easier to detect, as viscous delays can drastically flatten the peak of the light curve, as shown in Figure 4.
Dynamic Reprocessing Layer
TDEs can result in highly super-Eddington mass fallback rates (De Colle et al. 2012), and therefore we expect excess debris surrounding the black hole to reprocess light from the various dissipation regions (Loeb & Ulmer 1997;Ulmer et al. 1998).
This is particularly true for the events discussed in this work, as most of them are near full disruption (β fd = 1.8 for 4/3 polytropes and β fd = 0.9 for 5/3 polytropes), with large fractions of the disrupted star remaining bound to the black hole, as shown in Figure 6. Fraction of the total stellar mass that remains bound to the black hole versus the fraction of the Eddington limit the peak luminosity reaches.
As our model caps the luminosity of each flare to be no greater than the Eddington limit, our maximum radiated luminosities do not exceed Eddington for any of the modeled flares. As black holes near their Eddington limit, it becomes much more difficult to discern how much mass they are actually accreting as the luminosity depends little on the Eddington excess. This is reflected in the larger error bars of the events that are close to their Eddington limit.
The peak luminosities of most events are > 10% of their Eddington luminosities, and the peak bolomet-ric luminosity of the fitted events increases with black hole mass, suggesting the luminosities of the events are Eddington limited. Although this runs contrary to the inverse relationship between L peak and M h given by the peak luminosity scaling relation (Equation (1)), this is what we expect for Eddington limited events as L edd ∝ M h . Figure 7 shows the relationship between the radius and temperature of this reprocessing layer and the luminosity of the fits. In our fits where we have assumed that the size of the photosphere followsṀ to some power, the temperature we get from the emitting photosphere is comparable with that which has been derived from both fitting blackbodies to the photometry and from spectral observations, with peak values between 2 × 10 4 − 10 5 K (Arcavi et al. 2014;Blagorodnova et al. 2017;Brown et al. 2018;Cenko et al. 2016;Chornock et al. 2014;Gezari et al. 2008Gezari et al. , 2012Gezari et al. 2015;Holoien et al. 2014Holoien et al. , 2016bHung et al. 2017;van Velzen et al. 2011;Wyrzykowski et al. 2017). For the events that we fit, a single blackbody photosphere proved sufficient to match the optical and UV data.
Although we required the photosphere size to scale as a power law ofṀ , the parameter range used allowed the exponent of the power law to be zero, which would signify no correlation betweenṀ and R photo . Instead we found that for all fits the exponent was > 1/2 -the fits required that R photo be a strong function ofṀ . A similar power law relationship was used to fit the photospheric radius of simulations of TDEs in Jiang et al. (2016), and the power law exponent in that work was found to be ∼ 1, similar to what we find for some of the event fits presented here.
In Section 3.3 we discussed how our model for a growing and shrinking photosphere can help explain additional late time components in TDE light curves. This behavior can also help explain the minimal color evolution present in the light curves. Assuming that the size of the photosphere was set by the tidal radius or the last stable orbit (Loeb & Ulmer 1997;Ulmer et al. 1998), one might expect the temperature to fluctuate as the luminosity varied, as T ∝ L 1/4 . However, if the radius of the reprocessing layer increases with luminosity, then T ∝ L 1/4 /R 1/2 ∝ L 1/4 /L l/2 = L 1/4−l/2 where l is a power law exponent relating L and R (see Equation (7)). As can be seen in Table 1, we find that most fits prefer l > 1/2. Instead of the temperature increasing with luminosity, it decreases slightly near peak and then gradually increases as the luminosity decreases (Figure 7). Because the photosphere temperature is at a local minimum near peak, it can easily match observations that find approximately constant temperature at those times.
This can be interpreted as the result of reprocessing Photosphere temperature (K) Figure 7. Bolometric luminosity, photosphere radius, and photosphere temperature curves as a function of time since discovery. Each event's curves are colored distinctly and the shaded regions represents the 68% confidence intervals. The photosphere is approximated as a power law of L bol (see Equation 7), and the temperature plotted is the blackbody temperature of the photosphere.
the radiation by a layer of material with optical depth τ ∼ 1 in the accretion structures formed by the tidal disruption. The source of this material can be naturally explained by high-entropy material generated by the circularization process, of which only a fraction needs to be ejected to obscure the accretion disk (Guillochon et al. 2014). Just as prompt circularization allows the luminosity to follow the fallback rate, it might explain why the reprocessing radius follows the luminosity provided that the obscuring material drains into the black hole on timescales that are short enough to prevent a significant build-up of material. Another possible explanation is that the reprocessing layer is generated by a wind or an outflow (Ulmer et al. 1998;Strubbe & Quataert 2009;Miller 2015;Metzger & Stone 2016). This is described recently in Jiang et al. (2016), and we find that the temperature evolution seen in Figure 7 is reminiscent of the evolution they predict, although the exact power law relations we find betweeṅ M and the photosphere properties show a wider variety of solutions. The Jiang et al. model also predicts temperatures that decrease near peak, because the photospheric radius of the outflow grows with luminosity, and then temperatures that subsequently increase after peak as the ejecta eventually becomes transparent.
Summary and Future Prospects
• Black hole masses can be accurately measured using tidal disruption events. While the relationship between the time of peak of a TDE and the disrupting black hole's mass was first noted in Rees (1988) -t peak ∝ M 1/2 h , it remained unclear until this work if the luminous output of a disruption could be used to measure masses accurately. And although the black hole mass can be estimated from t peak alone, fitting multi-band light curves yields an increased precision of the measurement and makes it possible to learn about other key disruption parameters. Our measurements generally match previous values presented in the literature, as shown in Figure 3, but we do find some exceptions where the black hole masses acquired from light curve fitting disagree from those derived from galaxy scaling relations.
• All of the events in this sample have luminosity curves that almost directly follow the fallback of the stellar debris. This requires that the massenergy distribution remains frozen until it begins to radiate, which can be accomplished through rapid circularization (Hayasaki et al. 2013;Guillochon & Ramirez-Ruiz 2015a). However, it is unlikely that all TDEs experience rapid circularization (Guillochon & Ramirez-Ruiz 2015a), and there is still likely to be a class of TDEs that are viscously delayed and are thus generally overlooked in UV/optical surveys.
• These events are Eddington limited and in most cases significant fractions (∆M/M * > 0.1) of the disrupted stars are bound to the black holes (see Figure 6). In these cases there was likely a large amount of stellar debris surrounding the black hole after circularization that could reprocess light from the event.
• A reprocessing layer that evolves with the bolometric luminosity provides a good match to the optical and UV observations. This could be interpreted as high-entropy material that was generated during the circularization process and then quickly drained into the black hole on timescales short enough to avoid significant build-up. It could also be interpreted as an outflow of material that grows at early times and eventually becomes transparent (Jiang et al. 2016;Metzger & Stone 2016). Both of these scenarios could hide the accretion disk from view at early times, preventing X-rays from escaping until the reprocessing layer recedes and/or becomes transparent.
• Our results suggest that we are (unsurprisingly) biased towards observing the brightest TDEs, which are biased towards the largest black holes when the luminosity is Eddington-limited (but below ∼ 10 8 M as most stars are swallowed whole after that point). We find that events in our sample exhibit rapid circularization with no viscous delays lowering the peak luminosity, have luminosities that peak at a significant fraction of their Eddington limits, and are on the high mass end of potential host black holes for tidal disruptions.
While we are able to reliably obtain black hole masses from our analysis of light curves, we find the star and orbit properties are more difficult to determine uniquely. This is likely because the timescale at peak is insensitive to the star's mass, and also because the amount of mass that falls back onto the black hole is degenerate with the efficiency of the radiative process, which we remained agnostic about in this work. As a result, we are often able to find local solutions of similar quality even for radically different efficiency/star mass combinations. While the light curve fits are similar, we suspect that higher efficiency, lower mass solutions are preferable given their improved odds of occurring: low mass stars are significantly more likely to be disrupted than high mass stars. This degeneracy could be broken by a more complete library of stellar disruptions that accounts for relativistic effects (such as black hole spin, Tejeda et al. 2017) and stellar evolution (which affects composition, rotation, and central concentration) on the debris. By determining the stellar properties uniquely, we could reduce our systematic error in our black hole mass estimates from the range of values of the model estimates shown in Table 2, typically a factor of ∼ 2, to the statistical error bars of an individual model, ∼ 0.1 dex.
Our current model provides a solid basis for understanding events that radiate most of their energy in the optical/UV. In the future we plan to add an accretion disk component to our model, which will enable fits of TDEs that emit in the X-rays. We also plan to transition to a more realistic library of tidal disruption simulations (e.g. Law-Smith et al. in prep) that utilize MESA models of stars to account for their evolution. As explained above, we expect that this will break the current degeneracy between the mass of the star and the efficiency parameter and allow us to further refine our black hole mass estimates. Table 2. Comparison between test PS1-10jh runs with M * parameter set to different constant values: 0.1, 1.0, 10.0 M . While all runs converged with similar scores, we expect the run with M * = 0.1M to be the most likely true solution as these stars are much more common and are more likely to be disrupted.
Figure 1 .
1), PTF09djl (Arcavi et al. 2014), PTF09ge (Arcavi et al. 2014), iPTF16fnl (Blagorodnova et al. 2017; Brown et al. 2018), iPTF16axa (Hung et al. 2017), ASASSN-14li (Holoien et al. 2016b; Brown et al. 2017), ASASSN-15oi (Holoien et al. 2016a), Ensembles of TDE light curves each constructed from the posterior parameter distribution. The multicolor detections and associated upper limits are plotted for all selected TDEs.
Figure 3 .
3Comparison between the black hole mass estimates we derive from our model fits and those derived using the bulk properties of the host galaxy. The M h measurements from galactic properties come from the following sources: Arcavi et al. (2014); Blagorodnova et al. (2017); Brown et al. (2018); Chornock et al. (2014); Gezari et al. (2008); Guillochon et al. (2014); Holoien et al. (2014, 2016b,a); Hung et al. (2017); Mendel et al. (2014); van Velzen et al. (2011); Wevers et al. (2017); Wyrzykowski et al. (2017).Measurements are averaged and errors are added in quadrature where multiple measurements using the same method exist for a single black hole.
Figure 4.
Figure 5 .
5The dashed lines show Rt/Rg as a function of M h for differing M * . Because Rt/Rg ∝ M
Figure 6 .
6Figure 6.
2.1.1. Fallback EngineParameter
Prior
Min
Max
M h (M )
Log
10 5 5 × 10 8
b (scaled impact parameter a )
Flat
0
2
M * (M )
Kroupa 0.01
100
(efficiency)
Flat
0.005
0.4
R ph0 (photosphere power law constant)
Log
10 −4
10 4
l (photosphere power law exponent)
Flat
0
4
t first fallback (days since first detection bc )
Flat
-500
0
Table 1 .
1Here we list the parameters and priors used in our model.
https://tde.space
Figure 2. Posterior distributions of model parameters in the fit for each event as a function of M h . The plot shows that, for most events, t peak (not itself a model parameter) correlates strongly with M h .
Gezari et al. (2012), 2 Gezari et al. (2015), 3 Chornock et al. (2014), 4 Arcavi et al. (2014), 5 Blagorodnova et al. (2017), 6 Brown et al. (2018), 7 Hung et al. (2017), 8 Holoien et al. (2016b), 9 Holoien et al. (2016a), 10 Holoien et al. (2014), 11 Wyrzykowski et al. (2017), 12 Gezari et al. (2008), 13 van Velzen et al. (2011)Table 1. Here we list best fit parameters for all light curves with 1σ error bars.
We would like to thank Jamie Law-Smith and Nathaniel Roth for insightful input.We would also like to thank
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|
[] |
[
"Contribution to the Formal Specification and Verification of a Multi-Agent Robotic System",
"Contribution to the Formal Specification and Verification of a Multi-Agent Robotic System"
] |
[
"Nadeem Akhtar [email protected] ",
"Assistant Professor,Malik M Saad Missen [email protected] ",
"\nDepartment of Computer Science & IT\nIRISA -University of South Brittany\nFRANCE\n",
"\nThe Department of Computer Science & IT\nThe Islamia University of Bahawalpur Bahawalpur\n63100PAKISTAN\n",
"\nThe Islamia University of Bahawalpur Bahawalpur\n63100PAKISTAN\n"
] |
[
"Department of Computer Science & IT\nIRISA -University of South Brittany\nFRANCE",
"The Department of Computer Science & IT\nThe Islamia University of Bahawalpur Bahawalpur\n63100PAKISTAN",
"The Islamia University of Bahawalpur Bahawalpur\n63100PAKISTAN"
] |
[
"European Journal of Scientific Research"
] |
It is important to have multi-agent robotic system specifications that ensure correctness properties of safety and liveness. As these systems have concurrency, and often have dynamic environment, the formal specification and verification of these systems along with step-wise refinement from abstract to concrete concepts play a major role in system correctness. Formal verification is used for exhaustive investigation of the system space thus ensuring that undetected failures in the behavior are excluded. We construct the system incrementally from subcomponents, based on software architecture. The challenge is to develop a safe multi-agent robotic system, more specifically to ensure the correctness properties of safety and liveness. Formal specifications based on model-checking are flexible, have a concrete syntax, and play vital role in correctness of a multi-agent robotic system. To formally verify safety and liveness of such systems is important because they have high concurrency and in most of the cases have dynamic environment. We have considered a case-study of a multi-agent robotic system for the transport of stock between storehouses to exemplify our formal approach. Our proposed development approach allows for formal verification during specification definition. The development process has been classified in to four major phases of requirement specifications, verification specifications, architecture specifications and implementation. 36 reinforced during the analysis, design, and implementation of a multi-agent robotic system? These properties can be satisfied by having a multi-agent robotic system development approach based on formal methods and languages, having major phases of requirement specifications, verification specifications, architecture specifications, and implementation[Akhtar, 2010].Our area of research is formal methods for the specification and verification of a multi-agent robotic system. Model-checking has a degree of formalization that gives the flexibility to apply formal methods according to our implementation requirements. Our research domain focuses on formal methods, multi-agent systems, and robotics as shown infig.1.An agent is considered as a computer system situated in some environment, capable of autonomous actions in this environment in order to meet its design objectives[Wooldridge and Jennings, 1995]. Multiple agents are necessary to solve a problem, especially when the problem involves distributed data, knowledge, or control. A multi-agent system is a collection of several interacting agents in which each agent has incomplete information or capabilities for solving the problem[Jennings, Sycara and Wooldridge, 1998]. These are complex systems and their specifications involve many levels of abstractions. They have concurrency, and often have dynamic environments. A multi-agent robotic system is distributed. A distributed system along with the interactions between its components presents a high-level of complexity, and results into complex possible system behavior in different scenarios. Complete understanding of the system behavior is required for the analysis, design, and implementation of such a system. To overcome the complexity problems in them and get significant results with formal analysis, we must cope with complexity at every stage of development: from the specification phase to the analysis, design and verification phase. The formal specification and verification of a multi-agent robotic system along with its step-wise refinement from abstract to concrete concepts plays an important role in system correctness. Safety and liveness properties have to be enforced during each development phase of requirement specifications, verification specifications, architecture specifications, and implementation. Fig 1: Research domainOne of the most challenging tasks in software specification engineering for multi-agent robotic systems is to ensure correctness properties of safety and liveness, especially as these systems have high concurrency and in most cases have dynamic environment. Finite automata based model-checking of safety and liveness properties play major role in system correctness i.e. to verify that the code matches its requirement and design specifications is important.
| null |
[
"https://arxiv.org/pdf/1604.05577v1.pdf"
] | 595,841 |
1604.05577
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cae1536fee776a4259c777e9f3e67f6db5839af2
|
Contribution to the Formal Specification and Verification of a Multi-Agent Robotic System
January, 2014
Nadeem Akhtar [email protected]
Assistant Professor,Malik M Saad Missen [email protected]
Department of Computer Science & IT
IRISA -University of South Brittany
FRANCE
The Department of Computer Science & IT
The Islamia University of Bahawalpur Bahawalpur
63100PAKISTAN
The Islamia University of Bahawalpur Bahawalpur
63100PAKISTAN
Contribution to the Formal Specification and Verification of a Multi-Agent Robotic System
European Journal of Scientific Research
1171January, 2014Formal methodsCorrectness propertiesSafety propertyLiveness propertyFormal verificationMulti-agent robotic systemFormal architectureFinite State Process (FSP)Labelled Transition System (LTS)Architecture Description Language (ADL)
It is important to have multi-agent robotic system specifications that ensure correctness properties of safety and liveness. As these systems have concurrency, and often have dynamic environment, the formal specification and verification of these systems along with step-wise refinement from abstract to concrete concepts play a major role in system correctness. Formal verification is used for exhaustive investigation of the system space thus ensuring that undetected failures in the behavior are excluded. We construct the system incrementally from subcomponents, based on software architecture. The challenge is to develop a safe multi-agent robotic system, more specifically to ensure the correctness properties of safety and liveness. Formal specifications based on model-checking are flexible, have a concrete syntax, and play vital role in correctness of a multi-agent robotic system. To formally verify safety and liveness of such systems is important because they have high concurrency and in most of the cases have dynamic environment. We have considered a case-study of a multi-agent robotic system for the transport of stock between storehouses to exemplify our formal approach. Our proposed development approach allows for formal verification during specification definition. The development process has been classified in to four major phases of requirement specifications, verification specifications, architecture specifications and implementation. 36 reinforced during the analysis, design, and implementation of a multi-agent robotic system? These properties can be satisfied by having a multi-agent robotic system development approach based on formal methods and languages, having major phases of requirement specifications, verification specifications, architecture specifications, and implementation[Akhtar, 2010].Our area of research is formal methods for the specification and verification of a multi-agent robotic system. Model-checking has a degree of formalization that gives the flexibility to apply formal methods according to our implementation requirements. Our research domain focuses on formal methods, multi-agent systems, and robotics as shown infig.1.An agent is considered as a computer system situated in some environment, capable of autonomous actions in this environment in order to meet its design objectives[Wooldridge and Jennings, 1995]. Multiple agents are necessary to solve a problem, especially when the problem involves distributed data, knowledge, or control. A multi-agent system is a collection of several interacting agents in which each agent has incomplete information or capabilities for solving the problem[Jennings, Sycara and Wooldridge, 1998]. These are complex systems and their specifications involve many levels of abstractions. They have concurrency, and often have dynamic environments. A multi-agent robotic system is distributed. A distributed system along with the interactions between its components presents a high-level of complexity, and results into complex possible system behavior in different scenarios. Complete understanding of the system behavior is required for the analysis, design, and implementation of such a system. To overcome the complexity problems in them and get significant results with formal analysis, we must cope with complexity at every stage of development: from the specification phase to the analysis, design and verification phase. The formal specification and verification of a multi-agent robotic system along with its step-wise refinement from abstract to concrete concepts plays an important role in system correctness. Safety and liveness properties have to be enforced during each development phase of requirement specifications, verification specifications, architecture specifications, and implementation. Fig 1: Research domainOne of the most challenging tasks in software specification engineering for multi-agent robotic systems is to ensure correctness properties of safety and liveness, especially as these systems have high concurrency and in most cases have dynamic environment. Finite automata based model-checking of safety and liveness properties play major role in system correctness i.e. to verify that the code matches its requirement and design specifications is important.
Introduction
Today multi-agent robotic systems are not safe. Human lives can be lost due to errors in these systems, therefore it is important to have multi-agent robotic systems that are safe. Here by safe the emphasis is on correctness properties on the behavior of multi-agent robotic systems, the correctness properties that can be described by a combination of safety and liveness. How can safety and liveness properties be 37
Nadeem Akhtar and Malik M. Saad Missen
Our system consists of small robotic agents that work in a closed environment. Labelled Transition System (LTS) [Magee and Kramer, 2006] specifications based on Finite State Process (FSP) language have been used for specification definition of our multi-agent robotic system. These automata-based specifications are flexible, rigorous, executable and practical and play vital role in ensuring correctness properties. Therefore by using this automata-based model-checking approach we are able to obtain a concurrent system in which there are processes working in parallel and there are synchronizations between different processes. The LTS and its associated analysis tool LTSA have an incremental and interactive approach to the development of component based systems. Consequently, components can be designed and debugged before composing them into larger systems.
The Problem Statement, Objectives, and Contributions
Our problem statement is; How can a safe multi-agent robotic system be developed? Here by safe the focus is on correctness properties which can be described by a combination of safety and liveness. Thus the core question is how can safety and liveness properties be enforced during the development of a multi-agent robotic system?
The most challenging task in software specifications definition for robotic multi-agent systems is to ensure correctness. Safety and liveness properties are critical for system correctness. As these systems have concurrency, often have dynamic environments, the formal specification and verification of these systems; the step-wise refinement from abstract to concrete concepts play an important role in system correctness. It is important to address the following issues:
1. The formal specification of our multi-agent robotic system which has a dynamic architecture i.e. which can change during run-time; 2. To support the property-preserving transformations of agents from abstract to concrete specifications to code generation by stepwise refinement; 3. To support system verification using formal model-checking approaches; and to formally check the safety and liveness properties of the system. In order to address the above issues, a formal approach is required which does not rely solely on immediate software development, but on continuous engineering, adaptation, and evolution of the software system.
Our objective is to propose a development approach that provides formal verification of safety and liveness properties, architecture description, and a service-oriented simulation based system implementation. It results into the development of a multi-agent robotic system that satisfies correctness properties of safety and liveness. Another objective is the formal specification, architecture, and implementation by considering the functional properties; by refining in stepwise phases from abstract to concrete specifications along with the formal verification of these specifications.
This approach supports the efficient formal requirement specifications, verification specifications, architecture specifications, transformations, refinement from abstract to concrete concepts, and implementation of the system. The work aims to define and develop a formal architecture-based approach for the engineering of a multi-agent robotic system. The formal verification specifications i.e. verifying correctness properties of safety and liveness have been defined by labelled transition system based on finite state processes. For Formal architecture specifications, p-ADL dot NET [Oquendo, 2004] based formal architecture is specified. The system is implemented by Service-Oriented Architecture (SOA) based simulation.
Our contributions are;
1. An approach based on a combination of methods to allow for formal verification and evaluation during development phases of requirement specifications, verification specifications, architecture specifications, and implementation; 2. Checking correctness properties of safety and liveness at each development phase; 3. A multi-agent robotic system case study to exemplify each phase of this approach; 4. A combination of process algebra and finite automata based techniques to define the formal specifications of our system and verifying each flow of concurrent executions.
Background Studies
Formal Methods
Formal methods are based on a solid mathematical foundation. Formal specification has a precise mathematical semantics which in turn support formal verification. Formal verification allows mathematical rigorous proofs that specifications are according to the objectives, code is according to the specification, and code produces only the results that are required. These methods can achieve complete exhaustive coverage of the system thus ensuring that undetected failures in behavior are excluded. The core objective of a solid formal approach is to provide unambiguous and precise specification [George and Vaughn, 2003]. The requirements model based on mathematics create precise specification of the software, and ensure correctness. The formal representation of software requirements provides a way for logical reasoning about the construct produced and this achieves precise description and allows a stronger design that satisfies the required properties. As formal specification and verification techniques are getting more accomplished and mature, our capabilities to design and develop complex systems are also maturing and growing quickly. Formal notations are used to produce a complete detailed representation of the system that helps in the understanding, design, and development of the system. The requirements for distributed, large, and complex systems are complicated, problematic at the initial stages and evolve periodically throughout the life cycle. This creates a need for the method of requirement implementation to be flexible and robust, so that it can easily accommodate the continuous versions of change [Luqi and Goguen, 1997].
To overcome the complexity problems in multi-agent systems and get significant results with formal specifications, we must cope with complexity at each phase: requirement specification phase, architecture specification phase to design and implementation phases. We must assure formal verification during all phases. Formal verification can achieve complete exhaustive coverage of the system thus ensuring that undetected failures in the behavior are excluded. We can prove the correctness of agent software systems by formalizing critical components in the multi-agent development life-cycle. The reasons to have formal software engineering methods are:
• Rigorous analysis of system properties;
• Property-preserving transformations and error-free implementation • High quality of each phase of the development process;
• Firm foundation for the adaptation and evolution process;
• Continuous correctness especially as multi-agent robotic systems are concurrent and often have dynamic environments; • Formal specification and modeling of a multi-agent system architecture which can change at run-time; • Specification according to the functional and non-functional properties;
• Property-preserving step by step transformations from abstract to concrete concepts, then stepwise refinement to implementation code; • Improved documentation and understanding of specifications. Model-checking [Berard et al., 2001] [Clarke, Grumberg, and Peled, 2000] is a type of formal method used to verify concurrency properties; it can be viewed as exhaustive investigation of a system state space to prove certain correctness properties. Process calculi based symbolic techniques such as π-ADL [Oquendo, 2004], CSP [Hoare, 1978], CCS [Milner, 1980], ACP [Bergstra and Klop, 1987], and LOTOS [Van Eijk et al., 1989] provide formal specifications for complex systems. Here complex means a system with a large number of independent interacting components, with concurrency between components and constant evolution.
Correctness Properties: Safety and Liveness
The safety property is an invariant which asserts that "something bad never happens", which means that an acceptable degree of system working state is maintained. [Magee and Kramer, 2006] have defined safety property S = {a1, a2, … , an} as a deterministic process that asserts that any trace having actions in the alphabet of S, is accepted by S. ERROR conditions are like exceptions which state what is not required, as in complex systems we specify safety properties by directly stating what is required. The liveness property asserts that "something good happens" which describes the states of a system that an agent can bring under certain conditions. Progress property P = {a1, a2, … , an} defines a property P which asserts that in an infinite execution of the system, at least one of the actions a1, a2, … , an will be often executed infinitely [Giannakopoulou, Magee, and Kramer, 1999]. These properties play a vital role in system verification. Safety and liveness properties are complementary to each other, and both together are vital to ensure system correctness.
Gaia Multi-Agent Method
The Gaia [Zambonelli, Jennings, and Wooldridge, 2003] requirement specifications recognize the organizational structure as the core concept for the development of an agent system. A suitable choice of this organizational structure is required to meet the functional requirements. It is based on organizational abstractions to drive the analysis and design of a multi-agent system, and it considers a multi-agent system as a computational organization consisting of interacting roles. These organizational abstractions play a significant role in the analysis, design, and implementation of a multi-agent system in a complex environment. For Gaia, the word method is used instead of methodology, as we consider the term methodology to be used for the study of methods. It has a concrete syntax, which can be extended to deal with the formal specification aspects of a multi-agent system, and it generates a number of models and specifications that can be used by different software development methods for implementation. After the completion of the design phase, we have a well-defined collection of agent roles to implement, and can define the agent and service model. During the specification definition we move from abstract to concrete concepts, these abstract concepts conceptualize the system while the concrete concepts are used during the design phase, and are related to implementation.
The result of the design phase could be easily implemented in a technology neutral way. It captures the organizational structure of the system which allows for going systematically from the requirement analysis to a comprehensive design. For precise and unambiguous specifications, we need to formalize the specifications of each component and process. [Wooldridge and Jennings, 1995]. b. Once we have designed the specification, we must be able to implement a correct system with respect to this specification. The next issue is to propose an approach to move from an abstract specification towards a concrete model. Manually refine the specification into an executable form via some formal refinement process.
1. Directly execute the abstract specification along with its animation 2. Translate the specification into a concrete model using automatic translation techniques. c. It can play a significant role in the analysis and design of dynamic and open systems. In these systems components can join and leave the environment at runtime, and are composed of sub-components that may be different at design time and run time. They are a complicated class of systems to engineer [Gasser, 1991] [Hewitt, 1991]. d. The organization structure is implicitly defined in the role and interaction models. These structures capture and represent the organization's communication and control structures.
Labelled Transition System (LTS)
Labelled transition systems [Magee and Kramer, 2006] are mathematical objects for the formal verification and evaluation of concurrent systems. It is founded on model-checking for the verification of concurrency properties; it represents the system as a set of interacting finite state machines along with their properties; it exhaustively explores the system state space to prove correctness properties of safety and liveness, and it performs compositional reachability analysis to exhaustively search for violations of these properties. [Magee and Kramer, 2006] proposed an analysis tool LTSA [LTSA, 2006] shown in fig.2, that generates labelled transition system consisting of parallel composition of asynchronous processes, interleaving interaction-shared actions. As a result we are able to obtain a concurrent system in which there are processes working in parallel and there are synchronizations between different processes. LTSA also provides specification animation for an interactive exploration of system states.
FSP is a process algebra notation having finite state processes used for the concise description of component behavior particularly for concurrent systems. It is a finite-automata based method that provides construct to formalize specifications of software components and architecture. Each component consists of processes; each process has a finite number of states and is composed of one or more actions. There exists concurrency between elementary calculator activities for which there is a need to manage the interactions, communication and synchronization between processes. 41 Nadeem Akhtar and Malik M. Saad Missen
π-ADL
The π-ADL [Oquendo, 2004] provides the software engineer with the fundamental structure and behaviors constructions for describing static as well as dynamic software architectures. It is an executable formal specification language and supports automated analysis as well as refinement of dynamic architectures. The π-ADL has as mathematical foundation the higher-order typed π-calculus [Sangiorgi, 1992] [Milner, Parrow and Walker, 1992]. It is a well-formed higher-order calculus for defining dynamic and mobile architectural elements, which takes its foundation in work related to the use of π-calculus as a semantic foundation for architecture description languages ] ]. According to [Milner, 1999], a natural solution for specifying dynamic behavior is π-calculus as it provides a computation model which is Turing-complete. It is an ideal choice for describing concurrent processes that communicate through message passing. In πcalculus every computation can take place but it is not always easy to demonstrate and express. π-ADL is a language having both structural and behavioral architecture-centric constructs, defined as an extensive version of the higher-order typed π-calculus. Fig-3 shows the architectural concepts of π-ADL.
Fig 3:
Architectural concepts in π-ADL [Oquendo, 2004] It achieves high architecture expressiveness and Turing completeness with the help of a simple formal syntax notation. As with any design of a language, the design of π-ADL makes tradeoffs between competing requirements and constituencies:
1. Making the language well suited for machine-automated processing for enactment, analysis, refinement and evolution vs. as a stand-alone language for humans: π-ADL is specified in a layered-approach with a core canonical abstract syntax and formal semantics, and different concrete human-oriented notations [Oquendo, 2005]. 2. Making the language well suited for software architects to design large-scale software vs.
making it automatic semantically tractable: π-ADL is based on a compositional approach [Oquendo, 2005]. According to [Oquendo, 2005] the π-ADL design follows the following language design principles [Morrison, 1979] [Sangiorgi, 1992] [Strachey, 1967] [Tennent, 1977]:
1. Correspondence principle: the uses of names are consistent in π-ADL. Particularly there is a one to one relationship between the method of introducing names in declarations and parameter lists; 2. Abstraction principle: all major syntactic structures have abstractions defined over them e.g. π-ADL supports abstractions over behaviors as well as abstractions over data; 3. Data-type completeness principle: each data-type is a first-class citizen having no restrictions on its use.
The Proposed Approach
An approach has been proposed for the formal specification and verification of multi-agent robotic system. The requirements are specified, formally verified on the basis of safety and liveness properties, the architecture is specified, and the system is implemented. Our proposed approach is a combination of multi-agent methods, languages, and techniques; which takes into account the safety and liveness properties at each phase of development. This approach is exemplified by a case study of a multi-agent robotic system [Akhtar, 2010]. Our approach starts by the identification of components and sub-components of the system i.e. each and every part of the system that can be formally defined. Each component is formally verified and validated, particularly the critical components. The approach consists of four main development phases of requirement specification, requirement verification, architecture specification, and system implementation as shown in fig.4.
It is exemplified by a case study which is a multi-agent system composed of robotic agents. The mission is to transport goods from one storehouse to another. The robotic agents named as carrier agents transport goods from one storehouse to another. There is a possibility of collision between these carrier agents and collision resolution techniques are applied to avoid system deadlock. The requirements are specified by using Gaia [Zambonelli, Jennings, and Wooldridge, 2003]. Gaia has a concrete syntax to express properties, and it is suitable to model behaviors. Formal verification of correctness properties of safety and liveness is done by defining the system as a labelled transition system which uses FSP as input. FSP based on process algebra is a formal language which is specifically useful for specifying concurrent behavior. LTS has processes executing concurrently, with each process having one or more actions and synchronization between parallel processes by action sharing. During verification each sub-portion of the system is formally verified to make it consistent with the rest of the system, and at the end the system is verified as a whole. Transformations are made from the Gaia requirement specifications to LTS verification specifications for formal verification of the system. The architecture is specified by π-ADL dot NET language which provides a formal executable architecture model consisting of abstractions and behaviors. These architecture specifications describe the static, as well as dynamic aspects of architecture. The system is implemented by service-oriented based C# simulation implementation.
Requirement Specification
The requirement specification phase starts with the identification of early requirements. It is followed by the specification of a multi-agent system as an organization. In this organization, there are multiple abstraction levels. The organizational rules are defined, which put forth the global system properties; global relationships between roles; global relationships between protocols; global relationships between roles and protocols; and constraints within which the system has to work. The environmental model, which studies the environment and entities related to it, is defined. The role model has responsibilities and permissions, the responsibilities are expressed by safety and liveness properties. Agent roles are also defined. A single agent can have one or more roles but a single role cannot be performed by more than one agent. Safety and liveness properties are defined in this initial phase along with the definition of agent roles. At this phase these properties can be defined by regular expressions or by first-order predicate logic. Protocols are defined between agent roles which define the interactions between agents. A services model is defined, where each service is defined according to input, output, pre-condition, and post-condition. These requirement specifications have a well-defined formal semantics. They capture the organizational structure of the system. They allow for going systematically from the requirement analysis to a comprehensive design. For precise and unambiguous specifications, we need precise mathematical semantics. Organization structures are defined implicitly inside these requirement specifications, within the role and interaction models. It is important to have a precise knowledge of the terms and concepts of a method. Once we specify the system, we would be able to implement a system that is correct with respect to our specifications. The next step is to move from abstract specifications to a concrete computational model.
During requirement specification phase, the safety properties are defined using first-order predicate logic, while liveness properties are defined using regular expressions. Here, it is to be noted that the Gaia role model does not have constructs for the formal verification of safety and liveness properties; therefore the formal verification of these properties is carried out in the next phase of our approach. The agent model identifies the agent instances; and at the end, the acquaintances model is defined which gives a global picture of agents, their environment along with their interactions. The major emphasis throughout this phase is on the safety and liveness properties.
Requirement Verification
Major emphasis is put on the requirement verification and the safety and liveness properties defined in the requirement specification phase are verified in this phase. The system is broken down into subcomponents. Each component is verified by a formal model-checking exhaustive method. This involves exhaustive verification of all the states, processes, and actions of each component along with its sub-component. After that all the sub-components are assembled together and the system is verified as a whole [Akhtar, Guyadec, and Oquendo, 2009].
Finite state process is a process algebra notation used for the concise description of component behavior particularly for the concurrent systems. It has strong artifacts for construction of concurrent processes, and therefore it is ideal for concurrent systems. It provides the constructs to formalize the specification of software components, each component consists of processes and each process has a finite number of states and is composed of one or more actions. The processes are modelled as a sequence of actions, and formal specification of dynamic behavioral aspects of the multi-agent robotic system are provided, correctness properties of safety and liveness are verified, along with progress property, deadlock freedom, and sequencing constraints. Concurrency exists between elementary calculator activities; processes are sequential or concurrent and there is management of the interactions, communication, and synchronization between processes.
The correctness properties of safety and liveness defined in requirement specification phase along with the multi-agent system environment are now specified as a labelled transition system. There are actions, processes, states, and transitions between states. The verification specifications developed are a discrete system with a trace of actions; there are parallel processes with synchronization between them by action sharing.
Moving from Requirement Specification to Requirement Verification
There is a satisfaction relation between requirement specification and requirement verification. This satisfaction relation is the formal verification of the two correctness properties of safety and liveness. This satisfaction relation is exemplified by a case study. The Gaia role model liveness and safety properties along with the organizational rules are specified in the form of finite automates for verification.
Architecture Specification
A formal architecture [Akhtar, Guyadec, and Oquendo, 2012] has been proposed which specifies the static, as well as the dynamic aspects of the system. In this architecture, the architectural elements are identified. All these architectural elements are separately specified and then connected together to represent the system as one unit. The system architecture is based on π-ADL dot NET which is a dot NET extension of π-ADL [Oquendo, 2004]. We have an architecture consisting of abstractions and behaviors that is formal, consisting of components and connectors, that executes and that can change dynamically during the executions.
1. These architecture specifications provide a formal system having a mathematical foundation that can be used to describe static as well as the dynamic software architecture. 2. They have as formal foundation the higher-order typed π-calculus [Sangiorgi, 1992]. It is a well-formed higher-order calculus for defining communicating and mobile architectural elements. 3. They focus on the formal specification of architecture from the run-time perspective: the run-time structure, the run-time behavior, and the evolution of architecture over time. 4. They are executable i.e. a virtual machine executes the software architectures specifications. 5. They support multiple concrete syntaxes: both textual and graphical notations. 6. They support automated verification of properties by model checking.
Moving from Requirement Specification to Architecture Verification
The architecture specifications are based on requirement specifications. When we move from requirement to architecture then the safety and liveness properties should be preserved. The whole system is represented in the form of π-ADL dot NET with emphasis on safety and liveness properties.
Moving from Architecture Specifications to Simulation Implementation
The system is implemented as a simulation which reflects the architecture specifications; from π-ADL based system to Service-Oriented Architecture (SOA) based robotic simulation system. There should be conformations between the properties of architecture specification and simulation implementation.
System Implementation
The system is simulated based on SOA with each component as a service, with components having one or more sub-components with every sub-component implemented by a service. These services are loosely integrated and are orchestrated together by an orchestration service. As a result, we have a system that has reusable components.
This services based simulation is implemented by programming C# based Microsoft Robotics Developer Studio (MRDS) services. A refinement relation has been defined between the architecture specifications and these C# based services. It is an implementation of the LTS specifications in a simulation environment. In our system, each and every application is a service. An application is a composition of loosely-coupled concurrently executing components. For example: the carrier robot consists of a number of services orchestrated together. It has two wheels with a motor, sensors comprising of two bumpers for collision detection, and infrared laser for distance measurement and collision avoidance. Each of these motor, bumper, and infrared lasers is implemented by a service. There is a service for the orchestration of these sensors, motor, and actuator.
Moving from Implementation to Verification Specification
The robotics simulation implementation specifications must satisfy the finite automata based LTS system. Both implementation and verification specifications should preserve the safety and liveness properties. These LTS properties should also be preserved during the simulation implementation.
The simulation is continuous with a continuous flow of actions. Each part of the simulation is a service along with the orchestration of services. On the other hand, the LTS based system is a much lower abstraction level; has concurrent processes; each process having discrete actions. There are discrete states and the system moves from one state to another. The simulation which is continuous system must satisfy the discrete LTS system. The safety and liveness properties should be preserved in both, and there should be a clear relationship between the two systems. The simulation specifications are able to satisfy the verification specifications. In our simulation model, we create a trace of actions that is equivalent to the trace of actions created by LTS specifications. A mapping of activities provides trace equivalence among requirement specification, verification specification, architecture specification, and implementations.
Case Study: Multi-Agent Robotics Transport System
In this section we present a case study of multi-agent robotics system. It is a system composed of robotic transporting agents. The objective is to specify our system and then verify the correctness properties of safety and liveness. The mission is to transport stock from one storehouse to another. They move in their environment which in this case is static i.e. topology of the system does not evolve at run time. There is a possibility of collision between agents during the transportation. Collision resolution techniques are applied to avoid system deadlocks. We have specified each and every part of the system i.e. agents along with the environment in the form of LTS.
Types of Agents
There are three types of agents 1. Carrier agent: It transports stock from one store-house to another; can be loaded or unloaded and; can move both forward and backward direction. Each road section is marked by a sign number and the carrier agent can read this number. 2. Loader / Un-loader agent: It receives/delivers stock from the storehouse, can detect if a carrier is waiting (for loading or unloading) by reading the presence sensor, it ensures that the carrier waiting to be loaded is loaded and the carrier waiting to be unloaded is unloaded. 3. Store-manager agent: manages the stock count in the storehouse and it also transports the stock between storehouse and loader/un-loader.
Environment
There is a road between storehouse-A and storehouse-B which is composed of a sequence of interconnected sections of fixed length. Each road section has a numbered sign, which is readable by carrier agents. There are three types of road sections depending upon the topology of the road as shown in fig.9. Each of the three types of road sections has a unique numbered sign. The road is single lane and there is a possibility of collision between agents. There is a roundabout at storehouse-A and storehouse-B.
Scenario
In this case study we have used a road topology consisting of nine road partitions to represent all states and processes as shown in fig.9. It is the smallest circuit (i.e. combination of road partitions) that allows us to study all properties that would be in a much larger circuit. We have considered the case in which initially storehouse-A is full and storehouse-B is empty. The carrier task is to transport stock from storehouse-A to storehouse-B until the storehouse A is empty. Loader at the storehouse-A loads, and the un-loader at the store-house-B unloads the carrier agent. The store-manager keeps a count of stock in each storehouse. In this case the environment is static. At the central section (3,4,5) there is a possibility of collision between carrier agents coming from the opposite directions. Priority is given to the loaded carriers i.e. if there is a collision between a loaded and an empty carrier than the empty carrier moves back and waits at the parking region during which the loaded carrier passes and unloads. The parking region as shown in the fig.9 consists of the road partition 8.
Gaia Based Requirement Specifications
The major part of the work is to take the Gaia specifications and then use them in a way that they can be verified by using FSP language. Gaia method as described in section-4 consists of a number of models, we may be looking into only the roles model and interaction model which constitutes the analysis phase of Gaia.
Agent Roles
The role of an agent defines what it is expected to do in the organization, both in concert with other agents and in respect of the organization itself. Often an agent's role is simply defined in terms of the specific task that it has to accomplish in the context of the overall organization. Organizational role 48 models precisely describe all the roles that constitute the computational organization. They do this in terms of their functionalities, activities, responsibilities as well as in terms of their interaction protocols and patterns. In the role model the liveness and safety expressions play important role for system verification.
In our system for the carrier agent there are move_full and move_empty roles. These roles are better adapted for this type of route where priority is given to the loaded carriers. Here in this paper due to space constraints we present the Move_full role of our system i.e. role of a loaded carrier agent moving from Storehouse-A to Storehouse-B. Here activities (underlined) are ReadSign, MovetoNext, CollisionSensorTrue, CarrierWait, and ReadUnloadSign. And there are two protocols WaitforUnloading and UnloadCarrier WaitforUnloading: when a loaded carrier reads the unload sign i.e. it reaches the unload road partition, it stops there and waits until it is unloaded.
Consider the Liveness property of the Move_full role. It shows all the activities and protocols that make up the role. The carrier has two choices, first it can read sign and move to the next road partition, second it detects the collision sensor then it waits, at the end it reads the unload sign i.e. at the road partition in front of the un-loader, and in this case the carrier stops and waits for being unloaded, so now it's no more a loaded carrier and is no more part of the Move_full role. The safety property is an invariant which states that any carrier playing that role schema is currently loaded. Here next_position identifies the direction of the loaded carrier at the route.
Interaction Model
There are dependencies and relationships between the various roles in a multi-agent organization which are the set of protocol definitions, one for each type of inter-role interaction. Here table-1 shows the protocol definitions related to Move_full and Move_empty role.
Fig 2 :
2The toolkit LTSA
Fig 4 :
4A detailed view of the proposed approach
Fig 5 :
5Requirement specification phase
Fig 6 :
6Requirement verification phase
Fig 7 :
7Moving
Fig 8 :
8Satisfaction relation from implementation to requirement verification
Fig 9 :
9Environment consisting of road partitions (a) N is the unique numbered sign. P is the parking Flag (TRUE or FALSE) e.g. the section that can be used as a parking.
Table 1 :
1Gaia: Abstract and concrete concepts [Zambonelli, Jennings, and Wooldridge a. A successful correct method has a well-defined formal basis. Formal basis provides the precise understanding of the terms and concepts used in a methodAbstract concepts
Concrete concepts
Roles
Permissions
Responsibilities
Liveness properties
Protocols
Activities
Safety properties
Agent types
ServicesAcquaintances
Table 2 :
2Move_full Role Schema: Move_full Description: Role of a loaded carrier moving from storehouse A to storehouse B. Protocols and Activities: readSign, movetoNext, collisionSensorTrue, carrierWait, readUnloadSign, waitforUnloading, unloadCarrier next_position (external) /// (True or False) checks if the next position is available Responsibilities: Liveness: Move_full = Move.(readUnloadSign.waitForUnloading.unloadCarrier) Move = (readSign. movetoNext)+ | (collisionSensorTrue.Wait).(readSign.movetoNext)+ Wait = carrierWait+ Safety: (sign number Є {2,…,6} ⇒ isLoaded) ˄ (sign number Є {2,…,6} ∧ next position = sign number+1 ⇒ isLoaded)Permissions:
reads:
sign_number (external)
collision_sensor (internal)
changes:
position (internal)
Table :
:Move_full role protocols waitForUnloading Move_full Unload sign_number The full carrier agent waits for the un-loader agent position Unloading Move_full Unload sign_number The full carrier agent waits for the un-loader agent position The empty carrier agent waits for the loader agent position loadCarrier Move_empty load sign_number The empty carrier agent is loaded by the loader agent positionTable:
Move_empty role protocols
waitforLoading
Move_empty
load
sign_number
AcknowledgementWe are grateful to Prof. Dr. Muhammad Mukhtar, Vice Chanceller, The Islamia University of Bahawalpur for motivating us for doing research projects, and his support and encouragement for applied research. This work has been possible due to the support of The Department of Computer Science & IT, The Islamia University of Bahawalpur, Pakistan.LTS VerificationRoad -System EnvironmentIn our case study the road is environment and each carrier has its particular route. The route is the path taken by carrier agents on the road to transfer stock from one storehouse to another. The route has been classified in two types the FULL_ROUTE path taken by loaded carriers and the EMPTY_ROUTE path taken by the empty carriers. The carrier agents move on the route in a clockwise direction. Here below are the FSP specifications for the route.Carrier AgentThe next step is to specify the carrier agents i.e. specify the empty-carrier and full-carrier agents. Here only one carrier agent is taken to represent all the possible states of the system that can arise.Loader & Un-loader AgentsLoader and un-loader agent loads and un-loads the carrier agents respectivelyStock ManagementStock management ensures that the stock at the beginning of the case study at storehouse A is equal to the stock at the end of the case study at storehouse B.Contribution to the Formal Specification and Verification of a Multi-Agent Robotic SystemNOLOSS PropertySafety property NOLOSS of Carrier agent infers that there is no loss of stock during the carrier load, unload, and movements between the storehouses. To represent the LTS here with all its states, we have taken a mini-route with only three road partitions. The carrier is loaded and then the carrier is full, there is no loss of stock during the carrier agent's trajectory between storehouse A and B. Safety property specifies every trace that satisfies the property for a particular action alphabet. If the system produces traces that are not accepted by the property automata then a violation is detected during reachability analysis. Making graphical tools which may lead to an easy drag and drop graphical programming interface for a robotic application development based on π-ADL dot NET language. This programming interface allows novice programmers to program graphically without having knowledge of the underlying rigorous formal methods and languages.Concluding NotesThe major contribution is the development of an approach for a multi-agent robotic system that satisfy qualities of correctness i.e. safety and liveness property. An approach based on a combination of methods, formal languages, and techniques is proposed to support the efficient formal description; requirements gathering; formal specification definition; transformation; refinement from abstract to concrete concepts; and verification of multi-agent robotic systems. This approach is exemplified by a case study of a multi-agent robotic system. The proposed formal approach phases have key aspects of: organizational abstractions, organizational rules, requirement specifications, role model specifications, protocol definitions, formal requirement verification on the basis of correctness properties, LTS creation, formal static architecture specifications, formal dynamic architecture specifications, and Service Oriented Architecture based simulation implementation. The approach has models based on formal methods and it revolves around Contribution to the Formal Specification and Verification of a Multi-Agent Robotic System 54 formal verification of correctness properties in each phase from early requirements to the implementation i.e. Gaia method based requirements, Finite state process based finite automata formal verification, π-ADL dot NET language based formal architecture, and services based C# simulation implementation.The major goal is to facilitate greater assurance to component's correctness. The complete system is specified as a parallel composition of processes and each process synchronizes by means of shared actions. Our system has concurrency, synchronization, correctness, and deadlock issues to be handled and formal model-checking automata-based development methods offer solutions for these issues. Another objective is the use of formal analysis during analysis and design to discover correctness and safety problems early in the development cycle.
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"J K Zettler ",
"@ T Flissikowski ",
"P Koenraad ",
"⊥ H T Grahn ",
"L Geelhaar ",
"S Fernández-Garrido ",
"O Brandt [email protected] ",
"\n†Paul-Drude-Institut für Festkörperelektronik\nLeibniz-Institut im Forschungsverbund Berlin e. V\nEPFL Innovation Park, Bât. D\nAttolight AG\nHausvogteiplatzBerlin, LausanneGermany, Switzerland\n",
"\nDepartamento de Física de Materiales\nABB Corporate Research\nBaden-DättwilSwitzerland\n",
"\n⊥Department of Applied Physics, TU Eindhoven, AZ Eindhoven\nUniversidad Complutense de Madrid\nMadridSpain, The Netherlands\n",
"\nAutónoma de Madrid\nLayTec AG\nSeesener Str. -, C/ Francisco Tomás y ValienteBerlin, MadridGermany, Spain\n"
] |
[
"†Paul-Drude-Institut für Festkörperelektronik\nLeibniz-Institut im Forschungsverbund Berlin e. V\nEPFL Innovation Park, Bât. D\nAttolight AG\nHausvogteiplatzBerlin, LausanneGermany, Switzerland",
"Departamento de Física de Materiales\nABB Corporate Research\nBaden-DättwilSwitzerland",
"⊥Department of Applied Physics, TU Eindhoven, AZ Eindhoven\nUniversidad Complutense de Madrid\nMadridSpain, The Netherlands",
"Autónoma de Madrid\nLayTec AG\nSeesener Str. -, C/ Francisco Tomás y ValienteBerlin, MadridGermany, Spain"
] |
[] |
Several of the key issues of planar (Al,Ga)N-based deep-ultraviolet light emitting diodes could potentially be overcome by utilizing nanowire heterostructures, exhibiting high structural perfection and improved light extraction. Here, we study the spontaneous emission of GaN/(Al,Ga)N nanowire ensembles grown on Si(111) by plasmaassisted molecular beam epitaxy. The nanowires contain single GaN quantum disks embedded in long (Al,Ga)N nanowire segments essential for efficient light extraction. These quantum disks are found to exhibit intense emission at unexpectedly high energies, namely, significantly above the GaN bandgap, and almost independent of the disk thickness. An in-depth investigation of the ac-tual structure and composition of the nanowires reveals a spontaneously formed Al gradient both along and across the nanowire, resulting in a complex core/shell structure with an Al deficient core and an Al rich shell with continuously varying Al content along the entire length of the (Al,Ga)N segment. This compositional change along the nanowire growth axis induces a polarization doping of the shell that results in a degenerate electron gas in the disk, thus screening the built-in electric fields. The high carrier density not only results in the unexpectedly high transition energies, but also in radiative lifetimes depending only weakly on temperature, leading to a comparatively high internal quantum efficiency of the GaN quantum disks up to room temperature. arXiv:1905.04090v1 [cond-mat.mes-hall]
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10.1021/acs.nanolett.9b01521
|
[
"https://arxiv.org/pdf/1905.04090v1.pdf"
] | 150,374,044 |
1905.04090
|
a1ba2c4d98510b1c9e604430808a7240a615896a
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Absence of quantum-confined Stark effect in GaN quantum disks embedded in (Al,Ga)N nanowires grown by molecular beam epitaxy
10 May 2019
C Sinito
P Corfdir
¶ C Pfüller
G Gao
J Bartolomé Vílchez
S Kölling
⊥ A Rodil Doblado
# U Jahn
J Lähnemann
T Auzelle
J K Zettler
@ T Flissikowski
P Koenraad
⊥ H T Grahn
L Geelhaar
S Fernández-Garrido
O Brandt [email protected]
†Paul-Drude-Institut für Festkörperelektronik
Leibniz-Institut im Forschungsverbund Berlin e. V
EPFL Innovation Park, Bât. D
Attolight AG
HausvogteiplatzBerlin, LausanneGermany, Switzerland
Departamento de Física de Materiales
ABB Corporate Research
Baden-DättwilSwitzerland
⊥Department of Applied Physics, TU Eindhoven, AZ Eindhoven
Universidad Complutense de Madrid
MadridSpain, The Netherlands
Autónoma de Madrid
LayTec AG
Seesener Str. -, C/ Francisco Tomás y ValienteBerlin, MadridGermany, Spain
Absence of quantum-confined Stark effect in GaN quantum disks embedded in (Al,Ga)N nanowires grown by molecular beam epitaxy
10 May 2019‡Present address: ¶Present address: §Present address: Materials Science and Nanoengineering Department, Rice University, Houston, Texas , United States of America Present address: #Present address: Biomedical Neuroengineering Research Group (nBio), Bioengineering Institute, Universidad Miguel Hernández, Elche , Spain @Present address: Present address: Grupo de electrónica y semiconductores, Dpto. Física Aplicada, Universidad
Several of the key issues of planar (Al,Ga)N-based deep-ultraviolet light emitting diodes could potentially be overcome by utilizing nanowire heterostructures, exhibiting high structural perfection and improved light extraction. Here, we study the spontaneous emission of GaN/(Al,Ga)N nanowire ensembles grown on Si(111) by plasmaassisted molecular beam epitaxy. The nanowires contain single GaN quantum disks embedded in long (Al,Ga)N nanowire segments essential for efficient light extraction. These quantum disks are found to exhibit intense emission at unexpectedly high energies, namely, significantly above the GaN bandgap, and almost independent of the disk thickness. An in-depth investigation of the ac-tual structure and composition of the nanowires reveals a spontaneously formed Al gradient both along and across the nanowire, resulting in a complex core/shell structure with an Al deficient core and an Al rich shell with continuously varying Al content along the entire length of the (Al,Ga)N segment. This compositional change along the nanowire growth axis induces a polarization doping of the shell that results in a degenerate electron gas in the disk, thus screening the built-in electric fields. The high carrier density not only results in the unexpectedly high transition energies, but also in radiative lifetimes depending only weakly on temperature, leading to a comparatively high internal quantum efficiency of the GaN quantum disks up to room temperature. arXiv:1905.04090v1 [cond-mat.mes-hall]
Abstract
Several of the key issues of planar (Al,Ga)N-based deep-ultraviolet light emitting diodes could potentially be overcome by utilizing nanowire heterostructures, exhibiting high structural perfection and improved light extraction. Here, we study the spontaneous emission of GaN/(Al,Ga)N nanowire ensembles grown on Si(111) by plasmaassisted molecular beam epitaxy. The nanowires contain single GaN quantum disks embedded in long (Al,Ga)N nanowire segments essential for efficient light extraction. These quantum disks are found to exhibit intense emission at unexpectedly high energies, namely, significantly above the GaN bandgap, and almost independent of the disk thickness. An in-depth investigation of the ac-tual structure and composition of the nanowires reveals a spontaneously formed Al gradient both along and across the nanowire, resulting in a complex core/shell structure with an Al deficient core and an Al rich shell with continuously varying Al content along the entire length of the (Al,Ga)N segment. This compositional change along the nanowire growth axis induces a polarization doping of the shell that results in a degenerate electron gas in the disk, thus screening the built-in electric fields. The high carrier density not only results in the unexpectedly high transition energies, but also in radiative lifetimes depending only weakly on temperature, leading to a comparatively high internal quantum efficiency of the GaN quantum disks up to room temperature. (Al,Ga)N is the material of choice for the fabrication of deep ultraviolet (UV) solid-state emitters with a wide range of applications in biological, medical, and communication areas. , However, planar (Al,Ga)N-based heterostructures suffer from a number of problems, among them high densities of threading dislocations, low dopant activation and carrier injection, poor light extraction, and strong internal electrostatic fields in quantum wells. As a result, the currently existing (Al,Ga)Nbased UV light emitting diodes (LEDs) still exhibit comparatively low external quantum efficiencies, and their performance degrades with decreasing wavelength. , -(Al,Ga)N-based heterostructures in the form of three-dimensional nanostructures have the potential to resolve several of these issues challenging conventional planar devices. Consequently, the fabrication of such nanostructures is currently the subject of worldwide research activities, utilizing a wide range of techniques ranging from basic topdown and bottom-up , approaches to hybrid schemes combining top-down processes and subsequent regrowth. , A particularly active field of research is the synthesis of GaN/(Al,Ga)N nanowires by molecular beam epitaxy (MBE). These nanowires tend to form on a variety of structurally and chemically dissimilar substrates, invariably yielding material of high structural perfection, since dislocations do not propagate along the nanowire growth axis.
In addition, the efficient elastic strain relaxation at the nanowire sidewalls facilitates the fabrication of GaN/(Al,Ga)N heterostructures with coherent interfaces. Moreover, the elastic strain relief leads to reduced polarization fields as compared to planar layers, which is expected to result in higher internal quantum efficiencies. Finally, the high geometric aspect ratio of nanowires offers superior light extraction.
In most previous works utilizing MBE, the structures under investigation consisted of axial GaN/(Al,Ga)N nanowires containing an active region formed by GaN or (Al,Ga)N quantum disks clad by comparatively thin (Al,Ga)N or AlN barriers. Ultimately, a nanowire heterostructure serving as efficient deep UV emitter should be free of GaN segments to avoid any reabsorption of the light emitted by the quantum disk. At the same time, a GaN quantum disk embedded in a pure (Al,Ga)N nanowire or a long (Al,Ga)N segment will exhibit a significantly larger internal field than the same disk in a GaN nanowire with short (Al,Ga)N segments. , Since the radiative recombination rate of charge carriers confined in the disk critically depends on the magnitude of the internal electrostatic field, it is of high relevance for future applications to investigate the spontaneous emission of GaN quantum disks embedded in (Al,Ga)N nanowires.
In this Letter, we examine the carrier recombination of GaN/(Al,Ga)N nanowires synthesized by plasma-assisted MBE (PAMBE) on Si(111) substrates. The nanowires contain a single GaN quantum disk embedded in an . µm long (Al,Ga)N segment. The disks have a thickness ranging from . to . nm and emit intense luminescence at an energy invariably above the bandgap of bulk GaN, suggesting that the quantum-confined Stark effect in the disk is essentially absent. A detailed investigation of the composition of the (Al,Ga)N segment reveals a core/shell microstructure formed spontaneously during growth, with a continuously varying Al content both across the nanowire diameter and along the growth direction. The latter induces a high electron density in the shell via polarization doping, leaving the quantum disk in a degenerately doped state with screened internal electrostatic fields thus resulting in the high transition energies regardless of the disk thickness. A quantitative analysis of the transition energies and the temperature dependence of the radiative lifetime allows us to extract not only the mean value of the electron density in the disks, but also its distribution. Further consequences of the degenerate doping level are a short radiative lifetime and a comparatively high internal quantum efficiency up to room temperature. long GaN base grown on Si(111), followed by an -nm-long (Al,Ga)N segment and a GaN quantum disk embedded between the (Al,Ga)N segment and a -nm-long (Al,Ga)N cap. The samples under investigation differ in the thickness of the disk, with nominal values of . , . , and . nm. Both the (Al,Ga)N segment and the (Al,Ga)N cap have a nominal Al content of . . For comparison, we also include a sample nominally identical to those above, but without an embedded GaN quantum disk.
Figure (b) depicts room temperature photoluminescence (PL) spectra of the four nanowire ensembles under continuous-wave excitation at nm together with that of a bare GaN nanowire ensemble under continuous-wave excitation at nm. The PL spectrum of the GaN nanowires shows a line at . eV originating from the recombination of free A excitons in strain-free GaN. The PL spectrum of the bare (Al,Ga)N nanowires exhibits a single band at . eV. A band at essentially the same spectral position ( . eV) is also observed in the PL spectra of the (Al,Ga)N nanowires with an embedded GaN disk, and we thus attribute this highenergy band to emission from the (Al,Ga)N segment. Consequently, the low-energy band in the spectra is associated with the GaN quantum disks. In all cases, the corresponding transition energy is notably higher than the bandgap energy of GaN and depends only weakly on the thickness of the disk itself. The same is true for the integrated PL intensity of this band. In addition, the PL intensity drops by only a factor of two to three between and K (see Supporting Information) and is thus about two orders of magnitude higher than that of the bare GaN nanowires at K. These results suggest that the GaN quantum disks exhibit a very high internal quantum efficiency up to room temperature. As shown in the Supporting Information, the low thermal quenching is in part due to an efficient transfer of photoexcited carriers from the (Al,Ga)N segment to the quantum disk, but the actual internal quantum efficiency re-mains significantly higher (about a factor of ) than that of GaN nanowires of comparable length and density.
For the nominal parameters of our nanowires [see Fig. (a)], we would expect very different transition energies for both the (Al,Ga)N segment and the GaN quantum disk. Specifically, near-band edge emission of Al 0.3 Ga 0.7 N should occur above eV,while the energy observed ( . eV) is rather characteristic for an Al content of . . Moreover, the large internal electrostatic fields in GaN/Al 0.3 Ga 0.7 N quantum disks should result in a quantum-confined Stark shift of typically -meV/nm, leading to a strong dependence of the transition energies on disk thickness and emission energies well below the band gap of GaN for the two thicker quantum disks. , , To confirm our assignment of the PL bands in Fig. (b), we perform spatially resolved cathodoluminescence (CL) measurements at room temperature on the ensemble with the thickest GaN disk. Figures (c) and (d) show monochromatic CL maps collected at emission energies of . and . eV, respectively, superimposed on the same bird's eye view secondary electron micrograph of the nanowire ensemble. The maps clearly show the complementary nature of the two emission bands, in that the band at . eV arises from the top of the nanowires, where the GaN disk is located, while the homogeneous emission at . eV originates from the region corresponding to the (Al,Ga)N segment.
The high transition energies of the GaN quantum disks could be explained if their were actually much thinner than nominally. In this case, the quantum-confined Stark shift would be outweighed by quantum confinement, resulting in energies larger than the GaN band gap. To test this hypothesis, we perform a detailed structural and chemical analysis of the nanowire ensemble with the . -nmthick GaN disk, whose transition energy depends most sensitively on the internal electrostatic fields. Figure (a)-(c) show scanning transmission electron microscopy (STEM) images acquired in high-angular annular dark-field (HAADF) mode for three representative single nanowires from this ensemble. The GaN quantum disk is clearly visible in all three micrographs and has a thickness of (6 ± 1) nm, confirming that the actual thickness is consistent with the nominal value. This result rules out the possibility that the high transition energies are caused by strong quantum confinement in thin disks.
The HAADF-STEM images also reveal that the quantum disks are not delimited by flat (0001) planes, but by facets with semi-polar orientation, as most clearly seen in Fig. (c). The angles between the growth axis and the direction perpendicular to the semi-polar plane lie in the range of -• , which thus correspond to (101 ) planes with ranging from to . The internal electrostatic field in semipolar strained GaN/(Al,Ga)N quantum wells with these orientations is reduced, but only to about % of the one in a GaN/(Al,Ga)N{0001} quantum well. Fields of this strength (≈ 2 MV/cm) would still lead to transition energies well below the band gap of GaN, and the occurrence of semipolar facets therefore does not explain the discrepancy between the expected and experimentally observed transition energies.
What we would expect from the various shapes of the GaN quantum disks, however, is a significant variation in emission energy and thus a strong broadening of ensemble spectra such as those depicted in Fig. (b). Compared to the full width at half maximum (FWHM) of meV for the room temperature PL line of the GaN nanowire ensemble, the PL bands of the GaN quantum disks are certainly broader, but with an FWHM of meV not nearly as much as expected from the variation in thickness and orientation. The surprising uniformity of the emission energy of the GaN disks is also evident in the CL map displayed in Fig. (d), where the majority of the nanowires are seen to emit at . eV.
The contrast obtained by HAADF-STEM is primarily determined by the composition of the material. For several nanowires, we have observed a contrast along the nanowire crosssection as most clearly visible in Fig. (a), namely, a brighter core and a darker shell, indicating the presence of a core/shell structure with an Al-deficient inner core and an Al-rich outer shell which wraps around the core.
(c) (a) (b)(d)
To investigate this finding in more detail, we analyze individual nanowires with chemically sensitive techniques. Figure (d) shows the (radially averaged) Al content along the growth axis for two representative single nanowires as obtained by energy-dispersive xray spectroscopy (EDX) performed in our analytical scanning electron microscope (SEM) (see Methods). The inset illustrates the direction of the line scan on an exemplary nanowire. From the bottom to the top of the nanowire, the average axial Al content of the two nanowires increases from . to . up to a length of nm, stays then constant for about nm, and decreases from . to . for the top nm of the nanowire. Given that the length of the (Al,Ga)N segment amounts to only nm, the initial increase of the average Al content for the first nm is likely caused by the formation of an AlN shell (with decreas-ing thickness toward the bottom) wrapping around the GaN base. The average Al content in proximity of the GaN disk is about . , which is in fact close to the nominal value.
The presence of this axial gradient in the radially averaged Al content cannot explain the constant, low emission energy observed along the (Al,Ga)N segment [ Fig. (c)]. To resolve also the radial composition, we utilize atom probe tomography (APT). Figure (e) shows the Ga and Al profiles across the diameter of a single nanowire in proximity of the GaN disk extracted from full three-dimensional APT data. Several profiles have been extracted at different heights of the same nanowire (not shown), and they are all consistent with the profiles shown in Fig. (e), which demonstrates the presence of an Al-deficient inner core and a complex Al-rich shell with a continuously varying Al content. Figure (f) shows the Al content across a two-dimensional section of the same nanowire in proximity of the GaN disk. The inner blue region represents the Al-deficient core, which is surrounded by a thin shell with high Al content and an outer shell with lower Al content. The core appears compressed in the APT image due to well-known image aberration effects. When correcting for the compression, the dimensions of the core would change slightly but the anisotropic shape would remain. However, the exact shape of the core may be affected by the dependence of the evaporation rate of (Al,Ga)N on the Al content, which may potentially result in a change of the curvature of the surface of the specimen during evaporation. Note that the cation concentrations given in Figs. (e) and (f) correspond to the raw measured data without correcting for the loss of N atoms common in APT, which is why the sum of the Al and Ga content does not equal one. The actual concentrations will thus be slightly higher. Considering this fact, the Al content in the core should be close to the value of . deduced from the transition energy in the PL and CL experiments shown in Fig. . Combining the results of APT with EDX and assuming that the Al content in the core is constant along the length of the (Al,Ga)N seg-ment as suggested by the uniform emission energy observed in CL [ Fig. (c)], we deduce that the shell essentially consists of pure AlN at the bottom of the (Al,Ga)N segment [i. e., at about -nm in Fig. (d)]. Subsequently, the shell composition decreases toward the nanowire top to a value of about . close to the position of the disk at the top. Figure (a) visualizes the resulting complex core/shell structure that forms spontaneously during growth despite a constant temperature and constant fluxes, much like reported previously by Allah et al. for nm short (Al,Ga)N nanowire segments grown by PAMBE. The drastic deviation from the intended uniform alloy was ascribed to the different diffusion lengths of Ga and Al on the nanowire sidewalls and the geometric shadowing effect occurring during the growth of dense nanowire ensembles. An axial gradient of the Al content in an (Al,Ga)N segment followed by a constant composition after nm of growth was also reported by Pierret et al. In fact, growth would be expected to approach a steady-state regimeleading to a radial composition profile that does not change anymore along the nanowire axis-once the nanowire diameter and density have become constant. In the present case, Fig. (d) clearly shows that we do not reach this steady-state conditions despite the considerable length of the (Al,Ga)N segment. The origin of this extended growth under nonstationary conditions is the one order of magnitude lower nanowire density compared to the ensembles studied in Refs. and (for the density, see Ref. ), the correspondingly weaker influence of shadowing, and the resulting continuous tapering of the nanowires (see the top-and sideview secondary electron micrographs in the Supporting Information).
The complex core/shell structure has important consequences for the electronic and optical properties of the nanowires, which have not been elucidated so far. In particular, due to the spontaneous polarization of the group-III nitrides, the compositional grading of the (Al,Ga)N segment along the nanowire axis induces a bulk polarization charge, which in turn leads to the accumulation of mobile charge carriers of the opposite sign. This phenomenon, known as polarization doping, has been exploited for n-type as well as p-type doping of GaN/(Al,Ga)N heterostructures, and is the rationale for the nanowire diode design proposed in Refs. and . Obviously, the radial and axial compositional gradients as well as the effects of polarization doping are essential for an understanding of the radiative transitions observed in PL and CL [cf. Figs. (b) and (d)]. The low energy of the emission band attributed to the (Al,Ga)N segment, and the absence of any signal above . eV either in the PL [Fig. (b)] or CL spectra (not shown here) is a direct consequence of the nanowires' core/shell struc-ture in conjunction with an efficient transfer of photo-or cathodogenerated carriers from the Al-rich shell into the Al-deficient core. Likewise, electrons induced in the shell by polarization doping transfer to the core and from there to the GaN quantum disk, thus screening its internal electrostatic field. This charge accumulation potentially accounts for the absence of the quantum-confined Stark effect demonstrated by the spectra in Fig. (b). For a quantitative determination of the electron density and the resulting transition energy, a full three-dimensional simulation of the complex core/shell structure would be desirable. However, when attempting to simulate such a core/shell nanowire including the axial and radial gradients in composition using NextNano™, we found that the solution depends sensitively on details of the structure, such as the precise thickness of the shell and the presence or absence of semipolar facets. To illustrate the physical principles, we thus employ simple one-dimensional simulations , as discussed in the following (for further details, see the Methods section).
To estimate the electron density induced by polarization doping, we consider a planar heterostructure with a . -nm-wide GaN well embedded in an (Al,Ga)N layer with a negative Al gradient along the growth axis (the [0001] direction). Figure (b) shows the band profile of this heterostructure where the Al content is assumed to vary linearly from to . from the bottom to the top of the structure, similar to the variation of the Al content in the Al-rich shell of the top segment of our nanowires containing the GaN quantum disk [cf. Fig. (d)]. The negative gradient of the Al content along the [0001] direction leaves a three-dimensional slab of positive bound charge, which is compensated by free electrons that diffuse inward from surface donor states. This polarization-induced doping manifests itself by the change of band gap occurring entirely by a variation of the valence band edge, while the conduction band edge remains close to the Fermi level.
The dashed line in the inset of Fig. (b) shows the polarization-induced free electron density in a graded (Al,Ga)N segment without GaN disk to be on the order of 5 × 10 17 cm −3 . The solid line in the inset shows the electron concentration in a structure with a graded (Al,Ga)N segment and additional GaN quantum disk, corresponding to the sample whose band profile is shown in the main figure. The charge transfer taking place in this structure results in a degenerately doped GaN disk with an electron sheet density n s of 4 × 10 12 cm −2 .
To quantitatively determine the impact of this high electron density on the transition energies in the GaN quantum disk, we next calculate the electron and hole states in GaN quantum wells with a width between and nm embedded in Al 0.1 Ga 0.9 N, the approxi-mate composition of the nanowire core. The calculations are done for undoped wells, for which the polarization results in an internal electrostatic field of around MV/cm, and for wells with an electron density as determined by the simulations discussed above. We also take into account the compressive strain in the GaN quantum well imposed by a coherent interface with the Al 0.1 Ga 0.9 N layer. Figure (c) shows the transition energies for these two cases together with the experimental peak energies of the low-energy PL bands obtained by PL spectroscopy at K (not shown here). The transition energies for the undoped wells are governed by the quantum confined Stark effect, i. e., they depend linearly on width with a slope given by the internal electrostatic field. As expected, they fall below the GaN band gap ( . eV) for a well width exceeding . nm. In contrast, the quantum wells with an electron density equal to that induced in the disks by polarization doping in the shell exhibit a transition energy above the GaN band gap even for the widest well, in close agreement with the experimental results.
The insets of Fig. (c) illustrate that the screening of the internal fields increases the electron-hole overlap in the quantum disk. Whereas electrons and holes are spatially largely separated in the undoped disk (bottom left), they have an appreciable overlap for the disks with a high electron sheet density (right top). This enhanced electron-hole overlap is expected to affect the recombination dynamics of the electron-hole pairs within the GaN disk and consequently the internal quantum efficiency. Even more importantly, carrier recombination in these structures is affected not only quantitatively, but also qualitatively: the electron density in the disk is well above the Mott density, , and recombination is thus no longer excitonic, but occurs between the two-dimensional electron gas and photoexcited holes.
To investigate the consequences of this high electron density for the recombination dynamics in the GaN disk, we perform timeresolved PL measurements on the nanowire ensemble with the . nm-thick GaN disk at temperatures between and K. The disks were excited directly using an excitation wavelength of nm, corresponding to an energy below the bandgap of the (Al,Ga)N core. The PL transients presented below are thus not affected by carrier transfer processes. Figure (a) shows a representative streak camera image at K. The emission band of the GaN quantum disk is observed to redshift by meV during the initial ns of the PL decay. For polar GaN/(Al,Ga)N quantum wells, such a temporal redshift may occur as a consequence of a dynamical screening of the internal electrostatic fields induced by a high excitation density. , In the present case, this phenomenon can be safely ruled out, since the photogenerated carrier density ∆n inside the disk is at most 2 × 10 10 cm −2 , i. e., more than two orders of magnitude smaller than the electron sheet density resulting from polarization doping (n s ≈ 4 × 10 12 cm −2 ).
Another possible reason for this gradual redshift is the progressive relaxation of photoexcited excitons within a band of localized states, leading to their spectral diffusion. , This phenomenon is most frequently observed in ternary layers and quantum wells, for which compositional fluctuations induce the disorder responsible for exciton localization. -In the present case, compositional fluctuations in the barriers or steps at the GaN/(Al,Ga)N interfaces could potentially induce localization as well. However, the transfer of carriers from higher to lower energy states manifests itself in a characteristic temporal delay of the emission at lower energies, which is not observed here. Instead, the rise time of the emission is found to be the same regardless of energy. We also do not observe any other signature of localization effects, such as the typical S shape of the transition energy with temperature (not shown). -An alternative explanation for the spectral shift suggests itself when considering that we are not dealing with a spatially uniform emitter, but a large ensemble of individual emitters (in the present experiments, we excite about 10 4 nanowires). In other words, the temporal redshift may simply result from a spectral superposition of GaN quantum disks emitting at different energies. In fact, for the present nanowire ensembles with a spatially random arrangement, it is not conceivable that the Al gradient in the shell is identical for each nanowire. Differences must occur because of both shadowing effects , and the different diameters of individual nanowires, affecting the diffusion of the group-III adatoms on the nanowire sidewalls. These differences in the Al gradient in turn result in different electron densities induced by polarization doping. The higher the electron density in the quantum disk, the lower the internal electrostatic field due to screening, which translates into higher emission energies and shorter lifetimes. The most heavily doped quantum disks thus dominate the high-energy emission at short times. In more lightly doped disks, the residual field leads to longer lifetimes and lower emission energies, which lead to the long-living and redshifted tail of the emission. The net result of this superposition of an ensemble of disks with different transition energies and lifetimes is the apparent spectral shift of the emission band as seen in Fig. (a). Figure (b) shows PL transients spectrally integrated over the entire emission band between and K. The transients are characterized by a continuously changing slope on a semilogarithmic scale and can be adequately described only by the sum of at least three exponentials. This finding is in full agreement with the hypothesis discussed above: the spectrally integrated emission is a superposition of the individual (exponential) transients of nanowires with different electron densities, and thus different lifetimes. Approximating the fast and slow components of the transient at K by single exponentials, we obtain decay times of about . and ns, respectively. For comparison, planar GaN/(Al,Ga)N quantum wells of similar width and average doping density were found to exhibit a lifetime of ns. The maximum electron density in our nanowire ensemble is thus significantly higher than the average value of n s determined from the average transition energy. K. (c) Temperature dependence of the radiative lifetime determined from the peak intensity of the transients (solid circles). The dashed lines show the calculated temperature dependence for electron sheet densities in the quantum disk of 0.2, 0.6, and 1.6 × 10 12 cm −2 (from left to right). The solid line represents the radiative lifetime of a nanowire ensemble with an ensemble distribution of the electron volume density as shown in the inset.
For temperatures above K, the decay is seen to accelerate accompanied by a continuously decreasing integrated intensity [cf. Fig. (b)], revealing an increasing participation of nonradiative decay channels (see also the comparison of resonant and nonresonant excitation in the Supporting Information). To separate radiative and nonradiative channels, we examine the peak intensity of the transients, which is directly proportional to the radiative decay rate. -Figure (c) shows the temperature dependence of the radiative lifetime determined in this way. The radiative lifetime starts to increase already at temperatures as low as K, but does not enter the linear temperature dependence characteristic for nondegenerate two-dimensional systems in general and GaN/(Al,Ga)N quantum wells in particular. , To quantitatively understand this peculiar temperature dependence, we calculate the spontaneous recombination rate using Fermi's golden rule for the general case of Fermi-Dirac statistics valid also for the high carrier density in the GaN quantum disks (for details, see the Supporting information). -Note that this calculation ignores any residual electric fields in the disk. Consequently, we do not compare the absolute transition rate in the following, but only its temperature dependence.
The three dashed lines in Fig. (c) show the inverse radiative rates obtained by our calculations for electron sheet densities of n s 0.2, 0.6, and 1.6 × 10 12 cm −2 . In contrast to the strictly linear dependence of a nondegenerate quantum well, the radiative rate for high electron sheet densities is constant at low temperatures, for which the system becomes degenerate. The higher the electron density, the higher the temperature up to which degeneracy prevails, and thus the larger the temperature range of constant radiative lifetime (for more details, see the Supporting Information).
The theoretical dependence for an electron density of n s 6 × 10 11 cm −2 is closest to the experimental data, but it does not describe them well, particularly at low temperatures. The agreement can be much improved when allowing for a distribution of the electron density as qualitatively discussed above in the context of Figs. (a) and (b). Quantitatively, we obtain the solid line in Fig. (c) by assuming a log-normal distribution of n s with a mean of 2 × 10 12 cm −2 or a volume density of 3 × 10 18 cm −3 as shown in the inset of Fig. (c), not too far from the average value estimated above as a result of polarization doping (n s 4 × 10 12 cm −2 ). This agreement suggests that despite the crudeness of our approach concerning the energy and the lifetime of carriers in the GaN quantum disks in our nanowires, we have reached a fairly coherent understanding of both the static and dynamical electronic properties of these disks. In particular, the modest thermal quenching of the emission from the GaN quantum disks (see Supporting Information) is understood to be a result of the short radiative lifetime due to the high electron density and not a particularly high material quality.
To summarize and conclude, we have shown that, during the synthesis of GaN/(Al,Ga)N nanowire ensembles by molecular beam epitaxy, a complex core/shell structure forms spontaneously, characterized by a continuously varying Al composition both along the growth axis and across the diameter of the nanowires. This phenomenon has been observed in similar form by other groups, which shows that it is not caused by specific growth conditions, but the fundamental growth mechanisms of (Al,Ga)N nanowires by PAMBE. The peculiar structure resulting from this self-assembly process has important consequences for the optoelectronic properties of the nanowires. First of all, the axial Al gradient in the Al-rich shell induces a threedimensional polarization charge, leading to the accumulation of electrons in the shell. Second, due to the Al deficient core, these electrons in the shell are efficiently transferred to the GaN quantum disk. The resulting high electron sheet density inside the GaN disk screens the internal electrostatic field, effectively canceling the quantum-confined Stark effect, and manifesting itself in both transition energies almost independent of thickness and short radiative lifetimes up to room temperature that result in a comparatively high internal quantum efficiency. The latter is obviously attractive for the realization of efficient light emitting devices in the ultraviolet range. However, we emphasize that for taking ad-vantage of the effects reported here in any application, one would have to achieve a high level of control over the formation of the compositional gradients. To reach this aim, further studies are required to fully understand the mechanisms leading to the self-assembly of (Al,Ga)N nanowires in PAMBE. Moreover, this compositional self-assembly is likely to occur also for other mixed cation nanowires synthesized by diffusion-controlled growth techniques such as MBE. Hence, our findings may be relevant not only for deep UV emitters based on (Al,Ga)N nanowires, but for example also for MBE-grown (In,Ga)N nanowires for emission in the visible spectrum and solar energy harvesting.
Methods
Nanowire synthesis
The GaN/(Al,Ga)N nanowire ensembles are synthesized utilizing a DCA Instruments P molecular beam epitaxy system equipped with two radio frequency plasma sources for active N and solid-source effusion cells for Al and Ga. First, GaN nanowires are grown on Si(111) substrates relying on their spontaneous formation on this substrate under suitable experimental conditions. , A two-step growth approach is employed in order to minimize the incubation time. After the formation of a GaN nanowire ensemble with a nominal length of nm, the Ga flux is reduced and an Al flux is added for the synthesis of a nominally nm long (Al,Ga)N segment with a nominal Al content of . using the same V/III ratio as for the GaN nanowires underneath. For the formation of the GaN quantum disk, the Al shutter is simply closed for a duration corresponding to the intended disk thickness. Finally, the disk is capped by a nominally nm long (Al,Ga)N segment. Three samples are grown in this way, differing only in the thickness of the GaN quantum disk with nominal values of . , . , and . nm. A sample without quantum disk, but otherwise identical (Al,Ga)N nanowires on a GaN base serves as comparison. The nanowires thus obtained have an average length of . µm (close to the expected value) and an average equivalent disk diameter at their top of nm. The mean density of nanowires in the ensemble is 3 × 10 9 cm −2 . Top-and sideview secondary electron micrographs of the sample with a .nm-thick-disk are shown in the Supporting Information.
Morphology and microstructure
The morphological properties of the as-grown GaN/(Al,Ga)N nanowires are studied by scanning electron microscopy carried out in a Hitachi S-field emission microscope using an acceleration voltage of kV (see Supporting Information). Single nanowires in cross-sectional specimens are investigated by transmission electron microscopy. The crosssections are prepared by mechanical grinding and dimpling followed by Ar-ion beam milling down to electron transparency. Scanning transmission electron microscopy is performed by using a JEOL F field emission instrument equipped with a bright-field and a dark-field detector and operated at kV. High-angle annular dark-field micrographs of the nanowires reflect the chemical contrast between the GaN quantum disk and the surrounding (Al,Ga)N segments.
Composition
The composition of single nanowires is investigated by energy dispersive x-ray spectroscopy and atom probe tomography. The former is performed using an EDAX Apollo XV silicon drift detector attached to a Zeiss Ultra field emission scanning electron microscope operated at kV. The latter is carried out using a Cameca LEAP X-HR system. Single nanowires from the ensemble with the . nm thick quantum disks are isolated in an FEI Nova Nanolab i. Welds are created by electron-induced metal deposition of Pt or Co. The selected nanowire is mounted on a tip, and the evaporation of atoms is triggered by a laser generating picosecond pulses at a wavelength of nm. Since the (Al,Ga)N nanowire absorbs only weakly at this wavelength, a comparatively high pulse energy of pJ is used. The field of view probed by the technique is about nm.
Spectroscopy
For the investigation of the optical properties of the nanowire ensembles, the as-grown samples are mounted onto the cold finger of a liquid He cryostat allowing continuous control of the sample temperature between and K. Cathodoluminescence spectroscopy is carried out at room temperature with a Gatan Mon-oCL system fitted to a Zeiss Ultra fieldemission scanning electron microscope operated at kV with a probe current of . nA. The signal is spectrally dispersed by a monochromator and detected using a photomultiplier tube. Continuous-wave photoluminescence spectroscopy is performed by exciting the ensembles with the nm ( . eV) line of a Coherent Innova C FreD Ar + ion laser focused to a spot of about . µm diameter yielding an excitation density on the order of W/cm 2 . The signal is spectrally dispersed by a monochromator and detected with a cooled charge-coupled device array. Time-resolved PL measurements are performed by exciting the ensemble with the second harmonic ( nm) of -fs-pulses from an APE optical parametric oscillator synchronously pumped by a Coherent Mira Ti:sapphire laser, which itself is pumped by a Coherent Verdi V frequency-doubled Nd:YVO 4 laser. The pulses with an energy of pJ are focused onto the sample with a -mm plano-convex lens to a spot with a diameter of µm. The transient PL signal is dispersed by a monochromator and detected by a Hamamatsu C streak camera with a temporal resolution set to about ps.
Simulations
The band profiles and electron densities are obtained by one-dimensional Schrödinger-Poisson simulations performed with 1DPoisson. For the polarization of Al x Ga 1−x N, we assume a value of −2×10 −6 (1+ 2.1x + 0.475x 2 ) C/cm 2 (the negative sign is due to the fact that the nanowires grow in the [0001] direction). This value is about half of that predicted theoretically, but in agreement with the majority of experimental results. For the conduction band offset between GaN and (Al,Ga)N, we take the value reported by Tchernycheva et al. The spontaneous recombination rate is calculated from Fermi's golden rule assuming k conservation as R sp R 0
∫ ∞ 0 ρ r (E) f c (E)[1− f v (E)]dE,
where ρ r (E) is the reduced density of states, f c (E) and f v (E) are the Fermi-Dirac distributions for the conduction and the valence bands, respectively, and R 0 is a prefactor. Details of this calculation are presented in the Supporting Information.
Supporting Information Available
The following files are available free of charge. Top-and sideview secondary electron micrographs of the sample with a . -nm-thick disk, temperature dependence of the integrated photoluminescence intensity for excitation wavelengths of and nm, calculation of the spontaneous recombination rate.
Acknowledgement
The authors thank Jesús Herranz for a critical reading of the manuscript. P. C. is grateful to the Fonds National Suisse de la Recherche Scientifique for funding through project . J. K. Z. and T. A. acknowledge the financial support received by Deutsche Forschungsgemeinschaft within SFB and by Bundesministerium für Bildung und Forschung through Project No. FKZ: N , respectively. S. F. G. acknowledges the partial financial support received through the Spanish program Ramón y Cajal (cofinanced by the European Social Fund) under Grant No. RYC--from Ministerio de Ciencia, Innovación y Universidades.
Figure (a) shows the nominal structure of our nanowires, which consist of a -nm-3.55 eV Figure : Spectrally and spatially resolved luminescence of the GaN/(Al,Ga)N nanowires under investigation at room temperature. (a) Schematic representation of the nominal structure of the nanowires (not to scale). (b) PL spectra of the four GaN/(Al,Ga)N nanowire ensembles compared to the PL spectrum of a GaN nanowire ensemble. The spectra are normalized and vertically shifted for clarity. The numbers indicate the thickness of the GaN quantum disk for the samples containing one. (c) and (d) Monochromatic CL maps at . (c) and . eV (d) superimposed to bird's eye view secondary electron micrographs of the GaN/(Al,Ga)N nanowire ensemble with a . nm-thick GaN disk.
the center (nm) Distance from the center (nm) Figure : Structural properties of the nanowire ensemble with the . -nm-thick GaN disk. (a)-(c) HAADF-STEM images of three representative nanowires. The micrographs were acquired in the 1120 zone axis. (d) Radially averaged Al content along the nanowire growth axis of two individual nanowires recorded by EDX line scans in an SEM. Inset: secondary electron micrograph of a representative nanowire with the scale bar having a length of nm. Most dispersed nanowires exhibit a length between . and . µm, indicating that they break off at their coalescence joints (see sideview in Supporting Information) upon ultrasonic harvesting. The arrow indicates the [0001] growth direction. (e) Ga and Al profiles across the nanowire diameter in proximity of the GaN disk extracted from APT data on a single nanowire. (f) Al content across a two-dimensional section extracted from the same data.
Figure :
:Calculation of the transition energy of the GaN quantum disks with an electron density determined by polarization doping. (a) Schematic representation of the spontaneously formed core/shell structure of the GaN/(Al,Ga)N nanowires under investigation as determined by HAADF-STEM [cf. Figs. (a)-(c))], EDX [cf. Fig. (d)] and APT [cf. Figs. (e) and (f)]. The Al content is color-coded according to the scale in the figure. (b) Axial band profile of a planar heterostructure with a . nm wide GaN quantum well embedded in an (Al,Ga)N layer with a linear grading in the Al content from . at the surface ( ) to at the bottom ( nm). The inset shows the electron density in this graded (Al,GaN) layer without (blue dashed line) and with an inserted GaN quantum well (red solid line) close to the surface of the structure. A sketch of the conduction band profile is superimposed as thin green line to visualize the position of the GaN disk. (c) Transition energy as a function of the quantum well width due to the combined effect of polarization doping and the Al-deficient core with an Al content of . (circles). For comparison, the transition energy expected without polarization doping is shown by squares. The lines are guides to the eye. The insets show the band profiles and the electron and hole wave functions obtained for these two cases. The experimental peak energies of the PL band of the GaN disks at K are indicated by triangles. The error bars denote the FWHM of the band.
Figure :
:Recombination dynamics of the nanowire ensemble with a . nm thick GaN disk. (a) Streak camera image obtained upon pulsed excitation at nm and K. The dashed line highlights the spectral diffusion of the emission band from the GaN quantum disk. (b) Spectrally integrated PL intensity transients obtained from streak camera images acquired at temperatures between and
Advances in group III-nitride-based deep UV light-emitting diode technology. M Kneissl, T Kolbe, C Chua, V Kueller, N Lobo, J Stellmach, A Knauer, H Rodriguez, S Einfeldt, Z Yang, N M Johnson, M Weyers, Semicond. Sci. Technol. Kneissl, M.; Kolbe, T.; Chua, C.; Kueller, V.; Lobo, N.; Stellmach, J.; Knauer, A.; Rodriguez, H.; Einfeldt, S.; Yang, Z.; Johnson, N. M.; Weyers, M. Ad- vances in group III-nitride-based deep UV light-emitting diode technology. Semicond. Sci. Technol.
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[
"Large-x PDFs and the Drell-Yan Process",
"Large-x PDFs and the Drell-Yan Process"
] |
[
"J C Peng \nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIL\n"
] |
[
"University of Illinois at Urbana-Champaign\n61801UrbanaIL"
] |
[] |
Dimuon production has been studied in a series of fixed-target experiments at Fermilab during the last two decades. Highlights from these experiments, together with recent results from the Fermilab E866 experiment, are presented. Future prospects for studying the parton distributions in the nucleons and nuclei using dimuon production are also discussed.
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10.1063/1.3631530
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[
"https://arxiv.org/pdf/1102.0060v1.pdf"
] | 118,638,556 |
1102.0060
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dc576aca6b20d5aad0eb7ad8ce481802cb045bb0
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Large-x PDFs and the Drell-Yan Process
1 Feb 2011
J C Peng
University of Illinois at Urbana-Champaign
61801UrbanaIL
Large-x PDFs and the Drell-Yan Process
1 Feb 2011Drell-Yanquarkonium productionparton distributions PACS: 1385Qk1420Dh2485+p1388+e
Dimuon production has been studied in a series of fixed-target experiments at Fermilab during the last two decades. Highlights from these experiments, together with recent results from the Fermilab E866 experiment, are presented. Future prospects for studying the parton distributions in the nucleons and nuclei using dimuon production are also discussed.
INTRODUCTION
The Drell-Yan process [1], in which a charged lepton pair is produced in a hadron-hadron interaction via the electromagnetic qq → l + l − process, has provided unique information on parton distributions. In particular, the Drell-Yan process has been used to determine the antiquark contents of nucleons and nuclei [2], as well as the quark distributions of pions, kaons, and antiprotons [3]. Such information is difficult, if not impossible, to obtain from DIS experiments. As the Drell-Yan process can be well described by nextto-leading order QCD calculations [4], a firm theoretical framework exists for utilizing the Drell-Yan process to extract the parton distributions.
During the last two decades, a series of fixed-target dimuon production experiments (E772, E789, E866) have been carried out using 800 GeV/c proton beam at Fermilab. At 800 GeV/c, the dimuon data contain Drell-Yan continuum up to dimuon mass of ∼ 15 GeV as well as quarkonium productions (J/Ψ, Ψ ′ , and ϒ resonances). The Drell-Yan process and quarkonium productions often provide complementary information, since Drell-Yan is an electromagnetic process via quark-antiquark annihilation while the quarkonium production is a strong interaction process dominated by gluon-gluon fusion at this beam energy.
The Fermilab dimuon experiments covers a broad range of physics topics. The Drell-Yan data have provided informations on the antiquark distributions in the nucleons [5,6,7,8] and nuclei [9,10]. These results showed the surprising results that the antiquark distributions in the nuclei are not enhanced [9,10], contrary to the predictions of models which explain the EMC effect in term of nuclear enhancement of exchanged mesons. Moreover, the Drell-Yan cross section ratios p + d/p + p clearly establish the flavor asymmetry of thed andū distributions in the proton, and they map out the x-dependence of this asymmetry [6,7,8]. Pronounced nuclear dependences of quarkonium productions have been observed for J/Ψ, Ψ ′ , and ϒ resonances [11,12,13,14].
Several review articles covering some of these results are available [2,15,16]. In this article, we will focus on the recent results from experiment E866 and future prospect of dimuon experiments at Fermilab and J-PARC.
MRS S0' CTEQ6 MSTW2008 GJR08 F x ) F R(x FIGURE 1. Prediction of the ratio (p + p → W + + x)/(p + p → W − + x) at
√ s of 500 GeV using various PDFs.
FLAVOR STRUCTURE OF LIGHT-QUARK SEA
In the CERN NA51 [17] and Fermilab E866 [6,7,8] experiments on proton-induced dimuon production, a striking difference was observed for the Drell-Yan cross sections between p + p and p + d. As the underlying mechanism for the Drell-Yan process involves quark-antiquark annihilation, this difference has been attributed to the asymmetry between the up and down sea quark distributions in the proton. From the σ (p + d) DY /2σ (p + p) DY ratios the Bjorken-x dependence of the sea-quarkd/ū flavor asymmetry has been extracted [6,7,8,17].
Future fixed-target dimuon experiments have been proposed at the 120 GeV Fermilab Main Injector (FMI) and the 50 GeV J-PARC facilities. The Fermilab proposal [18], E906, has been approved and is expected to start data-taking around 2011. Two dimuon proposals (P04 [19] and P24 [20]) have also been submitted to the J-PARC for approval. The lower beam energies at FMI and J-PARC present opportunities for extending thē d/ū and the nuclear antiquark distribution measurements to larger x (x > 0.25). For given values of x 1 and x 2 , the Drell-Yan cross section is proportional to 1/s, hence a gain of ∼ 16 times in the Drell-Yan cross sections can be obtained at the J-PARC energy of 50 GeV. Since the perturbative process gives a symmetricd/ū while nonperturbative processes are necessary to generate an asymmetricd/ū sea, it would be very important to extend the Drell-Yan measurements to kinematic regimes beyond the current limits. Another advantage of lower beam energies is that a much more sensitive study of the partonic energy loss in nuclei could be carried out using the Drell-Yan nuclear dependence [21]. To disentangle thed/ū asymmetry from the possible charge-symmetry violation effect [22,23], one could consider W boson production in p + p collision at RHIC. An interesting quantity to be measured is the ratio of the p + p → W + +x and p + p → W − +x cross sections [24]. It can be shown that this ratio is very sensitive tod/ū. An important feature of the W production asymmetry in p + p collision is that it is completely free from the assumption of charge symmetry. Another advantage of W production in p + p collision is that no nuclear effects need to be considered. Finally, the W production is sensitive tod/ū flavor asymmetry at a Q 2 scale of ∼ 6500 GeV 2 /c 2 , significantly larger than all existing measurements. This offers the opportunity to examine the QCD evolution of the sea-quark flavor asymmetry. Figure 1 shows the predictions [25] for p + p collision at √ s = 500 GeV. The MRS S0 ′ corresponds to thed/ū symmetric parton distributions, while the other three parton distribution functions are from recent global fits with asymmetricd/ū sea-quark distributions. Figure 1 clearly shows that W asymmetry measurements at RHIC could provide an independent determination ofd/ū. First results from the LHC are also expected soon, and will provide additional information [25].
Unlike the electromagnetic Drell-Yan process, quarkonium production is a strong interaction dominated by the subprocess of gluon-gluon fusion at 800 GeV beam energy. Therefore, the quarkonium production cross sections are primarily sensitive to the gluon distributions in the colliding hadrons. The ϒ production ratio, σ (p + d → ϒ)/2σ (p + p → ϒ), is expected to probe the gluon content in the neutron relative to that in the proton [26].
The σ (p + d)/2σ (p + p) ratios for ϒ(1S + 2S + 3S) production are shown in Fig. 2 as a function of x 2 [27]. These ratios are consistent with unity, in striking contrast to the corresponding values for the Drell-Yan process, also shown in Fig. 2. The difference between the Drell-Yan and the ϒ cross section ratios clearly reflect the different underlying mechanisms in these two processes. For ϒ production, the dominance of the gluon-gluon fusion subprocess at this beam energy implies that σ (p
+ d → ϒ)/2σ (p + p → ϒ) ≈ 1 2 (1 + g n (x 2 )/g p (x 2 )
). Figure 2 shows that the gluon distributions in the proton (g p ) and neutron (g n ) are very similar over the x 2 range 0.09 < x 2 < 0.25. The overall σ (p + d → ϒ)/2σ (p + p → ϒ) ratio, integrated over the measured kinematic range, is 0.984 ± 0.026(stat.) ± 0.01(syst.). The ϒ data indicate that the gluon distributions in the proton and neutron are very similar.
The upcoming Fermilab E906 experiment is expected to provide a precise measurement of the σ (p + d → J/Ψ)/2σ (p + p → J/Ψ) ratio. These data should further test the equality of the gluon distributions in the proton and neutron. Moreover, these data could identify EMC effects for deuteron. It is interesting to note that analogous ratios measured in DIS and the Drell-Yan can not be used to determine the EMC effects for deuteron, since different parton distributions are being probed for the hydrogen and the deuterium targets. If one assumes that the gluon distributions are identical in proton and neutron, then any deviation of the σ (p + d → J/Ψ)/2σ (p + p → J/Ψ) ratio from unity would be attributed to the EMC effects of gluon distributions in deuteron.
FLAVOR DEPENDENCE OF THE EMC EFFECT
Despite a quarter century of significant experimental and theoretical effort, the specific origins of the observed A dependence of the nuclear quark distributions have yet to be unambiguously identified. Attempts to explain the EMC effect have led to a large collection of theoretical models [28,29], many of which are capable of describing the essential features of the data, however the underlying physics mechanisms in each model are often very different.
A new calculation of the modifications of nucleon quark distributions in the nuclear medium has recently been reported [30,31]. In this approach by Cloët, Bentz and Thomas (CBT), the Nambu-Jona-Lasinio model is used to describe the coupling of the quarks in the bound nucleons to the scalar and vector mean fields inside a nucleus. These nucleon quark distributions are then convoluted with a nucleon momentum distribution in the nucleus to generate the nuclear quark distributions [30]. An important feature of this model is that for N = Z nuclei (where N and Z refer to the number of neutrons and protons) the isovector-vector mean field (usually denoted by ρ 0 ) will affect the up quarks differently from the down quarks in the bound nucleons. Therefore, this model has a novel prediction that the u and d quarks have distinct nuclear modifications for N = Z nuclei.
Semi-inclusive DIS (SIDIS) on heavy nuclear targets, in which the flavor of the struck quark is tagged by the detected hadron, is a promising experimental tool to search for the flavor-dependent EMC effect. Recently, Lu and Ma [32] pointed out that charged lepton SIDIS off nuclear targets and the deuteron can be used to probe the flavor content of the nuclear quark sea, which can help distinguish between the various models of the EMC effect. Indeed, a SIDIS experiment [33] aiming at a precise determination of flavor dependence of the EMC effect has also been proposed at the upgraded 12 GeV JLab facility.
The flavor dependence of the EMC effect is a promising experimental observable to distinguish among the plethora of models that can describe the EMC effect. As discussed earlier, the nuclear dependence of proton-induced Drell-Yan process measured at Fermilab has shown the surprising results that the antiquark distributions in the nuclei are not enhanced [9,10], contrary to the predictions of many EMC models. Pion-induced Drell-Yan process provides another experimental tool with which search for flavordependent effects in the nuclear modification of the nucleon structure functions [34].
To explore the sensitivity of pion-induced Drell-Yan processes to a flavor-dependent EMC effect, we consider the three ratios σ DY (π + +A) σ DY (π − +A) , σ DY (π − +A) σ DY (π − +D) and σ DY (π − +A) σ DY (π − +H) , where A represents a nuclear, D a deuteron and H a hydrogen target. Assuming isospin symmetry, which implies u π + = d π − ,ū π − =d π + ,ū π + =d π − , u π − = d π + and keeping only the dominant terms in each cross-section, one readily obtains
R ± = σ DY (π + + A) σ DY (π − + A) ≈ d A (x) 4u A (x) (1) R − A/D = σ DY (π − + A) σ DY (π − + D) ≈ u A (x) u D (x) (2) R − A/H = σ DY (π − + A) σ DY (π − + H) ≈ u A (x) u p (x) ,(3)
The up and down nuclear quark distributions are labeled by u A and d A respectively, u D is the up quark distribution in the deuteron and u p the up quark distribution in the proton. Eqs. 1-3 demonstrate that these Drell-Yan cross-section ratios are very sensitive to the flavor dependence of the EMC effect. Fig. 3 shows the comparison [34] of the calculations of the pion-induced Drell-Yan cross-section ratios with the existing data. The top left panel shows the ratio of σ DY (π − +W ) σ DY (π − +D) from the NA10 experiment [35]. These plots contain both the P beam = 286 and 140 GeV data sets, which are very similar. The calculations are performed at P beam = 286 GeV, since most of the data was obtained at this energy. The PDFs of the CBT model [30,31] at a fixed Q 2 of 25 GeV 2 , which is approximately the mean Q 2 of the NA10 experiment, were used. The top right panel shows the ratio σ DY (π − +Pt) σ DY (π − +H) from the NA3 experiment [36]. The data were collected using a 150 GeV π − beam and the Q 2 range covered was 16.8 ≤ Q 2 ≤ 70.6 GeV 2 . The calculations are performed for P beam = 150 GeV and Q 2 = 25 GeV 2 .
The solid curves in Fig. 3 are calculations using the flavor-dependent u A (x) and d A (x) from the CBT model with N/Z = 1.5, corresponding approximately to the N/Z values for the Au, W and Pt nuclei. The dashed curves correspond to the calculated ratios using the nuclear PDFs from the CBT model for N = Z. Since u A /u D = d A /d D in this case, the dashed curves are representative of the predictions for flavor-independent EMC models. Figure 3 shows that the NA10 data do not exhibit a clear preference for the flavor-dependent versus flavor-independent nuclear PDFs. In contrast, the NA3 data strongly favor the calculations using flavor-dependent nuclear PDFs. The existing Drell-Yan data are not sufficiently accurate yet, although the NA3 data clearly favor the flavordependent over the flavor-independent nuclear PDFs. Precise future pion-induced Drell-Yan experiments can provide unique constraints that will help distinguish the various theoretical models and most importantly shed new light on the origins of the EMC effect.
TRANSVERSE SPIN AND DRELL-YAN PROCESS
The study of the transverse momentum dependent (TMD) parton distributions of the nucleon has received much attention in recent years as it provides new perspectives on the hadron structure and QCD [37]. These novel TMDs can be extracted from semiinclusive deep-inelastic scattering (SIDIS) experiments. Recent measurements of the SIDIS by the HERMES [38] and COMPASS [39] collaborations have shown clear evidence for the existence of the T-odd Sivers functions. These data also allow the first determination [40] of the magnitude and flavor structure of the Sivers functions and the nucleon transversity distributions.
The TMD and transversity parton distributions can also be probed in Drell-Yan experiments. As pointed out [41] long time ago, the double transverse spin asymmetry in polarized Drell-Yan, A T T , is proportional to the product of transversity distributions, h 1 (x q )h 1 (xq). The single transverse spin asymmetry, A N , is sensitive to the Sivers function [42], f ⊥ 1T (x) of the polarized proton (beam or target). Even unpolarized Drell-Yan experiments can be used to probe the TMD distribution function, since the cos2φ azimuthal angular dependence is proportional to the product of two Boer-Mulders functions [43],
h ⊥ 1 (x 1 )h ⊥ 1 (x 2 )
. A unique feature of the Drell-Yan process is that, unlike the SIDIS, no fragmentation functions are involved. Therefore, the Drell-Yan process provides an entirely independent technique for measuring the TMD functions. Furthermore, the proton-induced Drell-Yan process is sensitive to the sea-quark TMDs and can lead to flavor separation of TMDs when combined with the SIDIS data. Finally, the intriguing prediction [44] that the T-odd TMDs extracted from DIS will have a sign-change for the Drell-Yan process remains to be tested experimentally.
No polarized Drell-Yan experiments have yet been performed. However, some information on the Boer-Mulders functions have been extracted recently from the azimuthal angular distributions in the unpolarized Drell-Yan process. The general expression for the Drell-Yan angular distribution is [45] dσ dΩ ∝ 1 + λ cos 2 θ + µ sin 2θ cos φ + ν 2 sin 2 θ cos 2φ ,
where θ and φ are the polar and azimuthal decay angle of the l + in the dilepton rest frame. Boer showed that the cos 2φ term is proportional to the convolution of the quark and antiquark Boer-Mulders functions in the projectile and target [46]. This can be understood by noting that the Drell-Yan cross section depends on the transverse spins of the annihilating quark and antiquark. Therefore, a correlation between the transverse spin and the transverse momentum of the quark, as represented by the Boer-Mulders function, would lead to a preferred transverse momentum direction. Pronounced cos 2φ dependences were indeed observed in the NA10 [47] and E615 [48] pion-induced Drell-Yan experiments, and attributed to the Boer-Mulders function. The first measurement of the cos 2φ dependence of the proton-induced Drell-Yan process was recently reported for p + p and p + d interactions at 800 GeV/c [49]. In contrast to pion-induced Drell-Yan, significantly smaller (but non-zero) cos2φ azimuthal angular dependence was observed in the p + p and p + d reactions. While the pion-induced Drell-Yan process is dominated by annihilation between a valence antiquark in the pion and a valence quark in the nucleon, the proton-induced Drell-Yan process involves a valence quark in the proton annihilating with a sea antiquark in the nucleon. Therefore, the p + p and p + d results suggest [49,50] that the Boer-Mulders functions for sea antiquarks are significantly smaller than those for valence quarks.
FUTURE PROSPECTS
Future fixed-target dimuon experiments have been proposed at the 120 GeV Fermilab Main Injector and the 50 GeV J-PARC facilities. As discussed earlier, the Fermilab E906 experiment will extend thed/ū asymmetry measurement to larger x region. Another goal of this experiment is to determine the antiquark distributions in nuclei at large x using nuclear targets. New information on the quark energy loss in nuclei is also expected. As discussed earlier, an advantage of lower beam energies is that a much more sensitive study of the partonic energy loss in nuclei could be carried out [21].
With the possibility to accelerate polarized proton beams at J-PARC [20], the spin structure of the proton can also be investigated with the proposed dimuon experiments. In particular, polarized Drell-Yan process with polarized beam and/or polarized target at J-PARC would allow a unique program on spin physics complementary to polarized DIS experiments and the RHIC-Spin programs. Specific physics topics include the measurements of T-odd Boer-Mulders distribution function in unpolarized Drell-Yan, the extraction of T-odd Sivers distribution functions in singly transversely polarized Drell-Yan, the helicity distribution of antiqaurks in doubly longitudinally polarized Drell-Yan, and the transversity distribution in doubly transversely polarized Drell-Yan. It is worth noting that polarized Drell-Yan is one of the major physics program at the GSI Polarized Antiproton Experiment (PAX). The COMPASS experiment at CERN will also measure π − -induced Drell-Yan on transversely polarized targets to extract Sivers functions from the Single-Spin-Asymmetry. Two recent Letter-of-Intent have also been submitted to RHIC for measuring polarized Drell-Yan at RHIC-Spin, one in collider mode and the other utilizing internal targets. It is clear that the Drell-Yan process will continue to provide unique information on parton distributions in the forseeable future.
FIGURE 2 .
2The E866 σ (p + d)/2σ (p + p) cross section ratios for ϒ resonances as a function of x 2 . The corresponding ratios for Drell-Yan cross sections are also shown. The error bars are statistical only.
FIGURE 3 .
3The existing data for the ratios σ DY (π − +W ) σ DY (π − +D) (upper left), σ DY (π − +Pt) σ DY (π − +H) (upper right) andσ DY (π + +W ) σ DY (π − +W ) (lower) versus the predictions using nuclear PDFs of the CBT model for tungsten (red solid) and N = Z nuclear matter (blue dashed).
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[] |
[
"Two Decades of Colorization and Decolorization for Images and Videos",
"Two Decades of Colorization and Decolorization for Images and Videos"
] |
[
"Shiguang Liu "
] |
[] |
[] |
Colorization is a computer-aided process, which aims to give color to a gray image or video. It can be used to enhance black-and-white images, including black-and-white photos, old-fashioned films, and scientific imaging results. On the contrary, decolorization is to convert a color image or video into a grayscale one. A grayscale image or video refers to an image or video with only brightness information without color information. It is the basis of some downstream image processing applications such as pattern recognition, image segmentation, and image enhancement. Different from image decolorization, video decolorization should not only consider the image contrast preservation in each video frame, but also respect the temporal and spatial consistency between video frames. Researchers were devoted to develop decolorization methods by balancing spatial-temporal consistency and algorithm efficiency. With the prevalance of the digital cameras and mobile phones, image and video colorization and decolorization have been paid more and more attention by researchers. This paper gives an overview of the progress of image and video colorization and decolorization methods in the last two decades.
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10.48550/arxiv.2204.13322
|
[
"https://arxiv.org/pdf/2204.13322v2.pdf"
] | 248,427,172 |
2204.13322
|
d7c98574d7ff32b03a08c93bda3c55342078f9ec
|
Two Decades of Colorization and Decolorization for Images and Videos
Shiguang Liu
Two Decades of Colorization and Decolorization for Images and Videos
1Index Terms-ColorizationDecolorizationImage editingVideo editing
Colorization is a computer-aided process, which aims to give color to a gray image or video. It can be used to enhance black-and-white images, including black-and-white photos, old-fashioned films, and scientific imaging results. On the contrary, decolorization is to convert a color image or video into a grayscale one. A grayscale image or video refers to an image or video with only brightness information without color information. It is the basis of some downstream image processing applications such as pattern recognition, image segmentation, and image enhancement. Different from image decolorization, video decolorization should not only consider the image contrast preservation in each video frame, but also respect the temporal and spatial consistency between video frames. Researchers were devoted to develop decolorization methods by balancing spatial-temporal consistency and algorithm efficiency. With the prevalance of the digital cameras and mobile phones, image and video colorization and decolorization have been paid more and more attention by researchers. This paper gives an overview of the progress of image and video colorization and decolorization methods in the last two decades.
I. INTRODUCTION
I Mage manipulation is an important research topic in both computer graphics and image processing. Until now, many image manipulation topics have attracted attention from researchers, including image appearance transfer [47], [48], [49], image synthesis [46], [9], image analysis [24], [55], [56], [45], image quality assessment [95], [26], [25], image2audio [57], [20], image segmentation [91], [92], [93], [94], [64], image inpainting [86], etc. This paper focuses on the techniques about colorization and decolorization.
There are a large number of gray-scale or black-and-white images and video materials in various film, television, picture archives, medical, and other fields. Coloring them can greatly enhance the detail features and help one better identify and use them. Traditional manual coloring method consumes a lot of manpower and material resources, and may not get satisfactory results. Given a source image or video, colorization methods aim to automatically colorize the target gray image or video reasonably and reliably, which thereby greatly improves the efficiency of this work.
Image or video decolorization, also known as grayscale transformation, converts a three-channel color image or video into a single-channel grayscale one. Decolorization is actually a process of dimension reduction, so that the resulting grayscale image or video often only contains the most Shiguang Liu is with College of Intelligence and Computing, Tianjin university, Tianjin 300350, P.R. China. (e-mail: [email protected]).
important information, which greatly saves storage space. A grayscale image or video can better display the texture and contour of objects. Decolorization can also be widely applied in the field of image compression, medical image visualization, and image or video art stylization. Black and white digital printing of images, with the advantages of low cost and fast printing, is common in daily life, one important process of which is decolorization, i.e., a color image sent to a monochrome printer must undergo a color-to-grayscale transformation.
Below we will summarize various image and video colorization and decolorization methods in the last two decades.
II. IMAGE COLORIZATION
Colorization refers to adding colors to a grayscale image or video, which is a ill-posed task duet to that it is ambiguous to assign colors to a grayscale image or video without any prior knowledge. So, at the early stage, user intervention is usually involved in image colorization. Later, automatic image colorization methods and deep-learning based colorization methods emerged.
A. Semi-Automatic Colorization
Semi-automatic colorization methods require some amounts of user interactions. Among them, color transfer methods ( [87], [73], [1] and image analogy methods ( [22] in Chapter 2 are widely used. In this case, a source image is provided as an example for coloring a given grayscale image, i.e., the target image. When the source image and the grayscale image share similar contents, impressive colorization results can be achieved. Nevertheless, these methods are labor intensive, since the source image and the target image should be manually matched.
A luminance keying based method for transferring color to a grayscale image is described in Gonzalez and Woods [17]. Color and grayscale values are matched with a pre-defined look-up table. When assigning different colors for a same gray level, a few luminance keys should be simultaneously manipulated by the user for different regions, making it a tiresome process. As an extension of color transferring method between color images [73], Welsh et al. [87] proposed to transferring from a source color image to a target grayscale image. It matches color information between the two images with swatches.
Levin et al. [38] presented an efficient colorization method which allows users to interact with a few scribbles. With the observation that neighboring pixels in space-time share Fig. 1. Illustration of the colorization method using optimization [38]. From left to right: an input grayscale image marked with color scribbles by the user, the colorization result by [38] (middle), and the ground truth. similar intensities should have similar colors, they formulate colorization as an optimization problem for a quadratic cost function. As shown in Fig. 1, with a few color scribbles by the user, the indicated colors can be automatically propagated in the grayscale image. Nie et al. [69] developed a colorization method by a local correlation based optimization algorithm. This method depends on the color correlativity between pixels in different regions, limiting its practical application. Nie et al. [70] presented an efficient grayscale image colorization method. This method achieved comparable colorization quality with [38] with much less computation time by quadtree decomposition based non-uniform sampling. Furthermore, this method greatly reduces the problem of color diffusion among different regions via designing a weighting function to represent intensity similarity in the cost function. This is an interactive colorization method, where the user need provide color hints by scribbling or seed pixels.
Irony et al. [29] presented a novel colorization method by transferring color from an exampler image. This method uses a strategy of higher-level context of each pixel instead of independent pixel-level decisions in order to achieve better spatial consistency in the colorization result. Specifically, with a supervised classification scheme, they estimate the best example segment for each pixel to learn color from. Then, by combining a neighborhood matching metric and a spatial filter for high spatial coherence, each pixel is assigned a color from the corresponding region in the example image. It is reported that this approach requires considerably less scribbles than previous interactive colorization methods (e.g., [38]). Yatziv and Sapiro [90] proposed an image colorization method via chrominance blending. This scheme is based on the concept of color blending derived from a weighted distance function that is computed from the luminance channel. This method is fast and allows the user to interactively colorize a grayscale image by providing a reduced set of chrominance scribbles. This method can also be extended for recolorization and brightness change (Fig. 2). As shown in Fig. 2, given a target color image (a), the goal is to recolorize the yellow car into a darker one. Firstly, the blending medium is defined by simply marking areas to be changed and unchanged with scribbles (b). Then, the marks are propagated to be a grayscale matte (c). The brightness of the target image is changed by subtracting the grey-level matte from the intensity channel (d). (e) and (f) show more recolorization results by adding the grey-level matte to the C b and C r channels, respectively.
Image color can be viewed as a highly correlated vector space. Abadpour and Kasaei [1] realized grayscale colorization by applying the PCA (Principal Component Analysis) based transformation. They propose a category of colorizing methods that generate the color vector corresponding to the grayscale as a function. This method is significantly faster than previous approaches while producing visually acceptable colorization results. It can also be extended for recolorization. Nevertheless, this method is restricted by complicate segmentation that is tiresome by using the magic select tool in Adobe Photoshop. Luan et al. [63] proposed an interactive system for users to easily colorize natural images. The colorization procedure consists of two stages: color labeling and color mapping. In the first stage, pixels that should have similar colors are grouped into coherent regions in the first stage, while in the second stage color mapping is applied to generate vivid colorization effect through assigning colors to a few pixels in the region. It is very tedious to colorize texture by previous methods since each tiny region inside the texture need a new color. In contrast, this method handles texture by grouping both neighboring pixels sharing similar intensity and remote pixels with similar texture (see Fig. 3). This method is effective for natural image colorization. However, the user should usually provide multiple stokes on similar patterns with different orientation and scales in order to produce fine colorization results.
Liu et al. [58] proposed an example-based colorization method that is aware of illumination differences between the target grayscale image and the source color image. Firstly, an illumination-independent intrinsic reflectance map of the target scene is recovered from multiple color references collected by web search. Then, the grayscale versions of the reference images are employed for decomposing the target grayscale image into its intrinsic reflectance and illumination components. The color is transferred from the color reflectance map to the grayscale reflectance image. By relighting with the illumination component of the target image, the final colorization result can be produced. This method needs to search suitable source images for reference by web search.
Liu et al. [42] presented a gray-scale image colorization method by control of single-parameter. The polynomial fitting model of the histograms of the source image and the grayscale image are computed by linear regression, respectively. With the user-assigned order of the polynomials, the source image and the grayscale images can be automatically matched. By transferring between the corresponding regions of the source image and the gray-scale image, colorization can be finally achieved. Quang et al. [72] proposed an image and video colorization method based on the kernel Hilbert space (RKHS). This method can produce impressive colorization results. Nevertheless, it requires initialization for different regions by manual, that is time-consuming if there are many different contents in the grayscale image.
B. Automatic Colorization
The above colorization methods require the user to perform colorization by manual, either providing a source image or using scribbles and color seeds for interaction. Since there is usually no suitable correspondence between color and local texture, automatic colorization is necessary.
Li and Hao [39] proposed an automatic colorization approach by locally linear embedding. Given a source color image and a target grayscale image, this method clips both of them into overlapping patches, which are supposed to be distributed on a manifold [13], [5]. For each patch, its neighborhood in the training patches is estimated and its chromatic information is predicted by the manifold learning [74]. With multimodality, Charpiat et al. [7] predict the probability distribution of all possible colors for each pixel of the image to be colored,, rather than selecting the probable color locally. Then, the technique of graph cut is employed to maximize the probability of the whole colored image globally. Morimoto et al. [66] proposed an automatic colorization method using multiple images collected from the web. Firstly, this method chooses images with similar scene structure with the target grayscale image I m as the source images. The gist scene descriptor, a feature vector expressing the global scene in a lower dimension is used to aggregate oriented edge responses at multi-scales into coarse spatial bins. Then the distance between the gist of I m and that of the images from the web are computed. The most similar images are chosen as source images, which are used for colorization. Here, the transferring method of [87] was used for colorization. However, this method restricts from the searching results from the images collected from the web, which may produce unnatural results due to the source images that are structurally similar but semantically different.
To this end, Liu and Zhang [50] proposed an automatic grayscale image colorization method via histogram regression. Given a source image and a target grayscale image, the locally weighted regression is performed on both images to obtain the feature distributions of them. Then, these features are automatically matched by aligning zero-points of the histogram. Thus, the grayscale image is colorized in a weighted manner. Figure 4 shows a colorization result by this method. Although this method can achieve confident colorization results, it may fail for images with strong texture patterns or varied lighting effects (e.g., shadows and highlight).
Liu and Zhang [51] further proposed a colorization method based on texture map. Assuming that a source color image with the similar content with the target grayscale image can be provided by the user, this method is aware of both the luminance and texture information of images so that more convincing colorization results can be produced. Specifically, given a source color image and a target grayscale image, their respective spatial maps are computed. Note that the spatial map is a function of the original image, indicating the luminance spatial distribution for each pixel. Then, by Fig. 5. An example of stroke-preserving manga colorization [71]. (a) the target manga drawing with user scribbles, (b) the colorization result, and (c) the enlarged views. Note that a color-bleeding algorithm is utilized here, so that even if the user provides careless scribbles, the leave region can still be accurately separated from tree branches.
performing locally weighted linear regression on the histogram of the quantized spatial map, a series of spatial segments are computed. Within each segment, the luminance of target grayscale image is automatically mapped to color values. Finally, colorization results can be yielded through local luminance-color correspondence and global luminance-color correspondence between the source color image and the target grayscale image.
Beyond natural images, Visvanathan et al. [84] automatically colorized pseudocolor images by gradient-based value mapping. This method targets for visualizing pixel values and their local differences for scientific analysis.
C. Cartoon Colorization
Some researchers also extended the colorization technique to cartoon images. Sýkora et al. [81] proposed a semiautomatic, fast and accurate segmentation method for black and white cartoons. It allows the user to efficiently apply ink on the aged black and white cartoons. The inking process is composed of four stages, namely segmentation, marker prediction, color luminance modulation, and final composition of foreground and background layers.
Qu et al. [71] proposed a method for colorizing black-andwhite manga (comic books in Japanese) containing a large number of strokes, hatching, halftoning, and screening. Given scribbles by the user on the target grayscale manga drawing, Gabor wavelet filters are employed to measure the patterncontinuity and thereby a local, statistical based pattern feature can be estimated. Then, with the level set technique, the boundary is propagated to monitor the pattern continuity. In this way, areas with open boundaries or multiple disjointed regions with similar patterns can be well segmented. Once the segmented regions are obtained, conventional colorization methods can be used to color replacing, color preservation as well as pattern shading. Figure 5 shows an example of strokepreserving manga colorization by this method.
D. Deep Colorization
Cheng et al. [8] proposed a deep neural network model to achieve fully automatic image colorization by leveraging Fig. 6. The framework of the automatic colorization method via learning representations [35]. a large set of source images from different categories (e.g., animal, outdoor, indoor) with various objects (e.g., tree, person, panda, and car). This method consists of two stages, (1) training a neural network, and (2) colorizing a target grayscale image with the learned neural network.
Larsson et al. [35] trained a model to predict per-pixel color histograms for colorization. This method trains a neural architecture in an end-to-end manner by considering semantically meaningful features of varying complexity. Then, a color histogram prediction framework is developed to treat uncertainty and ambiguities inherent in colorization so as to avoid jarring artifacts. As shown in Fig. 6, given a grayscale image, with a deep convolutional architecture (VGG), spatially localized multilayer slices are chosen as per-pixel descriptors. The system then estimates hue and chroma distributions for each pixel p with its hypercolumn descriptor. Finally, at test time, the estimated distributions are used for color assignment.
Zhang et al. [97] treat image colorization as a classification problem considering the underlying uncertainty of this task. They leverage class-rebalancing during training to increase the diversity of colors. At test time, this method is performed as a feed-forward pass in a CNN with a million color images. This method demonstrates that with a deep CNN and a carefullytuned loss function, the colorization task can generate results close to real color photos.
Iizuka et al. [28] proposed an automatic, CNN-based grayscale image colorization method by combining both global priors and local image features. The proposed network architecture is able to jointly extract global and local features from an image and fuse them for colorization. Specifically, their model is composed of four parts, namely a low-level features network, a mid-level features network, a global features network, and a colorization network. Various evaluation experiments were performed to verify this method with user study and many historical hundred-year-old black-and-white photographs. Figure 7 shows an example of this method.
Zhang et al. [98] propose a CNN framework for userassisted image colorization. Given a target grayscale image, and sparse, local user edits, this method can automatically produce convincing colorization results. By training on a large amount of image data, this method learns to propagate user edits by merging both low-level cues and high-level semantic information. This method has help non-professionals to design a colorful work, since it has great ability to achieve fine colorization results even with random user inputs.
Deshpande et al. [11] learned a low-dimensional smooth embedding of color fields with a variational autoencoder (VAE) for grayscale image colorization. A multi-modal conditional model between the gray-level features and the lowdimensional embedding is learned to produce diverse colorization results. The loss functions are specially designed for the VAE decoder to avoid blurry colorization results and respect uneven distribution of pixel colors. This method has potential to handle other ambiguous problems, since the lowdimensional embeddings has ability to predict diversity with multi-modal conditional models. However, high spatial detail is not taken into account in this method.
III. IMAGE DECOLORIZATION
Image decolorization is often used as a preprocessing for downstream image processing tasks such as segmentation, recognition, and analysis. Recently, decolorization has attracted more and more attention of researchers. In the early stage, the three channels R, G, and B are represented by a single channel or only the brightness channel information is used to represent the grayscale image. However, these simple color removal methods suffer from contrast loss in the gray image. To this end, researchers have proposed local and global decolorization methods in order to preserve the contrast of color images in the resulting grayscale images.
A. Early Decolorization Methods
The early image decolorization method is simple, which directly processes the (R, G, B) channels of a color image in the RGB color space. These methods include the component method, the maximum method, the average method, and the weighted average method.
The component method uses one of the (R, G, B) in the color image as the corresponding pixel value in the grayscale image, written as
G 1 (i, j) = R(i, j), G 2 (i, j) = G(i, j), G 3 (i, j) = B(i, j),(1)
where (i, j) is the pixel coordinate in an image. Note that any one of G 1 , G 2 , G 3 can be selected as needed.
The maximum method takes the maximum value of (R, G, B) in the color image as the gray value of the grayscale image.
GRAY (i, j) = max{R(i, j), G(i, j), B(i, j)}.(2)
The average method is to average the three component values of (R, G, B) in the color image to obtain a gray value.
GRAY (i, j) = (R(i, j), G(i, j), B(i, j))/3.(3)
The weighted average method uses the weighted average of three components with different weights as the grayscale image.
GRAY (i, j) = 0.299R(i, j) + 0.578G(i, j) + 0.114B(i, j).
(4) In addition to using the color component of the RGB space, it is also common to employ the brightness channel of other color spaces to represent the gray value of a grayscale image. For example, Hunter [27] uses the L channel of the Lαβ space to represent a grayscale image, while Wyszecki and Stiles [88] adopt the Y component in the Y U V color space to represent the grayscale image. In the Y U V color space, the Y component is the brightness of pixels, reflecting the brightness level of an image. According to the relationship between the RGB color space and the Y U V color space, the mapping between the brightness y and three color components can be established as
y = 0.3r + 0.59g + 0.11b.(5)
The luminance value y is used to represent the gray value of the image. Based on this observation, Nayatani [67] proposed a color mapping model with independent input, i.e., input three components independently and set the weights of the corresponding components as needed. These early methods are easy to implement, however, they would cause the loss of image contrast, saturation, exposure, etc. To this end, researchers explored decolorization methods with higher accuracy and efficiency, including local decolorization methods, global decolorization methods, and deep learning based decolorization methods.
B. Local Decolorization Methods
Local decolorization methods usually use different strategies in solving the mapping model from a color image to a grayscale one. The strategy deals with different pixels or color blocks, and increases the local contrast by strengthening the local features.
Bala and Eschbach [4] proposed a decolorization method that locally enhances the edge and contours between adjacent colors through adding high-frequency chrominance information into the luminance channel. Specifically, a spatial highpass filter weighting the output with a luminance-dependent term is applied to the chrominance channels. Then the result is added to the luminance channel. Figure 8 shows a flow chart of this method.
Neumann et al. [68] view the color and luminance contrasts of an image as a gradient field and solve the inconsistency of the field. They chose locally consistent color gradients and performed 2D integration to produce the grayscale image. Since its complexity is linear in the number of pixels, this method is simple yet very efficient, which is suitable for handling high-resolution images. Smith et al. [76] proposed Fig. 8. The flow chart of the spatial color-to-grayscale transform method [4].
Here the Lab color space is taken as an example. Note that "HPF" represents high-pass filter. a perceptually accurate decolorization method for both images and videos. This approach consists of two steps: (1) globally assigning gray values and determine color ordering, and (2) locally improving the grayscale to preserving the contrast in the input color image. The Helmholtz-Kohlrausch color appearance effect is introduced to estimate distinctions between isoluminant colors. They also designed a multiscale local contrast enhancement strategy to produce a faithful grayscale result. Note that this method makes a good balance between a fully automatic method (first step) and user assist (second step), making it suitable for dealing with various images (e.g., natural images, photographs, artistic works, and business graphics). Figure 9 shows that, for a challenging image consists of equiluminant colors, this method is able to predict the H-K effect that makes a more colorful blue appear lighter than the duller yellow. A limitation of this approach comes from the locality of the second step, which may fail to preserve chromatic contrast between non-adjacent regions and lead to temporal inconsistencies.
Lu et al. [60] proposed a decolorization method aiming to preserving the original color contrast as far as possible. A bimodal contrast preserving function is designed to constrain local pixel differences and a parametric optimization approach is employed to preserve the original contrast. Owing to weak color order constraint, they relax the color order constraint and seek to better maintain color contrast and enhance the visual distinctiveness for edges. Nevertheless, this method cannot greatly preserve the global contrast in the image. Moreover, since the gray image is produce by solving the energy equation in an iterative manner, the efficiency of this algorithm is relatively low. Zhang and Liu [96] presented an efficient image decolorization method via perceptual group difference (PGD) enhancement. They view the perceptual group instead of individual image pixels as the human perception elements. Based on this observation, they perform decolorization for different groups in order to maximumly maintain the contrast between different visual groups. A global color to gray mapping is employed to estimate the grayscale of the whole image. Experimental results showed that, with PGD enhancement, this approach is capable of achieving better visual contrast effects.
The local decolorization methods may distort appearance for regions with constant colors and therefore lead to undesired haloing artifacts.
C. Global Decolorization Methods
Global decolorization methods perform decolorization on the whole image in a global manner, including linear declorization and nonlinear decolorization techniques.
Linear declorization methods. Gooch et al. [18] proposed Color2Gray, a saliency-preserving decolorization method. This method is performed in the CIE L * a * b * color space instead of the traditional RGB color space. Considering that the human visual system is sensitive to change, they preserve relationships between neighboring pixels rather than representing absolute pixel values. The chrominance and luminance changes in a source image are transferred to changes in the target grayscale image so as to produce images maintaining the salience of the source color images. Grundland and Dodgson [19] proposed an efficient, linear decolorization approach by adding a fixed amount of chrominance to lightness. To achieve a perceptually plausible decolorization result, Kuk et al. [34] proposed a color to grayscale conversion method by taking into account both local and global contrast. They encode both local and global contrast into an energy function via a target gradient field, which is constructed from two types of edges: (1) edges connecting each pixel to neighboring pixels, and (2) edges connecting each pixel to predetermined landmark pixels. Finally, they formulate the decolorization problem as reconstructing a grayscale image from the gradient field, that is solved by a fast 2D Poisson solver.
Nonlinear declorization methods. Kim et al. [33] presented a fast and robust decolorization algorithm via a global mapping that is a nonlinear function of the lightness, chroma, and hue of colors. Given a color image, the parameters of the function are optimized to make resulting grayscale image respect the feature discriminability, lightness, and color ordering in the input color image. Ancuti et al. [2] introduced a fusion-based decolorization technique. The input of their method include three independent RGB channels and an additional image that conserves the color contrast. The weights are based on three different forms of local contrast: a saliency map to preserve the saliency of the original color image, a second weight map taking advantages of well-exposed regions, and a chromatic weight map enhancing the color contrast. By enforcing a more consistent gray-shades ordering, this strategy can better preserve the global appearance of the image. Ancuti et al. [3] further presented a color to gray conversion method aiming to enhance the contrast of the images while preserving the appearance and quality in the original color image. They intensify the monochromatic luminance with a [76], the decolorization result of Kim et al. [33], Kim et al. [33], the decolorization result of Lu et al. [60], the decolorization result of Lu et al. [61], and the decolorization result of Liu et al. [43]. Fig. 11. A comparison among different decolorization result with running time [61]. From left to right: the input color image, the result of the built-in Matlab function rgb2gray (8 ms), the result of [60] (1,102 ms), and the result of [61] (30 ms). Note that all the methods were implemented in Matlab. mixture of saturation and hue channels in order to respect the original saliency while enhancing the chromatic contrast. In this way, a novel spatial distribution can be produced which is capable of better discriminating the illuminated regions and color features. Liu et al. [43] developed a decolorization model based on gradient correlation similarity (Gcs) so as to reliably maintain the appearance of the source color image. The gradient correlation is employed as a criterion to design a nonlinear global mapping in the RGB color space. Figure 10 shows a comparison result between this method and other image decolorization methods including Smith et al. [76], Kim et al. [33], Lu et al. [60], and Lu et al. [61]. It can be seen from the results that this method is able to better preserve features in the source color image which are more discriminable in the grayscale image; and it also has good ability to maintain a desired color ordering in color-to-gray conversion. Liu et al. [44] further proposed a color to grayscale method by introducing the gradient magnitude [89].
Song et al. [77] regard decolorization as a labeling problem to maintain the visual cues of a color image in the resulting [78]. The top row is the original color images. The middle row shows the failure results of current decolorization methods (from left to right: Gooch et al. [18], Gundland and Dodgson [19], [76], Kim et al. [33], Ancuti et al. [3], Lu et al. [60], and Lu et al. [61]). The bottom row are results by Song et al. [78] which are produced by modifying rgb2gray() with adjusted weights for R, G, and B channels. grayscale image. They define three types of visual cues, namely color spatial consistency, image structure information, and color channel perception priority, that can be extracted from a color image. Then, they cast color to gray as a visual cue preservation process based on a probabilistic graphical model, which are solved via integral minimization.
Most of the above image decolorization methods attempt to preserve as much as possible visual appearance and color contrast, however, little attention was devoted to the speed issue of decolorization. The efficiency of most method is lower than the standard procedure (e.g., Matlab built-in rgb2gray function). To this end, Lu et al. [61] proposed a real-time contrast preserving decolorization method. They achieved this goal by three main ingredients: a simplified bimodal objective function with linear parametric grayscale model, a fast non-iterative discrete optimization, and a sampling based P -shrinking optimization strategy. The running time of this method is a constant O(1), independent of image resolutions. As shown in Fig. 11, this method takes only 30ms (the rightmost result) to decolorize an one megapixel color image, that is comparable with the built-in Matlab rgb2gray function (the left second result), but achieving a better color to gray conversion result which is visually similar to a compelling contrast preserving decolorization method [60] (the right second result).
Lu et al. [62] further presented an optimization framework for image decolorization to preserve color contrast in the original color image as much as possible. A bimodal objective function is used to reduce the restrictive order constraint for color mapping. Then, they design a solver to realizing automatic selection of suitable grayscales via global contrast constraints. They also propose a quantitative perceptualbased metric, E-score, to measure contrast loss and content preservation in the resulting grayscale images. The E-score is to jointly consider two measures CCPR (Color Contrast Preserving Ratio) and CCFR (Color Content Fidelity Ratio), written as E score = 2 · CCP R · CCF R CCP R + CCF R
It is reported that this is among the first attempts in the color to gray field to quantitatively evaluate decolorization results. Considering that the above decolorization methods suffer from the robustness problem, i.e., may fail to accurately convert iso-luminant regions in the original color image, while the rgb2gray() function in Matlab works well in practice applications. Song et al. [78] proposed a robust decolorization method by modifying the rgb2gray() function. Figure 12 shows that this method is able to realize color to gray conversion for iso-luminance regions in an image, while previous methods, including Gooch et al. [18], Gundland and Dodgson [19], [76], Kim et al. [33], Ancuti et al. [3], Lu et al. [60], and Lu et al. [61] fail in this task. In this method, they avoid indiscrimination in iso-luminant regions by adaptively selecting channel weights with respect to specific images rather than using fixed channel weights for all cases. Therefore, this method is able to maintain multi-scale contrast in both spatial and range domain.
Sowmya et al. [80] presented a color to gray conversion algorithm with a weight matrix corresponding to the chrominance components. The weight matrix is obtained by reconstructing the chrominance data matrix through singular value decomposition (SVD). Ji et al. [31] presented a global image decolorization approach with a variant of difference-of-Gaussian band-pass filter, called luminance filters. Typically, the filter has high responses on regions of which colors differ from their surroundings for a certain band. Then, the grayscale value can be produced after luminance passing a series of band-pass filters. Due to that this approach is linear in the number of pixels, it is efficient and easy to implement,
D. Deep Learning Based Decolorization Methods
By training partial differential equations (PDEs) on 50 input/output image pairs, Lin et al. [40] constructed a mapping model for the task of color to gray conversion. It is reported that their learned PDEs can yield similar decolorization results with those of Gooch et al. [18].
Hou et al. [23] proposed the Deep Feature Consistent Deep Image Transformation (DFC-DIT) framework for oneto-many mapping image processing tasks (e.g., downscaling, decolorization, and tone mapping). The DFC-DIT achieves transformation between images with a CNN as a non-linear mapper respecting the deep feature consistency principle that is enforced with another pretrained and fixed deep CNN. As shown in Fig. 13, this system is comprised of two networks, a transformation network and a loss network. The former is used to convert an input to an output, and the later servers as computing the feature perceptual loss for the training of the transformation network. [99]. This framework is composed of four parts: a low-level features network, a local semantic feature network, a global feature network, and a decolorization network. The four components are tightly coupled so as to learn a complex color-to-gray mapping. The low-level features network uses four groups of convolution layers to extract low-level features from the input image. With the FCN (Fully Convolutional Networks) structure, the local semantic feature network acquires instance semantic information with semantic tags of an image, such as dog and airplane. The global feature network serves to produce global image features by processing low-level features with several convolution layers. Finally, the decolorization network with the Euclidean loss outputs the resulting grayscale image.
Considering that the local decorization methods are less accurate enough to process local pixel leading to local artifacts, while the global methods may fail to treat local color blocks, Zhang and Liu [99] proposed a novel image color to gray conversion method by combining local semantic features and global features. In order to preserve color contrast between adjacent pixels, a global feature network is developed to learn the global features and spatial correlation of an image. On the other hand, in order to preserve the contrast between different object blocks, they take care of local semantic features of images and fine classification of pixels during learning deep image features. Finally, with fusion of both the local semantic features and global features, this method performs better in terms of contrast preservation than the state-of-theart decolorization approaches. Figure 14 gives a flow chart of this method.
According to the human visual mechanism, exposure plays a critical role in human visual perception, e.g., low-exposure and overexposure areas usually easily catch the attention of an observer. However, exposure is missed in existing decolorization methods. To this end, Liu and Zhang [54] proposed an image decolorization approach by fusion of local features and exposure features with a CNN framework. This framework consists of a local feature network and a rough classifier. The local feature network aims to learn the local semantic features of the color so as to maintain the contrast among different color blocks, while the rough classifier classifies three types of exposure states: low-exposure, normal-exposure, and overexposure features of an image. Figure 15 shows the ability of this method to treat images with different exposures.
IV. VIDEO COLORIZATION
People are willing to watch a colorful film instead of a grayscale one. Gone with the Wind in 1939 is one of the first colorized films [16] which is popular with the audience. However, it is challenging to obtain a convincing video colorization because of its multimodality in the solution space and the requirement of global spatiotemporal consistency [36] Fig. 15. A comparison of the results with and without the exposure feature network [54]. From left to right are input images (a), results without (b) and with (c) the exposure feature network. From top to bottom row represent low-exposure, over-exposure, and normal-exposure.
is also inherently more challenging than Unlike single image colorization, video colorization should also satisfy temporal coherence. In view of this point, the above single image colorization cannot be used for video colorization. Currently, researches [30], [85], [52], [65], [36] realized video colorization by propagating the color information either from a color reference frame or sparse user scribbles to the whole target grayscale video.
Vondrick et al. [85] regard video colorization as a selfsupervised learning problem for visual tracking. To this end, they learn to colorize gray-scale videos by copying colors from a reference frame by exploiting the temporal coherency of color, rather than predicting the color directly from the grayscale frame. Jampani et al. [30] proposed Video Propagation Network (VPN), processes video frames in an adaptive manner. The VPN consists of a temporal bilateral network (TBN) and a spatial network (SN). The TBN aims for dense and video adaptive filtering, while the SN is used for refining features and increasing flexibility. This method propagates information forward without accessing future frames. Experiments showed that, given the source color image for the first video frame, this method can propagate the color to the whole target grayscale video. Given the color image for the first video frame, the task of this method is to propagate the color to the entire video. This method can also be used for video processing tasks requiring the propagation of structured information (e.g., video object segmentation and semantic video segmentation).
Meyer et al. [65] proposed a deep learning framework for video color propagation. This method consists of a short range propagation network (SRPN), a longer range propagation Fig. 16. An overview of the deep learning framework for video color propagation [65]. Both a short range network and a long range color propagation network are used to propagate colors in a video. The results of these two networks and the target grayscale image together constitute the input to the fusion and refinement network to output the final color frame. Fig. 17. The framework of the automatic video colorization method with self-regularization and diversity [36]. This model consists of a colorization network f and a refinement network g. f is used to colorize each grayscale video frame and outputs candidate colorization images. By inputting the i-th colorized candidate images and two confidence maps, g produces a refined video frame. network (LRPN), and a fusion and refinement network (FRN). The SRPN aims to propagate colors frame-by-frame ensuring temporal stability. The input to SRPN are two consecutive gray scale frames and it outputs an estimated warping function that is used to transfer the colors of the previous frame to the next one. The LRPN introduces semantical information by matching deep features extracted from the frames, which are then used to sample colors from the first frame. Except long range color propagation, this strategy also contributes to restore missing colors because of occlusion. With a CNN, the SRPN is used to combine the above two stages for fusion and refinement. Figure 16 gives an overview of the framework of this method.
Lei and Chen [36] proposed a fully automatic, selfregularized approach to video colorization with diversity. As shown in Fig. 17, this method is comprised of a colorization network f for video frame colorization and a refinement network g for spatiotemporal color refinement. A diversity loss is designed to allow the network to generate colorful videos with diversity. Moreover, the diversity loss can also make the training and process more stable.
V. VIDEO DECOLORIZATION
As for video decolorization, people mostly extend image decolorization methods to process video frames, which would easily lead to the flicker phenomenon due to the spatiotemporal inconsistency. Video decolorization should take into account Fig. 18. An overview of the video decolorization using visual proximity coherence optimization [83]. Firstly, the decolorization proximity for each frame is estimated. The DC-GMM classifier is then used to select a specific decolorization strategy, and finally decolorize the frame into grayscale using the selected strategy. Secondly, with DC-GMM, video frames are classified into three categories, i.e., high-proximity, median-proximity, and low-proximity. Finally, a salience C2G method is employed to maintain temporal coherence and alleviate flickering between frames.
both the contrast preservation of each video frame and the temporal consistency between video frames.
Since the method of Smith et al. [76] can preserve consistency avoiding changes in color ordering, they extended their two-step image grayscale transformation method to treat video decolorization. Owing to the ability to maintain consistency over varying palettes, Ancuti et al. [2] applied their fusionbased decolorization technique for video cases. Given a video, Ancuti et al. [3] searched in the entire sequence for the color palette that appears in each image (mostly identified with the static background). In this way, they extend their saliencyguided decolorization approach to video decolorization. For a video with relatively constant color palette, they computed a single offset angle value for the middle frame in a video.
Song et al. [79] proposed a real-time video decolorization method using bilateral filtering. Considering that human visual system is more sensitive to luminance than the chromaticity values, they recover the color contrast/detail loss in the luminance. They represent the loss as residual image by the bilateral filter. The resulting grayscale image is a sum of the residual image and the luminance of the original color image. Since the residual image is robust to temporal variations, this method can preserve the temporal coherence between video frames. Moreover, as the kernel of the bilateral filter can be set as large as the input image, this method is efficient and can run in real time on a 3.4 GHz i7 CPU.
Tao et al. [82], [83] defined decolorization proximity to measure the similarity of adjacent frames and presented a temporal-coherent video decolorization method using proximity optimization. They then respectively treat frames with low, medium, and high proximities in order to better preserve the quality of these three types of frames. Finally, with a decolorization Gaussian mixture model (DC-GMM), they classify the frames and assign appropriate decolorization strategies to them via their corresponding decolorization proximity. Figure 18 shows an overview of this method.
Most of the existing video decolorization methods directly apply image decolorization algorithms to treat video frames, which would easily causes temporal inconsistency and flicker Fig. 19. The framework of the video decolorization method based on the CNN and LSTM neural network [53]. Given a video sequence Ct|t = 1, 2, 3, ..., N , it is processed into sequence images. Then the local semantic content encoder extracts deep features of these sequence images, adjusts the scale of the feature maps, and inputs them to the temporal features controller. After the output feature maps are fed into the deconvolution-based decoder, the resulting grayscale video sequence Gt|t = 1, 2, 3, ..., N is produced.
phenomenon. Moreover, there may be similar local content features between video frames, which can be used to avoid redundant information. To this end, Liu and Zhang [53] introduced deep learning into the field of video decolorization by using CNN and a long short-term memory neural network. To the best of our knowledge, this is among the first attempts to perform video decolorization using deep learning techniques. A local semantic content encoder was designed to learn the same local content of a video. Here, the local semantic features were further refined by a temporal feature controller via a bi-directional recurrent neural network with long short-term memory units. Figure 19 shows an overview of this method.
VI. CONCLUSION AND FUTURE WORK This paper summarized the progress of colorization and decolorization methods for image and videos in the last two decades. According to that if user interaction is involved, we classified the image coloriztion methods into two categories, semi-automatic colorization methods and automatic colorization methods. As for image decolorization methods, we first discussed the early image decolorization methods, including the component method, the maximum method, the average method, and the weighted average method. We then summarized the existing image decolorization methods from the perspective of global decolorization and local decolorization. Finally, we also introduced the latest deep learning based colorization and decolorization approaches for images and videos.
Although convincing results can be achieved by the current colorization and decolorization methods. We think some challenges still remains. For example, a user-friendly image and video colorization and decolorization system is still needed. It is necessary to further improve the computational efficiency of the colorization and decolorization methods, especially for high-definition images and videos. Moreover, more objective metrics specific to colorization and decolorization assessment are required. Finally, large-scale datasets are needed for deep learning based image colorization and decolorization techniques. In the future, we believe researchers will pay more and more attention to this field.
Fig. 2 .
2The application of the chrominance blending based colorization method for image recolorization[90].
Fig. 3 .
3Results of colorizing texture with multiple colors[63]. Note that the strokes are shown in (a), (c), and (e), and the corresponding colorization results are presented in (b), (d), and (f). Colorizing texture with multiple colors is the unique property of this method.
Fig. 4 .
4A colorization result of an animal image using the automatic grayscale image colorization method via histogram regression[50].
Fig. 7 .
7A result of the real-time user-guided deep colorization method with learned deep priors[98]. Given a target grayscale image (the left image) and sparse user edits (the second left image), multiple plausible colorization results (the right three images). Photograph of Migrant Mother by Dorothea Lange, 1936 (Public Domain).
Fig. 9 .
9A decolorization result showing that isoluminant colors can be mapped to unique, properly ordered gray values[76].
Fig. 10 .
10A comparison on the dataset[6] among different methods[43].(a) through (f) are the input color image, the decolorization result of Smith et al.
Fig. 12 .
12A comparison of the robust decolorization method by Song et al. with conventional methods
Fig. 13 .
13An overview of the deep feature consistent deep image transformation (DFC-DIT) framework[23].
Fig. 14 .
14Overview of the contrast preserving image decolorization method combining global features and local semantic features
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|
[] |
[
"Classical Dimensional Transmutation and Confinement",
"Classical Dimensional Transmutation and Confinement"
] |
[
"Gia Dvali \nDepartment für Physik\nASC\nLMU\nMünchen Theresienstr. 3780333MünchenGermany\n\nMPI für Physik Föhringer Ring 6\n80805MünchenGermany\n\nCERN\nTheory Division\n1211Geneva 23Switzerland\n\nDepartment of Physics\nCCPP\nNYU 4 Washington Place10003New YorkNYUSA\n",
"Cesar Gomez \nInstituto de Física Teórica UAM-CSIC\nUniversidad Autónoma de Madrid\nC-XVI, 28049Cantoblanco, MadridSpain\n",
"Slava Mukhanov \nDepartment für Physik\nASC\nLMU\nMünchen Theresienstr. 3780333MünchenGermany\n\nMPI für Physik Föhringer Ring 6\n80805MünchenGermany\n"
] |
[
"Department für Physik\nASC\nLMU\nMünchen Theresienstr. 3780333MünchenGermany",
"MPI für Physik Föhringer Ring 6\n80805MünchenGermany",
"CERN\nTheory Division\n1211Geneva 23Switzerland",
"Department of Physics\nCCPP\nNYU 4 Washington Place10003New YorkNYUSA",
"Instituto de Física Teórica UAM-CSIC\nUniversidad Autónoma de Madrid\nC-XVI, 28049Cantoblanco, MadridSpain",
"Department für Physik\nASC\nLMU\nMünchen Theresienstr. 3780333MünchenGermany",
"MPI für Physik Föhringer Ring 6\n80805MünchenGermany"
] |
[] |
We observe that probing certain classical field theories by external sources uncovers the underlying renormalization group structure, including the phenomenon of dimensional transmutation, at purely-classical level. We perform this study on an example of λφ 4 theory and unravel asymptotic freedom and triviality for negative and positives signs of λ respectively.We derive exact classical β function equation. Solving this equation we find that an isolated source has an infinite energy and therefore cannot exist as an asymptotic state. On the other hand a dipole, built out of two opposite charges, has finite positive energy. At large separation the interaction potential between these two charges grows indefinitely as a distance in power one third.
|
10.1007/jhep12(2011)103
|
[
"https://arxiv.org/pdf/1107.0870v1.pdf"
] | 40,946,134 |
1107.0870
|
b8c338806958a469776b976f659bdf2231409bd5
|
Classical Dimensional Transmutation and Confinement
5 Jul 2011
Gia Dvali
Department für Physik
ASC
LMU
München Theresienstr. 3780333MünchenGermany
MPI für Physik Föhringer Ring 6
80805MünchenGermany
CERN
Theory Division
1211Geneva 23Switzerland
Department of Physics
CCPP
NYU 4 Washington Place10003New YorkNYUSA
Cesar Gomez
Instituto de Física Teórica UAM-CSIC
Universidad Autónoma de Madrid
C-XVI, 28049Cantoblanco, MadridSpain
Slava Mukhanov
Department für Physik
ASC
LMU
München Theresienstr. 3780333MünchenGermany
MPI für Physik Föhringer Ring 6
80805MünchenGermany
Classical Dimensional Transmutation and Confinement
5 Jul 2011
We observe that probing certain classical field theories by external sources uncovers the underlying renormalization group structure, including the phenomenon of dimensional transmutation, at purely-classical level. We perform this study on an example of λφ 4 theory and unravel asymptotic freedom and triviality for negative and positives signs of λ respectively.We derive exact classical β function equation. Solving this equation we find that an isolated source has an infinite energy and therefore cannot exist as an asymptotic state. On the other hand a dipole, built out of two opposite charges, has finite positive energy. At large separation the interaction potential between these two charges grows indefinitely as a distance in power one third.
I. INTRODUCTION
The discovery of asymptotic freedom in QCD [1] opened a new era in particle physics.
Besides its direct relevance for understanding the nature of strong interactions, it introduced a concept of dimensional transmutation or equivalently a dynamical generation of scale in a seemingly scale-free theory. Both phenomena, asymptotic freedom and dimensional transmutation, are usually perceived as intrinsically quantum phenomena, as they are both deeply rooted in the renormalization group properties of the quantum theory. The question we would like to address in this paper is, how much of these phenomena is captured by classical physics? This is a fully legitimate question, since usually quantum effects have classical precursors, which sometimes appear in the form of uncontrollable growth of the classical fields.
In order to illustrate our ideas, we will consider a simple example, namely, scalar theory with negative λφ 4 , which is known to be renormalizable and has a negative β function [2], but has an unbounded from below potential. However, our classical renormalization group treatment delivers a natural prescription which allows to self-consistently work with this theory in the presence of external sources, and isolate our findings from the issue of potential instability in a pure λφ 4 theory.
In the presence of the external sources, the requirement of independence of physical observable from the regulator scale of sources implies the running of the effective coupling.
This running is the main reason behind the whole renormalization group structure and the subsequent classical dimensional transmutation. This scale dependence is the key to why the renormalization group results can be safely disentangled from the instability issues.
Simply speaking, because of the emerging scale-dependence we always perform calculations on time-scales shorter than would-be instability time in a sourceless theory.
Surprisingly, by probing λφ 4 theory by the large external source, we uncover the whole built-in RG structure already at the classical level, with fully-fledged counterparts of asymptotic freedom as well as dimensional transmutation phenomena, in which an analog of QCDscale appears as a result of classical RG invariance. We derive the exact classical β function equation from which we extract non perturbative information about the infrared region.
Solving this equation in strong coupling regime we find that the energy of the isolated external charge is infinite and positive and hence it cannot exist as a free asymptotic state.
Moreover, considering a dipole, build out of two charges with opposite signs, we find that its energy is positive and finite. When charges in this dipole are separated their interaction potential grows indefinitely as distance in power one third, thus confining the charges. These findings indicate that there may exist a classical counterpart of confinement.
II. CLASSICAL SOLUTION
After the discovery of asymptotic freedom in the non-Abelian Yang Mills theories, the physics underlying the negative sign of the β function was understood as an anti-screening effect due to self-interactions of the gauge fields. From the classical point of view we can try to understand this anti-screening considering how self-interaction modifies the field created by an external point-like source Q at large distances. This modification compared to the case of free fields can be used to define the effective charge Q ef f (r) at distance r or, equivalently, the running coupling constant α (r). This can be done without invoking quantum theory and the result will only depend on the particular classical features of self-interactions. For the large external charge one can expect that the classical contribution to anti-screening will dominate over the one due to the vacuum polarization effects.
In this section we will solve perturbatively the classical equations of motion for a given external charge and show how the anti-screening effect (the growth of Q ef f (r) with r) can naturally be achieved.
A. Anti-screening
Let us consider λ 0 φ 4 theory with negative λ 0
S = 1 2 ∂ µ φ∂ µ φ − 1 4 λ 0 φ 4 + 4πQφ d 4 x,(1)
where the signature is taken to be +, − − −, and Q is the external charge. In the case of a point-like charge the field equation for the static spherically symmetric field φ (r) reduces
to 1 r 2 d dr r 2 dφ dr − λ 0 φ 3 = −4πQδ (x) .(2)
If one neglects the nonlinear term in this equation then its solution is
φ 0 = Q r .(3)
Notice that for Q ≫ 1 the amplitude of the classical scalar field on scales r is much larger than the typical amplitude of the quantum fluctuations, which is of order 1/r. If the coupling constant λ 0 is small enough, i.e. λ 0 Q 2 ≪ 1, then the corrections to solution (2) due to the self-interaction λ 0 φ 3 can be treated perturbatively. The leading order correction to the solution φ 0 can be obtained by solving equation
1 r 2 d dr r 2 dφ dr = −4πQδ (x) + λ 0 Q 3 r 3 ,(4)
where we have substituted φ 0 in the nonlinear term. The last term in this equation can be treated as the contribution to the effective charge induced by the nonlinear self-interaction.
As it was noticed above, for Q ≫ 1, the vacuum polarization contribution to the induced charge is much smaller than the classical contribution and therefore can be neglected. As one can easily see from (4) the induced charge for positive λ 0 has a sign which is opposite to the sign of the source and the nonlinear interaction leads to screening. For negative λ 0 the charges have the same sign and we have an anti-screening effect similar to the one of the non-Abelian gauge theories. Since we are mainly interested in asymptotic freedom and confinement we will consider only the case of negative λ 0 .
B. The perturbative expansion
Let us look for the solution to equation (2) in the following form
φ (r) = Qf (r) r .(5)
Integrating equation (2) and substituting this ansatz we can rewrite the equation for the scalar field as
f (r) = 1 + α 0 r ∞ r r ′ r 0 f 3 (r ′′ ) r ′′ dr ′′ dr ′ r ′2 − N (α 0 ) ,(6)
where
α 0 ≡ −λ 0 Q 2 > 0,(7)
is the effective coupling constant and we have introduced the ultraviolet cutoff scale r 0 to regularize the integral, which otherwise would diverge. The function N (α 0 ) , which depends only on α 0 , is fixed by the normalization condition: f (r 0 ) = 1. It is clear that in the limit α 0 → 0 it must vanish and therefore in the absence of self-interaction, the solution (5) with f (r) = 1 exactly satisfies equation (2). As it follows from (5), the function f defines the anti-screened effective charge Q ef f (r) = Qf (r) or, equivalently, the running coupling
constant α ef f (r) = α 0 f 2 (r)(8)
Assuming that α 0 ≪ 1 we can solve the integral equation (7) by iterations in powers of α 0 .
With this purpose it is convenient to rewrite it as
f (x) = 1 + α 0 e x ∞ x x ′ 0 f 3 (x ′′ ) dx ′′ e −x ′ dx ′ − N (α 0 ) ,(9)
where we have introduced x = ln (r/r 0 ) instead of r. Substituting f (x) = 1 into the right hand side of equation (9) and taking into account that N (
α 0 ) = α 0 + O (α 2 0 ) we find f (x) = 1 + α 0 x + O α 2 0 .(10)
Next we take this solution, substitute it again in (9) and take N (α 0 ) = α 0 + 3α 2 0 + O (α 3 0 ) . Keeping only the terms up to second order in α 2 0 leads to
f (x) = 1 + α 0 x + α 2 0 3 2 x 2 + 3x + O α 3 0 .(11)
This procedure can be iterated giving us at each step the next order term in α 0 . The result up to order α 6 0 is
f (x) =1 + α 0 x + α 2 0 3 2 x 2 + 3x + α 3 0 5 2 x 3 + 12x 2 + 24x + α 4 0 35 8 x 4 + 71 2 x 3 + 285 2 x 2 + 285x + α 5 0 63 8 x 5 + 93x 4 + 1143 2 x 3 + 2142x 2 + 4284x + α 6 0 231 16 x 6 + 9129 40 x 5 + 7665 4 x 4 + 10 521x 3 + 37 989x 2 + 75 978x + O α 7 0 .(12)
The function N (α 0 ) to the same order in perturbations should be taken to be
N (α 0 ) = α 0 + 3α 2 0 + 24α 3 0 + 285α 4 0 + 4284α 5 0 + 75 978α 6 0 + O α 7 0 .(13)
The effective running coupling as a function of distance can be written then as perturbative series in powers of α 0
α ef f (x) = α ef f (r) = α 0 f 2 (r) = ∞ n=0 α n+1 0 g n (x) ,(14)
where g 0 (x) = 1, g 1 (x) = 2x, g 2 (x) = 4x 2 + 6x, g 3 (x) = 8x 3 + 30x 2 + 48x, g 4 (x) = 16x 4 + 104x 3 + 342x 2 + 570x,
g 5 (x) = 32x 5 + 308x 4 + 1572x 3 + 4998x 2 + 8568x, g 6 (x) = 64x 6 + 4176 5 x 5 + 5880x 4 + 27 612x 3 + 86 832x 2 + 151 956x,(15)
etc. The calculation of g n (x) is straightforward and we did it until g 10 (x) . However, to simplify the formulae we present here the result only up to g 6 (x) . Note that the running coupling constant depends on r only logarithmically with the coefficients g n power series of x = ln (r/r 0 ) with the highest power n. Moreover, the series (14) can be rearranged and partially resummed. In particular, collecting together leading powers of logarithms, next-to-leading and next-to-next leading powers we get
α ef f (x) = α 0 1 + 2x + 4x 2 + 8x 4 + 16x 5 + 32x 6 + O x 7 + α 2 0 6x + 30x 2 + 104x 3 + 308x 4 + 4176 5x 5 + 10 704 5x 6 + O x 7 + α 3 0 48x + 342x 2 + 1572x 3 + 5880x 4 + 97 248 5x 5 + 59 248x 6 + O x 7 + O α 4 0 ...(16)
wherex = α 0 x. In the second and third brackets we have also included higher order terms compared to (15) . The series in the bracket can be resummed. In particular, it is obvious
that 1 + 2x + 4x 2 + 8x 4 + 16x 5 + 32x 6 + ... = 1 1 − 2x ,(17)
Much less obvious are the following results
6x + 30x 2 + 104x 3 + 308x 4 + 4176 5x 5 + ... = 3 ln (1 − 2x) (1 − 2x) 2 ,(18)
and 48x + 342x 2 + 1572x 3 + 5880x 4 + 97 248 5x 5 + 59 248x 6 ...
= 9 (ln (1 − 2x)) 2 − 9 ln (1 − 2x) + 30x (1 − 2x) 3 ,(19)
which the reader can verify just expanding the appropriate expressions in powers ofx.
One may wonder how did we manage to resum these last two series? The answer to this question is in the next section where we uncover the renormalization group structure of our entirely classical theory and derive the β function which generates the resummation of the perturbative expansion to the appropriate powers of α 0 .
III. RENORMALIZATION GROUP AND ASYMPTOTIC FREEDOM
In the Wilsonian approach [3] the renormalization group sets how the couplings of the quantum theory should change under re-scalings of the UV cutoff . The equations governing this dependence are known as the renormalization group equations. This general notion of renormalization group can be extended to the classical field theory with external point-like sources in the following sense. Let us introduce an UV cutoff r 0 setting the way we smear the source. The classical field created by such source will generically depends on the regulator r 0 and the self-coupling λ of the theory. One can ask under which circumstances we can require that the classical theory must be invariant under re-scaling of r 0 . This is possible only if the corresponding classical theory incorporates the renormalization group structure.
In this case the dependence of the coupling on the smearing cutoff λ(r 0 ) also captures the screening and anti-screening effects. Moreover, using the effective running coupling we can associate with an external source, a physical length scale R c by the standard procedure of dimensional transmutation.
The reason why the classical solution captures the renormalization group structure is easy to understand. Any regularization scheme in quantum field theory give rise to logarithmic contributions which even in a scale invariant theory lead to anomalous scaling.
These logarithmic contributions are of the type log (p/Λ) with Λ the UV cutoff. Since the divergent contribution log Λ is absorbed by renormalization, we are free to choose the scale p at which the logarithmic contribution to the self-energy vanishes. As a consequence the scaling of Λ should be accompanied by finite renormalizations (the RG transformations) of the coupling constants. In the classical theory under consideration we have found the same type of logarithmic contributions to the field created by the external source. In this case the role of cutoff Λ is played by the smearing scale r 0 . One can renormalize the classical theory by subtracting the log r 0 contributions as it is done in quantum field theory. However, if we want physics to be independent on the method of removing this infinity, we need to change the couplings, exactly as it is done in quantum field theory. Both renormalization group structures, the classical and the quantum ones, are structurally identical for λφ 4 theory because both have the same type of parent logarithmic contributions.
A. Perturbative β function
Once we have obtained the classical expression (14) for the effective coupling constant α ef f (r) we can check whether taking the bare coupling constant α 0 as a function of r 0 we can make α ef f (r) independent of r 0 . As we have said, this is possible only in the theories with associated renormalization group structure, which in turn imposes rather severe conditions on the functions g n (x) in the perturbative expansion (14) . Let us first derive these conditions, which do not depend on the origin (classical or quantum) of the renormalization group, and then verify whether they are satisfied by the functions in (15).
On general grounds the expansion of the dimensionless running coupling constant α ef f (r)
in powers of α 0 = α (r 0 ) , normalized at r = r 0 , can be written as
α ef f (r) = α (r 0 ) + α 2 (r 0 ) g 1 r r 0 + ... = ∞ n=0 α n+1 (r 0 ) g n r r 0 ,(20)
where we use the spatial scale r instead of the usually used energy scale k ∼ 1/r. It is clear that g 0 (r/r 0 ) = 1 and since α (r) = α (r 0 ) at r = r 0 , we have
g n (1) = 0,(21)
for n ≥ 1. Invariance under changes of the ultraviolet regulator r 0 , implies
d dr 0 ∞ n=0 α n+1 (r 0 ) g n r r 0 = 0,(22)
which in turn imposes severe restrictions on g n (r/r 0 ) . Taking the derivative and rearranging the terms in (22) leads to
dα (r 0 ) d ln r 0 = α 2 (r 0 ) ∞ k=0 g ′ k+1 (x) α k (r 0 ) ∞ k=0 (k + 1) g k (x) α k (r 0 ) ,(23)
where x = ln (r/r 0 ) and prime denotes the derivative with respect to x. The ratio of sums in the right hand side of (23) should not depend on x because the left hand side of this equation is x-independent. Therefore setting x = 0 (which corresponds to r = r 0 ) and taking into account (21) we find that
∞ k=0 g ′ k+1 (x) α k (r 0 ) ∞ k=0 (k + 1) g k (x) α k (r 0 ) = ∞ k=0 g ′ k+1 (0) α k (r 0 ) ,(24)
from where it follows that the functions g k (x) should satisfy the following recursion relations:
dg n+1 (x) dx = n k=0 (k + 1) g ′ n+1−k (0) g k (x) .(25)
Note that only if these conditions are satisfied then there exists a function α (r 0 ) for which the sum in the right hand side of (20) does not depend on r 0 . Nicely enough the unambiguous solution of these recursion relations with "initial conditions" (21) is given by
g n (x) = n k=1 c k x k ,(26)
where c k are completely determined by the numerical values of g ′ 1 (0) , g ′ 2 (0) , .. which in principle can be arbitrary. For instance, the coefficient in front of the leading logarithm x n = ln n (r/r 0 ) in g n is equal to c n = (g ′ 1 (0)) n . At this point it is quite rewarding to confirm that the set of classical functions (15) in fact satisfies the recursion relations (25) . This can be done by direct calculation to any order in perturbation theory (we did it up to g 10 ). Thus, taking α 0 in (14) to be the function of r 0 we uncover the renormalization group structure of the classical λφ 4 theory. We would like to stress that in distinction from the quantum field theory, where the renormalization group is normally checked by direct calculations only to the leading logarithms (and postulated otherwise), we verified it also for all subleading logarithms.
To take the advantage of renormalization group for partial resummation of the perturbative expansion (14) we note that from (23) and (24) it follows
dα (r 0 ) d ln r 0 = α 2 (r 0 ) ∞ k=0 g ′ k+1 (0) α k (r 0 ) .(27)
The running constant α ef f (r) depends on r in the same way that α (r 0 ) depends on r 0 .
Hence α ef f (x) satisfies the well known Gell-Mann-Low equation [4]
dα ef f (x) dx = α 2 ef f (x) ∞ k=0 g ′ k+1 (0) α k ef f (x) .(28)
The β function is normally defined as the derivative of α ef f with respect to the logarithm of the energy squared. For us it is more convenient to define it as
β ≡ dα ef f (x) dx ,(29)
which (up to factors 4π due to the choice of charge units) is related to the standard β st function as β st = −β/2. According to (28) and (15) the classical perturbative β function is equal to
β (α) = ∞ k=1 β k α k+1 = 2α 2 + 6α 3 + 48α 4 + 570α 5 + 8568α 6 + 151956α 7 + ...,(30)
where α ≡ α ef f (x) and β i ≡ g ′ i (0) . Obviously, we should not expect the numerical coefficients β i of this classical beta function to be identical to the ones derived in the quantum field theory. In the last case β i are determined by the loop contributions and they will depend, beyond two loops, on the particular renormalization scheme used to segregate a finite part of the divergent loop integrals. The classical beta function accounts for the anti-screening effects due to the classical self-interaction. A potential quantum theory check of the numerical coefficients derived above will require to work in the presence of large external charge where we have to modify the Green functions in order to account for the effect of the external charge. Because for Q ≫ 1 the quantum fluctuations are subdominant we expect that classical contribution dominates. Although the direct check of this expectation is obviously important we will not follow that path. Instead we will restrict ourselves to the physical consequences of the underlying renormalization group structure of the classical theory.
B. Partial resummations
In equation (16) we have separately collected the contribution of the leading and subleading logarithms to α (x) and presented the result of their resummation. For the subleading logarithm the result was derived using Gell-Mann-Low equation. For α ≪ 1 we can first neglect all terms in β function besides of the "one loop" contribution. Equation (28) then
reduces to dα (x) dx = 2α 2 (x) ,(31)
and its solution, with initial condition
α (0) = α 0 ≡ −λ 0 Q 2 , is α (r) = α 0 1 − 2α 0 x = −λ 0 Q 2 1 + 2λ 0 Q 2 ln (r/r 0 ) ,(32)
where λ 0 = λ (r 0 ) < 0. It is easy to see that this solution gives us the resummation of the leading logarithms in the expansion (16) , (17) .
We can repeat the same analysis keeping in Gell-Mann-Low equation the contribution up to two loops,
dα (x) dx = 2α 2 (x) + 6α 3 .(33)
Integrating this equation with initial condition α (0) = α 0 , we obtain
1 α (x) − 3 ln 1 + 3α (x) 1 + 3α 0 × α 0 α (x) = 1 − 2α 0 x α 0 .(34)
Solving this equation in terms of the perturbative expansion in α 0 one gets
α (x) = α 0 1 − 2α 0 x − 3 α 0 1 − 2α 0 x 2 ln (1 − 2α 0 x) + O α 3 0 (35)
Note that the second term agrees with resummation (18) of the next to the leading order logarithms. The same is true at three loop order, where the solution to
dα (x) dx = 2α 2 (x) + 6α 3 + 48α 4 ,(36)
which is
α (x) = α 0 1 − 2α 0 x − 3 α 0 1 − 2α 0 x 2 ln (1 − 2α 0 x)(37)+ 9 α 0 1 − 2α 0 x 3 ln 2 (1 − 2α 0 x) − ln (1 − 2α 0 x) + 30 9 α 0 x + O α 4 0 ,
also accounts for the resummation of next-to-next subleading logarithms. In other words the solutions to the classical renormalization group equation give us the resummation of the perturbative series (14) taking care in every step about next logarithms in g n (x) . It is clear that when the running coupling constant becomes of order unity (strong coupling regime) all terms in expansion (37) are of the same order and the series (37) should be further resummed. It is not a priori clear whether the singularity in this expansion (Landau pole [5]) will survive after this resummation. We will answer this question in the next section using nonperturbative methods.
C. Dimensional transmutation and asymptotic freedom
One important consequence of the renormalization group is dimensional transmutation.
We can easily understand this phenomenon using the result of the one loop resummation of perturbative expansion
α (r) = −λ 0 Q 2 1 + 2λ 0 Q 2 ln (r/r 0 ) .(38)
In this expression λ 0 depends on the regulator r 0 in such a way that α (r) is r 0 -independent to the corresponding order. Therefore we can define the renormalization group invariant
physical scale R c via ln R c r 0 = − 1 2λ 0 Q 2 .(39)
Note that this scale
R c = r 0 e − 1 2λ(r 0 )Q 2 ,(40)
does not depend on the particular value of regulator r 0 at one loop level. Using this dynamically generated scale we can rewrite the physical running coupling as
α (r) ≡ −λ (r) Q 2 = 1 2 ln (R c /r) .(41)
The physical meaning of this expression is obvious. Perturbatively the theory can be defined in the ultraviolet region corresponding to length scales r ≪ R c , where it becomes effectively free. Thus, we have found asymptotic freedom in the classical λφ 4 theory with negative λ.
In the infrared at length scales of order R c the theory becomes strongly coupled and non-perturbative. What is the potential meaning of this dynamically generated scale? From the point of view of the classical theory the existence of this scale is quite surprising since it is independent of the UV regulator. On the top of that R c is a very non-perturbative scale. Obviously it is tempting to think of R c as setting the natural confinement scale of the theory. A way to check this claim is to derive an exact classical β function equation and to read off the previous perturbative expansion from the corresponding solution of this equation. We address these issues below.
IV. BEYOND PERTURBATION THEORY AND ASYMPTOTIC BEHAVIOR
The usual way to address the non-perturbative phenomena within perturbation theory is to study the convergence of the perturbative series. In [6] it was found that the numerical coefficients in the perturbative expansion of β function asymptotically grow as β k ∼ k!β k 1 , where β 1 is one loop β function and k denotes the perturbative order in coupling constant.
Such behavior sets the limit of perturbation theory and fixes the uncertainty of the computations to be of order exp (−1/β 1 α) . In the theories with asymptotic freedom this uncertainty is extremely small in the deep UV region contrary to what happens in the theories with UV Landau pole. Normally in quantum field theory is hard to prove this factorial asymptotic behavior of the coefficients in β function. It can have different origin: either the growth of the number of diagrams contributing to a given order in perturbation theory (instanton effect) or the contribution of multi-bubble diagrams (renormalons).
In the classical approach to the renormalization group, however, there is an opportunity to convert the classical equations of motion into exact equation for the β function. This equation be can used afterwards to check the similarity between classical and quantum renormalization groups. In particular, as we will see, one can use the exact equation to derive the asymptotic behavior of the coefficients in the perturbative expansion of β function. Interestingly enough the asymptotic behavior anticipated by the exact classical β function agrees with the quantum filed theory expectations. In addition this allows us to clarify the origin of renormalons as well as the generic form of the non-perturbative uncertainties. Moreover, the non-perturbative contributions will be naturally defined in terms of the dynamically generated scale R c , as it should be.
A. Exact classical β function
To derive an exact equation for the classical β function we begin with equation for static scalar field outside an external source
1 r 2 d dr r 2 dφ dr − λ 0 φ 3 = 0.(42)Substituting φ = Qf (r) r ,(43)
we can rewrite the equation above as
f ′′ − f ′ + α 0 f 3 = 0,(44)
where α 0 = −λ 0 Q 2 and prime denotes the derivative with respect to x = ln (r/R c ) . Multiplying this equation by α 0 f and defining running coupling constant as before
α (x) = α 0 f 2 (x) ,(45)
we obtain the following second order differential equation for α (x)
α ′′ − α ′2 2α − α ′ + 2α 2 = 0.(46)
Recalling the definition of β function, β ≡ α ′ , and taking into account that
α ′′ = α ′ dα ′ dα = β dβ dα ,(47)
equation (46) reduces to the first order differential equation:
β = 2α 2 + 1 2 dβ 2 dα − β 2 α ,(48)
which determines the exact classical β function.
B. Weak coupling expansion and renormalons
Let us first use the exact equation for β function to reproduce our perturbative results above. In order to do that we substitute in (48)
β (α) = ∞ k=1 β k α k+1 .(49)
This leads to the following recursion relations for the unknown numerical coefficients β k
β 1 = 2, β k = k−1 m=1 m + 1 2 β k−m β m for k ≥ 2,(50)
Using this relations we find β 1 = 2, β 2 = 6, β 3 = 48, β 4 = 570, β 5 = 8568, β 6 = 151956, ...
in complete agreement with (30) to an arbitrary order in α. Thus we have proven that equation (48) yields the exact β function which is in complete agreement with the perturbative β function.
Let us now find the asymptotic behavior of the perturbative series. One can easily see that for large k the main contribution to the sum in (50) comes from the terms with m = k − 1 and m = 1 and the recursion relation reduces to
β k ≃ (k + 1) β 1 β k−1 ,(52)
the solution of which is
β k ≃ (k + 1)!β k 1(53)
with β 1 = 2. Nicely enough this is the same type of factorial behavior we expect in quantum field theory. Moreover, since the coefficient of the factorial is the one-loop β function it is natural to identify the origin of this behavior with a renormalon.
In order to clarify the meaning of this renormalon let us consider the perturbative solution of equation (48) assuming that α ≪ 1. Substituting
β = 2α 2 (1 + ε) ,(54)
in (48) we find that ε (α) satisfies the equation
2α 2 dε dα = ε 1 + ε − 3α (1 + ε) = ε − 3α + O ε 2 , εα .(55)
Because both ε and α are much less than unity we can neglect nonlinear terms. Solving the resulting linear equation one obtains
ε (α) = 3 β 1 Ei 1 β 1 α + C e − 1 β 1 α + O e − 1 β 1 α 2 ,(56)
where β 1 = 2, Ei (z) is the exponential-integral function and C is the constant of integration. Now using the asymptotic expansion of the exponential-integral function at large argument
Ei (z) = e z z n k=0 k! z k + O 1 z n+1 ,(57)
we find that asymptotically the coefficients of the β function in the perturbative expansion grow as β k ∼ k!β k 1 . This completely clarifies the origin of the renormalon, which is an artifact of the asymptotic expansion of non-analytic function. It also follows from (56) that the accuracy of the Borel resummation does not exceed
e − 1 β 1 α ∼ r R c ,(58)
which is the expected non-perturbative uncertainty!
V. THE NONPERTURBATIVE SOLUTION AND CONFINEMENT
Finally let us use the exact β function equation to determine what happens in the infrared
region r > R c , where the perturbation theory is completely out of the control. Obviously the nonperturbative effective coupling constant can be used to define a static inter-quark potential. Regarding the nature of the classical sources we will leave beyond the scope of paper and concentrate mostly on the calculation of the behavior of the running coupling constant in the infrared region. As we will see this coupling constant determines-similar to QCD-the confining potential between sources. In particular we will compute the energy of an isolated source and a dipole, built out of two opposite charges. In the first case we find the divergent energy which is an indication of confinement and can be interpreted as the absence of isolated sources. The energy of the dipole is finite and positive. Its typical size is of order R c and the binding energy ∼ R −1 c in case of sources with negligible masses. This is an encouraging hint towards explaining the colorless hadron states in QCD.
A. The infrared coupling constant
Equation (48) for the exact β function can rewritten in the form quartic potential in the presence of negative friction. Such particle "oscillates" and if we neglect for a moment the friction term, the typical "period of oscillation" can be estimated on dimensional grounds as
dβ dα = 2αβ − 4α 3 + β 2 2αβ ,(59)∆x ∼ 1 α 0 f 2 ∼ 1 α (x) ,
where f is the typical amplitude of oscillations. It is clear that this estimate is valid only if ∆x ≪ 1 because otherwise friction term dominates and completely damps the oscillations.
However, for α (x) ≫ 1, when ∆x ≪ 1, the friction is not so crucial and the system undergoes oscillations with the amplitude slowly growing due to this negative friction. In order to find how fast this amplitude grows we multiply equation (44) by f and rewrite it in the form
(f f ′ ) ′ − f ′2 − 1 2 f 2 ′ + α 0 f 4 = 0.(60)
Averaging this equation over the period of oscillations we find
f ′2 = α 0 f 4 .(61)
Multiplying now (44) by f ′ leads to the equation
1 2 f ′2 + 1 4 α 0 f 4 ′ = f ′2 ,(62)
which after averaging and taking into account (61) gives us
d f 4 dx = 4 3 f 4 .(63)
Finally solving this equation we obtain
f 4 = C exp 4x 3 = C r R c 4/3 ,(64)
leading to the following nonperturbative behavior of the running coupling constant
α (r) = α 0 f 2 ≃ α 0 f 4 cos 2 4 α 2 0 f 4 dx ≃ O (1) r R c 2/3 cos 2 r R c 2/3 ,(65)
for r ≫ R c . In Fig. 2 we summarize the behavior of the running coupling. As it was at scales larger than the confinement scale grows as r 2/3 . This is the main non-perturbative result we can extract from the exact β function equation.
B. Confinement
As already discussed we can mimic confinement identifying the classical sources as static quarks and defining a quench approximation to the static inter-quark potential in terms of the effective coupling in the IR region.
Since in our case the self-interaction contribution to the energy goes like φ 4 ∼ f 4 ∼ α(r) 2
we can define the static potential as:
V (r) ∼ α 2 (r) r(66)
Using the non-perturbative value of the running coupling (65) we get
V (r) ∼ O (1) R −1 c r R c 1/3(67)
To check this qualitative result let us compute the energy of the field created by a static external charge. Since the field is static and spherically symmetric, the total energy is given by the expression
E = 1 2 (∇φ) 2 + λ 0 2 φ 4 d 3 x = 2π (∂ r φ) 2 + λ 0 2 φ 4 r 2 dr = 2πQ 2 (r∂ r f − f ) 2 − 1 2 α 0 f 4 dr r 2(68)
Note that although the contribution of the second term is negative for negative λ 0 , the total energy is positive because the gradient term dominates. The integral above diverges when r → 0. This divergence has an ultraviolet origin and it is the same as well known divergence of the self-energy of classical point-like electric charge. Therefore it can be removed using standard methods. We will focus instead on the IR contribution to the energy. Taking into account that at r ≫ R c ,
(r∂ r f ) 2 = f ′2 = α 0 f 4 ,(69)
the following expression for the infrared contribution to the total energy is obtained
E ≃ πQ 2 r Rc α 0 f 4 dr r 2 ∼ O (1) Q 2 1 R c r R c 1/3 ,(70)
in agreement with the qualitative expectations. The energy of the isolated charge diverges as r 1/3 and therefore it cannot exist as a free asymptotic state. This can be interpreted as a hint of confinement of isolated sources.
One can also build "colorless configuration" using two opposite charges Q and −Q separated by distance l. At distances r ≫ l, the field φ decreases as r −2 and equation (3) becomes
1 r 2 d dr r 2 dφ dr = −4πQδ (x) + λ 0 Q 3 l 3 r 6 ,(71)
Since in this case the anti-screening effect, determined by the last term in this equation, is completely irrelevant at large distances we conclude that the total energy of the dipole system is infrared convergent. When the distance between charges exceeds the confinement scale R c the infrared contribution of the scalar field becomes essential and the total energy
is E ∼ O (1) R −1 c l R c 1/3 .(72)
Hence, the interaction potential between two charges grows as distance in power one third.
This can be interpreted as a confining potential leading to a natural estimate for the mass scale of the dipole configuration to be of order m ∼ O (1) R −1 c .
VI. DISCUSSION AND SPECULATIONS
We have shown that certain essential properties of the quantum field theory usually considered as having quantum origin can be revealed already at the classical level. In particular, the renormalization group structure of the theory including the phenomenon of dimensional transmutation is already encoded in the classical equations.
So far we have considered only the self-interacting scalar field with negative coupling constant and external sources. One can naturally ask up to what extent the qualitative results obtained in this paper are useful in application to gauge theories, such as QCD. An encouraging sign, that classical RG treatment can be generalized for such theories is provided by the following simple scaling argument. As we have found, the logarithmic effect of antiscreening comes from the term of φ 3 in the equation for the scalar field. In the perturbation theory this term represents the density of the charge induced by self-interaction and it drops as r −3 as distance r grows. In QCD the gauge field equations for gluons contain two kinds of self-interaction terms which drop in a similar way, namely, A 3 and A∂A. Only A∂A gives the negative contribution to the β function. This term leads to anti-screening effect inducing the density of the colored charge decaying as r −3 similar to the case of scalar field. Because the structure of the self-interaction terms is different (in one case it is φ 4 and in the other A 2 ∂A) the interaction potential between two sources in gauge theories can grow with the distance not necessarily as r 1/3 , but as r α , where 0 < α ≤ 1. So, the linear growth is not excluded. However, the linear growth, although leading to confined charges does not necessarily imply the formation of QCD flux tube (see Appendix B).
Appendix A: On the triviality of λφ 4 theory with positive λ.
We can use the exact β function equation to check the triviality of λ 0 φ 4 theory in the case of positive λ 0 in four dimensions (this triviality was rigorously proved in five and higher dimensions in [8]). In the case of positive λ 0 it is convenient to change the signs in the definitions of α 0 and x, so that,
α 0 ≡ λ 0 Q 2 > 0, x ≡ ln (r 0 /r) .(A1)
In this case the β function defined in (29) is related to the standard β st function used in the literature as β st = β/2. With these redefinitions the equation for the exact β function is obtained from (48) by substituting β → β and α → −α :
β = 2α 2 − 1 2 dβ 2 dα − β 2 α . (A2)
For α ≪ 1 the perturbative solution of this equation is
β = 2α 2 − 6α 3 + 48α 4 − 570α 5 + ...(A3)
The Gell-Mann-Low equation to one loop,
dα (x) dx = 2α 2 (x) ,(A4)
gives us
α (r) = α 0 1 − 2α 0 x = λ 0 Q 2 1 − 2λ 0 Q 2 ln (r 0 /r) .(A5)
According to this result the coupling constant blows up at the Landau pole
r L = r 0 e − 1 2λ 0 Q 2 .(A6)
The essence of the proof of λφ 4 triviality can be reduced to showing that this UV pole will survives at nonperturbative level. This is not so obvious because as we have seen the IR one loop Landau pole (for negative λ 0 ) disappears after resummation of the perturbative expansion. To find out whether the pole survives or not for positive λ, let us solve equation (A2) in strong coupling regime, α ≫ 1. We will do it perturbatively in terms of the inverse powers of α. Neglecting the linear β term in (A2) we have
dβ 2 dα − β 2 α ≃ 4α 2 (A7)
and the corresponding solution of this equation is
β = √ 2α 3/2 (A8) Rewriting (A2) as dβ 2 dα − β 2 α = 4α 2 − 2β,(A9)
substituting in the right hand side of this equation the result (A8) and solving the obtained inhomogeneous linear equation for β 2 we obtain
β = √ 2α 3/2 1 − 2 √ 2 3 α −1/2 1/2 = √ 2α 3/2 1 − √ 2 3 1 √ α + O 1 √ α 2 (A10)
This procedure can be repeated recursively giving us higher order power corrections in the expansion in 1/ √ α ≪ 1. Thus we see that for very large α the solution (A8) becomes more and more accurate and the behavior of β functions confirms the expectations in [7]. The solution (A8) exactly matches the one loop result in (A3) at α = 1/2. Using this to fix the integration constant in the Gell-Mann-Low equation
dα (x) dx = √ 2α 3/2 ,(A11)
we obtain the following non-perturbative result valid for α ≫ 1
α (r) = 8 α 0 1 − 2α 0 (x − 1) 2 = 8 λ 0 Q 2 1 − 2λ 0 Q 2 ln (r 0 /re) 2 (A12)
Thus we see that even after complete resummation of the perturbative expansion the Landau pole survives and its non-perturbative location is at (Fig. 3):
r L = r 0 e e − 1 2λ 0 Q 2 .(A13)
This leads to the triviality of λφ 4 theory. As we have shown in the paper the potential between the external sources grows unbounded with the separation as a third power of it. In such a picture, charge and anticharge are confined but the confinement in not due to formation of a flux tube (string-type object) but rather due to formation of a finite energy dipole. One may think, that this is a peculiarity of not having a linear growth of the potential. In this appendix we will discuss this issue and show that even the linear potential does not necessarily imply the existence of a string.
To demonstrate this we begin with an SO(3) sigma model of an isotriplet scalar field φ a (a = 1, 2, 3) with Lagrangian:
L = ∂ µ φ a ∂ µ φ a − λ 2 (φ a φ a − v 2 ) 2 .(B1)
In this case the equation of motion has the static spherically-symmetric solution (see e.g. [9])
φ a = f (r) x a r ,(B2)
where x a are Cartesian space coordinates and f (r) is the function with the following asymp-totic properties
f (0) = 0, f (r)| r≫ (λv) −1 → v (B3)
The size of the region where f (r) is different from v is of order r c = (λv) −1 . The solution above is the 't Hooft -Polyakov monopole in the limit of zero gauge coupling. Because in this limit the gauge fields become massless and decouple from φ this solution is often referred to as a global magnetic monopole.
The energy of this monopole can be easily estimated by considering separately the contribution from core (r < r c ) and from the rest. The core contribution is of order v, and can be neglected compared to the energy in the region r > r c , where f (r) can be set to be equal v. Then the contribution of the gradients of the angles (Nambu-Goldstone modes) is divergent and we have to cut-off the integral at some R ≫ r c . The resulting energy of the isolated "charge",
E r > rc ≃ 4π R rc drv 2 ≃ 4πRv 2 ,(B4)
is linearly divergent. This implies that the potential between two opposite charges is linear.
In fact, let us consider an anti-monopole placed at distance R from the monopole. The effect of anti-monopole is to cut the divergent integral at r = R and the resulting potential is
V R > rc ≃ R/r 2 c ,(B5)
thus confining monopole-antimonopole configuration! This picture is very different from the QCD flux-tube (string) confinement. To understand this difference let us confront them considering heavy quark-anti-quark pair placed at distance R apart. In the absence of light quarks this distance R can be much larger than the QCD scale, r QCD ≡ Λ −1 QCD . In string picture the force between this pair is mediated by a stretched string (electric flux tube) of constant tension ∼ Λ 2 QCD , giving the potential
V R > r QCD ≃ R/r 2 QCD ,(B6)
similar to the monopole-anti-monopole potential. However, in the monopole case the flux is not confined to a string and for monopole-anti-monopole it has a dipole configuration (see The difference between these two pictures can be stressed even more if we notice that the theory (B1) allows to be "deformed" to the theory in which true strings connecting monopoles appear. The appearance of "open-color" monopole-antimonopole configuration is due to S(3)/U(1) topology of the vacuum manifold with nontrivial π 2 . To change this topology we can further deform the vacuum manifold by spontaneous breaking of the remaining U(1) symmetry. This can be done by introducing additional scalar field χ α (α = 1, 2) in a doublet representation of the SO(3) group. The Lagrangian then becomes
L = ∂ µ φ∂ µ φ + ∂ µ χ * ∂ µ χ − λ 2 (φ 2 − v 2 ) 2 − λ 2 1 (χ * χ − v 2 1 ) 2 + h χ * φχ + h ′ * φ 2 χ ,(B7)
where the contraction of indices is obvious. The parameters (λ, λ 1 , v, v 1 , h, h ′ ) are chosen in such a way that the field χ develops an expectation value v ′ ≪ v. In this limit, core of the monopole remains nearly unchanged. However at very large distances the field χ dramatically changes the monopole field. The presence of the second field with nonzero expectation value leads to the following hierarchical symmetry-breaking pattern,
SO(3) → U(1) → 1 ,(B8)
and the vacuum manifold becomes topologically-trivial. As a result, the static isolated monopoles can not exist anymore. However, due the hierarchy of symmetry breaking, v ≫ v ′ , monopoles do not simply disappear from the spectrum, but rather get connected by the strings. Either strings or monopoles are stable in two limits: v = ∞ and finite v ′ or v finite
and v ′ = 0, respectively. However, when both v and v ′ are finite, they can only exist as hybrid configuration, namely, monopoles connected by strings. This picture is more close to the usual string confinement because here the monopole magnetic flux gets confined into the string. The thickness of this string is ∼ 1/v ′ ≫ r c , and its tension is v ′ 2 ln L, where L is the string's length. In other words, the tension of the string is logarithmically divergent.
For example, for the string oriented in z direction the field χ near the string but far away from the monopoles, is
χ α ≃ δ 1 α f (ρ) e iθ ,(B9)
where f (ρ) vanishes at ρ = 0 and approaches constant for ρ > 1/v ′ in the cylindric coordinates ρ, θ. The energy of this configuration is logarithmically divergent with the natural cut-off scale of order string size.
The picture above can be summarized as follows. For R ≪ 1/v ′ , the potential between monopoles is linear and field configuration is of dipole type, but for R ≫ 1/v ′ the flux is not spread anymore and becomes confined by a string. This leads to the modification of the potential (B5), which becomes
V (R) ∼ v ′2 R ln R .(B10)
This is not such a dramatic change in the potential, but more important is the qualitative change of the physical picture, because now the monopoles are becoming confined by the string. The consideration above illustrates a very important point, that the confinement can have very distinct physical origin for the same growing potential between charges.
Finally, we will consider here one more theory, where confining potential is due to usual electric flux. Let us consider U(1) theory with Lagrangian
L = (F µν F µν ) α + A µ j µ .(B11)
Unlike (B1) , which describes healthy theory, the legitimacy of this theory as a quantum field theory of U (1) gauge field is much less obvious. However, since we are interested only in geometric properties of the classical electric fluxes, we will use it for our purposes. In this case the equations of motion are
∂ µ 2 α(F 2 ) α−1 F µν = j ν .(B12)
For the static charge j µ = δ 0 µ δ(r)Q, which produces spherically-symmetric electric field,
F j0 ≡ E j (r) = E(r) x j r ,(B13)
they become,
∂ j 2 αE(r) 2α−2 E j (r) = δ(r)Q.(B14)
It immediately follows from here that
2 αE(r) 2α−1 = Q r 2 ,(B15)
and hence
E(r) = Q 2αr 2 1 2α−1 .
(B16)
The energy of an isolated charge smeared over a sphere r 0 diverges as
E charge = R r 0 r 2 dr Q 2αr 2 2α 2α−1 ∼ Q 2α 2α−1 R 2α−3 2α−1 ,(B17)
for either α > 3/2 or α < 1/2 when exponent is positive. Thus, in both these cases the energy of an isolated charge diverges. However, in this case the finite energy of the charge-anticharge configuration is not for granted automatically! The situation is much more subtle than in the usual case because of very strong non-linearity, which make superposition principle not applicable to E j . However, the superposition principle in this case is valid for E(r) 2α−2 E j (r). Therefore, the electric field of the dipole of size D is given by,
E(r) D ≃ QD 2αr 3 1 2α−1 .
(B18)
The corresponding energy is
E charge ∼ (DQ) 2α 2α−1 R −3 2α−1 ,(B19)
and it is finite only for α > 3/2.
FIG. 1 :
1and can be fully investigated using the phase diagram method. The particular solution we need is determined by the perturbative initial condition (30) . The resulting nonperturbative β function, shown inFig. 1, reaches a maximum value about 0.64 for α ≃ 0.69 and after that decreases and vanishes at α ≈ 0.98. Such behavior of the β function in nonperturbative regime is quite nontrivial. The most dramatic effect is the absence of any divergence for the coupling constant, meaning that in case of exact β function, which accounts for all resummations, the IR Landau pole is absent. This nonperturbative resummations leads to finite running coupling for any finite x interpolating smoothly between the asymptotically free UV-regime and the IR-region r ≫ R c .Beyond the first zero of the β function it is more convenient to draw α and β separately as functions of x = ln (r/R c ) because both α and β become oscillating functions of the scale. In order to find the solution in this region it is more convenient to work directly with equation (44) instead of (59) . This is the equation for a particle "moving" Numerical evaluation of nonperturbative β function
FIG. 2 :
2Nonperturbative running coupling said above although according to perturbation theory the coupling constant should become infinite at r = R c , in reality this does not happen. The coupling constant remains finite and
B: Confinement with and without strings.
Fig. 4 )FIG. 4 :
44. As a result the "monopole color" is not bounded within the string of width r c and can be probed everywhere in the space around the monopole at r > r c . On the other hand, in the case of string the only possibility to probe the color of the charge is to penetrate within the string of width r QCD . String vs. dipole
AcknowledgmentsWe are grateful to L. Alvarez-Gaume, C. Bachas, A. Barvinski, M. Henneaux and I. Sachs for discussions and valuable comments. We would like to thank T. Hofbaur for the help with
-07908, CPAN (CSD2007-00042) and HEPHACOS P-ESP00346. V.M. is supported by TRR 33 "The Dark Universe" and the Cluster of Excellence EXC 153 "Origin and Structure of the Universe. -07908, CPAN (CSD2007-00042) and HEPHACOS P-ESP00346. V.M. is supported by TRR 33 "The Dark Universe" and the Cluster of Excellence EXC 153 "Origin and Structure of the Universe".
1343; Politzer, H. Reliable perturbative results for strong interactions?. D Gross, F Wilczek, Phys. Rev. Lett. 301346Phys. Rev. Lett.Gross, D., Wilczek, F. Ultraviolet behavior of non-Abelian gauge theories. Phys. Rev. Lett., 30 (1973), 1343; Politzer, H. Reliable perturbative results for strong interactions? Phys. Rev. Lett., 30 (1973), 1346.
Deep inelastic scattering in a field theory with computable largemomenta behavior. K Symanzik, G Parisi, Lett. al Nuovo Cimento. 684Lett. al Nuovo CimentoSymanzik, K. A field theory with computable large momenta behavior. Lett. al Nuovo Cimento, 6 (1973), 77; Parisi, G. Deep inelastic scattering in a field theory with computable large- momenta behavior. Lett. al Nuovo Cimento, 7 (1973), 84.
The renormalization group and the ε-expansion. K Wilson, K Wilson, J Kogut, Physical Review, D. 376Phys. Rep.Wilson, K. RG and strong interactions. Physical Review, D 3 (1971), 1300; Wilson, K., Kogut, J. The renormalization group and the ε-expansion. Phys. Rep. 12 (1974), 76; 't Hooft, G. Nucl.
The renormalization group and Quantum Field Theory. Doorworth61455Phys. B61 (1973) 455 ; The renormalization group and Quantum Field Theory, Doorworth, 1988.
Quantum Electrodynamics at Small Distances. M Gell-Mann, F E Low, Physical Review. 9551300Gell-Mann, M., Low, F. E. Quantum Electrodynamics at Small Distances. Physical Review, 95 (5) (1954), 1300.
. L D Landau, A A Abrikosov, I M Khalatnikov, Dokl. Akad. Nauk SSSR. 951177Landau, L. D., Abrikosov, A. A., and Khalatnikov, I. M. Dokl. Akad. Nauk SSSR, 95, 497, 773, 1177 (1954).
. L N Lipatov, Zh.Eksp.Teor.Fiz. 72411Sov.Phys. JETPLipatov, L. N. Zh.Eksp.Teor.Fiz. 72 (1977) 411, [Sov.Phys. JETP 45 (1977) 216];
Perturbation theory at large order. E Brezin, J.-C Le Guillou, J Zinn-Justin, G Parisi, Phys. Rev. D. 15382Phys.Lett.Brezin, E., Le Guillou, J.-C., Zinn-Justin, J. Perturbation theory at large order Phys. Rev. D 15 (1977), 1544, 1558; Parisi, G. Phys.Lett. 66B (1977) 382.
. L D Landau, I Pomeranchuk, Ya, Dokl. Akad. Nauk SSSR. 102489Landau, L. D., Pomeranchuk, I. Ya., Dokl. Akad. Nauk SSSR, 102 (1955) 489;
. I Pomeranchuk, Ya, Dokl. Akad. Nauk SSSR. 1031005Pomeranchuk, I. Ya. Dokl. Akad. Nauk SSSR, 103 (1955) 1005.
On the triviality of λφ 4 theories and the approach to the critical point in d(-) > 4 dimensions. M Aizenman, J Froehlich, Comm. Math. Phys. 86281Nuclear Phys. BAizenman, M. Geometric analysis of φ 4 fields and Ising models, I, II. Comm. Math. Phys. 86 (1982) 1; Froehlich, J. On the triviality of λφ 4 theories and the approach to the critical point in d(-) > 4 dimensions, Nuclear Phys. B 200 (1982) 281.
A Vilenkin, E P Shellard, Cosmic strings and other topological defects. Cambridge University PressVilenkin A., Shellard E.P.S. Cosmic strings and other topological defects. Cambridge University Press, 1994
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[] |
[
"Absorption features caused by oscillations of electrons on the surface of a quark star",
"Absorption features caused by oscillations of electrons on the surface of a quark star"
] |
[
"R X Xu \nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n",
"S I Bastrukov \nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n\nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"F Weber \nDepartment of Physics\nSan Diego State University\n92182San DiegoCaliforniaUSA\n",
"J W Yu \nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n",
"I V Molodtsova \nJoint Institute for Nuclear Research\n141980DubnaRussia\n"
] |
[
"State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Department of Physics\nSan Diego State University\n92182San DiegoCaliforniaUSA",
"State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"Joint Institute for Nuclear Research\n141980DubnaRussia"
] |
[] |
If quark stars exist, they may be enveloped in thin electron layers (electron seas), which uniformly surround the entire star. These layers will be affected by the magnetic fields of quark stars in such a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this paper. Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea. The frequency spectrum of these oscillations is derived in analytic form. If the thermal X-ray spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model.
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10.1103/physrevd.85.023008
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[
"https://arxiv.org/pdf/1110.1226v2.pdf"
] | 119,102,102 |
1110.1226
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13b4fdfcaa80598e257929c68c40ee2377842ca6
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Absorption features caused by oscillations of electrons on the surface of a quark star
9 Jan 2012
R X Xu
State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
S I Bastrukov
State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
Joint Institute for Nuclear Research
141980DubnaRussia
F Weber
Department of Physics
San Diego State University
92182San DiegoCaliforniaUSA
J W Yu
State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
I V Molodtsova
Joint Institute for Nuclear Research
141980DubnaRussia
Absorption features caused by oscillations of electrons on the surface of a quark star
9 Jan 2012(Dated: February 3, 2013)numbers: 2660-c9760Jd9760Gb
If quark stars exist, they may be enveloped in thin electron layers (electron seas), which uniformly surround the entire star. These layers will be affected by the magnetic fields of quark stars in such a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this paper. Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea. The frequency spectrum of these oscillations is derived in analytic form. If the thermal X-ray spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model.
I. INTRODUCTION
The spectral features of thermal X-ray emission are essential for us to understand the real nature of pulsar-like compact stars. Calculations show that atomic spectral lines form in the atmospheres of neutron stars. From the detection and identification of atomic lines in thermal X-ray spectra one can infer neutron star masses (M ) and radii (R), since the redshift and broadening of the spectral lines depend on M/R and M/R 2 , respectively. Atomic features are expected to be detectable with the spectrographs on board of Chandra and XMM-Newton. No atomic features have yet been discovered with certainty, however. This may have it origin in the very strong magnetic fields carried by neutron stars. An alternative explanation could be that the underlying compact star is not a neutron star but a bare strange (quark matter) star [1,2]. The surface of such an object does not consist of atomic nuclei/ions, as it is the case for a neutron star, but of a sea of electrons which envelopes the quark matter.
Strange stars are quark stars made of absolutely stable strange quark matter [3][4][5][6][7][8]. They consist of essentially equal numbers of up, down and strange quarks as well as of electrons [9][10][11]. The latter are needed to neutralize the electric charges of the quarks, rendering the interior of strange stars electrically neutral. Quark matter is bound by the strong interaction, while electrons are bound to quark matter by the electromagnetic interaction. Since the latter is long-range, some of the electrons in the surface region of a quark star reside outside of the quark matter boundary, leading to a quark matter core which is surrounded by a fairly thin (thousands of femtometer thick) sea of electrons [8,9,12]. Due to the enormous advances in X-ray astronomy, more and more so-called dead pulsars are discovered, whose thermal radiation dominates over a very weak or negligibly small magnetospheric activity. The best absorption features (at ∼ 0.7 keV and ∼ 1.4 keV) were detected for the central compact object (CCO) 1E 1207.4-5209 in the center of supernova remnant PKS 1209-51/52 (see Table I). Initially, these features were thought to be associated with [13,19,23], with P the spin period, B the magnetic field (B10 = B/10 10 G) derived by magnetodipole braking, T the effective thermal temperature detected at infinity, and Ea the absorption energy. We do not list the B-fields of XDINSs for which the propeller braking could be significant because of their long periods. the atomic transitions of ionized helium in a stellar atmosphere where a strong magnetic field is present [14]. Soon thereafter, however, it was noted that these lines are of electron-cyclotron origin [15]. The spectrum of 1E 1207.4-5209 shows two more features that may be caused by resonant cyclotron absorption, one at ∼ 2.1 keV and another, but of lower significance, at ∼ 2.8 keV [16]. These features vary in phase with the star's rotation. Although the detailed mechanism which causes the absorption features is still a matter of debate, timing observations predict a rather weak magnetic field for this CCO,
in agreement to what is obtained under the assumption that the lowest-energy line at 0.7 keV is the electroncyclotron fundamental, favoring the electron-cyclotron interpretation [17,18]. Besides 1E 1207.4-5209, broad absorption lines have also been discovered in other dead pulsars (listed in Table I), especially in so-called X-ray dim isolated neutron stars (XDINSs), between about 0.3 and 0.7 keV [19].
In this paper, we re-investigate the physics of these absorption features. The key assumption that we make here is that these features originate from the electron seas on quarks stars rather than from neutron stars, whose surface properties are radically different from those of strange stars [8,9,12]. Of key importance is the magnetic field carried by a quark star, which critically affects the global properties (hydrodynamic surface fluctuations) of the electron sea at the surface of the star. We study this problem in the framework of classical electrodynamics in terms of cyclotron resonances of electrons in weak magnetic fields, since the magnetic fields of dead pulsars are much lower than the critical field, B q = 4.414 × 10 13 G, at which the quantization of the cyclotron orbits of electrons into Landau levels occurs.
II. HYDRO-CYCLOTRON WAVES
A. Governing equations
For what follows we restrict ourselves to a discussion of the large-scale oscillations of an electron sea subjected to a stellar magnetic field. We will be applying the semiclassical approach of classical electron theory of metals and making use of standard equations of fluid-mechanics. The electrons are viewed as a viscous fluid of uniform density ρ = n m e (where n is the electron number density) whose oscillations are given in terms of the mean electron flow velocity δv. This implies that the fluctuation current-carrying flow is described by the density of the convective current δj = ρ e δv, where ρ e = e n is the electron charge density. The equations of motions of a viscous electron fluid are then given by [20]
ρ ∂δv ∂t = 1 c [δj × B] + η∇ 2 δv,(1)j = ρ e δv, ρ = m e n, ρ e = e n ,(2)
where e and n are the change and number densities of electrons, respectively. We emphasize that δj stands for the convective current density and not for Ampére's j = (c/4π)∇ × δB, as it is the case for magnetohydrodynamics. This means that the hydrodynamic oscillations in question are of non-Alfvén type. In Eq. (1), η denotes the effective viscosity of the electron fluid, which originates from collisions of electrons with the magnetic field lines at the stellar surface. It is worth noting that the cyclotron waves can be regarded as an analogue of the inertial waves in a rotating incompressible fluid, as presented in Eq. (III.56) of Chandrasekhar's book [21].
The governing equation, Eq. (1), can be represented as
∂δv ∂t + ω c [n B × δv] − ν∇ 2 δv = 0,(3)ω c = eB m e c , n B = B B , ν = η ρ . (4)
where ω c is the cyclotron frequency. In the Appendix, we show that the electron sea can transmit macroscopic perturbations in the form of rotational hydro-cyclotron waves which are characterized by the following dispersion relation,
ω = ± ω c cos θ 1 1 − (νk 2 /ω) 2 + i (νk 2 /ω) 1 − (νk 2 /ω) 2 ,(5)
where ω and k denote the frequency and wave vector of the perturbations, respectively. The Larmor radius of an electron in a strong magnetic field, r L ≃ m e c 2 /(eB) ∝ B −1 , is very small for pulsar-like compact stars, and we neglect the viscosity term in the following analysis of the motion of collective electrons. In the collision-free regime, ν = 0, the hydro-cyclotron electron wave is described as a transverse, circularly polarized wave whose dispersion relation and propagation speed are given by
ω = ±ω c cos θ and V = ±(ω c /k) cos θ,(6)
respectively. Here, θ is the angle between the magnetic field B and the wave vector k. If k B one has ω = ±ω c . In metals these kind of oscillations are observed as electron-cyclotron resonances. There are two possible resonance states, one for ω = ω c and the other for ω = −ω c . These resonances correspond to the two opposite orientations of circularly polarized electron cyclotron waves.
B. Hydro-cyclotron oscillations of electrons on bare strange quark stars
We restrict our analysis to the collision-free regime of vortex hydro-cyclotron oscillations. Using spherical coordinates, equation (3) then takes the form
∂δv ∂t + ω c [n B × δv] = 0.(7)
Taking the curl of both sides of Eq. (7), we obtain
∂δω ∂t = ω c (n B · ∇) δv, δω = ∇ × δv.(8)
Let the magnetic field B be directed along the z-axis, so that in Cartesian coordinates n B = (0, 0, 1). We then have n r = cos θ, n θ = − sin θ, n φ = 0.
From a mathematical point of view, the problem can be considerably simplified if one expresses the velocity δv, which obeys the condition ∇ · δv = 0, in terms of Stokes' stream function, χ(θ, φ). This leads to
δv r = 0, δv θ = 1 r sin θ ∂χ(θ, φ) ∂φ , δv φ = − 1 r ∂χ(θ, φ) ∂θ .(10)
The depth of the electron layer near the star is much smaller than the stellar radius so that r ≈ R to a very good approximation. Equation (8) then simplifies to
∂δω r ∂t = −ω c n θ δv θ R ,(11)
with the radial component of the vortex given in terms of χ,
δω r = 1 R ∂δv φ ∂θ − 1 sin θ ∂δv θ ∂φ = − 1 R 2 ∇ 2 ⊥ χ(θ, φ),(12)∇ 2 ⊥ = 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂φ 2 .(13)
Substituting Eq. (12) and Eq. (13) into Eq. (11) leads to
∇ 2 ⊥ ∂χ ∂t + ω c ∂χ ∂φ = 0.(14)
The fact that free electrons undergo cyclotron oscillations in the planes perpendicular to the magnetic field suggests that the stream function χ can be written in the following separable form,
χ(θ, φ) = ψ(θ) cos(φ ± ωt).(15)
The "+" sign allows for cyclotron oscillations which are induced by the clockwise polarized wave, and the "−" sign allows for cyclotron oscillations induced by the count-clockwise polarized wave. Substituting (15) into (14) leads to
∇ 2 ⊥ ψ(θ) ± ω c ω ψ(θ) = 0.(16)
In the reference frame where the polar axis is fixed, Eq. (16) is identical to the Legendre equation for the surface spherical function,
∇ 2 ⊥ P ℓ (θ) + ℓ(ℓ + 1)P ℓ (θ) = 0,(17)
where P ℓ (cos θ) denotes the Legendre polynomial of degree ℓ. Hence, setting ψ(θ) = P ℓ (θ) we obtain
ω ± (ℓ) = ± ω c ℓ(ℓ + 1)
, ω c = eB m e c , ℓ ≥ 1.
From this relation we can read off the frequency of a surface hydro-cyclotron oscillation of a given order ℓ.
C. Characteristic features of cyclotron frequencies
Let us consider the spectrum of the positive branch ω(ℓ) = ω + (ℓ) of Eq. (18),
ω(ℓ) ω c = 1 ℓ(ℓ + 1) , ℓ ≥ 1.(19)
TABLE II. Comparison between single-particle (Landau level) and hydro-wave results for the cyclotron frequencies (B12 = B/10 12 G, ω1 = ω(ℓ = 1), ωc denotes the cyclotron frequency).
ω(ℓ = 1) ω(ℓ = 2) ω(ℓ = 3) ω1/keV
Landau level ωc 2ωc 3ωc 11.6B12 Hydro-wave ωc/2 ωc/6 ωc/12
5.8B12
From
ω(ℓ) ω(ℓ + 1) = ℓ + 2 ℓ , ℓ ≥ 1,(20)
it follows that this ratio becomes a constant for ℓ ≫ 1. Such a spectral feature is notably different from the one of electron-cyclotron resonances of transitions between different Landau levels. From the energy eigenvalues, E n , of an electron in a strong magnetic field, which are found by solving the Dirac equation (see Ref. [22]), one may approximate the value of E n for a relatively weak magnetic field, B ≪ B q , by E n = mc 2 + n ω c , n ≥ 0. (21) Therefore, in the framework of a single-particle approximation, the emission/absorption frequencies, which are given by ω(ℓ) = E n+ℓ − E n , should occur at Most notably, it follows that for the hydro-cyclotron wave model one obtains ω(ℓ = 2)/ω(ℓ = 3) = 2, in contrast to the cyclotron resonance model for single electrons which predicts this ratio for ω(ℓ = 2)/ω(ℓ = 1) = 2..
ω(ℓ) = ℓω c , ℓ ≥ 1.(22)
III. 1E 1207.4-5209 AND OTHER COMPACT OBJECTS
As already mentioned in the Introduction, 1E 1207.4-5209 (or J1210-5226) in PKS 1209-51/52 is one of the central compact objects in supernova remnants [23], where broad absorption lines, near (0.7, 1.4) keV [14], and possibly near (2.1, 2.8) keV [16] were detected for the first time. The interpretation of the absorption feature at ∼ 2.8 keV is currently a matter of debate, in contrast to the feature at ∼ 2.1 keV which is essentially unexplained. Intriguingly, an absorption feature with the same energy, 2.1 keV, has also been detected in the accretion-driven X-ray pulsar 4U 1538-52 [24].
For what follows, we assume that 1E 1207.4-5209 is a strange quark star and that (some of) these absorption features are produced by the hydro-cyclotron oscillations of the electron sea at the surface of such an object. Assuming a magnetic surface field of B ≃ 7 × 10 11 G and thus ω(ℓ = 3) = 0.7 keV, we obtain the oscillation frequencies shown in Table III. A magnetic field of ∼ 7×10 11 G is compatible with the magnetic fields inferred for 1E 1207.4-5209 from timing solutions [18] (9.9 × 10 10 G or 2.4 × 10 11 G), since 1E 1207.4-5209 shows no magnetospheric activity and theṖ -value would be overestimated if one applies the spin-down power of magnetic-dipole radiation [25,26]. We note that the absorption feature at ω(ℓ = 1) = 4.2 keV shown in Table III may not be detectable since the stellar temperature is only ∼ 0.2 keV (see Table I), which will suppress any thermal feature in that energy range. Aside from 1E 1207.4-5209, one may ask what would be the magnetic fields of other dead pulsars (e.g., radioquiet compact objects) if their spectral absorption features would also be of hydro-cyclotron origin? Intriguingly, the hydro-cyclotron wave model predicts magnetic fields that are twice as large as those derived from the electron cyclotron model if the absorption feature is at ω(ℓ = 1) = ω c /2; these fields could be ∼ 10 times greater (see Table II) if the absorption feature is at ω(ℓ = 2) = ω c /6 or ω(ℓ = 3) = ω c /12. The absorption lines at (0.3 ∼ 0.7) keV may indicate that the fields of XDINSs are on the order of ∼ 10 10 to 10 11 G, if oscillation modes with ℓ ≥ 4 are not significant.
As noted in [15], unique absorption features on compact stars are only detectable with Chandra and XMM-Newton if the stellar magnetic fields are relatively weak (10 10 G to 10 11 G), since the stellar temperatures are only a few 0.1 keV. The fields of many pulsar-like objects are generally greater than this value, with the exception of old millisecond pulsars whose fields are on the order of 10 8 G. Central compact objects, on the other hand, seem to have sufficiently weak magnetic fields (see Table I) so that absorption features originating from their surfaces should be detectable by Chandra and XMM-Newton. Arguments favoring the interpretation of compact central objects as strange quark stars have been put forward in [27], where it was shown that the magnetic field observed for some CCOs could be generated by small amounts of differential rotation between the quark matter core and the electron sea.
Besides dead pulsars, anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are enigmatic objects which have become hot topics of modern astrophysics. Whether they are magnetars/quark stars is an open question [28]. In case that AXPs/SGRSs should be bare strange stars, the absorption lines detected from SGR 1806-20 could be understood in the framework IV. Comparison between the frequencies, ω(ℓ), and the absorption frequencies detected, ω obs , for SGR 1806-20 (with an assumption of B ≃ 1.86 × 10 13 G). The data of ω obs are from [29]. Both ω(ℓ) and ω obs are in keV. of the hydro-cyclotron oscillation model. Assuming a normal magnetic field of B ≃ 1.86 × 10 13 G so that ω c /12 = 18 keV, one sees that the oscillation model predicts hydro-cyclotron frequencies which coincide with the observed listed in Table IV.
IV. SUMMARY
In this paper, we study the global motion of the electron seas on the surfaces of hypothetical strange quark stars. It is found that such electron seas may undergo hydro-cyclotron oscillations whose frequencies are given by ω(ℓ) = ω c /[ℓ(ℓ + 1)], where ℓ ≥ 1 and ω c the cyclotron frequency. We propose that some of the absorption features detected in the thermal X-ray spectra of dead (e.g., radio silent) compact objects may have their origin in excitations of these hydro-cyclotron oscillations of the electron sea, provided these stellar objects are interpreted as strange quark stars. The central compact object 1E 1207.4-5209 appears particularly interesting. It shows an absorption feature at 0.7 keV which is not much stronger than the another absorption feature observed at 1.4 keV. This can be readily explained in the framework of the hydro-cyclotron oscillation model, since two lines with ℓ and ℓ + 1 could essentially have the same intensity. This is very different for the electron-cyclotron model, for which the oscillator strength of the first harmonic is much smaller than the oscillator strength of the fundamental.
where ω c = eB m e c , n B = B B , ν = η ρ , ρ = m e n, ρ e = en, ω c is the cyclotron frequency, and η stands for the effective viscosity of an electron fluid originating from collisions between electrons. To make the problem analytically tractable, we treat the electron sea as an incompressible fluid and assume a uniform magnetic field. Equation (23) can then be written as
∂δv ∂t = −ω c [n B × δv] + ν ∇ 2 δv.(24)
Upon applying to Eq. (24) the operator ∇×, we arrive at
∂ ∂t [∇ × δv] = ω c (n B · ∇) δv − ν ∇ × ∇ × ∇ × δv,(25)
where ∇ · δv = 0, and
∂δω ∂t = ω c (n B · ∇) δv − ν∇ × ∇ × δω,(26)
where δω = ∇ × δv. Considering a perturbation in the form of δv = v ′ exp[i(kr − ωt)], we have
[k × δv] = i ω c ω (n B · k)δv + i νk 2 ω [k × δv],(27)1 − i νk 2 ω [k × δv] = i ω c ω (n B · k)δv,(28)
where (k · δv) = 0. It is convenient to rewrite the last equation as
[k × δv] = i ω c ω (n B · k) δv 1 + i(νk 2 /ω) 1 − (νk 2 /ω) 2 . (29) Multiplication of both sides of Eq. (29) with k leads to − ωδvk 2 = iω c (n B · k) 1 + i(νk 2 /ω)
1 − (νk 2 /ω) 2 [k × δv].(30)
Inserting the left-hand-side of Eq. (29) into the righthand-side of Eq. (30) gives
ω 2 = ω 2 c (n B · k) 2 k 2 1 + i(νk 2 /ω) 1 − (νk 2 /ω) 2 2 ,(31)
or ω = ±ω c (n B · k) k 1 + i(νk 2 /ω)
1 − (νk 2 /ω) 2 ,
which is Eq. (5).
.5 ± .5 11.2 ± .4 7.5 ± .3 5.0 ± .2 − −
APPENDIX
Here we derive the dispersion relation characterizing the propagation of hydro-cyclotron electron wave in the slab-geometry approximation. The governing equation of viscous electron fluid under the action of Lorentz force is given byρ ∂δv ∂t = ρ e c [δv × B] + η∇ 2 δv,which can be written as ∂δv ∂t + ω c [n B × δv] − ν∇ 2 δv = 0,
TABLE I .
IDead pulsars (CCOs and XDINSs) with observed spectral absorption lines
Table II
IIcompares the results of Eq. (22) with the results
of Eq. (19) obtained for the hydro-cyclotron wave model.
TABLE III .
IIIThe frequencies, ω(ℓ), at which hydro-cyclotron oscillations occur for 1E 1207.4-5209 with effective temperature T ≃ 0.2 keV, assuming a magnetic field of B ≃ 7 × 10 11 G.ℓ
1
2
3
4
5
6
ω(ℓ)/keV 4.2 1.4 0.7 0.4 0.3 0.2
TABLE
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[] |
[
"Systematic Generation of Algorithms for Iterative Methods Revised Edition",
"Systematic Generation of Algorithms for Iterative Methods Revised Edition"
] |
[
"B.ScHenrik Barthels ",
"Prof Paolo Supervised ",
"Ph.DBientinesi ",
"ProfGeorg May "
] |
[] |
[] |
The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as the Cholesky decomposition and the solution of Sylvester equations. In this thesis, we present an extension of the FLAME methodology to tackle iterative methods such as Conjugate Gradient. As a starting point, we use a formal description of the iterative method in matrix form. The result is a family of provably correct pseudocode algorithms. We argue that all the intermediate steps are sufficiently systematic to be fully automated.iiiConclusion 73A. Matrix Properties 75vii
| null |
[
"https://arxiv.org/pdf/1703.00279v1.pdf"
] | 4,304,674 |
1703.00279
|
e8bff459bc098f7866db9c1d9d476ce4dff2c7d0
|
Systematic Generation of Algorithms for Iterative Methods Revised Edition
1 Mar 2017
B.ScHenrik Barthels
Prof Paolo Supervised
Ph.DBientinesi
ProfGeorg May
Systematic Generation of Algorithms for Iterative Methods Revised Edition
1 Mar 2017Master's Thesis This is a revised edition of the author's thesis. It differs from the original version in corrections of typographical errors and clarifications of some passages. i
The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as the Cholesky decomposition and the solution of Sylvester equations. In this thesis, we present an extension of the FLAME methodology to tackle iterative methods such as Conjugate Gradient. As a starting point, we use a formal description of the iterative method in matrix form. The result is a family of provably correct pseudocode algorithms. We argue that all the intermediate steps are sufficiently systematic to be fully automated.iiiConclusion 73A. Matrix Properties 75vii
Introduction
The goal of this thesis is to simplify the development of algorithms for iterative methods. We present a methodology for the systematic derivation of such algorithms and lay the foundations for a system that automates the generation of algorithms and code for iterative solvers for linear systems.
Over the last few decades, iterative methods have become an indispensable tool for solving sparse linear systems. Such systems commonly occur in science and engineering, for instance when discretized partial differential equations have to be solved. While direct methods are a reliable tool to solve linear systems, their ability to use the sparsity of a matrix to their advantage is limited to specific sparsity patterns. Often enough, those sparse systems are so large that using direct methods is impractical.
Since the introduction of the Conjugate Gradient (CG) method in 1952 [9], much progress has been made in the field (for an overview, consider [1,11,13]). However, the way from an expert's idea for an algorithm to a working implementation is still a long one. At first, a formal description of the algorithm has to be derived on paper. Then, it has to be shown that the algorithm is correct, something that ideally follows from the derivation. Furthermore, to assess how useful the algorithm is in practice, it is desirable to prove that it is numerically stable. Finally, the algorithm has to be translated into code. Usually, that is done multiple times, for different languages, or potentially using different libraries. Every one of those steps takes time and is a potential source of errors.
To speed up this process and eliminate those sources of error, one may try to automate some or all of the steps described above. Automating the ingenuity of the iterative methods expert certainly lies in the distant future. In contrast, automatically translating a sufficiently formal description of an algorithm into code is a lot more feasible.
This thesis covers the systematical derivation of provably correct (pseudocode) algorithms, based on an abstract, formal description of an iterative method.
Background: FLAME
The Formal Linear Algebra Methods Environment (FLAME) is a project with the goal to automate the derivation of linear algebra algorithms, as well as their implementation. In [2], it was shown that it is possible to systematically derive algorithms for dense linear algebra in a number of well defined steps.
Introduction
The starting point of this derivation is a formal description of the input, consisting of a precondition (P pre ) and a postcondition (P post ). As an example, the description of a linear system where A is lower triangular is shown below:
x := Φ(A, b) ≡ P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ LowerTriangular[A]∧ Input[b] ∧ Vector[b]∧ Output[x] ∧ Vector[x]} P post : {Ax = b}
The function x := Φ(A, b) is used to abstract from the details of this representation.
At the core of the derivation lies the concept of a Partitioned Matrix Expression (PME). A PME describes all parts of the output operands of a linear algebra operation in terms of parts of the input operands. For the lower triangular system, the equation Ax = b is partitioned as
A T L 0 A BL A BR x T x B = b T b B ,
where A T L and A BR are square. Then, the PME is
x T := Φ (A T L , b T ) x B := Φ (A BR , b B − A BL x T ) .
The approach described in [2] has a number of advantages: (1) The resulting algorithms are built around a proof of correctness, so they are correct by construction. (2) It naturally leads to multiple variants of algorithms for the same operation. While they are all correct in exact arithmetic, they potentially behave very different in presence of round-off errors. (3) Furthermore, it was shown that this derivation can be combined with a systematic stability analysis. The author of [2] presented evidence that this derivation, based on a formal description of the operation and the PME, is systematic enough to be automated. A system that automates this process, including the generation of PMEs, was presented in [6].
All of the efforts described above focused on direct methods. Naturally, the question arises if this methodology extends to iterative methods. In [5], Eijkhout et al. introduced a matrix representation of the CG method and showed preliminary evidence that a systematic derivation of algorithms with a FLAME-like methodology is possible. However, the approach presented there heavily relied on guidance by a human expert. This thesis can be seen as a continuation of this work.
Challenges
The systematic derivation of algorithms for iterative methods introduces new challenges, especially for deriving the PME.
Contributions
-For direct methods, the sizes of all operands are constant. In case of iterative methods, a variable number of iterations can be performed. In the matrix representation, this is reflected by the fact that some operands have variable sizes.
-The matrix representation of iterative methods introduces new types of operands. Usually, an operand is either known or unknown, so it is input or output, respectively. Now, there are operands that are initially partially known and partially unknown.
-Quite often, equations have to be solved by using properties of certain expressions. Consider the following equation as an example:
−Pu + p = r P, p and r are known, and the goal is to find an assignment for u. It can be solved by using the fact that P T AP is lower triangular and P T Ap is zero (A is known as well). Multiplying P T A from the left to both sides of the equation results in −P T APu = P T Ar, which is a triangular system that can easily be solved.
This introduces two challenges: The properties of P T AP and P T Ap are not explicitly part of the matrix representation, so they have to be derived from it by algebraic manipulations and deductive reasoning. Then, to enable a system to solve this equation, it must be capable of identifying that the properties of those expressions can be used to do so.
-To ensure that the derived algorithms can be used in practice, it is desirable to derive the exact same algorithms as used today. The difficulty in achieving this goal lies in the fact that some of those algorithms are the result of nontrivial rewritings of easily derivable formulas. Those rewritings are hard to automate.
Introduction
for deriving those properties from the description, using a set of inference rules that encode linear algebra knowledge.
Systematic derivation of PMEs for iterative methods. The derivation of PMEs for iterative methods is more complex compared to direct methods. As part of this derivation, equations have to be solved using the derived matrix properties.
The presented methodology is systematic enough to be executed mechanically, that is, without any human intervention, thus setting the ground for a system that indeed automates its application.
Having made clear what is accomplished with this thesis, it should also be pointed out what is beyond its scope: The input to the derivation process is a representation of an iterative method that still has to be derived by an expert in the field. In case of direct methods, the formal description that is input immediately follows from the operation itself, as shown above for the lower triangular system. In contrast, the way from a linear system Ax = b to a formal description of an iterative method that is suitable as input for the presented approach is much more complicated.
Outline of the Thesis
The thesis is structured as follows: Chapter 2 serves as an introduction to familiarize the reader with the FLAME approach for deriving algorithms for direct methods. In Chapter 3, we lay the foundations for applying a similar approach to iterative methods. The matrix representation for iterative methods is introduced in Section 3.1. In Section 3.2, we describe a method for systematically deriving matrix properties from this representation. At the end of this chapter, in Section 3.3, the implications of the matrix representation for the derivation of algorithms are discussed. In Chapter 4, we explain how to derive algorithms for iterative methods. Finally, Chapter 5 summarizes the results of this thesis and points out opportunities for future research.
Derivation of Algorithms for Direct Methods
The purpose of this chapter is to introduce the reader to the systematic derivation of loop-based, blocked algorithms for direct methods using the FLAME methodology.
For the derivation, we will follow the systematic approach and the notation of [6]. We begin with the simple example of a triangular linear system in Section 2.1. In Section 2.2, we proceed with a somewhat more elaborate example to go into some details that are not covered in the first one. Next, in Section 2.3, a case will be demonstrated where it is not possible to derive algorithms that compute the solution for every input of the operation. Finally, we discuss the equivalence of loop invariants that are obtained with this approach in Section 2.4.
Notation
Throughout this thesis, two different notations are used for the indexing of matrix blocks and elements. The first one is the standard notation that uses numerals. The second one uses capital letters, as shown below:
• A L A R • A T A B • A T L A T R A BL A BR • A L A M A R • A T A M A B •
A T L A T M A T R A ML A MM A MR A BL A BM A BR
The subscript letters T , B, L, M and R stand for Top, Bottom, Left, Middle and Right, respectively.
Triangular Linear System
We begin with deriving algorithms for the linear system Ax = b where A is triangular. The starting point of the derivation is a complete description of the problem, consisting of two logical predicates, a precondition (P pre ) and a postcondition (P post ). The precondition lists the properties of all quantities that are part of the operation, while the postcondition consists of the equation (or equations) that constitute the operation.
Derivation of Algorithms for Direct Methods
Algorithm: . . . The precondition is true before the execution of the algorithm, and our goal is to find an algorithm that makes the postcondition true upon termination. Additionally, a function is introduced which abstracts from the details of the operation, and instead describes the output as a function of the input. For reasons that will become apparent later, we useb to denote the initial content of b. The description for a linear system with A lower triangular is:
P pre Partition { P inv } While G do {( P inv ) ∧ ( G )} Repartition { P before } Update { P after } Continue with { P inv } endwhile {( P inv ) ∧ ¬ ( G )} P postx := Φ(A,b) ≡ P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ LowerTriangular[A]∧ Input[b] ∧ Vector[b]∧ Output[x] ∧ Vector[x]} P post : {Ax =b}
The actual algorithms are then constructed by filling out a so-called "worksheet", a template for a loop-based algorithm [2,6], shown in Figure 2.1.
PME Generation
As a first step towards an algorithm, the PME for the operation is generated. In order to do so, we partition the postcondition. For direct methods, partitioning operands along each dimension in at most two parts proved to be sufficient (this is further discussed in Section 2.4). Since A is lower triangular, and to preserve the triangularity of the resulting objects, a 2 × 2 partitioning is chosen for it where A T L (and A BR ) is square.
A T L 0 A BL A BR x T x B = b T b B
Next, in the so called Matrix Arithmetic step, we compute the symbolic multiplication and distribute the equality over the partitioned objects, resulting in the following expression:
A T L x T =b T A BL x T + A BR x B =b B
In the final Pattern Matching step, we try to find ways to solve the given equations by writing them as known operations. We begin with the top part. A T L and b T are known, while x T is unknown. Since A T L is lower triangular, we recognize that A T L x T =b T is a lower triangular system. Thus, we can rewrite this expression as x T := Φ A T L ,b T , utilizing the function introduced earlier. Doing so, we can from now on consider x T as known, which is necessary to solve the second equation. A BL x T is a matrix-vector product of two known quantities, and if we subtract it on both sides of the equation, we identify A BR x B =b B − A BL x T as another triangular system. Rewriting this as x B := Φ A BR ,b B − A BL x T provides us with the PME
x T := Φ A T L ,b T x B := Φ A BR ,b B − A BL x T .
Loop Invariant Identification
The next step consists of finding loop invariants (P inv ). A loop invariant is a logical predicate that is true at certain points of a loop in an algorithm. It is true before the loop is entered and after it is left, as well as at the beginning and the end of the loop body. It allows to formally reason about the correctness of the loop [8].
As a first step, the PME is decomposed into basic operations, for example a matrix multiplication, or the function representing the operation we are deriving algorithms for. In general, one can think of those basic operations in terms of BLAS-like functions. For the lower triangular system, the following three operations are obtained:
1. x T := Φ A T L ,b T 2. b B :=b B − A BL x T 3. x B := Φ (A BR , b B )
Then, from those operations, a dependency graph is constructed. Every node represents one operation, while the edges of this directed graph represent the data dependencies between those operations. If one operation requires the output of another operation, the former depends on the latter. Thus, in the graph, there are edges from the node that computes a quantity to those which need this quantity as input. Consider Figure 2.2 for the resulting dependency graph. Subsets of nodes from this graph are now selected as loop invariants. For reasons that will be explained later, those subsets can neither be empty, nor can they contain all nodes. Furthermore, if a node is part of the subset, then all preceding nodes are also in the set. It follows that there are two subsets of nodes of the graph, {1} and {1, 2}, that can be used as loop invariants.
{1} : P 1 inv = x T := Φ A T L ,b T {1, 2} : P 2 inv = x T := Φ A T L ,b T ∧ b B :=b B − A BL x T
Now, we can also identify the loop guard G. The loop is supposed to terminate when all operands are traversed and the complete solution is computed. Thus, the loop guard depends on how the operands are traversed. The first assignment of the dependency graph operates on x T , A T L and b T , so we can conclude that x and b are traversed from the top to the bottom, and A is traversed from the top-left to the bottom-right corner. In the algorithm, this has the result that initially, A T L , x T and b T are empty. A T L grows with every iteration until it has the same size as A. Similarly, x T and b T grow to the size of x and b. Hence, one possible loop guard is "size(A T L ) < size(A)". How A T L , x T and b T change their sizes is explained in further detail in the next section.
Algorithm Construction
Having found a number of loop invariants, we can proceed to the final step, the derivation of the updates, using the worksheet (Figure 2.1). For each loop invariant, a separate worksheet is filled out. Every loop invariant is rewritten in two different ways, resulting in two predicates per loop invariant. The first predicate specifies the state of all operands before the update (P before ). Analogously, the second predicate P after represents the situation after the update. The update is then found by identifying the operations that transform the predicate P before into P after . Intuitively speaking, the difference between both predicates is determined. The rewritings that are applied to the loop invariant are repartitionings of the already partitioned operands. They are chosen in a way that ensures that the operation makes progress and eventually terminates in combination with a suitable loop guard: Those blocks of the output that are already computed grow, and those parts that are not computed yet shrink. This is done by splitting off parts of some quantities by applying the "Repartition" rules and merging them with others with the "Continue with" repartitioning.
We begin with repartitioning the partitioned operands. To obtain a blocked algorithm, we use the following "Repartition" rules:
A T L 0 A BL A BR → A 00 0 0 A 10 A 11 0 A 20 A 21 A 22 x T x B → x 0 x 1 x 2 b T b B → b 0 b 1 b 2
Doing so, during every iteration, some parts of A BR are split off. To continue after the update, those parts are merged into A T L . Thus, A BR shrinks and A T L grows until the entire matrix, and at the same time both vectors b and x, are traversed. This "Continue with" partitioning is shown below.
A T L 0 A BL A BR ← A 00 0 0 A 10 A 11 0 A 20 A 21 A 22 x T x B ← x 0 x 1 x 2 b T b B ← b 0 b 1 b 2
To get to the predicates P before and P after , we have to rewrite the loop invariant using the repartitioned operands. We demonstrate this for the second loop invariant, P 2 inv . For P before , we replace all quantities by their counterparts according to the first partitioning (from 2 × 2 to 3 × 3), resulting in
x 0 := Φ A 00 ,b 0 b 1 b 2 := b 1 b 2 − A 10 A 20 x 0 .
Then, we flatten the expression, that is, perform all algebraic operations, distribute the assignments and decompose the result into its parts. This yields the following predicate:
P before = {x 0 := Φ A 00 ,b 0 ∧ b 1 :=b 1 − A 10 x 0 ∧ b 2 :=b 2 − A 20 x 0 }
The same is done using the "Continue with" partitioning. To repartition the function Φ, the PME is used.
x 0
x 1 := Φ A 00 0 A 10 A 11 , b 0 b 1 b 2 :=b 2 − A 20 A 21 x 0 x 1
Thus, we obtain the predicate below:
P after = {x 0 := Φ A 00 ,b 0 ∧ x 1 := Φ A 11 ,b 1 − A 10 x 0 ∧ b 2 :=b 2 − A 20 x 0 − A 21 x 1 }
Now, by comparing the predicates P before and P after , we determine the update. Highlighted in red are those parts of P after that do not appear in P before .
P after = {x 0 := Φ(A 00 ,b 0 )∧ x 1 := Φ(A 11 ,b 1 − A 10 x 0 )∧ b 2 :=b 2 − A 20 x 0 − A 21 x 1 }
This provides us with the following update:
x 1 := Φ (A 11 , b 1 ) b 2 := b 2 − A 21 x 1
We have now derived a complete algorithm. Consider Figure 2.3 for the filled out worksheet with updates for both loop invariants (omitting some of the repartitionings and all predicates in the interest of visual clarity). Using the loop guard and the loop invariants, it is easy to see that the derived algorithms are correct and at the end of the computation, the linear system is solved. The algorithms terminate when the loop guard becomes false. The negation of "size(A T L ) < size(A)" implies that A T L has the same size as A, as it can not be larger. Since A T L is a part of A, this means that A T L is equal to A. Similarly, x T equals x.b T and b T are equal 2.2. Symmetric Positive Definite Linear System tob and b, respectively. As we updated b, we have to distinguish between its initial and its current content to reason about the correctness of the derived algorithm. All other parts have either size 0 × n, 0 × 0 or n × 0, so they disappear. Plugging that in the loop invariants, in both cases we only get x := Φ A,b , which proves that both algorithms compute the solution to a triangular system.
Algorithm: x := Φ (A, b) Partition A → A T L 0 A BL A BR where A T L is 0 × 0 While size(A T L ) < size(A) do Repartition A T L 0 A BL A BR → A 00 0 0 A 10 A 11 0 A 20 A 21 A 22 where A is k × k Variant 1 Variant 2 b 1 :=b 1 − A 10 x 0 x 1 := Φ (A 11 , b 1 ) x 1 := Φ (A 11 , b 1 ) b 2 := b 2 − A 21 x 1
Continue with
A T L 0 A BL A BR ← A 00 0 0 A 10 A 11 0 A 20 A 21 A 22 endwhile
Symmetric Positive Definite Linear System
As a second example, we derive algorithms for a symmetric positive definite (SPD) linear system to show in greater detail how the feasibility of loop invariants is checked. Again, the starting point is the formal description of the operation, which is shown below:
x := Σ (A, b) ≡ P pre : {Input[A] ∧ Matrix[A] ∧ SPD[A]∧ Input[b] ∧ Vector[b]∧ Output[x] ∧ Vector[x]} P post : {Âx =b}
Because of the symmetry ofÂ, we apply a 2 × 2 partitioning where T L (and BR ) is square. Furthermore, because of the symmetry, it holds that BL = T T R . The following partitioned postcondition is obtained.
 T L x T + T R x B =b T A BL x T + BR x B =b B
Unfortunately, none of the two equations matches the function x := Σ (A, b). If we rewrite T L x T + T R x b =b T as T L x T =b T − T R x b , the right-hand side of this equation is not completely known, as x b is not known. For the same reason, the second equation does not match the function either. In order to proceed, we need to perform a number of steps that are not part of the systematic approach presented in [6]. As is SPD and T L is square, we know that T L (and BR ) is also SPD, so we can rewrite the top part of the partitioned postcondition as:
x T =Â −1 T L b T −Â T R x B
By replacing x T in the second equation with the right-hand side of the equation above and performing further manipulations, it is possible to obtain an equation that matches the pattern of a SPD linear system. Since is SPD, we can infer that A BR − BL −1 T L T R is SPD too [12].
 BR − BL −1 T L T R x B =b B − BL −1 T Lb T
Thus, the following PME is derived:
x T := Σ Â T L ,b T −Â T R x B x B := Σ Â BR −Â BLÂ −1 T LÂ T R ,b B −Â BLÂ −1 T Lb T
Libraries for linear algebra operations like BLAS usually do not offer operations for products of more than two quantities. Thus, to decompose this PME into its basic buildings blocks, we need to introduce auxiliary variables. In general, there are mutliple ways to compute expressions like BL −1 T L T R , for example by solving BLÂ
−1 T L or −1 T L T R first.
To ensure that the dependency graph is as general as possible and does not impose an ordering on expressions like these, multiple auxiliary variables would have to be introduced. Furthermore, their sizes would have to be left unspecified. To keep this example simple, we compute BL −1 T L first, because it appears twice in the PME. While in that case, we only need one auxiliary variable that has the same size as BL , it is useful to formally introduce a complete matrix Z and partition it in the same way we partitionÂ. Doing so, we can treat the auxiliary variable just like all the other operands. Using Z BL , we obtain the following decomposition. The corresponding dependency graph is shown in Figure 2.4. The next step is again to select subsets of the dependency graph to use them as loop invariants and determine the loop guard.
1. Z BL :=Â BLÂ −1 T L 2. A BR :=Â BR − Z BLÂT R 3. b B :=b B − Z BLbT 4. x B := Σ (A BR , b B ) 5. b T :=b T −Â T R x B 6. x T := Σ Â T L , b T
Here, is traversed from the bottom right to the top left, and both x andb are traversed from the bottom to the top. Thus, the loop guard G is "size(A BR ) < size(A)". If we had solved BL x T + BR x B =b B to x B earlier, as opposed to x T , we would have obtained an algorithm that proceeds in the opposite direction.
For this operation, to select loop invariants, the simplified rule that the subsets can neither be empty nor contain all nodes is not sufficient anymore. We now have to consider the full constraints. In general, it has to be checked whether a loop invariant is feasible, that is, if it leads to an algorithm that actually computes the operation. In [6], the author lists the following two constraints:
1. There must exist a basic initialization of the operands, i.e., an initial partitioning, that renders the predicate P inv true:
{P pre } Partition {P inv } 2
. P inv and the negation of the loop guard, G, must imply the postcondition, P post :
P inv ∧ ¬G ⇒ P post
The empty subset always fails to satisfy the second constraint, as it translates to an empty predicate. The empty predicate in conjunction with the negation of the loop guard can never imply the postcondition. Similarly, the set that contains all nodes can not satisfy the first condition. In this case, the solution to the operation would already be computed even before the loop is entered. However, merely partitioning the operands does not render such a loop invariant true. In addition to that, predicates P before and P after would be identical, so no update would be derived. 2. {1, 2, 3, 4, 5} (P 2 inv ) As mentioned before, the algorithms proceed from the bottom right to the top left, so the repartitionings are different compared to the ones used for the lower triangular system. First, parts of A T L are split off, which are then merged with A BR after the update. Thus, the "Repartition" statement for A (and Z) is the following:
A T L A T R A BL A BR → Ab T b B → b 0 b 1 b 2 b T b B ← b 0 b 1 b 2
x is repartitioned in the same way. In the interest of brevity, the derivation of the updates will not be shown here. There are, however, two important points that should be mentioned. When deriving the predicate P before , the expression appears. To flatten it, the PME of a different SPD system, namely XA = B, has to be used. Then, in order to find out which expressions appear both in P before and P after and to identify the differences, it is necessary to rewrite the expressions of both predicates first. Doing so, the auxiliary variables are eliminated. Finally, the following two updates are found:
P 1 inv : b 1 :=b 1 − A 12 x 2 P 2 inv : x 1 := Σ(A 11 , b 1 ) x 1 := Σ(A 11 , b 1 ) b 0 := b 0 − A 01 x 1
General Linear System
In this section, we apply the FLAME methodology to a linear system where A is a general, nonsingular matrix.
x := Λ (A, b) ≡ P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Input[b] ∧ Vector[b]∧ Output[x] ∧ Vector[x]} P post : {Âx =b}
Since is not lower triangular, there is more than one possibility for the initial partitioning. The different partitionings are shown in Table 2.1.
At this point, we are again not able to make any further progress with the usual approach. No (sub)expression of the three expressions matches the initial operation. In case of the first two, the parts of are not square. Some or all of the four parts of # Partitioned postcondition Flattened expression in the third expression could be square, but since the partitioning does not prescribe any sizes, we can in general not assume that this is the case for any part.
1  LÂR x T x B =b L x T + R x B =b 2  T A B x = b T b B  T x =b T A B x =b B 3  T LÂT R A BLÂBR x T x B = b T b B  T L x T + T R x B =b T A BL x T + BR x B =b B
If we use the third partitioning and assume that T L (and BR ) is square, we are in a similar situation as with the SPD system. Recall that to proceed, we had to rewritê
A T L x T +Â T R x B =b T as x T =Â −1 T L b T −Â T R x B .
This was possible because is SPD, so we were able to infer that T L is SPD, and thus nonsingular, as well. The difference now is that the T L part of a nonsingular matrix is in general not nonsingular. While we could assume that it is, proceed as we did in the previous section and derive similar algorithms, those algorithms would not solve general linear systems.
Equivalence of Loop Invariants
For direct methods, one observes that finer partitionings lead to additional loop invariants. A finer partitioning here means that we partition along one dimension in more than the usual two parts. Some of those loop invariants lead to truly new algorithms that cannot be derived with a coarser partitioning. Intuitively, one can say that those algorithms expose intermediate steps that are not exposed with a coarser partitioning. Others are in some sense redundant, because they lead to algorithms that are computationally equivalent to algorithms derived from different loop invariants. We then consider those loop invariants to be equivalent. Those loop invariants may or may not have the same granularity. This means that two loop invariants from a 3 × 3 partitioning can be equivalent, but it is also possible that a 2 × 2 loop invariant is equivalent to a loop invariant obtained from a 3 × 3 partitioning. This is relevant for the derivation of algorithms for iterative methods because we compare different partitionings in Section 3.3. Even more important, the partitioning that is used for iterative methods behaves differently in this regard, as discussed in Section 4.3.1.
In this section, we show what equivalence means in this context and under which conditions this behavior occurs. We begin by introducing the notion of equivalence for algorithms and then extend the results to loop invariants.
Equivalence of Algorithms
The triangular continuous-time Sylvester equation, which will serve as an example, is defined as AX + XB = C with A and B being triangular. In the following, we assume A and B to be lower and upper triangular, respectively:
X := Ψ(A, B, C) ≡ P pre : {Input[A] ∧ Matrix[A] ∧ LowerTriangular[A]∧ Input[B] ∧ Matrix[B] ∧ UpperTriangular[B]∧ Input[C] ∧ Matrix[C]∧ Output[X] ∧ Matrix[X]} P post : {AX + XB =Ĉ}
Applying a 1 × 3 partitioning to X results in the following PME:
X L = Ψ(A, B T L ,Ĉ L ) X M = Ψ(A, B MM , C M − X L B T M ) X R = Ψ(A, B BL , C R − X L B T R − X M B MR )
At this point, we introduce a new notation for loop invariants taken from [6]. Instead of a logical predicate, the PME is used, leaving out those (sub)expressions that are not part of the loop invariant. If entire blocks are not included, we use the symbol = to express that no constraints are imposed on this part. We now choose the following two loop invariants and derive the corresponding updates, skipping the intermediate steps.
P inv = X L = Ψ(A, B T L ,Ĉ L ) = = P ′ inv = X L = Ψ(A, B T L ,Ĉ L ) X M = Ψ(A, B MM ,Ĉ M − X L B T M ) =
The updates are for P inv and P ′ inv are:
P inv : C 1 :=Ĉ 1 − X 0 B 01 X 1 := Ψ(A, B 11 , C 1 ) P ′ inv : C 2 :=Ĉ 2 − X 0 B 02 − X 1 B 12 X 2 := Ψ(A, B 22 , C 2 )
Obviously, the updates for X 1 and X 2 only differ in the indices, which are shifted by one. B 22 is the next block following B 11 on the main diagonal of B, and C 2 is the next set of columns following C 1 in C. Considering that those updates happen inside a loop, as the computation unfolds, they will both point to the same parts of B and C, respectively. Written as above, the updates for C do not immediately appear equivalent. Using BLAS operations, C 1 would be updated with one call to GEMM, C 2 with two. However, when we rewrite the update for C 2 as follows, it is easy to see that it is equivalent to just one GEMM operation.
C 2 :=Ĉ 2 − X 0 X 1 B 02 B 12
Similarly to C 1 and C 2 , X 0 and X 0 X 1 will, at some point during the computation, contain the same parts of X. The same is true for B 01 and
B 02 B 12 .
Hence, we consider both algorithms to be equivalent. Note that in general, the equivalence can be much less obvious, requiring much more elaborate rewriting. In the example above, it is easy to see that simply by replacing quantities in one update with the corresponding quantities of the other update, it is possible to transform one update into the other. The only constraint is that if one quantity replaces another, both must have matching sizes. Since we do not specify any concrete dimension and the size of some parts changes during the computation, we have to compare them using a symbolic representation of those sizes. Let us begin with listing the dimensions of all operands of the Sylvester equation:
-X and C are of size n × m.
-A is of size n × n. -B is of size m × m.
After the repartitioning, we specify the sizes as follows.
bi b b bj+c n X 0 X 1 X 2 X 3 bi b b bj+c n C 0 C 1 C 2 C 3 bi b b bj+c bi B 00 B 01 B 02 B 03 b 0 B 11 B 12 B 13 b 0 0 B 22 B 23 bj+c 0 0 0 B 33 b
is the block size, bi and bj are unspecified multiples of the block size. c is an additional constant that is nonzero if the block size b does not divide m and/or n. Blocks with constant sizes, that is, any combination of n, m and b, can only be replaced with blocks of the corresponding quantities that have the exact same sizes. For example, it is possible to replace X 1 with X 2 , or B 22 with B 11 . Similarly, the dimensions have to match for quantities with variable sizes. If one dimension is a multiple of the block size, any other multiple of the block size matches. As an example, bi matches bi + b, and bj + c + 2b matches bj + c. It is important to note that bi does not match b, because the former is variable, and the latter is constant. Thus, two valid replacements are:
B 00 → B 00 B 01 0 B 11 C 2 C 3 → C 3
Returning to the initial example, the replacement to transform the update for the loop invariant P inv into the one for the loop invariant P ′ inv is the following:
B 01 → B 02 B 12 B 11 → B 22 C 1 → C 2 X 0 → X 0 X 1 X 1 → X 2
Equivalence of Loop Invariants
As mentioned earlier, we consider two loop invariants to be equivalent if the resulting algorithms are equivalent. To determine this equivalence, however, it is not necessary to derive algorithms. The same method of replacing quantities can be performed directly on the loop invariants. If, with such a replacement, one loop invariant can be transformed into another one, both would result in the same update. Consequently, the loop invariants themselves are equivalent. Let us demonstrate this for the loop invariants P inv and P ′ inv . We choose the follow-2. Derivation of Algorithms for Direct Methods ing replacements:
X L → X L X M C L → C L C M B T L → B T L B T M 0 B MM Applying those to the loop invariant P inv = X L = Ψ(A, B T L ,Ĉ L ) = = ,
the equation for computing X L becomes
X L X M = Ψ A, B T L B T M 0 B MM , Ĉ LĈM .
Flattening the expressions, we obtain the second loop invariant, P ′ inv :
P ′ inv = X L = Ψ(A, B T L ,Ĉ L ) X M = Ψ(A, B MM ,Ĉ M − X L B T M ) =
Equivalence of Loop Invariants of Different Granularities
The presented method of term rewriting can also be used to decide whether an algorithm or loop invariant is equivalent to a different one obtained with a finer or coarser partitioning. So far, the quantities on both sides of the replacements originated from the same partitioned object. Now, each operand is partitioned twice, using partitionings of two different granularities. Parts obtained from one are replaced with parts of the other. Apart from that, exactly the same rules for the replacement hold. We demonstrate this with a more involved example: The inverse of a lower triangular matrix. A 2 × 2 partitioning yields the following PME, usingX andL to avoid confusion:
X T L :=L −1 T L 0 X BL := −L −1 BRL BLXT LXBR :=L −1 BR (2.1)
This is the 3 × 3 PME:
X T L := L −1 T L 0 0 X ML := −L −1 MM L ML X T L X MM := L −1 MM 0 X BL := −L −1 BR L BM X ML − L −1 BR L BL X T L X BM := −L −1 BR L BM X MM X BR := L −1 BR
We will show that the two loop invariants below are equivalent:
X T L :=L −1 T L 0 X BL := −L −1 BRL BLXT L = X T L := L −1 T L 0 0 X ML := −L −1 MM L ML X T L = 0 X BL := −L −1 BR L BM X ML − L −1 BR L BL X T L = =
To do so, we first need to choose an appropriate replacement. The replacement used in this example will transform the 2 × 2 loop invariant into the 3 × 3 loop invariant, so some parts of the coarser loop invariant will be replaced with multiple parts of the finer one.X
T L → X T L (2.2) L T L → L T L (2.3) X BL → X ML X BL (2.4) L BL → L ML L BL (2.5) L BR → L MM 0 L BM L BR (2.6)
To make sure that it is a valid replacement, it is necessary to check if the dimensions of the objects on both sides of the replacements match. The sizes of L, and likewise X, are as follows: Applying this replacement to the coarser loop invariant, we obtain two assignments:
X T L := L −1 T L (2.7) X ML X BL := − L MM 0 L BM L BR −1 L ML L BL X T L (2.8)
Clearly, the first one already has the exact same shape as in the finer loop invariant. The second one, however, requires some rewriting. To be able to compute the product on the right-hand side, we first have to find the symbolic inverse of
L MM 0 L BM L BR .
We obtain it from the PME of the inverse of a lower triangular matrix, (2.1), by eliminating all occurrences of parts of X and replacing the corresponding quantities:
L MM 0 L BM L BR −1 = L −1 MM 0 −L −1 BR L BM L −1 MM L −1 BR Assignment (2.8) then becomes X ML := −L −1 MM L ML X T L (2.9) X BL := L −1 BR L BM L −1 MM L ML X T L − L −1 BR L BL X T L . (2.10)
The assignment for X ML is now identical to the one in the 3 × 3 loop invariant. While the assignments for X BL still differ, we observe that it is possible to replace a subexpression of (2.10) with −X ML :
X BL := L −1 BR L BM L −1 MM L ML X T L −X ML −L −1 BR L BL X T L
Thus, we obtain the following expression, which is the same as the assignment for X BL in the finer loop invariant:
X BL := −L −1 BR L BM X ML − L −1 BR L BL X T L
Derivation of Algorithms for Iterative Methods: Foundations
To be able to use a FLAME-like methodology to derive algorithms for iterative methods, a matrix representation of those methods is required. Such a representation was used in [5] for CG and the Krylov sequence, and in [4] for some CG variants. This chapter begins with the introduction of an additional notation. The details of the matrix representation are explained in the first section. In the second part, a systematic method for deriving properties of matrices from this representation is presented. Finally, different possible partitionings are discussed, in addition to their implications for the derivation of algorithms.
Notation
In this thesis, we use a notation that deviates slightly from the one used in [4] and [5]. We use e 0 to denote the unit vector that is one in the first position and e r for the unit vector that is one in the last position. Both are column vectors. The matrix J is a square matrix with ones on the lower diagonal:
J = 0 0 . . . 1 0 0 1 . . . . . . 0 . . .
As usual, I is the identity matrix. The dimension of those matrices and vectors will not be given explicitly if they are clear from the context. If X is a matrix, we use X to indicate that the right-most column of this matrix is omitted. Thus, I and J are both lower trapezoidal matrices with one more row than columns.
Matrix Representation
In this section, we discuss the details of the matrix representation for iterative methods used in this thesis. Note that the representation for CG, which will serve as an example, is a slightly modified version of the one introduced in [5]. The difference lies in the use of the underline. The three governing equations are shown below. Deriving those equations is not trivial and beyond the scope of this thesis.
APD = R I − J (3.1) P (I − U) = R (3.2) PD = X I − J (3.3)
The operands have the following properties:
-A ∈ R n×n is the coefficient matrix of the linear system that is supposed to be solved. It is nonsingular. Depending on whether A is symmetric or not, different algorithms can be derived.
-P ∈ R n×m is the matrix of search directions, that is, each column represents the search direction vector during one iteration. It is initially unknown.
-D ∈ R m×m is an unknown diagonal matrix.
-R ∈ R n×(m+1) is the residual matrix. Initially, only the first column r 0 is known. It is computed as r 0 = Ax 0 − b, where x 0 is an initial guess for the solution.
Additionally, it is orthogonal.
-U ∈ R m×m is unknown and upper diagonal 1 if A is symmetric. Otherwise, it is strictly upper triangular.
-X ∈ R n×(m+1) is the matrix of approximated solution vectors. Similar to R, only the first column is initially known.
While it might seem unusual that the same matrix (R) appears twice in the governing equations with varying sizes, this is necessary to ensure the correctness of the last column of R and X. The formula for computing the residual, in indexed notation, is r i+1 = r i − Ap i δ i . Without the additional column of R in equation (3.1), the incorrect equation Ap i δ i = r i would be obtained for the last iteration. Similarly, from equation (3.3), we would obtain p i δ i = x i , which is not correct either.
One of the fundamental differences compared to direct methods is that the dimensions of some matrices are not fixed. Usually, all dimensions are determined by the sizes of input operands. In case of the LU-factorization, for example, we know that L and U have the same size as A. If not, the equation LU = A is not valid. If A is of size n × n, and the LU factorization of a k × k block of A, with k < n, is computed, then the postcondition is not rendered true.
Due to the orthogonality of R, m + 1 can not be larger than n, as the number of n-dimensional, orthogonal vectors is at most n. However, for every m < n, the equations above can be satisfied by performing the corresponding number of iterations.
For the systematic derivation of algorithms, this introduces the problem that it is not possible to derive a loop guard exclusively from the equations. It makes no sense to compare the size of a block of R to R itself, because R grows too. For those iterative methods that are used to find solutions for linear systems, the goal is usually to minimize the residual in some norm [1]. Thus, the loop guard typically is a predicate like " r i ε", where ε is a threshold chosen by the user. We make sure that the postcondition of CG correctly represents the situation at the end of the operation by adding Re T r < ε to it. This also allows us to derive a suitable loop guard from it. For stationary iterative methods, the stopping criterion quite often is x i − x i−1 < ε. Translating that in our notation, we obtain Xe T r − Xe T r−1 < ε. For iterative methods where those criteria are not applicable, we will add an expression to the postcondition that fixes the number of columns of a matrix with variable size. Such a predicate could be "size(Y) = n × k". 2
Systematic Derivation of Matrix Properties
In order to derive algorithms for CG from its description in matrix form, it is necessary to use properties of matrices and expressions that are not explicitly part of the initial description. Those properties have to be derived from the description by means of algebraic manipulation and deductive reasoning. To automate the process of finding algorithms, an automatic method for the derivation of properties is necessary. Thus, this systematic approach should replicate the steps performed by a human expert, without actually requiring any human guidance. In this section, we describe such a method.
Preliminaries
To begin, it is useful to formalize the notion of properties and equations. We start with the most basic building blocks, terms and expressions.
Definition 3.2.1 (Terms and Expressions)
Terms are inductively defined as follows:
1. Every matrix, vector and scalar is a term.
2. If t is a term, then t T , (−t) and t −1 are terms.
3. If t 1 and t 2 are terms, then (t 1 + t 2 ) and (t 1 · t 2 ) are terms.
Every term is also an expression. Furthermore, if t 1 and t 2 are terms, then t 1 = t 2 is an expression.
Note that this is a simplified definition, as it does not include functions, nor does it make any statement about the validity of terms. A product of two matrices with dimensions that do not match is still a valid term. Nonetheless, this definition is sufficient for our purposes. Furthermore, we usually use a simplified notation, omitting parentheses, if they are unnecessary, as well as the multiplication dot. We can now define equations and properties:
Definition 3.2.2 (Equations)
Let t 1 , t 2 be terms. An equation is an expression of the form t 1 = t 2 .
Definition 3.2.3 (Properties)
Let t be a term and P be a boolean predicate. Then P[t] is a property.
Note that properties are always predicates, even if they can also be expressed as equations. Take the property Orthogonal[R] as an example. It implies Diagonal[R T R]. Applying the partitioning that is also applied to the postcondition,
R → R L r M R R ,
to R T R, we find out, among other things, that R T L R L is diagonal and r T M R L equals zero. Instead of considering the equation r T M R L = 0 as a property, we use Zero[r T M R L ]. This allows us to use a more consistent notation and simplifies the systematic derivation.
Representing Knowledge about Linear Algebra
A human expert who derives properties of matrices inevitably applies some basic knowledge about linear algebra. To allow a system to replicate the expert reasoning, it needs a knowledgebase that encodes this knowledge. We define five different types of implications that will be included in the knowledgebase.
1. P 1 [t] ∧ . . . ∧ P i [t] → P[t]
This type of implication allows to reason about the combination of properties of one single term. One example is:
LowerTriangular[t] ∧ Symmetric[t] → Diagonal[t] (3.4) 2. P 1 [t 1 ] ∧ . . . ∧ P i [t i ] ∧ ∃t → P[t]
Here, t 1 , . . . , t i are subterms of t. Thus, it allows the system to infer the properties of a product or sum of multiple quantities with different properties. The ∃t is used to avoid deriving properties for terms that do not occur anywhere.
Consider two examples:
Diagonal[t 1 ] ∧ LowerTriangular[t 2 ] ∧ ∃t 1 t 2 → LowerTriangular[t 1 t 2 ] (3.5) StrictlyUpperTriangular[t] ∧ ∃I + t → UpperTriangular[I + t] (3.6) 3. P[t 1 ] ∧ t 1 = t 2 → P[t 2 ]
This implication enables the system to propagate properties across equalities. P now is a pattern that matches any property.
4. P 1 [t] → P 2 [f(t)]
f is a function, for example transposition. Thus, this kind of implication allows to derive properties of transposed or inverted quantities. In addition to that, it is used reason about orthogonal or orthonormal matrices. Let us look at three examples:
LowerTriangular [t] → UpperTriangular t T (3.7) LowerTriangular [t] → LowerTriangular t −1 (3.8) Orthogonal[t] → Diagonal t T t (3.9) 5. t → f(t)
For one of the steps of the method presented in this chapter, it is necessary that properties have a canonical form: The unary operators −1 and T will not be applied to products. However, some of the other types of implications may produce such terms. To transform those terms into the canonical form, implications are necessary that distribute those unary operators across products. Those implications are:
(t 1 t 2 ) T → t T 2 t T 1 (3.10) (t 1 t 2 ) −1 → t −1 2 t −1 1 (3.11)
Note that all the terms in the implications above may match not only a single operand, but every term that has the required property. So if the product AB is lower triangular, implication (3.7) tells us that (AB) T is upper triangular, and using (3.10), we find out that B T A T is upper triangular.
Derivation of Properties
In the following, we will assume that all equations have the form
1 i n t i = 1 j m t ′ j .
In case of a sum X + Y, the product consists of just one term t = X + Y. This assumption is no restriction as the only sums that appear in the matrix representations of iterative methods are of the form I − Y and are used exclusively to emphasize the structure of those matrices.
Initialization We start with two sets, P and E. P is a subset of the precondition P pre of the description of an operation. It only contains those properties that describe operand types, like Square[X], Matrix [X] or Diagonal(X). Not included are properties that specify what is input and output, as they are superfluous for the derivation. E is a set of expressions which contains the equations of the corresponding postcondition P post .
Derivation of Properties
The implications of the knowledgebase K are used to derive all possible properties at this stage. This is done as follows:
-Given P 1 [t] ∧ . . . ∧ P i [t] → P[t] ∈ K, the property P[t] is added to P if P k [t] ∈ P for all k ∈ {1, . . . , i}. -Given P 1 [t] ∧ . . . ∧ P i [t] ∧ ∃t → P[t] ∈ K, the property P[t] is added to P if 1.
there is an equation e ∈ E that contains the term t, and
2. P k [t] ∈ P for all k ∈ {1, . . . , i}. -Given P[t 1 ] ∧ t 1 = t 2 → P[t 2 ] ∈ K, the property P[t 2 ] is added to P if P[t 1 ] ∈ P and (t 1 = t 2 ) ∈ E. -Given P 1 [t] → P 2 [f(t)] ∈ K, the property P 2 [f(t)] is added to P if P 1 [t] ∈ P.
-Given t → f(t) ∈ K and P[t] ∈ P, the property P[f(t)] is added to P.
Matrix Inversion Then, new expressions are added to E.
1. For every equation e ∈ E with e = (t 1 · · · t i = u 1 · · · u j ), it is checked whether t 1 , t i , u 1 or u j is invertible.
2. If t 1 or u 1 is invertible, t −1 1 or u −1 1 , respectively, is multiplied from the left to e, eliminating the invertible term on the one side and adding its inverse on the other. We proceed analogously with t i and u j , then multiplying the inverse of the term from the right.
3. The resulting, new equation e ′ is added to E.
Those three steps are repeated until no new expressions are found.
Application of Properties
In the final step, we apply known properties to expressions to derive new properties. Intuitively, we multiply quantities to both sides of an equation in order to recreate subexpressions that are known to present some property, which are then used to infer new properties.
For every property
P[t] ∈ P with t = t 1 · · · t i , i > 1 a set of tuples S(t) = {(t 1 · · · t k , t k+1 · · · t i ) | 1 k < i}
is generated. Intuitively, this set contains the term t, split into two parts in all possible ways.
28 3.2. Systematic Derivation of Matrix Properties 2. Let (t L , t R ) ∈ S(t).
If there is an equation e ∈ E where t L is the rightmost (sub)term in any term, then t R is multiplied from the right. t L is multiplied from the left in the analogous case. Let e ′ be the resulting equation. As an example, let e be ABC = D and (BC, F) ∈ S(t). Since BC appears as the rightmost subterm in e, t R = F is multiplied from to right, resulting in ABCF = DF.
3. The knowledgebase is used to derive new properties using e ′ that are added to P (see step "Derivation of Properties"). e ′ is not added to E.
Design Considerations
One observes that the presented method is not goal-oriented. The derived properties are mainly used to solve equations, so it might seem more natural to derive properties starting with an equation that has to be solved. Based on this equation, expressions would be selected, and in a second step, the properties of those expression would be derived. Those properties would then be used to solve the equation. Instead, the presented method derives a large number or properties, irrespective of the question whether they might be useful or not.
The problem with a goal-oriented approach is that quite often, it is not obvious from the equation which property could be used to solve it. This leaves us with the much more challenging task of identifying which expression might have relevant properties. Take the following equation as an example, which appears when deriving algorithms for nonsymmetric CG. P L , p M , r M and A are known.
−P L u T M + p M = r M
It is solved for u T M by using the fact that P T AP is lower triangular. Thus, P T L AP L is lower triangular as well and P T L Ap M is zero. Multiplying P T L A from the left to both sides of the equation gives us
−P T L AP L u T M = P T L Ar M .
This now is a triangular system that can easily be solved. While it is possible to individually derive that P T L AP L is lower triangular and P T L Ap M is zero, the initial equation gives us little to no indication to inspect the properties of those expressions in the first place.
One the other hand, the advantage of a method that is not goal-oriented is that it may find properties that we do not expect to find.
Orthogonality of the Residual Matrix
For some iterative methods, for example CG, the residual matrix R is orthogonal. In the postcondition of those methods, R usually appears multiple times, either as a whole, or without the last column (R). Unfortunately, for the derivation of properties, this poses a problem.
If R is orthogonal, then R T R and R T R are diagonal. R T R and its transpose are rectangular and all entries except for the ones on the main diagonal are zero. Thus, in some sense, they are diagonal as well.
It turns out that for deriving certain properties, R T R is needed. Unfortunately, using the described method, neither Orthogonal[R] nor Orthogonal[R] implies any property of R T R. One way to solve this would be to treat any matrix that appears with and without the last column in a special way, such that properties of R T R and R T R are found as well. This, however, would require significant modifications of the derivation process.
The simpler solution is to consider R and R to be two distinct objects, and properties of R T R and its transpose are added to the precondition.
Substituting Equations
Note that we deliberately avoid substituting quantities in one equation by expressions obtained from others. While this might be a very natural approach if deriving properties by hand, doing this systematically is difficult. Plugging in equations quickly leads to arbitrarily large expressions unless some heuristics are applied to terminate this process. Apart from that, it is possible to achieve the same results using the approach presented in this section. Consider a short example to get an intuition why this is the case. Let us assume we have two equations t 1 = t 2 t 3 t 4 t 2 = t 5 and we want to derive a property for t 1 . Properties of t 3 , t 4 and t 5 are known (colored green). As the properties of t 2 are not known as well, we have to use the second equation to proceed. If t 4 is invertible, we can solve to t 2 = t −1 4 t 5 . Instead of substituting t 2 in t 1 = t 2 t 3 , yielding t 1 = t −1 4 t 5 t 3 , and then reason about properties of t −1 4 t 5 , we first derive all properties of t −1 4 t 5 , which are also properties of t 2 . In the final step, we derive all properties of t 2 t 3 , obtaining the same properties for t 1 we would find by plugging one equation into the other.
Termination
The disadvantage of this approach is that it may not terminate either, and increasingly long properties are derived. This is not unexpected, since this approach aims at replicating the process of substituting equations. The difference of the presented method is that the set of equations is finite, its size does not even change anymore after the matrix inversion step. 3 Furthermore, for most iterative methods, no properties of products of more than three quantities are used. While it might be possible to use significantly longer properties to solve equations, most likely, they result in algorithms that use unnecessarily large expressions to compute certain quantities. This naturally leads to the solution of introducing a maximum length for properties, similar to a recursion limit, with a reasonable default value that can be changed by the user. This way, there is only a finite number of properties that can be derived.
Unfortunately, if we just refrained from adding properties of products of more than three quantities to P, the derivation process would cease to work in certain cases (this will be explained in the following section). To avoid that, there are multiple options. In both cases, we initially add longer properties to P as well. Then, one solution is to never use them to derive further properties, that is, we never construct S(t) if t is a product of more than three quantities. Alternatively, those longer properties are removed from P when e ′ is discarded. A third option would be to construct an additional set of temporary properties.
Example: Nonsymmetric CG
We demonstrate the method presented in this chapter by deriving some properties for nonsymmetric CG. In the interest of brevity, we only derive a small number of selected properties, in addition to limiting properties to a maximum length of three quantities. The knowledgebase K is not shown here due to its size. Properties used in the following which are not self-explanatory are defined in Appendix A. The preand postcondition of nonsymmetric CG are shown below. For the sake of simplicity, we treat I − J as one distinct matrix, as I and J do not appear separately.
P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Output[P] ∧ Matrix[P]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ DiagonalR[R T R] ∧ DiagonalR[R T R]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Output[U] ∧ Matrix[U] ∧StrictlyUpperTriangular[t 1 ] ∧ I + t 1 → UpperTriangular[I + t 1 ]
is used to infer that (I − U) is upper triangular. Orthogonal [R] and Orthogonal [R] imply that R T R and R T R are diagonal. Thus, this step yields
P = P ∪ {Diagonal[R T R], Diagonal[R T R], UpperTriangular[I − U]}.
Systematic Derivation of Matrix Properties
Matrix Inversion
From the set of properties P it follows that A, D and (I − U) are nonsingular. The equation APD = R I − J is inspected first. Two invertible objects, namely A and D, occur in it. Since A is the leftmost quantity of the product APD, it can be eliminated by multiplying its inverse from the left-hand side to both sides of the equation. This yields PD = A −1 R I − J , which is added to E. By multiplying D −1 from the right to this new equation and the original APD = R I − J , two additional equations are obtained. By applying the same procedure to the two remaining equations, E becomes the following set:
E = {APD = R I − J , AP = R I − J D −1 , PD = A −1 R I − J , P = A −1 R I − J D −1 , P (I − U) = R, P = R (I − U) −1 , PD = X I − J , P = X I − J D −1 }
At this point, it is not possible to find any new equations by multiplying inverted quantities, so we proceed to the next step.
Application of Properties
Initially, the only properties
P[t] ∈ P with t = t 1 · · · t i , i > 1 are Diagonal[R T R] Diagonal[R T R] DiagonalR[R T R] DiagonalR[R T R]. From Diagonal[R T R], the set S(R T R) = {(R T , R)} is obtained. For every (t L , t R ) ∈ S(R T R)
, it now has to be checked if t L or t R appears in any equation contained in E. Since (R T , R) is the only element in S(R T R), the system just searches for t L = R T and t R = R. The following four equations, all containing R, are found:
APD = R I − J AP = R I − J D −1 PD = A −1 R I − J P = A −1 R I − J D −1
Only in the first two equations, t L = R appears at the rightmost position in a product. Hence, t R = R T is multiplied just to those two:
R T APD = R T R I − J R T AP = R T R I − J D −1
The knowledgebase is now used to derive new properties. In this example, we just look at the equation on the right-hand side, and show the steps that the system would perform.
1. R T R I − J is a product of a diagonal and a lower trapezoidal matrix, so it is lower trapezoidal.
2. D is diagonal, so D −1 is diagonal as well.
3. R T R I − J D −1 is a product of a lower trapezoidal matrix (R T R I − J ) and a diagonal matrix (D −1 ), so it is lower trapezoidal too.
4. The right-hand side of the equation is lower trapezoidal, so the left-hand side of the equation, R T AP, is lower trapezoidal as well.
With every step, the new properties are added to P. Thus, the set becomes:
P := P ∪ {LowerTrapezoidal[R T R I − J ], Diagonal[D −1 ], LowerTrapezoidal[R T R I − J D −1 ], LowerTrapezoidal[R T AP]}
Now, it becomes apparent why it is not possible to simply set a limit on the length of properties and never add any longer properties to P. R T R I − J D −1 is a product of four quantities. To infer that R T AP is lower trapezoidal, the property
LowerTrapezoidal[R T R I − J D −1 ]
has to be derived. If this property is never added to P, it is not possible to derive that R T AP is lower trapezoidal. Hence, it must be possible to derive properties of any length, even though they are only needed temporarily. Using the newly added properties, immediately some more are found because of the implications
LowerTrapezoidal [t] → UpperTrapezoidal t T (t 1 t 2 ) T → t T 2 t T 1 .
The following properties are added to the set:
P := P ∪ {UpperTrapezoidal[ I − J T R T R], Diagonal[D −T ], UpperTrapezoidal[D −T I − J R T R],
UpperTrapezoidal[P T AR]}
At this point, we cut this derivation short. In practice, many more properties would be derived. Most of them may not be of any use for the derivation of algorithms, but a few are crucial. For nonsymmetric CG, such an important property is that P T AP is lower triangular. In [5], this property is derived manually. Let us shortly illustrate how it is derived.
1. Because of DiagonalR[R T R], R T is multiplied from the left to P = R (I − U) −1 , yielding R T P = R T R (I − U) −1 .
(I − U) −1 is upper triangular and R T R is rectangular diagonal, so R T P is upper triangular and rectangular. It follows that its transpose P T R is lower triangular and rectangular.
2. Because of the property LowerTriangularR[P T R], P T is multiplied from the left to APD = R I − J . Based on the resulting equation
P T APD = P T R I − J ,
we derive that P T R I − J is square.
3. For the same reason, P T is multiplied from the left to AP = R I − J D −1 , resulting in P T AP = P T R I − J D −1 .
P T R is lower triangular and rectangular and I − J is lower trapezoidal. P T R I − J is square, so it is lower triangular. Since D −1 is diagonal, P T AP is lower triangular as well.
Initial Partitionings
The first step towards the derivation of algorithms in the FLAME methodology is to partition the operands. The matrix representation of iterative methods gives rise to a new type of operands, namely those which are initially partially known and partially unknown. Thus, we have to address the question of how to partition them. To some extent, it is easy to answer. Standard CG algorithms always compute full vectors r i and x i [1,11,13], so refraining from partitioning R and X horizontally, that is, into a top and a bottom part, is a very natural choice. For partitioning vertically, there are multiple possibilities. Their advantages and disadvantages will be discussed in this section.
While not strictly necessary to derive algorithms, describing the partitioned operands in terms of functions proved to be useful for the systematic derivation [7]. Hence, if and how different partitionings permit to match functions is an important criterion for their evaluation.
We begin this section with a discussion of what such a function should look like. Then, we investigate different possible partitionings, starting with partitionings that are also used for direct methods and continuing with some that are tailored to properties of algorithms for iterative methods.
Functions
On the highest level, the input of a CG algorithm are vectors b and x 0 , as well as the matrix A, so a function would have the form x i := CG(A, b, x 0 ). As we are using the recurrence relation to derive algorithms, which does not involve b, but a number of other quantities, for example the residual r, a function like this is not helpful.
From direct methods, we remember that the function is uniquely 4 defined by the precondition. Every quantity that is initially known has the property Input[X], initially unknown quantities have the property Output[X]. Take the LU factorization as an example. The governing equation is LU = A. A is input, L and U are output. This naturally leads to a function {L, U} = Ψ(A). It is important to note that this function is defined prior to any derivation steps, solely based on the abstract description of the operation. It then happens, due to the recursive nature of the operations, and a partitioning that reveals it, that this function matches expressions obtained by partitioning the postcondition. In some cases, the expressions have to be rewritten first. The remaining expressions can be decomposed into basic buildings blocks.
Applying this scheme to CG, the function {R, U, P, D, X} := CG(A, Re 0 , Xe 0 ) is obtained. For simplicity, we omit I and J as input, since they are constant.
Standard 2 x 2 Partitioning
Naturally, the first choice for a partitioning is the one that is usually used for direct methods. Partitioning R into R L R R implies the following partitioning for equations (3.1) and (3.2). Because of its similarity to the first equation, we will usually omit equation (3.3) in this section.
A P L P R D T L 0 0 D BR = R L R R I − J 0 −H I − J P L P R I − U T L −U T R 0 I − U BR = R L R R
Here, H is a matrix with one more row than columns that is one in the top right corner and zero everywhere else. Flattening the expressions, we obtain
AP L D T L = R L (I − J) − R R H AP R D BR = R R I − J (3.12) P L (I − U T L ) = R L −P L U T R + P R (I − U BR ) = R R . (3.13)
In this form, the expressions are not matched by the pattern of the CG function. While both the right equation of (3.12) and the left one of (3.13) have the correct shape, one contains R L and P L , while R R and P R appear in the other. In the original equations, those quantities are the same. If it is possible to match the function at all, then either all parts on the left or all parts on the right match (or both). Rewriting the equation on the left in (3.12) as
AP L D T L = R L R R I − J −H ,
we recognize similarities to the corresponding equation of the recurrence relation (3.1). However,
I − J −H
does not have one more row than columns. While we know that only the first row of H has a nonzero entry, formally, this equation does not match the pattern.
x Partitioning
The problem above can be solved by applying a 1 × 3 partitioning to R where the middle part is a single column. This has the effect that the first row of H becomes a separate block. As CG proceeds by one column per iteration, exposing a single column appears to be a suitable choice. The partitioned operands and the resulting expressions are shown below.
P R (I − U BR ) = R R + P L U T R + p M u MR ,
such that R R is updated. Here, this is not possible. While the first column of R R can be considered known at this point, neither the first column of P L U T R nor p M u MR is known, because neither U T R nor u MR are known. Apart from that, R R can not be updated, as this would have influences on the other two equations. Finally, if quantities are updated, they are usually updated in their entirety, before they are used as input for a function. Here, U T R and u MR are not known, and we would expect them to be the output of said function, leading to circular data dependencies.
Splitting off the First Column
Since the first column of R and X, respectively, plays a special role, an entirely different approach could be to apply a partitioning that splits off this first column. R is partitioned into r 0 R ′ , and for the actual derivation of algorithms, R ′ is used. The advantage is that there is a clear distinction between input and output, and it would be possible to write CG as {R ′ , . . . , X ′ } := CG(A, r 0 , x 0 ). As usual, partitioning R and X like that also implies a matching partitioning for the remaining operands:
A p 0 P ′ δ 0 0 0 D ′ = r 0 R ′ 1 0 −e 0 I − J p 0 P ′ 1 −u ′ 0 I − U ′ = r 0 R ′
Flattening those expression, we immediately obtain a value for p 0 :
Ap 0 δ 0 = r 0 − R ′ e 0 AP ′ D ′ = R ′ I − J p 0 = r 0 −p 0 u ′ + P ′ (I − U ′ ) = R ′
Using the orthogonality of R, it is also possible to find an assignment for δ 0 . Thus, this partitioning allows to compute all quantities of the first iteration and declare them as known. Unfortunately, splitting off r 0 forces us to also split off the first row of U. At this point, it is not possible to compute it in its entirety. Furthermore, because of −p 0 u ′ , the equations on the right-hand side do not have the same shape as the original description of CG. While it is of course still possible to compute u ′ , it is only possible entry by entry, adding an additional assignment to any update we can derive. Consequently, it will not be possible to derive the updates for CG usually found in textbooks.
Divide and Conquer
So far, no partitioning enabled us to describe one CG operation as multiple, smaller CG operations, if necessary, with updated quantities as input, similar to how the PME of the triangular system in Section 2.1 contains the function Φ two times. This is, however, possible, if we define a generalized version of CG. The disadvantage is that it requires a deeper understanding of the algorithm, in addition to some knowledge that is initially not available when deriving algorithms solely based on their matrix representation.
To derive such a representation, we partition R into R 0 r 1 r 2 R 3 . Since the resulting partitioned postcondition is very large, it is not shown here. The generalized version of CG requires some parts of P as an additional argument. Initially, it is only the first column P, here denoted with Pe 0 : {R, U, P, D, X} := CG(A, Re 0 , Xe 0 , Pe 0 ). Since it is equal to the first column of R, it can be considered known. For now, we will assume that A is nonsymmetric.
Our goal is now to write this CG operation as two separate ones, one covering R 0 r 1 , and one for r 2 R 3 . Clearly, the first one is R 0 r 1 r 2 , . . . , P 0 p 1 , . . . , X 0 x 1 x 2 := CG (A, R 0 e 0 , X 0 e 0 , P 0 e 0 ) , omitting some of the output in the interest of legibility. Now, to find the correct arguments for the second one, we need to know how p 2 is computed: p 2 = r 2 + P 0 u 02 + p 1 ν 12 u 02 and ν 12 are not part of the output of the function above, but they can in turn be computed using known quantities only: To ensure that this function computes the correct sequence of search directions, it also has to use the ones computed by the first function, which are P 0 p 1 . This is the reason why not just p 2 , but P 0 p 1 p 2 is input. This situation is slightly different if A is symmetric. In that case, U is upper diagonal and P T AP is diagonal, so p 2 is computed as
u 02 = −P T 0 AP 0 −1 · P Tp 2 = r 2 − p 2 p T 1 Ar 2 p T 1 Ap 1 .
Now, each search direction is computed using only the last one, so P 0 p 1 is not needed as input. Thus, the second function simplifies to {R 3 , . . . , P 3 , . . . , X 3 } := CG (A, r 2 , x 2 , p 2 ) .
As mentioned before, the disadvantage is that we already need to know how some quantities are computed to derive this representation, while it is actually our goal to find those updates.
Splitting off the Last Column
Among those presented in this section, the 3 × 3 partitioning that exposes a single column is the only one that resulted in expressions that were naturally matched by the CG function. The problem of this partitioning is that there is no obvious way how to deal with the right-hand side parts. To find one that better suits iterative methods, it is helpful to again inspect the differences to direct methods. After all, the 3 × 3 partitioning came to existence as a modification of the standard partitioning used for direct methods. As mentioned before, with direct methods, the sizes of all operands are initially known. Thus, at any point during the computation, there are (potentially empty) parts of operands that are already computed, and (potentially empty) parts that are not computed yet. The loop invariant describes those parts that are already computed at the beginning of the loop body. Conversely, one can think of those parts of the PME that are not part of the loop invariant as those parts that are not computed yet.
To see that this is consistent, recall that the reason that the full set of nodes can never be a feasible loop invariant is that it would imply that the solution is already computed before the loop is entered. Clearly, this is equivalent to saying that there are no parts left that are not computed yet, even before the loop is entered.
In case of iterative methods, there is little use in talking about parts of operands that are not yet computed beyond the current iteration, as each iteration might be the last. We can conclude that it makes little sense to use a partitioning where the right-hand side is more than a single column.
The solution is to use a partitioning that is a hybrid of the standard 2 × 2 partitioning and the 3 × 3 partitioning that exposes a single column: A partitioning that splits off the last column: P is partitioned into P L p R . Because of the additional column of R, it is partitioned into R L r R r + . Thus, for CG, we obtain
A P L p R D T L 0 0 δ BR = R L r R r + I − J 0 −e T r 1 0 −1 P L p R I − U T L −u T R 0 1 = R L r R P L p R D T L 0 0 δ BR = X L x R x + I − J 0 −e T r 1 0 −1 .
Flattening those expressions yields
AP L D T L = R L (I − J) − r R e T r Ap R δ BR = r R − r + P L (I − U T L ) = R L −P L u T R + p R = r R P L D T L = X L (I − J) − x R e T r p R δ BR = x R − x + .
The left-hand side parts are now matched by
R L r R , U T L , P L , D T L , X L x R := CG A, R L r R e 0 0 , X L x R e 0 0 .
Using some of the properties derived with the approach presented in Section 3.2, the equations on the right can be solved to all remaining unknowns. Thus, we obtain a PME with assignments for every unknown quantity. The systematic derivation of loop-based algorithms, using this partitioning, is presented in the next chapter. Note that this PME can be interpreted as an "inductive PME": Assuming it is possible to compute an arbitrary number of previous iterations, it is possible to compute one additional iteration. The previous iterations are represented by the CG function, and the additional iteration is computed using the remaining, explicit assignments of the PME. The base case is obtained by assuming all left and top left parts to be empty.
Derivation of Algorithms for Iterative Methods
After having laid the foundations in the previous chapter, the actual approach for deriving algorithms for iterative methods is presented in this chapter. The approach itself can be found in Section 4.1, followed by two examples in Section 4.2 and 4.3. In the final Section 4.4, scope and limitations of the approach are discussed.
Derivation of Algorithms
The derivation of algorithms for iterative methods mainly follows the same three basic steps as for direct methods. First, one or more PMEs are generated. In the second step, dependency graphs are constructed, which are then used to select loop invariants. In the third and final step, from each loop invariant, an algorithm is constructed.
In this section, we will present this process for iterative methods. There is, however, a new fourth step. In this step, some postprocessing is applied to the derived algorithms to generate a number of variants that may behave differently in floating point arithmetic or vary in their performance. If applicable, we follow the same structure as [6].
PME Generation
The first stage towards the generation of algorithms is to find PMEs. The necessary steps are explained in the following. There are a number of differences compared to direct methods: It is necessary to derive properties of matrices, which then enable the system to solve equations. Additionally, different partitionings are needed to deal with new types of operands.
Derivation of Properties
The approach presented in Section 3.2 is used to derive properties.
Initial Partitioning
Operands of the postcondition are partitioned depending on their shape and properties. For objects that are completely known or unknown, the applicable partitionings are similar to the ones used for direct methods. The difference lies in the sizes of the resulting objects. Only the top, left, and top left parts are matrices, the remaining ones are either vectors or scalars:
B → B B → B L b R B → B T b B B → B T L b T R b BL β BR
Just as with direct methods, triangular or symmetric matrices are either not partitioned at all, or the 2 × 2 partitioning is used. In case of the latter, the top left part is required to be square, such that it inherits the property of the matrix. Separate partitionings are necessary to deal with matrices where initially, only the first column is known. If the last column of those matrices is omitted, the usual 1 × 2 partitioning is applied:
B → B L b R
In case of the complete matrix, an additional column is obtained:
B → B L b R b +
The constant matrices J and I pose a special case. They have one more row than columns, so J is not lower diagonal and I is not an identity matrix. To derive algorithms, it is important to utilize their specific structure, so we will provide the partitionings explicitly:
J → J 0 e T r 0 0 1 I → I 0 0 1 0 0
The same partitioning that is applied to the operands in the postcondition is also applied to the operands in the set of properties. Then, properties of expressions of partitioned operands are derived, similar to how partitioned operands inherit properties. If for example B is partitioned into B L b R , and Diagonal B T B is contained in the set of properties, the system obtains
Diagonal B T L b T R B L b R = Diagonal B T L B L B T L b R b T R B L b T R b R .
Thus, it is possible to derive that B T L B L is diagonal as well, and B T L b R and b T R B L are zero. 44
Derivation of Algorithms
Finding the PME The first part of this step consists of performing symbolic arithmetic and distributing equalities across the partitionings. Consider the Krylov sequence as an example:
K := KS(A, Ke 0 ) ≡ P pre : {Input(A) ∧ Matrix(A) Matrix[J] ∧ LowerDiagonalR[J]∧ FirstColumnInput(K)} P post : {AK = KJ size(K) = n × m}
The partitioned postcondition looks as follows:
A K L k R = K L k R k + J 0 e T r 0 0 1
It can be rewritten as the following expression:
AK L = K L J + k R e T r Ak R = k +
In the second part, the goal is to find a representation of this expression where the value of each unknown quantity is determined by an assignment, using known operations. The quantities on the right-hand side of that assignment either have to be known, or their value is determined by another assignment. Intuitively, one could say the goal is to make this expression computable. This is done by matching patterns of known functions and operations. Rewriting the equation on the left-hand side as
AK L = K L k R J e T r ,
it is easy to see that it describes the computation of a Krylov sequence, so it is possible to use the function from the description of the operation:
K L k R := KS A, K L k R e 0 0
Now, K L and k R can be considered known, so all that remains is to find an assignment for k + . In this example, the equation on the right hand side is already solved to k + , so k + := Ak R is immediately obtained. The PME then is K L k R := KS A, K L k R e 0 0 k + := Ak R .
In general, finding assignments is not that straightforward. Usually, it is necessary to apply known properties. Similar to the "Application of Properties" step in Section 3.2.3, parts of expressions with known properties are multiplied to both sides of equations, with the intention to recreate said expressions. Unfortunately, it is not easy to determine beforehand if the application of a property allows to solve an equation. Thus, in the manner of an exhaustive search, all matching properties are applied. If by this means, multiple assignments for the same expression are found, separate PMEs are derived for each variant.
Remark on Recursive Algorithms
The PMEs for direct methods immediately lead to recursive algorithms [2,6]. This is also true for iterative methods. The difference is that in case of direct methods, those algorithms are usually divide and conquer algorithms. For iterative methods, the PME naturally leads to a "head recursive" implementation, that is, the recursive function call is the first operation in the function body.
Loop Invariant Identification
For the second step, the identification of loop invariants, constraints for the feasibility of loop invariants are introduced that differ from the ones used for direct methods. To understand why they are introduced, it is helpful to get an intuition for what those loop invariants express, and how they differ from the ones for direct methods. The construction of the dependency graph is even simpler compared to direct methods. The assignments of the PME are not decomposed into basic building blocks. Each assignment is represented by one node in the dependency graph. The dependencies are established as usual.
Subset Selection
Just as with direct methods, subsets of nodes of the dependency graph are selected as candidates for loop invariants. Again, for every node that is contained in a subset, all preceding nodes have to be in that set, too. To assess the feasibility of loop invariants, however, slightly modified constraints have to be imposed:
1. There must exist a basic initialization of the operands, that is, an initial partitioning, followed by some preprocessing operations, that renders the predicate P inv true:
{P pre } Partition Preprocessing {P inv } 2
. P inv and the negation of the loop guard, G, must imply the postcondition, P post :
P inv ∧ ¬G ⇒ P post
We begin with discussing the first condition. How the operands are partitioned was already established in the Section "Initial Partitioning", but the initial sizes of the partitioned operands were not specified. The iterative methods covered in this thesis proceed through those matrices where initially, only the first column is known, from the left to the right. As a consequence, the left parts of those matrices are initially empty. The initial block sizes for all partitionings are given in Table 4.1.
Recall that for direct methods, the full set of nodes of the dependency graph can never be a feasible loop invariant. Formally, the reason is that there is no initial partitioning that renders this loop invariant true. Alternatively, one can say that it implies that the complete solution is already computed even before the loop is entered. For iterative methods, this is different. Due to the initial partitioning, which does not expose blocks that represent the remaining iterations, there is no subset implying that the entire solution is already computed. Thus, the full set is a feasible loop invariant.
However, further loop invariant candidates for iterative methods are not rendered true by the initial partitioning for a different reason. This is the case for some subsets that contain more than the first node. The reason is that the additional nodes may Initial Partitioning Dimensions compute quantities that are not empty in the initial partitioning. Consider the Krylov sequence as an example: k R and k + remain n × 1 vectors in the initial partitioning. Thus, the assignment k + := Ak R computes a non-empty quantity, even if the initial partitioning is applied. However, at the beginning of the operation, only k R is known, which is the fist column of K in the initial partitioning, while k + is the second column, which is unknown. Fortunately, the equations from those additional nodes can be rendered true by computing those quantities with some preprocessing operations. Naturally, the necessary preprocessing operations are obtained by applying the initial partitioning to the operations of those nodes. In this example, the preprocessing operation is k + := Ak R . Those preprocessing operations can consist of the same type of operations as the actual update operations.
B → B L b R B L is n × 0 B → B T b R B T is 0 × n B → B T L b T R b BL β BR B T L is 0 × 0 B → B L b R B L is n × 0 B → B L b R b + B L is n × 0
To demonstrate how the first constraint is checked, we return to the example of the Krylov sequence. We start with the following loop invariant candidate:
K L k R := KS A, K L k R e 0 0 =
If K L has size n × 0, the expression on the left-hand side becomes k R := KS (A, k R ). Initially, the first column of K, which is now k R , is known. Thus, since both sides of this assignment are known, this expression is considered to be true. This implies that the loop invariant satisfies the first constraint. The second candidate for a loop invariant is
K L k R := KS A, K L k R e 0 0 k + := Ak R .
Note that in the interest of simplicity, we usually just write K L e 0 instead of
K L k R e 0 0 ,
if it is sufficient. The following expression is obtained if the initial partitioning is applied, where K L is empty:
k R := KS (A, k R ) k + := Ak R
As k + is not known, the initial partitioning alone does not render this expression true. This, however, can be solved with a preprocessing operation. Here, the operation is k + := Ak R . It follows that this loop invariant satisfies the first constraint as well.
To check if the second condition for the feasibility of loop invariants is satisfied, we first need to determine the loop guard. As discussed in Section 3.1, comparing the size of a growing block of a matrix to the size of the entire matrix does not work, as the matrix grows as well. For this reason, the additional predicates were added to the postcondition (by hand), which are now easily translated into loop guards (automatically). How this is done is shown in Note that even though Be T r in the predicate Be T r < ε refers to the last column of B, b R in the loop guard b R ε and B L e T r in B L e T r ε, respectively, are the second to last columns. Similarly, the other loop guards omit the last column as well. To understand why this is necessary, we have to look at the loop invariant candidates again. Just as with direct methods, the empty set can never be a valid loop invariant because it corresponds to the empty predicate. Thus, all remaining candidates contain at least the first node of the dependency graph. For iterative methods, this first node always represents the operation itself, that is, it contains the original function. Which parts of the operands are output of that function depends on the properties of the operands:
Be T r < ε b R ε B L e T r ε size(B) = n × k size B L b R < n × k size (B L ) < n × k Be T r − Be T r−1 < ε b R − B L e T r ε B L e T r − B L e T r−1 ε
-If initially, the first column of B is known (FirstColumnInput[B]), B L and b R are output of the function.
-If B is initially unknown (Output[B]), just B L is output.
Consider nonsymmetric CG as an example: The operation of the first node is R L r R , U T L , P L , D T L , X L x R := CG (A, R L e 0 , X L e 0 ) .
The first columns of R and X, here denoted by R L e 0 and X L e 0 , are initially known and the function computes R L , r R , X L and x R . In contrast, the property of P is Output[P], and only P L is computed. Intuitively, the reason is that the function always computes the same number of columns of all operands. If initially, one column of an operand is already known, in the end, one additional column is obtained.
Since only the first node is guaranteed to be part of the loop invariant, only those parts computed in the first node are guaranteed to be known at the beginning and at the end of the loop. Thus, for FirstColumnInput[B], checking the norm of b R in the loop guard is always possible, but using b + is not.
While there are variants which compute b + (or b R in case of Output[B]), and those would allow different loop guards, we refrain from using those to keep the derivation simple. This, however, means that if b + (or b R ) are computed, they will not be regarded as part of the solution. That is possible because B does not have a fixed size. Unfortunately, this has the effect that some algorithms compute results that are subsequently discarded. If, however, the loop body of those algorithms has a reduced computational complexity, this is an acceptable tradeoff.
We demonstrate how the second condition for the feasibility of loop invariants is checked using the following loop invariant candidate P inv as an example:
R L r R ,
P T L Ar R
The loop guard G is r R ε, so its negation ¬G is r R < ε. Now, it has to be checked whether P inv and ¬G imply the postcondition P post :
APD = R I − J P (I − U) = R PD = X I − J Re T r < ε
To do that, we rewrite P inv as
AP L D T L = R L r R I − J 0 −e T r 1 P L (I − U T L ) = R L P L D T L = X L x R I − J 0 −e T r 1 u T R = − P T L AP L −1 P T L Ar R .
Clearly, by rewriting R L r R as R, P L as P and so on, one can see that the first three equations above are the same as the equations in the postcondition. Furthermore, the negation of the loop guard, r R < ε, refers to the last column of R L r R , just like Re T r < ε refers to the last column of R. Thus, P inv ∧ ¬G implies the postcondition. While P inv ∧ ¬G also implies the equation
u T R = − P T L AP L −1 P T L Ar R ,
this equation is not needed to render the postcondition true. In the algorithm, u T R will be discarded. Note that the initial partitioning exposed one additional column of each operand. R for example was partitioned into R L r R r + . In the postcondition, R just consists of R L and r R . As mentioned before, this is possible because the number of columns of R is variable.
Algorithm Construction
The algorithm is constructed in the third step. In addition to the update, preprocessing operations have to be determined. Furthermore, in line with different rules for the initial partitioning, the repartitioning is modified. The process of identifying the update operations does not change.
Preprocessing
As mentioned before, the preprocessing operations are obtained by applying the initial partitioning (Table 4.1) to all but the initial node contained in the loop invariant. The first node is excluded because when the initial partitioning is applied to it, it always reduces to an expression similar to k R := KS (A, k R ) for the Krylov sequence. Since the output of this function is already known, nothing has to be computed. A FLAME worksheet, extended by the preprocessing, is shown in Figure 4.1.
Derivation of Algorithms for Iterative Methods
Algorithm: . . . The skeleton of a FLAME worksheet, extended by some preprocessing.
P pre Partition Preprocessing { P inv } While G do {( P inv ) ∧ ( G )} Repartition { P before } Update { P after } Continue with { P inv } endwhile {( P inv ) ∧ ¬ ( G )} P post
Repartitioning the Operands
To ensure that the resulting algorithms make progress, the operands have to be repartitioned. The sizes of some operands depend on the number of iterations that is computed, so with every iteration, their sizes have to grow. This is done by adding rows and/or columns in the "Continue with" repartitioning. The rules are shown in Table 4.3.
Initial Partitioning Repartition
Continue with To obtain the predicates P before and P after , the repartitioned operands are plugged into the loop invariant. The resulting expressions are flattened, using the PME if necessary. What is obtained by applying the "Repartition" rules to the loop invariant becomes P before . Applying the "Continue with" partitioning result in P after . We demonstrate this for the loop invariant K L k R := KS (A, K L e 0 ) = .
B L b R B 0 b 1 B 0 b 1 b 2 B T b R B 0 b 1 B 0 b 1 b 2 B T L b T R b BL β BR BB L b R b + B 0 b 1 b 2 B 0 b 1 b 2 b 3
The "Repartition" and "Continue with" rules for K are shown below, introducing the newly added k 3 :
K L k R k + → K 0 k 1 k 2 K L k R k + ← K 0 k 1 k 2 k 3 .
Applying the "Repartition" rules to the loop invariant yields the following predicate P before :
K 0 k 1 := KS (A, K 0 e 0 )
Using the "Continue with" repartitioning, we obtain the expression
K 0 k 1 k 2 := KS (A, K 0 e 0 ) .
To flatten this expression, the PME is used. This results in the following P after :
K 0 k 1 := KS (A, K 0 e 0 ) k 2 := Ak 1
Finding the Updates
The difference between P after and P before now is the update. Identifying the differences is much easier compared to direct methods because they are always entire equations, not subexpression. The reason is that no quantities are updated. For the Krylov sequence, k 2 := Ak 1 can easily be identified as the difference between P after and P before .
Refinement
In practice, algorithms would not, and depending on the language, can not be implemented exactly like they are derived with the presented approach. Consider the following three assignments as an example. They are part of the update of one nonsymmetric CG algorithm. To translate the assignments above to C for example, they have to be decomposed into basic operations that are implemented in a library like BLAS. Since such libraries usually do not include functions for general products of more than two quantities, auxiliary variables have to be introduced. While the assignments could immediately be translated into Matlab code, this would not result in efficient code. Clearly, some subexpressions appear multiple times, so it is preferable to introduce auxiliary variables for those and compute their values just once. This is referred to as common subexpression elimination. While this concept is well known in the domain of compiler construction [10], to the author's knowledge there is no research on the elimination of overlapping common subexpressions. Overlapping common subexpression are common subexpression that can not be eliminated at the same time. Consider the three terms P T 0 AP 0 , p T 1 Ar 2 and p T 1 AP 0 for a simple example. P T 0 AP 0 and p T 1 AP 0 have AP 0 in common, p T 1 Ar 2 and p T 1 AP 0 share p T 1 A. In p T 1 AP 0 , those two common subexpression overlap since A is part of both. Finding a replacement of subexpressions that is optimal in the sense that it has the lowest computational cost is not trivial. Inspecting the problem, one observes that a simplified version of it can be mapped to a maximum weight matching problem, which is solvable in polynomial time [3]. Again, consider the terms P T 0 AP 0 , p T 1 Ar 2 and p T 1 AP 0 as an example. Every expression becomes a node in a graph. Every possible replacement of a common subexpression is represented by an edge. The resulting graph is shown in Figure 4 cost of the expression that is replaced. The problem then is to find a set of edges, such that each node is attached to at most one of the selected edges. At the same time, the sum of the weights should be minimized. The requirement that each node is attached to at most one of the selected edges represents the fact that multiple replacements are not possible because the expressions overlap. Clearly, the actual problem is more complex. A common subexpression might be replaced in more than two expressions. A graph representing this is a hypergraph. In addition to that, in longer expressions, some subexpressions do not overlap, so they can both be replaced.
In practice, trying to solve this problem is probably not necessary. The expressions encountered in most iterative methods are rarely as complex as in the example above. Furthermore, simple heuristics may already produce good results. In products of more than two quantities, matrix-vector products should always be computed first. Then, if one subexpression is replaced, all other occurrences of this expression should be replaced as well.
Alternatively, to generate as many variants as possible, all possible replacements could be constructed.
Example: Nonsymmetric CG
In this section, as a more elaborate example, we will show how the derivation of an algorithm for nonsymmetric CG proceeds.
PME Generation
The derivation of some properties for nonsymmetric CG was already shown in Section 3.2.5 and will not be repeated here. The postcondition is shown below:
APD = R I − J P (I − U) = R PD = X I − J Re T r < ε
The initial partitioning is determined based on the properties of the operands. It is applied to the postcondition as well as all derived properties. The expression that is obtained from the postcondition is flattened, yielding
AP L D T L = R L (I − J) − r R e T r Ap R δ BR = r R − r + P L (I − U T L ) = R L −P L u T R + p R = r R P L D T L = X L (I − J) − x R e T r p R δ BR = x R − x + .
The left-hand side is now matched by the CG function, so
R L r R , U T L , P L , D T L , X L x R := CG (A, R L e 0 , X L e 0 )
is obtained. Thus, all quantities on the left-hand side of that assignment are considered to be known. To find assignments for the remaining unknown quantities, the three equations on the right have to be solved. One of the derived properties is that P T AP is lower triangular, so P T L AP L is lower triangular as well and P T L Ap R is zero. The system would recognize that both P L and p R appear in −P L u T R + p R = r R on the left-hand side of products. Hence, P T L A is multiplied from the left to both sides of the equation to recreate those properties. The resulting equation −P T L AP L u T R = P T L Ar R contains only one unknown quantity, so it is solvable. Since P T L AP L is lower triangular, a triangular system is identified, which can also be written as
u T R := − P T L AP L −1 P T L Ar R .
Having found an assignment for u T R , it is considered known as well. By instead using that R T AP is lower triangular, a different equation for u T R would have been found, resulting in a different algorithm. In practice, two separate derivation processes would be executed for both variants; here we continue just with the first one. Now, there is only one unknown quantity left in −P L u T R + p R = r R , so the following formula is determined for p R : p R := r R + P L u T R Because r T R r + is zero, r T R is multiplied from the left to both sides of Ap R δ BR = r R − r + . There is only one unknown in the resulting equation r T R Ap R δ BR = r T R r R , so another assignment is found: δ BR := r T R r R r T R Ap R Alternatively, the fact that P T R is lower triangular and rectangular could be used. Finally, Ap R δ BR = r R − r + and p R δ BR = x R − x + can be solved to r + and x + , respectively, completing the PME: R L r R , U T L , P L , D T L , X L x R := CG (A, R L e 0 , X L e 0 ) u T R := − P T L AP L −1 P T L Ar R p R := r R + P L u T R δ BR := r T R r R r T R Ap R r + := r R − Ap R δ BR
x + := x R − p R δ BR
Loop Invariant Identification
Based on this PME, the dependency graph is constructed. Since there are six assignments in the PME, there are six nodes: To determine the loop guard G, the additional predicate in the postcondition is inspected. It is Re T r < ε. Since the precondition contains FirstColumnInput[R], according to Table 4.2, the loop guard r R ε is selected.
In this example, we show the derivation for the set {1, 2, 3}, so the loop invariant is P inv = R L r R , U T L , P L , D T L , X L x R := CG (A, R L e 0 , X L e 0 ) ∧ u T R := − P T L AP L −1 P T L Ar R ∧ p R := r R + P L u T R .
Algorithm Construction
In a first step, the preprocessing operations are determined. The relevant assignments are the ones contained in the loop invariant, except for the one obtained from the first node. They are shown below:
u T R := − P T L AP L −1 P T L Ar R p R := r R + P L u T R
According to the initial partitioning, P L is empty, that is, it has the size n × 0. Consequently, the right-hand side of the first assignment is empty, too, so it disappears. The second assignment reduces to
p R := r R ,
which is the only preprocessing operation. Next, the update is derived. The "Repartition" rules are determined using Table 4.3:
R L r R r + → R 0 r 1 r 2 X L x R x + → X 0 x 1 x 2 U T L u T R 0 0 → U 00 u 01 0 0 D T L 0 0 δ BR → D 00 0 0 δ 11 P L p R → P 0 p 1
Applying those rules to the loop invariant yields the following predicate P before : P before = R 0 r 1 , U 00 , P 0 , D 00 , X 0 x 1 := CG (A, R 0 e 0 , X 0 e 0 ) ∧ u 01 := − P T 0 AP 0 −1 P T 0 Ar 1 ∧ p 1 := r 1 + P 0 u 01
The "Continue with" repartitioning is R L r R r + ← R 0 r 1 r 2 r 3
X L x R x + ← X 0 x 1 x 2 x 3 U T L u T R 0 0 ← U 00 u 01 u 02 0 0 ν 12 0 0 0 D T L 0 0 δ BR ← D 00 0 0 0 δ 11 0 0 0 δ 22 P L p R ← P 0 p 1 p 2 .
Plugging that into the function, the following expression is obtained:
R 0 r 1 r 2 , U 00 u 01 0 0 , P 0 p 1 ,
D 00 0 0 δ 11 , X 0 x 1 x 2 := CG (A, R 0 e 0 , X 0 e 0 )
It it flattened by using the PME, resulting in six assignments:
R 0 r 1 , U 00 , P 0 , D 00 , X 0 x 1 := CG (A, R 0 e 0 , X 0 e 0 ) u 01 := − P T 0 AP 0 −1 P T 0 Ar 1 p 1 := r 1 + P 0 u 01 δ 11 := r T 1 r 1 r T 1 Ap 1 r 2 := r 1 − Ap 1 δ 11
x 2 := x 1 − p 1 δ 11
For the second assignment of the loop invariant, the PME of a lower triangular system is needed (see Section 2.1). Algorithm: nonsymmetric CG
Partition R → R L r R r + where R L is n × 0 p R := r R While r R ε do
Repartition R L r R r + → R 0 r 1 r 2 t 1 := Ap 1 δ 11 := r T 1 r 1 r T 1 t 1 r 2 := r 1 − t 1 δ 11 x 2 := x 1 − p 1 δ 11 t 2 := Ar 2 u 02 := −P T 0 AP 0 −1 · P T 0 t 2 t 3 := P 0 u 02 ν 12 := − p T 1 t 2 + p T 1 At 3 p T 1 t 1 p 2 := r 2 + t 3 + p 1 ν 12 Continue with R L r R r + ← R 0 r 1 r 2 r 3 endwhile section, we now proceed at a slightly higher pace. Pre-and postcondition are shown below.
P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Output[P] ∧ Matrix[P]∧ Output[P] ∧ Matrix[P]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ Diagonal[R TR ] ∧ Diagonal[R TR ]∧ DiagonalR[R TR ] ∧ DiagonalR[R TR ]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Output[U] ∧ Matrix[U] ∧ UpperDiagonal[U]∧ Matrix[I − J] ∧ LowerTrapezoidal[I − J]} P post : {APD = R I − J A TP D =R I − J P (I − U) = R P (I − U) =R PD = X I − J Re T r < ε}
Following from the precondition, the function representing the operation is R,R, U, P,P, D, X := BiCG (A, Re 0 , Xe 0 ) .
Derivation of Properties
For BiCG, neither R norR is orthogonal. Instead, they are mutually orthogonal, which means that R TR is diagonal. Except for that, the derivation of properties is mostly the same to the one shown in the example in Section 3.2.5. For this reason, it will not be shown here. Instead. we give a short overview of the relevant properties:
-R T AP and R T A TP are lower triangular.
-P T AP (and thus P T A TP ) is diagonal.
-P TR andP T R are lower triangular and rectangular.
PME Generation
As a first step towards the PME, the operands are partitioned. Initially, only the fist columns of R andR are known, so they are partitioned into R L r R r + and R LrRr+ , respectively. The remaining operands are partitioned accordingly:
A P L p R D T L 0 0 δ BR = R L r R r + I − J 0 −e T r 1 0 −1 A T P LpR D T L 0 0 δ BR = R LrRr+ I − J 0 −e T r 1 0 −1 P L p R I − U T L −u T R 0 1 = R L r R P LpR I − U T L −u T R 0 1 = R LrR P L p R D T L 0 0 δ BR = X L x R x + I − J 0 −e T r 1 0 −1
After the execution of the Matrix Arithmetic step, those expressions become
AP L D T L = R L (I − J) − r R e T r Ap R δ BR = r R − r + A TP L D T L =R L (I − J) −r R e T r Ap R δ BR =r R −r + P L (I − U T L ) = R L −P L u T R + p R = r R P L (I − U T L ) =R L −P L u T R +p R =r R P L D T L = X L (I − J) − x R e T r p R δ BR = x R − x + .
The left-hand sides of those expressions can now be replaced by the BiCG function:
R L r R , R LRR , U T L , P L ,P L , D T L , X L x R := BiCG (A, R L e 0 , X L e 0 )
Updates for the remaining unknown quantities are obtained by solving the equations on the right-hand using the derived properties. For this example, to find an assignment for u T R , we use thatP T AP is diagonal. MultiplyingP T L A from the left to both sides of −P L u T R + p R = r R results in −P T L AP L u T R =P T L Ar R . SinceP T L AP L is diagonal, and thus is invertible, the following assignment is obtained:
u T R := − P T L AP L −1P T L Ar R
Note that it would have been also possible to use −P L u T R +p R =r R to get to an assignment for u T R . Now, the following assignments are obtained for p R andp R :
p R := r R + P L u T R p R :=r R +P L u T R
Similarly to u T R , δ BR can be computed in several different ways. Here, we use that P T R is lower triangular and rectangular, in combination with the equation Ap R δ BR = r R − r + . Multiplyingp T R from the left and solving to δ BR yields δ BR :=p T R r R p T R Ap R Finally, the following assignments are derived for r + ,r + and x + :
r + := r R − Ap R δ BR r + :=r R − Ap R δ BR x + := x R − p R δ BR
The complete PME is shown below (already in form of a list, in anticipation of the construction of the dependency graph):
1. R L r R , R LRR , U T L , P L ,P L , D T L , X L x R := BiCG (A, R L e 0 , X L e 0 ) 2. u T R := − P T L AP L −1P T L Ar R 3. p R := r R + P L u T R 4.p R :=r R +P L u T R 5. δ BR :=p T R r R p T R Ap R 6. r + := r R − Ap R δ BR 7.r + :=r R − Ap R δ BR 8. x + := x R − p R δ BR
Loop Invariant Identification
The dependency graph is shown in Figure 4.5. Of all the subsets of this graph that respect the dependencies, only the empty one fails to satisfy the conditions for the feasibility of loop invariant. The remaining 13 subsets are feasible loop invariants. We will continue the derivation in this example with the loop invariant that corresponds to the full set.
Because of the additional predicate in the postcondition, Re T r < ε, and the property FirstColumnInput [R] in the precondition, the loop guard r R < ε is determined.
Algorithm Construction
The first part of this step consists of finding the preprocessing operations. This is done by taking all assignments of the loop invariant except for the one from the first node, and eliminating all expression that are empty in the initial partitioning. The relevant assignments are shown below:
u T R := − P T L AP L −1P T L Ar R p R := r R + P L u T R p R :=r R +P L u T R δ BR :=p T R r R p T R Ap R r + := r R − Ap R δ BR r + :=r R − Ap R δ BR x + := x R − p R δ BR
Initially, P L andP L have the size n × 0 and u T R is of size 0 × 1. As a result, the first assignment is eliminated, while the second and third are reduced to p R := r R p R :=r R .
Derivation of Algorithms for Iterative Methods
The remaining four assignments do not change.
To determine the update, the predicates P before and P after have to be constructed by repartitioning the loop invariant. The "Repartition" rules are shown below:
R L r R r + → R 0 r 1 r 2 R LrRr+ → R 0r1r2 U T L u T R 0 0 → U 00 u 01 0 0 D T L 0 0 δ BR → D 00 0 0 δ 11 P L p R → P 0 p 1 P LpR → P 0p1 X L x R x + → X 0 x 1 x 2
Applying that to the loop invariant yields the following equations which constitute P before : For the "Continue with" repartitioning it is important to note that U is upper diagonal. Thus, u 02 is zero:
RR L r R r + ← R 0 r 1 r 2 r 3 R LrRr+ ← R 0r1r2r3 U T L u T R 0 0 ← U 00 u 01 0 0 0 ν 12 0 0 0 D T L 0 0 δ BR ← D 00 0 0 0 δ 11 0 0 0 δ 22 P L p R ← P 0 p 1 p 2 P LpR ← P 0p1p2 X L x R x + ← X 0 x 1 x 2 x 3
This repartitioning transforms R L r R , R LRR , U T L , P L ,P L , D T L , X L x R := BiCG (A, R L e 0 , X L e 0 )
into those assignments that are also contained in the predicate P before . Applying it to Now that the P before and P after are completely determined, the update is found by identifying those assignments that are contained in P after , but not in P before . For this example, no common subexpressions are replaced. The worksheet for this algorithm, together with a second one for the algorithm obtained from the set that only contains the first node, is shown in Figure 4.6.
u
Remark on the Equivalence of Loop Invariants and Algorithms
Comparing those algorithms, one notes that the updates are very similar. In fact, according to the criteria established in Section 2.4, those algorithms are considered equivalent. With a suitable replacement, the loop invariant for one can be transformed into the loop invariant of the other. The differences in the shape of some assignments stem from the fact that u 01 is zero except for the last position. In the algorithm on the left in Figure 4.6, this is revealed by the repartitioning, while this is not the case on the right. In general, one observes that the presented approach always produces two equivalent loop invariants, namely the subset of the dependency graph that only contains the first node, and the full set. For direct methods, the coarsest possible, that is, the standard 2 × 2 partitioning, never results in equivalent loop invariants. Finer partitionings result in new loop invariants than can not be found with a coarser one.
For the iterative methods covered in this thesis, this is different. The partitioning that is used for the derivation is the coarsest possible one, as a coarser one would only partition those matrices that are initially partially known. All other matrices would not be partitioned at all, so it would not be possible to derive a PME. On the other hand, for quite a few of the methods presented in Appendix B, a finer partitioning does not result in new loop invariants. This is the case for those methods where either there is no matrix U or it is upper diagonal. The reason is that for those methods, finer partitionings do not partition quantities that are computed in one iteration in multiple parts. All they do is expose additional quantities that are fully computed in different iterations.
Scope and Limitations
The presented method extends to a lot more iterative methods than those shown as examples throughout the thesis. Matrix representations of further methods are shown in Appendix B. Note that this includes stationary iterative methods (B.2). For those, the derivation is even simpler because no properties have to be derived.
With the presented approach, it is possible to derive a large number of algorithms for most iterative methods. In case of CG for example, by using the derived properties to solve equations in different ways, four PMEs are found. From each PME, seven loop invariants are obtained, resulting in 28 algorithms. Another considerable factor is added by the elimination of common subexpressions. Similar numbers can be expected for other, comparably complex iterative methods.
There are, however, some limitations. They are explained in the following.
Normalized Vectors
Some iterative methods construct a set of orthonormal vectors. With the presented approach, it is not possible to derive algorithms for those methods. Consider the Arnoldi iteration as an example. The matrix representation is shown below. It is based on the description in [11]: The postcondition is repartitioned as follows:
{QA Q L q R = Q L q R q + H T L h T R h ML η MR 0 η BR
After that, the Matrix Arithmetic step yields the following expression:
AQ L = Q L H T L + q R h ML Aq R = Q L h T R + q R η MR + q + η BR As usual, the left-hand side is matched by the function:
Q L q R , H T L h ML := AI(A, Q L e 0 )
Using that Q is orthonormal, the following assignments are obtained for h T R and η MR : h T R := Q T L Aq R η MR := q T R Aq R
The problem is now to compute the normalized vector q + and η BR . From Aq R = Q L h T R + q R η MR + q + η BR , the following equation can be obtained, where all quantities on the right-hand side are known:
q + η BR = Aq R − Q L h T R − q R η MR
Since we know that q + is normalized, the scalar η BR has to be computed as
η BR := Aq R − Q L h T R − q R η MR . (4.2)
Then, q + is determined by the following assignment:
q + := Aq R − Q L h T R − q R η MR η BR
With the presented method of solving equations by applying properties, the assignment (4.2) can not be obtained. To ensure that algorithms for such methods can be derived, the presented approach for solving equations must be expanded.
Conclusion
This thesis introduces a methodology that allows the systematic derivation of algorithms for iterative methods; the starting point for this methodology is a formal description of an iterative method in matrix form. In addition, we presented an approach for deriving properties of matrices and matrix expressions from this representation; those properties are necessary for the derivation of algorithms. The actual derivation of algorithms consists of four major steps. First, PMEs are generated by partitioning the operands of the matrix representation and applying the derived properties to solve equations. Then, from those PMEs, loop invariants are obtained. In the third step, from each loop invariant, one loop-based algorithm is constructed. Finally, common subexpressions are eliminated, generating an even larger number of algorithms.
One of the most important aspects, and indispensable for the automatic generation of libraries, is that the derived algorithms are provably correct. This is ensured by constructing them around a proof of correctness, based on the loop invariants generated in the second step.
A conscious effort was made to ensure that the entire process is systematic, that is, each step is performed according to well defined rules and no guidance by a human expert is required. This allows the approach to be implemented as a tool that automatically generates algorithms based on a formal description of the operation. We consider this to be another important step towards the automatic generation of linear algebra libraries as envisioned by the founders of the FLAME project.
As for future work, there are a number of ways to build on the results of this thesis: Implementation Executing the presented approach by hand is a laborious and thus error-prone task, not least because it was not designed to be executed by hand.
To be used productively, the presented approach should be implemented as a computer program.
Stability Analysis
To asses the usefulness of the derived algorithms in practice, a stability analysis is indispensable. In [2], it was shown that the FLAME methodology can be combined with a systematic stability analysis. The presented method should be extended in a similar way.
Performance Analysis While it is desirable to derive a large number of algorithms to find new, potentially faster variants, the task of identifying them should not be left to the user. Thus, similar to a systematic stability analysis, the system should also be able to reason about the performance of the generated algorithms and select the best ones.
Matrix Representations
In this thesis, only a small number of matrix representations for iterative methods is presented. Clearly, it would be desirable to find representations of many more methods. Additionally, it might be interesting to find out if this representation reveals new insights about different iterative methods and their relations to each other.
In the following, we define those matrix properties used throughout the thesis that are not self-explanatory. Let A ∈ R n×m be a matrix. The elements of this matrix are denoted as a ij with i ∈ {0, . . . , n − 1} and j ∈ {0, . . . , m − 1}.
-Upper diagonal (UpperDiagonal): a ij = 0 for i + 1 = j with n = m. Consider the matrix below as an example.
B.2. Stationary Iterative Methods
Let Ax = b be a linear system. e is a column vector where all entries are one. We write A as A = D − L − U, where D contains the entries on the main diagonal of A, L contains the entries below the main diagonal and U the ones above the main diagonal. The representations for the Gauss-Seidel, Jacobi and Successive Overrelaxation method are based on the descriptions (in indexed notation) in [1]. The one for the Richardson iteration is based on the the description (in indexed notation) in [5].
B.2.1. Gauss-Seidel Method
B.2.2. Jacobi Method
The precondition of the Jacobi method is identical to the one for the Gauss-Seidel method.
Figure 2 .
21.: The skeleton of a FLAME worksheet.
Figure 2 .
22.: Dependency graph for a lower triangular system.
Figure 2 .
23.: Worksheet for a lower triangular linear system.
Figure 2
2.4.: Dependency graph for a SPD system.
For SPD linear systems, there are additional subsets that fail to satisfy the second constraint. Let us look at {1} for example. The corresponding loop invariant is P inv = {Z BL := BL −1 T L }. The negation of the the loop guard "size(A BR ) < size(A)" implies that A BR is equal to A. Then, BL and T L are empty, and the empty predicate does not imply the postcondition. If we take the subset {1, 2}, the loop invariant becomes {A :=Â} at the end of the operation, which does not imply the postcondition either. Doing this for all subsets, two feasible loop invariants are obtained: 1. {1, 2, 3, 4} (P 1 inv )
The replacements (2.2) and(2.3) are easily seen to be correct, as all parts have size bi × bi. In case of (2.4) and (2.5), the left-hand sides have the dimensions (bj + c) × b, and the right-hand sides (bj + c + b) × b. The number of rows matches because both are a multiple of the block size b, plus a constant c. Similarly, the last replacement (2.6) is valid, as the sizes (bj + c) × (bj + c) and (bj + c + b) × (bj + c + b) are equivalent.
step consists of initializing P and E. The former contains all the properties of the precondition P pre that describe operand types. P = {Matrix[A], NonSingular[I − J], LowerTrapezoidal[I − J]} E contains the equations of the postcondition P post . E = {APD = R I − J , P (I − U) = R, PD = X I − J } Derivation of Properties During this step, only a small number of new properties can be derived. The implication
D T L = R L (I − J) − r M e T r Ap M δ MM = r M − R R e 0 AP R D BR = R R I − J P L (I − U T L ) = R L −P L u T M + p M = r M −P L U T R − p M u MR + P R (I − U BR ) = R R Now, a similar rewriting as in the previous section yields AP L D T L = R L the constant matrix has one more row than columns, and R L r M has one more column than R L in P L (I − U T L ) = R L . Hence, those equations, together with the additional one for X, are matched by the pattern of the CG function. equations in the middle are not matched by the function, they can be solved to u T M , p M , δ MM and R R e 0 , leading to computable assignments. Unfortunately, the function does not match the equations on the right-hand side. For direct methods, in a comparable situation, an equation like −P L U T R − p M u MR + P R (I − U BR ) = R R would be rewritten as
Loop
Invariants for Iterative Methods For iterative methods, there is usually a (partial) ordering in which parts of different operands can be computed. Let us assume there are three matrices B, C and D. During the execution of the algorithm, the following sequence is computed, where b k , c k and d K are columns of B, C and D, respectively:.. . b i c i d i b i+1 c i+1 d i+1 . . .A variant derived from one loop invariant may compute c i d i b i+1 in one iteration. In contrast, one derived from another loop invariant may compute d i b i+1 c i+1 . Since no quantities are overwritten, a loop invariant only expresses at which point in the sequence above an iteration starts. In contrast, different loop invariants for direct methods may result in algorithms that compute the solution by row or by column.
U T L , P L , D T L , X L x R := CG (A, R L e 0 ,
Figure 4 .
41.:
. 2 .
2The weight of each edge would be the computational
Figure 4 .
42.: Elimination of common subexpressions represented as a graph.
Figure 4 .
43.: Dependency graph for nonsymmetric CG.1.R L r R , U T L , P L , D T L , X L x R := CG (A, R L e 0 , r + := r R − Ap R δ BR 6. x + := x R − p R δ BRThe corresponding dependency graph is shown inFigure 4.3. All nonempty subsets of this graph that respect the dependencies are feasible loop invariants. Thus, the following seven loop invariants are obtained: {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 4, 5, 6}
Figure 4 .
44.: Worksheet for a nonsymmetric CG algorithm.
Figure 4 .
45.: Dependency graph for BiCG.
0 r 1
1, R 0R1 , U 00 , P 0 ,P 0 , D 00 , X 0 x 1 := BiCG (A,
12 . Because of p R := r R + P L u T R p R :=r R +P L u T R , the following two expressions are obtained and added to P after : p 2 := r 2 + p 1 ν 12 p 2 :=r 2 +p 1 ν 12For the remaining assignments of the loop invariant, only the indices change:
,
H} := AI(A, Qe 0 ) H] ∧ Matrix[H] ∧ UpperHessenberg[H]} P post : {AQ = QH size(Q) = n × m}
--
Lower diagonal (LowerDiagonal): a ij = 0 for i − 1 = j with n = m. -Diagonal and rectangular (DiagonalR): a ij = 0 for i = j with n = m. -Upper diagonal and rectangular (UpperDiagonalR): a ij = 0 for i + 1 = j with n = m. -Lower diagonal and rectangular (LowerDiagonalR): a ij = 0 for i − 1 = j with n = m. -Upper trapezoidal (UpperTrapezoidal): a ij = 0 for i > j with n < m. Thus, the following matrix is upper trapezoidal: Lower trapezoidal (LowerTrapezoidal): a ij = 0 for i < j with n > m. -Upper triangular and rectangular (UpperTriangularR): a ij = 0 for i > j with n > m. An example of such a matrix is shown below. chapter, a collection of matrix representations of iterative methods is provided. The representations for the Krylov sequence, Steepest Descent, symmetric CG and nonsymmetric CG are modifications of the ones introduced in [5]. The differences lie in the use of the underline. The representation for BiCG is based on the one for symmetric CG. B.1. Krylov Subspace Methods B.1.1. Krylov Sequence P pre : {Input(A) ∧ Matrix(A)∧ Matrix[J] ∧ LowerDiagonalR[J]∧ FirstColumnInput(K)} P post : {AK = KJ size(K) = n × m}
− J] ∧ LowerTrapezoidal[I − J]} P post : {APD = R I − J A TP D =R I − J P (I − U) = RP (I − U) =R PD = X I − J Re T r < ε}
Table 2 .
21.: Possible partitionings for a general linear system.
Table 4 .
41.: Initial sizes of partitioned operands.
Table 4 .2.
4Loop guard
Table 4 .
42.: Look-up table for determining loop guards for iterative methods. The row
is selected according to the additional predicate in the postcondition. The
column is selected depending on the property of the operand that appears
in the position of B in that predicate. Example: The predicate is Re T
r < ε.
The precondition contains the property FirstColumnInput[R]. Thus, the
loop guard is r R
ε. To allow for other loop guards, this table has to be
extended manually.
Table 4 .
43.: "Repartition" and "Continue with" rules for iterative methods.
52
B.1.2. Steepest DescentP pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R]∧ ZeroDiagonal[R T RJ] ∧ ZeroDiagonal[R T RJ T ]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Matrix[I − J] ∧ LowerTrapezoidal[I − J]} P post : {ARD = R I − J RD = X I − J Re T r < ε}B.1.3. Conjugate Gradient (symmetric)P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A] ∧ Symmetric[A]∧ Output[P] ∧ Matrix[P]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ DiagonalR[R T R] ∧ DiagonalR[R T R]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Output[U] ∧ Matrix[U] ∧ UpperDiagonal[U]∧ Matrix[I − J] ∧ LowerTrapezoidal[I − J]} P post : {APD = R I − J P (I − U) = R PD = X I − J ReT r < ε} B.1.4. Conjugate Gradient (nonsymmetric) P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Output[P] ∧ Matrix[P]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ FirstColumnInput[R] ∧ Matrix[R] ∧ Orthogonal[R]∧ DiagonalR[R T R] ∧ DiagonalR[R T R]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Output[U] ∧ Matrix[U] ∧ StrictlyUpperTriangular[U]∧ Matrix[I − J] ∧ LowerTrapezoidal[I − J]} P post : {APD = R I − J P (I − U) = R PD = X I − J Re T r < ε} B.2. Stationary Iterative Methods B.1.5. BiCG P pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Output[P] ∧ Matrix[P]∧ Output[P] ∧ Matrix[P]∧ Output[D] ∧ Matrix[D] ∧ Diagonal[D]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ FirstColumnInput[R] ∧ Matrix[R]∧ Diagonal[R TR ] ∧ Diagonal[R TR ]∧ DiagonalR[R TR ] ∧ DiagonalR[R TR ]∧ FirstColumnInput[X] ∧ Matrix[X]∧ Output[U] ∧ Matrix[U] ∧ UpperDiagonal[U]∧ Matrix[I
P pre : {Input[D] ∧ Matrix[D] ∧ Diagonal[D]∧ Input[L] ∧ Matrix[L] ∧ LowerTriangular[L]∧ Input[U] ∧ Matrix[U] ∧ UpperTriangular[U]∧ Input[b] ∧ Vector[b]∧ FirstColumnInput[X] ∧ Matrix[X]}P post : {(D − L)XJ = UX + be TXe T
r − Xe T
r−1 < ε}
P post :
post{DXJ = (L + U)X + be T P pre : {Input[D] ∧ Matrix[D] ∧ Diagonal[D]∧ Input[L] ∧ Matrix[L] ∧ LowerTriangular[L]∧ Input[U] ∧ Matrix[U] ∧ UpperTriangular[U]∧ Input[b] ∧ Vector[b]∧ Input[ω] ∧ Scalar[ω] FirstColumnInput[X] ∧ Matrix[X]} P post : {(D − ωL)XJ = (ωU + (1 − ω)D)X + ωbe T Xe T r − Xe T r−1 < ε}B.2.4. Richardson IterationP pre : {Input[A] ∧ Matrix[A] ∧ NonSingular[A]∧ Input[b] ∧ Vector[b]∧ Input[α] ∧ Scalar[α] FirstColumnInput[X] ∧ Matrix[X]∧ Output[R] ∧ Matrix[R]} P post : {αR = X I − J R = AX − be TXe T
r − Xe T
r−1 < ε}
B.2.3. Successive Overrelaxation Method
Re T
r < ε}
This property is defined in Appendix A.
It is not uncommon to combine multiple criteria[1]. For the sake of simplicity, we do not use more than one at a time.
This is why we refrain from adding e ′ to E in "Application of Properties", step 3.
Except for the ordering of input and output.
AcknowledgmentsI would like to thank Paolo Bientinesi and Diego Fabregat Traver for their support, guidance, and taking their time for all those lengthy discussions. I want to thank Georg May for agreeing to be the second supervisor of this thesis. Last but not least, I thank Friederike and Robert for proofreading.Derivation of Algorithms for Iterative MethodsThe third equation, p 1 := r 1 + P 0 u 01 , becomes p 2 := r 2 + P 0 u 02 + p 1 ν 12 after the application of the "Continue with" partitioning. Now that the complete predicate P after is determined, the update is found by comparing it to P before . The assignments that are contained in P after , but not in P before , constitute the update. They are shown below. δ 11 := r T 1 r 1 r T 1 Ap 1 r 2 := r 1 − Ap 1 δ 11· P T 0 Ar 2 ν 12 := − p T 1 Ar 2 + p T 1 AP 0 u 02 p T 1 Ap 1 p 2 := r 2 + P 0 u 02 + p 1 ν 12PostprocessingFor this example, we use the heuristics explained in Section 4.1.4 for the elimination of common subexpressions. The first expression that is identified is Ap 1 . The auxiliary variable t 1 := Ap 1 is introduced and all occurrences of Ap 1 are replaced with t 1 . The resulting assignments are shown below. · P T 0 Ar 2 ν 12 := − p T 1 Ar 2 + p T 1 AP 0 u 02 p T 1 t 1 p 2 := r 2 + P 0 u 02 + p 1 ν 12Further auxiliary variables are introduced for Ar 2 and P 0 u 02 . The update that is obtained at the end of this step is shown in the filled out worksheet inFigure 4Example: BiCGAs a second example, we show the derivation of two algorithms for the biconjugate gradient method (BiCG). Since we already showed a full example in the previous Continue with R L r R r + ← R 0 r 1 r 2 r 3 endwhile In some cases, the presented method is not able to generate those assignments that are commonly found in literature. This is for example the case for symmetric CG. The derived updates for ν 12 always have a shape like this:Algorithm: BiCGUsually, the following formula is used[1,11,13]:The advantage is that the matrix-vector product p T 1 A is eliminated. It is obtained as follows. We begin with rewriting r 2 = r 1 − Ap 1 δ 11 , which is the update for r 2 , as Ap 1 = (r 1 − r 2 )δ −1 11 . Then, both sides of this equation are transposed, resulting in p T 1 A = δ −1 11 (r 1 − r 2 ) T . Now, this equation is used to replace p T 1 A in the numerator of equation (4.1): By itself, this transformation is not particularly difficult. Since all steps follow well defined algebraic rules, it is not even difficult to design a system that is able to perform the individual steps of this rewriting. The problem is that based on the initial equation (4.1), there is no indication that it is possible to reduce the number of matrixvector products. Even if that is known, there is no indication which steps to perform. Thus, any system that is supposed to rewrite the assignment has to perform some sort of exhaustive search. Unfortunately, since expressions are substituted, the search space is infinite. Heuristics have to be applied to guarantee the termination of this search. However, they must not limit the capability of the system to find such rewritings.
Lower triangular and rectangular (LowerTriangularR): a ij = 0 for i < j with n < m. -Elements on the diagonal are zero (ZeroDiagonal): a ij = 0 for i = j with n = m. Lower triangular and rectangular (LowerTriangularR): a ij = 0 for i < j with n < m. -Elements on the diagonal are zero (ZeroDiagonal): a ij = 0 for i = j with n = m.
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Richard Barrett, Michael W Berry, Tony F Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, Henk A Van Der, Vorst, 43SiamRichard Barrett, Michael W. Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk A. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for It- erative Methods, volume 43. Siam, 1994.
Mechanical Derivation and Systematic Analysis of Correct Linear Algebra Algorithms. Paolo Bientinesi, Department of Computer Sciences, University of Texas at AustinPh.D. ThesisPaolo Bientinesi. Mechanical Derivation and Systematic Analysis of Correct Linear Algebra Algorithms. Ph.D. Thesis, Department of Computer Sciences, University of Texas at Austin, July 2006.
Maximum Matching and a Polyhedron with 0, l-Vertices. Jack Edmonds, J. Res. Nat. Bur. Standards B. 69Jack Edmonds. Maximum Matching and a Polyhedron with 0, l-Vertices. J. Res. Nat. Bur. Standards B, 69(1965):125-130, 1965.
FLAME Derivation of CG Variants. Victor Eijkhout, Paolo Bientinesi, Robert Van De Geijn, The University of Texas at Austin. Technical reportVictor Eijkhout, Paolo Bientinesi, and Robert van de Geijn. FLAME Derivation of CG Variants. Technical report, Texas Advanced Computing Center, The Uni- versity of Texas at Austin, 2010.
Proof-Driven Derivation of Krylov Solver Libraries. Victor Eijkhout, Paolo Bientinesi, Robert Van De Geijn, AICES- 2010/06-3Aachen Institute for Computational Engineering Science, RWTH AachenTechnical ReportVictor Eijkhout, Paolo Bientinesi, and Robert van de Geijn. Proof-Driven Deriva- tion of Krylov Solver Libraries. Technical report, Aachen Institute for Computa- tional Engineering Science, RWTH Aachen, June 2010. Technical Report AICES- 2010/06-3.
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|
[] |
[
"Doping evolution of the phonon density of states and electron-lattice interaction in Nd 2−x Ce x CuO 4+δ",
"Doping evolution of the phonon density of states and electron-lattice interaction in Nd 2−x Ce x CuO 4+δ"
] |
[
"H J Kang \nDepartment of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee\n",
"Pengcheng Dai \nDepartment of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee\n\nSolid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee\n",
"D Mandrus \nDepartment of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee\n\nSolid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee\n",
"R Jin \nSolid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee\n",
"H A Mook \nSolid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee\n",
"D T Adroja \nISIS facility\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK\n",
"S M Bennington \nISIS facility\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK\n",
"S.-H Lee \nNIST Center for Neutron Research\nNational Institute of Standards and Technology\n20899GaithersburgMaryland\n",
"J W Lynn \nNIST Center for Neutron Research\nNational Institute of Standards and Technology\n20899GaithersburgMaryland\n"
] |
[
"Department of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee",
"Department of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee",
"Solid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee",
"Department of Physics and Astronomy\nThe University of Tennessee\n37996-1200KnoxvilleTennessee",
"Solid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee",
"Solid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee",
"Solid State Division\nOak Ridge National Laboratory\n37831Oak RidgeTennessee",
"ISIS facility\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK",
"ISIS facility\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK",
"NIST Center for Neutron Research\nNational Institute of Standards and Technology\n20899GaithersburgMaryland",
"NIST Center for Neutron Research\nNational Institute of Standards and Technology\n20899GaithersburgMaryland"
] |
[] |
We use inelastic neutron scattering to study the evolution of the generalized phonon density of states (GDOS) of the n-type high-Tc superconductor Nd2−xCexCuO 4+δ (NCCO), from the halffilled Mott-insulator (x = 0) to the Tc = 24 K superconductor (x = 0.15). Upon doping the CuO2 planes in Nd2CuO 4+δ (NCO) with electrons by Ce substitution, the most significant change in the GDOS is the softening of the highest phonon branches associated with the Cu-O bond stretching and out-of-plane oxygen vibration modes. However, the softening occurs within the first few percent of Ce-doping and is not related to the electron doping induced nonsuperconducting-superconducting transition (NST) at x ≈ 0.12. These results suggest that the electron-lattice coupling in the n-type high-Tc superconductors is different from that in the p-type materials.One of the most remarkable properties of hightransition-temperature (high-T c ) copper-oxide (cuprate) superconductors is their close proximity to an antiferromagnetic (AF) phase. The parent compounds of the high-T c cuprates are AF insulators characterized by a simple doubling of the crystallographic unit cell in the CuO 2 planes [1]. When holes [2] or electrons [3] are doped into these planes, the long-range AF-ordered phase is destroyed, and the copper-oxide materials become metallic and superconducting with persistent short-range AF spin correlations (fluctuations). Much effort over the past decade has focused on understanding the nature of the interplay between magnetism and superconductivity [1], mainly because spin fluctuations may contribute a major part of the superconducting condensation energy[4,5]. On the other hand, the role of phonons in the microscopic mechanism of superconductivity is still largely unknown even though phonons in cuprates also display a variety of unusual properties[6,7,8,9,10]. The key question is whether magnetism and electron-electron correlations alone are sufficient to induce electron pairing that leads to superconductivity in high-T c cuprates, or electron-lattice coupling also plays an important role.From the analysis of high-resolution angle-resolved photoemission (ARPES) data in conjunction with those from neutron, optics and local structural probes, Shen and co-workers[11]suggest that phonons must also play an essential role in electron pairing for high-T c cuprates. The key evidence for electron-lattice coupling, they argue[12], is that the kink (or the change of slope) seen in the electronic dispersion of the holedoped (p-type) Bi 2 Sr 2 CaCu 2 O 8 (Bi2212), Bi 2 Sr 2 CuO 6 (Bi2201), and La 2−x Sr x CuO 4 (LSCO) from the ARPES data[13,14,15,16]occurs at an energy (∼70 meV) very close to the phonon anomalies observed by inelastic neutron scattering[7,8,9]. These phonon anomalies include the break in the dispersion of the oxygen halfbreathing mode in La 1.85 Sr 0.15 CuO 4 [7] and the abrupt development of new oxygen lattice vibrations near the doping-induced metal-insulator transition (MIT) in the generalized phonon density of states (GDOS) of LSCO[8,17]. Since the change of slope in the electronic dispersion indicates a dramatic drop in the "quasiparticle" scattering rate[11], their observation in hole-doped cuprate superconductors[13,14,15,16]suggests a strong coupling between the quasiparticles and a sharp collective spin or lattice mode. Although the neutron magnetic resonance [5] could be the collective spin mode coupled to the quasiparticles[16,18], Shen et al. argue that electron-lattice interaction is ultimately responsible for the quasiparticle velocity change and thus is crucial to the high-T c superconductivity[11,12]. Furthermore, since the dispersion of the electron-doped superconducting Nd 1.85 Ce 0.15 CuO 4+δ does not have such a kink, the authors[11]predict that the n-type materials have much weaker electron-lattice coupling and thus lower T c 's.If this hypothesis were correct, one would expect the exotic lattice dynamics seen in the p-type LSCO[7,8,9]to be reduced in the n-type Nd 2−x Ce x CuO 4+δ (NCCO)[11]. For LSCO, the abrupt development of the new oxygen lattice vibrations across the doping induced nonsuperconducting-superconducting transition (NST) was interpreted as evidence for strong electron-lattice coupling in the superconducting cuprates that is not present in nonsuperconducting materials[8,17]. Specifically, the new lattice mode at ∼70 meV in the GDOS is believed to be at least partly comprised of the anomalous Cu-O bond-stretching (oxygen half-breathing) mode[8]. Although the GDOS for NCCO with x = 0, 0.15 were studied by Lynn and co-workers[19,20], no systematic doping dependent measurements are available. If the quasiparticle velocity drop seen in the ARPES data of
|
10.1103/physrevb.66.064506
|
[
"https://arxiv.org/pdf/cond-mat/0204007v1.pdf"
] | 37,438,515 |
cond-mat/0204007
|
22aa8e5b1358e027d3a7fc14d3bf1a50da23d3d5
|
Doping evolution of the phonon density of states and electron-lattice interaction in Nd 2−x Ce x CuO 4+δ
29 Mar 2002
H J Kang
Department of Physics and Astronomy
The University of Tennessee
37996-1200KnoxvilleTennessee
Pengcheng Dai
Department of Physics and Astronomy
The University of Tennessee
37996-1200KnoxvilleTennessee
Solid State Division
Oak Ridge National Laboratory
37831Oak RidgeTennessee
D Mandrus
Department of Physics and Astronomy
The University of Tennessee
37996-1200KnoxvilleTennessee
Solid State Division
Oak Ridge National Laboratory
37831Oak RidgeTennessee
R Jin
Solid State Division
Oak Ridge National Laboratory
37831Oak RidgeTennessee
H A Mook
Solid State Division
Oak Ridge National Laboratory
37831Oak RidgeTennessee
D T Adroja
ISIS facility
OX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK
S M Bennington
ISIS facility
OX11 0QXRutherford Appleton Laboratory, Chilton, DidcotUK
S.-H Lee
NIST Center for Neutron Research
National Institute of Standards and Technology
20899GaithersburgMaryland
J W Lynn
NIST Center for Neutron Research
National Institute of Standards and Technology
20899GaithersburgMaryland
Doping evolution of the phonon density of states and electron-lattice interaction in Nd 2−x Ce x CuO 4+δ
29 Mar 2002(Dated: November 4, 2018)
We use inelastic neutron scattering to study the evolution of the generalized phonon density of states (GDOS) of the n-type high-Tc superconductor Nd2−xCexCuO 4+δ (NCCO), from the halffilled Mott-insulator (x = 0) to the Tc = 24 K superconductor (x = 0.15). Upon doping the CuO2 planes in Nd2CuO 4+δ (NCO) with electrons by Ce substitution, the most significant change in the GDOS is the softening of the highest phonon branches associated with the Cu-O bond stretching and out-of-plane oxygen vibration modes. However, the softening occurs within the first few percent of Ce-doping and is not related to the electron doping induced nonsuperconducting-superconducting transition (NST) at x ≈ 0.12. These results suggest that the electron-lattice coupling in the n-type high-Tc superconductors is different from that in the p-type materials.One of the most remarkable properties of hightransition-temperature (high-T c ) copper-oxide (cuprate) superconductors is their close proximity to an antiferromagnetic (AF) phase. The parent compounds of the high-T c cuprates are AF insulators characterized by a simple doubling of the crystallographic unit cell in the CuO 2 planes [1]. When holes [2] or electrons [3] are doped into these planes, the long-range AF-ordered phase is destroyed, and the copper-oxide materials become metallic and superconducting with persistent short-range AF spin correlations (fluctuations). Much effort over the past decade has focused on understanding the nature of the interplay between magnetism and superconductivity [1], mainly because spin fluctuations may contribute a major part of the superconducting condensation energy[4,5]. On the other hand, the role of phonons in the microscopic mechanism of superconductivity is still largely unknown even though phonons in cuprates also display a variety of unusual properties[6,7,8,9,10]. The key question is whether magnetism and electron-electron correlations alone are sufficient to induce electron pairing that leads to superconductivity in high-T c cuprates, or electron-lattice coupling also plays an important role.From the analysis of high-resolution angle-resolved photoemission (ARPES) data in conjunction with those from neutron, optics and local structural probes, Shen and co-workers[11]suggest that phonons must also play an essential role in electron pairing for high-T c cuprates. The key evidence for electron-lattice coupling, they argue[12], is that the kink (or the change of slope) seen in the electronic dispersion of the holedoped (p-type) Bi 2 Sr 2 CaCu 2 O 8 (Bi2212), Bi 2 Sr 2 CuO 6 (Bi2201), and La 2−x Sr x CuO 4 (LSCO) from the ARPES data[13,14,15,16]occurs at an energy (∼70 meV) very close to the phonon anomalies observed by inelastic neutron scattering[7,8,9]. These phonon anomalies include the break in the dispersion of the oxygen halfbreathing mode in La 1.85 Sr 0.15 CuO 4 [7] and the abrupt development of new oxygen lattice vibrations near the doping-induced metal-insulator transition (MIT) in the generalized phonon density of states (GDOS) of LSCO[8,17]. Since the change of slope in the electronic dispersion indicates a dramatic drop in the "quasiparticle" scattering rate[11], their observation in hole-doped cuprate superconductors[13,14,15,16]suggests a strong coupling between the quasiparticles and a sharp collective spin or lattice mode. Although the neutron magnetic resonance [5] could be the collective spin mode coupled to the quasiparticles[16,18], Shen et al. argue that electron-lattice interaction is ultimately responsible for the quasiparticle velocity change and thus is crucial to the high-T c superconductivity[11,12]. Furthermore, since the dispersion of the electron-doped superconducting Nd 1.85 Ce 0.15 CuO 4+δ does not have such a kink, the authors[11]predict that the n-type materials have much weaker electron-lattice coupling and thus lower T c 's.If this hypothesis were correct, one would expect the exotic lattice dynamics seen in the p-type LSCO[7,8,9]to be reduced in the n-type Nd 2−x Ce x CuO 4+δ (NCCO)[11]. For LSCO, the abrupt development of the new oxygen lattice vibrations across the doping induced nonsuperconducting-superconducting transition (NST) was interpreted as evidence for strong electron-lattice coupling in the superconducting cuprates that is not present in nonsuperconducting materials[8,17]. Specifically, the new lattice mode at ∼70 meV in the GDOS is believed to be at least partly comprised of the anomalous Cu-O bond-stretching (oxygen half-breathing) mode[8]. Although the GDOS for NCCO with x = 0, 0.15 were studied by Lynn and co-workers[19,20], no systematic doping dependent measurements are available. If the quasiparticle velocity drop seen in the ARPES data of
We use inelastic neutron scattering to study the evolution of the generalized phonon density of states (GDOS) of the n-type high-Tc superconductor Nd2−xCexCuO 4+δ (NCCO), from the halffilled Mott-insulator (x = 0) to the Tc = 24 K superconductor (x = 0.15). Upon doping the CuO2 planes in Nd2CuO 4+δ (NCO) with electrons by Ce substitution, the most significant change in the GDOS is the softening of the highest phonon branches associated with the Cu-O bond stretching and out-of-plane oxygen vibration modes. However, the softening occurs within the first few percent of Ce-doping and is not related to the electron doping induced nonsuperconducting-superconducting transition (NST) at x ≈ 0.12. These results suggest that the electron-lattice coupling in the n-type high-Tc superconductors is different from that in the p-type materials. One of the most remarkable properties of hightransition-temperature (high-T c ) copper-oxide (cuprate) superconductors is their close proximity to an antiferromagnetic (AF) phase. The parent compounds of the high-T c cuprates are AF insulators characterized by a simple doubling of the crystallographic unit cell in the CuO 2 planes [1]. When holes [2] or electrons [3] are doped into these planes, the long-range AF-ordered phase is destroyed, and the copper-oxide materials become metallic and superconducting with persistent short-range AF spin correlations (fluctuations). Much effort over the past decade has focused on understanding the nature of the interplay between magnetism and superconductivity [1], mainly because spin fluctuations may contribute a major part of the superconducting condensation energy [4,5]. On the other hand, the role of phonons in the microscopic mechanism of superconductivity is still largely unknown even though phonons in cuprates also display a variety of unusual properties [6,7,8,9,10]. The key question is whether magnetism and electron-electron correlations alone are sufficient to induce electron pairing that leads to superconductivity in high-T c cuprates, or electron-lattice coupling also plays an important role.
From the analysis of high-resolution angle-resolved photoemission (ARPES) data in conjunction with those from neutron, optics and local structural probes, Shen and co-workers [11] suggest that phonons must also play an essential role in electron pairing for high-T c cuprates. The key evidence for electron-lattice coupling, they argue [12], is that the kink (or the change of slope) seen in the electronic dispersion of the holedoped (p-type) Bi 2 Sr 2 CaCu 2 O 8 (Bi2212), Bi 2 Sr 2 CuO 6 (Bi2201), and La 2−x Sr x CuO 4 (LSCO) from the ARPES data [13,14,15,16] occurs at an energy (∼70 meV) very close to the phonon anomalies observed by inelastic neutron scattering [7,8,9]. These phonon anomalies include the break in the dispersion of the oxygen halfbreathing mode in La 1.85 Sr 0.15 CuO 4 [7] and the abrupt development of new oxygen lattice vibrations near the doping-induced metal-insulator transition (MIT) in the generalized phonon density of states (GDOS) of LSCO [8,17]. Since the change of slope in the electronic dispersion indicates a dramatic drop in the "quasiparticle" scattering rate [11], their observation in hole-doped cuprate superconductors [13,14,15,16] suggests a strong coupling between the quasiparticles and a sharp collective spin or lattice mode. Although the neutron magnetic resonance [5] could be the collective spin mode coupled to the quasiparticles [16,18], Shen et al. argue that electron-lattice interaction is ultimately responsible for the quasiparticle velocity change and thus is crucial to the high-T c superconductivity [11,12]. Furthermore, since the dispersion of the electron-doped superconducting Nd 1.85 Ce 0.15 CuO 4+δ does not have such a kink, the authors [11] predict that the n-type materials have much weaker electron-lattice coupling and thus lower T c 's.
If this hypothesis were correct, one would expect the exotic lattice dynamics seen in the p-type LSCO [7,8,9] to be reduced in the n-type Nd 2−x Ce x CuO 4+δ (NCCO) [11]. For LSCO, the abrupt development of the new oxygen lattice vibrations across the doping induced nonsuperconducting-superconducting transition (NST) was interpreted as evidence for strong electron-lattice coupling in the superconducting cuprates that is not present in nonsuperconducting materials [8,17]. Specifically, the new lattice mode at ∼70 meV in the GDOS is believed to be at least partly comprised of the anomalous Cu-O bond-stretching (oxygen half-breathing) mode [8]. Although the GDOS for NCCO with x = 0, 0.15 were studied by Lynn and co-workers [19,20], no systematic doping dependent measurements are available. If the quasiparticle velocity drop seen in the ARPES data of FIG. 1: Temperature dependences of the AC magnetic susceptibility χ ′ (real part) for the NCCO powder samples used in the neutron measurements. The background was subtracted using the χ ′ for x = 0.09, which shows Curie-Weisslike behavior down to the lowest temperature measured. The diamagnetic signal first appears for x ≥ 0.11.
p-type cuprates [11] is related to the anomalous lattice vibrational modes [8], the absence of such a drop in the electron-doped superconducting NCCO with x = 0.15 would suggest a weak (or no) phonon anomaly for NCCO.
In this paper, we present inelastic neutron scattering measurements of the GDOS in NCCO spanning electron doping concentrations from the half-filled Mott-insulator (Nd 2 CuO 4+δ or NCO) to optimally doped NCCO superconductor (x ≈ 0.15). Upon doping electrons to the CuO 2 planes by Ce substitution, the most significant change in the GDOS is the anomalous softening of the ∼ 70 meV phonon branches associated with the oxygen half-breathing and out-of-plane vibrational modes. However, in contrast to LSCO [8], the anomaly only occurs within the first few percent of Ce-doping and there is no evidence for any new lattice modes in the GDOS across the electron doping induced NST at x ≈ 0.12. Our results indicate that doping-induced phonon anomalies in holedoped cuprates are different from these in electron-doped materials, thus suggesting that electron-lattice coupling is unrelated to the superconductivity of NCCO.
Our experiments were performed on the MARI chopper spectrometer at ISIS facility, Rutherford Appleton Laboratory, UK. The detectors on MARI cover a wide scattering angle from 3 • to 135 • . For the experiments, we used a Fermi chopper to choose an incident beam energy of 110 meV. The energy resolution is between 1-2% of the incident energy. The powder samples were mounted inside the aluminum sample can on the cold head of a helium closed-cycle refrigerator and all measurements were performed at T = 30 K. The incident neutron beam size is 5 × 5 cm 2 and the unexposed area of the sample was covered by Cd sheets. To normalize the scattering from NCCO on an absolute scale, we used the elastic incoherent scattering from a vanadium standard. In addition to measurements at MARI, we have also collected data on the BT-4 filter analyzer spectrometer at the National Institute of Standards and Technology research reactor. To within the error of the measurements, the results of these two experiments are identical.
We prepared the ceramic samples of NCCO with Ceconcentrations of x = 0.00, 0.04, 0.08, 0.09, 0.10, 0.11, 0.12, 0.13, 0.15 by the conventional solid state reaction [3]. The as-grown samples have excess oxygen (δ > 0) and are nonsuperconducting. Various annealing procedures have been developed to remove the excess oxygen needed to produce superconductivity. However, the properties of the samples and the resulting electronic phase diagrams are different depending on the details of the annealing procedure used. In the original work of Tagaki et al. [3], it was found that samples treated in flowing Ar at temperatures in excess of 1100 • C followed by heating in air at 500 • C produced metallic samples with a sharp superconducting transition. However, it was also found that this procedure resulted in some decomposition of the sample as well as probable loss of Cu from the surfaces of the polycrystalline grains [3]. This procedure also results in the electronic phase diagram showing an abrupt NST around x = 0.14 with only half of the superconducting "dome" [3], different from hole-doped LSCO [1]. We have followed the annealing procedure developed by Maple's group [21], where the samples are treated in flowing Ar at temperature of about 900 • C. The resulting phase diagram shows the NST around x = 0.12 with the almost complete superconducting dome [21] as compared to the half-dome from [3]. To characterize the materials, bulk magnetization and resistivity measurements were performed for all the samples. Figure 1 shows the doping dependence of the AC susceptibility for x = 0.1, 0.11, 0.13, and 0.15. Superconductivity is clearly seen for NCCO with x ≥ 0.13, thus confirming that the NST in NCCO occurs around x ≈ 0.12 [21].
In an unpolarized neutron experiment, the major difficulty in obtaining the reliable GDOS is to separate phonons from magnetic scattering. For NCCO, the largest magnetic signal originates from single-ion crystalline electric field (CEF) excitations of the Nd ions [19,20]. The CEF excitations of NCCO with x = 0, and 0.15 have been studied in great detail and their level scheme has peaks aroundhω ≈ 12-16, 20.5, 27, and 93.3 meV at low temperatures [22]. We performed careful wave vector (Q) dependent analysis of the excitation intensities athω = 20.5, 27, and 93.3 meV for 2 A −1 < Q < 10Å −1 . The outcome confirms the earlier results that these three peaks are magnetic in origin and the phonon cutoff energy of NCCO is around 83 meV [20]. We also checked the strength of the multiple scattering and multi-phonon scattering using the Monte Carlo simulation program MSCAT, but found such multi-phonon scattering contributes negligibly to the total scattering intensity in the energy region of interest (hω ≥ 50 meV). To reduce the magnetic scattering contribution to the GDOS, we replaced the intensities of the 20.5 and 27 meV peaks with scattering from the highest measured wave vectors (9Å −1 < Q < 11Å −1 ). Although this procedure may not eliminate all the magnetic intensity, there are no magnetic contributions to the GDOS for 50 meV≤hω ≤ 80 meV. After subtraction of the empty aluminum sample can, multiple and multi-phonon scattering, single-phonon GDOS with Q values integrated from 3 to 11Å −1 were calculated by multiplying ω/[n(ω) + 1], where n(ω) is the Bose population factor. The total area of each GDOS was then normalized to 1 over the energy range from 15 to 80 meV. Figure 2 shows the GDOS for NCCO with x = 0.0, 0.04, 0.08, 0.10, 0.11, 0.12, 0.13, and 0.15. Consistent with previous measurements on NCCO for x = 0.0 and 0.15 [19,20], the spectra contain clear peaks at ∼36, 42, 48, and 65-70 meV. On moving from an insulator to a metal with increasing Ce-concentration, the largest observed effect is the softening and sharpening of the broad ∼70 meV phonon-band in the undoped NCO.
We systematically fit the 70 meV phonon band with two Gaussians on a sloping background for various x. The solid lines in Fig. 2 show the outcome of the fits. Although the precise functional form of the GDOS for the 70 meV phonon band is not known, the systematic Gaussian fits allow a quantitative determination for the magnitude of the phonon softening. For the undoped NCO, the 70 meV mode shows a flattish top and can be best fitted by two Gaussians centered at 67 and 71 meV, respectively. On increasing the Ce-concentration to x = 0.04, the 71 meV mode softens to 67 meV (6% softening) and shows less of a flattish top. Furthermore, the GDOS gains intensity at 65 meV at the expense of the 71 meV peak. At x = 0.08, the GDOS peaks more sharply at 65 meV. On further increasing x and across the NST at x = 0.12, the GDOS show essentially no change from that for x = 0.08 to within the error of the measurements. Figure 3 shows the comparison plots of the GDOS at various x. For NCCO with 0.0 ≤ x ≤ 0.08, the 70 meV phonon band shows significant softening while all other modes display no visible change with increasing x (Figs. 3a and 3b). The GDOS for x from 0.08 to 0.15 (Figs. 3c and 3d) overlap completely in the probed energy range (15 ≤hω ≤ 80 meV) and show no changes across the NST.
To understand the atomic displacement patterns of the phonon modes contributing to the 70 meV band in NCO, we consider its experimentally determined phonon dispersion curves [23]. For NCO, the highest energy phonon bands are around 70 meV [23]. These include the highest energy in-plane Cu-O bond-stretching mode with ∆ 1 -symmetry at Q = (0.5, 0, 0) (the oxygen halfbreathing mode) along the [ζ, 0, 0] direction, the out-ofplane (c-axis polarized) oxygen breathing mode with Λ 1symmetry along [0, 0, ζ], and the in-plane oxygen breathing mode with Σ 1 -symmetry at Q = (0.5, 0.5, 0) along [ζ, ζ, 0]. The inset of Fig. 2 shows the oxygen displacement patterns for these three modes. Since these three modes are at ∼70 meV in the dispersion curves [23], the 70 meV peak in the GDOS of NCO must consist, at least partially, of these modes. As a consequence, the electrondoping induced softening in NCCO must also occur in these modes.
In a very recent inelastic X-ray scattering study of longitudinal optical phonons in NCCO with x = 0.14, d'Astuto et al. discovered anomalous phonon softening in the two highest longitudinal branches associated with the Cu-O bond-stretching and out-of-plane oxygen vibrations [24]. By comparing their data on NCCO with undoped NCO, the authors concluded that strong electronphonon coupling is also present in electron-doped NCCO. From their work [24], it becomes clear that the significant softening of the 70 meV phonon band with x in Figs. 2 and 3 is mostly due to the softening of the oxygen halfbreathing and out-of-plane vibrational modes. For holedoped LSCO, the oxygen half-breathing modes display anomalous behavior [7,9] and show up as new lattice modes in the superconducting side of the phase diagram across the NST [8]. While the oxygen half-breathing modes also exhibit anomalous behavior [24] and soften with increasing electron-doping, our results indicate that the softening occurs within the first few percent of Cedoping in the nonsuperconducting regime and therefore is not associated with the electron doping induced NST in NCCO.
For p-type cuprates, previous investigations have established a clear correlation between superconducting properties of the materials and special features of the phonon spectrum. While such correlation is seen as anomalous phonon modes across the NST in LSCO [8], systematic studies of the GDOS in YBa 2 Cu 3 O 6+x show that the phonon cut-off energy softens across the NST and is closely related to T c (see Fig. 41 of [6]). In general, these phonon anomalies are related to the dielectric screening properties of metals and thus suggest a strong electron-lattice coupling in the superconductivity of the p-type materials. Although phonon softening is also ob-served in the n-type NCCO, our data indicate that these anomalies occur in the nonsuperconducting regime and are not directly related to the NST. Therefore, it becomes clear that the electron-lattice coupling in the electrondoped NCCO is different from that in the hole-doped materials.
It is interesting to compare our results with that of the ARPES on NCCO. In principle, the strong electronlattice coupling and large softening of the optical oxygen vibrational modes in NCCO with x = 0.04 should reveal themselves as distinctive features in the ARPES spectra [11]. If the kink in the electronic band dispersion in the hole-doped materials is due to the anomalous softening of the 70 meV oxygen half-breathing mode [11], its absence in electron-doped NCCO would suggest no softening of such oxygen modes in NCCO. Clearly, this is inconsistent with the results of [24] and the present work. On the other hand, if the kink in the ARPES spectra is not related to the softening of the 70 meV modes but to the changes of such modes across the hole-(electron-) induced NST, our data would be consistent with a weak electron-lattice coupling in NCCO. An unambiguous test of this idea will require comparison of the neutron data with the doping dependence of the electronic structure of NCCO. Although systematic ARPES investigations have been carried out very recently on NCCO [25], the evolution of the electronic dispersions across the MIT or NST is unavailable and therefore cannot be compared yet with the neutron results.
We thank Z.-X. Shen and Y. Ando for helpful discussions. This work was supported by NSF DMR-0139882 and by DOE under contract DE-AC05-00OR22725 with UT-Battelle, LLC.
PACS numbers: 74.25.Kc, 63.20.Kr, 71.30.+h, 74.20.Mn
FIG. 2 :
2The GDOS of NCCO as a function of x at T = 30 K. Each GDOS is displaced along the vertical axis for clarity and the solid lines are Gaussian fits discussed in the text. The nonsuperconducting-superconducting transition as a function of x is schematically shown on the right. The insets show the polarizations of the oxygen half-breathing (left), breathing (middle), and out-of-plane (right) vibrational modes.
FIG. 3 :
3Comparison of the GDOS of NCCO as a function of electron-doping, x. (a) The GDOS of NCCO for x = 0.0, and 0.04; (b) x = 0.04, and 0.08; (c) x = 0.08, and 0.11; and (d) x = 0.11, and 0.15.
Electronic address: [email protected]. Electronic address: [email protected]
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[
"HOMOLOGICAL PROPERTIES OF BINOMIAL EDGE IDEAL OF GRAPHS",
"HOMOLOGICAL PROPERTIES OF BINOMIAL EDGE IDEAL OF GRAPHS"
] |
[
"Himadri Mukherjee ",
"Priya Das "
] |
[] |
[] |
In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.
| null |
[
"https://export.arxiv.org/pdf/2209.01201v2.pdf"
] | 252,070,490 |
2209.01201
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36b5e89336f7ec3866dc98d98fc2768052020506
|
HOMOLOGICAL PROPERTIES OF BINOMIAL EDGE IDEAL OF GRAPHS
Sep 2022
Himadri Mukherjee
Priya Das
HOMOLOGICAL PROPERTIES OF BINOMIAL EDGE IDEAL OF GRAPHS
Sep 2022
In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.
Introduction
Let G be a finite simple graph on n vertices with set of edges, E(G) with no isolated vertices. Let S = K[x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n ] be the polynomial ring over a field K and f ij = x i y j − x j y i , {i, j} ∈ E(G), the ideal J G in S generated by f ij , 1 ≤ i < j ≤ n is called the binomial edge ideal (BEI) of G. It was introduced by Herzog-Hibi-Kahle-Rauh in the contest of study of conditional independence ideals [19] and independently by [51]. It can be also seen as a special case of the generalized binomial edge ideal as defined by [54], the collection of 2-minors of the generic (2 × n)-matrix.
In [60], Madani has given an overview of Gröbner bases, primary decomposition, and minimal graded free resolution of binomial edge ideal. The survey by Madani loc. cit. is a comprehensive survey of all the works till the year of its publication (2018), related to the binomial edge ideal in the context of homological studies. In this article, we will provide a survey of all the works mainly those that are related to the regularity of the binomial edge ideal, depth, Betti number, generalised binomial edge ideal, and parity binomial edge ideal, since [60].
Gröbner bases of binomial edge ideal are studied in [19], [51], the authors of loc. cit. proved that the generators of the binomial edge ideal form a Gröbner basis for closed graphs. In [15], the authors proved that S/J G is Koszul when G is a closed graph. The primary decomposition of the binomial edge ideal is studied in [19], using the primary decomposition the authors have described the minimal prime ideals of J G .
The study of the minimal free resolution of binomial edge ideals is one of the important topics in combinatorial commutative algebra. as the binomial ideals are geometrically and algebraically an important class. Also since the work of [4] has a very thorough discussion for the monomial ideals, it makes sense to look for a combinatorial understanding of the situation in the binomial ideals. Among the binomial ideals, binomial edge ideals of a graph is a major stream of work currently. Also a number of works have been done in the binomial toric ideals associated to distributive lattices, namely the Hibi ideal [12]. The authors in [10] have given the generators of the first syzygy of the Hibi ideal of a planar lattice in terms of four types of sublattices. In [61], the authors proved that BEI of a graph has a linear resolution if G is a complete graph and vice versa. The question of a pure resolution of a BEI is discussed in [35].
One of the most important invariants of an ideal is the Castelnuovo-Mumford regularity. It is proved in [49], for any graph G, l ≤ reg(S/J G ) ≤ n − 1, where n is the number of vertices of G and l is the length of the largest induced path in G. Also, in [37], authors proved that reg(S/J G ) = n − 1 if and only if G is a path. Madani-Kiani in [62] has given the following conjecture Madani and Kiani proved the above conjecture for closed graphs and generalized block graphs in [61], and [34]. The conjecture in general remains as one of the important open problem in this area. For a block graph, c(G) coincides with the number of blocks in G. Trees being a subclass of the class of block graphs and combining it with above result we have that, for a tree T , which has n vertices c(T ) = n − 1. In [30], authors have shown that reg(S/J G ) ≤ n − 1, thus proved the conjecture for some classes of trees. Also, this conjecture is proved for the class of chordal graphs and for fan graphs of complete graphs in [28]. Independently, the authors in loc. cit. prove this conjecture for chordal graphs in [59]. Recently, this conjecture has been proved for P 4 -free graphs in [32]. For the classes of block graphs and semi block graph the conjecture has been settled in [40]. But for a general graph the conjecture remains open and a good motivation for the works regarding the regularity of BEI.
In the article [14], the authors conjectured that the regularity of the binomial edge ideal of a graph coincides with the regularity of its initial ideal. If the conjecture holds for two graphs G 1 , G 2 then it also holds for the join of two graphs, in Theorem 2.1 in [63]. The above result was used by the authors to completely characterize the BEI of regularity 3, (Theorem 3.2 [63]). Threshold graphs which are also known as (2K 2 , C 4 , P 4 )-free graphs are the graphs which have regularity 3. For more details about threshold graph, see [45]. Whereas authors have characterized the binomial edge ideal of regularity 2 in [61]. A new general upper bound for the regularity of S/J G has been given in terms of the dimension of a simplicial complex in [17]. In [38], authors of loc. cit. obtained an improved lower and upper bound for the regularity of the BEI of a generalized block graph. Generalized block graphs are the generalization of block graphs and it is introduced in [43]. In [24], Hibi-Matsuda conjectured the following
reg(S/J G ) ≤ deg h S/J G (t).
They proved that this conjecture is true if S/J G has a unique extremal Betti number. In [64], Schenzel and Zafar have shown complete bipartite graphs have unique extremal Betti number. Also, Zafar and Zahid proved that n-cycle, C n has a unique extremal Betti number, see [66]. In [21], Herzog-Rinaldo characterized block graphs which admits unique extremal Betti number. In [38], the author characterized a generalized block graph which admits a unique extremal Betti number. In [40], authors provide a sufficient condition for this conjecture. The depth of an ideal is another commutative algebraic invariant of primary importance, in the context of BEI there have been a number of works in this area, but it is far from being complete. In [3], authors proved that depth of S/J G is n + 1, for a connected block graph G. In [66], the authors showed that depth S/J G is n, for n > 3. In [36], authors gave the depth of the BEI of generalised block graphs. For a connected graph G on n-vertices the depth of binomial edge ideal is at most n and further, if G is a connected graph on n vertices such that S/J G is Cohen-Macaulay then depth of it is exactly n + 1, see, [3]. Also, in the same article, the authors provided upper bounds for the depth of binomial edge ideal of a general graph in terms of certain graph invariants. Depth of binomial edge ideal for a cone graph remains invariant under the process of taking cone on connected graph, see [44]. In the same article they have given a formula for the depth of binomial edge ideal of join of two graphs. Recently, in [46] they have completely characterized the graphs for which is depth of the BEI is 5.
Betti numbers of binomial edge ideal of cycle, complete bipartite graphs, trees, unicyclic graphs, and cone graph are known, see [66], [64], [29], [44]. In [29], the authors have given the generators the first syzygy of binomial edge ideal of cycle and unicyclic graphs explicitly.
Generalized binomial edge ideal is introduced by Rauh in [54] as a generalization to encompass a variety of binomial ideals. As a result it is an important ideal to look at and the algebraic invariants of it are of primary importance. There are only relatively few results on this ideal, apart from the results in the particular case of graph binomial ideals. The generalized binomial edge ideal is defined as the ideal generated by a collection of 2-minors of a generic 2 × n matrix. The determinantal ideal is of classical importance to algebraic geometry and commutative algebra. Apart from the geometric significance, another motivation to study these ideals comes from their connection to conditional independence ideals, a topic in algebraic statistics. Let m, n ≥ 2 be integers and G be a simple graph on the vertex set [n]. Let X = (x ij ) be an m × n matrix of indeterminates and S = K[X] be the polynomial ring in the variables x ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n, over a field K. For 1 ≤ k < l ≤ m and {i, j} ∈ E(G) with 1 ≤ i < j ≤ n we set,
p kl ij = [k, l][i, j] = x ki x lj − x li x kj . The ideal J G = (p kl ij : 1 ≤ k < l ≤ m, {i, j} ∈ E(G))
is called the generalized binomial edge ideal. In [54], it is proved that J G is a radical ideal and its minimal primes are determined by the sets with the cut point property of G. Regarding the minimal free resolution of generalized binomial edge ideal, not much is known. But there have been a few key works in this area. Madani and Kiani in [62], proved that J G has linear resolution if and only if m = 2 and G is a complete graph. In [7], the authors proved the equality of the depth of generalized binomial edge ideal of generalized block graph and the depth of its initial ideal. Also they found the regularity of these two classes of generalized block graphs. Since a tree is a block graph, one can get an equivalent condition, which is given in [7]. Also, for a path, regularity of generalized binomial edge ideal is computed in [7], and independently computed in [39]. Regularity of the generalized binomial edge ideal for the complete graph is known in [39]. In [2], the authors have proved that the generalized binomial edge ideal is Cohen-Macaulay when the graph is a complete graph.
Another class of binomial ideals related to the binomial edge ideal is the parity binomial edge ideal which was introduced in [33]. The permanental binomial edge ideal and Lovasz-Saks-Schrjiver (in short, LSS) ideal in [20] are also important classes of binomial edge ideals associated to graphs. Not much work has been done on these classes of ideals and the authors of the present article feels that much work on these are also important to further understanding of the homological properties of the binomial ideals. The parity binomial edge ideal has generated some recent interest among the combinatorial commutative algebra community. Let G be a simple graph with edge set E(G) and S = K[x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n ] be a polynomial ring, the parity of binomial edge ideal of S is denoted by I G and is defined as I G = (x i x j −y i y j : {i, j} ∈ E(G)), whereas, the permanental edge ideal is denoted by Π G and is defined as Π G = (x i y j + x j y i : {i, j} ∈ E(G)). Let d ≥ 1 be an integer, then the Lovasz-Saks-Schrjiver ideal is denoted by L K G (d) and is defined as,
L K G (d) = ( d l=1 x il x jl : {i, j} ∈ E(G)) in the polynomial ring K[x kl : 1 ≤ k ≤ n, 1 ≤ l ≤ d]. For d = 1, L K G (d) it coincides with the edge ideal of graph G. For d = 2, LSS ideal of G is the binomial ideal defined as L G = (x i x j +y i y j : {i, j} ∈ E(G)) ⊂ K[x 1 , x 2 , . .
. , x n , y 1 , y 2 , . . . , y n ]. It is known that L G is a radical ideal if char(K) = 2, in [20]. Also, the authors in loc. cit. determined the primary decomposition of L G when √ −1 / ∈ K and char(K) = 2. In [33], the authors proved that the parity binomial edge ideal is radical if and only if the graph is bipartite. Also, in the same paper they have determined the minimal primes and the primary decomposition. In [9], the authors proved that L K G (2) is complete intersection if and only if G does not contain a claw or an even cycle (Theorem 1.4). Also they have shown that L K G (3) is prime if and only if G does not contain a claw or C 4 . In [20], authors computed a Gröbner basis of the permanental edge ideal of graphs and also they have shown that it is a radical ideal. In [41], the author characterizes the graphs whose parity binomial edge ideal is a complete intersection, see Theorem 3.2 and 3.5 of [41]. In the loc. cit., the author characterized the graphs for LSS ideals, parity binomial edge ideal and permanental edge ideals that are complete intersections. In [42], the authors have given a lower bound for the regularity of parity binomial edge ideal of graphs, and also classified the graphs whose parity binomial edge ideal has regularity 3. In the same article they have characterized the graphs whose parity binomial edge ideal have pure resolutions. In [11], the authors have given an explicit formula for the Hilbert-Poincare series of the parity binomial edge ideal of a complete graph.
Preliminaries
In this section, we recall basic definitions, notations, and terminologies of graphs and algebra which will be needed further.
2.1. Basic notions from Algebra. Let S = K[x 1 , x 2 , . . . , x n ] be a polynomial ring in n variables over a field K. Then S is called a graded ring, graded by degree, if S = ⊕ m≥0 S m , where S m is generated by all monomials of degree m, and if I is a graded ideal of S, then R = S/I is called graded algebra.
A graded free resolution of a finitely generated S-module M is an exact sequence
F : . . . F i φ i −→ F i−1 −→ . . . −→ F 1 φ 1 −→ F 0 φ 0 −→ M −→ 0 where F i = ⊕ j∈Z S(−j) β ij ,If m = (x 1 , x 2 , . . . , x n ) is the maximal ideal of S, then free resolution is called minimal if φ i+1 (F i+1 ) ⊂ mF i , ∀i.
Minimal free resolution of a finitely generated module is finite which follows from the following theorem.
Theorem 2.1 (Hilbert Syzygy Theorem). Let S = K[x 1 , x 2 , . .
. , x n ] be a polynomial ring of n variables over a field K. Then the graded minimal free resolution of a graded finitely generated S-module is finite, and its length is at most n.
A finitely generated S-module M has a d-linear resolution if the minimal free resolution is of the form
0 → S(−d − i) β i → . . . → S(−d − 1) β 1 → S(−d) β 0 → M → 0
where the degree of each generators of M is d.
The module M has a pure resolution if the minimal free resolution is of the following form
0 → S(−d i ) β i → . . . → S(−d 1 ) β 1 → S(−d 0 ) β 0 → M → 0 such that 0 < d 0 < d 1 < . . . < d i are all integers.
An ideal I is said to be Cohen-Macaulay if the Krull dimension coincide with the depth.
2.2.
Basic notions of Graph. Let G be a finite simple graph on n-vertices without isolated vertices. A subgraph H of G is called induced, if for every vertices u and v in H whenever {u, v} is an edge of G, it is an edge of H. A graph G is called chordal graph if every induced subgraph of G has cycle of length 3, and it is called co-chordal if its complement is chordal. A complete graph is called clique. The largest number of the clique is called clique number of G. A vertex v of G whose deletion from the graph gives a graph with more connected components, then v is called a cut point of G. A subset T ⊂ [n] is said to have a cut point property (in short, cut point set) for G, if for every v ∈ T , c(T \ {v}) < c(T ), where c(T ) is the number of connected components of the restriction of G to [n] \ T . A cut set of a graph G is a subset of the vertices whose deletion increase the number of connected components of G.
Let G 1 and G 2 be two graphs with vertex sets V 1 and V 2 and edge sets E 1 and E 2 respectively. The join of two graphs G 1 and G 2 denoted by G 1 * G 2 , is a graph defined on the vertex set V 1 ∪ V 2 and on the edge set E 1 ∪ E 2 ∪ {{x, y} : x ∈ V 1 , y ∈ V 2 }. For a graph G, a maximal subgraph of G without cut vertex is called a block of G. A graph G is said to be a block graph if each block of G is a clique. [40]) a graph G is said to be a quasi-block graph if G satisfies the following:
Definition 2.2. (See,(1) Each block of G is either a clique or a quasi-block (2) If v is an internal vertex of a quasi-block B, then for any u ∈ N G (v) \ V (B), u is not an internal vertex for any block. A graph G is called an interval graph if every vertex v ∈ V (G) can be labelled with a real closed interval I v = [a v , b v ] in such a way that two distinct vertices v, w ∈ V (G)
are adjacent if their corresponding intervals have non-empty intersection. The vertices of the interval graph G, are intervals namely
V (G) = {I 1 , . . . , I r }, where I j = [a j , b j ] with a j ≤ b j , for all 1 ≤ j ≤ r. Let N 0 = N ∪ {0}. Also, let J 0 = [0], J i = [i − 1, i], i = 1, . . . , k and I j = [a j , b j ], such that a j ∈ N 0 and I j ⊆ [0, k]
{k}, for all j = 1, 2, . . . , r. Then we call the interval graph on the vertex set {J i } k i=0 ∪ {I j } r j=1 a connected strong interval graph or simply a strong interval graph.
Regularity of binomial edge ideal of graphs
In this section, we will provide an overview of regularity of binomial edge ideal of graphs. In [61] and [63], authors have characterized regularity of binomial edge ideal of 2 and 3 respectively and its initial ideal. In the next theorem the authors characterize when binomial edge ideal and its initial ideal have a linear resolution that is reg(J G ) = 2. (1) J G has a linear resolution.
(2) in < (J G ) has a linear resolution.
(3) G is a complete graph.
In [14], authors conjectured that the regularity of binomial edge ideal of a graph is equal to the regularity of its initial ideal. This conjecture holds for the join of two graphs.
Theorem 3.2 ([63], Theorem 2.1). Let G 1 and G 2 be graphs on disjoint vertex sets V 1 and V 2 respectively, not both complete, let < be any term order on ring S. Then
(1) reg(J G 1 * G 2 )=max{reg(J G 1 ), reg(J G 2 ), 3} (2) reg (in < J G 1 * G 2 )=max{reg(in < J G 1 ), reg(in < J G 2 ), 3}
As an application of Theorem 3.2, they have characterized binomial edge ideal of regularity 3.
(1) G = K r ⊔ K s , with r, s ≥ 2 and r + s = n, or (2) G = G 1 * G 2 ,
where G i is a graph with n i < n vertices such that n 1 + n 2 = n and reg(J G i ) ≤ 3, for i = 1, 2.
In [62], Madani
1 = a 1 < a 2 < . . . < a s < a s+1 = m such that F i = [a i , a i+1 ], 1 ≤ i ≤ s and F 1 , F 2 , . . . , F S is a leaf order of △(H). Let F s+1 = [m, n] ∪ {1}. The graph on the vertex set [n] and edge set E(G) = E(H) ∪ {{i, j} : i = j, i, j ∈ F s+1 } is called a semi-cycle graph associated with H. Definition 3.4. A block B of a graph G is said to be a semi-block if B is a semi-cycle with B = K 3 . A
graph G is said to be a semi-block graph if all except one block are cliques and the block which is not a clique is a semi-block.
In [40], the author proved this conjecture for quasi block graphs and for semi block graphs. In [40], the author proved the same conjecture independently for chordal graph see, Theorem 3.15 and for Jahangir graphs see, Theorem 4.5. In [24], Hibi-Matsuda conjectured that for a graph G on n-vertices, reg(S/J G ) ≤ degh S/J G (t). Characterization of generalized block graphs which admit unique extremal Betti number is given in [38]. In [40], author provide a sufficient condition for this conjecture. In [40], Example 5.3 author has given a counter example for this conjecture. Also the author posed a question which is the following:
Question 3.8
. When does the binomial edge ideal of a graph admit a unique extremal Betti number?
In [30], authors proved that regularity of S/J G is less than or equal to n − 1, for some classes of trees. An improved upper bound for the regularity of binomial edge ideal of trees is given as a corollary of the
I = I 0 ∼ I 1 ∼ . . . ∼ I m = J.
I is said to be linkage class complete intersection, in short licci, if J is a complete intersection ideal.
If I and J are linked, then if one of them is Cohen-Macaulay then other is too. In particular, any licci ideal is always Cohen-Macaulay. A necessary condition for a homogeneous ideal in a polynomial ring which is Cohen-Macaulay as well as licci is given in [26].
I m ⊂ R = S m is licci, then reg(S/I) ≥ (heightI − 1)(indegI − 1)
where indegI is the initial degree of the ideal I, that is indegI = min{i : I i = 0}.
In [17], the authors have given a new upper bound for the regularity of S/J G .
reg(S/J G ) ≤ n − dim△(G) where △(G) is a simplicial complex of G.
For a disconnected graph G, the authors have found an upper bound, and it is the following Next theorem has improved the upper bound given by Matsuda and Murai in [49].
(1) (J G ) m ⊂ R = S m is licci.
(2) J G is Cohen-Macaulay and n − 2 ≤ reg(S/J G ) ≤ n − 1 (3) G is a path graph or it is isomorphic to one of the graphs in Figure 1, where r, s, t are non-negative integers. In other words, G is a triangle with possibly some paths connected to some of its vertices.
An equivalent statement of 3.14 for chordal graph is the following.
(1) (J G ) m ⊂ R is licci. (2) J G is Cohen-Macaulay and n − 2 ≤ reg(S/J G ) ≤ n − 1. (3) J G is unmixed and n − 2 ≤ reg(S/J G ) ≤ n − 1.
(4) G is a path graph or it is isomorphic to a graph in Figure 1.
An explicit formula for the regularity of binomial edge ideal for the block graphs in terms of combinatorics of the graph is an open problem. An algorithm for dimension of binomial edge ideal for block graph is given in [47], Algorithm 2.5.
For a flower graph, extremal Betti number is given in [47], Theorem 3.4. As a consequence, one can get regularity of binomial edge ideal for flower graph F (v). and G 2 , . . . , G c are flower free graphs.
G if G = G 1 ∪ . . . ∪ G c , where c = cdeg(v), such that G i ∩ G j = {v}, for all 1 ≤ i < J ≤ c
The formula for the regularity of binomial edge ideal of block graphs. where c is the number of connected components of H r which are not isolated vertices.
In [38], the author has given improved lower and upper bound for the regularity of binomial edge ideal of generalized block graphs. In the same article, they have also characterized generalized block graph in which binomial edge ideal admits a unique extremal Betti number.
Theorem 3.20 ([38], Theorem 3.11). Let G be a connected indecomposable generalized block graphs. Then the following are equivalent
(1) S/J G admits a unique extremal Betti number.
(2) For any v ∈ V (G), F h,k (v) is not an induced subgraph of G, for any h, k ≥ 0 with h + k ≥ 3.
In this case, reg(S/J G ) = m(G) + 1.
where F h,k (v) is a flower graph obtained by gluing of h copies of the complete graph K 3 and k copies of the star graph K 1,3 at a common free vertex v and m(G) is the number of minimal cut sets of the graph G.
Let deg G (v) denote the degree of a vertex v of a graph G. If deg G (v) = 1, then we say v is a pendant vertex. Let for v ∈ V (G), the number of maximal cliques of G which contain v, be denoted by cdeg G (v) and the number of pendant vertices adjacent to v be denoted by
pdeg G (v). A vertex v ∈ V (G) such that cdeg G (v) = pdeg G (v) + 1 with pdeg G (v) ≥ 1 is said to be of type 1, and of type 2 if cdeg G (v) ≥ pdeg G (v) + 2.
Let α(G) be the number of vertices of type 1, and pv(G) be the number of pendant vertices of G.
With the same notations as in above, in [38], the author obtained upper bound for the regularity of binomial edge ideal of connected indecomposable generalized block graphs.
E(G) ∪ {{u, w} : {u, w} ⊆ N G (v)}
In [58], the authors give a general upper bound for regularity of binomial edge ideal. For that they have introduced a map, called compatible map, and the definition is as the following:
Definition 3.22. Let G be the set of all graphs. We call a map φ : G → N 0 , compatible, if it satisfies the following conditions
(1) φ(Ĝ) ≤ φ(G), for all G ∈ G (2) If G = ⊔ t i=1 K n i , n i ≥ 2, for every 1 ≤ i ≤ t, then φ(G) ≥ t. (3) If G = ⊔ t i=1 K n i , then there exists v ∈ V (G) such that (a) φ(G − v) ≤ φ(G) and (b) φ(G v ) < φ(G).
Theorem 3.23 ([58], Theorem 2.4). Let G be a graph on [n] and φ a compatible map. Then
reg(S/J G ) ≤ φ(G).
Next we will see a new upper bound for regularity of binomial edge ideal which is clique disjoint edge set in graph.
Depth of Binomial edge ideal
In general, it is hard to compute another algebraic invariant such as the depth of binomial edge ideals. Regarding the depth of binomial edge ideal not much have been done. In the next result we will see the formula for the depth of cone graph. The definition of cone graph is the following
In particular, if H is Cohen-Macaulay, then G is almost Cohen-Macaulay.
For the binomial edge ideal of cone graph, depth formula is the following. [66], authors have shown that depth (S/J G ) = n, for n > 3. In [14], authors proved that depth (S/J G ) = n + 1, for a connected block graph G. Later in [36], authors computed the depth of generalized block graphs. In [44], authors have given the formula for depth of join product of two graphs G 1 and G 2 , which is denoted by G 1 * G 2 . In [3], the authors has given an upper bound in terms of graphs invariant for the depth of binomial edge ideal of a graph. They proved that for a non-complete connected graph G, depth (S/J G ) ≤ n − κ(G) + 2, where κ(G) denotes the vertex connectivity of G.
In [57], authors proved Hochster type formula for the local cohomology modules of binomial edge ideal, which is based on [1],Theorem 3.9.
H i m (S/J G ) ∼ = ⊕ q∈G H dq m (S/q) ⊕M i,q where d q = dimS/q and M i,q = dim K H˜i −dq−1 ((q, 1 Q G ); K).
In the next theorem, the authors have given a lower bound for the depth of binomial edge ideal.
(1) depth(S/J G ) = 4 (2) G = G ′ * 2K 1 , for some graph G ′ ,
where 2K 1 is the graph consisting of two isolated vertices.
In [3], authors proved that for a connected non-completed graph G,
depth(S/J G ) ≤ n + 2 − κ(G),
where κ(G) is the graph connectivity of G. In [46], authors proved that
f (G) + d(G) ≤ depth(S/J G ).
In [27], authors have characterized graphs for which depth of S/J G attain the lower bound. Next question arises is, whether there exists any graph G such that f (G) + d(G) = n + 2 − κ(G) or not. Existence of such graphs are characterized by Hibi and Saeedi Madani in [23]. Also, in [27], author characterizes connected non-complete graphs G with the property that f (G) + d(G) + 1 = n + 2 − κ(G). (1) If κ(G) = 1, then either G is chordal or G has precisely one induced C 4 and has no induced C l , for l ≥ 5. In [29], authors computed the first Betti number of binomial edge ideal of trees (see, Theorem 3.1) and second Betti number of unicyclic graphs (see, Theorem 3.4). In the same paper, they have calculated first syzygy of binomial edge ideal for cycle C n , Theorem 3.5. Next, they have calculated first syzygy of binomial edge ideal for unicyclic graphs (Theorem 3.6, 3.7) and for trees.
Let A ⊂ [n], and i ∈ A, define P A (i) = |{j ∈ A : j ≤ i}|. Here, P A denotes the position of an element in A when the elements are arranged in the ascending order. Now we are ready to state the following theorem.
(2) (−1) P A (j) f k,l e {i,j} +(−1) P A (k) f j,l e {i,k} +(−1) P A (l) f j,k e {i,l} , where A = {i, j, k, l} ∈ C G with center at i. where C G = v∈V (G) deg G (v) 3 .
Rees algebra of an ideal I is R(I) = ⊕ n≥0 I n t n . The generators of the defining ideal of the Rees algebra of a graph is described by Villarreal in [65]. He proved that I(G) is of linear type, that is the Rees algebra is isomorphic to the symmetric algebra if and only if G is either a tree or an odd unicyclic graph. But almost nothing is known about the Rees algebra of binomial edge ideal of a graph. It is known that for a connected graph G, J G is complete intersection if and only if G is a path [14].
Let G be a graph on n-vertices and J G be its binomial edge ideal.
Let R = S[T {i,j} ∈ E(G), i < j]. Let δ : R → S[t]
be the S-algebra homomorphism given by δ(T {i,j} ) = f i,j t.
Then Im(δ) = R(J G ) and ker(δ) is called the defining ideal of R(J G ).
The following theorem characterizes the trees whose binomial edge ideal are almost complete intersection.
Theorem 5.2 ([57], Theorem 4.3). If G is a tree which is not a path, then J G is an almost complete intersection ideal if and only if G is obtained by adding an edge between two vertices of two paths.
In general, they have characterized for connected graph which is not a tree in Theorem 4.4. Also they have shown that the associated graded ring and the Rees algebra of almost complete intersections binomial edge ideal are Cohen-Macaulay. . If G is a graph such that J G is an almost complete intersection ideal, then gr S (J G ) and R(J G ) are Cohen-Macaulay.
Since, complete intersections are of linear type, binomial edge ideal of paths are linear type. In [57], Proposition 4.9, authors proved that binomial edge ideal of K 1,n is of linear type. In the same paper authors proved that in the polynomial ring over an infinite field, almost complete intersection homogeneous ideals are generated by d-sequence, Proposition 4.10. As a corollary of the above result we have the following. . Classify all bipartite graphs whose binomial edge ideal are of linear type.
The above question is not true for tree. So, one can prove or disprove the following, Conjecture 5.6 ([57], Conjecture 4.17). If G is a tree or a unicyclic graph, then J G is of linear type.
5.2.
Construction of a graph. In [22], Hibi et. al. constructed a graph G such that for 1 ≤ b ≤ r, the regularity of monomial edge ideal of G is r, and the number of its extremal Betti number is b. Also in [24], for a given pair (r, s) with 1 ≤ r ≤ s, Hibi and Matsuda constructed a graph G such that reg(S/J G ) = r and the degree of the h-polynomial of S/J G is s. In [44], authors constructed the following. Cohen-Macaulay binomial edge ideals are studied by many authors [14], [53]. Also, Cohen Macaulay binomial edge ideal for bipartite graphs and block graphs were studied in [14], [3], [5], [36]. But the classification of it in terms of underlying graph is not known. In [55], authors have classified Cohen Macaulay and unmixed binomial edge ideal with deviation. As an extension of these results, in [56], authors classified Cohen-Macaulay and unmixed binomial edge ideal J(G), where G is a cactus graph i.e. a graph whose blocks are cycles.
In [14], Ene, Herzog and Hibi proved that if G is a closed graph then S/J G is Gorenstein if and only if G is a path. Motivated by this result, author in [18] obtained the similar result for connected graph G. 5.4. Hilbert series. The Hilbert series of binomial edge ideal of cycles and quasi cycles is computed in [50], [66]. Authors in [48] have computed the Hilbert-Poincare series of the binomial edge ideal of some Cohen-Macaulay bipartite graphs, Fan graphs (see Proposition 3.8). In [43], authors obtained the Hilbert series for decomposable graphs in terms of indecomposable components.
Definition 5.10. A graph G is said to be decomposable, if there exist induced subgraphs
G 1 and G 2 , such that G = G 1 ∪ G 2 , V (G 1 ) ∩ V (G 2 ) = {v},
and v is a free vertex of G 1 and G 2 .
Hilb S/J G (t) = (1 − t) 2 Hilb S 1 /J G 1 (t)Hilb S 2 /J G 2 (t) ,
where S i = K[x j , y j : j ∈ V (G i )], for i = 1, 2
They also have obtained dimension and multiplicity of S/J G . As a corollary of 5.11, they have computed Hilbert series for a connected graph. In the same paper, authors computed Hilbert series for join of two graphs.
Hilb S/J G (t) = Hilb S H /J H (t) + Hilb S H ′ /J H ′ (t) + (p+r−1)t+1 (1−t) p+q+1 − (p−1)t+1 (1−t) p+1 − (q−1)t+1
(1−t) q+1 5.5. Symbolic power. In general, symbolic power and ordinary power of an ideal do not coincide. But for some classes of homogeneous ideals in polynomial rings it coincide, e.g. for a bipartite graph symbolic power and ordinary powers of an edge ideal coincide.
In [13], Ene and Herzog proved that for any binomial edge ideal with quadratic Gröbner basis, the symbolic powers and ordinary powers of J G coincide. Authors have shown that based on the condition of in < (J G ) these two powers coincide. G = J k G , for k ≥ 1 As a consequence of the above result one can get the equality of these two powers for a closed graph, see, [13], Corollary 3.4. 5.6. Hochster Type formula. Hochster formula originally appeared in [25], it provides a decompostion of the local cohomology modules H r m (A/I) of the Stanley-Reisner ring R/I, associated to a squarefree monomial ideal I ⊆ A = K[x 1 , . . . , x n ]. It also describes the Hilbert series of the local cohomology modules of A/I. In [1] authors have given a Hochster type decomposition for the local cohomology modules associated to binomial edge ideal and is as the following. where M r,q = dim k Hr −dq−1 ((q, 1 Q J G ); K). Moreover, we have a decomposition as graded K-vector spaces.
In [8], when authors comparing invariant of Cartwright-Strumfels ideals with its generic initial ideal they got the following In [1], the author has proved the conjecture 5.15 for binomial edge ideal.
dim K H r m (A/J G ) a = dim K H r m (A/gin(J G )
) a , for every r ∈ N and every a ∈ Z n .
Generalized Binomial edge ideal
Generalized binomial edge ideal is introduced by Rauh in [54]. It is the ideal generated by a collection of 2-minors in generic matrix. Motivation to study these ideals comes from their connection to conditional independence ideals.
p kl ij = [k, l][i, j] = x ki x lj − x li x kj The ideal J G = (p kl ij : 1 ≤ k < l ≤ m, {i, j} ∈ E(G) is called the generalized binomial edge ideal.
In [54], it was proved that J G is a radical ideal and its minimal primes are determined by the sets with the cut point property of G. Not much is known about minimal free resolution of generalized binomial edge ideal. Madani and Kiani in [62], proved that J G has linear resolution if and only if m = 2 and G is a complete graph.
Let G be a generalized block graph. Let A i (G) be the collection of cut sets of G of cardinality i, i = 1, 2, . . . , ω(G) − 1. We denote a i (G) = |A i (G)|. In [7], authors proved the equality of depth of generalized binomial edge ideal of generalized block graph and depth of its initial ideal, and also found the regularity of these two generalized block graph.
(1) depth (S/J G )=depth (S/in < (J G )) = n + (m − 1) − ω(G)−1 i=2 a i (G) (2) If m ≥ n, then reg S/J G =reg S/in < (J G ) (3) If m < n, then reg S/J G ≤ reg S/in < J G ≤ n − 1.
As a consequence, one can get similar statements for block graph, as we have for block graph a i (G) = 0, for all i > 1. Since tree is a block graph, we can have equivalent result of the theorem 6.2 for tree. For a path graph regularity of generalized binomial edge ideal is the following Independently, in [39] author has given regularity bound for generalized binomial edge ideal. The author has defined in the following way Let G 1 and G 2 be graphs on the vertex set [m] and [n] respectively. Let e = {i, j} ∈ E(G 1 ), and e ′ = {k, l} ∈ E(G 2 ). Assign 2-minor p e,e ′ = x i,k x j,l − x i,l x j,l to the pair (e, e ′ ). The binomial edge ideal of the pair (G 1 , G 2 ) is, J G 1 ,G 2 = (p e,e ′ : e ∈ E(G 1 ), e ′ ∈ E(G 2 )).
The generalized binomial edge ideal of a graph G is the binomial edge ideal of the pair (K m , G). When m = 2, the ideal J K 2 ,G = J G is the classical binomial edge ideal of G.
In [62], Madani and Kiani conjectured that, if G is a graph on vertex set [n], then reg(S/J Km,G ) ≤ min { m 2 c(G), e(G)}, where c(G) denote the number of maximal cliques of G, and e(G) denote the number of edges in G. They have proved the conjecture for closed graphs. In [39], author proved this conjecture for chordal graphs. In [39], author computed regularity of generalized binomial edge ideal for complete graph K n . Also, for a connected graph G, in the same paper author obtained an upper bound for the regularity of generalized binomial edge ideal. For chordal graph, the upper bound for regularity is the following. In [39], author conjectured for generalized binomial edge ideal for a connected graph, for m < n. One can ask the following question for pair of a graphs G 1 and G 2 . In [7], [16], for some special cases unmixedness and Cohen-Macaulayness of binomial edge ideal of a pair of graphs are characterized. It is well-known that all Cohen-Macaulay ideal are unmixed. In [2], author characterizes all the unmixed ideal J G,m . Before that we will see the definition of power-cycle.
Let S ⊆ {1, 2, . . . , ⌊ n 2 ⌋}. The circulant graph G = C n (S) is a simple graph with V (G) = Z n = {1, 2, . . . , n − 1} and E(G = {{i, j} : |j − i| n ∈ S}), where |k| n = min{|k|, n − |k|}. C n (S) is the circulant graph of order n with generating set S and |k| n is the circular distance modulo n. It is known that if the generating set S = {±1, ±2, . . . , ±d}, where 1 ≤ d ≤ ⌊ n 2 ⌋ is a given integer, then the circulant graph C n (S) is equivalent to the d-th power of C n , where two vertices are adjacent if and only if their distance is at most d. In this case, we will denote C n (S) by C n (1, 2, . . . , d), and is called dth power-cycle. Theorem 6.10 ([2], Theorem 3.14). Let G = C n (1, 2, . . . , r) be a non-complete powercycle. Then J G,m is unmixed if and only if m is odd, n ∈ {m + 1, . . . , 3m+1 2 } and r = m−1 2 . In the same paper the author characterizes when J G,m is Cohen-Macaulay.
Other Binomial Edge Ideals
In this section, we will give an overview of other binomial ideals namely the parity binomial edge ideal, the permanental edge ideal and the Lovasz-Saks-Schrjiver ideal. In [33], the authors have given Gröbner basis for parity binomial edge ideal I G , see Theorem 3.6 and minimal primes in Theorem 4.15. Also they have proved when I G is radical ideal. In the same article they have calculated primary decomposition of parity binomial edge ideal. In [41], the author has given an alternative form of the results from [9], for LSS ideals in terms of parity binomial edge ideal. They have also characterized the non-bipartite graphs whose LSS ideals, permanental edge ideal and parity binomial edge ideals are complete intersection. In the same article, they have computed projective dimension of the parity binomial edge ideal of an odd unicyclic graphs. Also, they have computed second Betti number and first syzygy of LSS ideals for odd unicyclic graphs in Theorem 5.3 and 5.4. In [42], the author has given upper bound for the regularity of parity binomial edge ideal of a graph and for an odd cycle. In the next theorem, the author has characterized the graphs whose parity binomial edge ideal have regularity three. In the following theorem, the author has characterizes graphs whose parity binomial edge ideal have pure resolution. In particular, depth(S/I Kn ) ≥ 3 and reg(S/I Kn ) ≤ 3.
Conjecture 1. 1 .
1Let G be a simple graph and c(G) the number of maximal cliques of G. Then reg (S/J G ) ≤ c(G).
Conjecture 1.2 ([24], Conjecture 0.1). Let G be a graph on n vertices. Then
∀j are the free S-modules. The numbers β ij = β ij (M) are called the graded Betti numbers of M, and β i = j β ij is called the total ith Betti number of M. The projective dimension of M is defined by pd(M) = max{i : β ij = 0, for some i} The regularity of M is defined by reg(M) = max{j − i : β ij = 0}. The depth of M is defined by depth(M) = n − proj dim(M)
Theorem 3.1 ([6], Theorem 1.4 and [61], Theorem 2.1). Let G be a simple graph, and < be any term order. Then the following are equivalent.
.
Let G be a non-complete graph with n vertices, and no isolated vertices. Then reg(J G ) = 3 if and only if either
Theorem 3.5 ([40], Theorem 3.7 and Theorem 3.11). For a quasi-block graph as well as semi block graph G, reg(S/J G ) ≤ c(G).
Definition 3. 6 .
6The Jahangir graph denoted by J m,n is a graph on the vertex set [m, n + 1], m ≥ 1, n ≥ 3 such that the induced subgraph on [mn] is C mn and the neighbourhood of the vertex mn + 1 is {1, m + 1, . . . , m(n − 1) + 1}.
Theorem 3.7 ([40], Theorem 5.1). Let G be a connected graph on vertex set [n]. If S/J G admits a unique extremal Betti number, then reg(S/J G ) ≤ degh S/J G (t).
Theorem 3 .
311 ([26], Corollary 5.13). Let I be a Cohen-Macaulay homogeneous ideal in a standard graded polynomial ring S = K[x 1 , . . . , x n ] with the graded maximal ideal m. If
Theorem 3 .
312 ([17], Theorem 2.1). Let G be a connected graph on vertex set [n]. Then
Theorem 3 .
313 ([17], Corollary 2.2). Let G be a graph on n vertices with the connected components G 1 , . . . , G c . Then reg(S/J G ) ≤ n − (dim△(G 1 ) + . . . + dim△(G c )).
Figure 1 .
1Licci graphs
.
Let G be a connected graph on the vertex set [n]. Then the following are equivalent
Theorem 3 .
315 ([17], Theorem 4.2). Let G be a connected chordal graph on the vertex set [n]. Then the following are equivalent
Corollary 3 .
316 ([47],Corollary 3.5). Let F (v) be a flower graph, thenregS/J F (v) = i(F (v)) + cdeg(v) − 1where cdeg(v) is the clique degree of v, is the number of maximal cliques to which v belongs and i(F (v)) is the inner vertices of F (v).Definition 3.17. Let G be a block graph. If G has no flower graph as an induced subgraph, then G is called flower free graph. Definition 3.18. Let G be a block graph and F (v) be a flower graph as an induced subgraph of G. F (v) is called an end-flower of
Theorem 3 .
319 ([47], Theorem 4.2). Let G be a block graph, v 1 , . . . , v r ∈ V (G) H j = G {v 1 , . . . , v j } j = 1, 2, . . . , r and H 0 = G. If
( 1 )
1F (v) is an end-flower for H j−1 , for all j = 1, . . . , r (2) H r is flower-free then regS/J G = regS/J Hr = c + i(H r )
Theorem 3 .
321 ([38],Theorem 4.5). Let G be a connected indecomposable generalized block graph on [n] vertices, which is not a star graph. Thenreg(S/J G ) ≤ c(G) + α(G) − pv(G)where c(G) is number of maximal cliques of G.Let G be a graph on the vertex set V (G), and for subset T ⊆ V (G) the induced subgraph of G on the vertex set V (G) \ T , be denoted by G − T . Let N G (v) denote the set of neighbours of the vertex v in G. We setĜ = G \ Is(G), where Is(G) denotes the set of isolated vertices of G and K n denote the complete graph on n vertices. Associated to a vertex v in V (G), there is a graph denote by G v , with the vertex set V (G) and the edge set
Definition 3 . 24 .
324Let G be a graph and H ⊆ E(G) with the property that no two element of H belong to a clique of G. Then we call the set H, a clique disjoint edge set in G. we set η(G) = max{|H| : His a clique disjoint edge set inG} Theorem 3.25 ([58], Corollary 2.7). Let G be a graph on vertex set [n]. Then reg(S/J G ) ≤ η(G).So, by above result we get a sharp bound for regularity of binomial edge ideal, as η(G) ≤ c(G), for every graph G. In[59], authors proved that regularity of strong interval graph of G coincide with L(G) as well as c(G), where L(G) is the sum of the lengths of the longest induced paths of connected components of G.
Theorem 3 .
326 ([59], Corollary 4.3). Let G be a chordal graph. Then the following are equivalent (1) reg(S/J G ) = L(G) = c(G) (2) G is a strong interval graph.
Definition 4. 1 .
1Let H be a graph on vertex set[n].The cone of v on H is denote by v * H, is a graph with vertex set V (v * H) = V (H)⊔{v} and edge set E(v * H) = E(H)⊔{{u, v} : u ∈ V (H)}. Let G = v * H and S H = K[x i , y i : i ∈ V (H)] and S = S H [x v , y v ].In[44], authors has proved that under the process of taking cone on connected graph depth remains invariant .
Theorem 4. 2
2([44], Theorem 3.4). Let H be a connected graph on the vertex set [n] and let G = v * H be the cone graph. Then depth S (S/J G ) = depth S H (S H /J H )
.
Let G = v * H, where H is a disconnected graph on vertex set [n]. Then depth S S/J G = min{depth S H (S H /J H ), n + 2}.Depth formula of binomial edge ideal for join of two graphs can be found in[44]i m (S/J G ) denote the ith local cohomology module of S/J G , where maximal ideal associated to it is m = (x 1 , . . . , x n , y 1 , . . . , y n ). Then depth(S/J G ) = min{i : H i m (S/J G ) = 0}. In
Theorem 4.4. ([57], Theorem 3.6) Let G be a graph on [n], and Q(G) be the poset associated to J G . Then we have the K-isomorphism
Theorem 4.5 ([57],Theorem 5.2). Let G be a graph on[n]. Thendepth(S/J G ) ≥ 4r + n i=1 r i (i + 1), where r is the number of non-complete connected components of G, and r i is the number of complete connected components of G of size i, for every 1 ≤ i ≤ n.Next, the authors characterized the graphs for depth(S/J G ) = 4.
Theorem 4.6 ([57], Theorem 5.3). Let G be a graph on [n] with n ≥ 4. Then the following are equivalent.
Theorem 4. 7
7([27],Corollary 5.6). Let G be a unicyclic graph or a quasi-cycle graph such that depth(S/J G ) = d(G) + f (G). Then either G is chordal or G has an induced C 4 .
Theorem 4. 8
8([27], Theorem 3.9). Let G be a graph on [n] such that d(G) + f (G) + 1 = n + 2 − κ(G).
( 2 )
2If κ(G) ≥ 2 and d(G) = 2, then G is a chordal graph.(3) If κ(G) = 2 and d(G) = 3, then either G is chordal or G has precisely one induced C 4 , and has no induced C l , for l ≥ 5.The authors in ([27], Section 4) described the structure of the graphs with the property f (G) + d(G) + 1 = n + 2 − κ(G) and calculated the depth of S/J G . Also in the same paper authors posed the following question.
Question 4. 9
9([27], Question 5.5). If G is a graph containing an induced cycle of length at least 5, then is depth(S/J G ) ≥ d(G) + f (G + 1)?
Corollary 5. 4
4([57],Corollary 4.11). Let G be a graph on n-vertices. If J G is an almost complete intersection ideal, then J G is generated by a d-sequence. In particular, J G is of linear type.Also, they have obtained the defining ideal of Rees algebra of binomial edge ideal of cycles as a corollary, see [[57], Corollary 4.13]. Authors have asked the following question: Question 5.5 ([57], Question 4.16)
.
Let r and b be two positive integers with 1 ≤ b ≤ r −1. Then there exist a graph G = G r,b such that reg(S/J G ) = r, and the number of extremal Betti number of S/J G is b. Question 5.8. Does there exists a graph G such that the projective dimension is bounded by a linear function of b and r, where r = reg(S/J G ), and b is the number of extremal Betti number of S/J G ? 5.3. Cohen-Macaulay Binomial edge ideal.
Theorem 5. 9
9([18], Theorem 5.3). Let S = K[x 1 , . . . , x n , y 1 , . . . , y n ]. Suppose the char(K) = p > 0. Let G be a connected graph such that S/J G is Gorenstein. Then G is a path. The above result also holds for char(K) = 0 ([18], Theorem 5.4).
Theorem 5 .
511 ([43], Theorem 3.2). Let G = G 1 ∪ G 2 be a graph on vertex set[n]. Then
Theorem 5 .
513 ([13], Theorem 3.3). Let G be a connected graph on the vertex set [n]. If in < (J G ) is a normally torsion-free ideal, then J (k)
Theorem 5 .
514 ([1], Theorem 3.9). Let A = K[x 1 , . . . , x n , y 1 , . . . , y n ] be a polynomial ring over a field K. Let J G be the binomial edge ideal associated to a graph G on the vertex set [n]. Let Q J G be the poset associated to a minimal primary decomposition of J G . Then the local cohomology modules with respect to m of A/J G admit the following decompostion as K-vector spaces H r m (A/J G ) ∼ = ⊕ q∈Q J G H dq m (A/I q ) ⊕Mr,q
Conjecture 5 .
515 ([8], Conjecture 1.14). Let I ⊆ A be a Z m -graded Cartwright-Strumfels ideal and gin (I) its Z m -graded generic initial ideal, with m ≤ n. Then one has dim K H r m (A/I) a = dim K H r m (A/gin(I)) a . for every r ∈ N and every a ∈ Z m .
Theorem 5.16([1],Theorem 4.5). Let A = K[x 1 , . . . , x n , y 1 , . . . , y n ] be a polynomial ring over K and m be its homogeneous maximal ideal. Let J G be binomial edge ideal associated to a graph G on the vertex set[n]. Then
Definition 6 . 1 .
61Let m, n ≥ 2 be integers. Let G be a simple graph on the vertex set[n]. Let X = (x ij ) be an m × n matrix of indeterminate and denote S = K[X] the polynomial ring in the variables x ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n, over a field K. For 1 ≤ k < l ≤ m and {i, j} ∈ E(G) with 1 ≤ i < J ≤ n we set
Theorem 6.2 ([7], Theorem 3.3). Let m, n ≥ 2, and let G be a connected generalized block graph on the vertex set [n]. Then
.
Let m, n ≥ 2. If G is a path graph on the vertex set [n], then reg S/J G =reg S/in < (J G ) = n − 1.
Theorem 6.4 ([39], Theorem 3.13). Let G be a connected chordal graph on the vertex set [n]. If m < n, then reg(S/J Km,G ) ≤ min{(m − 1)c(G), n − 1}.
Proposition 6.5 ([39], Proposition 3.3). Let G = K n . Then reg (S/J Km,Kn ) = min{m − 1, n − 1}.
Theorem 6.6 ([39], Theorem 3.6). Let G be a connected graph on vertex set [n] and m ≥ 2. Then reg(S/J Km,G ) ≤ n − 1.
Theorem 6.7 ([39], Theorem 3.11). Let G be a connected chordal graph on the vertex set [n]. Then reg(S/J Km,G ) ≤ (m − 1)c(G).
Conjecture 6.8 ([39], Conjecture 3.15). Let G be a connected graph of vertex set [n]. Then reg(S/J Km,G ) ≤ min{(m − 1)c(G), n − 1}, if m < n.
Question 6.9 ([39], Question 3.16). Let G 1 and G 2 be graphs on the vertex set [m] and [n] respectively. Then reg(S/J G 1 ,G 2 ) ≤ min{(m − 1)c(G 2 ), (n − 1)c(G 1 )}.
Theorem 6 .
611 ([2], Corollary 4.3). Let J G,m be a generalized binomial edge ideal with m ≥ 3. Then J G,m is Cohen-Macaulay if and only if G is the complete graph.
Theorem 7.1 ([33], Theorem 5.5). Let G be a graph. If char(K) = 2, then I G is a radical ideal.
Theorem 7.2 ([41], Theorem 3.2). Let G be a bipartite graph on [n]. Then L G is complete intersection, if and only if I G is complete intersection, if and only if G is a disjoint union of paths.
Theorem 7.3 ([41], Theorem 3.5). Let G be a non-bipartite graph on [n]. Then L G is complete intersection, if and only if G is an odd cycle, if and only if I G is complete intersection.
.
For a connected odd unicyclic graph on [n], projective dimension of I G is pd(S/I G ) = n. Next, they have computed second Betti number and first syzygy of LSS ideals of trees.
.
Theorem 7.5 ([41], Theorem 5.1). Let G be a tree on[n]. Then Let G be a tree on[n]. If {e {i,j} : {i, j} ∈ E(G)} is standard basis for S n−1 , then the first syzygy of L G is generated by the following elements(1) g i,j e {k,l} − g k,l e {i,j} , where {i, j} = {k, l} ∈ E(G) and (2) (−1) p A (j) f k,l e {i,j} + (−1) p A (k) f j,l e {i,k} + (−1) p A (l) f j,k e {i,l} ,where A = {i, j, k, l} ∈ C G , with center at i.
.
Let G be a connected bipartite graph on n-vertices. Then reg(S/I G ) ≤ n − 1. Moreover, reg(S/I G ) = n − 1, if and only if G is a path graph.Theorem 7.8 ([42], Theorem 3.4). Let G = C n , where n is odd. Then reg(S/I G ) = n.
Theorem 7. 9
9([42], Theorem 3.9). Let G be a graph on n-vertices with no isolated vertex. Then reg(S/I G ) = 2 if and only if either G = K 2 ⊔ K 2 or G is a complete bipartite graph other than K 2 .
Theorem 7 .(
710 ([42],Theorem 3.14). Let G be a graph on n-vertices. Then S/I G has pure resolution if and only if G is one of the following (1) G is a complete bipartite graph.(2) G is a disjoint union of some odd cycles and some paths.Let H S/I G be the Hilbert function of S/I G . Then the Hilbert-Poincare series of the Smodule S/I G isHP S/I G (t) = i≥0 H S/I G (i)t i .From ([52], Theorem 16.2), the series has the following expressionHP S/I G (t) = P S/I G (t)(1−t) n . P S/I G (t) is the Hilbert-Poincare polynomial of S/I G and has the following formP S/I G (t−1) i β i,j (S/I G )t j .In the following theorem the authors have calculated Hilbert-Poincare polynomial of S/I Kn .
-Kiani conjectured that the regularity of binomial edge ideal of a graph is at most c(G) + 1, where c(G) is the number of maximal cliques of G. This conjecture is proved for closed graphs, generalized block graphs, some class of chordal graphs and for fan graphs of complete graph. Let H be a connected closed graph on [n] such that S/J H is Cohen-Macaulay. Then by ([14] Theorem 3.1), there exist integers 1 = a 1 < a 2 . . . < a s < a s+1 = n such that F i = [a i , a i+1 ], for 1 ≤ i ≤ s and F 1 , . . . , F s is a leaf order of △(H). Set e = {1, n}. The graph G = H ∪ {e} is called the quasi-cycle associated with H.Let H be a connected closed graph on the vertex set [m] such that S H /J H is Cohen-Macaulay. Then by ([14], Theorem 3.1), there exist integer
Let T be a tree on [n] with spine P of length l. Let e 2 denote the number of edges that are not in P and with both end points having degree at most 2 and d 3 denote the number of vertices, not in P , and having degree at least 3. Then reg(S/J T ) ≤ e 2 + l + 2d 3 . Definition 3.10. Let I and J be two proper ideals of a local regular ring R, they are called directly linked, we denote by I ∼ J, if there exists a regular sequence z = z 1 , . . . , z n in I ∩ J such that J = (z) : I and I = (z) : J. We say that I and J belong to the same linkage class if there exist a sequence a direct linksTheorem 0.1, [31]
Corollary 3.9 ([31], Corollary 0.1).
5 .
5Results on the other algebraic invariants 5.1. Betti number. Betti number is one of the homological invariant which will help us to know about its structure. Researchers have computed Betti number of binomial edgeideals. Zofar and Zahid calculated for cycles in [66], Schenzel and Zafar calculated for
complete bipartite graphs in [64] and Jayanthan et. al calculated for trees and unicycle
graphs in [29]. In [44] authors have calculated Betti number for cone graph [see, Theorem
3.10]. As a consequence of this, they have calculated Betti number for wheel graph [see,
[44], Corollary 3.11].
Theorem 5.1([29]). Let G be a tree on n-vertices. Then the first syzygy of J G is minimally generated by elements of the form(1) f i,j e {k,l} − f k,l e {i,j} , where {i, j}, {k, l} ∈ E(G) and {e {i,j} : {i, j} ∈ E(G)} is the standard basis of S(−2) n−1 .
and [q], respectively. Let G = H * H ′ be the join of H and H ′ . ThenTheorem 5.12 ([43], Theorem 4.2). Let H and H
′ be two disconnected graphs on vertex
sets [p]
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"An Efficient Generation Algorithm for Lexicalist MT",
"An Efficient Generation Algorithm for Lexicalist MT"
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[
"Victor Poznafiski \nSHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom\n",
"John L Beaven \nSHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom\n",
"Pete Whitelock \nSHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom\n"
] |
[
"SHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom",
"SHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom",
"SHARP Laboratories of Europe Ltd. Oxford Science Park\nOX4 4GAOxfordUnited Kingdom"
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The lexicalist approach to Machine Translation offers significant advantages in the development of linguistic descriptions. However, the Shake-and-Bake generation algorithm of (Whitelock, 1992) is NPcomplete. We present a polynomial time algorithm for lexicalist MT generation provided that sufficient information can be transferred to ensure more determinism.
|
10.3115/981658.981693
| null | 1,571,038 |
cmp-lg/9504027
|
3c527f5e9b04aa8968e53b159ef5efabdf2908d2
|
An Efficient Generation Algorithm for Lexicalist MT
Victor Poznafiski
SHARP Laboratories of Europe Ltd. Oxford Science Park
OX4 4GAOxfordUnited Kingdom
John L Beaven
SHARP Laboratories of Europe Ltd. Oxford Science Park
OX4 4GAOxfordUnited Kingdom
Pete Whitelock
SHARP Laboratories of Europe Ltd. Oxford Science Park
OX4 4GAOxfordUnited Kingdom
An Efficient Generation Algorithm for Lexicalist MT
The lexicalist approach to Machine Translation offers significant advantages in the development of linguistic descriptions. However, the Shake-and-Bake generation algorithm of (Whitelock, 1992) is NPcomplete. We present a polynomial time algorithm for lexicalist MT generation provided that sufficient information can be transferred to ensure more determinism.
Introduction
Lexicalist approaches to MT, particularly those incorporating the technique of Shake-and-Bake generation (Beaven, 1992a;Beaven, 1992b;Whitelock, 1994), combine the linguistic advantages of transfer (Arnold et al., 1988;Allegranza et al., 1991) and interlingual (Nirenburg et al., 1992;Dorr, 1993) approaches. Unfortunately, the generation algorithms described to date have been intractable. In this paper, we describe an alternative generation component which has polynomial time complexity.
Shake-and-Bake translation assumes a source grammar, a target grammar and a bilingual dictionary which relates translationally equivalent sets of lexical signs, carrying across the semantic dependencies established by the source language analysis stage into the target language generation stage.
The translation process consists of three phases: 1. A parsing phase, which outputs a multiset, or bag, of source language signs instantiated with sufficiently rich linguistic information established by the parse to ensure adequate translations. 2. A lexical-semantic transfer phase which employs the bilingual dictionary to map the bag *We wish to thank our colleagues Kerima Benkerimi, David Elworthy, Peter Gibbins, Inn Johnson, Andrew Kay and Antonio Sanfilippo at SLE, and our anonymous reviewers for useful feedback and discussions on the research reported here and on earlier drafts of this paper. of instantiated source signs onto a bag of target language signs.
A generation phase which imposes an order on
the bag of target signs which is guaranteed grammatical according to the monolingual target grammar. This ordering must respect the linguistic constraints which have been transferred into the target signs.
The Shake-an&Bake generation algorithm of (Whitelock, 1992) combines target language signs using the technique known as generate-and-test. In effect, an arbitrary permutation of signs is input to a shift-reduce parser which tests them for grammatical well-formedness. If they are well-formed, the system halts indicating success. If not, another permutation is tried and the process repeated. The complexity of this algorithm is O(n!) because all permutations (n! for an input of size n) may have to be explored to find the correct answer, and indeed must be explored in order to verify that there is no answer.
Proponents of the Shake-and-Bake approach have employed various techniques to improve generation efficiency. For example, (Beaven, 1992a) employs a chart to avoid recalculating the same combinations of signs more than once during testing, and (Popowich, 1994) proposes a more general technique for storing which rule applications have been attempted; (Brew, 1992) avoids certain pathological cases by employing global constraints on the solution space; researchers such as (Brown et al., 1990) and (Chen and Lee, 1994) provide a system for bag generation that is heuristically guided by probabilities. However, none of these approaches is guaranteed to avoid protracted search times if an exact answer is required, because bag generation is NPcomplete (Brew, 1992).
Our novel generation algorithm has polynomial complexity (O(n4)). The reduction in theoretical complexity is achieved by placing constraints on the power of the target grammar when operating on instantiated signs, and by using a more restrictive data structure than a bag, which we call a target language normalised commutative bracketing (TNCB). A TNCB records dominance information from derivations and is amenable to incremental updates. This allows us to employ a greedy algorithm to refine the structure progressively until either a target constituent is found and generation has succeeded or no more changes can be made and generation has failed.
In the following sections, we will sketch the basic algorithm, consider how to provide it with an initial guess, and provide an informal proof of its efficiency.
A Greedy Incremental Generation Algorithm
We begin by describing the fundamentals of a greedy incremental generation algorithm. The cruciM data structure that it employs is the TNCB. We give some definitions, state some key assumptions about suitable TNCBs for generation, and then describe the algorithm itself.
TNCBs
We assume a sign-based grammar with binary rules, each of which may be used to combine two signs by unifying them with the daughter categories and returning the mother. Combination is the commutative equivalent of rule application; the linear ordering of the daughters that leads to successful rule application determines the orthography of the mother. Whitelock's Shake-and-Bake generation algorithm attempts to arrange the bag of target signs until a grammatical ordering (an ordering which allows all of the signs to combine to yield a single sign) is found. However, the target derivation information itself is not used to assist the algorithm. Even in (Beaven, 1992a), the derivation information is used simply to cache previous results to avoid exact recomputation at a later stage, not to improve on previous guesses. The reason why we believe such improvement is possible is that, given adequate information from the previous stages, two target signs cannot combine by accident; they must do so because the underlying semantics within the signs licenses it.
If the linguistic data that two signs contain allows them to combine, it is because they are providing a semantics which might later become more specified. For example, consider the bag of signs that have been derived through the Shake-and-Bake process which represent the phrase:
(1) The big brown dog Now, since the determiner and adjectives all modify the same noun, most grammars will allow us to construct the phrases:
(2) The dog (3) The big dog (4) The brown dog as well as the 'correct' one. Generation will fail if all signs in the bag are not eventually incorporated in tile final result, but in the naive algorithm, the intervening computation may be intractable.
In the algorithm presented here, we start from observation that the phrases (2) to (4) are not incorrect semantically; they are simply under-specifications of (1). We take advantage of this by recording the constituents that have combined within the TNCB, which is designed to allow further constituents to be incorporated with minimal recomputation.
A TNCB is composed of a sign, and a history of how it was derived from its children. The structure is essentially a binary derivation tree whose children are unordered. Concretely, it is either NIL, or a triple:
TNCB = NILlValue × TNCB x TNCB Value = Sign I INCONSISTENT I UNDETERMINED
The second and third items of the TNCB triple are the child TNCBs. The value of a TNCB is the sign that is formed from the combination of its children, or INCONSISTENT, representing the fact that they cannot grammatically combine, or UN-DETERMINED, i.e. it has not yet been established whether the signs combine.
Undetermined TNCBs are commutative, e.g. they do not distinguish between the structures shown in Figure 1.
Figure 1: Equivalent TNCBs
In section 3 we will see that this property is important when starting up the generation process.
Let us introduce some terminology. A TNCB is • well-formed iff its value is a sign,
• ill-formed iff its value is INCONSISTENT, • undetermined (and its value is UNDETER-MINED) iff it has not been demonstrated whether it is well-formed or ill-formed.
• maximal iff it is well-formed and its parent (if it has one) is ill-formed. In other words, a maximal TNCB is a largest well-formed component of a TNCB.
Since TNCBs are tree-like structures, if a TNCB is undetermined or ill-formed then so are all of its ancestors (the TNCBs that contain it). We define five operations on a TNCB. The first three are used to define the fourth transformation (move) which improves ill-formed TNCBs. The fifth is used to establish the well-formedness of undetermined nodes. In the diagrams, we use a cross to represent ill-formed nodes and a black circle to represent undetermined ones.
Deletion: A maximal TNCB can be deleted from its current position. The structure above it must be adjusted in order to maintain binary branching. In figure 2, we see that when node 4 is deleted, so is its parent node 3. The new node 6, representing the combination of 2 and 5, is marked undetermined. Conjunction: A maximal TNCB can be conjoined with another maximal TNCB if they may be combined by rule. In figure 3, it can be seen how the maximal TNCB composed of nodes 1, 2, and 3 is conjoined with the maximal TNCB composed of nodes 4, 5 and 6 giving the TNCB made up of nodes 1 to 7. The new node, 7, is well-formed. Adjunction: A maximal TNCB can be inserted inside a maximal TNCB, i.e. conjoined with a non-maximal TNCB, where the combination is licensed by rule. In figure 4, the TNCB composed of nodes 1, 2, and 3 is inserted inside the TNCB composed of nodes 4, 5 and 6. All nodes (only 8 in figure 4) which dominate the node corresponding to the new combination (node 7) must be marked undetermined --such nodes are said to be disrupted. Movement: This is a combination of a deletion with a subsequent conjunction or adjunction. In figure 5, we illustrate a move via conjunction.
In the left-hand figure, we assume we wish to move the maximal TNCB 4 next to the maximal TNCB 7. This first involves deleting TNCB 4 (noting it), and raising node 3 to replace node 2. We then introduce node 8 above node 7, and make both nodes 7 and 4 its children. Note that during deletion, we remove a surplus node (node 2 in this case) and during conjunction or adjunction we introduce a new one (node 8 in this case) thus maintaining the same number of nodes in the tree. Evaluation: After a movement, the TNCB is undetermined as demonstrated in figure 5. The signs of the affected parts must be recalculated by combining the recursively evaluated child TNCBs.
Suitable Grammars
The Shake-and-Bake system of (Whitelock, 1992) employs a bag generation algorithm because it is assumed that the input to the generator is no more than a collection of instantiated signs. Full-scale bag generation is not necessary because sufficient information can be transferred from the source language to severely constrain the subsequent search during generation.
The two properties required of TNCBs (and hence the target grammars with instantiated lexicM signs) are:
1. Precedence Monotonicity. The order of the orthographies of two combining signs in the orthography of the result must be determinate -it must not depend on any subsequent combination that the result may undergo. This constraint says that if one constituent fails to combine with another, no permutation of the elements making up either would render the combination possible. This allows bottom-up evaluation to occur in linear time. In practice, this restriction requires that sufficiently rich information be transferred from the previous translation stages to ensure that sign combination is deterministic.
2. Dominance Monotonicity.
If a maximal TNCB is adjoined at the highest possible place inside another TNCB, the result will be wellformed after it is re-evaluated. Adjunction is only attempted if conjunction fails (in fact conjunction is merely a special case of adjunction in which no nodes are disrupted); an adjunction which disrupts i nodes is attempted before one which disrupts i + 1 nodes. Dominance monotonicity merely requires all nodes that are disrupted under this top-down control regime to be well-formed when re-evaluated. We will see that this will ensure the termination of the generation algorithm within n-1 steps, where n is the number of lexical signs input to the process.
We are currently investigating the mathematical characterisation of grammars and instantiated signs that obey these constraints. So far, we have not found these restrictions particularly problematic.
The Generation Algorithm
The generator cycles through two phases: a test phase and a rewrite phase. Imagine a bag of signs, corresponding to "the big brown dog barked", has been passed to the generation phase. The first step in the generation process is to convert it into some arbitrary TNCB structure, say the one in figure 6. In order to verify whether this structure is valid, we evaluate the TNCB. This is the test phase. If the TNCB evaluates successfully, the orthography of its value is the desired result. If not, we enter the rewrite phase.
If we were continuing in the spirit of the original Shake-and-Bake generation process, we would now form some arbitrary mutation of the TNCB and retest, repeating this test-rewrite cycle until we either found a well-formed TNCB or failed. However, this would also be intractable due to the undirectedness of the search through the vast number of possibilities. Given the added derivation information contained within TNCBs and the properties mentioned above, we can direct this search by incrementally improving on previously evaluated results.
We enter the rewrite phase, then, with an illformed TNCB. Each move operation must improve p lg Figure 6: An arbitrary right-branching TNCB structure it. Let us see why this is so.
The move operation maintains the same number of nodes in the tree. The deletion of a maximal TNCB removes two ill-formed nodes (figure 2). At the deletion site, a new undetermined node is created, which may or may not be ill-formed. At the destination site of the movement (whether conjunction or adjunction), a new well-formed node is created.
The ancestors of the new well-formed node will be at least as well-formed as they were prior to the movement. We can verify this by case:
1. When two maximal TNCBs are conjoined, nodes dominating the new node, which were previously ill-formed, become undetermined. When re-evaluated, they may remain ill-formed or some may now become well-formed.
2. When we adjoin a maximal TNCB within another TNCB, nodes dominating the new wellformed node are disrupted. By dominance monotonicity, all nodes which were disrupted by the adjunction must become well-formed after re-evaluation. And nodes dominating the maximal disrupted node, which were previously ill-formed, may become well-formed after reevaluation.
We thus see that rewriting and re-evaluating must improve the TNCB.
Let us further consider the contrived worst-case starting point provided in figure 6. After the test phase, we discover that every single interior node is ill-formed. We then scan the TNCB, say top-down from left to right, looking for a maximal TNCB to move. In this case, the first move will be PAST to bark, by conjunction ( figure 7).
Once again, the test phase fails to provide a wellformed TNCB, so we repeat the rewrite phase, this time finding dog to conjoin with the ( figure 8 shows the state just after the second pass through the test phase).
After further testing, we again re-enter the rewrite phase and this time note that brown can be inserted in the maximal TNCB the dog barked adjoined with dog ( figure 9). Note how, after combining dog and the, the parent sign reflects the correct orthography Figure 9: The TNCB after "dog" is moved to "the" After finding that big may not be conjoined with the brown dog, we try to adjoin it within the latter. Since it will combine with brown dog, no adjunction to a lower TNCB is attempted.
The final result is the TNCB in figure 11, whose orthography is "the big brown dog barked".
We thus see that during generation, we formed a basic constituent, the dog, and incrementally refined it by adjoining the modifiers in place. At the heart of this approach is that, once well-formed, constituents can only grow; they can never be dismantled. Even if generation ultimately fails, maximal wellformed fragments will have been built; the latter may be presented to the user, allowing graceful degradation of output quality.
the b~
PAST bXark d'og b~o.n ~he ~'bfg, Figure 10: The TNCB after "brown" is moved to "dog" the big brown dog barked PA k he Figure 11: The final TNCB after "big" is moved to "brown dog"
Initialising the Generator
Considering the algorithm described above, we note that the number of rewrites necessary to repair the initial guess is no more than the number of ill-formed TNCBs. This can never exceed the number of interior nodes of the TNCB formed from n lexical signs (i.e. n-2). Consequently, the better formed the initial TNCB used by the generator, the fewer the number of rewrites required to complete generation. In the last section, we deliberately illustrated an initial guess which was as bad as possible. In this section, we consider a heuristic for producing a motivated guess for the initial TNCB. Consider the TNCBs in figure 1. If we interpret the S, O and V as Subject, Object and Verb we can observe an equivalence between the structures with the bracketings: (S (V O)), (S (O V)), ((V O) S), and ((O V) S). The implication of this equivalence is that if, say, we are translating into a (S (V O)) language from a head-finM language and have isomorphic dominance structures between the source and target parses, then simply mirroring the source parse structure in the initial target TNCB will provide a correct initiM guess. For example, the English sentence (5):
(5) the book is red has a corresponding Japanese equivalent (6):
(6) ((hon wa) (akai desu)) ((book TOP) (red is))
If we mirror the Japanese bracketing structure in English to form the initial TNCB, we obtain: ((book the) (red is)). This will produce the correct answer in the test phase of generation without the need to rewrite at all.
Even if there is not an exact isomorphism between the source and target commutative bracketings, the first guess is still reasonable as long as the majority of child commutative bracketings in the target language are isomorphic with their equivalents in the source language. Consider the French sentence:
(7) ((le ((grandchien) brun)) aboya) (8) ((the ((big dog) brown)) barked)
The TNCB implied by the bracketing in (8) is equivalent to that in figure 10 and requires just one rewrite in order to make it well-formed. We thus see how the TNCBs can mirror the dominance information in the source language parse in order to furnish the generator with a good initial guess. On the other hand, no matter how the SL and TL structures differ, the algorithm will still operate correctly with polynomial complexity. Structural transfer can be incorporated to improve the efficiency of generation, but it is never necessary for correctness or even tractability.
4
The Complexity of the Generator
The theoretical complexity of the generator is O (n4), where n is the size of the input. We give an informal argument for this. The complexity of the test phase is the number of evaluations that have to be made. Each node must be tested no more than twice in the worst case (due to precedence monotonicity), as one might have to try to combine its children in either direction according to the grammar rules. There are always exactly n -1 non-leaf nodes, so the complexity of the test phase is O(n). The complexity of the rewrite phase is that of locating the two TNCBs to be combined. In the worst case, we can imagine picking an arbitrary child TNCB (O(n)) and then trying to find another one with which it combines (O(n)). The complexity of this phase is therefore the product of the picking and combining complexities, i.e. O(n2). The combined complexity of the test-rewrite cycle is thus O(n3). Now, in section 3, we argued that no more than n -1 rewrites would ever be necessary, thus the overall complexity of generation (even when no solution is found) is O(n4). Average case complexity is dependent on the quality of the first guess, how rapidly the TNCB structure is actually improved, and to what extent the TNCB must be re-evaluated after rewriting. In the SLEMaT system (Poznarlski et al., 1993), we have tried to form a good initial guess by mirroring the source structure in the target TNCB, and allowing some local structural modifications in the bilingual equivalences.
Structural transfer operations only affect the efficiency and not the functionality of generation. Transfer specifications may be incrementally refined and empirically tested for efficiency. Since complete specification of transfer operations is not required for correct generation of grammatical target text, the version of Shake-and-Bake translation presented here maintains its advantage over traditional transfer models, in this respect.
The monotonicity constraints, on the other hand, might constitute a dilution of the Shake-and-Bake ideal of independent grammars. For instance, precedence monotonicity requires that the status of a clause (strictly, its lexical head) as main or subordinate has to be transferred into German. It is not that the transfer of information per se compromises the ideal --such information must often appear in transfer entries to avoid grammatical but incorrect translation (e.g. a great man translated as un homme grand). The problem is justifying the main/subordinate distinction in every language that we might wish to translate into German. This distinction can be justified monolingually for the other languages that we treat (English, French, and Japanese). Whether the constraints will ultimately require monolingual grammars to be enriched with entirely unmotivated features will only become clear as translation coverage is extended and new language pairs are added.
Conclusion
We have presented a polynomial complexity generation algorithm which can form part of any Shakeand-Bake style MT system with suitable grammars and information transfer. The transfer module is free to attempt structural transfer in order to produce the best possible first guess. We tested a TNCB-based generator in the SLEMaT MT system with the pathological cases described in (Brew, 1992) against Whitelock's original generation algorithm, and have obtained speed improvements of several orders of magnitude. Somewhat more surprisingly, even for short sentences which were not problematic for Whitelock's system, the generation component has performed consistently better.
Figure 2:4 is deleted, raising 5
Figure 3:1 is conjoined with 4 giving 7
Figure 4:1 is adjoined next to 6 inside 4
Figure 5 :
5A conjoining move from 4 to 7
Figure 7 :
7The
Figure 8 :
8The TNCB after "PAST" is moved to "bark" even though they did not have the correct linear precedence.
Linguistics for Machine Translation: The Eurotra Linguistic Specifications. V Allegranza, P Bennett, J Durand, F Van Eynde, L Humphreys, P Schmidt, E Steiner, C. Copeland, J. Durand, S. Krauwer, and B. MaegaardThe Eurotra Formal Specifications. Studies in MachineV. Allegranza, P. Bennett, J. Durand, F. van Eynde, L. Humphreys, P. Schmidt, and E. Steiner. 1991. Linguistics for Machine Translation: The Eurotra Linguistic Specifications. In C. Copeland, J. Du- rand, S. Krauwer, and B. Maegaard, editors, The Eurotra Formal Specifications. Studies in Machine
|
[] |
[
"A SYMMETRY-INCLUSIVE ALGEBRAIC APPROACH TO GENOME REARRANGEMENT",
"A SYMMETRY-INCLUSIVE ALGEBRAIC APPROACH TO GENOME REARRANGEMENT"
] |
[
"Venta Terauds ",
"Joshua Stevenson ",
"Jeremy Sumner "
] |
[] |
[] |
Of the many modern approaches to calculating evolutionary distance via models of genome rearrangement, most are tied to a particular set of genomic modelling assumptions and to a restricted class of allowed rearrangements. The "position paradigm", in which genomes are represented as permutations signifying the position (and orientation) of each region, enables a refined model-based approach, where one can select biologically plausible rearrangements and assign to them relative probabilities/costs. Here, one must further incorporate any underlying structural symmetry of the genomes into the calculations and ensure that this symmetry is reflected in the model. In our recently-introduced framework of genome algebras, each genome corresponds to an element that simultaneously incorporates all of its inherent physical symmetries. The representation theory of these algebras then provides a natural model of evolution via rearrangement as a Markov chain. Whilst the implementation of this framework to calculate distances for genomes with 'practical' numbers of regions is currently computationally infeasible, we consider it to be a significant theoretical advance: one can incorporate different genomic modelling assumptions, calculate various genomic distances, and compare the results under different rearrangement models. The aim of this paper is to demonstrate some of these features.
|
10.1142/s0219720021400151
|
[
"https://arxiv.org/pdf/2106.09927v2.pdf"
] | 235,485,132 |
2106.09927
|
d876ffeda407303de815a28e8de58f5471ef2cb9
|
A SYMMETRY-INCLUSIVE ALGEBRAIC APPROACH TO GENOME REARRANGEMENT
Venta Terauds
Joshua Stevenson
Jeremy Sumner
A SYMMETRY-INCLUSIVE ALGEBRAIC APPROACH TO GENOME REARRANGEMENT
Of the many modern approaches to calculating evolutionary distance via models of genome rearrangement, most are tied to a particular set of genomic modelling assumptions and to a restricted class of allowed rearrangements. The "position paradigm", in which genomes are represented as permutations signifying the position (and orientation) of each region, enables a refined model-based approach, where one can select biologically plausible rearrangements and assign to them relative probabilities/costs. Here, one must further incorporate any underlying structural symmetry of the genomes into the calculations and ensure that this symmetry is reflected in the model. In our recently-introduced framework of genome algebras, each genome corresponds to an element that simultaneously incorporates all of its inherent physical symmetries. The representation theory of these algebras then provides a natural model of evolution via rearrangement as a Markov chain. Whilst the implementation of this framework to calculate distances for genomes with 'practical' numbers of regions is currently computationally infeasible, we consider it to be a significant theoretical advance: one can incorporate different genomic modelling assumptions, calculate various genomic distances, and compare the results under different rearrangement models. The aim of this paper is to demonstrate some of these features.
Introduction
Genome rearrangement modelling has historically been approached as a combinatorial problem, with the aim of developing fast algorithms to compute pairwise distances between genomes. Although permutations have long been used to represent genomes, it is only this century that serious consideration has been given to the algebraic frameworks that form the theoretical basis for the models. Beginning with the work of Meidanis and Dias [21], and continuing with many others [13,15,12,4], it has been recognised that developing the algebraic formalism is key to making progress in genome rearrangement modelling, in particular in enabling more refined model-based approaches.
As classified by Bhatia et al [5] in their excellent overview, algebraic frameworks for modelling genome rearrangement tend to use either the "content" or the "position" paradigm. In the former paradigm, genomes are represented in terms of the adjacencies between regions; in the latter, positions as well as regions are labelled, and genomes are denoted by maps that link regions to positions. Whilst the latter necessarily applies a choice of reference frame in the position labelling (unless the genome possesses no symmetry), the content paradigm has the advantage of being 'orientation free', since it only 'notices' which regions are adjacent, and in which orientation.
To model rearrangements in the content paradigm, the "double cut and join" [4] and, more generally, "k-break" [13] operations are widely used. These operations cover inversions, fissions, fusions and translocations, and the ability of a single operation to model a range of events, over multiple chromosomes, has been considered an advantage [3]. The double cut and join framework continues to be adapted in various ways, for example to include insertions and deletions [7,6], incorporate intergenic regions [26,14], and limit rearrangements to very small scale events [24]. However, these approaches almost always utilise minimum distance as the distance measure (or its generalisation to the median distance [32]), and the classification of rearrangements in these approaches remains coarse.
Biologically, the relative probabilities of different rearrangements are likely to differ according to their type (for example, inversions or translocations), size, and position on the genome [10,1]. Utilising the position paradigm framework enables a fine-grained approach to rearrangement models that can incorporate such information: rearrangements can be represented as permutations of positions, that is, operators that switch around the regions that are in particular positions, whatever the regions may be.
The implicit choice of reference frame in this paradigm means that (usually) more than one permutation will represent the same genome, due to inherent symmetry. The inclusion of genome symmetry in the theoretical framework has been previously considered [12], but in practice this has been added in as a separate element of calculations, thus increasing the computational complexity, whether treating genomes with unsigned regions or signed [25,17]. That is, the the process has generally been to (i) perform calculations for the genomes as 'fixed orientation entities" (usually group elements) and then (ii) repeat for each of the symmetries to get the result.
Less comprehensively considered has been any corresponding symmetry of rearrangement models. Although one chooses a reference frame in order to represent the genomes and rearrangements of interest, the set of allowed rearrangements should be independent of the reference frame. Rearrangement models used in practice have tended to have this property as a consequence of being quite general, for example models consisting of all inversions, all inversions of size k, all inversions with some probability p and all transpositions with probability q, and so on. However, until recently, [29,30] this has not been included in the theoretical framework.
The approach we present here incorporates the symmetry of genomes and rearrangements in the unified framework of genome algebras; here each genome and each rearrangement corresponds to a single mathematical object that simultaneously incorporates all of its inherent physical symmetries. Our framework easily incorporates different genomic modelling assumptions and different sets of allowed rearrangements; further, it naturally facilitates the calculation of different measures of evolutionary distance.
The aim of this paper is to demonstrate some of these features. We outline the construction of the relevant genome algebras for signed circular single-chromosome genomes with and without an origin of replication, and then give some results for computations of genomic distanceas estimated via minimum distance, mean first passage time, and the maximum likelihood estimate of time elapsed -under various rearrangement models.
Genome instances and permutation clouds
The genome algebra framework was presented in Terauds and Sumner [30], along with details of the construction for the case of unsigned circular genomes with dihedral symmetry. Here we consider the algebra for circular single-chromosome genomes with oriented regions, both with and without an origin of replication. We note that the former construction may also be applied to model linear genomes.
Modelling genomes with oriented regions as elements of the hyperoctahedral group is standard; we refer to Bhatia et al [5] and Egri-Nagy et al [12] for detailed treatments. We consider that the genomes of interest share n regions in common, where each region is a contiguous section of DNA (these may also be referred to as synteny blocks or conserved regions). Labelling the positions and regions both by 1, . . . , n, we represent an instance of a genome by a signed permutation σ, mapping regions to positions, where σ(i) = ±j ⇐⇒ region i is in position j with positive/negative orientation .
Setting σ(−i) = −σ(i) for all i makes σ an element of the hyperoctahedral group H n , modelled as a subgroup of the symmetric group on {±1, . . . , ±n}. We shall henceforth use the convention of writing i instead of −i.
With the term instance, we are emphasising that a single permutation represents an observation of the genome with a fixed physical orientation and a choice of position labelling. The labelling of the regions (including region orientation) is chosen once and is immutable, however, the labelling of the positions reflects a choice. For a genome with an origin of replication, we may decide on a labelling such that the origin lies between positions 1 and n. However, there remains a choice: we may label the positions either clockwise or anticlockwise. Thus there are two distinct permutations that may represent any given genome. For a fixed reference frame, this corresponds to the two physical orientations of the genome obtained by flipping it over in space. Thus, in this case, the symmetry group corresponding to the genomes has size two (it's S 2 = {e, f } -a "do nothing" and a "flip"), and we equivalently say that each genome has two instances, represented by the group elements {σ, f σ} = S 2 σ. A circular genome with no distinguished position has dihedral symmetry (one can rotate as well as flip); the symmetry group is a copy of the dihedral group, D n and each genome thus has 2n distinct instances, corresponding to elements of a coset D n σ.
Similarly, we model an instance of a rearrangement as a signed permutation that acts on a genome instance on the left, mapping signed positions to signed positions. For example, an inversion of the regions in positions 1 and 2 would be a = [2, 1, 3, 4, . . . , n] (expressed in oneline notation). As with genomes, there are |Z| instances of any given rearrangement, where Z is the relevant symmetry group. 1 In earlier work [29],we considered model symmetry as a two step process: if a above were an allowed rearrangement of genomes with origin of replication between positions n and 1, then the rearrangement that swaps the regions in positions n and n − 1, namely [1, 2, 3, . . . , n, n − 1], should be allowed and assigned the same probability.
In the genome algebra, all of the instances for a given genome (or rearrangement) are combined into a single element: a genome corresponds to the sum of its instances, each weighted by 1/(#symmetries). Such elements are the basis elements of the genome algebra, meaning that everything in the genome algebra may be expressed as a linear combination of genomes. In the genome algebra, elements may be added, multiplied by scalars and multiplied together; in particular, the latter is how we model a rearrangement acting on a genome.
To formally construct the genome algebra we begin with a group G, whose elements represent instances of the genomes, and a subgroup Z ⊆ G, representing their symmetries, and form the symmetry element
z := 1 |Z| z∈Z z of the group algebra C[G]. 2 The genome algebra of G with Z is A := zC[G].
The distinct genomes with instances in G correspond to the distinct elements of the set {zσ : σ ∈ G}, which forms a basis for A. For example, taking G = H n and Z = S 2 , the symmetry element is z = 1 2 e + 1 2 f and the distinct genomes have the form zσ = 1 2 σ + 1 2 f σ for σ ∈ H n . One can think of a genome as existing as the average of its instances, where there is an equal probability of observing any particular instance. Now rearrangements also have the form za, for a ∈ G, and rearrangement occurs via left action on a genome, zσ → za · zσ, which we can think of as "all orientations of (a acting on (all orientations of σ))". This results in a linear (convex) combination of genomes; in fact, this is a probability distribution of the genomes that may result.
As an explicit example, consider the following reference genome (with instance e) in the genome algebra A for the group H 6 with symmetry group S 2 : 1 Note that, as rearrangements, these need not all act distinctly. 2 Recall that the group algebra is formed from all finite linear combinations of group elements.
ze = z = 1 2
Choosing the rearangement instance a = [2, 1, 3, 4, . . . , 6] and applying the rearrangement za to the reference genome z in the genome algebra, we obtain
(za) · z = 1 4 = 1 2 z · 6 5 4 3 1 2 + z · 1 2 3 4 6 5 = 1 2 za + 1 2 zσ ,
where σ = af . This means that, by applying the rearrangement za to the reference genome z, one would obtain either the genome za or the genome zσ, with equal probabllities.
In order to compute evolutionary distance via rearrangement, we begin by fixing a model. Formally, a model is a set
M := {za 1 z, . . . , za q z} ⊆ A
for some a 1 , . . . a q ∈ G, along with a probability distribution w : M → (0, 1]. We note that, whilst there are exactly |G| |Z| distinct genomes zσ that have instances in G and symmetry group Z (for example, this is 2 n−1 n! for signed genomes with an origin of replication and 2 n−1 (n−1)! for those with no distinguished position), there are fewer distinct rearrangements. In particular, since z is an idempotent (z · z = z), rearrangments za and zb have the same left action on genomes whenever zaz = zbz. This motivates the above formulation of the model, since unintentionally duplicating a particular rearrangement action could unintentionally skew the probability distribution. In Section 4, we expand on this and provide an example.
Given a model (M, w), we form the model element in the genome algebra:
s := zaz∈M w(zaz)za .
Since w is a probability distribution, s is a convex sum, and thus, in a direct extension of the above, we see that acting on a genome on the left with the model element, zσ → s · zσ, results in a convex combination of genomes. These are exactly the genomes that may be obtained from zσ via one rearrangement chosen from the model, with their respective probabilities given by the coefficients. The left regular representation of the model element in A, ρ( s), summarises this information in the form of a K × K matrix (where K := |G| |Z| ): labelling the distinct genomes {zσ 1 , . . . , zσ K } ,
ρ( s) ij := coefficient of zσ i in expansion of s · zσ j , and thus ρ( s) k ij = ρ( s k ) ij = probability of zσ j → zσ i via k rearrangement events .
We thus see evolution via rearrangement explicitly as a discrete Markov process, with transition matrix ρ( s). We define the path probabilities for a genome zσ i with respect to the reference genome z via α k (zσ i ) := probability of obtaining zσ i from z via k rearrangement events . Now, these can be read from the first column of the matrix ρ( s) k , or obtained via the trace of a modified matrix [30]:
(1) α k (zσ) = |Z| |G| tr(ρ( s k zσ −1 )) = 1 K χ( s k zσ −1 ) .
To compute this more efficiently, one would usually utilise the algebra's irreducible modules and representations. As with the regular representation, each irreducible representation is a linear mapping that assigns to each algebra element a matrix, in such a way that preserves multiplication. A representation ρ : A → Mat m (C) defines an action of A on the module C m ; the module and the representation are irreducible if C m contains no non-trivial subspace invariant under this action. The irreducible representations always exist and enable the computation of the path probabilities via matrices of much smaller dimension than the regular representation 3 .
We note that we would usually assume that the model is sufficient to generate the set of genomes, that is, that the Markov chain is irreducible. When one considers genomes and rearrangements simply as group elements, the analogue of this condition is that the permutations in the model generate the entire group [12,25]. In fact, these formulations are equivalent [27]. The only other condition that we put on the model is model reversibility, that is, that whenever zaz ∈ M, one must have za −1 z ∈ M with w(zaz) = w(za −1 z). This is not required for any of the above, however it ensures that the Markov process is reversible [30], so that σ −1 can be replaced by σ in (1). Further, it allows the path probabilities to be more efficiently computed via diagonalising the respective matrices.
The path probabilities, or more generally, the Markov matrix, can be utilised to compute various types of evolutionary distance measures. We consider three of these in the next section.
Evolutionary distance
Genome evolution via rearrangement is most commonly modelled as a discrete process -that, is a sequence of discrete rearrangement events -and the most ubiquitous measure of evolutionary distance is the minimum distance. This usually takes the form of the minimum number of rearrangements that can be applied to the reference genome to obtain the target, but also includes minimum weighted distance, where different costs are applied to different types of rearrangement [2,23]. The issues with minimum distance as a proxy for true evolutionary distance have been widely discussed in the literature and many alternative measures have been proposed; these are often based either on modifying minimum distance or breakpoint distance in some way [31,20].
The maximum likelihood estimate of time elapsed (MLE), introduced by Serdoz et al [25], takes a completely different approach to evolutionary distance. Here, evolution is modelled as a discrete sequence of rearrangement events occuring in continuous time. The Poisson distribution is a natural choice for the distribution of events in time, since combining it with the discrete time Markov chain above produces the corresponding continuous time Markov chain,
P (t) := e (ρ( s)−I)t
which encapsulates the likelihoods. To be specific, for a genome zσ, the likelihood function at a time value t is the probability that the reference genome z evolved into zσ in time t, that is,
L(zσ|t) = P (z → zσ in time t) = k≥0 α k (zσ)P (k events in time t) .
The MLE distance from z to zσ is then the value of t for which the likelihood function attains a maximum.
As has been noted [29,30], one could choose a different distribution to combine with the path probabilities and calculate likelihood functions. However, utilising the above, the likelihood function for a given genome zσ can be found via the matrix trace [28,30]. We note that the MLE does not always exist. This may be seen as a feature [25,29] -it is biologically realistic that not all pairs of genomes display an evolutionary signal, and hence a finite evolutionary distance, under a given model -or a fault; see Francis and Wynn [16] for more discussion of this.
One of the motivators for considering the MLE is that it takes into account all possible paths via rearrangement between two genomes, along with their probabilities. For example, there may be a minimum path of length k rearrangements between two given genomes, but many more possible paths of length between them, making evolution via rearrangements the more likely evolutionary scenario. The discrete Markov model and accompanying depiction of the genome space as an edge-weighted graph (an algebra version of a Cayley graph) allow other distance measures that incorporate this information to be considered. Such graphs have previously been utilised for genomes modelled as group elements [22,12,8,16] and our framework extends this to algebra elements, where symmetry is automatically included.
One such measure is mean first passage time (MFPT), a well-studied concept in Markov chain theory whose application to genome rearrangement was recently considered by Francis and Wynn [16]. The mean first passage time is the expected length of a random walk on the edge-weighted graph that starts at the reference and ends when it first encounters the target genome. Extending Francis and Wynn's treatment from the group case to the genome algebra framework, the mean first passage time distance may be calculated directly from the Markov matrix ρ( s) via a simple row replacement and matrix inversion [16].
Some example computations
To demonstrate the flexibility one has in applying the genome algebra framework, we present some results from computations for genomes with six oriented regions. We claim no algorithmic sophistication, and have not applied any numerical methods to speed up computation. Thus, due to the size of the matrices involved, the results available at the time of submission are primarily for genomes modelled with no distinguished position (there are 23, 040 = 2 5 6! distinct genomes with a distinguished position on 6 regions, and 3840 = 2 5 5! without). Our small sample of results shows that changing either the rearrangement model or the distance measure can greatly affect the relative genomic distances obtained. Thus, whilst computations for large numbers of regions remain currently out of reach, the framework has the potential to provide insights in this vein.
With the introduction of the MLE distance measure in Serdoz et al [25], examples were provided of the MLE and the minimum difference measures ordering genomes with unsigned regions and dihedral symmetry differently, in terms of their distance from the reference. In our subsequent work [29], we gave further examples of this, along with an example of the MLEs calculated under two different models ordering (unsigned) genomes differently. Here we investigate the differences betwen the following models:
(i) inversions of one and two regions; equally likely (ii) inversions of one and two regions; single region twice as likely (iii) "all inversions equally likely" (iv) inversions of one, two and three regions; equally likely (v) inversions of one and two regions and one region translocations; equally likely.
By "all inversions" in model (iii), we mean all inversions of regions of up to size 5 (since an inversion of all six regions is just a flip). This may naively seem correct; however this is in fact a duplication of rearrangements, and we include it here in this way to demonstrate this effect. Note that any instance of an inversion of four regions, written as a ∈ H 6 , is the 'flip' of an inversion of 6 − 4 = 2 regions, that is, a = db (where d ∈ D 6 ) for an instance b ∈ H 6 of an inversion, and thus we include za = zb twice in the model. The same applies to inversions of sizes 1 and 5. We can think of this as the inversions a and b being complementary; in any case, including these duplicate rearrangements skews the probablity distribution, and rather than obtaining the intended uniform distribution, the result is a model that has inversions of sizes 1 and 2 with twice the probability of inversions of size 3. Thus, for 6 regions, model (iv) is the correct implementation of an "all inversions equally likely" model.
All of our computations were performed on an instance of the Nectar Research Cloud running Ubuntu 18.04 with 32 gigabytes of available RAM. We found the 3840 × 3840 Markov matrix ρ( s) for each model using SageMath [11], and used this to calculate the MFPT distances (matrix computation in SageMath, assuming a uniform mean inter-arrival time) and the minimum distances (via the Cayley graph of the matrix and the nx.shortest path length function in the Python package Networkx [19]). For the MLEs, we found the relevant irreducible representation matrices of H 6 via SageMath (utilising the Gap [18] package repsn [9]) and projected these onto the irreducibles of the genome algebra in order to compute the likelihood functions via the irreducible characters (see our paper [30] for more details); we then used an optimising function to find the MLE (or that there wasn't one).
Since the genomes form equivalence classes -in particular, all of these distances are the same for genomes zσ and zτ whenever σ = τ −1 or zσz = zτ z [30] -we needed only calculate the distances from the reference genome to 250 representatives to have the pairwise distances between all pairs of genomes. (As usual, if the distance between the reference and zσ is d(z, zσ), then the distance between zσ 1 and zσ 2 is d(z, zσ 1 σ −1 2 ).) We highlight a few observations from these results. Considering the MFPT distances, we observe that adding weights to model (i) to favour the smaller inversions (model (ii)), changes the ordering of genomes zσ 1 and zσ 2 with respect to their distance from the reference genome z = ze. Similarly, there are different relative orderings under the 'unintended' skewed probability distribution of model (iii) and the 'true' all inversions equally likely model (iv). Model (v), which includes small translocations, seems to differentiate the MFPT distances the most. We note that the relatively large values of the MFPT distances compared to the MLE and min distances reflect that they are calculated as the mean (weighted) length of a random walk on a quite sparsely connected graph with 3840 nodes. Thus, mean path lengths of the order of the number of nodes are quite reasonable.
d model(s) d(z, zσ 1 ) d(z, zσ 2 ) d(zσ 1 , zσ 2 ) (i), (ii) 5 4 5 min (iii), (iv) 4 3 3 (v) 3 4 3 (i) − − − (ii) − − − MLE (iii
For the MLE distance, we observe that there is no detectable evolutionary relationship for any of these pairs of genomes under models (i) and (ii). Under model (iii), the MLE distances of genomes zσ 1 and zσ 2 from the reference are similar (as are the MFPTs), but these two measures order them differently than the minimum distance. Under each of the models, we obtained an MLE value for between approximately thirty five and forty per cent of the genomes (although not the same ones in each case). For comparison, previous calculations of MLE distances [25,29] for genomes with unoriented regions have found a detectable evolutionary signal for around forty five percent of genomes (compared to a fixed reference). We note that the determination of whether or not an MLE exists is highly sensitive to the optimisation function applied.
It is perhaps also interesting that, under model (iv), there is the most 'agreement' between the distance measures, in that the MLE, MFPT and minimum distance all order genomes zσ 1 and zσ 2 the same in terms of their distance from the reference. Although the minimum distance clearly has the least resolution (with maximum minimum distances respectively 7, 6 and 5 for each of the three cases), it nonetheless gave different orderings of genomic distances under different models.
Overall, we found both MLE and MFPT to generally increase with minimum distance, with variation between the models. The MLE displays much more variance than the MFPT overall, which we would assume is due to the inclusion of the stochastic component. We provide plots to illustrate this in A. Further, in B we present constructions and results for all genomes with three regions under different symmetry assumptions.
Conclusion
The position paradigm approach to genomic modelling enables a fine-grained consideration of rearrangement models, allowing different types of rearrangements, of different scales, and at different genomic positions, to be included in models with different relative probabilities. In this framework, any structural symmetry of the genomes needs to be incorporated into the modelling, which has previously necessitated an extra step in calculations. Here, we have presented a genome algebra framework that provides a unified approach to the symmetry of genomes and rearrangement models. Our approach reflects biological reality -objectively, a genome is a physical object that exists with all of its possible symmetries simultaneouslyand reduces the complexity of the computation process, since the symmetry of the objects is included from the start.
The sample computations we have provided here are intended to demonstrate the flexibility of our approach in incorporating different models of genomic structure and different rearrangement models, enabling comparisons of the results obtained for different models and different measures of genomic distance. We consider that our symmetry-inclusive approach represents a significant theoretical advance in genome rearrangement modelling. Much work remains, however, to implement the theory in practically useful computations of genomic distance. Whilst the genome algebras represent a dimensionality reduction from previous group-based approaches, the number of distinct genomes is still factorial (2 n−1 (n − 1)! for signed genomes with n regions and dihedral symmetry, for example). This means that the dimension of the matrices we use for computation is very large, although here, as observed in Francis and Wynn [16] the Markov matrices are quite sparse, which should make more efficient matrix methods applicable.
Along with pursuing efficient algorithmic and numerical methods, in particular for the calculation of MLEs, we are interested in broadening the application of our framework. In future work, we aim to incorporate insertions and deletions into the framework by extending it to include semigroups, and investigate the potential to model multiple chromosomes and intergenic regions. The generally lower values taken by the MFPT distance under model (v) reflect the larger number of rearrangements under this model, meaning that the graph is more connected and there are more shorter paths available.
In Table 1 to have an origin of replication). Under this assumption, we further calculated the MLE distances between genomes with the above instances under each of the five models, and found an evolutionary signal in only two cases: an MLE of 5.20 between z and zσ 2 under model (iii) and an MLE of 5.52 for the same genomes under model (iv). For reference, the values assuming circular symmetry are in Table 2 below. To provide more detail on genomes, the genome algebras, the rearrangement models, and their construction under the different symmetry assumptions, we consider explicitly the case of genomes with three regions, where one can easily list all genomes.
model(s) d(z, zσ 1 ) d(z, zσ 2 ) d(zσ 1 , zσ 2 ) (i) − − − (ii) − − − (iii
We again consider the genomes to have signed regions, so instances of genomes will be elements of the group H 3 , which has 2 3 3! = 48 elements. As above, we use one-line notation to represent these signed permutations. Recall that when representing genome instances as permutations, we are mapping regions to positions; specifically, representing a genome instance by [a, b, c] means that "region 1 is in position a, region 2 is in position b and region 3 is in position c with negative orientation". The same notation to represent a rearrangement or a symmetry is a mapping from positions to positions; specifically, applying [a, b, c] to a genome instance means that "the region that was in position 1 moves to position a, the region that was in position 2 moves to position b and the region that was in position 3 moves to position c and reverses its orientation". Then, performing calculations for distances of each genome from the reference, with the distance measures of minimum distance, MLE and MFPT, we obtained the results listed in Table 3. (We note that for these dimensions, each set of calculations took a matter of seconds.) For each genome, we list a single instance as representative.
The coincidences of values in the table are due to the genomes forming equivalence classes. There are, in this case, 12 distinct equivalence classes of genomes; members of any equivalence class are equidistant from the reference. For more details, we refer the reader to Section 4 and the references therein.
The regular representation of the model element in the genome algebra, that is, the Markov matrix representing rearrangement via this model, is The corresponding graph, with edges of weight 1 3 and 1 6 coloured grey and black respectively is in Figure 3 below. The distances of each genome from the reference, via the distance measures of minimum distance, MLE and MFPT, are listed in Table 4. Again, there are coincidences in the values due to the equivalence classes of genomes; in this case, the genomes form four equivalence classes.
ρ(s) = .
Appendix A .
.Further results for the 6-region caseThe following plots are included to give an idea of the range of values taken by the MLE and MFPT distance measures. Since there are (c.f. Section 4) 250 equivalence classes of genomes, with the members of each class equidistant from the reference, there are 250 values for the MFPT distance for each model. There are fewer values for the MLE distance, since this does not always exist. We note again that the determination of MLEs is highly dependent on the optimisation function used.
Figure 1 .
1Plot of MLE distances (y-axis) between each genome and the reference, genomes ordered by min distance and MLE value for model (iii). Key: model (i) blue; model (ii) red; model (iii) yellow; model (iv) green; model (v) orange.
Figure 3 .
3Rearrangement paths under an all inversions model for genomes with three oriented regions; genomes modelled with an origin of replication.
Figure 4 .
4Rearrangement paths under an all inversions model for genomes with three oriented regions; genomes modelled without an origin of replication.
Table 1 .
1Pairwise distance estimates via each of minimum distance, MLE and MFPT, between genomes with instances e = [1, 2, 3, 4, 5, 6], σ 1 =
Table 2 .
2Pairwisedistance estimates via MLE between genomes with in-
stances e = [1, 2, 3, 4, 5, 6], σ 1 = [3, 4, 1, 2, 6, 5], σ 2 = [6, 3, 4, 5, 2, 1], assuming
dihedral symmetry, under the five models.
Suppose firstly that the (circular) genomes have an origin of replication between positions 1 and 3, so the appropriate symmetry group isZ = {e, f } = {[1, 2, 3],[3, 2, 1]}. Thus there are |H 3 | |Z| = 24 distinct genomes here, with each genome having exactly two instances. For example, [1, 2, 3] and [3, 2, 1] are both instances of the reference genome; [1, 3, 2] and[3,1,2] are both instances of another genome. To represent these in the genome algebra, we set z = 1 2 (e + f ) and then the reference genome is z = 1We consider a model consisting of all inversions of one and two regions, equally weighted. Note that for three regions, this is all inversions. Formally, our model isM = {z[1, 2, 3]z, z[1, 2, 3]z, z[2, 1, 3]z} ,with each rearrangement given a probability of1 3 . Note that this covers all of the relevant rearrangements since f ·[1, 2, 3]·f =[1,2,3]; f ·[2, 1, 3]·f =[1,3,2]; and we disallow inversions over the origin of replication.2 [1, 2, 3] + [3, 2, 1] ; the second genome
is z[1, 3, 2] = 1
2 [1, 3, 2] + [3, 1, 2] .
Table 3 .
3Pairwise distance estimates between each genome with three oriented regions and the reference; genomes modelled with an origin of replication.genome instance min MLE MFPT
[1, 2, 3]
0
0.00
0.00
[1, 2, 3]
1
1.40
24.52
[1, 3, 2]
3
12.89
29.35
[1, 3, 2]
2
7.12
28.20
[1, 2, 3]
1
1.20
19.50
[1, 2, 3]
2
3.77
27.29
[1, 3, 2]
2
7.12
28.20
[1, 3, 2]
1
1.32
24.98
[2, 1, 3]
3
12.89
29.35
[2, 1, 3]
2
−
30.03
[2, 3, 1]
2
−
29.26
[2, 3, 1]
1
1.32
24.98
[2, 1, 3]
2
7.12
28.20
[2, 1, 3]
3
−
30.01
[2, 3, 1]
3
−
30.01
[2, 3, 1]
2
7.12
28.20
[3, 1, 2]
2
−
29.26
[3, 1, 2]
3
−
30.01
[3, 2, 1]
3
−
29.81
[3, 2, 1]
2
3.77
27.29
[3, 1, 2]
3
−
30.01
[3, 1, 2]
2
−
30.03
[3, 2, 1]
2
−
29.11
[3, 2, 1]
1
1.40
24.52
Table 4 .
4Pairwise distance estimates between each genome with three signed regions and the reference; genomes modelled without an origin of replication.For this case, the regular representation of the model element in the genome algebra, that is, the Markov matrix representing rearrangement via this model, isgenome instance min MLE MFPT
[1, 2, 3]
0
0.00
0.0
[1, 2, 3]
1
1.65
7.0
[1, 3, 2]
3
−
10.0
[1, 3, 2]
2
−
9.0
[1, 2, 3]
1
1.65
7.0
[1, 2, 3]
2
−
9.0
[1, 3, 2]
2
−
9.0
[1, 3, 2]
1
1.65
7.0
ρ(s) =
0 1
although in this case the algebra is not isomorphic to to a direct sum of its irreducible modules; see[30] for full details
Alternatively, one can model the genomes without an origin of replication; we now turn to this case. The symmetry group now consists of all flips and rotations, that is,Accordingly, each genome has 6 instances in H 6 and there are |H 6 | |Z| = 8 distinct genomes here.We wish to again consider a model consisting of all inversions of one and two regions, equally weighted. Now, one may obtain any one-region inversion from another by conjugation with a rotation (and, in general, this is true for any k-region inversion). Further, here the oneand two-region inversions are complementary: a two-region inversion is just a rotated flip of a one-region inversion. For example, whith the rotation r = [2, 3, 1] and the flip, as above,
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Sorting by restricted-length-weighted reversals. Hieu Trung Thach Cam Nguyen, Nguyen Bao Ngo, Nguyen, Proteomics & Bioinformatics. 32GenomicsThach Cam Nguyen, Hieu Trung Ngo, and Nguyen Bao Nguyen. Sorting by restricted-length-weighted reversals. Genomics, Proteomics & Bioinformatics, 3(2):120-127, May 2005.
Super short reversals on both gene order and intergenic sizes. Andre Rodrigues Oliveira, Géraldine Jean, Guillaume Fertin, Ulisses Dias, Zanoni Dias, Advances in Bioinformatics and Computational Biology. Ronnie AlvesSpringer International PublishingAndre Rodrigues Oliveira, Géraldine Jean, Guillaume Fertin, Ulisses Dias, and Zanoni Dias. Super short reversals on both gene order and intergenic sizes. In Ronnie Alves, editor, Advances in Bioinformatics and Computational Biology, pages 14-25, Cham, 2018. Springer International Publishing.
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Distancebased genome rearrangement phylogeny. Li-San Wang, Tandy Warnow, M E Bernard, Robert K Moret, Linda A Jansen, Raubeson, J Mol Evol. 634Li-San Wang, Tandy Warnow, Bernard M. E. Moret, Robert K. Jansen, and Linda A. Raubeson. Distance- based genome rearrangement phylogeny. J Mol Evol, 63(4):473-483, Oct 2006.
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[
"A NOTE ON KOSMANN-LIE DERIVATIVES OF WEYL SPINORS",
"A NOTE ON KOSMANN-LIE DERIVATIVES OF WEYL SPINORS"
] |
[
"R A Sharipov "
] |
[] |
[] |
Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.
| null |
[
"https://arxiv.org/pdf/0801.0622v1.pdf"
] | 19,013,824 |
0801.0622
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10b33d081d99af34be93367e651bc58ae2fd4e4d
|
A NOTE ON KOSMANN-LIE DERIVATIVES OF WEYL SPINORS
4 Jan 2008
R A Sharipov
A NOTE ON KOSMANN-LIE DERIVATIVES OF WEYL SPINORS
4 Jan 2008
Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.
Introduction.
Lie derivatives arise in studying continuous symmetries of various geometric structures on manifolds. They are also used in symmetry analysis of ordinary and partial differential equations (see [1]). In general relativity the bundle of Weyl spinors SM is a special geometric structure built over the space-time manifold M . The main goal of this paper is to clarify the procedure of applying Lie derivatives to the basic attributes of this geometric structure, i. e. to the basic spin-tensorial fields associated with the bundle of Weyl spinors.
Lie derivatives of spatial structures.
Let M be a space-time manifold of general relativity. This means that it is a four-dimensional orientable manifold equipped with a Minkowski type metric g and with a polarization. A polarization, which is typically not mentioned, is a geometric structure that marks the future half light cone in the tangent space T p (M ) for each point p ∈ M (see more details in [2]). A Lie derivative L X is usually given by some vector field X in M . Once such a vector field X is fixed, it produces a one-parametric local group of local diffeomorphisms:
ϕ ε : M → M.
(2.1)
The letter t is typically used for the parameter of this local group (see [3]), but here we use the Greek letter ε since t in physics is reserved for the time variable. The local diffeomorphisms (2.1) induce the local diffeomorphisms Let's study the diffeomorphisms (2.1) and (2.2) in more details. Assume that p and q are two points of the space-time manifold M such that q = ϕ ε (p). Then p = ϕ −ε (q) and we have the following commutative diagram:
T p (M ) ϕ ε * − −−− → T q (M ) π π p ϕ ε − −−− → q π π T * p (M ) ϕ * −ε − −−− → T * q (M ).
(2.5)
Assume that we have some local chart with the coordinates x 0 , x 1 , x 2 , x 3 in some neighborhood of the point q. Assume also that ε is small enough so that the point p = ϕ −ε (q) in (2.5) is covered by the same local chart. Then the coordinates of the points p and q in this chart are related to each other as follows:
x 0 = u 0 (ε, y 0 , y 1 , y 2 , y 3 ),
x 1 = u 1 (ε, y 0 , y 1 , y 2 , y 3 ),
x 2 = u 2 (ε, y 0 , y 1 , y 2 , y 3 ),
x 3 = u 3 (ε, y 0 , y 1 , y 2 , y 3 ), (2.6)
y 0 = u 0 (−ε, x 0 , x 1 , x 2 , x 3 ),
Using (2.6), we define the matrices Φ(ε) and Φ(−ε) with the components Φ i j (ε) = ∂u i (ε, y 0 , y 1 , y 2 , y 3 )
∂y j , Φ i j (−ε) = ∂u i (−ε, x 0 , x 1 , x 2 , x 3 ) ∂x j . (2.7)
Let Y i1... ir j1... js (x 0 , x 1 , x 2 , x 3 ) be the components of some tensorial field Y of the type (r, s) in the local coordinates x 0 , x 1 , x 2 , x 3 and let ϕ ε (Y) i1... ir j1... js (x 0 , x 1 , x 2 , x 3 ) be the components of its image ϕ ε (Y) under the mapping (2.3). Then we have
ϕ ε (Y) i1... ir j1... js (x 0 , x 1 , x 2 , x 3 ) = 3 ... 3 h1, ... , hr k1, ... , ks Φ i1 h1 (ε) . . . Φ ir hr (ε) × × Φ k1 j1 (−ε) . . . Φ ks js (−ε) Y h1... hr k1.
.. ks (y 0 , y 1 , y 2 , y 3 ).
(2.8)
According to [3], the Lie derivative L X applied to Y is defined as follows:
L X (Y) = lim ε→ 0 Y − ϕ ε (Y) ε = − dϕ ε (Y) dε ε=0 .
(2.9) Let X 0 , X 1 , X 2 , X 3 be the components of the vector field X in the local coordinates x 0 , x 1 , x 2 , x 3 . Then for ε → 0 we have the following Taylor expansions of the functions (2.6) representing the mappings ϕ ε and ϕ −ε :
u 0 (ε, y 0 , y 1 , y 2 , y 3 ) = y 0 + X 0 (y 0 , y 1 , y 2 , y 3 ) ε + . . . , u 1 (ε, y 0 , y 1 , y 2 , y 3 ) = y 1 + X 1 (y 0 , y 1 , y 2 , y 3 ) ε + . . . , u 2 (ε, y 0 , y 1 , y 2 , y 3 ) = y 2 + X 2 (y 0 , y 1 , y 2 , y 3 ) ε + . . . , u 3 (ε, y 0 , y 1 , y 2 , y 3 ) = y 3 + X 3 (y 0 , y 1 , y 2 , y 3 ) ε + . . . ,
(2.10) u 0 (−ε, x 0 , x 1 , x 2 , x 3 ) = x 0 − X 0 (x 0 , x 1 , x 2 , x 3 ) ε + . . . , u 1 (−ε, x 0 , x 1 , x 2 , x 3 ) = x 1 − X 1 (x 0 , x 1 , x 2 , x 3 ) ε + . . . , u 2 (−ε, x 0 , x 1 , x 2 , x 3 ) = x 2 − X 2 (x 0 , x 1 , x 2 , x 3 ) ε + . . . , u 3 (−ε, x 0 , x 1 , x 2 , x 3 ) = x 3 − X 3 (x 0 , x 1 , x 2 , x 3 ) ε + . . . . (2.11)
Applying (2.10) and (2.11) to (2.7), we derive (2.14)
Φ i j (ε) = δ i j + ∂X i (x 0 , x 1 , x 2 , x 3 ) ∂x j ε + . . . , (2.12) Φ i j (−ε) = δ i j − ∂X i (x 0 , x 1 , x 2 , x 3 ) ∂x j ε + . . . . (2.13) Here X i (x 0 , x 1 , x 2 , x 3 ) are the components of the vector field X in the local coordi- nates x 0 , x 1 , x 2 , x 3 . Now if we denote by L X (Y) i1
The Lie derivative L X given by the formula (2.14) possesses the following properties:
(1) the Lie derivative L X preserves the type of a tensor field, i. e. Y and L X (Y) are tensor fields of the same type;
(2) L X (Y 1 ⊗ Y 2 ) = L X (Y 1 ) ⊗ Y 2 + Y 1 ⊗ L X (Y 2 ) for arbitrary two tensorial fields Y 1 and Y 2 ; (3) L X (C(Y)) = C(L X (Y)), i. e. L X commute with contractions.
The properties (1)-(3) are easily derived with the use of the formula (2.14) itself.
Lie derivatives in frames formalism.
Let x 0 , x 1 , x 2 , x 3 be the local coordinates of some local chart of the space-time manifold M . The coordinates x 0 , x 1 , x 2 , x 3 induce the frame X 0 , X 1 , X 2 , X 3 of the coordinate vector fields in the domain of these coordinates:
X 0 = ∂ ∂x 0 , X 1 = ∂ ∂x 1 , X 2 = ∂ ∂x 2 , X 3 = ∂ ∂x 3 . (3.1)
The frame composed by the vector fields (3.1) is a holonomic frame since these vector fields commute with each other:
[X i , X j ] = 0.
However, one can consider some non-holonomic frame Υ 0 , Υ 1 , Υ 2 , Υ 3 , i. e. a frame with non-commuting vector fields:
[Υ i , Υ j ] = 3 k=0 c k ij Υ k . (3.2)
The commutation coefficients c k ij in (3.2) are uniquely determined by the frame vector fields Υ 0 , Υ 1 , Υ 2 , Υ 3 since they are linearly independent at each point of their domain. Our nearest goal is to derive the formula analogous to (2.14) for the case where all tensorial fields are represented by their components is some non-holonomic frame Υ 0 , Υ 1 , Υ 2 , Υ 3 .
Let ϕ be a scalar field. Then the Lie derivative L X of ϕ is reduced to the differentiation of the function ϕ along the vector X:
L X (ϕ) = 3 k=0 X k ∂ϕ ∂x k . (3.3)
The formula (3.3) is easily derived by substituting Y = ϕ with r = 0 and s = 0 into (2.14). Similarly, if Y is a vector field, from (2.14) we derive
L Y (Y) = [X, Y]. (3.4)
Now assume that both of the vector fields X and Y are represented by their expansions in a non-holonomic frame Υ 0 , Υ 1 , Υ 2 , Υ 3 :
X = 3 k=0 X k Υ k , Y = 3 i=0 Y i Υ i . (3.5)
Then, substituting (3.5) into (3.4), we derive
L X (Y) k = 3 i=0 X i L Υi (Y k ) − 3 i=0 Y i L Υi (X k ) + 3 i=0 3 j=0 c k ij X i Y j . (3.6)
The Lie derivatives L Υi (Y k ) and L Υi (X k ) in (3.6) are calculated according to the formula (3.3), i. e. we substitute ϕ = Y k and ϕ = X k into (3.3). Note that X k in (3.3) differ from that of (3.5). The components of X in the formula (3.3) are taken from the expansion of the vector field X in the holonomic frame (3.1). The non-holonomic version of the formula (3.3) looks like
L X (ϕ) = 3 k=0 X k L Υi (ϕ).
Let's denote by η 0 , η 1 , η 2 , η 3 the dual frame for Υ 0 , Υ 1 , Υ 2 , Υ 3 . This means that η 0 , η 1 , η 2 , η 3 are four covectorial fields such that
η i (Υ j ) = η i , Υ j = C(η i ⊗ Υ j ) = δ i j . (3.7)
Using the properties (1)-(3) from Section 2 and using (3.4), from (3.7) we derive
L Υi (η k ) = − 3 j=0 c k ij η j . (3.8)
The commutation coefficients c k ij in (3.8) are the same as in (3.2). Assume that Y is a covectorial field expanded in the frame η 0 , η 1 , η 2 , η 3 :
Y = 3 j=1 Y i η i .L X (Y) k = 3 i=0 X i L Υi (Y k ) + 3 i=0 Y i L Υ k (X i ) − 3 i=0 3 j=0 X i Y j c j ik . (3.10)
Now, using the properties (1)-(3) again and combining (3.6) with (3.10), we can extend the formula (3.6) to the case of an arbitrary tensor field Y of the type (r, s): (3.11)
L X (Y) i1... ir j1... js = 3 i=0 X i L Υi (Y i1... ir j1... js ) + + s m=1 3 km=0 L Υj m (X km ) − 3 i=0 c km ijm X i Y i1... ... ... ir j1... km... js − − r m=1 3 km=0 L Υ km (X im ) − 3 i=0 c im ikm X i Y i1
The formulas (3.6) and (3.10) are special cases of the formula (3.11). If the frame Υ 0 , Υ 1 , Υ 2 , Υ 3 coincides with the holonomic frame (3.1), then c k ij = 0 and the formula (3.11) reduces to (2.14). Now let's return back to the formula (2.8). This formula is valid in frame presentation of tensor fields too. However, the matrices Φ i j (ε) and Φ i j (−ε) in this case are not given by the formulas (2.7). Here we use the formulas
Φ i j (ε) = δ i j + L Υj (X i ) − 3 m=0 X m c i mj ε + . . . , (3.12) Φ i j (−ε) = δ i j − L Υj (X i ) − 3 m=0 X m c i mj ε + . . . . (3.13)
The formulas (3.12) and (3.13) are analogous to (2.12) and (2.13).
Tangent vector fields on vector bundles.
Let V M be an n-dimensional vector bundle over the space-time manifold M . Any trivialization of V M is given by n sections Υ 1 , . . . , Υ n linearly independent at each point of their domain U . Let v be a vector of the fiber V q (M ):
v = v 1 Υ 1 + . . . + v n Υ n . (4.1) Thenq = (q, v) is a point of V M . If x 0 , x 1 , x 2 , x 3 are some local coordinates within the domain U ⊂ M and v 0 , v 1 , v 2 , v 3 are taken from (4.1), then x 0 , . . . , x 3 , v 1 , . . . , v n (4.2)
are the coordinates of the pointq = (q, v). The coordinates (4.2) are naturally subdivided into two groups -the base coordinates x 0 , . . . , x 3 and the fiber coordinates v 1 , . . . , v n . Let X be a tangent vector field on V M . In the local coordinates (4.2) it is represented as the following differential operator:
X = 3 i=0 X i ∂ ∂x i + n i=1 V i ∂ ∂v i . (4.3)
Under the canonical projection π : V M → M the vector field (4.3) is mapped to
X = 3 i=0 X i ∂ ∂x i . (4.4)
The vector field (4.3) produces the one-parametric local group of diffeomorphisms
ϕ ε : V M → V M (4.5)
that extends the local group (2.1) produced by the vector field (4.4). Due to (4.5) the functions (2.6) are complemented with the functions In the case of a concordant vector field X the local diffeomorphisms (4.5) break into the series of linear mappings
v 1 = U 1 (ε, y 0 , . . . , y 3 , w 1 , . . . , w n ), . . . . . . . . . . . . . . . . . . . . . . . . . v n = U n (ε, y 0 , . . . , y 3 , w 1 , . . . , w n ), (4.6) w 1 = U 1 (−ε, x 0 , . . . , x 3 , v 1 , . . . , v n ), . . . . . . . . . . . . . . . . . . . . . . . . . w n = U n (−ε, x 0 , . . . , x 3 , v 1 , . . . , v n ).ϕ ε : V p (M ) → V q (M ),(4.8)
where q = ϕ ε (p). The functions (4.6) present the diffeomorphisms (4.8) in local coordinates. In the case of a concordant vector field they are given by the formulas
U i (ε, y 0 , . . . , y 3 , w 1 , . . . , w n ) = n j=1 U i j (ε, y 0 , . . . , y 3 ) w j . (4.9)
The vertical components of the tangent vector field (4.3) are given by the derivatives
V i = dU i (ε, x 0 , . . . , x 3 , v 1 , . . . , v n ) dε ε=0 .
(4.10)
Substituting (4.9) into (4.10), we derive
V i = n j=1 V i j (x 0 , . . . , x 3 ) v j , where V i j = U i j ε=0
. For the matrices U i j in (4.9), which represent the linear mappings (4.8) in the frame Υ 1 , . . . , Υ n , the formulas (4.10) and (4.11) yield
U i j (ε) = δ i j + V i j (x 0 , . . . , x 3 ) ε + . . . , U i j (−ε) = δ i j − V i j (x 0 , . . . , x 3 ) ε + . . . . (4.12)
The expansions (4.12) are similar to (2.12), (2.13), (3.12), and (3.13). Let Y be a tensor field of the type (r, s) associated with the vector bundle V M , i. e. let Y be a section of V r s M , where V r s M is the following tensor bundle:
V r s M = r times V M ⊗ . . . ⊗ V M ⊗ V * M ⊗ . . . ⊗ V * M s times . (4.13) Assume that Y i1... ir j1... js (x 0 , x 1 , x 2 , x 3 )
are the components of the tensor field Y in the frame Υ 1 , . . . , Υ n . Then, using the quantities V i j from the expansions (4.12), we define the Lie derivative L X (Y) of the field Y:
L X (Y) i1... ir j1... js = 3 i=0 X i L Υi (Y i1... ir j1... js ) + + s m=1 n km=1 V km jm Y i1... ... ... ir j1... km... js − r m=1 n km=1 V im km Y i1... km... ir j1... ... ... js .
(4.14)
The Lie derivative (4.14) is called natural if the quantities V i j are expressed in some natural way through the components of the vector field X in (4.4).
Natural liftings and Kosmann liftings.
In this section we apply the results of the previous section 4 to the tangent bundle T M , i. e. we set V M = T M . Comparing the formula (4.13) with (2.4) and the formula (4.14) with (3.11), we find that
V i j = L Υj (X i ) − 3 m=0 X m c i mj . (5.1)
Now we substitute (5.1) into (4.11) and then we substitute (4.11) into (4.3):
X N = 3 i=0 3 m=0 X m Υ i m ∂ ∂x i + 3 i=0 3 j=0 L Υj (X i ) − 3 m=0 X m c i mj v j ∂ ∂v i . (5.2)
The tangent vector field (5.2) on T M is a natural lifting of the vector field
X = 3 i=0 3 m=0 X m Υ i m ∂ ∂x i = 3 m=0 X m Υ m (5.3)
from M to T M . The Lie derivative (4.14) determined by the vector field (5.3) and by its lifting (5.2) is a natural Lie derivative coinciding with (3.11) and (2.14). Now let's recall that the the space-time manifold M is equipped with the metric g. Its signature is (+, −, −, −). For this reason each fiber T p (M ) of the tangent bundle T M is a pseudo-Euclidean linear vector space. The proof is trivial. By definition, Killing vector fields are those whose local diffeomorphisms preserve the metric tensor g. Hence, the mapping ϕ ε * from (2.2) restricted to any fiber T p (M ) is a linear isometry.
Let's study the isometry condition from the definition 5.1 in more details. Applying the linear mappings (4.8) to the metric tensor, we get the equality
g ij (x 0 , . . . , x 3 ) = 3 r=0 3 s=0 U r i (−ε) U s j (−ε) g rs (y 0 , . . . , y 3 ). (5.4)
Differentiating the formula (5.4) with respect to ε, we take into account the formulas (4.12), (2.6), (2.10), (2.11), (4.10), and (4.11). As a result we get
0 = 3 r=0 V r i g rj + 3 r=0 V r j g ir + 3 m=0 X m L Υm (g ij ). (5.5)
The equality (5.5) can be simplified to
L X (g) = 0,(5.6)
where the Lie derivative L X is calculated according to the formula (4.14). We shall treat the equality (5.6) neither as a condition for X nor as a condition for g, but as a condition for V i j . For this purpose we denote
V ij = 3 r=0 V r i g rj . (5.7)
Then the equality (5.5), which is equivalent to (5.6) is written as follows:
V ij + V j i = − 3 m=0 X m L Υm (g ij ). (5.8)
The equality (5.8) fixes the symmetric part of V ij for a Kosmann lifting of a vector field. The skew-symmetric part of V ij can be obtained by alternating (5.1). As a result we get the following two formulas:
V sym ij = − 1 2 3 m=0 X m L Υm (g ij ), (5.9) V skew ij = 1 2 3 r=0 L Υi (X r ) g rj − 1 2 3 r=0 L Υj (X r ) g ri − − 1 2 3 r=0 3 m=0 X m c r mi g rj + 1 2 3 r=0 3 m=0 X m c r mj g ri . (5.10)
Adding the formulas (5.9) and (5.10), we derive
V ij = − 1 2 3 m=0 X m L Υm (g ij ) + 1 2 3 r=0 L Υi (X r ) g rj − − 1 2 3 r=0 L Υj (X r ) g ri − 1 2 3 r=0 3 m=0 X m c r mi g rj + 1 2 3 r=0 3 m=0
X m c r mj g ri .
(5.11)
In order to get back to V j i we need to raise the index j in (5.11):
V j i = − 1 2 3 m=0 3 r=0 X m L Υm (g ir ) g rj − 1 2 3 r=0 3 s=0 L Υs (X r ) g ri g sj + + 1 2 L Υi (X j ) − 1 2 3 m=0 X m c j mi + 1 2 3 r=0 3 s=0 3 m=0
X m c r ms g ri g sj .
(5.12)
Note that in order to fit (4.11) we should exchange the indices i and j in (5.12):
V i j = − 1 2 3 m=0 3 r=0 g ir X m L Υm (g rj ) − 1 2 3 r=0 3 s=0 g is L Υs (X r ) g rj + + 1 2 L Υj (X i ) − 1 2 3 m=0 X m c i mj + 1 2 3 r=0 3 s=0 3 m=0
g is X m c r ms g rj . As a result, substituting (5.13) into (4.11), we derive the formula
V i = − 1 2 3 m=0 3 r=0 3 j=0 g ir X m L Υm (g rj ) v j + 1 2 3 j=0 L Υj (X i ) v j − − 1 2 3 r=0 3 s=0 3 j=0 g is L Υs (X r ) g rj v j − 1 2 3 m=0 3 j=0 X m c i mj v j + + 1 2 3 r=0 3 s=0 3 m=0 3 j=0
g is X m c r ms g rj v j .
(5.14)
Applying (5.14) to (4.3) and taking into account that we choose V M = T M , we get the following tangent vector field on the tangent bundle: We can calculate the standard Kosmann lifting (5.15) using the holonomic frame (3.1) instead of the non-holonomic frame Υ 0 , Υ 1 , Υ 2 , Υ 3 . Then (5.15) reduces to
X K = 3 i=0 3 m=0 X m Υ i m ∂ ∂x i + 1 2 3 i=0 3 j=0 L Υj (X i ) v j − − 3 m=0 3 j=0 X m c i mj v j + 3 r=0 3 s=0 3 m=0 3 j=0 g is X m c r ms g rj v j − − 3 r=0 3 s=0 3 j=0 g is L Υs (X r ) g rj v j − 3 m=0 3 r=0 3 j=0 g ir X m L Υm (g rj ) v j ∂ ∂v i .X K = 3 i=0 X i ∂ ∂x i + 1 2 3 i=0 − 3 r=0 3 s=0 3 j=0 g is ∂X r ∂x s g rj v j + + 3 j=0 ∂X i ∂x j v j − 3 m=0 3 r=0 3 j=0 g ir X m ∂g rj ∂x m v j ∂ ∂v i . (5.16)
Now let's remember that the metric g is associated with the metric connection Γ. It is known as the Levi-Civita connections. The components of the Levi-Civita connection in a holonomic frame (3.1) are given by the formula
Γ k ij = 3 r=0 g kr 2 ∂g rj ∂x i + ∂g ir ∂x j − ∂g ij ∂x r . (5.17)
The formula (5.17) is easily derived from the following two conditions (see [8]):
Γ k ij = Γ k ji , ∇ k g ij = 0. (5.18)
Using the connection components Γ k ij , now we express the partial derivatives in the formula (5.16) through the corresponding covariant derivatives: .17), we get the following formula:
∂X r ∂x s = ∇ s X r − 3 m=0 Γ r sm X m , ∂X i ∂x j = ∇ j X i − 3 m=0 Γ i jm X m .(5.X K = 3 i=0 X i ∂ ∂x i + 3 i=0 3 j=0 − 1 2 3 r=0 3 s=0 g is ∇ s X r g rj + + 1 2 ∇ j X i − 3 m=0 X m Γ i mj v j ∂ ∂v i . (5.21)
The components of the natural lifting (5.2) also can be expressed through the connection components Γ k ij and covariant derivatives in the holonomic frame (3.1):
X N = 3 i=0 X i ∂ ∂x i + 3 i=0 3 j=0 ∇ j X i − 3 m=0 X m Γ i mj v j ∂ ∂v i . (5.22)
Two different liftings (5.21) and (5.22) are associated with two different Lie derivatives L X and L X respectively. Here L X is the regular Lie derivative, while L X is called the standard Kosmann-Lie derivative. Both of them are differentiations of the algebra of tensor fields. Comparing (5.21) and (5.22), we get the relationship
L X = L X + S X . (5.23)
Here S X is a degenerate differentiation in the sense of the proposition 3.3 in Chapter I of [3]. From (5.21) and (5.22) we derive that the degenerate differentiation S X in (5.23) is given by the tensor field S X with the following components:
S i j (X) = ∇ j X i + ∇ i X j 2 = 1 2 ∇ j X i + 1 2 3 r=0 3 s=0 g is ∇ s X r g rj (5.24)
The tensor field S X with the components (5.24) is equal to zero if and only if X is a Killing vector field. This fact is easily derived from the theorem 5.1 or from the equality (5.6), which is fulfilled identically by definition.
Another special case for the formula (5.15) is the case of a non-holonomic, but orthonormal frame Υ 0 , Υ 1 , Υ 2 , Υ 3 . In such a frame the components of the metric tensor are constants. They are given by the Minkowski matrix:
g ij = g ij = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 . (5.25)
For the components of the Levi-Civita connection in such a frame we have
Γ k ij = c k ij 2 − 3 r=0 3 s=0 c s ir 2 g kr g sj − 3 r=0 3 s=0 c s j r 2 g kr g si . (5.26)
The formula (5.26) is derived from the general formula (6.3) in [9]. From the formulas (5.25) and (5.26) we derive the following equalities:
L Υm (g rj ) = 0, Γ k ij − Γ k j i = c k ij . (5.27) Moreover, from ∇ m g ij = 0 in this case we derive 3 r=0 3 s=0 g is Γ r ms g rj = −Γ i mj . (5.28)
In the non-holonomic frame Υ 0 , Υ 1 , Υ 2 , Υ 3 the formulas (5.19) are replaced by
L Υs (X r ) = ∇ s X r − 3 m=0 Γ r sm X m , L Υj (X i ) = ∇ j X i − 3 m=0 Γ i jm X m .
(5.29) Applying (5.27), (5.28), and (5.29) to (5.15), we get the formula
X K = 3 i=0 3 m=0 X m Υ i m ∂ ∂x i + 3 i=0 3 j=0 1 2 ∇ j X i − − 1 2 3 r=0 3 s=0 g is ∇ s X r g rj − 3 m=0 X m Γ i mj v j ∂ ∂v i . (5.30)
Similarly, applying (5.29) to (5.2) and taking into account (5.27), we get
X N = 3 i=0 3 m=0 X m Υ i m ∂ ∂x i + 3 i=0 3 j=0 ∇ j X i − 3 m=0 X m Γ i mj v j ∂ ∂v i . (5.31)
The formulas (5.30) to (5.31) coincide with (5.21) to (5.22) respectively, though Γ i mj are not symmetric with respect to m and j in this case. The formulas (5.30) and (5.31) again lead to the formula (5.23), which is valid irrespective to the choice of a holonomic or non-holonomic frame in M . Note that the formulas (5.29) are the same as the formula (3.10) in [10]. The formula (5.29) resembles the formulas (3.11) and (3.12) in [10]. However, it doesn't coincide with them. Unlike [10] and [11], for the sake of simplicity in this paper I do not use principal fiber bundles at all.
Commutation relationships for Kosmann-Lie derivatives.
Regular Lie derivatives acting upon tensorial fields form a representation of the Lie algebra of vector fields in M . This fact is expressed by the formula
[L X , L Y ] = L [X,Y] . (6.1)
Commutation relationships for Kosmann-Lie derivatives are different from (6.1):
[L X , L Y ] = L [X,Y] + S X,Y . (6.2)
Here S X,Y is a degenerate differentiation given by the tensor field S X,Y , where
S X,Y = L X (S Y ) − L Y (S X ) − S [X,Y] + [S X , S Y ]. (6.3)
By means of direct calculations the formula (6.3) can be reduced to
S X,Y = −[S X , S Y ]. (6.4)
Applying (6.4) to (6.2), we get
[L X , L Y ] = L [X,Y] − [S X , S Y ]. (6.5)
The commutator [S X , S Y ] in the formulas (6.3), (6.4), and (6.5) is understood as a commutator of two operator-valued tensorial fields:
[S X , S Y ] = S X • S Y − S X • S Y . (6.6)
In the coordinate representation the commutator (6.6) turns to the commutator of two matrices whose components are calculated according to the formula (5.24). Note that in general case the commutator (6.6) is not zero. For this reason
[L X , L Y ] = L [X,Y] .
This fact is pointed out in [10].
Kosmann-Lie derivatives for Weyl spinors.
The bundle of Weyl spinors is a two-dimensional complex vector bundle over the space-time manifold M . We denote it SM . The spinor bundle SM is related to the tangent bundle T M in some special way. The relation of SM and T M is formulated in terms of frames. It is based on the well-known group homomorphism φ : SL(2, C) → SO + (1, 3, R).
(7.1) Let Υ 0 , Υ 1 , Υ 2 , Υ 3 be a positively polarized right orthonormal frame of the tangent bundle T M . By definition (see Section 5 in [12] or Section 1 in [13]) it is canonically associated with some frame Ψ 1 , Ψ 2 of SM in such a way that if two positively polarized right orthonormal frames Υ 0 , Υ 1 , Υ 2 , Υ 3 andΥ 0 ,Υ 1 ,Υ 2 ,Υ 3 are bound with the transition matrices S and T = S −1 in the formulas
Υ i = 3 j=0 S j i Υ j , Υ i = 3 j=0 T j iΥ j ,
then their associated spinor frames Ψ 1 , Ψ 2 andΨ 1 ,Ψ 2 are bound with the transition matrices S and T = S −1 in the formulas
Ψ i = 2 j=1 S j i Ψ j , Ψ i = 2 j=1
T j iΨ j and the spacial transition matrices S and T are produced from the spinor transition matrices S and T through the group homomorphism (7.1):
S = φ(S), T = φ(T).
A spinor frame Ψ 1 , Ψ 2 canonically associated with some positively polarized right orthonormal tangent frame Υ 0 , Υ 1 , Υ 2 , Υ 3 is called an orthonormal frame of the spinor bundle SM . In order to visualize this canonical frame association in SM and T M we use the following diagram:
Orthonormal frames → Positively polarized right orthonormal frames . The spin-tensorial fields d and G are introduced by means of the explicit formulas for their components in canonically associated frame pairs (7.2). Let Υ 0 , Υ 1 , Υ 2 , Υ 3 be some positively polarized right orthonormal frame in T M and let Ψ 1 , Ψ 2 be its associated orthonormal spinor frame in SM . The components of the Infeld-van der Waerden field G in such a frame pair composed by two canonically associated frames are given by the following Pauli matrices:
G iī 0 = 1 0 0 1 = σ 0 , G iī 2 = 0 −i i 0 = σ 2 , (7.4) G iī 1 = 0 1 1 0 = σ 1 , G iī 3 = 1 0 0 −1 = σ 3 .
The skew-symmetric metric tensor d is given by the matrix
d ij = 0 1 −1 0 (7.5)
in any orthonormal spinor frame Ψ 1 , Ψ 2 . Unlike (7.4), the choice of the associated frame Υ 0 , Υ 1 , Υ 2 , Υ 3 is inessential for the components of the matrix (7.5) since d has no spatial indices at all. The dual metric tensor for d is given by the matrix
d ij = 0 −1 1 0 . (7.6)
The matrix (7.6) is inverse to the matrix (7.5). Let S be some matrix from the group SL(2, C) and let S = φ(S) be its image under the homomorphism (7.1). Then we have the relationships
2 i=1 2 ī =1 S a i σ iī m Sā i = 3 k=1 S k m σ aā k , (7.7) 2 i=1 2 j=1 S i a d ij S j b = d ab ,(7.8)
where σ iī m and σ aā k are the components of the Pauli matrices (7.4). In essential, the relationships (7.7) form a definition of the group homomorphism (7.1) (see [13] for more details). The relationships (7.8) are fulfilled due to S ∈ SL(2, C). Now let's take some arbitrary vector field X on M . Then X K is its Kosmann lifting to the tangent bundle T M given by the formula (5.30). It induces local one-parametric group of local diffeomorphisms where q = ϕ ε (p). According to the definition 5.1 the mappings (7.10) are isometries. Therefore, taking some positively polarized right orthonormal frame Υ 0 , Υ 1 , Υ 2 , Υ 3 in T M and applying ϕ ε to it, we get another orthonormal frame ϕ ε (Υ 0 ),
ϕ ε (Υ 1 ), ϕ ε (Υ 2 ), ϕ ε (Υ 3 ) in T M .
Note that ϕ ε is homotopic to the identical mapping. For this reason it preserves the discrete properties like polarization and orientation, i. e. ϕ ε (Υ 0 ), ϕ ε (Υ 1 ), ϕ ε (Υ 2 ), ϕ ε (Υ 3 ) a positively polarized right orthonormal frame. Assume that ε is small enough so that both points p and q = ϕ ε (p) belong to the domain of the frame Υ 0 , Υ 1 , Υ 2 , Υ 3 . Then at the point q we have
ϕ ε (Υ j ) = 3 i=0 V i j (ε) Υ i . (7.11)
The Lorentzian matrix V with the components V i j (ε) ∈ SO + (1, 3, R) in (7.11) is a coordinate presentation of the linear mapping (7.10) in the frame Υ 0 , Υ 1 , Υ 2 , Υ 3 .
Each of the two positively polarized right orthonormal frames Υ 0 , Υ 1 , Υ 2 , Υ 3 and ϕ ε (Υ 0 ), ϕ ε (Υ 1 ), ϕ ε (Υ 2 ), ϕ ε (Υ 3 ) is associated with some orthonormal frame in SM . Using this fact, we define a linear mapping
ϕ ε : S p (M ) → S q (M ) (7.12)
closing the following commutative diagram of frame associations:
Ψ 1 , Ψ 2 − −−− → Υ 0 , Υ 1 , Υ 2 , Υ 3 ϕε ϕε ϕ ε (Ψ 1 ), ϕ ε (Ψ 2 ) − −−− → ϕ ε (Υ 0 ), ϕ ε (Υ 1 ), ϕ ε (Υ 2 ), ϕ ε (Υ 3 ) (7.13)
In the coordinate form the linear mapping (7.12), which closes the diagram (7.13), is presented by some matrix W ∈ SL(2, C):
ϕ ε (Ψ j ) = 2 i=1 W i j (ε) Ψ i . (7.
14)
The horizontal arrows in the diagram (7.13) are canonical frame associations. For this reason the components of the matrices V and W in (7.11) and (7.14) should satisfy the relationships (7.7) and (7.8):
2 i=1 2 ī =1 W a i (ε) σ iī m Wā i (ε) = 3 k=1 V k m (ε) σ aā k , (7.15) 2 i=1 2 j=1 W i a (ε) d ij W j b (ε) = d ab . (7.16)
According to the general recipe (4.12), now we pass from the matrices W (ε) and V (ε) to their expansions as ε → 0, i. e. we write
W i j (ε) = δ i j + W i j (x 0 , x 1 , x 2 , x 3 ) ε + . . . , V i j (ε) = δ i j + V i j (x 0 , x 1 , x 2 , x 3 ) ε + . . . . (7.17)
Note that the quantities V i j in (7.17) are already known. They are taken from (5.13). In our special case, where Υ 0 , Υ 1 , Υ 2 , Υ 3 is an orthonormal frame, we can use a more simple formula for V i j . It is extracted from (5.30):
V i j = − 1 2 3 r=0 3 s=0 g is ∇ s X r g rj + 1 2 ∇ j X i − 3 m=0 X m Γ i mj . (7.18)
As for the quantities W i j , they should be calculated by substituting (7.17) back into (7.15) and (7.16). Like in the case of (5.7), we denote
W ij = 2 s=1 W s i d sj . (7.19)
Then, applying (7.17) to (7.16), we derive the following formula:
W ij − W j i = 0. (7.20)
Due to the formula (7.20) the quantities (7.19) are symmetric with respect to the indices i and j. Our next goal is to resolve the relationships (7.15) with respect to these quantities W ij .
Let's substitute the expansions (7.17) into (7.15) and collect the first order terms with respect to the parameter ε. As a result we get
2 i=1 W a i σ iā m + 2 ī =1 σ aī m Wā i = 3 k=0 V k m σ aā k . (7.21)
Keeping in mind that we use the frames Υ 0 , Υ 1 , Υ 2 , Υ 3 and Ψ 1 , Ψ 2 canonically associated to each other in the sense of the diagram (7.2), we replace the components of Pauli matrices by the components of Infeld-van der Waerden field in (7.21):
2 i=1 W a i G iā m + 2 ī =1 G aī m Wā i = 3 k=0 V k m G aā k . (7.22)
In order to transform the formula (7.22) we multiply it by G m uū and sum over the index m. Doing it, we use one of the identities
3 m=0 G aā m G m uū = 2 δ a u δā u , 2 u=1 2 ū=1 G uū m G n uū = 2 δ n m ,(7.23)
where G m uū are the components of the inverse Infeld-van der Waerden field. They are produced from G aā n by lowering upper spinor indices a amdā and by raising lower spacial index n according to the following formula:
G m uū = 2 a=1 2 ā=1 3 n=0 G aā n d audāū g nm . (7.24)
The components of the conjugate spinor metricd in (7.24) are produced from the components of d by means of the complex conjugation:
dīj = dīj,dīj = dīj.
The identities (7.23) are taken from [14]. Applying them to (7.22), we get
2 W a u δā u + 2 δ a u Wā u = 3 k=0 3 m=0 V k m G aā k G m uū . (7.25)
In order to use the symmetry (7.20) we need to lower the indices a andā in (7.25):
W uadūā + d ua Wūā = 1 2 3 k=0 3 m=0 2 s=1 2 s=1 V k m G ss k d sadsā G m uū . (7.26)
Note that Wūā in (7.26) is symmetric with respect to the indicesū andā, whilē dāū is skew-symmetric with respect to these indices. Therefore, we have As a result we get the following formula for W ua :
W ua = 1 4 3 k=0 3 m=0 2 s=1 2 s=1 G ss k V k m G m us d sa . (7.28)
Now let's return back to the quantities W i j by raising the index a in (7.28). As a result of this standard procedure we obtain
W i j = 1 4 3 k=0 3 m=0 2 s=1 G is k V k m G m js . (7.29)
The next step is to substitute (7.18) into (7.29). Doing it, let's recall that the metric connection Γ has the unique extension (Γ, A,Ā) to the spinor bundle SM . Its spinor components are given by the formula
A i rj = 3 k=0 3 m=0 2 s=1 G is k Γ k rm G m js 4 − − 2 s=1 3 q=0 L Υr (G is q ) G q js 4 − 2 ī =1 2 j =1 L Υr (djī)dīj δ i j 4 . (7.30)
The formula is derived in [14] and is verified in [15]. In our special case the frames Υ 0 , Υ 1 , Υ 2 , Υ 3 and Ψ 1 , Ψ 2 form a canonically associated pair in the sense of the diagram (7.2). In this case the formula (7.30) reduces to
A i rj = 1 4 3 k=0 3 m=0 2 s=1 G is k Γ k rm G m js . (7.31)
The formulas (7.31) and (7.29) are very similar. Therefore, substituting (7.18) into (7.29), due to the presence of Γ i mj in (7.18) we can write
W i j = − 1 8 3 q=0 3 p=0 3 k=0 3 m=0 2 s=1 G is k g kp ∇ p X q g qm G m js + + 1 8 3 k=0 3 m=0 2 s=1 G is k ∇ m X k G m js − 3 m=0 X m A i mj .
(7.32)
Knowing the quantities (7.18) and (7.32) is sufficient to construct a spinor extension of the Kosmann-Lie derivative L X . Let Y be a spin-tensorial field of the type (ε, η|σ, ζ|e, f ). Then for the components of the field L X (Y) we have the formula It is easy to see that (7.33) is a version of (4.14). In the case of a purely tensorial field Y, i. e. if ε = 0, η = 0, σ = 0, and ζ = 0, the Kosmann-Lie derivative (7.33) reduces to (5.23). However, in general case we cannot use this formula (5.23) since the regular Lie derivative L X has no spinor extension yet. For this reason, instead of the formula (5.23), in this case we write
L X (Y) a1L X = ∇ X + S X . (7.34)
Like in (5.23), by S X in (7.34) we denote a degenerate differentiation. According to the results of [12], each degenerate differentiation extended to spinors is defined by three spin-tensorial field of the types (1, 1|0, 0|0, 0), (0, 0|1, 1|0, 0), and (0, 0|0, 0|1, 1). We denote them S X ,S X , and S X respectively. Here are the components of S X :
S i j (X) = ∇ i X j − ∇ j X i 2 .
(7.35) Comparing (7.35) with (5.24), we see that S X in (7.34) is different from that of (5.23). The formula (7.35) is extracted from (7.18). Similarly, looking at (7.32), we find the components of the spin-tensorial field S X :
S i j (X) = 3 k=0 3 m=0 2 s=1 G is k ∇ k X m − ∇ m X k 8 G m js . (7.36)
The components ofS X are produced from (7.36) by means of the complex conjugation:Sī j (X) = Sī j (X). For this components we derive the formulā
Sī j (X) = 3 k=0 3 m=0 2 s=1 G sī k ∇ k X m − ∇ m X k 8 G m sj . (7.37)
The formula (7.34) complemented with (7.35), (7.36), and (7.37) is equivalent to the formula (7.33).
Let's consider a particular example of applying the formula (7.33). Assume that ψ is a spinor field, i. e. a field with the spin-tensorial type (1, 0|0, 0|0, 0). Then for the components of the spinor field L X (ψ) we have
L X (ψ) i = 3 m=0 X m ∇ m ψ i + 3 k=0 3 m=0 2 s=1 2 j=1 G sī k ∇ k X m − ∇ m X k 8 G m sj ψ j .
This formula resembles the formula (3.19) in [10] and the formula (5.5 ′ ) in [11]. Note that the formulas (7.18) and (7.32) were derived under the assumption that Υ 0 , Υ 1 , Υ 2 , Υ 3 is a positively polarized right orthonormal frame in T M and Ψ 1 , Ψ 2 its canonically associated orthonormal frame in SM . However, these formulas remain valid for an arbitrary frame pair provided we use the general formula for Γ i mj in (7.18) instead of (5.26) and the general formula (7.30) for A i mj in (7.32) instead of (7.31). The formula (7.33) is also valid for an arbitrary frame pair under the same provisions.
Kosmann-Lie derivatives of the basic fields.
There are three basic field in the theory of Weyl spinors. Two of them d and G are listed in the table (7.3). The third is the metric tensor g. Now we shall apply the Kosmann-Lie derivative (7.33) to these basic fields. For this purpose it is convenient to choose some canonically associated pair of frames Υ 0 , Υ 1 , Υ 2 , Υ 3 and Ψ 1 , Ψ 2 . In such a frame pair the components of all basic fields are constants. Indeed, they are given by the formulas (5.25), (7.4), and (7.5). Therefore we have
L X (d) ij = 2 s=1 W s i d sj + 2 s=1 W s j d is = W ij − W ji = 0. (8.4)
Similarly, applying (7.33) to G and taking into account (8.2) and (7.22), we get
L X (G) aā m = − 2 i=1 W a i G iā m − 2 ī =1 G aī m Wā i + 3 k=0 V k m G aā k = 0. (8.5)
And finally we apply the formula (7.33) to the metric tensor g. As a result, taking into account ( 9. Some concluding remarks.
Note that the quantities V i j for (7.33) are taken from (7.18). However, they could be taken from (5.1) either. In the latter case the equality L X (d) = 0 would be preserved, but the equality L X (g) = 0 would be replaced by L X (g) = L X (g).
As for the formula (8.5), it would be replaced by the following one:
L X (G) aā m = 3 k=0 ∇ m X k + ∇ k X m 2 G aā k .
This choice of V i j is preferred in [11]. As for our choice of V i j in this paper, in [11] it is referred to as the "metric Lie derivative" introduced by Bourguignon and
ϕ ε * : T M → T M, ϕ * −ε : T * M → T * M (2.2) in tangent and cotangent bundles respectively. These induced diffeomorphisms (2.2) act as linear mappings in fibers of T M and T * M . For this reason they can be extended to local diffeomorphisms of tensor bundles: ϕ ε : T r s M → T r s M. (2.3) 2000 Mathematics Subject Classification. 53B30, 81T20, 22E70. Typeset by A M S-T E X Here in (2.3) through T r s M we denote the following tensor product of r copies of the tangent bundle T M and s copies of the cotangent bundle T * M : ⊗ . . . ⊗ T M ⊗ T * M ⊗ . . . ⊗ T * M
y 1 =
1u 1 (−ε, x 0 , x 1 , x 2 , x 3 ), y 2 = u 2 (−ε, x 0 , x 1 , x 2 , x 3 ), y 3 = u 3 (−ε, x 0 , x 1 , x 2 , x 3 ).
. 1 .
1The tangent vector field (4.3) on V M is called concordant with the bundle structure if the functions (4.6) and (4.7) are linear with respect to their arguments v 1 , . . . , v n and w 1 , . . . , w n .
. 1 .
1The tangent vector field (4.3) on a vector bundle V M is concordant with the bundle structure if and only if its vertical components are linear functions with respect to v 1 , . . . , v n given by the formula (4.11)
Definition 5.1. A lifting X of a vector field X from M to T M is called a Kosmann lifting if the linear mappings (4.8) associated with this lifting are isometries. Kossman liftings were first introduced by Yvette Kosmann in [4-7]. Theorem 5.1. The natural lifting (5.2) of a vector field X is a Kosmann lifting if and only if X is a Killing vector field.
. 2 .
2The tangent vector field (5.15) on the tangent bundle T M is called the standard Kosmann lifting of the vector field (4.4) from M to T M .
for SM is similar to that of the metric tensor g for T M . The spintensorial type in the table (3.1) specifies the number of indices in coordinate representation of fields. The first two numbers are the numbers of upper and lower spinor indices, the second two numbers are the numbers of upper and lower conjugate spinor indices, and the last two numbers are the numbers of upper and lower tensorial indices (they are also called spacial indices).
(2.1), but does not coincide with (2.2). The Kosmann lifting X K of the vector field X is concordant with the bundle structure of the tangent bundle T M in the sense of the definition 4.1. For this reason the local diffeomorphisms (7.9) break into the series of linear mappings ϕ ε : T p (M ) → T q (M ),(7.10)
7.27), we multiply (7.26) bydāū and sum up over the indicesū andā.
L
Υi (g rj ) + L Υj (g ir ) − L Υr (g ij ) kr g si .
to d and taking into account (8.1),(7.19), and (7.20), we obtain
ir = V ij + V ji = 0. (8.6) The formulas (8.4), (8.5), and (8.6) are summarized in the following theorem.Theorem 8.1. For any vector field X in M the basic tensorial and spin-tensorial fields g, d, and G associated with the bundle of Weyl spinors SM are constant with respect to the Kosmann-Lie derivative L X .
... ir j1... js the components of L X (Y), then, applying (2.12) and (2.12) to (2.8) and taking into account (2.9), we obtain L X (Y) i1... ir j1... js = ∂X im ∂x km Y i1... km... ir j1... ... ... js +s
m=1
3
km=0
∂X km
∂x jm Y i1... ... ... ir
j1... km... js −
−
r
m=1
3
km=0
3
k=0
X k ∂Y i1... ir
j1... js
∂x k .
... km... ir j1... ... ... js .
... aεā1...āσc1... ce b1... bηb1...b ζ d1... d f = m=0 X m L Υm (Y a1... aεā1...āσc1... ce b1... bηb1...b ζ d1... d f a1... vµ ... aεā1...āσc1... ce b1... ... ... bηb1...b ζ d1... d f wµ bµ Y a1... ... ... aεā1...āσc1... ce b1... wµ ... bηb1...b ζ d1... d f a1... aεā1... vµ ...āσc1... ce b1... bηb1... ... ...b ζ d1... d f wμ bµ Y a1... aεā1... ... ...āσ c1... ce b1... bηb1... wµ ...b ζ d1... d f a1... aεā1...āσc1... vµ ... ce b1... bηb1...b ζ d1... ... ... d f wµ bµ Y a1... aεā1...āσc1... ... ... ce b1... bηb1...b ζ d1... wµ ... d f3
) −
−
ε
µ=1
2
vµ=1
W aµ
vµ Y
+
+
η
µ=1
2
wµ=1
W
−
−
σ
µ=1
2
vµ=1
Wā µ
vµ Y
+
+
ζ
µ=1
2
wµ=1
W
−
−
e
µ=1
3
vµ=0
V cµ
vµ Y
+
+
f
µ=1
3
wµ=0
V
.
(7.33)
Gauduchon in[16]. Since there are various approaches, I should regretfully conclude that there is no canonical definition of the Lie derivative for spinors thus far.
N Ibragimov, Kh, Transformation groups in mathematical physics., Nauka publishers, Moscow. Ibragimov N. Kh., Transformation groups in mathematical physics., Nauka publishers, Mos- cow, 1983.
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L Fatibene, M Ferraris, M Francaviglia, M Godina, A geometric definition of Lie derivative for Spinor Fields, e-print gr-qc/9608003 in Electronic Archive. Fatibene L., Ferraris M., Francaviglia M., Godina M., A geometric definition of Lie derivative for Spinor Fields, e-print gr-qc/9608003 in Electronic Archive http://arXiv.org.
M Godina, P Matteucci, The Lie derivative of spinor fields: theory and applications, e-print math.DG/0504366 in Electronic Archive. Godina M., Matteucci P., The Lie derivative of spinor fields: theory and applications, e-print math.DG/0504366 in Electronic Archive http://arXiv.org.
Spinor functions of spinors and the concept of extended spinor fields, e-print math. R A Sharipov, DG/0511350 in Electronic Archive. Sharipov R. A., Spinor functions of spinors and the concept of extended spinor fields, e-print math.DG/0511350 in Electronic Archive http://arXiv.org.
A note on Dirac spinors in a non-flat space-time of general relativity, e-print math. R A Sharipov, DG/0601262 in Electronic Archive. Sharipov R. A., A note on Dirac spinors in a non-flat space-time of general relativity, e-print math.DG/0601262 in Electronic Archive http://arXiv.org.
A note on metric connections for chiral and Dirac spinors, e-print math. R A Sharipov, DG /0602359 in Electronic Archive. Sharipov R. A., A note on metric connections for chiral and Dirac spinors, e-print math.DG /0602359 in Electronic Archive http://arXiv.org.
R A Sharipov, arXiv:0801.0008A cubic identity for the Infeld-van der Waerden field and its application. e-printElectronic ArchiveSharipov R. A., A cubic identity for the Infeld-van der Waerden field and its application, e-print arXiv:0801.0008 in Electronic Archive http://arXiv.org.
operateurs de Dirac et variations de metriques. J P Bourguignon, P Gauduchon, Spineurs, Comm. Math. Phys. 1443Bourguignon J. P., Gauduchon P., Spineurs, operateurs de Dirac et variations de metriques, Comm. Math. Phys. 144 (1992), no. 3, 581-599.
Russia Cell Phone: +7(917)476 93 48 E-mail address: r-sharipov@mail. Rabochaya street, 450003 Ufa. Rabochaya street, 450003 Ufa, Russia Cell Phone: +7(917)476 93 48 E-mail address: [email protected] R [email protected] URL: http://www.geocities.com/r-sharipov http://www.freetextbooks.boom.ru/index.html
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[
"RESTRICTION OF THE OSCILLATOR REPRESENTATION TO DUAL PAIRS : SOME PROJECTIVE CASES",
"RESTRICTION OF THE OSCILLATOR REPRESENTATION TO DUAL PAIRS : SOME PROJECTIVE CASES"
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[
"Sabine Jessica Lang "
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We study here the restriction of the oscillator representation of the symplectic group Sp(2p(m + n), R) to two different subgroups, namely O(m, n; R) and Sp(2p, R). We use the duality correspondence introduced by Howe to analyze these restrictions, and determine sufficient conditions on m, n and p so that the modules obtained are projective. The duality correspondence gives a description of the restriction in terms of lowest and highest modules, and we conclude by using gradings and filtrations to identify the modules.
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10.2140/pjm.2020.308.393
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"https://arxiv.org/pdf/1711.10562v3.pdf"
] | 119,155,688 |
1711.10562
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04e7520c5e08809efc81f87a2145a641b8f9ee25
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RESTRICTION OF THE OSCILLATOR REPRESENTATION TO DUAL PAIRS : SOME PROJECTIVE CASES
23 Aug 2018
Sabine Jessica Lang
RESTRICTION OF THE OSCILLATOR REPRESENTATION TO DUAL PAIRS : SOME PROJECTIVE CASES
23 Aug 2018
We study here the restriction of the oscillator representation of the symplectic group Sp(2p(m + n), R) to two different subgroups, namely O(m, n; R) and Sp(2p, R). We use the duality correspondence introduced by Howe to analyze these restrictions, and determine sufficient conditions on m, n and p so that the modules obtained are projective. The duality correspondence gives a description of the restriction in terms of lowest and highest modules, and we conclude by using gradings and filtrations to identify the modules.
Introduction
A very classical problem in representation theory is the understanding of the restriction of a representation Π of a group G to one of its subgroups H. In that setting, it is often useful to analyze Hom H (Π, π), where π is a representation of H. For this purpose, one may use the derived functors Ext n H (Π, π) to understand Hom H (Π, π) itself. Calculating Ext n H (Π, π) is not necessarily easier than Hom H (Π, π), but their Euler characteristic might be. This difficult part can become much simpler when we have a projective representation of G. In this case, Ext n H (Π, π) vanishes for every n > 0. This is one of the basic motivations here: the projectivity of a representation is an extremely powerful property. The link between Euler characteristic and projectivity is emphasized in [2], for example.
We focus on dual pairs, an approach introduced in the framework of the duality correspondence for the oscillator representation. A dual pair is a pair (G, G ′ ) of subgroups of a symplectic group Sp(V ), such that G is the centralizer of G ′ in Sp(V ). Two dual pairs (G, G ′ ) and (H, H ′ ) of Sp(V ) together are called a seesaw dual pair if (G ⊃ H, H ′ ⊃ G ′ ).
More precisely, we consider the oscillator representation ω of Sp(2p(m + n), R) with the seesaw dual pair (U(m, n), U(p)), (O(m, n; R), Sp(2p, R) . We restrict ω to O(m, n; R), and to Sp(2p, R) respectively, and analyze the cases when these restrictions are projective. Because U(p) is a compact group, the restriction of ω to U(m, n) is discrete. It is therefore enough to analyze each U(m, n)-summand. We show in theorems 1 and 2 that imposing a relation between the variables m, n and p that determine the size of these groups is sufficient to force the projectivity of 1 these restrictions. The restriction to O(m, n; R) becomes projective under a slighlty stronger condition than being in the stable range:
Theorem. The restriction of the oscillator representation of Sp(2p(m + n), R) to O(m, n; R) is projective if p > m + n.
Using (O(m, n; R), Sp(2p, R)), we obtain a result related to the semistable range, namely:
Theorem. The restriction of the oscillator representation of Sp(2p(m + n), R) to Sp(2p, R) is projective if min(n, m) > 2p.
It might seem unusual to focus on only one representation of one chosen group. Due to the importance of the oscillator representation in many different topics, this is however not surprising. This representation also appears in some books and papers as (Segal-Shale)-Weil representation, harmonic or metaplectic representation, among many other names. As mentioned in [10], for example, it is a fundamental object for the study of the minimal representations of classical groups, not only the symplectic group. Many different models of the oscillator representation can be found in the literature. Lecture 2 of [10], and Adams' notes from [1] present several realizations, and provide explanations of which model is the most appropriate depending on the context. Seesaw dual pairs appear in the work of Kudla for the first time, in [9], and have been extensively used since that. Howe gives many results about dual pairs and their use together with the oscillator representation, for example in [4] and [5]. We focus here on compact dual pairs, i.e., dual pairs with one compact group, since it allows us to decompose representations under the action of the compact member.
The theory of duality correspondence, first introduced by Howe and also called theta correspondence, describes explicitly the subrepresentations that appear in the decomposition of the oscillator representation after restriction to a dual pair. For our case of interest, namely irreducible dual pairs of real reductive groups, the duality correspondence can be found in Adams' notes [1]. The description of the restriction is made in terms of highest (and lowest) weight modules, whose theory is used in the technical part of our result.
I would like to thank my advisor Gordan Savin, without whom none of this work would be possible.
Generalities
In this section, we introduce the mathematical objects that we use, and we recall some well-known results. The main goal is to introduce most of the notations, and to make a list of the different tools that are used here.
The basic set-up will be the following: let G be a Lie group with complexified Lie algebra g, and let K ⊂ G be a maximal compact Lie subgroup. We denote by k the complexified Lie algebra of K, and we choose a Cartan subalgebra t of both g and k.
2.1. Graded algebras. Let A be a ring with filtration A n , and corresponding graded algebra Gr(A) = ⊕ n A n /A n−1 . Let M, N be two A-modules with filtrations denoted by M n and N n . Assume that A n M m ⊂ M m+n for all m, n, and similarly for N m . We write Gr(M) = ⊕ n M n /M n−1 and Gr(N) = ⊕ n N n /N n−1 for the corresponding graded Gr(A)-modules Proposition 1. Let T : M → N be a morphism of A-modules preserving the filtrations M n and N n and such that the corresponding graded morphism of Gr(A)-modules
T G : Gr(M) → Gr(N) is surjective. If dim(M n ) = dim(N n ) for all n, then T is an isomorphism of A-modules.
This basic result is extremely useful to conclude the proof of our results. Indeed, the graded pieces of our modules are easier to describe than the whole modules themselves, so we use filtrations to obtain isomorphisms.
Froebenius reciprocity.
Let A, B be two rings with A ⊂ B. We recall the definition of a projective module, which is one of the central notions:
Definition. We say that a B-module P is projective if for any B-modules M, N and homomorphisms f :
N ։ M, g : P → M with f onto, there exists a homomorphism h : P → N such that f • h = g.
When we work with tensor products of (g, K)-modules, projectivity can be directly deduced from a corollary of the Froebenius reciprocity, recalled here. Proposition 2 (Froebenius reciprocity). Let M be an A-module and N be a Bmodule. We have a vector space isomorphism
Hom B (B ⊗ A M, N) ∼ = Hom A (M, N).
Corollary 1. Let Q be an A-module, and let
P = B ⊗ A Q. If Q is a projective A-module, then P is a projective B-module.
2.3. Highest weight modules. We do not give details about the theory of highest weight modules, as it can be found in many textbooks, as [7] for example. We mainly introduce our notations for these objects here.
We fixed a Cartan subalgebra t of g, and we can therefore define the root system ∆ of g with respect to t. Fixing a Borel subalgebra b of g determines the positive and negative roots in ∆, we write ∆ + for the positive roots. We denote by q the parabolic subalgebra q = k + b obtained by summing the Lie algebra of K and the fixed Borel subalgebra.
For a weight λ of g, we write F λ for the irreducible k-module with highest weight λ, and E λ for the irreducible g-module with highest weight λ. We use N(λ) to denote the U(g)-module U(g) ⊗ U(q) F λ , where U(a) is the universal enveloping algebra of a for any Lie algebra a.
2.4. Irreducibility criterion. We state here an irreducibility criterion for U(g)module of the form N(λ) = U(g) ⊗ U(q) F λ . This allows us to make a crucial identification that leads to the main result. We write ∆ c for the compact roots, namely the roots of k with respect to t, and ∆ n = ∆\∆ c denotes the non-compact roots. We also define ∆ + c = ∆ c ∩ ∆ + , ∆ + n = ∆ n ∩ ∆ + . The product 2 < λ, α > < α, α > , for any α ∈ ∆ and λ ∈ t * is denoted by (λ) α . As usual, ρ is half of the sum of the positive roots and we write s α for the reflection through the hyperplane determine by the root α.
Proposition 3. Assume for any
α ∈ ∆ + n with (λ + ρ) α ∈ Z >0 , there is γ ∈ ∆ n with (λ + ρ) γ = 0 and s α (γ) ∈ ∆ c . Then N(λ) = U(g) ⊗ U(q) F λ is irreducible. Moreover, if g is of type A n ,
it is a necessary and sufficient condition.
This appears as Corollary 6.3 and Theorem 6.4 in [3].
2.5. (g, K)-modules. The modules that we use in this paper are (g, K)-modules. We give here the definition and state a basic but fundamental result. More details can be found in [8].
Definition.
A (g, K)-module is a complex vector space V with an action of g and an action of K such that
(1) for all v ∈ V, k ∈ K, X ∈ g, we have k · (X · v) = (Ad(k)X) · (k · v), (2) V is K-finite, i.e., for every v ∈ V , K · v is a finite dimensional vector space, (3) for all v ∈ V, Y ∈ k, we have ( d dt exp(tY ) · v) | t=0 = Y · v.
In this definition, (1) is a compatibility condition between the action of K on V , the action of g on V and the action of K on g. Part (3) forces the compatibility between the action of k on V as a Lie subalgebra of g and the action of k on V as the complexified Lie algebra of K.
As a consequence of Froebenius reciprocity, we have the following result:
Proposition 4. Let V be a (g, K)-module. Then U(g) ⊗ U(k) V is a projective U(g)- module.
Proof. By K-finiteness, every (g, K)-module is U(k)-projective. Now the result is a direct application of corollary 1.
2.6. Oscillator representation. We are interested in a particular representation of the symplectic group Sp(2p, R) on L 2 (R p ), called the oscillator representation. We follow here the construction presented in [6], where more details can be found. Note that we never use explicitly this construction, but only the duality correspondence, which specifically applies to this representation. The oscillator representation for Sp(2p, R) will be constructed from a representation of the Lie algebra sp(2p, R) on the space of Schwarz functions on R p .
Definition. The space of Schwarz functions on
R n , denoted S(R n ), is the set of functions f ∈ C ∞ (R n ) such that sup (x 1 ,...,xn)∈R n |x α 1 1 . . . x αn n ∂ β 1 +···+βn f (x 1 , . . . , x n ) ∂x β 1 1 . . . ∂x βn n | < ∞
for all (α 1 , . . . , α n ), (β 1 , . . . , β n ) ∈ Z n >0 . When p = 1, the group Sp(2p, R) is isomorphic to SL(2, R), and it is easy to describe the explicit construction of the oscillator representation. If we pick a standard basis (h, e, f ) of sl(2, R), we can define a representation ω on S(R) by:
ω(h) = x d dx + 1 2 , ω(e) = i 2 x 2 , ω(f ) = i 2 d 2 dx 2 .
Exponentiating this sl(2, R)-module, we obtain a unitary representation of the double cover of SL(2, R) on the space L 2 (R).
A similar construction can be done for any p, but we do not explain the details here. We therefore obtain structure of sp(2p, R)-module on S(R p ), which is a derived representation of the double cover of Sp(2p, R) on L 2 (R p ). Several constructions and different models for p > 1 can be found in Lecture 2 in [10] or in [1]. We denote this representation by ω and call it the oscillator representation of the symplectic group Sp(2p, R). Note that ω is a (g, k)-module for G = Sp(2p, R) and K its maximal compact subgroup. 2.7. Reductive dual pairs. When we restrict the oscillator representation to a subgroup of the symplectic group, it is useful to use another group to decompose this restriction. This can be done using pairs of groups called dual pairs.
Definition. A pair (G, G ′ ) of subgroups in a symplectic group Sp(2q, R) is a reductive dual pair if (1) G and G ′ act reductively on R 2q , (2) G and G ′ are centralizers of each other inside Sp(2q, R). Moreover, if G is compact, we say that (G, G ′ ) is a compact dual pair.
It is also useful to consider two dual pairs with a particular relation, as introduced by Kudla in [9]:
Definition. Two dual pairs (G, G ′ ) and (H, H ′ ) form a seesaw dual pair if we have the inclusions H ⊂ G and G ′ ⊂ H ′ . We denote it by (G, G ′ ), (H, H ′ ) .
2.8. Duality correspondence. This section introduces briefly the idea of duality correspondence, which can be used to calculate the restriction of the oscillator representation to some subgroups.
The duality correspondence is a decomposition of the oscillator representation ω of a symplectic group Sp(q, R), under the action of a subgroup. We assume that we have a compact dual pair (G, G ′ ), so that we can decompose ω under the action of G.
Recall that if G is a compact group with finite dimensional representation π, we can decompose π as
π = σ (Hom G (σ, π) ⊗ σ),
where the sum is taken over all the irreducible representations σ of G. Indeed, if T ∈ Hom G (σ, π) and v ∈ σ, then T (v) ∈ π and we have a map
Hom G (σ, π) × σ → π, (T, v) → T (v).
This map can be extended to a map Hom G (σ, π) ⊗ σ → π, and it is injective when σ is irreducible. Since G is compact, π is completely reducible, hence π = σ (Hom G (σ, π) ⊗ σ). Note that here, G does not act on Hom G (σ, π); this only denotes the multiplicity of σ in π.
We apply the same method here, in the sense that we use the action of a compact group G on ω, and obtain a decomposition
ω = σ (Hom G (σ, ω) ⊗ σ),
where the sum is taken over all the irreducible representations σ of G. We denote Hom G (σ, ω) by θ(σ) in this case. The Lie algebra g ′ of G ′ acts naturally on θ(σ) = Hom G (σ, ω) as follows: for X ∈ g ′ , T ∈ Hom G (σ, ω) and v ∈ σ, we have (X · T )(v) = X · (T (v)), where X · (T (v)) comes from the action of g ′ on ω.
The duality correspondence gives an explicit description of θ(σ) in some specific cases. In general, we know that θ(σ) is a highest weight module, and we denote its highest weight by τ . We use, as before, E τ to denote the irreducible g ′ -module with highest weight τ . Note that τ is also a dominant weight for k ′ , the Lie algebra of a maximal compact subgroup K ′ of G ′ , so τ is also the highest weight of a finite dimensional representation of k ′ : we write F τ for the irreducible k ′ -module with highest weight τ . In most cases, the duality correspondence does not give us the highest weight τ directly from σ, but it produces a lowest weight τ ′ related to τ .
We give here the correspondence for the two cases that we use. This correspondence can be found with more details in [1].
Since gl m+n (C) is the complexified Lie algebra of U(m, n), the duality correspondance for (U(p), U(m, n)) can be expressed in term of U(p)-modules and gl m+n (C)modules:
Proposition 5. The duality correspondence for the pair (U(p), U(m, n)) is given by
σ = (a 1 + m − n 2 , . . . , a k + m − n 2 , m − n 2 , . . . , m − n 2 , b 1 + m − n 2 , . . . , b l + m − n 2 ) → τ ′ = (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . , p 2 ) ⊕ (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 ),
where σ defines an irreducible highest weight U(p)-module and τ ′ defines an irreducible lowest weight gl m+n (C)-module. All such weights occur, with the constraints
k + l ≤ p, k ≤ m, l ≤ n.
For O(n, R), it is not obvious how we can describe a representation using highest weights, since it is disconnected. However, we can use the embedding O(n, R) = U(n) ∩ GL(n, R) of O(n, R) into U(n) and the highest weights of U(n). Given a highest weight λ of U(n) and some parameter ǫ = ±1, we say that the representation of O(n, R) with highest weight (λ, ǫ) is the irreducible summand of the representation of U(n) with highest weight λ that contains the highest weight vector, tensored with the sgn representation of O(n, R) if ǫ = −1.
Proposition 6. The duality correspondence for the pair
(O(n, R), Sp(2p, R)) is given by σ = (a 1 , . . . , a k , 0, . . . , 0; ǫ) → τ ′ = (a 1 + n 2 , . . . , a k + n 2 , 1−ǫ 2 (n−2k) n 2 + 1, . . . , n 2 + 1, n 2 , . . . , n 2 ),
where σ defines an irreducible highest weight O(n, R)-module in the sense explained previously and τ ′ defines an irreducible lowest weight sp(2p, C)-module. All such weights occur, with the constraints k ≤ [ n 2 ], and k + 1 − ǫ 2 (n − 2k) ≤ p.
Restrictions
Set-up.
We consider the seesaw dual pair
(U(m, n), U(p)), (O(m, n; R), Sp(2p, R)
inside Sp(2p(m + n), R), with oscillator representation ω. We want to understand the restriction of ω to O(m, n; R) and to Sp(2p, R), and analyze the cases when these restrictions are projective. This will be done by first restricting to U(m, n) using the dual pair (U(p), U(m, n)), and then restricting further to O(m, n; R). In the second case, we will use the action of O(m, R) × O(n, R) to decompose ω and then focus on a pair of the form (O(n, R), Sp(2p, R)).
We recall that our notations are as follows :
• (G, G ′ ) a dual pair with G compact, • G, G ′ real Lie groups, with complexified Lie algebras g, g ′ , • K ′ a maximal compact subgroup of G ′ , with complexified Lie algebra k ′ , • t ′ a Cartan subalgebra of both g ′ and k ′ , • b ′ a Borel subalgebra of g ′ , • q ′ = k ′ + b ′ a parabolic subalgebra of g ′ .
In order to understand the restriction of ω to G ′ , we let G act and we obtain a decomposition of the form ω = ⊕(σ ⊗ θ(σ)) = ⊕(σ ⊗ E τ ), where σ is an irreducible representation of G with highest weight σ, E τ is an irreducible representation of G ′ with highest weight τ and the sum is taken over all the possible σ.
We know that E τ is irreducible and it is a quotient of
N(τ ) = U(g ′ ) ⊗ U(q ′ ) F τ . So if N(τ ) is also irreducible, this forces N(τ ) = E τ .
In this case, we have an explicit description of the restriction of ω, which makes it easier to analyze. The goal is now to determine which N(τ ) are irreducible, and which σ they correspond to. We will do this in two different settings, depending which restriction of ω we want to understand.
Restriction to O(m, n; R).
Our very first step uses the dual pair (G, G ′ ) = (U(p), U(m, n)) to restrict ω to U(m, n). Once we understand the restriction to U(m, n), we will be able to restrict further to O(m, n; R).
We first decompose ω under the action of the compact group U(p). We therefore obtain a decomposition of the form ω = σ (σ ⊗ θ(σ)) = σ (σ ⊗ E τ ), where σ is an irreducible representation of U(p) and E τ is a representation of U(m, n). Explicitly, the correspondence is given between σ and the lowest weight τ ′ of E τ as follows :
σ = (a 1 + m − n 2 , . . . , a k + m − n 2 , m − n 2 , . . . , m − n 2 , b 1 + m − n 2 , . . . , b l + m − n 2 ) → τ ′ = (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . , p 2 ) ⊕ (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 ), with the constraints k + l ≤ p, k ≤ m, l ≤ n.
The issue is that we have a correspondence between the highest weight σ for U(p) and the lowest weight τ ′ for U(m, n), but our irreducibility criterion works for a highest weight. We can solve this problem using a conjugation by the longest element of the Weyl group of G ′ , denoted by w 0 . This conjugation will send U(m, n) to U(n, m), but it will also switch positive and negative roots, so we will be able to work with a highest weight. To avoid unnecessary confusion of notation, we will still denote our group by G ′ after conjugation by w 0 . Now, instead of working with the lowest weight
τ ′ = (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . , p 2 ) ⊕ (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 )
on U(m, n), we can equivalently work with the highest weight
τ = (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 ) ⊕ (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . ,p 2 )
on U(n, m). Since we start with a highest weight σ for U(p) expressed as
σ = (a 1 + m − n 2 , . . . , a k + m − n 2 , m − n 2 , . . . , m − n 2 , b 1 + m − n 2 , . . . , b l + m − n 2 ),
we have the conditions a 1 ≥ · · · ≥ a k ≥ 0 and 0 ≥ b 1 ≥ · · · ≥ b l .
We can now apply the irreducibility criterion given by proposition 3 to determine in which cases N(τ ) is irreducible. We are working with G ′ = U(n, m) and K ′ = U(n) × U(m). The complexified Lie algebra of G ′ is g m+n (C), so we have a root system of type A n , which implies that this criterion is both necessary and sufficient for the irreducibility of N(τ ).
We apply the criterion to
τ = (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 ) ⊕ (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . ,p 2 )
.
Therefore, we need to carefully analyze the root systems occurring here :
• The roots for G ′ have the form e i − e j with 1 ≤ i, j ≤ n + m and i = j. If i ≤ j, we have a positive root. • The non-compact positive roots for G ′ are the roots not coming from K ′ , i.e., roots of the form e i − e j with 1 ≤ i ≤ n and n + 1 ≤ j ≤ n + m. We will write α ij = e i − e j for the corresponding non-compact root. • We can calculate ρ as half the sum of the positive roots, and we obtain :
ρ = ( m + n − 1 2 , . . . , i-th coordinate m + n − 2i + 1 2 , . . . , −m − n + 1 2 ). We look at τ = (− p 2 , . . . , − p 2 , b 1 − p 2 , . . . , b l − p 2 ) ⊕ (a 1 + p 2 , . . . , a k + p 2 , p 2 , . . . ,p 2 )
, with 0 > b 1 ≥ · · · ≥ b l and a 1 ≥ · · · ≥ a k > 0. For simplicity of notations, we will assume that a i and b j can be equal to zero, and we will rewrite τ as
τ = (b 1 − p 2 , . . . , b n − p 2 ) ⊕ (a n+1 + p 2 , . . . , a n+m + p 2 )
with b n ≤ · · · ≤ b 1 ≤ 0 and 0 ≤ a n+m ≤ · · · ≤ a n+1 .
Here we obtain (τ +ρ) α ij = b i −a j +j −i−p with 1 ≤ i ≤ n and n+1 ≤ j ≤ n+m. Since we know that b i − a j ≤ 0 for all i, j, we conclude that if p ≥ m + n − 1, then (τ + ρ) α ij is non-positive for all i, j, so there is nothing to check using this criterion. So for p ≥ m + n − 1, we have the irreducibility of N(τ ).
However, we cannot improve this condition on p without being specific about the values of a i and b j , as we can see by working out some small examples. So we will only use the case where p ≥ m + n − 1 for further work.
Indeed, if we look at the case m = n = p = 2, where p ≥ m + n − 1, we can play with values of a i and b j to get an irreducible N(τ ) or a reducible N(τ ):
(1) b 2 = −2 ≤ b 1 = −1 ≤ 0 ≤ a 4 = 1 ≤ a 3 = 2 gives τ = (−2, −3) ⊕ (3, 2), and (τ + ρ) α ij < 0 for all i, j, meaning that N(τ ) is irreducible. (2) b 2 = −1 ≤ b 1 = 0 ≤ 0 ≤ a 4 = 0 ≤ a 3 = 1 gives τ = (−1, −2) ⊕ (2, 1), and (τ + ρ) α ij < 0 except for i = 1, j = 4.
However, there is no non-compact root α ij such that (τ + ρ) α ij = 0. By the irreducibility criterion, and since we have a root system of type A n , N(τ ) is reducible.
The first example shows that p ≥ m + n − 1 is a sufficient but not necessary condition for the irreducibility of N(τ ) and therefore for the projectivity of the restriction of the oscillator representation to O(m, n; R). However, the second example does not mean that it is not possible to improve this criterion: in this case, N(τ ) is reducible and our method cannot be used. But this does not mean that the restriction will not necessarily be projective.
Conclusion for O(m, n; R).
We just saw that if p ≥ m + n − 1, then N(τ ) is irreducible for any τ . In particular, we have
E τ = N(τ ) = U(gl m+n (C)) ⊗ U(q ′ ) F τ ,
since the complexified Lie algebra of U(m, n) is equal to gl m+n (C). However, this is a restriction to U(m, n), and we would like to restrict further to O(m, n; R). Now we obtain
E τ | o(m,n,C) = U(gl m+n (C)) ⊗ U(q ′ ) F τ | o(m,n,C) = U(o(m, n, C)) ⊗ U(om(C)×on(C)) (F τ | om(C)×on(C) ).
This identification of the restriction is not obvious, but will be proved in the next section, namely in theorem 3. Now, Proof. Under the action of U(p), we had the decomposition ω = σ (σ ⊗E τ ). We saw previously that if p ≥ m + n − 1, then E τ = N(τ ) is a projective o(m, n, C)-module.
Since o(m, n, C) does not act on σ, which is a finite dimensional space, we obtain that σ ⊗ E τ is projective as an o(m, n, C)-module. We decomposed ω as a direct sum of such spaces, and the direct sum of projective modules is also projective, so we conclude that the restriction of ω is a projective o(m, n, C)-module. Sp(2p, R). We want to apply a similar method to understand the restriction of the oscillator representation to Sp(2p, R). To do so, we will first
Restriction to
ω = ω m ⊗ ω * n = σ (σ ⊗ E τ ) ⊗ σ ( σ ⊗ E τ ) = ((σ ⊗ σ) ⊗ (E τ ⊗ E τ )) .
We can now look closer at one of the dual pairs, say (O(n, R), Sp(2p, R)). Here, the correspondence between σ and the lowest weight τ ′ of E τ is given by: (a 1 , . . . , a k , 0, . . . , 0; So we use G ′ = Sp(2p, R) and K ′ = U(p). Here we do not have a root system of type A n , so the criterion is sufficient (but not necessary) for the irreducibility of N(τ ). We have a correspondence between the highest weight σ for O(n, R) and the lowest weight τ ′ for Sp(2p, R), but, as before, we need to write the corresponding highest weight τ . We can again conjugate by the longest element of the Weyl group of G, which will switch positive and negative roots but not change G.
σ =
So instead of working with the lowest weight
τ ′ = (a 1 + n 2 , . . . , a k + n 2 , 1−ǫ 2 (n−2k) n 2 + 1, . . . , n 2 + 1, n 2 , . . . , n 2 ),
we can work with the highest weight
τ = (− n 2 , . . . , − n 2 , 1−ǫ 2 (n−2k) − n 2 − 1, . . . , − n 2 − 1, −a k − n 2 , . . . , −a 1 − n 2
).
Since we start with a highest weight σ for O(n, R), we have a 1 ≥ · · · ≥ a k ≥ 0 and ǫ = ±1.
We will now apply proposition 3 to τ on G ′ = Sp(2p, R). The root system occurring here has the following properties :
• The roots for G ′ have the form • e i − e j with 1 ≤ i, j ≤ p and i = j. If i ≤ j, it is a positive root.
• ±(e i + e j ) with 1 ≤ i, j ≤ p and i = j, and e i + e j is a positive root.
• ±2e i with 1 ≤ i ≤ p, and 2e i is a positive root. • The non-compact positive roots for G ′ are the roots not coming from K ′ , i.e., roots of the form e i + e j with 1 ≤ i, j ≤ p, i = j and roots of the form 2e i with 1 ≤ i ≤ p. • We can calculate ρ as half the sum of the positive roots, and we obtain :
ρ = (p, . . . , i-th coordinate p + 1 − i , . . . , 1).
Depending on the value of the parameter ǫ, we have different values of τ . We will therefore look at the two cases separately. Therefore, the different products between τ + ρ and a non-compact positive root are as follows:
(τ + ρ) 2e i = p + 1 − i − n 2 if 1 ≤ i ≤ p − k p + 1 − i − n 2 − a p+1−i if p − k < i ≤ p , (τ + ρ) e i +e j = 2p + 2 − i − j − n if 1 ≤ i, j ≤ p − k 2p + 2 − i − j − n − a p+1−j if 1 ≤ i ≤ p − k < j ≤ p 2p + 2 − i − j − n − a p+1−i − a p+1−j if p − k < i, j ≤ p .
If we take n ≥ 2p, all these products are non-positive, and by proposition 3, there is nothing to check : N(τ ) is irreducible. But we can do slightly better: the condition n ≥ 2p − 1 is also sufficient. Indeed, for n = 2p − 1, we have p + 1 − i − n 2 = 3 2 − i, which is not an integer. All the other products are still non-positive, so the criterion can be apply without any further checking, and N(τ ) is irreducible. And as before, we cannot improve this condition without being specific about the values a k .
3.3.2.
Case ǫ = −1. If ǫ = −1, we have 1 − ǫ 2 (n−2k) = n−2k, so the highest weight τ is more complicated. It can be written as
τ = (− n 2 , . . . , − n 2 , (n−2k) − n 2 − 1, . . . , − n 2 − 1, −a k − n 2 , . . . , −a 1 − n 2 ).
The different products between τ + ρ and a non-compact positive root are:
(τ + ρ) 2e i = p + 1 − i − n 2 if 1 ≤ i ≤ p + k − n p − i − n 2 if p + k − n < i ≤ p − k p + 1 − i − n 2 − a p+1−i if p − k < i ≤ p , (τ + ρ) e i +e j = 2p + 2 − i − j − n if 1 ≤ i, j ≤ p + k − n 2p − i − j − n if p + k − n < i, j ≤ p − k 2p + 2 − i − j − n − a p+1−i − a p+1−j if p − k < i, j ≤ p 2p + 1 − i − j − n if 1 ≤ i ≤ p + k − n and p + k − n < j ≤ p − k 2p + 1 − i − j − n − a p+1−j if p + k − n < i ≤ p − k and p − k < j ≤ p 2p + 2 − i − j − n − a p+1−j if 1 ≤ i ≤ p + k − n and p − k < j ≤ p .
As before, if we take n ≥ 2p, all these products are non-positive, and by proposition 3, N(τ ) is irreducible. We can again refine this, since for n = 2p − 1, we have p + 1 − i − n 2 = 3 2 − i which is not an integer, and all the other products are non positive. So N(τ ) is irreducible for all n ≥ 2p − 1, and this is the best that we can do to stay in a general case. Sp(2p, R). As we saw by writing the oscillator representation as
Conclusion for
ω = ω m ⊗ ω * n = σ (σ ⊗ E τ ) ⊗ σ ( σ ⊗ E τ ) ,
we can analyze the situation using the pairs (O(m, R), Sp(p, R)) and (O(n, R), Sp(p, R)).
We now need to put our results together. For this purpose, we write q + = k ′ + b + and q − = k ′ + b − , where b + and b − are opposite choices of Borel subalgebras in sp(2p, C) and k ′ is the complexified Lie algebra of K ′ . The fact that ω m is a highest weight module means that E τ is a quotient of N + (τ ) = U(sp(2p, C)) ⊗ U(q + ) F τ but E τ is a quotient of N − ( τ ) = U(sp(2p, C)) ⊗ U(q − ) F τ since ω * n is a lowest weight module. However, our irreducibility criterion from proposition 3 can be applied to any choice of Borel subalgebra, so we can use our previous calculation for both cases.
If n, m ≥ 2p − 1, we have the identifications E τ = N + (τ ) = U(sp(2p, C)) ⊗ U(q + ) F τ and E τ = N − ( τ ) = U(sp(2p, C)) ⊗ U(q − ) F τ . Therefore, we can write ω as
ω = σ, σ ((σ ⊗ σ) ⊗ (E τ ⊗ E τ )) = σ, σ (σ ⊗ σ) ⊗ ( U(sp(2p, C)) ⊗ U(q + ) F τ ⊗ U(sp(2p, C)) ⊗ U(q − ) F τ ) .
Again, as in the first case, Sp(2p, R) does not act on the finite dimensional space σ ⊗ σ. In the next section, we will show that the tensor product of E τ and E τ is a projective sp(2p, C)-module. So we can write ω as a direct sum of such objects, and consequently we proved: Theorem 2. If min(n, m) > 2p, the restriction of the oscillator representation ω of Sp(2p(m + n), R) to Sp(2p, R) is projective.
Identifications and projectivity
This technical section is here to complete the proof of both theorems 1 and 2, through identifications of some tensor products. O(m, n; R). The goal here is to prove that we have an identification U(gl m+n (C)) ⊗ U(q ′ ) F τ | o(m,n,C) = U(o(m, n, C)) ⊗ U(om(C)×on(C)) (F τ | om(C)×on(C) ), with notations as in the previous section. We will analyze this restriction for a more general case, i.e., for a module of the form U(gl m+n (C)) ⊗ U(q ′ ) E.
Restriction to
Definitions and notations.
We keep the notations used in the previous section. So we have G ′ = U(m, n), with corresponding complex Lie algebra g ′ . We consider the Cartan decomposition g ′ = k ′ + p ′ . We can identify g ′ = gl m+n (C),
k ′ = { X 0 0 Y | X ∈ gl m (C), Y ∈ gl n (C)} ∼ = gl m (C) × gl n (C)
and
p ′ = p + + p − where p ′ = { 0 A B 0 | A ∈ M m,n (C), B ∈ M n,m (C)}, p + = { 0 A 0 0 | A ∈ M m,n (C)} and p − = { 0 0 B 0 | B ∈ M n,m (C)}.
We will also write q ′ = k ′ + p − . Note that using q ′ for this sum is not misleading, since we can choose the Borel subalgebra b ′ so that this new definition agrees with the first one as q ′ = k ′ + b ′ . We will also consider the subgroup of G ′ given by H = O(m, n, R), with complexified Lie algebra h = o(m, n, C) and Cartan decomposition h = k + p. Here we have
k = { X 0 0 Y | X ∈ o m (C), Y ∈ o n (C)} ∼ = o m (C) × o n (C) and p = { 0 A A T 0 | A ∈ M m,n (C)}.
Note that here we do not have a decomposition of p as a sum p + + p − .
Lemma 1. The map p ι −→ p ′ π −→ p + is a bijection, where ι : p → p ′ is the inclusion and π : p ′ → p + is the projection.
We consider a finite dimensional k ′ -module E. By letting p − act trivially on E, this will become a q ′ -module and we can form
V + E = U(g ′ ) ⊗ U(q ′ ) E, which is a U(g ′ )- module. Note that as vector spaces, we have the isomorphism V + E ∼ = S(p + ) ⊗ E, with S(p + ) the symmetric algebra on p + .
We can also use E to form a U(h)-module. By restriction, we can see E as a kmodule, denoted E | k , and form the tensor product V E = U(h) ⊗ U(k) (E | k ). Similarly, there is an isomorphism of vector spaces V E ∼ = S(p) ⊗ (E | k ).
Gradings and filtrations. We can define a filtration
V E = ⊕ n (V E ) n /(V E ) n−1 by setting (V E ) n = r≤n S(p)[r]U(k) ⊗ U(k) (E | k ).
We therefore have (V E ) 0 = 1 ⊗ (E | k ) and
(V E ) n /(V E ) n−1 ∼ = S(p)[n]U(k) ⊗ U(k) (E | k ).
By a similar construction on V + E , we have the filtration
V + E = ⊕ n (V + E ) n /(V + E ) n−1 , where (V + E ) n = r≤n S(p + )[r]U(k ′ ) ⊗ U(q ′ ) E And we obtain (V + E ) 0 = 1 ⊗ E, and (V + E ) n /(V + E ) n−1 ∼ = S(p + )[n]U(k ′ ) ⊗ U(q ′ ) E.
Identification of the restriction V + E | h . By Froebenius reciprocity, we have a map
T : V E → V + E , 1 ⊗ e → 1 ⊗ e for any e ∈ E. This map is extended to the whole V E by looking at the action of an element of S(p) on 1 ⊗ e. It is enough to look at the action of S(p)[n] and extend by linearity.
Recall that using lemma 1, if we start with an element x ∈ p, we can use ι to see x as an element of p b and then we can decompose x = y + z where y ∈ p + b and z ∈ p − b . We will write {x 1 , . . . , x r } for a basis of p. So an element of S(p)[n] can be written x 1 . . . x n , with possible repetitions in the indices. Recall that we have an inclusion of p in p ′ , and p is in bijection with p + (we could do the same with p − ) so we can choose a basis {y 1 , . . . , y r } of p + and a basis of {z 1 , . . . , z r } of p − such that
x i = y i + z i in p ′ .
Consequently, we extend the map T so that
T (x 1 . . . x n ⊗ e) = (y 1 + z 1 ) . . . (y n + z n ) ⊗ e
for any e ∈ E.
Using this definition of T , we want to show the following result :
Proposition 7. The map T : V E → V + E defined previously preserves the filtrations. This is just a consequence of the following lemma, whose proof consists only of technical calculations and is therefore omitted here: Lemma 2. The action of (y 1 + z 1 ) . . . (y n + z n ) on 1 ⊗ e is given by (y 1 + z 1 ) . . . (y n + z n ) ⊗ e = y 1 . . . y n ⊗ e + elements of (V + E ) n−1 . We can now prove the main result, namely :
Theorem 3. The map T : V E → V + E is
an isomorphism of U(h)-modules, and it is induced by an isomorphism of S(p)-modules on the graded spaces T
G : Gr(V E ) → Gr(V + E ) by the identification p ∼ = p + . Proof. By proposition 7, we know that T ((V E ) n ) ⊂ (V + E ) n .
We will now show that the restriction T | (V E )n is surjective onto (V + E ) n . Indeed, a basis of (V E ) n is given by elements of the form x 1 . . . x r ⊗ e with r ≤ n, x i ∈ p and e ∈ E, and a basis of (V E ) + is given by elements of the form y 1 . . . y s ⊗ e with s ≤ n, y i ∈ p + and e ∈ E. So the description by lemma 2 of the image x 1 . . . x n ⊗ e as T (x 1 . . . x n ⊗ e) = y 1 . . . y n ⊗ e + elements of (V + E ) n−1 is enough to show the surjectivity of T : (V E ) n → (V + E ) n , by induction on n and using the linearity of T .
We also need to compare the dimensions of (V E ) n and (V + E ) n , as complex vector spaces. We have the vector space identifications
(V E ) n = r≤n S(p)[r]U(k) ⊗ U(k) E | k ∼ = r≤n S(p)[r] ⊗ E and (V + E ) n = r≤n S(p + )[r]U(k ′ ) ⊗ U(q ′ ) E ∼ = r≤n S(p + )[r] ⊗ E.
Using these descriptions and the bijection between p and p + , we deduce that the dimensions of (V E ) n and (V + E ) n have to be equal. By proposition 1, we conclude that T is an isomorphism. This shows that
U(g ′ ) ⊗ U(q ′ ) E | h ∼ = U(h) ⊗ U(k) (E | k ),
and concludes the proof of theorem 1.
4.2.
Tensor product on Sp(2p, R). Our goal is now to show that the tensor product
U(sp(2p, C)) ⊗ U(q + ) F τ ⊗ U(sp(2p, C)) ⊗ U(q − ) F τ ′
is a projective U(sp(2p, C)-module. We will show this in a more general case, with
U(sp(2p, C)) ⊗ U(q + ) E ⊗ U(sp(2p, C)) ⊗ U(q − ) F ,
for some modules E and F .
Definitions and notations.
We fix now, as in the previous section, g ′ = sp 2p (C), and we have the corresponding Cartan decomposition of g ′ as g ′ = k ′ + p ′ . By choice of g ′ = sp 2p (C), we therefore have
k ′ = { X 0 0 −X T | X ∈ gl p (C)} ∼ = gl p (C) and p ′ = { 0 Y Z 0 | Y = Y T , Z = Z T },
that we can decompose further as
p + = { 0 Y 0 0 | Y = Y T } and p − = { 0 0 Z 0 | Z = Z T }.
We note that p + and p − are both commutative Lie algebras, but they do not commute with each other. Indeed, we have 0 = [p + , p − ] ⊂ k. We will let {α i } denote an ordered basis of p − and {β j } denote an ordered basis of p + .
We fix a Cartan subalgebra t ′ of g ′ that is also a Cartan subalgebra for k ′ . We define two more subalgebras of g ′ as follows : q + = k ′ + p + and q − = k ′ + p − We will now consider two finite dimensional k ′ -modules E and F . We can let p + act on E by zero, so that E becomes a q + -module. Similarly, we let p − act on F by zero and obtain a q − -module. We can therefore define
V E = U(g ′ ) ⊗ U(b + ) E and V F = U(g ′ ) ⊗ U(b − ) F,
that are both (g ′ , K ′ )-modules by construction. We also define
V = U(g ′ ) ⊗ U(k ′ ) (E ⊗ F ).
Gradings and filtrations.
By Poincaré-Birkhoff-Witt theorem, we have a grading on both U(p + ) and U(p − ), defined using a basis of p + (resp. p − ). Since p + and p − are commutative, we have U(p + ) = S(p + ) and U(p − ) = S(p − ), i.e., the universal envelopping algebra is the same as the symmetric algebra. We can therefore identify the graded piece of a degree n, denoted U(p + )[n] with the space of homogeneous polynomials of degree n, written as S(p + )[n] (and similarly for p − ).
We write M n for the subspace of elements of degree less or equal to n in U(p ′ ), i.e., we have
M n = r+s≤n (S(p − )[r] ⊗ S(p + )[s]) ∼ = ⊕ i≤n S(p ′ )[i].
This allows us to define a filtration on V : we can write V = ⊕ n V n /V n−1 where
V n = M n U(k ′ ) ⊗ U(k ′ ) (E ⊗ F ).
Note that V 0 = E ⊗ F . By the description of M n as ⊕ i≤n S(p ′ )[i], we observe that the quotient M n /M n−1 can be identified with S(p ′ )[n]. Therefore we obtain
V n /V n+1 = S(p ′ )[n] ⊗ U(k ′ ) (E ⊗ F ).
We can define a similar filtration on V E ⊗ V F :
(V E ⊗ V F ) n = r+s≤n M r U(k) ⊗ U(b + ) E ⊗ M s U(k) ⊗ U(b − ) F .
We observe that, as vector spaces, this is equivalent to
(V E ⊗ V F ) n = r+s≤n S(p − )[r] ⊗ U(b + ) E ⊗ S(p + )[s] ⊗ U(b − ) F .
We obtain
(V E ⊗ V F ) n /(V E ⊗ V F ) n−1 = r+s=n S(p − )[r] ⊗ U(b + ) E ⊗ S(p + )[s] ⊗ U(b − ) F .
4.2.3.
Identification of the tensor product V E ⊗ V F . Since E ⊗ F = V 0 is naturally a subset of V , we can use Froebenius reciprocity to extend this inclusion to a map
T : V → V E ⊗ V F , defined by 1 ⊗ (e ⊗ f ) → (1 ⊗ e) ⊗ (1 ⊗ f ),
for all e ∈ E and f ∈ F . We extend this map so that it is is compatible with the module structure and we want to show that it preserves the filtrations defined previously. Note that we will sometimes write e ⊗ f instead of 1 ⊗ (e ⊗ f ) to simplify notations.
Since {α i } is a basis of p − and {β j } a basis of p + , we can write any basis element of S(p ′ ) in the form α 1 . . . α k β 1 . . . β l with possible repetitions in the indices.
In V , the action of the element α 1 . . . α k β 1 . . . β l on 1 ⊗ (e ⊗ f ) is given by α 1 . . . α k β 1 . . . β l ⊗ (e ⊗ f ), which is an element in V k+l in the filtration described above. This element should be mapped by T to (α 1 . . . α k β 1 . . . β l ) · T (e ⊗ f ) = (α 1 . . . α k β 1 . . . β l ) · ((1 ⊗ e) ⊗ (1 ⊗ f )) .
The next lemma implies that the filtrations are preserved.
Lemma 3. With the notations above, we have
T (α 1 . . . α k β 1 . . . β l )(e ⊗ f ) = (α 1 . . . α k β 1 . . . β l ) · T (e ⊗ f ) = α 1 . . . α k e ⊗ β 1 . . . β l f + v, with v ∈ (V E ⊗ V F ) k+l−1 .
Proof. This can be checked by direct computation and induction on l and k.
Our next step will be given by the following theorem : Proof. We saw that T (V n ) ⊂ (V E ⊗ V F ) n . We will show that T G : Gr(V ) → Gr(V E ⊗ V F ) is in fact surjective and that both V n and (V E ⊗ V F ) n have the same dimension, for every n.
Recall that we can write the graded piece of V E ⊗ V F of degree n as when T is considered as a map on the graded pieces of degree n. By definition, we know that α 1 . . . α k β 1 . . . β l ⊗ (e ⊗ f ) is an element of degree n = k + l, so it is an element of V n /V n−1 . This shows that
(V E ⊗ V F ) n /(V E ⊗ V F ) n−1 = k+l=n S(p − )[k] ⊗ U(b + ) E ⊗ S(p + )[l] ⊗ U(b − ) F .T : V n /V n−1 → (V E ⊗ V F ) n /(V E ⊗ V F ) n−1
is a surjective map for every n.
Looking at the dimensions, we recall that we had
(V E ⊗ V F ) n = r+s≤n S(p − )[r] ⊗ U(b + ) E ⊗ S(p + )[s] ⊗ U(b − ) F , and V n = i≤n S(p ′ )[i] ⊗ U(k ′ ) E ⊗ F = r+s≤n S(p − )[r] ⊗ S(p + )[s] ⊗ U(k ′ ) E ⊗ F.
Considered as C-vector spaces, these two spaces have the same dimension, namely
dim(V n ) = dim(E) dim(F ) r+s≤n dim(S(p − )[r]) dim(S(p + )[s]) = dim((V E ⊗ V F ) n ).
A direct application of proposition 1 concludes the proof.
Corollary 2.
The tensor product V E ⊗ V F is a projective U(g ′ )-module.
Proof. The previous theorem implies that
V E ⊗ V F ∼ = V ∼ = U(g ′ ) ⊗ U(k ′ ) (E ⊗ F ).
We can then apply proposition 4, since E ⊗ F is a (g ′ , K ′ )-module.
This concludes the proof of theorem 2.
F τ | om(C)×on(C) is an (o(m, n, C), O(m, R) × O(n, R))-module, so proposition 4 implies that E τ | o(m,n,C) is a projective o(m, n, C)module. This is enough to prove: Theorem 1. If p > m + n, the restriction of the oscillator representation ω of Sp(2p(m + n), R) to O(m, n; R) is projective.
decompose ω under the action of O(m, R) × O(n, R). We can then write ω as ω = ω m ⊗ ω * n , where ω m is a highest weight module for O(m, R) and ω * n is a lowest weight module for O(n, R). Since O(m, R) and O(n, R) are compact, we can decompose each piece as before. So we have ω m = σ (σ ⊗ θ(σ)) = σ (σ ⊗ E τ ), summing over all the irreducible representations σ of O(m, R). Similarly, ω * n = σ ( σ⊗θ( σ)) = σ ( σ⊗E τ ), summing over all the irreducible representations σ of O(n, R). Here E τ , E τ denote representations of Sp(2p, R), since (O(m, R) × O(n, R), Sp(2p, R) × Sp(2p, R)) is a dual pair in Sp(2p(m + n), R). We can therefore express ω as
3. 3 . 1 .
31Case ǫ = 1. If ǫ = 1, we have 1 − ǫ 2 (n − 2k) = 0, so the highest weight τ can be written as τ = (− n 2 , . . . , − n 2 , −a k − n 2 , . . . , −a 1 − n 2 ).
Theorem 4 .
4The map T : V → V E ⊗ V F , induced by an isomorphism of S(p ′ )modules on the graded spaces T G : Gr(V ) → Gr(V E ⊗ V F ), is an isomorphism of U(g ′ )-modules.
So any element in (VE ⊗V F ) n /(V E ⊗V F ) n−1 is of the form (α 1 . . . α k ⊗e)⊗(β 1 . . . β l ⊗f ) with k + l = n. The surjectivity of T : V n /V n−1 → (V E ⊗ V F ) n /(V E ⊗ V F ) n−1is then clear from the work done before : if k + l = n, then we saw thatT (α 1 . . . α k β 1 . . . β l ⊗ (e ⊗ f )) = (α 1 . . . α k ⊗ e) ⊗ (β 1 . . . β l ⊗ f ) + v, with v ∈ (V E ⊗ V F ) n−1 , so T (α 1 . . . α k β 1 . . . β l ⊗ (e ⊗ f )) = (α 1 . . . α k ⊗ e) ⊗ (β 1 . . . β l ⊗ f ) (mod (V E ⊗ V F ) n−1 )
The theta correspondence over R. Workshop at the university of Maryland. J Adams, Adams, J. The theta correspondence over R. Workshop at the university of Maryland (May 1994).
Euler poincare characteristic for the oscillator representation. J D Adams, D Prasad, G Savin, Representation Theory, Number Theory, and Invariant Theory: In Honor of Roger Howe on the Occasion of His 70th Birthday. J. Cogdell, J.-L. Kim, and C.-B. ZhuAdams, J. D., Prasad, D., and Savin, G. Euler poincare characteristic for the oscillator representation. In Representation Theory, Number Theory, and Invariant Theory: In Honor of Roger Howe on the Occasion of His 70th Birthday, J. Cogdell, J.-L. Kim, and C.-B. Zhu, Eds. Birkhäuser, Basel, 2017, pp. 1-22.
A classification of unitary highest weight modules. T Enright, R Howe, N Wallach, Representation Theory of Reductive Groups, Proceedings of the University of Utah Conference. C. Trombi, Ed. BirkhäuserBoston, Basel, StuttgartEnright, T., Howe, R., and Wallach, N. A classification of unitary highest weight mod- ules. In Representation Theory of Reductive Groups, Proceedings of the University of Utah Conference 1982, P. C. Trombi, Ed. Birkhäuser, Boston, Basel, Stuttgart, 1983, pp. 97-143.
Reciprocity laws in the theory of dual pairs. R Howe, Representation Theory of Reductive Groups: Proceedings of the University of Utah Conference. C. Trombi, Ed. BirkhäuserBostonHowe, R. Reciprocity laws in the theory of dual pairs. In Representation Theory of Reductive Groups: Proceedings of the University of Utah Conference 1982, P. C. Trombi, Ed. Birkhäuser, Boston, 1983, pp. 159-175.
Remarks on classical invariant theory. R Howe, Transactions of the American Mathematical Society. 3132Howe, R. Remarks on classical invariant theory. Transactions of the American Mathematical Society 313, 2 (June 1989).
Non-abelian harmonic analysis: applications of SL(2, R). R Howe, E.-C Tan, Springer-VerlagNew YorkHowe, R., and Tan, E.-C. Non-abelian harmonic analysis: applications of SL(2, R). Springer-Verlag, New York, 1992.
Representation Theory of Semisimple Groups, An Overview Based on Examples. A W Knapp, Princeton University PressPrinceton, New JerseyKnapp, A. W. Representation Theory of Semisimple Groups, An Overview Based on Exam- ples. Princeton University Press, Princeton, New Jersey, 1986.
Lie Groups, Lie Algebras, and Cohomology. A W Knapp, PrincetonNew JerseyKnapp, A. W. Lie Groups, Lie Algebras, and Cohomology. Princeton University Press, Prince- ton, New Jersey, 1988.
Seesaw dual reductive pairs. S S Kudla, Automorphic Forms of Several Variables: Taniguchi Symposium. I. Satake and Y. MoritaBostonBirkhäuserKudla, S. S. Seesaw dual reductive pairs. In Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, I. Satake and Y. Morita, Eds. Birkhäuser, Boston, 1984, pp. 244-268.
Minimal representations and reductive dual pairs. J.-S Li, Representation Theory of Lie Groups. J. Adams and D. VoganUSAAmerican Mathematical SocietyLi, J.-S. Minimal representations and reductive dual pairs. In Representation Theory of Lie Groups, J. Adams and D. Vogan, Eds. American Mathematical Society, USA, 2000, pp. 291- 340.
. Sabine Jessica , Lang , Salt Lake City, UT 84112Department of Mathematics, University of UtahSabine Jessica Lang, Department of Mathematics, University of Utah, Salt Lake City, UT 84112
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"Valuations on Log-Concave Functions",
"Valuations on Log-Concave Functions"
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"Fabian Mussnig "
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A classification of SL(n) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as SL(n) and translation invariant valuations on log-concave functions.2000 AMS subject classification: 26B25 (46B20, 52A21, 52A41, 52B45)
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10.1007/s12220-020-00539-3
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"https://arxiv.org/pdf/1707.06428v1.pdf"
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1707.06428
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5c95bb19525c4896d8db523202e15040ace304f4
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Valuations on Log-Concave Functions
20 Jul 2017
Fabian Mussnig
Valuations on Log-Concave Functions
20 Jul 2017arXiv:1707.06428v1 [math.MG]
A classification of SL(n) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as SL(n) and translation invariant valuations on log-concave functions.2000 AMS subject classification: 26B25 (46B20, 52A21, 52A41, 52B45)
A function Z defined on a lattice (L, ∨, ∧) and taking values in an abelian semigroup is called
a valuation if Z(f ∨ g) + Z(f ∧ g) = Z(f ) + Z(g),(1)
for all f, g ∈ L. A function Z defined on a set S ⊂ L is called a valuation if (1) holds whenever f, g, f ∨g, f ∧g ∈ S. In the classical theory, valuations on the set of convex bodies (non-empty, compact, convex sets), K n , in R n are studied, where ∨ and ∧ denote union and intersection, respectively. Valuations played a critical role in Dehn's solution of Hilbert's Third Problem and have been a central focus in convex geometric analysis. In many cases, well known functions in geometry could be characterized as valuations. For example, a first classification of the Euler characteristic and volume as continuous, SL(n) and translation invariant valuations on K n was established by Blaschke [5]. The celebrated Hadwiger classification theorem [17] gives a complete classification of continuous, rotation and translation invariant valuations on K n and provides a characterization of intrinsic volumes. Alesker [2] obtained classification results for translation invariant valuations. Since several important geometric operators like the Steiner point and the moment vector are not translation invariant, also translation covariance played an important role. In particular, Hadwiger & Schneider [18] characterized linear combinations of quermassvectors as continuous, rotation and translation covariant vector-valued valuations.
In addition to the ongoing research on real-valued valuations on convex bodies [1,14,19,22,28,29], valuations with values in K n have attracted interest. Such a map is called a Minkowski valuation if the addition in (1) is given by Minkowski addition, that is K + L = {x + y : x ∈ K, y ∈ L} for K, L ∈ K n . The first results in this direction were established by Ludwig [23,24]. See [13,15,21,40,43] for some of the pertinent results.
More recently, valuations were defined on function spaces. For a space S of real-valued functions we denote by f ∨ g the pointwise maximum of f and g while f ∧ g denotes their pointwise minimum. For Sobolev spaces [25,26,30] and L p spaces [27,35,41,42] complete classifications of valuations intertwining with the SL(n) were established. For definable functions, an analog to Hadwiger's theorem was proven [4]. Valuations on convex functions [3,7,11,12] and quasi-concave functions [9,10] were introduced and classified.
The aim of this paper is to establish a classification of SL(n) and translation covariant valuations on log-concave functions, that is functions of the form e −u , where u is a convex function. Let LC(R n ) denote the space of log-concave functions f : R n → [0, +∞) that are not identically 0, upper semicontinuous and vanish at infinity. Here a function is said to vanish at infinity if lim
|x|→+∞ f (x) = 0,
where |x| denotes the Euclidean norm of x. Furthermore, we equip LC(R n ) with the topology associated to hypo-convergence (see Section 1.2). It is easy to see that for any K ∈ K n the characteristic function χ K is an element of LC(R n ). Since
χ K∪L = χ K ∨ χ L χ K∩L = χ K ∧ χ L ,
for all K, L ∈ K n such that also K ∪ L ∈ K n , valuations on LC(R n ) can be seen as a generalization of valuations on K n . For f ∈ LC(R n ) the level set body [f ] is given by
h([f ], z) = +∞ 0 h({f ≥ t}, z) dt, for z ∈ R n . Here, h(K, z) = max{z · x : x ∈ K},
where z · x is the standard inner product of x, z ∈ R n , denotes the support function of a convex body K ∈ K n and uniquely describes K. Moreover, we set h(∅, z) = 0 for every z ∈ R n . The level set body was recently introduced for a more general class of quasi-concave functions in [12], where it appeared in the classification of SL(n) covariant Minkowski valuations on convex functions. Note, that [χ K ] = K for every K ∈ K n . Furthermore, the moment vector m(f ) of a log-concave function f ∈ LC(R n ) is defined by
m(f ) = R n x f (x) dx,
which is an element of R n (for details see Section 4). For functions in L 1 (R n , |x| dx), the moment vector was introduced and characterized as a Minkowski valuation by Tsang [42]. For x ∈ R n , let τ x denote the translation z → z + x on R n and let S be a space of real-valued functions defined on R n . We call an operator Z :
S → K n translation covariant if there exists a function Z 0 : S → R associated with Z such that Z(f • τ −1 x ) = Z(f ) + Z 0 (f )x for every f ∈ S and x ∈ R n . Moreover, Z is said to be SL(n) covariant if Z(f • φ −1 ) = φ Z(f )
for every φ ∈ SL(n) and f ∈ S. Furthermore, a function Z : LC(R n ) → K n is called homogeneous of degree q ∈ R if Z(sf ) = s q Z(f ) for every s > 0 and f ∈ LC(R n ). We say that a map Z : LC(R n ) → K n is equi-affinely covariant if it is translation covariant, SL(n) covariant and homogeneous. For the following result let n ≥ 3.
Theorem 1. An operator Z : LC(R n ) → K n is a continuous, equi-affinely covariant Minkowski valuation if and only if there exist constants c 1 , c 2 ≥ 0, c 3 ∈ R and q > 0 such that
Z(f ) = c 1 [f q ] + c 2 (−[f q ]) + c 3 m(f q )
for every f ∈ LC(R n ).
Here, −K denotes the reflection of the body K ∈ K n at the origin, that is h(−K, z) = h(K, −z) for z ∈ R n . We will see that Theorem 1 corresponds to a result for valuations on K n (see Corollary 1.3). Denoting by D K = K + (−K) the difference body of K ∈ K n , we immediately obtain the following corollary, which again corresponds to a result in the theory of valuations on K n [24, Corollary 1.2.].
Corollary. An operator Z : LC(R n ) → K n is a continuous, SL(n) covariant, translation invariant and homogeneous Minkowski valuation if and only if there exist constants c ≥ 0 and q > 0 such that
Z(f ) = c D[f q ],
for every f ∈ LC(R n ).
Here, we say that a map Z defined on a space S of real-valued functions on R n is translation invariant if Z(f • τ −1 x ) = Z(f ) for every f ∈ S and x ∈ R n . We remark that this corollary also follows from [12,Theorem 2], where SL(n) covariant Minkowski valuations on convex functions were characterized.
In order to prove Theorem 1 we will need a classification of real-valued valuations on LC(R n ) that corresponds to the result by Blaschke mentioned above. A map Z :
S → R is called SL(n) invariant if Z(f • φ −1 ) = Z(f )
for every φ ∈ SL(n) and f ∈ S. We call an operator Z : LC(R n ) → R equi-affinely invariant if Z is translation invariant, SL(n) invariant and homogeneous, where homogeneity is defined as before. Let n ≥ 2.
Theorem 2. An operator Z : LC(R n ) → R is a continuous, equi-affinely invariant valuation if and only if there exist constants c 0 , c n ∈ R and q ∈ R, with q > 0 if c n = 0, such that
Z(f ) = c 0 max x∈R n f (x) q + c n R n f q (x) dx,
for every f ∈ LC(R n ).
In Section 2 we will see that max x∈R n f (x) and R n f (x) dx can be seen as functional versions of the Euler characteristic and volume, respectively (see also [6]).
Preliminaries
We collect some properties of convex bodies and convex functions. Basic references are the books by Schneider [39] and Rockafellar & Wets [37]. In addition, we recall definitions and classification results on Minkowski valuations. We work in R n and denote the canonical basis vectors by e 1 , . . . , e n . Furthermore, let conv(A) be the convex hull and pos(A) the positive hull of A ⊂ R n . The space of convex bodies, K n , is equipped with the Hausdorff metric, which is given by δ(K, L) = sup y∈S n−1 |h(K, y) − h(L, y)| for K, L ∈ K n , where h(K, z) = max{z · x : x ∈ K} is the support function of K at z ∈ R n . The subspace of convex bodies in R n containing the origin is denoted by K n o . Moreover, let P n denote the space of convex polytopes in R n and P n o the space of convex polytopes containing the origin. All these spaces are equipped with the topology coming from the Hausdorff metric.
For p ≥ 0, a function h : R n → R is p-homogeneous if h(t z) = t p h(z) for t ≥ 0 and z ∈ R n . It is sublinear if it is 1-homogeneous and h(x + y) ≤ h(x) + h(y) for x, y ∈ R n . Every sublinear function is the support function of a unique convex body. Note that for the Minkowski sum of K, L ∈ K n , we have
h(K + L, z) = h(K, z) + h(L, z)
and in particular h(K + x, z) = h(K, z) + z · x for x, z ∈ R n .
SL(n) Covariant Minkowski Valuations on Convex Bodies
The moment body M K of K ∈ K n is defined by
h(M K, z) = K |x · z| dx
for every z ∈ S n−1 . Furthermore, the moment vector m(K) of K is given by
h(m(K), z) = K x · z dx
for every z ∈ S n−1 . Note that the moment vector is an element of R n . We require the following result where the support function of certain moment bodies and moment vectors is calculated for specific vectors. Let n ≥ 2.
h(T λ , e 1 ) = λ h(−T λ , e 1 ) = 0 h(m(T λ ), e 1 ) = λ 2 (n+1)! h(M T λ , e 1 ) = λ 2 (n+1)! .
The first classification of SL(n) covariant Minkowski valuations was established by Ludwig [24], where also the difference body operator was characterized. The following result is due to Haberl. We say that a Minkowski valuation Z : K n → K n is translation covariant if there exists a function Z 0 : K n → R associated with Z such that Z(K + x) = Z(K) + Z 0 (K)x, for every K ∈ K n and x ∈ R n . Since several important geometric operators have this property, translation covariant valuations have attracted interest. For example, the identity on K n and the reflection K → −K are translation covariant. Furthermore, for z ∈ S n−1 we have
h(m(K + x), z) = K+x y · z dy = K (y + x) · z dy = K y · z dy + V n (K) (z · x),(2)
and hence m(K + x) = m(K) + V n (K)x for every K ∈ K n and x ∈ R n . Based on Schneider's characterization of the Steiner point [38], Hadwiger & Schneider [18] proved that the quermassvectors form a basis of the space of continuous, rotation and translation covariant vector-valued valuations. In [31], McMullen characterized weakly continuous and translation covariant vector-valued valuations on convex polytopes, extending a previous result by Hadwiger [16]. In his result the intrinsic moment vectors of the faces of a polytope appear. For further results on translation covariant valuations see [32,33].
1 , c 2 ≥ 0 and c 3 ∈ R such that Z K = c 1 K + c 2 (−K) + c 3 m(K),(3)
for every K ∈ K n .
Proof. We have already seen in Theorem 1.2 and (2) that (3) defines a continuous, SL(n) and translation covariant Minkowski valuation. Conversely, let Z be a continuous Minkowski valuation on K n that is SL(n) and translation covariant. Obviously, the restriction of Z to K n o is a continuous, SL(n) covariant Minkowski valuation. Hence, by Theorem 1.2 there exist constants c 1 , c 2 , c 4 ≥ 0 and c 3 ∈ R such that
Z K = c 1 K + c 2 (−K) + c 3 m(K) + c 4 M K,(4)
for every K ∈ K n o . Define the polytope P as P := [−e 1 , 2e 1 ] + [0, e 2 ] + · · · [0, e n ] and note that P, P + e 1 , P − e 1 ∈ K n o . By the translation covariance of Z we obtain
Z(P ) + Z 0 (P )e 1 = Z(P + e 1 ) = c 1 P + c 1 e 1 + c 2 (−P ) − c 2 e 1 + c 3 m(P ) + V n (P )e 1 + c 4 M(P + e 1 ), Z(P ) − Z 0 (P )e 1 = Z(P − e 1 ) = c 1 P − c 1 e 1 + c 2 (−P ) + c 2 e 1 + c 3 m(P ) − V n (P )e 1 + c 4 M(P − e 1 ).
Adding these equations shows that 2 Z(P ) = Z(P + e 1 ) + Z(P − e 1 ) = 2c 1 P + 2c 2 (−P ) + 2c 3 m(P ) + c 4 (M(P + e 1 ) + M(P − e 1 )).
On the other hand by (4) 2 Z(P ) = 2c 1 P + 2c 2 (−P ) + 2c 3 m(P ) + 2c 4 M P.
Evaluating and comparing the support functions of these two representations of 2 Z(P ) at e 1 gives 2c 4 5 2 = c 4 ( 9 2 + 5 2 ), and therefore c 4 = 0. Furthermore, this shows that Z 0 (K) = c 1 − c 2 + c 3 V n (K) for every K ∈ K n o . Now, fix an arbitrary K ∈ K n . Then, there exist K o ∈ K n o and x ∈ R n such that K = K o + x. By the properties of Z this gives
Z(K) = Z(K o + x) = Z(K o ) + Z 0 (K o )x = c 1 K o + c 2 (−K o ) + c 3 m(K o ) + (c 1 − c 2 + V n (K o ))x = c 1 K + c 2 (−K) + c 3 m(K).
Convex and Log-Concave Functions
Let f = e −u ∈ LC(R n ). Then, u : R n → (−∞, +∞] is convex, lower-semicontinuous, proper and coercive. Here we say that u is proper if u = +∞. Furthermore, u is called coercive if lim |x|→+∞ u(x) = +∞.
The set of all such functions u will be denoted by Conv(R n ). Obviously,
LC(R n ) = {e −u : u ∈ Conv(R n )}.
Furthermore, for a function u ∈ Conv(R n ), the domain, dom u = {x ∈ R n : u(x) < +∞}, of u is a convex subset of R n . Moreover, its epigraph
epi u = {(x, t) ∈ R n × R : u(x) ≤ t},
is a closed convex subset of R n × R. Hence, every function u ∈ Conv(R n ) attains its minimum and min x∈R n u(x) > −∞. Furthermore, for t ∈ R, the sublevel set
{u ≤ t} = {x ∈ R n : u(x) ≤ t},
is either a convex body or empty. Note that for u, v ∈ Conv(R n ) and t ∈ R
{u ∧ v ≤ t} = {u ≤ t} ∪ {v ≤ t} and {u ∨ v ≤ t} = {u ≤ t} ∩ {v ≤ t},
where for u∧v ∈ Conv(R n ) all occurring sublevel sets are either empty or in K n . Equivalently, every function f ∈ LC(R n ) attains its maximum and for 0 < t ≤ max x∈R n f (x) the superlevel set
{f ≥ t} = {x ∈ R n : f (x) ≥ t},
is a convex body. Moreover, for every f, g ∈ LC(R n ) and t > 0
{f ∧ g ≥ t} = {f ≥ t} ∩ {g ≥ t} and {f ∨ g ≥ t} = {f ≥ t} ∩ {g ≥ t}.
We equip Conv(R n ) with the topology associated to epi-convergence, which is the standard topology for a space of extended real-valued convex functions. Here a sequence u k :
R n → (−∞, ∞] is epi-convergent to u : R n → (−∞, ∞]
if for all x ∈ R n the following conditions hold:
(i) For every sequence x k that converges to x,
u(x) ≤ lim inf k→∞ u k (x k ).
(ii) There exists a sequence x k that converges to x such that
u(x) = lim k→∞ u k (x k ).
In this case we write u = epi-lim k→∞ u k and u k epi −→ u. We remark that epi-convergence is also called Γ-convergence. Correspondingly, we say that a sequence
f k in LC(R n ) is hypo- convergent to f ∈ LC(R n ) if there exist u k , u ∈ Conv(R n ) such that f k = e −u k for every k ∈ N, f = e −u and u k epi −→ u. In this case we write f = hypo-lim k→∞ f k and f k hypo −→ f .
The following results connect epi-convergence and Hausdorff convergence of sublevel sets. We say that
{u k ≤ t} → ∅ as k → ∞ if there exists k 0 ∈ N such that {u k ≤ t} = ∅ for all k ≥ k 0 . Lemma 1.4 ( [11], Lemma 5). Let u k , u ∈ Conv(R n ). If u k epi −→ u k , then {u k ≤ t}→{u ≤ t} for every t ∈ R with t = min x∈R n u(x). Lemma 1.5 ([37], Proposition 7.2). Let u k , u ∈ Conv(R n ). If for each t ∈ R there exists a sequence t k of reals convergent to t with {u k ≤ t k } → {u ≤ t}, then u k epi −→ u.
Furthermore, the so-called cone property and uniform cone property will be useful in order to show that certain integrals converge.
Lemma 1.6 ([8], Lemma 2.5). For u ∈ Conv(R n ) there exist constants a, b ∈ R with a > 0 such that u(x) > a|x| + b
for every x ∈ R n .
Lemma 1.7 ([11], Lemma 8). Let u k , u ∈ Conv(R n ). If u k epi −→ u, then there exist constants a, b ∈ R with a > 0 such that u k (x) > a|x| + b and u(x) > a|x| + b,
for every k ∈ N and x ∈ R n .
Next, we introduce some special elements of Conv(R n ). For K ∈ K n o , we define the convex function ℓ K : R n → [0, +∞] by
epi ℓ K = pos(K × {1}).
This means that the epigraph of ℓ K is a cone with apex at the origin and {ℓ K ≤ t} = t K for all t ≥ 0. It is easy to see that ℓ K is an element of Conv(R n ) for K ∈ K n o . Also the (convex) indicator function I K for K ∈ K n belongs to Conv(R n ), where
I K (x) = 0, if x ∈ K +∞, if x / ∈ K.
Observe, that e −I K = χ K for every K ∈ K n . Let f, g ∈ LC(R n ) be such that f ∨ g ∈ LC(R n ) and let Z : LC(R n ) → A, + , where A, + is an abelian semigroup. By definition, there exist functions u, v ∈ Conv(R n ) such that u ∧ v ∈ Conv(R n ) and f = e −u and g = e −v . Since The next result, which is based on [26], shows that a valuation on Conv(R n ) is uniquely determined by its behaviour on certain functions. Lemma 17). Let A, + be a topological abelian semigroup with cancellation law and let Y 1 ,
Lemma 1.8 ([11],Y 2 : Conv(R n ) → A, + be continuous valuations. If Y 1 (ℓ •τ −1 x ) = Y 2 (ℓ •τ −1
x ) for every ℓ ∈ {ℓ P + t : P ∈ P n o , t ∈ R} and every x ∈ R n , then Y 1 ≡ Y 2 on Conv(R n ).
We remark, that in [11,Lemma 17] it is assumed that the valuations are translation invariant. However, translation invariance itself is not needed for the proof and it is easy to see that this more general statement holds.
SL(n) Invariant Real-Valued Valuations on LC(R n )
We denote by V 0 the Euler characteristic, that is V 0 (K) = 1 for every K ∈ K n and V 0 (∅) = 0. Since for f ∈ LC(R n ) the level sets {f ≥ t} are convex bodies for every 0 < t ≤ max x∈R n f , it makes sense to consider
V 0 (f ) := +∞ 0 V 0 ({f ≥ t}) dt = max x∈R n f (x).
Furthermore, denoting by V n the n-dimensional volume or Lebesgue measure, and assuming that the integrals converge, we have by the layer-cake principle
V n (f ) := +∞ 0 V n ({f ≥ t}) dt = R n f (x) dx.
We remark that this notion for the volume of a (log-concave) function is frequently used and there are several examples of functional counterparts of geometric inequalities, in which the volume V n (K) of a convex body K is replaced by the integral f of a function f . For example, the Prékopa-Leindler inequality is the functional analog of the Brunn-Minkowski inequality [20,36]. Furthermore, functional versions of quermassintegrals where recently introduced for quasi-concave functions [6,34].
We need the following result where the volume operator of a specific function is calculated. Let n ≥ 2.
Lemma 2.1. For λ ≥ 0, q > 0 and T λ = conv{0, λ e 1 , e 2 , . . . , e n },
V n (e −qℓ T λ ) = λ q n .
Proof. By definition we have
V n (e −qℓ T λ ) = 1 0 V n ({e −qℓ T λ ≤ t}) dt = 1 0 V n ({ℓ T λ ≤ − log t q }) dt.
Using the substitution s = − log t q we have dt = −qe −qs ds and therefore
V n (e −qℓ T λ ) = q +∞ 0 V n ({ℓ T λ ≤ s}) e −qs ds.
By definition, {ℓ T λ ≤ s} = s T λ for every s ≥ 0. Hence, The following results are mostly deduced from [11], where analogs of V 0 and V n on Conv(R n ) were studied.
V n ({ℓ T λ ≤ s}) = s n V n (T λ ) = s n λ n! .
Lemma 2.2. For every q ∈ R, the map
f → V 0 (f ) q
is a continuous, equi-affinely invariant valuation on LC(R n ) that is homogeneous of degree q.
Proof.
Since
max x∈R n s f (x) q = s q max x∈R n f (x) q ,
for every f ∈ LC(R n ) and s > 0, the map f → V 0 (f ) q is homogeneous of degree q. By [11,Lemma 12] it is a continuous, SL(n) and translation invariant valuation.
Lemma 2.3. For every q > 0, the map f → V n (f q )
is a continuous, equi-affinely invariant valuation on LC(R n ) that is homogeneous of degree q.
Proof. By [11,Lemmas 15 & 16], the map f → V n (f q ) is a well-defined continuous, SL(n) and translation invariant valuation on LC(R n ). Since
R n (sf ) q (x) dx = s q R n f q (x) dx,
for every f ∈ LC(R n ) and s > 0, it is homogeneous of degree q.
Classification of SL(n) Invariant Real-Valued Valuations
In the following Lemma we collect some results that were proved in [11]. Let n ≥ 2.
Lemma 3.1. If Y : Conv(R n ) → R is a continuous, SL(n) and translation invariant valuation, then there exist continuous functions ζ 0 , ζ n , ψ n : R → R such that
Y(ℓ K + t) = ζ 0 (t) + ψ n (t)V n (K), Y(I K + t) = ζ 0 (t) + ζ n (t)V n (K),
for every K ∈ K n o and t ∈ R. Furthermore, lim t→+∞ ψ n (t) = 0 and
ζ n (t) = (−1) n n! d n dt n ψ n (t),
for every t ∈ R. Moreover, Y is uniquely determined by ζ 0 and ζ n .
Proof of Theorem 2
By Lemmas 2.2 and 2.3, the operator
f → c 0 V 0 (f ) q + c n V n (f q ),
defines a continuous, equi-affinely invariant valuation on LC(R n ) for every c 0 , c n ∈ R and q ∈ R, when q > 0 if c n = 0.
Conversely, let Z : LC(R n ) → R be a continuous, equi-affinely invariant valuation and let Y be the corresponding valuation on Conv(R n ), that is Y(u) = Z(e −u ) for every u ∈ Conv(R n ). Then Y is continuous, SL(n) and translation invariant. Furthermore,
Y(u + t) = Z(e −u−t ) = (e −t ) q Z(e −u ) = e −qt Y(u),
for every u ∈ Conv(R n ) and t ∈ R, where q ∈ R denotes the degree of homogeneity of Z. Let ζ 0 , ζ n , ψ n be the functions from Lemma 3.1. We have,
ζ 0 (t) = Y(I {0} + t) = e −qt Y(I {0} ),
for every t ∈ R. Hence, there exists a constant c 0 ∈ R such that ζ 0 (t) = c 0 e −qt for every t ∈ R.
Furthermore, let K ∈ K n o such that V n (K) > 0. Then, e −qt Y(ℓ K ) = Y(ℓ K + t) = ζ 0 (t) + ψ n (t)V n (K) = c 0 e −qt + ψ n (t)V n (K),
for every t ∈ R. Hence, there exists a constant c n ∈ R such that ψ n (t) = c n e −qt for every t ∈ R. Since lim t→+∞ ψ n (t) = 0, we must have q > 0 or c n = 0. Moreover,
ζ n (t) = (−1) n n! d n dt n ψ n (t) = c n q n n! e −qt =: c n e −qt ,
for every t ∈ R. For t ∈ R, let s = e −t . We have
Z(s χ K ) = Y(I K + t) = c 0 e −qt + c n e −qt V n (K) = c 0 s q + c n s q V n (K) = c 0 +∞ 0 V 0 ({s χ K ≥ r}) dr q + c n +∞ 0 V n ({(s χ K ) q ≥ r}) dr = c 0 V 0 (s χ K ) q + c n V n ((s χ K ) q ),
for every K ∈ K n . Since Y is uniquely determined by its values on indicator functions and
f → c 0 V 0 (f ) q + c n V n (f q )
defines a continuous, equi-affinely invariant valuation, the proof is complete.
SL(n) Covariant Minkowski Valuations on LC(R n )
In this section we discuss the operators that appear in Theorem 1 and show that they are continuous, equi-affinely covariant Minkowski valuations. In [6] it is proposed to generalize a function Φ :
K n → [0, +∞) to LC(R n ) via Φ(f ) = +∞ 0 Φ({f ≥ t}) dt,
for f ∈ LC(R n ). Note, that this construction implicitly uses the general convention Φ(∅) = 0. Following this approach, the level set body [f ] of f ∈ LC(R n ) is the convex body that is defined via
h([f ], z) = +∞ 0 h({f ≥ t}, z) dt,
for every z ∈ R n .
Lemma 4.1. For every q > 0, the map f → [f q ](5)
is a continuous, equi-affinely covariant Minkowski valuation on LC(R n ) that is homogeneous of degree q.
Proof. By [12,Lemma 7.2], the map f → [f q ] is a well-defined, continuous, SL(n) covariant Minkowski valuation on LC(R n ). Furthermore, for s > 0, x, z ∈ R n and f ∈ LC(R n ), we have
h([(s f • τ −1 x ) q ], z) = +∞ 0 h({(s f • τ −1 x ) q ≥ t}, z) dt = s q +∞ 0 h(τ x {f q ≥ t}, z) dt = s q +∞ 0 h({f q ≥ t}, z) dt + s q max x∈R n f q (x) (x · z) = s q h([f q ], z) + s q V 0 (f q ) (x · z).
Hence, (5) is homogeneous of degree q and translation covariant.
The next lemma will allow us to give a definition of the moment vector for functions in LC(R n ). Proof. Observe, that for K ∈ K n and z ∈ S n−1
|h(m(K), z)| = K x · z dx ≤ V n (K) max y∈S n−1 |h(K, y)|.
Fix f ∈ LC(R n ) and let u ∈ Conv(R n ) be such that f = e −u . By Lemma 1.6, there exist constants a, b ∈ R with a > 0 such that
u(x) > v(x) = a|x| + b,
for every x ∈ R n . Hence, for g = e −v ∈ LC(R n ) we have f < g pointwise and therefore
{f ≥ t} ⊂ {g ≥ t} = x : |x| ≤ − log t−b a
for every 0 < t ≤ e −b . This gives
|h(m({f ≥ t}), z)| ≤ V n ({g ≥ t}) max y∈S n−1 |h({g ≥ t}, y)| = vn a n+1 − log t − b n+1 ,
for every 0 < t ≤ e −b and z ∈ S n−1 , where v n denotes the volume of the n-dimensional unit ball. Thus, using the substitution t = e −s , we obtain
+∞ 0 |h(m({f ≥ t}), z)| dt ≤ vn a n+1 e −b 0 (− log t − b) n+1 dt ≤ vn a n+1 0 b (s − b) n+1 e −s ds < +∞,
for every z ∈ R n .
By Lemma 4.2, the integral
+∞ 0 h(m({f ≥ t}), z) dt(6)
converges for every f ∈ LC(R n ) and z ∈ R n . Since each of the support functions
z → h(m({f ≥ t}), z)
is sublinear, it is easy to see that (6) defines a sublinear function in z and thus is the support function of a convex body m(f ) ∈ K n . Using the definition of the moment vector and the layer-cake principle, we obtain
h(m(f ), z) = +∞ 0 {f ≥t} x · z dx dt = +∞ 0 R n χ {f ≥t} (x) (x · z) dx dt = R n f (x) (x · z) dx.
Hence,
m(f ) = R n f (x) x dx
is an element of R n and will be called the moment vector of f ∈ LC(R n ).
f → m(f q )(7)
is a continuous, equi-affinely covariant Minkowski valuation on LC(R n ) that is homogeneous of degree q.
Proof. Since f q ∈ LC(R n ) for every f ∈ LC(R n ) and q > 0, the map f → m(f q ) is welldefined. For φ ∈ SL(n) we have
m((f • φ −1 ) q ) = R n (f q • φ −1 )(x) x dx = R n f q (x) φx dx = φ m(f q ),
which shows SL(n) covariance. Furthermore, for x ∈ R n we obtain
m((f • τ −1 x ) q ) = R n f q (y − x) y dy = R n f q (y) y dy + x R n f q (y) dy = m(f q ) + V n (f q ) x,
and for s > 0
m((sf ) q ) = R n (sf ) q (x) x dx = s q R n f q (x) x dx = s q m(f q ).
Hence, (7) is equi-affinely covariant. In order to show the valuation property, let f, g ∈ LC(R n ) such that f ∨ g ∈ LC(R n ). Then,
m((f ∧ g) q ) = {f ≤g} f q (x) x dx + {f >g} g q (x) x dx m((f ∨ g) q ) = {f ≤g} g q (x) x dx + {f >g} f q (x) x dx. Hence, m((f ∧ g) q ) + m((f ∨ g) q ) = m(f q ) + m(g q ).
It remains to show continuity.
For f k , f ∈ LC(R n ) such that f k hypo −→ f , there exist u k , u ∈ LC(R n ) such that f k = e −u k for every k ∈ N, f = e −u and u k epi −→ u. By Lemma 1.7, there exist a > 0 and b ∈ R such that u k (x) > a|x| + b and u(x) > a|x| + b,
for every k ∈ N and x ∈ R n . Similar as in the proof of Lemma 4.2, this gives
|h(m({f ≥ t}), ·)| ≤ vn a n+1 (− log t − b) n+1 |h(m({f k ≥ t}), ·)| ≤ vn a n+1 (− log t − b) n+1 ,
which shows that these functions are dominated by an integrable function. Furthermore, Lemma 1.5 and the continuity of the moment vector on K n imply that h(m({f k ≥ t}), ·) → h(m({f ≥ t}), ·) pointwise for every t = max x∈R n f (x). Hence, by the dominated convergence theorem, we have
h(m(f k ), ·) = +∞ 0 h(m({f k ≥ t}, ·) dt −→ +∞ 0 h(m({f ≥ t}, ·) dt = h(m(f ), ·),
pointwise, which implies Hausdorff convergence of m(f k ) to m(f ). The claim now follows,
since f → f q is continuous and f q k hypo −→ f q .
Classification of SL(n) Covariant Minkowski Valuations
The next result extends the basic observation that the associated function Z 0 : K n → R n of a translation covariant Minkowski valuation Z : K n → K n is a translation invariant real-valued valuation. See for example [33,Lemma 10.5] for a corresponding result on vector-valued valuations. Similarly, SL(n) covariance of Z implies SL(n) invariance of Z 0 . Hence, it is no coincidence that the associated function of the Minkowski valuation described in Corollary 1.3 is a linear combination of the Euler characteristic and volume.
Lemma 5.1. If Z : LC(R n ) → K n is a continuous, equi-affinely covariant Minkowski valuation, then its associated function Z 0 : LC(R n ) → R is a continuous, equi-affinely invariant valuation. Furthermore, Z and Z 0 have the same degree of homogeneity.
Proof. Let x ∈ R n \{0} and f, g ∈ LC(R n ) be such that f ∨ g ∈ LC(R n ). Since
(f • τ −1 x ) ∨ (g • τ −1 x ) = (f ∨ g) • τ −1 x , (f • τ −1 x ) ∧ (g • τ −1 x ) = (f ∧ g) • τ −1 x ,
it follows from the translation covariance and the valuation property of Z that
Z(f • τ −1 x ) + Z(g • τ −1 x ) = Z((f ∨ g) • τ −1 x ) + Z((f ∧ g) • τ −1 x ) = Z(f ∨ g) + Z(f ∧ g) + Z 0 (f ∨ g)x + Z 0 (f ∧ g)x.
On the other hand,
Z(f • τ −1 x ) + Z(g • τ −1 x ) = Z(f ) + Z 0 (f )x + Z(g) + Z 0 (g)x = Z(f ∨ g) + Z(f ∧ g) + Z 0 (f )x + Z 0 (g)x.
Hence, Z 0 is a valuation. Now, for arbitrary y ∈ R n , we have
Z(f ) + Z 0 (f )x + Z 0 (f )y = Z(f • τ −1 x+y ) = Z(f • τ −1 y • τ −1 x ) = Z(f • τ −1 y ) + Z 0 (f • τ −1 y )x = Z(f ) + Z 0 (f )y + Z 0 (f • τ −1 y )x,
and therefore Z 0 (f ) = Z 0 (f • τ −1 y ). For φ ∈ SL(n) observe that
(τ −1 x • φ −1 )(z) = φ −1 z − x = φ −1 (z − φx) = (φ −1 • τ −1 φx )(z)
for every z ∈ R n and therefore
φ Z(f ) + Z 0 (f )φx = φ Z(f • τ −1 x ) = Z(f • τ −1 x • φ −1 ) = Z(f • φ −1 • τ −1 φx ) = Z(f • φ −1 ) + Z 0 (f • φ −1 )φx = φ Z(f ) + Z 0 (f • φ −1 )φx.
Hence, Z 0 is SL(n) invariant. Moreover, for s > 0 we have
s q Z(f ) + s q Z 0 (f )x = s q Z(f • τ −1 x ) = Z(s(f • τ −1 x )) = Z((sf ) • τ −1 x ) = s q Z(f ) + Z 0 (sf )x.
Lastly, if f k , f ∈ LC(R n ) are such that hypo-lim k→∞ f k = f , then also hypo-lim k→∞ f k •τ −1
x = f • τ −1
x . Hence, by the continuity of Z,
Z(f k ) + Z 0 (f k )x = Z(f k • τ −1 x ) −→ Z(f • τ −1 x ) = Z(f ) + Z 0 (f )x.
For the remainder of this section, let n ≥ 3.
Lemma 5.2. Let Z : LC(R n ) → K n be a continuous, equi-affinely covariant Minkowski valuation. There exist constants c 1 ,
c 2 , d 1 , d 2 , d 4 ≥ 0, c 3 , d 3 ∈ R and q ∈ R, with q > 0 if c 3 = 0, such that Z(s e −ℓ K ) = s q (d 1 K + d 2 (−K) + d 4 m(K) + d 3 M(K)),
for every K ∈ K n o and s > 0 and
Z(s χ K ) = s q (c 1 K + c 2 (−K) + c 3 m(K)),
for every K ∈ K n and s > 0. Furthermore,
Z 0 (f ) = (c 1 − c 2 )V 0 (f ) q + c 3 V n (f q ),
for every f ∈ LC(R n ).
Proof. Since for K, L ∈ K n o such that K ∪ L ∈ K n o we have
ℓ K∪L = ℓ K ∧ ℓ L , ℓ K∩L = ℓ K ∨ ℓ L , the map K → Z(e −ℓ K )(8)
defines a Minkowski valuation on K n o . Furthermore, ℓ φK = ℓ K • φ −1 for every φ ∈ SL(n) and ℓ K k epi −→ ℓ K for every sequence K j that converges to K in K n o by Lemma 1.5. Hence, (8) defines a continuous, SL(n) covariant Minkowski valuation on K n o . It follows from Theorem 1.2 that there exist constants d 1 , d 2 , d 4 ≥ 0 and d 3 ∈ R such that Z(s e −ℓ K ) = s q Z(e −ℓ K ) = s q (d 1 K + d 2 (−K) + d 3 m(K) + d 4 M(K)) for every K ∈ K n o and s > 0, where q ∈ R denotes the degree of homogeneity of Z. Similarly, K → Z(χ K ) defines a continuous, SL(n) and translation covariant Minkowski valuation on K n . Hence, by Corollary 1.3 there exist constants c 1 , c 2 ≥ 0 and c 3 ∈ R such that Z(s χ K ) = s q (c 1 K + c 2 (−K) + c 3 m(K)), for every K ∈ K n and s > 0.
For K ∈ K n , x ∈ R n \{0} and s > 0 let f := s χ K ∈ LC(R n ) and observe that
Z(f ) + Z 0 (f )x = Z(f • τ −1 x ) = Z(s χ K+x ) = s q (c 1 K + c 2 (−K) + c 3 m(K) + (c 1 − c 2 + c 3 V n (K))x) = Z(f ) + s q (c 1 − c 2 + c 3 V n (K))x.
On the other hand, by Lemma 5.1 and Theorem 2, there exist c 0 , c n ∈ R and q ∈ R, with q > 0 if c n = 0, such that Z 0 (g) = c 0 V 0 (g) q + c n V n (g q ),
for every g ∈ LC(R n ). Noting, that V 0 (f ) q = s q and V n (f q ) = s q V n (K), a comparison shows that
(c 1 − c 2 )s q V 0 (K) + c 3 s q V n (K) = Z 0 (s χ K ) = c 0 s q V 0 (K) + c n s q V n (K),
for every s > 0 and K ∈ K n . Choosing K = {0} and s = 1 gives c 1 − c 2 = c 0 . With the same K and arbitrary s > 0 we have q = q and with any full-dimensional K ∈ K n we obtain c n = c 3 .
u h ∧ ℓ h = ℓ [0,e 1 /h] , u h ∨ ℓ h = I {e 1 } + h.
Similar as in the proof of Theorem 2, let Y be the valuation on Conv(R n ) that corresponds to Z, that is Y(u) = Z(e −u ) for every u ∈ Conv(R n ). Then Y is continuous, SL(n) and translation covariant and Y(u + t) = e −qt Y(u) for every u ∈ Conv(R n ) and t ∈ R. We have
Y(ℓ h ) = e −c 1 = h(Y(I [0,e 1 ] ), e 1 ) = lim h→0 + h(Y(u h ), e 1 ) = lim h→0 + (h(Y(ℓ [0,e 1 /h] ), e 1 ) + h(Y(I {e 1 } + h), e 1 ) − h(Y(ℓ h ), e 1 )) = lim h→0 + ( d 1 h + (c 1 − c 2 )e −qh − e −qh d 1 h − (c 1 − c 2 )e −qh ) = lim h→0 + d 1 1−e −qh h = q d 1 .
Similarly, evaluating the support functions at −e 1 shows that c 2 = q d 2 .
In the following we say that a Minkowski valuation Z :
LC(R n ) → K n is trivial if Z(f ) = {0} for every f ∈ LC(R n ).
Using the earlier highlighted relation between valuations on LC(R n ) and valuations on Conv(R n ), we obtain the following result from [12,Lemma 8.7], where SL(n) covariant and translation invariant valuations on Conv(R n ) were studied. Proof. Let d 1 , d 2 , c 1 , c 2 , c 3 and q denote the constants from Lemma 5.2 and suppose that q ≤ 0. It follows from Lemma 5.2 that c 3 = 0. Furthermore, since c 1 , c 2 , d 1 and d 2 are non-negative, Lemma 5.3 yields that also c 1 = c 2 = 0. Hence, Z 0 ≡ 0 and Z is translation invariant. Moreover, Z(s χ K ) = {0} for every s > 0 and K ∈ K n . Thus, Lemma 5.4 shows that Z is trivial.
Lemma 5.6. For a, b ∈ R and q > 0 the following holds:
lim h→0 + a 1 − e −qh h 2 − b e −qh h = +∞ if b < q a q 2 b if b = q a −∞ if b > q a.
Proof. Since,
a 1 − e −qh h 2 − b e −qh h = a (1 − e −qh ) − b h e −qh h 2 , and lim h→0 + a (1 − e −qh ) − b h e −qh = 0,
we can apply L'Hospital's rule to obtain
lim h→0 + a (1 − e −qh ) − b he −qh h 2 = lim h→0 + q a e −q h − b e −qh + q b he −qh 2h = lim h→0 + e −qh 2h (q a − b) + q 2 b.
The claim now follows since e −qh 2h → +∞ as h → 0 + . for every s ∈ R. Thus, denoting Y(u) = Z(e −u ) for u ∈ Conv(R n ), this gives Y(u h ) + Y(ℓ T 1/h • τ −1 e 1 + h) = Y(ℓ T 1/h ) + Y(ℓ conv{0,e 2 ,...,en} • τ −1 e 1 + h). Conversely, let Z : LC(R n ) → R be a continuous, equi-affinely covariant Minkowski valuation. For arbitrary K ∈ K n and s > 0, let f = s χ K . By Lemma 5.2, there exist constants c 1 , c 2 ≥ 0 and c 3 , q ∈ R such that Z(f ) = s q (c 1 K + c 2 (−K) + c 3 m(K)) and by Lemma 5.5 we may assume that q > 0. Since
Lemma 1.1 ([12], Lemma 2.3). For λ > 0 and T λ = conv{0, λ e 1 , e 2 , . . . , e n },
Theorem 1. 2
2([13], Theorem 6). For n ≥ 3, a map Z : K n o → K n is a continuous, SL(n) covariant Minkowski valuation if and only if there exist constants c 1 , c 2 , c 4 ≥ 0 and c 3 ∈ R such that Z K = c 1 K + c 2 (−K) + c 3 m(K) + c 4 M(K), for every K ∈ K n o .
Corollary 1. 3 .
3For n ≥ 3, a map Z : K n → K n is a continuous, SL(n) and translation covariant Minkowski valuation if and only if there exist constants c
f
∨ g = e −(u∧g) and f ∧ g = e −(u∨g) , the map Z is a valuation if and only if Y : Conv(R n ) → A, + is a valuation, where Y(u) = Z(e −u ), for every u ∈ Conv(R n ). Hence, studying valuations on LC(R n ) is equivalent to studying valuations on Conv(R n ) and it will be convenient for us to switch between these points of view. By the definition of hypo-convergence on LC(R n ), the valuation Z is continuous if and only if Y is continuous. Furthermore, for x ∈ R n we have f • τ −1 x = e −u•τ −1 x . Hence, Z is translation invariant if and only if Y is translation invariant. Similarly, translation covariance, SL(n) invariance and SL(n) covariance are equivalent for valuations on LC(R n ) and their counterparts on Conv(R n ).
e −r dr = λ q n .
Lemma 4. 2 .
2For every f ∈ LC(R n ) and z ∈ S n−1 , +∞ 0 |h(m({f ≥ t}), z)| dt < +∞.
Lemma 4. 3 .
3For every q > 0, the map
qh Y(ℓ [0,e 1 /h] ) + (c 1 − c 2 )e −qh e 1 and furthermore
Lemma 5 . 4 .
54Every continuous, SL(n) covariant and translation invariant Minkowski valuation Z : LC(R n ) → K n is uniquely determined by the values Z(s χ K ) with s > 0 and K ∈ K n . Lemma 5.5. Every continuous, equi-affinely covariant Minkowski valuation Z : LC(R n ) → K n is either homogeneous of a positive degree or trivial.
h.
ℓ T 1/h • τ −1 e 1 + h) = e −qh Y(ℓ T 1/h ) + e −qh ((c 1 − c 2 ) + c 3 h q n )e 1 Y(ℓ conv{0,e 2 ,...,en} • τ −1 e 1 + h) = e −qh Y(ℓ conv{0,e 2 ,...,en} ) + e −qh (c 1 − c 2 )e 1 .Furthermore, using Lemma 1.1 we obtain for the support functionsh(Y(ℓ T 1/h ), e 1 ) = d 1 h + d 3 +d 4 h 2 (n+1)! , h(Y(ℓ T 1/h • τ −1 e 1 + h), e 1 ) = e −qh d 1 h + d 3 +d 4 h 2 (n+1)! + (c 1 − c 2 ) + c 3 h q n , h(Y(ℓ conv{0,e 2 ,...,en} ), e 1 ) = e −qh (c 1 − c 2 ). Observe, that for h → 0 + we have u h epi −→ v. Hence, by the continuity of Y and ((1 − e −qh ) + d 3 +d 4 h 2 (n+1)! (1 − e −qh ) − c 3 h q n e −qhSince this expression must be finite, it follows from Lemma 5.6 thatc 3 q n = q d 3 +d 4 (n+1)! .Similarly, repeating the calculations above but evaluating the support functions at −e 1 givesc 3 q n = q d 3 −d 4 (n+1)! .Hence, d 4 = 0 and c 3 = q n+1 (n+1)! d 3 .By Lemma 1.8, every continuous, equi-affinely covariant Minkowski valuation Z on LC(R n ) is uniquely determined by the constants c 1 , c 2 , c 3 , d 1 , d 2 , d 3 , d 4 and q from Lemma 5.2. By Lemmas 5.3 & 5.7 we have d1 = c 1 q , d 2 = c 2 q , d 3 = (n+1)! q n+1 c 3 and d 4 = 0.Hence, Z is completely determined by the constants c 1 , c 2 , c 3 and q. Thus, we have the following result.
Lemma 5. 8 .f
8Every continuous, equi-affinely covariant Minkowski valuation Z : LC(R n ) → K n is uniquely determined by the values Z(s χ K ) with s > 0 and K ∈ K n . → c 1 [f q ] + c 2 (−[f q ]) + c 3 m(f q ), defines a continuous, equi-affinely covariant Minkowski valuation on LC(R n ) for every c 1 , c 2 ≥ 0, c 3 ∈ R and q > 0.
{s q χ K ≥ t}, z) dt = s q h(K, z) h(m(f q ), z) = R n s q χ K (x)(x · z) dx = s q h(m(K), z) we have Z(f ) = c 1 [f q ] + c 2 (−[f q ]) + c 3 m(f q ). Thus, Lemma 5.8 completes the proof of the theorem.
Lemma 5.3. Let Z : LC(R n ) → K n be a continuous, equi-affinely covariant Minkowski valuation. If c 1 , c 2 , d 1 , d 2 , q denote the constants from Lemma 5.2, then c 1 = q d 1 and c 2 = q d 2 . Proof. For h > 0 let u h ∈ Conv(R n ) be defined via epi u h = epi ℓ [0,e 1 /h] ∩{x 1 ≤ 1}. Lemma 1.5 shows that u h epi −→ I [0,e 1 ] as h → 0. Moreover, for ℓ h := ℓ [0,e 1 /h] • τ −1 e 1 + h we have
Lemma 5.7. Let Z : LC(R n ) → K n be a continuous, equi-affinely covariant Minkowski valuation. If c 3 , d 3 , d 4 , q denote the constants from Lemma 5.2, then c 3 = q n+1 (n+1)! d 3 and d 4 = 0. Proof. By Lemma 5.5, we can assume without loss of generality that q > 0. Define v ∈ Conv(R n ) via{v < 0} = ∅, {v ≤ s} = [0, e 1 ] + conv{0, s e 2 , . . . , s e n }, for every s ≥ 0. Now, for h > 0 let T 1/h be defined as in Lemmas 1.1 & 2.1 and define the function u h via{u h ≤ s} = {ℓ T 1/h ≤ s} ∩ {x 1 ≤ 1},for every s ∈ R. It is easy to see that u h ∈ Conv(R n ) and furthermore,{u h ≤ s} ∪ {ℓ T 1/h • τ −1 e 1 + h ≤ s} = {ℓ T 1/h ≤ s} {u h ≤ s} ∩ {ℓ T 1/h • τ −1 e 1 + h ≤ s} = {ℓ conv{0,e 2 ,...,en} • τ −1 e 1 + h ≤ s},
AcknowledgmentsThe author was supported, in part, by Austrian Science Fund (FWF) Project P25515-N25.
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Austria e-mail: [email protected]. Wien, Wien, Austria e-mail: [email protected]
|
[] |
[
"Software Reuse in Cardiology Related Medical Database Using K-Means Clustering Technique",
"Software Reuse in Cardiology Related Medical Database Using K-Means Clustering Technique"
] |
[
"M Bhanu Sridhar [email protected] \nDepartment of Computer Science and Engineering\nRaghu Engineering College\nVisakhapatnamIndia\n",
"Y Srinivas [email protected] \nDepartment of Information Technology\nGITAM University\nVisakhapatnamIndia\n",
"M H M Krishna Prasad \nDepartment of Information Technology\nJawaharlal Nehru Technological University\nKakinadaIndia\n"
] |
[
"Department of Computer Science and Engineering\nRaghu Engineering College\nVisakhapatnamIndia",
"Department of Information Technology\nGITAM University\nVisakhapatnamIndia",
"Department of Information Technology\nJawaharlal Nehru Technological University\nKakinadaIndia"
] |
[
"Journal of Software Engineering and Applications"
] |
Software technology based on reuse is identified as a process of designing software for the reuse purpose. The software reuse is a process in which the existing software is used to build new software. A metric is a quantitative indicator of an attribute of an item/thing. Reusability is the likelihood for a segment of source code that can be used again to add new functionalities with slight or no modification. A lot of research has been projected using reusability in reducing code, domain, requirements, design etc., but very little work is reported using software reuse in medical domain. An attempt is made to bridge the gap in this direction, using the concepts of clustering and classifying the data based on the distance measures. In this paper cardiologic database is considered for study. The developed model will be useful for Doctors or Para-medics to find out the patient's level in the cardiologic disease, deduce the medicines required in seconds and propose them to the patient. In order to measure the reusability K-means clustering algorithm is used.
|
10.4236/jsea.2012.59081
|
[
"https://arxiv.org/pdf/1311.1197v1.pdf"
] | 5,289,360 |
1311.1197
|
27a17a2447dc159f284969b62886a0fba68c2ecc
|
Software Reuse in Cardiology Related Medical Database Using K-Means Clustering Technique
2012
M Bhanu Sridhar [email protected]
Department of Computer Science and Engineering
Raghu Engineering College
VisakhapatnamIndia
Y Srinivas [email protected]
Department of Information Technology
GITAM University
VisakhapatnamIndia
M H M Krishna Prasad
Department of Information Technology
Jawaharlal Nehru Technological University
KakinadaIndia
Software Reuse in Cardiology Related Medical Database Using K-Means Clustering Technique
Journal of Software Engineering and Applications
5201210.4236/jsea.2012.59081Received May 1 st , 2012; revised June 6 th , 2012; accepted June 25 th , 2012ReuseCardiologySoftware MetricsClustering, K-MeansCardiac
Software technology based on reuse is identified as a process of designing software for the reuse purpose. The software reuse is a process in which the existing software is used to build new software. A metric is a quantitative indicator of an attribute of an item/thing. Reusability is the likelihood for a segment of source code that can be used again to add new functionalities with slight or no modification. A lot of research has been projected using reusability in reducing code, domain, requirements, design etc., but very little work is reported using software reuse in medical domain. An attempt is made to bridge the gap in this direction, using the concepts of clustering and classifying the data based on the distance measures. In this paper cardiologic database is considered for study. The developed model will be useful for Doctors or Para-medics to find out the patient's level in the cardiologic disease, deduce the medicines required in seconds and propose them to the patient. In order to measure the reusability K-means clustering algorithm is used.
Introduction
Software Reuse is currently one of the most active and creative research areas in Software Engineering. It offers a solution to reduce repeated work and improve efficiency and quality in software development and management. It makes use of the experience obtained in the past development process. In the proposed article we have considered the database of the heart patients from [1] to focus on the cardiologic situations. Reuse is vital in medical field because the previous information is very handy in deducing a patient's current health position and save the precious life [2].
Cardiology is a medical specialty dealing with human heart disorders. This field includes diagnosis and treatment of disorders like heart defects, heart failure and other heart diseases. According to World Health Organization, India has the highest number of coronary heart disease deaths in the world [3]. This can be deduced not only due to lack of resources but also due to concentration of resources at places like cities and towns. By usage of Internet and cardiology database component reuse, the Para-medics, can deduce the medicines or methods to be used for the patients at remote places to temporarily put them out of danger. From the reuse of available data, the required medicines may also be deduced and proposed to the patients.
In this article we propose a methodology using the clustering technique together with classification technique where the heart patients' data is clustered, depending on the health conditions, into three categories: normal, pro-cardiac and cardiac. We use the Euclidean distance measure to classify the patients' disease level conditions into the three specified categories. The paper is organized as follows: Section 1 of the paper deals with introduction; in Section 2, categories of the heart patients is presented; K-means algorithm is presented in Section 3; Section 4 deals with the methodologies together with experimental results and finally the conclusion is presented in Section 5.
Our future work, which is at a research stage now would be very useful in aiding to the ailing patients and become an important part in the general usage of the Doctors.
Categories of Heart Patients
The heart is a myogenic (cell-related) muscular organ with a circulatory system (including all vertebrates), that is responsible for pumping blood throughout the blood vessels by repeated, rhythmic contradictions [4]. Among the problems related to heart, the major problem is car-683 diac arrest, which is the cessation of normal blood circulation due to failure of the heart to contract effectively. It should be effectively realized that cardiac arrest is different from a heart attack where blood supply is interrupted to a part of the heart which may/may not lead to the patient's death.
The patients who approach a doctor can be classified into three categories taking into consideration results of different tests conducted with the existing symptoms. The properties taken into consideration are Atherosclerosis (due to Cholesterol), Myocardial Infarction (heart attack), different medical signs like blood cell count and skin rashness, various symptoms like head ache and body pain, and other facts like Diabetes, Triglyceride, Migraine and so on [5].
Armed with all this information, the concerned patient is placed in one of the quoted three categories below.
Normal
A patient can be declared "normal" when no signs or symptoms of a cardiovascular/coronary disease are found within the results of various tests conducted. The general factors considered are the blood pressure (BP), sugar level in blood, results of Electrocardiography (ECG), Cholesterol level, Triglyceride, and other sensations. A normal patient should have the BP within control (<120/80 mmHg), Blood sugar level on waking up with an empty stomach between 80 to 120 mg/dl [6], normal output from the cardiac stress test conducted with the ECG [7], and no other notable problems. It should also be noted that a now normal patient might suffer from a heart stroke soon or later since he had inherited the problem, of which the reports wouldn't mention.
Pro-Cardiac
Pro-cardiac category keeps the account of those patients who are suspected to have some signs and/or symptoms of heart-problems. These can be observed from the BP tests slightly exceeding the normal levels, sugar levels in blood also rising, ECG suspecting (though not deducing) problems in future and some signs and symptoms like light chest pain, high cholesterol, severe head aches often turning up etc. do surface.
A pro-cardiac becomes a suspect of cardiac problems in near future and is advised by the Doctor not only to take a bit of medicines but also to consider doing regular exercises like light running, and other methods to bring his yet-controllable level to the normal state.
Cardiac
As it might be suspected, a cardiac is surely in the problematic range: prone to abnormal BP conditions, having severe pain the chest region, burning sensations, sweat-ing, pain along the left arm and finally having already had a light heart attack. A cardiac must be immediately taken into consideration for regular treatment with constant observation of all concerned positions in and around the heart and those that affect the heart. A cardiac is also advised to taken high-power medicines and conduct long walks every day so as to keep the bloodpumping in the heart at a normal position.
After mentioning and discussing all the classification parts, it should also be noted that effective medical data of the patient should be readily available for the Doctors which also should be frequently updated. This data forms the backbone of the patient's classification level, severity level and the chance of saving his/her life. An attempt is made in this paper, by bringing into picture the reuse of data, to correctly judge the patient's position.
K-Means Clustering Algorithm
Clustering in data mining is the process of grouping a set of objects into classes of similar objects [8]. Many clustering algorithms are discussed in the literature and the most important of these are partitioning and hierarchical algorithms. K-means remains one of the most popular clustering algorithms used in practice [9]. The main reasons are it is simple to implement, fairly efficient, results are easy to interpret and it can work under a variety of conditions. The steps to be followed for effective clustering using K-means algorithm are:
Step 1. Begin with a decision on the value of K = number of segments
Step 2. Put any initial partition that classifies the data into K segments. We can arrange the training samples randomly, or systematically as follows:
1) Take the first K training samples as a single-element Segment.
2) Assign each of the remaining (N-K) training samples to the segment with the nearest centroid. Let there be exactly K segments (C1, C2-CK) and n patterns to be classified such that, each pattern is classified into exactly one segment. After each assignment, re-compute the centroid of the gaining segment.
Step 3. Take each sample in sequence and compute its distance from the centroid of each of the segments. If the sample is not currently in the cluster with the closest centroid switch this sample to that segment and update the centroid of the segment gaining the new sample and cluster losing the sample.
Step 4. Repeat step 3 until convergence is achieved, that is until a pass through the training sample causes no new assignments. After determining the final value of the K (number of regions) we obtain the estimates the parameters μ i , σ i and α i for the i th region using the segmented regions.
Methodology and Experimental Results
In this article a novel methodology for cardiac medical data reusability is proposed. A database from archives [11] is considered for carrying out our proposed work. In this method, we have first categorised the data into 3 groups namely, normal, pro-cardiac and cardiac. We have considered the scenario of Chintapalli, a remote tribal village in Andhra Pradesh, India, where no superspeciality services for treating cardiac patients are available. It is necessary in such conditions to supplement the patient with sufficient primary aid so that he can sustain for the minimum period of shifting. Depending upon the clinical reports of the patient's data, he is to be categorized into one of the levels presented in Sections 2.1-2.3. A dissimilarity matrix is constructed with the readings from the clinical observations and identifying the most leading factors that may be prone to the cardiac diseases as per the experts' references. The various readings considered are categorized into the above mentioned three groups and a database is formulated from the realistic data obtained from medical patients from the data referred in [10]. The predominant features considered in the database are: blood pressure (BP), heartbeat (HB), pulse rate (PR), ECG (normal/abnormal), pain in the left shoulder region, sweating, nausea/vomiting, over weight, chest pain and breathlessness.
For the testing purpose in this paper, we have used a database of ten patients with the above mentioned ten features; if the reading is present we have represented it by using a value 1 else 0 (binary). Following this procedure for the other inputs, a binary matrix [11] is obtained and this matrix is to be categorized; K-Means algorithm is utilized for the same. Now within the clusters, the homogenous data is obtained. To classify a patient, the dissimilarity matrix is again formulated and is classified by calculating the minimum distance between the posed query data and the retrieved data by using the clustering technique.
Reuse Metrics
The reuse components for partitioning the data are divided into 4 steps performed at each phase in preparation to the next phase. These steps are: 1) Developing a reuse plan or strategy after studying the problem and available solutions to the problem.
2) Identifying a solution structure for the problem following the reuse plan or strategy.
3) Reconfiguring the solution structure to improve the possibility of using predefined components available at the next phase. 4) Evaluating the system. The major tasks under the first step are to understand the problem about the cardiac patients, build-up the knowledge for categorizing them into groups and de-velop a plan or strategy for their treatment. In the second step, apply the knowledge to develop a solution structure that is best suited for the problem following the reuse plan or strategy developed in the above phase. In the next step, reconfigure the solution in order to optimize the reuse both at both the current phase and next phase. Finally the computed components are to be classified using test features.
The data of 10 patients, from the archives [10] is converted into a binary matrix as above. The concepts in the clustering partition in reusable components [8] are utilized to construct a Java program that takes in the data from the Table 1. The program constructs the clusters by classifying the data using the Euclidean distance. After the K-Means clustering, the data is divided based on the binary clustering, into three groups. The patients with Ids (P4, P7, P3, P9, P10) belong to the first cluster, patients with Ids (P8, P2, P1) belong to the second cluster and patients with Ids (P5, P6, P10) belong to the third cluster.
The basic aim in this context is to assist the patients with minimum first aid for sustainability till he/she is shifted to the nearest multi-speciality clinic from the remote place Chintapalli considered here. In order to categorize the patients, it is necessary to identify the exactness of the category and thereby suggesting the minimum essential supportive drugs to maintain or better the current condition. It becomes clear by now that it is necessary to find the exactness of the disease if we are to achieve our goals.
To find the most exact solution in this concept, an auto-correlation model is used to find the exact correlation and categorization of the patients. The auto-correlation formula used here is given by
1 2 1 n t tk t k k n t t Y Y Y Y r Y Y
where t is the patient with the first symptom, K + 1 is the patient with the second symptom and so on.
In this model, we try to correlate the data to each patient by considering the auto-correlation model and the results obtained are tabulated (Figure 1).
From the above considered data, it can be clearly seen that the patient with R 6 is having highest auto-correlation factor and is likely to have symptoms of a cardiac. The value obtained here is 0.9. The patient with Ids P5 and P6 i.e. R5 and R6 have the next immediate ranges and they are also likely to be cardiac-prone. The values obtained by using the above quoted autocorrelation formula are given under: Here R6 is maximum, which specifies that the person is more likely to belong to the category cardiac; R1, R3, R4, R7, R8, R9 are at minimum risk and they belong to normal case and R2, R5 belong to the category pro-cardiac.
R1
We have also tried to estimate the significance of each symptom for each patient over the other symptoms using auto-correlation and could identify the symptom that would be leading to cardiac problems.
We now input a new patient's data to check out the cluster where it belongs to; the Java program promptly supplies us the answer. The output of the Java program is given in Figure 2.
From the screenshot Figure 2, it can be easily identified that the given test data belongs to a particular cluster. Utilizing the classification given in Section 2, we obtain the concerned category.
Conclusion
In this paper a new methodology for software reuse in cardiac domain is presented. A database is considered or generated with 10 patients and is categorised into 3 categories depending upon the health conditions. The readings for these categories are obtained from the super speciality doctors, and are used for checking the reus- ability. The dissimilarity matrix is generated and the clustering is performed on the binary data. Classification is carried out on the test data by finding the minimum distance using Euclidean distance, and the reusability for partitioning is carried out as prescribed by Boris Delibasic et al. [8] are presented in Section 4.1.
The results obtained from the K-Means algorithm are given as inputs to the auto-correlation model to categorize the patients more accurately to be declared a cardiac.The model developed will be immensely useful for the Doctors to prescribe the medicines used for the previous patients of the respective cluster to the new patient immediately without spending time in checking conditions. It may be much more valuable for the Para-medics at remote places who can save the life of the patient.
Figure 1 .
1= 0.3, R2 = 0.3, R3 = 0.1, R4 = 0.0023, R5 = 0.7, 6 = 0.9, R7 = 0.11, R8 = 0.3, R9 = 0.1, R10 = 0.72 R The results obtained from Autocorelation.
Figure 2 .
2The results of classification.
Table 1 .
1The Symptoms (→) of the patients.Patient ID
(↓)
BP
Heart beat
(HB)
Pulse Rate
(PR)
ECG
Left
Shoulder pain
Sweating Vomiting Over Weight Chest Pain Breathlessness
Software Reuse in Cardiology Related Medical Database Using K-Means Clustering Technique
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|
[] |
[
"Robust Joint Precoder and Equalizer Design in MIMO Communication Systems",
"Robust Joint Precoder and Equalizer Design in MIMO Communication Systems"
] |
[
"Saeed Kaviani [email protected] \nUniversity of Alberta and TRLabs\nT6G 2V4EdmontonAlbertaCanada\n",
"Witold A Krzymień \nUniversity of Alberta and TRLabs\nT6G 2V4EdmontonAlbertaCanada\n"
] |
[
"University of Alberta and TRLabs\nT6G 2V4EdmontonAlbertaCanada",
"University of Alberta and TRLabs\nT6G 2V4EdmontonAlbertaCanada"
] |
[] |
We address joint design of robust precoder and equalizer in a MIMO communication system using the minimization of weighted sum of mean square errors. In addition to imperfect knowledge of channel state information, we also account for inaccurate awareness of interference plus noise covariance matrix and power shaping matrix. We follow the worst-case model for imperfect knowledge of these matrices. First, we derive the worst-case values of these matrices. Then, we transform the joint precoder and equalizer optimization problem into a convex scalar optimization problem. Further, the solution to this problem will be simplified to a depressed quartic equation, the closed-form expressions for roots of which are known. Finally, we propose an iterative algorithm to obtain the worst-case robust transceivers.
|
10.1109/wcnc.2012.6214273
|
[
"https://arxiv.org/pdf/1304.4624v1.pdf"
] | 14,618,604 |
1304.4624
|
a56eb600181c7d649f44f9189a7866ed9d7a1f63
|
Robust Joint Precoder and Equalizer Design in MIMO Communication Systems
16 Apr 2013
Saeed Kaviani [email protected]
University of Alberta and TRLabs
T6G 2V4EdmontonAlbertaCanada
Witold A Krzymień
University of Alberta and TRLabs
T6G 2V4EdmontonAlbertaCanada
Robust Joint Precoder and Equalizer Design in MIMO Communication Systems
16 Apr 2013arXiv:1304.4624v1 [cs.IT]
We address joint design of robust precoder and equalizer in a MIMO communication system using the minimization of weighted sum of mean square errors. In addition to imperfect knowledge of channel state information, we also account for inaccurate awareness of interference plus noise covariance matrix and power shaping matrix. We follow the worst-case model for imperfect knowledge of these matrices. First, we derive the worst-case values of these matrices. Then, we transform the joint precoder and equalizer optimization problem into a convex scalar optimization problem. Further, the solution to this problem will be simplified to a depressed quartic equation, the closed-form expressions for roots of which are known. Finally, we propose an iterative algorithm to obtain the worst-case robust transceivers.
I. INTRODUCTION
Deployment of multiple antennas promises significant capacity gains in wireless systems [1]- [4], which motivates construction of pragmatic signalling strategies exploiting these gains. Nevertheless, these improvements are severely degraded by interference [5]. Therefore, combination of precoding at the transmitter and equalization at the receiver is often employed to reduce the interference and maximize the performance of system. In this setup, linear strategies have attracted more attention due to their simplicity and robustness. Although various multiple-input multiple-output (MIMO) linear precoding and equalization methods have been proposed [6]- [9], they assume that the channel state information (CSI) and interference plus noise covariance matrix are perfectly known at the transmitter and receiver. Unavailability of exact information of channel matrices may diminish the performance of the transceivers significantly. This motivates studying of the robust linear transmitter and/or receiver design problem [10]- [21].
There are normally two philosophies to consider imperfect CSI and to design robust transmission strategies: Worst-case deterministic scenario and stochastic scenario. In stochastic scenario, the CSI errors are modeled probabilistically and the average performance is optimized [20]- [23]. In this paper, we consider the worst-case scenario since it can characterize the instantaneous imperfect knowledge of system matrices. In this approach, the actual system matrices are assumed to lie within a so-called uncertainty region around the estimated values known by the transmitter. A worst-case robust design is a design which achieves a particular performance level for any channel realization staying in the corresponding uncertainty region [10]- [19], [24].
Worst-case robust transceiver design has been recently considered in [12], [15]. In [12], joint optimization of transceivers is addressed using semi-definite program (SDP) reformulation. However, even for the case of perfect channel knowledge it only gives a suboptimal solution [25]. Moreover, SDP-based approaches (see for example [10], [11]) do not give a closedform solution and the resultant algorithms require solving a SDP at each iteration. Closed-form solution for the worst-case robust MMSE precoder assuming pre-fixed equalizer is given in [13] and it is extended to design of worst-case robust MMSE transceivers in [15]. Nevertheless, the proposed algorithm is based on alternative optimization between precoder and equalizer (rather than joint optimization). It also involves solving a quintic equation, for which a closed-form solution is unknown and it is only solved numerically.
Conventionally, the robust transmission strategies in the MIMO communication systems consider imperfect knowledge of channel gains between the transmit and receive antennas. Here, we consider a wider range of system parameters all known inaccurately to the system. In addition to CSI, we consider imperfect knowledge of interference plus noise covariance matrix, and power shaping matrix. Moreover, our objective is weighted sum of mean square error minimization which is a more general performance metric [26]. We avoid SDP reformulation to solve the problem and consequently the proposed algorithm has lower complexity. First, we obtain the worst-case system matrices within the uncertainty region. Then, we jointly optimize precoder and equalizer where the resultant matrix-valued optimization problem is reduced to a scalar convex problem. Further, the solution to this problem can be simplified to a depressed quartic equation. Interestingly, the solution of a quartic equation can be expressed in terms of radicals. Hence, a closed-form expression for the precoder and equalizer matrices can be obtained. Finally, we propose an iterative algorithm to find the optimal transceivers. Note that the analysis and design approach for point-to-point MIMO systems presented in this paper can be extended to multiuser MIMO systems [27].
Notation: We denote positive semi-definite matrices as A 0. Capital bold letters represent matrices and small bold letters represent vectors. We denote conjugate transpose (Hermitian) operator with (·) H . A − 1 2 represents the inverse square root of positive definite matrix A. tr {·} is matrix trace operator and · demonstrates Frobenius norm of a matrix. λ max (·) is reserved to denote the maximum eigenvalue of a matrix.
II. SYSTEM MODEL AND PRELIMINARIES
We consider a MIMO communication system, where the transmitter is equipped with n t antennas and the receiver employs n r antennas (n r ≤ n t ). The transmitter broadcasts a data vector denoted by u ∈ C nr using the linear precoding matrix F ∈ C nt×nr . The channel between the transmitter and the receiver is characterized by the matrix H ∈ C nr ×nt . The receiver observes the signal
y = HFu + n,(1)
where n represents the correlated interference plus noise vector. We denote the interference plus noise covariance matrix as Ω = E nn H ∈ C nr ×nr . The linear processing at the receiver can be characterized by the equalizer matrix G ∈ C nr ×nr . Hence, the estimated symbol vector at the receiver can be described asû = Gy.
Let us define the estimation error covariance matrix as
E =E (û − u) (û − u) H =GHFF H H H G H − GHF − F H H H G H + GΩG H + I.(3)
which is referred to as mean square error (MSE)-matrix [8].
We are specifically interested in the following problem:
minimize G,F tr {WE} subject to tr ΦFF H ≤ P(4)
where the optimization is over precoding and equalization matrices with given diagonal weight matrices W ∈ C nr ×nr where the main diagonal of W is denoted by [w 1 , ..., w nr ] with non-negative weights w j ≥ 0. Since the diagonal elements of the MSE-matrix are MSE values of the estimated symbol vector, the problem (4) is often called weighted sum of mean square error minimization (WMMSE) problem (see [25], [26], [28], [29] for details). We also account for a linear power constraint tr ΦFF H ≤ P and specifically refer to the weight matrix Φ in the power constraint as the power shaping matrix. This matrix also can characterize the direction, in which the transmitted power can propagate while reducing the interference in other directions (e.g. to other users in a multiuser case). Additionally, we assume that the matrix Φ is full rank and square with size of n t . This assumption is a practical constraint due to the fact that if Φ is a rank deficient matrix then one can always transmit infinite power in one direction without violating the power constraint. Notice that when Φ = I, the sum power constraint emerges. We refer to the matrices H, Φ, and Ω as system matrices.
It is shown that any performance metric characterized by some particular function f (E) of the MSE-matrix E, can be approximated using the problem (4) [6], [25], [28]. The approach is that at each iteration, we select W = ∇ E f (Ē) T at the operating point E, then solve the optimization problem (4) 1 . The algorithm iterates until the convergence is achieved. For example, to adopt sum rate maximization one can select W =Ē −1 at each iteration. This approach has been extensively used in [6], [25], [26], [28], [30] to optimize any performance function of the MSE-matrices (e.g. sum-rate), since the resultant problem becomes convex by fixing any optimization variable.
III. PERFECT KNOWLEDGE OF SYSTEM MATRICES
We begin with the case of perfect channel knowledge which will be employed for the robust design. [26], [28] have considered WMMSE problem with perfect CSI, but their solutions for each of the precoding and equalization matrices are interdependent (requires alternative optimization). Here, we give an extension of the results in [6], [8] when power shaping matrix is present. Detailed discussion of this problem with more general constraints is available in [25]. Here, a special case of this result where the number of transmitted data streams is equal to n r is given.
Lemma 1 [25]: For any channel matrix H and given the full rank and square matrices Φ and Ω, the optimum precoding and equalization matrices of the problem (4) have the following structure
F =Φ − 1 2 VΣ,(5)G =ΛU H Ω − 1 2 .(6)
where Σ and Λ are diagonal matrices with the diagonal elements σ i ≥ 0 and λ i ≥ 0, i = 1, . . . , n r , respectively. U ∈ C nr×nr and V ∈ C nt×nr are obtained by performing the singular value decomposition (SVD) of the following matrix
Ω − 1 2 HΦ − 1 2 = U [Γ 0 nr×nt−nr ] VV H ,(7)
in which Γ contains its n r nonzero eigenvalues γ 1 ≥ . . . ≥ γ nr andV ∈ C nt×(nt−nr ) contains the right singular vectors corresponding to the zero eigenvalues 2 .
Proof: The proof can be found in [27].
IV. IMPERFECT KNOWLEDGE OF SYSTEM MATRICES
In this case, only estimated matrices H and Ω and Φ, are available at the transmitter and receiver. Therefore, the actual value of these matrices can be described as a sum of the estimated matrices and the error matrices:
H = H + ∆ H ,(8)Ω = Ω + ∆ Ω ,(9)Φ = Φ + ∆ Φ .(10)
We are interested in the joint optimization of the precoder and equalizer, while the unknown actual system matrices are guaranteed to fit in the (norm-based) uncertainty region. Hence, the error region can be described as
U = (∆ H , ∆ Ω , ∆ Φ ) : ∆ H ≤ ε H , ∆ Ω ≤ ε Ω , Ω + ∆ Ω 0, ∆ Φ ≤ ε Φ , Φ + ∆ Φ 0 . (11)
Consequently, the worst-case transceiver design can be expressed as
minimize F,G max (∆H ,∆Φ,∆Ω)∈U tr {WE} subject to tr ΦFF H ≤ P.(12)
Please note that we consider a case, in which the uncertainty of the system matrices is the same at the transmitter and the receiver. We leave the scenario, under which the uncertainty of the system matrices at the transmitter is much higher than at the receiver to future work.
A. Finding Least Favorable System Matrices
We proceed by finding the worst-case estimation errors for the system matrices. First, we expand the objective function of (12) in terms of the estimated and error system matrices using the definitions (3) and (8)-(10)
B =FF H ,
C =FF H H H G H WG − FWG.(15)
Least Favorable Matrices ∆ Ω and ∆ Φ : Since the error matrices are independent of each other, the least favorable interference plus noise covariance matrix can be obtained from the problem
maximize ∆Ω ≤εΩ tr G H WG∆ Ω subject to Ω + ∆ Ω 0.(17)
We assume that ε Ω is small enough to ignore the positive semi-definite condition of the problem (17). Nevertheless, we will see that this relaxation gives us a solution, which also satisfies a positive semi-definite condition. Using Cauchy-Schwartz inequality, we can obtain
tr G H WG∆ Ω ≤ G H WG · ∆ Ω ≤ ε Ω G H WG(18)
and the upper bound occurs when
∆ ⋆ Ω = ε Ω G H WG G H WG .(19)
which gives the worst-case estimation error matrix ∆ ⋆ Ω .
We continue by finding the worst-case estimation error of the interference direction matrix, i.e. ∆ Φ . It is trivial that the worst-case happens when the maximum allowed power is minimized. Consequently, we are interested in this optimization problem: maximize
∆Φ ≤εΦ tr ∆ Φ FF H subject to Φ + ∆ Φ 0(20)
Similarly to the problem (17), we can obtain the worst-case error matrix as
∆ ⋆ Φ = ε Φ FF H FF H .(21)
Substituting these worst-case estimation errors ∆ ⋆ Φ and ∆ ⋆ Ω into the problem (12) results in the terms ε Ω G H WG and ε Φ FF H . Since we are interested in the worst-case scenario, we can use upper bounds of these terms alternatively. Hence, we can write inequalities
ε Ω G H WG ≤ε Ω W 1 2 G 2 = ε Ω tr G H WG (22) ε Φ FF H ≤ε Φ F 2 = ε Φ tr FF H .(23)
where we have used the inequality XY ≤ X · Y which can be proved by utilizing the Cauchy-Schwartz inequality [31]. Since the Frobenius norm is invariant under the Hermitian operation, we get XX H ≤ X 2 . Note that the approximations given in (22) and (23) give an upper bound of the objective function of (12). By minimizing this upper bound, we also minimize the objective function. Now, we replace the terms tr G H WG∆ Ω and tr FF H ∆ Φ in the robust transceiver problem (13) with the upper bounds defined in (22) and (23) respectively. This is equivalent to setting
Ω ⋆ = Ω + ε Ω I,(24)Φ ⋆ = Φ + ε Φ I.(25)
With these fixed worst-case matrices, the robust transceiver design problem reduces to
minimize F,G max ∆H ≤εH tr {WE} subject to tr ΦFF H ≤ P Ω = Ω + ε Ω I, Φ = Φ + ε Φ I.(26)
Least Favorable Channel Error Matrix ∆ H : Now, we find the worst-case channel estimation error ∆ H following the maximization problem for given Ω and Φ: Lemma 2: The least favorable channel estimation error for any given Φ and Ω has the following structure:
∆ ⋆ H = Ω 1 2 U ∆ V H Φ 1 2 ,(28)
where U ∈ C nr×nr and V ∈ C nr×nt are defined in the SVD
Ω − 1 2 HΦ − 1 2 = U Γ 0 nr ×nt−nr VV H ,(29)
and ∆ ∈ R nr×nr is a diagonal matrix with elementsδ i ≥ 0.
Proof: Due to limited space of this paper the detailed proof is included in [27]. In order to explain our proposed transceiver optimization algorithm, here we only summarize the approach and results. The problem (27) can be categorized as a trust-region subproblem [32], [33]. The matrix-form restatement of this problem is given in [13]. It has been shown that the solution to this problem can be found by a minimization problem over an auxiliary variable ϑ ≥ λ max (A)λ max (B) [32], [33]. The worst-case channel matrices coincides with the structure of the precoding and equalization matrices given in (5) and (6) using the worst-case interference plus noise and power shaping matrices defined in (24) and (25). As a result, δ i s are given bỹ
δ i = w i λ i σ i (γ i λ i σ i − 1) ϑ − w i λ 2 i σ 2 i , i = 1, . . . , n r .(30)
Note that γ i , i = 1, . . . , n r are the diagonal elements of Γ in (29).
Recognizing j = argmax i w i λ 2 i σ 2 i , if ϑ > w j λ 2 j σ 2 j , then ϑ is the root of equation nr i=1 w 2 i λ 2 i σ 2 i (γ i λ i σ i − 1) 2 (ϑ − w i λ 2 i σ 2 i ) 2 = ε 2 H ,(31)
where
ε H = εH Ω 1 2 · Φ 1 2
. If ϑ = w j λ 2 j σ 2 j ,δ j cannot be found from equation (30). Let
ρ(ϑ) = i =j w 2 i λ 2 i σ 2 i (γ i λ i σ i − 1) 2 (ϑ − w i λ 2 i σ 2 i ) 2 .(32)
Therefore, if ρ(w j λ 2 j σ 2 j ) < ε 2 H , thenδ j = − ε 2 H − ρ(ϑ). Otherwise, ϑ > w j λ 2 j σ 2 j and it can be uniquely determined by (31).
B. Robust Transceiver Design
Now, we can use the worst-case system matrices descriptions (24), (25), and (28) and substitute into the robust transceiver design problem. Note that using the trust-region subproblems [32], [33] the resultant problem of finding worstcase channel estimation error ∆ H becomes a minimization problem over an auxiliary variable ϑ. The result can be compiled as follows:
Theorem 1: The robust precoding and equalization matrices have the following structure:
F = Φ + ε Φ I − 1 2 VΣ (33) G =Λ U H Ω + ε Ω I − 1 2(34)
where (i) U ∈ C nr ×nr and V ∈ C nt×nr are orthonormal matrices defined by the thin SVD [34] Ω + ε Ω I
− 1 2 H Φ + ε Φ I − 1 2 = U Γ V H(35)
where Γ ∈ C nr ×nr is a diagonal matrix with diagonal elements of γ i ≥ 0, (ii) Λ and Σ are diagonal matrices of size n r with the diagonal elements of λ i , i = 1, . . . , n r and σ i , i = 1, . . . , n r , respectively and they are obtained through solving the scalar optimization problem minimize λi,σi,ϑ 1≤i≤nr
nr i=1 ϑwi(σiλiγi−1) 2 ϑ−wiλ 2 i σ 2 i + nr i=1 w i λ 2 i + ϑ ε 2 H subject to ϑ ≥ w i λ 2 i σ 2 i , i = 1, . . . , n r nr i=1 σ 2 i ≤ P(36)
(iii) The optimum solutions for λ i and σ i can be obtained with respect to ϑ and a Lagrangian multiplier µ as follows
λ i = X i µ w i(37)σ i = X i w i µ(38)
where X i is a positive real root of a depressed quartic equation
ϕ i (X) = √ µw 2 i X 4 − (2w i ϑ √ µ + w i √ w i γ i ϑ)X 2 (γ 2 i ϑ + w i ) √ w i ϑX + ϑ 2 ( √ µ − γ i √ w i ) = 0. (39)
If there is no real positive root, then X i = 0. The closedform solutions for the roots of the quartic equation (39) can be obtained using the Ferrari's method [35] and can be found in [27].
Proof: Due to limited space of this paper the detailed proof is included in [27]. Here, we summarize briefly the approach used. We first substitute (24), (25), and (28) into the original problem (12), and hence we simplify it to a minimization problem with respect to F, G and the auxiliary variable ϑ. Now, using Lemma 1, the optimal expressions for the precoding and equalization matrices are given by (5) and (6) for any values of error matrices. Substituting expressions for F and G and the worst-case system matrices (24), (25), and (28) into (12), we can convert the problem into a scalar optimization problem, which can be simplified to (36). Notice that the maximization preserves the convexity, therefore this problem is a convex optimization problem with respect to G and F and consequently in λ i and σ i , i = 1, . . . , n r . By fixing ϑ, we can solve the problem in λ i and σ i . Next, the auxiliary variable ϑ will be updated following the Lemma 2.
Our closed-form solutions are functions of the auxiliary variables ϑ and µ. Using dual decomposition concept from [36], we can decompose the problem with outer loop optimization problems with respect to µ and ϑ. These values can be updated using a subgradient algorithm [37]. By differentiating the objective function in problem (36) with respect to ϑ, we can obtain the subgradient direction for ϑ as
∆ ϑ = ε 2 H − nr i=1 wiλ 2 i σ 2 i (λiσiγi−1) 2 (ϑ−wiλ 2 i σ 2 i ) 2 ϑ > w j λ 2 j σ 2 j ε 2 H − ρ(ϑ) ϑ = w j λ 2 j σ 2 j(40)
Similarly, by differentiation of the Lagrangian function of (36), we can get the subgradient direction for µ as ∆ µ = Initialize σ i s and λ i s and µ > 0, ϑ > max i w i λ 2 i σ 2 i . Perform thin SVD (35) to obtain γ i s.
Repeat (subgradient loop of ϑ)
Update ϑ ← ϑ + δ ϑ ∆ ϑ using (40).
Repeat (subgradient loop of µ)
Form the quartic equation (39) for i = 1, . . . , nr. Find its positive real root. Find σ i and λ i using (37) and (38). (31) Replace λ i s and σ i s into (33) and (34) and find F and G.
Update µ ← µ + δµ∆µ. Until nr i=1 σ 2 i − P ≤ ǫ 0 Until satisfaction ofnr i=1 σ 2 i − P.
The robust transceiver optimization algorithm is summarized in Table I. The algorithm consists of two loops. The inner loop solves the convex scalar problem with respect to λ i and σ i and therefore it is convergent. The outer loop updates the auxiliary variable ϑ using a subgradient method, which is based on the strong duality of the trust region subproblem [32], [33] and consequently it is also convergent. The objective function of problem (36) is bounded and it is reduced in each iteration. Therefore, the algorithm in Table I is convergent.
V. NUMERICAL RESULTS
In this section, the performance of robust transceivers is evaluated numerically. The robust design guarantees a performance level for any point within the uncertainty region. Hence, the performance is displayed by the worst-case sum of MSE values, which are averaged over different system realizations. Each system realization is a result of a random generation of elements of the estimated system matrices (i.e. H, Ω 1 2 , Φ 1 2 ), which are i.i.d. Gaussian with zero mean. The uncertainty region is characterized by a parameter 0 ≤ ε ≤ 1. In our simulations, it is assumed that √ ε is the radius of the uncertainty region for each of the system matrices, when they are normalized by their Frobenius norms (i.e. ε 2 H = ε H 2 , ε 2 Ω = ε Ω 2 , and ε 2 Φ = ε Φ 2 ). The non-robust transceivers consider the estimated system matrices as the actual system matrices and are discussed in Lemma 1 and [25]. The worstcase estimation error matrices have been given in Section IV. Note that only a solution of a special case of our problem is available in the literature, which includes uncertainty of the channel matrices H only. For this special case, our algorithm performs as the algorithm in [15] while it is less complex (by optimizing the precoder and equalizer jointly and reducing the problem to a quartic equation). It is shown for the special case (uncertainty of H) that our algorithm performs as well as SDP methods with much lower complexity using an iterative approach. Fig. 1 shows the comparison of robust and non-robust designs for different values of ε, i.e. the size of uncertainty regions. The sum-MSE of the transceivers obtained with perfect knowledge of system matrices is also given as a baseline. Fig. 2 explicitly illustrates the performance of the robust and non-robust design with respect to the size of the Fig. 1. Comparison of the proposed robust design, the non-robust design [27], and the transceiver design when system matrices are perfectly known (perfect CSI) for nt = nr = 2. uncertainty region. As expected, the performance of the robust transceivers degrades at a much lower pace with increase of signal-to-noise-ratio (SNR) and the size of uncertainty region ε compared to the non-robust transceivers.
VI. CONCLUSIONS
We have designed the robust transceivers when the channel matrix, interference plus noise covariance matrix, and power shaping matrix (system matrices) are all imperfectly known to the transmitter. The closed-form expressions for the precoder and equalizer have been found. This involves finding the worstcase system matrices first and then simplifying the problem to a scalar convex form. The solution to this optimization problem can be described in a form of a depressed quartic equation, the closed-form expressions for roots of which are known. Finally, we have proposed an iterative algorithm to obtain robust transceivers, which is significantly less complex compared to SDP-based alternating optimizations. Moreover, accounting for imperfect knowledge of all system matrices enables for the extension of our approach to the multiuser scenario, which is the subject of our current work [27].
maximize ∆H ≤εH tr A∆ H B∆ H H + 2Re {tr {C∆ H }} . (27)
Fig. 2 .
2Comparison of different transceiver designs with respect to the size of uncertainty region s for nt = nr = 2.
and simplify the worst-case problem as maximize (∆H ,∆Φ,∆Ω)∈U tr W E + tr G H WG∆ Ω + tr A∆ H B∆ H H + 2Re {tr {C∆ H }} subject to tr ΦFF H ≤ P − tr ∆ Φ FF H (13) where E is the MSE-matrix defined in (3) based on the estimated matrices H and Ω and A =G H WG,
TABLE I ROBUST
ITRANSCEIVER OPTIMIZATION ALGORITHM
SNR(dB)Worst-case sum-MSE0
2
4
6
8
10
12
14
16
18
20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Robust
non-Robust
Perfect CSI
s=0.05
s=0.1
s=0.01
The optimal forms of precoder and equalizer diagonalize the MSEmatrix E[25] and consequently the updated weight matrix W is diagonal.2 The matrix Ω − 1 2 HΦ − 1 2 with probability one has a rank of nr, due to the random nature of the channel matrix H and the fact that nr ≤ nt.
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[] |
[
"Reconstruction of a multidimensional scenery with a branching random walk",
"Reconstruction of a multidimensional scenery with a branching random walk"
] |
[
"Heinrich Matzinger ",
"Serguei Popov ",
"Angelica Pachon "
] |
[] |
[] |
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to[22]where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.
|
10.1214/16-aap1183
|
[
"https://arxiv.org/pdf/1511.00973v1.pdf"
] | 119,166,299 |
1511.00973
|
7ec178a290611bef984cdff1f15f1fdbc025108c
|
Reconstruction of a multidimensional scenery with a branching random walk
3 Nov 2015 November 4, 2015
Heinrich Matzinger
Serguei Popov
Angelica Pachon
Reconstruction of a multidimensional scenery with a branching random walk
3 Nov 2015 November 4, 2015
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to[22]where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.
Introduction
The classical scenery reconstruction problem considers a coloring ξ : Z → {1, 2, . . . , κ} of the integers Z. To ξ one refers as the scenery and to κ as the number of colors of ξ. Furthermore, a particle moves according to a a recurrent random walk S on Z and at each instant of time t observes the color χ t := ξ(S t ) at its current position S t . The scenery reconstruction problem is formulated through the following question: From a color record χ = χ 0 χ 1 χ 2 . . ., can one reconstruct the scenery ξ? The complexity of the scenery reconstruction problem varies with the number of colors κ, it generally increases as κ decreases.
In [10] it was shown that there are some strange sceneries which can not be reconstructed. However, it is possible to show that a.s. a "typical" scenery, drawn at random according to a given distribution, can be reconstructed (possibly up to shift and/or reflection).
In [13], [14], and [15] it is proved that almost every scenery can be reconstructed in the one-dimensional case. In this instance combinatorial methods are used, see for example [9], [12], [16], [17], [18], [19] and [20]. However, all the techniques used in one dimension completely fail in two dimensions, see [7].
In [11] a reconstruction algorithm for the 2-dimensional case is provided. However, in [11] the number of colors in the scenery is very high (order of several billions). The d-dimensional case is approached in [22] using a branching random walk and assuming that at each instant of time t, each particle is seeing the observations contained in a box centered at its current location.
In the present article we show that we can reconstruct a d-dimensional scenery using again a branching random walk but only observing at each instant of time t, the color at the current positions of the particles. For this we only need at least 2d + 1 colors. Our method exploits the combinatorial nature of the d-dimensional problem in an entirely novel way.
The scenery reconstruction problem goes back to questions from Kolmogorov, Kesten, Keane, Benjamini, Perez, Den Hollander and others. A related well known problem is to distinguish sceneries: Benjamini, den Hollander, and Keane independently asked whether all non-equivalent sceneries could be distinguished. An outline of this problem is as follows: Let η 1 and η 2 be two given sceneries, and sssume that either η 1 or η 2 is observed along a random walk path, but we do not know which one. Can we figure out which of the two sceneries was taken? Kesten and Benjamini in [2] proved that one can distinguish almost every pair of sceneries even in two dimensions and with only two colors. For this they take η 1 and η 2 both i.i.d. and independent of each other. Prior to that, Howard had shown in [3], [4], and [5] that any two periodic one dimensional non-equivalent sceneries are distinguishable, and that one can a.s. distinguish single defects in periodic sceneries. Detecting a single defect in a scenery refers to the problem of distinguishing two sceneries which differ only in one point. Kesten proved in [6] that one can a.s. recognize a single defect in a random scenery with at least five colors. As fewer colors lead to a reduced amount of information, it was conjectured that it might not be possible to detect a single defect in a 2-color random scenery. However, in [15] the first named author showed that the opposite is true -a single defect in a random 2-color scenery can be detected. He also proved that the whole scenery can be reconstructed without any prior knowledge.
One motivation to study scenery reconstruction and distinguishing problems was the T, T −1 -problem that origins in a famous conjecture in ergodic theory due to Kolmogorov. He demonstrated that every Bernoulli shift T has a trivial tail-field and conjectured that also the converse is true. Let K denote the class of all transformations having a trivial tailfield. Kolmogorov's conjecture was shown to be wrong by Ornstein in [21], who presented an example of a transformation which is in K but not Bernoulli. His transformation was particularly constructed to resolve Kolmogorov's conjecture. In 1971 Ornstein, Adler, and Weiss came up with a very natural example which is K but appeared not to be Bernoulli, see [23]. This was the so called T, T −1 -transformation, and the T, T −1 -problem was to verify that it was not Bernoulli. It was solved by Kalikow in [8], showing that the T, T −1transformation is not even loosely Bernoulli. A generalization of this result was recently proved by den Hollander and Steif [1].
Model and Statement of Results
We consider a random coloring of the integers in d-dimension. Let ξ z be i.i.d. random variables, where the index z ranges over Z d . The variables ξ z take values from the set {0, 1, 2, . . . , κ − 1} where κ ≥ 4, and all the values from {0, 1, 2, . . . , κ − 1} have the same probability. A realization of ξ = {ξ z } z∈Z d thus is a (random) coloring of Z d . We call this random field ξ a d-dimensional scenery.
Now assume that a branching random walk (BRW) on Z d observes the d-dimensional scenery ξ, i.e., i.e. that each particle of the BRW observes the color at its current position.
Formally a branching random walk in Z d is described as follows. The process starts with one particle at the origin, then at each step, any particle is substituted by two particles with probability b and is left intact with probability 1 − b, for some fixed b ∈ (0, 1). We denote by N n the total number of particles at time n. Clearly (N n , n = 0, 1, 2, . . .) is a Galton-Watson process with the branching probabilitiesp 1 = 1 − b,p 2 = b. This process is supercritical, so it is clear that N n → ∞ a.s. Furthermore, each particle follows the path of a simple random walk, i.e., each particle jumps to one of its nearest neighbors chosen with equal probabilities, independently of everything else.
To give the formal definition of the observed process we first introduce some notation. Let η t (z) be the number of particles at site z ∈ Z d at time t ≥ 0, with η 0 (z) = 1{z = 0}. We denote by η t = (η t (z)) z∈Z d the configuration at time t of the branching random walk on Z d , starting at the origin with branching probability b.
Let G be the genealogical tree of the Galton-Watson process, where G t = {v t 1 , . . . , v t Nt } are the particles of t-th generation. Let S(v t j ) be the position of v t j in Z d , i.e., S(v t j ) = z if the j-th particle at time t is located on the site z. Recall that we do not know the position of the particles, only the color record made by the particles at every time, as well as the number of particles at each time.
According to our notations,
{z ∈ Z d : η t (z) ≥ 1} = {∃j; S(v t j ) = z, j = 1, . . . , N t }.
The observations χ, to which we also refer as observed process, is a coloring of the random tree G. Hence the observations χ constitute a random map:
χ : G t → {0, 1, 2, . . . , κ − 1} v t i → χ(v t i ) = ξ(S(v t i )), i = {1, . . . , N t }.
In other words, the coloring χ of the random tree G yields the color the particle i at time t sees in the scenery from where it is located at time t.
Denote by Ω 1 = {(η t ) t∈N } the space of all possible "evolutions" of the branching random walk, by Ω 2 = {0, 1, 2, . . . , κ − 1} Z d the space of all possibles sceneries ξ and by Ω 3 = {(χ t ) t∈N } the space of all possible realizations of the observed process. We assume that (η t ) t∈N and ξ are independent and distributed with the laws P 1 and P 2 , respectively.
Two sceneries ξ and ξ ′ are said to be equivalent (in this case we write ξ ∼ ξ ′ ), if there exists an isometry ϕ :
Z d → Z d such that ξ(ϕx) = ξ ′ (x) for all x.
Now we formulate the main result of this paper. The measure P designates the product measure P 1 ⊗ P 2 .
Theorem 2.1 For any b ∈ (0, 1) and κ ≥ 2d + 1 there exists a measurable function Λ : Ω 3 → Ω 2 such that P (Λ(χ) ∼ ξ) = 1.
The function Λ represents an "algorithm", the observations χ being its input and the reconstructed scenery ξ (up to equivalence) being its output. The main idea used to prove Theorem 2.1 is to show how to reconstruct a finite piece of the scenery (close to the origin).
Main Ideas
We start by defining a reconstruction algorithm with parameter n denoted by Λ n , which works with all the observations up to time n 2 to reconstruct a portion of the scenery ξ close to the origin. The portion reconstructed should be the restriction of ξ to a box with center close to the origin. This means closer than √ n, which is a lesser order than the size of the piece reconstructed. We will show that the algorithm Λ n works with high probability (whp for short, meaning with probability tending to one as n → ∞). Let us denote by K x (s) the box of size s centered at x in Z d , i.e., x + [−s, s] d , and by K(s) the box of size s centered at the origin of Z d . For a subset A of Z d , we designate by ξ A the restriction of ξ to A, so ξ Kx(s) denotes the restriction of ξ to K x (s).
In what follows, we will say that w is a word of size k of ξ A , if it can be read in a straight manner in ξ A . This means that w is a word of ξ A if there exist an x ∈ Z d and a canonical vector e (it defined to be one that has only one non-zero entry equal to +1 or −1.), so that x + i e ∈ A, for all i = 0, 1, 2, . . . , k − 1, and w = ξ x ξ x+ e ξ x+2 e . . . ξ x+(k−1) e .
Let A and B be two subsets of Z d . Then we say that ξ A and ξ B are equivalent to each other and write ξ A ∼ ξ B , if there exists an isometry ϕ :
A → B such that, ξ A • ϕ = ξ B .
2.2
The algorithm Λ n for reconstructing a finite piece of scenery close to the origin.
The four phases of this algorithm are described in the following way:
1. First phase: Construction of short words of size (ln n) 2 . The first phase aims at reconstructing words of size (ln n) 2 of ξ K(n 2 ) . The set of words constructed in this phase is denoted by SHORT W ORDS n . It should hopefully contain all words of size (ln n) 2 in ξ K(4n) , and be contained in the set of all words of size (ln n) 2 in ξ K(n 2 ) . The accurate definition of SHORT W ORDS n is as follows: a word w 2 of size (ln n) 2 is going to be selected to be in SHORT W ORDS n if there exist two strings w 1 and w 3 both of size (ln n) 2 and such that:
(a) w 1 w 2 w 3 appears in the observations before time n 2 , and (b) the only word w of size (ln n) 2 such that w 1 ww 3 appears in the observations up to time n 4 is w 2 .
Formally, let W (ξ K(4n) ) and W (ξ K(n 2 ) ) be the sets of all words of size (ln n) 2 in ξ K(4n) and ξ K(n 2 ) respectively, then
W (ξ K(4n) ) ⊆ SHORT W ORDS n ⊆ W (ξ K(n 2 ) ). (2.1)
We prove that (2.1) holds whp in Section 3.1.
2.
Second phase: Construction of long words of size 4n. The second phase assembles the words of SHORT W ORDS n into longer words to construct another set of words denoted by LONGW ORDS n . The rule is that the words of SHORT W ORDS n to get assembled must coincide on (ln n) 2 − 1 consecutive letters, and it is done until getting strings of total size exactly equal to 4n + 1. In this phase, let W 4n (ξ K(4n) ) and W 4n (ξ K(n 2 ) ) be the set of all words of size 4n in ξ K(4n) and ξ K(n 2 ) respectively, then
W 4n (ξ K(4n) ) ⊆ LONGW ORDS n ⊆ W 4n (ξ K(n 2 ) ). (2.2)
We achieve that (2.2) holds whp in Section 3.2.
3. Third phase: Selecting a seed word close to the origin. The third phase selects from the previous long words one which is close to the origin. For that Λ n applies the previous two phases, but with the parameter being equal to n 0.25 instead of n. In other words, Λ n choses one (any) word w 0 of LONGW ORDS n 0.25 , and then choses the word w L in LONGW ORDS n which contains w 0 in such a way that the relative position of w 0 inside w L is centered. (The middle letters of w 0 and w L must coincide). Λ n places the word w L so that the middle letter of w 0 is at the origin. See Section 3.3.
4.
Forth phase: Determining which long words are neighbors of each other. The fourth phase place words from LONGW ORDS n in the correct relative position to each other, thus assembling the scenery in a box near the origin. For this, Λ n starts with the first long-word which was placed close to the origin in the previous phase. Then, words from the set LONGW ORDS n are placed parallel to each other until a piece of scenery on a box of size 4n is completed.
Let us briefly explain how Λ n choses which words of LONGW ORDS n are neighbors of each other in the scenery ξ K(n 2 ) , (i.e., they are parallel and at distance 1). Λ n estimates that the words v and w which appear in ξ K(n 2 ) are neighbors of each other iff the three following conditions are all satisfied:
(a) First, there exist 4 words v a , v b , v c and w b having all size (ln n) 2 except for v b which has size (ln n) 2 − 2, and such that the concatenation v a v b v c is contained in v, whilst up to time n 4 it is observed at least once v a w b v c .
(b) Second, the word w b is contained in w.
(c) Finally, the relative position of v b in v should be the same as the relative position of w b in w. By this we mean that the middle letter of v b has the same position in v as the middle letter of w b in w has.
See the precise definition in Section 3.4.
Let B be the set of all finite binary trees, and let χ t designates all the observations made by the branching random walk up to time t. That χ t is the restriction of the coloring χ to the sub-tree ∪ i≤t G i . The next result means that the algorithm Λ n works whp.
Theorem 2.2 Assume that the number of colors κ satisfies the condition that κ ≥ 2d + 1.
Then, for every n ∈ N large enough, the map
Λ n : {0, 1, . . . , κ − 1} B → {0, 1, . . . , κ − 1} K(4n) ,
defined above as our algorithm satisfies
P ∃x ∈ K( √ n) so that Λ n (χ n 4 ) ∼ ξ Kx(4n) ≥ 1 − exp(−C(ln n) 2 ), (2.3)
where C > 0 is a constant independent of n.
In words, the algorithm Λ n manages to reconstruct whp a piece of the scenery ξ restricted to a box of size 4n close to the origin. The center of the box has every coordinate not further than √ n from the origin. The reconstruction algorithm uses only observations up to time n 4 . We will give the exact proof of the above theorem in the next section, but, before we present the main ideas in a less formal way in the remainder of this section. We first want to note that the algorithm Λ n reconstructs a piece of the scenery ξ in a box of size 4n, but the exact position of that piece within ξ will in general not be known after the reconstruction. Instead, the above theorem insures that whp the center of the box is not further than an order √ n from the origin.
For what follows we will need a few definitions: Let [0, k − 1] designate the sequence {0, 1, 2, 3 . . . , k − 1} and R be a map R : [0, k − 1] → Z d such that the distance between R(i) and R(i+1) is 1 for every i = 0, 1, 2, . . . , k −2. We call such a map a nearest neighbor path of length k − 1. Let x and y be two points in Z d . If R(0) = x and R(k − 1) = y we say that R goes from x to y. We also say that R starts in x and ends in y. Let w be a string of colors of size k, such that w = ξ(R(0))ξ(R(1)) . . . ξ(R(k − 1)).
In that case, we say that R generates w on the scenery ξ.
The DNA-sequencing trick
We use the same trick as is used in modern methods for DNA-sequencing where instead of reconstructing the whole DNA-sequence at once, one tries to reconstruct smaller pieces simultaneously. Then to obtain the entire piece one puzzles the smaller pieces together. In this paper the scenery is multidimensional unlike DNA-sequences. Nonetheless, we first present the reconstruction method for DNA-sequencing in this subsection, because it is easiest to understand. The trick goes as follows. Assume that you have a genetic sequence D = D 1 D 2 . . . D n α , where α > 0 is a constant independent of n, and D is written in an alphabet with κ > 0 equiprobable letters. For instance in DNA-sequences this alphabet is {A, C, T, G}. To determine the physical order of these letters in a sequence of DNA, modern methods use the idea of do not go for the whole sequence at once, but first determine small pieces (subsequences) of order at least C ln(n) and then assemble them to obtain the whole sequence. (Here C > 0 is a constant independent of n, but dependent of α). Let us give an example. Which one are to be read forward and which one backward is not known to the biologist. He is only given all these subsequences in one bag without extra information. If he was given all the subsequences of size 5 appearing in D, he could reconstruct D by assembling these subsequences using the assembly rule that they must coincide on a piece of size 4. Note that each of these appears only once. Hence, four consecutive letters determine uniquely the relative position of the subsequences of size 5 with respect to each other, and given the bag of subsequences of size 5 is possible to assemble theme one after the other. How? Start by picking any subsequence. Then put down the next subsequence from the bag which coincides with the previous one on at least 4 letters. For example take GACT A and put it down on the integers in any position, for instance
G A C T A 0 1 2 3 4.
Then, take another subsequence from the bag which coincides with the previously chosen one, on at least 4 contiguous letters. For example, T GACT satisfies this condition. Now superpose the new subsequence onto the previous one so that on 4 letters they coincide:
T G A C T G A C T A −1 0 1 2 3 4
.This leads to:
T G A C T A −1 0 1 2 3 4
Next, observe that the subsequence ACT AT coincides on at least 4 consecutive letters, with what has been reconstructed so far. So, put ACT AT down in a matching position:
A C T A T T G A C T A −1 0 1 2 3 4 5
.This leads to:
T G A C T A T −1 0 1 2 3 4 5
The final result is the sequence T GACT AT which is D read in reverse order.
But how do we know that this method works? in the example we saw the sequence D before hand, and could hence verify that each subsequence of size 4 appears in at most one position in D. However, the biologist can not verify this condition. He only gets the bag of subsequences as only information. So, the idea is that if we know the stochastic model which generates the sequence, we can calculate that whp each subsequence of size C ln(n) appears only once in D, provided the constant C > 0 is taken large enough. Take for example the i.i.d. model with κ > 1 equiprobability letters. Then the probability that two subsequences located in different and non-intersecting parts of D be identical is given by:
P (E n+ i,j ) = 1 κ C ln n = n −C ln(κ) ,(2.4)
where E n+ i,j is the event that Similarly, let E n− i,j be the event that
D i+1 . . . D i+C ln n = D j−1 . . . D j−C ln n . (2.6)
Note that even if the subsequences given on both sides of (2.5) or (2.6) intersect, as long as they are not exactly in the same position, we get that (2.4) still holds. To see this take for example the subsequence w = D i+1 D i+2 . . . D i+C(ln n) and the subsequence v = D i+2 D i+3 . . . D i+C(ln n)+1 . These two subsequences are not at all independent of each other since up to two letters they are identical to each other. However we still have
P (w = v) = P (D i+1 D i+2 . . . D i+C(ln n) = D i+2 D i+3 . . . D i+C(ln n)+1 ) = 1 κ C ln n
To see why this holds, simply note that D i+2 is independent of D i+1 , so w and v agree on the first letter with probability 1/κ. Then, D i+3 is independent of D i+2 and D i+1 , so the second letter of v has a probability of 1/κ to be identical to the second letter of w, and so on. Thus, to get that whp no identical subsequence appears in any two different positions in D, we need to get a suitable upper bound of the right side of (2.4). If we take i and j both in n α , we find that the probability to have at least one subsequence appearing in two different positions in D, can be bounded in the following way:
P (∪ i =j E n+ i,j ) ≤ i =j P (E n+ i,j ) ≤ n 2α · n −C ln(κ) (2.7)
The same type of bound also holds for P (E n− i,j ) using (2.6). We take C strictly larger than 2α/ ln κ in order to get the right side of (2.7) be negatively polynomially small in n.
For our algorithm Λ n we will take subsequences (the short words) to have size (ln n) 2 instead of being linear in ln n. Thus we do not have to think about the constant in front of ln n. Then, the bound we get for the probability becomes even better than negative polynomial in n. It becomes of the type n −βn where β > 0 is a constant independent of n.
Applying the DNA-sequencing method to a multidimensional scenery
In our algorithm Λ n we do not have the task to reconstruct a sequence, instead we have to reconstruct a multidimensional scenery restricted to a box. We use the same idea as the DNA-sequencing method except that we will reconstruct several long words instead of reconstructing just one long word (one sequence). This corresponds to the second phase of Λ n where with a bag of short words, it constructs a collection of long words. These words will be the different restrictions of the scenery ξ to straight lines-segments parallel to some direction of coordinates. This long words of course do not yet tell us exactly how the scenery looks, we will still need to position these long words correctly with respect to each other. (This is done in the fourth phase of Λ n ). Let us explain that with another example. Here we assume that the alphabet has size κ = 10 and that we would be given a bag of all short words of size 4 appearing in (2.8). That is words which can be read horizontally or vertically in either direction: from up to down, down to up, left to right, right to left and of size 4. This bag of short-words is: and their reverses. As assemble rule we use that words must coincide on at least 3 consecutive letters. We can for example assemble 1943 with 9437 to get 19437. Similarly we assemble 1546 with 5462 and obtain 15462. So, we apply the DNA-puzzling trick, but instead of reconstruct only one long word, we will reconstruct several long words. In this example we reconstruct the set of 10 long words and their reverses: where again each of the above long words could be its own reverse.
{ 1943,{
Thus, the previous example shows how the second phase of the algorithm works: From a set of shorter words SHORT W ORDS n , we obtain a set of longer words LONGW ORDS n in the same manner how we obtained the set (2.10) from the bag of words (2.9). Note that the long words are successfully reconstructed in this example, because in the restriction of the scenery in (2.8), any word of size 3 appears at most in one place.
The differences in the numeric example presented above and how the second phase of Λ n works are the following: the short words in Λ n have length (ln n) 2 and the long words have length 4n, instead of 4 and 5. Moreover, Λ n will need to assemble in the second phase many short words to get one long word, despite in this example where we just used two short words to get each long word. On the other side, in the previous example is given to us a bag of all words of size 4 of the restriction of ξ to the box [0, 4] × [0, 4]. However, the second phase of Λ n has the bag of short words SHORT W ORDS n , which is not exactly equal to all words of size (ln n) 2 of ξ restricted to a box. Instead it is the bag of words that contains all words of size (ln n) 2 of ξ, restricted to the box K(4n), but augmented by some other words of the bigger box K(n 2 ).
The bag of short words SHORT W ORDS n is obtained in the first phase of Λ n . The reason why the first phase is not able to identify which words are in the box K(4n) and which are in K(n 2 ) is as follows: the observations in the first phase of Λ n are taken up to time n 2 . Since we assume that the first particle starts at time 0 at the origin, by time n 2 all particles must be contained in the box K(n 2 ), and the probability for one given particle at time n 2 to be close to the border of K(n 2 ) is exponentially small. Since we have many particles, a few will be close to the border of K(n 2 ) by time n 2 , and these will be enough particles to provide some few words close to the border of K(n 2 ) by time n 2 , and selected in the first phase of the algorithm.
There is an important consequence to this in the second phase of the algorithm, since some reconstructed long words in LONGW ORDS n might also be "far out in the box K(n 2 )" and not in K(4n).
The diamond trick to reconstruct all the words of ξ K(4n)
In the previous subsection, we showed how to assemble shorter words to get longer ones, but we have not yet explained the main idea of how we manage to obtain short words in the first phase of Λ n , being given only the observations. The basic idea is to use the diamonds associated with a word appearing in the scenery. Let us look at an example. 4 3 7 4 7 5 0 7 6 1 1 7 4 3 9 1 2 1 8 6 4 4 0 4 3 2 2 7 8 0 3 9 (2.11)
Consider the word w = 43912 = ξ (1,2) ξ (2,2) ξ (3,2) ξ (4,2) ξ 5,2
which appears in the above piece of scenery in green. That word "appears between the points x = (1, 2) and y = (5, 2)". We only consider here words which are written in the scenery "along the direction of a coordinate". In blue we highlighted the diamond associated with the word 43912. More precisely the diamond consists of all positions which in the above figure are green or blue.
The formal definition is that if x and y are two points in Z d so thatxy is parallel to a canonical vector e, then the diamond associated with the word w = ξ x ξ x+ e ξ x+2 e . . . ξ y− e ξ y consists of all points in Z d which can be reached with at most (|x − y|/2) − 1 steps from the point (y −x)/2. We assume here that the Euclidean distance |x−y| is an odd number.
The useful thing will be that whp for a given non-random point z outside the diamond associated with w, there is no nearest neighbor walk path starting at z and generating as observations w. So, whp a word w in the scenery can only be generated as observations by a nearest neighbor-walk path starting (and also ending) in the diamond associated with w. (At least if we restrict the scenery to a box of polynomial size in the length of w). To see this take for instance the following path R:
((0, 0), (1, 0), (1, 1), (1, 2), (2, 2))
In the previous example a random walker which would follow this little path would observe the sequence of colors given by
ξ • R = (2, 2, 6, 4, 3) (2.12)
Note that the path R and the straight path from x to y intersect. So, (2.12) and the green word w = 43912 are not independent of each other. However, because the path R starts outside the diamond, we get the probability of the event that (2.12) and the word w are identical, has the same probability as if they would be independent. That is assuming that R is a (non-random) nearest neighbor path starting (or ending) at a given point z outside the diamond associated with a word w,
P (ξ • R = ξ x ξ x+ e ξ x+2 e . . . ξ y− e ξ y ) = 1 κ m ,
where m designates the size of the word. In fact looking at our example, R starts at (0, 0) which is outside the diamond. So, the starting point of R is different from x = (1, 2) and hence by independence
P (ξ(R(0)) = ξ x ) = 1 κ .
Then the second letter in w, that is ξ x+ e independent of the first two letters of the observations along the path, ξ(R 0 )ξ(R 1 ), since both points R(0) = (0, 0) and R(1) = (1, 0) are different from x + e = (2, 2). Thus, we get the probability that the first two letters of w coincide with the first two observations made by R is equal to:
P (ξ(R(0))ξ(R(1)) = ξ x ξ x+ e ) = 1 κ 2 . (2.13)
The proof goes on by induction: the kth letter in the word w x+k e is independent of the first k observations made by R. The reason is that the first k positions R(0)R(1) . . . R(k − 1) visited by R do never contain the point x + k e, since "the walker following the path R never catches up with the walker going straight from x to y." On the other hand observe that in our example, we see a path starting inside the diamond and producing as observation the same green word w. Take the path (2, 1), (2, 2), (3, 2), (4, 2), (5, 2). Thus, this "counterexample" illustates how "easy" it is for a path starting "inside" the diamond associated with a word w, to generate the same word. This a second path different to the straight path "going from x to y". Now, using (2.13) we can calculate an upper bound for the probability that for a given non-random point z outside the diamond, there exists at least one non-random path R starting at z and producing as observation w. Observe that in d-dimensional scenery, for a given starting point z ∈ Z d , there are (2d) k−1 nearest neighbor walk paths of length k. So, we find that the probability that there exists a nearest neighbor path R starting at z, with z outside the diamond associated with a word w, and R generating w, has a probability bounded from above as follows:
P (∃ a nearest neighbor path R starting at z with ξ • R = w) ≤ 2d κ k−1 (2.14)
Note that the bound above is negatively exponentially small in k as soon as 2d < κ. The inequality 2d < κ is precisely the inequality given in Theorem 2.2 which makes our reconstruction algorithm work).
Thus, in the next section we will define the event B n 3 as follows: Let w be a word of size (ln n) 2 in ξ K(n 2 ) , and R a nearest neighbor path, R : [0, k − 1] → K(n 4 ), so that ξ • R = w, and R begins and ends in the diamond associated with w.
Note then that P (B nc 3 ) is bounded from above by (2.14) times the number of points in the box ξ K(n 2 ) . But expression (2.14) is negatively exponentially small in the size of the word (ln n) 2 and hence dominates the polynomial number of points in the box, i.e., it goes to zero as n goes to infinity. Thus, B n 3 holds whp. (To see the exact proof, go to lemma 3.3.)
Now, we know that with high probability the words can only be generated by a nearest neighbor walk path starting (and ending) in the diamond associated with w. (At least when we restrict ourselves to a box of polynomial size in the length of w). But, how can we use this to reconstruct words? The best is to consider an example. 1 9 4 3 7 4 1 2 5 2 2 7 8 0 6 9 7 5 0 7 6 1 1 8 2 5 8 6 7 4 0 4 2 7 4 3 9 1 2 1 7 8 4 7 6 1 7 7 7 4 8 6 4 4 0 4 3 5 3 6 7 5 1 9 9 9 1 2 2 7 8 0 3 9 4 3 7 2 1 9 4 5 7 0 (2.15) Let the word which appears in green be denoted by w 1 so that
w 1 = 43912
Let the word written in brown be denoted by w 3 so that
w 3 = 61777
Finally let the word which is written when we go straight from the green word to the brown word be denoted by w 2 so that w 2 = 17847
In the current example the diamond D 1 associated with the green word w 1 is given in blue and the diamond D 3 associated with the brown word w 3 is highlighted in red. Note that there is only one shortest nearest neighbor path to go from D 1 to D 3 , walking straight from the point (5, 2) to the point (11,2) in exactly six steps. There is not other way to go in six steps from D 1 to D 3 . When doing so a walker will see the word w 2 . Now assume that the size of our short words is 5, (that is the size which in the algorithm is given by the formula (ln n) 2 ). Assume also that the rectangle [0, 16] × [0, 4] is contained in K(n 2 ).
Remember that if B n 3 holds, we have that within the box K(n 2 ) a nearest neighbor walk can only generate a short word of ξ K(n 2 ) if it starts and ends in the diamond associated with that word. Using this to the words w 1 and w 3 , we see in the observations the following pattern w 1 * * * * * w 3
where * is a wild card which stands for exactly one letter, then whp the walker between w 1 and w 3 was walking in a straight manner from D 1 to D 3 . Hence, the wild card sequence * * * * * must then be the word w 2 . Of course, we need at least one particle to follow that path in order to observe w 1 w 2 w 3 up to time n 2 . This will be taken care in Λ n by the event B n 2 which stipulates that any nearest neighbor path of length 3(ln n) contained in K(4n), will be followed by at least one particle up before time n 2 . In other words we have proven that if B n 2 and B n 3 both hold, then w 2 gets selected by the first phase of the algorithm as a short word of SHORT W ORDS n . The argument of course works for any short word of ξ K(4n) and hence we have that
B n 2 ∩ B n 3 =⇒ W (ξ K(4n) ) ⊂ SHORT W ORDS n
Thus the previous example shows how the first phase of the algorithm manages to reconstruct all short words in ξ K(4n) .
How to eliminate junk observation-strings which are not words of ξ K(n 2 )
In the previous subsection we have shown how to reconstruct enough words. But now the next question is "how do we manage to not reconstruct too many words"? By this we mean, how do we make sure that observations which do not correspond to words of ξ K(n 2 ) do not get selected by the first phase of our algorithm? This means that we have to be able to eliminate observations which do not correspond to a word of ξ K(n 2 ) ! The best is again to see an example. 7 6 1 1 8 2 5 8 6 7 4 0 4 2 7 4 3 9 1 2 1 7 8 4 7 6 1 7 7 7 4 8 6 4 4 0 4 3 5 3 6 7 5 1 9 9 9 1 2 2 7 8 0 3 9 4 3 7 2 1 9 4 5 7 0 (2.16)
Observe this is the same piece of scenery as was shown in (2.15), but the brown word was moved two units to the left. So, let again w 1 denote the green word w 1 = 43912 and let this time w 3 be the "new" brown word:
w 3 = 47617
Now a particle in between following the green word and then the brown word could for example do the following little dance step: right, up, right, down, right, right, and then produce the observation string w 2 given by
w 2 = 11878.
How can we make sure the observation string w 2 which is not a word (it does not follow a straight path) of our piece of scenery, does not get selected by the first phase of our algorithm as a short word? To see how w 2 gets eliminated in the first phase of our algorithm consider the following dancing step:
right, down, right, up, right, right.
When doing this nearest neighbor path, a particle would produce the observationsw 2 wherē
w 2 = 13578
Let B n 5 be the event that up to time n 4 every nearest neighbor path of length 3(ln n) 2 in K(n 2 ) gets followed at least once. Then, assuming our piece of scenery (2.16) is contained in K(n 2 ), we would have that: Up to time n 4 we will observe both strings
w 1 w 2 w 3 and w 1w2 w 3 ,
at least once. Since w 2 =w 2 the second short-word-selection criteria of the first phase assures that the words w 2 andw 2 do not get selected in the first phase of Lambda n . The crucial thing here was that from the point (5, 2) to (8, 2) there were two different 6 step nearest neighbor paths generating different observations, w 2 andw 2 . In subsection 3.1, we will show that whp in the box K(n 2 ) for any pair of points x and y so that a nearest neighbor walk goes in (ln n) 2 − 1 steps from x to y, either one of the two following things hold:
1. The segmentxy is parallel to a direction of a coordinate and the distance |x − y| = (ln n) 2 − 1, or 2. there exist two different nearest neighbor walk paths going from x to y in (ln n) 2 − 1 steps and generating to different observation-strings.
So, this implies that whp, the first phase of our algorithm can eliminate all the strings as it does in the example with w 2 andw 2 which are not words of ξ K(n 2 ) .
Why we need a seed
The second phase produces a bag of long words which whp contains all long words of ξ K(4n) . Unfortunately, this bag is likely to contain also some long words of ξ K(n 2 ) which are "not close to the origin". Note that the size of the long words is 4n, so if such a long word appears close to the border of K(n 2 ), then it wouldn't serve to our reconstruction purpose. The reason is that the algorithm Λ n aims to reconstruct the scenery in a box close to the origin. That is why in the third phase of Λ n we apply the first two phases but with the parameter n being replaced by n 0.25 . We then take any long word w 0 from the bag of words produced by the second phase of the algorithm Λ n 0.25 . That long word w 0 of size 4n 0.25 is then whp contained in ξ K((n 0.25 ) 2 ) = ξ K(n 0.5 ) .
In other words, the long word w 0 chosen as seed in the third phase of the algorithm is whp not further away than √ n from the origin. Since it is likely that any word of that size appears only once in K(n 2 ), then we can use this to determine one long word w L from the bag created by Λ n which is not further from the origin than √ n. We simply chose w L to be any word from the bag of long words created by Λ n which contains w 0 . Finally in the fourth phase of the algorithms we will then add neighboring long words to that first chosen long word. If the one long word which gets chosen in the third phase is close to the origin, we can determine which long words are neighbor on each other, then this will ensure that the other long words used for the reconstruction in the fourth phase are also close to the origin. and v b = ξ (6,2) ξ (7,2) ξ (8,2) = 178.
Finally let w b designate the word "one line higher" than v b , so
w b := ξ (5,3) ξ (6,3) ξ (7,3) ξ (8,3) ξ (9,3) = 11825.
Note that w b has two digits more than v b , and the middle letter of w b has the same xcoordinate than the last letter of v b in (2.16). Furthermore, in the piece of scenery (2.16) we designate the third line by v so that v := ξ (0,2) ξ (1,2) ξ (2,2) . . . ξ (16,2) Assume that the two words v and w have already been reconstructed by the two first phases of our algorithm Λ n . Consider next the straight path R 1 :
(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2), (11,2), (12,2), (13, 2), so a particle following this straight path generates the observations ξ
• R 1 = v a v b v c .
Consider now a second path R 2 that is similar to R 1 but at the end of the word v a , it goes one step up to read w b and then, at the end of w b it goes one step down to read v c . So, the path R 2 is defined as follows:
(
), (10, 2), (11,2), (12,2), (13,2),
and generates the observations ξ • R 2 = v a w b v c .
If now up to time n 4 there is at least one particle following the path R 1 and another particle following R 2 , then in the observations before time n 4 , we will observe once v a v b v c and also v a w b v c . So, v a , v b , v c and w b pass all the criteria in the fourth phase of our algorithm together with the long words v and w. So, v and w are detected (correctly) as being neighboring long words. Again, for this to happen we only need particles to go on the path R 1 and R 2 .
So, what the previous example tell us is that to recognize correctly which words in LONGW ORDS n are neighbors in the fourth phase of our algorithm, we need first to guarantee that all nearest neighbor paths of length 3(ln n) 2 within the box K(n 2 ), being followed by at least one particle up to time n 4 . The event B n 3 guaranties this and we prove it holds whp in the first subsection of the next section.
On the other side, we still need the forth phase of Λ n not to classify pairs of long words as neighbors if they are not in the scenery. This problem of "false positives" is solved as soon as the previously defined diamond property holds, that is when the event B n 3 holds and also short words do not appear in different places in ξ K(n 2 ) . The proof of this is a little intricate and is given as proof of Lemma 3.10 in the next section.
2.2.7
Placing neighboring long words next to each other in the 4-th phase of our algorithm.
Here we show how the long words are located next to each other in the fourth phase, in order to reconstruct a piece of the scenery ξ.
Example 2.7 Let us assume that the scenery is 2-dimensional and that the seed word is w 0 = 012. To start we place w 0 at the origin, and it would be the first part of the "reconstruction done by the algorithm Λ n ", thus w 0 is the restriction of Λ n (χ n 4 ) to the set {(−1, 0), (0, 0), (1, 0)}. Say then that we find a long word w L which is 60123. This long word contains the seed word 012 centered in its middle, so, if we superpose w L over w 0 so that it matches we get again 60123, that corresponds to the restriction of Λ n (χ n 4 ) to the set {(−2, 0), (−1, 0), (0, 0), (1, 0), (2, 0)}.
Next we find that the two long words 01111 and 02222 of LONGW ORDS n are neighbors of w L . In that case, we place these two long words in a neighboring position to get the following piece of scenery: 01111 60123 02222, and assume that in our bag LONGW ORDS n of long words we find the word 03333 to be a neighbor of 01111 and 04444 to be neighbor of 02222. Then our reconstruction yields the following piece of scenery: 03333 01111 60123 02222 04444
(2.17)
This is then the "end product" produced by the algorithm Λ n . So, this final output of the algorithm is a piece of scenery on a box of 5 × 5 given in (2.17). We assumed the size of the long words to be 5.
Overview over the events which make the algorithm Λ n work.
A handful of events ensure that the algorithm Λ n works as already mentioned. These events will all be defined again in a more exhaustive way in the next section. But, here let us list them in a somewhat informal way. The first phase works through the following events:
• The event B n 2 guaranties that up to time n 2 every nearest neighbor path of length 3(ln n) 2 contained in K(4n) gets followed at least once by at least one particle.
• The event B n 3 asserts that up to time n 4 for every word w of ξ K(n 2 ) , any nearest neighbor walk path R in K(n 4 ) generating w must start and end in the diamond associated with w. This event is needed to guaranty that all the words of ξ K(4n) of length (ln n) 2 get selected in the first phase of the algorithm, and put into the bag of words SHORT W ORDS n .
• The event B n 4 states that for any two points x and y in K(n 2 ) so that there is a nearest neighbor path going in (ln n) 2 − 1 steps from x to y, either one of the following holds:
-The points x and y are at distance (ln n) 2 − 1 from each other and the segment xy is parallel to a direction of coordinate. In that case, there can only be one nearest neighbor walk in (ln n) 2 −1 steps from x to y, and that nearest neighbor walk goes "straight".
-There exist two different nearest neighbor walk paths going from x to y in exactly (ln n) 2 − 1 steps and generating different observations.
• The event B n 5 guaranties that up to time n 4 every nearest neighbor path of length 3(ln n) 2 − 1 contained in K(n 2 ) gets followed at least once. Furthermore, together with the event B n 4 , these make sure that observations which are not words of ξ K(n 2 ) get eliminated in the first phase of Λ n .
In Lemma 3.6, the combinatorics of the first phase is proven. It is shown that when all the events B n 2 , B n 3 , B n 4 , B n 5 hold, then the first phase works. In the first subsection of the next section it is also proven that all these events hold whp, which implies that the first phase works whp.
For the second phase of the algorithm, we only need that any word of ξ K(n 2 ) of size (ln n) 2 − 1 appears in only one place in K(n 2 ). That is given by the event C n 1 and needed to assemble short words into longer words.
The third phase of the algorithm is just using the first two phases of the algorithm but with a parameter different from n. Instead the parameter is n 0.25 . So, we don't need any other special event to make this phase work. We only need what we have proven for the first two phases of the algorithm.
Finally, the forth phase of the algorithm needs the diamond property to hold for the short words in ξ K(n 2 ) , that is the event B n 3 to hold. Furthermore, the event C n 1 which guaranties that short words can not appear in two different places of ξ K(n 2 ) is also needed. These events are defined already for the first two phases of the algorithm.
The next section gives all the detailed definitions of these events, the proofs for their high probability, and also the rigorous proofs that these events make the different phases of the algorithm work. Although most of the main ideas are already given in the present section, and it might seem a little redundant, we feel that presenting them once informally but with all the details in a rigorous manner, will be very useful to understand better the algorithm.
Proof of Theorem 2.2
In what follows we will we say that the Branching random walk BRW visits z ∈ Z d at time t if η t (z) ≥ 1.
First phase
In this phase we will construct the set of SHORT W ORDS n . Recall that a string w 2 of size (ln n) 2 is in SHORT W ORDS n if there exist two sequences w 1 and w 3 both of size (ln n) 2 and such that:
1. w 1 w 2 w 3 appears in the observations before time n 2 .
2. The only string w of size (ln n) 2 such that w 1 ww 3 appears in the observations up to time n 4 is w 2 .
Let W (ξ K(4n) ) and W (ξ K(n 2 ) ) be the sets of all words of size (ln n) 2 in ξ K(4n) and ξ K(n 2 ) respectively, then we are going to show that with high probability the set SHORT W ORDS n satisfies that W (ξ K(4n) ) ⊆ SHORT W ORDS n ⊆ W (ξ K(n 2 ) ).
We need the following results:
Lemma 3.1 Let B n 1 be the event that up to time n 2 all the sites in K(4n) are visited by the BRW more than exp(cn) times, where c is a constant independent of n, then there exists C > 0 such that
P (B n 1 ) ≥ 1 − e −Cn 2 .
Proof. We only sketch the proof: it is elementary to obtain that by time n 2 /2 the process will contain at least e δn 2 particles (where δ is small enough), with probability at least 1 − e −Cn 2 . Then, consider any fixed x ∈ K; for each of those particles, at least one offspring will visit x with probability at least cn −(d−2) , and this implies the claim of Lemma 3.1.
Lemma 3.2 Let B n 2 be the event that up to time n 2 for every nearest neighbor path of length 3(ln n) 2 − 1 contained in K(4n), there is at least one particle which follows that path. Then, for all n large enough we have:
P (B n 2 ) ≥ 1 − exp[−c 1 n],
where c 1 > 0 is constant independent of n.
Proof. Let R z be a nearest neighbor path of length 3(ln n) 2 − 1 in K(4n) starting at z ∈ K(4n). By Lemma 3.1 we know that with high probability up to time n 2 all the sites in K(4n) are visited by the BRW more than exp(cn) times, where c is a constant independent of n. Suppose we have been observing the BRW up to time n 2 , then define for any z ∈ K(4n) the following variables;
Y ij = 1 after the i-th visit to z, the corresponding particle follows the path R z j 0 otherwise, where i = 1, . . . , exp(cn) and j = 1, . . . , (2d) 3(ln n) 2 −1 . Note that the variables Y ij 's are independent because all the particles are moving independently between themselves. We are interested on the event
(2d) 3(ln n) 2 −1 j=1 e cn i=1 {Y ij = 1} , (3.1)
i.e, for every path of length 3(ln n) 2 − 1 starting at z ∈ K(4n), up to time n 2 , there is at least one visit to z, such that the corresponding particle on z follows it. Let Z j = i Y ij , thus Z j counts the number of times R j is followed, and it is binomially distributed with expectation and variance given by
E[Z j ] = exp(cn) 1 2d 3(ln n) 2 −1 and V [Z j ] = exp(cn) 1 2d 3(ln n) 2 −1 1 − 1 2d 3(ln n) 2 −1 . Observe that (3.1) is equivalent to (2d) 3(ln n) 2 −1 j=1
{Z j ≥ 1} , then by Chebyshev's inequality we have Now note that for any constant 0 < c 1 < c, the right side of (3.3) is less than exp(−c 1 n), for n large enough. Thus P (B nc 2 ) → 0 as n → ∞. In what follows we will denote by T w the diamond associated with a word w appearing in a certain place in the scenery. For the definition of diamond associated with a word see (2.2.3).
P (Z j ≤ 0) ≤ V [Z j ] E 2 [Z j ] < exp[(3(ln n) 2 − 1) ln 2d − cn],
Lemma 3.3 Let B n
3 be the event that for any word w of size (ln n) 2 contained in ξ K(n 2 ) , every nearest neighbor walk path R generating w on ξ K(n 4 ) must start and end in the diamond associated to w, i.e., R(0) ∈ T w and R((ln n) 2 − 1) ∈ T w . Then, P (B n 3 ) ≥ 1 − 2 2d+1 exp 8d ln(n + 1) + (ln n) 2 ln(2d/κ) , which goes to 1 as n → ∞ because we have assumed κ > 2d.
Proof. Take without loss of generality a sequence read in a straight way from left to right, i.e., w = ξ(x)ξ(x + e 1 )ξ(x + 2 e 1 ) . . . ξ(x + ((ln n) 2 − 1) e 1 ), and let R z be a nearest neighbor walk (non-random) of length (ln n) 2 − 1 starting at z with z ∈ K(n 4 ) \ T w . Since a nearest neighbor path at each unit of time only make steps of length one, then it follows that w i is independent of ξ(R(0)), ξ(R(1)), . . . , ξ(R(i − 1)), as well as of w 1 , . . . , w i−1 . Hence,
P [w = ξ(R(0))ξ(R(1)) . . . ξ(R((ln n) 2 − 1))] = (1/κ) (ln n) 2 (3.4)
(recall that κ is the number of colors). Let P n z be the set of all nearest neighbor paths of length (ln n) 2 − 1 starting at z, and note that P n z contains no more than (2d) (ln n) 2 −1 elements. For a fix z ∈ K(n 4 ) \ T w , let B n 3,z be the event that there is no nearest neighbor path R z of length (ln n) 2 −1 generating w. By (3.4), it follows that
P (B nc 3,z ) = P [∃R z ; ξ(R z ) = w] ≤ R z ∈P n z P (ξ(R z ) = w) ≤ (2d) (ln n) 2 −1 κ (ln n) 2 ≤ 2d κ (ln n) 2
. Now for any z ∈ K(n 4 ) \ T w , let B n 3S be the event that there is no nearest neighbor path R z of length (ln n) 2 − 1 generating w. Hence,
P (B nc 3S ) = P z∈K(n 4 )\Tw B nc 3,z ≤ z∈K(4n)\Tw P (B nc 3,z ) ≤ (2n 4 + 1) 2d 2d κ (ln n) 2 Now by symmetry P (B nc 3 ) ≤ 2P (B nc 3S ), so that P (B nc 3 ) ≤ 2(2n 4 + 1) 2d 2d κ (ln n) 2 ≤ 2(2(n + 1) 4 ) 2d 2d κ (ln n) 2
= 8d ln(n + 1) + (ln n) 2 ln(2d/κ) .
Lemma 3.4
Let B n 4 be the event that for every two points x, y ∈ K(n 2 ) and such that there is a nearest neighbor path R of length (ln n) 2 − 1 going from x to y, one of the following alternatives holds:
(a) x and y are at distance (ln n) 2 − 1 and on the same line, that means, along the same direction of a coordinate, or (b) there exists two different nearest neighbor paths R 1 and R 2 of length (ln n) 2 − 1 both going from x to y but generating different observations, i.e., ξ(R 1 ) = ξ(R 2 ).
Then, we have
P (B 4 n ) ≥ 1 − exp[−0.5(ln n) 2 ln κ].
Proof. Let x and y be two points in K(n 2 ) and such that there is a nearest neighbor path R of length (ln n) 2 − 1 going from x to y, so the distance between x and y, d(x, y) ≤ (ln n) 2 − 1.
Suppose that d(x, y) = (ln n) 2 − 1, which is defined as the length of the shortest path between x and y. If x and y are not on the same line, then, there exists two paths R 1 and R 2 of length (ln n) 2 − 1 both going from x to y and not intersecting anywhere, except in x and y. This means that R 1 (0) = R 2 (0) = x and R 1 ((ln n) 2 − 1) = R 2 ((ln n) 2 − 1) = y, but for all j 1 , j 2 strictly between 0 and (ln n) 2 − 1, we find R 1 (j 1 ) = R 2 (j 2 ). Since the scenery ξ is i.i.d, it thus follows that
P [ξ(R 1 (0)) = ξ(R 2 (0)), . . . , ξ(R 1 ((ln n) 2 − 1)) = ξ(R 2 ((ln n) 2 − 1))] = 1 κ (ln n) 2 −2 .
(3.5)
Now suppose that d(x, y) < (ln n) 2 − 1. Let R 1 be a path which makes a cycle from x to x and then going in shortest time from x to y, and R 2 be a path which follow first a shortest path between x and y − 1, next makes a cycle from y − 1 to y − 1 and then go to y. If neither the cycle from x to x intersects the shortest path which makes part of R 2 nor the cycle from y − 1 to y − 1 intersects the shortest path which makes part of R 1 , then we have that for i = 0, . . . , (ln n) 2 − 1, the positions taken by R 1 and R 2 are different, i.e., R 1 (1) = R 2 (1), . . . , R 1 ((ln n) 2 − 2) = R 2 ((ln n) 2 − 2). Hence, ξ(R 1 (i)) is independent of ξ(R 2 (i)) for i = 1, 2, . . . , (ln n) 2 − 2 and (3.5) holds again.
Let B n 4xy be the event that there exist two nearest neighbor paths S and T going from x to y with d(x, y) ≤ (ln n) 2 − 1, but generating different observations, and let B n 4 = x,y B n 4xy , where the intersection is taken over all x, y ∈ K(n 2 ) such that d(x, y) ≤ (ln n) 2 − 1. By (3.5) it follows that and observe that the right side of (3.7) is for all n large enough less than exp(−0.5(ln n) 2 ln κ). This finishes this proof.
P (B nc 4xy ) ≤ exp[−((ln n) 2 − 2) ln κ],(3.
Lemma 3.5 Let B n 5 be the event that up to time n 4 every path of length 3(ln n) 2 − 1 in K(n 2 ) gets followed at least once by a particle. Then
P (B n 5 ) ≥ 1 − exp[−c 2 n 2 ],
where c 2 > 0 is a constant not depending on n.
Proof. Note that the event B m 2 from Lemma 3.2, where we take m = n 2 gives that up to time n 4 , every path in K(4n 2 ) of length 12 ln n − 1 gets followed by at least one particle. So, this implies that B n 5 holds and hence
B n 2 2 ⊂ B n 5 .
The last inclusion above implies
P (B n 5 ) ≥ P (B n 2 2 ). (3.8)
We can now use the bound from Lemma 3.2 for bounding the probability of P (B n 2 2 ). Together with (3.8), this yields P (B n 5 ) ≥ 1 − exp[d ln(8n 2 + 1) + (6(ln n 2 ) 2 − 2) ln 2d − cn 2 ]. (3.9) Now note that for any constant c 2 > 0 for which c > c 2 , we have that: for all n large enough we have that the bound on the right side of inequality (3.9) is less than exp(−c 2 n 2 ) which finishes this proof.
Lemma 3.6 [The first phase works.] Let B n designate the event that every word of size (ln n) 2 in ξ K(4n) is contained in the set SHORT W ORDS n , and also all the strings in SHORT W ORDS n belong to W (ξ K(n 2 ) ), then
B n 1 ∩ B n 2 ∩ B n 3 ∩ B n 4 ∩ B n 5 ⊂ B n .
Proof. We start by proving that every word in W (ξ K(4n) ) of size (ln n) 2 , say w 2 , is contained in SHORT W ORDS n . Then there exist two integer points x, y ∈ K(4n) on the same line, i.e., along the same direction of a coordinate, and at a distance 3(ln n) 2 − 1 such that "w 2 appears in the middle of the segmentxy". By this we mean that there exists a canonical vector e (such a vector consists of only one non-zero entry which is 1 or −1), so that w 1 w 2 w 3 = ξ x ξ x+ e ξ x+2 e . . . ξ y ,
where w 1 and w 2 are words of size (ln n) 2 and w 1 w 2 w 3 designates the concatenation of w 1 , w 2 and w 3 . By the event B n 2 , up to time n 2 there is at least one particle which will go from x to y in exactly 3(ln n) 2 − 1 steps. That is up to time n 2 there is a particle which follows the "straight path from x to y" given by
x, x + e, x + 2 e, . . . , y.
When doing so, we will see in the observations the string w 1 w 2 w 3 . Thus the triple (w 1 , w 2 , w 3 ) satisfies the first criteria of the first phase of Λ n . It needs also to pass the second criteria to be selected. To see that (w 1 , w 2 , w 3 ) satisfies second the criteria, let us assume that w is a word of size (ln n) 2 so that the concatenation w 1 ww 2 appears in the observation before time n 4 . Then there exists a nearest neighbor walk path R of length 3(ln n) 2 − 1 generating w 1 ww 2 on ξ K(n 4 ) . By this we mean that imR ⊂ K(n 4 ) and ξ • R = w 1 ww 2 . By the event B n 3 we have that R((ln n) 2 − 1) is in the diamond T w 1 associated with w 1 and R(2(ln n) 2 ) is in the diamond T w 3 associated with w 3 . So, when we take the restriction of R to the time interval [(ln n) 2 −1, 2(ln n) 2 ] we get a nearest neighbor walk going in (ln n) 2 steps from T w 1 to T w 3 . The only way to do this is to go in a straight way from the point x + ((ln n) 2 − 1) e to x + (2(ln n) 2 ) e. (The reason being that the distance between T w 1 and T w 3 is (ln n) 2 and the only pairs of points (x ′ , y ′ ) so that x ′ ∈ T w 1 and y ′ ∈ T w 3 and located at that distance (ln n) 2 from each other, are x ′ = x + ((ln n) 2 − 1) e and y ′ = x + (2(ln n) 2 ) e.) So, during the time interval [(ln n) 2 − 1, 2(ln n) 2 ] we have that R is walking in a straight way on the middle part of the segmentxy, that is walking in a straight way from x ′ to y ′ . Hence, during that time R is generating in the observation the word w 2 . This prove that w = w 2 . Hence, the triple (w 1 , w 2 , w 3 ) also passes the second criteria of the first phase of our algorithm, which implies that w 2 ∈ SHORT W ORDS n , and hence W (ξ K(4n) ) ⊂ SHORT W ORDS n .
It remains to show that if the triple (w 1 , w 2 , w 3 ) gets selected through the first phase of our algorithm (hence passes the two selection criteria given there), then indeed w 2 is a word of ξ K(n 2 ) . Now, to pass the first selection criteria, we have that the concatenation w 1 w 2 w 3 must appear before time n 2 in the observations. Since the first particle starts at time 0 in the origin, by time n 2 all the particles must be still contained in the box K(n 2 ). Hence, there must exist a nearest neighbor path R of length 3(ln n) 2 − 1 which generates w 1 w 2 w 3 on ξ K(n 2 ) . Hence imR ⊂ K(n 2 ) and w 1 w 2 w 3 = ξ • R.
Assume that the restriction of R to the time interval [(ln n) 2 − 1, 2(ln n) 2 ] would not be a "straight walk" on a line. Then, by the event B n 4 there would exist a modified nearest neighbor walk R ′ of length 3(ln n) 2 −1 for which the following two conditions are satisfied:
1. Restricted to the time interval [(ln n) 2 − 1, 2(ln n) 2 ], we have that R ′ generates a string w different from w 2 on ξ.
2. Outside that time interval, R ′ generates the same observations w 1 and w 3 as R.
Summarizing: the nearest neighbor walk R ′ generates w 1 ww 3 on ξ K(n 2 ) , where w = w 2 . But by the event B n 5 , every nearest neighbor-walk of length 3(ln n) 3 − 1 in K(n 2 ) gets followed at least once by a particle up to time n 4 . Hence, at least one particle follows the path of R ′ before time n 4 . Doing so it produces the string w 1 ww 3 with w = w 2 before time n 4 in the observations. This implies however that the triple (w 1 , w 2 , w 3 ) fails the second selection criteria for phase one of our algorithm. This is a contradiction, since we assumed that (w 1 , w 2 , w 3 ) gets selected through the first phase of Λ n (and hence passes the two selection criteria given there). This proves by contradiction, that R restricted to the time interval [(ln n) 2 − 1, 2(ln n) 2 ] can only be a "straight walk". Hence the sequence generated during that time, that is w 2 can only be a word of the scenery ξ in K(n 2 ). This proves that w 2 is in W (ξ K(n 2 ) ) and then SHORT W ORDS n ⊂ W (ξ K(n 2 ) ).
Second phase
In this phase the words of SHORT W ORDS n are assembled into longer words to construct the set of LONGW ORDS n using the assembling rule: To puzzle two words together of SHORT W ORDS n , the words must coincide on at least (ln n) 2 − 1 consecutive letters. To get a correct assembling we will need that the short words could be placed together in a unique way. Lemma 3.7 Let C n 1 be the event that, for all x, y ∈ K(n 2 ), the words ξ x ξ x+ e i . . . ξ x+((ln n) 2 −1) e i and ξ y ξ y+ e j . . . ξ y+((ln n) 2 −1) e j are identical only in the case x = y and e i = e j . Then,
P (C n 1 ) ≥ 1 − exp[2d ln(2n 2 + 1) − ((ln n) 2 − 1) ln κ].
Proof. Let x and y be two points in K(n 2 ) and define the event
C x,y = {ξ x ξ x+ e i ξ x+2 e i . . . ξ x+((ln n) 2 −1) e i = ξ y ξ y+ e j ξ y+2 e j . . . ξ y+((ln n) 2 −1) e j },
with e i and e j two canonical vectors in Z d . Clearly,
C n 1 = x,y C x,y ,
where the intersection above is taken over all (x, y) ∈ K(n 2 ), and it leads to . Hence,
P C nc 1 ≤ x,y P (C c x,y ).P C nc 1 ) ≤ (2n 2 + 1) 2d 1 κ (ln n) 2 −1 < exp[2d ln(2n 2 + 1) − ((ln n) 2 − 1) ln κ]. (3.11)
Lemma 3.8 [The second phase works.] Let C n designate the event that every word of size 4n in ξ K(4n) is contained in LONGW ORDS n , and all the words in LONGW ORDS n belong to W 4n (ξ K(n 2 ) ), i.e.,
W 4n (ξ K(4n) ) ⊆ LONGW ORDS n ⊆ W 4n (ξ K(n 2 ) ),
where W 4n (ξ K(4n) ) and W 4n (ξ K(n 2 ) ) are the set of all words of size 4n in ξ K(4n) and ξ K(n 2 ) respectively, then, B n ∩ C n 1 ⊂ C n . Proof. Let W (ξ K(4n) ) and W (ξ K(n 2 ) ) be the set of all words of size (ln n)2 in ξ K(4n) and ξ K(n 2 ) respectively. Once the first phase has worked, i.e., when B n occurs we have
W (ξ K(4n) ) ⊆ SHORT W ORDS n ⊆ W (ξ K(n 2 ) ).
If the short words can be placed together in a unique way using some assembling rule, and it is done until getting strings of total exactly equal to 4n, then every word of size 4n in ξ K(4n) is contained in the set of assembled words, i.e., in LONGW ORDS n . On the other hand, if the assembled process is made using all words in SHORT W ORDS n , then all the words in LONGW ORDS n belong to W 4n (ξ K(n 2 ) ) because SHORT W ORDS n ⊆ W (ξ K(n 2 ) ).
Under the assembling rule given in Lemma 3.7 we conclude that B n ∩ C n 1 ⊂ C n .
Third phase
In this phase we use the previous two phases of Λ n but with the parameter n 0.25 instead of n. The idea is to take one long word from LONGW ORDS n 0.25 , say v, which will be of size 4n 0.25 instead of 4n, then, select any long word from LONGW ORDS n , say w, which contains v in its middle. In this manner we should hopefully get a word which has its middle not further than √ n from the origin.
In the next lemma we show that when the first two phases of our algorithm with parameter n as well as n 0.25 both work, then the third phase must work as well.
Lemma 3.9 [The third phase works.] Let D n be the event that the third phase of Λ n works. This means that the long-word of LONGW ORD n selected by the third phase has its center not further than n 0.5 from the origin. Thus
C n 1 ∩ C n ∩ C n 0.25 ⊂ D n .
Proof. Consider any word from LONGW ORD n that contains in its middle a word from LONGW ORDS n 0.25 , i.e., take w = ξ x ξ x+ e i ξ x+2 e i . . . ξ x+4n−1 e i in LONGW ORD n such that v = ξ x+2n e i , . . . , ξ x+(2n+4n 0.25 −1) e i belongs to LONGW ORDS n 0.25 .
By C n 0.25 the first two phases of the algorithm with parameter n 0.25 work. This implies that v is a word (of size 4n 0.25 ) contained in ξ K(n 0.5 ) . It is not difficult to see that C n 1 implies that any word of that size which appears in ξ K(n 2 ) , appears in a "unique position" therein. By C n all the words of LONGW ORD n are contained in ξ K(n 2 ) . Thus, when a word w of LONGW ORD n contains a word v of LONGW ORD n 0.25 , then the two words must lie (in K(n 2 )) on top of each other in a unique way, which implies that the middle of w is also the middle of v. By C n 0.25 we have that the middle of v (the way v appears in ξ K(n 0.5 ) ) is not further from the origin than n 0.5 . Hence, the middle of w (in where it appears in ξ K(n 2 ) ) is also not further than n 0.5 from the origin. This finishes our proof.
Fourth phase
For the fourth and last phase to work correctly, we need to be able to identify which words contained in ξ K(n 2 ) are neighbors of each other. Let us give the definition of neighboring words.
Let I be a box of Z d and w and v be two words (of the same length) contained in ξ I . We say that w and v are neighbors of each other if there exist x ∈ I and u, s ∈ {± e i , i = 1, . . . , d} such that: w = ξ x ξ x+ u ξ x+2 u . . . ξ k u and v = ξ x+ s ξ x+ u+ s ξ x+2 u+ s . . . ξ x+k u+ s , where s is orthogonal to u and all the points x + s + i u and x + i u are in I for all i = 0, 1, 2, . . . , k.
In other words, two words w and v contained in the restriction ξ I are called neighbors if we can read them in positions which are parallel to each other and at distance 1.
Lemma 3.10 [The fourth phase works.] Let F n denote the event that for the words of LONGW ORDS n the three conditions in the forth phase of Λ n allows to correctly identify and chose neighbors. Interestingly, the events B n 2 , B n 3 and C n 1 are enough for F n to occur. That is, B n 2 ∩ B n 3 ∩ C n 1 ⊂ F n II) Starting at x, make one step in some direction orthogonal to u, say with respect to s (in 2 dimensions, say one step up) then all steps with respect to u (to the right), and once making one step with respect to − s (one step down) in order to reach y.
III)
Starting at x + s − u instead of x and arriving in y + s + u instead of y.
Since the nearest neighbor walk R between time (ln n) 2 and 2(ln n) 2 +1 must be walking from the diamond T a to the diamond T c in (ln n) 2 + 1 steps, then it must satisfy during that time one of the three conditions above.
If condition II holds, then w b is the word which is "written" in the scenery ξ between x + s and y + s, i.e., w b = ξ x+ s ξ x+ s+ u ξ x+ s+2 u . . . ξ y+ s .
That shows that the line between the two points x+ s and y+ s is where the word w b is written in the scenery. Now observe that w appears in ξ K(n 2 ) because w is in LONGW ORDS N , and by the second condition w must contain the word w b , so w must contain the points x + s and y + s. By the event C n 1 , any word of size (ln n) 2 appears only in one place in ξ K(n 2 ) , then as w b has size (ln n) 2 , the place where the word w is written in ξ K(n 2 ) is a line parallel to the linexy and at distance 1. This means that the word w is a neighbor of the word v in ξ K(n 2 ) .
When condition III) holds, a very similar argument leads to the same conclusion that w and v are neighbors in ξ K(n 2 ) .
Finally, when condition I) holds, then we would have that v a w b v c appears in ξ K(n 2 ) , but we have also that v a v b v c appears in ξ K(n 2 ) (since we take the words w and v from the set LONGW ORDS n and since by the event C n 1 these are words from the scenery ξ K(n 2 ) ). However the word w b is longer than v b . This implies that either v a or v c (or both together) must appear in two different places in ξ K(n 2 ) . But this would contradict the event C n 1 . Hence, we can exclude this case.
Thus, we have proven that if the events B n 3 and C n 1 all hold, then the words selected to be put next to each other in phase 4 of the algorithm are really neighbors in the way they appear in the restricted scenery ξ K(n 2 ) . So, when, we use the criteria of the 4th phase of the algorithm to determine the words which should be neighbors in the scenery we make no mistake, by identifying as neighbors whilst they are not.
The next question is whether for all the neighboring words v and w in ξ K(4n) we recognize them as neighbors, when we apply the fourth phase of our algorithm. This is indeed true, due to the event B n 2 . Let us explain this in more detail. Let v and w be to neighboring words in ξ K(4n) . We also assume that both v and w have length 4n, are written in the direction of u, and w is parallel and at distance 1 (in the direction s perpendicular to u) of v. Now, take anywhere approximately in the middle of v, three consecutive strings v a , v b and v c . Take them so that v a and v c have size (ln n) 2 , but v b has size (ln n) 2 − 2. Hence, we assume that the concatenation v a v b v c appears somewhere in the middle of v. Let x ∈ Z d designate the point where v a ends and let y be the point where v c starts.
Hence v c = ξ y ξ y+ u ξ y+2 u . . . ξ y+((ln n) 2 −1) u Also the points x and y are at distance (ln n) 2 −1 from each other and on the line directed along u. Finally, v a = ξ x−((ln n) 2 +1) u . . . ξ x−2 u ξ x− u ξ x .
So, the word v is written on the line passing through x and y. Note that the word w is written on the line which passes through the points x + s and y + s. Now, because of the event B n 2 , for every nearest neighbor walk path of length 3(ln n) 2 − 1 contained in the region K(4n) there is at least one particle following that path up to time n 2 .
Take the nearest neighbor walk path R on the time interval [1, 3(ln n) 2 ] which starts in x − ((ln n) 2 − 1) u and then goes in the direction of u (ln n) 2 steps. Next goes one step in the direction of s (and hence reaches the line where w is written), and then, R takes (ln n) 2 steps in the direction of u and reads part of the word w. That part we designate by w b . From there one step in the direction (− s), to reach the point y and then all remaining (ln n) 2 steps are taken in the direction u. During those remaining steps the walk generates the color record v c . Such a nearest neighbor walk generates thus the color record v a w b v c :
ξ • R = v a w b v c
Hence, by B n 2 at least one particle up to time n 2 follows R and generates the color record v a w b v c . Similarly we can chose a neighbor path that generates v a v b v c . Since w b and v b are contained in w and v, respectively, then w and v get selected as neighbors by the forth step of our algorithm.
Lemma 3.11 [The algorithm Λ n works.] Let A n be the event that Λ n works. This means that the outcome after the fourth phase of Λ n is "correct". Thus B n ∩ C n ∩ D n ∩ F n ⊂ A n Proof. Recall we said our algorithm works as a whole correctly, if there exists a box I with size 4n with center closer than n 0.5 from the origin and such that the restriction ξ I is equivalent to our reconstructed piece of scenery. If the last phase of the algorithm works correctly, then we would like to see that outcome. The event that the outcome after the fourth phase is "correct", is denoted by A n (as already mentioned). Now, assume that we have the correct short and long words constructed in phase one and two, that is, assume that the events B n and C n occur. Assume also that the third phase of our algorithm works correctly and we get the one long word picked at the end of phase 3 to be close enough to the origin, hence the event D n occurs. When these three events occur (phase 1, 2 and 3 work) for the final phase of the algorithm to work, we then only need to identify which words of LONGW ORDS n are neighbors of each other (remember that the words in LONGW ORDS n are words of ξ K(n 2 ) ), so, what we really need after is finding a way to determine which words of ξ K(n 2 ) are neighbors of each other, and that holds by F n .
Already at this point we observe that, if we have a collection of objects to be placed in a box of Z d (in a way that all the sites of this box become occupied by exactly one object), and we know which objects should be placed in neighboring sites, then there is a unique (up to translations/reflections) way to do it. This will assure that the reconstruction works correctly once we identified the neighboring words.
Proof. [Proof of Theorem 2.2.] The last lemma above, Lemma 3.11, tells us that for the algorithm Λ n to work correctly, we just need the events How do we now reconstruct the infinite scenery ξ? So far we have seen only methods to reconstruct a piece of ξ on a box of size 4n close to the origin. The algorithm was denoted by Λ n , and the event that it works correctly is designated by the event A n . By working correctly, we mean that the piece reconstructed in reality is centered not further than √ n from the origin. In general, it is not possible to be sure about the exact location. So, instead we are only able to figure out the location up to a translation of order √ n. Also, the piece is reconstructed only up to equivalence, which means that we do not know in which direction it is rotated in the scenery or if it was reflected. Now, the probability that the reconstruction algorithm Λ n at level n does not work is small in the sense that it is finitely summable over n: n P (A nc ) < ∞.
So, by Borel-Cantelli lemma, when we apply all the reconstruction algorithms Λ 1 , Λ 2 , . . ., we are almost sure that only finite many of them will not work. We use this to assemble the sceneries and get the whole scenery a.s. Let us call ξ n the piece reconstructed by Λ n . Hence, ξ n := Λ n (χ 0 χ 1 . . . χ n 4 ),
where ξ n is a piece of scenery on a box K(4n). The next task is to put the reconstructed pieces ξ n together so that their relative position to each other is the same in the scenery ξ.
For this we will use the following rule. We proceed in an inductive way in n:
1. Letξ n designate the piece ξ n which has been moved so as to fit the previously placed pieces. Hence,ξ n is equivalent to ξ n .
2. We place ξ n by making it coincide with the previously placed ξ n−1 on a box of side length at least √ n. In other words,ξ n is defined to be any piece of the scenery equivalent to ξ n , and such that on a restriction to a box of size √ n it coincides with ξ n−1 . If no such box of size √ n can be found inξ n−1 which is equivalent to a piece of the same size in ξ n , then we "forget" about the previously placed pieces and just put the piece ξ n on the box K(4n), that is, we center it around the origin.
3. The end result after infinite time is our reconstructed scenery denoted bȳ ξ : Z d → {0, 1, . . . , κ}.
We will prove that a.s.ξ is equivalent to ξ. So,ξ represents our reconstructed ξ (since reconstruction in general is only possible up to equivalence). Forξ we simply take the point-wise limit of theξ n :ξ ( z) := lim n→∞ξ n .
For the above definition to be meaningful, takeξ n to be the extension ofξ n to all Z d by adding 0's whereξ n is not defined.
To conclude the proof of Theorem 2.1, it is enough to prove that the above algorithm defines a sceneryξ which is almost surely equivalent to ξ: Theorem 4.1 We have that P (ξ ≈ξ) = 1.
Proof. Let G n be the event that in the restriction of ξ to the box K(2n + 2) any two restrictions to a box of size √ n are different of each other. By this we mean, that if V 1 and V 2 are two boxes of size √ n contained in K(2n + 1), then if the restriction ξ V 1 is equivalent to ξ V 2 then V 1 = V 2 . Also, for G n to hold, we require that for any box V 1 ⊂ K(2n + 2) of size √ n the only reflection and/or rotation which leaves ξ V 1 unchanged is the identity. Now note that when the event G n occurs, and the piece ξ n+1 and ξ n are correctly reconstructed in the sense defined before, then our placing them together works properly. This means that in that case, the relative position ofξ n+1 andξ n is that same as the corresponding pieces in ξ. It is elementary to obtain that the event G n has probability at least 1−c 1 n d e −c 2 n d/2 . This means that n P (G nc ) < ∞, and so, by Borel-Cantelli lemma, all but a finite number of the events G n occur. Also, we have seen that the algorithm at level n has high probability to do the reconstruction correctly, and n P (A nc ) < ∞.
Hence, again by Borel-Cantelli lemma, all but a finite number of the reconstructed scenery ξ n will be equivalent to a restriction of ξ to a box close to the origin. We also see that the close to the origin for the box of the n-th reconstruction means not further than √ n. Thereof we have that all but a finite number of the piecesξ n are positioned correctly with respect to each other. Since we take a limit for getting ξ, a finite number ofξ n 's alone have no effect on the limit, and so the algorithm works.
Example 2. 1
1Let the sequence be equal to D = T AT CAGT, and suppose a biologist in his laboratory is able to find all subsequences of size 5. Typically, the direction in which these subsequences appear in D is not known. The set of all subsequences of size 5 which appear in D is T AT CA, AT CAG, T CAGT, as well as their reverses ACT AT, GACT A, T GACT.
Why? Consider the set of all subsequences of size 4: T AT C, AT CA, T CAG, CAGT and their reverses CT AT, ACT A, GACT, T GAC.
D
i+1 . . . D i+C ln n = D j+1 . . . D j+C ln n . (2.5)
Example 2. 2
2Let us assume d = 2 and the scenery ξ restricted to the box [0, 4] × [0, 4] be given by:
Example 2. 3
3Take the following piece of a two dimensional scenery which would be the restriction of ξ to the [0,6]
Example 2. 4
4For this let the restriction of the scenery ξ to [0, 16] × [0, 4]
2.2. 6
6Finding which long words are neighbors in the 4th phase of the algorithm Λ n . The fourth phase of the algorithm is then concerned with finding the relative position of the longer reconstructed words to each other. More precisely it tries to determine which long words are neighbors of each other. For the exact definition of neighboring long words check out subsection 3.4. Let us explain it through another example.Example 2.6 Consider the piece of scenery ξ [0,16]×[0,4] given in (2.16) and let us designate by v a the green word, by v c the brown word and by v b the word between v a and v c , i.e.,v a = ξ (1,2) ξ (2,2) ξ (3,2) ξ (4,2) ξ (5,2) = 43912, v c = ξ (9,2) ξ (10,2) ξ(11,2) ξ(12,2) ξ(13,2) = 47617,
by w the long word written one line above, i.e., w := ξ (0,3) ξ (1,3) ξ(2,3) . . . ξ(16,3) = 75076118258674042.
j ≤ 0} < exp[(6(ln n) 2 − 2) ln 2d − cn].(3.2) Since the number of sites in K(4n) is (8n + 1) d , by (3.2) it follows that P (B nc 2 ) < (8n + 1) d exp[(6(ln n) 2 − 2) ln 2d − cn]. (3.3)
<
[2(ln n) 2 − 3] d (2n 2 + 1) d exp[−((ln n) 2 − 2) ln κ] = exp[d ln(2(ln n) 2 − 3) + d ln(2n 2 + 1) − ((ln n) 2 − 2) ln κ],(3.7)
( 3 . 10 )
310If x = y and e i = e j , observe that the words intersect themselves only at the first position and since the scenery is i.i.d., then P (C c x,y ) = 1 κ (ln n) 2 −1 . On the other hand, whenx = y, it is somewhat similar to the previous case, i.e., the words intersect themselves at most by only one position, thus P (
Proof. We need to show that when all the events B n 2 , B n 3 and C n 1 occur, then we identify correctly and chose the long words from LONGW ORDS n which are neighbors of each other.If two words v and w belonging to LONGW ORDS N were selected by the fourth phase of our algorithm to be put on top of each other, this means that Λ n "estimated" that v and w were neighbors, is because the following three conditions were satisfied:1. There exist 4 words v a , v b , v c and w b having all size (ln n) 2 except for v b which has size (ln n) 2 − 2 and such that,3. The position of the middle letter of v b in v should be the same as the middle letter position of w b in w.Assume that the three previous conditions are satisfied, and let u be the direction of the word v, so, we have two points x and y in Z d such that y = x + ((ln n) 2 − 1) u and• to the left (with respect to the direction of u) from x we read v a :• between x and y we read v b :v b = ξ x+ u ξ x+2 u . . . ξ y− u , and• to the right of y we read v c : v c = ξ y ξ y+ u ξ y+2 u . . . ξ y+((ln n) 2 −1) u .By the first condition we know that up to time n 4 we observe at least once the concatenated word v a w b v c . Hence, there exists a nearest neighbor walk R on the time interval [1, 3(ln n) 2 ] which generated v a w b v c on ξ K(n 4 ) , so thatNote that after time 2(ln n) 2 R generates v c , then by B n 3 we have that R(2(ln n) 2 + 1) must be in the diamond T c associated with v c . Similarly, we get that R((ln n) 2 ) must be in the diamond T a associated with v a . Now observe that to go in (ln n) 2 + 1 steps with a nearest neighbor walk from the diamond T a to the diamond T c , there are only 3 possibilities (remember that x and y are at distance (ln n) 2 − 1 from each other): I) Going from x to y always making steps with respect to u (in 2 dimensions, say to the right), and once making one step with respect to − u (one step to the left). No steps in the directions orthogonal to u.
Mixing properties of the generalized T, T −1 -process. F Hollander, J E Steif, J. Anal. Math. 72den Hollander, F. and Steif, J.E. (1997) Mixing properties of the generalized T, T −1 -process. J. Anal. Math. 72 165-202.
Distinguishing sceneries by observing the scenery along a random walk path. I Benjamini, H Kesten, J. Anal. Math. 69Benjamini, I. and Kesten, H. (1996) Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69 97-135.
Detecting defects in periodic scenery by random walks on. C D Howard, Z. Random Structures Algorithms. 81Howard, C.D. (1996) Detecting defects in periodic scenery by random walks on Z. Random Structures Algorithms. 8 (1), 59-74.
Orthogonality of measures induced by random walks with scenery. C D Howard, Combin. Probab. Comput. 53Howard, C.D. (1996) Orthogonality of measures induced by random walks with scenery. Combin. Probab. Comput. 5 (3), 247-256.
Distinguishing certain random sceneries on Z via random walks. C D Howard, Statis. Probab. Lett. 342Howard, C.D. (1997) Distinguishing certain random sceneries on Z via random walks. Statis. Probab. Lett. 34 (2), 123-132.
Detecting a single defect in a scenery by observing the scenery along a random walk path. Ito's Stochastic Calculus and Probability theory. H Kesten, SpringerTokyoKesten, H. (1996) Detecting a single defect in a scenery by observing the scenery along a random walk path. Ito's Stochastic Calculus and Probability theory. Springer, Tokyo. 171-183.
Distinguishing and reconstructing sceneries from observations along random walk paths. Microsurveys in Discrete Probability: DIMACS. H Kesten, Amer. Math. Soc41Providence, RIKesten, H. (1998) Distinguishing and reconstructing sceneries from observations along random walk paths. Microsurveys in Discrete Probability: DIMACS. Amer. Math. Soc., Providence, RI. 41, 75-83.
T,T −1 transformation is not loosely Bernoulli. S A Kalikov, Annals of Math. 115Kalikov, S.A. (1982) T,T −1 transformation is not loosely Bernoulli. Annals of Math. 115, 393-409.
Information recovery from a randomly mixed up message-text. J Lember, H Matzinger, Electronic Journal of Probability. 13Lember, J. and Matzinger, H. (2008) Information recovery from a randomly mixed up message-text. Electronic Journal of Probability. 13, 396-466.
Indistinguishable sceneries. Random Structures Algorithms. E Lindenstrauss, 14Lindenstrauss, E (1999) Indistinguishable sceneries. Random Structures Algo- rithms. 14, 71-86.
Scenery reconstruction in two dimensions with many colors. Löwe, M Matzinger, Ann. Appl. Probab. 124Löwe, M and Matzinger, M. (2002) Scenery reconstruction in two dimensions with many colors. Ann. Appl. Probab. 12 (4), 1322-1347.
Reconstructing a multicolored random scenery seen along a random walk path with bounded jumps. M Löwe, H Matzinger, F Merkl, Electronic Journal of Probability. 15Löwe, M. Matzinger, H. and Merkl, F. (2004) Reconstructing a multicolored random scenery seen along a random walk path with bounded jumps. Electronic Journal of Probability. 15, 436-507.
Reconstructing a 2-color scenery by observing it along a simple random walk path. Ann.Appl.Probab. 15Reconstructing a 2-color scenery by observing it along a simple random walk path. Ann.Appl.Probab. 15, 778-819.
Reconstructing a three-color scenery by observing it along a simple random walk path. H Matzinger, Random Structures Algorithms. 152Matzinger, H. (1999) Reconstructing a three-color scenery by observing it along a simple random walk path. Random Structures Algorithms. 15 (2), 196-207.
Reconstructing a 2-color scenery by observing it along a simple random walk path with holding. H Matzinger, Cornell UniversityMatzinger, H. (1999) Reconstructing a 2-color scenery by observing it along a simple random walk path with holding. Cornell University.
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Reconstructing a random scenery observed with random errors along a random walk path. H Matzinger, S W W Rolles, Probab. Theory Related Fields. 1254Matzinger, H. and Rolles, S.W.W. (2003) Reconstructing a random scenery observed with random errors along a random walk path. Probab. Theory Related Fields. 125 (4), 539-577.
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A Kolmogorov automorphism that is not a Bernoulli shift. D Ornstein, Matematika. 151In RussianOrnstein, D. (1971) A Kolmogorov automorphism that is not a Bernoulli shift. Matematika. 15 (1), 131-150 (In Russian).
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[] |
[
"A NOTE ON THE DIFFERENTIAL OPERATOR ON GENERALIZED FOCK SPACES",
"A NOTE ON THE DIFFERENTIAL OPERATOR ON GENERALIZED FOCK SPACES"
] |
[
"Tesfa Mengestie "
] |
[] |
[] |
It has long been known that the differential operator D represents a typical example of unbounded operators on many Banach spaces including the classical Fock spaces, the Fock-Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits boundedness, compactness and membership in the Schatten S p class spectral structures. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk D in the complex plane.2010 Mathematics Subject Classification. Primary 47B32, 30H20, 46E20; Secondary 46E22,47B33 .
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10.1016/j.jmaa.2017.10.008
|
[
"https://arxiv.org/pdf/1606.09144v2.pdf"
] | 119,271,192 |
1606.09144
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bd81e0c7dcd01333ea13982c3ce690b134ce4278
|
A NOTE ON THE DIFFERENTIAL OPERATOR ON GENERALIZED FOCK SPACES
4 Oct 2017
Tesfa Mengestie
A NOTE ON THE DIFFERENTIAL OPERATOR ON GENERALIZED FOCK SPACES
4 Oct 2017
It has long been known that the differential operator D represents a typical example of unbounded operators on many Banach spaces including the classical Fock spaces, the Fock-Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits boundedness, compactness and membership in the Schatten S p class spectral structures. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk D in the complex plane.2010 Mathematics Subject Classification. Primary 47B32, 30H20, 46E20; Secondary 46E22,47B33 .
Introduction
Various order differential operators play fundamental rolls in many part of mathematics including in the study of differential equations. Nevertheless, the operator Df = f ′ often appears as a canonical example of unbounded operators in many Banach spaces including the very classical Hilbert space L 2 (R), the space of continuous functions C([a, b]) with the supremum norm, and the likes. Its unboundedness on Fock spaces with the classical Gaussian weight e −|z| 2 and on generalized Fock spaces where the weight decays faster than the Gaussian weight was recently verified in [9]. The same conclusion was also drawn in [11] on the Fock-Sobolev spaces which are typical examples of generalized Fock spaces with weight decaying slower than the Gaussian weight. A natural question to consider is whether there could exist spaces of Fock type where this operator admits richer operator-theoretic properties. Said differently, we would like to know how the function-theoretic properties of the weight functions generating the spaces are related to the operator-theoretic properties of D. The central aim of this note is to investigate this and identify Fock type spaces where the operator D admits some basic spectral properties.
In view of the above discussion, if there could exist generalized Fock spaces on which the operator D acts in a bounded fashion, then the associated weight must decay slower than the k th order Fock-Sobolev spaces with weight e − |z| 2 2 +k log(1+|z|) , where k is a nonnegative integer; see [4,10,11] for further information on these spaces. Keeping this in mind, we consider the following setting.
Let m > 0, 0 < p < ∞, and F (m,p) be a class of generalized Fock spaces consisting of all entire functions f for which
f p (m,p) = C |f (z)| p e −p|z| m dA(z) < ∞,
where dA denotes the usual Lebesgue area measure on C. With this, we plan to find conditions on m (equivalently on the growth of ψ m (z) = |z| m ) under which D admits boundedness, compactness, and other operator-theoretic structures while acting between the spaces F (m,p) . It turns out that such structures do happen to exist only if the inducing weight function ψ m grows at a rate much slower than the corresponding weight function in the classical Gaussian case ψ 2 (z) = |z| 2 . We precise this in our first main result to follow.
Theorem 1.1. (i) Let 0 < p ≤ q < ∞ and m > 0. Then D : F (m,p) → F (m,q) is (a) bounded if and only if m ≤ 2 − pq pq + q − p . (b) compact if and only if m < 2 − pq pq + q − p .
(ii) Let 0 < q < p < ∞ and m > 0. Then the following statements are equivalent.
(a) D :
F (m,p) → F (m,q) is bounded; (b) D : F (m,p) → F (m,q) is compact; (c) It holds that m < 1 − 2 1 q − 1 p .
The result effectively identifies the generalized Fock spaces on which the differential operator admits boundedness and compactness operator-theoretic structures. In particular, when p = q, the operator D enjoys any of the basic spectral structures on F (m,p) only if the corresponding weight functions ψ m grow at most polynomials of degree not exceeding one. If p < q, then ψ m could grow a bite faster as pq pq + q − p < 1.
On the other hand, if p > q, then ψ m grows slower than a polynomials of degree one.
We note in passing that if we replace both the domain and target spaces by the corresponding growth type spaces F (m,∞) which consist of entire functions f for which
f (m,∞) = sup z∈C |f (z)|e −|z| m < ∞,
the same conclusion, m ≤ 1, follows which can be also seen for example in [1,7] as a particular instance. Our next main result gives a condition on the growth of |z| m under which D belongs to the Schatten S p (F (m,2) ) class and also identifies its spectrum.
Theorem 1.2.
(i) Let 0 < p < ∞, m > 0, and D : F (m,2) → F (m,2) is compact. Then it belongs to the Schatten S p (F (m,2) ) class for all p.
Preliminaries
In this section we collect a few basic facts which will be used in the proofs of the main results. From [5], the Littlewood-Paley type estimate 1
f p (m,p) ≃ |f (0)| p + C |f ′ (z)| p e −p|z| m (1 + |z|) p(m−1) dA(z) (2.1)
holds for functions f in the space F (m,p) . Such a formula characterizes the spaces in terms of derivatives, and plays a significant roll specially in the study of integral operators on the spaces. This is a key estimation result which helps us obtain our main result on the spectrum of the operator D in Theorem 1.2.
Proof. The proof of the lemma follows from some ideas stemmed in the proof of Proposition 1 in [5]. We argue in the direction of contradiction and assume that (2.2) fails to hold. Then, we can find a sequence of entire functions (f n ) satisfying f n e λ ∈ F (m,p) , C |f n (z)e λz | p e −p|z| dA(z) = 1 and
C |f ′ n (z)e λz | p e −p|z| dA(z) < 1 n .
Now, if K is a compact subset of C, the point evaluation estimate for functions in F (m,p) ( see the analysis in [6]) gives that
|f ′ n (z)e λz | C K |f ′ n (z)e λz | p e −p|z| dA(z) ≤ C 1 n p
for some positive constant C that depends only on K. From this it follows that the sequence f ′ n converges to zero uniformly on compact subset of C. This shows 1 The notation U (z) V (z) (or equivalently V (z) U (z)) means that there is a constant C such that U (z) ≤ CV (z) holds for all z in the set of a question. We write
U (z) ≃ V (z) if both U (z) V (z) and V (z) U (z).
that f n also converges to zero uniformly on the compact subsets. We may rewrite
1 = C |f n (z)e λz | p e −p|z| dA(z) = |z|≤r |f n (z)e λz | p e −p|z| dA(z) + |z|>r |f n (z)e λz | p e −p|z| dA(z). (2.3)
Now the first integral on the right-hand side of (2.3) tends to zero when n → ∞ since f n → 0 uniformly on {z ∈ C : |z| ≤ r}. On the other hand, the second integral is the tile of a convergent integral and hence tend to zero when r → ∞, and the contradiction follows.
We denote by K (m,w) the reproducing kernel of the space F (m,2) at the point w ∈ C. Because of the reproducing property of the kernel and Parseval identity, it holds that
K (m,w) (z) = ∞ n=1 K (m,w) , e n e n (z) and K (m,w) 2 (m,2) = ∞ n=1 |e n (w)| 2
for any orthonormal basis (e n ) n∈N of F (m,2) . An immediate consequence of this representation is that
∂ ∂w K (m,w) (z) = ∞ n=1 e n (z)e ′ n (w), and ∂ ∂w K (m,w) 2 (m,2) = ∞ n=1 |e ′ n (w)| 2 . (2.4)
An explicit expression for the reproducing kernel K (w,m) in the weighted space F (m,2) is still unknown apart from the case when m = 2. From [2], we already have an estimate for the norm
K (m,w) 2 (m,2) ≃ |w| m−2 e 2|w| m . (2.5)
As a consequence of this, we obtain the following useful estimate for our further consideration.
Lemma 2.2. For each w ∈ C, we have the asymptotic estimate ∂ ∂w K (m,w) 2 (m,2) ≃ K (m,w) 2 (m,2) |w| 2m−2 ≃ |w| 3m−4 e 2|w| m . (2.6)
Proof. For simplicity, setting Ψ(r) = r m−2 2 e r m and
f (z) = ∞ n=0 z n z n (m,2)
,
then we have that M 2 (r, f ) 2 ≃ π −π |f (re iθ )| 2 dθ ≃ (Ψ(r)) 2 .
If we show that lim sup r→∞ Ψ ′′ (r)Ψ(r) (Ψ ′ (r)) 2 < ∞ and Ψ ′ (r) ≃ Ψ(r)r m−1 , then our conclusion will follow from Lemma 21 of [6] as
M 2 (r, f ′ ) ≃ ∂ ∂w K (m,w) (m,2) ≃ Ψ ′ (r) ≃ Ψ(r)r m−1 .
To this end, we compute
Ψ ′ (r) = m − 2 2 r m−4 2 e r m + mr 3m−4 2 e r m = e r m r m−2 2 −1 m − 2 2 + mr m ≃ e r m r m−2 2 −1+m ≃ Ψ(r)r m−1 .
Furthermore, a computation shows that
(Ψ ′ (r)) 2 ≃ e 2r m r m−2+2(m−1) and Ψ ′′ (r) ≃ e r m r m−2 2 +2(m−1) from which we have lim sup r→∞ Ψ ′′ (r)Ψ(r) (Ψ ′ (r)) 2 ≃ lim sup r→∞ e 2r m r m−2 2 +2(m−1) r m−2 2 e 2r m r m−2+2(m−1) ≃ 1.
It has been a fairly common practice to test many operator-theoretic properties on the reproducing kernels for the spaces. In the present setting, no explicit expression is known for the kernel function. Thus, for proving our mains results, we will rather use another sequence of test function which replaces the role of the reproducing kernel. Such a sequence was first constructed in [3] and has been further used by several authors for example [6,13,9]. We introduce the sequence of test functions as follows. We set
τ m (z) = 1, 0 ≤ |(m 2 − m)z| < 1 |z| 2−m 2 |m 2 −m| 1 2 , |(m 2 − m)z| ≥ 1.
For a sufficiently large positive number R, there exists a number η(R) such that for any w ∈ C with |w| > η(R), there exists an entire function f (w,R) such that (i)
|f (w,R) (z)|e −|z| m ≤ C min 1, min{τ m (w), τ m (z)} |z − w| R 2 2 (2.7)
for all z ∈ C and for some constant C that depends on |z| m and R. In particular when z ∈ D(w, Rτ m (w)), the estimate becomes
|f (w,R) (z)|e −|z| m ≃ 1. (2.8)
(ii) f (w,R) belongs to F (m,p) and its norm is estimated by
f (w,R) p (m,p) ≃ τ 2 m (w), η(R) ≤ |w| (2.9)
for all p in the range 0 < p < ∞.
Another important ingredient in our subsequent consideration is the pointwise estimate for subharmonic functions |f | p , namely that
|f (z)| p e −p|z| m 1 σ 2 τ 2 m (z) D(z,στm(z)) |f (w)| p e −p|w| m dA(w) (2.10)
for all finite exponent p and a small positive number σ. The estimate follows from Lemma 2 of [13]. Next, we recall the notion of covering for the space C. We denote by D(w, r) the Euclidean disk centered at w and radius r > 0. Then, we record the following useful covering lemma which is essentially from [6,12]. Lemma 2.3. Let τ m be as above. Then, there exists a positive σ > 0 and a sequence of points z j in C satisfying the following conditions.
(i) z j ∈ D(z k , στ m (z k )), j = k;
(ii) C = j D(z j , στ m (z j )); (iii) z∈D(z j ,στm(z j )) D(z, στ m (z)) ⊂ D(z j , 3στ m (z j )); (iv) The sequence D(z j , 3στ m (z j )) is a covering of C with finite multiplicity N max .
Lemma 2.4. Let R be a sufficiently large number and η(R) be as before. If (z k ) is the covering sequence from Lemma 2.3, then the function
F = z k :|z k |>η(R) a k f (z k ,R) τ 2 p m (z k ) belongs to F (m,p) for every sequence (a k ) in ℓ p and also F (m,p) (a k ) ℓ p .
The proof the Lemma follows from a simple variant of the proof of Proposition 9 in [6] or Proposition 1 in [13].
Proof of the Main results
3.1. Proof of Theorem 1.1-Part (i). Let us first prove the necessity of the condition in part (i), and assume that D : F (m,p) → F (m,q) is bounded. Then, making use of the estimates in (2.8), (2.9), (2.7) and (2.10), we have
D q τ − 2q p m (w) Df (w,R) q (m,q) = τ − 2q p m (w) C |f ′ (w,R) (z)| q e −q|z| m dA(z) ≥ τ − 2q p m (w) D(w,δτm(w)) |f ′ (w,R) (z)| q e q|z| m dA(z) τ 2− 2q p m (w) |f ′ (w,R) (w)| q e q|w| m ≃ m q τ 2− 2q p m (w)|w| q(m−1)
for all w ∈ C. It follows that
D |m 2+p − m 1+p | 1 p sup w∈C 1 + |w| (m−1)+ (q−p)(m−2) qp , m = 1. 1, m = 1 (3.1)
which holds only if pq(m − 1) + (q − p)(m − 2) ≤ 0 as asserted, and it also gives a one sided estimate for the norm of D.
We now turn to the proof of the sufficiency of the condition in part (i). We use the covering sequences approach along with Lemma 2.3, where the original idea goes back to [12]. Applying (2.1) and (2.10), we estimate
Df q (m,q) = C |f ′ (z)| q e −q|z| m dA(z) ≤ j D(z j ,στm(z j )) |f ′ (z)| q e −q|z| m dA(z) j D(z j ,στm(z j )) 1 τ 2 m (z) D(z,στm(z)) |f ′ (w)| p e −p|w| m dA(w) q p dA(z) =: S
Now for each point z ∈ D(w, στ m (w)), observe that 1 + |z| ≃ 1 + |w|. Taking this into account, we further estimate
S ≃ j D(z j ,στm(z j )) m p (1 + |z|) p(m−1) τ 2 m (z) D(z,στm(z)) |f ′ (w)| p e −p|w| m m p (1 + |w|) p(m−1) dA(w) q p dA(z) j D(z j ,3στm(z j )) |f ′ (w)| p e −p|w| m m p (1 + |w|) p(m−1) dA(w) q p D(z j ,στm(z j )) m q (1 + |z|) q(m−1) τ 2q p m (z) dA(z)
Since q ≥ p, applying Minkowski inequality and the finite multiplicity N max of the covering sequence D(z j , 3στ m (z j )), we obtain
j D(z j ,3στm(z j )) |f ′ (w)| p e −p|w| m m p (1 + |w|) p(m−1) dA(w) q p D(z j ,στm(z j )) m q (1 + |z|) q(m−1) τ 2q p m (z) dA(z) ≤ j D(z j ,3στm(z j )) |f ′ (w)| p e −p|w| m m p (1 + |w|) p(m−1) dA(w) q p D(z j ,στm(z j )) m q (1 + |z|) q(m−1) τ 2q p m (z) dA(z) f q (m,p) sup w∈C D(w,στm(w)) m q (1 + |z|) q(m−1) τ 2q p m (z) dA(z) f q (m,p) sup w∈C m q (1 + |w|) q(m−1) τ 2 m (w) τ 2q p m (w) ≃ f q (m,p) |m 2+p − m 1+p | q p sup w∈C (1 + |w|) q(m−1)+ q−p p (m−2)
from which the sufficiency of the condition and the reverse side of the estimate in (3.1) follow. Thus we estimate the norm by
D ≃ |m 2+p − m 1+p | 1 p sup w∈C 1 + |w| (m−1)+ (q−p)(m−2) qp , m = 1. 1, m = 1
To prove the compactness, we first assume that the condition m < 2 − pq pq+q−p holds. Then for each positive ǫ, there exists N 1 such that
|m 2+p − m 1+p | 1 p sup |w|>N 1 (1 + |w|) q(m−1)+ (q−p)(m−2) qp < ǫ. (3.2)
Next, we let f n to be a uniformly bounded sequence of functions in F (m,p) that converges uniformly to zero on compact subsets of C. Then applying (2.1) and
arguing in the same way as in the series of estimations made above, and invoking eventually (3.2) it follows that
Df n q (m,q) |z|≤N 1 |f ′ n (z)| q e q|z| m dA(z) + |z j |>N −1 D(z j ,στm(z j )) |f ′ n (z)| q e q|z| m dA(z) sup |w|≤N 1 |f n (w)| q + + |z j |>N −1 D(z j ,στm(z j )) m p (1 + |z|) p(m−1) τ 2 m (z) D(z,στm(z)) |f ′ n (w)| p e −p|w| m m p (1 + |w|) p(m−1) dA(w) q p dA(z) sup |w|≤N 1 |f n (w)| q + f n q (m,q) sup |w|>N 1 m q (1 + |w|) q(m−1) τ 2 m (w) τ 2q p m (w) sup |w|≤N 1 |f n (w)| q + |m 2+p − m 1+p | 1 p sup |w|>N 1 (1 + |w|) q(m−1)+ q−p p (m−2) ǫ as n → ∞.
Conversely, assume that D is compact, and observe that the normalized sequence f * (w,R) = f (w,R) / f (w,R) (m,p) converges to zero as |w| → ∞, and the convergence is uniform on compact subset of C. Then applying (2.10) and (2.8), we find
|w| q(m−1) τ 2 q−p p m (w) ≃ (1 + |w|) q(m−1) τ 2 p−q p + 2q p m (w)e −q|w| m |f * (w,η(R)) (w)| q D(w,στm(w)) (1 + |z|) q(m−1) |f * (w,η(R)) (z)| q e −q|z| m dA(z) Df * (w,η(R))
q (m,q) → 0, as |w| → ∞.
We note in passing that in particular when p = q the necessary of the conditions in part (i) could be also established using the sequence of the polynomials (z n ) as test functions. Such polynomials belong to the spaces F (m,p) for all p. Because arguing with polar coordinates and subsequently substitution, we could easily observe that
z n p (m,p) = C |z n | p e −p|z| m dA(z) = 2π ∞ 0 r pn+1 e −pr m dr = 2πp − pn+2 m ∞ 0 t pn+2 m −1 e −t dt = 2πp − pn+2 m Γ pn + 2 m < ∞.
For the case p < q, an application of such polynomials only gives the condition
m ≤ 2 − 2(pq − 3(q − p)) p − q + 2pq
which is weaker than the condition in the result since 2(pq − 3(q − p)) p − q + 2pq < pq pq + q − p , for p < q.
3.2.
Proof of Theorem 1.1-Part (ii). We assume 0 < q < p < ∞. As (b) obviously implies (a), we plan to show (a) implies (c) and (c) implies (b). For the first, we follow this classical technique where the original idea goes back to Luecking [8]. Let 0 < q < ∞ and R be a sufficiently large number and (z k ) be the covering sequence as in Lemma 2.3. Then by Lemma 2.4,
F = z k :|z k |≥η(R) a k f (z k ,R) τ 2 p
m (z k ) belongs to F (m,p) for every ℓ p sequence (a k ) with norm estimate F (m,p) (a k ) ℓ p . If (r k (t)) k is the Radmecher sequence of function on [0, 1] chosen as in [8], then the sequence (a k r k (t)) also belongs to ℓ p with (a k r k (t)) ℓ p = (a k ) ℓ p for all t. This implies that the function
F t = z k :|z k |≥η(R) a k r k (t) f (z k ,R) τ 2 p m (z k ) belongs to F (m,p) with norm estimate F t (m,p) (a k ) ℓ p .
Then, an application of Khinchine's inequality [8] yields
z k :|z k |≥η(R) |a k | 2 |f ′ (z k ,R) (z)| 2 τ 4 p m (z k ) q 2 1 0 z k :|z k |≥η(R) a k r k (t) f ′ (z k ,R) (z) τ 2 p m (z k ) q dt. (3.3)
Making use of (3.3), and subsequently Fubini's theorem, we have
C z k :|z k |≥η(R) |a k | 2 |f ′ (z k ,R) (z)| 2 τ 4 p m (z k ) q 2 e −q|z| m dA(z) C 1 0 z k :|z k |≥η(R) a k r k (t) f ′ (z k ,R) (z) τ 2 p m (z k ) q dte −q|z| m dA(z) = 1 0 C z k :|z k |≥η(R) a k r k (t) f ′ (z k ,R) (z) τ 2 p m (z k ) q e −q|z| m dA(z)dt ≃ 1 0 DF t q F (m,q) dt (a k ) q ℓ p .
Now arguing with this, the covering lemma, and (2.8) leads to the series of estimates
z k :|z k |≥η(R) |a k | q τ 2q p m (z k ) D(z k ,3στm(z k )) (1 + |z|) q(m−1) dA(z) ≃ z k :|z k |≥η(R) |a k | q τ 2q p m (z k ) D(z k ,3στm(z k )) |f ′ (z k ,R) (z)| q e −q|z| m dA(z) ≃ C z k :|z k |≥η(R) |a k | q τ 2q p m (z k ) χ D(z k ,3στm(z k )) (z)|f ′ (z k ,R) (z)| q e −q|z| m dA(z) max{1, N 1−q/2 max } C z k :|z k |≥η(R) |a k | 2 |f ′ (z k ,R) (z)| 2 τ 4 p m (z k ) q 2 e −q|z| m dA(z) (a k ) q ℓ p .
Applying duality between the spaces ℓ p/q and ℓ p/(p−q) , we again get
z k :|z k |≥η(R) 1 τ 2 m (z k ) D(z k ,3στm(z k )) (1 + |z|) q(m−1) dA(z) p p−q τ 2 m (z k ) ≃ z k :|z k |≥η(R) (1 + |z k |) qp(m−1) p−q τ 2 m (z k ) < ∞.
On the other hand, we can find a positive number r ≥ η(R) such that whenever a point z k of the covering sequence (z j ) belongs to {|z| < η(R)}, then D(z k , στ m (z k )) belongs to {|z| < η(R)}. In view of this we estimate
|w|≥r (1 + |w|) qp(m−1) p−q dA(w) ≤ |z k |≥η(R) D(z k ,στm(z k )) (1 + |w|) qp(m−1) p−q dA(w) |z k |≥η(R) D(z k ,στm(z k )) (1 + |w|) qp(m−1) p−q τ 2 m (z k )dA(w) ≃ |z k |≥η(R) (1 + |z k |) qp(m−1) p−q τ 2 m (z k ) < ∞.
It also follows that
|w|<r 1 τ 2 m (w) D(w,3δτm(w)) (1 + |z|) q(m−1) dA(z) p p−q dA(w) < ∞
Taking into account the range of the above estimates we find
C (1 + |z|) qp p−q (m−1) dA(w) = |z|≤r (1 + |w|) qp p−q (m−1) dA(w) + |w|>r (1 + |w|) qp p−q (m−1) dA(w) < ∞,
which holds only if qp p−q (m − 1) < −2 as claimed. To prove (c) implies (b), we argue as follows. Let f n to be a uniformly bounded sequence of functions in F (m,p) that converges uniformly to zero on compact subsets of C, and by the given condition, for each ǫ > 0, there exists a positive number r 1 such that
|z|>r 1 (1 + |z|) qp p−q (m−1) dA(z) < ǫ. (3.4)
Since p/q > 1, applying Hölder's inequality, (2.1) and (3.4), we have
|z|>r 1 |f ′ n (z)| q e −q|z| m dA(z) = |z|>r 1 |f ′ n (z)| q e −q|z| m (1 + |z|) q(m−1) (1 + |z|) q(m−1) dA(z) f q (m,p) |z|>r 1 (1 + |z|) qp p−q (m−1) dA(z) p−q p f q (m,p) ǫ ǫ.
On the other hand when |z| ≤ r 1 , then
|z|≤r 1 |f ′ n (z)| q e −q|z| m dA(z) |z|≤r 1 |f n (z)| q (1 + |z|) q e −q|z| m dA(z) sup |z|≤r 1 |f n (z)| q |z|≤r 1 (1 + |z|) q e −q|z| m dA(z) sup |z|≤r 1
|f n (z)| q → 0 as n → ∞ from which our claim follows.
3.3. Proof of Theorem 1.2. Part (i). Let us now turn to the Schatten S p (F (m,2) ) membership of D. We recall that a compact D belongs to the Schatten S p (F (m,2) ) class if and only if the sequence of the eigenvalues of the positive operator (D * D) 1/2 is ℓ p summable. It suffices to prove the statement for p ≥ 1. The remaining case for 0 < p < 1 follows by the monotonicity property S p (F (m,2) ) ⊇ S q (F (m,2) ) for p ≤ q.
If p > 1, then D belongs to S p (F (m,2) ) if and only if ∞ n=0 | De n , e n | p < ∞, (3.5) for any orthonormal basis (e n ) of F (m,2) (see [14,Theorem 1.27]). Note that the sequence of the polynomials (z n / z n (m,2) ) constitutes an orthonormal basis to F (m,2) . Since Part (ii). Recall that the spectrum σ(T ) of a bounded operator T is the set containing all λ ∈ C for which λI − T fails to be invertible, where I is the identity operator. The complement of the spectrum is referred as the resolvent set. A simple computation shows that the function f * (z) = ce λz solves the differential equation λf = Df = f ′ , where c is a constant. Then we analyze depending on the size of m. Let us first assume that m = 1. Then, the integral in (3.6) converges for each λ ∈ C such that |λ| < 1. This means that the function f * belongs to F (m,p) , and can be chosen in such a way that c = 0. From this we deduce D ⊆ σ(D) or {λ ∈ C : e λz ∈ F (m,p) } ⊆ σ(D).
(3.7)
To prove the reverse inclusion in (3.7), observe that the integral in (3.6) fails to converge for each |λ| ≥ 1 and c = 0. It means that λI − D is injective whenever |λ| ≥ 1. On the other hand, for such values of λ, a simple computation again shows that the equation λf − Df = h has a unique analytic solution f (z) = R λ h(z) = Ce λz − e λz z 0 e −λw h(w)dA(w), (3.8) where R λ is the resolvent operator of D at point λ, C = f (0) is a constant value. We remain to show that the operator R λ given by the explicit expression in (3.8) is bounded on F (m,p) . To this end, applying Lemma 2.1, we have We now turn to the case m < 1. For this, part (b) of our result forces D to be a compact operator. Furthermore, we observe that the integral in (3.6) converges only if c = 0 and hence f * (z) = 0, which clarifies that D has no point spectrum.
To this effect, σ(D) = {0}.
(ii) Let 1 ≤ p < ∞ and m > 0, and D : F (m,p) → F (m,p) is bounded, i.e m ≤ 1. Then its spectrum σ(D) = {0} whenever m < 1 and when m = 1; σ(D) = {λ ∈ C : e λz ∈ F (m,p) } = D.
Lemma 2. 1 .
1Let λ ∈ C and 0 < p < ∞. Then for each entire function f for which f e λ ∈ F (m,p) , we have C |f (z)e λz | p e −p|z| dA(z) C |f ′ (z)e λz | p e −p|z| dA(z).(2.2)
n, from which (3.5) easily follows.
λz | p e −p|z| m dA(z) = |c| p C e pℜ(λz)−p|z| m dA(z) (3.6)
z)| p e −p|z| dA(w)dA(z) = h p (m,p) .
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Integral Operators, Embedding Theorems and a Littlewood-Paley Formula on Weighted Fock Spaces. O Constantin, Joséángel Peláez, J. Geom. Anal. O. Constantin and JoséÁngel Peláez, Integral Operators, Embedding Theorems and a Littlewood-Paley Formula on Weighted Fock Spaces, J. Geom. Anal., (2015), 1-46.
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T Mengestie, 10.4134/JKMS.j160671Spectral properties of Volterra-type integral operators on Fock-Sobolev spaces. T. Mengestie, Spectral properties of Volterra-type integral operators on Fock-Sobolev spaces, Korean Journal of Mathematical Society, https://doi.org/10.4134/JKMS.j160671.
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Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weightes. J Pau, J A Peláez, J. Funct. Anal. 25910J. Pau and J. A. Peláez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weightes. J. Funct. Anal., 259 (10)(2010), 2727-2756.
Operator theory on function spaces, Second Edition. K Zhu, Norway E-mail address: Tesfa.Mengestie@hvl. 138American Mathematical SocietyDepartment of Mathematical Sciences, Western Norway University of Applied SciencesMath. Surveys and MonographsK. Zhu, Operator theory on function spaces, Second Edition, Math. Surveys and Monographs, Vol. 138, American Mathematical Society: Providence, Rhode Island, 2007. Department of Mathematical Sciences, Western Norway University of Ap- plied Sciences, Klingenbergvegen 8, N-5414 Stord, Norway E-mail address: [email protected]
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"Advection-enhanced kinetics in microtiter plates for improved surface assay quantitation and multiplexing capabilities",
"Advection-enhanced kinetics in microtiter plates for improved surface assay quantitation and multiplexing capabilities"
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"Iago Pereiro \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"Anna Fomitcheva Khartchenko \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"Robert D Lovchik \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"Govind V Kaigala \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"DrI Pereiro \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"DrA Fomitcheva Khartchenko \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"DrR D Lovchik \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n",
"DrG V Kaigala \nIBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland\n"
] |
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"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland",
"IBM Research -Europe\nSäumerstrasse 4CH-8803RüschlikonSwitzerland"
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[] |
Surface assays such as ELISA are pervasive in clinics and research and predominantly standardized in microtiter plates (MTP). MTPs provide many advantages but are often detrimental to surface assay efficiency due to inherent mass transport limitations. Microscale flows can overcome these and largely improve assay kinetics. However, the disruptive nature of microfluidics with existing labware and protocols has narrowed its transformative potential. We present WellProbe, a novel microfluidic concept compatible with MTPs. With it, we show that immunoassays become more sensitive at low concentrations (up to 9× signal improvement in 12x less time), richer in information with 3-4 different kinetic conditions, and can be used to estimate kinetic parameters, minimize washing steps and non-specific binding, and identify compromised results. We further multiplex single-well assays combining WellProbe's kinetic regions with tailored microarrays. Finally, we demonstrate our system in a context of immunoglobulin subclass evaluation, increasingly regarded as clinically relevant.
|
10.1002/anie.202107424
|
[
"https://arxiv.org/pdf/2202.03080v1.pdf"
] | 236,200,102 |
2202.03080
|
0ba2d96849b2a038779fad346d8c68b14e2f846a
|
Advection-enhanced kinetics in microtiter plates for improved surface assay quantitation and multiplexing capabilities
Iago Pereiro
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
Anna Fomitcheva Khartchenko
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
Robert D Lovchik
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
Govind V Kaigala
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
DrI Pereiro
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
DrA Fomitcheva Khartchenko
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
DrR D Lovchik
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
DrG V Kaigala
IBM Research -Europe
Säumerstrasse 4CH-8803RüschlikonSwitzerland
Advection-enhanced kinetics in microtiter plates for improved surface assay quantitation and multiplexing capabilities
1
Surface assays such as ELISA are pervasive in clinics and research and predominantly standardized in microtiter plates (MTP). MTPs provide many advantages but are often detrimental to surface assay efficiency due to inherent mass transport limitations. Microscale flows can overcome these and largely improve assay kinetics. However, the disruptive nature of microfluidics with existing labware and protocols has narrowed its transformative potential. We present WellProbe, a novel microfluidic concept compatible with MTPs. With it, we show that immunoassays become more sensitive at low concentrations (up to 9× signal improvement in 12x less time), richer in information with 3-4 different kinetic conditions, and can be used to estimate kinetic parameters, minimize washing steps and non-specific binding, and identify compromised results. We further multiplex single-well assays combining WellProbe's kinetic regions with tailored microarrays. Finally, we demonstrate our system in a context of immunoglobulin subclass evaluation, increasingly regarded as clinically relevant.
Introduction
Solid phase assays to detect and quantify analytes are essential in diagnostics and screening [1,2], drug discovery [3], environmental monitoring [4] and quality control [5,6], among others. The use of a solid phase to immobilize ligands or analytes enables separation steps, while a flat interface further facilitates optical reading and multiplexing. As a result, many bioassays today rely on solid surfaces, here referred to as "surface assays", including almost all immunoassays. [7] Well-known examples of surface assays are enzyme-linked immunosorbent assays (ELISA), immunohistochemistry, immunofluorescence and microarrays. In research, just these four assay types account for more than 100k new publications per year (results of a keyword search for these assay names in Scopus for the year 2019). In clinical practice, ELISA alone has become the gold standard for diagnosing many viral, bacterial and parasitic diseases, including HIV, Hepatitis B and C, influenza, paratuberculosis and Lyme disease. [1] Most surface assays are carried out in multi-well microtiter plates (MTPs), particularly in the 96-well format, as they allow multiple parallel tests in independent wells with minimal cross-contamination. It is noteworthy that MTPs have barely changed since they started being injection-molded in the 1980s. Their standardization in the early 2000s [8] and the emergence of dedicated equipment in the form of dispensers, washers, readers and automated on-bench instrumentation has further helped consolidate their use. And yet, this consolidation has occurred in spite of notable intrinsic drawbacks for surface assays. Notably, incubation steps typically require 1-8 hours or overnight to enable good sensitivity while compensating this with relatively high analyte concentrations, often making reagent/sample use inefficient. This is further compromised by evaporation, particularly for high yield plates with many (>1000) wells.
These drawbacks are associated with inefficient mass transport within the assay, or more specifically, a slow diffusion process of the analytes in the liquid towards the reaction surface. This is particularly prominent in the case of biomolecular assays, as proteins, [9] long DNA/RNA strands [10] and nano/micro-particles are often associated with low diffusivities. In the duration of the assay, the zone of the liquid adjacent to the reaction surface can become depleted in analyte. As this zone grows in size the reaction rate diminishes, significantly increasing the required incubation times. To overcome this, two approaches are possible: (i) using a higher analyte concentration in the liquid to accelerate diffusion by increasing the concentration gradients, albeit consuming more analyte, or (ii) generating flows (advection) within the liquid to constantly bring fresh analyte close to the surface, thereby reducing the thickness of the depletion zones. Overcoming transport limitations in the latter way can lead to gains in the reaction rate of several orders of magnitude. [11] Therefore, to obtain advection in surface assays, MTPs are often placed on orbital shakers to slosh the liquid in the wells. However, the obtained flows are of limited intensity and mostly present in the liquid bulk far from the surface. Thus, signal gains are limited and often coupled to non-uniformity and low reproducibility, resulting in suboptimal quantitation accuracy. [12] In contrast, a more precise control of flows can be obtained at the microscale, where hydrodynamics are dominated by deterministic laminar regimes. By integrating and miniaturizing many analytical functions in small devices, microfluidics has enabled notable applications in genomics, [13] cancer diagnostics, [14] single cell studies [15] or material discovery [16], to name a few. In the case of surface assays, microfluidics can significantly enhance kinetics and largely improve the time to results and reproducibility. [17][18][19] However, few commercially available systems exist to enhance surface assays with microfluidics, despite three decades of development. A key reason for this is that non-standardized, customdesigned and non-transferable microfluidic systems offer little versatility and compatibility with existing and well-established surface assay labware, protocols and practices. To try to overcome this limitation, a few attempts have been made to create microfluidic devices with an external geometry resembling the MTP format. The primary aim was to make them compatible with common plate readers and dispensers. For example, Witek et al. presented a nucleic acid purification device, fabricated using hot embossing, containing 96 purification beds consisting of individual chambers with microposts. [20] Hou et al. reported a PDMS microfluidic chip with channels to dilute the sample at different concentrations and let it react in chambers arranged as wells of a 48 well plate. [21] To avoid the need for external fluidic pumping, Kai et al. presented a device with an external geometry identical to a 96-well plate, incorporating a spiral microfluidic channel at the bottom of each well to drive the dispensed liquid towards an absorbent pad. [22] Similarly, Sanjay et al. presented a PMMA/paper hybrid device in which the dispensed reagents in the wells were transferred onto a paper-based assay. [23] In other cases, the microtiter plate format was miniaturized, to benefit from its parallelization in a portable format. [24] All these systems have in common that the resulting devices need to be customized, are still of limited compatibility with standard equipment and are not compatible with commercially available and well-established MTPs. Recently, an MTPcompatible cover comprising microfluidic channels was reported, allowing liquid exchange in MTP wells. [25] Such a system is useful for medium renewal in cell cultures, but unfortunately of limited use for biomolecular surface assays, as it does not enable controlled advection at the well surface.
Here, we present a new microfluidic concept to address standard microtiter plates while enhancing surface assays with accurate advection conditions. We call this concept "WellProbe". Performing MTP-based surface assays with WellProbe provides the following advantages: (i) it generates flowenhanced kinetics, which considerably reduce bioassay time and increase sensitivity, (ii) it provides several local kinetic conditions within one well, expanding the linear quantitation range of the assay, (iii) it provides kinetics-based information with a single test, which can serve to identify saturated conditions, identify artefacts and false positives or false negatives, and strengthen quantitation accuracy, (iv) it enables rapid liquid switching for multi-step protocols while avoiding crosscontamination and making intermediate washing steps unnecessary, (v) coupled with arrays of radially distributed spots, it allows high-level multiplexing in single-well tests, and, (vi) it is compatible with commercially available MTP-based kits for surface assay applications, as well as the instrumentation for reading results or dispensing additional liquids. We demonstrate the functionality of our system in the context of IgG binding, a step virtually compulsory in all ELISA tests, and illustrate its application for subclass IgG (IgGSc) testing. Measuring the four subclasses of IgG (IgG1, IgG2, IgG3 and IgG4) can provide a more complete view of the function of the humoral immune system than the more common testing of immunoglobulin classes IgG, IgA and IgM [26]. However, the concentration range of each subclass in adult human plasma differs strongly (IgG: 767-1590 mg/dL IgG2: 171-632 mg/dL, IgG4: 2,4-121 mg/dL). [27] Such variations can represent a challenge for accurate quantitation with standard ELISA assays, where the quantifiable range is obtained by blind sample dilution and pre-defined assay timing for each analyte. We thus regard IgGSc as a particularly pertinent target for more accurate and multiplexed testing, which could contribute to both increase our understanding of IgGSc deficiency and make its testing more readily available in the clinics. (d) appropriate injection/aspiration flowrates result in a radial flow that creates different kinetic regions on the surface, which can either be used for monoplex assays or multiplexed assays with spots; (e) the kinetic regions provide an expanded quantifiable range.
Results and Discussion
Working principle Figure 1a illustrates a typical surface assay in a well of a static MTP. The reaction surface consumes analyte in the adjacent liquid, creating a depletion zone of low analyte concentration that gradually increases in size. The remaining analyte in the liquid bulk needs to traverse this zone by diffusion to reach the reaction surface, making the reaction rate decay with time. Figures 1b-c show the concept of WellProbe, which consists of a cylindrical microfluidic device that can enter individual MTP wells to address the bottom surface. In contrast to previous open microfluidic probes that localize reagents in microscale regions and need to scan the surface to address larger areas [28][29][30], WellProbe creates instant circular flow confinements in the mm-and cm-scale, covering most of the well bottom. At its tip, a central injection opening is surrounded by circular flat areas, or mesas, of increasing height (Figure 1d). The liquid exits the injection opening radially and exhibits a decreasing flow velocity regime and shear rate below each section of the mesa. In transport-limited surface reactions -common for biomolecules e.g. relatively large proteins such as antibodies, enzymes, hormones or globular proteins -an increased shear typically translates into reduced depletion layers and higher binding rates. [31] As illustrated in Figure 1d, under each mesa a region of particular kinetic conditions is created where analytes can bind to ligands on a uniform surface or in discrete spots. Thus, assay dynamics are locally unique, with higher binding rates in the inner regions ( Figure 1e). Two working modes are possible: (i) monoplex, with standard MTP wells containing one specific ligand and (ii) multiplex, combining the kinetic regions with a tailoreddistribution of surface spots. In all cases, the system condenses several traditional assay tests into one single test, provides a larger quantitation range and kinetics-based information and shows when the signal is saturated or has been compromised. WellProbe-generated surface hydrodynamics Figure 2a illustrates the internal structure of WellProbe. The aluminum body (diameter 5.5 mm, length 20 mm, see SI-S6.2 for details) contains an internal channel network, created by sealing sidedrilled channels and surface grooves with a polymer thermal shrinking tube, thus becoming closed channels. In the configuration shown, WellProbe contains four aperture inlets for sequential liquid switching between up to four reagents and a single outlet aperture. The channels originating at the aperture inlets join 5 mm upstream of the tip aperture (the red arrows in Figure 2a follow one path). The distance between WellProbe and the well bottom is defined by peripheral feet (50 μm). After flowing over the processing area below the three mesas surrounding the central aperture (with increasing heights of 50, 100 and 500 μm), the reagent is collected by two symmetrically opposed apertures in a fourth external mesa of 1.2 mm height. By collecting the liquid using a 1 mm deep recession and two aspiration apertures, we ensure that the aspiration at the well bottom is constant and the resulting streamlines are perfectly radial with symmetrical shear stress in all directions (Figures 2b-c and SI-S2). This recession further acts as a bubble trap [32], collecting any undesired air coming through the aperture, increasing system robustness and reproducibility (see SI-S6.1). Using an aspiration flow-rate Qa slightly higher than the injection flow-rate Qi (Qi = 40 μL/min and Qa = 45 μL/min in Figure 2b) results in a 5 mm wide circular confinement of the injected reagent, 80% of the total well diameter. This confinement is entirely surrounded by a hydrodynamic wall of external immersion liquid (buffer) entering WellProbe through a circular groove near the tip edge, which sets the boundary of the flow confinement and prevents cross-contamination (Figure 2b). The channels with the liquid aspirated through the two apertures (reagent plus external immersion liquid) join within WellProbe and lead to a single outlet aperture (blue arrows in Figure 2a). The inlet channels join close to the tip of WellProbe, and if these channels are purged before the experiment, the reagents in the confinement can be switched very quickly by using upstream valves for each inlet. In Figures 2d and e, two colored reagents are sequentially switched with a transition time of less than 5 s at 40 μL/min. This allows for efficient multi-step assays.
Advection-enhanced kinetics enabled by WellProbe
A key feature of WellProbe is that it allows to enhance the kinetics of surface bioassays by creating high and accurate shear rate (and shear stress) conditions at the well bottom, which reduces and stabilizes the depletion layers. This shifts the kinetics from being transport-limited to reactionlimited. The shear rate (γ ) at any point of the well bottom below WellProbe is calculated as the local derivative of the flow profile between two infinite parallel plates, a solution of the Navier Stokes equations (FEM simulations in Figure 3a):
̇= 3 (1)
where Q is the injection flow-rate, R is the radius or horizontal distance between the injection and the position of interest and H is the vertical distance between the well bottom and WellProbe. The shear rate obtained under mesa 1 is 500× higher than under mesa 3 (e.g. for Q = 40 μL/min, γ _1= 765 s-1 and γ _3= 1.5 s-1, calculated at the middle diameter of each mesa). The sharp transition between mesa heights ensures that shear rate conditions under contiguous mesas have a well defined boundary.
The value of the shear rate indicates the magnitude of the advective flows close to the surface. Higher values result in improved transport and thus smaller depletion layers. The ratio between the advective and diffusive transport rates, or Peclet number, is high (Pe >> 1) in the working flow-rate regimes of WellProbe (> 1 μL/min). Therefore, using Eq. 1, we estimated the thickness of the depletion layer (δ) on the well bottom as the approximate distance from the bottom at which a particle requires equal time to reach the bottom by diffusion or escape the region by advection. [17] For this, we took into account the transport under each mesa more central than the mesa of interest n (e.g. mesas 1 and 2 when measuring mesa 3, see details in SI-S3):
= 3 ( − )(2)
where Ri is the external radius of each mesa considered, Hi its height and D is the diffusivity of the molecule being transported. Thus, e.g. at Q = 40 μL/min and a common IgG diffusivity of D = 6.5 x 10-5 mm2/s [33,34] we estimate the depletion layer thickness under mesas 1-3 to be approximately 2.3, 11 and 66 μm, respectively. Additionally, we performed 2D radially symmetric FEM simulations of the expected concentration profiles, as shown in Figure 3c. For the same Q and D, the obtained depletion layer thicknesses were 2.2, 13 and 84 μm under mesas 1, 2 and 3, respectively (Figure 3c), which is in good agreement with the analytical values.
If we consider the case of transport-limited kinetics, so that we assume that all analytes reaching the surface also bind to it, and further neglecting the saturation of the binding sites on the surface, we can approximate the local binding rate (b ) as being equal to the rate of diffusion through the depletion layer:
̇ ≈ (3)
where c0 is the concentration of the molecule of interest in the injected reagent. The potential difference in binding kinetics between mesas 1 and 3 (more precisely the mesa's central radius R1 = 0.33 mm, R2 = 1 mm, R3 = 1,67 mm) is b _1/b _3 ~30 (precisely 29.2) and between mesas 1 and 2 b _1/b _2 ~5 (precisely 4.8). The resulting binding kinetics can thus cover more than one order of magnitude with three points similarly spaced on a logarithmic scale.
Single-well antibody binding with 3 simultaneous kinetic conditions
As seen in Figure 3b, the resulting kinetics under each mesa translate into proportional signals in an IgG -anti-IgG reaction. To observe the temporal kinetic behavior (Figure 3d, experimental details explained in SI-S1 and supporting chemical characterization in SI-S5), we injected two fluorescent-IgG concentrations (at 1 and 10 µg/mL) for up to 6 minutes (incubation time) at 40 μL/min. As observed, the inner regions consistently exhibited a higher signal intensity than the outer ones, except when reaching saturation. At incubation times 3 and 6 minutes and 10 µg/mL, similar signal levels in mesas 1&2 indicate that saturation is being reached. As expected, in conditions far from saturation and well above LOD levels (e.g. 6 min with 1 μg/mL) the signals from mesas 1 and 3 differ by approximately one order of magnitude. Depending on the incubation times tested, quantification was possible in either the areas with fast kinetics (mesas 1 & 2) for challenging low concentrations or within the slow kinetic areas (mesa 3) when the signal approached saturation. Figure 3d additionally shows dashed red curves obtained from a classical on-bench assay, in which all conditions are kept identical expect that transport is purely diffusion based (no WellProbe used). As can be seen, even zone 3, the zone with the lowest flow velocities, exhibits significantly higher signal intensities than on-bench results. To better quantify the benefits of using the WellProbe system, we can estimate the gains in assay time (ε) with the following expression:
= ≈ 2 3 (1 + )(4)
where K = kon/koff (the ratio of the association rate to the dissociation rate) , bm is the density of binding sites on the surface, τdiffusion and τreaction are the approximate times to reach signal saturation (kinetic equilibrium) by relying purely on transport-limited diffusion (experiment onbench) or using an ideal system with negligible transport limitations (an efficient use of WellProbe, see SI-S3 for how we obtain this expression). This equation particularly reveals how the expected gains increase with decreasing analyte concentrations. For example, with our Ab model system, we would expect that for 10 μg/mL and 1 μg/mL, WellProbe could reach kinetic equilibrium at 12× (2 instead of 22 min) and 93× (13 min instead of 21 h) shorter assay times respectively, compared to on-bench protocols. Note that equilibrium does not need to be reached for assays to provide quantitation. In SI-S5.1 we performed a calibration of our model system, confirming that for lower concentrations (<1 μg/mL) WellProbe provides statistically better signal than equivalent on bench protocols of 20× the incubation time (3 min instead of 1 h) and an order of magnitude lower LOD for an equivalent assay time. We further confirmed this by performing ELISA tests at a lower concentration of 100 ng/mL (SI-S5.2), showing ~9× better signal with WellProbe in 5 min than on bench for 1 h of primary antibody incubation. Additionally, non-specific binding is also significantly reduced due to the constant flow-rate (SI-S5.3). In summary, WellProbe provides an increased signal level at low concentrations and improves the limit of detection (LOD) as compared to standard assays.
Hydrodynamic reconfigurability expands the quantitation range
As with any assay, sometimes quantitation may be compromised. WellProbe provides the advantage of revealing such situations by analyzing the obtained signal patterns in mesas 1-3 (see SI-S4). In one scenario, samples with a high concentration of analytes can lead to signal saturation (see SI-S5.4 for upper range of model system). To further increase the upper range of measurable analyte concentrations, it is possible to use the dynamic reconfigurability of the flow confinement to address an additional zone 4 of the surface for a different incubation time. As shown in Figure 4a, by increasing the ratio of outlet aspiration (Qo) to inlet injection (Qi), the area containing the confined reagent reduces its width, covering mesas 1, 2 and partially 3. In contrast, a ratio close to 1 (similar injection to aspiration) tends to cover the entire area below WellProbe (Fig. 4b, covering mesas 1, 2, 3 and 4). In Figure 4c we demonstrate the use of this dynamic feature to measure an IgG concentration of 50 µg/mL. While mesas 1, 2 and 3 were exposed to the analyte for 3 minutes, the area below mesa 4 was only exposed for the last 10% of this time (18 s). While these experimental conditions lead to saturation levels being reached in mesas 1-3, mesa 4 exhibits signal within the measurable range. Thus, coupling this dynamic reconfigurability to the system of fixed mesas allows to extend the quantitation range of surface assays by at least two orders of magnitude.
Coupling kinetic zones and spots to multiplex for IgG subclass quantitation
Lastly, we performed single-well multiplexing by coupling WellProbe with custom-made radial arrays of spots deposited by inkjet printing in a standard MTP (Figure 5a). Microarrays in MTPs offer the advantage of being easier to automate and adapt to biochemical protocols [36], but the concept has been little explored primarily due to diffusion limitations [37]. Here, we inkjetprinted arrays consisting of three concentric rings of spots, each ring containing 9 spots with 3 replicates for each tested IgG, for a 3plex. The spatial distribution of each ring of spots coincides with one of the 3 mesas. The first circle of spots is located closer to the inlet mesa radius (at R = 0.33 mm) to leverage the locally higher shear rate (see SI-S6.3 for information on spot configuration). We applied this multiplexing to IgG subclass testing. For certain subclasses, notably IgG2 and IgG4, concentration values below the normal healthy range are correlated with an impaired response to infections in the respiratory tract [38], allergic asthma, rhinitis and autoimmune conditions. [26] In the case of IgG4, elevated levels are associated with autoimmune pancreatitis, aspirin-exacerbated respiratory disease, nasal polyps, eosinophilia and celiac disease. [39] In spite of their increasing clinical relevance, a standardization of IgGSc assays is still lacking. [27] In this work, the spots contain antibodies specific to the capture of total IgG and subclasses IgG2 and IgG4. As sample, we used purified human IgG containing a representative mix of IgG subclasses in human plasma. This sample was injected with two different incubation times (1 and 5 min) at 40 μL/min and three dilutions in PBS: 1, 10 and 50 μg/mL. This was followed by liquid switching to inject the secondary fluorescent antibody for 5 min (enough to ensure saturation and avoid zone-dependent binding). A washing step between reagents was deemed unnecessary as the constant flow ensures that non-bound species are removed from the spots. Figure 5b shows representative results extracted from spot signals of a single well/array (c = 1 µg/mL, t = 1 min). A descending trend is observed between zones 1 and 3 for IgG2 and IgG4, indicative of spot signal in the measurable range and good assay quality. The approximately constant values for IgG-T evidence that saturation has been reached. This is easier to visualize in the form of heatmaps, as depicted in Figure 5c, where each square represents the average spot intensity of spots of the same zone and IgG subtype. When the signal level is less than 3 times the noise and the square color shows a saturation plateau (values of zones 1-2 or 1-3 were not statistically different), those values are considered inadequate for quantitation. To estimate the quantitation accuracy of the system, we used all selected values of Fig. 5c (marked in green) and, assuming a linear behavior, normalized with respect to the results of one condition (c = 10 µg/mL, t = 1 min) and multiplied by a factor corresponding to their difference in dilution or time: e.g., c = 1 µg/mL, t = 5 would be multiplied by 10/5=2. The resulting equivalent values in Figure 5d show that in spite of the large range studied, all estimated values fall within a 50% confidence interval for all IgG subtypes. Thus, a single test can offer multiplexed results, quality assessment and an indication of the measurable range of quantitation, while enabling replicates within the same experimental conditions.
Conclusion
Compared to on-bench diffusion-based tests, we have shown that signal enhancements of up to an order of magnitude are possible while supplying 3-4 times more information per well in the monoplex configuration. Additionally, the 3-mesa design shown here can be expanded to more mesas if an increased resolution in shear rate conditions is desired.
Higher flow-rates result in enhanced kinetics until transport ceases to be the limiting factor. For example, the 40 μL/min used for the experimental results in Figure 3 result in a Damköhler number Da = 0.4, very close to ideal reaction-limited kinetics. In this case, a higher flow-rate would lead to unnecessary waste of sample. Eq. 2 and 3 show that the binding rate is approximately proportional to the cube root of the flow-rate. Thus, reducing the flow-rate by half only results in a 20% loss in signal, something the user can leverage to optimize signal and reagent use for precious samples. That said, the results illustrated in this work used sample volumes of 60-250 μL, comparable or smaller than typical volumes in standard ELISA assays, with all the added benefits. Aspirating the sample through WellProbe prior to injection can ensure that only the necessary volume is used, thereby avoiding partial loss of the sample in internal channels.
The WellProbe concept is adaptable to MTPs of any size and shape. The specific probe shown here is reusable and inert to biological substrates due to the aluminum oxide layer that naturally covers its structure, but similar results could be expected with plastic, titanium or steel. For sequential chemistry, the design shown here provides four inlets. While intermediate washing steps are less needed with advection, some protocols still require a high number of reagents. Although a higher number of inlets is possible within the shown device, external junctions containing multiple inlets could be placed upstream of WellProbe to accommodate complex protocols requiring many reagents. Additionally, while WellProbe has the benefit of providing the results of many wells in a single well-test, increased throughput can be obtained with multiple WellProbes working in parallel, if needed. Furthermore, the same concept could be envisioned in integrated and disposable massmanufacturable devices. In this case we foresee that pipetting robots could be used for direct liquid injection into the inlets, to facilitate liquid handling automation.
The predominant use of MTPs in life science settings is due to so far unchallenged benefits in ease of use, accessibility, parallelization, low cross-contamination and compatibility with a myriad of now standard laboratory readers, washers or pipettors. Thirty years of microfluidic alternatives have had limited impact on this continued use of MTPs, which have barely been replaced for such ubiquitous bioassays as ELISA. Instead, new robotic and automated platforms have been created to operate and automate MTP-based assay workflows. Here, we combine the benefits of microfluidics and the intrinsic advantages of existing MTPs for the first time, thereby increasing their multiplexing capability, sensitivity and accuracy. Undeniably, the current demand for better quantitation, data acquisition and data quality will continue in the next decades to feed the new needs of the biological and biomedical communities. We see WellProbe as a key step to adapt many of the current workflows to the requirements of next generation diagnostics and quantitative biology, presenting a low implementation barrier at the same time.
Figure 1 :
1Working principle of WellProbe. (a) Common case of a surface assay in a well of a static MTP, with creation of a depletion zone that leads to reaction rate decay; (b) WellProbe positioned over an MTP well and (c) cross section of WellProbe within a well;
Figure 2 .
2WellProbe radial hydrodynamics and liquid exchange for sequential chemistry. (a) Flow paths within WellProbe and bottom view with mesas; (b) FEM hydrodynamic simulation of the flow confinement area; (c) experimental streamlines with fluorescent particles; (d, e) intermittent switching of colored solutions.
Figure 3 :
3Shear rate-dependent IgG binding dynamics. (a) Simulation of the shear rate at the well bottom with indicated position of mesas; (b) Fluorescent signal corresponding to 3 mesas after one test (40 μL/min, 10 μg/mL, 1 min); (c) simulation of analyte concentration profile and depletion layers in section view of confinement; (d) binding curves of an Ab-Ab system for different incubation times and antibody concentrations at an injection flow-rate of 40 μL/min (n = 3).From such curves, obtained from a few wells, it is possible to estimate the values of some kinetic constants. In SI-S2.3, we use the results fromFigure 3dat 10 µg/mL to obtain fitting curves with a finite element model based on first-order surface kinetics. Considering the value of D previously mentioned and a typical dissociation constant koff = 1 x 10-3 s-1, we estimated that the concentration of binding sites on the well surface is approximately bm = 3 x 104 mol/cm2 and the affinity constant kon = 4 x 105 L/mol•s, a value comparable to those reported for Ab-Ab interactions in the literature[35].
Figure 4 :
4Expanded quantitation range with confinement reconfigurability (a) Shrunk flow confinement with an injection:aspiration ratio of 1:3 (Qi = 40 μL/min, Qo = 120 μL/min). (b) Flow confinement covering the entire tip of WellProbe with a ratio close to 1 (Qi = 40 μL/min, Qo = 41 μL/min). (c) Results of an assay with IgG at 50 μg/mL (n = 3) for a total of 3 minutes under mesas 1-3 and the last 18 seconds additionally under mesa 4 (Qi = 40 μL/min, initial Qo = 41 μL/min and final Qo = 100 μL/min); the insert shows a fluorescence image of the obtained assay signal.
Figure 5 :
5Multiplexing combining kinetic areas and circular distribution of spots. (a) Radial array of spots after an assay with WellProbe, containing triplicate spots per zone and specificity for all IgG subtypes and subtypes 2 and 4. (b) Results obtained from a single well/array (sample was human IgG protein at 10 μg/mL, incubation time 1 min), with standard deviations obtained from in-well spot triplicate. (c) Heatmaps of individual arrays indicating average spot intensity for each spot type and zone at different dilutions of same sample or incubation time: dotted green line indicates measurable spots, neither in saturation nor close to LOD. (d) Results from all measurable averaged spots obtained from concentrations 1, 10 and 50 μg/mL and incubation times of primary antibody of 1 and 5 min (2 wells per condition): all results normalized to results from well at 10 μg/mL and 1 min.
AcknowledgementsWe thank A. Zulji for device fabrication and L. Petrini for detailed discussions. We acknowledge funding by ERC-PoC CellProbe (842790) and SNF Spark program (CRSK-2_190877). We thank Dr. E. Delamarche and Dr. H. Riel for continuous support.Keywords: surface assays, open-space microfluidics, microtiter plate, multiplexing, advection, mass transport
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|
[] |
[
"XMM-Newton campaign on the ultraluminous X-ray source NGC 247 ULX-1: outflows",
"XMM-Newton campaign on the ultraluminous X-ray source NGC 247 ULX-1: outflows"
] |
[
"C Pinto \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n\nESTEC/ESA\nKeplerlaan 12201AZNoordwijkThe Netherlands\n",
"R Soria \nCollege of Astronomy and Space Sciences\nUniversity of the Chinese Academy of Sciences\n100049BeijingChina\n",
"D J Walton \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUnited Kingdom\n",
"A D'aì \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n",
"F Pintore \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n",
"P Kosec \nMIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA\n",
"W N Alston \nESAC/ESA European Space Astronomy Center\nP.O. Box 7828691Villanueva de la Canada, MadridSpain\n",
"F Fuerst \nESAC/ESA European Space Astronomy Center\nP.O. Box 7828691Villanueva de la Canada, MadridSpain\n",
"M J Middleton \nPhysics & Astronomy\nUniversity of Southampton\nSO17 1BJSouthamptonHampshireUK\n",
"T P Roberts \nCentre for Extragalactic Astronomy\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"M Del Santo \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n",
"D Barret \nUniversité de Toulouse\nCNRS\nIRAP\n9 Avenue du colonel Roche, BP 4434631028, Cedex 4ToulouseFrance\n",
"E Ambrosi \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n",
"A Robba \nINAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly\n",
"H Earnshaw \nCahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n",
"A C Fabian \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUnited Kingdom\n"
] |
[
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"ESTEC/ESA\nKeplerlaan 12201AZNoordwijkThe Netherlands",
"College of Astronomy and Space Sciences\nUniversity of the Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUnited Kingdom",
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"MIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA",
"ESAC/ESA European Space Astronomy Center\nP.O. Box 7828691Villanueva de la Canada, MadridSpain",
"ESAC/ESA European Space Astronomy Center\nP.O. Box 7828691Villanueva de la Canada, MadridSpain",
"Physics & Astronomy\nUniversity of Southampton\nSO17 1BJSouthamptonHampshireUK",
"Centre for Extragalactic Astronomy\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"Université de Toulouse\nCNRS\nIRAP\n9 Avenue du colonel Roche, BP 4434631028, Cedex 4ToulouseFrance",
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"INAF -IASF Palermo\nVia U. La Malfa 153I-90146PalermoItaly",
"Cahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUnited Kingdom"
] |
[
"Mon. Not. R. Astron. Soc"
] |
Most ULXs are believed to be powered by super-Eddington accreting neutron stars and, perhaps, black holes. Above the Eddington rate the disc is expected to thicken and to launch powerful winds through radiation pressure. Winds have been recently discovered in several ULXs. However, it is yet unclear whether the thickening of the disc or the wind variability causes the switch between the classical soft and supersoft states observed in some ULXs. In order to understand such phenomenology and the overall super-Eddington mechanism, we undertook a large (800 ks) observing campaign with XMM-Newton to study NGC 247 ULX-1, which shifts between a supersoft and classical soft ULX state. The new observations show unambiguous evidence of a wind in the form of emission and absorption lines from highly-ionised ionic species, with the latter indicating a mildly-relativistic outflow (−0.17c) in line with the detections in other ULXs. Strong dipping activity is observed in the lightcurve and primarily during the brightest observations, which is typical among soft ULXs, and indicates a close relationship between the accretion rate and the appearance of the dips. The latter is likely due to a thickening of the disc scale-height and the wind as shown by a progressively increasing blueshift in the spectral lines.
|
10.1093/mnras/stab1648
|
[
"https://arxiv.org/pdf/2104.11164v2.pdf"
] | 233,346,964 |
2104.11164
|
eb79ca51fbbf591ae8a4a2254edcd3169bee5504
|
XMM-Newton campaign on the ultraluminous X-ray source NGC 247 ULX-1: outflows
2018
C Pinto
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
ESTEC/ESA
Keplerlaan 12201AZNoordwijkThe Netherlands
R Soria
College of Astronomy and Space Sciences
University of the Chinese Academy of Sciences
100049BeijingChina
D J Walton
Institute of Astronomy
Madingley RoadCB3 0HACambridgeUnited Kingdom
A D'aì
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
F Pintore
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
P Kosec
MIT Kavli Institute for Astrophysics and Space Research
02139CambridgeMAUSA
W N Alston
ESAC/ESA European Space Astronomy Center
P.O. Box 7828691Villanueva de la Canada, MadridSpain
F Fuerst
ESAC/ESA European Space Astronomy Center
P.O. Box 7828691Villanueva de la Canada, MadridSpain
M J Middleton
Physics & Astronomy
University of Southampton
SO17 1BJSouthamptonHampshireUK
T P Roberts
Centre for Extragalactic Astronomy
Department of Physics
Durham University
South RoadDH1 3LEDurhamUK
M Del Santo
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
D Barret
Université de Toulouse
CNRS
IRAP
9 Avenue du colonel Roche, BP 4434631028, Cedex 4ToulouseFrance
E Ambrosi
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
A Robba
INAF -IASF Palermo
Via U. La Malfa 153I-90146PalermoItaly
H Earnshaw
Cahill Center for Astronomy and Astrophysics
California Institute of Technology
91125PasadenaCAUSA
A C Fabian
Institute of Astronomy
Madingley RoadCB3 0HACambridgeUnited Kingdom
XMM-Newton campaign on the ultraluminous X-ray source NGC 247 ULX-1: outflows
Mon. Not. R. Astron. Soc
0002018Printed 7 June 2021 Accepted 2021 June 3. Received 2021 May 28; in original form 2021 April 22.(MN L A T E X style file v2.2)Accretion discs -X-rays: binaries -X-rays: individual: NGC 247 ULX-1
Most ULXs are believed to be powered by super-Eddington accreting neutron stars and, perhaps, black holes. Above the Eddington rate the disc is expected to thicken and to launch powerful winds through radiation pressure. Winds have been recently discovered in several ULXs. However, it is yet unclear whether the thickening of the disc or the wind variability causes the switch between the classical soft and supersoft states observed in some ULXs. In order to understand such phenomenology and the overall super-Eddington mechanism, we undertook a large (800 ks) observing campaign with XMM-Newton to study NGC 247 ULX-1, which shifts between a supersoft and classical soft ULX state. The new observations show unambiguous evidence of a wind in the form of emission and absorption lines from highly-ionised ionic species, with the latter indicating a mildly-relativistic outflow (−0.17c) in line with the detections in other ULXs. Strong dipping activity is observed in the lightcurve and primarily during the brightest observations, which is typical among soft ULXs, and indicates a close relationship between the accretion rate and the appearance of the dips. The latter is likely due to a thickening of the disc scale-height and the wind as shown by a progressively increasing blueshift in the spectral lines.
INTRODUCTION
There is a consensus that the majority of ultraluminous Xray sources (ULXs) are stellar-mass compact objects (neutron stars and perhaps black holes) accreting above the critical Eddington rate (see, e.g., King et al. 2001, Poutanen et al. 2007Middleton et al. 2011, Kaaret et al. 2017. The spectral curvature in the X-ray band, with a characteristic downturn above ∼5 keV, and the presence of residuals at energies 1 keV when modelled with featureless continuum models, are two characteristic ULX features (see, e.g., Soria et al. 2004, Goad et al. 2006, E-mail: [email protected] Gladstone et al. 2009) that are naturally explained if the primary X-ray photons are seen through a disc wind, as expected for systems accreting in the super-Eddington regime (e.g., Shakura & Sunyaev 1973, Poutanen et al. 2007). The thickness of the wind and, as a result, the "softness" of the observed ULX spectra in the ∼0.1-20 keV band, are likely a function of two main parameters: the mass outflow rate in the wind (which is related to the accretion rate), and our viewing angle, with softer sources being observed closer to the disc plane (see, e.g., Sutton et al. 2013, Middleton et al. 2015a, Pinto et al. 2017). In particular, ULXs with a power-law photon index Γ > 2 in the ∼0.3-5 keV band are empirically classified in a "soft ultraluminous" (SUL) state.
Supersoft ultraluminous sources
Among the ULXs, a special sub-class is represented by ultraluminous supersoft sources (ULSs), also referred to as sources in the "supersoft ultraluminous" (SSUL) state (Feng et al. 2016). This state is defined by a dominant, cool blackbody component (T bb 140 eV), with very weak or completely absent hard component at higher energies. Observationally, sources in the SSUL state show essentially no photons >1 keV. Their bolometric luminosity is typically a few 10 39 erg s −1 . They should not be confused with the "classical" supersoft sources (see, e.g., Krautter et al. 1996), which are usually interpreted as nuclear burning on the surface of a white dwarf. Classical supersoft sources also have blackbody spectra, but at lower luminosities (L bb 10 38 erg s −1 ), and generally with a smaller bb radius: R bb ∼ 5000 − 10000 km, consistent with a white dwarf, while SSUL spectra can have blackbody radii as large as 10 5 km (Urquhart & Soria 2016).
Blackbody modelling of SSUL spectra at different epochs shows that the characteristic radius is not constant, which rules out a hard surface as the origin of the thermal emission (Feng et al. 2016). It also shows a clear anti-correlation between blackbody radius and temperature, which rules out a standard accretion disc. Instead, such behaviour is consistent with emission from the photosphere of an optically-thick wind. The increase in the photospheric radius corresponds to an enhancement of the wind thickness.
The apparent non-periodic spectral variability in the SSUL state (Liu 2008) is very likely due to a thickening of the wind along our line of sight. One question we want to address is whether the short-term variations in optical depth are just stochastic variability (at a given mass accretion ratė M ) due to the clumpy nature of the wind ("weather", see, e.g., Takeuchi et al. 2013), or instead each variation is driven by a change in the underlyingṀ ("climate"), which then affects the wind density and launching radius.
A deep absorption edge was detected at ≈1.0-1.1 keV in several SSUL sources (e.g., those in M 51, NGC 6946 and M 101: Urquhart & Soria 2016, Earnshaw & Roberts 2017. This feature clearly appears when the source softens, progressively losing all X-ray photons with energy above 1 keV. There are no strong absorption edges predicted at the observed energy of those dips. Viable solutions are blueshifted O viii ionisation edges (871eV at rest), which was observed in novae during their supersoft phase (e.g., Pinto et al. 2012), or a combination of high-ionisation Fe L or Ne ix-x absorption lines with velocities of ∼0.1-0.2c, supporting the case for an optically-thick, mildly-relativistic wind.
NGC 247 ULX-1
Previous X-ray studies (see, e.g., Feng et al. 2016), based on two short XMM-Newton observations, showed that NGC 247 ULX-1 switched from a supersoft state with hardly any flux above 1 keV (in 2009) to a much brighter state (in 2014) consistent with the soft end of the "standard" ULX population (i.e., those with a bright hard X-ray spectral tail).
NGC 247 ULX-1 is also among the most variable ULXs. It exhibits strong dips in its X-ray lightcurve, which last several ks, and during which the flux decreases by an order of magnitude (Feng et al. 2016). Both the low and high flux spectra are characterised by strong spectral features. Pinto et al. (2017) found similarities between the narrow X-ray spectral features of NGC 247 ULX-1 in the high flux state with those of NGC 55 ULX, using observations taken with the high-resolution gratings aboard XMM-Newton. Despite the short duration (33 ks) of the observations available, two remarkable absorption features at 7.5Å and 16.2Å were found along with other weaker features, which can be modelled with absorption from photoionised gas outflowing at ≈ 0.14c. The observation taken in the low-flux (supersoft) state is far too short to provide any useful data. Thus, we proposed and were awarded a 800 ks deep XMM-Newton programme to characterise the properties of the outflows and the origin of the spectral residuals, and correlate them with the continuum flux variability and state changes.
This paper is the first in a series of intriguing results from our XMM-Newton campaign. Here, we focus on the search for wind signatures in the time-average spectrum, taking full advantage of the high-spectral-resolution data, and on the variability of the spectral features around 1 keV. We detail our observing campaign in Sect. 2 and present the results of our spectral analysis in Sect. 3. We discuss the results in Sect. 4, and outline some conclusions in Sect. 5.
NGC 247 XMM-NEWTON CAMPAIGN
We observed the NGC 247 galaxy between December 2019 and January 2020. The roll angle was similar throughout the whole campaign and avoided strong contamination along the dispersed grating spectra from the nearby brightest Xray sources (see Fig. 1 and Appendix A1 for more detail). Seven observations were expected to occur but, owing to an issue with the RGS instrument that occurred during the last observation (id:0844860701), an additional final observation was taken shortly afterwards (id:0844860801) to recover the lost exposure. In Table 1, we report the detail of our observations. We performed the spectral analysis with the SPEX code (Kaastra et al. 1996), we used C-statistics (C-stat, Cash 1979) for spectral fits, which was proved to be efficient in comparing models similarly to χ 2 statistics (Kaastra 2017), and we adopted 1-σ confidence intervals.
Data preparation
In this work we used data from the European Photon-Imaging Camera (EPIC), the Reflection Grating Spectrometer (RGS) and the Optical Monitor (OM) aboard XMM-Newton. The primary science was carried out with the RGS which can detect and resolve narrow spectral features. The broadband cameras (EPIC MOS 1,2 and pn) were mainly used to determine the spectral continuum and cover the hard X-ray band missed by the RGS.
EPIC cameras
We reduced the 8 new XMM-Newton observations with the Science Analysis System (SAS v18.0.0, CALDB available on March, 2020). EPIC-pn and MOS data were reduced with the EPPROC and EMPROC tasks, respectively. Following the recommened procedures, we filtered the MOS and pn event lists for bad pixels, cosmic-ray events outside the field of view, photons in the gaps (FLAG=0), and applied standard grade selections (PATTERN 12 for MOS and PATTERN 4 for pn). We corrected for contamination from high background by selecting background-quiescent intervals on the lightcurves for MOS 1,2 and pn in the 10−12 keV energy band. These lightcurves were extracted in time bins of 100 s and all those with a count rate above 0.4 c/s for pn and 0.2 c/s for MOS were rejected. The MOS 1-2 and pn clean exposure times are reported in Table 1.
We extracted EPIC MOS 1-2 and pn images in the 0.3−10 keV energy range and stacked them with the EMO-SAIC task (see Fig. 1). We also extracted EPIC MOS 1-2 and pn lightcurves from within a circular region of 20" radius centred on the source position (α, δ=00:47:04.0,-20:47:45.7). We used the task EPICLCCORR, which corrects for vignetting, bad pixels, chip gaps, PSF, and quantum efficiency. The background (BKG) lightcurves were extracted from within a larger circle in a nearby region on the same chip. In order to measure the variations in the spectra hardness, we extracted EPIC-pn (for its larger effective area) lightcurves in the soft (0.3-1 keV) and hard (1-10 keV) energy bands (with 1 ks time bins to increase the signal-to-noise ratio of each bin). The PN lightcurves are shown in Fig. 2 (top-left panel) with a length above 800 ks (despite an effective clean exposure of 600ks) due to the large 1ks bin size which does not show the time gaps of high BKG lasting few 100s. The lightcurves were also glued together for displaying purposes but with vertical dotted lines separating them. The hardness ratio was defined as the fraction of hard photons with respect to the total (H/(H + S)). The boundary energy was adopted owing to the strong spectral curvature of supersoft sources exhibited above 1 keV (see Sect. 2.1.4).
We extracted EPIC MOS 1-2 and pn spectra in the same regions used for the lightcurves. The EPIC-pn spectra of the individual observations are shown in Fig. 3. We avoided over-plotting the MOS spectra for displaying purposes.
RGS cameras
The RGS data reduction was performed with the RGSPROC pipeline. We filtered out periods affected by contamination from Solar flares by selecting background-quiescent intervals in the lightcurves of the RGS 1,2 CCD 9 (i.e., 1.7 keV) with a count rate below 0.2 c/s. As usual, Solar flares affected the RGS data on a much lower level than EPIC. The total clean exposure times are quoted in Table 1. We extracted the 1 st and 2 nd -order RGS spectra in a cross-dispersion region of 0.8' width, centred on the source coordinates and the background spectra by selecting photons beyond the 98% of the source point-spread-function. The background regions do not overlap with bright sources. After inspecting the 2 nd -order spectra, we decided not to use them as they were highly affected by the background. We stacked the 1 st -order RGS 1 and 2 spectra from all observations with RGSCOMBINE and the EPIC-pn, MOS 1 and MOS 2 spectra, using EPICSPECCOMBINE. The stacking provided 4 time-averaged high signal-to-noise spectra for RGS, MOS 1, MOS 2, and pn detectors.
All XMM-Newton spectra were grouped in channels of at least 1/3 of the spectral resolution, for optimal binning and to avoid over-sampling, and at least 25 counts per bin, using SAS task SPECGROUP. This has also the advantage to smooth the background spectra in the energy range with low statistics, avoiding narrow spurious features introduced by the background subtraction. This also enabled us to check our results with the χ 2 statistics. The stacked spectra have many counts with the binning affecting only the spectral range at the rather low and high energies (outside the 0.6-1.7 keV RGS band and above 4 keV in EPIC spectra), where line detection is not crucial. We found no significant effect onto our line or continuum modelling by decreasing the binning to just 1/3 of the spectral resolution.
Optical Monitor
We used OM data to search for a possible optical/UV counterpart to NGC 247 ULX-1. To increase the signal-to-noise ration we stacked all the internally aligned full-frame sky images per filter and per observation, using the SAS tool OM-MOSAIC. Each observation contains at least one image in one of these filters: V, UVW1, UM2 and UVW2 and the final total exposures corresponded to 105 ks, 105 ks, 115 ks and 220 ks, respectively. We ran the OMDETECT task on these stacked images with a limit on the detection threshold of 2 σ. At the position of the ULX (Section 2.1.1), no source was detected in any of the filters. This is unsurprising since the ULX region is very close to a bright association of OB stars, which makes such a detection challenging. To derive an upper limit for the ULX emission in these bands, we computed the total background rate for a circular region of 6" around the position of the ULX. This provided 3 σ upper limits for the ULX flux, which are comparable to previous measurements obtained by Feng et al. (2016) using deep observations with Hubble Space Telescope (HST, UV-optical fluxes ∼ 10 −14 erg s −1 cm −2 ). In the far-UV the OM flux upper limits are slightly below the HST detections, suggesting long-term flux variability and that a substantial fraction of the UV flux originates within the accretion disc rather than the stellar companion (see Fig. A3, top panel).
A final OM image obtained by stacking the data from all the filters of all the observations is shown in Fig. 1, bottom panel, where the bright association of OB stars can be seen on the right side of the X-ray source centroid.
Data investigation
The XMM-Newton 0.3-10 keV lightcurve shows a strong dipping behaviour with the source flux approaching zero c/s during time interval of less than 15 ks, as found by Feng et al. (2016), see . The dips have variable duration, with the shortest ones being of a few hundred seconds, which was estimated extracting finer lightcurves (D'Aì et al. in prep). During the dips, the flux drops by an order of magnitude and then returns to the previous level, which makes it difficult to believe it is due to an intrinsic flux change rather than to a temporary obscuration phenomenon. This behaviour causes the multiple peaks present in the histogram of the count rates (Fig. 2, right panel), which disagree with a single-peaked log-normal trend.
The hardness ratio decreases during the dips (Fig. 2, bottom-left panel), which is very similar to the soft source NGC 55 ULX-1 (Pinto et al. 2017). This was interpreted as evidence of temporary obscuration of the inner hot regions from a clumpy disc wind (e.g., Stobbart et al. 2006).
The lightcurve also shows that NGC 247 ULX-1 undergoes a long-term flux variability with observations 4-to-6 exhibiting significantly higher fluxes. Importantly, the frequency of the dips was higher during these observations.
In order to evaluate the source variability during each observation, we calculated the fractional excess variance of the EPIC-pn lightcurve of each observation and the rootmean-square (RMS), following standard formulae (see, e.g., Nandra et al. 1997, Vaughan et al. 2003, and Allevato et al. 2013). We adopted time bins of 1 ks and, as time length, the duration of each exposure (∼ 100 ks, with the exception of obsid:0844860801). The computed RMS values are reported in Table A1 and range from about 10 to 55 %.
The EPIC spectra of the individual observations in Fig. 3 show a variability pattern that is common to ULXs, with the harder band (> 1 keV) exhibiting the largest variation (see, e.g., Middleton et al. 2015a, Brightman et al. 2016, and Walton et al. 2018c. A thorough analysis of the variability pattern involving a careful sampling of time interval with similar flux and hardness ratio, and the study of the power density spectra, will be done in two separate papers (D'Aì et al. in prep, Alston et al. 2021).
SPECTRAL ANALYSIS
In this section we present the spectral analysis of NGC 247 ULX-1. We first show the time evolution of the main spectral residuals around 1 keV through the modelling of the spectral continuum in EPIC spectra from different observations (see Sect. 3.1). Then we will perform a thorough analysis of the high-statistics, time-averaged, stacked spectrum in order to identify the lines in the RGS (Sect. 3.2 and 3.3), to build the spectral energy distribution (SED), and to use physical plasma models for the wind detection and modelling (Sect. 3.4 and 3.5). The final best-fit models are shown in Sect. 3.6 and the statistical significance of our findings in Sect. A5.
ULX spectra require up to three components to obtain a satisfactory description of the spectral continuum. Two blackbody-like components are often used to model the soft (0.3-1 keV) and hard (1-10 keV) X-ray energy bands (see, e.g., Stobbart et al. 2006, Pintore et al. 2015. The availability of high-statistics spectra and broadband data reveals the presence of a third harder component, which dominates the continuum above 10 keV (see, e.g., Walton et al. 2018a). In the framework of super-Eddington accretion, the cooler, soft, component corresponds to the X-ray emission of the wind and the disc around the spherisation radius. The hard component refers to the inner accretion flow. The hard (> 10 keV) tail is either due to Compton scattering in the innermost regions or from an accretion column (see, e.g., Middleton et al. 2015a). The two hard components, especially the hard tail, are weak in supersoft ULXs.
Time evolution of the ∼1 keV residuals
For the modelling of EPIC spectra of individual observations we adopted a simple continuum model consisting of two blackbody (bb) components, which is often used as a proxy for more complex models (see, e.g., Walton et al. 2014, Pinto et al. 2017, Koliopanos et al. 2017, and Gúrpide et al. 2021. We did not model the hard tail in the individual spectra because it is so weak that any model would be highly unconstrained, but for the time-averaged SED modelling we took it into account (see Sect. 3.2). The emission components are corrected for absorption by the Galactic interstellar medium and the circumstellar medium near the ULX using the hot model in SPEX with a low temperature (kT = 0.2 eV, e.g. Pinto et al. 2013), at which the gas is neutral. In the spectral fits, we coupled the column density of the hot model across all observations as it is unlikely that the amount of neutral gas along our line of sight towards the ULX would change on time scales of a few days. In some Galactic Xray binaries (XRB) variable obscuration was found, but it is not clear whether these findings are related to ionised gas (i.e. winds) or uncertainties in the continuum rather than neutral gas (see, e.g., Miller et al. 2009. We simultaneously applied the hot (bb + bb) continuum model to the EPIC MOS 1,2 and pn spectra of all eight observations. The results are shown in Fig. 3 and Table A1. We obtained an average column density of NH = (3.4±0.1)× 10 21 cm −2 . This phenomenological model provides a good description of the broadband spectra, but the C-statistics are high when compared to the corresponding degrees of freedom (Cν ∼ 3−9) due to the well-known strong and sharp residuals in the form of absorption and emission features around 1 keV (see Fig. 3).
It is possible to understand the nature of the residuals by tracking their temporal evolution from one observation to another. In Middleton et al. (2015b) and Pinto et al. (2017), the three main spectral features observed around 1 keV in the spectra of NGC 1313 ULX-1 and NGC 55 ULX-1 were modelled with a positive (emission) and two negative Gaussians (absorption) lines. The availability of deeper observations allow us to fit the three Gaussian components independently, but we fixed the line broadening to zero km/s (i.e. only instrumental broadening) in order to minimise the degeneracy that can be produced by the low spectral resolution of EPIC. We chose three Gaussian lines as previous work on high-resolution RGS data identified Fe / Ne emission lines at 1 keV, O viii absorption lines around 0.7-0.8 keV, and Fe / Ne absorption lines above 1 keV (Pinto et al. 2016). Using only one or two Gaussian lines always resulted in significantly worse fits during alternative tests.
The hot (bb + bb) + (gaus + gaus + gaus) spectral model improves the fits for all observations with respect to the hot (bb + bb) model. In Table A1, we report the best-fitting parameters for each observation. The reduced C-stat are still rather high due to additional residuals that require more complex and physical models. Moreover there is evidence of a weak, broad, hard tail in all spectra above 3 keV, which cannot be explained with atomic lines (Fig. 3).
In Fig. 4, we compare the energy centroids (left panel) and the fluxes (right panel) as measured for the three Gaussian lines in the EPIC spectra of the eight observations. The point size was coded according to the value of their RMS estimate in Sect. 2.1.4. Both the energies and fluxes of the Gaussians lines vary with the time, showing a higher blueshift and lower flux (in absolute value) during the dipping observations with enhanced variability.
Time-averaged continuum: spectral modelling
The RGS spectra of the individual observations do not provide statistics sufficient to detect and resolve the spectral residuals with high significance. The 750 ks RGS 1+2 stacked spectrum instead has a much higher quality and enables line detection, despite the low source flux.
We simultaneously fitted the time-averaged stacked EPIC MOS 1,2 and pn, and the RGS spectra using the absorbed double-blackbody continuum model (hot (bb + bb)) adopted for the spectra of the individual observations. The hard tail above 3 keV is more evident in the stacked data and, therefore, we accounted for it using a third, hotter (kT ∼ 1 keV), blackbody as to mimic additional hard Xray photons down-scattered through the disc photosphere (and/or the wind). The three-blackbody model brings the C-stat from 4671 down to 4488 for four additional degrees of freedom. In all fits the parameters of the blackbody and Table 2. Time-averaged EPIC+RGS spectral fits.
Parameter
Units
hot (bb + bb) hot (bb + bb + bb) Area, bb1 10 19 cm 2 3.4 ± 0.3 4.5 ± 0.3 Area, bb2 10 16 cm 2 1.0 ± 0.1 1.7 ± 0.2 Area, bb3 10 13 cm 2 - 1.7 ± 0.5 kT, bb1 keV 0.120 ± 0.001 0.116 ± 0.001 kT, bb2 keV 0.382 ± 0.003 0.342 ± 0.005 kT, bb3 keV - 1.05 ± 0.07 L X,tot 10 39 erg/s 5.3 ± 0.5 6.2 ± 0.5 N H 10 21 cm −2 3.64 ± 0.05 3.84 ± 0.05 C-stat/d.o.f. 4671/1136 4488/1132
L X (0.3−10 keV) luminosities are calculated assuming a distance of 3.3 Mpc and are corrected for absorption (or de-absorbed).
The best-fit three-blackbody model and the spectra are shown in Fig. 6. Both models are shown in the SED modelling in Fig. A3.
the ISM absorber were coupled among the EPIC and RGS models. We left the overall normalisations of the MOS 1,2 and RGS models free to vary with respect to pn in order to account for the typical 5-10% cross-calibration uncertainties among their effective areas. Details on the spectral fits for both models are reported in Table 2 and Fig. 5. In order to avoid a crowded plot, we did not include the more noisy RGS spectrum in this plot, while it is shown later in Figs. 6 and 7 where the EPIC data below 1.77 keV was ignored. The blackbody models used so far are the simplest available and allowed us to constrain the parameters. We tested various combination of two-components models to possibly improve the description of the spectral continuum before accounting for narrow features. These consisted of the cool blackbody component plus either a disc blackbody (dbb) or a disc blackbody modified by coherent Compton scattering (mbb) or Comptonisation (comt). These did not provide improvements with respect to the three blackbody model. Similar results were provided by more complex three-component continuum models which anyway over-fit the data and lead to degeneracy among model parameters due to the weak hard (> 1 keV) continuum. Alternatively, one could use the common blackbody plus powerlaw emission model. However, the powerlaw would be very steep with a consequent unphysical divergence in the soft band, which would badly affect the ionisation balance calculation (see, e.g., Pinto et al. 2020a).
Time-averaged continuum: Gaussian line scan
In Fig. 6, we show the stacked RGS and EPIC spectra, indicating the dominant H-/He-like transitions of the X-ray band often found in the spectra of X-ray binaries. A zoom over the RGS spectrum with ad-hoc linear Y-axis can be found in Fig. 7. The RGS exhibits substantial residuals at the same energies as the EPIC residuals but resolves them in a structure of lines, although the former have lower count rates. Strong emission-like features appear near the transition energies of the most relevant neon K and iron L lines. Additional features may be related to N vii and O vii-viii, although the background starts to be important in the RGS below 0.6 keV. Some possible absorption-like features are indicated with vertical dotted lines. The very good agreement between the positions of the RGS, MOS 1,2 and pn residuals rules out instrumental dominant features.
Following the approach used in Pinto et al. (2016) searched for narrow spectral features by scanning the spectra with Gaussian lines. We adopted a logarithmic grid of 1000 points with energies between 0.3 (41Å) and 10 keV (1.24Å). This choice provided a spacing that is comparable to the RGS and EPIC resolving power in the energy range we are investigating (R RGS ∼ 100 − 500 and R EPIC ∼ 20 − 60). We tested line widths (σG=FWHM/2.355) of 100, 250, 500 and 1000 km/s, which are comparable to the RGS resolution. At each energy we recorded the ∆C improvement to the best-fit continuum model and expressed the significance as the square root of the ∆C. This provides the maximum significance for each line (as it neglects the number of trials). We multiplied √ ∆C by the sign of the Gaussian normalisation to distinguish between emission and absorption lines.
In Fig. 8 we show the results of the line scan obtained for the time-averaged stacked RGS+EPIC spectra using the three-blackbody continuum model. We performed the line scan in two ways: the first run using all RGS data (0.3-2 keV) and EPIC data (0.3-10 keV) and the second one ignoring the EPIC data between 0.33 and 1.77 keV, where the RGS effective area is well calibrated. When fitting only RGS in the 0.33-1.77 keV energy band we always fixed the temperatures of the blackbody components to the best-fit values obtained using the EPIC data in the whole 0.3-10 keV. This is due to the low count rate of the RGS spectra that limited our capability to constrain the overall continuum level and shape (see Pinto et al. 2020b).
The line scan of the EPIC + RGS spectra data picked out the strong emission-like features near 1 keV and other two around 0.6 and 1.5 keV. Broad absorption features were also found around 0.7 and 1.2-1.3 keV as previously done in the spectrum of each observation (see Fig. 4 and Table A1).
Owing to the low spectral resolution and high count rate of EPIC, the features appear very broad (∼ 0.1 keV) in the EPIC+RGS Gaussian scan preventing us from identifying them. This becomes easier above 2 keV due to the increasingly higher EPIC spectral resolution. The Gaussian scan performed using only RGS between 0.33 and 1.77 keV resolved the features into a forest of lines. Multiple lines are responsible for the 1 keV emission-like and the 1.2-1.3 keV absorption-like features. Interestingly, most absorption features are consistent with some Lyman α transitions also seen in emission, if we assume a systemic blueshifted absorption of about 0.17c (see vertical ticks in Fig. 8). The 0.6 keV features might either be interpreted as a blueshifted O vii α triplet or redshifted O vii β + O viii Ly α emission lines. The emission lines found between 0.9-1 keV are most likely from Ne ix-x and Fe xviii-xxiv ions. Alternative interpretations correspond to different velocities of the X-ray emitting (and absorbing) plasmas. The use of physical models is necessary to distinguish among the several interpretations. The most simple physical models involve the adoption of plasma in either collisionally-ionisation equilibrium (CIE) or photoionisation equilibrium (PIE).
The single-trial line significance ("σST ") of the individual strongest features is around 5 σ, which of course is smaller if we take into account the look-elsewhere effect. However, plasma models are able to model multiple lines, combining their individual ∆C improvements to the best-fit continuum, and boost the overall significance (see below). Table 2). The bottom panel shows the residuals calculated with respect to the continuum model. The rest-frame energies of the most common strong transitions in the X-ray band (red for RGS band and blue for EPIC) and the absorption features (dotted lines) are also shown. All spectra were rebinned for displaying purposes.
Collisional-ionisation jet modelling
Pinto et al. (2020b) performed automated scan models using either CIE or PIE plasmas. This technique prevents the fits from getting stuck in local minima, although is computationally expensive (lasting a few hours on one CPU).
Collisionally-ionised emitting gas
Following Kosec et al. (2018b) and Pinto et al. (2020b), we performed a multidimensional automated scan with an emission model that assumes collisional ionisation equilibrium (cie model in SPEX). We adopted a logarithmic grid of temperatures between 0.1 and 5 keV (50 points), and lineof-sight velocities, vLOS, between −0.3c (blueshifted jet) and +0.3c (redshifted jet, with steps of 500 km/s). We tested several values of velocity dispersion (from 100 to 10000 km/s), finding comparable results as already shown by the Gaussian line scan in Fig. 8, with the best fit achieved at vRMS ∼ 3000 km/s. Abundances were chosen to be Solar (to limit the computing time) and the emission measure EM = ne nH V was the only free parameter of the CIE in the spectral fit.
We applied the automated routine scanning CIE models onto the NGC 247 ULX-1 time-averaged RGS (0.33−2 keV) and EPIC MOS 1,2-pn (1.77−10 keV) spectra. We adopted the three-blackbody continuum model shown in Sect. 3.2 and Table 2 (see also black line in Fig. 6). The results are shown in Fig. 9 (left panel). The best-fit corresponds to a large improvement with respect to the continuum model (∆C = 82, for 4 additional degrees of freedom) and was achieved for a CIE temperature of 0.9 keV and a small blueshift of around 6500 km/s (∼ 0.022c). These results were driven by the strong lines that can be seen in the RGS spectrum 1 keV in Fig. 8 (bottom two panels).
We have checked our method by testing an identical CIE scan on the well-known Galactic X-ray source SS 433. This source is not ultraluminous in the X-ray band due to obscuration of the accretion disk from local circumstellar gas, but exhibits a persistent and bright (10 40 erg/s) radio jet, which is powering a luminous optical super-bubble (Brinkmann Fabrika 2004;Medvedev et al. 2020). SS 433 is therefore considered to be viewed edge-on. Were it observed face-on, it would likely appear as a ULX (Begelman et al. 2006;Poutanen et al. 2007;Middleton et al. 2018). SS 433 also shows a relativistic jet in the form of blueshifted lines from multi-temperature plasma in collisional-ionisation equilibrium (Marshall et al. 2002
Collisionally-ionised absorbing gas
It is uncommon to adopt absorption models of gas in collisional-ionisation equilibrium in accreting objects as 1) it is difficult to distinguish between photoionisation and collisional ionisation on the sole basis of the dominant resonant absorption lines and 2) we hardly expect any jet to absorb the X-ray source continuum along our line of sight. However, we cannot exclude that shocks are produced by interaction between the ULX wind and the stellar companion or the surrounding bubble (or within the wind itself). Therefore, we also performed a model scan with the hot model in SPEX, which works just like cie but assumes absorbing gas.
In Fig. 9 (right panel) we show the results using the hot model over the same kT range used for the emitting gas, adopting a velocity dispersion of 500 km/s and line-ofsight velocities, vLOS, ranging between −0.3c and zero, i.e. only Doppler blueshifts or outflows rather than inflows. The best-fit solution is obtained for a −0.17c blueshifted with a remarkable ∆C = 48 as suggested by the detection of several negative Gaussians blueshifted by similar values in Fig. 8.
Photoionisation wind modelling
The emission and absorption lines can be produced by winds rather than by jets as expected in the case of super-Eddington accretion discs and, therefore, in ULXs. Accurate photoionisation models require knowledge of the radiation field, i.e. the SED from optical to hard X-ray energies.
SED and photoionisation balance
Following Pinto et al. (2020a,b), we built the time-averaged SED of NGC 247 ULX-1 using data from the XMM-Newton campaign and archival HST observations (as taken from Feng et al. 2016). For issues regarding the non simultaneity of HST and XMM observations, see Appendix A4. For the X-ray band (0.3−10 keV or ∼ 10 16−18 Hz) we used the bestfit three-blackbody continuum model, estimated in Sect. 3.2, Fig. 5 and Table 2. As shown in Sect. 2.1.3, the OM filters were not sensitive enough to detect the optical counterpart, but their flux upper limits in the optical and UV energy bands are however comparable to the HST measurements (see Fig. A3 top panel). We therefore modelled the optical/UV portion of the SED with the two-blackbody model of Feng et al. (2016), which together with the three-blackbody X-ray model formed our five-blackbody SED model.
We can describe the photoionisation equilibrium with the ionisation parameter, ξ, defined as ξ = Lion/(nH R 2 ) (see, e.g., Tarter et al. 1969), where Lion is the ionising luminosity (measured between 13.6 eV and 13.6 keV), nH the hydrogen volume density and R the distance from the ionising source. The ionisation balance was calculating with the SPEX pion model, which calculates the transmission and the emission of a thin slab of photoionised gas, self-consistently.
Following Pinto et al. (2020b), we also computed the stability (or S) curve, which is the relationship between the temperature (or the ionisation parameter) and the ratio between the radiation and the thermal pressure, which can be expressed as Ξ = F/(nHckT ) = 19222 ξ/T (Krolik et al. 1981). The stability curve is shown in Fig. A3 (bottom panel). The branches of the S curve with a negative gradient are characterised by thermally unstable gas. In this work, we assumed that the wind is seeing the same SED that we observe and, therefore, adopted the five-blackbody model SED and ionisation balance. Systematic effects from the SED choice are discussed in Sect. 4 and Appendix A4.
Photoionised emitting gas
Once the SED and the ionisation balance were computed, we scanned the time-average EPIC+RGS spectra with the SPEX pion model with the same multi-dimensional routine used for the cie model in Sect. 3.4.1, and a similar parameter space. We adopted a logarithmic grid of ionisation parameters (log ξ [erg/s cm] between 0 and 6 with 0.1 steps). The only free parameter for the pion is the column density, NH.
Unlike NGC 1313 ULX-1, the RGS spectrum of NGC 247 ULX-1 does not exhibit well resolved emission line triplets. This is perhaps due to the longer integration time required and the variability of the line centroid (see Fig. 4), which could wash out the triplets when stacking all the spectra. Additionally, the crucial O vii complex is affected by the background noise. The lack of He-like triplets means that the volume density and the luminosity of the photoionised gas are degenerate. Fitting both parameters results in poor constraints and much higher computation time. We therefore chose not to fit the volume density and adopted nH = 10 10 cm −3 , which is a lower limit found for NGC 1313 ULX-1 (Pinto et al. 2020b). This would only slightly affect the overall flux and column density of the pion component.
The pion covering fraction is set to zero (i.e pion only produces emission lines) and the solid angle Ω = 4π. Fitting additional parameters such as Ω might provide even better fits but would significantly increase the computing time, without altering the velocity and ionisation parameters.
In Fig. 10 (left panel) we show the results obtained using a pion line width of 1000 km/s. The confidence level (CL) is expressed in σ, which is constrained using Monte Carlo simulations (see Sect. A5). A peak (∆C = 77) corresponding to a solution of blueshift emission is seen around 0.02 − 0.03c in agreement with the CIE model scans (see Sect. 3.4 and Fig. 9). However, the different ionisation balance and types of emission lines in the photoionisation equilibrium detected another, stronger (∆C = 102), peak corresponding to a redshift of ∼ +0.05c.
Photoionised absorbing gas
In principle, we could just use pion for both emission and absorption. However, this model re-calculates the ionisation balance at every iteration and therefore is computationally expensive. Therefore, for the absorbing gas we chose to use the faster xabs model, which is optimized for absorption and adopts the pre-calculated ionisation balance (see Sect. 3.5.1).
The xabs model shares several parameters with pion except the opening angle of the line emission which is zero since no emission is present in this model. We adopted a covering fraction equal to unity in order to avoid degeneracy and reduce the computing time. We calculated the grid of photoionised xabs models in the same way as the pion models, but assuming line-of-sight velocities, vLOS, ranging between −0.3c and zero, i.e. only Doppler blueshifts as for the CIE hot absorption models in Sect. 3.4.2.
In Fig. 10 (right panel) we show the probability distributions from scans of the RGS and EPIC spectra with vRMS = 1000 km/s. As expected, our code confirmed the v = −0.17c solution (∆C = 46).
Final fits with physical plasma models
In order to check the inter-dependence of the emitting and absorbing plasma components and to test for any variations in the values of their parameters we performed two more fits of the RGS and EPIC data (with EPIC still excluded Figure 11. Time-averaged XMM-Newton RGS (0.33−2 keV) and EPIC-pn (1.77−10 keV) spectra. Overlaid are two alternative models (jet -CIE, red line and the wind -PIE, black line). The spectra were regrouped and the plot zoomed onto the RGS data and the 0.4-2 keV energy band for displaying purposes. between 0.33-1.77 keV) adding onto the 3-blackbody continuum two alternative plasma models. The first one was a wind model that used the pion in emission and the xabs in absorption. The second model was an approximation of jets and shocks in the form of cie component in emission and hot component in absorption. The results for the two models are shown in Table 3 and Fig. 11 (zoomed onto the RGS). The absorption components provided very similar results, especially for the column densities and the line-of-sight velocities. As previously noted, the emission components show some differences. The results are discussed in Sect. 4.
The temperatures of the three blackbody components were always fixed to the EPIC 0.3-10 keV fits (Table 2) each time we ignored EPIC data between 0.33-1.77 keV, resulting in parameters consistent with the continuum modelling.
By alternatively excluding one particular plasma component from the spectral model we estimated the relative contribution to the spectral fit and the minimum ∆C-stat improvement of each component for both the wind and jet model (see ∆C e values in Table 3). Following Pinto et al. (2020b), we compared the minimum ∆C-stat values with the Table 3. NGC 247 ULX-1: alternative plasma models.
Model 1 Parameter
Emission Absorption
PIE L X (E) , N H (A) 1.4 ± 0.2 2.8 ± 0.1 (wind) log ξ (erg/s cm) 3.7 ± 0.1 4.3 ± 0.1 v LOS (c) +0.042 ± 0.004 −0.166 ± 0.002 v RMS (km/s) 3000 +2700 −800 400 ± 200 ∆C a ( e )
104 (97) Montecarlo simulations to obtain an approximate estimate of the minimum significance of each wind or jet component (parameter σ e ), which is highest for the emission phases.
Finally, to further test the strength of our results we performed a fit of the RGS and EPIC data including the whole EPIC energy band but fixing the plasma models to the best-fit results of the hybrid RGS (0.33-2 keV) plus EPIC (1.77-10 keV) fits. In Sect. 3.5.3, we noted that the inclusion of the photoionised xabs absorption component significantly decreased the C-stat from 4488 (of the simple 3-blackbody model) to 3033. The addition of the photoionised pion emission component further lowered the C-stat to 2431 (with a χ 2 = 1830 and a total of 1132 degrees of freedom). We performed the same fit by using instead the RGS jet model with the hot collisionally-ionised component fixing the parameters to those in Table 3. This decreased the C-stat from 4488 to 3043, similarly to the photoionised absorber. The addition of a cie emission component (with fixed plasma parameters) implied a final C-stat = 2542, which is slightly worse than pion due to some positive residuals left around 0.9 keV that can also be seen in the RGS spectral modelling in Fig. 11. We notice that all the spectral fits shown here are not formally acceptable (see Table 4), although the winds components provide significant improvements. One reason is the variability of the features, both of their centroids and relative strength (see Fig. 3 and 4). This means that more complex models would be required to correctly fit the lines. On the other hand, the winds are likely multiphase as shown by the low-temperature O vii clearly missed by our single phase model (see Fig. 11). This was already shown in Pinto et al. (2020b) and occurs in SS 433 too (see Appendix A3). Finally, some bad cross-calibration below 0.6 keV between the EPIC and RGS cameras further prevent us from obtaining fully acceptable fits (see Fig. 5).
DISCUSSION
It is still unclear how does the wind vary with the accretion rate and whether it has a major role in shaping ULX spectra. Pinto et al. (2020a,b) showed that the wind evolves with the changes in the continuum from the fainter, harder states to the brighter states, which implies a tight relationship between the source's spectral continuum and wind appearance as observed by comparing winds in different ULXs.
Among the ULXs, the supersoft ultraluminous X-ray (SSUL) sources are particularly fascinating objects. The fact that such sources reach very high luminosities (several 10 39 erg/s) but always exhibiting very soft (kT ∼ 0.1 keV) spectra indicates that they are being observed at moderate-tohigh inclination angles as also suggested by the presence of dips in their lightcurves (see, e.g., Feng et al. 2016). In fact, Urquhart & Soria (2016) modelled the soft X-ray residuals and the ∼ 1 keV drop found in several CCD spectra of ULSs with a model of thermal emission and an absorption edge, which they interpreted as a result of absorption and photon reprocessing by an optically-thick wind which obscures the innermost regions where most hard X-rays are produced.
Time evolution of the 1 keV residuals
The stacked XMM-Newton lightcurve (see Fig. 2) shows different pattern of source variability such as a long-term overall change in the average flux on daily time scales followed by abrupt drops in the flux where the source becomes softer (the dips) on timescales between 100s and a few hours. The dipping activity seems also to enhance during observations with higher flux peaks. The higher flux might be associated with a higher local accretion rate, which then would increase the radiative force and launch optically-thick wind cloudlets in the line of sight, thereby obscuring the innermost regions of the disc responsible for the hard X-ray emission (as suggested by Urquhart & Soria 2016). More insights on the nature of the dips will be provided by Alston et al. (2021). This work shows that the dips in the higher flux observations preferentially occur on 5 and 10 ks timescales, which suggests that they are caused from obscuration at ∼ 10 4−5 RG, where RG is the Gravitational radius (if the timescales are associated with keplerian motion around a NS or a stellar-mass BH). Such a range is comparable to the distance that the 0.17c wind would travel on a time scale of 1 ks, suggesting a possible connection between them.
The high-quality EPIC spectra of the individual observations show a remarkable flux variability in the features around 1 keV (see Fig. 3). In order to quantify such variability, we modelled the two strongest absorption features around 0.7 and 1.2 keV, and the dominant emission-like feature at 1 keV with three Gaussian lines for the EPIC (MOS and pn) spectral of the individual observations. All lines show a distinct pattern with their energy centroids significantly blueshifted during the brightest observations (which also exhibit most dips and the highest variability, see Fig. 4). Interestingly, the fluxes of the high-energy lines (1 and 1.2 keV) significantly decrease during the dipping observations while the 0.7 keV line seems to strengthen (see Table A1). This would either suggest a different location of the three lines, with the 0.7 keV line coming from the outer and less obscured regions, or a change in the ionisation state of the absorber during the high-flux periods. This is similar to what was observed in NGC 1313 ULX-1 (Pinto et al. 2020b, Middleton et al. 2015b. A detailed study of the broadband spectra and residuals evolution will be shown by D' Aì et al. (in prep). The fact that the location and strength of the residuals vary on hourly timescales with the source flux provides strong evidence in support for a disc wind rather than emission from the local ULX bubble or the galactic ISM.
A wind or a jet?
Emission lines
The time-average stacked RGS spectrum showed strong emission residuals near the transition energies of several ions such as O vii-viii, Ne ix-x and Fe xviii (see Fig. 7 and 8). The agreement between RGS and EPIC is corroborated by applying the wind model constrained using only RGS in the 0.33 − 1.77 keV band to the whole EPIC MOS and pn timeaverage spectra (see Table 4). Unfortunately, the He-like emission triples of e.g. O vii and Ne ix are not well resolved likely due to the stacking of RGS spectra from different observations that clearly showed some variability in the line centroid as discussed above. This limited our capabilities of distinguishing between collisional and photoionisation, but the use of full plasma models provided some constraints.
By performing automated searches of plasma models in a large parameter space, we built probability contours for both collisionally-ionised and photoionised plasma emission models. The properties of the line-emitting gas are very similar to those of the Galactic super-Eddington source SS 433 with a low velocity along the line of sight and a mild 1 keV temperature which is expected by the strong Ne K and Fe L emission around 1 keV (see Fig. 9). It is well established that the emission lines of SS 433 are from the jet with the low velocity indicating that the precessing jet was at very high angle, close to 90 degrees, in the analysed observation. If the lines of NGC 247 ULX-1 were also from a variable jet, the observed low velocity would suggest that it is being viewed at high angle in agreement with the presence of dips.
The photoionisation emission models (pion component in SPEX, see Fig. 10) however provided a significantly higher improvement to the spectral fits and a better description of the emission lines (see Table 3 and Fig. 11). This together with the evolution of the lines with the source continuum would favour photoionisation equilibrium similarly to the emission lines in NGC 1313 ULX-1 (Pinto et al. 2020b).
Regardless of the adopted equilibrium state, the luminosity of the line-emission component is remarkably high (L 0.3−10 keV > 10 38 erg/s), similarly to NGC 1313 ULX-1, NGC 5408 ULX-1 (Pinto et al. 2016), NGC 55 ULX-1 (Pinto et al. 2017), NGC 5204 ULX-1 (Kosec et al. 2018a) and other ULXs (e.g., Wang et al. 2019). This is about 2-3 orders of magnitude higher than the emission lines in SS 433 and those producing the winds in classical supergiant X-ray binaries (sub-Eddington neutron stars accreting from supergiant OB stars, e.g. El Mellah & Casse 2017) or the lines from accretion disc coronae of low-mass X-ray binaries (see, e.g., Psaradaki et al. 2018). The luminosity of 1.4 × 10 38 erg/s is instead comparable to the extended X-ray emission recently found around the extremely bright pulsating NGC 5907 ULX-1 (Belfiore et al. 2020), which suggests that the wind might be energetic enough to mechanically drive the ∼ 100-pc super bubbles (see also Pinto et al. 2020a).
Similarly to NGC 1313 ULX-1, the O vii emission lines cannot be reproduced with the emission component responsible for the Fe L and Ne K emission (see Fig. 11). A second component (either photo-or collisionally-ionised) with a low blueshift of 6000 km/s would be required.
Absorption lines
In this work we also reported a highly significant detection of mildly-relativistic, ultrafast, outflows. Both collisional and photoionisation (see Fig. 9, 10 and 11) plasma models identified a high velocity outflow (−0.17c) in the range of the velocities found in other ULX winds.
The ionisation parameter is rather high (log ξ = 4.3) which is not surprising given the soft SED adopted for this source (see Fig. A3). If the wind at the launch is seeing a different SED (e.g., the hot innermost regions presumably obscured along our line of sight) the overall ionisation balance might be significantly different. This subject was extensively discussed in Pinto et al. (2020a) who found larger instability branches in the S curves of harder ULXs. Therefore, as a test, we performed an additional fit with the photoionised pion + xabs wind model (as previously done in Sect. 3.6) by adopting the ionisation balance calculated for the hard state of NGC 1313 ULX-1 in Pinto et al. (2020b) to estimate the systematic effects on the wind parameters. The fit was statistically indistinguishable from the one performed with the ionisation balance computed for NGC 247 ULX-1, with the exception of the ionisation parameters which, as expected, turned out to be significantly lower by about ∆ log ξ ∼ 1.
The absorption lines are generally weaker than the emission lines in the RGS spectrum of NGC 247 ULX-1 which could be due to the low source continuum (Kosec et al. submitted). This was also predicted by Pinto et al. (2017) as the lines are normally seen on top of the continuum from the innermost regions which in this case is likely obscured.
Statistically we cannot distinguish photoionisation from collisional ionisation, but the former is favoured by the photoionised nature of the emitting plasma and the unusual detection of collisionally-ionised absorption in XRB winds.
Accretion disc and wind physics
In the framework of super-Eddington accretion the luminosity scales with the logarithm of the accretion rate in Eddington units (ṁ =Ṁ /ṀE) times the geometrical beaming of the funnel created by the height of the disc around the spherisation radius and by the wind itself (see Fig. 12). Following King & Lasota (2020), the apparent luminosity can be expressed with Lapp = L/b = LE(1 + lnṁ)/b, where L is the intrinsic luminosity, LE the luminosity in Eddington units, and b = 73/ṁ 2 the geometrical beaming.
To estimate the bolometric luminosity of NGC 247 ULX-1 we integrated the broadband SED between 1 eV and 10 keV (or 2.4 × 10 14−18 Hz, see Fig. A3) and obtained 9.4×10 39 erg/s. NGC 247 ULX-1 luminosity could therefore be explained by assuming a black hole accreting above 10 times the Eddington rate or a neutron star accreting abovė m = 25. Atṁ ∼ 10 the spherisation radius, i.e. the base of the wind, would be R sph = 27/4ṁRG = 68RG. Interestingly, this is very close to the escape radius for a −0.17c wind (Re = 2GM/v 2 = 2RGc 2 /v 2 = 73RG), which would indeed suggest that we detected a wind launched from the spherisation radius of a compact object above 10ṀE.
From Eq. (38) in Poutanen et al. (2007), assuming MBH = 10M andṁ = 10, we estimated a temperature for the spherisation radius T sph ∼ 0.3 keV, which is comparable to the warm blackbody component in our fits (see Table 2), with the cooler (∼ 0.1 keV) blackbody associated with the outer disc and, likely, the wind photosphere as suggested by recent work (see, e.g., Qiu & Feng 2021, Gúrpide et al. 2021. We notice, however, that the source is being seen at high inclination with a substantial fraction of the hard X-ray photons obscured by the funnel. The intrinsic luminosity of NGC 247 ULX-1 might therefore be higher than the value estimated above with a higher accretion rate, implying T sph ∼ 0.1 − 0.2 keV, closer to the cooler blackbody component, and a slightly larger R sph . It is also possible that the wind is launched with lower velocity at radii larger than 73RG and it gets accelerated by radiation pressure from the inner accretion flow (see, e.g., Takeuchi et al. 2013).
Similar considerations would apply to a non-magnetar (B 10 12 G) neutron star withṁ = 25 since the spherisation radius (in cm) would be of the same order of magnitude as a 10M black hole as both R sph and T sph scale with thė M and the mass of the compact object, whose trends nearly cancel out. This was briefly discussed in Pinto et al. (2020a).
The kinetic power of the wind can be written as Lw = 1/2Ṁw v 2 w = 2 π mp µ Ω C v 3 w /ξ Lion ∼ 4 × 10 40 erg/s, wherė Mw = 4 π R 2 ρ v 2
w Ω C is the outflow rate, Ω and C are the solid angle and the volume filling factor (or clumpiness), respectively, which were adopted equal to 0.3 as conservative values from MHD simulations of winds driven by radiation pressure in super-Eddington winds (Takeuchi et al. 2013), ρ is the density and R is the distance from the ionising source.
Here we have used the ξ definition to get rid of the R 2 ρ factor where ρ = nH mp µ with mp the proton mass and µ = 0.6 the average particle weight of a highly ionised plasma.
The filling factor of the wind might be much smaller. Using Eq. (23) in Kobayashi et al. (2018) and assuming that the outflow rate is comparable to the accretion rate, we obtain C ∼ 3 × 10 −2 . Systematics would tend to cancel out when also accounting for the uncertainty on the ionisation parameter in the case for a harder SED (∆ log ξ ∼ 1). In the pessimistic case the wind power would be of the order of 10 39 erg/s, which means still high enough to affect the surrounding medium and inflate ISM cavities.
The spectral shape, strong wind features, and presence of dips suggest that NGC 247 ULX-1 is likely being observed at high inclination (see Fig. 12, left panel) where the funnel is already obscuring the innermost, hot, hard X-ray emitting regions. As mentioned in Sect. 4.1, the increase of the average flux level during the intermediate observations (3,4,5) might be caused by a higher local accretion rate. Such a climate change would however affect the properties of both the disc and the wind. The scale-height is already relevant around the Eddington limit (see, e.g., Shakura & Sunyaev 1973, Poutanen et al. 2007. A further increase in the localṀ might push the optically-thick funnel further upwards (see Fig. 12, right panel) thereby obscuring the regions emitting photons with temperatures higher than that of the spherisation radius ( 0.3 keV), causing the very soft dips shown in Fig. 2 (see also Urquhart & Soria 2016).
During the dips the high-ionisation portion of the wind could be hard to see as its absorption lines were primarily affecting the (now) obscured hard X-ray continuum. In fact, the strength of the high-ionisation (1.2-1.3 keV) absorption lines clearly decreases during the dipping observations (see Fig. 4), while the 0.7 keV O viii absorption line seems constant in flux if not even stronger. This might also suggest a stratification in the wind. Outside the dips, an overall increase in the accretion rate would also imply a stronger radiative force and, therefore, a slightly faster wind which seems to be confirmed by the higher blueshift of the lines (see Fig. 3 and 4). The 1 keV emission lines also weaken during the bright / dipping observations, indicating that they should be produced in the inner regions in agreement with their overall larger broadening (see Table 3).
A similar picture was proposed by Guo et al. (2019) who argued that the ∼100s transitions can be explained by the viscous timescale with the X-ray flux variability driven by accretion rate fluctuations (atṁ 10). However, local fluctuations in theṀ might also cause variations in the winds, which could alter the source appearance (Feng et al. 2016).
Although fascinating and self-consistent, this scenario might be not the only one able to explain all the observables.
Additional, alternative, and (ideally) model-independent approaches could be considered. For instance, another phenomenon which might explain the nature of the dips might be the propeller effect due to a strong magnetic field. Such scenario would imply a decreasingṀ and a geometrical beaming to cause the observed brightening. More insights on the temporal evolution of the spectral residuals accounting for the different spectral states that the source shows inside / outside the dips will be given in D' Aì et al. (in prep). Similarly, the Fourier analysis of the characteristics timescales in NGC 247 ULX-1 and the corresponding association with the dipping activity will be shown by Alston et al. (2021). Here, in particular, we argue that the alternation of the dips might be due to azimuthally-dependent structures.
We plan to investigate the variability of the RGS spectral lines to place more constraints onto their nature. However, the low count rate of the grating spectra currently prevent us from trying to study them during the dips and on timescales shorter than 100 ks in the bright states outside the dips. Future missions like XRISM and Athena will revolutionise the study of ULX thanks to their high effective area, high spectral resolution, and low background (see, e.g., Barret et al. 2018, Guainazzi & Tashiro 2018. Pinto et al. (2020b) simulated NGC 1313 ULX-1 microcalorimeter spectra for these two missions and showed 1) how XRISM will strengthen the identification of lines in the 1 − 10 keV band and 2) how Athena / X-IFU will be able to detect winds in observations with just 1 ks of exposure time. The latter is primarily due to the fact that X-IFU will have two orders of magnitude higher effective area than RGS.
CONCLUSIONS
Most ULXs are believed to be powered by super-Eddington accreting neutron stars and, perhaps, black holes. The disc is expected to thicken at accretion rates above the Eddington rate and to launch powerful winds through radiation pressure and/or magnetic fields. Evidence of winds has been found in several ULXs through high-resolution X-ray spectrometers. It is yet unclear whether the switch between the classical soft and supersoft state -which is observed in supersoft ULXs -is due to the thickening of the disc and/or the optically-thick part of the wind. In order to better understand such phenomenology and the overall super-Eddington mechanism, we undertook a large observing campaign with XMM-Newton to study NGC 247 ULX-1, which is the brightest (in flux) of all supersoft ULXs.
The new observations showed for the first time unambiguous evidence of a wind in the form of emission and absorption lines from highly-ionised ionic species, with the absorption phase exhibiting a mildly-relativistic outflow (−0.17c) in line with the other ULXs whose grating spectra had sufficient quality to detect and identify spectral lines. Remarkable variability was observed in the source flux with strong dipping activity during the brightest observations, which is typical among soft ULXs such as NGC 55 ULX-1, and indicate a close relationship between the accretion rate and the appearance of the dips. The latter are likely due to a thickening of the disc scale-height and the wind as shown by a progressively increasing blueshift in the spectral lines. Figure 12. A possible scenario for the dips and ULX-ULS transitions in NGC 247 ULX-1. The source is observed at a viewing angle that is high enough that the inner disc is already partly obscured by the wind (soft ULXs, left panel). An increase of the accretion rate pushes up the scale-height of the disc and the optically-thick base of the wind, causing an near-total obscuration of the inner regions and the source appears as an ultraluminous supersoft source (ULS, right panel, see also Pinto et al. 2017, 2020b, Guo et al. 2019).
A1 Nearby bright X-ray source
The RGS extraction region includes a few faint sources with the brightest one being XMMU J004710.0-204708 (X-2 hereafter, see Fig. 1). We extracted its EPIC spectra from all observations and stacked them similarly to ULX-1. The spectrum of X-2 is much flatter than that of the ULX-1 and can be well modelled with a powerlaw model (Γ = 1.60±0.03), a moderate column density NH = (1.0 ± 0.1) × 10 21 cm −2 , and an intrinsic luminosity L 0.3−10keV = (1.45±0.04)×10 38 erg/s (assuming a distance of 3.3 Mpc). This corresponds to the Eddington limit for a Solar-mass star and, given the spectral slope, the source X-2 could be a common XRB near the NGC 247 centre. At 1 keV its spectrum is remarkably featureless and 40-50 times fainter than ULX-1 implying that it will have no significant effects on the RGS spectral lines. Table A1 shows the results of the EPIC spectral modelling and the root-mean square estimated from the EPIC-pn data of each observation (see Sect. 3.1 and 2.1.4).
A2 Modelling of individual EPIC observations
A3 CIE model scan for the SS 433 RGS spectrum
We analysed the XMM-Newton RGS spectrum of SS 433 from the observation id:0694870201 (2012-10-03), which provides the longest (∼130 ks) and best-exposed RGS spectrum of SS 433. We reduced the RGS spectrum of SS 433 obsid 0694870201 identically to that of the NGC 247 ULX-1 data shown in Sect. 2.1.2. After removing the very little Solar flares we are left with 129.4 ks for both RGS 1 and 2 cameras. We used the 6-25Å range because at higher wavelengths the source emission is significantly absorbed and the background noise dominates the RGS spectrum (see Fig. A1). No significant pile up was found in the RGS spectra.
Strong emission lines were observed close to the restframe energies of the most relevant transitions such as Si xiii, Mg xi, Ne x, Ne ix, Fe xvii, and O viii, which is very similar to NGC247 ULX-1, albeit at higher significance because SS 433 is much closer (∼ 5 kpc) and brigther. We modelled the RGS spectral continuum with an absorbed powerlaw model obtaining results similar to Marshall et al. (2013) and Medvedev et al. (2018) such as a column density NH = (1.14 ± 0.02) × 10 22 cm −2 , a slope Γ = 2.53 ± 0.06 and an X-ray unabsorbed luminosity L [0.3−10 keV] = (1.03 ± 0.05)×10 36 erg/s. We obtained high C-stat/d.o.f = 2393/618 as expected, due to the strong, unmodelled, emission lines.
We tested onto the SS 433 RGS spectrum the same routine used for the NGC 247 ULX-1 data in Sect. 3.4.2 to check the robustness of our method. We adopted collisional ionisation equilibrium (cie model in SPEX) to model the jet emission. The velocity dispersion was fixed to 500 km/s, i.e. close to the RGS spectral resolution, the abundances were chosen to be Solar (to limit the computing time). We found a dominant low-velocity solution with an average temperature of 1 keV (see dotted horizontal and vertical lines in Fig. A2). 10 19 cm 2 10 −1 keV 10 16 cm 2 10 −1 keV 10 39 erg/s 10 46 ph/s 10 −1 keV 10 46 ph/s 10 −1 keV 10 45 ph/s 10 −1 keV (%) 0844860101 0.9 ± 0.1 1.38 ± 0.02 1.1 ± 0.3 3.7 ± 0.1 2.9 ± 0.4 −1.2 ± 0.2 6.9 ± 0.1 1.5 ± 0.2 9.5 ± 0.1 −6.9 ± 0.8 12.4 ± 0.1 321/160 16.1 ± 0.1 0844860201 1.0 ± 0.1 1.31 ± 0.02 0.7 ± 0.2 3.5 ± 0.2 2.3 ± 0.3 −0.9 ± 0.2 6.7 ± 0.1 1.8 ± 0.2 9.3 ± 0.1 −5.1 ± 0.6 12.6 ± 0.1 341/140 11.7 ± 0.1 0844860301 1.8 ± 0.2 1.21 ± 0.02 0.6 ± 0.2 3.3 ± 0.2 2.8 ± 0.4 −1.3 ± 0.2 6.6 ± 0.1 2.6 ± 0.2 9.1 ± 0.1 −6.6 ± 0.5 12.4 ± 0.1 441/128 9.5 ± 0.1 0844860401 0.6 ± 0.1 1.50 ± 0.03 1.0 ± 0.1 4.3 ± 0.1 2.7 ± 0.3 −1.4 ± 0.2 7.5 ± 0.1 0.8 ± 0.2 9.9 ± 0.1 −2.8 ± 1.0 12.1 ± 0.2 412/192 47.7 ± 0.2 0844860501 0.6 ± 0.1 1.53 ± 0.03 1.0 ± 0.2 4.2 ± 0.1 2.9 ± 0.3 −1.7 ± 0.2 7.4 ± 0.1 0.6 ± 0.2 9.8 ± 0.2 −3.0 ± 0.9 13.0 ± 0.2 391/188 26.9 ± 0.2 0844860601 0.5 ± 0.1 1.47 ± 0.03 0.6 ± 0.2 4.1 ± 0.2 2.2 ± 0.3 −1.1 ± 0.2 7.2 ± 0.2 0.6 ± 0.2 9.8 ± 0.2 −3.7 ± 0.9 12.6 ± 0.2 271/154 54.7 ± 0.2 0844860701 1.8 ± 0.2 1.20 ± 0.02 0.2 ± 0.1 3.9 ± 0.3 2.8 ± 0.3 −1.1 ± 0.3 6.5 ± 0.2 2.2 ± 0.2 9.0 ± 0.1 −5.8 ± 0.6 11.8 ± 0.2 324/122 11.3 ± 0.1 0844860801 1.8 ± 0.2 1.21 ± 0.02 0.5 ± 0.3 3.4 ± 0.3 2.8 ± 0.4 −1.9 ± 0.3 6.6 ± 0.1 2.0 ± 0.2 9.2 ± 0.1 −7.2 ± 0.7 12.2 ± 0.1 298/109 11.3 ± 0.1 L X (0.3−10 keV) luminosities are calculated assuming a distance of 3.3 Mpc and are corrected for absorption (or de-absorbed, see Fig. 3). The velocity is consistent with the dynamical range found by Medvedev et al. (2018) using the Fe K lines from the EPIC-pn spectrum, but showed a lower temperature, which is expected given that the RGS spectrum is more sensitive to the cooler gas phase of the multi-temperature jet.
A4 SED modelling and systematics effects
The non detection of the optical and UV counterpart of NGC 247 ULX-1 (Sect. 2.1.3) prevent us from building a simultaneous multi-wavelength SED for our source. This might have systematic effects on the calculation of the photoionisation balance. Moreover, the flux upper limits obtained with the OM suggest that at least in the far-UV en- Figure A3. SED (top panel) and thermal-stability curve (bottom panel) computed for the time-averaged spectrum of NGC 247 ULX-1. The baseline SED (solid black line) consists of five unabsorbed blackbody models that account for emission in the optical, UV and X-ray bands. An alternative, simpler SED model uses just the X-ray two-blackbody model (dashed-dotted line).
ergy band the source flux was lower than the levels measured in the archival HST observations that we used here. Pinto et al. (2020a) showed that a lower IR-to-UV flux in moderately-hard sources, such as NGC 1313 ULX-1, mainly strengthens thermal instabilities at intermediate temperatures and ionisation parameters. In Fig. A3 we show the SED adopted here (solid black curve) consisting of a 5blackbody model along with the simple 2-blackbody model (dashed-dotted black line, see also Sect. 3.5.1). The lower panel shows the stability curves computed for these models. Some deviations are mainly seen at log ξ from 1.5−2.5, which is well below the values measured in this work (see Table 3). This is not surprising given the shortage of hard X-ray in our spectra which are the primary responsible of thermal instabilities. This suggests that the ionisation balance is not significantly affected even if the optical / UV fluxes were 2 orders of magnitude lower than our assumptions. Occurrences [ν] 4.0σ 4.5σ 5.0σ
Log[ν] = 6.57 -0.22*∆C 2 × 10 6 simulations forecast 2 × 10 5 simulations forecast 2 × 10 4 simulations histo-fit
A5 Monte Carlo simulations and significance
The ∆Cmax improvement to the continuum model does not necessarily yield the significance of the corresponding emission or absorption line models. This is due to the large parameters space that was explored and the possibility of detecting random spectral features (the look-elsewhere effect). Among our physical model searches, the one for the absorption lines provided the smallest ∆Cmax due to their strength being lower than that typical of the emission lines. We therefore focused on the results obtained with the xabs component and used them as a proxy for the pion.
Following the method used in Pinto et al. (2020b), we simulated 20 000 RGS and EPIC spectra adopting the 3-blackbody continuum model. Each faked spectrum was scanned with the same xabs grids used in Sect. 3.5.3. The results of our MC simulations are shown in Fig. A4. No outlier was found with ∆C ∆Cmax = 46, which suggests a significance > 4σ for the absorbing gas.
We compared the simulations histogram for NGC 247 ULX-1 with those obtained for different sources by adopting a similar approach: 20k simulations of NGC 1313 ULX-1 (Pinto et al. 2020b), 2k for NGC 5204 ULX-1 (Kosec et al. 2018a) and 50k for the same data with a new, faster, crosscorrelation method (Kosec et al. submitted), 20k for ULX NGC 7793 P13 (Pinto et al. in prep), and 1k for AGN PG 1448 (Kosec et al. 2020). We fit the histograms of the logarithm of the occurrences with straight lines and found an average slope Γ = −0.225 ± 0.015, which agrees with the simulations of NGC 247 ULX-1 (Γ = −0.218 ± 0.006).
We used the best-fit straight lines to estimate the overall shape of the ∆C-stat distribution and forecast the results of larger numbers of simulations thanks to the agreement between the trends from 1 000 to 50 000 simulations. We therefore scaled the histogram fit of NGC 247 ULX-1, assuming a constant slope and multiplying the intercept of the straight line by a number equal to the ratio of the parameter space that we want to forecast for a given number of simulations and the one we obtained with 20 000 simulations. In Fig. A4 we show the predictions for 2 × 10 5 (dash-dotted line) and 2×10 6 (dashed line) simulations. This would suggest 4.5 and 5 σ detection probabilities for ∆C-stat above 35 and 40, respectively, in the data with an uncertainty of 0.2σ according to the spread in the slope of the other histograms.
We finally retrieved the various ∆C-stat values that correspond to confidence levels ranging from 2.0, 2.5, ..., 5.0 σ and plot them as black contours in Fig. 10. The σ contours for the pion model scan were calculated by scaling the parameter space in the histogram of the xabs simulations in the same way used for the forecast.
Figure 1 .
1XMM-Newton image of the NGC 247 field obtained by combining all the data available for EPIC-pn and MOS 1,2 zooming onto the ULX-1 region (top panel). The black strip and circle indicate the RGS and EPIC extraction regions, respectively. The bottom panel shows the time-averaged image obtained by stacking all the data from the Optical Monitor. A small circle with 4" radius shows the X-ray source centroid.
Figure 2 .
2Top left panel: cumulated XMM-Newton (0.3-10 keV) pn lightcurve of NGC 247 ULX-1 with the individual observations separated by vertical dotted lines. Time bin size is 1 ks. Obsid 3-to-7 are separated by 20-60 ks each. Bottom left panel: hardness ratio (H/(H+S)) estimated from the lightcurves in the soft (0.3-1 keV) and hard (1-10 keV) X-ray bands. Right panel: count rate histogram.
Figure 3 .
3XMM-Newton EPIC-pn spectra of NGC 247 ULX-1. The top panel shows the pn spectra of the individual observations with overlaid the 2-blackbody continuum models. The bottom panel shows the corresponding residuals. Both the emission feature below 1 keV and the absorption feature above 1 keV vary in centroid and strength according to the continuum flux and shape. This is qualitatively similar to the intermediate-hard source NGC 1313 ULX-1 and the soft source NGC 55 ULX-1 (Middleton et al. 2015b; Pinto et al. 2017).
Figure 4 .
4Evolution of the XMM-Newton EPIC spectral residuals around 1 keV. The left and the right panels show the energy centroid and normalisation of the Gaussian lines used to fit the three main (unresolved) spectral residuals, respectively. The shaded grey areas highlight the observations of high spectral variability and dipping. 'Abs' and 'Emi' refer to absorption and emission features, respectively.
Figure 5 .
5Time-averaged XMM-Newton EPIC-pn and MOS 1,2 spectra. Overlaid are two alternative continuum models consisting of two (red line) and three (black line) blackbody components.
Figure 6 .
6Top panel: time-averaged XMM-Newton RGS (0.33−2 keV) and EPIC MOS 1,2-pn (1.77−10 keV) spectra. Overlaid is the baseline 3-blackbody continuum model (solid black line, see
Figure 7 .
7Time-averaged XMM-Newton RGS (0.33−2 keV) and EPIC-pn (1.77−10 keV) spectra. Overlaid is the baseline continuum model. This is a zoom ofFig. 6onto the RGS data.
Figure 8 .Figure 9 .
89Gaussian line scan performed on the time-averaged XMM-Newton EPIC and RGS spectra of NGC 247 ULX-1. The top panels show the case when the EPIC spectra are used throughout the whole 0.3-10 keV band, while the bottom two panels show the results obtained with RGS between 0.33−2 keV and EPIC-pn from 1.77−10 keV (see alsoFig. 6). The results for four different line widths are shown. No remarkable difference is observed among the adopted widths. The single-trial line significance ("σ ST ") is calculated as square root of the ∆C times the sign of the Gaussian normalisation (positive/negative for emission/absorption lines). Labels are red for RGS, green for strong EPIC features and blue for the faint Fe K. The grey shaded areas show the 3 and 5 σ ST limits for individual lines. Multi-dimensional scans of collisional-ionisation emission (left panel) and absorption models (right panel) for NGC 247 ULX-1 time-averaged EPIC+RGS spectra. The X-axis shows the line-of-sight velocity (negative means blueshift, i.e. motion towards the observer). The color is coded according to the ∆C fit improvement to the spectral continuum model.
Figure 10 .
10Scans of photoionisation emission (left) and absorption (right) models for the time-averaged EPIC and RGS spectra. Labels are same as inFig. 9. The black contours refer to the (2.0, 2.5, ... 4.5, 5.0) σ confidence levels estimated by Monte Carlo simulations.
of the plasma components for two alternative models. Model 1 or PIE/wind: photoionised emission (pion) and absorption (xabs, see Fig. 10 and Fig. 11). Model 2 or CIE/jet: collisionally-ionised emission (cie) and absorption (hot, see Fig. 9). The column densities, N H , are in 10 22 cm −2 . The lineof-sight velocities, v LOS , are in units of light speed c; the velocity dispersion, v RMS , is in km s −1 . The 0.3-10 keV luminosities of the emitting plasmas, L X (E), are defined in units of 10 38 erg/s. The ∆C a ( e ) refer to the ∆C-statistics of each component computed when the component is the only one in model (a) or when the other one is included (e). The same applies to the detection significances, σ a ( e ), evaluated with Monte Carlo simulations.
Figure A1 .Figure A2 .
A1A2SS 433 RGS spectrum and continuum model. Multi-dimensional scan of collisional-ionisation emission model for the SS 433 RGS spectrum. The X-axis shows the line-of-sight velocity. Labels are same as inFig. 9.
Figure A4 .
A4Histogram and corresponding power-law fit of the 20k Monte Carlo simulations of NGC 247 ULX-1 and forecast for 200 000 and 2 million simulations.
Table 1 .
1XMM-Newton campaign on NGC 247 ULX-1.OBS ID
Date
t RGS1 t RGS2 t MOS1 t MOS2 tpn
(ks)
(ks)
(ks)
(ks)
(ks)
0844860101 2019-12-03 110.5 110.1 104.6 105.0 76.6
0844860201 2019-12-09 110.9 110.6 109.3 109.2 90.3
0844860301 2019-12-31 117.4 117.0 112.6 113.9 76.8
0844860401 2020-01-02 112.3 112.0 110.6 110.5 93.5
0844860501 2020-01-04 115.9 115.6 113.3 113.3 94.4
0844860601 2020-01-06 102.3 101.7 83.3
83.2
57.4
0844860701 2020-01-08 28.2
28.3
96.1
97.5
70.1
0844860801 2020-01-12 61.0
60.8
59.3
59.4
41.7
Total [ks]
758.5 756.1 789.1 792.0 600.8
Total [kcnts]
7.2
10.4
56.7
57.2 186.5
Notes: exposure times account for high background removal.
Source counts are in the whole energy band for each detector.
Table 4 .
4NGC 247 ULX-1: continuum and plasma modelsModel
RGS+EPIC (> 1.77 keV) RGS+EPIC (full band)
3 bb
2019/994
4488/1132
3 bb * hot
1971/990
3043/1132 gas par fixed
3 bb * hot + cie
1895/886
2542/1132 gas par fixed
2233/1124
3 bb * xabs
1973/990
3033/1132 gas par fixed
3 bb * xabs + pion 1877/886
2431/1132 gas par fixed
2191/1124
C-stat/d.o.f. values for spectral continuum and plasma models.
'gas par fixed' means that for those fits the parameters of both
emission and absorption components were fixed to the best-fit
values obtained excluding EPIC data below 1.77 keV.
Table A1 .
A1Broadband properties of the individual observations. Spectra Model : hot (bb + bb) + (gaus + gaus + gaus) LightcurveBlackbody 1
Blackbody 2
0.3-10keV
Gaussian 1
Gaussian 2
Gaussian 3
0.3-10keV 0.3-10keV
Parameter
Area
Temp
Area
Temp
L X,tot
Norm
Energy
Norm
Energy
Norm
Energy
C/d.o.f.
RMS
Units
ACKNOWLEDGMENTSThis work is based on observations obtained with XMM-Newton, an ESA science mission funded by ESA Member States and USA (NASA). We acknowledge support from ESA Research Fellowships. We thank the XMM-Newton SOC for support in optimising our observing campaign and J. M. Miller, D. Proga and M. Parker for useful discussion regarding winds and absorption in Galactic X-ray binary. AD, MDS, EA acknowledge financial support from the agreement ASI-INAF n.2017-14-H.0 and INAF main-stream. We thank the anonymous referee for their very useful suggestions.DATA AVAILABILITYAll of the data and software used in this work are publicly available from ESA's XMM-Newton Science Archive (XSA 1 and NASA's HEASARC archive 2 . Our codes are publicly available and can be found on the GitHub 3 .
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APPENDIX A: TECHNICAL DETAILS In this section we put technical detail, plots, and tables. that were excluded from the main body of the paperAPPENDIX A: TECHNICAL DETAILS In this section we put technical detail, plots, and tables, that were excluded from the main body of the paper.
|
[] |
[
"Confirming the 3:2 Resonance Chain of K2-138",
"Confirming the 3:2 Resonance Chain of K2-138"
] |
[
"Mariah G Macdonald \nDepartment of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA\n\nDepartment of Physics\nThe College of New Jersey\n2000 Pennington Road08628EwingNJUSA\n",
"Leonard Feil \nDepartment of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA\n",
"Tyler Quinn \nDepartment of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA\n",
"David Rice \nDepartment of Physics & Astronomy\nUniversity of Nevada\nLas Vegas, Las Vegas89154NVUSA\n"
] |
[
"Department of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA",
"Department of Physics\nThe College of New Jersey\n2000 Pennington Road08628EwingNJUSA",
"Department of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA",
"Department of Astronomy & Astrophysics\nCenter for Exoplanets and Habitable Worlds\nThe Pennsylvania State University\n16802University ParkPAUSA",
"Department of Physics & Astronomy\nUniversity of Nevada\nLas Vegas, Las Vegas89154NVUSA"
] |
[] |
The study of orbital resonances allows for the constraint of planetary properties of compact systems. K2-138 is an early K-type star with six planets, five of which have been proposed to be in the longest chain of 3:2 mean motion resonances. To observe and potentially verify the resonant behavior of K2-138's planets, we run N -body simulations using previously measured parameters. Through our analysis, we find that 99.2% of our simulations result in a chain of 3:2 resonances, although only 11% of them show a five-planet resonance chain. We find we are able to use resonances to constrain the orbital periods and masses of the planets. We explore the possibility of this system forming in situ and through disk migration, and investigate the potential compositions of each planet using a planet structure code.
|
10.3847/1538-3881/ac524c
|
[
"https://arxiv.org/pdf/2201.12687v1.pdf"
] | 246,430,639 |
2201.12687
|
6abd0220707b7ec2748a780be07f11b2edac4875
|
Confirming the 3:2 Resonance Chain of K2-138
February 1, 2022
Mariah G Macdonald
Department of Astronomy & Astrophysics
Center for Exoplanets and Habitable Worlds
The Pennsylvania State University
16802University ParkPAUSA
Department of Physics
The College of New Jersey
2000 Pennington Road08628EwingNJUSA
Leonard Feil
Department of Astronomy & Astrophysics
Center for Exoplanets and Habitable Worlds
The Pennsylvania State University
16802University ParkPAUSA
Tyler Quinn
Department of Astronomy & Astrophysics
Center for Exoplanets and Habitable Worlds
The Pennsylvania State University
16802University ParkPAUSA
David Rice
Department of Physics & Astronomy
University of Nevada
Las Vegas, Las Vegas89154NVUSA
Confirming the 3:2 Resonance Chain of K2-138
February 1, 2022Draft version Typeset using L A T E X twocolumn style in AASTeX63Exoplanet dynamics (490)Exoplanet migration (2205)Exoplanet structure (495)
The study of orbital resonances allows for the constraint of planetary properties of compact systems. K2-138 is an early K-type star with six planets, five of which have been proposed to be in the longest chain of 3:2 mean motion resonances. To observe and potentially verify the resonant behavior of K2-138's planets, we run N -body simulations using previously measured parameters. Through our analysis, we find that 99.2% of our simulations result in a chain of 3:2 resonances, although only 11% of them show a five-planet resonance chain. We find we are able to use resonances to constrain the orbital periods and masses of the planets. We explore the possibility of this system forming in situ and through disk migration, and investigate the potential compositions of each planet using a planet structure code.
INTRODUCTION
The Kepler and K2 missions allowed for the study of worlds other than our own and the systems they inhabit. Since the launch in 2009, we have confirmed more than 4,000 exoplanets, with several thousands of other candidates being investigated. This catalog of planets has enabled the expansion of various sub-fields of astronomy, including astrobiology, the study of atmospheres, and orbital dynamics and evolution.
Some of the systems we have discovered exhibit resonant behavior, including Kepler-223 (Mills & Fabrycky 2017), Kepler-80 (MacDonald et al. 2016), and TRAPPIST-1 (Luger et al. 2017); mean motion resonance occurs when two or more planets repeatedly exchange angular momentum and energy as they orbit their host star, often seen as a repeated geometric configuration. We can predict a system's resonances by observing the orbital periods of the planets, as planets in or near mean motion resonance have period ratios that reduce to a ratio of small numbers. However, a period ratio near commensurability does not guarantee a resonance; we must study the system's dynamics and resonance angles to confirm resonance. We describe a two-body resonance by the libration, or oscillation, of the two-body angle: Θ b,c = j 1 λ b + j 2 λ c + j 3 ω b + j 4 ω c + j 5 Ω b + j 6 Ω c (1) where λ is the mean longitude, ω is the argument of periapsis, Ω is the longitude of the ascending node, planet b is interior to planet c, and the coefficients j i sum to zero.
A system can have more than two planets in resonance, either in a chain of librating two-body resonances, or in a three-or more body resonance. The three-body angle is the difference between the associated two-body resonance angles: φ = pλ 3 − (p + q)λ 2 + qλ 1
where λ i is again the mean longitude and p and q are integers. It is easier, and therefore more common, to constrain the libration of three-body resonance angles than that of two-body angles since the two-body angles depend on the longitudes of periapsis which are challenging to constrain for low-eccentricity exoplanets. By observing a potentially resonant system with high enough precision photometry or high enough cadence, we are able to measure the resonance angles and confirm resonance. However, such data do not yet exist for most systems, and therefore we must model the system across all planetary and orbital parameters that are consistent with the data; if all parameters lead to solutions with librating angles, we are able to confirm the resonance(s) of the system. K2-138 is a relatively bright (V = 12.21) K-dwarf, hosting six confirmed planets 1 . These inner five planets orbit their star fairly rapidly, with orbital periods ranging from 2.4 days to 12.8 days. In addition, the period ratios of adjacent planets suggest that the system could be locked in a five planet chain of 3:2 mean motion resonances, the longest 3:2 resonance chain known if confirmed. Both Christiansen et al. (2018) and suggest this chain, however, no study has yet performed an in-depth study of the orbital dynamics of K2-138 to confirm such a chain.
Here, we perform such a study with the aim of confirming the resonance chain and constraining the system's formation and dynamical evolution. In Section 2, we discuss our N -body simulations. We then present our results in Section 3. We use the system's resonances to constrain the planetary masses and orbital periods and discuss various pathways for forming the chain in Section 4. We also discuss the planetary compositions before summarizing and concluding our work in Section 5.
METHODS
To observe the long term behavior of K2-138 and to constrain the dynamics of the system, we run N -body simulations using REBOUND (Rein & Liu 2012). We model the system with a stellar mass of M = 0.93M (Christiansen et al. 2018), and use the orbital parameters as constrained by . We do not model the outer-most planet K2-138g since it is most likely dynamically decoupled with an orbital period of 42 days Hardegree-Ullman et al. 2021). For each simulation, we draw planetary masses, inclinations, and orbital periods from normal distributions centered on the nominal values from with standard deviations equal to the uncertainties. We use the WHFast integrator (Rein & Tamayo 2015) and integrate the system for 8Myr with a timestep of 5% of the innermost planet's period. We summarize the initial conditions of our simulations in Table 1.
RESULTS
We run 3000 N -body simulations for 8Myr and analyze the results of each to confirm a resonance chain. We look for libration of all of the two-body angles, all of the three-body angles, or any combination of angles that leads to all planets participating in the chain. We summarize the dynamical results of our simulations in Table 2.
1 K2-138g was most recently confirmed by Hardegree-Ullman et al. (2021) We find that 99.2% of our simulations result in a chain of 3:2 resonances, although only 11.0% of our simulations result in a a five-planet resonance chain; in 87.1% of the simulations, planet f is dynamically decoupled from the other planets. Overall, we find that 68.5% of the simulations result in a four-planet resonance chain, and 19.6% of the simulations result in only a threeplanet resonance chain. Of the 0.8% of our simulations where the planets are not interacting via a resonance chain, 76% of the simulations result in no three-body angles librating and only the two-body angle between K2-138b and K2-138c and the angle between K2-138d and K2-138e librating; the remaining 24% result in libration of only the two-body angle between K2-138d and K2-138e. We show an example of a fully librating five-planet resonance chain in Figure 1.
Given these results, we are able to confirm that the planets of K2-138 are indeed in a chain of 3:2 mean motion resonances, but we are not able to confirm a five-planet chain. The middle three planets -K2-138c, K2-138d, and K2-138e -are in resonance with oneanother, but K2-138b and K2-138f do not need to be in resonance with other planets for the system to be stable or to be consistent with the data.
DISCUSSION
Our motivation behind developing this method stems from the need to confirm more planetary candidates and diversify the planetary catalogue. In compact systems, resonant behavior could be a common means for maintaining stability (e.g., Tamayo et al. 2017). Finding these configurations allows further constraints on the mass of candidates, which we can use to confirm their planetary identity.
In the following subsections, we constrain the planetary masses and orbital properties using the resonances, explore why K2-138f is not part of the chain, and discuss both the resonance chain's formation and the composition of K2-138's planets.
Using Resonance to Constrain Mass
Since we run our simulations to explore a large range of mass and orbital properties, we analyze our results to see which planetary parameters lead to the libration of the resonance angles. We first compare the distributions of planetary mass, eccentricity, and orbital period for simulations where each angle is librating to distributions of the same parameter from simulations where each angle is not librating using a two-sample Kolmogorov-Smirnov test. In this test, the null hypothesis is that the two samples (parameter from librating simulations and parameter from circulating simulations) Note-Parameters for the N -body simulations of K2-138. Here, P is the orbital period, t0 represents the transit epoch (BJD−2457700), i is the sky-plane inclination, and Mp is the planetary mass. We use the values published by for all parameters, and a stellar mass of 0.9310 +0.0700 −0.0640 (Christiansen et al. 2018). Note-Resulting three-body and two-body angles from the RE-BOUND N -body simulations. We find that 99.2% of our simulations result in a resonance chain of 3:2 resonances, although only 11.0% result in a five-planet resonance chain. Planet f is dynamically decoupled from the other planets in most of the simulations (87.1%), while planet b is dynamically decoupled in 21.4% of the simulations, planet c is dynamically decoupled in 0.2% of the simulations, and planet e is dynamically decoupled in 0.17% of the simulations.
Θ c−b = 3λc − 2λ b −
are drawn from the same population, and the resulting p-value is the probability that the null hypothesis is correct. Therefore, a small p-value (p < 0.05) allows us to reject this hypothesis and suggests that the two distributions are statistically distinct. We find that the libration of the three-body angle between the inner three planets depends on the eccentricities and orbital periods of the three planets but only depends on the mass of K2-138b. Similarly, the libration of the three-body angle between the outer three planets depends on the eccentricities of K2-138d and f, the orbital periods of the three planets, and the mass of K2-138f. The three-body angle between the middle planets and the four two-body angles also depend on some mixture of masses, eccentricities, and orbital periods. We show Kernel Density Estimations of the distributions of some of the more interesting dependencies in Figure 2 and report the p-values resulting from our K-S tests involving mass and orbital period in Table 3.
For each angle and planet property pair with a small K-S p-value, we report the median and 68.3% confidence interval of the parameter in the simulations where the angle is librating in Table 3. To further quantify which planet properties are statistically distinct between our simulations with librating resonance angles and the initial distributions, we perform a T-test on each parameter with a small K-S p-value. The T-test null hypothesis states that there is no statistical difference between two groups, and a small p-value indicates that an observed difference is not due to chance. We find that the masses of K2-138b, K2-138c, and K2-138f are statistically distinct between our simulations with librating resonance angles and the estimates from . Our results suggest that these three planets are slightly more massive than the radial velocity data can constrain, with masses of 3.46 +1.07 −1.01 , 6.53 +1.17 −1.13 , and 2.65 +1.90 −1.62 M ⊕ , respectively. If we constrain the mass of K2-138f using the two-body angle between it and K2-138e, we would recover an even larger mass of 3.01 +2.58 −1.98 M ⊕ , but the libration of this angle also depends on the orbital period of K2-138e.
The periods of all five planets shift slightly within the beginning of our simulations, even when they do not lock into resonance, and this shift results in a distribution of final orbital periods that is statistically distinct from those measured by for all planets except K2-138f. Because of this, we compare the orbital periods of the planets between simulations with librating resonance angles and simulations without libration via the T-test. We recover large p-values for all orbital periods except K2-138e, suggesting that our small K-S p-values are due to chance. For the two-body angle between K2-138e and K2-138f to librate, K2-138e requires a slightly larger period of 8.262 ± 0.002 days than that measured by (8.2615 ± 0.0002 days).
The differences in the masses of K2-138b, K2-138c, and K2-138f and in the orbital period of K2-138e could explain why we do not recover a five-planet resonance Figure 1. Evolution of the orbital periods, eccentricities, period ratios, and three-body resonance angles in an example simulation of K2-138. Here, we define resonance as the libration of a resonance angle, which occurs if the angle oscillates between two values. We say that the system contains a resonance chain if three or more planets are in resonance with one another. chain in all of our simulations. We discuss potential reasons why K2-138f is dynamically decoupled in the majority of our simulations below in Section 4.2.
Ideally, we would constrain the masses and orbital parameters of the planets by fitting the TTVs of the planets or by photodynamically fitting the system. The TTVs, although estimated to be on the order of 2-5min (Christiansen et al. 2018;, have so-far been illusive, indicating that the cadence or precision of the data is insufficient to tease out the perturbations.
We summarize our photodynamic fitting efforts in the Appendix.
Resonance of K2-138f
We hypothesize three reasons why K2-138f is not part of the resonance chain in the majority of our simulations:
1. The planet is in resonance, but its orbital parameters, mass, and/or the masses or orbits of the other planets fall in the part of parameter space that is narrower than the data currently constrain (H1) Figure 2. Kernel Density Estimations of various planetary masses and orbital periods, separated by whether a resonance angle was librating. For K-S p-values and parameter estimates from this analysis, see Table 3. Table 3. p-values and constraints from K-S Tests
Parameter Angle K-S p-value Estimate M b φ 1 1.00E-16 *3.46 +1.07 −1.01 M b Θ c−b 1.1837E-66 † *3.36 +1.03 −1.02 Mc Θ c−b 1.36E-32 † *6.53 +1.17 −1.13 Mc Θ d−c 3.00E-04 6.33 +1.23 −1.17 M d φ 2 3.48E-02 7.93 +1.40 −1.37 M d Θ d−c 2.65E-05 7.96 +1.37 −1.40 Me φ 2 1.80E-03 13.05 +1.96 −2.00 M f φ 3 1.19E-04 *2.65 +1.90 −1.62 M f Θ f −e 4.84E-09 *3.01 +2.58 −1.98 P b φ 1 9.14E-09 *2.3535 +0.0007 −0.0006 Pc φ 1 1.64E-03 *3.5592 +0.0010 −0.0009 Pc φ 2 1.30E-04 *3.559 ± 0.001 Pc Θ c−b 9.09E-03 P d φ 1 5.47E-03 *5.405 ± 0.002 P d φ 2 5.96E-03 P d φ 3 9.24E-02 Pe φ 2 3.37E-05 *8.262 ± 0.002 Pe φ 3 2.19E-03 *8.262 +0.001 −0.002 Pe Θ f −e 1.19E-02 *8.262 ± 0.002 P f φ 3 4.53E-07 12.757 ± 0.005 P f Θ f −e 2.48E-08
Note-Resulting p-values from our two-sample Kolmogorov-Smirnov test, where one sample is the distribution of values from simulations where the resonance angle (Angle) is librating and the other sample is the values from circulating simulations. Here, the null hypothesis states that these two samples are drawn from the same population. We include the median and 68.3% confidence intervals of each parameter from the simulations where the resonance angle is librating (Estimate 2. The planet was once in resonance but this resonance has since been broken or disrupted (H2) 3. The planet is not and was never in resonance (H3)
We explore each of these hypotheses below.
H1: Insufficient data
The libration of a resonance angle depends on the masses and orbits of the participating planets, but also on the libration of other resonance angles in the system. For example, if three planets are near a chain of two MMRs, the libration of the two-body angle between the inner two planets will affect the libration of the twobody angle between the outer two planets. Because of this affect, an angle's libration might depend on other planets in the system.
To test this hypothesis, we run four additional suites of 500 N -body simulations each to explore the likelihood of Θ f −e and φ 3 librating: one suite where we alter K2-138f's mass, one suite where we alter its orbital period, one suite where we increase the period of K2-138e, and one suite where we model the outermost planet K2-138g. We describe the results of each below.
Mass of f: We increase and narrow the mass range of K2-138f to 2.65 +1.90 −1.62 M ⊕ (compared to 1.63 +2.12 −1.98 M ⊕ ). This increase in mass nearly triples the likelihood of both Θ f −e and φ 3 librating, as the angles librate in 28.9% and 33.3% of the simulations, respectively. In addition, we find that the other three-body angles librate in a greater fraction of the simulations-φ 1 librates in 42.1% of the simulations (compared to 28.9%) and φ 2 librates in 83.8% of our simulations (compared to 72.6%)-suggesting that the libration of one angle increases the probability of other angles in the system librating. Overall, this change results in a five-planet chain in 27.7% of our simulations and a 3:2 chain in 99.2% of our simulations. We compare these percentages to those from our original simulations in Table 4, referring to this suite as Suite M f .
Period of f: We change the orbital period of K2-138f from 12.7576 ± 0.0005 days to 12.757 ± 0.005 days, the final periods of our resonant simulations. We find that this change has a similar effect to altering the mass: planet f is more likely to participate in the chain either through the libration of Θ f −e or φ 3 (21.2% and 27.7% respectively), and the other three-body angles are more likely to librate (39.9% and 82.4% for φ 1 and φ 2 , respectively). This change in orbital period results in a five-planet chain in 24.0% of our simulations and a 3:2 chain in 99.8% of our simulations. We compare these percentages to those from our original simulations in Table 4, referring to this suite as Suite P f .
Period of e: Following our K-S and T-test results (see Section 4.1), we change the orbital period of K2-138e from 8.2615 ± 0.0002 days to 8.262 ± 0.002 days. This change of period increases the percentage of simulations with librating Θ f −e or φ 3 to 22% and 26.5%, respectively, more than doubling the values. Our simulations result in a 3:2 resonance chain 98.8% of the time, and 23.4% of the simulations result in a five-planet chain. We find that the numbers of simulations with librating φ 1 and φ 2 also increase, to 40.5% and 79.2% respectively. We compare these percentages to those from our original simulations in Table 4, referring to this suite as Suite P e .
Adding K2-138-g: We did not originally model the outermost planet K2-138g because its sub-Neptune mass (4.32 +5.26 −3.03 M ⊕ ) and large orbital period compared to K2-138f (P g /P f = 3.29) suggest that it is dynamically decoupled from the rest of the planets in the system. To explore the validity of this assumption, we model K2-138g with a mass of 4.32 +5.26 −3.03 M ⊕ and an orbital period of 41.966 ± 0.006 days . We find that the addition of planet g indeed increases the chances of planet f participating in the chain, as Θ f −e and φ 3 librate in 31.7% and 33.5% of our simulations, respectively. Similar to the two previous changes, modeling all six planets increases the probability of φ 1 and φ 2 librating to 38.9% and 82.2%. However, we find that this additional planet decreases the number of simulations with librating Θ c−b and Θ d−c to 48.9% and 85.0% respectively, resulting in the dynamical decoupling of K2-138b in 30.0% of the simulations. Overall, we find that 98.0% of our simulations result in a 3:2 resonance chain: 24.4% in a five-planet chain and 56.7% in a fourplanet chain. We compare these percentages to those from our original simulations in Table 4, referring to this suite as Suite K2-138g.
Although these four changes significantly increase the percentage of simulations where K2-138f is part of the resonance chain, the resulting percentages are still too low to confirm its participation in the chain. Our suite of simulations with increased mass led to the largest percentage of five-planet chains, yet still resulted in K2-138f being dynamically decoupled from the other planets 62.9% of the time. It is possible that some combination of these effects, or increased precision in the other planets' orbits and masses, will further improve these values. As it stands, we require more data to verify whether or not K2-138f is part of the resonance chain.
H2: The resonance was broken
If the K2-138e and K2-138f are not presently in resonance, then perhaps they were at some time but have since been pushed just wide of the resonance. The gravitational interactions between the two planets and their disk, specifically between a planet and the wake of its companion, can reverse convergent migration, increasing the period ratio between the two planets beyond the resonance width (e.g. Baruteau & Papaloizou 2013). Turbulence in the disk can also prevent planets from staying in resonance, sometimes destabilizing the system altogether (Adams et al. 2008;Rein & Papaloizou 2009;Hühn et al. 2021). After the gas disk dispersal, the planetesimal disk or rouge planets are also capable of disrupting the resonant state of planet pairs, whereas less massive planets-such as K2-138f with its mass of M p = 1.6 2.1 1.2 M ⊕ -are more readily disrupted (Quillen et al. 2013;Chatterjee & Ford 2015;Raymond et al. 2021).
The loss of energy through tidal dissipation is also an effective means to avoiding or disrupting resonance (Lithwick & Wu 2012;Lee et al. 2013;, although the result of the system depends on the balance of dissipation in both planets . Tidal dissipation, when combined with secular interactions, can also cause migration of only the innermost planets, leading to divergence; this affect becomes even more dramatic when coupled with a single giant planet on a longer orbital period, although this oftentimes leads to instability (Hansen & Murray 2015).
Lastly, it is possible that systems that lock into resonance quickly destabilize after the dispersal of the gas disk without any additional forces (e.g., Izidoro et al. 2017Izidoro et al. , 2019. If the planets' eccentricities are originally damped and the librations are overstable, then a planet pair can readily escape resonance (Goldreich & Schlichting 2014;Delisle et al. , 2015. The ultimate fate of a resonant system's stability is a function of the planet masses, the spacing between the planets, and the number of resonant planets (Matsumoto et al. 2012;Deck & Batygin 2015;Pichierri & Morbidelli 2020), and might also depend on whether the resonance was formed through inward or outward migration (Lee et al. 2009).
Although there are numerous ways for resonances to be disrupted, studies of resonance chains show that any perturbations that are disruptive enough to break a resonance typically lead to chaotic evolution and instability, resulting in a final system architecture that is very different from what we observe (Esteves et al. 2020;Hühn et al. 2021;Raymond et al. 2021). We therefore find it unlikely that K2-138f could have been removed from the resonance chain.
H3: K2-138f was never in resonance
Although convergent migration will generally trap planets into resonance, migration in the absence of effective eccentricity damping can complicate matters. Eccentricities larger than ∼ 0.01-which are comparable to typical eccentricities measured in Kepler planets (e.g. Hadden & Lithwick 2014;Van Eylen et al. 2019)can make resonance capture more difficult and sometimes impossible for super-Earths (Batygin 2015;Pan & Schlichting 2017). Pan & Schlichting (2017) also find that planet pairs that avoid resonance capture are more likely to migrate past and away from each other than they are to collide, leading to a pile-up of planet pairs wide of resonance instead of planet pairs that simply go unstable; however, Hühn et al. (2021) find that such a crossing in resonance chains often leads to rapid eccentricity excitation, which in turn breaks the other resonances in the system. Note-Percentage of simulations where each angle librates (φ i , Θ i−j ); percentage of simulations with a five-, four-, or three-planet 3:2 resonance chain; and percentage of simulations without a 3:2 resonance chains. We compare our results from the original suite of 3000 N -body simulations (Section 2) and the additional four suites (see Section 4.2). See Table 2 for angle definitions.
Suite φ 1 φ 2 φ 3 Θ c−b Θ d−c Θ e−d Θ f −
Forming the resonance chain
While K2-138's resonance chain is interesting, we now question how the resonances formed. Although longscale migration -forming the planets more widely spaced and further from their star than observed -often results in resonance pairs (e.g., Snellgrove et al. 2001;Papaloizou & Terquem 2005;Rein & Tamayo 2015) and resonance chains (Cossou et al. 2013) and is often used as the explanation of such chains (e.g., Kepler-223, Mills & Fabrycky 2017), resonance chains can potentially form in situ 2 , with only small changes to their semi-major axes. MacDonald & Dawson (2018) explored three different pathways for forming resonance chains, including long-scale migration and two pathways consistent with in situ formation: short-scale migration (where the planets form outside of resonance and small shifts to their orbital periods lock them into resonance) and eccentricity damping. In addition, Morrison et al. (2020) find that close-in super-Earths and mini-Neptunes, such as those in K2-138, can lock into resonance chains due to dissipation from a depleted gas disk and maintain resonance once the gas disk is fully dissipated. Since all currently confirmed resonance chains are consistent with both in situ formation and longscale migration (MacDonald & Dawson 2018), we test all three different pathways for forming the resonance chain of K2-138.
Following the methods of MacDonald & Dawson (2018), we simulate the formation of this resonance chain via long-scale migration, short-scale migration, and eccentricity damping only. Here, long-scale mi-gration (e.g., Mills & Fabrycky 2017) assumes that the planets form far from their star and migrate inwards until they reach their currently observed semimajor axes; short-scale migration (e.g., MacDonald et al. 2016) assumes that the planets do not undergo significant changes to their semi-major axes during or after the giant impact phase; and eccentricity damping only (e.g., Dong & Dawson 2016) assumes constant angular momentum. For these three pathways, we apply an inward migration force and/or eccentricity damping forces on timescales τ a ∼ 10 4 -10 6 and τ e ∼ 10 3 -10 5 years, respectively, following the prescription in Papaloizou & Larwood (2000). We draw these timescales from independent log-uniform distributions (MacDonald et al. 2021). We apply damping forces to only the outer planet 3 for long-scale and short-scale migration, and damp the eccentricities of all planets in the eccentricity damping only simulations. We start all simulations with all planets wide of their observed commensurability and with no librating resonance angles. For each formation pathway, we run 500 simulations, damping the semi-major axes and eccentricities where applicable using the modify orbits forces routine in the REBOUNDx library. We use the stellar and planetary properties as defined in Table 1.
We find that we are able to form this resonance chain via all three pathways described above, as each suite of simulations contains numerous sets of initial conditions that lead to fully librating resonance chains. We summarize the resulting centers and amplitudes for the threebody resonance angles in Table 5. Since both short-scale migration and eccentricity damping are consistent with in situ formation, we find that the resonance chain of Note-For each three-body angle φ i , we include the percentage of the stable simulations where the angle librated and characterize the angle by the center and amplitude of its libration. We report the median and the 68.3% confidence interval. For each formation mechanism (short-scale migration, eccentricity damping, or long-scale migration), we also report the number of stable simulations and the percentage of those that resulted in a five planet resonance chain.
K2-138 could have formed in situ. We show an example simulation from the short-scale migration suite in Figure 3. We caution against using the values in Table 5 to draw any stronger conclusions, as this study is only able to say what is possible and not what is more likely.
Compositions
The compositions of planets can also be used to constrain the formation and evolution of a system. As a first-order approximation of the compositions of K2-138's planets, we first explore the planet's bulk densities, comparing them to Earth-like and less dense compositions. We draw 1000 pairs of mass and radius estimates for the confirmed planets from normal distributions of the parameters in , while obeying the 99% credibility intervals of density. In Figure 4, we plot 200 of these samples with mass-radius curves for compositions of pure water, Earth-like, and 1% H/He envelopes. We see that K2-138b is consistent with a terrestrial composition while K2-138c, K2-138d, and K2-138e are less dense and require large volatile layers. K2-138f also has a low density, below 2.068 g cm −3 , and most likely requires the largest atmosphere envelope.
We model the interiors of the four inner planets (b-e) using the planet structure code MAGRATHEA 4 (Huang et al. submitted, 2022), which calculates the pressure, density, temperature, and radius of a spherically symmetric planet with defined mass in each differentiated layer in 1-D using the fourth order Runge-Kutta method. We assume a surface pressure of one bar and use MAGRATHEA's default model. The default model uses equations of state for solid (hexagonal-closepacked 5 ) and liquid iron in the core, bridgmanite and post-perovskite silicate in the mantle, and water, ice-VII, ice-VII', and ice-X in the hydrosphere (Oganov & Ono 2004;Sakai et al. 2016;Dorogokupets et al. 2017;Smith et al. 2018;Grande et al. 2019). We model the atmosphere as an ideal gas with a mean molecular weight of 3 g cm −3 , similar to a hydrogen-helium mixture.
To mitigate the degeneracy between water mass fraction and atmospheric mass fraction with core mass fraction, we separate our analysis of each planet into two suites of 1000 models: one where we explore the water mass fraction and one where we explore the atmospheric mass fraction. For our water analysis, we limit the hydrosphere to liquid and ice phases, although the planets' equilibrium temperatures would suggest a vapor layer. We assume an isentropic temperature profile with a surface temperature of 300 K (e.g., Hakim et al. 2018). For our atmosphere analysis, we assume an ideal gas using an isentropic temperature profile and set the surface temperatures equal to the equilibrium temperatures derived from . A real gas would require less atmosphere mass, while a less steep temperature profile would require more. It is important to note that models such as ours are inherently limited for large, hot planets with interior conditions beyond our experimentally-determined equations of state, but although limited, our models are efficient at exploring the range of possible interior solutions.
For each of the 1000 samples of mass and radius, we use a secant method and vary the mass percentage in each layer until the simulated radius matches the sample to 0.01%. We calculate the water mass fraction (WMF) uniformly across the range of core mass fractions while fixing the core-to-mantle mass ratio. For the atmosphere mass fraction (AMF), we fix the core-to-mantle mass ratio to 1:2 similar to Earth and calculate the AMF uniformly across WMF. Figure 3. A five planet resonance chain of K2-138 is consistent with in situ formation. Here, we show the evolution of one of our short-scale migration simulations. We start the planets out of resonance and then apply a small damping to the semi-major axis of the outer planet with a timescale of τa = 4.6 × 10 4 years. We force the planets to undergo this small migration for 1.5 × 10 4 years, before stopping the migration and allowing the system to evolve for an additional 10 5 years to confirm long-term stability and resonance. We show the evolution of the orbital periods, eccentricities, period ratios, and three-body resonance angles.
We show the resulting WMF in Figure 5. We find that K2-138b most likely has a WMF between 9.0 and 47%, depending on the core-to-mantle mass ratio, and an estimated water fraction of 24.3 +39.0 −22.0 % with an Earth-like core-to-mantle mass ratio. 32% of the samples result in hydrosphere-free solutions, meaning the planet could be composed of only core and mantle and no water. With Earth-like core-to-mantle mass ratios, only 18.3% of the K2-138c samples and 25.5% of the K2-138d samples have three-layer solutions with less than 90% WMF, suggesting appreciable atmospheres. For reference, models of , with standard deviations equal to the uncertainties. We overplot composition curves of Earth-like (solid), 1% H/He atmosphere (short dash), and 100% water (long dash). We find that K2-138b is consistent with an Earth-like and terrestrial compositions, but that planets c-f require at least 1% H/He envelopes to satisfy their densities.
Neptune suggest at least 80% mass in a water-dominated fluid layer (Scheibe et al. 2019).
In Figure 6, we show the AMF needed to match samples of the planets' masses and radii across WMF. Across all possible water mass fractions, we predict AMF of the four planets of: 1.7 +9.3 −0.9 × 10 −3 %, 5.3 +7.9 −3.8 × 10 −4 %, 7.0 +13 −5.6 × 10 −4 %, 0.022 +0.016 −0.011 %. K2-138b's mass and radius could be explained with a hot, inflated H/He atmosphere layer, but many solutions require atmospheric masses of less than 10 −6 M ⊕ , 1.25% of Venus's atmospheric mass. With WMF = 0, K2-138c and K2-138d require an AMF of over 0.001%. The pressure and temperature under the atmosphere of K2-138c with 50% WMF is around 10 bar and 4000 K which, in our model, creates a small layer of liquid water before transitioning to high-pressure ices in the model. This temperature and pressure suggest that the water would be gaseous, but understanding this boundary requires a model that couples the atmosphere and interior. K2-138e requires an AMF around 15-30 times more than K2-138c and d at zero WMF.
Aside from the similar inferred compositions of K2-138c and K2-138d, the possible compositions of the plan-ets of K2-138 have little overlap. Their densities decrease and inferred volatile content increases with orbital period. Other systems with resonances, such as TRAPPIST-1, and other compact multi-planet systems have inter-planetary similarities in their sizes and masses and therefore in their inferred compositions (Weiss et al. 2018;Millholland & Winn 2021;Agol et al. 2021). This similar sizing within systems, especially when paired with the regular orbital spacing these systems also exhibit, could be telling of the formation history and/or the subsequent dynamical evolution (e.g., Adams 2019;MacDonald et al. 2020;Mishra et al. 2021), and K2-138's lack of intra-system uniformity could be just as telling. (green) and d (orange) require an atmosphere and so we include only the lower 1σ bound of WMF. K2-138e requires an atmosphere to satisfy its density and is therefore not included. We use python-ternary by Harper et al. (2015).
CONCLUSION
The K-dwarf K2-138 hosts six confirmed planets, all with period ratios of adjacent planets near the 3:2 commensurability. We run numerical N -body simulations using REBOUND (Rein & Liu 2012), drawing the planetary parameters from normal distributions centered on the results from , and modeling the K2-138b K2-138c K2-138d K2-138e Median ±1σ Figure 6. The water and atmosphere mass fractions needed to match samples of mass and radius for each planet. We fix the core-to-mantle mass ratio to 1:2, similar to Earth. Thin background lines are solutions to each sample, and thick lines are the mean (solid) and 1σ bounds (dashed) of samples with solutions at the given water mass. We use an isentropic temperature profile with the surface temperature set to the equilibrium temperature from . We only include the statistics for K2-138b until more than 50% of the samples result in solutions that require less than 10 −4 % atmospheric mass. 19% of K2-138b samples are too dense to require an atmospheric mass of more than 10 −4 % and are not shown.
system for 8Myr. We analyze our resulting simulations, finding that nearly all of our simulations (99.2%) result in a chain of 3:2 resonances, although few (11.0%) result in a five planet chain. We find that K2-138b and K2-138f do not need to be in resonance for the system to be stable, as 87.1% of our simulations result in K2-138f being dynamically decoupled from the other planets and 21.4% of the simulations result in K2-138b being dynamically decoupled. We are, however, able to confirm a resonance chain of 4:6:9 between K2-138c, K2-138d, and K2-138e, as 99.0% of our simulations result in the libration of the angle φ 2 = 2λ c − 5λ d + 3λ e and/or the libration of both two-body resonance angles.
Although numerous mechanisms exist for breaking a potential past resonance between K2-138e and K2-138f, resonance breaking within resonance chains usually leads to breaking the other resonances in the systems and oftentimes to instability. We argue that it is then more likely that K2-138f was never in resonance or that it is resonant and we simply have insufficient data to prove the resonance. Additional photometry could tease out the TTVs-which are expected to be on the order of 2-5 minutes but continue to be elusive-that would produce a stronger signal to be fit, further constraining the planets' masses and orbital parameters. The resulting decrease in parameter uncertainties could be sufficient to confirm additional resonances in this system, including the three-body resonance between planets d, e, and f.
We analyze our simulations for links between initial planetary parameters and resulting resonance or stability. We find that all resonance angles aside from Θ e−d show preference for specific masses and orbital periods. K2-138b, K2-138c, and K2-138f require slightly larger masses than those estimated by for K2-138b and K2-138f to be in resonance, and K2-138e requires a slightly larger period for K2-138f to part of the chain. Through additional N -body simulations, we find that we can increase the number of simulations where K2-138f is part of the resonance chain by increasing its mass, by altering its orbital period, by increasing the orbital period of K2-138e, and by modeling K2-138g; however, none of these changes alone is sufficient in confirming that K2-138f is in resonance.
Resonance chains are often seen as the hallmark of disk migration, but previous studies have found that such dynamical configurations can also arise from in situ formation (MacDonald et al. 2016;Dong & Dawson 2016;MacDonald & Dawson 2018). We therefore explore whether a five-planet chain of 3:2 resonances could have formed in situ as well as through migration. We find that K2-138 and its dynamics are consistent with in situ formation, but could have also been formed through long scale migration. Additional data of the system could potentially constrain the formation history, depending on the resulting resonance centers and amplitudes.
Using the planet structure code MAGRATHEA, we then explore the potential compositions of the planets along the uncertainties in their masses and radii. We find that K2-138b is consistent with a terrestrial composition and that any atmosphere would be less than half the mass of Venus's atmosphere. K2-138c and K2-138d have similar compositions; both planets require a minimum of ∼80% of their mass to be water, or an atmosphere, with over 0.01% of their mass as atmospheres. The bulk density of K2-138e is inconsistent with a nonatmospheric model; without a hydrosphere, we estimate an atmospheric mass fraction of 0.5%, or 0.065M ⊕ .
The confirmation of additional resonance chains could help us constrain the formation history of the individual systems and identify indicators of formation history in other systems. With the new planets discovered by TESS and those to be discovered in the near future, we will soon have a sufficient number of resonance chain systems to leave the area of small-number statistics and begin a full scale study of the formation history of exoplanetary systems.
We thank the referee for their constructive review that significantly improved this work and Kevin Hardegree-Ullman for sharing his detrended lightcurve. DR would like to thank Chenliang Huang and Jason H. Steffen for their advice and discussion. MGM acknowledges that this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE1255832. Any opin-ions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The authors acknowledge use of the ELSA high performance computing cluster at The College of New Jersey for conducting the research reported in this paper. This cluster is funded in part by the National Science Foundation under grant numbers OAC-1826915 and OAC-1828163.
APPENDIX
A planetary mass estimate requires radial velocity follow-up or for the planets to be gravitationally perturbing each other's orbits enough that we can detect significant variations in the transit times, or TTVs. Using HARPS spectra in combination with the Kepler photometry, constrained the masses of the planets of K2-138, and both Christiansen et al. (2018) and estimate TTVs on the order of 2-5 minutes. However, the TTVs have so far been elusive and might require higher cadence observations.
One step beyond fitting the system's TTVs would be to forward model and fit the lightcurve itself, in a manner that is both contained and self-consistent. Mills & Fabrycky (2017) fit the lightcurve of Kepler-444 using a photodynamic model, constraining two of the planet masses and the orbital elements for all of the planets. Such a method can be employed to other systems. By directly forward modeling and fitting the lightcurve of this system, following Mills & Fabrycky (2017) and using PhoDymm (Ragozzine et al., in prep.), we aim to determine the masses and orbits for all five planets hosted by K2-138. We follow the methods outlined in Mills & Fabrycky (2017) which we summarize below.
We integrate the Newtonian equations of motion for K2-138 and its five confirmed planets. We generate a synthetic lightcurve from a limb-darkened lightcurve model to compare to the K2 photometry reduced by Hardegree-Ullman et al. (2021) and perform Bayesian parameter inference using differential equation Markov Chain Monte Carlo (DEMCMC, Ter Braak 2006). We fit the orbital period, the mid-transit time, the eccentricity, the argument of periapse, the sky-plane inclination, the radius, and the mass of each planet. In addition, we fit the star's mass and radius, the two limb-darkening coefficients, and the amount of dilution from other nearby stars. We employ Gaussian priors on the stellar mass and radius based on values from Christiansen et al. (2018) (M = 0.93 ± 0.06M , R = 0.86 ± 0.08R ). We also fix Ω = 0 for all planets, given that the system has small mutual inclinations and employ flat priors on all other parameters.
We run a 96-chain DEMCMC for 450,000 generations, recording every 1000th generation and removing a burn-in of 20,000 generations. We include the median and 68.3% confidence intervals from the phoyodynamic model in Table 6.
Unfortunately, we are unable to constrain the planet masses to any useful precision. We suspect this is due to the low signal of the gravitational perturbations between planets that also drives the missing TTVs. We also find that the planetary radii are greatly overestimated but with small precision -for example, our estimate of the radius of K2-138c is R c = 8.1 ± 0.2R ⊕ compared to 2.299 +0.12 −0.087 R ⊕ from Lopez et al. (2019) -suggesting that the values are overfit. Such a study should be repeated once higher precision or higher cadence data are available.
Figure 4 .
4Mass-radius diagram with the radius and mass estimates of K2-138's planets. These estimates were pulled from normal distributions centered on the values from
Figure 5 .
5Ternary diagram showing solutions from MA-GRATHEA to 1000 samples of each planet's observed mass and radius. Here the axes are the percentage of mass in a core, mantle, and hydrosphere. Thin lines show the WMF needed to match the observed radius across core-to-mantle mass ratios. The thick, solid lines correspond to the median WMF while dashed lines are the 1σ bounds. The grey dashed line shows a constant Earth-like core-to-mantle ratio. K2-138b (blue) is the only planet where all 1000 samples have non-atmosphere solutions. The median samples of K2-138c
Table 1 .
1Planetary Properties of K2-138K2-138 b
K2-138 c
K2-138 d
K2-138 e
K2-138 f
P [d]
2.3531 ± 0.0002
3.5600 ± 0.0001
5.4048 ± 0.0002 8.2615 ± 0.0002 12.7576 ± 0.0005
t0 [d]
73.3168 ± 0.0009 40.32182 ± 0.0009 43.1599 ± 0.0009 40.6456 +0.0009
−0.0008
38.7033 +0.0009
−0.0009
i [ • ]
87.2 +1.2
−1.0
88.1 ± 0.7
89.0 ± 0.6
88.6 ± 0.3
88.8 ± 0.2
Mp [M⊕]
3.1 ± 1.1
6.3 +1.1
−1.2
7.9 +1.4
−1.3
13.0 ± 2.0
1.6 +2.1
−1.2
Table 2 .
2Resonance Angles of K2-138Angle
% Res
Center [ • ]
Amplitude [ • ]
φ 1 = 2λ b − 5λc + 3λ d
28.56
179.76 +5.05
4.45
57.39 +19.15
18.29
φ 2 = 2λc − 5λ d + 3λe
72.60
179.96 +3.75
3.94
49.31 +22.89
16.09
φ 3 = 2λ d − 5λe + 3λ f
11.46
180.21 +5.94
5.15
57.94 +26.99
27.33
Table 4 .
4Percentage of resonant simulations
e Five-pl Four-pl Three-pl No chainOrig.
28.6 72.6 11.5
64.3
95.6
98.9
9.9
11.0
68.5
19.6
0.9
M f
42.1 83.8 33.3
61.1
87.5
98.2
28.9
27.7
60.1
11.4
0.8
P f
39.9 82.4 27.7
63.7
89.4
98.8
21.2
24.0
62.3
13.4
0.2
Pe
40.5 79.2 26.5
58.5
90.2
97.8
22.0
23.4
58.3
17.0
1.2
K2-138g 38.9 82.2 33.5
48.9
85.0
98.4
31.7
24.4
56.7
16.8
2.0
Table 5 .
5Resonance Angles from Chain Formation SimulationsAngle
% Res Center [ • ] Amplitude [ • ]
Short-scale migration, 415/500 stable, 25% in 5pl resonance
φ 1 = 2λ b − 5λc + 3λ d
36.63 179.9 +2.7
2.4
45.0 +26.2
−20.2
φ 2 = 2λc − 5λ d + 3λe
51.81 180.0 +3.8
3.4
35.9 +23.1
−13.8
φ 3 = 2λ d − 5λe + 3λ f
17.83 180.4 +2.4
1.7
33.8 +24.4
−11.7
Eccentricity Damping, 497/500 stable, 10% in 5pl resonance
φ 1 = 2λ b − 5λc + 3λ d
14.3
179.9 +1.8
2.7
46.8 +20.8
−26.7
φ 2 = 2λc − 5λ d + 3λe
32.4
180.0 +2.9
2.0
39.0 +35.3
−32.8
φ 3 = 2λ d − 5λe + 3λ f
10.3
180.0 +1.3
2.2
41.6 +34.8
−30.8
Long-scale migration, 316/500 stable, 53% 5pl resonance
φ 1 = 2λ b − 5λc + 3λ d
58.9
179.8 +1.6
0.6
18.3 +40.1
−14.2
φ 2 = 2λc − 5λ d + 3λe
64.6
179.7 +0.9
0.9
16.1 +24.5
−12.6
φ 3 = 2λ d − 5λe + 3λ f
57.0
177.9 +2.4
5.5
9.8 +19.6
−8.3
Table 6 .
6Photodynamic fitting results from PhoDyMMK2-138b
K2-138c
K2-138d
K2-138e
K2-138f
Mp (M⊕) 6.3 +13.5
−5.0
4.3 +10.9
−3.3
9.0 +10.3
−5.9
5.0 +7.0
−3.8
10.6 +12.2
−7.4
Rp (R⊕)
5.4 +0.1
−0.1
8.1 +0.2
−0.2
8.6 +0.2
−0.2
10.9 +0.2
−0.2
9.6 +0.2
−0.2
P (d)
2.353 +0.0003
−0.0003
3.561 +0.001
−0.001
5.404 +0.001
−0.002
8.263 +0.002
−0.001
12.759 +0.001
−0.001
t0 (d)
773.317 +0.003
−0.003
740.309 +0.009
−0.012
743.162 +0.009
−0.005
740.642 +0.004
−0.006
738.700 +0.003
−0.003
i ( • )
87.02 +1.41
−0.93
88.22 +0.85
−0.63
89.48 +0.37
−0.57
88.83 +0.33
−0.29
88.88 +0.20
−0.19
e
0.002 +0.04
−0.04
0.002 +0.03
−0.03
0.001 +0.02
−0.02
0.007 +0.03
−0.02
0.011 +0.02
−0.03
ω ( • )
74.0 +45.2
−54.0
-100.2 +52.0
−48.0
107.8 +46.8
−47.3
-112.7 +49.6
−42.9
Here, we use the term "in situ" to distinguish the evolution after the giant impact phase from long-scale migration (e.g.,Lee & Peale 2002) and to align with other works exploring the in situ formation of super-Earths (e.g.,Dawson et al. 2015). The local growth of these planets does not require that the planetesimals that form them be local, as they could have also accumulated by radial drift.
We do not know the migration rates for the planets since they depend on the conditions of the disk, so we implicitly assume that the migration of the inner planets is on a much longer timescale than that of the outer planet.
We used a preliminary version; the final version will be available at https://github.com/Huang-CL/Magrathea. 5 Under the pressure and temperature of Earth's inner core, the iron most likely takes this structure(Vočadlo et al. 1999)
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|
[
"https://github.com/Huang-CL/Magrathea."
] |
[] |
[
"Pan-Jun Kim \nDepartment of Physics\nKorea Advanced Institute of Science and Technology\nDaejeonKorea\n",
"Tae-Wook Ko \nNational Creative Research Initiative Center for Neuro-dynamics and Department of Physics\nKorea University\nSeoulKorea\n",
"Hawoong Jeong \nDepartment of Physics\nKorea Advanced Institute of Science and Technology\nDaejeonKorea\n",
"Kyoung J Lee \nNational Creative Research Initiative Center for Neuro-dynamics and Department of Physics\nKorea University\nSeoulKorea\n",
"Seung Kee Han \nDepartment of Physics\nChungbuk National University\nCheongjuChungbukKorea\n"
] |
[
"Department of Physics\nKorea Advanced Institute of Science and Technology\nDaejeonKorea",
"National Creative Research Initiative Center for Neuro-dynamics and Department of Physics\nKorea University\nSeoulKorea",
"Department of Physics\nKorea Advanced Institute of Science and Technology\nDaejeonKorea",
"National Creative Research Initiative Center for Neuro-dynamics and Department of Physics\nKorea University\nSeoulKorea",
"Department of Physics\nChungbuk National University\nCheongjuChungbukKorea"
] |
[] |
Chaotic itinerancy is a universal dynamical concept that describes itinerant motion among many different ordered states through chaotic transition in dynamical systems. Unlike the expectation of the prevalence of chaotic itinerancy in high-dimensional systems, we identify chaotic itinerant behavior from a relatively simple ecological system, which consists only of two coupled consumer-resource pairs. The system exhibits chaotic bursting activity, in which the explosion and the shrinkage of the population alternate indefinitely, while the explosion of one pair co-occurs with the shrinkage of the other pair. We analyze successfully the bursting activity in the framework of chaotic itinerancy, and find that large duration times of bursts tend to cluster in time, allowing the effective burst prognosis. We also investigate the control schemes on the bursting activity, and demonstrate that invoking the competitive rise of the consumer in one pair can even elongate the burst of the other pair rather than shorten it.
|
10.1103/physreve.76.065201
|
[
"https://arxiv.org/pdf/nlin/0612004v2.pdf"
] | 1,036,880 |
nlin/0612004
|
0900a2db23538e5233f21ba78cced59b1fd724d8
|
16 Dec 2007 (Dated: February 4, 2008)
Pan-Jun Kim
Department of Physics
Korea Advanced Institute of Science and Technology
DaejeonKorea
Tae-Wook Ko
National Creative Research Initiative Center for Neuro-dynamics and Department of Physics
Korea University
SeoulKorea
Hawoong Jeong
Department of Physics
Korea Advanced Institute of Science and Technology
DaejeonKorea
Kyoung J Lee
National Creative Research Initiative Center for Neuro-dynamics and Department of Physics
Korea University
SeoulKorea
Seung Kee Han
Department of Physics
Chungbuk National University
CheongjuChungbukKorea
16 Dec 2007 (Dated: February 4, 2008)arXiv:nlin/0612004v2 [nlin.CD] Emergence of Chaotic Itinerancy in Simple Ecological Systems
Chaotic itinerancy is a universal dynamical concept that describes itinerant motion among many different ordered states through chaotic transition in dynamical systems. Unlike the expectation of the prevalence of chaotic itinerancy in high-dimensional systems, we identify chaotic itinerant behavior from a relatively simple ecological system, which consists only of two coupled consumer-resource pairs. The system exhibits chaotic bursting activity, in which the explosion and the shrinkage of the population alternate indefinitely, while the explosion of one pair co-occurs with the shrinkage of the other pair. We analyze successfully the bursting activity in the framework of chaotic itinerancy, and find that large duration times of bursts tend to cluster in time, allowing the effective burst prognosis. We also investigate the control schemes on the bursting activity, and demonstrate that invoking the competitive rise of the consumer in one pair can even elongate the burst of the other pair rather than shorten it.
Since attractors determine the long-term behavior of dynamical systems, the concept of attractors is central to the analysis of many natural and artificial systems [1]. In general, the phase space of a nonlinear dynamical system is partitioned into various basins of attraction from which states evolve towards the respective attractors. These stable attractors can lose their stability with a change of the system condition such that the basin of attraction of each attractor becomes connected to each other through unstable manifolds. Hence, a dynamical state which sequentially traces out all of the destabilized attractor ruins emerges. This is referred to as a chaotic itinerant state [2]. The notion of chaotic itinerancy has received considerable attention in studying the adaptability of complex systems, especially in relation to brain information processing [2,3].
To embed a chaotic itinerant state, a system is expected to have a high degree of complexity; therefore, models of chaotic itinerancy are mostly built on high-dimensional phase space [4]. Albeit relatively lowdimensional systems, two coupled Morris-Lecar neural oscillators were found to exhibit chaotic itinerancy [5], the result seems to be limited to a rather special case obtained by using sophisticated forms of model equations in neurobiological systems. In the present work, we report that low-dimensional chaotic itinerancy exists and arises naturally in simple ecological systems, of which consumer-resource dynamics has broad relevance in metabolic, immune, social, and economical systems. The wide variety of related disciplines aside, the mathematical simplicity of our low-dimensional system renders the global organization of a chaotic itinerant state tractable with a detailed illustration.
At the outset, we suggest the equations of two consumer-resource pairs coupled via resource sharing [6]:
dC 1(2) dt = aC 1(2) R 1(2) κ + R 1(2) + DR 2(1) κ + R 2(1) − bC 1(2) , dR 1(2) dt = R 1(2) − R 2 1(2) − [C 1(2) + DC 2(1) ]R 1(2) κ + R 1(2) ,(1)
where C 1(2) and R 1(2) represent the population of consumer 1 (2) and resource 1 (2), respectively. a and b denote the growth and decay rates of the population of consumers. κ concerns the satiability level of the consumers taking resources. For simplicity in our analysis, we do not distinguish the parameter sets of the two consumerresource pairs. In this equation, R 1(2) is taken by C 1 (2) primarily, as well as by C 2(1) with a relative small uptake rate D, which ranges from 0 to 1.
When D is equal to zero, Eq. (1) splits into two Holling type-II forms of Lotka-Volterra equations [7], and the populations of each consumer-resource pair can exhibit a limit cycle oscillation. If D takes a nonzero value close to 0 or 1, synchronous limit cycle oscillation between the two consumer-resource pairs arises. Fig. 1(c) show the bursting behaviors, and those of R 1 and R 2 show the similar patterns as well. C 1 and C 2 , or R 1 and R 2 , fire bursts in an antiphase-synchronized way, such that the explosion of C 1 or R 1 co-occurs with the shrinkage of C 2 or R 2 , and vice versa. It is worth noting that other equations with similar systems to Eq. (1) also support the existence of such antiphase-synchronized irregular bursts [8]. For numerical simulations, we use the parameters a = 2, b = 0.82, κ = 0.33, and D = 0.57 unless specified.
To check whether the apparent irregularity of the bursts implies their initial condition sensitiveness, we evaluate
M 1(2) (t) = 1 t t 0 H[C 1(2) (t ′ )]H[C ′ 1(2) (t ′ )]dt ′ ,(2)
where
H(X) = 1 if X is in the burst mode, other- wise H(X) = −1.
The antiphase synchronization of the bursts enables us to define the burst mode unambiguously, such that once C 1(2) exceeds C 2(1) , C 1(2) enters the burst mode, and C 2(1) enters the shrinkage mode. C ′ 1(2) (t) is calculated in the same way as
C 1(2) (t), but is initially perturbed from C 1(2) (t) with ε = |C ′ 1(2) (0)−C 1(2) (0)|/C 1(2) (0) ≪ 1. Therefore, Eq.
(2) gives the similarity between the bursting times of C 1(2) (t) and those of C ′ 1(2) (t) with slightly different initial conditions. If the bursting times of C 1(2) (t) and C ′ 1(2) (t) are in complete agreement, M 1(2) (t) = 1, whereas with no correlation between them, M 1(2) (t) = 0. In the following, we drop the subscript of M 1(2) (t) due to the statistical equivalence of C 1 (t) and C 2 (t). One can employ M (t) for determining the necessary time for the discrepancy to be significant between the bursting times of C 1(2) (t) and of C ′ 1(2) (t). It is observed that M (t) evolves rapidly from 1 to 0 in the irregular bursting regime; thus, the half-life time τ h of M (t) can serve as the characteristic time scale of the discrepancy growth. Figure 1(d) shows that τ h scales logarithmically to ε, and using R 1(2) (t) and (2) does not alter the current result. This logarithmic scaling reveals that the bursting is sensitive to the initial conditions, i.e., behaves chaotically [9].
R ′ 1(2) (t) instead of C 1(2) (t) and C ′ 1(2) (t) in Eq.
To address such antiphase-synchronized chaotic bursts in detail, we divide a period of bursts into four stages -S1, S1 ′ , S2, and S2 ′ , as in Fig. 1(e). In stage S1, C 1 and R 1 dominate C 2 and R 2 , while C 1 , which is supported primarily by R 1 , depresses severely the growth of R 2 , and thereby of C 2 . Nonetheless, R 2 increases gradually in the negligible presence of C 2 , and R 1 comes to decline with overpopulated C 1 which takes both R 1 and R 2 . In stage S1 ′ , the resultant shrinkage of R 1 ensures that C 1 depends mostly on R 2 for survival. Meanwhile, R 2 can boost the increase of C 2 , which then suppresses both R 2 and R 1 , thereby leading to the drastic decay of C 1 in stage S2 [10]. The dominance of C 2 and R 2 in stage S2 is totally symmetric to that of C 1 and R 1 in stage S1. Accordingly, stage S2 ′ analogous to stage S1 ′ follows, and leads to stage S1 for the next period.
Each stage occupies finite time-span, forming a quasistable dynamical state. The alternating dominance of each species along the stages may be equivalent to the switching events among the sets of attractor ruins. In order to elucidate the underlying attractor ruin for a given stage, we consider an invariant subspace of (C 1 , R 1 , C 2 , R 2 ) which contains only the species governing the stage [see Fig. 1(e)]: at stage S1, (C 1 , R 1 , 0, 0); at stage S1 ′ , (C 1 , 0, 0, R 2 ); at stage S2, (0, 0, C 2 , R 2 ); at stage S2 ′ , (0, R 1 , C 2 , 0). The populations confined within each invariant subspace approach their own asymptotic solution. For instance, the limit cycle oscillation of C 1 and R 1 characterizes the asymptotic solution in the invariant subspace of stage S1 and thus underlies the bursting activity at this stage. Figure 2(a) shows an actual time trajectory of the populations in phase space, which also embeds the asymptotic solutions in the invariant subspaces. Near an invariant asymptotic solution, the trajectory remains there for a long time, but finally escapes towards another invariant solution at the next stage.
This escaping event is due to an existence of unstable manifolds outward from an invariant subspace. Along the transverse direction of the invariant subspace, we then perform a linear stability analysis to find the unstable manifolds: S1 :
dδR 2 dt ∼ = 1 − D C 1 κ δR 2 , S1 ′ : dδC 2 dt ∼ = min b D , a 1 + κ − b δC 2 ,(3)
where C 1 of stage S1 is evaluated in the absence of C 2 and R 2 . Stages S2 and S2 ′ take the formula obtained simply by interchanging C 1 and C 2 , R 1 and R 2 of stages S1 and S1 ′ in Eq. (3). The formula of unstable manifolds reveals which species causes the instability of a given stage; the instability of stages S1 and S1 ′ is invoked by the increase of R 2 and C 2 , as described above in the ecological argument. From Eq. (3), we recognize that increase of D tends to reduce the coefficients in the right-hand sides, i.e., to decrease the escape rates along the unstable manifolds. The resultant relaxation of the switching events among attractor ruins is identified by comparing Figs. 2(a) and 2(b), where consistently the trajectory between the invariant solutions looks less intermingled with increased D. The most evident effect of elongated residence in attractor ruins appears in Fig. 2(c): the distribution of burst duration T shifts to large value as D increases. Moreover, the larger D is, the higher the right peak of the bimodal duration distribution is, relatively to the left peak. We conclude that in the chaotic bursting regime, the enhanced coupling strength induces either of the consumer-resource pairs to dominate the other for a long time by an elongated bursting activity.
To investigate the bimodality appearing in Fig. 2(c), we plot a return time map for burst initiations of consumers by considering the terms between the initiation of stage S1 and that of stage S2, and between the initiation of stage S2 and that of stage S1, and so on. Figures 3(a) and 3(b) display nearly one-dimensional curves of such return time maps, where stepwise jumps between the upper and lower extremes are responsible for the bimodality in Fig. 2(c). The relative expansion of upper extremes in Fig. 3(b) induces the right side of the duration distribution in Fig. 2(c) to be highly peaked.
Referring to the return time maps, we find that mapping trajectories are frequently trapped in boxed area B in Figs. 3(a) and 3(b). The distribution of residual times N (total number of iteration) within boxed area B is indeed more right-skewed than those of any other areas (e.g., B ′ ) with the same size [Figs. 3(c) and 3(d)]. The distribution within area B follows exponential fit This indicates the residual dynamics within area B follows a poissonian process, but with considerable survivability p per iteration. Since area B corresponds to relatively long durations of bursts, large duration times of bursts tend to cluster in time, as partially observed in Fig. 1(c). In this regard, the known information about the past duration could be beneficial to improve the burst prognosis efficiently.
C 1 C 2 C 1 C 2 C 1 C 2 R 1 R 2 R 1 R 2 R 1 R 2 t (a) (c) (e)
An important outcome of the stability analysis on attractor ruins is the application to a control scheme on burst duration. Since a rise of R 2(1) destabilizes the dominance of C 1(2) and R 1 (2) , manually repressing the growth of R 2(1) might elongate the bursts of C 1(2) and R 1 (2) . As shown in Figs. 4(c) and 4(d), the repressed growth of R 2 prevents the overpopulation of C 1 for a while and thus delays the shrinkage of R 1 as well as of C 1 . It should be noticed that the effect of delayed rise of C 2 herein is not so essential to elongating the burst duration, despite the resource competition between C 2 and C 1 . Counterintuitively, even promoting the growth of C 2 could be helpful to the burst duration, if resulting in the depression of R 2 . Figures 4(e) and 4(f) indeed illustrate the possibility that a sufficiently large perturbation to increase C 2 withdraws transiently R 1 and C 1 , but also delays the growth of R 2 thereby elongating the bursting activity [11].
In summary, we investigated a simple dynamical system, which consists only of two consumer-resource pairs but exhibits chaotic itinerancy naturally. The mathematical simplicity of the system gives rise to a clear view of the organization of a chaotic itinerant state, where each consumer-resource relationship underlies its corresponding attractor ruin as a dynamical 'building block'. Such concept of building blocks could be generically utilized when one designs other systems exhibiting chaotic itinerancy. In addition, analysis on a chaotic itinerant state was found to be applicable to the prognosis and control of the dynamical system, in the rather counterintuitive way. Beyond the suggested ecological system, any dynamical system which shows antiphase-synchronized chaotic bursts might be analyzable in the framework of chaotic itinerancy via our methodology. We expect that host-parasitoid systems with whiteflies and their parasitic wasps could be employed for experimental validation of our results, since parasitic wasps (consumers) are known to have overlapped hosts (resources) in a manner similar to the present model [12]. where C 1(2) and R 1(2) represent the population of con-
Complicated dynamics develop at intermediate range of D, where we can observe irregular bursting activities as in Figs. 1(a)-1(c). Time trajectories of C 1 and C 2 in
FIG. 1 :
1(a) and (b): parameter regime of irregular bursting activities in Eq. (1) with a = 2, when (a) b = 0.82 or (b) κ = 0.33. (c) Time series of C1 and C2 in bursting activity. The arrowed time interval is magnified in (e). (d) Half-life time τ h of burst reproducibility with initial condition difference ε. (e) Magnification of the arrowed time interval in (c). Solid lines denote C1 and R1, and dotted lines C2 and R2.
FIG. 2 :
2(a) Projection of time trajectory onto (R1 − R2, C1 − C2) for ∆t = 3000, D = 0.57 (thin line, red in color). Overlapped is invariant solution at each stage (black). (b) Depicted by the similar way with (a), when D = 0.4. (c) Distribution of burst duration T of C1 with stages S1 and S1 ′ , or of C2 with stages S2 and S2 ′ , for different values of D (C1 and C2 show the identical distribution). Similar results are also observed with each of the stages.
FIG. 3 :
3(a) and (b): return time map for burst initiations when (a) D = 0.5 or (b) D = 0.57. (c) and (d): distribution of residual times N within boxed area in (a) [(c)] or in (b) [(d)].
FIG. 4 :
4(a) and (b): unperturbed time-series of populations. (c)-(f): initially the same as (a) and (b), but subjected to the perturbation during the arrowed time interval by reducing R2 − R 2 2 in half [(c), (d)] or doubling aC2 [(e), (f)], given the terms in Eq. (1). P (N ) ∝ p N with p = 0.64 for D = 0.5 and p = 0.68 for D = 0.57, and shows significantly larger p than the surrogated data (p = 0.33 for D = 0.5, p = 0.28 for D = 0.57) with the same distribution of burst duration time.
dC 1(2) dt = aC 1(2) » R 1(2) κ + µR 1(2) + DR 2(1) κ + µR 2(1) -− bC 1(2) , dR 1(2) dt = σR 1(2) − γR 2 1(2) − λ [C 1(2) + DC 2(1) ]R 1(2) κ + µR 1(2) ,
S H Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics. New YorkAddison-WesleyChemistry, and EngineeringS. H. Strogatz, Nonlinear Dynamics and Chaos: with Ap- plications to Physics, Biology, Chemistry, and Engineer- ing (Addison-Wesley, New York, 1994).
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. S K Han, D E Postnov, Chaos. 131105S. K. Han and D. E. Postnov, Chaos 13, 1105 (2003).
Equation (1) is obtained by rescaling of the following equation: sumer 1 (2) and resource 1 (2), respectively. The parameters include a, κ, µ, D, b, σ, γ, λ, of which µ, σ, γ, λ can be safely removed to give Eq. (1) without loss of generality. Equation (1) is obtained by rescaling of the following equation: sumer 1 (2) and resource 1 (2), respectively. The param- eters include a, κ, µ, D, b, σ, γ, λ, of which µ, σ, γ, λ can be safely removed to give Eq. (1) without loss of generality.
. C S Holling, Canadian Entomol. 91385C. S. Holling, Canadian Entomol. 91, 385 (1959).
. J Vandermeer, Am. Nat. 141687J. Vandermeer, Am. Nat. 141, 687 (1993).
we can hardly capture whether the chaotic behavior emerges at slow bursting scales or at fast oscillating scales. To clarify the chaotic activity in slow time scales, we instead utilize Eq. (2), which gives eventually the divergence rate of nearby initial conditions at bursting time scales. From Fig. 1(d), one can obtain constant ∼ = εe qτ h where q is expected to be the rough estimation of the maximum Lyapunov exponent λmax. Because the conventional Lyapunov exponent does not resolve the time scales involved in the present dynamics. as proven by q = 0.016 and λmax = 0.023Because the conventional Lyapunov exponent does not resolve the time scales involved in the present dynam- ics, we can hardly capture whether the chaotic behav- ior emerges at slow bursting scales or at fast oscillat- ing scales. To clarify the chaotic activity in slow time scales, we instead utilize Eq. (2), which gives eventu- ally the divergence rate of nearby initial conditions at bursting time scales. From Fig. 1(d), one can obtain constant ∼ = εe qτ h where q is expected to be the rough estimation of the maximum Lyapunov exponent λmax, as proven by q = 0.016 and λmax = 0.023.
If D is small enough that R2 alone cannot maintain C1 as nonzero, C1 becomes already decayed at stage S1 ′ . Otherwise, with sufficiently large D at stage S1 ′ , C1 can show either of stationary or oscillatory behavior depending on the parameters. If D is small enough that R2 alone cannot maintain C1 as nonzero, C1 becomes already decayed at stage S1 ′ . Oth- erwise, with sufficiently large D at stage S1 ′ , C1 can show either of stationary or oscillatory behavior depending on the parameters.
4(f), let tp denote the initial time of perturbation, ∆tp the duration time of perturbation, and fp the relative decrease of R2 − R 2 2. Figs. 4(c) and 4(d)For Figs. 4(c)-4(f), let tp denote the initial time of per- turbation, ∆tp the duration time of perturbation, and fp the relative decrease of R2 − R 2 2 [Figs. 4(c) and 4(d)]
We change each of tp. ∆tp, and fp while the others remain fixed as in Figs. 4(c)-4(f), and observe the burst being elongated for Figs. 4(e). with Eq. and 4(f) under the limitations tp ≤ 5573, 4.5 ≤ ∆tp, 0.8 ≤ fp. Either decreasing tp, increasing ∆tp, or increasing fp tends to elongate the burst durations throughout Figs. 4(c)-4(for increase of aC2 [Figs. 4(e) and 4(f)] with Eq. (1). We change each of tp, ∆tp, and fp while the others remain fixed as in Figs. 4(c)-4(f), and observe the burst being elongated for Figs. 4(e) and 4(f) under the limitations tp ≤ 5573, 4.5 ≤ ∆tp, 0.8 ≤ fp. Either decreasing tp, in- creasing ∆tp, or increasing fp tends to elongate the burst durations throughout Figs. 4(c)-4(f).
. S M Greenberg, W A Jones, T. -X Liu, Environ. Entomol. 31397S. M. Greenberg, W. A. Jones, and T. -X. Liu, Environ. Entomol. 31, 397 (2002).
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[] |
[
"THE CLASSIFICATION OF NORMALIZING GROUPS",
"THE CLASSIFICATION OF NORMALIZING GROUPS"
] |
[
"João Araújo ",
"Peter J Cameron ",
"ANDJames Mitchell ",
"Max Neunhöffer ",
"João Araújo ",
"João Araújo ",
"Peter J Cameron ",
"ANDJames Mitchell ",
"Max Neunhöffer "
] |
[] |
[] |
Let X be a finite set such that |X| = n. Let Tn and Sn denote the transformation monoid and the symmetric group on n points, respectively. Given a ∈ Tn \ Sn, we say that a group G Sn is a-normalizing ifwhere a, G and g −1 ag | g ∈ G denote the subsemigroups of Tn generated by the sets {a} ∪ G and {g −1 ag | g ∈ G}, respectively. If G is a-normalizing for all a ∈ Tn \ Sn, then we say that G is normalizing.The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.
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10.1016/j.jalgebra.2012.08.033
|
[
"https://arxiv.org/pdf/1205.0450v3.pdf"
] | 53,124,336 |
1205.0450
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a36a78f23ca588b645155f84830d29bbab317c5d
|
THE CLASSIFICATION OF NORMALIZING GROUPS
2010 Mathematics Subject Classification: 20B30, 20B35, 20B15, 20B40, 20M20, 20M17
João Araújo
Peter J Cameron
ANDJames Mitchell
Max Neunhöffer
João Araújo
João Araújo
Peter J Cameron
ANDJames Mitchell
Max Neunhöffer
THE CLASSIFICATION OF NORMALIZING GROUPS
2010 Mathematics Subject Classification: 20B30, 20B35, 20B15, 20B40, 20M20, 20M17Corresponding author:and phrases: Transformation semigroupspermutation groupsprimitive groupsGAP
Let X be a finite set such that |X| = n. Let Tn and Sn denote the transformation monoid and the symmetric group on n points, respectively. Given a ∈ Tn \ Sn, we say that a group G Sn is a-normalizing ifwhere a, G and g −1 ag | g ∈ G denote the subsemigroups of Tn generated by the sets {a} ∪ G and {g −1 ag | g ∈ G}, respectively. If G is a-normalizing for all a ∈ Tn \ Sn, then we say that G is normalizing.The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.
Introduction and Preliminaries
For notation and basic results on group theory we refer the reader to [8,11]; for semigroup theory we refer the reader to [18]. Let T n and S n denote the monoid consisting of mappings from [n] := {1, . . . , n} to [n] and the symmetric group on [n] points, respectively. The monoid T n is usually called the full transformation semigroup. In [24], Levi and McFadden proved the following result. Theorem 1.1. Let a ∈ T n \ S n . Then (1) g −1 ag | g ∈ S n is idempotent generated;
(2) g −1 ag | g ∈ S n is regular.
Using a beautiful argument, McAlister [30] proved that the semigroups g −1 ag | g ∈ S n and a, S n \ S n (for a ∈ T n \ S n ) have exactly the same set of idempotents; therefore, as g −1 ag | g ∈ S n is idempotent generated, it follows that g −1 ag | g ∈ S n = a, S n \ S n .
Later, Levi [25] proved that g −1 ag | g ∈ S n = g −1 ag | g ∈ A n (for a ∈ T n \ S n ), and hence the three results above remain true when we replace S n by A n . The following list of problems naturally arises from these considerations.
(1) Classify the groups G S n such that for all a ∈ T n \ S n we have that the semigroup g −1 ag | g ∈ G is idempotent generated. (2) Classify the groups G S n such that for all a ∈ T n \ S n we have that the semigroup g −1 ag | g ∈ G is regular. (3) Classify the groups G S n such that for all a ∈ T n \ S n we have a, G \ G = g −1 ag | g ∈ G .
The two first questions were solved in [4] as follows: Theorem 1.2. If n 1 and G is a subgroup of S n , then the following are equivalent:
(i) The semigroup g −1 ag | g ∈ G is idempotent generated for all a ∈ T n \ S n .
(ii) One of the following is valid for G and n: (a) n = 5 and G is AGL(1, 5); (b) n = 6 and G is PSL (2,5) or PGL(2, 5); (c) G is A n or S n . Theorem 1.3. If n 1 and G is a subgroup of S n , then the following are equivalent: (i) The semigroup g −1 ag | g ∈ G is regular for all a ∈ T n \ S n . (ii) One of the following is valid for G and n:
(a) n = 5 and G is C 5 , D 5 , or AGL(1, 5); (b) n = 6 and G is PSL (2,5) or PGL(2, 5); (c) n = 7 and G is AGL(1, 7); (d) n = 8 and G is PGL(2, 7); (e) n = 9 and G is PSL (2,8)
or PΓL(2, 8); (f) G is A n or S n .
These results leave us with the third problem. Given a ∈ T n \ S n , we say that a group G S n is a-normalizing if
a, G \ G = g −1 ag | g ∈ G .
If G is a-normalizing for all a ∈ T n \ S n , then we say that G is normalizing. Recall that the rank of a transformation f is just the number of points in its image; we denote this by rank(f ). For a given k such that 1 k < n, we say that G is k-normalizing if G is a-normalizing for all rank k maps a ∈ T n \ S n .
Levi, McAlister and McFadden [23, p.464] ask for a classification of all pairs (a, G) such that G is a-normalizing, and in [4] is proposed the more tractable problem of classifying the normalizing groups. The aim of this paper is to provide such a classification. Theorem 1.4. If n 1 and G is a subgroup of S n , then the following are equivalent:
(i) The group G is normalizing, that is, for all a ∈ T n \ S n we have
a, G \ G = g −1 ag | g ∈ G ;
(ii) One of the following is valid for G and n: (a) n = 5 and G is AGL(1, 5); (b) n = 6 and G is PSL (2,5) or PGL(2, 5); (c) n = 9 and G is PSL (2,8) or PΓL (2,8);
(d) G is {1}, A n or S n .
Main result
The goal of this section is to prove Theorem 1.4 for all groups of degree at least 10. This proof is carried out in a sequence of lemmas. The groups of degree less than 10 will be handled in the next section. The results of this section hold for all n unless otherwise stated.
If G is trivial, then G is obviously normalizing, so we always assume that G is non-trivial.
We start by stating an easy lemma whose proof is self-evident, and that will be used without further mention. A subset X of [n] is said to be a section of a partition P of [n] if X contains precisely one element in every class of P. The kernel of a ∈ T n is the equivalence relation ker(a) = {(x, y) ∈ [n] : (x)a = (y)a}.
Lemma 2.1. Let G be a subgroup of S n and let a ∈ T n \ S n . Then, if for some g, h ∈ G we have rank(h −1 ahg −1 ag . . .) = rank(a), then exists h 1 := hg −1 ∈ G such that h 1 maps the image of a to a section of the kernel of a.
The following lemma is probably well-known: it is an easy generalization of a result of Birch et al. [7]. Lemma 2.2. Let G be a transitive permutation group on X, where |X| = n. Let A and B be subsets of X with |A| = a and |B| = b. Then the average value of |Ag ∩ B|, for g ∈ G, is ab/n. In particular, if |Ag ∩ B| = c for all g ∈ G, then c = ab/n.
Proof. Count triples (x, y, g) with x ∈ A, y ∈ B, and xg = y. There are a choices for x and b choices for y, and then |G|/n choices for g. Choosing g first, there are |Ag ∩ B| choices for (x, y) for each g. The result follows. Lemma 2.3. Let G ≤ S n be normalizing and non-trivial. Then
(i) G is transitive; (ii) G is primitive.
Proof. Regarding (i), let A be an orbit of G which is not a single point, and suppose that |A| < n. Let a be an (idempotent) map which acts as the identity on A and maps the points outside A to points of A in any manner. Then a fixes A pointwise, and hence so does any G-conjugate of a, and so does any product of G-conjugates: that is, a G fixes A pointwise. On the other hand, if g ∈ G acts non-trivially on A, then so does ag, and ag ∈ a, G \ G. So these two semigroups are not equal, and G is not normalizing. Regarding (ii) suppose that G is imprimitive and let B be a non-trivial Ginvariant partition of {1, . . . , n}. Choose a set S of representatives for the B-classes, and let a be the map which takes every point to the unique point of S in the same B-class. Then a fixes all B-classes (in the sense that it maps any B-class into itself), and hence so does any G-conjugate of a, and so does any product of G-conjugates. On the other hand, the transitivity of G implies that there exists g ∈ G that does not fix all B-classes, so that neither does the element ag ∈ a, G \ G. As before, it follows that G is not normalizing. Now we are ready to prove the main lemma of this section. But before that we introduce some terminology and results. For natural numbers i, j n with i j, a group G S n is said to be (i, j)-homogeneous if for every i-set I contained in [n] and for every j-set J contained in [n], there exists g ∈ G such that Ig ⊆ J. This notion is linked to homogeneity since an (i, i)-homogeneous group is an i-homogeneous (or i-set transitive) group in the usual sense.
The goal of next lemma is to prove that a normalizing group is (k − 1, k)homogeneous, for all k such that 1 k ⌊ n+1 2 ⌋. But before stating our next lemma we state here two results about (k − 1, k)-homogeneous groups. (We denote the dihedral group of order 2p by D(2 * p).) Theorem 2.4. (See [1]) If n 1 and 2 k ⌊ n+1 2 ⌋ is fixed, then the following are equivalent: These groups admit an analogue of the Livingstone-Wagner [29] result about homogeneous groups.
(i) G is a (k − 1, k)-homogeneous subgroup of S n ; (ii) G is (k − 1)-homogeneous or G isCorollary 2.5. (See [1]) Let n 1, let 3 k ⌊ n+1 2 ⌋ be fixed, and let G S n be a (k − 1, k)-homogeneous group. Then G is a (k − 2, k − 1)-homogeneous group, except when n = 9 and G ∼ = ASL(2, 3) or AGL(2, 3), with k = 5.
Now we state and prove the main lemma in this section. Lemma 2.6. Let G S n be a normalizing group such that n 10. Then, for all k such that 2 k ⌊ n+1 2 ⌋, the group G is (k − 1, k)-homogeneous. Proof. Suppose that G fails to have the (k − 1, k)-homogenous property, for some k < ⌊ n+1 2 ⌋. Then it follows that G fails to be (m − 1, m)-homogeneous, for m = ⌊ n+1 2 ⌋, that is, there exist two sets, I and J, such that Ig ⊆ J, for all g ∈ G. Without loss of generality (since we can replace G by some appropriate g −1 Gg Observe that (for all g ∈ G) we have rank(aga) < rank(a), because there is no set in the orbit of {a 1 , . . . , a m } that contains {1, . . . , m − 1}; therefore there is only one chance for G to normalize a:
(∀g ∈ G)(∃h ∈ G) ag = h −1 ah.(1)
On the other hand, or not. We start by the second case. We are going to build a map ah ∈ T n \ S n and pick a permutation h −1 g ∈ G such that (ah)h −1 g is not normalized by G.
By assumption there exists g ∈ G such that Then, by the observation above, rank((ah) 2 ) = d + 1 and so the rank of any one of its conjugates is also d + 1: for all h 1 ∈ G we have rank((h −1 1 (ah)h 1 ) 2 ) = d + 1. On the other hand, rank((ah · h −1 g) 2 ) = c + 1(> d + 1) so that (∀h 1 ∈ G)ah · h −1 g = h −1 1 (ah)h 1 and hence by (1) ah · h −1 g ∈ (ah) h1 | h 1 ∈ G , a contradiction. It is proved that if the size of the following intersection |{a 1 , . . . , a m }g ∩ {1, . . . , m − 1}| varies with g ∈ G, then it is possible to build a map that is not normalized by G.
Now we turn to the first possibility, namely, exists a constant c such that, for all g ∈ G, we have |{a 1 , . . . , a m }g ∩ {1, . . . , m − 1}| = c. First observe that if c = 1, then m(m − 1) = n, which holds only when n = 6 (see Lemma 2.2 and recall that m = ⌊ n+1 2 ⌋). Since n 10 we have c 2. As |{a 1 , . . . , a m }g ∩ {1, . . . , m − 1}| = c, for all g ∈ G, it follows that (for g = 1) we have |{a 1 , . . . , a m } ∩ {1, . . . , m − 1}| = c. Without loss of generality (in order to increase the readability of the map a below), we will assume that a i = i, for i = 1, . . . , c. Now, as G is transitive, pick g ∈ G such that 1g = 2, and suppose there exists h ∈ G such that ag = a h , with It is proved that if G fails to be (k − 1, k)-homogeneous, for some k such that 1 k ⌊ n+1 2 ⌋, then G is not normalizing. The result follows. We have now everything needed in order to prove Theorem 1.4 regarding the groups of degree at least 10. In fact, if G is normalizing, then G is (k − 1, k)homogenous for all k such that 1 < k ⌊ n+1 2 ⌋ and hence the group (of degree at least 10) is (k − 1)-homogeneous (by Theorem 2.4). A primitive group (of degree n) is proper if it does not contain the alternating group of degree n. Therefore, if n = 10, then a proper primitive normalizing group must be (k = ⌊ n−1 2 ⌋ = 4)homogenous, but there are no such groups of degree 10. For n = 11, a proper primitive normalizing group must be (k = ⌊ n−1 2 ⌋ = 5)-homogenous, but there are no such groups of degree 11. If n = 12, then the group must be (k = ⌊ n−1 2 ⌋ = 5)homogenous, whose unique example (of degree 12) is M 12 . However M 12 , as the group of permutations of {1, . . . , 12} generated by the following permutations In fact, it is easily checked (using GAP [12]) that no element of M 12 maps {1, . . . , 6} to a section for the kernel of this map a. So, by Lemma 2.1, we only have to check whether, for every g ∈ M 12 , there exists h ∈ M 12 such that ag = h −1 ah. This fails for g = (132)(465)(798).
For n > 12, the group must be (k = ⌊ n−1 2 ⌋ 6)-homogenous, but for k 6 there are no proper primitive k-homogeneous groups [11, Theorem 9.4B, p. 289].
Therefore the unique groups that can be normalizing are the trivial group, the symmetric and alternating groups, and some primitive groups of degree at most 9.
In the next section we explain how we used GAP [12], orb [32] and Citrus [31], to check these groups of small degree. That the symmetric and the alternating groups are normalizing is already well known.
Computational considerations
In this section we describe the computational methods used to find the normalizing groups of degree at most 9. Regarding primitive groups of degree at most 3 they contain the alternating group and the result follows by Theorem 2.7. Therefore, from now on we assume that 4 n 9. We know that a normalizing group G S n is primitive and (k − 1, k)-homogeneous for all k ⌊ n+1 2 ⌋. By Theorem 2.4 we have two situations:
(1) G is (⌊ n−1 2 ⌋)-homogeneous and hence (by inspection of the GAP library of primitive groups) is one of the groups below: (2) or G is one of the groups in Theorem 2.4 (C 5 and D(2 * 5) of degree 5; AGL(1, 7) of degree 7; ASL(2, 3) and AGL(2, 3) of degree 9). To check that a group G ≤ S n is a-normalizing for some a ∈ T n \ S n it is enough to check that aG ⊆ g −1 ag | g ∈ G , since the latter is closed under conjugation with elements from G. So we only have to enumerate the G-orbit of a with right multiplication as action and check membership in the semigroup g −1 ag | g ∈ G for all its elements. This is essentially achieved by the following GAP-commands using the packages orb (see [32]) and Citrus (see [31]): However, for the larger examples on 9 points checking this for all a ∈ T n \ S n would have taken too long. Fortunately, this was not necessary, since if G is anormalizing, then it is of course a g -normalizing for all g ∈ G. So we only have to check this property for representatives of the G-orbits on T n \ S n under the conjugation action.
To compute a set of representatives we first implemented an explicit bijection of T n to the set {i ∈ N | 1 ≤ i ≤ n n }. Then we organised a bitmap of length n n and enumerated all conjugation G-orbits in T n , crossing off the transformations we had already encountered in the bitmap. Having the representatives as actual transformations then allowed us to perform the test explained above.
A slight speedup was achieved by actually verifying a stronger condition, namely that aG is a subset of the R-class of a in the semigroup a g | g ∈ G , which turned out to be the case whenever G was normalizing. Testing membership in the R-class of a in the transformation semigroup S := a g | g ∈ G can be done by computing the strong orbit of the image of a under the action of S and the permutation group induced by the elements of S that stabilise the image of a setwise; as described in [27]. This method is implemented in the Citrus package [31] These computational results complete the proof of our main Theorem 1.4.
Problems
Regarding this paper, the main problem that has to be tackled now should be the classification of the k-normalizing groups.
Problem 1. Let k be a fixed number such that 1 < k < ⌊ n+1 2 ⌋. Classify the knormalizing groups, that is, classify the groups that satisfy a, G \G = a g | g ∈ G , for every rank k map.
To solve this problem is necessary to use the results of [1], but that will be just a starting point since many delicate considerations will certainly be required.
The theorems and problems in this paper admit linear versions that are interesting for experts in groups and semigroups, but also to experts in linear algebra and matrix theory. For the linear case, we already know that any singular matrix with any group containing the special linear group is normalizing [5,6] (see also the related papers [14,33,34]). Problem 2. Classify the linear groups G GL(n, q) that, together with any singular linear transformation a, satisfy a, G \ G = h −1 ah | h ∈ G .
A necessary step to solve the previous problem is to solve the following.
Problem 3. Classify the groups G GL(n, q) such that for all rank k (for a given k) singular matrix a we have that rank(aga) = rank(a), for some g ∈ G.
To handle this problem it is useful to keep in mind the following results. Kantor [21] proved that if a subgroup of PΓL(d, q) acts transitively on k-dimensional subspaces, then it acts transitively on l-dimensional subspaces for all l ≤ k such that k + l ≤ n; in [22], he showed that subgroups transitive on 2-dimensional subspaces are 2-transitive on the 1-dimensional subspaces with the single exception of a subgroup of PGL(5, 2) of order 31 · 5; and, with the second author [9], he showed that such groups must contain PSL(d, q) with the single exception of the alternating group A 7 inside PGL(4, 2) ∼ = A 8 . Also Hering [15,16] and Liebeck [26] classified the subgroups of PGL(d, p) which are transitive on 1-spaces. (See also [21,22].) Problem 4. Solve analogues of the results (and problems) in this paper for independence algebras (for definitions and fundamental results see [2,3,10,13]).
one of the following groups (a) n = 5 and G ∼ = C 5 or D(2 * 5), k = 3; (b) n = 7 and G ∼ = AGL(1, 7), with k = 4; (c) n = 9 and G ∼ = ASL(2,3) or AGL(2, 3), with k = 5.
S n ) we can assume that I = {1, . . . , m − 1}, J = {a 1 , . . . , a m } and hence there is no g ∈ G such that {1, . . . , m − 1}g ⊆ {a 1 , . . . , a m }. Now pick a ∈ T n such that a = {1} . . . {m − 1} [n] \ {1, . . . , m − 1} a 1 . . . a m−1 a m .
|{a 1 , . . . , a m } ∩ {1, . . . , m − 1}| = r,implies that rank(a 2 ) = r + 1, and hence rank((h −1 ah) 2 ) = r + 1 as well.Now we have two situations: either there exists a constant c such that for all g ∈ G we have |{a 1 , . . . , a m }g ∩ {1, . . . , m − 1}| = c,
|{a 1 , . . . , a m }g ∩ {1, . . . , m − 1}| = c and there exists h ∈ G such that |{a 1 , . . . , a m }h ∩ {1, . . . , m − 1}| = d < c.
a = {1} . . . {c} {c + 1} . . . {m − 1} [n] \ {1, . . . , m − 1} 1 . . . . . . {c} {c + 1} . . . {m − 1} [n] \ {1, . . . , m − 1} 1g = 2 . . . cg a c+1 g . . . a m−1 g a m g and a h = {1}h . . . {c}h {c + 1}h . . . {m − 1}h [n] \ {1, . . . , m − 1}h 1h . . . In ag, 2 is not a fixed point and |2(ag) −1 | = 1. Therefore 2 is not a fixed point of a h and |2(a h ) −1 | = 1. As the possible non-fixed points of a h with singleton inverse image (under a h ) are contained in {a c+1 h, . . . , a m−1 h}, it follows there must be an element a j ∈ {a c+1 , . . . , a m−1 } such that a j h = 2. But this means that h does not permute {1, . . . , m − 1} and hence {{1}, . . . , {m − 1}}h = {{1}, . . . , {m − 1}} yielding that the kernel of a h and ag are different, a contradiction.
Theorem 2.7. ([23, Theorem 5.2]) The groups S n and A n are normalizing.
gap> o := Orb(G,a,OnRight);; Enumerate(o);; gap> o2 := Orb(G,a,OnPoints);; Enumerate(o2);; gap> s := Semigroup(o2);; gap> ForAll(o,x->x in s); true
for GAP. For degree 8, all three groups AGL(1, 8), AΓL(1, 8) and ASL(3, 2) fail to normalize the map and finally the group PGL(2, 7) fails to normalize the mapFor degree 5, only AGL(1, 5) is normalizing, since the group C 5 fails to normalize
the map
a =
{1, 2, 5} {3} {4}
1
3
4
,
and the group D(2 * 5) fails to normalize the map
a =
{1, 2, 3} {4} {5}
1
3
2
.
For degree 6, both groups PSL(2, 5) and PGL(2, 5) are normalizing.
For degree 7, we only had to check AGL(1, 7), which fails to normalize the map
a =
{1, . . . , 5} {6} {7}
1
2
3
.
a =
{1, . . . , 5} {6} {7} {8}
1
2
3
4
,
the group PSL(2, 7) fails to normalize the map
a =
{1, . . . , 5} {6} {7} {8}
1
2
3
5
,
a =
{1, . . . , 5} {6} {7} {8}
1
2
4
7
.
For degree 9, the two groups PSL(2, 8) and PΓL(2, 8) are normalizing, whereas
both groups ASL(2, 3) and ASL(2, 3) fail to normalize the map
a =
{1, 8} {2, 3, 7} {4} {5} {6} {9}
7
8
6
9
4
5
.
AcknowledgementsThe authors would like to express their gratitude to the referee for a very careful review and for suggestions that prompted a much simplified paper.The first author was partially supported by FCT through the following projects: PEst-OE/MAT/UI1043/2011, Strategic Project of Centro deÁlgebra da Universidade de Lisboa; and PTDC/MAT/101993/2008, Project Computations in groups and semigroups .The second author is grateful to the Center of Algebra of the University of Lisbon for supporting a visit to the Centre in which some of this research was done.
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|
[] |
[
"Mid-Infrared Microscopy via Position Correlations of Undetected Photons",
"Mid-Infrared Microscopy via Position Correlations of Undetected Photons"
] |
[
"Inna Kviatkovsky \nInstitut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany\n",
"Helen M Chrzanowski \nInstitut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany\n",
"Sven Ramelow \nInstitut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany\n\nIRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany\n"
] |
[
"Institut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany",
"Institut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany",
"Institut für Physik\nHumboldt-Universität zu Berlin\nBerlinGermany",
"IRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany"
] |
[] |
Quantum imaging with undetected photons (QIUP) has recently emerged as a new powerful imaging tool. Exploiting the spatial entanglement of photon pairs, it allows decoupling of the sensing and detection wavelengths, facilitating imaging in otherwise challenging spectral regions with mature silicon-based detection technology. All existing implementations of QIUP have so far utilised the momentum correlations within the biphoton state. Here, for the first time, we implement and examine theoretically and numerically the complementary scenario -utilising the tight position correlations formed within photon pair at birth. This image plane arrangement facilitates high resolution imaging with comparative experimental ease, and we experimentally show resolutions below 10µm at a sensing wavelength of 3.7 µm. Moreover, imaging a slice of mouse heart tissue at the mid-IR to reveal morphological features on the cellular level, we further demonstrate the viability of the technique for the life sciences. These results offer new perspectives on the capabilities of QIUP for label-free wide-field microscopy, enabling new real-world applications in biomedical as well as industrial imaging at inaccessible wavelengths.
|
10.1364/oe.440534
|
[
"https://arxiv.org/pdf/2106.06435v1.pdf"
] | 235,417,194 |
2106.06435
|
ef64ea6a882c2fd491090739f352cc1b3e1021ab
|
Mid-Infrared Microscopy via Position Correlations of Undetected Photons
11 Jun 2021
Inna Kviatkovsky
Institut für Physik
Humboldt-Universität zu Berlin
BerlinGermany
Helen M Chrzanowski
Institut für Physik
Humboldt-Universität zu Berlin
BerlinGermany
Sven Ramelow
Institut für Physik
Humboldt-Universität zu Berlin
BerlinGermany
IRIS Adlershof
Humboldt-Universität zu Berlin
BerlinGermany
Mid-Infrared Microscopy via Position Correlations of Undetected Photons
11 Jun 2021
Quantum imaging with undetected photons (QIUP) has recently emerged as a new powerful imaging tool. Exploiting the spatial entanglement of photon pairs, it allows decoupling of the sensing and detection wavelengths, facilitating imaging in otherwise challenging spectral regions with mature silicon-based detection technology. All existing implementations of QIUP have so far utilised the momentum correlations within the biphoton state. Here, for the first time, we implement and examine theoretically and numerically the complementary scenario -utilising the tight position correlations formed within photon pair at birth. This image plane arrangement facilitates high resolution imaging with comparative experimental ease, and we experimentally show resolutions below 10µm at a sensing wavelength of 3.7 µm. Moreover, imaging a slice of mouse heart tissue at the mid-IR to reveal morphological features on the cellular level, we further demonstrate the viability of the technique for the life sciences. These results offer new perspectives on the capabilities of QIUP for label-free wide-field microscopy, enabling new real-world applications in biomedical as well as industrial imaging at inaccessible wavelengths.
The functionality to image samples in the mid-infrared (mid IR) offers a new perspective for problems of tremendous biological and industrial relevance. By exploiting the highly specific vibrational and rotational 'fingerprints' of molecules as contrast mechanisms, one can obtain insights into the chemical and molecular structure inaccessible in traditional microscopy [1][2][3]. The principle limitation, however, remains one of detection, with mid-IR imaging technology being prohibitively expensive while suffering from limited spectral response, spatial resolution or sensitivity. This absence of suitable detection options has lead to Raman-based imaging techniques [4,5], which require raster scanning, often making the technique too slow for many applications. Other approaches employ nonlinear wavelength conversion to the visible regime, where one can enjoy the comparable maturity of charge-coupled device (CCD) and complementary metal-oxide-semiconductor (CMOS) technology driven by the life sciences. Frequency up-conversion imaging shifts the detection frequency from the IR light to the desired visible while retaining the spatial and spectral information. This up-conversion imaging technique has been realised in the near-and mid-IR [6][7][8].
An alternative approach and that which we consider here exploits quantum imaging with undetected photons (QIUP) [9]. Though its initial realisation was based on induced coherence without induced emission [10], it can equivalently be described as an application of nonlinear interferometry [11]. In QIUP, two nonlinear crystals are pumped sequentially and coherently with laser light, generating photon pairs through spontaneous parametric down conversation (SPDC). When the two processes are aligned such that any information distinguishing whether the biphoton was born on the first or second crystal is erased, the two processes interfere. The strength (visibility) and phase of this interference signature serve as the contrast mechanism for imaging -analogously so for similar applications in varied tasks including optical co-herence tomography [12][13][14], spectroscopy [15][16][17][18][19] and polarimetry [20]. In contrast to ghost imaging schemes [21][22][23], in QIUP the detection of only one of the photons of the pair suffices to yield the information imprinted on the other. The strong spatial entanglement shared between the signal and idler pair, allows inference of the idler via the measurement of the signal [24,25], facilitating multi-mode (widefield) imaging. Crucially, this allows for the decoupling of the sensing and detection wavelengths when using a non-degenerate SPDC process [9]. An appealing way to use this advantage is to realise sensing at wavelengths for which advanced multi-pixel detection technologies are lacking. In this way, QIUP can shift the detection into the visible or near-IR wavelength ranges, where the far superior CMOS-and CCD-based sensor technologies operate.
This potential of QUIP for varied imaging applications, spanning wavelengths and applications [26,27] motivates its relevance for the life sciences. Recently, QIUP was tailored to image at the microscopic length scale, so-called quantum microscopy with undetected photons (QMUP) [28,29]. All prior realisations of QIUP and QMUP have exclusively exploited the momentum anti-correlations, imaging in the far-field of the crystal [9, 24-26, 28, 29]. Here, we shift focus (both literately and figuratively) to instead examine imaging with the complementary position correlations that exist in the image plane [30]. We use these position correlations in the context of QIUP for the first time and demonstrate their advantage for microscopy in the mid-IR. In addition to demonstrating the viability of position correlations for imaging, we show that for microscopy, this approach dramatically simplifies the required optical overhead, obtaining sufficiently high resolutions with relative ease. arXiv:2106.06435v1 [quant-ph] 11 Jun 2021 THEORY All existing QIUP and QMUP experimental implementations have utilised the momentum anti-correlations that arise due to transverse momentum conservation in the pair production process. However, the same spatial entanglement that gives rise to the aforementioned anticorrelations in momentum space, equally gives rise to strong correlations in the conjugate position space [31]. Physically, these correlations stem from the tight position localisation created when the signal and idler pairs are born from the annihilation of a pump photon. Accordingly, the field of view (FoV) of the imaging system is then specified by the waist of the pump beam that illuminates the crystal, essentially defining an aperture within which the SPDC process can occur. For a Gaussian pump beam, the FoV accordingly is a Gaussian distribution with a full width at half maximum (FWHM) given by
F OV IP = √ 2ln2 w p M ,(1)
where w p is the pump waist at the crystal and M is the optical magnification of the setup that scales the FoV after the crystal. The resolution, defined as the FWHM of the point-spread-function (PSF) of the system, is given by res IP = 0.51 λi N A lim . The limiting numerical aperture N A lim is the minimum of the limit given by the optical components in the setup, or, more typically, given by the SPDC emission angle of the sensing (undetected) wavelength emitted from the crystal [31]. For our system, the emission angle of the (undetected) idler light, scaled by subsequent magnification, is the dominant contribution. The resulting resolution in the image plane is
res IP = 0.41 λ i θ i M .(2)
Here, θ i is half the idler emission angle at FWHM, which relates to the crystal length through: θ i = λ i 2.78 πL nsni nsλi+niλs . One can equivalently and perhaps, more intuitively, consider the resolution to be a limitation arising from the thickness of the down-conversion crystal itself; the longer the crystal the more the ambiguity that arises regarding the birthplace of the signal and idler pair. The ratio of the FoV and resolution allows us to approximate the number of spatial modes per direction,
m IP = √ 2ln2 0.41 w p θ i λ i ∝ ω p √ L(3)
Optimisation of the imaging capabilities requires maximising the pump waist and minimising the crystal length, while also balancing the required illumination per mode. When inserting our experimentally determined values of w p = 431 ± 6µm, θ i = 0.0491 ± 0.0003rad, λ i = 3.74 ± 0.02µm into Eqns. 1,2 and 3, we obtain F OV IP = 127 ± 2µm, res IP = 7.9 ± 0.1µm, m IP = 16.1 ± 0.3. By comparison, the corresponding far-field implementation [28] theoretically predicts almost 50 % more spatial modes per axis for the same pump light and crystal specifications. This is a consequence of some of the entanglement directly accessible in the far-field migrating into the imaginary part of the amplitude in the near-field, rendering it inaccessible with intensity measurements alone [32][33][34]. Despite this reduction in available spatial modes, here we show that image plane imaging offers an advantage: imaging at microscopic length scales with large reduction in optical complexity. Requiring less magnification, fewer optical elements and consequently shorter interferometric arms, the demonstrated resolution in the image plane is thrice superior to that of achieved in the far field [28].
EXPERIMENT
The experimental setup is detailed in Fig. 1 and exploits a Michelson-type configuration to realise the nonlinear interferometer. A 660 nm CW pump laser illuminates the ppKTP crystal, with a 431 µm waist maximally covering the crystal aperture. The ppKTP crystal is quasi-phase matched for a collinear type-0 process and specifically engineered [35] to produce simultaneously highly non-degenerate and broadband photon pairs. The broadband SPDC emission is due to the group velocity matching of the signal and idler, with spectral widths of 780-830 nm and 3.4-4.3 µm respectively at room temperature [35]. After the crystal, an off-axis parabolic mirror (OPM) is placed at its focal distance for achromatic collimation (focusing) of the emerging (returning) signal and idler. Using a dichroic mirror (DM), the idler is then split from the signal and pump, with the pump subsequently back-reflected using a cold mirror to preserve the desired imaging condition. The idler and signal fields are then each focused to align the image plane of the crystal with their respective end mirrors. A sample for imaging is placed on the low-e slide that serves as an end mirror for the idler arm. The reflected idler and signal fields then back-propagate and are focused into the crystal, interfering with signal and idler fields generated upon the second pass of the pump through the crystal. The idler and pump light emerging from the second pass of the crystal are then discarded and the signal light is imaged onto a CMOS camera. Prior to detection, the signal field is further filtered using a band-pass filter (3.5 nm FWHM) and a telescopic arrangement is used to position the CMOS camera in the image plane of the crystal.
It is crucial to carefully align the setup to ensure optimal spatial overlap between the biphoton fields generated in the two passes through the crystal. This ensures indistinguishably that will be manifested in the visibility of the interferometric image captured. This requirement is achieved by simultaneously matching the interferometric arms within the coherence length of the detected sig-nal light and carefully tuning the optical components to fulfil the imaging conditions. Any deviation from those requirements will result in departure from optimal visibility and potentially, resolution, of the imaging system. The signal (800 nm) and idler (3.7 µm) fields generated on the first pass are split via a dichroic mirror, allowing the idler to probe the sample, before being recombined and travelling collinear with the coherent pump field back into the crystal. The pump is independently reflected back via a separate cold mirror in the signal arm. The signal field emerging after the second pass of the crystal is spectrally filtered via a band-pass filter (BPF) and then imaged onto a CMOS camera, revealing the spatial information obtained by the idler when probing the sample. Fig. 2 presents the characterisation of our imaging system at a magnification of M = 4. The resolution, obtained via an edge knife response, is 9 ± 1µm (FWHM) and the obtained FoV is 161 ± 1µm (FWHM). These values compare favourably to the theoretical values for the resolution and FoV of 7.8µm and 127µm respectively, albeit at a lower than anticipated magnification, as the corresponding number of spatial modes -18 ± 2 per axis -is in agreement with the theoretical value.
RESULTS
The experimental results are summarised in Table 1. We attribute the systematic deviation of the larger obtained FoV and resolution to a reduced magnification realised in practice for the 4f system, when compared to the anticipated magnification of exactly 4. As predicted by the theory, QMUP via position correlations allows one to access high resolutions with a simplified optical system, notably when compared to realisations in the Fourier plane [28]. This is an inherent character- istic of Fourier plane imaging due to the divergence of the biphoton field at the crystal exit, resulting in a large illumination spot (FoV) in the far field. Due to the interferometric nature of the imaging technique, a distortion of the biphoton wavefront can result in ambiguities; for a single image it can be ambiguous whether a dark region references absorption, or rather, destructive interference. This is particularly relevant for biological or industrial applications, where complex morphologies underpin the imaging motivation. By scanning axially within the coherence length, one can obtain a pixel-wise visibility of the sample, allowing pixel-wise reconstruction of both the transmissivity and the phase. This has a secondary advantage of increasing the 'effective' FoV of the image; in contrast to the intensity distribution of the illumination spot itself, the visibility distribution is considerably flatter. The effective number of spatial modes in the visibility images is here approximately double, resulting in roughly 900 spatial modes for the 2D wide field imaging arrangement.
Using this scanning technique, a thin unstained slice of a mouse heart tissue was mounted on a low-e slide and imaged. Complementary to the above characterised imaging system (M=4), a lower magnification (M=2) configuration was also used to acquire a larger scale absorption image (Fig. 3(B)). To enable comparison, a bright field image acquired with a standard visible microscope is presented in Fig. 3(A). The left ventricle of the mouse heart and surrounding structure is visible in the images, revealing various morphological features. In Fig. 3(c) two smaller regions in the sample were characterised using the larger magnification arrangement (M=4), with this increased resolution revealing additional features that are not visible in the lower magnification images. The additional morphological information revealed in the mid-IR image is inaccessible in the bright field image, which can be attributed to the considerably reduced scattering at mid-IR wavelengths.
Experiment Theory
FoV (µ m) 161 ± 1 127 ± 2 Resolution (µ m) 9 ± 1 7.9 ± 0.1 Spatial modes 18 ± 2 16.1 ± 0.3 Table I. QMUP in the image plane -experiment vs. theory
DISCUSSION
One technical disadvantage of QMUP in the image plane is the reduced homogeneity of the illumination distribution. In the image plane, we illuminate with the photon birth zone itself, revealing inhomogeneities arising from small defects in the crystal, dust or imperfect pump modes. By contrast, the Fourier plane provides a very homogeneous illumination distribution, its uniformity being a consequence of the smoothness of the phasematching curve itself. This disadvantage is reminiscent of classical microscopy, where schemes usually avoid illumination in the image plane of the light source to circumvent the analogous noise contributions [36]. Here, where the correlations between the planes are indispensable, it cannot be easily avoided, but can be eliminated with care towards the crystal and pump. The higher noise in the image plane is one potential contributor to the reduction of visibility, as the background noise in the image plane is inverted on the second pass (and thus does not cancel out). A second source of potential visibility reduction stems from the high magnification, which results in a lower depth of focus and thus a higher sensitivity to alignment errors. Both of these sources of reduced visibility are technical in nature and not fundamental limitations.
In contrast to our prior work in the Fourier plane microscopy, we suffer no degradation from the theoretical performance of our system despite imaging at considerably smaller length scales. While imperfect matching of the image plane at the sample and camera will likely degrade the resolution of the imaging system, the short depth of focus aids in matching these conditions very precisely. The 9 µm resolution presented here was realised in a simple 4f setup with a standard aspheric lens (f = +25 mm, Thorlabs) serving as the magnifying lens. This basic 4f arrangement affords superior compactness and stability, and with targeted optical engineering, should attain diffraction-limited resolutions at the state-of-theart of mid-IR imaging.
The clear success of this approach for imaging and microscopy in the mid-IR motivates the question of its limitations -can we envisage this approach to wide field imaging stretching beyond its near and mid-IR, towards the far IR and even terahertz wavelengths [19], where detection technologies are even more limited? Fig. 4 presents a theoretical analysis of this question in terms of available spatial modes. It shows a decrease in wide field imaging capacity QIUP with increasing (undetected) wavelength. The increasing non-degeneracy between the signal and idler energy that enables 'silicon imaging' at silicon incompatible wavelengths, must be traded off against a decrease in the available spatial entanglement and thus the wide field imaging capacity. Wide field QIUP applications remain promising into the far IR, but become increasingly unfavourable as we approach the terahertz regime. Furthermore, this analysis does not consider factors including material absorption, parasitic seeding and the necessity for long crystals that complicate the imaging at very long wavelengths. This analysis, however, does not preclude single-pixel based scanning approaches to imaging tasks at these wavelengths.
The approach presented here also opens up the possibility of an imaging regime previously inaccessible at far wavelengths: facilitating shot-noise limited imaging with high quantum efficiencies at exceptionally low illumination powers. This is the consequence of the intrinsic (theoretical unity) efficiency of the nonlinear interferometer in the low-gain regime, where any (mid-IR) idler photon has its (silicon-compatible) partner. Accordingly, any imaging information carried by that idler photon can be transferred perfectly to the signal photon. Therefore, in the absence of additional loss and mode mismatch, the noise performance of the mid-IR imaging is determined by the properties of the silicon camera, where shot noise-limited images are accessible with only a few 1000 of photons per pixel per second or less. Here, the images were obtained at mid-IR illumination levels of only a 15 pW -with the light within our current detection bandwidth amounting to only 2-3 pW. Such low illuminations are significant for the understanding of photoreceptive samples, where the sensing illumination itself can invariably interfere with the cellular and molecular mechanisms one seeks to understand.
CONCLUSION
In conclusion, we have presented the first experimental realisation of QIUP via position correlations. We have shown that imaging with position correlations presents advantages over its predecessors for the task of microscopy, allowing access to resolutions below 10 µm at a sensing wavelength of 3.7 µm with no observable deviation from theoretical predictions. This improved resolution permitted mid-IR imaging of an unstained tissue from a mouse heart with several orders of magnitude less light than any comparative methods. The presented results further extend the growing toolbox of QMUP and take us another step closes toward real-world applications.
Method
Slide Preparation: 9-12 weeks old C57BL/6J mice were sacrificed by cervical dislocation. Hearts were removed, rinsed in ice-cold saline and placed in 4 % formalin. After 48 to 72 hrs fixation the tissue was rinsed with (Phosphate-buffered saline) PBS and then embedded in paraffin. The paraffin-embedded hearts were cut in transverse sections to a thickness of 2 to 3 µm and transferred to a low-e slide.
acknowledgements
The authors want to acknowledge Ellen G. Avery and Hendrik Bartolomaeus for providing the bio-samples and helping with the interpretation of the results. The authors also thank Sergey Berezinski for assistance with preparation of the figures. This work was funded by Deutsche Forschungsgemeinschaft (RA 2842/1-1).
Figure 1 .
1Experimental Setup: A continuous wave pump light at 660 nm stimulates a highly non-degenerate, collinear SPDC process in a Michelson-type interferometer.
Figure 2 .
2Characterization of the imaging arrangements. (A) FoV. The measured data (pink points) was fitted with a Gaussian (purple line) yielding a FoV of 161µm (FWHM). (B) Edge response in the top row, the measured data (pink points) was fitted with an error function (purple line), differentiation of the error function gives a Gaussian shaped (marked in blue) point spread function (PSF). The resolution, determined by the FWHM of the PSF is 9µm.
Figure 3 .
3Histology sample of a mouse heart imaged with (A) bright field microscopy with visible light for illustration which part of the sample we investigated with our method. (B) Mid-IR microscopy of the same sample with undetected photons for absorption imaging with a 2-fold magnification. (C) Higher resolution absorption images, taken with the a 4fold magnification arrangement. Images are formed by stitching roughly 7 wide-field absorption images (translating the sample transversely), each absorption image reconstructed by averaging 6 images at 1 s integration time for 6 axial positions within the coherence length of the biphoton (longitudinal scan).
Figure 4 .
4Numerical simulation of the number of spatial modes available for wide field imaging (as characterised by the Schmidt number) as a function of increasing idler wavelength for four different 'effective' crystal apertures, V. For instance, the four decreasing values of V (= 1.0, 0.71, 0.45 and 0.3 mm −1/2 ) correspond to increasing values of crystal length, L (= 1, 2, 4 and 11 mm) for a pump waist of ωp = 1 mm. The pump wavelength of the pump light is fixed at 660 nm and for simplicity the refractive indices are assumed to be 1.5 across the examined range. The pump wavelength and degeneracy point (1.32 µm) are highlighted by vertical dashed lines. The dashing of curves lines indicates a parameter regime where these results, derived under the paraxial approximation, may no longer faithfully describe the system. The imaging capacity of the system presented here (ωp = 430 µm and L = 2 mm) is indicated by a star.
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[
"Spectral Index in Curvaton Scenario",
"Spectral Index in Curvaton Scenario"
] |
[
"Qing-Guo Huang [email protected] \nSchool of physics\nInstitute for Advanced Study\n207-43, Cheongryangri-Dong, Dongdaemun-Gu130-722SeoulKorea, Korea\n"
] |
[
"School of physics\nInstitute for Advanced Study\n207-43, Cheongryangri-Dong, Dongdaemun-Gu130-722SeoulKorea, Korea"
] |
[] |
A red tilted primordial power spectrum is preferred by WMAP five-year data and a large positive local-type non-Gaussianity f N L might be observed as well.In this short note we find that a red tilted and large non-Gaussian primordial power spectrum cannot be naturally obtained in curvaton model, because f N L is related to the initial condition of inflation.
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10.1103/physrevd.78.043515
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[
"https://arxiv.org/pdf/0807.0050v2.pdf"
] | 15,967,225 |
0807.0050
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88699c2cad9863c0781ca4b5fb576a569a79f1f4
|
Spectral Index in Curvaton Scenario
6 Jul 2008
Qing-Guo Huang [email protected]
School of physics
Institute for Advanced Study
207-43, Cheongryangri-Dong, Dongdaemun-Gu130-722SeoulKorea, Korea
Spectral Index in Curvaton Scenario
6 Jul 2008
A red tilted primordial power spectrum is preferred by WMAP five-year data and a large positive local-type non-Gaussianity f N L might be observed as well.In this short note we find that a red tilted and large non-Gaussian primordial power spectrum cannot be naturally obtained in curvaton model, because f N L is related to the initial condition of inflation.
Inflation [1] provides an elegant mechanism to solve many puzzles in the Hot Big Bang model. The wrinkles in the cosmic microwave background radiation and the largescale structure of the Universe are seeded by the quantum fluctuations generated during inflation [2]. The shape of the primordial quantum fluctuations is characterized by its amplitude P ζ and tilt n s which can be measured by experiments. WMAP five-year data [3] combined with the distance measurements from the Type Ia supernovae (SN) and the Baryon Acoustic Oscillations (BAO) in the distribution of galaxies indicates P ζ,obs = 2.457 +0.092 −0.093 × 10 −9 ,
n s = 0.960 +0.014 −0.013 .
Gravitational wave perturbations are also generated during inflation and its amplitude P T is only determined by the inflation scale. For convenience, we define a new quantity, named tensor-scalar ratio r = P T /P ζ , to measure the amplitude of gravitational wave perturbations. The primordial gravitational wave perturbation has not been detected. Present limit on the tensor-scalar ratio is r < 0.20 (95% CL). The blue tilted primordial power spectrum (n s > 1) is disfavored even when gravitational waves are included.
The non-Gaussianity is characterized by the non-Gaussianity parameter f N L which is defined as follows
Φ(x) = Φ L (x) + f N L [Φ 2 L (x) − Φ 2 L (x) ],(3)
where Φ L (x) denotes the linear Gaussian part of the perturbation in real space. The simplest model of inflation predicts a closely Gaussian distribution of primordial fluctuations [4,5], namely |f N L | < O(1). The most general density perturbation is a superposition of an isocurvature density perturbation and an adiabatic density perturbation. We introduce a new parameter α −1 = f 2 iso /(1 + f 2 iso ) to measure the isocurvature density perturbation, where f iso is the ratio of the isocurvature and adiabatic amplitudes at the pivot scale. A Gaussian and adiabatic power spectrum of primordial perturbation is still consistent with WMAP five-year data:
− 9 < f local N L < 111 and − 151 < f equil N L < 253 (95%CL),(4)α −1 < 0.0037 (95%CL),(5)
where "local" and "equil" denote the shapes of the non-Gaussianity. In [6] the authors reported that a positive large non-Gaussianity
27 < f local N L < 147(6)
is detected at 95% C.L.. A large non-Gaussianity is not a conclusive result from experiments, but it is still worthy for us to discussing the theoretical probabilities of the large non-Gaussianity. If it is confirmed by the forthcoming cosmological experiments, it strongly shows up some very important new physics of the early Universe.
An attractive model for a large positive local-type non-Gaussianity is curvaton model [7,8] in which the primordial power spectrum is generated by a light scalar field, called curvaton σ, but not the inflaton φ, even though the dynamics of inflation is governed by the inflaton. Recently many issues about curvaton model were discussed in [9][10][11][12]. In [9] we considered the case in which the Hubble parameter is roughly a constant during inflation and found that f N L is bounded by the tensor-scalar ratio r from above. Or equivalently a large non-Gaussianity gives a lower bound on the amplitude of the tensor perturbation in curvaton scenario. In [3] the authors pointed out that a large positive f N L cannot be obtained by considering the bound on the isocurvature perturbation α −1 < 0.0037 in curvaton model. However if the cold dark matter was produced after the curvaton decays completely, the curvaton model is free from the constraint of isocurvature perturbation [12]. Some other related topics on the large non-Gaussianity are discussed in [13] recently.
The spectral index of primordial power spectrum in curvaton scenario is given by
n s = 1 − 2ǫ + 2η σσ ,(7)
where η σσ = 1
3H 2 d 2 V (σ) dσ 2 and ǫ ≡ −Ḣ/H 2(8)
denotes how fast the Hubble parameter varies during inflation. Usually in curvaton model we assume that the mass of curvaton d 2 V (σ) dσ 2 is much smaller than the Hubble parameter during inflation. Therefore n s ≃ 1 − 2ǫ. For n s = 0.96, ǫ ≃ 0.02. Such a large value of ǫ implies the variation of inflaton is larger than Planck scale, which might be inconsistent with string theory [14][15][16][17]. Usually a small value of ǫ and a closely scale-invariant power spectrum are expected in curvaton scenario. However, maybe the effective tran-Planckian excursion of inflation can be achieved in string landscape [18] or the monodromies [19].
In this short note we will extend our previous work [9] to more general cases and investigate whether a red tilted primordial power spectrum is naturally compatible with a large positive f N L in curvaton model.
Let us consider a simple curvaton model with potential
V (φ, σ) = V (φ) + 1 2 m 2 σ 2 ,(9)
where V (φ) ≫ 1 2 m 2 σ 2 , φ and σ denote the inflaton and curvaton respectively. The slowroll equations of motion are obtained by neglecting the kinetic term and the energy density of curvaton in the Friedmann equation, and the second time derivative in the inflaton field equation, namely
H 2 = V (φ) 3M 2 p ,(10)3Hφ = −V ′ (φ),(11)
where V ′ (φ) = dV (φ)/dφ and M p is the reduced Planck scale. Once inflation is over, the energy density of inflaton is converted into radiation and H 2 goes like a −4 . The value of curvaton field, denoted as σ * , does not change until the Hubble parameter becomes the same order of curvaton mass. After that curvaton oscillates around its minimum σ = 0 and its energy density decreases as a −3 . Once the Hubble parameter goes to the same order of the curvaton decay rate Γ σ , the curvaton energy is converted into radiations. Before primordial nucleosynthesis, the curvaton field is supposed to completely decay into radiation and thus the perturbations in the curvaton field are converted into curvature perturbations. The amplitude of the perturbations caused by curvaton is given in [8] by
P 1 2 ζσ = 1 3π Ω σ,D H * σ * ,(12)
where
Ω σ,D ≡ ρ σ ρ tot D(13)
is the fraction of curvaton energy density in the energy budget at the time of H = Γ σ .
Here the subscript * denotes that the quantities are evaluated at horizon exit during inflation. A large positive non-Gaussianity is obtained only when Ω σ,D ≪ 1 and f N L is given by
f N L = 2 3 − 5 6 Ω σ,D + 5 4Ω σ,D ≃ 5 4Ω σ,D .(14)
In the case of Ω σ,D ≪ 1 the Universe is dominated by radiation before the time of H = Γ σ .
In [8] the value of Ω σ,D is estimated as
Ω σ,D ≃ σ 2 * 6M 2 p m Γ σ 1 2 .(15)
Keeping m and Γ σ fixed, f N L ∝ M 2 p /σ 2 * . In the literatures σ * is treated as a free parameter and then a large f N L is naturally expected for σ * < M p . However this treatment seems too naive. The curvaton mass is much smaller than the Hubble parameter during inflation, which means the Compton wavelength is large compared to the curvature radius of the de Sitter space H −1 . So the gravitational effects may play a crucial role on the behavior of the light scalar field in such a scenario. In [20] the authors explicitly showed that the quantum fluctuation of the light scalar field σ with mass m in de Sitter space gives it a non-zero expectation value of the square of a light scalar field
σ 2 = 3H 4 8π 2 m 2 ,(16)
where the Hubble parameter H is assumed to be a constant. In [9] we estimated the value of curvaton as σ * ∼ H 2 /m and we found f N L < 522r 1 4 . It is more complicated to estimate the value of curvaton in the models, such as chaotic inflation, where the Hubble constant cannot be regarded as a constant. Fortunately, how to estimate the value of curvaton field in this case has been discussed by Linde and Mukhanov in [21]. According to the long-wave quantum fluctuation of a light scalar field (m ≪ H) in inflationary universe, its behavior can be taken as a random walk [22]:
σ 2 = H 3 4π 2 t.(17)
This equation is only valid for the case of a constant H. During inflation the Hubble parameter H is not a constant exactly. We don't explicitly know the behavior of σ 2 when H depends on t. However, for a short period ∆t (≪ H −1 ), Eq. (17) can be written as
∆ σ 2 ≃ H 3 4π 2 (1 − 3ǫH∆t)∆t.(18)
Since 3ǫH∆t ≪ 1, we suppose that the differential form of Eq.(17)
d σ 2 dt ≃ H 3 4π 2(19)
can be generalized to the case in which the Hubble parameter slowly varies (ǫ = −Ḣ/H 2 ≪ 1). On the other hand, a massive scalar field cannot grow up to arbitrary large vacuum expectation value because it has a potential. The long wavelength modes of the light scalar field are in the slow-roll regime and obey the slow-roll equation of motion, i.e.
3H dσ dt = − dV (σ) dσ = −m 2 σ.(20)
Combining these two considerations, Linde and Mukhanov proposed in [21] d
σ 2 dt = H 3 4π 2 − 2m 2 3H σ 2 .(21)
For a constant Hubble parameter, σ 2 stabilizes at the point of σ 2 = 3H 4 8π 2 m 2 which is just the same as Eq. (16). Classically the scalar field is stable at σ = 0. In the inflationary universe the scalar field σ gets a non-zero expectation value due to the gravitational effects. Integrating over Eq. (21) with the initial condition σ 2 (t = t i ) = 0, we obtain
σ 2 (t) = t t i dt 1 H 3 (t 1 ) 4π 2 exp − t t 1 dt 2 2m 2 3H(t 2 )
.
We will use this solution to estimate the value of curvaton field.
Let's take into account the chaotic inflation which is driven by the potential
V (φ) = 1 p λφ p M 4−p p ,(23)
where λ is a small dimensionless parameter (λ ≪ 1) and p > 0. From now on we work in the unit of M p = 1. The equations of motion for the slow-roll inflation are given by
H 2 = λφ p 3p ,(24)3Hφ = −λφ p−1 .(25)
The value of inflaton at the time of N e-folds before the end of inflation is related to N by
φ N = 2pN.(26)
Now the slow-roll parameter ǫ becomes
ǫ = −Ḣ H 2 = 1 2 V ′ V 2 = p 2 2φ 2 N = p 4N .(27)
The number of e-folds corresponds to the CMB scale is roughly N c = 60. For n s = 0.96, p ≃ 4.8. The amplitude of scalar primordial power spectrum caused by inflaton at CMB scale is
P φ = H 2 8π 2 ǫ = 1 12π 2 p 3 λφ p+2
Nc .
In curvaton scenario P φ ≤ P ζ,obs which induces an upper bound on λ, namely
λ ≤ 12π 2 p 3 P ζ,obs φ −(p+2) Nc .(29)
For p = 2 the mass of inflaton √ λ should be smaller than 6.36 × 10 −6 in unit of Planck scale. On the other hand, the curvaton mass is smaller than Hubble parameter which says
m 2 ≤ λφ p 3p ≃ λ 3p .(30)
In the last step we consider that φ ∼ M p at the end of inflation. Combing with Eq. (29),
we have m 2 ≤ 4π 2 p 2 P ζ,obs φ −(p+2) Nc .(31)
For p = 2, the curvaton mass satisfies m ≤ 2.6 × 10 −6 .
It is time to estimate the vacuum expectation value of curvaton in chaotic inflationary universe. Using the equations of motion for the slow-roll chaotic inflation (24) and (25), we simplify Eq. (21) to be
σ 2 (t) ≃ λ 12π 2 p 2 φ i φ(t) dφ 1 φ p+1 1 exp 2m 2 λ φ φ 1 dφ 2 φ −(p−1) 2 ≃ λ 12π 2 p 2 (p + 2) φ p+2 i ,(32)
where we ignore the contribution from the exponential function because its exponent
is proportional to m 2 /m 2 ef f (φ) ∼ m 2 /(λφ p−2 ) ≪ 1.
This vacuum expectation value of curvaton mainly comes from the perturbation mode with wavelength
H −1 (φ i ) exp( φ 2 i 2p ).
Since the wavelength is much larger than the Hubble horizon, this fluctuation mode is frozen to be a classical one and provides a non-zero classical configuration for curvaton field. The typical value of curvaton field in such a background is estimated as
σ 2 * = λ 12π 2 p 2 (p + 2) φ p+2 i ,(33)
which is obviously dependent on the initial value of inflaton. Requiring the curvaton energy density be much smaller than inflaton energy density during inflation yields a upper bound on the curvaton mass
m 2 ≪ 24π 2 p(p + 2)φ −(p+2) i .(34)
Here we also ignore a factor φ p on the right hand side of the above inequality because φ ∼ 1 at the end of inflation. Since the total number of e-folds of chaotic inflation is not very large, the constraint in Eq.(31) is much more stringent than Eq.(34).
In curvaton scenario the primordial power spectrum comes from the quantum fluctuation of curvaton during inflation. Substituting σ * in Eq.(33) into (12), we get
P ζσ = 4p(p + 2) 9 Ω 2 σ,D φ p N φ p+2 i .(35)
Now the spectral index of primordial power spectrum is given by
n s ≡ 1 + d ln P ζσ d ln k ≃ 1 − d ln P ζσ dN = 1 − p 2N .(36)
Using Eq. (14), we write down f N L as follows
f N L = 5 p(p + 2) 6 P −1/2 ζ,obs φ p/2 N φ p/2+1 i .(37)
We need to stress that f N L depends on the initial condition of inflation or the total number of e-
folds N t = φ 2 i /(2p)! For p = 2, f N L = 1.84 × 10 5 /N t . If f N L = 100, N t = 1.84 × 10 3 .
The problem is why inflation has such an initial condition. It is very hard to give a physical explanation on it. In this sense, curvaton model can not naturally explain a large non-Gaussian and red tilted primordial power spectrum.
Is it possible that eternal chaotic inflation [23] offers a proper initial condition? During the period of inflation, the evolution of the inflaton field φ is also influenced by quantum fluctuations, which can also be pictured as a random walk of inflaton with a step δφ = H 2π on a horizon scale per Hubble time H −1 . During the same epoch, the variation of the classical homogeneous inflaton field rolling down its potential is ∆φ = |φ| · H −1 . Eternal chaotic inflation happens when δφ = ∆φ, namely
φ = φ E = 12π 2 p 3 /λ 1 p+2(38)
for the chaotic inflation with potential Eq. (23). Naively we take φ E as the initial value of inflaton φ i . Therefore we have
σ 2 * = p p + 2 ,(39)
and the primordial power spectrum generated by curvaton
P ζσ = p + 2 27π 2 p 2 λφ p Nc Ω 2 σ,D .(40)
Considering that λ is bounded from above by (29) and P ζσ = P ζ,obs , we find
ǫ ≥ 9 8 Ω −2 σ,D − 1 2N c .(41)
The slow-roll parameter ǫ must be larger than one because Ω σ,D ≤ 1 and the slow-roll condition is violated. The reason is that the energy scale of eternal chaotic inflation is quite high and curvaton gets a large vacuum expectation value which strongly suppresses the amplitude of primordial power spectrum to be smaller than P ζ,obs . So we conclude that inflation should start at an energy scale lower than eternal inflation scale for the validity of curvaton model.
Before the end of this note, we ignore the initial condition problem and investigate the possible parameter space for curvaton model with a large non-Gaussianity. Following [24], the decay rate of curvaton Γ σ should be greater than the gravitional-strength decay rate m 3 in the unit of Planck scale. Combining with Eq. (14) and (15), we obtain
m ≤ 2 15 f N L σ 2 * .(42)
Using Eq. (29), (33) and (37), after a straightforward calculation we find
σ 2 * ≤ 25p 72N c f −2 N L .(43)
Combining the above two inequalities leads to an upper bound on the curvaton mass m ≤ 7.7 · 10 −4 · p/f N L .
(44)
If f N L ∼ 100, this constraint is roughly the same order as Eq.(31). In order to obtain a large non-Gaussianity the curvaton mass should be smaller than 10 12 GeV in the curvaton model combined with chaotic inflation. On the other hand, the curvaton should decay before neutrino decoupling [24]; otherwise the curvature perturbations may be accompanied by a significant isocurvature neutrino perturbation. The temperature of the universe at the moment of neutrino decoupling is roughly T nd = 1 MeV. The curvaton decay rate is bounded by the Hubble parameter at the time of neutrino decoupling from below, i.e. Γ σ > Γ 0 = 1.68 × 10 −43 in the unit of M p = 1. This requirement leads to a loose lower bound on the curvaton mass m ≥ 2.8 × 10 −37 f 2 N L /p 2 . To summarize, the vacuum expectation value of curvaton is sensitive to the physics in the very early universe. In order to get a red-tilted and large non-Gaussian primordial power spectrum in curvaton model we need to choose a suitable initial condition for inflation. As we know, one of the advantage of inflation is that its observational consequences are independent on the initial condition, which make the predictions of inflation strong. Unfortunately the curvaton model we discussed in this note loses this nice point. We need to figure out a more reasonable model with a red-tilted and large non-Gaussian primordial power spectrum in the future.
AcknowledgmentsWe would like to thank P. Chingangbam and M. Sasaki for useful discussions.
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|
[] |
[
"Multiobjective Combinatorial Optimization with Interactive Evolutionary Algorithms: the case of facility location problems",
"Multiobjective Combinatorial Optimization with Interactive Evolutionary Algorithms: the case of facility location problems"
] |
[
"Maria Barbati \nDepartment of Economics\nUniversitá Cá Foscari Venezia\n30121VeniceItaly\n\nPortsmouth Business School\nCentre of Operations Research and Logistics (CORL)\nUniversity of Portsmouth\nPortsmouthUnited Kingdom\n",
"Salvatore Corrente \nDepartment of Economics and Business\nUniversity of Catania\nCorso Italia\n5595129CataniaItaly\n",
"Salvatore Greco \nDepartment of Economics and Business\nUniversity of Catania\nCorso Italia\n5595129CataniaItaly\n\nPortsmouth Business School\nCentre of Operations Research and Logistics (CORL)\nUniversity of Portsmouth\nPortsmouthUnited Kingdom\n"
] |
[
"Department of Economics\nUniversitá Cá Foscari Venezia\n30121VeniceItaly",
"Portsmouth Business School\nCentre of Operations Research and Logistics (CORL)\nUniversity of Portsmouth\nPortsmouthUnited Kingdom",
"Department of Economics and Business\nUniversity of Catania\nCorso Italia\n5595129CataniaItaly",
"Department of Economics and Business\nUniversity of Catania\nCorso Italia\n5595129CataniaItaly",
"Portsmouth Business School\nCentre of Operations Research and Logistics (CORL)\nUniversity of Portsmouth\nPortsmouthUnited Kingdom"
] |
[] |
We consider multiobjective combinatorial optimization problems handled by means of preference driven efficient heuristics. They look for the most preferred part of the Pareto front on the basis of some preferences expressed by the Decision Maker during the process. In general, what is searched for in this case is the Pareto set of efficient solutions. This is a problem much more difficult than optimizing a single objective function. Moreover, obtaining the Pareto set does not mean that the decision problem is solved since one or some of the solutions have to be chosen. Indeed, to make a decision, it is necessary to determine the most preferred solution in the Pareto set, so that it is also necessary to elicit the preferences of the user. In this perspective, what we are proposing can be seen as the first structured methodology in facility location problems to search optimal solutions taking into account preferences of the user. With this aim, we approach facility location problems using a recently proposed interactive evolutionary multiobjective optimization procedure called NEMO-II-Ch. NEMO-II-Ch is applied to a real world multiobjective location problem with many users and many facilities to be located. Several simulations considering different fictitious users have been performed. The results obtained by NEMO-II-Ch are compared with those got by three algorithms which know the user's true value function that is, instead, unknown to NEMO-II-Ch. They show that in many cases NEMO-II-Ch finds the best subset of locations more quickly than the methods knowing, exactly, the whole user's true preferences.
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[
"https://arxiv.org/pdf/2203.03922v1.pdf"
] | 247,315,378 |
2203.03922
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45918b16f75b736fc982d80d05318d1807f8cee1
|
Multiobjective Combinatorial Optimization with Interactive Evolutionary Algorithms: the case of facility location problems
8 Mar 2022
Maria Barbati
Department of Economics
Universitá Cá Foscari Venezia
30121VeniceItaly
Portsmouth Business School
Centre of Operations Research and Logistics (CORL)
University of Portsmouth
PortsmouthUnited Kingdom
Salvatore Corrente
Department of Economics and Business
University of Catania
Corso Italia
5595129CataniaItaly
Salvatore Greco
Department of Economics and Business
University of Catania
Corso Italia
5595129CataniaItaly
Portsmouth Business School
Centre of Operations Research and Logistics (CORL)
University of Portsmouth
PortsmouthUnited Kingdom
Multiobjective Combinatorial Optimization with Interactive Evolutionary Algorithms: the case of facility location problems
8 Mar 2022Multiobjective OptimizationCombinatorial OptimizationPreferencesNEMOFacility Location problems
We consider multiobjective combinatorial optimization problems handled by means of preference driven efficient heuristics. They look for the most preferred part of the Pareto front on the basis of some preferences expressed by the Decision Maker during the process. In general, what is searched for in this case is the Pareto set of efficient solutions. This is a problem much more difficult than optimizing a single objective function. Moreover, obtaining the Pareto set does not mean that the decision problem is solved since one or some of the solutions have to be chosen. Indeed, to make a decision, it is necessary to determine the most preferred solution in the Pareto set, so that it is also necessary to elicit the preferences of the user. In this perspective, what we are proposing can be seen as the first structured methodology in facility location problems to search optimal solutions taking into account preferences of the user. With this aim, we approach facility location problems using a recently proposed interactive evolutionary multiobjective optimization procedure called NEMO-II-Ch. NEMO-II-Ch is applied to a real world multiobjective location problem with many users and many facilities to be located. Several simulations considering different fictitious users have been performed. The results obtained by NEMO-II-Ch are compared with those got by three algorithms which know the user's true value function that is, instead, unknown to NEMO-II-Ch. They show that in many cases NEMO-II-Ch finds the best subset of locations more quickly than the methods knowing, exactly, the whole user's true preferences.
Introduction
Multiple Objective Combinatorial Optimization (MOCO) problems (for a survey see [28]) are very complex and difficult to be solved. They can be approached with different aims but, in general, one focuses on the computation of all the efficient solutions (see [72] for a discussion on the different concepts of solutions of a MOCO problem). In general, the number of efficient solutions grows exponentially with the size of the problem [28,76]. This, together with the intrinsic complexity related to the "nonsmothness" of the optimization problems, requires a huge computational effort, much greater than that one involved in the resolution of the single objective cases [2]. The high number of efficient solutions and the required very high computational effort are considered the main bottleneck of the MOCO problem [2,17]. These considerations have triggered the development of a certain number of approaches using heuristics that are able to determine an approximation of the whole set of nondominated or efficient solutions involving less computational effort than that one involved in exact algorithms [29]. However, observe that if from a mere theoretical point of view these can be seen as the main critical issues of a MOCO problem, from the point of view of real life applications, there are other difficulties. Indeed, it would be hard to say that a problem has been solved even in case the whole set of efficient solutions has been computed. This set can contain even several thousands of elements so that, finally, the Decision Maker (DM) who should choose one or some of them could feel himself lost [2,17]. Therefore, beyond the technical limitations related to the computational aspects, there are a little more practical questions related to the support given to the DM. From this point of view, the algorithms can take advantage from the integration of preferences expressed by the DM guiding the search to the part of the Pareto front most interesting for him. Considering different moments in which the DM is asked to provide his preferences, in literature one distinguishes between a priori, interactive and a posteriori methods [28]:
• in a priori methods, the preferences of the DM are articulated at the beginning of the process,
• in the interactive methods, the DM expresses his preferences during the search,
• in the a posteriori methods the DM is presented with the set of all efficient solutions that is therefore analyzed w.r.t. his preferences.
On the one hand, the use of a priori methods asks the DM to define at the beginning of the procedure his preferences that are translated by some particular utility function. This assumes that the DM is rational and that his decisions are taken on the basis of some pre-existing preferences that, consequently, have only to be discovered. However, this is not always true since the DM not only is uncertain about his preferences at the beginning of the search but, even more, these a priori preferences are in general absent and have to be constructed during the decision process [65,66]. On the other hand, in the a posteriori methods the DM is often presented with many solutions. This approach has some drawbacks too since:
• the DM has to choose the best solution(s) analyzing the tradeoffs among objectives [17],
• showing the whole set of solutions can cause an information overload on the DM who may have difficulty in selecting the best one(s) [46].
From what we said above, using interactive methods seems the best choice [16,72,79]. Therefore, for MOCO problems, a reasonable approach seems the use of specific heuristics that instead of approximating the whole set of efficient solutions, look for some efficient solutions being the most preferred by the DM. This implies that the heuristics used to explore the feasible set of solutions incorporate few preference information supplied by the DM permitting to drive the search towards some regions of the Pareto front containing the most preferred solutions for the DM. This is possible using some recently proposed heuristics [8] that combine the capacity of a "smart" exploration of the set of feasible solutions (typical of multiobjective optimization oriented heuristics such as NSGA-II [20] or SPEA [84]) with the capacity to build a decision model representing DM's preferences (typical of some Multiple Criteria Decision Aiding (MCDA) approaches such as ordinal regression [38,44]). One example of such composite methodology is given by the recently proposed NEMO-II-Ch algorithm [9] that combines the search procedure of NSGA-II with the preference representation obtained by the nonadditive robust ordinal regression [3]. This approach that drives the search of optimal solutions guided by a preference model incorporating the preferences expressed by the DM, seems us a very promising approach to MOCO problems in real life applications. Indeed, it can give appropriate answers to all the limitations of MOCO problems that we have described:
• it handles the big number of efficient solutions of a MOCO problem by looking only to small subsets of efficient solutions that are well appreciated by the DM,
• it handles the computational effort by using heuristics that have proved to be very effective in complex multiobjective problems,
• it handles the request of a decision support by driving the whole search algorithm by the preferences step by step expressed by the DM in an interactive procedure.
To test the usefulness of such an approach in this paper we consider a typical MOCO problem that is the Facility Location Problem (FLP) [33].
In FLPs we aim to locate a set of facilities in a space optimizing some objective functions and satisfying some constraints. Historically FLPs have been modeled using a mono objective approach in which a single objective function has been adopted. Many have been the contributions in this sense with a multitude of objectives adopted [32] for describing several and very different applications [51]. However, in the real world, DMs deal with several conflicting objectives at the same time so that it is advisable that also the adopted algorithms take into account a multiobjective formulation of the problem at hand [28,67].
The classical approach for choosing the position of a facility consists in describing a function of the distances between the potential users of the facility and the facility itself [31]. The objective function becomes a linear mathematical expression of the distances to optimize. Introducing several constraints, a combinatorial optimization model is built and the optimal solution can be found from the resolution of the model, as described in what is considered the seminal paper by [39]. Therefore the main aim becomes the theoretical development and the description of properties of the models and their solutions [51]. Some reviews gather the basic knowledge on location science as in [60] and in the recent books of [27] and [51].
Multiple Objectives Facility Location Problems (MOFLPs) have captured attention from the researchers especially in the last decade. Many objectives can be used: from the classical distance related objectives, to the environmental and ecological criteria (for a list see [33]). The majority of the methodologies aims to find the whole Pareto front or a part of it implying a considerable computational effort [2]. To this aim several methodologies can be adopted: from exact approaches (e.g. [40,58]) to multiobjective evolutionary algorithms (e.g. [19]) for complex problems.
Behind all these approaches there is the strong assumption that the DM is able to select the alternative that is the best for him which implies that the DM has clear and well defined preferences and is completely rational. In most of the practical problems these assumptions are not very realistic [2,53]. Moreover, very few papers take into account directly the opinion of the DMs. Often the objectives described are derived from considerations related to the specific problem without investigating further the opinion of DMs. Therefore, handling complex MOFLPs with an optimization algorithm guided by DM's preferences seems an interesting approach to be explored and, in this perspective, our contribution can be considered the first structured methodology in MOFLPs to search optimal solutions taking into account preferences of the user. For this reason we propose to deal with MOFLPs by using NEMO-II-Ch. In this way, interactively the DM provides his preferences on some pairs of possible facility locations assignments guiding therefore the search to the part of the Pareto front most interesting w.r.t his preferences and avoiding to loose time in looking for other solutions not matching his expectation. We shall present our proposal considering a case study introduced in [26].
To underline the efficiency of NEMO-II-Ch to MOCO problems, we simulated different user's value functions and we compared the algorithm's performances with the ones of three algorithms, denoted by EA-UVF [9], EA-UVF1 and EA-UVF2, based on the knowledge of the user's true value function. We observed that, quite often, NEMO-II-Ch performs better than the algorithms knowing the user's preferences. To test how the convergence of NEMO-II-Ch to the preferred solution is dependent on the preference information provided by the DM with the related cognitive burden, we considered three different variants asking him to compare one pair of solutions every 5, 10 and 20 generations, respectively. The results proved that asking preference information in a parsimonious way is better than requiring an unrealistic cognitive effort to the user. Therefore, this sheds light on the necessity to carefully study how often the user should be queried with a pairwise comparison of solutions to ensure and speed the convergence of the algorithm.
The paper is structured as follows. In Section 2, an overview of location problems is provided; MCDA and, in particular, NEMO-II-Ch are presented in Section 3; the particular MOLFP to which we applied NEMO-II-Ch is described in Section 4, while the three algorithms based on the complete knowledge of the user's preference with which NEMO-II-Ch is compared are presented in Section 5. The experimental setup and the numerical results are detailed in Section 6; in Section 7 we discuss the obtained results; finally, the last section provides some conclusions together with possible avenues of research.
Review on recent approaches to location problems
According to [32] three types of objectives can be adopted when locating facilities. The mini − max problems, also known as center problems, aim to minimize the maximum distance between a user and its assigned facility [18]. Several variants of the center problems can be identified (see e.g. [11]). For instance, recently, [70] proposed a new formulation to address a situation in which the k-th largest weighted distance between the users and the facilities needs to be minimized. The mini−sum problems minimize the sum of the distances between users and facilities; this objective is well known and much studied and called the median problem [39,47,54]. Among the median problems let us recall the Discrete Ordered Median Problem (DOMP) [23,24,56], where the objective is the minimization of an ordered weighted average of the distances of the users to the facilities. Therefore, in this variant of the problem, each user can be seen as an objective. Lastly, the covering models aim to find solutions in which the maximum number of users is covered, i.e. users are positioned within a given threshold distance from a facility [4,14].
MOFLPs have been generated from these classical location problems optimizing at the same time more objectives. The very first example was proposed by [75] that optimized the median and the center objectives. Indeed, it proposes to use the median together with other objectives. [12] proposed a multiobjective model in which the classical median problem is integrated with a robustness measure that considers potential demand changes. [6] adopted as additional objective the maximization of the distance from the nearest affected region in order to decrease the impact of the facility on the population. On a similar topic, [62] described a model in which the total number of users that are affected by the facility is minimized. [45] modified the median problem in presence of more DMs considering that the evaluation of the distances between users and facilities is different for every DM. Finally, covering objectives are combined with median objectives as in [58].
Minimizing the distances from the facility (e.g. a disposal site) and the users is also defined in [17]. They generate solutions containing one of the two objectives adopted (minimizing the distance from the container) and imposing a threshold distance that counts as dissatisfaction from the users.
In addition to the types of objectives described, equality measures can be adopted as objective functions in FLPs [52]. These measures are often used in combination with an efficient objective (e.g., median) to avoid inefficient solutions very much far from all the users [31]. For example, [59] minimized the sum of the absolute differences, the equality measure, and the sum of squared users-facility distances, either to be minimized or maximized for a desirable or obnoxious facility, respectively. [26] included all the different types of objectives that have been described so far. They model how to choose the location for a given number of casualty collection points in the California State. They adopt five objectives: the median, the center objective, the covering objectives (using two different distance thresholds) and the variance as equality measure.
It can be noted that many models also include location costs that can depend on several parameters for different potential locations as for example construction costs or maintenance costs [50]. Other MOFLPs adopting several and different objectives can be retrieved in the recent survey by [33], often related to the particular case study. They also categorized the MOFLPs on the basis of the methodology developed identifying both exact and heuristic approaches. Beyond that, also several metaheuristics have been applied. Among these, we focus our attention on the evolutionary algorithms [83]. One first group of applications uses NSGA-II [20]. For example, in [80] NSGA-II is adopted for the choice of the location of depots in the Colombian coffee supply network maximizing the cover provided by the depots, minimizing the costs of locating the depots and minimizing the distances from purchasing centers to the depot. Similarly, in [5] the NSGA-II methodology is implemented for the location of warehouses and distribution centers in the supply chain perspective, optimizing cost of locating warehouses and cost of transportation from these. Another case for the location of the warehouses in supply chain is reported in [74]. In addition to that, some specific applications are approached in [22] for the location of public services in high risk tsunami areas or in [42] for the selection of the best raster points in a Geographical Information System. Finally, [61] proposed a generic problem in which the first objective function minimizes total setup cost of facilities while the second one minimizes the total expected traveling and waiting time for the customers.
Other examples of evolutionary algorithms include the application of SPEA2 in [41] for deciding the location of depots that serve a single product type to several customers. Furthermore, the Swarm Optimization has been used as in [82] for approximating the Pareto front in a bi-objective FLP.
While several applications are tackled with evolutionary algorithms, very few examples have been proposed in the literature in which interactive methods have been implemented [29]. In [57], for some generic objective functions of the distances between users and the facility, the DM is asked to indicate some reference levels to be introduced as constraints in the model. Many years later, [46] proposed for the two objectives mini − max and mini − sum an interactive geometrical branch and bound algorithm in which good regions for the location of the facility are selected through the interaction with the DM. In [21] a memetic algorithm integrates DM's preferences. In particular the DM can choose to indicate reference levels for the objectives, or he can provide the upper bound on the objective function levels. The algorithm can be adapted for several MOFLPs. A useful tool to help DMs in the interactive phase can be the use of the Geographical Information System (GIS) to help DMs to visualize the potential solutions as in [1]. Recently, [34] developed a Decision Support System for a bi-objective problem; in the computation phase the lexicographic optima and the ideal point are found, while in the dialog phase the DM can choose the area in which looking for more non dominated solutions, analyzing maps provided in a GIS environment. This process can be repeated until the DM is satisfied of the final position for the facilities [2].
Brief Introduction to MCDA and NEMO-II-Ch
MCDA and the Choquet integral
As observed in the previous sections, the use of evolutionary multiobjective optimization methods permits to solve complex multiobjective optimization problems by using evolutionary algorithms. Anyway, the application of these algorithms will give back to the user a set of potentially optimal solutions that will be well-distributed along the Pareto front. The user is therefore asked to choose among them the best one(s) with respect to his preferences. This choice can be very difficult since, in general, the number of non-dominated solutions is quite big and, therefore, the DM could feel himself uncomfortable in performing it.
In order to avoid this, in recent years the interactive methods have been spread out [7]. Their aim is the inclusion of some preference information from the part of the DM addressing the search to the subset of the Pareto front more interesting for him. In order to do that, MCDA methods are used together with evolutionary algorithms (for an updated state of the art survey on MCDA see [38]).
Given a set of alternatives A = {a, b, . . .} evaluated on a set of n evaluation criteria G = {f 1 , . . . , f n } 1 , MCDA methods deal with ranking, choice and sorting problems. In this case, we will be more interested in ranking and choice problems. In ranking problems, all considered alternatives have to be rank ordered from the best to the worst, while, in choice problems, the best alternative (eventually more than one) has to be chosen, removing all the others. Since the dominance relation 2 stemming from the evaluations of the alternatives on the criteria at hand is too poor, several aggregation methods can be considered. In this paper, we will use as aggregation method the Choquet integral [13] (see [35] for a survey on the use of the Choquet integral in MCDA), a method that can be included under the family of Multiattribute Value Theory (MAVT) [48]. MAVT methods are based on value functions U : A → R such that the greater the value assigned to an alternative a by U, that is U(a), the better a can be considered. In particular, a preference (≻) and an indifference (∼) relations can be defined such that
a ≻ b iff U(a) > U(b), while a ∼ b iff U(a) = U(b).
The most common value function U is the additive one
U(a) = U(f 1 (a), . . . , f n (a)) = n j=1 u j (f j (a)),(1)
where, u j : A → R are non-decreasing functions of the evaluations f j (a) for all f j ∈ G. Moreover, due to its simplicity, the additive value function most used in applications is the weighted sum w j · f j (a) (2) where w j are the weights attached to criteria f j ∈ G such that w j 0 for all f j ∈ G and n j=1 w j = 1. However, the use of an additive value function assumes that the set of criteria is mutually preferentially independent [48,81] even if, in real world applications, this assumption is not always verified. Indeed, the evaluation criteria can present a certain degree of positive or negative interaction. On the one hand, two criteria are positively interacting if the importance assigned to them (together) is greater than the sum of the importance assigned to the two criteria taken alone. On the other hand, two criteria are negatively interacting if the importance assigned to them (together) is lower than the sum of the importance assigned to the two criteria singularly. In literature, interaction between criteria is dealt by using non-additive integrals [35,37] and, among them, the most well known is the Choquet integral. The Choquet integral is based on a capacity, being a set function µ : 2 G → [0, 1] such that the following constraints are satisfied: 1a) µ(∅) = 0 and µ(G) = 1 (normalization), 1 Let us observe that the criteria in MCDA will be the objective functions of the considered multiobjective optimization problem on which the different solutions have to be evaluated. 2 An alternative a dominates an alternative b iff a is at least as good as b for all considered criteria and better for at least one of them. 2a) µ(S) µ(T ) for all S ⊆ T ⊆ G (monotonicity).
Given a ∈ A, the Choquet integral of (f 1 (a), . . . , f n (a)) with respect to µ (in the following, for the sake of simplicity, we shall write "the Choquet integral of a w.r.t. µ") is computed as follows
C µ (a) = C µ (f 1 (a), . . . , f n (a)) = n j=1 f (j) (a) − f (j−1) (a) µ({f i ∈ G : f i (a) f (j) (a)})(3)
where (·) is a permutation of the indices of criteria such that 0
= f (0) (a) f (1) (a) . . . f (n) (a) 3 .
To make things easier, a Möbius transformation of the capacity µ [64,73] and 2-additive capacities [36] are used in practice:
• the Möbius transformation of the capacity µ is a set function m : 2 G → R such that µ(S) = In this case, the Choquet integral of a w.r.t. µ can be written as follows:
C µ (a) = C µ (f 1 (a), . . . , f n (a)) = T ⊆G m(T ) min f j ∈T f j (a);(4)
• a capacity µ is said k-additive if its Möbius transformation m is such that m(T ) = 0 for all T ⊆ G such that |T | > k.
By using the Möbius transformation of the capacity µ and a 2-additive capacity, the Choquet integral can be written in the following linear form
C µ (a) = C µ (f 1 (a), . . . , f n (a)) = f j ∈G m({f j })f j (a) + {f i ,f j }⊆G m({f i , f j }) min{f i (a), f j (a)}(5)
while monotonicity 1b) and normalization constraints 2b) become
1c) m(∅) = 0, f i ∈G m({f i }) + {f i ,f j }⊆G m({f i , f j }) = 1, 2c) m({f i }) 0, for all f i ∈ G, m({f i }) + f j ∈T m({f i , f j }) 0, for all f i ∈ G and for all T ⊆ G \ {f i }, T = ∅.
3 As observed in [9], if some evaluations f j (a) are lower than zero, then it is enough performing a translation f j (a) → f * j (a) = f j (a) + c where c −min fj∈G,a∈A f j (a) for all f j ∈ G and for all a ∈ A.
NEMO-II-Ch
NEMO-II-Ch [9] is an interactive multiobjective optimization method aiming to address the search to the region of the Pareto front most interesting for the DM. The method belongs to the family of NEMO 4 methods [8] which, on the basis of NSGA-II, integrate some preferences provided by the DM during the iterations of the algorithm. The aim is getting points focused in a particular region of the Pareto front avoiding to waste time in surfing through regions not interesting for the DM. At the beginning, the model uses a simple weighted sum (2) as preference function and, if necessary, passes to the 2-additive Choquet integral (5) when the preference function is not able to replicate the preferences provided by the DM. if Time to ask the DM then
5:
Elicit user's preferences by asking DM to compare two randomly selected non-dominated solutions 6: if there is no value function remaining compatible with the user's preferences then 7: if Current preference model = LINEAR then Remove information on pairwise comparisons, starting from the oldest one, until feasibility is restored and reintroduce them in the reverse order as long as feasibility is maintained 11: end if 12: end if 13: Rank solutions into fronts by iteratively identifying all solutions that are most preferred for at least one compatible value function. Rank within each front using crowding distance 14: end if 15: Select solutions for mating 16: Generate offspring using crossover and mutation and add them to the population 17: Rank solutions into fronts by iteratively identifying all solutions that are most preferred for at least one compatible value function. Rank within each front using crowding distance 18: Reduce population size back to initial size by removing worst solutions 19: until Stopping criterion met In the following, we shall describe the different steps in Algorithm 1:
1: As mentioned above, at the beginning a linear value function is used to represent the preferences of the DM; 2: We generate an initial population of solutions and we evaluate them with respect to the considered objective functions; 4-5: If it is time to ask the DM for preference information, we order the solutions in fronts using the dominance relation, exactly as done in NSGA-II. The non-dominated solutions are put in the first front. Once removed from the population, the other non-dominated solutions are put in the second front and so on, until all solutions have been ordered in different fronts. Inside the same front, the solutions are ordered using the crowding distance [20]. The DM is therefore presented with two non-dominated solutions. They are taken in a random way from the first front (if there are at least two solutions in it) or from the following ones having at least two non-dominated solutions. In the extreme case in which there is only one solution for each front (therefore we have a complete order of the solutions), the DM is not presented with any pair of solutions and we can pass to step 15:. Let us suppose that solutions a and b have been chosen to be presented to the DM. He is therefore asked to pairwise compare the two objective functions vectors (f 1 (a), . . . , f n (a)) and
(f 1 (b), . . . , f n (b)) stating if a is preferred to b (a ≻ b), b is preferred to a (b ≻ a) or a and b are indifferent (a ∼ b).
A linear constraint will be used to translate this preference information. In
particular, a ≻ DM b is translated to the constraint U(a) > U(b) and, a ∼ DM b iff U(a) = U(b).
Let us observe that U is the function in (2) if the current preference model is the linear one, while U is the function in (5) if the current preference model is the 2-additive Choquet integral;
6: Checking if there exists at least one value function compatible with the preferences provided by the DM:
-If the current preference model is the linear one (2), then one has to solve the following LP problem:
ε linear DM = max ε subject to U(a) U(b) + ε, if a ≻ DM b, U(a) = U(b), if a ∼ DM b, n j=1 w j = 1, w j 0, for all j = 1, . . . , n. E linear DM Let us observe that one constraint U(a) U(b) + ε should be included for all pairs (a, b) ∈ A × A for which the DM states that a is preferred to b (a ≻ DM b), while one constraint U(a) = U(b) should be included for all pairs (a, b) ∈ A × A for which the DM states that a is indifferent to b (a ∼ DM b). If E linear DM
is feasible and ε linear DM > 0, then there is at least one linear value function compatible with the preferences provided by the DM.
-If the current preference model is the 2-additive Choquet integral in (5), then one has to solve the following problem:
ε Ch DM = max ε subject to C µ (w 1 f 1 (a), . . . , w n f n (a)) C µ (w 1 f 1 (b), . . . , w n f n (b)) + ε, if a ≻ DM b, C µ (w 1 f 1 (a), . . . , w n f n (a)) = C µ (w 1 f 1 (b), . . . , w n f n (b)), if a ∼ DM b, w j 0, for all j = 1, . . . , n, n j=1 w j = 1, m(∅) = 0, and f i ∈G m({f i }) + {f i ,f j }⊆G m({f i , f j }) = 1, m({f j }) 0, for all, j = 1, . . . , n, m({f j }) + f i ∈T m({f i , f j }) 0, for all j = 1, . . . , n, and for all T ⊆ {f 1 , . . . , f n } \ {f j }, T = ∅. E Ch DM
Let us underline that in the set of constraints above, we need to introduce a set of weights (w 1 , . . . , w n ) so that w j 0 and n j=1 w j = 1 since the Choquet integral application implies that all objectives are expressed on the same scale. The set of weights is therefore necessary to put the objectives on the same scale and, for this reason, they become unknown of our model [9]. If E Ch DM is feasible and ε Ch DM > 0, then there is at least one value function, being a 2-additive Choquet integral, compatible with the preferences provided by the DM. Let us observe that the previous problem is not linear anymore and, consequently, we use the Nelder-Mead method [55] to get the set of weights and the Möbius parameters optimizing it. It is a numerical algorithm used to solve non-linear optimization problems that, iteratively, evaluates solutions belonging to a simplex. At each iteration this simplex is transformed and the procedure continues until a stopping criterion is met (see [9] for a description of the application of the method in this context). The non-linearity of the problem comes from the constraints translating the preferences of the DM since, for all a ∈ A,
C µ (w 1 f 1 (a), . . . , w n f n (a)) = n j=1 w j f j (a) · m ({f j }) + {f i ,f j }⊆G m ({f i , f j }) · min{w i f i (a), w j f j (a)} and, consequently, a ≻ DM b is translated into the constraint n j=1 w j f j (a) · m ({f j }) + {f i ,f j }⊆G m ({f i , f j }) · min{w i f i (a), w j f j (a)} n j=1 w j f j (b) · m ({f j }) + {f i ,f j }⊆G m ({f i , f j }) · min{w i f i (b), w j f j (b)}.
Let us underline that in the programming problems above, the strict inequalities have been converted into weak inequalities by using an auxiliary variable ε which maximization is the objective of our problems. For example, the strict inequality U(a) > U(b) has been converted into the weak inequality U(a) U(b) + ε; 7-10: If there is not any model compatible with the preferences provided by the DM, we have to distinguish the case in which the current preference model is the linear one from the case in which the current preference model is the 2-additive Choquet integral. In the first case, since there does not exist any linear value function able to replicate the preferences of the DM, we increase the complexity of the model passing to the 2-additive Choquet integral. Having more degrees of freedom, it is more flexible and, therefore, it can better adapt itself to the preferences of the DM. In the second case, if we already passed to the 2-additive Choquet integral but there is not any model (therefore weights and Möbius parameters) compatible with the preferences of the DM, we remove some pieces of this preference information starting from the oldest one until the feasibility is restored. Let us observe that the removal of a piece of preference information should be performed only if the DM agrees on it. This is a relevant aspect since the DM could be very convinced about a certain comparison and, consequently, he doesn't want to remove it;
13: In order to use the information gathered until now from the DM and, consequently, to address the search to the most interesting region of the Pareto front, we shall order the solutions in fronts in a different way than before. For each solution x in the current population (we shall denote by A the current set of solutions), we have to check if there is at least one compatible function such that x is strictly preferred to all other solutions in A. Again, we have to distinguish two cases:
-If the current preference model is the linear one, the following LP problem has to be solved:
ε linear x = max ε subject to, U(x) U(a) + ε, for all a ∈ A \ {x}, E linear DM . E linear x If E linear
x is feasible and ε linear x > 0, then x is put in the first front.
-If, instead, the current preference model is the 2-additive Choquet integral preference model, then the following programming problem has to be solved:
ε Ch x = max ε subject to, C µ (w 1 f 1 (x), . . . , w n f n (x)) C µ (w 1 f 1 (a), . . . , w n f n (a)) + ε, for all a ∈ A \ {x}, E Ch DM . E Ch x If E Ch x is feasible and ε Ch x > 0, then x is put in the first front.
Once the first front has been built, all solutions contained in it are removed from the current population and the same procedure is used with the remaining solutions to build the second front. We shall continue in this way until all solutions have been ordered in different fronts. Inside the same front, solutions are ordered using the crowding distance.
In the rare case in which there is not any solution that can be preferred to the others for any compatible model, all solutions are retained equally preferable and, therefore, they are put in the same front;
15-18: The usual evolution of the population is performed by using the selection, crossover and mutation operators together with the ordering of the population described above;
3-19: Repeat steps 4-18 until the stopping condition has been met.
Using Interactive Evolutionary Multiobjective Optimization in location problems: a case study
We test our approach on a well known multiobjective location problem introduced in [25] and later in [26]. The problem, considered as a reference in its domain, consists in choosing the location of a given number p of facilities among a set of potential locations optimizing five different classical objective functions for FLPs. More in detail, the facilities are Casualty Collection Points (CCPs) to which people can go if they need help in case disasters have happened. These centers should operate where a huge amount of people need to be provided with emergency service. In [25] a comparison of the different objectives is proposed and also a first multiobjective version, including only three objectives, is formulated; whereas in [26] a multiobjective heuristic has been introduced adopting the five objective functions described later. The problem is of particular interest among the MOFLPs because at least one mini − max objective is selected, one for the mini − sum, and one equality measure are optimized.
We define:
• I = {1, . . . , q}: the set of demand points,
• L = {1, .
. . , m}: the set of potential locations for the facilities,
• d ij : the distance between demand point i and potential facility j,
• pop i : the population at the demand point i,
• p: the total number of facilities to locate,
• P ⊆ L: a vector of p selected facilities in L,
• D i (P ): the distance from a demand point i to the closest facility in P ,
D i (P ) = min k∈P {d ik }.
We consider five objectives:
1. The median objective, minimizes the sum of the distances between the demand points and the closest facility [39,63]:
min P f 1 (P ) = min P 1 q q i=1 D i (P ) , 2.
The maximum distance objective, minimizes the distance of the farthest demand point [39]:
min P f 2 (P ) = min P {max i {D i (P )}},
3. The maximum covering objectives, maximize the population inside two different distance thresholds S 1 and S 2 [14]: max
P f 3 (P ) = max P i: D i (P ) S 1 pop i , max P f 4 (P ) = max P i: D i (P ) S 2 pop i .
4.
The minimum variance objective, balances the distances between demand points and the closest facility, minimizing the variance of the closest distances for all the demand points [52]:
min P f 5 (P ) = min P q i=1 [D i (P ) − f 1 (P )] 2 q .
The case study has I = {1, . . . , 577} demand points and L = {1, . . . , 141} potential sites for the facilities located in the Orange County in California, an area where the careful planning for the location of CCPs represents an essential requirement due to frequent earthquakes. The data, that include coordinates and associated weights for the demand points and coordinates for the potential facilities, are available upon request to the authors of [26].
Let us point out that this is just one of the possible examples that our methodology can handle. Our approach is very flexible and we could adopt many different objective functions.
Algorithms used for the comparison
As already observed above, the use of a heuristic not taking into account preferences of the DM, such as NSGA-II, gives back the user with a set of non-dominated vectors of p-facilities from which he has to choose the best with respect to his preferences. For this reason, we proposed to apply NEMO-II-Ch to address the search not to the entire Pareto front but to the most interesting part for the user. We shall consider the full size problem in which the 141 different locations will be taken into account choosing the best p among them with p = 4, 5. Moreover, we will simulate different users' value functions. On the one hand, we will show that, in most of the cases, NEMO-II-Ch is able to find the best subset of p locations for the user by asking few preference information. On the other hand, to test its performances, we will compare them to the performances of three algorithms, denoted by EA-UVF, EA-UVF1 and EA-UVF2. These are based on the knowledge of the user's true value function that is, instead, unknown to the NEMO-II-Ch algorithm. While the EA-UVF algorithm has been presented in [9], its two variants, namely EA-UVF1 and EA-UVF2, are presented in this paper for the first time. The three algorithms are briefly presented in the following sections.
EA-UVF: Evolutionary Algorithm based on User's Value Function
This algorithm has been presented in [9] and its main steps, which are listed in Algorithm 2 are detailed in the following lines: Select solutions for mating 6: Generate offspring using crossover and mutation and add them to the population 7:
Rank the solutions into fronts with respect to their true value 8: Reduce population size back to initial size by removing worst solutions Select solutions for mating 5: Generate offspring using crossover and mutation and add them to the population 6: Rank solutions into fronts by dominance and inside each front order them using their true value 7:
Reduce population size back to initial size by removing worst solutions 8: until Stopping criterion met 4-7: Evolve the population; 3-8: Repeat steps 4-7 until the stopping condition has not been met.
The EA-UVF1 implements exactly the NSGA-II method with the replacement of the crowding distance used to diversify solutions inside the same front with the value assigned to the solutions by the user's true value function.
EA-UVF2: NSGA-II with a roulette wheel driven by User's Value Function
The steps of the EA-UVF2 algorithm are shown in Algorithm 4 and detailed in the following lines: Assign a probability to be parent to each solutions by using their true value 4: Select solutions for mating 5: Generate offspring using crossover and mutation and add them to the population 6: Rank solutions into fronts by dominance and inside each front order them by the crowding distance 7: Reduce population size back to initial size by removing worst solutions 8: until Stopping criterion met 1: Generate an initial population of solutions and evaluate them with respect to the considered objective functions;
3: A probability to be parent of the next generation is assigned to each solution in the population. This probability, denoted by P rob(P ), is computed as
P rob(P ) = U(P ) P ∈P OP U(P ) if U has to be maximized,(6)P rob(P ) = 1 U (P ) P ∈P OP 1 U(P )
if U has to be minimized (7) and P OP denotes the current population of solutions; 4-7: Evolve the population; 2-8: Repeat steps 3-7 until the stopping condition has not been met.
The EA-UVF2 algorithm implements, therefore, all steps of the NSGA-II method, while the user's true value function is used to assign a probability to be parent of the next generation to each solution. The better the value assigned by the user's true value function to a solution, the higher its probability to become parent of the next generation.
Let us conclude this section by underlining that the EA-UVF represents the ideal situation in which the algorithm knows exactly how the user chooses among two whichever solutions and, consequently, it has the maximal theoretical availability of preference information. At the same time, the EA-UVF1 and the EA-UVF2 use this information, on the one hand, to select solutions within non-dominated fronts the generated population and, on the other hand, to decide which solutions are the best to be parents of the next generation. However, all of them use the whole preference information that the DM could theoretically provide by preferentially ranking all solutions at all iterations of the evolutionary algorithm. Of course, in real life the DM can not be able to provide all these preferences because of the unrealistic huge cognitive burden related to the request of so many preference comparisons at each iteration. For this reason, a methodology being much more parsimonious in asking preferences to the DM is requested for any real world application. To study the amount of preference information necessary to get reasonably acceptable solutions, we investigate the relation between, on the one hand, the frequency of asking preferences to the user and, on the other hand, the quality of results and the speed of the algorithms' convergence. To this aim, in the following simulations, we run NEMO-II-Ch asking the DM one preference every 5, 10 and 20 generations, respectively.
Experimental setup and numerical results
The parameters and the technical details used in the simulations are the following: • The mating selection is performed by tournament selection in all methods apart from EA-UVF2 where it is performed by a roulette wheel selection:
-Tournament selection: Let us denote by P 1 , . . . , P 30 the solutions in the current population. To each solution P s is associated the front it belongs to (F s ). Moreover, in all methods each solution is associated with a second score. In NEMO-II-Ch and in EA-UVF2 this second score is the crowding distance (CD s ) 5 , while, in EA-UVF1 the second score is the true value. We create a random permutation of the solutions in the population denoted by P (1) , . . . , P (30) . Then, a tournament is performed between P s and P (s) for each s = 1, . . . , 30, to choose which solution has to be selected as parent of the next generation. The tournament is won from the solution being in the lowest front (P s iff F s < F (s) or P (s) iff F (s) < F s ) or, if they belong to the same front (F s = F (s) ), from the solution having the greatest second score. If P s and P (s) belong to the same front and they have the same second score, the winner is chosen randomly. 30 tournaments will therefore be performed and, consequently, 30 solutions will become parents of the next generation. Denoting by P ′ s the winner of the tournament between P s and P (s) , the pairs of parents which will generate the offsprings of the next generation are, therefore, (P ) have to be chosen, for each k = 1, . . . , 30, a solution is sampled in a random way from the probability distribution given by eq. (6) if the user's true value function U has to be maximized or by eq. (7) if the same function as instead to be minimized; the sampled solution becomes, therefore, the parent P ′ k of the next generation;
• Each pair of parents generate two offspring by one-point crossover with probability of 1 and random resetting mutation 6 with probability of 1 p [30]; in particular, since each solution can contain a certain location at most once, the one-point crossover has to be slightly modified if the two considered solutions have some common locations. In this case, the common potential location(s) are inherited by both offspring, while the one-point crossover is performed on the two vectors composed by uncommon potential locations for both parents. For example, let us suppose that the two parents solutions are (10,15,21,30) and (6,10,20,50). In this case, the potential location labeled by 10 is present in both parents and, therefore, it is inherited by the two offspring. The remaining vectors composed of uncommon locations are (15,21,30) and (6,20,50). To these two vectors the one-point crossover is applied exchanging the two tails. Supposing that the cut point is the second integer, exchanging the two tails we obtain the vectors (15,21,50) and (6,20,30). The two offspring will therefore be the vectors (10,15,21,50) and (6,10,20,30). Let us underline that the evolution of the population is performed in such a way that if a new offspring is exactly the same as another solution in the current population, it is "killed". Therefore, it is not possible having multiple copies of the same solutions in the population;
• Considering the set L of potential locations and a solution P composed of p of these potential locations, we assumed the following different forms of user's preferences described as follows:
U D ) the maximal deviation from the optimal objective values [26] is computed as follows
U D (P ) = max k∈{1,...,5} {∆ k (P )} where ∆ k (P ) = f k (P )−f * k f * k , if the objective f k is to be minimized, f * k −f k (P ) f * k ,
if the objective f k is to be maximized, that is, f * k is the optimal value for the objective f k , k = 1, . . . , 5; a solution P is preferred to a solution P ′ if U D (P ) < U D (P ′ ); U D v ) On the basis of the U D defined above, we considered the function U D v computed as follows: U N ) the value is computed as follows 1, 0.15, 0.2, 0.25, 0.3), and a solution P is preferred to a solution P ′ if U N (P ) < U N (P ′ );
U D v (P ) = max k∈v {∆ k (P )} where v ∈ {{1,U N (P ) = 5 k=1 w k · f k (P ) where f k (P ) = f k (P )−f min k f max k −f min k , if the objective f k is to be minimized, f max k −f k (P ) f max k −f min k , if the objective f k is to be maximized, w = (0.U N v ) the value is computed as follows U N v (P ) = k∈v w ′ k · f k (P )
where w ′ = (0.
if U N v (P ) < U N v (P ′ ).
For all considered user's value functions, the best subset of p locations is P b ⊆ L, such that |P b | = p and U(P b ) = min
P ⊆L: |P |=p U(P ) where U ∈ {U D , U D v , U N , U N v };
• All algorithms are run for a maximum of 1,000 generations. In particular, for NEMO-II-Ch we asked the user to provide one preference comparison every 5, 10 and 20 generations. The resulting algorithms are therefore denoted by NIICh 5, NIICh 10 and NIICh 20. All the algorithms stop as soon as P b is present in the current population or when the maximum number of generations has been reached.
After we described the setup of the simulations, let us present the results of the application of the compared methods to the considered full-size problem. This means that we shall check for the best subset of p locations, with p = 4, 5, among the 141 taken into account. Of course, this problem is quite difficult since the possible subsets of p locations from which the best has to be discovered are = 432, 295, 143, respectively. Therefore, we would like to prove that the method is able to deal with big-size problems in which a huge number of solutions is involved. We performed 50 independent runs (changing, therefore, the starting population), and we applied the three NEMO-II-Ch variants (NIICh 5, NIICh 10 and NIICh 20) as well as the three algorithms knowing the user's true value function (EA-UVF, EA-UVF1 and EA-UVF2).
In the tables below, to present the results of the performed simulations, we used the following notation: 7 Let us observe that in the computation of U N v (P ) the functions f k have a weight increasing with k. For example,
if v = {1, 3, 4, 5}, then, U N v (P ) = 0.1 · f 1 (P ) + 0.2 · f 3 (P ) + 0.3 · f 4 (P ) + 0.4 · f 5 (P ).
• #SR: number of runs (over the 50 considered), in which the algorithm was able to discover the best subset P b of possible locations;
• M#G: mean number of generations necessary to the algorithm to discover P b ;
• S#G: standard deviation of the number of generations necessary to the algorithm to discover P b ;
• A#P : mean number of pairwise comparisons asked to the user necessary to discover P b . We did not include this data for EA-UVF and EA-UVF1 since they are only used as benchmark and a comparison between the number of pairwise comparisons asked from the NEMO-II-Ch versions and the one involved in the application of both algorithms is meaningless. Of course, the number of times the user is queried by NEMO-II-Ch is only a small portion of the number of times the user has to provide a pairwise comparison in the two algorithms. Just to give an example, let us underline that in EA-UVF and EA-UVF1, where solutions are ranked with respect to the user's true value function, to rank order p solutions it is necessary to perform p(p−1) 2 pairwise comparisons 8 . This means that to rank order 30 solutions in the population, the user has to provide 435 pairwise comparisons in a single iteration and, as will be clear in the next section, this number is much higher than the number of pairwise comparisons asked from the three NEMO-II-Ch versions in whichever considered test problem. With respect to EA-UVF2, the user is not asked to provide any pairwise comparison. However, the algorithm can never be applied in practice since it is assumed that the user is able to assign a utility to each solution, utility that needs to be used to implement the roulette wheel selection described above. Of course, this is not realistic at all;
• S#P : standard deviation of the number of pairwise comparisons asked to the user necessary to discover P b ;
• MT : mean time necessary to the algorithm to discover P b ; all simulations have been performed using the commercial software MATLAB2019 but on different PCs. For each method and each user's value function, the 50 runs have been performed on the same machine. In the tables presenting the results, we reported the characteristics of the PCs used to perform the different simulations;
• ST : standard deviation of the time necessary to the algorithm to discover P b 9 ;
• A BRSD: this is the average distance of the best solution in the final population from the optimal solution P b .The distance, denoted by BRSD(U), is computed only for the simulations in which the algorithm was not able to discover P b (in the case in which the algorithm is able to discover P b the distance is zero). Denoting by P Best the best solution in the final population, following [77], BRSD(U) is computed as
BRSD(U) = |U(P Best ) − U(P b )| U(P b ) .(8)
The less BRSD(U), the better the performance of the algorithm. The value A BRSD is then obtained by averaging BRSD(U) over the number of runs in which the algorithm was not able to discover P b .
6.1. Comparison with EA-UVF, EA-UVF1 and EA-UVF2 In Tables 1-4 we reported the results of the application of the three versions of NEMO-II-Ch as well as those obtained by the three algorithms knowing the user's true value function. We have Table 2: Results for functions U N and U N v considering p = 5. All simulations have been performed with four different PCs which characteristics and labels are the following: (PC1) intel core i7 3.6GHz; (PC2) intel core i5 2.5GHz; (PC3) intel core i7 2.7GHz; (PC4) intel core i7 1.9GHz. Table 3: Results for functions U D and U D v considering p = 4. All simulations have been performed with four different PCs which characteristics and labels are the following: (PC1) intel core i7 3.6GHz; (PC2) intel core i5 2.5GHz; (PC3) intel core i7 2.7GHz; (PC4) intel core i7 1.9GHz. Table 4: Results for functions U D and U D v considering p = 5. All simulations have been performed with four different PCs which characteristics and labels are the following: (PC1) intel core i7 3.6GHz; (PC2) intel core i5 2.5GHz; (PC3) intel core i7 2.7GHz; (PC4) intel core i7 1.9GHz. considered the twelve different user's true value functions defined in the previous section and the cases p = 4 and p = 5 for the number of best locations to be discovered. In the following, by (U, p) we denote the case in which the user's true value function is U and the number of best locations is p. The following can be observed:
• U N , 4 and U N v , 4 :
-Convergence: the three variants of NEMO-II-Ch as well as EA-UVF and EA-UVF1 are always able to find the best solution in the 50 runs. This is not the case for EA-UVF2 that, with respect to U N is not able to converge in one of the 50 runs, while, with respect to U N 1245 , quite surprisingly, it is able to find the best subset of 4 locations only in 5 of the 50 runs;
-Convergence speed: As can be observed from the data in Table 1, apart from the U N case in which the EA-UVF1 converges more quickly (in terms of number of generations necessary to find P b ) than all the other algorithms, the EA-UVF is the quickest among the considered algorithms. As to the comparison between the three NEMO-II-Ch variants, in average, NIICh 5 converges more quickly than NIICh 10 in four of the six considered cases, while NIICh 20 is always the slowest. However, as already observed before, the number of pairwise comparisons asked from EA-UVF and EA-UVF1 is tremendously higher than the one involved in whichever NEMO-II-Ch version. For this reason, it is more meaningful giving a more in depth analysis of the NEMO-II-Ch variants to understand if and how the number of times the user is queried with a pairwise comparison affects the convergence speed of the algorithm. It can be observed that the lowest number of pairwise comparison is asked in correspondence of NIICh 20, followed by NIICh 10 and, then, by NIICh 5 (see values in italics). This means that not only NIICh 20 is efficient in finding P b but it is able to find it asking very few pairwise comparisons to the user;
-Distance from P b : Considering EA-UVF2 and assuming that the best solution in the final population is the optimal one, the user makes an error, in average, of the 40.6% in the U N case, and of the 23.3% in the U N 1245 one;
• U N , 5 and U N v , 5 :
-Convergence: The three variants on NEMO-II-Ch are able to find P b in all considered runs for all test problems apart from the case U N 1235 in which NIICh 10 and NIICh 20 are not always able to find P b . In particular, NIICh 10 does not find the best subset of five locations in one of the 50 runs, while NIICh 20 does not find the same subset of best locations in 3 out of the 50 runs. As to the three algorithms knowing the user's true value functions, EA-UVF1 is always able to find the best subset of five locations, while this is not true for the other two. In particular, EA-UVF does not find P b in five of the fifty runs in the U N 1235 case, while EA-UVF2 has its best performances when U N is considered (49/50) and its worst one in the case U N 1235 is the user's true value function (11/50). This suggests that using the user's true value function to assign a probability to become parent of the next generation is worse than using the same function to rank the solutions belonging to the same front;
-Convergence speed: As in the p = 4 cases, it results that NIICh 20 is the quickest among the three NEMO-II-Ch versions to reach P b since, it asks to the user to provide almost half of the pairwise comparisons asked by NIICh 10 and almost one third of the pairwise comparisons asked by NIICh 5.
Regarding EA-UVF and its two variants, once again EA-UVF2 is the worst among them. Moreover, we would like to underline that the number of generations necessary to get P b is lower for EA-UVF1 than for EA-UVF. In particular, it is meaningful observing that the number of pairwise comparisons asked to the user by EA-UVF1 is not greater than the number of times the user is queried with a pairwise comparison in the EA-UVF. In fact, the application of the EA-UVF1 implies the same number of pairwise comparisons of EA-UVF only in case all solutions are non-dominated and, therefore, they are in one nondominated front only. This suggests once again that a parsimonious preference information is beneficial for the convergence of the algorithms to P b ;
-Distance from P b : In the U N 1235 case, in average, the error done in assuming that the best solution in the last population is the optimal one is almost 7% for NIICh 10, NIICh 20 and EA-UVF, while it is 21.3% for the EA-UVF2. An higher error is also done by EA-UVF2 in the U N 1234 , U N 1245 and U N 2345 cases. • U D , 4 and U D v , 4 :
-Convergence: The three variants of NEMO-II-Ch are always able to find P b in all considered runs. This is not the case for the three algorithms knowing the user's true value function. In particular, EA-UVF1 finds in all 50 runs the best subset of four locations for all user's true value functions apart from U D 1245 in which it finds P b in 39 of the 50 runs; the EA-UVF never finds the best subset of locations in all runs. The same holds for EA-UVF2 that in the U D 1234 and U D 1245 cases finds P b 49 and 47 times, respectively. Considering all other user's true value functions, it is able to find the best subset of four locations more or less half of the times;
-Convergence speed: NIICh 20 is confirmed as the best among the three variants of the NEMO-II-Ch since it asks a lower number of pairwise comparisons than the other two maintaining always the best possible convergence since, as observed in the previous item, it is always able to find P b . Comparing NIICh 10 and NIICh 5, the first is better than the second in terms of number of pairwise comparisons asked to the DM;
-Distance from P b : Assuming that the best solution in the last population is optimal, one makes an error ranging from 7.9% to 40.8% considering the EA-UVF, from 7.4% to 25.8% considering the EA-UVF2 and of the 7.8% considering the EA-UVF1.
• U D , 5 and U D v , 5 :
-Convergence: For all considered cases, one of the three variants of NEMO-II-Ch finds P b more often than the algorithms based on the knowledge of the user's true value function. Even more, in all cases the worst among the three NEMO-II-Ch variants performs at least as well as all three algorithms knowing the user's true value function in terms of number of runs in which it converges to P b ;
-Convergence speed: Looking at the average number of pairwise comparisons asked to the user, once more we have the confirmation that NIICh 20 is the best among the three variants of NEMO-II-Ch since it finds P b asking less pairwise comparisons than NIICh 5 and NIICh 10. However, differently from the previous cases, the doubt is now related to the fact that NIICh 20 is not able to find the best subset of locations as frequently as NIICh 5 and NIICh 10 and, therefore, it could be better to ask more pairwise comparisons to increase the probability to converge to the best solution.
-Distance from P b : Comparing the three versions of NEMO-II-Ch one can observe that, apart from U D 1235 and U D 1245 cases, NIICh 5 presents the best A BRSD. In particular, the maximum average error is equal to 6.7% for NIICh 5, while it is 9.3% for NIICh 10 and even 10.4% for NIICh 20. The situation is even worse for the three algorithms based on the full knowledge of the user's true value function since, apart from the U D 1245 case in which the average error done assuming as optimal solution the best solution in the final population is 1.9% considering EA-UVF1 and 4.6% considering EA-UVF, in all the other cases, this average error is at least equal to 9.8% with a pick of 51.8% done by EA-UVF2 in the U D 1234 case. This means that, in the case in which the EA-UVF algorithm and the other two variants are not able to find P b , they are very far from the area of the Pareto front most interesting with respect to the user's preferences.
To evaluate the significance of the data provided above we performed the Mann-Whitney U test with 5% significance level [43] to two different indicators:
1. considering BRSD of each of the six algorithms in each of the 50 considered runs, 2. considering the number of pairwise comparisons asked to the user in each run for algorithms NIICh 5, NIICh 10 and NIICh 20.
Regarding the BRSD, we performed the test only for problems where at least one algorithm did not converge in at least one of the 50 runs. Indeed, if all methods had converged to the optimal solution in all runs, the BRSD would be always equal to 0 and, consequently, the comparison between the algorithms would be absolutely meaningless. Regarding the number of pairwise comparisons asked to the user, we performed the test on the NIICh 5, NIICh 10 and NIICh 20 only, since the number of pairwise comparisons asked to the user in EA-UVF, EA-UVF1 and EA-UVF2 is only virtual due to the unrealistic applicability of the algorithms. In particular, in the case in which the algorithm did not converge to the optimal solution, for that run, we considered the maximum number of pairwise comparisons asked to the user being 200 for NIICh 5, 100 for NIICh 10 and 50 for NIICh 20 since each of them asks one pairwise comparison every 5, 10 and 20 generations, respectively, and the maximum number of admitted generations is 1,000.
In the supplementary material we included the results of the two tests. For brevity, we report here just the tables for the U D , 5 case obtained performing the Mann-Whitney U test with 5% significance level on the BRSD (Table 5) and on the number of pairwise comparisons asked to the user (Table 6). In both tables, we give the p-value together with the difference between the A BRSD of each ordered pair of algorithms in Table 5 and the difference between A#P of each ordered pair of algorithms in Table 6. Red values represent significant values considering the performed test. Table 5: Mann-Whitney U test with 5% significance level performed on BRSD for the U D , 5 . In the table the p-value is provided as well as the difference between the A BRSD of each ordered pair of algorithms. In red the significant values. In Table 5 one can observe that the difference in the BRSD between the NEMO variants is not significant, while the difference between the BRSD of each NEMO variant and each of the algorithms based on the knowledge of the user's true value function is significant apart from the comparison between NIICh 20 and EA-UVF1 for which the difference between their BRSD is not significant Table 6: Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in algorithms NIICh 5, NIICh 10 and NIICh 20 for the U D , 5 case. In the table the p-value is provided as well as the difference between A#P of each ordered pair of algorithms. In red the significant values.
NIICh 10
0.0105 (46.12−28.8) for the Mann-Whitney U test. This means that, on the one hand, the NEMO-II-Ch variants can be considered equivalent, while each of them is better than the three algorithms knowing the user's true value function. On the other hand, one can conclude that EA-UVF1 is better than EA-UVF that, in turn, is better than EA-UVF2.
Going to the data in Table 6, one can see that the difference between the distributions of the number of pairwise comparisons asked to the user in each pair of NEMO variants is significant. This means that the number of pairwise comparisons asked to the user by NIICh 20 to converge to the optimal solution is retained significantly smaller than the one involved in NIICh 10 and NIICh 5 and, consequently, with respect to the required preference information, NIICh 20 is better than NIICh 10 that, in turn, is better than NIICh 5.
Similar conclusions can be gathered looking at all the other tables included in the supplementary material. Once again they confirm that the difference in the BRSD between the NEMO variants and the algorithm based on the user's true value function is considered significant and that with respect to the three NEMO variants, the difference between the number of pairwise comparisons asked to the user from each algorithm is significant. The last fact proves that asking to the user a lower number of information does not affect, in general, the capacity of NEMO-II-Ch to converge to the optimal solution.
Discussion
In this paper we faced Multiobjective Combinatorial Optimization (MOCO) problems by using Interactive Evolutionary Multiobjective Optimization (IEMO) methods. In particular, we applied an IEMO method, namely NEMO-II-Ch, to Facility Location Problems (FLPs). When facility location is considered from a multiobjective perspective, that is the different facility location options are evaluated simultaneously by several conflicting aspects, the problem becomes quite difficult and the user is presented with a huge number of Pareto optimal solutions from which he is asked to choose the best w.r.t. his preferences. Anyway, this set can be composed of thousands and even millions of different possible facility locations options and, therefore, no user can cope with such a type of problem and in a reasonable time. Moreover, the user has not clear defined preferences at the beginning of the search process, so that, the already difficult multiobjective optimization problem, is coupled with a not less complex problem of preference elicitation. In fact, what we are proposing can be considered as the first structured methodology in FLPs to search optimal solutions taking into account preferences of the user. For these reasons, we proposed to apply NEMO-II-Ch that is a state of the art algorithm permitting to conjugate the efficiency of evolutionary multiobjective optimization methods with the parsimonious user's preferences elicitation of most advanced multiple criteria decision aiding models, with the aim of searching the best solution w.r.t. the preferences of the user in the part of the Pareto front most appealing for him. In this way, focusing on a single region of the Pareto front, the algorithm reaches the best option in a limited number of generations avoiding to lose time in searching solutions in parts of the Pareto front not really interesting for the user.
To prove the efficiency of the method to this setting, we considered a classical FLP very wellknown in literature [26] based on the most typical objective functions adopted in the domain. We performed different simulations running NEMO-II-Ch and comparing its performances with those of other three algorithms, namely EA-UVF, EA-UVF1 and EA-UVF2, based on the knowledge of the user's true value function that is, instead, unknown to NEMO-II-Ch. In the comparison we tested twelve different types of users' value functions and two different values for the number of facilities p that need to be located (p = 4 and p = 5). Moreover, to investigate how the number of comparisons asked to the user influences the convergence of the algorithm, we considered three different versions of the NEMO-II-Ch method, namely NIICh 5, NIICh 10 and NIICh 20, asking the user to compare one pair of non-dominated solutions every 5, 10 and 20 generations, respectively.
The results obtained should be read as an answer to the question: "is there any methodological tool to handle real world multiobjective facility problems"? The considered problem is very complex for the following reasons:
1. There is a plurality of objectives to be optimized, 2. Some of these objectives are quite complex in itself (this is, in particular, the case of f 5 (P ) [26]), 3. The preferences of the user have to be considered, 4. The preference information has to be collected maintaining tolerable the cognitive burden for the DM, 5. The computation time should be acceptable for real world operational applications.
The data of the conducted simulations allow to conclude that the proposed methodology answers positively the research question. Indeed, NEMO-II-Ch is able to find the optimal solution in most of the considered FLPs taking into account the preferences of the user, without requiring too much preference information and involving a computational time definitely admissible. The performances of the "control" procedures having full preference information, that is, EA-UVF, EA-UVF1 and EA-UVF2, can be used to measure the task complexity in the sense that the worse their performances, the more complex the task. In particular, let us evaluate the complexity of the problem taking into account EA-UVF1, having the best performances between the three algorithms knowing the user's true value function. Considering as object of the performance the number of runs in which the optimal solution was obtained, that is indicator #SR, we can see that almost always NIICh 20 obtains performances at least as good EA-UVF1 and sometimes even better. More precisely NEMO-II-Ch is obtaining better performances for the cases Instead, there is only one case in which EA-UVF1 is performing better than NIICh 20 with respect to the number of runs in which the optimal solution is found, that is U N 1235 , 5 (NIICh 20 found the optimal solution in 47 runs, while EA-UVF1 in all 50 runs). In all other cases, both algorithms where able to find the optimal solution in all runs.
Considering the number of pairwise comparison requested by NEMO-II-Ch we have to conclude that it is definitely acceptable. Indeed,in terms of comparison the pairwise comparisons requested by one of the most well-known and most adopted MCDA method, that is AHP [69]. Let us consider the didactic example presented in [68] in which three schools (alternatives) are evaluated with respect to six different aspects (criteria). In terms of FLPs, it would be a really easy problem that will concern the selection of a single facility among three potential locations to optimize six different objectives. Since the DM must provide a pairwise comparison in terms of a qualitative judgment on a nine point scale for each non-ordered pair of criteria and a comparison for each non-ordered pair of alternatives with respect to each criterion, the decision maker has to provide 6 2 + 6 3 2 = 15 + 6 · 3 = 33 pairwise comparisons in total. This means that in a didactic example of, probably, the most adopted MCDA method [78], the DM is asked to give 33 pairwise comparisons. Looking again at the performances of NIICh 20, one can see that with a single exception, in all our cases the algorithm was able to find the best solution with a number of pairwise comparison much smaller than 33. Observe also that very often the required average number of pairwise comparison asked to the user by NIICh 20 is lower than 15 (in 17 out of 24 considered cases). In addition, observe that while the pairwise comparisons of AHP require to give an evaluation on a nine point scale, the pairwise comparisons considered in NEMO-II-Ch require simply to say which solution is the preferred among two. To have a more fair comparison between the judgments required by AHP and the information required by NEMO-II-Ch, consider that for each pair of items α and β being alternatives (α, β ∈ A) or criteria (α, β ∈ G) AHP requires, in fact, two comparisons: the first related to which one between α and β has the greatest priority and the second, expressed on the nine point scale, related to how much greater is the priority of the item with the greatest priority with respect to the other. In general, it seems reasonable that the second comparison of AHP (the one on the nine point scale) is more demanding than the pairwise comparison of NEMO-II-Ch related to which solution is the preferred among two. Consequently, to each pairwise comparison asked from AHP on a pair of non-ordered items one should assign a cognitive burden at least double with respect to the pairwise comparison required by NEMO-II-Ch. In conclusion, we can say that, in average, NEMO-II-Ch can handle a quite challenging problem with a complexity comparable to that one of the most demanding real world problems asking the user a cognitive burden much smaller than the one required by the most adopted MCDA method in a very didactic example. Coming to the computational time, even considering the case taking more time, NIICh 20 is almost always (apart from one case only) achieving the optimal solution, in average, in less than three hours and, very often, in less than one hour (quite frequently in the U N and U N v cases in some minutes). This seems a very reasonable running time for a so complex problem. Observe also that our simulations were performed with a non dedicated programming language and with computer machine commonly available on the market as will be further underlined below.
Beyond the specific interest for the multiobjective facility location problems, the results we obtained are relevant also from the general point of view of the multiobjective optimization algorithms. In fact, the procedure that has been proposed can be seen as a parsimonious exploration of the space of solutions and of the DM's preferences. The parsimony of the multiobjective optimization procedure we have applied can be decomposed in two components:
• a component related to the optimization procedure: it is based on the evaluations of combinations of most promising solutions maintaining a certain level of diversification typical of the evolutionary algorithms,
• a component related to the preference learning procedure: it is based on a "dynamical" induction of the DM's utility function on the basis of few preference comparisons, typical of the ordinal regression approach [44] that is properly applied in an "incremental" version adding time by time preferences related to new solutions discovered by the optimization algorithm.
The parsimony of the optimization algorithm permits to select the most promising directions in the search of the optimal solution avoiding to be trapped in some local optimum, while the parsimony of the preference learning algorithm permits to escape from a possible overfitting originated by an excess of preference information that could bring to an inappropriate generalization at global level of preferences that hold only at local level. With respect to the parsimony of the preference learning algorithm, let us point out that the obtained results have an autonomous interest. Indeed, the comparisons of the results obtained by NIICh 20 with the three algorithms based on a complete knowledge of the DM's preferences, proves that a quite limited use of preference information gives better results than the use of the complete preference information. In this sense, it is particularly meaningful that, in general, the best performances are obtained by EA-UVF1. Indeed, while EA-UVF, based only on the complete and perfect preference information, is operating as a single objective evolutionary algorithm without taking into account diversity, and EA-UVF2, beyond considering the multiobjective nature of the problem through dominance front ranking, is based on the maintenance of the diversity by means of crowding distance, EA-UVF1 is obtained as a "mutiobjectivization" of EA-UVF building dominance front ranking without any consideration of the diversification. This means that, according to a concordant literature in the evolutionary optimization domain [49,71], the multiobjecivization of the optimization problem is beneficial, while the diversification, in general, does not give any contribution to improve the performances of the algorithm. In particular, taking into consideration the number of runs in which the best solution was discovered, observe that, on the one hand, EA-UVF2 is performing better than EA-UVF only in two cases ( U D 1234 , 4 and U D 1245 , 4 ), while EA-UVF is performing better than EA-UVF2 in 17 cases and they have the same performances on the remaining 5 cases. On the other hand, EA-UVF1 is performing better than EA-UVF in 12 cases, while EA-UVF is performing better than EA-UVF1 only in one case ( U D 1245 , 5 with 5 successful runs for EA-UVF and 4 successful runs for EA-UVF1). Observe that if the multiobjectivization of the EA-UVF1 algorithm permits to improve the performance of EA-UVF, however, it is not enough to attain the same efficiency of NIICh 20. In fact, taking into consideration the number of runs in which the optimal solution was discovered, NIICh 20 is able to obtain better results than EA-UVF1 in 6 cases, while EA-UVF1 is able to perform better than NIICh 20 in one case only. We believe that this can be interpreted in the sense that the parsimony in the required preference information of NIICh 20 permits to obtain better performances of an algorithm using the whole preference information as EA-UVF1. In conclusion, the results we obtained on the multiobjective facility location problem seems to suggest that in very complex combinatorial optimization problems a smart approach based on an evolutionary algorithm and a limited elicitation of the DM's preference information can be an appropriate approach. Of course, this hypothesis needs to be tested on other multiobjective combinatorial problems and, more in general, on other complex multiobjective problems (not necessarily combinatorial), to obtain a more precise and definitive confirmation.
Conclusions
We considered a very complex problem resulting from the combination of two other complex problems already quite challenging in themselves. The combination of the two problems highly exacerbates the difficulty. The two problems are the facility location problem and the search of optimal solutions in multiobjective decision problems taking into account the user's preferences. In this perspective, the research question of the paper is: "is it possible to give an adequate answer, especially taking into account real world applications, to the so complex problem resulting from the combination of the above-mentioned problems?" Technically the answer to the problem is obtained from the application of a state-of-the-art multiobjective optimization procedure to the standard formulation of multiobjective facility location problem. The contribution of the paper is in handling the question and in providing a surprisingly very positive answer: the two complex problems can be solved together with a reasonable cognitive burden (comparable and even smaller than the cognitive burden required from didactic examples of the most adopted MCDA methods) and with reasonable computational times (especially considering the use of non-specialised programming languages and the computation on common laptops daily used). Beyond the application of the presented methodology to other complex multiobjective combinatorial optimization problems in order to collect further evidence on its effectiveness and reliability in so complex decision problems, the following possible avenues of research can be underlined:
• research should be addressed on studying how often the user should be asked to provide preference information to speed the convergence of the algorithm and how techniques investigating which solutions should be presented to the user to maximize the learning capabilities of the algorithm [10,15] could improve the same convergence;
• to make applicable to big size real world problems, a better implementation of NEMO-II-Ch should be provided. Indeed, analyzing in detail the computational time necessary to run the algorithm, it is evident that almost 93% of the time is taken by the execution of the Nelder-Mead method. Of course, implementing other methods to solve non-linear optimization problems could speed the algorithm and, therefore, making it more applicable in practice,
• on the basis of the good results obtained by NEMO-II-Ch applied to location problems, we think that it could be interesting applying it to other classical combinatorial optimization problems that can be formulated in a multiobjective perspective such as the one presented in [22] and [40]. In this section we performed the Mann-Whitney U test with 5% significance level on the BRSD as defined by eq. (8) of the paper and as recalled in the following:
BRSD(U) = |U(P Best ) − U(P b )| U(P b ) .
Let us remind that P b is the optimal solution to be discovered by each algorithm, while P Best is the best solution in the last population obtained when the algorithm stopped. Of course, if the algorithm in the considered run converged to the optimal solution, then, BRSD(U) = 0. In each table, we provided the p-value as well as the difference between the average BRSD, that is, A BRSD, of each ordered pair of algorithms. Red values in each table represent the fact that the difference between the A BRSD of the two algorithms is significant for the performed test. For example, looking at the data in Table 1 and considering the U N 1235 , on the one hand, the p-value in the comparison between NIICh 5 and EA-UVF is 0.0229 and, therefore, the difference between the A BRSD of the two algorithms (0 and 0.0004) is not significant. On the other hand, the p-value between NIICh 5 and EA-UVF is 0.0229 and, therefore, the difference between the A BRSD of the two algorithms (0 and 0.0007) is significant. Moreover, we performed the considered test only for the problems in which at least one of the six considered algorithms did not converge for at least one of the 50 runs because, otherwise, a whole table of NaN would be obtained. Results are reported in Tables 1-4. 2. Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user In this section we performed the Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in NIICh 5, NIICh 10 and NIICh 20. Of course, for the nature of EA-UVF, EA-UVF1 and EA-UVF2, it is not possible to give a similar number of pairwise comparisons. For the cases in which the algorithm did not converge, we assume that the user has to provide the maximum number of pairwise comparisons being 200 for NIICh 5, 100 for NIICh 10 and 50 for NIICh 20 since each of the three methods asks a pairwise comparison every 5, 10 and 20 generations, respectively and the maximum number of admitted generations for each algorithm is 1,000. As in the Table 1: Mann-Whitney U test with 5% significance level performed on BRSD for functions U N and U N v considering p = 4. In the table the p-value is provided as well as the difference between the A BRSD of each ordered pair of algorithms. In red the significant values. previous case, in each table we provided the p-value as well as the difference between the A#P (the average number of pairwise comparisons asked to the user) of each algorithm. Red values represent the fact that the difference between the A#P of the two algorithms is significant for the performed test. For example, looking at the data in Table 5 and considering U N , the p-value in the comparison between NIICh 5 and NIICh 10 is 2.2610 −5 and, therefore, the difference between the A#P of the two algorithms (16.8 and 8.6) is significant. As one can see in all tables, all values are significant. Results are reported in Tables 5-8. Table 2: Mann-Whitney U test with 5% significance level performed on BRSD for functions U N and U N v considering p = 5. In the table the p-value is provided as well as the difference between the A BRSD of each ordered pair of algorithms. In red the significant values. Table 3: Mann-Whitney U test with 5% significance level performed on BRSD for functions U D and U D v considering p = 4. In the table the p-value is provided as well as the difference between the A BRSD of each ordered pair of algorithms. In red the significant values. Table 4: Mann-Whitney U test with 5% significance level performed on BRSD for functions U D and U D v considering p = 5. In the table the p-value is provided as well as the difference between the A BRSD of each ordered pair of algorithms. In red the significant values. Table 6: Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in algorithms NIICh 5, NIICh 10 and NIICh 20 for functions U N and U N v considering p = 5. In the table the p-value is provided as well as the difference between A#P of each ordered pair of algorithms. In red the significant values. Table 7: Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in algorithms NIICh 5, NIICh 10 and NIICh 20 for functions U D and U D v considering p = 4. In the table the p-value is provided as well as the difference between A#P of each ordered pair of algorithms. In red the significant values. Table 8: Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in algorithms NIICh 5, NIICh 10 and NIICh 20 for functions U D and U D v considering p = 5. In the table the p-value is provided as well as the difference between A#P of each ordered pair of algorithms. In red the significant values.
U
(a) = U(f 1 (a), . . . , f n (a)) = n j=1
T
⊆S m(T ) for all S ⊆ G (conversely, m(S) = T ⊆S (−1) |S−T | µ(T ) for all S ⊆ G) and constraints 1a) and 2a) are replaced by the following ones: for all f j ∈ G and for all S ⊆ G \ {f j }, T ⊆S m(T ∪ {f j }) 0.
Algorithm 2
2Evolutionary Algorithm User's Value Function (EA-UVF) algorithm 1: Generate initial population of solutions and evaluate them 2: Compute the utility of each solution by the user's true value function 3: Rank the solutions into fronts with respect to their true value 4: repeat 5:
9: until Stopping criterion met 1: Generate an initial population of solutions and evaluate them with respect to the considered objective functions; 2: Compute the utility of each solution by using the user's true value function; 3: Rank the solutions into fronts by using the values assigned to them from the user's true value function and computed at the previous step. The solution having the best utility value (the minimum [maximum] value if the user's true value function has to be minimized [maximized]) is put in the first front; the solution having the second best utility value is put in the second front and so on until the solution having the worst utility value that is included in the last front. Solutions having the same utility value are included in the same front; 5:8 Evolve the population; 4-9: Repeat steps 5-8 until the stopping condition has not been met. 5.2. EA-UVF1: NSGA-II with diversification replaced by User's Value Function The steps of the EA-UVF1 algorithm are shown in Algorithm 3 and detailed in the following lines: 1: Generate an initial population of solutions and evaluate them with respect to the considered objective functions; 2: Rank solutions in non-dominated fronts. Then, inside each front, compute the true value of all solutions and rank them by these utility values;
Algorithm 4
4NSGA-II with a roulette wheel driven by User's Value Function (EA-UVF2) 1: Generate initial population of solutions and evaluate them 2: repeat 3:
•
The population P OP is composed of 30 solutions where each solution is a vector P of p different integer values taken in the interval [1, m];
PP
⊆L: |P |=p f k (P ), if the objective f k is to be minimized, ⊆L: |P |=p f k (P ), if the objective f k is to be maximized,
NIICh 10 (PC2) NIICh 20 (PC2) EA-UVF EA-UVF1 EA-UVF2
NIICh 10 (PC3) NIICh 20 (PC1) EA-UVF EA-UVF1 EA-UVF2
NIICh 10 (PC2) NIICh 20 (PC3) EA-UVF EA-UVF1 EA-UVF2
• U D 1245 , 4 (
4NIICh 20 found the solution in all the 50 runs, while EA-UVF1 in 39 runs), • U D , 5 (NIICh 20 found the solution in 40 runs, while EA-UVF1 in 33 runs), • U D 1235 , 5 (NIICh 20 found the solution in 40 runs, while EA-UVF1 in 30 runs), • U D 1245 , 5 (NIICh 20 found the solution in 9 runs, while EA-UVF1 in 4 runs), • U D 1345 , 5 (NIICh 20 found the solution in 39 runs, while EA-UVF1 in 27 runs), • U D 2345 , 5 , (NIICh 20 found the solution in 40 runs, while EA-UVF1 in 33 runs).
09·10 −5 (27.56−15.7) 2.30·10 −13 (27.56−8.72) NIICh 10 2.70·10 −9 (15.7−8.72)
Also in this case we consider a subset composed of four of the five objective functions and a solution P is preferred to a solution P ′1, 0.2, 0.3, 0.4) and v ∈ {{1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}} 7 .
Table 1 :
1Results for functions U N and U N v considering p = 4. All simulations have been performed with four different PCs which characteristics and labels are the following: (PC1) intel core i7 3.6GHz; (PC2) intel core i5 2.5GHz; (PC3) intel core i7 2.7GHz; (PC4) intel core i7 1.9GHz.U N
NIICh 5 (PC3) NIICh 10 (PC3) NIICh 20 (PC3) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
49/50
M#G
80.92
80.44
113.16
79.7
77.54
152.78
S#G
55.28
46.99
76.45
50.34
49.16
137.43
A#P
16.80
8.60
6.16
S#P
11.04
4.69
3.83
MT
51.71s
36.4s
46.87s
ST
43.99s
24.29s
33.16s
A BRSD
0.406
U N
1234
NIICh 5 (PC3) NIICh 10 (PC2) NIICh 20 (PC2) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
50/50
M#G
73.98
74.04
85.06
65.6
65.90
134.54
S#G
61.49
40.43
72.10
51.15
49.1
153.43
A#P
15.42
7.98
4.78
S#P
12.27
4.00
3.65
MT
57.05s
1.85m
56.79s
ST
1.13m
1.41m
52.01s
U N
1235
NIICh 5 (PC2) NIICh 10 (PC2) NIICh 20 (PC2) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
50/50
M#G
140.88
138.02
152.5
86.66
116.64
225.54
S#G
100.30
106.56
109.93
68.12
95.84
196.90
A#P
28.76
14.34
8.06
S#P
20.05
10.64
5.58
MT
3.70m
2.83m
2.16m
ST
4.04m
2.98m
2.05m
U N
1245
NIICh 5 (PC3) NIICh 10 (PC4) NIICh 20 (PC1) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
5/50
M#G
122.66
143.56
189.36
89.86
117.34
117.2
S#G
61.13
88.49
103.44
77.42
73.69
63.14
A#P
25.12
14.92
10.02
S#P
12.24
8.81
5.20
MT
2.1m
1.9m
2.16m
ST
1.47m
1.9m
1.5m
A BRSD
0.233
U N
1345
NIICh 5 (PC4) NIICh 10 (PC4) NIICh 20 (PC4) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
50/50
M#G
76.92
82.86
112.22
63.08
76.48
162.52
S#G
51.36
48.87
79.03
40.21
60.42
150.01
A#P
16.08
8.88
6.14
S#P
10.26
4.90
3.99
MT
1.25m
59.62s
1.14m
ST
1.07m
43.73s
52.14s
U N
2345
NIICh 5 (PC4) NIICh 10 (PC4) NIICh 20 (PC4) EA-UVF EA-UVF1 EA-UVF2
#SR
50/50
50/50
50/50
50/50
50/50
50/50
M#G
77.46
83.02
108.2
68.86
84.56
119.76
S#G
53.44
51.36
72.09
45.84
62.00
97.92
A#P
16.16
8.86
5.94
S#P
10.67
5.12
3.61
MT
1.26m
59.47s
1.04m
ST
1.18m
48.9s
43.31s
Supplementary material to the paper: Multiobjective Combinatorial Optimization with Interactive Evolutionary Algorithms: the case of facility location problems Maria Barbati b , Salvatore Corrente a , Salvatore Greco a,b a Department of Economics and Business, University of Catania, Corso Italia, 55, 95129 Catania, Italy b Portsmouth Business School, Centre of Operations Research and Logistics (CORL), University of Portsmouth, Portsmouth, United Kingdom 1. Mann-Whitney U test with 5% significance level on BRSD
Table 5 :
5Mann-Whitney U test with 5% significance level on the number of pairwise comparisons asked to the user in algorithms NIICh 5, NIICh 10 and NIICh 20 for functions U N and U N v considering p = 4. In the table the p-value is provided as well as the difference between A#P of each ordered pair of algorithms. In red the significant values.2.72·10 −5 (16.16−8.86) 1.95·10 −9 (16.16−5.94)U N
NIICh 5 NIICh 10 NIICh 20
NIICh 5
2.26·10 −5
(16.8−8.6)
1.04·10 −8
(16.8−6.16)
NIICh 10
0.0047
(8.6−6.16)
U N
1234
NIICh 5 NIICh 10 NIICh 20
NIICh 5
0.0002
(15.42−7.98)
2.80·10 −10
(15.42−4.78)
NIICh 10
9.04·10 −6
(7.98−4.78)
U N
1235
NIICh 5 NIICh 10 NIICh 20
NIICh 5
6.11·10 −6
(28.76−14.34)
2.42·10 −12
(28.76−8.06)
NIICh 10
0.0002
(14.34−8.06)
U N
1245
NIICh 5 NIICh 10 NIICh 20
NIICh 5
1.51·10 −6
(25.12−14.92)
2.68·10 −11
(25.12−10.02)
NIICh 10
0.0012
(14.92−10.02)
U N
1345
NIICh 5 NIICh 10 NIICh 20
NIICh 5
0.0001
(16.08−8.88)
2.17·10 −8
(16.08−6.14)
NIICh 10
0.0025
(8.88−6.14)
U N
2345
NIICh 5 NIICh 10 NIICh 20
NIICh 5
NIICh 10
0.0010
(8.86−5.94)
2.62·10 −7 (37.8−21.48) 1.04·10 −11 (37.8−15.66) NIICh 10 1.71·10 −19 (97.58−48.08)U D
NIICh 5 NIICh 10 NIICh 20
NIICh 5
1.82·10 −5
(98.68−46.12)
2.12·10 −9
(98.68−28.8)
NIICh 10
0.0105
(46.12−28.8)
U D
1234
NIICh 5 NIICh 10 NIICh 20
NIICh 5
NIICh 10
0.0029
(21.48−15.66)
U D
1235
NIICh 5 NIICh 10 NIICh 20
NIICh 5
2.63·10 −5
(97.04−46.02)
2.12·10 −9
(97.04−28.8)
NIICh 10
0.0105
(46.02−28.8)
U D
1245
NIICh 5 NIICh 10 NIICh 20
NIICh 5
1.15·10 −13
(181.7−97.58)
9.77·10 −18
(181.7−48.08)
U D
1345
NIICh 5 NIICh 10 NIICh 20
NIICh 5
7.64·10 −5
(96.98−48.1)
4.63·10 −10
(96.98−27.92)
NIICh 10
0.0012
(48.1−27.92)
U D
2345
NIICh 5 NIICh 10 NIICh 20
NIICh 5
1.82·10 −5
(98.68−46.12)
2.12·10 −9
(98.68−28.8)
NIICh 10
0.0105
(46.12−28.8)
NEMO: Necessary preference enhanced Evolutionary Multiobjective Optimizer
Citing[20], the crowding distance is ..."the average distance of two points on either side of a particular solution along each of the objectives" and it is computed to maintain the diversification of the population. The higher the crowding distance of a solution P s , the more isolated is the solution in the considered population.
"...in each position independently, with probability p m , a new value is chosen at random from the set of permissible values"[30]
The best solution is found after p − 1 comparisons, the second after p − 2 comparisons and so on.9 The mean and the standard deviation are computed for the runs in which P b is discovered.
AcknowledgementsThe authors are grateful to Professor Tammy Drezner for making available the data concerning the real world problem. The second and the third authors wish to acknowledge the support of theNIICh 20NaN NaN
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|
[] |
[
"Charged Dilatonic AdS Black Branes in Arbitrary Dimensions",
"Charged Dilatonic AdS Black Branes in Arbitrary Dimensions"
] |
[
"Per Berglund [email protected] \nDepartment of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA\n",
"Jishnu Bhattacharyya [email protected] \nDepartment of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA\n",
"David Mattingly \nDepartment of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA\n"
] |
[
"Department of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA",
"Department of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA",
"Department of Physics\nUniversity of New Hampshire\n03824DurhamNHUSA"
] |
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We study electromagnetically charged dilatonic black brane solutions in arbitrary dimensions with flat transverse spaces, that are asymptotically AdS. This class of solutions includes spacetimes which possess a bulk region where the metric is approximately invariant under Lifshitz scalings. Given fixed asymptotic boundary conditions, we analyze how the behavior of the bulk up to the horizon varies with the charges and derive the extremality conditions for these spacetimes. arXiv:1107.3096v2 [hep-th] 17 Oct 2012 Contents 1 Introduction 1 2 The action and the equations of motion 4 3 The global Lifshitz and the Lifshitz-Schwarzschild solutions 7 4 Asymptotically AdS solutions 9 5 Near horizon analysis 11 5.1 The non-extremal solution 13 5.2 The extremal solution 18 5.3 The near horizon Lifshitz-Schwarzschild solution 21 6 Thermodynamics 25 7 Summary and conclusions 26 A Restrictions on the dilatonic potential for the existence of exact global Lifshitz solutions 28 B The Hawking-Horowitz Mass 31with λ being an arbitrary positive constant. Now, the metric (1.1) is isometric under the scale transformations (1.2), in addition to being trivially invariant under constant translations of t and x i1 . In particular, for z > 1 we have different scalings of the space and time which is usually the case with non-relativistic fixed points 2 . Therefore, based on our general intuition about AdS/CFT, we expect the metric (1.1) to holographically capture physics of non-relativistic fixed points. Spacetimes with a metric of the form (1.1) are called global Lifshitz spacetimes. Subsequently, with "non-relativistic holography" in mind, global Lifshitz and Lifshitz-Schwarzschild spacetimes as exact solutions of Einstein-Maxwell-dilatonic gravity with negative cosmological constant in four and higher dimensions were found[25][26][27][28]. Note that the Lifshitz-Schwarzschild solution has been called "Lifshitz black brane" in the recent literature. We call it the Lifshitz-Schwarzschild solution instead, in order to avoid certain confusions (see below). Our nomenclature is justified in section 3.The gravitational systems considered in[25][26][27]3 are very closely related to the one considered in the classic work of [3], in that, they all have an additional negative cosmological constant term in the action. The existence of global Lifshitz and Lifshitz-Schwarzschild solutions in such systems clearly make them candidate holographic duals of non-relativistically scale invariant quantum field theories. However, the presence of the negative cosmological constant in the action indicates that such systems should also posses asymptotically AdS solutions[26,29,30]4 . Moreover,[26]argued that the near horizon geometry of such an asymptotically AdS black brane, at extremality, should possess Lifshitz like behaviour 5 . As we will also see, the near horizon geometry of a certain subclass of the general non-extremal asymptotically AdS solutions is Lifshitz-Schwarzschild-like. We refer to such asymptotically AdS solutions with Lifshitz/Lifshitz-Schwarzschild like behaviour near the horizon, as the near horizon Lifshitz black brane and the near horizon Lifshitz-Schwarzschild black brane, respectively. The AdS completions of the near horizon Lifshitz/Lifshitz-Schwarzschild solutions have already been found in[26,29,30]. Note, in this regard, that the near horizon Lifshitz black brane solution has exact scaling symmetry 1 The metric (1.1) does not have boost invariance however. For the construction of holographic duals of quantum field theories with non-relativistic boost invariance in addition to scale invariance of the above type, see [23], [24]. 2 When z = 1 (1.1) actually describes the Poincaré patch of (n + 2)-dimensional AdS (AdSn+2). We then have additional isometries of the metric corresponding to the Lorentz boosts and the special conformal transformations of the dual relativistic CFT. 3 Note that [25] considers other kinds of gravitational systems too. 4 In this work, by AdS we will always mean the Poincaré patch of AdS and not the global AdS spacetime. 5 This is consistent with our general understanding of charged extremal black hole solutions, e.g., asymptotically flat/AdS extremal Reissner-Nordström black holes, extremal D3, M2, M5 branes etc. In all these examples the respective Einstein's equations are solved exactly by the corresponding (extremal) near horizon solutions.
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10.1007/jhep08(2012)042
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"https://arxiv.org/pdf/1107.3096v2.pdf"
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1107.3096
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7d0981a0b0df16dadb5874bf07551485614310d6
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Charged Dilatonic AdS Black Branes in Arbitrary Dimensions
17 Oct 2012
Per Berglund [email protected]
Department of Physics
University of New Hampshire
03824DurhamNHUSA
Jishnu Bhattacharyya [email protected]
Department of Physics
University of New Hampshire
03824DurhamNHUSA
David Mattingly
Department of Physics
University of New Hampshire
03824DurhamNHUSA
Charged Dilatonic AdS Black Branes in Arbitrary Dimensions
17 Oct 2012Prepared for submission to JHEP
We study electromagnetically charged dilatonic black brane solutions in arbitrary dimensions with flat transverse spaces, that are asymptotically AdS. This class of solutions includes spacetimes which possess a bulk region where the metric is approximately invariant under Lifshitz scalings. Given fixed asymptotic boundary conditions, we analyze how the behavior of the bulk up to the horizon varies with the charges and derive the extremality conditions for these spacetimes. arXiv:1107.3096v2 [hep-th] 17 Oct 2012 Contents 1 Introduction 1 2 The action and the equations of motion 4 3 The global Lifshitz and the Lifshitz-Schwarzschild solutions 7 4 Asymptotically AdS solutions 9 5 Near horizon analysis 11 5.1 The non-extremal solution 13 5.2 The extremal solution 18 5.3 The near horizon Lifshitz-Schwarzschild solution 21 6 Thermodynamics 25 7 Summary and conclusions 26 A Restrictions on the dilatonic potential for the existence of exact global Lifshitz solutions 28 B The Hawking-Horowitz Mass 31with λ being an arbitrary positive constant. Now, the metric (1.1) is isometric under the scale transformations (1.2), in addition to being trivially invariant under constant translations of t and x i1 . In particular, for z > 1 we have different scalings of the space and time which is usually the case with non-relativistic fixed points 2 . Therefore, based on our general intuition about AdS/CFT, we expect the metric (1.1) to holographically capture physics of non-relativistic fixed points. Spacetimes with a metric of the form (1.1) are called global Lifshitz spacetimes. Subsequently, with "non-relativistic holography" in mind, global Lifshitz and Lifshitz-Schwarzschild spacetimes as exact solutions of Einstein-Maxwell-dilatonic gravity with negative cosmological constant in four and higher dimensions were found[25][26][27][28]. Note that the Lifshitz-Schwarzschild solution has been called "Lifshitz black brane" in the recent literature. We call it the Lifshitz-Schwarzschild solution instead, in order to avoid certain confusions (see below). Our nomenclature is justified in section 3.The gravitational systems considered in[25][26][27]3 are very closely related to the one considered in the classic work of [3], in that, they all have an additional negative cosmological constant term in the action. The existence of global Lifshitz and Lifshitz-Schwarzschild solutions in such systems clearly make them candidate holographic duals of non-relativistically scale invariant quantum field theories. However, the presence of the negative cosmological constant in the action indicates that such systems should also posses asymptotically AdS solutions[26,29,30]4 . Moreover,[26]argued that the near horizon geometry of such an asymptotically AdS black brane, at extremality, should possess Lifshitz like behaviour 5 . As we will also see, the near horizon geometry of a certain subclass of the general non-extremal asymptotically AdS solutions is Lifshitz-Schwarzschild-like. We refer to such asymptotically AdS solutions with Lifshitz/Lifshitz-Schwarzschild like behaviour near the horizon, as the near horizon Lifshitz black brane and the near horizon Lifshitz-Schwarzschild black brane, respectively. The AdS completions of the near horizon Lifshitz/Lifshitz-Schwarzschild solutions have already been found in[26,29,30]. Note, in this regard, that the near horizon Lifshitz black brane solution has exact scaling symmetry 1 The metric (1.1) does not have boost invariance however. For the construction of holographic duals of quantum field theories with non-relativistic boost invariance in addition to scale invariance of the above type, see [23], [24]. 2 When z = 1 (1.1) actually describes the Poincaré patch of (n + 2)-dimensional AdS (AdSn+2). We then have additional isometries of the metric corresponding to the Lorentz boosts and the special conformal transformations of the dual relativistic CFT. 3 Note that [25] considers other kinds of gravitational systems too. 4 In this work, by AdS we will always mean the Poincaré patch of AdS and not the global AdS spacetime. 5 This is consistent with our general understanding of charged extremal black hole solutions, e.g., asymptotically flat/AdS extremal Reissner-Nordström black holes, extremal D3, M2, M5 branes etc. In all these examples the respective Einstein's equations are solved exactly by the corresponding (extremal) near horizon solutions.
Introduction
Dilatonic black hole/brane solutions in (super)gravity have been studied extensively in the past, see e.g., [1][2][3][4][5][6][7]. There has also been a good deal of work on dilatonic black holes/branes in asymptotically (anti-)de Sitter spaces; see e.g., [8][9][10][11][12][13]. More recently, electromagnetically charged dilatonic black holes/branes have found applications in an offshoot of the AdS/CFT correspondence [14][15][16] (for reviews see [17,18]), the so-called AdS/condensed matter field theory or AdS/CMT correspondence; for some reviews on AdS/CMT, see [19][20][21].
The idea that spacetimes in general relativity with a metric of the form ds 2 = −(r/r 0 ) 2z dt 2 + (r 0 /r) 2 dr 2 + (r/r 0 ) 2
n i=1 (dx i ) 2 (1.1)
can holographically describe non-relativistic fixed points of quantum field theories was put forward in [22]; here r 0 is a fixed length scale, while the constant z 1, known as the scaling exponent, depends on the non-relativistic fixed point in question.
To appreciate the proposed correspondence, consider a system near a non-relativistic fixed point, with a mass gap ∆. The energy scale ∆ of such a system is related to the correlation length ξ through ∆ −1 ∼ ξ z . As the critical point is approached, the correlation length diverges, the mass gap vanishes and the system has an effective scale invariance under
x i → λx i ; r → r λ ; t → λ z t , (1.2) under (1.2) near the horizon. Since we call the so-called "Lifshitz black brane" solution as the Lifshitz-Schwarzschild solution in this paper, there should hopefully be no confusion henceforth.
In the current paper, we expand upon the recent work [26,29,30] on finding asymptotically AdS black brane solutions to Einstein-Maxwell-dilaton gravity with negative cosmological constant in arbitrary dimensions 6 . There are several reason why such solutions are interesting to study: first of all, as already mentioned above, the gravitational system which admits these solutions are simple generalizations of those studied in [3]. It is thus important to investigate the nature of the asymptotically AdS counterparts the asymptotically flat dilatonic black hole solution of [3]. It is also useful to compare these solutions with the well known AdS-Reissner-Nordström black brane solutions, admitted by the same gravitational system when the dilatonic interaction is switched off; we will make some comments on the similarities and the differences among these two classes of solutions below. Finally, since the solutions we are going to discuss are all electrically charged black branes with AdS asymptotics, by the AdS/CFT correspondence they are gravitational duals of (some currently unknown) relativistic CFTs at finite temperature and chemical potential. In particular, the radial evolution of the near horizon Lifshitz black brane solution describes a renormalization group flow between a relativistic conformal fixed point in the UV and a Lifshitz-like fixed point in the IR. In contrast, the near horizon geometry of the (n + 2)-dimensional extremal AdS-Reissner-Nordström black brane solution is AdS 2 × R n . Thus, the radial evolution of the extremal AdS-Reissner-Nordström solution also describes a holographic renormalization group flow between two fixed points, but the IR fixed point in this case is very different from the Lifshitz-like one. In other words, the dilatonic black brane solutions allow us to obtain new kinds of (holographic) renormalization group flows. Embedding the global Lifshitz solution into an asymptotically AdS solution then allows us to interpret the IR fixed point as having an emergent scale invariance just like in the AdS 2 × R n case.
Our central result is the general non-extremal, static, asymptotically AdS black nbrane solution, admitted by Einstein-Maxwell-dilatonic gravity with negative cosmological constant in (n + 2)-dimensions, with n 2. The solution has a regular horizon and is characterized by three asymptotic charges, namely, the mass, the electric charge and the dilatonic charge. In contrast to their asymptotically flat cousins [3], the dilatonic charge in our case is not a function of the electric charge and the mass, but is a free parameter in itself. We also derive the appropriate extremality conditions satisfied by this general non-extremal solution. Quite interestingly, the extremality condition, which is related to the vanishing of the surface gravity at the horizon, takes the same form as that in [3]. These extremality conditions furthermore allow us to identify the appropriate limits in 6 Certain generalizations of these gravitational systems have also appeared in the literature. Among these, [31][32][33] have cosidered replacing the pure cosmological constant term with a (or a sum of) Liouville potential(s), and have also considered even higher dimensional generalizations involving higher rank form fields, such that the previous generalizations appear as Kaluza-Klein reductions of the latter. Another possible generalization involving multiple U (1) fields has been considered in [34]. Note that these references do not necessarily deal with asymptotically AdS solutions.
which the general non-extremal solution reduces to the near horizon Lifshitz-Schwarzschild black brane and the near horizon Lifshitz black brane solutions, respectively. In particular, we are able to verify explicitly that the near horizon Lifshitz black brane solution is indeed the extremal solution of the system. To help the reader keep track of the relevant solutions, table 1 in section 7 provides a summary of all the asymptotically AdS solutions in terms of the appropriate asymptotic charges.
In addition, we study the thermodynamics of these solutions. Since our solutions are asymptotically AdS, such relations also describe the thermodynamics of the dual CFT via holography. Comparing the situation with [3], we notice that unlike the discontinuous drop in the temperature of an asymptotically flat dilatonic black hole in the extremal limit, the temperature of the non-extremal black brane in our case smoothly goes to zero as extremality is approached. Furthermore, the area of the horizon in the case of the extremal (near horizon Lifshitz) solution vanishes, and therefore so does the Bekenstein-Hawking entropy of the brane. This is, however, in stark contrast with the case of the extremal AdS-Reissner-Nordström (near horizon AdS 2 × R n ) solution, where the entropy is finite. In other words, although the latter solution can be found from the former in the limit of vanishing dilatonic coupling, even for very small values of the dilatonic coupling the entropy is strictly zero in the former case. This suggests that not all physical quantities (although some are as we are going to show) pertaining to these two cases are smoothly related as the dilatonic coupling is varied.
We conclude the introduction by outlining the rest of the paper: in section 2 we present the action and the equations of motion that we study. We next collect some relevant facts about the global Lifshitz and Lifshitz-Schwarzschild solutions, particularly in the context of our gravitational system, in section 3. Our main results including the various solutions and the extremality conditions are contained in sections 4 and 5, and in section 6 we discuss the thermodynamics of these solutions. We summarize our results and make some final concluding remarks in section 7. We have also included a detailed proof of a no-go result in appendix A: here we show that the only kind of potential for the dilaton allowing an exact, global Lifshitz solution is the negative cosmological constant. Finally, in appendix B we discuss in some detail the Hawking-Horowitz prescription [35] to determine the mass of solutions in general relativity, which we have used to find the mass of the various solutions we discuss.
The action and the equations of motion
Following the general remarks above, we want to pursue a study of (n + 2)-dimensional (n 2) static geometries supporting an electromagnetic (two-form) field F µν and a scalar dilaton φ in the presence of negative cosmological constant 7 . A suitable action for this system is 8
S = d n+2 x − det g R 2κ 2 eh + n(n + 1) 2κ 2 eh 2 − 1 2 (∂φ) 2 − e 2αφ F 2 2 + boundary terms , (2.1) where F 2 = 1 2 F µν F µν . (2.2)
The boundary terms in the action ensure a well defined variational problem and determine the conserved charges, but the equations of motion are, of course, unaffected by them.
For algebraic simplicity, we can absorb the normalization of the Einstein-Hilbert term by making the redefinitions 9 :
φ → φ 2κ 2 eh ; α → α 2κ 2 eh ; A → A 2κ 2 eh . (2.3)
Note that the product αφ is unaffected by the redefinition while F µν is redefined in the same way as A µ . In terms of these redefined fields and parameters, the Einstein's equations are
R µν = − n + 1 2 + e 2αφ F 2 2n g µν + 1 2 e 2αφ F µλ F λ ν + 1 2 (∂ µ φ)(∂ ν φ) . (2.4a)
The dilaton's equation of motion is
∇ 2 φ = α e 2αφ F 2 (2.4b)
and the Maxwell's equations are
∇ ν (e 2αφ F µν ) = 0 . (2.4c)
In order to find static Lifshitz-like solutions (1.1) to the above equations, we want the spacetime to be locally a product of a 2-dimensional space spanned by a global time function t and a radial coordinate r, and a flat R n along the transverse directions. We also assume that the various fields in the problem can only depend on the radial coordinate. The most general form of a metric with these properties is
ds 2 = −A(r)dt 2 + C(r)dr 2 + B(r) n i=1 (dx i ) 2 (2.5)
where i, j = 1, ..., n labels the directions along the R n . The functions A(r), C(r) and B(r) are not independent components of the metric, i.e., we can relate one of them to the other two by a suitable redefinition of r. For the dilaton we assume φ = φ(r). Finally, we only consider the case when the electromagnetic field is purely electric in nature through the following ansatz F µν = −f e (r) A(r)C(r) µνx 1 ...x n , (2.6) 8 We let κ 2 eh = 8πG (n+2) n , with G (n+2) n the (n + 2)-dimensional Newton's constant. As expected, the physical charges do not depend on this choice. 9 However, in order to define physical quantities (e.g., mass) associated with the solutions of (2.1) we will need to use the canonically normalized counterparts of the various fields and parameters according to (2.3).
where µ 1 µ 2 ν 1 ...νn is the completely antisymmetric symbol 10 , normalized such that trx 1 ...x n = 1. From the Maxwell's equation (2.4c), we then have
f e (r) = q e −2αφ(r) B(r) n 2 , (2.7)
where q is a dimensionless constant related to the physical charge. This can be seen as follows: the n-form dual field strength of F is given bỹ
F n = ε λ 1 ...λnµν e 2αφ F µν n! 2! dx λ 1 ∧ ... ∧ dx λn = q dx 1 ∧ ... ∧ dx n . (2.8)
Note that the dual field strength is not the simple Hodge dual but includes a factor of exp(2αφ). This ensures that the Maxwell term in the action (2.1) has the canonical form when expressed as the integral of the exterior product of F andF n , i.e.,
S em = − 1 4 d n+2 x − det g e 2αφ F µν F µν = − 1 2 F ∧F n . (2.9)
Now, the physical electric charge density per unit n-volume is defined as the (infinite) flux of the canonically normalized (see (2.3)) counterpart ofF n , through a transverse Gaussian surface located at any arbitrary value of the radial coordinate, divided by the (infinite) nvolume of the transverse space. This is a good opportunity to point out, that here and later, whenever we come across an "extensive" quantity, i.e., one that depends on the volume of the transverse space, one needs to consider densities per unit n-volume. This is because, owing to the infinite volume of the transverse space, the actual extensive quantities are all infinite. With this prescription, the physical electric charge density per unit n-volume is related to q through q phys = q 2κ 2 eh .
(2.10)
Upon these simplifications, the Einstein's equations (2.4a) take the following form: the (t, t) components are
∇ 2 log A(r) = n − 1 n q 2 2 e −2αφ(r) B(r) n + 2(n + 1) 2 .
(2.11a)
The non-trivial part of the Einstein's equations along (i, j) is
∇ 2 log B(r) = − 1 n q 2 2 e −2αφ(r) B(r) n + 2(n + 1) 2 . (2.11b)
The linear combination of the (t, t) and the (r, r) equations become
n 4 B (r) B(r) A (r) A(r) + C (r) C(r) − B (r) B(r) − n 2 d dr B (r) B(r) = φ (r) 2 2 (2.11c)
and finally, the dilaton equation of motion (2.4b) becomes
∇ 2 φ(r) = − αq 2 2 e −2αφ(r) B(r) n . (2.11d)
Analyzing the solutions to the above equations is the main focus of the current work. In particular, we want to obtain the general, asymptotically AdS non-extremal black brane solution of (2.11) and seek its connection to the special cases of asymptotically AdS but near horizon Lifshitz/Lifshitz-Schwarzschild solutions, recently found in [26,29,30].
The global Lifshitz and the Lifshitz-Schwarzschild solutions
Before discussing the asymptotically AdS solutions, we want to briefly review the asymptotically Lifshitz solutions of (2.11), because of their relevance in the near horizon geometry of the special solutions mentioned at the end of the last section. There exists [25][26][27] a one parameter family of exact solutions of (2.11) given as follows:
A(r) = (r/ Lif ) 2z f (r); C(r) = (r/ Lif ) −2 f (r) −1 ; B(r) = (r/ Lif ) 2 , (3.1) where f (r) = 1 − 2m ls n+z Lif r n+z ,(3.2)
with m ls being a free parameter (see below), and the dilaton having a logarithmic behaviour
φ(r) = − n α log(r/ Lif ) . (3.3)
The scaling exponent z is given by
z = 1 + n 2α 2 ,(3.4)
while q = ±q Lif , where q Lif and Lif are given by q Lif = 2n(n + 1) 2α 2 + 1 ; Lif = (n + z − 1)(n + z) n(n + 1) .
(3.5)
For m ls = 0 we recover the scale invariant global Lifshitz solution (1.1). When m ls = 0, we will call this solution the Lifshitz-Schwarzschild solution (for reasons to be explained in a moment). The Lifshitz-Schwarzschild solution has a curvature singularity at r = 0 which can be seen from the expressions of the curvature invariants, e.g., the Ricci scalar
R = − 2 2 Lif n(n + 1) 2 + nz + z 2 + n(z − 1) 2 2m ls n+z Lif r n+z . (3.6)
To prevent a naked singularity in the solution, we must therefore insist m ls > 0 so that there is a horizon at r = r ls , where r ls is given by
r n+z ls = 2m ls n+z Lif . (3.7)
It should also be noted that the curvature invariants are all finite when m ls = 0 11 , i.e., the global Lifshitz solution is free of curvature singularities at r = 0 12 .
The parameter m ls is related to the mass per unit n-volume of the Lifshitz-Schwarzschild solution (see appendix B) M ls = n m ls κ 2 eh Lif .
(3.8)
Also, as a black object in general relativity, the Lifshitz-Schwarzschild solution has a temperature associated with the horizon given by
T ls = (n + z)(2m ls ) z n+z 4π Lif = (2m ls ) z n+z 4π n(n + 1)(n + z) n + z − 1 . (3.9)
The temperature vanishes in the limit m ls → 0 when we also get the global Lifshitz solution back; this is consistent with the scale invariance of the Lifshitz solution. For m ls = 0, but z = 1, the expression for the temperature matches that of the AdS-Schwarzschild solution. We also have an entropy density per unit n-volume associated with the Lifshitz-Schwarzschild solution, given by the Bekenstein-Hawking formula
S ls = 2π κ 2 eh r n ls n Lif = 2π(2m ls ) n n+z κ 2 eh .
(3.10)
From (3.8), (3.9) and (3.10) we then have the Smarr's formula [38] M ls = n n + z T ls S ls . For z = 1, we get back the corresponding formula for the AdS-Schwarzschild solution. Based on our discussion so far, the general Lifshitz-Schwarzschild solution for z = 1 can be seen to be a straight-forward generalization of the AdS-Schwarzschild solution, just like the global Lifshitz solution (for z = 1) itself is a generalization of the AdS n+2 solution. Stated differently, any expression pertaining to these solutions, reduces to the corresponding one pertaining to the AdS n+2 /AdS-Schwarschild solutions, as we set z = 1. We are therefore justified in calling the "Lifshitz black brane" solution as the Lifshitz-Schwarzschild solution.
As already mentioned before, these solutions, apart from being exact solutions to (2.11), also arise as near horizon approximations of certain asymptotically AdS solutions of (2.11). It is then natural to ask, as well, about the relationship of these special solutions to the general, non-extremal asymptotically AdS black brane solution of (2.11). We conclude the current section by presenting the Lifshitz and the Lifshitz-Schwarzschild solutions in a way suitable to address these issues, and in sections 5.2 and 5.3 below, we address these in details.
As an artifact of the gauge fixing condition (4.2) that we impose on the metric in our analysis of the asymptotically AdS solutions, the Lifshitz/Lifshitz-Schwarzschild solutions appear, as near horizon solutions, in a different radial coordinate. If we temporarily denote the new radial coordinate byr, then it is related to r, appearing in (3.1), througĥ
r(r) =ˆ Lif (r/ Lif ) n+z ;ˆ Lif = Lif n + z . (3.12)
Treatingr as a new radial coordinate, the metric (3.1) takes the form
ds 2 = −(r/ˆ Lif ) γf (r)dt 2 + (r/ˆ Lif ) −2 f (r) dr 2 + (r/ˆ Lif ) 2δ n i=1 (dx i ) 2 ;f (r) = 1 − 2m lsˆ Lif r , (3.13) where δ = 1 n + z ; γ = 2z n + z .
(3.14)
In this new coordinate the metric components satisfy (− det g) =Â(r)Ĉ(r)B(r) n = 1. The horizon in this new coordinate is atr ls = 2m lsˆ Lif (3.15) and the dilaton isφ
(r) = − n α(n + z) log(r/ˆ Lif ) . (3.16)
The temperature of the brane, being independent of the choice of coordinate, is still given by (3.9). In particular, this is one way to relate the arbitrary constant inf (r) to 2m ls .
Asymptotically AdS solutions
We now turn to the class of solutions of the equations of motion (2.11) that admit AdS boundary conditions asymptotically. One can scale out the dependence in (2.11) by using a dimensionless radial coordinate R, related to r 13 through
R = r . (4.1)
It is convenient at this stage to fix the redundancy in the metric components which persists so far. We will impose the following
− det g = A(R)C(R)B(R) n = R 2n ,(4.2)
so that C(R) is known in terms of A(R) and B(R). The asymptotically AdS boundary conditions now read
lim R→∞ A(R) R 2 = A 0 , lim R→∞ B(R) R 2 = B 2 0 , lim R→∞ φ(R) = φ ∞ ,(4.3)
where A 0 , B 0 and φ ∞ are constants. The gauge choice (4.2) and the boundary conditions (4.3) imply that R 2 C(R) goes to a constant as R → ∞, which is consistent with asymptotic AdS-ness. Note however, that we have departed from the standard choice of setting A 0 = B 0 = 1. Also, as can be seen from the equations of motion (2.11), if φ(r) is a solution for the dilaton, then so is φ(r) + φ 0 , φ 0 being any constant, provided the q is replaced by q exp(αφ 0 ). We could have utilized this freedom to set φ ∞ to zero. As we are going to argue later, using the more general form of boundary condition (4.3) will help us with our analysis.
Under these assumptions, the equations of motion (2.11) take the following form d dR
A(R)B(R) n R n A (R) A(R) = q(n − 1) n E(R) + 2(n + 1)R n , (4.4a) d dR A(R)B(R) n R n B (R) B(R) = − q n E(R) + 2(n + 1)R n , (4.4b) d dR A(R)B(R) n R n φ (R) = −αqE(R) , (4.4c) n d dR B (R) B(R) + n(n + 1) 2 B (R) B(R) 2 − n 2 R B (R) B(R) + φ (R) 2 = 0 , (4.4d)
where we have introduced the electrostatic field E(R) as follows
E(R) ≡ −F tR (R) = q e −2αφ(R) R −n B(R) n . (4.5)
Let us first review the solutions to (4.4) for the special cases of q = 0 (with α not necessarily zero), and q = 0, α = 0. For the first case, we have the AdS-Schwarzschild black brane solution
A(R) = R 2 B 2n 0 1 − 2m R n+1 ; B(R) = B 2 0 R 2 ; φ(R) = φ ∞ , (4.6)
where m 0 is a constant. We should highlight that the equations of motion (specifically, the normalization of the cosmological constant term in the equations) force the constraint A 0 B 2n 0 = 1 upon us, and this is true for all the solutions we have obtained. When m = 0 we have the pure AdS n+2 solution; for m > 0, the mass per unit n-volume M of the AdS-Schwarzschild black brane is related to m (see (B.14)) through:
M = n m κ 2 eh . (4.7)
The solution has a curvature singularity at R = 0, but the singularity is hidden behind a horizon located at R = R sch (Schwarzschild radius), where
R n+1 sch = 2m . (4.8)
The other case, namely q = 0, α = 0 leads to the AdS-Reissner-Nordström black brane solution. Defineq = q B n 0 e αφ∞ (4.9) in terms of which the AdS-Reissner-Nordström solution takes the form
A(R) = R 2 B 2n 0 1 − 2m R n+1 +q 2 2n(n − 1)R 2n ; B(R) = B 2 0 R 2 ; φ(R) = φ ∞ . (4.10)
The parameter m has the same interpretation (4.7) as that in the AdS-Schwarzschild solution ((4.10) reduces to (4.6) as q → 0). A horizon (and hence a physical solution to hide the curvature singularity at R = 0) exists only if the extremality condition m m ext is met, where
m ext = n n − 1 R n+1 h,ext ; |q| = 2n(n + 1) R n h,ext . (4.11)
For the more general case of q = 0, α = 0 the above equations (4.4) cannot be solved in a closed form analytically. In the following, we will instead present a solution to (4.4) as a power series in R −1 . The solution accurate up to O(R −(2n+2) ) is
A(R) = R 2 B 2n 0 1 − 2m R n+1 +q 2 2n(n − 1)R 2n − 2α 2 µ 2 φ nR 2n+2 + O R −(2n+3) , (4.12a) B(R) = B 2 0 R 2 1 − 2α 2 µ 2 φ nR 2n+2 + O R −(2n+3) , (4.12b) φ(R) = φ ∞ + 2αµ φ R n+1 − αq 2 2n(n − 1)R 2n + 2αmµ φ R 2n+2 + O R −(2n+3) . (4.12c)
This is the "large R form" of the asymptotically AdS dilatonic black brane solution we are after in the present work.
A few comments about this form of the solution are in order: first, the parameter m has the same interpretation as above, i.e., the energy density per unit n-volume of the brane is still given by (4.7). Secondly, apart from m and q, the solution is characterized by a third constant µ φ , which appears in (4.12) as a constant of integration, and is proportional (up to unimportant constants) to the dilatonic charge of the brane 14 . From the α dependence of the above solution (at least up to the order shown), it can be seen that in the limit α → 0, the solution reduces to the AdS-Reissner-Nordström solution. As we will show in the next section, q = 0 ⇔ µ φ = 0, so that when q = 0 (even if α = 0), the solution reduces to the AdS-Schwarzschild solution. From the asymptotic analysis it can be concluded that (4.12) is the unique, static, asymptotically AdS solution to (4.4).
Near horizon analysis
By computing the curvature invariants (in powers of R −1 ), it can be seen that (4.12) has a curvature singularity at R = 0. Therefore, there must be an event horizon at a positive value of R which hides the singularity. We will denote the radial location of the outermost event horizon by R h . Being static, the system admits a timelike Killing vector χ t = {1, 0, ..., 0} 15 , and by the usual requirement of the vanishing of the norm of χ t on a regular horizon, we have A(R h ) = 0.
To study the nature of a solution near the horizon, it is necessary to expand the various functions as a power series in (r − r h ). The analysis is facilitated when done in terms of the dimensionless radial coordinate w defined as
w = r − r h r h = R − R h R h ⇔ R = R h (1 + w) . (5.1)
We include in our definition the restriction w 0, such that the coordinate w is well suited to study only the spacetime outside the horizon. Manifestly, w = 0 is where the horizon is located, and by "near horizon" we mean small w, i.e., 0 < w 1 (equivalently R h < R 2R h ). The gauge fixing condition for the metric (4.2) now reads
A(w)C(w)B(w) n = R 2n h (1 + w) 2n . (5.2)
We now need to make appropriate series expansion ansätze for the various functions, to be valid in the near horizon region. Since A(w) must vanish on the horizon, we make the following ansatz (factors of R h have been included in the following expressions for future convenience)
A(w) = a 0 R 2 h w γ (w);Â(w) = 1 + a 1 w + a 2 w 2 + O(w 3 ) , (5.3)
where a 0 > 0 and γ 1, with γ = 1 (at least) for the non-extremal case. For the case of B(w) we make a similar ansatz
B(w) = b 2 0 R 2 h w 2δB (w);B(w) = 1 + b 1 w + b 2 w 2 + O(w 3 ) ,(5.4)
where b 0 > 0 and δ 0, so that B(w) is allowed to vanish on the horizon. When we solve the equations of motion (4.4) starting from the horizon, we need to specify boundary conditions on the horizon. The possible vanishing of A(w) and B(w) are already captured through their postulated dependence on w γ and w 2δ , respectively. The constants a 0 and b 0 in (5.3) and (5.4) respectively, similar to A 0 and B 0 in (4.3), are to be specified as boundary conditions on the horizon, and cannot be solved through the equations of motion.
When the preceding ansatz for B(w) is used in (4.4d) we find the need to allow for a logarithmic divergence in φ(w) near the horizon; in other words, we need the following ansatz for φ(w)
φ(w) = φ 0 log w + φ c +φ(w);φ(w) = φ 1 w + φ 2 w 2 + O(w 3 ) ,(5.5)
where φ 0 and φ c are constants. Plugging in the above ansatz for φ(w) in (4.4d) we obtain
2nδ{δ(n + 1) − 1} + φ 2 0 w 2 + 2nδ w (n + 1)B (w) B(w) − n 1 + w + n d dw B (w) B(w) + n(n + 1) 2 B (w) B(w) 2 − n 2 1 + w B (w) B(w) +φ (w) 2 = 0 . (5.6)
The lowest order term in the equation imposes the following relation between δ and φ 0
2nδ [δ(n + 1) − 1] + φ 2 0 = 0 . (5.7)
In particular, if δ vanishes (as is expected for the non-extremal solution) then so does φ 0 and vice-versa 16 . We can now express the equations of motion (4.4a) through (4.4c) 16 There is another case when φ0 must vanish, namely when δ = (n + 1) −1 . As we are going to argue, this can only be true when q = 0, that is, for the trivial case of the AdS-Schwarzschild solution.
entirely in terms of the functionsÂ(w),B(w) andφ(w) as follows:
k d dw w γ+2nδ−1Ŷ (w) γ + w (w) A(w) =q 2 (n − 1) n w −2(αφ 0 +nδ)Ê (w) + 2(n + 1)(1 + w) n , (5.8a) k d dw w γ+2nδ−1Ŷ (w) 2δ + wB (w) B(w) = −q 2 n w −2(αφ 0 +nδ)Ê (w)+2(n+1)(1+w) n , (5.8b) k d dw w γ+2nδ−1Ŷ (w) φ 0 + wφ (w) = −αq 2 w −2(αφ 0 +nδ)Ê (w), (5.8c) where k = a 0 b 2n 0 ;q = q e −αφc b n 0 R n h ⇔ q =qe αφc b n 0 R n h (5.9) andÊ (w) = e −2αφ(w) (1 + w) n B(w) n ;Ŷ (w) = (w)B(w) n (1 + w) n . (5.10)
Clearly we have the same number of unfixed "boundary parameters", namely B 0 and φ ∞ on the asymptotic boundary versus b 0 and φ c on the horizon (by (5.9) the value of a 0 is fixed in terms of b 0 ). Whichever side we start from, choosing the values for one pair fixes the values for the other. We will leave this "normalization freedom" manifest in all the expressions to follow, except in those cases where we make an explicit choice. The linear combination formed by adding (1/α) times (5.8c) to the difference of (5.8a) and (5.8b) takes the following form
k d dw w γ+2nδ−1Ŷ (w) γ − 2δ + φ 0 α + w  (w) A(w) −B (w) B(w) +φ (w) α = 0 . (5.11)
When (5.11) is considered to the lowest order, we face two possibilities: either γ = 1 − 2nδ or γ = 1 − 2nδ. In the following section 5.1 we will consider the former case, and then in section 5.2 we will separately analyse the latter possibility.
The non-extremal solution
The first case we consider is when γ = 1 − 2nδ. As we will show below, this choice gives the non-extremal solution. Expanding the equations of motion (5.8) to the first subleading order, we further impose φ 0 = −nδ/α since otherwiseq = 0. Butq = 0 corresponds to the AdS-Schwarzschild black brane solution. From (5.7) we then find either δ = 0 or δ = (n + z) −1 where z is the scaling exponent as in (3.4). In particular, this rules out the possibility φ 0 = 0, δ = (n + 1) −1 . Furthermore, given the near horizon ansätze for A(w) (5.3) and B(w) (5.4), the near horizon behaviour of the expansion of radial null geodesics is [39]
θ(w) = na 0 b n 0 R n+1 h 2 w −nδ [1 + O(w)] ,(5.12)
where we have used that γ = 1 − 2nδ. The expansion θ(w) should vanish on a true horizon, but if we choose δ = 0 for the present case, we see it actually diverges. Therefore the appropriate choices for the exponents are
δ = φ 0 = 0; γ = 1 . (5.13)
In other words, A(w) vanishes linearly on the horizon while B(w) and φ(w) are regular on the horizon. By defining a new near horizon radial coordinate ρ = 2 w/k, the near horizon metric can now be brought to the Rindler form ds 2 = −κ 2 ρ 2 dt 2 + dρ 2 + ds 2 ⊥ , where ds 2 ⊥ is the metric on the transverse space and
κ = kR h 2b n 0 (5.14)
is the surface gravity. From the Rindler form of the metric, we note ∂N/∂ρ = κ, where N = κρ is the lapse function. Therefore, the horizon is a regular horizon [39], and the solution thus describes a non-extremal black brane. The coefficients to determine the functionsÂ(w) (5.3),B(w) (5.4) andφ(w) (5.5) accurately up to O(w 2 ) are given as follows:
a 1 = (2n − 1)q 2 − 2n(n 2 − 1) 2nk + n 2 ; b 1 = − q 2 − 2n(n + 1) nk ; φ 1 = − αq 2 k , (5.15) a 2 = 2{(3n − 2)α 2 + 9n − 5}q 4 − 8n(3n 2 + n − 2)q 2 + 12n 2 (n + 1) 2 (n − 1) 12nk 2 + (2n − 1)q 2 − 2n(n 2 − 1) 2k + n(n − 1) 6 , b 2 = − (2nα 2 + n − 1)q 4 − 4n(n 2 − 1)q 2 + 4n 2 (n + 1) 2 (n − 1) 4n 2 k 2 +q 2 − 2n(n + 1) 2k , (5.16) φ 2 = − α(2nα 2 + n + 1)q 4 − 2nα(n + 1)(2n + 1)q 2 4nk 2 + nq 2 α 2k .
To be able to connect the near horizon solution found above to the asymptotic solution (4.12), we need to relate the parameters k andq characterizing near horizon form of the solution to m, µ φ and q characterizing the asymptotic form. Our goal in the remainder of this section will be to accomplish that. To that end, consider integrating the equations (4.4a), (4.4b) and (4.4c) from R = R h to R = ∞. The analysis is facilitated by the following observation: from the expression of E(R) (4.5) the electrostatic potential at the horizon, Φ h , is
Φ h = − Rh ∞ dR E(R) = q ∞ Rh dR e −2αφ(R) R −n B(R) n . (5.17)
Note that (q −1 Φ h ) > 0 since the integrand is positive 17 . Upon performing the above mentioned integration 18 and rearranging the equations a bit, we obtain
2m + 2 − k n − 1 R n+1 h = qΦ h n ; 2m − R n+1 h = qΦ h 2n ; 2(n + 1)µ φ = qΦ h . (5.18)
These are three equations in three unknowns: R h , k and Φ h . The last equation in fact gives Φ h in terms of µ φ and q. Since q 2 (q −1 Φ h ) > 0, we can further conclude
µ φ 0 (5.19)
and the equality holds iff q = 0 (i.e., only for the AdS-Schwarzschild solution). Solving the remaining equations, we find
R n+1 h = 2m − n + 1 n µ φ = 2m − qΦ h 2n = R n+1 sch 1 − n + 1 2n µ φ m ,(5.20)
where R n+1 sch = 2m (4.8), and
k = 2(n + 1)(m − µ φ ) R n+1 h = (n + 1) 1 − µ φ m 1 − n + 1 2n µ φ m . (5.21)
Since k > 0 by definition (5.9), we must have m > µ φ for the non-extremal case. Also, for a fixed m, µ φ can increase until it equals m at which point k vanishes. Therefore,
m µ φ (5.22)
represents one of the extremality conditions. It is very interesting to note that the above condition is identical to the equivalent one in [3]. When we discuss the thermodynamics of this solution in section 6, we will show that the above condition, when saturated, is equivalent to the vanishing of the surface gravity at the horizon. This is then consistent with the expectations of an extremality condition. The bounds (5.19) and (5.22) on µ φ are equivalent to the following bounds on k 0 k (n + 1) .
(5.23)
The upper-bound on k is true when µ φ = 0, i.e., when q = 0. We then have the AdS-Schwarzschild solution (4.6) and it can be easily checked that A(w) = (n+1)wR 2 sch +O(w 2 ) in this case. The left plot in figure 1 shows the behaviour of k as a function of µ φ /m.
There seems to be an apparent mismatch in the number of parameters labeling a given solution. From the near horizon side these parameters are (naïvely) k andq, whereas from the asymptotic side they are m, q and µ φ . This, as we are going to argue now, is not conflicting. The black hole solutions we describe include a static matter field (the dilaton) outside the horizon. It is thus natural to expect that the location of the horizon should not be solely determined by the total mass and the total electric charge carried by the spacetime, but also by the configuration of the dilaton in the bulk. In other words, the location of the horizon, R h , should be an independent parameter that needs to be specified in addition to m and q to completely describe the solution 19 . From (5.21), (5.20) and (5.9) this is equivalent to choosing m, q and µ φ . We discuss this issue in more details at the end of section 5.3.
On the other hand, starting from near the horizon, the relevant parameters to choose are k,q, and m. It can be shown from (4.4) that two solutions with the same values for the ratios µ φ m and q m n n+1 are related to each other by a transformation involving a constant scaling of the radial coordinate. This is also naturally contained in the fact that for the 19 It then seems to be more of a coincidence, that the dilatonic charge is determined by the total electromagnetic charge and the mass of an asymptotically flat dilatonic black holes [3], and in spite of the presence of a non-trivial dilaton in the bulk, the location of the horizon is still determined just by these two parameters.
class of solutions with the same values for the parameters k andq So far we have only dealt with the approximate forms of the solution near the horizon and in the asymptotic region. To find the solution in the bulk of the spacetime we numerically integrated the equations (5.8) to solve for the functionsÂ(w),B(w) andφ(w). We used the NDSolve routine of Mathematica to perform the numerical integration. Stating from slightly outisde the horizon (w min ∼ 10 −17 ), we set the boundary conditions onÂ(w), B(w) andφ(w) by evaluating the functions and their derivatives at this point using the near horizon coefficients (5.15) and (5.16) and ran the integration up to w max ∼ 10 10 .
µ φ m = 2n(n + 1 − k) (n + 1)(2n − k) ; q (2m) n n+1 =q n − 1 2n − k n n+1 ,(5.
To relate the hatted functions to the actual metric coefficients and the dilaton through (5.3), (5.4) and (5.5), we need to pick values for b 0 and φ c -we made the most natural choice of setting b 0 = 1, φ c = 0. In figure 2 we show some of our numerical results, with the caption of the figure explaining the details. Based on our earlier remarks on relating the near horizon parameters k andq with the asymptotic ones i.e., m, µ φ and q, we could also check that the asymptotic forms of the functions agree very well with the corresponding numerically evauated functions in the large w region.
Although the numerical results shown in figure 2 were for a specific value of n and α, we found similar results for other values of these parameters as well. In (5.23) we already found upper and lower bounds on the values of the parameter k; this naturally narrows down the allowed parameter space to explore. The bound (5.23) on k, as noted earlier, is equivalent to the extremality bound (5.22). However, µ φ being independent of m and q in the present situation, we do not have a bound on q. This, as we already pointed out, is contrary to the usual charged black hole/brane solutions, and particularly the asymptotically flat dilatonic black hole solution [3]. There the dilatonic charge is related to the total mass and the electromagnetic charge, so that the extremality condition implies an upper bound on the total electromagnetic charge. In the following two sections, we will find that there actually exists a bound on the parameterq (5.9) (although not on q itself) and identify the value of this bound.
The extremal solution
The other case to consider is when γ = 1 − 2nδ, which as we will see corresponds to the extremal case. Analyzing the equations of motion, (5.8) and (5.11) (to lowest order), we find the complete lowest order solution 20
A ext (w) = k ext R 2 h,ext b 2n 0,ext w γ ; B ext (w) = b 2 0,ext R 2 h,ext w 2δ ; φ ext (w) = φ c,ext + φ 0 log w (5.25)
where the various exponents and coefficients (as defined in (5.3), (5.4) and (5.5)) take the values δ = 1 n + z ; γ = 2z n + z ; φ 0 = − n α(n + z)
, (5.26) with z, the scaling exponent (3.4), given by
z = 1 + n 2α 2 . (5.27)
while the constants b 0,ext and φ c,ext stay unconstrained. Furthermore, the constants k and q, defined in (5.9), are also fixed by (5.8) and (5.11), and take the following values
k ext = n(n + 1)(n + z) n + z − 1 = 2 2 Lif ; |q ext | = q Lif = 2n(n + 1) 1 + 2α 2 ,(5.28)
where q Lif andˆ Lif were defined in (3.5) and (3.12), respectively. It is easy to check that The near horizon solution satisfies A(w)C(w)B(w) n = R 2n h,ext (as opposed to a one on the right hand side because of the rescaling of t, x i -compare with (3.13)). This certainly agrees with the gauge fixing condition (5.2) near the horizon, but the Lifshitz solution is an exact solution everywhere only if the right hand side of (5.2) equals R 2n h,ext everywhere. This is a virtue of our gauge fixing condition (5.2), in that, it distinguishes between the solution which is Lifshitz all the way out, from the one which is Lifshitz only near the horizon.
The change in the behaviour of the solution from Lifshitz to AdS is effected by the correction functions ext (w),B ext (w) andφ ext (w) as defined in (5.3), (5.4) and (5.5). The series expansion coefficients of these functions about w = 0, up to O(w 2 ), are found by solving (5.8) to the same order and are given as follows a 1,ext = n(n + 2α 2 ) n + 2(n + 1)α 2 ; b 1,ext = 2nα 2 n + 2(n + 1)α 2 ; φ 1,ext = − n 2 α n + 2(n + 1)α 2 , (5.29) a 2,ext = n(n + 2α 2 )(n(7n − 4) + 2(n − 1)(4 + n)α 2 ) 12(n + 2(n + 1)α 2 ) 2 , b 2,ext = nα 2 (n(n − 4) + 2(n − 1)(n + 4)α 2 ) 6(n + 2(n + 1)α 2 ) 2 , (5.30) φ 2,ext = − (n − 4)n 2 α 12(n + 2(n + 1)α 2 ) .
To connect the near horizon solution for this case to the asymptotic solution (4.12) we integrate (4.4a), (4.4b) and (4.4c), as before, from R = R h,ext to R = ∞. Defining Φ h,ext analogous to (5.17) we again obtain a set of three equations, which are however nothing but those in (5.18) with k = 0 22 . Solving these we obtain, first
µ φ,ext = m . (5.31)
This shows that the extremality bound (5.22) is saturated by this solution. One should also note that this particular solution exists only for a special value of the parameterq, i.e., when |q ext | = q Lif (5.28). This is very similar to general extremal black hole/brane solutions, which can also carry only a specific amount of charge. In the following section, we will argue that the absolute value ofq for a non-extremal solution is bounded from above precisely by q Lif . The present solution therefore saturates both the extremality bounds, prompting us to call it the extremal solution. Next, from integrating (4.4a), (4.4b) and (4.4c) from R = R h,ext to R = ∞ we also find R n+1 h,ext = n − 1 n m , (5.32) 21 Rescaling of t and x i however affects the lapse and the measure on the transverse space; we will discuss the consequences of this when we consider the thermodynamics of our solutions. 22 Warning: the constant k (5.21) does not reduce to kext in the extremal limit µ φ → m, but vanishes.
which is the µ φ → m limit of (5.20). This is also to the analogous relation (4.11) for the extremal AdS-Reissner-Nordström black brane. Finally, similar to (5.18), we get q ext Φ h,ext = 2(n + 1)µ φ,ext = 2(n + 1)m , (5.33) where q ext is related toq ext according to (5.9) (since R h = R h,ext for the present case), and the last equality follows from the extremality bound (5.31). One can check that the near horizon Lifshitz solution reduces to the AdS 2 × R n near horizon limit of the extremal AdS-Reissner-Nordström solution when α → 0 23 , which also happens to be an exact solution to (2.11) when α = 0. This is consistent with the fact that the whole system reduces to the AdS-Reissner-Nordström solution as α → 0. In the bulk of the spacetime, the nature of the solution can be obtained by numerically integrating 24 the equations (5.8). The procedure is similar to that performed for the non-extremal solution. As boundary conditions on the horizon, we set b 0,ext = k δ/2 ext and 23 The other special limit, namely α → ∞, does give the pure AdS solution, but in an unusual radial coordinate. 24 The numerical solution in the bulk for this case has been previously obtained in [26], [29] and [30]. φ c,ext = φ 0 log √ k ext , so that, as discussed above, the Lifshitz nature of the solution is most clear. In figure 3 we show that the solutions, for various allowed values of the parameters, are all asymptotically AdS while in figure 4 we compare the near horizon behaviour of the same set of solutions to the corresponding global Lifshitz solutions.
The near horizon Lifshitz-Schwarzschild solution
A natural question to ask, is, where does the Lifshitz-Schwarzschild solution (the m ls = 0 case of (3.13)) fits in this set-up. Indeed, one finds that the following ansatz solves (5.8) and (5.6) exactly [25][26][27]
A(w) = k ext R 2 h b 2n 0 w(w+w 0 ) γ−1 ; B(w) = b 2 0 R 2 h (w+w 0 ) 2δ ; φ(w) = φ c +φ 0 log(w+w 0 ) , (5.34)
where δ, γ, φ 0 and k ext are as in the extremal solution, (5.26) and (5.28), while φ c and b 0 are unconstrained as before. It also follows from analyzing (5.8) that the solution (5.34) has the same value |q| = q Lif as the extremal brane (5.28). The constant w 0 in (5.34) is a free parameter such that w 0 = 0 corresponds to the near horizon Lifshitz solution (5.25).
Since A(w) in (5.34) vanishes linearly in w, this solution must correspond to a special case 25 of the non-extremal brane solution studied in section 5.1. Our first task is to show that this solution is indeed the Lifshitz-Schwarschild solution. As with the near horizon Lifshitz solution, this can be done most easily by choosing the 25 Since |q| is not arbitrary but takes a particular value. boundary conditions b 0 = k δ/2 ext and φ c = φ 0 log √ k ext on the horizon, rescale t and x i to absorb the factors of R h and identify (w + w 0 ) √ k ext with the Lifshitz radial coordinate in (3.12). This then allows us to identify w 0 with the mass parameter of the Lifshitz-Schwarzschild (3.13) through
w 0 = 2m ls √ k ext = 2m lsˆ Lif , (5.35)
with the last equality following from (5.28). We should stress that the above identification depends on the particular boundary condition we choose on the horizon. On the other hand, to find out the corrections that take the near horizon Lifshitz-Schwarzschild solution to AdS asymptotically, we introduce the correction functions nh-ls (w), B nh-ls (w) andφ nh-ls (w), in terms of which the asymptotically AdS function can be expressed as
A nh-ls (w) = k nh-ls R 2 h w b 2n 0,nh-ls 1 + w w 0 γ−1 +Â nh-ls (w) , (5.36a) B nh-ls (w) = b 2 0,nh-ls R 2 h 1 + w w 0 2δ +B nh-ls (w) , (5.36b) φ nh-ls (w) = φ c,nh-ls + φ 0 log 1 + w w 0 +φ nh-ls (w) , (5.36c)
where the subscript "nh-ls" stands for near horizon Lifshitz-Schwarzschild. In (5.36) above, we have also reverted back to a general boundary condition on the horizon (i.e., arbitrary b 0 and φ c ) and introduced b 0,nh-ls = b 0 w δ 0 ; k nh-ls = k ext w 0 ; φ c,nh-ls = φ c + φ 0 log w 0 . Taken together with (5.22), these two equations furnish two extremality conditions on the system. Thus, as expected, we have obtained non-extremal solutions when at least one of the two extremality conditions is in the form of a strict inequality and the extremal solution is only obtained when both of them are strict equalities. In figure 6 we give a pictorial representation of the extremality conditions. In particular, it is clear from this plot thatq is an independent parameter which can be varied independently of m and µ φ (except at an isolated set of points). It is very interesting that in going to non-extremality starting from the extremal solution, one always ends up on the near horizon Lifshitz-Schwarzschild solution first, after which one can decrease the value ofq to obtain a more general non-extremal solution. This seems reasonable from the perspective of an observer near the horizon of the extremal solution. For such an observer the spacetime is Lifshitz, and the only way to overcome extremality is to throw in matter into the brane to make the solution non-extremal, which necessarily makes the solution Lifshitz-Schwarzschild-like near the horizon. Equivalently, there exists no smooth limit to the extremal solution starting from a general non-extremal solution i.e., for which |q| < q Lif . We can take µ φ as close to m as we want, but it is impossible to make µ φ = m without simultaneously making |q| = q Lif . Figure 6. The parameter space of solutions. The horizontal axis corresponds to (µ φ /m) and ranges from 0 to 1, while the vertical axis corresponds toq and ranges between −q Lif to q Lif . We have chosen n = 2 and α = 1 (i.e., q Lif = 2) in order to obtain a definite number for q Lif for the purpose of plotting, but otherwise the plot is applicable for any n and α. The dashed red lines, excluding the blue points at (0, 0) and (1, ±q Lif ), indicate regions where there are no physical solutions. The blue points correspond to the special cases of the AdS-Schwarzschild solution and the extremal solutions, respectively. The solid blue lines atq = ±q Lif with (µ φ /m) ranging between 0 < µ φ /m < 1 correspond the the near horizon Lifshitz-Schwarzschild solutions. The interiour light blue region corresponds the all the possible values of (µ φ /m,q) for which we have general non-extremal solutions.
m ls = k nh-ls 2 √ k ext = n + 1 2 1 − µ φ m 1 − n + 1 2n µ φ m ˆ Lif .
Thermodynamics
The solutions we have obtained are black objects in general relativity and hence are thermal objects governed by the laws of thermodynamics. We discuss such aspects of our solutions in this section. Let us first consider the non-extremal solutions. Given the near horizon expressions for the metric we can compute the temperature of the black brane directly by transforming to a local Rindler coordinate and reading off the surface gravity (5.14). The temperature is then the surface gravity divided by 2π 28 , i.e.,
T = kR h 4πb n 0 . (6.1)
Note that we have not set any specific boundary condition on the horizon which is why there is an explicit appearance of b 0 in the formula. For comparison among the various asymptocially AdS solutions, if we choose the same value for b 0 for all of them, then the temperature can only depend on m and µ φ through their dependence on k (5.21) and R h (5.20). In particular, the temperature does not depend onq, so that a near horizon Lifshitz-Schwarzschild can be hotter or colder than a general non-extremal solution (i.e., for which |q| < q Lif ). On the other hand, the extremal solution corresponds to k = 0 and hence its temperature is always zero. This is consistent with our expectation of an extremal solution, as well as with the fact that owing to the Lifshitz nature (scale invariance) of the extremal solution near the horizon, there cannot be a temperature (scale) associated with it. Note, that unlike the asymptotically flat dilatonic black brane solutions [3], the temperature (6.1) vanishes smoothly as extremality is approached in the limit k → 0.
On the other hand, to compare the temperature of the near horizon Lifshitz-Schwarzschild solution with that of the global Lifshitz-Schwarzschild solution, we should choose the boundary condition b 0 = k δ/2 ext . Using (6.1) for non-extremal solutions along with the relations (5.35), (5.37) and (3.12), we obtain
T nh-ls = (n + z)(2m ls ) z n+z R h 4π Lif . (6.2)
Once we recall that the lapse functions for the near horizon Lifshitz-Schwarzschild (5.34) and the global Lifshitz-Schwarzschild (3.13) differ by a factor of R h we find perfect agreement between (6.2) and (3.9). Next, we obtain the entropy density per unit n-volume of the general non-extremal black brane from the Bekenstein-Hawking proposal Perfect agreement with (3.10) is achieved once we recall that the transverse space "area densities" in (5.34) and (3.13) differ by a factor of R n h . Note that the entropy density vanishes for the extremal (near horizon Lifshitz) as well as the global Lifshitz solutions. Even though we noted earlier that in the limit α → 0, the global Lifshitz solution reduces to AdS 2 × R n , the entropy density is a physical quantity which does not have a smooth behaviour in this limit. Stated differently, the entropy density of the near horizon Lifshitz solution is zero as long as α = 0, but has a finite jump at α = 0.
From (5.18) we can now write down Smarr's formula [38] for the general non-extremal black brane
m = kR n+1 h + qΦ h 2(n + 1) = 2 κ 2 eh T S + qΦ h 2(n + 1) . (6.5)
Note that neither T S nor the Smarr's formula itself depends on the choice of boundary condition (b 0 ) on the horizon. In fact, Smarr's formula can be recast in more physical terms. In particular, since all the solutions we have are asymptotically AdS, by AdS/CFT all such solutions correspond to thermal states in the dual CFT. The Smarr's formula (6.5) especially corresponds to the thermodynamic identity of the CFT equivalent to the well known "E = T S + pV " for an ideal gas. By the standard rules of AdS/CFT, the temperature (6.1), entropy density (6.3) and the mass density (4.7) of the brane are identified with the temperature T cft , entropy S cft and the energy E cft , respectively, of the dual CFT. Next, the physical electric charge density (2.10) is identified with the number density n cft of the dual CFT, i.e., n cft = q 2κ 2 eh (6.6) and its conjugate chemical potential is related to the electrostatic potential with canonical dimensions (2.3)
µ cft = Φ h 2κ 2 eh . (6.7)
Putting all these together, the Smarr's formula (6.5) becomes [29,30] E cft = n n + 1 (T cft S cft + µ cft n cft ) . (6.8)
Comparison with (3.11) shows that the global Lifshitz-Schwarzschild solution has quite different thermodynamic properties compared to the near horizon Lifshitz-Schwarzschild solution.
Summary and conclusions
In this paper, we have studied static, asymptotically AdS n-brane solutions of Einstein-Maxwell-dilaton gravity with negative cosmological constant in (n + 2)-dimensions where n 2. Depending on the values of the various parameters labeling a solution, there are five different cases to consider, as summarized in table 1. Once the boundary conditions, either at the asymptotic infinity or at the horizon, are imposed appropriately, each solution is in fact labelled by three parameters: the mass density m, the electric charge density q and the dilatonic charge density µ φ carried by the brane 29 . However, instead of q, it is actually convenient to use the parameterq as a label, whereq is related to q through (5.9); note that the parameterq, being a ratio of q and R n h (5.20) is a function of the asymptotic charges only once an appropriate boundary condition is imposed.
Case m µ φq Solution 1 m = 0 µ φ = 0q = 0 Poincaré patch of AdS. 2 m > 0 µ φ = 0q = 0 AdS-Schwarzschild black brane.
3 m > 0 µ φ < mq < q Lif asymptotically AdS non-extremal dilatonic black brane.
4 m > 0 µ φ < mq = q Lif
asymptotically AdS non-extremal dilatonic black brane, near horizon Lifshitz-Schwarzschild.
5
m > 0 µ φ = mq = q Lif asymptotically AdS extremal dilatonic black brane, near horizon Lifshitz. With the exception of cases 1 and 2 in table 1 (the Poincaré patch of AdS and the AdS-Schwarzschild black brane, respectively) all the solutions describe electrically charged black n-branes, and none of them admit a closed analytic form. We have therefore expressed the solutions corresponding to cases 3 to 5 as power series (4.12) in the inverse power of the radial coordinate R (4.1) valid in the asymptotic (R 1) region, as well as in a power series in the radial coordinate with respect to the horizon, w (5.1), valid near the horizon (w 1) -see (5.3), (5.4), (5.5) for the general forms of the near horizon solutions; we will refer to the appropriate equations for the expressions of the various exponents and other parameters as we discuss each case separately.
The central result of this paper is the general non-extremal solution (case 3, table 1) discussed in section 5.1. This is a black brane with a regular horizon (g tt vanishing linearly in w) and the dilaton being finite at the horizon 30 . The asymptotic charges m, q and µ φ are related to the near horizon parameters k andq (5.9) through the relations (5.21) and (5.20).
We also derived the appropriate extremality conditions satisfied by all the solutions, and they are conveniently expressed as (5.22), (5.39)
m µ φ ; |q| q Lif ,(7.1)
where q Lif is defined in (3.5); see figure 6 for a pictorial representation of the extremality conditions. The first condition is in fact the actual extremality condition related to the vanishing of the surface gravity at the horizon, and this is very similar to the corresponding 29 To be precise, the physical densities with the correct dimensions are proportional to m, q and µ φ ; see e.g., (4.7) and (2.10). Also, we consider densities because the actual charges, being proportional to the horizon area, are infinite due to the infinite volume of the transverse space. 30 The higher order corrections to the metric away from the horizon can be found in (5.15) and (5.16).
condition in [3]. When the second extremality bound is saturated (i.e., |q| = q Lif but m > µ φ ), we obtain a special case of the non-extremal solution which is Lifshitz-Schwarzschildlike near the horizon (case 4, table 1 -see section 5.3). On the other hand, when both the bounds are saturated we obtain the extremal solution which is Lifshitz-like near the horizon (case 5, table 1 -see section 5.2). These near horizon solutions are however driven away from their Lifshitz/Lifshitz-Schwarzschild-like near horizon nature towards an AdS behaviour asymptotically by the higher order corrections as given in (5.15), (5.16) and (5.29), (5.30). These solutions have already been studied earlier in [26], [27], [29] and [30]. Finally, we have also studied the thermodynamics of the solutions in section 6. In particular, we observe that the temperature of the non-extremal solutions vanish smoothly in the extremal limit. The entropy of the extremal solution, being Lifshitz near the horizon, also vanishes. This is consistent with the third law of thermodynamics, but the behaviour is very different from the AdS-Reissner-Nordström solution. In this regard, the latter is an exact solution to this system when the dilatonic coupling is switched off, and the functional form of the near horizon Lifshitz solution reduces to AdS 2 × R n , which is the near horizon metric of the extremal AdS-Reissner-Nordström solution.
One important issue, particularly in the light of [40], is the stability of the solutions found in this paper, against small perturbations. It would also be interesting to holographically study e.g., transport properties of the CFT duals of our solutions (especially the general non-extremal solution) with condensed matter physics applications in mind. We hope to return to these questions in the near future.
Acknowledgments
PB would like to thank the theory group at CERN for their hospitality during various stages of this project. JB would like to thank Arnab Kundu and Robert de Mello Koch for useful discussions at the early stages of this project. DM thanks Sayan Basu for useful discussions. The work of PB and JB is supported by the NSF CAREER grant PHY-0645686 and by the University of New Hampshire through its Faculty Scholars Award Program.
A Restrictions on the dilatonic potential for the existence of exact global Lifshitz solutions
Note added in version 2: The results of this appendix has some overlap with section 1 of [36]. We than Eric Permutter for bringing this to our notice. In this appendix, we prove the following no-go result: by considering the following generalization of the action (2.1)
S = d n+2 x − det g R 2κ 2 eh − V (φ) − 1 2 (∂φ) 2 − e 2αφ F 2 2 (A.1)
we show that to obtain an exact, global Lifshitz solution of the form
A(r) = (r/ Lif ) 2z ; C(r) = (r/ Lif ) −2 ; B(r) = (r/ Lif ) 2 (A.2)
V (φ) has to be constant and negative. As already noted in the introduction of this paper, the existence of global Lifshitz and Lifshitz-Schwarzschild solutions in a gravitational system makes it a candidate holographic dual of a non-relativistically scale invariant quantum field theory. Our search for possible gravitational systems involving a dilaton and a gauge field, which gives rise to an exact Lifshitz solution, is therefore conveniently narrowed down to (2.1), the one we consider in this paper 31 . In (A.2) z and Lif depend upon the various parameters of the theory provided a solution of the above form exists, as we show below. After performing the redefinition (2.3) to get rid of factors of κ 2 eh , the equations of motion from (A.1) take the following form under the same assumptions as those in section 2 ( is an arbitrary length scale at this point)
∇ 2 log A(r) = n − 1 n q 2 2 e −2αφ(r) B(r) n − 2 n V (φ) (A.3a) n 4 B (r) B(r) A (r) A(r) + C (r) C(r) − B (r) B(r) − n 2 d dr B (r) B(r) = φ (r) 2 2 (A.3b) ∇ 2 log B(r) = − 1 n q 2 2 e −2αφ(r) B(r) n − 2 n V (φ) (A.3c)
and the dilaton equations of motion is
∇ 2 φ(r) = − αq 2 2 e −2αφ(r) B(r) n + dV dφ (A.3d)
To begin with, note that the Laplacian of any function ψ(r), for the metric (A.2) is
∇ 2 ψ(r) = r 2 2 Lif ψ (r) + ψ (r) r (z + n + 1) (A.4)
Furthermore, if ψ(r) has a logarithmic behaviour, i.e., ψ(r) = ψ c + ψ 0 log(r/ Lif ) where ψ c and ψ 0 are constants, then
∇ 2 ψ(r) = ψ 0 2 Lif (z + n) (A.5)
Subtracting (A.3c) from (A.3a) and using the above expression for the Laplacian on the left hand side
∇ 2 log A(r) B(r) = 2(z − 1)(z + n) 2 Lif = q 2 2 e −2αφ(r) (r/ Lif ) −2n (A.6)
Since the left hand side (term in the middle) is constant, consistency demands e −2αφ(r) (r/ Lif ) −2n is a constant as well. Writing this constant as e −2αφc , where φ c is another real constant, we have
φ(r) = φ c − n α log(r/ Lif ) (A.7)
31 Through a straight-forward generalization of the following proof, we also arrive at the same conclusion by considering an addition scalar field in (A.1), which is allowed to interact with itself as well as with the dilaton but is not coupled to the gauge field.
This also means that the dilatonic coupling terms in (A.3) are constant
q 2 2 e −2αφ(r) B(r) n =q 2 2 Lif ;q = q Lif e −αφc (A.8)
Getting back to (A.6), we findq to be restricted as followŝ
q 2 = 2(z − 1)(n + z) (A.9)
Next, if we use the ansatz for the metric components (A.2) and the logarithmic behaviour of the dilaton just derived in (A.7) into (A.3b), we find that the scaling exponent is given by
z = 1 + n 2α 2 (A.10)
Now, using (A.5), (A.9) and (A.10) in (A.3d), we can show that the radial variation of the potential vanishes as well
dV dφ = ∇ 2 φ(r) + αq 2 2 Lif = 0 ⇒ dV dr = dV dφ φ (r) = 0 (A.11)
Thus the only potential consistent with a scaling solution of the form (A.2) is a constant.
We are now going to fix the arbitrary length scale by setting it equal to the scale of the constant potential/vacuum energy as follows
V (φ) = kn(n + 1) 2 (A.12)
where k is either −1 or 0 or 1. Using (A.3c) now, we find
q 2 = − 2kn(n + 1) 2α 2 + 1 2 Lif 2 (A.13)
Sinceq is real, we are forced to choose k = −1. From (A.8) The charge q is then given by q 2 = 2n(n + 1) 2α 2 + 1 (A.14)
where, the constant φ c , which remains unconstrained up to this point, has been set to zero without any loss in generality. Note that q is independent of both Lif and . Finally, to satisfy (A.3c) (and (A.3a)) the constant Lif needs to be fixed at Lif = (n + z − 1)(n + z) n(n + 1) = (1 + 2α 2 )(n + 2(n + 1)α 2 ) 2α 4 (n + 1) (A. 15) We can regard Lif as the length scale associated with the Lifshitz geometry (like in AdS n+2 ) and call it the Lifshitz scale. Note that Lif in this particular solution. Also, Lif → as α → ∞ i.e., as z → 1, and the scaling solution becomes identical to the Poincaré patch of AdS n+2 with as its radius.
B The Hawking-Horowitz Mass
In this appendix, we calculate the mass of n-brane solutions in asymptotically AdS and Lifshitz spacetimes via the prescription of [35]. As with any Hamiltonian formulation, we first need to consider a foliation of the spacetime by constant time slices Σ t . In our case, we can choose the coordinate time t as the global time function, and parametrize the time slices (which are also the Cauchy surfaces) Σ t with t. The unit timelike normal on Σ t is The above construction allows us to obtain the lapse function and the shift vector. Let t µ (not to be confused with the coordinate time t) be the vector field generating time translations and satisfying t µ ∇ µ t = 1. The natural choice for a t µ is the timelike Killing vector itself t µ = (χ t ) µ = {1, 0, ..., 0} (B.4)
The lapse function N and the shift vector N µ are defined in the usual way [39] as the decomposition of t µ t µ = N (n t ) µ + N µ (B.5)
We can read off the lapse and the shift from (B.2) N = A(r); N µ = 0 (B.6)
We also have a "boundary near infinity", Σ ∞ , on which we specify the asymptotic behaviour of the various fields. To obtain Σ ∞ , consider a foliation of the spacetime through the hypersurfaces Σ r , which are slices of the spacetime at fixed values of the radial coordinate r 32 . We can then identify Σ ∞ with the asymptotic boundary of the spacetime located at r = ∞, i.e., Σ ∞ ≡ Σ r=∞ . The unit spacelike normal on Σ r , denoted byn r , is given by (n r ) µ = C(r)∂ µ r = C(r){0, 1, 0, ..., 0} ⇔ (n r ) µ = 1 C(r) {0, 1, 0, ..., 0}
(B.7) The fact thatn t ·n r = 0 clearly shows thatn r is tangential to Σ t whilen t is tangential to Σ r . The induced metric on Σ r , to be denote by h(Σ r ) µν , is h(Σ r ) µν = g µν − (n r ) µ (n r ) ν = diag {−A(r), 0, B(r), ..., B(r)} (B.8) 32 We have used r itself to parametrize the hypersurfaces Σr
The hypersurfaces Σ t and Σ r are themselves foliated by the n dimensional transverse spaces Σ t,r with induced metric h(Σ t,r ) µν = h(Σ t ) µν − (n r ) µ (n r ) ν = h(Σ r ) µν + (n t ) µ (n t ) ν = g µν + (n t ) µ (n t ) ν − (n r ) µ (n r ) ν = diag {0, 0, B(r), ..., B(r)} (B.9)
The final expression for the energy involves the trace of the extrinsic curvature of Σ t,r due to its embedding in Σ t evaluated at r = ∞. For a finite value of r, the extrinsic curvature is given by 33 K (Σ t,r ) µν = 1 2 £n r h(Σ t,r ) µν = 1 2 £n r g µν + 1 2 £n r (n t ) µ (n t ) ν − 1 2 £n r (n r ) µ (n r ) ν (B.10)
The trace of the this extrinsic curvature is then K (Σ t,r ) = g µν K (Σ t,r ) µν = nB (r) 2B(r) C(r) (B.11)
The Hamiltonian for our case is now given by
H = 1 2κ 2 eh Σt N H con − 1 κ 2 eh Σt,r=∞ N K (Σ t,r ) (B.12)
where N is the lapse (B.6), H con is the Hamiltonian constraint and we have dropped terms involving the shift vector since it vanishes for our case (B.6) (see [35] for the complete expression). Also, as emphasized in [35], in case of spatially non-compact geometries (as with flat n-branes) the Hamiltonian might be divergent when evaluated on a solution. In such cases, one needs to consider a reference background which is a static solution of the field equations, such that, the reference background is asymptotically approached by any physical solution whose energy we want to compute. The physical Hamiltonian is then obtained by subtracting the Hamiltonian evaluated on the reference background from the usual Hamiltonian (B.12), and the energy of a given solution is the value of the physical Hamiltonian evaluated on the solution
E = − 1 κ 2 eh Σ∞ N [K (Σ t,r ) − K 0 (Σ t,r )] r=∞ (B.13)
where we have used the fact that that H con vanishes on a physical solution and have denoted the parts evaluated on the reference background by a subscript 0. It should be emphasized that for the above prescription to work, it is crucial for the lapse of the solution of interest and the background solution to agree asymptotically (which allowed us to factor the same out from the expression inside the integral). The square bracketed quantity inside the integral above secretly contains the volume factor of the transverse space.
For the special class of asymptotically AdS solutions that we studied in this paper, we must choose the reference background to be the Poincaré patch of the pure AdS n+2 solution with a constant dilaton. However, we need to remember that the integral over 33 The extrinsic curvature of Σt,r due to its embedding in Σr vanishes. Σ ∞ gives an infinite contribution owing to the infinite n-volume of the flat transverse space. Therefore, the physical quantity to work with is the energy density per unit nvolume, obtained through dividing the energy computed from (B.13) by the volume of the transverse space. In what follows, we denote by E this energy density (as opposed to the infinite total energy) with the hope that there will be no confusion after our clarifying remarks above. Now, all the solutions discussed in this paper, including the special cases of the AdS-Schwarzschild and the AdS-Reissner-Nordström solutions as well as the various cases of the dilatonic black brane solutions, satisfy the metric fixing condition (4.2). We can then employ this condition to eliminate the function C(R). Furthermore, all such solutions have a series expansion in 1/R analogous to (4.12), where for the AdS-Schwarzschild and the AdS-Reissner-Nordström solutions the series truncate after a finite number of terms. Using the explicit expression (B.11) for the trace of the extrinsic curvature in (B.13) the energy is E = nm κ 2 eh (B.14)
Similarly, to find the energy of the global Lifshitz-Schwarzschild solution discussed in section 3, we use the global Lifshitz solution as the reference background. Following the same procedure as discussed above in the context of asymptotically AdS solutions, we employ (B.13) to obtain (3.8).
Figure 1 .
1k (left) and Rh R sch (right) plotted as functions of µ φ m for n = 2 to 5. The parameter k starts at (n + 1) when µ φ = 0 and goes to zero as µ φ → m. Correspondingly, the ratio Rh R sch starts at one for all n at µ φ = 0 and reaches the extremal value of the ratio when m = µ φ (dashed vertical line).
Figure 2 .
2from (5.21),(5.20) and(5.9). The infinite class of such solutions are however distinguishable by the fact that they all have different values for R h (5.20) and thus, even though the solutions are same as functions of w, they are different as functions of R. A(R), B(R) and φ(R) for n = 2, α = 1, k = 2.4 (µ φ = 0.5 m) andq = 0.5 q Lif . The curves for A(R) and φ(R), from left to right, correspond to m = 0.01, 0.1, 1.0, 10 and 100, respectively. The curves for B(R) follow the same colour pattern, and for each mass, B(R) ends at the corresponding value of R h shown by the same coloured dotted line. The square roots of A(R) and B(R) were plotted to show their linear (asymptotic AdS) nature as R becomes large. The coefficients A 0 and B 0 (4.3) are both very close to one in all the cases shown in the figure.
( 5 .
525) solves the equations of motion (5.8) and (5.6) exactly. Furthermore, (5.25) has the same form as the global Lifshitz solution (3.13) (when m ls = 0). In fact, these two metrics are identical if we set b 0,ext = k δ/2 ext and φ c,ext = φ 0 log √ k ext as boundary conditions on the horizon, identify w √ k ext withr/ˆ Lif in (3.12) and rescale both t and x i in (3.13) by R h,ext 21 .
Figure 3 .
3The R) and φ(R) in the bulk for n = 2 and α = 1. The plots for B −n 0 A(R) and φ(R), from left to right, correspond to m = 0.01, 0.1, 1.0, 10 and 100, respectively, while the curves for B −1 0 B(R) follow the same colour pattern. The functions A(R) and B(R) were divided by the normalization constants A 0 and B 2 0 (4.3) and their square roots were plotted to show their linear (asymptotic AdS) nature as R becomes large. Compared with the non-extremal case (figure 2), the functions asymptote to AdS much slowly.
Figure 4 .
4Comparing A(w), B(w) and φ(w) near the horizon (solid) with the corresponding global Lifshitz solutions (dotted). We have set n = 2 and α = 1 (q Lif = 2). The three plots in each sub-figure correspond to m = 0.1, m = 1.0 and m = 10, respectively, with the corresponding values R h,ext = 0.37, R h,ext = 0.79 and R h,ext = 1.7 for the horizon radii.
and (5.5), it is clear that the solution is nothing but a nonextremal solution (with a specific value of |q|). In particular, we need to choose the correction functions nh-ls (w),B nh-ls (w) andφ nh-ls (w) such that the leading order terms in all of them vanish linearly on the horizon. Then the complete functions in the big square brackets in (5.36) are precisely the hatted functions for the non-extremal case, with the coefficients of series expansion in w being those in (5.15) and (5.16) up to O(w 2 ).Since we are dealing with a special case of the non-extremal solution, connecting the near horizon form of the solution to the asymptotic form of the same is analogous to what we did in the context of the general non-extremal solution, see the discussion after equation (5.16) in section 5.1. In particular, by integrating the equations (4.4a), (4.4b) and (4.4c) from R = R h to R = ∞, we obtain (5.20) and (5.21) as the expressions for the horizon radius R h,nh-ls and k nh-ls , respectively, in terms of m and µ φ . Owing to (5.37) and (5.35) (by choosing the appropriate boundary conditions on the horizon) we can also express the Lifshitz mass parameter m ls to m and µ φ
Figure 5 .
5Comparing A(w), B(w) and φ(w) near the horizon (solid) with the corresponding Lifshitz-Schwarzschild solutions (dotted). We have set n = 2, α = 1 (q Lif = 2) and have chosen k = 2.4 (µ φ = 0.5 m). The three plots in each sub-figure correspond to m = 0.1, m = 1.0 and m = 10, respectively, with the corresponding values R h = 0.5, R h = 1.1 and R h = 2.3 for the horizon radius. B(R) and φ(R) abruptly halts at the respective values based on the boundary conditions chosen; in particular, φ c,nh-ls = −0.5 for all value of the mass, as can be verified from the plot.As consistency demands, m ls vanishes in the extremal limit m → µ φ , leaving us with the extremal solution.Finding the solution in the bulk through numerical analysis 26 for this case is similar to the previous two cases discussed. For the present case, one important verification is whether the numerical analysis, set up for the general non-extremal solution, indeed shows the near horizon Lifshitz-Schwarzschild behaviour for the appropriate values of the parameters. We verify this infigure 5, with the following choice of boundary conditions on the horizon:b 0 = k δ/2 ext and φ c = φ 0 log √k ext (see the caption of the figure for other necessary details). Furthermore, the numerical analysis shows 27 that there are no physical solutions for the general non-extremal case, with |q| > q Lif and with different values k within the bound (5.23), i.e., we require |q| q Lif . (5.39)
28
The surface gravity divided by 2π is the inverse of the period of the compact Euclidean time.
(
n t ) µ = − A(r)∂ µ t = − A(r){1, 0, ..., 0} ⇔ (n t ) µ = 1 A(r) {1, 0, ..., 0} (B.1)Note that the timelike Killing vector (χ t ) µ = {1, 0, ..., 0} is related to the normal vector throughχ t = A(r)n t (B.2)Let us denote by h(Σ t ) µν the (spatial) metric induced by g µν on Σ t . The induced metric is related to the metric through h(Σ t ) µν = g µν + (n t ) µ (n t ) ν = diag {0, C(r), B(r), ..., B(r)} (B.3)
Table 1 .
1List of all static asymptotically AdS solutions discussed in this paper.
As is shown in appendix A the case of a more general potential V (φ) for the dilaton reduces to the current situation by demanding that global Lifshitz solutions are admitted.
The completely antisymmetric (Levi-Civita) tensor ε is related to through ε = √ − det g .
This is true for the invariants Ricci 2 and Riemann 2 as well. Also, when z = 1, the only curvature invariant that is singular when mls = 0 is Riemann 2 .12 As has been observed in[19,22], when z = 1 the horizon at r = 0 does have a pp singularity which renders these spacetimes geodesically incomplete. However it is expected that stringy effects will resolve this mild singularity[37].
This radial coordinate is different from the one appearing in the Lifshitz-Schwarzschild metric (3.1).
This can be shown, for instance, by comparing the first subleading term (O(R −(n+1) ) in our case) in the asymptotic expansion of the dilaton (4.12c) to that of[3].15 The normalization of the Killing vector at infinity is irrelevant for our present argument.
Work done to bring a unit charge (of the same type the black brane carries) to the horizon is positive.18 The R n terms on the right hand side of (4.4a) and (4.4b) are divergent when the upper limit of the integration is taken to ∞. There are however identical divergent pieces on the left hand side of both the equations coming from A(R) ∼ R 2 and B(R) ∼ R 2 when R is large. Thus, to make things well defined one should first integrate up to R = R large , cancel out the potentially divergent pieces from both sides of the equations and finally take R large → ∞.
We put a subscript "ext" on the quantities pertaining to this case, to distinguish them from the corresponding ones in the non-extremal case.
The numerical solution in the bulk for this case has been previously obtained in[29] and[30].27 To be precise, the numerical code shows divergent behaviour of the various metric components/dilaton very near the horizon for |q| > q Lif and with different values k within the bound (5.23).
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|
[] |
[
"Combining RGB and Points to Predict Grasping Region for Robotic Bin-Picking",
"Combining RGB and Points to Predict Grasping Region for Robotic Bin-Picking"
] |
[
"Quanquan Shao \nTong University Shanghai\nChina\n",
"Jie Hu \nTong University Shanghai\nChina\n",
"Shanghai Jiao \nTong University Shanghai\nChina\n"
] |
[
"Tong University Shanghai\nChina",
"Tong University Shanghai\nChina",
"Tong University Shanghai\nChina"
] |
[] |
This paper focuses on a robotic picking tasks in cluttered scenario. Because of the diversity of objects and clutter by placing, it is much difficult to recognize and estimate their pose before grasping. Here, we use U-net, a special Convolution Neural Networks (CNN), to combine RGB images and depth information to predict picking region without recognition and pose estimation. The efficiency of diverse visual input of the network were compared, including RGB, RGB-D and RGB-Points. And we found the RGB-Points input could get a precision of 95.74%.
| null |
[
"https://arxiv.org/pdf/1904.07394v2.pdf"
] | 118,623,854 |
1904.07394
|
2d2881c2b43c81038cd168f6cf55744b7bb5ceba
|
Combining RGB and Points to Predict Grasping Region for Robotic Bin-Picking
Quanquan Shao
Tong University Shanghai
China
Jie Hu
Tong University Shanghai
China
Shanghai Jiao
Tong University Shanghai
China
Combining RGB and Points to Predict Grasping Region for Robotic Bin-Picking
component; RGB-Pointssuction graspnueral networksbin-picking
This paper focuses on a robotic picking tasks in cluttered scenario. Because of the diversity of objects and clutter by placing, it is much difficult to recognize and estimate their pose before grasping. Here, we use U-net, a special Convolution Neural Networks (CNN), to combine RGB images and depth information to predict picking region without recognition and pose estimation. The efficiency of diverse visual input of the network were compared, including RGB, RGB-D and RGB-Points. And we found the RGB-Points input could get a precision of 95.74%.
I. INTRODUCTION
Robotic picking objects in clutter are widely researched in recently years. Relative technologies have a wide range of application demands in material transportation, waste sorting and logistics automation. However, it remains a big challenge because of the variety of scenarios, the diversity of objects, clutter by placing and complicated background. Vision based robotic manipulation have been successfully used in various applications in manufacturing industry and logistics industry. Normally, vision based robotic grasping methods mainly include object detection, object segmentation, pose estimation and the selection of grasping point [1]. 2D features are used to estimate the pose of target objects [2][3] [4]. With the development of 3D vision technology, 3D visions are also used in robotic grasp in cluttered scene to detect the objects and estimate poses [5] [6]. Matching 3D models for object recognition and pose estimation is difficult with various target objects and even impossible for unknown objects because of the less of 3D models beforehand [7]. Self-occlusion and disordered placing also weaken the performance of RGB-D based part 3D matching in robotic picking tasks. Traditional technical routes are weak to deal with grasp tasks in cluttered scenario, especially with unknown objects.
Compared with model-based technologies, data-driven methods have a great success in many vision tasks. Deep Convolution Neural Networks (CNN) got a great performance in image classification tasks [8]. Deep Neural Networks (DNN) are also used in robotic grasp tasks. Vision based system with DNN could choose the grasp point in cluttered scenario directly without 3D models or pose estimation [9] [10]. These methods treat the grasp point detection as classification problems or regress the coordinates of grasp point directly after training. DNN based grasp point detection could not only grasp known objects but also grasp unknown objects with a high performance of generalization. There are mainly three types of hands involved in robotic grasp tasks, namely, dexterous hand, three or two fingers grippers and suction gripper. Most of grasping point detection are based on parallel-jaw grasping configuration. Other gripper configurations are not studied deeply, especially suction gripper. In this paper, we focus on suction grasps which is also widely applied in robotic grasp and object manipulation. A region prediction approach was proposed to detect suction point in cluttered scenario with known or unknown target objects. Visual information got by a RGB-D camera was inputted into a U-net structure convolution neural network and the output is a probability map which means the successful probability of each pixel as a suck point. After some smoothing and other post process with this probability map, we could choose a best suck point in bin-picking like cluttered environment. We expanded the input of region detection networks with RGB-D and RGB-Points The result are shown below and we could find the RGB-Points have the best performance.
The paper is organized as follows: we first review related works in section 2, and then elaborately present our region suction point detection method with a deep neural network in U-net framework in section 3. Finally, we show our experiments and conclusion in section 4 and section 5.
II. RELATED WORK
Object grasp has a long history in robotics research and it is still an active research topic until now. Early studies mainly research the pose estimation and the grasp plan with forceclosure and form closure. Rahardja used 2D image feature to recognize the objects and estimate their poses in plane coordinates in bin-picking situations [2]. RANSAC [11] is widely used in pose estimation with image feature or point clouds for object picking in cluttered scene [6] [12][13] [14]. ICP [15] are also often used in object recognition and pose estimation with point clouds [16] [17].
Most of researches are based on force-closure and form closure in grasp plan field in past decades [18]. The stability of grasped objects in force unclosed situation is also studied [19]. Force-close with the effect of friction is also explored [20]. More details of grasping configuration could be found in [21]. Nevertheless, All these methods are model-based and need the template to match. They have weak performance in complicated background and lack the ability to deal with unknown object. Learning based methods are also used in synthesizing grasp configuration. These methods could predict the grasp point directly without pose estimation or 3D model of target objects [22]. With the great success of deep neural network in vision tasks, most of learning based grasp detection methods use neural networks to process vision information in recent research [9][10] [23].
Rectangles were used stand for grasp configuration and image features were learned automatically to detect robotic grasps with RGB-D information in situation with objects placed separately [23]. Some variant was proposed to improve the speed of this learning-based method. Pinto used the same grasp configuration and studied grasp detection with self-supervision type, which was low efficient and needed 700 hours [9]. Levine combined reinforcement learning with vision robotic grasp with millions grasp trials. As a result, the robot could grasp objects placed in a bin without hand-eye calibration [24]. Konstantinos improved this method with GAN and domain adaption using simulation to speed up the learning speed [25]. However, learning from scratch is low efficient and time-consuming. Levine's method and its variant are all only using the RGB images without depth information.
Pas used point clouds got by depth sensor to generate lots of grasp candidates with some projection process and evaluated these candidates with a convolution neural network [10]. These methods described above are all in situation with parallel-jaw or multi-finger grippers. Mahler introduced a Grasp Quality Convolution Neural Network (GQ-CNN) for estimating the quality of suction with point clouds, which used 1500 3D object models to train the network [26]. Zeng applied fully convolutional networks to suction grasp detection and took the first place in Amazon Robotics Challenge [27]. However, it needed reprojection before inputting prediction network and used the height map to predict grasp region. Inspired by above research, we directly use the perception information of RGB-D camera to input prediction network in suction gripper configuration. At the same time, we compare the performance of RGB images, RGB image with depth maps and RGB-Points. And we could find that combining RGB images with point clouds have the best performance. The perception network has a U-net framework which combines down-pooling operation and up-sample operation shown as Figure 1. Left-to-right grey arrow means a layer of convolution neural network. Top-to-bottom green arrow means a layer of max-pooling down-sampling features and bottom-totop brown arrow means a layer of deconvolution operation which could expand the size of feature map. The input of U-net is a RGB-D camera perception information, which includes RGB image, depth image or point clouds. The output of this neural network is the successful suction region probability map. Each pixel value of this probability map denotes predicted successful possibility while sucking vertically at the 3D point corresponding to this pixel point.
The U-net framework had been successfully used in image segmentation and region detection with images in different situations. In this paper, we use U-net to predict the suction grasp region map with visual information directly in robotic picking system. The framework is mainly composed by convolutional layers which could learn multifold scale features expression by itself avoiding the complex artificial feature design. The detail architecture of this region prediction neural network is shown in Table 1. The size of input images is 128×128 while the size of output is 64×64. Some operations of batch normalization are executed after each convolutional layer, which are not expressed in the table. To train the U-net, we collect training data with an Intel Realsense camera, which gets color image and depth map with a 30Hz frequency. The points could also be obtained after a simple process with projection matrix of the sensor. The relation between point clouds and depth map is as follows:
[ ] = −1 [ 1 ] (1)
where Σ is projection matrix of the RGB sensor, [ , , ] is coordinate of point in RGB local frame, z is the value in the depth map and [ , , 1] stands for generalized coordinate of each point in depth map. What we need to point out is that the depth map has been registered into the RGB frame with official RealSense SDK. And then we labeled these images as mask maps manually (shown in Figure 2).
Figure 2: Labeled images and prediction
Graspable region was labeled as one while other area was labeled as zero. We totally collected 950 labeled images and 800 images were used for training this grasp region prediction network. And the rest of these images were used for testing and validation purpose. As the labeled data was limited, we also used some data augmentation tricks such as flip and rotation to expand the labeled data in training phase. And then U-net could be trained with these labeled images. A weight cross-entropy loss function with weight decay was chosen in this situation:
Loss = αy(− log(̃)) − (1 − ) log(1 −̃) + 2 (2)
where α is weight of positive in cross entropy, is weight of regularization, is parameter of the network while y, ̃ is label value and prediction value respectively. The next step was choosing the suction point with the predict map. As shown before, the map meant the probability of graspable region. The graspable region was partly disconnected. We used a gaussian filter to smooth the result and focus on the center of the graspable region. The gaussian filter was followed:
After that normalized operation was executed. At last we chose the max value point of the processed map as the grasp point.
IV. EXPERIMENT AND RESULTS
The network was structured with Tensorflow1.0, a machine learning system published by Google. And the hardware is a notebook with a 2.6GHz Intel Core i7-6700HQ CPU and a NVDIA GTX 965 GPU. The learning rate is 0.001 with a decay 0.8. The weight of positive pixel, namely α in the loss function, is 5, 4, 2 respectively with different inputs. And the weight of regularization ( ) is 10 −4 . The sensor is Intel Realsense SR300 RGB-D camera. The resolution of color image is 1920 × 1080 while the resolution of infrared camera is 640 × 480. The detect range of depth is 0.2m to 1.5m with a precision 0.125 millimeter.
We trained the U-net with different inputs and compared the performance of these inputs. There was color image (RGB), color image with depth (RGB-D) and color image with point clouds (RGB-Points). The point clouds were calculated with the depth image and the projection matrix of the visual sensor. As the point clouds were special inputs, we first chose a boundary of these points and normalized coordinates of these points to 0 ~ 1 respectively. The points beyond this boundary were set to zeros. Because of the precision of the depth sensor, there are many points having null depth value. We set x-coordinate, ycoordinate and z-coordinates of these points are all zeros. These points were all in range from 0 to 1 and could be processed like images.
Traditionally, PR-curves was used to evaluate the performance of the saliency detection and segmentation However, recall rate was not important. We chose the precision rate as the evaluation index. After the normalization of prediction map, different threshold value was chosen to evaluate the performance.
= ( >= ℎ ℎ ) ( =1)(4)
The results of these experiments were shown in Table 2 and Figure 3. We could find that color images with point clouds have the best performance compared with RGB and RGB-D inputs. The depth dimension could also improve the performance of the prediction of grasp region in cluttered scene. We also compared the performance of gaussian filter post process. The results could be found in Figure 3. The result with gaussian smooth was shown in Figure 4 and Table 3. Compared Table 2 with Table 3, we could find that post process of prediction map improved the performance of the prediction in We also constructed a suction grasping platform with a real robot in ROS environment. The framework of grasping platform was shown in Figure 5. The camera got images and depth maps and they were sent to ROS with a ROS message. Grasping prediction node operated these images to get the suction point with tensorflow toolkits. After got the suction point, ROS executed motion plan with MoveIt software. At last the ROS driven the real robot to execute the suction manipulation. 7489 We used color images and point clouds as the input of U-net and executed some experiments in this platform. The hardware platform was a WidowX Robot Arm (Produced by Trossen Robotics, USA), which was shown in Figure 6. The results showed that the robot could suck and move all the cylinder objects in cluttered scenario efficiently. This paper proposed a new method which used a region detection convolution framework to combine RGB and Points information to detect picking point in clutter. Experimental results demonstrated that U-net framework could also be used for picking point detection with a high efficiency as it revealed in image segmentation. This method predicted grasping region without recognition and pose estimation which was very appropriate in object-agnostic scenario. The results were also shown that combining color image and point clouds could get a high performance with the same structure of neural networks.
Figure 1 :
1The Framework of U-net III. MATHODOLOGY
Figure 4 :Figure 3 :
43The Results with Gaussian Filter The Results of Different Inputs all three different inputs. With the threshold 0.98, the precision value of color image with point clouds was 95.74%.
Figure 5 :
5The Framework of Suction Platform
Figure 6 :
6The Suction Grasp Platform V. CONCLUSION
Table 1 :
1The Structure of U-netNumber
Layers
Parameters
1.
Input
Images, Depth, Points
2.
convolution1
Kernel: 28; Size: 11×11
3.
convolution2
Kernel: 64; Size: 7×7
4.
max-pooling1
Stride: 2; Size: 2×2
5.
convolution3
Kernel: 64; Size: 5×5
6.
max-pooling2
Stride: 2; Size: 2×2
7.
convolution4
Kernel: 128; Size: 3×3
8.
max-pooling3
Stride: 2; Size: 2×2
9.
convolution5
Kernel: 192; Size: 3×3
10.
deconvolution1
Kernel: 192; Size: 2 × 2
Image
Label
Prediction
Table 2 :
2The Results of Different InputsThreshold
RGB
RGB-D
RGB-Points
0.98
0.7597
0.7643
0.8046
0.85
0.6157
0.6960
0.7134
Table 3 :
3The Results with Gaussian FilterThreshold
RGB
RGB-D
RGB-Points
0.98
0.7853
0.8110
0.9574
0.85
0.5970
0.6717
0.
ACKNOWLEDGMENT This work was supported by National Natural Science Foundation of China.
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Grasp pose detection in point clouds. Ten Pas, Andreas, The International Journal of Robotics Research. 36ten Pas, Andreas, et al. "Grasp pose detection in point clouds." The International Journal of Robotics Research 36.13-14 (2017): 1455-1473.
Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Martin A Fischler, Robert C Bolles, Communications of the ACM. 24Fischler, Martin A., and Robert C. Bolles. "Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography." Communications of the ACM 24.6 (1981): 381- 395.
Object recognition and full pose registration from a single image for robotic manipulation. Alvaro Collet, IEEERobotics and Automation, 2009. ICRA'09Collet, Alvaro, et al. "Object recognition and full pose registration from a single image for robotic manipulation." Robotics and Automation, 2009. ICRA'09. IEEE International Conference on. IEEE, 2009.
Rigid 3D geometry matching for grasping of known objects in cluttered scenes. Chavdar Papazov, The International Journal of Robotics Research. 31Papazov, Chavdar, et al. "Rigid 3D geometry matching for grasping of known objects in cluttered scenes." The International Journal of Robotics Research 31.4 (2012): 538-553.
An efficient ransac for 3d object recognition in noisy and occluded scenes. Chavdar Papazov, Darius Burschka, Asian Conference on Computer Vision. Berlin, HeidelbergSpringerPapazov, Chavdar, and Darius Burschka. "An efficient ransac for 3d object recognition in noisy and occluded scenes." Asian Conference on Computer Vision. Springer, Berlin, Heidelberg, 2010.
Method for registration of 3-D shapes. Paul J Besl, Neil D Mckay, Sensor Fusion IV: Control Paradigms and Data Structures. 1611Besl, Paul J., and Neil D. McKay. "Method for registration of 3-D shapes." Sensor Fusion IV: Control Paradigms and Data Structures. Vol. 1611. International Society for Optics and Photonics, 1992.
Efficient bin-picking and grasp planning based on depth data. Dirk Buchholz, IEEE. IEEERobotics and Automation (ICRA)Buchholz, Dirk, et al. "Efficient bin-picking and grasp planning based on depth data." Robotics and Automation (ICRA), 2013 IEEE International Conference on. IEEE, 2013.
Fast graspability evaluation on single depth maps for bin picking with general grippers. Yukiyasu Domae, IEEE. IEEERobotics and Automation (ICRA)Domae, Yukiyasu, et al. "Fast graspability evaluation on single depth maps for bin picking with general grippers." Robotics and Automation (ICRA), 2014 IEEE International Conference on. IEEE, 2014.
On the closure properties of robotic grasping. Antonio Bicchi, The International Journal of Robotics Research. 14Bicchi, Antonio. "On the closure properties of robotic grasping." The International Journal of Robotics Research 14.4 (1995): 319-334.
On the stability of grasped objects. W Howard, Vijay Stamps, Kumar, IEEE transactions on robotics and automation. 12Howard, W. Stamps, and Vijay Kumar. "On the stability of grasped objects." IEEE transactions on robotics and automation 12.6 (1996): 904- 917.
Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming. Yun-Hui Liu, IEEE Transactions on Robotics and Automation. 15Liu, Yun-Hui. "Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming." IEEE Transactions on Robotics and Automation 15.1 (1999): 163-173.
An overview of 3D object grasp synthesis algorithms. Anis Sahbani, Sahar El-Khoury, Philippe Bidaud, Robotics and Autonomous Systems. 60Sahbani, Anis, Sahar El-Khoury, and Philippe Bidaud. "An overview of 3D object grasp synthesis algorithms." Robotics and Autonomous Systems 60.3 (2012): 326-336.
Empty the basket-a shape based learning approach for grasping piles of unknown objects. David Fischinger, Markus Vincze, iros. 2012Fischinger, David, and Markus Vincze. "Empty the basket-a shape based learning approach for grasping piles of unknown objects." iros. 2012.
Deep learning for detecting robotic grasps. Ian Lenz, Honglak Lee, Ashutosh Saxena, The International Journal of Robotics Research. 34Lenz, Ian, Honglak Lee, and Ashutosh Saxena. "Deep learning for detecting robotic grasps." The International Journal of Robotics Research 34.4-5 (2015): 705-724.
Learning hand-eye coordination for robotic grasping with deep learning and large-scale data collection. Sergey Levine, The International Journal of Robotics Research. 37Levine, Sergey, et al. "Learning hand-eye coordination for robotic grasping with deep learning and large-scale data collection." The International Journal of Robotics Research 37.4-5 (2018): 421-436.
Using simulation and domain adaptation to improve efficiency of deep robotic grasping. Konstantinos Bousmalis, IEEE International Conference on Robotics and Automation (ICRA). IEEEBousmalis, Konstantinos, et al. "Using simulation and domain adaptation to improve efficiency of deep robotic grasping." 2018 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2018.
Dex-Net 3.0: Computing Robust Robot Vacuum Suction Grasp Targets in Point Clouds using a New Analytic Model and Deep Learning. Jeffrey Mahler, arXiv:1709.06670arXiv preprintMahler, Jeffrey, et al. "Dex-Net 3.0: Computing Robust Robot Vacuum Suction Grasp Targets in Point Clouds using a New Analytic Model and Deep Learning." arXiv preprint arXiv:1709.06670 (2017).
Robotic pick-and-place of novel objects in clutter with multi-affordance grasping and cross-domain image matching. A Zeng, S Song, K.-T Yu, E Donlon, F R Hogan, M Bauza, ICRAA. Zeng, S. Song, K.-T. Yu, E. Donlon, F. R. Hogan, M. Bauza, et al., "Robotic pick-and-place of novel objects in clutter with multi-affordance grasping and cross-domain image matching," ICRA, 2018
|
[] |
[
"Mixed reality technologies for people with dementia: Participatory evaluation methods",
"Mixed reality technologies for people with dementia: Participatory evaluation methods"
] |
[
"Shital Desai [email protected] \nSchool of Arts Media Performance and Design\nKITE UHN Toronto Rehabilitation Institute\nSocial and Technological Systems (SaTS) lab\nYork University\n550 University Ave #12Toronto, TorontoCanada, Canada\n",
"Arlene Astell [email protected] \nSchool of Arts Media Performance and Design\nKITE UHN Toronto Rehabilitation Institute\nSocial and Technological Systems (SaTS) lab\nYork University\n550 University Ave #12Toronto, TorontoCanada, Canada\n"
] |
[
"School of Arts Media Performance and Design\nKITE UHN Toronto Rehabilitation Institute\nSocial and Technological Systems (SaTS) lab\nYork University\n550 University Ave #12Toronto, TorontoCanada, Canada",
"School of Arts Media Performance and Design\nKITE UHN Toronto Rehabilitation Institute\nSocial and Technological Systems (SaTS) lab\nYork University\n550 University Ave #12Toronto, TorontoCanada, Canada"
] |
[] |
Technologies can support people with early onset dementia (PwD) to aid them in Instrumental Activities of Daily Living (IADL). The integration of physical and virtual realities in Mixed reality technologies (MRTs) could provide scalable and deployable options in developing prompting systems for PwD. However, these emerging technologies should be evaluated and investigated for feasibility with PwD. Survey instruments such as SUS, SUPR-Q and ethnographic methods that are used for usability evaluation of websites and apps are used to evaluate and study MRTs. However, PwD who cannot provide written and verbal feedback are unable to participate in these studies. MRTs also present challenges due to different ways in which physical and virtual realities could be coupled. Experiences with physical, virtual and the couplings between the two are to be considered in evaluating MRTs. This paper presents methods that we have used in our labs -DATE and SaTS, to study the use of MRTs with PwD. These methods are used to understand the needs of PwD and other stake holders as well as to investigate experiences and interactions of PwD with these emerging technologies.
| null |
[
"https://arxiv.org/pdf/2107.07336v1.pdf"
] | 235,899,098 |
2107.07336
|
25df147c9be5cffde98cbbd5314c51267a04b5ab
|
Mixed reality technologies for people with dementia: Participatory evaluation methods
Shital Desai [email protected]
School of Arts Media Performance and Design
KITE UHN Toronto Rehabilitation Institute
Social and Technological Systems (SaTS) lab
York University
550 University Ave #12Toronto, TorontoCanada, Canada
Arlene Astell [email protected]
School of Arts Media Performance and Design
KITE UHN Toronto Rehabilitation Institute
Social and Technological Systems (SaTS) lab
York University
550 University Ave #12Toronto, TorontoCanada, Canada
Mixed reality technologies for people with dementia: Participatory evaluation methods
CCS Concepts • Human-centred computing • Interaction design • Interaction design process and methods • User centered design Keywords Mixed realityDementiaExperienceEvaluation
Technologies can support people with early onset dementia (PwD) to aid them in Instrumental Activities of Daily Living (IADL). The integration of physical and virtual realities in Mixed reality technologies (MRTs) could provide scalable and deployable options in developing prompting systems for PwD. However, these emerging technologies should be evaluated and investigated for feasibility with PwD. Survey instruments such as SUS, SUPR-Q and ethnographic methods that are used for usability evaluation of websites and apps are used to evaluate and study MRTs. However, PwD who cannot provide written and verbal feedback are unable to participate in these studies. MRTs also present challenges due to different ways in which physical and virtual realities could be coupled. Experiences with physical, virtual and the couplings between the two are to be considered in evaluating MRTs. This paper presents methods that we have used in our labs -DATE and SaTS, to study the use of MRTs with PwD. These methods are used to understand the needs of PwD and other stake holders as well as to investigate experiences and interactions of PwD with these emerging technologies.
INTRODUCTION
Technologies can support people with early onset Dementia (PwD) to participate in Instrumental Activities of Daily Living (IADL) such as making a cup of tea, cooking and laundry. IADL is a list of activities related to independent living that health care professionals use to assess PwD for the level of impairment and their ability to care for themselves. PwD are unable to sequence tasks in an activity which makes it difficult for them to finish the task. Intelligent prompting systems can support PwD in completing IADL through prompts generated when PwD lose track of the activity (for example, A. Astell et al., 2009;Orpwood et al., 2008). Blended environments such as Mixed Reality Technologies (MRTs) could offer scalable and reconfigurable solutions that can be easily adopted and deployed. MRTs consist of augmentations of physical and virtual elements and they come in various configurations [3]. Augmented reality and virtuality are two main categories of augmentations depending on whether physical is augmented with virtual (augmented reality) or virtual is augmented with physical (augmented virtuality). Use of MRTs as intelligent devices have been explored with Microsoft Kinect [4], augmented reality (AR) HoloLens [5] and projection based systems [6]. However, for MRTs to be used as prompts, these technologies need to be studied and evaluated with PwD. Understanding the experiences and interactions of PwD with MRTs is important for adoption and acceptance of these technologies by PwD. Our research on designing MRTs for PwD has thus focused on investigating experiences of PwD with MRTs through the concept of presence in blended environments [7] and identifying interaction modalities that work for PwD using perception action model [8]. Ethnographic methods such as participatory and codesign methods, observations, interviews, focus groups andsurveys can provide useful insights into the needs and experiences of people. Experience is evaluated through observations of users carrying out certain tasks with the technology. Standardised measures such as the System Usability Scale (SUS) for apps and SUPR-Q for websites involves users reflecting on their experiences with the technology with open ended and detailed questions about features [9], [10]. The same measures are also used in the design and evaluation of apps and websites for PwD [11]. Designing for experiences with MRTs revolves around creating an illusion of being in a certain place or environment when you are physically situated in another place [12]. So, attempts are made to make the digital world ubiquitous to the user. However, all realities in the design should be observable and detectable by PwD. They should be aware of the reality with which they are interacting for successful perception and action loops, thus contributing to positive experiences with the technology [7]. Creating illusions or the feeling of being somewhere else creates confusion rather than enhanced positive experiences in the context of PwD using MRTs as assistive technologies. Desai et al further emphasise that studying experiences with MRTs involves understanding people's experiences with physical and digital space as well as the correspondences or couplings between the two. Direct access to elements or objects in these spaces and the natural flow of actions on these elements is important. The challenge is to facilitate all of these while keeping the mediating technology ubiquitous to the user. Ethnographic methods could present challenges in eliciting information from PwD and thus in evaluating technologies to be designed for them. Some PwD may be unable to provide verbal or written feedback in interviews and surveys. Studies such as [7], [8], [13] have successfully used observation methods to investigate experiences and interactions of PwD with MRTs. We are developing research methods in our labs: Social and Technological Systems (SaTS) lab and Dementia Ageing Technology and Engagement lab (DATE)where the primary objective is to allow vulnerable populations or those who cannot provide verbal and written feedback due to their impairments, to have a say in the entire design and developmental process of technologies. We will discuss these methods and our experiences with these methods at the workshop -'Evaluating User Experiences in Mixed Reality'.
Cocreating experiences using Tungsten
We have used TUNGSTEN TM (Tools for User Needs Gathering to Support Technology Engagement) (http://tungsten-training.com), a set of practical tools for researchers and technology developers to involve older adults as experts in the technology development, testing and implementation process, from conception of ideas to adoption of products . We have used these tools in half day and full day workshop settings to allow participants to share their experiences with technologies with all stakeholders [14], [15]. Older adults with dementia, their care givers, technology developers and health care professionals engaged in three TUNGSTEN co-creation activities ( Figure 1): (i) Technology Interaction -activity designed to determine factors that influence older adults' impressions of new technologies from a 'mystery box' and that will enable them to persevere with trying to get them working and not abandon them, (ii) Show and Tell -activity designed to understand what makes people love or abandon technology that they have owned in the past or they own currently and (iii) Scavenger Hunt -is used to gather early feedback on a prototype, make it ready for market release and want to understand how users interact with products that are under development.
Observation method
We used off the shelf MRTs -HoloLens and XBOX Kinect from Microsoft, Osmo from Tangible Play and ARkit from Apple (IphoneX) in our studies. Using off the shelf existing technologies is an effective way to understand technology needs of people and their perception action behaviour [7], [16]- [18]. We have used game play as a probe to elicit natural behaviour in the participants when they interact with MRTs. Games can also be easily integrated in the day programs of PwD. Play also acts as an ice breaker and makes participants feel more comfortable around emerging technologies such as MRTs. PwD played Tangram on Osmo, Young Conker on HoloLens and a game of bowling on XBOX Kinect and Stack AR IphoneX ARkit (Figure 2).
(a) (c) (b) (d)
Figure 2 (a) Tangram on Osmo (b) Bowling on Kinect XBOX (c) Young Conker on HoloLens (d) Stack AR on IphoneX
Cognitive impairment of the participant is recorded using assessment tools such as MoCA before the game play sessions. The observations are video recorded for analysis in Noldus Observer XT, a software for analysis of behavioral data. It facilitates coding and description of participant behaviour over a period of observation time. The coding heuristics can either be determined deductively before the data collection, based on a theoretical framework or determined inductively during the analysis from the data. The coded data is then analysed either qualitatively using visualisations in Observer XT or quantitively using statistics or both. Figure 3 shows the coding environment in Observer XT, where data collected simultaneously from maximum four sources can be analysed at a given time.
Figure 3 Coding multiple video sources in Observer XT
Triangulation with biological and egocentric data
In our studies with children [17], we have successfully used retrospective interviews [19] and concurrent and retrospective verbal protocols [20] in observational studies to reliably identify the behaviour codes in the data for thematic analysis. With PwD, some participants provided limited verbal protocols during the game play, but most did not provide verbal feedback. Thus, we are exploring use of additional data sources such as gaze data using eye tracking glasses, facial emotions using a face reader software and biological signals using EEG in addition to behavioural data. Figure 4 shows gaze information captured using eye tracking glasses.
Figure 4 Gaze information captured using eye tracking glasses in IADL: making cup of tea
At any given time during participants' use of MRTs, data is captured from five sources: (1) video cameras capture behavioral data to determine actions and perceptions with the technology (2) eye tracking glasses capture gaze and pupil data to determine where participants are looking (3) FaceReader module from Observer XT indicates emotions of participants (4) EEG data provides quantitative information about neurological processes in the brain. (5) a task assessment tool created using Assessment of Motor and Process Skills (AMPS) [21] and the Perceive: Recall: Plan: Perform (PRPP) [22] is used to assess the execution of tasks in PwD with or without MRT support. Triangulation of all these data in Observer XT environment helps us to develop an exhaustive coding scheme for thematic analysis and also helps us to reliably code behaviours and interactions of participants with MRTs ( Figure 5)
Conclusion
Emerging technologies such as MRTs can support PwD in carrying out IADL. However, these technologies should be studied and evaluated with primary users and other stake holders. The impairments of PwD and the dual reality experienced in MRTs present challenges to the use of conventional methods in studying and evaluating MRTs with PwD. We have presented some of the methods that we use in DATE and SaTS lab to study MRTs with PwD. These methods are unique in the way that they can be adapted to the participant's abilities and impairments.
ACKNOWLEDGMENTS
The research described is funded by AGE-WELL. We are very thankful to all participants and the staff at Alzheimers Society of Durham, Memory and Company and Carefirst for their ongoing support in our studies. Thanks to Noldus for supporting our research with remote Observer licenses during the pandemic.
Figure 1
1People with early onset dementia participating in (a) Technology Interaction (b) Show and Tell (c) and (d)Scavenger Hunt
Figure 5
5Observer XT environment showing simultaneous visualisations of coding for data from video cameras, eye tracking glasses, FaceReader module and EEG
Working with people with dementia to develop technology: The CIRCA and Living in the Moment projects. A , PSIGE Newsl. 64A. Astell et al., "Working with people with dementia to develop technology: The CIRCA and Living in the Moment projects," PSIGE Newsl., vol. 64, 2009.
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A Projection-based Augmented Reality for Elderly People with Dementia. H Ro, Y J Park, T.-D Han, H. Ro, Y. J. Park, and T.-D. Han, "A Projection-based Augmented Reality for Elderly People with Dementia," 2019.
Designing for experiences in blended reality environments for people with dementia. S Desai, D Fels, A Astell, S. Desai, D. Fels, and A. Astell, "Designing for experiences in blended reality environments for people with dementia," 2020.
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Usability measurement of mobile applications with system usability scale (SUS)," in Industrial Engineering in the Big Data Era. A Kaya, R Ozturk, C A Gumussoy, SpringerA. Kaya, R. Ozturk, and C. A. Gumussoy, "Usability measurement of mobile applications with system usability scale (SUS)," in Industrial Engineering in the Big Data Era, Springer, 2019, pp. 389-400.
SUPR-Q: A Comprehensive Measure of the Quality of the Website User Experience. J Sauro, 1019J. Sauro, "SUPR-Q: A Comprehensive Measure of the Quality of the Website User Experience," vol. 10, no. 2, p. 19, 2015.
Using the TUNGSTEN Approach to Co-design DataDay: A Selfmanagement App for Dementia. A Astell, E Dove, C Morland, S Donovan, 10.1007/978-3-030-32835-1_11A. Astell, E. Dove, C. Morland, and S. Donovan, "Using the TUNGSTEN Approach to Co-design DataDay: A Self- management App for Dementia," no. 2020, pp. 171-185, 2020, doi: 10.1007/978-3-030-32835-1_11.
Measuring presence in virtual environments: A presence questionnaire. B G Witmer, M J Singer, Presence. 73B. G. Witmer and M. J. Singer, "Measuring presence in virtual environments: A presence questionnaire," Presence, vol. 7, no. 3, pp. 225-240, 1998.
The Kinect Project: group motionbased gaming for people living with dementia. E Dove, A , Dementia. 186E. Dove and A. Astell, "The Kinect Project: group motion- based gaming for people living with dementia," Dementia, vol. 18, no. 6, pp. 2189-2205, 2019.
Users as 'experts': co-creating experiences for people with dementia using TUNGSTEN. A Astell, S Desai, Health and Wellbeing Promotion for People Living with Dementia through Human-Centred Technologies. A. Astell and S. Desai, "Users as 'experts': co-creating experiences for people with dementia using TUNGSTEN," Int. J. Environ. Res. Public. Health, no. Health and Wellbeing Promotion for People Living with Dementia through Human-Centred Technologies, 2021.
Age-Friendly Technologies: Interaction Design with and for Older People. S Desai, C Mcgrath, H Mcneil, J Mcmurray, H Sveistrup, A J , Int. J. Environ. Res. Public. Health, no. Special Issue. 2021Experiential value-based framework for older adults' use of technologyS. Desai, C. McGrath, H. McNeil, J. McMurray, H. Sveistrup, and A. J. Astell, "Experiential value-based framework for older adults' use of technology," Int. J. Environ. Res. Public. Health, no. Special Issue "Age- Friendly Technologies: Interaction Design with and for Older People", 2021.
Intuitive Interaction Framework in User-product Interaction for People Living with Dementia," in HCI and Design in the Context of Dementia, G. Brankaert, Rens, Kenning. A Blackler, C Li-Hao, S Desai, A Astell, 2020A. Blackler, C. Li-Hao, S. Desai, and A. Astell, "Intuitive Interaction Framework in User-product Interaction for People Living with Dementia," in HCI and Design in the Context of Dementia, G. Brankaert, Rens, Kenning, Ed. 2020.
Children's embodied intuitive interaction--Design aspects of embodiment. S Desai, A Blackler, V Popovic, Int. J. Child-Comput. Interact. S. Desai, A. Blackler, and V. Popovic, "Children's embodied intuitive interaction--Design aspects of embodiment," Int. J. Child-Comput. Interact., 2019.
Age, familiarity, and intuitive use: An empirical investigation. S Lawry, V Popovic, A Blackler, H Thompson, 10.1016/j.apergo.2018.08.016Appl. Ergon. 74S. Lawry, V. Popovic, A. Blackler, and H. Thompson, "Age, familiarity, and intuitive use: An empirical investigation," Appl. Ergon., vol. 74, pp. 74-84, 2019, doi: https://doi.org/10.1016/j.apergo.2018.08.016.
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Development, Standardisation, and Administration Manual, 2 vols. A G Fisher, K B Jones, Three Star Press Inc1ColoradoAMPS: Assessment of Motor and Process SkillsA. G. Fisher and K. B. Jones, AMPS: Assessment of Motor and Process Skills., 7th ed., vol. Volume 1: Development, Standardisation, and Administration Manual, 2 vols. Colorado: Three Star Press Inc., 2012.
The perceive, recall, plan, perform (PRPP) system of task analysis. C Chapparo, J Ranka, Occup. Perform. Model Aust. Monogr. 1C. Chapparo and J. Ranka, "The perceive, recall, plan, perform (PRPP) system of task analysis," Occup. Perform. Model Aust. Monogr., vol. 1, 1997.
|
[] |
[
"Application of Quantitative Systems Pharmacology to guide the optimal dosing of COVID-19 vaccines",
"Application of Quantitative Systems Pharmacology to guide the optimal dosing of COVID-19 vaccines"
] |
[
"Mario Giorgi \nCertara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK\n",
"Rajat Desikan \nCertara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK\n",
"Piet H Van Der Graaf \nCertara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK\n",
"Andrzej M Kierzek \nCertara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK\n"
] |
[
"Certara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK",
"Certara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK",
"Certara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK",
"Certara QSP\nCertara UK Limited\n1 Concourse WayS1 2BJSheffieldUK"
] |
[] |
Optimal use and distribution of Covid-19 vaccines involves adjustments of dosing. Dueto the rapidly-evolving pandemic, such adjustments often need to be introduced before full efficacy data are available. As demonstrated in other areas of drug development, quantitative systems pharmacology (QSP) is well placed to guide such extrapolation in a rational and timely manner. Here we propose for the first time how QSP can be applied
|
10.1002/psp4.12700
|
[
"https://arxiv.org/pdf/2102.07610v1.pdf"
] | 231,924,906 |
2102.07610
|
204b9332b79dc1323370341a6bf08f80181f622d
|
Application of Quantitative Systems Pharmacology to guide the optimal dosing of COVID-19 vaccines
Mario Giorgi
Certara QSP
Certara UK Limited
1 Concourse WayS1 2BJSheffieldUK
Rajat Desikan
Certara QSP
Certara UK Limited
1 Concourse WayS1 2BJSheffieldUK
Piet H Van Der Graaf
Certara QSP
Certara UK Limited
1 Concourse WayS1 2BJSheffieldUK
Andrzej M Kierzek
Certara QSP
Certara UK Limited
1 Concourse WayS1 2BJSheffieldUK
Application of Quantitative Systems Pharmacology to guide the optimal dosing of COVID-19 vaccines
Optimal use and distribution of Covid-19 vaccines involves adjustments of dosing. Dueto the rapidly-evolving pandemic, such adjustments often need to be introduced before full efficacy data are available. As demonstrated in other areas of drug development, quantitative systems pharmacology (QSP) is well placed to guide such extrapolation in a rational and timely manner. Here we propose for the first time how QSP can be applied
real time in the context of COVID-19 vaccine development.
The SARS-CoV-2 pandemic has catalysed a remarkable mobilisation in vaccine development. The virus genome was sequenced almost instantly after the first cases were identified and new vaccines entered clinical trials within a couple of months, followed by regulatory approval and rollout of national vaccination programs within a year. Most of these vaccines use platform modalities, in some cases approved for the first time, which will enable even more rapid updates following the discovery of new variants.
In the initial stages of Covid-19 vaccine development, there was little time for extensive optimisation of treatment regimen (i.e. dose amount, number of doses and dosing intervals).
To date, most vaccines have progressed successfully from first-in-human studies to demonstration of efficacy in the wider population within months. However, often only after regulatory approval and roll-out in the real world does the critical importance of optimisation of dosing regimens become apparent, mainly due to the challenges of balancing limited supply with near-universal demand in the context of epidemiological and health-economical outcomes at local and international levels. For example, the United Kingdom Joint Committee on Vaccination and Immunisation (UK JCVI) recommended to extend the interval between the primary and booster doses from the originally-approved 3 or 4 weeks to 12 weeks (which at the time of the recommendation had not been tested), thus allowing single dose vaccination of twice the number of people in the first phase of the rollout 1 . Another potential example of a possible area for dose optimisation, both in terms of efficacy and supply-chain management, is the increased response reported for an arm of AZD1222 trial where half of the primary dose, followed by a booster dose was tested 2 . In addition, there is growing realisation that different vaccines may have to be combined, but it will not be possible to test all possible combinations in actual clinical trials in a timely manner 3 .
We anticipate that the requirement for dose optimisation will also remain when the focus will shift to sustaining long-term Covid-19 vaccination programs in the light of emerging new strains for the virus. In addition, due to the wide-spread roll-out of vaccination programs and expected drop in Covid-19 incidence, it would become more difficult to run clinical trials in a timely manner. Recently, we described how quantitative systems pharmacology (QSP) is being used in immuno-oncology (IO) drug development to address similar challenges, i.e., decreasing access to sufficient number of clinical trial participants and the inability to explore all possible combination therapies and dosing regimens, in a timely manner 4 . We now propose that QSP can be used in a similar manner in Covid-19 vaccine development and present the first results demonstrating proof-of-principle. Since 2017, the Immunogenicity QSP Consortium 6 has focussed on modelling formation of anti-drug antibodies (ADA), an unwanted immunological response to therapeutic proteins. We used the seminal model of Chen, Hickling and Vicini 7 as a starting point, expanded the physiological compartment structure, and created a platform model which has now been validated with ~20 clinical compounds. In the wake of the Sars-Cov-2 pandemic, we repurposed this model to Covid-19 vaccines. Since the basic biology of the humoral immune response is the same regardless of whether we simulate an unwanted ADA response to therapeutic proteins or desired immunogenicity to a vaccine antigen, we could quickly repurpose and expand the model by developing a vaccine administration module (lipid nanoparticle (LNP) mRNA in first instance). This illustrates an important and at times underestimated feature of QSPmechanistic platform models can be quickly applied across seemingly unrelated therapeutic areas, which share the same underlying fundamental biology.
Likewise, pre-clinical and clinical data collected in seemingly unrelated projects can be integrated within a single QSP platform and contribute to confidence in its application. Figure 1A shows both the calibration result and a virtual trial showing extrapolation beyond 120 days to examine response durability, as well as the predicted effect of a second 100 ug dose on antibody levels. Importantly, it is predicted that vaccination 11 months after the second dose still produces a burst of antibody synthesis, characteristic of a so-called booster effect, rather than a new, primary response. The calibrated model can then be used to examine different intervals between the first and second dose ( Figure 1B). In agreement with the expectation of many immunology experts 1 , and preliminary evidence from a subset of the AZD1222 Phase III trial 2 , expansion of the dosing interval is predicted to increase antibody responses. Our model predicts a bell-shaped response, with an optimum between 7 and 8 weeks ( Figure 1B). The 12 weeks interval proposed by UK JCVI is predicted to lead to lower immune response than this theoretical optimum, but the expected response is still higher than with the original 28 days interval. Importantly, in the model, the second dose given after 12 weeks still acts as a booster rather than a new primary dose. A potential downside of extending the time of the dosing interval is that antibodies may drop to low levels before the booster dose is administered. To explore this issue in a quantitative manner, we used convalescent plasma concentration as a reference ( Figure 1A and 1B). While the question whether IgG level is a correlate of protection remains subject to debate and investigation, we note that the median convalescent IgG level is very close to the level observed in Covid-19 vaccine Phase III trials between day 10 and 14, the earliest timepoints where placebo and vaccine incidence curves separate 10 . The prolonged exposure of the virus to relatively low level of antibodies may also increase concerns related to the selection of vaccine-escaping mutant strains 1 , although the modelling of in vivo virus mutation rates over 12 weeks would be needed before drawing conclusions. Figures 1C and 1D illustrate an application of the QSP model to examine the effects of varying dose amounts in different age groups. While QSP is in principle applicable towards mechanistically modelling the aging immune system, we adopted a more phenomenological approach here and created two virtual populations calibrated by the clinical data of Walsh et al. 11 , collected separately for younger (18-55) and older (65-85) adults. Using median convalescent concentration as a threshold, we calculated the percent of responding subjects at different time points (Figures 1C and 1D). Our results show that the antibody response is similar in age groups for 30 and 100 ug doses, consistent with observed high efficacy in older adults. However, lowering the dose to 10ug would have a larger negative impact in older adults ( Figure 1D).
Another important application of large-scale mechanistic models is to generate virtual trials that enable the investigation of biomarkers, including (virtual) ones, which were not measured in the actual clinical trials. For example, Figure 2 shows the time profiles of memory B cells and memory CD4 T-cells in plasma simulated by the model calibrated with mRNA-1273 data ( Figure 1A). These results can be used in two ways. First, the model can be subject to additional calibration by other biomarkers than IgG, thus increasing confidence in results.
Second, model predictions can be used to guide selection of most informative biomarkers to be clinically investigated.
In summary, we believe that dose regimen optimisation will become increasingly important in ongoing and future development of Covid-19 vaccines. Is seems clear that the old "trial and error" vaccine development paradigm is inadequate to meet the worlds urgent needs.
We therefore propose that, similar to other areas of drug development like IO 4 , running virtual trials ahead of and in parallel with actual clinical trials using QSP models like the one presented here should become standard practise in vaccine development. Extrapolation to different intervals between primary and booster dose. The model calibrated for mRNA-1273 vaccine was used to predict antibody response for 100 ug dose administered at intervals of 1-9 and 12 weeks. We plot median IgG ratio of 85 virtual subjects. Administration of second dose leads to burst of antibody production, with maximum amount following bell-shaped curve. C,D) Extrapolation from Phase I/II data on BNT162b2 vaccine to different dose amount in younger (C) and older (D) adults. Two doses were given with 21 day interval and amounts of 10, 30, 100 ug. The 246 and 121 virtual patients were simulated in older and younger age groups. Plots show percent of virtual patients with anti-RBD amount above median convalescent plasma concentration at each time point. Response durability depends on the dose. The 10 ug dose results in substantially lower antibody response in older individuals.
QSP focusses on supporting drug development with mechanistic modelling and simulation of underlying biology. A typical QSP model consists of a pharmacokinetic module, describing absorption, distribution, and elimination of the drug, connected to a systems biology model quantitatively describing biology of the disease and mechanisms of drug action. The model (usually expressed as a set of Ordinary Differential Equations), is first parameterised with diverse literature and preclinical data usually available before the start of a drug development project. The model then extrapolates from these data and produces a first hypothesis about efficacious dosing regimens, often before clinical data are available. When a stage of clinical trial is completed, the model is validated, refined, and then applied for extrapolation, thus informing the next stage of the program. Recently, an increasing number of models have reached the maturity required to inform regulatory submission 5 , with most applications in combination dose selection in immune-oncology 4 . In terms of regulatory acceptance, QSP follows the trajectory of physiologically-based pharmacokinetics (PBPK), where system-wide mechanistic models of physiology underlying pharmacokinetics are now routinely used in lieu of clinical trials on drug-drug interactions and recently also other fields. In an analogous manner, QSP models informed by a fast-expanding volume of pre-clinical and clinical data on Covid-19 immunology and vaccination are useful tools for optimisation of Covid-19 vaccine dosing regimens, especially in the context of increasing challenge of clinical subject recruitment and confounding factors.
Figure 1
1illustrates possible application of the new QSP model to dose regimen selection in mRNA Covid-19 vaccines. Our example focusses on extrapolation of longitudinal antibody response from Phase I/II clinical trial data to dosing intervals, dose amounts and longterm vaccination, which have not yet been tested in actual clinical trials. We use the virtual population methodology 8 to generate ensembles of parameter sets (typically referred to as virtual patients), which fall within the range of observed patient variability in 120 days long anti-RBD IgG titer profiles collected by Widge et al. 9 , for individual subjects treated with two 100 ug doses of mRNA-1273 vaccine administered with an interval of 28 days.
Figure 1 .
1Example application of QSP vaccine model to extrapolate from Phase I/II data to different dosing regimens and long term vaccination. Plots A and B show ratio of anti-RBD IgG to the median of convalescent plasma concentrations, plotted by red horizontal line. A) Calibration with mRNA-1273 data and extrapolation to annual vaccination. Black lines show simulation results for 85 virtual patients. Coloured lines show clinical data available for first 120 days. A 100 ug dose was given at days 0, 28 and 365. B)
Figure 2 .
2Example application of QSP model calibrated by Phase I/II data for investigation of biomarkers which were not observed in the clinic. The QSP model was calibrated by clinical data for anti-RBD IgG titer following administration of 100 ug mRNA-1273 vaccines to younger adults at day 0 and 28. Calibrated mechanistic model simulates not only antibodies, but also other biomarkers of interest. Here, we plot A) Memory B-cells and B) Memory CD4 T-cells in plasma compartment. Plots show ratios of the number of cells in plasma compartment, to the median number of cells at day 28, before booster dose was administered. Administration of booster dose increases both B and T cell memory. The model predicts considerable variability of individual responses, especially for T-cells.
Covid-19 vaccination: What's the evidence for extending the dosing interval?. G Iacobucci, E Mahase, BMJ. 37218Iacobucci G, Mahase E. Covid-19 vaccination: What's the evidence for extending the dosing interval? BMJ 372 n18. (2021)
Safety and efficacy of the ChAdOx1 nCoV-19 vaccine (AZD1222) against SARS-CoV-2: an interim analysis of four randomised controlled trials in Brazil, South Africa, and the UK. M Voysey, Lancet. 397Voysey M, et al. Safety and efficacy of the ChAdOx1 nCoV-19 vaccine (AZD1222) against SARS-CoV-2: an interim analysis of four randomised controlled trials in Brazil, South Africa, and the UK. Lancet 397 99-111. (2021)
Could mixing COVID vaccines boost immune response?. H L , Nature. H L. Could mixing COVID vaccines boost immune response? Nature. (2021)
Quantitative Systems Pharmacology Approaches for Immuno-Oncology: Adding Virtual Patients to the Development Paradigm. V Chelliah, Clin Pharmacol Ther. Chelliah V, et al. Quantitative Systems Pharmacology Approaches for Immuno-Oncology: Adding Virtual Patients to the Development Paradigm. Clin Pharmacol Ther. (2020)
Quantitative Systems Pharmacology: A Regulatory Perspective on Translation. I Zineh, CPT Pharmacometrics Syst Pharmacol. 8Zineh I. Quantitative Systems Pharmacology: A Regulatory Perspective on Translation. CPT Pharmacometrics Syst Pharmacol 8 336-339. (2019)
A Quantitative Systems Pharmacology Consortium Approach to Managing Immunogenicity of Therapeutic Proteins. A M Kierzek, CPT Pharmacometrics Syst Pharmacol. 8Kierzek AM, et al. A Quantitative Systems Pharmacology Consortium Approach to Managing Immunogenicity of Therapeutic Proteins. CPT Pharmacometrics Syst Pharmacol 8 773-776. (2019)
A mechanistic, multiscale mathematical model of immunogenicity for therapeutic proteins: part 2-model applications. X Chen, T P Hickling, P Vicini, CPT Pharmacometrics Syst Pharmacol. 3134Chen X, Hickling TP, Vicini P. A mechanistic, multiscale mathematical model of immunogenicity for therapeutic proteins: part 2-model applications. CPT Pharmacometrics Syst Pharmacol 3 e134. (2014)
Efficient Generation and Selection of Virtual Populations in Quantitative Systems Pharmacology Models. R J Allen, T R Rieger, C J Musante, CPT Pharmacometrics Syst Pharmacol. 5Allen RJ, Rieger TR, Musante CJ. Efficient Generation and Selection of Virtual Populations in Quantitative Systems Pharmacology Models. CPT Pharmacometrics Syst Pharmacol 5 140- 146. (2016)
Durability of Responses after SARS-CoV-2 mRNA-1273 Vaccination. A T Widge, N Engl J Med. 384Widge AT, et al. Durability of Responses after SARS-CoV-2 mRNA-1273 Vaccination. N Engl J Med 384 80-82. (2021)
Efficacy and Safety of the mRNA-1273 SARS-CoV-2 Vaccine. L R Baden, N Engl J Med. 384Baden LR, et al. Efficacy and Safety of the mRNA-1273 SARS-CoV-2 Vaccine. N Engl J Med 384 403-416. (2021)
Safety and Immunogenicity of Two RNA-Based Covid-19 Vaccine Candidates. E E Walsh, N Engl J Med. 383Walsh EE, et al. Safety and Immunogenicity of Two RNA-Based Covid-19 Vaccine Candidates. N Engl J Med 383 2439-2450. (2020)
|
[] |
[
"CYCLOSTATIONARY IN THE TIME VARIABLE UNIVERSE",
"CYCLOSTATIONARY IN THE TIME VARIABLE UNIVERSE"
] |
[
"Jiang Dong "
] |
[] |
[
"Yun Nan Astronomical Observatory"
] |
Cyclostationary processes are those signals whose have vary almost periodically in statistics. It can give rise to random data whose statistical characteristics vary periodically with time although these processes not periodic functions of time. Intermittent pulsar is a special type in pulsar astronomy which have period but not a continuum. The Rotating RAdio TransientS (RRATs) represent a previously unknown population of bursting neutron stars. Cyclical period changes of variables star also can be thought as cyclostationary which are several classes of close binary systems. Quasi-Periodic Oscillations (QPOs) refer to the way the X-ray light from an astronomical object flickers about certain frequencies in high-energy (X-ray) astronomy. I think that all above phenomenon is cyclostationary process. I describe the signal processing of cyclostationary, then discussed that the relation between it and intermittent pulsar, RRATs, cyclical period changes of variables star and QPOs, and give the perspective of finding the cyclostationary source in the transient universe.
| null |
[
"https://arxiv.org/pdf/0911.0580v1.pdf"
] | 118,227,256 |
0911.0580
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0813017f6d3cba658695c487214119bf8d9a4afe
|
CYCLOSTATIONARY IN THE TIME VARIABLE UNIVERSE
3 Nov 2009 Draft version November 4, 2009 November 4, 2009
Jiang Dong
CYCLOSTATIONARY IN THE TIME VARIABLE UNIVERSE
Yun Nan Astronomical Observatory
NAOs, CAS3 Nov 2009 Draft version November 4, 2009 November 4, 2009Preprint typeset using L A T E X style emulateapj v. 04/20/08 Draft versionSubject headings: Cyclostationary -methods: data analysis: Intermittent pulsarRRATsCyclical period changes of variables starQPOs
Cyclostationary processes are those signals whose have vary almost periodically in statistics. It can give rise to random data whose statistical characteristics vary periodically with time although these processes not periodic functions of time. Intermittent pulsar is a special type in pulsar astronomy which have period but not a continuum. The Rotating RAdio TransientS (RRATs) represent a previously unknown population of bursting neutron stars. Cyclical period changes of variables star also can be thought as cyclostationary which are several classes of close binary systems. Quasi-Periodic Oscillations (QPOs) refer to the way the X-ray light from an astronomical object flickers about certain frequencies in high-energy (X-ray) astronomy. I think that all above phenomenon is cyclostationary process. I describe the signal processing of cyclostationary, then discussed that the relation between it and intermittent pulsar, RRATs, cyclical period changes of variables star and QPOs, and give the perspective of finding the cyclostationary source in the transient universe.
INTRODUCTION
It was first mentioned by the design of synchronization algorithms for communications systems that cyclostationary process (Serpedin et al. 2005). Many processes encountered in nature arise from periodic phenomena. For example, in telecommunications, telemetry, radar, and sonar applications, periodicity is due to modulation, sampling, multiplexing, and coding operations. In mechanics it is due, for example, to gear rotation. In econometrics, it is due to seasonality; and in atmospheric science it is due to rotation and revolution of the earth (Gardner et al. 2006;Serpedin et al. 2005;Gardner 1994). Cyclostationary processes are named in multiple different ways such as periodically correlated, periodically nonstationary, periodically nonstationary or cyclic correlated processes in literature (Hurd 1989(Hurd , 1997.
In astronomy, now just in removing cyclostationary radio frequency interferences (RFI) from radio astronomical data (Leshem et al. 2000;Bretteil & Weber 2005) and the galactic white-dwarfs background that will be observed by LISA (Edlund et al. 2005) cyclostationary process be mentioned. Some radio pulsar have been discovered the phenomenon of intermittent period in recently . The nulling of the pulsar be thought a characteristic of the older pulsars. In X-ray pulsar, the phenomenon of intermittent far-flung exist for the effect of companion. It also show almost period in time series. The Rotating RAdio TransientS (RRATs) demonstrate a previously unknown population of bursting neutron stars (McLaughlin et al. 2006). Cyclical period changes of variables star that including Algol, WUrsaeMajoris, and RS Canum Venaticorum systems and the cataclysmic variables, and RR Lyrae star et al. also can be thought as cyclostationary. The fastest variability components in X-ray binaries are the kilohertz quasi-periodic oscillations (kHz QPOs), which occur in a wide variety of low magnetic-field neutron Electronic address: [email protected] star systems(van der Klis 2005). The signal process of the above phenomenon, in a statistical sense, as a periodic function of time. These kind of processes have been studied for many years, and are usually referred to as cyclostationary random processes (see (Gardner et al. 2006;Serpedin et al. 2005;Gardner 1994) for a comprehensive overview of the subject and for more references).
In what follows we will briefly summarize the properties of cyclostationary processes, then discuss the cyclostationary signal process in some astronomical phenomenon, in the end give the perspective of finding cyclostationary processes in the time variable universe.
THE BASIC OF CYCLOSTATIONARY PROCESSES
The continuous stochastic process X (t) having finite second order moments is said to be cyclostationary with period T if the following expectation values
E[X (t)] = m(t) = m(t + T ), (1) E[X (t ′ )X (t)] = C(t ′ , t) = C(t ′ + T, t + T )(2)
are periodic functions of period T , for every (t ′ , t) ∈ R × R. We will assume m(t) = 0 for simplicity now (Gardner & Franks 1975;Hurd 1989Hurd , 1997. The important special case of cyclostationary signals are those which exhibit cyclostationary in second-order statistics (e.g., the autocorrelation function). It is called widesense cyclostationary signals, and is analogous to widesense stationary processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series (Gardner 1978). If X (t) is cyclostationary, then the function B(t, τ ) ≡ C(t + τ, t) for a given τ ∈ R is periodic with period T , and it can be represented by the following Fourier series
B(t, τ ) = ∞ r=−∞ B r (τ )e i2π rt T ,(3)
where the functions B r (τ ) are given by
B r (τ ) = 1 T T 0 B(t, τ )e −i2πr t T dt .(4)
The Fourier transforms g r (f ) of B r (τ ) are the so called "cyclic spectra" of the cyclostationary process X (t) (Hurd 1989)
g r (f ) = ∞ −∞ B r (τ )e −i2πf τ dτ .(5)
If a cyclostationary process is real, the following relationships between the cyclic spectra hold
B −r (τ ) = B * r (τ ) , (6) g −r (−f ) = g * r (f ) ,(7)
where the symbol * means complex conjugation. This implies that, for a real cyclostationary process, the cyclic spectra with r ≥ 0 contain all the information needed to characterize the process itself. The function σ 2 (τ ) ≡ B(0, τ ) is the variance of the cyclostationary process X (t), and it can be written as a Fourier decomposition as a consequence of Eq. (4)
σ 2 (τ ) = ∞ r=−∞ H r e i2π rτ T ,(8)
where H r ≡ B r (0) are harmonics of the variance σ 2 . From Eq. (6) it follows that H −r = H * r . For a discrete, finite, real time series X t , t = 1, . . . , N the cyclic spectra can be estimated by generalizing standard methods of spectrum estimation used with stationary processes. Assuming again the mean value of the time series X t to be zero, the cyclic autocorrelation sequences are defined as
s r l = 1 N N −|l| t=1 X t X t+|l| e − i2πr(t−1) T .(9)
The cyclic autocorrelations are asymptotically (i.e. for N → ∞) unbiased estimators of the functions B r (τ ) (Hurd 1989). The Fourier transforms of the cyclic autocorrelation sequences s r l are estimators of the cyclic spectra g r (f ). These estimators are asymptotically unbiased, and are called "inconsistent estimators" of the cyclic spectra, i.e. their variances do not tend to zero asymptotically. In the case of Gaussian processes (Hurd 1989) consistent estimators can be obtained by first applying a lag window to the cyclic autocorrelation and then perform a Fourier transform.
The alternative procedure for identifying consistent estimators of the cyclic spectra is to first take the Fourier transform (Gardner et al. 2006;Gardner 1994),X (f ), of the time series X (t)
X (f ) = N t=1 X t e −i2πf (t−1)(10)
and then estimate the cyclic periodograms g r (f )
g r (f ) =X (f )X * (f − 2πr T ) N .(11)
By finally smoothing the cyclic periodograms, consistent estimators of the spectra g r (f ) are then obtained. The estimators of the harmonics H r of the variance σ 2 of a cyclostationary random process can be obtained by first forming a sample variance of the time series X t . The sample variance is obtained by dividing the time series X t into contiguous segments of length τ 0 such that τ 0 is much smaller than the period T of the cyclostationary process, and by calculating the variance σ 2 I over each segment. Estimators of the harmonics are obtained either by Fourier analyzing the series σ 2 I or by making a least square fit to σ 2 I with the appropriate number of harmonics. Note that the definitions of (i) zero order (r = 0) cyclic autocorrelation, (ii) periodogram, and (iii) zero order harmonic of the variance, coincide with those usually adopted for stationary random processes. Thus, even though a cyclostationary time series is not stationary, the ordinary spectral analysis can be used for obtaining the zero order spectra. Note, however, that cyclostationary random processes provide more spectral information about the time series they are associated with due to the existence of cyclic spectra with r > 0 (Edlund et al. 2005).
For stationary and cyclostationary time-series, there is two alternative philosophical frameworks for the two problems of estimating the time-invariant or time-variant autocorrelation function and its Fourier transform. One is based on the stochastic process model, and the other is based on the nonstochastic time-series model. Gardner, W.A. compared it, and explained that results on estimator bias and variance for these two problems couched within the stochastic process framework have analogs within the nonstochastic framework. The bias and variance results for cyclostationary time-series that are available within these two frameworks (Gardner 1991).
As an important and practical application, assuming consider a time series y t consisting of the sum of a stationary random process, n t , and a cyclostationary one X t (i.e. y t = n t + X t ). Let the variance of the stationary time series n t be ν 2 and its spectral density be E(f ). It is easy to see that the resulting process is also cyclostationary. If the two processes are uncorrelated, then the zero order harmonic Σ 2 0 of the variance of the combined processes is equal to
Σ 2 0 = ν 2 + σ 2 0 ,(12)
and the zero order spectrum, G 0 (f ), of y t is
G 0 (f ) = E(f ) + g 0 (f ) .(13)
The harmonics of the variance as well as the cyclic spectra of y t with r > 0 coincide instead with those of X t . In other words, the harmonics of the variance and the cyclic spectra of the process y t with r > 0 contain information only about the cyclostationary process X t , and are not "contaminated" by the stationary processn t (Edlund et al. 2005).
CYCLOSTATIONARY IN PULSAR, QPOS AND
VARIABLES STAR
Recently, Kramer, M. et al. discovered one class of neutron stars that are seemingly ordinary radio pulsars, but which are only active for some short time and in a quasi-periodic fashion. So they call these "Intermittent Pulsars". These pulsar that is only periodically active. It appears as a normal pulsar for about a week and then "switches off" for about one month before emitting pulses again. The pulsar, called PSR B1931+24, is unique in this behaviour and affords astronomers an opportunity to compare its quiet and active phases. Most surprisingly, the pulsar rotation slows down faster when the pulsar is on than when it is off ). In the Figure 1, part a) show a typical sequence of observations covering a 20-month interval is indicated by the black lines. It shows respectively the times of observation and the times when PSR B1931+24 was on. It is clear that the pulsar is not visible for ∼80% of the time. part b) give the appearance of the pulsar is quasi-periodic nature, demonstrated by the power spectrum of the intensity obtained from the Fourier Transform of the autocorrelation function of the mean pulse flux density obtained over the same 20-month interval. part c) is histograms of the durations of the on (solid) and off (hatched) phases. In off phases, integration over several weeks shows that any pulsed signal has a mean flux density of less than 2 µJy at 1400 MHz ..
McLaughlin, M.A. et al. discovered a class of Rotating Radio Transients(RRATS) which are identified as rotating neutron stars that send out very short flashes of radio light. These flashes are very short and very rare: one hundredth of a second long, the total time the objects are visible amounts to only about one tenth of a second per day. The isolated flashes last for between 2 and 30 milliseconds. In between, for times ranging from 4 minutes to 3 hours, the new stars are silent. They current estimates suggest that these objects are four times more common in the Galaxy than radio pulsars. In the Figure 2, from top to bottom, it is show that the original detections of J1317-5759, J1443-60 and J1826-14 in the Parkes Multibeam Survey data (McLaughlin et al. 2006). It is show that RRATs still have period in short time scale. They have identified periodicities in the range of 0.4 to 7 s in 10 of the 11 objects (Lyne 2007). Camilo, F. et al. show that XTE J18102197 emits bright, narrow, highly linearly polarized radio pulses, observed at every rotation, thereby establishing that magnetars can be radio pulsars. There is no evidence of radio emission before the 2003 X-ray outburst (unlike ordinary pulsars, which emit radio pulses all the time), and the flux varies from day to day. The flux at all radio frequencies is approximately equal-and at > 20GHz .
In wide-sense, if one pulsar have nulling or giant pulse phenomenon, it will not have strict period function with time, just in statistics have periodic. Another pulsar cyclostationary process is Shabanova, T.V. find that the nature of the observed cyclical changes in the timing residuals from PSR B1642 03 is a continuous generation of peculiar glitches whose amplitudes are modulated by a periodic large-scale sawtooth-like function. As the modulation function is periodical, the picture of cyclical timing residuals will be exactly repeated in each modulation period or every 60 years (Shabanova 2009).
QPOs were first identified in white dwarf systems(van der Klis et al. 1985) and then in neutron star systems(van der Klis 2005). Two QPO peaks (the 'twin peaks') occur in the power spectrum of the X-ray flux variations. They move up and down in fre-quency together in the 300-1200Hz range in correlation with source state and often, luminosity. The typically 300-Hz peak separation usually decreases by a few tens of Hz when both peaks move up by hundreds of Hz(van der Klis 2005). In the Figure 3, show that Keck II spectroscopy of optical mHz quasi-periodic oscillations (QPOs) in the light curve of the X-ray pulsar binary Hercules X-1 (O'Brien et al. 2001).
Year-to decade-long cyclic orbital period changes have been observed in several classes of close binary systems, including Algol, W Ursae Majoris, and RS Canum Venaticorum systems and the cataclysmic variables. The origin of these changes is unknown, but mass loss, apsidal motion, magnetic activity, and the presence of a third body have all been proposed. In the Figure 4, show that cyclical period changes in the dwarf novae V2051 Oph.
The other two statistics periodic phenomenon also relate with stellar. One is Hallinan, G. et al. de-tected periodic bursts of extremely bright, circularly polarized, coherent radio emission from the ultracool dwarf (Hallinan et al. 2007). Another is Double Periodic Variables (DPVs) that are blue stars characterized by a short periodicity (1-16 days) and a long periodicity (50-600 days) in their light curves. They were discovered in the Magellanic Clouds after a search for Be stars in the OGLE variable star catalog (Mennickent et al. 2003)
DISCUSSION AND CONCLUSIONS
Signal detection techniques designed for cyclostationary signals take account of the periodicity or almost periodicity of the signal autocorrelation function. Singlecycle and multicycle detectors exploit one or multiple cycle frequencies, respectively (Gardner et al. 2006). Some search-efficient methods of detection of cyclostationary signals (Yeung & Gardner 1996) and higher-order cyclostationary for weak-signal detection (Spooner & Gardner 1992) be developed. These will benefit of search transient source in the time variable universe, especially that have periodic in statistics.
Although no periodicities were detected in any of the sources using standard Fourier or folding methods, for ten of the sources(RRATs) McLaughlin, M.A. et al. identify a periodicity from the arrival times of the individual bursts that used search techniques similar to those described in (Cordes & McLaughlin 2003). In short, the 35-minute time series were dedispersed for a number of trial values of DM. The time series were smoothed by convolution with boxcars of various widths to increase sensitivity to broadened pulses, with a maximum boxcar width of 32 ms. Because the optimal sensitivity is achieved when the smoothing window width equals the burst width, our sensitivity is lower for burst durations greater than 32 ms. Each of these time series was then searched for any bursts above a threshold of five standard deviations, computed by calculating a running mean and root-mean-square deviation of the noisy time series. All bursts detected above a 5-σ threshold are plotted as circles, with size proportional to the signal-to-noise ratio of the detected burst. The abcissa shows arrival time while the ordinate shows the DM. Because of their finite width, intense bursts are detected at multiple DMs and result in vertical broadening of the features. Bursts which are strongest at zero DM and therefore likely to be impulsive terrestrial interference are not shown. In general these were easily identified by their detection in multiple beams of the 13-beam receiver (McLaughlin et al. 2006). More recently, Deneva, J.S. et al. use the above method and a friends-of-friends algorithm perform the ongoing Arecibo Pulsar ALFA (PALFA) survey of the Galactic planethen discover seven objects (Deneva et al. 2009) .
The discovery of RRATs increases the current Galactic population estimates of radio pulsar by at least several times. It seems also that there will be many candidates for whom it will be impractical or impossible to follow up at present with current observing facilities. These will require followup with instruments like LOFAR, FAST or the SKA. Keane, E.F. et al. note that these instruments will produce extraordinarily large volumes of data so that searching for transient RRAT-like sources will necessitate the development of automated algorithms which will use the steps as outlined above (Keane et al. 2009). So I think we should use the detection methods of cyclostationary process integrate with DM search, that will be one automated and effective algorithms to seek RRATs.
If we do similar things in find the other astronomical cyclostationary sources that include Intermittent Pulsars, QPOs et al., it will lead to discover more unusual astronomical phenomenon. Fig. 1.-Time variation of the radio emission of PSR B1931+24. During the on phases, the pulsar is easy to detect and has the stable long-term intrinsic flux density associated with most normal pulsars. Since 1998, the pulsar has been observed as frequently as twice a day.(see Kramer et al. 2006)
DJ thanks
Fig. 2 .
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The dashed and dot-dashed lines in the lower panel show the best-fit 11 yr cycle period sinusoidal ephemeris, respectively, for the data in the first and the second halves of the time interval. Fig. 4.-The dashed and dot-dashed lines in the lower panel show the best-fit 11 yr cycle period sinusoidal ephemeris, respectively, for the data in the first and the second halves of the time interval. (see Baptista et al. 2003)
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[] |
[
"A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities",
"A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities"
] |
[
"Fang Wang \nCollege of Science\nHunan Agricultural University\n410128ChangshaP. R. China\n\nDepartment of Mathematics and Statistics\nUniversity of New Brunswick\nE3B 5A3FrederictonNBCanada\n",
"Lin Wang ",
"Yuming Chen \nDepartment of Mathematics and Statistics\nUniversity of New Brunswick\nE3B 5A3FrederictonNBCanada\n\nDepartment of Mathematics\nWilfrid Laurier University\nN2L 3C5WaterlooONCanada\n"
] |
[
"College of Science\nHunan Agricultural University\n410128ChangshaP. R. China",
"Department of Mathematics and Statistics\nUniversity of New Brunswick\nE3B 5A3FrederictonNBCanada",
"Department of Mathematics and Statistics\nUniversity of New Brunswick\nE3B 5A3FrederictonNBCanada",
"Department of Mathematics\nWilfrid Laurier University\nN2L 3C5WaterlooONCanada"
] |
[
"Scientific RepoRts |"
] |
On the basis of detrended fluctuation analysis (DFA), we propose a new bivariate linear regression model. This new model provides estimators of multi-scale regression coefficients to measure the dependence between variables and corresponding variables of interest with multi-scales. Numericaltests are performed to illustrate that the proposed DFA-bsaed regression estimators are capable of accurately depicting the dependence between the variables of interest and can be used to identify different dependence at different time scales. We apply this model to analyze the PM2.5 series of three adjacent cities (Beijing, Tianjin, and Baoding) in Northern China. The estimated regression coefficients confirmed the dependence of PM2.5 among the three cities and illustrated that each city has different influence on the others at different seasons and at different time scales. Two statistics based on the scale-dependent t-statistic and the partial detrended cross-correlation coefficient are used to demonstrate the significance of the dependence. Three new scale-dependent evaluation indices show that the new DFA-based bivariate regression model can provide rich information on studied variables.In recent years, air pollution has become a more and more serious problem around the world. The new air quality model presented by the World Health Organization in 2016 confirmed that 92% of the world's population lives in areas where air quality levels exceed their limits 1 . Fortunately, more and more governments have realized the importance of managing air pollution and some actions have been placed. Nowadays, a common topic around the world is the governance of the air pollution source such as smog (the main ingredient is fine particulate matter). Many researchers have been involved in the study on the cause and propagation of smog 2-9 . Modern statistic methods provide some new perspectives to assess smog trends and propagation characteristics. Among them, most studies have focused on studying the correlations among various air pollution indicators including air pollution index (API), air quality index (AQI), fine particulate matter of PM2.5 (diameter ≤ 2.5 μm) concentrations, and PM10 (diameter ≤ 10 μm) concentrations, and very limited studies considered the correlations among neighboring areas. A common sense is that smog produced at one source place can spread to surrounding areas 6-10 . Therefore, it is more practical to explore the dependence of air pollution indicators among adjacent cities as it helps assess the causes of local smog and its spread behavior. It has been found by a newly proposed time-lagged cross-correlation coefficient in ref. 10 that there are different degrees of correlation for PM2.5 series between four neighboring cities in Northern China. However, what has not been investigated is how the PM2.5 series of one city depends on those of the neighbouring cities. In this work, we will develop a detrended fluctuation analysis (DFA)-based bivariate regression model to investigate this dependence.The simplest and maturest method to describe the dependence of variables is the linear regression. However, the information gained from the traditional linear regression cannot fully meet our need of investigation on the dependence among different variables at different time periods. On the other hand, note that the DFA proposed in 1990s 11,12 performs excellently in analyzing the long-range correlations 13 of a nonstationary series with fractality and multifractality 14,15 at different time-scales. To obtain the cross-correlation between two nonstationary series, DFA was extended to the detrended cross-correlation analysis (DCCA) 16 . By defining scale-dependent detrended Published: xx xx xxxx OPEN www.nature.com/scientificreports/ 2 Scientific RepoRts | (2018) 8:7475 | fluctuation functions, the methods of DFA and DCCA together with their extensions have been applied in a wide range of disciplines 17-30 . Since the ordinary least squares (OLS) method expresses the estimated parameters of standard regression framework as a form of variances and covariances, it builds a bridge between the regression framework and the family of DFA/DCCA as the latter can also produce variances and covariances. Then, the idea of estimating multiple time scale regression coefficients can be achieved by the DFA/DCCA. Recently, Kristoufek 31 constructed a simple DFA-based regression framework exactly by this bridge. The selected examples show the relationship between the pair of variables varies strongly across scales.In this work, we focus on the interaction of PM2.5 series of three adjacent cities in Northern China, namely, Beijing, Tianjin, and Baoding. The three cities form a triangular shape in the map. The distances between Beijing and Tianjin, Beijing and Baoding, and Tianjin and Baoding are about 115 km, 140 km, and 150 km, respectively. All three cities have a population of more than 10 million and have been greatly affected by heavy smog in recent years. The real-time data of PM2.5 series of these three cities from
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10.1038/s41598-018-25822-w
| null | 13,701,990 |
1905.10297
|
202fac79a6e78fe9153f87499d64f561b5a14b2a
|
A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities
2018
Fang Wang
College of Science
Hunan Agricultural University
410128ChangshaP. R. China
Department of Mathematics and Statistics
University of New Brunswick
E3B 5A3FrederictonNBCanada
Lin Wang
Yuming Chen
Department of Mathematics and Statistics
University of New Brunswick
E3B 5A3FrederictonNBCanada
Department of Mathematics
Wilfrid Laurier University
N2L 3C5WaterlooONCanada
A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities
Scientific RepoRts |
87475201810.1038/s41598-018-25822-wReceived: 24 October 2017 Accepted: 30 April 2018 Published: xx xx xxxx OPEN1 Correspondence and requests for materials should be addressed to F.W. (
On the basis of detrended fluctuation analysis (DFA), we propose a new bivariate linear regression model. This new model provides estimators of multi-scale regression coefficients to measure the dependence between variables and corresponding variables of interest with multi-scales. Numericaltests are performed to illustrate that the proposed DFA-bsaed regression estimators are capable of accurately depicting the dependence between the variables of interest and can be used to identify different dependence at different time scales. We apply this model to analyze the PM2.5 series of three adjacent cities (Beijing, Tianjin, and Baoding) in Northern China. The estimated regression coefficients confirmed the dependence of PM2.5 among the three cities and illustrated that each city has different influence on the others at different seasons and at different time scales. Two statistics based on the scale-dependent t-statistic and the partial detrended cross-correlation coefficient are used to demonstrate the significance of the dependence. Three new scale-dependent evaluation indices show that the new DFA-based bivariate regression model can provide rich information on studied variables.In recent years, air pollution has become a more and more serious problem around the world. The new air quality model presented by the World Health Organization in 2016 confirmed that 92% of the world's population lives in areas where air quality levels exceed their limits 1 . Fortunately, more and more governments have realized the importance of managing air pollution and some actions have been placed. Nowadays, a common topic around the world is the governance of the air pollution source such as smog (the main ingredient is fine particulate matter). Many researchers have been involved in the study on the cause and propagation of smog 2-9 . Modern statistic methods provide some new perspectives to assess smog trends and propagation characteristics. Among them, most studies have focused on studying the correlations among various air pollution indicators including air pollution index (API), air quality index (AQI), fine particulate matter of PM2.5 (diameter ≤ 2.5 μm) concentrations, and PM10 (diameter ≤ 10 μm) concentrations, and very limited studies considered the correlations among neighboring areas. A common sense is that smog produced at one source place can spread to surrounding areas 6-10 . Therefore, it is more practical to explore the dependence of air pollution indicators among adjacent cities as it helps assess the causes of local smog and its spread behavior. It has been found by a newly proposed time-lagged cross-correlation coefficient in ref. 10 that there are different degrees of correlation for PM2.5 series between four neighboring cities in Northern China. However, what has not been investigated is how the PM2.5 series of one city depends on those of the neighbouring cities. In this work, we will develop a detrended fluctuation analysis (DFA)-based bivariate regression model to investigate this dependence.The simplest and maturest method to describe the dependence of variables is the linear regression. However, the information gained from the traditional linear regression cannot fully meet our need of investigation on the dependence among different variables at different time periods. On the other hand, note that the DFA proposed in 1990s 11,12 performs excellently in analyzing the long-range correlations 13 of a nonstationary series with fractality and multifractality 14,15 at different time-scales. To obtain the cross-correlation between two nonstationary series, DFA was extended to the detrended cross-correlation analysis (DCCA) 16 . By defining scale-dependent detrended Published: xx xx xxxx OPEN www.nature.com/scientificreports/ 2 Scientific RepoRts | (2018) 8:7475 | fluctuation functions, the methods of DFA and DCCA together with their extensions have been applied in a wide range of disciplines 17-30 . Since the ordinary least squares (OLS) method expresses the estimated parameters of standard regression framework as a form of variances and covariances, it builds a bridge between the regression framework and the family of DFA/DCCA as the latter can also produce variances and covariances. Then, the idea of estimating multiple time scale regression coefficients can be achieved by the DFA/DCCA. Recently, Kristoufek 31 constructed a simple DFA-based regression framework exactly by this bridge. The selected examples show the relationship between the pair of variables varies strongly across scales.In this work, we focus on the interaction of PM2.5 series of three adjacent cities in Northern China, namely, Beijing, Tianjin, and Baoding. The three cities form a triangular shape in the map. The distances between Beijing and Tianjin, Beijing and Baoding, and Tianjin and Baoding are about 115 km, 140 km, and 150 km, respectively. All three cities have a population of more than 10 million and have been greatly affected by heavy smog in recent years. The real-time data of PM2.5 series of these three cities from
2 , where r 12,3 denotes the partial correlation coefficient between the first and second variables eliminating the effects of the third one, N − 3 is the degree of freedom) of the partial correlation coefficients to assess the statistical significance at the given significance level. Unsurprisingly, Table 1 shows that the correlations of PM2.5 between per two cities are of statistical significance. It explains that the air quality in one city of Northern China cannot be irrelevant to that of its neighbouring cities, which implies potential dependence among the three cities. However, we also note in Table 1 that the degrees of relevance are different among different cities and in different seasons though all of them are significant.
To fully detect and quantify the dependence among the PM2.5 series of the above-mentioned three cities, in this work, we construct a new bivariate regression framework which prevails the DFA method and allows us to investigate the dependence of three nonstationary series with multiple time scales. With the DFA-based variance instead of the standard variance, this new DFA bivariate regression model provides more information on the dependence among variables at different time scales. We organize the rest of this paper as follows. The performance of the proposed DFA regression model and the results on the application to PM2.5 series analysis are reported and discussed in the following section, which is followed by our conclusions. The methodologies including the standard regression method, the DFA method, and the DFA-based regression method are introduced at the end of this paper.
Results and Discussions
Performance of DFA estimators. The bivariate DFA-based regression model produces two time scalebased regression coefficients. This allows us to detect the dependence of a response variable and two independent variables at different time scales. In order to examine the validity of the model and show its advantages, in this section, we perform two numerical tests on the non-stationary bivariate regression frameworks Y = β 0 + β 1 X 1 + β 2 X 2 + ε.
In the first test, we investigate the performance of the DFA estimators under different levels of long-term dependence in X 1 , X 2 , and Y. According to 31 , the setting I is given as below: two artificial series X 1 and X 2 with length 10000 are generated by ARFIMA(0, d, 0) process with identical fractional integration parameter (d) and independent Gaussian noises (ξ i (t), i = 1 and 2) as
ξ = ∑ − = ∞ X t a d t n ( ) ( ) ( ) i n n i 0 .
The quantity a n (d) is defined by a n (d) = Γ(n − d)/[Γ(−d)Γ(n + 1)], where Γ(⋅) is the Gamma function. The error-term ε is set as a standard Gaussian noise so that the response variable Y has the same parameter d as the two independent variables. The regression coefficients are set as β 0 = β 1 = 1 and β 2 = 2. Figure 1 shows mean values and standard deviation of the two DFA estimators βˆi DFA (i = 1 and 2) for the generated series with d ranging from −0.5 to 0.5 (at the step size of 0.1). The estimators are averaged over scales between 10 and 1000 with a logarithmic isometric step. Each case is run 1000 times to eliminate the noise interference. It is clear that the two estimators locate the two given regression coefficients of 1 (Fig. 1a) and 2 ( Fig. 1b) unbiasedly, and are independent of the value of d. In addition, the standard deviations of both estimators decrease with the increasing memory. The good performance shows that the method is feasible. On the other hand, to investigate the performance of the DFA estimators faced with a long-range dependent error-term ε, we use setting II given as: the memory parameter d is fixed at 0.4 for both X 1 and X 2 , and the ε is produced by an ARFIMA process with d ε varying from −0.5 to 0.5. Other settings are as those in setting I. Figure 2 records similar information as that in Fig. 1. Although the fluctuation of DFA estimators increases with d ε , which is expected due to an increasing weight of the error-term in the dynamics of Y with the increasing memory of the error-term, we are satisfied to find that the two estimators are still unbiased pointing to the given values with a narrow range for each level of memory of the error-terms. Our second numerical test aims to show that the DFA estimators are able to identify the dependence of studied variables at different time scales whereas the classical method cannot. To this end, a binomial multifractal series (BMFs) is employed to be regarded as the independent variable X 1 , which is constructed as −1] , k = 1, 2, …, 2 n , where the parameter p ∈ (0, 0.5) (We take p = 0.3 in our test), n[k] denotes the number of digit 1 in the binary representation of the index k. The variable X 2 is a Gauss variable with 0 mean and 0.0001 standard deviation. Both X 1 and X 2 are of length 2 15 . The bivariate regression framework Y = β 0 + β 1 X 1 + β 2 X 2 + ε is set with the same coefficients as the first test (β 0 = β 1 = 1 and β 2 = 2). The error-term ε is the Gauss noise of the same strength as X 2 . For the BMFs X 1 , we remove all values smaller than 0.00001 so that only a few of the largest elements are left. In their places, we substitute Gaussian distributed random numbers with 0 mean and 0.0001 standard deviation. Then we obtain a binomial cascade series embedded in random noise. We analyze the dependence between the response variable Y and two independent variables and find that the estimated βˆD FA 2 is unbiased at 2 with a desirable error bar for every time scale, as shown in Fig. 3. However, the performance of βˆD FA 1 has changed a lot. The dependence between Y and X 1 is obviously less than the given value at the smaller scales contrary to the larger ones. This is because in the smaller scales, the dependency has been destroyed by the random noise. Our DFA estimators have the capability to recognize this effect while the classical estimators fail to do so (see the errorbar with circle symbol in Fig. 3). Performance of the three models' regression coefficients. As mentioned above, air pollution in Northern China is very serious in recent years. Fine particulate matter from industrial exhaust and smoke dust forms smog to fill in the air. We now apply our DFA regression model to investigate the dependence of PM2.5 series in these three cities. We build three bivariate models for Beijing, Tianjin, and Baoding, respectively. In Model I, the dependent variable (Y) is the PM2.5 series of Beijing while the two independent variables are the PM2.5 series of Tianjin (X 1 ) and Baoding (X 2 ); in Model II, Y is the PM2.5 series of Tianjin, X 1 is the PM2.5 series of Beijing and X 2 is the PM2.5 series of Baoding; in Model III, Y is the PM2.5 series of Baoding, X 1 and X 2 stand for the PM2.5 series in Beijing and Tianjin, respectively. In this section, we first show the performance of the regression coefficients at different scales in the three models and then make two statistical tests for the two regression coefficients in each model. Some evaluations for the DFA-based regression and the standard regression are conducted at the end of this section.
X 1 = p n−n[k−1] (1 − p) n[k
The two regression coefficient estimators together with their standard deviations of the three models are sketched in Figs 4-6, respectively. As expected, the effect is obviously positive. However, a strong variation across scales is found in different seasons. More specifically, is close to 0 from the smaller scale to the larger scale at about 50 days (1200 hours), which implies that the positive correlation between Tianjin and Baoding can last less than 50 days. In addition, the two coefficients are less than 0.5 in most days, which indicates that Beijing and Baoding have little impact on the PM2.5 in Tianjin. (c) For the model of Baoding, the effect of Tianjin (X 2 ) to Baoding is similar to that of Baoding to Tianjin in model II. However, the fact that after approximately 17 days (408 hours) the effect reaches the value greater than 1 indicates that an increase of 1 unit PM2.5 concentration of Tianjin will lead to the increase of more than 1 unit PM2.5 concentration in Baoding. In this regard, Tianjin has more impact on Baoding. In addition, the narrow confidence intervals and low standard deviations (less than 0.02) shown in all sub-plots suggest satisfied reliability of the estimates. Statistic significance tests of regression coefficients. As mentioned above, the estimated βˆn ( ) DFA is able to theoretically describe the dependence between the impulse variables and the response variables at different time scales. In theory, as long as βˆn ( ) j DFA is not equal to zero, the independent variable X j will affect Y. However, for finite time series, βˆn ( ) j DFA is not always equal to 0 even in the absence of relationship between X j and Y due to the size limitation. Therefore, we perform a hypothesis test for the estimated βˆn ( ) DFA to ensure the significance. The standard regression analysis provides a so-called t statistic defined as
= β β β −t j var( ) j j j (j = 1, 2) for this purpose. We have ∼ − t t N ( 3) j for the bivariate regression model as β β β ∼N var ( ,( ))
j j j . In general, if |t j | > t 1−α/2 (N − 3) with a given α, we should reject the null hypothesis of β j = 0 and the dependence between X j and Y is considered to be statistically significant. However, since lots of time scales are taken accounted in the DFA regression model, using a single critical value of t 1−α/2 (N − 3) is inappropriate. A correct way is to generate a critical value t c (n) for each time scale. To this end, inspired by the idea proposed by Podobnik et al. 32 , we shuffle the considered PM2.5 series and repeat the DFA regression calculations for 10,000 times. Then let the integral of probability distribution function (PDF) from −t c (n) to t c (n) be equal to 1 − α (here, we take α = 0.01). As an example, we show the PDF of t c (n) with five given n's produced by the shuffled PM2.5 series of fall in Fig. 7.
As expected, the symmetrical PDF of t c (n) converges to a Gaussian distribution according to the central limit theorem. In addition, the critical value increases as n increases. This implies that large time scale may strengthen dependence between two variables. By using t c (n), we can determine whether the dependence between the impulse variable and the response variable is significant or not. In practice, the dependence between X j and Y is present when The partial DCCA coefficient ρ PDCCA (n) is recently developed to uncover the intrinsic relation for two nonstationary series at different time scales. We also calculate the partial DCCA coefficients ρ PDCCA (n) of Beijing and Tianjin, Beijing and Baoding, and Tianjin and Baoding, respectively, and present the results in Fig. 9. For the same purpose of testing the statistical significance, we also produce a critical value for the four seasons. Similarly, the PM2.5 data are shuffled 10,000 times in the PDCCA calculations repeatedly, and thus ρ n ( ) PDCCA c for 99% confidence level is obtained, which is also shown in Fig. 9.
= β β β − t n ( )
Comparing results in Figs 8 and 9 gives amazing similarities, which are also in agreement with the results shown in Figs 4-6. Based on the results, we can draw the following three main points. However, the dependence between the two cities is lower than other cities. This finding uncovers that the reason for the serious air pollution in these two cities are mainly due to their own heavy smog or are impacted by other cities. (b) The dependence between Beijing and Baoding (the green triangle line) is significant in spring, summer, and fall. In winter, however, the dependence disappears at long time scale, which implies that the two cities can only affect each other at relatively short term. Moreover, compared to winter and fall, the dependence is much stronger in spring and summer, especially at long time scales, which indicates that they affect much longer in warm weather. (c) In spring and summer, the t(n)-statistics and ρ PDCCA (n) of Tianjin vs. Baoding (the red circle line) go down through the critical lines of t c (n) and ρ n ( ) PDCCA c at about 800 hours, respectively. This suggests that the dependence between Tianjin and Baoding will disappear when it's more than one month. However, the exact opposite occurs in winter and fall. In these two seasons, both t(n)-statistic and ρ PDCCA (n) increase with the increasing time scales, which demonstrates that the interaction of bad air quality between the two cities will last longer in cold days. , and the beta coefficient β* DFA (n) and the average elasticity coefficient η DFA (n) in Fig. 10, Figs 11-13, respectively.
To show the new model provides more information than the standard regression model does, we also include the three corresponding coefficients of standard bivariate regression model in these figures. As seen from Fig. 10 that R n ( ) DFA 2 is superior to the standard R 2 at most time scales. The good performance illustrates that one will gain richer information in explaining the response variable when using our DFA-based regression model. On the other hand, we can conclude from Figs 11-13 that (1) Baoding has more influence than Tianjin on Beijing in all seasons except for winter. (2) Tianjin is more sensitive to Baoding's changes in air quality than Beijing's in winter and fall. , respectively, at smaller scales but much smaller at larger scales, which shows that the sensitivity of Y to X 2 (Baoding) is greater than that of Y to X 1 (Tianjin) for short term (≤300 hours) but Tianjin is Fig. 11. Here the subscripts 1 and 2 denote Beijing and Tianjin, respectively.
Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w more sensitive to Beijing at the long term. This can help air quality inspectors make the correct analysis for Beijing's PM2.5 at different periods.
Conclusions
The study of dependence between variables helps expose the causal relationship and correlation of the variables of interest in the real world. The linear regression model is undoubtedly one of the simplest methods among many approaches. However, single variety of regression coefficient and evaluation index cannot show all aspects of the dependence between independent variables and dependent variable. As a meaningful extension, we design a new framework for bivariate regression model using the prevailing DFA method. The proposed bivariate DFA regression model allows us to estimate multi-scale regression coefficients and other corresponding scale-dependent evaluation indicators. It has been shown via two artificial tests that these DFA-based regression coefficients are able to describe the dependence between the response variable and two independent variables exactly; and can capture different dependence at different time scales.
An application of the new model to the study of dependence of PM2.5 series among three heavily air polluted cities in Northern China unveils that huge difference of the dependence exists in per two cities in different seasons and at different periods. Three new indicators of the scale-dependent determination coefficient, the scale-dependent beta coefficient, and the scale-dependent elasticity coefficient are proposed, which turned out to be more practical than those in standard regression models. Three main points can be concluded as (1) Beijing and Baoding have little impact on the PM2.5 in Tianjin while Tianjin takes more impact on Baoding and the air quality of Beijing is more sensitive to the changes in Baoding. (2) In contrast, the air quality in Beijing and Tianjin is not significantly relevant, while the air quality in Tianjin and Baoding has a very significant impact on each other especially in the cold weather. (3) In comparison, the fluctuation of PM2.5 in Baoding has the greatest impact on the other two cities in most days. While Baoding's air quality is more sensitive to Beijing's changes in spring and summer, and is more sensitive to Tianjin's changes in winter and fall. These findings may provide some useful insights on understanding air pollution sources among cities in Northern China.
Methods
The standard bivariate regression model. To study the dependence of air quality among three neighboring cities, we consider a bivariate linear regression model as
β β β ε = + + + Y X X ,(1)0 1 1 2 2
where Y is a dependent variable, X 1 and X 2 are two independent variables, ε is a Gaussian error term with zero mean value, and β j (j = 1, 2) is the partial regression coefficient characterizing the dependence on X j . The most critical work in empirical studies is to estimate β 1 and β 2 . The OLS method gives
β β = ∼ = ∼ σ σ σ σ σ σ σ σ σ σ σ σ σ σ ∑ ⋅ ∑ − ∑ ⋅ ∑ ∑ ⋅ ∑ − ∑ ⋅ − ⋅ ⋅ − ∑ ⋅ ∑ − ∑ ⋅ ∑ ∑ ⋅ ∑ − ∑ ⋅
x y x xy x x where 〈·〉 denotes the mean value of the whole time period, x 1t = X 1t − 〈X 1 〉, x 2t = X 2t − 〈X 2 〉, and y t = Y t − 〈Y〉.
x x x x x y x xy x x x x x x 1 ( )( ) ( )( ) ( ) ( ) ( ) 2 ( )( ) ( )( ) ( ) ( ) ( ) t N t t t N t t N t t t N t t t N t t N t y N t t X Y X X Y XX X X X X t N t t t N t t N t t t N t t t N t t N t t N t t X Y X X Y XX X X X X
Then the estimator of residuals can be determined by
β β β β = − − −〈 − − 〉ˆˆˆê Y X X Y X X t t t t t t t 1 1 2 2 1 1 2 2 .
With it one can obtain the estimators of variance of the two regression coefficients as The variance illustrates the accuracy of the estimated parameters. The estimated regression coefficients together with their variances can be further employed for hypothesis test and model evaluation. As an important indicator to evaluate the regression model, the determination coefficient R 2 is defined by
β β = ∼ ⋅ = ∼ ⋅ . σ σ σ σ σ σ σ σ σ σ ∑ ⋅ ∑ ⋅ ∑ − ∑ − ⋅ ⋅ − ∑ ⋅ ∑ ⋅ ∑ − ∑ − ⋅ ⋅ − ε ε = ∑ = − = = = = ∑ = − = = = ˆˆv ar var ( ) , ( ) x x x x x N x x x x x N 1 ( ) ( ) () ( ) 1 3 2 ( ) ( ) ( ) ( ) 1 3t N t t N e t N t N t t N t t N t t X X X X X t N t t N e t N t N t t N t t N t t X X X X Xσ σ = − ∑ ∑ = − ε = = R e y 1 1 ,(3)
j j j which can explain the relative importance of variables X 1 and X 2 to Y. According to 31 , the advantage of translating the standard notation into variance and covariance shown on the right-hand side of Eqs (2)-(5) is available to use the DFA/DCCA methods based on the same idea.
The DFA-based variance and DCCA-based covariance functions. DFA into N n = [N/n] nonoverlapping segments of equal length n, denoted as Z j,k , k = 1, 2, …, n. The same procedure is repeated starting from the opposite end to avoid disregarding a short part of the series in the end and thus 2N n segments are obtained altogether. In the j th segment, we have Z j,k = Z (j−1)n+k for j = 1, 2, …, N n and = − − + Z Z j k N j N n k , ( ) n for j = N n + 1, N n + 2, …, 2N n , where k = 1, 2, …, n. In each segment, the local linear (or other) trend 33,34 can be fitted as X j k , (in our work, we use 2 nd order polynomial to fit the trend in each segment). Fluctuation function f n j ( , ) The scale-characteristic fluctuation F n ( ) Z Z 2 1 2 is the so-called DCCA-based covariance, which expresses the cross-correlation fluctuations between the series of {z 1t } and {z 2t }. Thus we have obtained all objects to create the DFA-based regression model. But for purpose of testing, we need some accessories of the DFA process. The DCCA cross-correlation coefficient ρ(n), proposed by Zebende 35 , can measure the cross-correlation between two nonstationary series at multiple time scales, which is defined as To access intrinsic relations between the two time series on time scales of n, Yuan et al. 36 and Qian et al. 37 developed a so-called partial DCCA coefficient independently, which applies partial correlation technique to delete the impact of other variables on the two currently studied variables. This coefficient is defined as
ρ = − Z Z n C n C n C n ( , , ) ( ) ( ) ( ) ,(11)
where C is the inverse matrix of the cross-correlation matrix produced by ρ DCCA (n) of Z 1 , Z 2 , …, and subscripts j 1 and j 2 stand respectively for the row and column of the location of ρ DCCA (Z 1 , Z 2 , n).
The DFA-based bivariate regression model. We now translate the standard bivariate regression process described above into the DFA-based bivariate regression model. The two estimators in Eq. (2) can be extended to the scale-dependent estimators in the following way using the scale-dependent variance and covariance defined in Eqs (7) and (9)
Figure 1 .
1Estimated two DFA Regression coefficients with setting I of ARFIMA model. Mean values of the DFA estimators and standard deviation are shown as solid line (left axis) and dashed line (right axis), respectively. X 1 and X 2 are two independent variables generated by ARFIMA model with the same changing fractional integration parameter (x-axis) and independent Gaussian noise. Y = 1 + X 1 + 2X 2 + ε, ε is a standard Gaussian noise error-term. Both of them are of length 10000 and repeated 1000 times. (a) is the result for the estimated βˆD FA 1 and (b) is for the estimated βˆD
FA 2 .
2The DFA estimators β 1 and β 2 are unbiased at 1 and 2 with the errorterm ±0.002 and ±0.005, respectively, and their standard deviations decrease with the memory strength.
Figure 2 .
2Estimated two DFA Regression coefficients with setting II of ARFIMA model with the same legend as inFig. 1. X 1 and X 2 are generated by ARFIMA series with the same fixed d = 0.4. ε is an ARFIMA process with changing parameter d ε (x-axis). The remaining settings are the same as those in case I. Results show that the DFA estimators β 1 and β 2 are unbiased at 1 and 2 with the error-term ±0.002 and ±0.003, respectively, while the standard deviations increase with the error-term memory strength.Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w
Figure 3 .
3Estimated two DFA Regression coefficients with BMFs model.
Figure 4 .Scientific
4Bivariate DFA regression of Beijing. Main planes of subplots (a-d) show estimated DFA regression coefficients β 1 (n) and β 2 (n) of winter, spring, summer, and fall, respectively. Gray zones denote 95% confidence intervals. Inserts are standard deviations of βˆn ( ) RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w
Figure 5 .
5Bivariate DFA regression of Tianjin with the same legend as inFig. 4. Here, subscripts 1 and 2 denote Beijing and Baoding, respectively.
Figure 6 .
6Bivariate DFA regression of Baoding with the same legend as in Fig. 4. Here, subscripts 1 and 2 denote Beijing and Tianjin, respectively. Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w (a) In the Beijing's model, Tianjin (X 1 ) has strongly positive effect in every season, especially for the larger time scales. On the contrary, Baoding (X 2 ) has different effects on Beijing. Compared to spring and summer, the effect is quite weak in the other two seasons, especially in winter, βˆn ( ) DFA 2 is nearly 0 when the scale is more than 800 hours. (b) In the Tianjin's model, Baoding (X 2 ) presents more unstable effect at different scales. Particularly in summer, βˆn ( ) DFA 2
Figure 7 .
7PDF of critical points t-statistics critical values at different scales for the statistical test with 10000 times of the shuffled PM2.5 series of fall.
Figure 8 .
8t-statistical test of the estimated DFA-based bivariate regression coefficients. (a-d) Are for winter, spring, summer, and fall, respectively. The dashed line represents the t c (n) with 0.01 significant levels. Above this line means to decline the null hypothesis β j = 0. Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w
DFA 1 ( 1 (
11than t c (n). For the four seasons, the scale-dependent t-statistics of regression coefficient together with the scale-dependent critical value t c (n) are presented inFig. 8.Note that in Model I (for Beijing), the t(n)-statistics of βˆn ( ) Tianjin's coefficient) is equal to that of βˆn ( )DFA 1 (Beijing's coefficient) in Model II (for Tianjin), the t(n)-statistics of βˆn ( ) DFA 2 (Baoding's coefficient) is equal to that of βˆn ( ) DFA Beijing's coefficient) in Model III (for Baoding), and in Model II, the t(n)-statistics of Baoding's coefficient βˆn ( ) DFA 2is equal to that of Tianjin's coefficient βˆn ( )DFA 2 in Model III (for Baoding). Here the three colored lines with different symbols represent the t(n)-statistics between each per two cities while the black dashed line stands for t c (n).
Figure 9 .
9Statistical test of DPCCA coefficients among the three cities. (a-d) Are for winter, spring, summer, and fall, respectively. The dashed line represents the critical value of ρ DPCCA which is obtained from 10000 times Monte-Carlo simulations with 99% confidence level. Below this line suggests no cross-correlated significance. Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w (a) The dependence between Beijing and Tianjin (the blue square line) gradually increases with the increasing time scales in all seasons.
Figure 10 .
10Determination coefficients of bivariate DFA and standard regression model. (a-d) Are for models of Beijing, (e-h) are for models of Tianjin, and (i-l) are for models of Baoding. The solid line denotes R n ( ) DFA 2 and the dashed line denotes R 2 .
Figure 11 .
11Beta coefficients and elasticity coefficients of bivariate DFA and standard regression model of Beijing. The four columns from left to right are for winter, spring, summer, and fall, respectively. The subscript 1 of β* and η denotes Tianjin and 2 denotes Baoding. Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-w Evaluations of DFA-based regression model. To evaluate our estimated DFA-based bivariate regression model, we plot the scale-dependent determination coefficient R n ( ) DFA 2
( 3 )
3Tianjin affects Baoding more than Beijing does in winter and fall, but less in the other two seasons. In addition, Figs 11-13 illustrate that the standard β ⁎ j and η j can be seen as the mean values of the DFA-based β ⁎ n ( ) j DFA and η n ( ) j DFA , respectively. This means that β ⁎ n ( ) j DFA and η n ( ) j DFA are able to measure the dependence degree of the studied independent variable on the dependent variable in all directions. Thus one can access the measurement according to his/her needs. For example, in winter of Model I, we find that the β ⁎ n ( )
Figure 12 .
12Beta coefficients and elasticity coefficients of bivariate DFA and standard regression model of Tianjin with the same legend as inFig. 11. Here the subscripts 1 and 2 denote Beijing and Baoding, respectively.
Figure 13 .
13Beta coefficients and elasticity coefficients of bivariate DFA and standard regression model of Baoding with the same legend as in
the so-called DFA-based scale-dependent variance function. To obtain the scale-dependent covariance of two equal length series {z 1t } and {z 2t }, t = 1, 2, …, N, we only need to translate the univariate fluctuation function in each segment and average fluctuation into the bivariate case directly,
Table 1. Partial correlation coefficients and t-statistics between per two cities of Beijing, Tianjin, and Baoding in four seasons. Note: *Indicates statistical significance with 0.01 significance level.Winter
Spring
Summer
Fall
Beijing vs. Tianjin
0.3048
0.3072
0.2625
0.1660
t-statistics
25.8011*
26.2714*
22.1366*
13.2672*
Beijing vs. Baoding
0.2745
0.4545
0.4815
0.4468
t-statistics
23.0124*
41.5216*
44.7079*
39.3711*
Tianjin vs. Baoding
0.5570
0.4517
0.3461
0.5992
t-statistics
54.0752*
41.1961*
30.0154*
58.9941*
with the range of [0, 1]. R 2 measures a proportion of variance of Y explained by X 1 and X 2 and higher value of R 2 implies better model interpretation ability. Moreover, to quantify sensitivity of explained variable to each explaining variable, two quantities, namely, the beta coefficient (denoted as β ⁎ j ) and the average elasticity coefficient (denoted as η j ), are defined Scientific RepoRts | (2018) 8:7475 | DOI:10.1038/s41598-018-25822-wt
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© The Author(s) 2018
AcknowledgementsThis work was partially supported by National Natural Science Foundation of China (No. 31501227, No. 11401577), and NSERC of Canada. The authors would like to thank two anonymous reviewers and the handling editor for their comments and suggestions, which led to a great improvement to the presentation of this work.Similarly, the scale-dependent residuals arewith zero mean value. Inserting the calculated ê n ( ) t into the DFA process, we obtain the fluctuation ε F n ( )2to estimate the variances of βˆn ( )DFA 1and βˆn ( )DFA XThen Eqs(3)-(5)can be translated into the DFA regression form asComparing to the standard R 2 , β*, and η, the scale-dependent R n ( )DFA 2, β* DFA (n), and η DFA (n) express more abundant information on model interpretation from multiple time scales.Author ContributionsAdditional 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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On spurious and corrupted multifractality: The effects of additive noise, short-term memory and periodic trends. J Ludescher, M I Bogachev, J W Kantelhardt, A Y Schumann, A Bunde, Physica A. 39013Ludescher, J., Bogachev, M. I., Kantelhardt, J. W., Schumann, A. Y. & Bunde, A. On spurious and corrupted multifractality: The effects of additive noise, short-term memory and periodic trends. Physica A 390(13), 2480-2490 (2011).
Dcca cross-correlation coefficient, quantifying level of cross-correlation. G Zebende, Physica A. 390Zebende, G. Dcca cross-correlation coefficient, quantifying level of cross-correlation. Physica A 390, 614-618 (2011).
Detrended partial-cross-correlation analysis: a new method for analyzing correlations in complex system. N M Yuan, Scientific Reports. 58143Yuan, N. M. et al. Detrended partial-cross-correlation analysis: a new method for analyzing correlations in complex system. Scientific Reports 5, 8143 (2015).
Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces. X Y Qian, Phys. Rev. E. 9162816Qian, X. Y. et al. Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces. Phys. Rev. E 91, 062816 (2015).
|
[] |
[
"Structural and electronic properties of LaO 0.85 F 0.15 FeAs superconductor modified under neutron irradiation",
"Structural and electronic properties of LaO 0.85 F 0.15 FeAs superconductor modified under neutron irradiation"
] |
[
"A Gerashenko \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"S Verkhovskii \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"A Karkin \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"V Voronin \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"A Kazantsev \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"B Goshchitskii \nInstitute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia\n",
"J Werner \nInstitute for Solid State and Materials Research IFW Dresden\n01069DresdenGermany\n",
"G Behr \nInstitute for Solid State and Materials Research IFW Dresden\n01069DresdenGermany\n"
] |
[
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute of Metal Physics\nUral Branch of Russian Academy of Sciences\n620041EkaterinburgRussia",
"Institute for Solid State and Materials Research IFW Dresden\n01069DresdenGermany",
"Institute for Solid State and Materials Research IFW Dresden\n01069DresdenGermany"
] |
[] |
The effect of atomic disorder induced by neutron irradiation on the crystal structure and electronic states near EF of the lightly overdoped LaO0.85F0.15FeAs (Tc = 21K) was studied by X-ray diffraction and 75 As NMR. The irradiation of the polycrystalline sample by "moderate" neutron fluence of Φ = (0; 0.5; 1.6) · 10 19 cm −2 at T = 50 • C leads to the suppression of superconductivity. It is shown that neutron irradiation produces an anisotropic expansion of the tetragonal lattice almost due to an increase of the Fe-As distance. A partial loss of the 2D character of the FeAs layer is accompanied with a suppression of the gap-like feature in temperature dependence of the spin susceptibility. In the most disordered state the 75 As spin-lattice relaxation rate follows the Korringa law 75 T −1 1 ∼ T , the thermal behavior being typical for an isotropic motion of the conducting electrons.
| null |
[
"https://arxiv.org/pdf/0911.2127v2.pdf"
] | 119,294,453 |
0911.2127
|
e847d674bd5f702e193993058b4b035ce5edef37
|
Structural and electronic properties of LaO 0.85 F 0.15 FeAs superconductor modified under neutron irradiation
12 Nov 2009 (Dated: November 12, 2009)
A Gerashenko
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
S Verkhovskii
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
A Karkin
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
V Voronin
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
A Kazantsev
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
B Goshchitskii
Institute of Metal Physics
Ural Branch of Russian Academy of Sciences
620041EkaterinburgRussia
J Werner
Institute for Solid State and Materials Research IFW Dresden
01069DresdenGermany
G Behr
Institute for Solid State and Materials Research IFW Dresden
01069DresdenGermany
Structural and electronic properties of LaO 0.85 F 0.15 FeAs superconductor modified under neutron irradiation
12 Nov 2009 (Dated: November 12, 2009)PACS numbers: 7470-b, 7660-k
The effect of atomic disorder induced by neutron irradiation on the crystal structure and electronic states near EF of the lightly overdoped LaO0.85F0.15FeAs (Tc = 21K) was studied by X-ray diffraction and 75 As NMR. The irradiation of the polycrystalline sample by "moderate" neutron fluence of Φ = (0; 0.5; 1.6) · 10 19 cm −2 at T = 50 • C leads to the suppression of superconductivity. It is shown that neutron irradiation produces an anisotropic expansion of the tetragonal lattice almost due to an increase of the Fe-As distance. A partial loss of the 2D character of the FeAs layer is accompanied with a suppression of the gap-like feature in temperature dependence of the spin susceptibility. In the most disordered state the 75 As spin-lattice relaxation rate follows the Korringa law 75 T −1 1 ∼ T , the thermal behavior being typical for an isotropic motion of the conducting electrons.
1
of 75 As and 57 Fe respectively probing the Fermi-liquid (q ≈ 0) and staggered (q(π, π)) components of spin susceptibility χ(q) give 9 no evidence of any q-space structure in spin susceptibility χ(q) 7 , that might be expected in the presence of antiferromagnetic spin correlations. of any determined almost by the uniform (q = 0) component of spin susceptibility 8 . It is suggested that pseudogap behavior of spin susceptibility is more relevant to rather complex topology of the Fermi surface including both electron and hole pockets which filling can be varied differently whether electron or holes are doped into the FeAs layer. The 75 As NMR study of the overdoped LaO 1−x F x FeAs (x = 0.14) show 10 clearly an increase of the density of states at the Fermi energy with applying an external pressure additional to the internal "chemical" one.
In this report the structural X-ray and 75 As NMR results are presented for light overdoped polycrystalline LaO 0.85 F 0.15 FeAs (T c =21K) affected by neutron irradi- ation. In fact the neutrons present an unique to create atomic-scale defects uniformly distributed in the lattice and acting like "negative chemical pressure" that expands crystal lattice with negligible variation of the concentration of carriers. It is shown that neutron irradiation produces an anisotropic expansion of the tetragonal lattice mainly due to the displacement of arsenic from equilibrium atomic positions, thus creating structural disorder resulting in a partial loss of the 2D character of the FeAs layer. Polycrystalline sample of LaO 0.85 F 0.15 FeAs was synthesized as a pellet using two-step solid state reaction and subsequent annealing in vacuum 11 . The pellet was sliced into the plate-like samples which were irradiated with neutron fluence fluence Φ = (0; 0.5; 1.6) · 10 19 cm −2 at T irr = 50 ± 10 • C. After irradiation both virgin and irradiated samples were moderately crushed into pow-der (∼200mess) for X-ray, ac and dc susceptibility, and 75 As NMR measurements. The superconducting transition temperature T c was determined as an onset of the diamagnetic response in the ac susceptibility measurements performed at driving 10 Hz magnetic field of 4 Oe with the MPMS-9 device from Quantum Design. Unirradiated sample The ac susceptibility curves plotted in Fig.1 show superconducting transition of ordered sample at T c (Φ = 0)=21K which shifts rather fast down to T c (Φ = 0.5 · 10 19 cm −2 )=4 K at the light structural disorder induced by neutron irradiation. The sample acquired rather moderate neutron fluence Φ = 1.6 · 10 19 cm −2 does not show superconducting transition down to 2 K, the lowest temperature available in our experimental setup.
The structural characterization of irradiated samples was performed at room temperature by a powder Xray diffraction technique using Cu − K α radiation. The diffraction patterns obtained in the range of 2θ = (25 − 5) • confirm a single phase tetragonal structure for each sample studied in this work. The subsequent Rietveld refinement performed within space group P4/nmm results in structural parameters listed in Table. An insufficient broadening of the Bragg peaks observed even in the X-ray pattern of the sample with the highest neutron fluence is indicative that structural defects in the sublattices of iron, arsenic are lanthanum are almost relevant to the displacement of atoms from their positions in the ordered material. The main features of the radiation-induced structural disorder are shown in Fig.2. With the neutron fluence increasing a growth of the tetragonal unit cell (Fig.2b) occurs mainly along c direction due to the thicker Fe-As layer in irradiated samples. While the structural parameters of the La-O layer (Table I) do not show any sufficient variation. It is evident (Fig.2c) that a thickness of the Fe-As layer is growing due to an increase of the Fe-As interatomic distance, whereas the distance d (Fe-Fe) remains practically unchanged. Under neutron irradiation the structural changes of the nearly optimal electron-doped LaO 0.85 F 0.15 FeAs are developed in the way entirely opposite to those using an effect of "chemical pressure" to optimize T c in the electron-doped LaO 1−x F x FeAs (x > 0.6) 1,12 and LaO 1−x FeAs 13,14 compositions. The cell volume V =144Å of the most disordered nonsuperconducting sample is found above an upper limit of the cell volume values which were reported somewhere for the Fe-based superconducting pnictides 2 . The 75 As NMR measurements were carried out on the home-built pulse-coherent spectrometer in magnetic field of 94 kOe over the temperature range 10-300 K. Each quadrupole broadened 75 As ( 75 I = 3/2) NMR spectrum was obtained by summing the Fourier-transformed halfecho signals acquired at equidistant operating frequencies. peaked line shape of the transition originates in an interaction of the 75 As nuclear quadrupole moment with the electric field gradient created at arsenic by electronic environment, and the high-frequency peak corresponds to the crystallites with c crystal axis oriented perpendicular (θ = 90 • ) to the magnetic field direction. It is remarkable that under neutron irradiation the peak does not show any additional broadening due apparently to induced charge disorder. This NMR observation is completely consistent with X-ray results thus evidencing that local charge symmetry at the As sites does not deviate in average from axial symmetry of the As site in the ordered (Φ = 0) material.
The 75 As spin-lattice relaxation rate T −1 1 was measured to trace thermal behavior of the spin susceptibility χ s (q) in the normal state of the irradiated LaO 0.85 F 0.15 FeAs samples.
We measured the 75 As spin-lattice relaxation rate T −1 1 using an inversion recovery method.
The nuclear magnetization 75 m(t) was measured by integrating spectral intensity within ±50 kHz around a peak (θ = 90 • ) of the central transition (Fig. 3).
The recovery curve of nuclear magnetization was fitted with an expression {m(∞) − m(t)} ∼ 0.1·exp(−t/T 1) + 0.9·exp(−6t/T 1) presuming that hyperfine magnetic interaction of the nuclear spin 75 I = 3/2 with electronic spin environment is dominating.
The temperature dependence of the 75 As nuclear spinlattice relaxation rate measured in the neutron irradiated LaO 0.85 F 0.15 FeAs samples is presented in Fig. 4 as a product (T 1 T ) −1 . In the ordered (Φ = 0) and lightly irradiated (Φ = 0.5 · 10 19 cm −2 ) superconducting samples (T 1 T ) −1 show gradual decrease from room temperature with nearly constant behavior below 30 K, which is above T c . Such pseudogap behavior of (T 1 T ) −1 was observed in all electron-doped Fe-based superconductors 5,6,7 . Following 4 we have used an expression a + b · exp(−∆/T ) to fit the (T 1 T ) −1 data. The corresponding fitting curves are plotted by solid lines in Fig. 4. As a result, the magnitude of the pseudogap is estimated to be ∆(Φ = 0)=168(30) K and ∆(Φ = 0.5 · 10 19 cm −2 )=108(20) K with a = 0.035(6) (sK) −1 independent on Φ in superconducting samples. The decrease of ∆ with increase of neutron fluence Φ is indicative of that pseudogap behavior of (T 1 T ) −1 originates in the specific 2D band structure near the Fermi energy 9 . In fact, recently reported 7 scaling of (T 1 ) −1 (T ) measured in LaO 0.9 F 0.1 FeAs at 57 Fe, 75 As, 139 La and 19 F nuclei of atoms probing spin fluctuations in different areas of the q -space 9 gives compelling evidence, that dynamic spin susceptibility does not have any strong qdependence in the optimally electron-doped oxypnictide.
In the most disordered nonsuperconducting sample of LaO 0.85 F 0.15 FeAs the 75 As spin-lattice relaxation rate follows the Korringa law 75 T −1 1 ∼ T , the thermal behavior being typical for an isotropic spectrum of the quasiparticle excitations near EF. The Curie-like upturn of T −1 1 below 30 K is addressed to an additional contribution to T −1 1 due accumulated structural defects, including themselves the localized magnetic moments. The magnetism of localized magnetic moments is seen clearly in the Curie term of the bulk magnetic susceptibility at low temperature. It was found that corresponding Curie constant increases proportionally to the neutron fluence.
In conclusion, an influence of structural disorder induced by neutron irradiation up to the fluence Φ = 1.6 · 10 19 cm −2 on the spin susceptibility was studied in normal state of the lightly overdoped superconducting LaO 0.85 F 0.15 FeAs by measuring nuclear spin-lattice relaxation of 75 As. According to the X-ray diffraction data the radiation-induced structural defects remain unchanged the tetragonal symmetry of the irradiated by neutrons LaO 0.85 F 0.15 FeAs. The accumulated disorder results in a growth of the cell volume, almost due to an increase of the Fe-s interatomic distances. A partial loss of the 2D character of the FeAs layer is accompanied with a suppression of the gap-like feature in temperature dependence of the spin susceptibility. In the most disordered state the 75 As spin-lattice relaxation rate follows the Korringa law 75 T −1 1 ∼ T , the thermal behavior being typical for an isotropic motion of the conducting electrons
FIG. 1 :
1The ac susceptibility χac superconducting transition curves for ordered (Φ = 0) and neutron-irradiated (Φ = 1.6 · 10 19 cm −2 ) samples of LaO0.85F0.15FeAs.
FIG. 2
2: a)The tetragonal (P 4/nmm) lattice parameters; b) cell volume V , the ratio c/a; c) the interatomic distances d(F e − As) and d(F e − As) in the FeAs layer versus neutron fluence Φ acquired by the n-irradiated LaO0.85F0.15FeAs.
FIG. 3 :
3Figure 3show representative spectral patterns of the central transition (m I = -1/2 -+1/2) measured at room temperature in the LaO 0.85 F 0.15 FeAs powder samples irradiated by different neutron fluence Φ. The two-Room-temperature 75 As NMR spectra (transition m = − 1 2 ↔ + 1 2 ) measured at magnetic field of 94 kOe in the powder sample LaO0.85F0.15FeAs irradiated by neutron fluence Φ. The vertical arrow points spectral peak which at spin-lattice relaxation rate 75 T1 data were collected.
FIG. 4 :
4The temperature dependence of the 75 As (T1T ) −1 for LaO0.85F0.15FeAs irradiated by neutron fluence Φ = 0( ), 0.5 · 10 19 cm −2 ( ) and 1.6 · 10 19 cm −2 (•). The solid curves are fits to an expression a + b · exp(−∆/T ) of the corresponding (T1T ) −1 data.
TABLE I :
IRoom-temperature structural data of LaO0.85F0.15FeAs irradiated by neutron fluence Φ. Atomic posititions refined in space group P 4/nmm are for La( 14 , 1
4 , z),
Fe( 3
4 , 1
4 , 1
4 ), As( 1
4 , 1
4 , z) and O/F( 3
4 , 1
4 , 0).
Fluence
0
5×10 18 cm −2 1.6×10 19 cm −2
Tc,K
n.s
4
21
a,Å
4.0251(5)
4.0251(6)
4.0323(6)
c,Å
8.7017(16) 8.7207(16)
8.7710(18)
V ,Å 3
140.98(4)
141.29(4)
142.61(4)
c/a
2.162
2.167
2.175
zLa
0.1446(7)
0.1453(21)
0.1457(12)
zAs
0.6501(11) 0.6522(38)
0.6639(20)
La-O/F,Å 2.374(3)×4 2.381(9)×4
2.388(6)×4
Fe-Fe,Å
2.8462(3)×4 2.8462(3)×4 2.8513(3)×4
Fe-As,Å
2.400(3)×4 2.410(19)×4 2.477(10)×4
AcknowledgmentsThe authors are grateful to C. Hess, R. Klingeler, B. Buechner, A.N. Vasiliev and O.S. Volkova for valuable discussions and for providing a high quality sample of LaO 0.85 F 0.15 FeAs. This work is supported in part by the Programme Basic Researches RAS "Condensed Matter Quantum Physics" under project No.4 UB RAS.
. Y J Kamihara, T Watanabe, M Hirano, H Hosono, J. Am. Chem. Soc. 1303296Y.J. Kamihara, T. Watanabe, M. Hirano and H.Hosono, J. Am. Chem. Soc. 130 (2008) 3296.
. M V Sadovskii, arXiv:0812.0302Physics Uspekhi. 51122M.V. Sadovskii, Physics Uspekhi 51 (2008) 122; arXiv:0812.0302.
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. K Ishida, arXiv:0906.2045K. Ishida et al., arXiv:0906.2045.
. K Ahilan, Phys. Rev.B. 78100501R)K. Ahilan et al., Phys. Rev.B 78 (2008) 100501(R).
. Y Nakai, arXiv:0810.4569Y. Nakai et al., arXiv:0810.4569.
. H.-J Grafe, arXiv:0811.4508H.-J. Grafe et al., arXiv:0811.4508.
. H Mukuda, arXiv:0904.4301H. Mukuda et al., arXiv:0904.4301.
. N Terasaki, J.Phys. Soc. Jpn. 7813701N. Terasaki et al., J.Phys. Soc. Jpn. 78 (2009) 013701.
. T Nakano, arXiv:0909.0318v1T. Nakano et al., arXiv:0909.0318v1.
. H Luetkens, Phys.Rev. Letters. 10197009H. Luetkens et al., Phys.Rev. Letters 101, 097009 (2008).
. H Luetkens, Nat. Mater. 8305H. Luetkens et al., Nat. Mater. 8 (2009) 305.
. H Mukuda, arXiv:0904.4301v2H. Mukuda et al., arXiv:0904.4301v2.
. H Kito, J. Phys. Soc. Jpn. 7763707H. Kito et al., J. Phys. Soc. Jpn., 77 (2008) 063707.
|
[] |
[
"NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS",
"NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS"
] |
[
"Eddine Bedr' ",
"Younes Ainseba ",
"Lahcen Echarroudi ",
"Maniar "
] |
[] |
[] |
In this paper, we are concerned with the null controllability of a linear population dynamics cascade systems (or the so-called prey-predator models) with two different dispersion coefficients which degenerate in the boundary and with one control force. We develop first a Carleman type inequality for its adjoint system, and then an observability inequality which allows us to deduce the existence of a control acting on a subset of the space domain which steers both populations of a certain age to extinction in a finite time.2000 Mathematics Subject Classification. 35K65, 92D25, 93B05, 93B07.
| null |
[
"https://arxiv.org/pdf/1701.04083v1.pdf"
] | 119,320,867 |
1701.04083
|
833aa2b7a161cd77c45b587dbbbc66d349cfd16e
|
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS
15 Jan 2017
Eddine Bedr'
Younes Ainseba
Lahcen Echarroudi
Maniar
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS
15 Jan 2017
In this paper, we are concerned with the null controllability of a linear population dynamics cascade systems (or the so-called prey-predator models) with two different dispersion coefficients which degenerate in the boundary and with one control force. We develop first a Carleman type inequality for its adjoint system, and then an observability inequality which allows us to deduce the existence of a control acting on a subset of the space domain which steers both populations of a certain age to extinction in a finite time.2000 Mathematics Subject Classification. 35K65, 92D25, 93B05, 93B07.
Introduction
We consider the coupled population cascade system
∂y ∂t + ∂y ∂a − (k 1 (x)y x ) x + µ 1 (t, a, x)y = ϑχ ω in Q, (1.1) ∂p ∂t + ∂p ∂a − (k 2 (x)p x ) x + µ 2 (t,
a, x)p + µ 3 (t, a, x)y = 0 in Q, y(t, a, 1) = y(t, a, 0) = p(t, a, 1) = p(t, a, 0) = 0 on (0, A) × (0, T ), y(0, a, x) = y 0 (a, x); p(0, a, x) = p 0 (a, x) in Q A , y(t, 0, x) = where Q = (0, T ) × (0, A) × (0, 1), Q A = (0, A) × (0, 1), Q T = (0, T ) × (0, 1) and we will denote q = (0, T ) × (0, A) × ω. The system (1.1) models the dispersion of a gene in two given populations which are in interaction. In this case, x represents the gene type and y(t, a, x) and p(t, a, x) as the distributions of individuals of age a at time t and of gene type x of both populations. The parameters β 1 (t, a, x) (respectively β 2 (t, a, x)), µ 1 (t, a, x) (respectively µ 2 (t, a, x)) are respectively the natural fertility and mortality rates of individuals of age a at time t and of gene type x of the population whose distribution is y (respectively p), µ 3 can be interpreted as the interaction coefficient between two populations (cancer cells and healthy cells for instance) which depends on x, t and a, the subset ω is the region where a control ϑ is acting. Such a control corresponds to an external supply or to removal of individuals on the subdomain ω. Finally, A 0 β 1 (t, a, x)y(t, a, x)da and A 0 β 2 (t, a, x)p(t, a, x)da are the distributions of the newborns of the two populations that are of gene type x at time t.
The control problems of (1.1) or in general of coupled systems take an intense interest and are widely investigated in many papers, among them we find [3], [7], [17] and the references therein. In fact, in [3] the authors studied a coupled reaction-diffusion equations describing interaction between a prey population and predator population. The goal of this work was to look for a suitable control supported on a small spatial subdomain which guarantees the stabilization of the predator population to zero. In [17], the objective was different. More precisely, the authors considered an age-dependent prey-predator system and they proved the existence and uniqueness for an optimal control (called also "optimal effort") which gives the maximal harvest via the study of the optimal harvesting problem associated to their coupled model. However, the previous results were found in the case when the diffusion coefficients are constants. This leads Ait Ben Hassi et al. in [7] to generalize the model of [3] and investigate a semilinear parabolic cascade systems with two different diffusion coefficients allowed to depend on the space variable and degenerate at the left boundary of the space domain. Moreover, the purpose of this paper was to show the null controllability via a Carleman type inequality of the adjoint problem of the associated linearized system using the results of [8] (or [12]) and with the help of the Schauder fixed point theorem. On the other hand, a massive interest was given to the question of null controllability of the population dynamics models in the case of one equation both in the case without diffusion (see for example [9]) and with diffusion (see for instance [1,2,4,5,15] in the case of a constant diffusion coefficient). Recently, a more general case was investigated by B. Ainseba and al. in [6] and [13]. Indeed, in [6] the authors allowed the dispersion coefficient to depend on the variable x and verifies k(0) = 0 (i.e, the coefficient of dispersion k degenerates at 0) and they tried to obtain the null controllability in such a situation with β ∈ L ∞ basing on the work done in [8] for the degenerate heat equation to establish a new Carleman estimate for a suitable full adjoint system and afterwards his observability inequality. However, the main controllability result of [6] was shown under the condition T ≥ A (as in [9]) and this constitutes a restrictiveness on the "optimality" of the control time T since it means, for example, that for a pest population whose the maximal age A may equal to a many days (may be many months or years) we need much time to bring the population to the zero equilibrium. In the same trend and to overcome the condition T ≥ A, L. Maniar et al in [13] suggested the fixed point technique implemented in [15] and which requires that the fertility rate must belong to C 2 (Q) and consists briefly to demonstrate in a first time the null controllability for an intermediate system with a fertility function b ∈ L 2 (Q T ) instead of cascade systems both in degenerate and nondegenerate cases to our knowledge and the work done in this paper will address to such a control problem and it will be a generalization of the results established in [6] and [13]. More precisely, following the strategy of [7] we expect in this contribution to prove the null controllability of system (1.1) when T ∈ (0, δ) where δ ∈ (0, A) small enough in the case of one control force. That is, we show that for all y 0 , p 0 ∈ L 2 (Q A ) and δ ∈ (0, A) small enough, there exists a control ϑ ∈ L 2 (q) such that the associated solution of (1.1) verifies y(T, a, x) = 0, a.e. in (δ, A) × (0, 1), p(T, a, x) = 0, a.e in (δ, A) × (0, 1).
(
1.2)
Such a result is gotten under the conditions that all the natural rates possess an L ∞ −regularity (see (2.4) beneath) and the dispersion coefficients are different and depend on the gene type with a degeneracy in the left hand side of its domain, i.e k i (0) = 0; i = 1, 2 (e.g k i = x α , α > 0). In this case, we say that (1.1) is a degenerate population dynamics cascade system. Genetically speaking, such a property is natural since it means that if each population is not of a gene type, it can not be transmitted to its offspring. The remainder of this paper is organized as follows: in Section 2, we give the wellposedness result of system (1.1) and we bring out a Carleman inequality for an intermediate trivial adjoint system which helps us to prove the main Carleman estimate for the full adjoint model. With the aid of this inequality, we establish in Section 3 the observability inequality and show the main result of the null controllability of (1.1). The last section takes the form of an appendix wherein we will give the proofs of some basic tools.
2. Well-posedness and Carleman estimates 2.1. Well-posedness result. For this section and for the sequel, we assume that the dispersion coefficients k i , i = 1, 2 satisfy the hypotheses
k i ∈ C([0, 1]) ∩ C 1 ((0, 1]), k i > 0 in (0, 1] and k i (0) = 0, ∃γ ∈ [0, 1) : xk ′ i (x) ≤ γk i (x), x ∈ [0, 1]. (2.3)
The last hypothesis on k i means in the case of k(x) = x α i that 0 ≤ α i < 1. Similarly, all results of this paper can be obtained also in the case of 1 ≤ α i < 2 taking, instead of Dirichlet condition on x = 0, the Newmann condition (k i (x)u x )(0) = 0. On the other hand, we assume that the rates µ 1 , µ 2 , µ 3 , β 1 and β 2 verify
µ 1 , µ 2 , µ 3 , β 1 , β 2 ∈ L ∞ (Q), µ 1 , µ 2 , µ 3 , β 1 , β 2 ≥ 0 a.e in Q, β i (., 0, .) ≡ 0 a.e. in (0, T ) × (0, 1), for i = 1, 2. (2.4)
The third assumption in (2.4) on the fertility rates β 1 and β 2 is natural since the newborns are not fertile. As in [13], we discuss the well-posedness of (1.1) by introducing the weighted spaces H 1 k i (0, 1) and H 2 k i (0, 1) defined by H 1 k i (0, 1) := {u ∈ L 2 (0, 1) : u is abs. cont. in [0, 1] :
√ k i u x ∈ L 2 (0, 1), u(1) = u(0) = 0},H 2 k i (0, 1) := u ∈ H 1 k (0, 1) : k i (x)u x ∈ H 1 (0, 1) ,
endowed respectively with the norms
u 2 H 1 k i (0,1) := u 2 L 2 (0,1) + √ k i u x 2 L 2 (0,1) , u ∈ H 1 k i (0, 1), u 2 H 2 k i := u 2 H 1 k i (0,1) + (k i (x)u x ) x 2 L 2 (0,1) , u ∈ H 2 k i (0, 1),
with i = 1, 2 (see [7], [8], [12] or the references therein for the properties of such a spaces). We recall from [11,12] that the operators C i u := (k i (x)u x ) x , u ∈ D(C i ) = H 2 k i (0, 1), i = 1, 2 are closed self-adjoint and negative with dense domains in L 2 (0, 1). On the other hand, in the Hilbert space H = (L 2 ((0, A) × (0, 1))) 2 , the system (1.1) can be rewritten abstractly as an inhomogeneous Cauchy problem in the following way
X ′ (t) = AX(t) + B(t)X(t) + f (t),X(0) = y 0 p 0 , (2.5) where X(t) = y(t) p(t) , A = A 1 0 0 A 2 ; D(A) = D(A 1 ) × D(A 2 ), f (t) = ϑ(t, ., ·)χ ω (.) 0 , B(t) = M µ 1 (t) 0 M µ 3 (t) M µ 2 (t) , where M µ j (t) w = −µ j (t)w, the operators A 1 : L 2 ((0, A) × (0, 1)) → L 2 ((0, A) × (0, 1)) and A 2 : L 2 ((0, A) × (0, 1)) → L 2 ((0, A) × (0, 1)
) are defined respectively by:
A 1 θ(a, x) = − ∂θ ∂a + (k 1 (x)θ x ) x , ∀θ ∈ D(A 1 ), D(A 1 ) = {θ(a, x) : θ, A 1 θ ∈ L 2 ((0, A) × (0, 1)), θ(a, 0) = θ(a, 1) = 0, θ(0, x) = A 0 β 1 (a, x)θ(a, x)da},(2.
6) and
A 2 θ(a, x) = − ∂θ ∂a + (k 2 (x)θ x ) x , ∀θ ∈ D(A 2 ), D(A 2 ) = {θ(a, x) : θ, A 2 θ ∈ L 2 ((0, A) × (0, 1)), θ(a, 0) = θ(a, 1) = 0, θ(0, x) = A 0 β 2 (a, x)θ(a, x)da}.
(2.7) It is well-known, from [16] and the references therein that the operators A 1 and A 2 defined above generate a C 0 −semigroups. On the other hand, one can see that the operator A is diagonal and B(t) is a bounded perturbation. Therefore, the following well-posedness result holds (see for instance [7] for a similar result of cascade parabolic equations).
Theorem 2.1. i) The operator A generates a C 0 −semigroup.
ii) Under the assumptions (2.3) and (2.4) and for all ϑ ∈ L 2 (Q) and (y 0 , p 0 ) ∈ (L 2 (Q A )) 2 , the system (1.1) admits a unique solution (y, p). This solution belongs to E :
= C([0, T ], (L 2 ((0, A)× (0, 1))) 2 ) ∩C([0, A], (L 2 ((0, T ) ×(0, 1))) 2 ) ∩L 2 ((0, T ) ×(0, A), H 1 k 1 (0, 1) ×H 1 k 2 (0, 1)).
Moreover, the solution of (1.1) satisfies the following inequality
sup t∈[0,T ] (y(t), p(t)) 2 L 2 (Q A )×L 2 (Q A ) + sup a∈[0,A]
(y(a), p(a)) 2
L 2 (Q T )×L 2 (Q T ) + 1 0 A 0 T 0 (( k 1 y x ) 2 + ( k 2 p x ) 2 )dtdadx ≤ C q ϑ 2 dtdadx + (y 0 , p 0 ) 2 L 2 (Q A )×L 2 (Q A ) . (2.8) 2.2.
Carleman inequality results. In this paragraph, we show a Carleman type inequality for the following adjoint system of (1.1)
∂u ∂t + ∂u ∂a + (k 1 (x)u x ) x − µ 1 (t, a, x)u − µ 3 (t, a, x)v = −β 1 (t, a, x)u(t, 0, x) in Q, (2.9) ∂v ∂t + ∂v ∂a + (k 2 (x)v x ) x − µ 2 (t, a, x)v = −β 2 (t, a, x)v(t, 0, x) in Q, u(t, a, 1) = u(t, a, 0) = v(t, a, 1) = v(t, a, 0) = 0 on (0, T ) × (0, A), u(T, a, x) = u T (a, x) in Q A , v(T, a, x) = v T (a, x) in Q A , u(t, A, x) = v(t, A, x) = 0 in Q T .
To do this, we prove firstly the Carleman estimate for the following intermediate system 2 and h 1 , h 2 ∈ L 2 (Q). Such a system can be rewritten in the following way
∂u ∂t + ∂u ∂a + (k 1 (x)u x ) x − µ 1 (t, a, x)u − µ 3 (t, a, x)v = h 1 in Q, (2.10) ∂v ∂t + ∂v ∂a + (k 2 (x)v x ) x − µ 2 (t, a, x)v = h 2 in Q, u(t, a, 1) = u(t, a, 0) = v(t, a, 1) = v(t, a, 0) = 0 on (0, T ) × (0, A), u(T, a, x) = u T (a, x) in Q A , v(T, a, x) = v T (a, x) in Q A , u(t, A, x) = v(t, A, x) = 0 in Q T , with (u T , v T ) ∈ (L 2 (Q A ))∂u ∂t + ∂u ∂a + (k 1 (x)u x ) x − µ 1 (t, a, x)u = h 1 + µ 3 (t, a, x)v in Q, (2.11) u(t, a, 1) = u(t, a, 0) = 0 on (0, T ) × (0, A), u(T, a, x) = u T (a, x) in Q A , u(t, A, x) = 0 in Q T ,
where v is the solution of ∂v ∂t
+ ∂v ∂a + (k 2 (x)u x ) x − µ 2 (t, a, x)v = h 2 in Q, (2.12) v(t, a, 1) = v(t, a, 0) = 0 on (0, T ) × (0, A), v(T, a, x) = v T (a, x) in Q A , v(t, A, x) = 0 in Q T .
Classically, the proof of such a kind of estimates is based tightly on the choice of the so-called weight functions. In our case, these functions are set in the following way
ϕ i (t, a, x) := Θ(t, a)ψ i (x), i = 1, 2, Θ(t, a) := 1 (t(T − t)) 4 a 4 , ψ i (x) := λ i x 0 r k i (r) dr − d i , φ(t, a, x) = Θ(a, t)e κσ(x) , Φ(t, a, x) = Θ(a, t)Ψ(x), Ψ(x) = e κσ(x) − e 2κ σ ∞ , (2.13)
where σ is the function given by
σ ∈ C 2 ([0, 1]), σ(x) > 0 in (0, 1), σ(0) = σ(1) = 0, σ x (x) = 0 in [0, 1]\ω 0 ,
(2.14)
ω 0 ⋐ ω is an open subset. The existence of this function is proved in [14, Lemma 1.1]. λ i , d i for i = 1, 2 and κ are supposed to verify following assumptions
d 1 > 1 k 1 (1)(2−γ) , λ 1 λ 2 ≥ d 2 d 1 − 1 0 r k 1 (r) dr , κ ≥ 4 ln 2 σ ∞ , d 2 ≥ 5 k 2 (1)(2−γ) , (2.15) with λ 2 ∈ I = [ k 2 (1)(2−γ)(e 2κ σ ∞ −1) d 2 k 2 (1)(2−γ)−1 , 4(e 2κ σ ∞ −e κ σ ∞ ) 3d 2
) which can be shown not empty (see Theorem 2.2. Assume that k i satisfy the hypotheses (2.3) and let A > 0 and T > 0 be given. Then, there exist two positive constants C and s 0 , such that every solution (u, v) of (2.10) satisfies, for all s ≥ s 0 , the following inequality
Q s 3 Θ 3 x 2 k 1 (x) u 2 + sΘk 1 (x)u 2 x e 2sϕ 1 dtdadx + Q s 3 Θ 3 x 2 k 2 (x) v 2 + sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx ≤ C Q (h 2 1 + h 2 2 )e 2sΦ dtdadx + q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx . (2.16)
The proof of Theorem 2.2 needs two basic results. These results are concerned with Carleman type inequalities in both cases degenerate and nondegenerate. The first one is stated in the following proposition Proposition 2.3. Consider the following system with h ∈ L 2 (Q), µ ∈ L ∞ (Q) and k verifies the hypotheses (2.3)
∂u ∂t + ∂u ∂a + (k(x)u x ) x − µ(t, a, x)u = h, (2.17) u(t, a, 1) = u(t, a, 0) = 0, u(T, a, x) = u T (a, x), u(t, A, x) = 0.
Then, there exist two positive constants C and s 0 , such that every solution of (2.17) satisfies, for all s ≥ s 0 , the following inequality
s 3 Q Θ 3 x 2 k(x) u 2 e 2sϕ dtdadx + s Q Θk(x)u 2 x e 2sϕ dtdadx (2.18) ≤ C Q | h | 2 e 2sϕ dtdadx + sk(1) A 0 T 0 Θu 2 x (a, t, 1)e 2sϕ(a,t,1) dtda ,
where ϕ and Θ are the weight functions defined by ϕ(t, a, x) := Θ(t, a)ψ(x) with :
Θ(t, a) := 1 (t(T −t)) 4 a 4 , ψ(x) := c 1 ( x 0 r k(r) dr − c 2 ). (2.19) with c 2 > 1 k(1)(2−γ) , c 1 > 0 and γ is the parameter defined by (2.3).
For the proof of this proposition, we refer the reader to [13,Proposition 3.1]. The second result is the following Proposition 2.4. Let us consider the following system
∂z ∂t + ∂z ∂a + (k(x)z x ) x − c(t, a, x)z = h in Q b , (2.20) z(t, a, b 1 ) = z(t, a, b 2 ) = 0 on (0, T ) × (0, A), where Q b := (0, T ) × (0, A) × (b 1 , b 2 ), (b 1 , b 2 ) ⊂ [0, 1], h ∈ L 2 (Q b ), k ∈ C 1 ([0, 1])
is a strictly positive function and c ∈ L ∞ (Q b ). Then, there exist two positive constants C and s 0 , such that for any s ≥ s 0 , z verifies the following estimate
Q b (s 3 φ 3 z 2 + sφz 2 x )e 2sΦ dtdadx ≤ C Q b h 2 e 2sΦ dtdadx + ω A 0 T 0 s 3 φ 3 z 2 e 2sΦ dtdadx , (2.21)
where φ, Θ and Φ are defined by (2.13) and σ by (2.14).
For the proof of Proposition 2.4, a careful computations allow us to adapt the same procedure of [2, Lemma 2.1] to show (2.21) in case where k is a positive general nondegenerate coefficient, with our weight function Θ(t, a) = 1 t 4 (T −t) 4 a 4 and the source term h.
Besides the two Propositions 2.3 and 2.4, we must bring out another important result Lemma 2.5. Under assumptions (2.15), the functions ϕ 1 , ϕ 2 and Φ defined by (2.13) satisfy the following inequalities
ϕ 1 ≤ ϕ 2 , 4 3 Φ < ϕ 2 ≤ Φ. (2.22)
Proof. By the definitions of ϕ 1 , ϕ 2 and Φ and taking into account that Θ is positive, showing the results of (2.22) is equivalent to show
ψ 1 ≤ ψ 2 , 4 3 Ψ < ψ 2 ≤ Ψ.
(2.23)
The first inequality in (2.23) is assured by the second assumption in (2.15) while the second one is deduced from
λ 2 ∈ I = [ k 2 (1)(2−γ)(e 2κ σ ∞ −1) d 2 k 2 (1)(2−γ)−1 , 4(e 2κ σ ∞ −e κ σ ∞ )3d 2
) and this achieves the proof. Now, we can address the proof of Theorem 2.2.
Proof. Let us introduce the smooth cut-off function ξ : R → R defined as follows
0 ≤ ξ(x) ≤ 1, x ∈ R, ξ(x) = 1, x ∈ [0, 2x 1 +x 2 3 ], ξ(x) = 0, x ∈ [ x 1 +2x 2 3 , 1].
(2.24)
Let u and v be respectively the solutions of (3.75) and (3.76). Set w := ξu, z := ξv and put ω
′ = ( 2x 1 +x 2 3 , x 1 +2x 2 3
). Then, (w, z) satisfies the following system
∂w ∂t + ∂w ∂a + (k 1 (x)w x ) x − µ 1 (t, a, x)w = µ 3 (t, a, x)z + ξh 1 + (k 1 ξ x u) x + ξ x k 1 u x in Q, (2.25) ∂z ∂t + ∂z ∂a + (k 2 (x)z x ) x − µ 2 (t, a, x)z = ξh 2 + (k 2 ξ x v) x + ξ x k 2 v x in Q, w(t, a, 1) = w(t, a, 0) = z(t, a, 1) = z(t, a, 0) = 0 on (0, T ) × (0, A), w(T, a, x) = w T (a, x) in Q A , z(T, a, x) = z T (a, x) in Q A , w(t, A, x) = z(t, A, x) = 0 in Q T . Using Proposition2.3 for the inhomogeneous term ξ(h 1 + µ 3 v) + (k 1 ξ x u) x + ξ x k 1 u x ,
the definition of ξ and Young inequality, we get the following inequality
Q (sΘk 1 w 2 x + s 3 Θ 3 x 2 k 1 w 2 )e 2sϕ 1 dtdadx ≤ C( Q [ξ 2 (h 1 + µ 3 v) 2 + ((k 1 ξ x u) x + ξ x k 1 u x ) 2 ]e 2sϕ 1 dtdadx +sk 1 (1) A 0 T 0 Θw 2 x (t, a, 1)e 2sϕ 1 (t,a,1) dtda) ≤ C Q [µ 2 3 z 2 + ξ 2 h 2 1 + ((k 1 ξ x u) x + ξ x k 1 u x ) 2 ]e 2sϕ 1 dtdadx. (2.26)
Thanks again to the definition of ξ, we have
1 0 ((k 1 ξ x u) x + ξ x k 1 u x ) 2 e 2sϕ 1 dx ≤ ω ′ (8(k 1 ξ x ) 2 u 2 x + 2((k 1 ξ x ) x ) 2 u 2 )e 2sϕ 1 dx ≤ C ω ′ (u 2 + u 2 x )e 2sϕ 1 dx. (2.27)
On the other hand, since x 2 k 2 (x) is non-decreasing, with the help of Hardy-Poincaré inequality stated in [8] and since ϕ 1 ≤ ϕ 2 we get
1 0 µ 2 3 z 2 e 2sϕ 1 dx ≤ µ 3 2 ∞ k 2 (1) 1 0 k 2 (x) x 2 (ze sϕ 2 ) 2 dx ≤ C µ 3 2 ∞ k 2 (1) 1 0 k 2 (x)((ze sϕ 2 ) x ) 2 dx.
Thus, from the definition of ψ 2 , we obtain
1 0 µ 2 3 z 2 e 2sϕ 1 dx ≤ C 1 0 k 2 (x)z 2 x e 2sϕ 2 dx + C 1 0 s 2 Θ 2 x 2 k 2 (x) z 2 e 2sϕ 2 dx.
Hence, for s quite large we get
1 0 µ 2 3 z 2 e 2sϕ 1 dx ≤ 1 2 1 0 sΘk 2 (x)z 2 x e 2sϕ 2 dx + 1 2 1 0 s 3 Θ 3 x 2 k 2 (x) z 2 e 2sϕ 2 dx. (2.28)
Combining (2.26), (2.27) and (2.28), for s quite large the following inequality holds
Q (sΘk 1 w 2 x + s 3 Θ 3 x 2 k 1 w 2 )e 2sϕ 1 dtdadx (2.29) ≤ C Q h 2 1 e 2sϕ 1 dtdxda + 1 2 Q (sΘk 2 (x)z 2 x + s 3 Θ 3 x 2 k 2 (x) z 2 )e 2sϕ 2 dtdadx +C 1 ω ′ A 0 T 0 (u 2 + u 2 x )e 2sϕ 1 dtdadx.
Applying the same way with
ξh 2 + (k 2 ξ x v) x + ξ x k 2 v x we obtain Q (sΘk 2 z 2 x + s 3 Θ 3 x 2 k 2 z 2 )e 2sϕ 2 dtdadx ≤ C 2 Q h 2 2 e 2sϕ 2 dtdxda + C 3 ω ′ A 0 T 0 (v 2 + v 2 x )e 2sϕ 2 dtdadx. (2.30)
Therefore, for s quite large we conclude by inequalities (2.29) and (2.30) and again ϕ 1 ≤ ϕ 2 that
Q (sΘk 1 w 2 x + s 3 Θ 3 x 2 k 1 w 2 )e 2sϕ 1 dtdadx + Q (sΘk 2 z 2 x + s 3 Θ 3 x 2 k 2 z 2 )e 2sϕ 2 dtdadx ≤ C 4 Q (h 2 1 + h 2 2 )e 2sϕ 2 dtdadx + C 5 ω ′ A 0 T 0 (u 2 + v 2 + u 2 x + v 2 x )e 2sϕ 2 dtdadx.
Using Caccioppoli's inequality (4.87), the last inquality becomes
Q (sΘk 1 w 2 x + s 3 Θ 3 x 2 k 1 w 2 )e 2sϕ 1 dtdadx + Q (sΘk 2 z 2 x + s 3 Θ 3 x 2 k 2 z 2 )e 2sϕ 2 dtdadx ≤ C 6 Q (h 2 1 + h 2 2 )e 2sϕ 2 dtdadx + C 7 q s 2 Θ 2 (u 2 + v 2 )e 2sϕ 2 dtdadx. (2.31)
Now, let W := ηu and Z := ηv with η = 1 − ξ. Then W and Z are supported in (x 1 , 1) and verify the following system ∂W ∂t 1). Then, the system satisfied by W and Z is nondegenerate. Hence, applying Proposition 2.4 on the first equation of (2.32) for b 1 = x 1 , b 2 = 1 and h := η(h 1 + µ 3 v) + (k 1 η x u) x + η x k 1 u x , with the aid of Caccioppoli's inequality stated in [13,Lemma 5.1], thanks to the definition of η and Young inequality and taking s quite large we obtain the following estimate
+ ∂W ∂a + (k 1 (x)W x ) x − µ 1 (t, a, x)W = µ 3 (t, a, x)Z + ηh 1 + (k 1 η x u) x + η x k 1 u x in Q x 1 , (2.32) ∂Z ∂t + ∂Z ∂a + (k 2 (x)Z x ) x − µ 2 (t, a, x)Z = ηh 2 + (k 2 η x v) x + η x k 2 v x in Q x 1 , W (t, a, 1) = W (t, a, x 1 ) = Z(t, a, 1) = Z(t, a, x 1 ) = 0 on (0, T ) × (0, A), W (t, a, x) = W T (a, x) in Q A , Z(t, a, x) = Z T (a, x) in Q A , W (t, A, x) = Z(t, A, x) = 0 in Q T , where, Q x 1 := (0, T ) × (0, A) × (x 1 ,Q (s 3 φ 3 W 2 + sφW 2 x )e 2sΦ dtdadx ≤ C Q (η(h 1 + µ 3 v) + (kη x u) x + kη x u x ) 2 e 2sΦ dtdadx + ω A 0 T 0 s 3 Θ 3 u 2 e 2sΦ dtdadx ≤ C Q η 2 (h 1 + µ 3 v) 2 e 2sΦ + ((kη x u) x + kη x u x ) 2 e 2sΦ dtdadx + ω A 0 T 0 s 3 Θ 3 u 2 e 2sΦ dtdadx ≤ C( Q η 2 (h 1 + µ 3 v) 2 e 2sΦ dtdadx + ω ′ A 0 T 0 (8(kη x ) 2 u 2 x + 2((kη x ) x ) 2 u 2 )e 2sΦ dtdadx + ω A 0 T 0 s 3 Θ 3 u 2 e 2sΦ dtdadx) ≤ C 1 Q η 2 (h 1 + µ 3 v) 2 e 2sΦ dtdadx + ω ′ A 0 T 0 (u 2 x + u 2 )e 2sΦ dtdadx + ω A 0 T 0 s 3 Θ 3 u 2 e 2sΦ dtdadx ≤ C 2 Q η 2 (h 1 + µ 3 v) 2 e 2sΦ dtdadx + ω A 0 T 0 s 3 Θ 3 u 2 e 2sΦ dtdadx ≤ C 3 Q (h 2 1 + µ 2 3 Z 2 )e 2sΦ dtdadx + q s 3 Θ 3 u 2 e 2sΦ dtdadx ,(2.33)
with Φ and φ are defined in (2.13) and ω ′ is defined in the beginning of the proof. On the other hand, using the fact that x → x 2 k 2 (x) is non-decreasing, Hardy-Poincaré inequality for the function Ze sΦ and the definition of ψ 2 we have for s quite large the following inequality
Q µ 2 3 Z 2 e 2sΦ dx ≤ c Q k 2 (x)Z 2 x e 2sΦ dtdadx + Q s 2 Θ 2 x 2 k 2 (x) Z 2 e 2sΦ dtdadx ≤ 1 2 Q (s 3 φ 3 Z 2 + sφZ 2 x )e 2sΦ dtdadx.
(2.34) Therefore, injecting (2.34) in (2.33) we get
Q (s 3 φ 3 W 2 + sφW 2 x )e 2sΦ dtdadx (2.35) ≤ C Q h 2 1 e 2sΦ dtdadx + q s 3 Θ 3 u 2 e 2sΦ dtdadx + 1 2 Q (s 3 φ 3 Z 2 + sφZ 2 x )e 2sΦ dtdadx.
Replying the same argument for the source term h :
= ηh 2 + (k 2 η x v) x + η x k 2 v x we infer that Q (s 3 φ 3 Z 2 + sφZ 2 x )e 2sΦ dtdadx ≤ C 8 Q h 2 2 e 2sΦ dtdadx + q s 3 Θ 3 v 2 e 2sΦ dtdadx .([s 3 φ 3 (W 2 + Z 2 ) + sφ(W 2 x + Z 2 x )]e 2sΦ dtdadx ≤ C 9 Q (h 2 1 + h 2 2 )e 2sΦ dtdadx + q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx .
(2.37)
Using the fact that u = w + W and v = z + Z, ϕ 1 ≤ ϕ 2 ≤ Φ, the estimates (2.31) and (2.37)lead to estimate (2.16).
Using the Theorem 2.2 for a special functions h 1 and h 2 , we are ready to deduce the following result Theorem 2.6. Assume that the assumptions (2.3) and (2.4) hold. Let A > 0 and T > 0 be given such that T ∈ (0, δ) with δ ∈ (0, A) small enough. Then, there exist positive constants C (independent of δ) and s 0 such that for all s ≥ s 0 , every solution (u, v) of (2.9) satisfies
Q s 3 Θ 3 x 2 k 1 (x) u 2 + sΘk 1 (x)u 2 x e 2sϕ 1 dtdadx + Q s 3 Θ 3 x 2 k 2 (x) v 2 + sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx ≤ C q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx . (2.38) Proof. Let h 1 := −β 1 (t, a, x)u(t, 0, x) and h 2 := −β 2 (t, a, x)v(t, 0, x).
Therefore, thanks to (2.16) and (2.4) we have the existence of two positive constants C and s 0 such that, for all s ≥ s 0 , the following inequality holds
s 3 Q Θ 3 x 2 k 1 (x) u 2 e 2sϕ 1 + x 2 k 2 (x) v 2 e 2sϕ 2 dtdadx + s Q Θ k 1 (x)u 2 x e 2sϕ 1 + k 2 (x)v 2 x e 2sϕ 2 dtdadx ≤ C Q ((β 1 ) 2 u 2 (t, 0, x) + (β 2 ) 2 v 2 (t, 0, x))e 2sΦ dtdadx + q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx ≤ C 1 1 0 T 0 (u 2 (t, 0, x) + v 2 (t, 0, x))dtdadx + q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx (2.39) Set U(t, a, x) = u(T − t, A − a, x) and V (t, a, x) = v(T − t, A − a, x). Then, one has ∂U ∂t + ∂U ∂a − (k 1 (x)U x ) x + µ 1 (T − t, A − a, x)U + µ 3 (T − t, A − a, x)V = β 1 (T − t, A − a, x)U(t, A, x), U(t, a, 1) = U(t, a, 0) = 0, (2.40) U(0, a, x) = U 0 (a, x) = u T (A − a, x), U(t, 0, x) = 0, where V is the solution of ∂V ∂t + ∂V ∂a − (k 2 (x)V x ) x + µ 2 (T − t, A − a, x)V = β 2 (T − t, A − a, x)V (t, A, x),
V (t, a, 1) = V (t, a, 0) = 0, (2.41)
V (0, a, x) = V 0 (a, x) = v T (A − a, x), V (t, 0, x) = 0.
Integrating along the characteristic lines, we get respectively the implicit formulas for the solutions U of (2.40) and V of (2.41) given by
U(t, a, ·) = a 0 S(a − l)(β 1 (T − t, A − l, ·)U(t, A, ·) − µ 3 (T − t, A − l, ·)V (t, l, ·))dl, if t > a U(t, a, ·) = S(t)U 0 (a − t, ·) + t 0 S(t − l)(β 1 (T − l, A − a, ·)U(l, A, ·) − µ 3 (T − l, A − a, ·)V (l, a, ·))dl, if t ≤ a,
(2.42) and
V (t, a, ·) = a 0 L(a − l)β 2 (T − t, A − l, ·)V (t, A, ·)dl, if t > a V (t, a, ·) = L(t)V 0 (a − t, ·) + t 0 L(t − l)β 2 (T − l, A − a, ·)V (l, A, ·)dl, if t ≤ a,
(2.43) where (S(t)) t≥0 and (L(t)) t≥0 are the bounded semigroups generated respectively by the operators
A 4 U = −(k 1 U x ) x + µ 1 (T − t, A − a, x)U and A 7 V = −(k 2 V x ) x + µ 2 (T − t, A − a, x)V .
Hence, after a careful computations, (2.42) and (2.43) become respectively
u(t, a, ·) = A−a 0 S(A − a − l)(β 1 (t, A − l, ·)u(t, 0, ·) − µ 3 (t, A − l, ·)v(t, A − l, ·))dl, if a > t + (A − T ) u(t, a, ·) = S(T − t)u T (T + (a − t), ·) + T t S(l − t)(β 1 (l, a, ·)u(l, 0, ·) − µ 3 (l, a, ·)v(l, a, ·))dl, if a ≤ t + (A − T ), (2.44) v(t, a, ·) = A−a 0 L(A − a − l)β 2 (t, A − l, ·)v(t, 0, ·)dl, if a > t + (A − T ) v(t, a, ·) = L(T − t)v T (T + (a − t), ·) + T t L(l − t)β 2 (l, a, ·)v(l, 0, ·)dl, if a ≤ t + (A − T ),
(2.45) Thus, by the third hypothesis in (2.4) on β 1 and β 2 one has
u(t, 0, ·) = S(T − t)u T (T − t, ·) − T t S(l − t)µ 3 (l, 0, ·)v(l, 0, ·)dl, v(t, 0, ·) = L(T − t)v T (T + (a − t), ·).
(2.46) Subsequently, by (2.39) we deduce that
s 3 Q Θ 3 x 2 k 1 (x) u 2 e 2sϕ 1 + x 2 k 2 (x) v 2 e 2sϕ 2 dtdadx + s Q Θ k 1 (x)u 2 x e 2sϕ 1 + k 2 (x)v 2 x e 2sϕ 2 dtdadx ≤ C 1 q s 3 Θ 3 (u 2 + v 2 )e 2sΦ dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx ,(2.47)
since (S(t)) t≥0 and (L(t)) t≥0 are a bounded semigroups, µ 3 ∈ L ∞ (Q) and T ∈ (0, δ). Then the thesis follows.
We come now to the more challenging point and the novelty of this contribution which is the following ω-Carleman type inequality. Such an estimate plays a crucial role to obtain the null controllability of population dynamics cascade system with one control force.
Theorem 2.7. Let (2.3) and (2.4) be verified. Let A > 0 and T > 0 be given such that T ∈ (0, δ) with δ ∈ (0, A) small enough. Assume that there exists a positive constant ν such that
µ 3 ≥ ν on [0, T ] × [0, A] × ω 1 for some ω 1 ⋐ ω,(2.
48)
Then every solution (u, v) of (2.9) satisfies
Q s 3 Θ 3 x 2 k 1 (x) u 2 + sΘk 1 (x)u 2 x e 2sϕ 1 dtdadx + Q s 3 Θ 3 x 2 k 2 (x) v 2 + sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx ≤ C δ q u 2 dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx . (2.49)
This inequality is an immediate outcome of Theorem 2.6 applied to ω 1 and the following lemma (see for instance [7] and the references therein).
Lemma 2.8. Assume that (2.3) and (2.4) hold and let A > 0 and T > 0 be given such that T ∈ (0, δ) with δ ∈ (0, A) small enough. we suppose also that (2.48) holds. Then, for all ǫ > 0 there exist two positive constants C and M ǫ such that for every solution (u, v) of (2.9) the following inequality is satisfied
ω 1 A 0 T 0 s 3 Θ 3 v 2 e 2sΦ dtdadx ≤ ǫC Q s 3 Θ 3 x 2 k 2 v 2 e 2sϕ 2 dtdadx + Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx +M ǫ ω A 0 T 0 u 2 dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx . (2.50)
Proof. Let χ : R → R be the non-negative cut-off function defined as follows
χ ∈ C ∞ (0, 1), supp(χ) ⊂ ω, χ ≡ 1 on ω 1 .
(2.51)
Recall that ω = (x 1 , x 2 ). Multiplying the first equation of (2.9) by χs 3 Θ 3 ve 2sΦ and after an integration by parts, we get
Q χs 3 Θ 3 ve 2sΦ u t dtdadx = − Q (3 + 2sΦ)χs 3 Θ t Θ 2 uve 2sΦ dtdadx − Q χs 3 Θ 3 uv t e 2sΦ dtdadx. Q χs 3 Θ 3 ve 2sΦ u a dtdadx = − Q (3 + 2sΦ)χs 3 Θ a Θ 2 uve 2sΦ dtdadx − Q χs 3 Θ 3 uv a e 2sΦ dtdadx. Q χs 3 Θ 3 ve 2sΦ (k 1 u x ) x dtdadx = − Q χs 3 Θ 3 k 1 e 2sΦ u x v x dtdadx + Q s 3 Θ 3 k 1 (χe 2sΦ ) x uv x dtdadx + Q s 3 Θ 3 (k 1 (χe 2sΦ ) x ) x uvdtdadx. − Q χs 3 Θ 3 ve 2sΦ µ 1 udtdadx = − Q χs 3 Θ 3 µ 1 uve 2sΦ dtdadx. − Q χs 3 Θ 3 ve 2sΦ µ 3 vdtdadx = − Q χs 3 Θ 3 µ 3 v 2 e 2sΦ dtdadx.
Then, summing all these identities side by side, using the second equation of (2.9) and integrating again by parts Q χs 3 Θ 3 µ 3 v 2 e 2sΦ dtdadx = I 1 + I 2 + I 3 + I 4 + I 5 , (2.52) where, I 1 := Q χs 3 Θ 3 β 1 vu(t, 0, x)e 2sΦ dtdadx,
I 2 := − Q ((3 + 2sΦ)s 3 Θ t Θ 2 + (3 + 2sΦ)s 3 Θ a Θ 2 + µ 1 s 3 Θ 3 + µ 2 s 3 Θ 3 ) χe 2sΦ uvdtdadx + Q s 3 Θ 3 (k 1 (χe 2sΦ ) x ) x uvdtdadx, I 3 := Q χs 3 Θ 3 β 2 uv(t, 0, x)e 2sΦ dtdadx, I 4 := Q s 3 Θ 3 (k 1 − k 2 )(x)uv x (χe 2sΦ ) x dtdadx, I 5 := − Q χs 3 Θ 3 (k 1 + k 2 )(x)u x v x e 2sΦ dtdadx.
On one hand, we have by Young inequality and definition of χ
I 5 ≤ ǫ Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx + 1 4ǫ Q χ 2 s 5 Θ 5 (k 1 + k 2 ) 2 u 2 x e 2s(2Φ−ϕ 2 ) k 2 dtdadx ≤ ǫ Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx + max [0,1] (k 1 + k 2 ) 2 4ǫ min ω k 2 Q χs 5 Θ 5 u 2 x e 2s(2Φ−ϕ 2 ) dtdadx. (2.53) Put L := Q χs 5 Θ 5 u 2 x e 2s(2Φ−ϕ 2 ) dtdadx.
To increase I 5 , we will find an upper bound of L. To do this, we multiply the first equation of (2.9) by χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 u and after integration by parts
Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 uu t dtdadx = − 1 2 Q s 5 χ k 1 Θ 4 Θ t (5 + 2s(2Φ − ϕ 2 ))e 2s(2Φ−ϕ 2 ) u 2 dtdadx. Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 uu a dtdadx = − 1 2 Q s 5 χ k 1 Θ 4 Θ a (5 + 2s(2Φ − ϕ 2 ))e 2s(2Φ−ϕ 2 ) u 2 dtdadx. Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 u(k 1 u x ) x dtdadx = − Q χs 5 Θ 5 u 2 x e 2s(2Φ−ϕ 2 ) dtdadx + 1 2 Q s 5 Θ 5 k 1 χe 2s(2Φ−ϕ 2 ) k 1 x x u 2 dtdadx. − Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 uµ 1 udtdadx = − Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 µ 1 u 2 dtdadx. − Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 uµ 3 vdtdadx = − Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) k 1 µ 3 uvdtdadx.
Hence, adding these equalities side by side we get
L = L 1 + L 2 + L 3 ,(2.k 1 µ 1 + 1 2 s 5 χ k 1 Θ 5 Θ t ( 5 Θ + 2s(2Ψ − ψ 2 )) + 1 2 s 5 χ k 1 Θ 5 Θ a ( 5 Θ + 2s(2Ψ − ψ 2 )) e 2s(2Φ−ϕ 2 ) u 2 dtdadx + 1 2 Q s 5 Θ 5 k 1 χe 2s(2Φ−ϕ 2 ) k 1 x x u 2 dtdadx.
The assumptions in (2.4) on β 1 together with Young inequality, Lemma ??, the definitions of χ and Θ, the fact that the function x → k 2 x 2 is non-increasing, |Θ t | ≤ CΘ 2 and |Θ a | ≤ CΘ 2 and sup (t,a,x)∈Q s p Θ p e 2s(2Φ−ϕ 2 ) < +∞ for p ∈ R, (2.55) lead to
L 1 ≤ 1 4ǫ Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) (k 1 ) 2 u 2 dtdadx + ǫ Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) (β 1 ) 2 u 2 (t, 0, x)dtdadx ≤ K 1 4ǫ Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) u 2 dtdadx + ǫK 1 1 0 A 0 T T −δ χu 2 (t, 0, x)dtdadx ≤ K 1 4ǫ Q χs 5 Θ 5 e 2s(2Φ−ϕ 2 ) u 2 dtdadx + ǫK 2 1 0 δ 0 χu 2 T (a, x)dadx (2.56)
and
L 2 ≤ ǫ 2 Q x 2 k 2 s 3 Θ 3 e 2sϕ 2 v 2 dtdadx + 1 4ǫ 2 Q χ 2 s 7 Θ 7 (k 1 ) 2 e 2s(4Φ−3ϕ 2 ) k 2 x 2 (µ 3 ) 2 u 2 dtdadx ≤ ǫ 2 Q x 2 k 2 s 3 Θ 3 e 2sϕ 2 v 2 dtdadx + K 4 4ǫ 2 ω A 0 T 0 s 7 Θ 7 e 2s(4Φ−3ϕ 2 ) u 2 dtdadx,(2.L ≤ ǫ 2 Q x 2 k 2 s 3 Θ 3 e 2sϕ 2 v 2 dtdadx + K ǫ ω A 0 T 0 s 7 Θ 7 e 2s(4Φ−3ϕ 2 ) u 2 dtdadx +ǫK 2 1 0 δ 0 u 2 T (a, x)dadx. (2.60)
Hence, by (2.53) and (2.60) we deduce
I 5 ≤ ǫC Q x 2 k 2 s 3 Θ 3 e 2sϕ 2 v 2 dtdadx + Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx +K 1 ǫ ω A 0 T 0 s 7 Θ 7 e 2s(4Φ−3ϕ 2 ) u 2 dtdadx + K 2 1 0 δ 0 u 2 T (a, x)dadx. (2.61)
where K 1 ǫ is a positive constants that depend on ǫ. Similarly, we will find an upper bounds of I 1 , I 2 , I 3 and I 4 . Firstly, we will start by I 2 . One has the following relations
Q χ(3 + 2sΦ)s 3 Θ t Θ 2 e 2sΦ uvdtdadx ≤ Q χ|3 + 2sΦ|s 3 |Θ t |Θ 2 e 2sΦ |uv|dtdadx ≤ C Q χ|3 + 2sΦ|s 3 Θ 4 e 2sΦ |uv|dtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C ǫ ω A 0 T 0 s 5 Θ 5 e 2s(2Φ−ϕ 2 ) u 2 dtdadx, (2.62) Q χ(3 + 2sΦ)s 3 Θ a Θ 2 e 2sΦ uvdtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 1 ǫ ω A 0 T 0 s 5 Θ 5 e 2s(2Φ−ϕ 2 ) u 2 dtdadx, (2.63) Q χ(µ 1 + µ 2 )s 3 Θ 3 e 2sΦ uvdtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 2 ǫ ω A 0 T 0 s 3 Θ 3 e 2s(2Φ−ϕ 2 ) u 2 dtdadx, (2.64) Q s 3 Θ 3 (k 1 (χe 2sΦ ) x ) x uvdtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + 1 4ǫ Q s 3 Θ 3 k 2 x 2 (k 1 (χe 2sΦ ) x ) 2 x e −2sϕ 2 u 2 dtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 2 4ǫ Q s 3 Θ 3 k 2 x 2 (χ 2 + χ 2 x + χ 2 xx )e 2s(2Φ−ϕ 2 ) u 2 dtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 3 ǫ ω A 0 T 0 s 3 Θ 3 e 2s(2Φ−ϕ 2 ) u 2 dtdadx,(2.I 2 ≤ 4ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 4 ǫ ω A 0 T 0 s 5 Θ 5 e 2s(2Φ−ϕ 2 ) u 2 dtdadx. (2.66)
For the rest of integrals,
I 1 = Q χs 3 Θ 3 β 1 vu(t, 0, x)e 2sΦ dtdadx ≤ ǫ Q s 3 Θ 3 x 2 k 2 e 2sϕ 2 v 2 dtdadx + C 5 ǫ 1 0 δ 0 u 2 T (a, x)dadx.
(2.67)
I 3 = Q χs 3 Θ 3 β 2 uv(t, 0, x)e 2sΦ dtdadx ≤ ǫ 1 0 δ 0 v 2 T (a, x)dadx + 1 4ǫ ω A 0 T 0 s 7 Θ 7 e 2s(2Φ−ϕ 2 ) u 2 dtdadx.
(2.68)
I 4 = Q s 3 Θ 3 (k 1 − k 2 )(x)uv x (χe 2sΦ ) x dtdadx = Q s 3 Θ 3 (k 1 − k 2 )(x)uv x (χ x + 2sΦ x χ)e 2sΦ dtdadx ≤ ǫ Q sΘk 2 v 2 x e 2sϕ 2 dadx + 1 4ǫ Q s 5 Θ 5 (k 1 − k 2 ) 2 k 2 (χ x + 2sΦ x χ) 2 e 2s(2Φ−ϕ 2 ) u 2 dtdadx ≤ ǫ Q sΘk 2 v 2 x e 2sϕ 2 dadx + C 6 ǫ ω A 0 T 0 s 7 Θ 7 e 2s(2Φ−ϕ 2 ) u 2 dtdadx. (2.69)
Subsequently, combining (2.61), (2.66), (2.67), (2.68), (2.69) and using again (2.59)
Q χs 3 Θ 3 µ 3 v 2 e 2sΦ dtdadx ≤ ǫC 7 Q s 3 Θ 3 x 2 k 2 v 2 e 2sϕ 2 dtdadx + Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx +C 8 ǫ ω A 0 T 0 s 7 Θ 7 e 2s(4Φ−3ϕ 2 ) u 2 dtdadx + C 9 ǫ 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx.
Finally, the hypothesis (2.48), the definition of χ and the relation
sup (t,a,x)∈Q s p Θ p e 2s(4Φ−3ϕ 2 ) < +∞ for p ∈ R, (2.70) yield ω 1 A 0 T 0 s 3 Θ 3 v 2 e 2sΦ dtdadx ≤ ǫC 10 Q s 3 Θ 3 x 2 k 2 v 2 e 2sϕ 2 dtdadx + Q sΘk 2 (x)v 2 x e 2sϕ 2 dtdadx +C 11 ǫ ω A 0 T 0 u 2 dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx , (2.71)
which finishes the proof.
The above Carleman estimate can be used in a standard way to obtain the null controllability of the cascade system with one control force. This will be reached showing an observability inequality of the adjoint system.
Observability inequality and null controllability results
This paragraph is devoted to the observability inequality of system (2.9) and then the null controllability result of system (1.1). We start to show our observability inequality whose proof is based essentially on Carleman estimate (2.49) and Hardy-Poincaré inequality.
Proposition 3.1. Assume that (2.3) and (2.4) hold. Suppose also that (2.48) is fulfilled and let A > 0 and T > 0 be given such that T ∈ (0, δ) with δ ∈ (0, A) small enough. Then, there exists a positive constant C δ such that for every solution (u, v) of (2.9), the following observability inequality is satisfied
1 0 A 0 (u 2 (0, a, x) + v 2 (0, a, x))dadx ≤ C δ q u 2 dtdadx + 1 0 δ 0 (u 2 T (a, x) + v 2 T (a, x))dadx . (3.72)
Proof. Then for κ > 0 to be defined later, u = e κt u and v = e κt v are respectively a solutions of ∂ u ∂t + ∂ u ∂a
+ (k 1 (x) u x ) x − µ 1 (t, a, x) u = µ 3 (t, a, x) v − β 1 u(t, 0, x) in Q, (3.73)
u(t, a, 1) = u(t, a, 0) = 0 on (0, T ) × (0, A),
u(T, a, x) = e κT u T (a, x) in Q A , u(t, A, x) = 0 in Q T , and ∂ v ∂t + ∂ v ∂a + (k 2 (x) v x ) x − µ 2 (t, a, x) v = −β 2 v(t, 0, x) in Q, (3.74) v(t, a, 1) = v(t, a, 0) = 0 on (0, T ) × (0, A), v(T, a, x) = e κT v T (a, x) in Q A , v(t, A, x) = 0 in Q T ,
where, u and v are respectively the solutions of
∂u ∂t + ∂u ∂a + (k 1 (x)u x ) x − µ 1 (t, a, x)u = µ 3 (t, a, x)v − β 1 u(t, 0, x) in Q, (3.75) u(t, a, 1) = u(t, a, 0) = 0 on (0, T ) × (0, A), u(T, a, x) = u T (a, x) in Q A , u(t, A, x) = 0 in Q T , and ∂v ∂t + ∂v ∂a + (k 2 (x)v x ) x − µ 2 (t, a, x)v = −β 2 v(t, 0, x) in Q, (3.76) v(t, a, 1) = v(t, a, 0) = 0 on (0, T ) × (0, A), v(T, a, x) = v T (a, x) in Q A , v(t, A, x) = 0 in Q T .
Multiplying the first equations of (3.73) and (3.74) respectively by u and v and integrating by parts on Q t = (0, t) × (0, A) × (0, 1) one obtains
1 2 Q A u 2 (0, a, x)dadx + 1 2 1 0 t 0 u 2 (τ, 0, x)dτ dx +κ 1 0 A 0 t 0 u 2 (τ, a, x)dτ dadx ≤ β 1 2 ∞ + 1 4ǫ ′ 1 0 A 0 t 0 u 2 (τ, a, x)dτ dadx +ǫ ′ A 1 0 t 0 u 2 (τ, 0, x)dτ dx + ǫ ′ Qt µ 2 3 v 2 dτ dadx + 1 2 Q A u 2 (t, a, x)dadx. (3.77) and 1 2 Q A v 2 (0, a, x)dadx + 1 2 1 0 t 0 v 2 (τ, 0, x)dτ dx +κ 1 0 A 0 t 0 v 2 (τ, a, x)dτ dadx ≤ β 2 2 ∞ + 1 4ǫ ′ 1 0 A 0 t 0 v 2 (τ, a, x)dτ dadx +ǫ ′ A 1 0 t 0 v 2 (τ, 0, x)dτ dx + 1 2 Q A v 2 (t, a, x)dadx. (3.78)
Summing (3.77) and (3.78) side by side and taking κ = max( β 1 2
∞ +1 4ǫ ′ , β 2 2 ∞ +1 4ǫ ′ + ǫ ′ µ 3 2 ∞ ) and ǫ ′ < 1 2A , on gets Q A u 2 (0, a, x)dadx + Q A v 2 (0, a, x)dadx ≤ Q A u 2 (t, a, x)dadx + Q A v 2 (t, a, x)dadx. (3.79)
Arguing as in [2]and integrating over ( T 4 , 3T 4 ) we conclude
Q A u 2 (0, a, x)dadx + Q A v 2 (0, a, x)dadx ≤ C 12 e 2κT 1 0 δ 0 u 2 T (a, x)dadx + 1 0 δ 0 v 2 T (a, x)dadx + 2e 2κT T 1 0 A δ 3T 4 T 4 u 2 (t, a, x)dtdadx + 1 0 A δ 3T 4 T 4 v 2 (t, a, x) dtdadx. (3.80)
Hence, Hardy-Poincaré inequality and the definitions of ϕ i , i = 1, 2 stated in (2.13) lead to
Q A u 2 (0, a, x)dadx + Q A v 2 (0, a, x)dadx ≤ C 12 e 2κT 1 0 δ 0 u 2 T (a, x)dadx + 1 0 δ 0 v 2 T (a, x)dadx +C 13 δ 1 0 A δ 3T 4 T 4 sΘk 1 (x)u 2 (t, a, x)e 2sϕ 1 dtdadx + 1 0 A δ 3T 4 T 4 sΘk 2 (x)v 2 (t, a, x)e 2sϕ 2 dtdadx .
Finally, using the Carleman estimate (2.49) we deduce the observability inequality (3.72). and then the proof is finished. Now, obtaining our observability inequality, following a standard argument, we are now ready to prove our main result.
Theorem 3.2. Assume that (2.3) and (2.4) are verified. Let A > 0 and T > 0 be given such that T ∈ (0, δ) with δ ∈ (0, A) small enough. Then, for all (y 0 , p 0 ) ∈ L 2 (Q A ) × L 2 (Q A ), there exists a control ϑ ∈ L 2 (q) such that the associated solution of (1.1) verifies y(T, a, x) = 0, a.e. in (δ, A) × (0, 1), p(T, a, x) = 0, a.e in (δ, A) × (0, 1). (3.81)
Proof. Let ε > 0 and consider the following cost function
J ε (ϑ 1 , ϑ 2 ) = 1 2ε 1 0 A δ (y 2 (T, a, x) + p 2 (T, a, x))dadx + 1 2 q ϑ 2 (t, a, x)dtdadx.
We can prove that J ε is continuous, convex and coercive. Then, it admits at least one minimizer ϑ ε and we have
ϑ ε = −u ε (t, a, x)χ ω (x) in Q, (3.82)
with u ε is the solution of the following system ∂u ε ∂t + ∂u ε ∂a
+ (k 1 (x)(u ε ) x ) x − µ 1 (t, a, x)u ε − µ 3 v ε = −β 1 u ε (t, 0, x) in Q,(3.
83)
u ε (t, a, 1) = u ε (t, a, 0) = 0 on (0, T ) × (0, A),
u ε (T, a, x) = 1 ε y ε (T, a, x)χ (δ,A) (a) in Q A , u ε (t, A, x) = 0 in Q T ,
where v ε is the solution of
∂v ε ∂t + ∂v ε ∂a + (k 2 (x)(v ε ) x ) x − µ 2 (t, a, x)v ε = −β 2 v ε (t, 0, x) in Q, (3.84) v ε (t, a, 1) = v ε (t, a, 0) = 0 on (0, T ) × (0, A), v ε (T, a, x) = 1 ε p ε (T, a, x)χ (δ,A) (a) in Q A , v ε (t, A, x) = 0 in Q T ,
and (y ε , p ε ) is the solution of the system (1.1) associated to the control ϑ ε .
Multiplying the first equation of (3.83) by y ε and the second equation of (1.1) by v ε , integrating over Q, using (3.82) and the Young inequality we obtain
1 ε 1 0 A δ (y 2 ε (T, a, x) + p 2 ε (T, a, x))dadx + q ϑ 2 ε (t, a, x)dtdadx = Q A (y 0 (a, x)u ε (0, a, x) + p 0 (a, x)v ε (0, a, x))dadx ≤ 1 4C δ Q A (u 2 ε (0, a, x) + v 2 ε (0, a, x))dadx + C δ Q A (y 2 0 (a, x) + p 2 0 (a, x))dadx,
with C δ is the constant of the observability inequality (3.72). Hence, using relation (3.82), the observability inequality leads to
1 ε 1 0 A δ (y 2 ε (T, a, x) + p 2 ε (T, a, x))dadx + 3 4 q ϑ 2 ε (t, a, x)dtdadx ≤ C δ Q A (y 2 0 (a, x) + p 2 0 (a, x))dadx. (3.85)
Hence, it follows that
1 0 A δ y 2 ε (T, a, x)dadx ≤ C δ ε Q A (y 2 0 (a, x) + p 0 (a, x))dadx, 1 0 A δ p 2 ε (T, a, x)dadx ≤ C δ ε Q A (y 2 0 (a, x) + p 0 (a, x))dadx, q ϑ 2 ε (t, a, x)dtdadx ≤ 4C δ 3 Q A (y 2 0 (a, x) + p 0 (a, x))dadx.
(3.86)
Then, we can extract two subsequences of (y ε , p ε ) and ϑ ε denoted also by ϑ ε and(y ε , p ε ) that converge weakly towards ϑ and (y, p) in L 2 (q) and L 2 ((0, T ) × (0, A); H 1 k 1 (0, 1) × H 1 k 2 (0, 1)) respectively. Now, by a variational technic, we prove that (y, p) is a solution of (1.1) corresponding to the controls ϑ and, by the first and second estimates of (3.86), (y, p) satisfies (1.2).
Appendix
As is mentioned in the introduction, this section is devoted to the proofs of some intermediate results useful to show the Carleman type inequality (2.49). Firstly, we begin by the Caccioppoli's inequality stated in the following lemma Lemma 4.1. Let ω ′ be a subset of ω such that ω ′ ⊂⊂ ω. Then, there exists a positive constant C such that
ω ′ A 0 T 0 (u 2 x +v 2 x )e 2sϕ i dtdadx ≤ C q s 2 Θ 2 (u 2 + v 2 )e 2sϕ i dtdadx + q (h 2 1 + h 2 2 )e 2sϕ i dtdadx ,(4.2 (ϕ i ) t (u 2 + v 2 )e 2sϕ i dtdadx +2 1 0 A 0 T 0 ζ 2 u(−(k 1 u x ) x − u a + h 1 + µ 1 u + µ 3 v)e 2sϕ i dtdadx +2 1 0 A 0 T 0 ζ 2 v(−(k 2 v x ) x − v a + h 2 + µ 2 v)e 2sϕ i dtdadx.
Then, integrating by parts we obtain
2 Q ζ 2 (k 1 u 2 x + k 2 v 2 x )e 2sϕ i dtdadx = −2s Q ζ 2 (u 2 + v 2 )ψ i (Θ a + Θ t )e 2sϕ i dtdadx −2 Q ζ 2 (uh 1 + vh 2 )e 2sϕ i dtdadx − 2 Q ζ 2 (µ 1 u 2 + µ 2 v 2 )e 2sϕ i dtdadx + Q (k 1 (ζ 2 e 2sϕ i ) x ) x u 2 dtdadx + Q (k 2 (ζ 2 e 2sϕ i ) x ) x v 2 dtdadx −2 Q ζ 2 µ 3 uve 2sϕ i dtdadx.
On the other hand, by the definitions of ζ, ψ and Θ, using Young inequality and taking s quite large there is a constant c such that 2 Q ζ 2 (k 1 u 2 x + k 2 v 2 x )e 2sϕ i dtdadx ≥ 2 min(min
x∈ω ′ k 1 (x), min x∈ω ′ k 2 (x)) ω ′ A 0 T 0 (u 2 x + v 2 x )e 2sϕ i dtdadx, Q (k 1 (ζ 2 e 2sϕ i ) x ) x u 2 dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 u 2 e 2sϕ i dtdadx, Q (k 2 (ζ 2 e 2sϕ i ) x ) x v 2 dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 v 2 e 2sϕ i dtdadx, −2s Q ζ 2 (u 2 + v 2 )ψ i (Θ a + Θ t )e 2sϕ i dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 (u 2 + v 2 )e 2sϕ i dtdadx, −2 Q ζ 2 uh 1 e 2sϕ i dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 u 2 e 2sϕ i dtdadx + ω A 0 T 0 h 2 1 e 2sϕ i dtdadx , −2 Q ζ 2 vh 2 e 2sϕ i dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 v 2 e 2sϕ i dtdadx + ω A 0 T 0 h 2 2 e 2sϕ i dtdadx , −2 Q ζ 2 (µ 1 u 2 + µ 2 v 2 )e 2sϕ i dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 (u 2 + v 2 )e 2sϕ i dtdadx, −2 Q ζ 2 µ 3 uve 2sϕ i dtdadx ≤ c ω A 0 T 0 s 2 Θ 2 (u 2 + v 2 )e 2sϕ i dtdadx.
Combining all these inequalities, we can see that there is C > 0 such that
ω ′ A 0 T 0 (u 2 x + v 2 x )e 2sϕ i dtdadx ≤ C q s 2 Θ 2 (u 2 + v 2 )e 2sϕ i dtdadx + q (h 2 1 + h 2 2 )e 2sϕ i dtdadx .
Thus, the proof is achieved.
Remark 4.2. In Lemma 4.1, the set ω ′ is chosen so that 0 which is exactly the point of degeneracy of the dispersion coefficients k 1 and k 2 does not belong to ω ′ . More generally, if the degeneracy occurs at a point x 0 ∈ (0, 1), one must take x 0 out of ω ′ in the case of interior degeneracy to establish a Caccioppoli's type inequality (see [10] for more details in this context).
We close this section by the following result is not empty.
Proof. Indeed, one has 4(e 2κ σ ∞ − e κ σ ∞ ) 3d 2 − k 2 (1)(2 − γ)(e 2κ σ ∞ − 1) d 2 k 2 (1)(2 − γ) − 1 = 4(e 2κ σ ∞ − e κ σ ∞ )(d 2 k 2 (1)(2 − γ) − 1) − 3d 2 k 2 (1)(2 − γ)(e 2κ σ ∞ − 1) 3d 2 (d 2 k 2 (1)(2 − γ) − 1) = e 2κ σ ∞ (d 2 k 2 (1)(2 − γ) − 4) − 4e κ σ ∞ (d 2 k 2 (1)(2 − γ) − 1) 3d 2 (d 2 k 2 (1)(2 − γ) − 1) + k 2 (1)(2 − γ) d 2 k 2 (1)(2 − γ) − 1 = e κ σ ∞ [e κ σ ∞ (d 2 k 2 (1)(2 − γ) − 4) − 4(d 2 k 2 (1)(2 − γ) − 1)] 3d 2 (d 2 k 2 (1)(2 − γ) − 1) + k 2 (1)(2 − γ) d 2 k 2 (1)(2 − γ) − 1 .
Using the fact that d 2 ≥ 5 k 2 (1)(2−γ) , we can conclude that 4(d 2 k 2 (1)(2−γ)−1) d 2 k 2 (1)(2−γ)−4) ≤ 16. Since κ ≥ 4 ln 2 σ ∞ , then we have e κ σ ∞ ≥ 16. Therefore, the previous difference is positive and subsequently I = ∅.
(t, a, x)p(t, a, x)da in Q T ,
Lemma 4.3 in the appendix). On other hand, in the light of the first and the fourth conditions in (2.15) on d 1 and d 2 , one can observe that ψ i (x) < 0 for all x ∈ [0, 1], and Θ(t, a) → +∞ as t → 0 + , T − and a → 0 + . Now, we state the first result of this section which is the intermediate Carleman estimate satisfied by solution of system (2.10).
1 ) 2 minω k 1 . On the other hand, by Lemma 2.5 we have e 2s(2Φ−ϕ 2 ) ≤ e 2s(4Φ−3ϕ 2 ) . (2.59) Then, combining relations (2.54), (2.56), (2.57) and (2.58) we conclude
87)where (u, v) is the solution of (2.10) and the weight functions ϕ i , i = 1, 2 are defined by (2.13).Proof. The proof of this result is similar to the one of [13, Lemma 5.1]. Indeed, consider the cut-off function ζ defined by 0 ≤ ζ(x) ≤ 1, x ∈ R, ζ(x) = 0, x < x 1 and x > x 2 , ζ(x) = 1, x ∈ ω ′ , (ϕ i ) t (u 2 + v 2 )e 2sϕ i dtdadx +
Lemma 4. 3 .
3Assume that the conditions (2.15) hold. Then, I = [ k 2 (1)(2−γ)(e 2κ σ ∞ −1)
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|
[] |
[
"Constrained Classification and Policy Learning *",
"Constrained Classification and Policy Learning *"
] |
[
"Toru Kitagawa [email protected]. \nDepartment of Economics\nUniversity College London\n\n",
"Shosei Sakaguchi [email protected]. \nDepartment of Economics\nGeneva School of Economics and Management\nUniversity College London\nUniversity of Geneva\n\n",
"Aleksey Tetenov [email protected] "
] |
[
"Department of Economics\nUniversity College London\n",
"Department of Economics\nGeneva School of Economics and Management\nUniversity College London\nUniversity of Geneva\n"
] |
[] |
Modern machine learning approaches to classification, including AdaBoost, support vector machines, and deep neural networks, utilize surrogate loss techniques to circumvent the computational complexity of minimizing empirical classification risk. These techniques are also useful for causal policy learning problems, since estimation of individualized treatment rules can be cast as a weighted (cost-sensitive) classification problem. Consistency of the surrogate loss approaches studied in Zhang(2004)andBartlett et al. (2006)crucially relies on the assumption of correct specification, meaning that the specified set of classifiers is rich enough to contain a first-best classifier. This assumption is, however, less credible when the set of classifiers is constrained by interpretability or fairness, leaving the applicability of surrogate loss based algorithms unknown in such second-best scenarios. This paper studies consistency of surrogate loss procedures under a constrained set of classifiers without assuming correct specification. We show that in the setting where the constraint restricts the classifier's prediction set only, hinge losses (i.e., 1 -support vector machines) are the only surrogate losses that preserve consistency in second-best scenarios. If the constraint additionally restricts the functional form of the classifier, consistency of a surrogate loss approach is not guaranteed even with hinge loss. We therefore characterize conditions for the constrained set of classifiers that can guarantee consistency of hinge risk minimizing classifiers. Exploiting our theoretical results, we develop robust and computationally attractive hinge loss based procedures for a monotone classification problem.
| null |
[
"https://arxiv.org/pdf/2106.12886v1.pdf"
] | 235,356,702 |
2106.12886
|
8deab163aa482174fe658fb243f256285d60d7c4
|
Constrained Classification and Policy Learning *
June 25, 2021 24 Jun 2021
Toru Kitagawa [email protected].
Department of Economics
University College London
Shosei Sakaguchi [email protected].
Department of Economics
Geneva School of Economics and Management
University College London
University of Geneva
Aleksey Tetenov [email protected]
Constrained Classification and Policy Learning *
June 25, 2021 24 Jun 20211Surrogate losssupport vector machinemonotone classificationfairness in machine learningstatistical treatment choicepersonalized medicine *
Modern machine learning approaches to classification, including AdaBoost, support vector machines, and deep neural networks, utilize surrogate loss techniques to circumvent the computational complexity of minimizing empirical classification risk. These techniques are also useful for causal policy learning problems, since estimation of individualized treatment rules can be cast as a weighted (cost-sensitive) classification problem. Consistency of the surrogate loss approaches studied in Zhang(2004)andBartlett et al. (2006)crucially relies on the assumption of correct specification, meaning that the specified set of classifiers is rich enough to contain a first-best classifier. This assumption is, however, less credible when the set of classifiers is constrained by interpretability or fairness, leaving the applicability of surrogate loss based algorithms unknown in such second-best scenarios. This paper studies consistency of surrogate loss procedures under a constrained set of classifiers without assuming correct specification. We show that in the setting where the constraint restricts the classifier's prediction set only, hinge losses (i.e., 1 -support vector machines) are the only surrogate losses that preserve consistency in second-best scenarios. If the constraint additionally restricts the functional form of the classifier, consistency of a surrogate loss approach is not guaranteed even with hinge loss. We therefore characterize conditions for the constrained set of classifiers that can guarantee consistency of hinge risk minimizing classifiers. Exploiting our theoretical results, we develop robust and computationally attractive hinge loss based procedures for a monotone classification problem.
Introduction
Binary classification, the prediction of a binary dependent variable Y ∈ {−1, +1} based on covariate information X ∈ X , is one of the most fundamental problems in statistics and econometrics. Many modern machine learning algorithms build on statistically and computationally efficient classification algorithms, and their application has had a sizeable impact on various fields of study and in society in general, e.g., pattern recognition, credit approval systems, personalized recommendation systems, to list but a few. Since estimation of an optimal treatment assignment policy can be cast as a weighted (cost-sensitive) classification problem (Zadrozny (2003)), methodological advances in the study of the classification problem apply to the causal problem of designing individualized treatment assignment policies. As the allocation of resources in both business and public policy settings has become more evidence-based and dependent on algorithms, so too has there been increasingly active debate on how to make allocation algorithms respect societal preferences for interpretability and fairness (Dwork et al. (2012)). Understanding the theoretical performance guarantee and efficient implementation of classification algorithms under interpretability or fairness constraints is a problem of fundamental importance with a strong connection to real life.
In the supervised binary classification problem, the typical objective is to learn a classification rule that minimizes the probability of false prediction. We denote the distribution of (Y, X) by P and a (non-randomized) classifier by f : X → R that predicts Y ∈ {−1, +1} based on sign(f (X)), where sign(α) = 1{α ≥ 0} − 1{α < 0}. We denote the 0-level set of f by G f ≡ {x ∈ X : f (x) ≥ 0} ⊂ X , and refer to G f as the prediction set of f . The goal is to learn a classifier that minimizes classification risk : R(f ) ≡ P (sign(f (X)) = Y ) = E P [1{Y · sign(f (X)) ≤ 0}].
(1)
Given a training sample {(Y i , X i ) ∼ iid P : i = 1, . . . , n}, the empirical risk minimization principle of Vapnik (1998) recommends estimating the optimal classifier by minimizing empirical classification risk,
f ∈ arg inf f ∈F R(f ),(2)R(f ) ≡ 1 n n i=1
1{Y i · sign(f (X i )) ≤ 0}, over a class of classifiers F = {f : X → R}. If the complexity of F is properly constrained, the empirical risk minimizing (ERM) classifierf has statistically attractive properties including risk consistency and minimax rate optimality. See, for example, Devroye et al. (1996) and Lugosi (2002).
Despite the desirable performance guarantee of the ERM classifer, the computational complexity of solving the optimization in (2) becomes a serious hurdle to practical implementation, especially when the dimension of covariates is moderate to large. To get around this issue, the existing literature has offered various alternatives to the ERM classifier, including support vector machines (Cortes and Vapnik (1995)), AdaBoost (Freund and Schapire (1997)), and neural networks. Focusing on optimization, each of these algorithms can be viewed as targeting the minimization of surrogate risk,
R φ (f ) ≡ E P [φ(Y f (X))],(3)
where φ : R → R is called the surrogate loss function, a different specification of which corresponds to a different learning algorithm. Convex functions make for a desirable choice of surrogate loss function as, combined with some functional form specification for f , the minimization problem for the empirical analogue of the surrogate risk in (3) is a convex optimization problem. This insight and the computational benefit that it yields has been pivotal to learning algorithms being able to handle large scale problems with high-dimensional features.
Can surrogate risk minimization lead to an optimal classifier in terms of the original classification risk? The seminal works of Zhang (2004) and Bartlett et al. (2006) provide theoretical justification for the use of surrogate losses by clarifying the conditions under which surrogate risk minimization also minimizes the original classification risk. A crucial assumption for this important result is correct specification of the classifiers, requiring that the class of classifiers F over which the surrogate risk is minimized contains a classifier that globally minimizes the original classification risk, i.e., a classifier that is identical to or performs as well as the Bayes classifier f * Bayes (x) ≡ 2P (Y = 1|X = x) − 1 in terms of its classification risk.
The credibility of the assumption of correct specification is, however, limited if the set of implementable classifiers is constrained exogenously, independently of any belief concerning the underlying data generating process. Such a situation is becoming more prevalent due to the increasing need for interpretability or fairness of classification algorithms. Given that f determines the classification rule only through G f , such constraints can be represented by shape restrictions on the prediction set of f , i.e., the class of feasible f is represented by F G ≡ {f ∈ F : G f ∈ G}, where G is a restricted class of sets in X satisfying the requirements for interpretability and fairness. To the best of our knowledge, how the validity of a surrogate loss approach is affected if F G misses the first-best classifier is not known.
The main contribution of this paper is to establish conditions under which a surrogate loss approach is valid without assuming correct specification. We first characterize those conditions on surrogate loss such that minimization of the surrogate risk can lead to a second-best rule (i.e., constrained optimum) in terms of the original classification risk. Specifically, we show that hinge losses φ h (α) = c max{0, 1 − α}, c > 0, are the only surrogate losses that guarantee consistency of the surrogate risk minimization for a secondbest classifier. An important implication of this result is that 1 -support vector machines are the only surrogate loss based methods that are robust to misspecification.
The computational attractiveness of a surrogate loss approach crucially depends not only on the convexity of the surrogate loss function φ but also on the functional form restrictions on the classifer f that lead to convex F. We hence investigate how additional constraints on f on top of G f ∈ G can affect the consistency of the hinge risk minimization. As the second contribution of this paper, we characterize a simple-to-check sufficient condition for consistency of the hinge risk minimization in terms of the additional functional form restrictions we can impose on F G . We term a subclass of classifiers of F G satisfying the sufficient condition a classification-preserving reduction of F G .
Exploiting our main theoretical results, we develop novel procedures for monotone classification. In monotone classification, prediction sets are constrained to
G M ≡ {G ⊂ X : x ∈ G ⇒ x ∈ G ∀x ≤ x},
where x ≤ x is an element-wise weak inequality. Since G M coincides with the class of prediction sets spanned by the class of monotonically decreasing bounded functions F M ≡ {f : f decreasing in x, −1 ≤ f ≤ 1}, hinge loss based estimation for monotone classification can be performed by solvinĝ
f M ∈ arg inf f ∈F M R φ h (f ),(4)R φ h (f ) ≡ 1 n n i=1 φ h (y i f (x i )).
We show that the class of monotone classifiers F M is a constrained classification-preserving reduction of F G M , guaranteeing consistency of the hinge-risk minimizing classifierf M . Furthermore, we show that convexity of F M reduces the optimization of (4) to a finite dimensional linear programming problem and hence delivers significant computational gains relative to minimization of the original empirical classification risk. We also consider approximating F M using a sieve of Bernstein polynomials and estimating a monotone classifier by solving (4) over the Bernstein polynomials. In either approach, an application of our main theorems guarantees
R(f M ) − inf f ∈F M R(f ) − → p 0,
as n → 0, and this convergence is valid regardless of whether F M attains the first-best risk, i.e., inf f ∈F M R(f ) = inf f ∈F R(f ), or not, whereF is the class of measurable functions f : X → R. We also derive the uniform upper bound of the mean of R(f M )−inf f ∈F M R(f ) to characterize the regret convergence rate attained byf M .
Connection and contributions to causal policy learning
For simplicity of exposition, this paper mainly focuses on the prototypical setting of binary classification. The main theoretical results can easily be extended to weighted (cost-sensitive) classification, where the canonical representation of the population risk criterion is given by
R ω (f ) ≡ E P [ω · 1{Y · sign(f (X)) ≤ 0}].(5)
Here, ω is a non-negative random variable defining the cost of misclassifying Y that typically depends on (Y, X). The cost of misclassification ω may represent the decisionmaker's economic cost (Lieli and White (2010)) or welfare weights over the individuals to be classified, as considered in Rambachan et al. (2020) and Babii et al. (2020). The surrogate risk for weighted classification can be defined similarly to (3), as
R ω φ (f ) = E P [ω · φ(Y f (X))].(6)
As discussed in Kitagawa and Tetenov (2018), there are fundamental conceptual differences between the prediction problem of classification and the causal problem of treatment choice. Nevertheless, if the training sample is obtained from a randomized control trial (RCT) or an observational study satisfying unconfoundedness (selection on observables), we can view minimization of the weighted classification risk in (5) as being equivalent to the maximization of the additive welfare criterion commonly specified in treatment choice problems. To see this equivalence, let {(Z i , D i , X i ) : i = 1, . . . , n} be an independent and identically distributed RCT sample of n experimental subjects, where Z i ∈ R is subject i's observed outcome, D i ∈ {−1, +1} is an indicator for his assigned treatment, and X i ∈ X is a vector of pretreatment covariates, and let (Z i (d) : d ∈ {−1, +1}) be i's potential outcomes satisfying Z i = Z i (+1) · 1{D i = +1} + Z i (−1) · 1{D i = −1}. We denote the propensity score in the RCT sample by e(x) ≡ P (D = +1|X = x) and assume that e(x) is bounded away from 0 and 1 for all x ∈ X . We denote the joint distribution of (Z i (+1), Z i (−1), D i , X i ) by P and assume P satisfies unconfoundedness, (Z(+1), Z(−1)) ⊥ D|X.
Similar to our consideration of classification, we represent a (non-randomized) treatment assignment rule by the sign of f : X → R, i.e., the 0-level set G f = {x ∈ X : f (x) ≥ 0} ⊂ X specifies the subgroup of the population assigned to the treatment +1. Following Manski (2004), consider evaluating the welfare performance of the assignment policy f by the average outcomes attained under its associated assignment rule:
W (f ) ≡ E P [Z(+1) · 1{X ∈ G f } + Z(−1) · 1{X / ∈ G f }]
Relying on unconfoundedness of the experimental data and employing the inverse propensity score weighting technique, we can express this welfare in terms of the observable variables as 1
W (f ) = E P Z De(X) + (1 − D)/2 · 1{D = sign(f (X))} = E P max 0, Z De(X) + (1 − D)/2 − E P [ω p · 1{sign(Z) · D · sign(f (X)) ≤ 0}] ,(7)
where ω p ≡ |Z| De(X) + (1 − D)/2 ≥ 0.
Provided that ω p has finite first moment, maximization of W (f ) is equivalent to minimization of the weighted classification risk R ω (f ) defined in (5) with ω = ω p and Y = sign(Z) · D. As a result, optimal treatment assignment rules can be viewed as optimal classifiers for D in terms of weighted classification risk. This equivalence also holds for other methods of policy learning, such as the offset-tree learning of Beygelzimer and Langford (2009) and the doubly-robust approaches of Swaminathan and Joachims (2015) and Athey and Wager (2021), which correspond to different ways of constructing or estimating the weighting term ω p . Due to its equivalence to weighted classification, a surrogate loss approach to policy learning proceeds by minimizing the empirical analogue of (6) with ω = ω p and Y = sign(Z) · D. Section 7 of this paper shows that our main theoretical results established for constrained binary classification carry over to the setting of policy learning in which feasible treatment assignment policies are constrained exogenously due to fairness and legislative considerations. This paper therefore offers valuable and novel contributions to current research and public debate regarding how to make use of machine learning algorithms to design individualized policies. If treatment assignment rules are constrained to be monotone, our concrete proposals for monotone classification algorithms can be applied to policy learning, which yields significant gains in computational efficiency relative to the mixed integer programming approaches considered in Kitagawa and Tetenov (2018) and Mbakop and Tabord-Meehan (2021).
Related literature
This paper is closely related to the literature of consistency and performance guarantees for surrogate risk minimization. Notable works in this literature include Mannor et al. (2003), Jiang (2004), Lugosi and Vayatis (2004) , Zhang (2004), Steinwart (2005Steinwart ( , 2007, Bartlett et al. (2006), Nguyen et al. (2009), andScott (2012). Under the assumption of correct specification, Zhang (2004) and Bartlett et al. (2006) derive quantitative relationships between excess classification risk and excess surrogate risk, and then provide general conditions for surrogate risk minimization to achieve risk consistency. Bartlett et al. (2006) show that the classification-calibration property of surrogate loss, defined in Section 3 below, guarantees risk consistency. Zhang (2004) and Bartlett et al. (2006) show that many commonly used surrogate loss functions, including hinge loss, exponential loss, and truncated quadratic loss, satisfy the conditions needed for risk consistency. In a classification problem different from ours, where a pair comprising a quantizer and a classifier is chosen, Nguyen et al. (2009) study sufficient and necessary conditions for surrogate risk minimization to yield risk consistency. Nguyen et al. (2009) show that only hinge loss functions satisfy the conditions required for risk consistency in their problem. Correct specification of the class of classifiers is an essential condition for consistency in all of the surrogate risk minimization approaches studied in the literature. The key contribution of our paper is to relax the assumption of correct specification and to clarify the conditions that are required for the surrogate loss function to yield a consistent surrogate risk minimization procedure.
Relaxing the assumption of correct specification connects this paper to classification problems with exogenous constraints. Such problems are studied in machine learning and statistics, and include interpretable classification (e.g., Zeng et al. (2017), andZhang et al. (2018)), fair classification (e.g., Dwork et al. (2012)), and monotone classification (e.g., Cano et al. (2019)). Some works in the existing literature adopt a surrogate loss approach. Donini et al. (2018) use the 1 -support vector machine in fair classification, where the hinge risk minimization is subject to a statistical fairness constraint. Chen and Li (2014) use the 1 -support vector machine with a monotonicity constraint, which constrains the class of feasible classifiers to a class of certain monotone functions. However, neither paper shows the risk consistency of their hinge risk minimization procedures.
Focusing on optimization, ERM classification and maximum score estimation (Manski (1975), Manski and Thompson (1989)) share the same objective function. Horowitz (1992) proposes smooth maximum score estimation, where kernel smoothing is performed on the 0-1 loss to obtain a differentiable objective function. However, the smoothed objective function remains non-convex and does not offer the computational gains that the surrogate risk minimization approach with convex surrogates can deliver. This paper also contributes to a growing literature on statistical treatment rules in econometrics, including Manski (2004), Dehejia (2005), Hirano and Porter (2009), Stoye (2009, Chamberlain (2011), Bhattacharya andDupas (2012), Tetenov (2012), Kasy (2018), Tetenov (2018, 2021), Viviano (2019), Kitagawa and Wang (2020), Athey and Wager (2021), Mbakop and Tabord-Meehan (2021), Sakaguchi (2021), among others. As discussed above, the policy learning methods of Kitagawa and Tetenov (2018), Athey and Wager (2021), and Mbakop and Tabord-Meehan (2021) build on the similarity between empirical welfare maximizing treatment choice and ERM classification. Mbakop and Tabord-Meehan (2021) propose penalization methods to control the complexity of treatment assignment rules, and derive relevant finite sample upper bounds on regret of the estimated treatment rules. Athey and Wager (2021) apply doubly-robust estimators to estimate the weight ω in (6), and show that an 1/ √ n-upper bound on regret can also be achieved in the observational study setting. These works optimize an empirical welfare objective involving an indicator loss function. Hence, the practical implementation of their methods is sometimes discouraging, especially when the sample size or number of covariates is moderate to large.
Estimation of individualized treatment rules is a topic of active research in other fields including medical statistics, machine learning, and computer science. Notable works in these fields include Zadrozny (2003), Beygelzimer and Langford (2009), Qian andMurphy (2011), Zhao et al. (2012), Swaminathan and Joachims (2015), Zhao et al. (2015), and Kallus (2021), among others. Zhao et al. (2012) propose using 1 -support vector machines to solve the weighted classification with individualized treatment choice problem, and show risk consistency. They specify a rich class of treatment choice rules that is a reproducing kernel Hilbert space, and assume correct specification. Zhao et al. (2015) extend this approach to estimate optimal dynamic treatment regimes.
Constrained classification with surrogate loss
Consider the binary classification problem of ascribing a binary label Y ∈ {−1, +1} based upon covariates X ∈ X , which are collectively distributed according to a joint distribution P . We let X be a d x -dimensional vector, d x < ∞, and denote its marginal distribution by P X . We denote the conditional probability of Y = +1 given X = x by η(x) ≡ P (Y = +1|X = x) and otherwise maintain the notation introduced in the Introduction. The ultimate objective is to minimize the classification risk of (1).
We study constrained classification problems where an optimal classifier is searched for over a restricted class of functions. Section 2.1 studies the consistency of surrogate risk minimization in the special case that the prespecified class of classifiers contains a classifier whose prediction set agrees with the prediction set of the Bayes classifier. Section 2.2 introduces a classification problem that embeds a constraint on the prediction sets, which is a central problem throughout the paper.
Misspecification in constrained classification
Let F be a constrained class of classifiers f : X → R. If the set of classifiers were unconstrained, it is well known that the Bayes classifier defined by
f * Bayes = 2η(x) − 1
minimizes the classification risk. Due to the constraints on the class of classifiers, however, the minimized classification risk on F can be strictly larger than the first-best minimal risk R(f * Bayes ). We refer to this situation as R-misspecification of F, which we formally define in the following definition.
Definition 2.1 (R-misspecification). F is R-misspecified if inf f ∈F R(f ) > R(f * Bayes ).
If the inequality instead holds with equality, we say that F is R-correctly specified.
Because the 0-1 loss function is neither convex nor continuous, minimizing the empirical analog of R(f ) is computationally challenging and often infeasible given the scale of the problems that we encounter in practice. Commonly used classification algorithms, such as boosting and support vector machines, replace the 0-1 loss with a surrogate loss function, φ : R → R, and aim to minimize the surrogate risk Table 1 below lists some commonly used surrogate loss functions including the hinge loss φ h (α) = c max{0, 1 − α}, which corresponds to 1 -support vector machines, and the exponential loss φ e (α) = exp(−α), which corresponds to AdaBoost.
R φ (f ) ≡ E P [φ(Y f (X))].
We also introduce the concept of misspecification of F in terms of surrogate risk as follows.
Definition 2.2 (R φ -misspecification). Let f * φ,F B be a minimizer of R φ over the unconstrained class of classifiers, i.e., the class of all measurable functions f :
X → R. A constrained class F is R φ -misspecified if inf f ∈F R φ (f ) > R φ (f * φ,F B ).
If the inequality instead holds with equality, we say that F is R φ -correctly specified.
The seminal theoretical results that guarantee consistency of surrogate-risk classification (Zhang (2004), Bartlett et al. (2006), andNguyen et al. (2009)) crucially rely on the assumption that F is both R-correctly specified and R φ -correctly specified in the sense of Definitions 2.1 and 2.2, respectively. The central question that this paper poses is how is a surrogate loss approach affected if F is R-misspecified or R φ -misspecified? This misspecification is a likely scenario especially when the origins of the constraints have nothing to do with the assumptions on P , as is the case in the examples discussed in the next subsection.
Throughout the paper, we limit our analysis to the class of classification-calibrated loss functions defined in Bartlett et al. (2006).
Definition 2.3 (Classification-calibrated loss functions). For a ∈ R and 0 ≤ b ≤ 1, define C φ (a, b) ≡ φ(a)b + φ(−a)(1 − b). A loss function φ is classification-calibrated if for any b ∈ [0, 1]\{1/2}, inf {a∈R : a(2b−1)<0} C φ (a, b) > inf {a∈R : a(2b−1)≥0} C φ (a, b).
Noting that the surrogate risk can be expressed as
E P [φ(Y f (X))] = E P X [C φ (f (X), η(X))],(8)
the definition of classification-calibrated loss functions implies that at every x ∈ X with η(x) = 1/2, every f (x) that minimizes C φ (f (x), η(x)) has the same sign as the Bayes classifier, sign(2η(x) − 1)). Bartlett et al. (2006) shows that many commonly used surrogate loss functions including those listed in Table 1 are classification-calibrated. 2 Having introduced two notions of misspecification, we now clarify the relationship between R-misspecification and R φ -misspecification.
Proposition 2.1. Let F be a constrained class of classifiers and f * φ ∈ F be a minimizer of R φ over F. Suppose φ is a classification-calibrated loss function. (i) For any distribution P on {−1, 1} × X , if F is R φ -correctly specified, then F is Rcorrectly specified and R(f * φ ) = R(f * Bayes ) holds; (ii) If φ is, in addition, convex, there exist a distribution P on {−1, 1} × X and a class of classifiers F under which F is R-correctly specified but R φ -misspecified, and R(f * φ ) > R(f * Bayes ) holds.
Proof. See Appendix A.
Proposition 2.1 (i), which rephrases Claim 3 of Theorem 1 in Bartlett et al. (2006), implies that surrogate risk minimization on the R φ -correctly specified class F leads to (first-best) optimal classification in terms of the classification risk. An equivalent statement following Theorem 1 in Bartlett et al. (2006) is that for any P and every sequence of measurable functions {f i : X → R},
R φ (f i ) → inf f ∈F R φ (f ) implies that R(f i ) → inf f ∈F R(f ).
This result justifies the approach of surrogate risk minimization when F is a sufficiently rich class of classifiers (e.g., the reproducing kernel Hilbert space of functions with a large number of features as used in support vector machines), since R φ -correct specification, which is a credible assumption to make given a rich class of classifiers, guarantees Rcorrect specification.
Proposition 2.1 (ii), in contrast, shows that R-correct specification of F does not guarantee R φ -correct specification. 3 R φ -misspecification of F can lead to the selection of a suboptimal classifier in F in terms of the classification risk, which illustrates the pitfall of adopting a surrogate loss approach with constrained classifiers. Even when we are confident that the constrained class F is R-correctly specified, we cannot justify the use of F in the surrogate risk minimization.
G-constrained classification
In this section, we consider restricting the class of classifiers by requiring that their prediction sets belong to a prespecified class of sets, G ⊂ 2 X . See Examples 2.4-2.6 below for motivating examples.
We denote by
F G ≡ {f : G f ∈ G, f (·) ∈ [−1, 1]}
the class of classifiers whose prediction sets are constrained to G. In this definition, we restrict f to be bounded and, without loss of generality, normalize its range to [−1, 1]. Other than on the shape of the 0-level set and on the range, F G does not impose any constraint on the functional form of f ∈ F G . The goal of the constrained classification problem is then to find a best classifier, in the sense that it minimizes the classification risk R (·) over F G . We refer to F G as the G-constrained class of classifiers and to the classification problem over F G as G-constrained classification.
The specification of the class of prediction sets G represents the fairness, interpretability, and other exogenous requirements that are desired for classification rules. Some examples follow.
Example 2.4 (Interpretable classification). Decision-makers may prefer simple decision or classification rules that are easily understood or explained even at the cost of harming prediction accuracy. This concept, often referred to as interpretable machine learning, has been pursued, for instance, in the prediction analysis of recidivism (Zeng et al. (2017)) and the decision on medical intervention protocol (Zhang et al. (2018))). An example is a linear classification rule, in which G is a class of half-spaces with linear boundaries in X ,
G = {x ∈ R dx : x T β ≥ 0, β ∈ R dx }.
Note that f ∈ F G is not restricted to be a linear function. Any function f , including nonlinear functions, is included in F G as long as its prediction set G f is a hyperplane in X . A classification tree is another type of classification rule that is interpretable. See, e.g., Breiman et al. (1984).
Example 2.5 (Monotone classification). The framework we study can accommodate monotonicity constraints on classification. Formally, a monotonicity constraint corresponds to a partial order on X , and any prediction set G f has to respect this partial order in the sense that if x 1 x 2 and x 1 ∈ G f , then x 2 ∈ G f . Monotonicity constraints have been utilized in the classification of credit rating (Chen and Li (2014)), and in the assignment of job training in the context of policy learning (Mbakop and Tabord-Meehan (2021)).
Example 2.6 (Fair classification). Specification of G can accommodate some fairness constraints introduced in the literature on fair classification. Let A = {0, 1} be an element of X indicating a binary protected group variable (e.g., race, gender). The decision-maker wants to ensure fairness of classification by, for instance, equalizing the raw positive classification rate (known as statistical parity):
P X (f (x) ≥ 0 | A = 1) = P X (f (x) ≥ 0 | A = 0).
The classification problem embedding this constraint is equivalent to G-constrained classification with
G = G ∈ 2 X : P X (X ∈ G | A = 1) = P X (X ∈ G | A = 0) ,
where G depends on P X in this case. This fairness constraint is studied by Calders and Verwer (2010), Kamishima et al. (2011), Dwork et al. (2012, Feldman et al. (2015), among others. Some other forms of fairness constraint, such as equalized odds and equalized positive predictive value as reviewed by Chouldechova and Roth (2018), can be accommodated in our framework as well via an appropriate construction of G.
In the G-constrained classification problem, R-correct specification of F G is necessary and sufficient for the surrogate risk minimizer f * φ to achieve the first-best minimum risk.
Proposition 2.2. Suppose φ is a classification-calibrated loss function. Let G ⊆ 2 X be a class of measurable subsets of X and f * φ ∈ F G be a minimizer of R φ over F G . Then, for any distribution P on
{−1, 1} × X , R(f * φ ) = R(f * Bayes ) holds if and only if F G is R-correctly specified. Proof. See Appendix A.
Proposition 2.2 shows that if φ is classification-calibrated, f * φ ∈ F G that minimizes the surrogate risk over F G leads to a globally optimal classifier in terms of the classification risk if and only if F G is R-correctly specified. A comparison of Proposition 2.1 (ii) and Proposition 2.2 clarifies a special feature of the G-constrained class of classifiers. Specifically, Proposition 2.1 (ii) establishes that, in general, R-correct specification of a constrained class of classifiers F does not guarantee R(f * φ ) = R(f * Bayes ). In contrast to the seminal results about surrogate risk consistency shown in Zhang (2004) and Bartlett et al. (2006), our claim does not require R φ -correct specification of F G .
If constraints defining G are motivated by some considerations that are independent of any belief on the underlying data generating process (e.g., Examples 2.4-2.6 above), R-correct specification of F G is hard to justify. Therefore, an important question for our analysis to consider is whether or not surrogate risk minimization procedures can yield a classifier achieving inf f ∈F G R(f ) without requiring R-correct specification of F G .
Calibration of G-constrained classification
This section investigates the risk consistency of a surrogate risk minimization approach over F G , where F G is now allowed to be R-misspecified. Let f * be an optimal classifier that minimizes the classification risk over F G :
f * ∈ arg inf f ∈F G R(f ).
Similarly, we denote a best classifier among F G in terms of the surrogate risk by f * φ ,
f * φ ∈ arg inf f ∈F G R φ (f ), .
To begin our analysis, let us first perform a simple numerical example to assess the influence of misspecification in constrained classification. Here, G imposes monotonicity of the prediction sets in a way that is compatible with Example 2.5. We specify P X to be uniform on X and P (Y = +1 | X = 0) = 0.9, P (Y = +1 | X = 1) = 0.3, and P (Y = +1 | X = 2) = 0.2. The Bayes classifier therefore predicts Y = +1 at x = 0 and Y = −1 at x = 1 and 2, but such a prediction set is excluded from G. That is, F G is R-misspecified. Under this specification, the second-best (constrained optimum) classifier f * has its prediction set equal to ∅, and the attained classification risk R(f * ) = 0.47.
For each of hinge loss φ h with c = 1, exponential loss φ e , and truncated quadratic loss φ q , we compute the classifier minimizing the surrogate risk f * φ and the classification risk at the surrogate optimal classifier R(f * φ ). Figure 1 illustrates each computed classifier with each loss function. We obtain
R(f * φ h ) = 0.47 = R(f * ), R(f * φe ) = R(f * φq ) = 0.53, G f * φ h = ∅ = G f * , G f * φe = G f * φq = {2, 1, 0}.
In this specification, the hinge risk optimal classifier agrees with the second best optimal classifier, whereas that is not the case for the exponential or truncated quadratic loss.
This example illustrates that hinge loss is robust to R-misspecification of F G , but exponential and truncated quadratic losses are not. To what extent, can we generalize this finding? What conditions do we need to guarantee that surrogate risk minimizing classifiers are consistent to the second-best (constrained optimal) classification rule in terms of the classification risk? We answer these questions below.
For any classifier f , we define the G-constrained excess risk of f as
R(f ) − inf f ∈F G R(f ),
which is the regret of f relative to a constrained optimum f * in terms of the classification risk. Similarly, we define the G-constrained excess φ-risk of f as Fix G ∈ G and let
R φ (f ) − inf f ∈F G R φ (f ).F G ≡ {f : G f = G, f (·) ∈ [−1, 1]}
be the class of classifiers that share the prediction set G. Then {F G : G ∈ G} forms a partition of F G indexed by the prediction set, and satisfies F G = ∪ G∈G F G and F G ∩F G = ∅ for G, G ∈ G with G = G . With this definition to hand, choosing a classifier from F G can be decomposed into two steps: choosing a prediction set G from G and, then, choosing a classifier f from F G . Denote the classification risk evaluated at a prediction set G by R(G) ≡ inf f ∈F G R(f ). Note that any f ∈ F G attains the same level of classification risk, so R(G) = R(f ) holds for all f ∈ F G . R(G) can be written as
R(G) = X [η(x)1{x / ∈ G} + (1 − η(x))1{x ∈ G}] dP X (x), = X (1 − 2η(x)) · 1{x ∈ G}dP X (x) + P (Y = 1).(9)
Similarly, we define the surrogate risk evaluated at G by R φ (G) ≡ inf f ∈F G R φ (f ), which can be written as
R φ (G) = inf f ∈F G X [η(x)φ(f (x)) + (1 − η(x))φ(−f (x))] dP X (x) = G inf 0≤f (x)≤1 C φ (f (x), η(x))dP X (x) + G c inf −1≤f (x)<0 C φ (f (x), η(x))dP X (x),
where the second line follows from the fact that f ∈ F G is unconstrained other than via its prediction set and that the minimization over f ∈ F G can be performed pointwise at each x. For f ∈ F G with x ∈ G, f (x) is constrained to [0, 1], and with x ∈ G c , f (x) is constrained to [−1, 0). To simplify the notation, we define
C + φ (η(x)) ≡ inf 0≤f (x)≤1 C φ (f (x), η(x)), C − φ (η(x)) ≡ inf −1≤f (x)<0 C φ (f (x), η(x)), ∆C φ (η(x)) ≡ C + φ (η(x)) − C − φ (η(x)),
where C + φ (η(x)) and C − φ (η(x)) are the minimized surrogate risks conditional on X = x under the constraints f (x) ∈ [0, 1] and f (x) ∈ [−1, 0), respectively. Using these definitions, the surrogate risk at G can be written as
R φ (G) = X C + φ (η(x)) · 1{x ∈ G} + C − φ (η(x)) · 1{x / ∈ G} dP X (x) = X ∆C φ (η(x)) · 1{x ∈ G}dP X (x) + X C − φ (η(x))dP X (x).(10)
By comparing the expressions of the risks in (9) and (10), we obtain the first main theorem that clarifies the condition for the surrogate risk R φ (G) to calibrate the global ordering of the classification risk R(G) over G ∈ G.
Theorem 3.2 (Global calibration of the G-constrained excess risk). Let P be an arbitrary distribution on {−1, 1} × X and G ⊆ 2 X be a class of measurable subsets of X . For G, G ∈ G, the risk ordering R(G) ≥ R(G ) in terms of the classification risk is equivalent to
G\G (1 − 2η(x))dP X (x) ≥ G \G (1 − 2η(x))dP X (x),(11)
while the risk ordering
R φ (G) ≥ R φ (G ) in terms of the surrogate risk is equivalent to G\G ∆C φ (η(x))dP X (x) ≥ G \G ∆C φ (η(x))dP X (x).(12)Hence, if ∆C φ (η(x)) is proportional to 1 − 2η(x) up to a positive constant, i.e., ∆C φ (η(x)) = c(1 − 2η(x)) for some c > 0,(13)
the risk ordering over G in terms of the surrogate risk R φ (G) agrees with the risk ordering over G in terms of the classification risk R(G) for any distribution P on {−1, 1} × X .
In particular, when φ is the hinge loss φ h (α) = c max{0, 1 − α}, c > 0,
∆C φ (η(x)) = c(1 − 2η(x))
holds, establishing that hinge risk preserves the risk ordering of the classification risk.
Proof. By equation (9),
R(G) − R(G ) = X (1 − 2η(x)) · [1{x ∈ G} − 1{x ∈ G }]dP X (x) = X (1 − 2η(x)) · [1{x ∈ G \ G } − 1{x ∈ G \ G}]dP X (x) = G\G (1 − 2η(x))dP X (x) − G \G (1 − 2η(x))dP X (x).
This proves (11), the first claim of the theorem. Given the representation of the surrogate risk shown in (10), a similar argument yields (12), the second claim of the theorem.
For the hinge loss φ h (α) = c max{0, 1 − α} and f ∈ F G , we have
C φ h (f (x), η(x)) = c(1 − 2η(x))f (x) + c.
Hence, we obtain
C + φ h (η) = c(1 − 2η) + c for η > 1/2, c for η ≤ 1/2, C − φ h (η) = c for η > 1/2, 2cη for η ≤ 1/2. Hence, ∆C φ h (η) = c(1 − 2η) holds for all η ∈ [0, 1].
Theorem 3.2 does not exploit the condition that φ is classification-calibrated, but if a surrogate loss function satisfies condition (13), it is automatically classification-calibrated. Another remark follows.
Remark 3.3. Many commonly used surrogate loss functions do not satisfy condition (13) in Theorem 3.2. Table 1 shows the forms of ∆C φ (η) for the hinge loss, exponential loss, logistic loss, quadratic loss, and truncated quadratic loss functions. With the exception of the hinge loss function, none of these functions satisfy condition (13). That is, among the surrogate loss-based algorithms that are commonly used in practice, the 1 -support vector machine corresponding to hinge loss is the only algorithm whose surrogate risk preserves the classification risk.
Loss function φ(α) ∆C φ (η) 0-1 loss 1{α ≤ 0} 1 − 2η Hinge loss c max{0, 1 − α} c(1 − 2η) Exponential loss e −α −2 η(1 − η) + 1 2 η(1 − η) − 1 if 0 ≤ η < 1/2 if 1/2 ≤ η ≤ 1 Logistic loss log(1 + e −α ) log(2η η (1 − η) 1−η ) − log(2η η (1 − η) 1−η ) if 0 ≤ η < 1/2 if 1/2 ≤ η ≤ 1 Quadratic loss (1 − α) 2 (1 − 2η) 2 −(1 − 2η) 2 if 0 ≤ η < 1/2 if 1/2 ≤ η ≤ 1 Truncated quadratic loss (max{0, 1 − α}) 2 (1 − 2η) 2 −(1 − 2η) 2 if 0 ≤ η < 1/2 if 1/2 ≤ η ≤ 1
The well known inequality by Zhang (2004) relates the excess surrogate risk to the excess classification risk under R-correct specification. As a corollary of Theorem 3.2, if we set φ = φ h , we can generalize Zhang's inequality by allowing R-misspecification of the classifiers. To formally state this generalization, we let G * ∈ arg inf G∈G R(G), and set G = G * in Theorem 3.2. Let f ∈ F G be arbitrary and G f = {x ∈ X : f (x) ≥ 0} ∈ G. The alignment of the risk ordering between the classification and hinge risks implies that the minimizers of R(·) also minimize R
φ h (·), i.e., inf f ∈F G R φ h (f ) = inf G∈G R φ h (G) = R φ h (G * ).
Theorem 3.2 therefore implies that the G-constrained excess classification risk of f satisfies the following inequality:
R(f ) − inf f ∈F G R(f ) = R(G f ) − R(G * ) = G f \G * (1 − 2η(x))dP X (x) − G * \G f (1 − 2η(x))dP X (x) = c −1 [R φ h (G f ) − R φ h (G * )] = c −1 inf f ∈F G f R φ h (f ) − inf f ∈F G R φ h (f ) ≤ c −1 R φ h (f ) − inf f ∈F G R φ h (f ) ,(14)
where the second equality follows by equation (9); and the third equality follows by equation (10) and ∆C φ h (η) = c(1 − 2η). That is, when φ = φ h , Zhang's inequality holds without requiring the R-correct specification of the classifiers.
Corollary 3.4. For any distribution P on {−1, 1} × X and class of measurable subsets
G ⊆ 2 X , if ∆C φ (η(x)) is proportional to 1 − 2η(x) with a proportionality constant c > 0, i.e., ∆C φ (η(x)) = c(1 − 2η(x))
, then the following inequality holds
c(R(f ) − inf f ∈F G R(f )) ≤ R φ (f ) − inf f ∈F G R φ (f ) for any f ∈ F G .
Proof. See equation (14).
Corollary 3.4 shows that if the surrogate loss φ satisfies condition (13), then the classifier f * φ that minimizes the surrogate risk over F G also minimizes the classification risk over F G . Importantly, this result holds without assuming the R-correct specification of F G . It justifies the use of hinge loss in the constrained classification problem irrespective of whether or not F G is correctly R-specified. Note, however, that the result relies on the fact that at every x ∈ X we can choose any f (x) ∈ [−1, 1] as long as the prediction set constraint is satisfied: G f ∈ G. We relax this requirement in the next section.
Further analysis can show that the condition (13) in Theorem 3.2 is not only sufficient but also necessary. To formally show this, we adopt the concept of universal equivalence of loss functions introduced by Nguyen et al. (2009) to the current setting.
Definition 3.5 (Universal equivalence). Loss functions φ 1 and φ 2 are universally equivalent, denoted by φ 1 u ∼ φ 2 , if for any distribution P on {−1, 1} × X and class of measurable subsets G ⊆ 2 X ,
R φ 1 (G 1 ) ≤ R φ 1 (G 2 ) ⇔ R φ 2 (G 1 ) ≤ R φ 2 (G 2 )
holds for any G 1 , G 2 ∈ G.
Universally equivalent loss functions φ 1 and φ 2 lead to the same risk ordering over G. Hence, if a loss function φ is universally equivalent to the 0-1 loss, the φ-risk shares the same risk ordering with the classification risk.
The following theorem establishes a necessary and sufficient condition for two classificationcalibrated loss functions to be universally equivalent.
Theorem 3.6. Let φ 1 and φ 2 be classification-calibrated loss functions. Then φ 1 u ∼ φ 2 if and only if ∆C φ 2 (η) = c∆C φ 1 (η) for some c > 0 and any η ∈ [0, 1], i.e., ∆C φ 2 is proportional to ∆C φ 1 up to a positive constant.
Proof. See Appendix A.
The 'if' part of the theorem is a generalization of Theorem 3.2 in that it does not assume that either of φ 1 or φ 2 is the 0-1 loss function.
When we set φ 2 to the 0-1 loss function, Theorem 3.6 yields the class of loss functions that are universally equivalent to the 0-1 loss functions. This class exactly coincides with the class of loss functions that satisfy the condition (13) in Theorem 3.2. Hence, the following corollary holds.
Corollary 3.7. A classification-calibrated loss function φ is universally equivalent to the 0-1 loss function if and only if φ satisfies condition (13) for any η(x) ∈ [0, 1]. That is, the class of hinge loss functions {φ(α) = a max{0, 1 − α} + b : a > 0, b ≥ 0} agrees with the class of loss functions that are universally equivalent to the 0-1 loss function.
In the following sections, without loss of generality, we maintain the assumption that c = 1 in the definition of the hinge loss function where it is convenient to do so. We conclude this section with a remark to compare our constrained classification framework to that of Nguyen et al. (2009).
Remark 3.8. Nguyen et al. (2009) show that, for the classification problem in which an optimal pair comprising a quantizer and a classifier is to be chosen, the hinge loss function is also the only surrogate loss function that preserves the consistency of surrogate loss classification. In their framework, the quantizer is a stochastic mapping Q ∈ Q : X → Z, where Z is a discrete space and Q is a possibly constrained class of conditional distributions of Z given X, Q (Z | X). The classifier is a function γ ∈ Γ : Z → R, where Γ is the set of all measurable functions on Z. The motivation for using Z as an input, instead of X, is to reduce the dimension of X, which might be a high-dimensional vector. Nguyen et al. (2009) propose estimating the pair (Q, γ) ∈ Q × Γ that minimizes the risk R (γ, Q) ≡P (Y = sign (γ (Z))), by solving the surrogate loss classification problem:
inf (Q,γ)∈Q×Γ R φ (Q, γ), where R φ (Q, γ) = Eφ(Y γ(Z))
. They show that, among the commonly used surrogate loss functions, only hinge loss classification leads to the optimal pair of (Q, γ).
The framework we study is different from that of Nguyen et al. (2009), and neither nests the other. The framework Nguyen et al. (2009) study constrains the mapping Q : X → Z, whereas the framework we study constrains prediction sets G f for all classifiers f . Furthermore, the class of classifiers Γ considered in Nguyen et al. (2009) contains the Bayes classifier, whereas the class of classifiers F G we consider may not contain the Bayes classifier.
Consistency of hinge risk classification with functional form constraints
The previous section considers F G , the class of all functions whose prediction sets are in G. The generalized Zhang's inequality shown in Corollary 3.4 heavily relies on the richness of F G . This richness, however, limits the computational attractiveness of a surrogateloss approach, since convexity in optimization of an empirical analogue of the surrogate risk does not directly follow from F G , and typically requires additional functional form restrictions for f .
Unfortunately, once a functional form restriction on f is imposed on top of the prediction set constraint G f ∈ G, the global calibration property of the hinge risk shown in Theorem 3.2 breaks down. The following example illustrates this phenomenon. as in Example 3.1. We here consider choosing a classifier from the following class of nondecreasing linear functions:
F L = {f (x) = c 0 + c 1 x : c 0 ∈ R, c 1 ∈ R + , f (x) ∈ [−1, 1] for all x ∈ X }.
Note that the class of prediction sets {G f : f ∈ F L } agrees with G; hence, F L is a subclass of F G . We set X to be uniformly distributed on X and Y to have conditional probabilities P (Y = 1 | X = 0) = 0.6, P (Y = 1 | X = 1) = 0.2, and P (Y = 1 | X = 2) = 0.8.
The Bayes classifier predicts positive Y at x = 0 and 2. Hence, no classifier in F L shares the prediction set with the Bayes classifier, and F L is R-misspecified. Figure 2 illustrates the computed classifiers, f * and f * φ h , that minimize the classification and hinge risks, respectively, over F L . The optimal classification risk R(f * ) over
F L (equivalently, over F G since {G f : f ∈ F L } agrees with G) is R(f * ) = 0.33 with G f * = {2}, while the classification risk at f * φ h is R(f * φ h ) = 0.54 with G f * φ h = {2, 1}.
Thus, in contrast to Example 3.1 where f is unconstrained other than via the constraint G f ∈ G, adding the linear functional form constraint to F G invalidates the calibration property of the hinge risk, and the hinge risk minimization is no longer consistent to the second-best (constrained optimal) classifier in terms of the classification risk.
This example illustrates that even with hinge loss, consistency to the second best classifier becomes a fragile property once the functional form of f is constrained in addition to the prediction set constraint G f ∈ G. Consequently, it is natural to ask what additional functional form restriction we can safely introduce to F G without threatening consistency, i.e., for which subclass F G ⊂ F G does minimizing the hinge risk R φ h (f ) over f ∈ F G lead to a classifier that minimizes the classification risk R(f ) over f ∈ F G ? Formally, we introduce the following definition of classification-preserving reduction of F G .
Definition 4.2 (Classification-preserving reduction). Letf * ∈ arg inf f ∈ F G R φ h (f ). A subclass of classifiers F G (⊆ F G ) is a classification-preserving reduction of F G if R(f * ) = inf f ∈F G R(f ) holds for any P , distribution on {−1, 1} × X .
To start with the heuristic, consider a simple case where F G consists of piecewise constant functions with at most J jumps, J ≥ 1, of the following form:
F G,J = f (·) = 2 J j=1 c j 1{· ∈ G j } − 1 : G j ∈ G and c j ≥ 0 for j = 1, . . . , J; G J ⊆ · · · ⊆ G 1 ; J j=1 c j = 1 .(15)
By construction, any function in F G,J is a step function bounded in [−1, 1] and its sublevel sets {x ∈ X : f (x) ≤ t} belong to G for any t ∈ [−1, 1].
Let
G * ≡ arg inf G∈G R(G)
be the collection of best prediction sets in G, and
R * ≡ inf G∈G R(G)
be the optimal classification risk. For any G ∈ G, we definef G (x) ≡ 2·1{x ∈ G}−1, a step function over X that indicates x ∈ G * and x / ∈ G * with values +1 and −1, respectively. The following lemma shows that F G,J is a classification-preserving reduction of F G . Lemma 4.3. Let G ⊆ 2 X be a class of measurable subsets of X . The following two claims hold:
(i) F G,J is a classification-preserving reduction of F G . (ii) For any distribution P on {−1, 1} × X and G * ∈ G * ,f G * is a minimizer of R φ h (·) over F G,J , and inf f ∈ F G,J R φ h (f ) = 2R * holds. Proof. See Appendix A.
Characteristic features of F G,J are (i) sublevel sets of any f ∈ F G,J are in G, and (ii) F G,J containsf G for any G ∈ G. It transpires that these two features are the key features that need to be maintained for F G to generalize Lemma 4.3.
The next theorem is the second main theorem of the paper that extends Lemma 4.3 to a more general class of classifiers that can accommodate continuous ones.
Theorem 4.4 (Consistency under classification-preserving reduction). Given a class of measurable subsets G ⊆ 2 X and F G = {f : G f ∈ G, f (·) ∈ [−1, 1]}, suppose F G ⊂ F G satisfies the following two conditions:
(A1) For every f ∈ F G , {x ∈ X : f (x) ≤ t} ∈ G for all t ∈ [−1, 1]; (A2) For any G ∈ G,f G ∈ F G .
Then the following claims hold:
(i) F G is a classification-preserving reduction of F G ; (ii) For any distribution P on {−1, 1} × X and G * ∈ G * ,f G * is a minimizer of R φ h (·) over F G , and inf f ∈ F G R φ h (f ) = 2R * holds. Proof. See Appendix A.
The theorem establishes that the two conditions (A1) and (A2) are sufficient for F G to be a classification-preserving reduction of F G . This result holds regardless of whether F G is correctly R-specified or not. Examples 4.6 and 4.7 at the end of this section give examples of classification-preserving reductions for linear classification and monotone classification.
The conditions (A1) and (A2) in Theorem 4.4 are simple to interpret and guarantee the consistency of the hinge risk minimization, but they do not imply that the empirical hinge risk minimization over F G can be reduced to a convex optimization. We are unaware of a general way to construct a classification-preserving reduction that makes the empirical hinge risk minimization a convex program. For monotone classification, analyzed in Section 6, we propose two constructions of F G M , one of which is exactly a classification-preserving reduction of F G M while the other is approximately classificationpreserving. We show that for both cases, minimization of the empirical hinge risk is a linear programming problem.
Although Theorem 4.4 shows the consistency of the hinge risk minimization over F G , it does not lead to the generalized Zhang's (2004) inequality in Corollary 3.4. Instead, the following corollary gives proportional equality between the G-constrained excess classification risk and the F G -constrained excess hinge risk with an extra term added.
Corollary 4.5. Assume F G is a subclass of F G satisfying conditions (A1) and (A2) in Theorem 4.4. If ∆C φ (η) = c(1 − 2η) holds for some c > 0 and any η ∈ [0, 1], c(R(f ) − inf f ∈F G R(f )) = 1 2 R φ (f ) − inf f ∈ F G R φ (f ) + 1 2 R φ (f G f ) − R φ (f )(16)
for any classifier f : X → [−1, 1]. Moreover, the following holds:
c(R(f ) − inf f ∈F G R(f )) ≤ 1 2 R φ (f ) − inf f ∈ F G R φ (f ) + 1 2 R φ (f ) − inf f ∈F G R φ (f ) (17) for any f ∈ F G . Proof. See Appendix A.
The extra term (the right-most term) in (16) measures the difference in the hinge risks between a classifier f and the step function indicating the prediction set of f by the values +1 or −1. Due to the fact that some of the best classifiers are of the form Theorem 4.4 (ii)), if f closely approximates such a classifier, the extra term is close to zero. In the following section, we use equation (16) to derive the statistical properties of the hinge risk minimization in terms of the Gconstrained excess classification risk. Equation (17) implies that the G-constrained excess classification risk is bounded from above by the average of the two F G -constrained excess hinge risks. One is over F G and the other is over F G . We are unable to determine if the excess hinge risk over F G can be bounded from above by a term that is proportional to the excess hinge risk over F G . As such, the constrained-classification-preserving reduction F G cannot replace F G in Zhang's inequality, shown in Corollary 3.4.
f * (·) = 1 {· ∈ G * } − 1 {· / ∈ G * } for G * ∈ G * (
We conclude this section by presenting examples of classes of classifiers that approximately or exactly satisfy the conditions for classification-preserving reduction.
Example 4.6 (Linear classification with a class of transformed logistic functions). Suppose that the prediction sets are subject to the linear index rules:
G L = {x ∈ R dx : x T β ≥ 0 : β ∈ R dx }, where X = R dx . Let π(α, k) ≡ (1 − e −kα )/(1 + e −kα ) = 2/(1 + e −kα )
− 1 be a transformed logistic function and define a class of classifiers
F Logit = {π(x T β, k) : β ∈ R dx and k ∈ R + },
where k is a tuning parameter that determines the steepness of the logistic curve. F Logit satisfies the condition (A1) in Theorem 4.4. 4 Since F Logit at fixed k < ∞ rules out any step function, the condition (A2) in Theorem 4.4 is not exactly met. Fix G ∈ G, and letβ be such that {x ∈ X : x Tβ ≥ 0} = G. Then, as k → ∞, π(x Tβ , k) approximates sign(x Tβ ), so the condition (A2) is approximately met for large k. Every function in F Logit is smooth and depends on a finite number of parameters. Hence, the empirical hinge risk becomes a smooth and continuous function with finite number of parameters, although it is not generally convex.
Example 4.7 (Monotonic classification with a class of monotone functions). Hinge risk minimization embedding a monotonicity restriction remains consistent when we use a class of monotone functions. Let be a partial order on X , and let G be the collection of all G ∈ 2 X that respect monotonicty (i.e., if x 1
x 2 and x 1 ∈ G, then x 2 ∈ G). Define F G as a class of functions f : X → [−1, 1] that are weakly monotonic in (i.e.,
satisfying f (x 1 ) ≤ f (x 2 ) if x 1 x 2 ). Then the prediction set of any f ∈ F G respects the partial order (i.e., if x 1 x 2 and x 1 ∈ G f , then x 2 ∈ G f ). For any t ∈ [−1, 1] and f ∈ F , {x : f (x) ≤ t} = {x : x
x for anyx such that f (x) = t} ∈ G holds, satisfying condition (A1) in Theorem 4.4. In addition, for any G ∈ G , sincef G (x) = 2·1{x ∈ G} is weakly monotonic in ,f G ∈ G holds, satisfying condition (A2) in Theorem 4.4. Hence F G is a classification-preserving reduction of F G . Therefore, according to Theorem 4.4, hinge risk minimization over F yields the optimal classifier in terms of the classification risk. Section 6 focuses on monotone classification and investigates its statistical and computational properties.
Statistical properties
The analyses presented so far concern the consistency of a surrogate loss approach in terms of the population risk criterion. It is important to note that Theorems 3.2 and 4.4 do not impose any restriction on the underlying distribution of (Y, X). Accordingly, equivalence of the risk orderings and risk minimizing classifiers between the classification and hinge risks remains valid even if we consider empirical analogues of the risks constructed from the empirical distribution of the sample. Theorems 3.2 and 4.4 hence guarantee that a classifier minimizing the empirical hinge risk over F G or over a classification-preserving reduction F G also minimizes the empirical classification risk.
In this section, we assess the generalization performance of hinge risk minimizing classifiers, allowing for general misspecification of the constrained class of classifiers. Towards that goal, let G be fixed and considerF, a class of classifiers whose members satisfy −1 ≤ f ≤ 1.F may or may not be a subclass of F G , while in our analysis of monotone classification below,F corresponds to an approximation of a classification-preserving reduction F G . Let {(Y i , X i ) : i = 1, . . . , n} be a sample of observations that are independent and identically distributed (i.i.d.) as (Y, X). We denote the joint distribution of a sample of n observations by P n and the expectation with respect to P n by E P n [·]. We define the empirical classification risk and empirical hinge risk, respectively, aŝ
R(f ) ≡ 1 n n i=1 1 {Y i · sign(f (X i )) ≤ 0} , R φ h (f ) ≡ 1 n n i=1 max{0, 1 − Y i f (X i )} = 1 n n i=1 (1 − Y i f (X i )) ,
where the max operator in the hinge loss is redundant if we constrain f (·) to [−1, 1]. Let f be a classifier that minimizesR φ h (·) overF. We evaluate the statistical properties of f in terms of the excess classification risk relative to the minimal risk over
F G , R(f ) − inf f ∈F G R(f ).
In particular, we later derive a distribution-free upper bound on the mean of the excess classification risk. Let F G be a subclass of F G that satisfies conditions (A1) and (A2) in Theorem 4.4. F G is a classification-preserving reduction of F G (Definition 4.2). Following Corollary 4.5, we have
R(f ) − inf f ∈F G R(f ) = 1 2 R φ h (f ) − inf f ∈ F G R φ h (f ) + 1 2 R φ h f Gf − R φ h (f ) .(18)
WhenF = F G , evaluating each term in the right hand side of (18) gives an upper bound on the mean of the G-constrained excess classification risk off . Let H B 1 ( , F, P X ) be the L 1 (P X )-bracketing entropy of a class of functions F and H B 1 ( , G, P X ) be that of a class of prediction sets G. 5 For definitions of these two terms, see Definition B.1 in Appendix B. WhenF coincides with F G , the following theorem gives a non-asymptotic distribution-free upper bound on the mean of the G-constrained excess classification risk in terms of the bracketing entropy.
Theorem 5.1. Let F G be a subclass of F G that satisfies conditions (A1) and (A2) in Theorem 4.4. Suppose that P is a class of distributions on {−1, 1} × X such that there exist positive constants C and r for which
H B 1 ( , G, P X ) ≤ C −r(19)
holds for any P ∈ P and > 0, or
H B 1 , F G , P X ≤ C −r(20)
holds for any P ∈ P and > 0. Define τ n = n −1/2 if r < 1, τ n = log (n) / √ n if r = 1, and
τ n = n −1/(r+1) if r ≥ 2. Let q n = √ nτ n . Then, forf ∈ arg inf f ∈ F G R φ h (f ), the following holds: sup P ∈P E P n R(f ) − inf f ∈F G R(f ) ≤ L C (r, n),(21)
where
L C (r, n) = 2D 1 τ n + 4D 2 exp (−D 2 1 q 2 n ) 2D 3 τ n + 2n −1 D 4 if r ≥ 1 if r < 1
for some positive constants D 1 , D 2 , D 3 , D 4 , which depend only on C and r. to zero at the rate of τ n , which depends on r in the bracketing entropy conditions (19) and (20). Dudley (1999) shows many examples that satisfy these bracketing entropy conditions. In particular, the class G ⊆ 2 X that is compatible with monotone classification and introduced in Example 4.7, satisfies condition (19) with r equal to d x −1 (see Theorem 8.3.2 in Dudley (1999)). We next consider the case whenF does not coincide with F G . This case corresponds to a scenario where minimizing the empirical hinge risk over F G is difficult but minimizing overF, a class approximating F G , is feasible.
A (18) leads to
further decomposition of R φ h (f ) − inf f ∈ F G R φ h (f ) inR(f ) − inf f ∈F G R(f ) = 1 2 R φ h (f ) − inf f ∈F R φ h (f ) + 1 2 inf f ∈F R φ h (f ) − inf f ∈ F G R φ h (f ) + 1 2 R φ h (f Gf ) − R φ h (f ) .(22)
Hence the G-constrained excess classification risk is decomposed into three terms. We call the first term estimation error, the second term approximation error to a best classifier, and the third term approximation error to a step classifier. Evaluating each error gives an upper bound on the G-constrained excess classification risk.
The following theorem evaluates the estimation error in terms of bracketing entropy.
Theorem 5.2. Let F G be a subclass of F G that satisfies conditions (A1) and (A2) in Theorem 4.4. Suppose that P is a class of distributions on {−1, 1} × X such that there exist positive constants C and r for which
H B 1 ,F, P X ≤ C −r(23)
holds for any P ∈ P and > 0.
Letf ∈ arg inf f ∈F R φ h (f ). Then sup P ∈P E P n R(f ) − inf f ∈F G R(f ) ≤ L C (r , n) + 1 2 inf f ∈F R φ h (f ) − inf f ∈ F G R φ h (f ) + 1 2 R φ h (f Gf ) − R φ h (f ) ,(24)
where L C (r , n) is as defined in Theorem 5.1.
Proof. See Appendix B.
Remark 5.3 (Approximation errors). Evaluating each approximation error (the final two terms on the right-hand side) in (24) depends on the functional form restriction placed on f ∈F. IfF grows and approaches F G as n → ∞, each approximation error converges to zero. In Section 6.2 below, where we consider the monotone classification problem and setF being a sieve of Bernstein polynomials, we characterize convergence of these two approximation errors. We then apply Theorem 5.2 to obtain the regret convergence rate of the estimated monotone classifier.
Applications to monotone classification
This section applies the general theoretical results shown in Sections 3-5 to the monotone classification problem (Example 2.5). By Theorem 3.2, we limit our analysis to hinge loss. We assume that X is compact in R dx , d x < ∞, and without loss of generality, we represent X as the d x -dimensional unit hypercube (i.e., X = [0, 1] dx ). To be specific, we consider the class of monotone prediction sets G M such that, for any G ∈ G M and x,x ∈ X , x ∈ G and x ≤x impliesx ∈ G holds 6 (i.e., G M respects the partial order ≤ on X ). Accordingly, the class of monotonically increasing classifiers can be represented as
F M ≡ {f : f (x) ≤ f (x) for any x,x ∈ X with x ≤x ; f (·) ∈ [−1, 1]} .
In this section, we first study the monotone classification problem on F M . Note that F M is a classification-preserving reduction of F G M (see Example 4.7). As an alternative to F M , we next consider using a sieve of Bernstein polynomials to approximate a hinge risk minimizing classifier on F M . The Bernstein polynomial is known for its capability to accommodate bound constraints and various shape constraints on functions (e.g., monotonicity or convexity). The class of Bernstein polynomials becomes a classificationpreserving reduction only at the limit with a growing order of polynomials.
Nonparametric monotone classification
We first consider hinge risk minimization given the class of monotonically increasing classifiers F M . Letf M be a minimizer ofR φ h (·) over F M . Since the hinge risk for classifiers constrained to −1 ≤ f (x) ≤ 1 gives the linear loss φ h (yf (x)) = 1 − yf (x), minimization of the empirical hinge risk can be formulated as the following linear programming:
max f 1 ,...,fn n i=1 Y i f i (25) s.t. f i ≥ f j for any X i = X j with X i ≥ X j for 1 ≤ i ≤ j ≤ n; − 1 ≤ f i ≤ 1 for 1 ≤ i ≤ n,
where the first inequality constraints correspond to the monotononicity constraint on F M , and the second inequality constraints correspond to the range constraint for f ∈ F M . Solving this linear programming yields the values off M at the values of x observed in the training sample. Let f M (X 1 ) , . . . ,f M (X n ) be the solution of (25). Then any function in F M that passes the points X 1 ,f M (X 1 ) , . . . , X n ,f M (X n ) minimizes the empirical hinge risk over F M . 7 Since F M is a classification-preserving reduction of F G M , Theorem 4.4 with P replaced by P n shows that any solution to (25) exactly minimizeŝ
R φ h (·) over F G M .
We investigate the statistical properties of this procedure. Since F M is a classificationpreserving reduction of F G M , we can apply Theorem 5.1. Towards this goal, we first characterize an upper bound on the bracketing entropy number of the class of monotone prediction sets G M . The next lemma, which we borrow from Theorem 8.3.2 in Dudley (1999), gives an upper bound on the L 1 (P X )-bracketing entropy of G M . Here, we assume that X is continuously distributed with bounded density.
Lemma 6.1. Suppose that P X is absolutely continuous with respect to the Lebesgue measure on X and has a density that is bounded from above by a finite constant A > 0. Then there exists a constant C, which depends only on A, such that
H B 1 ( , G M , P X ) ≤ C 1−dx .
holds for all > 0.
Proof. See Appendix C.
With this lemma to hand, setting r = 1 − d x in Theorem 5.1 yields a finite sample uniform upper bound on the G M -constrained excess classification risk off M . The following theorem shows that the excess risk off M obtained from the linear program in (25) attains the same convergence rate as the welfare regret of monotone treatment rules shown by Mbakop and Tabord-Meehan (2021).
Theorem 6.2. Let P be a class of distributions on {−1, 1} × X such that the marginal distribution P X is absolutely continuous with respect to the Lebesgue measure on X and 7 All classifiers obtained from this procedure predict a unique label at each point of x observed in the training sample, whereas they may not give a unique prediction at a point of x not observed in the training sample. One possible way to predict a label at an unobserved point of x without violating the monotonicity constraint is to predict its label by the largest label among those predicted by all classifiers in arg inf f ∈F MR φ h (f ). Let X be a set of x observed in the training sample. Given anŷ f M ∈ arg inf f ∈F MR φ h (f ), this way of predicting a label is equivalent to predicting the label of x ∈ X \ X as the sign of min{f M (x) :x ∈ X ,x ≥ x} if there existsx ∈ X such thatx ≥ x, and as 1 otherwise. has a density that is bounded from above by some finite constant A > 0. Define
τ n = n −1/2 if d x = 1, τ n = log (n) / √ n if d x = 2, and τ n = n −1/dx if d x ≥ 3. Let q n = √ nτ n . Then, forf M ∈ arg inf f ∈F MR φ h (f ), sup P ∈P E P n R(f M ) − inf f ∈F G M R(f ) ≤ 2D 1 τ n + 4D 2 exp (−D 2 1 q 2 n ) 2D 3 τ n + 2n −1 D 4 if d x ≥ 2 if d x = 1
for some positive constants D 1 , D 2 , D 3 , D 4 , which depend only on d x and A.
Proof. Since F M satisfies conditions (A1) and (A2) in Theorem 4.4 with G equal to G M (Example 4.7), the result follows from Theorem 5.1 and Lemma 6.1.
This theorem guarantees the consistency of monotone classification using hinge loss and the class of monotone classifiers F M . The rate of convergence corresponds to τ n .
Monotone classification with Bernstein polynomials
To illustrate our theoretical results for monotone classification, the second approach we consider is to use multivariate Bernstein polynomials to approximate a best classifier in
F M . Let b kj (x) = k j x j (1 − x) k−j beB k (θ, x) = k 1 j 1 =0 · · · k dx j dx =0 θ j 1 ...j d · b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) ,
where k = (k 1 , . . . , k dx ) T is a vector collecting the orders of the Bernstein polynomial bases specified by the analyst, θ ≡ θ j 1 ...j dx j 1 =0,...,k 1 ;··· ;j dx =0,...,k dx is a (k 1 + 1) × · · · × (k dx + 1)dimensional vector of the parameters to be estimated, and x j denotes the j-th element of the d x -dimensional vector x. If −1 ≤ θ j 1 ...j dx ≤ 1 for all (j 1 , . . . , j dx ), the range of the function B k (θ, ·) is bounded in [−1, 1]. Moreover, if θ j 1 ...j dx ≥ θj 1 ...j dx for all (j 1 , . . . , j dx ) ≥ j 1 , . . . ,j dx , B k (θ, ·) is non-decreasing in x. 8 Hence, to preserve the bound and nondecreasing constraints on F M , we constrain the class of Bernstein polynomials to
B k = B k (θ, ·) : θ ∈ Θ ,
where Θ is the set of θ such that θ j 1 ...j dx ∈ [−1, 1] for all (j 1 , . . . , j dx ) and θ j 1 ...j dx ≥ θj 1 ...j dx for all (j 1 , . . . , j dx ) ≥ j 1 , . . . ,j dx . An appropriate choice of k is discussed later.
Noting that some hinge risk minimizing classifiers on F M have the form of step functions taking only the values −1 and 1 (Theorem 4.4), we propose approximating such a step function using the sieve of Bernstein polynomials. To this end, we propose the following two steps:
1. Minimize the empirical hinge riskR φ h (f ) over f ∈ B k and obtainf B ∈ arg inf f ∈B kR φ h (f ).
2. Let {θ j 1 ...j dx } j 1 =0,...,k 1 ;··· ;j dx =0,...,k dx be the vector of coefficients inf B . Compute a modified classifier
f † B (x) ≡ k 1 j 1 =1 · · · k dx j dx=1 sign θ j 1 ...j dx · b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) ,
which converts each estimated coefficientθ j 1 ...j dx to either −1 or 1 depending on its sign.
Our proposal is to usef
† B rather thanf B . Lemma C.3 in Appendix C shows thatf † B also minimizes R φ h (f ) over f ∈ B k .
With respect to the first step, since the hinge loss of a classifier f constrained on [−1, 1] has the linear from φ h (yf (x)) = 1 − yf (x), any function in B k is linear in the parameters θ, and the parameter space Θ is a polyhedron, minimization ofR φ h (·) over B k can be formulated as the following linear programming:
max θ n i=1 Y i · k 1 j 1 =0 · · · k dx j 1 =dx θ j 1 ...j dx · b k 1 j 1 (X i1 ) × · · · × b k dx j dx (X idx ) s.t. θ j 1 ...j dx ≥ θj 1 ...j dx for any (j 1 , . . . , j dx ) ≥ j 1 , . . . ,j dx ; − 1 ≤ θ j 1 ...j dx ≤ 1 for all (j 1 , . . . , j dx ) ,(26)
where X ij denotes the j-th element of X i . 9 The first inequality constraints restrict the feasible classifiers to a class of non-decreasing functions. The second inequality constraints bound the feasible classifiers to [−1, 1].
We then consider applying the general result for the excess risk bound in Theorem 5.2 withF = B k . Lemma C.2 in Appendix C gives finite upper bounds on two approximation errors:
inf f ∈B k R φ h (f ) − inf f ∈F M R φ h (f ) and R φ h (1{· ∈ Gf † B } − 1{· / ∈ Gf † B }) − R φ h (f † B ) in (24) upon setting (F, F,f ) = (B k , F M ,f † B )
. The binarized coefficients inf † B help us to make the second approximation error shrink to zero. Moreover, Lemma C.1 in Appendix C gives a finite upper bound on the bracketing entropy of B k . Combining these results, the following theorem gives a finite sample upper bound on the mean of the G M -constrained excess classification risk off † B .
Theorem 6.3. Let P be a class of distributions on {−1, 1} × X that satisfy the same conditions as in Theorem 6.2. Letτ n = log (n) / √ n if d x = 1 andτ n = n −1/dx if d x ≥ 2. Defineq n = √ nτ n . Then the following holds:
sup P ∈P E P n R(f † B ) − inf f ∈F G M R(f ) ≤ 2D 1τn + 4D 2 exp −D 2 1q 2 n + 4A dx j=1 log k j k j + dx j=1 8 k j ,(27)
where D 1 and D 2 are some positive constants, which depend only on d x and A.
Proof. From the fact that B k ⊆ F M and Lemma C.1 in Appendix C, we have H B 1 ( , B k , P X ) ≤ C −dx for some positive constant C, which depends only on A, and all > 0 . Then the result follows by combining Theorem 5.2 and Lemma C.2.
The upper bound in (27) converges to zero as the sample size n and the order of the Bernstein polynomials k j (j = 1, . . . , d x ) increase. Note that the rate of convergence for the estimation error in this theorem,τ n , is slower than that in Theorem 6.2, τ n . The difference in the rates of convergence are due to the different orders in the upper bounds on H B 1 ( , G M , P X ) and H B 1 ( , F M , P X ) in Lemmas 6.1 and C.1. To achieve the convergence rate ofτ n for the mean of the excess risk off † B , Theorem 6.3 suggests the tuning parameters k j , for j = 1, . . . , d x , should be set sufficiently large so that log k j /k j = O (τ n ).
In practice, one may want to select the complexity of the Bernstein polynomials by minimizing penalized empirical surrogate risk. The classification and treatment choice literature (Koltchinskii (2006), Mbakop and Tabord-Meehan (2021), and references therein) analyze the regret properties and oracle inequalities for penalized risk minimizing classifiers. We leave theoretical investigation of the applicability of penalization methods to the current hinge risk minimization using Bernstein polynomials to future research.
Extension to individualized treatment rules
This section extends the primary results obtained in Sections 3 and 4 for binary classification to the weighted classification introduced in Section 1.1, and to causal policy learning.
Extensions of the results in Sections 5 and 6 to weighted classification are presented in Appendix D. We use the same notation and definitions introduced in Section 1.1. We term R ω and R ω φ , defined in (5) and (6), weighted classification risk and weighted φ-risk, respectively. Throughout this section, with some abuse of notation, we denote by P a distribution on R + × {−1, 1} × X and suppose that (ω, Y, D) ∼ P .
Consistency of weighted classification with hinge loss
We first show consistency of weighted classification with hinge risk by adapting the analyses in Sections 3 and 4. Given a prespecified G, let F G be as in Section 2. Analogous to R(G) and
R φ (G), we define R ω (G) ≡ inf f ∈F G R ω (f ), the weighted-classification risk evaluated at G, and R ω φ (G) ≡ inf f ∈F G R ω φ (f ), the weighted φ-risk evaluated at G. Note that R ω (G) = R ω (f ) for all f ∈ F G . Let R w * ≡ inf G∈G R ω (G) = inf f ∈F G R ω (f )
be the optimal weighted risk, and G * ≡ arg inf G∈G R ω (G) be the collection of best prediction sets.
For the non-negative weight variable ω, we define
ω + (x) ≡ E P [ω | X = x, Y = +1] ω − (x) ≡ E P [ω | X = x, Y = −1] . Let C φ (a, b, c, d) ≡ aφ (c) d + bφ (−c) (1 − d), and C w+ φ (ω + , ω − , η) ≡ inf 0≤f ≤1 C φ (ω + , ω − , f, η) , C w− φ (ω + , ω − , η) ≡ inf −1≤f <0 C φ (ω + , ω − , f, η) , ∆C ω φ (ω + , ω − , η) ≡ C w+ φ (ω + , ω − , η) − C w− φ (ω + , ω − , η) ,
which are analogous to C + φ , C − φ and ∆C φ defined in Section 3. The next theorem generalizes Theorems 3.2, 3.6, and Corollary 3.7 to weighted classification, giving a necessary and sufficient condition for equivalence of the risk ordering among surrogate loss functions. In particular, we show that hinge loss functions share a common risk ordering with the 0-1 loss function.
Theorem 7.1. Let φ 1 and φ 2 be classification-calibrated loss functions in the sense of Definition 2.3. Then
R ω φ 1 (G 1 ) ≤ R ω φ 1 (G 2 ) ⇔ R ω φ 2 (G 1 ) ≤ R ω φ 2 (G 2 )
holds for any distribution P on R + × {−1, 1} × X , any class of measurable subsets G ⊆ 2 X , and any G 1 , G 2 ∈ G if and only if there exists c > 0 such that ∆C ω φ 2 (ω + , ω − , η) = c∆C ω φ 1 (ω + , ω − , η) holds for any (ω + , ω − , η) ∈ R + × R + × [0, 1]. In particular, the 0-1 loss function, φ 01 (α) = 1{α ≤ 0}, satisfies
∆C ω φ 01 (ω + , ω − , η) = −ω + η + ω − (1 − η) ,(28)
and the hinge loss function φ h (α) = c max{0, 1 − α} satisfies
∆C ω φ h (ω + , ω − , η) = c (−ω + η + ω − (1 − η)) = c∆C ω φ 01 (ω + , ω − , η) .
Proof. See Appendix E.
Theorem 7.1 and inequalities similar to (14) lead to a generalized Zhang's (2004) inequality for weighted classification, as shown in the next corollary.
Corollary 7.2. For any distribution P on R + × {−1, 1} × X and any surrogate loss
function φ satisfying ∆C ω φ (ω + , ω − , η) = c (−ω + η + ω − (1 − η)), c(R ω (f ) − inf f ∈F G R ω (f )) ≤ R ω φ (f ) − inf f ∈F G R ω φ (f )(29)
holds for any f ∈ F G .
Proof. See Appendix E.
Remark 7.3. Table 2 shows the forms of ∆C ω φ (ω + , ω − , η) for the hinge loss, exponential loss, logistic loss, quadratic loss, and truncated quadratic loss functions, where µ + ≡ ω + η and µ − ≡ ω − (1−η). With the exception of the hinge loss function, none of these functions satisfy ∆C ω φ (ω + , ω − , η) = c(−µ + + µ − ) for some positive constant c > 0. That is, similar to the standard binary classification, hinge losses also have a special status in weighted classification, since they are the only surrogate losses that preserve classification risk.
Similar to the analysis in Section 4, we consider adding functional form restrictions to the class of classifiers F G . Let F G be a subclass of F G . We suppose that the non-negative weight variable ω is bounded from above.
Loss function
φ(α) ∆C ω φ (ω + , ω − , η) 0-1 loss 1{α ≤ 0} −µ + + µ − Hinge loss c max{0, 1 − α} c (−µ + + µ − ) Exponential loss e −α ( √ µ + − √ µ − ) 2 −( √ µ + − √ µ − ) 2 if µ + ≤ µ − if µ + > µ − Logistic loss log(1 + e −α ) −µ + log 2µ+ µ++µ− − µ − log 2µ− µ++µ− µ + log 2µ+ µ++µ− + µ − log 2µ− µ++µ− if µ + ≤ µ − if µ + > µ − Quadratic loss (1 − α) 2 (µ+−µ−) 2 µ++µ− − (µ+−µ−) 2 µ++µ− if µ + ≤ µ − if µ + > µ − Truncated quadratic loss (max{0, 1 − α}) 2 (µ+−µ−) 2 µ++µ− − (µ+−µ−) 2 µ++µ− if µ + ≤ µ − if µ + > µ − Note: µ + = ω + η and µ − = ω − (1 − η).
In causal policy learning, Condition 7.4 holds if the outcome variable Z has bounded support and the propensity score e(x) satisfies a strict overlap condition. For example, if the support of Z is contained in [−M ,M ] for someM < ∞, and the propensity score satisfies κ < e(x) < 1 − κ for some κ ∈ (0, 1/2) and all x ∈ X , then the weight variable for the causal policy learning ω p defined in (7) is bounded from above byM /κ a.s.
The following theorem, which is analogous to Theorem 4.4, shows that the two conditions (A1) and (A2) in Theorem 4.4 remain sufficient for F G to guarantee the consistency of the hinge risk minimization approach to weighted classification.
Theorem 7.5. Given a distribution P on R + × {−1, 1} × X and a class of measurable subsets G ⊆ 2 X , suppose that F G ⊂ F G satisfy the conditions (A1) and (A2) in Theorem 4.4 and that the weight variable ω satisfies Condition 7.4. Then the following claims hold:
(i)f * ∈ arg inf f ∈ F G R ω φ h (f ) minimizes the weighted-classification risk R ω (·) over F G . (ii) For G * ∈ G * ,f G * is a minimizer of R ω φ h (·) over F G .
Proof. See Appendix E.
Conclusion
This paper studies consistency of surrogate risk minimization approaches to classification and weighted classification given a constrained set of classifiers, where weighted classification subsumes policy learning for individualized treatment assignment rules. Our focus is on how surrogate risk minimizing classifiers behave if the constrained class of classifiers violates the assumption of correct specification. Our first main result shows that when the constraint restricts classifiers' prediction sets only, hinge losses are the only loss functions that secure consistency of the surrogate-risk minimizing classifier without the assumption of correct specification. When the constraint additionally restricts the functional form of the classifiers, the surrogate risk minimizing classifier is not generally consistent even with hinge loss. Our second main result is to show that, in this case, the set of conditions (A1) and (A2) in Theorem 4.4 becomes a sufficient condition for the consistency of the hinge risk minimizing classifier. This paper also investigates the statistical properties of hinge risk minimizing classifiers in terms of uniform upper bounds on the excess regret. Exploiting hinge loss and the class of monotone classifiers in the monotone classification problem, we show that the empirical surrogate-risk minimizing classifier can be computed using linear programming. All of the results obtained in the standard classification setting are naturally extended to the weighted classification problem, so that our contributions carry over to causal policy learning and related applications.
Appendix A Proofs of the results in Sections 2-4
This appendix provides proof of the results in Sections 2-4 alongside some auxiliary lemmas. We first give the proofs of Propositions 2.1 and 2.2.
Proof of Proposition 2.1. The statement (i) follows from Claim 3 of Theorem 1 of Bartlett et al. (2006). To prove the statement (ii), let z 1 , z 2 ∈ R + be such that φ(z 2 ) < φ(z 1 ). Such a pair (z 1 , z 2 ) exists in a neighborhood of zero because, from Theorem 2 of Bartlett et al. (2006), φ is differentiable at 0 and φ (0) < 0 if φ is classificationcalibrated and convex. Then the pair (z 1 , z 2 ) satisfies arg min
(b 1 ,b 2 )∈{(−1,0),(0,1)} φ(−b 1 z 1 ) + φ(b 2 z 2 ) = (0, 1). Suppose that F is a constrained class such that F = F 1 ∪ F 2 with F 1 = {f (x) = x T b : (b 1 , b 2 ) = (−1, 0), b ∈ R dx } and F 2 = {f (x) = x T b : (b 1 , b 2 ) = (0, 1), b ∈ R dx },
where b j denotes the j-th element of b. Let x 1 = (z 1 , 0, . . . , 0) and x 2 = (0, z 2 , 0, . . . , 0) be two points in X , and let P be a distribution such that η(x 1 ) = 0, η(x 2 ) = 1, and P X (x 1 ) = P X (x 2 ) = 1/2. Given the pair (P, F), any classifier f 1 in F 1 has the same sign as the Bayes classifier, P X -almost everywhere, because f 1 (x 1 ) < 0 and f 1 (x 2 ) = 0 while η(x 1 ) < 1/2 and η(x 2 ) ≥ 1/2. This means that F, as well as F 1 , is R-correctly specified for such P . On the other hand, any classifier f 2 in F 2 does not have the same sign as the Bayes classifier at x 1 because f 2 (x 1 ) = 0 while η(x 1 ) < 1/2. f * φ must be in F 2 because for any f 1 ∈ F 1 and f 2 ∈ F 2 , R φ (f 2 ) = φ(z 2 )/2 < φ(z 1 )/2 = R φ (f 1 ). Hence R(f * φ ) > R(f * bayes ) holds. F is also R φ -misspecified because any classifier that minimizes R φ over all measurable functions takes a negative value at x 1 whereas f * φ (x 1 ) > 0.
Proof of Proposition 2.2 Assume R-correct specification of F G . Then F G includes a classifier f * that is identical to or shares the same sign as f * Bayes (x) = 2η(x) − 1, P Xalmost everywhere. Since f ∈ F G is unconstrained except for G f ∈ G and −1 ≤ f (·) ≤ 1, the classification-calibrated property of φ and the representation of the surrogate risk
R φ (f ) = E P X [C φ (f (X), η(X))] implies f * φ (x) ∈ arg min a:(2η(x)−1)a≥0,|a|≤1
C φ (a, η(x)), P X -almost everywhere, because otherwise f * dominates f * φ in terms of the surrogate risk. This means that f * φ (x) has the same sign as f * Bayes (x) , P X -almost everywhere, i.e.,
R(f * φ ) = R(f * Bayes ) holds. Assume conversely that F G is R-misspecified.
Then sign(f * φ ) has to differ from sign(f * Bayes (x)) for some x with positive measure in terms of P X . Failure to find such a value x contradicts the assumption of R-misspecification of F G . The difference in signs then implies
R(f * φ ) > R(f * Bayes ).
We here let φ be any surrogate loss function. Before proceeding to the proofs of the results in Sections 3 and 4, we note that if φ is classification-calibrated, ∆C φ has the same sign as the Bayes classifier: for any η ∈ [0, 1]\{1/2},
∆C φ (η) > 0 < 0 if η > 1/2 if η < 1/2 ,(30)
which will be used in the following proofs.
Proof of Theorem 3.6.
('If' part) For any class of measurable subsets G ⊆ 2 X and any G 1 , G 2 ∈ G, we show in Theorem 3.2 that R φ 1 (G 2 ) ≥ R φ 1 (G 1 ) is equivalent to
G 2 \G 1 ∆C φ 1 (η(x)) dP X (x) ≥ G 1 \G 2 ∆C φ 1 (η(x)) dP X (x).
This inequality does not change if we replace ∆C φ 1 (η(x)) with ∆C φ 2 (η(x)) = c∆C φ 1 (η(x)) with c > 0. From Theorem 3.2, R φ 2 (G 2 ) ≥ R φ 2 (G 1 ) if ∆C φ 1 (η(x)) is replaced by ∆C φ 2 (η(x)) in the above inequality. Therefore, if ∆C φ 2 (·) = c∆C φ 1 (·) with c > 0, φ 1 u ∼ φ 2 holds.
('Only if' part) We prove the 'only if' part of the theorem by exploiting a specific class of data generating processes (DGPs). Suppose X = {1, 2} and G = {∅, G 1 , G 2 , X } with G 1 = {1} and G 2 = {2}. Let α = P (X = 1) (= 1 − P (X = 2)) and (η 1 , η 2 ) = (η (1) , η (2)). The DGP varies depending on the values of (α, η 1 ,
η 2 ) ∈ [0, 1] 3 .
In what follows, we will show that
∆C φ 1 (η 1 ) ∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) ∆C φ 2 (η 2 )(31)
holds for any (η 1 , η 2 ) ∈ ([0, 1]\{1/2}) 2 . Then applying Lemma A.1 below proves the 'only if' part of the theorem. In the current setting, for G ∈ G, R φ (G) can be written as
R φ (G) = P (X = 1) ∆C φ (η 1 ) 1 {1 ∈ G} + P (X = 2) ∆C φ (η 2 ) 1 {2 ∈ G} + 2 x=1 P (X = x) C − φ (η(x)) = α∆C φ (η 1 ) 1 {1 ∈ G} + (1 − α) ∆C φ (η 2 ) 1 {2 ∈ G} + C α,η 1 ,η 2 ,
where C α,η 1 ,η 2 ≡ α∆C φ (η 1 ) + (1 − α) ∆C φ (η 2 ), which does not depend on G. Thus, we have
R φ (∅) = C α,η 1 ,η 2 R φ (G 1 ) = α∆C φ (η 1 ) + C α,η 1 ,η 2 , R φ (G 2 ) = (1 − α) ∆C φ (η 2 ) + C α,η 1 ,η 2 , R φ (X ) = α∆C φ (η 1 ) + (1 − α) ∆C φ (η 2 ) + C α,η 1 ,η 2 .
In what follows, we will show that (31) holds, separately, in four cases: (i) η 1 > 1/2 and η 2 > 1/2; (ii) η 1 < 1/2 and η 2 < 1/2; (iii) η 1 < 1/2 and η 2 > 1/2; (iv) η 1 > 1/2 and η 2 < 1/2. First, we consider case (i): η 1 > 1/2 and η 2 > 1/2. Because we assume φ 1 u ∼ φ 2 ,
R φ 1 (G 1 ) ≤ R φ 1 (G 2 ) ⇔ R φ 2 (G 1 ) ≤ R φ 2 (G 2 ) ,
holds for any (α, η 1 , η 2 ) ∈ (0, 1) × (1/2, 1] 2 . This is equivalent to
α∆C φ 1 (η 1 ) ≤ (1 − α) ∆C φ 1 (η 2 ) ⇔ α∆C φ 2 (η 1 ) ≤ (1 − α) ∆C φ 2 (η 2 )
for any (α, η 1 , η 2 ) ∈ (0, 1) × (1/2, 1] 2 . Let γ + ≡ (1 − α) /α, which can take any value in (0, +∞) by varying α on (0, 1). From the classification-calibrated property (30), both ∆C φ 1 (η) and ∆C φ 2 (η) are positive for η ∈ (1/2, 1]. Thus, it follows for any (γ + , η 1 , η 2 ) ∈ (0, +∞) × (1/2, 1] 2 that
∆C φ 1 (η 1 ) ∆C φ 1 (η 2 ) ≤ γ + ⇔ ∆C φ 2 (η 1 ) ∆C φ 2 (η 2 ) ≤ γ + ,(32)
where both ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) and ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) are positive. Since (32) holds for any value of η + ∈ (0, +∞), ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) holds for any (η 1 , η 2 ) ∈ (1/2, 1] 2 . Similarly, for case (ii): η 1 < 1/2 and η < 1/2, the following equivalence statement holds for any (γ + , η 1 , η 2 ) ∈ (0, +∞) × [0, 1/2) 2 :
∆C φ 1 (η 1 ) ∆C φ 1 (η 2 ) ≥ γ + ⇔ ∆C φ 2 (η 1 ) ∆C φ 2 (η 2 ) ≥ γ + ,(33)
where both ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) and ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) are positive. Thus, varying the value of γ + on (0, +∞) in (33) shows that ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) holds for any (η 1 , η 2 ) ∈ [0, 1/2) 2 . Next, we consider case (iii): η 1 < 1/2 and η 2 > 1/2. Because we assume φ 1 u ∼ φ 2 , it follows for any (α, η 1 , η 2 ) ∈ (0, 1) × [0, 1/2) × (1/2, 1] that
R φ 1 (∅) ≤ R φ 1 (X ) ⇔ R φ 2 (∅) ≤ R φ 2 (X ) ,
which is further equivalent to
0 ≤ α∆C φ 1 (η 1 ) + (1 − α) ∆C φ 1 (η 2 ) ⇔ 0 ≤ α∆C φ 2 (η 1 ) + (1 − α) ∆C φ 2 (η 2 ) .
Let γ − ≡ (α − 1) /α, which can take any value in (−∞, 0) by varying the value of α on (0, 1). Because ∆C φ 1 (η 1 ) < 0 and ∆C φ 2 (η 2 ) > 0 hold for (η 1 , η 2 ) ∈ [0, 1/2) × (1/2, 1] due to the classification-calibrated property (30),
∆C φ 1 (η 1 ) ∆C φ 1 (η 2 ) ≥ γ − ⇔ ∆C φ 2 (η 1 ) ∆C φ 2 (η 2 ) ≥ γ −(34)
for any (γ − , η 1 , η 2 ) ∈ (−∞, 0) × [0, 1/2) × (1/2, 1], where both ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) and ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) are negative. Thus, varying the value of γ − on (−∞, 0) in (34) shows that ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) holds for any (η 1 , η 2 ) ∈ [0, 1/2)× (1/2, 1]. Similarly, in case (iv): η 1 > 1/2 and η 2 < 1/2, the following equivalence statement holds for any (γ − , η 1 , η 2 ) ∈ (−∞, 0) × (1/2, 1] × [0, 1/2):
∆C φ 1 (η 1 ) ∆C φ 1 (η 2 ) ≤ γ − ⇔ ∆C φ 2 (η 1 ) ∆C φ 2 (η 2 ) ≤ γ − ,(35)
where both ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) and ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) are negative. Therefore, varying the value of γ − in (35) shows that ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) for any (η 1 , η 2 ) ∈ (1/2, 1] × [0, 1/2). Combining these four results, we have ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) for any (η 1 , η 2 ) ∈ ([0, 1] \ {1/2}) 2 . Then the proof follows from Lemma A.1 below.
Lemma A.1. Let φ 1 and φ 2 be classification-calibrated loss functions.
If ∆C φ 1 (η 1 ) /∆C φ 1 (η 2 ) = ∆C φ 2 (η 1 ) /∆C φ 2 (η 2 ) holds for any (η 1 , η 2 ) ∈ ([0, 1] \ {1/2}) 2 , then there exists some con- stant c > 0 such that ∆C φ 2 (η) = c∆C φ 1 (η) for any η ∈ [0, 1]. Proof. For η ∈ [0, 1] \ {1/2}, let c (η) be a value such that ∆C φ 2 (η) = c (η) ∆C φ 1 (η) .(36)
Because φ 1 and φ 2 are classification-calibrated, c (η) must be positive from (30). We will show that c (η) is constant over η ∈ [0, 1] \ {1/2} by contradiction.
To obtain a contradiction, suppose there exists (η 1 , η 2 ) ∈ ([0, 1] \ {1/2}) 2 such that c (η 1 ) = c (η 2 ). From the assumption, the following equations hold
∆C φ 2 (η 1 ) = ∆C φ 2 (η 2 ) ∆C φ 1 (η 2 ) ∆C φ 1 (η 1 ) , ∆C φ 2 (η 2 ) = ∆C φ 2 (η 1 ) ∆C φ 1 (η 1 ) ∆C φ 2 (η 2 ) .
Combining these equations with equation (36), we have ∆C φ 2 (η 2 ) = c (η 1 ) ∆C φ 1 (η 2 ) and ∆C φ 2 (η 2 ) = c (η 2 ) ∆C φ 1 (η 2 ). However, this contradicts the assumption that c (η 1 ) = c (η 2 ). Therefore, c (η) must be constant over η ∈ [0, 1] \ {1/2}.
When η = 1/2, ∆C φ 1 (η) = ∆C φ 2 (η) = 0 holds by definition. In this case, ∆C φ 2 (η) = c∆C φ 1 (η) holds for any c.
The following expression of the hinge risk will be used in the proofs of Lemma 4.3 and Theorem 4.4:
R φ h (f ) = X [η(x)(1 − f (x)) + (1 − η(x))(1 + f (x))] dP X (x) = X (1 − 2η(x)) f (x)dP X (x) + 1.(37)
Proof of Lemma 4.3. Fix a distribution P on {−1, 1} × X , and letf ∈ F G,J .f has the formf
(x) = 2 J j=1 c j 1 {x ∈ G j } − 1(38)
for some G 1 , . . . , G J ∈ G such that G J ⊆ · · · ⊆ G 1 and some c j ≥ 0 for j = 1, . . . , J such that J j=1 c j = 1. Thus, substitutingf into (37) yields
R φ h (f ) = 2 J j=1 c j G j (1 − 2η(x)) dP X (x) + 2P (Y = 1).(39)
Comparing (39) with equation (9) leads to
R φ h (f ) = 2 J j c j R(G j ).(40)
From this expression and the assumption that J j=1 c j = 1, R φ h (f ) ≥ 2R * holds for anỹ f ∈ F G,J .
For G * ∈ G * , a functionf G * (x) = 2 · 1{x ∈ G * } − 1 can be extracted from F G,J by setting G 1 = G * and c 1 = 1. Then, from equation (40), R φ h (f G * ) = 2R * holds; that is, f G * minimizes R φ h (·) over F G,J . This proves statement (ii) of the lemma.
We will next show that F G,J is a classification-preserving reduction of F G (statement (i) of the lemma). To obtain a contradiction, suppose that a classifierf with the form of (38) minimizes R φ h (·) over F G,J but does not minimize R(·) over F G . Asf does not minimize R(·) over F G , Gf / ∈ G * holds. Let m be the smallest integer in {1, . . . , J} such that m j=1 c j ≥ 1/2 in (58). Then G m = Gf . It then follows that
R φ h (f ) = 2 J j=1 c j R(G j ) = 2c m R(G m ) + j∈{1,...,m−1,m+1,...,J} c j R(G j ) ≥ 2c m R(f ) + 2(1 − c m )R * > 2R * ,
where the last line follows from c m > 0 and Gf / ∈ G * . Hence R φ h (f ) does not take the minimum value of R φ h (·) over F G,J (i.e., R φ h (f ) > 2R * ), which contradicts the assumption thatf minimizes R φ h (·) over F G,J . Therefore,f minimizes R(·) over F G . Since this discussion is valid for any distribution P on {−1, 1} × X , we conclude F G,J is a classification-preserving reduction of F G . equality follows fromf * J ∈ F * J ; the last inequality follows from F * J ⊆ F G,J . Lemma 4.3 shows that inff ∈ F G,J R φ h (f ) = 2R * for any J. Hence, from equation (42), we have
R φ h (f * ) ≥ lim J→∞ inf f ∈ F G,J R φ h (f ) ≥ 2R * ,
On the other hand, Lemma 4.3 also shows that R φ h (f G * ) = 2R * . Therefore,f G * minimizes R φ h (·) over F G , which proves statement (ii) of the theorem.
Next we will show that F G is a classification-preserving reduction of F G (statement (i) of the theorem). To obtain a contradiction, suppose thatf * does not minimize R(·) over F G (i.e., Gf * / ∈ G * ). Then, from the proof of Lemma 4.3, for any J andf ∈ F * J , R φ h (f ) > 2R * . Therefore, from equation (41),
R φ h (f * ) ≥ lim J→∞ inf f ∈ F * J R φ h (f ) >2R * .
This contradicts the assumption thatf * minimizes R φ (·) over F G because, as we have seen,f G * (∈ F G ) achieves R φ h (f G * ) = 2R * . Therefore, Gf * ∈ G * must hold, i.e.,f * ∈ arg inf f ∈F G R(f ). Since this discussion holds for any distribution P on {−1, 1} × X , F G is a classification-preserving reduction of F G .
Proof of Corollary 4.5. Note that, for G ∈ G,
R(G) = X [η(x)1 {x / ∈ G} + (1 − η(x))1 {x ∈ G}] dP X (x) = X [η(x)1 {x ∈ G c } + (1 − η(x)) (1 − 1 {x ∈ G c })] dP X (x) = − G c (1 − 2η(x)) dP X (x) + P (Y = −1) .(43)
By equations (9) and (43), cR(f ) can be written as
cR(f ) = 1 2 X c(1 − 2η(x)) (1 {x ∈ G f } − 1 {x / ∈ G f }) dP X (x) + c .
By equation (37), the term inside the braces equals R φ (f G f ). Combining this result with Theorem 4.4 (i) leads to equation (16). When F G = F G , by equation (16) and Corollary 3.4,
R φ (f G f ) − R φ (f ) ≤ R φ (f ) − inf f ∈F G R φ (f )
holds. Combining this with equation (16) leads to the second result.
B Proof of Theorems 5.1 and 5.2
This appendix provides proof of the results in Section 5 with some auxiliary results.
The results below are related to the theory of empirical processes. We refer to Alexander (1984), Mammen and Tsybakov (1999), Tsybakov (2004), and Mbakop and Tabord-Meehan (2021) for the general strategy of the proof. Given G ∈ G, let 1 G be an indicator function on X such that 1 G (x) = 1{x ∈ G}. We first give the definition of bracketing entropy for a class of functions and a class of sets.
let f p,Q ≡ X |f (x)| p dQ(x) 1/p . · p,Q is the L p (Q)-metric on X , where Q is a measure on X . Given a pair of functions (f 1 , f 2 ) with f 1 ≤ f 2 , let [f 1 , f 2 ]≡{f ∈ F : f 1 ≤ f ≤ f 2 } be the bracket. Given > 0, let N B p ( , F, Q) be the smallest k such that there exist pairs of functions f L j , f U j , j = 1, . . . , k, with f L j ≤ f U j that satisfy f U j − f L j p,Q < and F ⊆ ∪ k j=1 f L j , f U j .
We term N B p ( , F, Q) the L p (Q)-bracketing number of F, and H B p ( , F, Q) ≡ log N B p ( , F, Q) the L p (Q)-bracketing entropy of F. We also refer to [f L j , f U j ] as the -bracket with respect to L p (Q) if and only if f U j − f L j p,Q < holds. (ii) Given a class of measurable subsets G ⊆ 2 X , let H G ≡ {1 G : G ∈ G}. We define H B p ( , G, Q) ≡ H B p ( , H G , Q) and term this the L p (Q)-bracketing entropy of G.
Note that in the definition of N B p ( , F, Q), the functions f L j and f U j do not have to belong to F. Note also that if
F ⊆ F, H B p ( , F, Q) ≤ H B p , F, Q holds. When 1 G ∈ F for all G ∈ G, H B p ( , G, Q) ≤ H B p ( , F, Q)
holds. The following theorem gives a finite-sample upper bound on the mean of the estimation error in Section 5, auxiliary results of which are provided below.
Theorem B.2. LetF be a class of classifiers whose members satisfy −1 ≤ f ≤ 1.
Suppose that P is a class of distributions on {−1, 1} × X such that there exist positive constants C and r for which H B 1 ,F, P X ≤ C −r holds for any P ∈ P and > 0. Let q n and τ n be as in Theorem 5.1. Letf minimizê R φ h (·) overF. Then the following holds:
sup P ∈P E P n R φ h (f ) − inf f ∈F R φ h (f ) ≤ 4D 1 τ n + 8D 2 exp (−D 2 1 q 2 n ) 4D 3 τ n + 4n −1 D 4
for r ≥ 1 for 0 < r < 1 for some positive constants D 1 , D 2 , D 3 , D 4 , which depend only on C and r.
Proof. Fix P ∈ P. Letf * minimize R φ h (·) overF. Define a class of functionsḞ ≡ (f + 1)/2 : f ∈F , which normalizesF so that 0 ≤ f ≤ 1 for all f ∈Ḟ. A standard argument gives
E P n R φ h (f ) − inf f ∈F R φ h (f ) ≤ E P n R φ h (f ) −R φ h (f ) +R φ h (f * ) − R φ h (f * ) ∵R φ h (f ) ≤R φ h (f * ) = 2E P n R φ h f + 1 2 −R φ h f + 1 2 + 2E P n R φ h f * + 1 2 − R φ h f * + 1 2 ≤ 4 sup f ∈Ḟ E P n R φ h (f ) −R φ h (f ) .(44)
Since R φ h (f ) −R φ h (f ) can be seen as the centered empirical process indexed by f ∈Ḟ, we can apply results in empirical process theory to (44) to obtain a finite-sample upper bound on the mean of the excess hinge risk. We follow the general strategy of Theorem 1 in Mammen and Tsybakov (1999) and Proposition B.1 in Mbakop and Tabord-Meehan (2021). Note that
sup f ∈Ḟ E P n R φ h (f ) −R φ h (f ) = sup f ∈Ḟ E P n E P (Y f (x)) − 1 n n i=1 Y i f (X i )(45)
and that
sup f ∈Ḟ E P (Y f (x)) − 1 n n i=1 Y i f (X i ) ≤ 2
with probability one. We first prove the result for the case of r ≥ 1. For any f ∈Ḟ and D > 0,
√ n q n sup f ∈Ḟ E P n E (Y f (x)) − 1 n n i=1 Y i f (X i ) ≤ D + √ n q n P n sup f ∈Ḟ √ n q n E (Y f (x)) − 1 n n i=1 Y i f (X i ) > D .
We apply Corollary B.3 by setting Z = (Y, X), g (z 1 ) = z 1 and H =Ḟ as appear in the statement of the corollary. Note that, sinceḞ is an affine transformation multiplying 1/2 to f ∈F, H B 1 ( ,F, P 2 ) = H B 1 (2 , H, P 2 ) holds. Then, by Corollary B.3 shown below with K = 2 −r C, there exist D 1 , D 2 , D 3 > 0, depending only on C and r, such that
P n sup f ∈Ḟ √ n q n E (Y f (x)) − 1 n n i=1 Y i f (X i ) > D ≤ D 2 exp −D 2 q 2 n ,
for D 1 ≤ D ≤ D 3 √ n/q n . Thus when r ≥ 1, we have
τ −1 n E P n R φ h (f ) − inf f ∈F R φ h (f ) ≤ 4τ −1 n sup f ∈Ḟ E P n E (Y f (x)) − 1 n n i=1 Y i f (X i ) ≤ 4D 1 + 8τ −1 n D 2 exp −D 2 1 q 2 n ,
which leads to the result for the case of r ≥ 1.
The result for the case of 0 < r < 1 follows immediately by applying Lemma B.4 to equation (45), where we set Z = (Y, X), g (z 1 ) = z 1 , and H =Ḟ.
We now give the proofs of Theorems 5.1 and 5.2.
Proof of Theorem 5.2. The result in Theorem 5.2 follows by combining equation (22) and Theorem B.2.
Proof of Theorem 5.1. Let P ∈ P be fixed. Definef † ≡f Gf . Since R(f † ) = R(f ), it follows from equation (18) that
R(f ) − inf f ∈F G R(f ) = R(f † ) − inf f ∈F G R(f ) = 1 2 R φ h (f † ) − inf f ∈ F G R φ h (f ) .(46)
When (19) holds for all > 0, the result follows by applying Theorem B.2 to (46). We consider the case when (20) holds for all > 0. Define a class of step functions
I G ≡ {f G : G ∈ G}. We now show that (A)f † minimizesR φ h (·) over I G , and that (B) inf f ∈ F G R φ h (f ) = inf f ∈I G R φ h (f ).
If both hold, we can apply Theorem B.2 to the excess hinge rink in (46) withF replaced by I G .
We first prove (B). Since
inf f ∈I G R φ h (f ) = 2 inf G∈G R(G) = 2R * , Theorem 4.4 (i) shows that inf f ∈ F G R φ h (f ) = inf f ∈I G R φ h (f ).
We next prove (A). Note first thatf † ∈ I G holds. Let P n be the empirical distribution of the sample {(Y i , X i ) : i = 1, . . . , n}. Replacing P with P n in Theorem 4.4 shows that f minimizesR(·) over F G . HenceR(G) ≡ inf f ∈F GR (f ) is minimized by Gf over G. Then Theorem 4.4 (ii) with P replaced by P n shows that the new classifierf † also minimizeŝ R φ h (·) over F G . Then replacing R φ h withR φ h in the statement of (B),f † minimizesR φ h (·) over I G .
From the definitions of H B 1 ( , G, P X ) and I G , we have H B 1 ( , I G , P X ) = H B 1 (2 , G, P X ). Therefore, we can apply Theorem B.2, withF replaced by I G , to (46) and then obtain the inequality in (21).
The following corollary is similar to Corollary D.1 in Mbakop and Tabord-Meehan (2021). The difference is that the class of functions H in the following corollary does not need to be a class of binary functions.
Corollary B.3. Let Z = (Z 1 , Z 2 ) ∼ P , and {Z i } n i=1
be a sequence of random variables that are i.i.d. as Z. Denote by P 2 the marginal distribution of Z 2 . Suppose P 2 is absolutely continuous with respect to Lebesugue measure and its density is bounded from above by a finite constant A > 0. Let F be a class of real-valued functions of the form f for some constants K > 0 and r ≥ 1 and for all > 0. Then there exist positive constants D 1, D 2 , D 3 , depending only on K and r, such that for n ≥ 3:
(z) = f (z 1 , z 2 ) = g (z 1 ) · h (z 2 ), where h ∈ H,P n sup f ∈F 1 √ n n i=1 (f (Z i ) − E P [f (Z i )]) > xq n ≤ D 2 exp −x 2 q 2 n , for D 1 ≤ x ≤ D 3 √ n/q n , where q n = log n n (r−1)/2(r+2) r = 1 r > 1 .
Proof. Let h L j , h U j , j = 1, . . . , N B 1 ( , H, P 2 ), be a set of -brackets of H with respect to L 1 (P 2 ) such that h U j − h L j 1,P 2 < and H ⊆ ∪
N B 1 ( ,H,P 2 ) j=1 h L j , h U j . Since h U j − h L j < 1, h U j − h L j 2 2,P 2 ≤ h U j − h L j 1,P 2 < holds. We hence have H B 2 ( , H, P 2 ) ≤ H B 1 2 , H, P 2 ≤ K −2r .
The result immediately follows by applying Proposition B.1.
Lemma B.4. Maintain the same definitions and assumptions as in Corollary B.3 with r ≥ 1 replaced by 0 < r < 1. Then, there exist positive constants D 3 and D 4 , depending only on K and r, such that:
sup f ∈F E P n 1 n n i=1 f (Z i ) − E P [f (Z)] ≤ D 3 √ n + D 4 n .
Proof. We look to apply Proposition 3.5.15 in Giné and Nickl (2016). Note first that |f | ≤ 1 and f 2,P ≤ 1 for all f ∈ F. Then we can apply Proposition 3.5.15 in Giné and Nickl (2016), with F = 1 and δ = 1, and obtain
sup f ∈F E P n 1 n n i=1 f (Z i ) − E P [f (Z)] ≤ 58 √ n + 1 3n 2 0 log (2N B 2 ( , F, P ))d × 2 0 log (2N B 2 ( , F, P ))d . ≤ 58 √ n + 2 3n + 1 3n 2 0 H B 2 ( , F, P )d × 2 3 + 1 3 2 0 H B 2 ( , F, P )d .(47)
By combining the arguments from the proofs of Corollary B.3 and Proposition B.1 below, we have
H B 2 ( , F, P ) ≤ K −2r .
Substituting this upper bound into (47) yields
sup f ∈F E P n 1 n n i=1 f (Z i ) − E P [f (Z)] ≤ 58 √ n + 2 3n + 1 3n 2 0 K −r d × 2 3 + 1 3 2 0 K −r d = 58 √ n + 2 3n + 2 1−r K 3n(1 − r) 2 3 + 2 1−r K 3(1 − r) .
Therefore, setting
D 3 ≡ 116 3 + 29 · 2 2−r K 3(1 − r) , D 4 ≡ 2 3 + 2 1−r K 3(1 − r) 2 ,
leads to the result.
Proposition B.1. Let Z = (Z 1 , Z 2 ) ∼ P , and {Z i } n i=1 be a sequence of random variables that are i.i.d. as Z. Denote by P 2 the marginal distribution of Z 2 . Let F be a class of real-valued functions of the form f (z) = f (z 1 , z 2 ) = g (z 1 ) · h (z 2 ), where h ∈ H, H is a class of functions with values in [0, 1], and g takes values in [−1, 1]. Suppose H satisfies
H B 2 ( , H, P 2 ) ≤ K −r(48)
for some constants K > 0 and r ≥ 2 and for all > 0. Then there exist positive constants C 1, C 2 , C 3 , depending only on K and r, such that if
ξ ≤ √ n 128 (49) and ξ ≥ C 1 n (r−2)/2(r+2) C 2 log max (n, e) r ≥ 2 r = 2 ,(50)
then
P n sup f ∈F 1 √ n n i=1 (f (Z i ) − E P [f (Z i )]) > ξ ≤ C 3 exp −ξ 2 .
Proof. We follow the general strategy of Theorem 2.3 and Corollary 2.4 in Alexander (1984) and Proposition D.1 in Mbakop and Tabord-Meehan (2021).
Define
v n (f )≡ 1 √ n n i=1 [f (Z i ) − E (f (Z i ))] .
We start by giving some definitions. Let δ 0 > δ 1 > · · · > δ N > 0 be a sequence of real numbers where {δ j } N j=0 and N are specified later. For each δ j , there exists a set of δ j -brackets H B j of H with respect to L 2 (P 2 ) such that H B j = N B 2 (δ j , H, P 2 ). Define the function H(·) : (0, ∞) → [0, ∞) as follows:
H (u) = Ku −r 0 if u < 1 if u ≥ 1 .
Note that by Assumption (48) and the fact that H has unit diameter by definition, N B 2 (δ j , H, P 2 ) ≤ exp (H (δ j )) for all δ j > 0. For each 0 ≤ j ≤ N and any f = g · h ∈ F,
define f L j ≡g · h L j 1 {g ≥ 0} + g · h U j 1 {g < 0} and f U j ≡g · h U j 1 {g ≥ 0} + g · h L j 1 {g < 0} for some h L j , h U j that forms a δ j -bracket for h with respect to L 2 (P 2 ) such that h ∈ h L j , h U j and h L j , h U j ∈ H B j . From the construction, f U j , f L j
is a δ j -bracket for f with respect to L 2 (P ). Let f j = f L j , and let F j = {f j : f ∈ F}. We have |F j | ≤ exp (H (δ j )) and f − f j 2,P < δ j for every f ∈ F.
By a standard chaining argument,
P sup f ∈F |v n (f )| > ξ ≤ |F 0 | sup f ∈F P |v n (f )| > 7 8 ξ + N −1 j=0 |F j | |F j+1 | sup f ∈F P (|v n (f j − f j+1 )| > η j ) + P sup f ∈F |v n (f N − f )| > ξ 16 + η N ,
where {η j } N j=0 are to be chosen so as to satisfy N j=0 η j ≤ ξ/16. Define
R 1 = |F 0 | sup f ∈F P |v n (f )| > 7 8 ξ , R 2 = N −1 j=0 |F j | |F j+1 | sup f ∈F P (|v n (f j − f j+1 )| > η j ) , R 3 = P sup f ∈F |v n (f N − f )| > ξ 16 + η N .
We now choose {δ j } N j=0 , {η j } N j=0 and N to make the three terms sufficiently small. First we study R 1 . Set δ 0 such that H (δ 0 ) = ξ 2 /4. Then, applying Hoeffdings's inequality, we have
R 1 ≤ 2 |F 0 | exp −2 7 8 ξ 2 ≤ 2 exp −ξ 2 ,
where we use the fact that |F 0 | ≤ exp (H (δ 0 )) = exp (ξ 2 /4) in the second inequality. Next, we study R 2 . Since f j − f j+1 2,P ≤ 2δ j by construction, applying Bennet's inequality (Lemma B.5) to each sup f ∈F P (|v n (f j − f j+1 )| > η j ) in R 2 leads to
R 2 ≤ N −1 j=0 2 exp (2H (δ j+1 )) exp −ψ 1 η j , n, 4δ 2 j ,
where ψ 1 satisfies the properties described in Lemma B.5.
Next, we study R 3 . Given the construction of F N ,
|v n (f N − f )| ≤ v n f U N − f L N + 2 √ n f U N − f L L 1,P ≤ v n f U N − f L N + 2 √ nδ N . The last inequality holds because f U N − f L N 1,P ≤ h U N − h L N 1,P 2 and h U N − h L N 1,P 2 ≤ h U N − h L N 2,P 2 ≤ δ N ,
which holds from Hölder's inequality. Set δ N ≤ s≡ξ/ (32 √ n). Then, by the above derivation and Bennet's inequality,
R 3 ≤ P sup f ∈F v n f U N − f L N > η N ≤ 2 |F N | exp −ψ 1 η N , n, δ 2 N .
To develop upper bounds on R 2 and R 3 , we consider two distinct cases. First we consider the case δ 0 ≤ s. Set N = 0 and η 0 = ξ/16. Then we have that R 2 = 0 and
R 3 ≤ 2 |F 0 | exp −ψ 1 η 0 , n, δ 2 0 .
Since Assumption (49) and δ 0 ≤ s hold, we have
2η 0 = ξ 8 ≥ 4 √ n ξ 32 √ n 2 ≥ 4 √ nδ 2 0 .
Hence by the properties of ψ 1 specified in Lemma B.5,
ψ 1 η 0 , n, δ 2 0 ≥ 1 4 ψ 1 2η 0 , n, δ 2 0 ≥ 1 4 η 0 √ n.
Using η 0 = ξ/16 and Assumption (49), we obtain
ψ 1 η 0 , n, δ 2 0 ≥ 1 4 η 0 √ n = ξ 64 √ n ≥ 2ξ 2 .
By the definition of δ 0 , we also have |F 0 | ≤ exp (ξ 2 /4) . Therefore, combining these results gives
R 2 + R 3 ≤ 2 exp −ξ 2 .
Next we consider the case δ 0 > s. We here apply Lemma B.6, where we let N and {δ j } N j=0 be as in Lemma B.6 and t = δ 0 and s be as defined above. Let η j = 8 √ 2δ j H (δ j+1 ) 1/2 for 0 ≤ j < N and η N = 8 √ 2δ N H (δ N ) 1/2 . Then Lemma B.6 leads to
N j=0 η j = 8 √ 2 N j=0 H (δ j+1 ) 1/2 ≤ 64 √ 2 δ 0 s/4 H (u) 1/2 du.
We have that for 0 < s < t,
t s H (u) 1/2 du ≤ K 1/2 log (1/s) 2K 1/2 (r − 2) −1 s (2−r)/2 r = 2 r > 2.
Combining this with Assumption (50), where C 1 and C 1 are set to be sufficiently large, we have
N j=0 η j ≤ ξ 16 ,
which is consistent with our choice of {η j } j . Setting C 1 and C 2 sufficiently large, it follows from Assumption (50) that
H (s) ≤ ξ √ n 16 .
Hence we have
η j 4δ 2 j √ n 2 < 8H (s) ns 2 ≤ 16.
Then from the properties of ψ 1 ,
ψ 1 η j , n, 4δ 2 j ≥ η 2 j 16δ 2 j .
Using our bound on R 2 , we obtain
R 2 ≤ N −1 j=0 2 exp 2H (δ j+1 ) − η 2 j 16δ 2 j ≤ N −1 j=0 2 exp −4 j+1 H (δ 0 ) .
Similarly, we obtain
R 3 ≤ 2 exp −4 N +1 H (δ 0 ) .
Putting these results together and using Assumption (50), we have
R 2 + R 3 ≤ ∞ j=0 2 exp −4 j+1 H (δ 0 ) ≤ C exp −ξ 2 ,
where C is a constant that depends only on K and r.
Lemma B.5 (Bennet's inequality: see Theorem 2.9 in Boucheron et al. (2013)). Let
{Z i } n i=1
be a sequence of independent random vectors with distribution P . Let f be some function taking values in [0, 1] and define
v n (f )≡ 1 √ n n i=1 [f (Z i ) − E P (f (Z i ))] .
Then, for any ξ ≥ 0, the following holds:
P n (|v n (f )| > ξ) ≤ 2 exp (−ψ 1 (ξ, n, a)) ,
where a = var (v n (f )) and
ψ 1 (ξ, n, a) = ξ √ nh ξ √ nα ,
with h(x) = (1 + x −1 ) log (1 + x) − 1. Furthermore, ψ 1 has the following two properties:
ψ 1 (ξ, n, α) ≥ ψ 1 (Cξ, n, ρα) ≥ C 2 ρ −1 ψ 1 (ξ, n, α)
for C ≤ 1 and ρ ≥ 1, and
ψ 1 (ξ, n, α) ≥ ξ 2 4α ξ 2 √ n if ξ < 4 √ nα if ξ ≥ 4 √ nα .
Lemma B.6 (Lemma 3.1 in Alexander (1984)). Let H : (0, t] → R + be a decreasing function, and let 0 < s < t. Set δ 0 ≡t, δ j+1 ≡s ∨ sup {x ≤ δ j /2 : H(x) ≥ 4H (δ j )} for j ≥ 0, and N ≡ min {j : δ j = s}. Then
C Proof of the results in Section 6
This appendix provides proof of the results in Section 6. Throughout this appendix, we suppose X = [0, 1] dx as in Section 6. We first provide the proof of Lemma 6.1.
Proof of Lemma 6.1. Let µ X be the Lebesgue measure on X . From Theorem 8.3.2
in Dudley (1999), H B 1 ( , G M , µ X ) ≤ K dx−1 holds for some positive constant K and for all > 0. Since P X is absolutely continuous with respect to µ X and has a density that is bounded from above by A,
we have H B 1 (A −1 , G M , P X ) ≤ H B 1 ( , G M , µ X ). Thus result (i) follows by setting C = A −dx K.
The following lemma is used in the proof of Theorem 6.3.
Lemma C.1. Suppose that P X is absolutely continuous with respect to the Lebesgue measure on X and has a density that is bounded from above by a finite constant A > 0. Then there exists a constant C, which depends only on A, such that
H B 1 ( , F M , P X ) ≤ C −dx .
holds for all > 0. Gao and Wellner (2007) to F M , in which we setC = 2 −dx C 2 , where C 2 is the same constant that appears in Corollary 1.3 in Gao and Wellner (2007). Note that this corollary requires that P X is absolutely continuous with respect to the Lebesgue measure on X and has a bounded density.
Proof. Transform F M into F M = {(f + 1) /2 : f ∈ F M },
The following lemma gives finite upper bounds for two approximation errors:
inf f ∈B k R φ h (f ) − inf f ∈F M R φ h (f ) and R φ h (1{· ∈ Gf † B } − 1{· / ∈ Gf † B }) − R φ h (f † B ) in (24) with (F, F,f ) = (B k , F M ,f † B ).
Lemma C.2. Let k j ≥ 1, for j = 1, . . . , d x , be fixed. Suppose that the density of P X is bounded from above by some finite constant A > 0 .
(i) The following holds for the approximation error to the best classifier:
inf f ∈B k R φ h (f ) − inf f ∈F M R φ h (f ) ≤ 2A dx j=1 log k j k j + dx j=1 4 k j . (ii) Forf B ∈ arg inf f ∈B kR φ h (f )
such that the associated coefficients of the Bernstein bases take values in {−1, 1}, the following holds for the approximation error to the step function:
R φ h 1 · ∈ Gf B − 1 · / ∈ Gf B − R φ h (f B ) ≤ 2A dx j=1 log k j k j + dx j=1 4 k j .
The two approximation errors have the same upper bound which converges to zero as k j (j = 1, . . . , d x ) increases. The convergence rate is max j=1,...,dx (log k j )/k j . Note also that the upper bound on the approximation error to the step function does not depend on the sample size n.
The following two lemmas will be used in the proof of Lemma C.2.
Lemma C.3. Letf B ∈ arg inf f ∈B kR φ h (f ), andθ≡ θ j 1 ...j dx j 1 =1,.
..,k 1 ;...;j dx =1,...,k dx be the vector of the coefficients of the Bernstein bases inf B . Let r + 1 and r − 1 be the smallest non-negative value and the largest negative value inθ, respectively. (i) If all non-negative elements inθ take the same value r + 1 , let r + 2 be 1; otherwise, let r + 2 be the second smallest non-negative value inθ. Propose a (k 1 + 1) × · · · × (k dx + 1)dimensional vectorθ≡ Θ j 1 ...j dx j 1 =1,...,k 1 ;...;j dx =1,...,k dx such that, for all j 1 , . . . , j dx , ifθ j 1 ...j dx = r + 1 , Θ j 1 ...j dx = r + 2 ; otherwise, Θ j 1 ...j dx =θ j 1 ...j dx . Then a new classifier (ii) Similarly, if all negative elements inθ take the same value r − 1 , let r − 2 be −1; otherwise, let r − 2 be the second largest negative value inθ. Propose a (k 1 + 1) × · · · × (k dx + 1)-dimensional vectorθ≡ θ j 1 ...j dx j 1 =1,...,k 1 ;...;j dx =1,...,k dx such that, for all j 1 , . . . , j dx ,
f B (x)≡ k 1 j 1 =1 · · · k dx j dx =1 Θ j 1 ...j dx (b k 1 j 1 (x 1 ) × · · · × b k 1 j 1 (x dx )) minimizesR φ h (·) over B k .ifθ j 1 ...j dx = r − 1 ,θ j 1 ...j dx = r − 2 ; otherwise,θ j 1 ...j dx =θ j 1 ...j dx .
Then a new classifieř
f B (x)≡ k 1 j 1 =1 · · · k dx j dx =1θ j 1 ...j dx (b k 1 j 1 (x 1 ) × · · · × b k 1 j 1 (x dx )) minimizesR φ h (·) over B k . (iii) A classifier f † B (x)≡ k 1 j 1 =1 · · · k dx j dx=1 sign θ j 1 ...j dx · b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) minimizesR φ h (·) over B k .
Proof. First, note thatθ,θ ∈ Θ holds by construction. We now prove (i). The proof of (ii) follows using a similar argument. Define
L n (θ) ≡ n i=1 Y i · k 1 j 1 =1 · · · k dx j dx =1 θ j 1 ...j dx n i=1 b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) .
Minimization ofR φ h (·) over B k is equivalent to the maximization of L n (·) over Θ. Thus, θ maximizes L n (·) over Θ. We prove the result by contradiction. Supposeθ / ∈ arg max θ∈ Θ L n (θ). Let
J 1 ≡ (j 1 , . . . , j dx ) :θ j 1 ...j dx = r + 1 .
Then,
L n θ − L n θ = (j 1 ,...,j dx )∈J 1 r + 2 − r + 1 n i=1 Y i b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) < 0.
Since r + 2 − r + 1 ≥ 0, the above inequality implies that there exists some (j 1 , . . . , j dx ) ∈ J 1 such that n i=1 Y i b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) < 0. For such (j 1 , . . . , j dx ), settinĝ θ j 1 ...j dx to r − 1 can increase the value of L n θ without violating the constraints in Θ. But this contradicts the requirement thatθ j 1 ...j dx is non-negative. Therefore,θ maximizes L n (·) over Θ, or equivalentlyf B minimizesR φ h (·) over B k .
Result (iii) is shown by applying Lemma C.3 (i) and (ii) repeatedly tof B .
Lemma C.4. Fix G ∈ G and k j ≥ 1 for j = 1, . . . , d x . Define a classifier
f G (x)≡ k 1 j 1 =1 · · · k dx j dx =1 θ j 1 ...j dx b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) ,
such that, for all j 1 , . . . , j dx ,
θ j 1 ...j dx = 1 −1 if (j 1 /k 1 , . . . , j dx /k dx ) ∈ G if (j 1 /k 1 , . . . , j dx /k dx ) / ∈ G .
Then the following holds:
|R φ h (f G ) − R φ h (1 {· ∈ G} − 1 {· / ∈ G})| ≤ 2A dx j=1 log k j k j + dx j=1 4 k j .
Proof. Define
J k ≡ {(j 1 , . . . , j dx ) : (j 1 /k 1 , . . . , j dx /k dx ) ∈ G} ,
which is a set of grid points on G. It follows that
R φ h (f G ) − R φ h (1 {· ∈ G} − 1 {· / ∈ G}) = [0,1] dx (2η(x) − 1) (1 {x ∈ G} − 1 {x / ∈ G} − B k (θ, x)) dP X (x) = [0,1] dx (2η(x) − 1) 1 {x ∈ G} dP X (x) − [0,1] dx (2η(x) − 1) 1 {x / ∈ G} dP X (x) − [0,1] dx (2η(x) − 1) B k (θ, x) dP X (x) (I) .(51)
(I) can be written as
(I) = [0,1] dx (2η(x) − 1) (j 1 ,...,j dx )∈J k dx v=1 b kvjv (x v ) dP X (x) − [0,1] dx (2η(x) − 1) (j 1 ,...,j dx ) / ∈J k dx v=1 b kvjv (x v ) dP X (x).
Thus,
= [0,1] dx (2η(x) − 1) 1 {x ∈ G} − (j 1 ,...,j dx )∈J k dx v=1 b kvjv (x v ) dP X (x)(51)+ [0,1] dx (2η(x) − 1) (j 1 ,...,j dx ) / ∈J k dx v=1 b kvjv (x v ) − 1 {x ∈ G c } (III) dP X (x).(II)
Let Bin (k j , x j ), j = 1, . . . , d x , be independent binomial variables with parameters (k j , x j ). Then, both (II) and (III) are equivalent to Pr ( (Bin(k 1 , x 1
), . . . , Bin(k dx , x dx )) ∈ J c k ) 1 {x ∈ G} − Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) 1 {x ∈ G c } .
Hence,
(51) = 2 G (2η(x) − 1) Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x). − 2 G c Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x),
and therefore
|R φ h (f G ) − R φ h (1 {· ∈ G} − 1 {· / ∈ G})| ≤ 2 G Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x) (IV ) + 2 G c Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x) (V )
.
We first evaluate (V). Let be a small positive value which converges to zero as k v → ∞.
For small ∆ v ≤ / √ d x , v = 1, . . . , d x , which converges to zero as k v → ∞, definẽ G c ≡ {x ∈ G c : (x 1 + ∆ 1 , . . . , x dx + ∆ dx ) ∈ G c } .
This set is either nonempty or empty. We consider these cases separately. First, suppose thatG c is nonempty. For each x ∈G c , let (j 1 (x), . . . , j dx (x)) ∈ arg min (j 1 ,...,j dx )∈J k :j 1 /k 1 ≥x 1 +∆ 1 ,...,j dx /k dx ≥x dx +∆ dx
x − (j 1 /k 1 , . . . , j dx /k dx ) .
Then
(V ) ≤ G c \G c
Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x)
+ Gc dx v=1 Pr (Bin(k dx , x dx ) ≥ j v (x)) dP X (x) ≤ A · (∆ 1 + · · · + ∆ dx ) + Gc dx v=1 exp −2k v j v (x) k v − x v 2 dP X (x) ≤ A · (∆ 1 + · · · + ∆ dx ) + dx v=1 Gc exp −2k v ∆ 2 v dP X (x).
To obtain the second inequality, we apply Hoeffding's inequality to Pr (Bin(k dx ,
x dx ) ≥ j v (x)), which is applicable since k v x v ≤ j v (x)
for each x ∈G c , and use the following:
G c \G c Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x) ≤ A G c \G c dx ≤ A · (∆ 1 + · · · + ∆ dx ) ,
where the second inequality holds because G c \G c dx is bounded from above by dx
v=1 ∆ v − (d x − 1) dx v=1 ∆ v , which is derived when G c = X . The last inequality follows from the fact that j v (x)/k v − x v ≥ ∆ v for all v = 1, . . . , d x and x ∈G .
Next, we consider the case thatG c is empty. In this case,
(V ) = G c Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x) ≤ G c dP X (x) ≤ A · max v=1,...,dx ∆ v .
The inequality follows because G c dx is bounded from above by max v=1,...,dx ∆ v whenG c is empty. Therefore, regardless of whetherG c is empty or not, we have
(V ) ≤ A · (∆ 1 + · · · + ∆ dx ) + dx v=1 Gc exp −2k v ∆ 2 v dP X (x). Set ∆ v = √ log k v / 2 √ k v for each v = 1, . . . , d x . Then we have (V ) ≤ A 2 dx v=1 log k v k v + dx v=1 exp − 1 2 log k v = A 2 dx v=1 log k v k v + dx v=1 1 √ k v .
Next, we evaluate (IV). For small ∆ v ≤ / √ d x , v = 1, . . . , d x , which converges to zero
as k v → ∞, defineG ≡ {x ∈ G : (x 1 − ∆ 1 , . . . , x dx − ∆ dx ) ∈ G} .
We again separately consider the two cases:G is nonempty or empty. First, suppose that G is nonempty. For each x ∈G, let
j 1 (x), . . . ,j dx (x) ∈ arg min (j 1 ,...,j dx )∈(J k ) c :j 1 /k 1 ≤x 1 −∆ 1 ,...,j dx /k dx ≤x dx −∆ dx x − (j 1 /k 1 , . . . , j dx /k dx ) .
Then,
(IV ) ≤ G\G Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x) + G dx v=1 Pr Bin(k dx , x dx ) ≤j v (x) dP X (x) ≤ A · (∆ 1 + · · · + ∆ dx ) + G dx v=1 exp −2k v x v −j v (x) k v 2 dP X (x) ≤ A · (∆ 1 + · · · + ∆ dx ) + dx v=1 G exp −2k v ∆ 2 v dP X (x).
The second inequality follows from Hoeffding's inequality and the fact that G\G Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x)
≤ A G\G dx ≤ A · (∆ 1 + · · · + ∆ dx ) ,
where the inequality holds because G\G dx attains its largest value, dx v=1 ∆ v − dx v=1 ∆ v , when G = X . The last inequality follows from the fact that
j v (x)/k v ≤ x v − ∆ v for all v = 1, . . . d x and x ∈G .
Next, we consider the case thatG is empty. In this case,
(IV ) = G Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x) ≤ G dP X (x) ≤ A · max v=1,...,dx ∆ v ,
where the inequality follows because G dx is bounded from above by max v=1,...,dx ∆ v when G is empty. Therefore, regardless of whetherG is empty or not, we have
(IV ) ≤ A · (∆ 1 + · · · + ∆ dx ) + dx v=1 G exp −2k v ∆ 2 v dP X (x). Set ∆ v = √ log k v / 2 √ k v for each v = 1, . . . , d x . Then, we have (IV ) ≤ A 2 dx v=1 log k v k v + dx v=1 exp − 1 2 log k v = A 2 dx v=1 log k v k v + dx v=1 1 √ k v .
Consequently, combining above the results, we obtain
|R φ h (f G ) − R φ h (1 {· ∈ G} − 1 {· / ∈ G})| ≤ 2A dx v=1 log k v k v + dx v=1 4 √ k v .
Finally, the following provides the proof of Lemma C.2.
Proof of Lemma C.2. We first prove (i). Let G * minimize R(·) over G M . From Theorem 4.4, a classifierf * (x)≡1 {x ∈ G * }−1 {x ∈ (G * ) c } minimizes the hinge risk R φ h (·) over F M . Define a vector θ * = θ * j 1 ...j d j 1 =0,...,k 1 ;...;j d =0,...,k d such that for each j 1 , . . . , j d ,
θ * j 1 ...j d = 1 −1 if (j 1 /k 1 , . . . , j d /k d ) ∈ G * otherwise.
Note that θ * is contained by Θ. Thus, it follows that
inf f ∈B k R φ h (f ) − R φ h (f * ) ≤ R φ h (B k (θ * , ·)) − R φ h (f * ).
Then, applying Lemma C.4 to R φ h (B k (θ * , ·)) − R φ h (f * ) yields result (i).
Result ( Let P n denote the joint distribution of the sample of n observations. Given the sample, the empirical weighted classification risk and hinge risk for a classifier f are defined aŝ
R ω (f ) ≡ n −1 1 i=1 ω i 1{Y i · sign(f (X i )) ≤ 0}, R ω φ h (f ) ≡ n −1 1 i=1 ω i max{0, 1 − Y i f (X i )},
respectively. LetF be a subclass of F G , on which we learn a best classifier, and F G be a constrained classification-preserving reduction of F G in the sense of Definition 4.2.
As an analogue of Theorems 5.1 and 5.2, the following theorem gives general upper bounds on the mean of the G-constrained excess weighted classification risk.
Theorem D.1. Suppose that F G is a subclass of F G satisfying conditions (A1) and (A2) in Theorem 4.4. Letf ∈ arg inf f ∈FR ω φ h (f ), and (q n , τ n , L C (r, n)) be as in Theorem 5.1. (i) Let P be a class of distributions on R + × {−1, 1} × X such that, for any P ∈ P, Condition 7.4 holds and there exist positive constants C and r for which condition (19) holds for all > 0 or condition (20) holds for all > 0. Then ifF = F G ,
sup P ∈P E P n R ω (f ) − inf f ∈F G R ω (f ) ≤ M L C (r, n).
(ii) Suppose that P is a class of distributions on R + × {−1, 1} × X such that, for any P ∈ P, Condition 7.4 holds and there exist positive constants C and r for which condition (23) holds for all > 0. Then the following holds:
sup P ∈P E P n R ω (f ) − inf f ∈F G R ω (f ) ≤ M L C (r , n) + 1 2 inf f ∈F R ω φ h (f ) − inf f ∈ F G R ω φ h (f ) + 1 2 R ω φ h (f Gf ) − R ω φ h (f ) .(52)
Proof. See Appendix E.
Similar comments as in Remark 5.3 apply to Theorem D.1 (ii). The two approximation errors in (52) are small asF is a good approximation of F G .
D.2 Monotone weighted classification
Finally, we extend the results for monotone classification in Section 6 to weighted classification. Let F G M , F M , and B k be as in Section 6, and suppose X = [0, 1] dx . Our aim is to find a best classifier that minimizes R ω (·) over F G M . We again consider using the whole class of monotone classifiers F M and sieve of Bernstein polynomials B k in the empirical hinge risk minimization for weighted classification.
The following theorem shows the statistical properties of monotone weighted classification using F M .
Theorem D.2. Let P be a class of distributions of (ω, Y, X) such that Condition 7.4 holds for any P ∈ P, and that for any P ∈ P the marginal distribution P X is absolutely continuous with respect to the Lebesgue measure on X and has a density that is bounded from above by some finite constant A > 0. Let q n and τ n be as in Theorem 6.2, and let f M ∈ arg inf f ∈F MR ω φ h (f ). Then the following holds:
sup P ∈P E P n R ω (f M ) − inf f ∈F G M R ω (f ) ≤ 2M D 1 τ n + 4M D 2 exp (−D 2 1 q 2 n ) 2M D 3 τ n + 2M n −1 D 4 if d x ≥ 2 if d x = 1
for some positive constants D 1 and D 2 , which depend only on d x and A.
Proof. Since F M satisfies conditions (A1) and (A2) in Theorem 4.4 with G being G M (Example 4.7), the result follows from Theorem D.1 (i) and Lemma 6.1.
The classifierf M that minimizes R ω φ h (·) over F M can be obtained by solving the linear program in (25)
with n i=1 Y i f i replaced by n i=1 ω i Y i f i .
For monotone weighted classification using Bernstein polynomials B k , similarly to Section 6.2, we propose usinĝ
f † B (x) ≡ k 1 j 1 =1 · · · k dx j dx=1 sign θ j 1 ...j dx · b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) ,
where {θ j 1 ...j dx } j 1 =0,...,k 1 ;··· ;j dx =0,...,k dx is the vector of coefficients characterizing somef B ∈ arg inf f ∈B kR ω φ h (f ).f † B converts each estimated coefficientθ j 1 ...j dx inf B to either −1 or 1 depending on its sign. The following theorem states the statistical properties of monotone weighted classification usingf † B .
Theorem D.3. Let P be a class of distributions of (ω, Y, X) that satisfies the same conditions as in Theorem D.2. Letq n andτ n be as in Theorem 6.3 . Then the following holds:
sup P ∈P E P n R ω (f † B ) − inf f ∈F G M R ω (f ) ≤ 2M D 1τn + 4M D 2 exp −D 2 1q 2 n + 4M A dx j=1 log k j k j + dx j=1 8M k j ,
where D 1 and D 2 are the same constants as in Theorem D.2, which depend only on d x and A.
Proof. The result follows by combining Theorem D.1 (ii), Lemma C.1 in Appendix C, and Lemma E.5 in Appendix E.
Similar comments as those in Section 6 apply to Theorems D.2 and D.3. Using F M leads to the faster convergence rate than using the Bernstein polynomials B k . When using the Bernstein polynomials B k , to achieve the convergence rate ofτ n for the excess risk, Theorem D.3 suggests setting the tuning parameters k j , j = 1, . . . , d x , sufficiently large so that log k j /k j = O (τ n ).
E Proof of the results in Section 7 and Appendix D
This section provides proof of the results in Section 7 and Appendix D for weighted classification. Most of the various proofs are natural extensions of the proofs of the results in Sections 3-6. For simplicity of notation, define a function
L(ω + , ω − , η) ≡ −ω + η + ω − (1 − η),
the right hand side of which appears in (28). Similar to (9) and (10), the following expressions are useful for the proofs in this appendix:
R ω (G) = G L(ω + (x), ω − (x), η(x))dP X (x) + X ω + (x)η(x)dP X (x),(53)R ω φ (G) = G ∆C ω φ (ω + (x), ω − (x), η(x)) dP X (x) + X C w− φ (ω + (x), ω − (x), η(x)) dP X (x).(54)
We first state the proofs of Theorem 7.1 and Corollary 7.2.
Proof of Theorem 7.1. We first prove the 'if' part of the first statement. Fix P , G, and G 1 , G 2 ∈ G. By equation (54),
R ω φ (G 1 ) − R ω φ (G 2 ) = G 1 \G 2 ∆C ω φ (ω + (x), ω − (x), η(x)) dP X (x) − G 2 \G 1 ∆C ω φ (ω + (x), ω − (x), η(x)) dP X (x). Thus, R ω φ 1 (G 1 ) ≥ R ω φ 1 (G 2 ) is equivalent to G 1 \G 2 ∆C ω φ 1 (ω + (x), ω − (x), η(x)) dP X (x) ≥ G 2 \G 1 ∆C ω φ 1 (ω + (x), ω − (x), η(x)) dP X (x).
Replacing ∆C ω φ 1 by ∆C ω φ 2 = c∆C ω φ 1 with c > 0 does not change the above inequality. Moreover, replacing ∆C ω φ 1 with ∆C ω φ 2 in the above inequality is equivalent to R ω φ 2 (G 1 ) ≥ R ω φ 2 (G 2 ). Therefore, since the above discussion holds for any P , G, and G 1 , G 2 ∈ G, the 'if' part of the first statement holds.
The 'only if' part of the first statement of the theorem follows directly from Theorem 3.6 upon setting ∆C ω
φ (ω + (x), ω − (x), η(x)) = ∆C φ (η(x)), or equivalently ω + (x) = ω − (x) = 1, for all x ∈ X .
We next prove the second statement, or equivalently that ∆C ω φ 01 (ω + , ω − , η) = L(ω + , ω − , η) holds for all (ω + , ω − , η) ∈ R + × R + × [0, 1]. For f ∈ F G , it follows that
C ω φ 01 (ω + , ω − , f, η) = ω + 1{f ≤ 0}η + ω − 1{f > 0} (1 − η) , ∆C w+ φ 01 (ω + , ω − , f, η) = ω − (1 − η), ∆C w− φ 01 (ω + , ω − , f, η) = ω + η.
Thus, we have ∆C ω φ 01 (ω + , ω − , f, η) = ∆C w+ φ 01 (ω + , ω − , f, η) − ∆C w− φ 01 (ω + , ω − , f, η) = L(ω + , ω − , η).
Finally, we prove the last statement. For the hinge loss function φ h (α) = c max {0, 1 − α} and f ∈ F G , we have
C φ h (ω + , ω − , f, η) = c (ω + (1 − f ) η + ω − (1 + f ) (1 − η)) .
Hence, we obtain
∆C w+ φ h (ω + , ω − , η) = 2cω − (1 − η) c (ω + η + ω − (1 − η))
for L (ω + , ω − , η) < 0 for L (ω + , ω − , η) ≥ 0 , ∆C ω φ h (ω + , ω − , η) = c (ω + η + ω − (1 − η)) 2cω + η for L (ω + , ω − , η) < 0 for L (ω + , ω − , η) ≥ 0 . Therefore, ∆C ω φ h (ω + , ω − , η) = cL(ω + , ω − , η) holds for all (ω + , ω − , η) ∈ R + × R + × [0, 1].
Proof of Corollary 7.2. Equation (29) follows from
R ω (f ) − inf f ∈F G R ω (f ) = R ω (G f ) − R ω (G * ) = X L(ω + (x), ω − (x), η(x)) (1{x ∈ G f } − 1{x ∈ G * }) dP X (x) = c −1 X ∆C ω φ (ω + (x), ω − (x), η(x)) (1{x ∈ G f } − 1{x ∈ G * }) dP X (x) = c −1 R ω φ (G f ) − R ω φ (G * ) = c −1 inf f ∈F G f R ω φ (f ) − inf f ∈F G R ω φ (f ) ≤ c −1 R ω φ (f ) − inf f ∈F G R ω φ (f ) ,
where the first equality follows from (53); the second equality follows from the assumption; the third equality follows from (54).
We next provide the proof of Theorem 7.5. Beforehand, note that the weighted hinge risk can be expressed as
R ω φ h (f ) = X (ω + (x) (1 − f (x)) η(x) + ω − (x)
(1 + f (x)) (1 − η(x))) dP X (x) = X L(ω + (x), ω − (x), η(x))f (x)dP X (x) + E P [ω].
Moreover, for G ∈ G, R(G) can be written as
R ω (G) = X (ω + (x)η(x)1 {x ∈ G c } + ω − (x) (1 − η(x)) 1 {x ∈ G}) dP X (x) = − G c L(ω + (x), ω − (x), η(x)) (1 − η(x)) dP X (x) + X ω − (x) (1 − η(x)) dP X (x).(56)
The following lemma, which is an analogue of Lemma 4.3, will be used in the proof of Theorem 7.5.
Lemma E.1. Let G ⊆ 2 X be a class of measurable subsets and F G,J be defined as in (15). (i) Letf * be a minimizer of the weighted hinge risk R ω φ h (·) over F G,J . Thenf * minimizes where the last line comes from c − statement (ii) of the theorem. We will next prove statement (i) of the theorem. Letf * ∈ arg inf f ∈ F G R ω φ h (f ). To obtain a contradiction, suppose thatf * does not minimize R ω (·) over F G , or equivalently Gf * / ∈ G * . Then, from equation (57), for any J andf ∈ F * J , R ω φ h (f ) > 2R w * holds. Therefore, from equation (59),
R ω φ h (f * ) ≥ lim J→∞ inf f ∈ F * J R ω φ h (f ) > 2R w * .
This contradicts the assumption thatf * minimizes R ω φ h over F G because R ω φ h (f G * ) = 2R w * .
The following corollary shows a similar relationship between the G-constrained excess weighted-classification risk and F G -constrained excess weighted-hinge risk as is present in Corollary 4.5.
Hence, to obtain the inequality in (52), we need to prove that
R ω φ h (f ) − inf f ∈F R ω φ h (f ) ≤ L C (r, n).
We follow the same strategy as the proof of Theorem B.2. Letf * minimizes R ω φ h (·) oveř F. A standard argument gives
E P n R ω φ h (f ) − inf
H =Ḟ, shows that there exist D 1 , D 2 , D 3 > 0, depending only on r and C, such that
P n sup f ∈F √ n q n E P ω M Y f (x) − 1 n n i=1 ω i M Y i f (X i ) > D ≤ D 2 exp −D 2 q 2 n ,
for D 1 ≤ D ≤ D 3 √ n/q n . Therefore, when r ≥ 1, we have
τ −1 n E P n R ω φ h (f ) − inf f ∈F R ω φ h (f ) ≤ 4M D 1 + 8M τ −1 n D 2 exp −D 2 1 q 2 n .
Combining this result with (60) leads to the inequality in (52) for the case of r ≥ 1. The inequality in (52) for the case of r < 1 follows immediately by applying Lemma B.4 to equation (61).
Proof of Theorem D.1 (i). Let P ∈ P be fixed. We follow the same strategy as in the proof of Theorem 5.1. Definef † (x) = 1{x ∈ Gf } − 1{x / ∈ Gf }. Then equation (60) becomes
R ω (f ) − inf f ∈F G R ω (f ) = R ω (f † ) − inf f ∈F G R ω (f ) = 1 2 R ω φ h (f † ) − inf f ∈ F G R ω φ h (f ) .
It follows that
R ω φ h (f † ) − inf f ∈ F G R ω φ h (f ) = E P [ωYf † (X)] − inf f ∈ F G E P [ωY f (X)] ≤ M E P [Yf † (X)] − inf f ∈ F G E P [Y f (X)] = M R φ h (f † ) − inf f ∈ F G R φ h (f ) ,
where the third line follows because F G is a classification-preserving reduction of F G and, accordingly,
E P [Yf † (X)] ≥ inf f ∈ F G E P [Y f (X)] holds. Thus we have R ω (f ) − inf f ∈F G R ω (f ) ≤ M R φ h (f † ) − inf f ∈ F G R φ h (f ) .
Then the result follows by applying the same argument in the proof of Theorem 5.1 to the above equation.
The following are extensions of Lemmas C.3, C.4, and C.2.
Lemma E.3. Letf B ∈ arg inf f ∈B kR ω φ h (f ), andθ≡ θ j 1 ...j dx j 1 =1,...,k 1 ;...;j dx =1,...,k dx be the vector of the coefficients characterizingf B . Let r + 1 and r − 1 be the smallest non-negative value and the largest negative value inθ, respectively. (i) If all non-negative elements inθ take the same value r + 1 , let r + 2 be 1; otherwise, let r + 2 be the second smallest non-negative value inθ. Propose a (k 1 + 1) × · · · × (k dx + 1)dimensional vectorθ≡ Θ j 1 ...j dx j 1 =1,...,k 1 ;...;j dx =1,...,k dx such that for all j 1 , . . . , j dx ifθ j 1 ...j dx = r + 1 , Θ j 1 ...j dx = r + 2 ; otherwise, Θ j 1 ...j dx =θ j 1 ...j dx . Then, a classifier (ii) Similarly, if all negative elements inθ take the same value r − 1 , let r − 2 be −1; otherwise, let r − 2 be the second largest negative value inθ. Propose a (k 1 + 1) × · · · × (k dx + 1)-vectoř θ≡ θ j 1 ...j dx j 1 =1,...,k 1 ;...;j dx =1,...,k dx such that for all j 1 , . . . , j dx ifθ j 1 ...j dx = r − 1 ,θ j 1 ...j dx = r − 2 ; otherwise,θ j 1 ...j dx =θ j 1 ...j dx . Then, a classifieř
f B (x)≡ k 1 j 1 =1 · · · k dx j dx =1 Θ j 1 ...j dx (b k 1 j 1 (x 1 ) × · · · × b k 1 j 1 (x dx )) minimizesR φ h over B k .f B (x)≡ k 1 j 1 =1 · · · k dx j dx =1θ j 1 ...j dx (b k 1 j 1 (x 1 ) × · · · × b k 1 j 1 (x dx )) minimizesR ω φ h over B k . (iii) A classifier f † B (x)≡ k 1 j 1 =1 · · · k dx j dx=1 sign θ j 1 ...j dx · b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) minimizesR φ h (·) over B k .
Proof. First, note thatθ,θ ∈ Θ holds by construction. We here prove (i). The proof of (ii) follows using a similar argument. Define
L n (θ) = n i=1 ω i Y i k 1 j 1 =1 · · · k dx j dx =1 θ j 1 ...j dx n i=1 b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) .
Minimization ofR φ h over B k is equivalent to the maximization of L n (θ) over Θ. Thus,θ maximizes L n (θ) over Θ.
We prove the result by contradiction. Supposeθ / ∈ arg max θ∈ Θ L n (θ). Let J 1 ≡ (j 1 , . . . , j dx ) :θ j 1 ...j dx = r + 1 .
Then, L n θ − L n θ = (j 1 ,...,j dx )∈J 1
r + 2 − r + 1 n i=1 ω i Y i b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) < 0
holds. Since r + 2 − r + 1 ≥ 0, the above equation implies that there exists some (j 1 , . . . , j dx ) in J 1 such that n i=1 ω i Y i b k 1 j 1 (X 1i ) × · · · × b k dx j dx (X dxi ) < 0. For such (j 1 , . . . , j dx ), settingθ j 1 ...j dx to r − 1 increases the value of L n θ without violating the constraints in Θ. But this contradicts the requirement thatθ j 1 ...j dx is non-negative. Therefore,θ maximizes L n (θ) over Θ, or equivalentlyf B minimizesR φ h over B k .
Result (iii) follows by applying results (i) and (ii) repeatedly tof B .
Lemma E.4. Fix G ∈ G and k 1 , . . . , k dx . Define a classifier
f G (x) = k 1 j 1 =1 · · · k dx j dx =1 θ j 1 ...j dx b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx ) ,
such that, for all j 1 , . . . , j dx , θ j 1 ...j dx = 1 if (j 1 /k 1 , . . . , j dx /k dx ) ∈ G, and θ j 1 ...j dx = −1 otherwise. Then the following holds:
R ω φ h (f G ) − R ω φ h (1 {· ∈ G} − 1 {· / ∈ G}) ≤ 2M A dx j=1 log k j k j + dx j=1 4M k j .
Proof. Define J k ≡ {(j 1 , . . . , j dx ) : (j 1 /k 1 , . . . , j dx /k dx ) ∈ G} , which is a set of grid points on G, and L(x) ≡ −ω + (x)η(x) + ω − (x)(1 − η(x)).
It follows that
R ω φ h (f G ) − R ω φ h (1 {· ∈ G} − 1 {· / ∈ G}) = [0,1] dx L(x) (1 {x ∈ G} − 1 {x / ∈ G} − B k (θ, x)) dP X (x) = [0,1] dx L(x)1 {x ∈ G} dP X (x) − [0,1] dx L(x)1 {x / ∈ G} dP X (x) − [0,1] dx L(x)B k (θ, x) dP X (x) (I) .(62)
(I) can be written as
(I) = [0,1] dx L(x) (j 1 ,...,j dx )∈J k dx v=1 b kvjv (x v ) dP X (x) − [0,1] dx L(x) (j 1 ,...,j dx ) / ∈J k dx v=1 b kvjv (x v ) dP X (x).
Thus, Let Bin (k j , x j ), j = 1, . . . , d x , be independent binomial variables with parameters (k j , x j ). Then (II) and (III) are equivalent to Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) 1 {x ∈ G} − Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) 1 {x ∈ G c } . (k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x)
− 2 G c L(x) Pr ((Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x), and therefore
R ω φ h (f G ) − R ω φ h (1 {· ∈ G} − 1 {· / ∈ G}) ≤ 2M G
Pr ( (Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J c k ) dP X (x)
(IV ) + 2M G c
Pr ( (Bin(k 1 , x 1 ), . . . , Bin(k dx , x dx )) ∈ J k ) dP X (x) (V ) , because |L(x)| < M for all x ∈ X . The proof of Lemma C.4 shows that
(IV ) ≤ A 2 dx v=1 log k v k v + dx v=1 1 √ k v , (V ) ≤ A 2 dx v=1 log k v k v + dx v=1 1 √ k v .
Therefore,
R ω φ h (f G ) − R ω φ h (1 {· ∈ G} − 1 {· / ∈ G}) ≤ 2M A dx v=1 log k v k v + dx v=1 4M √ k v .
Lemma E.5. Let k j ≥ 1, for j = 1, . . . , d x , be fixed. Suppose that the density of P X is bounded from above by A > 0 . Suppose further that k = (k 1 , . . . , k dx ) satisfies d x log k j / 2 k j ≤ for all j = 1, . . . , d x and some > 0. (i) The following holds for the approximation error to the best classifier:
inf f ∈B k R ω φ h (f ) − inf f ∈F M R ω φ h (f ) ≤ 2AM dx j=1 log k j k j + dx j=1 4M k j .
(ii) Forf B ∈ arg inf f ∈B kR ω φ h (f ) such that its coefficients take values in {−1, 1}, the following holds for the approximation error to the step function:
R ω φ h 1 · ∈ Gf B − 1 · / ∈ Gf B − R ω φ h (f B ) ≤ 2AM dx j=1 log k j k j + dx j=1 4M k j .
Proof. We first prove (i). Let G * minimize R ω (·) over G M . From Theorem 7.5, a classifier f * (x)≡1 {x ∈ G * } − 1 {x ∈ (G * ) c } minimizes the hinge risk R ω φ h (·) over F M . Define a vector θ * = θ * j 1 ...j d j 1 =0,...,k 1 ;...;j d =0,...,k d such that for each j 1 , . . . , j d ,
θ * j 1 ...j d = 1 −1
if (j 1 /k 1 , . . . , j d /k d ) ∈ G * otherwise.
Note that θ * is contained in Θ. Thus, it follows that
inf f ∈B k R ω φ h (f ) − inf f ∈F M R ω φ h (f ) ≤ R ω φ h (B k (θ * , ·)) − R ω φ h (f * ).
Then, applying Lemma E.4 to R ω φ h (B k (θ * , ·)) − R ω φ h (f * ) establishes result (i). The inequality in Lemma E.5 (ii) follows immediately from Lemma E.4. Applying Lemma E.3 (iii) to anyf B ∈ arg inf f ∈B kR φ h (f ) shows that a classifier
f † B (x) = k 1 j 1 =1 · · · k dx j dx =1 sign θ j 1 ...j dx b k 1 j 1 (x 1 ) × · · · × b k dx j dx (x dx )
minimizesR ω φ h (·) over B k , which proves the existence off B ∈ arg inf f ∈B kR ω φ h (f ) such that its coefficients take values in {−1, 1}.
Example 3. 1 (
1Numerical example 1). Let X = {0, 1, 2} and G = {∅, {2}, {2, 1}, {2, 1, 0}}.
Figure 1 :
1Monotone classifiers minimizing classification and surrogate risksNotes: The square points corresponds to the values of f * (x) at x = 0, 1, and 2. The circular points correspond to the values of each of f * φ h (x), f * φe (x), and f * φq (x) at x = 0, 1, and 2.
Example 4 . 1 (
41Numerical example 2). Maintain X = {0, 1, 2} and G = {∅, {2}, {2, 1}, {2, 1, 0}}
Figure 2 :
2Linear monotone classifiers minimizing classification and hinge risksNote: The orange and blue lines are the graphs of the computed classifiers, f * and f * φ h , respectively.
Proof. See Appendix B.The upper bound on the mean of the G-constrained excess classification risk converges 5 With a slight abuse of notation, we denote by H B 1 ( , G, P X ) the bracketing entropy number of the class of indicator functions, H B 1 ( , H G , P X ), where H G ≡ {1{· ∈ G} : G ∈ G}.
Condition 7 . 4 (
74Bounded weight variable). There exists M < ∞ such that 0 ≤ ω ≤ M a.s.
Definition B. 1 (
1Bracketing entropy). (i) Let F be a class of functions on X . For f ∈ F,
H is a class of functions with values in [0, 1], and g takes values in [−1, 1]. Suppose H satisfies H B 1 ( , H, P 2 ) ≤ K −r
H (δ j+1 ) 1/2 ≤ 8 t s/4 H(x) 1/2 dx.
which is a class of monotonically increasing functions taking values in [0, 1]. Following this transformation, N B 1 ( , F M , P X ) = N B 1 /2, F M , P X holds. Then the result follows by applying Corollary 1.3 in
ii) follows immediately from Lemmas C.3 (iii) and C.4. D Statistical properties of weighted classification and their application to weighted classification D.1 Statistical properties of weighted classification with hinge loss This section extends the analysis of Section 5 to weighted classification with hinge losses. Let {(ω i , Y i , X i ) : i = 1, . . . , n} be a sample of observations that are i.i.d. as (ω, Y, X).
{x ∈ G} − (j 1 ,...,j dx )∈J k b kvjv (x v ) 1 ,...,j dx ) / ∈J k b kvjv (x v ) − 1 {x ∈ G c
Table 1 :
1Surrogate loss functions and their associated forms for ∆C φ
the Bernstein basis. The Bernstein polynomial for a d x -dimensional function takes the following form:
Table 2 :
2Surrogate loss functions and their associated forms for ∆C wφ
Kitagawa and Lin (2021) makes use of this transformation of the welfare objective function to develop an Adaboost algorithm for treatment choice.
Bartlett et al. (2006) also show that any convex loss function φ is classification-calibrated if and only if it is differentiable at 0 and φ (0) < 0.
Given a convex classification-calibrated loss function φ, our proof of Proposition 2.1(ii) in Appendix A constructs a pair comprising a R-correctly specified class of classifiers F and a distribution P that leads to R φ -misspecification. In the construction, we assume that x 1 = x 2 ∈ X supported by P X on which φ(f (x 1 )) < φ(−f (x 2 ))) holds for all f ∈ F and that f (x 1 ) < 0 ≤ f (x 2 ) holds for some f ∈ F, and consider P that specifies a value of η(x 2 ) close to 1, and a value of η(x 1 ) slightly below 1/2. Such a construction of P is not pathological or limited to the specific class of classifiers considered in the proof.
Fix β ∈ R dx and k ∈ R + . The condition (A1) is satisfied as, for any t ∈ [−1, 1], {x : π(x T β, k) ≤ t} = {x : x T β ≤ π −1 (t, k)} ∈ G, where π −1 (·, k) is an inverse function of π(·, k) with fixed k.
We define the partial order ≤ on X as follows. For any x = (x 1 , . . . , x d ) T andx = (x 1 , . . . ,x d ) T , we say x ≤x if x j ≤x j for every j = 1, . . . , d. We further say x <x if x ≤x holds and for some j ∈ {1, . . . , d}, x j <x j holds.
See, e.g.,Wang and Ghosh (2012) for the bound and shape preserving properties of the multivariate Bernstein polynomials.
The linear program in (25) for the nonparametric monotone classification problem has n-decision variables, whereas the linear program in (26) has (k 1 + 1) × · · · × (k dx + 1)-decision variables. Hence when the dimension of X is small to moderate relative to the sample size n, the linear programming for the Bernstein polynomials would be easier to compute. The reverse is also true.
R φ h (f ),(42)where the first and third equalities follow from equation(37); the second equality follows from the dominated convergence theorem, which holds because both (1 − 2η)f * J (x) → (1 − 2η(x))f * (x) and (1 − 2η(x))f * J (x) < 1 hold P X -almost everywhere; the first in-
Proof. Letf ∈ F G,J . The classifierf has the form of f (x) = 2 J j=1 c j 1{x ∈ G j } − 1 for some G 1 , . . . , G J ∈ G such that G J ⊆ · · · ⊆ G 1 and some c j ≥ 0 for j = 1, . . . , J with J j=1 c j = 1. Substitutingf into (55) yieldsComparing this expression with equation(53),From this expression and the assumption thatFor G * ∈ G * ,f is equivalent tof G * when G 1 = G * and c j = 1. Thus, from equation(57)This proves statement (ii) of the lemma.We will next prove that the minimizerf * of R ω φ h (·) over F G,J also minimizes R ω (·) over F G,J . Letf * be denoted bỹfor some G 1 , . . . , G J ⊆ G such that G J ⊆ · · · ⊆ G 1 and some c j ≥ 0 for j = 1, . . . , J such that J j=1 c j = 1. To obtain a contradiction, supposef * does not minimizeWe are now equipped to give proof of Theorem 7.5.Proof of Theorem 7.5. such thatWe show in the proof of Theorem 4.4 thatf * J →f * as J → ∞, P X -almost everywhere. Thenwhere the first and third equalities follow from (55). The second equality follows from the dominated convergence theorem, which holds as bothhold P X -almost everywhere, where the second condition is satisfied by Condition 7.4. The first inequality follows fromf * J ∈ F * J , and the last inequality follows from F * J ⊆ F G.J . Lemma E.1 shows that inff ∈ F G,J R ω φ h (f ) = 2R w * for any J. Thus, we haveThis means that the minimal value of R ω φ h on F G is at least 2R w * . Lemma E.1 also shows thatf G * leads to R ω φ h (f G * ) = 2R w * . Therefore,f G * minimizes R ω φ h over F G , which provesProof. By equations(53)and(56), R ω (f ) can be written asFrom equation(55), the right-hand side is equal to 2 −1 R ω φ (f G f ).We now give the proof of Theorem D.1, which is an extension of the proof of Theorem 5.1.Proof of Theorem D.1 (ii). For convenience of notation, we prove the result with C and r replaced by C and r, respectively. Let P ∈ P be fixed. First of all, Corollary E.2 and decomposing R ω
weighted classification risk R ω (·) over F G,J , and leads to R ω φ h (f * ) = 2R w *. weighted classification risk R ω (·) over F G,J , and leads to R ω φ h (f * ) = 2R w * .
. G For, J * ∈ G * ,F G * Minimizes R Ω Φ H (·) Over F G, For G * ∈ G * ,f G * minimizes R ω φ h (·) over F G,J .
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|
[] |
[
"Skew hook formula for d-complete posets",
"Skew hook formula for d-complete posets"
] |
[
"Hiroshi Naruse ",
"Soichi Okada "
] |
[] |
[] |
Peterson and Proctor obtained a formula which expresses the multivariate generating function for P -partitions on a d-complete poset P as a product in terms of hooks in P . In this paper, we give a skew generalization of Peterson-Proctor's hook formula, i.e., a formula for the generating function for (P \ F )-partitions for a d-complete poset P and its order filter F , by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant K-theory of Kac-Moody partial flag varieties. This generalization provides an alternate proof of Peterson-Proctor's hook formula.Mathematics Subject Classification (MSC2010): 05A15 (primary), 06A07, 14N15, 19L47 (secondary).
| null |
[
"https://arxiv.org/pdf/1802.09748v1.pdf"
] | 119,120,547 |
1802.09748
|
89bfa17f32c7fb247662856a56ba36f110974209
|
Skew hook formula for d-complete posets
27 Feb 2018
Hiroshi Naruse
Soichi Okada
Skew hook formula for d-complete posets
27 Feb 2018arXiv:1802.09748v1 [math.CO]d-complete posetshook formulasP -partitionsSchubert calculusequivariant K-theory
Peterson and Proctor obtained a formula which expresses the multivariate generating function for P -partitions on a d-complete poset P as a product in terms of hooks in P . In this paper, we give a skew generalization of Peterson-Proctor's hook formula, i.e., a formula for the generating function for (P \ F )-partitions for a d-complete poset P and its order filter F , by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant K-theory of Kac-Moody partial flag varieties. This generalization provides an alternate proof of Peterson-Proctor's hook formula.Mathematics Subject Classification (MSC2010): 05A15 (primary), 06A07, 14N15, 19L47 (secondary).
Introduction
One of the most elegant formulas in combinatorics is the Frame-Robinson-Thrall hook formula [2,Theorem 1] for the number of standard tableaux. Given a partition λ of n, a standard tableaux of shape λ is a filling of the cells of the Young diagram D(λ) of λ with numbers 1, 2, . . . , n such that each number appears once and the entries of each row and each column are increasing. The Frame-Robinson-Thrall hook formula asserts that the number f λ of standard tableaux of shape λ is given by
f λ = n! v∈D(λ) h D(λ) (v) ,(1)
where h D(λ) (v) denotes the hook length of the cell v in D(λ). Similar formulas hold for the number of shifted standard tableaux ( [13,5.1.4,Exercise 21], see also [30, § 40] and [35,Theorem 1]) and the number of increasing labeling of rooted trees ( [13,5.1.4,Exercise 20]). These tableaux and labelings can be regarded as linear extensions of certain posets.
Stanley [31] introduced the notion of P -partitions for a poset P , and found a relationship between the univariate generating function and the number of linear extensions of P . Given a poset P , a P -partition is an order-reversing map σ from P to N, the set of nonnegative integers. We denote by A(P ) the set of all P -partitions. For a P -partition σ, we write |σ| = v∈P σ(x). Then Stanley [31,Corollaries 5.3 and 5.4] proved that, for a poset P with n elements, there exists a polynomial W P (q) satisfying σ∈A(P )
q |σ| = W P (q) n i=1 (1 − q i ) ,(2)
and that W P (1) is equal to the number of linear extensions of P . Also in [32,Proposition 18.3] he proved that, if P is the Young diagram D(λ) of a partition λ, viewed as a poset, the generating function of D(λ)-partitions (also called reverse plane partitions of shape λ) is given by
σ∈A(D(λ)) q |σ| = 1 v∈D(λ) (1 − q h D(λ) (v) )
.
Combining (2) and (3) and taking the limit q → 1, we obtain the Frame-Robinson-Thrall hook formula (1). Gansner [3,Theorem 5.1] gave a multivariate generalization of (3).
Proctor [27], [28] introduced a wide class of posets, called d-complete posets, enjoying "hook-length property", as a generalization of Young diagrams, shifted Young diagrams and rooted trees. d-Complete posets are defined by certain local structural conditions (see Section 2 for a precise definition). Peterson and Proctor obtained the following theorem, which is a far-reaching generalization of the hook formulas (1) and (3). Theorem 1.1. (Peterson-Proctor, see [29]) Let P be a d-complete poset. The multivariate generating function of P -partitions is given by
σ∈A(P ) z σ = 1 v∈P (1 − z[H P (v)]) .(4)
(Refer to Section 2 for undefined notations.)
However the original proof of this theorem is not yet published, though an outline of their proof is given in [29]. Different proofs are sketched by Ishikawa-Tagawa [8], [9] and Nakada [23], [24]. Our skew generalization (Theorem 1.2 below) provides an alternate proof of Theorem 1.1. In the univariate case, a full proof is given by Kim-Yoo [11].
Another direction of generalizing the Frame-Robinson-Thrall hook formula (1) is to consider skew shapes. For partitions λ ⊃ µ, a standard tableau of skew shape λ/µ is a filling of the cells of the skew Young diagram D(λ/µ) = D(λ)\D(µ) satisfying the same conditions as standard tableaux of straight shape. However one cannot expect a nice product formula for the number f λ/µ of standard tableaux of skew shape λ/µ in general. Naruse [25] presented and sketched a proof of a subtraction-free formula for f λ/µ :
f λ/µ = n! D∈E D(λ) (D(µ)) 1 v∈D(λ)\D h λ (v) ,(5)
where n = |λ/µ|, and D runs over all excited diagrams of D(µ) in D(λ). Morales-Pak-Panova [21] gave a q-analogue of Naruse's skew hook formula for the univariate generating functions for P -partitions on P = D(λ/µ).
The main result of this paper is the following skew generalization of Peterson-Proctor's hook formula (Theorem 1.1): Theorem 1.2. Let P be a connected d-complete poset and F an order filter of P . Then the multivariate generating function of (P \F )-partitions, where P \F is viewed as an induced subposet of P , is given by
σ∈A(P \F ) z σ = D∈E P (F ) v∈B(D) z[H P (v)] v∈P \D (1 − z[H P (v)]) ,(6)
where D runs over all excited diagrams of F in P . (See Sections 2 and 3 for undefined notations.)
Taking an appropriate limit, we see that the number of linear extensions of P \ F is given by
n! D∈E P (F ) 1 v∈P \D h P (v) ,(7)
where n = #(P \ F ) and h P (v) is the hook length of v in P . (See Corollary 5.6(b).) If F = ∅, then our main theorem (Theorem 1.2) gives Peterson-Proctor's hook formula (Theorem 1.1). If P = D(λ) and F = D(µ) are the Young diagrams of partitions λ ⊃ µ, then (6) reduces Morales-Pak-Panova's q-hook formula [21, Corollary 6.17] after specializing z i = q for all i ∈ I, and (7) is nothing but Naruse's skew hook formula (5).
This paper is organized as follows. In Section 2, we review a definition and basic properties of d-complete posets. In Section 3, we introduce the notion of excited diagrams for d-complete posets, which is the key ingredient of the formulation of our main theorem, and study their properties. Our proof of the main theorem uses the Billey-type formula and the Chevalley-type formula for the equivariant K-theory of Kac-Moody partial flag varieties. In Section 4, we recall some properties of the equivariant K-theory and translate the Billey-type formula and the Chevalley-type formula in terms of combinatorics of d-complete posets. We will give a proof of our main theorem (Theorem 1.2) and derive some corollaries in Section 5.
d-Complete posets
In this section we review a definition and some properties of d-complete posets and explain their connections to Weyl groups. See [27], [28], [29] and [34] for details.
Combinatorics of d-complete posets
For an integer k ≥ 3, we denote by d k (1) the poset consisting of 2k − 2 elements u 1 , · · · , u k−2 , x, y, v k−2 , · · · , v 1 with covering relations
u 1 ⋗ u 2 ⋗ · · · ⋗ u k−2 , u k−2 ⋗ x ⋗ v k−2 , u k−2 ⋗ y ⋗ v k−2 , v k−2 ⋗ · · · ⋗ v 2 ⋗ v 1 .
Note that z and w are incomparable. The poset d k (1) is called the double-tailed diamond. The Hasse diagram of d k (1) is shown in Figure 1. (1). Then v and u are called the bottom and top of [v, u] respectively, and the two incomparable elements of [v, u] are called the sides. A subset I of P is called convex if x < y < z in P and x, z ∈ I imply y ∈ I. A convex subset I is called a d − k -convex set if it is isomorphic to the poset obtained by removing the top element from d k (1). Definition 2.1. A poset P is d-complete if it satisfies the following three conditions for every k ≥ 3:
t t t t t t t t ❅ ❅ ❅ ❅ u 1 u 2 u k−2 x y v k−2 v 2 v 1 Figure 1: Double-tailed diamond d k (1) Let P be a poset. An interval [v, u] = {x ∈ P : v ≤ x ≤ u} is called a d k - interval if it is isomorphic to d k(D1) If I is a d −
k -convex set, then there exists an element u such that u covers the maximal elements of I and I ∪ {u} is a d k -interval.
(D2) If I = [v, u] is a d k -interval and the top u covers u ′ in P , then u ′ ∈ I.
(D3) There are no d − k -convex sets which differ only in the minimal elements. It is clear that rooted trees, viewed as posets with their roots being the maximum elements, are d-complete posets.
Example 2.2. For a partition λ, let D(λ) be the Young diagram of λ given by
D(λ) = {(i, j) ∈ Z 2 : 1 ≤ j ≤ λ i }.
For a strict partition µ, let S(µ) be the shifted Young diagram of µ given by
S(µ) = {(i, j) ∈ Z 2 : i ≤ j ≤ µ i + i − 1}.
We endow Z 2 with a poset structure by defining
(i, j) ≥ (i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ .(8)
Then we regard the Young diagram D(λ) and the shifted Young diagram S(µ) as induced subposets of Z 2 . The resulting posets are called a shape and a shifted shape respectively. It can be shown that shapes and shifted shapes are d-complete posets. Figure 2 illustrates the Hasse diagrams of D(5, 4, 2, 1) and S(5, 4, 2, 1). (1, 1), (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (4,7), (4,8) .
t t t t t t t t t t t t ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ (a) D(5, 4, 2, 1) t t t t t t t t t t t t ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ (b) S(5, 4, 2, 1)
If we regard P as an induced subposet of Z 2 with ordering given by (8), then P is a d-complete poset, called a swivel. See Figure 3 for the Hasse diagram of P .
A poset P is called connected if its Hasse diagram is a connected graph. It is easy to see that, if P is a d-complete poset, then each connected component of P is d-complete. Hence there is no harm in assuming that a d-complete poset is connected.
Proposition 2.4. ([27, §3])
Let P be a d-complete poset. If P is connected, then P has a unique maximal element.
Let P be a poset with a unique maximal element. The top tree Γ of P is the subgraph of the Hasse diagram of P , whose vertex set consists of all elements x ∈ P such that every y ≥ x is covered by at most one other element. Let P be a connected d-complete poset and Γ its top tree. Let I be a set of colors whose cardinality is the same as Γ. Then a bijective labeling c : Γ → I can be uniquely extended to a map c : P → I satisfying the following three conditions:
(C1) If x and y are incomparable, then c(x) = c(y).
(C2) If an interval [v, u] is a chain, then the colors c(x) (x ∈ [v, u]) are distinct. (C3) If [v, u] is a d k -interval then c(v) = c(u).
Moreover this map c satisfies (C4) If x covers y, then the nodes labeled by c(x) and c(y) are adjacent in Γ.
(C5) If c(x) = c(y) or the nodes labeled by c(x) and c(y) are adjacent in Γ, then x and y are comparable.
Such a map c : P → I is called a d-complete coloring.
Let P be a connected d-complete poset and c : P → I a d-complete coloring. Let z = (z i ) i∈I be indeterminates. Given an order filter F of P , we regard P \ F as the induced subposet. For a (P \ F )-partition σ ∈ A(P \ F ), we put
z σ = v∈P \F z σ(v) c(v) .
We are interested in the multivariate generating function σ∈A(P \F ) z σ of (P \ F )-partitions. For a subset D of P , we write
z[D] = v∈D z c(v) .
Instead of giving a definition of hooks H P (u) ⊂ P for a general d-complete poset P , we define associated monomials z[H P (u)] directly by induction as follows:
Definition 2.6. Let P be a connected d-complete poset with d-complete coloring c : P → I.
(i) If u is not the top of any d k -interval, then we define
z[H P (u)] = w≤u z c(w) . (ii) If u is the top of a d k -interval [v, u], then we define z[H P (u)] = z[H P (x)] · z[H P (y)] z[H P (v)] ,
where x and y are the sides of [v, u].
Example 2.7. Let P = D(λ) be the shape corresponding to a partition λ. Then the top tree Γ of D(λ) is given by
Γ = {(1, j) : 1 ≤ j ≤ λ ′ 1 } ∪ {(i, 1) : 1 ≤ i ≤ λ 1 },
where λ ′ 1 is the number of cells in the first column of the Young diagram D(λ). A d-complete coloring c :
D(λ) → I = {−(λ ′ 1 − 1), . . . , −1, 0, 1, . . . , λ 1 − 1} is given by c(i, j) = j − i. The hook H D(λ) (i, j) of u = (i, j) in D(λ) is the subset of D(λ) defined by H D(λ) (i, j) = {(i, j)} ∪ {(i, l) ∈ D(λ) : l > j} ∪ {(k, j) ∈ D(λ) : k > i}.
Example 2.8. Let P = S(µ) be the shifted shape corresponding to a strict partition µ of length ≥ 2.. Then the top tree Γ of S(µ) is given by
Γ = {(1, j) : 1 ≤ j ≤ µ 1 } ∪ {2, 2},
and a d-complete coloring c : S(µ) → I = {0, 0 ′ , 1, 2, . . . , µ 1 − 1} is given by
c(i, j) = j − i if i < j, 0 if i = j and i is odd, 0 ′ if i = j and i is even. The (shifted) hook H S(µ) (i, j) of u = (i, j) in S(µ) is the subset of S(µ) defined by H S(µ) (i, j) = {(i, j)} ∪ {(i, l) ∈ S(µ) : l > j} ∪ {(k, j) ∈ S(µ) : k > j} ∪ {(j + 1, l) ∈ S(µ) : l > j}.
d-Complete posets and Weyl groups
Let P be a connected d-complete poset with top tree Γ. We regard Γ as a (simplylaced) Dynkin diagram with node set I and the d-complete coloring as a map c : P → I. Let A = (a ij ) i,j∈I be the generalized Cartan matrix of Γ given by We fix the following data associated to A:
a ij = 2 if i = j,
• a free Z module Λ, called the weight lattice,
• a linearly independent subset Π = {α i : i ∈ I} of Λ, called the simple roots,
• a subset Π ∨ = {α ∨ i : i ∈ I} of the dual lattice Λ * = Hom Z (Λ, Z), called the simple coroots,
• a subset {λ i : i ∈ I} of Λ, called the fundamental weights,
such that α ∨ i , α j = a ij , α ∨ i , λ j = δ ij , where ,
: Λ * × Λ → Z is the canonical pairing. Let W be the corresponding Weyl group generated by the simple reflections {s i : i ∈ I}, where s i acts on Λ and Λ * by the rule
s i (λ) = λ − α ∨ i , λ α i (λ ∈ Λ), s i (λ ∨ ) = λ ∨ − λ ∨ , α i α ∨ i (λ ∨ ∈ Λ * ).
Then W is a Coxeter group, and we have the length function l and the Bruhat order < on W . The set of real roots Φ and the set of real coroots Φ ∨ are defined by Φ = W Π and Φ ∨ = W Π ∨ respectively. The set of simple roost Π (resp. the set of simple coroots Π ∨ ) determines the decomposition of Φ (resp. Φ ∨ ) into the positive system Φ + (resp. Φ ∨ + ) and the negative system Φ − (resp. Φ ∨ − ). We introduce the standard partial ordering on Φ + (resp. Φ ∨ + ) by setting α > β if α − β is a sum of simple roots
{α i : i ∈ I} (resp. α ∨ > β ∨ if α ∨ − β ∨ is a sum of simple coroots {α ∨ i : i ∈ I}).
For p ∈ P , we put
α(p) = α c(p) , α ∨ (p) = α ∨ c(p) , s(p) = s c(p) .
Let α P and λ P be the simple root and the fundamental weight corresponding to the color i P of the maximum element of P . Take a linear extension and label the elements of P with p 1 , · · · , p N (N = #P ) so that p i < p j in P implies i < j. Then we construct an element w P ∈ W by putting w P = s(p 1 )s(p 2 ) · · · s(p N ).
A Weyl group element w ∈ W is called λ-minuscule if there exists a reduced expression w = s i 1 · · · s i l such that
α ∨ i k , s i k+1 · · · s i l λ = 1 (1 ≤ k ≤ l),
or equivalently
s i k · · · s i l λ = λ − α i k − · · · − α i l .
A element w ∈ W is called fully commutative if any reduced expression of w can be obtained from any other by using only the Coxeter relations of the form st = ts.
Proposition 2.9. (See [27] and [34, Proposition 2.1]) The Weyl group element w P ∈ W is λ P -minuscule and hence fully commutative.
If p = p k ∈ P , then we define
β(p k ) = s(p 1 ) · · · s(p k−1 )α(p k ), γ(p k ) = s(p N ) · · · s(p k+1 )α(p k ), γ ∨ (p k ) = s(p N ) · · · s(p k+1 )α ∨ (p k ).
It follows from Proposition 2.9 that, for each p ∈ P , the roots β(p), γ(p) and the coroot γ ∨ (p) are independent of the choices of linear extensions. For a Weyl group element w ∈ W , we put
Φ(w) = Φ + ∩ wΦ − , Φ ∨ (w) = Φ ∨ + ∩ wΦ ∨ − .
Then it is well-known (see [5, §5.6]) that Let W λ P be the stabilizer of λ P in W . Then W λ P is the maximal parabolic subgroup corresponding to I \ {i P }. Let W λ P be the set of minimum length coset representatives of W/W λ P . A subset F of P is called an order filter if x < y in P and x ∈ F imply y ∈ F . For a subset D = {p i 1 , · · · , p ir } (i 1 < · · · < i r ) of P , we define w D = s(p i 1 ) · · · s(p ir ).
Φ(w P ) = {β(p) : p ∈ P }, Φ ∨ (w −1 P ) = {γ(p) ∨ : p ∈ P }.
Moreover we have
Since w P is fully commutative (Proposition 2.9), we see that w D is independent of the choices of linear extensions of P . The map F → w F gives a poset isomorphism from the set of all order filters of P ordered by inclusion to the Bruhat interval [e, w P ] in W λ P .
(b) (See [34, Remark 2.7 (b)]) If F is an order filter of P , then w F is λ P -minuscule, and w F λ P = λ P − p∈F α(p).
Remark 2.12. Let W be an arbitrary symmetrizable Kac-Moody Weyl group corresponding to a Dynkin diagram Γ with node set I. Given a (not necessarily reduced) expression s i 1 s i 2 . . . s i N of an element w ∈ W in simple reflections, we can define a poset H, called the heap, as follows (see [33]). The poset H consists of the ground set {1, 2, . . . , N } and the partial ordering obtained by taking the transitive closure of the relations given by
a ≺ b if a < b and either s ia s i b = s i b s ia or i a = i b .
The heap H has a natural labeling (coloring) c : H → I given by c(a) = i a . If w ∈ W is fully commutative, then the haep defined by a reduced expression of w is independent of the choices of reduced expressions. In this case we denote the resulting heap by H(w). Every d-complete poset P is isomorphic to the simply-laced heap H(w P ). In general, if w ∈ W is dominant minuscule, i.e., λ-minuscule for some dominant weight λ, the corresponding heap H(w) is isomorphic (as a unlabeled poset) to a d-complete poset. See [34,Sections 3 and 4].
The propositions in this section hold literally for heaps H(w) of dominant minuscule elements, except for Proposition 2.5 and Proposition 2.10 (b). The latter half of Proposition 2.5 holds for heaps, i.e., the labeling c : H(w) → I satisfies (C4) and (C5). And we adopt Proposition 2.10 (b) as a definition of the hook monomial for H(w).
Example 2.13. Let µ = (µ 1 , . . . , µ l ) be a strict partition of length l. Then the shifted shape S(µ) can be regarded as a heap associated to a dominant minuscule element of the Weyl group of type B. Put m = µ 1 and let W be the Weyl group generated by s 0 , s 1 , . . . , s m−1 subject to the relations
s 2 i = 1 (i = 0, 1, . . . , m − 1), s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 , s i s i+1 s i = s i+1 s i s i+1 (i = 1, 2, . . . , m − 2),s i s j = s j s i (|i − j| ≥ 2).
Then we define an element w µ ∈ W by putting
w µ = (s µ l −1 · · · s 1 s 0 ) · · · (s µ 2 −1 · · · s 1 s 0 )(s µ 1 −1 · · · s 1 s 0 ).
Then it can be shown that w µ is λ 0 -minuscule, where λ 0 is the fundamental weight corresponding to s 0 , and that the map
S(µ) ∋ (i, j) → µ l + · · · + µ i − j + i ∈ H(w µ )(10)
gives a poset isomorphism. We identify the ground set of H(w µ ) with S(µ) via the isomorphism (10). Then the natural labeling c ′ :
S(µ) → {0, 1, . . . , m − 1} of H(µ) is given by c ′ (i, j) = j − i, which is different from the d-complete coloring of S(µ) given in Example 2.8. And the "hook" H ′ S(µ) (v) (see [23, Definition 4.8]) is defined by H ′ S(µ) (i, j) = {(i, j)} ∪ {(i, l) ∈ S(µ) : l > j} ∪ {(k, j) ∈ S(µ) : k > i} ∪ {(i, i)} ∪ {(j, l) ∈ S(µ) : l ≥ j} if i < j and (j, j) ∈ S(µ), ∅ otherwise,
which is also different from the shifted hook given in Example 2.8.
Excited diagrams
In this section we introduce the notion of excited and K-theoretical excited diagrams in a d-complete poset and study their properties.
Excited diagrams
First we generalize the notion of exited diagrams for Young diagram and shifted Young diagram, which were introduced by Ikeda-Naruse [6] and Kreiman [14], [15] independently, to a general d-complete posets. And we give a generalization of backward movable positions or excited peaks introduced in [7], [12] and [21]. Let P be a connected d-complete poset with top tree Γ and d-complete coloring c : P → I. For a subset D ⊂ P and a color i ∈ I, we put
D i = {x ∈ D : c(x) = i}.
For i ∈ I, let N i be the subset of P consisting of element x ∈ P whose color c(x)
is adjacent to i in the Dynkin diagram Γ. Note that, if [v, u] is a d k -interval, then [v, u] ∩ N c(u) consists of elements x ∈ [v, u]
such that x is covered by u or covers v. Definition 3.1. Let P be a connected d-complete poset and let F be an order filter of P .
(a) Let D be a subset of P and u ∈ D. We say that u is D-active if there exists an
element v ∈ (P \ D) c(u) such that v < u, [v, u] is a d k -interval and [v, u] ∩ D ∩ N c(u) = ∅.
(b) Let D be a subset of P and u ∈ D. If u is D-active, then we define α u (D) to be the subset of P obtained by replacing u ∈ D by the bottom element v of the d k -interval [v, u]. We call this replacement an (ordinary) elementary excitation.
(c) An excited diagram of F in P is a subset of P obtained from F after a sequence of elementary excitations on active elements. Let E P (F ) be the set of all excited diagrams of F in P .
− −− → × × − −− → × × − −− → × × − −− → × Figure 4: Excited diagrams of D(3, 1) in D(5, 4, 2, 1) (d) To an excited diagram D ∈ E P (F ) we associate a subset B(D) ⊂ P as follows: If D = F , then B(F ) = ∅.
If D is an excited diagram with an active element u, then we define
B(α u (D)) = B(D) \ ([v, u] ∩ N c(u) ) ∪ {u}, where [v, u] is the d k -interval with top element u. We call B(D) the set of excited peaks of D. (We will show that B(D) is a well-defined subset of P \ D in Proposition 3.8.) In general, if two elements v and u of P with v < u have the same color i = c(v) = c(u) and satisfy [v, u] ∩ D ∩ N i = ∅, then D \ {u} ∪ {v} is obtained from D by a sequence of elementary excitations.
If P is a shape or a shifted shape, our definition above coincides with the definitions of elementary excitations in [6], [7], or ladder moves in [14], [15], and backward movable positions in [7], [12] or excited peaks in [21] (only for a shape).
Example 3.2. If P = D(5, 4, 2, 1) is the shape corresponding to a partition (5, 4, 2, 1) and F = D(3, 1), then there are 6 excited diagrams in E P (F ) shown in Figure 4. In Figure 4 (and Figures 5, 6), the shaded cells form an exited diagram and a cell with × is an excited peak. And the arrow D → D ′ means that D ′ is obtained from D by an elementary excitation. Example 3.3. If P = S(5, 4, 2, 1) is the shifted shape corresponding to a strict partition (5, 4, 2, 1) and F = S(3, 1), then there are 5 excited diagrams in E P (F ) shown in Figure 5. Let W be the Weyl group corresponding to the top tree Γ viewed as a Dynkin diagram, and fix a labeling of the elements of P with p 1 , . . . , p N so that p i < p j in P implies i < j. Then we can associate to a subset D of P a well-defined element w D ∈ W as in (9). The following proposition gives a characterization of excited diagrams. To prove this proposition, we prepare two lemmas.
− −− → × × − −− → × × − −− → ×
Lemma 3.6. Let D be a subset of P and u and v elements of P such that v < u and
c(u) = c(v) = i. Suppose v = p k and u = p l . If [v, u] ∩ D ∩ N i = ∅, then we have s(p j )s(p l ) = s(p l )s(p j ) for any p j ∈ D with k < j < l
Proof. follows from Property (C5) in Proposition 2.5. Let v ∈ W be a fully commutative element and v = s i 1 · · · s ir = s j 1 · · · s jr be its reduced expressions. Then we have (a) Let i, j be adjacent nodes in the Dynkin diagram. Then the subsequence of (i 1 , · · · , i r ) consisting of i and j is identical with the subsequence of (j 1 , · · · , j r ) consisting of i and j.
(b) Let i be a node in the Dynkin diagram. Then the number of occurrence of i in (i 1 , · · · , i r ) is equal to the number of occurrence of i in (j 1 , · · · , j r ).
We use these lemmas to prove the characterization of excited diagrams.
Proof of Proposition 3.5. We follow the same idea used in the proof of [4, Proposition 4.8]. We denote by R P (F ) the set of all subsets D ⊂ P satisfying #D = #F and
w D = w F . First we prove that E P (F ) ⊂ R P (F ). Since F ∈ R P (F ), it is enough to show that, if D ′ ∈ E P (F ) is obtained from D ∈ E P (F ) by an elementary excitation, then w D ′ = w D . Let u, v ∈ P be elements such that [v, u] is a d k -interval, [v, u] ∩ D ∩ N c(u) = ∅ and D ′ = α u (D) = D \ {u} ∪ {v}. If v = p k , u = p l and {j : k < j < l, p j ∈ D} = {j 1 , . . . , j m } (j 1 < · · · < j m ), then we have w D = · · · s(p j 1 ) · · · s(p jm )s(p l ) · · · , w D ′ = · · · s(p k )s(p j 1 ) · · · s(p jm ) · · · .
Then by using Lemma 3.6, we have Figure 6: Excited diagrams in a swivel
w D ′ = w D . × × × − −− → × × × × − −− → × × × × ×
Next we prove that R P (F ) ⊂ E P (F ). Since F is an order filter, we can take a linear extension of P such that F = {p n+1 , · · · , p N }, where n = #(P \ F ). We define the energy of any subset D ⊂ P with #D = #F by putting
e(D) = p k ∈F k − p k ∈D k.
Then, by the assumption on our linear extension, we see that e(D) ≥ 0 and that e(D) = 0 if and only if D = F .
We proceed by induction on e(D) to prove D ∈ R P (F ) implies D ∈ E P (F ). Let D ∈ R P (F ). If e(D) = 0, then we have D = F ∈ E P (F ).
Suppose that e(D) > 0. Then there exists an element u ∈ F such that u ∈ D. Let u be the maximal element satisfying u ∈ F and u ∈ D. Since w D = w F , it follows from Lemma 3.7 (b) that there exists an element v ∈ D with the same color i as u. Let v be the maximal element satisfying v ∈ D and c(v) = c(u) = i. Then, by applying Lemma 3.7 (a), we see that [v, u] Lemma 3.6 and e(D ′ ) < e(D). Then by the induction hypothesis we have D ′ ∈ E P (F ). Since D is obtained from D ′ by a sequence of elementary excitations, we obtain D ∈ E P (F ).
∩ D ∩ N i = ∅. If we put D ′ = D \{v}∪ {u}, then w D ′ = w F by
Next we give a non-recursive description of the set of excited peaks B(D), which implies that B(D) is well-defined, i.e., it is independent of the choices of elementary excitations to reach D from F .
Proposition 3.8. Let D ∈ E P (F ) be an excited diagram.
(a) The following are equivalent for x ∈ P :
(i) x ∈ B(D). (ii) There exists an element y ∈ D c(x) such that y < x and [y, x]∩D∩N c(x) = ∅. (b) We have D ∩ B(D) = ∅.
In the proof of this proposition, we utilize the following lemma, which will be used also in the sequel of this section.
Lemma 3.9. Let x, y, z, u and v be elements of P such that c(x) = x(y) = c(z) = i and c(u) = c(v) = j. Let D be a subset of P . Then we have (a) If [z, y] ∩ D ∩ N i = ∅ and [y, x] ∩ D ∩ N i = ∅, then we have [z, x] ∩ D ∩ N i = ∅.∩ N i = ([z, y] ∩ N i ) ∪ ([y, x] ∩ N i ). (b) Assume to the contrary that [y, x]∩(D∪{v})∩N i = ∅. Since [y, x]∩D∩N i = ∅, we have v ∈ [y, x] ∩ N i . Thus j = c(u) = c(v) is adjacent to i = c(x) in Γ. Since u ∈ D and [y, x] ∩ D ∩ N i = ∅, we have u ∈ [y, x] ∩ N i .
Hence, by using Property (C5) in Proposition 2.5, we see that y < v < x < u and x ∈ [v, u] ∩ N j , which contradicts to the assumption.
(c) By an argument similar to (b), we can show that, if [y, x] ∩ D ∩ N i = ∅, then v < y < u < x, which contradicts to y ∈ [v, u] ∩ N j . We begin with considering the case where D = F . Let x and y be elements of F with the same color i satisfying y < x. Then it follows from Properties (C4) and (C5) in Proposition 2.5 that an element z covered by x or covers y belongs to
[y, x] ∩ F ∩ N i . Hence we have B ′ (F ) = ∅ = B(F ) and F ∩ B ′ (F ) = ∅.
We prove that B(α u (D)) = B ′ (α u (D)) and α u (D) ∩ B ′ (α u (D)) = ∅ for D ∈ E P (F ) and a D-active element u ∈ D. Let v be the element such that v < u, Next, in order to show B ′ (α u (D)) ⊂ B(α u (D)), we take an element x ∈ B ′ (α u (D)) such that x = u and prove
[v, u] ∩ D ∩ N c(u) = ∅ and α u (D) = D \ {u} ∪ {v}. First we show B(α u (D)) ⊂ B ′ (α u (D)). Since v ∈ α u (D) and [v, u]∩D∩N c(u) = ∅, we have u ∈ B ′ (α u (D)). Let x ∈ B(D) such that x ∈ [v, u] ∩ N c(u) . Since B(D) = B ′ (D)x ∈ B(D) \ ([v, u] ∩ N c(u) ). Then there exists y ∈ (α u (D)) c(x) such that y < x and [y, x] ∩ α u (D) ∩ N c(x) = ∅. If y = v, then u ∈ D and [u, x]∩ D ∩ N c(x) ⊂ [v, x]∩ D ∩ N c(x) = ∅, hence x ∈ B(D) and x ∈ N c(u)
. We consider the case where y = v. In this case, y ∈ D and it follows from Lemma 3.9 (c) that
[y, x]∩ D ∩ N c(x) = ∅. Also we have x ∈ [v, u]∩ N c(u) . In fact, if x ∈ [v, u]∩ N c(u) , then c(y) = c(x) is adjacent to c(u) and it follows from y ∈ D and [v, u] ∩ D ∩ N c(u) = ∅ that y ∈ [v, u].
Hence, by using Property (C5) in Proposition 2.5, we have y < v < x < u and this contradicts to
[y, x] ∩ α u (D) ∩ N c(x) = ∅. Therefore we have x ∈ B(D) \ ([v, u] ∩ N c(u) ).
Finally we show that α u (D) ∩ B ′ (α u (D)) = ∅. Let x and y be elements of α u (D) with the same color i satisfying y < x. Since α u (D) = D \ {u} ∪ {v}, it is enough to consider the following three cases:
Case 1. x, y ∈ D, Case 2. y = v, Case 3. x = v.
In Case 1, assume that [y, x]∩α u (D)∩N i = ∅. Since [y, x]∩D∩N i = ∅ by the induction hypothesis, we have v ∈ [y, x] ∩ N i , i.e., y < v < x and c(u) = c(v) is adjacent to i in Γ. Since v ∈ α u (D), we have v ∈ [y, x] ∩ N i . Hence by using Property (C5) in Proposition 2.5 we see that v < y < u < x, which contradicts to the D-activity of u.
In Case 2, we have v < u < x because [v, u] is a d k -interval. Since [u, x] ∩ D ∩ N i = ∅ by the induction hypothesis, we have [v, x] ∩ α u (D) ∩ N i = ∅. In Case 3, we have [v, u] ∩ D ∩ N i = ∅ (u is D-active) and [y, u] ∩ D ∩ N i = ∅ (the induction hypothesis).
Hence by using Lemma 3.9 (a) we obtain [y, v] ∩ α u (D) ∩ N i = ∅. Therefore we see that any element satisfying the condition (ii) in (a) for α u (D) does not belong to α u (D), and α u (D) ∩ B ′ (α u (D)) = ∅. This completes the proof.
K-theoretical excited diagrams
We define K-theoretical excited diagrams and study their properties. For shapes and shifted shapes, these diagrams were introduced in [4]. (a) Let D be a subset of P and u ∈ D. If u is D-active and [v, u] is a d k -interval, then we define α * u (D) to be the subset of P obtained by adding v to D. We call this operation a K-theoretical elementary excitation.
(b) A K-theoretical excited diagram of F in P is a subset of P obtained from F after a sequence of ordinary and K-theoretical elementary excitations on active elements. Let E * P (F ) be the set of all K-theoretical excited diagrams of F in P . For a fixed linear extension of P and a subset D = {p i 1 , · · · , p ir } (i 1 < · · · < i r ) of P , we define an element w * D ∈ W by putting
w * D = s(p i 1 ) * s(p i 2 ) * · · · * s(p ir ),
where * : W × W → W is the associative product, called the Demazure product, defined by
s i * w = s i w if l(s i w) = l(w) + 1, w if l(s i w) = l(w) − 1.
Since w P is fully commutative (Proposition 2.9), the element w * D is independent of the choices of linear extensions of P
The following proposition is a key to rephrase the Billey-type formula for equivariant K-theory in terms of combinatorics of d-complete posets (see Proposition 4.7). Let v ∈ W and v = s i 1 * · · · * s ir with i 1 , . . . , i r ∈ I.
(a) There is a increasing sequence 1 ≤ k 1 < k 2 < · · · < k l ≤ r such that v = s i k 1 s i k 2 . . . s i k l is a reduced expression of v. In particular, we have l(v) ≤ r.
(b) If l(v) = r, then v = s i 1 · · · s ir .
(c) If v is fully commutative and l(v) < r, then there exist a < b such that s ia = s i b and commutes with s ic for every a < c < b.
Proof of Proposition 3.11. We denote by R * P (F ) the set of all subsets D ⊂ P satisfying w * D = w F . First we prove E * P (F ) ⊂ R * P (F ). Since F ∈ R P (F ), it is enough to show that, if D ′ ∈ E P (F ) is obtained from D ∈ E P (F ) by an ordinary or K-theoretical elementary excitation, then w D ′ = w D . Let u be a D-active element and [v, u] be the d k -interval with top element u. Let v = p k , u = p l and {j : k < j < l, p j ∈ D} = {j 1 , . . . , j m } (j 1 < · · · < j m ). If D ′ = α u (D) = D \ {u} ∪ {v}, then we have w * D = · · · * s(p j 1 ) * · · · * s(p jm ) * s(p l ) * · · · , w * D ′ = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * · · · .
If D ′ = α * u (D) = D ∪ {v}, then we have w * D = · · · * s(p j 1 ) * · · · * s(p jm ) * s(p l ) * · · · , w * D ′ = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * s(p l ) * · · · .
Since u is D-active, it follows from Lemma 3.6 and s(p k ) * s(p l ) = s(p l ) * s(p l ) = s(p l ) that w * D ′ = w * D in both cases. Next we prove R * P (F ) ⊂ E * P (F ). We proceed by induction on #D to prove D ∈ R * P (F ) implies D ∈ E * P (F ). Since w * D = w F , we have #D ≥ l(w F ) = #F by Lemma 3.12 (a). If #D = #F , then w * D = w D by Lemma 3.12 (b), thus we have D ∈ E P (F ) ⊂ E * P (F ) by Proposition 3.5. If #D > #F , then it follows from Lemma 3.12 (c) that there exist u, v ∈ D with v < u such that c(u) = c(v) and s(u) = s(v) commutes with every elements between u and v in the expression of w * D . If we put D ′ = D \ {v}, then we see that w * D ′ = w * D . Hence by the induction hypothesis we have D ′ ∈ E * P (F ). Since D ′ is obtained from D by a sequence of ordinary and K-theoretical elementary excitations, we obtain D ′ ∈ E * P (F ).
The following proposition plays a crucial role in the proof of our main theorem (see the proof of Theorem 5.3).
In order to prove this proposition, we prepare several lemmas.
i = c(y) in Γ and w ∈ D ∩ N c(x) . Since [y, x] ∩ D ∩ N c(x) = ∅, we have w ∈ [y, x].
Hence by using Property (C5) in Proposition 2.5 we see that w < y < z < x, which contracts to
[w, z] ∩ D ∩ N c(z) = ∅.
For E ∈ E * P (F ), we define a subset S(E) of E by putting
S(E) = x ∈ E : there exists y ∈ E c(x) such that y < x and [y, x] ∩ E ∩ N c(x) = ∅ .(12)
It follows from Property (C4) in Proposition 2.5 that S(F ) = ∅ for an order filter F of P .
Lemma 3.15. Let E ∈ E * P (F ) and u ∈ E an E-active element. Then we have
S(α u (E)) = S(E) \ {u} ∪ {v} if u ∈ S(E), S(E) if u ∈ S(E),(13)S(α * u (E)) = S(E) ∪ {v} if u ∈ S(E), S(E) ∪ {u} if u ∈ S(E),(14)
where [v, u] is the d k -interval with top element u. In particular, we have Proof. Since α u (E) = E \ {u} ∪ {v}, the equality (13) follows from the following three claims:
(i) S(α u (E)) \ {v} ⊂ S(E). (ii) If u ∈ S(E), then v ∈ S(α u (E)). (iii) S(E) \ {u} ⊂ S(α u (E)).
To prove (i), we take x ∈ S(α u (E)) such that x = v. Then, by the definition (12), there exists y ∈ (α u (E)) c(x) such that y < x and [y,
x] ∩ α u (E) ∩ N c(x) = ∅. If y = v, then [u, x] ∩ E ∩ N c(x) ⊂ [v, x] ∩ E ∩ N c(x) = ∅, hence x ∈ S(E)
. We consider the case where y = v. In this case, y ∈ E and it follows from [v, u] ∩ E ∩ N c(u) = ∅ that y ∈ [v, y] ∩ N c(u) . Now we can use Lemma 3.9 (c) to obtain [y, x] ∩ E ∩ N c(x) = ∅, hence x ∈ S(E).
Next we prove (ii). Since u ∈ S(E), there exists z ∈ E c(u) such that z < u and
[z, u] ∩ E ∩ N c(u) = ∅. Then we have [z, v] ∩ α u (E) ∩ N c(v) ⊂ [z, u] ∩ α u (E) ∩ N c(u) = ∅, hence v ∈ S(α u (E)).
To prove (iii), we take x ∈ S(E) such that x = u. By the definition (12), there exists y ∈ E c(x) such that y < x and [y, x] ∩ E ∩ N c(x) = ∅. Then we have y = u or y ∈ α u (E). If y = u, then [u, x] ∩ E ∩ N c(x) = ∅ and [v, u] ∩ α u (E) ∩ N c(u) = ∅ by the E-activity of u. Hence by using Lemma 3.9 (a) we have [v, x] ∩ α u (E) ∩ N c(x) = ∅ and x ∈ S(α u (E)). We consider the case where y ∈ α u (E). Since [v, u]
∩ E ∩ N c(u) = ∅ and x ∈ E, we have x ∈ [v, u] ∩ N c(u) . Hence by using Lemma 3.9 (b) we see [y, x] ∩ α u (E) ∩ N c(x) = ∅ and x ∈ S(α u (E)).
Since α * u (E) = E ∪ {v}, the equality (14) follows from the following three claims:
(iv) S(α * u (E)) \ {u, v} ⊂ S(E). (v) If u ∈ S(E), then v ∈ S(α * u (E)). (vi) S(E) \ {u, v} ⊂ S(α * u (E)
). To prove (iv), we take x ∈ S(α * u (E)) such that x = u, v. Then there exists y ∈ (α * u (E)) c(x) such that y < x and [y,
x] ∩ α * u (E) ∩ N c(x) = ∅. If y = v, then [u, x] ∩ E ∩ N c(x) ⊂ [v, x] ∩ E ∩ N c(x) = ∅, hence x ∈ S(E). If y ∈ E, then [y, x] ∩ E ∩ N c(x) ⊂ [y, x] ∩ α * u (E) ∩ N c(x) = ∅, hence x ∈ S(E).
Next we prove (v). Since u ∈ S(E), there exists z ∈ E c(u) such that z < u and (u) . Hence by using Lemma 3.9 (b), we see [y, x]∩α * u (E)∩N c(x) = ∅ and x ∈ S(α * u (E)). Proof. First we proceed by induction on #S to prove D ⊔ S ∈ E * P (F ). If S = ∅, then we have D ∈ E P (F ) ⊂ E * P (F ). If S = ∅, we take an element x ∈ S and put S ′ = S \ {x}. By the induction hypothesis and Proposition 3.11, we have D ⊔ S ′ ∈ E * P (F ) and w * D⊔S ′ = w F . Using Proposition 3.8 (a), we see that there exists y ∈ D c(x) such that y < x and [y, x] ∩ D ∩ N c(x) = ∅. Then it follows from Lemma 3.14 that
[z, u] ∩ E ∩ N c(u) = ∅. Then we have [z, v] ∩ α * E ∩ N c(v) ⊂ [z, u] ∩ α * (E) ∩ N c(x) = ∅, hence x ∈ S(α * u (E)). To prove (vi), we take x ∈ S(E) such that x = u, v. Then there exists y ∈ E c(x) such that y < x and [y, x] ∩ E ∩ N c(x) = ∅. Since [v, u] ∩ E ∩ N c(u) = ∅ and x ∈ E, we have x ∈ [v, u]∩N c[y, x] ∩ (D ⊔ S) ∩ N c(x) = ∅. If y = p k , x = p l and {j : k < j < l, p j ∈ D ∪ S} = {j 1 , . . . , j m } (j 1 < · · · < j m ), then we have w * D⊔S = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * s(p l ) * · · · , w * D⊔S ′ = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * · · · .
By using Lemma 3.6 and s(p k ) * s(p l ) = s(p k ) * s(p k ) = s(p k ), we obtain w * D⊔S = w * D⊔S ′ = w F . Hence by Proposition 3.11 we have D ⊔ S ∈ E * P (F ). Next we put E = D ⊔ S and prove that S = S(E). In order to show the inclusion S ⊂ S(E), we take x ∈ S ⊂ B(D). Then by Proposition 3.8 (a), there exists y ∈ D c(x) such that [y, x] ∩ D ∩ N c(x) = ∅. Hence by using Lemma 3.14 we see that [y, x] ∩ E ∩ N c(x) = ∅ and x ∈ S(E). In order to show the reverse inclusion S(E) ⊂ S, we take x ∈ S(E) and prove x ∈ B(D). By the definition (12), there exists y ∈ E c(x) such that y < x and [y, x]∩E ∩N c(x) = ∅. Since D ⊂ E, we have [y, x]∩D∩N c(x) = ∅. If y ∈ D, then by Proposition 3.8 (a) we have x ∈ B(D). If y ∈ S, then there exists z ∈ D c(y) such that z < y and [z, y] ∩ D ∩ N c(y) = ∅. Then by using Lemma 3.9 (a) we have [z, x] ∩ D ∩ N c(x) = ∅ and x ∈ B(D).
Lemma 3.17. Let E ∈ E * P (F ) and z ∈ S(E). If we put E ′ = E \ {z}, then we have (a) E ′ ∈ E * P (F ). (b) S(E ′ ) = S(E) \ {z}.
Proof. By the definition (12), there exists w ∈ E c(z) such that w < z and [w, z] ∩ E ∩ N c(z) = ∅.
(a) If w = p k , z = p l and {j : k < j < l, p j ∈ E} = {j 1 , . . . , j m } (j 1 < · · · < j m ), then we have w * E = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * s(p l ) * · · · , w * E ′ = · · · * s(p k ) * s(p j 1 ) * · · · * s(p jm ) * · · · . By using Lemma 3.6 and s(p k ) * s(p l ) = s(p k ), we obtain w * E = w * E ′ . Hence it follows from Proposition 3.11 that E ′ ∈ E * P (F ). (b) First we prove that S(E ′ ) ⊂ S(E). Let x ∈ S(E ′ ). Then there exists y ∈ (E ′ ) c(x) such that y < x and [y,
x] ∩ E ′ ∩ N c(x) = ∅. Since y ∈ E and [w, z] ∩ E ∩ N c(z) = ∅, we have y ∈ [w, z] ∩ N c(z)
. Hence by using Lemma 3.9 (d) we have [y, x] ∩ E ∩ N c(x) = ∅ and x ∈ S(E).
Next we prove that S(E) \ {z} ⊂ S(E ′ ). We take an element x ∈ S(E) such that x = z. Then there exists y ∈ E c(x) such that y < x and [y,
x] ∩ E ∩ N c(x) = ∅. If y = z, then y ∈ E ′ and [y, x] ∩ E ′ ∩ N c(x) = ∅, hence x ∈ S(E ′ ). We consider the case where y = z. In this case [z, x] ∩ E ′ ∩ N c(x) = ∅. Since [w, z] ∩ E ′ ∩ N c(z) = ∅, it follows from Lemma 3.9 (a) that [w, x] ∩ E ′ ∩ N c(x) = ∅ and x ∈ S(E ′ ).
Now we are in position to give a proof of Proposition 3.13.
Proof of Proposition 3.13. By using Lemma 3.16, it is enough to show that any E ∈ E * P (F ) can be written as E = D ⊔ S with D ∈ E P (F ) and S ⊂ B(D). Given E ∈ E * P (F ), we put D = E \ S(E) and prove that D ∈ E P (F ) and S(E) ⊂ B(D). We proceed by induction on #S(E). If S(E) = ∅, then E ∈ E P (F ) by Lemma 3.15. We consider the case where S(E) = ∅. Then we take z ∈ S(E) and put
E ′ = E \ {z}. Since S(E ′ ) = S(E)\{z} by Lemma 3.17 (b), we see that D = E \S(E) = E ′ \S(E ′ ).
By the induction hypothesis, D ∈ E P (F ) and S(E ′ ) ⊂ B(D). It remains to show that z ∈ B(D). By the definition (12), there exists w ∈ E c(z) such that w < z and [w, z] ∩ E ∩ N c(z) = ∅. Let w be the minimal such element. If w ∈ D, i.e., w ∈ S(E), then by definition there exists w ′ ∈ E c(w) such that [w ′ , w] ∩ E ∩ N c(w) = ∅. Then it follows from Lemma 3.9 (a) that [w ′ , z] ∩ E ∩ N c(z) = ∅, which contradicts to the minimality of w. Therefore we have w ∈ D and z ∈ B(D). (c) If u = (i, j) ∈ D is D-active, then we define
B(α u (D)) = B(D) \ {(i, j + 1), (i + 1, j)} ∪ {(i, j)} if i < j, B(D) \ {(i, j + 1)} ∪ {(i, j)} if i = j.
This notion of excited diagrams is the same as Ikeda-Naruse's excited diagrams of type I introduced in [6]. For example, if P = S(5, 4, 2, 1) and F = S(3, 1), then there are 10 excited diagrams of F in P as a heap for the type B Weyl group. See Figure 7.
− −− → × × − −− → × × − −− → × − −− → × × × − −− → × × × − −− → × × − −− → × ×
Equivariant K-theory and localization
Let A = (a ij ) i,j∈I be a symmetrizable generalized Cartan matrix, and Γ the corresponding Dynkin diagram with node set I. Then the associated Kac-Moody group over C is constructed from the following data: the weight lattice Z-module Λ, the simple roots Π = {α i : i ∈ I}, the simple coroots Π ∨ = {α ∨ i : i ∈ I}, and the fundamental weights {λ i : i ∈ I} (see the beginning of Subsection 2.2).
In what follows, we fix a subset J of I. Let B be a Borel subgroup corresponding to the positive system Φ + and T ⊂ B a maximal torus. Let P − be the opposite parabolic subgroup corresponding to the subset J, which contains the opposite Borel subgroup B − such that B ∩ B − = T . Then we can introduce the Kashiwara thick partial flag variety X = G/P − . (We refer the readers to [10] for a construction of X .)
Let W J be the parabolic subgroup of W corresponding to J and W J be the set of minimum length coset representatives of W/W J . For each element v ∈ W J , we put X • v = BvP − /P − and X v = X • v , the Zariski closure of X • v , which are called the Schubert cell and the Schubert variety respectively. Then X v has codimension l(v) in X and
X v = w∈W J , w≥v X • w .
Let K T (X ) be the T -equivariant K-theory of X . Then K T (X ) has a commutative associative K T (pt)-algebra structure. Here the T -equivalent K-theory K T (pt) of a point is isomorphic to the group algebra Z[Λ] with basis {e λ : λ ∈ Λ}, and to the representation ring R(T ) of T . For each v ∈ W J , let [O v ] be the class of the structure sheaf O v of X v in K T (X ) and call it the equivariant Schubert class. Then we have
K T (X ) ∼ = v∈W J K T (pt)[O v ].
Any elements of K T (X ) is a (possibly infinite) K T (pt)-linear combination of the equivariant Schubert classes.
Each w ∈ W J gives a T -fixed point e w = wP − /P − ∈ X . Then the inclusion map ι w : {e w } → X induces the pull-back ring homomorphism, called the localization map at w, ι * w : K T (X ) → K T (e w ) ∼ = Z[Λ]. If L λ is the line bundle on X corresponding to a weight λ ∈ Λ, then the image of the class [L λ ] under the localization map is given by ι * w ([L λ ]) = e wλ . For two elements v, w ∈ W J , we denote by ξ v | w the image of the T -equivariant Schubert class ξ v = [O v ] ∈ K T (X ) under the localization map ι * w :
ξ v | w = ι * w ([O v ]).
Then the Billey-type formula for the equivariant K-theory can be stated as follows:
ξ v | w = (k 1 ,...,kr) (−1) r−l(v) r a=1 1 − e β (ka) ,(15)
where the summation is taken over all sequences (k 1 , . . . , k r ) such that 1 ≤ k 1 < k 2 < · · · < k r ≤ N and s i k 1 * · · · * s i kr = v (with respect to the Demazure product), and β (k) is given by β (k) = s i 1 . . . s i k−1 (α i k ) for 1 ≤ k ≤ N . By using Lemma 3.12 (a), we can deduce the following corollary from Proposition 4.1.
Corollary 4.2. (a) For w ∈ W J , we have ξ w | w = N k=1 1 − e β (k) .(16)
In particular, ξ w | w = 0.
(b) Let v, w ∈ W J . If ξ v | w = 0, then we have v ≤ w in the Bruhat order.
Equivariant K-theoretical Littlewood-Richardson coefficients
We consider the structure constants for the multiplication in K T (X ) with respect to the equivariant Schubert classes. For u, v, w ∈ W J , we denote by c w u,v ∈ K T (pt) the structure constant determined by
[O u ][O v ] = w∈W J c w u,v [O w ]. Lemma 4.3. If c w u,v = 0, then u ≤ w and v ≤ w.
Proof. We use the induction on l(w) to prove that, if u ≤ w or v ≤ w, then c w u,v = 0. Assume that u ≤ w or v ≤ w. By apply the localization map ι * w to
[O u ][O v ] = x∈W J c x u,v [O x ]
and then by using Corollary 4.2 (b), we have
(ξ u | w ) · (ξ v | w ) = x≤w c x u,v ξ x | w .
If there exists an element x ∈ W J satisfying x < w and c x u,v = 0, then we have u ≤ x and v ≤ x by the induction hypothesis, and hence u ≤ w and v ≤ w, which contradicts to the assumption. Hence, by using Corollary 4.2 (b) and the assumption, we have 0 = c w u,v ξ w | w . Since ξ w w = 0 (Corollary 4.2 (a)), we obtain c w u,v = 0.
Proposition 4.4. For v, w ∈ W J , we have c w v,w = ξ v | w .(17)
Proof. By apply the localization map ι
* w to [O v ][O w ] = x∈W J c x u,v [O x ]
and then by using Corollary 4.2 (b), we have
(ξ v | w ) · (ξ w | w ) = x≤w c x v,w ξ x | w .
By Lemma 4.3, we see that c x v,w = 0 unless w ≤ x. By Corollary 4.2 (b), we see that ξ x | w = 0 unless x ≤ w. Hence we have
(ξ v | w ) · (ξ w | w ) = c w v,w ξ w | w .
Since ξ w w = 0 (Corollary 4.2 (a)), we obtain the desired equality. The following lemma gives a recurrence of the equivariant K-theoretical Littlewood-Richardson coefficients c w u,v . We use the same idea as [20,Corollary 6.5] and [26, Proposition 3.1].
Lemma 4.5. Let u, v, w ∈ W J and s ∈ W J a simple reflection. If c w s,w = c u s,u , then we have
c w u,v = 1 c w s,w − c u s,u u<x≤w c x s,u c w x,v − u≤y<w c w s,y c y u,v .
In particular, we have
c w u,w = 1 c w s,w − c u s,u u<x≤w c x s,u c w x,w .(18)
Proof. Consider the associativity
([O s ][O u ]) [O v ] = [O s ] ([O u ][O v ]) .
Taking the coefficients of [O w ] in the both hand sides and using Lemma 4.3, we have
c u s,u c w u,v + u<x≤w c x s,u c w x,v = c w s,w c w u,v + u≤y<w c w s,y c y u,v ,
from which we get the conclusion.
The Chevalley formula give a combinatorial expression of c w s,v for a simple reflection s. To state the Chevalley formula of [19] we need several notations. For a dominant weight λ ∈ Λ, we put
H λ = {(γ ∨ , k) : γ ∨ ∈ Φ ∨ + , 0 ≤ k < γ ∨ , λ }.
Fix a total order on I so that I = {i 1 < · · · < i r }, and define a map ι :
H λ → Q r+1 by ι r j=1 c i α ∨ i j , k = 1 γ ∨ , λ (k, c 1 , · · · , c r ) .
Then it is known that ι is injective. We define a total ordering < on H λ by
h < h ′ ⇐⇒ ι(h) < lex ι(h ′ ),
where < lex is the lexicographical ordering on Q r+1 . For h = (γ ∨ , k), we define affine transformations r h and r h on Λ by
r h (µ) = µ − γ ∨ , µ γ, r h (µ) = r h (µ) + γ ∨ , λ − k γ.
Note that r h = s γ . Now we can state the Chevalley formula for the equivariant K-theory of the partial flag variety X .
Proposition 4.6. ( [19,Theorem 4.8 (4.12) and (4.13)], see also [17,Corollary 7.1]) Let s be a simple reflection such that s ∈ W J and v, w ∈ W J . If s = s i and λ = λ i is the corresponding fundamental weight, then we have
c w s,v = 1 − e λ−vλ if w = v, (h 1 ,··· ,hr) (−1) r−1 e λ−v r h 1 ··· r hr λ if w > v, 0 otherwise,(19)
where the summation is taken over all sequences (h 1 , · · · , h r ) of length r ≥ 1 satisfying the following two conditions:
(H1) h 1 > h 2 > · · · > h r in H λ , (H2) v ⋖ vr h 1 ⋖ vr h 1 r h 2 ⋖ · · · ⋖ vr h 1 · · · r hr = w is a saturated chain in W J .
Connection to d-complete posets
In this subsection we rephrase the Billey-type formula and the Chevalley-type formula in terms of combinatorics of d-complete posets. Let P be a connected d-complete poset with top tree Γ. We regard Γ as a simplylaced Dynkin diagram with node set I. Let α P and λ P be the simple root and the fundamental weight corresponding to the color i P of the maximum element of P . We apply the above argument to the Kashiwara thick partial flag variety X = G/P − , where P − is the maximal parabolic subgroup corresponding to J = I \ {i P }. In this case, the parabolic subgroup W J coincides with the stabilizer W λ P of λ P in W , and the minimum length coset representatives W J is denoted by W λ P .
By using a labeling of the elements of P with p 1 , · · · , p N (N = #P ) so that p i < p j in P implies i < j, we can associate to each subset D = {i 1 , . . . , i r } (i 1 < · · · < i r ) of P a well-defined element w D = s(p i 1 ) · · · s(p ir ) ∈ W . Then the following formula is obtained from the Billey-type formula.
Proposition 4.7. Let P be a connected d-complete poset and F an order filter of P . Then we have
ξ w F | w P = E∈E * P (F ) (−1) #E−#F p∈E (1 − z[H P (p)]) ,(20)
under the identification z i = e α i (i ∈ I).
Proof. Follows from Proposition 4.1 by using Proposition 2.10 (b) and Proposition 3.11.
Also the following explicit expression is obtained from the Chevalley-type formula.
Proposition 4.8. Let P be a connected d-complete poset and put s = s i P . For two order filters F and F ′ of P , we have
c w F ′ s,w F = 1 − z[F ] if F ′ = F , (−1) #(F ′ \F )−1 z[F ] if F ′ F and F ′ \ F is an antichain, 0 otherwise,(21)
under the identification z i = e α i (i ∈ I).
First we consider the case r = 1 in Proposition 4.6.
Lemma 4.9. Let F be an order filter of P and h = (γ ∨ , k) ∈ H λ P . If w F r h ∈ W λ P and w F ⋖ w F r h ≤ w P , then there exists p ∈ P such that F ⊔ {p} is an order filter of P , w F r h = w F ′ and γ = γ(p). In this case k = 0 and r h λ P = λ P .
Proof. Since the interval [e, w P ] in W λ P is isomorphic to the poset of order filters of P (Proposition 2.11 (a)), there exists a unique order filter F ′ of P such that F ′ ⊃ F , #F ′ = #F + 1 and w F ′ = w F r h . Hence we have p ∈ P such that F ′ = F ⊔ {p} and w F ′ = s(p)w F . We take a linear extension of P such that F = {p n+1 , · · · , p N } with N = #P and n = #(P \ F ). If p = p m , then p is incomparable with p m+1 , · · · , p n , hence s(p) is commutative with s(p m+1 ), · · · , s(p n ) by Property (C5) in Proposition 2.5. Hence we have
γ = w −1 F α(p m ) = s(p N ) · · · s(p n+1 )α(p m ) = s(p N ) · · · s(p n+1 )s(p n ) · · · s(p m+1 )α(p m ) = γ(p m ).
By Proposition 2.10 (c), we see that k = 0 and r h λ P = λ P . Now we deduce Proposition 4.8 from the Chevalley-type formula.
Proof of Proposition 4.8. It follows from Proposition 2.11 (b) that
c w F s,w F = 1 − z[F ].
Suppose that there exists a sequence (h 1 , · · · , h r ) of elements in H λ P satisfying Conditions (H1) and (H2) in Proposition 4.6. Then by Lemma 4.9, we have a sequence (q 1 , · · · , q r ) of elements of P such that F i = F ⊔ {q 1 , · · · , q i } is an order filter of P , h i = (γ ∨ (q i ), 0) for 1 ≤ i ≤ r and r h 1 · · · r hr λ P = λ P . Now we show that {q 1 , · · · , q r } is an antichain. Assume to the contrary that there exist i and j such that i < j and q i and q j are comparable. Since q i is maximal in P \ (F ⊔ {q 1 , · · · , q i−1 }) and q j ∈ P \ (F ⊔ {q 1 , · · · , q i−1 }), we have q i > q j . Then by Proposition 2.10 (a), we see that γ ∨ (q i ) < γ ∨ (q j ). Hence by the definition of the total order on H λ P , we have h i < h j , which contradicts to Condition (H1). Moreover it follows from Proposition 2.11 (b) that
e λ P −w F r h 1 ··· r hr λ P = z[F ′ ].
Conversely, suppose that F ′ F and F ′ \ F is an antichain. For q ∈ F ′ \ F , we put h(q) = (γ ∨ (q), 0) ∈ H λ P . Since F ′ \ F is an antichain, we can label the elements of F ′ \ F so that h(q 1 ) > · · · > h(q r ). Then (h(q 1 ), · · · , h(q r )) is the unique sequence satisfying Conditions (H1) and (H2) in Proposition 4.6.
Proof and corollaries of Main Theorem
In this section, we give a proof of the main theorem (Theorem 1.2 in Introduction) and derive several consequences.
Proof of the Main Theorem
Recall the main theorem of this paper:
Theorem 5.1. Let P be a connected d-complete poset and F an order filter. Then the multivariate generating function of (P \ F )-partitions, where P \ F is viewed as an induced subposet of P , is given by
σ∈A(P \F ) z σ = D∈E P (F ) v∈B(D) z[H P (v)] v∈P \D (1 − z[H P (v)]) ,(22)
where D runs over all excited diagrams of F in P .
Theorem 5.1 is a direct consequence of the following two theorems, which describe the ratio ξ w F | w P ξ w P | w P of the localizations of elements in the equivariant K-theory K T (X ) in two ways.
Theorem 5.2. For a connected d-complete poset P and its order filter F , we have
ξ w F | w P ξ w P | w P = σ∈A(P \F ) z σ ,(23)
under the identification z i = e α i (i ∈ I).
Theorem 5.3. For a connected d-complete poset P and its order filter F , we have
ξ w F | w P ξ w P | w P = D∈E P (F ) q∈B(D) z[H P (q)] p∈P \D (1 − z[H P (p)]) ,(24)
under the identification z i = e α i (i ∈ I).
First we prove Theorem 5.2 by using the Chevalley-type formula (Proposition 4.8).
Proof of Theorem 5.2. For an order filter F of P , we put
Z P/F (z) = ξ w F | w P ξ w P | w P , G P/F (z) = σ∈A(P \F ) z σ .
It is clear that Z P/P (z) = G P/P (z) = 1. Hence it is enough to show that Z P/F (z) and G P/F (z) satisfy the same recurrences:
Z P/F (z) = 1 1 − z[P \ F ] F ′ (−1) #(F ′ \F )−1 Z P/F ′ (z),(25)G P/F (z) = 1 1 − z[P \ F ] F ′ (−1) #(F ′ \F )−1 G P/F ′ (z),(26)
where F ′ runs over all order filters such that F F ′ ⊂ P and F ′ \ F is an antichain. First we prove (25). Under the isomorphism of posets given in Proposition 2.11 (a), the interval (w F , w P ] = {z ∈ W λ P : w F < z ≤ w P } corresponds to {F ′ : F ′ is an order filter of P and F F ′ ⊂ P }. Then by using the recurrence (18) and Proposition 4.8, we see that
ξ w F | w P = 1 (1 − z[P ]) − (1 − z[F ]) F ′ (−1) #(F ′ \F )−1 z[F ]ξ w F ′ | w P = 1 1 − z[P \ F ] F ′ (−1) #(F ′ \F )−1 ξ w F ′ | w P .
Next we prove (26). Let M be the set of maximal elements of P \ F . Then we have
F ′ (−1) #(F ′ \F )−1 G P/F ′ (z) = I⊂M,I =∅ (−1) #I−1 G P/(F ⊔I) (z). For I ⊂ M , we put A(P \ F ) I = {σ ∈ A(P \ F ) : σ(x) = 0 for all x ∈ I}.
Then we have G P/(F ⊔I) (z) = σ∈A(P \F ) I z σ .
By the Inclusion-Exclusion Principle, we have
F ′ (−1) #(F ′ \F )−1 G P/F ′ (z) = σ∈A ′ (P \F ) z σ ,
where we put
A ′ (P \ F ) = {σ ∈ A(P \ F ) : σ(x) = 0 for some x ∈ M }.
Given σ ∈ A(P \ F ), let m = min{σ(x) : x ∈ P \ F } and define σ ′ ∈ A(P \ F ) by
σ ′ (x) = σ(x) − m (x ∈ P \ F ).
Then the map σ → (m, σ ′ ) gives a bijection from A(P \ F ) to N × A ′ (P \ F ), and
z σ = z[P \ F ] m · z σ ′ .
Hence we have σ∈A(P \F )
z σ = 1 1 − z[P \ F ] σ∈A ′ (P \F ) z σ .
This completes the proof.
Next we derive Theorem 5.3 from the Billey-type formula (Proposition 4.7). By dividing the both sides by ξ w P | w P = p∈P (1 − z[H P (p)]), we obtain the desired identity (24).
Corollaries of the Main Theorem
First we derive the equivariant cohomology version of Theorem 5.1. This corollary is a skew generalization of Nakada's colored hook formula [22,Corollary 7.2].
Corollary 5.4. Let P be a connected d-complete poset with d-complete coloring c : P → I and F its order filter. Let a = (a i ) i∈I be indeterminates. We put a(p) = a c(p) (p ∈ P ) and define a linear polynomial a H P (u) as follows: Then we have (q 1 ,...,qn) 1 a(q 1 )(a(q 1 ) + a(q 2 )) . . . (a(q 1 ) + · · · + a(q n ))
= D∈E P (F ) v∈P \D 1 a H P (v) ,(27)
where the summation is taken over all linear extensions of P \ F , i.e., all labelings of the elements of P \ F with q 1 , . . . , q n so that q i < q j in P \ F implies i < j.
Proof. Any (P \ F )-partition σ ∈ A(P \ F ) is determined by a nonnegative integer r, an increasing sequence i 1 < · · · < i r of positive integers and an increasing sequence F ⊂ F 0 F 1 · · · F r = P of order filters of P , by the condition
σ(x) = 0 if x ∈ F 0 \ F , i k if x ∈ F k \ F k−1 and 1 ≤ k ≤ r.
Hence we have
σ∈A(P \F ) z σ = F ⊂F 0 F 1 ··· Fr=P 0<i 1 <···<ir z[F 1 \ F 0 ] i 1 z[F 2 \ F 1 ] i 2 · · · z[P \ F r−1 ] ir = F ⊂F 0 F 1 ··· Fr=P z[P \ F 0 ] 1 − z[P \ F 0 ] z[P \ F 1 ] 1 − z[P \ F 1 ] · · · z[P \ F r−1 ] 1 − z[P \ F r−1 ]
. Now by using Theorem 5.1, we have
F ⊂F 0 F 1 ··· Fr=P z[P \ F 0 ] 1 − z[P \ F 0 ] z[P \ F 1 ] 1 − z[P \ F 1 ] · · · z[P \ F r−1 ] 1 − z[P \ F r−1 ] = D∈E P (F ) v∈B(D) z[H P (v)] v∈P \D (1 − z[H P (v)])
. By substituting z i = t a i (i ∈ I) and multiplying the both sides by (1 − t) n with n = #(P \ F ), and then by taking the limit t → 1, we obtain
F =F 0 F 1 ··· Fn=P 1 a P \ F 0 a P \ F 1 · · · a P \ F n−1 = D∈E P (F ) v∈P \D 1 a H P (v) ,
where the summation on the left hand side is taken over all increasing sequences F = F 0 F 1 · · · F n = P of order filters of length n, and a D = p∈D a c(p) for a subset D ⊂ P . Such increasing sequences of order filters are in one-to-one correspondence with linear extensions (q 1 , · · · , q n ) of P \ F by the relation F k = F ∪ {q n , · · · , q n−k+1 } (0 ≤ k ≤ n).
Hence we obtain the desired result. By specializing z i = q for all i ∈ I in (22), and a i = 1 for all i ∈ I in (27), we obtain Corollary 5.6. Let P be a connected d-complete poset and F an order filter of P . We define the hook length h P (u) at u ∈ P as follows:
(i) If u is not the top of any d k -interval, then we define h P (u) = #{w ∈ P : w ≤ u}. Then we have (a) The univariate generating function of (P \ F )-partitions is given by
σ∈A(P \F ) q |σ| = D∈E P (F ) v∈P \D 1 1 − q h P (v) .(28)
(b) The number of linear extensions of P \ F is given by
n! D∈E P (F ) v∈P \D 1 h P (v) ,(29)
where n = #(P \ F ).
If P = D(λ) and F = D(µ) are shapes corresponding to partitions λ ⊃ µ, Equations (28) and (29) reduce to Morares-Pak-Panova's q-hook formula [21, Corollary 6.17] and Naruse's hook formula [25] respectively. The trace generating function of revers plane partitions of skew shape [21, Corollary 6.20] is obtained from Theorem 5.1 by specializing z i = tq if i is the color of the maximum element of D(λ), q otherwise.
Remark 5.7. Theorem 5.1 and its corollaries hold for heaps H(w) associated to dominant minuscule elements w in any symmetrizable Kac-Moody Weyl groups, after suitable modifications are made. See Remarks 2.12 and 3.18.
Example
Example 5.8. Let P = S(3, 2, 1) ⊃ F = S(2) be the shifted shapes corresponding to strict partitions (3, 2, 1) and (2). If we regard P as a d-complete poset with a dcomplete coloring c : P → {0, 0 ′ , 1, 2} given in Example 2.8, then the hook monomials in z = (z 0 , z 0 ′ , z 1 , z 2 ) are given by Since we have
E P (F ) = , × ,
we apply Theorem 5.1 to obtain π∈A(P \F )
z π = 1 (1 − z 0 z 1 z 2 )(1 − z 0 z 0 ′ z 1 )(1 − z 0 z 1 )(1 − z 0 ) + z 0 z 0 ′ z 1 z 2 (1 − z 0 z 0 ′ z 1 z 2 )(1 − z 0 z 1 z 2 )(1 − z 0 z 0 ′ z 1 )(1 − z 0 )
Figure 2 :
2Shape and shifted shape Example 2.3. Let P be a subset of Z 2 given by P =
Figure 3: Swivel
For
example, if µ = (5, 4, 2, 1) and (i, j) = (1, 2), then the corresponding hook is given by H S(5,4,2,1) (1, 2) = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (3, 3), (3, 4)}.
−1 if i = j and i and j are adjacent in Γ, 0otherwise.
Proposition 2 . 10 .
210Let P be a connected d-complete poset. Then we have (a) (See [34, Proposition 3.1 and Theorem 5.5]) The poset P is isomorphic to the order dual of Φ ∨ (w −1 P ) with the standard coroot ordering on Φ ∨ + . (b) (See [29, Lemma IV]) Under the identification z i = e α i (i ∈ I), we have z[H P (p)] = e β(p) (p ∈ P ). (c) (See [34, Proposition 5.1]) We have γ ∨ (p), λ P = 1 (p ∈ P ).
Proposition 2 . 11 .
211Let P be a connected d-complete poset. Then we have (a) (See [29, Proposition I])
Example 3. 4 .
4If P is the swivel given in Example 2.3 and F is the order filter consisting of three elements, then there are 8 excited diagrams in E P (F ) shown inFigure 6.
Figure 5 :
5Excited diagrams of S(3, 1) in S(5, 4, 2, 1) Proposition 3.5. Let P be a connected d-complete poset and let F be an order filter of P . Then a subset D ⊂ P is an excited diagram of F in P if and only if #D = #F and w D = w F .
( b )
bSuppose u ∈ D and v ∈ D. If [y, x] ∩ D ∩ N i = ∅ and x ∈ [v, u] ∩ N j , then we have [y, x] ∩ (D ∪ {v}) ∩ N i = ∅.(c) Suppose u ∈ D and v ∈ D. If [y, x]∩ (D \{u}∪ {v})∩ N i = ∅ and y ∈ [v, u]∩ N j , then we have [y, x] ∩ D ∩ N i = ∅. (d) Suppose that u ∈ D and v ∈ D. If [y, x] ∩ D ∩ N i = ∅ and y ∈ [v, u] ∩ N j , then we have [y, x] ∩ (D ∪ {u}) ∩ N i = ∅. Proof. (a) By using Property (C5) in Proposition 2.5, we have [z, x]
(d) By an argument similar to (b), we can show that, if [y, x] ∩ (D ∪ {u}) ∩ N i = ∅, then v < y < u < x, which contradicts to y ∈ [v, u] ∩ N j .Proof of Proposition 3.8. We denote by B ′ (D) the subset of P consisting of elements x ∈ P satisfying the condition (ii) in (a), and prove B(D) = B ′ (D) and D ∩ B ′ (D) = ∅. We proceed by induction on the number of elementary excitations to reach D from F .
by the induction hypothesis, there exists y ∈ D c(x) such that y < x and [y, x]∩D∩N c(x) = ∅. Then, by using Lemma 3.9 (b), we have [y, x]∩α u (D)∩N c(x) = ∅, hence x ∈ B ′ (α u (D)).
Definition 3 . 10 .
310Let P be a connected d-complete poset and let F be an order filter of P .
Proposition 3 . 11 .
311Let P be a connected d-complete poset and F an order filter of P . Then a subset D ⊂ P is a K-theoretical excited diagram of F in P if and only if w * D = w F . We follow the same idea as the proof of [4, Proposition 4.8]. Lemma 3.12. ([4, Lemma 3.1 and Proposition 3.4])
Proposition 3 . 13 .
313Let P be a connected d-complete poset and F an order filter of P . Then we have E * P (F ) = D∈E P (F ) {D ⊔ S : S ⊂ B(D)}.
Lemma 3 . 14 .
314Let x, y be elements of P such that c(x) = x(y) = i and D a subset of P . If [y, x] ∩ D ∩ N i = ∅ and y ∈ D, then we have [y, x] ∩ B(D) ∩ N i = ∅. Proof. Assume to the contrary that [y, x] ∩ B(D) ∩ N i = ∅ and take an element z ∈ [y, x] ∩ B(D) ∩ N i . By Proposition 3.8 (a), there exists w ∈ D c(z) such that w < z and [w, z] ∩ D ∩ N c(z) = ∅. Then c(w) = c(z) is adjacent to
#S(α u (E)) = #S(E), #S(α * u (E)) = #S(E) + 1, and S(E) = ∅ if and only if E ∈ E P (F ).
Lemma 3 . 16 .
316If D ∈ E P (F ) and S ⊂ B(D), then we have D ⊔ S ∈ E * P (F ) and S = S(D ⊔ S). In particular, if E ∈ E * P (F ) is expressed as E = D ⊔ S = D ′ ⊔ S ′ with D, D ′ ∈ E P (F ) and S ⊂ B(D), S ′ ⊂ B(D ′ ), then we have D = D ′ and S = S ′ .
Remark 3 . 18 .
318We can define the notion of D-active elements and ordinary and K-theoretical elementary excitations for a dominant minuscule heap H(w) just by replacing d k -intervals with intervals [v, u] such that c(u) = c(v) and [v, u]∩H(w) c(u) = {u, v}. Then the arguments in this section work as well for H(w). In particular, Propositions 3.11 and 3.13 holds for H(w).
Example 3 . 19 .
319Let µ be a strict partition and regard the shifted shape S(µ) as a heap for the Weyl group of type B (see Example 2.13). Then the notion of active elements, (ordinary and K-theoretical) elementary excitations and excited peaks are modified as follows. Let D be a subset of S(µ).
element u = (i, j) ∈ D is D-active if either i < j and (i, j + 1), (i + 1, j), (i + 1, j + 1) ∈ S(µ) \ D, or i = j and (i, i + 1), (i + 1, i + 1) ∈ S(µ) \ D.
( b )
bIf u = (i, j) ∈ D is D-active, then we define an ordinary and K-theoretical elementary excitation by puttingα u (D) = D \ {(i, j)} ∪ {(i + 1, j + 1)}, α * u (D) = D ∪ {(i + 1, j + 1)}, respectively.
Figure 7 :
7Excited diagrams in S(5, 4, 2, 1) viewed as a type B heap 4 Equivariant K-theory of Kac-Moody partial flag varieties In this section we review the basic properties of the equivariant K-theory of thick flag varieties following [18, Section 3], and rephrase the Billey-type formula and the Chevalley-type formula in terms of combinatorics of d-complete posets.
Proposition 4. 1 .
1([18, Proposition 2.10]) Let v, w ∈ W J , and fix a reduced expression w = s i 1 s i 2 . . . s i N of w. Then we have
( i )
iIf u is not the top of any d k -interval, then we define a H P (u) = w≤u a c(w) .(ii) If u is the top of a d k -interval [v, u], then we define a H P (u) = a H P (x) + a H P (y) − a H P (v) , where x and y are the sides of [v, u].
Remark 5. 5 .
5Corollary 5.4 can be proved by using the Billey formula [1, Theorem 4] and the Chevalley formula [16, Theorem 11.1.7 and Corollary 11.3.17] for the equivariant cohomology along the same line as Theorem 5.1.
(ii) If u is the top of a d k -interval [v, u], then we define h P (u) = h P (x) + h P (y) − h P (v), where x and y are the sides of[v, u].
z[H P (1, 1)] = z 0 z 0 ′ z 2 1 z 2 , z[H P (1, 2)] = z 0 z 0 ′ z 1 z 2 , z[H P (1, 3)] = z 0 z 1 z 2 , z[H P (2, 2)] = z 0 z 0 ′ z 1 , z[H P (2, 3)] = z 0 z 1 , z[H P (3, 3)] = z 0 .
AcknowledgmentsThe first author was partially supported by the JSPS Grants-in-Aid for Scientific Research No. 16H03921. The second authors gratefully acknowledges the support and hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics, where part of this work was carried out.whereExample 5.9. Let P = S(3, 2, 1) ⊃ F = S(2) be the same as in Example 5.8. If we regard P as the heap H(w (3,2,1) ) for the Weyl group of type B 3 (see Example 2.13), then the hook monomials in z = (z 0 , z 1 , z 2 ) are given bySince we havewe apply a heap version of Theorem 5.1 to obtain π∈A(P \F )where z π = z π(1,1)+π(2,2)+π(3,3) 0 z π(1,2)+π(2,3) 1 z π(1,3) 2.Note that Equation(31)is obtained from (30) by putting z 0 ′ = z 0 .
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|
[] |
[
"Precursors and BRST Symmetry",
"Precursors and BRST Symmetry"
] |
[
"Jan De Boer [email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n",
"Ben Freivogel [email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n",
"Laurens Kabir [email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n",
"Sagar F Lokhande [email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands\n"
] |
[
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands",
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands",
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands",
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam Science\nPark 9041098 XHAmsterdamThe Netherlands"
] |
[] |
In the AdS/CFT correspondence, bulk information appears to be encoded in the CFT in a redundant way. A local bulk field corresponds to many different non-local CFT operators (precursors). We recast this ambiguity in the language of BRST symmetry, and propose that in the large N limit, the difference between two precursors is a BRST exact and ghost-free term. Using the BRST formalism and working in a simple model with global symmetries, we re-derive a precursor ambiguity appearing in earlier work. Finally, we show within this model that this BRST ambiguity has the right number of parameters to explain the freedom to localize precursors within the boundary of an entanglement wedge order by order in the large N expansion.
|
10.1007/jhep07(2017)024
|
[
"https://arxiv.org/pdf/1612.05265v1.pdf"
] | 76,656,978 |
1612.05265
|
01be903248461d98e944afefea93cf617a42eac7
|
Precursors and BRST Symmetry
15 Dec 2016
Jan De Boer [email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam Science
Park 9041098 XHAmsterdamThe Netherlands
Ben Freivogel [email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam Science
Park 9041098 XHAmsterdamThe Netherlands
Laurens Kabir [email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam Science
Park 9041098 XHAmsterdamThe Netherlands
Sagar F Lokhande [email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam Science
Park 9041098 XHAmsterdamThe Netherlands
Precursors and BRST Symmetry
15 Dec 2016
In the AdS/CFT correspondence, bulk information appears to be encoded in the CFT in a redundant way. A local bulk field corresponds to many different non-local CFT operators (precursors). We recast this ambiguity in the language of BRST symmetry, and propose that in the large N limit, the difference between two precursors is a BRST exact and ghost-free term. Using the BRST formalism and working in a simple model with global symmetries, we re-derive a precursor ambiguity appearing in earlier work. Finally, we show within this model that this BRST ambiguity has the right number of parameters to explain the freedom to localize precursors within the boundary of an entanglement wedge order by order in the large N expansion.
Introduction
The AdS/CFT correspondence is the most precise non-perturbative definition of quantum gravity. A central problem is how local bulk physics emerges from CFT data. This question has been studied extensively and is reasonably well-understood at large N , for small perturbations around vacuum AdS [1,2]. In this limit, a bulk field Φ at a point X is defined by integrating a local CFT operator O over the boundary with an appropriate smearing function K [3]:
Φ(X) = dt d d−1 x K(X|t, x)O(x) + O 1 N . (1.1)
This CFT operator can subsequently be time evolved to a single timeslice using the CFT Hamiltonian, which gives a non-local operator P in the CFT corresponding with the field Φ(X) in the bulk. This type of operator is called a 'precursor' [4][5][6].
The study of precursors is fundamental to understanding a concrete realization of holography. There are several unresolved questions one can ask, such as how to construct precursors that correspond to bulk fields behind a black hole horizon. Here we focus on two particular puzzles that are related to each other. At large N, bulk locality requires the precursor to commute with all local CFT operators at a fixed time, while basic properties of quantum field theory demand that only trivial operators can commute with all local operators at a given time [7]. Another is that a local bulk operator corresponds with many different precursors with different spatial support in the CFT, because the bulk field can be reconstructed in a particular spatial region of the CFT as long as it is contained in the corresponding entanglement wedge of that region.
Both of these apparent paradoxes can be resolved by requiring that different precursors are not equivalent as true CFT operators [7]. In particular, the difference between two precursors corresponding to the same bulk field seems to have no clear physical meaning, and must act trivially some class of states. In what follows, we will refer to this perplexing feature as the 'precursor ambiguity'.
In [7] and [8] some progress was made in giving a guiding principle for constructing the ambiguity between two precursors corresponding to the same bulk field. The former approach recasts the AdS/CFT dictionary in the language of quantum error correction (QEC). From this viewpoint, the ambiguity is an operator which acts trivially in the code subspace of QEC, which in this case is naturally thought of as the space of states dual to low-energy excitations of the bulk. The latter work, on the other hand, proposed that gauge symmetry in the CFT can give a prescription to construct the precursor ambiguity. Moreover, they claimed that the code subspace is the full space of gauge invariant states.
In this paper, we start in section 2 by proposing the language of BRST symmetry as a tool for making the precursor ambiguity concrete. In section 3, we show that this approach nicely reduces to an already identified precursor ambiguity in the presence of a global SO(N ) symmetry [8]. Furthermore, it has the added benefit that it generalizes to arbitrary gauge theories at any N . In section 4 we show in a particular toy model how this precursor ambiguity has the right number of parameters to enable us to localize precursors in the boundary of the entanglement wedge order by order in 1/N.
Proposal: Precursor Ambiguities from BRST
In most of the known examples of holography, the boundary theory has some gauge symmetry. The presence of these 'unphysical' degrees of freedom renders the naive path integral for gauge theories divergent. One approach to deal with these problems while covariantly quantizing the gauge theory is the BRST formalism [9,10]. The rough idea is to replace the original gauge symmetry with a global symmetry, by enlarging the theory and introducing additional fields. This new rigid symmetry, the BRST symmetry, will still be present after fixing the gauge. Since the generator of the BRST symmetry Q BRST is nilpotent of order two, we can construct its cohomology which will describe the gauge invariant observables of the original theory.
We propose that the natural framework to understand precursor ambiguities is the language of BRST symmetry. In particular, we claim that if P 1 and P 2 are two precursors in the large N -limit corresponding with the same local bulk field Φ(X), then
P 1 − P 2 = O where • O is BRST exact: O = {Q BRST ,Õ}
• O does not contain any (anti-)ghosts.
By construction this leaves any correlation function of gauge invariant operators in arbitrary physical states invariant
O 1 · · · O i · · · O n = O 1 · · · (O i + {Q BRST ,Õ}) · · · O n (2.1) since [Q BRST , O i ] = 0
for a gauge invariant operator O i , and Q BRST |ψ = 0 for a gauge invariant state |ψ .
As an example, we will show in section 3 that in the case of N free scalars with a global SO(N ) symmetry, we can reproduce the results of [8]. That means, there exists an operatorÕ such that
{Q BRST ,Õ} ∼ L ij A ij (2.2)
where L ij is the generator of the SO(N ) symmetry, and A ij is any operator in the adjoint.
Note that while the BRST ambiguity is well-defined for any gauge theory and even at finite N , the notion of bulk locality only makes sense perturbatively in 1/N . In order to connect the abstract BRST ambiguity to concrete equivalences between different CFT operators, we need to make use of the large N expansion. Thus the precursor ambiguity we find is valid within states where the number of excitations is small compared to N .
BRST Symmetry of N Real Scalars
In this section we will apply the BRST formalism to a theory of N real scalars. The Lagrangian for this gauge theory in the covariant gauge is given by
L = − 1 4 (F a µν ) 2 + 1 2 D µ φ i D µ φ i + ξ 2 (B a ) 2 + B a ∂ µ A a µ + ∂ µca (D µ c) a (3.1)
where the auxiliary field B a can be integrated out using ξB a = −∂ µ A a µ . We take the φ i in the fundamental representation of SO(N ), while the ghost c a , anti-ghostc a and the gauge field A a µ are in the adjoint. The (anti-)ghosts are scalar fermion fields. The covariant derivatives are given by
(D µ c) a = ∂ µ c a + gf abc A b µ c c (3.2)
and
(D µ φ) i = ∂ µ φ i − igA a µ (T a ) ij φ j . (3.3)
Note that D µ φ i is real since the matrices (T a ) ij are purely imaginary for SO(N ). The field strength F is given by
F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν . (3.4)
This Lagrangian is invariant under the following BRST symmetry:
δ B A a µ = (D µ c) a δ B φ i = ig c a (T a ) ij φ j δ B c a = − 1 2 g f abc c b c c (3.5) δ Bc a = B a δ B B a = 0.
where is a constant Grassmann parameter.
The BRST Charge
In order to compute the BRST charge, we start by constructing the Noether current associated to this symmetry
J µ = α δL δ(∂ µ Φ α ) δ B Φ α (3.6)
where the sum runs over all possible fields in the Lagrangian. The BRST charge is then defined via
Q B = d d−1 x J 0 B (3.7)
and generates the BRST transformations on the fields via
δ B Φ α = [Φ α , Q BRST ] ± . (3.8)
Let's start by computing the variations and defining the conjugate momenta
δL δ(∂ µ φ i ) = D µ φ i Π i ≡ D 0 φ i [φ i (x), Π j (y)] = δ ij δ (d−1) (x − y) (3.9) δL δ(∂ µ c a ) = (∂ µc ) a π a c ≡ (∂ 0c ) a {c a (x), π b c (y)} = δ ab δ (d−1) (x − y) (3.10) δL δ(∂ µc a ) = (D µ c) a π ā c ≡ (D 0 c) a {c a (x), π b c (y)} = δ ab δ (d−1) (x − y) (3.11)
and finally for the gauge field
δL δ(∂ µ A a ν ) = −F a µν + η µν B a Π a ν ≡ −F a 0ν + η 0ν B a (3.12) with commutation relation [A a µ (x), Π b ν (y)] = η µν δ ab δ (d−1) (x − y). (3.13)
That gives the following Noether current
J µ = −F a µν + η µν B a (D ν c) a + igD µ φ i c a (T a ) ij φ j − 1 2 g(∂ µc a )f abc c b c c + (D µ c) a B a . (3.14)
The BRST charge is then given by
Q BRST = dx d−1 Π a ν (D ν c) a + igΠ i c a (T a ) ij φ j − 1 2 gf abc π a c c b c c + B a π ā c (3.15) = dx d−1 Π a ν (∂ ν c) a − gf abc A b ν Π c ν c a + igΠ i c a (T a ) ij φ j − 1 2 gf abc π a c c b c c + B a π ā c .
We can define the generators of the SO(N ) symmetry, as the Noether currents associated with the gauge transformations. The current has two contributions, one from the Yang-Mills parts F 2 and one from the matter part (Dφ) 2 :
J a matter ≡ i Π i (T a ) ij φ j J a gauge ≡ −f abc A b µ Π c µ (3.16) J a ≡ J a matter + J a gauge . (3.17)
This finally leads to the the BRST charge:
Q BRST = dx d−1 gc a J a − 1 2 gf abc π a c c b c c + B a π ā c + Π a ν (∂ ν c) a . (3.18)
Reduction to a Global SO(N ) Symmetry
In order to connect with previous work on precursors [8], we are interested in degrading the SO(N ) gauge symmetry to a global symmetry. One crude way of accomplishing this, is by setting the gauge fields A a µ = 0 (and also B a = 0 since B a ∼ ∂ µ A a µ ). In this case, the ghosts become quantum mechanical (position independent) and the BRST charge reduces to
Q BRST = dx d−1 gc a J a − 1 2 gf abc π a c c b c c J a = i Π i (T a ) ij φ j (3.19)
where the global SO(N ) generator is given by
L a = d d−1 x J a (x).
Now consider an operator of the form π a c O a and compute the anti-commutator with the BRST charge:
{Q BRST , π d c O d } = dx d−1 g{c a J a , π d c O d } − 1 2 gf abc {π a c c b c c , π d c O d } (3.20) = g dx d−1 O a J a = gL a O a (3.21)
where we used that the generator of global SO(N ) transformations rotates the operator
O as [J a , O b ] = f abc O c .
This expression is BRST exact by construction, and ghost-free. Adding this to a CFT operator will have no effect whatsoever within correlation functions in physical states. It is exactly the precursor ambiguity found in [8].
Localizing Precursors in a Holographic Toy Model
In the previous section, we computed the ambiguous part of the precursors as a BRST exact and ghost-free operator. This ambiguity can be viewed as the redundant, quantum error correcting part of the precursors. Once it has been identified, the physical information contained in the precursors becomes clear. In this section we will study the particular ambiguity (3.21) in a toy model. We will show that this ambiguity has the structure of an HKLL series, and that it contains enough freedom to localize bulk information in a particular region of the CFT by setting the smearing function to zero in that region.
The Model
The model is a CFT containing N free scalar fields in 1 + 1 spacetime dimensions:
L = N i=1 − 1 2 ∂ µ φ i ∂ µ φ i . (4.1)
It was first considered by [8] and refined in [11]. There is a ∆ = 2 primary operator O = ∂ µ φ i ∂ µ φ i which we take to be dual to a massless scalar Φ in AdS 2+1 .
Following [8] and (3.21), the precursor ambiguity is given by L ij A ij where A ij is any operator in the adjoint of SO(N ) and L ij is the generator of global SO(N ) transformations. Note that we only kept the global part of the SO(N ) transformations by setting A a µ = B a = 0 in the full gauge theory discussed in section 3.
Expanding the boundary field φ in terms of left/right-moving creation and annihilation modes, one can compute the generator of global rotations
L ij = dk 2 k α † [i k α j] k +α † [i kα j] k (4.2)
where the tilde denotes a right-moving polarization of the creation or annihilation modes and any zero modes are left out. If there is no confusion what momentum a given mode has, we will omit the subscript k.
Precursor Ambiguity and Bulk Localization Perturbatively in 1/N
The bulk field Φ in global AdS 3 can be constructed at large N by smearing quadratic operators of the form O ∼ α kαk over a particular region of the CFT [3]:
Φ(X) = d 2 x K 1 (X|x) O(x) + O 1 √ N (4.3)
where the smearing function K obeys the bulk free wave equation
AdS 3 K 1 (X|x) = 0. (4.4)
This procedure correctly reproduces the bulk two-point function. The precursor can be obtained from (4.3) by time evolving the CFT operator to a single timeslice. Extending the HKLL procedure perturbatively in 1/N will look schematically as follows [12,13]:
Φ(X) = K 1 O + 1 √ N K 2 OO + O 1 N (4.5)
where the expansion parameter is 1/ √ N instead of 1/N because we are dealing with a vector-like theory [14].
In [11] it was shown that, at leading order in 1/N , the spatial support of the smearing function K 1 (and hence the information of the bulk field) can be localized in a particular Rindler wedge of the CFT due to an ambiguity in the smearing function. This freedom can be understood by noting that the term α †i k 1α i k 2 can be added to O within two-point functions since it annihilates the vacuum in both directions. While this two-parameter family of freedom is enough to localize the bulk field at leading order in N , one can see that it generically will be insufficient to set K 2 to zero in particular region, because this requires a four-parameter family of freedom. Since changing the smearing function corresponds with picking a different precursor, we would like to identify the aforementioned freedom in the smearing function with the precursor ambiguity. In what follows, we will explain how the precursor ambiguity L ij A ij has enough freedom to localize bulk information order by order in 1/N .
Start by considering the following quadratic (adjoint) operator
A ij 2 ≡ α †i k 1α j k 2 . (4.6)
A possible ambiguity of the precursor will be given by L ij A ij 2 . Normal ordering yields
1 N 3 2 L ij A ij 2 = 1 N 3 2 dk 2 k α † [i k α j] k +α † [i kα j] k α †i k 1α j k 2 = (1 − N ) N 3 2 α †i k 1α i k 2 + 1 √ N α †i k 1 L ijα j k 2 N ∼ O + 1 √ N OO (4.7)
where O denotes an operator quadratic in the α's and normalized by 1/ √ N such that it is O(1) in N -scaling. Note that the LHS of (4.7), by construction, is zero in physical states (and hence can be added to the precursor without changing any of its correlation functions).
The piece quadratic in the α s in (4.7) is exactly the ambiguity needed to localize the precursor in the CFT to leading order in N , as was shown in detail in [11]. One can now also see that one generically needs a four-parameter ambiguity if we want to be able to set K 2 in (4.5) to zero in certain regions. Even though the term OO/ √ N in (4.7) has the right structure to fit in the HKLL series, it does not have enough freedom to set K 2 to zero (it has only 2 free parameters, while we need 4). It can be done, however, by constructing a new operator which annihilates SO(N )-invariant states and is quartic in the α's:
A ij 4 ≡ A ij 2 − 1 N A ij 2 α † m k 3 α m k 4 . (4.8)
The ambiguity in the precursor to order 1 √ N is then given by L ij A ij 4 . Normal ordering yields
L ij A ij 4 = L ij A ij 2 + T 4 + T 6 (4.9)
where
T 4 = α † i k 1 α † i k 3α m k 2 α m k 4 − α † i k 3α i k 2 α † m k 1 α m k 4 + (1 − N ) α † i k 1α i k 2 α † m k 3 α m k 4
(4.10)
T 6 = α † i k α † i k 1 α † m k 3α j k 2 α j k α m k 4 − α † j k α † i k 1 α † m k 3α j k 2 α i k α m k 4 +α † i k α † i k 1 α † m k 3α j k 2α j k α m k 4 −α † j k α † i k 1 α † m k 3α j k 2α i k α m k 4 (4.11)
and repeated momenta are integrated over appropriately. By T 4 we denote the ambiguity to quartic order in L ij A ij 4 and similarly with T 6 to hexic order. As before, T 4 and T 6 scale the same with respect to N in any gauge invariant state. Also they do not contribute in three-point functions of the bulk field.
Again we find that all the terms nicely arrange themselves in the right structure of an HKLL series
L ij A ij 4 ∼ O + 1 √ N OO + 1 N OOO (4.12)
where O schematically denotes an operator quadratic in the α's and normalized by 1/ √ N such that it is O(1) in N -scaling. The main difference with L ij A ij 2 is that the term quartic in the α's now gets a contribution from T 4 , which does have four independent parameters, and hence has enough freedom to localize the smearing function K 2 .
Doing so also introduced a term like α 6 . The connected piece of this will be down in 1/N relative to α 4 . If T 4 fixes the ambiguity at order 1/ √ N , T 6 will contribute towards fixing it at order 1/N . Thus, by choosing a proper operator A ij , we will be able to fix the ambiguity in the precursor to any order in 1/N perturbatively.
We can now summarize how this recursive procedure works to localize bulk information order by order in N . When the operator we want to smear A ij 2 is quadratic, the ambiguity in the precursor to the quadratic order is given by (1 − N ) α †i k 1α j k 2 . These modes are labeled by two different momenta. Since we are working in two spacetime dimensions, they are able to fix all the ambiguity in the precursor up to quadratic level.
But fixing the quadratic level, introduces a quartic piece: α †i k 1 L ijα j k 2 . This piece has insufficient freedom to localize the precursor up to 1/ √ N effects. To fix the ambiguity to the quartic level, one introduces a quartic ambiguity L ij A ij 4 . This gives a piece T 4 which has four independent momenta and hence can now fix any ambiguity in the precursor up to quartic order. However, doing so also introduced a hexic piece T 6 . This hexic term makes the precursor ambiguous to order six. We can repeat the procedure, smear a different A ij and then fix the ambiguity in the precursor up to order six. Surprisingly, each term at a higher order is 1 √ N relative to the current order. Hence, this procedure can be carried out order by order in 1 √ N and thus fixes all the ambiguity in the interacting HKLL series in this toy model. While it is not explicitly demonstrated in this paper, a similar story should hold when the matter fields are in the adjoint.
One should note that, while the quadratic and quartic piece in the ambiguity (4.7) (and similarly for the quartic and hexic piece in the ambiguity (4.12)) have the correct 'naive' N -scaling (α ∼ N 1 4 ) to be arranged in an HKLL series, their real N -scaling is the same. This means that neither term in (4.7) or (4.12) is smaller compared to the other. For clarity, we will elaborate on this a bit more in the next section 4.3.
N -Scaling
Within physical states, both terms on the RHS of (4.7) will be equal and opposite. In particular, they must have the same N -scaling (in contrary to what was claimed in [8]), even though naive N -counting would suggest otherwise. In order to explicitly see that both terms have the same N -scaling in SO(N )-invariant states, we pick the following three states and label the operators as follows:
States
Operators
|ψ 1 = 1 √ N α †m k 3 α †m k 4 |0 O 1 = α †i k 1 L ijα j k 2 /N 3 2 |ψ 1 = 1 √ Nα †m k 3α †m k 4 |0 O 2 = α †i k 1α i k 2 / √ N |ψ 2 = 1 √ Nα †m k 5 α †m k 6 |0
In order to assign a N -scaling to O 2 , one could check its two-point function. However, since this operator has vanishing two-point functions, we investigate the three-point function and find that it goes like 1/ √ N . This justifies us to call assign an O(1) N -scaling to O 2 . We will estimate the size of O 1 and O 2 in the subspace spanned by the three states above. Let us denote the matrix elements of an arbitrary operator O in the above subspace as
O = ψ 1 |O|ψ 1 ψ 1 |O|ψ 1 ψ 1 |O|ψ 2 ψ 1 |O|ψ 1 ψ 1 |O|ψ 1 ψ 1 |O|ψ 2 ψ 2 |O|ψ 1 ψ 2 |O|ψ 1 ψ 2 |O|ψ 2 .
Then we get the following matrix elements for O 1 and O 2
O 1 = 1 √ N 0 0 1 0 0 0 0 1 0 O 2 = 1 √ N 0 0 1 0 0 0 0 1 0 . (4.13)
We can see that both the pieces in L ij A ij 2 scale in the same way with respect to N , as expected. Naively, one could expect the part quartic in the α's to be down to part quadratic in the α's by a factor 1/ √ N . For these particular operators that doesn't happen, because the disconnected piece in O 1 enhances its N -scaling.
Applying similar arguments to (4.12), we conclude T 6 must have the same N -scaling as T 4 . Again, the reason why this does not agree with naive N -scaling, is due to the contribution from the disconnected piece in T 6 .
Outlook
In this paper we have presented preliminary evidence that precursors are related to BRST invariance and hence to the underlying gauge symmetry of the field theory. There are several interesting follow-up directions to explore. One could for example study precursors in the toy model in non-trivial states (such as thermal states), but more importantly, one would like to generalize the construction to a proper gauge theory with local gauge invariance. Perhaps the simplest example of a field theoretic precursor ambiguity is to consider the field theoretic dual of the bulk operator one obtains by integrating a bulk field over a symmetric minimal surface. Such operators were studied in [15,16], and to lowest order in the 1/N expansion in the field theory for a bulk scalar they are given by
Q O (x, y) = C D(x,y) d d ξ (y − ξ) 2 (ξ − x) 2 −(y − x) 2 (∆ O −d) 2 O(ξ) (5.1)
where the integral is over the causal diamond D(x, y) with past and future endpoints x and y, and ∆ O is the scaling dimension of the primary operator O. The constant C is a normalization constant which at this point is arbitrary. The past light-cone of y and the future light-cone of x intersect at a sphere B, which is the boundary of the bulk minimal surface.
If the field theory is defined on S d−1 ×R, then there are two equivalent choices of causal diamonds for a given symmetric minimal surface. Together, they contain a full Cauchy slice for the field theory. Hence, there are two inequivalent boundary representations of the same bulk operator, and the difference between these two is an example of a precursor ambiguity. We would therefore like to conjecture that there exists an operator Y such that
{Q BRST , Y } = D(x,y) d d ξ (y − ξ) 2 (ξ − x) 2 −(y − x) 2 (∆ O −d) 2 O(ξ) (5.2) − D (x,ȳ) d dξ (ȳ −ξ) 2 (ξ −x) 2 −(ȳ −x) 2 (∆ O −d) 2 O(ξ) + O(1/N ) (5.3)
Here, the second complimentary causal diamond is denoted by D(x,ȳ) with past and future endpointsx andȳ. It would be very interesting to construct an operator Y for which (5.2) holds, and we hope to come back to this in the near future.
AcknowledgmentsWe thank Vladimir Rosenhaus for helpful discussions. This work is part of the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). SFL would like to acknowledge financial support from FOM, which is part of the NWO.
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|
[] |
[
"Scattering approach to Impurity Thermodynamics",
"Scattering approach to Impurity Thermodynamics"
] |
[
"Pankaj Mehta \nCenter for Materials Theory\nRutgers University\n08854PiscatawayNJ\n",
"Natan Andrei \nCenter for Materials Theory\nRutgers University\n08854PiscatawayNJ\n"
] |
[
"Center for Materials Theory\nRutgers University\n08854PiscatawayNJ",
"Center for Materials Theory\nRutgers University\n08854PiscatawayNJ"
] |
[] |
Recently the authors developed a scattering approach that allows for a complete description of the steady-state physics of quantum-impurities in and out of equilibrium. Quantum impurities are described using scattering eigenstates defined ab initio on the open, infinite line with asymptotic boundary conditions imposed by the leads. The scattering states on the open line are constructed for integrable quantum-impurity models by means of a significant generalization of the Bethe-Ansatz which we call the Scattering Bethe-Ansatz (SBA). The purpose of the paper is to present in detail the scattering approach to quantum-impurity models and the SBA and show that they reproduce well-known thermodynamic results for several widely studied models: the Resonance Level model, Interacting Resonance Level model and the Kondo model. Though the SBA is more complex than the traditional Thermodynamic Bethe Ansatz (TBA) when applied to thermodynamical questions, the scattering approach (SBA) allows access to an array of new questions that cannot be addressed otherwise, ranging from scattering of electrons off magnetic impurities to nonequilibrium dynamics.
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[
"https://arxiv.org/pdf/cond-mat/0702612v2.pdf"
] | 119,385,638 |
cond-mat/0702612
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05616b56d535bd2b973e3063cbb64212ae1ad457
|
Scattering approach to Impurity Thermodynamics
26 Feb 2007
Pankaj Mehta
Center for Materials Theory
Rutgers University
08854PiscatawayNJ
Natan Andrei
Center for Materials Theory
Rutgers University
08854PiscatawayNJ
Scattering approach to Impurity Thermodynamics
26 Feb 2007
Recently the authors developed a scattering approach that allows for a complete description of the steady-state physics of quantum-impurities in and out of equilibrium. Quantum impurities are described using scattering eigenstates defined ab initio on the open, infinite line with asymptotic boundary conditions imposed by the leads. The scattering states on the open line are constructed for integrable quantum-impurity models by means of a significant generalization of the Bethe-Ansatz which we call the Scattering Bethe-Ansatz (SBA). The purpose of the paper is to present in detail the scattering approach to quantum-impurity models and the SBA and show that they reproduce well-known thermodynamic results for several widely studied models: the Resonance Level model, Interacting Resonance Level model and the Kondo model. Though the SBA is more complex than the traditional Thermodynamic Bethe Ansatz (TBA) when applied to thermodynamical questions, the scattering approach (SBA) allows access to an array of new questions that cannot be addressed otherwise, ranging from scattering of electrons off magnetic impurities to nonequilibrium dynamics.
I. INTRODUCTION
Recent advances in nanotechnology have allowed extensive experimental study of quantum impurity systems out of equilibrium in controlled, tunable settings 1 . The impurities are typically realized as quantum dots, tiny islands of two-dimensional electron gas attached to leads via tunnel junctions. The number of electrons on the dot can be controlled using a gate voltage since the hopping of electrons is impeded by a large charging energy U . When there is an odd-number of electrons on the dot the upper-most energy level contains only a single, unpaired electron, which behaves effectively as an Anderson or Kondo impurity coupled to the two (or more) leads playing the role of electron baths. Applying a potential difference between the leads results in a current flowing across the dot. A wealth of new experimental data has been collected in recent years on quantum-impurities out of equilibrium in this setting including current vs voltage curves and nonequilibrium density of states (DOS) on the quantum dots 2 . Nonetheless, a comprehensive theoretical understanding of the physics of these models is lacking.
Quantum impurity systems are also the simplest examples of strongly correlated electron systems, wherein interactions between electrons are strong enough to result in new collective behaviors which require a new set of degrees of freedom for their description-the Kondo effect being a canonical example 3 . The strongly-correlated behavior is typically characterized by a low energy scale such as the Kondo temperature below which strong correlation physics dominates and perturbative descriptions break down. One of the most fascinating new frontiers in strongly-correlated systems is the study of such systems in out-of-equilibrium situations. Quantum impurities are an ideal experimental and theoretical setting for exploring the interplay between nonequilibrium-and strongly-correlated dynamics due to the relative simplicity of these models and the wealth of experimental data available.
New theoretical questions arise in this context. Do sufficiently large voltages suppress strong-correlations and thus kill the Kondo effect? Do new scales, such as the decoherence scale, arise? Does voltage effectively behave as a temperature? How should one handle intrinsically non equilibrium phenomena such as nonequilibrium particle and energy currents or entropy production. What is the effect of strong correlation the entropy production?
Currently, the most commonly used technique to treat quantum-impurities out of equilibrium is Keldysh perturbation theory 4 . The perturbative methods, however, are applicable only in the high voltage regime and break down precisely where strong correlations become important. As such, they are unable to answer the interesting questions proposed above. A variety of non-perturbative techniques have been developed in order to capture the strong correlation physics of quantum impurity models, mainly in the context equilibrium physics. These include renormalization group methods, techniques from bosonization and conformal field theory, and exact solutions using the Bethe-Ansatz 3 . Most of these methods are no longer applicable when the influence of nonequilibrium dynamics is comparable to the strong correlations in the problem. This highlights the need for new theoretical approaches that can probe the interesting nonperturbative regimes 5,6,7,8,9,10 .
Recently we have introduced such a non-perturbative framework that allows the description of a steady state out-of-equilibrium quantum impurity system in terms of a time-independent scattering formulation 11 . A steady state ensues 12 when the system is open. Open systems must be defined directly on the infinite line to allow an in-flow and out-flow of electrons and energy from the system. The infinite volume limit, which needs to be taken ab initio, provides a dissipation mechanism. Under these circumstances the non-equilibrium steady state can be described by a scattering eigenstate of the full hamiltonian, an eigenstate defined on the infinite line with its asymptotic behavior specified at the incoming infinity. In most cases the asymptotic boundary conditions are determined by the electron leads 11 .
We have subsequently also introduced a method, the Scattering Bethe Ansatz (SBA), to construct those scattering eigenstates on the infinite line for the Kondo model and other integrable impurity models. The traditional Bethe Ansatz, on the other hand, which has been extensively applied to these models, is defined with periodic boundary condition with periodicity L (with L subsequently sent to infinity). This approach is appropriate to closed systems and allows an efficient calculation of the thermodynamic properties of the systems. However, it does not give access to their scattering properties, nor to the non-equilibrium physics.
The scattering approach can also be applied under equilibrium conditions, when all baths are held at the same chemical potential, or in the case when only one lead is present. The purpose of this paper is to study the Scattering Approach under these simpler circumstances and confront it with the conventional approach which can also be applied here. We will show that the SBA reproduces known thermodynamical results for quantum-impurity models. Nonetheless, as mentioned above, the algebraic Bethe-Ansatz and its finite temperature counterpart the Thermodynamic Bethe Ansatz, prove technically easier when calculating thermodynamics of quantum-impurity models. The real advantage of the SBA is that it allows us to harness the power of integrability to explore new questions about electron S-matrices and T-matrices important for understanding quantum-mechanical coherence and dephasing due to magnetic impurities 13,14 . And, in a context which we will not further explore in this paper, SBA allows us to understand nonequilibrium steady-states in these models.
The paper is organized as follows. We start with a formal introduction to the scattering approach to quantumimpurity models and the scattering Bethe-Ansatz (SBA). Subsequently, we demonstrate our ideas on the Resonant Level model where the physics is particularly simple since the Hamiltonian is quadratic. Finally, we use the SBA to construct scattering states to reproduce well-known T = 0 equilibrium results for Interacting Resonant Level and Kondo models. We conclude the paper with some conjectures about the Kondo model that significantly simplify the calculation of some impurity properties.
II. THE SCATTERING FORMALISM
The basic idea underlying the scattering formalism is the observation that a quantum impurity can be viewed as a localized dynamical scatterer off which electrons from the attached leads or host metal scatter. The scattering changes the internal state of both the impurity and host electrons and leads to the generation of strong correlations. The standard procedure for treating such problems is to set-up an initial state with wavepackets that represent the incoming particles in the far past and evolve this initial state for a very long time with a time-evolution operator U (t, t o ) = T exp(−i priate interacting field-theory; H is the Hamiltonian that describes the particles and dynamical scatterer -in this case the quantum impurity. As the impurity is local, the interaction switches off far away from the impurity and we can define 'in' or 'out' states by specifying the asymptotic behavior in the far past or the far future. Namely, the 'in' , respectively, 'out' states are eigenstates of the total Hamiltonian, satisfy the boundary conditions that they tend to plane waves representing free incoming particles in the t → −∞ and t → ∞ limits respectively. The cross-section for a particular process is then obtained by calculating the overlap of the "in" state with an appropriate "out" state. A recent application of these ideas to quantum impurities is given in 15,16 . While scattering states are designed to allow access to the scattering properties of the system, they also allow calculation of the thermodynamic properties.
The Hamiltonian for a quantum impurity attached to a bath of free electrons is of the form:
H = H 0 + H int = α, k ǫ k ψ † α k ψ α k + H int(1)
with ǫ k the full three-dimensional dispersion of the electrons and α denoting the internal degrees of freedom of the electrons. The term H int describes the impurity and its interaction with the bath of electron. Examples include the Kondo interaction,
H int = J a k ψ † a k ( σ) aa ′ a ′ k ′ ψ a ′ k ′ · S,
with S a localized spin representing the impurity, and the resonance level model (RLM),
H int = t k ( ψ † k d + h.c.) + ǫ d d † d,
describing a local level at energy ǫ d which hybridizes with the bath electrons. Standard manipulations 17 allow us to rewrite the Hamiltonian as chiral 1-d field theories. Since only the combination a k ψ † a k enters into the interaction we can rewrite the theory in terms of the field
ψ † αǫ = d 3 k δ(ǫ k − ǫ)ψ † a k
as (D denotes the bandwidth, namely the cut-off)
H 0 = D −D dǫ ǫ ψ † αǫ ψ αǫ
while the interaction terms take the form, J dǫ ψ † aǫ ( σ) aa ′ dǫ ′ ψ a;ǫ ′ · S or t( dǫ ψ † ǫ d+h.c.)+ǫ d d † d for the Kondo or the IRLM Model respectively. Finally introducing a chiral fermion field
ψ † α (x) = D −D e iǫx ψ † αǫ ν(ǫ) −1/2 the kinetic term becomes H 0 = −i D −D dx ψ † α (x)∂ x ψ α (x)
while the field enters locally into H int , in the form ψ † α (0). As we are interested in the physics on energy scales much smaller than the cut-off D, we consider only universal results obtained in the limit D → ∞.
Open vs Closed Boundary Conditions: The scattering approach to quantum impurity problems, by its very nature, is defined in infinite systems, without boundaries. Physically, this is equivalent to requiring that once incoming electrons scatter off the impurity they do not return and scatter off the impurity again. We refer to infinite size systems with no boundaries as "open systems". The infinite size of the electron bath assures that the host metal or lead is a good thermal bath. Real life systems are not infinite but possess boundaries; our treatment is valid as long as the return time for the electrons is much smaller than the system size. The infinite size of the system implies that scattering states are no longer normalizable and in particular, the Feynman-Hellman theorem no longer holds 18 . This will be important in understanding the results of later sections when we construct eigenstates for the IRLM and Kondo models.
In this open system framework the nature of the incoming particles that scatter off the impurity is specified by asymptotic boundary conditions. The incoming particles, far from the impurity, are eigenstates of the freeelectrons Hamiltonian H 0 and any eigenstate of H 0 is a possible boundary condition describing what the incoming particles look like. Two different boundary conditions are of primary interest: (i) when the incoming particles are a Fermi sea, typically representing the host metal and (ii) when the incoming particles are a Fermi sea and an excited quasi-particle. The former allows for us to calculate thermodynamical properties from scattering while the latter allows us to compute, in principle, single particle S and T matrices. These boundary conditions are depicted in Figure 1.
Time Dependent and Time Independent Formalisms: T = 0: There are two different descriptions for scattering processes. In the time-dependent description, the interaction between the conduction electrons and quantum impurity is turned on in the far past, at t = t o and then adiabatically time-evolved using the Hamiltonian
H = H o + e ηt H int θ(t − t o ),(2)
with H o describing the free electron bath and H int the interactions between the quantum impurity and the incoming electrons, i.e., between the dot and the leads. Before t = t o the quantum impurity is decoupled from the electron bath and the system is described by an asymptotic boundary condition, an eigenstate of H 0 which we denote |Φ o (at T > 0 the system is described by some density matrix ρ o describing decoupled leads and the dot). The incoming particles in the chiral picture correspond to the particles on the left of the impurity and the outgoing scattered particles are on the right. We take the convention that the impurity is at x = 0. Using this convention, incoming particles are located on the negative x-axis, x < 0, and outgoing particles to the positive x-axis, x > 0. We typically consider two types of open boundary conditions for incoming particles. In the first, the incoming particles describe a filled Fermi sea. This is useful for calculating thermodynamics. In the second, the incoming particles are a Fermi sea and an excited quasi-particle. This allows us to calculate the single-particle S and T matrices.
At later times, as the baths and the quantum-impurity evolve, the interaction is turned on adiabatically 28 from the state |Φ o under the action of the time evolution operator
|Ψ(t) = U (t, t o )|Φ o U (t, t o ) = T {exp (−i t to dt ′ H(t ′ ))}(3)
We now wish to establish that a steady-state ensues after sufficiently long time-long enough that all transients die out. For this purpose one must show that the limit t o → −∞ exists, free of infra-red divergences. This has been shown to be the case for the Kondo model 12 under the condition that the infinite volume limit is taken first, i.e. the system is open and the impurity is coupled to good thermal baths. The "openness" of the system provided the dissipation mechanism necessary for the steady state, allowing the high-energy electrons to relax and escape to infinity. The adiabatic limit η → 0 is taken last, allowing the smearing of the bath levels (level separation δ ∼ 1/L to take place turning the poles in the Green's function into a cut).
Under these circumstances |Ψ(t) become timeindependent and describes a time independent eigenstate which we denote |Ψ s . Thus, we can also describe our state in a time-dependent picture. In the timeindependent picture time is traded for space and -for the chiral unfolded picture-the far past corresponds to incoming particles located at distances x ≪ 0 and the far future to outgoing particles located at x ≫ 0. (see Figure 1). Under both equilibrium and non-equilibrium conditions (e.g. coupling to baths at different chemical potentials), the expectation value of any operatorÔ
Ô = Ψ|Ô|Ψ s Ψ|Ψ s(4)
is time independent. A stronger conclusion can be deduced which is central to our construction: the state |Ψ(0) = |Ψ s becomes, by the Gellman-Low theorem 19 , an eigenstate of the full Hamiltonian H, specified by the initial condition |Φ o that describes the electrons in the far past (x ≪ 0). In other words , the state |Ψ s is a scattering eigenstate of the full hamiltonian H = H 0 + H int , satisfying the Lippman-Schwinger equation,
|Ψ s = |Φ o + 1 E − H 0 ± iη H int |Ψ s(5)
with |Φ o -the incoming state playing the role of a boundary condition imposed asymptotically. The scattering state |Ψ s can be viewed as consisting of incoming particles (commonly taken to be a bath of free electrons) described by |Φ o , and of scattered outgoing particles given by the second term in the above equation. Once again two elements are required to fully determine the system: a hamiltonian, H, and a boundary condition, |Φ o , which describes the incoming scattering state far from the impurity. Note that previously, in the timedependent picture, |Φ o played the role of an initial condition rather than a boundary condition. As the impurity is short ranged the scattering state |Ψ s must reduce to the eigenstate |Φ o when all the particles are far to the left of the impurity. This gives a prescription for calculating scattering eigenstates for quantum-impurity problems. We must construct an eigenstate of the full Hamiltonian H with the requirement that when all the electrons are to the left of the impurity the eigenstate reduces to a prescribed eigenstate of H 0 describing the free decoupled two baths and the impurity. It is worth emphasizing that we never explicitly solve (5). Instead, we directly construct eigenstates of the full-Hamiltonian with |Ψ s that are of the form described above. Scattering formalism at finite Temperatures: The above discussion can be generalized to finite temperatures. Once again there are two equivalent frameworks for quantum-impurity problems, a time-dependent and time-independent. In the former, we proceed as in the zero temperature case. We consider the quantum impurity and the baths to be decoupled in the far past, at t = t o → −∞, adiabatically turning on the coupling. The Hamiltonian is again given by (2). The change from zero temperature is that the system is no longer described by a single eigenstate but must instead by described by a density matrix. At t = −∞, the quantum impurity is decoupled from the electron bath and the system is described by the density matrix
ρ 0 = exp (−βH o ) β = T −1 .(6)
At later times, the system is described by time evolving the density matrix ρ 0 with the time evolution operator
ρ(t) = U † (t, −∞) ρ 0 U (t, −∞)(7)
with U (t, −∞) being understood as the limit U (t, t o → −∞). The expectation value of an operatorÔ can be calculated in the usual manner by
Ô = Tr(ρ(t)Ô) Trρ(t)(8)
Again a time independent description can be given. Now the boundary-conditions for our evolved density matrix ρ s is provided by ρ 0 : to the left of the impurity, we know that the scattering density matrix ρ s must reduce to ρ 0 . Further, the finite temperature analogue of our zero temperature condition that our scattering state |Ψ be an eigenstate of H is requirement that the density matrix ρ s commute with the full Hamiltonian in the limit η → 0.
Thus, at T > 0 we consider the incoming states, {|φ, m }, the complete set of eigenstates of H o with energies E m , distributed with the probability of each state given by the Boltzman weight, e −βE 0 m ,
ρ 0 = e −βHo = m e −βE 0 m |φ, m φ, m|,(9)
Using (7), the time-independent density matrix ρ s is
ρ s = U (0, −∞)ρ o U † (0, −∞) = m e −βE 0 m U (0, ∞)|φ, m φ, m|U † (0, −∞) = m e −βE 0 m |Ψ, m Ψ, m|(10)
where we have used (9) and in the second line we have defined the scattering state |Ψ, m = U (0, −∞)|Φ, m with incoming particles describe by |Φ, m . The steady state physics is captured by the operator ρ s which describes an ensemble of scattering states weighted by the Boltzman factors determined by the energy of the incoming electrons. This form for ρ s is consistent with the requirement that ρ s commute with the Hamiltonian and reduce to ρ 0 to the left of the impurity. We can calculate the expectation value of an operator as in (8)
Ô = Tr(ρ sÔ ) Trρ s(11)
III. THE SCATTERING BETHE-ANSATZ
We have shown in the previous section that the thermodynamic properties can be obtained from scattering eigenstates defined directly on the infinite line with incoming boundary conditions imposed by the lead. In general, constructing such eigenstates is a formidable task due to the strong correlations between particles, and is only carried out approximately. But for a special class of models, many of which have important direct physical application, the many-particle eigenstates can be explicitly constructed using the Bethe-Ansatz wavefunction form.
The Bethe-Ansatz approach has a long history stretching back to Bethe's study of the Heisenberg model 20 . The approach has been typically implemented on systems defined with periodic boundary conditions with respect to some finite length L. Subsequently the thermodynamic limit is achieved sending L to infinity maintaining a constant density. If a field theory limit is to be taken, then a further scaling (or universality) limit is required. By means of this "Traditional Bethe Ansatz" (TBA) approach the thermodynamics of several impurity models was studied in great detail, 17,21,22 .
Scattering, on the other hand, must by definition take place in infinite systems with no boundaries -open systems in our terminology. To compute scattering properties a different formulation is required. This can be seen from several points of view. To begin with, particles must come in from asymptotic regions and after scattering occurs, escape again. Thus the system must be open to allow the flow of particles and energy in and out of the system. Furthermore, there must be a way to distinguish between the incoming particles, typically bare particles, eigenstates of H o but not of H, and the renormalized quasiparticles that are the eigenstates of the latter but not of the former. Expressing the bare particles in terms of the renormalized quasiparticles and vice versa lies at the heart of the scattering theory.
Thus the traditional (or closed) Bethe-Ansatz (TBA) and the Scattering (or open) Bethe-Ansatz (SBA) naturally address different sets of questions. The natural questions that can be addressed using the first are thermodynamical. The TBA, by using a periodic system and requiring wave-functions to be self-consistent, reproduces the full renormalized excitation spectrum of a quantumimpurity model. With the knowledge of the spectrum, one can use statistical mechanics arguments to calculate the thermodynamic quantities. Boundary conditions (periodic or otherwise) imposed on a finite length system are essential to this approach. But all knowledge of the bare theory is lost. As such, the TBA is unable to tackle questions about the scattering properties of the quantum impurities. Scattering relies on working in systems with bare particles and open boundary conditions-namely systems of infinite extent with no boundaries.
There is a price to pay for working in open systems. The wavefunctions are no longer normalizable and one does not have recourse to thermodynamic concepts such as free energy. Thus, while the SBA is essential for analyzing scattering properties of quantum impurity models, it is more difficult to extract the thermodynamics using it. Table I
A. The Bethe-Ansatz Wavefunction
The central objective of the SBA is to construct on the infinite line eigenstates of the Hamiltonian (12) with the condition that far away from the quantumimpurity the incoming sector of the eigenstate reduces to a prescribed eigenstate, |Φ , of the free-electron Hamiltonian H 0 . As such, any scattering state must have a welldefined sense of incoming and outgoing particles, with the incoming electrons being to the left of the impurity (x < 0) and the outgoing scattered electrons those to its right. The state |Φ can be any eigenstate of H 0 . We focus in this paper on the case where |Φ is a Fermisea of incoming particles. However, many other choices are possible. In particular, to calculate S-matrices and T-matrices of quantum-impurity Hamiltonian one can choose |Φ to be a Fermi-sea with one incoming particle above the Fermi-sea (see Figure 1). The choice of |Φ describing incoming particles imposes an asymptotic boundary condition on the full scattering eigenstate. In general, imposing boundary conditions on our scattering states is quite difficult. However, when the incoming particles are a free Fermi sea, imposing the boundary-condition simplifies greatly. The key to this simplification is the observation that the natural basis for Bethe-Ansatz wavefunctions is not the Fock basis, but a new "Bethe basis" described extensively below.
H = H 0 + H int = −i ∞ −∞ dxψ † α (x)∂ x ψ α (x) + H int
The SBA constructs eigenstates of the Hamiltonian using wave-functions of Bethe-Ansatz type 20 . The Bethe-Ansatz utilizes the integrability of the Hamiltonian H to divide multi-particle scattering events into two-particle scattering events characterized by the two-particle Smatrices, S ij derived from H. The integrability of the Hamiltonian translates in this language into a selfconsistency condition on the two-particle S-matrices known as the Yang-Baxter Equation (YBE) 23 ensuring that all multi-particle interactions can be consistently broken-up into pair-wise interactions. The consistent wavefunctions of the Bethe form, which we collectively refer to as Bethe-Ansatz wavefunctions, are eigenstates of the Hamiltonian.
We restrict our analysis to quantum-impurities coupled to non-interacting electrons. We further assume that particle number is conserved, the Bethe-Ansatz wavefunctions all have a definite number of particles, N , and there are only local interactions: two particles can interact only if they are at the same point. To write a Bethe-Ansatz wavefunction, it is necessary to divide the configuration space into N ! regions according to the ordering of the particles on the infinite line. For example, we can consider a region where particle 5 is to the left of particle 7 which is to the left particle 9 etc., (x 5 < x 7 < x 9 . . .). Each such region is labelled by a permutation Q in the symmetric group, S N +1 . Since a particle i and j can only interact when they occupy the same position x i = x j , there are no interactions in the interior of these regions. Within each region, the Hamiltonian H reduces to H 0 and the eigenfunctions are sums of plane waves. The most general wave-function of the above form is (with x 0 = 0 the position of the impurity)
|BA, {p} = dx 1 . . . dx N e i P j pj xj (13) Q A Q α1...αN ,α0 θ(x Q ) N j=1 ψ † αj (x j )|0, α 0 where θ(x Q ) = θ(x Q(1) < x Q(2) . . . x Q(N ) < x Q(0) )
and Q runs over all N + 1! permutations. The state |0, α 0 denotes the drained Fermi sea (ψ αj (x j )|0 = 0) and the state of the impurity. When a boundary between two regions is crossed, two particles interact (multi-particle interactions forbidden by Fermi statistics) and hence the amplitude in the regions across the boundary are related by a two particle S-matrix determined by solving the two-particle Schrodinger Equation for the relevant Hamiltonian. The amplitude in a region Q, A α1...αN (Q), is related to the amplitude in an adjacent region, Q ′ , differing from it by the exchange of neighboring particles i and j, via the S-matrix S ij , (14) where in the second equality we have used the fact the two-particle S-matrix S ij acts non-trivially only on the sectors of the Hilbert space corresponding to particles i and j. In general, the matrix relating the amplitude in the region Q = I, defined by (
A Q ′ α1...αN = (S ij ) β1...βN α1...αN A Q β1...βN = (S ij ) βiβj αiαj A Q β1...βNx 1 < x 2 < . . . < x N < x 0 ) is related to the amplitude in region Q, (x Q(1) < x Q(2) < . . . < x Q(N ) < x Q(0)
), by an S-matrix S Q given by a product of two-particle exchange S-matrices S ij along the path leading from I to Q. Since many paths can lead from I to Q consistency requires that S Q be uniquely defined in a path independent way. This is assured by the Yang-Baxter condition 21 . Thus, the Bethe-Ansatz wavefunction can be written in terms of a single amplitude A = A I in the region Q = I and the S-matrices S Q ,
|BA, {p} = dx 1 . . . dx N e P j pj xj (15) Q (S Q A) α1...αN θ(x Q ) N j=1 ψ † αj (x j )|0 .
The energy of a Bethe-Ansatz wavefunction (15) is given by E = j p j . Note, however, that the Bethe-Ansatz wavefunction with Bethe-Ansatz momenta, {p j } is degenerate in energy with all other Bethe-Ansatz wave-
functions {p ′ j } with j p ′ j = E = j p j .
Thus, there are an infinite number of degenerate Bethe-Ansatz wavefunctions of the same energy for any Hamiltonian. Generically, a scattering state with energy E is a sum of Bethe-Ansatz wavefunctions (14)
|Ψ = {p}; P j pj =E C {p} |BA, {p} ,(16)
with C {p} the amplitude in the scattering state of the Bethe-Ansatz wavefunction |BA, {p} and the sum running over all sets of Bethe-Ansatz momenta {p} with energy E.
B. The Bethe-Ansatz Basis
To construct scattering eigenstates for integrable quantum models the Bethe-Ansatz wavefunction exploits the large degeneracy of the linearized free electron gas. As taught in standard chapters on degenerate perturbation theory the correct basis of states in a degenerate subspace to perturb from is the one that diagonalizes the perturbation, or equivalently, the one to which the system returns once the perturbation is turned off. This is precisely the intuition behind the "Bethe basis" of a non interacting field theory. A Bethe basis for a free electron gas is the basis inherited from the interacting quantumimpurity theory when the impurity is removed, or when the system is studied far from the short range impurity. The basis is defined by the presence of a non-trivial two particle S-matrix S ij between the right moving free electrons in H 0 . Indeed, a moment's reflection shows that as the particles move with the same velocity (to the right with v F = 1) an S-matrix does not indicate interaction but a choice of basis.
We now discuss the Bethe basis in more detail. For a quantum-impurity model, there are two kinds of twoparticle S-matrices: those that describe electron-electron scattering, which we denote S ij , and those that describe impurity-electron scattering which we denote S 0j . The S-matrices S ij and S 0j are determined by the impurity interaction term H int in (12) and the Yang-Baxter consistency conditions. Imagine turning off the coupling to the impurity in (12) so that H int = 0. Then (12) reduces to the free-field Hamiltonian H 0 and the electron-impurity S-matrix, S 0j reduces to the identity, S 0j → 1. The electron-electron S-matrix S ij , however, does not change. This leads to the somewhat surprising conclusion that Bethe-Ansatz wavefunctions of the form (15) with S ij = 1 are eigenstates of the free field Hamiltonian H 0 .
This can be understood as follows. Consider the first quantized version of H 0 . In the two-particle sector, the first quantized H 0 is given by H 0 = −i(∂ x1 +∂ x2 ). Notice that any wavefunction of the form
|2; p 1 , p 2 ; q = dxA q α1α2 e i(p1+q)x1+(p2+q)x2 ψ † α1 (x 1 )ψ † α2 (x 2 )|0(17)
is an eigenfunction of H 0 with energy E = p 1 + p 2 (the {α i } label the internal degrees of freedom of the free electrons). Since q can take on any value, there is an infinite number of such states. Any sum of eigenfunctions of the above form is also an eigenfunction of the H 0 with energy E,
|2; p 1 , p 2 = (18) q dx 1 dx 2 e ip1x1+ip2x2 A q α1α2 e iq(x1−x2) ψ † α1 (x 1 )ψ † α2 (x 2 )|0 = dx 1 dx 2 e ip1x1+ip2x2 f α1α2 (x 1 − x 2 )ψ † α1 (x 1 )ψ † α2 (x 2 )|0
where to go from the first line to the second line we have used the fact that q A q α1α2 e iq(x1−x2) is the general expression for the Fourier transform of an arbitrary function, f (x 1 − x 2 ), of x 1 − x 2 . Thus, due to the large symmetry of the free electron problem, there is an infinite number of degenerate two-particle eigenstates for H 0 . The above argument easily generalizes to more than two particles: any function of the form
|N = dx 1 . . . dx N e P N s=1 psxs i<j f αiαj (x i − x j ) j ψ † αj (x j )|0 (19) is an N -particle eigenstate of H 0 . Since θ(x Q ) = i<j θ(x Q(i)−Q(j) )
, is of that form 19 we conclude that the most general N -particle Bethe-Ansatz wavefunction with S Q a product of electron-electron S-matrices, S ij ,
|N, BA = d xe i P j pj xj (S Q A) α1...αN θ(x Q ) ψ † α1 (x 1 ) . . . ψ † αN (x N ).(20)
is an eigenstates of H 0 . However, for S Q = 1 (which implies S ij = 1), it is clearly not of the usual Fock-basis form,
|N, F = d xe i P j pj xj A α1...αN ψ † α1 (x 1 ) . . . ψ † αN (x N )|0 .(21)
The different choices for S ij , and in turn S Q , correspond to different 'Bethe-Ansatz' bases for free electrons. The choice of S ij imposed by the impurity interaction corresponds to working in a particular "Bethe-Ansatz" basis for the problem. The usual Fock basis corresponds to the choice S ij = 1.
We now proceed to discuss the relationship between the Bethe basis, with S ij = 1 and the Fock basis S ij = 1. We denote, for a particular choice of a consistent set of matrices S ij , the resulting Bethe-Ansatz wavefunctions by {|BA n } where n enumerates all possible choices for the {p} and A α1...αN in (20). The set of Bethe-Ansatz wavefunctions {|BA n } form a complete basis for our Hilbert space of H 0 in the limit of infinite size and particle number. In quantum mechanics, different basis for the Hilbert space are related by unitary transformation. Thus, we can formally define an operator U that relates the Fock basis {|F m } to the BA basis {|BA n }. U maps states in the Fock basis (21) to states in the Bethe-Ansatz basis (20). In general, the matrix U relating the two basis is quite complicated since a single state in the Fock basis |F i maps onto a sum of wavefunctions of the Bethe-Ansatz form |F n → m U nm |BA m . For example, in (19) we saw that the two particle eigenstate is actually a sum over many Fock states of type (17).
However, U simplifies greatly if we restrict ourselves to asking how the ground state of H 0 in the Bethe-Ansatz and Fock basis are related. For a systems with unique ground-states, U must map the Fock basis ground state, |N, F gs to the ground state in the Bethe-Ansatz basis |N, BA gs . Thus, the ground state in the Fock basis maps to a single wavefunction of the Bethe-Ansatz form (20). Since the ground state of H 0 is a free Fermi-sea, it follows that a Fermi-sea can be represented by a single Bethe-Ansatz wavefunction. In the sections that follow, we will restrict ourselves to this case where we represent a free Fermi sea, the ground state of H o in both basis.
C. Imposing Asymptotic Boundary-Conditions
The goal of the SBA is to construct eigenstates of the Hamiltonian (12) satisfying the asymptotic boundarycondition that the incoming particles are a prescribed eigenstate, |Φ , of H 0 . We focus on the simplest case when incoming particles come from a bath and are a free Fermi-sea. Central to the imposition of any boundarycondition on the fully interacting Bethe-Ansatz wavefunctions is the observation that these wave functions pick a particular Bethe-Ansatz basis for the free Hamiltonian H 0 . Thus, the boundary condition, typically formulated in the Fock basis, must be reformulated in the natural basis for the scattering state wavefunctions, the Bethe-Ansatz basis. The antagonism between the Fock basis, natural for boundary-conditions, and the Bethe-Ansatz basis, natural for wavefunction is at the heart of many of the SBA. We discuss only the zerotemperature case. The generalization to finite tempera-tures is straightforward.
Recall that the incoming electrons in our chiral picture are electrons to the left of the impurity, x < 0 (see Figure ??). Thus, the asymptotic boundary condition requires that the scattering state reduce to the eigenstate of H 0 , |Ψ → |Φ o = |Φ baths ⊗ |α d , when all particles are to the left of the impurity, {x j } < 0, with |Φ bath a state describing a Fermi sea of free electrons. In general, the scattering state |Ψ is a sum of wavefunctions of the Bethe-Ansatz form (16). The amplitudes of the different Bethe-Ansatz wavefunctions C {pj } are determined by the asymptotic boundary condition. It was argued in the last section that the |Φ baths can be written using a single Bethe-Ansatz wavefunction of the form (20). Thus, in the case where the incoming particles are described by |Φ baths , our scattering state |Ψ can also be described by a single Bethe-Ansatz wavefunction. The incoming electron corresponds to the regions in the wavefunctions of the form θ(
x Q ′ ; x 0 ) ≡ θ(x Q ′ (1) < x Q ′ (2) < . . . < x Q ′ (N ) < x 0 ) with Q ′ a
permutation of the N e electrons in the problem. Since there are no electron-impurity scattering events in these regions, S Q ′ can be written entirely in terms of the electron-electron scattering matrix S ij and the scattering state |Ψ reduces to |Ψ − when all electrons are to the left of the impurity,
|Ψ → |Ψ − = dx 1 . . . dx N e i P j kj xj (22) ′ Q S Q ′ A α1...αN e ;α0 θ(x Q ′ ; x 0 ) N j=1 ψ † αj (x j )|0 .
The right hand side is precisely of the form (20). We therefore conclude that |Ψ reduces to eigenstate of H 0 in the Bethe-Ansatz basis when all particles are to the left of the impurity. This leads to the observation that when the incoming particles are a free Fermi sea, imposing the asymptotic boundary conditions corresponds to choosing the amplitude A α1...αN and the Bethe-Ansatz momenta {p j } for a single wavefunction of the form (20) such that |Ψ − describes a Fermi sea. As is usual in the Bethe-Ansatz, we do not seek to determine the BA momenta {p j } in the thermodynamic limit, computing, instead, the distribution function for the BA momenta, ρ(p). For an infinite system, the distribution ρ(p) and the amplitude A α1...αN are independent of the procedure used to arrive at them 21 . This observation allows us to find ρ(p) and A α1...αN using an auxiliary Algebraic Bethe Ansatz problem for a system of free electrons on a finite ring of length L ′ with Hamiltonian H 0 and two-particle S-matrices, S ij e . In the limit L ′ → ∞, the distribution function for the BA momenta and amplitude in the auxiliary problem will coincide with those of the physical system. ρ(p) and A α1...αN are obtained in the auxiliary problem in the usual way by requiring that the wavefunction be periodic. In particular, the amplitude A α1...αN and the BA momenta {p j } must satisfy the auxiliary Bethe-Ansatz equations,
e ipj L ′ A α1...αN = S jj−1 . . . S j1 S jN . . . S jj+1 A α1...αN
This program is carried out explicitly for the IRLM and Kondo models in the appendix.
To summarize, the imposition of the asymptotic boundary condition on the incoming particle greatly simplifies in the special case where the incoming particles are a free Fermi-sea. The scattering state |ψ can be described by a single Bethe-Ansatz wavefunction and the imposition of the boundary condition is reduced to finding the amplitude and BA momenta for this BA wavefunction. These are found by using the TBA to treat the auxiliary problem of free electrons on a finite ring of length L ′ in the appropriate Bethe-Ansatz basis. The amplitude and the BA momenta of the auxiliary problem coincide with those of the scattering state in the limit L ′ → ∞.
D. Computing with Scattering States
Thus far, we have discussed the explicit construction of the scattering states |Ψ for integrable quantum impurity models. We proceed now to compute the expectation values of physical quantities in the scattering eigenstates using (4)
Ô = Ψ|Ô|Ψ Ψ|Ψ .(23)
The calculations of expectation values are greatly simplified because we work directly with infinite systems. Technically, this is because for strictly infinite systems, we can ignore all but one term in the Slater-determinants occurring in the above expression. This simplification is the mathematical expression of the physics for infinite systems: electrons that scatter off the impurity "leave" the system and never return to scatter off the impurity again. Consider first the overlap between two Bethe-Ansatz wavefunctions.
Since they are given as a sum of plane waves in each region Q, the overlap of two suchwavefunctions, BA, {p j }|BA, {k j } , is (suppressing the internal index α j for notational brevity)
Q,Q d xd y e i P j (kj xj −pj yj) θ(x Q )θ(yQ) A(Q)A(Q) 0| N s=1 ψ(y s ) N j=1 ψ † (x j )|0(24)
The Fermions field give rise to a Slater determinant
Q,Q,S (−1) sgn(S) d xd y e i P j (kj xj −pj yj ) θ(x Q )θ(yQ) A(Q)A(Q) N j=1 δ(x S(j) − y j )(25)
where S is a permutation of the N particles. Integrating over y, we have
BA, {p j }|BA, {k j } = (26) = Q,S (−1) sgn(S) d x e i P j (kj xj−pj x S(j) ) θ(x Q )A(Q)A(QS −1 ) = Q,S (−1) sgn(S) d x e i P j (kj −p S −1 (j) )xj θ(x Q )A(Q)A(QS −1 )
Thus, we see that this expression is the norm of plane waves integrated over a region θ(x Q ). As is usual we regularize plane waves by first placing the system in a box of size L whose size is then taken to infinity at the end of the calculation. This allows us to consider the simpler problem of plane-waves
lim L→∞ L/2 −L/2 dxe i(kj −pj )xj θ(x 1 < x 2 . . . < x N ).(27)
It is straightforward to show that the leading order contribution in L to this integral is L N /N ! which occurs only if the two sets of Bethe-Ansatz momenta are identical {k j } = {p j }. This is the statement that plane waves are 'orthogonal' even on a region θ(x Q ) for an infinite system. Thus, for infinite size systems we can ignore all terms in (26) where the k j = p S −1 (j) for all j.. This leads to great technical simplifications as we only need to keep terms in the sum (26) where Q = 1. Similar, simplifications occur when computing the expectation value of an operatorÔ between Bethe-Ansatz wavefunctions.
IV. SCATTERING APPROACH TO THE RESONANT LEVEL MODEL
In this section, we will apply the scattering framework to a quadratic model, the Resonance Level Model (RLM). Despite its simplicity there is much interest in this model because it describes the strong coupling physics of the Kondo model. It will be shown that our results agree with other approaches to this model such as Keldysh or Landauer which can be carried out completely in this quadratic case. In the next section we shall apply our approach to a fully interacting model with strong correlations.
The Hamiltonian for the RLM, the chiral picture, the free incoming electrons are located to the left of the impurity, x < 0, and the scattered outgoing electrons are to the right of the impurity, x > 0. The RLM serves as a good pedagogical introduction to the scattering framework for quantum impurity models since it is quadratic model and we will not have to resort to the full machinery of the scattering Bethe-Ansatz.
H RL = H 0 + H RLint(28)H 0 = −i dx ψ † (x)∂ x ψ(x) H RLint = t(ψ † (0)d + h.c.) + ǫ d d † d,
A. RLM at T = 0: Thermodynamical Properties
Consider first the zero temperature thermodynamics. We must construct a 'in' scattering state, |Ψ s , describing incoming electrons from the host metal scattering off the impurity. The scattering state |Ψ s is an eigenstate of the full Hamiltonian (28) such that when all the particles are to the left of the impurity |Ψ s reduces to an eigenstate |Φ o of H 0 describing a Fermi see (see Figure2).
The RLM Hamiltonian (28) conserves total particle number. Hence, we can work in a sector of the Hilbert space with a definite number of particles, N . Beginning with N = 1, the most general single particle eigenstate is of the form
|1p s = ∞ −∞ dxe ipx g p (x)ψ † (x) + e p d † |0(29)
Applying the Hamiltonian leads to Schrodinger equation
∂ x g p (x) + V e p δ(x) = pg p (x) (30) tg p (0) + ǫ d e p = pe p .(31)
Taking the ansatz that g p (x) is of the form g p (x) = Aθ(−x) + Bθ(x) and inserting this into the above equation, one has, using the regularization scheme δ(x)θ(x) = 1 2 δ(x) 21 , that
B A = 1 + i t 2 2(p−ǫ d ) 1 − i t 2 2(p−ǫ d ) ≡ e iδp(32)
Thus, the most general single particle eigenstate is given by
|1p s = dxe ipx A(θ(−x) + e iδp θ(x)) + e p d † |0 e p = tg(0) (p − ǫ d ) = t(1 + e iδp ) 2(p − ǫ d )(33)
where to get the second equation we have used (31) and the aforementioned regularization scheme. For future reference it will be helpful to define the single particle scattering state creation operator
α † p (x) = (θ(−x) + e iδp θ(x))ψ † (x) + δ(x)e p d †(34)
Since the Hamiltonian (28) is quadratic, a N -particle eigenstate is given by a tensor product of single particle eigenstates. The most general N -particle eigenstate is of the form
|Ψ = N j=1 ⊗|1p j = N j=1 d xe i P j pj xj α † pj (x j )|0 (35)
Notice that we have not yet specified the momenta {p j } of the state. Since we wish to construct a scattering eigenstate, these momenta must be chosen to satisfy the boundary condition that when all particles are to the left of the impurity our eigenstate reduces to an eigenstate of H o describing the incoming electrons of the host metal or lead at thermal equilibrium (see Figure 2). At zero temperature, this means that the scattering state must reduce to |Φ o = |Φ baths ⊗ |φ d , when all particles are to the left of the impurity. Here |φ d describes some impurity state and |Φ baths describes a free Fermi sea
|Φ baths = dx 1 . . . dx N e i P N j=1 pj xj ψ † (x 1 ) . . . ψ † (x N )|0
(36) under the additional condition that the momenta of the particles {p j } be distributed according to the Fermi-Dirac distribution function. Since we are interested in the limit where the number of particles goes to infinity, it is sufficient to specify the distribution of the momenta instead of the individual values of the momenta themselves.
The single particle eigenstate (33) consists of an incoming particle, dxθ(−x)e ipx ψ † (x)|0 , and an outgoing scattered wave, dxθ(x)e i(px+δp ) ψ † (x)|0 . Since the multi-particle scattering state (35) is a tensor product of the single particle state, when all particles are to the left of the impurity |ψ s reduces to
|Ψ s → (37) dx 1 . . . dx N N s=1 θ(−x s )e i P N j=1 pj xj ψ † (x 1 ) . . . ψ † (x N )|0 .
If we choose the momenta {p j } to be distributed according to the Fermi-Dirac distribution, (35) reduces to the expression for |Φ baths . Hence, our scattering state is given by (35) with the requirement that the momentum distribution of the electrons be chosen according to the Fermi-Dirac distribution. Expectation values We can calculate the expectation value of operators for the RL model using (4). We are interested in calculating the dot occupation n d = d † d . Since the multi-particle eigenstate (35) is a tensor product of single particle eigenstates (33), it is useful to prove some identities about single-particle scattering states. We regularize our system, as is usual in scattering theory, by placing the system in a box of length L.
The physical system corresponds only to the L = ∞ limit and finite L properties are not well defined. A straightforward calculations yields (without loss of generality setting A = 1 in (33)) 1k|1p = (38)
Lδ pk + |e p | 2 δ pk + 1 − e i(δp−δ k )L i(p − k) (1 − δ pk ) and 1k|d † d|1p = e * k e p(39)
Thus the overlap of states with the same momenta is of higher order in L than those with different momenta, so that plane waves are an orthogonal basis for infinite size systems. In the scattering framework which works directly with infinite size systems, it is sufficient to consider overlaps only of single-particle states with the same momenta. Consider now the dot occupation. To leading order in L, one finds, combining (39), (38), (35), and (4), that the occupation is given by
n d = 1 L N j=1 |e pj | 2 |1 + e iδp j | 2 = 1 L N1 j=1 2∆ ∆ 2 + (p j − ǫ d ) 2 (40)
with ∆ = t 2 /2, where to go from the first to the second line we have used the explicit forms of e p and e iδp . Since, we are interested in the infinite size limit N, L → ∞, we can replace the sum by an integral over the distribution of incoming electrons which is given by the Fermi-Dirac distribution function, θ(ǫ f − p) to yield
n d = dp θ(ǫ f − p) 2∆ (p j − ǫ d ) 2 + ∆ 2(41)
.
We compare this result to the one from the traditional Bethe-Ansatz, defined with periodic boundary conditions on a ring of length L. In the usual Bethe-Ansatz, one puts the system on a circle and imposes the self consistency condition that |ψ at x = 0 equals |ψ at x j = L 21 . This leads to the B.A. equations. For this model where the two-particle S-matrices are trivial, the B.A. equations yield for the energy
E = p j = j 2πn j L + 1 L δ pj .(42)
The {n j } are integers corresponding to the energy of a free electron and the {δ pj } the shift in the energies due to the impurity. For future reference, define the 'impurity' energy as E imp = lim L→∞ 1 L j δ pj = dpρ(p)δ(p) with ρ(p) the distribution that describes the free electrons in the Bethe-Ansatz basis. From the Feynman Hellman theorem 18 , we know that
n d = ∂E ∂ǫ d = ∂E imp ∂ǫ d = 1 L j 2∆ (p j − ǫ d ) 2 + ∆ 2 = dp θ(E f − p) 2∆ (p j − ǫ d ) 2 + ∆ 2 (43)
in agreement with the expression we computed using the scattering state formalism (41). It is helpful to define an operator that directly yields the impurity energy using scattering states. This is done by considering the overlap of the outgoing scattered waves with the unscattered Fermi-sea. Define a state |Φ + that describes a bath of outgoing particles (i.e. all particles are to the right of the impurity)
|Φ + o = n i dx i N s=1 θ(x s )e i P N j=1 pj xj ψ † (x 1 ) . . . ψ † (x N )|0
Then, we can define impurity energy alternatively in terms of an impurity energy operator,Ê imp , that acts on scattering state |ψ
Ê imp |Ψ = lim L→∞ −i L log Φ + |Ψ Φ + |Φ + |Ψ .
A straightforward calculation shows that the expectation value of impurity-energy operator
Ê imp = lim L→∞ Ψ|Ê imp |Ψ Ψ|Ψ = 1 L j δ pj = dpρ(p)δ(p)
(44) agrees with the expression derived from traditional methods. The virtue of this operator is that it can be generalized in a straightforward manner to all integrable quantum-impurity models. This object is closely related to the many-body T -matrix for the quantum-impurity model.
RLM at Finite Temperatures: Thermodynamical Properties Consider now the finite temperature case. At finite temperature, T > 0, the system is no longer described by single scattering eigenstate. Instead, we must consider a density matrix of the form (10) composed of scattering states weighted by the thermal Boltzmann distribution. Label the set of N -particle scattering states by the energy of the incoming electrons {|ψ, m }, with m labelling all possible sets of energies for the particles p 1 < p 2 < . . . < p N . We expect that these scattering states are a complete basis for the Hilbert space, and indeed find that this assumption reproduces known thermodynamic results correctly.
We calculate the finite temperature properties of the RLM using (11). The dot expectation value is calculated using dot occupation operatorn d = d † d: The above expressions simplify when we work in the infinite physical L limit since we can keep only leading order terms in L. Recall, that m, m ′ and n are shorthand labels for the ordered set of energies of the N electrons p 1 < p 2 < . . . < p N . Thus, if m = n, there is at least one electron in each state with different energy. Furthermore,notice that the overlaps of a single particle eigenstates (33) given by (38) are leading order in L only if the energies of the two single particle eigenstates coincide. Hence, we conclude from (35) that the leading order in L contribution to the overlap of multi-particle eigenstates comes from states where all particles have the same energy, or in other words, when the two states are identical. Thus, for the infinite L limit, we can set m = m ′ = n in the above expressions to get
n d = n e −βE 0 n m ′ e −βE 0 m ′ ψ, n|n d |ψ, n L N ≡ n P (n) ψ, n|n d |ψ, n L N = n P (n) L pj ∈{pj }n 2∆ ∆ 2 + (p j − ǫ d ) 2 ≡ n P (n) L pj ∈{pj }n n d (ǫ)(45)
where P (n) is the Boltzman probability for the state labelled by n. We can now use a standard trick of statistical mechanics and replace the sum over all configurations by an integral over the average occupancy of a level of energy p, N (p), which in this case is given by the finitetemperature Fermi-Dirac distribution function, f (p, T ). This yields
n d = dp f (p, T ) 2∆ (p j − ǫ d ) 2 + ∆ 2 .(46)
Thus, the effect of temperature is then incorporated by requiring that the momentum distribution of the incoming electrons be chosen according to the finite temperature Fermi-Dirac distribution for free electrons. This expression is in agreement with known results. We will see that the above argument is quite general and that the effect of temperature can be generically incorporated by integrating over finite-temperature distribution functions instead of their zero-temperature counterparts. An almost analogous calculation using the impurity energy operatorÊ imp yields that the finite energy impurity energy is
Ê imp = dp f (p, T )δ p .(47)
The great limitation of the scattering formalism is that though we can calculate the finite temperature energy, calculating the free energy is much trickier. A free energy operator can also be defined for these models though this is much trickier and will not be discussed in this paper 25 .
V. SCATTERING APPROACH TO THE INTERACTING RESONANCE MODEL (IRLM) THERMODYNAMICS
In this section, we compute the zero temperature thermodynamic properties of the interacting Resonance Level Model (IRLM) within the scattering framework. The IRLM Hamiltonian,
H IRLM = H 0 + H I = −i dxψ † (x)∂ x ψ(x) + H I = −i dx ψ † (x)∂ x ψ(x) + t(ψ † (0)d + h.c.) +U ψ † (0)ψ(0)d † d + ǫ d d † d
describes a local level, d † , onto which spinless electrons hop on and off. There is an additional Coulomb interaction between the level and electrons. We consider only the case where ǫ d > 0, where the level is above the Fermi energy of the electrons. Unlike the RLM considered earlier, this model is no longer quadratic and we must use the full Scattering Bethe-Ansatz (SBA) technology to construct scattering states.
We construct the scattering states. They satisfy the Lippman-Schwinger equation (5), and specifying the boundary condition on the incoming particles, |Φ , and the Hamiltonian (48), uniquely determine the corresponding scattering state |Ψ . In this section, we restrict ourselves to scattering states where the incoming particles are a Fermi-sea at zero temperature |Φ o . Such scattering states are sufficient to describe the zero temperature thermodynamic properties of the IRLM such as the dot occupation and impurity energy.
In principle, the scattering formalism can also be used to describe quasi-particle S and T matrices. We defer these topics to future publications as they require treating more complicated boundary condition for incoming particles that includes quasi-particle excitations above the Fermi-sea.
A. Construction of the scattering state
The scattering states for the IRLM are constructed using the SBA, directly in open systems of infinite size, L → ∞. The most general N-particle eigenstate is of the Bethe-Ansatz form
|{p} = A d xe i 2 P i<j sgn(xi−xj )Φ(pi,pj ) N j=1 α † pj (x j )|0 (48) with Φ(p.k) = tan −1 U (p − k) 2(p + k − 2ǫ d )(49)
and δ p and e p given in (32) and (33) 26 . Note that α † is the operator that creates a single-particle eigenstate (34) in the non-interacting RLM. The states |{p} are a complete set of states in terms of which a particular scattering state can be constructed as a linear combination of by the set ({p}) determined by the boundary conditions. In our case the boundary condition requires that the incoming particles look like a free Fermi sea. As discussed previously, for this boundary condition a single state |{p} with appropriately chosen set {p} suffices to determine |Ψ s . In more detail, when all the particles are to the left on the impurity, {x i } < 0, |Ψ s must reduce to an eigenstate of H 0 , |Φ o , describing a zero temperature Fermi sea and a decoupled impurity. When all the {x j } < 0, the operators {α † pj (x j )} reduce to {e ipj xj ψ † (x j )} and the eigenstate |Ψ s reduces to
|ψ → d xAe i 2 P i<j sgn(xi−xj )Φ(pi,pj ) e i P j xj pj N j=1 ψ † (x j )|0 .
Thus, we must choose the {p j } in such a manner that the above expression describes a free Fermi sea. Despite its appearance the expression on the right hand side is an eigenstate of H o . This can be seen by applying h o = −i N j=1 ∂ xj to the wave function. Indeed, since all particles are right mover the scattering S-matric S = e iΦ(pi,pj ) describes the choice of a Bethe basis in the infinitely degenerate energy subspace of free electrons. Thus, for {x j } < 0, |Ψ s reduces to an eigenstate expressed in the Bethe basis characterized by the two-particle S-matrix S = e iΦ(pi,pj ) . This Bethe basis is the natural basis for our problem since, as discussed previously, degenerate perturbation theory demands that we choose the basis for the free electron eigenstates by "turning off" the perturbation, in this case the coupling to the quantum impurity. It is worth emphasizing that the momenta {p j } should coincide with the usual Fock momenta of quasi-particles only when U = 0 and the S ij = 1.
The boundary-condition on incoming particles must be implemented in the Bethe-Ansatz basis with a nontrivial two particle electron S-matrix S = e iΦ(pi,pj ) . As discussed previously, the requirement that the incoming particles be a Fermi sea translates in this Bethe basis into the condition that in |Φ o the incoming particles be an eigenstate of H 0 of the form (20)
|Φ bath = A d xe i P j pj xj e i 2 P i<j sgn(xi−xj)Φ(pi,pj ) ψ † (x 1 ) . . . ψ † (x N )|0 (50)
with the additional condition that the distribution for the BA momenta of the incoming particles, ρ(p), satisfy a set of free Bethe-Ansatz equations for an auxiliary problem of free electrons on a ring of length L ′ with a two particle S-matrix, S = exp (iΦ(p, k)). These equations are derived in the appendix (A9) and are given by,
ρ(p) = 1 2π − dkρ(k)K(p, k) (51) K(p, k) = 1 2π ∂Φ(p, k) ∂p = U π (ǫ d − k) (p + k − 2ǫ d ) 2 + U 2 4 (p − k) 2 .
The desired scattering state, |Ψ s , is given by (48) with the additional requirement that the distribution of the BA momenta, ρ(p), solves the Bethe-Ansatz equation above. The simplicity of the equation follows from the fact the ground states in the Fock basis and in the Bethe basis are unique. This is no longer the case for excited excited states. It is also worth emphasizing that (52) correspond to a free Hamiltonian H 0 and thus differ from the usual Bethe-Ansatz equations for the IRLM 26 in that they contain no impurity contribution.
B. Zero Temperature Properties
Having constructed scattering states, we now use them to calculate the thermodynamic properties of the IRLM. In particular, we will use scattering states to calculate the zero-temperature dot occupation n d = d † d and the impurity energy E imp defined as using the impurity energy operator (44). At zero temperature, E imp plays the role of the free-energy for all dot thermodynamic properties. We then show that our results agree with those derived using traditional Bethe-Ansatz techniques.
To calculate the impurity dot occupation we use (4) which yields
n d = Ψ|d † d|Ψ Ψ|Ψ .(52)
with |Ψ as in (48). As is usual in scattering theory, we regularize our calculations by placing the system in a box of size L. Since scattering is defined only for open systems, the physical system correspond to the infinite L limit and finite L properties are not well defined. From the definition of α † (34), it follows that d † dα † ps (x s )|0 = δ(x s )e ps d † |0 . Thus,
Ψ|d † d|Ψ = (−1) s A 2 d y d x δ(x s ) × e i 2 P i<j (sgn(xi−xj )−sgn(yi−yj ))Φ(pi,pj ) × e ps N j ′ ,j=1,j =s 0|α j ′ (y j ′ )d † α † j (x j )|0(53)
A very similar calculation yields
Ψ|Ψ =A 2 d y d x e i 2 P i<j (sgn(xi−xj )−sgn(yi−yj ))Φ(pi,pj ) N j ′ ,j=1 0|α j ′ (y j ′ )α † j (x j )|0(54)
To proceed with the calculation we note that from (34), one has the relations
{α j (x j ), α † s (x s )} = e i(ps−pj ) [θ(−x s ) + e i(δp s −δp j ) θ(x s )] × δ(x s − x j ) + e pj e ps δ(x s )δ(x j ) {d, α † s (x s )} = e ps δ(x s )(55)
The right hand side of the first equation has two terms: the first term proportional to δ(x s − x j ) comes from the anti-commutation of the fermionic field ψ while the second comes from d. When calculating (53) and (54) keeping only the first term is sufficient to get the leading order in L in the dot occupation since the first term contains only one delta function where as the second contains two. In explicitly open systems where L in infinite, it is sufficient to treat the anti-commutation relation as
{α j (x j ), α † s (x s )} ≈ e i(ps−pj ) (56) ×[θ(−x s )e i(δp s −δp j ) θ(x s )]δ(x s − x j )
Then, the norm to leading order in L is given by
Ψ|Ψ = A 2 σ∈SN (−1) sgnσ d y d x e i 2 P i<j (sgn(xi−xj )(Φ(pi,pj )−Φ(p σ(i) ,p σ(j) )) N s=1 e i(ps−p σ(s) )xs [θ(−x s ) + e i(δp s −δp σ(s) ) θ(x s )]δ(y σ(s) − x s )
The integral over y is trivial. As explained in the last section, the leading order in L contribution to such an integral comes when e i(ps−p σ(s) )xs = 1 or precisely when the permutation σ = 1. In this case the integral is performed trivially and yields Ψ|Ψ = A 2 L N . An analogous calculation using (53) yields to leading order in L that Ψ|d † d|Ψ = A 2 L N −1 N j=1 |e ps | 2 . Combining these two results yields
n d = Ψ|d † d|Ψ Ψ|Ψ = 1 L N s=1 |e ps | 2 = 1 L N s=1 2∆ ∆ 2 + (p − ǫ d ) 2
where we have defined the hybridization ∆ = t 2 /2. In the, infinite L, infinite N limit, we can replace the sum by an integral over the distribution of BA momenta for the incoming particles, ρ(p) given by (A9) to get n d = dp ρ(p)
2∆ ∆ 2 + (p − ǫ d ) 2(57)
We can also compute the impurity-energy using the impurity energy operator
E imp = −i L log Φ + |Ψ Φ + |Φ +(58)
where |Φ + is the eigenstate of the free bath given by (59) with the additional requirement that all particles be to the right of the impurity:
|Φ + = A d x N s=1 θ(x s )e i P j pj xj e i 2 P i<j sgn(xi−xj)Φ(pi,pj ) ψ † (x 1 ) . . . ψ † (x N )|0 .(59)
These correspond to outgoing free Fermi-sea of scattered electrons. In this case,
Φ + |ψ = d y d x e i 2 P i<j (sgn(xi−xj )−sgn(yi−yj ))Φ(pi,pj ) × N j ′ ,j=1 e −ip j ′ y j ′ θ(y j ′ ) 0|ψ(y j ′ )α † j (x j )|0 = A 2 σ∈SN (−1) sgnσ d y d x e i 2 P i<j (sgn(xi−xj )(Φ(pi,pj)−Φ(p σ(i) ,p σ(j) )) × N s=1
e i(ps−p σ(s) )xs e iδp s θ(x s )δ(y σ(s) − x s ) (60)
Once again the integral over y is trivial and the leading order in L contribution comes from when the permutation σ = 1 This yields Φ + |ψ = A 2 (L/2) N e i P N s=1 δp s . An almost identical calculation to the one used to calculate ψ|ψ gives Φ + |Φ + = A 2 (L/2) N . Combining these results and substituting in (58) gives
Ê imp = −i L log Φ + |ψ Φ + |Φ + = 1 L N s=1 δ ps(61)
We can once again replace the sum by integrals over ρ(p) to get E imp = dp ρ(p)δ p .
These results can be checked with those arrived at using the traditional Bethe-Ansatz (TBA) 26 . The TBA results are almost identical to those from the SBA except that the distribution ρ(p) must be replaced by TBA distributions ρ I (p) that include a contribution from the impurity,
E imp = dp ρ I (p)δ p n d = dp ρ I (p) 2∆ ∆ 2 + (p − ǫ d ) 2(63)
Since, as pointed out in 26 , the distributions for the TBA, ρ I (p), differs from the distribution from the SBA, ρ(p), by a term proportional to N −1 where N is the number of particles, in the L, N → ∞ limit, the SBA and TBA expressions coincide.
VI. SCATTERING APPROACH TO THE KONDO THERMODYNAMICS
In this section, we discuss how the scattering Bethe-Ansatz could be used to calculate interesting physical quantities in the Kondo model. Due to the complexity of the scattering state for the Kondo model, doing concrete calculations requires the generalization of many mathematical methods described in the context of spin chains. In particular, we discuss the tantalizing possibility that many of the methods of Maillet, Terras, and collaborators 27 can be generalized to the Kondo model where they may allow exact calculation of as yet inaccessible interesting physical quantities such as the impurity T-matrix. The section starts with a brief discussion of the scattering state that captures the thermodynamics of the Kondo problem. In the next subsection, we discuss a possible mapping between the Kondo problem and an auxiliary 'abelian' problem similar to the IRLM model. Finally, we discuss how to calculate quantities in this auxiliary problem and discuss how this formalism may be generalized. We concentrate only on the zero temperature properties of the Kondo model. The generalization to finite temperatures will be presented later.
A. The Scattering State
The scattering state for the Kondo model is significantly more complicated than that for the IRLM. These extra complications stem from the non-abelian nature of the electron two-particle S-matrices in the Kondo model, S ij e = P ij . This is already evident in the appendix where we represent the free-Fermi seas in the Kondo Bethe-Ansatz basis. We focus on constructing scattering states where the incoming particles are a free Fermi-sea at zero temperature. Such scattering states, using a conjecture discussed below, allow one to recover the zero temperature thermodynamics of the Kondo model using scattering states.
It was shown earlier that for scattering states with the asymptotic boundary conditions that the incoming particles are a Fermi sea, that the scattering state |Ψ can be described by a single Bethe-Ansatz wavefunction. The most general Bethe-Ansatz wavefunctions is of the form
|Ψ = d x e P j pj xj Q S Q A α1...αN α0 θ(x Q ) N j=1 ψ † αj (x j )|0 .(64)
with S Q the product of two-particle S-matrices in the Kondo model, S ij = P ij for electron-electron scattering and S i0 = 1+iJP i0 1+iJ for electron impurity scattering with P ij is the permutation matrix that exchanges the spins of particles i and j 21 .
The asymptotic boundary conditions that the incoming particles be a filled Fermi-sea now reduce to choosing the Bethe-Ansatz momenta {p j } and the amplitude A α1...αN α0 so that when all the particles are to the left of the impurity are scattering state reduces to eigenstate of H 0 in the Kondo Bethe-Ansatz basis describing a filled Fermi-sea. This state, |Φ baths is extensively discussed in the appendix and is described by a wavefunction of the form (A10) with A b1...bN given by (A13)and BA momenta {p j } of the form 2πnj L with n j integers running from −N to 0. The amplitude is written in terms of solutions to (A12), the spin rapidities {Λ γ }.
When all particles are to the left of the impurity, the scattering state (64) reduces to
|Ψ → (65) d x e P j ipj xj Q ′ S Q ′ A α1...αN α0 θ(x Q ′ ; x 0 ) N j=1 ψ † αj (x j )|0 .
with Q ′ a permutation of the N e electrons, θ(x Q ′ ; x 0 ) = θ(x Q ′ (1) < θ Q ′ (2) < . . . < x Q ′ (N e ) < x o ) with x 0 the position of the impurity. Since reaching the regions Q ′ involves no electron-impurity scattering, the S Q is a product of the electron-electron scattering matrix S ij = P ij only. If we choose the momenta {p j } and amplitude A b1...bN as in the paragraph above, (??) reduces to the desired eigenstate of H 0 (A10). Thus, the imposition of the boundary-conditions follows directly from the representation of the filled Fermi-sea in the Kondo Bethe-Ansatz basis. Summarizing, the full scattering state is described by (??) with the additional conditions that A b1...bN be of the form (A13) with the {Λ γ } solutions to (A12) whose density is given by (A16) and BA momenta {p j } of the form 2πnj L with n j integers running from −N to 0. Choosing the amplitude and BA momenta in this way ensure the scattering state |Ψ reduces to a state describing a filled Fermi sea |Φ baths for incoming particles.
B. Can we map the Kondo to an abelian quantum-impurity problem?
Having constructed the scattering state, the next task is to compute quantum-impurity properties using this state. This task is significantly more difficult than in the IRLM since the amplitude A b1...bN is written in terms of lowering B operators of the quantum-inverse scattering method. These operators do non commute but instead satisfy a complicated algebra. This makes it difficult to manipulate them 21 . For this reason, it is quite desirable to explore the intriguing possibility that the Kondo problem is in fact equivalent to an auxiliary quantumimpurity problem similar to the IRLM. The central difference between the Kondo scattering state and the IRLM is that the scattering states for the Kondo problem are constructed using non-abelian two particle S-matrices where as the two-particle S-matrix for the IRLM is an abelian phase. We call models with abelian two-particle S-matrices, abelian quantum impurity problems. In this section, we conjecture that the Kondo problem can indeed be mapped to a very particular 'abelian' quantumimpurity problem. This abelian quantum-impurity problem correctly reproduces the thermodynamics of the Kondo model. We conjecture that arguments similar to those given by Maillet et al will show that the abelianization of the problem extends to all quantities allowing an easy computation of the scattering properties.
The starting point for the conjecture are the Bethe-Ansatz equations for the Kondo model. These Bethe-Ansatz equations are derived using the TBA by considering a quantum impurity on a finite ring of length L and imposing periodic boundary conditions. They are given by
e ipj L = M γ=1 Λγ −1+ic/2 Λγ −1+ic/2 (66) M δ=1,δ =γ Λ δ − Λ γ + ic Λ δ − Λ γ − ic = Λγ −1−ic/2 Λγ −1+ic/2 N e Λγ −ic/2 Λγ +ic/2 N i
with the additional information that the energy of the Bethe-Ansatz wavefunction is E = j p j . The Λ are known as the spin rapidity and parameterize the M spindown particles. We also need the log of these equations which yields
p j = 2π L n j + 1 L M γ=1 [θ 2 (Λ γ − 1) − π] (67) N e θ 2 (Λ γ − 1) + N i θ 2 (Λ γ ) = −2πI γ + M δ=1 θ 1 (Λ γ − Λ δ )
with θ n (x) = −2 tan −1 nx/c and n j and I j integers coming from the logarithm and are the charge and spin quantum numbers respectively. The energy of the eigenstate is given
E = j p j = j 2π L n j + N e L M γ=1 [θ 2 (Λ γ − 1) − π] = j 2π L n j + 1 L M γ=1 −2πI γ − N i θ 2 (Λ γ ) + M δ=1 θ(Λ γ − Λ δ ) = j 2π L n j + M γ=1 − 2π L I γ + N i L M γ=1 −θ 2 (Λ γ )(68)
The first two terms are the energy of a free-electron gas in the spin-charge decoupled Kondo basis and the last term is the shift in the ground state energy due to the Kondo impurity. Previously, we have defined this as the impurity energy E imp . Thus, for the Kondo problem we can write (suggestively)
E imp = 1 L M j=1 δ K (Λ γ )(69)
with δ K (Λ) = −θ 2 (Λ γ ) = 2 tan −1 (2Λ/c). We can also define a phase Φ K (Λ γ −Λ δ ) = θ 1 (Λ γ −Λ δ ) and a functioñ k(Λ) = Dθ 2 (Λ − 1) with D = N e /L. Then, the second Bethe-Ansatz equation in(67) can be derived from the equation
e ik(Λγ )L = e iδK (Λγ ) M δ=1 e iΦK (Λγ −Λ δ )
by taking the natural logarithm of both sides. This suggestive notation is illuminating because the above equation is of the form of the self-consistency monodromy equation in the TBA that leads to the BAE e ik(Λ)L A = Z j A = S jj−1 . . . S j1 . . . S jM S j0 . . . S jj+1 A with S js = e iΦK (Λj −Λs) (j, s = 0) and S j0 = e iδK (Λγ ) . Thus, viewing the Λ ′ s as a function of thek ′ s, we see that the BAE for the Kondo problem could be derived from another abelian quantum impurity problem of M electrons with an electron-electron scattering matrix given by S js = e iΦK (Λj −Λs) and electron-impurity scattering matrix given by S j0 = e iδK (Λγ ) .
The SBA can be applied to this auxiliary quantumimpurity problem in the abelian formulation. The scattering states are analogous to those of the IRLM model with {Φ, δ} → {Φ K , δ K }. A straight-forward construction and calculation using the SBA for this abelian problem yield the correct Kondo thermodynamic properties. This opens up the possibility that scattering properties of the Kondo model can be alternatively calculated in this abelian quantum-impurity model where manipulations of the scattering states are much easier. The scattering states constructed in the last section are unwieldily because they are defined in terms of complicated algebras found in the ABA.
The open problem in this conjecture is how to map operators in the original Kondo problem to this new abelian quantum-impurity problems. Such a mapping has been worked out for the Heisenburg spin-chain by Terras and collaborators 27 . Due to the close analogy of the Bethe-Ansatz equations for the Heisenburg spin chain equations, we expect that a similar mapping of operators can be performed for the Kondo model. If such a mapping can be fully flushed out, the SBA should lead to exact solutions for many impurity properties such as the impurity T and S-matrices.
VII. CONCLUSIONS
This paper outlines a scattering framework for quantum-impurity models. Generally, constructing scattering states for interacting impurity models is quite difficult. However, if the model is integrable, these states can be constructed using the Scattering Bethe Ansatz developed in this paper. The SBA correctly reproduces the zero temperature thermodynamic properties of both the Kondo model and the IRLM. In addition, it raises the exciting possibility that the Kondo model may be equivalent to an abelian quantum-impurity problem.
The scattering framework also gives us insight into how the Bethe-Ansatz works. The impurity physics in any Bethe-Ansatz basis, always looks like single-particle impurity phase shifts, δ. This suggests that the Bethe-Ansatz basis diagonalizes the lead electrons so that the impurity T -matrix is a phase shift. The complexity of the problem is shifted from the impurity-electron interaction to finding an appropriate basis for free electrons. This observation is essential when using the SBA to calculate nonequilibrium properties of the Kondo model. We feel that this new perspective on the Bethe-Ansatz may lead to new physical insights and is worth exploring in greater detail.
new 'Bethe-Ansatz' basis. Central to constructing scattering eigenstates, is the requirement that far away for the impurity, the incoming electrons look like a free Fermi sea. In this section, we show how to represent a Fermisea in the Bethe-Ansatz basis appropriate to the IRLM and Kondo models. For these models, the impurity forces two-particle S-matrices S ij to be 21,26
S IRLM = e iΦ(pi,pj ) = exp i tan −1 U (p i − p j ) 2(p i + p j − 2ǫ d ) S ij Kondo = P ij (A1)
In this appendix we show how to represent free-electrons in the Bethe-Ansatz basis for each of these models. Denote these two-basis the IRLM basis and the Kondo basis respectively.
Free Electrons in the IRLM Bethe-Ansatz Basis
We first focus on the IRLM. Particles in the IRLM are spinless and labelled by their B.A. momenta p j . The IRLM Bethe-Ansatz basis has a electron-electron Smatrices of the form (A1). A Bethe-Ansatz wavefunction in the IRLM Bethe-Anatz basis is given by (up to an overall multiplicative phase independent of the {x j })
|N, BA = A d xe i P j pj xj e i 2 P i<j sgn(xi−xj)Φ(pi,pj ) ψ † (x 1 ) . . . ψ † (x N )|0 .(A2)
with sgn(x) the sign function which is equal to ±1 if x > 0/x < 0. In writing the above expression, we have used the identity that (θ(−x) + e iΦ θ(x)) = e −i i 2 Φ e i 2 Φsgn(x) . As discussed in the main text, to find the B.A. momenta {p j } we consider an auxiliary problem of free electrons living on a finite ring of size L ′ . In the L ′ → ∞ the momenta of the physical and auxiliary problem coincide.
We restrict ourselves to the zero temperature case and when ǫ d is greater than the Fermi-level of the electrons. This is the case considered in 26 . To derive the Bethe-Ansatz for the BA momenta distribution functions, we must equate the wavefunction for the auxiliary problem on a circle when a particle j is at x j = 0 and at x j = L ′ . This gives rise to a Bethe-Ansatz condition of the form
S 1j . . . S j−2j S j−1j A = S jN . . . S jj+1 e ipj L ′(A3)
which implies that
S jj−1 . . . S j1 S jN . . . S jj+1 A = e −ipj L ′ A.(A4)
Plugging in the explicit form of the two-particle S-matrix for the IRLM from (A1), this equation gives rise to an equation for the BA momenta {p j } of the form (noting that we can cancel A from both sides since it is a constant) e ipj L ′ = e i P N s=1 Φ(pj ,ps) .
Taking the log and multiplying by −i one has
p j = 1 L ′ N s=1 Φ(p j , p s ) + 2πn j L ′ (A6)
with n j an integer. Notice that the amplitude A has dropped out of the equation implying that it may be taken to be any constant. Notice that the 'free' Bethe-Ansatz equations (A6) for the BA momenta of H 0 in the IRLM basis can be obtained form the Bethe-Ansatz equations for the full IRLM Hamiltonian (including impurity interactions) 26
p j = 1 2L ′ N s=1 Φ(p j , p s ) + 2πn j L ′ + N i L ′ δ p ,(A7)
by setting the impurity contribution proportional to N i equal to zero.
As is usual, we will not be concerned with solving the discrete version of this equation but instead will solve for the distribution function, ρ(p) describing the density of solutions to the equations in an interval (p, p + dp). It is worth emphasizing that such distributions make sense only in the limit L ′ → ∞. In this limit, we can replace the sum by an integral to get
p j = dkρ(k)Φ(p j , k) + 2πn j L ′ .(A8)
In the usual way, an equation for the zero temperature density ρ(p) is obtained by subtracting the equation for p j from that for p j+1 and expanding in the difference ∆p = p j+1 − p j which yields 21 ρ(p) = 1 2π − dkρ(k)K(p, k) (A9) K(p, k) = 1 2π
∂Φ(p, k) ∂p = U π (k − ǫ d ) (p + k − 2ǫ d ) 2 + U 2 4 (p − k) 2
This equations are valid as long as ǫ d is greater than the Fermi energy of the lead electrons. Though we do not do it here, we could also find the distribution of the BA momenta at finite temperatures by considering the free Thermodynamic Bethe Ansatz (TBA) equations for H 0 corresponding to the free zero temperature BA equation (A9) for H 0 . Summarizing, in the IRLM basis, there is a non-trivial two particle S-matrix between free electrons of the form (A1). The presence of this matrix corresponds to working in a Bethe-Ansatz basis for the IRLM that is distinct from the usual Fock basis. In this basis, the eigenstates of H 0 are of the form (A2) with the multi-particle Smatrices S Q given as products of two particle S-matrices of the form (A1). For a free Fermi-sea at zero temperature, the distribution for the BA momenta, ρ(p), is given by (A9) not the Fermi-Dirac distribution functions.
Free Electrons in the Kondo Bethe-Ansatz Basis
We now concentrate of the wavefunction of a free-Fermi sea at zero temperature in the Kondo basis. In the Kondo basis, free-electrons have a two-particle S-matrix S ij = P ij where P ij is the permutation matrix acting on the spins of electrons i and j 21 . The Bethe-Ansatz wavefunction for the Kondo Bethe-Ansatz basis is
|N = d xe i P j pj xj (S Q ) b1...bN a1...aN A b1...bN θ( x Q ) (A10) ψ † a1 (x 1 ) . . . ψ † aN (x N )|0
with S Q an appropriate product of two particle Smatrices P ij . Note that the amplitude in the region Q = 1, A b1...bN and the choice of BA momenta {p j } are still unspecified. We will once again have to choose these appropriately by considering an auxiliary problem defined on a circle of length L ′ . In the limit where L ′ → ∞, the expressions from the auxiliary problem coincide with those for the infinite-size open system. Thus, we can use the beautiful quantum-inverse scattering technology 21 .
Once again the BA equations for the auxiliary problem are derived by equating the wavefunction for the auxiliary problem on a circle when a particle j is at x j = 0 and at x j = L ′ . This gives rise to a Bethe-Ansatz condition of the form We must choose A b1...bN such that it is eigenvector for the equation Z j A = z j A with eigenvalue z j = e −ipj L . Note, that in general there are many solutions to this equation. We will be concerned with a single eigenvector, namely the ground-state. A general method called the quantum-inverse scattering method has been developed to solve this problem. Let N and M denote the total number of particles and the total number of spin down particles respectively. Let us denote the spin Hilbert space of particle j by V j . Let V N ⊗ N j=1 V j be the N -particle spin-space. The Bethe-Ansatz equations for the ferromagnetic vacuum are 21 where the usual spin amplitude notation A b1...bM is written as A j1...jM by specifying the position of the M down spins and the operators B(Λ γ +ic/2) are defined in the as is usual in the quantum-inverse scattering matrix 21 . The B are best thought of as generalized lowering operators σ − that lower the spin of particle j. The BA momenta {p j } in (A10) on the other hand are trivially of the from p j = 2πnj L . For the ground state, it runs from some lower cut-off K = − 2πN L to the Fermi energy at zero. Just as in the IRLM case, we can talk about the density of {λ j }, σ o (λ j ) rather than the individual λ themselves. This approach is valid in the limit: N, L → ∞ with D = N/L held fixed. To get the distribution for the density, we take the logarithm of both sides of the second equation in (A12) to get To summarize, a Free Fermi sea in the Kondo Bethe-Ansatz basis is captured by a state of the form (A10) with A b1...bN given by (A13), the {Λ γ } solutions to (A12) whose density is given by (A16). The BA momenta {p j } are of the trivial
(Z j ) b1z j = λ(α j , β 1 . . . β M ) = M γ=1 Λ γ + i c 2 Λ γ − i c 2 M δ=1,δ =γ Λ δ − Λ γ + ic Λ δ − Λ γ − ic = Λ γ − i c 2 Λ γ + i c
FIG. 1: The incoming particles in the chiral picture correspond to the particles on the left of the impurity and the outgoing scattered particles are on the right. We take the convention that the impurity is at x = 0. Using this convention, incoming particles are located on the negative x-axis, x < 0, and outgoing particles to the positive x-axis, x > 0. We typically consider two types of open boundary conditions for incoming particles. In the first, the incoming particles describe a filled Fermi sea. This is useful for calculating thermodynamics. In the second, the incoming particles are a Fermi sea and an excited quasi-particle. This allows us to calculate the single-particle S and T matrices.
a local level d † onto which electrons can hop on and off. The energy of the level (relative to the Fermi energy) is controlled by ǫ d , related to the magnetic field in the anisotropic Kondo 24 . Notice, that we have already projected to one dimension and there are only right moving chiral electrons. As explained in the last sections, in N Free Electrons FIG. 2:The scattering state |ψ describes a quantumimpurity where the incoming particles ({xi < 0}) are a free Fermi-sea with N particles.
n e −βE 0 m ψ, n|ψ, m ψ, m|n d |ψ, n m ′ ,n e −βE 0 m ′ | ψ, n|ψ, m ′ | 2
of solutions to the Bethe-Ansatz equations {λ j } corresponds to a different eigenstate of H 0 . Different choices of M correspond to eigenstates of spin N/2 − M . Since, We are interested in the ground state configuration with zero spin we restrict ourselves to the sector where M = N 2 . Let the solutions of the BA equations with M = N/2 for the ground state be given by {Λ gs j }. Define β γ = Λ gs γ + ic and denote the ferromagnetic vacuum in the space V N by |ω = the amplitude A b1...bN in (A10) is given by A(Λ 1 . . . Λ M ) b1...bN = B(Λ 1 + ic/2) . . . B(Λ M + ic/2)|ω = j1...jM A j1...jM σ − j1 . . . σ − jM |ω > (A13)
(x) = −2 tan −1 (x/c) and I γ an integer. Since we are interested in the {λ j } for the groundstate, we set M = N/2. We then consider σ o (Λ) describing the number of solutions in an interval (Λ, Λ + dΛ). Standard manipulations yield the equation 21 ′ σ o (Λ ′ )K(Λ − Λ ′ ) be easily solved by Fourier transform to yield the equation for the density of {λ j } in the groundstate,
summarizes the comparison between the two approaches.SBA
TBA
System
Infinite
Finite
Boundary condition asymptotic (open)
periodic
Wavefunctions
used explicitly
not used
Thermodynamics
difficult
easy
Scattering Properties
possible not possible
Nonequilibrium Generalization
Yes
No
TABLE I :
ISummary of differences between the Scattering
Bethe-Ansatz (SBA) and Algebraic Bethe Ansatz (ABA).
...bN a1...aN A b1...bN = (A11) S jj−1 ..S j1 S jN ..S jj+1 b1...bN a1...aN A b1...bN = e −ipj L ′ A a1...aN .
t to H(t ′ )dt ′ ) of the appro-
In general constructing scattering eigenstates is a formidable task. However, for integrable quantumimpurity models such as the IRLM and Kondo Models, scattering states can be constructed using a generalization of the Algebraic Bethe-Ansatz and quantum Inverse scattering methods. Consequently, the natural basis for these scattering states is not the Fock basis, but rather a
AcknowledgmentsWe would like to thank Edouard Boulat, Achim Rosch and especially Avi Schiller, Sung-Po Chao and Eran Lebanon for many illuminating discussions.This is equivalent to saying that one can actually turn on the interaction infinitely slowly, with the limit η → 0 taken after the L → ∞ limit. This is the definition we use here. The second, stronger meaning associated with adiabatic turning on, is that the interaction is turned on so slowly that there are no level crossings in the spectrum and there exists a one-to-one mapping between the spectrum H0 and H, now the η → 0 limit is taken before the infinite volume limit. We do not use adiabaticity in this stronger sense.
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There are commonly two meanings of 'adiabaticity' in the literature. In the first, adiabaticity means that when computing physical quantities the limit η → 0 is well definedThere are commonly two meanings of 'adiabaticity' in the literature. In the first, adiabaticity means that when com- puting physical quantities the limit η → 0 is well defined.
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