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[
"Parsimonious Hierarchical Modeling Using Repulsive Distributions",
"Parsimonious Hierarchical Modeling Using Repulsive Distributions"
]
| [
"José Quinlan [email protected] \nDepartamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile\n",
"Fernando A Quintana [email protected] \nDepartamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile\n",
"Garritt L Page [email protected] \nDepartamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile\n"
]
| [
"Departamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile",
"Departamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile",
"Departamento de Estadística Pontificia\nDepartamento de Estadística Pontificia Universidad Católica de Chile\nDepartment of Statistics Brigham Young University\nUniversidad Católica\nChile"
]
| []
| Employing nonparametric methods for density estimation has become routine in Bayesian statistical practice. Models based on discrete nonparametric priors such as Dirichlet Process Mixture (DPM) models are very attractive choices due to their flexibility and tractability. However, a common problem in fitting DPMs or other discrete models to data is that they tend to produce a large number of (sometimes) redundant clusters. In this work we propose a method that produces parsimonious mixture models (i.e. mixtures that discourage the creation of redundant clusters), without sacrificing flexibility or model fit. This method is based on the idea of repulsion, that is, that any two mixture components are encouraged to be well separated. We propose a family of d-dimensional probability densities whose coordinates tend to repel each other in a smooth way. The induced probability measure has a close relation with Gibbs measures, graph theory and point processes. We investigate its global properties and explore its use in the context of mixture models for density estimation. Computational techniques are detailed and we illustrate its usefulness with some well-known data sets and a small simulation study. | null | [
"https://arxiv.org/pdf/1701.04457v2.pdf"
]
| 88,521,976 | 1701.04457 | e55bed22ad0e7de70409103aeb1272f62c487c17 |
Parsimonious Hierarchical Modeling Using Repulsive Distributions
July 3, 2017
José Quinlan [email protected]
Departamento de Estadística Pontificia
Departamento de Estadística Pontificia Universidad Católica de Chile
Department of Statistics Brigham Young University
Universidad Católica
Chile
Fernando A Quintana [email protected]
Departamento de Estadística Pontificia
Departamento de Estadística Pontificia Universidad Católica de Chile
Department of Statistics Brigham Young University
Universidad Católica
Chile
Garritt L Page [email protected]
Departamento de Estadística Pontificia
Departamento de Estadística Pontificia Universidad Católica de Chile
Department of Statistics Brigham Young University
Universidad Católica
Chile
Parsimonious Hierarchical Modeling Using Repulsive Distributions
July 3, 2017Gibbs measuresgraph theorymixture modelsrepulsive point processes
Employing nonparametric methods for density estimation has become routine in Bayesian statistical practice. Models based on discrete nonparametric priors such as Dirichlet Process Mixture (DPM) models are very attractive choices due to their flexibility and tractability. However, a common problem in fitting DPMs or other discrete models to data is that they tend to produce a large number of (sometimes) redundant clusters. In this work we propose a method that produces parsimonious mixture models (i.e. mixtures that discourage the creation of redundant clusters), without sacrificing flexibility or model fit. This method is based on the idea of repulsion, that is, that any two mixture components are encouraged to be well separated. We propose a family of d-dimensional probability densities whose coordinates tend to repel each other in a smooth way. The induced probability measure has a close relation with Gibbs measures, graph theory and point processes. We investigate its global properties and explore its use in the context of mixture models for density estimation. Computational techniques are detailed and we illustrate its usefulness with some well-known data sets and a small simulation study.
Introduction
Hierarchical mixture models have been very successfully employed in a myriad of applications of Bayesian modeling. A typical formulation for such models adopts the basic form
y i | θ i ind.
∼ k(y i ; θ i ), θ 1 , . . . , θ n i.i.d.
∼ N k=1 π k δ φ k , φ 1 , . . . , φ N i.i.d. ∼ G 0 ,(1.1)
where k( · ; θ) is a suitable kernel density indexed by θ, 1 ≤ N ≤ ∞, component weights π 1 , . . . , π N are nonnegative and N k=1 π k = 1 with probability 1, and G 0 is a suitable probability distribution. Here N could 1 arXiv:1701.04457v2 [stat.ME] 29 Jun 2017 be regarded as fixed or random and in the latter case a prior p(N ) would need to be specified. Depending on the modeling goals and data particularities, the model could have additional parameters and levels in the hierarchy. The generic model (1.1) includes, as special cases, finite mixture models (Frühwirth-Schnatter 2006) and species sampling mixture models (Pitman 1996;Quintana 2006), in turn including several well-
known particular examples such as the Dirichlet Process (DP) (Ferguson 1973) and the Pitman-Yor Process (Pitman and Yor 1997).
A common feature of models like (1.1) is the use of i.i.d. atoms φ 1 , . . . , φ N . This choice seems to have been largely motivated by the resulting tractability of the models, specially in the nonparametric case (N = ∞). There is also a substantial body of literature concerning important properties such as wide support, posterior consistency, and posterior convergence rates, among others. See, for instance, Ghosal and van der Vaart (2007) and Shen et al. (2013).
While the use of i.i.d. atoms in (1.1) is technically (and practically) convenient, a typical summary of the induced posterior clustering will usually contain a number of very small clusters or even some singletons. As a specific example, we considered a synthetic data set of n = 300 independent observations simulated from the following mixture of 4 bivariate normal distributions: The left panel in Figure 1 shows the original data and clusters, labeled with different numbers and colors.
y ∼ 0.2N 2 (µ 1 , Σ 1 ) + 0.3N 2 (µ 2 , Σ 2 ) + 0.3N 2 (µ 3 , Σ 3 ) + 0.2N 2 (µ 4 , Σ 4 ),(1.
We fit to these data the variation of model (1.1) implemented in the function DPdensity of DPpackage (Jara et al. 2011), which is the bivariate version of the DP-based model discussed in Escobar and West (1995). The right panel of Figure 1 shows the same data but now displays the cluster configuration resulting from the least squares algorithm described in Dahl (2006). The estimated partition can be thought of as a particular yet useful summary of the posterior distribution of partitions for this model. What we observe is a common situation in the application of models like (1.1): we find 6 clusters (the simulation truth involved 4 clusters), one of which is a singleton. Such small clusters are very hard to interpret and a natural question arises, is it possible to limit and ideally, avoid such occurrences? In an example like what is described above, our main motivation is not pinning down the "true" number of simulated clusters. What we actually want to accomplish is to develop a model that encourages joining such small clusters with other larger ones. This would certainly facilitate interpretation of the resulting clusters. Doing so has another conceptual advantage, which is sparsity. The non-sparse behavior shown in the right panel of Figure 1 is precisely facilitated by the fact that the atoms in the mixture are i.i.d. and therefore, can move freely with respect to each other. Thus to achieve our desired goal, we need atoms that mutually repel each other.
Colloquially, the concept of repulsion among a set of objects implies that the objects tend to separate rather than congregate. This notion of repulsion has been studied in the context of Point Processes. For example, Determinantal Point Processes (Lavancier et al. 2015), Strauss Point Processes (Mateu and Montes 2000;Ogata and Tanemura 1985) and Matérn-type Point Processes (Rao et al. 2016) are all able to generate point patterns that exhibit more repulsion than that expected from a Poisson Point Process (Daley and Vere-Jones 2002). Given a fixed number of points within a bounded (Borel) set, the Poisson Point Process can generate point configurations such that two points can be very close together simply by chance. The repulsion in Determinantal, Strauss and Matérn-type Processes discourages such behavior and is controlled by a set of parameters that inform pattern configurations. Among these, to our knowledge, only Determinantal Point Processes have been employed to introduce the notion of repulsion in statistical modeling (see Xu et al. (2016)).
An alternative way to incorporate the notion of repulsion in modeling is to construct a probability distribution that explicitly parameterizes repulsion. Along these lines Fúquene et al. (2016) develop a family of probability densities called Non-Local Priors that incorporates repulsion by penalizing small relative distances between coordinates. Our approach to incorporating repulsion is to model coordinate interactions through potentials (functions that describe the ability to interact) found in so called (second order) Gibbs measures. As will be shown, this allows us to control the strength of repulsion and also consider a large variety of types of repulsion.
Gibbs measures have been widely studied and used for describing phenomena from Mechanical Statistics (Daley and Vere-Jones 2002). Essentially, they are used to model the average macroscopic behavior of particle systems through a set of probability and physical laws that are imposed over the possible microscopic states of the system. Through the action of potentials, Gibbs measures can induce attraction or repulsion between particles. A number of authors have approached repulsive distributions by specifying a particular potential in a Gibbs measure (though the connections to Gibbs measures was not explicitly stated). For example, Petralia et al. (2012) use a Lennard-Jones type potential (Jones 1924) to introduce repulsion. Interestingly, there is even a connection between Gibbs measures and Determinantal Point Processes via versions of Papangelou intensities (Papangelou 1974). See Georgii and Yoo (2005) for more details. It is worth noting that in each of the works just cited, the particles (following the language in Mechanical Statistics) represent location parameters in mixture models.
Similar to the works just mentioned, we focus on a particular potential specification that introduces repulsion via a joint distribution. There are at least three benefits to employing the class of repulsive distributions we develop for statistical modeling:
(i) The repulsion is explicitly parameterized in the model and produces a flexible and smooth repulsion effect.
(ii) The normalizing constant and induced probability distribution have closed forms, they are (almost) tractable and provide intuition regarding the presence of repulsion.
(iii) The computational aspects related to simulation are fairly simple to implement.
In what follows, we discuss theoretical and applied aspects of the proposed class of repulsive distributions and in particular we emphasize how the repulsive class of distributions achieves the three properties just listed.
The remainder of this chapter will be organized as follows. In Section 2 we formally introduce the notion of repulsion in the context of a probability distribution and discuss several resulting properties. In Section 3, we detail how the repulsive probability distributions can be employed in hierarchical mixture modeling for density estimation. Section 4 contains results from a small simulation study that compares the repulsive mixture model we develop to DPM and finite mixture models. In Section 5 we apply the methodology to two well known datasets. Proofs of all technical results and computational strategies are provided in Appendix A-I.
Probability Repulsive Distributions
We start by providing contextual background and introducing notation that will be used throughout.
Background and Preliminaries
We will use the k-fold product space of
R d denoted by R d k = k i=1 R d and B(R d k )
its associated σ-algebra as the reference space on which the class of distributions we derive will be defined. Here, k ∈ N (k ≥ 2) and d ∈ N. Let x k,d = (x 1 , . . . , x k ) with x 1 , . . . , x k ∈ R d . The k-tuple x k,d can be thought of as k ordered objects of dimension d jointly allocated in R d k . We add to the measurable space
(R d k , B(R d k )) a σ-finite measure λ k d ,
that is the k-fold product of the d-dimensional Lebesgue measure λ d . To represent integrals with respect to λ k d , we will use dx k,d instead of dλ k d (x k,d ). Also, given two metric spaces (Ω 1 , d 1 ) and (Ω 2 , d 2 ) we denote by C(Ω 1 ; Ω 2 ) the class of all continuous functions f : Ω 1 → Ω 2 . In what follows we use the term repulsive distribution to reference a distribution that formally incorporates the notion of repulsion.
As mentioned previously, our construction of non-i.i.d. distributions depends heavily on Gibbs measures where dependence (and hence repulsion) between the coordinates of x k,d is introduced via functions that model interactions between them. More formally, consider ϕ 1 : R d → [−∞, ∞] a measurable function and
ϕ 2 : R d × R d → [−∞, ∞] a measurable and symmetric function. Define ν G k i=1 A i = k i=1 Ai exp − k i=1 ϕ 1 (x i ) − k r<s ϕ 2 (x r , x s ) dx k,d ,(2.1)
where k i=1 A i is the cartesian product of Borel sets A 1 , . . . , A k in R d . Here, ϕ 1 can be thought of as a physical force that controls the influence that the environment has on each coordinate x i while ϕ 2 controls the interaction between pairs of coordinates x r and x s . If ϕ 1 and ϕ 2 are selected so that ν G (R d k ) is finite, then by Caratheodory's Theorem ν G defines a unique finite measure on (R d k , B(R d k )). The induced probability measure corresponding to the normalized version of (2.1), is called a (second order) Gibbs measure. The
normalizing constant (total mass of R d k under ν G ) ν G (R d k ) = R d k exp − k i=1 ϕ 1 (x i ) − k r<s ϕ 2 (x r , x s ) dx k,d
is commonly known as partition function (Pathria and Beale 2011) and encapsulates important qualitative information about the interactions and the degree of disorder present in the coordinates of x k,d . In general,
ν G (R d k ) is (almost) intractable mainly because of the presence of ϕ 2 .
Note that symmetry of ϕ 2 (i.e., ϕ 2 (x r , x s ) = ϕ 2 (x s , x r )) means that ν G defines a symmetric measure.
This implies that the order of coordinates is immaterial. If ϕ 2 = 0 then ν G reduces to a structure where coordinates do not interact and are only subject to environmental influence through ϕ 1 . When ϕ 2 = 0, it is common that ϕ 2 (x, y) only depends on the relative distance between x and y (Daley and Vere-Jones 2002).
More formally, let ρ :
R d × R d → [0, ∞) be a metric on R d and φ : [0, ∞) → [−∞, ∞] a measurable function.
To avoid pathological or degenerate cases, we consider metrics that do not treat singletons as open sets in the topology induced by ρ. Then letting ϕ 2 (x, y) = φ{ρ(x, y)}, interactions will be smooth if, for example,
φ ∈ C([0, ∞); [−∞, ∞])
. Following this general idea, Petralia et al. (2012) use φ(r) = τ (1/r) ν : τ, ν ∈ (0, ∞)
to construct repulsive probability densities, which is a particular case of the Lennard-Jones type potential (Jones 1924) that appears in Molecular Dynamics. Another potential that can be used to define repulsion is the (Gibbs) hard-core potential φ(r) = +∞I [0,b] (r) : b ∈ (0, ∞) (Illian et al. 2008), which is a particular case of the Strauss potential (Strauss 1975). Here, I A (r) is the indicator function over a Borel set A in R.
This potential, used in the context of Point Processes, generates disperse point patterns whose points are all separated by a distance greater than b units. However, the threshold of separation b prevents the repulsion from being smooth (Daley and Vere-Jones 2002). Other examples of repulsive potentials can be found in Ogata andTanemura (1981, 1985). The key characteristic that differentiates the behavior of the potentials provided above is the action near 0; the faster the potential function goes to infinity as relative distance between coordinates goes to zero, the stronger the repulsion that the coordinates of x k,d will experiment when they are separated by small distances. Even though Fúquene et al. (2016) do not employ a potential to model repulsion, the repulsion that results from their model is very similar to that found in Petralia et al. (2012) and tends to push coordinates far apart.
It is often the case that ϕ 1 and ϕ 2 are indexed by a set of parameters which inform the types of patterns produced. It would therefore be natural to estimate these parameters using observed data. However,
ν G (R d k )
is typically a function of the unknown parameters which makes deriving closed form expressions of ν G (R d k ) practically impossible and renders Bayesian or frequentist estimation procedures intractable. To avoid this complication, pseudo-maximum likelihood methods have been proposed to approximate ν G (R d k )
when carrying out estimation (Ogata and Tanemura 1981;Penttinen 1984). We provide details of a Bayesian approach in subsequent sections.
Rep
k,d (f 0 , C 0 , ρ) Distribution
As mentioned, our principal objective is to construct a family of probability densities for x k,d that relaxes the i.i.d. assumption associated with its coordinates and we will do this by employing Gibbs measures that include an interaction function that mutually separates the k coordinates. Of all the potentials that might be considered in a Gibbs measure, we seek one that permits modeling repulsion flexibly so that a soft type of repulsion is available which avoids forcing large distances among the coordinates. As noted by Daley and Vere-Jones (2002) and Ogata and Tanemura (1981) the following potential
φ(r) = − log{1 − exp(−cr 2 )} : c ∈ (0, ∞) (2.2)
produces smoother repulsion compared to other types of potentials in terms of "repelling strength" and for this reason we employ it as an example of interaction function in a Gibbs measure. A question that naturally arises at this point relates to the possibility of specifying a tractable class of repulsive distributions that incorporates the features discussed above. Note first that connecting (2.2) with ν G is straightforward:
if we take
ϕ 2 (x, y) = − log[1 − C 0 {ρ(x, y)}], C 0 (r) = exp(−cr 2 ) : c ∈ (0, ∞)
then ν G will have a "pairwise-interaction term" given by
exp − k r<s ϕ 2 (x r , x s ) = k r<s [1 − C 0 {ρ(x r , x s )}]. (2.3)
The right-hand side of (2.3) induces a particular interaction structure that separates the coordinates of x k,d , thus introducing a notion of repulsion. The degree of separation is regulated by the speed at which C 0 decays to 0. The answer to the question posed earlier can then be given by focusing on functions C 0 : [0, ∞) → (0, 1] that satisfy the following properties:
A1. C 0 ∈ C([0, ∞); (0, 1]). A2. C 0 (0) = 1. A3. C 0 (r) → 0 (right-side limit) when x → ∞. A4. For all r 1 , r 2 ∈ [0, ∞), if r 1 < r 2 then C 0 (r 1 ) > C 0 (r 2 ).
For future reference we will call A1 to A4 the C 0 -properties. The following Lemma guarantees that the type of repulsion induced by the C 0 -properties is smooth in terms of x k,d .
Lemma 2.1. Given a metric ρ : R d × R d → [0, ∞) such that singletons are not open sets in the topology induced by ρ, the function R C : R d k → [0, 1) defined by R C (x k,d ) = k r<s [1 − C 0 {ρ(x r , x s )}] (2.4) belongs to C(R d k ; [0, 1)) for all d ∈ N and k ∈ N (k ≥ 2).
Through out the article we will refer to (2.4) as the repulsive component. We finish the construction of repulsive probability measures by specifying a distribution supported on R d which will be common for all the coordinates of x k,d . Let f 0 ∈ C(R d ; (0, ∞)) be a probability density function, then under ϕ 1 (x) = − log{f 0 (x)}, ν G will have a "base component term" given by
exp − k i=1 ϕ 1 (x i ) = k i=1 f 0 (x i ). (2.5) Incorporating (2.3) and (2.5) into (2.1) we get ν G k i=1 A i = k i=1 Ai k i=1 f 0 (x i ) R C (x k,d )dx k,d .
The following Proposition ensures that the repulsive probability measures just constructed are well defined.
Proposition 2.2. Let f 0 ∈ C(R d ; (0, ∞)) be a probability density function. The function
g k,d (x k,d ) = k i=1 f 0 (x i ) R C (x k,d ) (2.6)
is measurable and integrable for all d ∈ N and k ∈ N (k ≥ 2).
With Proposition 2.2 it is now straightforward to construct a probability measure with the desired repulsive structure; small relative distances are penalized in a smooth way. Notice that the support of (2.6)
is determined by the shape of the "baseline distribution" f 0 and then subsequently distorted (i.e. contracted)
by the repulsive component. The normalized version of (2.6) defines a valid joint probability density function which we now provide.
Definition 2.1. The probability distribution Rep k,d (f 0 , C 0 , ρ) has probability density function
Rep k,d (x k,d ) = 1 c k,d k i=1 f 0 (x i ) R C (x k,d ), (2.7) c k,d = R d k k i=1 f 0 (x i ) R C (x k,d )dx k,d . (2.8) Here x k,d ∈ R d k , f 0 ∈ C(R d ; (0, ∞)) is a probability density function, C 0 : [0, ∞) → (0, 1] is a function that satisfies the C 0 -properties and ρ : R d × R d → [0, ∞)
is a metric such that singletons are not open sets in the topology induced by it.
Rep
k,d (f 0 , C 0 , ρ) Properties
In this section we will investigate a few general properties of the Rep k,d (f 0 , C 0 , ρ) class. The distributional results are provided to further understanding regarding characteristics of (2.7) from a qualitative and analytic point of view. As a first observation, because of symmetry, Rep k,d (x k,d ) is an exchangeable distribution in
x 1 , . . . , x k . This facilitates the study of computational techniques motivated by
Rep k,d (f 0 , C 0 , ρ). However, it is worth noting that {Rep k,d (f 0 , C 0 , ρ)} k≥2 does not induce a sample-size consistent sequence of finite- dimensional distributions, meaning that R d Rep k+1,d (x k+1,d )dx k+1 = Rep k,d (x k,d ).
This makes predicting locations of new coordinates problematic. In Section 3 we address how this may be accommodated in modeling contexts. To simplify notation, in what follows we will use [m] = {1, . . . , m}, with m ∈ N.
Normalizing Constant
Because R C (x k,d ) is invariant under permutations of the coordinates of x k,d , an interaction's direction is immaterial to whether it is present or absent (i.e., x r interacts with x s if and only if x s interacts with x r ).
Therefore it is sufficient to represent the interaction between x r and x s as (r, s) ∈ I k where I k = {(r, s) : 1 ≤ r < s ≤ k}. In this setting, I k reflects the set of all pairwise interactions between the k coordinates of x k,d
and k = card(I k ) = k(k−1) 2 , where card(E) is the cardinality of a set E. Now, expanding (2.4) term-by-term results in R C (x k,d ) = 1 + k l=1 (−1) l A⊆I k card(A)=l (r,s)∈A C 0 {ρ(x r , x s )} .
(2.9)
The right-side of (2.9) is connected to graph theory in the following way: A ⊆ I k can be interpreted as a non-directed graph whose edges are (r, s) ∈ A.
Using (2.9), it can be shown that expression (2.8) in Definition 2.1 has the following form:
c k,d = 1 + k l=1 (−1) l A⊆I k card(A)=l Ψ k,d (A) (2.10) Ψ k,d (A) = R d k k i=1 f 0 (x i ) (r,s)∈A C 0 {ρ(x r , x s )} dx k,d . (2.11)
Note that representing A as a graph or Laplacian matrix can help develop intuition on how each summand contributes to the expression (2.10). Figure 2 shows one particular case of how 3 of k = 4 coordinates in R d might interact by providing the respective Laplacian matrix together with the contribution that (2.11) brings to calculating c 4,d according to (2.10).
Equation (2.10) retains connections with the probabilistic version of the Inclusion-Exclusion Principle.
This result, which is very useful in Enumerative Combinatorics, says that in any probability space (Ω, F, P) (unattainable) extreme case C 0 = 0 (i.e., the coordinates x 1 , . . . , x k are mutually independent and share a common probability law f 0 ).
P k i=1 A c i = 1 + k l=1 (−1) l I⊆[k] card(I)=l P i∈I A i ,
The tractability of c k,d depends heavily on the number of coordinates k since the cost of evaluating (2.11) becomes prohibitive as it requires carrying out (at least) 2 k − 1 numerical calculations. In Subsection 3.1
we highlight a particular choice of f 0 , C 0 and ρ that produces a closed form expression for (2.11). through a Gaussian density N d ( · ; θ j , Λ j ) with location θ j ∈ R d and scale Λ j ∈ S d . Here, S d is the space of real, symmetric and positive-definite matrices of dimension d × d. We let θ k,d = (θ 1 , . . . , θ k ) ∈ R d k and
Λ k,d = (Λ 1 , . . . , Λ k ) ∈ S d k where S d k is the k-fold product space of S d . Next let π k,1 = (π 1 , . . . , π k ) ∈ ∆ k−1 , where ∆ k−1 is the standard (k − 1)-simplex (∆ 0 = {1})
, denote a set of weights that reflect the probability of allocating y i : i ∈ [n] to a cluster. Then the standard Gaussian Mixture Model is
y i | π k,1 , θ k,d , Λ k,d i.i.d. ∼ k j=1 π j N d (y i ; θ j , Λ j ). (3.1)
It is common to restate (3.1) by introducing latent cluster membership indicators z 1 , . . . , z n ∈ [k] such that y i is drawn from the jth mixture component if and only if z i = j:
y i | z i , θ k,d , Λ k,d ind. ∼ N d (y i ; θ zi , Λ zi ) (3.2) z i | π k,1 i.i.d. ∼ P(z i = j) = π j . (3.3)
after marginalizing over the z i indicators. The model is typically completed with conjugate-style priors for all parameters.
Specifying a prior distribution for k ∈ N is possible. For example, DPM models by construction induce a prior distribution on the number of clusters k. Alternatively, Reversible Jump MCMC (Green 1995;Richardson and Green 1997) or Birth-Death Chains (Stephens 2000) could be employed after assigning a particular prior for k. These methods do not translate well to the non-i.i.d. case and so we employ a case-specific upper bound k ≥ 2.
In the above mixture model, the location parameters associated with each mixture component are typically assumed to be independent a priori. This is precisely the assumption that facilitates the presence of redundant mixture components. In contrast, our work focuses on employing Rep k,d (f 0 , C 0 , ρ) as a model for location parameters in (3.1) which promotes reducing redundant mixture components without sacrificing goodness-of-fit, i.e, more parsimony relative to alternatives with independent locations. Moreover, the responses will be allocated to a few well-separated clusters. This desired behavior can be easily incorporated in the mixture model by assuming
θ k,d ∼ Rep k,d (f 0 , C 0 , ρ) f 0 (x) = N d (x; µ, Σ) : µ ∈ R d , Σ ∈ S d (3.4) C 0 (r) = exp(−0.5τ −1 r 2 ) : τ ∈ (0, ∞) (3.5) ρ(x, y) = {(x − y) Σ −1 (x − y)} 1/2 . (3.6)
The specific forms of f 0 , C 0 and ρ are admissible according to Definition 2.1. The repulsive distribution parameterized by (3.4)-(3.6) will be denoted by
NRep k,d (µ, Σ, τ ). Because NRep k,d (µ, Σ, τ ) introduces
dependence a priori (in particular, repulsion) between the coordinates of θ k,d , they are no longer conditionally independent given (y n,d , z n,1 , Λ k,d ), with y n,d = (y 1 , . . . , y n ) ∈ R d n and z n,1 = (z 1 , . . . , z n ) ∈ [k] n . The parameter τ in (3.5) controls the strength of repulsion associated with coordinates in θ k,d via (3.6): as τ → 0 (right-side limit), the repulsion becomes weaker. The selection of (3.4) mimics the usual i.i.d. multivariate
normal assumption.
To facilitate later reference we state the "repulsive mixture model" in its entirety:
y i | z i , θ k,d , Λ k,d ind. ∼ N d (y i ; θ zi , Λ zi ) (3.7) z i | π k,1 i.i.d. ∼ P(z i = j) = π j (3.8)
together with the following mutually independent prior distributions:
π k,1 ∼ Dir(α k,1 ) : α k,1 ∈ (0, ∞) k (3.9) θ k,d ∼ NRep k,d (µ, Σ, τ ) : µ ∈ R d , Σ ∈ S d , τ ∈ (0, ∞) (3.10) Λ j i.i.d. ∼ IW d (Ψ, ν) : Ψ ∈ S d , ν ∈ (0, ∞). (3.11)
In what follows we will refer to the model in (3.7)-(3.11) as the (Bayesian) Repulsive Gaussian Mixture
Model (abbreviated as RGMM).
Parameter Calibration
We briefly discuss stategies of selecting values for parameters that control the prior distributions in (3.9)-(3.11). We select values for µ, Σ and τ of the RGMM instead of treating them as unknown and assigning them hyperprior distributions because of computational cost. First notice that (µ, Σ) acts as a location/scale parameter: if Σ = CC is the corresponding Cholesky decomposition for Σ, then θ k,d ∼ NRep k,d (0 d , I d , τ )
implies that
1 k ⊗ µ + (I k ⊗ C)θ k,d ∼ NRep k,d (µ, Σ, τ ),
where I d is the d×d identity matrix and 0 d , 1 d ∈ R d are d-dimensional vectors of zeroes and ones, respectively.
Although a Gaussian hyperprior for µ is a reasonable candidate (the full conditional distribution is also Gaussian), it is not straightforward how to select its associated hyperparameters. A slightly more complicated problem occurs with Σ, since this parameter participates in the repulsive component and no closed form is available for its posterior distribution. Even more problematic, the induced full conditional distribution for τ turns out to be doubly-intractable (Murray et al. 2006) and as a result the standard MCMC algorithms do not apply. To see this, it can be shown using (2.10), (2.11) and the Gaussian integral that the normalizing
constant of NRep k,d (µ, Σ, τ ) is c k,d = 1 + k l=1 (−1) l A⊆I k card(A)=l det(I k ⊗ I d + L A ⊗ τ −1 I d ) −1/2 ,
where I k is the k × k identity matrix, L A denotes the Laplacian matrix associated to the set of interactions A ⊆ I k (see Subsection 2.3.1) and ⊗ is the matrix Kronecker product, making it a function of τ .
To facilitate hyperparameter selection we standardize the y i 's (a common practice in mixture models see, e.g. Gelman et al. 2014). Upon standardizing the response, it is reasonable to assume that µ = 0 d and Σ = I d . Further Gelman et al. (2014) argue that setting α k,1 = k −1 1 d produces a weakly informative prior for π k,1 . Selecting ν and Ψ is particularly important as they can dominate the repulsion effect. Setting ν = d + 4 and Ψ = 3ψI d with ψ ∈ (0, ∞) guarantees that each scale matrix Λ j is centered on ψI d and that their entries possess finite variances. The value of ψ can be set to a value that accommodates the desired variability.
To calibrate τ , we follow the strategy outlined in Fúquene et al. (2016). Their approach consists of first specifying the probability that the coordinates of θ k,d are separated by a certain distance u and then set τ to the value that achieves the desired probability. To formalize this idea, suppose first that θ 1 , . . . , θ k are a random sample coming from N d (0 d , I d ). To favor separation among these random vectors we can use (3.5) and (3.6) with Σ = I d to choose τ such that for all r = s ∈ [k]
P[1 − exp{−0.5τ −1 (θ r − θ s ) (θ r − θ s )} ≤ u] = p,
for fixed values u, p ∈ (0, 1). Letting w(u) = − log(1 − u) for u ∈ (0, 1), standard properties of the Gaussian distribution guarantee that the previous relation is equivalent to
P{G ≤ w(u)τ } = p, G = 1 2 (θ r − θ s ) (θ r − θ s ) ∼ G(d/2, 1/2). (3.12)
Creating a grid of points in (0, ∞) it is straightforward to find a τ that fulfills criterion (3.12). This criterion allows the repulsion to be small (according to u), while at the same time preventing it with probability p from being too strong. This has the added effect of avoiding degeneracy of (3.10), thus making computation numerically more stable. In practice, we apply the procedure outlined above to the vectors coming from the repulsive distribution (3.10), treating them as if they were sampled from a multivariate Gaussian distribution.
This gives us a simple procedure to approximately achieve the desired goal of prior separation with a prespecified probability.
Theoretical Properties
In this section we explore properties associated with the support and posterior consistency of (3.1) under (3.9)-(3.11). These results are based on derivations found in Petralia et al. (2012). However, we highlight extensions and generalizations that we develop here. Consider for k ∈ N the family of probability densities
F k = {f ( · ; ξ k ) : ξ k ∈ Θ k }, where ξ k = π k,1 × θ k,1 × {λ} = (π 1 , . . . , π k ) × (θ 1 , . . . , θ k ) × {λ}, Θ k = ∆ k−1 × R 1 k × (0, ∞) and f ( · ; ξ k ) = k j=1 π j N( · ; θ j , λ).
Let B p (x, r) with x ∈ R 1 k and r ∈ (0, ∞) denote an open ball centered on x, and with radius r, and D p (x, r)
its closure relative to the Euclidean L p -metric (p ≥ 1) on R 1 k .
The following four conditions will be assumed to prove the results stated afterwards.
B1. The true data generating density f 0 ( · ; ξ 0 k0 ) belongs to F k0 for some fixed
k 0 ≥ 2, where ξ 0 k0 = π 0 k0,1 × θ 0 k0,1 × {λ 0 } = (π 0 1 , . . . , π 0 k0 ) × (θ 0 1 , . . . , θ 0 k0 ) × {λ 0 }.
B2. The true locations θ 0 1 , . . . , θ 0 k0 satisfy min(|θ 0 r − θ 0
s | : r = s ∈ [k 0 ]) ≥ v for some v > 0.
B3. The number of components k ∈ N follows a discrete distribution κ on the measurable space (N, 2 N ) such that κ(k 0 ) > 0.
B4. For k ≥ 2 we have ξ k ∼ Dir(k −1 1 k ) × NRep k,1 (µ, σ 2 , τ ) × IG(a, b). In the case that k = 1, ξ k ∼ δ 1 × N(µ, σ 2 )×IG(a, b) with δ 1 a Dirac measure centred on 1. In both scenarios µ ∈ R and σ 2 , τ, a, b ∈ (0, ∞)
are fixed values.
Condition B2 requires that the true locations are separated by a minimum (Euclidian) distance v, which favors disperse mixture component centroids within the range of the response. For condition B4, the sequence {ξ k : k ∈ N} can be constructed (via the Kolmogorov's Extension Theorem) in a way that the elements are mutually independent upon adding to each Θ k an appropriate σ-algebra. This guarantees the existence of a prior distribution Π defined on F = ∞ k=1 F k which correspondingly connects the elements of F with ξ = ∞ k=1 ξ k . To calculate probabilities with respect to Π, the following stochastic representation will be useful
ξ | K = k ∼ ξ k , K ∼ κ. (3.13)
Our study of the support of Π employs the Kullback-Leibler (KL) divergence to measure the similarity between probability distributions. We will say that f 0 ∈ F k0 belongs to the KL support with respect to Π if, for all ε > 0
Π f ∈ F : R log f 0 (x; ξ 0 k0 ) f (x; ξ ) f 0 (x; ξ 0 k0 )dx < ε > 0,(3.14)
where ξ ∈ ∞ k=1 Θ k . Condition (3.14) can be understood as Π's ability to assign positive mass to arbitrarily small neighborhoods around the true density f 0 . A fundamental step to proving that f 0 lies in the KL support of Π is based on the following Lemmas.
Lemma 3.1. Under condition B1, let ε > 0. Then there exists δ > 0 such that
R log f 0 (x; ξ 0 k0 ) f (x; ξ k0 ) f 0 (x; ξ 0 k0 )dx < ε for all ξ k0 ∈ B 1 (θ 0 k0,1 , δ) × B 1 (π 0 k0,1 , δ) × (λ 0 − δ, λ 0 + δ).
Lemma 3.2. Assume condition B2 and let θ k0,1 ∼ NRep k0,1 (µ, σ 2 , τ ). Then there exists δ 0 > 0 such that
P{θ k0,1 ∈ B 1 (θ 0 k0,1 , δ)} > 0.
for all δ ∈ (0, δ 0 ]. This result remains valid even when replacing B 1 (θ 0 k0,1 , δ) with D 1 (θ 0 k0,1 , δ).
Using Lemmas 3.1 and 3.2 we are able to prove the following Proposition.
Proposition 3.3. Assume that conditions B1-B4 hold. Then f 0 belongs to the KL support of Π.
We next study the rate of convergence of the posterior distribution corresponding to a particular prior distribution (under suitable regularity conditions). To do this, we will use arguments that are similar to those employed in Theorem 3.1 of Scricciolo (2011), to show that the posterior rates derived there are the same here when considering univariate Gaussian Mixture Models and cluster-location parameters that follow condition B4. First, we need the following two Lemmas.
Lemma 3.4. For each k ≥ 2 the coordinates of θ k,1 ∼ NRep k,1 (µ, σ 2 , τ ) share the same functional form.
Moreover, there exists γ ∈ (0, ∞) such that
P(|θ i | > t) ≤ 2 (2π) 1/2 c k−1 c k σ(|µ| + 1) −1 exp − (4σ 2 ) −1 t 2 for all t ∈ [γ, ∞) and i ∈ [k].
Here, c k = c k,1 is the normalizing constant of NRep k,1 (µ, σ 2 , τ ) with c 1 = 1.
Lemma 3.5. The sequence {c k : k ∈ N} defined in Lemma 3.4 satisfies
0 < c k−1 c k ≤ A 1 exp(A 2 k)
for all k ∈ N (k ≥ 2) and some constants A 1 , A 2 ∈ (0, ∞).
These results permit us to adapt certain arguments found in Scricciolo (2011) that are applicable when the location parameters of each mixture component are independent and follow a common distribution that is absolutely continuous with respect to the Lebesgue measure, whose support is R and with tails that decay exponentially. Using Lemmas 3.4 and 3.5, we now state the following Proposition 3.6. Assume that conditions B1, B2 and B4 hold. Replace condition B3 with:
B3 . There exists B 1 ∈ (0, ∞) such that for all k ∈ N, 0 < κ(k) ≤ B 1 exp{−B 2 k}, where B 2 > A 2 and A 2 ∈ (0, ∞) is given by Lemma 3.5.
Then, the posterior rate of convergence relative to the Hellinger metric is ε n = n −1/2 log(n).
Sampling From NRep k,d (µ, Σ, τ )
Here we describe an algorithm that can be used to sample from NRep k,d (µ, Σ, τ ). Upon introducing component labels, sampling marginally from the joint posterior distribution of θ k,d , Λ k,d , π k,1 and z n,1 can be done with a Gibbs sampler. However, the full conditionals of each coordinate of θ k,d are not conjugate but they are all functionally similar. Because of this, evaluating these densities is computationally cheap making it straightforward to carry out sampling from NRep k,d (µ, Σ, τ ) via a Metropolis-Hastings step inside the Gibbs sampling scheme. In Appendix A we detail the entire MCMC algorithm (Algorithm RGMM), but here we focus on the nonstandard aspects.
To begin, the distribution (θ k,d | · · · ) is given by
(θ k,d | · · · ) ∝ k j=1 N d (θ j ; µ j , Σ j ) k r<s [1 − exp{−0.5τ −1 (θ r − θ s ) Σ −1 (θ r − θ s )}] where µ j = Σ j (Σ −1 µ + Λ −1 j s j ), s j = n i=1 I {j} (z i )y i , Σ j = (Σ −1 + n j Λ −1 j ) −1 and n j = card(i ∈ [n] : z i = j). Now, the complete conditional distributions (θ j | θ −j , · · · ) for j ∈ [k] and θ −j = (θ l : l = j) ∈ R d k−1 ,
have the following form
f (θ j | θ −j , · · · ) ∝ N d (θ j ; µ j , Σ j ) k l =j [1 − exp{−0.5τ −1 (θ j − θ l ) Σ −1 (θ j − θ l )}].
The following pseudo-code describes how to sample from f (θ k,d | · · · ) by way of (θ j | θ −j , · · · ) via a random walk Metropolis-Hastings step within a Gibbs sampler:
1. Let θ (0) k,d = (θ (0) 1 , . . . , θ(0)
k ) ∈ R d k be the actual state for θ k,d .
2. For j = 1, . . . , k:
(a) Generate a candidate θ
(1) j from N d (θ (0) j , Γ j ) with Γ j ∈ S d . (b) Set θ (0) j = θ (1) j with probability min(1, β j ), where β j = N d (θ (1) j ; µ j , Σ j ) N d (θ (0) j ; µ j , Σ j ) k l =j 1 − exp{−0.5τ −1 (θ (1) j − θ (0) l ) Σ −1 (θ (1) j − θ (0) l )} 1 − exp{−0.5τ −1 (θ (0) j − θ (0) l ) Σ −1 (θ (0) j − θ (0) l )} .
The selection of Γ j can be carried out using adaptive MCMC methods (Roberts and Rosenthal 2009) so that the acceptance rate of the Metropolis-Hastings algorithm is approximately 50% within the burn-in period for each j ∈ [k]. One approach that works well for the RGMM is to take
Γ j = 1 B B t=1 {Σ −1 + n (t) j (Λ (t) j ) −1 } −1 : n (t) j = card(i ∈ [n] : z (t) i = j), (3.15) where t ∈ [B]
is the tth iteration of the burn-in period with length B ∈ N.
Simulation Study
To provide context regarding the proposed method's performance in density estimation, we conduct a small simulation study. In the simulation we compare density estimates from the RGMM to what is obtained using an i.i.d. Gaussian Mixture Model (GMM) and a Dirichlet Process Gaussian Mixture Model (DPMM). This is done by treating the following as a data generating mechanism: 25N(1, 0.3 2 ) + 0.4N(4, 0.8 2 ).
y ∼ f 0 = 0.3N(−5, 1.0 2 ) + 0.05N(0, 0.3 2 ) + 0.
(4.1)
Using (4.1) we simulate 100 data sets with sample sizes 500, 1000 and 5000. For each of these scenarios, we compare the following 4 models (abbreviated by M1, M2, M3 y M4) to estimate f 0 :
M1. GMM corresponding to (3.7)-(3.8) with prior distributions given by (3.9)-(3.11), replacing (3.10) by
θ 1 , . . . , θ k i.i.d.
∼ N d (µ, Σ). In this case:
• k = 10, d = 1, α k,1 = 10 −1 1 10 , µ = 0, Σ = 1, Ψ = 0.06 and ν = 5.
We collected 10000 MCMC iterates after discarding the first 1000 as burn-in and thinning by 10.
M2. RGMM with τ = 5.45. This value came from employing the calibration criterion from Section 3.1.1 and setting u = 0.5 and p = 0.95. The remaining prior parameters are:
• k = 10, d = 1, α k,1 = 10 −1 1 10 , µ = 0, Σ = 1, τ = 5.45, Ψ = 0.06 and ν = 5.
We collected 10000 MCMC iterates after discarding the first 5000 as burn-in and thinning by 20.
M3. RGMM with τ = 17.17. This value came from employing the calibration criterion from Section 3.1.1 and setting u = 0.2 and p = 0.95. Since τ is bigger here than in M2, M3 has more repulsion than M2.
The remaining prior parameters are the same as in M2:
• k = 10, d = 1, α k,1 = 10 −1 1 10 , µ = 0, Σ = 1, τ = 17.17, Ψ = 0.06 and ν = 5.
We collected 10000 MCMC iterates after discarding the first 5000 as burn-in and thinning by 20.
M4. DPMM given by:
y i | µ i , Σ i ind. ∼ N d (µ i , Σ i ) (4.2) (µ i , Σ i ) | H i.i.d. ∼ H (4.3) H | α, H 0 ∼ DP(α, H 0 ) (4.4)
where the baseline distribution H 0 is the conjugate Gaussian-Inverse Wishart
H 0 (µ, Σ) = N d (µ; m 1 , k −1 0 Σ) IW d (Σ; Ψ 1 , ν 1 ) : ν 1 ∈ (0, ∞). (4.5)
To complete the model specification given by (4.2)-(4.5), the following independent hyperpriors are assumed:
α | a 0 , b 0 ∼ G(a 0 , b 0 ) : a 0 , b 0 ∈ (0, ∞) (4.6) m 1 | m 2 , S 2 ∼ N d (m 2 , S 2 ) : m 2 ∈ R d , S 2 ∈ S d (4.7) k 0 | τ 1 , τ 2 ∼ G(τ 1 /2, τ 2 /2) : τ 1 , τ 2 ∈ (0, ∞) (4.8) Ψ 1 | Ψ 2 , ν 2 ∼ IW d (Ψ 2 , ν 2 ) : Ψ 2 ∈ S d , ν 2 ∈ (0, ∞). (4.9)
In the simulation study we set d = 1. The selection of hyperparameters found in (4.6)-(4.9) was based on similar strategies as outlined in Escobar and West (1995) which produced:
• a 0 = 2, b 0 = 5, ν 1 = 4, ν 2 = 4, m 2 = 0, S 2 = 1, Ψ 2 = 1, τ 1 = 2.01 and τ 2 = 1.01.
We collected 10000 MCMC iterates after discarding the first 1000 as burn-in and thinning by 10.
Models M2 and M3 were fit using the Algorithm RGMM which was implemented in Fortran. For model M4, density estimates were obtained using the function DPdensity which is available in the DPpackage of R (Jara et al. 2011).
To compare density estimation associated with the four procedures just detailed we employ the following metrics:
• Log Pseudo Marginal Likelihood (LPML) (Christensen et al. 2011) which is a model fit metric that takes into account model complexity. This was computed by first estimating all the corresponding conditional predictive ordinates (Gelfand et al. 1992) using the method in Chen et al. (2000).
• Mean Square Error (MSE).
• L 1 -metric between the estimated posterior predictive density and f 0 .
Additionally, to explore how the repulsion influences model parsimony in terms of the number of occupied mixture components, we wecorded the following numeric indicators:
• Average number of occupied mixture components.
• Standard deviation of the average number of occupied mixture components. and M4 require many more occupied mixture components to achieve the same goodness-of-fit, a trend that persists when the sample size grows.
Data Illustrations
We now turn our attention to two well known data sets. The first is the Galaxy data set (Roeder 1990), and the second is bivariate Air Quality (Chambers 1983). Both are publicly available in R. For the second data set we removed 42 observations that were incomplete. We compare density estimates available from the DPMM to those from the RGMM. For each procedure we report the LPML as a measure of goodness-of-fit, a brief summary regarding the average number of occupied components, and posterior distribution associated with the number of clusters. It is worth noting that both data sets were standardized prior to model fit. We now provide more details on the two model specifications.
1. DPMM: We employed the R function DPdensity available in DPpackage (Jara et al. 2011). Decisions on hyperprior parameter values for both data sets were again guided by Escobar and West (1995). In both cases the model is specified by (4.2)-(4.9). We collected 10000 MCMC iterates after discarding the first 1000 (5000) as burn-in for Galaxy (Air Quality) data and thinning by 10. Specific details associated with model prior parameter values are now provided:
(a) Galaxy: d = 1, a 0 = 2, b 0 = 2, ν 1 = 4, ν 2 = 4, m 2 = 0, S 2 = 1, Ψ 2 = 0.15, τ 1 = 2.01 and τ 2 = 1.01.
(b) Air Quality: d = 2, a 0 = 1, b 0 = 3, ν 1 = 4, ν 2 = 4, m 2 = 0 2 , S 2 = I 2 , Ψ 2 = I 2 , τ 1 = 2.01 and τ 2 = 1.01. (b) Air Quality: k = 10, d = 2, α k,1 = 10 −1 1 10 , µ = 0 2 , Σ = I 2 , τ = 116.76, Ψ = 3I 2 and ν = 6.
RGMM: We coded Algorithm RGMM in
Results of the fits are provided in Table 1. Notice that the fit associated with RGMM is better relative to the DPMM, which corroborates the argument that RGMM sacrifices no appreciable model fit for the sake of model parsimony. Figure 8 further reinforces the idea that RGMM is more parsimonious relative to DPMM. This can be seen as the posterior distribution of the number of clusters (or non-empty components)
for RGMM concentrates on values that are smaller relative to the DPMM. Graphs of the estimated densities (provided in Figure 9) show that the cost of parsimony is negligible as density estimates are practically the same.
Data
Discussion and Future Work
We have created a class of probability models that explicitly parametrizes repulsion in a smooth way. In addition to providing pertinent theoretical properties, we demonstrated how this class of repulsive distributions can be employed to make hierarchical mixture models more parsimonious. Acompelling result is that this added parsimony comes at essentially no goodness-of-fit cost. We studied properties of the models, adapting the theory developed in Petralia et al. (2012) to accommodate the potential function we considered.
Moreover, we generalized the results to include not only Gaussian Mixtures of location but of also of scale (though the scale is constrained to be equal in each mixture component).
Our approach shares the same modeling spirit (presence of repulsion) as in Petralia et al. (2012), Xu et al. (2016 and Fúquene et al. (2016). However, the specific mechanism we propose to model repulsion differs from these works. Petralia et al. (2012) employ a potential (based on Lennard-Jones type potential) that introduces a stronger repulsion than our case, in the sense that in their model, locations are encouraged to be further apart. Xu et al. (2016) is based on Determinantal Point Processes, which introduces repulsion through the determinant of a matrix driven by a Gaussian covariance kernel. By nature of the point process, this approach allows a random number of mixture components (similar to DPM models) something that our approach lacks. However, our approach allows a direct modeling of the repulsion that is easier to conceptualize. Finally, the work by Fúquene et al. (2016) defines a family of probability densities that promotes well-separated location parameters through a penalization function, that cannot be re-expressed as a (pure) repulsive potential. However, for small relative distances, the penalization function can be identified as an interaction potential that produces repulsion similar to that found in Petralia et al. (2012).
Presently
A Algorithm RGMM
In what follows we describe the Gibbs Sampler for the RGMM in its entirety. Let B, S, T ∈ N be the total number of iterations during the burn-in, the number of collected iterates, and the thinning, respectively.
• (Start) Choose initial values z (0) i : i ∈ [n], π(0)
k,1 and θ
(0) j , Λ (0) j : j ∈ [k]. Set Γ j = O d : j ∈ [k], where O d is the null matrix of dimension d × d.
• (Burn-in phase) For t = 0, . . . , B − 1:
1. (z (t+1) i | · · · ) ∼ P(z (t+1) i = j) = π (t,i) j independently for each i ∈ [n], where π (t,i) j = k l=1 π (t) l N d (y i ; θ (t) l , Λ (t) l ) −1 π (t) j N d (y i ; θ (t) j , Λ (t) j ) : j ∈ [k]. 2. (π (t+1) k,1 | · · · ) ∼ Dir(α (t) k,1 ), where α (t) k,1 = (α 1 + n (t+1) 1 , . . . , α k + n (t+1) k ) n (t+1) j = card(i ∈ [n] : z (t+1) i = j) : j ∈ [k].
3. For j = 1, . . . , k:
3.1. Generate a candidate θ ( ) j from N d (θ (t) j , Ω (t) j ), where Ω (t) j = {Σ −1 + n (t+1) j (Λ (t) j ) −1 } −1 . 3.2. Update θ (t) j → θ (t+1) j = θ ( ) j with probability min(1, β j ), where β j = N d (θ ( ) j ; µ (t) j , Σ (t) j ) N d (θ (t) j ; µ (t) j , Σ (t) j ) k l =j 1 − exp{−0.5τ −1 (θ ( ) j − θ (t) l ) Σ −1 (θ ( ) j − θ (t) l )} 1 − exp{−0.5τ −1 (θ (t) j − θ (t) l ) Σ −1 (θ (t) j − θ (t) l )} .
In the above expression for β j Σ (t)
j = {Σ −1 + n (t+1) j (Λ (t) j ) −1 } −1 µ (t) j = Σ (t) j {Σ −1 µ + (Λ (t) j ) −1 s (t) j } : s (t) j = n i=1 I {j} (z (t+1) i )y i . Otherwise, set θ (t+1) j = θ (t) j . 3.3. Update Γ j → Γ j + B −1 Ω (t) j . 4. (Λ (t+1) j | · · · ) ∼ IW d (Ψ (t) j , ν (t) j ) independently for each j ∈ [k], where ν (t) j = ν + n (t+1) j and Ψ (t) j = Ψ + n i=1 I {j} (z (t+1) i )(y i − θ (t+1) j )(y i − θ (t+1) j
) .
• (Save samples) For t = B, . . . , ST + B − 1: Repeat steps 1, 2 and 4 of the burn-in phase. As for step 3 ignore 3.3, maintain 3.2 and replace 3.1 with 3.1a. Generate a candidate θ
( ) j from N d (θ (t) j , Γ j ).
Finally, save the generated samples every T th iteration.
• (Posterior predictive estimate) With the T saved samples, compute
f (y | y 1 , . . . , y n ) ≈ 1 T T t=1 k j=1 π (t) j N d (y; θ (t) j , Λ (t) j ) .
B Proof of Lemma 2.1.
Assign to R d k and [0, 1) the metrics d 1 (x k,d , y k,d ) = max{ρ(x i , y i ) : i ∈ [k]} and d 2 (x, y) = |x−y|, respectively. Continuity of R C : R d k → [0, 1) follows from condition A1 of C 0 -properties and the following inequality:
|ρ(x r , x s ) − ρ(y r , y s )| < 2d 1 (x k,d , y k,d ).
C Proof of Proposition 2.2.
Notice that g k,d ∈ C(R d k ; (0, ∞)) by construction (see Lemma 2.1). Because of the continuity, measurability follows. Using conditions A1-A4 of C 0 -properties it follows that for all x ∈ [0, ∞), {1 − C 0 (x)} ∈ [0, 1). By Tonelli's Theorem
R d k g k,d (x k,d )dx k,d ≤ R d f 0 (x)dx k = 1.
The upper bound only proves that g k,d is integrable. However, this does not guarantee that g k,d is well defined, i.e. λ k d (g k,d > 0) = 0. For this, it is sufficient to show that
R d k g k,d (x k,d )dx k,d > 0 because for all x k,d ∈ R d k , g k,d (x k,d ) ≥ 0 by construction. To prove the above inequality, fix x 0 k,d ∈ R d k such that x 0 r = x 0 s for r = s ∈ [k]. Then g k,d (x 0 k,d ) > 0. Because g k,d is a continuous function on R d k , there exists r 0 ∈ (0, ∞) such that for all x k,d ∈ B(x 0 k,d , r 0 ) g k,d (x k,d ) > 0, where B(x 0 k,d , r 0 ) is the cartesian product of B 2 (x 0 1 , r 0 ), . . . , B 2 (x 0 k , r 0 ). Further, B(x 0 k,d , r 0 ) ∈ B(R d k ) and λ k d {B(x 0 k,d , r 0 )} = (π kd/2 r kd 0 )Γ(1 + d/2) −k ∈ (0, ∞) by the Volume Formula, where Γ( · ) is the Gamma function. Thus R d k g k,d (x k,d )dx k,d ≥ B(x 0 k,d ,r0) g k,d (x k,d )dx k,d > 0.
D Proof of Lemma 3.1.
Using the initial assumptions
| log{f (x; ξ 0 k0 )}| ≤ h 1 (x) : x ∈ (−∞, t 0 1 ) | log{f (x; ξ k0 )}| ≤ h 2 (x) : (x, ξ k0 ) ∈ (−∞, t 0 1 ) × V 0 | log{f (x; ξ 0 k0 )}| ≤ h 3 (x) : x ∈ (t 0 2 , ∞) | log{f (x; ξ k0 )}| ≤ h 4 (x) : (x, ξ k0 ) ∈ (t 0 2 , ∞) × V 0 .
Taking into account the existence of second order moments of a Gaussian distribution
I 3 = (−∞,t 0 1 ) {h 1 (x) + h 2 (x)}f 0 (x; ξ 0 k0 )dx ∈ (0, ∞) I 4 = (t 0 2 ,∞) {h 3 (x) + h 4 (x)}f 0 (x; ξ 0 k0 )dx ∈ (0, ∞).
Again, using the Triangle Inequality
(−∞,t 0 1 )∪(t 0 2 ,∞) | log{f 0 (x; ξ 0 k0 )} − log{f (x; ξ k0 )}|f 0 (x; ξ 0 k0 )dx ≤ I 3 + I 4 ∈ (0, ∞).
The previous arguments show that | log{f 0 (x; ξ 0 k0 )} − log{f (x; ξ k0 )}|f 0 (x; ξ 0 k0 ) for all (x, ξ k0 ) ∈ R × V 0 is bounded above by a positive and integrable function that depends only in x ∈ R. As a consequence of Lebegue's Dominated Convergence Theorem
R log f 0 (x; ξ 0 k0 ) f (x; ξ k0 ) f 0 (x; ξ 0 k0 )dx → 0 as ξ k0 → ξ 0 k0 .
In other words, for all ε > 0 there exists δ > 0 such that
R log f 0 (x; ξ 0 k0 ) f (x; ξ k0 ) f 0 (x; ξ 0 k0 )dx < ε
provided that ξ k0 ∈ B 1 (θ 0 k0,1 , δ) × B 1 (π 0 k0,1 , δ) × (λ 0 − δ, λ 0 + δ).
E Proof of Lemma 3.2.
Set δ 00 = 0.25vk 0 with v > 0 specified by condition B2. Notice that
θ k0,1 ∈ B δ = k0 i=1 θ 0 i − δ k 0 , θ 0 i + δ k 0 ⊆ B 1 (θ 0 k0,1 , δ).
for all δ ∈ (0, δ 00 ]. Using the definition of NRep k0,1 (µ, σ 2 , τ ) and denoting c k0 = c k0,1 the associated normalizing constant, we have that
P{θ k0,1 ∈ B 1 (θ 0 k0,1 , δ)} ≥ 1 c k0 B δ k0 i=1 N(θ i ; µ, σ 2 ) k0 r<s 1 − exp − (θ r − θ s ) 2 2τ σ 2 dθ k0,1 .
Choose δ = min(δ 0 , δ 1 ) where δ 0 > 0 is given by Lemma 3.2. Now p 1 = P{θ k0,1 ∈ B 1 (θ 0 k0,1 , δ)} > 0. The same holds for p 2 = P{π k0,1 ∈ B 1 (π 0 k0,1 , δ)) and p 3 = P{λ ∈ (λ 0 − δ, λ 0 + δ)}. Thus, independence between π k0,1 , θ k0,1 and λ implies P ξ k0 ∈ Θ k0 : R log f 0 (x; ξ 0 k0 ) f (x; ξ k0 ) f 0 (x; ξ 0 k0 )dx < ε ≥ p 1 p 2 p 3 > 0.
G Proof of Lemma 3.4.
As already mentioned at the beginning of Subsection 2.3, θ k,1 ∼ NRep k,1 (µ, σ 2 , τ ) is an exchangeable distribution in θ 1 , . . . , θ k for k ≥ 2. This implies that the probability laws of each θ i : i ∈ [k] are the same.
To prove the desired inequality, observe that for all t ∈ (0, ∞) P(|θ i | > t) ≤ c k−1 c k At N(x; µ, σ 2 )dx = c k−1 c k Bt N(s; 0, 1)ds.
where A t = {x ∈ R : |x| > t} and B t = {s ∈ R : |µ + σs| > t}. Now B t ⊆ {s ∈ R : |µ| + σ|s| > t} = s ∈ R : |s| > t − |µ| σ = C t .
Set γ = max{2|µ| + 1, (2 + √ 2)|µ|} ∈ (0, ∞). By Mill's Inequality, for all t ∈ [γ, ∞) Ct N(s; 0, 1)ds ≤ 2 (2π) 1/2 σ(t − |µ|) −1 exp{−(2σ 2 ) −1 (t − |µ|) 2 } ≤ 2 (2π) 1/2 σ(|µ| + 1) −1 exp{−(4σ 2 ) −1 t 2 }.
Using the previous information P(|θ i | > t) ≤ 2 (2π) 1/2 σ(|µ| + 1) −1 exp{−(4σ 2 ) −1 t 2 } for all t ∈ [γ, ∞) and i ∈ [k].
H Proof of Lemma 3.5.
By the Change of Variables Theorem and Fubini's Theorem, it can be shown that for all k ≥ 2 (k ∈ N)
c k = R 1 k−1 F k−1 (θ −1,1 ) k i=2 N(θ i ; 0, 1) k 2≤r<s 1 − exp − (θ r − θ s ) 2 2τ dθ −1,1
where θ −1,1 = (θ i : i = 1) ∈ R 1 k−1 and F k−1 : R 1 k−1 → (0, 1) is given by
F k−1 (θ −1,1 ) = R N(θ 1 ; 0, 1) k j=2 1 − exp − (θ 1 − θ j ) 2 2τ dθ 1 .
Notice that F k−1 ∈ C(R 1 k−1 ; (0, 1)) (as a consequence of Lebesgue's Dominated Convergence Theorem) and F k−1 (θ −1,1 ) → 1 as ||θ −1,1 || → ∞. By Jensen's Inequality, for all θ −1,1 ∈ R 1 k−1 log{F k−1 (θ −1,1 )} ≥ k j=2 R N(θ 1 ; 0, 1) log 1 − exp − (θ 1 − θ j ) 2 2τ dθ 1 . Now R N(θ 1 ; 0, 1) log 1 − exp − (θ 1 − θ j ) 2 2τ dθ 1 ≤ −2 τ 1/2 π 1/2 ∞ 0 log{1 − exp(−θ 2 1 )}dθ 1 .
Using the substitution θ 1 (z) = z 1/2 : z ∈ (0, ∞) and then integrating by parts where Γ( · ) and ζ( · ) are the Gamma and Riemann Zeta functions, respectively. The previous information implies that R N(θ 1 ; 0, 1) log 1 − exp − (θ 1 − θ j ) 2 2τ dθ 1 ≤ 2.6124τ 1/2 ∈ (0, ∞).
With this bound, defining A 2 = 2.6124τ 1/2 and A −1 1 = exp(A 2 ) the following holds: for all θ −1,1 ∈ R 1 k−1 log{F k−1 (θ −1,1 )} ≥ −(k − 1)A 2 which implies F k−1 (θ −1,1 ) ≥ A −1 1 exp(−A 2 k). To conclude the proof, notice that Using the previous equation it follows that for all k ≥ 2 (k ∈ N) c k ≥ A −1 1 exp(−A 2 k)c k−1 > 0, the above being equivalent to 0 < c k−1 c k ≤ A 1 exp(A 2 k).
I Proof of Proposition 3.6.
Following Theorem 3.1 in Scricciolo (2011) p = 2 induce a (finite) Gaussian Mixture Model, λ ∼ IG(a, b) : a, b ∈ (0, ∞) satisfy (i) and π k,1 ∼ Dir(k −1 1 k ) satisfy (iii). Condition B3 is equivalent to (ii). However, (iv) does not apply because the cluster-location parameters are not i.i.d. in our framework. Along the proof of Theorem 3.1 we identified those steps that can be adapted by the assumption θ k,1 ∼ NRep k,1 (µ, σ 2 , τ ). It is important to mention that Theorem 3.1 appeals to conditions (A.1), (A.2) and (A.3) in Theorem A.1 (Appendix of Scricciolo's paper) which is a powerful result given by Ghosal and van der Vaart (2001). We will check that (A.1) to (A.3) are satisfied:
(A.1) The proof is the same as the arguments presented at page 277 and the first paragraph in page 278.
The reason for this is that it only depends on the structure of the mixture, leaving aside the prior distributions for all the involved parameters.
(A.2) What needs to be modified on the first inequality found on page 278 is the term E(K)Π([−a n , a n ] c ). This quantity is part of the chain of inequalities kn i=1 ρ(i) i j=1 P(|θ j | > a n ) = kn i=1 iρ(i)Π([−a n , a n ] c ) ≤ E(K)Π([−a n , a n ] c ) exp{−ca ϑ n } under the conditions (ii) and (iv). In our case, ρ(i) = κ(i) for i ∈ N. By way of Lemma 3.4 i j=1 P(|θ j | > a n ) ≤ 2i (2π) 1/2 c i−1 c i σ(|µ| + 1) −1 exp{−(4σ 2 ) −1 a 2 n } under the convention that c 0 = 1 and n ∈ N is big enough. Thus, kn i=1 ρ(i) i j=1 P(|θ j | > a n ) ≤ 2 (2π) 1/2 σ(|µ| + 1) −1 exp{−(4σ 2 ) −1 a 2 n } kn i=1 iρ(i) c i−1 c i and by Lemma 3.5
kn i=1 iρ(i) c i−1 c i ≤ A 1 B 1 ∞ i=1 i exp{−(B 2 − A 2 )i} ∈ (0, ∞).
Finally, we obtain the following upper bound (in order), which is analogous to that obtain in Scricciolo (2011):
kn i=1 ρ(i) i j=1
P(|θ j | > a n ) exp{−(4σ 2 ) −1 a 2 n }.
(A.3) We only need to adapt the following inequality found on page 279, whose validity is deduced from (iv):
P{θ k0 ∈ B(θ 0 k0 ; ε)} = Π ⊗k0 {B(θ 0 k0 ; ε)} exp{−d 1 k 0 log(1/ε)} In our case, θ k0 = θ k0,1 , θ 0 k0 = θ 0 k0,1 and B(θ 0 k0 ; ε) = D 1 (θ 0 k0,1 , ε). At the end of the proof of Lemma 3.2 it is shown that for every δ = ε ∈ (0, δ 0 ] P{θ k0,1 ∈ D 1 (θ 0 k0,1 , ε)} ≥ R 0 c k0 k0 i=1 S 0 i 2 exp{−k 0 log(1/ε)}.
With this information, we obtain a lower bound (in order) analogous to that obtained in Scricciolo (2011): P{θ k0 ∈ B(θ 0 k0 ; ε)} exp{−k 0 log(1/ε)}.
Figure 1 :
1Data simulated from the mixture of 4 bivariate normal densities in (1.2). The left panel shows the original n = 300 data points with colors and numbers indicating the original cluster. The right panel shows the clustering resulting from applying Dahl's least squares clustering algorithm to a DPM.
Figure 2 :
2with A 1 , . . . , A k events on F and A c i denoting the complement of A i . With this in mind, c k,d is the result of adding/substracting all the contributions Ψ k,d (A) that emerge for every non-empty set A ⊆ I k . If we think of c k,d as an indicator of the strength of repulsion, Ψ k,d (A) provides the specific contribution from the interactions (r, s) ∈ A. Moreover, it quantifies how distant a Rep k,d (f 0 , C 0 , ρ) distribution is from the The graph and Laplacian matrix for a possible interaction for k = 4 coordinates.
Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :
34567Boxplots that resume the behavior of LPML for each of the four models. Boxplots that resume the behavior of MSE for each of the four models. Boxplots that resume the behavior of L 1 -metric for each of the four models.Figures 3, 4 and 5 contain side-by-side boxplots of the LPML, MSE and L 1 -metric respectively as the sample size grows. Notice that trends seen here indicate that M1 and M4 tend to fit better, but M2 and M3 are very competitive with the advantage of being more parsimonious. In other words, very little model fit was sacrificed for the sake of parsimony. Side-by-side boxplots of the average number of occupied mixture components for each of the procedure.Figures 6 and 7show that the average number of occupied mixture components is much smaller for M2 and M3 relative to M1 and M4. This pattern persists (possibly becomes more obvious) as the number of observations grows. The number of occupied mixture components for M2 and M3 are also highly concentrated around 3, 4 and 5 (recall that the data were generated using a mixture of four components). Conversely, M1 Side-by-side boxplots that display the average standard deviation associated with the posterior distribution of occupied mixture components for each of the four procedures.
Fortran to generate posterior draws for this model. For both data sets, we collected 10000 MCMC iterates after discarding the first 5000 as burn-in and thinning by 50. The values of τ were selected using the procedure outlined in Subsection 3.1.1: (u, p) = (0.5, 0.95) and (u, p) = (0.05, 0.95) for Galaxy and Air Quality data respectively. Parameter selection for model components (3.9)-(3.11) were carried out according to the methods in Subsection 3.1.1. Specific details now follow:(a) Galaxy: k = 10, d = 1, α k,1 = 10 −1 1 10 , µ = 0, Σ = 1, τ = 5.45, Ψ = 0.15 and ν = 5.
Figure 8 :
8Posterior distribution for the active number of clusters in (a) Galaxy and (b) Air Quality data. Black (gray) bars correspond to RGMM (DPMM).
Figure 9 :
9Posterior predictive densities for (a) Galaxy and (b) Air Quality data. Black solid (gray dashed) curves correspond to RGMM (DPMM).
we are pursuing a few directions of continued research. First, Propositions 3.3 and 3.6 were established for Gaussian mixtures of dimension d = 1 with mixture components sharing the same variance. Extending results to the general d dimensional case would be a natural progression. Additionally, we are exploring the possibility of relaxing the assumption of common variance between mixture components and adapting the mentioned theoretical results to a larger class of potential functions. Studying the influence of the metric on the repulsive component in Definition 2.1 and allowing the number of mixture components to be random are also topics of future research. Rousseau and Mengersen (2011) developed some very interesting results that explore statistical properties associated with mixtures when k is chosen to be conservatively large (overfitted mixtures) with decaying weights associated with these extra mixture components. They did so using a framework that is an alternative to what we developed here. Under some restrictions on the prior and regularity conditions for the mixture component densities, the asymptotic behavior of the posterior distribution on the weights tends to empty the extra mixture components. We are currently exploring connections between these two approaches. Stephens, M. (2000), "Bayesian Analysis of Mixture Models with an Unknown Number of Components An Alternative to Reversible Jump Methods," The Annals of Statistics, 28, 40-74. Strauss, D. J. (1975), "A model for clustering," Biometrika, 62, 467-475. Xu, Y., Müller, P., and Telesca, D. (2016), "Bayesian Inference for Latent Biological Structure with Determinantal Point Processes (DPP)," Biometrics, 72, 955-964.
exp − (θ r − θ s ) 2 2τ dθ −1,1 .
1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Gaussian Mixture Models and NRep k,d (µ, Σ, τ ) Distribution In this section we will briefly introduce Gaussian Mixture Models, which are very popular in the context of density estimation(Escobar and West 1995) because of their flexibility and computational tractability. Then we show that repulsion can be incorporated by modeling location parameters with the repulsion distribution described previously.3.1 Repulsive Gaussian Mixture Models (RGMM)Consider n ∈ N experimental units whose responses y 1 , . . . , y n are d-dimensional and assumed to be exchangeable. Gaussian mixtures can be thought of as a way of grouping the n units into several clusters, say k ∈ N, each having its own specific characteristics. In this context, the jth cluster (j ∈ [k]) is modeled
For any x ∈ R we have that |f 0 (x; ξ 0 k0 ) − f (x; ξ k0 )| ≤ ||π 0 k0,1 − π k0,1 || 1 (2πλ 0 ) 1/2 + ||θ 0 k0,1 − θ k0,1 || 1 {2π exp(1)} 1/2 λ 0 + u(λ, θ k0,1 ; x, λ 0 )|λ − λ 0 | and u(λ, θ k0,1 ; x, λ 0 ) = 1 (2π) 1/2 k 0 λλ 1/2 0k0 . The last statement is equivalent to the condition that. By condition B2, we can assume that θ 0 1 < · · · < θ 0 k0 (possibly after an appropriate relabeling). Choose t 0 1 , t 0 2 ∈ R and l 0 1 , l 0 2 ∈ (0, ∞) such that λ 0 ∈ [l 0 1 , l 0 2 ] and, for allBy the Triangle InequalityOn the other hand, define the following continuous functions:for all δ ∈ (0, δ 00 ]. Nowfor all θ k0,1 ∈ B δ , with v 0 = (v − 2δ 00 k −1 0 ) 2 and k0 = 0.5k 0 (k 0 − 1). Using this information and Fubini's Theoremas δ → 0 (right-side limit), there exists δ 0i > 0 such thatfor all δ ∈ (0, δ 0 ]. Remark: The previous inequality also applies replacing B 1 (θ 0 k0,1 , δ) by D 1 (θ 0 k0,1 , δ).F Proof of Proposition 3.3.We will follow the proof of Lemma 1 inPetralia et al. (2012)with a few variations. For this, let ε > 0 and defineUsing the stochastic representation (3.13),By condition B3, κ(k 0 ) > 0. In this case, to guarantee (3.14) it is sufficient to show that P ξ k0 ∈ Θ k0 :Lemma 3.1 guaranties the existence of δ 1 > 0 such that for all ξ k0 ∈ B 1 (θ 0 k0,1 , δ 1 ) × B 1 (π 0 k0,1 , δ 1 ) × (λ 0 − δ 1 , λ 0 + δ 1 ) R log f 0 (x; ξ 0 k0 ) f (x; ξ k0 ) f 0 (x; ξ 0 k0 )dx < ε.
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[
"Nonperturbational \"Continued-Fraction\" Spin-offs of Quantum Theory's Standard Perturbation Methods",
"Nonperturbational \"Continued-Fraction\" Spin-offs of Quantum Theory's Standard Perturbation Methods"
]
| [
"Steven Kenneth Kauffmann "
]
| []
| []
| The inherently homogeneous stationary-state and time-dependent Schrödinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for scattering, the preferred permitted asymptotic behavior. For bound states it provides sufficient focus to exclude all but one of the homogeneous version's solutions. Because of their unique solutions, such inhomogeneous versions of Schrödinger equations have long been the indispensable basis for a solution scheme of successive perturbational corrections which are anchored by their inhomogeneous term. Here it is noted that every such perturbational solution scheme for an inhomogeneous linear vector equation spins off a nonperturbational continued-fraction scheme. Unlike its representation-independent antecedent, the spin-off scheme only works in representations where all components of the equation's inhomogeneous term are nonzero. But that requirement seems to confer theoretical physics robustness heretofore unknown: for quantum fields the order of the perturbation places a bound on unperturbed particle number, the spin-off scheme contrariwise has only basis elements of unbounded unperturbed particle number. It furthermore is difficult to visualize such a continued-fraction spin-off scheme generating infinities, since its successive iterations always go into denominators. | null | [
"https://arxiv.org/pdf/1301.1647v1.pdf"
]
| 18,671,069 | 1301.1647 | d9a907bc81ea69b622a4d7533671e8bab87d5cf9 |
Nonperturbational "Continued-Fraction" Spin-offs of Quantum Theory's Standard Perturbation Methods
31 Dec 2012
Steven Kenneth Kauffmann
Nonperturbational "Continued-Fraction" Spin-offs of Quantum Theory's Standard Perturbation Methods
31 Dec 2012
The inherently homogeneous stationary-state and time-dependent Schrödinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for scattering, the preferred permitted asymptotic behavior. For bound states it provides sufficient focus to exclude all but one of the homogeneous version's solutions. Because of their unique solutions, such inhomogeneous versions of Schrödinger equations have long been the indispensable basis for a solution scheme of successive perturbational corrections which are anchored by their inhomogeneous term. Here it is noted that every such perturbational solution scheme for an inhomogeneous linear vector equation spins off a nonperturbational continued-fraction scheme. Unlike its representation-independent antecedent, the spin-off scheme only works in representations where all components of the equation's inhomogeneous term are nonzero. But that requirement seems to confer theoretical physics robustness heretofore unknown: for quantum fields the order of the perturbation places a bound on unperturbed particle number, the spin-off scheme contrariwise has only basis elements of unbounded unperturbed particle number. It furthermore is difficult to visualize such a continued-fraction spin-off scheme generating infinities, since its successive iterations always go into denominators.
Introduction
Schrödinger equations, whether stationary-state or time-dependent, are homogeneous, and as such can suffer from solution nonuniqueness. In order to be able to effectively apply standard successive approximation schemes, such as perturbational ones, to Schrödinger equations, they are often first recast into a specialized inhomogeneous linear form which has a unique solution. Denoting its state vector as |ψ , a simple generic presentation of a Schrödinger equation in inhomogeneous form is,
|ψ = |ψ 0 − V|ψ .(1a)
The state vector |ψ 0 , which comprises the inhomogeneous term of Eq. (1a) usually shares a key feature with |ψ , such as asymptotic behavior in a region of infinite extent or exact value at an initial time, or, alternately, is a passable approximation to |ψ . Given that the Hamiltonian operator of the homogeneous Schrödinger equation which underlies Eq. (1a) is H, we in addition assume the existence of an exactly diagonalizable Hamiltonian operator H 0 that has |ψ 0 as one of its eigenstates, and for which the linear operator V of Eq. (1a) is of no less than first order in V def = ( H − H 0 ). If we now take the viewpoint that the inhomogeneous term |ψ 0 on the right-hand side of Eq. (1a) is a sufficiently good approximation to |ψ on its left-hand side that the remainder term − V|ψ is only a small perturbation, then the perturbational successive approximations,
|ψ (n+1) = |ψ 0 − V|ψ (n) ,(1b)
where, of course, |ψ (0) = |ψ 0 , would be expected to converge reasonably rapidly. Additional insight into the Eq. (1b) perturbational iteration scheme can be obtained from taking the second term on the right-hand side of Eq. (1a) to its left-hand side, which yields,
|ψ + V|ψ = |ψ 0 .(1c)
Eq. (1c) has the formal operator solution,
|ψ = [1 + V] −1 |ψ 0 .(1d)
This presents the issue of how to practically evaluate the formal operator [1 + V] −1 . One way is to try to use its formal geometric series expansion,
[1 + V] −1 = 1 − V + ( V) 2 + · · · + (− V) n + · · · ,(1e)
which is readily verified to yield the exactly the same result when inserted into Eq. (1d) as is obtained from successively applying the Eq. (1b) perturbational iteration scheme. Given the well-known convergence and divergence characteristics of the geometric series, it is apparent that the Eq. (1b) perturbational iteration scheme for the inhomogeneous Eq. (1a) is unlikely to be greatly useful unless |ψ 0 is substantially dominant over V|ψ 0 . That naturally raises the question of whether there might be an alternate general iteration approach to the inhomogeneous Eq. (1a) that could come to the rescue when its Eq. (1b) perturbational iterations converge too slowly or diverge. A Lippmann-Schwinger nonrelativistic potential-scattering variant of the inhomogeneous Eq. (1a), considered for arbitrarily strong potentials in a recent article [1], has revealed that if a complete representation basis set { ρ i |} can be found such that every component ρ i |ψ 0 of the inhomogeneous-term state vector |ψ 0 is nonzero, then a robust nonperturbational iteration scheme can be devised from the explicit presentation of Eq. (1c) in that representation, namely from the specific equalities,
ρ i |ψ + ρ i | V|ψ = ρ i |ψ 0 .(2a)
If we now make the assumption that ρ i |ψ is, like ρ i |ψ 0 , nonzero, we can factor the left-hand side of Eq. (2a) into ρ i |ψ and the resulting nonzero cofactor . We then proceed to divide both sides of Eq. (2a) by that nonzero cofactor , which changes the appearance of Eq. (2a) to,
ρ i |ψ = ρ i |ψ 0 / 1 + ( ρ i |ψ ) −1 ρ i | V|ψ ,(2b)
from which we straightforwardly devise the manifestly nonperturbational successive-approximation scheme,
ρ i |ψ (n+1) = ρ i |ψ 0 / 1 + ρ i |ψ (n) −1 ρ i | V|ψ (n) ,(2c)
where, of course, ρ i |ψ (0) = ρ i |ψ 0 . If ρ i |ψ (n) is nonzero, then because ρ i |ψ 0 is nonzero, barring the pathological occurrence of a divergence in ρ i | V|ψ (n) , ρ i |ψ (n+1) will in turn be nonzero. We also note that the iteration scheme of Eq. (2c) has the desirable nonperturbational character of a continued fraction. Thus the Eq. (1a) generic specialized inhomogeneous form of the Schrödinger equation always spawns not only the perturbational successive-approximation scheme of Eq. (1b), but as well the nonperturbational continued-fraction successive-approximation scheme of Eq. (2c) that can regarded as its spin-off.
We shall now survey some of the well-known circumstances where a Schrödinger equation with Hamiltonian operator H for a state |ψ is combined both with a state |ψ 0 which has a crucial similarity to |ψ and as well with an exactly diagonalizable Hamiltonian operator H 0 that includes |ψ 0 as one of its eigenstates to produce an inhomogeneous linear equation for |ψ which has the generic form given by Eq. (1a), where the linear operator V is of at least first order in V def = ( H − H 0 ). We shall as well mention some of the complete representation basis sets { ρ i |} for which certain of the surveyed inhomogeneous terms |ψ 0 have exclusively nonzero components ρ i |ψ 0 -note that the eigenstate set of the Hamiltonian operator H 0 is entirely unacceptable as such a complete representation basis set { ρ i |} because all of its members aside from ψ 0 | itself are orthogonal to |ψ 0 .
The Lippmann-Schwinger equation for nonrelativistic potential scattering As in Ref. [1] we consider the coordinate-representation nonrelativistic Schrödinger equation for an eigenstate
r|ψ E of positive energy E, −h 2 ∇ 2 r /(2m) + V (r) r|ψ E = E r|ψ E ,(3a)
where,
lim |r|→∞ V (r) = 0. (3b)
Thus for sufficiently large |r|, the Schrödinger equation of Eq. (3a) reduces to,
−h 2 ∇ 2 r /(2m) r|ψ E = E r|ψ E ,(3c)
which is satisfied by any plane wave e ip·r/h for which |p| = (2mE) 1 2 and by any linear superposition of these plane waves. Among those linear superpositions are all the angularly modulated ingoing and outgoing spherical waves that have wave number k = (2mE) 1 2 /h. Now a scattering experiment at energy E > 0 is described by a particular solution of the Schrödinger Eq. (3a) which at sufficiently large |r| that Eq. (3a) is well-approximated by Eq. (3c) consists of only a single specified plane wave e ip·r/h of momentum p plus only outgoing spherical waves [2]. We denote this scattering solution of the Schrödinger Eq. (3a) as r|ψ + p . It turns out that an inhomogeneous modification of the Schrödinger Eq. (3a) describes r|ψ + p uniquely, namely the following nonrelativistic Lippmann-Schwinger equation for potential scattering [2],
r|ψ + p = e ip·r/h − r|( H 0 − E p − iǫ) −1 V |ψ + p ,(3d)
where H 0
def = | p| 2 /(2m) = −h 2 ∇ 2 /(2m) is the kinetic energy operator and E p def = |p| 2 /(2m)
is the kinetic energy c-number scalar that corresponds to the c-number momentum vector p.
Taking r|p def = e ip·r/h , we can write Eq. (3d) in the form of Eq. (1a), i.e.,
|ψ + p = |p − ( H 0 − E p − iǫ) −1 V |ψ + p ,(3e)where |ψ 0 = |p and V = ( H 0 − E p − iǫ) −1 V . Furthermore, since H 0 |p = E p |p ,
we can recover the Schrödinger Eq. (3a) from Eq. (3e) by multiplying the latter through by ( H 0 − E p ), followed by rearrangement of the resulting terms between the left-hand and right-hand sides. The negative imaginary infinitesimal −iǫ that appears in the Lippmann-Schwinger Eqs. (3d) and (3e) ensures that only outgoing spherical waves are present for sufficiently large |r|, in addition, of course, to the single specified plane wave e ip·r/h of momentum p.
The usual perturbational iteration of the Lippmann-Schwinger Eq. (3d) is,
r|ψ (n+1)+ p = e ip·r/h − r|( H 0 − E p − iǫ) −1 V |ψ (n)+ p ,(3f)
which with r|ψ (0)+ p = e ip·r/h generates the familiar perturbational geometric Born series for nonrelativistic potential scattering [3].
For nonperturbational "continued-fraction" iteration of the Lippmann-Schwinger Eq. (3d) we happen to be in the extraordinarily fortunate situation that coordinate representation of the inhomogeneous term |ψ 0 = |p is always nonzero because r|p = e ip·r/h = 0. Therefore in the manner of Eq. (1c) and Eqs. (2a)-(2c), the Lippmann-Schwinger Eq. (3d) in coordinate representation gives rise to the nonperturbational "continued fraction" iteration scheme,
r|ψ (n+1)+ p = e ip·r/h / 1 + r|ψ (n)+ p −1 r|( H 0 − E p − iǫ) −1 V |ψ (n)+ p ,(3g)
where we of course have that r|ψ
(0)+ p = e ip·r/h .
A bound-state inhomogeneous equation suitable for iteration solution
The just-discussed Lippmann-Schwinger equation is an inhomogeneous Schrödinger-equation variant that is of the form of our generic Eq. (1a). With an arbitrary plane wave as its inhomogeneous term; the Lippmann-Schwinger equation enables completely prescribed iteration refinement of that approximating plane wave toward its unique associated "asymptotically outgoing-only spherical-wave" exact scattering wave function.
The key ingredient that is needed to fashion the inhomogeneous Lippmann-Schwinger equation from its underlying homogeneous Schrödinger equation is of course the "outgoing-only spherical-wave" free-particle propagator, which is constructed from the selfsame free-particle Hamiltonian for which that approximating plane wave is an eigenstate.
In contrast, the standard Schrödinger approach to the perturbational refinement of an approximating bound state |ψ 0 j fails to explicitly present the inhomogeneous Schrödinger-equation variant which links that approximating bound state to a unique associated exact bound-state solution |ψ j of the underlying homogeneous stationary-state Schrödinger equation H|ψ j = E j |ψ j . Due to the absence of the information which is inherent to that inhomogeneous linking equation, the standard Schrödinger bound-state perturbational approach can't offer minutely prescribed iteration instructions for its perturbational refinement procedure [4]. More importantly, if that inhomogeneous linking equation is not in hand, there is no obvious way to devise alternate nonperturbational methods for the iteration refinement of approximating bound states.
Fortunately, we can follow the blueprint that is provided to us by the Lippmann-Schwinger equation to devise its missing bound-state analog. A major difference between the two, however, is that bound states certainly don't feature outgoing, ingoing or any other type of traveling wave. Therefore the kind of propagator we need for bound states is of the standing-wave type. Nor can a propagator suited to an approximating bound state possibly be constructed from the free-particle Hamiltonian operator as it properly is for the approximating plane waves of the Lippmann-Schwinger equation. The construction of the standing wave propagator we need must be from a Hamiltonian operator H 0 which has that approximating bound state |ψ 0 j as one of its bound-state eigenstates, i.e., H 0 must satisfy,
H 0 |ψ 0 j = E 0 j |ψ 0 j .(4a)
Of course, since that approximating bound state |ψ 0 j is indeed bound , we can normalize it to unity, i.e.,
ψ 0 j |ψ 0 j = 1. (4b)
Finally, the construction of the required standing-wave propagator from the Hamiltonian operator H 0 can't be accomplished in practice unless H 0 is exactly diagonalizable. If all the eigenstates and eigenvalues of H 0 are actually available, its corresponding standing-wave propagator can be obtained in the schematic form,
P/( H 0 − E 0 j ) def = lim ǫ→0 H 0 − E 0 j H 0 − E 0 j 2 + ǫ 2 −1 = i =j |ψ 0 i ψ 0 i |/(E 0 i − E 0 j ),(4c)
where the letter P in the standing-wave propagator's definition denotes "principal value". The exactly diagonalizable Hamiltonian operator H 0 is frequently called the "unperturbed" Hamiltonian operator, its particular bound eigenstate |ψ 0 j the "unperturbed" approximating bound state, and that state's H 0 eigenvalue E 0 j the corresponding "unperturbed" approximating bound-state energy. By the same token, one could call the operator P/( H 0 − E 0 j ) the "unperturbed" standing-wave propagator. Note that P/( H 0 − E 0 j ) has a special feature which has no analog for traveling-wave propagators such as the outgoing-wave propagator of the Lippmann-Schwinger equation, namely that,
ψ 0 j | P/( H 0 − E 0 j ) = 0.(4d)
The final ingredient we need to obtain the bound-state analog of the inhomogeneous Lippmann-Schwinger equation is the homogeneous stationary-state Schrödinger equation for the exact bound state |ψ j that is uniquely linked to the "unperturbed" approximating bound state |ψ 0 j . Denoting the exact Hamiltonian operator for the physical system we are studying as H, this homogeneous stationary-state Schrödinger equation is of course,
H|ψ j = E j |ψ j ,
which it can sometimes be more convenient to express as,
H − E j |ψ j = 0.
Since the physical system's exact Hamiltonian operator H won't in general be exactly diagonalizable (unlike the somewhat artificial "unperturbed" Hamiltonian operator H 0 ), we have no more a priori knowledge of the particular exact bound-state energy E j than we do of its corresponding exact bound-state eigenstate |ψ j of the exact Hamiltonian operator H. In order to avoid having the exact bound-state energy E j become a completely independent unknown for which we need to solve, we choose to write the physical system's exact homogeneous stationary-state Schrödinger equation for the bound state |ψ j in the more cumbersome form,
H − ψ j | H|ψ j / ψ j |ψ j |ψ j = 0,(4e)
which entirely supplants E j by |ψ j and H. Eqs. (4a)-(4e) supply all the ingredients needed to devise the bound-state analog of the Lippmann-Schwinger equation. Inspection of the Lippmann-Schwinger equation strongly suggests that we multiply the physical system's homogeneous Schrödinger Eq. (4e) from the left by the "unperturbed" standing-wave propagator P/( H 0 − E 0 j ) as the first step toward that analog. That operator multiplication produces the physical Schrödinger-equation variant,
P/( H 0 − E 0 j ) H − ψ j | H|ψ j / ψ j |ψ j |ψ j = 0,(4f)
which is not yet the inhomogeneous Lippmann-Schwinger equation analog that we seek. To complete our task we now develop an identity involving |ψ j , P/( H 0 − E 0 j ) and − H 0 + E 0 j to which we shall then add the Eq. (4f) physical Schrödinger-equation variant. One might naively expect that,
|ψ j + P/( H 0 − E 0 j ) − H 0 + E 0 j |ψ j = 0,
but if we multiply the left-hand side of this proposed equality by ψ 0 j | and take note of Eq. (4d), we see that the result is ψ 0 j |ψ j rather than zero. From Eqs. (4c) and (4b) it is indeed seen that,
|ψ j + P/( H 0 − E 0 j ) − H 0 + E 0 j |ψ j = |ψ 0 j ψ 0 j |ψ j .(4g)
We now add the Eq. (4g) identity to the physical Schrödinger-equation variant given by Eq. (4f) to obtain,
|ψ j + P/( H 0 − E 0 j ) H − H 0 + E 0 j − ψ j | H|ψ j / ψ j |ψ j |ψ j = |ψ 0 j ψ 0 j |ψ j . (4h)
If we now define the "interaction Hamiltonian operator" V as,
V def = ( H − H 0 ),(4i)
we can rewrite Eq. (4h) as,
|ψ j + P/( H 0 − E 0 j ) V − ψ j |( V + H 0 − E 0 j )|ψ j / ψ j |ψ j |ψ j = |ψ 0 j ψ 0 j |ψ j .(4j)
If we move the term involving the "unperturbed" standing-wave propagator P/( H 0 − E 0 j ) to the right-hand side of Eq. (4j), we obtain,
|ψ j = |ψ 0 j ψ 0 j |ψ j − P/( H 0 − E 0 j ) V − ψ j |( V + H 0 − E 0 j )|ψ j / ψ j |ψ j |ψ j ,(4k)
which is the inhomogeneous bound-state analog of the Lippmann-Schwinger equation presented in Eq. (3e). The exact energy E j of the exact physical bound state |ψ j is of course given by,
E j = ψ j |( V + H 0 )|ψ j / ψ j |ψ j .(4l)
The complete detailed prescription for the perturbational iteration of Eq. (4k) is then obviously,
|ψ (n+1) j = |ψ 0 j ψ 0 j |ψ (n) j − P/( H 0 − E 0 j ) V − ψ (n) j |( V + H 0 − E 0 j )|ψ (n) j / ψ (n) j |ψ (n) j |ψ (n) j , (4m)
where, of course, |ψ (0) j = |ψ 0 j . In addition, entirely as a byproduct of |ψ (n) j , we have,
E (n+1) j = ψ (n) j |( V + H 0 )|ψ (n) j / ψ (n) j |ψ (n) j . (4n)
As was pointed out earlier, the complete detailed prescription of Eqs. (4m) and (4n) for the perturbational iteration of a bound state approximation |ψ 0 j cannot be obtained within the confines of the standard Schrödinger bound-state perturbation approach [4] because that approach makes no attempt to work out the inhomogeneous Eq. (4j) or (4k) variant of the stationary-state Schrödinger equation for the exact bound state |ψ j . Of course having Eq. (4j) in hand is also absolutely critical for the development of nonpertubational "continued fraction" iteration of ρ i |ψ 0 j , where ρ i |ψ 0 j must be nonzero for every member of the complete orthogonal basis set { ρ i |}. The prescription for that nonperturbational "continued fraction" iteration clearly comes out to be,
ρ i |ψ (n+1) j = ρ i |ψ 0 j ψ 0 j |ψ (n) j / 1 + ρ i |ψ (n) j −1 ρ i | P/( H 0 − E 0 j ) V − ψ (n) j |( V + H 0 − E 0 j )|ψ (n) j / ψ (n) j |ψ (n) j |ψ (n) j ,(4o)
where, of course, ρ i |ψ
(0) j = ρ i |ψ 0 j , and the approximation E (n+1) j
to the energy eigenvalue is given by Eq. (4n).
For a bound state approximation |ψ 0 j , finding a complete orthogonal basis set { ρ i |} such that every ρ i |ψ 0 j is nonzero may not be simple because, aside from the ground state, bound states normally have nodes (where they of course vanish) in both coordinate and momentum representation. However if that bound state approximation |ψ 0 j is an eigenstate of an "unperturbed" harmonic oscillator Hamiltonian H 0 , then one can find a related "harmonic-oscillator coherent false ground-state with false excitations" orthogonal basis { ψ(c) 0 i |} such that every ψ(c) 0 i |ψ 0 j is indeed nonvanishing. Here c is a complex number such that |c| 2 is a transcendental positive real number, e.g., the natural-logarithm base e or the constant π, and the "false ground state" is the minimum-uncertainty coherent state that is characterized by the complex number c, i.e., it is annihilated by one of the harmonic-oscillator annihilation operators with c subtracted from it . Successive mutually orthogonal states |ψ(c) 0 i are then generated from this particular coherent "false ground state" characterized by c through the repeated action of the corresponding harmonic-oscillator creation operator with the complex conjugatec of c subtracted from it-these are the "false excitations" of the coherent "false ground state". It turns out that ψ(c) 0 i |ψ 0 j can't vanish if |c| 2 is a positive transcendental number because it only vanishes when |c| 2 is a zero of an appropriate polynomial that has rational coefficients. We note that the nontranscendental (i.e., the "algebraic") real numbers, being a countable set , are of measure zero. Furthermore, choosing to use harmonic-oscillator states as bound state approximations is likely a viable proposition, e.g., for a bound state of a three-dimensional system, its approximation by a suitable three-dimensional harmonic oscillator state could probably be made to work out quite well.
Time-dependent Dirac-picture successive approximation methods
It is often the case that the Hamiltonian operator H for a time-dependent Schrödinger equation, ihd(|ψ(t) )/dt = H|ψ(t) ,
can be written in the form H = H 0 + V , where H 0 is exactly diagonalizable and |ψ(t 0 ) is specified to equal one of the eigenstates |ψ 0 of H 0 . In that case it can be useful to reexpress the time-dependent Schrödinger Eq. (5a) in the Dirac picture [5],
|ψ(t) = e −i H0(t−t0)/h |ψ D (t) ,(5b)
which yields,
ihd(|ψ D (t) )/dt = V D (t)|ψ D (t) ,(5c)
where,
V D (t) def = e +i H0(t−t0)/h V e −i H0(t−t0)/h ,(5d)
and |ψ D (t 0 ) = |ψ 0 . Through integration, Eq. (5c) can be reexpressed in an inhomogeneous form that incorporates the eigenstate |ψ 0 of H 0 as the inhomogeneous term,
|ψ D (t) = |ψ 0 − (i/h) t t0 V D (t ′ )|ψ D (t ′ ) dt ′ .(5e)
The inhomogeneous Eq. (5e) has the form of our generic Eq. (1a), and therefore is subject to either standard perturbational iteration, |ψ
(n+1) D (t) = |ψ 0 − (i/h) t t0 V D (t ′ )|ψ (n) D (t ′ ) dt ′ ,(5f)
where of course |ψ (0) D (t) = |ψ 0 , or, if we can find a complete orthogonal basis set { ρ i |} such that ρ i |ψ 0 is never zero, we can carry out nonperturbational "continued-fraction" iteration,
ρ i |ψ (n+1) D (t) = ρ i |ψ 0 / 1 + (i/h) ρ i |ψ (n) D (t) −1 t t0 ρ i | V D (t ′ )|ψ (n) D (t ′ ) dt ′ ,(5g)
where of course ρ i |ψ (0) D (t) = ρ i |ψ 0 . A prime application of the Dirac-picture Eq. (5e) is to quantum field theories [6], where it has invariably so far been pursued by using the standard perturbational iteration scheme of Eq. (5f), no doubt most famously by Dyson to systematically extract the quintessentially perturbational Feynman diagrams [6]. It would certainly be interesting if the nonperturbational "continued-fraction" iteration scheme of Eq. (5g) could conceivably be applied to quantum field theories.
In quantum-field applications |ψ 0 is a small number of unperturbed boson and fermion particles. Now unperturbed boson systems can be regarded as a massive collection of quantized simple harmonic oscillators, while unperturbed fermion systems can be regarded as an equally massive collection of quantum two-state entities. In the previous section we have pointed out that systems which are initially harmonic oscillators can be made compatible with continued-fraction iteration by using a "coherent false ground state" and related "false excitation" orthogonal basis (in quantum field theories perhaps better termed a "coherent false vacuum" and related "false Bose-particle" orthogonal basis), wherein quantized oscillator annihilation operators are translated by a complex number, and creation operators by that number's complex conjugate, which, of course can also be described as a simple real-valued c-number vector translation of the quantum coordinate-momentum operator phase space. The reason that such c-number operator translations simply generate additional orthogonal bases is, of course, that c-number translations have no effect on the crucial commutator algebra of those operators, i.e., they are canonical transformations, indeed unitary ones.
Systems which are initially collections of two-state systems would seem in principle even easier to make compatible with "continued-fraction" iteration. In Pauli-matrix language, if the natural basis for the initial two-state system consists of the two eigenstates of 1 2 (1 + σ z ) (which has the two eigenvalues 0 and 1, namely the two natural fermionic-state occupation numbers), then for this particular purpose of "continued fraction" iteration, we can instead adopt the two eigenstates of 1 2 (1 + σ y ), for example, since each state of the natural basis indeed has a a nonzero inner product with each one of the states of this adopted basis. The example just given can be described as a spinor basis rotation of ninety degrees, and other rotation angles ought to achieve the desired result as well. Note that the natural unperturbed-particle basis has here once again been subjected to a unitary transformation.
The fact that the continued-fraction iteration scheme forces on one a basis that has no zero-valued inner products with the members of the natural particle occupation-number basis of the quantum field system's unperturbed Hamiltonian operator H 0 implies that this iteration scheme will very rapidly indeed force an unbounded number of unperturbed particles into active participation. That is in very sharp contrast with the perturbational iteration scheme: the intrinsic nature of Feynman diagrams immediately reveals that no more than a certain maximum number of unperturbed particles can ever participate through a given order of the perturbational iteration. (The only exception to this iron Feynman-diagram rule occurs when individual diagrams fail by yielding infrared divergences -whose cure is widely agreed to involve coherent photon states, which have no upper bound on the possible number of photons which they describe). The continued-fraction iteration scheme, contrariwise, forces on one coherent states (i.e., "false vacua") and their related mutually orthogonal "false Bose-particle" brethren that all have an unbounded number of unperturbed particles. It as well forces on one the fact that all the fermionic basis states have a greater than zero probability to actually be occupied by unperturbed fermions. Feynman diagrams reveal "virtual particles", but continuedfraction iteration reveals immense "virtual clouds" which have no upper bound on the number of unperturbed particles involved . Landing in the thick of such "virtual clouds" within an iteration or two suggests much better convergence for the continued-fraction scheme than for the perturbational one.
Feynman diagrams can diverge. In the continued-fraction scheme, everything that is calculated gets pushed into a denominator , which makes actual divergence quite difficult to visualize mathematically. What diverges in the continued-fraction scheme would seem to be unperturbed-particle participation rather than the iteration expressions themselves.
S K Kauffmann, viXra:1212.0128 Mathematical Physics. S. K. Kauffmann, viXra:1212.0128 Mathematical Physics, http://vixra.org/pdf/1212.0128v1.pdf (2012).
M L Goldberger, K M Watson, Collision Theory. New YorkJohn Wiley & SonsM. L. Goldberger and K. M. Watson, Collision Theory (John Wiley & Sons, New York, 1964), pp. 197- 199.
. M L Goldberger, K M Watson, Op , M. L. Goldberger and K. M. Watson, op. cit., pp. 306-313.
L I Schiff, Quantum Mechanics. New YorkMcGraw-HillL. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 151-154.
. L I Schiff, L. I. Schiff, op. cit., pp. 195-205.
S S , Schweber An Introduction to Relativistic Quantum Field Theory. New YorkHarper & RowS. S. Schweber An Introduction to Relativistic Quantum Field Theory (Harper & Row, New York, 1961).
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"Charged jets in p-Pb collisions at √ s NN = 5.02 TeV measured with the ALICE detector",
"Charged jets in p-Pb collisions at √ s NN = 5.02 TeV measured with the ALICE detector"
]
| [
"Rüdiger Haake \nCERN\nCH-1211Geneva 23Switzerland\n"
]
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"CERN\nCH-1211Geneva 23Switzerland"
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| Highly energetic jets are sensitive probes for the kinematics and the topology of nuclear collisions. Jets are collimated sprays of charged and neutral particles, which are produced in the fragmentation of hard scattered partons in the early stage of the collision. The measurement of nuclear modification of charged jet spectra in p-Pb collisions provides an important way of quantifying the effects of cold nuclear matter in the initial state on jet production, fragmentation, and hadronization. Unlike in Pb-Pb collisions, modifications of jet production due to hot nuclear matter effects are not expected to occur in p-Pb collisions. Therefore, measurements of nuclear modifications in charged jet spectra in p-Pb collisions (commonly known as R pPb ) can be used to isolate and quantify cold nuclear matter effects. Potential nuclear effects are expected to be more pronounced in more central p-Pb collisions due to a higher probability of an interaction between the proton and the lead-nucleus. To measure the centrality dependence of charged jet spectra, it is crucial to use a reliable definition of event centrality, which ALICE developed utilizing the Zero-Degree Calorimeter (ZDC). | null | [
"https://arxiv.org/pdf/1605.03367v1.pdf"
]
| 118,511,553 | 1503.06441 | 65adbfe9897d3e4e912031b344b0eb7864d25fb5 |
Charged jets in p-Pb collisions at √ s NN = 5.02 TeV measured with the ALICE detector
Rüdiger Haake
CERN
CH-1211Geneva 23Switzerland
Charged jets in p-Pb collisions at √ s NN = 5.02 TeV measured with the ALICE detector
Highly energetic jets are sensitive probes for the kinematics and the topology of nuclear collisions. Jets are collimated sprays of charged and neutral particles, which are produced in the fragmentation of hard scattered partons in the early stage of the collision. The measurement of nuclear modification of charged jet spectra in p-Pb collisions provides an important way of quantifying the effects of cold nuclear matter in the initial state on jet production, fragmentation, and hadronization. Unlike in Pb-Pb collisions, modifications of jet production due to hot nuclear matter effects are not expected to occur in p-Pb collisions. Therefore, measurements of nuclear modifications in charged jet spectra in p-Pb collisions (commonly known as R pPb ) can be used to isolate and quantify cold nuclear matter effects. Potential nuclear effects are expected to be more pronounced in more central p-Pb collisions due to a higher probability of an interaction between the proton and the lead-nucleus. To measure the centrality dependence of charged jet spectra, it is crucial to use a reliable definition of event centrality, which ALICE developed utilizing the Zero-Degree Calorimeter (ZDC).
Introduction
Jets can conceptually be described as the final state produced in a hard scattering. Therefore, jets are an excellent tool to access a very early stage of the collision. The jet constituents represent the final state remnants of the fragmented and hadronized partons that were scattered in the reaction. While all the detected particles have been created in a non-perturbative process (i.e. by hadronization), ideally, jets represent the kinematic properties of the originating partons. Thus, jets are mainly determined by perturbative processes due to the high momentum transfer between two partons and the cross sections can be calculated with pQCD. This conceptual definition is descriptive and very simple, the technical analysis of those objects is complicated though.
This article presents minimum bias and centrality-dependent results for charged jets measured in proton-lead collisions at √ s NN = 5.02 TeV with the ALICE experiment. Due to lack of space, the focus will be on the presentation of the results, leaving out most technical details that are summarized very briefly in a single section. A detailed description of the analyses presented in these proceedings can be found in the respective publication 1 2 . arXiv:1605.03367v1 [nucl-ex] 11 May 2016
The data were recorded with ALICE, the dedicated heavy-ion experiment at the LHC studying properties of the quark-gluon plasma and the QCD phase diagram in general. The detector is designed as a general-purpose heavy-ion detector 3 to measure and identify hadrons, leptons, and also photons down to very low transverse momenta.
For the charged jet analysis, mainly data from the ALICE TPC 4 -a time projection chamber -, and the ITS 5 -a six-layered silicon detector -is used to form charged-particle tracks that serve as the basic ingredient to jet reconstruction. For event triggering, the VZERO 6 scintillation counters are utilized. The centrality estimation method makes use of the Zero-Degree Calorimeter (ZDC), a quartz fibers sampling calorimeter located 116 m from the interaction point. Details on the centrality selection can be found here 7 .
To identify jets, the anti-k T algorithm 8 implemented in the FastJet 9 package is used. The track cuts for those particles are chosen in order to obtain a uniform charged track distribution in the full η − φ plane and only tracks with |η| < 0.9 and p T > 150 MeV/c are accepted for the jet finding procedure.
To avoid edge effects, only jets fully contained within the acceptance are used for further analysis.
Two corrections are applied to the raw jets after reconstruction: background correction, which includes the subtraction of the mean event background density and the consideration of the background fluctuations within the event, and the correction for detector effects.
While the background density is subtracted on a jet-by-jet basis, region-to-region fluctuations of the background are measured by probing the transverse momentum in randomly distributed cones and comparing it to the average background. While the background subtraction can be applied for each jet, the background fluctuations can only be taken into account on a probabilistic basis in an unfolding procedure.
Like the background fluctuations, detector effects -e.g. from the limited single-particle tracking efficiency -are considered in a Singular Value Decomposition (SVD) unfolding procedure 10 .
Results
Figure 1a
shows the charged jet nuclear modification factor for R = 0.2 and R = 0.4, respectively. It has been measured with 20 and 120 GeV/c jet momentum and shows no significant modification within the uncertainties. A good agreement with NLO pQCD calculations (not shown) is observed for both resolution parameters 1 .
The centrality-dependent results are shown in Figs. 1b and c -Fig. 1b shows the cross section measurement in bins of centrality, in Fig. 1c the nuclear modification factor is depicted in bins of centrality. Like for the minimum bias results, no significant modification and therefore also no centrality dependence has been observed in the measured range between 20 and 120 GeV/c. In addition, a comparison to ATLAS data on full jet measurements in Fig. 2 (left) shows a good agreement in the comparable region of transverse momentum and pseudo rapidity.
As a very simple measure for the jet collimation, the jet cross section ratio was measured for R = 0.2/R = 0.4, see Fig. 2 (right). Within the uncertainties, it shows no significant modification in bins of centrality. This observable was also calculated in pp collisions and several Monte Carlo simulations were performed (not shown here). A good agreement has been observed.
Summary
In the two data analyses presented here, charged jets were measured in p-Pb collisions at √ s NN = 5.02 TeV at the LHC with the ALICE experiment both for minimum bias events and in bins of centrality. A good agreement of the minimum bias cross sections with NLO pQCD calculations was observed for both analyzed resolution parameters R = 0.2 and R = 0.4. The nuclear modification factors show no significant effect, neither pointing to a strong nuclear modification nor to a strong centrality dependence. In addition, the result is compatible with the ATLAS full jet measurement.
The charged jet cross section ratio does not point to any (strong) change in the jet structure compared to pp collisions and several Monte Carlo simulations. Additionally, the ratio exhibits no significant centrality dependence.
Figure 1 -
1(a) Charged jet nuclear modification factor for R = 0.2 and R = 0.4 for minimum bias p-Pb collisions. (b) Centrality-dependent charged jet cross sections in p-Pb collisions. Note that the spectra are not corrected for the number of binary collisions. (c) Centrality-dependent charged jet nuclear modification factor for R = 0.2 and R = 0.4 for p-Pb collisions.
Figure 2 -
2Left: Comparison of the ALICE and ATLAS charged jet nuclear modification factors for R = 0.4 and different centralities for p-Pb collisions. Right: Centrality-dependent jet cross section ratios R = 0.2/R = 0.4.
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. JINST. 53003ALICE Collaboration: JINST 5 (2010) 3003.
. Phys. Rev. Lett. 105252301ALICE Collaboration: Phys. Rev. Lett. 105 (2010) 252301.
. Phys. Rev. C. 9164905ALICE Collaboration: Phys. Rev. C 91 (2015) 064905.
. M Cacciari, G P Salam, G Soyez, JHEP. 080463M. Cacciari, G.P. Salam, and G. Soyez: JHEP 0804 (2008) 063.
. M Cacciari, G P Salam, Phys. Lett. B. 641M. Cacciari and G.P. Salam: Phys. Lett. B 641 (2006) 57-61.
. A Hoecker, V Kartvelishvili, Nucl. Instrum. Meth. 372A. Hoecker and V. Kartvelishvili: Nucl. Instrum. Meth. A372 (1996) 469-481.
| []
|
[
"Adiabatic loading of a Bose-Einstein condensate in a 3D optical lattice",
"Adiabatic loading of a Bose-Einstein condensate in a 3D optical lattice"
]
| [
"T Gericke \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n",
"F Gerbier \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n",
"A Widera \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n",
"S Fölling \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n",
"O Mandel \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n",
"I Bloch \nInstitut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany\n"
]
| [
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany",
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany",
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany",
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany",
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany",
"Institut für Physik\nJohannes Gutenberg-Universität\n55099MainzGermany"
]
| []
| We experimentally investigate the adiabatic loading of a Bose-Einstein condensate into an optical lattice potential. The generation of excitations during the ramp is detected by a corresponding decrease in the visibility of the interference pattern observed after free expansion of the cloud. We focus on the superfluid regime, where we show that the limiting time scale is related to the redistribution of atoms across the lattice by single-particle tunneling.The observation of the superfluid to Mott insulator (MI) transition [1] undergone by an ultracold Bose gas in an optical lattice has triggered a lot of experimental and theoretical activities (see[2,3,4]). This system allows to experimentally produce strongly-correlated quantum systems in a well controlled environment, with applications ranging from the realization of novel quantum phases in multi-component systems (see, e.g., [4] and references therein) to the implementation of collisional quantum gates[5,6,7].Most of these proposals rely on producing a system very close to its ground state. However, in experiments so far, the successful production of an ultracold gas in the optical lattice relies on adiabatic transfer. The condensate is first produced and evaporatively cooled in order to minimize the thermal fraction. The almost pure condensate is subsequently transferred into the lattice potential by ramping up the laser intensities as slow as possible in order to approach the adiabatic limit. For too fast a ramp, excitations are generated in the system, which eventually results in heating after the cloud has equilibrated at the final lattice depth.It is important in this respect to note the existence of two energy scales in this problem, related to the singleparticle band structure on the one hand or to the manybody physics within the lowest Bloch band on the other hand. Adiabaticity with respect to the band structure is associated with the absence of interband transitions. The characteristic time scales are on the order of the inverse recoil frequency, typically hundreds of microseconds. Experimentally, adiabaticity with respect to the band structure is easily ensured, and can be checked by detecting atoms in the higher Bloch bands[8,9]. Adiabaticity with respect to the many-body dynamics of the system involves considerably longer time scales (on the order of tens or even hundreds of milliseconds). Theoretical studies of the loading dynamics have been reported in[10,11,12,13,14,15]. It is clear that non-adiabatic effects are more pronounced in the superfluid phase. The insulator phase is indeed expected to be quite insensitive to such effects, due to the presence of an energy gap. In this paper, we focus on the superfluid phase. Our goal is to to clarify experimentally how slowly the loading has to proceed to minimize unwanted excitations and heating.To this aim, we make use of long range phase properties characteristic to a BEC. A key observable for ultracold Bose gases in optical lattices is the interference pattern observed after releasing the gas from the lattice and letting it expand for a certain time of flight[1,8,16,17,18,19]. The contrast of this interference pattern is close to unity when most atoms belong to the condensate, but diminishes with the condensed fraction as the cloud temperature increases. Hence, the visibility of this interference pattern can be used to investigate the dynamical loading of a condensate into the optical lattice. We focus here on a specific lattice depth in the superfluid regime. We find a characteristic time scale of ∼ 100 ms above which the visibility of the interference pattern appears to be stationary. We show how it relates to the redistribution of atoms in the lattice through single-particle tunneling.In our experiment, a 87 Rb Bose-Einstein condensate is loaded into an optical lattice created by three orthogonal pairs of counter-propagating laser beams (see[1]for more details). The superposition of the lattice beams, derived from a common source at a wavelength λ L = 850 nm, results in a simple cubic periodic potential with a lattice spacing d = λ L /2 = 425 nm. The lattice depth V 0 is controlled by the laser intensities, and is measured here in units of the single-photon recoil energy, E r = h 2 /2mλ 2 L ≈ h × 3.2 kHz. The optical lattice is ramped up in a time τ ramp , using a smooth waveform that minimizes sudden changes at both ends of the ramp. The ramp form is calculated numerically. The program imposes that the lattice depth is initially zero and reaches its final value after a time τ ramp . In between (for times 0 ≤ t ≤ τ ramp ), it tries to match the functional formwith α = 20, by a piecewise linear approximation. The function (1) has been proposed in[12]and is shown inFig. 1a. After this ramp, the cloud is held in the lattice potential for a variable hold time t hold (seeFig. 1a), during which the system can re-thermalize. After switching | 10.1080/09500340600777730 | [
"https://arxiv.org/pdf/cond-mat/0603590v1.pdf"
]
| 119,104,847 | cond-mat/0603590 | 9b5796e9fb057661d61f9344d4c6a9877af84011 |
Adiabatic loading of a Bose-Einstein condensate in a 3D optical lattice
22 Mar 2006 (Dated: October 29, 2018)
T Gericke
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
F Gerbier
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
A Widera
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
S Fölling
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
O Mandel
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
I Bloch
Institut für Physik
Johannes Gutenberg-Universität
55099MainzGermany
Adiabatic loading of a Bose-Einstein condensate in a 3D optical lattice
22 Mar 2006 (Dated: October 29, 2018)numbers: 0375Lm0375Hh0375Gg
We experimentally investigate the adiabatic loading of a Bose-Einstein condensate into an optical lattice potential. The generation of excitations during the ramp is detected by a corresponding decrease in the visibility of the interference pattern observed after free expansion of the cloud. We focus on the superfluid regime, where we show that the limiting time scale is related to the redistribution of atoms across the lattice by single-particle tunneling.The observation of the superfluid to Mott insulator (MI) transition [1] undergone by an ultracold Bose gas in an optical lattice has triggered a lot of experimental and theoretical activities (see[2,3,4]). This system allows to experimentally produce strongly-correlated quantum systems in a well controlled environment, with applications ranging from the realization of novel quantum phases in multi-component systems (see, e.g., [4] and references therein) to the implementation of collisional quantum gates[5,6,7].Most of these proposals rely on producing a system very close to its ground state. However, in experiments so far, the successful production of an ultracold gas in the optical lattice relies on adiabatic transfer. The condensate is first produced and evaporatively cooled in order to minimize the thermal fraction. The almost pure condensate is subsequently transferred into the lattice potential by ramping up the laser intensities as slow as possible in order to approach the adiabatic limit. For too fast a ramp, excitations are generated in the system, which eventually results in heating after the cloud has equilibrated at the final lattice depth.It is important in this respect to note the existence of two energy scales in this problem, related to the singleparticle band structure on the one hand or to the manybody physics within the lowest Bloch band on the other hand. Adiabaticity with respect to the band structure is associated with the absence of interband transitions. The characteristic time scales are on the order of the inverse recoil frequency, typically hundreds of microseconds. Experimentally, adiabaticity with respect to the band structure is easily ensured, and can be checked by detecting atoms in the higher Bloch bands[8,9]. Adiabaticity with respect to the many-body dynamics of the system involves considerably longer time scales (on the order of tens or even hundreds of milliseconds). Theoretical studies of the loading dynamics have been reported in[10,11,12,13,14,15]. It is clear that non-adiabatic effects are more pronounced in the superfluid phase. The insulator phase is indeed expected to be quite insensitive to such effects, due to the presence of an energy gap. In this paper, we focus on the superfluid phase. Our goal is to to clarify experimentally how slowly the loading has to proceed to minimize unwanted excitations and heating.To this aim, we make use of long range phase properties characteristic to a BEC. A key observable for ultracold Bose gases in optical lattices is the interference pattern observed after releasing the gas from the lattice and letting it expand for a certain time of flight[1,8,16,17,18,19]. The contrast of this interference pattern is close to unity when most atoms belong to the condensate, but diminishes with the condensed fraction as the cloud temperature increases. Hence, the visibility of this interference pattern can be used to investigate the dynamical loading of a condensate into the optical lattice. We focus here on a specific lattice depth in the superfluid regime. We find a characteristic time scale of ∼ 100 ms above which the visibility of the interference pattern appears to be stationary. We show how it relates to the redistribution of atoms in the lattice through single-particle tunneling.In our experiment, a 87 Rb Bose-Einstein condensate is loaded into an optical lattice created by three orthogonal pairs of counter-propagating laser beams (see[1]for more details). The superposition of the lattice beams, derived from a common source at a wavelength λ L = 850 nm, results in a simple cubic periodic potential with a lattice spacing d = λ L /2 = 425 nm. The lattice depth V 0 is controlled by the laser intensities, and is measured here in units of the single-photon recoil energy, E r = h 2 /2mλ 2 L ≈ h × 3.2 kHz. The optical lattice is ramped up in a time τ ramp , using a smooth waveform that minimizes sudden changes at both ends of the ramp. The ramp form is calculated numerically. The program imposes that the lattice depth is initially zero and reaches its final value after a time τ ramp . In between (for times 0 ≤ t ≤ τ ramp ), it tries to match the functional formwith α = 20, by a piecewise linear approximation. The function (1) has been proposed in[12]and is shown inFig. 1a. After this ramp, the cloud is held in the lattice potential for a variable hold time t hold (seeFig. 1a), during which the system can re-thermalize. After switching
We experimentally investigate the adiabatic loading of a Bose-Einstein condensate into an optical lattice potential. The generation of excitations during the ramp is detected by a corresponding decrease in the visibility of the interference pattern observed after free expansion of the cloud. We focus on the superfluid regime, where we show that the limiting time scale is related to the redistribution of atoms across the lattice by single-particle tunneling. The observation of the superfluid to Mott insulator (MI) transition [1] undergone by an ultracold Bose gas in an optical lattice has triggered a lot of experimental and theoretical activities (see [2,3,4]). This system allows to experimentally produce strongly-correlated quantum systems in a well controlled environment, with applications ranging from the realization of novel quantum phases in multi-component systems (see, e.g., [4] and references therein) to the implementation of collisional quantum gates [5,6,7].
Most of these proposals rely on producing a system very close to its ground state. However, in experiments so far, the successful production of an ultracold gas in the optical lattice relies on adiabatic transfer. The condensate is first produced and evaporatively cooled in order to minimize the thermal fraction. The almost pure condensate is subsequently transferred into the lattice potential by ramping up the laser intensities as slow as possible in order to approach the adiabatic limit. For too fast a ramp, excitations are generated in the system, which eventually results in heating after the cloud has equilibrated at the final lattice depth.
It is important in this respect to note the existence of two energy scales in this problem, related to the singleparticle band structure on the one hand or to the manybody physics within the lowest Bloch band on the other hand. Adiabaticity with respect to the band structure is associated with the absence of interband transitions. The characteristic time scales are on the order of the inverse recoil frequency, typically hundreds of microseconds. Experimentally, adiabaticity with respect to the band structure is easily ensured, and can be checked by detecting atoms in the higher Bloch bands [8,9]. Adiabaticity with respect to the many-body dynamics of the system involves considerably longer time scales (on the order of tens or even hundreds of milliseconds). Theoretical studies of the loading dynamics have been reported in [10,11,12,13,14,15]. It is clear that non-adiabatic effects are more pronounced in the superfluid phase. The insulator phase is indeed expected to be quite insensitive to such effects, due to the presence of an energy gap. In this paper, we focus on the superfluid phase. Our goal is to to clarify experimentally how slowly the loading has to proceed to minimize unwanted excitations and heating.
To this aim, we make use of long range phase properties characteristic to a BEC. A key observable for ultracold Bose gases in optical lattices is the interference pattern observed after releasing the gas from the lattice and letting it expand for a certain time of flight [1,8,16,17,18,19]. The contrast of this interference pattern is close to unity when most atoms belong to the condensate, but diminishes with the condensed fraction as the cloud temperature increases. Hence, the visibility of this interference pattern can be used to investigate the dynamical loading of a condensate into the optical lattice. We focus here on a specific lattice depth in the superfluid regime. We find a characteristic time scale of ∼ 100 ms above which the visibility of the interference pattern appears to be stationary. We show how it relates to the redistribution of atoms in the lattice through single-particle tunneling.
In our experiment, a 87 Rb Bose-Einstein condensate is loaded into an optical lattice created by three orthogonal pairs of counter-propagating laser beams (see [1] for more details). The superposition of the lattice beams, derived from a common source at a wavelength λ L = 850 nm, results in a simple cubic periodic potential with a lattice spacing d = λ L /2 = 425 nm. The lattice depth V 0 is controlled by the laser intensities, and is measured here in units of the single-photon recoil energy,
E r = h 2 /2mλ 2 L ≈ h × 3.2 kHz.
The optical lattice is ramped up in a time τ ramp , using a smooth waveform that minimizes sudden changes at both ends of the ramp. The ramp form is calculated numerically. The program imposes that the lattice depth is initially zero and reaches its final value after a time τ ramp . In between (for times 0 ≤ t ≤ τ ramp ), it tries to match the functional form
V 0 (t) = V max 1 + exp −α t τramp ,(1)
with α = 20, by a piecewise linear approximation. The function (1) has been proposed in [12] and is shown in Fig. 1a. After this ramp, the cloud is held in the lattice potential for a variable hold time t hold (see Fig. 1a), during which the system can re-thermalize. After switching off the optical and magnetic potentials simultaneously and allowing for typically t = 10 − 22 ms of free expansion, standard absorption imaging of the atom cloud yields a two-dimensional map of the density distribution n (integrated along the probe line of sight).
To extract quantitative information from time-of-flight pictures, we use the usual definition of the visibility of interference fringes,
V = n max − n min n max + n min .(2)
In this work, we follow the method introduced in [18] and measure the maximum density n max at the first lateral peaks of the interference pattern, (i.e. at the center of the second Brillouin zone). The minimum density n min is measured along a diagonal with the same distance from the central peak (see inset in Fig. 1b). In this way, the Wannier envelope is the same for each term and cancels out in the division (see [18]). For a given twodimensional absorption image, four such pairs exist and the corresponding values are averaged to yield the visibility. In addition, it is worth pointing out that this is an essentially model-independent characterization of the many-body system. For comparison, we reproduce here measurements of the visibility as a function of lattice depth, similar to those presented in [18]. The data corresponds to a given total atom number N ≈ 3 × 10 5 atoms (grey circles in Fig. 1b). For lattice depths larger than 12.5 E r , the system is in the insulating phase, as demonstrated in [1]. Yet, the visibility remains finite well above this point. For example, at a lattice depth of 15 E r , the contrast is still around 30%, reducing to a few percent level only for a rather high lattice depth of 30 E r . In [18], we have shown that such a slow loss in visibility is expected in fact even in the ground state of the system, being a consequence of the admixture of particle/hole pairs to the ground state for finite lattice depths. For these experiments, a ramp time of 160 ms followed by a hold time of 40 ms were used. We will show later on that this is slow enough to ensure adiabaticity.
We now focus on the superfluid regime, at a lattice depth V 0 = 10 E r . To investigate how the system is af-fected by the ramping procedure, we fix the ramp time τ ramp and record how the interference pattern evolves as a function of hold time τ hold . Examples of such measurements are shown in Fig. 2a-d. For the slowest ramp shown, τ ramp = 160 ms, the visibility decreases with a time constant ∼ 500 ms. In fact, a very similar behavior is observed as soon as the ramp time exceeds τ ramp > 100 ms. Since this behavior is essentially independent of the ramp speed, it points to the presence of heating mechanisms which significantly degrade the visibility on a time scale of several hundred ms. The source of heating could be technical, due for instance to intensity or pointing fluctuations of the lattice beams, or intrinsic processes. An unavoidable process is for instance atomic spontaneous emission following excitations by one of the lattice beams. Here we attempt a crude estimate of the effect of spontaneous emission on visibility as follows. The heating rate Γ heat ∼ Γ sp × (h 2 /2mλ 2 0 ) is given by the rate Γ sp at which such events happen due to all lattice beams, times the recoil energy h 2 /2mλ 2 0 , where λ 0 ≈ 780 nm is the resonant wavelength of the D 2 transition. To estimate the time scale T V over which the visibility vanishes, we compute the time over which the energy gain per particle, Γ heat T V , is on the order of the critical temperature T c times Boltzmann's constant k B . In a lattice potential with approximately one atom per site, we have k B T c ∼ zJ, where J is the tunneling matrix element and z = 6 the number of nearest neighbors in a three dimensional cubic lattice. This yields a final estimate
T V ∼ Γ −1 sp zJ E r .(3)
For a lattice depth of 10 E r , we calculate a tunneling energy zJ/E r ∼ 0.12 and a total scattering rate Γ sp ∼ 0.2 s −1 , giving T V ∼ 0.6 s, in good agreement with the observed time scale. We conclude that on the time scales considered here, photon scattering contributes significantly to the observed heating rate. Although heating due to technical noise is not excluded, the bound above implies that it is not much more severe than photon scattering. We note that in principle, the decay in visibility could be used to measure the heating rate, provided the dependance of the visibility on temperature is known. When the ramp times is decreased below τ ramp = 100 ms, the visibility decreases in a qualitatively different way. As shown in Fig. 2b-d, the decay occurs with a much shorter time constant. Note that at long hold times τ hold = 800 ms, the visibility has dropped to V ∼ 0.2, a value almost independent of the ramp time. This is consistent with our earlier claim that heating dominates at long times. We attribute the short-times decay to the generation of excitations by a finite ramp speed. To compare the different ramp times, we plot in Fig. 3 the measured visibility as a function of ramp time for a fixed hold time of 300 ms, long enough for the excitations generated during the ramp to relax, but short enough that heating effects do not blur all differences between the different ramps. A time scale of ∼ 100 ms clearly emerges, above which the ramp time can be increased without noticeable effect. To check whether this could be an artifact of our plotting method, we have fitted the data to an empirically chosen Lorentzian form. The half-width at half-maximum of the Lorentzian was taken as the visibility decay time. As seen from Fig. 4, the same behavior is found, with a characteristic 1/e ramp time ≈ 60 ms. This time scale can be understood using elementary arguments. Adiabatic evolution in a quantum system with time-dependent hamiltonian requires the condition
|Ḣ| ≪ |ω f i | 2 ,(4)
whereḢ is the derivative of the hamiltonian and where ω f i is the Bohr frequency of transition between the (instantaneous) eigenstates |i and |f . Eq. (4) should be fulfilled at all times. For an ultracold gas in an optical lattice, three energy scales appear: (i) the tunneling matrix element J defining the rate of hopping from site to site, (ii) the on-site interaction energy U and, (iii) the energy associated to the "external" confinement potential usually present on top of the lattice. This potential V ext results from the combination of the magnetic trapping potential in which the condensate is initially produced, and of an additional confinement due to the Gaussian shape of the lasers producing the optical lattice. In practice, V ext is nearly harmonic, with a trapping frequency Ω = ω ext ≈ ω 2 m + 8V 0 /mw 2 , where ω m = 2π × 16 Hz is the oscillation frequency in the magnetic trap and where w ≈ 136 µm is the laser beam size. With a proper choice of the laser beam sizes, the increase in U and Ω are such that the equilibrium size of the condensate as calculated in the Thomas-Fermi approximation varies little during the ramp [20]. In addition, when the lattice depth is increased, J drops exponentially fast whereas U and Ω increase slowly. Hence, the question of adiabaticity essentially reduces to whether the atoms can redistribute through tunneling in order to adapt the size of the system to the instantaneous Thomas-Fermi shape. This suggests to take |Ḣ| ∼ |J| and ω f i ∼ J/ in Eq. (4), giving the following definition for the adiabaticity parameter A,
A = Max 0≤t≤τramp |J| J 2 .(5)
For a given ramp, we require A ≪ 1 to ensure an adiabatic loading of the cloud into the lattice. In the inset of Fig. 5, we show the quantity |J|/J 2 calculated for τ ramp = 50 ms. We have calculated this curve for a final lattice depth of 10E R using the lattice ramp given in Eq. (1), and the analytical estimate [2],
J E r ≈ 8 π (V 0 /E r ) 3/4 exp(−2 V 0 /E r ).(6)
The sharp decrease for short times in the inset of Fig. 5 follows from the inadequacy of the tight-binding approximation under which Eq. (6) is derived. We ignore this feature, and calculate A from the peak value occurring near t ≈ τ ramp /2, where the rate of change of the lattice depth is highest. We have repeated the calculation for several ramp times (see Fig. 5). We find that the critical A = 1 corresponds to a ramp time τ ramp ≈ 80 ms, close to the experimental findings. This suggests that for our experimental parameters, the loading is indeed limited by single-particle tunneling.
In conclusion, we have studied how the visibility of the interference pattern was affected by the speed at which the lattice was ramped up. A time scale of ∼ 100 ms was found for adiabatic loading in the optical lattice . In this paper, we focused on the dynamics properties in the superfluid regime. Even more interesting is the dynamical evolution of the system as the superfluid-to-Mott-insulator transition is crossed. An important and still open question is in particular how reversible this transition is. Experiments [1, 16, 17? ] found that one could ramp up the lattice intensity and reach the regime where phase coherence is lost, then ramp down the lattice and regain it back. To what extent the initial phase coherent state can be recovered, and what are the limiting mechanisms has however not been studied. Also, the recent availability of virtually exact methods for one-dimensional systems [12] could allow for a precise comparison with the experimental data, in a strongly non-equilibrium regime where theory is still in progression. Finally, new dynamical effects are predicted, such as oscillation of the order parameter and vortices-driven relaxation analog to the Kibble-Zurek mechanism [21]. Our work is supported by the Deutsche Forschungsgemeinschaft (SPP1116), AFOSR and the European Union under a Marie-Curie Excellence grant (OLAQUI). FG acknowledges support from a Marie-Curie Fellowship of the European Union.
PACS numbers: 03.75.Lm,03.75.Hh,03.75.Gg
FIG. 1 :
1(a)Sketch of the time profile used to ramp up the lattice depth to its maximum value Vmax. (b) Evolution of the visibility of the interference pattern as the lattice depth is increased. The set of data shown corresponds to ∼ 3 × 10 5 atoms (grey circles).
FIG. 2 :
2Time evolution of the visibility for a fixed final depth Vmax = 10 Er and various ramp times: (a): τramp = 160 ms, (b): τramp = 80 ms, (c): τramp = 40 ms, and (d): τramp = 20 ms.
FIG. 3 :FIG. 4 :
34Measured visibility versus ramp time, for a fixed 300 ms hold time. Measured visibility decay time versus ramp time. The time constant is the half-width at half-maximum of a Lorentzian fit to the data. Error bars indicate the statistical error. The solid line is a fit to an inverted Gaussian, returning a time scale of ∼ 60 ms for the visibility decay time to become independent of the ramp.
FIG. 5 :
5Adiabaticity parameter (see text) plotted versus ramp time. In the calculation, we have used the actual ramp form as generated in the experiment to describe the lattice depth increase. In the inset, we show how the adiabaticity parameter changes in time as the lattice depth is ramped up. A ramp time of τramp = 50 ms has been used for this plot.
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| []
|
[
"Skyrmion-string defects with arbitrary topological charges in spinor Bose-Einstein condensates",
"Skyrmion-string defects with arbitrary topological charges in spinor Bose-Einstein condensates"
]
| [
"R Zamora-Zamora \nInstituto de Física\nUniversidad Nacional Autónoma de México Apartado Postal\n20-36401000Cd. MéxicoMexico\n",
"V Romero-Rochín \nInstituto de Física\nUniversidad Nacional Autónoma de México Apartado Postal\n20-36401000Cd. MéxicoMexico\n"
]
| [
"Instituto de Física\nUniversidad Nacional Autónoma de México Apartado Postal\n20-36401000Cd. MéxicoMexico",
"Instituto de Física\nUniversidad Nacional Autónoma de México Apartado Postal\n20-36401000Cd. MéxicoMexico"
]
| []
| Under the presence of external magnetic fields with cylindrical symmetry, Skyrmion-string defects with arbitrary topological charges are shown to appear in spinor F = 1 Bose-Einstein condensates.We show that, depending on the magnetic field boundary condition, the topological spin texture, at the planes perpendicular to the cylindrical axis, can take zero, half integer, or arbitrary values between −1/2 and 1/2. We argue that these are true topological defects since their charge is independent of the spatial location of the singularity and since the total Skyrmion charge is the sum of the individual charges of the defects present. Our findings are obtained by numerically solving the corresponding fully coupled Gross-Pitaevskii equations without any symmetry assumptions.We analyze, both, polar 23 Na and ferromagnetic 87 Rb condensates. * [email protected] 1 arXiv:1703.03795v1 [cond-mat.quant-gas] | 10.1088/1361-6455/aaa324 | [
"https://arxiv.org/pdf/1703.03795v1.pdf"
]
| 119,192,080 | 1703.03795 | e31ff979aa079392d1bb4f89e314cdeb91c63bac |
Skyrmion-string defects with arbitrary topological charges in spinor Bose-Einstein condensates
10 Mar 2017
R Zamora-Zamora
Instituto de Física
Universidad Nacional Autónoma de México Apartado Postal
20-36401000Cd. MéxicoMexico
V Romero-Rochín
Instituto de Física
Universidad Nacional Autónoma de México Apartado Postal
20-36401000Cd. MéxicoMexico
Skyrmion-string defects with arbitrary topological charges in spinor Bose-Einstein condensates
10 Mar 2017(Dated: September 27, 2018)
Under the presence of external magnetic fields with cylindrical symmetry, Skyrmion-string defects with arbitrary topological charges are shown to appear in spinor F = 1 Bose-Einstein condensates.We show that, depending on the magnetic field boundary condition, the topological spin texture, at the planes perpendicular to the cylindrical axis, can take zero, half integer, or arbitrary values between −1/2 and 1/2. We argue that these are true topological defects since their charge is independent of the spatial location of the singularity and since the total Skyrmion charge is the sum of the individual charges of the defects present. Our findings are obtained by numerically solving the corresponding fully coupled Gross-Pitaevskii equations without any symmetry assumptions.We analyze, both, polar 23 Na and ferromagnetic 87 Rb condensates. * [email protected] 1 arXiv:1703.03795v1 [cond-mat.quant-gas]
I. INTRODUCTION
The appearance of Skyrmions [1] range from systems in nuclear physics [2], superconductivity [3], magnetic solid state physics [4,5], liquid crystals [6], and Bose-Einstein condensates (BEC), thus becoming a topological unifying framework for seemingly different physical phenomena. In BEC systems, starting from early proposals to create and observe topological defects [7][8][9][10][11] to recent ones [12][13][14][15][16] carried on several experimental realizations already been achieved [17][18][19][20][21][22]. In particular, for the spinor F = 1 BEC that concerns us here, an intense recent activity on Skyrmions has developed [16,18,23,24].
Skyrmions are topological defects of a given spatial vector field or order-parameter of the system in question, that can be classified in terms of homotopy groups of the corresponding field [25][26][27]. There are versions in two-and three-dimensions and, typically, the Skyrmion topological charge is either an integer or a half-integer [7,28,29]. For a three-dimensional spinor F = 1 Bose-Einstein condensate, the field that develops the Skyrmion defects is the spin texture, a real measurable quantity. We analyze different phases in the mentioned spinor BEC, that show a two-dimensional, and the same, Skyrmion topological defect in all planes perpendicular to a privileged axis; thus our naming of "Skyrmion-string" [30]. While this system has been the focus of great attention [10,16,[22][23][24][31][32][33], we report here a novel aspect regarding the value of the acquired Skyrmion charge, namely, that depending particularly on boundary conditions of the external magnetic field that nucleates the vortices and Skyrmions in the spinor BEC, the 2D Skyrmion charge in each plane may take arbitrary values. As we discuss in detail below, while the appropriate topological identification of the defect shown may need further elucidation in terms of topology theory, we support our claim that the defects are of a topological nature since their charge is independent of the location of the defect and, when there are more than one defect present, the total charge is the sum of the individual charges.
Our study is based on the full numerical solution of the 3D spinor F = 1 Gross-Pitaevskii (GP) equations, without assuming any symmetry of the solution [34]. We analyze both polar and ferromagnetic condensates, with parameters corresponding to actual values of 87 Rb and 23 Na [35]. In addition to the numerical analysis, we discuss analytic results to support our discussion. Regarding the numerical study, we search for stationary states of N confined, weakly interacting bosons of spin F = 1, in the presence of an external magnetic field, whose energy functional is,
E[Ψ] = d 3 r 2 2m ∇Ψ * n · ∇Ψ n + V ext ( r)Ψ * n Ψ n + c 0 2 Ψ * n Ψ n Ψ * k Ψ k + c 2 2 Ψ * k F kn Ψ n · Ψ * j F jl Ψ l + p B · Ψ * n F nk Ψ k .(1)
The confining potential, V ext ( r) = mω 2 (x 2 + y 2 + z 2 )/2 is an isotropic harmonic optical trap with frequency ω = 2π × 130 Hz; c 0 and c 2 are the usual two-body interaction parameters as defined by Ho [36], with c 2 < 0 for ferromagnetic phases and c 2 > 0 for polar ones, in the absence of the external field B. The vector F are the F = 1 angular momentum matrices.
The latin subindices run over the three components of spin F = 1. The last term is the linear Zeeman coupling to the external magnetic field B, with strength p < 0.
We consider external magnetic fields of the form,
B = B 0 ((x − x 0 )x − (y − y 0 )y) + B z (r)z, with r = ((x − x 0 ) 2 + (y − y 0 ) 2 ) 1/2
and (x 0 , y 0 ) an arbitrary location inside the condensate planes. Two different types of the z-component are studied. In one case, B z (r) = constant, which can take any value, including zero. In the second case, B z (r) = B z r, with B z = 0. The main difference of these two fields is their behavior as r → ∞. In the first case, the direction of the field B/| B| → x cos φ − y sin φ, with tan φ = y/x the planar polar angle;
that is, if B z = constant, the B field lies always on the xy-plane as r → ∞. However, in the second case, the asymptotic direction of the B field no longer points on the plane, but in a direction defined by the values of B 0 and B z ,
B | B| → B 0 (x cos φ − y sin φ) + B z z (B 2 0 + B 2 z ) 1/2 as r → ∞.(2)
Our numerical analysis shows that in the presence of the B field, the polar and ferromagnetic character is lost as r → ∞, the gas behaving as "paramagnetic" with the spin texture pointing along the direction of the field. Thus, as we show below, the first case yields Skyrmions with charges 0, and ±1/2 always, while the second one gives rise to Skyrmions with arbitrary non-integer charge. In both cases, due to the zero of B along the line (x 0 , y 0 ), there appear well defined vortices with charges 0, ±1, and ±2. [37] This is discussed in Section II. In Section III, we present the analysis concerning the Skyrmions topological charges, and we conclude with some remarks in Section IV.
II. VORTICES AND SKYRMIONS IN SPINOR CONDENSATES
As already known [21,[37][38][39], magnetic fields of the type here considered, generate quantized vortices on the spin condensate components, along the line (x 0 , y 0 ). As it is shown in
Ref. [37], since the line or lines of zero field are at our disposal, one can create vortices, of the Mermin-Ho type [28], at arbitrary locations. Here, we show that a Skyrmion-string is also created at the same lines of zero field and, hence, they can also be externally created on demand.
Our numerical solutions of the corresponding GP equations indicate the existence of two or three stationary states [40], that may be called "ground" and "excited" states, depending on their corresponding chemical potential values. In order to classify those states we use the notation for Mermin-Ho vortices, namely, we find states (1, 0, −1), (0, −1, −2) and (+2, +1, 0), whose notation (l, m, n) represents vortices of charge l, m and n in the spinor components m F = +1, 0, −1 of the fieldΨ. on, the states (+1,0,-1) exist for both signs of B z , while (+2,+1,0) exists always for B z < 0, becoming (numerically) unstable for value of B z > 0 above a threshold. The opposite is true for (0,-1,-2). We defer a discussion of the stability of the states to the last part of the article.
The vortex classification of the above spinor BEC states can be fairly understood from the following considerations. The solution of the three GP equations for a F = 1 BEC can be written, in general, asΨ
( r) = ρ( r) ζ +1 ( r)e iθ + ( r) ζ 0 ( r)e iθ 0 ( r) ζ −1 ( r)e iθ − ( r) (3)
where ρ( r) =Ψ † ( r)Ψ( r) is the total particle density of the condensate. The amplitudes ζ m ( r) are real functions, obeying ζ 2 +1 ( r) + ζ 2 0 ( r) + ζ 2 −1 ( r) = 1 everywhere. An analysis of the vortex solutions, if they exist and if single valuedness is imposed, leads to the following general results [37]: (1) Only two components can show a vortex, say ζ α → 0 and ζ β → 0 as r → 0, with θ α = 0 and θ β = 0, while the third one, ζ γ → 1 as r → 0, and θ γ = 0. And, (2) the phases differences obey θ m − θ m−1 = φ, with φ the polar angle. These two conditions imply the appearance of three vortex solutions, with boundary conditions at r → 0 and with charges θ m = Q m φ, given by
(1, 0, −1) ζ +1 → 0 , ζ 0 → 1 , ζ −1 → 0 as r → 0 Q +1 = +1 , Q 0 = 0 , Q −1 = −1 .(4)
(+2, +1, 0) Fig. 2 shows the velocity field v k = m ∇Φ k of the condensate spinor components Ψ k , with tan Φ k = ImΨ k /ReΨ k , as well as the their particle density ρ k = (Ψ * k Ψ k ) 1/2 , in the z = 0 plane. The vortices and the corresponding density spikes, ζ m → 1 as r → 0, can be clearly observed. For the other cases, B z =constant and B z = B z r, these vortex structures remain and, at first sight, look as in Fig. 2. Certainly, as shown below, one can tailor external fields with more than one vortex per component. We now turn to the spin texture description. For this, it first appears convenient to factorize the phase of the 0-component of the full solutionΨ( r), Eq. (3), and writê
ζ +1 → 0 , ζ 0 → 0 , ζ −1 → 1 as r → 0 Q +1 = +2 , Q 0 = 1 , Q −1 = 0 . (5) ζ +1 → 1 , ζ 0 → 0 , ζ −1 → 0 as r → 0 Q +1 = 0 , Q 0 = −1 , Q −1 = −2 .(6)Ψ( r) = ρ( r)e iθ 0 ( r)ζ ( r) (7) whereζ ( r) = ζ +1 ( r)e iδ + ( r) ζ 0 ( r) ζ −1 ( r)e iδ − ( r) (8)
with δ m = θ m − θ 0 . We shall leave the r dependence implicit in the foregoing analysis. As it is common, a 3D complex vector a = (a x , a y , a z ) can be introduced, with a
x = 1 √ 2 (ζ −1 e iδ + − ζ +1 e iδ − ), a y = −i √ 2 (ζ −1 e iδ + + ζ +1 e iδ − )
, and a z = ζ 0 , obeying a · a * = 1. This vector can be further decomposed in its real and imaginary parts, a = a R + i a I .
The magnetization, a physical observable, is given byΨ † FΨ = ρ f , with f =ζ † Fζ the spin texture. Using the above decomposition of the state, the spin texture can be expressed as f = 2 a R × a I . Let us analyze this expression for different cases. The simplest one is when the full Zeeman contribution is zero, namely B = 0 in Eq. (1). The solution, as shown by Ho [36], is that for the polar case, c 2 < 0, a is real and f = 0. For the ferromagnetic case, c 2 > 0, f = f 0 , a constant vector everywhere, which by an appropriate rotation can be brought to the case ζ +1 = 1, ζ 0 = ζ −1 = 0, namely f = z. For B = 0, the polar and ferromagnetic characters are overridden and the three types of quantum-vortex phases appear.
The above spinor vortex structures have an additional associated topological defect that one may call a "string-Skyrmion" due to the presence of the external magnetic field. That is, while the trap imposes its spherical symmetry on the total density ρ( r), the external magnetic field imposes its additional cylindrical symmetry on the texture field f , as one should expect on physical grounds. Hence, the spin texture does not depend on the z coordinate, f = f (x, y), and furthermore, it shows cylindrical symmetry around the location of the zero line of the B field. Thus the "string" qualifier. In other words, the spin texture shows the same geometric and topological structure in all z−planes .
The connection between the vortex solutions and the Skyrmions can be found as follows.
An important step lies in the restriction on the phases of the vortex structure, θ m −θ m−1 = φ.
This restriction yields the phase requirement θ +1 + θ −1 − 2θ 0 = 0, which in turn, and only in this case, makes the set ( a R , a I , f ) an orthogonal triad (group O(3)). Their explicit form is
a R = ζ −1 − ζ +1 √ 2 ρ + ζ 0 z (9) a I = − ζ −1 + ζ +1 √ 2 δ(10)
and, hence,
f = (ζ 2 +1 − ζ 2 −1 )z + √ 2ζ 0 (ζ +1 + ζ −1 )ρ.(11)
The unit vectors are ρ = x cos δ + y sin δ and δ = −x sin δ + y cos δ, with δ = δ − = −δ + .
The angle δ spans 0 → 2π, since it is actually the phase of the wavefunctions that yield the vortex charges, however, it is not the polar angle tan φ = y/x. Nevertheless, the triad unit vector (ρ, δ, z) together with r = (x 2 + y 2 ) 1/2 and z span the space with cylindrical symmetry. This is important, as we shall return below, since far from the singularity, ρ tends to align to the in-plane component of the B-field, namely, to (x cos φ − y sin φ). It is also important to mention that while a R and a I depend on the choice of the global phase, as shown in Eq. (7), f does not. This is because f is an observable and a R and a I are not.
Since the (real) spinor components depend only on r, ζ m = ζ m (r), the spin texture can be written as f = f z (r)z + f r (r)ρ, and the components f z and f r can be read off of Eq.
(11). With these, the existence or not of an Skyrmion structure in any plane z =constant, can be checked. To this end one recalls that the 2D Skyrmion charge is given by
Q 2D sky = 1 4π dxdy f · ∂ f ∂x × ∂ f ∂y ,(12)
which, with the cylindrical symmetry of f , can be cast as,
Q 2D sky = 1 2 ∞ 0 f r f z df r dr − f r df z dr dr.(13)
There exists, however, an additional important constraint for the cases studied in this work, except for the polar vortex (+1,0,-1), namely, that f · f = 1 everywhere. Although we have not been able to prove this constraint rigorously, our numerical solutions demonstrate it.
This restriction may be written as f 2 z + f 2 r = 1, which further implies that the Skyrmion charge, Eq.(13), can be cast as,
Q 2D sky = 1 2 (f z (0) − f z (∞)) if f · f = 1.(14)
That is, if f · f = 1 everywhere, the Skyrmion charge is given solely by (one-half) the difference of the boundary values of the z−component of the spin texture. The boundary value at r = 0 can always be found, as can be seen from Eqs. (4)- (6) and (11) in which the Zeeman term can be written as p B · Ψ * n F nk Ψ k = p| B|ρ( B/| B|) · f . Since p < 0 in our calculations, this term tends to minimize the energy when B/| B| and f are parallel.
We discuss separately the three general cases. Fig. 3 illustrate the spin texture f for several typical Skyrmions.
A. B z = 0
First, for completeness, we review the case in the absence of a z-component of the B-field, namely, B z = 0. For the vortex solution (+1,0,-1), by symmetry (and numerically verified)
ζ 2 +1 = ζ 2
−1 for all r, hence, f z = 0, see Eq. (11). That is, the vector f not only lies on the xy-plane, it becomes zero as r → 0. Its Skyrmion charge is thus zero, Q 2D sky (+1, 0, −1) = 0. This is the so-called polar coreless vortex [38]. For the vortices (+2,+1,0) and (0,-1,-2), Eqs. 6) and (11), show that f r → −1 and f r → +1, respectively, as r → 0. The numerical solutions further show that ζ 2 +1 = ζ 2 −1 as r → ∞, namely f r → 0 as r → ∞ for both cases. See Fig (+2,+1,0) as example of this case. Hence, one finds that the vortex (+2,+1,0) has an associated Skyrmion charge Q 2D sky (+2, +1, 0) = −1/2 and, analogously, the vortex (0,-1,-2) has Q 2D sky (0, −1, −2) = +1/2. We also calculated these charge values by integrating directly Q 2D sky , using the full equation (12), finding values very close to ±1/2.
B. B z = constant
We now consider a constant z−component of the external B-field, but different from zero.
If B z > 0, the vortex structure remains the same as before but, as seen in Fig. 1, the most stable case is now (+2,+1,0). For B z < 0, the situation is reversed and the stable phase is (0,-1,-2). A direct calculation of Q 2D sky using the definition given by Eq. (12) shows that the charge is apparently not zero for the polar Skyrmion (+1,0,-1) and different from ±1/2 for (+2,+1,0) and (0,-1,-2) (Results not shown here). This is a numerical artifact, however, because the cloud reaches out up to the Thomas-Fermi radius only, namely, the particle density is numerically negligible beyond it. That is, notwithstanding that the structure of the spin texture f is modified by the B z = 0 component, we assert that the topological charges remain 0 for the polar Skyrmion (+1,0,-1) and ±1/2 for (+2,+1,0) and (0,-1,-2).
To verify this, we calculated the spin texture for the cases where the B−field has its zero line at (x 0 = 0, y 0 = 0) and also at (x 0 = 0, y 0 = 0). As expected, the vortex structure remains the same, except that the vortices are centered now at the corresponding values with those of (x 0 = 0, y 0 = 0), with continuous lines. It can be clearly seen that as r grows, f z keeps diminishing with no bound. We take this as an evidence that as r → ∞, it must be true that f z → 0 and, therefore, that the topological Skyrmion charge is −1/2 in this case. Incidentally, the fact that the charge is the same, independently of the location of the defect, tell us that it is of a topological nature. For the case (+1,0,-1), lower panel in Fig. 4, we see that the spin texture structure changes strongly from B z = 0 to B z = 0, however, the topological charge remains zero. We conclude this from a direct calculation of the Skyrmion charge for larger values of the cutoff of the radial integral Eq. (13). The explanation that as r → ∞, f z (r) → 0, for all these cases, thus yielding charges 0 or ±1/2 independently of the value of B z =constant, is that far from the defect location, (x 0 , y 0 ), the spin texture This constitutes an indication that f z → 0 as r → ∞.
C. B z = B z r
We turn our attention now to the case with B z = B z r, which implies that, as r → ∞, the (+2,+1,0) and the second one to (+1,0,-1). Again, we have superimposed defects created at (x 0 = 0, y 0 = 0) and (x 0 = 0, y 0 = 0) to check that the asymptotic behavior is correctly inferred: In all cases, we find that as r → ∞, f z (r) → 0, as expected, due to fact that f aligns to the corresponding asymptotic direction of B-field. This yields topological charges of arbitrary values. Fig 7 summarizes the topological charges obtained for all cases, as the amplitude B z is varied. It shows that the topological charges not only take all values from -1/2 to +1/2, but they can go beyond to higher values, for the corresponding numerically stable cases B z < 0 and B z < 0) for (+2,+1,0) and (0,-1,-2) respectively. We have verified that for (+2,+1,0) and (0,-1-2), both formulae, Eq. (12) and (14), agree thus supporting our conclusions. The topological nature can again be assured from the standpoint of view that the charge does not depend on the spatial location of the defect and that the charges of several defects add up.
IV. FINAL COMMENTS
To the best of our knowledge, there are no previous reports on Skyrmions with arbitrary topological charges, thus, there remains as a task the full elucidation of whether this This constitutes an indication that f z → 0 as r → ∞.
property can be fully considered as of topological nature. Such a problem is beyond the scope of the present study. Nevertheless, we insist that this result follows very simple from expression (14) of the Q 2D sky Skyrmion topological charge, which tells us that the value can be anything, from −1/2 to −1/2, depending solely on the boundary values of the spin texture. Moreover, the topological nature of the defects also shows by finding that the values of the charges are independent of the location of the singularity axis and by checking that the charges of multiple defects add up. The simplest explanation is that the the texture becomes paramagnetic away from the location of the defect and, hence, the boundary values can be adjusted by appropriately tailoring the external magnetic field B. The "arbitrary" value of this topological charge is also reminiscent of the arbitrariness of Berry phases in spin systems.
An important question left to be addressed is the stability of the states here considered.
We have numerically tested this stability criterion as follows. First, we recall that the states are found through a minimization numerical procedure of the energy functional given by Eq. (1), which is equivalent to solving the time-independent GP equation. Part of this procedure allows to finding the corresponding chemical potential. If the state is stationary, it should not evolve under the propagation of the time-dependent GP. Thus, the states found were evolved for more than 100 time units, approximately 50 milliseconds for the systems here considered, and they remained completely stable for those times. Some of our cases, as mentioned above, have been shown to be stable in in Ref. [38]. We are thus confident of their stability.
ACKNOWLEDGMENTS
Fig. 1
1shows the observed vortex-Skyrmion phases, in terms of the equilibrium chemical potential µ as a function of z−component of the external fields analyzed. The left panel of Fig. 1 refer to the case of B z = constant, while the right one to B z = B z r, and in both cases we show the results for ferromagnetic ( 87 Rb) and polar ( 23 Na) phases. In both situations one sees that if B z = 0, (+1,0,-1) is the ground state and (+2, +1, 0) and (0, −1, −2) are degenerated excited states. As B z is turned
FIG. 1 .
1(Color online). Chemical potential µ as a function of B z = constant, left panel, and as a function of B z , right panel, for the case B z = B z r, both for a ferromagnetic condensate 87 Rb and a polar one 23 Na. The labels (+2,+1,0), (+1,0,-1) and (0,-1,-2), indicate the three different vortex phases. These states are numerically stable, see text.
FIG. 2 .
2Typical velocity v k and density fields ρ k of the condensate spinor components Ψ k , in the z = 0 plane. The density spike in the components with no vortices are due to the boundary condition of the spinor at r = 0, see Eqs. (4)-(6).
value at r → ∞ cannot be directly accessed since the confining harmonic trap allows for calculations up to the Thomas-Fermi radius of the atomic cloud only. Below, along the presentation of our results, we show how we circumvent this difficulty and how we find trustable values of the topological charges. III. SKYRMIONS IN MAGNETIC FIELDS WITH DIFFERENT BOUNDARY CONDITIONS As discussed in the previous section, the vortex structures (+2,+1,0), (+1,0,-1) and (0,-1,-2) each have associated their own spin textures and Skyrmion defects. Moreover, although the spin texture of the vortex (+1,0,-1) does not satisfy f · f = 1 everywhere, because f → 0 as r → 0, it is still clear that the boundary conditions are essential to determine the Skyrmion charges. The boundary conditions at r = 0 are completely determined by the corresponding vortex structure, as shown by Eqs. (4)-(6). However, supported by our calculations, the boundary values at r → ∞ depend on the external B-field. We find that far from the vortex singularity, f tends to align to the direction of the magnetic field B. This can be further understood by looking at the expression for the energy E[Ψ], Eq. (1),
FIG. 3 .
3Skyrmion spin textures f . (a) (+1,0,-1), Q 2D sky = 0, B z = 0. (b) (+2,+1,0), Q 2D sky = −1/2, B z = 0. (c) (+2,+1,0), Q 2D sky = 0.37 B z = −0.25r. (d) (+1,0,-1), Q 2D sky = −0.19. B z = −r.(e) Two Skyrmions (+2,+1,0) + (0,-1,-2), Q 2D sky = +1/2 − 1/2 = 0, B z = 0. (f) 2D view of two Skyrmions, B z = 0, the alignment of f to the B-field far from the singularities.
(x 0
0, y 0 ). To see the effect on the Skyrmions we show Fig. 4. The upper panel shows the z−component of the spin texture, f z (r), for vortices (+2,+1,0) with different values of B z , and in which we have superposed the solutions with (x 0 = 0, y 0 = 0), marked with dots,
f
becomes paramagnetic in the sense that it gets aligned to the direction of the external B-field. Since this field points along the xy-plane as r → ∞, hence f z (r) → 0. This insight suggests to consider the other case of the B-field, which can be made to point at arbitrary directions, such that the boundary value of f z (r), at r → ∞, can be also made to point to such a direction. For completeness and to reinforce the topological nature of the defects, we plot inFig. 5the profile of two Skyrmion defects of the same type in the cloud, one case with the same sign of the charges and the other with opposite signs, thus yielding the double of the charge and zero respectively.
FIG. 4 .
4Spin texture component f z as a function of r, for different values of B z = constant, for the vortex structure (+2,+1,0), upper panel, and (+1,0,-1), lower panel. The dotted lines corresponds to a magnetic field centered at (x 0 = 0, y 0 = 0), while the continuous ones to a (x 0 = 0, y 0 = 2).
B
-field can be made to point at any arbitrary direction, depending on the values of B 0 and B z , see Eq.(2). Figures 6 illustrate typical results of this case. The first refers to vortices FIG. 5.Spin texture components f z as a function of r, for two defects, one located (x 0 ≈ −3.6, y 0 = 0) and the other at (x 0 ≈ 3.6, y 0 = 0), with charges (+1/2, +1/2) and (+1/2-1/2), yielding Q 2D sky = 1 and Q 2D sky = 0, respectively.
vortex structure (+2,+1,0), upper panel, and (+1,0,-1), lower panel. The dotted lines corresponds to a magnetic field centered at (x 0 = 0, y 0 = 0), while the continuous ones to a (x 0 = 0, y 0 = 2).
FIG. 7 .
7(Color online) Skyrmion topological charges Q 2D sky as a function of the amplitude B z of the component B z = B z r of the external magnetic field. Upper panel corresponds to 87 Rb, lower one to 23 Na. vortex structures (+2,+1,0), (+1,0,-1), (0,-1-2) are indicated.
Acknowledgment is given to grants CONACYT 2555573 (Mexico) and PAPIIT-IN105217(UNAM). RZZ thanks CONACYT (Mexico) for a graduate scholarship.
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To perform our calculations we use state of the art GPU programming via the PyCUDA library in a GPU with 2304 CUDA cores. We do calculations with double and single precision on 128 3 and 256 3 mesh points. We point out that the structure of the stationary states can be numerically ensured with 128 3 within single precision calculationsTo perform our calculations we use state of the art GPU programming via the PyCUDA library in a GPU with 2304 CUDA cores. We do calculations with double and single precision on 128 3 and 256 3 mesh points. We point out that the structure of the stationary states can be numerically ensured with 128 3 within single precision calculations.
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"Andrzej Jarosz \nThe Henryk Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nRadzikowskiego 15231-342KrakówPoland\n"
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"The Henryk Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nRadzikowskiego 15231-342KrakówPoland"
]
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| Inspired by the theory of quantum information, I use two non-Hermitian random matrix models-a weighted sum of circular unitary ensembles and a product of rectangular Ginibre unitary ensemblesas building blocks of three new products of random matrices which are generalizations of the Bures model. I apply the tools of both Hermitian and non-Hermitian free probability to calculate the mean densities of their eigenvalues and singular values in the thermodynamic limit, along with their divergences at zero; the results are supported by Monte Carlo simulations. I pose and test conjectures concerning the relationship between the two densities (exploiting the notion of the N -transform), the shape of the mean domain of the eigenvalues (an extension of the single ring theorem), and the universal behavior of the mean spectral density close to the domain's borderline (using the complementary error function). Theory for model S solid model T dashed : Example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 R MC histograms solid theory dashed : Model W; example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 K 2, r2 0.4 a R MC histograms solid theory erfc dashed : Model W; example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 K 2, r2 0.4 b MC histograms solid theory dashed : Model W; example 2 w j1 2 w j2 2 1 ; singular values J 2 w11 0.2, w21 0.3 J 3 w11 0.2, w21 0.3, w31 0.4 J 5 w11 0.2, w21 0.3, w31 0.4, w41 0.5, w51 0.6 MC histograms solid : | null | [
"https://arxiv.org/pdf/1202.5378v1.pdf"
]
| 117,480,529 | 1202.5378 | ecd3566f25bfbf473f6793d4c1cf03ac798ccd7d |
24 Feb 2012
Andrzej Jarosz
The Henryk Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
Radzikowskiego 15231-342KrakówPoland
24 Feb 2012Generalized Bures products from free probabilitynumbers: 0210Yn (Matrix theory)0250Cw (Probability theory)0540Ca (Noise)0270Uu (Ap- plications of Monte Carlo methods) Keywords: random matrix theoryfree probabilityquaternionnon-Hermitianunitaryquantum entangle- mentdensity matrixsumproduct R
Inspired by the theory of quantum information, I use two non-Hermitian random matrix models-a weighted sum of circular unitary ensembles and a product of rectangular Ginibre unitary ensemblesas building blocks of three new products of random matrices which are generalizations of the Bures model. I apply the tools of both Hermitian and non-Hermitian free probability to calculate the mean densities of their eigenvalues and singular values in the thermodynamic limit, along with their divergences at zero; the results are supported by Monte Carlo simulations. I pose and test conjectures concerning the relationship between the two densities (exploiting the notion of the N -transform), the shape of the mean domain of the eigenvalues (an extension of the single ring theorem), and the universal behavior of the mean spectral density close to the domain's borderline (using the complementary error function). Theory for model S solid model T dashed : Example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 R MC histograms solid theory dashed : Model W; example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 K 2, r2 0.4 a R MC histograms solid theory erfc dashed : Model W; example 1; eigenvalues J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 K 2, r2 0.4 b MC histograms solid theory dashed : Model W; example 2 w j1 2 w j2 2 1 ; singular values J 2 w11 0.2, w21 0.3 J 3 w11 0.2, w21 0.3, w31 0.4 J 5 w11 0.2, w21 0.3, w31 0.4, w41 0.5, w51 0.6 MC histograms solid :
I. INTRODUCTION
This paper is a continuation of the work [1], whose focus was on the simplest characteristics (level density) of a weighted sum of unitary random matrices-a non-Hermitian random matrix model encountered e.g. in quantum information theory or the theory of random walks on trees. Inspired by the first of these areas of physics (Sec. I C), I introduce more general models (Sec. I A) and calculate-using the tools of free probability (Sec. I B)-their level densities (Sec. II).
A. Models
Definitions of the models
The goal of this article is a basic study of the following non-Hermitian random matrix models T, W, V, which I will call "generalized Bures products" for the reason explained at the end of Sec. I C 2. In order to introduce them, I recall two other matrix models, S and P, investigated recently in the literature:
(i) Let L ≥ 2 be an integer; let U l , for l = 1, 2, . . . , L, be independent N × N unitary random matrices belonging to the Wigner-Dyson "circular unitary ensemble" (CUE or "Haar measure"; i.e., their eigenvalues are on average uniformly distributed on the unit circle); let w l be arbitrary complex parameters. Then define a weighted sum,
S ≡ w 1 U 1 + w 2 U 2 + . . . + w L U L .
(1) * Electronic address: [email protected]
This non-Hermitian model may be referred to as the "generalized Kesten model" [2] (cf. also [3,4]), and has been considered in some detail in [1].
(ii) Let K ≥ 1 be an integer; let A k , for k = 1, 2, . . . , K, be rectangular (of some dimensions N k × N k+1 ) complex random matrices such that all the real and imaginary parts of the entries of each matrix are independent random numbers with the Gaussian distribution of zero mean and variance denoted by σ 2 k /(2(N k N k+1 ) 1/2 ), i.e., the joint probability density function (JPDF),
P ({A k }) ∝ K k=1 exp − N k N k+1 σ 2 k Tr A † k A k ;(2)
these are rectangular "Ginibre unitary ensembles" (GinUE) [5,6]. Then define their product, an N 1 × N K+1 random matrix,
P ≡ A 1 A 2 . . . A K .(3)
This model has been thoroughly investigated (cf. e.g. [7][8][9][10][11]). Now, having a number of copies of S and P (all of them assumed statistically independent from each other), one may use them as building blocks of various products:
(i) A product of J ≥ 1 weighted sums (1),
T ≡ S 1 S 2 . . . S J ,(4)
where each sum S j has arbitrary length L j and complex weights w jl .
(ii) A product of (4) and (3),
W ≡ TP,(5)
i.e., this is a string of J generalized Kesten ensembles and K Ginibre unitary ensembles. Note that one has to set N = N 1 , and then W is rectangular of dimensions N 1 × N K+1 . (iii) Finally, a most general string of generalized Kesten and Ginibre unitary ensembles is obtained by multiplying a number I ≥ 1 of random matrices (5),
V ≡ W 1 W 2 . . . W I .(6)
The dimensions of the terms (N i,1 × N i,Ki+1 , for i = 1, 2, . . . , I) must obey N i,Ki+1 = N i+1,1 , and then V is rectangular of dimensions N 1,1 × N I,KI +1 .
Thermodynamic limit
The techniques I apply to approach the above models are valid only in the "thermodynamic limit," i.e., for all the matrix dimensions infinite and their "rectangularity ratios" finite,
N = N 1 , N 2 , . . . , N K+1 → ∞, r k ≡ N k N K+1 = finite.(7)
[I use here the notation for the model W (5); for V (6), one should add an index i.]
Mean densities of the eigenvalues and singular values
Level densities. This paper is just a most basic study of the models X ∈ {T, W, V}, namely, I will be calculating only:
(i) The mean density of the eigenvalues ("mean spectral density" or "level density"),
ρ X (z, z) ≡ 1 N N i=1 δ (2) (z − λ i ) ,(8)
where the mean values are with respect to the JPDF of X and are evaluated at the complex Dirac delta functions at the (generically complex) eigenvalues λ i of X. Note that one must set N K+1 = N (i.e., r 1 = 1) for W to be a square matrix.
(ii) The mean density of the "singular values," which are the (real and non-negative) eigenvalues µ i of the N K+1 × N K+1 Hermitian random matrix H ≡ X † X,
ρ H (x) ≡ 1 N K+1 NK+1 i=1 δ (x − µ i ) ,(9)
where the Dirac delta is real. This time, r 1 may acquire any positive value. Future work-universal quantities. Certainly, this is but the first step in understanding the considered models. The level density of a random matrix is known not to be universal, i.e., it depends on the precise probability distribution of the matrix. (However, cf. the end of Sec. I B 3 for a hint in favor of a certain universality of the level densities of our models.) As a next step, it would be desirable to investigate some "universal" properties of our models, i.e., depending solely on their symmetries but not the specific probability distributions. Some basic universal observables would be a "two-point connected correlation function,"
ρ connected H (x, y) ≡ ρ H (x, y) − ρ H (x)ρ H (y), (10a) ρ H (x, y) ≡ 1 N 2 K+1 NK+1 i,j=1 δ (x − µ i ) δ (y − µ j ) (10b)
(here written in the Hermitian case), or the behavior of the level density/correlation function close to the borderline (for non-Hermitian models) or the edges (for Hermitian models) of the spectrum.
Universal erfc scaling. Even though I focus on the level densities in the bulk of the spectrum and not close to its borders, the models T, W, V share a certain property (their mean spectrum is rotationally-symmetric around zero; cf. Sec. I B 2) which permits an application of the so-called "erfc conjecture" (cf. Sec. I B 6), which describes the universal way in which the mean spectral density (8) is modified close to the borderline of the "mean spectral domain" D (i.e., the subset of the complex plane where the eigenvalues of an infinite random matrix fall). The pertinent form-factor (38) depends on one or two parameters, and I positively test this hypothesis on a number of examples for each class of the models T, W, V by fitting these parameters to match the Monte Carlo data. However, it still remains to prove this hypothesis, as well as analytically determine the form-factor parameters.
Future work-microscopic quantities. Another step in the research on our models could be to consider finite matrix dimensions, and to calculate the complete JPDF of the eigenvalues and singular values of S, P, T, W, V. This however is a much more involved task.
B. Tools
In this Section, I sketch the means by which the level densities [(8), (9)] of the models [(4), (5), (6)] will be evaluated in the thermodynamic limit (7). I do not delve into details, as they have been addressed e.g. in [1,12].
Basic language of random matrix theory
Mean spectral densities and Green functions. First of all, it is convenient to replace the mean spectral density of any (N × N , N → ∞) Hermitian random matrix H or non-Hermitian random matrix X with an equivalent but more tractable object in the following way: Since the definitions [(9), (8)] exploit the real (Hermitian case) or complex (non-Hermitian case) Dirac delta function, one uses their respective representations,
δ(x) = − 1 2πi lim ǫ→0 + 1 x + iǫ − 1 x − iǫ ,(11a)δ (2) (z) = 1 π ∂ z lim ǫ→0 z |z| 2 + ǫ 2 ,(11b)
which prompt one to introduce the following "holomorphic" or "nonholomorphic Green functions (resolvents)" [13][14][15][16][17],
G H (z) ≡ 1 N N i=1 1 z − µ i = = 1 N Tr 1 z1 N − H , (12a) G X (z, z) ≡ lim ǫ→0 lim N →∞ 1 N N i=1 z − λ i |z − λ i | 2 + ǫ 2 = = lim ǫ→0 lim N →∞ 1 N Tr· · z1 N − X † (z1 N − X) (z1 N − X † ) + ǫ 2 1 N . (12b)
Thanks to (11a)-(11b), the densities are straightforward to reproduce from them,
ρ H (x) = − 1 2πi lim ǫ→0 + (G H (x + iǫ) − G H (x − iǫ)) ,(13a)ρ X (z, z) = 1 π ∂ z G X (z, z),(13b)
while the resolvents prove to be handier from the computational point of view. Remark: Eqs. [(12b), (13b)] are valid for z inside the mean spectral domain D. Outside it, the regulator ǫ may be safely set to zero, and the nonholomorphic Green function reduces to its holomorphic counterpart (12a),
G X (z, z) = G X (z), for z / ∈ D.(14)
An implication is that by calculating the Green function both inside D and outside D, and then equating them according to (14), one arrives at an equation of the borderline ∂D. M -transforms. In addition to the resolvents, one often finds even more convenient to work with their simple modification, the "holomorphic" or "nonholomorphic Mtransforms,"
M H (z) ≡ zG H (z) − 1, (15a) M X (z, z) ≡ zG X (z, z) − 1.(15b)
The Hermitian M -transform (15a) has the interpretation of the generating function of the "moments" m H,n (if they exist), by expanding around z = ∞,
M H (z) = n≥1 m H,n z n , m H,n ≡ 1 N Tr H n .(16)
Rotational symmetry and N -transform conjecture
Rotational symmetry of the mean spectrum. In this paper, I investigate only non-Hermitian random matrix models X with a feature that their mean spectrum is rotationally-symmetric around zero, i.e., that their density (8) depends only on
R ≡ |z|.(17)
Equivalently, this means that their nonholomorphic Mtransform (15b) also depends only on the radius,
M X (z, z) = M X R 2 ,(18)
giving the relevant radial part of the density through
ρ rad. X (R) ≡ 2πR ρ X (z, z)| |z|=R = d dR M X R 2 . (19)
N -transform conjecture. For such ensembles, it has been suggested and numerically confirmed on a number of examples [1,11,12] that there exists a simple relationship between the mean densities of their eigenvalues and singular values (i.e., eigenvalues of X † X)-the "Ntransform conjecture." In order to express it, one needs to define the functional inverse of the M -transform. In the Hermitian case (15a), it can be directly done,
M H (N H (z)) = N H (M H (z)) = z,(20)
obtaining the "holomorphic N -transform." In a generic non-Hermitian case (15b), however, such an inversion is obviously impossible. But with the rotational symmetry present (18), the M -transform again depends on one argument, and the inversion becomes generically doable,
M X (N X (z)) = z, N X M X R 2 = R 2 ,(21)
which is called the "rotationally-symmetric nonholomorphic N -transform." [To be more precise, N X (z) defined by the left equation in (21) is a holomorphic continuation of the one defined by the right equation.] Now the hypothesis states that these N -transforms of X and H ≡ X † X remain in a simple relation,
N X † X (z) = z + 1 z N X (z).(22)
Typically, one of these random matrices will be much more accessible by analytical methods than the other, and therefore, Eq. (22) will provide a way to that more complicated model. I will henceforth make extensive use of this hypothesis; numerical tests of the results thus obtained will further support it. 3. Free probability and model P Algorithm for computing the densities of P and P † P. The model P (3) is a product of independent (rectangular) random matrices A k . If r 1 = 1 (i.e., P is rectangular), then one is interested in the eigenvalues of P † P only, while if r 1 = 1 (i.e., P is square), then its mean spectrum has rotational symmetry around zero (18) [10,11], and can therefore be related (22) to the spectrum of P † P. The level density of this latter matrix can now be expressed (by cyclic shifts of the terms) [11] through the density of the product of the Hermitian ("Wishart" [18]) matrices A † k A k . Hence, the situation is suitable for the employment of the "multiplication law" of free probability calculus of Voiculescu and Speicher [19,20].
Multiplication law in free probability. This theory is essentially a probability theory of noncommuting objects, such as large random matrices. Its foundational notion is that of "freeness," being a proper generalization of statistical independence. Qualitatively speaking, freeness requires the matrices H 1 , H 2 to not only be independent statistically, but also "rotationally" (i.e., no distinguished direction in their probability measures, i.e., dependence only on the eigenvalues)-independent random rotations from the CUE,
U 1 H 1 U † 1 and U 2 H 2 U † 2 , would ensure that condition if necessary.
Assuming freeness, there is a very useful prescription for computing the mean spectral density of the product H 1 H 2 (assuming it is still Hermitian, which obviously is not always the case; but the same procedure applies when all the matrices are unitary), once the densities of the terms are known: (i) Encode the densities by the holomorphic M -transforms (15a), M H1 (z) and M H2 (z), as outlined in Sec. I B 1. (ii) Invert them functionally to find the respective N -transforms (20), N H1 (z) and N H2 (z). (iii) The N -transform of the product follows from the "multiplication law,"
N H1H2 (z) = z z + 1 N H1 (z)N H2 (z).(23)
(iv) Invert the result functionally to find the Mtransform, and thus the mean spectral density of the product.
Eigenvalues and singular values of P. Implementing the above steps to P, one readily discovers [11],
N P † P (z) = σ 2 √ r 1 z + 1 z K k=1 z r k + 1 ,(24)
where for short, σ ≡ Furthermore, the authors of [10] conjecture and numerically affirm that (24) is to some degree universal, namely that it holds for the matrix elements of the A k not only IID Gaussian but also arbitrary IID obeying the Pastur-Lindeberg condition. Depending on a possible analogous universality of the level density of S, also the densities of T, W, V may exhibit a certain universality.
Let me close by saying that an algorithm parallel to the one described above will also be applied to derive the main results of this paper (cf. Sec. II A).
Quaternion free probability and model S
Although the topic of this work is multiplication rather than summation of random matrices, I should mention another [in addition to the technique outlined in Sec. I B 3 and Eq. (24)] pillar of this article-the results for the model S (1) derived in [1].
Addition law in free probability. In the realm of Hermitian random matrices, free probability provides-besides the multiplication law (23)-a rule for summing free matrices H 1 , H 2 . This time, one should invert functionally not the holomorphic M -transforms, but the Green functions (12a),
G H (B H (z)) = B H (G H (z)) = z,(25)
which is known as the "holomorphic Blue function" [21]. This objects then satisfies the "addition law" [19,20],
B H1+H2 (z) = B H1 (z) + B H2 (z) − 1 z ,(26)
upon which it remains to functionally invert the left-hand side, which leads to the holomorphic Green function of the sum, and consequently, its mean spectral density.
(Recall that in classical probability theory, an analogous algorithm makes use of the logarithms of the characteristic functions of independent random variables.) Addition law in quaternion free probability. For non-Hermitian random matrices, a construction parallel to (25)- (26) has been worked out in [22,23]. Basically, it consists of three steps:
(i) "Duplication" [24,25] (cf. also [26,27])-the nonholomorphic Green function (12b) has a denominator quadratic in X, which makes its evaluation hard, and one needs to linearize it by introducing the (2 × 2) "matrixvalued Green function," (27) where for short,
G X (z, z) ≡ lim ǫ→0 lim N →∞ 1 N bTr 1 Z ǫ ⊗ 1 N − X dupl. ,Z ǫ ≡ z iǫ iǫ z , X dupl. ≡ X 0 N 0 N X † ,(28)
and the "block-trace,"
bTr
A B C D ≡ TrA TrB TrC TrD .(29)
In other words, this object resembles the holomorphic Green function (12a), and thus can be approached by methods designed for Hermitian random matrices; the cost being the need to work with 2 × 2 matrices rather than complex numbers. The nonholomorphic Green function lies precisely on its upper left entry,
[G X (z, z)] 11 = G X (z, z).
(ii) The extension to the complete quaternion spaceby replacing in (27) the infinitesimal ǫ with a finite complex number,
Q ≡ c id id c , c, d ∈ C,(30)
one obtains a quaternion argument of the "quaternion Green function,"
G X (Q) ≡ 1 N bTr 1 Q ⊗ 1 N − X dupl. .(31)
The mean spectral density is reproduced from this quaternion function by approaching the complex plane (c = z, d = ǫ), just as in the Hermitian case, it is obtained from the complex Green function by approaching the real line (z = x ± iǫ) (13a).
(iii) The functional inversion can now be performed in the quaternion space,
G X (B X (Q)) = B X (G X (Q)) = Q(32)
(the "quaternion Blue function"), and the "quaternion addition law" for free non-Hermitian random matrices can be proven,
B X1+X2 (Q) = B X1 (Q) + B X2 (Q) − Q −1 .(33)
Eigenvalues and singular values of S. Formula (33) massively simplifies calculations of the mean spectral density of sums of free non-Hermitian ensembles. In particular, harnessing it to the weighted sum (1) implies that: (i) Its mean spectrum exhibits the rotational symmetry around zero (18). (ii) Its rotationally-symmetric nonholomorphic N -transform (21) is a solution to the following set of (L + 2) polynomial equations,
z = L l=1 M l , (34a) −C = z(z + 1) N S (z) , (34b) −C = M l (M l + 1) |w l | 2 , l = 1, 2, . . . , L,(34c)
where C ≥ 0 and M l are auxiliary unknowns. (iii) Its mean spectral domain D is either a disk or an annulus, whose external and internal radii are found from
R 2 ext. = N S (0),(35a)R 2 int. = N S (−1). (35b)
Furthermore, Eq. (22) means that the singular values are derived from an identical set of equations except (34b) which turns into
−C = (z + 1) 2 N S † S (z) .(36)
Single ring conjecture
Single ring conjecture. It is another hypothesis [1] (being a generalization of the "Feinberg-Zee single ring theorem" [27][28][29][30]) that for non-Hermitian random matrix models with the rotational symmetry (18) present, their mean spectral domain D is always a disk or an annulus. It has been proven for the models S and P; for T, W, V, I will assume it holds, and support it a posteriori by numerical simulations.
Radii of the disk/annulus. If this conjecture is true, then the external and internal radii of the annulus (becoming a disk if the internal radius shrinks to zero) are still given by (35a)-(35b), provided that there are no zero modes. Indeed, on the borderline of the domain D, the nonholomorphic Green function (from the inside of D) and the holomorphic one (from the outside of D) must match (14). But the holomorphic Green function compatible with the rotational symmetry (18) has the form G X (z) = (1 + M X (R 2 ))/z, hence, holomorphicity leaves one possibility, i.e., that M X (R 2 ) is a constant, G X (z) = (1 + M)/z. In the external outside of D (including z = ∞), it is well-known that G X (z) ∼ 1/z as z → ∞, which means that M = 0. In the internal outside (including z = 0), supposing there are no zero modes, the Green function must be just zero, i.e., M = −1. If there are zero modes, ρ zero modes X (z, z) = αδ (2) (z, z), they are obtained by applying (13b) to the Green function G zero modes X (z) = α/z regularized according to Eq. (11b); therefore, M = α − 1. In other words,
R 2 ext. = N X (0),(37a)R 2 int. = N X (α − 1). (37b)
6. erfc conjecture Calculations in the thermodynamic limit (7) are capable of reproducing the mean spectral density only in the bulk of the domain D, but not close to its borderline. However, based on earlier works [9,[31][32][33], it has been suggested [1,11,12] how to extend the radial mean spectral density (19) to the vicinity of the borderline, provided that the rotational symmetry (18) holds.
For each circle R = R b enclosing D (one or two; cf. Sec. I B 5), one should simply multiply the radial density ρ rad. X (R) by the universal form-factor,
f N,q b ,R b ,s b (R) ≡ 1 2 erfc q b s b (R − R b ) √ N ,(38)
where erfc(x) ≡ 2 √ π ∞
x dt exp(−t 2 ) is the complementary error function, while the sign s b is +1 for the external borderline and −1 for the internal borderline, and q b is a parameter dependent on the particular model, whose evaluation requires truly finite-size methods, but whose value I will adjust by fitting to the Monte Carlo data.
C. Motivation
Interesting mathematical properties
I find the models T, W, V mathematically interesting for the following reasons: (i) They are non-Hermitian, and the theory of such random matrices is richer and much less developed than for the Hermitian ones (cf. e.g. [33]). (ii) They belong to a special class of non-Hermitian matrices, namely, with rotationallysymmetric mean spectrum (18). As such, they conjecturally exhibit certain features, which demand testing (and eventually proofs): the N -transform conjecture (Sec. I B 2), which makes them reducible (at least concerning the level density) to Hermitian models; the single ring conjecture (Sec. I B 5); the erfc conjecture (Sec. I B 6). (iii) They contain two operations widely investigated in the literature on random matrices, albeit rather separately: summation (cf. e.g. [3,4,[22][23][24][25][26][27][34][35][36]) and multiplication (cf. e.g. [7][8][9][10][11][12][36][37][38][39][40][41][42][43][44][45][46][47][48]).
Applications to quantum information theory
The models T and W are encountered in the theory of quantum information. This application has been described in [49,50], as well as in Sec. I B 2 of [1], and I refer the reader to these sources for a more detailed exposition (the textbook [51] is an introduction to the subject). It is debatable whether one can find the model V in this theory; if not, the reader may treat it just as a mathematically natural generalization of W.
Density matrix. Fundamental objects in quantum information theory are "mixed states," i.e., statistical ensembles of quantum states. Such random states arise in a number of important settings: (i) If a quantum system interacts with its environment in a complicated (noisy) way, which may be regarded as random. (ii) If a quantum system is in thermal equilibrium. (iii) If the preparation history of a quantum system is unknown or uncertain (such as for quantum analogues of classically chaotic systems). (iv) If one investigates generic properties of an unknown complicated quantum system, one may assume it is random. (v) If a system consists of subsystems which are entangled, each of them must be described by a mixed state (and quantum entanglement is a central feature in the theory of quantum computers). A classic example is light polarization: a polarized photon can be written as a superposition of two helicities, right and left circular polarizations, (a|R + b|L ) (a "pure state"); whereas unpolarized light may be described as being |R or |L , each with probability 1/2 (a mixed state).
A mixed state cannot be represented by a single state vector-a proper formalism is that of a "density matrix." If one considers a statistical mixture of N pure states
|ψ i , each with probability p i ∈ [0, 1] ( N i=1 p i = 1), the density matrix is defined as ρ ≡ N i=1 p i |ψ i ψ i | (a con-
vex combination of pure states; called sometimes an "in-coherent superposition"), and it has this genuine property that the expectation value of any observable A is given by Tr(ρA). More generally, any operator which is Hermitian, positive-semidefinite (its eigenvalues are nonnegative) and has trace one (its eigenvalues sum to one) may be considered a density matrix.
I will be interested in complicated composite quantum systems. For example, for a bi-partite system consisting of a subsystem A of size N 1 and B of size N 2 (with orthonormal bases, {|i A } and {|j B }), a general pure state of the full system is "entangled," i.e., it cannot be written as a tensor product of pure states of A and B,
|ψ ≡ N1 i=1 N2 j=1 X ij |i A ⊗ |j B .(39)
In other words, A or B cannot be said to be in any definite pure state. But they can be characterized by a density matrix-consider any observable A on A; its expectation value in the state |ψ reads Tr(ρ A A), where the "reduced density matrix" of A,
ρ A ≡ XX † Tr (XX † ) ,(40)
and X is rectangular N 1 × N 2 complex matrix. (This may equivalently be obtained by performing the "partial trace" Tr B of the density matrix of the full system, |ψ ψ|, i.e., j B j|ψ ψ|j B , and normalizing it.) Therefore, the eigenvalues of ρ A (which are the properly normalized singular values of X) are precisely the above probabilities p i . This picture should be supplied by the discussed above basic notion of replacing the complicatedness of the full system by randomness-considering X to be some random matrix, and calculating the mean density of its singular values. An important measure for a mixed state is its "von Neumann entropy,"
S(ρ) ≡ −Tr(ρ log ρ) = − N i=1 p i log p i .(41)
It represents the degree of randomness (mixture) in the mixed state (thus, quantum information; thus, it also measures entanglement)-it is the larger, the more disperse the probabilities p i are; it is zero for a pure state, and reaches its maximum log N for all the probabilities equal [the full system is in the "maximally entangled (Bell) state"]. Moreover, a measurement can never decrease this entropy; consequently, a measurement may take a pure state to a mixed one, but not conversely (except there is a greater growth of entropy in the environment). I will focus on some physically motivated "structured ensembles" of random states in which an important role is played by the tensor product structure of the Hilbert spaces of the subsystems, i.e., which are invariant under local unitary transformations in the respective Hilbert spaces.
Model S. The following construction leads to the appearance of the model S in the above setting:
(i) Consider a bi-partite system (A, B) of size N × N in the Bell state,
|Ψ + AB ≡ 1 √ N N i=1 |i A ⊗ |i B .(42)
(ii) Consider the following type of randomness-apply to this Bell state L independent random local unitary transformations (belonging to the CUE) U l in the principal system A, and form a coherent superposition of the resulting (maximally entangled) states with some weights w l ,
|ψ ≡ L l=1 w l (U l ⊗ 1 N ) |Ψ + AB = = (S ⊗ 1 N ) |Ψ + AB = = 1 √ N N i,j=1 S ij |i A ⊗ |j B .(43)
(iii) Hence, the reduced density matrix for A is given by (40) with X = S. [The normalization constant is Tr(SS † ) = N L l=1 |w l | 2 + . . ., where the dots are much smaller than N in the large-N limit (7). So one may set for convenience, L l=1 |w l | 2 = 1, and investigate the singular values of S rescaled by N .]
Model T. It is a matter of a nested repetition of the above procedure to obtain the model T. For instance, for J = 2:
(i) Consider the above pure state (43), constructed according to a matrix S 2 of length L 2 , and take its L 1 copies.
(ii) Perform a random independent CUE rotation U 1l of each of these copies, and form their coherent superposition with some weights w 1l .
(iii) The reduced density matrix for A (40) will have X = T = S 1 S 2 . One may continue this process any J times.
Model P. This ensemble originates from a measurement in the product basis of Bell states, in the following way:
(i) Consider a composite system with 2K subsystems of sizes
A 1 , size N1 A 2 , A 3 size N2 , . . . , A 2K−2 , A 2K−1 size NK , A 2K . size NK+1(44)
(ii) Take an arbitrary product state |ψ 0 ≡ |0 A1 ⊗ |0 A2 ⊗ . . . ⊗ |0 A2K , and apply to it random unitary local transformations acting on the pairs of subsystems,
|ψ ≡ U A1A2 ⊗ U A3A4 ⊗ . . . ⊗ U A2K−1A2K |ψ 0 . (45)
The result is separable with respect to the above pairing, i.e., it can be expanded in the product basis as
|ψ = N1 i1=1 N2 i2,i ′ 2 =1 . . . NK iK ,i ′ K =1 NK+1 iK+1=1 [A 1 ] i1i2 [A 2 ] i ′ 2 i3 . . . [A K−1 ] i ′ K−1 iK [A K ] iK iK+1 · · |i 1 A1 ⊗ |i 2 A2 ⊗ |i ′ 2 A3 ⊗ . . . ⊗ |i K+1 A2K ,(46)
where the coefficients, gathered into K rectangular (N k × N k+1 ) matrices A k , may be assumed independent Gaussian (2). (iii) Consider the Bell states (42) on the pairs (A 2 , A 3 ), . . . , (A 2K−2 , A 2K−1 ), and project |ψ onto their product,
P ≡ 1 A1 ⊗ K k=2 |Ψ + A 2k−2 A 2k−1 Ψ + A 2k−2 A 2k−1 | ⊗ 1 A2K ,(47)
which leads to a random pure state describing the remaining two subsystems, A 1 and A 2K , through the matrix P,
|φ ≡ P|ψ ∝ ∝ N1 i1=1 NK+1 iK+1=1 [P] i1iK+1 |i 1 A1 ⊗ |i K+1 A2K .(48)
(iv) The reduced density matrix for A 1 (i.e., the normalized partial trace over A 2K ) is therefore (40) with X = P.
Model W. A direct combination of the above two algorithms leads to the model W-one should consider the random pure states corresponding to the model T on the 2K subsystems (44), and proceed as above.
One may find in the literature only an expression for the mean density of the singular values of W in the case of J = 1, L 1 = 2, w 1l = 1/ √ 2 (for l = 1, 2), K = 1, σ 1 = 1, r 1 = 1-the "Bures distribution" [52,53],
ρ W † W (x) = 1 4 √ 3π β x + β 2 x 2 − 1 2/3 − − β x − β 2 x 2 − 1 2/3 ,(49)
for x ∈ [0, β] and zero otherwise, where for short, β ≡ 3 √ 3. The chief purpose of this paper is to extend this result to arbitrary values of the parameters, even to a product of a number of matrices W, as well as to the mean density of the eigenvalues.
II. GENERALIZED BURES PRODUCTS
A. Generalized multiplication laws This paper deals with products of random rectangular or non-Hermitian matrices, hence, I will start from adjusting the free probability multiplication law (23) to such a situation. Consider a product of arbitrary independent random matrices,
X ≡ X 1 X 2 . . . X I ,(50)
where X i , i = 1, 2, . . . , I, is rectangular of dimensions T i × T i+1 , which tend to infinity in such a way that the rectangularity ratios
s i ≡ T i T I+1 ,(51)
remain finite. I will follow the steps sketched in Sec. I B 3 (cf. [11]) to derive the mean densities of its singular values and if s 1 = 1 also the eigenvalues.
Singular values of the product X via cyclic shifts and multiplication law
Let me for simplicity set I = 2 here. I begin from the eigenvalues of the Hermitian T 3 × T 3 random matrix,
X † X = X † 2 X † 1 X 1 X 2 .(52)
However, as a first step, consider instead the T 2 × T 2 matrix,
Y ≡ X † 1 X 1 X 2 X † 2 ,(53)
which differs from the previous one only by a cyclic shift of the terms. Therefore, as follows from (16), their Ntransforms are related by
N X † X (z) = N Y T 3 T 2 z .(54)
Now, Y is a product of two free Hermitian random matrices (their freeness is what I actually mean by the assumed "independence" of X 1 and X 2 ), and thus the multiplication law (23) can be applied,
N Y (z) = z z + 1 N X † 1 X1 (z)N X2X † 2 (z) = = z z + 1 N X † 1 X1 (z)N X † 2 X2 T 2 T 3 z ,(55)
where in the last step I used an analogue of (54). Finally, inserting (55) into (54), one obtains a multiplication law which allows to calculate the singular values of X from the singular values of X 1 and X 2 ,
N X † X (z) = z z + T2 T3 N X † 1 X1 T 3 T 2 z N X † 2 X2 (z).(56)
This formula may be generalized to any I,
N X † X (z) = z I−1 I i=2 (z + s i ) I i=1 N X † i Xi z s i+1(57)
[cf. Eq. (58) of the first position in [11]].
Eigenvalues of the product X assuming it is square and has rotationally-symmetric mean spectrum
If s 1 = 1, i.e., X is square, one may also ask about its eigenvalues. Assume that its mean spectrum has the rotational symmetry around zero (18). Combining the N -transform conjecture (22) with the multiplication law (57), one finds the appropriate formula,
N X (z) = I i=1 z z + s i N X † i Xi z s i+1 .(58)
Eigenvalues of the product X assuming all Xi are square and have rotationally-symmetric mean spectra
If moreover all s i = 1, i.e., all X i are square, and also the mean spectra of all X i are rotationally symmetric around zero, then the right-hand side of the multiplication law (58) may be expressed through the eigenvalues of X i by virtue of the N -transform conjecture (22), which yields simply
N X (z) = I i=1 N Xi (z).(59)
The models T, W, V all possess the property (18) since they are products of terms which separately exhibit this symmetry. Hence, one is allowed to use the multiplication law (59) for them. Eventually, the rotational symmetry will be confirmed by Monte Carlo simulations.
B. Product T
Master equations for T
Eigenvalues.
It is now straightforward to write down the master equations for the nonholomorphic Mtransform M ≡ M T (R 2 ) of the product T (4)-they are comprised of the multiplication law (59),
N T (z) = J j=1 N Sj (z),(60)
which links the J sets (34a)-(34c),
z = Lj l=1 M jl ,(61a)−C j = z(z + 1) N Sj (z) ,(61b)−C j = M jl (M jl + 1) |w jl | 2 , l = 1, 2, . . . , L j ,(61c)
for j = 1, 2, . . . , J; these are (1 + 2J + J j=1 L j ) polynomial equations. They are valid inside the mean spectral domain D, whose borderline (one or two centered circles) is given by (35a)-(35b).
Singular values. These are obtained from the same master equations, albeit with N T (z) in (60) replaced by z z+1 N T † T (z) (22).
I will now simplify the above equations and solve them either analytically or numerically in a number of special cases, comparing the findings with Monte Carlo simulations.
Example 0
Eigenvalues. Before that, however, let us remark that if all the terms S j have identical lengths, L j = L, and sequences of weights, w jl = w l , then (61a)-(61c) imply that all N Sj (z) are equal to each other. This along with (60) in turn imply that the nonholomorphic M -transforms of T and any S j are related by a simple rescaling of the argument,
M T R 2 = M S R 2/J .(62)
Singular values. Unfortunately, there is no such scaling relation here; it is clear from trying to follow the above argument in conjunction with (22).
Example 1
As the first example, take arbitrary J and L j , but suppose that the weights for each S j are equal to each other and denoted by
w jl = w j L j , l = 1, 2, . . . , L j ,(63)
for some J constants w j . Eigenvalues. Then, the master equations (61a)-(61c) give explicitly
N Sj (z) = |w j | 2 z + 1 z Lj + 1 , j = 1, 2, . . . , J,(64)
which inserted into (60) and after performing the functional inversion (21) leads to a polynomial equation of order J for M,
R 2 |w| 2 = (M + 1) J J j=1 M Lj + 1 ,(65)
where for short, w ≡ J j=1 w j . [Notice a certain similarity to the counterpart Eq. (24) for P; indeed, I will show (Sec. II C 2) that T of Example 1 is in some sense "inverse" to a certain class of models P.] Inserting M = 0 or M = −1 (35a)-(35b) (no zero modes here, α = 0) into (65), one finds that the mean spectral domain is a centered disk of radius
R ext. = |w|.(66)
In particular, if all L j = L, Eq. (65) can be solved explicitly, yielding (19),
ρ rad. T (R) = 2|w| 2 1 J 1 − 1 L R 2/J−1 |w| 2 − R 2/J L 2 ,(67)
for R ≤ |w|, and zero otherwise. This could also be obtained by using the scaling relation (62) and the proper result for S, i.e., Eq. (58) of [1]. Singular values. The master equation for the holomorphic M -transform M ≡ M T † T (z) is a small modification of (65) according to (22),
z |w| 2 = (M + 1) J+1 M J j=1 M Lj + 1 ,(68)
which is polynomial of order (J + 1). The above findings, along with the erfc conjecture (38), are tested against Monte Carlo data in Fig. 1 [(a), (b), (c)].
Remark 1: Divergence at zero. Let me note that from the above equations one may easily derive an interesting feature of the level densities-their divergence close to zero. I will apply the logic presented in [11] to this and the following models. Assume that for z → 0, also zG → 0 and zG → 0 (recall, M = zG − 1, M = zG − 1), which will be verified a posteriori. Then the denominators of the right hand sides of [(65), (68)] tend to nonzero constants, and consequently, G ∼ z 1/J−1 z 1/J , G ∼ z −J/(J+1) , i.e.,
ρ rad. T (R) ∼ R − d−2 d , R → 0, (69a) ρ T † T (x) ∼ x − d d+1 , x → 0,(69b)
where
d = J.(70)
[The initial suppositions thus hold true: zG ∼ R 2/J → 0 and zG ∼ z 1/(J+1) → 0.] Note that the mean density of the singular values diverges for any J, while for the eigenvalues, the radial mean spectral density is finite for J = 1 (i.e., the model S, for which it zeroes) or J = 2, in which case however, the proper density (8) diverges. L j terms (multiplied by L −1/2 ), each of which is a product of some J CUE matrices. Now, it is known that a product of CUE matrices still belongs to the CUE, hence, T looks like the model S of length L-except the fact that now the various terms are not statistically independent (free). So one might ask how relevant these correlations between the L terms are, i.e., how much the level densities of T and of S of length L differ. Figure 2 illustrates the theoretical mean spectral densities of these two random matrix models-they look completely different, showing that even though T may be recast as a sum of CUE matrices, correlations between them are important, and it is not our model S at all. Especially, the behavior at zero of the level densities in these two cases is entirely distinct, cf. Remark 1 above. Also, with growing L, the density of S tends to that of the square GinUE distribution (cf. [1]), ρ rad. S (R) → 2R inside the unit circle and zero otherwise, while the limit of T is very different.
Example 2
As the second example, take arbitrary J, and let all the lengths L j = 2, but with arbitrary weights w j1 , w j2 .
Eigenvalues. In this case, the master equations (61a)-(61c) can be transformed into Remark 1: Divergence at zero. Following the same line of reasoning as in Remark 1 in Sec. II B 3, one recognizes that the divergences close to zero of the level densities stemming from [(76), (78)]-disregarding their possible zero-mode part, which amounts to considering r 1 ≥ 1, which means that there cannot happen L j = 1/r 1 , i.e., one may disregard in our analysis the products over j in the above formulae-are governed by the number R ≡ # {2 ≤ k ≤ K : r k = 1}, and given again by (69a)-(69b) but with d = (J + 1)δ r1,1 + R.
z = 1 (|wj1|+|wj2|) 2 NS j (z) − 1 + 1 (|wj1|−|wj2|) 2 NS j (z) − 1 ,(71)
Remark 2: T as an "inverse" of P with integer rectangularity ratios. The above master equations of W imply the following peculiar joint property of the models T (of Example 1) and P: Assume that P (of length K; set for simplicity σ = 1) is such that a certain number 1 ≤ J ≤ K of the rectangularity ratios r k /r 1 are integers greater than 1. If one multiplies this P from the left they are the master equations for a model P ≡ P 1 P 2 , where P 1 ≡ B 1 . . . B J , with B j being the square GinUE random matrices (2), while P 2 is the model P with the terms A k for which r k /r 1 = L j removed from it.
J 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6 K 5,
In particular, if the model P is such that all of its rectangularity ratios r k /r 1 , k ≥ 2, are integers greater than 1, then multiplying it from the left by the appropriate matrix T [of length (K − 1) and the lengths of its terms L k = r k /r 1 ] yields M = R 2/K − 1, i.e., ρ rad.
W (R) = 2 K R 2/K−1 ,(82)
random matrices and the addition law for non-Hermitian ones) greatly simplifies otherwise difficult calculations of the mean densities of eigenvalues and singular values of the generalized Bures products T, W, V, which are random matrix models of relevance in quantum information theory.
B. Open problems
As already mentioned, this article is certainly only the initial step (nevertheless important, especially from the point of view of quantum information theory) in learning about the models T, W, V. A major endeavor would be to consider finite matrix dimensions and attempt a computation of the complete JPDF of the eigenvalues of these models. One could also investigate some of their universal properties (cf. Sec. I A 3). Actually, inspired by [10], one could check whether the level densities themselves show any sign of universality, just like for P.
A pressing challenge is to prove the three hypotheses which my derivation is founded upon: the N -transform conjecture (cf. Sec. I B 2), the single ring conjecture (cf. Sec. I B 5), and the erfc conjecture (cf. Sec. I B 6).
It would also be desirable to understand more about the features of our models important for quantum entanglement theory, e.g. analyze their von Neumann entropy (41).
k , from which stems both the mean density of the singular values [(20), (15a), (13a)] and (provided that r 1 = 1) the eigenvalues [(22), (21), (19)] of P [in the latter case, the mean spectral domain D is a centered disk of radius R ext. = σ, cf. (37a)-(37b)].
Remark 2 :
2Is T really different from S? Notice that if one opens up the brackets in the definition of the model T [(4), (1)] (let me focus on this Example 1 and all w j = 1), T = U jl ), one obtains a sum of L ≡ J j=1
FIG. 1 :
1L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L1 2, L2 3, L3 4 J 5 L1 2, L2 3,L3 4, L4 5, Theoretical level densities versus Monte Carlo data for the model T. Top row concerns the eigenvalues, middle row the eigenvalues plus the erfc form-factor, bottom row the singular values. Left column illustrates Example 1, right column Example 2.
FIG. 5 :
5L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, Numerical verification of the property described in Remark 2 in Sec. II C 2. The theoretical graphs are the level densities of the model P1, obtained from (24) with all r k = 1. W1 example 1:J1 5 L11 2, L12 3, L13 4, L14 5, L15 6 , K1 2 r11 1 0.7, r12 0.4 ; W2 example 1: J2 2 L21 2, L22 3 , K2 5 r21 0.7, r22 0.4, r23 1.2, r24 0.6, r25 1W1 example 1: J1 5 L11 2, L12 3, L13 4, L14 5, L15 6 ,K1 2 r11 1 0.7, r12 0.4 ; W2 example 1: J2 2 L21 2, L22 3 , K2 5 r21 0.7, r22 0.4, r23 1.2, r24 0.6, r25 1W1 example 1: J1 5 L11 2, L12 3, L13 4, L14 5, L15 6 , K1 2 r11 1 0.7, r12 0.4 ; W2 example 1: J2 2 L21 2, L22 3 , K2 5 r21 0.7, r22 0.4, r23 1.2, r24 0.6, r25 1.5 c FIG. 6: Theoretical level densities [the eigenvalues (a), the eigenvalues plus the erfc form-factor (b), the singular values (c)] versus Monte Carlo data for the model V.
R
MC histograms solid theory erfc dashed :FIG. 3: Theoretical level densities versus Monte Carlo data for the model W, with K = 2. Top row concerns the eigenvalues, middle row the eigenvalues plus the erfc form-factor, bottom row the singular values. Left column illustrates Example 1, right column Example 2.Model W;
example 2 w j1
2 w j2
2 1 ; eigenvalues
J 2 w11 0.2, w21 0.3
J 3 w11 0.2, w21 0.3, w31 0.4
J 5 w11 0.2, w21 0.3, w31 0.4,
w41 0.5, w51 0.6
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MC histograms solid theory dashed : Model W; example 1; eigenvaluesJ 2 L1 2, L2 3 J 3 L1 2, L2 3, L3 4 J 5 L1 2, L2 3, L3 4, L4 5, L5 6MC histograms solid theory erfc dashed : Model W; example 1; eigenvalues0.0
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J 2 w11 0.2, w21 0.3
J 3 w11 0.2, w21 0.3, w31 0.4
J 5 w11 0.2, w21 0.3, w31 0.4,
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AcknowledgmentsMy work has been partially supported by the Polish Ministry of Science and Higher Education Grant "Iuventus Plus" No. 0148/H03/2010/70. I acknowledge the financial support of Clico Ltd., Oleandry 2, 30-063 Kraków, Poland, while completing parts of this paper. I am grateful to KarolŻyczkowski for valuable discussions.FIG. 2: Numerical verification of the property described inRemark 2 in Sec. II B 3, that even though the model T may be rewritten as a sum of CUE matrices, it is very different from the model S due to the correlations between the thus obtained CUE terms. which are quadratic for N Sj (z); they are supplied by the multiplication law (60).The borderline of the mean spectral domain is in general a centered annulus of radii (35a)-(35b),which reduces to a disk only if the (absolute values of the) two weights in at least one term are equal to each other. Singular values. As above, with the left-hand side of (60) changed according to(22).These master equations are solved numerically and compared to the Monte Carlo eigenvalues inFig. 1Remark: Divergence at zero. Assume that there is a certain number 1 ≤ W ≤ J of terms S j whose (absolute values of the) two weights are equal to each other, |w j1 | = |w j2 |; then one may ask about a behavior of the above level densities close to zero. For each of these W terms, Eq. (71) becomes linear and yields (I already replace z by M = zG − 1 or M = zG − 1, respectively, and assume zG → 0 and zG → 0 for z → 0), N Sj (M) = 4|w j | 2 zG/(zG + 1), which tends to zero as zG for z → 0. For the remaining (J − W) terms in T, Eq. (71) implies that any N Sj (M) tends to a nonzero constant ||w j1 | 2 − |w j2 | 2 | for z → 0. Substituting these results into (60), one discovers that the densities diverge at zero again according to (69a)-(69b), but with(This finding is consistent with zG → 0 and zG → 0 for z → 0.) C. Product WMaster equations for WEigenvalues. The master equations for the mean spectral density of W (5) are easily obtained through the multiplication law (59), N W (z) = N T (z)N P (z), from formula (24) (obviously with r 1 = 1)-they readwhere the rotationally-symmetric nonholomorphic Ntransform of T is given by (60)-(61c); one should substitute here M ≡ M W (R 2 ) in the place of z. Singular values. Choose arbitrary r 1 , use the multiplication law (56) along with (24)-obtaining,Example 1Consider the situation described in Sec. II B 3 [i.e., arbitrary J and L j , weights for each S j equal to each other (63)] plus arbitrary rectangularity ratios r k .Eigenvalues. The master equation is polynomial of order (J + K),valid inside the disk of radius (37a)-(37b),[The internal radius vanishes because the density of the zero modes, α = 1 − min{r k }, hence, either M = −1 or at least one term in the product over k in (76) with M = α − 1 is zero. This same argument will be valid in all the models investigated henceforth.] Singular values. The master equation is polynomial of order (J + K + 1), by a product T of length J such that the lengths L j of its terms are equal to these integer rectangularity ratios (and w j = 1), then the master equations [(76), (78)] turn intoComparing [(80), (81)] with (24), one recognizes thatIn other words, the model T (of Example 1) is "inverse" (at least on the level of the mean densities of eigenvalues and singular values) to the model P with rectangularity ratios r k /r 1 , k ≥ 2, equal to the lengths L j , with the "unity" beingExample 2In the setting of Sec. II B 4 [arbitrary J, all L j = 2, arbitrary weights w j1 , w j2 ] with arbitrary r k , the master equations(74)An analytical result can be obtained for the mean spectral domain-it is a disk of radius (37a)-(37b),This should be contrasted with the analogous situation for the model T, where the domain is in general an annulus (72a)-(72b). Remark: Divergence at zero. Combining the reasonings from Secs. II B 4 and II C 2, one finds that for the mean spectral domains to touch zero, there must be a number W of weights obeying |w j1 | = |w j2 |, in which case the divergences at zero of the level densities (without taking into account possible zero modes) are still described by (69a)-(69b) but withNotice that this is consistent with (80), and even suggests how an expression for d might look like for a most arbitrary model W.D. Product VMaster equations for VThe mean densities of the eigenvalues or singular values of any product V (6) can be directly derived from (74) or (75) using the multiplication laws (58) [or (59)] or (57).ExampleI will consider in this paper just one instance of the model V, namely, any length I but all the terms W i = T i P i of the form of Example 1 above (Sec. II C 2) [i.e., the lengths J i of T i arbitrary, with arbitrary lengths L ij of S ij but with the weights w ijl = w ij / L ij independent of l; the lengths K i of P i , variances σ 2 ik and rectangularity ratios r ik of A ik also arbitrary], having moreover arbitrary rectangularity ratios (51),Eigenvalues. The master equation is polynomial of orderJi j=1 w ij , and it is valid inside the disk of radius (37a)-(37b),78)]. Also, setting I = 1, J 1 = 1, K 1 = 1, w = 1, σ = 1, s 1 = r 1 = 1, L 11 = 2 in (88) yields a cubic equation, whose solution is the original Bures distribution(49). Of course, an even broader generalization of the Bures model would be to include distinct weights w ijl in the S ij ; I refrain from explicitly writing down the pertinent master equations, even though it is a straightforward step from the above results.Singular values. The master equation is polynomial of orderRemark 2: Divergence at zero. The level densities stemming from [(86), (88)] diverge at zero (beyond possible zero modes) again according to (69a)-(69b) but withwhere for short, R i ≡ # {2 ≤ k ≤ K i : r ik = 1/s i+1 }. Formula (89) reduces to (79) for I = 1, as it should.III. CONCLUSIONSA. SummaryIn this paper, I attempted to show how free probability theory (the multiplication law in the realm of Hermitian
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| []
|
[
"A General Framework of Nonparametric Feature Selection in High-Dimensional Data",
"A General Framework of Nonparametric Feature Selection in High-Dimensional Data"
]
| [
"Hang Yu \nDepartment of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC\n",
"Yuanjia Wang \nDepartment of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC\n",
"Donglin Zeng [email protected] \nDepartment of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC\n"
]
| [
"Department of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC",
"Department of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC",
"Department of Biostatistics\nUniversity of North Carolina\nChapel Hill27599NC"
]
| []
| Nonparametric feature selection in high-dimensional data is an important and challenging problem in statistics and machine learning fields. Most of the existing methods for feature selection focus on parametric or additive models which may suffer from model misspecification. In this paper, we propose a new framework to perform nonparametric feature selection for both regression and classification problems. In this framework, we learn prediction functions through empirical risk minimization over a reproducing kernel Hilbert space. The space is generated by a novel tensor product kernel which depends on a set of parameters that determine the importance of the features. Computationally, we minimize the empirical risk with a penalty to estimate the prediction and kernel parameters at the same time. The solution can be obtained by iteratively solving convex optimization problems. We study the theoretical property of the kernel feature space and prove both the oracle selection property and the Fisher consistency of our proposed method. Finally, we demonstrate the superior performance of our approach compared to existing methods via extensive simulation studies and application to a microarray study of eye disease in animals. | 10.1111/biom.13664 | [
"https://arxiv.org/pdf/2103.16438v1.pdf"
]
| 232,417,592 | 2103.16438 | 1971b7ac418010247ffc57c9e86ed810839a91d1 |
A General Framework of Nonparametric Feature Selection in High-Dimensional Data
Hang Yu
Department of Biostatistics
University of North Carolina
Chapel Hill27599NC
Yuanjia Wang
Department of Biostatistics
University of North Carolina
Chapel Hill27599NC
Donglin Zeng [email protected]
Department of Biostatistics
University of North Carolina
Chapel Hill27599NC
A General Framework of Nonparametric Feature Selection in High-Dimensional Data
Hang Yu is PhD candidate at Department of Statistics and Operation Research, University of North Car-olina, Chapel Hill, NC 27599 ([email protected]), Yuanjia Wang is Professor at Department of Biostatistics, Columbia University, New York, NY 10032 ([email protected]), and Donglin Zeng is Professor atTensor product kernelReproducing kernel Hilbert spaceFisher consistencyOracle propertyVariable selection
Nonparametric feature selection in high-dimensional data is an important and challenging problem in statistics and machine learning fields. Most of the existing methods for feature selection focus on parametric or additive models which may suffer from model misspecification. In this paper, we propose a new framework to perform nonparametric feature selection for both regression and classification problems. In this framework, we learn prediction functions through empirical risk minimization over a reproducing kernel Hilbert space. The space is generated by a novel tensor product kernel which depends on a set of parameters that determine the importance of the features. Computationally, we minimize the empirical risk with a penalty to estimate the prediction and kernel parameters at the same time. The solution can be obtained by iteratively solving convex optimization problems. We study the theoretical property of the kernel feature space and prove both the oracle selection property and the Fisher consistency of our proposed method. Finally, we demonstrate the superior performance of our approach compared to existing methods via extensive simulation studies and application to a microarray study of eye disease in animals.
Introduction
With biotechnology advances in modern medicine, biomedical studies collecting complex data with a large number of features are becoming the norm. High-dimensional feature selection is an essential tool to allow using such data for disease prediction or precision medicine, for instance, to discover a set of diagnostic biomarkers from neuroimaging measures for early prediction of neurodegenerative diseases, or to determine predictive biomarkers for effective management of type 2 diabetic patients' healthcare. Accurately identifying the subset of true important features is even more crucial and challenging than before in the fields of statistics and machine learning.
High-dimensional feature selection has been extensively studied for linear or generalized linear models in the past decades, and many methods have been developed including Lasso (Tibshirani, 1996), SCAD (Fan and Li, 2001), MCP (Zhang, 2010) and (Wang and Kulasekera, 2012). In these parametric models, the importance of individual features is characterized by non-null coefficients associated with them, so proper penalization can identify those non-null coefficients with probability tending to one when the sample size increases. However, parametric model assumptions are likely to be incorrect for many biomedical data due to potential correlations and higher-order interactions among feature variables. In fact, applying these approaches to any simple transformation of feature variables may lead to very different feature selection results.
More recently, increasing efforts have been devoted to high-dimensional feature selection when parametric assumptions, especially linearity assumption, do not hold. Various approaches were proposed to select features based on measuring certain marginal dependency (Guyon and Elisseeff (2003), Fan and Lv (2008), Fan et al. (2011), Song et al. (2012), Yamada et al. (2014), Urbanowicz et al. (2018)). For example, nonparametric association between each feature and outcome was used for screening (Fan and Lv (2008), Fan et al. (2011), Song et al. (2012)). LI et al. (2012) adopted a a robust rank correlation screening method based on marginal Kendall correlation coefficient. Yamada et al. (2014) considered a feature-wise kernelized Lasso, namely HSICLasso, for capturing nonlinear dependency between features and outcomes. In this approach, after a Lasso-type regression of an output kernel matrix on each feature-wise kernel matrix, unimportant features with small marginal dependence in terms of a Hilbert-Schmidt independence criterion (HSIC) would be removed. However, all methods based on marginal dependence may fail to select truly important variables since marginal dependency does not necessarily imply the significance of a feature when other features are also included for prediction, which is the case even for a simple linear model.
Alternatively, other approaches were proposed to relax parametric model assumptions and perform feature selection and prediction simultaneously. Lin and Zhang (2006) proposed COmponent Selection and Smoothing Operator (COSSO) to perform penalized variable selection based on smoothing spline ANOVA. Ravikumar et al. (2009) studied feature selection in a sparse additive model (SpAM), which assumed an additive model but allowed arbitrary nonparametric smoothers such as approximation in a reproducing kernel Hilbert space (RKHS) for each individual component function. Huang et al. (2010) considered spline approximation in the same model and adopted an adaptive group Lasso method to perform feature selection.
Although both COSSO and SpAM allowed nonlinear prediction from each feature, they still imposed restrictive additive model structures, possible with some higher-order interactions. To allow arbitrary interactions among the features and perform a fully nonparametric prediction, Allen (2013) propsed a procedure named KerNel Iterative Feature Extraction (KNIFE), in which the feature input was constructed in a Gaussian RKHS in order to perform nonparametric prediction. Different weights were used for different features in the constructed Gaussian kernel function so that a larger weight implied a higher importance of the corresponding feature variable. However, due to high nonlinearity in the kernel function, estimating the weights was numerically unstable even when the dimension of the features was moderate.
In this paper, we propose a general framework to perform nonparametric high-dimensional feature selection. We consider a general loss function which includes both regression models and classification as special cases. To perform nonparametric prediction, we construct a novel RKHS based on a tensor product of kernels for individual features. The constructed tensor product kernel, as discussed in Gao and Wu (2012), can handle any high-order nonlinear relationship between the features and outcome and any high-order interactions among the features. More importantly, each feature kernel depends on a non-negative parameter which determines the feature importance, so for feature selection, we further introduce a l 1 -penalty of these parameters in the estimation. Computationally, coordinate descent algorithms are used for updating parameters and each step involves simple convex optimization problems. Thus, our algorithm is numerically stable and can handle high-dimensional features easily. Theoretically, we first derive the approximation property of the proposed RKHS and characterize the complexity of the unit ball in this space in terms of bracket covering numbers. We then show that the estimated prediction function from our approach is consistent and moreover, we show that under some regularity conditions, the important features can be selected with probability tending to one.
The rest of the paper is organized as follows. In Section 2, we introduce our proposed regularized tensor product kernel and lay out a penalized framework for both estimation and feature selection. We then provide detailed computational algorithms to solve the optimization problem. In Section 3, we provide two theorems studying the property of the proposed RKHS.
We then give the main result of this paper including the consistency of the estimated prediction function and the oracle property of the feature selection. In Section 4, two simulation studies for regression and classification problems are conducted and we compare our method to existing methods. Application to a microarray study is given in Section 5. We conclude the paper with some discussion in Section 6.
Method
Suppose data are obtained from n independent subjects and consist of (X i , Y i ), i = 1, ..., n, where we let X denote p n -dimensional feature variables and Y be the outcome which can be continuous, binary or ordinal. Our goal is to use the data to learn a nonparametric prediction function, f (X), for the outcome Y .
We learn f (X) through a regularized empirical risk minimization by assuming f (·) belongs to a RKHS associated with a kernel function, κ(X, X), which will be described later. Specifically, if we denote the RKHS generated by κ(X, X) by H κ , equipped with norm · Hκ , then the empirical regularized risk minimization on RKHS for estimating f (X) solves the following optimization problem:
min f P n l(Y, f (X)) + γ n f 2 Hκ ,
where l(y, f ) a pre-specified non-negative and convex loss function to quantify the prediction performance, P n denotes the empirical measure from n observations, i.e., P n g(Y, X) = n −1 n i=1 g(Y i , X i ), and γ n is a tuning parameter to control the complexity of f . For a continuous outcome, l(y, f ) is often chosen to be a L 2 -loss given as (y − f ) 2 , while for a binary outcome, it can be one of the large-margin losses such as exp{−yf } in Adaboost. There are many choices of kernel functions for κ(·, ·) so that the estimated f (X) is nonlinear. One of the most commonly used kernel functions in machine learning is the Gaussian kernel function given by κ(X, X) = exp − X − X 2 /σ 2 for some bandwidth σ, where · is the Euclidean norm. To handle high-dimensional features, SpAM considered an additive kernel function by assuming κ(X, X) = pn j=1 exp −|X j − X j | 2 /σ 2 . In the KNIFE procedure, the kernel function is defined as κ ω (X, X) = exp − pn j=1 ω j (X j − X j ) 2 /σ 2 , where ω j , j = 1, .., p n are the additional weights to determine the feature importance.
To achieve the goal of both nonparametric prediction and feature selection, we propose a tensor product kernel as follows. For any given nonnegative vector λ = (λ 1 , λ 2 , · · · λ pn ) , we define a λ-regularized kernel function as
κ λ,σn (X, X) = pn m=1 1 + λ m κ n (X m , X m ) ,(1)
where κ n (x, y) = exp {−(x − y) 2 /2σ 2 n } with a pre-defined bandwidth σ n in R. There are two important observations for this new kernel function. First, it is a product of a univariate kernel function for each feature variable, which is given by 1 + λ m κ n (X m , X m ). Thus, the RKHS generated by κ λ,σn is equivalent to the tensor product of the RKHS generated by each featurespecific space. Second, each univariate kernel function is essentially the same as the Gaussian kernel function when λ m = 0. Consequently, the resulting tensor product space is the same as the RKHS generated by the multivariate Gaussian kernel function from all features whose λ m 's are non-zero. Therefore, the closure for the RKHS generated by κ λ,σn consists of all functions that only depend on feature variables for which λ m = 0. In other words, non-negative parameters, λ m , completely capture and regularize the contribution of each feature X m . In this way, the feature selection can be achieved by estimating the regularization parameters, λ m 's, in the kernel function.
More specifically, using the proposed kernel function, we let H λ,σn denote the RKHS corresponding to κ λ,σn so we aim to minimize
L n (λ, f ) ≡ P n l(Y, f (X)) + γ 1n ||f || 2 H λ,σn + γ 2n P (λ) subject to M ≥ λ 1 , λ 2 , · · · , λ pn ≥ 0,(2)
where M is a pre-specified large constant. P (λ) = pn m=1 P (λ m ) = pn m=1 λ m I(λ m < M/2), which is a truncated Lasso, and γ 1n , γ 2n are tuning parameters. Here, we include an l 1 penalization term on the regularization vector to perform feature selection and restrict λ m to be bounded. The latter bound is useful for numerical convergence to avoid the situation that some λ m can diverge. Since our RKHS contains constant and based on the representation theory for RKHS, solution for (2) takes form
f (X) = n i=1 α i κ λ,σn (X, X i ) and f 2 H λ,σn = α T K λ,σn α,
where α = (α 1 , ..., α n ) T and K λ,σn is an n × n matrix with entry κ λ,σn (X i , X j ). Then the optimization becomes solving
min α 1 ,...,αn,λ P n l(Y, n i=1 α i κ λ,σn (X, X i )) + γ 1n α T K λ,σn α + γ 2n pn m=1 λ m I(λ m < M/2) subject to M ≥ λ 1 , λ 2 , · · · , λ pn ≥ 0.
We iterate between α and λ to solve the above optimization problem. At the k-th iteration,
α k+1 = min α n −1 n j=1 l(Y j , n i=1 α i κ λ k ,σn (X j , X i )) + γ 1n α K λ k ,σn α (3) λ k+1 = min 0≤λ≤M n −1 n j=1 l(Y j , n i=1 α k+1 i κ λ,σn (X j , X i )) + γ 1n (α k+1 ) K λ,σn α k+1 + γ 2n pn m=1 λ m I(λ m < M/2).(4)
Since the loss function is a convex loss, the optimization in (3) is a convex minimization problem, so many optimization algorithms can be applied. To solve (4) for λ, we adopt a coordinate descent algorithm to update each λ q (q = 1, 2, · · · , p n ) in turn. Specifically, to obtain λ k+1 q , we fix λ k+1 1 , λ k+1 2 , · · · , λ k q+1 , λ k q+2 , · · · , λ k pn and then after simple calculation, the objective function takes the following form,
min λq≥0 1 n n i=1 g(a iq + b iq λ q ) + d q λ q ,(5)where g(λ q ) is equal to l(Y j , n i=1 α k+1 i κ λ,σn (X j , X i )
) as a function of λ q , and a iq , b iq , d q 's are constants. By the construction of κ λ,σn , g(λ q ) is a convex function so each step in the coordinating descent algorithm is a constrained convex minimization problem in a bounded inteval, which is easy to solve. Thus, our algorithm guarantees that the objective function decreases over iterations and converges to a local minimum. We summarize the algorithm in the following table. At the convergence after k iterations, the final prediction function is given as
f λ k+1 (X) = n i=1 α k+1 i κ λ k+1 ,σn (X, X i ).
For classification problem, the classification rule is sign(
f λ k+1 (X)) = sign( n i=1 α k+1 i κ λ k+1 ,σn (X, X i )).
We give details of our algorithm below (Algorithm 1).
Algorithm 1 Algorithm for learning f (X)
Input: Data (X, Y); Regularization parameter γ 1n and γ 2n ; Former updating results,
α k , λ k , f λ k ;
Initialize For regression, λ 0 = 0; For classification, λ 0 = (0, · · · , 1, · · · , 0), where all elements equal to 0, expect the one having largest margin correlation with outcome.
Iterate until convergence (δ = |L n ( λ k+1 , f λ k+1 ) − L n ( λ k , f λ k )| ≤ c 1 , e = λ k+1 − λ k 1 ≤ c 2 ,
where c 1 and c 2 are given cut points):
(i) Update α k+1 for fix λ k , which can be solved explicitly for regression and via fminsearch function for classification.
(ii) Update λ k+1 for fixed α k+1 via coordinate descent algorithm.
(iii) δ = |L n ( λ k+1 , f λ k+1 ) − L n ( λ k , f λ k )| and e = λ k+1 − λ k 1 . Output: α k+1 , λ k+1 , f λ k+1 .
Remark 1. When updating α interatively, for regression, it can be solved in a closed form as
α k+1 = (K λ k ,σn K λ k ,σn + nγ 1n K λ k ,σn ) −1 K λ k ,σn Y .
For classification, we apply one-step Newton method for updating. Tuning parameters in the algorithm are chosen via cross-validation over a grid of 2 −15 , 2 −13 , · · · , 2 −13 , 2 15 . Although the kernel bandwidth, σ n , can also be tuned, to save computation cost, we follow Jaakkola et al. (1999) to set it to be the median value of the paired distances.
Theoretical Properties
In this section, we present some theoretical properties of our proposed method. Since our proposed kernel function is new, we first provide two theorems that describe the properties for the RKHS generated by this kernel function. In the first theorem, we show that this space is dense in L 2 (P ) subspace consisting of all measurable functions that only depend on the feature variables for which λ m = 0 in the kernel function. In the second theorem, we obtain the entropy number for the unit ball in this space. Both theorems are necessary to establish the asymptotic properties of the proposed estimator for f (X) as given in the previous section.
To state our results, we define f 0 (X) as the Bayesian prediction function, which is assumed to be unique. That is, E[l(Y, f )] attains its minimum when f = f 0 . We assume that feature variables X 1 , X 2 , · · · , X q are important in terms that f 0 (X) is only a function of X 1 , X 2 , ..., X q and for any 1 ≤ s ≤ q,
E f 0 (X) − E f 0 (X) X 1 , X 2 , X s−1 , X s+1 · · · , X q 2 > 0.
Finally, we let d 2 (f 0 , H λ,σn ) denote the L 2 (P )-distance between f 0 and the RKHS generated by κ λ,σn .
Theorem 1. For a vector λ n = (λ n1 , ..., λ npn ) with λ nm ≥ 0 for m = 1, ..., p n , the following results hold:
(i) If λ nm > 0 for m = 1, ..., q, i.e., λ n 's that are associated with the important features are strictly positive, then d 2 (f 0 , H λn,σn ) → 0.
(ii) If for some m ≤ q, λ nm = 0, then lim inf d 2 (f 0 , H λn,σn ) > 0.
Note: The Theorem holds for λ whose value depends on n and denoted as λ n .
Proof. To prove (i), we first note that after expansion, κ λn,σn (X, X) is the summation of a number of Gaussian kernels. In particular, one term of this summation is
λ n1 λ n2 · · · λ nq κ σn (X 1 , X 1 )κ σn (X 2 , X 2 ) · · · κ σn (X q , X q ) , where κ σ (x, y) = exp{−(x − y) 2 /σ 2 }. Since λ n1 , .
.., λ nq > 0, the kernel function associated with this term is proportional to the Gaussian kernel in the space of (X 1 , · · · , X q ) with bandwidth σ n for each domain k. Therefore, the closure of the RKHS generated by κ λn,σn includes the RKHS generated by the Gaussian kernel in the space of (X 1 , · · · , X q ). The result in (i) holds since the latter is asymptotically dense in the subspace of L 2 (P ) consisting of any functions depending on (x 1 , ..., x q ).
feature variables except X m . Therefore,
H λn,σn ⊂ {g(X −m ) : g ∈ L 2 (P )} ,
where X −m denotes all the feature variables excluding X m . On the other hand, the projection
of f 0 on the latter space is E[f 0 |X −m ]. Therefore, lim inf d(f 0 , H λn,σn ) ≥ d(f 0 , E[f 0 |X −m ]) > 0
since X m is one important variable for f 0 . We obtain the result.
Our next theorem studies the bracket covering number for a unit ball in H λn,σn . We consider Theorem 2. For a vector λ n = (λ n1 , ..., λ npn ) such that λ nm is uniformly bounded by a constant M for m = 1, ..., q and λ n(q+1) = ... = λ npn = 0, it holds
log N [] ( , B n , · L 2 (P ) ) ≤ Cσ −(1−v/4)q n −v ,
where v is any constant within (0, 2) and C only depends on M and q.
Proof. For any f ∈ B n with form
f (x) = ∞ i=1 α i κ λn,σn (x, x i ),
where x 1 , x 2 , ... are a sequence of given points. Using the expansion of κ λn,σn , we have
f (x) = {k 1 ,...,ks}⊂{1,...,q}∪φ λ nk 1 · · · λ nks ∞ i=1 α i exp − (x ik 1 − x k 1 ) 2 + · · · + (x iks − x ks ) 2 σ 2 n = {k 1 ,.
..,ks}⊂{1,...,q}∪φ λ nk 1 · · · λ nks f k 1 ...ks (x), where x ik and x k are respectively the kth component of x i and x, and
f k 1 ...ks (x) = ∞ i=1 α i λ nk 1 · · · λ nks exp − (x ik 1 − x k 1 ) 2 + · · · + (x iks − x ks ) 2 σ 2 n .
Here, if the index set if empty, then the exponential part in the summation is replaced by 1.
Clearly, if we denote H k 1 ...ks as the reproducing kernel Hilbert space generated by the Gaus-
sian kernel exp {−[( x k 1 − x k 1 ) 2 + · · · + ( x ks − x ks ) 2 ]/σ 2 n } , then f k 1 ...ks (x) ∈ H k 1 .
..ks and moreover,
f 2 H λn,σn = ∞ i=1 ∞ j=1 α i α j κ λn,σn (x i , x j ) = n i=1 n j=1 α i α j {k 1 ,...,ks}⊂{1,...,q}∪φ λ nk 1 · · · λ nks exp − (x ik 1 − x jk 1 ) 2 + · · · + (x iks − x jks ) 2 σ 2 n = {k 1 ,...,ks}⊂{1,...,q}∪φ ∞ i=1 ∞ j=1 α i α j λ nk 1 · · · λ nks exp − (x ik 1 − x jk 1 ) 2 + · · · + (x iks − x jks ) 2 σ 2 n = {k 1 ,...,ks}⊂{1,...,q}∪φ f k 1 ...ks 2 H k 1 ...ks .
Thus, f H λn,σn ≤ 1 implies f k 1 ...ks H k 1 ...ks ≤ 1 for any k 1 , ..., k s .
Consequently, since such f is dense in B n , we conclude
B n ⊆ {k 1 ,...,ks}⊂{1,...,q}∪φ f k 1 ...ks (x) λ nk 1 · · · λ nks : f k 1 ...ks 2 H k 1 ...ks ≤ 1 .
Thus, there exists a constant C only depending on M and q such that
log N [] (2 q M q/2 , B n , · L 2 (P ) ) ≤ {k 1 ,...,ks}⊂{1,...,q}∪φ log N [] ( , {f k 1 ...ks (x), f k 1 ...ks H k 1 ...ks ≤ 1}, · L 2 (P ) )
According to (Steinwart and Scovel (2007)), we know
log N [] ( , {f k 1 ...ks (x), f k 1 ...ks 2 H k 1 ...ks ≤ 1}, · L 2 (P ) ) ≤ Cσ −(1−v/4)s n −v ,
for any constant v ∈ (0, 2) and a constant C only depending on s. Therefore, log N ( , B n , · L 2 (P ) ) ≤ C(M, q) Our next theorem gives the main properties of the estimated prediction function. We show that the resulting prediction function from our method leads to Bayesian risk asymptotically.
Moreover, with probability tending to one, the variable selection based on non-zero λ n 's is oracle as if we knew which variables were important. Recall that ( λ n , f ) is the optimal solution of the objective function L n (λ n , f ) = P n l(Y, f (X)) + γ 1n f 2 H λn,σn + γ 2n P (λ n ),
where P (λ n ) is the truncated Lasso penalty for λ n . Equivalently, if we define for any λ n ,
f λn = arg min f L n (λ n , f ),
which exists due to the convexity of L n (λ n , f ) in f , then λ minimizes L n ( λ n , f λn ) and f = f λn .
For the main theorem, we assume (Y, X) to have a bounded support and need the following conditions.
(C1). The loss function l(y, f ) is convex and is Lipschtisz continuous with respect to f in any bounded set.
(C2). There exit δ > 0 and a constant c 1 > 0 such that
E[l(Y, f (X)) − l(Y, f 0 (X))] ≥ c 1 f (X) − f 0 (X) 2 L 2 (P )
whenever E[l(Y, f (X)) − l(Y, f 0 (X))] is smaller than δ.
(C3). Assume l 2 (Y, f (X)) − l 2 (Y, f 0 (X)) L 2 (P ) ≤ c 2 f (X) − f 0 (X) L 2 (P ) for a constant c 2 , where l 2 (y, x) = ∂l(y, x)/∂x.
(C4). For any λ n = (λ n1 , ..., λ npn ) such that λ nk = 0 for k > q, let Λ max (X −q ) and Λ min (X −q )
be the largest and smallest eigenvalues of the matrix E[K λn (X j , X)K λn (X l , X)|X −q ] where X −q denotes all unimportant variables. We assume that with probability one, there exists one constant c such that Λ max (X −q )/Λ min (X −q ) ≤ cσ −1/2 n and E[Λ min (X −q )κ n (x, X m ) 2 ] ≤ cσ 1/2 n for any m > q.
(C5). Assume log p n = o(n 1−(2+q)α 1 −α 2 −α 3 ). Moreover, we assume σ n = n −α 1 , γ 1n = n −α 2 , γ 2n = n −α 3 , where α k > 0 for k = 1, 2, 3 and they satisfy (i) 1 − (2 + q)α 1 − α 2 > 0 (ii) 0 < α 3 < min 1 4 (1 + α 1 q 2 + α 2 ), 1 − (2 + q)α 1 − α 2 , α 1 2 , α 2 2 .
Conditions (C1)-(C3) give the assumptions for the loss functions. It can be verified that they hold for l(y, f ) = (y − f ) 2 for a continuous y and for l(y, f ) = exp(−yf ) for a binary y.
Condition (C4) implies the equivalence between the Euclidean norm of the coefficients and the reproducing kernel Hilbert space norm, up to a scale proportional to σ −1/2 n . The second half of the condition in (C4) holds automatically if the important variables are independent of the unimportant variable when Λ min (X −q ) does not depend on X −q . We note that such a condition is analogue to the design matrix condition assumed in high dimensional linear model literature.
Finally, condition (C5) allows the dimensionality of the feature variable to be ultra-high and imposes additional constraints for the choices of the bandwidth and two tuning parameters.
Theorem 3. Under Conditions (C1)-(C5), there exists a local minimizer λ n for L n (λ n , f λn )
such that with probability tending to one,
(a) E[l Y, f λn ] converges to E[l Y, f 0 ].
(b) For m = 1, ..., q, λ nm > 0.
(c) For m = q + 1, q + 2, · · · , p n , λ nm = 0. We conducted two simulation studies, one for a regression problem with continuous Y and the other for classification with binary Y . In the first simulation study, we considered a continuous outcome model with total number of p correlated feature variables, which were generated from a multivariate normal distribution, each with mean zero and variance one. Furthermore, X 1 , X 2 , X 3 , X 4 were correlated with corr(X 1 , X 2 ) = 0.4, corr(X 1 , X 3 ) = −0.3, corr(X 2 , X 3 ) = 0.5 and corr(X 3 , X 4 ) = 0.2, while the others were all independent. The outcome variable, Y , was simulated from a linear model
Y = 0.9X 3 5 + 4X 1 X 2 X 3 + 2.3 exp(−X 3 ) + 4X 4 + ,
where ∼ N (0, 1). Thus, X 1 to X 5 were important variables but not any others. In the second simulation study, X's were generated similarly but with some different correlations:
corr(X 1 , X 2 ) = −0.2, corr(X 1 , X 4 ) = 0.2, corr(X 2 , X 3 ) = 0.5, corr(X 3 , X 4 ) = 0.3 and corr(X 3 , X 4 ) = −0.4. The binary outcome, Y , with values −1 and 1, were generated from a Bernoulli distribution with the probability of being one given by
1 + e −0.25+(X 2 −1.1X 3 +0.3X 4 ) 3 −1 ,
so only X 2 to X 4 were important variables. Since many biomedical applications (as well as our application in this work) have small to moderate sample sizes, in both simulation studies, we considered sample size n = 100, 200 and 400 and varied the feature dimension from p = 200, 400
to 1000. Each simulation setting was repeated 500 times.
For each simulated data, we used the proposed method to learn the prediction function.
Initial values, tuning parameters and optimization package used for binary case are chosen as in Remark 1 of Section 2, where 3-fold cross-validation was used for selecting the tuning parameters. The bound of regularized parameter M was chosen to be 10 5 . We also centerized continuous outcome and re-weighted class label controlled to be balanced before iteration to make numerical stable. We reported the true positive rates, true negative rates and the average number of the selected variables for feature selection. We also reported the prediction errors or misclassification rates using a large and independent validation data. For comparison, we compared our proposed method with HSICLasso and SpAM since both methods were able to estimate nonlinear functions in high dimensional settings. In addition, we also compared the performance with LASSO in the first simulation study and l 1 -SVM in the second simulation study, in order to study the impact due to model misspecification.
The results based on 500 replicates are summarized in Tables 1 and 2. From these tables, we observe that for fixed dimension, the performance of our method improves as sample size n becomes large in terms of the improved true positive and true negative rates for feature selection as well as decreasing prediction errors. In almost all cases, our true negative rate is close to 100%, which shows that noise variables can be identified with a very high chance. As expected, the performance deteriorates as the dimensionality increases. Interestingly, our method continues to select only a small number of feature variables. Comparatively, HSICLasso selected many more noise variables and had larger prediction errors, while SpAM also tended to select more features than our method. The performance of these methods become much worse when the feature dimension is 1000. Clearly, LASSO and l 1 -SVM did not yield reasonable variable selection results and their prediction errors are much higher due to model misspecification. We also give boxplots to visualize prediction performance of 500 replications in Figures 1 and 2.
Since Lasso cannot provide stable prediction errors, its prediction errors from many replicates are out of the bound as shown in Figure 1. Figure 1 and 2 further confirm that our method is superior to all other methods, even when the dimension is as large as 1000 and the sample size is as small as n = 100, which is of similar size as our real data analysis example in Section 5.
Application
We applied our proposed method to analyze a gene expression study in Scheetz et al. (2006).
This study analyzed microarrays RNAs of eye disease from 120 male rats, containing the expression levels from about 31, 000 gene probes. One interesting question was to determine which probes might be associated with the expression of gene TRIM32, which had been implicated Note. The plots give the distribution of misclassification rates among four competing methods.
The comparing methods from left to right in each plot are our proposed method, HSICLasso, SpAM and l 1 -SVM. in a number of diverse biological pathways and also known to be one of 14 genes linked to Bardet-Biedl syndrome (Locke et al., 2009). For this purpose, we dichotomized TRIM32 based on whether it was over expressed as compared to a reference sample in the dataset. We further restricted our feature variables to the top 1000 probe sets that were most correlated with TRIM32. All feature variables were on a log-scale and standardized in the analysis. To examine the performance of our method, we randomly divided the whole sample so that 70% were used for training and the rest were used for testing. This random splitting was then repeated 500 times to obtain reliable results. For each training data, we used 3-fold cross validation to choose tuning parameters. We also applied HSICLasso, SpAM and l 1 -SVM for comparison.
The analysis results are shown in Table 3. We notice that our method gives almost the same classification error as l 1 -SVM, which is the smallest on average. However, our method selects a much smaller set of feature variables with an average of 5 variables. SpAM selects 13 variables on average but its classification error is higher. In Table 4, we report the top 10 most-frequent selected features among all 500 replications for each method. We notice that some features such as Fbxo7 and LOC102555217 were selected by at least three methods. In addition, Gene Sirt 3 was identified by all three nonlinear feature selection methods, but not l 1 -SVM, indicating some possible nonlinear relationship between Sirt 3 and TRIM32. In fact, Figure 3 reveals some nonlinear relationship between Sirt 3 and Fbxo7 using 5-Nearest-Neighbors model. Our method also selected some genes that were not identified by any other method. We applied our method to analyze the whole sample and obtained a training error of 21.9% along five 5 genes identified (Fbxo7, Plekha6, Nfatc4, 1375872 and 1388656), which were all selected as the top 10 genes in the previous random splitting experiment.
Discussion
In this work, we have proposed a general framework for nonparametric feature selection for both regression and classification in high dimensional settings. We introduced a novel tensor product kernel for empirical risk minimization. This kernel led to fully nonparametric Note. The numbers are the mean of misclassification rates from 500 replicates. The numbers within parentheses are the median absolute deviations from 500 replicates. "min#" is the minimum number of the selected features, "max#" is the max number of the selected features, and "avg.#" is the average number of the selected features. Note. The numbers within parentheses are the frequencies to be selected in 500 random splittings. The genes also selected by the proposed method are highlighted in boldface. estimation for the prediction function but allowed the importance of each feature to be captured by a non-negative parameter in the kernel function. Our approach is computationally efficient because it iteratively solves a convex optimization problem in a coordinate descent manner.
We have shown that the proposed method has theoretical oracle property for variable selection.
The superior performance of the proposed method was demonstrated via simulation studies and a real data application with a large number of feature variables.
We considered l 2 loss function for regression and exponential loss function for classification as examples. Clearly, the proposed framework applies to feature selection under many different loss functions in machine learning field. Another extension is to incorporate structures of feature variables in constructing the kernel function. For example, in integrative data analysis, feature variables arise from many different domains such as clinical domain, DNA, RNA, imaging and nutrition. It will be interesting to construct a hieachical kernel function which can not only identify feature variables within each domain but also identify important domains at the same time.
Our framework of nonparametric feature selection can be generalized to precision medicine where one of the main goals is to identify predictive biomarkers for treatment response. We can adopt loss functions used for precision medicine in our proposed method to simultaneously accomplish variable selection and discovering optimal individual treatment rules. Extensions to categorical outcomes and multi-stage treatment rule estimation are also possible under our general framework, which can be pursued in future work.
B
n as the unit ball in H λn,σn , i.e., B n ≡ f (x) : f H λn,σn ≤ 1 , Then the -bracket covering number for B n , denoted as N [] ( , B n , · L 2 (P ) ), is defined as the minimal number of pairs [l(x), u(x)] such that any function u(X) − l(X) L 2 (P ) ≤ and any function f in B n is between one pair, i.e., l(x) ≤ f (x) ≤ u(x).
{k 1 ,...,ks}⊂{1,...,q}∪φσ −(1−v/4)s n −v ≤ C(M, q)σ −(1−v/4)q n −vfor a constant C(M, q). We have proved Theorem 2.
The first part of Theorem 3 implies that the loss of the estimated prediction function converges to the Bayes risk. The last two conclusions in Theorem 3 show that the λ nm 's associated with important feature variables should be non-zero, i.e., the estimated function does depend on important variables. More importantly, the proposed method can estimate the predicted function as if we knew which variables are important in the truth. The proof for Theorem 3 is given in the supplementary file. The proof of Theorem 3(a) entails careful examination of the stochastic variability of L n (λ n , f λn ), for which we first establish a preliminary bound for f λn and then appeal to some concentration inequalities for empirical processes with metric entropy as derived from Theorem 2. To prove Theorem 3(b) and (c) in the theorem, we examine the KKT conditions to show that the oracle estimators, i.e., λ nm is known to be zero for m > q, satisfies the KKT conditions with probability tending to one. Again, concentration inequalities for empirical processes are needed in technical arguments in the proof.
Figure 1 :
1Boxplots . The plots give the distribution of prediction errors among four competing methods. The comparing methods from left to right in each plot are our proposed method, HSICLasso, SpAM and Lasso.
Figure 2 :
2Boxplots
Figure 3
3Figure 3: 5-Nearest-Neighbor Plot of Sirt3 versus Fbxo7 in Real Data Study
Table 1 :
1Results from The Simulation Study with Continuous Outcome(a) Summary of Feature Selection Performance
Proposed Method
HSICLasso
SPAM
LASSO
p
n
TPR TNR Avg#
TPR
TNR Avg#
TPR
TNR Avg#
TPR
TNR Avg#
100 100 60.9% 97.3%
5.6
81.5% 78.5%
24.5
99.6% 34.6% 67.1
98.8% 1.3%
98.8
200 71.2% 99.0%
4.5
98.0% 60.4 % 42.5
100.0% 4.4%
95.8
100.0% 0.1%
99.9
400 82.7% 98.4%
5.7
99.6 % 78.0 % 25.8
100.0% 0.3%
99.7
100.0% 0.1%
99.9
200 100 57.2% 98.7%
5.5
75.6% 88.8%
25.6
99.1% 63.0% 77.1
84.0% 52.2% 97.5
200 66.6% 99.5%
4.2
94.0% 75.2%
53.1
100.0% 33.7% 134.1
99.1% 0.0% 198.6
400 78.1% 99.4%
5.0
99.8 % 84.2%
35.8
100.0% 5.4% 189.5
100.0% 0.12% 199.8
400 100 47.3% 99.3%
5.2
68.5% 90.4%
41.5
98.2% 80.8% 80.8
79.4% 76.4% 97.1
200 65.0% 99.7%
4.5
86.3% 89.0%
47.6
100.0% 62.1% 154.6
90.7% 51.4% 196.6
400 73.1% 99.8%
4.4
99.7% 87.6%
54.0
100.0% 34.0% 265.8
99.1% 0.7% 397.3
1000 100 40.7% 99.7%
5.0
56.0% 91.8%
84.5
93.7% 92.2% 82.6
73.6% 90.6% 97.2
200 61.2% 99.9%
4.5
78.2% 98.6%
18.4
99.9% 84.2% 162.0
85.5% 80.7% 196.1
400 70.7% 99.9%
4.0
99.4% 91.0%
94.9
100.0% 68.7% 316.4
94.5% 60.7% 395.6
(b) Summary of Prediction Errors
p
n
Proposed Method HSICLasso
SPAM
LASSO
Table 2 :
2Results from The Simulation Study with Binary Outcome(a) Summary of Feature Selection Performance
Proposed Method
HSICLasso
SPAM
l 1 -SVM
p
n
TPR
TNR Avg#
TPR
TNR Avg#
TPR
TNR Avg#
TPR
TNR Avg#
100 100 74.7% 99.0%
3.3
71.1% 79.4%
22.1
64.5% 89.6%
12.1
76.2% 75.1%
26.5
200 83.9% 99.9%
2.6
80.7% 89.7 % 12.4
53.4% 98.9%
2.6
92.5% 80.4%
21.8
400 86.0% 99.9%
2.6
87.8% 90.3%
12.1
50.6% 99.9%
1.5
98.8% 71.3%
30.8
200 100 70.4% 99.3%
3.5
71.3% 80.1%
41.3
63.6% 91.2%
19.2
71.3% 85.4%
31.0
200 84.1% 99.8%
2.9
78.3% 95.0 % 12.2
54.5% 98.7%
4.1
90.7% 80.2%
41.8
400 87.0% 100.0%
2.7
83.1 % 96.5%
9.3
50.6% 99.9%
1.6
89.3% 73.4%
55.0
400 100 68.5% 99.5%
3.9
70.9% 79.4%
84.0
63.7% 92.8%
30.6
65.9% 86.7%
54.7
200 84.5% 99.9%
3.0
76.9% 95.5 % 20.2
57.7% 98.3%
8.3
87.0% 91.0%
38.1
400 87.0% 100.0%
2.6
79.6 % 98.9%
6.8
51.9% 100.0%
1.8
99.1% 82.3%
73.0
1000 100 61.3% 99.8%
4.1
72.2% 77.4% 227.4
61.0% 95.7 % 45.0
58.4% 90.3 % 98.9
200 86.3% 99.9%
3.3
75.5 % 95.9 % 43.6
54.3% 98.9 % 12.8
79.4% 91.4%
87.4
400 87.7% 100.0%
2.8
73.9 % 99.6 %
6.6
50.0% 100.0%
1.9
96.8% 90.1 % 101.6
(b) Summary of Misclassification Errors
p
n
Proposed Method HSICLasso
SPAM
l 1 -SVM
100 100
0.314 (0.017)
0.345 (0.028) 0.343 (0.018) 0.359 (0.032)
200
0.290 (0.009)
0.307 (0.012) 0.312 (0.002) 0.305 (0.011)
400
0.283 (0.004)
0.292 (0.012) 0.297 (0.002) 0.292 (0.007)
200 100
0.316 (0.019)
0.351 (0.042) 0.344 (0.034) 0.352 (0.031)
200
0.280 (0.008)
0.302 (0.015) 0.302 (0.003) 0.321 (0.028)
400
0.270 (0.004)
0.282 (0.014) 0.297 (0.002) 0.326 (0.025)
400 100
0.331 (0.024)
0.372 (0.047) 0.369 (0.046) 0.390 (0.031)
200
0.286(0.010)
0.319 (0.018) 0.311 (0.003) 0.327 (0.026)
200
0.277 (0.004)
0.288 (0.014) 0.305 (0.001) 0.295 (0.010)
1000 100
0.352 (0.027)
0.397 (0.037) 0.390 (0.036) 0.416 (0.027)
200
0.287 (0.008)
0.335 (0.024) 0.315 (0.003) 0.381 (0.020)
400
0.277 (0.004)
0.294 (0.008) 0.305 (0.001) 0.353 (0.016)
Note. See Table 1.
Table 3 :
3Summary of Feature Selection Results in The Real Data Application min # max # avg # classification errorProposed Method
2
13
5.1
0.286 (0.057)
HSICLasso
1
1000
250.3
0.293 (0.046)
SpAM
1
26
12.3
0.316 (0.057)
l 1 -SVM
7
990
448.7
0.283 (0.058)
Table 4 :
4Top 10 Most Selected Genes for Each Method Based on 500 Random SplittingsProposed Method
HSICLasso
SpAM
l 1 -SVM
Fbxo7
(67.5%)
Ska1
(76.6%)
1388491
(46.2%)
1376747
(99.1%)
Plekha6
(47.3%)
Sirt3
(76.2%)
Fbxo7
(37.8%)
1390538
(98.9%)
LOC102555217
(24.5%)
Ddx58
(76.2%)
Slco1c1
(36.6%)
RragB
(98.6%)
Nfatc4
(22.7%)
1371610
(76.0%)
Stmn1
(35.4%)
Atl1
(97.9%)
1390538
(20%)
LOC100912578
(73.2%)
1373944
(32.4%)
Fbxo7
(97.3%)
1375872
(20%)
Ttll7
(70.4%)
Ufl1
(32.2%)
Plekha6
(95.1%)
RGD1306148
(13.4%)
Decr1
(70.4%)
LOC100912578
(31.0%)
1375872
(94.8%)
Sirt3
(11.6%)
Mff
(68.0%)
LOC100911357
(28.6%)
RGD1306148
(94.1%)
Prpsap2
(11.4%)
Pkn2
(67.0%)
LOC102555217
(26.8%)
Ska1
(93.6%)
1388656
(10.2%)
Taf11
(65.0%)
Sirt3
(22.4%)
LOC102555217
(93.2%)
TNR" is the true negative rate, and "Avg#" is the average number of the selected variables from 500 replicates. TPR" is the true positive rate. In (b), the numbers are the mean squared errors from prediction, and the numbers within parentheses are the medianNote. In (a), "TPR" is the true positive rate, "TNR" is the true negative rate, and "Avg#" is the average number of the selected variables from 500 replicates. In (b), the numbers are the mean squared errors from prediction, and the numbers within parentheses are the median
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| []
|
[
"Quark spectral density and a strongly-coupled QGP",
"Quark spectral density and a strongly-coupled QGP"
]
| [
"Si-Xue Qin \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n",
"Lei Chang \nInstitute of Applied Physics and Computational Mathematics\n100094BeijingChina\n",
"Yu-Xin Liu \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n\nCenter of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator\n730000LanzhouChina\n",
"Craig D Roberts \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n\nPhysics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n"
]
| [
"Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"Institute of Applied Physics and Computational Mathematics\n100094BeijingChina",
"Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"Center of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator\n730000LanzhouChina",
"Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA"
]
| []
| The maximum entropy method is used to compute the dressed-quark spectral density from the self-consistent numerical solution of a rainbow truncation of QCD's gap equation at temperatures above that for which chiral symmetry is restored. In addition to the normal and plasmino modes, the spectral function also exhibits an essentially nonperturbative zero mode for temperatures extending to 1.4 − 1.8-times the critical temperature, Tc. In the neighbourhood of Tc, this long-wavelength mode contains the bulk of the spectral strength and so long as this mode persists, the system may fairly be described as a strongly-coupled state of matter.It is believed that a primordial state of matter has been recreated by the relativistic heavy-ion collider (RHIC)[1]. This substance appears to behave as a nearly-perfect fluid on some domain of temperature, T , above that required for its creation, T c [2]. An ideal fluid has zero shear-viscosity: η = 0, and hence no resistance to the appearance and growth of transverse velocity gradients. A perfect fluid with near-zero viscosity is the best achievable approximation to that ideal. Graphene might provide a room temperature example[3]. From Newton's law for viscous fluid flow; viz., (v/d) = (1/η)(F/A), it is apparent that in near-perfect fluids a macroscopic velocity gradient is achieved from a microscopically small pressure. Strong interactions between particles constituting the fluid are necessary to achieve this outcome. Hence the primordial state of matter is described as a strongly-coupled quark gluon plasma (sQGP).Quantum chromodynamics (QCD) produces the bulk of the mass of normal matter. At T = 0 it is characterised by confinement and dynamical chiral symmetry breaking (DCSB), phenomena that are represented by a range of order parameters which all vanish in the sQGP. Understanding the sQGP therefore requires elucidation of the behaviour and properties of quarks and gluons within this state. Perturbative techniques have been developed for use far above T c ; viz., the hard thermal loop (HTL) expansion[4,5], which has enabled the computation of gluon and quark thermal masses m T ∼ gT and damping rates γ T ∼ g 2 T , with g = g(T ) being the strong running coupling. It also suggests the existence of a collective plasmino or "abnormal" branch to the dressed-quark dispersion relation, which is characterised by antiparticlelike evolution at small momenta[6].Owing to asymptotic freedom, the running coupling in QCD increases as T → T + c . Therefore, a simple interpretation of the HTL results suggests the plasmino should disappear before T c is reached because γ T increases more rapidly than m T and γ T /m T ∼ 1 invalidates a quasi-particle picture. On the other hand, lattice-regularised quenched-QCD suggests that the plasmino branch persists in the vicinity of T c [7]. It is necessary to resolve the active degrees of freedom in the neighbourhood of T c because the spectral properties of the dressed-quark propagator are intimately linked with light-quark confinement[8]and it is the long-range modes which might produce strong correlations.When addressing issues concerning the dressed-quark propagator it is natural to employ the gap equation, which is one of QCD's Dyson-Schwinger equations (DSEs)[9][10][11]. Equations of this type are ubiquitous in physics and in QCD the DSEs are distinguished by their ability to unify the analysis of confinement and DCSB within a single nonperturbative, Poincaré covariant framework. Extensive work within this approach has shown that below T c dressed-gluons and -quarks are confined and chiral symmetry is dynamically broken in the chiral limit[10,11], and that deconfinement and chiral symmetry restoration occur via coincident second-order phase transitions[12][13][14]. | 10.1103/physrevd.84.014017 | [
"https://arxiv.org/pdf/1010.4231v1.pdf"
]
| 119,222,761 | 1010.4231 | d0628865f660dad68ed14022ac60d099e8bb58b2 |
Quark spectral density and a strongly-coupled QGP
20 Oct 2010
Si-Xue Qin
Department of Physics and State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
Lei Chang
Institute of Applied Physics and Computational Mathematics
100094BeijingChina
Yu-Xin Liu
Department of Physics and State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
Center of Theoretical Nuclear Physics
National Laboratory of Heavy Ion Accelerator
730000LanzhouChina
Craig D Roberts
Department of Physics and State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
Physics Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
Quark spectral density and a strongly-coupled QGP
20 Oct 2010(Dated: 17 October 2010)numbers: 1110Wx1238Mh1115Tk2485+p
The maximum entropy method is used to compute the dressed-quark spectral density from the self-consistent numerical solution of a rainbow truncation of QCD's gap equation at temperatures above that for which chiral symmetry is restored. In addition to the normal and plasmino modes, the spectral function also exhibits an essentially nonperturbative zero mode for temperatures extending to 1.4 − 1.8-times the critical temperature, Tc. In the neighbourhood of Tc, this long-wavelength mode contains the bulk of the spectral strength and so long as this mode persists, the system may fairly be described as a strongly-coupled state of matter.It is believed that a primordial state of matter has been recreated by the relativistic heavy-ion collider (RHIC)[1]. This substance appears to behave as a nearly-perfect fluid on some domain of temperature, T , above that required for its creation, T c [2]. An ideal fluid has zero shear-viscosity: η = 0, and hence no resistance to the appearance and growth of transverse velocity gradients. A perfect fluid with near-zero viscosity is the best achievable approximation to that ideal. Graphene might provide a room temperature example[3]. From Newton's law for viscous fluid flow; viz., (v/d) = (1/η)(F/A), it is apparent that in near-perfect fluids a macroscopic velocity gradient is achieved from a microscopically small pressure. Strong interactions between particles constituting the fluid are necessary to achieve this outcome. Hence the primordial state of matter is described as a strongly-coupled quark gluon plasma (sQGP).Quantum chromodynamics (QCD) produces the bulk of the mass of normal matter. At T = 0 it is characterised by confinement and dynamical chiral symmetry breaking (DCSB), phenomena that are represented by a range of order parameters which all vanish in the sQGP. Understanding the sQGP therefore requires elucidation of the behaviour and properties of quarks and gluons within this state. Perturbative techniques have been developed for use far above T c ; viz., the hard thermal loop (HTL) expansion[4,5], which has enabled the computation of gluon and quark thermal masses m T ∼ gT and damping rates γ T ∼ g 2 T , with g = g(T ) being the strong running coupling. It also suggests the existence of a collective plasmino or "abnormal" branch to the dressed-quark dispersion relation, which is characterised by antiparticlelike evolution at small momenta[6].Owing to asymptotic freedom, the running coupling in QCD increases as T → T + c . Therefore, a simple interpretation of the HTL results suggests the plasmino should disappear before T c is reached because γ T increases more rapidly than m T and γ T /m T ∼ 1 invalidates a quasi-particle picture. On the other hand, lattice-regularised quenched-QCD suggests that the plasmino branch persists in the vicinity of T c [7]. It is necessary to resolve the active degrees of freedom in the neighbourhood of T c because the spectral properties of the dressed-quark propagator are intimately linked with light-quark confinement[8]and it is the long-range modes which might produce strong correlations.When addressing issues concerning the dressed-quark propagator it is natural to employ the gap equation, which is one of QCD's Dyson-Schwinger equations (DSEs)[9][10][11]. Equations of this type are ubiquitous in physics and in QCD the DSEs are distinguished by their ability to unify the analysis of confinement and DCSB within a single nonperturbative, Poincaré covariant framework. Extensive work within this approach has shown that below T c dressed-gluons and -quarks are confined and chiral symmetry is dynamically broken in the chiral limit[10,11], and that deconfinement and chiral symmetry restoration occur via coincident second-order phase transitions[12][13][14].
The maximum entropy method is used to compute the dressed-quark spectral density from the self-consistent numerical solution of a rainbow truncation of QCD's gap equation at temperatures above that for which chiral symmetry is restored. In addition to the normal and plasmino modes, the spectral function also exhibits an essentially nonperturbative zero mode for temperatures extending to 1.4 − 1.8-times the critical temperature, Tc. In the neighbourhood of Tc, this long-wavelength mode contains the bulk of the spectral strength and so long as this mode persists, the system may fairly be described as a strongly-coupled state of matter.
PACS numbers: 11.10.Wx, 12.38.Mh, 11.15.Tk,24.85.+p It is believed that a primordial state of matter has been recreated by the relativistic heavy-ion collider (RHIC) [1]. This substance appears to behave as a nearly-perfect fluid on some domain of temperature, T , above that required for its creation, T c [2]. An ideal fluid has zero shear-viscosity: η = 0, and hence no resistance to the appearance and growth of transverse velocity gradients. A perfect fluid with near-zero viscosity is the best achievable approximation to that ideal. Graphene might provide a room temperature example [3]. From Newton's law for viscous fluid flow; viz., (v/d) = (1/η)(F/A), it is apparent that in near-perfect fluids a macroscopic velocity gradient is achieved from a microscopically small pressure. Strong interactions between particles constituting the fluid are necessary to achieve this outcome. Hence the primordial state of matter is described as a strongly-coupled quark gluon plasma (sQGP).
Quantum chromodynamics (QCD) produces the bulk of the mass of normal matter. At T = 0 it is characterised by confinement and dynamical chiral symmetry breaking (DCSB), phenomena that are represented by a range of order parameters which all vanish in the sQGP. Understanding the sQGP therefore requires elucidation of the behaviour and properties of quarks and gluons within this state. Perturbative techniques have been developed for use far above T c ; viz., the hard thermal loop (HTL) expansion [4,5], which has enabled the computation of gluon and quark thermal masses m T ∼ gT and damping rates γ T ∼ g 2 T , with g = g(T ) being the strong running coupling. It also suggests the existence of a collective plasmino or "abnormal" branch to the dressed-quark dispersion relation, which is characterised by antiparticlelike evolution at small momenta [6].
Owing to asymptotic freedom, the running coupling in QCD increases as T → T + c . Therefore, a simple interpretation of the HTL results suggests the plasmino should disappear before T c is reached because γ T increases more rapidly than m T and γ T /m T ∼ 1 invalidates a quasi-particle picture. On the other hand, lattice-regularised quenched-QCD suggests that the plasmino branch persists in the vicinity of T c [7]. It is necessary to resolve the active degrees of freedom in the neighbourhood of T c because the spectral properties of the dressed-quark propagator are intimately linked with light-quark confinement [8] and it is the long-range modes which might produce strong correlations.
When addressing issues concerning the dressed-quark propagator it is natural to employ the gap equation, which is one of QCD's Dyson-Schwinger equations (DSEs) [9][10][11]. Equations of this type are ubiquitous in physics and in QCD the DSEs are distinguished by their ability to unify the analysis of confinement and DCSB within a single nonperturbative, Poincaré covariant framework. Extensive work within this approach has shown that below T c dressed-gluons and -quarks are confined and chiral symmetry is dynamically broken in the chiral limit [10,11], and that deconfinement and chiral symmetry restoration occur via coincident second-order phase transitions [12][13][14]. Herein we use the DSEs to elucidate the active fermion quasiparticles for T T c .
On the domain T > T c the chiral-limit dressed-quark propagator can be written
S(iω n , p) = −i γ · p σ A (ω n , p 2 ) − iγ 4 ω n σ C (ω n , p 2 ) ,(1)
where ω n = (2n + 1)πT , n ∈ Z, is the fermion Matsubara frequency. There is no Dirac-scalar part because chiral symmetry is realised in the Wigner mode. The retarded real-time propagator is found by analytic continuation
S R (ω, p) = S(iω n , p)| iωn→ω+iη +(2)
and from this one obtains the spectral density
ρ(ω, p) = −2ℑ S R (ω, p) .(3)
Equations (2) and (3) are equivalent to the statement:
S(iω n , p) = 1 2π +∞ −∞ dω ′ ρ(ω ′ , p) ω ′ − iω n .(4)
Notably, if one requires a nonnegative spectral density, then Eq. (4) is only valid for T > T c ; i.e., on the deconfined domain [12].
For an unconfined dressed-quark propagator of the form in Eq. (1), the spectral density can be expressed
ρ(ω, p) = ρ + (ω, p 2 )P + + ρ − (ω, p 2 )P − ,(5)
where P ± = (γ 4 ± i γ · u p )/2, u p · p = | p|, are operators which project onto spinors with a positive or negative value for the ratio H := helicity/chirality: H = 1 for a free positive-energy fermion. The spectral density is interesting and expressive because it reveals the manner by which interactions distribute the single-particle spectral strength over momentum modes; and the behaviour at T = 0 shows how that is altered by a heat bath. As with many useful quantities, however, it is nontrivial to evaluate ρ(ω, | p|). Nonetheless, if one has at hand a precise numerical determination of the dressed-quark propagator in Eq. (1), then it is possible to obtain the spectral density via the maximum entropy method (MEM) [15]. We obtain the chiral-limit dressed-quark propagator from the gap equation
S(iω n , p) −1 = Z A 2 i γ · p + Z 2 iγ 4 ω n + Σ ′ (iω n , p) ,(6)Σ ′ (iω n , p) = i γ · p Σ ′ A (iω n , p) + iγ 4 ω n Σ ′ C (iω n , p) ,(7) Σ ′ F (iω n , p) = T l d 3 q (2π) 3 g 2 D µν (ω n −ω l , p− q) × 1 3 tr D P F γ µ S(iω l , q)Γ ν (ω n , ω l , p, q),(8)
where: P A = −Z A 1 i γ · p/ p 2 , P C = −Z 1 iγ 4 /ω n ; D µν is the dressed-gluon propagator; Γ ν is the dressed-quark-gluon vertex; and Z 1,2 , Z A 1,2 are, respectively, the vertex and quark wave function renormalisation constants [12,16].
The gap equation is determined once the kernel is specified. Herein we work at leading-order in the symmetrypreserving truncation scheme of Ref. [17] and employ a phenomenologically-efficacious one-loop renormalizationgroup-improved interaction [16]. Namely:
g 2 D µν (ω n − ω l , p − q)Γ ν (ω n , ω l , p, q) = [P µν T (k Ω )D T (k Ω ) + P µν L (k Ω )D L (k Ω )]γ ν ,(9)
where
k Ω := (Ω, k) = (ω n − ω l , p − q); P µν T (k Ω ) = 0, µ and/or ν = 4 , δ ij − k i k j k 2 , µ, ν = 1, 2, 3 ,(10)with P µν L + P µν T = δ µν − k µ Ω k ν Ω /k 2 Ω ; and D T (k Ω ) = D(k 2 Ω , 0), D L (k Ω ) = D(k 2 Ω , m 2 g ) , (11) D(k 2 Ω , m 2 g ) = 4π 2 D s Ω ω 6 e −sΩ/ω 2 + 8π 2 γ m ln[τ +(1+s Ω /Λ 2 QCD ) 2 ] F (s Ω ) ,(12)
with F (s Ω ) = (1−exp(−s Ω /4m 2 t )/s Ω , s Ω = Ω 2 + k 2 +m 2 g , τ = e 2 − 1, m t = 0.5 GeV, γ m = 12/25, and Λ N f =4 QCD = 0.234 GeV. For pseudoscalar and vector mesons with masses 1 GeV, this interaction provides a uniformly good description of their T = 0 properties [18] when ω = 0.4 GeV, D = (0.96 GeV) 2 . In generalizing to T = 0, we have followed perturbation theory and included a Debye mass in the longitudinal part of the gluon propagator:
m 2 g = (16/5)T 2 .
A justification of the kernel is readily provided. At T = 0 it reproduces the results of perturbative QCD for p 2 2 GeV 2 , so any model-dependence appears only in the infrared, and provides a unified description of lightvector and -pseudoscalar mesons. It also predicts a momentum dependence for the dressed-quark propagator that is qualitatively in agreement with results from numerical simulations of lattice-QCD [19]. The extension to T > 0 preserves the agreement with perturbative QCD at large spacelike momenta. Finally, in employing the kernel we obtain coincident second-order deconfinement and chiral symmetry restoring transitions for two massless flavors at T c = 0.14 GeV, which is 10% smaller than that obtained in Ref. [20].
One insufficiency of the interaction defined above is that D, the parameter expressing its infrared strength, is assumed to be T -independent. Since the nonperturbative part of the interaction should be screened for T T c , we remedy that by writing D → D(T ) with
D(T ) = D , T < T p , a b + ln[T /Λ QCD ] , T ≥ T p ,(13)
where T p is a "persistence" temperature; i.e., a scale below which nonperturbative effects associated with confinement and dynamical chiral symmetry breaking are not materially influenced by thermal screening. Logarithmic screening is typical of QCD and with a = 0.028, b = 0.56 our numerical solutions yield m T = 0.8 T for T 2 T c ; viz., a thermal quark mass consistent with lattice-QCD [7]. We usually take T p = T c herein.
We have computed the spectral density by employing the MEM in connection with the solution of our gap equation. Notably, the behaviour changes qualitatively at T c . Indeed, employing a straightforward generalisation of the inflexion point criterion introduced in Refs. [11,21], one can readily determine that reflection positivity is violated for T < T c . This signals confinement. On the other hand, the spectral function is nonnegative for T > T c .
In Fig. 1 we depict the T > T c -dependence of the locations of the poles in ρ(ω, p = 0); i.e., the thermal masses. We anticipated that spectral strength would be located at ω + ( p = 0) and ω − ( p = 0) = −ω + ( p = 0), corresponding to the fermion's normal and plasmino modes at nonzero temperature. However, it is striking that on a measurable T -domain, spectral strength is also associated with a quasiparticle excitation described by ω 0 ( p = 0) = 0. The appearance of this zero mode is an essentially nonperturbative effect. It is an outgrowth of the evolution in-medium of the gap equation's T = 0 Wigner-mode solution and analogous to this solution's persistence at nonzero current-quark mass in vacuum [22]. The spectral density possesses support associated with this zero mode on T ∈ [0, T s ]. In fact: all the Wignerphase spectral strength is located within this mode at T = 0; it is the dominant contribution for T T c ; and, while it is dominant, it is the system's longest wavelength collective mode. On the other hand, as evident in the right panel of Fig. 1, the mode's spectral strength diminishes uniformly with increasing T and finally vanishes at T s ≈ 1.35 T c . Then, for T > T s the quark's normal and plasmino modes exhibit behavior that is broadly consistent with HTL calculations. This is apparent in Fig. 1 and in a comparison between the upper and lower panels of Fig. 2. Given these observations, we judge that the system should be considered a sQGP for T ∈ [T c , T s ], whereupon it contains a long-range collective mode.
We observe that the HTL approach is perturbative and only applicable for T ≫ T c . Hence it could not have predicted the zero mode's existence. Numerical simulations of lattice-QCD, on the other hand, are nonperturbative. However, it is practically impossible in contemporary computations to exactly preserve chiral symmetry. This can plausibly explain the absence of the zero mode in lattice simulations because any source of explicit chiral symmetry breaking heavily suppresses the mode [22].
The upper-left panel of Fig. 2 depicts the dispersion relations for all dressed-quark modes that exist for T < T s and the behavior of their associated residues. On this sQGP domain the dispersion relations are atypical, with
ω ± (| p|) p∼0 = m T −0.2 | p| , +0.3 | p| ,(14)
ω 0 (| p|)
p∼0 = 0.80 | p| .(15)
Notwithstanding this, all realise free-particle behaviour for | p| ≫ T . We note and emphasise that the usual spectral sum rules are satisfied. Indeed, the identity
Z 2 2 ∞ −∞ dω ′ 2π ω ′ ρ ± (ω ′ , | p|) = Z A 2 | p|(16)
assists in understanding the momentum-dependence in the upper-left panel of Fig. 2. The upper-right panel displays the momentum-dependence of the pole residues: spectral support is located completely in the normal mode for | p| ≫ T ; i.e., on the perturbative domain. The lower panel of Fig. 2 characterises the behaviour of ρ(ω, | p|) at on the high-T domain. In agreement with HTL analysis [5], expected to be valid thereupon, we find only normal and plasmino modes, with
ω ± (| p|) p∼0 = m T ± 0.33| p| .(17)
The plasmino dispersion law exhibits the expected minimum, in this case at | p|/T ≃ 1; and both ω ± (| p|) approach free-particle behaviour at | p| ≫ T , with that of the plasmino approaching this limit from below. The lower-right panel shows that the contribution to the spectral density from the plasmino is strongly damped and contributes little for p > 2T . These results are in-line with those obtained via simulations of lattice-QCD [7]. Equation (13) is a model and it is natural to enquire after its influence. None of our results are qualitatively altered by varying T p but, as one would expect, the width of the sQGP domain expands slowly with increasing T p ; e.g., a 50% increase in T p produces a 30% increase in T s .
Whilst we used the MEM to compute ρ(ω, | p|) from the completely self-consistent numerical solution of a rainbow truncation of QCD's gap equation, the appearance of a third and long-wavelength mode in the dressed-fermion spectral density on a material temperature domain above T c has also been observed in one-loop computations of the fermion self-energy, irrespective of the nature of the boson which dresses the fermion [23]. This mode appears for T > m G , where m G is an infrared mass-scale associated with the boson. In our case, m G = 0.12 GeV. Where a comparison is possible, the dependence of our spectral density on (ω, | p|, T ) is similar to that seen in the oneloop analyses of model gap equations. In analogy with a similar effect in high-temperature superconductivity [25], that behaviour has been attributed to Landau damping, an interference phenomenon known from plasma physics.
Notably, we find that a coupling to meson-like correlations in the gap equation is not a precondition for the appearance of the zero mode because such correlations are absent in the rainbow truncation [13]. On the other hand our gap equation's kernel is characterised by an interaction that features an infrared mass-scale m G T c and supports dynamical chiral symmetry breaking at T = 0. We anticipate that the zero mode will markedly affect colour-singlet vacuum polarisations on T ∈ [T c , T s ]. This could be explicated using the methods of Refs. [26].
Hadron physics experiment has presented us with the fascinating possibility that a near-perfect fluid might have been a key platform in the universe's evolution. We have sought to provide new insights into this possibility by employing the Dyson-Schwinger equations (DSEs), a tool used widely in many branches of physics and which in QCD provides a nonperturbative, continuum tool that unifies the treatment of confinement, dynamical chiral symmetry breaking and observable phenomena.
In pursuing this aim, we solved self-consistently a rainbow truncation of the gap equation for massless 2-flavour QCD, employing a kernel whose temperature-dependence is constrained by T = 0 hadron physics phenomenology and T = 0 lattice-QCD results, and found that chiralsymmetry restoration and deconfinement occur together at a temperature T c = 140 MeV. We subsequently computed the quark spectral density from the gap equation's solution for T > T c using the maximum entropy method, thereby demonstrating its potential when based on accurate input. Remarkably, on a significant domain T /T c ∈ [1, 1 + ∆], ∆ ≃ 0.4 − 0.8, the self-consistently determined Wigner phase supports a zero mode, despite the absence of meson-like correlations in our gap equation's kernel. This mode contains the bulk of the spectral strength for T T c and so long as this mode persists, the system may reasonably be described as a stronglyinteracting state of matter. If, as we argued, the existence of this long-wavelength mode is model-independent, then it is natural to anticipate that a strongly-interacting state of matter should precede the QCD phase transition.
Work supported by: National Natural Science Foundation of China, under contract nos. 10425521, 10705002 and 10935001; Major State Basic Research Development Program contract no. G2007CB815000; and U. S. Department of Energy, Office of Nuclear Physics, contract no. DE-AC02-06CH11357.
FIG. 1 .
1(Color online) Left panel -Temperature-dependence of the dressed-quark thermal masses. Notably, spectral strength is associated with a massless-mode. Right panel -T -dependence of the residue associated with that zero mode.
FIG. 2 .
2(Color online) Upper panel, left -Quasiparticle dispersion relations, ω±,0(p), at temperature T = 1.1Tc. The diagonal dotted line is the free-fermion dispersion relation at this T . Upper panel, right -The residues associated with these quasiparticle poles. Lower panel -Same information for T = 3Tc, whereat the zero mode has vanished.
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[
"Superframes, A Temporal Video Segmentation",
"Superframes, A Temporal Video Segmentation"
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| [
"Hajar Sadeghi Sokeh ",
"Vasileios Argyriou [email protected] ",
"Dorothy Monekosso [email protected] \nFaculty of Science, Engineering, and Computing\nThe Robot Vision Team\nLeeds Beckett University\nUK, Leeds\n",
"Paolo Remagnino [email protected] ",
"\nFaculty of Science, Engineering, and Computing\nThe Robot Vision Team\nKingston University\nLondonUK\n"
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| [
"Faculty of Science, Engineering, and Computing\nThe Robot Vision Team\nLeeds Beckett University\nUK, Leeds",
"Faculty of Science, Engineering, and Computing\nThe Robot Vision Team\nKingston University\nLondonUK"
]
| []
| The goal of video segmentation is to turn video data into a set of concrete motion clusters that can be easily interpreted as building blocks of the video. There are some works on similar topics like detecting scene cuts in a video, but there is few specific research on clustering video data into the desired number of compact segments. It would be more intuitive, and more efficient, to work with perceptually meaningful entity obtained from a low-level grouping process which we call it 'superframe'. This paper presents a new simple and efficient technique to detect superframes of similar content patterns in videos. We calculate the similarity of content-motion to obtain the strength of change between consecutive frames. With the help of existing optical flow technique using deep models, the proposed method is able to perform more accurate motion estimation efficiently. We also propose two criteria for measuring and comparing the performance of different algorithms on various databases. Experimental results on the videos from benchmark databases have demonstrated the effectiveness of the proposed method. | 10.1109/icpr.2018.8545723 | [
"https://arxiv.org/pdf/1804.06642v2.pdf"
]
| 4,937,677 | 1804.06642 | 2e3719ab12abc013750c651bd44bffa2ab216725 |
Superframes, A Temporal Video Segmentation
Hajar Sadeghi Sokeh
Vasileios Argyriou [email protected]
Dorothy Monekosso [email protected]
Faculty of Science, Engineering, and Computing
The Robot Vision Team
Leeds Beckett University
UK, Leeds
Paolo Remagnino [email protected]
Faculty of Science, Engineering, and Computing
The Robot Vision Team
Kingston University
LondonUK
Superframes, A Temporal Video Segmentation
The goal of video segmentation is to turn video data into a set of concrete motion clusters that can be easily interpreted as building blocks of the video. There are some works on similar topics like detecting scene cuts in a video, but there is few specific research on clustering video data into the desired number of compact segments. It would be more intuitive, and more efficient, to work with perceptually meaningful entity obtained from a low-level grouping process which we call it 'superframe'. This paper presents a new simple and efficient technique to detect superframes of similar content patterns in videos. We calculate the similarity of content-motion to obtain the strength of change between consecutive frames. With the help of existing optical flow technique using deep models, the proposed method is able to perform more accurate motion estimation efficiently. We also propose two criteria for measuring and comparing the performance of different algorithms on various databases. Experimental results on the videos from benchmark databases have demonstrated the effectiveness of the proposed method.
I. INTRODUCTION
In computer vision, many existing algorithms on video analysis use fixed number of frames for processing. For example optical flow or motion estimation techniques [1] and human activity recognition [2], [3]. However, it would be more intuitive, and more efficient, to work with perceptually meaningful entity obtained from a low-level grouping process which we call it 'superframe'. Similar to superpixels [4] which are key building blocks of many algorithms and significantly reduce the number of image primitives compared to pixels, superframes also do the same in time domain. They can be used in many different applications such as video segmentation [5], video summarization [6], and video saliency detection [7]. They are also useful in the design of a video database management system [8] that manages a collection of video data and provides content-based access to users [9]. Video data modeling, insertion, storage organization and management, and video data retrieval are among the basic problems that are addressed in a video database management system which can be solved more efficiently by using superframes. By temporal clustering of the video, it is easier to identify the significant segment of the video to achieve better representation, indexing, storage, and retrieval of the video data.
The main goal of this work is an automatic temporal clustering of a video by analyzing the visual content of the video and partitioning it into a set of units called superframes. This process can also be referred to as video data segmentation.
Each segment is defined as a continuous sequence of video frames which have no significant inter-frame difference in terms of their motion contents. Motion is the main criteria we use in this work, therefore we assume all the videos are taken from a single fixed camera.
There is little literature on specifically temporal segmenting of video, and some works on related areas. In this section, we briefly discuss the most relevant techniques to this work: temporal superpixexls, scene cut, and video segmentation.
The main idea of using superpixels as primitives in image processing was introduced by Ren and Malik in [10]. Using superpixels instead of raw pixel data is even beneficial for video applications. Although until recently, superpixel algorithms were mainly on the still images, researchers started to apply them to the video sequences. There are some recent works on using the temporal connection between consecutive frames. Reso et al. [11] proposed a new method for generating superpixels in a video with temporal consistency. Their approach performs an energy-minimizing clustering using a hybrid clustering strategy for a multi-dimensional feature space. This space is separated into a global color subspace and multiple local spatial subspaces. A sliding window consisting multiple consecutive frames is used which is suitable for processing arbitrarily long video sequences.
A generative probabilistic model is proposed by Chang et al. in [12] for temporally consistent superpixels in video sequences which uses past and current frames and scales linearly with video length. They have presented a low-level video representation which is very related to a volumetric voxel [13], but still different in such a way that temporal superpixels are mainly designed for video data, whereas supervoxels are for 3D volumetric data.
Lee et al. in [14] developed a temporal superpixel algorithm based on proximity-weighted patch matching. They estimated superpixel motion vectors by combining the patch matching distances of neighboring superpixels and the target superpixel. Then, they initialized the superpixel label of each pixel in a frame, by mapping the superpixel labels in the previous frames using the motion vectors. Next, they refined the initial superpixels by updating the superpixel labels of boundary pixels iteratively based on a cost function. Finally, they performed some postprocessing including superpixel splitting, merging, and relabeling.
In video indexing, archiving and video communication such [15], proposed a simple technique to detect sudden and unexpected scene change based on only pixel values without any motion estimation. They first screen out many non-scene change frames and then normalize the rest of the frames using a histogram equalization process. A fully convolutional neural network has been used for shot boundary detection task in [16] by Gygli. He considered this work as a binary classification problem to correctly predict if a frame is part of the same shot as the previous frame or not. He also created a new dataset of synthetic data with one million frames to train this network.
Video segmentation aims to group perceptually and visually similar video frames into spatio-temporal regions, a method applicable to many higher-level tasks in computer vision such as activity recognition, object tracking, content-based retrieval, and visual enhancement. In [5], Grundmann et al. presented a technique for spatio-temporal segmentation of long video sequences. Their work is a generalization of Felzenszwalb and Huttenlocher's [17] graph-based image segmentation technique. They use a hierarchical graph-based algorithm to make an initial over-segmentation of the video volume into relatively small space-time regions. They use optical flow as a region descriptor for graph nodes.
Kotsia et al. in [18] proposed using the 3D gradient correlation function operating at the frequency domain for action spotting in a video sequence. They used the 3D Fourier transform which is invariant to spatiotemporal changes and frame recording. In this work, the estimation of motion relies on the detection of the maximum of the cross-correlation function between two blocks of video frames [19].
Similar to all these tasks, in this work we propose a simple and efficient motion-based method to segment a video over time into compact episodes. This grouping leads to an increased computational efficiency for subsequent processing steps and allows for more complex algorithms computationally infeasible on frame level. Our algorithm detects major changes in video streams, such as when an action begins and ends.
The only input to our method is a video and a number which shows the desired number of clusters in that video. The output will be the frame numbers which shows the boundaries between the uniform clusters. Visual features like colors and textures per frame are of low importance compare to motion features. This motivates us to employ motion information to detect big changes over the video frames. We use FlowNet-2 [1], a very accurate optical flow estimation with deep networks, to extract the motion between every subsequent frame. We then use both the average and the histogram of flow over video frames and compare our results over a baseline.
II. THE PROPOSED SUPERFRAME TECHNIQUE
Our superframe segmentation algorithm detects the boundary between temporal clusters in video frames. The superframe algorithm takes the number of desired clusters, K, as input, and generates superframes based on the motion similarity and proximity in the video frames. In other words, the histogram of magnitude is used with the direction of motion per frame and the frame position in the video as features to cluster video frames. Our superframe technique is motivated by SLIC [4], a superpixel algorithm for images, which we generalize it for video data (see Figure 1).
A. The proposed algorithm
Our model segments the video using motion cues. We assume the videos are taken with a fixed camera. Therefore, one can represent the type of motion of the foreground object by computing features from the optical flow. We first apply FlowNet-2 [1] to get the flow per video frame, and then we make a histogram of magnitude (HOM) and direction (HOD) of flow per frame to initialize K cluster centers C 1 , . . . , C K . Therefore each cluster center has 20 feature values as C k = [HOM 1..11 , HOD 1. For a video with N frames, in the initialization step, there are K equally-sized superframes with approximately N/K frames. Since initially, the length of each superframe is S = N/K, like [4], we safely assume that the search area to find the best place for a cluster center is a 2S × 2S area around each cluster center over the location of video frames.
Following the cluster center initialization, a distance measure is considered to specify each frame belongs to which cluster. We use D s as a distance measure defined as follows:
Distance between cluster k and frame i is calculated by: After the initialization of the K cluster centers, we then move each of them to the lowest gradient position in a neighborhood of 3 frames. The neighbourhood of 3 frames is chosen arbitrarily but reasonable. This is done to avoid choosing noisy frames. The gradient for frame i is computed as Eq. 2:
d c = (X k − X i ) 2 , X = (x 1 . . . x f ) d s = f r k − f r i D s = ( dc m ) 2 + ( ds S ) 2(1)G(i) = (X(i + 1) − X(i − 1)) 2(2)
We associate each frame in the video with the nearest cluster center in the search area of 2S × 2S. When all frames are associated with the nearest cluster center, a new cluster center is computed as the average of flow values of all the frames belonging to that cluster. We repeat this process until convergence when the error is less than a threshold.
At the end of this process, we may have few clusters which their length is very short. So, we do a postprocessing step to merge these very small clusters to the closer left or right cluster. The whole algorithm is summarized in Algorithm 1 B. Evaluation criteria for the performance of the algorithm Superpixel algorithms are usually assessed using two error metrics for evaluation of segmentation quality: boundary recall and under-segmentation error. Boundary recall measures how good a superframe segmentation adhere to the groundtruth boundaries. Therefore, higher boundary recall describes better adherence to video segment boundaries. Suppose S = Assign the best matching frames from a 2S neighbourhood around the cluster center according the distance measure (Eq. 1). 8: end for 9:
Compute new cluster centers and error E {L1 distance between previous centers and recomputer centers} 10: until E ≤ threshold 11: Postprocessing to remove very short clusters where T P (S, G) is the number of boundary frames in G for which there is a boundary frame in S in range r and F N (S, G) is the number of boundary pixels in G for which there is no boundary frame in S in range r. In simple words, Boundary Recall, Rec, is the fraction of boundary frames in G which are correctly detected in S. In this work, the range r, as a tolerance parameter, is set to 0.008 times the video length in frames based on the experiments.
Under-segmentation error is another error metric which measures the leakage from superframes with respect to the ground truth segmentation. The lower under-segmentation error, the better match between superframes and the ground truth segments. We define under-segmentation error, U E(S, G), as follows:
U E(S, G) = 1 N L i=1
j|sj ∩gi>β min{|s j ∩ g i |, |s j − g i |} (4) Where N is the number of frames in the video, L is the number of ground truth superframes, |.| indicates the length of a segment in frames and S j − G i = {x ∈ S j |x / ∈ G i }. By doing some experiments, we found the best number of β as an overlap threshold, is 0.25 of each superframe s j .
III. EXPERIMENTS
This proposed algorithm requires two initialization: K (the number of desired clusters) and 'compactness'. In our work, we initialize compactness to 0.1 * K to make the evaluation of results easier.
Experiments are carried out using the 'MAD' 1 database and the 'UMN' databases. The MAD 2 database [20] is recorded using a Microsoft Kinect sensor in an indoor environment with a total of 40 video sequences of 20 subjects. Each subject performs 35 sequential actions twice and each video is about 4000-7000 frames. We also labelled superframes manually for each video. That shows in which frames there is a big change of motion for the frame, which illustrates the video segments with similar motions, superframes. UMN 3 is an unusual crowd activity dataset [21], a staged dataset that depicts sparsely populated areas. Normal crowd activity is observed until a specified point in time where behavior rapidly evolves into an escape scenario where each individual runs out of camera view to simulate panic. The dataset comprises 11 separate video samples that start by depicting normal behavior before changing to abnormal. The panic scenario is filmed in three different locations, one indoors and two outdoors. All footage is recorded at a frame rate of 30 frames per second at a resolution of 640×480 using a static camera.
Each frame is represented by 20 features, 11 values for HOM features, 8 values for HOD features, and one value for the frame location in the video. HOF features provide us with a normalized histogram at each frame of the video.
A dense optical flow [1] is used in this work to detect motion for each video frame. We have employed FlowNet-2 which has an improved quality and speed compared to other optical flow algorithms. With the MAD and UMN databases it took only about 50ms and 40ms respectively per frame to extract the flow using FlowNet-2. Figure 2 illustrates the results of FlowNet-2 on both databases. To compare our results against, we have used 'phase correlation' between two group of frames to determine relative translative movement between them as proposed in [18]. This method relies on estimating the maximum of the phase correlation, which is defined as the inverse Fourier transform Given two space-time volumes v a and v b , we calculate the normalized cross-correlation using Fourier transform and taking the complex conjugate of the second result (shown by * ) and then the location of the peak using equation 6.
F a = F(v a ), F b = F(v b ), R = F a • F * b | F a • F * b |(5)
where • is the element-wise product.
r = F −1 {R}, corr = argmax v {r}(6)
where corr is the correlation between two space-time volumes v a and v b . We calculate this correlation every 2nd frame of each video and then we conclude that there is a big change in the video when the cross-correlation is less than a threshold (see Figure 3). Figure. 4 illustrates how boundary recall and under-segmentation error changes over the number of clusters.
To test the performance of our method on video clustering to superframes, we consider two groups of features: first the averaged value of optical flow over u and v, i.e. the x and y components of the optical flow and second the histogram of magnitude and direction of flow. Figure. 5 shows the boundary recall with respect to the input number of desired superframes for one of the videos (sub01 seq01) in the MAD dataset. For this video, the number of superframes in the ground truth is 71 which for K > 70, HOF features has outstanding improvement on boundary recall over the averaged features.
As discussed before, the output number of clusters is usually less than the input number of desired clusters K. Since in the postprocessing step some of the clusters may get merged with other clusters as their length is too short. Figure. 6 illustrates the relation between the output number of clusters, K and the boundary recall for a video from the MAD database with 71 ground-truth clusters. The ground-truth number of clusters is shown using a red horizontal line in the figure. It is shown that when the output number of clusters for this video is 71 or more, K is more than 115 and the boundary recall is bigger than %91.
The ground-truth and the result boundaries for a video of MAD database are shown in Figure 7. The difference between them is also shown in this figure which is almost zero except for clusters between 50 to 62 between frames 1600 and 2100.
As stated before a combination of under-segmentation error and boundary recall is a good evaluation metric for the algorithm. Under-segmentation error accurately evaluates the quality of superframe segmentation by penalizing superframes overlapping with other superframes. Higher boundary recall also indicates less true boundaries are missed. Figure 8 shows the experimental results for both boundary recall and undersegmentation error which are the average values over all 40 videos in the MAD database. According to this figure, for K more than about 122, the boundary recall is %100 and the under-segmentation error is less than %50.
A quantitative evaluation of all videos in each dataset is done and the results are illustrated in Table I. These results are calculated by averaging over all the videos in each dataset when the number of clusters is about 115. The experimental results show that our model works quite well on both datasets. Regarding this table, the histogram of flow works as a better feature than just averaging the flow or using phase correlation between volumes in the video. There is a %22 improvement of boundary recall for the MAD dataset and %8 for UMN dataset. The boundary recall is quite low for the baseline on UMN dataset, although the segmentation error is still very low. An interesting point in this table is that, there a higher boundary recall for all methods on the MAD dataset, however lower under-segmentation error on UMN datasets.
IV. CONCLUSION
In this paper, we proposed using Histogram of Optical Flow (HOF) features to cluster video frames. These features are independent of the scale of the moving objects. Using these atomic video segments to speed up later-stage visual processing, has been recently used in some works. The number of desired clusters, K, is the input to our algorithm. This parameter is very important and may cause over/undersegmenting the video.
Our superframe method divides a video into superframes by minimizing a cost function which is distance-based and makes each superframe belong to a single motion pattern and not overlap with others. We used FlowNet-2 to extract motion information per video frame to describe video sequences and quantitatively evaluated out method over two databases, MAD and UMU.
One interesting trajectory for the future work is to estimate the saliency score for each of these superframes. This helps us to rank different episodes of a video by their saliency scores and detect the abnormalities in videos as the most salient segment.
This work was supported in part by the European Unions Horizon 2020 Programme for Research and Innovation Actions within IoT (2016): Large Scale Pilots: Wearables for smart ecosystem (MONICA). Fig. 6. Boundary recall and the number of output clusters for a video from MAD database in blue-circle and orange-square, respectively. The number of superframes in the ground truth for this video is 71 which is shown by a red line.
Fig. 1 .
1An example of superframes in a video from MAD database as rate control, scene change detection plays an important role. It can be very challenging when scene changes are very small and sometimes other changes like brightness variation may cause false change detection. Many various methods have been proposed for scene change detection. Yi and Ling in
.8 , f r] T with k = [1, K]. HOM 1..11 indicates 'Histogram of Magnitude' and HOD 1..8 indicates 'Histogram of Direction' for 8 different directions, and f r stands for the frame index in the video.
where x is a feature value and X is a feature vector of 19 values per video frame including 11 values for the histogram of the magnitude of flow and 8 values for the histogram of the direction of flow. We consider frame location separately as f r. S is the interval and m is a measure of compactness of a superframe which regarding the experiment we choose it as 10% of the input number of clusters in this work i.e. 0.1 * K to make the result comparison easier.
{S 1 , . . . , S H } be a superframe segmentation with H number of clusters and G = {G 1 , . . . , G L } be a ground-truth segmentation with L number of clusters. Then, boundary recall Rec(S, G) is defined as follows: Rec(S, G) = T P (S, G) T P (S, G) + F N (S, G)
Fig. 2 .
2Examples of optical flow results using FlowNet-2 on a frame of a video from MAD and UMN databases.
Fig. 3 .
3The relation between the threshold and the number of clusters in one of the videos of MAD database in PC-clustering technique[18].
Fig. 4 .
4Boundary recall and under-segmentation error based on the number of clusters for one of the videos of MAD database for the proposed algorithm.
Fig. 5 .
5Comparison of results when using averaged flow over components of flow and histogram of flow. The former is shown in pink-triangular and the latter is shown in green-circles.
Fig. 7 .
7Comparison of results when using averaged magnitude/direction and HOF. The number of ground-truth clusters for this video is 71 clusters and the boundary recall is 0.91. The averaged flow results are shown in pinktriangular and the HOF is shown in green-circles.
Fig. 8 .
8Quantitative evaluation of our algorithm by averaging over 40 videos in MAD dataset.
TABLE I QUANTITATIVE
IEVALUATION FOR VIDEO CLUSTERINGMAD database
UMN database
UE
Rec
UE
Rec
PC-clustering [18]
0.54
0.64
0.27
0.25
Averaged flow
0.52
0.77
0.19
0.63
Histogram of flow (proposed)
0.31
0.99
0.17
0.71
Multimodal Action Database 2 Videos available from www.humansensing.cs.cmu.edu/mad/
Videos available from mha.cs.umn.edu/projevents.shtml of the normalized cross-spectrum between two space-time volumes in the video sequence. We call this phase-correlation technique as PC-clustering in this work. We sampled every 2nd frame in each video and chose a sub-section of each frame. We use 240 × 240 pixels in the middle of each frame to carry out the phase correlation. Therefore, each space-time volume is considered to be 240 × 240 × 30, in which 30 is the number of frames and shows the temporal length of each volume.
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[
"Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -Even-mode perturbations",
"Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -Even-mode perturbations"
]
| [
"Kouji Nakamura \nGravitational-Wave Science Project\nNational Astronomical Observatory of Japan\n2-21-1181-8588, 30/5/2022OsawaMitaka, TokyoJapan\n"
]
| [
"Gravitational-Wave Science Project\nNational Astronomical Observatory of Japan\n2-21-1181-8588, 30/5/2022OsawaMitaka, TokyoJapan"
]
| [
"Prog. Theor. Exp. Phys"
]
| This is the Part II paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gaugeinvariant perturbation theory and the proposal on the gauge-invariant treatments for l = 0, 1 mode perturbations on the Schwarzschild background spacetime in the Part I paper [K. Nakamura, arXiv:2110.13508 [gr-qc]], we examine the linearized Einstein equations for even-mode perturbations. We discuss the strategy to solve the linearized Einstein equations for these evenmode perturbations including l = 0, 1 modes. Furthermore, we explicitly derive the l = 0, 1 mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for l = 0 mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the l = 0, 1-mode problem is physically reasonable due to the realization of the additional Schwarzschild mass parameter perturbation and the Kerr parameter perturbation in the Part I paper. † (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. works, perturbations on the Schwarzschild spacetime are decomposed through the spherical harmonics Y lm because of the spherical symmetry of the background spacetime, and l = 0 and l = 1 modes should be separately treated. Furthermore, "gauge-invariant" treatments for l = 0 and l = 1 even-modes were unknown.Owing to this situation, in the previous papers[29,30], we proposed the strategy of the gaugeinvariant treatments of these l = 0, 1 mode perturbations, which is declared as Proposal 2.1 in Sec. 2 of this paper below. One of important premises of our gauge-invariant perturbations is the distinction of the first-kind gauge and the second-kind gauge. The first-kind gauge is the choice of the coordinate system on the single manifold and we often use this first-kind gauge when we predict or interpret the measurement results of experiments and observation. On the other hand, the secondkind gauge is the choice of the point-identifications between the points on the physical spacetime M ε and the background spacetime M . This second-kind gauge have nothing to do with our physical spacetime M . The proposal in the Part I paper [30] is a part of our developments of the general formulation of a higher-order gauge-invariant perturbation theory on a generic background spacetime toward unambiguous sophisticated nonlinear general-relativistic perturbation theories[31][32][33][34][35][36]. This general formulation of the higher-order gauge-invariant perturbation theory was applied to cosmological perturbations[37][38][39][40][41][42][43][44]. Even in cosmological perturbation theories, the same problem as the above l = 0, 1-mode problem exists as gauge-invariant treatments of homogeneous modes of perturbations. In this sense, we can expect that the proposal in the previous paper [30] will be a clue to the same problem in gauge-invariant perturbation theory on the generic background spacetime.In addition to the proposal of the gauge-invariant treatments of l = 0, 1-mode perturbations on the Schwarzschild background spacetime, in the previous Part I paper, we also derived the linearized Einstein equations in a gauge-invariant manner following Proposal 2.1. From the parity of perturbations, we can classify the perturbations on the spherically symmetric background spacetime into even-and odd-mode perturbations. In the Part I paper [30], we also gave a strategy to solve the oddmode perturbations including l = 0, 1 modes. Furthermore, we also derived the explicit solutions for the l = 0, 1 odd-mode perturbations to the linearized Einstein equations following Proposal 2.1. This paper is the Part II paper of the series of papers on the application of our gauge-invariant perturbation theory to that on the Schwarzschild background spacetime. This series of papers is the full paper version of our short paper[29]. In this Part II paper, we discuss a strategy to solve the linearized Einstein equation for even-mode perturbations including l = 0, 1 mode perturbations. We also derive the explicit solutions to the l = 0, 1 mode perturbations with generic linear-order energy-momentum tensor. As the result, we show that the additional Schwarzschild mass parameter perturbation in the vacuum case. This is the realization of the Birkhoff theorem at the linearperturbation level in a gauge-invariant manner. This result is physically reasonable, and it also implies that Proposal 2.1 is also physically reasonable. The other supports for Proposal 2.1 are also given by the realization of exact solutions with matter fields which will be discussed in the Part III paper[46]. Furthermore, brief discussions on the extension to the higher-order perturbations are given in the short paper[45].The organization of this Part II paper is as follows. In Sec. 2, after briefly review the framework of the gauge-invariant perturbation theory, we summarize our proposal in Refs.[29,30]. Then, we also summarize the linearized even-mode Einstein equation on the Schwarzschild background spacetime which was derived in Ref.[30] following Proposal 2.1. In Sec. 3, following Proposal 2.1, we discuss a strategy to solve these even-mode Einstein equations including l = 0, 1 mode perturbations. In Sec. 4, we derive the explicit solutions to the linearized Einstein equation for the l = 0 mode 2/33 | null | [
"https://arxiv.org/pdf/2110.13512v4.pdf"
]
| 239,885,796 | 2110.13512 | 9fd4d0192f504698c46547d597701e374654aef9 |
Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -Even-mode perturbations
May 2022
Kouji Nakamura
Gravitational-Wave Science Project
National Astronomical Observatory of Japan
2-21-1181-8588, 30/5/2022OsawaMitaka, TokyoJapan
Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -Even-mode perturbations
Prog. Theor. Exp. Phys
20150May 202210.1093/ptep/0000000000
This is the Part II paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gaugeinvariant perturbation theory and the proposal on the gauge-invariant treatments for l = 0, 1 mode perturbations on the Schwarzschild background spacetime in the Part I paper [K. Nakamura, arXiv:2110.13508 [gr-qc]], we examine the linearized Einstein equations for even-mode perturbations. We discuss the strategy to solve the linearized Einstein equations for these evenmode perturbations including l = 0, 1 modes. Furthermore, we explicitly derive the l = 0, 1 mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for l = 0 mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the l = 0, 1-mode problem is physically reasonable due to the realization of the additional Schwarzschild mass parameter perturbation and the Kerr parameter perturbation in the Part I paper. † (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. works, perturbations on the Schwarzschild spacetime are decomposed through the spherical harmonics Y lm because of the spherical symmetry of the background spacetime, and l = 0 and l = 1 modes should be separately treated. Furthermore, "gauge-invariant" treatments for l = 0 and l = 1 even-modes were unknown.Owing to this situation, in the previous papers[29,30], we proposed the strategy of the gaugeinvariant treatments of these l = 0, 1 mode perturbations, which is declared as Proposal 2.1 in Sec. 2 of this paper below. One of important premises of our gauge-invariant perturbations is the distinction of the first-kind gauge and the second-kind gauge. The first-kind gauge is the choice of the coordinate system on the single manifold and we often use this first-kind gauge when we predict or interpret the measurement results of experiments and observation. On the other hand, the secondkind gauge is the choice of the point-identifications between the points on the physical spacetime M ε and the background spacetime M . This second-kind gauge have nothing to do with our physical spacetime M . The proposal in the Part I paper [30] is a part of our developments of the general formulation of a higher-order gauge-invariant perturbation theory on a generic background spacetime toward unambiguous sophisticated nonlinear general-relativistic perturbation theories[31][32][33][34][35][36]. This general formulation of the higher-order gauge-invariant perturbation theory was applied to cosmological perturbations[37][38][39][40][41][42][43][44]. Even in cosmological perturbation theories, the same problem as the above l = 0, 1-mode problem exists as gauge-invariant treatments of homogeneous modes of perturbations. In this sense, we can expect that the proposal in the previous paper [30] will be a clue to the same problem in gauge-invariant perturbation theory on the generic background spacetime.In addition to the proposal of the gauge-invariant treatments of l = 0, 1-mode perturbations on the Schwarzschild background spacetime, in the previous Part I paper, we also derived the linearized Einstein equations in a gauge-invariant manner following Proposal 2.1. From the parity of perturbations, we can classify the perturbations on the spherically symmetric background spacetime into even-and odd-mode perturbations. In the Part I paper [30], we also gave a strategy to solve the oddmode perturbations including l = 0, 1 modes. Furthermore, we also derived the explicit solutions for the l = 0, 1 odd-mode perturbations to the linearized Einstein equations following Proposal 2.1. This paper is the Part II paper of the series of papers on the application of our gauge-invariant perturbation theory to that on the Schwarzschild background spacetime. This series of papers is the full paper version of our short paper[29]. In this Part II paper, we discuss a strategy to solve the linearized Einstein equation for even-mode perturbations including l = 0, 1 mode perturbations. We also derive the explicit solutions to the l = 0, 1 mode perturbations with generic linear-order energy-momentum tensor. As the result, we show that the additional Schwarzschild mass parameter perturbation in the vacuum case. This is the realization of the Birkhoff theorem at the linearperturbation level in a gauge-invariant manner. This result is physically reasonable, and it also implies that Proposal 2.1 is also physically reasonable. The other supports for Proposal 2.1 are also given by the realization of exact solutions with matter fields which will be discussed in the Part III paper[46]. Furthermore, brief discussions on the extension to the higher-order perturbations are given in the short paper[45].The organization of this Part II paper is as follows. In Sec. 2, after briefly review the framework of the gauge-invariant perturbation theory, we summarize our proposal in Refs.[29,30]. Then, we also summarize the linearized even-mode Einstein equation on the Schwarzschild background spacetime which was derived in Ref.[30] following Proposal 2.1. In Sec. 3, following Proposal 2.1, we discuss a strategy to solve these even-mode Einstein equations including l = 0, 1 mode perturbations. In Sec. 4, we derive the explicit solutions to the linearized Einstein equation for the l = 0 mode 2/33
Introduction
Gravitational-wave observations are now carrying out through the ground-based detectors [1][2][3][4]. Furthermore, the projects of future ground-based gravitational-wave detectors [5,6] are also progressing to achieve more sensitive detectors. In addition to these ground-based detectors, some projects of space gravitational-wave antenna are also progressing [7][8][9][10]. Among them, the Extreme-Mass-Ratio-Inspiral (EMRI), which is a source of gravitational waves from the motion of a stellar mass object around a supermassive black hole, is a promising target of the Laser Interferometer Space Antenna [7]. To describe the gravitational wave from EMRIs, black hole perturbations are used [11]. Furthermore, the sophistication of higher-order black hole perturbation theories is required to support these gravitational-wave physics as a precise science. The motivation of this paper is in such theoretical sophistications of black hole perturbation theories toward higher-order perturbations for wide physical situations.
Although realistic black holes have their angular momentum and we have to consider the perturbation theory of a Kerr black hole for direct applications to the EMRI, we may say that further sophistications are possible even in perturbation theories on the Schwarzschild background spacetime. From the pioneering works by Regge and Wheeler [12] and Zerilli [13][14][15], there have been many studies on the perturbations in the Schwarzschild background spacetime [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In these perturbations in both the vacuum and the non-vacuum cases. In Sec. 5, we also derive the explicit solutions to the linearized Einstein equation for the l = 1 mode perturbations in both the vacuum and the non-vacuum cases. The final section (Sec. 6) is devoted to our summary and discussions.
We use the notation used in the previous papers [29,30,45] and the unit G = c = 1, where G is Newton's constant of gravitation and c is the velocity of light.
Brief review of the general-relativistic gauge-invariant perturbation theory
In this section, we review the premise of the series of our papers [29,30,46] and this paper. In Sec. 2.1, we briefly review the framework of the gauge-invariant perturbation theory [31,32]. This is an important premise of the series of our papers. In Sec. 2.2, we review the linear perturbation on spherically symmetric background spacetimes which includes our proposal in Ref. [29,30]. In Sec. 2.3, we review the linearized Einstein equations for even-mode perturbations on the Schwarzschild background spacetime which are to be solved in this paper.
General framework of gauge-invariant perturbation theory
In any perturbation theory, we always treat two spacetime manifolds. One is the physical spacetime (M ph ,ḡ ab ), which is identified with our nature itself, and we want to describe this spacetime (M ph ,ḡ ab ) by perturbations. The other is the background spacetime (M , g ab ), which is prepared as a reference by hand. Note that these two spacetimes are distinct. Furthermore, in any perturbation theory, we always write equations for the perturbation of the variable Q as follows:
Q("p") = Q 0 (p) + δ Q(p).
(2.1) Equation (2.1) gives a relation between variables on different manifolds. Actually, Q("p") in Eq. (2.1) is a variable on M ε = M ph , whereas Q 0 (p) and δ Q(p) are variables on M . Because we regard Eq. (2.1) as a field equation, Eq. (2.1) includes an implicit assumption of the existence of a point identification map X ε : M → M ε : p ∈ M → "p" ∈ M ε . This identification map is a gauge choice in general-relativistic perturbation theories. This is the notion of the second-kind gauge pointed out by Sachs [47]. Note that this second-kind gauge is a different notion from the degree of freedom of the coordinate transformation on a single manifold, which is called the first-kind gauge [30,43,44].
To compare with the variable Q on M ε and its background value Q 0 on M , we use the pull-back X * ε of the identification map X ε : M → M ε and we evaluate the pulled-back variable X * ε Q on the background spacetime M . Furthermore, in perturbation theories, we expand the pull-back operation X * ε and the variable Q with respect to the infinitesimal parameter ε for the perturbation as
X * ε Q = Q 0 + ε (1) X Q + O(ε 2 ). (2.2)
Eq. (2.2) are evaluated on the background spacetime M . When we have two different gauge choices X ε and Y ε , we can consider the gauge-transformation, which is the change of the point-identification X ε → Y ε . This gauge-transformation is given by the diffeomorphism Φ ε := (X ε ) −1 • Y ε : M → M . Actually, the diffeomorphism Φ ε induces a pull-back from the representation X * ε Q ε to the representation Y * ε Q ε as Y * ε Q ε = Φ * ε X * ε Q ε . From general arguments of the Taylor expansion [48], the pull-back Φ * ε is expanded as where ξ a (1) is the generator of Φ ε . From Eqs. (2.2) and (2.3), the gauge-transformation for the firstorder perturbation (1) Q is given by
Y * ε Q ε = X * ε Q ε + ε£ ξ (1) X * ε Q ε + O(ε 2 ),(2.(1) Y Q − (1) X Q = £ ξ (1) Q 0 .
(2. 4) We also employ the order by order gauge invariance as a concept of gauge invariance [41]. We call the kth-order perturbation
X Q = (k) Y Q (2.5)
for any gauge choice X ε and Y ε . Based on the above setup, we proposed a procedure to construct gauge-invariant variables of higher-order perturbations [31,32]. First, we expand the metric on the physical spacetime M ε , which was pulled back to the background spacetime M through a gauge choice X ε as
X * εḡab = g ab + ε X h ab + O(ε 2 ). (2.6)
Although the expression (2.6) depends entirely on the gauge choice X ε , henceforth, we do not explicitly express the index of the gauge choice X ε in the expression if there is no possibility of confusion. The important premise of our proposal was the following conjecture [31,32] for the linear metric perturbation h ab :
Conjecture 2.1. If the gauge-transformation rule for a tensor field h ab is given by Y h ab − X h ab = £ ξ (1) g ab with the background metric g ab , there then exist a tensor field F ab and a vector field Y a such that h ab is decomposed as h ab =:
F ab + £ Y g ab , where F ab and Y a are transformed into Y F ab − X F ab = 0 and Y Y a − X Y a = ξ a
(1) under the gauge transformation, respectively. We call F ab and Y a as the gauge-invariant and gauge-variant parts of h ab , respectively. The proof of Conjecture 2.1 is highly nontrivial [33], and it was found that the gauge-invariant variables are essentially non-local. Despite this non-triviality, once we accept Conjecture 2.1, we can decompose the linear perturbation of an arbitrary tensor field (1) X Q, whose gauge-transformation is given by Eq. (2.4), through the gauge-variant part Y a of the metric perturbation in Conjecture 2.1 as (1)
X Q = (1) Q + £ X Y Q 0 . (2.7)
As examples, the linearized Einstein tensor
(1) X G b a and the linear perturbation of the energymomentum tensor
(1) X T b
a are also decomposed as
(1) X G b a = (1) G b a [F ] + £ X Y G b a ,(1)X T b a = (1) T b a + £ X Y T b a ,(2.8)
where G ab and T ab are the background values of the Einstein tensor and the energy-momentum tensor, respectively. The gauge-invariant part (1) G b a of the linear-order perturbation of the Einstein tensor is given by
(1) G b a [A] := (1) Σ b a [A] − 1 2 δ b a (1) Σ c c [A] , (2.9) (1) Σ b a [A] := −2∇ [a H bd d] [A] − A cb R ac , H c ba [A] := ∇ (a A c b) − 1 2 ∇ c A ab ,(2.10)
where A ab is an arbitrary tensor field of the second rank. Then, using the background Einstein equation G b a = 8πT b a , the linearized Einstein equation
(1)
X G ab = 8π
(1) X T ab is automatically given 4/33 in the gauge-invariant form
(1) G b a [F ] = 8π (1) T b a (2.11)
even if the background Einstein equation is nontrivial. We also note that, in the case of a vacuum background case, i.e., G b a = 8πT b a = 0, Eq. (2.8) shows that the linear perturbations of the Einstein tensor and the energy-momentum tensor is automatically gauge-invariant of the second kind.
We can also derive the perturbation of the divergence of∇ aT a b of the second-rank tensorT a b on (M ph ,ḡ ab ). Through the gauge choice X ε ,T a b is pulled back to X * εT a b on the background spacetime (M , g ab ), and the covariant derivative operator∇ a on (M ph ,ḡ ab ) is pulled back to a derivative operator∇ a (= X * ε∇ a (X −1 ε ) * ) on (M , g ab ). Note that the derivative∇ a is the covariant derivative associated with the metric X εḡab , whereas the derivative ∇ a on the background spacetime (M , g ab ) is the covariant derivative associated with the background metric g ab . Bearing in mind the difference in these derivatives, the first-order perturbation of∇ aT a b is given by
(1) ∇ aT a b = ∇ a (1) T a b + H a ca [F ] T c b − H c ba [F ] T a c + £ Y ∇ a T a b .
(2.12)
The derivation of the formula (2.12) is given in Ref. [32]. If the tensor fieldT a b is the Einstein tensor G b a , Eq. (2.12) yields the linear-order perturbation of the Bianchi identity ∇ a
(1)
G a b [F ] + H a ca [F ] G c b − H c ba [F ] G a c = 0 (2.13)
and if the background Einstein tensor vanishes G b a = 0, we obtain the identity ∇ a
(1) G a b [F ] = 0.
(2.14)
By contrast, if the tensor fieldT a b is the energy-momentum tensor, Eq. (2.12) yields the continuity equation of the energy-momentum tensor
∇ a (1) T a b + H a ca [F ] T c b − H c ba [F ] T a c = 0, (2.15)
where we used the background continuity equation ∇ a T a b = 0. If the background spacetime is vacuum T ab = 0, Eq. (2.15) yields a linear perturbation of the energy-momentum tensor given by
∇ a (1) T a b = 0. (2.16)
We should note that the decomposition of the metric perturbation h ab into its gauge-invariant part F ab and into its gauge-variant part Y a is not unique [41,43,44]. As explained in the Part I paper [30], for example, the gauge-invariant part F ab has six components and we can create the gauge-invariant vector field Z a through these components of the gauge-invariant metric perturbation F ab such that the gauge-transformation of the vector field Z a is given by Y Z a − X Z a = 0. Using this gauge-invariant vector field Z a , the original metric perturbation can be expressed as follows:
h ab = F ab − £ Z g ab + £ Z+Y g ab =: H ab + £ X g ab .
(2.17)
The tensor field H ab := F ab − £ Z g ab is also regarded as the gauge-invariant part of the perturbation h ab because Y H ab − X H ab = 0. Similarly, the vector field X a := Z a + Y a is also regarded as the gauge-variant part of the perturbation h ab because Y X a − X X a = ξ a (1) . This non-uniqueness appears in the solutions derived in Secs. 4 and 5, as in the case of the l = 1 odd-mode perturbative solutions in the Part I paper [30]. These non-uniqueness of gauge-invariant variable can be regarded as the firstkind gauge as explained in Part I paper [30], i.e., degree of freedom of the choice of the coordinate system on the physical spacetime M ε . Since we often use the first-kind gauge when we predict and interpret the measurement results of observations and experiments, we should regard that this nonuniqueness of gauge-invariant variable of the second kind may have some physical meaning [30]. 5/33
Linear perturbations on spherically symmetric background
Here, we consider the 2+2 formulation of the perturbation of a spherically symmetric background spacetime, which originally proposed by Gerlach and Sengupta [20][21][22][23]. Spherically symmetric spacetimes are characterized by the direct product M = M 1 × S 2 and their metric is g ab = y ab + r 2 γ ab , (2.18)
y ab = y AB (dx A ) a (dx B ) b , γ ab = γ pq (dx p ) a (dx q ) b ,(2.19)
where x A = (t, r), x p = (θ , φ ), and γ pq is the metric on the unit sphere. In the Schwarzschild spacetime, the metric (2.18) is given by
y ab = − f (dt) a (dt) b + f −1 (dr) a (dr) b , f = 1 − 2M r , (2.20) γ ab = (dθ ) a (dθ ) b + sin 2 θ (dφ ) a (dφ ) b = θ a θ b + φ a φ b , (2.21) θ a = (dθ ) a , φ a = sin θ (dφ ) a . (2.22)
On this background spacetime (M , g ab ), the components of the metric perturbation are given by
h ab = h AB (dx A ) a (dx B ) b + 2h Ap (dx A ) (a (dx p ) b) + h pq (dx p ) a (dx q ) b . (2.23)
Here, we note that the components h AB , h Ap , and h pq are regarded as components of scalar, vector, and tensor on S 2 , respectively. In many literatures, these components are decomposed through the decomposition [49][50][51] using the spherical harmonics S = Y lm as follows:
h AB = ∑ l,mh AB S, (2.24) h Ap = r ∑ l,m h (e1)ADp S +h (o1)A ε pqD q S , (2.25) h pq = r 2 ∑ l,m 1 2 γ pqh(e0) S +h (e2) D pDq − 1 2 γ pqD rD r S + 2h (o2) ε r(pDq)D r S , (2.26)
whereD p is the covariant derivative associated with the metric γ pq on S 2 ,D p = γ pqD q , ε pq = ε [pq] = 2θ [p φ q] is the totally antisymmetric tensor on S 2 .
If we employ the decomposition (2.24)-(2.26) with S = Y lm to the metric perturbation h ab , special treatments for l = 0, 1 modes are required [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. This is due to the fact that the set of harmonic functions S,D p S, ε pqD q S, 1 2 γ pq S, D pDq − 1 2 γ pq S, 2ε r(pDq)D r S (2.27) loses its linear-independence for l = 0, 1 modes. Actually, the inverse-relation of the decomposition formulae (2.24)-(2.26) requires the Green functions of the derivative operators∆ :=D rD r and∆ + 2 :=D rD r + 2. Since the eigen modes of these operators are l = 0 and l = 1, respectively. This is the reason why the special treatments for these modes are required. However, these special treatments become an obstacle when we develop higher-order perturbation theory [52].
6/33
To resolve this l = 0, 1 mode problem, in Part I paper [29,30], we chose the scalar function S as
S = S δ = Y lm for l ≥ 2; k (∆+2)m for l = 1; k (∆) for l = 0.
(2. 28) and use the decomposition formulae (2.24)-(2.26), where the functions k (∆) and k (∆+2) satisfy the equation∆
k (∆) = 0, ∆ + 2 k (∆+2) = 0, (2.29)
respectively. As shown in Part I paper [30], the set of harmonic functions (2.27) becomes the linearindependent set including l = 0, 1 modes if we employ
k (∆) = 1 + δ ln 1 − cosθ 1 + cosθ 1/2 , δ ∈ R, (2.30) k (∆+2,m=0) = cos θ + δ 1 2 cos θ ln 1 + cosθ 1 − cosθ − 1 , δ ∈ R, (2.31) k (∆+2,m=±1) = sin θ + δ + 1 2 sin θ ln 1 + cosθ 1 − cosθ + cotθ e ±iφ .
(2.32)
These choices guarantee the one-to-one correspondence between the components {h AB , h Ap , h pq } and the mode coefficients After deriving the field equations such as linearized Einstein equations by using the harmonic functions S δ , we choose δ = 0 as regular boundary condition for solutions when we solve these field equations.
{h AB ,h (e1)A ,h (o1)A ,h (e0) ,h (e2) ,h (o2) } with
As shown in the Part I paper [30], once we accept Proposal 2.1, the Conjecture 2.1 becomes the following statement:
Theorem 2.1. If the gauge-transformation rule for a tensor field h ab is given by Y h ab − X h ab = £ ξ (1) g ab with the background metric g ab with spherically symmetry, there then exist a tensor field F ab and a vector field Y a such that h ab is decomposed as h ab =:
F ab + £ Y g ab , where F ab and Y a are transformed into Y F ab − X F ab = 0 and Y Y a − X Y a = ξ a
(1) under the gauge transformation, respectively.
Furthermore, including l = 0, 1 modes, the components of the gauge-invariant part F ab of the metric perturbation h ab is given by
F AB = ∑ l,mF AB S δ ,
(2.33)
F Ap = r ∑ l,mF A ε pqD q S δ ,D p F Ap = 0,(2.
Even-mode linearized Einstein equations
Since the odd-mode perturbations are discussed in Part I paper [30], we consider the linearized even-mode Einstein equations on the Schwarzschild background spacetime in this paper. The Schwarzschild spacetime is vacuum solution to the Einstein equation G b a = 0 = T b a . Since we proved Theorem 2.1 on the spherically symmetric background spacetime, the linearized Einstein equation is given in the following gauge-invariant form as Eq. (2.11). To evaluate the Einstein equation (2.11) through the mode-by-mode analysis including l = 0, 1, we also consider the modedecomposition of the gauge-invariant part (1) T ab := g bc
(1) T c a of the linear-perturbation of the energy momentum tensor through the set (2.27) of the harmonics as follows:
(1)
T ab = ∑ l,mT AB S δ (dx A ) a (dx B ) b + r ∑ l,m T (e1)ADp S δ +T (o1)A ε prD r S δ 2(dx A ) (a (dx p ) b) + ∑ l,m T (e0) 1 2 γ pq S δ +T (e2) D pDq S δ − 1 2 γ pqDrD r S δ +T (o2) 2ε r(pDq)D r S δ (dx p ) a (dx q ) b . (2.36)
Since the background spacetime is vacuum, the pull-backed divergence of the energy-momentum tensor is given by Eq. (2.16) and the even-mode components of Eq. (2.16) in terms of the mode coefficients defined by Eq. (2.36) are given bȳ
D CT B C + 2 r (D D r)T B D − 1 r l(l + 1)T B (e1) − 1 r (D B r)T (e0) = 0, (2.37) D CT (e1)C + 3 r (D C r)T (e1)C + 1 2rT (e0) − 1 2r (l − 1)(l + 2)T (e2) = 0. (2.38)
Owing to the linear-independence of the set (2.27) of the harmonics, we can evaluate the gaugeinvariant linearized Einstein equation (2.11) through the mode-by-mode analyses including l = 0, 1 modes. As summarized in the Part I paper [30], the traceless even part of the (p, q)-component of the linearized Einstein equation (2.11) is given bỹ
F D D = −16πr 2T (e2) . (2.39)
Using this equation, the even part of (A, q)-component, equivalently (p, B)-component, of the linearized Einstein equation (2.11) yields
D DF AD − 1 2D AF = 16π rT (e1)A − 1 2 r 2D AT(e2) =: 16πS (ec)A (2.40)
through the definition of the traceless partF AB of the variableF AB :
F AB :=F AB − 1 2 y ABF C C .(D DD D + 2 r (D D r)D D − (l − 1)(l + 2) r 2 F − 4 r 2 (D C r)(D D r)F CD = 16πS (F) , (2.42) S (F) :=T C C + 4(D D r)T D (e1) − 2r(D D r)D DT (e2) − (l(l + 1) + 2)T (e2) . (2.43)
On the other hand, the traceless part of the (A, B)-component of the linearized Einstein equation (2.11) is given by
−D DD D − 2 r (D D r)D D + 4 r (D DD D r) + l(l + 1) r 2 F AB + 4 r (D D r)D (AFB)D − 2 r (D (A r)D B)F = 16πS (F)AB , (2.44) S (F)AB := T AB − 1 2 y AB T C C − 2 D (A (rT (e1)B) ) − 1 2 y ABD D (rT (e1)D ) +2 (D (A r)D B) − 1 2 y AB (D D r)D D (rT (e2) ) + r D ADB − 1 2 y ABD DD D (rT (e2) ) +2 (D A r)(D B r) − 1 2 y AB (D C r)(D C r) T (e2) + 2y AB (D C r)T (e1)C − ry AB (D C r)D CT(e2) .
(2.45)
Equations (2.39), (2.40), (2.42), and (2.44) are all independent equations of the linearized Einstein equation for even-mode perturbations. These equations are coupled equations for the variablesF C C , F, andF AB and the energy-momentum tensor for the matter field. When we solve these equations, we have to take into account of the continuity equations (2.37) and (2.38) for the matter fields. We note that these equations are valid not only for l ≥ 2 modes but also l = 0, 1 modes in our formulation. For l ≥ 2 modes, we can derive the Zerilli equation, while we can derive formal solutions for l = 0, 1 modes. The derivations of these formal solutions for l = 0, 1 modes are the main ingredients of this paper.
Component treatment of the even-mode linearized Einstein equations
To summarize the even-mode Einstein equations, we consider the static chart of y AB as Eq. (2.20). On this chart, the components of the Christoffel symbolΓ C AB associated with the covariant derivativē D A is summarized as
Γ t tt = 0,Γ t tr = f ′ 2 f ,Γ t rr = 0,Γ r tt = f f ′ 2 ,Γ r tr = 0,Γ r rr = − f ′ 2 f , (3.1) where f ′ := ∂ r f . First, Eq. (2.39)
is a direct consequence of the even-mode Einstein equation. Here, we introduce the components X (e) and Y (e) of the traceless variableF AB bỹ
F AB =: X (e) − f (dt) A (dt) B − f −1 (dr) A (dr) B + 2Y (e) (dt) (A (dr) B) .
(3.2)
Through these components X (e) and Y (e) , t-and r-components of Eq. (2.40) are given by
∂ t X (e) + f ∂ r Y (e) + f ′ Y (e) − 1 2 ∂ tF = 16πS (ec)t , (3.3) 1 f ∂ t Y (e) + ∂ r X (e) + f ′ f X (e) + 1 2 ∂ rF = −16πS (ec)r . (3.4) 9/33
The source term S (ec)A is defined by
S (ec)A := rT (e1)A − 1 2 r 2D AT(e2) . (3.5)
Furthermore, the evolution equation (2.42) for the variableF is given by
−∂ 2 tF + f ∂ r ( f ∂ rF ) + 2 r f 2 ∂ rF − (l − 1)(l + 2) r 2 fF + 4 r 2 f 2 X (e) = 16πG f S (F) . (3.6)
The source term S (F) is defined by
S (F) :=T E E + 4(D D r)T D (e1) − 2r(D D r)D DT (e2) − (l(l + 1) + 2)T (e2) (3.7) = − 1 fT tt + fT rr + 4 fT (e1)r − 2r f ∂ rT(e2) − (l(l + 1) + 2)T (e2) . (3.8)
The
∂ 2 t X (e) − f ∂ r ( f ∂ r X (e) ) − 2(1 − 2 f ) f r ∂ r X (e) − (1 − f )(1 − 5 f ) − l(l + 1) f r 2 X e − (1 − 3 f ) f r ∂ rF = −16π S (F)tt + 2 f (3 f − 1) r S (ec)r , (3.9) ∂ 2 t Y (e) − f ∂ r f ∂ r Y (e) − 2(1 − 2 f ) f r ∂ r Y (e) − (1 − f )(1 − 5 f ) − l(l + 1) f r 2 Y (e) + 1 − 3 f r ∂ tF = 16π f S (F)tr − 2(1 − 2 f ) r S (ec)t . (3.10)
Here, we note that (rr)-component of Eq. (2.44) with the constraint (3.4) is equivalent to Eq. (3.9). The source terms S (F)tt and S (F)tr in Eqs. (3.9) and (3.10) are given by
S (F)tt = 1 2 T tt + f 2T rr − r∂ tT(e1)t − 3 f 2T (e1)r − r f 2 ∂ rT(e1)r + r 2 2 ∂ 2 tT(e2) + r 2 2 f 2 ∂ 2 rT(e2) + 3r f 2 ∂ rT(e2) + 2 f 2T (e2) , (3.11) S (F)tr =T tr − r∂ tT(e1)r − r∂ rT(e1)t −T (e1)t + 1 − f fT (e1)t +r 2 ∂ t ∂ rT(e2) + 2r∂ tT(e2) − r(1 − f ) 2 f ∂ tT(e2) . (3.12)
The components of Eq. (2.37) is given by
−∂ tTtt + f 2 ∂ rTrt + (1 + f ) f rT rt − f r l(l + 1)T (e1)t = 0, (3.13) −∂ tTtr + 1 − f 2r fT tt + f 2 ∂ rTrr + (3 + f ) f 2rT rr − f r l(l + 1)T (e1)r − f rT (e0) = 0, (3.14)
where Eq. (3.13) is the t-component and Eq. (3.14) is the r component, respectively. Furthermore, Eq. (2.38) is given by
−∂ tT(e1)t + f 2 ∂ rT(e1)r + (1 + 2 f ) f rT (e1)r + f 2rT (e0) − f 2r (l − 1)(l + 2)T (e2) = 0. (3.15) 10/33
From the time derivative of Eqs. (3.3) and (3.4), we obtain
∂ 2 t X (e) − f ∂ r ( f ∂ r X (e) ) − 2 1 − f r f ∂ r X (e) − (1 − 3 f )(1 − f ) r 2 X (e) − 1 2 ∂ 2 tF − 1 2 f ∂ r ( f ∂ rF ) − 1 − f 2r f ∂ rF −16π∂ t S (ec)t − 16π∂ r ( f 2 S (ec)r ) = 0. (3.16) −∂ 2 t Y (e) + f ∂ r ( f ∂ r Y (e) ) + 2 f 1 − f r ∂ r Y (e) + (1 − 3 f )(1 − f ) r 2 Y (e) − 1 − f 2r ∂ tF − f ∂ r ∂ tF − 16π∂ r ( f S (ec)t ) − 16π f ∂ t S (ec)r = 0. (3.17)
From Eqs. (3.9) and (3. 16), we obtain
4 f ∂ r ( f X (e) ) + 2 r l(l + 1)( f X (e) ) + r∂ 2 tF + r f ∂ r ( f ∂ rF ) + (5 f − 1) f ∂ rF = −32πr S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r . (3.18)
Furthermore, from Eqs. (3.10) and (3.17), we obtain
4 f ∂ r ( fY (e) ) + 2 r l(l + 1)( fY (e) ) − 2r f ∂ r ∂ tF − (5 f − 1)∂ tF = 32πr f S (F)tr + f ∂ t S (ec)r − 1 − 3 f r S (ec)t + f ∂ r S (ec)t . (3.19)
Equations (3.3) and (3.19) yields
l(l + 1)Y (e) = r∂ t 2X (e) + r∂ rF + 3 f − 1 2 f r∂ tF +16πr 2 S (F)tr + ∂ t S (ec)r − 1 − f r f S (ec)t + ∂ r S (ec)t . (3.20)
Similarly, Equations (3.18) and (3.6) yield
4 f ∂ r ( f X (e) ) + 2 r [l(l + 1) + 2 f ]( f X (e) ) + 2 f ∂ r (r f ∂ rF ) + (5 f − 1) f ∂ rF − (l − 1)(l + 2) r fF = −32πr S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r − f 2 S (F) . (3.21)
Thus, we may regard that the independent components of the Einstein equations for the even-mode perturbations are summarized as Eqs. (3.6), (3.9), (3.20), and (3.21).
As shown in many literatures [13][14][15][16][17][18], it is well-known that Eqs. where α, β , and γ may depend on r. Substituting Eq. (3.22) into Eq. (3.21), we obtain
0 = −4r f α ′ Φ (e) − 2[l(l + 1) + 2 f ]αΦ (e) − 4r f α∂ r Φ (e) + 4 γ − 1 2 r f ∂ r [r f ∂ rF ] + 4β + 4r f γ ′ + 2 {l(l + 1) + 2 f }γ − (5 f − 1) r f ∂ rF + 4r f β ′ + 2 {l(l + 1) + 2 f }β + (l − 1)(l + 2) f F −32πr 2 S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r − f 2 S (F) . (3.23)
Here, we choose
γ = 1 2 (3.24)
to eliminate the term of the second derivative ofF. Owing to this choice, we obtain
0 = −4r f α ′ Φ (e) − 2[l(l + 1) + 2 f ]αΦ (e) − 4r f α∂ r Φ (e) + [4β + Λ]r f ∂ rF + 4r f β ′ + 2 {l(l + 1) + 2 f }β + (l − 1)(l + 2) f F −32πr 2 S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r − f 2 S (F) . (3.25)
Here, we choose β as
β = − 1 4 Λ := − 1 4 [(l − 1)(l + 2) + 3(1 − f )], Λ := (l − 1)(l + 2) + 3(1 − f ). (3.26)
to eliminate the term of the first derivative ofF. Due to this choice, we obtain
l(l + 1)ΛF = −8r f ∂ r αΦ (e) − 4[l(l + 1) + 2 f ]αΦ (e) −64πr 2 S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r − f 2 S (F) . (3.27)
This equation yields that the variableF is determined by the single variable Φ (e) and the source terms if l = 0 and if the coefficient α is determined. At this moment, the variable Φ (e) is determined up to its normalization α as
αΦ (e) := f X (e) − 1 4 ΛF + 1 2 r f ∂ rF .
(3.28)
Eliminating X (e) in Eq. (3.6) through Eq. (3.28), we obtain
−∂ 2 tF + f ∂ r ( f ∂ rF ) + 1 r 2 3(1 − f ) fF + 4 r 2 f αΦ (e) = 16π f S (F) (3.29) 12/33
Similarly, eliminating X (e) in Eq. (3.9) through Eqs. (3.27)-(3.29), we obtain
0 = −α∂ 2 t Φ (e) + α f ∂ r f ∂ r Φ (e) + 2α α ′ α + 1 r + 1 rΛ 3(1 − f ) f 2 ∂ r Φ (e) + α ′′ f + α ′ 1 r (1 + f ) + α ′ 1 rΛ 3(1 − f )2 f + α 3(1 − f ) {l(l + 1) + 2 f } r 2 Λ −α (l − 1)(l + 2) + 1 + f r 2 f Φ (e) +16π f Λ + 3(1 − f ) Λ S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 f 2 r S (ec)r − f 2 S (F) −16π f S (F)tt − 32π 3 f − 1 r f 2 S (ec)r − 16π(− 1 4 Λ) f S (F) − 16πr 1 2 f ∂ r f S (F) . (3.30)
We determine α so that the terms proportional to ∂ r Φ (e) vanish. Then, we obtain the equation for α as
α ′ α + 1 r + 1 rΛ 3(1 − f ) = 0. (3.31)
From this equation, we obtain
1 α = Cr Λ , (3.32)
where C is a constant of integration. In this paper, we choose C = 1. Then, we obtain
Φ (e) := 1 α f X (e) − 1 4 ΛF + 1 2 r f ∂ rF = r Λ f X (e) − 1 4 ΛF + 1 2 r f ∂ rF . (3.33)
This is the Moncrief variable. From Eq. (3.32), we obtain
α ′ = − 1 r α − 1 Λ 3 1 − f r α, α ′′ = + 2 r 2 α + 12(1 − f ) Λr 2 α. (3.34)
Then, using
µ := (l − 1)(l + 2), Λ = µ + 3(1 − f ),(3.35)
Eq. (3.30) is given by
− 1 f ∂ 2 t Φ (e) + ∂ r f ∂ r Φ (e) − 1 r 2 Λ 2 µ 2 [(µ + 2) + 3(1 − f )] + 9(1 − f ) 2 (µ + 1 − f ) Φ (e) = 16π r Λ −∂ t S (ec)t − f 2 ∂ r S (ec)r + 2 f f − 1 r S (ec)r + r 2 f ∂ r S (F) + 1 2 S (F) − 1 4 ΛS (F) + 3(1 − f ) Λ −S (F)tt − ∂ t S (ec)t − f 2 ∂ r S (ec)r − 4 r f 2 S (ec)r + f 2 S (F) . (3.36)
This is the Zerilli equation for the Moncrief variable (3.33).
13/33
Here, we summarize the equations for even-mode perturbations. We derive the definition of the Moncrief variable as Eq. (3.33), i.e.,
Φ (e) := r Λ f X (e) − 1 4 ΛF + 1 2 r f ∂ rF , (3.37)
where Λ is defined by
Λ = µ + 3(1 − f ), µ := (l − 1)(l + 2). (3.38)
This definition of the variable Φ (e) implies that if we obtain the variables Φ (e) andF are determined, the component of X (e) of the metric perturbation is determined through the equation
f X (e) = 1 r ΛΦ (e) + 1 4 ΛF − 1 2 r f ∂ rF .l(l + 1)ΛF = −8 f Λ∂ r Φ (e) + 4 r [6 f (1 − f ) − l(l + 1)Λ]Φ (e) − 64πr 2 S (ΛF) , (3.40) l(l + 1)Y (e) = r∂ t 2X (e) + r∂ rF + 3 f − 1 2 f r∂ tF + 16πr 2 S (Y (e) ) ,(3.41)
where the source term S (ΛF) and S (Y (e) ) are given by
S (ΛF) := S (F)tt + ∂ t S (ec)t + f 2 ∂ r S (ec)r + 4 r f 2 S (ec)r − f 2 S (F) (3.42) =T tt + r f 2 ∂ rT(e2) + 2 f ( f + 1)T (e2) + 1 2 f (l − 1)(l + 2)T (e2) ,(3.1 f ∂ 2 t Φ (e) + ∂ r f ∂ r Φ (e) −V even Φ (e) = 16π r Λ S (Φ (e) ) ,(3.46)
where the potential function V even is defined by
V even := 1 r 2 Λ 2 µ 2 [(µ + 2) + 3(1 − f )] + (3(1 − f )) 2 (+µ + (1 − f )) = 1 r 2 Λ 2 Λ 3 − 2(2 − 3 f )Λ 2 + 6(1 − 3 f )(1 − f )Λ + 18 f (1 − f ) 2 ,
(3.47) 14/33 and the source term in Eq. (3.46) is given by
S (Φ (e) ) := −∂ t S (ec)t − f 2 ∂ r S (ec)r + 2 f f − 1 r S (ec)r + r 2 f ∂ r S (F) + 1 2 S (F) − 1 4 ΛS (F) + 3(1 − f ) Λ −S (F)tt − ∂ t S (ec)t − f 2 ∂ r S (ec)r − 4 r f 2 S (ec)r + f 2 S (F) (3.48) = 1 2 Λ 2 f − 1 T tt + 1 2 (2 − f ) − 1 2 Λ fT rr − 1 2 r∂ rTtt + 1 2 f 2 r∂ rTrr − f 2T (e0) − l(l + 1) fT (e1)r + 1 2 r 2 ∂ 2 tT(e2) − 1 2 f 2 r 2 ∂ 2 rT(e2) − 1 2 3(1 + f )r f ∂ rT(e2) − 1 2 (7 − 3 f ) fT (e2) + 1 4 (l(l + 1) − 1 − f )(l(l + 1) + 2)T (e2) − 3(1 − f ) Λ T tt + r f 2 ∂ rT(e2) + 1 2 (1 + 7 f ) fT (e2) .(0 = r 2 Λ∂ 2 t S (ΛF) − (5 − 3 f )Λ + 3(1 − f )(1 + f ) + 18 1 Λ f (1 − f ) 2 f S (ΛF) −2 [3(1 − f ) + 2Λ] f 2 r∂ r S (ΛF) − Λr 2 f ∂ r f ∂ r S (ΛF) + 1 4 [(1 − 3 f ) − Λ]Λ 2 f S (F) −2r f 2 Λ∂ r S (Φ (e) ) − [Λ + (1 + 3 f )]Λ f S (Φ (e) ) . (3.50)
This is an identity of the source terms. We have confirmed Eq. (3.50) is an identity due to the definitions (3.43)-(3.8) and the continuity equations (3.13)-(3.15) of the perturbative energy-momentum tensor. This means that the evolution equation (3.29) is trivial when l = 0. Thus, we have confirmed that the above strategy for l = 0 modes are consistent. Of course, this strategy is valid only when l = 0. In the l = 0 case, we have to consider the different strategy to obtain the variable X (e) , Y (e) , andF. This will be discussed Sec. 4.
Before going to the discussion on the strategy to solve l = 0 mode Einstein equation, we comment on the original equation derived by Zerilli [13,14] for l ≥ 2. We consider the original time derivative of the Moncrief master variable (3.37) as
∂ t Φ (e) = r Λ f ∂ t X (e) − 1 4 Λ∂ tF + 1 2 r f ∂ t ∂ rF .
(3.51)
15/33
On the other hand, Eq. (3.41) is given by
∂ t X (e) = l(l + 1) 2r Y (e) − r 2 ∂ t ∂ rF − 3 f − 1 4 f ∂ tF − 8πrS (Y (e) ) .
(3.52) Substituting Eq. (3.52) into Eq. (3.51), for l = 0 modes, we obtain
1 l(l + 1) ∂ t Φ (e) = 1 2Λ fY (e) − r 2 ∂ tF − 8πr 2 f 1 l(l + 1)Λ S (Y (e) ) .
(3.53)
Here, if we define the variable Ψ (e) by
Ψ (e) := 1 2Λ fY (e) − r 2 ∂ tF (3.54) = 1 l(l + 1) ∂ t Φ (e) + 8πr 2 f 1 l(l + 1)Λ S (Y (e) ) ,(3.
l = 0 mode perturbations on the Schwarzschild Background
Here, we consider the l = 0 mode perturbations based on the perturbation equations for the evenmode on Schwarzschild background which are summarized in Sec. 3. Since Proposal 2.1 enable us to carry out the mode-by-mode analyses including l = 0, 1 modes, all equations in Sec. 3 except for Eqs. (3.53) and (3.55) are valid even in l = 0 mode. However, the strategy to solve these equations is different from that l = 0 modes, because Eqs. (3.40) and (3.41) do not directly give the components (F,Y (e) ) of the metric perturbation for l = 0 mode. Before showing the strategy to solve even-mode Einstein equations for l = 0 mode, we note that
D p k (∆) = 0 =D pDq k (∆) (4.1)
if we impose the regularity δ = 0 to the harmonic function k (∆) . In this case, the only remaining components of the linearized energy-momentum tensor is
T ab =T AB k (∆) (dx A ) a (dx B ) b + 1 2 γ pqT(e0) k (∆) (dx p ) a (dx q ) b . (4.2)
Therefore, we can safely regard thatT Then, the Moncrief master variable Φ (e) is given by Eq. (3.37), i.e.,
Φ (e) := r 1 − 3 f f X (e) − 1 4 (1 − 3 f )F + 1 2 r f ∂ rF . (4.6)
This is equivalent to Eq. (3.39) with l = 0 as
f X (e) = 1 − 3 f r Φ (e) + 1 − 3 f 4F − 1 2 r f ∂ rF . (4.7)
As in the case of l = 0 mode, this equation yields the component X (e) of the metric perturbation is determined by (F, Φ (e) ).
The crucial difference between the l = 0 mode and l = 0 modes is Eqs. (3.40) and (3.41). In the l = 0 case, these equations yield
∂ r (1 − 3 f )Φ (e) = − 8πr 2 fT tt , (4.8) ∂ t (1 − 3 f )Φ (e) = −8πr 2 fT tr ,(4.9)
where we used Eq. (4.7) to derive Eq. (4.9). The components of the divergence of the energy momentum tensor are summarized as In the case of l = 0 mode, the evolution equation (3.46) has the same form, but the potential V even defined by Eq. (3.47) with l = 0 is given by
∂ tTtt − f 2 ∂ rTrt − (1 + f ) f rT rt = 0, (4.10) ∂ tTtr − 1 − f 2r fT tt − f 2 ∂ rTrr − (3 + f ) f 2rT rr = 0,(4.V even = 3(1 − f )(1 + 3 f 2 ) r 2 (1 − 3 f ) 2 (4.14)
and the source term in Eq. (3.49) is given by
S (Φ (e) ) = −T tt − r 2 ∂ rTtt + r 2 ∂ tTtr − 3(1 − f ) 1 − 3 fT tt . (4.15)
Through Eqs. (4.8) and (4.9), we obtain
− 1 f ∂ 2 t Φ (e) + ∂ r ( f ∂ r Φ (e) ) −V even Φ (e) = 16πr 1 − 3 f −T tt − r 2 ∂ rTtt + r 2 ∂ tTtr − 3(1 − f ) 1 − 3 fT tt . (4.16)
This coincides with the master equation (3.46) with l = 0. Thus, the master equation (3. 46) does not give us any information other than that of Eqs. (4.8) and (4.9).
17/33
As in the case of l = 0 modes, the metric component X (e) is determined by the variables (F, Φ (e) ) as seen in Eq. (4.7). AlthoughF is determined by Eq. (3.40) in the l = 0 case, this is impossible in the l = 0 case. Therefore, we have to consider Eq. (3.29) for the variableF which is trivial in the l = 0 case
− 1 f ∂ 2 tF + ∂ r ( f ∂ rF ) + 1 r 2 3(1 − f )F + 4 r 3 (1 − 3 f )Φ (e) = 16π − 1 fT tt + fT rr .∂ r ( fY (e) ) = 1 2 ∂ tF − ∂ t (X (e) ), (4.18) ∂ t ( fY (e) ) = − f ∂ r ( f X (e) ) − 1 2 f 2 ∂ rF . (4.19)
We may regard that Eqs. (4.18) and (4.19) are equations to obtain the variable Y (e) . Actually, the integrability of these equations is guaranteed by Eqs. . We may carry out the above strategy to obtain the l = 0 mode solution to the linearized Einstein equations, but it is instructive to consider the vacuum case where all components of the linearized energy-momentum tensor (1) T ab vanishes before the derivation of the non-vacuum case.
l = 0 mode vacuum case
Here, we consider the vacuum case of the above equations for l = 0 mode perturbations. First, we consider Eqs. (4.8) and (4.9) with the vacuum condition:
∂ r (1 − 3 f )Φ (e) = 0, (4.20) ∂ t (1 − 3 f )Φ (e) = 0. (4.21)
These equations are easily integrated as
(1 − 3 f )Φ (e) = −2M 1 , M 1 ∈ R. (4.22)
Furthermore, the variableF is determined by Eq. (4.17) with vacuum condition:
− 1 f ∂ 2 tF + ∂ r ( f ∂ rF ) + 1 r 2 3(1 − f )F −f X (e) = − 2M 1 r + 1 − 3 f 4F − 1 2 r f ∂ rF .
(4.24) 18/33 Moreover, the components Y (e) is obtain the direct integration of Eqs. (4.18) and (4.19), because the integrability is already guaranteed. Substituting Eq. (4.24) into Eqs. (4.18) and (4.19), we obtain
f ∂ r ( fY (e) ) = − 1 4 (1 − 5 f )∂ tF + 1 2 r f ∂ r ∂ tF , (4.25) ∂ t ( fY (e) ) = 2M 1 f r 2 − 3 4r f (1 − f )F − 1 4 f (1 − 3 f )∂ rF + 1 2 r∂ 2 tF , (4.26)
where we used Eq. (4.23).
Here, we assume the existence of the solution to Eq. (4.23) and we denote this solution bỹ
F =: ∂ t ϒ,(4.− 1 f ∂ 2 t ϒ + ∂ r ( f ∂ r ϒ) + 1 r 2 3(1 − f )ϒ − 8M 1 r 3 t + ζ (r) = 0. (4.28)
where ζ (r) is an arbitrary function of r. Using Eq. (4.27) and integrating by t, Eq. (4.26) yields
fY (e) = 2M 1 f r 2 t − 3 4r f (1 − f )ϒ − 1 4 f (1 − 3 f )∂ r ϒ + 1 2 r∂ 2 t ϒ + Ξ(r), (4.29)
where Ξ(r) is an arbitrary function of r. Substituting Eq. (4.29) into Eq. (4.25) and using Eq. (4.28), we obtain
ζ (r) = − 4 1 − 3 f ∂ r Ξ(r). (4.30)
In summary, we have obtained the components of X (e) , Y (e) , andF of the metric perturbations as follows: (4.33) and Ξ(r) is an arbitrary function of r.
f X (e) = − 2M 1 r + 1 − 3 f 4 ∂ t ϒ − 1 2 r f ∂ r ∂ t ϒ, (4.31) fY (e) = 2M 1 f r 2 t − 3 4r f (1 − f )ϒ − 1 4 f (1 − 3 f )∂ r ϒ + 1 2 r∂ 2 t ϒ + Ξ(r), (4.32) andF = ∂ t ϒ, − 1 f ∂ 2 t ϒ + ∂ r ( f ∂ r ϒ) + 1 r 2 3(1 − f )ϒ − 8M 1 r 3 t − 4 1 − 3 f ∂ r Ξ(r) = 0,
Here, we consider the covariant form F ab of the l = 0 mode metric perturbation. According to Proposal 2.1, we impose the regularity on S 2 to the harmonic function k (∆) so that
k (∆) = 1.
(4.34)
SinceF D D = 0 by Eq. (4.4) for l = 0 mode perturbations, the gauge-invariant metric perturbation F ab for the l = 0 mode is given by
F ab =F AB (dx A ) a (dx B ) b + 1 2 γ pq r 2F (dx p ) a (dx q ) b = −( f X (e) ) (dt) a (dt) b + f −2 (dr) a (dr) b + 2( fY (e) ) f −1 (dt) (A (dr) B) + 1 2 γ pq r 2F (dx p ) a (dx q ) b . (4.35)
As in the case of the l = 1 odd-mode perturbation in Part I paper [30], the solutions (4.31)-(4.33) may include the terms in the form of £ V g ab for a vector field V a . To find the term £ V g ab , we consider 19/33 the generator V a whose components are given by V a = V t (t, r)(dt) a +V r (r,t)(dr) a .
(4.36)
Then, the nonvanishing components of £ V g ab are given by
£ V g tt = 2∂ t V t − f f ′ V r , (4.37) £ V g tr = ∂ t V r + ∂ r V t − f ′ f V t , (4.38) £ V g rr = 2∂ r V r + f ′ f V r , (4.39) £ V g θ θ = 2r fV r , (4.40) £ V g φ φ = 2r f sin 2 θV r .V r = 1 4 f rF = 1 4 f r∂ t ϒ, £ V g θ θ = 1 sin 2 θ £ V g φ φ = 1 2 r 2F = 1 2 r 2 ∂ t ϒ.£ V g tt = 2∂ t V t − 1 4 (1 − f )∂ t ϒ, (4.43) £ V g tr = 1 4 f r∂ 2 t ϒ + ∂ r V t − 1 f r (1 − f )V t , (4.44) £ V g rr = − 1 4 f 2 (1 − 3 f )∂ t ϒ + 1 2 f r∂ r ∂ t ϒ. (4.45)
To identify the degree of freedom which expressed as £ V g ab in X (e) , we choose
∂ t V t = 1 4 f ∂ t ϒ + 1 4 r f ∂ r ∂ t ϒ (4.46)
so that
£ V g tt = − 1 4 (1 − 3 f )∂ t ϒ + 1 2 r f ∂ r ∂ t ϒ. (4.47)
Then, we obtain (4.48) where γ(r) is an arbitrary function of r. Substituting Eq. (4.48) into Eq. (4.44) and using the equation (4.33) for ϒ, we obtain
V t = 1 4 f ϒ + 1 4 r f ∂ r ϒ + γ(r),£ V g tr = 2M 1 r 2 t + r 2 f ∂ 2 t ϒ − 1 4 (1 − 3 f )∂ r ϒ − 3 4r (1 − f )ϒ + r 1 − 3 f ∂ r Ξ(r) + ∂ r γ(r) − 1 f r (1 − f )γ(r).F ab = 2M 1 r (dt) a (dt) b + f −2 (dr) a (dr) b + £ V g ab +2 1 f Ξ(r) − r 1 − 3 f ∂ r Ξ(r) − ∂ r γ(r) + 1 f r (1 − f )γ(r) (dt) (a (dr) b) . (4.50) 20/33
As a choice of the generator V a , we choose the arbitrary function γ(r) in V a such that
γ(r) = − r (1 − 3 f ) Ξ(r) + f dr 2 f (1 − 3 f ) 2 Ξ(r).
(4.51)
Then, we obtain
F ab = 2M 1 r (dt) a (dt) b + f −2 (dr) a (dr) b + £ V g ab ,(4.52)
where
V a = f 4 ϒ + r f 4 ∂ r ϒ − r 1 − 3 f Ξ(r) + f dr 2 f (1 − 3 f ) 2 Ξ(r) (dt) a + r 4 f ∂ t ϒ(dr) a . (4.53)
The function ϒ(t, r) is the solution to the second equation (4.33).
The solution (4.52) is the O(ε) mass parameter perturbation M + εM 1 of the Schwarzschild spacetime apart from the term the Lie derivative of the background metric g ab . Since l = 0 mode is the spherically symmetric perturbations, the solution (4.52) is the realization of the linearized gauge-invariant version of Birkhoff's theorem [56]. We also note that the vector field V a is also gauge-invariant in the sense of the second kind. Here, we have to emphasize that the generator (4.53) with the second equation in Eq. The integrability of Eqs. (4.8) and (4.9) was already confirmed in Eq. (4.13). Then, we obtain m 1 (t, r) = 4π dr r 2 fT tt + M 1 = 4π dtr 2 fT tr + M 1 . (4.55)
Eq. (4.7) yields the component X (e) of the metric perturbation as follows:
f X (e) = − 2m 1 (t, r) r + 1 − 3 f 4F − 1 2 r f ∂ rF . (4.56)
As discussed in above, the variableF is determined by Eq. (4.17). As in the vacuum case in Sec. 4.1, we introduce the function ϒ such that
F =: ∂ t ϒ, (4.57) − 1 f ∂ 2 t ϒ + ∂ r ( f ∂ r ϒ) + 3(1 − f ) r 2 ϒ − 8 r 3 dtm 1 (t, r) + ζ (r) = 16π dt − 1 fT tt + fT rr , (4.58)
where ζ (r) is an arbitrary function of r. Through the variable ϒ and Eq. (4.56), Eq. (4.19) is integrated as follows:
fY (e) = 2 f r 2 dtm 1 (t, r) + 8πr f 2 dtT rr − 3 f (1 − f ) 4r ϒ − f (1 − 3 f ) 4 ∂ r ϒ + r 2 ∂ 2 t ϒ + Ξ(r),ζ (r) = − 4 1 − 3 f ∂ r Ξ(r) (4.60)
as expected from the vacuum case in Sec. 4.1.
In summary, we have obtained the solution to the components of the metric perturbations X (e) , Y (e) , andF as follows:
f X (e) = − 2m 1 (t, r) r + 1 − 3 f 4 ∂ t ϒ − 1 2 r f ∂ r ∂ t ϒ, (4.61) fY (e) = 2 f r 2 dtm 1 (t, r) + 8πr f 2 dtT rr − 3 f (1 − f ) 4r ϒ − f (1 − 3 f ) 4 ∂ r ϒ + r 2 ∂ 2 t ϒ + Ξ(r), (4.62) F =: ∂ t ϒ, (4.63) ∂ 2 t ϒ − f ∂ r ( f ∂ r ϒ) − 3 f (1 − f ) r 2 ϒ + 8 f r 3 dtm 1 (t, r) + 4 f ∂ r Ξ(r) 1 − 3 f = 16π dt T tt − f 2T rr . (4.64)
Here, we consider the covariant form of the above l = 0 mode non-vacuum solutions. As in the vacuum case in Sec. 4.1, we show the expression (4.35) of the above non-vacuum solution
F ab = −( f X (e) ) (dt) a (dt) b + f −2 (dr) a (dr) b + 2( fY (e) ) f −1 (dt) (A (dr) B) + 1 2 γ pq r 2 ∂ t ϒ(dx p ) a (dx q ) b . (4.65)
The components of F ab are given by
F tt = 2m 1 (t, r) r − 1 − 3 f 4 ∂ t ϒ + 1 2 r f ∂ r ∂ t ϒ, (4.66) F tr = 2 r 2 dtm 1 (t, r) + 8πr f dtT rr − 3(1 − f ) 4r ϒ − (1 − 3 f ) 4 ∂ r ϒ + r 2 f ∂ 2 t ϒ + 1 f Ξ(r), (4.67) F rr = 2m 1 (t, r) r f 2 − 1 − 3 f 4 f 2 ∂ t ϒ + r 2 f ∂ r ∂ t ϒ, (4.68) F θ θ = r 2 2 ∂ t ϒ = 1 sin 2 θ F φ φ . (4.69)
As in the vacuum case, we consider the term in the form £ V g ab with the generator
V a = V t (t, r)(dt) a +V r (r,t)(dr) a .V r = 1 4 f r∂ t ϒ, £ V g θ θ = 1 sin 2 θ £ V g φ φ = 1 2 r 2 ∂ t ϒ,(4.71)
and we have
F θ θ = £ V g θ θ , F φ φ = £ V g φ φ .
(4.72)
Substituting the choice (4.71) into Eq. (4.39) and compare with Eq. (4.68), we obtain
£ V g rr = − 1 − 3 f 4 f 2 ∂ t ϒ + 1 2 f r∂ r ∂ t ϒ, F rr = 2m 1 (t, r) r f 2 + £ V g rr .
(4.73) 22/33 Substituting the choice V r in Eq. (4.71) into Eq. (4.37) and comparing with Eq. (4.66), we choose (4.74) and obtain
V t = 1 4 f ϒ + 1 4 r f ∂ r ϒ + γ(r),£ V g tt = − 1 − 3 f 4 ∂ t ϒ + 1 2 r f ∂ r ∂ t ϒ, F tt = 2m 1 (t, r) r + £ V g tt . (4.75)
Finally, from Eq. (4.38) with the choice (4.74) of V t and the choice (4.71) of V r , we obtain
£ V g tr = 1 4 f r∂ 2 t ϒ − 1 − 3 f 4 ∂ r ϒ + 1 4 r∂ r ( f ∂ r ϒ) + ∂ r γ(r) − 1 − f f r γ(r). (4.76)
Furthermore, using Eq. (4.64), we have
£ V g tr = 2 r 2 dtm 1 (t, r) − 4π r f dtT tt + 4πr f dtT rr + 1 2 f r∂ 2 t ϒ − 1 − 3 f 4 ∂ r ϒ − 3(1 − f ) 4r ϒ + ∂ r γ(r) − 1 − f f r γ(r) + r∂ r Ξ(r) 1 − 3 f . (4.77)
Through Eq. (4.67), we obtain
F tr = 4πr dt 1 fT tt + fT rr + £ V g tr + f 2 f (1 − 3 f ) 2 Ξ(r) − ∂ r r f (1 − 3 f ) Ξ(r) − ∂ r ( 1 f γ(r)) . (4.78)
The same choice of γ(r) in the generator V a as Eq. (4.51) yields F tr = 4πr dt 1 fT tt + fT rr + £ V g tr , (4.79)
Then, we have obtained
F ab = 2 r M 1 + 4π dr r 2 f T tt (dt) a (dt) a + 1 f 2 (dr) a (dr) a +2 4πr dt 1 fT tt + fT rr (dt) (a (dr) b) + £ V g ab ,(4.80)
where
V a = f 4 ϒ + r f 4 ∂ r ϒ − rΞ(r) (1 − 3 f ) + f dr 2Ξ(r) f (1 − 3 f ) 2 (dt) a + 1 4 f r∂ t ϒ(dr) a . (4.81)
The variable ϒ must satisfy Eq. (4.64). We also note that the expression of F ab is not unique, since we may choose different vector field V a . We can also choose the time component V t of the vector field V a so that F tr = £ V g tr . In this case, the additional terms appear in the component F tt .
We also note that the term £ V g ab in Eq. (4.80) is gauge-invariant of the second kind. Furthermore, unlike the vacuum case, the variable ϒ in this term includes information of the matter field through Eq. (4.64). In this sense, the term £ V g ab in Eq. (4.80) is physical. 23/33 5. l = 1 mode non-vacuum perturbations on the Schwarzschild Background In this section, we consider the l = 1 mode perturbations based through the variables defined in Secs. 2 and 3. Even in the case of l = 1 mode, the gauge-invariant variables given by Eqs. (2.33)-(2.35) are valid. Since the mode-by-mode analyses are possible in our formulation, we can consider l = 1 modes, separately. For the l = 1 even-mode perturbations, the component F Ap of the gaugeinvariant part of the metric perturbation vanishes and the other components are given by
F AB := 1 ∑ m=−1F AB k (∆+2)m , F pq := 1 2 γ pq r 2 1 ∑ m=−1F k (∆+2)m . (5.1)
We can also separate the trace partF D D and the traceless partF AB for the metric perturbationF AB as Eq. (2.41). We also consider the components of the traceless part F AB as Eq. (3.2).
Following Proposal 2.1, we impose the regularity to the harmonic function k (∆+2)m . Then, we have
D pDq − 1 2 γ pq∆ k (∆+2)m = ε r(pDq)D r k (∆+2)m = 0. (5.2)
In this case, the only remaining components of the linearized energy-momentum tensor (1) T ab are given by
(1)
T ab = 1 ∑ m=−1T AB k (∆+2) (dx A ) a (dx B ) b +2r 1 ∑ m=−1 T (e1)ADp k (∆+2)m +T (o1)A ε prD r k (∆+2)m (dx A ) (a (dx p ) b) + 1 2 r 2 γ pq 1 ∑ m=−1T (e0) k (∆+2)m (dx p ) a (dx q ) b . (5.3)
Therefore, for even-mode perturbations, we can safely regard that
T (Φ (e) := r 3(1 − f ) f X (e) − 3(1 − f ) 4F + 1 2 r f ∂ rF . (5.6)
In other words, the components X (e) is given by
f X (e) = 3(1 − f ) r Φ (e) + 3(1 − f ) 4F − 1 2 r f ∂ rF (5.7) 1 − f r 2 Φ (e) = 0. (5.22)
As in the case of l = 0 mode, we consider the problem whether the solution (5.18) with Eqs. (5.19)-(5.21) is described by £ V g ab for an appropriate vector field V a , or not. From the symmetry of the above solution, we consider the case where the vector field V a is given by
V a = V t (dt) a +V r (dr) a +V θ (dθ ) a , ∂ φ V t = ∂ φ V r = ∂ φ V θ = 0 (5.23)
and calculate all components of £ V g ab . We note that all components of F ab given by Eq. (5.18) are proportional to cos θ . Therefore, if we may identify some components of F ab with £ V g ab , the θ -dependence of the components in Eq. (5.23) should be given by
V a = v t (t, r) cosθ (dt) a + v r cos θ (dr) a + v θ sin θ (dθ ) a .
(5.24) 26/33 Then, the non-trivial components of £ V g ab are given by
£ V g tt = 2∂ t v t − f f ′ v r cosθ = 0, (5.25) £ V g tr = ∂ t v r + ∂ r v t − f ′ f v t cos θ = 0, (5.26) £ V g tθ = (∂ t v θ − v t ) sin θ = 0, (5.27) £ V g rr = 2 f −1/2 ∂ r f 1/2 v r cos θ = 0, (5.28) £ V g rθ = r 2 ∂ r 1 r 2 v θ − v r sin θ = 0, (5.29) £ V g θ θ = 2 (v θ + r f v r ) cos θ = 0, (5.30) £ V g φ φ = 2 (r f v r + v θ ) sin 2 θ cos θ = 0. (5.31)
From Eqs. (5.27) and (5.29), we obtain
r 2 v(t, r) := v θ , v t = ∂ t v θ = r 2 ∂ t v, v r = r 2 ∂ r 1 r 2 v θ = r 2 ∂ r v,(5.32)
i.e.,
V a = r 2 ∂ t v cos θ (dt) a + r 2 ∂ r v cos θ (dr) a + r 2 v sin θ (dθ ) a . (5.33)
Then, Eqs. (5.25)-(5.31) are summarized as
£ V g tt = r 2 2∂ 2 t v − f (1 − f ) r ∂ r v cos θ , (5.34) £ V g tr = ∂ t 2r 2 ∂ r v − 1 − 3 f f rv cosθ , (5.35)
£ V g rr = 2 f −1/2 ∂ r f 1/2 r 2 ∂ r v cos θ , (5.36)
£ V g θ θ = 2r 2 (r f ∂ r v + v)cos θ . (5.37)
As the first trial, we consider the correspondence
£ V g θ θ = F θ θ ,(5.38)
i.e.,
r f ∂ r v + v = − f ∂ r Φ (e) − 1 − f r Φ (e) . (5.39)
As the second trial, we consider the correspondence
£ V g rr = F rr ,(5.40)
i.e.,
− 1 − 5 f f r∂ r v + 2r 2 f ∂ r ( f ∂ r v) = 1 − f r f Φ (e) + 1 − f f ∂ r Φ (e) − 2r f ∂ r f ∂ r Φ (e) .(£ V g tt = −2r∂ 2 t Φ (e) − f (1 − f ) r Φ (e) + f (1 − f )∂ r Φ (e) cos θ = − f − 1 r (1 − f )Φ (e) − (1 − f )∂ r Φ (e) + 2r∂ r f ∂ r Φ (e) cos θ = F tt ,(5.44)
where we used Eq. (5.22). Then, we have shown that
F ab = £ V g ab , (5.45) where V a = −r∂ t Φ (e) cos θ (dt) a + Φ (e) − r∂ r Φ (e) cos θ (dr) a − rΦ (e) sin θ (dθ ) a . (5.46)
Thus, the vacuum solution of l = 1-mode perturbations described by the Lie derivative of the background metric through the master equation (5.22).
l = 1 mode non-vacuum case
Here, we consider the non-vacuum solution to the l = 1 even-mode linearized Einstein equations. In this non-vacuum case, we concentrate only on the m = 0 mode perturbations as in the vacuum case, because the extension to our arguments to m = ±1 modes is straightforward. The solution is given by the covariant form (5.18) as in the case of the vacuum case. The non-vacuum solutions for the variableF, Y (e) , and X (e) are given by Eqs. (5.8), (5.9), and (5.10), respectively. The master variable Φ (e) must satisfy the master equation (5.11) with the source term (5.12). We have to emphasize that the components of the linear perturbation of energy-momentum tensor satisfy the continuity equations (5.13)-(5.15). Then, the components of the gauge-invariant part F ab for l = 1 even-mode non-vacuum perturbations are summarized as follows:
F tt = f 1 r (1 − f )Φ (e) + (1 − f )∂ r Φ (e) − 2r∂ r f ∂ r Φ (e) cosθ + 8πr 2 3(1 − f ) 3(1 − 3 f )T tt − 2r f ∂ rTtt cosθ , (5.47) F tr = r∂ t 1 − f r f Φ (e) − 2∂ r Φ (e) − 16πr 2 3 f (1 − f )T tt cos θ + 8πr 2T tr cos θ , (5.48) F rr = 1 f 1 − f r Φ (e) + (1 − f )∂ r Φ (e) − 2r∂ r ( f ∂ r Φ (e) ) cos θ + 8πr 2 3 f 2 (1 − f ) 3(1 − 3 f )T tt − 2r f ∂ rTtt cos θ , (5.49) F θ θ = −2r r f ∂ r Φ (e) + (1 − f )Φ (e) + 8πr 3 3(1 − f )T tt cosθ ,(5.50)F φ φ = −2r r f ∂ r Φ (e) + (1 − f )Φ (e) + 8πr 3 3(1 − f )T tt sin 2 θ cos θ .
(5.51) 28/33 As seen in the vacuum case, if we choose the generator V a as Eq. (5.46), i.e., V a = V (vac)a := −r∂ t Φ (e) cos θ (dt) a + Φ (e) − r∂ r Φ (e) cos θ (dr) a −rΦ (e) sin θ (dθ ) a .
(5.52)
we obtain
£ V g tt = f − 2r f ∂ 2 t Φ (e) + (1 − f )∂ r Φ (e) − 1 − f r Φ (e) cos θ , (5.53) £ V g tr = r∂ t 1 − f r f Φ (e) − 2∂ r Φ (e) cosθ ,(5.54)£ V g rr = 1 f 1 − f r Φ (e) + (1 − f )∂ r Φ (e) − 2r∂ r f ∂ r Φ (e) cos θ , (5.55) £ V g θ θ = −2r r f ∂ r Φ (e) + (1 − f )Φ (e) cos θ , (5.56) £ V g φ φ = −2r r f ∂ r Φ (e) + (1 − f )Φ (e) sin 2 θ cos θ , (5.57) £ V g tθ = £ V g tφ = £ V g rθ = £ V g rφ = £ V g θ φ = 0. (5.58)
Through these formulae of the components £ V g ab and Eqs. (5.47)-(5.51) for the components of F ab , we obtain
F tt = £ V g tt − 16πr 2 f 2 3(1 − f ) 1 + f 2T rr + r f ∂ rTrr −T (e0) − 4T (e1)r cos θ , (5.59) F tr = £ V g tr − 16πr 3 3 f (1 − f ) ∂ tTtt − 3 f (1 − f ) 2rT tr cos θ , (5.60) F rr = £ V g rr − 16πr 3 3 f (1 − f ) ∂ rTtt − 3(1 − 3 f ) 2r fT tt cos θ , (5.61) F θ θ = £ V g θ θ − 16πr 4 3(1 − f )T tt cos θ , (5.62) F φ φ = £ V g φ φ − 16πr 4 3(1 − f )T tt sin 2 θ cos θ . (5.63)
where we used Eq. (5.11) with the source term (5.12) and the component (5.14) of the continuity equation in Eq. (5.59). Eqs. (5.59)-(5.63) are summarized as
F ab = £ V g ab − 16πr 2 3(1 − f ) f 2 1 + f 2T rr + r f ∂ rTrr −T (e0) − 4T (e1)r (dt) a (dt) b + 2r f ∂ tTtt − 3 f (1 − f ) 2rT tr (dt) (a (dr) b) + r f ∂ rTtt − 3(1 − 3 f ) 2r fT tt (dr) a (dr) b +r 2T tt γ ab cos θ . (5.64)
We note that there may be exist the term £ W g ab in the right-hand side of Eqs. (5.64) in addition to the term £ V g ab discussed above. Such term will depend on the equation of state of the matter field. This situation can be seen in the Part III paper [46]. Even if we consider such terms, we will not have a simple expression of the metric perturbation, in general. Therefore, we will not carry out such further considerations, here.
29/33
Summary and Discussion
In summary, after reviewing our general framework of the general-relativistic gauge-invariant perturbation theory and our strategy for the linear perturbations on the Schwarzschild background spacetime proposed in Refs. [29,30], we developed the component treatments of the even-mode linearized Einstein equations. Our proposal in Refs. [29,30] was on the gauge-invariant treatments of the l = 0, 1 mode perturbations on the Schwarzschild background spacetime. Since we used singular harmonic functions at once in our proposal, we have to confirm whether our proposal is physically reasonable, or not.
To confirm this, in the Part I paper [30], we carefully discussed the solutions to the Einstein equations for odd-mode perturbations. We obtain the Kerr parameter perturbations in the vacuum case, which is physically reasonable. In this paper, we carefully discussed the solutions to the evenmode perturbations. Due to Proposal 2.1, we can treat the l = 0, 1 mode perturbations through the equivalent manner to the l ≥ 2-mode perturbations. For this reason, we derive the equations for evenmode perturbations without making distinction among l ≥ 0 modes for even-mode perturbations.
To derive the even-mode perturbations, it is convenient to introduce the Moncrief variable. In this paper, we explain the introduction of the Moncrief variable through an initial value constraint (3.21) is regard as an equation for the componentF of the metric perturbation and the Moncrief variable Φ (e) . This consideration leads to the well-known definition of the Moncrief variable Φ (e) . Furthermore, from the evolution equation (3.9), we obtain the well-known master equation (3.46) for the Moncrief variable Φ (e) .
Moreover, we obtain the constraint equations (3.40) and (3.41) together with the definition (3.39) of the Moncrief variable. From their derivations, we have shown that these equations are valid not only for l ≥ 2 but also for l = 0, 1 modes. We also checked the consistency of these equations, and we derived the identity of the source terms which are given by the components of the linear perturbation of the energy-momentum tensor. This identity is confirmed by the components of the linear perturbation of the energy-momentum tensor.
In this paper, we also carefully discussed the l = 0, 1 mode solutions to the linearized Einstein equations for even-mode perturbations to check that Proposal 2.1 is physically reasonable.
The l = 0-mode solutions are discussed in Sec. 4. After summarizing the linearized Einstein equations and the linearized continuity equations for generic matter field for l = 0 mode, we first considered the vacuum solution of the l = 0-mode perturbations following Proposal 2.1. Then, we showed that the additional mass parameter perturbation of the Schwarzschild spacetime is the only solution apart from the terms of the Lie derivative of the background metric g ab in the vacuum case. This is the gauge-invariant realization of the linearized version of the Birkhoff theorem [56].
In the non-vacuum case, we use the method of the variational constant with the Schwarzschild mass constant parameter in vacuum case. Then, we obtained the general non-vacuum solution to the linearized Einstein equation for the l = 0 mode. As the result, we obtained the linearized metric (4.80). The solution (4.80) includes the additional mass parameter perturbation M 1 of the Schwarzschild mass and the integration of the energy density. Furthermore, in the solution (4.80), we have the 2(dt) (a (dr) b) term due to the integration of the components of the energy-momentum tensor. In the solution (4.80), we also have the term which have the form of the Lie derivative of the background metric g ab . The off-diagonal term of 2(dt) (a (dr) b) can be eliminate by the replacement of the generator V a of the term of the Lie derivative of the g ab . However, if we eliminate the off-diagonal term of 2(dt) (a (dr) b) through the replacement of the generator V a , we have additional 30/33 term to the diagonal components of the linearized metric perturbation (4.80). Since these diagonal components have complicated forms, we do not carry out this displacement.
We also discussed the l = 1-mode perturbations in Sec. 5. In this paper, we concentrated only on the m = 0 mode, since the extension to m = ±1 modes are straightforward. The solution of the l = 1 mode is obtained through the similar strategy to the case of l ≥ 2 modes that are discussed in Sec. 3. As in the case of l = 0-mode perturbations, we first discuss the vacuum solution for l = 1mode perturbations. As the result, l = 1-mode vacuum metric perturbations are described by the Lie derivative of the background metric g ab with an appropriate operator. On the other hand, in the non-vacuum l = 1-mode perturbations, the l = 1 mode metric perturbation have the contribution from the components of the energy-momentum tensor of the matter field in addition to the term of the Lie derivative of the background metric g ab which is derived as the above vacuum solution.
As the odd-mode solutions in the Part I paper [30], we also have the terms of the Lie derivative of the background metric g ab in the derived solutions in the l = 0, 1 even-mode solutions. We have to remind that our definition of gauge-invariant variables is not unique, and we may always add the term of the Lie derivative of the background metric g ab with a gauge-invariant generator as emphasized in Sec. 2.1. In other words, we may have such terms in derived solutions at any time, and we may say that the appearance of such terms is a natural consequence due to the symmetry in the definition of gauge-invariant variables. Furthermore, since our formulation completely excludes the second kind gauge through Proposal 2.1, these terms of the Lie derivative should be regarded as the degree of freedom of the first kind gauge, i.e., the coordinate transformation of the physical spacetime M ε as emphasized in the Part I paper [30]. This discussion is the consequence of our distinction of the first-and second-kind of gauges and the complete exclusion of the gauge degree of freedom of the second kind as emphasized in the Part I paper [30].
We also note that the existence of the additional mass parameter perturbation M 1 requires the perturbations ofF due to the linearized Einstein equations. In this sense, the term described by the Lie derivative of the background spacetime is necessary. The solutions derived in this paper and the Part I paper [30] are local perturbative solutions. If we construct the global solution, we have to use the solutions obtained in this paper and in the Part I paper [30] as local solutions and have to match these local solutions. We expect that the term of the Lie derivative derived here will play important roles in this case.
Besides the term of the Lie derivative of the background metric g ab , we have realized the Birkhoff theorem for l = 0 even-mode solutions and the Kerr parameter perturbations in l = 1 odd-mode solutions. These solutions are physically reasonable. This also implies that Proposal 2.1 is physically reasonable nevertheless we used singular mode functions at once to construct gauge-invariant variables and imposed the regular boundary condition on the functions on S 2 when we solve the linearized Einstein equations, while the conventional treatment through the decomposition by the spherical harmonics Y lm corresponds to the imposition of the regular boundary condition from the starting point.
the decomposition formulae (2.24)-(2.26) owing to the linear-independence of the set of the harmonic functions (2.27) when δ = 0. Then, the mode-by-mode analysis including l = 0, 1 is possible when δ = 0. However, the mode functions (2.30)-(2.32) are singular if δ = 0. When δ = 0, we have k (∆) ∝ Y 00 and k (∆+2)m ∝ Y 1m . Using the above harmonics functions S δ in Eq. (2.28), we propose the following strategy: Proposal 2.1. We decompose the metric perturbation h ab on the background spacetime with the metric (2.18)-(2.21) through Eqs. (2.24)-(2.26) with the harmonic function S δ given by Eq. (2.28). Then, Eqs. (2.24)-(2.26) become invertible including l = 0, 1 modes.
to the single master equation for a single variable. We trace this procedure. Equation (3.21) is an initial value constraint for the variables (X (e) ,F), while Eqs. (3.6) and (3.16) are evolution equations. Equation (3.20) directly yields that the variable Y (e) is determined by the solution (X (e) ,F) to Eqs. (3.21), (3.6) and (3.16), if l = 0. If the initial value constraint (3.21) is reduced to the equation of a variable Φ (e) andF, we may expect that Φ (e) linearly depends on f X (e) , 11/33 F , and r f ∂ rF . To show this, we introduce the variable Φ (e) as αΦ (e) := f X (e) + βF + γr f ∂ rF , (3.22)
initial value constraint for the variableF and Y (e) , we have Eqs. (3.27) and (3.20) as
S
(Y (e) ) := S (F)tr + ∂ t S (ec)r − 1 − f r f S (ec)t + ∂ r S (ec)t (3.44) =T tr + r∂ tT(e2) .(3.45) Equation (3.40) implies that the variableF of the metric perturbation is determined if the variable Φ (e) and source term S (ΛF) are specified. Equation (3.41) implies that the component Y (e) of the metric perturbation is determined if the variables X (e) ,F, and the source term S (Y (e) ) are specified. Thus, apart from the source terms, the componentF of the metric perturbation is determined through Eq. (3.40) if the Moncrief variable Φ (e) is specified. The component X (e) of the metric perturbation is determined through Eq. (3.39) if the variables Φ (e) andF are specified. Finally, the component Y (e) of the metric perturbation is determined through Eq. (3.41) if the variablesF and X (e) are specified. Namely, the components X (e) , Y (e) , andF of the metric perturbation are determined by the Moncrief variable Φ (e) . The Moncrief variable Φ (e) is determined by the master equation
−
Eq. (4.3), the trace of the perturbationF AB is determined by the Einstein equation (2.39), i.e.,F D D = 0. (4.4)In the case of l = 0 mode, Λ defined by Eq. (3.38) is given byΛ = 1 − 3 f .
we check the integrability condition of Eqs. (4.8) and (4.9). Differentiating Eq. (4.8) with respect to t and differentiating Eq. (4.9), we obtain the integrability condition of Eqs. (4.8) and (4.9) follows 0 = ∂ t −8π r 2 fT tt − ∂ r −8πr 2 fT tr = −8πr 2 1 f ∂ tTtt − f 2 ∂ rTtr − (1 + f ) f rT tr . (4.13) This coincides with the component (4.10) of the continuity equation of the matter field. Thus, Eqs. (4.8) and (4.9) are integrable and there exist the solution Φ (e) = Φ (e) [T tt , T tr ] to these equations.
has the same form of the inhomogeneous version of the Regge-Wheeler equation with l = 0, while the original Regge-Wheeler equation is valid only for the l ≥ 2 modes. If we solve this equation (4.17), we can determine the variableF which depends on the variable Φ (e) and the matter fieldsT tt andT rr . Then, through this solutionF =F[Φ (e) ,T tt ,T rr ] and the solution to Eqs. (4.8) and (4.9), we can obtain the variable X (e) through Eq. (4.7) as a solution to the linearized Einstein equation for the l = 0 mode. The remaining component to be obtained is the component Y (e) of the metric perturbation. To obtain the variable Y (e) , we remind the original initial value constraints (3.3) and (3.4). In the l = 0 mode case, the source term S (ec)t and S (ec)r are given by S (ec)t = S (ec)r = 0 from Eqs. (3.5) and (4.3). Then, the initial value constraints (3.3) and (3.4) are given by
(4.7), (4.8), (4.9), (4.11), and (4.17). Then, we can obtain the component Y (e) of the metric perturbation by the direct integration of Eqs. (4.18) and (4.19)
From
Eqs. (4.7) and (4.22), we obtain the component X (e) of the metric perturbation as follows:
. (4.42) into Eqs. (4.37)-(4.39), we obtain
solutions (4.31), (4.32), and (4.33), and the expression (4.35) of the gauge-invariant part of the metric perturbation, and the components (4.42), (4.45), (4.47), and (4.49) of £ V g ab , we obtain
(4.33) is necessary if we include the perturbative Schwarzschild mass parameter M 1 as the solution to the linearized Einstein equation in our framework. This can be seen from the second equation in Eq. (4.33). This equation indicates that M 1 = 0 if we choose ϒ = 0 for arbitrary time t. 4.2. l = 0 mode non-vacuum case Inspecting the above vacuum case, we apply the method of variational constants. In Eq. (4.22), the Schwarzschild mass parameter perturbation M 1 is an integration constant. Then, we choose the function m 1 (t, r) so that m 1 (t, r) := − 1 2 (1 − 3 f )Φ (e) . (4.54)
we obtain Eqs. (4.37)-(4.41). Comparing Eqs. (4.40), (4.41), and (4.69), we choose V r so that
From
Eqs. (2.39) and (5.4), the componentsF AB is traceless. Then, we may concentrate on the components X (e) and Y (e) defined by Eq. (3.2) and the componentF as the metric perturbations. Furthermore, all arguments in Sec. 3 are valid even in the case of l = 1 modes. Therefore, we may use Eqs. (3.37)-(3.50) when we derive the l = 1 mode solutions to the linearized Einstein equations. From the definition (3.38) of Λ, we obtain Λ = 3(1 − f ). (5.5) Then, the Moncrief variable Φ (e) defined by Eq. (3.37) is given by
B)-component of the linearized Einstein equation (2.11) is given by2.41)
Using Eqs. (2.39), (2.40), and the background Einstein, the trace part of (p, q)-component of the
linearized Einstein equation (2.11) yields Eq. (2.38).
8/33
Finally, through Eqs. (2.39) and (2.40) and the background Einstein equations, the trace part of
the (A,
can be transformed to the Regge-Wheeler equation. This transformation is called the Chandrasekhar transformation. Since the Regge-Wheeler equation can be solved by MST (Mano Suzuki Takasugi) formulation [53-55], we may say that the solution to the Zerilli equation (3.46) without the source term is obtained through MST formulation. Finally, we note that the solutions Φ (e)F satisfy the equation (3.29), as the consistency of the linearized Einstein equation. Here, the source term S (F) is explicitly given by Eq. (3.8). Here, we check this consistency of the initial value constraint (3.40) and the evolution equation (3.29).3.49)
To solve the master equation (3.46) we have to impose appropriate boundary conditions and solve as
the Cauchy problem. In the book [19], it is shown that the Zerilli equation (3.46) without the source
term, i.e., S (Φ (e) ) = 0, From
Eqs. (3.29) and (3.46), we obtain
corresponds to the time-derivative of the variable Φ (e) with additional source terms from the matter fields. Therefore, it is trivial Ψ (e) also satisfies the Zerilli equation with different source terms. In other words, the Zerilli equation for Ψ (e) is derived by the time derivative of the Zerilli equation for Φ (e) . This means that the solution to the Zerilli equation for Ψ (e) may include an additional arbitrary function of r as an "integration constants." This "integration constants" do not included in the solution Φ (e) for the Zerilli equation (3.46). In this sense, the restriction of the initial value of Eq. (3.46) for Φ (e) is stronger than that of Eq. (3.46) for Ψ (e) .55)
the variable Ψ (e) corresponds to original Zerilli's master variable. Roughly speaking, the variable
Ψ (e)
AcknowledgementsThe author deeply acknowledged to Professor Hiroyuki Nakano for various discussions and suggestions.where we used Eq. (5.7) and (5.8) in the derivation of Eq. (5.9). Under the given the componentsT tt andT tr of the linearized energy-momentum tensor, Eqs. (5.8) and (5.9) yield that the componentF and Y (e) are determined by Φ (e) . Furthermore, substituting Eq. (5.8) into Eq. (5.7), we obtainThis also yields that the component X (e) is determined by Φ (e) under the given components of the linearized energy-momentum tensor. Thus, the components X (e) , Y (e) , andF are determined by the single variable Φ (e) apart from the contribution from the components of the linearized energymomentum tensor. The determination of the Moncrief variable Φ (e) is accomplished by solving the master equation (3.46):11)and the source term in Eq. (3.46) is given byThe master variable Φ (e) is determined through the master equation (5.11) with appropriate initial conditions. Furthermore, we have to take into account of the perturbation of the divergence of the energymomentum tensor, which are summarized as follows:The expression of (5.12) for the source term S (Φ (e) ) in Eq. (5.12) was derived by using Eq. (5.14).l = 1 mode vacuum caseAs in the case of l = 0 modes, it is instructive to consider the vacuum case where all components of the linearized energy-momentum tensor vanish before the derivation of the non-vacuum case.25/33Here, we consider the covariant form F ab of the l = 1-mode metric perturbation as follows:The harmonic function k (∆+2)m is explicitly given by Eqs. (2.31) and (2.32). If we impose the regularity on these harmonics by the choice δ = 0, these harmonics are given by the spherical harmonics Y l=1,m with l = 1:Since the extension of our arguments to m = ±1 modes is straightforward, we concentrate only on the m = 0 modes. For the m = 0 mode, the gauge-invariant part F ab of the metric perturbation is given by Here, Φ (e) is a solution to the equation
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"The Highly Energetic Expansion of SN 2010bh Associated with GRB 100316D",
"The Highly Energetic Expansion of SN 2010bh Associated with GRB 100316D"
]
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"Filomena Bufano \nINAF Post-Doc Fellow\nINAF -Osservatorio Astronomico di Catania\n95123CataniaItaly\n",
"Elena Pian \nINAF -Osservatorio Astronomico di Trieste\nVia G.B. Tiepolo 11I-34143TriesteItaly\n\nScuola Normale Superiore di Pisa\nPiazza dei Cavalieri 756126PisaItaly\n\nINFN -Sezione di Pisa\nLargo Pontecorvo 356127PisaItaly\n",
"Jesper Sollerman \nDepartment of Astronomy\nThe Oskar Klein Centre\nAlbaNova, SE-106 91StockholmSweden\n",
"Stefano Benetti \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"Giuliano Pignata \nDepartamento de Ciencias Fisicas\nUniversidad Andres Bello\nAv. Republica 252SantiagoChile\n",
"Stefano Valenti \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"Stefano Covino \nINAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly\n",
"Paolo D'avanzo \nINAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly\n",
"Daniele Malesani \nDark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark\n",
"Enrico Cappellaro \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"MassimoDella Valle \nINAF -Osservatorio Astronomico di Capodimonte\nSalita Moiariello, 16I-8013NapoliItaly\n\nInternational Center for Relativistic Astrophysics Network\nPescaraItaly\n",
"Johan Fynbo \nDark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark\n",
"Jens Hjorth \nDark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark\n",
"Paolo A Mazzali \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n\nMax-Planck Institut fur Astrophysik\nKarl-Schwarzschildstr. 1D-85748GarchingGermany\n",
"Daniel E Reichart \nUniversity of North Carolina at Chapel Hill\nCampus, Chapel HillBox 325527599-3255NCUSA\n",
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"Guido Chincarini \nDip. Fisica G. Occhialini\nUniverisit Milano Bicocca\nP.zza della Scienza 320126MilanoItaly\n",
"Valerio D'elia \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly\n\nASI-Science Data Center\nVia Galileo GalileiI-00044FrascatiItaly\n",
"Antonio De Ugarte Postigo \nDark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark\n",
"Jinsong Deng \nNational Astronomical Observatories\nCAS\n20A Datun Road100012Chaoyang District, BeijingChina\n",
"Patrizia Ferrero \nInstituto de Astrofsica de Canarias (IAC)\n38200 La LagunaTenerifeSpain\n",
"Alexei V Filippenko \nDepartment of Astronomy\nUniversity of California\n94720-3411BerkeleyCAUSA\n",
"Paolo Goldoni \nLaboratoire Astroparticule et Cosmologie\n10 rue A. Domon et L. Duquet75205Paris Cedex 13France\n\nDSM/IRFU/SAp\nService d'Astrophysique\nCEA-Saclay\n91191Gif-sur-YvetteFrance\n",
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"Jochen Greiner \nMax-Planck Institut für extraterrestrische Physik\nGiessenbachstrasse 1D-85740GarchingGermany\n",
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"Lex Kaper \nCentre for Astrophysics and Cosmology\nScience Institute\nUniversity of Iceland\nDunhagi 5, 107 Reyk--3\n\nAstronomical Institute Anton Pannekoek\nUniversity of Amsterdam\nScience Park 904, 1098 XH Ams-terdamThe Netherlands\n",
"Koji S Kawabata \nHiroshima Astrophysical Science Center\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan\n",
"Sylvio Klose \nThuringer Landessternwarte Tautenburg\nSternwarte 5D-07778TautenburgGermany\n",
"Andrew J Levan \nDepartment of Physics\nUniversity of Warwick\nCV4 7ALCoventryUK\n",
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"Bo Milvang-Jensen \nDark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark\n",
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"Palle Møller ",
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"Silvia Piranomonte \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly\n",
"Ruben Salvaterra \nDipartimento di Fisica e Matematica\nUniversità dell'Insubria, via Valleggio 722100ComoItaly\n",
"Giulia Stratta \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly\n",
"Gianpiero Tagliaferri \nINAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly\n",
"Masaomi Tanaka \nInstitute for the Physics and Mathematics of the Universe\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaChibaJapan\n",
"Nial R Tanvir \nDepartment of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK\n",
"Ralph A M J Wijers \nINAF -Osservatorio Astronomico di Trieste\nVia G.B. Tiepolo 11I-34143TriesteItaly\n\nAstronomical Institute Anton Pannekoek\nUniversity of Amsterdam\nScience Park 904, 1098 XH Ams-terdamThe Netherlands\n"
]
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"INAF Post-Doc Fellow\nINAF -Osservatorio Astronomico di Catania\n95123CataniaItaly",
"INAF -Osservatorio Astronomico di Trieste\nVia G.B. Tiepolo 11I-34143TriesteItaly",
"Scuola Normale Superiore di Pisa\nPiazza dei Cavalieri 756126PisaItaly",
"INFN -Sezione di Pisa\nLargo Pontecorvo 356127PisaItaly",
"Department of Astronomy\nThe Oskar Klein Centre\nAlbaNova, SE-106 91StockholmSweden",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"Departamento de Ciencias Fisicas\nUniversidad Andres Bello\nAv. Republica 252SantiagoChile",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly",
"INAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly",
"Dark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Capodimonte\nSalita Moiariello, 16I-8013NapoliItaly",
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"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"Max-Planck Institut fur Astrophysik\nKarl-Schwarzschildstr. 1D-85748GarchingGermany",
"University of North Carolina at Chapel Hill\nCampus, Chapel HillBox 325527599-3255NCUSA",
"Department of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"INAF -Osservatorio Astronomico di Trieste\nVia G.B. Tiepolo 11I-34143TriesteItaly",
"INAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly",
"Department of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"INAF-Istituto di Astrofisica Spaziale e Fisica cosmica\nVia Gobetti 101I-40129BolognaItaly",
"Astrophysics Research Institute\nLiverpool John Moores University\n2 Rodney StL3 5UXLiverpoolUK",
"INAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly",
"Astrophysics Research Institute\nLiverpool John Moores University\n2 Rodney StL3 5UXLiverpoolUK",
"Instituto de Astrofisica de Andalucia (IAA-CSIC)\nGlorieta de la Astronomia s/n18008GranadaSpain",
"Dip. Fisica G. Occhialini\nUniverisit Milano Bicocca\nP.zza della Scienza 320126MilanoItaly",
"INAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly",
"ASI-Science Data Center\nVia Galileo GalileiI-00044FrascatiItaly",
"Dark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark",
"National Astronomical Observatories\nCAS\n20A Datun Road100012Chaoyang District, BeijingChina",
"Instituto de Astrofsica de Canarias (IAC)\n38200 La LagunaTenerifeSpain",
"Department of Astronomy\nUniversity of California\n94720-3411BerkeleyCAUSA",
"Laboratoire Astroparticule et Cosmologie\n10 rue A. Domon et L. Duquet75205Paris Cedex 13France",
"DSM/IRFU/SAp\nService d'Astrophysique\nCEA-Saclay\n91191Gif-sur-YvetteFrance",
"Instituto de Astrofisica de Andalucia (IAA-CSIC)\nGlorieta de la Astronomia s/n18008GranadaSpain",
"Max-Planck Institut für extraterrestrische Physik\nGiessenbachstrasse 1D-85740GarchingGermany",
"GEPI-Observatoire de Paris Meudon\n5 Place Jules JannsenF-92195MeudonFrance",
"Centre for Astrophysics and Cosmology\nScience Institute\nUniversity of Iceland\nDunhagi 5, 107 Reyk--3",
"Astronomical Institute Anton Pannekoek\nUniversity of Amsterdam\nScience Park 904, 1098 XH Ams-terdamThe Netherlands",
"Hiroshima Astrophysical Science Center\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan",
"Thuringer Landessternwarte Tautenburg\nSternwarte 5D-07778TautenburgGermany",
"Department of Physics\nUniversity of Warwick\nCV4 7ALCoventryUK",
"Institute for the Physics and Mathematics of the Universe\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaChibaJapan",
"INAF -Istituto di Astrofisica Spaziale e Fisica cosmica\nVia Gobetti 101, I-40129, Karl-Schwarzschild-Str. 285748Bologna, GarchingItaly, Germany",
"Dark Cosmology Centre\nNiels Bohr Institute\nUniversity of Copenhagen\nJuliane Maries Vej 30DK-2100CopenhagenDenmark",
"DSM/IRFU/SAp\nService d'Astrophysique\nCEA-Saclay\n91191Gif-sur-YvetteFrance",
"Institute for the Physics and Mathematics of the Universe\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaChibaJapan",
"INAF -Istituto di Astrofisica Spaziale e Fisica cosmica\nVia Gobetti 101, I-40129, Karl-Schwarzschild-Str. 285748Bologna, GarchingItaly, Germany",
"INAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly",
"Dipartimento di Fisica e Matematica\nUniversità dell'Insubria, via Valleggio 722100ComoItaly",
"INAF -Osservatorio Astronomico di Roma\nvia di Frascati 33, 00040 Monte Porzio CatoneRomeItaly",
"INAF -Osservatorio Astronomico di Brera\nVia Emilio Bianchi 46I-23807MerateItaly",
"Institute for the Physics and Mathematics of the Universe\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaChibaJapan",
"Department of Physics and Astronomy\nUniversity of Leicester\nUniversity RoadLE1 7RHLeicesterUK",
"INAF -Osservatorio Astronomico di Trieste\nVia G.B. Tiepolo 11I-34143TriesteItaly",
"Astronomical Institute Anton Pannekoek\nUniversity of Amsterdam\nScience Park 904, 1098 XH Ams-terdamThe Netherlands"
]
| []
| We present the spectroscopic and photometric evolution of the nearby (z = 0.059) spectroscopically confirmed type Ic supernova, SN 2010bh, associated with the soft, long-duration gamma-ray burst (X-ray flash) GRB 100316D. Intensive follow-up observations of SN 2010bh were performed at the ESO Very Large Telescope (VLT) using the X-shooter and FORS2 instruments. Owing to the detailed temporal coverage and the extended wavelength range (3000-24800Å), we obtained an unprecedentedly rich spectral sequence among the hypernovae, making SN 2010bh one of the best studied representatives of this SN class. We find that SN 2010bh has a more rapid rise to maximum brightness (8.0 ± 1.0 rest-frame days) and a fainter absolute peak luminosity (L bol ≈ 3 × 10 42 erg s −1 ) than previously observed SN events associated with GRBs. Our estimate of the ejected 56 Ni mass is 0.12 ± 0.02 M ⊙ . From the broad spectral features we measure expansion velocities up to 47,000 km s −1 , higher than those of SNe 1998bw (GRB 980425) and 2006aj (GRB 060218). Helium absorption lines He I λ5876 and He I 1.083 µm, blueshifted by ∼20,000-30,000 km s −1 and ∼28,000-38,000 km s −1 , respectively, may be present in the optical spectra. However, the lack of coverage of the He I 2.058µm line prevents us from confirming such identifications. The nebular spectrum, taken at ∼186 days after the explosion, shows a broad but faint [O I] emission at 6340Å. The light-curve shape and photospheric expansion velocities javík, Iceland | 10.1088/0004-637x/753/1/67 | [
"https://arxiv.org/pdf/1111.4527v2.pdf"
]
| 54,975,718 | 1111.4527 | dc01e401d89d6b3ee152aa59d23b7e8a05ca0f5e |
The Highly Energetic Expansion of SN 2010bh Associated with GRB 100316D
2 May 2012
Filomena Bufano
INAF Post-Doc Fellow
INAF -Osservatorio Astronomico di Catania
95123CataniaItaly
Elena Pian
INAF -Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11I-34143TriesteItaly
Scuola Normale Superiore di Pisa
Piazza dei Cavalieri 756126PisaItaly
INFN -Sezione di Pisa
Largo Pontecorvo 356127PisaItaly
Jesper Sollerman
Department of Astronomy
The Oskar Klein Centre
AlbaNova, SE-106 91StockholmSweden
Stefano Benetti
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
Giuliano Pignata
Departamento de Ciencias Fisicas
Universidad Andres Bello
Av. Republica 252SantiagoChile
Stefano Valenti
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
Stefano Covino
INAF -Osservatorio Astronomico di Brera
Via Emilio Bianchi 46I-23807MerateItaly
Paolo D'avanzo
INAF -Osservatorio Astronomico di Brera
Via Emilio Bianchi 46I-23807MerateItaly
Daniele Malesani
Dark Cosmology Centre
Niels Bohr Institute
University of Copenhagen
Juliane Maries Vej 30DK-2100CopenhagenDenmark
Enrico Cappellaro
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
MassimoDella Valle
INAF -Osservatorio Astronomico di Capodimonte
Salita Moiariello, 16I-8013NapoliItaly
International Center for Relativistic Astrophysics Network
PescaraItaly
Johan Fynbo
Dark Cosmology Centre
Niels Bohr Institute
University of Copenhagen
Juliane Maries Vej 30DK-2100CopenhagenDenmark
Jens Hjorth
Dark Cosmology Centre
Niels Bohr Institute
University of Copenhagen
Juliane Maries Vej 30DK-2100CopenhagenDenmark
Paolo A Mazzali
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
Max-Planck Institut fur Astrophysik
Karl-Schwarzschildstr. 1D-85748GarchingGermany
Daniel E Reichart
University of North Carolina at Chapel Hill
Campus, Chapel HillBox 325527599-3255NCUSA
Rhaana L C Starling
Department of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Massimo Turatto
INAF -Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11I-34143TriesteItaly
Susanna D Vergani
INAF -Osservatorio Astronomico di Brera
Via Emilio Bianchi 46I-23807MerateItaly
Klass Wiersema
Department of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Lorenzo Amati
INAF-Istituto di Astrofisica Spaziale e Fisica cosmica
Via Gobetti 101I-40129BolognaItaly
David Bersier
Astrophysics Research Institute
Liverpool John Moores University
2 Rodney StL3 5UXLiverpoolUK
Sergio Campana
INAF -Osservatorio Astronomico di Brera
Via Emilio Bianchi 46I-23807MerateItaly
Zach Cano
Astrophysics Research Institute
Liverpool John Moores University
2 Rodney StL3 5UXLiverpoolUK
Alberto J Castro-Tirado
Instituto de Astrofisica de Andalucia (IAA-CSIC)
Glorieta de la Astronomia s/n18008GranadaSpain
Guido Chincarini
Dip. Fisica G. Occhialini
Univerisit Milano Bicocca
P.zza della Scienza 320126MilanoItaly
Valerio D'elia
INAF -Osservatorio Astronomico di Roma
via di Frascati 33, 00040 Monte Porzio CatoneRomeItaly
ASI-Science Data Center
Via Galileo GalileiI-00044FrascatiItaly
Antonio De Ugarte Postigo
Dark Cosmology Centre
Niels Bohr Institute
University of Copenhagen
Juliane Maries Vej 30DK-2100CopenhagenDenmark
Jinsong Deng
National Astronomical Observatories
CAS
20A Datun Road100012Chaoyang District, BeijingChina
Patrizia Ferrero
Instituto de Astrofsica de Canarias (IAC)
38200 La LagunaTenerifeSpain
Alexei V Filippenko
Department of Astronomy
University of California
94720-3411BerkeleyCAUSA
Paolo Goldoni
Laboratoire Astroparticule et Cosmologie
10 rue A. Domon et L. Duquet75205Paris Cedex 13France
DSM/IRFU/SAp
Service d'Astrophysique
CEA-Saclay
91191Gif-sur-YvetteFrance
Javier Gorosabel
Instituto de Astrofisica de Andalucia (IAA-CSIC)
Glorieta de la Astronomia s/n18008GranadaSpain
Jochen Greiner
Max-Planck Institut für extraterrestrische Physik
Giessenbachstrasse 1D-85740GarchingGermany
Francois Hammer
GEPI-Observatoire de Paris Meudon
5 Place Jules JannsenF-92195MeudonFrance
Lex Kaper
Centre for Astrophysics and Cosmology
Science Institute
University of Iceland
Dunhagi 5, 107 Reyk--3
Astronomical Institute Anton Pannekoek
University of Amsterdam
Science Park 904, 1098 XH Ams-terdamThe Netherlands
Koji S Kawabata
Hiroshima Astrophysical Science Center
Hiroshima University
1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan
Sylvio Klose
Thuringer Landessternwarte Tautenburg
Sternwarte 5D-07778TautenburgGermany
Andrew J Levan
Department of Physics
University of Warwick
CV4 7ALCoventryUK
Keiichi Maeda
Institute for the Physics and Mathematics of the Universe
University of Tokyo
5-1-5 Kashiwanoha277-8583KashiwaChibaJapan
Nicola Masetti
INAF -Istituto di Astrofisica Spaziale e Fisica cosmica
Via Gobetti 101, I-40129, Karl-Schwarzschild-Str. 285748Bologna, GarchingItaly, Germany
Bo Milvang-Jensen
Dark Cosmology Centre
Niels Bohr Institute
University of Copenhagen
Juliane Maries Vej 30DK-2100CopenhagenDenmark
Felix I Mirabel
DSM/IRFU/SAp
Service d'Astrophysique
CEA-Saclay
91191Gif-sur-YvetteFrance
Palle Møller
Ken'ichi Nomoto
Institute for the Physics and Mathematics of the Universe
University of Tokyo
5-1-5 Kashiwanoha277-8583KashiwaChibaJapan
Eliana Palazzi
INAF -Istituto di Astrofisica Spaziale e Fisica cosmica
Via Gobetti 101, I-40129, Karl-Schwarzschild-Str. 285748Bologna, GarchingItaly, Germany
Silvia Piranomonte
INAF -Osservatorio Astronomico di Roma
via di Frascati 33, 00040 Monte Porzio CatoneRomeItaly
Ruben Salvaterra
Dipartimento di Fisica e Matematica
Università dell'Insubria, via Valleggio 722100ComoItaly
Giulia Stratta
INAF -Osservatorio Astronomico di Roma
via di Frascati 33, 00040 Monte Porzio CatoneRomeItaly
Gianpiero Tagliaferri
INAF -Osservatorio Astronomico di Brera
Via Emilio Bianchi 46I-23807MerateItaly
Masaomi Tanaka
Institute for the Physics and Mathematics of the Universe
University of Tokyo
5-1-5 Kashiwanoha277-8583KashiwaChibaJapan
Nial R Tanvir
Department of Physics and Astronomy
University of Leicester
University RoadLE1 7RHLeicesterUK
Ralph A M J Wijers
INAF -Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11I-34143TriesteItaly
Astronomical Institute Anton Pannekoek
University of Amsterdam
Science Park 904, 1098 XH Ams-terdamThe Netherlands
The Highly Energetic Expansion of SN 2010bh Associated with GRB 100316D
2 May 2012. 35 IAFE-CONICET-UBA. cc67, suc 28, Buenos Aires, Argentina 36 European Organization for Astronomical Research in the Southern Hemisphere (ESO),Subject headings: supernovae: general -supernovae: individual SN 2010bh, GRB 100316D
We present the spectroscopic and photometric evolution of the nearby (z = 0.059) spectroscopically confirmed type Ic supernova, SN 2010bh, associated with the soft, long-duration gamma-ray burst (X-ray flash) GRB 100316D. Intensive follow-up observations of SN 2010bh were performed at the ESO Very Large Telescope (VLT) using the X-shooter and FORS2 instruments. Owing to the detailed temporal coverage and the extended wavelength range (3000-24800Å), we obtained an unprecedentedly rich spectral sequence among the hypernovae, making SN 2010bh one of the best studied representatives of this SN class. We find that SN 2010bh has a more rapid rise to maximum brightness (8.0 ± 1.0 rest-frame days) and a fainter absolute peak luminosity (L bol ≈ 3 × 10 42 erg s −1 ) than previously observed SN events associated with GRBs. Our estimate of the ejected 56 Ni mass is 0.12 ± 0.02 M ⊙ . From the broad spectral features we measure expansion velocities up to 47,000 km s −1 , higher than those of SNe 1998bw (GRB 980425) and 2006aj (GRB 060218). Helium absorption lines He I λ5876 and He I 1.083 µm, blueshifted by ∼20,000-30,000 km s −1 and ∼28,000-38,000 km s −1 , respectively, may be present in the optical spectra. However, the lack of coverage of the He I 2.058µm line prevents us from confirming such identifications. The nebular spectrum, taken at ∼186 days after the explosion, shows a broad but faint [O I] emission at 6340Å. The light-curve shape and photospheric expansion velocities javík, Iceland
of SN 2010bh suggest that we witnessed a highly energetic explosion with a small ejected mass (E k ≈ 10 52 erg and M ej ≈ 3 M ⊙ ). The observed properties of SN 2010bh further extend the heterogeneity of the class of GRB supernovae.
Subject headings: supernovae: general -supernovae: individual SN 2010bh, GRB 100316D
Introduction
During the past decade, the link between long-duration gamma-ray bursts (GRBs) and type Ic core-collapse supernovae (SNe; e.g., Filippenko 1997) has been firmly established; see Woosley & Bloom (2006) and Hjorth & Bloom (2011) for reviews. The first clear case occurred in 1998, when the luminous SN 1998bw, at z = 0.0085, was found spatially and temporally coincident with GRB 980425 (Galama et al. 1998). The GRB-SN connection was supported in 2003 by two further associations between nearby GRBs and spectroscopically confirmed SNe: GRB 030329/SN 2003dh at redshift z = 0.17 (Hjorth et al. 2003;Stanek et al. 2003;Matheson et al. 2003) and GRB 031203/ SN 2003lw at z = 0.11 (Malesani et al. 2004; Thomsen et al. 2004;Gal-Yam et al. 2004;Cobb et al. 2004). The most recent case of a spectroscopic connection is GRB 060218/SN 2006aj (z = 0.033, Campana et al. 2006;Pian et al. 2006;Mirabal et al. 2006;Modjaz et al. 2006;Cobb et al. 2006;Ferrero et al. 2006). On average, SNe associated with classical GRBs appear to be more luminous at peak than SNe Ic not accompanied by GRBs, while SNe associated with X-ray flashes have maximum luminosities more similar to those of normal SNe Ic (see, e.g., Pian et al. 2006;Pignata et al. 2011;Drout et al. 2010). However, SNe associated with both GRBs and X-ray flashes exhibit broader features in their spectra, indicating unusually large expansion velocities. From the modeling of their light curves and spectra, very high explosion energies are inferred (∼ 10 52 erg, about 10 times higher than typical SNe), which made earn them the name of hypernovae (HNe; Paczyński 1998;Iwamoto et al. 1998). The GRB-SN connection has been best studied at low redshift (z < 0.2), where the clear, spectroscopically confirmed cases have been detected. Although GRBs at these low redshifts are rarely observed, the association between GRBs and SNe has been extended up to z ∼ 1, (corresponding to a look back time of about 60% the age of the Universe) in a number of GRB-SNe which have been identified through single epoch spectra characterized by the presence of SN features (Lazzati 2001;Della Valle et al. 2003Soderberg et al. 2005;Bersier et al. 2006;Cobb et al. 2010;Cano et al. 2011a;Sparre et al. 2011;Berger et al. 2011). The investigation of all confirmed GRB-SN associations is critical to understand the nature of their progenitors and the mechanism by which powerful stripped-envelope SNe produce ultra-relativistic jets (e.g., Zhang et al. 2003;Uzdensky & MacFadyen 2006;Mazzali et al. 2008;Fryer et al. 2009;Lyutikov 2011).
GRB 100316D ) is a low-redshift event (z = 0.059; Vergani et al. 2010;Starling et al. 2011; see also §3.1), whose prompt emission is characterized by a very soft spectral peak, similar to that of X-ray flashes (XRFs; GRBs with energy peak at low frequency; Heise et al. 2001), and a slowly decaying flux. A few days after its detection, an associated type Ic supernova was identified through the spectral features of the early optical counterpart: SN 2010bh (Chornock et al. 2010a,b;Wiersema et al. 2010;Bufano et al. 2010). Similar to the low-redshift X-ray flash GRB 060218, GRB 100316D had an unusually long duration (T 90 > 1300 s) and a spectral-hardness evolution with a stable and soft spectral shape throughout the prompt and late-time emission (Starling et al. 2011). The early X-ray spectrum of GRB 100316D, like that of GRB 060218 (Campana et al. 2006), is best described by a power law plus a thermal component (Starling et al. 2011;Fan et al. 2011). The latter may be the signature of either the shock breakout following core collapse (Campana et al. 2006;Waxman et al. 2007), or additional radiation from the central engine (e.g., Ghisellini et al. 2007;Li 2007;Chevalier & Fransson 2008).
We intensively monitored the optical and near-infrared (NIR) spectrophotometric evolution of SN 2010bh, starting ∼12 hr after the Swift/BAT GRB trigger, which occurred on 2010 March 16 at 12:44:50 ; UT dates are used throughout this paper), until about 2 months past the discovery, when solar constraints made no longer observable the SN. To enable accurate host-galaxy subtraction, we reobserved the field 0.5-1 yr after the explosion. This campaign was the outcome of the coordination of various observing programs at the European Southern Observatory (ESO, Chile) Very Large Telescope (VLT) 1 and at the Cerro Tololo Interamerican Observatory (CTIO, Chile). The spectral behavior of SN 2010bh from day 1.39 to day 21.2 has also been discussed by Chornock et al. (2010c) and photometry during the first 3 months after explosion has been presented by Cano et al. (2011b) and Olivares E. et al. (2011). The larger wavelength range and improved phase coverage of our observations allow us to analyze the early phases in more detail and to push the investigation into the late evolutionary stages.
In §2, we present the dataset and describe the data reduction methods, while in §3 we show the spectrophotometric evolution of SN 2010bh and compare it to that of the previously well-studied GRB-SNe. In §4, we discuss the derived properties of the progenitor star and the explosion parameters, and we summarize our conclusions.
Data Acquisition and Reduction
Photometry
UBV RI photometry of SN 2010bh was obtained with the FOcal Reducer and low dispersion Spectrograph (FORS2; field of view [FOV] 6.8 ′ ×6.8 ′ ; scale 0.25 ′′ pixel −1 ; Appenzeller et al. 1998) at the ESO VLT UT1 and with the 0.41-m Panchromatic Robotic Optical Monitoring and Polarimetry Telescopes (PROMPTs) 1 and 5 located at CTIO (FOV 10 ′ × 10 ′ , 0.6 ′′ pixel −1 ; Reichart et al. 2005). The R-band images from the X-shooter Acquisition Camera (FOV 1.47 ′ × 1.47 ′ ; 0.173 ′′ pixel −1 ; D' Odorico et al. 2006) were also used to cover the early evolution. The data were reduced with standard techniques using IRAF 2 tasks. Since the SN exploded in a region with a complex background, a template subtraction method, based on the ISIS package (Alard & Lupton 1998, Alard 2000, was applied to remove contamination from the host-galaxy light. As no pre-explosion observations were available of SN 2010bh, we used images obtained with VLT/FORS2 on September 17 (about 185 days past explosion; see Table 1), assuming that the SN flux contribution was negligible at this epoch (see §3). For PROMPTs observations, template images were acquired on 2011 February 2, with the exception of the I band at PROMPT5, for which the template image was acquired on 2011 January 26 (see Table 1).
A point-spread-function (PSF) fitting method was applied to measure the SN magnitudes in the difference images. In particular, the PSF was derived from field stars measured on either the template or target image, whichever had the worst seeing. The kernel size was scaled according to the seeing to avoid wings truncation. We also performed aperture photometry on the subtracted images obtaining very similar results. The uncertainties were estimated by means of artificial stars with the same magnitude as the SN: we firstly placed them close to the SN position (within few pixels) and then in the opposite side of the galaxy, where the background residuals were similar. The two approaches, intended to test for the effect of different background contributions, gave similar results. By observing several photometric standard fields, from Landolt (1992) for the U band and from Stetson (2000) for the BV RI filters, we obtained the color equations for each night and instrument, and used them to transform instrumental magnitudes to the standard photometric system. We calibrated the magnitudes of a local sequence of stars in the SN 2010bh field over four photometric nights (April 3, 5, 8 and 11) and used them to obtain the photometric zeropoints for the non-photometric nights ( Fig. 1 and Table 2). Because of the significant color term, we applied a calibration correction (S-correction) to the PROMPTs instrumental SN magnitudes to transform them into a standard photometric system following Pignata et al. (2008). Finally, a K-correction based on the nearly simultaneous spectra was applied to the observed BV RI magnitudes. For the U-band magnitudes, the signal-to-noise (S/N) ratio was too low to measure a reliable K-correction from the spectra. UBV RI magnitudes of the supernova are reported in Table 3 and plotted in Figure 2. We computed upper limit magnitudes (Table 3 and Fig. 2) placing an artificial star at the SN position on the background subtracted images and decreasing its magnitude down to the point of detectability over the background level. At template images epochs, limiting magnitudes were measured directly over the host galaxy level, reported in Table 3 but not in Fig. 2 because they do not set tight limits to the late phase decline.
Spectroscopy
We followed the spectroscopic evolution of SN 2010bh with VLT/X-shooter (12 epochs) and VLT/FORS2 (8 epochs), as reported in Table 4. When possible, the spectra were acquired with the slit positioned along the North-South direction, to minimize the host-galaxy contamination. For both instruments, the effects of atmospheric dispersion (Filippenko 1982) at high airmass have been reduced by using an atmospheric dispersion corrector. Simultaneous UV, VIS, and NIR spectra (∼ 3000-24800Å) were taken with X-shooter using slit widths of 1.0 ′′ , 0.9 ′′ , and 0.9 ′′ for each arm, respectively (D'Odorico et al. 2006). We used a nodding throw along the slit (nodding lengths of 2 ′′ and 4 ′′ ) to obtain better sky subtraction. The data were reduced using version 0.9.4 of the ESO X-shooter pipeline (Goldoni et al. 2006) with the calibration frames (biases, darks, arc lamps, and flatfields) taken during daytime. After reducing the data using more advanced versions of the software, no relevant changes were found. With FORS2 we used the 300V grism (3300-9000Å) and a slitwidth of 1 ′′ . Both X-shooter and FORS2 spectra were extracted using standard IRAF tasks. Spectrophotometric and telluric standard-star exposures taken on the same night as the SN 2010bh observations were used to flux-calibrate the extracted spectra and to remove telluric absorption features. We checked the absolute flux calibration of the spectra by using the nearly simultaneous R-band magnitudes. Figure 5 shows the spectral sequence after correcting for both Galactic and host galaxy reddening; both the wavelength scale and epochs are reported in the host-galaxy rest frame (see §3.1 and §3.3). The most prominent emission lines of the host galaxy have been removed. Figures 6 is a zoom-in of the spectral sequence in the optical range. The three spectra taken in the nebular phase (2010 September 28 to October 1; see Table 4) have been combined to improve the signal-to-noise ratio. The coadded spectrum is Magnitudes have not been corrected for Galactic and host galaxy extinction. K-corrections have been applied to the BV RI magnitudes. FORS2 and X-shooter R-band photometry is reported with black and red solid circles, respectively. Open symbols are used for V RI magnitudes obtained with PROMPT. For clarity, the light curves are vertically displaced by the amount reported in the legend for each filter. shown in Figure 9 (see §3.4). We estimated the host-galaxy extinction by measuring the total equivalent width (EW) of the interstellar Na I D absorption doublet (λλ5890.0, 5895.9) with the assumption of a gas-to-dust ratio similar to the average ratio in our Galaxy. Measurements were performed on a spectrum obtained combining the almost featureless early-epoch X-shooter spectra (phases 2.4d, 3.3d and 4.2d, see Table 4). We found EW(λ5890.0) host = 0.59 ± 0.05Å and EW(λ5895.9) host = 0.30 ± 0.02Å, giving a total EW(Na I D) host = 0.89 ± 0.07Å (Fig. 3). Applying the relation by Turatto et al. (2003), E(B − V ) = 0.16 × EW(Na I D), we obtained E(B − V ) host = 0.14 ± 0.01 mag, which is the value we adopt throughout this work. For the Milky Way extinction, we measured EW(λ5890.0) MW = 0.40 ± 0.10Å, while the second doublet component was not detectable (Fig. 3). Assuming a flux ratio 2:1 between the two absorption lines, we obtained a total EW(Na I D) MW = 0.60 ± 0.15Å, implying E(B − V ) MW = 0.10 ± 0.03 mag. This value is in agreement with that found by Schlegel et al. (1998), E(B − V ) MW = 0.12 mag. We decided to adopt the latter because of the large uncertainty in our estimate of EW(Na I D) MW .
Results
Host-Galaxy Properties
Recently, Poznanski et al. (2011) and Olivares E. et al. (2010) claimed that Na I D absorption may be a bad proxy for the extinction, especially if one uses low-resolution spectra where the two doublet lines cannot be resolved. Although we have a higher dispersion in the X-shooter spectra that allows us to separate the two components, we checked our result by estimating the reddening inside the host galaxy (along the line of sight) from the Balmer-line intensity ratios of the H II region coincident with the SN. Firstly, we corrected both X-shooter and FORS2 spectra for the Milky Way extinction, then, assuming Case B recombination (T = 10 4 K; Osterbrock 1989), we measured the Hα/Hβ ratios from each spectrum. We obtained an average value of E(B − V ) host = 0.18 ± 0.06 mag, in agreement with that used in this work (E(B − V ) host = 0.14 mag). An independent and consistent estimate of the reddening has been given by Cano et al. (2011b), who found a host galaxy color excess E(B − V ) host = 0.18 ± 0.08 mag comparing the SN 2010bh colors with those of the type Ibc SN sample studied by Drout et al. (2010). A higher value for the host galaxy reddening (E(B − V ) host = 0.39 ± 0.03 mag) has been found by Olivares E. et al. (2011), by fitting a broad-band SED constructed using GROND and Swift/XRT data. While Cano et al. (2011b) and Olivares E. et al. (2011) have estimated E(B − V ) host from indirect methods (statistics of Type Ic SNe and SED modeling, respectively), our procedure is based on a direct estimate of the dust amount in the line of sight of the SN from the optical spectra, the only necessary underlying assumption consisting in considering the dust-to-gas ratio as constant and equal to the Galactic one.
We also used the 12 X-shooter spectra to estimate the metallicity of the bright region underlying SN 2010bh. We measured both the N2 and O3N2 diagnostic ratios (Pettini et al. 2004), obtaining an average oxygen abundance 12 + log(O/H) = 8.20 ± 0.24, where the error is dominated by the uncertainties associated with the adopted linear relationships. The values of ∼8.2 reported by Chornock et al. (2010c) at the SN location, which is based on a spectrum at +3.3 days after the explosion, and 8.2 ± 0.1 by Levesque et al. (2011), which is based on a spectrum at +52 days, are in excellent agreement with our estimate. From the spectrum of the H II region located close to the SN, Starling et al. (2011) found an oxygen abundance of 8.23 ± 0.15. Figure 2 illustrates the SN 2010bh light curves. Our R-band light curve traces well the early evolutionary stages; the SN reaches maximum light (M R ≈ −18.5 mag) at 8.0±1.0 restframe days past the explosion, confirming the rise time independently found by Cano et al. (2011b) and Olivares E. et al. (2011). This is the steepest rise to maximum brightness ever found among both SNe associated with GRBs and broad-lined (BL) SNe for which explosion dates have been well constrained (e.g., SN 2002ap: R max at 12 days; Mazzali et al. 2007b;Foley et al. 2003;SN 2003jd: R max at ∼ 16 days; Valenti et al. 2008b;SN 2005nc: R max at Table 4) X-shooter spectra.
SN 2010bh Light Curves
∼ 12 days; Della Valle et al. 2006).
A decline of 0.056 ± 0.015 mag day −1 is measured between 0 and 15 days after R-band maximum. Good sampling of the post-maximum phases was also obtained in the UBV I bands (Fig. 2). In Figure 4, we compare the R-band light curve of SN 2010bh with those of two previous well-sampled GRB-SNe: SN 1998bw (Galama et al. 1998Patat et al. 2001) and SN 2006aj (Sollerman et al. 2006;Pian et al. 2006;Ferrero et al. 2006). The light curve of a more typical type Ic SN (SN 1994I; Richmond et al. 1996) is also shown.
After maximum brightness, SN 2010bh and SN 1998bw have a similar behavior, but the decay rate of SN 2010bh between the last two R-band points corresponds to ∼ 0.03 mag day −1 in the rest frame, to be compared with that of SN 1998bw (0.013 mag day −1 ). Using the decline rates found by Patat et al. (2001) for SN 1998bw, we estimate the magnitudes of SN 2010bh at the epoch when the VLT subtraction images were acquired (+166.5 rest-frame days from maximum; see Table 1): V ≈ 25.0, R ≈ 23.8, and I ≈ 23.6 mag. Considering this as an upper limit, if it was indeed the flux level of SN 2010bh, it would cause an oversubtraction and an underestimate of the SN fluxes in the V , R, and I bands by less than 0.03 mag around maximum and 0.15 mag in the latest epochs. For each epoch, the corresponding inferred possible contamination by a residual SN flux has been included in the error estimate (Table 3). SN 2010bh is less luminous than the other two GRB-SNe (Fig. 4), which suggests a smaller amount of ejected 56 Ni mass. Moreover, since the width of a light curve scales with the ratio between the total ejected mass M ej and the total explosion kinetic energy E k (Arnett 1982(Arnett , 1996, the fast evolution of SN 2010bh likely reveals a relatively highly energetic explosion and/or a small M ej . On the other hand, a highly asymmetric explosion may also result in a faster expansion in the polar direction, leading to a short diffusion time and a fast rise time (Maeda et al. 2006).
We note that our R-band photometry (neither corrected for Galactic or host extinction) at 0.5 days is ∼1 mag and ∼0.6 mag fainter than those of Cano et al. (2011b) and Olivares E. et al. (2011), respectively. This implies that we observe in our data the smooth rise in flux typical of a SN, and no evidence of the extra early component that they interpret as shock breakout. While we processed all raw data in a homogeneous way and consider this first flux point and its uncertainty formally correct, we caution that at these low flux levels the X-Shooter acquisition camera imaging data may depend very critically on the assumed background. (Galama et al. 1998;Patat et al. 2001) and SN 2006aj (Sollerman et al. 2006;Pian et al. 2006;Ferrero et al. 2006), both of which were associated with a GRB/XRF, and to that of the type Ic SN 1994I (Richmond et al. 1996), not accompanied by a high-energy event. All SN light curves have been corrected for total reddening along the line of sight and are reported in the host-galaxy rest frame. The epochs are given with respect to the burst detection. For SN 1994I, the explosion date was obtained from the light-curve models of Iwamoto et al. (1994).
Spectra in the Photospheric Phase
The spectral sequence of SN 2010bh (Figures 5 and 6) is unique among BL-SNe and HNe for its detailed temporal coverage and extended wavelength range (3000-24800Å). At early epochs, the spectral energy distribution can be fit with a blackbody spectrum with T bb ≈ 8500 K. Thereafter, the continuum, shaped by broad bumps and absorptions, becomes redder with time because of expansion and cooling. The two main minima at ∼ 5500Å and ∼ 7500Å (Fig. 6) can be identified with the blueshifted Si II (λ6355) and Ca II NIR triplet (gf -weighted line centroid λ8579) absorption lines, respectively. These lines are the most representative of the stratification of the expanding ejecta: their expansion velocities and evolution with time are shown in Figure 7. All velocities have been determined by fitting a Gaussian profile to the absorption features in the rest-frame spectra and measuring the blueshift of the minimum. The uncertainty that affects each line velocity has been taken equal to three times the standard deviation of the measured minimum positions. The velocity of Si II λ6355 ranges from ∼36,000 km s −1 at about 7 days from the explosion to ∼25,500 km s −1 at the last epoch. Ca II velocities are systematically about 20% higher than those of Si II. In Figure 7, we note that the velocity of the Si II λ6355 line in SN 2010bh is higher than in SNe 1998bw and 2006aj, while it is similar to that of SN 2003dh although with a shallower drop.
Considering its importance in constraining the nature and the evolutionary state of the progenitor star, we have searched for the spectroscopic signature of helium in our spectra. We observe weak absorption features in the optical and NIR that may be compatible with He I λ5876 and He I 1.083 µm blueshifted by ∼20,000-30,000 km s −1 and ∼28,000-38,000 km s −1 , respectively (see Fig. 6). The latter is in agreement with the velocities measured for Si II and Ca II. However, He I features may be blended with other species, like Na I in the optical and C I or Si I in the NIR (Mazzali & Lucy 1998;Millard et al. 1999;Sauer et al. 2006;Taubenberger et al. 2006). In particular, the contribution of C I cannot be ruled out. Indeed, by comparing the spectra of SN 2010bh to those of SNe 1998bw and 2007gr (Valenti et al. 2008a) at similar epochs, we can identify the broad absorption at ∼ 1.6 µm with the C I λ16,890 line. Such absorption becomes more prominent starting 8 days after explosion. A detection of the He I 2.058 µm line, typically not blended with other species, in the spectra of SN 2010bh is not possible, since the line, possibly blueshifted at any velocity up to 35,000 km s −1 , would lie in the observed range 1.9-1.95 µm, which is heavily affected by telluric absorptions. Consequently, we cannot confirm the identification of He I λ5876 and He I 1.083 µm. Spectral modeling may help in recognizing the different ions contributing to the spectral line formation. This will be the scope of a future paper.
In Bufano et al. (2010), based on the spectrum taken on March 23, we reported the presence of a significant flux deficit in the range 4500-5500Å. The flux density also seems low at wavelengths shorter than ∼ 3500Å. From the spectral evolution (Fig. 5), we can see that such deficits are seen only at this epoch. A careful analysis of the X-shooter spectrum has not identified any instrumental cause of these features. However, their time scale is too short to be explained physically, and therefore we will regard them as spurious.
In Figure 8, we compare the spectra of SN 2010bh with those of SNe 1998bw ) and 2006aj (Mazzali et al. 2006a) at similar phases after explosion. At ∼ 4 days from the burst, both SNe 2006aj and 2010bh present a featureless spectrum, with the exception in the latter SN of a weak and broad (∼ 47,000 km s −1 ) P-Cygni feature due to the Ca II NIR triplet. This line becomes more prominent at later phases and displays a decreasing velocity (see also Fig. 7). The Ca II triplet velocity remains significantly higher than in SN 2006aj, for which Mazzali et al. (2006a) measured ∼25,000 km s −1 roughly constant with time. In SN 1998bw spectra, Patat et al. (2001) found that the main contribution to the absorptions at ∼ 7000Å was given by O I (Fig.8), which is not obvious in our spectra of SN 2010bh. Absorptions at ∼ 3500Å and ∼ 4500Å in SN 2010bh spectra are likely due to Fe II and Ti II, as found for SN 2006aj through spectral modeling (Mazzali et al. 2006a).
Spectrum in the Nebular Phase
In the upper panel of Figure 9, we show the FORS2 nebular spectrum (∼186 days after explosion in the rest frame). At this time, no significant continuum flux contribution from the supernova photosphere is expected. Therefore, since the SN exploded on a bright region of the host galaxy, we fit the spectral continuum with a polynomial function to obtain and subtract the background flux. The final continuum-subtracted nebular spectrum in the range 5500-7000Å is plotted in the lower-left panel of Figure 9. It shows the [O I] narrow emission lines at 6300Å and 6363Å from the underlying galaxy region, and a broad but faint component, which, when fitted with a Gaussian function, peaks at 6340Å with a total flux of 1.3 × 10 −16 erg cm −2 s −1 .
From the comparison of SN 2010bh with SN 1998bw and SN 2006aj at similar rest-frame phases after the burst (214 and 206 days, respectively; Patat et al. 2001;Mazzali et al. 2007a; lower-right panel in Fig. 9), we find that the [O I] bump is very weak in SN 2010bh. The signal in the continuum-subtracted spectrum is too low to guarantee a secure measurement of the [O I] abundance and to perform spectral modeling. However, taking into account the lack of strong evidence of O I lines in SN 2010bh spectra at early epochs, this could indicate a very small amount of ejected oxygen, and, consequently, a less massive progenitor than for SNe 1998bw and 2006aj. On the other hand, it could also be explained as a lower nebular Mazzali et al. 2006a). Epochs are reported in rest-frame days after the explosion. flux, providing an indirect additional support for the M Ni , M ej , and E k values we found and discuss in the next Section (see also Table 5). Indeed, considering that the energy input scales roughly as M Ni × (τ γ + 0.035), where τ γ is the optical depth to radioactive gamma rays (τ γ ≈ 10 3 M 2 ej E −1 k t −2 : see, e.g., Maeda et al. 2003) and 0.035 is the positron contribution, we obtain a late-time flux in SN 2010bh which is about a factor of ∼7 and a factor of ∼2.5 smaller than that in SN 1998bw and SN 2006aj, respectively (for the same distance and reddening).
Bolometric Light Curve
Since photometry in the individual bands is affected by possible spectral lines and their time evolution, it is important to construct a bolometric light curve to estimate reliably the SN physical parameters. As a first approximation, we obtained a quasi-bolometric BVRI light curve using FORS2 photometry and the synthetic BVRI magnitudes obtained from X-shooter spectra. To this aim, BVRI magnitudes were firstly corrected for extinction and converted to flux density at the effective wavelength of the Johnson-Cousin filters. The spectral energy distribution was then integrated over the entire wavelength range and, finally, the integrated flux was converted into luminosity using the adopted distance (Sect. 3.1). The error bars of the bolometric luminosities obtained using X-shooter spectra are equal to the typical uncertainty of 10% that affects spectra flux calibration. In Figure 10, the pseudo-bolometric luminosity of SN 2010bh is compared to those of the GRB-SNe 1998bw and 2006aj. It displays an evolution similar to that of SN 2006aj, although with a fainter peak by about 0.2 dex (L bol,10bh ≈ 3 × 10 42 erg s −1 ). For the bolometric light curve fitting, we used a a simple model, that assumes, for the photometric phases, a concentration of the radioactive nickel ( 56 Ni) in the core, a homologous expansion of the ejecta and a spherical symmetry, following the prescriptions of Arnett (1982), and, for the nebular phases, includes the energy contribution from the 56 Ni− 56 Co− 56 Fe decay (Sutherland & Wheeler 1984;Cappellaro et al. 1997). For a detailed description of the model see Valenti et al. (2008b). The bolometric light curve model suggests a total ejected mass of radioactive 56 Ni of M Ni = 0.12 ± 0.02 M ⊙ . The lack of measurements during the nebular phase prevents us from verifying this value through the radioactive tail. Although a direct comparison with the quasi-bolometric curves created by Cano et al. (2011b) and Olivares E. et al. (2011) is not possible because of the difference in the wavelength ranges used to construct them, our M Ni estimate is in good agreement with that of Cano et al. Table 4). The blue line is the fit of the continuum we used to subtract the host-galaxy continuum flux contamination.
Discussion and Conclusions
In order to obtain further information on the explosion and stellar progenitor, we compared the observed properties of SN 2010bh with those of SN 2006aj. We find that the SN 2010bh R-band light-curve width was τ peak,10bh = (0.96 ± 0.11) × τ peak,06aj by "stretching" its time-scale in order to match SN 2006aj light curve (see Perlmutter et al. 1997) and its photospheric expansion velocity v ph,10bh = (1.74 ± 0.05) × v ph,06aj (assuming Si II λ6355 as a good tracer, see e.g. Valenti et al. 2008b). Using the relations between τ peak and v ph with the ejected mass M ej and the kinetic explosion energy E k (τ peak ∝ M Arnett 1982Arnett , 1996 and the M ej and E k estimates of SN 2006aj found by Mazzali et al. (2006a), we derive M ej ≈ (3.2 ± 1.6) M ⊙ and E k ≈ (9.7 ± 5.5) × 10 51 erg. From the analytical modeling of their pseudo-bolometric light curves, Cano et al. (2011b) and Olivares E. et al. (2011) found comparable ejected masses: M ej = 2.24 ± 0.08 M ⊙ and M ej = 2.6 ± 0.2 M ⊙ , respectively. While the kinetic energy found by Cano et al. is not too dissimilar from ours (E k = (1.39 ± 0.06) × 10 52 erg), the one derived by Olivares et al. is significantly larger, E k = (2.4 ± 0.7) × 10 52 erg. These discrepancies are related to the different species used for the velocity measurements and to the uncertainties of the measurements themselves, that are affected by line blending and by some arbitrariness in the choice of the line profiles. These problems are overcome by the use of a radiative transport model.
3/4 ej E −1/4 k ; v ph ∝ M −1/2 ej E 1/2 k ;
Although the photometric evolution of SN 2010bh was similar to that of SN 2006aj (i.e., similar light-curve width), SN 2010bh had a higher M ej and E k , explaining the faster expansion velocities measured from the spectra. This could suggest that spectra are more sensitive than light curves to possible effects of the viewing angle, in case of asymmetric explosion (a bipolar explosion is expected in the presence of a GRB; Piran 2004). Then we would expect that the weakest GRB (most off-axis) also has the lowest registered E k /M ej ratio. In Table 5, we report the M ej and E k values found for previous GRB-SNe and other broad-lined SNe Ic, as well as the corresponding E k /M ej ratio, and thus plot the latter versus the relative ejected M Ni (Fig. 11). While SN 2010bh has an intermediate E k /M ej ratio, it lies on the low Nickel mass tail of the energetic type Ic SN distribution (SN 2002ap, Mazzali et al. 2007bSN 2003jd, Valenti et al. 2008b). The main reasons for such a wide variety among GRB-SNe cannot rely only on differences in the viewing angle, but must be intrinsic (e.g., explosion collimation, progenitor mass, etc.). Indeed, GRB-SNe have been supposed to come from different explosion scenarios (see Woosley & Bloom 2006, and references therein), where the core collapse of a massive progenitor star (20-60 M ⊙ ) leads to the formation of different central engines (a magnetar or a black hole). On the other hand, this heterogeneity does not have an obvious correspondence in the properties of the GRB. In Figure 12 the intrinsic peak energy is plotted as a function of the isotropic emitted energy (see, e.g., Amati et al. 2009), showing that GRB-SNe events, including GRB 100316D, are consistent with the correlation in the E p,i − E iso plane holding for all long GRBs (with exception of GRB 980425). As noted by Starling et al. (2011), GRB 100316D has prompt gamma-ray spectral properties similar to GRB 060218.
The similarity between the light curves of SN 2010bh and SN 2006aj and between the observational characteristics of their associated GRBs suggests a common explosion mechanism for these two events. SN 2010bh would be produced by the core collapse of a relatively massive progenitor star (20-25 M ⊙ ), which leaves behind a magnetar, similar to SN 2006aj (Mazzali et al. 2006a). Indeed, by analyzing the spectral and temporal properties of the associated GRB 100316D, Fan et al. (2011) claimed that SN 2010bh was possibly powered by a magnetar with a spin period of P ≈ 10 ms and a magnetic field B ≈ 3 × 10 15 G. In this picture, the magnetar rotational energy would be injected in the expanding remnant on a timescale set by the magnetic dipole radiation. Following Eq. 2 of Kasen & Bildsten (2010), we would expect t 0 ≈ 12 hr; thus, even if a good fraction of the pulsar energy is converted into radiation, it will be lost by adiabatic expansion before escaping from the ejecta. Maeda et al. (2007) argued that in SN 2006aj the contribution from the magnetar to the light curve was likely negligible compared with the 56 Ni power, and this could be applicable in general to all magnetar powered GRBs/XRFs. We consider the detection at ∼ 12 hr after the burst (R ≈ 21.4 mag) as the contribution from the "cooling envelope" during the post-shock breakout phases (we may also have a nonthermal contribution from the afterglow at this phase). Following Chevalier & Fransson (2008), we estimate the progenitor radius: using their Eq. 4 and 5 and E k and M ej from our Table 5, we obtain the expected blackbody luminosity and temperature at 12 hr for different progenitor radii. Then the expected blackbody spectrum was converted to the observed frame (assuming the SN 2010bh redshift and total color excess) and convolved with the Rband filter. The resulting magnitude was fainter than the observed one only in the case of a progenitor radius R 10 11 cm. Even considering the 3-sigma upper limit flux, we would obtain a progenitor radius of the same order. This estimate is certainly rough, since it is based only on the single R-band image and uses simplified formulae (Chevalier & Fransson 2008), but it could provide support for a compact progenitor scenario and it is also consistent with the initial radius of 7 × 10 11 cm found by Olivares E. et al. (2011) analyzing the early X-ray-to-NIR emission.
On the other hand, we cannot exclude a scenario in which the outcome of GRB 100316D/ SN 2010bh was a black hole, which could explain the small ejected mass as a consequence of the possible fall-back of ejecta onto the BH. Recently, such small ejected masses have been inferred from the kinematics of black holes with >10 M ⊙ in our Galaxy, such as Cygnus X-1 (Mirabel & Rodrigues 2003;Gou et al. 2011), GRS 1915and V404 Cyg (Mirabel 2011 Fig. 11.-The ejected 56 Ni mass as a function of the ratio between the explosion energy and the ejected mass (E k /M ej ) for several broad-lined supernovae/hypernovae. Fig. 12.-Location of GRB 100316D in the E p,i − E iso plane. GRBs/XRFs connected with a spectroscopically confirmed SN are shown with red dots. Similar to GRB 060218, GRB 100316D is consistent with the correlation E p,i −E iso (solid line) derived by Amati et al. (2002). The two parallel dotted lines delimit the 2.5σ confidence region (Amati et al. 2009). ever, in this case it may be difficult to accommodate the higher expansion velocities measured for SN 2010bh than those of previous HNe with a collapsar progenitor (e.g., SN 1998bw).
As anticipated, the high v ph may likely be explained as an effect of the explosion geometry. Good indicators of the explosion geometry are the nebular emission lines of Fe II (a blend near 5200Å) and [O I] (λλ6300,6363; Maeda et al. 2002Maeda et al. , 2006Mazzali et al. 2005), whose profiles can reveal the presence of asymmetry. No such information on the SN 2010bh explosion geometry can be deduced from its nebular spectrum, because of the faintness of the emission lines.
Fig. 1 .
1-Close-up view of SN 2010bh field from the R-band image taken with VLT/FORS2 on 2010 April 3 (scale in the lower-left corner). SN 2010bh and the sequence of local reference stars(Table 2) are reported.
Fig. 2 .
2-UBV RI light curves of SN 2010bh. The abscissa represents the rest-frame time after the explosion, which is assumed to be coincident with the burst trigger (2010 March 16.53;Stamatikos et al. 2010).
SN 2010bh exploded at α = 07 h 10 m 30 s .53 and δ = −56 • 15 ′ 19. ′′ 78 (J2000;Starling et al. 2011) in a bright anonymous galaxy. We measured the host-galaxy redshift by calculating the average shift of the central wavelengths of its strongest emission lines ([O II] λ3727, Ne III λ3869, [O III] λλ4959, 5007, H I Balmer lines, He I λ5876, [O I] λ6300, [N II] λ6584, [Si II]λλ6716, 6731) in each of the 12 X-shooter spectra and correcting it for the radial component of the Earth's heliocentric motion. The weighted mean of the resulting heliocentric redshifts is z = 0.0592 ± 0.0001. The high precision of the redshift estimate was possible thanks to the accuracy of the wavelength solution over the whole wavelength range of the X-shooter spectra (2 km s −1 for the UV and VIS arms). This value is in good agreement with the values presented in previous works (z ≈ 0.059,Vergani et al. 2010; z = 0.0591 ± 0.0001,Starling et al. 2011; z = 0.0593, Chornock et al. 2010c). For a concordance cosmology (Hubble constant H 0 = 73 km s −1 Mpc −1 , Ω Λ = 0.73, and Ω m = 0.27), we obtained a luminosity distance of about 254 Mpc (i.e., distance modulus µ= 37.02 mag).
Fig. 3 .
3-Na I D doublet absorption lines in the SN 2010bh spectrum obtained by averaging the early-time (phases 2.4d, 3.3d and 4.2d, see
Fig. 4 .
4-Light curve of SN 2010bh compared to those of SN 1998bw
Fig. 5 .
5-SN 2010bh spectral evolution. The phase is given in rest-frame days after the explosion, assumed to be coincident with the GRB start time (2010 March 16.53;Stamatikos et al. 2010). The spectra are corrected for total (Milky Way + host-galaxy) reddening, shifted to the galaxy rest frame, vertically displaced and rebinned for clarity. Dot-dashed vertical lines indicate the wavelengths of the minima in the Si II and Ca II absorption lines on the April 18 spectrum. The most prominent emission lines of the host galaxy have been removed and telluric band positions indicated.
Fig. 6 .
6-Spectral evolution of SN 2010bh in the optical range. The positions of the tentative identifications of He I λ5876 and He I 1.083 µm features are marked with dot vertical lines at the wavelength of the minimum on April 11 and 18 spectrum, respectively. Further details are in the caption of Fig. 5.
Fig. 7 .
7-Temporal evolution of the expansion velocity of SN 2010bh measured from different ions. The Si II λ6355 line velocities fromChornock et al. (2010c) are reported with open circles. The SN 2010bh Si II λ6355 expansion velocities are compared to those of SNe 1998bw, 2006aj (measurements performed directly on the spectra published inPatat et al. 2001 and Mazzali et al. 2006a, respectively) and 2003dh(Hjorth et al. 2003).
Fig. 8 .
8-Spectral comparison of SN 2010bh with other GRB-SNe
(2011b) who found a 56 Ni mass of M Ni = 0.10 ± 0.01 M ⊙ . Olivares E. et al. (2011) derived a higher value (M Ni = 0.21 ± 0.03 M ⊙ ), likely because of the higher extinction correction.
Fig
. 9.-(Upper Panel) FORS2 spectrum of SN 2010bh at ∼186 rest-frame days after the explosion obtained by combining the spectra acquired in September and October 2010 (see
(Lower-left panel) Close-up view of the continuum-subtracted nebular spectrum in the wavelength interval 5500-7000Å, centered on the expected wavelength of the [O I] λλ6300, 6363 nebular emission lines. Lines from the host galaxy H II region dominate. The broad emission at ∼ 6300Å is fitted with a Gaussian curve (green line). (Lower-right panel) The spectra of SN 1998bw at ∼ 214 rest-frame days after explosion (red line; Patat et al. 2001) and SN 2006aj at 206 days (blue line; Mazzali et al. 2007a) are shown for comparison. The fluxes of SNe 1998bw and 2010bh have been normalized to that of SN 2006aj at the peak of the [O I] line (the narrow H II region component, in the case of SN 2010bh). An additional rescaling of SN 1998bw has been done for clarity.
Fig. 10 .
10-SN 2010bh pseudo-bolometric (BV RI) light curve compared with those of SNe 1998bw and 2006aj (open triangles and squares, respectively). SN 2010bh pseudo-bolometric luminosity obtained from FORS2 photometry is shown with filled circles, while open circles represent synthetic magnitudes obtained from X-shooter spectra.
). How-0
2
4
6
E
k /M
ej [10 51 erg/ M
sun ]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M
Ni [M
sun
]
98bw
03dh
03lw
06aj
10bh
02ap
03jd
09bb
Table 1 :
1Journal of late-epoch template observations. Phases from Swift/BAT trigger (2010 March 16.53; Stamatikos et al. 2010).UT
JD
Phase † Instr.
Bands
(+2400000)
2010/09/17
55456.8
184.8
FORS2
UBV RI
2011/01/26
55587.5
315.5
PROMPT5
I
2011/02/02
55594.5
322.5
PROMPT1
V RI
2011/02/02
55594.5
322.5
PROMPT5
V R
†
Table 2 :
2Optical magnitudes of the local reference stars in the field of SN 2010bh. 79±0.06 21.74±0.02 20.31±0.03 19.43±0.01 18.64±0.02 2 19.88±0.04 19.42±0.02 18.57±0.04 18.02±0.01 17.51±0.01 3 21.48±0.09 21.57±0.07 20.82±0.04 20.35±0.05 19.86±0.03 4 18.84±0.06 18.35±0.02 17.55±0.03 17.05±0.01 16.57±0.01 5 21.70±0.10 21.86±0.02 21.33±0.03 21.33±0.03 20.50±0.02 6 21.24±0.12 20.78±0.02 19.98±0.03 19.45±0.01 18.98±0.01 7 17.49±0.07 17.49±0.03 16.96±0.02 16.60±0.06 16.23±0.07 8 · · · 22.58±0.08 20.98±0.03 20.02±0.01 19.04±0.02 9 18.30±0.06 17.95±0.02 17.18±0.03 16.72±0.03 16.25±0.03 10 · · · 21.23±0.08 19.78±0.04 18.88±0.01 18.14±0.01 No corrections have been applied to the reported magnitudes.Star ID
U
B
V
R
I
1
22.
Table 3 .
3UBV RI observed magnitudes of SN 2010bh.UT
JD
Phase †
U
B
V
R
I
Instr.
+2400000
2010/03/17
55272.5
0.4
--
--
--
21.49 −0.37
+0.23
--
X-shooter
2010/03/19
55274.5
2.4
--
--
--
20.24 −0.24
+0.20
--
X-shooter
2010/03/20
55275.5
3.3
--
--
--
19.91 −0.14
+0.11
--
X-shooter
2010/03/21
55276.5
4.2
--
--
--
19.88 −0.14
+0.16
--
X-shooter
2010/03/24
55279.5
7.1
--
--
20.01 −0.11
+0.10
19.63 −0.06
+0.04
19.54 −0.33
+0.31
VLT observations were taken within the GTO programs 084.D-0265 and 085.D-0701 (P.I. S. Benetti) and 084.A-0260 and 085.A-0009 (P.I. J. Fynbo) at UT2/X-shooter, and GO program 085.D-0243 (P.I. E. Pian) at UT1/FORS2. SN 2010bh photometry on March 23 and 28 was obtained with GO program 084.D-0939 with UT1/FORS2 (P.I. K. Wiersema). Observations were performed in ToO mode.
IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc. under contract with the National Science Foundation.
(1) This paper; (2)Iwamoto et al. 1998; (3) Mazzali et al. 2001; (4) Mazzali et al. 2003; (5) Mazzali et al. 2006b; (6) Pian et al. 2006; (7) Mazzali et al. 2006a; (8) Mazzali et al. 2007b; (9) Valenti et al. 2008b; (10) Pignata et al. 2011.
Facilities: VLT/X-shooter, VLT/FORS2, PROMPTs.
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. U V Vlt/X-Shooter, 2400/2400/2400VLT/X-shooter UV/VIS/NIR 2400/2400/2400
. U V Vlt/X-Shooter, 2600/2600/3240VLT/X-shooter UV/VIS/NIR 2600/2600/3240
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16.53Stamatikos et al. 2010) in the host-galaxy rest frame. † Phases from Swift/BAT trigger† Phases from Swift/BAT trigger (2010 March 16.53; Stamatikos et al. 2010) in the host-galaxy rest frame.
‡ X- , shooter arm wavelength ranges are UV 3000-5600Å, VIS 5500-10200Å, and NIR 10200-24800Å. FORS2 Grism 300V is. ‡ X-shooter arm wavelength ranges are UV 3000-5600Å, VIS 5500-10200Å, and NIR 10200- 24800Å. FORS2 Grism 300V is 3300-9000Å.
| []
|
[
"Hippocampus Temporal Lobe Epilepsy Detection Using a Combination of Shape-based Features and Spherical Harmonics Representation",
"Hippocampus Temporal Lobe Epilepsy Detection Using a Combination of Shape-based Features and Spherical Harmonics Representation"
]
| [
"Zohreh Kohan ",
"Hamidreza Farhidzadeh ",
"Reza Azmi ",
"Behrouz Gholizadeh "
]
| []
| []
| Most of the temporal lobe epilepsy detection approaches are based on hippocampus deformation and use complicated features, resulting, detection is done with complicated features extraction and pre-processing task. In this paper, a new detection method based on shape-based features and spherical harmonics is proposed which can analysis the hippocampus shape anomaly and detection asymmetry. This method consisted of two main parts; (1) shape feature extraction, and (2) image classification. For evaluation, HFH database is used which is publicly available in this field. Nine different geometry and 256 spherical harmonic features are introduced then selected Eighteen of them that detect the asymmetry in hippocampus significantly in a randomly selected subset of the dataset. Then a support vector machine (SVM) classifier was employed to classify the remaining images of the dataset to normal and epileptic images using our selected features. On a dataset of 25 images, 12 images were used for feature extraction and the rest 13 for classification. The results show that the proposed method has accuracy, specificity and sensitivity of, respectively, 84%, 100%, and 80%. Therefore, the proposed approach shows acceptable result and is straightforward also; complicated pre-processing steps were omitted compared to other methods. | null | [
"https://arxiv.org/pdf/1612.00338v2.pdf"
]
| 464,532 | 1612.00338 | 987eb428c851a0c6203b5464675b4192bdd4b98f |
Hippocampus Temporal Lobe Epilepsy Detection Using a Combination of Shape-based Features and Spherical Harmonics Representation
Zohreh Kohan
Hamidreza Farhidzadeh
Reza Azmi
Behrouz Gholizadeh
Hippocampus Temporal Lobe Epilepsy Detection Using a Combination of Shape-based Features and Spherical Harmonics Representation
temporal lobe epilepsy · spherical harmonics · hippocampus shape- based features · classification · SVM
Most of the temporal lobe epilepsy detection approaches are based on hippocampus deformation and use complicated features, resulting, detection is done with complicated features extraction and pre-processing task. In this paper, a new detection method based on shape-based features and spherical harmonics is proposed which can analysis the hippocampus shape anomaly and detection asymmetry. This method consisted of two main parts; (1) shape feature extraction, and (2) image classification. For evaluation, HFH database is used which is publicly available in this field. Nine different geometry and 256 spherical harmonic features are introduced then selected Eighteen of them that detect the asymmetry in hippocampus significantly in a randomly selected subset of the dataset. Then a support vector machine (SVM) classifier was employed to classify the remaining images of the dataset to normal and epileptic images using our selected features. On a dataset of 25 images, 12 images were used for feature extraction and the rest 13 for classification. The results show that the proposed method has accuracy, specificity and sensitivity of, respectively, 84%, 100%, and 80%. Therefore, the proposed approach shows acceptable result and is straightforward also; complicated pre-processing steps were omitted compared to other methods.
one of the brain hemisphere or both. Therefore, unilateral and/or bilateral shape analysis of brain structures could lead to a better understanding of the abnormalities. There are evidences in the literature [1,2,3,4] showing that the brain structures deform when some diseases such as schizophrenia, Alzheimers, Parkinson, and epilepsy occur. So using precise representation is helpful to parameterize the deformation that will useful to diagnosis of the mentioned illnesses automatically. Hippocampus is a brain structure that belongs to the limbic system and is located in the medial temporal lobe. Hippocampus plays an important role in the formation of declarative, emotional, and long-term memories as well as language processing [5]. Hippocampus is one of the main targets of deformation in the brain in some disorders such as temporal lobe epilepsy. So, 3D representation and analysis of this structure could be useful in prognosis and diagnosis of those types of diseases. To characterize the deformation of the brain structures volumetric [6] or shape analysis is used. Volumetric analysis is easy to interpret deformation but shape analysis is more common because of its ability in geometric and morphometric features description. There are two groups of approaches used for shape representation; explicit and implicit shape representation. In explicit representation, the shape is illustrated as a parametric form and in explicit representation it represents as the level set of a scalar function. There are some examples of shape analysis methods in the literature in which either implicit or explicit shape representation is used. Statistical analysis of the anatomical shape deformations, which occur in epilepsy and other neurological disorders, require both global and local parameter based characterization of the anatomical shape that is deformed. Size-and volume-based analysis is the most popular method to achieve the parameterization of the shape deformations [8,9]. Many methods are proposed to analysis hippocampus are needed complicated pre-processing task. Most of them are reviewed in [8,9,10,11]. Since many of the shape-based features can illustrated hippocampus deformation and employ for detection. This features are straightforward and avoided complicated pre-processing steps. This method utilized spherical harmonics (SPHARM) that is a powerful mathematically tool that used for representation and analysis of 3D closed surfaces [15]. SPHARM can be utilized in 3D representation and analysis of the brain structures. However, in contrast to previous studies on hippocampus shape analysis using SPHARM-Based shape descriptor, this work has been focused on the detection of asymmetry in hippocampus shape using SPHARM coefficients as the shape features. A sub set of the coefficients that were able to detect the asymmetry in left and right hippocampus was used to classify the images into either normal or epilepsy patient groups. Moreover, the complexity of this method is less compared to the other presented methods. As a result, in this paper, Combination of shape-based features and spherical harmonics, this proposed method can analysis hippocampus shape and diagnosis temporal lobe epilepsy in MR images.
PREVOIUS WORKS
Shen et al. [7] have reported about deformation of the hippocampus in epilepsy patients. The deformation of hippocampus has led the research groups to study the shape of the hippocampus for diagnosis purposes. Gerig et al. [8] to analysis hippocampal shape use volume measurments and shape-based features. Shape-based feature that is used is based on Mean Square Distance (MSD) between left and right hippocampus surface shapes. Then Support Vector Machines (SVM) is employed to classify, also leave-one-out cross validation is applied to evaluation performance. Csernansky et al. [9] used registration to compare the hippocampal volume and shape characteristics in patient and control subjects. Computing transformation vector from the points on the hippocampus surface represents shape. Kodipaka et al. [10] used the Kernel Fisher Discriminant (KFD) algorithm for shape-based classification of hippocampal shapes. In this method, some landmarks manually placed on the hippocampus surface by an expert that indicated boundaries of the shape, then feature extraction is done by fitting a model to the landmarks using a deformable pedal surface, see detail about a deformable pedal surface in [13]. The result of this fitting process is a smooth surface that used in similarity alignment followed by a level-set non-rigid registration that describe in [14]. The output of this registration is local deformation between left and right hippocampus that used as input to the Kernel Fisher classifier. Esmaeilzadeh et al. employed spherical harmonics (SPHARM) to analysis hippocampus shape [11]. This method alignment hippocampus left and right to each other the extract features based on spherical harmonic confidents, finally using SVM for calssification. Shishegar et al. have used Laplace Beltrami operator for TLE diagnosis [12]. This method proposed a feature vector that described size and shape based on Laplace Beltrami eigenvalues and eigen-functions.Some of pervious methods need registration as pre-processing step [6,7,8]. Also in [10] using shape alone could not capture shape differences. Furthermore the most of reviewed methods is complicated and need pre-processing step. In this paper, a new method is presented that using straightforward shape-based features for detection and omitted complicated pre-processing step such as registration.
THE PROPOSED METHOD
The purposed method has two parts for feature extraction. In each part using statistical t-test most significant features is selected. Then a main feature vector is formed. Finally SVM is employed for classification hippocampus shapes. Fig.1 illustrate block diagram of the purposed method.
Materials
Our dataset contained 25 T1-weighted coronal brain MR images (MRI) from Henry Ford hospital. MRI data is very useful for imaging in different applications such as breast cancer [28] and sarcomas [29]. 20 of the images were acquired from temporal lobe epilepsy subjects with medically intractable seizures and the remaining five were from normal subjects used as control dataset. Ten images out of 25 (all from epilepsy patients) were acquired with 1.5 Tesla MRI scanners and the remaining ones (15 images from epilepsy patients and all the normal cases) were from 3.0 Tesla MRI scanners. Slice thickness of all the images was 2.0 mm and the 2015,inplane resolution of the pixels varied from 0.39 × 0.39mm to 0.75 × 0.75mm. For each image one expert provided a manual segmentation, then the segmentation labels were reviewed and confirmed by two more experts.
Feature extraction based on spherical harmonics
This part consists of pre processing steps that explained below.
Surface meshing
To generate a surface mesh for hippocampus shape, first, 3D manual segmentation was applied. For the further shape analysis and also for the classification of the hippocampus structure, it was necessary to have a mathematical representation of the 3D structure. One of the most common and simplest representations for explicit representation of the 3D surfaces is triangulation. In this method the shape is represented as a set of 3D discrete points that connected via triangular mesh. The marching cube algorithm [26] is a popular triangulation method applied to a 3D surface to obtain triangle surface mesh. After applying marching cube algorithm, the 3D shape is represented by coordinates of triangle mesh vertices as:
X = [x 1 , y 1 , z 1 , ..., xn, yn, zn] T(1)
Where n is the number of vertices.
Spherical parametrization
First step in SPHARM representation is mapping the 3D surface to unit sphere under bijective mapping with lower distortion in area and topology called spherical parameterization [25]. This mapping is applied to a 3D surface represented as a triangular mesh. After mapping the surface mesh on the unit sphere each vertices can be represented in spherical polar coordinate with two parameters, the inclination angle θ and the azimuth angle ϕ . In this paper, for spherical parametrization we use, CALD algorithm that proposed by Shen and Makedon [26]. This algorithm consists of two steps: first, initial parametrization for triangular mesh [25] and second, local smoothing and global smoothing parametrization improve the quality of the parametrization. The goal of local smoothing step is minimization of the area distortion at a local sub mesh, this goal is achieved by solving a linear system and controlling its worst length distortion simultaneously. The global smoothing step as the second step of the algorithm compute the distribution of the surface distortions for all the mesh vertices, to equalize them over the complete sphere. In an overall view, the CALD algorithm merges the local and global methods together and executes each method alternately until a best parametrization is achieved. Fig.2 shows the left part of a hippocampus mapped to the unit sphere through this 3D mapping approach.
SPHARM analysis
The SPHARM is a powerful mathematically tool that used for representation and analysis of 3D closed surface. The overall SPHARM analysis includes three main steps:
-Surface meshing -Surface parametrization -Spherical expansion The first two steps have been explained in section II.C. When the 3D surface is represented via two parameters (the inclination angle θ and the azimuth angle ϕ) the surface can be expanded into a complete set of SPHARM basis functions
Y m l . Y m l (θ, ϕ) = 2l + l(1 − m)! 4Π(1 + m)! p m l cos(θ)e imϕ(2)
where 0 < l < Lmax and −l < m < l and p m l cos(θ) is the associated Legendre polynomials defined by the differential equation as follows:
P m l (x) = (−1) m (2 i l!) (1 + x 2 ) m/2 (d l+m ) (dx l+m ) (x 2 − 1) l(3)
The surface is independently decomposed through SPHARM as: These terms can be combined into a single function:
x(θ, ϕ) = Lmax l=0 l m=−l C m lx Y m l (θ, ϕ) (4) y(θ, ϕ) = Lmax l=0 l m=−l C m ly Y m l (θ, ϕ) (5) z(θ, ϕ) = Lmax l=0 l m=−l C m lz Y m l (θ, ϕ)(6)v(θ, ϕ) = Lmax l=0 l m=−l C m l Y m l (θ, ϕ)(7)
Where v(θ, ϕ) = (x(θ, ϕ), y(θ, ϕ), z(θ, ϕ)) T
And
C m l = (C m lx , C m ly , C m lz ) T(9)
The coefficients C m l are computed using least-squares estimations. According to equation (7) these coefficients are determined up to user-desired maximum degree Lmax.
The original surface is indicated as X = (x 1 , x 2 , x 3 , ..., xn) T and (a 1 , a 2 , a 3 , ..., a k ) is an estimate for the coefficients which are Featureselection obtained by solving previous equation follow manner:
C m l ∼ = Y m l T Y m l −1 X(10)
As claimed by equation (7), after coefficients are estimated, the 3D surface can be reconstructed. Fig.3 illustrates reconstruct shape in four different degrees using higher degrees yield to more details in the reconstructed shape.
Featureselection
The real parts of SPHARM coefficients were used as features in this work. For each subject hippocampus has two parts; left and right. For each part the coefficients were computed up to degree 15. So the length of feature vector is 3×(15+1) 2 = 768 for each side. To make the coefficients comparable across left and right hippocampus, we needed to normalize them by cancelling out the translational and rotational misalignments. Rotation was achieved by alignment of reconstructed shape based on first degree coefficients only (i.e. an ellipse). We used the alignment algorithm that used in [26]. For that purpose we aligned the shortest and the longest axis of the ellipse along x-axis and z-axis, respectively. We also translated the shapes to the origin of the coordinate system by ignoring the coefficients of degree 0. Since we used the ratio of left and right coefficients for each hippocampus, no scaling adjustment was required. After normalizing the coefficients, we calculated sum of the power of each coefficients for all the degrees as shown in equation (11). This step yielded to a feature vector of (15 + 1) 2 = 256 elements. Since our aim was to detect hippocampal asymmetry we calculated the ratio of logarithm of left coefficients to logarithm of right coefficients. We used logarithm to makes the features of different orders more comparable. (Fig.4) Since only some of the listed features carried applicable information for the classification of shape (significantly differentiate between normal and abnormal shapes), we employed a statistical t-test to identify and select the most discriminative features. Fig.5 shows a schematic diagram of the feature selection algorithm. We run t-test on the each element (feature) of features vector to compare the element values of patients and normal subjects in a training set. For two sets of samples (in this paper one set is the feature values for normal cases and the second set is the feature values for the epilepsy cases) with averages of M1 and M2 the p-value indicates the probability that the observed difference between M1 and M2 is due to chance under the null hypothesis that M1 and M2 are the same. Therefore, a lower p-value shows statistically stronger difference that corresponds to a more significant feature. We obtained the associated p-value for each feature and selected those features with p-values of less than 0.05 as significantly discriminative features. Our hypothesis was that the significant features could help better in classification of hippocampus into healthy and epileptic classes. To select the best features that are able to differentiate between normal and epileptic subjects, first we randomly selected a subset of 12 images from our data set to form our training set. The training set contained both normal (two images) and epileptic (ten images) subjects. The remaining images were used as the test dataset in the classification explained in the next subsection. We extracted all SPHARM coefficients then created the feature vectors as described; we used t-test to select those features in which the value of the feature between normal and patient cases was detected significantly different (p − value < 0.05).
( m<|l| ||C m 0 || 2 , m<|l| ||C m 1 || 2 , m<|l| ||C m 2 || 2 , ..., m<|l| ||C m l || 2 )(11)
Feature extarction based on shape-based features
In this section we first introduced all the features we used to study shape characteristics. Table 1 shows a list of shape and size features we have used in this project to detect asymmetry between right and left hippocampus. Fig.6 shows a [26] 0.7054 Mesh size 0.0015** First-order border moments [25] 2.38 × 10 −4 ** Third-order border moments [25] 0.5088 Circumscribed sphere volume to 0.6028 shape volume ratio Curvature 0.075* general block diagram of our feature extraction method. Following we described these features.
Maximum shape diameter
The maximum possible Euclidian distance between two points of the surface of the shape. Since epilepsy disease could deform the surface of the hippocampus [25] it might changes maximum diameter of the shape.
Shape volume
To measure the shape volume of right and left hippocampus we multiply number of voxels by the volume of a single voxel. Surface area: To measure the surface area we calculated the summation of the outer surfaces of all the voxels that formed the hippocampus surface.
3D Compactness
Compactness is a unit less shape feature that describes how closely packed the shape is. In the other hand, it shows the roughness of a shape surface to its volume [26]. Sphere has the smallest value of 3D compactness i.e. about 113. As the sphere deforms towards a more complicated shape, compactness becomes larger and larger. 3D compactness (C) calculated as ratio of the cubed surface area (A) to the squared surface volume (V):
C = A 3 V 2(12)
Mesh Size
In this paper we measured the number of vertices in the shape surface mesh using marching cube meshing algorithm [26]. The marching cubes algorithm is a 3D iso-surface representation technique for generating mesh for a 3D surface. Moments are the statistical property of the shape that are evolved from the moment concept in physics. There are two types of moments, region moments and boundary moments. In this work, we use boundary moments [25]. The pth moment is defined as:
mp = 1 N N i=1 [z(i)] p(13)
where d(i) is Euclidian distance between ith voxels and the centroid, N is number of voxels. The pth central moment defined as:
Mp = 1 N N i=1 [z(i) − m 1 ] p(14)
In this paper we use low-order moments such as M1 and M3 that are defined as below:
F 1 = [ 1 N N i=1 [z(i) − m 1 ] 2 ] 1 2 1 N N i=1 z(i)(15)
and
F 3 = [ 1 N N i=1 [z(i) − m 1 ] 4 ] 1 4 1 N N i=1 z(i)(16)
where
m 1 = 1 N N i=1 z(i)(17)
3.7.6 Circumscribed sphere volume to shape volume ratio This is the ratio of the volume of the circumscribed sphere centred at the shape centroid to the volume of the shape.
Curvature
We defined the curvature of the shape as follows:
C = d C1 + d C2 d 12(18)
Where d C 1 and d C 2 are the Euclidian distances between centroid of the shape and two opposite tips of the shape that are located in the maximum distances to the centroid and d 1 2: is the Euclidian distance between the two tips of the shape. To analysis the shape and size of the left and right hippocampus, we extracted all the features for left and right hippocampus, separately. Then we measured the ratio of right feature values to left feature values as well as left feature values to right. We chose the ratio that is greater or equal to 1 as an indicator for hippocampal symmetry/asymmetry. In this part feature selection is done according to statistical feature selection test that describe in Feature Selection part.
Classification
Support vector machine (SVM) is a supervised learning models used for classification and regression analysis with works very well on similar studies [27,28,29]. Linear SVMs search for the optimal hyperplane that separate the dataset to groups, i.e. the hyperplane that makes the maximal margin between groups. For more information about SVM classifiers you can see [26]. We used SVM classifier to classify each test image to normal or patient categories using the selected features (Fig.7). We used our test dataset to evaluate the performance of our classification algorithm.
Evaluation metrics
To evaluate our detection algorithm performance we calculated specificity, sensitivity and accuracy metrics (equations 19 to 21) for the classifier results. These metrics have been largely used in medical image analysis [25,26].
Accuracy = T P + T N T P + F N + T N + F P × 100 (19) Sensitivity = T P T P + F N × 100 (20) Specif icity = T N F P + T N × 100(21)
TP , TN , FP and FN are, respectively, true positive, true negative, false positive and false negative.
Experimental results
We run our algorithm on 25 segmentation labels of hippocampus on T1-weighted MRI. We divided our image dataset to two subsets; training and test. Training images (12 images, 2 normal and 10 epileptic) were used for feature selection. Test image set consisted of 13 images; 3 normal and 10 epileptic images. All the implementations of this work were implemented in MATLAB platform and run on a Mac operating system.
Features selection
Feature selection consists of two parts. First, feature extraction based on spherical harmonics coefficients. Second, feature extraction based on shape-based features. In each part statistical t-test is used for feature selection. In both parts, the most significant features (associated with p-value < 0.05) is selected. Then combination two selected features create feature vector.
Classification
Leaveoneout cross validation methodology has been applied to divide the test image set to one test image and 12 training for SVM classifier. Having the 18 selected features used in our SVM classifier, the specificity, sensitivity, and accuracy of our detection algorithm were 84%, 100% and 80%, respectively.
Conclusion and future work
Our preparatory results show that combination SPHARM coefficients and shapebased features could be helpful to describe the hippocampus shape deformation and could be used in diagnosis of the temporal lobe epilepsy disease in MRI. Since the dataset we used in our study is formed based on some part of the dataset used in [10,11], not the whole, it is challenging to reach a fair comparison between presented results and ours. As an advantage, the complexity of our algorithm is less than the other presented algorithms. In contrast to other work, our method detects the asymmetry in hippocampus structure shape for diagnosis of epilepsy and the preliminary results showed that hippocampal asymmetry detection could be useful for diagnosis of the brain abnormalities such as temporal lobe epilepsy. However, small size of the dataset is a limitation in our work. As future work we are going to increase the number of images in the training and test dataset. We will also compare the results to other reported results in the literature using appropriate statistical tests to examine the significance of the results. We would like to improve the results of our work by adding more shape-based features that are able to significantly separate patients from control groups. It could be also considered to use and evaluate this algorithm for detecting other brain abnormalities.
Fig. 1
1Purposed method block diagram
Fig. 2
2A) Left hippocampus. B) The result of marching cube meshing algorithm. C) The result of spherical parameterization
Fig. 3
3A) 3D shape of left hippocampus and (B-E) its reconstruction through SPHARM based shape reconstruction using (B) L − max=1 (C) L − max =8 (D) L − max =16 and (E) L − max =24
Fig. 4
4Feature extraction
Fig. 5
5Feature extraction
Fig. 6
6Feature
Fig. 7
7Classification block diagram
Table 1
1Shape-based features and corresponding p-valuesFeature
p-value
Maximum shape diameter
0.7187
Shape volume
2.54 × 10 −4 **
Surface area
0.7690
Compactness
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"Eye Know You Too: A DenseNet Architecture for End-to-end Eye Movement Biometrics"
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| Eye movement biometrics (EMB) is a relatively recent behavioral biometric modality that may have the potential to become the primary authentication method in virtualand augmented-reality devices due to their emerging use of eye-tracking sensors to enable foveated rendering techniques. However, existing EMB models have yet to demonstrate levels of performance that would be acceptable for real-world use. Deep learning approaches to EMB have largely employed plain convolutional neural networks (CNNs), but there have been many milestone improvements to convolutional architectures over the years including residual networks (ResNets) and densely connected convolutional networks (DenseNets). The present study employs a DenseNet architecture for end-to-end EMB and compares the proposed model against the most relevant prior works. The proposed technique not only outperforms the previous state of the art, but is also the first to approach a level of authentication performance that would be acceptable for real-world use. | null | [
"https://arxiv.org/pdf/2201.02110v2.pdf"
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| 247,778,872 | 2201.02110 | 7fc6d118d26bed157414cafb93e433c1fed8ffa6 |
Eye Know You Too: A DenseNet Architecture for End-to-end Eye Movement Biometrics
AUGUST 2021 1
Journal Of L A T E X Class
Files
Eye Know You Too: A DenseNet Architecture for End-to-end Eye Movement Biometrics
148AUGUST 2021 1Index Terms-Eye trackinguser authenticationmetric learn- ingtemplate agingpermanence
Eye movement biometrics (EMB) is a relatively recent behavioral biometric modality that may have the potential to become the primary authentication method in virtualand augmented-reality devices due to their emerging use of eye-tracking sensors to enable foveated rendering techniques. However, existing EMB models have yet to demonstrate levels of performance that would be acceptable for real-world use. Deep learning approaches to EMB have largely employed plain convolutional neural networks (CNNs), but there have been many milestone improvements to convolutional architectures over the years including residual networks (ResNets) and densely connected convolutional networks (DenseNets). The present study employs a DenseNet architecture for end-to-end EMB and compares the proposed model against the most relevant prior works. The proposed technique not only outperforms the previous state of the art, but is also the first to approach a level of authentication performance that would be acceptable for real-world use.
I. INTRODUCTION
B IOMETRICS have become a part of everyday life due to the ubiquity of fingerprint and face recognition in smartphones. Most biometric modalities can be separated into two categories: physical and behavioral. Physical biometrics reflect the physical traits of a person, including face, fingerprint, iris, and retina. Behavioral biometrics reflect a person's patterns of behavior, with some of the most commonly studied modalities being gait, signature, and voice. Physical biometrics tend to be more distinctive and exhibit greater permanence, whereas behavioral biometrics tend to be less intrusive and more applicable for continuous authentication. An overview of these common biometric modalities is given in [1].
A more recent behavioral biometric modality is eye movement biometrics (EMB) [2]. Eye movements may be particularly spoof-resistant because the oculomotor system is controlled by a complex combination of neurological and physiological mechanisms, both voluntary and involuntary. Eye movements also enable liveness detection [3], [4] and continuous authentication [5], [6] and could easily be paired with other modalities like mouse dynamics [7] or iris recognition [8] in a multimodal biometrics system. Because eye movements have been shown to carry distinguishable information, studies have even begun to explore methods of deidentifying eye movement signals in an effort to preserve both the users'
The authors are affiliated with the Department of Computer Science, Texas State University, San Marcos, TX 78666 USA. Author emails: {djl70,ok11}@txstate.edu.
Manuscript received April 19, 2021; revised August 16, 2021. privacy and the utility of the eye movements as an input method [9]- [13].
There is an emerging use of eye-tracking sensors in both virtual-reality (VR) and augmented-reality (AR) devices (e.g., Vive Pro Eye [14], Magic Leap 1 [15], HoloLens 2 [16]) in part to enable foveated rendering [17] techniques which offer a significant reduction in overall power consumption without a noticeable impact to visual quality. In addition to foveated rendering, eye tracking also enables various applications in these devices including user interactions, analytics, and various display technologies [18]. Because the hardware required for EMB is already included with these devices, and because EMB offers continuous authentication capabilities, EMB has the potential to become the primary security method for these devices [19]. However, existing EMB models have yet to demonstrate levels of authentication performance that would be acceptable for real-world use, even when using eye-tracking signals with higher levels of signal quality than are available in current VR/AR devices. Deep learning models for EMB [4], [20]- [23] have largely been limited to plain convolutional neural networks (CNNs) which, despite being capable of outperforming more traditional statistical approaches, do not take advantage of milestone developments over the years in the area of convolutional architectures, including residual networks (ResNets) [24] and-the focus of the present studydensely connected convolutional networks (DenseNets) [25].
We propose a novel DenseNet-based architecture for end-toend EMB. The model is trained and evaluated on the Gaze-Base [26] dataset which contains eye movement data from 322 subjects each recorded up to 18 times over a 37-month period. Although the eye-tracking signals in GazeBase reflect a higher level of signal quality than is available in current VR/AR devices, it is important to first establish whether EMB is capable, even with high-quality data, of achieving a level of authentication performance that is acceptable for real-world use. We primarily evaluate our model in the authentication scenario (also commonly called verification) using equal error rate (EER), both because authentication performance has been shown to remain relatively stable regardless of population size [27] and because behavioral biometric traits are generally not distinctive enough for large-scale identification [28]. We also perform additional analyses that we have seen only a small portion of works do (e.g., [23]), including assessing the permanence of the learned features and estimating the false rejection rate (FRR) at a false acceptance rate (FAR) of 1-in-10000 (abbreviated FRR @ FAR 10 −4 ). 1. Overview of the process for embedding an eye-tracking signal using the proposed methodology. We primarily focus on the case where only the first 5-second window is embedded, but we explore aggregating embeddings across windows in § V-C.
The main contributions of the present study are:
• A novel, highly parameter-efficient, DenseNet-based architecture that achieves state-of-the-art EMB performance in the authentication scenario on high-quality data, namely 3.66% EER when enrolling and authenticating with just 5 s of eye movements during a reading task. For perspective, 5 s is somewhat comparable to the time it takes to enter a 4-digit pin or to calibrate an eye-tracking device. • The first to show that, using 30 s of eye movements during a reading task, it is possible to achieve an estimated 5% FRR @ FAR 10 −4 which approaches a level of authentication performance that would be acceptable for real-world use. • The first to report significantly better-than-chance FRR @ FAR 10 −4 with 60 s of eye movements at artificially degraded sampling rates as low as 50 Hz, suggesting that EMB has the potential to become suitable for deployment at the sampling rates present in existing VR/AR devices. • The first application of a more modern convolutional architecture for EMB.
II. PRIOR WORK
A. Convolutional neural networks (CNNs)
Since the seminal works of AlexNet [29] and VGGNet [30], CNNs have quickly become some of the most popular types of neural networks for image processing tasks. Such architectures also started being employed in time series domains like eye movement event classification [31] and audio synthesis [32]. In time series domains such as these, varieties of recurrent neural networks (RNNs) [33], [34] were once the most common, but CNNs have empirically shown to be capable of similar-orbetter performance while also being much faster to train [35]. Several pivotal architectural improvements have been made to CNNs since their infancy. We focus on two such improvements: ResNets [24] and DenseNets [25]. ResNets [24] introduce so-called "skip connections" that combine the output of each convolutional block with its input via summation. These skip connections improve gradient flow through the network, enabling the training of significantly deeper networks than was previously possible. DenseNets [25] include similar skip connections between each convolutional block and all subsequent blocks, using channel-wise concatenation instead of summation to facilitate even better information flow than ResNets. One study visualizing loss landscapes [36] showed that DenseNets have much smoother loss landscapes than ResNets which may lead to increased ease of convergence during training.
We acknowledge that there are more recent convolutional architectures than DenseNet that claim better performance on image processing tasks (e.g., ResNeXT [37], DSNet [38], EfficientNet [39], EfficientNetV2 [40]), not to mention the various transformer [41], [42] architectures that have seen success in domains including natural language processing (NLP) and image classification. Rather than using one of these more cutting-edge architectures, we base our architecture on DenseNet because of its parameter efficiency and relative simplicity. We find that this architecture is able to achieve state-of-the-art performance in the EMB domain.
B. Eye movement biometrics (EMB)
EMB has been studied extensively since the introduction of the modality in 2004 [2]. Most earlier works in the field [43]- [47] require explicit classification of eye movement signals into physiologically-grounded events, from which hand-crafted features are extracted and fed into statistical or machine learning models. The state-of-the-art statistical model is the approach by Friedman et al. [45] which centers around the use of principal component analysis (PCA) and the intraclass correlation coefficient (ICC).
Since the recent introduction of deep learning to the field of EMB [20], [21], end-to-end deep learning approaches have become more common [4], [20]- [23]. The current state-ofthe-art model is DeepEyedentificationLive (DEL) [4] which utilizes two convolutional subnets that separately focus on "fast" (e.g., saccadic) and "slow" (e.g., fixational) eye movements. Another recent model, Eye Know You (EKY) [23], uses exponentially dilated convolutions to achieve reasonable biometric authentication performance with a relatively small (∼475K learnable parameters) network architecture.
Both DEL and EKY employ plain CNN architectures that do not take advantage of the improvements made to CNNs over the years. The present study improves upon the previous state of the art by using a more modern DenseNet-based architecture to simultaneously increase expressive power and reduce parameter count.
III. NETWORK ARCHITECTURE
The proposed network architecture, which we call Eye Know You Too (EKYT), is visualized in Fig. 2. The network Fig. 2. The proposed pre-activation DenseNet-based network architecture, including the optional classification layer. Each convolution layer has kernel size k = 3, stride s = 1, and dilation rate d that varies by layer. Each convolution layer outputs 32 feature maps that are concatenated with the previous feature maps before being fed into the next convolution layer.
performs a mapping f : R C×T → R 128 , where C is the number of input channels, T is the input sequence length, and the output is a 128-dimensional embedding. It begins with a single dense block of 8 one-dimensional convolution layers, where the feature maps produced by each convolution layer are concatenated with all previous feature maps before being fed into the next convolution layer. The final set of concatenated feature maps is then sent through a global average pooling layer, flattened, and then fed into a fully-connected layer to produce a 128-dimensional embedding of the input sequence. When classification is required (e.g., for cross-entropy loss), an additional fully-connected layer is appended after the embedding layer that outputs class logits.
All convolution layers (except the first), the global average pooling layer, and the optional classification layer are all preceded by batch normalization (BN) [48] and the rectified linear unit (ReLU) [49] activation function (called a "pre-activation" architecture). We use a "growth rate" of 32, meaning each convolution layer outputs 32 feature maps to be concatenated with the previous feature maps. Because we use BN, there is no need for the convolution layers to learn an additive bias.
The convolution layers (labeled n = 1, . . . , 8) use constant kernel size k = 3 and stride s = 1, an exponentially increasing dilation rate d n = 2 (n−1) mod 7 , and enough zero padding p n = d n on both sides of the input to preserve the length along the feature dimension. The use of exponentially dilated convolutions produces an exponential growth of the receptive field of the network with only a linear increase in the number of learnable parameters. In general, assuming s = 1, the receptive field of layer n, denoted r n , is given by
r n = 1 + n i=1 d n (k n − 1).(1)
The final convolution layer of our network has a (maximum) receptive field of r 8 = 257 time steps from the input. Excluding the optional classification layer, our proposed architecture has ∼123K learnable parameters for C = 2 and any T . Weights are initialized in the following manner. Each convolutional layer uses He initialization [50] with a normal distribution and learns no additive bias. Each BN layer is initialized with a weight of 1 and a bias of 0. Each fullyconnected layer is initialized with a bias of 0, and weights are initialized using the default method of PyTorch 1.10.0.
In preliminary experiments, we experienced overfitting on the train set relative to the validation set when increasing the depth of the network and/or adding additional dense blocks (each separated by a transition block to optionally reduce the size of the channel dimension). Specialized dropout techniques have been proposed for DenseNet architectures to resolve such overfitting problems [51]; but in the interest of keeping our network small, we did not pursue such techniques. We also experienced worse performance when using global max pooling instead of global average pooling. We found no noticeable difference in performance when swapping the order of BN and ReLU, nor when using a "post-activation" architecture (i.e., applying BN and ReLU after each convolution before channelwise concatenation). Though, we note that pre-activation DenseNet (and ResNet) architectures generally produce lower errors than their post-activation counterparts [52].
IV. METHODOLOGY
A. Hardware & software
All models are trained inside Docker containers on a Lambda Labs workstation. The workstation is equipped with quad NVIDIA GeForce RTX A5000 GPUs (24 GB VRAM), an AMD Ryzen Threadripper PRO 3975WX CPU @ 3.
B. Dataset
We use the GazeBase [26] dataset consisting of 322 collegeaged participants, each recorded monocularly (left eye only) at 1000 Hz using an EyeLink 1000 eye tracker. Nine rounds of recordings (R1-9) were captured over a period of 37 months. Each subsequent round comprises a subset of participants from the preceding round (with one exception, participant 76, who was not present in R3 but returned for R4-5), with only 14 of the initial 322 participants present across all 9 rounds. Each round consists of 2 recording sessions separated by approximately 30 minutes. During each recording session, participants performed a battery of 7 tasks: random saccades (RAN), horizontal saccades (HSS), fixation (FXS), an interactive ballpopping game (BLG), reading (TEX), and two video-viewing tasks (VD1 and VD2). More details about each task can be found in the dataset's paper [26].
We create class-disjoint train and test sets by assigning the 59 participants present during R6 to the test set and the remaining 263 participants to the train set. In this way, the test set-which comprises nearly 50% of all recordings in GazeBase-can be used to assess the generalizability of our model both to out-of-sample participants and to longer testretest intervals than are present during training. The train set is further partitioned into 4 class-disjoint folds in a way that balances the number of participants and recordings between folds as well as possible (the fold assignment algorithm we use is described in [23]). These 4 folds are used for 4-fold cross-validation, where 1 fold acts as the validation set and the remaining folds act as the train set. We exclude the BLG task from the train and validation sets due to the large variability in its duration relative to the other tasks, but we include it in the test set to enable an assessment of our model on an out-of-sample task.
We used 4-fold cross-validation to manually tweak our network architecture and determine the final training parameters. Only at the very end of our experiments did we use the test set to get a final, unbiased estimate of our model's performance.
C. Data preprocessing
We start with a sequence of T record tuples t (i) , x (i) , y (i) , i = 1, . . . , T record , where t (i) is the time stamp (s) and x (i) , y (i) are the horizontal and vertical components of the monocular (left eye) gaze position ( • ). Next, we estimate the first derivative (i.e., velocity in • /s) of the horizontal and vertical channels using a Savitzky-Golay [58] differentiation filter with order 2 and window size 7, inspired by [45].
Each recording is then split into non-overlapping windows of 5 s (T = 5000 time steps) using a rolling window. Excess time steps at the end of a recording that would form only a partial window are discarded. Although previous studies like DEL [4] and EKY [23] use input sizes of around 1 s, we found in our experiments that our model performs better when using a larger input size of 5 s than when it has to learn from isolated samples of 1 s. We believe this is because the model can take advantage of longer-term patterns when given longer sequences. We note that 5 s is somewhat comparable to the amount of time it takes to enter a 4-digit pin or to calibrate an eye-tracking device.
Velocities are clamped between ±1000 • /s to limit the influence of noise. They are then z-score transformed using a single mean and standard deviation determined across both channels in the train set. Finally, any NaN values are replaced with 0 after z-scoring.
We observed during our experiments that estimating velocity with a Savitzky-Golay differentiation filter and scaling with a z-score transformation led to marginal improvements in performance metrics compared to the approach of [4], wherein velocity is computed with the two-point central difference method and velocities are transformed using the "fast" and "slow" transformations proposed in [22].
D. Training
Input samples consist of windows of T = 5000 time steps and C = 2 channels: horizontal and vertical velocity. Following [23], we primarily use multi-similarity (MS) [59] loss to train our model. MS loss encourages the learned embedding space to be well-clustered, meaning a sample from one class is closer to other samples from the same class than to samples from different classes. However, we observed during our experiments that a weighted sum of MS loss (using the output of the embedding layer) and categorical cross-entropy (CE) loss (using the output of the classification layer) led to marginal improvements in performance metrics compared to using MS loss alone.
Therefore, our loss function L is given by the following equations: x i,j is the predicted logit for sample i and class label j; and y i is the target class label for sample i. The above formulation for MS loss implicitly includes an online pair miner with an additional hyperparameter ε = 0.1. More details about MS loss can be found in [59]. Each minibatch consists of 256 samples constructed in the following manner. First, 16 unique subjects are selected at random from the train set. Next, 16 windows are randomly selected without replacement for each of the selected subjects. These windows could be selected from any of the rounds, sessions, and tasks that each subject was present for in the train set. This results in 256 windows per minibatch. Each training "epoch" iterates over as many minibatches as needed until a number of windows, equivalent to the total number of unique windows in the train set, has been sampled. Note that because of the nature of this minibatch construction method, windows from earlier rounds may be over-represented [23], and not every window from the train set may be included in any given epoch. One downside of using input windows of 5 s is that it greatly reduces the number of samples available for training compared to when using a smaller input window like 1 s. This is a problem because deep learning methodologies generally perform better when trained on larger datasets. We are able to mitigate this issue by training on all tasks (except BLG) simultaneously instead of training on a single task. Additionally, the use of varied tasks encourages learned features to be informative for all types of eye movements (particularly fixations, saccades, and smooth pursuits) and enables a single model to be applied on multiple tasks, instead of necessitating a separate model for different tasks as most prior works do.
L M S = 1 m m i=1 1 α log 1 + k∈Pi exp (−α (S ik − λ)) + 1 β log 1 + k∈Ni exp (β (S ik − λ)) ,(2)L CE = 1 m m i=1 − log exp (x i,yi ) N j=1 exp (x i,j ) ,(3)L = w M S L M S + w CE L CE ,(4)
We employ the Adam [60] optimizer with a one-cycle cosine annealing learning rate scheduler [61] (visualized in Fig. 3A). The learning rate starts at 10 −4 , gradually increases to a maximum of 10 −2 over the first 30 epochs, and then gradually decreases to a minimum of 10 −7 over the next 70 epochs. We found that, compared to using a fixed learning rate throughout training, this learning rate schedule both accelerated the training process and led to higher levels of performance. Training lasts for a fixed duration of 100 epochs, and the final weights of the model are saved.
We note that there is no feedback from the validation set when training in this way (in contrast to when early stopping is employed, for example). However, the validation set was used while manually tweaking the proposed architecture and training paradigm. The final architecture was chosen as the one that maximized Mean Average Precision at R (MAP@R) [62] on the validation set (using embeddings from TEX only). MAP@R is a clustering metric that we believe is more informative than EER for model selection. We visualize the progression of MAP@R throughout training in Fig. 3B to provide some insight into the values we achieve; but because it is not directly related to biometric authentication, we do not report MAP@R in our results.
E. Evaluation
Although the model is trained on samples from all tasks (except BLG), we primarily evaluate the model on only TEX due to the prevalent usage of reading data in the EMB literature.
We primarily evaluate the model using equal error rate (EER), which is the point where false rejection rate (FRR) is equal to false acceptance rate (FAR). Measuring EER requires a set of data used for enrollment and a separate set of data for authentication (also commonly called verification). The enrollment set is formed using the first window (5 s) of the session 1 TEX task from R1 for each subject in the test set. The authentication set is formed using the first window (5 s) of the session 2 TEX task from R1 for each subject in the test set.
To ensure a minimal level of sample fidelity at evaluation time, we discard windows with more than 50% NaNs. Subjects are effectively excluded from the enrollment or authentication sets if they have no valid windows in the respective set.
For each window in the enrollment and authentication sets, we compute the 128-dimensional embeddings with each of the 4 models trained with 4-fold cross-validation. We then concatenate these embeddings to form a single, 512-dimensional embedding for each window, effectively treating the 4 models as a single ensemble model. We compute all pairwise cosine similarities between the embeddings in the enrollment set and those in the authentication set. The resulting similarity scores are fed into a receiver operating characteristic (ROC) curve to measure EER.
Given the resolution of a particular ROC curve, there may not be a similarity threshold where FRR and FAR are exactly equal. In such cases, the EER needs to be estimated, and there are several ways this estimation can be done. The method we use is to linearly interpolate between the points on the ROC curve to estimate the point where FRR and FAR would be equal.
In addition to the primary evaluation setting described above, we can also change different parameters to evaluate our model under various conditions. We will explore our model's performance on different tasks, across longer testretest intervals, and using increasing amounts of data for enrollment and authentication. We will also examine how well our network adapts to data with lower sampling rates. These additional analyses will be described later.
V. RESULTS
Unless otherwise specified, presented results are measured on the held-out test set using an ensemble model evaluated under the primary evaluation setting, meaning we enroll and authenticate with 5 s of 1000 Hz data from R1 TEX.
Results on the test set for our primary evaluation setting are presented in the first row of Table I. The other results in that table are described in the upcoming subsections. Fig. 4 shows the similarity score distributions and ROC curve under the primary evaluation setting.
To visualize the embedding space, DensMAP [63] is used to create a low-dimensional representation of the embedding space in a way that attempts to globally and locally preserve structure and density. A subset of the embedding space is visualized in Fig. 5.
Although we focus on the authentication scenario, it is worth briefly mentioning for completeness how the model performs in the identification scenario. We employ the rank-1 identification rate which measures how often the correct identities have the highest similarity score between the enrollment and authentication sets. Under the primary evaluation setting, after removing any authentication subjects who are not present in the enrollment set, rank-1 identification rate is 91.38% (53 of 58 subjects are correctly identified).
A. Effect of task on authentication accuracy
For this analysis, we replace TEX with one of the other tasks during evaluation and repeat for each task. Note that we evaluate our single ensemble model across all tasks; we do not train a separate model for each task. Results are presented in Table I in the "Task" effect group. To assess our model's performance on an out-of-sample task, we also include results for BLG.
B. Effect of test-retest interval on authentication accuracy
For this analysis, we continue using the first session of R1 for the enrollment set, but for the authentication set we use the second session of one of the later rounds (R2-9) to assess how robust our model is to template aging after as many as 37 months. Results are presented in Table I in the "Test-retest interval" effect group.
C. Effect of recording duration on authentication accuracy
For this analysis, instead of limiting ourselves to the first 5-second window of a recording, we aggregate embeddings across the first n windows to form a new, centroid embedding. Since the model is trained to create a well-clustered embedding space, averaging multiple embeddings for a given class should lead to a better estimate of that class's central tendency in the embedding space. Results are presented in Table I in the "Duration" effect group.
D. Effect of sampling rate on authentication accuracy
For this analysis, instead of using 1000 Hz data, we downsample each recording to different target sampling rates using an anti-aliasing filter (SciPy's [64] decimate function) to assess how robust our network architecture is to lower sampling rates. The targeted sampling rates are the same as those in [23]: 500, 250, 125, 50, and 31.25 Hz. Input size is reduced by the same integer factor as the sampling rate and then truncated to remove any fractional components. For example, at 31.25 Hz (a downsample factor of 32), the input size becomes 5000 32 = 156 time steps. Since our network architecture contains a global pooling layer prior to the fully-connected layer(s), the network can be applied to time series of any length without any modifications. But, because features learned at one sampling rate would not likely translate well to different sampling rates, we opted to train a new ensemble of 4-fold cross-validated models for each degraded sampling rate to have the best chance at extracting meaningful information at each sampling rate. We do not adjust the Savitzky-Golay differentiation filter parameters for the lower sampling rates. It is also worth noting that the (maximum) receptive field of our network, 257 time steps, is larger than the input sizes at 50 and 31.25 Hz.
E. Estimating FRR @ FAR 10 −4
The ultimate goal of EMB is to enable the use of eye movements for biometric authentication in real-world settings. It is important to consider how EMB compares to existing security methods, because if it cannot outperform such methods then wide adoption would be unlikely. Like [23], we use the 4digit (10-key) pin as a representative for existing security methods, because it is one of the most common security methods in everyday life as both a primary and secondary security measure. The 4-digit pin effectively has a FAR of 1in-10000, assuming each of the 10 4 combinations of 4-digit 10-key pins is equally likely to be chosen by enrolled users.
For this analysis, to mimic the level of security afforded by a 4-digit pin, we provide estimates of FRR when FAR is fixed at 10 −4 (abbreviated FRR @ FAR 10 −4 ). Directly measuring FRR @ FAR 10 −4 requires at least N = 10000 impostor similarity scores, but we are limited to a maximum of 3422. Therefore, to enable the estimation of FRR @ FAR 10 −4 , we use bootstrapping (i.e., repeated random sampling with replacement) to resample our empirical genuine and impostor similarity score distributions to form new distributions with P = 20000 and N = 20000 scores. We repeat bootstrapping 1000 times and report the mean and standard deviation of the performance across those 1000 bootstrapped distributions. The FIDO Biometrics Requirements [65] suggest that a biometric system should have no higher than 3-5% FRR @ FAR 10 −4 , though we note that our bootstrapping technique differs from theirs and our test set population of 59 does not meet their minimum population requirements of 123-245.
We note that EKY [23] employs a different method to estimate FRR @ FAR 10 −4 involving the Pearson family of distributions. We propose the use of bootstrapping because it is simpler, makes fewer assumptions about the empirical distribution, and is more commonly used as a resampling tool. Bootstrapping largely preserves the shape of the empirical distribution, whereas the Pearson-based approach by [23] produces a new distribution that may not preserve characteristics of the region of interest where the genuine and impostor distributions overlap. A "failure case" of the Pearson-based Fig. 6. Comparison of genuine vs impostor similarity score distributions for (left) the empirical distributions, (center) resampled distributions using the Pearson-based method from [23], and (right) resampled distributions using bootstrapping. We note that bootstrapping more closely preserves the shape of the empirical distributions, particularly the overlapping region between the genuine and impostor distributions. In contrast, the Pearsonbased method produces significantly less overlap between the genuine and impostor distributions, leading to a significant reduction in FRR @ FAR 10 −4 compared to the bootstrapped distributions. Plotted scores are from evaluating our model on 31.25 Hz R1 TEX with 5 × 12 inputs. In this example, the Pearson-based method results in 6.26% FRR @ FAR 10 −4 which is significantly different from the bootstrapped result of 51.28%. approach is shown in Fig. 6. We believe the main source of this failure is that the empirical distribution of genuine scores is not unimodal (there appears to be a smaller second mode around 0.8 similarity), so we break the assumptions of the Pearson distribution.
F. Determining accept/reject threshold on validation set
As mentioned in [23], it is problematic to compute EER on the test set, because doing so leaks information from the test set into the decision of which accept/reject threshold to use. A more principled approach is to use the validation set to determine the accept/reject threshold and then apply that threshold to the test set.
For this analysis, we do exactly that. For each individual model from the ensemble, we build a ROC curve using similarity scores computed on that model's validation set and determine the threshold that yields the EER. Then, separately for each model, we apply the chosen threshold on the similarity scores from the test set.
Note that for this analysis, unlike the previous analyses, we are no longer treating the 4 models as a single ensemble model, because each model's validation set is present in the train set for the other 3 models. So, to provide a better understanding of performance without ensembling the individual models, we also measure EER for each individual model.
Results for this analysis are presented in Table IV. We label the folds F0, F1, F2, and F3 and match each model to the fold that was used as its validation set.
G. Comparison to previous state of the art
The statistical approach by Friedman et al. [45] reports 2.01% EER (P = 149, N = 22052) on the TEX task from GazeBase when enrolling with the full duration (approx. 60 s) of the first session of R1 and authenticating with the full duration (approx. 60 s) of the second session of R1. This result can be directly compared to our result of 0.58% EER (P = 59, N = 3422) for the 5 × 12 duration in Table I in the "Duration" group.
EKY [23] reports 14.88% EER (P = 59, N = 3422) on the TEX task from GazeBase when enrolling with the first Table I where we achieve 3.66% EER (P = 58, N = 3364) under the same condition (except we exclude windows with more than 50% NaNs). DEL [4] reports 10.0% EER (P = 25, N = 600) on the TEX task from GazeBase when enrolling with 24 s of data sampled from R1-2 and authenticating with 5 s of data sampled from R3-4. This result can be roughly compared to our results in Table I in the "Test-retest interval" group, particularly for R2-4, where we achieve between 7.43% and 8.71% EER (P = 58, N = 3364). Though, we note that we enroll with only the first 5 s of the first session of R1.
It is also worth briefly comparing against EKY and DEL at degraded sampling rates. DEL reports a mean EER of 9% with 5 s of 125 Hz data in [66]. This is similar to our result of 8.77% EER with 5 s of 125 Hz data, but it is difficult to directly compare across studies since we use a different dataset. EKY reports a mean EER of 18.75% with 10 s of 125 Hz data which is significantly worse than our results. We note that estimates of FRR @ FAR are not reported in [66].
For our final analysis, we evaluate our model on the JuDo1000 [67] dataset which is recorded with an eye-tracking device similar to the one used in GazeBase and which uses an eye-tracking task similar to the RAN task from GazeBase. JuDo1000 contains 150 subjects recorded across 4 sessions, each separated by at least 1 week. Each session contains 12 repetitions each of 9 different trial configurations. More details about the dataset can be found in [67]. From each recording, the 12 trials with the largest display area (grid = 0.25) and longest duration (dur = 1000) are selected, providing us with 12 windows of 5 s each. JuDo1000 is a binocular dataset but our model was trained on monocular data, so we combine the left and right eye gaze positions by averaging them. Gaze positions in each window are converted from pixels to degrees and then from position to velocity using the same Savitzky-Golay differentiation filter we used for GazeBase. Using the same trained model that produced the results shown in Table I and without any fine-tuning, we simply compute embeddings of these windows from JuDo1000. We enroll the embeddings from the first recording session and authenticate with the embeddings from the second recording session (a test-retest interval of approximately 1 week), excluding windows with more than 50% NaNs. Table V shows our results alongside the published results of DEL. Since there are more than 10000 negative pairs, FRR @ FAR 10 −4 is directly measured without any resampling. We note that this is not a one-to-one comparison with DEL for several reasons, including: DEL trained on JuDo1000 but we trained on GazeBase; DEL used a total of 12 input channels but we used 2; DEL enrolled using 24 s of data across 3 recording sessions but we enrolled with 5 s from only 1 session; and DEL averages results across all 9 trial configurations in JuDo1000 but we only use 1 trial configuration. That said, these results show the robustness of our model to completely out-of-sample data.
VI. DISCUSSION
The primary result that we present is 3.66% EER on a reading task with 5-second-long enrollment and authentication periods and with an approximately 30-minute test-retest interval. EKY [23] reports a mean EER of 14.88% under similar conditions. DEL [4] reports a mean EER of 3.97% on a jumping dot task with an enrollment period lasting 24 seconds over a 3-week period and a 5-second-long authentication period with an approximately 1-week test-retest interval.
Authentication accuracy is generally better for TEX than the other tasks, as is expected given the literature's predominant use of reading data for EMB. The EER for VD2 is slightly lower than for TEX, but this difference may not be statistically significant and the trend may not continue for longer durations. Unsurprisingly, authentication accuracy is the worst for FXS; but it is impressive that we manage to achieve below 10% EER given just 5 s of pure fixational data. We note that the FXS task was not well represented in the training set, because the task has a maximum duration of approx. 15 s (compared to 60-100 s for the other tasks) and all other tasks are likely to elicit several saccadic movements in any given 5-second period. What is quite surprising, however, is that our model achieves 5.49% EER on BLG despite that task not being present during training. BLG presumably elicits very different eye movement responses than the other tasks because it is an interactive game with many objects moving on the screen at once, but our model is still able to create meaningful embeddings of the eye movement signals. We also draw attention to the fact that the embedding space (Fig. 5B) appears to be fairly well-clustered across tasks, suggesting that it may be viable to enroll with one task and authenticate with another.
Our model exhibits high robustness to template aging, even with just 5 s of eye movement data. When authenticating on R6, which is approx. 1 year after R1 and is not represented in the train or validation sets, we still achieve 6.09% EER. In fact, EER remains consistently between 6-9% for all test-retest intervals from approx. 1 to 37 months.
It is still an open question as to how much eye movement data is necessary to adequately perform user authentication and whether there is a point beyond which additional data provides no new information. For our model, EER improves as the duration of enrollment and authentication increases from 5 s to 20 s, after which it starts to saturate around 0.4-0.6%. Estimates of FRR @ FAR 10 −4 improve with increasing duration up to 30 s before saturating around 5%. Our results suggest that there may not be much additional information to be gained beyond 30 s of eye movements during a reading task. Though, it must be noted that this claim is based on the TEX task from GazeBase wherein each subject read through each passage at different speeds. Perhaps the reason we do not see much improvement beyond 30 s is that most subjects may have finished reading after 30 s and did not have consistent behavior afterward.
Authentication accuracy remains relatively stable as the sampling rate is degraded from 1000 Hz down to 250 Hz, starts to noticeably worsen at 125 Hz, and then drops significantly starting at 50 Hz. It is unclear how much of this performance degradation is due to the use of an untuned differentiation filter. These results may reflect findings in the literature that saccade characteristics (e.g., peak velocity and duration) can be measured accurately at a sampling rate of 250 Hz [68] and begin to become less accurate at lower sampling rates [69], [70]. At 125 Hz, which is close to the 120 Hz sampling rate of the Vive Pro Eye [14], our model is still able to achieve 8.77% EER with just 5 s of data and an estimated 10.52% FRR @ FAR 10 −4 with 60 s of data. Depending on the degree of security necessary, these results suggest the present applicability of EMB at sampling rates present in current VR/AR devices. Another interesting observation is that we are able to achieve around 5.09% EER with 60 s of 31.25 Hz data, suggesting that there is still meaningful biometric information that can be extracted at such low sampling rates. While we acknowledge that simply degrading the sampling rate of highquality data is not a sufficient proxy for other eye-tracking devices, we note that gaze estimation pipelines could always be improved to produce higher levels of signal quality at a particular sampling rate, whereas it may not always be possible for a device to increase the sampling rate of its eye-tracking sensor(s) due to power constraints (though some efforts are being made to enable eye tracking at high sampling rates with lower power requirements [18]).
Most EMB studies, including the present study, report measures of EER directly on the test set, leaking information from the test set into the accept/reject decisions. When taking a more principled approach and fitting the accept/reject threshold on the validation set instead, we find that the thresholds become more strict, resulting in a lower FAR and a higher FRR. As such, these thresholds would be better for settings requiring a higher degree of security but may be more frustrating for users.
VII. CONCLUSION
We presented a novel, highly parameter-efficient, DenseNetbased architecture for end-to-end EMB that achieves state-ofthe-art biometric authentication performance. When enrolling and authenticating with just 5 s of eye movements during a reading task-a duration somewhat comparable to the time it takes to enter a 4-digit pin or to calibrate an eye-tracking device-we achieved 3.66% EER. With 30 s of data, we achieved an estimated 5.08% FRR @ FAR 10 −4 which approaches a level of authentication performance that would be acceptable for real-world use. At 125 Hz, which is close to the 120 Hz sampling rate of the Vive Pro Eye [14], we achieved 8.77% EER with just 5 s of data and an estimated 10.52% FRR @ FAR 10 −4 with 60 s of data.
Our embedding space visualizations suggest that it may be feasible to enroll with one (or several) tasks and authenticate with a different task. We are not aware of any study that has attempted this. It would also be interesting to see how well privacy-preserving models (e.g., [12]) can defend against more powerful EMB models like the one presented herein.
Since eye tracking is seeing increasing use in VR/AR devices due in part to the power-saving potential of foveated rendering, EMB may be an ideal biometric modality for such devices. EMB models would need to have low resource requirements when performing (continuous) user authentication on such consumer-grade devices. Our architecture (excluding the classification layer) has only 123K learnable parameters which is around 4x smaller than EKY (approx. 475K learnable parameters) and around 1700x smaller than DEL (approx. 209M learnable parameters according to [23]). Models with lower complexity such as ours may enable more powerefficient implementations that would make them a better fit for deployment on consumer-grade devices. Following works like [19], we encourage future studies to explore EMB directly on eye-tracking-enabled VR/AR devices.
where w M S = 1.0 and w CE = 0.1 are the weights for the respective loss functions; m = 256 is the size of each minibatch; α = 2.0, β = 50.0, and λ = 0.5 are hyperparameters for MS loss; P i and N i are the sets of indices of the mined positive and negative pairs for each anchor sample x i ; S ik is the cosine similarity between the pair of samples {x i , x k }; N is the number of classes (either 197 or 198) in the train set;
Fig. 3 .
3(A) A visualization of the learning rate schedule used during training. (B) Progression of MAP@R (measured on TEX only) throughout training.
Fig. 4 .
4Qualitative results for the primary evaluation setting: 5 s of R1 TEX. (A) Genuine and impostor similarity score distributions. (B) ROC curve for bootstrapped similarity score distributions (see § V-E for an explanation of how the bootstrapped distributions are made). The dashed black line shows where FRR and FAR are equal. The blue line is the mean ROC curve across 1000 bootstrapped distributions, and the shaded region represents ±1 standard deviation around the mean.
Fig. 5 .
5DensMAP[63] visualizations of the embedding space for 10 subjects present across all rounds. All embeddings of valid (≤50% NaNs) windows across all rounds R1-9 and both sessions are plotted together. A different mapping is fit for each plot. (A) Embeddings from only the TEX task. (B) Embeddings from all tasks (including BLG). We use umap-learn[57] parameters metric=cosine, n neighbors=30, min dist=0.1, and densmap=True.
0000-0000/00$00.00 © 2021 IEEE arXiv:2201.02110v2 [cs.CV] 28 Mar 2022Fig.
5 GHz (32 cores), and 256 GB RAM. Each Docker container runs Ubuntu 18.04 with the most notable packages being Python 3.7.11, PyTorch [53] 1.10.0, and PyTorch Metric Learning (PML) [54] 0.9.99. PyTorch Lightning [55] 1.5.0 is used to accelerate development. Experiments are logged using Weights & Biases [56] 0.12.1. For visualizing the embedding space, we employ umap-learn [57] 0.5.1. Our full source code and trained models are available on the Texas State Digital Collections Repository at [TO BE PUBLISHED AFTER ACCEPTANCE].
TABLE I BIOMETRIC
IAUTHENTICATION RESULTS FOR VARIOUS EVALUATION SETTINGS USING A SINGLE ENSEMBLE OF MODELS TRAINED WITH 4-FOLD CROSS-VALIDATION. DURATION IS GIVEN AS T × n, WHERE T IS THE LENGTH OF EACH SAMPLE AND n IS THE NUMBER OF SAMPLES. P AND N ARE THE NUMBERS OF POSITIVE AND NEGATIVE PAIRS, RESPECTIVELY. BLG was not included in the train or validation sets.Effect
Duration (s) Round Task
EER (%)
P
N
-
5 × 1
R1
TEX
3.66
58 3364
Task
( § V-A)
5 × 1
R1
HSS
5.08
59 3422
RAN
5.08
59 3422
FXS
9.38
59 3422
VD1
5.45
55 3135
VD2
3.39
59 3422
BLG*
5.49
59 3422
Test-retest interval
( § V-B)
5 × 1
R2
TEX
8.62
58 3364
R3
7.43
58 3364
R4
8.71
58 3364
R5
7.14
58 3306
R6
6.09
58 3364
R7
8.52
34 1996
R8
8.89
30 1710
R9
7.69
13
799
Duration
( § V-C)
5 × 2
R1
TEX
2.23
58 3364
5 × 3
0.76
59 3422
5 × 4
0.38
59 3422
5 × 5
0.58
59 3422
5 × 6
0.56
59 3422
5 × 7
0.58
59 3422
5 × 8
0.56
59 3422
5 × 9
0.56
59 3422
5 × 10
0.50
59 3422
5 × 11
0.41
59 3422
5 × 12
0.58
59 3422
*
TABLE II BIOMETRIC
IIAUTHENTICATION RESULTS AT DEGRADED SAMPLING RATES. A DIFFERENT ENSEMBLE OF MODELS IS TRAINED FOR EACH SAMPLING RATE.Sampling rate (Hz) EER (%)
P
N
500
5.66
53 3079
250
6.20
53 3079
125
8.77
57 3306
50
15.52
58 3364
31.25
23.37
58 3364
TABLE III BIOMETRIC
IIIAUTHENTICATION RESULTS USING BOOTSTRAPPED SIMILARITY SCORE DISTRIBUTIONS. RESULTS ARE REPORTED AS MEAN ±SD ACROSS 1000 BOOTSTRAPPED DISTRIBUTIONS. EACH BOOTSTRAPPED DISTRIBUTION CONTAINS P = 20000 POSITIVES AND N = 20000 NEGATIVES.Sampling rate (Hz) Duration (s)
EER (%)
FRR @ FAR (%)
10 −1
10 −2
10 −3
10 −4
1000
5 × 1
3.67 ±0.12
3.45 ±0.13
10.34 ±0.22
14.27 ±1.04
30.19 ±2.87
5 × 2
2.22 ±0.11
0.00 ±0.00
5.16 ±0.19
13.60 ±1.78
17.13 ±0.50
5 × 4
0.38 ±0.05
0.00 ±0.00
0.00 ±0.00
8.33 ±0.73
8.48 ±0.20
5 × 6
0.56 ±0.05
0.00 ±0.00
0.00 ±0.00
5.08 ±0.16
5.08 ±0.16
5 × 12
0.59 ±0.05
0.00 ±0.00
0.00 ±0.00
5.09 ±0.16
5.09 ±0.16
500
5 × 12
0.32 ±0.04
0.00 ±0.00
0.00 ±0.00
3.82 ±0.31
7.46 ±0.43
250
0.81 ±0.06
0.00 ±0.00
0.00 ±0.00
3.78 ±0.13
5.58 ±0.44
125
3.49 ±0.12
0.00 ±0.00
7.02 ±0.18
10.47 ±0.36
10.52 ±0.21
50
3.38 ±0.13
0.00 ±0.00
10.40 ±0.63
16.06 ±1.47
20.21 ±0.52
31.25
5.09 ±0.16
5.09 ±0.16
14.47 ±1.12
29.36 ±2.12
51.28 ±4.26
TABLE IV BIOMETRIC
IVAUTHENTICATION RESULTS ON THE TEST SET FOR EACH INDIVIDUAL MODEL FROM THE ENSEMBLE WHEN THE ACCEPT/REJECT THRESHOLD IS DETERMINED ON EITHER THE TEST SET OR A GIVEN MODEL'S VALIDATION SET.Fold
Fit on test set
Fit on validation set
threshold EER (%) threshold FRR (%) FAR (%)
F0
0.4231
5.71
0.5405
12.07
1.28
F1
0.3970
8.62
0.5850
15.52
0.65
F2
0.4505
5.17
0.5575
10.34
1.25
F3
0.4277
6.90
0.5024
8.62
2.82
5 s of the first session of R1 and authenticating with the first
5 s of the second session of R1. This result can be directly
compared to our primary result in the first row of
TABLE V BIOMETRIC
VAUTHENTICATION RESULTS ON THE JUDO1000 [67] DATASET. Our model was trained on GazeBase and evaluated on JuDo1000.Model
Duration (s) EER (%)
FRR @ FAR (%)
P
N
10 −2
10 −4
DEL [4]
1 × 5
3.97
25.67
-
25
600
1 × 10
3.01
22.01
-
25
600
Ours*
5 × 1
12.00
44.00
92.00
150 22 350
5 × 2
7.33
28.67
68.67
150 22 350
5 × 12
2.67
5.33
24.00
150 22 350
*
ACKNOWLEDGMENTSThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144466. The study was also funded by National Science Foundation grant CNS-1714623 to Dr. Komogortsev. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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| We have experimentally observed a new phenomenon in the surface flow of a granular material. A heap is constructed by injecting sand between two vertical glass plates separated by a distance much larger than the average grain size, with an open boundary. As the heap reaches the open boundary, "soliton-like" fluctuations appear on the flowing layer, and move "up the hill" (i.e., against the direction of the flow). We explain the phenomenon in the context of stop-and-go traffic models, and show that soliton-like behavior is allowed within a Saint-Venant description for the granular flow. | 10.1103/physreve.75.031303 | [
"https://arxiv.org/pdf/cond-mat/0601685v1.pdf"
]
| 29,269,529 | cond-mat/0601685 | c99a3d2f25f236dbaca3f5437c106aa299901dd3 |
Uphill solitary waves in granular flows
31 Jan 2006
E Martínez
C Pérez-Penichet
O Sotolongo-Costa
Physics Faculty
"Henri Poincaré" Group of Complex Systems
University of Havana
10400HavanaCuba
O Ramos
K J Måløy
Physics Department
University of Oslo
BlindernN-0316OsloNorway
S Douady
Laboratoire Matire et Systmes Complexes (MSC)
Universit Paris 7 -Denis Diderot / CNRS
7056 75251, Cedex 05ParisCCFrance
E Altshuler
Physics Faculty-IMRE
Henri Poincaré" Group of Complex Systems and Superconductivity Laboratory
University of Havana
10400HavanaCuba
Uphill solitary waves in granular flows
31 Jan 2006(Dated: September 13, 2018)numbers: 4570-n4570Mg4570Vn8105Rm8975-k
We have experimentally observed a new phenomenon in the surface flow of a granular material. A heap is constructed by injecting sand between two vertical glass plates separated by a distance much larger than the average grain size, with an open boundary. As the heap reaches the open boundary, "soliton-like" fluctuations appear on the flowing layer, and move "up the hill" (i.e., against the direction of the flow). We explain the phenomenon in the context of stop-and-go traffic models, and show that soliton-like behavior is allowed within a Saint-Venant description for the granular flow.
The rich dynamics of granular matter -studied for centuries by engineers-has attracted much attention from the Physics community since the early 1990's [1,2,3]. Granular flows, for example, have concentrated intense interest, due to their relevance to natural avalanches and industrial processes, and also because they make liquidlike and solid-like behaviors coexist. They have been theoretically described based on the existence of two phases: the rolling (or flowing) one, and the static one. Such idea has been casted into ad hoc phenomenological equations [4,5,6], into a Saint-Venant hydrodynamic approach conveniently modified to take into account the particularities of granular matter [7,8], and eventually by defining an order parameter characterizing the local state of the system [9,10]. Finally, granular flows have been described by "microscopic" equations based on the newtonian motion of individual grains submitted to gravity, shocks, and trapping events [11,12].
In particular, researchers have studied both experimentally and theoretically granular flows on inclined planes and tubes, and granular heaps, finding a whole jungle of patterns, spatio-temporal structures, and other nontrivial phenomena, such as fingering [13], avalanches extending both downhill and uphill [14], "logitudinal vortices" [15,16], "bubbling flows" [17,18],"revolving rivers" [19] and even "singing" dunes [20].
So it looks very unlikely that one can find further unexplored phenomena in experiments as simple as pouring sand on a heap with an open boundary. However, we report here the existence of soliton-like waves moving uphill in such experiments, i.e., the spontaneous appearance of bump-like instabilities in the flowing layer that propagate uphill, contrary to the flow of sand.
We used sand with a high content of silicon oxide and an average grain size of 100µm from Santa Teresa (Pinar del Río, Cuba) [19]. The sand was poured into a cell consisting in a horizontal base and a vertical wall, sand- wiched between two square glass plates with inner surfaces separated by a distance w in the range from 0.3cm to 3cm (Fig. 1). The lengths of the base and the vertical wall were approximately d ≈ 23cm. The sand was poured vertically into the cell near the vertical wall using tiny funnels with several hole diameters, in order to obtain different flux values. As the sand was poured into the cell, a heap grew until it reached the open boundary, where the grains were allowed to fall freely ( Fig. 1). Digital videos were taken using a High Speed Video Camera Photron FASTCAM Ultima-APX model 120K in the range from 50f ps to 4000f ps, with a resolution of 1024 × 1024 pixels. Fig. 2 contain images from experiments with w = 0.7cm and F = 0.6cm 3 /s. Fig. 2 (a) is a picture taken from a video recorded at 50f ps, and show a closeup of a section of the surface at the central region of the heap, where our basic findings are easily identified: "bumpshaped" instabilities appearing at random places of the surface move "up the hill", as indicated by the white arrows. These bumps could travel either alone or in tandem. The bumps maintain their "identity" within variable times, which can reach more than 10 seconds. After that, they flatten out. We have also observed that, if two of them move at speeds different enough to interfere with each other, they can be still identified after the interaction. However, these are rare events, and we have no video record of them. All in all, our observations hint at a soliton-like behavior in the observed instabilities. Fig. 2(b) shows the difference between two pictures from a single bump taken from a video recorded at 500f ps, which are separated 50ms from each other. The darker horizontal band allows to visualize the downhill motion. Careful examination of picture 2(b) allows to separate an approximately 1mm-depth band of "flying" grains, and an approximately 1mm depth layer of "flowing" grains underneath. The height of the perturbation above the unperturbed stream is difficult to measure, but it can be roughly estimated as 1/5 of the flowing layer. Fig. 2(c) shows a spatial-temporal diagram of the free surface. A horizontal line of the video record was taken just at the (average) free surface, so lower parts appear black, and higher parts appear white. From the picture it can clearly be seen four shock-waves, passing suddenly from black to white, moving upward with a well defined velocity (even if it fluctuates a little on the right). On the left the appearance of a new soliton can be seen, first increasing in height before starting suddenly to propagate.
The average speed of the uphill motion of the bumps as a function of the input flux, F was estimated from the analysis of the motion of several instabilities using pictures analogous to those used to obtain Fig 2(b). Fig. 3 presents the dependence of the average uphill speed versus input flux. The solid line corresponds to v up ∼ F 1/4 , which will be discussed below. Fig. 4 shows a sequence of pictures of a bump separated by intervals of 0.125s, extracted from a video taken at 4000 fps. Careful inspection of the video associated to Fig. 4(a), represents one of those grains, which is moving from left to right. From 4(b) to 4(d), the grain has incorporated into the static layer (deposition), while the bump is passing from right to left above it. In Fig. 4(e), the grain has reincorporated into the flowing layer (erosion), moving again from left to right.
Exchange of particles between the static and fluid phases has been considered in several models of surface flows since the 1990's [4]. We believe that our bumps form when a fluctuation implies an extra deposition of grains from the flowing layer on the static one, which then propagates backwards through an "stop-and-go" mechanism observed in traffic dynamics [22]. However, in our case the usual traffic models should be modified to take into account the free surface. A way to do it could be a two-lane system consisting in a fast lane where cars flow steadily, and a slow lane where cars stop and go, producing a backward wave (or train of waves) within the slow lane, well described in the literature [22].
All in all, the situation can be pictured as a solitary wave crawling uphill underneath a shallow stream of sand flowing downhill. In fact, the stop-and-go traffic model has been mapped into the "classical" Kortweg-de Vries (KdV) equation after a few approximations, resulting in soliton-like solutions [23]. In the context of the KdV equation, the speed of the soliton can be estimated as v s ∼ √ gh 0 where g and h 0 are the acceleration of gravity and the depth of the unperturbed stream, respectively [24]. If one assumes the well accepted result that the depth of the flowing layer is proportional to F 1/2 [21], we then get v s ∼ F 1/4 , which follows the experimental result shown in Fig. 3.
Knowing that the KdV equation can be derived from mass and momentum conservation equations applied to an incompressible fluid flowing down a shallow channel [24], we could expect that soliton solutions can be found if we describe our system through hydrodynamics Saint-Venant equations modified to take into account the particularities of granular matter [7,8] which we are writing relative to a reference frame parallel to the average free surface of the heap (see Fig. 5) . The first equation corresponds to mass conservation:
∂ t ζ + ∂ x (ΓH 2 /2) = 0 (1)
where ζ is the height of the free surface, H is the thickness of the flowing layer, and Γ is the velocity gradient transversal to the flowing layer, which is assumed constant. Then, the momentum q = ΓH 2 /2 evolves with H due to the erosion/deposition process, like the one illustrated for one grain in Fig. 4. The second equation corresponds to the conservation of momentum, which can be seen as an equation governing the evolution of the thickness of the flowing layer, H:
∂ t H + ∂ x (ΓH 2 /2) = g Γ (tanθ − µ(H))(2)
where µ(H) is the friction acting on the layer and tanθ = −∂ x ζ is the slope of the free surface.
Let us assume that that the friction coefficient depends on H in the general form:
µ = α + (β + γ H )(h 2 0 − H 2 ) δ − ε(h 2 0 − H 2 )(3)
where the values of constants α, β, γ, δ and ǫ can be expressed in terms of experimental parameters v up , v f low , h 0 , g, and a characteristic length L 0 . Then, equations (1) and (2) can be transformed into the "textbook" KdV equation [24]
L 2 0 v f low η ′′ + 6(v up − v f low )η + 30 v f low h 0 η 2 = 0(4)
In (4), η = ζ − h 0 and η ′′ = d 2 η dϕ 2 , where ϕ = x + v up t. One particular solution of that equation, is
ζ = h 0 + h 0 5 sech 2 ( x + v up t L 0 )(5)
where we have taken L 0 = 10h 0 , v f low = 1 2 Γh 0 = √ gh 0 , and v up = 1 3 v f low in order to match our experimental observations (i.e., mainly that the uphill speed is roughly 1/3 of the flow speed, and the height of the perturbation is approximately 1/5 of the thickness of the unperturbed flowing layer). Formula (5) describes a "textbook", bell-shaped soliton moving upward. This solution holds for a friction coefficient given by equation (3) which is a positive, decreasing function of H when reasonable experimental parameters are introduced, in agreement with previous models [8]. Although we do not discard that "non-textbook" solitonic equations might describe the observed bumps without imposing such strong constraints, we stress that our objective here is to show that "textbook" solitons can be derived if the system displays an appropriate µ(H).
We now present a number of additional experimental facts relative to our uphill soliton waves. First, they have been observed in the whole range of experimental parameters explored, i.e. 0.15cm 3 /s ≤ F ≤ 3cm 3 /s and 0.3cm ≤ w ≤ 3cm. As many phenomena in granular matter, the uphill waves depend, to some extent, on the way the heap is prepared, and show some degree of "memory". When the heap is prepared from scratch, uphill bumps nucleate at random places near the center of the heap, and, within a few seconds, they typically appear near the open boundary. If the experiment is stopped and re-started in such conditions, solitons reappear immediately. That happens even if a layer of sand less than 1mm-thick is removed from the surface. If a thicker layer is removed, one has to wait a few seconds to observe them. This is probably due to the formation of a "compactified" layer during the flow that is necessary for the formation of these "jammed" bumps. Finally, all the results we have presented here have been observed just for one type of sand that produces sandpiles through "revolving rivers" when dropped on a flat horizontal surface [19]. This fact supports the idea that a quite specific µ(H) dependence is needed to observe the solitary waves we report in this paper.
In summary, we have observed soliton-like instabilities in a flow of sand established on a heap with open boundaries, moving against the direction of the flow. The phenomenon can be understood in the light of stop-and-go traffic arguments, even for this free surface flow, and can be described by Saint-Venant equations adapted to granular flows. The "microscopic" mechanisms that make a certain sand more suitable to show uphill bumps remains a mystery, but it seems clear that the dependence of the friction coefficient of our particular sand on the flowing depth is appropriate for the appearance of solitons.
We thank Ø. Johnsen and C. Noda for help in experi-ments and image processing, and E. Clèment, A. Daerr, S. Franz, H. Herrmann, J. Marín, D. Martínez, R. Mulet, D. Stariolo and J.E. Wesfreid for useful discussions and comments. G. Quintero and J. Fernández collaborated in numerical calculations. We appreciate financial support from the "Abdus Salam" ICTP during the last stage of this project.
FIG. 1 :
1Simplified diagram of the experimental setup
FIG. 2 :
2Basic experimental findings. (a) Uphill bumps moving in tandem, identified by arrows. (b) Difference between two pictures of a single bump, separated by 50ms. In both pictures, the smallest ticks in the scales correspond to 1mm. (c) Spatial-temporal diagram of bump dynamics, where x grows to the right and t grows upward, spanning 20cm and 2s, respectively.
FIG. 3 :
3Uphill speeds of bumps measured as a function of input flux, for w = 0.7mm this sequence of images reveals the mechanism of movement of the instability at the single grain level. The grains involved in the flowing layer in picture (a) show two types of behavior: while most of the grains keep flowing from left to right at an average speed v f low ∼ 10cm/s as the bump passes underneath from right to left , a fraction of the grains at the lower part of the flowing layer show a different behavior. The white circle in
FIG. 4 :
4Sequence of pictures of an bump moving uphill, with the same parameters described for Fig. 2(e). The white circle represents a grain of sand (a) moving downhill as part of the flowing layer, (b)-(d) trapped in the static layer as the bump passes by from left to right, and (e) moving again as part of the flowing layer. The horizontal length of the picture corresponds to 22mm in reality
FIG. 5 :
5Diagram showing the neighborhood of a "bump" which includes some parameters of the Saint-Venant model described in the text.
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| []
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[
"Probing Accretion Disk Winds in AGN I. Asymmetric broad Balmer emission lines",
"Probing Accretion Disk Winds in AGN I. Asymmetric broad Balmer emission lines"
]
| [
"Cosmos Dumba \nDepartment of Physics\nFaculty of Science\nMbarara University of Science & Technology\nP. O. Box 1410MbararaUganda\n"
]
| [
"Department of Physics\nFaculty of Science\nMbarara University of Science & Technology\nP. O. Box 1410MbararaUganda"
]
| []
| The Broad Line Region of Active Galactic Nuclei is characterized by broad Balmer emission lines in their optical spectra. The broad Balmer emission lines are found to be asymmetric, some blue sided and others red sided in their asymmetry. One of the components behind the asymmetry is thought to be an accretion disk wind. We probe the accretion disk wind using the broad balmer emission line profiles.This asymmetry of the broad balma emission line profiles is measured in velocity space after a measurement of the line shift at percentiles from 0, in increaments of 10, up to 90. In addition, the Kurtosis Index is obtained at appropriate points of the emission lines' profiles. This study is based on many hundreds of SDSS spectra, starting with low redshift high signal to noise ratio spectra. We also consider a definite number in each bin of their FWHM, in bins of 1000 km/s (atleast 40 per bin), starting from 1000km/s to the very broad emission lines.We present how strong the asymmetry (by plotting Asymmetry Index as a function of percentile) of the broad/narrow lines (in percent) is, what the Kurtosis (R20, 80) is. We also present what the Asymmetry Index as a function of line width (FWHM), luminosity (V-band), core-radio flux and Ionization Degree. | null | [
"https://arxiv.org/pdf/1409.6174v1.pdf"
]
| 118,477,421 | 1409.6174 | 97e4868fb3e3489ce711390f01d0e42fb60908fa |
Probing Accretion Disk Winds in AGN I. Asymmetric broad Balmer emission lines
September 23, 2014
Cosmos Dumba
Department of Physics
Faculty of Science
Mbarara University of Science & Technology
P. O. Box 1410MbararaUganda
Probing Accretion Disk Winds in AGN I. Asymmetric broad Balmer emission lines
September 23, 2014
The Broad Line Region of Active Galactic Nuclei is characterized by broad Balmer emission lines in their optical spectra. The broad Balmer emission lines are found to be asymmetric, some blue sided and others red sided in their asymmetry. One of the components behind the asymmetry is thought to be an accretion disk wind. We probe the accretion disk wind using the broad balmer emission line profiles.This asymmetry of the broad balma emission line profiles is measured in velocity space after a measurement of the line shift at percentiles from 0, in increaments of 10, up to 90. In addition, the Kurtosis Index is obtained at appropriate points of the emission lines' profiles. This study is based on many hundreds of SDSS spectra, starting with low redshift high signal to noise ratio spectra. We also consider a definite number in each bin of their FWHM, in bins of 1000 km/s (atleast 40 per bin), starting from 1000km/s to the very broad emission lines.We present how strong the asymmetry (by plotting Asymmetry Index as a function of percentile) of the broad/narrow lines (in percent) is, what the Kurtosis (R20, 80) is. We also present what the Asymmetry Index as a function of line width (FWHM), luminosity (V-band), core-radio flux and Ionization Degree.
Introduction
Active Galaxies have been widely studied by many authors revealing many fascinating properties which among all include; compact nuclear emission (Clavel et al. 1990), nonthermal continuum emission
F ν ∼ ν −α
(1) (Bregman 1990).
In addition, authors notice that; the continuum stretches from the radio to Xray/Gamma rays (Mehdipour et al. 2011), the luminosity of the nucleus exceeds that of the host (Osterbrock 1989), they have strong emission lines (De Breuck et al. 2000), are highly variable (Ulrich et al. 1997), and also have X-ray emission (Turner & Pounds 1989). The energy source in the AGN is believed to be accretion (Blandford & Znajek 1977). This can be demonstrated through the relation :
E = ηmc 2(2)
where η is the efficiency. The luminosity can then be re-written as:
L = dE dt = η dM dt c 2 = ηṀ c 2(3)
whereṀ = dM dt is the accretion rate. For a typical AGN;
M = L ηc 2 ≈ 1.8 × 10 −3 L 37 η M sun yr −1 (4)
where L 37 is the luminosity in units of 10 37 W. This energy, at one point, through the principle of hydrostatic equilibrium, reaches a point at which gravitational forces (causing the accretion) balance with radiative forces (from the nucleus). This limit defines a characteristic luminosity, the Eddington luminosity (L Edd ) through the relations;
F grav = F rad (5) GM (m p + m e ) r 2 ≈ GM m p r 2 = σ T L Edd 4πcr 2 (6) L Edd = 4πGm p c σ T M (7) L Edd = 1.26 × 10 31 M M sun [W ](8)
The important parameter in AGN is L L Edd . One of the consequences of high L L Edd are winds, Accretion Disk Winds. Just like in Solar Winds....once a particle exceeds the escape velocity, we see this as a wind from the accretion disk. Accretion disk winds have been confirmed by some authors by studying BAL Quasars (Hamann 1998). Hamann (1998) observed strong absorption trouphs in the restframe UV spectrum and estimated outflows with 10,000 km/s.
In this work, we study the accretion disk Figure 1: The rest frame of the UV spectrum of a BAL Quasar showing a significant outflow in the absorption lines of up to 10,000 km/s Hamann (1998) winds by analysing broad Balmer line profile 2 asymmetry and steepness values. We define the two parameters in line shape analysis as:
A.I = S IP V (9) K.I = IP V U IP V L(10)
where S is the line shift, IPV is the interpercentile velocity, IP V U and IP V L are the interpercentile velocities for the upper and lower parts of the emission line profile. The following sections will outline what we did in more detail and explain the most important steps we carried out during the study.
Data
Introduction to Data Analysis
In order to study any sample of galaxies and quasars, it is important not to forget a few useful conditions that make the sample produce reliable information:
• the size of the sample and
• the quality of the spectra, both in terms of resolution and Signal to Noise ratio. (Whittle 1985a) Our data meets both criteria thanks to dedicated surveys like the SDSS that have made such homogeneous data sets available to the public.
We have obtained, in each sample, the top 600 high signal to noise galaxies and quasars from the seventh data release (DR7) (York et al. 2000). The samples have the highest spectroscopic quality in the release (Abazajian et al. 2009), and are uniform in terms of calibration let alone being complete. We have two samples for the same reasons of obtaining quality spectra in all ranges of FWHM from 1000 km/s to above 10,000 Km/s. This is so because as we are selecting the spectra with the highest signal to noise ratio, there is a tendency to obtain only spectra that are both from sources near (low redshift), thus neglecting those with higher redshift, and also biasing the data primarily on signal to noise ratio. This is avoided when we split the sample in two samples, one for those between 1000 km/s FWHM and 3000 km/s FWHM, and another for those having FWHM greater than 3000 km/s. We end up having spectra with very high signal to noise ratio across the whole spectrum of broad Balmer emission lines. Remember broad Balmer emission lines are those with FWHM greater than 1000 km/s. In this section, we explain how we download the data, how we treat the data, describing the software we use and all the tasks used.
We also explain all the steps we carry out, giving details of the output in each step and why it is carried out. In addition we show samples of the values extracted from the plots (the whole list being found in the appendices). Lastly we describe the secondary treatment of the extracted data from the plots, explaining how and why we carried out the treatment in such ways and show samples of the results in tables (the actual results being shown in the next chapter). But then, it is necessary to first briefly describe the database (SDSS) and thereafter the telescope and the surveys it has carried out so that one gets a feeling of the whole process from observing, collection of data, treatment of data and analysis.
SDSS Database
The Sloan Digital Sky Survey consists of three major surveys that together provide scientists with immense volumes of data obtained from a dedicated 2.5m telescope lo-cated at Apache Point Observatory in Southern New Mexico. This survey has been collecting data since the year 2000 (Abazajian et al. 2009;York et al. 2000). The three surveys are;
• Legacy
• SEGUE
• Supernova
SDSS Legacy Survey
The SDSS Legacy Survey provided a uniform, well-calibrated map in ugriz of more than 7,500 square degrees of the North Galactic Cap, and three stripes in the South Galactic Cap totaling 740 square degrees. The central stripe in the South Galactic Gap, Stripe 82, was scanned multiple times to enable a deep co-addition of the data and to enable discovery of variable objects. Legacy data supported studies ranging from asteroids and nearby stars to the large-scale structure of the universe. Almost all of these data were obtained in SDSS-I, but a small part of the footprint was finished in SDSS-II.
SEGUE -Sloan Extension for Galactic Understanding and Exploration
SEGUE was designed to explore the structure; formation history; kinematics; dynamical evolution; chemical evolution; dark matter distribution of the Milky Way. The images and spectra obtained by SEGUE allowed astronomers to map the positions and velocities of hundreds of thousands of stars, from faint, relatively near-by (within about 100 parsec or roughly 300 light-years) ancient stellar embers known as white dwarfs to bright stellar giants located in the outer reaches of the stellar halo, more than 100,000 light-years away.
Encoded within the spectral data are the composition and temperature of these stars, vital clues for determining the age and origin of different populations of stars within the Galaxy (Yann et al. 2009).
The SDSS Supernova Survey
The SDSS Supernova Survey was one of three components (along with the Legacy and SEGUE surveys) of SDSS-II, a 3-year extension of the original SDSS that operated from July 2005 to July 2008. The Supernova Survey was a time-domain survey, involving repeat imaging of the same region of sky every other night, weather permitting. The primary scientific motivation was to detect and measure light curves for several hundred supernovae through repeat scans of the SDSS Southern equatorial stripe 82 (about 2.5 • wide by 120 • long) (Frieman et al. 2008). The above three surveys have provided scientists with a catalog derived from the images obtained by the 2.5m telescope. These images include more than 350 million celestial objects, and spectra of 930,000 galaxies, 120,000 quasars, and 460,000 stars. This data is not only fully calibrated and reduced, carefully checked for quality, and publicly accessible through efficient databases, but has also been been publicly released in a series of annual data releases. It is through this effort that we are able to carry out this study on the asymmetry of the broad emission lines of active galactic nuclei. In this Data release, this study exploits the immense volume of spectra of galaxies and quasars (Abazajian et al. 2009;York et al. 2000). The sample obtained from the release includes around 600 spectra of both galaxies and quasars in the redshift range of 0 and 1, with broad Hβ lines from 1500km/s enabling us contain all groups of broad line emission objects.SDSS (2013) Figure 2: The 2.5m telescope located at Apache Point Observatory in Southern New Mexico BBC (2013)
Obtaining the Data
In order to obtain data from the SDSS database with your own constraints on the sample, one needs to write an SQL Query that generates a list of the sources that meets your requests.
In this study, we focus on the parameters of the Hα & Hβ emission line, looking for asymmetry in these lines.
Data Properties and Constraints Invoked
The data in our samples combined consists of around 300 objects, galaxies and quasars, restricted to a redshift range between 0 and 1. The samples consist of reduced spectra of these objects in fits files that we renamed in ascending order from those with the highest signal to noise ratio. This helps us analyze those with the highest signal to noise ratio first. This is important because the results obtained from high signal to noise ratio sources are more reliable for the derivation empirical relations. The redshift limit was chosen so that we have a good coverage of the Hβ line and its adjacent spectral regions. Figure presents the general properties of the sample in terms of redshift, signal to noise ratio and the broadening of the Hβline.
In addition, this data consists of around 283 spectra with measured Hβ profiles around 165 measured Hα profiles. This is because not all spectra could have both the Hβ emission line and Hα emission line due to redshifting at both ends of the optical window. However, most of the spectra with Hβ lines also have Hα lines but not vice versa. Of course spectra that are at the extreme end of our redshift selection are the ones affected with not having the Hα line visible, but this is not an issue for us to worry about since we were more interested in the Hβ line profiles.
Figures 3 show the distribution of the signal to noise ratio of our sample and the redshift. It is seen here in the signal to noise distribution that we have a fairly good signal to noise ratio, the minimum being 37 and maximum 79. The distribution is binned in 5 starting from 35. The figure therefore clearly shows that most of our sample spectra have a signal to noise ratio between 40 and 45, to be specific around 120 out of 300 (40%), with a few having more than that. This is not a problem since, given the nature of our study, a signal to noise ratio of even 30 would be sufficient for good measurements (Thorne et al. 1999).
In the same way, the redshift distribution is shown. Because we chose spectra with the highest signal to noise ratio, it seemed obvious that most of the sample spectra will be in the near end of the redshift window we chose with those below redshift 0.2 dominating. However, since the distribution does not fall rapidly as we move to higher redshift, the data will provide a sufficient study of AGN within this redshift bin chosen. In a later study, it would be good to also study other redshift ranges, possibly using Ultraviolet emission lines which can be obtained in the optical spectra after they have been red shifted.
Figures 4 show the distribution of the FWHM of all the Hβ emission lines measured. The distribution shows that there is a peak between 3000 km/s and 4000 km/s. But still it is not a big range from the bins at both ends which are at 50 each. Our intention is to have a good distribution across the whole range from 1000 km/s to over 10,000 km/s, and since we obtain enough counts for the first five bins, then the sample will provide a statistically sufficient analysis for our findings.
The other figure here is a cumulative distribution, which shows that from FWHM of 9000 km/s, there isn't any increase in counts as there are very few AGN with such extremely broad lines. Our study will obtain information about the broad lines basing almost entirely on the first six or seven bins with sufficient numbers.
The distribution of the FWHM in our sample is in agreement with the AGN statistics obtained by Hao et al. (2005) when they plotted the distribution of the FWHM values of the Hα emission line for over 40,000 emission line galaxies. Their distribution was bimodal since they included all AGN, narrow and broad line AGN. Their boarder line of broad line AGN was naturally placed at those with a FWHM of over 1200 km/s. To compare with our distribution, we only consider those over 1000 km/s, and it gives us the same distribution. In their study, defining broad line AGN as objects with a FWHM greater than 1200 km/s, they obtained 1,317 objects out of 42,435 emission line galaxies. This makes our sample of 300 objects not bad since it is a quarter of this value, let alone having been restricted to a redshift between 0 and 1.
Following this simple look at the general properties of the downloaded data, it is now necessary to explain the treatment of the data in order to obtain measurements that are needed for our later analysis. The detailed spectral properties of the sample will follow later in the forthcoming chapters.
Image Plotting and Analysis with IRAF
IRAF (an acronym for Image Reduction and Analysis Facility) is a collection of software written at the National Optical Astronomy Observatory (NOAO) geared towards the reduction of astronomical images in pixel array form. In this thesis, i used IRAF to plot spectra from our sample AGN and analyzed a few spectral lines needed to derive more information for analysis later (Tody 1986;Valdes 1986). To be specific, we plotted the Balmer lines, Hα and Hβ emission lines, although not all spectra contained both emission lines. Most of them had the Hβ emission line, a good number had both Hβ and Hα emission line, while very few(∼2%) had only the Hα emission line. In the following subsections, i will explain briefly the steps and tasks i used to extract the information i needed from the broad Balmer lines in my sample. This was principally the part of the thesis that occupied me most since it involved careful visual analysis of the spectrum first and accurate execution of the required commands for which a mistake with one of them means redoing the whole process from the beginning. Explained below are the steps i took during my data extraction.
Spectrum Visual Analysis
After downloading the data, (the fits files), and having renamed them, starting from the one having the highest signal to noise ratio, i plotted each spectrum and analyzed the Balmer emission lines. The reason why i needed to do this is because each spectrum is unique and the tasks to perform on each varied from one spectrum to another. Some of the things i looked out for were;
• The availability of both Hβ and Hα emission lines. • The function to use while subtracting the continuum. This was important because the uniqueness of each spectrum, and the quest to apply a good estimate to a flat spectrum, entails using different functions for each spectrum in order to obtain a flat spectrum from the power law spectra downloaded.
All the above mentioned reasons vary from spectrum to spectrum and each of them is important in order for me to obtain the best results from the spectrum, and for minimizing errors as much as i can. In our data, which has two samples, sample 1 being that of AGN with FWHM between 1000 km/s and 3000 km/s, and sample 2 being that of AGN with FWHM from 3000 km/s and above, this is what we observed from the visual analysis;
• The AGN in sample 1 have both Hα and Hβ, meaning they are found in the lower part of the redshift range from 0 to 1, preferably less than z = 0.6. There are less than 10 out of 81 that had only the Hβ emission line.
• Sample 2 has at least 40% of the AGN having both Hα and Hβ. This means 60% of them are found in the higher end of the redshift bin, preferably above z = 0.7. Recall that my redshift range is from z = 0 to z = 1.
Continuum Subtraction
This is the task done after visual analysis of the spectrum. Spectra with both the Hα and Hβ emission lines look similar to the ones in figure 5 although some with significant redshift will only have the Hβ emission line still available. The subtraction of the continuum is done in splot, with the keys "t" followed by a "-", followed with the appropriate function necessary for the selected region. The continuum is subtracted for each emission line separately and separate images are saved out of the original image having both emission lines. This can be seen in the sample plots in figure 6 on page 9. Continuum subtraction is the second step to spectrum analysis for the Balmer emission line features and it sets a vantage point for line normalization, as we shall discuss in the next subsection.
Line Normalization
To normalize the emission line simply means to have its base at a zero point and its peak Figure 5: Seen above are two samples of the spectra during visual analysis. It should be noted that not all spectra contain the Hα and Hβ emission lines as seen here, reason being that those red-shifted significantly will eliminate the Hα since it will be Doppler shifted to wavelengths outside the optical range. In addition, not all spectra are relatively flat like the ones shown due to the varying amounts of continuum emission at different wavelengths for many AGN Figure 6: The images show an example of extracted Balmer emission lines after continuum subtraction on each of them. One should note here that all the extracted emission lines have their continuum at zero, making it the starting point of the percentile ranges we need examine later.
at unity. It should be noted that we are normalizing the broad Balmer lines. This means that we either subtract the narrow components or simply leave them visible, but ignore their additional height as we simply place a unity value at the peak of the broad line. In the figure 7, examples on the diversity of broad and narrow components in our sample with some not having narrow components at all and others dominated by the narrow component are shown.
To perform the normalization of the broad Balmer emission lines, we use the arithmetic task, "imarith", which can be used anytime in IRAF. However, after continuum subtraction, the counts at the peak of the broad Balmer line are recorded in a table. Also in the same table is recorded the velocity at which the emission line is centered. This velocity will be of use while transforming to velocity space, centering the emission line peak at that velocity. Table 1 shows part of the table. The whole table can be viewed in the appendices section. It shows the Spectrum Number, the Balmer line heights used to normalize the profile to unity and the corresponding wavelengths at which they are identified. These wavelengths are the wavelengths at which we center the corresponding lines while transforming to velocity space.
Line Correction : Filtering out other lines and Smoothening out the noise
In any spectrum, there is always a probability that the emission line you are interested in is surrounded by other emission or absorption lines. For some emission lines, the neighboring emission lines or absorption lines are not a threat to any measurements pertaining the emission line under study, for example, for the Hα emission line, the neighboring Sulphur II emission lines do not pose a threat to any measurements needed from the Hα emission line. This can be seen in the examples below.
However, for the Hβ emission line, there is an undeniable problem. The neighboring emission lines are significant enough to pose a threat to measurements taken from it. These emission lines are; the Oxygen III lines and the Iron emission at both sides of the Hβ emission line. The Iron emission will increase on the error in the continuum subtraction since it creates a pseudo continuum and thus making it difficult to estimate a continuum level. To reduce on the error in this measurement, one has to subtract the iron emission, preferably using some already developed templates before the actual continuum level can be estimated. In our case, since it was a small fraction of spectra that had significant iron emission on both sides, we neglected this since it affects less than 5% of the data. The Oxygen III emission lines also increase on the uncertainty of the FWHM values measured since they are embedded in the broad Balmer Hβ emission lines. The most reliable way to deal with this uncertainty is to subtract them before any values are measured. However, since they are also broadened at their bases, their subtraction, using a Gaussian model, increases absorption features in the residue Hβ emission line left. Thus, since the emission lines are already broad, we decided to simply cut them out in order to leave a smoother profile for the residue for better measurements without introducing a smoothing factor. It is noticed that subtracting the OIII lines introduces absorption features which in turn adds an uncertainty to the measurements, so simply cutting them off and preventing this uncertainty seemed better because the uncertainty we deal with by not doing the subtraction itself is much less. When all other neighboring emission lines are removed, the spectrum is saved, and it is this spectrum that is used for all the other needed measurements. The images of the spectra that will be displayed will in most cases be those that have been cleaned of all their neighboring emission line components.
Display Transformation to Velocity Space : Centering velocity at Line center
The spectra downloaded from SDSS displays counts on the y-axis against wavelength in angstroms on the x-axis. The wavelength is logarithmic in scale. In order for us to measure the asymmetry in a clearer manner, we preferred to transform the wavelength scale to a velocity scale. This is easily done in IRAF using the command "disptrans" which is accessed in splot as well. However, since we need to measure the asymmetry, we also center the velocity at the emission line peak. This procedure displays the profile in velocity space but centered at the line peak making it possible to measure the asymmetry from the line center.
Spectrum Conversion to Text File : For further analysis of lines
To measure the asymmetry, there several ways one can use, some measure by hand, others prefer to use a program. In our case, we preferred to use IDL to measure asymmetry (Bowman 2005). This meant that we have to convert the spectrum image to a text file for the program to be able to read values and calculate accordingly the asymmetry. For IDL to perform this, we had to write a simple script with the necessary conditions and the desired output we needed. The output from this program are our results of the asymme-try of all the line profiles we plotted. Another advantage of converting the spectrum image to text was that we could easily plot the very image using any other plotting software like GNUplot, Python and QtiPlot (Russell & Cohn 2012;Beazley 2006;Dawson 2003;Fehily 2002;Mark 2009), making it possible to plot in any way we preferred other than the default plot of splot in Iraf.
The values or output data we obtained from the IDL program is by definition our results. By results i mean the Asymmetry Index, because from this, we can easily obtain the Kurtosis Index as well. It is the Asymmetry Index and Kurtosis Index from out data that was of prime importance from which we shall relate to other kinematic properties of the host galaxies or Ionizing regions of the AGN.
Data Treatment and Analysis
The data for our study is obtained from a series of treatment processes. The previous sections were explaining how the individual emission lines are treated in each spectrum to the moment when a text file is extracted from the fits image. This text file is convenient for further treatment and analysis since we can re-plot the emission line with any other plotting software like GnuPlot, QtiPlot and Python (Russell & Cohn 2012;Beazley 2006;Dawson 2003;Fehily 2002;Mark 2009). Further treatment on these extracted text files is done with IDL, a program we use to extract values of the velocities at the chosen percentiles and dump them out as the useful values for further analysis. This is done for each emission line, after which a text file having all the necessary percentile velocities for all the files is obtained. It is in this file that we shall start our analysis of the asymmetry of our emission lines. In other words, this file will contain all the information we need for each of our 447 broad Balmer emission lines in order to study the asymmetry.
Before we describe the process of extracting percentile velocities for all the emission lines, it is also necessary to briefly describe the observations made on the broad Balmer emission lines during the initial processes of obtaining the extracted individual plots.
Description of Obtained Data
A first analysis of the data samples showed a clear distinction between AGN with broad Balmer lines with FWHM between 1000 km/s and 3000 km/s, and those with their FWHM greater than 3000 km/s. It was observed that the former are predominantly in a lower redshift region, while the later are predominantly high red-shifted. This is evident from the fact that out of 80 spectra in the first sample, 67 contained both broad Balmer emission lines, while for sample 2, out of 200 spectra, 98 contained both emission lines, the others missing the Hα emission line due to being red-shifted.
From figures 9 and 10, we can notice that sample 2 is clearly unbiased in its redshift distribution as sample 1 is.
Other levels of the data
The data obtained initially is in the form of spectroscopic data, spectra downloaded from SDSSDR7 from which the asymmetry is measured using the IDL script we developed (Bowman 2005). The asymmetry values are then used to obtain the steepness of the profiles.
In order to relate the asymmetry and kurtosis to other kinematic properties, we needed to obtain more data pertaining the objects whose values of asymmetry and kurtosis we have. This other data includes:
• Flux values of [OIII], [OII] & [OI] for
obtaining the ionization degrees through their flux ratios.
• V-Band magnitudes for calculating the Luminosity.
• Radio Flux measurements for those sources whose radio fluxes have been databased already.
• Line width values of Hα and Hβ emission lines.
Values of the flux of the oxygen lines were obtained from the downloaded spectra. This was not an easy task as quite a number of sources were red-shifted in such a manner that either [OI] was missing or [OII] was missing. However, it was still possible to have objects in which we could obtain values from both emission lines.
The V-Band magnitudes were obtained using an sql script, which we wrote, to download spectroscopic values of ugriz photometry data, from which i transformed the values to the U BV RcIc system. The V-Band magnitude was then calculated as:
V = g − 0.58 * (g − r) − 0.01(11)
It then made it possible to transform the magnitude to Luminosity.
Radio flux values were obtained from the NASA/IPAC EXTRAGALACTIC DATABASE, in which i inputted the coordinates of all my sources. The query returned all the sources whose radio flux was databased. The radio flux was in milliJansky. Also to note is that since we are interested in the core radio flux, the radio flux measurements queried were for regions within a 5.0 arc-sec cone.
All the different sets of data were synchronized to match the asymmetry and kurtosis values for analysis. To note too is that the data was separated in two sets; that of Hα profiles and that of Hβ profiles. In the analysis to follow, it will be noted that plots of Hα and Hβ are placed besides each other, with the Hα to the left and Hβ to the right, where possible in different colors as well.
Plotting
Plotting in this work is done using quite a number of programs, from IDL (Bowman 2005), QtiPlot (Russell & Cohn 2012) and Python (Beazley 2006;Dawson 2003;Fehily 2002;Mark 2009). However, most of the plots in the Data section are plotted using Python, where made a script for each dataset so that it minimizes the size of the text file containing the data. It is beyond the scope of this section to give more details of the scripts. However, a sample will be attached in the Appendix section for more clarification. The reasons for using various programs to plot ranged from which type of plot was needed and which data i was analyzing as i always used the most efficient one for each type of plot i needed.
Results
In this section, we present the results obtained from the data analysis and a discussion about them. By results here, we focus on the Asymmetry Index and Kurtosis Index of the Balmer emission lines measured. We analyze the distribution of the Asymmetry Index across the whole profile from FWZI to its peak. In the later half of the chapter, we relate the Asymmetry Index to other kinematic properties such is ionization degree (De Robertis & Shaw 1990;Padovani & Rafanelli 1988), radio flux (Brotherton 1996), luminosity and the line width (FWHM) (Yu & Gan 2006).
Distribution of results
On obtaining the Asymmetry Index and Kurtosis Index, it was necessary to briefly show how they are distributed. We chose to plot a frequency distribution of the Asymmetry Index and Kurtosis index. The Asymmetry Index varies from -1 to +1, a value of 0 meaning the profile is symmetric while a deviation to either side of the zero implies asymmetry. The degree of asymme-try in each case will be probed by relating to other kinematic properties. It is to our interest to find out whether this positive asymmetry and kurtosis is related to either non-thermal or thermal radiation and/or even to the relative strength and kinematics of the BLR. If the degree of asymmetry is a measure of the radial flow of material from or to the BLR, then it will be possible to understand the structure of the accretion disk winds using asymmetry index.
In our distributions to follow, we separate the Hα emission line profile asymmetry from that of the Hβ emission lines. The Kurtosis Index follows the same pattern.
Asymmetry Index
The values obtained from the IDL script yielded values of the Asymmetric Index for both the Hα emission lines and Hβ emission lines. The bar graphs that follow reproduce the statistical distribution of the Asymmetry Index for both Balmer lines, with the Hα on the left of each double image figure (in blue) and Hβ to the right (in green). It is also important to review the meaning of a value of asymmetry index as seen on the plots. This is seen in 2 below Fig 11 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW20%I. It is noticed that the AI is almost symmetric and peaking in the red end for Hα but still red-shifted for Hβ. Fig 12 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW30%I. It is noticed that the AI is almost symmetric for Hα but still red-shifted for Hβ. Fig 13 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW40%I. It is noticed that the AI is almost symmetric for Hα but still red-shifted for Hβ. Fig 14 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW50%I. It is noticed that the AI is almost symmetric for Hα but still red-shifted for Hβ. Fig 15 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW60%I. It is noticed that the AI is almost symmetric for Hα but still red-shifted for Hβ. Fig 16 shows the Asymmetry Index distribution of Hα emission lines, on the left hand, and Hβ emission lines, in the right side at FW70%I. It is noticed that the AI is almost symmetric for Hα but still red-shifted for Hβ.
A few key features are noticed here, Hα profiles are almost symmetric, they tend to peak in the red end but the degree of asymmetry is negligible. For the Hβ profiles, it is noticed that most of them are clearly asymmetric, peaking predominantly in the positive side. This is maintained all through the profile from the base, FWZI, to the core of the profile at the higher percentiles.
A sample of symmetric profiles can be seen in figures 17 for Hα profiles and 19 for the Hβ profiles. Asymmetric profiles can be seen in figures 18 and 18 for the Hα and Hβ emission lines respectively. It is evident that the Hβ profiles display the highest degree of asymmetry.
Kurtosis Index
The Kurtosis Index is a measure of how steep the profile is. The Kurtosis parameters are all less than unity, with a decrease in value indicating an increase in steepness. The parame- Table 3 shows an overview of the Kurtosis parameters and their meaning. For the Kurtosis parameters, it is vital to note that this is a measure of how the profile shape changes from the top to the bottom. A value of close to unity signifies almost no change in the steepness. A value close to zero on the other hand will mean a very large change in the shape as one moves towards the base of the profile. This means the profile shape rapidly changes from narrow to broad. Some profiles may be broad but with low values of Kurtosis, that is if they are consistently broad from the upper part of the emission line to below the half maximum. Profiles that show this shape are mostly those in which the narrow component is not available or obscured. There are other broad emission line profiles that will display high values of Kurtosis, that is, having significantly extended wings. It is also noted that high values of asymmetry in a profile suffice low values in Kurtosis.
In dealing with the Kurtosis measure, three regions of the profile are chosen, the top measured with KI 1 , the middle, measured with KI 2 , and the bottom, measured with KI 3 . Thus each of these values shows a change in profile in the mentioned regions on the overall profile. To capture the whole profile, another parameter is defined, KI Gen , which measures the overall change in shape of the profile from the top to the bottom. Figures 23, 24, 25 and 27 show the change in profile shape of the Hα and Hβ profiles, with the ones to the left (in red) for Hα and the ones to the right (in blue) for the Hβ profiles.
It is shown in fig 23 that the change in profile shape for the top parts of the Hα and Hβ emission lines is quite high, peaking at values close to 0.25. Fig 24 shows that the change in profile shape for the center parts of the Hα and Hβ emission lines is moderate, peaking at values close to 0.6. Fig 25 shows that the change in profile shape for the lower parts of the Hα and Hβ emission lines is low, peaking at values close to 0.8.
The three measures of steepness, KI 1 , KI 2 and KI 3 show a pattern in profile shape change. The change in shape breaks down as one moves to the base. This can be an indication of a systematic change to Lorentzian from Gaussian, as the change should be smooth as one moves from around 60% of the profile downwards. One cannot expect an abrupt change from Gaussian to Lorentzian. It could also be due to the selection of the regions of low, center and bottom. Looking at only two regions, top and bottom, breaks the smooth transition since the measure of the center of the profile reflects the effects from the same physical conditions as the lower part of the profile as can be seen in 26.
Fig 27 shows that the change in overall profile shape of the Hα and Hβ emission lines is moderate, peaking at values close to 0.55.
The general trend of profile shape change is expected as not so many sources show high asymmetries, but the mere fact that there are some sources in the distribution with high measures in asymmetry and low measures in kurtosis justifies the study. It is now clearly vital to proceed and look up other kinematic properties and find out which of them correlate with the high values of asymmetry and low values of Kurtosis. A previous study on 90 emission line profiles,constituting 30 Balmer lines and 55 forbidden and 5 other permitted lines, from 31 objects comprising of S1, S2, S3 classes of Seyfert galaxies, H II regions and QSOs showed that Forbidden lines are found to be narrower and steeper, while Balmer lines and other permitted lines are broader and flatter (Basu 1994), meaning that the Kurtosis measure will be small as we also observed in the KI 1 .
Asymmetry Index relation to other kinematic properties
While we have the Asymmetry Index and Kurtosis Index, its scientifically worthwhile to relate this measure to some other kinematic properties (Whittle 1985b). Some of the properties we related Asymmetry Index to are; the FWHM, the luminosity in the V-Band, the Radio Flux, and Ionization degree using the Oxygen narrow emission lines.
Relation of Asymmetry Index to the Line Width(FWHM)
The Line width is a measure of the strength of the spectral properties in the source which can be used to obtain a number of estimates about the size, radius of the region. A relation of this measurement with line asymme-try is important in the quest to obtain the origin of the asymmetry and the nature of the mechanism. Thus the asymmetry here will play a great deal in probing the flow of material in and out of the BLR. This is in line with the relation here as the base of the profiles shows high asymmetries, thus more effects from the motions of the material than in the upper portions of the profile. This is a general trend, which is in agreement with the underlying physics. We also study individual relations with each percentile in the plots that follow.
The following ten plots are relations of Asymmetry Index and FWHM, those to the left are of Hα (in green) while those to the right are for Hβ (in blue).
In Fig30, the relation between the FWHM and AI20 is shown. The Hα does not appear to have a relation with the two variables, the points are heavily scattered in the plotBut also the relation observed with the Hβ tends to start breaking down from previous relations. We observe an increase in asymmetry with line width. In a nutshell, excluding the FWZI, all Hα profile percentiles display a negative correlation while Hβ profiles display positive correlations excluding their 00% and 90% percentile. The later correlations are tighter making the Hβ shape parameters better at analyzing effects causing profile shape in Balmer emission lines. It is also a reflection from the previous statistic on the distribution of asymmetry index where we observed that the Hβ profiles displayed more positive asymmetry as opposed to the Hα profiles that were statistically symmetric.
Relation of Asymmetry Index to the V Band Luminosity
The relationship of Luminosity with the asymmetry of broad Balmer lines is of great importance because luminosity of one of the properties that can easily be obtained from a source. Having a clear relationship with this property will help in further scientific relationships with the AGN Broad Line Region physics. Since the luminosity has already existing relationships with parameters like Black Hole mass, accretion rate and mor-phological type, it will help us relate asymmetry of profiles to all there other properties of the galaxies. One excellent correlation is that between the Black Hole mass and Optical luminosity obtained by Peterson et al. (2004). It was observed that Black Hole mass increased with optical luminosity. The correlation between asymmetry and V-Band luminosity therefore will also help us in probing this further. Kaspi et al. (2002) also found a tight correlation between the luminosity and the radius of the BLR. This will also help is relate our asymmetry with the radius of the BLR. The preceding relations are also divided into two, the relation to the left being that of Hα profiles and that to the right being that of the Hβ profiles. But generally, there seems not to be direct relationships between these parameters apart from that of AI00.
In all the relations of Luminosity with asymmetry, it is observed right from figure 36 through to 41 that the Hα asymmetry relations are non-existent or very weak, if found. However, the asymmetry in Hβ is observed to be consistent apart from the first and last percentiles. Hβ positive asymmetry is observed to rise with increasing Luminosity. Increasing luminosity correlates with increasing BLR ra- dius and increasing BH mass. This seems to point to a asymmetry arising from any of the factors that positively correlate with luminosity. This means a tendency of highly luminous sources giving rise to asymmetric Hβ profiles. The gradient in the relation is observed to fall as one moves from the low percentiles to the high ones, a 10% percentile having a ∼ 1.7 gradient and an 80% percentile having a gradient of ∼ 0.3.
Relation of Asymmetry Index to the Radio Flux (Core-Radio Flux)
One of the kinds of AGN are in radio galaxies (Urry & Padovani 1995;Peterson 1997). A bulk of the emission from such sources lies in the radio regime. A number of authors (Pearson & Readhead 1988;McCarthy et al. 1991) have studied the behavior of galaxies with radio emission and all have observed asymmetry in many emission lines from radio galaxies (De Breuck et al. 2000). Even a study on the behavior of jets in weak radio sources noted asymmetry in spectral lines observed (Laing et al. 1999). It is this strong foundation that encourages us to systematically study the relation of radio Flux to the asymmetry of the emission lines. Our study is important since the spectral line asymmetry is binned in percentiles, making it easy to notice while part of the emission line responds more to the radio emission. In this study, the core radio flux at 6.0 arc secs is obtained. This is because radio flux can be extended to several tens of arc secs yet our interest is the radio flux emanating or very close to the BLR, in which the broad Balmer lines are formed. The following plots are paired for each percentile, having a relation with the Hα emission line on the left and that of the Hβ emission line on the right. We analyze each percentile separately starting with the lowest percentile, the FWZI, to the highest part of te emission line possible, 90& of the broad Balmer line. Figure 42 is displaying the correlation between radio flux and asymmetry at 20% of the line profiles. The same behavior of Hβ profile asymmetry being positively correlated with radio flux. However too in this case, the Hα profile asymmetry correlation is flat. Figures 43, 44, 45, 46 and 47 all show the pattern of the most asymmetric Hα and Hβ profiles positively correlating with core radio flux.
From the relations of asymmetry with radio flux, it has been noted that binning the profile is important in finding out which section of the profile responds more to the radio emission. The top part of the profile does not respond to changes in radio emission as the centroid of the profile. It is also noted that the Hα and Hβ profile shape is consistent with its response to radio flux throughout all the percentiles, although it the relation is weaker in the Hα asymmetry.
Relation of Asymmetry Index to the Ionization degree
The Ionization degree is a measure of the excitation degree of the region from where the lines used to measure the ratio arise from. It is vital for us, since it will give us a clue to how asymmetric profiles relate to this ionization potential of the regions. A direct relation will give more insight to a wind scenario from the broad line region to the narrow line region in which these emission lines are obtained.
In the plots to follow, we have analyzed two measures of excitation, [OIII]/[OI] in red, and [OIII]/[OII] in blue. We go ahead to analyze the relations separately for each percentile. As in previous plots, Hα profile analysis is on the left and Hβ profile analysis on the right of each figure. For all the relations of ionization degree with asymmetry, it is maintained from figure 48 all through to 53 that it is very difficult to obtain a precise correlation of ionization degree with Balmer line asymmetry.
Statistical analyses show that the Hβ relations are more reliable than Hα measurements as seen from the P − values and correlation coefficients.
Kurtosis Index relation to other kinematic properties
The way a profile changes, or how steep it may be, the Kurtosis Index, is a useful tool to use in studying other kinematic properties of the source because its a visual property. Once one knows how this measure is related to other indirect kinematic properties, it becomes easier to infer such properties by a simple analysis of the profile shape.
Relation of Kurtosis Index with Line Width (FWHM)
Since the Line width is a measure of the strength of the spectral features in the emission line, relating the Kurtosis with line width is simply relating the strength of the spectral features to the steepness (shape) of a profile. Fig 54 shows how the Kurtosis varies with both Balmer emission lines. The Hβ providing a tighter correlation than the Hα. But still one thing in evident, higher values of Kurtosis, translate to higher line widths generally. This suggests that broader profiles are flatter as previous studies show (Basu (1994)). Basu (1994) found a good correlation between FWHM and kurtosis, in which he noted that Balmer lines and other permitted lines are broader and flatter. This means that very broad lines experience less change in their shape parameters than less broadened lines. As seen in tables 14 and 15, the correlations are reliable with the Hβ profile kurtosis index being tighter and steeper.
Relation of Kurtosis Index with Luminosity (V Band)
The relation of Kurtosis with Luminosity is also clearer with the Hβ profile kurtosis. Hα profile kurtosis relation is flat . This trend suggests that the most luminous sources will most likely have Hβ profiles with extended wings. Although they both show Kurtosis Index decreasing with increasing Luminosity, the P −values of the correlations are too high for us to rely on this trend only. However, this points to the direction that luminosity most likely is one tool to look at when studying accretion disk winds.
Relation of Kurtosis Index with Radio Flux (Core-Radio Flux)
Hα kurtosis displays a positive correlation while Hβ kurtosis shows an almost flat correlation. These relations of radio flux to kurtosis make is challenging to extract a meaningful trend of radio flux with profile kurtosis. Previous studies by Whittle (1985a) showed a that radio flux varied for different kinds of AGN. In our data, we did not separate the kind of AGN, thus making it impossible to see any relation. He found out that there was a significant difference between the profile kurtosis of linear sources and non-linear sources, with linear sources having steeper sided profiles. He went ahead to say that such observations could be naturally explained if the NLR gas at each end of the jet were radiating with opposite Doppler shifts relative to the synthetic(line-center) velocity. This means that the jets associated with the linear sources would perturb the outer parts of the velocity field, broadening the core to produce a high-kurtosis value, although the perturbation is not strong enough to influence the profile asymmetry.
Relation of Kurtosis Index with Ionization degree
The ionization degree here will be related to the excitation state of the NLR. The correlations between profile shape and the NLR excitation state are potentially very important because they reveal the presence of dynamical interactions between moving BLR moving gas(winds) and their environment. We have looked at two parameters of excitation degree, [OIII]/[OI] in red, and [OIII]/[OII] in blue. These values were obtained from the spectra of the data downloaded from SDSS. In general, the dispersion in excitation degree reduces with flatter profiles as observed in both Hα, on the left, and in Hβ to the right of figure 57. Since the flatter profiles are generally the more asymmetric ones, then there is a view that more asymmetric profiles have less excitation degrees. Whittle (1985b) found out that there was a correlation between dust and asymmetry, where it was noted that an increase in extinction matched higher asymmetric profiles. High asymmetry correlates with low values of kurtosis. This meant that dust not only causes profile asymmetry but also softens the ionizing radiation field which consequently produces lower excitation NLR. Thus in the presence of dust, it is not easy to obtain a direct correlation between Kurtosis and the BLR winds or moving clouds.
Discussion
In the last section, we displayed the results obtained from the analysis of the Asymmetry Index and Kurtosis Index. We also looked at how the Asymmetry Index and Kurtosis Index relates to some other kinematic properties such as line width, luminosity,radio flux and ionization degree. It this chapter we shall discuss some of the most striking relations that were observed and, if possible, study the most likely implications of the relation to understanding the structure and properties of the accretion disk winds in AGN.
Asymmetry Index
Astrophysical line profiles are observed to display a variety of shapes. This is because We shall not include relations obtained at 00% percentiles since errors in that measurement were higher than for other parts of the profile due to the difficulty in estimating the continuum levels at both ends of the broad Balmer lines. (Balcells 1991) studied asymmetric line profiles in merger remnants, the study in his models suggesting that the asymmetry is not due to a disk of accreted secondary material; it is intrinsic to the distribution of primary particles, and has been added during the merger. It is argued that the asymmetries are a consequence of the transfer of the orbital energy and angular momentum to the primary particles during the merger; for the mergers studied in his models, profile asymmetries are a relic of the formation dynamics rather than the signature of superimposed components. Relating this to our case of the BLR, the asymmetries may be due to the transfer of the orbital energy and angular momentum of the accreting material and it is a relic of the material falling in or leaving the accretion disk. This material flows in or out in form of a wind. According to Gaskell & Goosemann (2013), a line shifts, thus asymmetry, in the base of a profile is a reflection of high accretion rates of the AGN. They demonstrated that high accretion rate AGNs will show line blue shifts, both in their models and observational data. For our case, most of the Balmer emission line asymmetries were red-shifted, indicating outflow of material (wind) rather than inflow (accretion).
Line Width
Line width are a good measure in investigating the strength of the spectral features in a system. Some authors use it to differentiate Seyfert galaxies from other emission line galaxies (Feldman et al. 1982
Luminosity
Luminosity is another property that is widely studied, having many relations (Padovani & Rafanelli 1988) with other properties of host galaxies (La Mura et al. 2009) like among other properties, the BLR size and BH mass. Although the scatter was high, some plots appeared to suggest that highly asymmetric profiles correlated with high luminosities. This means these luminosities are being driven by the same mechanism giving rise to asymmetry. This means winds will create asymmetry and also rise the luminosity of the source as seen in such relations. The best relations were obtained in Hβ profile asymmetry for the 10%, 20%, 30%, and 40% percentiles. The moderately tight positive correlations point to a direction in which we notice that the most luminous sources have high asymmetries in their Hβ emission line profiles.
Radio Flux
The radio flux scaled in such a way that high Hβ emission line asymmetries positively correlated with radio flux. The clearest relations are in 20%, 30%, 40%, 50% and 60% Hβ emission line percentiles as seen in fig 60. (Corbin 1997), in his study of "The Emission-Line Properties of Low-Redshift Quasi-stellar Objects. II. The Relation to Radio Type", found out that FRS quasars have significantly wider and more red-ward asymmetric Hβ profiles. Studies on the Hα Balmer emission line in Solar Physics by (Ichimoto & Kurokawa 1984) noted red asymmetry of Halpha flare line profiles. They believed that the Red-shifted emission streaks of H-alpha line are found at the initial phase of almost all flares which occur near the disk center, and are considered to be substantial features of the asymmetry. It is found that a downward motion in the flare chromospheric region is the cause of the red-shifted emission streak. The downward motion abruptly increases at the onset of a flare, attains its maximum velocity of about 40 to 100 km/s shortly before the impulsive peak of the microwave burst, and rapidly decreases before the intensity of Halpha line reaches its maximum. This proves that high radio emission can cause asymmetry in emission line shapes and the asymmetry is a consequence of motion material that is giving rise to the emission line.
Ionization Degree
The ionization degree in the narrow line region has also been noted to respond with effects taking place in the broad line region. For an outflow,a wind from an accretion disk would drive gas and dust out of the broad line region (Kollatschny & Zetzl 2013). This would cause both cooling and heating consequences in the narrow line region and this has been observed in the high variance in ionization degree for higher asymmetries. An inflow would also rid the narrow line region of material that would contribute to either heating or cooling, thus having still a dispersion in ionization degree measurements.
In all the properties studied, it seems plausible to confirm that accretion disk winds are part of the reason to the asymmetry in emission line profiles, both directly and indirectly. It is also noted that whether its inflow or outflow, the results would be similar although the most studied relations favor outflows.
The clear relations in this case can be seen in figure 61 which shows those of the 10th, 20th, 30th and 40th percentile from top left to right.
Conclusion
In this study we investigated the asymmetry in the first two broad Balmer emission lines, (Hα & Hβ), with the aim of probing accretion disk winds in Active Galactic Nuclei (AGN). Our sample was chosen from the SDSS DR7 with slightly over 300 spectra. We extracted the individual broad Balmer emission lines from each spectrum, normalized them to have the emission line peak have a value of unity, and measured the line shifts at each chosen percentile after centering the profiles at the a wavelength value corresponding to the emission line peak. Line shift values were then used to calculate the Asymmetry Index (A.I) and Kurtosis Index (K.I) for each respective percentile, and statistical distribution of these measurements was analyzed, after which, both the AI and KI were plotted against some kinematic properties like line width (FWHM), Luminosity, Radio Flux, and ). According to our results, we come to the following conclusions:
• Asymmetry in Hβ profiles positively correlated with Line width, V-band Luminosity, Radio Flux and Ionization degree.
• Asymmetry in Hβ profiles shows a stronger correlation with Line width, Vband Luminosity, Radio Flux and Ionization degree than that of Hα emission line profiles. Its correlations were tighter, with steeper gradients.
• From the statistical distribution of the AI and KI, the asymmetry of the profiles is predominantly red-shifted.
• Broader lines showed more asymmetry than narrower ones.
• Overplots of Hα profiles with Hβ profiles from the same spectrum showed that the Hβ profiles had an extended red asymmetry from their Hα counterparts.
• The flux ratio [OIII]λ 5007 [OI]λ 6302 , as a measure of ionization degree is more reliable than [OIII]λ 5007
[OII]λ 3727 since [OII]λ 3727 values contained more errors in their values.
• The observed distribution of the asymmetry is consistent with previous studies on many other emission and absorption lines, most showing positive asymmetries.
• Percentiles of 20%, 30%, 40%, 50%, 60% and 80% contain the most reliable information about profile shape since they are least affected by errors due to continuum estimation and line peak estimation. • The line profile shape parameters carry much detailed information about kinematics of the BLR, which can be better understood by means of techniques exploiting the profile by region, rather than measuring general shape parameters.
Although this analysis may be an advancement in probing accretion disk winds, a study with more sources and incorporating many more other emission and absorption lines would be much better in drawing conclusive relations with the kinematic properties and thus key to understanding the structure of accretion disk winds in AGN.
Figure 3 :
3Signal to Noise ratio & Redshift distribution of Sample
Figure 4 :
4Full Width at Half Maximum distribution of Sample
Figure 7 :
7The images show an example of the normalized broad Balmer emission lines after continuum subtraction on each of them. It is clear from these examples, the diversity of broad and narrow components in AGN permitted emission lines
13 Figure 8 :
138A pie chart and bar graph showing the distribution of both Hα and Hβ emission lines in the sample
Figure 9 :Figure 10 :
910Redshift Redshift distribution of sample 2
Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 : 20 Figure 19 :Figure 20 : 21 Figure 21 :
11121314151617182019202121Frequency Distribution of the AI20 of Hα and Hβ emission line profiles Frequency Distribution of the AI30 of Hα and Hβ emission line profiles Frequency Distribution of the AI40 of Hα and Hβ emission line profiles Frequency Distribution of the AI50 of Hα and Hβ emission line profiles Frequency Distribution of the AI60 of Hα and Hβ emission line profiles Frequency Distribution of the AI70 of Hα and Hβ emission line profiles Some examples of symmetric Hα line profiles Some examples of asymmetric Hα line profiles Some examples of symmetric Hβ line profiles Some examples of asymmetric Hβ line profiles Some examples of symmetric Hα and Hβ line profiles overploted
Figure 22 :
22Some examples of asymmetric Hα and Hβ line profiles overploted ters will range from 0.0 to 1.0, but of course with very few profiles having values close to 1.0.(Whittle 1985a)
Figure 23 :
23Frequency Distribution of the KI 1 of Hα and Hβ emission line profiles
Figure 24 :Figure 25 :Figure 26 :
242526Frequency Distribution of the KI 2 of Hα and Hβ emission line profiles Frequency Distribution of the KI 3 of Hα and Hβ emission line profiles Images Showing the percentiles used to calculate values of Kurtosis and Asymmetry.
Figure 27 :
27Frequency Distribution of the KI Gen of Hα and Hβ emission line profiles.
Figures 31 , 26 Figure 28 :
31262832, 33 and 34 all show a decrease in line width with asymmetry for Hα and an increase in line width with asymmetry Kurtosis Index Regions for the Steep Profiles
Figure 29 :
29Kurtosis Index Regions for the Flatter Profiles
Figure 30 : 28 Figure 31 :Figure 32 :
30283132Relation of the AI20 of Hα and Hβ emission line profiles with Line width Relation of the AI30 of Hα and Hβ emission line profiles with Line width Relation of the AI40 of Hα and Hβ emission line profiles with Line width
29 Figure 33 :Figure 34 :Figure 35 :
29333435Relation of the AI50 of Hα and Hβ emission line profiles with Line width Relation of the AI60 of Hα and Hβ emission line profiles with Line width Relation of the AI70 of Hα and Hβ emission line profiles with Line width with Hβ.
Figure 36 :Figure 37 :
3637Relation of the AI20 of Hα and Hβ emission line profiles with V Band Luminosity Relation of the AI30 of Hα and Hβ emission line profiles with V Band Luminosity
Figure 38 :Figure 39 :
3839Relation of the AI40 of Hα and Hβ emission line profiles with V Band Luminosity Relation of the AI50 of Hα and Hβ emission line profiles with V Band Luminosity
Figure 40 :Figure 41 :
4041Relation of the AI60 of Hα and Hβ emission line profiles with V Band Luminosity Relation of the AI70 of Hα and Hβ emission line profiles with V Band Luminosity
35 Figure 42 :Figure 43 :
354243Relation of the AI20 of Hα and Hβ emission line profiles with Core Radio Flux Relation of the AI30 of Hα and Hβ emission line profiles with Core Radio Flux
Figure 44 :Figure 45 :
4445Relation of the AI40 of Hα and Hβ emission line profiles with Core Radio Flux Relation of the AI50 of Hα and Hβ emission line profiles with Core Radio Flux
Figure 46 :Figure 47 :Figure 48 :Figure 49 : 39 Figure 50 :Figure 51 :Figure 52 :Figure 53 :
464748493950515253Relation of the AI60 of Hα and Hβ emission line profiles with Core Radio Flux Relation of the AI70 of Hα and Hβ emission line profiles with Core Radio Flux Relation of the AI20 of Hα and Hβ emission line profiles with Ionization Degree Relation of the AI30 of Hα and Hβ emission line profiles with Ionization Degree Relation of the AI40 of Hα and Hβ emission line profiles with Ionization Degree Relation of the AI50 of Hα and Hβ emission line profiles with Ionization Degree Relation of the AI60 of Hα and Hβ emission line profiles with Ionization Degree Relation of the AI70 of Hα and Hβ emission line profiles with Ionization Degree
Figure 54 :Figure 55 :Figure 56 :
545556Relation of the Line Width of Hα and Hβ emission line profiles with Kurtosis Index Relation of the V Band Luminosity of with Kurtosis Index Relation of the core Radio Flux with Kurtosis Index
Figure 57 :
57Relation of the Ionization Degree with Kurtosis Index
Figure 60 :
60The Relation of the AI20 of Hβ and AI30 of Hβ emission line profiles with Core Radio Flux
Figure 61 :
61The Relation of the AI10, AI20, AI30, AI40 of Hβ emission line profiles with Ionization Degree 53
Table 1 :
1A table showing the initial values extracted from each spectrum plot for Line Normalization and Velocity Transformation.Spectrum Number Hα height Centering λ(Å) Hβ height Centering λ
00001(ha & hb.fit)
88.01
7353.861
00002(ha & hb.fit)
58.36
8599.299
00003(ha & hb.fit)
39.25
7235.383
00004(ha & hb.fit)
354.4
7365.362
87.10
5455.679
00005(ha & hb.fit)
258.2
8927.240
88.46
6612.635
00006(ha & hb.fit)
414.5
6785.756
140.0
5026.721
00007(ha & hb.fit)
22.12
6819.172
00008(ha & hb.fit)
173.7
6837.628
44.66
5064.256
00009(ha & hb.fit)
15.91
7756.598
00010(ha & hb.fit)
28.17
6809.348
Table 2 :
2Interpretation of Asymmetry IndexAsymmetry Index (x) Measure Interpretation
0<x ≤ 0.08
Weak
The distribution is relatively symmetrical.
0.08<x ≤ 0.15
Moderate The distribution is relatively asymmetrical.
x>0.15
Strong
The distribution is asymmetrical.
Table 3 :
3Interpretation of Kurtosis IndexKurtosis Index (x) Measure Interpretation
0<x ≤ 0.25
Strong
The change in shape is very large.
0.25<x ≤ 0.65
Moderate The change in shape is normal.
x>0.65
Weak
The change in shape in insignificant.
Table 4 :
4Correlation Coefficients for Line Width Verses Hα AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
-0.0338
0.6661
-0.0235
0.7642
-0.0155
0.7673
-0.0838 3.5049
30
-0.1500
0.0545
-0.1319
0.0914
-0.0896
0.0877
-0.3372 3.5151
40
-0.1604
0.0396
-0.1627
0.0368
-0.1076
0.0402
-0.3227 3.5162
50
-0.1507
0.0533
-0.1512
0.0525
-0.1032
0.0492
-0.2620 3.5137
60
-0.1323
0.0903
-0.1220
0.1185
-0.0878
0.0941
-0.2036 3.5110
70
-0.1004
0.1996
-0.0889
0.2559
-0.0627
0.2321
-0.1459 3.5082
Table 5 :
5Correlation Coefficients for Line Width Verses Hβ AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.2808
0.0000
0.2950
0.0000
0.2092
0.0000
0.4842 3.5448
30
0.3331
0.0000
0.3416
0.0000
0.2412
0.0000
0.5977 3.5481
40
0.3060
0.0000
0.2973
0.0000
0.2045
0.0000
0.5515 3.5719
50
0.2536
0.0000
0.2272
0.0001
0.1519
0.0002
0.4338 3.5919
60
0.2001
0.0008
0.1622
0.0066
0.1085
0.0069
0.3283 3.6045
70
0.1218
0.0421
0.1010
0.0921
0.0686
0.0876
0.1787 3.6210
Table 6 :
6Correlation Coefficients for V-Band Luminosity Verses Hα AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
-0.0808
0.3023
-0.0512
0.5141
-0.0313
0.5502
-0.4690 44.5500
30
-0.0746
0.3412
-0.0755
0.3350
-0.0445
0.3963
-0.3932 44.5504
40
-0.0763
0.3302
-0.0863
0.2705
-0.0534
0.3090
-0.3599 44.5510
50
-0.0897
0.2521
-0.0885
0.2585
-0.0584
0.2656
-0.3655 44.5515
60
-0.0898
0.2511
-0.0941
0.2294
-0.0593
0.2584
-0.3243 44.5494
70
-0.0548
0.4849
-0.0599
0.4449
-0.0368
0.4828
-0.1867 44.5431
Table 7 :
7Correlation Coefficients for V-Band Luminosity Verses Hβ AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.3415
0.0000
0.3679
0.0000
0.2536
0.0000
1.6778 44.6916
30
0.3075
0.0000
0.3288
0.0000
0.2283
0.0000
1.5719 44.7758
40
0.2292
0.0001
0.2489
0.0000
0.1686
0.0000
1.1768 44.8697
50
0.1901
0.0014
0.2033
0.0006
0.1366
0.0007
0.9263 44.9124
60
0.1589
0.0078
0.1695
0.0045
0.1130
0.0049
0.7429 44.9354
70
0.1313
0.0283
0.1360
0.0231
0.0902
0.0247
0.5489 44.9612
Table 8 :
8Correlation Coefficients for Hα relationsPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.0136
0.9270
0.0403
0.7858
0.0284
0.7761
0.1207 22.2540
30
-0.0481
0.7453
-0.0280
0.8501
-0.0266
0.7897
-0.3450 22.2752
40
-0.0698
0.6371
-0.0455
0.7589
-0.0355
0.7222
-0.4732 22.2829
50
0.0157
0.9156
0.0688
0.6421
0.0496
0.6187
0.0988 22.2544
60
0.1455
0.3238
0.1424
0.3342
0.0798
0.4238
0.8695 22.2000
70
0.2359
0.1065
0.2085
0.1549
0.1259
0.2069
1.2763 22.1829
36
Table 9 :
9Correlation Coefficients for Hβ relationsPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.2524
0.0300
0.2511
0.0309
0.1714
0.0307
2.4011 22.7527
30
0.3136
0.0065
0.2974
0.0101
0.1973
0.0129
3.2619 22.6989
40
0.2607
0.0249
0.2832
0.0145
0.1773
0.0254
2.5361 21.6919
50
0.2276
0.0512
0.2552
0.0282
0.1581
0.0463
2.0519 22.9814
60
0.2494
0.0321
0.2372
0.0418
0.1448
0.0680
2.3258 22.9817
70
0.2288
0.0499
0.2476
0.0335
0.1670
0.0353
1.9114 23.0574
37
Table 10 :
10Correlation Coefficients for [OIII] [OI] Ionization Degree Verses Hα AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.1357
0.0882
0.0955
0.2313
0.0708
0.1855
0.9175 1.5411
30
0.1497
0.0597
0.0780
0.3284
0.0525
0.3263
0.9266 1.5335
40
0.1243
0.1185
0.0543
0.4968
0.0418
0.4343
0.6905 1.5390
50
0.1343
0.0913
0.0574
0.4725
0.0415
0.4378
0.6414 1.5407
60
0.1224
0.1244
0.0453
0.5706
0.0305
0.5684
0.5182 1.5459
70
0.1046
0.1895
0.0281
0.7253
0.0187
0.7263
0.4225 1.5504
Table 11 :
11CorrelationCoefficients for [OIII]
[OII] Ionization Degree Verses Hα Asymmetry
Percentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.1247
0.1174
0.1484
0.0620
0.0986
0.0650
1.0134 0.8136
30
0.0471
0.5555
0.0882
0.2687
0.0612
0.2521
0.3505 0.8314
40
0.0172
0.8292
0.0540
0.4988
0.0385
0.4719
0.1151 0.8399
50
0.0087
0.9130
0.0365
0.6477
0.0270
0.6136
0.0501 0.8428
60
0.0047
0.9531
0.0207
0.7961
0.0157
0.7692
0.0239 0.8439
70
0.0218
0.7853
0.0301
0.7063
0.0227
0.6712
0.1057 0.8402
Table 12 :
12Correlation Coefficients for [OIII] [OI] Ionization Degree Verses Hβ AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.1898
0.0091
0.0980
0.1811
0.0680
0.1656
0.7812 1.4197
30
0.1889
0.0094
0.1394
0.0563
0.0939
0.0558
0.8271 1.4452
40
0.1922
0.0082
0.1332
0.0685
0.0873
0.0754
0.8391 1.4654
50
0.1522
0.0370
0.0824
0.2609
0.0536
0.2748
0.6199 1.4930
60
0.1492
0.0410
0.0716
0.3289
0.0481
0.3267
0.5974 1.4997
70
0.1494
0.0408
0.0927
0.2058
0.0675
0.1692
0.5168 1.5130
Table 13 :
13Correlation Coefficients for [OIII] [OII] Ionization Degree Verses Hβ AsymmetryPercentile
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
20
0.1288
0.0781
0.1113
0.1283
0.0766
0.1187
0.6798 0.7197
30
0.1289
0.0779
0.1484
0.0421
0.0967
0.0488
0.7236 0.7414
40
0.1279
0.0803
0.1728
0.0178
0.1106
0.0242
0.7158 0.7608
50
0.0837
0.2533
0.1367
0.0613
0.0860
0.0796
0.4372 0.7920
60
0.0274
0.7088
0.0716
0.3285
0.0473
0.3348
0.1407 0.8178
70
-0.0264
0.7195
-0.0087
0.9056
-0.0066
0.8930
-0.1170 0.8355
Table 14 :
14Correlation Coefficients for the Kinematic Properties Verses Hα Kurtosis IndexRelation
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
LW
0.2324
0.0032
0.1837
0.0205
0.1220
0.0224
0.8158 3.2392
RF
0.0969
0.5266
0.0262
0.8643
0.0061
0.9532
1.0674 29.9262
L
0.0649
0.4166
0.0576
0.4709
0.0378
0.4793
0.5398 44.3827
ID 1
-0.0982
0.2179
-0.0389
0.6267
-0.0286
0.5929
-0.9613 1.8726
ID 2
0.0315
0.6939
0.0558
0.4845
0.0362
0.4980
0.3699 0.7284
Table 15 :
15Correlation Coefficients for the Kinematic Properties Verses Hβ Kurtosis Index the shape of a line profile can depend on a number of parameters which include; shocks, inflows and outflows of a wind component, obscuration by dust and rotation. If we only take the effect of the velocity field, that is shocks, Doppler motions, turbulence, inflow and outflow wind components and rotation, we can separate the region of the line profile that suffers from each effect. Since the line width is an excellent measure of strength of the properties of the line profile, we shall keep its relation to asymmetry in mind while discussing the rest of the kinematic properties.Relation
ρ p
P p −value
ρ s
P s −value
ρ k
P k −value
m
b
LW
0.4766
0.0000
0.4527
0.0000
0.3264
0.0000
1.6614 3.1338
RF
-0.0666
0.6424
-0.0338
0.8141
-0.0353
0.7147
-0.8794 30.8045
L
-0.3269
0.0000
-0.3334
0.0000
-0.2224
0.0000
-2.4625 45.3767
ID 1
0.0326
0.6588
0.0397
0.5910
0.0246
0.6183
0.2657 1.4750
ID 2
nan
1.0000
0.0240
0.7448
0.0390
0.4293
nan
nan
). Others, use it to separate quasars in two populations, Population A and Population B in which the later have extremely broad Balmer emission lines exceeding a width of 3000 km/s. The line width of the emission line profiles might depend on the the velocity field, geometry of the line emiting gas, obscuring effects, the superposition of line emission from different regions and on isotropy/anisotropy of the line emission(Kollatschny & Zetzl 2012). In the relations with asymmetry, it was observed that Hα profiles showed a negative correlation, while the Hβ profiles displayed a positive correlation. This would mean that the line width as a measure of the spectral features strength of the line emitting region will provide better results when we use the Hβ line asymmetry relation to line width. High values in line width were observed to correlate with higher values in asymmetry for the Hβ profiles. This may suggest that systems with high values in Hβ profile asymmetry have within them lots of information that we can obtain on the velocity field and geometry of the line emitting gas, whereas those with less Hβ asymmetry carried less information that can be extracted about the geometry of the line emitting gas. 30%, 40%, 50% and 60% Hβ emission line percentiles displaying good correlations to support this.In
this
case
it
is
the
20%,
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"Acceleration and cooling of the corona during X-ray flares from the Seyfert galaxy I Zw 1"
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"D R Wilkins \nKavli Institute for Particle Astrophysics and Cosmology\nStanford University\n452 Lomita Mall94305StanfordCAUSA\n",
"L C Gallo \nDepartment of Astronomy and Physics\nSaint Mary's University\nB3H 3C3HalifaxNSCanada\n",
"E Costantini \nSRON\nNetherlands Institute for Space Research\nSorbonnelaan 23584 CAUtrechtThe Netherlands\n\nAnton Pannekoeck Institute for Astronomy\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n",
"W N Brandt \nDepartment of Astronomy and Astrophysics\nThe Pennsylvania State University\n525 Davey Lab16802University ParkPAUSA\n\nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA\n\nDepartment of Physics\nThe Pennsylvania State University\n104 Davey Lab16802University ParkPAUSA\n",
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"SRON\nNetherlands Institute for Space Research\nSorbonnelaan 23584 CAUtrechtThe Netherlands",
"Anton Pannekoeck Institute for Astronomy\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"Department of Astronomy and Astrophysics\nThe Pennsylvania State University\n525 Davey Lab16802University ParkPAUSA",
"Institute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA",
"Department of Physics\nThe Pennsylvania State University\n104 Davey Lab16802University ParkPAUSA",
"Kavli Institute for Particle Astrophysics and Cosmology\nStanford University\n452 Lomita Mall94305StanfordCAUSA"
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"MNRAS"
]
| We report on X-ray flares that were observed from the active galactic nucleus I Zwicky 1 (I Zw 1) in 2020 January by the NuSTAR and XMM-Newton observatories. The X-ray spectrum is well-described by a model comprised of the continuum emission from the corona and its reflection from the accretion disc around a rapidly spinning ( > 0.94) black hole. In order to model the broadband spectrum, it is necessary to account for the variation in ionisation across the disc. Analysis of the X-ray spectrum in time periods before, during and after the flares reveals the underlying changes to the corona associated with the flaring. During the flares, the reflection fraction drops significantly, consistent with the acceleration of the corona away from the accretion disc. We find the first evidence that during the X-ray flares, the temperature drops from 140 +100 −20 keV before to 45 +40 −9 keV during the flares. The profile of the iron K line reveals the emissivity profile of the accretion disc, showing it to be illuminated by a compact corona extending no more than 7 +4 −2 g over the disc before the flares, but with tentative evidence that the corona expands as it is accelerated during the flares. Once the flares subsided, the corona had collapsed to a radius of 6 +2 −2 g . The rapid timescale of the flares suggests that they arise within the black-hole magnetosphere rather than in the accretion disc, and the variation of the corona is consistent with the continuum arising from the Comptonisation of seed photons from the disc. | 10.1093/mnras/stac416 | [
"https://arxiv.org/pdf/2202.06958v3.pdf"
]
| 246,863,528 | 2202.06958 | a58e4bcb06c21aa99e5c9c7d402ad531e2522553 |
Acceleration and cooling of the corona during X-ray flares from the Seyfert galaxy I Zw 1
2022
D R Wilkins
Kavli Institute for Particle Astrophysics and Cosmology
Stanford University
452 Lomita Mall94305StanfordCAUSA
L C Gallo
Department of Astronomy and Physics
Saint Mary's University
B3H 3C3HalifaxNSCanada
E Costantini
SRON
Netherlands Institute for Space Research
Sorbonnelaan 23584 CAUtrechtThe Netherlands
Anton Pannekoeck Institute for Astronomy
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
W N Brandt
Department of Astronomy and Astrophysics
The Pennsylvania State University
525 Davey Lab16802University ParkPAUSA
Institute for Gravitation and the Cosmos
The Pennsylvania State University
16802University ParkPAUSA
Department of Physics
The Pennsylvania State University
104 Davey Lab16802University ParkPAUSA
R D Blandford
Kavli Institute for Particle Astrophysics and Cosmology
Stanford University
452 Lomita Mall94305StanfordCAUSA
Acceleration and cooling of the corona during X-ray flares from the Seyfert galaxy I Zw 1
MNRAS
0002022Accepted 2022 February 11. Received 2022 February 10; in original form 2021 September 29Preprint 6 April 2022 Compiled using MNRAS L A T E X style file v3.0accretion, accretion discs -black hole physics -galaxies: active -galaxies: Seyfert -X-rays: galaxies
We report on X-ray flares that were observed from the active galactic nucleus I Zwicky 1 (I Zw 1) in 2020 January by the NuSTAR and XMM-Newton observatories. The X-ray spectrum is well-described by a model comprised of the continuum emission from the corona and its reflection from the accretion disc around a rapidly spinning ( > 0.94) black hole. In order to model the broadband spectrum, it is necessary to account for the variation in ionisation across the disc. Analysis of the X-ray spectrum in time periods before, during and after the flares reveals the underlying changes to the corona associated with the flaring. During the flares, the reflection fraction drops significantly, consistent with the acceleration of the corona away from the accretion disc. We find the first evidence that during the X-ray flares, the temperature drops from 140 +100 −20 keV before to 45 +40 −9 keV during the flares. The profile of the iron K line reveals the emissivity profile of the accretion disc, showing it to be illuminated by a compact corona extending no more than 7 +4 −2 g over the disc before the flares, but with tentative evidence that the corona expands as it is accelerated during the flares. Once the flares subsided, the corona had collapsed to a radius of 6 +2 −2 g . The rapid timescale of the flares suggests that they arise within the black-hole magnetosphere rather than in the accretion disc, and the variation of the corona is consistent with the continuum arising from the Comptonisation of seed photons from the disc.
INTRODUCTION
The accretion of matter onto supermassive black holes powers some of the most luminous objects we observe in the Universe; active galactic nuclei, or AGN. A significant fraction of their energy output arises in the form of the X-ray continuum, produced by a compact corona of energetic particles associated with the black-hole magnetosphere and the inner parts of the accretion disc (Galeev et al. 1979;Haardt & Maraschi 1991).
It is hypothesised that the X-ray continuum is produced in the corona by the Comptonisation of ultraviolet seed photons emitted thermally from the accretion disc, producing the observed power law form of the continuum spectrum, cut off exponentially at an energy corresponding to the characteristic temperature of the corona. Much remains unknown, however, about the exact nature of the corona, its precise location and structure, and the physical processes by which it is energised from the accreting material, or even the spin of the black hole. Another outstanding question is the relationship between the X-ray emitting corona and the powerful jets that emanate from radioloud AGN. Recent studies have shown that in many radio-loud AGN, the X-ray emission still appears to come from a corona analogous to ★ Contact e-mail: [email protected] that found in radio-quiet AGN, rather than from the jet itself (e.g. Zhu et al. 2020Zhu et al. , 2021, but in radio-quiet AGN, there may be a component of the corona that can be associated with a failed jet Yuan et al. 2019b).
The X-ray emission from the corona is highly variable. Shorttimescale variability is frequently observed, with the count rate changing by factors of two or three on timescales of just hours. In addition, many AGN have been seen to transition from higher to lower flux 'states' or 'epochs' corresponding to changes in the structure of the corona (e.g. Fabian et al. 2012;. On top of this, transient phenomena, including bright X-ray flares, are observed. A particularly bright X-ray flare was observed from the AGN Markarian 335 in 2014, during which the corona was found to have been ejected and accelerated away from the accretion disc, akin to a failed jet-launching event .
The X-ray emission from the corona illuminates the inner regions of the accretion disc. In addition to seeing the direct continuum emission from the corona, we therefore see the characteristic 'reflection' spectrum, produced when the X-ray continuum is reprocessed by the plasma in the accretion disc (Ross & Fabian 2005). The reflection spectrum contains emission lines produced by the accretion disc, notably the iron K fluorescence line around 6.4 keV, a soft excess of emission below approximately 1 keV, and the 'Compton hump' that is observed to peak around 20 keV. The emission lines from the inner accretion disc are broadened by the combination of Doppler shifts (from their orbital motion) and gravitational redshifts (due to the proximity of the emission to the black hole), producing a blueshifted peak and characteristic redshifted wing (Fabian et al. 1989). The shape of the redshifted wing encodes a wealth of information including the spin of the black hole (Brenneman & Reynolds 2006), the location and geometry of the corona and the structure of the inner disc (Wilkins & Fabian 2011, 2012.
A further dimension is added to this picture by the measurement of reverberation time lags in the X-ray emission. Variability of the soft X-ray excess and iron K line that are reflected from the disc is found to lag behind correlated variations in the primary X-ray continuum, owing to the additional light travel time from the corona to the inner accretion disc (Fabian et al. 2009;Uttley et al. 2014). The time lags are short relative to the time for light to cross event horizon scales, indicating that the corona is compact and that the reflected X-ray emission is probing the innermost region around the black hole (De Marco et al. 2013;Kara et al. 2016). When coupled with measurements of the broad iron K line, the measured reverberation timescale provides additional constraints on the location and geometry of the corona, notably its scale height above the accretion disc (Wilkins & Fabian 2013;Cackett et al. 2014;Wilkins et al. 2016).
I Zwicky 1 (I Zw 1) is classified as a narrow-line Seyfert 1 (NLS1) galaxy (Gallo 2018) and is found at redshift = 0.06. NLS1 galaxies are AGN characterised by comparatively high mass accretion rates onto relatively less massive supermassive black holes between 10 6 and 10 7 M (Boller et al. 1996). The mass of the black hole in I Zw 1 is estimated to be between 8×10 6 and 3×10 7 M via the width of the H line and optical reverberation mapping (Vestergaard & Peterson 2006;Huang et al. 2019). I Zw 1 is classed as a 'complex' NLS1 galaxy, weaker in its X-ray emission compared to its optical and UV emission, relative to the 'simple' NLS1 galaxies, but exhibiting extreme variability, showing sharp peaks and drops in its X-ray light curve (Gallo 2006).
Detailed measurements of X-ray reverberation from the inner accretion disc in I Zw 1 reveal a corona that is composed of two components (Gallo et al. 2007a,b;Wilkins et al. 2017). Part of the corona extends over the inner accretion disc, varies relatively slowly over time, and is likely accelerated by magnetic field lines anchored to the inner accretion disc that generate the magneto-rotational instability (MRI) and provide the viscosity though which angular momentum is transferred away from the accreting material. The rapid variability of the X-ray emission in this model, though, originates in a collimated core within the corona that is energised at its base. Fluctuations in luminosity propagate upwards through this core. Although I Zw 1 is a radio-quiet AGN and does not exhibit a large-scale jet, this collimated core within the corona is reminiscent of a failed jet . In 2015, an X-ray flare was seen and was found to originate in the extended component of the corona, propagating inwards until it was finally observed in the emission from the collimated core ). The corona is accompanied by a multicomponent, variable outflow, detected through absorption lines it imprints on the soft X-ray spectrum (Costantini et al. 2007), and possibly found to vary over the course of the X-ray flare (Silva et al. 2018).
The first hard X-ray observations of I Zw 1 made by NuSTAR in 2020 show that the 3-50 keV X-ray spectrum is well described by the combination of the primary continuum emission from the corona and its reflection from the accretion disc around a spinning black hole ( = / > 0.75 −2 ). The detection of reflected X-rays from the inner accretion disc is further corroborated by the measurement of reverberation time lags between the continuum and both the soft X-ray excess and the iron K line. The measured time lag corresponds to an X-ray emitting corona located a height of approximately 4 g above the disc, where the gravitational radius g = / 2 is the radial coordinate of the event horizon in the equatorial plane about a maximally spinning black hole. During the 2020 observations, bright X-ray flares were observed, during which short flashes of redshifted iron K line emission were seen. These are consistent with the re-emergence of X-rays that were reflected from the back side of the disc, behind the black hole, and bent into our line of sight by the strong gravitational field, predicted when an X-ray flare is seen reverberating off the inner accretion disc (Wilkins et al. 2021).
We report on the 2020 observations of I Zw 1 made with the NuS-TAR and XMM-Newton X-ray observatories. We analyse the broadband X-ray spectrum and explore variability in the X-ray spectrum in the context of variations in the X-ray emitting corona over the course of the flare that shed light on the structure of the corona and the mechanisms by which it is powered and through which the flaring occurred.
OBSERVATIONS AND DATA REDUCTION
I Zw 1 was observed simultaneously by NuSTAR (Harrison et al. 2013) and XMM-Newton (Jansen et al. 2001) between 2020 January 11 and 2020 January 16. NuSTAR obtained a total of 233 ks exposure over a continuous period of 5.3 days. XMM-Newton observed I Zw 1 twice during this period, obtaining continuous exposures of 76 and 69 ks. The observations are outlined in Table 1, and the observed light curves are shown in Fig. 1.
Toward the end of the first XMM-Newton observation, 150 ks from the start of the NuSTAR observing period, flaring of the X-ray emission was observed. Two flares, most prominent in the soft X-ray band, were observed. Each lasted approximately 10 ks and the count rate peaked at 2.5 times the mean count rate that was seen before the flaring started. From the hard X-ray light curve obtained by NuSTAR, the flaring can be seen to continue after the first XMM-Newton observation ended, but had finished before the second XMM-Newton observation began.
NuSTAR data reduction
The NuSTAR observations were reduced using the NuSTAR dataanalysis system, , v1.9.2. The event lists from each of the focal plane module (FPM) detectors were reprocessed and filtered using the task, applying the most recent calibration available at the time of writing. We extracted the source photons from a circular region, 30 arcsec in diameter, centered on the point source. We find that a smaller 30 arcsec extraction region, suitable for X-ray sources that are fainter in the hard X-ray band, improves the signal to background ratio in the observations of I Zw 1 compared to the larger The overlaid line shows the light curve binned per spacecraft orbit. X-ray flaring was observed 150 ks from the beginning of the observations, toward the end of the first XMM-Newton exposure, which had subsided by the time the second XMM-Newton exposure had commenced. The lower panels show the hardness ratio (c) between the 0.3-1 and 2-10 keV bands, measured by XMM-Newton in 1000 s time bins, and (d) between the 3-10 and 10-50 keV bands, measured by NuSTAR, binned by spacecraft orbit. Vertical dashed lines indicate time time interval during which X-ray flaring was detected in the XMM-Newton observations. 60 arcsec region commonly employed for brighter targets. The background was extracted from a region, the same size, away from the point source, on each detector. Source and background spectra, along with their corresponding response matrices and ancillary responses (the effective area functions) were extracted using the tasks. In addition, was used to extract source and background light curves with all appropriate dead time and exposure corrections applied.
Initially, the separate spectra obtained from the FPMA and FPMB detectors were analysed separately. Fitting the spectra with a power law model, we find the best-fitting index to be consistent between the two detectors within statistical uncertainty. We therefore proceed to sum the spectra from FPMA and FPMB and analyse them using response matrices averaged between the two detectors, in order to maximise the signal-to-noise in the highest energy X-ray bands. Based upon the consistency between the FPMA and FPMB spectra, we expect that any uncertainty introduced by differences in the calibration and response between the two detectors and their respective telescopes to be dominated by statistical uncertainties arising from the limited photon count rate. I Zw 1 was significantly detected by NuSTAR above the background up to 50 keV.
XMM-Newton data reduction
We analyse primarily the data collected by the EPIC pn camera on board XMM-Newton, due to the instrument's superior sensitivity, particularly when analysing the variability of the X-ray emission. During the observations, the pn camera was operated in small window mode so as not to be impacted by photon pile-up given the average count rate observed from I Zw 1. The XMM-Newton observations were reduced using the ( ) v18.0.0. The event lists were reprocessed using the task, applying the latest available version of the calibration. Source photons were extracted from a circular region, 35 arcsec in diameter, and the background was extracted from a circular region of the same size, located on the same detector chip. The source and background spectra were extracted using the task and the corresponding response and ancillary response matrices were generated using and . Light curves were extracted from the XMM-Newton observations, also using , and were corrected to account for dead time and exposure variations using the task.
Assessing the X-ray variability
To obtain an initial estimate of the spectral variability, we calculate the X-ray hardness ratio as a function of time. The hardness ratio, between the count rates in a hard X-ray band and soft X-ray band , is defined as ( − )/( + ). From the XMM-Newton observations, we compute the hardness ratio between the 0.3-1 and 2-10 keV bands to assess variability in the shape of the soft X-ray spectrum. From the NuSTAR observations, we compute the hardness ratio between the 3-10 and 10-50 keV bands, to assess variability in the shape of the hard X-ray spectrum. The hardness ratios as a function of time are shown alongside the light curves in Fig. 1. Substantial variability can be seen in the soft Xray spectrum. Leading into the flares the X-ray spectrum can be seen to soften. At the beginning of the second XMM-Newton observation, once the flares had subsided, the spectrum hardens significantly. On the other hand, variability in the shape of the hard X-ray spectrum, as seen by NuSTAR, is much less. The hardness ratio in the NuSTAR bands remains much more constant, though a slight softening of the spectrum can be seen during the flares.
In Section 4, we present a detailed analysis of the variability in the X-ray spectrum to determine its underlying causes, as well as the cause of the flares.
MODELLING THE BROADBAND X-RAY SPECTRUM
The 3-50 keV X-ray spectrum of I Zw 1 is well-described by continuum emission from the corona, possessing a power law spectrum, and the reflection of this continuum emission from the accretion disc (Wilkins et al. 2021). In addition, time lags between the continuum emission, and both the soft X-ray emission and iron K line, suggest the reverberation of continuum variations. We therefore begin with this model to describe the full 0.3-50 keV band covered by the combination of XMM-Newton and NuSTAR. The continuum emission and the reflection from the accretion disc are described by the model , which includes reprocessing of the incident continuum emission by the plasma in the accretion disc, from the model (García & Kallman 2010;García et al. 2011García et al. , 2013, as well as the relativistic broadening of this reflection spectrum by Doppler shifts and gravitational redshifts from an accretion disc orbiting a spinning black hole. Reflection from the inner accretion disc contributes a relativistically broadened iron K line, a Compton hump, peaking between 20 and 30 keV, and a soft excess comprised of bremsstrahlung emission and soft X-ray emission lines that are blended together below around 1 keV. We here seek to extend the spectral model to the cover the full XMM-Newton and NuSTAR bandpass, from 0.3 to 50 keV.
The continuum model is described by the photon index (the slope of the power law) and the high energy cutoff, corresponding to the temperature of the corona, above which the power law spectrum transitions to an exponential cutoff. The model of the reflection from the accretion disc has parameters corresponding to:
• The iron abundance in the disc, Fe .
• The ionisation parameter, defined by the ratio of the ionising flux to the density, = 4 / .
• The emissivity profile of the disc, defined as the reflected flux as a function of radius, measured in the rest frame of the reflecting material. The emissivity profile depends upon the location and geometry of the corona that illuminates the disc and is parametrised as a broken power law.
• The inclination, , of the disc to the line of sight.
• The inner radius of the accretion disc, in , from which the spin of the black hole can be inferred if it coincides with the innermost stable orbit, ISCO .
• The reflection fraction, , defining the observed ratio of the reflected to continuum flux (integrated over the 0.3-100 keV energy band).
The soft X-ray spectrum exhibits absorption, not just from the interstellar medium along the line of sight in our own Galaxy, but from a multi-component, variable outflow intrinsic to I Zw 1 (Costantini et al. 2007). We model the absorption using the photoionisation code (Kallman & Bautista 2001). Since the outflows in I Zw 1 have been observed to vary in time, was used to generate grids of absorption models that span the ranges of column densities and ionisation parameters measured in previous observations of I Zw 1 (Silva et al. 2018). While the EPIC pn spectrum does not have the energy resolution required to resolve the absorption lines and accurately measure the column density, velocity and ionisation state of the outflows, these models account for the shape of the soft X-ray spectrum and ensure that uncertainty in the absorption is properly accounted for when estimating the parameters of the coronal X-ray emission and reflection from the accretion disc. We apply two multiplicative absorption models to the spectrum to describe the two outflow components that have been detected in I Zw 1 (Silva et al. 2018), but allow the column density, ionisation parameters and redshifts (which account for the velocities of the components) to vary freely as required to describe the observed spectrum. We then marginalise over uncertainties in the absorption parameters when estimating the uncertainties associated with the continuum and reflection model parameters. Preliminary analysis of the high-resolution spectrum obtained by the XMM-Newton Reflection Grating Spectrometer (RGS) in 2020 shows that the two warm absorber components are still present. The first component is measured with a column density of 1.3 × 10 21 cm −2 , an ionisation parameter of log( /erg cm s −1 ) = −1.1 and an outflow velocity of 1700 km s −1 , while the second component has a column density of 1.1 × 10 21 cm −2 , an ionisation parameter of log( /erg cm s −1 ) = −2.4 and an outflow velocity of 3200 km s −1 . For each component, the lines are broadened, corresponding to a turbulent velocity of 100 km s −1 . Although these measurements are made from the time-averaged spectrum across the entirety of the observations, the parameters of the warm absorbers measured by the RGS are roughly consistent with the constraints on the warm absorbers obtained from the lower resolution EPIC pn spectra. It should be noted, however, that the EPIC pn spectra resolve only the shape of the absorbed continuum and the most significant edges and cannot resolve the narrow absorption lines that enable detailed measurement of the absorbers with the RGS.
Due to the high degree of spectral variability we observe via the hardness ratio, we divide the observations into three time periods: before the flaring begins, the flares, and the period after the flares. The model was fit simultaneously to the spectra from the three time intervals. The spin of the black hole (and inner radius of the accretion disc), the inclination of the accretion disc to the line of sight and the iron abundance in the disc were tied between the three intervals, since these parameters should not have changed over the course of the observations. We allow all other parameters including the ionisation of the disc, the location and geometry of the corona (encoded in the emissivity profile of the reflection component observed from the disc), the reflection fraction, and the column densities, ionisation states and velocities of the warm absorbers to vary between the three time periods, should the data require these parameters to vary. Fitting the spectra from the three time periods simultaneously in this way determines not only the best-fitting model, but requires that the model reproduces the observed spectral variability in a physically meaningful manner.
We fit the model simultaneously to the XMM-Newton spectra over the 0.3-10 keV energy band and the NuSTAR spectra over the 3-50 keV band during the three time periods using (Arnaud 1996). The model applied to the NuSTAR spectrum is multiplied by a constant to account for calibration uncertainties between the instruments. We find the best-fitting value of the constant, representing the normalisation of the NuSTAR spectrum with respect to the XMM-Newton spectrum, to be 1.06 +0.03 −0.02 . The value of this cross-calibration constant was tied between the three time intervals.
The model parameters were optimised to minimise the modified -statistic, which is based upon the Cash statistic (Cash 1979) but modified such that when the number of counts is high, tends toward the 2 statistic. 1 Once the best-fitting parameters have been found, we estimate the parameter uncertainties using a Markov chain Monte Carlo (MCMC) calculation to explore the parameter space. MCMC chains were run using the algorithm of Goodman & Weare (2010), implemented in . Chains were run with 80 walkers for 10 7 iterations, discarding (or 'burning') the first 5000 iterations so as to remove any 'memory' of their starting positions.
We find that although the initial model of the continuum, reflection from the accretion disc, and absorption by two outflow components, provides a good description of the 3-50 keV spectrum, this model provides only a reasonable description of the spectrum over the full 0.3-50 keV band (Fig. 2), yielding a 2 statistic of 3380 for 3161 degrees of freedom ( 2 / = 1.07). The residuals between the model and the observed spectrum show that this simple model cannot simultaneously account for the shape of the soft excess and the amplitude of the iron K line. We therefore assess modifications to the model of reflection from the accretion disc.
In order to compare the descriptions of the observed spectrum provided by different models, we employ the deviance information The time-averaged spectrum over the whole of the observations is shown. The top panel shows the observed spectrum with the best-fitting model, consisting of the continuum, reflection from the accretion disc with a radial gradient in the ionisation parameter, and two warm absorption components. The lower panels show the ratio between the observed spectrum and four candidate models: a single, fixed-density reflection component, a single reflection component with variable density, a reflection component with additional soft excess component arising from warm Comptonization, and a reflection component with a radial ionization profile across the disc. Each of these models includes the continuum emission from the corona and the warm absorbers.
criterion, DIC (Spiegelhalter et al. 2002). DIC is a Bayesian model selection metric that factors in both the goodness of fit (from the likelihood function) and the number of free parameters. The DIC is a hierarchical generalization of the Akaike information criterion (AIC), which estimates the relative quality of a set of models in representing a specific set of data (in this case the observed spectrum), in terms of the information lost by each model. The DIC statistic is an asymptotic approximation to the model evidence, i.e. the probability that a given model will produce the observed data, integrated over the entire parameter space. When simply comparing fit statistics, such as the minimum 2 obtained between models, it is possible for a model to not only under-fit the data (thereby providing a poor fit), but also to over-fit the data by having too many free parameters. AIC and DIC statistics address this problem by penalising models with more free parameters, and represent a trade-off between the goodness of fit and the simplicity of each model. DIC can be computed directly from the results of MCMC calculations and has the advantage over other Bayesian model selection techniques and information criteria (e.g. the BIC or AIC) of defining an effective number of parameters. The DIC thereby does not penalise a model for 'nuisance' parameters that are not constrained by the data. In general, a model yielding a lower DIC is preferred by the data, and the ΔDIC between models quantifies the strength of the evidence in favour of one model over another. ΔDIC values between zero and two suggest only marginal evidence, while ΔDIC greater than 6 shows strong evidence for one model in favour of another (Kass & Raftery 1995). The mismatch between the shape of the soft excess and the iron K line could arise due to the accretion disc density being greater than the canonically assumed electron density of e = 10 15 cm −3 . Increasing the density of the accretion disc leads to the trapping of radiation in the upper layers, increasing the temperature and increasing the soft X-ray emission through bremsstrahlung .
We therefore use the model to describe the continuum and reflection in place of . models the reflection from an accretion disc with variable electron density. We find that this model improves the description of the soft X-ray spectrum, with Δ 2 = −10 compared to the fixed-density reflection model, with one fewer degree of freedom. The improvement of the fit is not enough, however, to offset the additional free parameter in this model, and is slightly disfavoured by the DIC statistic with ΔDIC = +6. The accretion disc in I Zw 1 is best-described with a slightly-enhanced electron density of log( e / cm 3 ) = 16.7 +0.3 −1.1 . It possible that the soft X-ray excess arises from a separate emission component from a 'warm corona' that extends over the inner regions of the accretion disc (e.g. Done et al. 2012;Petrucci et al. 2018). In this model, the energy liberated at small radii in the disc is not fully thermalised, but is split between both an optically thin corona (which produces the high energy X-ray continuum with a powerlaw spectrum), and a lower temperature (∼ 0.2 keV) optically-thick corona. Like the optically thin corona, this optically thick corona Comptonises the thermal photons emitted from the accretion disc, but instead of producing a power law spectrum extending to high energies, the warm corona produces an excess of soft X-ray emission, peaking below 1 keV. To test this model against the observed spectra of I Zw 1, we replace the power law continuum component of the spectral model with , an energetically self-consistent model which includes the power law continuum from the optically thin corona and the soft excess from the warm corona. We also include the reflection from the accretion disc, modelled by , to account for the broad iron K line and the Compton hump, and the warm absorbers. We fit the radius of the warm corona, its temperature and optical depth to the observed spectrum, and allow these parameters to vary between the three time intervals. It is necessary to apply a high-energy cutoff to the power law continuum spectrum in order to match the observed NuSTAR spectra. We find that while adding the additional soft excess component provides a much better description of the soft X-ray spectrum below 1 keV, providing ΔDIC= −50 compared with the original model, this model is still unable to simultaneously account for the shape of the soft excess and the strength of the observed iron K line. Broad residuals are still seen around 6 keV in the spectrum. Indeed, since we found that the simplest reflection model alone underestimates the iron K line with respect to the soft excess, adding further emission components to the soft X-ray band does not remedy this problem.
The shape of the soft excess can also be altered by the ionisation structure of the accretion disc. If the accretion disc is photoionised by the irradiating flux from the corona, we expect that the ionisation state of the disc varies as a function of radius. The ionisation parameter is defined = 4 / for a plasma with number density , receiving ionising flux . When the disc is illuminated by a compact corona close to the black hole, light bending toward the black hole focuses the irradiating flux onto the inner parts of the disc, causing it to fall off as steeply as −7 over the inner few gravitational radii, then following approximately −3 over the outer disc. On the other hand, the density of a thin accretion disc is expected to decrease as ∝ − 3 2 in regions where the gas pressure dominates, though increasing with radius as ∝ 3 2 in the innermost regions where radiation pressure becomes dominant (Novikov & Thorne 1973). Depending upon the precise density profile of the disc, the ionisation parameter can fall off as steeply as − 17 2 in the innermost regions, tending to − 3 2 over the outer disc (see e.g. Wilkins et al. 2020a).
The variant of the model approximates the variation in ionisation parameter across the accretion disc as a single power law, the index of which is fit to the data. We find that allowing the ionisation of the accretion disc to vary as a function of radius in this way, significantly improves the description of the observed spectrum, and simultaneously fits the soft X-ray spectrum, the iron K line and the Compton hump, providing ΔDIC = −110 compared to the original, fixed-density, constant-ionisation reflection model, and ΔDIC = −60 with respect to the model with an additional soft Xray emission component arising from warm Comptonisation. Before and after the flares, the ionsiation of the disc is relatively low, with log( in /erg cm s −1 ) ∼ 1. The ionisation parameter during this time periods was found to fall off as ∼ − 1 2 . During the flares, the ionisation parameter is not well constrained, but a relatively steeply falling profile with power law index greater than three is required to fit the spectrum. It should be noted that the power law index is expected to vary as a function of radius and the best-fitting index represents a flux-weighted average over the entire disc. In , the density of the accretion disc is frozen at the canonical value of e = 10 15 cm −3 .
We conclude that the model best-describing the observed spectra consists of the directly-observed continuum emission from the corona, as well as reflection from an accretion disc with a radial ionisation gradient, in addition to the two warm absorption components that were previously discovered in I Zw 1. This model provides an adequate description of the data, yielding a 2 statistic of 3253 for 3161 degrees of freedom ( 2 / = 1.03). The best-fitting parameters to the time-averaged spectrum obtained from the full observations are shown in Table 2
Ultra-fast outflows
Reeves & Braito (2019) report the detection of an ultra-fast outflow (UFO) in previous observations of I Zw 1 via the presence of P Cygnilike emission and absorption features in the iron K band of the X-ray spectrum. The presence of blueshifted absorption at 9 keV and excess emission around 7 keV suggests a highly ionised wind, launched from the accretion disc at a velocity of at least 0.25 . To determine if this wind was still present in the 2020 observations, we search for similar features in the X-ray spectrum by adding a narrow Gaussian emission or absorption line to the best-fitting model spectrum. The width of the line was frozen at 0.01 keV (i.e. below the energy resolution of the detectors so as to search for unresolved emission and absorption lines) and the line was scanned through the spectrum to determine the improvement in the 2 fit statistic as a function of line energy and normalisation (see Parker et al. 2017 for a discussion of the method).
The improvement in fit statistic with the addition of narrow emission and absorption features in time periods before, during and after the flares, is shown in Fig 4. We find similar absorption and emission features that could be attributed to an ultra-fast outflow, though in the 2020 observations we find that the absorption features are detected at only 2.7 significance. The emission feature is most strongly detected before the flares, at 3.7 significance. Interestingly, however, we note that the features in the spectrum attributable to the UFO vary. Before the flares, blueshifted absorption is detected at 9 keV, with an associated emission feature at 8.5 keV. During the flares, only the emission component is detected. After the flares, the absorption feature is detected, but at a lower energy of 8 keV, along with an associated emission feature at 7.5 keV. These results suggest that the flaring in the X-ray emission is accompanied by variations in the velocity and ionisation state of the UFOs launched from the inner disc, however there is insufficient signal to analyse this variability further with the present observations. We find that in addition to only being marginally detected, including these Gaussian emission and absorption features in the model does not alter the best-fitting parameters of the coronal X-ray emission and the reflection from the accretion disc. We therefore do not consider these features further in this work.
Constraining the geometry of the corona
The emissivity profile of the disc, that is the variation in reflected flux as a function of radius, corresponds to the flux received from the corona at each radius on the disc. The approximates the corona as a point source located on the rotation axis above the black hole. Under this approximation, the height of the corona above the disc plane can be estimated to be ℎ = 2.3 +1.2 −0.2 g , so as to provide the emissivity profile of the disc that best matches the observed reflection spectrum. Measuring the emissivity profile of the accretion disc directly, however, allows for the measurement of the location, geometry and spatial extent of the corona over the disc (Wilkins & Fabian 2012).
We therefore measured the emissivity profile of the accretion disc from the observed shape of the relativistically broadened iron K line in the spectrum. Varying the relative illumination of the inner and outer disc varies the relative contribution of redshifted line photons to the observed spectrum and hence alters the shape of the redshifted wing of the line. We measured the emissivity profile fitting the reflection spectrum in the 3-7 keV range (i.e. the iron K line), as the sum of the contributions from different radii (Wilkins & Fabian 2011). The flux received from each radius was fit to the spectrum as a free parameter in an MCMC calculation. In order to measure the emissivity profile in this way, it is necessary to have detected sufficient photon counts from the broad iron K line so as to accurately constrain the contribution from each radius on the disc. We therefore obtain a first estimate of the emissivity profile by fitting the shape of the line in the time-averaged spectra across the whole period of the observations, though noting that variability in both the broad line and underlying continuum during this period could skew the measured profile. The measured emissivity profile is shown in Fig. 5.
The emissivity is defined as the flux measured in the rest frame of the orbiting material in the disc and can be described by a twicebroken power law. Over the outer disc, the emissivity falls as −3 , as expected for the illumination of a disc by a central source in flat spacetime, with no gravity. The emissivity of the inner disc falls steeply (following −8 ). The combination of light bending, focusing X-rays towards the black hole and hence onto the inner disc, with the blueshifting of rays as they travel closer to the black hole, strongly enhances the flux received by observers on the inner disc (Wilkins & Fabian 2012).
The emissivity profile shows evidence of flattening over the middle region. Beyond 3 g the measured emissivity points lie above the expectation of the −8 power law observed over the innermost radii of the disc, and at around 8 g and 20 g , there is evidence of deviation from the ∼ −3 decrease that describes the emissivity of the outer disc. Flattening of the emissivity profile to a power law index less than 3 can be understood in the context of a point source, where the flux received is approximately constant for ℎ. For compact, point-like coronae higher above the black hole, the break radius from the flat middle portion of the profile to the −3 power law over the outer disc corresponds to the height of the corona. As the height increases, however, less illumination is received by the inner disc, so as the break radius moves out to larger radius, the steepness of the inner emissivity profile is expected to decrease. The simultaneous observation of an extended flat portion of the emissivity profile with a steep inner emissivity profile cannot be explained by a compact, point-like corona. The profile we observe can be explained by a corona at low height, but with a portion that is spatially extended over the surface of the disc. In this instance, the break radius corresponds to radial extent of the corona over the disc.
Variability of the emissivity profile and coronal geometry
If the geometry of the corona is variable over the time period of the observations, the emissivity profile we measure will be a superposition of the emissivity profiles at different times during the observation, where the outer break radius can move if the spatial extent of the corona over the accretion disc is changing. It is also possible that variation in the slope of the underlying X-ray continuum could be manifested as inaccuracies in the measured emissivity profile. We therefore measure the emissivity profile of the accretion disc during the three time periods, before, during and after the flares. These shorter time intervals do not have the required signal-to-noise to measure the emissivity profile as a free function of radius, however we are motivated by the approximate form of the emissivity profile measured from the time-averaged spectrum to fit the emissivity profile during each time period as a twice-broken power law, applying the 3 2 relativistic blurring kernel to the reflection model.
We find that the twice-broken power law model of the accretion disc emissivity profile is preferred over a model in which the emissivity profile is modelled by a single power law. The twice-broken power law model yields ΔDIC= −12 over the single power law model. During each of the time intervals, the index over the innermost part of the disc is constrained to be greater than 7 at the 90 per cent confidence limit, showing that even accounting for spectral variability between the time periods, the observed iron K line supports the focusing of the X-ray continuum emission towards the black hole and onto the innermost parts of the disc, as predicted when reflection is observed from an accretion disc is illuminated by a compact corona.
The best-fitting value of the outer break radius of the emissivity profile was found to be 7 +4 −2 g before the flares and 6 +2 −2 g after the flares subsided. During the flares, there is insufficient exposure to tightly constrain the outer break radius, but the measured value of 18 +6 −7 is suggestive that the corona expanded during the flares. Variation of the outer break radius can be explained either by a compact corona that varies in height above the accretion disc, or by the variation of the radial extent of the corona over the inner part of the disc (Wilkins & Fabian 2012). If the height of the corona above the disc increases, the increasing outer break radius of the emissivity profile is expected to be accompanied by a decrease in the slope of the inner part of the emissivity profile. The fact that the power law index over the innermost part of the disc is constrained to be > 7 at all times suggests that it is the radial extent of the corona over the accretion disc that varies. The corona expands from a radius of 7 +4 −2 g before the flares, to 18 +6 −7 during the flares, collapsing again to a radius of 6 +2 −2 g once the flares subside. The increase in the outer break radius between the before flare and flaring periods is detected only tentatively at 86.3 per cent (1.5 ) significance. The contraction after the flares is detected at 92.5 per cent (1.8 ) significance. Neither change in the outer break radius is detected to sufficient significance to be confident of variations in the coronal geometry, however hints of variation to the outer break radius might well be useful interpreting other aspects of the observed variability in the context of the corona. Fig. 6 illustrates how the profile of the relativistically broadened iron K line from the inner accretion disc constrains the geometry of the corona. The observed line profile is compared to models in which the emissivity profile of the accretion disc corresponds to illumination by a point source on the rotation axis of the black hole, and in which the emissivity profile is given by the best-fitting twicebroken power law, accounting for the spatial extent of the corona over the disc. The twice-broken power law is able to better explain the detailed shape of the redshifted wing of the line.
VARIABILITY DURING THE X-RAY FLARES
To understand the underlying changes that lead to the flares observed in the X-ray emission, we compare the best-fitting values of parameters that describe the accretion disc and corona in the time periods before, during and after the flares. The best-fitting parameters to each time period are shown in Table 2 and the variation in the properties of the corona are summarised in Fig. 8. Fig 7 shows the changes in the reflection from the accretion disc, illustrated as the ratio between the spectrum observed by NuSTAR in the 3-50 keV band and the 2 https://github.com/wilkinsdr/kdblur3 Energy / keV 10 5 Figure 6. The ratio of the 3-10 keV band of the time-averaged X-ray spectrum, measured by XMM-Newton, to the best-fitting power-law continuum component of the model. The ratio shows the profile of the relativistically broadened iron K line. The measured line profile is compared to models in which the emissivity profile of the accretion disc corresponds to illumination by a point source ( ), shown by the dashed line, and in which the emissivity profile is modelled by a twice-broken power law to account for the spatial extent of the corona ( 3), shown by the solid line. A model in which the corona is spatially extended is able to better-reproduce the shape of the redshifted wing of the line. best fitting power law that represents the directly-observed continuum emission. In addition to the tentative evidence that the corona expands during the flares from the change in the profile of the iron K line, we find that the strength of the reflection decreases relative to the continuum during the flares, the X-ray continuum spectrum significantly softens and steepens, and that during the flares the amplitude of the Compton hump decreases with respect to the amplitude of the iron K line.
The weakening of the reflection with respect to the continuum during the flares is manifested in the measured values of the reflection fraction. We find that the reflection fraction drops from 0.23 +0.04 −0.11 in the time period before the flares to just 0.07 +0.02 −0.04 during the flares, recovering to 0.45 +0.03 −0.15 once the flares have subsided. From the posterior distributions of the reflection fraction before, during and after the flares, we can estimate the significance level at which the drop in reflection fraction is detected by testing the null hypothesis that the reflection fraction during the flares is the same or greater than that before or after. We estimate that the null hypothesis that the reflection fraction does not drop as the flares begin can be rejected at the 99.7 per cent (3.0 ) confidence level, and that the null hypothesis that the reflection fraction does not increase again after the flares can be rejected at the 99.995 per cent (4.1 ) confidence level.
We see significant variation in the spectrum of the X-ray continuum emitted by the corona softens. The photon index steepens from 2.22 +0.04 −0.03 before the flares to 2.29 +0.06 −0.02 . After the flares subside, the continuum spectrum hardens, with the photon index dropping to 2.06 +0.03 −0.04 . The increase in photon index during the flares is detected at 99 per cent (2.6 ) confidence, and the drop in photon index after the flares in detected to at least 99.99999 per cent (5.4 ) confidence.
In addition to the drop in reflection fraction and the softening of the continuum spectrum, we find tentative evidence that during the flares, the temperature of the corona drops. We find that the cutoff energy of Table 2. The best-fitting model parameters. The model was applied simultaneously to the spectra obtained in time intervals before, during and after the flares in the 0.3-10 keV 3-50 keV bands measured by XMM-Newton and NuSTAR, respectively. The continuum emission from the corona is described by a power law with an exponential cutoff at high energy. The reflection from the accretion disc accounts for a radial varation of the ionization parameter, described by a power law. The reflection spectrum was initially modelled assuming a point-like corona on the spin axis of the black hole at variable height, defining the initial estimate of the emissivity profile. The emissivity profile was then measured in detail and fit using a twice-broken power law. The soft X-ray spectrum is modified by two warm absorption components from outflows. Parameter values represent the maximum of the likelihood function and uncertainties and upper/lower limits correspond to the 90 per cent confidence interval derived from MCMC calculations. Parameter values in the left-most column were tied between the three time periods. Due to the limited exposure available during the flares, the outer break radius of the emissivity profile was fit to the combined spectrum before and during the flares to better-constrain the variation of this parameter. Energy / keV 10 5 20 50 Figure 7. The ratio between the 3-50 keV spectrum measured by NuSTAR and the best-fitting power continuum, in time intervals before, during and after the flares. The ratio to the power law shows the reflection from the accretion disc: the relativistically broadened iron K line around 6.4 keV and the Compton hump. The shape of the reflection spectrum, as well as the reflection fraction (the strength of the reflection relative to the continuum) can be seen to vary.
Before flares
the continuum spectrum drops from 140 +100 −20 keV before the flares to 45 +40 −9 keV during the flares, reheating to a cutoff energy of 70 +40 −30 after the flares. The drop in the cutoff energy to ∼ 45 keV during the flares can be seen in Fig. 7. The cutoff energy approximately coincides with the peak of the Compton hump, resulting in the suppression of X-ray emission above this energy and a reduction of the hump relative to the strength of the iron K line during the flares. From the posterior distributions of the cutoff energies before, during and after the flares, we find that the drop in coronal temperature as the flares begin is detected at the 99.96 per cent (2.7 ) confidence level. As the corona recovers after the flares, the temperature of the corona appears to rise again, but the null hypothesis that the temperature does not increase as the flares end can only be rejected at the 73 per cent (1.1 ) confidence level.
While the ionisation parameter and power law index of the ionisation profile, along with the column densities and ionisation parameters of the warm absorption components, are allowed to vary between the intervals, we find that the data do not constrain significant variation in these parameters between the three time intervals. We marginalise over these parameters in estimating the uncertainty of each of the parameters of interest.
The black hole spin and the inner edge of the disc
The spin of the black hole can be estimated from the inner radius of the accretion disc assuming that the disc from which the broad iron K line seen extends down to the innermost stable circular orbit (ISCO) in the Kerr spacetime (Brenneman & Reynolds 2006). The spin of the black hole was linked between the three time intervals as the spectra were fit simultaneously. The extent of the redshifted wing of the iron K line indicated that the disc extends to the ISCO around a rapidly spinning black hole, with the spin parameter being at least > 0.94 −2 (90 per cent confidence limit).
As a further test, assuming that the black hole in I Zw 1 is maximally spinning ( = 0.998 −2 ), we allow the inner radius of the accretion disc to vary freely in each time interval. The posterior distributions of the inner radius of the disc in each of the time intervals allow us to test for truncation of the accretion disc outside the ISCO and to determine whether there is any variation to the inner edge of the disc. We find no evidence for significant variation of the inner edge of the accretion disc. Before the flares, the inner radius of the disc is found to be at a radial coordinate of 3.5 +2.6 −0.6 g . During the flares, the inner radius is most tightly constrained around the ISCO of a rapidly spinning black hole, at 2.1 +1.3 −0.4 g , and is least well constrained after the flares, when we can place an upper limit on the inner radius, lying within 15 g , though the inner radii during all three time intervals were statistically consistent with one another.
DISCUSSION
The reflection of X-rays from the inner regions of the accretion disc, measured in the X-ray spectrum of I Zw 1, offers a unique probe of the inner regions of the accretion flow and the corona that produces the X-ray continuum emission. We find that while the 3-50 keV Xray spectrum can be well-described by a simple model consisting of the X-ray continuum and reflection from a disc described by a single ionisation parameter at all radii (Wilkins et al. 2021), in order to describe the broadband 0.3-50 keV spectrum, it is necessary to account for the gradient in the ionisation of the disc as a function of radius. The observed reflection spectrum is consistent with a model in which the ionisation parameter falls following a power law in radius. Applying a constant ionisation with variable plasma density suggests that the electron density in the accretion disc may be as high as log( e / cm 3 ) = 16.7 +0.3 −1.1 , but this increased density over the canonical value of e = 10 15 cm −3 is not formally required in the model with a radial ionisation gradient.
Variation of the ionisation parameter with radius is expected when the accretion disc is radiatively ionised by the irradiation from a central, compact corona due to strong variations in ionising flux, as well as the density of the underlying disc, as a function of radius. We find in I Zw 1, however, that accounting for the ionisation gradient is required to provide a good description of the spectrum. In the past, phenomenological models consisting of two reflection components have been used to model the broadband spectra of AGN, and such models have been interpreted post hoc as accounting for the ionisation structure of the accretion disc. Comparing the DIC statistics, we find that in I Zw 1, the single reflection component with a radial ionisation gradient is preferred to the two-reflector model. Both models produce similar residuals, however the ionisation gradient model has fewer (effective) parameters (and a clearer physical interpretation).
From the profile of the iron K line, we estimate the illumination pattern of the accretion disc by X-rays from the corona (e.g. Wilkins & Fabian 2011). The emissivity profile of the accretion disc is bestdescribed by a twice-broken power law. Before and after the flares, the emissivity profile of the disc is consistent with illumination by a compact corona that extends corona that extends no more than 7 +4 −2 g over the inner disc before the flares, and 6 +2 −2 g after the flares. During the flares, the emissivity profile is consistent with the corona having expanded to a radius of 18 +6 −7 g over the inner parts of the accretion disc, however due to limited exposure time during the flares, this expansion is only tentatively detected and the data are formally consistent with no expansion during the flares. Combining measurements of the emissivity profile with measurements of the reverberation time delay (Wilkins et al. 2021), we may infer that the bulk of the corona extends to a radius of between 6 and 18 g over the accretion disc at an average height of approximately 4 g above the disc plane.
The precise mechanism by which the corona is produced remains unknown. Yuan et al. (2019b) show that a compact corona can be formed if the gradient in flux density arising from the disc exceeds a critical value. An instability develops in three-dimensional force-free electrodynamic simulations that causes magnetic field structures that would form a large-scale jet to collapse and dissipate energy in the black hole magnetosphere. This phenomenon can explain compact Xray sources close to black holes and may be responsible for generating the core of the corona previously inferred from X-ray spectral-timing measurements , or the compact corona seen in I Zw 1 before and after the flares. Zhu et al. (2020Zhu et al. ( , 2021, however, find the X-ray emission from most radio-loud AGN likely originates from a corona analogous to that found in radio-quiet AGN, suggesting that there must also be a component of the corona that does not require the destruction of the jet. We may speculate that a component of the corona extending over the innermost radii of the accretion disc (out to around 6 g before and after the flares, but potentially more extended as the flares are launched) can arise from the same magnetic fields that are posited to drive accretion via the disc (Balbus & Hawley 1998). If the field lines responsible for and amplified by the magneto-rotational instability (MRI) accelerate particles and undergo a significant number of reconnection events where the flux is most concentrated, above the innermost parts of the disc, they would naturally be expected to generate X-ray emission.
Variation in the coronal temperature
We find that during the X-ray flares, the temperature of the corona drops, with the high-energy cutoff of the continuum spectrum decreasing from 140 +100 −20 keV before the flares to 45 +40 −9 keV during the flares. Although the error bar on the measurement is large during the flares, the decrease is detected at 2.7 significance. After the flares, there is tentative evidence that the corona reheats, but increase in cutoff energy is only detected at 1.1 significance.
The temperature of the corona determines the high energy cutoff of the power law continuum spectrum and alters the energy balance in the accretion disc as it is heated by the X-ray continuum. We are able to detect the change in coronal temperature because this change in the energy balance within the disc changes the spectrum of the X-rays that are reflected (García et al. 2015). In particular, a lower temperature corona alters the ratio of the brightness of the iron K emission line to that of the Compton hump that are observed in the NuSTAR bandpass. A lower energy cutoff reduces the peak of the Compton hump relative to the peak of the line in a way that cannot be explained by variation in the ionisation state of the disc in these data (we marginalise over this parameter in the MCMC calculation). Such a drop in the Compton hump relative to the line can also be due to the iron abundance, however this cannot vary on the timescales upon which flares are observed.
If the coronal X-ray emission is produced by the Comptonisation of seed photons, e.g. from the accretion disc (Galeev et al. 1979), the drop in the coronal temperature during the flares could be caused either by an increase in the flux of seed photons entering the corona, reducing its temperature by Compton cooling, or by a reduction in the energy density that is deposited by the process that heats the corona. The cooling time of the corona is short, meaning that the continued emission of the X-ray continuum requires constant injection of energy . It is likely that the corona is produced by the dissipation of energy from reconnecting magnetic fields associated with the orbiting plasma in the accretion disc and the spinning black hole (Merloni & Fabian 2001;Yuan et al. 2019a,b;Bransgrove et al. 2021). The characteristic temperature of the corona is therefore likely to be highly sensitive to the same changes to the magnetic field configuration which leads to variation of the location and geometry of the corona. Observing this change in temperature of the corona during the flares gives us new insight into the mechanism by which the corona is formed and by which the flares occur.
In addition to reducing in the high-energy cutoff, the drop in coronal temperature is consistent with the observed softening of the X-ray continuum. Such behaviour, where the continuum spectrum becomes softer as the X-ray source becomes brighter has previously been observed in AGN (Markowitz et al. 2003;Wilkins et al. 2014;, however in these observations of I Zw 1, we are able to directly relate the softening of the continuum spectrum to variations in the temperature of the corona, following previous hints of such behaviour (e.g. Kang et al. 2021).
The photon index (Γ) increased from 2.22 +0.04 −0.03 before the flares to 2.29 +0.06 −0.02 during the flares. Following Sunyaev & Trümper (1979), we can estimate the spectrum produced by Comptonisation within a corona with characteristic temperature and through which the optical depth to Thomson scattering is . The spectral index (related to the photon index by Γ = 1 + ) is given by:
= − 3 2 + 9 4 + 1 2 ,(1)where = − 2 + e 2 B + 2 3 2
The constant is determined by the geometry of the system. Assuming a spherical geometry, = 3. While this relation is formally derived in the diffusion limit, appropriate for high optical depth, it has been shown to be a relatively good approximation in the optically thin limit (Titarchuk 1994).
Assuming that the X-ray continuum arises by Comptonisation in a thermal population, and notwithstanding any non-thermal contribution to the X-ray continuum emission during the short-timescale flares, we can use the temperature of the corona measured from the high-energy cutoff to estimate the optical depth that is required through the corona in order to produce a continuum spectrum with the observed photon index. Before the flares (noting the large upper error bar on the temperature measurement), we estimate that < 0.3 is required to produce the observed continuum spectrum, consistent with the typical assumption that the X-ray continuum is produced in an optically-thin corona. During the flares, producing the softer continuum spectrum with Γ = 2.29 +0.06 −0.02 from a cooler corona with = 45 +40 −9 keV requires = 0.8 +0.2 −0.4 . In the optically-thin limit, the probability of scattering seed photons will scale proportional to the optical depth. This means that, coupled with tentative evidence for the expansion of the corona over the inner accretion disc during the flares, increasing the rate at which seed photons are intercepted, the increase in optical depth by a factor is 2.6 is consistent with the factor 2.5 seen in X-ray count rate during the flares (although a further discussion of the seed photon population can be found below in §5.3). After the flares, the hardening of the continuum spectrum to Γ = 2.06 +0.03 −0.04 requires = 0.7 +0.4 −0.3 . Although the optical depth is greater than that before the flares, contraction of the corona to a smaller area over the disc after the flares reduces the X-ray count rate.
The low reflection fraction and motion of the corona
We find that during the X-ray flares, the reflection fraction drops from 0.23 +0.04 −0.11 to 0.07 +0.02 −0.04 , rising again to 0.45 +0.03 −0.15 after the flare. In a simple reflection scenario, in which a compact, isotropicallyemitting corona is located above the accretion disc, the reflection fraction is expected to be unity as half of the continuum flux is emitted downwards, toward the disc, and half is emitted upwards to be observed directly. While gravitational light bending focuses rays towards the black hole and inner parts of the disc, increasing the reflection fraction (Wilkins 2016), a reflection fraction below unity indicates that the inner accretion disc is under-illuminated. Such a drop in the reflection fraction can be understood in terms of the corona accelerating away from the accretion disc during the flares. As the corona is ejected away from the disc, relativistic beaming of the X-ray emission means that a greater fraction of the coronal X-rays are emitted upward, away from the disc, hence are observed directly as continuum emission rather than being reflected (Beloborodov 1999). After the flares, the reflection fraction rises again as the remaining corona slows.
A drop in the measured reflection fraction can also be explained by truncation of the inner accretion disc during the flare (e.g. De Marco et al. 2021). If the innermost part of the disc is either ejected or becomes optically thin such that no reflection is seen, the solid angle of the reflector subtended at the corona is reduced. Over-ionisation of the inner accretion disc can also cause the disc to appear truncated, if the iron K line emission from the inner disc is weakened and the reflection spectrum is smoothed to the extent that it blends into the continuum. This explanation, however, is inconsistent with the finding that even during the flare, redshifted reflection is detected from radii as far in as 2.1 +1.3 −0.4 g . Ray tracing simulations, following , show that to measure a reflection as low as 0.29 from a static X-ray source at a height of 4 g , the inner disc would need to be truncated at a radius of 18 g . The model of the relativistically broadened iron K line, however, probes the inner radius of the accretion disc via the maximal observed redshift of line photons emitted from close to the black hole. Measurements of the inner radius of the accretion disc before, during and after the flare are inconsistent with the disc becoming truncated at larger radius (or over-ionised) during the flare. Indeed, the data quality during the flare are such that the inner radius of the disc is most tightly constrained to be small during the flare.
Assuming that the measured reflection fraction over the 0.3-100 keV energy band is representative of total fraction of the 0.3-100 keV X-ray continuum that is intercepted by the disc, and that reprocessing of the X-ray continuum emission by the disc does not cause a significant fraction of the incident flux to be re-emitted outside this band, relativistic motion of the X-ray source away from the accretion disc remains the most plausible explanation of such low reflection fractions, and the drop in reflection fraction during the flares. A direct detection of the ejected corona, which would likely be similar in appearance to the ejection of a blob within a jet and be detectable via radio observations, would support this scenario, however no such radio monitoring has yet been conducted simultaneously with X-ray observations of I Zw 1 or of flaring in similar NLS1 AGN. The acceleration of the corona away from the disc during X-ray flares, followed by its subsequent collapse into a confined region around the black hole is consistent with the behaviour previously reported during X-ray flares observed in another NLS1 galaxy, Markarian 335 . Gonzalez et al. (2017) derive an empirical formula that predicts the reflection fraction as a function of the velocity and height, , of a point-like corona, accounting for not just special relativistic beaming of the emission due to the motion of the corona, but also the effect of light bending close to the black hole focusing emission towards the inner regions of the disc:
( , ) = out − in 1 − out (2)
Where the emission angles for the limiting rays reaching the inner and outer disc from the source are given by out = 2 /( 2 + 2 ) and in = (2− 2 )/( 2 + 2 ), and the prime symbols denote the correction applied to each angle to account for relativistic aberration due to the motion of the source for velocity = / , with = ( − )/(1− ).
When an accretion disc around a black hole is illuminated by a compact corona, up to 90 per cent of the reflected flux originates from the innermost 2 g of the disc (Wilkins & Fabian 2011). A significant fraction of the reflected flux will therefore be lost into the event horizon of the black hole and the observed reflected flux will be reduced to approximately 50 per cent of the total flux emitted form the disc when the X-ray source is between 2 and 10 g from the black hole (Wilkins et al. 2020b). This means that the observed reflection fraction will be approximately half of the 'intrinsic' reflection fraction (i.e. the ratio of the emitted flux from the corona that reaches the disc to that escaping to be observed directly). Assuming the corona to be a point source at a height of ℎ = 4 g above the disc plane, as inferred from measurements of the reverberation time delay in I Zw 1 (Wilkins et al. 2021), we can estimate for a reflection fraction of 0.07 to be measured, corresponding to an intrinsic reflection fraction of approximately 0.14, the velocity of the corona during the flare would need to reach 0.89 , accelerating from a velocity of 0.70 just before the flare. The measured reflection fraction after the flare corresponds to a velocity of 0.48 . It should be noted, however, that these represent upper limits on the velocity, since these estimates assume the corona is a compact point source. Extension of the corona over the inner accretion disc, away from the black hole, will reduce the fraction of rays bent towards the inner disc, thus will reduce the velocity required to produce the observed reflection fraction. The velocity of the corona during the flare exceeds the escape velocity from a radius of ℎ = 4 g from the black hole, and the plasma ejected from the corona can escape the black hole's gravitational influence. The evolution of the corona inferred from the variation of the spectrum during the flares is shown in Fig. 9.
If the reflection fraction drops due to the acceleration of the corona away from the disc, it is expected that the apparent redshifting of the disc in the rest frame of the accelerating corona causes the continuum to become photon-starved. This results in a decrease in the photon index as the continuum spectrum hardens (Beloborodov 1999). We, however, detect a softening of the continuum spectrum during the flares. This can be explained by the reduction of the coronal temperature we measure during the flares. Decreasing the temperature reduces the Compton amplification factor (the average fractional gain in energy per scattering during the Comptonisation process) and, thus, acts to soften the X-ray continuum.
Based upon the velocities inferred from the reflection fractions, and the relations derived by Beloborodov (1999), we expect a decrease in photon index by 0.15, corresponding to a factor 10 increase in amplification factor. The amplification factor scales as ( ) 2 , thus it would only take a factor of three drop in temperature to counteract this hardening of the spectrum, in line with the drop we measure. After the flare, we find that the continuum spectrum hardens. This is consistent with the measured contraction of the corona to a confined region close to the black hole. A compact corona will naturally be photon-starved as much of the solid angle subtended at such a corona is occupied by the event horizon, and relatively little is occupied by the source of seed photons, which is the accretion disc. Such short timescale variability in the temperature of the corona and the photon Outflowing portion of corona produces low reflection fraction Compact corona within 7 r g Coronal temperature 140 keV Acceleration of corona reduces reflection fraction Temperature drops to 45 keV Tentative expansion to ~18 r g Corona collapses and reheats Figure 9. The evolution of the corona during the X-ray flares, inferred from measurements of the reflection fraction, coronal temperature and profile of the broad iron K line over the course of the observations. index of the continuum spectrum it emits likely contributes to the scatter in the observed relationship between the photon index and the Eddington ratio observed in AGN (e.g. Shemmer et al. 2008;Brightman et al. 2013). While the average photon index can be related to the accretion rate onto the black hole, there is additional shorttimescale variability that is not trivially related to the mass accretion rate. The accretion rate likely varies on viscous timescales through the accretion disc, which is much longer than the timescale of the variability we observe.
On the decline of each of the flares, short flashes of photons were detected consistent with the re-emergence of iron K line photons reflected from the back side of the accretion disc, magnified and lensed around the black hole in the strong gravitational field (Wilkins et al. 2021). This interpretation requires that the reflection and reverberation from the accretion disc arises only in response to the first ∼ 2000 s of the flare, such that the flashes of photons re-emerging from the back side of the disc remain short. The measured evolution of the spectrum during the flares and the significant drop in the reflection fraction further supports this interpretation. The acceleration of the corona away from the disc and the resultant relativistic beaming of the continuum emission away from the disc means that the inner accretion disc is strongly illuminated only during the first part of the flare when the coronal velocity is low. Once the corona has accelerated at the peak of the flares, the majority of the continuum emission is beamed away from the disc and it no longer produces a strong response.
General relativistic magneto-hydrodynamic (GRMHD) simulations suggest a mechanism for the observed X-ray flares McKinney et al. (2012) find that, in certain circumstances, large-scale toroidal magnetic flux can be accreted inwards through the accretion disc until a critical flux density is reached in the inner regions. At this point, the inner disc becomes compressed by the magnetic field, which forms a barrier to further accretion, leading to a magnetically choked, or magnetically arrested disc (MAD). Further build-up of magnetic flux can lead to an inversion of the polarity of the field that threads the event horizon, during which magnetic flux is ejected and bursts of rapid accretion occur. The MAD state can also readily arise when geometrically thin discs (as suggested by the appearance of the X-ray reflection spectrum) are highly magnetised (Avara et al. 2016). The ejections of magnetic flux (that would form the corona) and bursts of accretion (with their corresponding radiative output) could plausibly be what we observe as the flares in the X-ray emission. The GRMHD simulations show bursts in mass accretion rate corresponding to the magnetic flux ejections lasting time periods around 400-800 on the accretion disc to a point at height 4 g on the spin axis. If the process driving the flare arises from activity on the inner accretion disc, it must therefore propagate through the magnetosphere at close to the speed of light. In a force-free magnetosphere, in which the kinetic and thermal energy densities of the plasma are negligible compared to the energy density associated with the electromagnetic fields, the Alfvén speed, associated with the propagation of waves along magnetic field lines, approaches the speed of light. The rapid nature of the flares points to their origin in the magnetic fields associated with the black hole and inner accretion disc, rather than changes in the accretion disc itself, which would propagate much more slowly on the viscous timescale.
The UV light curve and the seed photon population
Given that the total X-ray flux is observed to increase during the flares, as the coronal temperature decreases, it is necessary that if the X-ray continuum is produced by Comptonisation, that the flux of seed photons entering the corona increases. This requirement is strengthened if the corona is accelerated away from the disc. Relativistic beaming reduces the solid angle subtended by the accretion disc in the rest frame of the corona, reducing the effective cross section of the corona to photons from the disc, thus additional seed photons must be available to counteract this photon-starving. We can roughly estimate the seed photons available from the accretion disc via the UV light curve, which was recorded by the XMM-Newton Optical Monitor (OM) in the 'image+fast' timing mode. For the majority of the observations (including the flaring period), OM observations were obtained through the UVW1 filter, with a central wavelength of 2910Å. The UVW1 light curve is shown alongside the X-ray light curve in Fig. 10. For a standard accretion disc (Shakura & Sunyaev 1973) through which matter is accreting at approximately 20 per cent of the Eddington limit onto a black hole of mass 3 × 10 7 M , emission in the UVW1 band is dominated by regions of the disc around 200 g (see e.g. , which is likely not the dominant source of seed photons to a corona at smaller radii, within 28 g of the black hole (approximately 1 per cent of the black body emission at 30 g is expected to emerge in the UVW1 bandpass), so the light curve in this band is only approximate tracer of the available seed photon flux. We note, however, past studies of AGN with the Extreme Ultraviolet Explorer (EUVE) that show that the EUV varies approximately simultaneously with the near-UV, though with up to twice the variability amplitude in the EUV than the near-UV (Marshall et al. 1997). We see variability at the 4 per cent level in the UV light curve of I Zw 1. Much of the UV variability before the flares shows no obvious correlation with the X-ray flux, but we do note that both of the Xray flares are simultaneous with peaks in the UV emission, meaning that, in principle, there is a slight enhancement in seed photon flux to produce the X-ray flare through the cooler corona is available, in addition to the tentative evidence for the expansion of the corona such that it possesses a greater cross secion for intercepting seed photons form the disc during the flares.
It is also of interest that the increase in UVW1 flux seen peaking 8000 s following the peak of the first X-ray flare is consistent with the light travel time from a compact, central corona to a radius of ∼ 150 g on the disc. It is therefore plausible that this rise in UV flux between the peaks is due to the reprocessing of the flaring X-ray emission heating the surface of the disc.
CONCLUSIONS
Analysing the broadband X-ray spectrum of the narrow line Seyfert 1 galaxy I Zw 1, measured by NuSTAR and XMM-Newton, we find that the 0.3-50 keV spectrum is well-described by the X-ray continuum emission from the corona, the relativistically-broadened reflection of this X-ray continuum from the accretion disc around the rapidly spinning black hole, and soft X-ray absorption from the two warm, ionised outflows that were previously detected in this AGN.
In order to describe the reflection spectrum over the entire 0.3-50 keV energy band, it is necessary to account for the radial variation in ionisation across the accretion disc. Approximating the variation in ionisation parameter as a power law in radius provides a good fit to the observed spectrum.
During a series of X-ray flares, each lasting around 10 ks, the reflection fraction drops, consistent with the acceleration and ejection of the X-ray emitting corona away from the accretion disc. We find the first evidence that the temperature of the corona drops from 140 +100 −20 keV before the flares to 45 +40 −9 keV during the flares. This drop in the temperature of the corona is consistent with the observed softening of the X-ray continuum during the flares, and subsequent hardening after the flares subside. In a model in which the X-ray continuum is produced by the Comptonisation of thermal seed photons from the accretion disc, we infer changes in the optical depth by a factor of approximately 2.5
Before and after the flares, the emissivity profile of the accretion disc, measured via the profile of the broad iron K line, and the reverberation time lag between the continuum and the iron K line, are consistent with the inner accretion disc being illuminated by a compact corona, extending no more than 7 +4 −2 g over the disc before the flares, and 6 +2 −2 g over the disc after the flares. There is tentative evidence that during the flares the corona extends to 18 +6 −7 over the inner accretion disc, as it is accelerated away, increasing the cross section for scattering seed photons from the disc. The combination of the expansion and cooling of the corona, the increased optical depth and the peaks in the UV seed photon flux seen during the flares is consistent with the production of the flaring X-ray continuum by the Comptonisation of photons from the accretion disc as they pass through the corona.
From the reflection and reverberation of X-rays off of the inner regions of the accretion disc, a picture is starting to emerge of the extreme environment around the innermost stable orbit and just outside the event horizon of the black hole. The reflection spectrum and reverberation time delays between variations in the continuum and in the emission lines produced from the accretion disc reveal the properties of the X-ray source, including its location and extent. By observing the changes to the corona that underlie extreme episodes of variability, namely the bright X-ray flares emitted by I Zw 1, we can begin to understand the physical processes underlying its formation. Observing how the corona accelerates and cools during the flares and then collapses the flares subsides provides new constraints on the mechanism by which the corona is formed and energised by the accretion flow and black hole, and by which accretion onto supermassive black holes is able to power some of the most luminous objects in the Universe. The rapid timescale of the flares points to their origin in the magnetic fields associated with the accretion disc and black hole.
Figure 1 .
1X-ray light curves of I Zw 1 in 2020 January, (a) in the 0.3-10 keV band, measured with the XMM-Newton EPIC pn camera, and (b) in the 3-50 keV band, measured by NuSTAR and summed between the FPMA and FPMB detectors in 1000 s time bins.
Figure 2 .
2The 0.3-50 keV X-ray spectrum of I Zw 1, measured by XMM-Newton and NuSTAR.
and the model components are shown in Fig 3.
Figure 3 .
3The best-fitting model to the 0.3-50 keV X-ray spectrum of I Zw 1, fit simultaneously to the observations in three time intervals: before, during and after the flares. The model consists of the continuum emission from the corona (upper lines), relativistically-broadened reflection component from the accretion disc with radial ionisation gradient approximated by a power law (lower lines), and warm absorption from two outflow components (applied to each of the continuum and reflection components).
Figure 4 .
4The improvement in the 2 fit statistic when narrow Gaussian absorption and emission features are added at different energies in the iron K band, to search for signatures of ultra-fast outflows (UFOs). The model is applied simultaneously to the XMM-Newton and NuSTAR spectrum. Positive normalisations correspond to emission features, negative normalisation to absorption features. The detected features are shown from time intervals (a) before, (b) during, and (c) after the flares. Contours correspond to 1, 2 and 3 detection significance.
Figure 5 .
5The emissivity profile of the accretion disc, defined as the reflected flux from the accretion disc as a function of radius (defined in the rest frame of the emitting material). The emissivity profile was measured by fitting the time-averaged reflection spectrum in the 3-7 keV range (i.e. the iron K line), as the sum of the contributions from different radii. The emissivity profile shows the typical twice-broken power law form expected when the disc is illuminated by a compact corona. Solid lines show the best fitting power law indices to the inner and outer portions of the profile. The outer break radius corresponds to the extent of the corona over the accretion disc.
Figure 8 .
8Variation in parameters of the continuum emission and reflection from the accretion disc in time periods before, during and after the flares. The top panel shows the light curves obtained with the XMM-Newton EPIC pn camera. The variation is shown in the photon index of the power law continuum spectrum, the continuum cutoff energy, indicating the temperature of the corona, the ratio of the reflected to continuum flux (the reflection fraction), and the outer break radius of the accretion disc emissivity profile, which indicates the extent of the corona over the disc. The width of the bubbles represent the posterior probability density function at each parameter value. The central ticks show the best-fitting parameter values, and the bars represent the 1 uncertainties.
Figure 10 .
10X-ray and UV light curves of I Zw 1, obtained during the first XMM-Newton observation using the EPIC pn camera and Optical Monitor (OM), respectively. OM observations were obtained through the UVW1 filter, with effective wavelength 2910Å, in 'image+fast' timing mode. The raw UV light curve was extracted from the OM data in 10 s bins, and a 100-point moving average filter is applied to suppress noise. Vertical dashed lines show peaks in the UV emission that are simultaneous with the peaks of the two X-ray flares.
Table 1 .
1The NuSTAR and XMM-Newton observations of I Zw 1 obtained in 2020 that are incorporated in this analysis.Observatory
OBSID
Start Date
Exposure
NuSTAR
60501030002 2020-01-11
233 ks
XMM-Newton
0851990101
2020-01-12
76 ks
0851990201
2020-01-14
69 ks
− 3
3(where −3 is the light crossing time across one gravitational radius). In the light curve of I Zw 1 obtained by NuSTAR, we see that the flaring lasts 80-100 ks, corresponding to 500-700 (assuming a black hole mass of 3 × 10 7 M , Vestergaard & Peterson 2006; Wilkins et al. 2021), thus the flaring occurs on approximately the timescale expected in the MAD model. If the acceleration of the corona is to occur within 2000 s, the process by which it is driven must propagate rapidly through the black hole magnetosphere. 2000 s corresponds to a time period of 13 −3 for a 3 × 10 7 M black hole. From ray tracing calculations around a maximally spinning black hole, 13 −3 is the light travel time from a radius of 1.4 g−3
© 2022 The Authors
D. R.Wilkins et al.
MNRAS 000, 1-15(2022)
https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/ XSappendixStatistics.html
ACKNOWLEDGEMENTSThis work was supported by the NASA NuSTAR and XMM-Newton Guest Observer programs under grants 80NSSC20K0041 and 80NSSC20K0838. DRW received additional support from a Kavli Fellowship at Stanford University. WNB acknowledges support from the V.M. Willaman Endowment. Computing for this project was performed on the Sherlock cluster. DRW thanks Stanford University and the Stanford Research Computing Center for providing computational resources and support. We thank the anonymous referee for their valuable feedback on the original version of this manuscript.DATA AVAILABILITYThe data used in this study are available in the NuSTAR and XMM-Newton public archives. The NuSTAR observations can be accessed through the NASA HEASARC (https://heasarc. gsfc.nasa.gov) via observation ID 60501030002. The XMM-Newton observations can be accessed via the XMM Science Archive (http://nxsa.esac.esa.int/nxsa-web) via observation IDs 0851990101 and 0851990201. Analysis of the X-ray spectra was conducted using , distributed as part of the package (https://heasarc.gsfc.nasa.gov/docs/software/ heasoft). The and X-ray reflection models are available at http://www.sternwarte.uni-erlangen.de/ dauser/research/relxill. The 3 relativistic blurring kernel, using a twice-broken power law emissivity profile, is available at https://github.com/wilkinsdr/kdblur3. Scripts to conduct more detailed aspects of the analysis presented here, including the measurement of the accretion disc emissivity profile, are available upon request to the corresponding author.
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| [
"https://github.com/wilkinsdr/kdblur3",
"https://github.com/wilkinsdr/kdblur3."
]
|
[
"ON NON-DEGENERACY OF RIEMANNIAN SCHWARZSCHILD-ANTI DE SITTER METRICS",
"ON NON-DEGENERACY OF RIEMANNIAN SCHWARZSCHILD-ANTI DE SITTER METRICS"
]
| [
"Piotr T Chruściel ",
"Erwann Delay ",
"Paul Klinger "
]
| []
| []
| We prove that the T T -gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics with dimension-and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mass. | 10.4310/atmp.2019.v23.n5.a2 | [
"https://arxiv.org/pdf/1710.07597v1.pdf"
]
| 119,171,321 | 1710.07597 | 73274bb064b8c3c24aefb8a6c98fc515bce4dc3d |
ON NON-DEGENERACY OF RIEMANNIAN SCHWARZSCHILD-ANTI DE SITTER METRICS
Piotr T Chruściel
Erwann Delay
Paul Klinger
ON NON-DEGENERACY OF RIEMANNIAN SCHWARZSCHILD-ANTI DE SITTER METRICS
We prove that the T T -gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics with dimension-and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mass.
Introduction
There is currently considerable interest in the literature in spacetimes with a negative cosmological constant. In particular one is interested in existence of stationary black hole solutions of the Einstein equations with Λ < 0, with or without sources, and in properties thereof. Many such solutions have been constructed numerically, e.g. [6,7,22,23,26].
In [1,9] we showed how to construct infinite dimensional families of nonsingular stationary black hole spacetimes, solutions of the Einstein equations with a negative cosmological constant in vacuum or with various matter sources, assuming that a suitable linearised operator was an isomorphism. In [1,Proposition D.2] we proved the isomorphism property at Kottler solutions with negatively curved sections of conformal infinity for some dimension-dependent explicit ranges of the mass parameter (the whole range of masses in space-time dimension four, and the whole range of negative masses in all dimensions). However, the case of flat or positively curved conformal infinity has been open so far. The aim of this work is to prove the optimal result for all topologies in spacetime dimension four, and to provide partial answers to this problem in higher dimensions. This extends immediately the applicability of the existence theorems of [1,9] to the topologies and dimensions covered here.
Indeed, we prove:
Theorem A. Let us denote by P L the linearisation, at Riemannian Kottler metrics (2.1) with negative cosmological constant, of the T T -gauge-fixed Einstein operator. Then:
(1) P L has no L 2 -kernel in spacetime dimension n + 2 = 4 except for spherical black holes with mass parameter (2) P L has no L 2 -kernel for toroidal and higher genus black holes for all mass parameters µ in dimension n = 2, as well as for open ranges of parameters µ ∈ (0, µ(n)) for n > 2 and K = 0, where µ(n) > 0 solves a polynomial equation; cf. Table 1 in low dimensions.
Theorem A is a rewording of Theorem 5.1 below. In order to prove an equivalent of part (2) of Theorem A for black holes in higher-dimensions with K = −1 it remains to establish a higher-dimensional topology-independent linearised-Birkhoff-type theorem, which is likely to hold but which we did not attempt to prove.
We also propose a scheme, supported by numerical evidence, that could establish an equivalent of part (1) of Theorem A for all topologies and spacetime dimensions, except for the exceptional value (1.1) of the mass in the spherical case. We thus conjecture:
Conjecture B. P L has no L 2 -kernel except if K = 1 and µ is given by (1.1).
In order to establish the conjecture one would need to justify rigorously that our proposed scheme works, to prove the already alluded-to linearised Birkhoff theorem for K ∈ {±1} in dimensions n > 2, and to show the gauge character of l = 1 modes with K = 1 in dimensions n > 2.
We emphasise that point (1) of Theorem A holds regardless of the genus of the black hole. The proof in [1,Proposition D.2] of the higher-genus case K = −1 of (1) uses a completely different method, and leads to restricted ranges of masses in higher dimension. Our method here provides an alternative argument which, at the level of numerical tests, covers all masses. Now, the idea in [1,9] is to show that the construction of stationary Lorentzian solutions near a known static d-dimensional metricg can be carried-out using the implicit function theorem near a Riemannian d dimensional partner metricg. Consider then the operator P L obtained by linearising the Einstein equations at the "Wick rotated" metricg associated tog and imposing the T T gauge. The construction of [1,9] applies if P L has no L 2 -kernel; we then say that the Riemannian metricg is non-degenerate. We show here that non-degeneracy holds for Kottler metrics with spherical black hole horizons in four spacetime dimension, and for toroidal black hole horizons in all dimensions, for wide ranges of masses.
Some comments on the proof are in order. For this, let h be an element of the L 2 -kernel of the operator P L defined above. Our proof relies heavily on the remarkable construction of master functions of Ishibashi and Kodama [16]. Indeed, h can be decomposed into a sum of eigentensors of relevant operators, which we will refer to as modes. The modes split into two families, which we will refer to as exceptional modes and master modes. The master modes of h are controlled by the Ishibashi-Kodama master functions, which we show to be zero under the conditions above. We use the vanishing of the master modes to show existence of a vector field with controlled asymptotic behaviour which can be used to gauge-away the relevant part of h. Likewise we show that the special modes are pure gauge in the last, controlled, sense. Adding the associated vector fields and using the TTgauge condition, together with suitable Birkhoff-type linearised theorems, we establish that the kernel is trivial in the cases listed in Theorem A.
We note that a non-trivial kernel of P L implies existence of growing linear modes in the corresponding spacetime, except perhaps for variations in the direction of nearby stationary metrics. So proving that P L has no L 2 -kernel has implications for the dynamics of the associated Lorentzian solutions. However, non-degeneracy does not imply linear dynamical stability on the Lorentzian side, since non-degeneracy only excludes modes with a specific spectrum of frequencies.
Riemannian Kottler metrics
We start with a review of the Riemannian counterparts of the "generalized Kottler [19] metrics", also known as "Schwarzschild-anti de Sitter metrics", or "Birmingham metrics" [5].
In what follows we will be making extensive use of [16], in order to be consistent as much as possibly with their notations we set
n = d − 2 ,
where d is the dimension of spacetime. (The reader is warned that this does not coincide with our notations in [1,9], where n denotes space dimension d − 1.)
The metrics of interest read
(2.1)g = r 2 2 + K − 2µ r n−1 =:f (r) dt 2 + dr 2 r 2 2 + K − 2µ r n−1 + r 2 h K ,
where is a constant related to the cosmological constant Λ < 0 by the formula
= − n(n + 1) 2Λ > 0 ,
µ is a real constant, and K ∈ {0, ±1}. Furthermore, h K is a metric on an n-dimensional Einstein manifold n N . The metricg is Einstein if the Ricci tensor of h K equals (n − 2)Kh K . This will be assumed in what follows.
We require that f has positive zeros, we denote by r 0 > 0 the largest such zero, which we assume to be of first order (as will necessarily be the case if K ≥ 0 and µ > 0):
(2.2) r 2 0 2 + K − 2µ r n−1 0 = 0 .
After introducing a new coordinateř by the formula
(2.3)ř(r) = r r 0 1 s 2 2 + K − 2µ s n−1 ds ,
one can rewrite the metric (2.1) as
(2.4)g = dř 2 +ř 2 H(ř)dt 2 + r 2 h K ,
where H is obtained by dividing g tt byř 2 . Elementary analysis, using the fact that r 0 is a simple zero of F , shows that
H(0) = f (r 0 ) 2 4 = (n + 1) −2 r 2 0 + K(n − 1) 2r 0 2 .
This implies that a periodic identification of t with period
(2.5) T := 4π f (r 0 )
guarantees that dř 2 +ř 2 H(ř)dt 2 is a smooth metric on R 2 with a rotation axis atř = 0. As a result, (2.4) defines a smooth Riemannian metric on
(2.6) M := R 2 × n N .
The metric (2.1) can be smoothly conformally compactified by introducing, for large r, a coordinate ρ := 1/r and rescaling:
(2.7) ρ 2g = −2 + Kρ 2 − 2µρ n+1 dt 2 + dρ 2 −2 + Kρ 2 − 2µρ n+1 + h K .
Hence, the (n + 1)-dimensional conformal boundary ∂M := {ρ = 0} of M is diffeomorphic to S 1 × n N , with conformal metric
(2.8) −2 dt 2 + h K .
As already pointed out, it has been shown in [1,Appendix D] that all such solutions with n + 1 = 3 and K = −1 are non-degenerate. Furthermore, in that reference non-degenerate families of higher-dimensional such solutions have been described explicitly. We thus conjecture that the ranges of mass parameters there are not sharp, and that the arguments proposed here can be used to establish non-degeneracy for all values of µ when K ≤ 0.
In the analysis of the master equations below we assume that ( n N, h K ) is a maximally symmetric compact manifold. When K ≥ 0 the hypothesis that Λ < 0 and that of existence of a black hole with r 0 > 0 implies µ > 0 .
When K = −1 one needs instead (2.9) µ > µ min := − 1 n + 1 n + 1 2 (n − 1) 1−n 2 ,
with the corresponding outermost-horizon radius r 0 = 1/ n+1 2 (n−1) . The idea is to reduce the question of non-degeneracy to the Riemannian equivalent of the "master equations" of Ishibashi and Kodama [12,16] as follows:
(1) rewrite the master equations in a Riemannian form;
(2) work-out the asymptotic behaviour of the master fields corresponding to L 2 -elements of the kernel of the shifted Lichnerowicz operator P L defined below; (3) prove that all associated solutions of the master equations are trivial; (4) prove that elements of the L 2 -kernel with trivial master fields vanish identically. Point (1) above is straightforward once the impressive work in [16] has been carried out, but the remaining parts require some work. We note that point (4) captures the fact that the master equations contain the whole gauge-invariant information about the linearised gravitational field.
Consider, thus, an element h = h µν dx µ dx ν of the L 2 -kernel of (2.10) P L := ∆ L + 2(n + 1) ,
where ∆ L is the Lichnerowicz Laplacian, acting on symmetric two-tensor fields h µν as [4, § 1.143] [20] that |h|g = O(ρ n+1 ), or in local coordinates near the conformal boundary,
(2.11) ∆ L h µν = −∇ α ∇ α h µν + R µα h α ν + R να h α µ − 2R µανβ h αβ . It follows from(2.12) h µν = O(ρ n−1 ) .
Let us show, first, that h satisfies the linearised Einstein equations. For this, recall that the Hodge Laplacian acting on one-forms is defined as (2.13) ∆ H := d * g d + dd * g = ∇ * ∇ + Ric . If the Ricci tensor is covariantly constant, then [21] div
•∆ L = ∆ H • div .
This implies that L 2 -elements of the kernel of P L are divergence-free: Indeed, assume that h is in the L 2 -kernel of P L and let u = div h. We have 0 = div(∆ L + 2(n + 1))h, u = (∆ H + 2(n + 1))u, u , so |du| 2 + |d * g u| 2 + 2(n + 1)|u| 2 = 0, and u ≡ 0 follows. Note that there are no boundary terms in the integrationby-parts above by, e.g., [20].
Next, recall that we always have
Tr •∆ L = −∆ • Tr
(we use the convention ∆ = ∇ α ∇ α ). It follows that elements of the L 2 -kernel of P L are trace-free. The linearisation of the trace-shifted Ricci tensor reads
(2.14) D(Ric +(n + 1))h µν = 1 2 ∆ L h µν + (n + 1)h µν − (div * div grav h) µν , where (2.15) grav h = h− 1 2 Tr g hg, (div h) µ = −∇ ν h µν , (div * w) µν = 1 2 (∇ µ w ν +∇ ν w µ ) ,
and where Tr denotes the trace (note the geometers' convention to include a negative sign in the definition of divergence).
We have just seen that tensors in the kernel of P L are transverse and traceless. It follows from (2.14)-(2.15) that they are also in the kernel of the linearised vacuum Einstein operator. Now, it follows immediately from the analysis in [16] that, similarly to the Lorentzian case, the linearised Einstein operator for metrics (2.1) on manifolds as in (2.6) leaves invariant the subspaces of "scalar", "vector", and "tensor modes".
The modes l = 0, 1 require separate attention. A detailed analysis of this case, in Lorentzian signature, four spacetime dimensions, and spherical black-hole topology, can be found in [11]. 1 Indeed, it is shown there that the l = 0, 1 modes are, up to a gauge transformation, variations of the mass parameter of the Schwarzschild-anti de Sitter metrics (l = 0), or variations of the angular-momentum parameter when the Schwarzschild-anti de Sitter metric is viewed as a member of the Kerr-anti de Sitter family of metrics (l = 1). We show in Appendices F-H how to adapt the arguments of [11] to the cases of interest here.
As is well known, solutions of the linearised Einstein equations corresponding to variations of the mass parameter are not in L 2 , except if µ is given by (2.19) below, which is seen as follows: Replacing t in (2.1) by a 2π-periodic variable ϕ we find, in all spacetime dimensions d = n + 2,
(2.16)g = r 2 2 + K − 2µ r n−1 4dϕ 2 (f (r 0 )) 2 + dr 2 r 2 2 + K − 2µ r n−1 + r 2 h K , hence dg dµ = − 2f (r 0 ) r n−1 − 2 r 2 2 + K − 2µ r n−1 d(f (r 0 )) dµ 4dϕ 2 (f (r 0 )) 3 (2.17) + 2dr 2 r n−1 ( r 2 2 + K − 2µ r n−1 ) 2 .
It is then easily checked that dg/dµ ∈ L 2 if and only if
(2.18) d(f (r 0 )) dµ = 0 .
This has no solution with r 0 > 0 when K = 0 or K = −1, while if K = 1 this leads to
(2.19) r 0 = r c := n − 1 n + 1 , µ = µ c := n n + 1 n − 1 n + 1 n−1 .
In Appendix J we review the Riemannian Kerr-anti de Sitter metrics, and we prove there that variations of angular momentum lead to linearised solutions of the vacuum Einstein equations which are not in L 2 either.
Master functions
The master functions of Ishibashi and Kodama, which we denote by Φ S,I , Φ V,I , and Φ T,I , where the index I runs over all eigenfunctions of ∆ h K on n N (i.e. l = l(I)), are solutions of a two-dimensional Schrödinger equation
(3.1) ∆2gΦ i,I − V i,l Φ i,I = 0 , i ∈ {S, V, T } , where (3.2) 2g = f dt 2 + dr 2 f , with f = f (r) as in (2.1), while V S,k = 1 16r 2 (m + xn(n + 1)/2) 2 × (3.3) n 3 (n + 2)(n + 1) 2 x 2 − 12n 2 (n + 1)(n − 2)mx + 4(n − 2)(n − 4)m 2 −2 r 2 +n 4 (n + 1) 2 x 3 +n(n + 1) 4(2n 2 − 3n + 4)m + n(n − 2)(n − 4)(n + 1)K x 2 −12n (n − 4)m + n(n + 1)(n − 2)K mx +16m 3 + 4Kn(n + 2)m 2 , k 2 > nK , V V,k V = 1 r 2 k 2 V + K + n(n − 2) 4 K (3.4) + n(n − 2) 4 −2 r 2 − 3 n 2 µ 2r n−1 , k 2 V − (n − 1)K > 0 , V T,k T = 1 r 2 k 2 T + 2K + n(n − 2) 4 K (3.5) + n(n + 2) 4 −2 r 2 + n 2 µ 2r n−1 , where
x = 2µr 1−n , m = k 2 − nK . Note that our potentials Φ i,I differ from those of [16] by a multiplicative positive factor f (r). For a sphere there are no master potentials for l = 0, 1; for a torus no master potentials Φ S,I and Φ V,I for k = 0 = k V ; and no master potentials for k = 0 (scalar potential) in the higher genus case.
For K = 1 we have k 2 = l(l + n − 1) with l ≥ 0, k 2 V = l(l + n − 1) − 1 with l ≥ 1, and k 2 T = l(l + n − 1) − 2 with l ≥ 2. For K = 0 and a flat torus at infinity,
T n = S 1 × · · · × S 1 n factors , with each S 1 -coordinate of period 2π, and with h K=0 ≡ γ ij dx i dx j where ∂ µ γ ij = 0, we have (3.6) k 2 , k 2 V , k 2 T ∈ {γ ij k i k j } k i ∈N . Equation (3.6)
is an immediate consequence of decompositions into Fourier series.
After a periodic identification of t with period T given by (3.2), the metric (3.2) becomes a smooth rotation-invariant conformally-compactifiable metric on R 2 , with a smooth center of rotation atř = 0 (equivalently, at r = r 0 ).
Chasing through the definitions of [16] we show in Appendices D.1, D.2, and K, using the notation there, that for linearised solutions which are in L 2
we have H T = O(ρ n+1 ), |f a |g = O(ρ n+1 ), |f ab |g = O(ρ n+1 ), etc., resulting in Φ S,I = O(ρ n/2−1 ), n > 2; o(1), n = 2, (3.7) Φ V,I = O(ρ n/2 ) , (3.8) Φ T,I = O(ρ n/2+1 ) . (3.9)
3.1. Vanishing via the maximum principle. We want to prove the vanishing of the master functions for L 2 -elements of the kernel. The simplest case occurs when the potentials in (3.3)-(3.6) are non-negative. Since all the Φ i,I 's tend to zero as r tends to infinity, the vanishing of the corresponding master potential follows from the maximum principle. This leads to rigorous statements for restricted dimensions and masses. For the purposes of the proof of Theorem A, the main conclusions of the analysis that follows in this section are:
Proposition 3.1. Let h be an element of the L 2 -kernel of the operator P L defined in (2.10). Then:
(1) The associated tensor master functions vanish.
(2) The associated scalar master functions vanish if n = 2 and either K = 0 or K = 1 and l ≥ 2.
Remark 3.2. Note that the above concerns only the master modes, i.e. those which are controlled by the master functions Φ i,I . The scalar modes with k = 0 (in particular, the modes corresponding to the variation of mass), and the modes K = 1 and l = 1 (which include the variation of angular momentum) require separate attention.
Since we have already shown decay of the master functions, it remains to analyze positivity of the potentials occurring in the master equations. We have not attempted an exhaustive analysis of this, but we certainly have positivity under the following circumstances, keeping in mind that we are working in the region where f > 0:
• for V T,k T .
• When n = 2 the formula (3.3) for V S,k simplifies to (3.10)
V S,k = 1 3 6µ r 3 + k 2 −2 r 2 + 2 2 (k 6 −3k 4 +4)+216µ 2 ) (6µ+(k 2 −2)r) 2 , K = 1; V S,k = 1 3 6µ r 3 + k 2 r 2 + 2 2 k 6 +216µ 2 2 (6µ+k 2 r) 2 , K = 0; V S,k = 1 3 6µ r 3 + k 2 +2 r 2 + 2 2 (k 6 +3k 4 −4)+216µ 2 2 (6µ+(k 2 +2)r) 2 , K = −1.
So V S,k ≥ 0 if n = 2 and either K = 0, or K = −1 and µ ≥ 0, or K = 1 and l ≥ 1. Further, when K = 0 and n > 4 one also finds V S,k ≥ 0 when 0 < µ < µ(n) n−1 for a function µ(n) which solves a polynomial equation. Indeed, by inserting µ = 0 into (3.3), we see that for n > 4 the scalar potential is positive for small masses. As we also know that it is positive at large r, we find that the limiting value of µ is reached when V S,k has a minimum with value zero, i.e. when the resultant of the denominators of V S,k and ∂ r V S,k , after dividing each by a suitable power of r corresponding to the root r = 0, has a positive solution. This resultant is a polynomial in µ and k. Now, V S,k is positive for all r and sufficiently large k at any fixed value of µ, so the problem is solved by choosing the smallest zero of a finite number of polynomials in µ parameterised by k. Similarly when K = 1 and n ∈ {4, 5}. Note that
(3.11) V S,k → r→∞ V S,k (∞) := (n − 2)(n − 4) 4 2 ,
so that the dimensions n = 3, 4 (thus, spacetime dimensions five and six) require different considerations in any case.
• It holds that V V,k V ≥ 0 if (3.12) n(n+1) 2 (2µ 1−n ) 2/(n+1) ≤ k 2 V , K = 0
; l ≥ 2 and 0 < µ < , K = 1 and n = 2; l ≥ 2 and 0 < µ < 2 9 2 , K = 1 and n = 3;
k V ≥ k V,1 (n) and 0 < µ < µ 1 (k V , n) n−1 , K = 1 and n ≥ 4; k V ≥ k V,−1 (n) and µ min < µ < µ −1 (k V , n) n−1 , K = −1,
for some functions k V,K (n) and µ K (k V , n) > 0, where µ min is given by (2.9).
3.2.
Vanishing using the bottom of the spectrum. In this section we will prove further vanishing theorems for the master potentials by studying the first L 2 -eigenvalue λ 1 , and more precisely the kernel, of Schrödinger operators ∇ * ∇ + V with a smooth potential V and an asymptotically hyperbolic metric 2g on R 2 ,
2g = dř 2 +ř 2 H(ř)dt 2 = dr 2 f + f dt 2 ,
where, as before,
f = r 2 2 + K − 2µ r n−1 > 0 , df dr = 2 r 2 + (n − 1)
µ r n > 0 , withř is as in (2.3). Recall that r −s is in L 2 if and only if s > 1/2. Using Theorem C of [20] we have the following: Lemma 3.3. Let V be a smooth potential on R 2 . We assume that −∆ + V has a non-trivial indicial interval with smallest characteristic index s − . If −∆ + V has no L 2 -kernel, then this last operator has no non trivial function of order o(r −s − ) at infinity in its kernel.
We would like to apply this lemma to (3.1). We note the following indicial exponents for the master equations, which turn-out to depend only on the type of the mode: in obvious notation,
s S ± = 1 ± 1 + (n − 2)(n − 4) 2 , (3.13) s V ± = 1 ± 1 + n(n − 2) 2 , (3.14) s T ± = 1 ± 1 + n(n + 2) 2 . (3.15)
Remark 3.4. Note that for n = 3 we have s S ± = 1 2 so we can not use directly Theorem C of [20] as in Lemma 3.3, but this weight 1/2 is the critical weight to be in L 2 so the conclusion of Lemma 3.3 remains true if the replace o(r −s − ) with O(r −s − −ε ) for some ε > 0, which is the case in our applications.
We now study the L 2 -spectrum of our Schrödinger operators.
Lemma 3.5. Let X be a vector field in W 1,1 loc . The first L 2 -eigenvalue of ∇ * ∇ + V acting on functions is bigger or equal than the almost-everywhereinfimum of
(3.16)Ṽ := ∇ i X i − |X| 2 + V . Proof. Let δ X ∈ R ∪ {−∞} be the a.e.-infimum ofṼ . For any function u smooth with compact support we have ∇ i (u 2 X i ) = 0, so |∇u| 2 + |X| 2 u 2 ≥ −2 uX i ∇ i u = u 2 ∇ i X i .
We thus obtain
u(∇ * ∇ + V )u ≥ Ṽ u 2 ≥ δ X u 2 .
In order to apply Lemma 3.5, we need to find a vector field X ∈ W 1,1 loc such that we have, weakly, We look for X of the form X = S∂ r where S is a function on R 2 , then
V = div X − |X| 2 + V ≥ 0 ,(3.17)Ṽ = div X − |X| 2 + V = ∂ r S − S 2 f + V .
This should be compared with the Ishibashi-Kodama modified potential [12]Ṽ which coincides with ours up to a factor f (similarly to the potentials of the master equations). So whenever theirṼ is non negative with their choice of S, we obtain positivity of ourṼ by taking X = S∂ r . We can then apply Lemma 3.5 after verifying that X belongs to W 1,1 loc . This works very well for vector modes in any dimension, leading to:
Corollary 3.6. Under the condition k 2 V > (n − 1)K, the L 2 kernel of −∆ + V V,k V is trivial for all n ≥ 2 and K ∈ {−1, 0, 1}. We deduce that any function in the kernel which is of order o(r −s V − ) at infinity, where s V − is given by (3.14), is trivial.
Remark 3.7. Corollary 3.6 applies to K = 1 and l ≥ 2, since then we have
k 2 V − (n − 1)K = (l − 1)(l + n) > 0 .
Proof of Corollary 3.6: In [12, Equation (2.21)] one takes S = nf 2r , so we take X = nf 2r ∂ r , which is in W 1,1 loc . With this choice of X, the vector master potential (as in [12,Equation (2.17)
]) V V = 1 r 2 k 2 V − (n − 1)K + n(n + 2) 4 f − n 2 r df dr , is transformed toṼ V = 1 r 2 k 2 V − (n − 1)K > 0 .
Whence the triviality of the kernel, as claimed.
Let us turn our attention to the scalar potential. The following corollary of Lemma 3.5 gives an alternative proof of point (2) of Proposition 3.1, and extends that last proposition to the case K = −1 (cf., however, Remark 3.2):
Corollary 3.8. Under the condition k 2 > max(0, 2K), the L 2 kernel of −∆ + V S,k is trivial for n = 2 for any K = −1, 0, 1. We deduce that any function of order o(1) at infinity and in the kernel is trivial.
Proof. We use the function S chosen in [17,Equation (6.23)
], where here we have n = 2, Q = δ = 0, H = h − = H − = m + 6µ r , m = k 2 − 2K, that is S = f H dH dr , so X = S∂ r is in W 1,1 loc . Our scalar potential V S becomes (see [17, Equa- tion (6.24)] with a factor f removed) V S = k 2 m r 2 H , which is positive if k = 0 and m = k 2 − 2K > 0.
To continue, set
(3.18) λ := inf r≥r 0 d √ f dr ∈ {1}, K ∈ {−1, 0}; (0, 1), K = 1.
We have:
Corollary 3.9. The L 2 first eigenvalue of ∇ * ∇ acting on functions is bigger or equal than λ 2 /(4 2 ).
Proof. We choose X = c √ f ∂ r ∈ W 1,1 loc ,
where the positive constant c will be chosen later. It follows that |X| = c and div 2 is decreasing (recall that we have assumed (2.9) when K = −1) and tends to c 2 −2 , but for K = 1, this function has a positive infimum less than
X = c∂ r ( √ f ) > 0, with (div X) 2 = c 2 (∂ r f ) 2 4f = c 2 1 2 + (n − 1) µ r n+1 2 1 2 + K 1 r 2 − 2µ r n+1 −1 , where 2µ = r n+1 0 / 2 + Kr n−1 0 , r 0 > 0, r ≥ r 0 . If K = 0, −1 the function (div X)c 2 −2 , say λ 2 c 2 −2 . Thus div X − |X| 2 ≥ λc −1 − c 2 .
The right-hand side is maximized by c = λ −1 /2, giving div X − |X| 2 ≥ λ 2 /(4 2 ).
To apply Corollary 3.9, note that the L 2 -kernel of −∆ + V will be trivial if V + λ 2 /(4 2 ) ≥ 0 and is strictly positive somewhere. More generally, from the proof above with X = λ 2 √ f ∂ r , this kernel will be trivial if (1) Let K ∈ {0, 1}. There exists a function µ(K, n) > 0 such that for 0 < µ < µ(K, n) the corresponding scalar master functions vanish. (2) Let K = −1, and let λ 1 > 0 be the first non-zero eigenvalue of the scalar Laplacian of ( n N, h K ). There exists a function µ(λ 1 , n) > µ min and an ε(n) > 0 such that for λ 1 ≥ ε(n) and µ min < µ < µ(λ 1 , n) the corresponding scalar master functions vanish.
(3.19)Ṽ = V + λ 2 d √ f dr − λ 2 ≥ 0,
Remark 3.11. By a result of Schoen [24] we have, for the case K = −1, the lower bound λ 1 ≥ (n − 1) 2 /4 under the assumption that the volume of n N is sufficiently small, as described in detail there. We have checked in dimensions 2 ≤ n ≤ 10 that Schoen's estimate is sufficient to obtain µ(λ 1 , n) > 0, we give approximate values of an upper bound of the µ's allowed in Table 1. Thus the result is not empty.
The actual values of ε(n) are lower than this estimate, we give approximate values for dimensions 2 ≤ n ≤ 10 in Table 2.
Similarly to the discussion in the paragraph following (3.10), Equation (3.19) can be used to determine µ(K, n) as a root of a high order polynomial. These polynomials grow with dimension, being extremely long already in n = 4, so that they cannot be usefully displayed here.
As an illustration, we list approximate numerical values of µ(K, n) for dimensions n ≤ 10 in Table 1 and the corresponding values of r 0 in Table 3.
Let us return to the definition (3.17) ofṼ . It is tempting to try to choose S so thatṼ ≥ 0 globally, thus solving the equation
(3.20) ∂ r S − S 2 f + V =Ṽ ≥ 0 ,
for some given functionṼ (r) which is positive in an open set. For this, we numerically solved the ODE (3.17) for S using Mathematica, with V S set to zero, with initial value S(r 0 ) = 0. The integration was performed using the StiffnessSwitching method and a WorkingPrecision of 30-100 depending on the value of µ. If the resulting solutions have a zero at a point r S,0 such that the potential V S is positive for all r ≥ r S,0 , we extend S for r ≥ r S,0 by setting it to be equal to zero there, giving a non-negativeṼ S for all r > r 0 andṼ S > 0 for r ≥ r S,0 . This procedure works for all attempted mass parameters and dimensions n ≥ 5 for K = 0, 1, and all positive masses Table 1. For n−1 µ between n−1 µ min (where µ min is the lowest value giving positive r 0 , i.e. µ min = 0 for K ∈ {0, 1} and µ min < 0 for K = −1, given by (2.9)) and the values shown here we obtain trivial L 2 kernel of −∆ + V S . For K = −1 we have used the fact that the first non-zero eigenvalue of the Laplacian on n N is ≥ (n − 1) 2 /4, which holds by [24] under the assumption that the volume of n N is sufficiently small, cf. [24] for details. For the cases marked "-" there is no µ > 0 such that the inequality (3.19) is satisfied for all relevant k (i.e. those not covered by the linearised Birkhoff theorem or the discussion of l = 1 modes in appendix G). Table 2. This table shows approximate values of the function ε(n), appearing in Proposition 3.10, obtained from the inequality (3.19). 0.16 -1.14 10 0.14 -1.15 Table 3. This table is similar to Table 1, except that we list the value of r 0 , the largest zero of f , rather that the value of µ. Thus, for r 0 smaller than the values shown here we obtain a trivial L 2 kernel of −∆ + V S with = 1.
n K = 0 K = 1 K = −1 2 ∞ ∞ ∞ 3 1/48 - 1/24 4 4.4 × 10 −3 3.5 × 10 −1 0.11 5 4.7 × 10 −4 2.2 × 10 −2 0.12 6 3.9 × 10 −5 - 0.19 7 2.7 × 10 −6 - 0.26 8 1.6 × 10 −7 - 0.35 9 8.6 × 10 −9 - 0.45 10 4.0 × 10 −10 - 0.57n K = 0 K = 1 K = −1 2 ∞ ∞ ∞ 3 0.
for K = −1. Indeed, we tested masses in all orders of magnitude between the limits in Table 1 and µ = 10 30 (with = 1) for dimensions n = 5, 6, 100, and at least 5 values of the mass parameter within this range for each dimension 5 ≤ n ≤ 100.
In dimension n = 4, for K = 1, this construction of S works for µ small enough, giving a larger mass range than that obtained from (3.19) (up to µ ≈ 1.5 for = 1). A typical result for S andṼ S is shown in Figure 1. In dimensions n = 3 and 4 for all K, and for K = −1 with negative mass parameter in all dimensions, the numerical solution S obtained in this way does not have a zero (except for K = 1 with low mass) but appears to exist globally. Since we need positivity ofṼ somewhere to conclude, we instead numerically solved the ODE (3.17) withṼ = εr −2 for ε > 0 (for µ > 0 we can choose ε = 1, for µ < 0 the choice of ε depends on µ), with again initial value S(r 0 ) = 0. The resulting numerical solution S appears to grow asymptotically linearly for large r and we therefore cannot set it to zero at some r S,0 , as a negative jump in S would add a negative distributional component toṼ . Instead we just use the solution directly, without cutting off at finite distance. This works in fact for all masses, dimensions, and K = 0, 1, −1, but the results are less conclusive as one has then to rely on the behaviour of the solution up to r = ∞, while the numerical integration necessarily stops at some finite r. A typical result for S andṼ S for this approach is shown in Figure 2.
In any case, our numerical experiments strongly hint at the following conjecture:
Conjecture 3.12. There are no non-trivial scalar master modes associated with the L 2 -kernel of the operator P L given by (2.10).
Gauge vectors for vanishing master functions
In this section we study the consequences of the vanishing of the master functions. We consider perturbations of the (n + 2)-dimensional metric
g = g ab dx a dx b + r 2 γ ij dx i dx j ,
where g ab dx a dx b = f dt 2 + f −1 dr 2 , and where γ is the metric of an n-dimensional manifold with constant sectional curvature K. The indices a, b take values t, r, while i, j are "angular" indices. We will denote by D the covariant derivative associated to g and by D that of γ.
As emphasised in [16], a general metric perturbation h can be split into "scalar", "vector", and "tensor" parts as
(4.1) h = h S + h V + h T .
An analytic description of the splitting (4.1), without referring to a modedecomposition, is presented in Appendix C. We show in Appendix A below that the T T conditions are consistent with the splitting above.
The components in (4.
h S ab = I f S ab,I S I , h S ai = I rf S a,I S I i , h S ij = I 2r 2 (H S L,I γ ij S I + H S T,I S I ij ) , (4.2) h V ab = 0 , h V ai = I rf V a,I V I i , h V ij = I 2r 2 H V T,I V I ij , (4.3) h T ab = 0 , h T ai = 0 , h T ij = I 2r 2 H T T,I T I ij ,(4.4)
where the following holds: for K = 0, the above are obvious decompositions in Fourier series, with coordinate-independent tensor components leading to zero eigenvalues. For K = 1 (see, e.g. [8])
( ∆ n + k 2 )S I = 0 , k 2 = l(l + n − 1) , l = 0, 1, 2, . . . , (4.5)
( ∆ + k 2 V )V I i = 0 , k 2 V = l(l + n − 1) − 1 , l = 1, 2, . . . , (4.6) ( ∆ + k 2 T )T I ij = 0 , k 2 T = l(l + n − 1) − 2 , l = 2, . . . , n > 2 . (4.7)
For K = −1 there is no example of compact quotient of H n with explicit values of the whole spectrum, but we have a countable set of increasing eigenvalues of finite multiplicity:
k 2 = (λ 0 = 0), λ 1 , . . . , k 2 V = λ V,0 , .
. . , where λ V,0 ≥ (n − 1) by non-negativity of the Hodge Laplacian (2.13). The first non zero eigenvalue λ 1 is greater or equal than a computable constant [24] that can be chosen to be (n − 1) 2 /4 for sufficiently small volumes.
Whatever the value of K ∈ {−1, 0, 1} one sets 12) and denote by I S and I V their complements.
S I i = − 1 k D i S I , k = 0 , (4.8) S I ij = 1 k 2 D i D j S I + 1 n γ ij S I , k = 0 , (4.9) V I ij = − 1 2k V ( D i V I j + D j V I i ) = − 1 2k V L V I γ ij , k V = 0 .I V = {I | k V (I) > 0, k 2 V (I) > (n − 1)K} ,(4.
Note that k V never vanishes when K ∈ {±1}, and that those vector fields V I j for which l(I) in (4.6) equals one, are Killing vector fields of γ ij (cf., e.g., the eigenvalue λ 1 1 for the Hodge Laplacian in [8, Theorem 3.1]), so that the associated tensor V I ij vanishes. From now on we assume that the master potentials Φ S , Φ V , and Φ T vanish.
The vanishing of the tensor potential directly gives h T = 0 [16, (5.4)].
The vanishing of the vector potential implies [16, (5.10)-(5.13)], for modes such that k V = 0,
(4.13) f V a,I = − r k V D a H V T,I .
Let us define the vector field Y V as (4.14)
Y V := −r 2 I∈I V H V T,I k V (I) V I j γ ij ∂ i .
The convergence of the series is justified in Appendix K.
Let us denote by h V ij and h V ia tensors in which we have collected all those modes in h V ij and h V ia which are governed by the master equations, and by h V ij and h V ia whatever remains:
h V ai = I∈I V rf V a,I V I i , (4.15) h V ij = I∈I V 2r 2 H V T,I V I ij (4.16) h V ai = I∈I V rf V a,I V I i , (4.17) h V ij = I∈I V 2r 2 H V T,I V I ij (4.18)
From (K.10) and (K.15), Appendix K, we obtain
(4.19) D a (r −2 h V ij )dx i dx j H k ( n N ) ≤ 2r −2 h V ai dx i H k+1 ( n N ) , (4.20) h V ij = h V ij + r 2 (L Y V γ) ij = h V ij + (L Y V g) ij .
Using (4.13), the mixed components h V ai take the form (4.21)
h V ai = I rf V a,I V I i = h V ai − I∈I V r 2 k V V I i D a H V T,I = h V ai +D a Y V i = h V ai +(L Y V g) ai
(note thatg ij γ jk = r 2 δ k i ). We therefore have (4.22) h
V = h V + L Y V g .
We continue with the scalar variations. We define Using (4.24), the angular part of the variation can be written as
h S ij = I∈I S 2r 2 H S T,I k 2 D i D j S I − γ ij S I 1 r X(r) = r 2 L Y S γ ij + 2rY S (r)γ ij = L Y S (r 2 γ ij ) ,(4.27)
where (4.28)
Y S := I∈I S 1 k 2 H S T,I γ ij D j S I ∂ i − S I X a ∂ a .
The convergence of all series above can be justified in a way very similar to that of Appendix K, and will be omitted. Using (4.26) and (4.8) the mixed part is given by
h S ai = I∈I S D i S I −X a,I + r 2 k 2 D a H S T,I = D i Y S,a + D a Y S,i = (L Y S g) ai .
(4.29)
The r, t part of the variation is finally, using (4.25),
(4.30) h S ab = I∈I S f S ab,I S I = − I∈I S S I (L Y S g) ab = (L Y S g) ab .
In conclusion, we have
(4.31) h = h + L Y g , with (4.32) Y = Y S +Y V = I∈I S 1 k 2 H S T,I D i S I ∂ i − S I X a I ∂ a − I∈I V H V T,I k V V I .
Using the estimates in Appendix D.1 we obtain for the asymptotics of Y
|Y | 2 =g tt ((Y ) t ) 2 +g rr ((Y ) r ) 2 + r 2 |(Y ) i | 2 γ = O(r −2−2n+4 ) + O(r 2−2−2n ) + O(r 2−2−2n ) = O(r 2−2n ) .
(4.33)
Triviality of the kernel
We are ready now to pass to the proof of non-degeneracy:
Theorem 5.1. Consider an (n+2)-dimensional Riemannian Kottler metric g as in (2.1), with µ > 0 and an axis of rotation at r = r 0 > 0, as descrbied in Section 2. Suppose that (1) K = 0, n ≥ 2, µ as in Proposition 3.10; or (2) K = 1, n = 2, µ = µ c given by (2.19); or
(3) K = −1, n = 2.
Then (R 2 × n N,g) is non-degenerate in the sense described in the Introduction.
Proof. Let h ∈ L 2 be in the kernel of the operator P L . Consider the decomposition of h into master scalar, vectorial, and tensor modes, together with their non-master counterparts:
(5.1) h = h S + h S h S + h V + h V h V + h T h T ,
(note that all tensorial modes are controlled by master functions).
Suppose, first, that K = 0. We show in Appendix H below that all angleindependent modes h of h (in particular, all non-master modes) are pure gauge:
(5.2) h := h S + h V = L Y g , |Y |g = O(r −3 ) .
It follows from Proposition 3.10 together with the analysis in Section D.2 that, for n ≥ 2, all master modes are likewise pure gauge for a nontrivial range of mass parameters µ (for all µ > 0 when n = 2):
(5.3) h := h S + h V + h T = L Y g , |Y |g = O(r 1−n ) . Thus (5.4) h = L Yg , Y := Y + Y , |Y |g = O(r −1 ) .
Now, h in (4.31) is in T T -gauge, and the operator obtained by composing the divergence and the trace-free part of the Lie derivative is precisely the operator L * L considered in [20,Proposition G]. Keeping in mind that our n is shifted by one as compared to the parameter n used in [20], from [20, Proposition G] we find that the indicial radius of L * L is R = n+3 2 . From [20, Theorem C(c)], choosing δ there so that the Sobolev space C k,α δ contains only decaying fields, we see that any element Y in the L 2 -kernel of L * L has to decay at infinity at a rate as close to the lower end of the indicial interval as desired. (In fact a more careful analysis shows that an element of the L 2 -kernel must decay as
(5.5) |Y |g = O(r −( n+1 2 +R) ) = O(r −2 )
.) Since the L 2 -kernel of L * L is the same as the L 2 -kernel of L, one can invoke [2, Proposition 6.2.2] to conclude that the L 2 -kernel of L * L is always trivial.
Next, (5.4) shows that for our gauge field Y it holds that Y ρ δ is in L 2 for δ > 3−n 2 ; choosing δ ∈ ( 3−n 2 , n+3 2 ) we can thus use [20, Theorem C(b)] to conclude that Y ≡ 0. Hence (5.6) h ≡ 0 , which concludes the proof when K = 0. The argument for K = ±1 is very similar: when n = 2 there are no nonmaster tensor modes; the fact that non-master scalar modes are pure gauge is the contents of the linearised Birkhoff theorems of Appendices F and I, while the pure-gauge character of the l = 1, K = 1 modes is established in Appendix G.
Appendix A. Divergence of a symmetric two-tensor in coordinates
The object of this appendix is to show that the T T condition does not mix the scalar, vector, and tensor modes.
Let us consider a warped product metric of the following form (note that this allows for metrics more general than the metricg of the main body of the paper):
G = g ab (y)dy a dy b + r 2 (y)γ ij (x)dx i dx j ,
where the y a 's are local coordinates on a m-dimensional manifold and the x i 's are local coordinates in n-dimensions. The non trivial Christoffel symbols of G are:
Γ c ab = Γ c ab (g) , Γ k ij = Γ k ij (γ) , Γ c ij = −g cb r∂ b r γ ij , Γ k ib = r −1 ∂ b r δ k i . The divergence of a 2-tensor is ∇ α T αβ = ∂ α T αβ + Γ α ασ T σβ + Γ β ασ T ασ , or equivalently ∇ α T αβ = ( |G|) −1 ∂ α ( |G|T αβ ) + Γ β ασ T ασ , where |G| = |g| |γ|r n . We deduce ∇ α T αβ = ( |g|) −1 r −n ∂ a (r n |g|T aβ ) + ( |γ|) −1 ∂ i ( |γ|T iβ ) + Γ β ασ T ασ . In particular ∇ α T αb = ( |g|) −1 r −n ∂ a (r n |g|T ab )+( |γ|) −1 ∂ i ( |γ|T ib )+Γ b ac T ac −g bc r∂ c rγ ij T ij , so (A.1) ∇ α T αb = r −n D a (r n T ab ) + D i (T ib ) − g bc r∂ c rγ ij T ij ,
where D is the γ-connection and D is the g-connection.
Similarly
∇ α T αj = ( |g|) −1 r −n ∂ a (r n |g|T aj )+( |γ|) −1 ∂ i ( |γ|T ij )+Γ j kl T kl +r −1 ∂ b rT bj , so (A.2) ∇ α T αj = r −n D a (r n T aj ) + D i (T ij ) + r −1 ∂ b rT bj .
A.1. Divergence of a scalar variation. Let us write, as in [16, Equation (2.4)],
T ab = Sf ab , T ai = r −1 f a S i , T ij = 2r −2 (H L Sγ ij + H T S ij ) ,
Note that here S i = γ ij S j but T ai =g ij T a j . From (A.1) and (A.2)
∇ α T αa = Sr −n D b (r n f ab ) + r −1 f a D i S i − g ac ∂ c rγ ij 2r −1 (H L Sγ ij + H T S ij ) ∇ α T αj = r −n D b (r n−1 f b )S j + 2r −2 (H L D j S + H T D i S ij ) + r −2 ∂ b rf b S j
Assume moreover that (see [16], Equations (2.2) and (2.6))
∆ γ S = −k 2 S , S i = − 1 k ∂ i S , S ij = 1 k 2 D i ∂ j S + 1 n γ ij S, We obtain ∇ α T αa = Sr −n D b (r n f ab ) + r −1 f a kS − g ac ∂ c rγ ij 2r −1 (H L Sγ ij + H T S ij ) , = S[r −n D b (r n f ab ) + r −1 f a k − 2r −1 g ac ∂ c r nH L ] , ∇ α T αj = r −n D b (r n−1 f b )S j + 2r −2 (H L D j S + H T D i S ij ) + r −2 ∂ b rf b S j , = D j S − 1 k r −n D b (r n−1 f b ) + 2r −2 H L + H T (−1 + (n − 1)K k 2 + 1 n ) − 1 k r −2 ∂ b rf b .
A.2. Divergence of a vector variation. If, we have
T ab = 0 , T ai = r −1 f a V i , T ij = 2r −2 H T V ij ,
then from (A.1) and (A.2)
∇ α T αa = r −1 f a D i V i − g ac ∂ c rγ ij 2r −1 H T V ij , ∇ α T αj = r −n D b (r n−1 f b )V j + 2r −2 H T D i V ij + r −2 ∂ b rf b V j .
Assume moreover that (see [16], Equations (5.7a), (5.7b) and (5.9))
∆ γ V = −k 2 V V , V ij = − 1 2k V ( D i V j + D j V i ) , D i V i = 0 , where k V = 0, then ∇ α T αa = 0 , ∇ α T αj = V j r −n D b (r n−1 f b ) − 1 k V r −2 H T [−k 2 V + (n − 1)K] + r −2 ∂ b rf b .
A.3. Divergence of a tensor variation. If we assume that
T ab = 0 , T ai = 0 , T ij = 2r −2 H T T ij ,
then from (A.1) and (A.2)
∇ α T αa = −g ac ∂ c rγ ij 2r −1 H T T ij , ∇ α T αj = 2r −2 H T D i T ij .
Assume moreover that (see [16] equation (5.1a) and (5.1b))
γ ij T ij = 0 , D i T ij = 0 , then ∇ α T αµ = 0 . Appendix B. Divergence & double divergence of h S ij and h S aj
The aim of this appendix is to derive some divergence identities, as implicitly used in Appendix C.
We start by calculating the divergence of S I jk . Assuming k = 0 it holds that
D i (γ ij S I jk ) = 1 k 2 γ ij D i D j D k S I + 1 n D k S I = − D k S I + 1 k 2 γ m R mk D S I + 1 n D k S I = 1 n − 1 + (n − 1)K k 2 D k S I ,
where we have used R ij = (n − 1)Kγ ij and
∇ i ∇ j ∇ k f = ∇ k ∇ j ∇ i f + R jki ∂ f .
The double divergence of S I jk is then Similarly, for V I m , assuming k V = 0,
D D i (γ k γ ij S I jk ) = − k 2 1 n − 1 + (n − 1)K S I .D j (γ j V I m ) = − 1 2k V γ j D j D V I m + D j D m V I = − 1 2k V −k 2 V V I m + D m γ j D j V I + R j jm γ i V I i = 1 2 V I m (k V + (n − 1)K) ,
where we have used D j (γ jk V k ) = 0 [16, (5.7b)], and
D i D j (γ mi γ j V I m ) = (k V + (n − 1)K) D i (γ mi V m ) = 0 . Therefore (B.3) D i (γ ij δg V jk ) = I r 2 H I T,V V I m (k V + (n − 1)K) , and (B.4) D D i (γ k γ ij δg V jk ) = 0 .
Appendix C. The decomposition of h
In this appendix we wish to justify the decomposition (4.1) of h. Recall that we denote by x a the coordinates t and r, and by x i the coordinates on the compact boundaryless Riemannian manifold (N, γ). The metric functions h ab form obviously a family of t-and r-dependent scalar functions on N , similarly for the γ-trace γ ij h ij of h µν . Next, the fields h ai dx i form a family of t-and r-dependent covectors on N , while h ij dx i dx j forms a similar family of tensor fields on N . Removing the γ-trace of h ij dx i dx j , one obtains a family of trace-free tensor fields on N .
We have the standard two L 2 -orthogonal decompositions for vector or covector fields, and for trace-free symmetric two-tensor fields (see e.g. [3]):
C ∞ (N, T N ) = Im d ⊕ ker d * ,
C ∞ (N,S 2 N ) = ImL ⊕ ker div, whereL is the conformal Killing operator,
L(W ) ij = D i W j + D j W i − 2 n D k W k γ ij .
This allows to write any covector field uniquely as
W i = W S i + W V i , W S ∈ Im d , W V ∈ ker d * .
This decomposition can be applied to the fields h ai dx i .
Next, any trace free symmetric tensor field u ij dx i dx j can be written in unique way as
u ij = u S ij + u V ij + u T ij , u S ∈L(Im d) , u V ∈L(ker d * )
, u T ∈ ker div . We apply this last decomposition to the γ-trace-free part of h ij dx i dx j which, together with what been said above, results in (4.1).
Note that (minus) the divergence of u is also decomposed as
D j u ij = D j u S ij + D j u V ij . Next, if we use (compare Appendix B) divL = D * D g − Ric + n−2 2 dd * = ∆ Hodge − 2 Ric + n − 2 2 dd * = d * d + n 2 dd * − 2 Ric,
and if the metric γ is Einstein, we have div u S ∈ Im d , div u V ∈ ker d * .
More precisely, if Ric = K(n − 1) and u S =LdS then
div u S = d n 2 (d * dS) − 2K(n − 1)S , and if u V =LV , with d * V = 0 then div u V = d * dV − 2K(n − 1)V.
Appendix D. The asymptotics of scalar modes Whatever the value of k, for large r the potentials V S,k tend to
V S,k (∞) = (n − 2)(n − 4) 4 .
We find the characteristic exponents of the master equation by solving the equation
s(1 − s) + V S,k (∞) = 0, giving s ± = 1 ± |n − 3| 2 .
This has to be compared with the asymptotics
(D.1) Φ S,I = O(ρ (n−2)/2 ) ,
as obtained by directly translating the asymptotic behaviour of elements of the L 2 -kernel of the linearised Einstein operator into the asymptotic behaviour of the master fields. For n ≥ 4 the decay (D.1) corresponds to s + . For n = 3 there is only one index, implying logarithmic terms in an asymptotic expansion. For n = 2 the decay (D.1) corresponds to s − , which is zero, but we will improve this asymptotics below.
D.1. Estimate of Φ S,I in any dimension n. From |h|g = O(ρ n+1 ) we obtain for the components in (t, r, z j ) coordinates (j = 2, . . . , n + 1)
(D.2) h tt = O(r 1−n ) , h tr = O(r −1−n ) , h rr = O(r −3−n ) , h tj = O(r 1−n ) , h rj = O(r −1−n ) , h jk = O(r 1−n ) . By (4.2) f S tt,I = O(r 1−n ) , f S tr,I = O(r −1−n ) , f S rr,I = O(r −3−n ) , f S t,I = O(r −n ) , f S r,I = O(r −2−n ) , H S L/T,I = O(r −1−n )
. Then we have for X a,I , defined in (4.26), assuming k = 0, D.2. Estimate of Φ S in dimension n = 2. We revisit the preceding equations when n = 2. From [16, (2.11),(2.13)] we see that
X t,I = O(r 1−n ) , X r,I = r k (f r + r k D r H T ) = r k O(r −2−n ) + r k O(r −2−n ) = O(r −n ) ,F = F , F a a = 0 . Let us write H T = h t r −3 + O(r −3−ε ) , D r H T = −3h t r −4 + O(r −4−ε ),
where ε > 0. From the estimates in Section D.1,
X r = −3 1 k 2 h t r −2 + O(r −2−ε ) , F = −3 1 k 2 h t r −1 + O(r −1−ε ) F tt = O(r −1 ) , F tr = O(r −2 ), and F rr = 2D r (r 2 D r H T ) + O(r −3−ε ) = 12h t r −3 + O(r −3−ε )
. From F a a = 0 we see that h t = 0, so F and then X + Y decay faster. We deduce from the definitions of X and Y that there exists ε > 0 such that
F, X, Y = O(r −1−ε ) .
This information inserted into [16, (2.24d)] gives
Z = O(r −ε ) .
In conclusion Φ S,I = O(r −ε ), tends to zero at infinity.
Appendix E. The asymptotics of vector modes
In order to estimate the rate of decay of Φ V,I we start with [16, (5.10)]
F r = f r + r k V D r H T = O(r −2−n ) + O(r −1−n )) = O(r −1−n ) , F t = f t + r k V D t H T = O(r −n ) .
From [16, (5.12)] we have
r n−1 F a = ε a b D b Ω ,
where ε ab = O(1). Therefore
Appendix F. The linearised Birkhoff theorem
In the case K = 1, n = 2, a gauge transformation h µν → h µν + L Ygµν , with gauge vector Y , takes the form
h tt → h tt + Y r ∂ r f + 2f ∂ t Y t , (F.1) h tr → h tr + f −1 ∂ t Y r + f ∂ r Y t , (F.2) h rr → h rr + Y r ∂ r f −1 + 2f −1 ∂ r Y r , (F.3) h θϕ → h θϕ + r 2 sin 2 θ∂ θ Y ϕ + r 2 ∂ ϕ Y θ , (F.4) h ϕϕ → h ϕϕ + 2r 2 sin 2 θ(r −1 Y r + Y θ cot θ + ∂ ϕ Y ϕ ) , (F.5) h θθ → h θθ + 2rY r + 2r 2 ∂ θ Y θ , (F.6) h tθ → h tθ + f ∂ θ Y t + r 2 ∂ t Y θ , (F.7) h tϕ → h tϕ + f ∂ ϕ Y t + r 2 sin 2 θ∂ t Y ϕ , (F.8) h rθ → h rθ + f −1 ∂ θ Y r + r 2 ∂ r Y θ , (F.9) h rϕ → h rϕ + f −1 ∂ ϕ Y r + r 2 sin 2 θ∂ r Y ϕ . (F.10)
We consider an l = 0 linearised solution h µν of the Einstein equations, i.e.
(F.11) h ab = h ab (t, r) , h ia ≡ 0 , h ij = ψ(t, r)g ij .
We assume that h ∈ L 2 and is in the kernel of the operator P L . We set
(F.12) Y r = rψ/2 = O(r −2 ) , Y i ≡ 0 ,
which implies h ij = L Ygij . We define Y t by integrating (F.2) in r:
(F.13) Y t = − ∞ r f −1 (h rt − ∂ t Y r f −1 )dr = O(r −4 ) ,
so that h tr = L Ygtr . An analysis of the components of L Yg , using (F.1)-(F.10), proves that
|L Yg | 2 g = O(r −6 ), so L Yg ∈ L 2 . Set (F.14) h µν = h µν − L Ygµν ,
thus h µν is a solution of the linearised Einstein equations in L 2 with all components vanishing except possibly h tt and h rr . Now, a variation of the metric arising from a variation δµ of the mass in the coordinate system (2.1) takes the form
(F.15) 2 δµ r (−dt 2 + f −2 dr 2 ) ,
which suggests that it might be convenient to define new functionsh rr and h tt as (F.16)h rr := rf 2 h rr ,h tt := r( h tt + f 2 h rr ) .
Inserting (F.16) into the linearised Einstein tensor G [h] µν one finds
(F.17) G tr [h] = ∂ thrr r (−2µ + r 3 + r) ,
thush rr depends at most upon r. One can now eliminate the second radial derivative ofh tt between the G tt and G rr equations, obtaining
(F.18) ∂ r h tt rf = 0 .
Hence,
(F.19)h tt = C(t)rf .
for some function C depending only upon t. Inserting all this into the G ij = 0 equations gives ∂ rhrr = 0, and thush rr is a constant, say 2δµ. The tensor field h µν is in L 2 if and only if δµ = 0 = C(t). Hence h µν ≡ 0, and so
(F.20) h µν = L Ygµν , |Y |g = O(r −3 ) .
If we denote by h S the l = 0-part of h, and denote by Y S the vector field defined above, we can rewrite (F.20) as
(F.21) h S = L Y S g , |Y S |g = O(r −3 ) .
Appendix G. The l = 1 modes for K = 1, n = 2
In this section we analyze the l = 1 modes when ( n N, γ ij ) is a twodimensional round unit sphere. We follow the treatment of the Lorentzian case in [11], which requires only trivial modifications when adressing the Riemannian setting. We present the argument here because of the need of establishing estimates for the gauge vector fields.
As
(G.1) h (−) µν = 0 r 2 h (l,m) a ε i j ∇ j S (l,m) r 2 h (l,m) a ε i j ∇ j S (l,m) 2r 4 k (l,m) ε (i m ∇ j) ∇ m S (l,m) ,
where h (l,m) a and k (l,m) are functions of t and r, ∇ is the covariant derivative on the sphere, ε ij = sin θ(δ θ i δ φ j − δ φ i δ θ j ) and S (l,m) are the scalar spherical harmonics.
Restricting to l = 1 modes, the ij components of h
Y a = 0 , Y i = r 4 ε ij ∇ j f Y ,
for some function f Y , preserve the odd character of the perturbation and those with
f Y = 1 m=−1 f (m) Y S (l=1,m) ,
stay within the l = 1 modes. The effect of such a gauge transformation on the perturbation is given by [11, (73)]
h (−) ai → 1 m=−1 3 4π (h (l=1,m) a + r 2 ∂ a f (m) Y )r −2 J (m)i ,
with all other components unaffected.
Defining
h (−) by h (−) µν = h (−) µν + L Y −g with a gauge vector Y − defined as (Y − ) a := 0 and (G.3) (Y − ) i = −r 4 ε ij ∂ j 1 m=−1 S (l=1,m) h (l=1,m) r r −2 dr = O(r −4 ) , the components h (−) ri vanish, leaving only h (−)
ti . The norm of the gauge part is found to be |L Y −g| 2 g = O(r −6 ) , as before, and therefore L Y −g ∈ L 2 .
As the h
r 2 ∂ 2 r h (l=1,0) t − 2 h (l=1,0) t = 0 , (G.4) 2∂ t h (l=1,0) t − r∂ r ∂ t h (l=1,0) t = 0 . (G.5) Integrating (G.5) twice, we obtain h (l=1,0) t = C 1 r 2 t + g(r) ,
where C 1 has to vanish because of the periodicity of the t coordinate. Inserting this into (G.4) leads to
h (l=1,0) t = C 2 r 2 + C 3 r −1 ,
where C 2 has to vanish to ensure h tϕ is in L 2 , while C 3 has to vanish because the tensors dtdx i are not smooth at the axis of rotation r = r 0 .
Therefore h = 0, and if we denote by h 1− the odd part of the l = 1 modes, we obtain
(G.6) h 1− = L Y −g , |Y − |g = O(r −3 ) .
G.2. Even perturbations. Even l = 1 solutions of the linearised Einstein equations can be parameterised as [11,Equation (36)]
(G.7) (h (+) αβ ) = h ( =1) ab D i q ( =1) a D i q ( =1) b r 2 2 J ( =1) γ ij .
Under gauge transformations with gauge-vector Y of the form
(G.8) (Y α ) = (Y a , r 2 D i X) , (h (+) αβ ) transforms to ( h (+)
αβ ) given by [11,Equation (122)
] (G.9) h ( =1) ab +D a Y ( =1) b +D b Y ( =1) a D i (q ( =1) a + r 2D a X ( =1) + Y ( =1) a ) D i (q ( =1) b + r 2D b X ( =1) + Y ( =1) b ) r 2 2 γ ij (J ( =1) − 4X ( =1) + 4 r Y a ( =1)D a r)
.
According to [11,Section IV.A.2], it is convenient to define (X, Y a ) by solving the following system of equations:
r 2 D t X ( =1) + Y t = −q ( =1) t = O(r −1 ) , (G.10) r 2 D r X ( =1) + Y r = −q ( =1) r = O(r −3 ) , (G.11) D b Y b = − 1 2g ab h ( =1) ab = O(r −3 ) . (G.12) With this choice, h (+) satisfies (G.13) h (+) ai = 0 ,g ab h (+) ab = 0 . Note that (G.10)-(G.12) imply (G.14) D b (r 2 D b X) = 1 2g ab h ab − D b q b .
The homogeneous version of the equation (G.14) for X has no non-trivial solutions tending to zero at infinity by the maximum principle. The operator at the left-hand side of (G.14) has indicial exponents in {0, −3}, and therefore (G.14) has a unique solution X = O(r −3 ) which is a linear combination of l = 1 spherical harmonics. The conditions (G.13) do not fix the gauge uniquely: an additional gauge transformation satisfying
(G.15) r 2 D a X + Y a = 0 , D a Y a = 0 ,
preserves the form of h (+) . By a rotation we set J = J (1) S (l=1,m=1) . We define new variables C a as (G. 16)
h ( =1,T ) ab = 1 f C ( =1) a D b r + C ( =1) b D a r −g ab C ( =1) d D d r , C ( =1) a = h ( =1,T ) ab D b r,
and decompose them into modes
(G.17) C a = 3 m=1 C (m) a S (l=1,m) .
Eliminating C r between the m = 2, 3 components of G tr [h] and G tφ [h] we obtain
(G.18) ∂ r C (2,3) t + C (2,3) t r − 2µ + r 3 = 0 ,
and therefore, by integration from the axis of rotation at r 0 , the C
C (1) r = 6µ∂ r Z (1) r + r 2 ∂ r J (1) , C (1) t = Z (1) t − 3µ − r 2 ∂ t J (1) − rf 6µ∂ t ∂ r Z (1) r .
Note that this defines Z
r 2 f ∂ r D a (r 2 D a Z (1) r ) + r 4 ∂ r (r −2 f )D a (r 2 D a Z (1) r ) = 0 . (G.21) This implies (G.22) D a (r 2 D a Z (1) r ) = Cr 2 f ,
with a constant C which has to vanish for Z (1) r to be regular at r 0 . We now consider the remaining gauge freedom. We see from (G.15) that for any X satisfying
(G.23) D a (r 2 D a X) = 0
there exists an associated Y a giving a gauge transformation which preserves (G.13).
Inserting the definition of our new variables into (G.9) we find that Z (1) r and J (1) transform as 25) where X and Y a have been split into l = 1 modes as for C a in (G.17).
∂ r Z (1) r → ∂ r (Z (1) r + X (1) ) , (G.24) J (1) → J (1) − 4X (1) + 4f r Y (1) r , (G.
As Z
(1) r satisfies (G.23) we can set it to a constant by a gauge transformation with ∂ r X (1) = −∂ r Z (1) r , which preserves (G.13).
With ∂ r Z (1) r ≡ 0 we see from (G.20) that J (1) can only depend on t. From the remaining equations ∂ 2 t J (1) = 0, i.e. J (1) is constant. We can exploit the remaining freedom in X (1) to set (G.26)
X (1) = J (1) 4 , Y a = 0 ,
obtaining J (1) ≡ 0. This gives Z
(1) r = const and therefore C r ≡ C t ≡ 0. We arrive at h (+) = L Y +g where Y + is the combined gauge vector consisting of the part defined by (G.10)-(G.12), that given by (G.24) and that given by (G.26). From the asymptotics (D.2) of h and from (F.1)-(F.10) with the right-hand sides being zero now, we conclude that
(G.27) h 1+ ≡ h (+) = L Y +g , |Y + |g = O(r −1 ) .
An alternative, and somewhat simpler, proof can be given using [14]. Since the last reference is only available in Polish so far [25], we felt it more appropriate to provide the argument above.
Appendix H. A linearised Birkhoff-type theorem with K = 0
In the case K = 0 the gauge transformations take the form
h tt → h tt + Y r ∂ r f + 2f ∂ t Y t , (H.1) h tr → h tr + f −1 ∂ t Y r + f ∂ r Y t , (H.2) h rr → h rr + Y r ∂ r f −1 + 2f −1 ∂ r Y r , (H.3) h ij → h ij + δ ij 2rY r + r 2 (∂ i Y j + ∂ j Y i ) , (H.4) h ti → h ti + f ∂ i Y t + r 2 ∂ t Y i , (H.5) h ri → h ri + f −1 ∂ i Y r + r 2 ∂ r Y i . (H.6)
We consider a k = 0 = k V = k T linearised solution of the Einstein equations, i.e. one that only depends on t and r, without splitting into scalar, vector, and tensor components:
(H.7) h µν = h µν (t, r) .
We choose a gauge vector, which we denote by Y , to be independent of the "angular" coordinates x i . We define (Y ) i by integrating (H.6) to obtain h ri = L Ygri :
(H.8) (Y ) i = r −2 h ri − f −1 ∂ i (Y ) r dr = r −2 h ri dr = O(r −n−2 )
.
Setting (Y ) r to (H.9) (Y ) r = 1 2nr δ ij h ij = O(r −n ) ,
gives h ij δ ij = δ ij L Ygij . As for the K = 1 case we define (Y ) t = O(r −n−2 ) by integrating (H.2) so that h tr = L Y g tr . As before, equations (H.1)-(H.6) show that
|L Y g| 2 g = O(r −2n−2 ), so L Y g ∈ L 2 .
As in (F.14) we define h µν as h µν − L Y g, thenh rr ,h tt are defined as in (F. 16 as before. Inserting this into δ ij G ij [h] givesh rr = 2δµ, for some constant δµ. As in the K = 1 case, the h rr and h tt terms are in L 2 if and only if δµ = 0 = C(t).
The remaining linearised Einstein equations turn out to be
r 2 ∂ 2 r h ti + r(n − 2)∂ r h ti − 2(n − 1)h ti = 0 , (H.13) 2∂ t h ti − r∂ r ∂ t h ti = 0 , (H.14) ∆2g h ij + nf r ∂ r h ij = 0 . (H.15)
Integrating (H.14) twice gives
h ti = C 1 r 2 t + g(r) ,
where C 1 has to vanish because of the periodicity of the t coordinate. Inserting this into (H.13) leads to h ti = C 2 r 2 + C 3 r 1−n where C 2 has to vanish to ensure h ti is in L 2 , while C 3 has to vanish because the tensors dtdx i are not smooth at the axis of rotation r = r 0 .
Equation (H.15) gives h ij = 0 by the maximum principle, in view of h ij = O(r 1−n ).
Thus h µν ≡ 0 and, if we denote by h the part of h from which all higher modes have been removed (in the notation of (4.2), and setting S 0 (x i ) ≡ 1 for simplicity, (H. 16)
h µν dx µ dx ν = f S ab,0 dx a dx b + rf V ai,0 dx a dx i + 2r 2 (H S L,0 γ ij + H T T,0 T 0 ij )dx i dx j , ) we obtain (H.17) h = L Y g , |Y |g = O(r −n−1 ) .
Appendix I. A linearised Birkhoff-type theorem with K = −1
In the case K = −1 the gauge transformations take the form
h tt → h tt + Y r ∂ r f + 2f ∂ t Y t , (I.1) h tr → h tr + f −1 ∂ t Y r + f ∂ r Y t , (I.2) h rr → h rr + Y r ∂ r f −1 + 2f −1 ∂ r Y r , (I.3) h ij → h ij + δ ij Y r 2r (x n+1 ) 2 − Y n+1 2r 2 (x n+1 ) 3 (I.4) + r 2 x n+1 (∂ i Y j + ∂ j Y i ) , h ti → h ti + f ∂ i Y t + 4r 2 B 2 ∂ t Y i , (I.5) h ri → h ri + f −1 ∂ i Y r + 4r 2 B 2 ∂ r Y i , (I.6)
where we use the form h K = n+1 i=2 (dx i ) 2 /(x n+1 ) 2 of the hyperbolic metric. We consider a k = 0 = k V = k T solution of the linearised Einstein equations in dimension n = 2, i.e. one that only depends on t and r, without splitting into scalar, vector, and tensor components:
(I.7) h µν = h µν (t, r) .
We choose a gauge vector, denoted by Y , in the same way as for the K = 0 case: All components are chosen to be independent of the "angular" coordinates x i . We define (Y ) i by integrating (I.6) to obtain h ri = L Ygri :
(I.8) (Y ) i = r −2 h ri − f −1 ∂ i (Y ) r dr = r −2 h ri dr = O(r −4 ) .
Setting (Y ) r to
(I.9) (Y ) r = ϕ 2 4r δ ij h ij + r 2 ϕ Y ϕ = O(r −2 ) ,
gives h ij δ ij = δ ij L Ygij . The final component, (Y ) t , is obtained by integrating (I.2) so that h tr = L Y g tr , giving (Y ) t = O(r −4 ). As before, equations (I.1)-(I.6) show that
|L Y g| 2 g = O(r −6 ), so L Y g ∈ L 2 .
As in (F.14) we define h µν as h µν − L Y g, thenh rr ,h tt are defined as for K = 0 and 1, In contrast to the K = 0 case, the G ij [h] equations depend on h ti , so we first consider only the G ai [h] equations. These turn out to be
r 2 (2µ + r − r 3 )∂ 2 r h ti + (−4µ + 2r 3 )h ti = 0 , (I.12) 2∂ t h ti − r∂ r ∂ t h ti = 0 . (I.13)
Integrating (I.13) twice gives
h ti = C 1 r 2 t + g(r) ,
where C 1 has to vanish because of the periodicity of the t coordinate. Inserting this into (I.12) leads to
h ti = C 2 f + C 3 (6µ − r)(3µ + r) 2r(27µ 2 − 1) + X
where X is a function that vanishes at r = r 0 . Here C 2 has to vanish to ensure that h ti is in L 2 , while C 3 has to vanish because the tensors dtdx i are not smooth at the axis of rotation r = r 0 . Inserting this into the G ij [h] equations gives
(I.14) ∆2g h ij + 2f r ∂ r h ij = 0 .
and therefore h ij = 0 by the maximum principle, in view of h ij = O(r −1 ). Thus h µν ≡ 0 and, if we denote again by h the part of h from which all higher modes have been removed, we obtain
(I.15) h = L Y g , |Y |g = O(r −3 ) .
Appendix J. The Riemannian Kerr anti-de Sitter metrics
The Riemannian Schwarzschild-anti de Sitter metrics belong to the family of the Riemannian Kerr anti-de Sitter metrics, parameterised with m and an "angular momentum parameter" a. The variations of those metrics with respect to the parameter a provide non-trivial solutions of the linearised Einstein equations at the Schwarzschild-anti de Sitter metric, which we need to analyse. For this is it is convenient to start with a discussion of the family of the Riemannian Kerr anti-de Sitter metrics with small parameter a. Our presentation follows [10], where Kerr-Newman-de Sitter metrics were considered.
In Boyer-Lindquist coordinates, after the replacements a → ia and t → it the Kerr anti-de Sitter metric becomes
g = Σ ∆ r dr 2 + Σ ∆ θ dθ 2 + sin 2 (θ) Ξ 2 Σ ∆ θ (adt + (r 2 − a 2 )dϕ) 2 + 1 Ξ 2 Σ ∆ r (dt − a sin 2 (θ)dϕ) 2 , (J.1)
where, after setting λ = Λ/3, we have Σ = r 2 − a 2 cos 2 (θ) and ∆ r = (r 2 − a 2 ) 1 − λr 2 − 2µr , ∆ θ = 1 − λa 2 cos 2 (θ), and Ξ = 1 − λa 2 .
Keeping in mind that we are interested in the metric for small a, we consider µ > 0 and we assume that the largest zero of ∆ r , which we denote by r 0 , is positive. For r ∈ [r 0 , ∞) we introduce a new coordinate ρ defined as ρ = Smoothness at r = r 0 requires that t defines a 2πω-periodic coordinate, with (J.5) ω := 2Ξ r 2 0 − a 2 ∆ r (r 0 ) .
In order to guarantee regularity near the intersection of the axis {sin θ = 0} with the axis {∆ r = 0}, near θ = 0 we use a coordinate system (ρ, τ, θ, φ), with t = ωτ and φ defined through the formula (J.6) dϕ := αdφ + a a 2 − r 2 0 dt ≡ αdφ + aω a 2 − r 2 0 dτ , for some constants α, ω ∈ R * which will be determined shortly by requiring 2π-periodicity of τ and φ. In this coordinate system the metric takes the form g = Σ dρ 2 + 1 Ξ 2 Σ 2 κ 2 ω 2 Σ 2 4 r 2 0 − a 2 2 1 4 (ρ 2 , sin 2 (θ))ρ 2 dτ 2 + α 2 ∆ θ a 2 − r 2 2 + a 2 ∆ r sin 2 (θ) sin 2 (θ)dφ 2 + F (ρ 2 , sin 2 (θ))ρ 2 sin 2 (θ)dτ dφ + 1 ∆ θ dθ 2 , (J.7)
for some smooth functions 1 4 and F , with 1 4 (0, y) = 1. As is well known, when (ρ, τ ) are viewed as polar coordinates around ρ = 0, the one form ρ 2 dτ and the quadratic form dρ 2 + ρ 2 dτ 2 are smooth. Similarly when (θ, φ) are polar coordinates around θ = 0, the one form sin 2 (θ)dφ and the quadratic form dθ 2 + sin(θ) 2 dφ 2 are smooth. It is then easily inferred that the requirements of 2π-periodicity of τ and φ, together with (J.8) κ 2 ω 2 4Ξ 2 r 2 0 − a 2 2 = 1 , α 2 ∆ 2 θ a 2 − r 2 2 Ξ 2 (r 2 − a 2 cos 2 (θ)) 2 θ=0 ≡ α 2 = 1 , imply smoothness both of the sum of the diagonal terms of the metric g and of the off-diagonal term g τ φ dτ dφ on Ω := {(r, τ, θ, φ) ∈ [r 0 , ∞) × S 1 × [0, π) × S 1 } .
Here [r 0 , ∞) × S 1 is understood as R 2 with center of rotation at r 0 , similarly [0, π) × S 1 is understood as a disc D 2 of radius π.
The above calculations remain valid without changes near θ = π. When θ ∈ (0, π] we will denote by τ and φ the relevant angular coordinates, and ω, α the corresponding coefficients. Thus, for θ ∈ (0, π]:
(J.9) t = ω τ , dϕ = αd φ + a ω a 2 − r 2 0 d τ , with (J.10) ω = ±ω , α = ±1 .
We obtain likewise a smooth metric on the set Ω := {(r, τ , θ, φ) ∈ [r 0 , ∞) × S 1 × (0, π] × S 1 } ≈ R 2 × D 2 .
Disregarding issues of orientations, without loss of generality we can choose the plus signs above. The manifold M , obtained by patching together Ω with Ω, using the obvious identifications resulting from the formulae (J.11) ωdτ = ωd τ , αdφ + aω a 2 − r 2 0 dτ = αd φ + a ω a 2 − r 2 0 d τ ,
is diffeomorphic to R 2 × S 2 .
Note that while dϕ is a well defined one-form on M , the function ϕ is a well defined coordinate-modulo-2π on M if and only if aω a 2 −r 2 0 ∈ Z * . We emphasise that it is not necessary to impose this last restriction to obtain a well defined smooth Riemannian metric on M , and we will not impose it.
Differentiating (J.1) with respect to a in the (ρ, τ, θ, φ) coordinates we obtain dg da a=0 = −2αω The object of this appendix is to justify the convergences of the modedecomposition series in the vector sector. We thus consider the vector projection h V of h, which we decompose into a complete (cf., e.g., [8]) set of vector harmonics V I i :
(K.1) h V ab = 0 , h V ai = r I f V a,I V I i , h V ij = −r 2 I : k V (I) =0 H V T,I 1 k V (I) ( D i V I j + D j V I i ) ,
where k V = k V (I) in the last sum is determined by the corresponding eigenvalue of the vector Laplacian ∆ acting on V I i : (K.2) ∆V I i = −k 2 V (I)V I i . For k ∈ N let H k ( n N ) denote the space of tensor fields on n N of Sobolev regularity with k derivatives. Standard functional analysis shows that we have
(K.3) D i 1 · · · D i V I i L 2 ( n N ) ≈ (1 + k V (I)) ,
where we use ≈ to denote equivalence of norms, hence
(K.4) h V ai dx i 2 H k ( n N ) ≈ r 2 I (1 + k V (I) 2 ) k |f V a,I | 2 .
Similarly, for any j, From now on we assume that Φ V,I vanishes. From (K.7) we then obtain (K.10) D a (r −2 h V ij )dx i dx j H k ( n N ) ≤ 2r −2 h V ai dx i H k+1 ( n N ) , which furthermore justifies the convergence and equality (K.7) in the Φ V,I ≡ 0 case. Set, again formally
16 5 .
5Triviality of the kernel 19 Appendix A. Divergence of a symmetric two-tensor in coordinates 20 A.1. Divergence of a scalar variation 21 A.2. Divergence of a vector variation 22 A.3. Divergence of a tensor variation 22 Appendix B. Divergence & double divergence of h S ij and h S aj 22 Appendix C. The decomposition of h 23 Appendix D. The asymptotics of scalar modes 24 D.1. Estimate of Φ S,I in any dimension n 25 D.2. Estimate of Φ S in dimension n = 2 26 Appendix E. The asymptotics of vector modes 26 Appendix F. The linearised Birkhoff theorem 27 Appendix G. The l = 1 modes for K = 1, H. A linearised Birkhoff-type theorem with K = 0 32 Appendix I. A linearised Birkhoff-type theorem with K = −1 34 Appendix J. The Riemannian Kerr anti-de Sitter metrics 35 Appendix K. The master equation for vector perturbations 37 References 39 Date: June 21, 2018. Preprint UWThPh-2017-30.
1
Compare[15, Section 5]. The variable x used there is actually awkward for the l = 0 modes because its definition involves an operator which becomes singular at r = 3m precisely for this mode. The calculations there for spherically symmetric perturbations become clear if instead of x one uses B −1 x = (1 − 3m r )x. The operator B has been introduced there to obtain simple formulae for all remaining modes.
withṼ positive on an open set. Indeed, if such a vector field X exists, and if u is in the L 2 -kernel of −∆ + V , then (see the last inequality in the proof above) u has to vanish on the open set whereṼ > 0, so by unique continuation u = 0 everywhere.
and positive on an open set. This leads to (compare Proposition 3.1 and Remark 3.2): Proposition 3.10. Let h be an element of the L 2 -kernel of the operator P L defined in (2.10). Let n ∈ {4, 5} if K = 1, and n ∈ N otherwise.
Figure 1 .
1A typical numerical solution for S withṼ = 0, cut off at the third zero of S. Here n = 7, K = 0, k = 1, µ = 100, = 1.
Figure 2 .
2A typical numerical solution for S withṼ = 1/r 2 . Here n = 3, K = 1, l = 2, µ = 10, = 1.
the sets I S and I V of indices I corresponding to modes governed by the scalar and vector master equations:I S = {I | k(I) > 0, k 2 (I) > nK} ,(4.11)
,I S I , with obvious similar definitions for h S ai , etc. The vanishing of the scalar master potential implies[16,
For k = 0
0this gives for the divergence and double divergence of δg S ij D i (γ ij δg S jk ) = I 2r 2 H I L,S D k S I + H I T,S ( D i γ ij S
r −1 D a rX a,I = O(r −1−n ) + r −1 g rr X r,I = O(r −1−n ) + O(r 1−n ) = O(r 1−n ) , F ab,I := f S ab,I + D a X b,I + D b X a,I , F tt,I = O(r 1−n ) , F tr,I = O(r −n ) , F rr,I = O(r −n−1 ) .The rescaled quantities, defined as [16, (2.13)] F I := r n−2 F I ,F ab,I := r n−2 F ab,I , areF I = O(r −1 ) ,F tt,I = O(r −1 ) ,F tr,I = O(r −2 ) ,F rr,I = O(r −3 ) . Then [16, (2.20)] X I :=F t t,I −2F I = O(r −1 ) , Y I :=F r r,I −2F I = O(r −1 ) , Z I :=F r t,I = O(1) , and finally [16, (3.1)] Φ S,I := H −1 I r 1−n/2 (nZ I − r(X I + Y I )) = O(r 1−n/2 ) = O(ρ n/2−1 ) , where H I := m + xn(n + 1)/2 = O(1) , m := k(I) 2 − nK , x := 2µr 1−n andZ is of the same order as Z.
D t Ω = O( 1 )
1, D r Ω = O(r −3 ) , and Ω = O(1). By [16, (5.13)] we have Φ V,I = r −n/2 Ω = O(r −n/2 ) = O(ρ n/2 ) .
explained in [13, end of Section 2.2] (compare [11, Section II.A]), in the case at hand a general metric perturbation can be decomposed as h µν = h is the odd and h (+) µν the even part. The linearised Einstein equations decouple into separate equations for the odd and even parts. G.1. Odd perturbations. Odd perturbations take the form [11, Equation (35)]
J
(m)i where the J (m) 's form a basis of Killing vector fields on S 2 .Gauge transformations defined by a gauge vector Y of the form[11, (28)]
-components are of the form (G.2), we can set one of them to zero by a rotation, leaving h , r), as the only non-zero component of the perturbation.The linearised Einstein equations give
on r. The m = 2, 3 components of G tφ [h] show ∂ t C on r. Eliminating second derivatives between G tt [h] and G rr [h] shows that C (2,3) r vanishes as well. To handle the m = 1 equations we define new variables Z
up to an irrelevant constant. The m = 1 component of G tr [h] directly gives Z (1) t = 0. Eliminating third order derivatives from the remaining equations we obtain(G.20) ∂ r J (1) + 4rf ∂ 2 r Z (1) r + 12r 2 + 8 ∂ r Z (1) r = 0 .Differentiating the m = 1 equations by r and using (G.20) to express derivatives of J by Z (1) r gives two fifth order and one fourth order equation for Z (1) r . Eliminating higher derivatives we finally obtain a third order equation for Z (1) r
(
I.10)h rr := r n−1 f 2 h rr ,h tt := r n−1 ( h tt + f 2 h rr ) , and all this is inserted into the linearised Einstein tensor G µν [ h]. The G tt [h], G tr [h], G rr [h] components lead tõ h rr =h rr (r) ,h tt = C(t)rf , (I.11) as before. Inserting this into δ ij G ij [h] givesh rr = 2δµ, for some constant δµ. As in the K = 1 case, the h rr and h tt terms are in L 2 if and only if δµ = 0 = C(t).
:= ∆ r | r=r 0 = 0 ,and with a function 1 1 which is smooth near the origin and satisfies 1 1 1 2 , 1 3 which are smooth near the origin, with 1 2 (0) = 1 = 1 3 (0).
sin 2 (θ) dτ dφ . (J.12) This is in L 2 if and only if a = 0. Appendix K. The master equation for vector perturbations
(K. 5 )
5D a 1 ...a j (r −1 h V ai )dx i 2 H k ( n N ) ≈ I (1 + k V (I) 2 ) k |D a 1 ...a j f V a,I | 2 .The Ishibashi-Kodama master functions Φ V,I are defined, fork V (I) > (n − 1)K ,as solutions of the (integrable) system (cf., e.g., [12, Equations (2.13)-Note that r ≥ r 0 > 0 throughout, where r 0 is the location of the event horizon, so there is no issue of singularities arising in the equations at r = 0 in the current case. If Φ V,I is known we have, formally,D a (r −2 h V ij ) = − I : k V (I) =0 1 k V (I) D a H V T,I ( D i V I j + D j V I − r −n+1 ε a b D b (Φ V,I ( D i V I j + D j V I i ) .It is convenient to define the parts h V ij , respectively h V ai , of h V ij , respectively of h V ai , in which the low vector harmonics have been removed:h V ij := h V ij + r 2 I : 0< k V (I)≤(n−1)K 1 k V (I) H V T,I ( D i V I j + D j V I i ) , (K.8)h V ai := h V ai − r I : 0< k V (I)≤(n−1)K f V a,I V I i . (K.9)
( n N ) = h V ij H k−1 ( n N ) ,which justifies convergence in (K.11). Next, again formally (K.13) D a Y V i = −r 2 I : k V (I)>(n−1)K D a H V T,I k V (I) V I i = −r I : k V (I)>(n−1)K f V a,I V I i = h V ai .
), (H.10)h rr := r n−1 f 2 h rr ,h tt := r n−1 ( h tt + f 2 h rr ) , and all this is inserted into the linearised Einstein tensor G µν [ h]. The equations in [18, Appendix B] can be used to show that, in the current setting the terms G ab [ h] and δ ij G ij [h] depend only on the t, r components of h. The G tt [h], G tr [h], G rr [h] components lead tõ h rr =h rr (r) , (H.11)h tt = C(t)rf (H.12)
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"Phonological modeling for continuous speech recognition in Korean",
"Phonological modeling for continuous speech recognition in Korean"
]
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"Wonil Lee \nDepartment of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea\n",
"Geunbae Lee \nDepartment of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea\n",
"Jong-Hyeok Lee [email protected] \nDepartment of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea\n"
]
| [
"Department of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea",
"Department of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea",
"Department of Computer Science & Engineering\nPohang University of Science and Technology Pohang\n790-784Korea"
]
| []
| A new scheme to represent phonological changes during continuous speech recognition is suggested. A phonological tag coupled with its morphological tag is designed to represent the conditions of Korean phonological changes. A pairwise language model of these morphological and phonological tags is implemented in Korean speech recognition system. Performance of the model is verified through the TDNN-based speech recognition experiments. | null | [
"https://arxiv.org/pdf/cmp-lg/9607023v1.pdf"
]
| 11,849,562 | cmp-lg/9607023 | c793c72129cbe6d2e5b3d8eeba525318f117f7df |
Phonological modeling for continuous speech recognition in Korean
Jul 1996
Wonil Lee
Department of Computer Science & Engineering
Pohang University of Science and Technology Pohang
790-784Korea
Geunbae Lee
Department of Computer Science & Engineering
Pohang University of Science and Technology Pohang
790-784Korea
Jong-Hyeok Lee [email protected]
Department of Computer Science & Engineering
Pohang University of Science and Technology Pohang
790-784Korea
Phonological modeling for continuous speech recognition in Korean
Jul 1996arXiv:cmp-lg/9607023v1 18
A new scheme to represent phonological changes during continuous speech recognition is suggested. A phonological tag coupled with its morphological tag is designed to represent the conditions of Korean phonological changes. A pairwise language model of these morphological and phonological tags is implemented in Korean speech recognition system. Performance of the model is verified through the TDNN-based speech recognition experiments.
Introduction
The most widely used language models in speech recognition are word-level models, such as wordpairs and word-bigrams (Lee, 1989) (Bates et al., 1993) (Agnas et al., 1994). However, these models take too much space and need large corpus to be correctly trained. Also they are domain dependent, so it is hard to add new vocabularies. To cope with these problems, several category-level language models are suggested (Jardino, 1994) (Yang et al., 1994). These models include word-category models based on the fisrt and last syllables of the words, and models using an automatic categorization technique to reduce the perplexity. The category-level language models showed a reduction in space requirements and better domain independance. For the agglunative languages, several morpheme category/tag-level models are also suggested (Sakai, 1993) (Nakata, 1994). These models are basically the same as the ones used in text tagging systems, and use bigram/trigram statistics between tags.
However, Korean has many phonological changes which happen in a morpheme and between morphemes, and those changes result in the disparity be-tween phonetic and orthographic descriptions of the morphemes. To cope with the phonological changes during Korean speech recognition, we suggest a representation scheme for the phonological changes, and a morphological and phonological tag pair language model (we call it pairwise language model). A hierarchical morphological tag set derived from the one used in written text analysis (Lee and Lee, 1992) is used and a phonological tag set is constructed from the Korean standard pronounciation rules (of Education, 1991). Performance of the model is tested through an experimental TDNN(time-delayed neural network) speech recognition system. The proposed model is quite extensible to new vocabularies and new domains by adding new dictionary entries for the necessary morphemes, and can be refined to bigram or trigram probabilistic models to give better recognition results.
Declaritive modeling of Korean phonological rules
Phonological changes during speech recognition in Korean are modeled with phoneme-sequence-tomorpheme dictionary entries and a binary connectivity matrix.
Phoneme-sequence-to-morpheme dictionary Figure 1 shows a sample entry of the phonemesequence-to-morpheme dictionary. For a phoneme sequence [n u n], two morphemes are in the dictionary: one is an adnominal verbending and the other is a noun-ending. The figure shows a left and right morphological tag, and a left and right phonological tag for the adnominal verb-ending case. 1
The morphological tag "eCNMG" says that the 1 A phoneme sequence can be a sequence of morphemes due to contraction, especially in spontaneous dialogues, and can have different left and right morphological categories. morpheme is a verb-ending(e), makes a complex sentence(C), especially a inner sentence(N), through a noun phrase construction(M) and it is an adnominal verb-ending(G). The phonological tag "P-n" says that the morpheme is not changed at all(-) and the first(and the last for the right phonological tag) phoneme is [n]. "P" in "P-n" says that it is a phonological tag. A phonological tag of the form "Pa=b"(see figure 2 and 4) means that 'a' is pronounced as [b] and "Pa2b"(see figure 3 and 4) means 'a' is pronounced as [b] by the neutralization phenomenon.
Binary connectivity matrix While the dictionary keeps the information about how a single morpheme is changed phonologically, the phonological binary connectivity matrix keeps the collocational information of two morphemes' pronounciations. Figure 2 shows the connectivity matrix entries for consonant assimilation phenomenon in Korean. These entries say that a morpheme whose last consonant 't s ss' 2 are changed into [n], can be followed by a morpheme whose first phoneme is [n] or [m]. Wild characters(*, ?) can be used to reduce the number of entries in the matrix.
Generally, we apply the following two guidelines for the phonological rule modeling.
• Make a new dictionary entry for each morphologically conditioned phonological changes: Some phonological changes, such as vowel contraction and neutralization, happen only in the specific morphemes in a specific collocational relation. In these cases, registering all the phonologically changed morphemes or morphemesequences is prefered.
• Represent the final changes when more than one changes occur: When more than one phonological changes occur for a morpheme or between morphemes, register only the final form of each morpheme rather than all the intermediate forms. This strategy increases the number of dictionary entries but eliminates the successive rule application.
Representative phonological modeling examples
In this section, major Korean pronounciation rules (text-to-speech rules) (of Education, 1991) are explained and their modeling (for speech-to-text conversion) using the dictionary and the connectivity matrix is described. 3 Yale romanization is adoted to represent the Korean phonemes.
Neutralization 4 In Korean, only 7 consonants are pronounced as syllable coda. This is called neutralization or consonant cluster simplification and happens when the morpheme is followed by a pause or a consonant. Figure 3 shows the dictionary entry for the neutralized "talk"(chicken) and the corresponding connectivity matrix entry. "PEND" is a special tag for the pause. For each "Pa2b" tag, a connectivity with "PEND" is added in the matrix. Figure 6 shows the case of "noh-ko". The 'h' and 'k' are merged to [kh]. The phonological tag "Ph=X" means that "h" is disappeared Others "cye ccye chye" in a word's conjugational form, are pronounced as [ce cce che]. For example, "ka-ci+e" ⇒ "ka-cye"
[ka-ce] (have) 'cye' ⇒ [ce] "cci+e" ⇒ "ccye" [cce] (cook) 'ccye' ⇒ [cce]
Since the desyllabification is morphologically conditioned, the dictionary entries model the phenomenon according to our general guidelines. So, [k a c e] have the following many morphological forms in the dictionary: [k a c e] "ka-ci" (morpheme root form) "ka-ci+e" (root+sentential ending) "ka-ci+e" (root+connective verb-ending) "ka-ci+e" (root+aux. connective verb-ending) However, 'yey' not in syllables "yey lyey" can be pronounced as [ey]. In these cases, two distinct entries are made in the dictionary for each morpheme.
In this way, we modeled all the Korean pronounciation rules in about 1000 entries of phonemesequence-to-morpheme dictionary and more than 500 lines of binary phonological connectivity matrix. The phonological connectivity matrix developed in the previous section, coupled with the morphological connectivity matrix is used as a pair-wise language model for continuous Korean speech recognizer. The morphological connectivity matrix is constructed similarly to model the Korean morphotactics using the morphological tags in the dictionary (Lee and Lee, 1992). Figure 7 shows the architecture of the TDNNbased continuous speech recognizer. The TDNNbased phoneme recognizer gvies a sequence of phoneme vectors for the input speech, and this phoneme sequence is decoded by the Viterbi lexical decoder. Tree-structured phoneme-sequence-tomorpheme dictionary is used in the lexical decoding phase and a morpheme graph is extracted after the pair-wise language model is applied. The language model checks each adjacent pair of morphemes in the graph whether they are connectable morphologically and phonologically.
The suggested model using the connectivity matrices for the phonological tags and the morphological tags is easy to construct, easy to maintain, and domain independent. A new morpheme can be added by coding one or more dictionary entries corresponding to its phonological variations.
Experiments
Performance of the pairwise language model is tested using the TDNN-based phoneme recognizer. Input speech is sampled at 16Khz and the melscaled filterbank output is used as the recognizer's input. The TDNN phoneme recognizer is trained for all 39 Korean phonemes from the carefully selected 75 sentences (phone-balanced corpus). Using this recognizer, we do the Viterbi lexical decoding by em- ploying the tree-structured phoneme-sequence-tomorpheme dictionary, and apply the proposed pairwise language model. For new 321 sentences, applying the language model produces 92.6% correct morphemes under the 70% correct phoneme recognition performance (figure 8). The evaluation is based on the DP best matching of the morpheme graphs with the correct morpheme sequences.
Conclusion
In this paper, a new scheme to represent phonological changes in Korean is suggested. A pair-wise language model of morphological and phonological tags is proposed for continuous Korean speech recognition. The proposed model has the following advantages in phonological modeling for Korean speech recognition:
• domain independent,
• easy to construct,
• easy to maintain,
• easy to add a new vocabulary.
The pairwise language model integrates speech recognition and natural language processing at the morpheme-level, and the morpheme-level integration provides the full-fledged morphological/phonological processing which is essencial for agglunative and morphologically complex languages, such as Korean and Japanese. The model can be extended to categorial bigram models which are widely used in Korean text tagging systems.
Figure 1 :
1Sample
Figure 2 :
2Sample phonological connectivity matrix for consonant assimilation
Figure 6 :
6Dictionary entries for [n o] and [kh o], and the corresponding connectivity matrix entry (changed to X(nothing)).
Figure 7 :
7The TDNN-based speech recognizer 4 Pair-wise language model
Figure 8 :
8Morpheme recognition results for new 321 sentences
We will use 't s ss' notation to mean 't', 's', or 'ss' through out in this paper.
The pronounciation rules cover intra-word and interword phonological changes.
AcknowledgementsThis paper was partially supported by KOSEF (#941-0900-084-2) and Ministry of Information and Telecommunication, Information super-highway application project (#95-122).
Spoken language trnaslator: first year report. Agnas, Swedish Institute of Computer Science and SRI InternationalReferences [Agnas et al.1994] Agnas, M., H. Alshawi, I. Bretan, D. Carter, K. Ceder, M. Collins, R. Crouch, V. Di- galakis, B. Ekholm, B. Bamback, J. Kaja, J. Karl- gren, B. Lyberg, P. Price, S. Pulman, M. Rayner, C. Samuelsson, and T. Svensson. 1994. Spoken language trnaslator: first year report. Swedish Institute of Computer Science and SRI Interna- tional.
The BBN/HARC spoken language understanding system. Bates, Proceedings of the ICASSP-93. the ICASSP-93[Bates et al.1993] Bates, M., R. Bobrow, P. Fung, R. Ingria, F. Kubala, J. Makhoul, L. Nguyen, R. Schwartz, and D. Stallard. 1993. The BBN/HARC spoken language understanding sys- tem. In Proceedings of the ICASSP-93.
A class bigram model for very large corpus. Michael Jardino, Proceedings of the international conference on spoken language processing 94. the international conference on spoken language processing 94Jardino, Michael. 1994. A class bi- gram model for very large corpus. In Proceedings of the international conference on spoken language processing 94, pages 867-870.
Implementation of the morphological analyzer using hierarchical symbolic connectivity information. [ Lee, Euncheol Lee, Jong-Hyeok Lee, Proceedings of the Hangul and Korean Linguistic Processing 92. the Hangul and Korean Linguistic Processing 92[Lee and Lee1992] Lee, EunCheol and Jong-Hyeok Lee. 1992. Implementation of the morphologi- cal analyzer using hierarchical symbolic connec- tivity information. In Proceedings of the Hangul and Korean Linguistic Processing 92.
Automatic speech recognition. K F Lee, Kluwer Academic Publishers, IncLee, K. F. 1989. Automatic speech recog- nition. Kluwer Academic Publishers, Inc.
A stochastic morphological analyzer for spontaneously spoken languages. Misaaki Nakata, Proceedings of the international conference on spoken language processing 94. the international conference on spoken language processing 94Nakata, Misaaki. 1994. A stochas- tic morphological analyzer for spontaneously spo- ken languages. In Proceedings of the interna- tional conference on spoken language processing 94, pages 795-798.
Education1991, Korean Ministry. 1991. Korean orthographic rules. Samsung Publishers. Education1991] of Education, Korean Ministry. 1991. Korean orthographic rules. Samsung Pub- lishers.
Morphological categorial bigram: a single language model for both spoken language and text. Shinsuke Sakai, Proceedings of the international symposium on spoken dialogue. the international symposium on spoken dialogueSakai, Shinsuke. 1993. Morphological categorial bigram: a single language model for both spoken language and text. In Proceedings of the international symposium on spoken dialogue, pages 87-90.
An intelligent and efficient word-classbased Chinese language model for Mandarine speech recognition with very large vocabulary. Yang , Proceedings of the international conference on spoken language processing 94. the international conference on spoken language processing 94[Yang et al.1994] Yang, Yen-Ju, Sung-Chien Lin, Lee-Feng Chien, Keh-Jiann Chen, , and Lin-Shan Lee. 1994. An intelligent and efficient word-class- based Chinese language model for Mandarine speech recognition with very large vocabulary. In Proceedings of the international conference on spoken language processing 94, pages 1371-1374.
| []
|
[
"Nonautonomous Young Differential Equations Revisited",
"Nonautonomous Young Differential Equations Revisited"
]
| [
"Nguyen Dinh Cong ",
"· Luu ",
"Hoang Duc ",
"· Phan ",
"Thanh Hong "
]
| []
| [
"J Dyn Diff Equat"
]
| In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points. | 10.1007/s10884-017-9634-y | null | 53,670,267 | 1705.07473 | 65eb83abb21026291ad94ebb64e87a45b2802674 |
Nonautonomous Young Differential Equations Revisited
2018
Nguyen Dinh Cong
· Luu
Hoang Duc
· Phan
Thanh Hong
Nonautonomous Young Differential Equations Revisited
J Dyn Diff Equat
30201810.1007/s10884-017-9634-yReceived: 28 May 2017 / Revised: 6 December 2017 / Published online: 20 December 2017Stochastic differential equations (SDE) · Fractional Brownian motion (fBm) · Young integral · p-variation
In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points.
Introduction
This paper deals with the Young differential equation of the form
dx t = f (t, x t )dt + g(t, x t )dω t , t ≥ 0 (1.1)
where f : R × R d → R d and g : R × R d → R d×m are continuous functions, ω is a R m -valued function of finite p-variation norm for some 1 < p < 2. Such type of system is generated from stochastic differential equations driven by fractional Brownian noises, as seen e.g. in [20]. Equation (1.1) is understood in the integral form
x t = x 0 + t 0 f (s, x s )ds + t 0 g(s, x s )dω s , t ≥ 0,(1.2)
where the first integral is of Riemannian type, meanwhile the second integral can be defined in the Young sense [23]. The existence and uniqueness of the solution of (1.2) are studied by several authors. When f, g are time-independent, system (1.2) is proved in [23] and [22] and [18] to have a unique solution in a certain space of continuous functions with bounded p-variation. The result is then generalized for the case 2 ≤ p < 3 in Lyons' seminal paper [19] in which rough path theory is introduced to define the second integral in (1.2) and also the integration equation (see [8], [16] and [17]). An alternative theory of controlled paths in Gubinelli's work [10] simplifies and generalizes the concept of integration and differential equations, leading to the concept of rough differential equations (see recent works in [3] and [2] for 2 ≤ p < 3, or [14] for controlled differential equations as Young integrals). According to their settings, f, g are time independent and g is often assumed to be differentiable upto a certain order and bounded in its derivatives.
A different approach following Zähle [24] by using fractional derivatives can be seen in [21] which derives very weak conditions for time varying f and g in (1.1), in particular g need to be only C 1 with bounded and Hölder continuous first derivative, to ensure the existence and uniqueness of the solution in the space of Hölder continuous functions. Later, one finds that there is a connection between the rough path approach and the techniques in fractional calculus, see e.g. [11] and [12].
Our aim in this paper is to close the gap between the two methods for nonautonomous Young equations by proving that, under similar assumptions to those of Nualart and Rascanu [21], the existence and uniqueness theorem for system (1.1) still holds in the space of continuous functions with bounded p-variation norm. For that to work, we construct the socalled greedy sequence of times (see [4,Definition 4.7]) such that the solution can be proved to exists uniquely in each interval of the consecutive times of the greedy sequence, and is then concatenated to form a global solution. It is important to note that since we are using estimates for p-variation norms, we do not apply the classical arguments of contraction mappings, but use Shauder-Tychonoff fixed point theorem as seen in [18] and a Gronwall-type lemma.
Another issue is the generation of flow which was asserted in [17] for the autonomous systems. The idea is to construct the shift dynamical system in the extended space of finite p-variation norm for the whole real line time, and to prove that the system generates a nonautonomous dynamical system satisfying the cocycle property (see [3]). When applying to stochastic differential equations driven by fractional Brownian motions, by considering an appropriate probability space, one can prove that the system generates a random dynamical system (see [3,5,9]). However in the nonautonomous situation, one only expects a generation of a two-parameter flow on the phase space.
The paper is organized as follows. In Sect. 2, the Young integral is introduced and a version of greedy sequence of times is presented. In Sect. 3, we prove the existence and uniqueness of the global solution of system (1.2) in Theorem 3.6, for this we need to formulate a Gronwalltype lemma. Proposition 3.7 gives an estimate of q-var norm of solution via p-var norm of the driver ω. We also prove the existence and uniqueness of the solution of the backward equation (3.23) in Theorem 3.8. In Sect. 4, the fact in Theorem 4.1 that two trajectories do not intersect helps to conclude that the Cauchy operator or the Ito map of (1.2) generates a continuous two parameter flow. In the autonomous case it generates a continuous nonautonomous dynamical system which helps to form a topological skew product flow.
Preliminaries
Young Integral
In this section we recall some facts about Young integral, more details can be seen in [8].
Let C([a, b], R d ) denote the space of all continuous paths x : [a, b] → R d equipped with sup norm · ∞,[a,b] given by x ∞,[a,b] = sup t∈[a,b] |x t |, where | · | is the Euclidean norm in R d . For p ≥ 1 and [a, b] ⊂ R, a continuous path x : [a, b] → R d is of finite p-variation if |||x||| p-var,[a,b] := sup (a,b) n i=1 |x t i+1 − x t i | p 1/ p < ∞, (2.1)
where the supremum is taken over the whole class of finite partition of [a, b]. The subspace
C p ([a, b], R d ) ⊂ C([a, b], R d ) ofC ∞ ([a, b], R d ) in C p ([a, b], R d ) is a separable Banach space denoted by C 0, p ([a, b], R d ) which can be defined as the space of all continuous paths x such that lim δ→0 sup (a,b),| |≤δ i |x t i+1 − x t i | p = 0.
It is easy to prove (see [8,Corollary 5.33, p. 98]) that for 1 ≤ p < p we have
C p ([a, b], R d ) ⊂ C 0, p ([a, b], R d ).
Also, for 0 < α ≤ 1 denote by C α-Hol ([a, b], R d ) the Banach space of all Hölder continuous paths x : [a, b] → R d with exponential α, equipped with the norm x α-Hol, [a,b] := |x a | + |||x||| α-Hol, [a,b] = |x a | + sup
(s,t)∈ [a,b] |x t − x s | (t − s) α < ∞. (2.2) Clearly, if x ∈ C α-Hol ([a, b], R d ) then for all s, t ∈ [a, b] we have |x t − x s | ≤ |||x||| α-Hol,[a,b] |t − s| α .
Hence, for all p such that pα ≥ 1 we have
|||x||| p-var,[a,b] ≤ |||x||| α-Hol,[a,b] (b − a) α < ∞. (2.3) Therefore, C 1/ p-Hol ([a, b], R d ) ⊂ C p ([a, b], R d ). As introduced in [21], the Besov space W 1/ p,∞ b ([a, b], R d ) of measurable functions g : [a, b] → R d such that sup a<s<t<b |g t − g s | (t − s) 1/ p + t s |g y − g s | (y − s) 1+1/ p dy < ∞ is a subspace of C 1/ p-Hol ([a, b], R d ). Hence W Lemma 2.1 Let x ∈ C p ([a, b], R d ), p ≥ 1. If a = a 1 < a 2 < · · · < a k = b, then k−1 i=1 |||x||| p p-var,[a i ,a i+1 ] ≤ |||x||| p p-var,[a 1 ,a k ] ≤ (k − 1) p−1 k−1 i=1 |||x||| p p-var,[a i ,a i+1 ] .
Proof The proof is similar to the one in [8, p. 84], by using triangle inequality and power means inequality [s,t] , where x is of bounded p-variation norm on [a, b] and q ≥ p are some examples of control function. The following lemma gives a useful property of controls in relation with variations of a path (see [8] for more properties of control functions).
1 n n i=1 z i ≤ 1 n n i=1 z r i 1/r , ∀z i ≥ 0, r ≥ 1. Definition 2.2 A continuous map ω : [a, b] −→ R + is called a control if it is zero on the diagonal and superadditive, i.e (i), For all t ∈ [a, b], ω t,t = 0, (ii), For all s ≤ t ≤ u in [a, b], ω s,t + ω t,u ≤ ω s,u . The functions (s, t) −→ (t − s) θ with θ ≥ 1, and (s, t) −→ |||x||| q p-var,
Lemma 2.3 Let ω j be a finite sequence of control functions on
[0, T ], C j > 0, j = 1, k, p ≥ 1 and x : [0, T ] → R d be a continuous path satisfying |x t − x s | ≤ k i= j C j ω j (s, t) 1/ p , ∀s < t ∈ [0, T ], then |||x||| p-var,[s,t] ≤ k j=1 C j ω j (s, t) 1/ p , ∀s < t ∈ [0, T ].|x s i+1 − x s i | p 1/ p ≤ ⎡ ⎣ n i=0 ⎛ ⎝ k j=1 C j ω j (s i , s i+1 ) 1/ p ⎞ ⎠ p ⎤ ⎦ 1/ p ≤ k j=1 n i=0 C p j ω j (s i , s i+1 ) 1/ p ≤ k j=1 C j ω j (s, t) 1/ p .
This implies the conclusion of the lemma.
Now, consider x ∈ C q ([a, b], R d×m ) and ω ∈ C p ([a, b], R m ), p, q ≥ 1, if Riemann- Stieltjes sums for finite partition = {a = t 0 < t 1 < · · · < t n = b} of [a, b] and any ξ i ∈ [t i , t i+1 ] S := n i=0 x ξ i (ω t i+1 − ω t i ),(2.5)
converges as the mesh | | := max 0≤i≤n−1 |t i+1 − t i | tends to zero, we call the limit is the
Young integral of x w.r.t ω on [a, b] denoted by b a x t dω t .
It is well known that if p, q ≥ 1 and 1 p + 1 q > 1, the Young integral b a x t dω t exists (see [23, pp. 264-265]). Moreover, if x n and ω n are of bounded variation, uniformly bounded in C q ([a, b], R d×m ), C p ([a, b], R m ) and converges uniformly to x, ω respectively, then the sequence of the Riemann-Stieljes integral b a x n t dω n t approach b a x t dω t as n → ∞ (see [8]). This integral satisfies additive property by the construction, and the so-called Young-Loeve estimate [8,Theorem 6.8 [s,t] |||ω||| p-var, [s,t] , (2.6) where
, p. 116] t s x u dω u − x s [ω t − ω s ] ≤ K |||x||| q-var,K := 1 − 2 1−θ −1 , θ := 1 p + 1 q > 1. (2.7) Lemma 2.4 For 1 ≤ p, 1 ≤ q such that θ = 1 p + 1 q > 1 and x ∈ C q ([a, b], R d×m ), ω ∈ C p ([a, b], R m ), the following estimate holds . a x u dω u p-var,[a,b] ≤ |||ω||| p-var,[a,b] |x a | + (K + 1) |||x||| q-var,[a,b] , (2.8) where K is determined by (2.7).
Proof The conclusion is a direct sequence of (2.6) and [8, Proposition 5.10(i), p. 83].
Due to Lemma 2.4, the integral t → t a x s dω s is a continuous bounded p-variation path. Note that the definition of Young integral does depend on the direction of integration in a simple way like the Riemann-Stieltjes integral. Namely, it is easy to see that
a b x u dω u = lim (a,b),| |→0 n i=1 x ξ i (ω t i − ω t i+1 ) = − lim (a,b),| |→0 n i=1 x ξ i (ω t i+1 − ω t i ) = − b a x u dω u . (2.9)
The Greedy Sequence of Times
The original idea of a greedy sequence was introduced in [4, Definition 4.7]. Given α > 0, a compact interval I ∈ R and a control ω : (I ) → R + , the construction of such a sequence aims to have a "greedy" approximation to the supremum in the definition of the so-called accummulated α-local ω-variation (see [4,Definition 4.1])
M α,I (ω) = sup (I ),ω t i ,t i+1 ≤α t i ∈ (I ) ω t i ,t i+1 .
In particular, ω s,t is chosen to be |||·||| p p-var, [s,t] in [4]. A similar version for stopping times was developed before in [9] and then has been studied further recently by [7] for stability of the system. Here we propose another version of greedy sequence of times which matches with the nonautonomous setting.
1 + ω 1 − ω 2 p-var,[−n,n] .
Let n ∈ N, observe that metric d satisfies
d(ω 1 , ω 2 ) ≤ ω 1 − ω 2 p-var,[−n,n] + 2 −n , ω 1 − ω 2 p-var,[−n,n] ≤ 2 n d(ω 1 ,ω 2 ) 1−2 n d(ω 1 ,ω 2 ) , (2.10)
where the second inequality holds for any fixed n and ω 1 , ω 2 close enough such that 2 n d(ω 1 , ω 2 ) < 1. Hence every Cauchy sequence (ω k ) k w.r.t. metric d is also a Cauchy sequence when restricted to C p ([−n, n], R m ), thus converges to a limit ω * ∈
C p ([−n, n], R m ) which is uniquely defined pointwise, so ω * ∈ C p (R, R m ). Therefore, ( C p (R, R m ), d) is a complete metric space. Remark 2.5 (i) Truncation: Another consequence of (2.10) is that the truncated version of ω ∈ C p (R, R m ) in any C p ([−n, n], R m ) differs very little w.r.t. metric d from the original ω if we choose n large enough. Moreover, if a function is continuous w.r.t. ω on any restriction in C p ([−n, n], R m ) for any n > 0 then it is also continuous w.r.t. ω in C p (R, R m ) with respect to metric d. (ii) Concatenation: Let a < b < c. Suppose that ω 1 ∈ C p ([a, b], R m ), ω 2 ∈ C p ([b, c], R m ) and ω 1 b = ω 2 b . Then ω 1 .1 [a,b] + ω 2 .1 [b,c] belongs to C p ([a, c], R m )
. For any given λ, μ > 0 we construct a strict increasing sequence of times {τ n },
τ n : C p (R, R m ) −→ R + , such that τ 0 ≡ 0 and |τ i+1 (ω) − τ i (ω)| λ + |||ω||| p-var,[τ i (ω),τ i+1 (ω)] = μ. (2.11)
To do so, first define τ :
C p (R, R m ) −→ R + such that τ (ω) := sup{t ≥ 0 : t λ + |||ω||| p-var,[0,t] ≤ μ}.
Observe that the function κ(t) := t λ + |||ω||| p-var,[0,t] is continuous and stricly increasing w.r.t. t with κ(0) = 0 and κ(∞) = ∞, therefore due to the continuity there exists a unique τ = τ (ω) > 0 such that
τ λ + |||ω||| p-var,[0,τ ] = μ. (2.12)
Thus τ is well defined. Next, we construct the time sequence inductively as follows. Set τ 0 := 0, τ 1 (ω) := τ (ω). Suppose that we have defined τ n (ω) for n ≥ 1, looking at the following equality as an equation of δ n (ω) ∈ R + , like above we find an unique δ n (ω) such that
μ = δ λ n (ω) + |||ω(· + τ n (ω))||| p−var,[0,δ n (ω)] , hence we can set τ n+1 (ω) := τ n−1 (ω) + δ n (ω),(2.13)
where δ n (ω) is determined above. Thus we have defined a time sequence {τ n } for all n = 0, 1, 2, . . .. Such a sequence then satisfies (2.11). Now, we fix ω ∈ C p (R, R m ) and consider the number of times of the greedy sequence inside an arbitrary finite interval of R + . We write τ n for τ n (ω) to simplify the notation. For given T > 0, we introduce the notation
N (T, ω) := sup{n : τ n ≤ T } < ∞.
(2.14)
or more generally, for any 0 ≤ a < b < ∞,
N (a, b, ω) := sup{n : τ n ≤ b} − inf{n : τ n ≥ a}.(2.N (T, ω) ≤ 2 p −1 μ p T p λ + |||ω||| p p-var,[0,T ] . (2.16)
More generally,
N (a, b, ω) ≤ 2 p −1 μ p (b − a) p λ + |||ω||| p p-var,[a,b] . (2.17) Proof We have for all n ∈ N * nμ p = n−1 i=0 μ p = n−1 i=0 |τ i+1 − τ i | λ + |||ω||| p-var,[τ i ,τ i+1 ] p ≤ 2 p −1 ⎡ ⎣ n−1 i=0 |τ i+1 − τ i | p λ + n−1 i=0 |||ω||| p p-var,[τ i ,τ i+1 ] p / p ⎤ ⎦ ≤ 2 p −1 ⎡ ⎣ (τ n − τ 0 ) p λ + n−1 i=0 |||ω||| p p-var,[τ i ,τ i+1 ] p / p ⎤ ⎦ ≤ 2 p −1 τ p λ n + |||ω||| p p-var,[0,τ n ] . (2.18)
Consequently, we obtain
N (T, ω) ≤ 2 p −1 μ p T p λ + |||ω||| p p-var,[0,T ] .
Similarly, (2.17) holds.
Remark 2.7 (i) Since the left-hand side of (2.18) tends to infinity as n goes to ∞ its right hand side can not be bounded. This implies that τ n → ∞ as n → ∞. (ii) We can construct a time sequence starts at τ 0 = t 0 , an arbitrary point in R, and on (−∞, t 0 ] in a similar manner.
Existence and Uniqueness Theorem
In this section, we are working with the restriction of any trajectory ω in a given time interval [0, T ] by considering it as an element in C p ([0, T ], R m ), for a certain p ∈ (1, 2) (see Remark 2.5 for the relation between ω ∈ C p (R, R m ) and its restrictions). Consider the Young differential equation in the integral form as:
x t = x 0 + t 0 f (s, x s )ds + t 0 g(s, x s )dω s , t ∈ [0, T ]. (3.1)
We recall here a result in [21] on existence and uniqueness of solution of (3.1), which was proved using contraction mapping arguments with ω in a Besov-type space. In this paper we however would like to derive a proof in C p applying Shauder fixed point theorem and greedy sequence of times tool. First we need to formulate some assumptions on the coefficient functions f and g of (3.1).
(H 1 ) g(t,
x) is differentiable in x and there exist some constants 0 < β, δ ≤ 1, a control function h(s, t) defined on [0, T ] and for every N ≥ 0 there exists M N > 0 such that the following properties hold:
(H g ) : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) Lipschitz continuity |g(t, x) − g(t, y)| ≤ L g |x − y|, ∀x, y ∈ R d , ∀t ∈ [0, T ], (ii) Local Hölder continuity |∂ x i g(t, x) − ∂ y i g(t, y)| ≤ M N |x − y| δ , ∀x, y ∈ R d , |x|, |y| ≤ N , ∀t ∈ [0, T ], (iii) Generalized Hölder continuity in time |g(t, x) − g(s, x)| + |∂ x i g(t, x) − ∂ x i g(s, x)| ≤ h(s, t) β ∀x ∈ R d , ∀s, t ∈ [0, T ], s < t.
(H 2 ) There exists a > 0 and b ∈ L
1 1−α ([0, T ], R d ),
where 1 2 ≤ α < 1, and for every N ≥ 0 there exists L N > 0 such that the following properties hold:
(H f ) : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (i) Local Lipschitz continuity | f (t, x) − f (t, y)| ≤ L N |x − y|, ∀x, y ∈ R d , |x|, |y| ≤ N , ∀t ∈ [0, T ], (ii) Boundedness | f (t, x)| ≤ a|x| + b(t), ∀x ∈ R d , ∀t ∈ [0, T ].
(H 3 ) The parameters in H 1 and H 2 statisfy the inequalities
δ > p−1, β > 1− 1 p , δα > 1 − 1 p .
We would like to study the existence and uniqueness of the solution of (3.1) under the given conditions that
x ∈ C q ([0, T ], R d ) with appropriate constant q > 0.
By the assumption p ∈ (1, 2) and the condition H 3
, 1 − 1 p < min β, δα, δ p , 1 2 , we
can choose consecutively constants q 0 , q such that
1 − 1 p < 1 q 0 < min β, δα, δ p , 1 2 , (3.2) 1 q 0 δ ≤ 1 q < min α, 1 p . (3.3)
Then, we have
1 p + 1 q 0 > 1, q 0 β > 1, q 0 ≥ q 0 δ ≥ q > p, qα > 1. (3.4) We now consider x ∈ C q ([t 0 , t 1 ], R d ) with some [t 0 , t 1 ] ⊂ [0, T ].
Define the mapping given by
F(x) t = x t 0 + I (x) t + J (x) t := x t 0 + Introduce the notations M := max L g , aT 1−α , |g(0, 0)| + h(0, T ) β , b L 1 1−α , (3.6) M N := max{L N , M N , M}, ∀N > 0. (3.7)
It can be seen from the above assumptions that |g(t,
x)| ≤ |g(t, 0)| + L g |x| and |g(t, 0)| ≤ |g(0, 0)| + h(0, T ) β , hence |g(t, x)| ≤ |g(0, 0)| + h(0, T ) β + L g |x| ≤ M(1 + |x|). (3.8)
For the next proposition we need the following auxiliary lemma.
Lemma 3.1 Assume that H 1 − H 3 are satisfied. (i) If x ∈ C q ([t 0 , t 1 ], R d ) then g(·, x . ) ∈ C q 0 ([t 0 , t 1 ], R d×m ) and |||g(·, x . )||| q 0 -var,[t 0 ,t 1 ] ≤ M(1 + |||x||| q-var,[t 0 ,t 1 ] ). (3.9) (ii) For all s < t and for all x i ∈ R d such that |x i | ≤ N , i = 1, 2, 3, 4, then |g(s, x 1 ) − g(s, x 3 ) − g(t, x 2 ) + g(t, x 4 )| ≤ L g |x 1 − x 2 − x 3 + x 4 | + |x 2 − x 4 |h(s, t) β + M N |x 2 − x 4 |(|x 1 − x 2 | δ + |x 3 − x 4 | δ ). (iii) For any x, y ∈ C q ([t 0 , t 1 ], R d ) such that x t 0 = y t 0 and x ∞,[t 0 ,t 1 ] ≤ N , y ∞,[t 0 ,t 1 ] ≤ N we have |||g(·, x . ) − g(·, y . )||| q 0 -var,[t 0 ,t 1 ] ≤ M N |||x − y||| q-var,[t 0 ,t 1 ] 2 + |||x||| δ q-var,[t 0 ,t 1 ] + |||y||| δ q-var,[t 0 ,t 1 ] . (3.10) Proof (i) For s < t in [t 0 , t 1 ], we have |g(t, x t ) − g(s, x s )| ≤ |g(t, x t ) − g(t, x s )| + |g(t, x s ) − g(s, x s )| ≤ L g |x t − x s | + h(s, t) β . Let = (s i ) n+1 1 be an arbitrary finite partition of [t 0 , t 1 ], s 1 = t 0 , s n+1 = t 1 . Since q 0 ≥ q and q 0 β > 1 we have n i=1 |g(s i+1 , x s i+1 ) − g(s i , x s i )| q 0 1/q 0 ≤ L g n i=1 |x s i+1 − x s i | q 0 1/q 0 + n i=1 h(s i , s i+1 ) q 0 β 1/q 0 ≤ L g |||x||| q 0 -var,[t 0 ,t 1 ] + h(t 0 , t 1 ) β ≤ L g |||x||| q-var,[t 0 ,t 1 ] + h(0, T ) β ≤ M(1 + |||x||| q-var,[t 0 ,t 1 ] ) < ∞.
Take the superemum over the set of all finite partition we get g(·, x . ) ∈ C q 0 ([t 0 , t 1 ], R d×m ) and (iii) Note that q 0 β > 1 and q 0 δ ≥ q hence
|||g(·, x . )||| q 0 -var,[t 0 ,t 1 ] ≤ M(1 + |||x||| q-var,[t 0 ,t 1 ] ).|||g(·, x . ) − g(·, y . )||| q 0 -var,[t 0 ,t 1 ] := sup ([t 0 ,t 1 ]) i |g(s i+1 , x s i+1 ) − g(s i+1 , y s i+1 ) − g(s i , x s i ) + g(s i , y s i )| q 0 1/q 0 ≤ L g sup ([t 0 ,t 1 ]) i |x s i+1 − y s i+1 − x s i + y s i | q 0 1/q 0 + x − y ∞,[t 0 ,t 1 ] sup ([t 0 ,t 1 ]) i h(s i , s i+1 ) q 0 β 1/q 0 + M N x − y ∞,[t 0 ,t 1 ] sup ([t 0 ,t 1 ]) ⎡ ⎣ i |x s i+1 − x s i | q 0 δ 1/q 0 + i |y s i+1 − y s i | q 0 δ 1/q 0 ⎤ ⎦ ≤ L g |||x − y||| q-var,[t 0 ,t 1 ] + x − y ∞,[t 0 ,t 1 ] × h(t 0 , t 1 ) β + M N |||x||| δ q-var,[t 0 ,t 1 ] + |||y||| δ q-var,[t 0 ,t 1 ] ≤ M N |||x − y||| q-var,[t 0 ,t 1 ] 2 + |||x||| δ q-var,[t 0 ,t 1 ] + |||y||| δ q-var,[t 0 ,t 1 ] .
The lemma is proved.
For a proof of our main theorem on existence and uniqueness of solutions of an Young differential equation, we need the following proposition.
Proposition 3.2
Assume that H 1 − H 3 are satisfied. Let 0 ≤ t 0 < t 1 ≤ T be arbitrary, q be chosen as above satisfying (3.3) and F be defined by (3.5). Then for any
x ∈ C q ([t 0 , t 1 ], R d ) we have F(x) ∈ C q ([t 0 , t 1 ], R d ), thus F : C q ([t 0 , t 1 ], R d ) −→ C q ([t 0 , t 1 ], R d ).
Moreover, the following statements hold
(i) The q-variation of F(x) satisfies |||F(x)||| q-var,[t 0 ,t 1 ] ≤ M(K + 2) 1 + x q-var,[t 0 ,t 1 ] × (t 1 − t 0 ) α + |||ω||| p-var,[t 0 ,t 1 ] . (3.11) (ii) Let N ≥ 0 be arbitrary but fixed. Suppose that x, y ∈ C q ([t 0 , t 1 ], R d ) be such that x ∞,[t 0 ,t 1 ] ≤ N , y ∞,[t 0 ,t 1 ] ≤ N and x t 0 = y t 0 , then we have F(x) − F(y) q-var,[t 0 ,t 1 ] ≤ x − y q-var,[t 0 ,t 1 ] (t 1 − t 0 ) + |||ω||| p-var,[t 0 ,t 1 ] ×M N (K + 1) 2 + |||x||| δ q-var,[t 0 ,t 1 ] + |||y||| δ q-var,[t 0 ,t 1 ] .(3.
12)
Proof (i) Since 1 p + 1 q 0 > 1, by virtue of (3.9), the Young integral t 0 g(s, x s )dω s exists for all t ∈ [t 0 , t 1 ]. Using (2.8), (3.5) and (3.8) we get
||| J (x)||| q-var,[t 0 ,t 1 ] ≤ |||ω||| p-var,[t 0 ,t 1 ] |g(t 0 , x t 0 )| + (K + 1) |||g(., x . )||| q 0 -var,[t 0 ,t 1 ] ≤ |||ω||| p-var,[t 0 ,t 1 ] M(1 + |x t 0 |) + M(K + 1)(1 + |||x||| q-var,[t 0 ,t 1 ] ) ≤ |||ω||| p-var,[t 0 ,t 1 ] M (K + 2) + |x t 0 | + (K + 1) |||x||| q-var,[t 0 ,t 1 ] .
Now, by Hölder inequality and the assumption H 2 we have
t s |b(u)|du ≤ t s |b(u)| 1 1−α du 1−α t s 1du α ≤ b L 1 1−α (t − s) α ≤ M(t − s) α . Therefore, for s < t in [t 0 , t 1 ] using the assumption H 2 we have t s f (u, x u )du ≤ a x ∞,[s,t] (t − s) + b L 1 1−α (t − s) α ≤ (t − s) α aT 1−α x ∞,[t 0 ,t 1 ] + b L 1 1−α ≤ (t − s) α M 1 + |x t 0 | + |||x||| q-var,[t 0 ,t 1 ] .
This implies
|||I (x)||| q-var,[t 0 ,t 1 ] = . t 0 f (u, x u )du q-var,[t 0 ,t 1 ] ≤ M(t 1 − t 0 ) α 1 + |x t 0 | + |||x||| q-var,[t 0 ,t 1 ]
by [8, Proposition 5.10(i), p. 83] and the fact that the function (s, t) → (t − s) qα defined on [t 0 , t 1 ] is a control function for qα > 1. Since
|||F(x)||| q-var,[t 0 ,t 1 ] ≤ |||I (x)||| q-var,[t 0 ,t 1 ] + ||| J (x)||| q-var,[t 0 ,t 1 ]
(3.11) holds.
(ii) By virtue of (2.8), (3.10) and the condition x t 0 = y t 0 , we have
||| J (x) − J (y)||| p-var,[t 0 ,t 1 ] ≤ |||ω||| p-var,[t 0 ,t 1 ] |g(t 0 , x t 0 ) − g(t 0 , y t 0 )| + (K + 1) |||g(., x . ) − g(., y . )||| q 0 -var,[t 0 ,t 1 ] ≤ |||ω||| p-var,[s,t] (K + 1) |||g(., x . ) − g(., y . )||| q 0 -var,[t 0 ,t 1 ] ≤ (K + 1)M N |||ω||| p-var,[t 0 ,t 1 ] x − y q-var,[t 0 ,t 1 ] × 2 + |||x||| δ q-var,[t 0 ,t 1 ] + |||y||| δ q-var,[t 0 ,t 1 ] .
Similarly,
|[I (x) t − I (y) t ] − [I (x) s − I (y) s ]| ≤ t s | f (u, x u ) − f (u, y u )|du ≤ L N x − y q-var,[t 0 ,t 1 ] (t − s), hence |||I (x) − I (y)||| q-var,[t 0 ,t 1 ] ≤ M N x − y q-var,[t 0 ,t 1 ] (t 1 − t 0 ).
Inequality (3.12) is a direct consequence of these estimates for I (x) and J (x).
Before proving the existence and uniqueness theorem, we need the following lemma of Gronwall type. where [s,t] ). [u,v]
Lemma 3.3 (Gronwall-type Lemma) Let 1 ≤ p ≤ q be arbitrary and satisfy
A 0 = A 1/q 0,T . Proof Put c := max{a 1 , a 2 (K + 1)}, in which K is defined in (2.7). We have |y t − y s | ≤ A 1/q s,t + a 1 y ∞,[s,t] (t − s) + a 2 |||ω||| p-var,[s,t] ( y ∞,[s,t] + K |||y||| q-var,[s,t] ) ≤ A 1/q s,t + max a 1 y ∞,[s,t] , a 2 ( y ∞,[s,t] + K |||y||| q-var,[s,t] ) (t − s + |||ω||| p-var,
Fix the interval [s, t] ⊂ [0, T ] and apply the above inequality for arbitrary subinterval [u, v] ⊂ [s, t] we obtain
|y v − y u | ≤ A 1/q u,v + max{a 1 y ∞,[s,t] , a 2 ( y ∞,[s,t] + K |||y||| q-var,[s,t] )}(v − u + |||ω||| p-var,[u,v] ) ≤ A 1/q u,v + max{a 1 , a 2 (K + 1)}(|y s | + |||y||| q-var,[s,t] )(v − u + |||ω||| p-var,[u,v] ) ≤ A 1/q u,v + c(|y s | + |||y||| q-var,[s,t] )(v − u + |||ω||| p-var,(t i+1 − t i + |||ω||| p-var,[t i ,t i+1 ] ) = 1 2c .
Then, by (3.15) for all s, t ∈ [t i , t i+1 ], s < t, we have |||y||| q-var,[s,t] ≤ A 0 + 1 2 (|y s | + |||y||| q-var, [s,t] ), which implies |||y||| q-var, [u,v] where N (T, ω) is defined by (2.14), the sequence of inequalities 2 A 0 + |y t i+1 | ≤ 2(2 A 0 + |y t i |) ≤ · · · ≤ 2 i−k (2 A 0 + |y t k+1 |) ≤ 2 i−k+1 (2 A 0 + |y s |).
≤ 2 A 0 + |y u | for all u, v ∈ [t i , t i+1 ], u < v. Therefore, |y t i+1 | ≤ y ∞,[s,t i+1 ] ≤ 2(A 0 +|y s |) for all s ∈ [t i , t i+1 ]. By induction we obtain for any s ∈ [t k , t k+1 ], 0 ≤ k ≤ i, i ∈ {0, . . . , N (T, ω)},
Hence, where N (s, t, ω) is defined in (2.15). We have s ≤ t N < t N +1 < · · · < t N ≤ t and where C = 4 p c p ln 2. The proof is complete.
|||y||| q-var,[t i ,t i+1 ] ≤ 2 A 0 + |y t i | ≤ 2 i−k (2 A 0 + |y s |), ∀s ∈ [t k , t k+1 ], 0 ≤ k ≤ i.|||y||| q-var,[s,t N ] ≤ 2 A 0 + |y s |, |||y||| q-var,[t N +i ,t N +i+1 ] ≤ 2 i+1 (2 A 0 + |y s |), i = 0, . . . , N − 1, |||y||| q-var,[t N ,t] ≤ 2 A 0 + |y t N | ≤ 2 N (2 A 0 + |y s |).
By Lemma 2.1 we have
|||y||| q-var,[s,t] ≤ (N + 1) q−1 q |||y||| q q-var,[s,t N ] + N −1 i=1 |||y||| q q-var,[t i ,t i+1 ] + |||y||| q q-var,[t N ,t] 1/q ≤ (N + 1) q−1 q (2 A 0 + |y s |) ⎛ ⎝ N j=0 2 jq ⎞ ⎠ 1/q ≤ (N + 1)(2 A 0 + |y s |)2 N . In the case [s, t] ⊂ [t i , t i+1 ] with some i ∈ {0, 1, . . . , N (T, ω)},
Remark 3.4 (i) Gronwall
Lemma is an important tool in the theory of ordinary differential equations, and the theory of Young differential equations as well. Some versions of Gronwall-type lemma can be seen in [21] and [6]. . (3.18) We are now at the position to state and prove the main theorem of this section.
y ∞,[0,T ] ≤ (2 A 0 + |y 0 |)2 N (T,ω)+1 ≤ (2 A 0 + |y 0 |)2 (4cT ) p +1+(4c|||ω||| p-var,[s,t] ) p .
Corollary 3.5 If in Lemma
x t = x t 0 + t t 0 f (s, x s )ds + t t 0 g(s, x s )dω s , t ∈ [t 0 , T ], x t 0 ∈ R d .
with T being an arbitrary fixed positive number and x 0 ∈ R d being an arbitrary initial condition. Assume that the conditions H 1 −H 3 hold. Then, this equation has a unique solution
x in the space C q ([t 0 , T ], R d ),
where q is chosen as above satisfying (3.3). Moreover, the
solution is in C p ([t 0 , T ], R d ), where p = max{ p, 1 α }.
Proof The proof proceeds in several steps.
Step 1: In this step we will show the local existence and uniqueness of solution. Set
μ := 1 2M(K + 2)
, (3.19) where M is defined in (3.6) and K is defined in (2.7). Let s 0 ∈ [t 0 , T ) be arbitrary but fixed. We recall here the time sequence τ n with the parameters α, μ, i.e
τ 0 = 0, |τ i+1 − τ i | α + |||ω||| p-var,[τ i ,τ i+1 ] = μ.
Put r 0 = min{n : τ n > s 0 } and define s 1 = min{τ r 0 , T }. Then,
|s 1 − s 0 | α + |||ω||| p-var,[s 0 ,s 1 ] ≤ μ. (3.20)
We will show that the Eq. (3.1) restricted to [s 0 , s 1 ] has a unique solution.
Existence of local solutions.
Recall the mapping F defined by the formula (3.5) with t 0 , t 1 replaced by s 0 , s 1 , respectively. By Proposition 3. We show furthermore that if [s,t] Taking into account (3.12), F : B 1 → B 1 is continuous. We show that B 1 is a closed convex set in the Banach space C q ([s 0 , s 1 ], R d ), and F is a compact operator on B 1 . Indeed, for the former observation, note that if z = λx + (1 − λ)y for some x, y ∈ B 1 , λ ∈ [0, 1] then
x ∈ C q ([s 0 , s 1 ], R d ) then F(x) ∈ C (q− )-var ([s 0 , s 1 ], R d ) with small enough . Indeed, since q > p, qα > 1, we can choose > 0 such that q − ≥ p and (q − )α ≥ 1. For all s < t in [s 0 , s 1 ], using (3.11) we have |F(x) t − F(x) s | ≤ |||F(x)||| q-var,[s,t] ≤ M(K + 2) 1 + x q-var,[s 0 ,s 1 ] (t − s) α + |||ω||| p-var,[s,t] ≤ M(K + 2) 1 + x q-var,[s 0 ,s 1 ] × (t − s) (q− )α 1 q− + |||ω||| (q− ) p-var,z s 0 = λx s 0 + (1 − λ)y s 0 = λx s 0 + (1 − λ)x s 0 = x s 0 and z q-var,[s 0 ,s 1 ] = λx + (1 − λ)y q-var,[s 0 ,s 1 ] ≤ λ x q-var,[s 0 ,s 1 ] + (1 − λ) y q-var,[s 0 ,s 1 ] ≤ 2|x s 0 | + 1.
Now, we prove that for any sequence y n ∈ F(B 1 ), there exists an subsequence converges in p-var norm to an element y ∈ B 1 , i.e. F(B 1 ) is relatively compact in B 1 . To do that, we will show that (y n ) are equicontinuous, bounded in (q − )-var norm. Namely, take the sequence y n = F(x n ) ∈ F(S), x n ∈ B 1 . Then, by virtue of Lemma 2.3 we have
sup n y n (q− )-var,[s 0 ,s 1 ] ≤ |x s 0 | + 2M(K + 2)(1 + |x s 0 |)((s 1 − s 0 ) α + |||ω||| p-var,[s 0 ,s 1 ] ).
It means that y n are bounded in C([s 0 , s 1 ], R d ) with sup norm, as well as bounded in
C (q− )-var ([s 0 , s 1 ], R d ).
Moreover, for all n, s 0 ≤ s ≤ t ≤ s 1 ,
|y n t − y n s | ≤ 2M(K + 2)(1 + |x s 0 |) (t − s) α + |||ω||| p-var,[s,t] ,
which implies that (y n ) is equicontinuous. Applying Proposition 5.28 of [8], we conclude that y n converges to some y along a subsequence in C q ([s 0 , s 1 ], R d ). This proves the compactness Step 2: Next, by virtue of the additivity of the Riemann and Young integrals, the solution can be concatenated. Namely, let 0 < t 1 < t 2 < t 3 ≤ T . Let x t be a solution of the Eq. Furthermore, for all such that q − ≥ p the solution x belongs to [s,t] ) (t − s) α + |||ω||| p-var, [s,t] .
of F(B 1 ). Hence, F(B 1 ) is a relative compact set in C q ([s 0 , s 1 ], R d ). We conclude that F isC q− ([t i , t i+1 ], R d ), for all i = 0, N (T, ω). Hence, x ∈ C p ([t 0 , T ], R d ).1 = C 1 (T ), C 2 = C 2 (T ) such that x q-var,[t 0 ,T ] ≤ C 1 [1+(T − t 0 ) α ](1+|x 0 |)(1+ |||ω||| p-var,[t 0 ,T ] )e C 2 |||ω||| p p-var,[t 0 ,T ] , (3.22) where p = max{ p, 1 α }. Proof Since x is a solution, x = F x,+ M(K + 2)(|x s | + |||x||| q-var,
Use the arguments similar to that of the proof of Lemma 3.3 we conclude that there exist C 1 = C 1 (T ) and C 2 = C 2 (T ) such that (3.22) is satisfied.
In order to study the flow generated by the solution of system (3.1) in the next section, we need also to consider the backward version of (3.1) in the following form
x t = x T + T t f (s, x s )ds + T t g(s, x s )dω s , t ∈ [0, T ],(3.23)
where x T ∈ R d is the initial value of the backward equation (3.23), the coefficient functions
f : [0, T ] × R d → R d , g : [0, T ] × R d → R d × R mf (s, x s )ds = T t f (T − u, x T −u )ds = − 0 T −tf (u, y u )du = v 0f (u, y u )du.
Furthermore, by virtue of the property (2.9) of the Young integral we have
T t g(s, x s )dω s = T t g(T − u, x T −u )dω T −u = v 0ĝ (u, y u )dω u .
Therefore, the backward equation (3.23) is equivalent to the forward equation
y v = y 0 + v 0f (u, y u )du + v 0ĝ (u, y u )dω u , v ∈ [0, T ],(3.24)
where y 0 = x T ∈ R d . Now, we verify the conditions of Theorem 3.6 for the forward equation (3.24).
First note that if ω ∈ C p ([0, T ], R m ) thenω ∈ C p ([0, T ], R m )
. Furthermore, the condition (H 1 ) obviously holds forĝ and the condition (i) of (H 2 ) holds forf . For the condition (ii) of (H 2 ) we note that if it holds for f then
|f (v, x)| = | f (T − v, x)| ≤ a|x| + b(T − v) = a|x| +b(v), v ∈ [0, T ], whereb(v) = b(T − v) ∈ L 1 1−α (0, T ; R d ) because (H 2 )(ii)
is satisfied for f . Thus, (H 2 )(ii) is satisfied forf . Consequently, Theorem 3.6 is applicable to the forward equation (3.24) implying that (3.24) has unique solution y ∈ C q ([0, T ], R d ). Since (3.24) is equivalent to the backward equation (3.23) we have the theorem proved. Step 1 (Continuity w.r.t x 0 ): By Proposition 3.7, we can choose N 0 (depending on x 0 , ω) such that
X (t 0 , ·, ω , x 0 ) q-var,[0,T ] ≤ N 0 for all t 0 ∈ [0, T ], |x 0 − x 0 | ≤ 1, ω − ω p-var,[0,T ] ≤ 1.
We use here, for short, notation y . = X (t 0 , ·, ω , x 0 ), y . = X (t 0 , ·, ω , x 0 ). Using arguments similar to that of the proof of Proposition 3.2(ii), we have Therefore,
|(y − y ) t − (y − y ) s | ≤ t s | f (u, y u ) − f (u, y u )|du + t s g(u, y u ) − g(u, y u )dω u ≤ M N 0 (t − s) y − y ∞,|x t − y t | ≤ |x t 0 − y t 0 | + |||x − y||| q-var,[t 0 ,t] ≤ |x 0 − x 0 | C 3 e C 4 (1+ ω p-var,[0,T ] ) p + 1 .
Consequently, we find a positive constants C 1 (T, ω, x 0 ) such that for all t 0 , t ∈ [0, T ], all ω such that ω − ω p-var,[0,T ] < 1, we have
|X (t 0 , t, ω , x 0 ) − X (t 0 , t, ω , x 0 )| ≤ C 1 (T, ω, x 0 )|x 0 − x 0 |. (3.25)
Step 2 (Continuity w.r.t. ω):
Let ω ∈ C p ([0, T ], R m ) be such that ω − ω p-var,[0,T ] ≤ 1. We use here, for short, [s,t] ) ω − ω p-var, [s,t] + ω p-var, [s,t]
notation x . = X (t 0 , ·, ω, x 0 ), x . = X (t 0 , ·, ω , x 0 ). For all s < t in [0, T ], we have |(x − x) t − (x − x) s | = t s [ f (u, x u ) − f (u, x u )]du + t s [g(u, x u ) − g(u, x u )]dω u + t s g(u, x u )d(ω − ω) u ≤ L N 0 (t − s) x − x ∞,[s,t] + M(K + 1)(1 + x q-var,M N 0 (K + 1) |(x − x) s | + x − x q-var,[s,t] × 2 + x δ q-var,[0,T ] + |||x||| δ q-var,[0,T ] ≤ C 5 ω − ω p-var,[s,t] + C 6 t − s + ω p-var,[s,t] |(x − x) s | + x − x q-var,[s,t] ≤ C 5 ω − ω p-var,[s,t] + C 6 t − s + ω p-var,[s,t] |(x − x) s | + x − x q-var,[s,t] ,
where C 5 , C 6 depend on N 0 . Consequently, by virtue of Lemma 2.3 we get [s,t] .
x − x q-var,[s,t] ≤ C 3 ω − ω p-var,[s,t] + C 4 t − s + |||ω||| p-var,[s,t] × |(x − x) s | + x − x q-var,Now, since x t 0 − x t 0 = 0, using Collorary 3.5 on [t 0 , t] (or [t, t 0 ] and use backward equation if t < t 0 ) we find positive constant C 2 (T, ω, x 0 ) such that x − x q-var,[t 0 ,t] ≤ C 2 (T, ω, x 0 ) ω − ω p-var,[t 0 ,t] ≤ C 2 (T, ω, x 0 ) ω − ω p-var,[0,T ] .
Therefore, for all t 0 , t ∈ [0, T ],
|X (t 0 , t, ω , x 0 ) − X (t 0 , t, ω, x 0 )| ≤ C 2 (T, ω, x 0 ) ω − ω p-var,[0,T ] .
(3.26)
Step 3 (Continuity in all variables): Now we fix (t 1 , t 2 , ω, x 0 ) and let (t 1 , t 2 , ω , x 0 ) be in a neighborhood of (t 1 , t 2 , ω, x 0 ) such that
|t 1 − t 1 |, |t 2 − t 2 |, ω − ω p-var,[0,T ] , |x 0 − x 0 | ≤ 1.
By triangle inequality and (3.25), (3.26), we have
|X (t 1 , t 2 , ω , x 0 ) − X (t 1 , t 2 , ω, x 0 )| ≤ |X (t 1 , t 2 , ω , x 0 ) − X (t 1 , t 2 , ω , x 0 )| + |X (t 1 , t 2 , ω , x 0 ) − X (t 1 , t 2 , ω, x 0 )| + |X (t 1 , t 2 , ω, x 0 ) − X (t 1 , t 2 , ω, x 0 )| + |X (t 1 , t 2 , ω, x 0 ) − X (t 1 , t 2 , ω, x 0 )| ≤ (C 1 (T, ω, x 0 ) + C 2 (T, ω, x 0 ))(|x 0 − x 0 | + ω − ω p-var,[0,T ] ) + |X (t 1 , t 2 , ω, x 0 ) − X (t 1 , t 2 , ω, X (t 1 , t 1 , ω, x 0 ))| + |||X (t 1 , ·, ω, x 0 )||| q-var,[t 2 ,t 2 ]
It is obvious that when the triple ., ω, x 0 )||| q-var,[t 2 ,t 2 ] → 0. As for the remaining term, let |t 1 − t 1 | be small enough so that |X (t 1 , t 1 , ω, x 0 ) − x 0 | ≤ 1, using (3.25) again we obtain
(|x 0 − x 0 |, ω − ω p-var,[0,T ] , |t 2 − t 2 |) tends to 0 we have (C 1 (T, ω, x 0 ) + C 2 (T, ω, x 0 ))(|x − x 0 | + ω − ω p-var,[0,T ] ) → 0 and |||X (t 1 ,|X (t 1 , t 2 , ω, X (t 1 , t 1 , ω, x 0 )) − X (t 1 , t 2 , ω, x 0 )| ≤ C 1 (T, ω, x 0 )|X (t 1 , t 1 , ω, x 0 )) − x 0 | ≤ C 1 (T, ω, x 0 ) |||X (t 1 , ·, ω, x 0 )||| q-var,[t 1 ,t 1 ] ,
hence |X (t 1 , t 2 , ω, X (t 1 , t 1 , ω, x 0 )) − X (t 1 , t 2 , ω, x 0 )| → 0 as |t 1 − t 1 | → 0. Summing up the above arguments, we conclude that X is continuous.
Topological Flow Generated by Young Differential Equations
In this section we show that the solution of a nonautonomous Young differential equation generates a two-parameter flow on the phase space R d , thus we can study the long term behavior of the solution flow using the tools of the theory of dynamical systems. We also discuss the autonomous situation, in which we show that the solution then satisfies the cocycle property, thus generates a topological skew product flow. The reader is referred to the work [15] and [18], [17] for the smoothness and diffeomorphism property of the flow.
For simplicity of the presentation, we will assume from now on that for any given T > 0 all hypotheses H 1 − H 3 hold on [0, T ]. Following [13, p. 114], below we introduce the concept of two parameter flows.
Topological Two-Parameter Flows for Nonautonomous Systems
Definition 4.5 (Two-parameter flow)
A family of mappings X s,t : R d → R d depending on two real variables s, t ∈ [a, b] ⊂ R is call a two-parameter flow of homeomorphisms of R d on [a, b] if it satisfies the following conditions:
(i) For any s, t ∈ [a, b] the mapping X s,t is a homeomorphism of R d ; (ii) X s,s = id for any s ∈ [a, b]; (iii) X −1 s,t = X t,s for any s, t ∈ [a, b]; (iv) X s,t = X u,t • X s,u for any s, t, u ∈ [a, b]. Theorem 4.6 (Two-parameter flow generated by Young differential equations) Assume that the conditions H 1 − H 3 hold on any compact interval of R. The family of Cauchy operators of (1.1) generates a two parameter flow of homeomorphisms of R d . Namely, for −∞ <
The problem of generation of the random dynamical systems [1] from stochastic differential equations driven by fractional Brownian noise has been discussed in [3,5,9], to name a few, where they solve the stochastic equation in the path-wise sense as in (4.4) for each realization ω of the fractional Brownian motion. Here in our deterministic setting, due to the fact that the shift dynamical system θ and the solution Cauchy operator are continuous, it follows that the skew product flow defined by
: R × C p 0 (R, R m ) × R d → C p 0 (R, R m ) × R d t (ω, x 0 ) := (θ t ω, X (t, ω, x 0 )) (4.5)
is a continuous mapping which satisfies the group property, i.e. t+s = t • s , for all t, s ∈ R. Therefore it is a topological skew-product dynamical system.
Proof
Consider an arbitrary finite partition = (s i ), i = 0 . . . , n+1, of [s, t]. By assumption and Minskowski inequality we have n i=0
Denote by C p (R, R m ) the space of all continuous functions ω : R → R m such that for any T > 0 the restrictions of ω to [−T, T ] is of C p ([−T, T ], R m ). Equip C p (R, R m ) with the metric d(ω 1 , ω 2 ) := ∞ n=1 2 −n ω 1 − ω 2 p-var,[−n,n]
(
ii) This part is similar to [21, Lemma 7.1] with our function h(s, t) β playing the role of |t − s| β in [21, Lemma 7.1].
we estimate the q-var norm of y in an arbitrary but fixed interval [s, t] ⊂ [0, T ]. Recall the time sequence defined in (2.11). If there exists i such that s < t i < t, put N := sup{n : t n ≤ t}, N := inf{n : t n ≥ s}, N := N − N =N (s, t, ω),
(ii) The conclusion of Lemma 3.3 is still true if one replaces A 0 by A 1/q s,t . (iii) It can be seen from the proof that in the conditions of Lemma 3.3 we have
Theorem 3. 6 (
6Existence and uniqueness of global solution) Consider the Young differential equation(3.1), starting from an arbitrary initial time t 0 ∈ [0, T ),
2 and (3.19)-(3.20), for s 0 , s 1 determined above we have F : C q ([s 0 , s 1 ], R d ) −→ C q ([s 0 , s 1 ], R d ) and F(x) q-var,[s 0 ,s 1 ] = |F(x) s 0 | + |||F(x)||| q-var,[s 0 ,s 1 ] ≤ |x s 0 | + 1 2 1 + x q-var,[s 0 ,s 1 ] .
a compact operator from B 1 into itself. Therefore, by the Schauder-Tychonoff fixed point theorem (see e.g [25, Theorem 2.A, p. 56]), there exists a functionx ∈ B 1 such that F(x) =x, thus there exists a solutionx ∈ B 1 of (3.1) on the interval [s 0 , s 1 ]. Uniqueness of local solutions. Now, we assume that x, y are two solutions in C q ([s 0 , s 1 ], R d ) of the Eq. (3.1) such that x s 0 = y s 0 . It follows that F(x) = x and F(y) = y. Put N 0 = max{ x q-var,[s 0 ,s 1 ] , y q-var,[s 0 ,s 1 ] }, and z = x − y, we have z s 0 = 0 and x ∞,[s 0 ,s 1 ] , y ∞,[s 0 ,s 1 ] ≤ N 0 . By virtue of Proposition 3.2(ii), we obtain |||z||| q-var,[s,t] = |||x − y||| q-var,[s,t] = |||F(x) − F(y)||| q-var,[s,t]≤ M N 0 (K + 1)(1 + 2N δ 0 ) |z s | + |||z||| q-var,[s,t] (t − s + |||ω||| p-var,[s,t] ).(3.21) Applying Corollary 3.5 to the function z, since z s 0 = 0 we conclude that |||z||| q-var,[s 0 ,s 1 ] = 0. That implies z ≡ 0 on [s 0 , s 1 ]. The uniqueness of the local solution is proved.
(3.1) on [t 1 , t 2 ] and y t be a solution of the Eq. (3.1) on [t 2 , t 3 ] with y(t 2 ) = x(t 2 ). Define a continuous function z(·) : [t 1 , t 3 ] → R d by setting z(t) = x(t) on [t 1 , t 2 ] and z(t) = y(t) on [t 2 , t 3 ]. Then z(·) is the solution of the Young differential equation (3.1) on [t 1 , t 3 ]. Conversely, if z t is a solution on [t 1 , t 3 ] then its restrictions on [t 1 , t 2 ] and on [t 2 , t 3 ] are solutions of the corresponding equation with the corresponding initial values. Step 3: Finally, apply the estimates (2.17) to the case of μ being defined by (3.19), we can easily get the unique global solution to the Eq. (3.1) on [t 0 , T ]. Put n 0 = min{n : τ n > t 0 }. The interval [t 0 , T ] can be covered by N (T, ω) − n 0 + 1 intervals [t i , t i+1 ], i = 0, N (T, ω) − n 0 + 1, determined by times t i = τ n 0 +i−1 , i = 1, . . . , N (T, ω) − n 0 , with parameter μ being defined by (3.19) and t N (T,ω)+1 := T . The arguments in Step 1 are applicable to each of intervals [t i , t i+1 ], i = 0, N (T, ω), implying the existence and uniqueness of solutions on those intervals. Then, starting at x(t 0 ) = x t 0 the unique solution of (3.1) on [t 0 , t 1 ] is extended uniquely to [t 1 , t 2 ], then further by induction up to [t N (T,ω)−1 , t N (T,ω) ] and lastly to [t N (T,ω) , T ]. The solution x of (3.1) on [t 0 , T ] then exists uniquely.
Proposition 3. 7
7Assume that the conditions H 1 − H 3 are satisfied. Let 0 ≤ t 0 < T . Denote by x(·) = x(t 0 , ·, ω, x 0 ) the solution of the Eq. (3.1) on [t 0 , T ]. Then there exist positive constants C
are continuous functions, and the driven force ω : [0, T ] → R m belongs to C p ([0, T ], R m ).
Theorem 3. 8 (
8Existence and uniqueness of solutions of backward equation) Consider the backward equation (3.23) on [0, T ]. Assume that the conditions H 1 − H 3 hold. Then the backward equation (3.23) has a unique solution x ∈ C q ([0, T ], R d ), where q is chosen as above satisfying (3.3). Proof We make a change of variableŝ f (u, x) := f (T − u, x),ĝ(u, x) := g(T − u, x),ω(u) := ω(T − u), y u := x T −u , u ∈ [0, T ]. Then x T = y 0 , and by putting v = T − t and u = T − s we have T t
Theorem 3. 9
9Suppose that the assumptions of Theorem 3.6 are satisfied. Denote byX (t 0 , ·, ω, x 0 ) the solution of (3.1) starting from x 0 at time t 0 , i.e. X(t 0 , t 0 , ω, x 0 ) = x 0 .Then the solution mappingX : [0, T ] × [0, T ] × C p ([0, T ], R m ) × R d → R d , (s, t, ω, z) → X (s, t, ω, z),is continuous.Proof First observe that, fixing (ω, x 0 ) ∈ C p ([0, T ], R m ) × R d and looking at forward and backward equations (3.1) and (3.23), we can extend the solution X (t 0 , ·, ω, x 0 ) of (3.1), with the initial value x 0 at t 0 to the whole [0, T ]. The proof is divided into several steps.
Remark 3.10The time interval in Theorems 3.6 to 3.9 needs not be [0, T ]. It can be [t 0 , t 0 +T ] for any t 0 ∈ R, T > 0.
Theorem 4. 1 (
1Different trajectories do not intersect) Assume that the conditions H 1 − H 3 hold. Let x t andx t be two solutions of the Young differential equation (3.1) on [0, T ]. If x a =x a for some a ∈ [0, T ] then x t =x t for all t ∈ [0, T ]. In other words, two solutions of the differential equation (3.1) either coincide or do not intersect. Proof Suppose that x a =x a for some a ∈ [0, T ]. If a = 0 then by the uniqueness of the solution provided by Theorem 3.6, x t =x t for all t ∈ [0, T ]. Let a ∈ (0, T ]. Since the restrictions of the functions x t andx t on [a, T ] are solutions of the Eq. (3.1) starting from a, Theorem 3.6 implies that x t =x t for all t ∈ [a, T ]. Now, consider the restrictions of the functions x t andx t on [0, a]. They are solutions of the equations x t = x 0 + t 0 f (s, x s )ds + t 0 g(s, x s )dω s , t ∈ [0, a],
Assume that ω ∈ C p ([0, T ], R) and y ∈ C q ([0, T ], R d ) satisfy |y t − y s | ≤ A1
p + 1
q > 1.
1/q
s,t + a 1
t
s
y u du + a 2
t
s
y u dω u , ∀s, t ∈ [0, T ], s < t,
(3.13)
for some fixed control function A on [0, T ] and some constants a 1 , a 2 ≥ 0. Then there
exists a constant C independent of T such that for every s, t ∈ [0, T ], s < t,
|||y||| q-var,[s,t] ≤ (|y s | + A 0 )e C(|t−s| p +|||ω||| p
p-var,[s,t] ) ,
(3.14)
) .
)Therefore, by virtue of Lemma 2.3, we get|||y||| q-var,[s,t] ≤ A 1/q s,t + c(|y s | + |||y||| q-var,[s,t] )(t − s + |||ω||| p-var,[s,t] ) ≤ A 0 + c(|y s | + |||y||| q-var,[s,t] )(t − s + |||ω||| p-var,[s,t] ).(3.15)
Now we construct the time sequence t i with parameter {1, 1
2c } according to Sect. 2.2, that is
we already have |||y||| q-var,[s,t] ≤ 2 A 0 + |y s |. To sum up, for any [s, t] ⊂ [0, T ] we have the estimate|||y||| q-var,[s,t] ≤ (2 A 0 + |y s |)2 2N .
Combining with (2.17), we conclude that
|||y||| q-var,[s,t] ≤ (2 A 0 + |y s |)2 4 p c p (|t−s| p +|||ω||| p
p-var,[s,t] )
≤ (2 A 0 + |y s |)e C(|t−s| p +|||ω||| p
p-var,[s,t] ) ,
3.3 we replace the condition (3.13) by the condition |||y||| q-var,[s,t] ≤ A1/q
s,t + a 1 (|y s | + |||y||| q-var,[s,t] )(t − s + |||ω||| p-var,[s,t] )
(3.17)
for all s < t in [0, T ], a positive constant a 1 > 0 and ω ∈ C p ([0, T ], R m ). Then there exists
a constant C independent of T such that for every s < t in [0, T ]
|||y||| q-var,[s,t] ≤ (|y s | + A
1/q
s,t )e C(|t−s| p +|||ω||| p
p-var,[s,t] )
[s,t] + M N 0 (K + 1) ω p-var,[s,t] |y s − y s | + y − y q-var,[s,t] (2 + 2N δ 0 ) ≤ M N 0 (K + 1)(2 + 2N δ 0 ) |y s − y s | + y − y q-var,[s,t] t − s + ω p-var,[s,t] .Due to Corollary 3.5, there exist constants C 3 , C 4 depending on parameters of the equation (3.1) and N 0 , such that y − y q-var,[0,T ] ≤ |y 0 − y 0 |C 3 e C 4||| ω ||| p p-var,[0,T ] ≤ |y 0 − y 0 |C 3 e C 4 (1+ ω p-var,[0,T ] ) p .
/ p,∞ b ([a, b], R d ) ⊂ C p ([a, b], R d ).
t t 0 f (s, x s )ds + t t 0 g(s, x s )dω s , ∀t ∈ [t 0 , t 1 ]. (3.5)Note that a fixed point of F is a solution of (3.1) on [t 0 , t 1 ] with the boundary condition x(t 0 ) = x t 0 (the initial condition x t 0 of (3.1) is then not given).
q− , hence |||F(x)||| (q− )−var,[s 0 ,s 1 ] ≤ M(K + 2) 1 + x q-var,[s 0 ,s 1 ] (s 1 − s 0 ) α + |||ω||| p-var,[s 0 ,s 1 ] and the assertion follows by an application of Lemma 2.3. Consider the setB 1 := x ∈ C q ([s 0 , s 1 ], R d )| x(s 0 ) = x s 0 , x q-var,[s 0 ,s 1 ] ≤ 2|x s 0 | + 1 .
t 1 ≤ t 2 < +∞ and ω ∈ C p (R, R m ) we define X (t 1 , t 2 , ω, ·) according to Definition 4.3 and setting X (t 2 , t 1 , ω, ·) := X −1 (t 1 ,t 2 , ω, ·), then the family X (t 1 , t 2 , ω, ·), t 1 , t 2 ∈ [0, T ], is a two parameter flow of homeomorphisms of R d on [0, T ]. Furthermore, the flow is continuous.
Acknowledgements Open access funding provided by Max Planck Society. We would like to thank the anonymous referees for their careful reading and insightful remarks which lead to improvement of our manuscript.with the initial values x 0 andx 0 respectively. Since x a =x a , the two functions x t andx t are solutions of the same backward equation(4.1)with the same initial value x a . Theorem 3.8 asserts the uniqueness of solution of (4.1) on [0, a], hence x t must coincide withx t on [0, a] and the theorem is proved. Now, in analog with the theory of ordinary differential equation we give a definition of the Cauchy operator of the Eq. (1.1), which is an operator in R d acting along trajectoties of (1.1).Definition 4.3 (Cauchy operator) Suppose that on any compact interval of R the conditionsis the mapping along trajectories of (1.1) from time moment t 1 to time moment t 2 , i.e., for any vector For any −∞ < t 1 ≤ t 2 < +∞ the Cauchy operator X (t 1 , t 2 , ω, ·) of (1.1) is a homeomorphism. Moreover, X (t 1 , t 1 , ω, ·) = id.Proof By Theorem 4.1 the Cauchy operator X (t 1 , t 2 , ω, ·) is an injection. Using arguments of the proof of Theorem 4.1 we get that the equationwith the terminal value x t 2 ∈ R d and unknown initial value x t 1 , is equivalent to the following initial value problem for the backward equation on [t 1 , t 2 ]3)Proof Conditions (i)-(ii) of Definition 4.5 follow from Theorem 4.4. Condition (iii) of Definition 4.5 follows from the definition X (t 2 , t 1 , ω, ·) := X −1 (t 1 , t 2 , ω, ·) for t 1 ≤ t 2 . Actually, it is seen from the proof of Theorem 4.4 that the inverse X (t 2 , t 1 , ω, ·) satisfies the backward equation(4.3).Condition (iv) of Definition 4.5 follows from the definition of the Cauchy operators and Theorem 4.1.The continuity of the flow follows directly from Theorem 3.9.Topological Skew Product Flows for Autonomous SystemsIn this subsection we restrict the discussion to the autonomous systemwhere f, g are time independent and sastisfy the assumptions H 1 − H 3 . We consider ω in the space C p 0 (R, R m ) := {ω ∈ C p 0 (R, R m ), ω(0) = 0} and introduce the shift operator θ :It is easy to check that θ t+s ω = θ t • θ s ω for all t, s ∈ R and ω ∈ C p 0 (R, R m ). Moreover, it is followed from [3, Theorem 5] that θ is continuous w.r.t. (t, ω),is a continuous dynamical system. On the other hand, it follows from definition of Young integral that t+τ s+τ y u dω(u) = t s y u+τ dθ τ ω(u), ∀s, t, τ ∈ R (see[5]for a version using fractional derivatives). Hence from the existence and uniqueness theorem, the solution X (t, s, ω, x 0 ) of the Young equation (4.4) satisfies X (t, s, ω, x 0 ) = X (t − s, 0, θ s ω, x 0 ), ∀t, s ∈ R, therefore the mapping ϕ : R × C p 0 (R, R m ) × R d → R d defined by ϕ(t, ω)x 0 := X (t, 0, ω, x 0 ) possesses a cocycle property ϕ(t + s, ω)x 0 = ϕ(t, θ s ω) • ϕ(s, ω)x 0 , ∀x 0 ∈ R d , ω ∈ C p 0 (R, R m ), t, s ∈ R.
It is clear from the proof of Theorems 3.6 and 3.9 that the solutions of (1.1) depend continuously on the initial values. Therefore, the Cauchy operator X (t 1 , t 2 , ω, ·) acts continuously on R d. Consequently, the Cauchy operator X (t 1 , t 2 , ω, ·) is a surjection, thus a bijection. Similar conclusion holds for the inverse X −1 (t 1 , t 2 , ω, ·) by using backward equation. Hence X (t 1 , t 2 , ω, ·) is a homeomorphism and trivially X (t 1 , t 1 , ω, ·) = idConsequently, the Cauchy operator X (t 1 , t 2 , ω, ·) is a surjection, thus a bijection. It is clear from the proof of Theorems 3.6 and 3.9 that the solutions of (1.1) depend contin- uously on the initial values. Therefore, the Cauchy operator X (t 1 , t 2 , ω, ·) acts continuously on R d . Similar conclusion holds for the inverse X −1 (t 1 , t 2 , ω, ·) by using backward equation. Hence X (t 1 , t 2 , ω, ·) is a homeomorphism and trivially X (t 1 , t 1 , ω, ·) = id.
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"INFERRING GENE ASSOCIATION NETWORKS USING SPARSE CANONICAL CORRELATION ANALYSIS *",
"INFERRING GENE ASSOCIATION NETWORKS USING SPARSE CANONICAL CORRELATION ANALYSIS *"
]
| [
"Y X Rachel Wang ",
"Keni Jiang ",
"Lewis J Feldman ",
"Peter J Bickel ",
"Haiyan Huang ",
"\nDepartment of Statistics\nDepartment of Plant and Microbial Biology\nUniversity of California\nBerkeley\n",
"\nUniversity of California\nBerkeley §\n"
]
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"Department of Statistics\nDepartment of Plant and Microbial Biology\nUniversity of California\nBerkeley",
"University of California\nBerkeley §"
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| Networks pervade many disciplines of science for analyzing complex systems with interacting components. In particular, this concept is commonly used to model interactions between genes and identify closely associated genes forming functional modules. However, characterizing gene relationships remains a challenging problem due to the complexity of high dimensional biological data. In this paper, we develop a two-stage procedure for estimating gene association networks and inferring gene functional groups as tightly connected communities. We propose a novel way of computing edge weights in a gene network, which relies on sparse canonical correlation analysis with repeated subsampling and random partition. Based on the estimated network, community structures can be identified using the block model or hierarchical clustering. Our approach is conceptually appealing as it aims to capture gene relationships in a more realistic manner by taking into account higher level group interactions. It is also flexible enough to incorporate prior biological knowledge. Comparisons with several popular approaches using simulated and real data show our procedure improves both the statistical significance and biological interpretability of the results. In addition to achieving considerably lower false positive rates, our procedure shows better performance in detecting important biological pathways. | 10.1214/14-aoas792 | [
"https://arxiv.org/pdf/1401.6504v2.pdf"
]
| 88,514,036 | 1401.6504 | ea0363152757932d9edb0ad7f618d5e6351d08f1 |
INFERRING GENE ASSOCIATION NETWORKS USING SPARSE CANONICAL CORRELATION ANALYSIS *
Y X Rachel Wang
Keni Jiang
Lewis J Feldman
Peter J Bickel
Haiyan Huang
Department of Statistics
Department of Plant and Microbial Biology
University of California
Berkeley
University of California
Berkeley §
INFERRING GENE ASSOCIATION NETWORKS USING SPARSE CANONICAL CORRELATION ANALYSIS *
Submitted to the Annals of Applied Statistics
Networks pervade many disciplines of science for analyzing complex systems with interacting components. In particular, this concept is commonly used to model interactions between genes and identify closely associated genes forming functional modules. However, characterizing gene relationships remains a challenging problem due to the complexity of high dimensional biological data. In this paper, we develop a two-stage procedure for estimating gene association networks and inferring gene functional groups as tightly connected communities. We propose a novel way of computing edge weights in a gene network, which relies on sparse canonical correlation analysis with repeated subsampling and random partition. Based on the estimated network, community structures can be identified using the block model or hierarchical clustering. Our approach is conceptually appealing as it aims to capture gene relationships in a more realistic manner by taking into account higher level group interactions. It is also flexible enough to incorporate prior biological knowledge. Comparisons with several popular approaches using simulated and real data show our procedure improves both the statistical significance and biological interpretability of the results. In addition to achieving considerably lower false positive rates, our procedure shows better performance in detecting important biological pathways.
Networks pervade many disciplines of science for analyzing complex systems with interacting components. In particular, this concept is commonly used to model interactions between genes and identify closely associated genes forming functional modules. However, characterizing gene relationships remains a challenging problem due to the complexity of high dimensional biological data. In this paper, we develop a two-stage procedure for estimating gene association networks and inferring gene functional groups as tightly connected communities. We propose a novel way of computing edge weights in a gene network, which relies on sparse canonical correlation analysis with repeated subsampling and random partition. Based on the estimated network, community structures can be identified using the block model or hierarchical clustering. Our approach is conceptually appealing as it aims to capture gene relationships in a more realistic manner by taking into account higher level group interactions. It is also flexible enough to incorporate prior biological knowledge. Comparisons with several popular approaches using simulated and real data show our procedure improves both the statistical significance and biological interpretability of the results. In addition to achieving considerably lower false positive rates, our procedure shows better performance in detecting important biological pathways.
1. Introduction. Many complex systems in science and nature are composed of interacting parts. Such parts can be modeled as nodes and their relationships as edges in a network. Network modeling has found numerous applications in the studies of friendship networks in sociology, the Internet and the World Wide Web in information technology, predator-prey interactions in ecology, and protein-protein interactions, gene networks and many other biochemical networks in biology. More examples of networks and their basic properties can be found in Newman (2010). Close examinations of these networks can reveal important knowledge about the nature of the individual nodes, their connections, and most crucially, interesting connection patterns such as communities, where groups of nodes exhibit high internal connectivity. In this paper, we focus on the study of gene association networks, where communities correspond to genes with related functional groupings. Many of of these functional groups encode biological pathways. A major task in understanding biological processes is to identify these pathway genes and elucidate the relationships between them.
The study of community structure relies on knowing the relationships between pairs of nodes. In gene networks, direct observation of gene relationships by experimental approaches is extremely cost-prohibitive given that the typical size of the networks is in the tens of thousands. The gene expression levels, on the other hand, are easier to measure and can be regarded as sets of covariates associated with the nodes. The analysis of this type of network is essentially a two-fold problem. First, a network of relations needs to be learned in an unsupervised fashion using the covariates associated with each node. Community detection techniques can then be applied to identify tightly knit sub-structures. In this paper, we investigate this composite problem and aim to isolate gene functional groups through exploiting the community structure of a constructed gene network.
Constructing gene network using expression data has remained a challenging unsupervised learning problem in the statistics literature due to the complexity of data structure and the difficulty of finding an appropriate measure for characterizing gene relationships. Popular measures include gene co-expression estimating marginal relationships between pairs of genes and conditional measures taking into account the influence of other genes. Widely used co-expression measures include the Euclidean distance or the angle between vectors of observed expression levels, or most commonly, the marginal covariance or correlation. Conditional measures are based on the sparse inverse covariance matrix. This is equivalent to using partial correlations (i.e. the correlation between any two genes conditioned on one or several other genes) and leads to a graph built on conditional independence relationships.
A major concern of the two types of measures is their limited biological inference. Co-expression measures only consider pairwise relationships between genes and can easily miss higher level group interactions. In this regard, the inverse covariance matrix offers a more realistic way to represent complex gene networks due to its interpretation in terms of conditional correlations. However, the selection of appropriate conditional set of genes INFERRING GENE ASSOCIATION NETWORKS USING SCCA 3 is critical to ensure the correct inference of conditional relationships. In the current literature, partial correlation is usually calculated conditioned on either all of the available genes or a more or less arbitrary subset of them that may contain noisy (biologically unrelated) genes. de la Fuente et al. (2004) reported that conditioning on all genes simultaneously can introduce spurious dependencies which are not from a direct causal or common ancestors effect. There are also efforts on using lower order partial correlations (de la Fuente et al. (2004); Magwene and Kim (2004); Wille et al. (2004); Wille and Bühlmann (2006)). These methods, however, lose sensitivity for inferring higher level gene associations and cannot guarantee to eliminate the effect of noisy genes. Kim et al. (2012) proposed to minimize the impact of noisy genes by conditioning on a small set (3-5 genes) of known pathway genes, or "seed genes". Such prior biological information however, is not always available, especially in exploratory studies.
In this paper we tackle the problem of estimating gene relationships when the correct conditional set for partial correlation is unknown. As a result, gene functional groups can be found via a two-stage procedure. We first introduce a new way of estimating edge weights in a gene network using sparse canonical correlation analysis (SCCA) to look for sparse, maximally correlated subsets of genes. By noting the connection between partial correlations and linear regression, the advantage of partial correlations for detecting group interactions naturally carries over to our method. Our approach is flexible and can be adapted to work with or without prior biological knowledge. There has been a growing interest in applying SCCA to genomic datasets (Waaijenborg, Verselewe de Witt Hamer and Zwinderman (2008); Parkhomenko, Tritchler and Beyene (2009); Witten and Tibshirani (2009); Lee et al. (2011)) in the context of studying relationships between two or more sets of variables, such as gene expression levels, copy numbers and other phenotype variations, with measurements taken from the same sample. One novelty of our method lies in the application of SCCA to a single dataset facilitated by a random partition scheme. Using this construction, we aim to build an edge weight matrix for the whole gene network to more effectively represent gene functional relationships.
The second stage of our procedure involves identifying these densely connected communities within the network. We consider two well-known methods in the network literature to solve this problem: the stochastic block model (SBM) and hierarchical clustering (HC). The SBM, formally introduced by Holland, Laskey and Leinhardt (1983), generalize the Erdős-Rényi model and groups nodes into different classes with different connectivities. Parameter estimation for SBM remains an active area of research due to the graph intricacy of the model. Various algorithms have been proposed (e.g. Snijders and Nowicki (1997), Nowicki and Snijders (2001), Bickel and Chen (2009), J.-J. Daudin and Robin (2008), Rohe, Chatterjee and Yu (2011)), but many methods are not scalable and their theoretical properties remain only partially studied. In this paper, we adopt the pseudo-likelihood algorithm proposed by Amini et al. (2013). In addition to fitting the conventional SBM, a conditional variation of the algorithm also allows for fitting networks with varying node degrees within communities -an idea suggested in the degree-corrected block model by Karrer and Newman (2011). In this paper, we mainly evaluate the performance of our edge weight matrix by examples with non-overlapping functional groups. We give an example with overlapping groups in Section 5 of the supplementary information. In that case a different community detection method (e.g. mixed membership SBM )) can be applied to identify the overlapping structures.
The rest of the paper is organized as follows. In Section 2, we discuss the motivations behind our new scheme of computing edge weights in detail and provide an outline of the full procedure. In Section 3, comparisons are made between our procedure and correlation-based method, as well as between SBM and HC using simulations. We demonstrate that our procedure in general achieves a significant reduction in the rate of false discoveries. To test its performance in real data applications, our procedure is applied to an Arabidopsis thaliana microarray dataset obtained under oxidation stress. Finally in Section 4, we discuss the advantages and potential extensions of the present method.
2. Methods. As mentioned in Section 1, the conditional correlation interpretation of partial correlation suggests it is a more appropriate framework for modeling higher level interactions in gene networks, provided the conditional computation is carried out properly. In this section, we discuss some of the limitations of the partial correlation approach that arise due to its reliance on the correct selection of conditional sets of genes and how our SCCA based approach circumvents this difficulty. We then give detailed description of our new method of estimating an edge weight matrix using SCCA with subsampling and consider ways of recovering community structures from such a matrix.
2.1. Method motivation. Recall that when the gene expression levels follow a multivariate normal distribution, for a set of genes W , the partial INFERRING GENE ASSOCIATION NETWORKS USING SCCA 5 correlation between genes i and j can be expressed as
(2.1) ρ ij = cor(i, j|W \{i, j}) = − ω ij √ ω ii ω jj , i = j 1, i = j,
where ω ij are elements in the precision matrix (Σ G ) −1 with Σ G being the gene covariance matrix of the set W (see e.g. Edwards (2000)). Genes i and j being conditionally independent is equivalent to the corresponding partial correlation and element in the precision matrix being zero. As pointed out in de la Fuente et al. (2004) and Kim et al. (2012), the selection of a proper set of genes on which the correlation in (2.1) is conditioned determines the effectiveness of using partial correlation to measure gene interactions. The inclusion of noisy (biologically unrelated) genes in the set W \{i, j} may introduce spurious dependencies and consequently false edges in the estimated network. The use of partial correlation may also prove problematic when W contains multiple pathways. For example, suppose the set W has two pathways {x, y, z} and {u, v} with expression relationships
(2.2) z = x + y + 1 u + 2 v, u = δ 1 x + δ 2 y + δ 3 z + v,
where i and δ j are small positive constants so that the dependencies between the two pathways are negligible, and gene v is independent of genes x and y. Computing the partial correlations, we have the desired dependencies:
cor(z, x|W \{z, x}) = cor(z, y|W \{z, y}) = 1, cor(u, v|W \{u, v}) = 1, but also some spurious ones:
cor(u, x|W \{u, x}) = cor(u, y|W \{u, y}) = 1.
Using these partial correlations to construct an edge weight matrix would imply the two pathways are fully connected. The proper calculation should condition only on genes in the same pathway, but such information is usually hard to obtain in practice. A better edge weight measure should take into account the magnitude of the linear coefficients in (2.2) so that it reflects the amount of contribution each gene makes to a pathway and the two-block nature of the network. Recall that in a regression setting, the regression coefficients are multiplicative functions of the corresponding partial correlation. In this sense, the coefficients encompass more information and provide a better resolution on gene relationships than the partial correlations alone.
Motivated by these observations, we propose a two-stage procedure to look for gene functional groups. First we make direct use of the linear coefficients found by SCCA with subsampling to build an edge weight matrix reflecting the aggregated level of direct or partial gene interactions. More discussion on how CCA coefficients relate to partial correlations can be found in Section 4 of the supplementary information. Sparsity is imposed to reduce dimensionality and in particular in the example above, ensures the mixing of the two pathways is negligible on average. The second stage of our procedure concerns discovering densely connected communities in the network prescribed by the edge weight matrix.
2.2.
Review of sparse canonical correlation analysis and its implementation. Let X ∈ R n×q 1 be a matrix comprised of n observations on q 1 variables, and Y ∈ R n×q 2 a matrix comprised of n observations on q 2 variables. CCA introduced by Hotelling (1936) involves finding maximally correlated linear combinations between the two sets of variables. More explicitly, one seeks to find α ∈ R q 2 and β ∈ R q 1 that solve the optimization problem
(2.3) max α,β α T Σ Y X β subject to α T Σ Y Y α = 1, β T Σ XX β = 1,
where Σ (·,·) represent the correlation matrices. Note that provided the variables in X and Y have nonzero variances, this is equivalent to the usual CCA formulation in terms of covariance matrices.
In practice the population correlations are replaced with their sample counterparts. That is, S Y X = Y T X/(n − 1), S XX = X T X/(n − 1) and S Y Y = Y T Y /(n−1), assuming the columns of X and Y have been centered and scaled. Denote a and b the weight vectors obtained by using these sample correlations, then (a, b) solves
(2.4) max a,b a T S Y X b subject to a T S Y Y a = 1, b T S XX b = 1.
For high throughput biological data, q 1 and q 2 are typically much larger than n. It is thus natural to impose sparsity on a and b, and this can be done by including (typically convex) penalty functions in (2.4). A number of studies (Waaijenborg, Verselewe de Witt Hamer and Zwinderman (2008); Witten, Tibshirani and Hastie (2009); Parkhomenko, Tritchler and Beyene (2009)) have proposed various methods for formulating the penalized optimization problem and obtaining sparse solutions. Here we adopt the diagonal penalized CCA criterion given by Witten, Tibshirani and Hastie (2009), which treats the covariance matrices in (2.4) as diagonal and relaxes the equality constraints for convexity:
(2.5) max a,b a T Y T Xb subject to a T a ≤ 1, b T b ≤ 1, p 1 (a) ≤ c 1 , p 2 (b) ≤ c 2 ,
where p 1 and p 2 are convex penalty functions. In this paper, we consider an L 1 penalty and solve the above optimization using the modified NIPALS algorithm proposed by Lee et al. (2011), which is reported to yield better empirical performance than Witten et. al. (2009)'s algorithm. The modified NIPALS algorithm performs penalized regressions iteratively on X and Y with the penalty functions p λ 1 (·) = λ 1 · 1 and p λ 2 (·) = λ 2 · 1 . This is an equivalent formulation to iteratively optimizing (2.5) using the bounded constraints.
It is important to note that one more complication arises when SCCA is applied to gene expression data. In CCA, the estimation of the correlation matrix using sample correlations requires the data matrices X and Y have independent rows. However, given a gene expression matrix with genes in columns and experiments in rows, it is often the case that row-wise and column-wise dependencies co-exist. Row-wise dependencies, or experiment dependencies, can be defined as the dependencies in gene expression between experiments due to the similar or related cellular states induced by the experiments (Teng and Huang (2009)). When unaccounted for, they can introduce redundancies that overwhelm the important signals and lead to inaccurate estimates of gene correlation matrix. To decouple the effect of experiment dependencies from the estimation of gene correlations, we apply the Knorm procedure from Teng and Huang (2009). The Knorm model assumes a multiplicative structure for the gene-experiment interactions, and iteratively estimates the gene covariance matrix and experiment covariance matrix through a weighted correlation formula. In addition, row subsampling and covariance shrinkage are used to ensure robust estimation.
2.3. Constructing edge weight matrix by SCCA with subsampling. Suppose an observed dataset contains measurements of the expression levels of p genes in n experiments, where each experiment has a small number of replicates. We next describe a new procedure of computing edge weights in the gene network.
Summary of procedure (i) Data normalization by Knorm. A gene expression matrix Z b of dimension n × p can be generated from the full dataset by sampling one replicate from each experiment. Using the Knorm model in Teng and Huang (2009),
we normalize Z b as (2.6) Z * b = (Σ E ) −1/2 (Z b −M),
whereM is the estimated mean matrix andΣ E is the estimated experiment correlation matrix.
(ii) Subsampling and SCCA with random partition. For each normalized expression matrix Z * b , sample a fixed fraction s, say 70%, of the genes to obtain an n × sp submatrix Z sub b . For each partition t, randomly split the columns of Z sub b into two groups of equal size to form X sub b,t and Y sub b,t . Find sparse weight vectors a sub b,t and b sub b,t using the modified NIPALS algorithm (Lee et al. (2011)) with the L 1 penalty and tuning parameters λ = (λ 1 , λ 2 ), the choice of which will be discussed in Section 3. Combine them to form an unsigned weight vector c b,t and average over all the partitions to obtain the average weightsc b . Define edge weight matrix A b =c bc T b , setting diag(A b ) = 0 to exclude self loops. More insights into this step are provided in the remarks below.
(iii) Repeat step (ii) B times. DefineĀ = 1/B B b=1 A b and normalize by the maximum value inĀ. As we search through different subsets of genes, different signal groups are identified depending on the strengths of the individual functional group in the subset.Ā can be taken as an aggregated measure of direct or partial gene interactions. As will be shown empirically in Section 3.1, the averaged result leads to the formation of a distinct block structure with different connectivities in the matrix.
We make a few remarks here: a) During the random partition, the two sets of genes do not have to be exactly equal in size, but they need to be comparable in order to maximize the chance of separating any gene functional group of interest into two sets. b) Subsampling is necessary if we aim to identify multiple functional groups simultaneously, as there will be multiple groups with strong interactions and not all of them can be detected unless different subsets of genes are considered. On the other hand, if running time is not a concern, we can always run the whole procedure iteratively with no subsampling, each time identifying one dominating signal group and removing it from the subsequent analysis. For more discussion about the choice of subsampling levels, we refer to Section 5 in the supplementary information. c) Overall subsampling and random partition enable us to consider different subsets of the genes and ways to group them. Thus the elements inĀ can be interpreted as an aggregated measure of partial correlations of different orders as the algorithm steps through different conditional sets of genes. d) If it is known in advance that some genes operate in the same functional group, one may focus on the identification of this group first and incorporate 9 the prior knowledge by lowering the penalties associated with those genes in the SCCA algorithm. Examples involving using prior knowledge can be found in Section 5 of the supplementary information.
Below we show asymptotically the validity of our random partition scheme by considering a simple case where the entire gene set contains only one correlated functional group. In this case, we can quantify the asymptotic difference in the assigned weights between functional group genes and noisy genes.
Asymptotic behavior of the random partition scheme
Suppose Z ∈ R n×p is an matrix containing expression levels for p genes and n independent experiments, and its columns have been centered and scaled. We consider the case where there exists only one functional group and all the other genes are uncorrelated. Due to this simplification, no subsampling is needed, and the use of CCA without sparsity suffices since in the asymptotics we consider the regime n → ∞ with p fixed. Without loss of generality, in the entire gene set G = {1, 2, . . . , p} let the first k genes K = {1, 2, . . . , k} form one pathway.
For every partition t, let a t and b t be the solutions to (2.4) and c t be the list of the absolute values |a t | and |b t | ordered according to the gene list. Assuming Z follows a multivariate normal distribution and the inverse covariance matrix has a diagonal block structure (detailed assumptions are presented in Section 2 of the supplementary information), we have the following proposition regarding the asymptotic difference between the average values of {c i,t , i ∈ K} and {c j,t , j / ∈ K}. For convenience suppose p is even and denote q = p/2.
Proposition 2.1. Letc = N t=1 c t /N ,
where N is the number of partitions, then given 1 < k < q,
(2.7) lim N →∞ lim n→∞ (min i∈Kc i − max j / ∈Kc j ) = D for some positive constant D.
We give the proof with a lower bound on D in the supplementary information Section 2. The separation inc implies the genes in the graph characterized by the edge weight matrixĀ =cc T can be grouped into different clusters based on their connectivity. As will be demonstrated in Section 3.1, A as defined in Section 2.3 exhibits a natural block structure when there is one or multiple functional groups.
In addition, in supplementary information Section 2, we present asymptotic analysis of an example to theoretically understand the performance of our procedure when multiple functional groups exist, and highlight and explain the role of subsampling. One can see that analytical computations are tedious even in this explicit case. We deem the general proof out of scope of this paper. We next discuss two approaches used in this paper for identifying such community structures.
2.4. Identify community structures in edge weight matrix by SBM and HC. Community structures naturally exist in gene networks. Genes in the same pathway or with related functionalities are expected to have dense connections, whereas biologically unrelated (noisy) genes may be only sparsely connected. Despite the overall heterogeneity of connectivity in the network, within each community we may expect to find the node degrees more homogeneous. This is a situation where the SBM offers an appropriate modeling framework.
Definition 2.2. A SBM is a family of probability distributions for a graph with node set {1, 2, . . . , p} and Q node blocks defined as follows.
1. Let C = (C 1 , C 2 , . . . , C p ) denote the set of labels such that C i = k if the node i belongs to block k. C i.i.d ∼ Multinomial(γ),
where γ = (γ 1 , γ 2 , . . . , γ Q ) is the vector of proportions. 2. Let π = (π lk ) 1≤l,k≤Q be a symmetric matrix of block dependent edge probability matrix and A be the adjacency matrix. Conditioned on the block labels C, (A ij ) for i < j are independent, and
P (A ij |C) = P (A ij = 1|C i = l, C j = k) = π lk .
DiscretizingĀ defined in Section 2.3 into a 0-1 matrix, the class labels and the parameters γ and π are estimated using the pseudo-likelihood algorithm by Amini et al. (2013). The unconditional version of the algorithm fits the conventional SBM above, while the conditional version takes into account the variability of node degrees within blocks (Karrer and Newman (2011)). Potential functional groups are identified as classes having large diagonal entries in π.
Another widely used non-model-based technique for extracting communities, especially in the study of social networks (Scott (2000)), is agglomerative HC. Here we adopt the Ward's distance (Ward (1963)) for the computation of merging costs. Let g i be the nodes, the distance between two clusters M 1 , M 2 is defined as
d(M 1 , M 2 ) = n 1 n 2 n 1 + n 2 m 1 − m 2 2 = 1 2(n 1 + n 2 ) i,j∈M 1 ∪M 2 g i − g j 2 − 1 2n 1 i,j∈M 1 g i − g j 2 − 1 2n 2 i,j∈M 2 g i − g j 2
where n 1 and n 2 denote the sizes of M 1 and M 2 , m 1 and m 2 are the cluster centers of M 1 and M 2 respectively. A natural way to define the square of the pairwise distance is g i − g j 2 = 1 −Ā ij for i = j, and zero otherwise. Since
Ward's method minimizes the increase in the within group sum of squares at each merging and tends to merge clusters that are close to each other and small in size, a small cluster that manages to survive a long distance before coalescing is likely to be a tight cluster, indicating the genes it contains have high connectivity with each other. Thus at an appropriately chosen cutoff level Q, we identify the smallest few clusters as potential functional groups. Both SBM and HC require a priori knowledge of the number of clusters Q, and the proper selection of Q remains an open problem in literature. For SBM, we refer to some discussions in J.-J. Daudin and Robin (2008) and Channarond, Daudin and Robin (2012). For HC, a common way to choose the cutoff Q is to set it as the number just before the merging cost starts to rise sharply. Due to the scale and complexity of a typical gene expression dataset, this criterion is not very applicable. In this paper, for the HC approach we choose Q empirically based on the sizes of the clusters each Q produces. That is, Q is increased incrementally until small clusters start to emerge. A comparison between SBM and HC can be found in Section 3.1.
2.5. Flow chart summarizing the whole procedure. A comprehensive summary of the whole procedure including the tuning parameters needed in the two stages is provided in Figure 1. The choices of the parameters are explained in the paper, and summarized again in Section 3 of the supplementary information. and recall = TP/(TP+FN), as measures for evaluating classification performance. Here TP is the number of true positive findings of functional group genes, FP is the number of false positives and FN is the number of false negatives. In the context of this study, they can be regarded as a measure of exactness and completeness of our search results, respectively. The problem of choosing appropriate λ for sparsity is also discussed.
3.1. Simulation.
3.1.1. Generation of simulation datasets. We simulate a microarray dataset consisting of p = 150, 300 or 500 genes and n = 30 experiments, with 5 replicates for each experiment. To make the data more realistic, we introduce experiment dependencies, multiple functional groups and random noise. The simulation parameters are generated as follows:
(i) Experiment correlation matrix, Σ E . For illustrative purpose, we set the experiment correlation matrix to have 0, 33 and 67% dependencies. In the case of a 33% dependency, for example, 33% of the experiments have high dependencies (correlation between 0.5 and 0.6) while the remaining experiments are uncorrelated with one another.
(ii) Gene correlation matrix, Σ G . In each dataset, we introduce one or two functional groups with 15 genes in each. Genes in the same group are correlated, having either high correlations (0.5 -0.6) or low correlations (0.1 -0.2) with the other genes, and otherwise they are not.
Using the above parameters, we generate the simulation data as follows. First, we generate a 30 × 500 gene expression matrix Z, with vec(Z T ), from a multivariate normal distribution with mean zero and a covariance matrix Σ G ⊗ Σ E . To introduce linear relationships, within each group we take linear combinations of some genes to replace their original values. Using the final 30 × 500 gene expression matrix, we add random noise with a small SD (e.g. 0.01) to each row to generate the 5 replicates for each experiment.
3.1.2. EstimatedĀ and tuning parameter selection. Figure 2 shows the heatmaps of the matrixĀ for two datasets with different numbers of functional groups. For visual clarity, the genes are ordered according to their true group memberships. In both cases, the matrix demonstrates a clear block structure. In particular, in the two-group case both pathways are visible although the first one is more prominent. We remark here that the difference in signal strength between the two pathways is introduced by chance variation during data generation and the use of subsampling is necessary for the identification of the weaker group. λ is chosen such that the matrixĀ displays optimal contrast between the pathway and non-pathway groups, and we shall use this as a guidance for assessing the quality ofĀ and selecting λ.
Among the common approaches for the selection of optimal tuning parameters, cross-validation based methods are used in Waaijenborg, Verselewe de Witt Hamer and Zwinderman (2008), Parkhomenko, Tritchler and Beyene (2009) and Lee et al. (2011). However, all of their methods involve dividing a sample into multiple sets which is impractical for datasets with only a few tens of observations. Witten and Tibshirani (2009) proposed an alternative permutation-based method which estimates the p-value of the maximal correlation found by performing SCCA on permuted samples. Due to the large number of partitions and subsamplings required in our method, this approach would be very computationally expensive. Instead we measure the effectiveness of λ using the entropy ofĀ, defined as
(3.1) H(A) = − i<j,A ij >0 (A ij /S A ) log(A ij /S A ), where S A = i<j A ij .
The entropy quantifies the sharpness of its distribution and thus is indicative of the signal intensity. Figure 3 plots the contours of H(Ā) for the same two datasets used in Figure 2. Regions with low entropy correspond to λ leading to a matrix with better signal intensity. . . , 18} 2 using datasets with (a) p=150, 0% experiment dependency, one functional group and subsampling level 70%; (b) p=300, 0% experiment dependency, two functional groups and subsampling level 70%.
3.1.3. Performance comparison. Figure 4 compares the classification performance of our methods, scca.sbm and scca.hc, with four correlation-based methods, pearson.hc, pearson.sbm, module.dynamic and module.hybrid. The methods are named by cross-mixing the following to allow for comparisons in the two-stage procedure.
scca: CalculateĀ's with λ ∈ {9, 12, . . . , 27} 2 and select 10 of these with the smallest entropy values. The final cluster membership (after community detection) is decided by a majority vote based on the selectedĀ's so only stable clusters and cluster members are chosen.
pearson: Pearson's correlation matrix after the data is normalized using equation (2.6) and Knorm estimates.
module: Transformed Pearson's correlation matrix used in Langfelder and Horvath (2007).
sbm: Fit a SBM on a discretized edge weight matrix (at level {0.3, 0.4, . . . , 0.8}) using the unconditional pseudo-likelihood algorithm in Amini et al. (2013) with Q = 2 (or 3) initialized by spectral clustering with perturbation. Select the cluster with the highest internal connectivity based on the estimates.
hc: HC with the Ward's distance and cut the dendrogram when clusters of size less than 25 start to appear as the number of clusters Q increases. The choice of this upper bound is based on the size of the cluster selected in scca.sbm, and a range of reasonable numbers can be used without affecting the final results.
dynamic, hybrid : HC with dendrogram cutting methods in the R package dynamicTreeCut (Langfelder, Zhang and Horvath (2008)). It can be seen that using our SCCA approach to compute edge weights in general leads to higher precision across all experiment dependency levels. Of the two ways of community identification, scca.hc produces higher precision than scca.sbm at comparable recall levels. Table 1 shows the same performance measures obtained from datasets containing two independent functional groups for scca.hc, pearson.hc, module.dynamic and module.hybrid. The numbers are averages from 10 simulation datasets for each level of experiment dependency. Similar to the onegroup case, we choose the smallest Q that produces two clusters of size less than 25 as the cutoff in HC. We remark here that when multiple groups are present, scca.sbm tends to detect only the strongest signal group while failing to pick up the weaker one. This can be explained by considering the within-class homogeneity assumption in the SBM model and noting that the degree distribution is often less homogeneous in the weaker signal group (see e.g. Figure 2). Neither is the conditional pseudo-likelihood algorithm in Amini et al. (2013) sensitive enough to detect the finer distinctions. Results from pearson.sbm are also omitted as they are very noisy. In all the cases, scca.hc demonstrates the best precision at comparable, if not better recall. 3.2. Application to real data. We tested the performance of our procedure by applying it to Arabidopsis thaliana microarray expression data retrieved from AtGenExpress (http://www.arabidopsis.org/servlets/Tair Object?type=expression set&id=1007966941). Analyzed dataset included expression measurements collected from shoot tissues subject to oxidation stress for 22810 genes under 13 experiment conditions with two replicates for each experiment. In these experiments, the plants were treated with methyl viologen (MV), which led to the formation of reactive oxygen species (ROS). Various studies have shown that depending on the type of ROS, a different biological response is provoked. Thus by focusing on the ROS induced by MV, we were able to show and validate that the results of our pathway gene search were supported, in part, by other already published ROS-related microarray experiments.
A subset of all 22810 genes was selected for analysis based on the following criteria. (i) The experiment variance of the gene exceeds 0.1. An unvarying expression profile suggests the gene has an activity level unaltered by the particular stress condition and hence is unlikely to be part of any stress-induced pathway. The inclusion of such genes may cause problems in covariance estimation as well. We also removed genes with a suspiciously high experiment variance as it could suggest inaccuracy in measurements.
(ii) The discrepancy between the two replicates is smaller than 2 for each experiment. This ensures only genes with consistent measurements are included in our analysis. (iii) The minimum expression level exceeds 7. More active genes are likely to possess stronger signals, making our search easier. This requirement further trims down the dataset to a smaller size more desirable for our procedure. We note here that the inclusion of (iii) is optional -if running time is not a concern, the minimum expression level could be either lowered or entirely removed. The final subset for analysis contained 2718 genes.
Potential functional groups were found by scca.hc. Due to the complexity and noise level of the dataset, we did not expect the entropy (3.1) to have a clean-cut unimodal distribution. Furthermore, the presence of many groups with varying signal strengths implies each may need a different optimal λ for detection. For example, strong groups are likely to require more regularization, or in other words, larger λ. For this reason, we performed our search in multiple stages starting from large λ for stronger groups to smaller λ for weaker ones. At every stage, the groups found were removed from the original set before proceeding to the next stage. The upper bound on λ was found by increasing λ until the entropy stabilized. Searching down from this upper bound, we chose λ from three grids: {90, 100, 110} 2 , {60, 70, 80} 2 and {30, 40, 50} 2 . The cutoff level Q in HC was increased incrementally until at least five clusters of size less than 30 appeared. A reasonable range of numbers can be used to choose the cutoff and our results are not very sensitive to the choice of this number. The full procedure produced 13 groups of genes, the full list of which including annotations can be found in Section 6 of the supplementary information.
To test the biological significance of all 13 groups found (i.e., whether there is a functional relationship between genes within the various groups), we first examined for enrichment of gene product properties, collectively designated gene ontology (GO) annotations, within each group using information available at The Arabidopsis Information Resource (http://www.ar abidopsis.org/tools/bulk/index.jsp). We determined that 8 out of 13 groups were highly enriched with genes having the same GO annotation and calculated their p-values using Fisher's exact test to compare with the counts obtained from the full analyzed dataset (Table 2).
In addition to the GO enrichment approach for validating the groups, and in order to support the biological significance of the groups found, we also evaluated other forms of evidence. We were able to determine that for several Endomembrane system 3 out of 4 2.35 × 10 −3 groups, that the genes placed in the groups encode for known pathways. For example, group 2 genes encode steps in the phenylpropanoid-flavonoid (FB) biosynthesis pathway, and group 3 genes encode for steps in the glucosinolate (GSL) biosynthesis pathway. Both are well-studied secondary metabolic pathways. Flavonoids are compounds of diverse biological activities such as anti-oxidants, functioning in UV protection, in defense, in auxin transport inhibition, and in flower coloring ( (2007)). A considerable number of genes in both pathways are induced by broad environmental stresses, and regulated at the transcriptional level. Based on the lists of genes associated with these two pathways reported in Kim et al. (2012), our analyzed dataset contained 13 FB pathway genes and 26 GSL pathway genes. The precisions of our search are 75% and 100%, respectively. In order to assess the likelihood that genes in the remaining groups could also encode steps within specific pathways, we reviewed microarray data from plants subjected to other forms of oxidative stress (these experiments are similar to the experiment from which our dataset using MV was obtained). Using this approach we found that genes in each of the additional seven groups (1, 4,5,8,9,11,12) were strongly associated in these independent experiments (supplementary information Section 6).
Of all the groups found, groups 6, 7 and 13 remain uncharacterized in the literature. Nonetheless, using CoExSearch (part of the ATTD-II database (http://atted.jp/top search.shtml #CoexVersion)), all four genes in group 7 were correlated to some degree with abiotic stress conditions. We also found these genes were common anoxia-repressed genes (Loreti et al. (2005)). The lack of complete characterization for these groups in the current literature leaves potential scope for further biological examination. Located in plasma membrane 2 out of 5 65
Located in plasma membrane 3 out of 5 66
Pyridoxine biosynthetic process 2 out of 5
For comparison we applied pearson.hc, module.dynamic and module.hybrid to the same data. As the simulation study suggests the latter two methods in general have better performance than pearson.hc, particularly in the multigroup case, we will present the results from these two methods and refer to Section 6 in the supplementary information for pearson.hc based results. In order to compare with our results, we chose two cuts of the dendrogram such that the first cut produces the same number of groups as our method, and the second one leads to groups with sizes comparable to ours. The first cut results in 13 groups with sizes ranging from 60 to 293. We picked three most promising groups based on their annotations and the GO analysis is summarized in Table 3. Although all of them have statistically significant p-values, their precisions are quite low. In particular, group 11 contains our group 2 as a subset and includes 11 genes (out of 76) in the FB pathway and 5 genes are in isoprenoid biosynthesis pathway. These two pathways are derived from different initial precursors and and known to be unrelated. We note here that at this cut level, the GSL pathway cannot be identified by the method. The second cut produces 66 groups with sizes from 5 to 81. We picked five small groups for analysis and only one group with genes localized in chloroplast has significant GO enrichment (Table 4). Even so, these genes are unlikely to be functionally related. The comparison suggests our method can achieve better precision and lead to more biologically meaningful groupings of genes.
4. Discussion. In this paper, we focus on the problem of searching for functional groups in gene networks, where data are given in the form of nodes and their associated covariates and the true network needs to be estimated first. We propose a new method to construct an edge weight matrix for the full network by applying SCCA to sampled subsets of genes with random partitioning. The final community structure can be discovered by fitting SBM or applying HC. Our experience suggests HC is better suited for the study of large networks with multiple functional groups, as its running time scales better than most SBM algorithms and requires no discretization of the adjacency matrix. Although the work is presented under the setting of gene networks, we believe our approach can be generally applicable to answer similar questions in other biochemical networks and even networks in other fields that are sparse and have similar covariate features.
Compared to other popular ways of measuring gene interactions, our SCCA approach is more conceptually appealing. By seeking maximally correlated sets of genes among randomly sampled subsets, this approach provides an aggregated measure of gene partial correlations when the correct conditional set is unknown and thus gives us a better chance of capturing group interactions. As demonstrated in both simulation and real data applications, one of the main attractions of our procedure is its high precision. Although it does not seem to greatly improve recall, this is not a huge drawback in light of the search algorithm by Kim et al. (2012). Given the accuracy of our search results in general, one can use these identified genes as "seed genes" to initiate a more complete search and expand on the current lists.
Our approach can be modified to handle other practical situations. When it is known in advance that some genes operate in the same functional group, one may incorporate the prior knowledge by lowering the penalties associated with those genes in the SCCA algorithm. Although we have focused on the case with disjoint functional groups, our method of constructing an edge weight matrix is still applicable to the overlapping case as long as the shared genes possess strong direct or partial interactions with all the other functional genes (supplementary information Section 5). However, a different community detection method (e.g. mixed membership SBM )) should be applied to identify the overlapping structures.
The core of our procedure consists of an implementation of SCCA by LASSO regression, and this naturally opens room for further investigation.
For example, it would be interesting to find out if using other penalty functions yields different results; more importantly, whether SCCA can be implemented using a different optimization criterion or a more efficient algorithm to lessen the computational cost of our procedure. In the theoretical aspect, it would be desirable to incorporate sparsity into our asymptotic analysis.
On the community detection side, our use of SBM and HC also gives rise to other interesting extensions. As noted in Section 3.1, conventional SBM does not perform well when there are multiple groups, which is mainly caused by the heterogeneity of node degrees. However, fitting a degree-corrected model using the conditional pseudo-likelihood algorithm does not seem offer significant improvement. It would be desirable to carry out further study on the theoretical properties of the degree-corrected SBM and characterize its identifiability problem. Another possible extension is to modify these algorithms to take weighted adjacency matrices without discretization. Developing a practical but more systematic way of choosing the cutoff level for HC also invites future study.
SUPPLEMENTARY MATERIAL
Supplement: Supplementary Information
SUPPLEMENTARY INFORMATION
1. Summary of the supplementary information. The supplementary information includes:
• Section 2. Asymptotic behavior of subsampling • Section 3. Summary of tuning parameters • Section 4. Discussion on the edge weight matrix • Section 5. Simulation results under additional scenarios, discussing subsampling levels, overlapping functional groups and incorporating prior knowledge • Section 6. Tables and plots for the Arabidopsis data 2. Asymptotic behavior of subsampling.
2.1. One functional group. In this section we first present the assumptions and proofs needed to establish Proposition 2.1 in the paper. Recall that Z ∈ R n×p represents an expression matrix with p genes and n experiments, with centered and scaled columns. We have the following assumptions regarding the distribution of Z.
Assumption 2.1. Z = (z 1 , . . . , z n ) T , where z i are iid p−dimensional normal random variables with mean 0 and correlation matrix Σ that is invertible.
Assumption 2.2. The matrix Ω = Σ −1 is a diagonal block matrix,
(2.1) Ω = Ω 1 0 k×(p−k) 0 (p−k)×k Ω 2 ,
where Ω 2 = diag(1, . . . , 1).
Remark 2.3. Note that the diagonal block structure of Ω in Assumption 2.2 is mirrored in its inverse Σ, that is
(2.2) Σ = Σ 1 0 k×(p−k) 0 (p−k)×k Σ 2 ,
where Σ 1 = Ω −1 1 and Σ 2 = diag(1, . . . , 1). The structure of the correlation matrix implies dependencies only exist among pathway genes.
Partition the index set G into two sets J 1 and J 2 of equal size. Let I 1 = J 1 ∩ K and I 2 = J 2 ∩ K, that is, I 1 and I 2 represent the corresponding partition on the pathway gene set. For convenience, assume the indices in J 1 , J 2 , I 1 and I 2 are ordered. Compose submatrix X by selecting columns of Z whose indices lie in the set J 1 . Similarly compose submatrix Y based on the index set J 2 . In the population case CCA requires finding (α, β) that solves the optimization problem (2.3) in the paper. Note that when Σ Y X is a nonzero matrix, α and β are uniquely determined up to a sign. To eliminate this indeterminacy we require α 1 > 0, β 1 > 0.
We also assume the following is true regarding the singular value decomposition of Σ I 1 ,I 2 for every partition.
1 Assumption 2.4. For any partition, the nonzero singular values of Σ I 1 ,I 2 are all distinct.
Remark 2.5. Assumption 2.4 is equivalent to requiring the corresponding submatrix Σ Y X has distinct nonzero singular values. This assumption is common in literature for the purpose of establishing asymptotic theory for PCA or CCA.
Since in practice one always aims to solve the sample case (equation (2.4) in the paper), we first need to establish the asymptotic properties of a and b for a given partition.
Lemma 2.6. As n → ∞, (i) If I 1 = ∅, then a i ≤ 1 + o P (1) for k + 1 ≤ i ≤ q and b i ≤ 1 + o P (1) for 1 ≤ i ≤ q. Similar conclusions hold for the case I 2 = ∅.
(ii) If I 1 = ∅ and I 2 = ∅, we have a P −→ α and b P −→ β.
Proof of Lemma 2.6. We first show the constraints on a and b imply they are bounded with probability one. Letλ i be the eigenvalues of S Y Y and λ i be the eigenvalues of Σ Y Y , then
(2.3)λ i a.s. −→ λ i follows from the fact that (2.4) S Y Y a.s. −→ Σ Y Y . Writing S Y Y = U diag(λ i )U T , where U is an orthogonal matrix, a T S Y Y a = a T U diag(λ i )U T a = q i=1λ i (a i ) 2 = 1, (2.5)
with (a 1 , . . . , a q ) T = U T (a 1 , . . . , a q ) T . Noting that a 2 = a 2 and λ i > 0, one can conclude that a = O P (1). Thus
(2.6) a T S Y Y a = a T Σ Y Y a + o P (1) = 1,
and (i) follows from the structure (2.2) of Σ. The same argument applies to b.
In the case I 1 = ∅ and I 2 = ∅, rank(Σ Y X ) ≥ 1. One can show the convergence holds using Assumption 2.4, the fact that √ n(S (·,·) − Σ (·,·) ) has a limiting normal distribution and following the arguments in Anderson (1999).
We then proceed to prove Proposition 2.1.
Proof of Proposition 2.1. Consider the following possible partition configurations. (For convenience, the dependency of the coefficients on partition t is suppressed.) Case (i) I 2 = ∅. The probability of this configuration is (2.7) P 0 = P({I 2 = ∅}) = p−k q p q · 1 2 2 with q = p/2. By Lemma 2.6, c i ≤ 1 + o P (1) as n → ∞ for i / ∈ K. Case (ii) |I 2 | = 1, |I 1 | = k − 1. This happens with probability kq q−k+1 P 0 . Assume without loss of generality I 1 = {1, . . . , k − 1}, I 2 = {k}. Partition the pathway correlation matrix Σ 1 and its inverse Ω as (2.8) Σ 1 = Σ I 1 ,I 1 Σ I 1 ,k Σ k,I 1 1
, Ω = Ω I 1 ,I 1 Ω I 1 ,k Ω k,I 1 ω k,k .
It is easy to see in the population case the solution to (2.3) has the form α = (α 1 , α 2 ) with α 1 ∈ R, α 2 = 0 and β = (β 1 , β 2 ) with β 1 ∈ R k−1 , β 2 = 0. Furthermore, α 1 and β 1 solve the optimization problem (2.9) (α 1 , β 1 ) = arg max α 1 ,β 1 α 1 Σ k,I 1 β 1 subject to α 2 1 = 1, β T 1 Σ I 1 ,I 1 β 1 = 1.
The above condition implies α 1 = 1, β 1 = (1/ρ)Σ −1 I 1 ,I 1 Σ I 1 ,k α 1 and ρ 2 = Σ k,I 1 Σ −1 I 1 ,I 1 Σ I 1 ,k , where ρ is the maximal correlation. Noting that (2.10) Ω I 1 ,k = −Σ −1 I 1 ,I 1 Σ I 1 ,k ω k,k and (2.11)
ω k,k = (1 − Σ k,I 1 Σ −1 I 1 ,I 1 Σ I 1 ,k ) −1 ,
we can write β 1 as
(2.12) β 1 = − Ω I 1 ,k ω 2 k,k − ω k,k .
Generalizing this to any partition resulting in |I 2 | = 1 and using the convergence in Lemma 2.6, as n → ∞, c i = o P (1) for i / ∈ K and c i = C 1 + o P (1) for i ∈ K, where (2.13)
C 1 = 1 if I 2 = {i} |ω i,j | ω 2 j,j −ω j,j if I 2 = {j}, j = i.
Case (iii) |I 1 | > 1 and |I 2 | > 1. By the same argument as in Case (ii), c i = o P (1) for i / ∈ K, and a i = α i +o P (1) and b i = β i +o P (1) for i ∈ K with α 1 and β 1 solving the sub-problem (α 1 , β 1 ) = arg max α 1 ,β 1 α 1 Σ I 2 ,I 1 β 1 subject to α T 1 Σ I 2 ,I 2 α 1 = 1, β T 1 Σ I 1 ,I 1 β 1 = 1. (2.14)
Combining results from the above discussion,
lim N →∞ lim n→∞c i ≤ 2P 0 , i / ∈ K; (2.15) lim N →∞ lim n→∞c i ≥ 1 + (k − 1) min 1≤j =i≤k |ω i,j | ω 2 j,j − ω j,j × q q − k + 1 2P 0 , i ∈ K. (2.16)
The proposition holds with
(2.17) D ≥ 2P 0 k − 1 q − k + 1 + (k − 1) min 1≤j =i≤k |ω i,j | ω 2 j,j − ω j,j q q − k + 1 > 0 2.2.
Multiple functional groups. Next we extend the above analysis to the case with more than one functional group using an explicit example. As will be demonstrated below, analytical computations are tedious even in this easy case. While we decide to not provide a full proof in the paper, we hope the example will provide sufficient insights into the meaning of the weight matrix and the role of subsampling.
Suppose there are 20 genes in total with two independent functional groups of size 3 each. Let Z = (z 1 , . . . , z n ) T ∈ R n×20 represent an expression matrix with z i ∼ iid normal variables with mean 0 and correlation matrix Σ, where Σ = diag(Σ 1 , Σ 2 , 1, . . . , 1) with
Σ 1 = 1 1/ √ 2 1/ √ 2 1/ √ 2 1 0 1/ √ 2 0 1 and Σ 2 = 1 0.4 0.8 0.4 1 0.3 0.8 0.3 1 .
Note that the genes in the first group have a perfect linear relationship z i,1 = 1 √ 2 z i,2 + 1 √ 2 z i,3 while the second group does not.
We first compute the asymptotic value of A with no subsampling as the number of partitions goes to infinity using the population correlation matrix Σ (i.e. assuming we have infinite observations). Since no subsampling is involved, there is only one such edge weight matrix. The asymptotic values of the edge weight matrix can be calculated by summing the weights from CCA under different partition configurations, weighted by their respective probabilities. Here are more algebraic details for the computation of A. For every possible partition, split the index set {1, 2, . . . , 20} into two sets J 1 and J 2 of equal size. Let α and β be weight vectors solving
(α, β) = arg max α,β α T Σ J 1 ,J 2 β subject to α T Σ J 1 ,J 1 α = 1, β T Σ J 2 ,J 2 β = 1.
When Σ J 1 ,J 2 is a zero matrix, we adopt the convention which randomly chooses one gene in J 1 and one gene in J 2 and assigns them weight 1. As in the paper, suppose c is the list of the absolute values |α| and |β| ordered according to the gene list. For example, for the partition placing the 20 genes in J 1 and J 2 with genes 1, 4, 5, 6 in J 1 and genes 2, 3 in J 2 , c = (1, 1/ √ 2, 1/ √ 2, 0, . . . , 0). Now define the average weight vectorc = 1/N N t=1 c t , where N is the number of random partitions.c → E(c) as N → ∞ and E(c) can be computed explicitly from the four cases listed in the calculation part below. Then asymptotically,
(2.18) A =cc T → E(c)E(c) T .
The asymptotic values (setting the diagonal to 0, without normalization) are plotted in Figure 1 (a). We see that without subsampling, the second group is completely overwhelmed by the first 4 group (as demonstrated in configurations of type (iv)). Note also the signal strength within the second group is weaker than that of the interaction between the two groups. When agglomerative clustering is applied, the genes in group two will be merged with group one before merging among themselves, making it very difficult to identify the second group.
With the help of subsampling, we hope to create more subsamples in which the second group dominates the first, thus enhancing its signal strength inĀ when the averages are taken over different subsamples. In this small example, it is possible to compute the asymptotic value ofĀ as the number of random partitions and the number of subsamples go to infinity, again assuming we have infinite obeservations which allow us to use the population correlation matrix. Averaging over all subsamples b,Ā
= 1/B B b=1c bc T b (2.19) = 1/B B b=1 1/N N t=1 c b,t 1/N N t=1 c b,t T → E b (E tcb ) (E tcb ) T ,
as N → ∞ and B → ∞, where E t and E b denote expectation taken with respect to random partition and subsampling, respectively. The asymptotic values ofĀ (setting diagonal to zero, without normalization) are plotted in Figure 1 (b). We have set the subsampling level to 70% and more details of the calculations can be found below. The comparison with Figure 1 (a) demonstrates theoretically subsampling helps the identification of the weaker group. The theoretical analysis of the role of subsampling for multiple pathways under general settings can be carried out in a similar fashion. However, since one needs to consider all the possible subsamples and their corresponding partition configurations, the process is rather tedious even when the gene groups are very small as shown by the toy example above. We deem the full proof out of scope of the current paper. We remark here that subsampling is only necessary when we would like to identify multiple functional groups simultaneously. In practice, if running time is 5 not a concern, we can always run the whole procedure iteratively with no subsampling, each time identifying one dominating signal group and removing it from the subsequent analysis.
Detailed Calculations
We first show the calculations for the case with no subsampling. Let K 1 = {1, 2, 3} and K 2 = {4, 5, 6} denote the indices of the genes in these two functional groups. For each partition, let I i,j = K i ∩ J j for i, j = 1, 2. Consider the following scenarios:
(i) |I 1,j | = 3 and |I 2,l | = 3, j, l ∈ {1, 2}. This happens with probability 2 · 14 10 + 2 · 14 7 20 10 , and since the cross correlation matrix is zero in this case, we randomly choose two genes to assign weight 1.
(ii) |I 1,j | = 1 and |I 2,l | = 3, j, l ∈ {1, 2}. The probability of this type of partition is 2 · 3 · 14 6 + 2 · 3 · 14 9 20 10
. Clearly c i = 0 for 4 ≤ i ≤ 20. Depending on which gene is grouped into I 1,1 (or I 1,2 ),
(c 1 , c 2 , c 3 ) = (1, 1/ √ 2, 1/ √ 2) if gene 1 is chosen, ( √ 2, 1, 1) if gene 2 is chosen, ( √ 2, 1, 1) if gene 3 is chosen,
all of which are equally likely.
(iii) |I 1,j | = 3 and |I 2,l | = 1, j, l ∈ {1, 2}, which has probability 2 · 3 · 14 6 + 2 · 3 · 14 9 20 10
. In this case, c i = 0 for 1 ≤ i ≤ 3 and 7 ≤ i ≤ 20. Depending on which gene is selected by I 2,1 (or I 2,2 ), (iv) |I 1,j | = 1 and |I 2,l | = 1, j, l ∈ {1, 2}, which has probability 2 · 3 · 3 · 14 8 + 2 · 3 · 3 · 14 7 20 10
. Both functional groups have been split up by the partition, resulting in the cross correlation matrix having two non-zero diagnal blocks. It is easy to see the genes in the first group are collinear, thus possessing a stronger linear relationship than the second group. The non-zero block associated with 6
Type of subsample Probability |S1| = 3, |S2| = 3 0.0775 |S1| = 3, |S2| = 2 0.1550 |S1| = 2, |S2| = 3 0.1550 |S1| = 3, |S2| = 1 0.0775 |S1| = 2, |S2| = 2 0.2324 |S1| = 1, |S2| = 3 0.0775 |S1| = 3, |S2| = 0 0.0094 |S1| = 2, |S2| = 1 0.0845 |S1| = 0, |S2| = 3 0.0094 |S1| = 2, |S2| = 0 0.0070 |S1| = 1, |S2| = 1 0.0211 |S1| = 0, |S2| = 2 0.0070 |S1| = 1, |S2| = 0 0.0011 |S1| = 0, |S2| = 1 0.0011 |S1| = 0, |S2| = 0 0.0000 Table 1 Different subsamples created.
the first group produces the largest singular value, and it follows that the values in c are the same as in case (ii).
Combining the above four cases gives the values in Figure 1 (a). With subsampling, let S denote the indices of the selected genes. Further denote S 1 = K 1 ∩ S and S 2 = K 2 ∩ S. We can create subsamples listed in Table 1. For each type of subsample listed, one can carry out the same computation for E(c b ) by considering all the possible partitions for a subsample b.
3. Summary of tuning parameters. A comprehensive overview of the whole procedure is summarized in the flow chart Figure 1 in the paper. Here we explain again the selection of all the tuning parameters required.
Parameters in Stage 1: 1. Percentage of genes to be subsampled: First we comment that this step is necessary only when one is interested in finding multiple functional groups simultaneously. When the dataset is small and one can afford to isolate functional groups one at a time, this subsample level can be set to 100%. In general, in order to recover groups with complex structures simultaneously, a lower subsampling level is needed. On the other hand, if the ratio of signal genes to noise genes is small, we need to apply higher subsampling levels to reduce noise. We also refer to Response 2.3 for more explanations.
2. Penalty parameter λ in SCCA: We use the entropy of the matrixĀ produced by every fixed λ as a guidance for its performance. Smaller entropy values correspond to more contrast inĀ between the signals and background noise.
Parameters in Stage 2 (SBM): 3. Thresholding level to discretizeĀ for SBM: A range of thresholding levels were tried on the simulation data in the paper for comparison.
4. Number of blocks in SBM: As we only tried fitting SBM on simulation data, the number of blocks is assumed to be known. In practice, the issue of estimating the block number remains an open problem.
Parameter in Stage 2 (hierarchical clustering):
5. Cutting the dendrogram: Starting from the root of the dendrogram, we gradually increase the depth of the cut until small clusters begin to emerge. Knowing rough estimates of functional group sizes will help decision making at this step. The small clusters produced are tight and stable at a range of cutting levels.
Going through the whole procedure fixing a set of parameters produces candidate groups. For comparison we recommend running the procedure with different parameters. In the paper, we did this using different values of λ and the final candidate groups were determined by majority votes.
4. Discussion on the edge weight matrix. The edge weight matrixĀ provides an aggregated measure of gene interactions within one functional group as well as with different subsets of the group. Within one functional group, if there exists a perfect linear relation among the gene expression levels, for each partition splitting the functional group into two non-empty sets, the elements in c are always proportional to their corresponding linear coefficients. Thus the elements inĀ are proportional to the products of such linear coefficients. When only a subset of the functional group is selected, the weight vector c is still monotone with respect to the magnitude of the coefficients. Overall the edge weights inĀ provide a weighted average of the importance of a pair of genes in interacting with the whole functional group and all possible subsets of the group.
The connection with partial correlation can be understood in the regression setting. For a functional group (z 1 , . . . , z k ) with expression levels following a multivariate normal distribution, regressing z i on the other genes gives
z i = j =i β ij z j + i ,
where the coefficient β ij is proportional to the partial correlation between z i and z j , or the correlation between z i and z j conditioned on the other genes in the group. Thus for partitions leading to such configurations (1 vs. k − 1), the elements in the weight vector c are proportional to β ij , and therefore proportional to the correlations between pairs of genes conditioned on the other genes in the same group (and selected in the same subsample). Generalizing to other configurations (l genes in one set vs k − l genes in the other set), the weight vector is proportional to the correlation between a gene and a linear combination of the genes in the other set, conditioned on the rest of the genes in the same set. Overall the average weight vector provides an aggregated measure of partial correlations of different orders, as the process of random subsampling and partitioning enables us to consider all possible dependent sets.
5.
Simulation results under additional scenarios.
Subsampling levels.
Using the same simulated dataset that produced the heatmaps in Figure 1 of the paper, we performed the calculation ofĀ again at (a) 50% subsampling level and (b) 95% subsampling level. There are two gene groups at positions 1-15 and 16-30, respectively.
As can be seen in Figure 2, when almost no subsampling is applied,Ā is predominantly expressing signals from the first group, while the signal intensity of the second group is weaker than that of the between-group interaction (due to the product definition of A) and only marginally stronger than the background noise. This is because genes in the first group possess stronger linear relationships and the weights in SCCA are preferentially assigned to them under most partitions. Thus the only way to simultaneously capture both groups under our framework is to consider different subsamples whereby the first group does not dominate all the time. A low subsampling level, however, also increases the chance of selecting only noise genes in the subsample. As a result, more background noise is introduced in the final output and the signal ratio is diminished (Figure 2 of subsampling level is a trade-off between group structure complexity and signal to noise ratio. More complex group structures require lower subsampling levels. On the other hand, if the ratio of signal genes to noise genes is small, we need to apply higher subsampling levels to ensure the results do not include too much noise.
5.2.
Overlapping functional groups. In practice it is often the case there are genes actively participating in multiple pathways. Our edge weight matrix would reflect the overlapping structure as long as the overlapping genes possess strong direct or partial correlations with other genes in those pathways. At the second step, these overlapping blocks can be detected by, for example, fitting an overlapping SBM ). To test our method's performance under the overlapping setting, we simulate a dataset with 150 genes and 30 experiments with 5 replicates each. Genes 1-15 form one functional group and genes 11-25 form the second group. The two groups overlap by 5 genes and genes in the same group have correlations around 0.5-0.6. Figure 3 shows the heatmaps ofĀ when the procedure is run at 50% level of subsampling, with the matrix demonstrating the desired overlapping block structure. 5.3. Incorporating prior knowledge. For illustration, we incorporate prior knowledge for two simulated datasets used in the computation of Table 1 in the paper. Both have 500 genes, 30 experiments with five replicates each. The first set has the first 15 genes forming a functional group. Randomly selecting four genes in the group as prior knowledge and reducing their penalties by half, the procedure retains a perfect precision of 1, and recall improves from 0.533 to 0.733. The second set has two functional groups of size 15 each (genes 1-15 and genes 16-30). Randomly choosing four genes in the first group as prior knowledge, the recall of the first group improved from 0.533 to 0.733. Doing the same for the second group, the recall increased from 0.467 to 0.667. 6. Tables and plots for the Arabidopsis data. Continued on next page Figure 4 compares the classification performance of scca.hc and pearson.hc for the FB and GSL pathways. As a common practice in the biology literature, we modified pearson.hc slightly by adding a thresholding step to the Pearson's correlation matrix. For a chosen threshold level θ, elements in the matrix with an absolute value smaller than θ were set to 0. The plot shows the precision and recall of all the clusters obtained by applying a cutoff Q ranging from 2 to 500 in HC. In both cases, scca.hc achieves the best precision regardless of the parameter setting.
Fig 2 .Fig 3 .
23Heatmaps of the matrixĀ using datasets with (a) p = 150, 0% experiment dependency, one functional group, subsampling level 70% and (λ1, λ2) = (9, 9); (b) p = 300, 0% experiment dependency, two functional groups, subsampling level 70% and (λ1, λ2) = (9, 15). For clarity, only the first 100 × 100 entries are shown and the functional groups are placed at positions 1-15 and 16-30, respectively. Contour plots of the entropy of the upper triangular entries ofĀ on the grid (λ1, λ2) ∈ {0, 3, .
Fig 4 .
4Classification performance of different methods using datasets with p = 500, one pathway group, subsampling level 70%, and (a) 0%, (b) 33% and (c) 67% of experiment dependency. pearson.sbm and scca.sbm are applied to matrices at discretization levels {0.3, 0.4, . . . , 0.8} (from left to right on the curve).
Figure 4
4plots the average precision and recall of the above six methods calculated on 10 simulation datasets for each level of experiment dependency.
Gachon et al. (2005);Naoumkina et al. (2010);Taylor and Grotewold (2005); Woo, Jeong and Hawes(2005)), and GSLs are sulfur-rich amino acid-containing compounds which become active in response to tissue damage, and believed to offer a protective function(Sønderby, Geu-Flores and Halkier (2010);Verkerk et al. (2009); Yan and Chen
(http://???). Asymptotic analysis and additional explanations of the procedure, additional simulation and real data results.Y. X. R. WANG ET AL.Wille, A. and Bühlmann, P. (2006). Low-order conditional independence graphs for inferring genetic networks. Statistical Applications in Genetics and Molecular Biology 5 1. Wille, A., Zimmermann, P., Vranova, E., A., F., Laule, O., Bleuler, S., Hennig, L., Prelic, A., von Rohr, P., Thiele, L., Zitzler, E., Gruissem, W. andBühlmann, P. (2004). Sparse graphical Gaussian modeling of the isoprenoid gene network in Arabidopsis thaliana. Genome Biology 5 1-13. Witten, D. M. and Tibshirani, R. (2009). Extensions of sparse canonical correlation analysis with applications to genomic data. Statistical Applications in Genetics and Molecular Biology 8 1-27. Witten, D. M., Tibshirani, R. and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10 515-534. Woo, H. H., Jeong, B. R. and Hawes, M. C. (2005). Flavonoids: from cell cycle regulation to biotechnology. Biotechnology Letters 27 365-374. Yan, X. and Chen, S. (2007). Regulation of plant glucosinolate metabolism. Planta 226 1343-1352.
Figure 1 .
1Asymptotic values ofĀ with (a) no subsampling and (b) 50% subsampling.
which are equally likely.
Figure 2 .
2Heatmaps ofĀ at (a) 50% subsampling level and (b) 95% subsampling level.
Figure 3 .
3Heatmap ofĀ with 50% subsampling.
Figure 4 .
4Precision and recall for (a) the FB pathway and (b) the GSL pathway using scca.hc and pearson.hc. The Pearson's correlation matrix used in pearson.hc were thresholded at θ = 0.4, 0.5, 0, 6 and 0.7, and the plot includes all the clusters produced by stopping HC at level Q from 2 to 500.
Fig 1.Flow chart summarizing the whole procedure. Each numeric superscript in the diagram indicates the need for tuning parameters: 1. subsampling level, 2. penalty parameter λ, 3. discretization level, 4. number of blocks in SBM, 5. number of clusters in HC.expression
data
submatrices
and
unsigned weights
binary adjacency
matrix
dist=1-
community
structure
stochastic block model
hierarchical
clustering
loop, subsample
loop, partition
discretize
SCCA
sample genes (columns)
column partition
community
structure
normalized
expression data
sample replicates
subsample matrix
center, remove
experimental dependencies
by Knorm
1
2
3
4
5
Data
Pre-processing
Stage 1
Stage 2
Table 1
1Classification performance of different methods using datasets with p = 500,two pathway
Table 2
2GO enrichment of groupsGroup
ID
Enriched GO term
Number of genes with
enriched terms
P-values
1
Chloroplast organellar gene
10 out of 15 1
1.10 × 10 −4
2
Phenylpropanoid-flavonoid biosynthesis 3 out of 4
6.65 × 10 −7
3
Glucosinolate biosynthsis
7 out of 7
1.95×10 −14
4
Chloroplast organellar gene
3 out of 3
7.83 × 10 −3
5
Ribosome
10 out of 15
7.20×10 −13
8
Ribosome
5 out of 6
8.31 × 10 −8
10
Photosystem I or II
8 out of 10
2.87×10 −14
12
Table 3
3GO enrichment of groups -first cutGroup
ID
Enriched GO term
Number of genes with
enriched terms
P-values
9
Cell wall
16 out of 81
4.46 × 10 −6
10
Defense response
29 out of 78
1.58 × 10 −2
11
Phenylpropanoid-flavonoid biosynthesis 11 out of 76
5.42×10 −12
Table 4
4GO enrichment of groups -second cutGroup
ID
Enriched GO term
Number of genes with
enriched terms
62
NA
0 out of 6
63
Chloroplast
4 out of 6
64
Table 2 :
2List of gene groups foundProbe set ID Locus IdentifierAnnotation
Group 1
244939 at
ATCG00065
Ribosomal Protein S12A (RPS12A)
245024 at
ATCG00120
ATP Synthase subunit alpha (ATPA)
245025 at
ATCG00130
ATPase F subunit
244997 at
ATCG00170
RNA polymerase beta subunit-2 (RPOC2)
245008 at
ATCG00360
Encodes a protein required for photosystem I assembly and
stability (YCF3)
245009 at
ATCG00380
Chloroplast Ribosomal Protein S4 (RPS4)
245010 at
ATCG00420
NADH dehydrogenase subunit J (NDHJ)
245015 at
ATCG00490
Large subunit of RuBisCO (RBCL)
245016 at
ATCG00500
Acetyl-CoA carboxylase carboxyl transferase subunit beta
(ACCD)
245020 at
ATCG00540
Photosynthetic electron transfer A (PETA)
244903 at
ATMG00660
Hypothetical protein
265228 s at
AT2G07698;
ATMG01190
ATPase subunit 1
244912 at
AT2G07783;
ATMG00830
CCB382, Cytochrome c biogenesis 382
244951 s at
AT2G07723;
ATMG00180
Cytochrome c biogenesis orf452
Table 2 -
2Continued from previous page Probe set ID Locus Identifier
Table 3 :
3Groups that show co-expression in other oxidativestress-inducing conditions Experimental (oxidative-stress-inducing) conditionsGroup ID
Number of
genes with
changed
expression
SOD
knockdown 1
apx1
exposed to
high light 2
Ozone 3
alx8 4
High light
(time
course) 5
1
11 out of 15
Up
regulated 6
4
2 out of 3
Up
regulated
Up
regulated
Up
regulated
12
4 out of 4
Up
regulated
Down
regulated 7
Down
regulated
Up
regulated
5
15 out of 15
Co-
expression 8
8
6 out of 6
Co-
expression
11
8 out of 8
Co-
expression
9
5 out of 5
Co-
expression
. Results. In this section we evaluate the performance of the proposed method and other approaches using simulation data and real microarray datasets. In particular, we compare the quality of the estimated networks resulting from different ways of computing edge weights, and the two methods of community detection (SBM and HC) discussed in Section 2.4. We use precision and recall, defined as precision = TP/(TP+FP)
4 out of the 10 chloroplast genes are mitochondrial organellar genes.
The thylakoid-bound Cu/Zn superoxide dismutases ( Cu/ZnSOD, At2g28190) knockdown mutant was compared to wild type plants(Rizhsky, Liang and Mittler (2003)) 2 The knockout cytosolic ascorbate peroxidase (apx1, At1g07890) mutant was exposed to high light, and compared to untreated apx1 plants.(Davletova et al. (2005)) 3 Arabidopsis seedlings were exposed to 200 ppb ozone for 1 h. (http://affymetrix.arabidopsis.info /narrays/experimentpage.pl? experimentid526)
Continued on next page 12
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| []
|
[
"Computing the unknotting numbers of certain pretzel knots",
"Computing the unknotting numbers of certain pretzel knots"
]
| [
"Seph Shewell Brockway [email protected] "
]
| []
| []
| We compute the unknotting number of two infinite families of pretzel knots, P (3, 1, . . . , 1, b) (with b positive and odd and an odd number of 1s) and P (3, 3, 3c) (with c positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynomial, building on work of Lickorish and Millett. | 10.1016/j.topol.2015.08.006 | [
"https://arxiv.org/pdf/1312.4502v1.pdf"
]
| 119,700,839 | 1312.4502 | 0d2429217aefc70b1579c730597a8ac02fba7aaf |
Computing the unknotting numbers of certain pretzel knots
16 Dec 2013 17th December 2013
Seph Shewell Brockway [email protected]
Computing the unknotting numbers of certain pretzel knots
16 Dec 2013 17th December 2013
We compute the unknotting number of two infinite families of pretzel knots, P (3, 1, . . . , 1, b) (with b positive and odd and an odd number of 1s) and P (3, 3, 3c) (with c positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynomial, building on work of Lickorish and Millett.
Introduction
The unknotting number u(K) of a knot K is the minimal number of crossing changes (whereby one strand of the knot is passed through another) required to transform K into the unknot. Any diagram of K can be used to compute such an unknotting sequence for K, and thereby place an upper bound on u(K). Calculating u(K) exactly, or even computing a lower bound, is in general a hard problem.
The pretzel link P (a 1 , . . . , a n ) with a i ∈ Z \ 0 for all i is the link shown in Figure 1, with a i < 0 representing crossings in the opposite direction to that shown. Observe that P (a 1 , . . . , a n ) is a knot when n and all of the a i are odd, and also when exactly one of the a i is even; every pretzel knot is of one of these two types.
We will make use of the knot signature σ(K), originally defined in terms of a Seifert surface (an orientable surface embedded in S 3 bounded by K; see e.g. [7]) with σ(unknot) = 0. It is well known (see e.g. [1]) that if K − is obtained from K + by changing a positive crossing then σ(K − ) − σ(K + ) ∈ {0, 2}, so that, for any knot K, u(K) ≥ 1 2 |σ(K)|. The method of Gordon and a 1 a 2 a n Figure 1: The pretzel link P (a 1 , . . . , a n ).
Litherland [4] shows easily that σ(P (a 1 , . . . , a n )) = n − 1 whenever the a i are all positive and odd. That u(P (3, 1, 3)) = 2 was established by Lickorish [6]. Owens [10] later showed that P (3, 1, 3) could not be unknotted by changing one negative and any number of positive crossings, and in a separate paper [9] also showed that u(P (3, 1, 1, 1, 3)) = 3.
Traczyk [13] used the Jones polynomial to show that P (3, 3, 3) could not be unknotted by changing one positive and one negative crossing. Owens [9] used this work and an obstruction from Heegaard Floer theory to show that u(P (3, 3, 3)) = 3.
We extend the techniques of Owens and Traczyk to establish the following two theorems. Theorem 1. For K = P (3, 1, . . . , 1, b) with an odd number r of 1s and b positive and odd, u(K) = 1 2 (r + 3). More generally, K = P (a, 1, . . . , 1, b), with an odd number r of 1s and a and b positive and odd, cannot be unknotted by changing 1 2 σ(K) = 1 2 (r + 1) negative and any number of positive crossings. Theorem 2. We have u(P (3, 3, 3c)) = 3 for c positive and odd. In general, for K = P (3a, 3b, 3c) with a, b and c positive and odd, the unknotting number u(K) ≥ 3.
Both of these results are special cases of the following conjecture of Jablan and Sazdanović [5]. Conjecture 1.1. For n odd and a 1 , . . . , a n positive and odd, and with a 1 ≤ a 2 ≤ · · · ≤ a n , u(P (a 1 , . . . , a n )) =
n−1 i=1 a i 2 .
This paper is an expanded excerpt from a master's thesis [12] prepared at the University of Glasgow under the supervision of Brendan Owens, to whom many thanks are due for his advice and support.
Unknotting rational pretzel knots
In this section we consider pretzel knots of the form P (a, 1, . . . , 1, b), with a and b odd and at least 3, and an odd number r of 1s; knots of this form are also part of the family known as rational or two-bridge knots. The technique in this section was established for another family of rational knots, including P (3, 1, 3), by Owens [10], who also established the case of P (3, 1, 1, 1, 3) by other methods in [9].
We will require the following definition. We also make use of the following corollary of [9, Theorem 3] (see also [10, Theorem 2], [11, Theorem 1]). Proposition 2.2. Let K be a knot with signature σ, and suppose that K can be unknotted by changing p positive and n negative crossings, with n = 1 2 σ. Then the branched double cover Y of S 3 over K bounds some smooth, oriented, positive-definite 4-manifold X, with intersection form of half-integer surgery type.
See, e.g., [3, Section 1.2] for the definition of the intersection form q X . Recall that a lattice over the integers is a free abelian group L equipped with a non-singular bilinear form · : L ⊗ L → Z, and that given a sublattice M of L, the orthogonal complement of M is the sublattice M ⊥ = {l ∈ L : l · m = 0, ∀m ∈ M}. Let Z m denote the free abelian group on generators e 1 , . . . , e m , with the bilinear form defined by e i · e j = δ ij (the Kronecker delta). For convenience we introduce the notation Λ X for the lattice (H 2 (X)/Tor H 2 (X), q X ).
Theorem 1 (restated). For K = P (3, 1, . . . , 1, b) with an odd number r of 1s and a positive and odd, u(K) = 1 2 (r + 3). More generally, K = P (a, 1, . . . , 1, b), with an odd number r of 1s and a and b positive and odd, cannot be unknotted by changing 1 2 σ(K) = 1 2 (r + 1) negative and any number of positive crossings.
Proof. Let K = P (a, 1, . . . , 1, b), with a, b and r as above, and let Y be the twofold branched cover of S 3 over K. A well-known result [4, ] states that Y is obtained as the boundary of a smooth 4-manifold Z with intersection form q Z equal to the Goeritz form g K , having matrix representation
Q Z = A 0 α 0 B β α τ β τ −r − 2 , where A and B are respectively (a − 1) × (a − 1) and (b − 1) × (b − 1) matrices of the form −2 −1 0 0 −1 . . . . . . 0 0 . . . . . . −1 0 0 −1 −2
and α and β are column vectors of the form (−1, 0, . . . , 0). From Sylvester's criterion, we see that Z is negative-definite. We have two 4-manifolds, X and Z, with diffeomorphic boundaries. Consider the manifold −Z, with boundary −Y and intersection form q −Z = −q Z . We can join X and −Z along their boundaries such that their orientations are preserved. We denote this manifold W . Note that in general W is not unique: we have to make a choice of diffeomorphism of the boundaries; however, it doesn't matter which one we choose for the purposes of this argument.
We have established that W is a closed, smooth 4-manifold. Consider the Mayer-Vietoris sequence
H 2 (Y ) → H 2 (X) ⊕ H 2 (−Z) φ → H 2 (W ) → H 1 (Y ).
First, note that since the map φ is induced by the inclusion maps of X and −Z into W , it preserves intersection forms. Since K is a knot, H 2 (Y ) is trivial, so φ is a monomorphism. The cokernel coker φ ⊆ H 1 (Y ) is finite since K is a knot [7]. We conclude from this that W is positive-definite.
Donaldson's diagonalization theorem [2] tells us that there exists a basis such that the intersection form q W has the m × m matrix representation I m = diag(1, . . . , 1). In terms of lattices, this means that we can embed Λ −Z in Z m ∼ = Λ W . Let Z m be generated by e 1 , . . . , e m as above, and let Λ −Z have a basis {ξ 1 , . . . , ξ a−1 , η 1 , . . . , η b−1 , ζ}, over which −q Z has the matrix representation shown above with reversed signs. Up to changes of sign and permutations of the e i , such an embedding must have the form
ξ i → e i + e i+1 η j → e a+j + e a+j+1 ,
but ζ does not embed uniquely. We know that Λ X must embed as a finite-index sublattice into Λ ⊥ −Z . Since rk Λ −Z = a + b − 1, it follows that rk Λ ⊥ −Z = m − a − b + 1. A finite-index sublattice of half-integer surgery type must therefore have 1 2 (m − a − b + 1) generators x i with x i · x j = 2δ ij . Any element of Λ ⊥ −Z whose expression involves a non-zero multiple of e i , with 1 ≤ i ≤ a, must contain some multiple of e 1 − e 2 + e 3 − · · · + e a by the definition of the orthogonal complement. Similarly, if e a+j , with 1 ≤ j ≤ b, is involved in the expression we have to include e a+1 − e a+2 + · · · + e a+b . Therefore these elements with square 2 must come out of the sublattice of Λ W spanned by e a+b+1 , . . . , e m , of which there are m − a − b − 1; for brevity we write g i = e a+b+i . First let x 1 = g 1 + g 2 . We can't let x 2 = g 1 − g 2 because in that case x 1 · y 1 ≡ x 2 · y 1 modulo 2, and we need x 1 · y 1 = 1 and x 1 · y 2 = 0. Therefore we have to set x 2 = g 3 + g 4 , x 3 = g 5 + g 6 and so on. Therefore the greatest number of
x i we can embed in Λ ⊥ −Z is 1 2 (m − a − b − 1) = 1 2 (m − a − b + 1) − 1.
The second part of the result follows, and since for P (3, 1, . . . , 1, b) we have an explicit unknotting sequence of 1 2 (r + 3) (negative) crossing changes (see Figure 2) we also obtain the first part. Remark 2.3. If [11,Theorem 1] is used in place of Proposition 2.2, then we obtain versions of Theorem 1 with the 4-ball crossing number c * (K), the concordance unknotting number u c (K) or the slicing number u s (K) (see [11] for definitions of all of these) in place of the unknotting number. for a skein triple of links differing only inside a 3-ball as shown in Figure 3. A thorough treatment of the Jones polynomial, including a proof that this relation is indeed well-defined and sufficient to compute the Jones polynomial of any oriented link, may be found in Chapter 3 of [7]. Using the Jones polynomial in addition to the above results obtained using Donaldson's diagonalization theorem, it is possible to compute a lower bound on the unknotting number of P (3a, 3b, 3c), giving an explicit value for P (3, 3, 3c). We will require the following result of Lickorish and Millett. Proof. The symmetrized Seifert form corresponding to the pretzel link P (3a, b) has matrix representationŜ = (3a+b). Denote its modulo-3 versionŜ 3 = (β). This has nullity 1 if β = 0, that is if 3|b, and nullity 0 otherwise, and we apply Proposition 3.1. Define a skein triple (L + , L − , L 0 ) by L + = P (3a, 3b − 2), L − = P (3a, 3b) and L 0 = O. Noting that L ± have two components, we have
Unknotting more pretzels
V (L + ; ω) = (−1) s + i V (L − ; ω) = −(−1) s − √ 3 V (L 0 ; ω) = 1.
We can substitute q = ω into the Jones skein relation, although care must be taken, as ω 1 2 has two possible values. Here we take ω 1 2 = e iπ/6 , but we could equally take ω 1 2 = e 7iπ/6 ; the argument in the latter case is entirely parallel to the one given here. In any case, we have
1 = ω 1 2 − ω − 1 2 i = ±ω −1 − ω(−1) s − i √ 3 ±ω −1 − 1 iω √ 3 = (−1) s − .
Take the ± sign first to be positive. This yields
(−1) s − = −ω iω √ 3 = i √ 3
which obviously is not satisfied for any s − . Now let the ± be negative. Here,
(−1) s − = −ω − 1 2 √ 3 iω √ 3 = iω − 3 2 = 1.
We conclude that s − ≡ 0 modulo 2, so that V (P (3a, 3b); ω) = − √ 3 as required. Proof. The symmetrized Seifert form corresponding to P (3a, 3b, c) has matrix representationŜ
= 3(a + b) −3b −3b 3b + c with modulo-3 versionŜ 3 = 0 0 0 γ .
This has nullity 2 if γ = 0, that is if 3|c, and nullity 1 otherwise. Define a skein triple by L + = P (3a, 3b, 3c − 2), L − = P (3a, 3b, 3c), L 0 = P (3a, 3b). We have
V (L + ; ω) = (−1) s + i √ 3 V (L − ; ω) = −3(−1) s − V (L 0 ; ω) = − √ 3.
The skein relation gives
−i √ 3 = ±iω −1 √ 3 + 3ω(−1) s − − i √ 3(1 ± ω −1 ) 3ω = (−1) s − .
Take the ± sign to be positive. Thus
(−1) s − = − 3iω − 1 2 3ω = −iω − 3 2 = −1.
Taking the ± sign to be negative,
(−1) s − = − √ 3iω 3ω = − i √ 3 ,
which again has no solutions. Therefore s − ≡ 1 modulo 2, so V (P (3a, 3b, 3c); ω) = 3 as required.
We now use the following result of Traczyk.
Proposition 3.4. Let K be a knot with V (K; ω) = (−1) s (i √ 3) d that can be transformed into the unknot by changing n negative and p positive crossings, such that n + p = d. Then p ≡ s modulo 2. [13, Theorem 3.1] Theorem 2 (restated). We have u(P (3, 3, 3c)) = 3 for c positive and odd. In general, for K = P (3a, 3b, 3c) with a, b and c positive and odd, the unknotting number u(K) ≥ 3.
Proof. Since P (3, 1, 3) can be transformed into P (3a, 3b, 3c) by changing 3(a + b + c) − 7 positive and no negative crossings (to see this, it is more instructive to change negative crossings in the standard diagram of P (3a, 3b, 3c) in order to reach P (3, 1, 3)), an unknotting sequence for P (3a, 3b, 3c) with n ≤ 1 would induce an unknotting sequence for P (3, 1, 3) with the same value of n, which is impossible by Theorem 1. If u = 1 then obviously n ≤ 1. This rules out the case u = 1.
Assume that u = 2. By Lemma 3.3, s ≡ 1 modulo 2. Since 0 ≤ p ≤ 2, Proposition 3.4 tells us that p ≥ 1, so that n ≤ 1. As in the previous paragraph, Theorem 1 rules out an unknotting sequence for P (3a, 3b, 3c) with n ≤ 1.
In the case where a = b = 1, we have an explicit unknotting sequence of three crossing changes (Figure 4), so u = 3.
Definition 2. 1 .
1A bilinear form q on some free abelian group M of rank 2m is of half-integer surgery type if it has a matrix representation Q = 2I I I * over some basis {x 1 , . . . , x m , y 1 , . . . , y m }. [10, Definition 1]
Figure 2 :Figure 3 :
23Recall that the Jones polynomial V (L) is an oriented link invariant which takes values in the ring Z[q ± 1 2 ] of Laurent polynomials in a single indeterminate q1 2 with integer coefficients. The Jones polynomial is defined by V (O) = 1, where O denotes the unknot, and the V (L 0 ) = q −1 V (L + ) − qV (L − ) An unknotting sequence for P (3, 1, . . . , 1, A skein triple.
Proposition 3 . 1 .
31For any r-component link L, V (L; ω) = (−1) s i r−1 (i √ 3) d , where d is the nullity of the modulo-3 reduction of the symmetrized Seifert form of L and ω = e iπ/3 . [8, Theorem 3] Lemma 3.2. Let K = P (3a, 3b), where a and b are positive and odd. Then V (K; ω) = − √ 3.
Lemma 3 . 3 .
33With K = P (3a, 3b, 3c) with a, b and c positive and odd, V (K; ω) = 3.
Unknotting information from 4-manifolds. Tim D Cochran, W B Raymond Lickorish, Trans. Amer. Math. Soc. 2792Tim D. Cochran and W. B. Raymond Lickorish. Unknotting information from 4-manifolds. Trans. Amer. Math. Soc., 279(2):125-142, 1986.
The orientation of Yang-Mills moduli spaces and 4-manifold topology. K Simon, Donaldson, J. Differential Geom. 263Simon K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397-428, 1987.
Gompf and András I. Stipsicz. 4-Manifolds and Kirby Calculus. E Robert, American Mathematical SocietyRobert E. Gompf and András I. Stipsicz. 4-Manifolds and Kirby Calcu- lus. American Mathematical Society, 1999.
On the signature of a link. Cameron M Gordon, Richard Litherland, Invent. Math. 47Cameron M. Gordon and Richard Litherland. On the signature of a link. Invent. Math., 47:53-69, 1978.
Unlinking number and unlinking gap. Slavik Jablan, Radmila Sazdanović, J. Knot Theory Ramifications. 1610Figure 4: An unknotting sequence for P (3, 3, 3c)Slavik Jablan and Radmila Sazdanović. Unlinking number and unlinking gap. J. Knot Theory Ramifications, 16(10):1331-1355, 2007. Figure 4: An unknotting sequence for P (3, 3, 3c).
The unknotting number of a classical knot. W B Raymond Lickorish, Contemp. Math. 44W. B. Raymond Lickorish. The unknotting number of a classical knot. Contemp. Math., 44:117-121, 1985.
An Introduction to Knot Theory. W B Raymond Lickorish, SpringerW. B. Raymond Lickorish. An Introduction to Knot Theory. Springer, 1997.
Some evaluations of link polynomials. W B , Raymond Lickorish, Kenneth C Millett, Comment. Math. Helv. 611W. B. Raymond Lickorish and Kenneth C. Millett. Some evaluations of link polynomials. Comment. Math. Helv., 61(1):349-359, 1986.
Unknotting information from Heegaard Floer homology. Brendan Owens, Adv. Math. 217Brendan Owens. Unknotting information from Heegaard Floer homo- logy. Adv. Math., 217:2353-2376, 2008.
On slicing invariants of knots. Brendan Owens, Trans. Amer. Math. Soc. 3626Brendan Owens. On slicing invariants of knots. Trans. Amer. Math. Soc., 362(6):3095-3106, 2010.
Immersed disks, slicing numbers and concordance unknotting numbers. Brendan Owens, Sašo Strle, arXiv:1311.6702Brendan Owens and Sašo Strle. Immersed disks, slicing numbers and concordance unknotting numbers. arXiv:1311.6702, 2013.
On the unknotting numbers of certain pretzel knots. Seph Shewell Brockway, University of GlasgowMaster's thesisSeph Shewell Brockway. On the unknotting numbers of certain pretzel knots. Master's thesis, University of Glasgow, 2013.
A criterion for signed unknotting number. Pawe L Traczyk, Contemp. Math. 233Pawe l Traczyk. A criterion for signed unknotting number. Contemp. Math., 233:215-220, 1998.
| []
|
[
"Phase diagram of the three-band half-filled Cu-O two-leg ladder",
"Phase diagram of the three-band half-filled Cu-O two-leg ladder"
]
| [
"Sootaek Lee \nDepartment of Physics\nBrown University\n02912-1843Providence, Rhode IslandUSA\n",
"J B Marston \nDepartment of Physics\nBrown University\n02912-1843Providence, Rhode IslandUSA\n",
"J O Fjaerestad \nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCaliforniaUSA\n\nDepartment of Physics\nUniversity of Queensland\n4072BrisbaneQldAustralia\n"
]
| [
"Department of Physics\nBrown University\n02912-1843Providence, Rhode IslandUSA",
"Department of Physics\nBrown University\n02912-1843Providence, Rhode IslandUSA",
"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCaliforniaUSA",
"Department of Physics\nUniversity of Queensland\n4072BrisbaneQldAustralia"
]
| []
| We determine the phase diagram of the half-filled two-leg ladder both at weak and strong coupling, taking into account the Cu d x 2 −y 2 and the O px and py orbitals. At weak coupling, renormalization group flows are interpreted with the use of bosonization. Two different models with and without outer oxygen orbitals are examined. For physical parameters, and in the absence of the outer oxygen orbitals, the D-Mott phase arises; a dimerized phase appears when the outer oxygen atoms are included. We show that the circulating current phase that preserves translational symmetry does not appear at weak coupling. In the opposite strong-coupling atomic limit the model is purely electrostatic and the ground states may be found by simple energy minimization. The phase diagram so obtained is compared to the weak-coupling one. | 10.1103/physrevb.72.075126 | [
"https://arxiv.org/pdf/cond-mat/0503380v2.pdf"
]
| 27,176,875 | cond-mat/0503380 | ecc542a90f1ef80c41134ef77e04fd8421378b90 |
Phase diagram of the three-band half-filled Cu-O two-leg ladder
25 Jul 2005
Sootaek Lee
Department of Physics
Brown University
02912-1843Providence, Rhode IslandUSA
J B Marston
Department of Physics
Brown University
02912-1843Providence, Rhode IslandUSA
J O Fjaerestad
Department of Physics and Astronomy
University of California
90095Los AngelesCaliforniaUSA
Department of Physics
University of Queensland
4072BrisbaneQldAustralia
Phase diagram of the three-band half-filled Cu-O two-leg ladder
25 Jul 2005(Dated: July 3, 2018)numbers: 7110Fd7110Hf7130+h7420Mn
We determine the phase diagram of the half-filled two-leg ladder both at weak and strong coupling, taking into account the Cu d x 2 −y 2 and the O px and py orbitals. At weak coupling, renormalization group flows are interpreted with the use of bosonization. Two different models with and without outer oxygen orbitals are examined. For physical parameters, and in the absence of the outer oxygen orbitals, the D-Mott phase arises; a dimerized phase appears when the outer oxygen atoms are included. We show that the circulating current phase that preserves translational symmetry does not appear at weak coupling. In the opposite strong-coupling atomic limit the model is purely electrostatic and the ground states may be found by simple energy minimization. The phase diagram so obtained is compared to the weak-coupling one.
I. INTRODUCTION
Considerable effort has been expended to explain the phase diagram of the high temperature superconductors and the associated pseudogap phenomenon within the framework of quantum critical and competing order pictures. Among the several candidates for orders that could compete with superconductivity are the "circulating current (CC) phase," 1,2 the "staggered flux (SF) phase" 3,4,5,6 or "d-density wave (DDW) phase," 7,8 the "spin-Peierls phase," 9,10 "stripes" 11,12 or the "quantum liquid crystal phase," 13,14 and the antiferromagnetic phase. 15 The CC phase, like the SF phase, breaks time reversal symmetry and is characterized by circulating currents which produce local orbital magnetic moments. An important difference between the two, however, is that the CC phase preserves translational symmetry while the SF phase breaks it. It is a nontrivial task to ascertain phase diagrams of strongly correlated systems in two dimensions due to the lack of accurate, systematic, nonperturbative methods. Mean-field type approximations always favor from the outset particular types of orders. Exact diagonalization is constrained by small system size. Quantum Monte-Carlo methods have the notorious fermion sign problem. On the other hand, there are reliable nonperturbative methods available in one dimension such as bosonization 16,17,18 and the density matrix renormalization group (DMRG) method. 19,20,21 By studying ladder systems one can take a first step toward investigating some of the theoretical ideas that have been conjectured for two-dimensional systems, while remaining in an essentially one-dimensional setting. Moreover, studies of ladder systems have been strongly stimulated by synthetic compounds with the ladder structure. For example, Sr 14 Cu 24 O 41 exhibits superconductivity upon Ca doping and the application of pressure. 22 Soon after the discovery of high-temperature superconductivity, one-band models such as the Hubbard or t-J model were suggested as the simplest microscopic systems that could capture the interesting physics of the Cu-O system. 23 Indeed, most work on ladders has focused on one-band models that consider only the Cu d orbital. 24,25,26,27,28,29,30,31,32,33,34,35,36 However, there are a number of theoretical studies 37,38 that suggest that neither the two-dimensional Hubbard nor t-J models have a superconducting ground state. It is also impossible to capture some other types of potential order like the CC phase within a one-band model as the circulating current pattern in the CC phase involves the O orbitals in a crucial way. 39 A more complete model that incorporates these orbitals is the three band or Emery model. 1,40,41 The three-band model takes into account both strong correlations and the hybridization of Cu and O orbitals.
Relatively little work has been done on three-band ladder models so far. 39,42 In this paper we obtain phase diagrams of the half-filled Cu-O ladder both at weak coupling and at strong coupling. At weak coupling, we examine ladder systems of two different geometries: with and without outer oxygen orbitals. Renormalization group (RG) flows are interpreted with the use of bosonization followed by semiclassical energy minimization. Several gapped phases arise depending on the particular values of the parameters. Within the physically relevant region, the D-Mott phase, which upon doping has a tendency toward d-wave superconductivity, arises in the absence of the outer oxygen sites while a spin Peierls phase occurs when outer oxygens are added. We find that the CC phase does not appear at weak coupling in the Cu-O ladder system. At strong coupling, the model is purely electrostatic. Treating the hopping term as a weak perturbation, it is then possible to make connections between the weak and strong coupling phase diagrams.
The paper is organized as follows. In Sec. II, we define the models and discuss the weak-coupling continuum limit. We present the RG flow equations in Sec. III. In Sec. IV, we bosonize the Hamiltonian and the various local order parameters of interest. From these, in Sec. V The five orbital case: There are two Cu d orbitals and three O p orbitals in a unit cell. Hopping integrals t dp and t ⊥ dp are between Cu d and O p; tpp is the hopping between the oxygens. U d and Up are on-site Coulomb interactions, and V dp , V ⊥ dp and Vpp are nearest-neighbor Coulomb interactions. (b) The seven orbital case: There are two extra outer oxygen orbitals in a unit cell labeled 6 and 7. the phase diagrams of the Cu-O ladders are established at weak coupling. The volumes of the different phases in the high-dimensional parameter space is quantified by means of Monte-Carlo sampling. In Sec. VI, we present the phase diagram at strong coupling and compare our findings to those at weak coupling. Our results are summarized in concluding Sec. VII. Some technical details are relegated to the Appendix.
II. MODEL
We study half-filled Cu-O two-leg ladders with Cu d x 2 −y 2 and O p x and p y orbitals as shown in Fig. 1. We consider two different geometries: (1) the five or-bital case with two Cu d x 2 −y 2 orbitals (i = 1, 2), two O p x orbitals (i = 3, 4), and one O p y orbital (i = 5) in a unit cell [see Fig. 1(a)] and (2) the seven orbital case in which in addition to the five orbitals there are also two outer oxygen p y orbitals (i = 6, 7) [see Fig. 1(b)]. At half-filling, on the average, there are eight electrons in a unit cell in the five orbital model and twelve electrons in the seven orbital model.
The Hamiltonian we investigate in this paper is given by
H = H 0 + H I ,(2.1)
where
H 0 = x,i ǫ i n i (x) − x,i,j,σ t intra ij c † iσ (x)c jσ (x) +t inter ij c † iσ (x)c jσ (x − 1) + h.c. (2.2)
is the tight-binding Hamiltonian and
H I = x,i U i 2 n i (x) [n i (x) − 1] + x,i,j V intra ij n i (x)n j (x) +V inter ij n i (x)n j (x − 1) (2.3)
is the Coulomb interaction. Each site is labeled by two integers (i, x) where "i" labels atoms within a unit cell, and "x" identifies different cells. Operators c iσ (x) and c † iσ (x), respectively, annihilate and create either Cu d electrons (i = 1, 2) or O p electrons (i = 3, 4, 5 in the five orbital case and i = 3, 4, 5, 6, 7 in the seven orbital case) in xth unit cell with spin σ (σ =↑, ↓). The number operator n i (x) = σ c † iσ (x)c iσ (x) counts electrons at site (i, x).
Consider now the hopping amplitudes between neighboring Cu and O sites, t dp and t ⊥ dp , as well as between nearest O and O sites, t pp . Since there is no particlehole symmetry, the sign of each hopping term is relevant. Signs of the various hopping parameters are determined by the symmetry of the orbitals. By choosing appropriate phases for each orbital, the hopping parameters t intra ij and t inter ij in Eq. (2.2) for the five orbital case may be all be taken to be positive
t intra ij = t dp if (i, j) = (1, 3) or (2, 4) t ⊥ dp if (i, j) = (1, 5) or (2, 5) t pp if (i, j) = (3, 5) or (4, 5) 0 otherwise, (2.4a) t inter ij = t dp if (i, j) = (1, 3) or (2, 4) t pp if (i, j) = (3, 5) or (4, 5) 0 otherwise. (2.4b)
Hopping parameters in the seven orbital case follow a similar pattern. The on-site energy of an electron on a Cu or O site is given by ǫ d or ǫ p respectively. For guidance we choose on-site energies and hopping matrix elements as extracted from a density functional theory (DFT) calculation for YBCO by Andersen et al.: 43 ǫ = ǫ d − ǫ p = 3.0 eV, t dp = 1.6 eV, and t pp = 1.1 eV. Although the precise values of these parameters for real ladder compounds will differ from those of the full two-dimensional problem studied by Andersen et al., for concreteness we use the same values here for the ladder. Thus for example the hopping along the legs t dp and along the rungs t ⊥ dp are taken to be the same, though this is not required by symmetry.
Turning now to the Coulomb interactions, the on-site energies U i take two different values, one for Cu atoms, the other for the O atoms
U i = U d if i = 1, 2 U p if i = 3, 4, 5. (2.5) Nearest-neighbor interactions are V intra ij = V dp if (i, j) = (1, 3) or (2, 4) V ⊥ dp if (i, j) = (1, 5) or (2, 5) V pp if (i, j) = (3, 5) or (4, 5) 0 otherwise, (2.6a) V inter ij = V dp if (i, j) = (1, 3) or (2, 4) V pp if (i, j) = (3, 5) or (4, 5) 0 otherwise. (2.6b)
The case of seven orbitals again follows a similar pattern. The Hamiltonian has the usual U(1)×SU(2) global charge/spin symmetry. Furthermore, it is invariant under lattice translations, time reversal, parity, and chain interchange operations. As stated above, there is no particle-hole symmetry in this system in contrast to the one-band model. Since the symmetry differs, the RG equations also differ, as discussed below in Sec. III.
A. Band Structure
We first diagonalize the quadratic, noninteracting portion of the Hamiltonian H 0 to obtain the band structure of the Cu-O ladder. The band structure is shown in Fig. 2. Evident in Fig. 2 is the fact that only two bands cross the Fermi level so we can analyze both ladders (with and without extra oxygens) in a similar manner. Note, however, that for very different tight-binding parameters different numbers of bands may intersect the Fermi level. In the two band case, it can be easily shown that k F 1 + k F 2 = π at half filling, where k F n > 0 is a Fermi momentum defined by ǫ n (k F n ) = µ and µ is chemical potential. However, since there is no particlehole symmetry, the Fermi velocity v F n = ∂ǫ n /∂k| k=kF n at the two Fermi points differs even at half filling:
v F 1 = v F 2 .
The band operator ψ nσ (x) is a linear combination of the c iσ (x) where n = 1, 2, ..., 5, (6, 7) is a band index and i = 1, 2, ..., 5, (6, 7) is a site index in a unit cell. It is always possible to choose the matrix element a in to be real since the Hamiltonian is time-reversal invariant. Furthermore, due to the symmetry under chain exchange, we have the following relations between a in :
ψ nσ (x) = i a in c iσ (x),(2.|a 1n | = |a 2n |, |a 3n | = |a 4n |, |a 6n | = |a 7n |.
(2.8)
At weak coupling it suffices to focus only on the four Fermi points. Therefore, we will consider only the upper two bands (n = 1, 2) in the subsequent calculation, since the completely filled bands are not active. We linearize the bands around the Fermi points and decompose the band operators into right and left moving fermion fields (ψ nRσ /ψ nLσ ):
ψ nσ ∝ ψ nRσ e ikF nx + ψ nLσ e −ikF n x .
(2.9)
With this decomposition H 0 reduces to
H 0 = − n,σ dxv F n (ψ † nRσ i∂ x ψ nRσ − ψ † nLσ i∂ x ψ nLσ ).
(2.10)
B. Interaction Hamiltonian -Current Algebra
We now discuss the interactions between the electrons around the four Fermi points. We introduce the usual current operators 26,44
J nP =: ψ † nP α ψ nP α :, J nP = 1 2 : ψ † nP α σ αβ ψ nP β :, L P = ψ † 1P α ψ 2P α , L P = 1 2 ψ † 1P α σ αβ ψ 2P β , M nP = 1 2 ψ nP α σ y αβ ψ nP β , N P αβ = ψ 1P α ψ 2P β , (2.11)
where σ denotes Pauli matrices, P denotes the chirality (R or L), and : : means normal ordering (vacuum subtraction.) Normal ordering signs are suppressed in the subsequent discussion for notational convenience. We also follow the notation of Balents et al. 26 to permit comparison between the RG equations for the one and three-band ladder systems in the Sec. III.
Interaction Hamiltonian density H I can be expressed into three parts in terms of their nature. Let H I = H
(1)
I + H (2) I + H (3)
I . There are eight allowed interactions connecting left and right movers, with Hamiltonian densities H (1)
I : − H (1) I = g 1ρ J 1R J 1L + g 2ρ J 2R J 2L + g xρ (J 1R J 2L +J 2R J 1L ) + g 1σ J 1R · J 1L + g 2σ J 2R · J 2L + g xσ (J 1R · J 2L + J 2R · J 1L ) + g tρ (L R L L +L † R L † L ) + g tσ (L R · L L + L † R · L † L ). (2.12)
Six additional interactions are completely chiral − H
(2) I = λ 1ρ (J 2 1R + J 2 1L ) + λ 2ρ (J 2 2R + J 2 2L ) + λ xρ (J 1R J 2R + J 1L J 2L ) + λ 1σ (J 1R · J 1R +J 1L · J 1L ) + λ 2σ (J 2R · J 2R + J 2L · J 2L ) + λ xσ (J 1R · J 2R + J 1L · J 2L ).
(2.13)
The couplings in Eq. (2.13) just renormalize the velocities of the charge and spin modes, and can be neglected in our second order calculation. Additional M nP and N P αβ operators must be introduced to describe Umklapp processes − H
(3) I = g 1u (M † 1R M 1L + M † 1L M 1R ) + g 2u (M † 2R M 2L +M † 2L M 2R ) + g xu (M † 1R M 2L + M 1R M † 2L +M † 2R M 1L + M 2R M † 1L ) + g tu1 (N † Rαβ N Lαβ +N Rαβ N † Lαβ ) + g tu2 (N † Rαβ N Lβα +N Rαβ N † Lβα ) (2.14)
At half-filling the three interband Umklapp terms ( g xu , g tu1 , g tu2 ) are nonvanishing. The single-band Umklapp term, g nu , is nonzero only if k F n = π/2. The detailed relationships between g and U , V are listed in the Appendix.
III. RENORMALIZATION GROUP
We employ the RG approach combined with bosonization followed by semiclassical energy minimization to obtain the phase diagram at weak coupling. Comparison of the method to results of essentially exact DMRG calculations has shown it to be qualitatively reliable in cases where agreement has been checked; see for instance Ref. 35.
The current algebra as outlined in Balents et al. 26 may be used to obtain one-loop RG flow equations for coupling constants. At half-filling there are eleven coupling constants, and the full set of equations is given bẏ
g 1ρ = β g 2 tρ + 3 16 g 2 tσ − g 2 tu1 − g tu1 g tu2 − g 2 tu2 , g 2ρ = α g 2 tρ + 3 16 g 2 tσ − g 2 tu1 − g tu1 g tu2 − g 2 tu2 , g xρ = −g 2 tρ − 3 16 g 2 tσ − g 2 tu1 − g tu1 g tu2 − g 2 tu2 − g 2 xu , g 1σ = β 2g tρ g tσ − 1 2 g 2 tσ − 4g 2 tu1 − 4g tu1 g tu2 − αg 2 1σ , g 2σ = α 2g tρ g tσ − 1 2 g 2 tσ − 4g 2 tu1 − 4g tu1 g tu2 − βg 2 2σ , g xσ = −2g tρ g tσ − 1 2 g 2 tσ − 4g tu1 g tu2 − 4g 2 tu2 − g 2 xσ ,
g tρ = g 0ρ g tρ + 3 16 g 0σ g tσ − g xu (g tu1 − g tu2 ), g tσ = g 0σ g tρ + g 0ρ − 1 2 g 0σ − 2g xσ g tσ +4g xu (g tu1 + g tu2 ),
g xu = − 2g tρ − 3 2 g tσ g tu1 + 2g tρ + 3 2 g tσ g tu2 −4g xρ g xu , g tu1 = −(2g tρ − 1 2 g tσ )g xu − g tu2 g xσ − g tu1 2g xρ − 1 2 g xσ + αg 1ρ + βg 2ρ + 3 4 (αg 1σ + βg 2σ ) , g tu2 = (2g tρ + g tσ /2)g xu − g tu1 g xσ − g tu2 2g xρ + 3 2 g xσ + αg 1ρ + βg 2ρ − 1 4 (αg 1σ + βg 2σ ) ,(3.1)
where of v F 1 = v F 2 and g xσ = g tν = g xu = g tu2 = 0. Furthermore, previously derived RG equations for half-filled ladders 26 with particle-hole symmetry can be recovered by
g i ≡ g i /[π(v F 1 + v F 2 )], α ≡ (v F 1 + v F 2 )/(2v F 1 ), β ≡ (v F 1 + v F 2 )/(2v F 2 ), g 0ρ = αg 1ρ + βg 2ρ − 2g xρ , and g 0σ = αg 1σ + βg 2σ − 2g xσ .setting v F 1 = v F 2 .
We integrate Eq. (3.1) numerically starting from initial values of g i (0) determined by the bare interactions as presented in the Appendix. In general, the couplings diverge at some large length scale. We integrate Eq. (3.1) until the largest coupling constant equals 0.01, so that the one loop RG flow equations remain valid. We have checked that the phase diagram so obtained is not sensitive to the choice of infrared cut-off.
IV. BOSONIZATION
To elucidate nature of the phase, we use the Abelian bosonization technique to interpret the action semiclassically. The fermionic field operator ψ nP σ is first expressed in terms of dual Hermitian bosonic fields φ nσ and θ nσ by
ψ nP σ = 1 √ 2πǫ κ nσ exp[i(P φ nσ + θ nσ )], (4.1)
where ǫ is a short-distance cutoff, and again P = R or L = ±1 is chirality. The bosonic fields satisfy the usual commutation relations
[φ nσ (x), φ n ′ σ ′ (x ′ )] = [θ nσ (x), θ n ′ σ ′ (x ′ )] = 0, [φ nσ (x), θ n ′ σ ′ (x ′ )] = iπδ n,n ′ δ σ,σ ′ Θ(x − x ′ ), (4.2)
where Θ(x) is the Heaviside function. The Klein factor κ nσ is introduced to ensure the correct anticommutation relations of the original fermionic fields; κ nσ commutes with the bosonic fields, and satisfies
{κ nσ , κ n ′ σ ′ } = 2δ n,n ′ δ σ,σ ′ . (4.3)
A canonical transformation separates bosons into charge and spin pieces
(φ, θ) nρ = 1 √ 2 [(φ, θ) n↑ + (φ, θ) n↓ ], (φ, θ) nσ = 1 √ 2 [(φ, θ) n↑ − (φ, θ) n↓ ]. (4.4)
We also define (φ, θ) rν as the following:
(φ, θ) rν = 1 √ 2 [r(φ, θ) 1ν + (φ, θ) 2ν ],(4.5)
where r is either + or −. With these definitions, the noninteracting part of the Hamiltonian density has the following form:
H 0 = v F 1 + v F 2 2π r,ν [(∂ x φ rν ) 2 + (∂ x θ rν ) 2 ] − v F 1 − v F 2 2π ν [(∂ x φ +ν )(∂ x φ −ν ) + (∂ x θ +ν )(∂ x θ −ν )].
(4.6)
The momentum-conserving part of the interaction can be written as the sum of two terms, H
I = H (1a) I +H (1b) I , where H (1a) I = 1 2π 2 r,ν A rν [(∂ x φ rν ) 2 − (∂ x θ rν ) 2 ] (4.7) with A rρ = − 1 2 [g 1ρ + g 2ρ + 2rg xρ ], A rσ = − 1 8 [g 1σ + g 2σ + 2rg xσ ].(1)
And H (1b) I = − 1 (2πǫ) 2 [2Γg tσ cos 2θ −ρ cos 2φ +σ − (g 1σ + g 2σ ) × cos 2φ +σ cos 2φ −ρ − 2Γg xσ cos 2φ +σ cos 2θ −σ + cos 2θ −ρ (Γg + t cos 2φ −σ + g − t cos 2θ −σ )], (4.8) with g ± t = g tσ ∓ 4g tρ andΓ = κ 1↑ κ 1↓ κ 2↑ κ 2↓ . Finally, the bosonized form of the Umklapp interaction density reads
φ+ρ φ+σ θ−ρ θ−σ φ−σ CDW 0 0 π/2 0 * SP π/2 0 π/2 0 * SF 0 0 0 0 * DC π/2 0 0 0 * D-Mott 0 0 0 * 0 D ′ -Mott π/2 0 0 * 0 S-Mott 0 0 π/2 * 0 S ′ -Mott π/2 0 π/2 * 0H (3) I = 2 (2πǫ) 2 cos 2φ +ρ [2Γg xu cos 2θ −ρ + 2(g tu1 +g tu2 ) cos 2φ +σ + g + u cos 2φ −σ +Γg − u cos 2θ −σ ], (4.9)
with g ± u = (g tu1 + g tu2 ) ± 4(g tu1 − g tu2 ). Note that since the Hermitian operatorΓ obeysΓ 2 = I,Γ has eigenvalues Γ = ±1. At half-filling the system is either fully gapped (by far the most common case) or gapless in all sectors (referred to as C2S2 -see Ref. 26.) We do not find any partially gapped phases at half-filling. For the gapped phases the ground state configuration of the bosonic fields can be determined by minimizing the energy of the low-energy effective Hamiltonian at the end of the RG flow.
Once the ground state configuration of the bosonic fields is determined, the bosonized order parameters may be examined to determine the physical nature of the ground states. We consider the same order parameters studied in previous work on one-band ladder systems: the order parameter of the (π, π) charge density wave (CDW) phase, the (π, π) spin Peierls (SP) phase, the SF phase, the diagonal current (DC) phase, and the four quantum disordered phases (D-Mott, D ′ -Mott, S-Mott, and S ′ -Mott.) Order parameters expressed in terms of bosonic fields are as follows:
O CDW ∝ cos φ +ρ sin θ −ρ cos φ +σ cos θ −σ + sin φ +ρ cos θ −ρ sin φ +σ sin θ −σ ,(4.10a)
O SP ∝ cos φ +ρ cos θ −ρ sin φ +σ sin θ −σ + sin φ +ρ sin θ −ρ cos φ +σ cos θ −σ , (4.10b)
O SF ∝ cos φ +ρ cos θ −ρ cos φ +σ cos θ −σ + sin φ +ρ sin θ −ρ sin φ +σ sin θ −σ , (4.10c)
O DC ∝ cos φ +ρ sin θ −ρ sin φ +σ sin θ −σ + sin φ +ρ cos θ −ρ cos φ +σ cos θ −σ , (4.10d)
O D−Mott ∝ e iθ+ρ cos θ −ρ cos φ +σ cos φ −σ −ie iθ+ρ sin θ −ρ sin φ +σ sin φ +σ ,(4.10e) O S−Mott ∝ e iθ+ρ cos θ −ρ sin φ +σ sin φ −σ −ie iθ+ρ sin θ −ρ cos φ +σ cos φ −σ .(4.10f)
We emphasize that Klein factors have been taken into account in the derivation of these expressions, though we suppress the Klein factors for the sake of simplicity (see Refs. 29,30,31 for more details.) Note that O D−Mott is the order parameter for both D-Mott and D ′ -Mott phase (the same holds between S-Mott and S ′ -Mott.) These primed phases are half-cell translated states of the unprimed states since φ +ρ differs by π/2. 30 The bosonized form of the order parameters is essentially the same as in one band case except for the appearance of complicated band coefficients a ij . Table I lists the bosonic configurations for all the gapped phases. Note that since φ rν and θ rν are conjugate to each other, they cannot be pinned simultaneously. The nature of these phases was discussed in Refs. 27,29,30,31 in one band context. Transitions between the various phases have the same critical behavior as in the one-band case (see, for example, Fig. 5 in Ref. 30.) Note also that the order parameter of the circulating current phase that preserves translational symmetry does not appear in Eqs. (4.10). This is because the longwavelength part of the current is determined by gradients of the boson fields, ∂ x θ ±ρ , but these cannot acquire non-zero expectation values at the semiclassical minima. Field θ −ρ locks into the particular values specified in Table I, and field θ +ρ fluctuates with zero mean gradient. The bosonized form of the current along any link inside the unit cell, apart from the unimportant gradient terms, is given by
j ij (x) ∝ (−1) x t ij (a i1 a j2 − a i2 a j1 )
×(cos φ +ρ cos θ −ρ cos φ +σ cos θ −σ + sin φ +ρ sin θ −ρ sin φ +σ sin θ −σ ). (4.11)
It is clear from this expression that the current pattern, if it exists, must be staggered along the leg direction such that it breaks translational symmetry. Hence, the circulating current phase cannot occur at weak coupling in half-filled Cu-O two-leg ladder system. Furthermore, this is also true away from half-filling, at least as long as the Fermi level intersects only two bands. We sample U d , Up, V dp , V ⊥ dp , and V pp uniformly.
V. WEAK COUPLING PHASE DIAGRAMS
Equipped with the above considerations, we deduce the phase diagram of the three-band Cu-O ladder system in the weak coupling regime. By sampling the parameter space at random, we may quantify the volume of each phase. Table II lists the volume in the five orbital case. We find that the D-Mott, S ′ -Mott, S-Mott, and CDW phases occupy most of the parameter space. The SF phase also arises, though its volume is very small. However, the DC phase does not appear.
It is also useful to consider some slices through parameter space that may be relevant for real systems. In Fig. 3 we present the weak coupling phase diagram of the five orbital Cu-O ladder for U p = 0.38U d and V pp = 0.1V dp . In the absence of nearest neighbor Coulomb interactions the D-Mott phase appears for positive on-site Coulomb interactions; the S-Mott phase appears for negative onsite Coulomb interactions. 45 The CDW phase is found in the second quadrant bisecting the S-Mott phase. There is a small region of S-Mott phase located around 92 • between the D-Mott phase and the CDW phase. Note that direct transitions between the D-Mott and the CDW are not generically possible since it is necessary to unpin two bosonic field simultaneously: θ −ρ and φ −σ or θ −σ (In the one-band model, for example, the SF phase mediates the transition between the D-Mott and the CDW phase. 29 ). The SF phase arises next to the D-Mott phase in the fourth quadrant bisecting the D-Mott phase. A tiny portion of the D-Mott phase exists between the D ′ -Mott and the SF phases, which can be understood with the same argument used for the transition between the D-Mott and the CDW: a direct transition from the S ′ -Mott to the D-Mott or the SF phase is not possible since two bosonic fields must become unpinned simultaneously. Increasing V pp and holding other parameters fixed, the SF phase shrinks and then vanishes. Except for the disappearance of the SF phase, the phase diagram is qualitatively similar to Fig. 3 with slightly different phase boundaries.
The phase diagram of the seven orbital case is presented in Fig. 4. The phase diagram is topologically rather similar to the five orbital case, however, with two notable differences: the SP phase now appears inside the D-Mott phase, and the SF phase disappears altogether. Both of these aspects might be related to the anisotropy between the effective exchange integrals parallel and perpendicular to the legs. In a DMRG calculation on a two-leg Cu-O ladder system by Nishimoto et al., 42 it was found that in the absence of the outer O orbitals, the dimensionless anisotropy ratio R = S i ·S j rung / S i ·S j leg becomes significantly larger than one. Nishimoto et al. concluded that the inclusion of outer oxygens is crucial for a qualitatively correct description of two-leg Cu-O ladders. We can understand this behavior as follows. When R is smaller and the system is therefore more isotropic, the SP phase seems to be a more stable state than the D-Mott phase, as the D-Mott phase may be thought of as a product of rung singlets. For large values of the anisotropy ratio, the strong effective Heisenberg interaction along the rung makes rung singlets relatively strong, favoring the D-Mott phase. Vpp = 0.1V dp . Inset is an enlarged plot which is drawn from -20 to 20 eV. Phase S1 through S6 are defined in the text.
VI. STRONG COUPLING AND THE ROLE OF THE EXCHANGE ENERGIES
In the strong coupling, atomic, limit of the extended Hubbard model, hopping matrix elements may be ignored, and the system is purely electrostatic. The number of electrons on each orbital is restricted to be an integer 0, 1, or 2 by the Pauli exclusion principle. We assume periodic boundary conditions and take the number of sites to be large. The various charge-ordered ground states of the Cu-O ladder may then be found by energy minimization. The lowest energy charge configurations at most double the size of the unit cell in size. Results for the five orbital case are shown in Fig. 5.
For positive U and V we find a uniform phase in which each Cu site is occupied by one electron. This phase corresponds to a Mott insulator. Superexchange Heisenberg spin-spin interaction J and J ⊥ between nearestneighbor copper spins are induced, respectively, along the legs and rungs of the ladder perturbatively at fourth order in the hopping. 46 In the five orbital case
J = 4t 4 dp (ǫ + U d − U p + 3V dp − 4V pp ) 2 × 1 U d + 2 2ǫ + 2U d − U p + 4V dp − 8V pp ,(6.1)J ⊥ = 4t 4 dp (ǫ + U d − U p + 3V dp − 8V pp ) 2 × 1 U d + 2 2ǫ + 2U d − U p + 4V dp − 16V pp ,
and in the seven orbital case
J = J ⊥ = 4t 4 dp (ǫ + U d − U p + 7V dp − 8V pp ) 2 × 1 U d + 2 2ǫ + 2U d − U p + 8V dp − 16V pp , (6.2)
where the on-site energy difference between the d and p electrons is ǫ = ǫ d − ǫ p > 0. With these effective superexchange interactions, degeneracies in the spin degree of freedom are lifted with antiferromagnetic interactions appearing when U d > U p > 0 and V dp > V pp > 0. It is easy to see that the anisotropy ratio is larger than one in five orbital case within this simple calculation, qualitatively consistent with the previous DMRG calculation. 42 This phase can then be mapped to the D-Mott phase found at weak coupling in a similar region of parameter space. In the seven orbital case, the exchanges are isotropic, weakening tendencies towards a D-Mott phase. It should be noted, however, that if we set V pp = 0 in the five orbital case, the system becomes isotropic J = J ⊥ but the SP phase still does not arise. Another possible way to explain the SP phase is to consider frustration. For example, competition between spin exchange along the plaquette and along the diagonal can stablize the SP phase. Also, as pointed out by Kivelson et al. 47 other higher order interactions such as plaquette four-spin ring exchange interaction can be comparable in magnitude to the superexchange energy even at relatively large U , and other phases may arise due to the competition between superexchange and ring exchange interactions. For U < 0, the (π, π) CDW phase arises in a similar region as that found in the case of weak coupling. It is robust against perturbation due to the hopping term. We also find other charge ordered phases that have no direct relationship to the weak coupling phase diagram. All of these charge ordered phases have tendencies toward stripes or charge separation. Phase S1, near the negative U axis with positive V , has two holes occupying the O p y orbital. The ladder breaks into two decoupled completely filled chains. In the S2 regime, two holes occupy O p x orbitals on a leg in doubled unit cell. In the S3 regime, pairs of two holes occupy the Cu orbital and the O p x orbitals on a leg in doubled unit cell. In S4 and S5, holes gather around a Cu site. Finally, in the S6 regime holes gather along one leg of the ladder, and the other leg is completely filled.
VII. CONCLUSION
We determined the phase diagram of the half-filled Cu-O two-leg ladder both at weak and strong coupling. At weak coupling, perturbative RG flows are interpreted with the use of bosonization. Due to absence of particlehole symmetry the Fermi velocities of the two bands differ to each other and different RG flow equations govern the three band model than in the case of the simple one-band ladder. After bosonization, however, the Hamiltonian and the order parameters have essentially the same form as in the one-band case; it is the phase diagrams that differ qualitatively. By studying the Cu-O ladder with and without outer oxygen sites, four interesting conclusions may be drawn. First, in the physically relevant regime, the D-Mott phase arises when outer oxygen sites are absent. Second, the SP phase appears in the seven orbital case. Third, the SF phase only appears when the outer oxygen sites are absent. The appearance / disappearance of phases may be related to the anisotropy of the effective exchange energies in the system. Finally, we found that the CC phase which preserves lattice translational symmetry does not appear, at least at weak coupling in the Cu-O two-leg ladder. Local currents or charge modulation, when they arise, always have a staggered pattern. g tu1 = i,j,q f (i, j, 1, 1, 2, 2, 1, q) + f (i, j, 2, 2, 1, 1, 1, q), g tu2 = −2 i,j,q f (i, j, 1, 2, 2, 1, 1, q).
(A.
3)
The sum in each of the above equations runs over all site indices in the unit cell. Equations (A.3) are the most general form of coupling constants for the case of two bands cutting the Fermi level and short range Coulomb interactions. By modifying w (0) or w (1) further generalizations can be studied.
FIG. 1 :
1Schematic diagrams of the Cu-O ladder. (a)
FIG. 2 :
2Energy spectrum of the quadratic portion of the Hamiltonian in momentum space. (a) Five orbital case: Three low-lying bands are completely filled and only the upper two bands intersect the Fermi level. The Fermi velocities at kF 1 and kF 2 differ even at half filling unlike the one band model. We use ǫ d − ǫ p = 3.0 eV, t dp = t ⊥ dp = 1.6 eV, and t pp = 1.1 eV per the DFT calculation of Andersen et al. (see Ref. 43). (b) Seven orbital case for the same parameters as the five orbital case.
The dot indicates logarithmic derivative with respect to the length scale, i.e.,ġ i ≡ ∂g i /∂s, where s = ln l.The set of RG flow equations obtained in this system is different from those obtained in one band case, due to the absence of particle-hole symmetry in Cu-O ladder. Equation(3.1) is the most general form of RG flow equations for half-filled ladder systems with four Fermi points and with parity, time-reversal symmetry, chain interchange, and U (1) × SU (2) global charge/spin symmetry. Note that Eq. (3.1) is invariant under band interchange. Equation (3.1) also correctly reproduce the RG equations in the SU(4) limit (see Ref. 4 for example)
Furthermore, since [H,Γ] = 0, H andΓ can be simultaneously diagonalized. We choose Γ = 1 in this paper. See Refs. 29,30,31 for details regarding to the subtleties of the Klein factors. For the sake of simplicity we also suppress the Klein factors in what follows.
FIG. 3 :
3Weak coupling phase diagram in the five orbital case: Up = 0.38U d and Vpp = 0.1V dp .
FIG. 4 :
4Weak coupling phase diagram in the seven orbital case: Up = 0.38U d and Vpp = 0.1V dp .
FIG. 5 :
5Strong coupling phase diagram of the five orbital case: Parameter values are ǫ = 3.0 eV, Up = 0.38U d , and
TABLE I :
IPotential phases of the Cu-O two-leg ladder, and
their pinned configurations (a '*' means that the field is fluc-
tuating.)
TABLE II :
IIVolume of each phase over the entire parameter
space determined by Monte-Carlo sampling in the five orbital
case.
λ 1ρ = i,j,qf (i, j, 1, 1, 1, 1, −1, q), λ 2ρ = i,j,q f (i, j, 2, 2, 2, 2, −1, q),λ xρ = i,j,q f (i, j, 1, 1, 2, 2, −1, q) + f (i, j, 2, 2, 1, 1, −1, q) −f (i, j, 1, 2, 2, 1, −1, q), λ 1σ =λ 2σ = 0, λ xσ = −4 i,j,q f (i, j, 1, 2, 2, 1, −1, q), g 1ρ = i,j,q 2f (i, j, 1, 1, 1, 1, −1, q) − f (i, j, 1, 1, 1, 1, 1, q), g 2ρ = i,j,q 2f (i, j, 2, 2, 2, 2, −1, q) − f (i, j, 2, 2, 2, 2, 1, q), g xρ = i,j,q f (i, j, 1, 1, 2, 2, −1, q) + f (i, j, 2, 2, 1, 1, −1, q) −f (i,j, 1, 2, 2, 1, 1, q),g 1σ = −4 i,j,q f (i,j, 1, 1, 1, 1, 1, q),g 2σ = −4 i,j,q f (i, j, 2, 2, 2, 2, 1, q), g xσ = −4 i,j,q f (i, j, 1, 2, 2, 1, 1, q), g tρ = i,j,q2f (i, j, 1, 2, 1, 2, −1, q) − f (i, j, 1, 2, 1, 2, 1, q), g tσ = −4 i,j,q f (i, j, 1, 2, 1, 2, 1, q), g xu = i,j,q 2f (i, j, 1, 2, 1, 2, 1, q),
VIII. ACKNOWLEDGMENTSAPPENDIX: RELATION BETWEEN THE CONTINUUM COUPLINGS AND THE LATTICE MODEL INTERACTION PARAMETERSIn this appendix we discuss the relations between the continuum coupling constants λ and g and the lattice Coulomb interactions U and V . Define w (0) ij and w(1)ij as follows:, and V inter ij are given in Eqs. (2.5) and (2.6). Parameters w (0) and w (1) represent the intracell and intercell interactions, respectively. Double-counting is avoided by taking w (q) ij to not be symmetric. We introduce coefficients f (i, j, k, l, m, n, p, q) as f (i, j, k, l, m, n, p, q) = −a ik a il a jm a jn cos[(k F m + pk F n )q]w where i and j are site indices, k, l, m, and n are band indices, p = ±1, and q = 0, 1.With these definitions, the continuum coupling constantsλ andg are given bỹ
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Current operators in this section should not be confused with the physical current operators. Current operators in this section should not be confused with the physical current operators.
This happens for more general cases in which |U d | > |Up| and also in the absence of the nearest-neighbor Coulomb interactions. This happens for more general cases in which |U d | > |Up| and also in the absence of the nearest-neighbor Coulomb interactions.
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| []
|
[
"ZERO DISTRIBUTION OF HERMITE-PADÉ POLYNOMIALS AND CONVERGENCE PROPERTIES OF HERMITE APPROXIMANTS FOR MULTIVALUED ANALYTIC FUNCTIONS",
"ZERO DISTRIBUTION OF HERMITE-PADÉ POLYNOMIALS AND CONVERGENCE PROPERTIES OF HERMITE APPROXIMANTS FOR MULTIVALUED ANALYTIC FUNCTIONS"
]
| [
"Nikolay R Ikonomov ",
"ANDRalitza K Kovacheva ",
"Sergey P Suetin "
]
| []
| []
| In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class L . The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27]. | null | [
"https://arxiv.org/pdf/1603.03314v1.pdf"
]
| 10,198,720 | 1603.03314 | 1bbcb6c68fbdb6029a85c953266ae5d3d35de231 |
ZERO DISTRIBUTION OF HERMITE-PADÉ POLYNOMIALS AND CONVERGENCE PROPERTIES OF HERMITE APPROXIMANTS FOR MULTIVALUED ANALYTIC FUNCTIONS
Nikolay R Ikonomov
ANDRalitza K Kovacheva
Sergey P Suetin
ZERO DISTRIBUTION OF HERMITE-PADÉ POLYNOMIALS AND CONVERGENCE PROPERTIES OF HERMITE APPROXIMANTS FOR MULTIVALUED ANALYTIC FUNCTIONS
Bibliography: [59] items Figures: 14 items
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class L . The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
new conjectures (see Conjecture 1 and Conjecture 2) on convergence properties of Hermite approximants of multivalued analytic functions of Laguerre class L . Given the germ of a function f analytic at the point of infinity z = ∞, the Hermite Approximants H n,j , j = 0, 1, of order n are completely determined by the 3n + 2 initial Laurent coefficients of the given power series of f . The rational functions H n,j , j = 0, 1 are constructed on the basis of type I Hermite-Padé (HP) polynomials of the collection of three functions 1 [1, f, f 2 ]. All zeros and poles of HA are free. In this respect, they are very similar to Padé approximants (PA). From now on, we assume that f is a multivalued analytic function with a finite set of singular points. In [58], it was proven for a partial class of such multivalued analytic functions that the HA is interpolating approximately 2n times (at free nodes) some other branch 2 of the given function f . Furthermore, there exist limit distributions of the free zeros and poles of HA, as well as of the free nodes. The associated limit measures solve some special equilibrium problems for mixed Green logarithmic potentials with external fields. In some particular cases, it was proven that such HA possesses an alternating property which, as it turns out, is similar to the classical Chebyshëv's alternating property. All these properties make HA very similar to the best Chebyshëv rational approximants of analytic functions (see [21], [47]). We note that for the construction of the n-th HA H n,j merely the 3n + 2 initial Laurent coefficients suffice. In contrast to this, in order to find the best Chebyshëv approximant one needs the function f to be given in an explicit form. Recall once again that to construct the PA [n/n] f of order n of the function f , given by a power series, one should know the 2n + 1 initial Laurent coefficients of the power series (see [6], [2]). All zeros and poles of PA approximants are free, whereas the interpolation nodes are fixed at the point of infinity. This approach is very novel and may be considered as a very promising direction in the theory of the analytic continuation (see [8], [5]).
Let us now introduce the notion of HA of the analytic function f . Given a germ f
f (z) = ∞ k=0 c k z k(1)
of a function f analytic at the infinity point z = ∞, we assume that the three functions 1, f, f 2 are rationally independent over the field of rational functions C(z). Let n be fixed, n ∈ N. Let now Q n,0 , Q n,1 , Q n,2 ∈ C * n [z] := C n [z] \ {0} be the type I Hermite-Padé (HP) polynomials of order n for the collection of the three functions [1, f, f 2 ], that is functions H n,0 := Q n,0 /Q n,2 and H n,1 := Q n,1 /Q n,2 as Hermite approximants of the given analytic function f ∈ H (∞). Given now a finite set Σ ⊂ C (i.e. the set of finite cardinality, card Σ < ∞), we denote by A (C\Σ) the class of all functions f ∈ H (∞) which admit an analytical continuation from the infinity point along each path avoiding the given set Σ. Let A • (C \ Σ) := A (C \ Σ) \ H (C \ Σ). Up to the end of the current paper, we will restrict our attention, while discussing problems concerned with HA, only to the Laguerre class L of multivalued analytic functions. In other words, to the class of multivalued analytic functions given by the explicit representation
f (z) = p j=1 (z − a j ) α j , α j ∈ C \ Z, p j=1 α j = 0,(3)
where a j ∈ C, j = 1, . . . , p, and a j = a k , j = k. Thus if f ∈ L , then f ∈ A • (C \ Σ), where Σ = {a 1 , . . . , a p }. Let us fix the germ of f at the infinity point by the condition f (∞) = 1. We mainly restrict our attention to the partial subclass L of L R of functions given by the representation
f (z) = q j=1 z − e 2j−1 z − e 2j α j , α j ∈ R \ Z,(4)
where e j ∈ R, j = 1, . . . , 2q, and e 1 < · · · < e 2q .
1.2. New conjectures. The main purpose of the current paper is to explain how to use the Hermite Approximants (HA) in the constructive approximation theory, as well as to impose two new conjectures on the convergence of HA for the functions of Laguerre class L . To be more precise, we are interested in studying type I Hermite-Padé polynomials and the corresponding rational HA with free zeros and poles, as well as interpolation nodes. The main objectives of the current paper are the next conjectures.
Conjecture 1. Let f ∈ L and the functions 1, f, f 2 be rationally independent over the field C(z).
Then for z ∈ C \ F Q n,0 Q n,2 (z) cap −→ f 2 (z), n → ∞,(5)
where the compact set F consists of a finite number of closed analytic arcs, F = m j=1 F j . Conjecture 2. Let f ∈ L and all the exponents α j = ±1/2 (see (3)).
Then for z ∈ C \ F Q n,1 Q n,2 (z) cap −→ const ·f (z), n → ∞,(6)
where const = 0 and the compact set F is just the same as in Conjecture 1.
Conjectures 1 and 2 might be considered as a step towards the construction of a general convergence theory of Hermite Approximants. No doubt that the new theory should be much more complicated than Stahl's Theory about classical PA and Buslaev's Theory about multipoint PA. For other conjectures on the limit zero distribution (LZD) of HP polynomials, the reader is referred to [41], [54] and [1].
Conjectures 1 and 2 are based on the rigorous results from [38, Theorem 1.8] and [58], as well as on the numerical experiments produced by the authors in [26], [27]; for more details see §3.4 and Fig. 1-14. 2. Padé approximants 2.1. Padé approximants and J-fractions. Since Hermite approximants are a generalization of classical Padé approximants, we start from the basic definition of Padé polynomials P n,0 , P n,1 and of Padé approximants [n/n] f := −P n,0 /P n,1 .
We recall the definition of PA of an analytic function, given by the power (in fact, by Laurent) series (1) at the infinity point z = ∞. For the sake of convenience, we introduce the Padé polynomials P n,0 , P n,1 ∈ C * n [z] in the following way. There exist polynomials P n,0 and P n,1 of degree n such that (cf. (2))
(P n,0 · 1 + P n,1 · f )(z) = O 1 z n+1 , z → ∞.(7)
The rational function −P n,0 /P n,1 is uniquely determined and is called the diagonal Padé approximant [n/n] f = −P n,0 /P n,1 of the function f (at the infinity point).
In the "generic case" the relation (7) is equivalent to the relation
f − [n/n] (z) = O 1 z 2n+1 , z → ∞.(8)
Thus, from (8) follows that the n-th PA [n/n] f is the best local rational approximant of order n of the given power series (1). Notice that [n/n] f is a rational function with free poles and free zeros. Furthermore, it interpolates the given power series f at the fixed point z = ∞ up to the order 2n + 1. Hence,
[n/n] f (z) = c 0 + c 1 z + · · · + c 2n z 2n + O 1 z 2n+1 , z → ∞.(9)
Recall that by definition of the partial sums S 2n (z) of the power series (1)
S 2n (z) = c 0 + c 1 z + · · · + c 2n z 2n ,(10)
that is,
[n/n] f (z) − S 2n (z) = O 1 z 2n+1 , z → ∞.
Relations (9) and (10) together lead to a very natural question, namely: do the PA [n/n] f (z) have some real advantages over the partial sums S 2n (z)?
The answer is "yes" and comes from the classical J-fractions theory. This is the well-known classical way to evaluate an analytic function going out from its germ, a way which goes back to Gauss and Jacoby (see [6]). However, they used it to evaluate only special (in particular, hypergeometric) functions.
Recall that f is a multivalued analytic function with a finite set Σ of branch points, card Σ < ∞. To be more precise, we suppose that f is analytic in the domain C \ Σ, but not holomorphic in C \ Σ. We adopt the
notation f ∈ A • (C \ Σ) := A (C \ Σ) \ H (C \ Σ).
Let Q ∈ C[z] be an arbitrary complex polynomial. We denote the zerocounting measure of the polynomial Q by χ(Q), that is,
χ(Q) := ζ:Q(ζ)=0 δ ζ ,(11)
where the zeros of Q are counted with regards to their multiplicities; as usual, δ ζ denotes the Dirac measure, concentrated at the point ζ ∈ C.
Let f ∈ L be in a "generic case". Then we can use the functional analog of Euclid's algorithm to obtain the formal expansion (see [16], [6])
. We recall that J n is a rational function of order n, J n ∈ C n (z). Set J ∞ (z) := lim n→∞ J n (z). The problem of convergence of the J-fraction to f may be stated as the problem of equality
f (z) ? = J ∞ (z),(12)
in other words, it is the problem of evaluation of f (z) via J ∞ (z). Since f is a multivalued function and all the J n are single valued functions, two main questions arise in connection with Problem (12): in what sense this equality might be understood and in what domain does it hold true?
2.2. Problem of equality f (z) = J ∞ (z): case p = 2. Let in (3) p = 2, i.e. f (z) = z + 1 z − 1 α , α ∈ C \ Z.(13)
Let assume that α ∈ (−1/2, 1/2), α = 0. Then for Q n , where J n = P n /Q n , we easily obtain (see [16], [59])
1 −1 Q n (x) x k 1 + x 1 − x α dx = 0, k = 0, . . . , n − 1.(14)
Thus Q n (x) = P (−α,α) n (x) is the Jacobi polynomial of degree n with the parameters (−α, α), α ∈ (−1/2, 1/2), and orthogonal on E = [−1, 1]. It follows immediately from (14) that all zeros of Q n belong to the segment [−1, 1], furthermore, for χ(Q n ) the relation
1 n χ(Q n ) * −→ dx π √ 1 − x 2 , n → ∞.(15)
holds (see (11)). It is well-known that P (−α,α) n (z) solves the following linear differential equation of 2-nd degree
(z 2 − 1)w + 2(z − α)w − n(n + 1)w = 0.(16)
By applying the classical asymptotic Liouville-Steklov method [59, § 8.63] to the equation (16), we obtain a formula for the strong asymptotics of the Jacobi polynomials
P (−α,α) n (z) = z − 1 z + 1 α/2 (z + (z 2 − 1) 1/2 ) n+1/2 (z 2 − 1) 1/4 1 + O 1 n , z / ∈ ∆.
(17) Since for the numerator P n of J n = P n /Q n we have P n = P (α,−α) n , i.e. P n is Jacobi polynomial of order n with parameters (α, −α), a direct analog of the strong asymptotics formula (17) is also valid for P n . Thus after combining them, these two formulae provide a strong asymptotics formula for the rational function J n J n (z) = z + 1 z − 1
α 1 + O 1 n , z / ∈ ∆. Therefore, f (z) = J ∞ (z) for z ∈ C \ ∆.(18)
2.3.
Problem of equality f (z) = J ∞ (z): case p = 3. In 1885 Laguerre [32] made an attempt to solve Problem (12) for the partial case when p = 3 and f ∈ L (cf. (3)), i.e.,
f (z) = 3 j=1 (z − a j ) α j , α j ∈ C \ Z, 3 j=1 α j = 0,(19)
where the points a 1 , a 2 , a 3 are in a "general position"; in particular, they are pairwise distinct and don't belong to a straight line. Laguerre derived in 1885 the property of nonhermittian orthogonality for the denominators Q n of the rational function J n = P n /Q n , i.e.
Γ Q n (ζ)ζ k f (ζ) dζ = 0, k = 0, . . . , n − 1,(20)
where Γ is an arbitrary closed contour that separates the three points a 1 , a 2 , a 3 from the infinity point. He also proved (see also [44], [42], [38]) that the polynomial P n and the function Q n f solve the following linear differential equation of second order A 3 (z)Π n,1 (z)w + Π n,3 (z)w + Π n,2 (z)w = 0,
where A 3 (z) = 3 j=1 (z − a j ) and Π n,k ∈ C k [z], k = 1, 2, 3, are some polynomials of degree k. To be more precise,
Π n,1 (z) = z − z n , Π n,2 (z) = −n(n + 1)(z − b n )(z − v n ), Π n,3 (z) = (z − z n )B 2 (z) − A 3 (z), B 2 (z) = A 3 (z)f (z)/f (z).
Thus the polynomial coefficients in equation (21) are of fixed degrees, but depend on n. These polynomial coefficients contain three so-called accessory parameters z n , b n , v n , the behavior of which as n → ∞ is presently unknown. That is why Laguerre couldn't solve neither the problem about the asymptotic behavior of the polynomials P n and Q n , nor the Problem about the equality f (z) = J ∞ (z) as well.
For case Σ = {a 1 , a 2 , a 3 }, Problem (12), which is about the strong convergence of J-fraction, was solved by J. Nuttall in 1986 only in terms of PA, and on the basis of the seminal Stahl's Theorem [55] about the convergence in capacity of PA of an arbitrary multivalued analytic function with a finite set of branch points (for the strong asymptotics and strong convergence properties, see also [41], [43], [56], [7], [28], [4]).
In the "generic case" [n/n] f (z) = J n (z). Hence, the Problem about the equality f (z) = J ∞ (z) is in fact the problem about the convergence of the sequence of PA {[n/n] f (z), n = 0, 1, . . . } of the given analytic function f .
Nuttall proved (see [42]) that for the function f ∈ L , given by (19), the equality f (z) = J ∞ (z) holds true inside the domain D := C \ S, where S is Stahl's compact set, up to a unique arbitrary zero-pole pair (in other words, a spurious zero-pole pair, or a Froissart doublet; see [19], [56], [4]). To be more precise, there is a sequence z n ∈ C such that for each compact set K ⊂ D and for every positive ε > 0
sup z∈K\{z:|z−zn|<ε} f (z) − J n (z) → 0, n → ∞(22)
(cf. (18)). Notice that the convergence relation (22) does not result from Stahl's Theorem, since it is dealing with the LZD (the Limit Zero Distribution) of Padé polynomials and with the convergence of PA in capacity; for the strong convergence see also [7], [4], [36], [29].
Classical Padé approximants: Stahl's Theory. Let f ∈ H (∞) be a multivalued analytic function in the class
A • (C \ Σ), card Σ < ∞,
Given a positive Borel measure µ with a compact support supp µ ⊂ C, supp µ = C, let V µ (z) be the logarithmic potential (see [33], [48]) associated with µ, that is:
V µ (z) := supp µ log 1 |z − ζ| dµ(ζ).
We set V µ * (z) for the spherically normalized logarithmic potential of measure µ, i.e.
V µ * (z) := |ζ| 1 log 1 |z − ζ| dµ(ζ) + |ζ|>1 log 1 |1 − z/ζ| dµ(ζ).
Let f be the germ of a multivalued analytic function f with a finite set of branch points. Then the seminal Stahl's Theorem gives a complete answer to the problem about the limit zero-pole distribution of the classical PA of f . The keystone of Stahl's Theory is the existence of a unique "maximal domain" of holomorphy of f , i.e. of a domain D = D(f ) ∞ such that the given germ f can be continued as a holomorphic (i.e. analytic and single-valued) function from a neighborhood of the infinity point z = ∞ into D (i.e. the function f is continued analytically along each path belonging to D). "Maximal" means that ∂D is of "minimal capacity" among all compact sets ∂G such that G is a domain, G ∞ and f ∈ H (G). To be more precise, we have
cap ∂D = min{cap ∂G : domain G ∞, f ∈ H (G)}.
The "maximal" domain D is unique up to an arbitrary compact set of zero capacity.
The compact set S = S(f ) := ∂D is called "Stahl's compact set" or "Stahl's S-compact set" and D is called "Stahl's domain", respectively. The crucial properties of S for the theory of Stahl to be true are the following: the complement D = C \ S is a domain, S consists of a finite number of analytic arcs (in fact, the union of the closures of the critical trajectories of a quadratic differential), and finally, S possesses the following property of "symmetry" (compact sets of such type are usually called "S-compact sets" or "S-curves", see [45], [30]):
∂g S (z, ∞) ∂n + = ∂g S (z, ∞) ∂n − , z ∈ S • ;(23)
where g S (z, ∞) is the Green's function of the domain D with a logarithmic singularity at the point z = ∞, S • is the union of all open arcs of S (whose closures constitute S, i.e. S \ S • is a finite set), and ∂n + , ∂n − mean the inner (with respect to D) normal derivatives of g S (z, ∞) at a point z ∈ S • from the opposite sides of S • . Let λ = λ S be the unique equilibrium probability measure for S, i.e. V λ (z) ≡ const = γ S for z ∈ S; γ S is the Robin constant for S. Then, by the identity g S (z, ∞) ≡ γ S − V λ (z), the property of symmetry (23) is equivalent to the property
∂V λ ∂n + (z) = ∂V λ ∂n − (z), z ∈ S • .(24)
If
f (z) = 3 j=1 (z − a j ) α j ,
then we have that the compact set S consists of the critical trajectories of the quadratic differential
− z − v A 3 (z) dz 2 > 0, A 3 (z) := 3 j=1 (z − a j ).(25)
These trajectories emanate from the points a j and culminate at the so-called Chebotarëv's point z = v (see [31]). All points a 1 , a 2 , a 3 are the simple poles of the quadratic differential (25) and the Chebotarëv point is the simple zero of that differential. In general, Chebotarëv's point couldn't be found via elementary functions of the points a 1 , a 2 , a 3 . It is uniquely determined from the condition that both periods of the Abelian integral
z ζ − v A 3 (ζ) dζ(26)
are purely imaginary. Because of this, the function
z a 1 ζ − v A 3 (ζ) dζ(27)
is a single-valued harmonic function on the two-sheeted elliptic Riemann surface R 2 , given by the equation
w 2 = (z − v)A 3 (z)
. The Chebotarëv-Stahl compact set is given by the equality
S = z ∈ C : z a 1 ζ − v A 3 (ζ) dζ = 0 ,(28)
and the so-called g-function
g(z) := z a 1 ζ − v A 3 (ζ) dζ(29)
equals identically to the Green's function g S (z, ∞) of the domain D. From the above results it follows immediately that for the equilibrium measure λ (see (25)) the following representation holds:
dλ(z) = 1 πi z − v A 3 (z) dz > 0, z ∈ S.
One of the main results of Stahl's Theory (see [49]- [53], and also [55]) is
Stahl Theorem (H. .
Let the function f ∈ H (∞), f ∈ A • (C \ Σ), card Σ < ∞, let D = D(f ) be Stahl's "maximal" domain of f , S = ∂D be Stahl'
s compact set, and [n/n] f = −P n,0 /P n,1 be the n-th diagonal Padé approximant of the function f . Then the following statements are valid: 1) There exists a LZD of Padé polynomials P n,j , j = 0, 1, namely,
1 n χ(P n,j ) * −→ λ, as n → ∞, j = 0, 1,(30)
where λ = λ S is the unique probability equilibrium measure for the compact set S, i.e. V λ (z) ≡ γ S , z ∈ S, γ S -the Robin constant for S; 2) the n-th diagonal Padé approximants converge in capacity to the function f inside the domain D,
[n/n] f (z) cap −→ f (z), n → ∞, z ∈ D;(31)
3) the rate of the convergence in (31) is completely characterized by the equality
(f − [n/n] f )(z) 1/n cap −→ e −2g S (z,∞) , n → ∞, z ∈ D.(32)
In fact, for each f ∈ A • (C \ Σ) with card Σ < ∞, there is only a finite number of the so-called "spurious" zero-pole pairs, or Froissart doublets [19], which makes impossible the pointwise convergence of PA in Stahl's domain.
The numerical distributions of zeros and poles of PA for the some functions from Laguerre class are demonstrated on the four pictures (see Fig. 1, 2, 3, 4).
2.5. Multipoint Padé approximants: Buslaev's Theory. Let the set Σ with card Σ < ∞, the points z k ∈ C\Σ and functions f k ∈ A • (C\Σ), k = 1, . . . , m be given. We assume that f j ∈ H (z j ), j = 1, . . . , m. Let n ∈ N be fixed. Then there exists two polynomials P n , Q n ≡ 0 of degrees n each and such that the following characteristic relations
(Q n f j − P n )(z j ) = O (z − z j ) n j , z → z j , j = 1, . . . , m,(33)
hold, where m j=1 n j = 2n + 1, n j ∈ Z + , j = 1, . . . , m. Such polynomials P n and Q n are not unique, but the rational function B n = P n /Q n is uniquely determined by the relation (33) and is called a multipoint (or m-point) PA of the given set
f = {f 1 , . . . , f m } of the analytic functions f j ∈ A • (C \ Σ). In short, we will call the set f = {(f 1 , z 1 ), . . . , (f m , z m )} of m multivalued analytic functions f j ∈ H (z j ) the multi-germ or m-germ f.
In general, all functions of the m-germ f are supposed to be different, i.e. not even one of them, say f j , might be obtained as an analytic continuation of another germ, say f k , k = j, along paths, avoiding the set Σ.
In the generic case, (33) is equivalent to the relation
(f j − B n )(z) = O (z − z j ) n j , z → z j , j = 1, . . . , m.(34)
We now suppose that in (33) n j /n → 2p j as n → ∞, m j=1 p j = 1, p j 0, j = 1, . . . , m. According to Buslaev's Theory (2013-2015; see [11]- [12] and also [13], [14]), there exists (in the nondegenerate case) a unique (up to a set of zero capacity) compact set F = F Bus which is an S-curve weighted in the presence of the external field, which is generated by the unit negative charge −ν, ν = m j=1 p j δ z j concentrated at the interpolation points z 1 , . . . , z m . This compact set possesses the following properties: F consists of a finite number of analytic arcs, the complement C \ F of F consists of a finite number of domains D j z j , C \ F = m j=1 D j ; each of the functions f j is holomorphic (i.e. analytic and single-valued) in the corresponding domain D j , f j ∈ H (D j ); if for some k = j the domains coincide with each other, D j = D k , then the corresponding functions are also equal, f k = f j ; the compact set F possesses the property of "symmetry" in the external field V −ν * . Namely, the following relation holds
∂(V β F − V ν * ) ∂n + (z) = ∂(V β F − V ν * ) ∂n − (z), z ∈ F • ,(35)
where β F ∈ M 1 (F ) is a unique equilibrium probability measure concentrated on F and weighted in V −ν * . In other words, the identity
V β F (z) − V ν * (z) ≡ const = w F , z ∈ F,is
valid, where F • is the union of all open arcs which closures constitute the compact set F ; ∂/∂n ± are the normal derivatives to F at the point z ∈ F • from the opposite sides of F . It is worth noting that for the fixed m-germ f the compact set F depends on the numbers p j 0, m j=1 p j = 1. Therefore, the "optimal" (Buslaev's) partition of the Riemann sphere into domains D j also depends on p j .
Just as in Stahl's Theory, the existence of the V −ν -weighted S-curve F = F Bus is crucial for Buslaev's Theory. In accordance with the theory of Stahl, the weighted S-property of the compact set F (35) may be expressed in the following way
∂ m j=1 p j g D j (z, z j ) ∂n + = ∂ m j=1 p j g D j (z, z j ) ∂n − , z ∈ F • ,(36)
where g D j (z, z j ) is the Green's function for the domain D j (as usual, we set
g D j (z, z j ) ≡ 0 when z ∈ D k = D j ).
In what follows, for the sake of simplicity, we restrict our attention to the particular case m = 2 of Buslaev Theorem. Thus, we will discuss in details only the case of two-point Padé approximant.
Let z 1 = 0, z 2 = ∞ and f = {f 0 , f ∞ } be the set of two multivalued analytic functions, such that f 0 ∈ H (0) and f ∞ ∈ H (∞), and also f 0 , f ∞ ∈ A • (C \ Σ), where card Σ < ∞. Thus, each of the functions f 0 and f ∞ is a multivalued analytic function on the Riemann sphere, punctured at a finite set of points, each of which is a branch point of f 0 or of f ∞ or of both of them. In other words, f 0 and f ∞ are two germs of the multivalued analytic function, given at the point z 1 = 0 and z 2 = ∞, respectively. It is worth noting that they may be the two germs of the same analytic function, taken at two different points, namely at z 1 = 0 and z 2 = ∞.
The two-point (in the classical terminology, this is the n-th truncated fraction of the classical T -fraction) PA is defined as follows. Given a number n ∈ N, let P n , Q n ∈ C n [z], Q n ≡ 0, be polynomials of degree n, such that 3 the following relations hold
R n (z) := Q n f − P n (z) = O(z n ), z → 0, O(1/z), z → ∞.(37)
The pair of polynomials P n and Q n is not unique, but the rational function B n := P n /Q n is uniquely determined by (37), and is called the two-point diagonal PA of the set of 2-germ of the functions f = {f 0 , f ∞ }. In the generic case, it follows from (37) that
f − B n (z) = O(z n ), z → 0, O(1/z n+1 ), z → ∞.(38)
If it exists, then the rational function B n = B n (z; f ) ∈ C n (z) is uniquely determined by the relation (38). As for the classical Stahl's case, the existence of an S-curve, associated with the two-point PA and weighted in the external field V −ν * , ν = (δ 0 + δ ∞ )/2, is the crucial element of Buslaev's two-point convergence theorem. Such a weighted S-curve F = F Bus (f 0 , f ∞ ) exists 4 and realizes the "optimal" partition of the Riemann sphere into two domains D 0 0 and D ∞ ∞, such that C = D 0 F D ∞ , f 0 ∈ H (D 0 ) and f ∞ ∈ H (D ∞ ). The compact set F is a weighted S-curve, i.e. F consists of a finite number of analytic arcs and possesses the following property of "symmetry":
∂(V β − V ν * ) ∂n + (z) = ∂(V β − V ν * ) ∂n − (z), z ∈ F • ,(39)
where β = β F is the probability measure concentrated on F and the equilibrium measure in the external field V −ν * (z) = 1 2 log |z|, that is,
V β (z) − V ν * (z) ≡ const, z ∈ F(40)
(In fact, the equilibrium measure β is generated by the negative unit charge −ν, ν = (δ 0 + δ ∞ )/2). As before, F • is the union of all open arcs of F (the closures of which constitute F ) and ∂n + and ∂n − are the inner (with respect to D 0 and D ∞ ) normal derivatives at a point z ∈ F • from the opposite sides of F • . Clearly, β is the balayage of the measure ν from D 0 D ∞ onto F . It is worth noting that F itself is a union of the closures of the critical trajectories of a quadratic differential and the weighted equilibrium measure β = β F is given by (see [10])
dβ(ζ) = 1 2πi 1 ζ V p (ζ) A p (ζ) dζ > 0, ζ ∈ F.(41)
Here, for the sake of simplicity, we only consider the case of two-point PA, and we set z 1 = 0 and z 2 = ∞. In what follows, we also suppose that f 0 and f ∞ are the germs of the same multivalued analytic function f , and we denote them by f 0 ∈ H (0) and f ∞ ∈ H (∞). We suppose that the function f has a finite set of singular points in C.
Notice that the functions f 0 (z) = (1 − z 2 ) −1/2 ∼ 1, z → 0, and f ∞ = (z 2 − 1) −1/2 ∼ 1/z, z → ∞, are the germs of the same analytic function f , given by the equation (z 2 − 1)w 2 = 1. But the functions f 0 (z) = (1 − z 2 ) −1/2 and f ∞ = (z 2 − 1) −1/2 + 1 are not so. Thus, the latter case is the generic case, and hence D 0 ∩ D ∞ = ∅ (see Fig. 5, 6). Now we are ready to formulate the particular case of Buslaev Theorem for two-point PA (cf. Stahl Theorem).
Buslaev Two-Point Theorem (V. I. Buslaev, 2013. Let the function f ∈ H (0) ∩ H (∞), f ∈ A • (C \ Σ), card Σ < ∞, and let the pair of germs f 0 , f ∞ be in a general position 5 . Let D 0 F D ∞ = C be the optimal partition of the Riemann sphere into two domains D 0 0 and D ∞ ∞, such that f 0 ∈ H (D 0 ), f ∞ ∈ H (D ∞ ), D 0 ∩ D ∞ = ∅, and F possesses the weighted S-property with respect to the external field V −ν * , ν = (δ 0 + δ ∞ )/2. Then for the n-diagonal two-point PA B n of the set of the germs f = {f 0 , f ∞ } the following statements hold true: 1) there exists a limit zero-pole distribution for B n , namely,
1 n χ(P n ), 1 n χ(Q n ) * −→ β F , n → ∞;(42)
2) there is a convergence in capacity as n → ∞, namely,
B n (z) cap −→ f 0 (z), z ∈ D 0 , B n (z) cap −→ f ∞ (z), z ∈ D ∞ ;(43)
3) the rate of the convergence in (43) is completely characterized by the relations
f 0 (z) − B n (z) 1/n cap −→ e −g D 0 (z,0) , z ∈ D 0 , f ∞ (z) − B n (z) 1/n cap −→ e −g D∞ (z,∞) , z ∈ D ∞ .(44)
3. Hermite-Padé polynomials and Hermite approximants 3.1. Definition and uniqueness of Hermite approximants. Let us now suppose that the functions 1, f, f 2 are rationally independent and let us consider type I HP polynomials, i.e. Q n,0 , Q n,1 , Q n,2 ∈ C * n [z] and
(Q n,0 · 1 + Q n,1 · f + Q n,2 · f 2 )(z) = O 1 z 2n+2 , z → ∞.(45)
We are now facing two very natural questions. What kind of new results come out from Hermite-Padé polynomials? What can be said about the ratios Q n,0 /Q n,2 and Q n,1 /Q n,2 (cf. (7)), do they converge to analytic functions corresponding with the given f in some way, or do they not? If yes, then does the sequence H n,0 (z) := −Q n,0 /Q n,2 provide more detailed information about the analytic properties of the function f than the sequence of Padé approximants [n/n] f (z) = −P n,0 /P n,1 ? In general, the answer is unknown. However, in some special cases the answer is positive and appears to be very unusual for the HP polynomials theory. Hence, this problem is very promising for forthcoming investigations.
Lemma 1. Let two triples of polynomials Q n,0 , Q n,1 , Q n,2 ∈ C * n [z] and Q n,0 , Q n,1 , Q n,2 ∈ C * n [z] satisfy relation (45). Then the following equalities Q n,0
Q n,0 (z) ≡ Q n,1 Q n,1 (z) ≡ Q n,2 Q n,2 (z).(46)
are true.
Proof of Lemma 1. Indeed, the conditions of Lemma 1 yield
( Q n,0 · 1 + Q n,1 · f + Q n,2 · f 2 )(z) = O 1 z 2n+2 , z → ∞.(47)
After multiplying both sides of (45) by Q n,2 and both sides of (47) by Q n,2 , respectively and subtracting the new equations, we come to (Q n,0 Q n,2 − Q n,0 Q n,2 )(z) + (Q n,1 Q n,2 − Q n,1 Q n,2 )(z)f (z) =
= O 1 z n+2 , z → ∞.(48)
Just in the same way we obtain the equality ( Q n,0 Q n,1 − Q n,0 Q n,1 )(z) + (Q n,1 Q n,2 − Q n,1 Q n,2 )(z)f 2 (z) =
= O 1 z n+2 , z → ∞.(49)
It follows immediately from (48) and (49) that the polynomial P 2n := (Q n,1 Q n,2 − Q n,1 Q n,2 ) ∈ C 2n [z], being of degree 2n, is in fact a type II HP polynomial for the pair f, f 2 .
Since under the conditions of Lemma 1 the triple 1, f, f 2 is rationally independent over the field C(z), it follows that in both relations (48) and (49) the order of approximation at the infinity point should be O(1/z n+1 ) and not O(1/z n+2 ), unless P 2n ≡ 0. Lemma 1 is proved.
Definition 1.
In what follows, we call the uniquely defined rational functions Q n,0 /Q n,2 and Q n,1 /Q n,2 the Hermite Approximants (HA) H n,0 and H n,1 , respectively.
3.2.
Some theoretical results about Hermite approximants. Suppose that f ∈ L . Let Q nj , j = 1, 2, 3 be the HP polynomials for the collection [1, f, f 2 ], and H n,0 , H n,1 be the corresponding HA of the function f . The case (see (3)) p = 2 and a 1 = −1, a 2 = 1,
f (z) = z + 1 z − 1 α , f (∞) = 1,
where 2α ∈ R\Z, was treated by A. Martínez-Finkelshtein, E. A. Rakhmanov and S. P. Suetin, 2014-2015 (see [37], [38]). It was proven [38, Theorem 1.8] that for z ∈ C \ F and F := R \ [−1, 1], we have for n → ∞ (cf. (5) and (6))
Q n,1 Q n,2 (z) → −2 cos απ 1 + z 1 − z α , z / ∈ F, Q n,0 Q n,2 (z) → 1 + z 1 − z 2α = f 2 (z), z / ∈ F, f (0) = 1.(50)
Let now f ∈ L be given by the representation
f (z) = q j=1 z − e 2j−1 z − e 2j α = q j=1 z − e 2j−1 z − e 2j α , f (∞) = 1, α ∈ R \ Z,(51)
with e j ∈ R, −1 = e 1 < · · · < e 2q = 1. We set L R for this subclass of L .
Notice that for f ∈ L R the pair f, f 2 forms the so-called Nikishin's system (see [39], [40], [22], [18], [3], [34]).
Set E := q j=1 [e 2j−1 , e 2j ], D := C\E. Since E = S is the Stahl's compact set for the function f under consideration, then by Stahl's Theorem
[n/n] f (z) cap −→ f (z), n → ∞, z ∈ D,(52)
and
f (z) − [n/n] f (z) 1/n cap −→ e −2g E (z,∞) = e 2(γ E −V λ (z)) , n → ∞, z ∈ D,(53)where g E (z, ∞) ≡ γ E − V λ (z) is the Green's function of D, λ = λ E is the unique equilibrium measure of E, i.e. V λ (x) ≡ const, x ∈ E.
Let now f 2 (z) = const ·f (z), const = 0, be another "branch" (see [17]) of the function f , which is holomorphic in the domain G := C \ F , where F := R \ E, that is, G = D. In general, if f ∈ L is given by the equality
f (z) = p j=1
(z − a j ) α j , then both functions f 1 = f and f 2 solve the same differential equation
A p (z)w + B p−2 (z)w = 0, where A p (z) = p j=1 (z − a j ) and B p−2 (z) = −A p (z) p j=1 α j (z − a j ) −1
are polynomials of degrees p and p − 2, respectively. If p = 2, a 1 = −1,
a 2 = 1, f (z) := z + 1 z − 1 α , z / ∈ E = [−1, 1], f (∞) = 1,
then we have
f 2 (z) = −2 cos απ 1 + z 1 − z α , z / ∈ F, f 2 (0) = 1; see A.
Martínez-Finkelshtein, E. Rakhmanov and S. Suetin [38].
In wider sense, the following result is valid [58] (cf. [35]).
Theorem 1 ((S. Suetin, 2015)). Let f be of type (51) where α ∈ (−1/2, 1/2), α = 0, −1 = e 1 < · · · < e 2q = 1. Then 1) all zeros of Q n,0 , Q n,1 and Q n,2 , up to a finite number that is fixed and independent of n, belong to F ; there exists a LZD of HP Q n,j :
1 n χ(Q n,j ) * −→ η F , n → ∞,(54)
where 3V
η F * (y) + G η F E (y) + 3g E (y, ∞) ≡ const, y ∈ F, supp η F = F ;(55)
2) the rational function H n,1 := −Q n,1 /Q n,2 interpolates the function f 2 at least at 2n − m distinct ("free") nodes x n,j of E • := q j=1 (e 2j−1 , e 2j ) where m ∈ N does not depend on n, and there exist LZD of those free nodes x n,j , namely 1 2n
2n−m j=1 δ x n,j * −→ η E , n → ∞,(56)
where
3V η E (x) + G η E F (x) ≡ const, x ∈ E, supp η E = E;(57)
3) in the domain G := C \ F , the following relation is valid
H n,1 (z) cap −→ f 2 (z), z ∈ G, n → ∞;(58)
and the rate of convergence is completely characterized by the relations (cf.
(53))
f 2 (z) − H n,1 (z) 1/n cap −→ e −2G η E F (z) < 1, z ∈ G \ R, n → ∞, (59) lim n→∞ f 2 (x) − H n,1 (x) 1/n e −2G η E F (x) < 1, x ∈ E • ,(60)
where the measure η E solves problem (57).
In Theorem 1 G η E F (z) = E g E (x, z) dη E (x)
is the Green potential of the measure η E , supp η E = E, g E (x, z) is the Green function for D :
= C \ E, G η F E (z) = F g F (x, z) dη F (x)
is Green potential of measure η F , supp η F ⊂ F , g F (x, z) is the Green function for G := C \ F . Notice that the equilibrium problem (55) was introduced by S. Suetin and E. Rakhmanov in [46] (see also [57], [9], [15]) and is different from the problem that was studied before in papers [20], [39], [21], [23]; see also [40] and [22].
The case when we have (51) with q = 1 and e 1 = −1, e 2 = 1, that is,
f (z) = z + 1 z − 1 α , f (∞) = 1, where 2α ∈ C\Z,dη F dx (x) = √ 3 2π 1 3 √ x 2 − 1 1 3 |x| − 1 − 1 3 |x| + 1 , x ∈ R \ [−1, 1], dη E dx (x) = √ 3 4π 1 3 √ 1 − x 2 1 3 √ 1 − x + 1 √ 1 + x , x ∈ (−1, 1).
Recall the explicit representation of Chebyshëv-Robin equilibrium probability measure λ cheb for the unit segment [−1, 1]:
dλ cheb dx = 1 π √ 1 − x 2 , x ∈ (−1, 1).
Under the condition α = 1/3, i.e. for the function
f (z) = z + 1 z − 1 1/3
relation (60) from Theorem 1 might be improved in the following form. The Hermite approximation H n,1 (z) := −Q n,1 /Q n,2 (z) possesses the property of "almost Chebyshëv alternation" on the open interval (−1, 1) in the following sense. For each positive and arbitrary small θ > 0 on the interval (−1, 1) there exist at least N n = [2n(1 − θ)] consecutive points x j , −1 < x 1 < · · · < x Nn < 1, such that the following equality holds:
f 2 (x j ) − H n,1 (x j ) = (−1) j e −2nG η E F (x j ) 2 3 3 1 + x j 1 − x j 1 + ε n (x j ) ,(61)
where ε n (x) → 0 as n → ∞ with a geometrical rate locally uniformly in (−1, 1). Let
w n (z) := e 2nG η E F (x j ) 3 2 3 1 − x j 1 + x j
be the weight function. Then (61) implies the following weighted equality
w n (x j ) f 2 (x j ) − H n,1 (x j ) = (−1) j (1 + ε n (x j )), j = 1, . . . , N n . 3.3. Orthogonality relations. Let f ∈ H (∞), f (z) = p j=1 (z − a j ) α j , α j ∈ C \ Z, p j=1 α j = 0,(62)
where the points a j ∈ C are pairwise distinct, i.e. a j = a k when j = k.
Thus f ∈ A • (C \ Σ), where Σ = {a 1 , . . . , a p }. We have in the partial case f ∈ L R f (z) = q j=1 z − e 2j−1 z − e 2j α , α ∈ R \ Z,(63)
where −1 = e 1 < · · · < e 2q = 1. Let |α| ∈ (0, 1/2). Let E := q j=1 [e 2j−1 , e 2j ], E • := q j=1 (e 2j−1 , e 2j ), E j := [e 2j−1 , e 2j ]. We fix the branch of f at z = ∞ by f (∞) = 1 and fix a number n ∈ N. By definition (7) γ (P n,0 + P n,1 f )(ζ)q(ζ) dζ = 0 ∀q ∈ C n−1 [ζ],
where γ is an arbitrary contour separating the points e 1 , . . . , e 2q from the infinity point. Let f be given by (63); then it follows from (64) that
E P n,1 (x)x k ∆f (x) dx = 0, k = 0, . . . , n − 1,(65)
where ∆f (x) :
= f + (x) − f − (x), x ∈ E.
Since const ·∆f > 0 on E • for some const = 0, we conclude from (65) that: 1) all but some fixed and independent of n number of zeros of P n,1 belong to E;
2) by Stahl's Theorem, there exists LZD of Padé polynomials P n,1 :
1 n χ(P n,1 ) * −→ λ, n → ∞,(66)
where λ = λ E is a unique equilibrium probability measure concentrated on E, i.e.
V λ (x) ≡ const, x ∈ E;(67)
E = S is the Stahl's compact set of f . From definition (2) of HP polynomials, we may write
γ (Q n,0 + Q n,1 f + Q n,2 f 2 )(ζ)q(ζ) dζ = 0 ∀q ∈ C 2n [ζ],(68)
where γ is an arbitrary closed contour that separates points e 1 , . . . , e 2q from the infinity point. From (68) it follows that for q(z) = P n+k,1 (z) = P n+k,1 (z; f ) we have
E Q n,2 (x)P n+k,1 (x) f (x)∆f (x) dx = 0, k = 1, . . . , n,(69)where f (x) := f + (x) + f − (x),
x ∈ E, and const f (x)∆f (x) > 0 for x ∈ E • with some const = 0. From (69), it follows (see [58]) that: 1) all but some fixed and independent of n number of zeros of Q n,2 belong to F := R \ E;
2) there exists LZD of HP polynomials Q n,2 :
1 n χ(Q n,2 ) * −→ η F , n → ∞,
where η F is a unique special equilibrium probability measure concentrated on F , i.e.
3V η F * (x) + G η F E (x) + ψ(x) ≡ const, x ∈ F ; (70) here G µ E (z) := g E (ζ, z) dµ(ζ), ψ(z) := 3g E (z, ∞),(71)
g E (ζ, z) is the Green function for D := C \ E. The pair of compact sets E, F forms the so-called Nuttall condenser N := (E, F ) = (E; F, ψ). We call the corresponding special equilibrium measure η F from (71) the Nuttall equilibrium measure (see [46], [57], [29]). For LZD of HP polynomials, the notion of Nuttall's condenser plays a role, which is very similar to the role played by Stahl's compact set S in the case of Padé polynomials. In general, if the plates E, F ⊂ R, then they both possess some special "symmetry" property, see [46], [57], [29].
3.4. Discussion of some numerical results. We are going to discuss some numerical examples in order to demonstrate a numerical basis for Conjectures 1 and 2 and for the results of Theorem 1 as well.
From numerical experiments made by R. Kovacheva, N. Ikonomov, and S. Suetin [26], [27], it follows that the distribution of zeros of HP polynomials and the convergence of Hermite approximants itself are very sensitive to the type of branching of multivalued analytic function. More precisely, the situation becomes generally much more complicated, even if all branch points e j still belong to the real line, but in (51) instead of one parameter α we take different parameters α j , α j ∈ R \ Z (see (4)). To be more precise, let the multivalued analytic function f be given by the explicit representation
f (z) = q j=1 z − e 2j−1 z − e 2j α j ,(72)
where e 1 < · · · < e 2q , but α j = α k , j = k. Let us fix the germ of f by the relation f (∞) = 1. In the general situation (72), when there are different α j (instead of a single α) there should be membranes which separate the segments of the set F (see Fig. 9-14).
Case 2. Let q = 3 and f (z) = z + 2.5 z + 1.3 1/3 z + 0.8 z − 0.8 −1/3 z − 1.3 z − 2.5 1/3 .(74)
Thus in (72) α j = (−1) j+1 α. [26], [27] it follows that the distribution of zeros of HP polynomials for the collection [1, f, f 2 ] and the convergence of Hermite approximants H n,j , j = 0, 1, itself are very sensitive to the type of branching of the given multivalued analytic function f . By this reason, it might be very difficult to construct a general theory of limit zero distribution of HP polynomials of such type as Stahl's and Buslaev's theories are. But as surplus, this sensitivity makes Hermite approximants H n,j very powerful tool to recover the unknown properties of a multivalued analytic function given by a germ. Thus the genus of the Stahl's hyperelliptic Riemann surface is 2. Here we observe a single Froissart doublet located in the fourth quadrant. In full compliance with the Rakhmanov's model [45], the Froissart doublet attracts to itself the Stahl's S-compact set S 300 ; cf. Fig. 3.
f = {f 0 , f ∞ }, where f 0 = ((1 − 2z)(2 − z)) −1/2 , f 0 ∈ H (0), f ∞ = ((2z − 1)(z − 2)) −1/2 + 1, f ∞ ∈ H (∞)
. The germs f 0 and f ∞ result in two different multivalued analytic functions. Thus, this is a generic case and by Buslaev's Theorem the associated weighted S-curve divides the Riemann sphere into two domains. . Zeros of HP polynomials Q 320,0 (blue points) and Q 320,1 (red points) for the triple of functions for [1, f, f 2 ], where f (z) = z + 2.5 z + 1.3 3) come from an appropriate theoretical-potential equilibrium problem. By chance, there are no Froissart doublets at all. In case of the given function f , the numerical distribution of zeros of HP polynomial Q 320,0 is very different from the numerical distribution of zeros of HP polynomial Q 320,1 ; cf. Fig. 11 and also 14. There is a zero of polynomial Q 320,0 inside the membrane which correspond to the simple zero of the function f 2 at the point z = −0.3. 3) come from an appropriate theoretical-potential equilibrium problem. By chance, there are no Froissart doublets at all. In case of the given function f , the numerical distribution of zeros of HP polynomial Q 320,2 is very different from the numerical distribution of zeros of HP polynomial Q 320,1 ; cf. Fig. 11 and also 14. There is a zero of polynomial Q 320,2 inside the membrane which correspond to the simple pole of the function f 2 at the point z = 0.3. . There is a membrane which splits the Riemann sphere into two domains. One domain is simply connected, but the other is a doubly connected domain. The zeros of these HP polynomials are distributed in accordance with the description given in Fig. 11, 12
1/3 z + .8 z − .8 −1/3 z − 1.3 z − 2.5 1/3 .
.
Problem of equality f (z) = J ∞ (z): case p = 2 5 2.3. Problem of equality f (z) = J ∞ (z): case p = Description of the problem. The main goal of the current paper is to describe and illustrate the main features of Hermite approximants of multivalued analytic functions. The notion of Hermite approximants (HA) is very novel; it was introduced in an implicit form by A. Martínez-Finkelshtein, E. Rakhmanov, and S. Suetin in [38]. We also propose two Date: March 10, 2016. The third author was partially supported by the Russian Foundation for Basic Research (RFBR, grants 13-01-12430-ofi-m2 and 15-01-07531-a), and Russian Federation Presidential Program for support of the Leading Scientific Schools (grant NSh-2900.2014.1).
Case 1 .
1Let (73) all the exponents are equal to the same α = 1/3, the zeros of the associated HP polynomials Q 200,0 , Q 200,1 , Q 200,2 of the collection [1, f, f 2 ] should be distributed in accordance to Theorem 1. From figures 7-8, it follows that it is really the case. All zeros, except a pair of Froissart triplets, are distributed on the real line R on the complement of three real segments [−2.5, −1.3], [−0.8, 0.8], and [1.3, 2.5].
Case 3 .
3Figures 9-10 represent the numerical distribution of zeros of HP polynomials Q 320,0 , Q 320,1 , Q 320,2 of the collection of the functions [1, f, f 2 ]. In this case there is a membrane, which splits the complement of the segments [−2.5, −1.3], [−0.8, 0.8] and [1.3, 2.5] into two domains. The zeros of these HP polynomials are distributed on the real line R on the complement of the segments [−2.5, −1.3], [−.8, .8] and [1.3, 2.5] and on this membrane. The points of intersection of the membrane with the two segments are the Chebotarëv's points of zero-density for the equilibrium measure for a compact set F . By chance, there are no Froissart triplets at all (see Fig. 9-10). Let -14 represent the numerical distribution of zeros of HP polynomials Q 200,0 , Q 200,1 , Q 200,2 for the collections of functions [1, f, f 2 ]. There also exists a membrane, but of another type than in Case 2. This membrane splits the complement of the three segments [−2.5, −1.3], [−0.3, 0.3] and [1.3, 2.5] into two domains. The zeros of those HP polynomials are distributed on the real line R on the complement of the segments [−2.5, −1.3], [−0.3, 0.3] and [1.3, 2.5] on this new membrane. Just as in Case 2, the two points of intersection of the membrane with the segments are the Chebotarev's points of zero-density for the equilibrium measure for compact set F . 3.5. Final remarks about Hermite approximants. Thus, from the numerical experiments of R. Kovacheva, N. Ikonomov, and S. Suetin, see
Figure 1 .Figure 2 .
12Zeros (blue points) and poles (red points) of PA [130/130] f of the function f (z) = z−(−1.2+0.8i) genus of the corresponding Stahl's two-sheeted Riemann surface equals 1 (i.e. it is an elliptic Riemann surface), there might be at most a single "spurious" zero-pole pair, i.e. a single Froissart doublet. It is really present on the picture.22 NIKOLAY R. IKONOMOV, RALITZA K. KOVACHEVA, AND SERGEY PZeros (blue points) and poles (red points) of PA [267/267] f of the function f (z) = {(z + (4.3 + 1.0i))(z − (2.0 + 0.5i))(z + (2.0 + 2.0i))(z + (1.0 − 3.0i))(z − (4.0 + 2.0i))(z − (3.0 + 5.0i))} −1/6 . These zeros and poles are distributed in a plane, under fixed n = 267, accordingly to the electrostatic model by Rakhmanov[45]. There are 4 Chebotarëv points on the picture. Thus the genus of the Stahl's hyperelliptic Riemann surface is 4. By this reason for each n there might be no more than 4 Froissart doublets. Here are observed 4 Froissart doublets (cf. [58,Fig. 2]).
Figure 3 .
3Zeros (blue points) and poles (red points) of PA [300/300] f of the quadratic function f (z) = z − (
Figure 4 .
4Zeros (blue points) and poles (red points) of PA [300/300] f of the logarithmic function f (z) = log z − (−1.0 + 0.8i) z − (1.0 + 1.2i) + log z − (−1.0 + 1.5i) z − (−1.0 − 1.5i) . There are 2 Chebotarëv points on the picture.
Figure 5 .
5Numerical zeros (blue points) and poles (red points) distribution of two-point PA [120/120] f of the set of functions
Figure 6 .
6Numerical zeros (blue points) and poles (red points) distribution of two-point PA[195/195] f to the function f (z) = 4 (z − a 1 )/(z − a 2 ), where a 1 = 0.9 − 1.1i and a 2 = 0.1 + 0.2i. Here are selected two "quite different branches" of the function f , namely, f 0 = 4 (z − a 1 )/(z − a 2 ) and f ∞ = − 4 (z − a 1 )/(z − a 2 ). All, but one pair, zeros (blue points) and poles (red points) approximate numerically Buslaev's compact set. But there is a single Froissart doublet located in the domain D 0 (f ) 0; cf. [58,Fig. 3].
Figure 7 ..
7Zeros of HP polynomials Q 200,0 (blue points) and Q 200,1 (red points) for the triple of functions [1, f, f 2 ], where f (All but two pairs of zeros are distributed in accordance with Theorem 1 on the real line on the complement of the three closed segments [−2.5, −1.3], [−0.8, 0.8], and [1.3, 2.5]. There are two pairs of complex conjugate Froissart doublets; cf. Fig. 8.
Figure 8 ..
8Zeros of HP polynomials Q 200,0 (blue points), Q 200,1 (red points), and Q 200,2 (black points) for the triple of functions [1, f, f 2 ], where f (All but two pairs of zeros are distributed in accordance with Theorem 1 on the real line on the complement of the three closed segments [−2.5, −1.3], [−0.8, 0.8], and [1.3, 2.5]. There are two pairs of complex conjugate Froissart triplets; cf. Fig. 7.
Figure 9
9Figure 9. Zeros of HP polynomials Q 320,0 (blue points) and Q 320,1 (red points) for the triple of functions for [1, f, f 2 ], where f (z) = z + 2.5 z + 1.3
Figure 10 ..
10Zeros of HP polynomials Q 320,0 (blue points), Q 320,1 (red points), and Q 320,2 (black points) for the triple of functions for [1, f, f There is a mem-brane which splits the complement to the segments [−2.5, −1.3], [−0.8, 0.8] and [1.3, 2.5] into two domains. Both domains are simply connected. The zeros of these HP polynomials are distributed on real line R on the complement of the segments [−2.5, −1.3], [−0.8, 0.8] and [1.3, 2.5] and on this membrane. The points of intersection of the membrane with the two segments are the Chebotarëv's points of zero-density for the equilibrium measure for a compact set F . By chance, there are no Froissart triplets at all; cf. Fig. 9.
Figure 11 .
11Zeros of HP polynomial Q 320,1 (red points) for the triple of functions [1, f, f 2 ], where f (no membrane here. The zeros of HP polynomial Q 320,1 are distributed on the real line on the complement of four segments [−2.5, −1.3], [−a, −0.3], [0.3, a], and [1.3, 2.5] where a ∈ (0.3, 1.3) is an unknown parameter. This parameter should be evaluated from an appropriate theoreticalpotential equilibrium problem. By chance, there are no Froissart doublets at all. In case of the given function f , the numerical distribution of zeros of HP polynomial Q 320,1 is very different from numerical distribution of zeros of HP polynomial Q 320,0 and Q 320,2 ; cf.Fig. 12, 13, 14.
Figure 12 .
12Zeros of HP polynomial Q 320,0 (blue points) for the triple of functions [1, f, f 2 ], where f (a membrane which splits the Riemann sphere into two domains. The zeros of HP polynomial Q 320,0 are distributed on this membrane and on the real line on the complement of three segments [−2.5, −1.3], [−a, a], and [1.3, 2.5]. The membrane and a parameter a ∈ (0.3, 1.
Figure 13 .
13Zeros of HP polynomial Q 320,2 (black points) for the triple of functions [1, f, f 2 ], where f (a membrane which splits the Riemann sphere into two domains. The zeros of HP polynomial Q 320,2 are distributed on this membrane and on the real line on the complement of three segments [−2.5, −1.3], [−a, a], and [1.3, 2.5]. The membrane and a parameter a ∈ (0.3, 1.
Figure 14 .
14Zeros of HP polynomials Q 320,0 (blue points), Q 320,1 (red points), and Q 320,2 (black points) for the triple of functions [1, f, f 2
was treated by A. Martínez-Finkelshtein, E. A. Rakhmanov and S. P. Suetin in 2013-2015. The first version of Theorem 1 was established in [38, Theorem 1.8]; furthermore, the following explicit representation for both measures η E and η F were found, namely
. There are 2 Chebotarëv points on the picture.Thus the genus of the Stahl's hyperelliptic Riemann surface is 2. Here we observe a single Froissart doublet located in the second quadrant. In full compliance with the Rakhmanov's model[45], the Froissart doublet attracts to itself the Stahl's S-compact set S 300 ; cf.Fig. 4.1/2
There is a membrane which splits the complement of the segments [−2.5, −1.3], [−0.8, 0.8] and [1.3, 2.5] into two domains. Both domains are simply connected. The zeros of these HP polynomials are distributed on real line R on the complement of the segments [−2.5, −1.3], [−0.8, 0.8] and [1.3, 2.5] and on this membrane. The points of intersection of the membrane with the two segments are the Chebotarëv's points of zero-density for the equilibrium measure for a compact set F . By chance, there are no Froissart doublets at all; cf. Fig. 10.
, 13. The points of intersection of the membrane with the two segments [−1.3, −a] and [a, 1.3] are the Chebotarëv's points of zero-density for the equilibrium measure from an appropriate theoretical-potential equilibrium problem. By chance, there are no Froissart doublets at all.
For a fixed n ∈ N, we can also claim that the left side of (37) is O(z n+1 ) as z → 0 and O(1) as z → ∞, but this does not change the convergence theorem itself.4 In general there may exist some degenerated cases.12 NIKOLAY R. IKONOMOV, RALITZA K. KOVACHEVA, AND SERGEY P. SUETIN
Equivalently, we say that Buslaev's S-curve F divides the Riemann sphere into two domains.
NIKOLAY R. IKONOMOV, RALITZA K. KOVACHEVA, AND SERGEY P. SUETIN
NIKOLAY R. IKONOMOV, RALITZA K. KOVACHEVA, AND SERGEY P. SUETIN
NIKOLAY R. IKONOMOV, RALITZA K. KOVACHEVA, AND SERGEY P. SUETIN
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Gabor Szegö, xiii+432 pp. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences E-mail address: [email protected]. American Mathematical SocietyOrthogonal polynomialsGabor Szegö, Orthogonal polynomials, Fourth edition, Colloquium Publications, XXIII, American Mathematical Society, Providence, R.I., 1975, xiii+432 pp. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences E-mail address: [email protected]
| []
|
[
"Simultaneous Broadband Vector Magnetometry Using Solid-State Spins",
"Simultaneous Broadband Vector Magnetometry Using Solid-State Spins"
]
| [
"Jennifer M Schloss \nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n\nCenter for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA\n",
"John F Barry \nCenter for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA\n\nLincoln Laboratory\nMassachusetts Institute of Technology\n02420LexingtonMassachusettsUSA\n\nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n",
"Matthew J Turner \nCenter for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n",
"Ronald L Walsworth \nCenter for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n"
]
| [
"Department of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Center for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA",
"Center for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA",
"Lincoln Laboratory\nMassachusetts Institute of Technology\n02420LexingtonMassachusettsUSA",
"Harvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA",
"Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA",
"Center for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA",
"Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA",
"Center for Brain Science\nHarvard University\n02138CambridgeMassachusettsUSA",
"Harvard-Smithsonian Center for Astrophysics\n02138CambridgeMassachusettsUSA",
"Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA"
]
| []
| We demonstrate a vector magnetometer that simultaneously measures all Cartesian components of a dynamic magnetic field using an ensemble of nitrogen-vacancy (NV) centers in a single-crystal diamond. Optical NV-diamond measurements provide high-sensitivity, broadband magnetometry under ambient or extreme physical conditions; and the fixed crystallographic axes inherent to this solid-state system enable vector sensing free from heading errors. In the present device, multichannel lock-in detection extracts the magnetic-field-dependent spin resonance shifts of NVs oriented along all four tetrahedral diamond axes from the optical signal measured on a single detector. The sensor operates from near DC up to a 12.5 kHz measurement bandwidth; and simultaneously achieves ∼ 50 pT/ √ Hz magnetic field sensitivity for each Cartesian component, which is to date the highest demonstrated sensitivity of a full vector magnetometer employing solid-state spins. Compared to optimized devices interrogating the four NV orientations sequentially, the simultaneous vector magnetometer enables a 4× measurement speedup. This technique can be extended to pulsed-type sensing protocols and parallel wide-field magnetic imaging. | 10.1103/physrevapplied.10.034044 | [
"https://arxiv.org/pdf/1803.03718v2.pdf"
]
| 4,893,399 | 1803.03718 | aa8e4b035ad786573a19a29dfdc46ad6ca163700 |
Simultaneous Broadband Vector Magnetometry Using Solid-State Spins
Jennifer M Schloss
Department of Physics
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
Center for Brain Science
Harvard University
02138CambridgeMassachusettsUSA
John F Barry
Center for Brain Science
Harvard University
02138CambridgeMassachusettsUSA
Lincoln Laboratory
Massachusetts Institute of Technology
02420LexingtonMassachusettsUSA
Harvard-Smithsonian Center for Astrophysics
02138CambridgeMassachusettsUSA
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Matthew J Turner
Center for Brain Science
Harvard University
02138CambridgeMassachusettsUSA
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Ronald L Walsworth
Center for Brain Science
Harvard University
02138CambridgeMassachusettsUSA
Harvard-Smithsonian Center for Astrophysics
02138CambridgeMassachusettsUSA
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Simultaneous Broadband Vector Magnetometry Using Solid-State Spins
(Dated: March 28, 2018)
We demonstrate a vector magnetometer that simultaneously measures all Cartesian components of a dynamic magnetic field using an ensemble of nitrogen-vacancy (NV) centers in a single-crystal diamond. Optical NV-diamond measurements provide high-sensitivity, broadband magnetometry under ambient or extreme physical conditions; and the fixed crystallographic axes inherent to this solid-state system enable vector sensing free from heading errors. In the present device, multichannel lock-in detection extracts the magnetic-field-dependent spin resonance shifts of NVs oriented along all four tetrahedral diamond axes from the optical signal measured on a single detector. The sensor operates from near DC up to a 12.5 kHz measurement bandwidth; and simultaneously achieves ∼ 50 pT/ √ Hz magnetic field sensitivity for each Cartesian component, which is to date the highest demonstrated sensitivity of a full vector magnetometer employing solid-state spins. Compared to optimized devices interrogating the four NV orientations sequentially, the simultaneous vector magnetometer enables a 4× measurement speedup. This technique can be extended to pulsed-type sensing protocols and parallel wide-field magnetic imaging.
I. INTRODUCTION
A wide range of magnetometry applications require real-time sensing of a dynamic vector magnetic field, including magnetic navigation [1][2][3][4], magnetic anomaly detection [2,3], surveying [5], current and position sensing [2,3,6], and biomagnetic field detection and imaging [7][8][9][10][11][12][13][14]. Scalar magnetometers, such as vapor cell, proton precession, and Overhauser effect magnetometers, measure only the magnetic field magnitude [6]. Vector projection magnetometers, including SQUIDs, fluxgates, and Hall probes, measure the magnetic field projection along a specified axis in space; the determination of all three Cartesian field components then requires multiple sensors aligned along different axes. Uncertainty or drifts in the relative orientations or gains of these multiple sensors can result in heading errors, which limit the vector field reconstruction accuracy [15,16]. In contrast, the fixed crystallographic axes inherent to solid-state spinbased sensors allow complete vector field sensing while mitigating systematic errors from sensor axis misalignment and drifting gains [4,10,[17][18][19][20].
In particular, negatively charged nitrogen-vacancy (NV) centers in single-crystal diamond provide highsensitivity broadband magnetic sensing and imaging under ambient conditions [23]. The NV center is an atomicscale defect consisting of a substitutional nitrogen adjacent to a vacancy in the lattice. The NV center's electronic ground state has spin S = 1 with the lower-energy m s = 0 level separated from the m s = ±1 levels by a zerofield splitting D ≈ 2.87 GHz (see Fig. 1(a)). NV centers have symmetry axes aligned along one of four crystallographic orientations set by the tetrahedral symmetry of the diamond lattice (see Figs. 1(b) and 1(c)). In a bias magnetic field B 0 , the m s = ±1 energy levels of an NV center are Zeeman-shifted by ≈ ± γ e B 0 ·n for fields γ e B 0 2πD, where γ e = g e µ B / = 2π × 28.03 GHz/T is the NV electron gyromagnetic ratio andn is the NV symmetry axis. The NV spin state can be prepared and read out optically, mediated by an intersystem crossing through a set of singlet states with preferential decay to the |m s = 0 state, causing higher photoluminescence (PL) from the |m s = 0 than from the |m s = ±1 states [24,25]. By measuring the optically detected magnetic resonance (ODMR) features of an ensemble of NV centers, with NV symmetry axes distributed along all four crystallographic orientations ( Fig. 1(d)), the three Cartesian components of a vector magnetic field signal can be sensed using a monolithic diamond crystal [18].
To date, ensemble NV vector magnetometers measure the three Cartesian magnetic field components either by sweeping a microwave (MW) tone across the full ODMR spectrum [17,26,27] or by interrogating multiple ODMR features, either individually [10,18,23,[28][29][30] or in parallel [31][32][33], with near-resonant MWs. Although at least three ODMR features must be interrogated to determine the magnetic field vector, four or more are often probed to mitigate systematic errors from strain, electric fields, or temperature variation (see Appendix B) [21,34,35]. Regardless of implementation, these existing methods all reconstruct the three Cartesian magnetic field components from a series of field projection measurements along at least three predetermined axes.
In existing implementations, the projective field measurements are performed sequentially, and such magnetometers have so far only demonstrated sensing of static [11 N 1] [1 N 11] [111] [1 N 1 N 1] [110] [1 N 10] [001] or slowly varying fields fields [10,17,18,26,32,33,36]. A sequential vector magnetometer inherently exhibits suboptimal sensitivity, however, as the sensor is temporarily blind to magnetic field components transverse to the chosen axis during each projective measurement. In addition, any dead time associated with the vector field measurement, including time spent switching the MW frequency or driving far off resonance, reduces the measurement speed and bandwidth as well as the achievable sensitivity of a shot-noise-limited device.
To overcome these drawbacks, we demonstrate simultaneous measurement of all Cartesian magnetic field components using parallel addressing and readout from all four NV orientations. By performing four projective field measurements simultaneously, our technique enables high-sensitivity, broadband vector magnetometry without the inefficiency inherent to sequential projective measurement techniques. This method can decrease the time required to reconstruct a magnetic field vector with a given signal-to-noise ratio (SNR) by at least 4× compared to optimized sequential addressing of the NV orientations, resulting in at least √ 4 = 2× higher sensitivity for shot-noise-limited magnetometers.
II. MAGNETOMETRY METHOD
In many high-sensitivity measurements, technical noise such as 1/f noise is mitigated by moving the sensing bandwidth away from DC via up-modulation. One method, common in NV-diamond magnetometry experiments, applies frequency [12,33,37,38] or phase modulation [18,[39][40][41] to the MWs addressing a spin transition, which causes the magnetic field information to be encoded in a band around the modulation frequency. Here we demonstrate a multiplexed extension of this scheme, where information from multiple NV orientations is encoded in separate frequency bands and measured on a single optical detector. Lock-in demodulation and filtering then extracts the signal associated with each NV orientation, enabling concurrent measurement of all components of a dynamic magnetic field.
In this technique, four dedicated MW tones, each dithered at a unique modulation frequency, address a subset of four of the eight ODMR features shown in Fig. 1(d). The implementation here uses modulated continuous-wave (CW) ODMR, where the MWs are frequency modulated and the PL from all NV orientations is detected continuously on a single optical detector. Multichannel demodulation and filtering in software reveal the ODMR line center shifts in response to a change in the magnetic field vector. The Cartesian components of the dynamic magnetic field are reconstructed in real time utilizing an approximated linear transformation derived from the NV ground state spin Hamiltonian (see Appendix B 2). This simultaneous magnetometry method generalizes to addressing any number of ODMR features. Figure 2 depicts the experimental setup, including laser excitation, MW generation, and magnetic field detection. The diamond crystal is a 4 mm × 4 mm × 0.5 mm chip with 110 edges and a {100} front facet, with [ 14 N] ≈ 28 ppm and an N-to-NV − conversion efficiency of ∼ 10%. The diamond is adhered to a 2" diameter, 330 µm thick semi-insulating silicon carbide (SiC) wafer for thermal and mechanical stabilization (see Fig. 2 Excitation laser beam passes through polarizer, half waveplate (HWP), and focusing lens. After a mirror, the beam passes through a beam sampler, where a fraction is imaged onto a photodiode and digitized at the analog-to-digital converter (ADC); and the rest of the beam impinges on the diamond. The diamond PL is collected by the aspheric condenser, long-pass filtered at 633 nm, imaged onto a photodiode, and digitized. See Appendices C 1 and C 4 for additional information and Fig. S1 of the Supplemental Material for a detailed electronics schematic.
III. EXPERIMENT
a 633 nm long-pass filter, after which ∼ 52 mW of PL is imaged onto a photodiode and digitized, as shown in Fig. 2(b) (see Appendices C 1-C 4). A picked-off portion of the green excitation light (∼ 135 mW) is collected onto a second photodiode and digitized for software-based laser intensity noise cancellation (see Appendix C 5).
As shown in Fig. 2(b), four separate MW sources generate four carrier signals at frequencies ν λ , ν χ , ν ϕ , and ν κ resonant with the ODMR features respectively numbered 2, 4, 6, and 8 in Fig. 1(d). These MW tones are sinusoidally frequency modulated at corresponding modulation frequencies f λ , f χ , f ϕ , and f κ and frequency deviations δν λ , δν χ , δν ϕ , and δν κ , all of which are tabulated in Table I. A mixer for each carrier signal generates sidebands at ±2.158 MHz to address hyperfine subfeatures [12]. These MW signals are amplified and combined onto a copper wire loop made from a shorted end of semi-rigid nonmagnetic coaxial cable, which is placed in proximity to the diamond, as depicted in Fig. 2(a). The modulation frequencies f i , i = λ, χ, ϕ, κ are selected to balance measurement bandwidth and contrast [12,37] while ensuring that no frequency is an integer multiple of any other. The latter choice avoids cross-talk from the NVs responding nonlinearly to the modulated MWs at harmonics of f i . All f i are chosen to divide evenly into the overall sampling rate F s = 202.800 kSa/s. Deviations δν λ , δν χ , δν ϕ , and δν κ are empirically optimized for maximal demodulated signal contrast. The optimal δν i depends on ODMR linewidth, which varies among the addressed NV resonances, as shown in Fig. 1(d). The linewidth variations are the result of varying degrees of optical and MW power broadening, which arise from different laser and MW polarization angles with respect to the optical and magnetic transition dipole moments of the four NV orientations [43][44][45]. The demodulated signals S i are converted to ODMR line shifts ∆ν i via measured lock-in signal slopes dS i /d∆ν i reported in Table I. These slopes are determined by sweeping the modulated MW carrier frequencies by 20 kHz and fitting linear functions to the detected demodulated signals S i (see Appendix C 7).
As a demonstration, the NV-diamond sensor is placed in a bias magnetic field | B 0 | = 7.99 mT, B 0 = (3.54, 1.73, 6.95) mT, where the lab-frame coordinates (x, y, z) are defined with respect to the normal faces of the mounted diamond crystal, with unit vectorŝ x,ŷ, andẑ lying along [110], [110], and [001], respectively, as depicted in Fig. 1(c). In this coordinate system, the unit vectors parallel to the NV symmetry axes aren κ = 2 /3, 0, 1 /3
[111],n λ = 0, − 2 /3, − 1 /3 [111],n ϕ = 0, 2 /3, − 1 /3 [111], andn χ = − 2 /3, 0, 1 /3 [111]
. In this bias field B 0 , the ODMR spectrum is measured (see Fig. 1(d)), and the resonance line centers are tabulated (see Appendix B 1). By numerically fitting the NV ground-state Hamiltonian to the eight measured resonance line centers (see Appendix B 1), the bias field B 0 is determined, along with the value of D and the longitudinal electric/strain field coupling parame-
ters M z ≡ [M λ z , M χ z , M ϕ z , M ϕ z ].
From the fit, we obtain D = 2.8692 GHz, M z = [−20, −60, 50, 30] kHz, and B 0 as reported above, which is consistent with a Hall probe measurement of B 0 (see Appendix C 3).
Next, the Hamiltonian is linearized around the measured B 0 , D, and M z , to calibrate the expected frequency shifts of the four addressed NV resonance line centers in the presence of an additional small field B sens to be sensed. The result of the numerical linearization is a
4 × 3 matrix A, where ∆ν λ ∆ν χ ∆ν ϕ ∆ν κ sens = A B x B y B z sens .
(1) Table I), and encoded magnetic field signal around each fi. Third-order intermodulation products at 2028 Hz, 2366 Hz, 3042 Hz, and 4732 Hz, and the second harmonic of fχ at 5408 Hz arise due to the nonlinear response of the NV PL to frequency-modulated MW driving (see Supplemental Material) [42]. c-f) Spectral densities of magnetic-field-dependent frequency shifts ∆νi,RMS of each addressed NV ODMR feature from 1 second of PL detection, after demodulating at fi and filtering; (inset), time series of same demodulated and filtered data, showing magnetic-field-dependent shifts ∆νi, and cartoons depicting the NV orientation corresponding to each measured ∆νi trace. In the limit of small magnetic field and low strain, the rows of the matrix A are given, up to a sign, by the NV symmetry axis unit vectorsn i (see Appendix B 2). The left Moore-Penrose pseudoinverse A + is then numerically calculated at the measured B 0 , D, and M z and used to transform detected frequency shifts ∆ν i (t), i = λ, χ, ϕ, κ (displayed in Figs. 3(c)-3(f)) into a measured vector field B sens (t), shown in Fig. 4. This linearized matrix method provides a ∼ 25,000× speedup compared to unoptimized numerical least-squares fitting of the resonance frequencies ν i in the presence of B 0 + B sens . A comparison of the two methods yields good agreement, with fractional error 10 −5 for sensed fields 100 nT (see Appendix B 2).
IV. VECTOR SENSING DEMONSTRATION
Three orthogonal coils create the time-varying mag-
netic field B sens (t) = (B x (t), B y (t), B z (t)) at the diamond sensor, where B j (t) = √ 2B j,RMS · sin(2πf j · t + φ j ) for j = x, y, z.
Here f x = 67 Hz, f y = 32 Hz, f z = 18 Hz, and the phases φ x , φ y , φ z are chosen arbitrarily. The applied field amplitudes B x,RMS = 8.12 nT, B y,RMS = 9.56 nT, and B z,RMS = 9.86 nT are calibrated by conventional sequential NV magnetometry methods and are consistent with a priori calculations from the known coil geometries and applied currents. (See Supplemental Material for discussion of off-axis field nulling.) Figure 3(b) shows the voltage spectral density of the digitized, noise-canceled PL signal from 1 second of data acquisition. (See Fig. S4 of the Supplemental Material for semi-log plots of the same data over different frequency ranges.) The raw PL signal is high-pass filtered at 1690 Hz and demodulated at the four modulation frequencies f i , i = χ, κ, λ, ϕ, by mixing the PL signal with a normalized sinusoidal waveform at each f i . The four demodulated time traces are then band-pass filtered, notchstop filtered, and downsampled to 2.704 kSa/s, producing the data shown in Fig.s 3
(c)-3(f) (see Appendix C 6).
The single-sided equivalent noise bandwidth of each of the resulting time traces is f ENBW = 203 Hz. For applications requiring sensing at higher frequencies, measurement bandwidth can be greatly increased for a small (order unity) loss in sensitivity. (See Supplemental Material and Fig. S2 for a demonstration with higher frequency magnetic fields and ≈ 12.5 kHz measurement bandwidth.) Figure 4 displays the vector field components B x , B y , B z extracted from the measured frequency shifts of Figs. 3(c)-3(f). The extracted field components show good agreement with the amplitudes determined by sequential NV vector magnetometry, with differences at the 1% level or better (see Supplemental Material). Sensitivities η x , η y , and η z to magnetic field components alongx,ŷ, andẑ are determined from a series of magnetometry measurements with no applied magnetic signal. After multi-channel demodulation and filtering of these zero-signal traces, the detected frequency shifts ∆ν λ , ∆ν χ , ∆ν ϕ , ∆ν κ are extracted. These shifts are then transformed to B x , B y , and B z using the matrix A + . The sensitivity η j to fields along the j direction is given by
η j = σ Bj √ 2f ENBW(2)
for j = x, y, z, where σ Bj is the standard deviation of the zero-signal magnetic field time trace B j . Sensitivities η x = 57 pT/ √ Hz, η y = 46 pT/ √ Hz, and η z = 45 pT/ √ Hz are determined based on 1 second of recorded PL with f ENBW = 203 Hz. Photon shot-noise-limited sensitivities are calculated to be η shot
V. PROPOSED PULSED EXTENSION
This simultaneous vector magnetometry method should be extendable to pulsed-type measurement protocols, such as Ramsey [46], pulsed ODMR [47], and Hahn echo [48]. For example, Ramsey magnetometry typically employs MW phase modulation in a modified variant of the dual measurement scheme used in NV sensing protocols to mitigate systematic noise sources [18,39,49]. In this dual measurement scheme, otherwise identical pulse sequences alternately project the final spin state onto the |m s = ±1 and |m s = 0 basis states by varying the phase of the final π/2 pulse, as illustrated in Fig. 5(a). The magnetic field is then calculated using
B = α 2 S S 1 − S 2 ,(3)
where S i denotes the integrated PL signal resulting from the i th measurement, α is a proportionality constant between the magnetic field and the integrated PL, S denotes the mean integrated PL averaged over projections on the |m s = ±1 and |m s = 0 basis states, and the gray box (no box) indicates that the given measurement S i is projected onto the |m s = ±1 (|m s = 0 ) state. The dual measurement scheme effectively removes background PL offsets, mitigates laser intensity fluctuations, and protects against certain systematics causing long-term drifts of S i . For the proposed extension to pulsed vector magnetometry, simultaneous pulse sequences are applied to multiple spectrally-separated ODMR features, each with a separate near-resonant MW frequency and a distinct alternation pattern of final π/2-pulse phases. Use of orthogonal binary sequences such as Walsh codes [50][51][52][53][54] for the phase alternation patterns ensures the detected PL can be demodulated to separate out the magnetic field signal associated with each NV orientation. For example, with the encoding scheme illustrated in Fig. 5(b), these magnetic signals are given by After an initial π/2 pulse, the magnetic field is sensed for a duration τ followed by a final MW π/2 pulse of variable phase. Sequences denoted + and − differ in phase by 180 • and yield equal and opposite PL contrast signals from a DC magnetic field [18,39,49]. b) Four-channel modulated Ramsey scheme. Final MW π/2 pulse phases from each sequence are modulated according to the a set of orthogonal Walsh codes [50]. Detected PL signal is demodulated according to same Walsh codes to separate and extract magnetic-field-dependent shifts of four addressed NV orientations.
B 1 = α 1 8 S S 1 − S 2 + S 3 − S 4 + S 5 − S 6 + S 7 − S 8(4)B 2 = α 2 8 S S 1 + S 2 − S 3 − S 4 + S 5 + S 6 − S 7 − S 8(5)B 3 = α 3 8 S − S 1 + S 2 + S 3 − S 4 − S 5 + S 6 + S 7 − S 8(6)B 4 = α 4 8 S S 1 + S 2 + S 3 + S 4 − S 5 − S 6 − S 7 − S 8 .(7)
From the observed values of B 1 , B 2 , B 3 , and B 4 , the lab-frame magnetic field components B x , B y , and B z can be determined utilizing a linearized matrix as described previously (see Appendix B 2). The simultaneous scheme (Eqns. 4-7) achieves the same bandwidth but a 2× higher SNR than the scheme in Eqn. 3 applied sequentially to the four NV orientations, since both schemes require eight pulse sequences to reconstruct the magnetic field vector (see Supplemental Material). This pulsed implementation of simultaneous vector magnetometry is expected to allow improvements in both bandwidth and sensitivity compared to the demonstrated CW-ODMR implementation. In particular, sensing bandwidths up to ∼ 100 kHz are anticipated, based on arguments in Ref. [12]. The two main contributors to the expected sensitivity enhancement are (i) more effective noise rejection due to modulated pulsed-type protocols encoding magnetic information at higher frequencies than CW-ODMR, and (ii) enhanced PL contrast from avoiding laser and MW power broadening [12,47].
VI. OUTLOOK
The method presented here allows simultaneous recording of all three Cartesian components of a dynamic vector magnetic field using a solid-state spin sensor. The technique is a straightforward extension of established methods for broadband magnetometry using ensembles of solid-state defects, and implementation in an existing system requires only additional MW components. The method offers at least a 2× improvement in shot-noiselimited sensitivity, corresponding to a 4× reduction in measurement time to achieve a target SNR when compared to sequential vector magnetic field sensing. While the technique is demonstrated here for a single optical detector employing CW-ODMR, it is expected to be compatible with CW and pulsed-type measurements in both single-channel detectors and camera-based magnetic field imagers (see Appendix A).
This simultaneous vector magnetometry method is expected to be compatible with camera-based wide-field magnetic imagers using NV-diamond [10,14,18,26,[55][56][57][58][59][60][61], both for frequency-modulated CW-ODMR and for phase-modulated pulsed-type sensing schemes. We note that the demodulation and summation described in Section V and shown in Fig. 5 is a time-domain picture of the demodulation and low-pass filtering lock-in scheme used in the CW-ODMR demonstration. The same approach is expected to apply to camera-based parallel imaging, where the PL detection S n represents the n th camera exposure, and the signals B 1 , B 2 , B 3 , B 4 and reconstructed field components B x , B y , B z represent magnetic field image frames.
For frequency-modulated CW-ODMR magnetic imaging, square-wave modulation may enable increased SNR compared to sinusoidal modulation, as the adding and subtracting of image exposures amounts to square-wave demodulation of the detected signal. In both imaging and single-channel detection modalities, square-wave modulation and demodulation is expected to slightly increase measurement SNR by ensuring that the MWs always interrogate NV ODMR features at the points of steepest slope [12,38,62].
Appendix B: Vector Field Reconstruction
Bias Field Determination
The ODMR line center frequencies in the bias field B 0 are measured by sweeping a single MW tone from 2.65 to 3.10 GHz and recording the PL signal; this yields an ODMR spectrum as shown in Fig. 1(d). Using a leastsquares fit, we determine the line center of the middle hyperfine subfeature of each of the eight m s spin transitions. Averaging 10 3 sweeps yields the following set of line centers, which are used to fit for the static field parameters B = B 0 , D, and M z ≡ [M λ z , M χ z , M ϕ z , M ϕ z ]:
ν ODMR = ν κ − ν λ − ν ϕ − ν χ − ν χ + ν ϕ + ν λ + ν κ + = GHz.(B1)
A nonlinear least-squares (Levenberg-Marquardt) numerical minimization method is used to fit the differences between eigenvalues of the NV ground state spin Hamil-tonianĤ
i = h(D + M i z )(S i z ) 2 + g e µ B B · S i (B2)
to the measured line centers for the four NV orientations i = λ, χ, ϕ, κ, as described in the Supporting Information of Ref. [61]. Here the dimensionless spin-1 operator of the NV triplet ground state S i is defined in the NV body frame withẑ ≡n i ; D is the temperatureand strain-dependent zero-field splitting, defined specifically to be the coupling component ∝ (S i z ) 2 that is common to all four NV orientations [21]; and M i z is the additional anisotropic coupling, which differs between the four orientations and is attributed to longitudinal strain and electric fields [45,61,63]. Coupling of transverse strain and electric fields M i x and M i y to the NV spin is suppressed by the on-axis component of the bias field,
( geµ B h B i z M i x , M i y )
, and is therefore neglected [41,64]. In contrast, the components of B 0 transverse to the NV symmetry axes, (B i x and B i y ), contribute non-negligible shifts to the observed ODMR line centers.
Dynamic Magnetic Field Determination
In the present demonstration, a subset of the detected ODMR line centers from Eqn. B1 are selected for MW ad-
dressing ν MW = ν ODMR {2, 4, 6, 8} = [ν λ − , ν χ − , ν ϕ + , ν κ + ].
The + and − subscripts are dropped herein. The matrix A from Eqn. 1, reproduced here,
∆ν λ ∆ν χ ∆ν ϕ ∆ν κ sens = A B x B y B z sens ,(B3) B0,D, Mz ,(B4)
where B x , B y , and B z are the lab-frame magnetic field components. The values of D and M z are taken here to be constant during measurements, such that changes in the ODMR line centers are entirely attributed to magnetic field variations.
The assumption of constant M z is valid in the present device because strain in the diamond is fixed and electric fields couple only very weakly to the NV energy levels at typical values of the bias magnetic field B 0 [64,65]. Although temperature drifts couple to D with dD/dT = 74 kHz/K [35], these drifts occur on timescales of seconds to hours, and the associated changes in D are therefore outside the 5 Hz to 210 Hz measurement bandwidth of the present device. Furthermore, use of a SiC heat spreader attached to the diamond mitigates laser-induced temperature fluctuations (see Appendix C 2) [12]. In a vector magnetometer optimized for sensing lower-frequency magnetic fields ( Hz), a changing zero-field splitting D sens could also be determined along with B sens from the four measured ODMR line shifts. To additionally sense any dynamic changes in M z would require MW addressing of more than four ODMR features.
In the simple limiting case where, for each NV orientation i = λ, χ, ϕ, κ, the transverse components of the bias magnetic field are much smaller than the zero-field splitting, (B i are linearly proportional to the magnetic field projections B i z along the respective NV symmetry axes. In this linear Zeeman regime, the rows of the matrix A are given, up to a sign, by the NV symmetry axis unit vectorsn i :
A lin = g e µ B h n λ −n χ −n φ n κ = g e µ B h 0 − 2 /3 − 1 /3 2 /3 0 − 1 /3 0 − 2 /3 1 /3 2 /3 0 1 /3 .
(B5) Whether the unit vector is multiplied by +1 or −1 depends on the sign of B i z ≡ B 0 ·n i and on whether the addressed transition couples the |m s = 0 state to the |m s = +1 or |m s = −1 state.
Because the bias field B 0 has non-negligible components transverse to each NV symmetry axis, the present experiment does not satisfy the requirement that only magnetic field projections on the NV symmetry axes contribute to the measured ODMR frequency shifts, and consequently A differs from A lin . We determine A numerically by evaluating the partial derivatives using a step size of δB x = δB y = δB z = h geµ B · 10 Hz. The matrix A is calculated to be (B7) The entries of the matrix A + are robust to small variation in the bias magnetic field and other Hamiltonian parameters. A 10 µT change in B x , B y , or B z changes no entry of A + by more than 1% and some by less than 0.01%. Doubling the strain parameters M z also affects the entries of A + by less (and for most entries much less) than 1%. A 150 kHz change in D (corresponding to a temperature change of 2 K [35]) affects the entries of A + by 0.01% or less. Thus, drifts in temperature or in the bias electric, strain, or magnetic fields have a negligible effect on the reconstruction accuracy of B sens and can be ignored.
A = g e µ B h
This linearized matrix method was compared against a numerical least-squares Hamiltonian fit, which is identical to the eight-frequency minimization method for ν ODMR described in Appendix B 1 except that D and M z are held constant and B = B 0 + B sens is determined from only the four frequencies ν MW . The methods were compared for a range of simulated fields B sens ranging from 1 nT to 100 µT. The linear transformation using A + from Eqn. B7 agreed with the Hamiltonian fit to better than 0.001% for | B sens | 100 nT and to better than 0.3% for | B sens | 100 µT. When run on the same desktop computer, the linearized matrix method determines B sens from ν MW with ∼ 20 µs per measurement, whereas the least-squares Hamiltonian fit requires ∼ 500 ms per field measurement. This ∼ 25,000× speedup enables realtime vector magnetic field reconstruction from sensed frequency shifts in the present device.
Appendix C: Experimental Setup Figure S1 of the Supplemental Material shows the electronic equipment used in the present experiment. Agilent E8257D, E4421B, E4421B, and E4422B MW synthesizers generate the four modulated carrier frequencies ν λ , ν χ , ν ϕ , and ν κ , respectively. These carriers are sinusoidally frequency modulated at f λ , f χ , f ϕ , and f κ with frequency deviations δν λ , δν χ , δν ϕ , and δν κ using the two analog outputs of each of two National Instruments (NI) PXI 4461 cards within an NI PXIe-1062Q chassis. The chassis also contains another NI PXI-4461 card used to trigger MW sweeps on the Agilent E8275D and for digitizing the detected optical signals, and an NI PXI-4462 card also for digitizing optical signals. All cards within the chassis and all MW and RF sources are synchronized to the Agilent E8257D's 10 MHz clock using a distribution amplifier (Stanford Research Systems FS735).
Microwave Electronics
An Agilent E4430B synthesizer generates a 2.158 MHz RF tone for the MW sidebands. A power divider (TRM DL402) splits this 2.158 MHz signal four ways. Each of the split signals passes through a Mini-Circuits SLP-19+ low-pass filter and then is mixed with one of the four modulated carrier MW signals using a Relcom doublebalanced mixer, either M1J or M1K. Before the mixer, each carrier signal passes through a Teledyne 2-4.5 GHz isolator and a 10 dB directional coupler. The coupled portion of the carrier signal passes through a 3 dB attenuator and is then combined (Mini-Circuits ZX10-2-42-S+) with the sideband frequencies after the mixer to generate a MW signal resonant with all three hyperfine-split spin resonances for a given ODMR feature [12,21].
The four sets of modulated MWs are amplified using four Mini-Circuits ZHL-16W-43-S+ amplifiers. The amplifier outputs pass through Teledyne 2-4.5 GHz isolators and then circulators (either Pasternack PE 8401 or Narda 4923, each terminated with a 10 W 50 Ω terminator), before being combined with each other using three hybrid couplers (two Anaren 10016-3 and one Narda 4333, each terminated with a 10 W 50 Ω terminator). The combined MWs pass through a 20 dB directional coupler, which picks off a portion of the signal for monitoring on a spectrum analyzer, and the rest is sent to a copper wire loop near the diamond to drive the NV spin resonances. The loop is positioned so that MWs are polarized approxi-mately alongẑ in the lab frame and address the four NV orientations roughly equally.
Diamond Mounting
The diamond is a 4 mm × 4 mm × 500 µm chip with 110 edges and a {100} front facet, grown by Element Six Ltd. The diamond contains a bulk density of grownin nitrogen [ 14 N] ≈ 4.9×10 18 cm −3 and an estimated Nto-NV − conversion efficiency of ∼ 10% after irradiation and annealing. The diamond is affixed to a 2" diameter, 330 µm thick wafer of semi-insulating 6H silicon carbide (SiC) from PAM-XIAMAN to provide both thermal and mechanical stability to the diamond crystal. The SiC wafer is in turn affixed to a 0.04" thick tungsten sheet for additional mechanical stability, which is attached to an aluminum breadboard. The breadboard is mounted vertically so that the SiC and diamond surface ({100} face) make a 90 • angle with the optical table. A 1.5" hole cut in the center of the tungsten sheet and a 2" aperture in the center of the breadboard enable the SiC and diamond to be accessed from both sides. The axis normal to the diamond surface, aligned with the [001] crystal lattice vector, is defined to be the lab-frame zaxis; the vertical axis normal to the optical table and along crystal lattice vector [110] is the lab-frame y-axis, and the horizontal axis perpendicular to both the z-and y-axes, along [110], is the x-axis (see Fig. 1(c)).
Applied Magnetic Fields
The bias magnetic field B 0 is applied via a pair of 50 mm diameter, 30 mm thick N52 NdFeB neodymium magnets (Sunkee). The bias field is measured by sweeping a MW tone over the ODMR spectrum, as described in Appendix B 1, and is consistent with a Hall probe measurement of | B 0 | = 8.4 mT, which is itself limited by the Hall probe's precision and ∼ cm standoff distance from the diamond.
The magnetic field coils are displaced from the diamond along the x-, y-, and z-axes by 28 cm, −26.5 cm, and 28.5 cm respectively. The coils respectively contain 129, 130, and 142 turns and have diameters of 25 cm, 32 cm, and 25 cm. Sinusoidally varying currents with root-mean-square (RMS) amplitudes of 0.24 mA, 0.13 mA, and 0.28 mA were applied to the respective coils. Based on the coil geometries and placement, and assuming that the coils are centered on the diamond and that there is no distortion of the fields by the magnetizable steel optical table, (an incorrect assumption, as discussed in the Supplemental Material), we expect B x,RMS = 10.51 nT, B y,RMS = 8.71 nT, and B z,RMS = 12.86 nT. The magnetic fields at the diamond were also determined via conventional sequential vector magnetometry, which found B x,RMS = 8.12 nT, B y,RMS = 9.56 nT, and B z,RMS = 9.86 nT, as marked by the open circles in Fig. 4. The discrepancies between the measured fields and the fields predicted from the known geometry and applied currents are attributed to magnetic field distortion by the ferromagnetic optical table, which is confirmed by a numerical simulation (Radia) [66,67] (see Supplemental Material).
Optical Setup
The excitation laser source is a 532 nm Verdi V-5 outputting 4.3 W during typical operating conditions. The laser output passes through a Glan-Thompson polarizer, a half waveplate, and an f = 400 mm focusing lens. A silver mirror (Thorlabs PF10-03-P01) then directs the beam through a beam sampler (Thorlabs BSF10-A), after which 3.3 W impinges on the {100} diamond surface at an oblique angle ≈ 73 • to the normal. Reflections and scattered excitation light are reflected back toward the diamond using an aluminized mylar sheet opposite the excitation light entry side. The PL from the diamond is collected by an aspheric, aplanatic condenser (Olympus 204431), is long-pass filtered at 633 nm (Semrock LP02-633RU-25), and ∼ 52 mW is imaged onto a silicon photodiode (Thorlabs FDS1010), termed the signal photodiode. This photodiode is reversed biased at 25 volts with a voltage regulator (Texas Instruments TPS7A49) followed by two capacitance multipliers in series [68]. The photocurrent is terminated into R sig = 300 Ω. Before the diamond, the beam sampler picks off and directs ∼ 135 mW of the excitation light through a beam diffuser and onto a second identical photodiode, termed the reference photodiode (see Fig. 2(b)). This photodiode is powered from the same voltage source as the signal photodiode, and its photocurrent is terminated into R ref = 270 Ω.
Each photodiode voltage signal is simultaneously digitized by three analog-to-digital converters (ADCs), two of which are AC-coupled channels of an NI PXI-4462 digitizer. The signals from the AC-coupled channels are averaged in software to reduce digitization noise. Each photodiode voltage signal is also digitized by third ADC, which is a DC-coupled channel of an NI PXI-4461 digitizer. The signals from the DC-coupled channels are used to implement the laser noise cancellation. For the data shown in Figs. 3 and 4, all channels operate at sampling rate F s = 202.8 kSa/s.
Laser Noise Cancellation
Laser intensity noise is canceled by scaling and subtracting the green reference signal from the diamond PL signal. The digitized AC voltage from the reference photodiode, sampled over a 1-second-long interval, is scaled and subtracted from the AC voltage from the signal photodiode. The scaling factor for each interval is the ratio of the signal photodiode's mean DC value to the reference photodiode's mean DC value from that interval. This cancellation reduces the experimental noise in the 2-6 kHz frequency band by ∼ 30×, achieving a noise level (in the absence of MW noise) that is ∼ 1.5× above the expected level due to shot noise from both the signal and reference photocurrents (see Appendix D). Under operating conditions with all modulated MWs in use, the experimental noise floor is 2.5 -3× above this same expected shot-noise level.
Demodulation and Filtering
As described in the main text, lock-in demodulation is performed in software in the present implementation, although the technique is also compatible with hardware demodulation. The noise-canceled PL signal is first highpass filtered at 1690 Hz with a 10th-order Butterworth filter, then separately mixed with sinusoidal waveforms at the four modulation frequencies given in Table I.
The four demodulated traces are band-pass filtered (5 Hz to 210 Hz 10th-order Butterworth filter), which eliminates cross-talk from the other channels, upmodulated signal at 2f i , and environmental noise outside the sensing band. The filtered traces are then downsampled (decimated without averaging) by 75× from 202.8 kSa/s to 2.704 kSa/s. Finally, spurious signals at 49 Hz, 50 Hz, and 60 Hz and remaining cross-talk at 338 Hz not completely eliminated by the band-pass filter are removed with 1-Hz-wide FFT notch-stop filters. The resulting signals each have single-sided equivalent noise bandwidth f ENBW = 203 Hz [69,70].
Lock-In Signal Slope Measurement
The demodulated lock-in signals S i for channels i = λ, χ, ϕ, κ are converted to frequency shifts using the following calibration: to each channel (sequentially), a 20 kHz frequency chirp is applied to the modulated MWs. The PL voltage signal is measured, downmixed, and filtered as described in the main text, and the resulting signal is plotted vs. MW frequency and fit to a line, from which the slope is extracted. The slope for each channel in µV/kHz is averaged over ∼ 25 seconds. For the data shown in Fig. 3, the measured slopes for the four channels are tabulated in Table I.
Appendix D: Shot-Noise-Limited Vector Field Sensitivity
The photon-shot-noise-limited sensitivities to magnetic fields oriented alongx,ŷ, andẑ are calculated herein [71][72][73]. First we consider only photon shot noise on the detected PL signal from the diamond. The smallest detectable ODMR line shift (SNR = 1) due to a magnetic field in the presence of shot noise from photoelectrons on the signal photocurrent alone, ∆ν
where σ Isig i is the standard deviation of the frequency shifts (in Hz) on channel i due to shot noise on the PL photocurrent, R sig is the signal photodiode's termination in ohms, dSi d∆νi is the PL lock-in signal slope in V/Hz, q is the elementary charge, and ∆f is the single-sided measurement bandwidth.
When limited by photon shot noise, fluctuations on each of the four lock-in detection channels are uncorrelated, and the covariance matrix of ODMR frequency fluctuations is given by
Σ ∆ν = (σ Isig λ ) 2 0 0 0 0 (σ Isig χ ) 2 0 0 0 0 (σ Isig ϕ ) 2 0 0 0 0 (σ Isig κ ) 2 .(D2)
The sensitivities to fields alongx,ŷ, andẑ are found by transforming Σ ∆ν into a covariance matrix of lab-frame magnetic field shifts:
Σ B = A + Σ ∆ν A + T ,(D3)
where A + T is the transpose of the matrix A + . The diagonal elements of Σ B are the variances (σ Isig x ) 2 , (σ Isig y ) 2 , and (σ Isig z ) 2 , which have dimensions of tesla 2 . Offdiagonal elements of Σ B represent noise correlations; for diagonal Σ ∆ν , these correlations arise from mixing of the |m s = ±1 states with the |m s = 0 state due to the bias magnetic field's non-negligible projections transverse to the NV symmetry axes.
The sensitivity η Isig x is given by
η Isig x = σ Isig x √ T ,(D4)
where, for continuous readout as in CW-ODMR, the measurement time is T = 1/(2∆f ). Note this definition matches Eqn. 2, where ∆f = f ENBW , and with σ Isig x here replacing the measured standard deviation σ x . The yand z-sensitivities η Isig y and η Isig z are defined equivalently.
As described in Appendix C 5, the present experiment approaches shot-noise-limited sensing by canceling laser intensity fluctuations using a reference photodetector and a software-based noise cancellation protocol. Here we consider the limit wherein this method completely cancels laser intensity fluctuations. In this limit, shot noise from the reference photocurrent also contributes to the estimated sensitivity. Including this contribution as a separate uncorrelated noise source increases the minimum detectable ODMR line shift by a term T .
In the present experiment, the reference photodiode collects an average photocurrent I ref = 30.1 mA from the picked-off 532 nm beam, and the signal photodiode collects an average photocurrent I sig = 24.1 mA from the diamond PL. Inserting these values into Eqn. D1 along with R sig = 300 Ω and the slopes dSi d∆νi given in Table I
FIG. 1 .
1(a) Energy level diagram for the nitrogen-vacancy (NV) center in diamond, with zero-field splitting D between the ground-state electronic spin levels ms = 0 and ms = ±1.
FIG. 3 .
3Applied magnetic fields and simultaneous vector magnetometry data. a) Three coils generate magnetic field signals Bx(t), By(t), and Bz(t) at fx = 67 Hz, fy = 32 Hz, and fz = 18 Hz with root-mean-square (RMS) amplitudes Bx,RMS = 8.12 nT, By,RMS = 9.56 nT, and Bz,RMS = 9.86 nT. b) Spectral density of detected PL signal from 1 second of continuous acquisition. Shaded regions mark frequency bands containing four modulation frequencies fi, where i = λ, χ, ϕ, κ (see
FIG. 4 .
4Detected magnetic fields using simultaneous vector magnetometer, extracted from data inFig. 3. a) Bx time trace (inset) and spectral density showing detected signal at fx = 67 Hz. b) By time trace (inset) and spectral density showing detected signal at fy = 32 Hz. c) Bz time trace (inset) and spectral density showing detected signal at fz = 18 Hz. Dashed lines mark applied signal frequencies fx, fy, fz, and circles center on expected applied field amplitudes determined by sequential NV vector magnetometry. Cartoon recreations ofFig. 3(a) illustrate isolated detected components of dynamic vector magnetic field Bx(t), By(t), Bz(t).
.4 pT/ √ Hz, and η shot z = 17.5 pT/ √ Hz (see Appendix D). The 2.5 -3× factor above shot noise is attributed to uncanceled MW and laser intensity noise. The reported sensitivities are the highest demonstrated to date for any solid-state spin-based magnetometer performing broadband sensing of all magnetic field vector components.
FIG. 5 .
5Proposed pulsed implementation of simultaneous vector magnetometry. a) Set of two Ramsey sequences with modulated MW phase. Green and red rectangles depict periods of 532 nm laser excitation and PL collection, respectively, with NV-spin-state initialization time tI and spin readout time tR.
B D), and strain coupling is negligible, (M i x , M i y , M i z gµ B h B i z ), the marginal shifts ∆ν i ,
-Penrose left pseudoinverse A + used in the experiment to determine B sens (t) is numerically computed from A with the MATLAB function pinv and is found to be
min = 0.600 Hz. Using Eqns. D2-D4 with T = 1 s to calculate the photon-noise-limited sensitivities alongx,ŷ, andẑ, we find η better than the realized sensitivities of the present device.
Expanded view shows Zeeman shifts of the ms = ±1 energy levels in the presence of a magnetic field B for different projections along the NV symmetry axis. (b) Four crystallographic orientations of the NV center in diamond. For an ensemble of NV centers within a single crystal diamond, the NV symmetry axes are equally distributed along the four orientations. (c) NV symmetry axes and lab-frame directions (x,y,ẑ), defined in terms of diamond lattice vectors. A mag-
netic field B projects onto the four NV orientations, causing
the Zeeman shifts shown in (a). Shifts associated with off-
axis magnetic fields are ignored for simplicity. (d) Optically
detected magnetic resonance (ODMR) spectrum displaying
photoluminescence (PL) signal from an ensemble of NV cen-
ters in a bias magnetic field B = B0 = (3.54, 1.73, 6.95) mT,
| B0| = 7.99 mT. Resonance features numbered 1 -4 (5 -8) cor-
respond to |ms = 0 → |ms = −1 (|ms = 0 → |ms = +1 ) spin
transitions, and subfeatures arise from NV hyperfine structure
(see Supplemental Material and Fig. S3) [21, 22].
width impinges on the diamond chip's {100} facet at ≈ 73 • to the normal, exciting the NVs. An aspheric, aplanatic condenser collects the PL and directs it through(a)).
A 532 nm, ∼ 3.3 W beam with 400 µm Gaussian 1/e 2
MW source
#4
FM #1
FM #2
FM #3
FM #4
Amplifier
#4
Laser
Computer
PD
PD
Condenser
Filter
Lens
HWP
Polarizer
Sampler
MW loop
Diamond
Diffuser
Mirror
Hybrid
couplers
Mixers
RF source
MW source
#3
MW source
#2
MW source
#1
DAC
Amplifier
#3
Amplifier
#2
Amplifier
#1
MW loop
Excitation beam
SiC wafer
Diamond
Aspheric condenser
"
ADC
(a)
(b)
FIG. 2. a) Laser and microwave (MW) excitation of NV cen-
ters in diamond sensor crystal and photoluminescence (PL)
collection scheme. Diamond is affixed to one side of silicon
carbide (SiC) wafer for stabilization and heat sinking. MW
loop on reverse side of SiC provides modulated MW drive
to NV ensemble. Excitation light at 532 nm enters diamond
at ≈ 73 • to the normal and PL is collected by aspheric apla-
natic condenser shown below diamond. From this perspective,
the y-axis points primarily into the page. b) Schematic of
setup. Digital-to-analog converter (DAC) outputs MW fre-
quency modulation (FM) waveforms. MWs are generated
by four sources and mixed with radiofrequency (RF) signal
at 2.158 MHz to produce modulated carriers plus sidebands,
which are amplified, combined, and radiated by the MW loop.
TABLE I. Parameters detailing the four-axis simultaneous vector magnetometry implementation, as discussed in the main text.MW Source Carrier Freq. NV Axis Transition Mod. Freq. Freq. Deviation Lock-in Signal Slope
1
ν λ = 2.731 GHzn λ [111] |0 ↔ |−1 f λ = 4056 Hz δν λ = 832 kHz dS λ /d∆ν λ = 39.5 µV/kHz
2
νχ = 2.862 GHznχ [111] |0 ↔ |−1 fχ = 2704 Hz δνχ = 828 kHz dSχ/d∆νχ = 42.0 µV/kHz
3
νϕ = 2.966 GHznϕ [111] |0 ↔ |+1 fϕ = 5070 Hz δνϕ = 775 kHz dSϕ/d∆νϕ = 53.4 µV/kHz
4
νκ = 3.069 GHznκ [111] |0 ↔ |+1 fκ = 3380 Hz δνκ = 1178 kHz dSκ/d∆νκ = 41.6 µV/kHz
F ref = 1 + Isig I ref to ∆ν Isig,I ref i,min = F ref ∆ν Isig i,min . In the limit of high reference photocurrent, F ref approaches 1; when I sig = I ref , F ref = √ 2. The term F ref enters Eqns. D2-D4 in the form of a constant prefactor, such that the photon-noise-limited sensitivity, modified by the reference detection, is given by η Isig,I ref x = F ref σIsig
x
√
Appendix A: Imaging Implementation
leading execution of the experiments, analysis of the data, and paper writing. J M S , J F B J M S , J F B , M J T , R L W , J. M. S. and M. J. T. analyzed the results. J. M. S., J. F. B., M. J. T., and R. L. W.built the apparatus. reviewed all results and wrote the paper. R. L. W. supervised the projectJ. M. S. and J. F. B. contributed equally to this work, with J. F. B. leading initial conceptualization and J. M. S. leading execution of the experiments, analy- sis of the data, and paper writing. J. M. S., J. F. B., M. J. T., and R. L. W. contributed to conception of the experiments. J. M. S. and J. F. B. built the appara- tus. J. M. S. conducted the experiments. J. M. S. and M. J. T. analyzed the results. J. M. S., J. F. B., M. J. T., and R. L. W. reviewed all results and wrote the paper. R. L. W. supervised the project.
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| []
|
[
"Classification of Magnetic Vortices by Angular Momentum Conservation",
"Classification of Magnetic Vortices by Angular Momentum Conservation"
]
| [
"Kenji Fukushima \nDepartment of Physics\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n",
"Yoshimasa Hidaka \nInstitute of Particle and Nuclear Studies\nKEK\n1-1 Oho305-0801TsukubaIbarakiJapan\n\nRIKEN iTHEMS\nRIKEN\n2-1 Hirosawa351-0198WakoSaitamaJapan\n",
"Ho-Ung Yee \nDepartment of Physics\nUniversity of Illinois\n60607ChicagoIllinoisU.S.A\n"
]
| [
"Department of Physics\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan",
"Institute of Particle and Nuclear Studies\nKEK\n1-1 Oho305-0801TsukubaIbarakiJapan",
"RIKEN iTHEMS\nRIKEN\n2-1 Hirosawa351-0198WakoSaitamaJapan",
"Department of Physics\nUniversity of Illinois\n60607ChicagoIllinoisU.S.A"
]
| []
| Superfluid vortices are quantum excitations carrying quantized amount of orbital angular momentum in a phase where global symmetry is spontaneously broken. We address a question of whether magnetic vortices in superconductors with dynamical gauge fields can carry nonzero orbital angular momentum or not. We discuss the angular momentum conservation in several distinct classes of examples from crossdisciplinary fields of physics across condensed matter, dense nuclear systems, and cosmology. The angular momentum carried by gauge field configurations around the magnetic vortex plays a crucial role in satisfying the principle of the conservation law. Based on various ways how the angular momentum conservation is realized, we provide a general scheme of classifying magnetic vortices in different phases of matter. | 10.1103/physrevresearch.3.033009 | [
"https://arxiv.org/pdf/2011.06981v1.pdf"
]
| 226,955,898 | 2011.06981 | a7836abef631107c86c978ce4bad5c63112b164b |
Classification of Magnetic Vortices by Angular Momentum Conservation
Kenji Fukushima
Department of Physics
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Yoshimasa Hidaka
Institute of Particle and Nuclear Studies
KEK
1-1 Oho305-0801TsukubaIbarakiJapan
RIKEN iTHEMS
RIKEN
2-1 Hirosawa351-0198WakoSaitamaJapan
Ho-Ung Yee
Department of Physics
University of Illinois
60607ChicagoIllinoisU.S.A
Classification of Magnetic Vortices by Angular Momentum Conservation
Superfluid vortices are quantum excitations carrying quantized amount of orbital angular momentum in a phase where global symmetry is spontaneously broken. We address a question of whether magnetic vortices in superconductors with dynamical gauge fields can carry nonzero orbital angular momentum or not. We discuss the angular momentum conservation in several distinct classes of examples from crossdisciplinary fields of physics across condensed matter, dense nuclear systems, and cosmology. The angular momentum carried by gauge field configurations around the magnetic vortex plays a crucial role in satisfying the principle of the conservation law. Based on various ways how the angular momentum conservation is realized, we provide a general scheme of classifying magnetic vortices in different phases of matter.
the superfluid vortex can also be an important constituent in rotating nuclear matter found inside neutron star, where the extremely high matter density causes a nonzero order parameter that signifies spontaneous breaking of global baryon number U (1) B symmetry. In a more interesting scenario of dense quark matter in quantum chromodynamics (QCD), this order parameter is also responsible for the superconducting phase of color gauge interactions, most likely the color-flavor-locked (CFL) superconducting phase [1]. There exist highly nontrivial vortices in the CFL phase, called non-Abelian CFL vortices [2], that involve both dynamics of the global baryon and the local color gauge symmetries; see Ref. [3] for a comprehensive review.
As seen in many interesting examples including vortices in CFL quark matter, some of which we will study in later discussions, the symmetries involved in vortex contents are entirely or partially gauge symmetries. The prototypical example is of course the magnetic vortex in Type-II superconductors. In these cases the vortex profile is fundamentally different from that of the purely superfluid vortex; a magnetic flux is threaded into the vortex core. Among many differences between a superfluid vortex and a gauged magnetic vortex, one may specifically ask about the angular momentum they carry. Surprisingly to us, we find that this simple question has not been properly addressed in the literature. As we try to answer the question in various examples across different fields of physics, we discover surprisingly diverse situations. It is the purpose of this article to present a compelling list of examples where the answers are quite different from each other, and also to provide an overarching physics explanation of why the answers can be so diverse. We will demonstrate that the angular momentum conservation offers a key guiding principle to understand the physics origin of the different answers. Our detailed analysis in the main text shows that the angular momentum carried by not only the matter sector of the system but also the dynamical gauge fields surrounding the vortex should be considered in order to fulfill the principle of angular momentum conservation. Building upon this principle, we attempt a general classification scheme of magnetic vortices in different phases of matter, that can hopefully be applied to other physical systems.
A natural starting point of our discussion lies in the vortices in Type-II superconductors.
Quite generally, it is easy to see that the vortex should carry a nonvanishing orbital angular momentum. Due to one of the Maxwell equations, ∇ × B = j (where we chose a natural unit in which the magnetic constant µ 0 is the unity), a smooth and finite ranged profile of magnetic flux means the existence of azimuthal component of the current density j. Under a fairly general assumption that the charge and momentum carriers are nonrelativistic quasielectrons in the conduction band forming the Fermi surface with superconducting gap, the current and the momentum are linearly related as an operator relation, holding for all states:
j = − e m P ,
where P is the momentum density operator, m is the effective mass of conduction electrons, and −e is the charge of electrons. Since j = 0, we have P = 0 and the finite sized vortex should carry a finite angular momentum by L = x x × P = 0.
The linear relation between the current and the momentum for nonrelativistic electrons is a consequence of Galilean invariance, and is not necessarily universal. Even though the dispersion relation deviates from the nonrelativistic Galilean invariant one, the current and the momentum are still negatively correlated, and there is no reason to exclude nonvanishing angular momentum. This discussion also implies that the quantization of the angular momentum in units of may not be universal. To complicate the situation more nontrivially, some vortices may also carry a localized electric charge [4,5], and the resulting electromagnetic (EM) field around such a vortex gives rise to a nonvanishing Poynting vector around the vortex core axis. The total angular momentum should then include a contribution from the EM field around the vortex.
Although the above features are robust, one can consider the following thought-experiment, that is somewhat similar to Feynman's angular momentum paradox. One places a solenoid below a superconductor sample, and turns on an external magnetic field to create magnetic vortices piercing the superconductor. The process can be implemented in azimuthal symmetric way, and should not change the total angular momentum which is zero initially. Since the created vortices carry finite amount of angular momentum, where can the compensating angular momentum be found?
The answer to this question is easy to guess: the background of solid crystal and the electrons in filled valence bands should carry the compensating angular momentum. Their inertia is infinitely large and their rotation may not be detectable, but the torque acting on them during the vortex creation process should impart to them precisely the negative amount of the angular momentum of the created vortices. In the next section we will be able to confirm this quantitatively in a concrete model which is simple and yet general enough to carry out the analysis of charged magnetic vortices. In this case the angular momentum carried by EM field also needs to be counted in the total angular momentum, and the angular momentum conservation holds true quite nontrivially only after including this EM contribution. We note that the EM field is localized around and attached to the vortex, so one should think of it as a part of the vortex profile.
An obvious next question as a continuation of the above thought-experiment of creating the vortex by a hypothetical solenoid is: what would happen in a system that has no background matter to absorb the angular momentum? A concrete example of such system is provided by a relativistic field theory which is self-consistent by itself without any other degrees of freedom: it could be identified as the electroweak sector of the Standard Model with Higgs field condensate, or more simply, a theoretical model by Nielsen and Olesen [6]. For this class of examples, our previous argument of angular momentum conservation becomes powerful enough to dictate that any magnetic vortices, either charged or not, should carry zero angular momentum. We call them "spinless vortices." We will show that this statement is indeed true for the Nielsen-Olesen model. In showing this for the charged vortex case, it is again critical to include the EM or gauge contribution to the total angular momentum.
For a similar conclusion for the dyonic solitons, see Refs. [7][8][9]. We make a remark that Appendix C of a renowned paper, Ref. [10], contains an erroneous statement on this for the charged vortex case, which we will rectify. With this example we showcase the nontriviality of our argument of angular momentum conservation.
What would happen if a vortex consists of a combination of magnetic vortex of gauge symmetry and superfluid vortex of global symmetry? To answer this question, we take an example of the "non-Abelian vortex" [2,11] in the CFL superconducting phase of dense quark matter, which may be relevant for the physics of neutron star cores. The non-Abelian CFL vortices also play a role in the idea of quark-hadron continuity [12] in the high density region of QCD phase diagram [13][14][15][16] (see, however, Refs. [17][18][19] for recent debates on the idea of quark-hadron continuity). In this example, color symmetry is obviously (non-Abelian) gauged, and the global symmetry is associated with the U (1) B baryon number.
For such an object of composite nature, one can imagine a creation process by an external magnetic field for the gauge symmetry together with a physical rotation of the superfluid for the global symmetry. Our angular momentum conservation argument then predicts that the total angular momentum should be given only by the superfluid part of global symmetry without any contribution from the gauged symmetry part. We will explicitly confirm this expectation in a highly nontrivial manner.
There is a logical exception to the above argument for spinless vortices in a system with no background. During the above considered creation process by an external solenoid, a finite amount of angular momentum may diffuse away to spatial infinity, resulting in an opposite amount of angular momentum localized around the vortex. The total angular momentum is conserved and zero, but the part at infinity is not visible, and should not be thought of as a contribution to the angular momentum of the localized vortex. The vortex then carries a left-over angular momentum that is finite. What distinguishes this case from our first case with background matter is that the opposite angular momentum to the one carried by the vortex strictly resides at spatial infinity, or more precisely at the boundary of the system far away from the vortex. This makes a contrast to the previous case with background matter, where the bulk of the background absorbs a finite angular momentum.
An instructive example of this class of vortex is provided by the magnetic vortex on the surface of a time-reversal invariant (i.e., T -invariant) strong topological insulator (TI) in a setup recently studied in Ref. [20]. Although the authors of Ref. [20] considered an interface between TI and a superconductor, we will focus on the TI part to account clearly for the physics origin of the net fractional (in units of ) angular momentum of the vortex. We will show that the total angular momentum solely arises from the gauge field configuration surrounding the vortex on the TI surface, without any TI matter contributions. We will argue for this peculiar feature that the topological nature of the TI is responsible for moving apart a finite angular momentum to the (infinitely separated) boundary, which characterizes this class of example. Ubiquitous topological vortices with fractional angular momentum in topological phases of matter as found in Refs. [21][22][23] should belong to this class of magnetic vortex.
The final class of vortex in our classification is delineated by the last logical possibility: a vortex may not be created by our thought-experiment with external solenoid in a way that conserves angular momentum, and additional operations to violate angular momentum conservation must be performed to create a vortex. This class of vortex is rather exotic and rarely found in the literature: one example we address in this paper is an object called the "charged semilocal vortex" of Abraham [24]. Since this class of vortex simply falls outside of our principle of angular momentum conservation, they may or may not carry an angular momentum: in our example the charged semilocal vortex carries a finite angular momentum.
It is an inhomogeneous profile along the vortex axis that makes it impossible to create this kind of vortices by simply piercing an external magnetic flux: an additional "twisting" or "spinning" along the axis is needed to create such a vortex profile.
In summary, we have the following distinct classes of vortices in regard to angular momentum conservation and their creation processes:
• Class Ia (spinful vortices): They carry a finite angular momentum due to the existence of background matter that can absorb angular momentum. Examples are the vortices in Type-II superconductors. • Class III (exotic vortices): They have an inhomogeneous profile along the vortex axis, so that they cannot be created by a simple procedure of piercing magnetic flux.
They may or may not carry angular momentum. An example is the charged semilocal vortex.
In the following sections, we present detailed analysis on concrete examples that belong to each of the above classes, in order.
II. CLASS I: CASE STUDY OF MAGNETIC VORTICES WITH NONZERO
ANGULAR MOMENTUM
In this section we discuss the case of magnetic vortices that carry a nonzero angular momentum. Because the angular momentum should be conserved as long as rotational symmetry is preserved, the angular momentum of magnetic vortices, if it is nonzero, should be balanced with other contributions. According to the types of such balancing contributions, we further classify them into two distinct subclasses; namely, Class Ia and Class Ib.
A. Class Ia -Incomplete cancellation due to background matter
The most familiar magnetic vortices in Type-II superconductor belong to Class Ia. The magnetic vortices can carry a nonzero angular momentum but its value is not quantized in units of , unlike the angular momentum carried by superfluid vortices. Explicit calculations as shown below make clear where the difference appears.
For an explicit demonstration we shall consider a relativistic scalar field theory in the Higgs phase of U(1) symmetry, so that gauged magnetic vortices emerge. We then take the nonrelativistic reduction and find the equations of motion that are familiar in condensed matter physics describing magnetic vortices in Type-II superconductors. The Lagrangian density we study in the natural unit system ( = c = 1) reads as
L = (D µ φ) † (D µ φ) − U (φ) + 1 2 E 2 − 1 2 B 2 − qA 0 ,(2.1)
where D µ = ∂ µ + ien e A µ is the covariant derivative, with n e being the electric charge carried by φ in units of e > 0. As usual, E = −∇A 0 − ∂ 0 A and B = ∇ × A are electric and magnetic fields, respectively. We should choose n e = −2 for the Cooper pair in electron superconductivity. The last term, qA 0 , with a background charge density q(x), is introduced to keep the total electric charge neutrality, which we will simply refer to as the "background" in the following. For example, in a solid with conduction electrons, positively charged ions in the crystal and other electrons in valence bands neutralize the whole system. We also note that a finite chemical potential µ will be introduced by replacing ien e A 0 → ien e A 0 −iµ.
The potential, U (φ), is chosen to have a nonzero condensate of φ in the Higgs phase, the simplest choice of which would be a polynomial form:
U (φ) = −λ 2 |φ| 2 + λ 4 2 (|φ| 2 ) 2 . (2.2)
The equations of motion from the Lagrangian are given by
− (D µ D µ )φ + λ 2 φ − λ 4 |φ| 2 φ = 0 ,(2.
3)
− ∂ 0 E + ∇ × B + ien e (Dφ) † φ − φ † (Dφ) = 0 , (2.4) ∇ · E + ien e (D 0 φ) † φ − φ † D 0 φ − q = 0 ,(2.(µ − en e A 0 ) 2 + D 2 + λ 2 φ − λ 4 |φ| 2 φ = 0 . (2.6)
Instead of solving this equation directly, we would like to make the problem close to a more conventional situation in condensed matter physics, by considering the nonrelativistic reduction. Because the nonrelativistic energy is measured from the rest mass energy m, we should split the mass term and rescale the field as
µ → m +μ , λ 2 → −m 2 + 2mλ 2 , φ → ψ √ 2m , (2.7)
whereμ denotes the nonrelativistic chemical potential. Equation (2.6) multiplied by 1/ √ 2m then becomes
(λ 2 +μ − en e A 0 )ψ + D 2 2m ψ − λ 4 4m 2 |ψ| 2 ψ = 0 ,(2.λ 2 +μ → µ , λ 4 4m 2 → g , A 0 = µ en e a , ψ → µ g Ψ . (2.10)
Here, we note that this µ is different from the original one in Eq. (2.6). Together with the Maxwell equation for A in Eq. (2.4), our equations finally become
(1 − a)Ψ + 1 m 2 H (∇ − ien e A) 2 Ψ − |Ψ| 2 Ψ = 0 , (2.11) ∇ × (∇ × A) + m 2 V A|Ψ| 2 − i 2en e Ψ∇Ψ † − Ψ † ∇Ψ = 0 , (2.12) ∇ 2 a + 2m 2 m 2 V m 2 H (|Ψ| 2 +q) = 0 ,(2.13)
whereq ≡ (g/en e µ)q, and we also introduce the two typical mass scales as
m 2 H ≡ 2mµ , m 2 V ≡ (en e ) 2 µ mg .
(2.14)
Physically, 1/m H represents the coherent length of the field Ψ, while 1/m V represents the penetration length of the magnetic field. If the penetration length is smaller than the coherent length, m V > m H , the Meissner screening effect is dominant and the phase separation is more preferred than forming magnetic vortices, which corresponds to Type-I superconductivity. We are interested in Type-II superconductivity in the opposite regime with m H > m V .
The Ansatz for the vortex solution with the winding number ν is
Ψ = f (r) e iνϕ , a = a(r) , A i = − ν en e ε ij x j r 2 1 − h(r) , (2.15)
where r ≡ x 2 + y 2 and tan ϕ ≡ y/x. Introducing a dimensionless radial coordinate,
ρ ≡ m V r, we can rewrite the differential equations (with ≡ d dρ ) as − (ρ f ) + ν 2 h 2 ρ f + λ ρ f (f 2 − 1 + a) = 0 , (2.16) ρ h ρ − f 2 h = 0 , (2.17) 1 ρ (ρ a ) + 2 λ m 2 m 2 V (f 2 +q) = 0 , (2.18) where λ ≡ m 2 H /m 2 V > 1.
For the total charge neutrality condition, we impose the condition,
xq = − x f 2 . (2.19)
Here, x refers to the two-dimensional integration on the plane perpendicular to the vortex axis. This neutrality condition is demanded by the fact that the static potential would behave as a(ρ 1) = Q 2π log ρ if the total net charge Q is nonzero. The combination of (µ − en e A 0 ) appears in the equations of motion and it plays a role of an effective chemical potential. To have a well-defined effective chemical potential at spatial infinity, we should impose Q = 0.
We can numerically solve these differential equations with appropriate boundary conditions. Let us first consider the conventional "locally neutral" vortex solution without coupling to electric field, so that a(r) = 0 simply. This can be achieved by choosing a space dependent background charge densityq(x) that locally neutralizes the net charge; that is,
f (0) = 0 , f (∞) = 1 , h(0) = 1 , h(∞) = 0 . (2.20)
We can easily obtain the numerical solutions using the shooting method to satisfy these boundary conditions. The left panel of Fig. 1 shows an example of the profile of the magnetic vortex for λ = 1.5. We see that h(ρ) extends more widely than f (ρ), reflecting m H > m V .
As a nontrivial example where the local charge density and the electric field are nonvanishing, let us consider a constant background charge densityq, that is determined by the total charge neutrality condition (2.19) as
q = − 1 S x f 2 ,(2.21)
with S ≡ x is the transverse area. In the limit of infinitely large systemq would approach the negative unity. In the present case we should revise the boundary conditions accordingly.
That is, f needs not be unity at large ρ, but f 2 − 1 + a should be vanishing as ρ gets large.
Also, we physically require vanishing electric field at ρ = 0 and ρ → ∞. Therefore, we impose the following boundary conditions:
f (0) = 0 , f (∞) = 1 − a(∞) , h(0) = 1 , h(∞) = 0 , a (0) = 0 , a (∞) = 0 . (2.22)
Actually, these boundary conditions are not sufficient to determine the numerical solution uniquely, but a shift of a(ρ) → a(ρ)+c with a constant c still exists. This shift would change the value of µ, and the magnitude of condensate would also be modified, which would result in a different value ofq in Eq. (2.21). In other words, we can adjustq to make a constant shift on a(ρ). To fix this freedom, a natural condition to impose would be to set a → 0 at large ρ, so that the effective chemical potential at infinity, by definition, remains to be µ. We choose λ = 1.5 and m 2 /m 2 V = 1 to find that a(∞) → 0 is realized withq −0.985. In the right panel of Fig. 1 we present the numerical solution with these parameters. This explicitly demonstrates that nontrivial solutions with nonzero local charge density and electric field certainly exist. We see that the profile of condensate slightly shrinks as compared to the locally neutral case shown in the left panel.
Let us now compute the angular momenta carried by the matter and the EM fields. The matter part of the angular momentum per unit vortex length is
L matter z = x ψ † i D ϕ ψ , D ϕ ≡ ∂ ϕ − ien e A ϕ , A ϕ ≡ ij x i A j = ν en e [1 − h(r)] ,(2.23)
where we reinstate as a common unit for the angular momentum and also change the variables back to r and ϕ. We note that the boundary condition (2.22) guarantees D ϕ [f (r)e iνϕ ] → 0 as r → ∞, and the above integral is convergent. It should be mentioned that the above form of the angular momentum using D ϕ corresponds to the kinetic angular momentum, that is the angular momentum carried by matter alone. We could have defined the canonical angular momentum using ∂ ϕ . It is straightforward to find:
L can,matter z = x ψ † i ∂ ϕ ψ = ν(2π ) µ g R 0 dr r f 2 (r) = ν N ,(2.24)
where N ≡ µ g x f 2 is the total number of particles per unit vortex length, and R is the size of the system in radial direction. This expression is identical to the well-known one for the quantized angular momentum of a superfluid vortex. In a gauge theory, there is a contribution from the gauge field: and also a general consideration in Ref. [26].
L can,gauge z = x E · ∂ ϕ A + (E × A) z ,(2.
With the explicit forms of the vortex profile and the associated vector potential, the matter part of the angular momentum per unit vortex length becomes
L matter z = ν(2π ) µ g R 0 dr r h(r) f 2 (r) . (2.26)
The difference from the canonical expression (2.24) is the presence of h(r) in the integrand.
Because h(r) decays when f (r) increases as in Fig. 1, we see that L matter z is smaller than L can z . Let us next consider the EM contribution, i.e.,
L gauge z = x x × (E × B) z . (2.27)
Plugging the explicit forms of E and B into the above, we find L gauge
z as L gauge z = −(2π ) νµ (en e ) 2 R 0 dr r da(r) dr dh(r) dr .
(2.28)
In Fig. 2 the integrand corresponding to the local angular momentum density is plotted,
where the variables are made dimensionless again.
The angular momentum distribution is peaked around ρ ∼ 1 and decays at large ρ. We can perform an integration by part and use the equation of motion to transform the above expression into Comparing with the matter contribution L matter z in Eq. (2.26), the first term is remarkably equal to −L matter z , and the total kinetic angular momentum is thus,
L gauge z = −(2π ) νµ (en e ) 2 r da(r) dr h(r) R 0 +2 m 2 λ R 0 dr r h(r) f 2 (r) +q(r) .L total z = −ν(2π ) µ g R 0 dr r h(r)q(r) . (2.31)
We see that L total z is proportional toq and this nonzero value of the total angular momentum is attributed to the presence of the background. If we had no background,q = 0, then L matter z and L gauge z would have perfect cancellation, but we then allow for a "charged" magnetic vortex. This might be possible due to finiteness of the system bounded by R. A natural realization of this possibility will be discussed as Class II in the next section.
B. Class Ib -Incomplete cancellation due to topological boundary Our next example for incomplete cancellation has been motivated by the physical setup discussed in Ref. [20] where a fractional angular momentum in the units of is found to be carried by the magnetic vortex at the interface between a superconductor and a Tinvariant strong topological insulator (TI) 1 . We will consider a simplified situation that still demonstrates the essential physics involved; we will show that a nonzero and fractional angular momentum is localized around a magnetic flux on the boundary surface between a bulk TI and the vacuum outside. Let us think of this situation from a different perspective. In the same way we discussed Class Ia in the previous section, we can imagine a procedure to turn on the magnetic field gradually from zero, and yet the angular momentum conservation guarantees zero total angular momentum of see, for example, Refs. [21][22][23]. A deeper insight to the angular momentum conservation from our discussion should be useful for better understanding of these systems.
Let us consider a situation where we have a TI bulk in the z > 0 region and the vacuum in z < 0, with an interface at the z = 0 surface, as illustrated in Fig. 3. It is well known that the boundary of TI supports massless surface states that can be described by a single Dirac fermion field. For our purpose, let us assume that there are T -violating magnetic impurities on the surface, that opens a mass gap for the surface fermions. Integrating out the massive surface fermion gives rise to a new term in the effective action in the low energy limit for the EM fields, which is the Chern-Simons action with a half integer level, ν = 1 2 [27] (which should not be confused with the winding number in the previous subsection). To capture the essential physics of our discussion, we will consider an idealized situation that this is the only response of the TI surface (with T -violating impurities) to an externally applied EM field. At least in long wavelength and time limit, the Chern-Simons term becomes dominant over other higher derivative terms in the action.
From the Chern-Simons action, the charge current in response to an applied EM field is obtained as
j µ = − ν 2 e 2 h µνα F να = − e 2 8π µνα F να ,(2.32)
which in components reads as
Q = e 2 4π B z , j x = e 2 4π E y , j y = − e 2 4π E x . (2.33)
where Q and j x,y are the charge density and the quantum Hall effect (QHE) current, respectively. Here, j x,y , B z , and E x,y in Eq. (2.33) represent 3D vector components without distinction between upper and lower indices.
We consider a magnetic flux that is vertically piercing the interface and is cylindrically symmetric: B = B z (r)ẑ, where r is the radius in the x-y plane. We further assume that B z (r) is localized for r ≤ R, so we can regard it as a flux tube like a magnetic vortex. In fact,
we may realize such a magnetic profile by an external superconducting vortex as postulated in Ref. [20]. As seen from Eq. (2.33) the magnetic flux induces a surface charge density Q and this charge gives rise to a nonzero electric field according to the Gauss law. It is easy to understand that a nonzero angular momentum emerges from the resulting EM fields which are indicated by arrows in Fig. 3.
Before going into the computation of the angular momentum carried by the EM field, we first show that the angular momentum contribution from the TI matter part at z > 0 is generally vanishing. The easiest way to confirm this is to compute the angular momentum that may be transferred to the TI surface states as we increase the magnetic flux from zero. This is because the TI bulk is gapped, and only the surface states may carry angular momentum in response to the applied EM fields in the system. During the process of turning on the magnetic flux, we have a tangential electric field E ϕ = (xE y − yE x )/r from Faraday's law,
2πrE ϕ (r) = −2π r 0 dr r ∂B z (r , t) ∂t . (2.34)
Then, according to Eq. (2.33) in the cylindrical coordinates, we have the QHE current as j r = (xj x +yj y )/r = e 2 4π E ϕ . From this, we can compute the torque from the EM force acting on the surface states along the ϕ direction. The EM force reads,
F ϕ = QE ϕ − j r B z ,(2.35)
where the second term represents the Lorentz force of j × B. Using Q = e 2 4π B z and j r = e 2 4π E ϕ , we see that the force vanishes identically, that is, the surface states do not experience any tangential force, or torque, by the Chern-Simons term. In fact, it is easy to verify that this result generally holds for any geometry. We conclude that no angular momentum is carried by the TI matter and its surface states. The angular momentum of the whole system resides solely in the EM sector. Now let us return back to the angular momentum in the EM sector. For static fields satisfying ∇ × E = 0 and the vector potential A = A ϕφ /r satisfying ∇ · A = 0, we can rewrite the angular momentum of EM fields as
L z = x x × (E × B) z = x (∇ · E) A ϕ − A ϕ (E · dS) ,(2.(∇ · E) A ϕ = e 2 2 ∞ 0 dr rB z (r) r 0 dr r B z (r ) = e 2 4 ∞ 0 dr rB z (r) 2 = e 2 16π 2 Φ 2 0 ,(2.38)
where Φ 0 is the total magnetic flux. For the second term, we consider a cylindrical boundary at r = R, and the Stokes theorem leads to
2πA ϕ (R) = Φ 0 , (2.39)
which takes a constant value along the boundary. Then, the vector potential can be taken out from the integrand, which gives
− A ϕ (E · dS) = − Φ 0 2π E · dS = − Φ 0 2π Q tot = − e 2 8π 2 Φ 2 0 , (2.40)
where Q tot = e 2 4π Φ 0 from Eq. (2.33) is used. Summing the above two terms, we get the total angular momentum as
L z = − e 2 16π 2 Φ 2 0 (2.41)
with the right sign that can easily be confirmed. We note that the original integral of the angular momentum is convergent by itself as B z (r) is of finite range, and the above way to split it into two terms is just a mathematical manipulation for convenience.
We shall suppose that the magnetic flux is quantized as if it were provided by an adjacent superconducting vortex of the winding number ν considered in Ref. [20]. We note that the magnetic vortex in superconductivity does not carry a finite net angular momentum except for the background contribution, so that the total angular momentum of our interest is still given by the above formula. The flux quantization gives 2e Φ 0 = 2πν, where the factor 2 of We consider turning on the magnetic field adiabatically from zero, and the time-dependent magnetic field gives E ϕ as well as the QHE current j θ , but the net force on the surface states is vanishing as we saw before. Therefore, the total angular momentum resides in the EM fields only.
To compute the EM part of the angular momentum, we follow the same steps as before.
Previously we considered only the contribution from the incoming magnetic flux, but if we perform the same computation including the whole TI boundary surface, the total L z turns out to be zero as we show in the following, that is consistent with our angular momentum conservation argument. From the spherical symmetry of the TI geometry and the cylindrical symmetry of the vector potential, we have
2πA ϕ (θ) = 2πR 2 θ 0 dθ sin θ B r 3D (θ ) . (2.44)
The Gauss law gives
∇ · E = e 2 4π B r 3D (θ)δ(r 3D − R) . (2.45)
Then, we find the first term in Eq. (2.36) to be vortices then dictates that they should be spinless. We will confirm this claim also in a nontrivial example where the angular momentum carried by surrounding localized gauge fields is essential for the cancellation of the total angular momentum. We emphasize that these localized gauge fields around the vortex core should be considered as a part of the magnetic vortex configuration under consideration.
x (∇ · E) A ϕ = 2πR 4 e 2 4π π 0 dθ sin θ B r 3D (θ) θ 0 dθ sin θ B r 3D (θ ) = 2πR 4 e 2 4π 1 2 π 0 dθ sin θ B r 3D (θ) 2 = 0 ,(2.
A. Example 1: Relativistic Nielsen-Olesen vortices
Let us illustrate our main points in the simplest example of Nielsen-Olesen vortices in relativistic scalar theory that we already treated in the previous section. The formulation of the theory presented below is a standard one, but we would like to pay a special attention to the cases with nonvanishing charge density. Consequently, nonzero electric fields accompany the vortices; we then have a precise description of the charged vortices, taking proper account of the Gauss law constraint and the Coulomb energy contribution to the Ginzburg-Landau free energy to be minimized. This endeavor, that we did not find in the literature in full generality as we present here, turns out to be crucial to show the exact cancellation of total angular momenta carried by the matter and the gauge field parts.
This section has some redundancy with our discussions in the previous section, but to make our analysis as self-contained as possible, let us retain some calculational details.
Showing explicit terms for our later convenience, we write down the Lagrangian as
L = (D 0 Φ) † (D 0 Φ) − (DΦ) † (DΦ) − U (Φ † Φ) + 1 2 E 2 − 1 2 B 2 , (3.1)
which is Eq. (2.1) without background, i.e., q = 0. Here, we take n e = 1 and D µ Φ = (∂ µ + ieA µ )Φ and, as defined in Sec. II A, we adopt a convention of (D) i = D i . The EM
fields are E = −∇A 0 − ∂ 0 A and B = ∇ × A.
We use the unit system with c = = 1 in this section. We also take a conventional form of the potential same as in the previous section;
U (Φ † Φ) = −λ 2 Φ † Φ + λ 4 2 (Φ † Φ) 2 .
We reproduce the equations of motion and the Gauss law from this action as
− D 2 0 Φ + D 2 Φ − U (Φ † Φ)Φ = 0 , (3.2) − ∂ 0 E + ∇ × B + ie (DΦ) † Φ − Φ † (DΦ) = 0 , (3.3) ∇ · E + ie (D 0 Φ) † Φ − Φ † (D 0 Φ) = 0 , (3.4)
which are equivalent to Eqs. (2.3)-(2.5) with q = 0 and n e = 1. We will work in the Hamiltonian formulation of the theory to look into the dynamics further.
The canonical conjugate field is given by definition as
Π † = δL δ∂ 0 Φ = (D 0 Φ) † . (3.5)
It should be noted that in our convention the above expression defines Π † , not Π. The charge density from the Nöther method is 6) and the Gauss law takes the form of
Q = −i (D 0 Φ) † Φ − Φ † (D 0 Φ) = −i(Π † Φ − Φ † Π) ,(3.∇ · E = eQ = −ie(Π † Φ − Φ † Π) . (3.7)
The Hamiltonian density from the Legendre transformation (including the EM sector) is obtained as
H = Π † (∂ 0 Φ) + Π(∂ 0 Φ) † − E(∂ 0 A) − L = Π † Π + (DΦ) † (DΦ) + U (Φ † Φ) + 1 2 (E 2 + B 2 ) − ieA 0 (Π † Φ − Φ † Π) − A 0 ∇ · E = Π † Π + (DΦ) † (DΦ) + U (Φ † Φ) + 1 2 (E 2 + B 2 ) , (3.8)
where we dropped the total derivative term ∇ · (A 0 E) in the second line, and from the second to the last line, we used the Gauss law to have cancellation between the last two terms. This should be the case since A 0 is not a dynamical degrees of freedom in the Hamiltonian formulation of gauge theory.
For our convenience we introduce a chemical potential µ via the free energy to be minimized; F = H − µQ. This is equivalent to introducing µ in the covariant derivative, once F in this section is identified as the Hamiltonian density in the previous section. The free energy is explicitly given by
F = H − µQ = Π † Π + (DΦ) † (DΦ) + U (Φ † Φ) + 1 2 (E 2 + B 2 ) + iµ(Π † Φ − Φ † Π) . (3.9)
This, together with the Gauss law, constitutes a precise formulation of gauged Ginzburg-Landau description for the cases with nonzero charge distributions. From the Gauss law, we see that E is not independent but generated through Π and Φ, albeit in a nonlocal way.
The variables, Π, Φ, and A, are considered as independent degrees of freedom, with respect to which the free energy F should be extremized to obtain the equations of motion.
We are interested in the stationary configurations where magnetic field is static; ∂ 0 B = 0.
In this case, as is familiar in classical electromagnetism, we can introduce an auxiliary function or static potential A 0 such that E = −∇A 0 and, with a proper boundary condition at spatial infinity, the Gauss law can be solved nonlocally as
A 0 = ie 1 ∇ 2 (Π † Φ − Φ † Π) , A 0 (∞) = 0 . (3.10)
This boundary condition is necessary, since a nonzero A 0 (∞) would shift our definition of chemical potential µ, that is, the true chemical potential is µ − eA 0 (∞), as we have already seen in the previous section. Using this, one of the terms in F , that is, the electric field energy is expressed as
1 2 E 2 = e 2 2 (Π † Φ − Φ † Π) 1 ∇ 2 (Π † Φ − Φ † Π) , (3.11)
which is nothing but the Coulomb energy induced by the charge distributions. The resulting expression for the free energy F involves only the independent variables, Π, Φ, and A, from which we can proceed to obtain the equations of motion.
From the variation with respect to Π † , we get
Π + iµΦ + e 2 Φ 1 ∇ 2 (Π † Φ − Φ † Π) = 0 . (3.12)
Using the expression for A 0 , this can be written as
Π + iµΦ − ieA 0 Φ = 0 or Π = −i(µ − eA 0 )Φ .∂ 0 Φ = −iµΦ . (3.14)
Since F is quadratic in Π, one may choose to insert back the solution for Π from Eq. (3.12) into F to get a more conventional form of the free energy in terms of Φ and A only. It is explicitly given by
F = (DΦ) † (DΦ) + U (Φ † Φ) − µ(µ − eA 0 )Φ † Φ + 1 2 B 2 . (3.15)
One should keep in mind that A 0 in the above expression is a functional of Φ that should be obtained by solving the Gauss law (3.10) together with Eq. (3.12), which is in general nonvanishing for µ = 0 corresponding to nonzero charge distributions in a solution. A more practical way to approach this problem is indeed what we have described in the preceding paragraphs, i.e., keeping Π as an independent degree of freedom.
The variation of the free energy with respect to Φ † gives
− D 2 Φ + ΦU (Φ † Φ) − iµΠ − e 2 Π 1 ∇ 2 (Π † Φ − Φ † Π) = 0 ,(3.16)
which, upon using the expression for A 0 , is equivalent to
− D 2 Φ + U (Φ † Φ)Φ − i(µ − eA 0 )Π = 0 ,(3.17)
and using the solution for Π, we finally get to
− D 2 Φ + U (Φ † Φ)Φ − (µ − eA 0 ) 2 Φ = 0 . (3.18)
This, as it should, agrees with the equation of motion for Φ obtained from the Lagrangian, as written in Eq. (3.2), after using the Josephson relation, ∂ 0 Φ = −iµΦ. However, we again need to recall that A 0 is a solution of the Gauss law, which itself involves Π and Φ as
∇ 2 A 0 = ie(Π † Φ − Φ † Π) = −2e(µ − eA 0 )Φ † Φ . (3.19)
Equations (3.18) and (3.19) together with the Maxwell equation for A, i.e., which we will refer to as "charged Nielsen-Olesen vortices."
∇ × (∇ × A) + ie (DΦ) † Φ − Φ † (DΦ) = 0 ,(3.
We could estimate the angular momentum in the matter part using an expression like Eq. (2.23), but here, we shall show an alternative physical approach. To compute the matter contribution to the angular momentum, we need the linear momentum density, i.e., P , obtained from T 0i component of the energy-momentum tensor. From the Nöther method, we see that T 0i is given by
P i = T 0i = (D 0 Φ) † (D i Φ) + (D i Φ) † (D 0 Φ) = Π † (D i Φ) + (D i Φ) † Π . (3.21)
Using Eq. (3.12) for Π, we obtain,
P = −i (DΦ) † (µ − eA 0 )Φ − Φ † (µ − eA 0 )(DΦ) . (3.22)
The kinetic angular momentum carried by the matter part is then
L matter z = x x × P z . (3.23)
On the other hand, the gauge field contribution to the angular momentum is easily found from the Poynting vector, namely from Eq. (2.27).
The charge neutral case at µ = 0 makes the essential difference from the nonrelativistic case in the previous section. The above equations of motion for relativistic vortices are mathematically identical, up to trivial scaling of parameters, to those of "locally" neutral nonrelativistic vortices without coupling to electric field: both of them are known as Nielsen-Olesen vortices. For the charge neutral case, it is algebraically trivial to see Π = 0 and P = E = 0 from µ = A 0 = 0, and both the matter and gauge field contributions to the total angular momentum are zero, but this conclusion is physically nontrivial; vortices are circulating configurations and yet they have no angular momentum. Intuitively, the absence of matter contribution to the angular momentum in the relativistic theory can be understood as a cancellation between particle vortex and anti-particle anti-vortex as follows: recall that
Φ ∼ a + b † and Π ∼ i(a − b † )
where a and b are annihilation operators for particle and antiparticle, respectively. In the superfluid phase of a large occupation number, we can regard a and b as c-numbers as usual. A vortex profile with winding number ν can be viewed as a superposition of a particle vortex of a ∼ e iνϕ and an antiparticle antivortex of b ∼ e −iνϕ .
In the charge neutral case of Π = 0, their amplitudes are precisely equal, i.e., a = b † , and the antiparticle antivortex contribution to the angular momentum is precisely opposite to that of the particle vortex. The absence of antiparticles in the nonrelativistic vortex in the previous section is the major difference from the relativistic theory discussed in this section. For the charged case at µ = 0, the two systems of equations are different by terms that we previously neglected in the nonrelativistic reduction. In addition to this difference for the charged case, a background charge density that we introduced as q in the previous section is also absent here. This means that the net charge of a charged Nielsen-Olesen vortices is not zero, and the electric field grows logarithmically at large distance in two-dimensional space perpendicular to the vortex string in three dimensions. This implies that the line density of energy of a charged vortex is divergent in infinite space, and a sensible solution would exist only in a finite transverse volume. As we will focus on azimuthally symmetric vortex configurations to apply our angular momentum conservation argument, we consider a spatial cutoff in transverse space at a certain distance from the origin, that is, r ≤ R in the radial direction. We will show that the total angular momentum of a charged vortex within the volume r ≤ R is always zero for any cutoff R, when we sum the contributions of both matter part and the gauge fields.
Our conclusion rectifies a misleading statement in the Appendix C of the well-known literature, Ref. [10], that a charged Nielsen-Olesen vortex carries a nonzero angular momentum. Our result of vanishing angular momentum even for charged vortices is independent of the issue of diverging line energy density in infinite space. Later, we will more precisely point out where the misleading conclusion in Ref. [10] stems from.
The computation in the charged vortex case is more delicate than the neutral case, and the detailed mechanism for cancellation is similar to that in the previous section. First of all, since Π = 0, the particle vortex and the antiparticle antivortex have different amplitudes, and the net matter angular momentum no longer cancels to be zero. From the radial electric field, E = 0, the gauge fields also contribute to the total angular momentum. We take the following Ansatz,
Φ = f (r) e iνϕ , A 0 = a(r) , A ϕ = ν e 1 − h(r) (3.24)
with the boundary condition for vanishing magnetic flux, i.e., h(R) = 0 at sufficiently large boundary r = R. Then the equations of motion and the Gauss law become (with ≡ d dr ) We finish this subsection by pointing out that our finding of zero angular momentum for magnetic vortices in the U (1) Abelian Higgs model is consistent with the particle-vortex duality in 2+1 dimensions [22,23],
1 r (rf ) − ν 2 r 2 h 2 f + (λ 2 − λ 4 f 2 )f + (ea − µ) 2 f = 0 , (3.25) h r − 2e 2 r f 2 h = 0 , (3.26) 1 r (ra ) − 2e(ea − µ)f 2 = 0 .
B. Example 2: Non-Abelian CFL vortices
We can test our assertion in a more non-trivial example of non-Abelian vortices [2,11] in the CFL color-superconducting phase of QCD quark matter at high baryon density and low temperature. The diquark condensates in the CFL phase break both QCD gauge symmetry and the global U (1) B baryon number symmetry. The non-Abelian vortices arise from coupled dynamics of color fields and U (1) B superfluidity, and carry fractional winding numbers for both gauge and global symmetries, such that the total winding number for each color component of the diquark condensate field is an integer. One might think that the non-Abelian CFL vortex is peculiar to QCD, but similar structures can also be found in multi-component superconductivity, see Ref. [28] for example. The minimal non-Abelian vortex carries only 1/2 of the U (1) B winding number (that is equivalent to 1 2 × 2 3 = 1 3 winding number for the diquark field), so that the non-Abelian CFL vortices can be considered as fractionalized U (1) B vortices. In the hadronic phase, on the other hand, the minimal dibaryon Cooper-pair superfluid vortex also carries the same winding number 1 2 , so that across the two phases the dibaryon vortex should transmute to the non-Abelian CFL vortex [15], which is schematically illustrated in Fig. 5 (see also Ref. [16] for an alternative scenario). Since the angular momentum must be conserved during this transmutation process, we expect the angular momenta of the two vortices to be equal. The minimal dibaryon vortex of 1/2 of the U (1) B winding number is a usual superfluid vortex and carries the angular momentum L z = N B /2
where N B is the total baryon number. In contrast, the non-Abelian CFL vortex is also accompanied by color gauge fields, and in general, its total angular momentum receives contribution from these localized color fields. It is a nontrivial check to see that the total angular momentum of the non-Abelian CFL vortex from both matter part and the gauge fields is indeed L z = N B /2, i.e., the same as in the hadronic phase, as we will show below.
Essentially, this means that the color-magnetic part of the non-Abelian CFL vortex does not contribute to the angular momentum, and only the U (1) B superfluid part makes a finite contribution. This situation provides another example of confirming our assertion that the gauged magnetic vortex does not carry angular momentum.
The diquark condensate in the CFL phase is described by a 3 × 3 matrix field, Φ = Φ iα , where i and α are color and flavor indices, respectively. More precisely, there are two such fields for left-handed and right-handed diquarks, and we assume that they share the same configuration in a vortex solution. We can always perform suitable color rotations, such that the profile of the non-Abelian CFL vortex appears only in the global U (1) B and the eighth component of the color field A 8 µ with the generator t 8 = 1 √ 12 diag(−2, 1, 1) 2 . Therefore, we will show expressions only in these parts in the following. The QCD covariant derivative
with A 8 µ only is D µ Φ = ∂ µ − igA 8 µ t 8 Φ , (3.31)
where g is the QCD coupling constant. We will work with the gauge invariant Lagrangian in terms of the diquark field given by
L = tr (D 0 Φ) † (D 0 Φ) − (DΦ) † (DΦ) − V tr(Φ † Φ) + 1 2 E 8 · E 8 − 1 2 B 8 · B 8 ,(3.32)
where E 8 = −∇A 8 0 − ∂ 0 A 8 and B 8 = ∇ × A 8 . The concrete shape of the potential V is not important for our purpose. The ensuing analysis is very similar to that in the previous subsection, and we will highlight only the important differences and the major results.
The chemical potential µ B is introduced for the baryon charge density Q B , which is
Q B = − 2 3 i tr (D 0 Φ) † Φ − Φ † (D 0 Φ) . (3.33)
Here the coefficient is understood from the baryon charge 2/3 carried by the diquark field Φ. The color charge that appears in the Gauss law constraint is given by
∇ · E 8 = ig tr (D 0 Φ) † t 8 Φ − Φ † t 8 (D 0 Φ) ,(3.34)
which is easily obtained from the equation of motion for A 8 0 . Introducing the canonical conjugate field Π ≡ D 0 Φ and following the same steps in the previous section, we can find the Hamiltonian and the free energy as
F = tr Π † Π + (DΦ) † (DΦ) + V tr(Φ † Φ) + g 2 2 tr(Π † t 8 Φ − Φ † t 8 Π) 1 ∇ 2 tr(Π † t 8 Φ − Φ † t 8 Π) + 2 3 iµ B tr(Π † Φ − Φ † Π) + 1 2 B 8 · B 8 . (3.35)
Recall that the second line arises from the Coulomb energy of color field, 1 2 E 8 · E 8 [see Eq. (3.11)]. By minimizing F with respect to Π, Φ, and A 8 , we obtain the equations of motion. It is convenient to introduce an auxiliary variable A 8 0 to render the nonlocality into local equations of motion, which is defined by E 8 = −∇A 8 0 for static configuration, so that the Gauss law becomes and the Gauss law becomes
∇ 2 A 8 0 = −igtr Π † t 8 Φ − Φ † t 8 Π .∇ 2 A 8 0 = 2g tr Φ † t 8 (gA 8 0 t 8 + 2µ B /3)Φ . (3.38)
In the above the term ∝ µ B should be understood as the unity matrix in color space. The other equations of motion are
D 2 Φ − ΦV tr(Φ † Φ) + gA 8 0 t 8 + 2µ B /3 2 Φ = 0 ,(3.39)
and The non-Abelian CFL vortex solution has the following form [2]:
∇ × (∇ × A 8 ) = igtr (DΦ) † t 8 Φ − Φ † t 8 (DΦ) .A 8 0 = a(r) , A 8 ϕ = ν g √ 12 3 1 − h(r) (3.41) with A 8 = A 8 ϕφ /r and Φ = f (r) e iνϕ 0 0 0 b(r) 0 0 0 b(r) . (3.42)
The boundary condition is h(∞) = 0 which ensures,
− DΦ = (∇ + igA 8 t 8 )Φ → i ν 3φ r Φ as r → ∞ . (3.43)
This signifies that the vortex carries a superfluid winding number ν/3 with respect to the diquark global U (1) (which is equivalent to ν/2 with respect to U (1) B symmetry). To see how the color-magnetic vortex is embedded in the above solution, we can factorize it as follows,
Φ = e i ν 3 ϕ e i 2ν 3 ϕ 0 0 0 e −i ν 3 ϕ 0 0 0 e −i ν 3 ϕ f (r) 0 0 0 b(r) 0 0 0 b(r) , (3.44)
where the overall phase corresponds to the global U (1) and the middle matrix, e −iν
1 r (ra ) − g 2 3 a(2f 2 + b 2 ) − 4g 3 √ 3 µ B (−f 2 + b 2 ) = 0 , (3.45) 1 r (rf ) − ν 2 9r 2 (1 + 2h) 2 f − f V (f 2 + 2b 2 ) + − 2g √ 12 a + 2 3 µ B 2 f = 0 , (3.46) 1 r (rb ) − ν 2 9r 2 (1 − h) 2 b − bV (f 2 + 2b 2 ) + g √ 12 a + 2 3 µ B 2 b = 0 , (3.47) 1 r h + g 2 3 f 2 r (1 + 2h) − g 2 3r (1 − h)b 2 = 0 ,(3.48)
where Eq. (3.45) corresponds to the Gauss law.
To compute the matter part of the angular momentum, we need the momentum density,
P i = T 0i = tr (D 0 Φ) † (D i Φ) + (D i Φ) † (D 0 Φ) = tr Π † (D i Φ) + (D i Φ) † Π . (3.49)
Substituting the solution of Π for the above P i , we obtain,
P = P ϕφ = −i tr (DΦ) † (gA 8 0 t 8 + 2µ B /3)Φ − Φ † (gA 8 0 t 8 + 2µ B /3)(DΦ) = ν 3r (1 + 2h)f 2 − 4g √ 12 a + 4 3 µ B + 4ν 3r (1 − h)b 2 g √ 12 a + 2 3 µ B φ . (3.50)
Thus, the matter part of the angular momentum is
L matter z = 2π R 0 dr r(rP ϕ ) = 2πν 3 R 0 dr r (1 + 2h)f 2 − 4g √ 12 a + 4 3 µ B + 4(1 − h)b 2 g √ 12 a + 2 3 µ B .
(3.51)
The color gauge field contribution is as before
L gauge z = −(2π) √ 12ν 3g R 0 dr ra (r)h (r) . (3.52)
Integrating by part and using the Gauss law, we have,
L gauge z = (2πν) √ 12 3 R 0 dr r g 3 a(2f 2 + b 2 ) + 4 3 √ 3 µ B (−f 2 + b 2 ) h .L tot z = 2πν R 0 dr r f 2 − 2g 3 √ 3 a + 4 9 µ B + b 2 2g 3 √ 3 a + 8 9 µ B . (3.54)
One might think that the above result is an involved expression, but there is an elegant physical interpretation. To this end, we shall compute the baryon charge density as
Q B = − 2 3 i tr (D 0 Φ) † Φ − Φ † (D 0 Φ) = − 2 3 i tr(Π † Φ − Φ † Π) = 4 3 tr Φ † (gA 8 0 t 8 + 2µ B /3)Φ ,(3.55)
from which the total baryon charge per unit vortex length reads:
N B = 2π R 0 dr r f 2 − 4g 3 √ 3 a + 8 9 µ B + b 2 4g 3 √ 3 a + 16 9 µ B .
(3.56)
Comparing L tot z and N B , we see that the following relation holds:
L tot z = ν 2 N B . (3.57)
This confirms that the total angular momentum of the non-Abelian vortex in the CFL phase contains only the contribution from the global U (1) B vortex; the total angular momentum (in the unit of ) is ν times the number of the Cooper pairs. We note that this result completely agrees with Eq. (14) in Ref. [15], but in Ref. [15] only the U (1) B contribution to the angular momentum was postulated without rigorous justification.
IV. CLASS III: CASE STUDY WITHOUT ANGULAR MOMENTUM CONSERVATION
The last logical possibility in our classification is that magnetic vortices cannot be created by simply turning on external magnetic flux in an azimuthally symmetric way. What distinguishes this case from all previous cases is that the principle of angular momentum conservation does not naïvely apply in the vortex creation process. The vortices classified in this class are characterized by inhomogeneous profiles along the vortex axis, which means that not only the external magnetic flux but also something else are needed to create the vortices: roughly speaking, a kind of twisting along the axis would be required. Such vortices do exist as we discuss below, although they seem to be rare in the literature.
An example that belongs to this class is provided by an object called "charged semilocal vortex" as constructed by Abraham in Ref. [24]. This Abraham vortex is an extension of the semilocal vortex [29] that has been discussed in the context of electroweak strings in cosmology (see Ref. [30] for a review). They also appear quite commonly as topological BPS (Bogomol'nyi-Prasad-Sommerfield) objects in supersymmetric gauge theories. The simplest model of the Abraham vortex consists of two charged scalar fields, Φ a (a = 1, 2), with the equal charge, and a U (1) gauge field A µ . The Hamiltonian in the critical limit reads:
H = a=1,2 |D 0 Φ a | 2 + |DΦ a | 2 + g 2 2 a=1,2 |Φ a | 2 − v 2 2 + 1 2 (E 2 + B 2 ) ,(4.1)
where D µ Φ a = (∂ µ − igA µ )Φ a , and the Gauss law is
∇ · E = ig a=1,2 (D 0 Φ a ) † Φ a − Φ † a (D 0 Φ a ) . (4.2)
The vortex solution relies on the following Bogomol'nyi bound, Here, A = (A x , A y , A z ), E = (E x , E y , E z ), B = (B x , B y , B z ) and we imposed additional conditions that ∂ 3 Φ 2 = iαΦ 2 with a constant α and all other ∂ 3 is vanishing. We defined Q 2 as
H = a=1,2 |D 0 Φ a ± D 3 Φ a | 2 + |D 1 Φ a ± iD 2 Φ a | 2 + 1 2 |E x ∓ B y | 2 + 1 2 |E y ± B x | 2 + 1 2 B z ∓ g a=1,2 |Φ a | 2 − v 2 2 ∓ αQ 2 ∓ v 2 gB z ∓ ∇ · (EA z ) .Q 2 = i (D 0 Φ 2 ) † Φ 2 − Φ † 2 (D 0 Φ 2 ) . (4.4)
The equations we obtain from this, for the upper sign, are
D 0 Φ a +D 3 Φ a = 0 , D 1 Φ a +iD 2 Φ a = 0 , E i = ij B j , B z −g a=1,2
|Φ a | 2 −v 2 = 0 . (4.5)
It can be checked that these solve the original equations of motion. In Ref. [24] it was also shown that these equations admit nice solutions with zero net gauged U (1) charge but nonzero Q 2 , which are somewhat misleadingly called "charged" semilocal vortices. These solutions have finite line energy density, due to the fact that the total U (1) charge is zero.
The solutions are possible only for α = 0, that corresponds to a "twisting" along the vortex axis. Due to this, the vortex string carries a net linear momentum along the axis direction. Although the total U (1) charge is zero, the charge density profile in space is nonzero, and there exists nontrivial profile of local electric and magnetic fields. This leads to a nonvanishing contribution of the electromagnetic fields to the total angular momentum.
As pointed out in Ref. [24], the solutions carry nonzero total angular momentum, but we would not go into technicality here, and the readers can directly consult Ref. [24]. The "twisting", parametrized by α, can be considered as spinning the vortex to give a finite angular momentum. This is an extra operation that would be needed to create such a vortex profile by hand, and the angular momentum conservation cannot be applied to the situation.
In other words, in this peculiar system belonging to this class, the angular momentum conservation is not satisfied by L matter z or L gauge z or their sum.
V. CONCLUSION
In this work, we apply the principle of angular momentum conservation to understand the origin of angular momentum carried by magnetic vortices in various physical systems in condensed matter, high energy, nuclear physics, and cosmology. We find that this simple principle is powerful enough to allow us an overarching scheme of classifying the known examples, according to how the principle of angular momentum conservation is satisfied.
We find the four distinct classes of examples in our classification scheme; spinful (class Ia), topological (class Ib), spinless (class II) and exotic (class III) vortices, as already summarized in Introduction. We present detailed analyses for these examples, and emphasize that the angular momentum carried by localized gauge fields around the vortex core plays a crucial role in satisfying the angular momentum conservation. We believe that our study gives a
•
Class Ib (topological vortices): They carry a finite angular momentum, but no background matter exists in the bulk. The angular momentum resides only on the boundary. The angular momentum carried by the surrounding gauge fields must be counted for the total angular momentum. Examples are the vortices on the surface of topological phases of matter. • Class II (spinless vortices): They do not carry a net angular momentum due to the angular momentum conservation. The angular momentum carried by the surrounding gauge fields should be included. Examples are the vortices in relativistic field theories and cosmology.
5) with (D) i ≡ D i , where we note that ∂ i = ∂/∂x i = −∂/∂x i in our metric convention (+, −, −, −). The magnetic vortices we consider are the static solutions of the above equations of motion, so we drop the time derivative terms in the below. Then, Eq.
FIG. 1 .
1(Left panel) Profile of the conventional elementary (ν = 1) magnetic vortex; f and h without coupling to a for λ = 1.5. (Right panel) Profile of the elementary vortex with the electric field; f , h, and a for λ = 1.5 and m 2 /m 2 V = 1. f 2 +q = 0, leading to a(r) = 0 from Eq. (2.18). Most Type-II vortices behave this way, but there are examples where this does not happen in general; see Refs. [4, 5]. The regularity of Ψ at ρ = 0 requires f (0) = 0, and at infinity it should approach the vacuum value of f (∞) = 1. In the absence of a, then the boundary conditions should be
25) which vanishes in the vortex configuration(2.15). The sum of L can,matter z and L can,gauge z gives the total angular momentum L can z which is conserved. Alternatively we can consider the conserved angular momentum as the sum of L physical setup. In our present setup we can gradually turn on the magnetic field, so that the magnetic vortex emerges. In this case, it makes sense to consider L total z , not L can z . Although the difference between L can z and L total z is the only boundary term, it plays an essential role in the conservation of angular momentum as discussed in Sec. III A. For more discussions on the canonical angular momentum, see a concrete analysis in Ref.[25]
FIG. 2 .
2the boundary conditions (2.22), the surface contribution vanishes. Using λ = m 2 H /m 2 V we can simplify the above expression into L gauge z = −ν(2π ) µ g R 0 dr r h(r) f 2 (r) +q(r) . The integrand of Eq. (2.28) in terms of dimensionless variables for λ = 1.5 and m 2 /m 2 V = 1, which represents the local distribution of the EM angular momentum.
FIG. 3 .
3Interface between the TI and the vacuum with a localized flux of magnetic field B. The electric charge Q is stored at the interface which produces the electric field E.
the whole system. The only way our result of fractional angular momentum can be consistent with the angular momentum conservation is that the other compensating angular momentum should be located in the other part of the TI-vacuum boundary where the magnetic flux leaves out from the bulk TI. If this boundary region is far separated from the place where the original incoming flux enters the TI, we can reasonably neglect this far-away region, and focus only on the angular momentum localized on the incoming flux alone. This angular momentum indeed takes a fractional value, as we confirm in the following discussions. We can say that the fractional angular momentum is transported from the boundary at infinity to the incoming magnetic flux; this characterizes the magnetic vortices of Class Ib in our classification. Such magnetic vortices with fractional angular momenta are not peculiar, but rather ubiquitous in topological phases of matter;
36)where the last term is the surface integral on the exterior boundary. Using the Stokes theorem and the cylindrical symmetry, we find the vector potential with the boundary condition, A ϕ (with the Gauss law, ∇ · E = Qδ(z), the first term in the above expression of L z becomes x
2e originates from the Cooper pair. This finally leads to L z = − ν 2 16 . (2.42) Therefore, the EM field surrounding the magnetic vortex between a TI and a superconductor carries a nonzero angular momentum given in Eq. (2.42). The conservation of the total angular momentum during the process of turning on the magnetic flux requires the existence of an opposite and compensating angular momentum somewhere else. To identify where this compensating component is, let us consider a global geometry of the bulk TI and its closed boundary. For simplicity we assume that the bulk TI (which is a blue shaded region in Fig. 3) is a large ball of radius R and the boundary surface is a sphere of radius R. A localized magnetic tube with a flux Φ 0 enters the TI at θ = π, where θ is the polar angle in 3D spherical coordinates. The same flux leaves out from the TI at other places of the surface in cylindrically symmetric (i.e., ϕ independent) way. Let the radial component of the magnetic field at r 3D = R be B r 3D (θ) as a function of θ, where r 2 3D = r 2 + z 2 is the 3D radius. The flux conservation results in π 0 dθ sin θ B r 3D (θ) = 0 . (2.43)
emphasize that the original expression of the angular momentum is localized in the region where B = 0 and E = 0, that is, it is localized around the TI boundary where the magnetic flux either enters or leaves the TI. Therefore, the fractional angular momentum localized around the magnetic tube at θ = π is compensated by the angular momentum carried by the outgoing flux in other places of the TI boundary which can be taken infinitely away.III. CLASS II: CASE STUDY OF MAGNETIC VORTICES WITH ZERO ANGULAR MOMENTUM In this section we consider magnetic vortices in relativistic field theory as examples of self-consistent systems without any background matter or boundary that could absorb angular momentum. Such vortices could appear in the Standard Model and extensions of the Standard Model. They have been considered in the context of high energy physics and cosmology. A faithful application of our angular momentum conservation argument to these
relation, Π = D 0 Φ = (∂ 0 + ieA 0 )Φ, this gives the well-known Josephson relation;
final set of closed equations for Φ, A and, A 0 to be solved for classical configurations in relativistic theory. If µ = 0, then it is consistent with A 0 = 0, and there is no electric field. This situation at µ = 0 corresponds to charge neutral vortices in our problem. For µ = 0 there exists nonvanishing charge and the electric field in the solution,
FIG. 4 .
4Schematic illustration of the charge neutral Nielsen-Olesen vortex composed from a particle vortex and an antiparticle antivortex.
part of the angular momentum is computed from Eq. the last equality we performed the integration by part and used the boundary condition at r = R as in the previous section. Using the Gauss law (3.27) to replace [ra (r)] , we arrive at (r)f 2 (r)[−ea(r) + µ] = −L matter z , (3.30)which precisely cancels the matter contribution. As a result the total angular momentum is vanishing. In Appendix C of Ref.[10], the surface term of Eq. (C3) that was neglected is nonzero: this can be seen from the description of the solution below Eq. (C6) with Q = 0. It can be shown that Eq. (C3) precisely cancels Eq. (C4), so that the total angular momentum is zero. This cancellation has its origin in the angular momentum conservation, and it holds for any R regardless of an issue of infinite line energy density of the charged solution.
FIG. 5 .
5where the magnetic vortices in the Abelian Higgs model are mapped to the elementary excitations of a dual complex scalar field which are clearly spinless. Checking the spins of other excitations in the web of dualities [22] Schematic illustration of the continuity between the dibaryon vortex in the hadronic phase and the non-Abelian CFL vortex in CFL quark matter in QCD.
3.38), (3.39), and (3.40) form a closed set to solve for the vortex profile of Φ, A 8 , and A 8 0 .
SU (3), and this is why this configuration as implemented in Eq. (3.42) is called a "non-Abelian" vortex. The equations of motion, (3.38), (3.39), and (3.40), become, after some algebra,
. (3.51) and L gauge z in Eq. (3.53), we get the total angular momentum per unit vortex length to be
different Q tot . Previously we took account of Q tot around the incoming magnetic flux only, but if we sum up all the contributions from the whole TI surface, it should amount to Q tot = 0 due to Eq. (2.43). In this way we see that the second term is zero as well. We46)
using Eq. (2.43). For the second term in Eq. (2.36), we can still employ Eq. (2.40) with
We note that our result derived in the following is different by a factor 1/2 from Ref.[20]. We have identified where this difference stems from, but it is not essential for our present argument, so we will not go into that detail.
This matrix representation is an unconventional choice; in later discussions we will focus on the u-quark sector and this choice is good for that purpose.
answer to the seemingly confusing, but surprisingly rich, question of angular momentum. answer to the seemingly confusing, but surprisingly rich, question of angular momentum
is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI. K F , K. F. is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI
Grant Nos. 18H01211 and 19K21874. Y. H. is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Nos. 17H06462 and 18H01211. H.-U. Y. is supported by the. Grant NoU.S. Department of Energy, Office of Science, Office of Nuclear PhysicsGrant Nos. 18H01211 and 19K21874. Y. H. is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Nos. 17H06462 and 18H01211. H.-U. Y. is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Grant No.
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| []
|
[
"Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic",
"Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic"
]
| [
"Mihaly Barasz ",
"Paul Christiano ",
"Benja Fallenstein ",
"Marcello Herreshoff ",
"Patrick Lavictoire ",
"Eliezer Yudkowsky "
]
| []
| []
| We consider the one-shot Prisoner's Dilemma between algorithms with read-access to one anothers' source codes, and we use the modal logic of provability to build agents that can achieve mutual cooperation in a manner that is robust, in that cooperation does not require exact equality of the agents' source code, and unexploitable, meaning that such an agent never cooperates when its opponent defects. We construct a general framework for such "modal agents", and study their properties. | null | [
"https://arxiv.org/pdf/1401.5577v1.pdf"
]
| 14,329,044 | 1401.5577 | 5e15e6ff319242b1d270a562252bc1d82441e14f |
Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic
22 Jan 2014 January 23, 2014
Mihaly Barasz
Paul Christiano
Benja Fallenstein
Marcello Herreshoff
Patrick Lavictoire
Eliezer Yudkowsky
Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic
22 Jan 2014 January 23, 2014
We consider the one-shot Prisoner's Dilemma between algorithms with read-access to one anothers' source codes, and we use the modal logic of provability to build agents that can achieve mutual cooperation in a manner that is robust, in that cooperation does not require exact equality of the agents' source code, and unexploitable, meaning that such an agent never cooperates when its opponent defects. We construct a general framework for such "modal agents", and study their properties.
Introduction
Can cooperation in a one-shot Prisoner's Dilemma be justified between rational agents? Rapoport [18] argued in the 1960s that two agents with mutual knowledge of each others' rationality should be able to mutually cooperate. Howard [10] explains the argument thus:
Nonetheless arguments have been made in favour of playing C even in a single play of the PD. The one that interests us relies heavily on the usual assumption that both players are completely rational and know everything there is to know about the situation. (So for instance, Row knows that Column is rational, and Column knows that he knows it, and so on.) It can then be argued by Row that Column is an individual very similar to himself and in the same situation as himself. Hence whatever he eventually decides to do, Column will necessarily do the same (just as two good students given the same sum to calculate will necessarily arrive at the same answer). Hence if Row chooses D, so will Column, and each will get 1. However if Row chooses C, so will Column, and each will then get 2. Hence Row should choose C.
Hofstadter [9] described this line of reasoning as "superrationality", and held that knowledge of similar cognitive aptitudes should be enough to establish it, though the latter contention is (to say the least) controversial within decision theory. However, one may consider a stronger assumption: what if each agent has some ability to predict in advance the actions of the other?
This stronger assumption suggests a convenient logical formalism. In the 1980s, Binmore [3] considered game theory between Turing machines which had access to one anothers' Gödel numbers 1 :
...a player needs to be able to cope with hypotheses about the reasoning processes of the opponents other than simply that which maintains that they are the same as his own. Any other view risks relegating rational players to the role of the "unlucky" Bridge expert who usually loses but explains that his play is "correct" and would have led to his winning if only the opponents had played "correctly". Crudely, rational behavior should include the capacity to exploit bad play by the opponents. In any case, if Turing machines are used to model the players, it is possible to suppose that the play of a game is prefixed by an exchange of the players' Gödel numbers.
Howard [10] and McAfee [15] considered the Prisoner's Dilemma in this context 2 , and each presented an example of an algorithm which would always return an answer, would cooperate if faced with itself, and would never cooperate when the opponent defected. (The solution discussed in both papers was a program that used quining of the source code to implement the algorithm "cooperate if and only if the opponent's source code is identical to mine"; we represent it in this paper as Algorithm 3, which we call CliqueBot on account of the fact that it cooperates only with the 'clique' of agents identical to itself.)
More recently, Tennenholtz [20] reproduced this result in the context of other research on multi-agent systems, noting that CliqueBot can be seen as a Nash equilibrium of the expanded game where two players decide which code to submit to the Prisoner's Dilemma with source-code swap. This context (called "program equilibrium") led to several novel game-theoretic results, including folk theorems by Fortnow [7] and Kalai, Kalai, Lehrer and Samet [11], an answer by Monderer and Tennenholtz [16] to the problem of seeking strong equilibria (many-agent Prisoner's Dilemmas in which mutual cooperation can be established in a manner that is safe from coalitions of defectors), a Bayesian framework by Peters and Szentes [17], and more.
However, these approaches have an undesirable property: they restrict the circle of possible cooperators dramatically-in the most extreme case, only to agents that are syntactically identical! (In a moment, we will see that there are many examples of semantically distinct agents such that one would wish one's program to quickly cooperate with each of them.) Thus mutual cooperation is inherently brittle for CliqueBots, and an ecology of such agents would be akin to an all-out war between incompatible cliques. This problem can be patched somewhat, but not cured, by prescribing a list of agents with whom mutual cooperation is desirable, but this approach is inelegant and requires all of the relevant reasoning to happen outside of the system. We'd like to see agents that can reason on their own somewhat.
A natural-seeming strategy involves simulating the other agent to see what they do when given one's own source code. Unfortunately, this leads to an infinite regress when two such agents are pitted against one another.
One attempt to put mutual cooperation on more stable footing is the model-checking result of van der Hoek, Witteveen, and Wooldridge [8], which seeks "fixed points" of strategies that condition their actions on their opponents' output. However, in many interesting cases there are several fixed points, or none at all, and so this approach does not correspond to an algorithm as we would like.
Since the essence of this problem deals in counterfactuals-e.g. "what would they do if I did this"-it is worth considering modal logic, which was intended to capture reasoning about counterfactuals, and in particular the Gödel-Löb modal logic GL with provability as its modality. (See Boolos [4] and Lindström [13] for some good references on GL.) That is, if we consider agents that cooperate if and only if they can prove certain logical formulas, the structure of logical provability gives us a genuine framework for counterfactual reasoning, and in particular a powerful and surprising tool known as Löb's Theorem [14]: Theorem 1.1 (Löb's Theorem). Let S be a formal system which includes Peano Arithmetic. If φ is any well-formed formula in S, let φ be the formula in a Gödel encoding of S which claims that there exists a proof of φ in S; then whenever S ⊢ ( φ → φ), in fact S ⊢ φ.
We shall see that Löb's Theorem enables a flexible and secure form of mutual cooperation in this context. In particular, we first consider the intuitively appealing strategy "cooperate if and only if I can prove that my opponent cooperates", which we call FairBot. If we trust the formal system used by FairBot, we can conclude that it is unexploitable (in the sense that it never winds up with the sucker's payoff). When we play FairBot against itself (and give both agents sufficient power to find proofs), although either mutual cooperation or mutual defection seem philosophically consistent 3 , it always finds mutual cooperation (Theorem 3.1)! 4 Furthermore, we can construct another agent after the same fashion which improves on the main deficit of the above strategy: namely, that FairBot fails to correctly defect against CooperateBot. 5 We call this agent PrudentBot.
Moreover, the underpinnings of this result (and the others in this paper) do not depend on the syntactical details of the programs, but only on their semantic interpretations in provability logic; therefore two such programs can cooperate, even if written differently (in several senses, for instance if they use different Gödel encodings or different formal systems). Accordingly, we define a certain class of algorithms, such as FairBot and PrudentBot, whose behavior can be described in terms of a modal formula, and show that the actions these "modal agents" take against one another can be described purely in terms of these modal formulas. Using the properties of Kripke semantics, one can algorithmically derive the fixedpoint solutions to the action of one modal agent against another; indeed, the results of this paper have additionally been checked by a computer program written by two of the authors, hosted at github.com/klao/provability. We next turn to the question of whether a meaningful sense of optimality exists among modal agents. Alas, there are several distinct obstacles to some natural attempts at a nontrivial and non-vacuous criterion for optimality among modal agents. This echoes the impossibility-of-optimality results of Anderlini [2] and Canning [5] on game theory for Turing machines with access to each others' source codes.
All the same, the results on Löbian cooperation represent a formalized version of robust mutual cooperation on the Prisoner's Dilemma, further validating some of the intuitions on "superrationality" and raising new questions on decision theory. The Prisoner's Dilemma with exchange of source code is analogous to Newcomb's problem, and indeed, this work was inspired by some of the philosophical alternatives to causal and evidential decision theory proposed for that problem (see Drescher [6] and Altair [1]).
A brief outline of the structure of this paper: in Section 2, we define our formal framework more explicitly. In Section 3, we introduce FairBot, prove that it achieves mutual cooperation with itself and cannot be exploited (Theorem 3.1); we then introduce Prudent-Bot, and show that it is also unexploitable, cooperates mutually with itself and with FairBot, and defects against CooperateBot.
In Section 4, we develop the theory of modal agents, with a focus on showing that their action against one another is well-defined. We also show that a feature of PrudentBotnamely, that it checks its opponent's response against DefectBot-is essential to its functioning: modal agents which do not use third parties cannot achieve mutual cooperation with FairBot unless they also cooperate with CooperateBot. Then, in Section 5, we discuss several obstacles to proving nontrivial optimality results for modal agents. In Section 6, we will explain our preference for PrudentBot over FairBot, and speculate on some future directions, before closing in Section 7 with a list of open problems in this area.
Agents in Formal Logic
There are two different formalisms which we will bear in mind throughout this paper. The first formalism is that of algorithms, where we can imagine two Turing machines X and Y, each of which is given as input the code for the other, and which have clearly defined outputs corresponding to the options C and D. (It is possible, of course, that one or both may fail to halt, though the algorithms that we will discuss will provably halt on all inputs.) This formalism has the benefit of concreteness: we could actually program such agents, although the ones we shall deal with are often very far from efficient in their requirements. It has the drawback, however, that proofs about algorithms which call upon each other are generally difficult and untidy, relying upon delicate bounds on (e.g.) the length of proofs.
Therefore, we will do our proofs in another framework: that of logical provability in certain formal systems. More specifically, the agents we will be most interested in can be interpreted via modal formulas in Gödel-Löb provability logic, which is especially pleasant to work with. This bait-and-switch is justified by the fact that all of our tools do indeed have equivalently useful bounded versions; variants of Löb's Theorem for bounded proof lengths are well-known among logicians. The interested reader can therefore construct algorithmic versions of all logically defined agents in this paper, and with the right parameters all of our theorems will hold for such agents.
In particular, our "agents" will be formulas in Peano Arithmetic, and our criterion for action will be the existence of a finite proof in the tower of formal systems PA+n, where PA is Peano Arithmetic, and PA+(n+1) is the formal system whose axioms are the axioms of PA+n, plus the axiom that PA+n is consistent, i.e. that ¬ . . . ⊥ with n + 1 copies of .
Fix a particular Gödel numbering scheme, and let X and Y each denote well-formed formulas with one free variable. Then let X(Y ) denote the formula where we replace each instance of the free variable in X with the Gödel number of Y. If such a formula holds in the standard model of Peano Arithmetic, we interpret that as X cooperating with Y; if its negation holds, we interpret that as X defecting against Y. (In particular, we will prove theorems in PA+n to establish whether the agents we discuss cooperate or defect against one another.) Thus we can regard such formulas of arithmetic as decision-theoretic agents, and we will use "source code" to refer to their Gödel numbers.
Remark To maximize readability in the technical sections of this paper, we will use typewriter font for agents, which are formulas of Peano Arithmetic with a single free variable, like X and CooperateBot; we will use sans-serif font for the formal systems PA+n; and we will use italics for logical formulas with no free variables such as X(Y ). Furthermore, we will use [X(Y ) = C] and [X(Y ) = D] interchangeably with X(Y ) and ¬X(Y ).
Of course, it is easy to create X and Y so that X(Y ) is an undecidable statement in all PA+n (e.g. the statement that the formal system PA+ω is consistent). But the philosophical phenomenon we're interested in can be achieved by agents which do not present this problem, and whose finitary versions in fact always return an answer in finite time.
Two agents which are easy to define and clearly decidable are the agent which always cooperates (which we will call CooperateBot, or CB for short) and the agent which always defects (which we will call DefectBot, or DB). In pseudocode:
Input : Source code of the agent X Output: C or D return C ; Note further that PA ⊢ ¬ [DB(X) = C], but that PA+1 ⊢ ¬ [DB(X) = C]; this distinction is essential.
Howard [10], McAfee [15] and Tennenholtz [20] introduced functionally equivalent agent schemas, which we've taken to calling CliqueBot; these agents use quining to recognize self-copies and mutually cooperate, while defecting against any other agent. In pseudocode:
Input : Source code of the agent X Output: C or D if X=CliqueBot then return C; else return D; end Algorithm 3: CliqueBot By the diagonal lemma, there exists a formula of Peano Arithmetic which implements CliqueBot. (The analogous tool for computable functions is Kleene's recursion theorem [12]; in this paper, we informally use "quining" to refer to both of these techniques.)
CliqueBot has the nice property that it never experiences the sucker's payoff in the Prisoner's Dilemma. This is such a clearly important property that we will give it a name:
Definition We say that an agent X is unexploitable if there is no agent Y such that X(Y ) = C and Y (X) = D.
However, CliqueBot has a notable drawback: it can only elicit mutual cooperation from agents that are syntactically identical to itself. (If two CliqueBots were written with different Gödel-numbering schemes, for instance, they would defect against one another!) One might patch this by including a list of source codes (or a schema for them), and cooperate if the opponent matches any of them; one would of course be careful to choose only source codes that would cooperate back with this variant. But this is a brittle form of mutual cooperation, and an opaque one: it takes a predefined circle of mutual cooperators as given. For this reason, it is worth looking for a more flexibly cooperative form of agent, one that can deduce for itself whether another agent is worth cooperating with.
Löbian Cooperation
A deceptively simple-seeming such agent is one we call FairBot. On a philosophical level, it cooperates with any agent that can be proven to cooperate with it. In pseudocode:
Input : Source code of the agent X Output: C or D if PA ⊢ [X(FairBot)= C] then return C; else return D; end
Algorithm 4: FairBot (FB) FairBot references itself in its definition, but as with CliqueBot, this can be done via the diagonal lemma. By inspection, we see that FairBot is unexploitable: presuming that Peano Arithmetic is sound, FairBot will not cooperate with any agent that defects against FairBot.
The interesting question is what happens when FairBot plays against itself: it intuitively seems plausible either that it would mutually cooperate or mutually defect. As it turns out, though, Löb's Theorem guarantees that since the FairBots are each seeking proof of mutual cooperation, they both succeed and indeed cooperate with one another! (This was first shown by Vladimir Slepnev [19].) However, it is a tidy logical accident that the two agents are the same; we will understand better the mechanics of mutual cooperation if we pretend in this case that we have two distinct implementations, FairBot 1 and FairBot 2 , and prove mutual cooperation from their formulas without using the fact that their actions are identical.
Proof of Theorem 3.1 (Real Version):
Let A be the formula "F B 1 (F B 2 ) = C" and B be the formula "F B 2 (F B 1 ) = C". By inspection, PA⊢ A → B and PA⊢ B → A. This sort of "Löbian circle" works out as follows:
PA ⊢ ( A → B) ∧ ( B → A) (see above) PA ⊢ ( A ∧ B) → (A ∧ B) (follows from above) PA ⊢ (A ∧ B) → ( A ∧ B) (tautology) PA ⊢ (A ∧ B) → (A ∧ B) (previous lines) PA ⊢ A ∧ B (Löb's Theorem).
Remark One way to build a finitary version of FairBot is to write an agent FiniteFairBot that looks through all proofs of length ≤ N to see if any are a proof of [X(F initeF airBot) = C], and cooperates iff it finds such a proof. If N is large enough, the bounded version of Löb's Theorem implies the equivalent of Theorem 3.1.
Remark Unlike a CliqueBot, FairBot will find mutual cooperation even with versions of itself that are written in other programming languages. In fact, even the choice of formal system does not have to be identical for two versions of FairBot to achieve mutual cooperation! It is enough that there exist a formal system S in which Löbian statements are true, such that anything provable in S is provable in each of the formal systems used, and such that S can prove the above. (Note in particular that even incompatible formal systems can have this property: a version of FairBot which looks for proofs in the formal system PA+¬Con(PA) will still find mutual cooperation with a FairBot that looks for proofs in PA+1.)
However, FairBot wastes utility by cooperating even with CooperateBot. (See Section 6 for the reasons we take this as a serious issue.) Thus we would like to find a similarly robust agent which cooperates mutually with itself and with FairBot but which defects against CooperateBot.
Consider the agent PrudentBot, defined as follows: Proof. Unexploitability is again immediate from the definition of PrudentBot and the assumption that PA is sound, since cooperation by PrudentBot requires a proof that its opponent cooperates against it.
Input
In Remark It is important that we look for proofs of X(DB) = D in a stronger formal system than we use for proving X(P B) = C; if we do otherwise, the resulting variant of PrudentBot would lose the ability to cooperate with itself. However, it is not necessary that the formal system used for X(DB) = D be stronger by only one step than that used for X(P B) = C; if we use a much higher PA+n there, we broaden the circle of potential cooperators without thereby sacrificing safety.
Modal Agents
It is instructive to consider FairBot and PrudentBot as modal statements in Gödel-Löb provability logic (often denoted GL). Namely, if we consider the actions of ). There are a number of tools, including fixed-point theorems and Kripke semantics, which work for families of such modal statements; and thus we will define a class of modal agents for the purpose of study.
Informally, a modal agent is one whose actions are determined solely by the provability of statements regarding its opponent's actions against itself and against other simpler agents 6 . That is, if X is a modal agent, then there is a modal-logic formula 7 ϕ and a fixed set of simpler modal agents Y 1 , . . . , Y N such that, for any opponent Z,
[X(Z) = C] ↔ ϕ ([Z(X) = C], [Z(Y 1 ) = C], . . . , [Z(Y N ) = C]) .
(4.1)
Furthermore, since a modal agent does all of this via provability, the formula ϕ must be fully modalized : all instances of variables must be contained inside sub-formulas of the form ψ.
We must lay some groundwork (following Lindström [13]) before formally defining the class of modal agents. Write ϕ(p 1 , . . . , p n ) to denote a formula ϕ in the language of GL whose free (propositional) variables are included in the set {p 1 , . . . , p n }. (Note that this is different from Lindström's convention, who doesn't display the free variables.) Proof. This is Theorem 1 of Lindström.
Theorem 4.2 (Modal fixed point theorem).
Suppose that the formula ϕ(p, q 1 , . . . , q n ) in the language of GL is modalized in p. Then there is a modal formula ψ(q 1 , . . . , q n ) such that GL ⊢ ψ(q 1 , . . . , q n ) ↔ ϕ(ψ(q 1 , . . . , q n ), q 1 , . . . , q n ).
Moreover, if ϕ is modalized in q i , then so is ψ.
Proof. Except for the last sentence, this is Theorem 11 of Lindström. The last sentence is obvious from Lindström's proof, since the ψ constructed there differs from ϕ only inside boxes.
Write + ϕ to mean ϕ ∧ ϕ.
GL ⊢ + p ↔ ϕ(p, q 1 , . . . , q n ) ∧ + p ′ ↔ ϕ(p ′ , q 1 , . . . , q n ) → (p ↔ p ′ ).
Proof. This is Theorem 12 of Lindström.
Corollary 4.4 (Uniqueness of arithmetic fixed points).
Suppose that ϕ(p, q 1 , . . . , q n ) is modalized in p, and let ψ, ψ ′ , ψ 1 , . . . , ψ n be closed formulas in the language of PA. If
PA ⊢ ψ ↔ ϕ(ψ, ψ 1 , . . . , ψ n ) and PA ⊢ ψ ′ ↔ ϕ(ψ ′ , ψ 1 , . . . , ψ n ), then PA ⊢ ψ ↔ ψ ′ .
Proof. By applying Theorem 4.1 to the conclusion of Theorem 4.3, we obtain
PA ⊢ + ψ ↔ ϕ(ψ, ψ 1 , . . . , ψ n ) ∧ + ψ ′ ↔ ϕ(ψ ′ , ψ 1 , . . . , ψ n ) → (ψ ↔ ψ ′ ).
But for any formulaφ, if PA ⊢φ then PA ⊢ + (φ), so the conclusion follows.
Lemma 4.5. If ϕ(p 1 , . . . , p n ) is a modal formula and ψ 1 , ψ ′ 1 , . . . , ψ n , ψ ′ n are arithmetic formulas such that PA ⊢ ψ i ↔ ψ ′ i for each i, then PA ⊢ ϕ(ψ 1 , . . . , ψ n ) ↔ ϕ(ψ ′ 1 , . . . , ψ ′ n ). Proof. Lindström's Lemma 8 states that for any modal formulaφ(p, q 1 , . . . , q m ),
GL ⊢ + (p ↔ p ′ ) → φ(p, q 1 , . . . , q m ) ↔φ(p ′ , q 1 , . . . , q m ) .
The desired result is obtained by applying this n times and then appealing to Theorem 4.1.
Definition An agent is a well-formed formula in the language of PA of (at most) one free variable.
If X and Y are agents, we write [X(Y ) = C], or [X(Y )] for short, for the application of X to the Gödel number of Y; we interpret this logical formula as the assertion that X cooperates when playing against Y. Accordingly, we say that X cooperates with Y if [X(Y )] holds in the standard model of PA.
Definition An agent X is called a modal agent of rank k ≥ 0 if there are modal agents Y 1 , . . . , Y n of rank < k and a fully modalized formula ϕ(p, q 1 , . . . , q n ) such that such that for all agents Z,
PA ⊢ [X(Z)] ↔ ϕ([Z(X)], [Z(Y 1 )], . . . , [Z(Y n )]).
CooperateBot, DefectBot, FairBot and PrudentBot are all modal agents, but as we shall see, CliqueBot is not.
We now prove three theorems demonstrating that the notion of "modal agent" has good properties. First, we note that it makes no practical difference if we include proofs about the actions [X(Z)] and [Y i (Z)] in our definition:
Theorem 4.6. Suppose that X is an agent, Y 1 , . . . , Y n are modal agents of rank < k, and ϕ(p, p,q 1 , q 1 , . . . ,q n , q n ) is a fully modalized formula such that for all agents Z,
PA ⊢ [X(Z)] ↔ ϕ([X(Z)], [Z(X)], [Y 1 (Z)], [Z(Y 1 )], . . . , [Y n (Z)], [Z(Y n )]).
Then X is a rank-k modal agent.
Proof. Without loss of generality, assume that if the modal formula for Y i depends on a lower-rank modal agent Y ′ , then Y ′ = Y j for some j. Then for every i, there is a fully modalized formula ϕ i such that for all By Theorem 4.2, there is a fully modalized formula ψ(p, q 1 , . . . , q n ) such that GL proves ψ(p, q 1 , . . . , q n ) ↔ ϕ(ψ(p, q 1 , . . . , q n ), p, q 1 , . . . , q n ). Hence by Theorem 4.1, for all agents Z
Z, PA ⊢ [Y i (Z)] ↔ ϕ i ([Z(Y 1 )],PA ⊢ ψ([Z(X)], [Z(Y 1 )], . . . , [Z(Y n )]) ↔ ϕ ψ([Z(X)], [Z(Y 1 )], . . . , [Z(Y n )]), [Z(X)], [Z(Y 1 )], . . . , [Z(Y n )] .
Since [X(Z)] is also a fixed point of this equation, the conclusion follows by Corollary 4.4.
Next, we show that the actions two modal agents take against each other are described by a fixed point of their modal formulas, as one would expect. In particular, this shows that modal agents' actions against each other depend only on their modal formulas, not on other features of their source code.
Theorem 4.7. If X and Y are modal agents, define a closed modal formula ψ [X(Y )] by recursion on their rank, as follows. Write X i and Y i for the lower-rank agents X and Y depend on, respectively, and write ϕ X (p, q 1 , . . . , q m ) and ϕ Y (p, q 1 , . . . , q n ) for the modal formulas corresponding to X and Y. Now let ψ [X(Y )] be the formula provided by Theorem 4.2 satisfying
GL ⊢ ψ [X(Y )] ↔ ϕ X ϕ Y (ψ [X(Y )] , ψ [X(Y 1 )] , . . . , ψ [X(Yn)] ), ψ [Y (X 1 )] , . . . , ψ [Y (Xm)] .
Then
PA ⊢ [X(Y )] ↔ ψ [X(Y )] .
Proof. By induction, we assume that this already holds for lower ranks. By the definition of modal agent, the induction hypothesis, and Lemma 4.
5, PA shows that [X(Y )] is equivalent to ϕ X [Y (X)], ψ [Y (X 1 )] , . . . , ψ [Y (Xm)]
and that this is in turn equivalent to
ϕ X ϕ Y [X(Y )], ψ [X(Y 1 )] , . . . , ψ [X(Yn)] , ψ [Y (X 1 )] , . . . , ψ [Y (Xm)] .
Thus Proof. Suppose that X is a modal agent and Y and Z are behaviorally equivalent. Write X i for the lower-rank agents X depends on, and ϕ(p, q 1 , . . . , q n ) for the modal formula corresponding to X. Then
PA ⊢ [X(Y )] ↔ ϕ [Y (X)], [Y (X 1 )], . . . , [Y (X n )] and PA ⊢ [X(Z)] ↔ ϕ [Z(X)], [Z(X 1 )], . . . , [Z(X n )] .
But by the behavioral equivalence of Y and Z together with Lemma 4.5, the right-hand sides are equivalent, so PA ⊢ [X(Y )] ↔ [X(Z)] as desired.
Corollary 4.9. CliqueBot is not a modal agent.
Proof. CliqueBot cooperates with itself, but not with a syntactically different but logically (and therefore behaviorally) equivalent variant. Hence, CliqueBot is not a behavioral agent, and by Theorem 4.8 it is not a modal agent.
It feels a bit clunky, in some sense, for the definition of modal agents to include references to other, simpler modal agents. Could we not do just as well with a carefully constructed agent that makes no such outside calls (i.e. a modal agent of rank 0)?
Surprisingly, the answer is no: there is no modal agent of rank 0 that achieves mutual cooperation with FairBot and defects against CooperateBot. In particular:
Obstacles to Optimality
It is worthwhile to ask whether there is some meaningful sense of "optimality" for logical agents or modal agents in particular. For many natural definitions of optimality, this is impossible. For instance, there is no X such that for all Y, the utility achieved by X against Y is the highest achieved by any Z against Y. (To see this, consider Y defined so that Y (Z) = C if and only if Z =X.) More generally, an agent can "punish" or "reward" other agents for any arbitrary feature of their code.
Could we hope that PrudentBot might at least be optimal among modal agents in some meaningful sense? As it happens, there are at least three different kinds of impediments to optimality among modal agents, which together make it very difficult to formulate any nontrivial and non-vacuous definition of optimality.
Most directly, for any modal agents X and Y, either their outputs are identical on all modal agents, or there exists a modal agent Z which cooperates against X but defects against Y. (For an enlightening example, consider the agent TrollBot which cooperates with X if and only if PA ⊢ X(DB) = C.) Thus any nontrivial and nonvacuous concept of optimality must be weaker than total dominance, and in particular it must accept that third parties could seek to "punish" an agent for succeeding in a particular matchup.
Another issue is illustrated by the following agent:
Input : Source code of the agent X Output: C or D if PA ⊢ X(FairBot)= C then return C; else return D; end Algorithm 6: JustBot (JB) That is, JustBot cooperates with X if and only if X cooperates with FairBot. (Note that JustBot has different source code from FairBot; in particular, it can use a hard-cooded reference to FairBot's code, where FairBot must use a quine.) Clearly, JustBot is exploitable by some algorithm (in particular, consider the non-modal algorithm which cooperates only with the corresponding FairBot and with nothing else), but since it is behaviorally equivalent to FairBot, by Theorem 4.8 it cannot be exploited by any modal agent.
A third problem is that a modal agent has a finite amount of predictive power, and it can fail to act optimally against sufficiently complicated or slow-moving opponents. Explicitly, consider the family of modal agents WaitFairBot K , defined by
[W aitF airBot K (X) = C] ↔ ((¬ K+1 ⊥) ∧ (¬ K ⊥ → [X(W aitF airBot K ) = C])).
As it happens 8 , any modal agent which defects against DefectBot will fail to achieve mutual cooperation with WaitFairBot K for all sufficiently large K.
Despite these reasons for pessimism, we have not actually ruled out the existence of a nontrivial and non-vacuous optimality criterion which corresponds to our philosophical intuitions about "correct" decisions. Additionally, there are a number of ways to depart only mildly from the modal framework (such as allowing quantifiers over agents), and these could invalidate some of the above obstacles.
Philosophical Digressions
One might ask (on a philosophical level) why we object to FairBot in the first place; isn't it a feature, not a bug, that this agent offers up cooperation even to agents that blindly trust it? We suggest that it is too tempting to anthropomorphize agents in this context, and that many problems which can be interpreted as playing a Prisoner's Dilemma against a 8 We originally tried to include a proof of this claim, but it ballooned this section out of proportion. The interested reader should start by proving the folk theorem that Kripke frames for GL correspond to ordinal chains, and then show that for any modal agent X which defects against DefectBot, there is a number n such that X(DB) = D holds in every world above height n within a Kripke frame, and then induct on agents and subformulas of agents to show that X(W F B K ) = D for any K > n.
CooperateBot are situations in which one would not hesitate to "defect" in real life without qualms.
For instance, consider the following situation: You've come down with the common cold, and must decide whether to go out in public. If it were up to you, you'd stay at home and not infect anyone else. But it occurs to you that the cold virus has a "choice" as well: it could mutate and make you so sick that you'd have to go to the hospital, where it would have a small chance of causing a full outbreak! Fortunately, you know that cold viruses are highly unlikely to do this. 9 If you map out the payoffs, however, you find that you are in a Prisoner's Dilemma with the cold virus, and that it plays the part of a CooperateBot. Are you therefore inclined to "cooperate" and infect your friends in order to repay the cold virus for not making you sicker?
The example is artificial and obviously facetious, but not entirely spurious. The world does not come with conveniently labeled "agents"; entities on scales at least from viruses to governments show signs of goal-directed behavior. Given a sufficiently broad counterfactual, almost any of these could be defined as a CooperateBot on a suitable Prisoner's Dilemma. And most of us feel no compunction about optimizing our human lives without worrying about the flourishing of cold viruses. 10 In a certain sense, PrudentBot is actually "good enough" among modal agents that one might expect to encounter: there are bound to be agents (CooperateBot and DefectBot) whose action fails to depend in any sense upon predictions of other agents' behavior, and other agents (FairBot, PrudentBot, etc) whose action depends meaningfully on such predictions. One should not expect to encounter a TrollBot or JustBot arising naturally! But it's worth pondering if this reasoning can be made formal in any elegant way.
Do these results imply that sufficiently intelligent and rational agents will reach mutual cooperation in one-shot Prisoner's Dilemmas? In a word, no, not yet. 11 Many things about this setup are notably artificial, most prominently the perfectly reliable exchange of source code (and after that, the intractably long computations that might perhaps be necessary for even the finitary versions).
Nor does this have direct implications among human beings; our abilities to read each other psychologically, while occasionally quite impressive, bear only the slightest analogy to the extremely artificial setup of modal agents. Governments and corporations may be closer analogues to our agents (and indeed, game theory has been applied much more successfully on that scale than on the human scale), but the authors would not consider the application of these results to such organizations to be straightforward, either. The theorems herein are not a demonstration that a more advanced approach to decision theory (i.e. one which does not fail on what we consider to be common-sense problems) is practical, only a demonstration that it is possible.
Open Problems
In particular, here are some open problems we have come across in this area:
• Is there a natural, nonvacuous, and nontrivial definition of optimality among modal agents?
• Are there tractable ways of studying agents which can incorporate quantifiers as well as modal operators? For example, we might consider the non-modal agent X such that X cooperates with Y iff some formal system proves both that Y is unexploitable (given the consistency of some other formal system) and that Y cooperates with X.
• What different dynamics arise when we consider the analogues of modal agents in more complicated games than the Prisoner's Dilemma? In particular, there are issues raised by games with more than one "superrational equilibrium", like the Coordination Game. The natural analogues of FairBot and PrudentBot transform any finite game between themselves into a coordination or bargaining game, but do not provide insight on how to resolve those sorts of conflicts.
• What happens if we apply probabilistic reasoning rather than provability logic? As this allows for mixed strategies, it introduces all of the complexities of bargaining games, as well as new ones.
• What differs in games with more than two players; in particular, what might coordination and bargaining among coalitions look like? It is easier to imagine two agents with each others' source code agreeing to cooperate in a Prisoner's Dilemma than it is to imagine three agents with each others' source code agreeing on how to subdivide a fixed prize (which they lose if they do not have a majority agreeing on an acceptable split).
MIRI's hospitality and support throughout the workshop.
Patrick LaVictoire was partially supported by NSF Grant DMS-1201314 while working on this project.
Algorithm 1 :
1CooperateBot (CB) Input : Source code of the agent X Output: C or D return D; Algorithm 2: DefectBot (DB) Remark In the Peano Arithmetic formalism, CooperateBot can be represented by a formula that is a tautology for every input, while DefectBot can be represented by the negation of such a formula. For any X, then, PA ⊢ [CB(X) = C] and PA ⊢ [DB(X) = D].
Theorem 3. 1 .
1PA ⊢ [FairBot(FairBot)= C]. Proof (Simple Version): By inspection of FairBot, PA⊢ ( [F B(F B) = C]) → [F B(F B) = C]. By Löb's Theorem, Peano Arithmetic does indeed prove that FairBot(FairBot)=C.
particular, PA+1 ⊢ [P B(DB) = D] (since PA ⊢ [DB(P B) = D], PA+1 ⊢ ¬ [DB(P B) = C]). It is likewise clear that PA+2 ⊢ [P B(CB) = D]. Now since PA+1 ⊢ [F B(DB) = D] and therefore PA ⊢ (¬ ⊥ → [F B(DB) = D]), we again have the Löbian cycle where PA ⊢ [P B(F B) = C] ↔ [F B(P B) = C], and of course vice versa; thus PrudentBot and FairBot mutually cooperate. And as we have established PA+1 ⊢ [P B(DB) = D], we have the same Löbian cycle for PrudentBot and itself.
FairBot and any other agent X against one another, then the definition of FairBot is simply [F B(X) = C] ↔ [X(F B) = C], and the definition of PrudentBot is [P B(X)] ↔ ( [X(P B)] ∧ (¬ ⊥ → ¬[X(DB)])
Theorem 4. 1 (
1Arithmetic soundness of GL). Suppose that GL ⊢ ϕ(p 1 , . . . , p n ), and that ψ 1 , . . . , ψ n are closed formulas in the language of PA. Then PA ⊢ ϕ(ψ 1 , . . . , ψ n ).
Theorem 4. 3 (
3Uniqueness of modal fixed points). Suppose that ϕ(p, q 1 , . . . , q n ) is modalized in p. Then
. . . , [Z(Y n )]). Thus, by Lemma 4.5, we may assume that ϕ is of the form ϕ([X(Z)], [Z(X)], [Z(Y 1 )], . . . , [Z(Y n )]). It remains to eliminate the dependency on [X(Z)].
,
PA shows that both [X(Y )] and ψ [X(Y )] are fixed points of the same formula, and hence that [X(Y )] ↔ ψ [X(Y )] by Corollary 4.4. Finally, we show that modal agents' actions depend only on their opponents' behavior, not on other features of their source code. Definition Two agents X and Y are called behaviorally equivalent if for every agent Z, PA proves [X(Z)] ↔ [Y (Z)]. A behavioral agent is an agent X such that for any pair of behaviorally equivalent agents Y and Z, PA proves [X(Y )] ↔ [X(Z)].
Theorem 4 . 8 .
48Modal agents are behavioral.
Theorem 4 . 10 .
410Any modal agent X of rank 0 such that PA ⊢ [X(F B) = C] must also have PA ⊢ [X(CB) = C]. Proof. Writing ϕ(·) for the modal formula defining X, by Lemma 4.5 we see that PA proves [X(CB) = C] ↔ ϕ(⊤) and [X(F B) = C] ↔ ϕ( [X(F B) = C]); since by assumption it also proves [X(F B) = C], we have PA ⊢ [X(F B) = C] and hence PA ⊢ [X(F B) = C] ↔ ⊤, so with another application of Lemma 4.5 we obtain PA ⊢ ϕ( [X(F B) = C]) ↔ ϕ(⊤), whence PA ⊢ [X(F B) = C] ↔ [X(CB) = C] and finally PA ⊢ [X(CB) = C].
Theorem 3.2. PrudentBot is unexploitable, mutually cooperates with itself and with FairBot, and defects against CooperateBot.: Source code of the agent X
Output: C or D
if PA ⊢ [X(PrudentBot)=C] and PA+1 ⊢ [X(DefectBot)=D] then
return C;
end
return D;
Algorithm 5: PrudentBot (PB)
Binmore's analysis, however, eschews cooperation in the Prisoner's Dilemma as irrational! 2 One can consider the usual one-shot strategies (always cooperate, always defect) as Turing machines that return the same output regardless of the given input; we denote these algorithms as CooperateBot and DefectBot in order to distinguish them from the outputs Cooperate and Defect.
As we shall see, the symmetry between mutual cooperation and mutual defection is broken by the positive criterion for action.4 This result was proved by Vladimir Slepnev in an unpublished draft[19], and the proof is reproduced later in this paper with his permission.
For a philosophical discussion of why this is the obviously correct response to CooperateBot, see Section 6.
For instance, PrudentBot tries to prove that its opponent defects against DefectBot. As for the requirement that these secondary agents be "simpler", we want to avoid the possibility of infinite regress using something like Russell's theory of types.7 That is, a formula built from , ⊤, the logical operators ∧, ∨, →, ↔ and ¬, and the input variables.
Incidentally, the reason that cold viruses do not pursue this strategy is that, throughout human history and most of the world today, making the host sicker gives the virus fewer, not more, chances to spread.10 Note that it would, in fact, be different if a virus were intelligent enough to predict the macroscopic behavior of their host and base their mutations on that! In such a case, one might well negotiate with the virus. Alternatively, if one's concern for the well-being of viruses reached a comparable level to one's concern for healthy friends, that would change the payoff matrix so that it was no longer a Prisoner's Dilemma. But both of these considerations are far more applicable to human beings than to viruses.11 However, to borrow what Randall Munroe said about correlation and causation, this form of program equilibrium does waggle its eyebrows suggestively and gesture furtively (toward cooperation in the Prisoner's Dilemma) while mouthing 'look over there'.
AcknowledgmentsThis project was developed at a research workshop held by the Machine Intelligence Research Institute (MIRI) in Berkeley, California, in April 2013. The authors gratefully acknowledge Thanks to everyone who has commented on various partial results and drafts, in particular Alex Altair, Stuart Armstrong, Andrew Critch, Wei Dai, Daniel Dewey, Gary Drescher, Kenny Easwaran, Cameron Freer, Bill Hibbard, Vladimir Nesov, Vladimir Slepnev, Jacob Steinhardt, Nisan Stiennon, Jacob Taylor, and Qiaochu Yuan. Further thanks to readers on the blog LessWrong for their comments on a preprint of this article.
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| []
|
[
"Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems",
"Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems"
]
| [
"Getachew K Befekadu [email protected] \nDepartment of Electrical Engineering\nDepartment of Electrical Engineering\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n",
"Panos J Antsaklis [email protected] \nDepartment of Electrical Engineering\nDepartment of Electrical Engineering\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n",
"P J Antsaklis \nUniversity of Notre Dame\nNotre Dame46556INUSA\n"
]
| [
"Department of Electrical Engineering\nDepartment of Electrical Engineering\nUniversity of Notre Dame\nNotre Dame\n46556INUSA",
"Department of Electrical Engineering\nDepartment of Electrical Engineering\nUniversity of Notre Dame\nNotre Dame\n46556INUSA",
"University of Notre Dame\nNotre Dame46556INUSA"
]
| []
| In this paper, we first draw a connection between the existence of a stationary density function (which corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that strategically interacts in a game-theoretic framework. In particular, we show that there exists a set of (game-theoretic) equilibrium feedback operators such that the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, by a means of a stationary density function (i.e., a common fixed-point) for a family of Frobenius-Perron operators, how the dynamics of the system together with the equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a small random perturbation in the system.Index TermsEquilibrium feedback operators; Frobenius-Perron operators; game theory; maximum entropy; multichannel system; resilient behavior; small random perturbation. | null | [
"https://arxiv.org/pdf/1312.5168v3.pdf"
]
| 117,490,291 | 1312.5168 | 905cbcc933f15d18b58e990fe25311fadc331bec |
Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems
14 Jan 2014
Getachew K Befekadu [email protected]
Department of Electrical Engineering
Department of Electrical Engineering
University of Notre Dame
Notre Dame
46556INUSA
Panos J Antsaklis [email protected]
Department of Electrical Engineering
Department of Electrical Engineering
University of Notre Dame
Notre Dame
46556INUSA
P J Antsaklis
University of Notre Dame
Notre Dame46556INUSA
Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems
14 Jan 20141 The first author acknowledges support from the College of Engineering, University of Notre Dame. G. K. Befekadu is with the Version -December 19, 2013. 2
In this paper, we first draw a connection between the existence of a stationary density function (which corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that strategically interacts in a game-theoretic framework. In particular, we show that there exists a set of (game-theoretic) equilibrium feedback operators such that the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, by a means of a stationary density function (i.e., a common fixed-point) for a family of Frobenius-Perron operators, how the dynamics of the system together with the equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a small random perturbation in the system.Index TermsEquilibrium feedback operators; Frobenius-Perron operators; game theory; maximum entropy; multichannel system; resilient behavior; small random perturbation.
I. INTRODUCTION
The main purpose of this paper is to draw a connection between the existence of a stationary density (that corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that interacts strategically in a game-theoretic framework. We first specify a game in a strategic form over an infinite-horizon -where, in the course of the game, each feedback operator generates automatically a feedback control in response to the action of other feedback operators through the system (i.e., using the current state-information of the system) and, similarly, any number of feedback operators can decide on to play their feedback strategies simultaneously. However, each of these feedback operators are expected to respond in some sense of best-response correspondence to the strategies of the other feedback operators in the system. In such a scenario, it is well known that the notion of (game-theoretic) equilibrium will offer a suitable framework to study or characterize the robust property of all equilibrium solutions under a family of information structures -since no one can improve his payoff by deviating unilaterally from this strategy once the equilibrium strategy is attained (e.g., see [1], [26] or [28] on the notions of optimums and strategic equilibria in games). 1 In view of the above arguments, we present in this paper an extension of game-theoretic formalism for multi-channel systems that tend to move towards an equilibrium or "maximum entropy" state in the sense of statistical mechanics (e.g., see Lanford [23, pp 77-95] or Ruelle [29]) -when the criterion is to minimize the relative entropy between any two density functions, for large-time, with respect to the control channels or the class of admissible control functions (i.e., the set of feedback operators). This further allows us to establish a connection between the existence of a stationary density function (which corresponds to a unique equilibrium state) and a set of feedback operators that strategically interacts in the system.
Moreover, based on a common fixed-point for a family of Frobenius-Perron operators, we provide a sufficient condition on the existence of a set of (game-theoretic) equilibrium feedback operators such that when the composition of the multi-channel system with this set of equilibrium feedback operators, described by density functions, evolves towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, how the dynamics of the system 1 In this paper, we consider this set of feedback operators as noncooperative agents (or players), but fully-rational entities, over an infinite-horizon, in a game-theoretic sense. Further, at each instant-time, each feedback operator knows that the others will look for feedback strategies, but they are not necessarily informed about each others strategies. together with these equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a random perturbation in the system. We, in particular, establish sufficient conditions, based on the convergence of invariant measures (i.e., stochastic stability -in the sense of deterministic limit (e.g., see [35], [2], [20] or [12] for related discussions), that will guarantee the resilient behavior for the set of equilibrium feedback operators with respect to random perturbations in the system.
Here, we hasten to add that such a study, which involves evidence of systems exhibiting resilient behavior, would undoubtedly provide a better understanding of reliability or prescribing an optimal (sub-optimal) degree of redundancy in decentralized control systems. Finally, in the information theoretic-games, we also note that the notions of entropy, game-theoretical equilibrium and complexity, based on the Maximum Entropy Principle (MaxEnt) of Jaynes [17], [18] and the Idivergence metric of Csiszár [6], [7], have been investigated in the context of zero-sum games by Topsøe (e.g., see [30] or [31]), Grünwald and Dawid [14] and, similarly, by Haussler [16]. Moreover, we observe that the notion of entropy (and its variants) has been well discussed in systems theory literature in the context of robustness analysis and/or synthesis for systems with uncertainties (e.g., see [27] and [4]).
The remainder of the paper is organized as follows. In Section II, we recall the necessary background and present some preliminary results that are relevant to our paper. Section III introduces a family of mappings for multi-channel systems that will be used for our main results. In Section IV we present our main results -where we establish a three way connection between the existence of an equilibrium state (i.e., the maximum entropy in the sense of statistical mechanics), a common stationary density function for the family of Frobenius-Perron operators, and a set of (game-theoretic) equilibrium feedback operators. This section also discusses an extension of the resilient behavior (to these equilibrium feedback operators), when there is a small random perturbation in the system.
II. BACKGROUND, DEFINITIONS, AND NOTATIONS
In the following, we provide the necessary background and recall some known results from measure theory that will be useful in the sequel. The results are standard (and will be stated without proof); and they can be found in standard graduate books (e.g., see [15], [13] and [34] on the measure theory; and see also [24] or [20] on the stochastic aspects of dynamical systems).
Definition 1: Let (X, A , µ) be a measure space and L 1 (X, A , µ) be the space of all possible real-valued measurable functions ϑ : X → R satisfying X |ϑ(x)|µ(dx) < ∞.
(1)
If S : X → X is a measurable nonsingular transformation, i.e., µ(S −1 (A)) = 0 for all A ∈ A such that µ(A) = 0, then the operator P :
L 1 (X, A , µ) → L 1 (X, A , µ) defined by A P ϑ(x)µ(dx) = S −1 (A) ϑ(x)µ(dx), ∀A ∈ A ,(2)
is called the Frobenius-Perron operator with respect to S.
Definition 2: Let (X, A , µ) be a measure space. Define D(X, A , µ) = ϑ(x) ∈ L 1 (X, A , µ) ϑ(x) ≥ 0 and ϑ(x) L 1 (X,A ,µ) = 1 .(3)
Then, any continuous function ϑ(x) ∈ D(X, A , µ) is called a density function.
Definition 3: Let (X, A , µ) be a measure space. If S : X → X is a nonsingular transformation and
ζ(x) ∈ L ∞ (X, A , µ). Then, the operator U : L ∞ (X, A , µ) → L ∞ (X, A , µ) defined by U ζ(x) = ζ S(x) ,(4)
is called the Koopman operator with respect to S.
Note that for every ζ(x) ∈ L ∞ (X, A , µ)
U ζ(x) L ∞ (X,A ,µ) ≤ ζ(x) L ∞ (X,A ,µ) .(5)
Moreover, for every ϑ(x) ∈ L 1 (X, A , µ) and ζ(x) ∈ L ∞ (X, A , µ), then we have
P ϑ, ζ = ϑ, U ζ ,(6)
so that the operator U is an adjoint to the Frobenius-Perron operator P . 2
Remark 1:
The transformation S is said to be measure preserving if µ S −1 (A) = µ A for all A ∈ A .
Note that the property of measure preserving depends both on S and µ.
Definition 4: Let ϑ(x) ∈ L 1 (X, A , µ) and ϑ(x) ≥ 0. If the measure
µ ϑ (A) = A ϑ(x)µ dx ,(7)
2 P ϑ, ζ X P ϑ(x)ζ(x)µ(dx).
is absolutely continuous with respect to the measure µ, then ϑ(x) is called the Radon-Nikodym derivative of µ ϑ with respect to µ.
Theorem 1: Let (X, A , µ) be a measure space, S : X → X be a nonsingular transformation, and let P be the Frobenius-Perron operator with respect to S. Consider a nonnegative function ϑ(x) ∈ L 1 (X, A , µ),
i.e., ϑ(x) > 0, ∀x ∈ X. Then, a measure µ ϑ given by
µ ϑ (A) = A ϑ(x)µ(dx), ∀A ∈ A ,(8)
is invariant, if and only if, ϑ(x) is a stationary density function (i.e., a fixed-point) of P .
Theorem 2:
Let (X, A , µ) be a measure space and S : X → X be a nonsingular transformation. S is ergodic, if and only if, for every measurable function ϑ :
X → R ϑ(S(x)) = ϑ(x),(9)
for almost all x ∈ X, implies that ϑ(x) is constant almost everywhere.
Definition 5: Convergence of sequences of functions (e.g., see [21] or [10]).
(i) A sequence of functions ϑ n (x) , ϑ n (x) ∈ L 1 (X, A , µ), is weakly Cesàro convergent to ϑ(x) ∈ L 1 (X, A , µ) if lim n→∞ 1 n n k=1 ϑ n , ζ = ϑ, ζ , ∀ζ ∈ L ∞ (X, A , µ).(10)
(ii) A sequence of functions ϑ n (x) , ϑ n (x) ∈ L 1 (X, A , µ), is weakly convergent to ϑ(x) ∈ L 1 (X, A , µ)
if lim n→∞ ϑ n , ζ = ϑ, ζ , ∀ζ ∈ L ∞ (X, A , µ).
(iii) A sequence of functions ϑ n (x) , ϑ n (x) ∈ L 1 (X, A , µ), is strongly convergent to ϑ(x) ∈ L 1 (X, A , µ) if lim n→∞ ϑ n (x) − ϑ(x) L 1 (X,A ,µ) = 0.(11)
Theorem 3 (Chebyshev's inequality): Let (X, A , µ) be a measure space and let V : X → R + be an arbitrary nonnegative measurable function. Define
E V (x) ϑ(x) = X V (x)ϑ(x)µ(dx), ∀ϑ(x) ∈ D(X, A , µ).(13)If G α = x ∈ X V (x) < α , then Gα V (x)µ(dx) = 1 − E V (x) ϑ(x) .(14)
Theorem 4: Let (X, A , µ) be a finite measure space (i.e., µ(X) < ∞) and let S : X → X be a measure preserving and ergodic. Then, for any integrable function ϑ * (x), the average of ϑ(x) along the trajectory of S is equal to almost everywhere to the average of ϑ(x) over the space X, i.e., 3
lim n→∞ 1 n n−1 k=1 ϑ(S k (x)) = 1 µ(X) X ϑ(x)µ(dx).(15)
Corollary 1: Let (X, A , µ) be a finite measure space and S : X → X be measure preserving and ergodic. Then, for any set A ∈ A , µ(A) > 0, and for almost all x ∈ X, the fraction of the points
S k (x) in A ∈ A as k → ∞ is given by µ(A)/µ(X).
Corollary 2:
Let (X, A , µ) be a normalized measure space (i.e., µ(X) = 1) and let S : X → X be measure preserving. Suppose that P is the Frobenius-Perron with respect to S. Then, S is ergodic if and
only if lim n→∞ 1 n n−1 k=1 P n ϑ(x) = 1,(16)
for every ϑ(x) ∈ D(X, A , µ).
Theorem 5:
Let (X, A , µ) be a measure space, S : X → X be a nonsingular transformation, and let P be the Frobenius-Perron operator with respect to S. If S is ergodic, then there exist at most one stationary density function ϑ * (x) of P (i.e., a fixed-point of P ϑ * (x) = ϑ * (x)). Furthermore, if there is a unique stationary density function ϑ * (x) of P and ϑ * (x) > 0 almost everywhere, then S is ergodic.
Corollary 3:
Let (X, A , µ) be a measure space, S : X → X be a nonsingular transformation, and let
P : L 1 (X, A , µ) → L 1 (X, A , µ)
be the Frobenius-Perron operator P with respect to S. If, for some 3 Theorem (Birkhoff's individual ergodic theorem) Let (X, A , µ) be a measure space, S : X → X be a measurable transformation, and let ϑ : X → R be an integrable function. If the measure µ is invariant, then there exists an integrable
function ϑ * (x) such that ϑ * (x) = lim n→∞ 1 n n−1 k=1 ϑ(S k (x)),
for almost all x ∈ X.
ϑ(x) ∈ D(X, A , µ), there is an ℓ(x) ∈ D(X, A , µ) such that P n ϑ(x) ≤ ℓ(x), ∀n ≥ 0.(17)
Then, there is a stationary density function ϑ * (x) ∈ D(X, A , µ) such that P ϑ * (x) = ϑ * (x).
Definition 6: If ϑ(x) ∈ D(X, A , µ), then the entropy of ϑ(x) is defined by H(ϑ(x)) = − X ϑ(x) ln ϑ(x)µ(dx).(18)
Remark 2: The following integral inequality (which is useful for verifying the extreme properties of
H(ϑ(x))) holds for any ϑ(x), ξ(x) ∈ D(X, A , µ) − X ϑ(x) ln ϑ(x)µ(dx) ≤ − X ϑ(x) ln ξ(x)µ(dx).(19)
Note that, in general, we have the following Gibbs inequality
ϑ(x) − ϑ(x) ln ϑ(x) ≤ ξ(x) − ϑ(x) ln ξ(x),
for any two nonnegative measurable functions ϑ(x), ξ(x) ∈ L 1 (X, A , µ).
ϑ 0 (x) = 1/µ(X),(20)
and for any other density function ϑ(x), the entropy is strictly less than H(ϑ 0 (x)), i.e.,
− X ϑ(x) ln ϑ(x)µ(dx) ≤ − ln 1 µ(X)
.
Proof: Take any ϑ(x) ∈ D(X, A , µ). Then, the entropy of ϑ(x) is given by
H(ϑ(x)) = − X ϑ(x) ln ϑ(x)µ(dx).
Using Equation (19), we have the following 8 and the equality is satisfied only, when ϑ(x) = ϑ 0 (x). Note that the entropy for ϑ 0 (x) is also given by
H(ϑ(x)) ≤ − X ϑ(x) ln ϑ 0 (x)µ(dx), = − ln 1 µ(X)H(ϑ 0 (x)) = − X 1 µ(X) ln 1 µ(X) µ(dx), = − ln 1 µ(X)
.
Hence, H(ϑ(x)) ≤ H(ϑ 0 (x)) for all ϑ(x) ∈ D(X, A , µ). ✷ Definition 7: Let ϑ(x), ξ(x) ∈ L 1 (X, A , µ) be two nonnegative measurable functions such that supp ξ(x) ⊂ supp ϑ(x). Then, the relative entropy of ξ(x) with respect to ϑ(x) is defined by 4 H r (ξ(x) | ϑ(x)) = X ξ(x) ln ξ(x) ϑ(x) µ(dx), = X ξ(x) ln ξ(x) − ξ(x) ln ϑ(x) µ(dx).(21)
Remark 4: Note that the relative entropy H r (ξ(x) | ϑ(x)), which measures the deviation of ξ(x) from the density function ϑ(x), has the following properties.
(i) If ξ(x), ϑ(x) ∈ D(X, A , µ), then H r (ξ(x) | ϑ(x)) ≥ 0 and H r (ξ(x) | ϑ(x)) = 0 if and only if ξ(x) = ϑ(x).
(ii) If ϑ(x) is constant density and ϑ(x) = 1, then H r (ξ(x) | 1) = H(ξ(x)). Thus, the relative entropy is a generalization of entropy. Voigt (1981)]) Suppose that P is a Markov operator, then
Remark 5: For any ϑ(x) ∈ L 1 (X, A , µ), the support of ϑ(x) is defined by supp ϑ(x) = x ∈ X | ϑ(x) = 0 . 4 Lemma ([33,Hr(P n ξ(x) | P n ϑ(x)) ≥ Hr(ξ(x) | ϑ(x)), ∀ϑ(x) ∈ D(X, A , µ),
for any nonnegative measurable function ξ(x).
Remark 3: Notice that any linear operator P :
L 1 (X, A , µ) → L 1 (X, A , µ) satisfying (i) P ϑ(x) ≥ 0 and (ii) P ϑ(x) L 1 (X,A ,µ) = ϑ(x) L 1 (X,A ,µ)
for any nonnegative measurable function ϑ(x) ∈ L 1 (X, A , µ) is called a Markov operator.
III. A FAMILY OF MAPPINGS FOR MULTI-CHANNEL SYSTEMS
Consider the following continuous-time multi-channel systeṁ
x(t) = A(t)x(t) + j∈N B j (t)u j (t), x(t 0 ) = x 0 , t ∈ [t 0 , +∞),(22)
where A(·) ∈ R d×d , B j (·) ∈ R d×rj , x(t) ∈ X is the state of the system, u j (t) ∈ U j is the control input to the jth -channel and N {1, 2, . . . , N } represents the set of control input channels (or the set of feedback operators) in the system.
Moreover, we consider the following class of admissible control strategies that will be useful in Section IV (i.e., in a game-theoretic formalism)
U L ⊆ u(t) ∈ j∈N L 2 (R + , R rj ) ∩ L ∞ (R + , R rj ) Uj ,(23)
where u(t) is given by
u(t) = u 1 (t), u 2 (t), . . . , u N (t) .
In what follows, suppose there exists a set of feedback operators L * 1 , L * 2 , . . . , L * N from a class of linear operators L :
X → U L (i.e., (L j x)(t) ∈ U j for j ∈ N ) with strategies L * j x (t) ∈ U j for t ≥ t 0 and for j ∈ N . Further, let φ j t; t 0 , x 0 , u j (t), u * ¬j (t) ∈ X be the unique solution of the jth -subsysteṁ x j (t) = A(t) + i∈N¬j B i (t)L * i (t) x j (t) + B j (t)u j (t),(24)
with an initial condition x 0 ∈ X and control inputs given by
u j (t), u * ¬j (t) u * 1 (t), u * 2 (t), . . . , u * j−1 (t), u j (t), u * j+1 (t), . . . , u * N (t) ∈ U L ,(25)where u * i (t) = L * i (t)x j (t) for i ∈ N ¬j N \{j} and j ∈ N .
Furthermore, we may require that the control input for the jth -channel to be u j (t) = L j x j (t) ∈ U j and with this set of linear feedback operators
L * 1 , . . . , L * j−1 , L j , L * j+1 , . . . , L * N Lj,L * ¬j ∈ L .
Then, the unique solution φ j t; t 0 , x 0 , u j (t), u * ¬j (t) will take the form
φ j t; t 0 , x 0 , u j (t), u * ¬j (t) = Φ L * ¬j (t, t 0 ) Φ Lj (t, t 0 ) Φ (L j ,L * ¬j ) t x 0 , ∀t ∈ [t 0 , +∞),(26)
where
∂Φ L * ¬j (t, τ ) ∂t = A(t) + i∈N¬j B i (t)L * i (t) Φ L * ¬j (t, τ ),(27)∂Φ Lj (t, τ ) ∂t = B * j (t)Φ Lj (t, τ ),(28)
with both Φ L * ¬j (τ, τ ) and Φ Lj (τ, τ ) are identity matrices; and B * (t) is given by
B * j (t) = Φ L * ¬j (t, τ ) −1 B j (t)L j (t)Φ L * ¬j (t, τ ),(29)
for each j ∈ N . 5
In the following, we assume that X is a topological Hausdorff space and A is a σ -algebra of Borel set,
i.e., the smallest σ -algebra which contains all open, and thus closed, subsets of X. With this, for any t ≥ 0 (assuming that t 0 = 0) and (L j , L * ¬j ) ∈ L , we can consider a family of continuous mappings (or
transformations) Φ (Lj,L * ¬j ) t t≥0 on X satisfying (R + × X) ∋ (t, x) → Φ (Lj,L * ¬j ) t x ∈ X.(30)
Note that, for each fixed t ≥ 0, the transformation Φ
(Lj,L * ¬j ) t is measurable, i.e., we have Φ (Lj,L * ¬j ) t −1 (A) ∈ A for all A ∈ A, where Φ (Lj,L * ¬j ) t −1 (A) denotes the set of all x such that Φ (Lj,L * ¬j ) t
x ∈ A (e.g., see [25] for invariant measures on topological spaces; see also [3] for topological properties of measure spaces).
Then, we can introduce the following definitions. 5 Note that, with t0 = τ , if we take the partial derivative of Φ (L j , L * ¬j ) t with respect to t and make use of Equations (6) and (7) together with Equation (8), then we have
∂ ∂t Φ (L j , L * ¬j ) t = ∂ ∂t Φ L * ¬j (t, τ ) Φ L j (t, τ ) + Φ L * ¬j (t, τ ) ∂ ∂t Φ L j (t, τ )) , = A(t) + i∈N ¬j Bi(t)L * i (t) Φ L * ¬j (t, τ ) Φ L j (t, τ ) + Φ L * ¬j (t, τ ) Φ L * ¬j (t, τ ) −1 Bj (t)Lj(t) Φ L * ¬j (t, τ ) Φ L j (t, τ )), = A(t) + i∈N ¬j Bi(t)L * i (t) + Bj (t)Lj(t) Φ (L j , L * ¬j ) t .
Moreover, we note that Φ L * ¬j (t, τ ) satisfies the following
Φ L * ¬j (t2, t1) Φ L * ¬j (t1, τ ) = Φ L * ¬j (t2, τ ), ∀t1, t2 ∈ [τ, +∞),
(e.g., see [5] for such a decomposition that arises in differential equations).
Definition 8: A measure µ is called invariant under the family of measurable transformations
Φ (Lj,L * ¬j ) t t≥0 if µ Φ (Lj,L * ¬j ) t −1 (A) = µ A , ∀A ∈ A.(31)
Next, we assume that the family of transformations Φ (Lj,L * ¬j ) t t≥0
are nonsingular and, for each fixed t ≥ 0, the unique Frobenius-Perron operator P
(Lj,L * ¬j ) t : L 1 (X, A , µ) → L 1 (X, A , µ) is then defined by A P (Lj,L * ¬j ) t ϑ(x)µ(dx) = Φ (L j ,L * ¬j ) t −1 (A) ϑ(x)µ(dx), ∀A ∈ A .(32)
Definition 9: Let (X, A , µ) be a measure space, then the family of operators P (Lj,L * ¬j ) t t≥0
, ∀(L j , L * ¬j ) ∈ L , ∀t ≥ 0 and ∀j ∈ N , satisfies the following properties.
(P1) For all ϑ 1 (x), ϑ 2 (x) ∈ L 1 (X, A , µ) and λ 1 ,
λ 2 ∈ R P (Lj,L * ¬j ) t λ 1 ϑ 1 (x) + λ 2 ϑ 2 (x) = λ 1 P (Lj,L * ¬j ) t ϑ 1 (x) + λ 2 P (Lj,L * ¬j ) t ϑ 2 (x). (P2) P (Lj,L * ¬j ) t ϑ(x) ≥ 0 if ϑ(x) ≥ 0. (P3) For all ϑ(x) ∈ L 1 (X, A , µ), X P (L * j ,L * ¬j ) t ϑ(x)µ(dx) = X ϑ(x)µ(dx),
is called a semigroup; and it is uniformly continuous, if
lim t→t0 P (L * j ,L * ¬j ) t ϑ(x) − P (L * j ,L * ¬j ) t0 ϑ(x) L 1 (X,A ,µ) = 0, ∀t 0 ≥ 0,(33)
for each ϑ(x) ∈ L 1 (X, A , µ) with (L * j , L * ¬j ) ∈ L .
IV. MAIN RESULTS
A. Game-theoretic formalism
In the following, we specify a game in a feedback strategic form -where, in the course of the game, each feedback operator generates automatically a feedback control in response to the action of other feedback operators via the system state x(t) for t ∈ [t 0 , +∞). For example, the jth -feedback operator can generate a feedback control u j (t) = L j x j (t) in response to the actions of other feedback operators u * i (t) = L * i x j (t) for i ∈ N ¬j , where u j (t), u * ¬j (t) ∈ U L , and, similarly, any number of feedback operators can decide on to play feedback strategies simultaneously. Hence, for such a game to have a set of stable (game-theoretic) equilibrium feedback operators (which is also robust to small perturbations in the system or strategies played by others), then each feedback operator is required to respond (in some sense of best-response correspondence) to the others strategies.
To this end, it will be useful to consider the following criterion functions Then, the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards the unique equilibrium state, i.e., the entropy of the whole system will be maximized (see Lanford [23, pp 1-113] for an exposition of equilibrium states and entropy in statistical mechanics).
L ∋ (L j , L * ¬j ) → H r P (Lj,L * ¬j ) t ϑ(x) ϑ(x) ∈ R − ∪ {−∞}, ∀t ≥ 0, ∀j ∈ N ,(34)
Therefore, more formally, we have the following definition for the set of equilibrium feedback operators
(L * 1 , L * 2 , . . . , L * N ) ∈ L .
Definition 10: We shall say that a set of linear system operators (L * 1 , L * 2 , . . . , L * N ) ∈ L is a set of (game-theoretic) equilibrium feedback operators, if they produce control responses given by
u * j (t) = L * j x (t) ∈ U j , ∀j ∈ N ,(35)
for t ∈ [0, ∞] and satisfy further the following conditions
H r P (Lj,L * ¬j ) t ϑ(x) ϑ(x) ≥ H r P (L * j ,L * ¬j ) t ϑ(x) ϑ(x) , ∀t ≥ 0, ∀(L j , L * ¬j ) ∈ L , ∀j ∈ N , D(X, A , µ) ∋ P (L * j ,L * ¬j ) t ϑ(x) → ϑ * (x) ∈ D(X, A , µ) as t → ∞, H P (Lj,L * ¬j ) t ϑ(x) ≤ H P (L * j ,L * ¬j ) t ϑ * (x) , ∀t ≥ 0, ∀j ∈ N , (36)
for each ϑ(x) ∈ D(X, A , µ).
Remark 6:
We remark that the relative entropy H r (·|·) in Equation (36) is determined with respect to L j for each j ∈ N ; while the others L * ¬j remain fixed.
Then, we formally state the main objective of this paper.
Problem 1: Provide a sufficient condition for the existence of a set of equilibrium feedback operators in the multi-channel system (that interacts strategically in a game-theoretic framework) such that when the composition of the multi-channel system with this set of equilibrium feedback operators, described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized.
B. Existence of a set of (game-theoretic) equilibrium feedback operators
In the following, we provide a sufficient condition for the existence of a unique equilibrium state that is associated with a stationary density function (i.e., a common fixed-point) for the family of Frobenius-
Perron operators P (L * j ,L * ¬j ) t t≥0
.
Proposition 2: Let B(X, A , µ)
be an open ball in D(X, A , µ) of center ϑ 0 (x) ∈ D(X, A , µ) and radius β, i.e,,
B(X, A , µ) = ϑ(x) ∈ D(X, A , µ) ϑ(x) − ϑ 0 (x) L 1 (X,A ,µ) ≤ β .(37)
Suppose that there exists a set of feedback operators L * 1 , L * 2 , . . . , L * N ∈ L such that the family of
Frobenius-Perron operators P (Lj,L * ¬j ) t t≥0 with respect to Φ (Lj,L * ¬j ) t satisfies sup (Lj,L * ¬j )∈L P (Lj,L * ¬j ) t ϑ 2 (x) − P (Lj,L * ¬j ) t ϑ 1 (x) L 1 (X,A ,µ) ≤ κ ϑ 2 (x) − ϑ 1 (x) L 1 (X,A ,µ) , ∀t ≥ 0, ∀j ∈ N ,(38)
for any two ϑ 1 (x), ϑ 2 (x) ∈ B (X, A , µ), where κ is a positive constant which is less than one.
Then, if sup (Lj,L * ¬j )∈L P (Lj,L * ¬j ) t ϑ 0 (x) − ϑ 0 (x) L 1 (X,A ,µ) ≤ β 1 − κ , ∀t ≥ 0, ∀j ∈ N ,(39)
there is at least one stationary density function (i.e., a common fixed-point)
ϑ * (x) ∈ B(X, A , µ) such that P (L * j ,L * ¬j ) t ϑ * (x) = ϑ * (x), ∀t ≥ 0, ∀j ∈ N .(40)
Furthermore, there exists a unique equilibrium state, which corresponds with ϑ * (x), if the measure µ *
µ * (A) = A ϑ * (x)µ(dx), ∀A ∈ A ,(41)
is invariant with respect to Φ (L * j ,L * ¬j ) t for each fixed t ≥ 0. 6 Proof: Observe that P (Lj,L * ¬j ) t is continuous for each t ≥ 0 and for any ϑ(x) ∈ D(X, A , µ) (cf.
Equation (33)). For ϑ * (x) ∈ B(X, A , µ), we will show that there exists a convergent sequence of functions ϑ n (x) such that
B(X, A , µ) ∋ ϑ n (x) = P (Lj,L * ¬j ) t ϑ n−1 (x) − ϑ 0 (x), ∀n ≥ 0, ∀j ∈ N , ∀(L j , L * ¬j ) ∈ L ,
We see that if ϑ p (x) is defined in B(X, A , µ) for 1 ≤ p ≤ n, i.e., P (Lj,L * ¬j ) t ϑ p−1 (x) − ϑ 0 (x) ∈ B(X, A , µ), ∀p ∈ [1, n], then we have followings (X,A ,µ) ,
ϑ p (x) − ϑ p−1 (x) = P (Lj,L * ¬j ) t ϑ p−1 (x) − P (Lj,L * ¬j ) t ϑ p−2 (x). and ϑ p (x) − ϑ p−1 (x) L 1 (X,A ,µ) ≤ κ sup (Lj,L * ¬j )∈L P (Lj,L * ¬j ) t ϑ p−1 (x) − P (Lj,L * ¬j ) t ϑ p−2 (x) L 1∀t ≥ 0, ∀j ∈ N .
With (L * j , L * ¬j ) ∈ L , we conclude that
ϑ p (x) − ϑ p−1 (x) L 1 (X,A ,µ) ≤ κ p−1 ϑ 1 (x) L 1 (X,A ,µ) ,
which further gives us (X,A ,µ) ,
ϑ p (x) L 1 (X,A ,µ) ≤ (1 + κ + κ 2 + · · · + κ p−1 ) ϑ 1 (x) L 1and ϑ p (x) L 1 (X,A ,µ) ≤ 1 1 − κ ϑ 1 (x) L 1 (X,A ,µ) < β.
Hence, this agrees with our claim, i.e.,
sup (Lj,L * ¬j )∈L P (Lj,L * ¬j ) t ϑ 0 (x) − ϑ 0 (x) L 1 (X,A ,µ) < β(1 − κ), ∀t ≥ 0, ∀j ∈ N .
Note that, for any n ≥ 0, we have
ϑ n (x) − ϑ n−1 (x) L 1 (X,A ,µ) ≤ κ n−1 ϑ 1 (x) L 1 (X,A ,µ) ,
which is strongly convergent (i.e., lim n→∞ ϑ n (x)−ϑ n−1 (x) L 1 (X,A ,µ) = 0). Then, by passing to a limit, we conclude that there exists a common fixed-point (or a stationary density function) ϑ * (x) ∈ B(X, A , µ)
for the family of Frobenius-Perron operators P
(L * j ,L * ¬j ) t t≥0 that satisfies P (L * j ,L * ¬j ) t ϑ * (x) = ϑ * (x), ∀t ≥ 0,
which also corresponds to the unique equilibrium state, in the sense of statistical mechanics, for the multi-channel system together with (L * j , L * ¬j ) ∈ L . 7
Moreover, from Theorem 1, we see that the measure µ * , i.e.,
µ * (A) = A ϑ * (x)µ(dx), ∀A ∈ A , is invariant with respect to Φ (L * j ,L * ¬j ) t for each fixed t ≥ 0. ✷
The above proposition (i.e., Proposition 2) is important because of the three way connection it draws between the existence of a common stationary density function ϑ * (x) ∈ D(X, A , µ) for P (L * j ,L * ¬j ) t t≥0 7 Note that the following also holds true
lim t→∞ P (L * j ,L * ¬j ) t ϑ(x) = ϑ * (x),
for any ϑ(x) ∈ B(X, A , µ).
(i.e., the unique equilibrium state), the invariant measure µ * (i.e., the measure preserving property of Φ (L * j ,L * ¬j ) t for all t ≥ 0) and the set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N ∈ L . Moreover, the corresponding maximum entropy H max ϑ * (x) is given by
H max ϑ * (x) = − X ϑ * (x) ln ϑ * (x)µ(dx).
Remark 7:
We remark that, in the above proposition, a fixed-point theorem is implicitly used for deriving a sufficient condition for the existence of a common stationary density function for the family of Frobenius-Perron operators (e.g., see Dunford and Schwartz [10, pp 456] or Dieudonné [9, pp 261]).
C. Asymptotic stability of the family of Frobenius-Perron operators
Here, we provide a connection between the stationary density function ϑ * (x) ∈ D(X, A , µ) (which corresponds to the equilibrium state) and the asymptotic stability of the family of Frobenius-Perron operators P
(L * j ,L * ¬j ) t t≥0
. Note that, from Proposition 2, any initial density function ϑ(x) ∈ D(X, A , µ)
under the action of the family of Frobenius-Perron operators P (Lj,L * ¬j ) t t≥0
will only converge to a unique stationary density function ϑ * (x) ∈ D(X, A , µ), if the relative entropy
sup (Lj,L * ¬j )∈L H r P (Lj,L * ¬j ) t ϑ(x) ϑ * (x) , ∀j ∈ N ,(42)
tends zero as t → ∞, and when the set of feedback operators attains a (game-theoretic) equilibrium.
Then, we have the following corollary that exactly establishes the connection between the relative entropy and the stationary density function (where the latter corresponds to the unique equilibrium state).
Corollary 4:
Suppose that the set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N ∈ L satisfies Proposition 2. Then,
lim t→∞ H r P (L * j ,L * ¬j ) t ϑ(x) ϑ * (x) = 0, ∀j ∈ N ,(43)
for each ϑ(x) ∈ D(X, A , µ) such that H r ϑ(x) ϑ * (x) is finite.
Proof: From Proposition 2, if L * 1 , L * 2 , . . . , L * N ∈ L is a set of equilibrium feedback operators. Then, there is a common fixed-point density function ϑ * (x) ∈ D(X, A , µ) such that
ϑ * (x) ∈ t≥0 P (Lj,L * ¬j ) t ϑ(x) = ∅, ∀(L j , L * ¬j ) ∈ L , ∀j ∈ N ,
∀ϑ(x) ∈ D(X, A , µ),
and P (L * j ,L * ¬j ) t ϑ * (x) = ϑ * (x), ∀t ≥ 0, with (cf. Equation (36) or Footnote 7) lim t→∞ P (L * j ,L * ¬j ) t ϑ(x) = ϑ * (x), ∀ϑ(x) ∈ D(X, A , µ). Further, if H r ϑ(x) ϑ * (x) is finite, then we have H r P (L * j ,L * ¬j ) t ϑ(x) ϑ * (x) → 0 as t → ∞,
for any ϑ(x) ∈ D(X, A , µ). ✷
D. Resilient behavior of a set of (game-theoretic) equilibrium feedback operators
Here, we consider the following systems with a small random perturbation term
dZ j ǫ (t) = A(t) + i∈N¬j B i (t)L * i (t) Z j ǫ (t)dt + B j (t)L i (t)Z j ǫ (t)dt + √ ǫ σ(t, Z j ǫ (t)) dW (t), Z j ǫ (0) = x 0 , (L j , L * ¬j ) ∈ L , j ∈ N ,(44)
where σ(t, Z j ǫ (t)) ∈ R d×d is a diffusion term, W (t) is a d-dimensional Wiener process and ǫ is a small positive number, which represents the level of perturbation in the system. Note that we assume here there exists a set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N ∈ L , when ǫ = 0 (which corresponds to the unperturbed multi-channel system). Then, we investigate, as ǫ → 0, the asymptotic stability behavior of an invariant measure for the family of Frobenius-Perron operators P (Lj,L * ¬j ) ǫ,t t≥0
, with (L j , L * ¬j ) ∈ L , which corresponds to the multi-channel system with a small random perturbation. 8
Remark 8: Note that, in general, the evolution of the density function is given by
P (L * j ,L * ¬j ) ǫ,t ϑ(x) = X Γ ǫ,t (x, y)ϑ(y)µ(dy), ∀t ≥ 0,
where Γ ǫ,t (·, ·) is the kernel (i.e., the fundamental solution), which is independent of the initial density function ϑ(x) ∈ D(X, A , µ). Moreover, it is well known that the solution, which is associated with 8 We remark that such a solution for Equation (44) is assumed to have continuous sample paths with probability one (see Kunita [22] for additional information).
Cauchy problem, satisfies the Fokker-Planck (or Kolmogorov forward) equation that is completely specified, with some additional regularity conditions, by A(t) + j∈N B j (t)L * j (t) Z ǫ (t) and √ ǫ σ(t, Z ǫ (t)) (e.g., see also [13] or [22]).
In what follows, we provide additional results, based on the asymptotic stability of an invariant measure, that partly establish the resilient behavior for the set of equilibrium feedback operators with respect to the random perturbation in the system.
Proposition 3:
For any continuous density function ϑ(x) ∈ D(X, A , µ), suppose that
sup (Lj,L * ¬j )∈L P (Lj,L * ¬j ) ǫ,t ϑ(x) − P (L * j ,L * ¬j ) t ϑ(x) L 1 (X,A ,µ) , ∀t ≥ 0, ∀j ∈ N ,(45)
tends to zero in a weak* topology on X as ǫ → 0. Then, the weak limit of invariant measure µ ǫ * of P
(L * j ,L * ¬j ) ǫ,t t≥0
is absolutely continuous with respect to the invariant measure µ * , where the latter corresponds to the family of Frobenius-Perron operators P (L * j ,L * ¬j ) t t≥0
.
Proof: Note that, from the standard perturbation arguments for linear operators, if the set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N ∈ L and the fixed-point density function ϑ * (x) ∈ D(X, A , µ) (i.e., P (L * j ,L * ¬j ) t ϑ * (x) = ϑ * (x), ∀t ≥ 0) satisfy Proposition 2. Then, the following holds
lim ǫ→0 sup x∈X X P (L * j ,L * ¬j ) ǫ,t ϑ(x) − P (L * j ,L * ¬j ) t ϑ(x) µ(dx) = 0, ∀ϑ(x) ∈ D(X, A , µ),
for any fixed t ≥ 0. This further implies the following
lim ǫ→0 ϑ ǫ * (x) − ϑ * (x) L 1 (X,A ,µ) = 0, where ϑ ǫ * (x) is invariant of P (L * j ,L * ¬j ) ǫ,t
for each fixed t ≥ 0.
In order for µ ǫ * to be absolutely continuous with respect to µ * , i.e., µ ǫ * ≪ µ * and µ ǫ * (A) = A ϑ ǫ * (x)µ ǫ * (dx), ∀A ∈ A , it is suffice to show that, for any fixed t ≥ 0, the family of Frobenius-Perron operators P (Lj,L * ¬j ) ǫ,t , with respect to (L j , L * ¬j ) ∈ L for all j ∈ N , should not be too different from P (L * j ,L * ¬j ) t for small ǫ ≥ 0 (cf. Remark 9 below).
On the other hand, under the game-theoretic framework (cf. Proposition 2), each of these feedback operators are required to respond in some sense of best-response correspondence to the others feedback strategies in the system. As a result of this, the following will hold true sup (Lj,L * ¬j )∈L (X,A ,µ) → 0 as ǫ → 0, for each j ∈ N , when only the set of feedback operators attains a robust/stable (game-theoretic) equilibrium solution (L * j , L * ¬j ) ∈ L . Note that, in the above equation, the supremum is computed with respect to L j with (L j , L * ¬j ) ∈ L for each j ∈ N , while others L * ¬j remain fixed, and when there is also a small random perturbation in the system.
P (Lj,L * ¬j ) ǫ,t ϑ * (x) − P (L * j ,L * ¬j ) t ϑ * (x) =ϑ * (x), ∀t≥0 L 1
Then, we see that µ ǫ * tends to µ * weakly as ǫ → 0. This completes the proof. (with respect to the set of equilibrium feedback operators (L * j , L * ¬j ) ∈ L ) as well as on the measure space L 1 (X, A , µ) (see also [2] and [19]).
We conclude this subsection with the following corollary, which is concerned with the resilient behavior of the set of equilibrium feedback operators, when there is a small random perturbation in the system.
The proof follows similar arguments as in the proofs of Proposition 3 and Corollary 4, and therefore will be omitted.
Corollary 5:
For ǫ > 0 and supp ϑ ǫ * (x) ⊂ supp ϑ * (x), if the relative entropy of the multi-channel system, with a random perturbation term, satisfies the following condition
where θ (L * j ,L * ¬j ) ǫ is a small positive number that depends on ǫ (and also tends to zero as ǫ → 0). Then, the set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N ∈ L exhibits a resilient behavior.
The above corollary states that the set of equilibrium feedback operators exhibits a resilient behavior, when the contribution of the perturbation term, to move away the system from the invariant measure µ * , is bounded from above for all t ≥ 0.
We also note that the following holds true (see Equation (45))
lim t→∞ P (L * j ,L * ¬j ) ǫ,t ϑ(x) − ϑ * (x) L 1 (X,A ,µ) → 0 as ǫ → 0,(47)
for any ϑ(x) ∈ D(X, A , µ). Therefore, such a bound in Equation (46) is an immediate consequence of this fact. 9 9 For small ǫ ≥ 0, notice that lim t→∞ X P (L * j ,L * ¬j ) ǫ,t ϑ(x)µ(dx) = X ϑ ǫ * (x)µ(dx), ∀ϑ(x) ∈ D(X, A , µ), when the stochastic semigroup P (L * j ,L * ¬j ) ǫ,t is asymptotically stable for each fixed t ≥ 0 (cf. [24,Sec. 11.9]). Remark 10: Finally, we note that although we have not discussed the limiting behavior, as ǫ → 0, of the family of measures µ ǫ * on the space L 1 (X, A , µ). It appears that the theory of large deviations can be used to estimate explicitly the rate at which this family of measures converges to the limit measure µ * , where the latter is invariant with respect to Φ (L * j ,L * ¬j ) t for each fixed t ≥ 0 (e.g., see [32], [11] or [8] for a detailed exposition of this theory).
Proposition 1 :
1Let (X, A , µ) be a finite measure space. Consider all (nonnegative) possible density functions ϑ(x) defined on X. Then, for such a family of density functions, the maximum entropy occurs for a constant density function
over the class of admissible control functions U L (or the set of linear feedback operators L ) and for any ϑ(x) ∈ D(X, A , µ). Note that, under the game-theoretic framework, if there exists a set of equilibrium feedback operators L * 1 , L * 2 , . . . , L * N (from the class of linear feedback operators L ). Then, this set of equilibrium feedback operators decreases the relative entropy between any two density functions from D(X, A , µ) for all t ≥ 0. On the other hand, if there exists a unique stationary density function (i.e., a common fixedpoint) ϑ * (x) ∈ D(X, A , µ) for the family of Frobenius-Perron operators P (L * j ,L * ¬j ) t for each fixed t ≥ 0.
✷ Remark 9 :
9We remark that, in general, the relation between P
,
∀t ≥ 0, ∀ϑ(x) ∈ D(X, A , µ),
Note that the supremum in Equation (38) (and also in Equation(39)) is computed with respect to Lj with (Lj, L * ¬j ) ∈ L for each j ∈ N , while L * ¬j remains fixed.
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| []
|
[
"Fast Nonlinear Risk Assessment for Autonomous Vehicles Using Learned Conditional Probabilistic Models of Agent Futures",
"Fast Nonlinear Risk Assessment for Autonomous Vehicles Using Learned Conditional Probabilistic Models of Agent Futures"
]
| [
"Ashkan Jasour ",
"Xin Huang ",
"Allen Wang ",
"Brian C Williams "
]
| []
| []
| This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev's Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events. | 10.1007/s10514-021-10000-1 | [
"https://arxiv.org/pdf/2109.09975v2.pdf"
]
| 237,581,217 | 2109.09975 | 6fbe283bb2705baa23962fc3f5facf6799329824 |
Fast Nonlinear Risk Assessment for Autonomous Vehicles Using Learned Conditional Probabilistic Models of Agent Futures
Ashkan Jasour
Xin Huang
Allen Wang
Brian C Williams
Fast Nonlinear Risk Assessment for Autonomous Vehicles Using Learned Conditional Probabilistic Models of Agent Futures
This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev's Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.
I. INTRODUCTION
Prediction under uncertainty plays a key role in safety of autonomous vehicles. More precisely, in order for autonomous vehicles to drive safely on public roads, they need to predict the future states of other agents (e.g. human-driven vehicles, pedestrians, cyclists) and plan accordingly. Predictions, however, are inherently uncertain, so it is desirable to represent uncertainty in predictions of possible future states and reason about this uncertainty while planning. This desire is motivating ongoing work in the behavior prediction community to go beyond single mean average precision (MAP) prediction and develop methods for generating probabilistic predictions [1]- [4]. In the most general sense, this involves learning joint distributions for the future states of all the agents conditioned on their past trajectories and other context specific variables (e.g. an agent is at a stop light, lane geometry, the presence of pedestrians, etc). However, learning such a distribution can All authors are with the Computer Science and Artificial Intelligence Laboratory (CSAIL), Massachusetts Institute of Technology (MIT) {jasour, huangxin, allenw, williams} @ mit.edu *These authors contributed equally to the paper. often be intractable, so current works use a wide variety of different simplified representations for probabilistic predictions. For example [3] trains a conditional Variational Autoencoder (CVAE) to generate samples of possible future trajectories. Other works use generative adversarial networks (GANs) to generate multiple trajectories with probabilities assigned to each of them [4], [5]. As a discrete alternative, [6], [7] train a DNN to generate a probabilistic occupancy grid map with a probability assigned to each cell. However, such grid-based approaches effectively treat possible agents' trajectories as belonging to a discrete space, while, in reality, agents may be at an uncountable number of points in continuous space. Many recent papers try to account for the continuous nature of uncertainty in space by learning Gaussian mixture models (GMMs) for vehicle positions [1], [6], [8] or coefficients of polynomials in R 2 that represent the vehicles' positions [9]. Since learning uncertain models for position or pose can also sometimes produce results that are inconsistent with basic kinematics, some recent works develop DNNs that predict future control inputs which are then propagated through a kinematic model to predict future positions [2], [10].
Given a probabilistic prediction, an autonomous vehicle still needs to be able to rapidly evaluate the probability of a given plan resulting in a collision or, more generally, a constraint violation. We will refer to this problem as risk assessment and it is particularly challenging in the context of autonomous driving as 1) autonomous vehicles need to reason about low probability events to be safer than human drivers and 2) there are hard real time constraints on algorithm latency. Latency is a critical consideration for safety and will be a major consideration motivating the methods presented in this paper. While an algorithm with a latency of, for example, one second would often be acceptable in other robotics applications, it would be unacceptable for an autonomous vehicle traveling at 20 m/s on public roads. This requirement of low latency while retaining the ability to reason about low probability events makes naive Monte Carlo computationally intractable. To address this problem, adaptive and importance sampling methods have been proposed to estimate these probabilities with fewer samples [11], [12]. However, such methods i) do not provide any bound on the risk and any safety guarantees, ii) are not suitable for real-time risk assessment without parallelization and running on GPU, and iii) lead to performance that is highly sensitive to algorithm parameters and proposal distributions.
Non-sampling based methods are widely used in risk assessment problems. Although existing non-sampling based methods are fast, they are limited to a particular class of uncertainties and safety constraints. For example, Gaussian-Linear methods use Boole's inequality [13]- [15] to estimate the probability of violation of linear constraints in the presence of Gaussian uncertainties. More precisely, the probability of a convex polytope is calculated as:
Prob{∩ N j=1 a i X ≤ b j } = 1− Prob{∪ N j=1 a i X ≥ b j } ≤ i Prob{a i X ≥ b j })
. This results in a conservative upper bound on the risk. Chebyshev inequality based methods provide conservative bound on the risk of linear safety constraints using first two moments of uncertainties [16], [17]. Conditional-value-at-risk (CVaR) based methods use a conservative approximation of the indicator functions of the safety constraints, [18], [19]. To evaluate the CVaR based risk constraints sampling-based methods or Gaussian uncertainties are used, [20], [21].
Statement of Contributions:
We present fast methods to assess the risk of trajectories for both Gaussian and non-Gaussian position and control models of other agents. In section IV, we begin by addressing the case when GMMs are used for agent position predictions. We show this particular case can be reduced to the problem of computing the CDF of a quadratic form in a multivariate Gaussian (QFMVG)a well-studied problem in the statistics community for which methods exist that can rapidly solve it to arbitrary accuracy. To address the more general case when potentially non-Gaussian mixture models are used for agent position predictions, we apply statistical moment-based approaches to determine upper bounds on risk, which we will refer to as risk bounds.
Namely, we propose using nonlinear Chebyshev's Inequality and a univariate sums-of-squares (SOS) program that can be seen as a generalization of Chebyshev's Inequality; the former is faster, while the latter can provide arbitrarily tight risk bounds. These moment-based approaches have the feature of being distributionally robust, producing risk bounds that are true for all possible distributions that take on the value of the given moments. To address uncertain models for control inputs, in Section V, we propagate the moments of predicted probabilistic control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planing horizon. For this purpose, given the nonlinear motion dynamics and moments of uncertain control inputs, we construct new deterministic moment-state linear systems that govern the exact time evolution of the moments of uncertain position. This enables the application of our non-Gaussian position risk assessment methods to the problem of risk assessment when models are learned for agent control inputs. Figure 1 illustrates our framework in this case. In Section VI, we present an encoder-decoder-based conditional predictor generating GMM predictions given the observed data and scene context. In Section VII, we demonstrate our methods on realistic predictions generated by DNNs trained on the Argoverse and CARLA datasets [22], [23]. Source code can be found at https://github.com/allen-adastra/risk assess.
II. NOTATION
Let S n ++ denote the set of n × n positive definite matrices. For any matrix Q ∈ S n ++ and vector x ∈ R n , let Q(x) := x T Qx. Let Q ij denote the element in the i th row and j th column of Q. For any θ ∈ R, let R(θ) be the 2D rotation matrix parameterized by θ. For a vector x ∈ R n and multiindex α ∈ N n , let
x α = n i=1 x αi i . For n ∈ N, let [n] = {k ∈ N : k ≤ n}. For a vector valued function f , f i denotes the i th component of f . Polynomials: Let R[x]
be the set of real polynomials in the variables x ∈ R n . Given polynomial P (x) : R n → R, we represent P as α∈N n pαx α using the standard monomial basis {x α } α∈N n of R[x], and p = {pα} α∈N n denotes the coefficients and α ∈ N n . Also, let R d [x] ⊂ R[x] denotes the set of polynomials of degree at most d ∈ N.
Sum of Squares Polynomials: Polynomial P (x) is a sum of squares (SOS) polynomial if it can be written as a sum of finitely many squared polynomials, i.e., P (x) = m j=1 hj(x) 2 for some m < ∞ and h j (x) ∈ R[x] for 1 ≤ j ≤ m, SOS condition is a convex constraint that can be represented as a linear matrix inequality (LMI) in terms of coefficients of polynomial, i.e., P (x) ∈ SOS → P (x) = xT Ax, where x is the vector of standard basis and A is a positive semidefinite matrix in terms of the coefficients of the polynomial, Moments of Probability Distribution: For a random vector w and any d ∈ N, let µ wt , Σ wt denote its mean vector and covariance matrix respectively, and Φ w denote its characteristic function. Moments of random variables, are the generalization of mean and covariance and are defined as expected values of monomials of random variables. More precisely, given (α 1 , ..., α n ) ∈ N n where α = n i=1 α i , moment of order α of random vector w is defined as E[Π n i=1 w αi i ]. Hence, the sequence of all moments of order α is defined as expected values of all monomials of order α. For example, sequence of the moments of order α = 2 for n = 3 is defined as
E[w 2 1 ], E[w1w2], E[w1w3], E[w 2 2 ], E[w2w3], E[w 2 3 ]
. Moment of order α can be computed by applying n partial derivatives of the characteristic function as follows:
E[w α 1 1 w α 2 2 ...w αn n ] = i −(α 1 +...+αn) ∂ α 1 +...+αn ∂t α 1 1 ...∂t αn n Φx(t1, ..., tn) (1)
for t 1 = 0, ..., t n = 0. Note that for any probability density function, the characteristic function always exists [24]. We will use finite sequence of the moments to represent Non-Gaussian probability distributions.
III. PROBLEM STATEMENT
We define risk as the probability of an agent entering an ellipsoid around the ego vehicle. Thus, we are interested in computing the probability of other agents entering:
x ∈ R 2 : Q(x) ≤ 1 , Q ∈ S 2 ++ (2)
We argue ellipsoids are a useful representation as: 1) they can be fit relatively tightly to the profiles of vehicles, and 2) the sizes of both the ego vehicle and agent can be accounted for by properly scaling the size of the ellipsoid around the vehicle. Throughout the paper, agent positions at each time step are always defined in the frame of the planned future poses of the ego vehicle unless stated otherwise; Section IV-A shows how moments of distributions can be expressed in different frames. Given this formulation, the ellipsoid is parameterized by a constant matrix Q ∈ S 2 ++ in the ego vehicle frame. In practice, multiple ellipsoids can be defined around the vehicle and an appropriate one selected at run-time. We restrict our focus to the single agent case and note that the risk in a multi-agent setting can be upper bounded by summing the risk associated with each agent.
If x t = [x t , y t ] T is some random vector for the position of the agent at time t, then the risk associated with an agent across the whole T step time horizon is:
R := P T t=1 {Q(x t ) ≤ 1}(3)
By the inclusion-exclusion principle, the probability (3) can be computed as the sum of the probabilities of the marginal events and the probabilities of all possible intersections of events:
R = J∈P([T ]) (−1) |J|+1 P j∈J {Q(x j ) ≤ 1} (4)
In many works, the random variables are assumed to be independent across time or can be made to be independent across time by conditioning on a discrete mode [1], [6], [8]. If there is dependence across time, one would need the conditional distributions of the events which require additional information to be learned. As most work on behavior prediction currently assumes independence across time, this paper restricts its focus to the time independent case, and so:
R = 1 − t∈[T ] (1 − P(Q(x t ) ≤ 1))(5)
Thus, the problem of risk assessment along the trajectory can be solved by computing the marginals at each time step t, so the rest of the paper restricts its focus to the marginals.
IV. RISK ASSESSMENT
In this section, we present solutions for both Gaussian and non-Gaussian risk assessment when moments of the random vector for agent position x t are known. We begin by addressing the problem of determining moments of agent positions in different frames to account for the ego vehicles planned trajectory. We then present our solution for the GMM case using numerical approximations of the CDFs of QFMVGs. To address the non-Gaussian case, we present methods based off Chebyshev's Inequality and SOS programming. We assume basic knowledge of SOS programming; We refer the reader to [25], [26] for an overview of SOS programming and [26]- [31] for moment-SOS based planning under uncertainty. Throughout this section, we assume the necessary moments of x t are known.
A. Changing Frames
Predictions are usually given in a global frame, so this section provides a method for transforming the global frame distribution moments into the ego vehicle frame. More generally, we are concerned with computing moments of x t in a new frame offset by v ∈ R 2 and rotated by −θ ∈ R. As shown in the appendix, if x t is a mixture model, its moments can be computed in terms of moments of its components, so, in this section, let x t be a component of a mixture model. We propose only translating the moments and then accounting for the rotation by using Q * = R(θ) T QR(θ) instead of Q. The rotation can be accounted for by using Q * instead of Q because:
x T t Q * x t = x T t R(θ) T QR(θ)x t (6) = (R(θ)x t ) T Q(R(θ)x t )(7)
The translated moments can be computed by applying the binomial theorem to (x t − v) n (here, the power is applied element-wise). Note that applying the binomial theorem to (x t − v) n requires moments of x t up to order n.
B. Risk Assessment for GMM Position Models
In this section, we provide a method to solve the risk assessment problem when the uncertain prediction is represented as a sequence of GMMs, x t , of the agents position with discrete modes determined by the Multinoulli Z t . Many works currently learn GMMs for vehicle position as they express both multi-modal and continuous uncertainty [1], [6], [8]. As shown in Figure 2, they provide an intuitive representation of An example risk assessment scenario. One standard deviation confidence ellipses (in blue) of a multi-modal GMM prediction are shown with mode probabilities. The observed agent trajectory and planned ego vehicle trajectory are also shown in red with different markers. uncertainty in both the drivers high level decisions and low level execution. With time independence, the risk is:
n z=1 1 − t∈[T ] 1 − P(Q(x t ) ≤ 1 : Z t = z) P(Z t = z) (8)
Note that the above expression can be easily modified for the case when there is a single Multinoulli random variable that is constant across all time, an assumption used in, for example, [1]. The probabilities P(Z t = z) are learned parameters of GMMs, so the problem of risk assessment can be solved by computing P (Q(x t ) ≤ 1 : Z t = z) for each agent, time step, and mode. Note that this is exactly the CDF of Q(x t ) conditioned on Z t = z which is a quadratic form in a multivariate Gaussian (QFMVG). Unfortunately, there does not exist a known closed form solution to exactly evaluate the CDF of QFMVGs, but fast approximation methods with bounded errors have been studied within the statistics community [32]- [36]. Several of these methods have been implemented in the R package CompQuadForm [37]. Of particular interest is the method of Imhof, which produces results with bounded approximation error by numerical inversion of the characteristic function of the QFMVG [36]. A faster, but less accurate, alternative is the method of Liu-Tang-Zhang which involves approximating the CDF of the QFMVG with the CDF of a non-central chi square distribution with parameters chosen to minimize the difference in kurtosis and skew between the approximate and target distributions [32].
C. Non-Gaussian Risk Assessment with Chebyshevs Inequality
As a consequence of the one-tailed Chebyshev's Inequality, for any measurable function g, whenever E[g(x t )] > 0, we have that:
P(g(x t ) ≤ 0) ≤ E[g(x t ) 2 ] − E[g(x t )] 2 E[g(x t ) 2 ](9)
That is, the first two moments of g(x t ) are sufficient to establish a bound on the risk that the constraint g(x t ) ≤ 0 is violated. We note that the requirement E[g(x t )] > 0 is not particularly restrictive because E[g(x t )] ≤ 0 means the average case involves collision, thus corresponding to what is usually an unacceptable level of risk. 1) Applying Chebyshev's Inequality to the Quadratic Form: To apply Chebyshev's inequality to P(Q(x t ) − 1 ≤ 0), we would need the first two moments of Q(x t ) − 1 which can be expressed in terms of the first two moments of Q(x t ). The first moment can be expressed in terms of the mean vector and covariance matrix of x t [38]:
E[Q(x t )] = Tr(QΣ xt ) + µ T xt Qµ xt(10)
We can determine an expression for E[(Q(x t ) 2 ] via an alternate representation for the quadratic form:
E[Q(x t ) 2 ] = (i,j,k,l)∈[2] 4 Q ij Q kl E x ti x tj x t k x t l(11)
Thus, to compute the second moment of Q(x t ), we would need the moments of x t of order up to four.
2) Conservative Approximation with Half-Spaces: It's possible to reduce the order of the moments that need to be propagated to two by instead approximating the ellipsoid as the intersection of n h half-spaces parameterized by a i ∈ R 2 and b i ∈ R. The approximated set is thus:
X Approx = ∩ n h i=1 {x ∈ R 2 : a T i x + b i ≤ 0}(12)
Since the probability of any individual event is greater than the probability of the intersection of events, we have that:
P(∩ n h i=1 {a T i x t + b i ≤ 0}) ≤ min i∈[n h ] P(a T i x t + b i ≤ 0) (13)
So if we determine an upper bound on the probability of each P(a T i x t + b i ≤ 0) with Chebyshev's Inequality, the minimum of the Chebyshev bounds will be an upper bound on our risk. Since a T i x t +b i is an affine transformation of x t , its mean and variance can be expressed with the mean vector and covariance matrix of x t .
D. Non-Gaussian Risk Assessment with SOS Programming
When tighter risk bounds are desired than those obtained via Chebyshev's Inequality, for any measurable function g, an univariate SOS program can be used to upper-bound P(g(x t ) ≤ 0) -the SOS program is univariate in the sense that it searches for a polynomial in a single indeterminant, not in the sense that there is only one decision variable [28]. The fact that the SOS program is univariate is significant because the key disadvantages of SOS, scalability and conservatism, are not as limiting for univariate SOS because: 1) the number of decision variables in the resulting SDP scales quadratically w.r.t. the order of the polynomial we are searching for and 2) the set of nonnegative univariate polynomials is equivalent to the set of univariate SOS polynomials, allowing univariate SOS to explore the full space of possible solutions.
We begin by noting that the probability of constraint violation is equivalent to the expectation of the indicator function of the sub-level set of g:
P(g(x t ) ≤ 0) = {xt:g(xt)≤0} pr(x t )dx t = E[1 g(xt)≤0 ](14)
where, pr(x t ) is probability density function of x t and 1 g(xt)≤0 is the indicator function of the sub-level set of g defined as
1 g(xt)≤0 = 1 if x ∈ {x t : g(x t ) ≤ 0}, and 0 otherwise.
The expectation of the indicator function, however, is not necessarily easily computable. To solve this problem, we find some polynomial with a more easily computable expectation that upper bounds the indicator function. If we can find some univariate polynomial, p : R → R of order d in some indeterminant x ∈ R with coefficients c k , k = 0, ..., d that upper bounds the indicator function, then clearly the following implication holds by substitution: (15) Given the coefficients c k , if we apply the expectation w.r.t. the density function of x t to both sides, then we can reduce the problem of finding an upper bound on P(g(x t ) ≤ 0) to that of computing moments of the random variable g(x t ):
p(x) := d k=0 c k x k ≥ 1 x≤0 ⇒ d k=0 c k g(x t ) k ≥ 1 g(xt)≤0d k=0 c k E[g(x t ) k ] ≥ E[1 g(xt)≤0 ] = P(g(x t ) ≤ 0)(16)
where, E[g(x t ) k ] is the moment of order k of random variable g(x t ). The moments of g(x t ), in turn, are computable in terms of the moments of x t , i.e., E[x α t ], α ∈ N, by expanding out the polynomial power and applying the linearity of expectation, [28]. For example, if g(x t ) = x 2 t + y 2 t , then:
E[g(x t ) 3 ] = E[x 6 t ] + 3E[x 4 t y 2 t ] + 3E[x 2 t y 4 t ] + E[y 6 t ] (17)
The moments of x t can be computed using the moment generating function as in (1). In this section, we assume that we know the necessary moments of x t to compute the moments E[g(x t ) k ], ∀k ∈ [d]. We also normalize the moments, as doing so improves the numerical conditioning of the problem 1 . Now consider the following univariate SOS program in the indeterminant x which can search for upper bound polynomial indicator function, i.e., p(x) := d k=0 c k x k ≥ 1 x≤0 , which minimizes the upper bound on risk [28]:
min p,s1,s2 d k=0 c k E[g(x t ) k ] (18a) p(x) − 1 = s 1 (x) − xs 2 (x) (18b) p(x), s 1 (x), s 2 (x) SOS (18c)
If the order of the polynomial is chosen to be d = 2n for some n ∈ N, then we should have that deg(s 1 ) = d and deg(s 2 ) = d − 2. If d = 2n + 1 for some n ∈ N, then we should have that deg(s 1 ) = 2n and deg(s 2 ) = 2n. Note that 1 normalization is valid because P(X ≤ 0) = P(cX ≤ 0) for c > 0 constraints (18b) and (18c) are the nonnegativity constraints of the indicator function 1 x≤0 , i.e., 1 x≤0 = 1 if x ≤ 0, and otherwise 0. More precisely, the constraint (18b) enforces:
p(x) ≥ 1 ∀x ≤ 0(19)
Also, according to the constraint (18c), p(x) is constrained to be SOS; So it is globally nonnegative, i.e, p(x) ≥ 0. Hence, polynomial p(x) is an upper bound polynomial approximation of the indicator function, i.e., p(x) ≥ 1 x≤0 , ∀x ∈ R. Thus, according to (15) and (16), the optimal objective value of this SOS program yields an upper bound on P(g(x t ) ≤ 0). Also, note that SOS optimization (18) is a convex optimization. More precisely, it has a linear cost function in terms of the coefficients of the polynomial p(x) and convex constraints in the form of linear matrix inequalities in terms of the coefficients of the polynomials p(x), s 1 (x), and s 2 (x). Hence, we can solve the SOS optimization efficiently using the off-the-shelf LMI optimization solvers.
(x t ) ≤ 0) ≤ E[p(g(x t ))] = d k=0 c k E[g(x t ) k ].
For more information see [28] and https://github.com/jasour/Nonlinear-Risk-Assessment Remark 2: The provided non-Gaussian risk assessment approaches in Sections IV.C and IV.D are less conservative than the existing non-sampling based methods. This is because, we are using higher order moments of uncertainties and also tight polynomial approximation of the indicator functions of safety constraints. One can improve the risk assessment results by increasing the number of the moments in SOS programming based approach.
V. MOMENT PROPAGATION
While directly learning distributions for agents future positions can be an effective strategy, one major disadvantage is it can produce physically unrealistic predictions. [2], [10] address this by learning distributions for control inputs and then propagating samples through a kinematic model. While the Kalman filter and its variants, such as the extended and unscented Kalman filters, can be used to propagate mean and covariance, they are not exact and do not immediately apply to higher order moments [39]- [41].
In this section, we provide an approach for nonlinear moment propagation that can, in principle, work for moments up to arbitrary order [30], [42]. Given a nonlinear motion model and a random vector for control inputs, w t , this section is concerned with the problem of computing statistical moments of the uncertain position x t s.t. the non-Gaussian risk assessment methods presented in Section IV can be applied. More precisely, we are looking for the moments of uncertain position (x t , y t ) of the form E[x α t y β t ] where α, β ∈ N. We use a stochastic version of the discrete-time Dubin's car to both demonstrate the general approach and to address the problem of agent risk assessment:
x t+1 = x t + v t cos(θ t ) (20a) y t+1 = y t + v t sin(θ t ) (20b) v t+1 = v t + w vt (20c) θ t+1 = θ t + w θt (20d)
Above, the control vector is w t = [w vt , w θt ] where w vt and w θt are random variables describing the agent's acceleration and steering at time t and are assumed to be independent.
x t = [x t , y t ] is the position of some reference point on the agent in a fixed frame, v t is its speed, and θ t is the angle of its velocity vector with respect to the fixed frame. The time steps ∆t for discretization are omitted for brevity; the values of the variables can simply be scaled accordingly.
To obtain the moments of the uncertain position over the planning horizon, we will propagate the moments of uncertain control inputs w t through the nonlinear stochastic Dubin's model. To this end, we construct deterministic linear dynamical systems, i.e., mapping between the moments at time t + 1 and t, that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. By obtaining such dynamical systems in terms of the moments, we can recursively propagate the initial moments of the uncertain position over the planning horizon.
A. Motivating Example
To show how our moment propagation algorithm works, we begin by showing how the dynamics of the first order moment for the state x t in system (20) can be found manually, i.e., mapping between E[x t+1 ] and E[x t ]. Our proposed algorithm is essentially an automated version of this process. By substituting the equations (20) in and applying the linearity of expectation, we arrive at the dynamics of the moment:
E[x t+1 ] = E[x t ] + E[v t cos(θ t )](21)
To complete the obtained mapping between E[x t+1 ] and E[x t ], we need to compute the update rule for the term E[v t cos(θ t )].
By substituting the equations (20) in and applying the linearity of expectation, we arrive at the update rule of the moment as:
E[vt+1cos(θt+1)] = E[cos(ω θ t )]E[vtcos(θt)] + E[ωv t ]E[cos(ω θ t )]E[cos(θt)] − E[sin(ω θ t )]E[vtsin(θt)] − E[ωv t ]E[sin(ω θ t )]E[sin(θt)]
where, E[ω vt ] is the first order moment of uncertain control input ω vt and can be computed using it's characteristic function as in (1 Similarly, we can obtain the dynamics of the higher order moments of the uncertain position, i.e., E[x α t y β t ]. This process, however, is tedious and is easily subject to human error, especially for larger moment orders α, β. To address these issues, in [43], [44], we developed TreeRing algorithm that uses a dependency graph to identify all the slack moment states needed to construct the moment dynamical systems.
In this paper, we use a different approach and provide a general framework to construct the moment dynamical systems. The main idea is to transform the nonlinear stochastic Dubbin's model in (20) into a equivalent new augmented linear-state system. In this case, due to the linear relation of the states of the augmented linear-state system at time t and t + 1, moments of order α of the states at time t + 1 can be described only in terms of the moments of order α of the states at time t. Hence, we do not need to look for a set of slack moment states as described in (21).
B. Equivalent Augmented Linear-State System
In [42], we show that nonlinear stochastic motion dynamics can be transformed in to equivalent linear-state dynamical systems by introducing suitable new state variables. In this section, we define such equivalent augmented linear-state system for stochastic nonlinear Dubin's model (20) as follows:
x augt+1 = A t (ω θt , ω vt )x augt (22) where x aug = [x, y, vcos(θ), vsin(θ), cos(θ), sin(θ)] T is the augmented state vector in terms of the position (x, y) and a set of nonlinear functions of the states of the original nonlinear model in (20). Also, matrix A t (ω θt , ω vt ) is defined only in terms of the nonlinear functions of the uncertain control inputs ω θt and ω vt as follows:
A t (ω θt , ω vt ) = 1 0 1 0 0 0 1 0 0 1 0 0 0 0 cos(ω θt ) −sin(ω θt ) ω vt cos(ω θt ) −ω vt sin(ω θt ) 0 0 sin(ω θt ) cos(ω θt ) ω vt sin(ω θt ) ω vt cos(ω θt ) 0 0 0 0 cos(ω θt ) −sin(ω θt ) 0 0 0 0 sin(ω θt ) cos(ω θt )
Note that the obtained augmented system in (22) is linear in terms of the states x aug and also equivalent to the original stochastic nonlinear Dubin's model in (20). We use the obtained augmented linear-state system in (22) to obtain moment-state linear dynamical systems that govern the exact time evolution of the moments of the uncertain position states (x, y) in nonlinear stochastic system (20).
C. Moment-State Linear Systems
We define the moment-state linear systems for the obtained augmented linear-state system in (22) as follows [42]:
E[x α augt+1 ] = A momα t E[x α augt ](23)
where, E[x α augt ] is the vector of all moments of order α of the augmented state vector x aug and A momα is a matrix in terms of the moments of the uncertain control inputs. To construct the moment-state linear system in (23), we need the expected values of the monomials of order α of the vector x augt+1 , e.g.,
E[x α augt+1 ]
. Since x augt+1 is a linear function of x augt as in (22), we can describe the moments of order α of x augt+1 completely in terms of the moments of order α of x augt . Hence, we do not need to look for a set of slack moment states as described in the motivating example of Section V-A.
For example, we obtain the moment-state linear system of is the vector of all moments of order α = 1 of x augt . Also, matrix A mom1 t is described in terms the first order moments of uncertain control input ω vt and first order trigonometric moments of uncertain control input ω θt as follows:
order α = 1 of the form E[x augt+1 ] = A mom1 t E[x augt ] where E[xaug t ] = [E[xt],Amom 1t = 1 0 1 0 0 0 1 0 0 1 0 0 0 0 E[cos(ωθ t )] −E[sin(ωθ t )] E[ωv t ]E[cos(ωθ t )] −E[ωv t ]E[sin(ωθ t )] 0 0 E[sin(ωθ t )] E[cos(ωθ t )] E[ωv t ]E[sin(ωθ t )] E[ωv t ]E[cos(ωθ t )] 0 0 0 0 E[cos(ωθ t )] −E[sin(ωθ t )] 0 0 0 0 E[sin(ωθ t )] E[cos(ωθ t )]
We can compute the moments of uncertain control inputs in terms of the characteristic functions as shown in (1) and Appendix.B. The obtained first order moment-state linear dynamical system describes the exact time evolution of the first order moments of the uncertain position (x, y) of the original stochastic nonlinear system in (20). Similarly, we can obtain the moment-state linear system of the form (23) for higher moment order α to describe the exact time evolution of the higher order moments of the uncertain position. Note that we can construct the deterministic linear moment-state systems of the form (23) for different moment order α in the offline step. Hence, we can use the obtained moment systems to propagate the moments of the uncertain initial states over the planning horizon in real-time. More precisely, given initial moments E[x α aug0 ] and A momα t | N −1 t=0 , the moments at time step N can be obtained by recursion
of E[x α augt+1 ] = A momα t E[x α augt ], t = 0, ..., N − 1.
Similarly, we can describe the moments by the solution of the linear moment system as
E[x α aug N ] = Π N −1 t=0 A momα t E[x α aug0 ].
Remark 3:
The provided approach is not limited to the motion dynamics in (20). We can use different motion dynamics with different sources of uncertainties to model the uncertain behaviour of the agent vehicles. For more information see [42] and https://github.com/jasour/Uncertainty-Propagation
VI. DEEP NEURAL NETWORK PREDICTOR
To obtain probabilistic future states of agents for risk assessment, we propose a conditional deep neural network that generates Gaussian mixture model parameters for positions or controls for a given sequence of observed positions over the past 20 time steps and scene context. Although our framework works with any conditional prediction model that outputs GMM parameters for positions, we aim to generate accurate and realistic predictions by selecting an encoder-decoder-based predictor that utilizes long short-term memory (LSTM) units, because of the recent success of recurrent neural networks in trajectory prediction on different benchmarks [4], [8], [10], [45].
A. Input
The input consists of observed vehicle positions x −19:0 for the past 20 timesteps, as well as scene context c representing driving context such as lanes coordinates c M and the future trajectory of the ego car c E . Since the observed trajectory is usually collected in global coordinate, which can lead to bias in prediction, we normalize the past trajectory of the target vehicle, so that the last and first observed positions are at origin and the x-axis, respectively, as shown in Figure 3.
The conditioning scene context is represented by a set of coordinates of the lanes that are close to the target vehicle (e.g., within 50 meters) and also the future planned trajectory of the ego car. The coordinates of the lanes and the ego car trajectory are normalized into the local coordinate of the target car. Such scene context provides additional cues on the possible future location of the target vehicle. Note that these contexts are usually not available across all driving platforms. Thus, we design a prediction model that is flexible in terms of the inputs, as we show in Section VI-C.
B. Output
The output Y consists of a sequence of GMM parameters over the prediction horizon T as follows:
Y = {(w 1 t , µ 1 t , Σ 1 t ), . . . , (w N t , µ N t , Σ N t )} T t=1
where N represents the number of components in the GMM and w i t represents the weight of ith component at time step t such that N i=1 w i t = 1. Also, mean µ and covariance Σ represent the predicted uncertain position and predicted uncertain control inputs in GMM position predictor and GMM control predictor, respectively.
C. Model Architecture
The encoder-decoder predictor is shown in Figure 4. The encoder is a sequence of LSTM units taking observed trajectories of the target agent and scene context as input and outputs a latent vector encoding agent hidden state. The input Fig. 4. Architecture diagram of GMM predictor, including an LSTMbased encoder and an LSTM-based decoder. We introduce extra multilayer perceptrons (MLPs) to process the inputs in the encoder and process the predicted hidden states before generating predicted parameters in the decoder. modalities are processed with separate multilayer perceptrions (MLPs) before being fed into the LSTM unit. This allows our model to handle different input options given their availability. The decoder is also a sequence of LSTM units that takes the latent vector and generates a set of GMM parameters from each LSTM unit with MLPs. For simplicity, we use three component mixture models. For each component, we generate a weight value, a mean vector, and a covariance matrix representing uncertainties of predictions.
D. Losses
Given the prediction Y and the groundtruth position or control valuesŶ , the loss function is composed of 2 terms as follows:
L = αL L2 (Y,Ŷ ) + L N LL (Y,Ŷ ),(24)
where L L2 measures the L 2 loss between the predicted mean values and ground truth observations over the future trajectories, L N LL measures the negative log-likelihood loss between the predicted distributions and groundtruth observations, and α is the weight coefficient for the L 2 loss.
VII. EXPERIMENTS In this section, we demonstrate the performance of our system through two learning-based predictors that predict stochastic position and control for the target agent. For each predictor, we describe its network details and training procedure, before presenting risk assessment results. All computations were performed on a desktop with an Intel Core i9-7980XE CPU at 2.60 GHz. All Monte Carlo (MC) methods are implemented with vectorized NumPy operations to have a realistic assessment of run times for naive MC.
A. GMM Position Predictor 1) Training Details: To obtain probabilistic trajectory distributions of agents, we trained an encoder-decoder DNN described in Section VI, with the following details. Each MLP in Encoder is a 2-layer MLP followed by a ReLU activation function, where each layer consists of 32 hidden units. The LSTM units in both Encoder and Decoder has a hidden size of 32 with one layer. The MLP in Decoder is a single layer MLP that produces the desired output parameters. The model is trained and validated on a subset of the Argoverse dataset [22]. During training, we use a batch size of 32, learning rate of 0.0001, and α = 0.1. 2) Prediction Experiments: In order to validate the prediction performance of our model, we perform an ablation study and compare our model with baseline models, as summarized in Table I. The performance is evaluated over standard Argoverse trajectory forecasting metrics [22] such as minimum-of-N average displacement error (MoN ADE) and minimum-of-N final displacement error(MoN FDE), which measures the best prediction error over the entire prediction horizon and at the end of prediction horizon, respectively.
We first present prediction results from two physics-based baselines that assume constant velocity and constant acceleration when generating predictions. Although working well over short horizons such as a few seconds, physics-based models fail to generate accurate predictions over 3 seconds. Furthermore, we compare our model with other models whose inputs are i) unnormalized global trajectories, ii) normalized trajectories, iii) normalized trajectories and ego car trajectory, and iv) normalized trajectories and both map context and ego trajectory. We observe that normalizing target car's past trajectory improve prediction performance by 12.67% in terms of MoN ADE metric. On the other hand, ego car's trajectory is not as helpful as map context in terms of improving the prediction error.
3) Risk Assessment Experiments: On a dataset of 500 scenarios similar to that shown in Figure 2, predictions were made and the risk was evaluated along a predefined trajectory for the ego vehicle as shown in Table II. To evaluate QFMVG's, we tested both the methods of Imhof and Liu-Tang-Zhang. The methods proposed are much faster than naive Monte Carlo with far lower error. The method of Imhof with an error tolerance of 10 −10 was used as ground truth [36]. Only 170 scenarios were used for error computation as results from scenarios with computed ground truth errors within tolerances (i.e: 10 −10 ) were neglected for error computation. We note that the method of Liu-Tang-Zhang empirically produces results with very small errors while being several times faster than the method of Imhof, which may prove useful in certain contexts.
B. GMM Control Predictor 1) Training Details: We use a similar DNN as the GMM position predictor, but the output becomes instead a set of GMM parameters for control signals defined in (20). When training and validating our model, instead of using the Argo- verse dataset which has noisy differentiated control data due to perception noise, we use our own data collected from a naturalistic driving simulator called CARLA [23] that provides accurate ground truth control values. The model is trained and validated on 10k samples collected in CARLA.
2) Chebyshev Experiments: In this section the half-space approximation method with 12 half-spaces was tested. The initial state of the agent vehicles was assumed to be known and deterministic. Random variables for control obtained using the DNN and moment-state dynamical systems were then used to compute the mean and covariance matrix of position at each time step. Over 50 scenarios, the mean time to evaluate the risk for a given trajectory for the Chebyshev method was 80ms while the Monte Carlo method with 10 6 samples took 140 seconds. The average worst-case conservatism of the Chebyshev risk estimate for a given time step along a trajectory was 0.012 (assuming the Monte Carlo results represent ground truth). Figure 5 shows the risk for both methods.
3) Comparing SOS + Chebyshev: Experiments were run to test and compare the Chebyshev and SOS methods described in (IV-C1) and (IV-D). For this experiment, higher order moments were obtained by automatic differentiation of the MVG moment generating function and the resulting moments of Q(x t )−1 were normalized. YALMIP was used to transcribe the SOS programs into Semidefinite programs, and SeDuMi was used to solve the resulting semidefinite programs [46], [47]. As shown in Figure 6, we observe that 1) Chebyshev bound produces nearly the same result as the second order SOS program and 2) the SOS program with higher order moments can yield significantly better bounds, especially in the tails. The solve times for each time step only marginally increased for the higher order SOS programs; the mean solve times were 42, 44, and 49 ms for the second, fourth, and sixth order SOS formulations, respectively. While these solve times obtained by solving univariate SOS programs are much better than those often encountered with multivariate SOS programs, further advances in performance are needed for this to be used online.
VIII. CONCLUSIONS
In this paper, we provided fast non-sampling based methods to propagate uncertainties and assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks. Our experimental results with risk assessment methods for learned Gaussian mixture models of position and the Chebyshev and SOS program based risk assessment methods for learned non-Gaussian position and control models suggest that high performance implementations would be immediately practical for use in online applications. As future work, we will incorporate uncertainty propagation and risk assessment into motion planning algorithms, but we note that it may be easily incorporated into standard algorithms with "collision check" primitives such as RRTs and PRMs [26]. The SOS method does significantly improve upon the risk bounds from the Chebyshev method at the cost of additional computation; future research should further develop methods to make the SOS step offline to improve runtimes for online applications as proposed in [28]. Future work in both prediction and risk assessment should also work towards relaxing assumptions such as time independence.
A. Moments and Characteristic Functions of Mixture Models
Let f X denote the pdf of a K-component mixture model X, with pdf components f Xi , ∀i ∈ [K] and let f Z denote the pdf of the K category Multinoulli. Then, by definition f X (x) = m i=1 f Xi (x)f Z (i). For any measurable function g, by interchanging the order of integration and summation, the following holds true
E[g(X)] = g(x)f X (x)dx (25) = K i=1 f Z (i) g(x)f Xi (x)dx(26)
By letting g(X) = X n or g(X) = e itX , the moments and characteristic function of X can both be computed as the weighted sum of those of their components.
B. Moments of Trigonometric Random Variables
In this section, we show how trigonometric moments of the form E[cos n (X)], E[sin n (X)], and E[cos m (X) sin n (X)] can be computed in terms of the characteristic function of the random variable X, denoted by Φ X [42], [43]. We begin by applying Euler's Identity to the definition of the characteristic function as follows:
Φ X (t) = E[e itX ] = E[cos(tX)] + iE[sin(tX)](27)
Thus, we have that E[cos(tX)] = Re(Φ X (t)) and E[sin(tX)] = Im(Φ X (t)). This immediately gives us the ability to compute the first moments of our trigonometric random variables. For higher moments, the trigonometric power formulas can be used to express quantities of the form cos n (X) as the sum of quantities of the form cos(mX) where m ∈ N and similarly for sin n (X) [48]. Thus, higher moments of sin(X) and cos(X) can be computed using Φ X (t). More precisely, given n ∈ N, trigonometric moments of order n of the forms E[cos n (X)] and E[sin n (X)] reads as [42]:
can also ultimately be computed in terms of Φ X (t). This can be seen if we make the substitutions cos(X) = 1 2 (e ix + e −ix ) and sin(X) = 1 2i (e ix − e −ix ), then (30) can be expressed as:
E 1 i n 2 m+n (e iX + e −iX ) m (e iX − e −iX ) n(31)
By applying the binomial theorem to both expressions in parentheses, and multipying the resulting expressions, we find the entire expression in the expectation operator can be expressed as a polynomial in e iX and e −iX . Thus, the entire expression can be written as the sum of terms of the form E[e itX ] for t ∈ Z which is in the definition of Φ X (t). More precisely, given (n, m) ∈ N 2 , trigonometric moment of the form E [cos m (X)sin n (X)] reads as [42]:
E [cos m (X)sin n (X)] = (32) (−i) n 2 m+n (m,n) (k1,k2)=(0,0) m k1 n k2 (−1) n−k2 Φ X (2(k 1 + k 2 ) − m − n)
Fig. 1 .
1Illustration of our risk assessment framework when control distributions are used for predictions. When position distributions are used, the nonlinear moment propagation step is skipped.
Fig. 2 .
2Fig. 2.
Remark 1 :
1We can solve the optimization in(18) to obtain the polynomial indicator function i.e., p(x) = d k=0 c k x k , in the offline step. Hence, we can compute the upper bound of the risk in terms of the obtained polynomial indicator function and moments of the uncertain states of the agent vehicle, in real-time, i.e., P(g
Fig. 3 .
3Illustration of trajectory normalization. Left: an observed trajectory in global coordinate. Right: an observed trajectory in its local coordinate, after normalization.
Fig. 5 .
5Risk estimates across time for an example scenario using random variables from the GMM control predictor.
Fig. 6 .
6Risk bounds computed with the SOS formulation from section IV-D compared with Chebyshev's inequality without half-space approximations.
of the form: E[cos m (X) sin n (X)]
This will produces a closed form set of equations that can recursively compute E[x t+1 ] in terms of the moments of the initial uncertain states and moments of uncertain control inputs at each time step.). Also, E[cos(ω θt )] and E[sin(ω θt )] are the
first order trigonometric moments of uncertain control input
ω θt and can be computed using it's characteristic function as
shown in Appendix.B. To complete the obtained update rule,
we need to compute the update rule of the terms E[v t sin(θ t )],
E[cos(θ t )], and E[sin(θ t )]. By doing so, we can complete
the dynamics of the moment E[x t ] in (21) in terms of a set
of slack moments E[v t sin(θ t )], E[v t cos(θ t )], E[cos(θ t )], and
E[sin(θ t )].
TABLE I RESULTS
IFROM GMM POSITION PREDICTOR AND BASELINES OVER A PREDICTION HORIZON OF THREE SECONDS. FOR EACH PREDICTOR, THREE PREDICTION SAMPLES ARE SELECTED AND THE ERROR OF THE BEST SAMPLE IS REPORTED.Model
MoN ADE (m) MoN FDE (m)
Constant Velocity
1.75
3.96
Constant Acceleration
3.17
8.12
DNN Unnormalized
1.50
2.80
DNN Normalized
1.31
2.64
DNN Normalized+Ego
1.30
2.62
DNN Normalized+Ego+Map 1.22
2.49
TABLE II RISK
IIEVALUATION RESULTS FOR 500 SCENARIOS INVOLVING THIRTY TIME STEPS THREE-MODE GMM PREDICTIONS. ERRORS CORRESPOND TO THE TIME STEP WITH THE MAXIMUM ERROR.Method
Mean Time
(ms)
Mean Max.
Absolute Error
Mean Max.
Relative Error
Imhof
91.21
0.0
0.0
Liu-Tang-Zhang 26.67
2.7 × 10 −6
2.3 × 10 −4
MC 10 4
106.9
6.7 × 10 −4
0.38
MC 5 × 10 4
422.5
2.7 × 10 −4
0.13
MC 10 5
1329
1.9 × 10 −4
0.12
ACKNOWLEDGEMENTSThis work was supported in part by Boeing grant MIT-BA-GTA-1 and by the Masdar Institute grant 6938857. Allen Wang was supported in part by a NSF Graduate Research Fellowship.IX. APPENDIX
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The unscented kalman filter for nonlinear estimation. E A Wan, R Van Der Merwe, Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373). the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373)IeeeE. A. Wan and R. Van Der Merwe, "The unscented kalman filter for nonlinear estimation," in Proceedings of the IEEE 2000 Adaptive Sys- tems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373). Ieee, 2000, pp. 153-158.
New results in linear filtering and prediction theory. R E Kalman, R S Bucy, R. E. Kalman and R. S. Bucy, "New results in linear filtering and prediction theory," 1961.
Unscented filtering and nonlinear estimation. S J Julier, J K Uhlmann, Proceedings of the IEEE. 923S. J. Julier and J. K. Uhlmann, "Unscented filtering and nonlinear estimation," Proceedings of the IEEE, vol. 92, no. 3, pp. 401-422, 2004.
Moment-based exact uncertainty propagation through nonlinear stochastic autonomous systems. A Jasour, A Wang, B C Williams, arXiv:2101.12490A. Jasour, A. Wang, and B. C. Williams, "Moment-based exact uncer- tainty propagation through nonlinear stochastic autonomous systems," arXiv:2101.12490, 2021.
Fast risk assessment for autonomous vehicles using learned models of agent futures. A Wang, RSSX Huang, RSSA Jasour, RSSB C Williams, RSSRobotics: Science and System. 2020A. Wang, X. Huang, A. Jasour, , and B. C. Williams, "Fast risk assessment for autonomous vehicles using learned models of agent futures," Robotics: Science and System (RSS), 2020.
Non-gaussian chanceconstrained trajectory planning for autonomous vehicles in the presence of uncertain agents. A Wang, A Jasour, B C Williams, IEEE Robotics Automation Letters (RA-L). 54A. Wang, A. Jasour, , and B. C. Williams, "Non-gaussian chance- constrained trajectory planning for autonomous vehicles in the presence of uncertain agents," IEEE Robotics Automation Letters (RA-L), vol. 5(4), pp. 6041-6048, 2020.
Social LSTM: Human trajectory prediction in crowded spaces. A Alahi, K Goel, V Ramanathan, A Robicquet, L Fei-Fei, S Savarese, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionA. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. Savarese, "Social LSTM: Human trajectory prediction in crowded spaces," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 961-971.
YALMIP: A toolbox for modeling and optimization in MAT-LAB. J Löfberg, Proceedings of the CACSD Conference. the CACSD ConferenceTaipei, TaiwanJ. Löfberg, "YALMIP: A toolbox for modeling and optimization in MAT- LAB," in In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. J F Sturm, Optimization methods and software. 11J. F. Sturm, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones," Optimization methods and software, vol. 11, no. 1-4, pp. 625-653, 1999.
D Zwillinger, CRC standard mathematical tables and formulae. Chapman and Hall/CRC. D. Zwillinger, CRC standard mathematical tables and formulae. Chap- man and Hall/CRC, 2002.
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"Extreme Climate Variations from Milankovitch-like Eccentricity Oscillations in Extrasolar Planetary Systems Climate Change at the Eve of the Second Decade of the Century",
"Extreme Climate Variations from Milankovitch-like Eccentricity Oscillations in Extrasolar Planetary Systems Climate Change at the Eve of the Second Decade of the Century"
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| Although our solar system features predominantly circular orbits, the exoplanets discovered so far indicate that this is the exception rather than the rule. This could have crucial consequences for exoplanet climates, both because eccentric terrestrial exoplanets could have extreme seasonal variations, and because giant planets on eccentric orbits could excite Milankovitch-like variations of a potentially habitable terrestrial planets eccentricity, on timescales of thousandsto-millions of years. A particularly interesting implication concerns the fact that the Earth is thought to have gone through at least one globally frozen, "snowball" state in the last billion years that it presumably exited after several million years of buildup of greenhouse gases when the ice-cover shut off the carbonatesilicate cycle. Water-rich extrasolar terrestrial planets with the capacity to host life might be at risk of falling into similar snowball states. Here we show that if a terrestrial planet has a giant companion on a sufficiently eccentric orbit, it can undergo Milankovitch-like oscillations of eccentricity of great enough magnitude to melt out of a snowball state.ProceedingEven very mild astronomical forcings can have dramatic influence on the Earth's climate. Although the orbital eccentricity varies between ∼0 and only ∼0.06, and the axial tilt, or obliquity, between ∼22.1 • and 24.5 • , these slight quasi-periodic changes are sufficient to help drive the Earth into ice ages at regular intervals. Milankovitch articulated this possibility in his astronomical theory of climate change. Specifically, Milankovitch posited a causal connection between three astronomical cycles (precession -23 kyr period, and variation of both obliquity and eccentricity -41-kyr and 100-kyr periods, respectively) and the onset of glaciation/deglaciation. Though much remains to be discovered about these cycles, often in the literature referred to as "Milankovitch cycles," 1 they are now generally acknowledged to 1 Or as "Croll-Milankovitch cycles" (1; 2). | null | [
"https://arxiv.org/pdf/1010.2197v1.pdf"
]
| 118,226,943 | 1010.2197 | 16a34eb69eb6ae57310b9c3d7359b50ed9b03e6a |
Extreme Climate Variations from Milankovitch-like Eccentricity Oscillations in Extrasolar Planetary Systems Climate Change at the Eve of the Second Decade of the Century
2009
David S Spiegel [email protected]
Extreme Climate Variations from Milankovitch-like Eccentricity Oscillations in Extrasolar Planetary Systems Climate Change at the Eve of the Second Decade of the Century
Proceedings of the Milutin Milankovitch Anniversary Symposium
the Milutin Milankovitch Anniversary Symposium2009
Although our solar system features predominantly circular orbits, the exoplanets discovered so far indicate that this is the exception rather than the rule. This could have crucial consequences for exoplanet climates, both because eccentric terrestrial exoplanets could have extreme seasonal variations, and because giant planets on eccentric orbits could excite Milankovitch-like variations of a potentially habitable terrestrial planets eccentricity, on timescales of thousandsto-millions of years. A particularly interesting implication concerns the fact that the Earth is thought to have gone through at least one globally frozen, "snowball" state in the last billion years that it presumably exited after several million years of buildup of greenhouse gases when the ice-cover shut off the carbonatesilicate cycle. Water-rich extrasolar terrestrial planets with the capacity to host life might be at risk of falling into similar snowball states. Here we show that if a terrestrial planet has a giant companion on a sufficiently eccentric orbit, it can undergo Milankovitch-like oscillations of eccentricity of great enough magnitude to melt out of a snowball state.ProceedingEven very mild astronomical forcings can have dramatic influence on the Earth's climate. Although the orbital eccentricity varies between ∼0 and only ∼0.06, and the axial tilt, or obliquity, between ∼22.1 • and 24.5 • , these slight quasi-periodic changes are sufficient to help drive the Earth into ice ages at regular intervals. Milankovitch articulated this possibility in his astronomical theory of climate change. Specifically, Milankovitch posited a causal connection between three astronomical cycles (precession -23 kyr period, and variation of both obliquity and eccentricity -41-kyr and 100-kyr periods, respectively) and the onset of glaciation/deglaciation. Though much remains to be discovered about these cycles, often in the literature referred to as "Milankovitch cycles," 1 they are now generally acknowledged to 1 Or as "Croll-Milankovitch cycles" (1; 2).
have been the dominant factor governing the climate changes of the last several million years (3; 4; 5; 6; 7).
The nonzero (but, at just 0.05, nearly zero) eccentricity of Jupiter's orbit is the primary driver of the Earth's eccentricity Milankovitch cycle. Were Jupiter's eccentricity greater, it would drive larger amplitude variations of the Earth's eccentricity. This same mechanism might be operating in other solar systems. In the last 15 years, roughly ∼500 extrasolar planets have been discovered around other stars, where an object is here defined as a "planet" by the condition that it will not burn significant amounts of deuterium, which corresponds approximately to 13 Jupiter masses (8). (This might not be the best way to define exoplanets, but it is probably the most widely used.). Among these ∼500, there are many that have masses comparable to Jupiter's and that are on highly eccentric orbits; ∼20% of the known exoplanets have eccentricities greater than 0.4, including such extreme values as 0.93 and 0.97 (HD 20782b; HD 80606). Furthermore, tantalizing evidence suggests that lower mass terrestrial planets might be even more numerous than the giant planets that are easier to detect. Therefore, it seems highly likely that many terrestrial planets in our galaxy experience exaggerated versions of the Earth's eccentricity Milankovitch cycle.
These kinds of cycles could have dramatic influence on life that requires liquid water. Since the seminal work of Milankovitch several decades ago, a variety of theoretical investigations have examined the possible climatic habitability of terrestrial exoplanets. Kasting and collaborators emphasized that the habitability of an exoplanet depends on the properties of the host star (9). Several authors have considered how a planet's climatic habitability depends on the properties of the planet, as well. In particular, two recent papers have focused on the climatic effect of orbital eccentricity. Williams & Pollard used a general circulation climate model to address the question of how the Earth's climate would be affected by a more eccentric orbit (10). Dressing et al. used an energy balance climate model (11) to explore the combined influences of eccentricity and obliquity on the climates of terrestrial exoplanets with generic surface geography (see also (12) and (13; 14) for further description of the model). A more eccentric orbit both accentuates the difference between stellar irradiation at periastron and at apoastron, and increases the annually averaged irradiation. Thus, periodic oscillations of eccentricity will cause concomitant oscillations of both the degree of seasonal extremes and of the total amount of starlight incident on the planet in each annual cycle. Since these oscillations depend on gravitational perturbations from other companion objects, the present paper can be thought of as examining how a terrestrial planet's climatic habitability depends not just on its star, not just on its own intrinsic properties, but also on the properties of the planetary system in which it resides.
There is evidence that, at some point in the last billion years, Earth went through a "Snowball Earth" state in which it was fully (or almost fully) covered with snow and ice. The high albedo of ice gives rise to a positive feedback loop in which decreasing surface temperatures lead to greater ice-cover and therefore to further net cooling. As a result, the existence of a low-temperature equilibrium climate might be a generic feature of water-rich terrestrial planets, and such planets might have a tendency to enter snowball states. The icealbedo feedback makes it quite difficult for a planet to recover from such a state. In temperate conditions, the Earth's carbonate-silicate weathering cycle acts as a "chemical thermostat" that tends to prevent surface temperatures from straying too far from the freezing point of water. A snowball state would interrupt this cycle. The standard explanation of how the Earth might have exited its snowball state is that this interruption of the weathering cycle would have allowed carbon dioxide to build up to concentrations approaching ∼1 bar over a million-to-10-million years, at which point the greenhouse effect would have been sufficient to melt the ice-cover and restore temperate conditions. 2 However, an exoplanet in a snowball state that is undergoing a large excitation of its eccentricity might be able to melt out of its globally frozen state in significantly less time, depending on the magnitude of the eccentricity variations and on other properties of the planet. Exploring this possibility is the primary focus of (17), in which, using an energy balance climate model, we searched for orbital configurations that would lead to an icecovered planet melting out of the snowball state. In brief, we found that orbital configurations that are not unlikely could cause a snowball-Earth-analog to melt out by dint of increased eccentricity. Figure 1 shows the temperature evolution of two cold-start planet models, one of which (on the right) has a crude approximation of a carbonate-silicate cycle incorporated in the infrared cooling term, and the other (on the left) does not. Both model planets have orbital semimajor axis 1 AU, and are initialized to very cold temperatures. The high orbital eccentricity of these models (0.8) causes them to intercept more stellar irradiation over the annual cycle than would a model on a circular orbit. They therefore heat rapidly and, with a crude accounting of the latent heat of melting/freezing water (17), are eventually able to melt through the ice layer. Figure 2 shows two different compressed Milankovitch-like cycles. In each, a cycle that might take 10,000 -400,000 years is compressed to 25 years, for computational feasibility and visualization purposes. In one (the top row), the planet is at semimajor axis 1 AU and has eccentricity varying sinusoidally between 0 and 0.83. In the other (bottom row), the planet is at semimajor axis 0.8 AU and has eccentricity varying between 0.1 and 0.33. In each case, after several years, a "catastrophic event" dramatically increases the albedo for several years, so as to plunge the model planet into a snowball state. The increasing eccentricity, then, eventually leads the planet to melt out of the snowball state. Finally, see Figures 3 and 4 of (17) for exemples of the magnitudes and frequencies of Milankovitch-like eccentricity oscillations that can result from gravitational interactions between an eccentric giant planet and a terrestrial planet. Though these kinds of oscillations might be rare, they are not impossible. Entirely prosaic planetary system architectures can lead to less dramatic, but still highly important, variations of a terrestrial planet's eccentricity.
In the coming years, as new observatories such as the James Webb Space Telescope come online, exploring the atmospheres and atmospheric dynamics of exoplanets will become an increasingly tractable research problem. Already, planets of the hot Jupiter class have been amenable to investigation with the Spitzer Space Telescope, Kepler, and various groundbased observatories (see, e.g., (18; 19; 20; 21; 22; 23; 24), and more). It might even be possible to probe the atmospheric composition of even extremely distant exoplanets, in the Galactic bulge (25). Increasingly, it is possible to learn about the properties of Neptunemass exoplanets (26; 27; 28). Discerning the spectral signatures of habitability and of life on terrestrial planets will be the next frontier (29). As the field of exoplanets matures, it will be important to keep in mind that the long-term climatic habitability of a planet might depend not just on the intrinsic properties of the host star and of the planet itself, but also on the detailed architecture of the planetary system in which the planet resides. Temperature is initialized to 100 K, and quickly rises to near 273 K. The melting of the ice-cover is handled in accordance with the prescription of (17). Left: CO 2 partial pressure is held constant at 0.01 bars. In this model, once the equatorial region melts, the region of surface that has melted ice-cover grows steadily until the entire planet has melted, and temperatures eventually grow to more than 400 K over much of the planet (not shown). Right: CO 2 partial pressure varies with temperature, in a crude simulation of a "chemical thermostat". In this model, the climate reaches a stable state with equatorial melt regions and polar ice-cover. (K) 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 Fig. 1, except the eccentricity varies sinusoidally between 0 and 0.83 with a 25-year period, to simulate a time-acceleration (by a factor of ∼10 2 to ∼10 4 ) of a Milankovitch-like cycle. When the eccentricity falls below 0.05, the planet's albedo spikes to 0.8, simulating a catastrophic event that plunges the planet into a snowball state, with the latent heat prescription of (17). In the bottom row (0.8 AU), the eccentricity varies between 0.1 and 0.33, also with a 25-year period. Left: CO 2 partial pressure is held fixed at 0.01 bars. As in the left panel of Fig. 1, these planets do not establish a temperate equilibrium. Right: CO 2 partial pressure varies with temperature. Here, temperature increases are muted by reduced greenhouse effect once the ice-cover has melted somewhere.
( 7 )
7Berger, A., Mélice, J. L. & Loutre, M. F. On the origin of the 100-kyr cycles in the astronomical forcing. Paleoceanography 20, A264019+ (2005). (8) Spiegel, D. S., Burrows, A. & Milsom, J. A. The Deuterium-Burning Mass Limit for Brown Dwarfs and Giant Planets. ArXiv e-prints (2010). arXiv:1008.5150. (9) Kasting, J. F., Whitmire, D. P. & Reynolds, R. T. Habitable Zones around Main Sequence Stars. Icarus 101, 108-128 (1993). (10) Williams, D. M. & Pollard, D. Extraordinary climates of Earth-like planets: threedimensional climate simulations at extreme obliquity. International Journal of Astrobiology 2, 1-19 (2003). (11) Dressing, C. D., Spiegel, D. S., Scharf, C. A., Menou, K. & Raymond, S. N. Habitable Climates: The Influence of Eccentricity. ApJ 721, 1295-1307 (2010). 1002.4875. (12) Williams, D. M. & Kasting, J. F. Habitable Planets with High Obliquities. Icarus 129, 254-267 (1997). (13) Spiegel, D. S., Menou, K. & Scharf, C. A. Habitable Climates. ApJ 681, 1609-1623 (2008). arXiv:0711.4856. (14) Spiegel, D. S., Menou, K. & Scharf, C. A. Habitable Climates: The Influence of Obliquity. ApJ 691, 596-610 (2009). 0807.4180. (15) Nordhaus, J. & Blackman, E. G. Low-mass binary-induced outflows from asymptotic giant branch stars. MNRAS 370, 2004-2012 (2006). arXiv:astro-ph/0604445. (16) Nordhaus, J., Spiegel, D. S., Ibgui, L., Goodman, J. & Burrows, A. Tides and tidal engulfment in post-main-sequence binaries: period gaps for planets and brown dwarfs around white dwarfs. MNRAS 1164-+ (2010). 1002.2216. (17) Spiegel, D. S., Raymond, S. N., Dressing, C. D., Scharf, C. A. & Mitchell, J. L. Generalized Milankovitch Cycles and Long-Term Climatic Habitability. ApJ 721, 1308-1318 (2010). 1002.4877. (18) Harrington, J. et al. The Phase-Dependent Infrared Brightness of the Extrasolar Planet Upsilon Andromedae b. Science 314, 623-626 (2006). astro-ph/0610491.
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19Knutson, H. A. et al. A map of the day-night contrast of the extrasolar planet HD 189733b. Nature 447, 183-186 (2007). arXiv:0705.0993. (20) Fortney, J. J., Lodders, K., Marley, M. S. & Freedman, R. S. A Unified Theory for the Atmospheres of the Hot and Very Hot Jupiters: Two Classes of Irradiated Atmospheres. ApJ 678, 1419-1435 (2008). 0710.2558.
Fig. 1 .
1-Temperature evolution maps for cold-start models at 1 AU. Both models have orbital eccentricity of 0.8 along with Earth-like 23.5 • polar obliquity and 1 bar surface pressure.
Fig. 2 .
2-Compressed Milankovitch-like evolution of eccentricity and temperature at 1 AU and at 0.8 AU. Planets are initialized with warm equator and cold poles, similar to presentday Earth. In the top row (1 AU), the model planets are the same as in
Even an ice-encrusted planet in the habitable zone will eventually melt, due to the post-main sequence red-giant evolution of a Sun-like star, as the star grows larger and brighter. The planet will not enjoy temperate conditions for long, however, as the continued growth in size and luminosity of the giant will eventually sterilize it of any water-based surface life. Whether the Earth will be engulfed by the Sun in its giant phase (either by direct expansion or by tidal decay of its orbit), or will survive through the planetary nebula phase, remains an open question (15; 16).
This preprint was prepared with the AAS L A T E X macros v5.2.
AcknowledgmentsThe author would like to thank the contributions of Sean Raymond, Courtney Dressing, Caleb Scharf, Kristen Menou, and Jonathan Mitchell. Furthermore, DSS gratefully acknowledges the participants and organizers of the Milutin Milankovitch Anniversary Symposium, 2009, in particular Fedor Mesinger and Andre Berger.
Climate and time in their geological relations: a theory of secular changes of the earth's climate (Daldy, Tsbister & co. J Croll, LondonCroll, J. Climate and time in their geological relations: a theory of secular changes of the earth's climate (Daldy, Tsbister & co., London, 1875).
Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem. M Milankovitch, Belgrade, Mihaila CurcicaMilankovitch, M. Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeiten- problem (Belgrade, Mihaila Curcica, 1941).
Long-period variations of the earth's orbital elements -A question of precision in astronomical theory of paleoclimates. I. A Berger, Ciel et Terre. 91Berger, A. Long-period variations of the earth's orbital elements -A question of precision in astronomical theory of paleoclimates. I. Ciel et Terre 91, 261-277 (1975).
Variations in the Earth's Orbit: Pacemaker of the Ice Ages. J D Hays, J Imbrie, N J Shackleton, Science. 194Hays, J. D., Imbrie, J. & Shackleton, N. J. Variations in the Earth's Orbit: Pacemaker of the Ice Ages. Science 194, 1121-1132 (1976).
Obliquity and precession for the last 5 000 000 years. A L Berger, A&A. 51Berger, A. L. Obliquity and precession for the last 5 000 000 years. A&A 51, 127-135 (1976).
Long-Term Variations of Daily Insolation and Quaternary Climatic Changes. A L Berger, Journal of Atmospheric Sciences. 35Berger, A. L. Long-Term Variations of Daily Insolation and Quaternary Climatic Changes. Journal of Atmospheric Sciences 35, 2362-2367 (1978).
Can TiO Explain Thermal Inversions in the Upper Atmospheres of Irradiated Giant Planets?. D S Spiegel, K Silverio, A Burrows, 0902.3995ApJ. 699Spiegel, D. S., Silverio, K. & Burrows, A. Can TiO Explain Thermal Inversions in the Upper Atmospheres of Irradiated Giant Planets? ApJ 699, 1487-1500 (2009). 0902.3995.
Atmospheric Circulation of Hot Jupiters: Coupled Radiative-Dynamical General Circulation Model Simulations of HD 189733b and HD 209458b. A P Showman, 0809.2089ApJ. 699Showman, A. P. et al. Atmospheric Circulation of Hot Jupiters: Coupled Radiative- Dynamical General Circulation Model Simulations of HD 189733b and HD 209458b. ApJ 699, 564-584 (2009). 0809.2089.
A Temperature and Abundance Retrieval Method for Exoplanet Atmospheres. N Madhusudhan, S Seager, 0910.1347ApJ. 707Madhusudhan, N. & Seager, S. A Temperature and Abundance Retrieval Method for Exoplanet Atmospheres. ApJ 707, 24-39 (2009). 0910.1347.
Atmosphere and Spectral Models of the Kepler-field Planets HAT-P-7b and TrES-2. D S Spiegel, A Burrows, 1006.1660ApJ. 722Spiegel, D. S. & Burrows, A. Atmosphere and Spectral Models of the Kepler-field Planets HAT-P-7b and TrES-2. ApJ 722, 871-879 (2010). 1006.1660.
Can We Probe the Atmospheric Composition of an Extrasolar Planet from Its Reflection Spectrum in a High-Magnification Microlensing Event?. D S Spiegel, M Zamojski, A Gersch, J Donovan, Z Haiman, arXiv:astro-ph/0501107ApJ. 628Spiegel, D. S., Zamojski, M., Gersch, A., Donovan, J. & Haiman, Z. Can We Probe the Atmospheric Composition of an Extrasolar Planet from Its Reflection Spec- trum in a High-Magnification Microlensing Event? ApJ 628, 478-486 (2005). arXiv:astro-ph/0501107.
Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations. B Demory, 0707.3809A&A. 475Demory, B. et al. Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations. A&A 475, 1125-1129 (2007). 0707.3809.
Models of Neptune-Mass Exoplanets: Emergent Fluxes and Albedos. D S Spiegel, A Burrows, L Ibgui, I Hubeny, J A Milsom, 0909.2043ApJ. 709Spiegel, D. S., Burrows, A., Ibgui, L., Hubeny, I. & Milsom, J. A. Models of Neptune-Mass Exoplanets: Emergent Fluxes and Albedos. ApJ 709, 149-158 (2010). 0909.2043.
The dayside atmosphere of the hot-Neptune GJ 436b. N Madhusudhan, S Seager, 1004.5121ArXiv e-printsMadhusudhan, N. & Seager, S. The dayside atmosphere of the hot-Neptune GJ 436b. ArXiv e-prints (2010). 1004.5121.
Deciphering Spectral Fingerprints of Habitable Exoplanets. L Kaltenegger, 0906.2263Astrobiology. 10Kaltenegger, L. et al. Deciphering Spectral Fingerprints of Habitable Exoplanets. As- trobiology 10, 89-102 (2010). 0906.2263.
| []
|
[
"IEEE TRANSACTIONS ON MULTIMEDIA 1 RGBT Salient Object Detection: A Large-scale Dataset and Benchmark",
"IEEE TRANSACTIONS ON MULTIMEDIA 1 RGBT Salient Object Detection: A Large-scale Dataset and Benchmark"
]
| [
"Zhengzheng Tu ",
"Yan Ma ",
"Zhun Li ",
"Chenglong Li ",
"Jieming Xu ",
"Yongtao Liu "
]
| []
| []
| Salient object detection in complex scenes and environments is a challenging research topic. Most works focus on RGB-based salient object detection, which limits its performance of real-life applications when confronted with adverse conditions such as dark environments and complex backgrounds. Taking advantage of RGB and thermal infrared(RGBT) images becomes a new research direction for detecting salient objects in complex scenes, since the thermal infrared spectrum provides the complementary information and has been used in many computer vision tasks. However, current research for RGBT salient object detection is limited by the lack of a large-scale dataset and comprehensive benchmark. This work contributes such a RGBT image dataset named VT5000, including 5000 spatially aligned RGBT image pairs with ground truth annotations. VT5000 has 11 challenges collected in different scenes and environments for exploring the robustness of algorithms. With this dataset, we propose a powerful baseline approach, which extracts multilevel features of each modality and aggregates these features of all modalities with the attention mechanism for accurate RGBT salient object detection. To further solve the problem of blur boundaries of salient objects, we also use an edge loss to refine the boundaries. Extensive experiments show that the proposed baseline approach outperforms the state-of-the-art methods on VT5000 dataset and other two public datasets. In addition, we carry out a comprehensive analysis of different algorithms of RGBT salient object detection on VT5000 dataset, and then make several valuable conclusions and provide some potential research directions for RGBT salient object detection. Our new VT5000 dataset is made publicly available at https://github.com/lz118/RGBT-Salient-Object-Detection. | 10.1109/tmm.2022.3171688 | [
"https://arxiv.org/pdf/2007.03262v6.pdf"
]
| 220,381,440 | 2007.03262 | d6c973302dfa2518e129097e9bf5df780bce5ab4 |
IEEE TRANSACTIONS ON MULTIMEDIA 1 RGBT Salient Object Detection: A Large-scale Dataset and Benchmark
23 May 2022
Zhengzheng Tu
Yan Ma
Zhun Li
Chenglong Li
Jieming Xu
Yongtao Liu
IEEE TRANSACTIONS ON MULTIMEDIA 1 RGBT Salient Object Detection: A Large-scale Dataset and Benchmark
23 May 2022Index Terms-Salient object detectionAttentionVT5000 dataset
Salient object detection in complex scenes and environments is a challenging research topic. Most works focus on RGB-based salient object detection, which limits its performance of real-life applications when confronted with adverse conditions such as dark environments and complex backgrounds. Taking advantage of RGB and thermal infrared(RGBT) images becomes a new research direction for detecting salient objects in complex scenes, since the thermal infrared spectrum provides the complementary information and has been used in many computer vision tasks. However, current research for RGBT salient object detection is limited by the lack of a large-scale dataset and comprehensive benchmark. This work contributes such a RGBT image dataset named VT5000, including 5000 spatially aligned RGBT image pairs with ground truth annotations. VT5000 has 11 challenges collected in different scenes and environments for exploring the robustness of algorithms. With this dataset, we propose a powerful baseline approach, which extracts multilevel features of each modality and aggregates these features of all modalities with the attention mechanism for accurate RGBT salient object detection. To further solve the problem of blur boundaries of salient objects, we also use an edge loss to refine the boundaries. Extensive experiments show that the proposed baseline approach outperforms the state-of-the-art methods on VT5000 dataset and other two public datasets. In addition, we carry out a comprehensive analysis of different algorithms of RGBT salient object detection on VT5000 dataset, and then make several valuable conclusions and provide some potential research directions for RGBT salient object detection. Our new VT5000 dataset is made publicly available at https://github.com/lz118/RGBT-Salient-Object-Detection.
Abstract-Salient object detection in complex scenes and environments is a challenging research topic. Most works focus on RGB-based salient object detection, which limits its performance of real-life applications when confronted with adverse conditions such as dark environments and complex backgrounds. Taking advantage of RGB and thermal infrared(RGBT) images becomes a new research direction for detecting salient objects in complex scenes, since the thermal infrared spectrum provides the complementary information and has been used in many computer vision tasks. However, current research for RGBT salient object detection is limited by the lack of a large-scale dataset and comprehensive benchmark. This work contributes such a RGBT image dataset named VT5000, including 5000 spatially aligned RGBT image pairs with ground truth annotations. VT5000 has 11 challenges collected in different scenes and environments for exploring the robustness of algorithms. With this dataset, we propose a powerful baseline approach, which extracts multilevel features of each modality and aggregates these features of all modalities with the attention mechanism for accurate RGBT salient object detection. To further solve the problem of blur boundaries of salient objects, we also use an edge loss to refine the boundaries. Extensive experiments show that the proposed baseline approach outperforms the state-of-the-art methods on VT5000 dataset and other two public datasets. In addition, we carry out a comprehensive analysis of different algorithms of RGBT salient object detection on VT5000 dataset, and then make several valuable conclusions and provide some potential research directions for RGBT salient object detection. Our new VT5000 dataset is made publicly available at https://github.com/lz118/RGBT-Salient-Object-Detection.
Index Terms-Salient object detection, Attention, VT5000 dataset.
I. INTRODUCTION
S ALIENT object detection aims to find the object that human eyes pay much attention to in an image. Salient object detection has been extensively studied over the past decade, but still faces many challenges in complex environment, e.g., when appearance of the object is similar to the surrounding, the algorithms of salient object detection for RGB images Z. Tu often perform not well. Researches on adopting different modalities to assist salient object detection have attracted more and more attentions. Many works [1], [2] have achieved good results on salient object detection by combining RGB images with depth information. However, depth image has its limitations, for example, when the object is perpendicular to the shot of depth camera, the depth map shows inconsistent values of the same surface. Integrating RGB and thermal infrared (RGBT) data has also shown its effectiveness for some computer vision tasks, such as moving object detection, person Re-ID, and visual tracking [3], [4], [5]. The imaging principle of thermal infrared camera is based on thermal radiation from the object surface, and different places of object surface have almost same thermal radiation. Existing RGB and Depth(RGBD) salient object detection methods mainly focus on how to explore depth information to complement RGB SOD. However, RGBT salient object detection treats RGB and thermal modalities equally and its target is to leverage complementary advantages to discover common conspicuous objects in both modalities.
Recently, RGBT salient object detection has become attractive. The first work of RGBT salient object detection [5] proposes a multi-task manifold ranking algorithm for RGBT image saliency detection, and creates an unified RGBT dataset called VT821. In addition, VT821 has several limitations: (1) RGB and thermal imaging parameters are completely different, and there might thus be some alignment errors; (2) The operators for modalities alignment will introduce the black background in image boundaries; (3) Most of these scenarios are very simple, and thus it is not so challenging. The second important work of RGBT image saliency detection [6] contributes a more challenging dataset named VT1000 and proposes a novel collaborative graph learning algorithm. Compared with VT821, VT1000 has its advantages but also have several limitations: (1) As RGB and thermal infrared imaging have different sighting distances, thermal infrared image and visible light image look different and need to be aligned, as shown as the left image pair in Fig. 3; (2) RGB image and thermal infrared image are still not automatically aligned, inevitably introducing errors in the process of manually aligning them; (3) Although VT1000 is larger than VT821, the complexity and diversity of the scenes have not been greatly improved.
In this paper, we construct a more comprehensive benchmark for RGBT salient object detection based on the demands of large-scale, good resolution, high diversity and low bias. First, existing RGBT datasets are not enough to train a good deep network, thus we collect 5000 pairs of RGB and thermal images in different environments, in which each pair of RGBT images is automatically aligned. Second, as most of the backgrounds or scenes are simple in existing datasets, our dataset considers different sizes, categories, surroundings, imaging quantities and spatial locations of salient objects, and we also give a statistical analysis to show the diversity of objects. To analyze the sensitivity of different methods for various challenges, we annotate 11 different challenges in the consideration of above factors. Finally, we annotate not only attributes of challenges but also imaging quality of objects in the dataset. The annotations of imaging quality of objects provide the labels for weakly supervised RGBT salient object detection in the future researches.
Although significant progress has been made in terms of RGBT saliency detection [5], [6], [7], the performance might be limited by three main problems: (1) Both of these works are based on traditional methods, but they simply merge the features of the two modalities without proposing an effective way for feature fusion ; (2) Background noise is introduced when the features of the two modalities are fused; (3) The problem of unclear object boundaries is not solved effectively. To provide a powerful baseline for RGBT salient object detection, we design an end-to-end trained CNN-based framework. In specific, the two-stream CNN architecture employs VGG16 [8] as the backbone network to extract multi-scale RGB and thermal infrared features separately. To obtain task-related features, we use channel-wise and spatial-wise attention based Convolution Block Attention Module (CBAM) [9] to selectively collect features from RGB and thermal infrared branches. Then we perform a pixel-wise addition on RGB and thermal infrared features to fuse them, and pass the merged features from first convolutional block of VGG16 to next convolutional block. To obtain global guidance information, we input the fused RGB and thermal infrared features from last convolutional block into the Pyramid Pooling Module (PPM) [10]. We also use the average pooling to capture the global context information, thus can locate salient objects accurately. To make better use of characteristics of different layers, we upsample the features processed by each block of VGG16 with different sampling rates, and then combine them with the features processed by PPM. We also utilize the Feature Aggregation Module (FAM) [10] after feature fusion to capture the local context information. Finally, to solve the problem of unclear object boundaries, we employ an edge loss to guide the network to learn more details around boundaries, and further refine the boundary of the object.
In summary, the main contributions of this work are summarized as follows:
• We create a large-scale RGBT dataset containing 5000 pairs of RGB and thermal images for salient object detection, with manually labeled ground truth annotations. We hope that this dataset would promote the research progress for deep learning techniques on RGBT salient object detection. This dataset and its all annotations will be released to public for free academic usage.
• We propose a novel deep CNN architecture to provide a powerful baseline approach for RGBT salient object detection. In particular, we propose to utilize a convolu-tional block attention to selectively collect features from RGB and thermal infrared branches and an edge loss to guide the network to learn more details that can reserve the boundaries of salient objects.
• Extensive experiments show that the designed approach outperforms the state-of-the-art methods on VT5000 dataset and other two public datasets, i.e., VT821 and VT1000. In addition, a comprehensive analysis for different algorithms of RGBT salient object detection is performed on VT5000 dataset. Through the analysis, we make several valuable conclusions and provide some potential research directions for RGBT salient object detection.
II. RELATED WORK
A. Multi-modal Salient Object Detection Datasets
With the emergence of multi-modal data, RGBD salient object detection (SOD) has been proposed, and the related RGBD datasets have been constructed.
NJU2K [12] collects 1,985 RGBD image pairs which are from the Internet and 3D movies or taken by a Fuji W3 stereo camera. NLPR [13] uses Microsoft Kinect captures 1,000 RGBD image pairs. DES [14] also uses Microsoft Kinect to collect 135 RGBD image pairs in indoor scenes, and it is also called RGBD135. SSD [15] contains 80 image pairs picked up from three stereo movies. STERE [16] has 1000 image pairs collected in real-world scenes and virtual scenes. More specifically, GIT [1] and LFSD [11] datasets are designed for the specific purposes, such as generic object segmentation based on saliency map or saliency detection in the light field.
Subsequently, Li et al. [5] construct the first RGBT dataset VT821 with 821 pairs of RGBT images. Tu et al. [6] contribute a more challenging dataset VT1000 for RGBT image saliency detection.
B. Attention Mechanism
Attention mechanism is first proposed by Bahdanau et al. [17] for neural machine translation, the attention mechanisms in deep neural networks have been studied widely recently. Attention mechanisms are proven to be useful in many tasks, such as scene recognition [18], [19], question answering [20], caption generation [21] and pose estimation [22]. For example, Chu et al. [22] propose a network based on multi-context attention mechanism and apply it to the end-toend framework of pose estimation. Zhang et al. [23] propose a progressive attention guidance network, which generates attention features successively through the channel and the spatial attention mechanisms for salient object detection. In PiCANet [24], Liu et al. propose a novel pixel-wise contextual attention network. In specific, the network generates the attention map with the contextual information of each pixel. With the learned attention map, the network selectively incorporates the features of useful contextual locations, thus contextual features can be constructed. Then the pixel-wise contextual attention mechanism is embedded into the pooling and convolution layers to bring in the global or local contextual information.
As performing quite well on feature selection, the attention mechanism is also suitable for salient object detection. Some methods adopt effective strategies, such as progressive attention [23] and gate function [25]. Inspired by the above, we utilize a lightweight and general attention module [9], which decomposes the learning process of channel-wise attention and spatial-wise attention. The separate attention generation process of the 3D feature map has much less parameters and computational cost.
Moreover, in order to enhance the ability of feature representation, then channel-wise attention mechanism assigns the weight to the channel that is highly responsive to the salient object. Some details in the background are inevitably introduced when the saliency map is generated with low-level features. Taking advantages of high-level features, spatial-wise attention mechanism removes some backgrounds, thus highlights foreground area, which benefits salient object detection.
C. Multi-modal SOD methods
In recent years, with the popularity of thermal sensors, integrating RGB and thermal infrared data has applied to many tasks of computer vision [26], [27], [28], [5], [29].
In addition to RGBT SOD, there are many methods adopting different modality to obtain multiple cues for better detection, such as depth images. In order to combine multiple modalities well, many RGBD methods utilize the better modality to assist the other modality. For example, Qu et al. [30] design a novel network to automatically learn the interaction mechanism for RGBD salient object detection. Han et al. [31] design a twostream architecture, combining the depth representation to make the collaborative decision through a joint full connection layer. Wang et al. [59] also propose a two-stream network, in which they design a saliency fusion module to learn a switch map for fusing the saliency maps adaptively. Chen et al. [58] realize cross-modal interactions by designing a multiscale multi-path fusion network. They diversify the fusion path to perform the global reasoning and the local capturing. Compared with two-stream architectures, this method has more adaptive and flexible fusion flows. Chen et al. [62] also propose to learn complementary information by designing a novel complementarity-aware fusion module with cross-modal residual functions and complementarity-aware supervisions. They cascade this module and add the level-wise supervision, which enables more sufficient fusion results. Piao et al. [57] design a depth-induced multi-scale recurrent attention network. They use residual connections to extract and fuse multilevel paired complementary cues. Then they combine depth cues with multi-scale context features for accurate location. Finally, they use a recurrent attention module to generate more accurate results. Piao et al. [60] further consider that existing two-stream methods have extra costs, bringing difficulties to practical applications. So they propose a depth distiller to transfer the depth knowledge from the depth stream to the RGB stream, thus construct a lightweight architecture that is practically usable. Inspired by the non-local model, Liu et al. [61] propose to integrate the self-attention and the mutual attention to propagate long range contextual dependencies,thus incorporate multi-modal information for more accurate results. Compared with the thermal infrared camera, depth imaging has the limitation that the objects with the same distance to the camera have the same gray level, which is an obvious weakness of depth images. In addition, RGB imaging is usually influenced by various illuminations or bad weathers. To avoid the above problems, more and more researches focus on how to fuse RGB and thermal infrared images, and take full advantages of complementary advantages to discover common conspicuous objects in both modalities. For example, Wang et al. [5] propose a multi-task manifold ranking algorithm for RGBT image saliency detection, and at the same time build up an unified RGBT image dataset. Tu et al. [6] propose an effective RGBT saliency detection method by taking superpixels as graph nodes, moreover, they use hierarchical deep features to learn the graph affinity and the node saliency in a unified optimization framework. With this benchmark [5], Tang et al. [32] propose a novel approach based on a cooperative ranking algorithm for RGBT salient object detection, they introduce a weight for each modality to describe the reliability and design an efficient solver for multiple subproblems. All of the above methods are based on traditional methods, and time-consuming. Therefore, we propose an end-to-end deep network for RGBT salient object detection.
III. VT5000 BENCHMARK
In this work, for promoting the research of RGBT salient object detection(SOD) and considering the insufficiency of existing RGBT datasets, we capture 5000 pairs of RGBT images. In this section, we will introduce our new dataset in detail.
A. Capture Platform
The equipment used to collect RGB and thermal infrared images are FLIR (Forward Looking Infrared) T640 and T610, as shown in Fig. 3, equipped with the thermal infrared camera and CCD camera. The two cameras have same imaging parameters, thus we don't need to manually align RGB and thermal infrared images one by one, which reduces errors from manual alignment.
B. Data Annotation
In order to evaluate RGBT SOD algorithms comprehensively, after collecting more than 5500 pairs of RGB images and corresponding thermal infrared images, we first select 5500 pairs of RGBT images as different as possible, each of image pairs contains one or more salient objects. Similar to many popular SOD datasets [33], [34], [35], [36], [37], we ask six viewers to choose the most salient objects they saw at the first sight for the same image. Because different person might look at different salient object in the same image, 5000 pairs of RGBT images with same selection for salient objects are finally retained. Finally, we use Adobe Photoshop to manually segment the salient objects in each image to obtain pixel-level ground truth masks. I DISTRIBUTION OF ATTRIBUTES AND IMAGING QUALITY IN VT5000 DATASET, SHOWING THE NUMBER OF COINCIDENT ATTRIBUTES ACROSS ALL RGBT IMAGE PAIRS. THE LAST TWO ROWS AND TWO COLUMNS IN THE TABLE.I INDICATE THE POOR PERFORMANCE OF RGB AND T RESPECTIVELY,DUE TO LOW LIGHT, OUT OF FOCUS AND THERMAL CROSSOVER, ETC. CHALLENGE BSO CB CIB IC LI MSO OF SSO SA TC BW RGB T BSO 1746 371 590 446 211 159 96 3 99 244
C. Dataset Statistics
The image pairs in our dataset are recorded in different places and environments, moreover, our dataset records different illuminations, categories, sizes, positions and quantities of objects, as well as the backgrounds, etc. In specific, the following main aspects are considered when creating VT5000 dataset.
Size of Object: We define the size of the salient object as the ratio of number of pixels in the salient object to sum of all pixels in the image. If this ratio is more than 0.26, the object belongs to the big salient object.
Illumination Conditions: We create the image pairs under different light conditions(e.g., low-illumination, sunny or cloudy). Low-illumination and illumination variation under different environments usually bring great challenges to visible light images.
Center Bias: Previous studies on visual saliency show that the center bias has been identified as one of the most significant biases in the saliency datasets [38], which involves a phenomenon that people pay more attentions to the center of the screen [39]. As described in [40], the degree of center bias cannot be described by simply overlapping all the maps in the dataset.
Amounts of Salient Object: It is called multiple salient objects that the salient objects in an image are more than one. We find that the images have less salient objects in existing RGBT SOD datasets. In VT5000 dataset, we capture 3 to 6 salient objects in an image for the challenge of multiple salient objects.
Background factor: We take two factors related to background into consideration. Firstly, it is a big challenge that the temperature or appearance of the background is similar to the salient object. Secondly, it is difficult to separate salient objects accurately from cluttered background.
Considering above-mentioned factors, together with the challenges in existing RGBT SOD datasets [5], [6], we annotate 11 challenges for testing different algorithms, including big salient object (BSO), small salient object (SSO), multiple salient object (MSO), low illumination (LI), center bias (CB), cross image boundary (CIB), similar appearance (SA), thermal crossover (TC), image clutter (IC), out of focus (OF) and bad weather (BW). Descriptions for these challenges are as follows: BSO: size of the object is the ratio of number of pixels in the salient object to sum of all pixels in the image. If the ratio is more than 0.26, the object belongs to the big salient object. SSO: size of the object is the ratio of number of pixels in the salient object to sum of all pixels in the image.
If the ratio is less than 0.05, the object belongs to the small salient object. LI: the images are collected in cloudy days or at night. MSO: there are more than one salient object in an image. CB: the salient object is far away from the center of the image. CIB: the salient object crosses the boundaries of the image, therefore the image always contains part of the object. SA: the salient object has a similar color to the background surroundings. TC: the salient object has a similar temperature to other objects or its surroundings. IC: the scene around the object is complex or the background is cluttered. OF: the object in the image is out-of-focus, and the whole image is blurred. BW: the images collected in rainy days or greasy weather. In addition, we also label those images with good or bad imaging quality of objects in RGB modality (RGB) or Thermal modality (T) in the dataset for researches in the future. RGB: the objects are not clear in RGB modality. T: the objects are not clear in Thermal infrared modality. We also show the attribute distributions on the VT5000 dataset as shown in Table I. We use this 2D array to present the number of samples with one or two attributes, as many samples have multiple attributes(challenges). For example, the number '371' in the first row and second column represents that there are 371 samples with big salient object(BSO) and center bias(CB) challenges simultaneously. There are also a few samples with more than two kinds of challenges, we do not further show them here.
The comparisons of our VT5000 with VT821 and VT1000 on the challenge distribution are shown in Fig. 2. In addition, some challenging RGB and thermal infrared images in our dataset and the corresponding ground truths are shown in Fig. 1. Here, we also give another statistic result that is size distribution in Fig. 4. It shows the distribution of size of the salient object in the training set and the testing set respectively. We can see that the big salient objects in the training set are more than those in the testing set, meanwhile the small salient objects in the testing set are more than those in the training set, which means our traing set is more generic while our testing contains more hard samples.
D. Advantages of Our Dataset
Generally speaking, it is possible to train a usable SOD model with 1000 or so samples. Therefore, our VT5000 Fig. 3. The image on the left shows a sample of RGBT image pairs from VT1000 dataset captured by FLIR (Forward Looking Infrared) SC620, the one on the right is a sample from VT5000 dataset captured by FLIR T640 and T610. with 2500 samples in training set and in testing set can be considered as a large scale dataset. Although existing VT821 and VT1000 are sufficient in quantity, their qualities are not satisfactory. The VT821 has many defective samples which catch unnatural noise and black padding caused by manual aligning. Image pairs in VT1000 are captured by a new equipment so that VT1000 avoids the influence of aligning. However, VT1000 contains too many simple samples. Our main purpose of constructing VT5000 is to boost the research for deep learning based RGBT SOD methods since existing VT821 and VT1000 are not suitable and sufficient for training a robust deep model and doing experiment analysis. Different from VT821 and VT1000, we carefully set 11 challenges and collect samples in different environments. In addition, we further annotate the quality of two modalities, which may be useful for weakly supervised methods in future researches. We divide all the data into training set and testing set for unifying experiment settings, which can effectively avoid unfair comparative experiments and reduce repetitive experiments for subsequent researches. In sum, our VT5000 is more normative and valuable for studying and analyzing RGBT SOD methods.
Compared with existing RGBT datasets VT821 and VT1000, our VT5000 dataset has the following advantages: (1) Being different from previous thermal infrared camera as shown in Fig. 3, RGBT image pairs in our dataset do not require manual alignment, thus errors brought by manual alignment can be reduced; (2) Our thermal infrared camera can automatically focus, which enhances the accuracy of long-distance shooting and captures image texture information effectively; (3) Since these images were captured in summer and autumn, so our dataset has more thermal infrared images with severe thermal crossover; (4) We provide a large scale dataset with more RGBT image pairs and more complex scenes.
IV. ATTENTION-BASED DEEP FUSION NETWORK
In this section, we will introduce the architecture of the proposed Attention-based Deep Fusion Network (ADFNet), and describe the details of RGBT salient object detection.
A. Overview of ADFNet
We build our network architecture based on [8] as shown in Fig. 5, and employ a two-stream CNN architecture, which first extracts RGB and thermal infrared features separately and then proceeds RGBT salient object detection. We use the pretrained VGG16 model as initialization, which provides a great ability for feature representation. To make the network focus on more informative regions, we utilize a series of attention modules to extract weighted features from RGB and thermal infrared branches before fusion of these features. From the second block of VGG16, the fused features of each layer are transmitted from the lower-level to the high-level in turn.
Although high-level semantic information could facilitate the location of salient objects [41], [42], [43], low-level and midlevel features are also essential to refine deep level features. Therefore, we add two complementary modules (Pyramid Pooling Module and Feature Aggregation Module) [10] to accurately capture the exact position of a prominent object while sharpening its details.
B. Convolutional Block Attention Module
As illustrated in Fig. 5, RGB and thermal infrared images respectively generate five different levels of features through five blocks of the backbone network VGG16, expressed by X R i and X T i ∈ R C×H×W respectively, where i represents VGG16 i-th block. As most of complex scenes contains cluttered background, which will bring lots of noises to feature extraction, we expect to selectively extract the features with less noises from RGB and thermal infrared branches. Therefore, we adopt Convolutional Block Attention Module (CBAM) [9] with channel-wise attention and spatial-wise attention shown in Fig. 6. As shown in Fig. 7, with CBAM, the proposed network can capture the spatial details around the objects, especially at the shallow layer, which is conducive to saliency refinement. If without CBAM, the network will have some redundant information that is helpless for saliency refinement.
The channel-wise attention focuses on what makes sense for an input image. Currently, most of methods typically uses average pooling operations to aggregate spatial information. In addition to previous works [44], [45], we think that the maxpooling collects discriminative characteristics of the object to infer finer channel-wise attention. Therefore, we use both of average pooling and max pooling features. The RGB branch is described here as an example, just as the thermal infrared branch does. Firstly, we aggregate the spatial information from a feature map with the average pooling and max pooling operations to generate two different spatial context information, X Ravg i and X Rmax i , which represents the features after average pooling and max pooling respectively. Secondly, these features are forwarded to two convolution layers of 1 * 1 to generate channel attention map M CR i , and we merge the outputted feature vectors with pixel-wise summation. Finally, the channel attention weight vector is obtained by a sigmoid function. The specific process can be expressed as:
M CR i =(σ(Conv(AvgP ool(X R i )) + Conv(M axP ool(X R i )))) * X R i(1)
where σ denotes the sigmoid function, Conv denotes the convolutional operation and * denotes multiply operation.
The spatial-wise attention is complementary to the channel attention. Different from channel-wise attention, spatial-wise attention focuses on structural information and it highlights the informative spatial positions in the features. In specific, we first apply average pooling and max pooling operations to features along the channel axis and connect these features to produce efficient descriptors. Next, we obtain a two-dimensional feature map with a standard convolution layer. Finally, the spatial attention weight vector is obtained by a sigmoid function. The specific process can be expressed as:
M SR i = (σ(f k * k ([AvgP ool(M C i ), M axP ool(M C i )]))) * M CR i
(2) where σ denotes the sigmoid function and f k * k represents a convolution operation with the filter size of k * k.
C. Multi-modal Multi-layer Feature Fusion
The previous works [46], [47] show that fusion of multimodal features at the shallower layer or the deeper layer might not take good advantage of the useful features from multiple modalities. To obtain rich and useful features of RGB and thermal infrared images during downsampling, we adopt a strategy of multi-layer feature fusion here. Specifically, we use two VGG16 networks to extract RGB and thermal infrared features respectively, which can preserve RGB and thermal infrared features before upsampling. Each branch provides a set of feature maps from each block of VGG16. After passing through each Conv block, the two features are processed by CBAM respectively and then added for fusion on the pixel level. Here, we add the features of two modalities directly in the first layer, and add the features of the current layer after the convolution operation with the features of the previous layer. In this way, both of low-level features and high-level features are extracted, and the corresponding formula is expressed as:
F i = M SR i + M ST i , if i is 1 Conv(F i−1 ) + M SR i + M ST i , if i is 2,3,4,5(3)
D. Pyramid Pooling Module
A classic encoder-decoder classification architecture generally follows the top-down pathway. However, the top-down pathway is built upon the bottom-up backbone. Higher-level features will be gradually diluted when they are transferred to shallower layers, therefore, loss of useful information inevitably happens. The receptive field of CNN will become smaller and smaller when the number of network layers increases [48], so the receptive field of the whole network is not large enough to capture the global information of the image. Considering fine-level feature map lacks of high-level semantic information, we use a Pyramid Pooling Module (PPM) [10] to process features for capturing global information with different sampling rates. Thus we can clearly identify position of the object at each stage.
More specifically, the PPM includes four sub-branches to capture context information of the image. The first and fourth branches are the global average pooling layer and the identity mapping layer, respectively. For the two intermediate branches, we use adaptive averaging pooling to ensure that sizes of output feature maps are 3*3 and 5*5 respectively. The guidance information generated by PPM will be properly integrated with feature maps of different levels in the topdown pathway, and high-level semantic information can be easily passed to the feature map of each level by a series of up-sampling operations. Providing global information to the feature of each level makes sure locating salient objects accurately.
E. Feature Aggregation Module
As shown in Fig. 5, with the help of global guidance flow, the global guidance information can be passed to the feature at different pyramid level. Next, we want to perfectly integrate the coarse feature map with the feature at different scale by the global guidance flow. At first, the input image passes through five convolution blocks of VGG16 in sequence, thus feature maps corresponding to F = {F 2 , F 3 , F 4 , F 5 } in the pyramid have been downsampled with downsample rate of {2, 4, 8, 16} respectively. In the original top-down pathway, RGB and thermal infrared features with coarser resolution are upsampled by a factor of 2. After the merging operation, we use a convolutional layer with kernel size 3×3 to reduce the aliasing effect of upsampling.
Here, we adopt a series of feature aggregation modules [10], each feature aggregation module contains four branches. In the process of forward pass, with different downsampling rates, the input feature maps are first converted to the features with different scales by feeding them into an average pooling layer. Then we combine the features from different branches through upsampling, followed by a 3×3 convolutional layer, which helps our model reduce aliasing effects caused by upsampling operations, especially when the upsampling rate is large.
F. Loss Function 1) Cross Entropy Loss:
The cross entropy loss is usually used to measure the error between the final saliency map and the ground truth of salient object detection. The cross entropy loss function is defined as:
L C = size(Y ) i=0 (Y i log(P i ) + (1 − Y i )log(1 − P i ))(4)
Where Y represents the ground truth, P represents the saliency map output by the network and N represents the number of pixels in an image.
2) Edge Loss:
The cross-entropy loss function provides a general guidance for the generation of the saliency map. Nevertheless, edge blur is an unsolved problem for saliency detection. Inspired by [44] and different from it, we use a simpler strategy to sharpen the boundary around the object. In specific, we use Laplace operator [54] to generate boundaries of ground truth and the predicted saliency map, and then we use the cross entry loss to supervise the generation of boundaries of salient object.
∆f = ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 (5) ∆f = abs(tanh(conv(f, K laplace ))) (6) L E = − size(Y ) i=0 (∆Y i log(∆P i )+(1−∆Y i )log(1−∆P i )) (7)
The Laplace operator is defined as the divergence of the gradient ∆f . Since the second derivative can be used to detect edges, we use the Laplace operator to obtain boundaries of salient object. Laplacian uses the gradient of image, which is actually calculated with convolution. In Eq. 5, x and y are the standard Cartesian coordinates of the XY -plane. Next, we use the absolute value operation followed by tanh activation in Eq. 6 to map the value to a range of 0 to 1. Then, we use the edge loss(Eq. 7) to measure the error between real boundaries of salient object and its generated boundaries. The total loss can be represented as:
L S = L C + L E (8) V. EXPERIMENTS
In this section, we first introduce our experiment setups, including the experimental details, the training set, the testing set, and the evaluation criteria. Then we conduct a series of ablation studies to prove the effect of each component in the proposed benchmark method. Finally, we show the performance of our method and compare it with the stateof-the-art methods.
To provide a comparison platform, Table II presents the baseline methods about the main technique, book title and published time. We take RGB and thermal images as the input to these ten state-of-the-art methods to achieve RGBT salient object detection, including PoolNet [10], RAS [49], BASNet [52], CPD [51], R3Net [50], PFA [44], PiCANet [24], EGNet [53], MTMR [5] and SGDL [6]. These methods utilize deep features except for MTMR [5]. Furthermore, only MTMR [5] and SGDL [6] are traditional models. In our method, we combine the deep features extracted from RGB and thermal branches and compare with the above-mentioned methods.
A. Experiment Setup
Implementation Details. In this work, the proposed network is implemented based on the PyTorch and hyperparameters are set as follows. We train our network on single Titan Xp GPU. We use Adam [55] with a weight decay of 5e-4 to optimize parameters and train 25 epochs. The initial learning rate is 1e-4, after the 20th epoch, the learning rate is reduced to 1e-5. For the preprocessing of samples, we do data augmentation with simple random horizontal flipping. The original size of input image is 640*480. To improve the efficiency of training stage, we resize the input image to 400*400.
In our code, we randomly initialize the parameters of network except of VGG16 backbone. We also set fixed random seeds for all random operations. So theoretically, the initialisation setups of each run are consistent. Although the results of different runs are not identical since the error of floating-point calculation, the difference of these results are very slight. So in subsequent experiments, we report the results of single run, which is representative and reasonable.
Evaluation Metrics. Similar to RGB dataset MSRA-B [56], we use the 2500 pairs of RGBT images in VT5000 dataset as the training set, and take the rest in VT5000 together with VT821 [5] and VT1000 [6] as the test set. We evaluate performances of different methods on three different metrics, including Precision-Recall (PR) curves, F-measure and Mean Absolute Error (MAE). The PR curve is a standard metric to evaluate saliency performance, which is obtained by binarizing the saliency map using thresholds from 0 to 255 and then comparing the binary maps with the ground truth. The Fmeasure can evaluate the quality of the saliency map, by computing the weighted harmonic mean of the precision and recall, II LIST OF THE BASELINE METHODS WITH THE MAIN TECHNIQUES AND THE PUBLISHED INFORMATION Baseline Technique Book Title Year RAS [49] residual learning and reverse attention ECCV 2018 PiCANet [24] pixel-wise contextual attention network CVPR 2018 R3Net [50] recurrent residual refinement network IJCAI 2018 MTMR [5] multi task manifold ranking with cross-modality consistency IGTA 2018 SGDL [6] collaborative graph learning algorithm TMM 2019 PFA [44] context-aware pyramid feature extraction module CVPR 2019 CPD [51] multi-level feature aggregate CVPR 2019 PoolNet [10] global guidance module and feature aggregation module CVPR 2019 BASNet [52] predict-refine architecture and a hybrid loss CVPR 2019 EGNet [53] integrate the local edge information and global location information ICCV 2019 where β 2 is set to 0.3 as suggested in [33]. MAE is a complement to the PR curve and quantitatively measures the average difference between predicted S and ground truth G at the pixel level,
F β = (1 + β 2 ) · P recision · Recall β 2 · P recision + Recall(9)M AE = 1 W × H W x=1 H y=1 | S(x, y) − G(x, y) |(10)
where W and H is the width and height of a given image.
B. Comparison with State-of-the-Art Methods
We include eight deep learning-based and two traditional state-of-the-art methods in our benchmark for advanced evaluations, including PoolNet [10], RAS [49], BASNet [52], CPD [51], R3Net [50], PFA [44], PiCANet [24], EGNet [53], MTMR [5], SCGL [6]. It is worth mentioning that results are obtained by testing the corresponding method on RGBT data without any post-processing, and evaluated with the same evaluation code. The results of all methods are obtained with the published codes. For a fair comparison, the deep learning methods take the same training set and testing set as ours.
Challenge-sensitive performance. To dispaly and analyze the performance of our method on the challenge-sensitivity and imaging quality of objects compared with other methods, we give a quantitative comparison in Table III. We evaluate our method on eleven challenges and bad imaging quality for objects in two modality (i.e., BSO, SSO, MSO, LI, CB, CIB, SA, TC, IC, OF, BW, RGB, T) in VT5000 dataset. Notice that our method is significantly better than other methods, showing that our method is more robust for these challenges. Compared with PoolNet [10], our method outperforms 10.8% and 14.8% in F-measure on SA and SSO challengs, respectively. This results show that the thermal infrared data can provide effective information to help the network distinguish the object and the background when the object is similar to the background in RGB modality. And the small object is a challenge for every modality. Our network can locate salient object well with the help of the global guidance flow derived from PPM, even for the small object.
Quantitative Comparisons. We compare the proposed methods with others in terms of F-measure scores, MAE scores, and PR-curves. And we have verified the effectiveness of our method on three datasets, and the quantitative results are shown in Fig. 8. We take PoolNet [10] as our baseline. Fig. 8 shows the results on VT821, VT1000 and VT5000, and our method performs best in F-measure. Compared with the baseline PoolNet, with the synergy of thermal infrared branch, our model outperforms PoolNet by a large margin of 5.5%-9.9% on three RGBT datasets(VT821, VT1000 and VT5000). Compared with the method PiCANet [24] that also uses attention mechanism, our F-measure value achieves 1.5% gains and MAE value is 0.1% more than it on VT821 Fig. 8. Precision-recall curves of our model compared with PoolNet [10], RAS [49], BASNet [52], CPD [51], R3Net [50], PFA [44], PiCANet [24], EGNet [53], MTMR [5], SCGL [6]. Our model can deliver state-of-the-art performance on three datasets. The numbers in first column are the values of F-measure, and the second column represents the value of MAE.
dataset. On VT1000 dataset, our F-measure value outperforms PiCANet [24] with 5.9% and MAE value is 2.6% less than it. On VT5000 dataset, our F-measure value outperforms PiCANet [24] with 7.5% and MAE value is 1.8% less than it. As a method with high performance, CPD [51] proposes a new Cascaded Partial Decoder framework for salient object detection, through integrating features of deeper layers and discarding larger resolution features of shallower layers to achieve fast and accurate salient object detection. Our Fmeasure value outperforms CPD [51] with 1.8% and MAE value is 0.2% less than it on VT821 dataset, our F-measure value outperforms it with 0.9% and MAE value is 0.3% less than it on VT1000 dataset. On VT5000 dataset, our F-measure value outperforms CPD [51] with 1.6% and MAE value is 0.1% more than it. The EGNet [53] is the latest approach among the compared methods, composed of three parts: edge feature extraction, feature extraction of salient object and oneto-one guidance module. The edge feature can help to locate the object and make the object boundary more accurate. From Fig. 8, we can see the results on VT1000 dataset. And Fmeasure value of our method is 0.6% higher than EGNet and its MAE value is 0.1% lower than our method. Same as above, the results of our method and other state-of-the-arts are shown in Fig. 8(right). Compared with the baseline PoolNet, our Fmeasure value achieves 9.9% gains and MAE value is 3.3% less, and compared with best method EGNet [53] , F-measure value of our method is 2.4% higher than EGNet [53] and its MAE value is 0.3% lower than our method on VT5000. Compared with EGNet [53], our method has these merits: (1) Our method can refine the edge of the salient object without using additional edge detection model; (2) The global guidance flow derived from PPM can make good use of the global context information and better locate the salient object; (3) With the help of the thermal infrared branch, we can make use of the complementary information of the two modalities to better deal with various challenges in salient object detection. This shows that our method is still optimal in general for RGBT SOD.
PR Curves. In addition to showing the results of F-measure and MAE, we also show the PR curves on three datasets. As shown in Fig. 8, it can be seen that the PR curve (red) obtained by our method is particularly prominent compared with all previous methods. When the recall score is close to 1, our accuracy score is much higher than compared methods. This also shows that the truth-positive rate of our saliency maps is higher than compared methods.
Our main purpose of constructing this dataset is to boost the researches on RGBT SOD since existing datasets are limited in scale so that they are not enough to train a robust deep learning model. Although we try our best to collect hard samples, there are still some normal samples. So the reported results do not show huge differences compared to VT821 and VT1000. From Fig. 8, we can observe that the difficulty of VT5000 is at middle level among three datasets where the VT821 is hardest and the VT1000 is easiest, which is reasonable because the VT1000 collects an amount of simple samples while the samples in VT821 are randomly added noise and their quality is not well. Our VT5000 contains lots of challenging samples without any degradation operation.
Visual Comparison. To qualitatively evaluate the proposed method on the new RGBT dataset, we visualize and compare some results of our method with other state-of-the-art methods in Fig. 9. These examples are from various scenarios, including big salient object(BSO) (row 1, 3, 4, 7), multiple salient objects(MSO) (row 2, 5, 13), small salient object(SSO) (row 13), cross image boundary(CIB) (row 3,6,9), cluttered background(IC)(row 3,7,9), low illumination(LI) (row 12), center bias(CB) (row 8), out-of-focus(OF) (row 2, 13), bad weather(BW) (row 3,9), similar appearance(SA) (row 8) and Thermal Crossover(TC) (row 11,13). Each row includes at least one challenge in Fig. 9. It is easy to see that our method obtains best results in various challenging scenes. Specifically, the proposed method not only clearly highlights the objects, but also suppresses the background, and the objects have welldefined contours. Fig. 9. Saliency maps produced by the PoolNet [10], RAS [49], BASNet [52], CPD [51], R3Net [50], PFA [44], PiCANet [24], EGNet [53], MTMR [5], SGDL [6]. Our model can deliver state-of-the-art performance on three datasets VI. ABLATION ANALYSIS In this part of ablation analysis, we respectively investigate the effectiveness of CBAM and edge loss of the proposed method. The experiments are performed on VT5000. We use the training set of VT5000 to train the model. Then we predict saliency maps of all the samples in testing set of VT5000 and compute the max F-measure and MAE for reporting. As shown in Table IV, firstly, we run the basic network without CBAM and edge loss and the result is not well. Then if we only add CBAM into the basic network PoolNet [10], the value of Fmeasure increases by 2.1% and the value of MAE decreases by 0.5%, and if we only add the edge loss, our performance is degraded. In the course of experiment, we find that if we only add the edge loss, the value of loss is downward overall, but fluctuates greatly during training stage. In addition, although adding CBAM to the basic network can effectively suppress the noise, but if too much redundant noises appear, extracted edges are unsatisfactory, influencing greatly on the stability during training stage. These observations mean that without the attention mechanism, salient object can not be located accurately. As shown in Table IV, with the help of CBAM for locating the salient object, and edge loss for refining edges, our network obtains best performances. We further study the time cost of all the modules(CBAM,PPM and FAM) used in our proposed model. We also do an ablation study on three modules and record corresponding training time and testing time. The experiments are performed on our VT5000 dataset. And we report the training time of one epoch where 2500 samples are included in calculation. The testing time is the time for computing VT5000's test set which also has 2500 samples. As reported in Table V, the training time of CBAM,PPM and FAM separately increase about 0.095, 0.079 and 0.051 second per sample. And their testing time separately increase about 0.061, 0.042 and 0.025 second per sample. Generally speaking, these modules will not bring too much time cost while boosting the performance.
VII. EXTENSION EXPERIMENTS ON MULTI-MODAL SOD
In this section, we further make analysis on multi-modal SOD that mainly includes RGBT SOD and RGBD SOD. On one hand, we first apply existing RGBD SOD methods to our VT5000 dataset. Then we report max F-measure and MAE to show the performance of those methods. On the other hand, we use our ADFNet to perform RGBD SOD task on existing RGBD datasets.
A. The performance of RGBD SOD methods on RGBT datasets
In recent years, RGBD SOD has been widely studied and many robust algorithms have been proposed. The depth information is introduced as an assistant for boosting salient object detection in challenging scenes. And RGBD SOD algorithm receives the inputs of two modalities, which is formally same to RGBT SOD task. Therefore, to verify the effectiveness of our proposed method, we also need to compare with the modal-modal methods. We select five advanced RGBD SOD methods which are publicly available, and train them on our VT5000 testing set with their default settings. These five methods are DMRA [57], MMCI [58], AFNet [59], A2dele [60] and S2MA [61].
The performance comparison of these RGBD methods are shown in TableVI. The advanced RGBD methods can also work well on RGBT data and our ADFNet achieves optimal or suboptimal results. We don't make any specifical design for thermal image, and just propose a standard architecture with attention mechanism for multi-modal information aggregation. The performances of some RGBD methods such as DMRA and A2dele are inferior to average level, which design a module for depth map or use the distillation strategy.
B. The performance of our method on RGBD SOD datasets
For further exploring the effectiveness of our method on multi-modal task, we also conduct experiments on existing RGBD SOD datasets and compare our method with advanced RGBD SOD methods. In this part, the compared methods are PCF [62], CTMF [63], MMCI [58], AFNet [59], TANet [64], D3Net [65] and S2MA [61]. The codes or saliency maps of these methods are publicly available. We use DES [14], LFSD [66], SSD [15] and STERE [16] as the test datasets. As same as the common setting of existing RGBD methods, we train our ADFNet on a data collection that contains randomly sampled 1485 image pairs and 700 image pairs from the NJU2K [12] and NLPR [13] datasets. The introduction of above datasets are written in Section II-A.
Compared with our ADFNet, the max F-measure and MAE scores of RGBD methods are reported in Table VII. On three test datasets, our ADFNet shows similar performance to SOTA RGBD SOD methods, meaning that our methods can also be applied to other multi-modal SOD tasks with competitive results.
VIII. CONCLUDING REMARKS AND POTENTIAL DIRECTIONS
In this work, we propose a novel attention-based deep fusion network for RGBT salient object detection. Our network consists of a basic feature extraction network, convolutional block attention modules, pyramid pooling modules and feature aggregation modules. The comparison experiments demonstrate our method performs best over all the state-of-the-art methods with most evaluation metrics. We also create a new largescale RGBT dataset for deep learning based RGBT salient object detection methods, with the attribute annotations for 11 challenges and the imaging quality annotations for salient objects in RGB and thermal images. From the evaluation results, taking advantage of the thermal image can boost the results of salient object detection in the scenarios of big salient objects, far away from center of the images, crossing the image boundaries, cluttered background, low illumination and similar temperature with background. Cluttered background and low illumination are common scenes but bring big challenges to salient object detection, while thermal infrared images can provide complementary information to RGB images to improve SOD results. However, when thermal crossover occurs, thermal data become unreliable, but visible spectrum imaging will not be influenced by temperature.
According to the evaluation results, we observe and draw some inspirations which are essential for boosting RGBT SOD in the future. Firstly, deep learning-based RGBT SOD methods need to be explored further. For example, how to design a suitable deep network which takes the special properties of RGB and thermal modalities into considerations for RGBT SOD is worth studying. How to make best use of attention mechanisms and semantic information is still important for improving feature representations of salient objects and can prevent salient objects from being gradually diluted. Secondly, the attribute-based feature representations could be studied for handling the problem of lacking sufficient training data. Comparing with object detection and classification, the scale of annotated data for training networks of RGBT SOD are very small. We annotate various attributes in our VT5000 dataset and these attribute annotations should be used to study the attribute-based feature representations, which can model different visual contents under certain attributes to reduce network parameters. Thirdly, unsupervised and weakly supervised RGBT SOD are valuable research directions. The task of RGBT SOD needs pixel-level annotations, thus an-notating large-scale datasets needs unacceptable manual cost. Therefore, no and less relying on large-scale labeled dataset is a potential research direction for RGBT SOD. It should be noted that we have annotated some weakly supervised labels for our VT5000, i.e., imaging quality of different modalities, and we believe it would be beneficial to the related researches of unsupervised and weakly supervised RGBT SOD. Finally, the alignment-free methods would make RGBT SOD more popular and practical in real-world applications. We find that existing datasets contain some misaligned image pairs even though we adopt several advanced techniques to perform the alignment for images. However, the images recorded from most of existing RGBT imaging platforms are non-aligned. Therefore, alignment-free RGBT SOD is also worth being explored in the future.
, Y. Ma, Z. Li, C. Li, J. Xu, and Y. Liu are with Anhui Provincial Key Laboratory of Multimodal Cognitive Computation, School of Computer Science and Technology, Anhui University, Hefei, China, Email: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]. C. Li is also with Institute of Physical Science and Information Technology, Anhui University, Hefei 230601, China. (Corresponding author is Chenglong Li) This research is jointly supported by Natural Science Foundation of Anhui Higher Education Institution of China(No.KJ2020A0033), Anhui Provincial Natural Science foundation(No.2108085MF211,2020 Anhui Energy Internet Joint Fund Project: No.2008085UD07),National Natural Science Foundation of China (No.61876002), Anhui Provincial Key Research and Development Program (No.202104d07020008), NSFC Key Project of International (Regional) Cooperation and Exchanges (No. 61860206004).
Fig. 1 .
1Sample image pairs with annotated ground truth and challenges from our RGBT dataset. (a) and (b) indicate the RGB image and its corresponding thermal image respectively. (c) is the corresponding ground truth of RGBT image pair.
Fig. 2 .
2Challenge distribution of VT821, VT1000 and VT5000.
Fig. 4 .
4Size Distribution: the horizontal axis represents the proportion of pixels of objects to total pixels of the image, the vertical axis represents the number of corresponding images.
Fig. 5 .
5The overall architecture of our method. VGG16 is our backbone network, in which different color blocks represent different convolution blocks of VGG16.
Fig. 6 .
6Convolutional Block Attention Module(CBAM), (a) channel-wise attention module and (b) spatial-wise attention module Fig. 7. Visualization of features from different fusion layers in the proposed network without CBAM (shown in the first row) and with CBAM (shown in the second row). From left to right, there are the saliency map, the fused feature from layer 1 to 4, respectively.
TABLE
TABLE
TABLE III THE
IIIVALUE OF F-MEASURE IN EACH CHALLENGE OF OUR METHOD AND TEN COMPARISON METHODSChallenge
PoolNet
BASNet
CPD
PFA
R3Net
RAS
PiCANet
EGNet MTMR
SCGL ADFNet
BSO
0.800
0.858
0.872
0.802
0.831
0.768
0.804
0.873
0.667
0.754
0.880
CB
0.725
0.808
0.845
0.748
0.794
0.669
0.796
0.838
0.575
0.703
0.854
CIB
0.740
0.822
0.860
0.742
0.822
0.688
0.790
0.854
0.582
0.694
0.860
IC
0.721
0.775
0.812
0.735
0.745
0.672
0.752
0.818
0.564
0.681
0.835
LI
0.757
0.832
0.840
0.749
0.790
0.707
0.783
0.848
0.695
0.742
0.868
MSO
0.706
0.794
0.826
0.729
0.774
0.655
0.777
0.815
0.620
0.710
0.837
OF
0.762
0.816
0.821
0.754
0.759
0.738
0.758
0.817
0.707
0.738
0.837
SA
0.727
0.762
0.825
0.726
0.728
0.673
0.748
0.791
0.653
0.665
0.835
SSO
0.658
0.718
0.767
0.695
0.663
0.535
0.676
0.701
0.698
0.753
0.806
TC
0.720
0.791
0.811
0.762
0.729
0.711
0.745
0.791
0.570
0.675
0.841
BW
0.750
0.768
0.795
0.671
0.753
0.701
0.773
0.774
0.606
0.643
0.804
RGB
0.733
0.785
0.804
0.731
0.736
0.690
0.743
0.785
0.670
0.671
0.817
T
0.719
0.787
0.802
0.755
0.719
0.699
0.736
0.776
0.564
0.664
0.833
TABLE IV THE
IVIMPACT OF EACH COMPONENT OF THE NETWORK ON THEPERFORMANCE
CBAM
Edge Loss
max F β
MAE
0.836
0.057
0.857
0.052
0.825
0.063
0.863
0.049
TABLE V THE
VTIME COST OF EACH MODULE USED IN THE NETWORKCBAM
PPM FAM
Training time
Testing time
744.63s
229.01s
786.32s
277.78s
854.72s
321.32s
982.65s
383.30s
TABLE VI THE
VIPERFORMANCE OF RGBD METHODS ON RGBT SOD DATASETSMethods
VT821
VT1000
VT5000
max F β
MAE max F β
MAE max F β
MAE
DMRA
0.702
0.216
0.824
0.124
0.631
0.184
MMCI
0.723
0.089
0.875
0.040
0.797
0.056
AFNet
0.735
0.069
0.887
0.033
0.818
0.050
A2dele
0.644
0.074
0.813
0.061
0.725
0.065
S2MA
0.812
0.081
0.923
0.029
0.837
0.055
ADF
0.804
0.077
0.923
0.034
0.863
0.048
TABLE VII THE
VIIQUANTITATIVE COMPARISON ON RGBD SOD DATASETS Methods DES LFSD SSD STERE max F β MAE max F β MAE max F β MAE max F β MAEPCF
0.842 0.049 0.828 0.112 0.843 0.062 0.875 0.064
CTMF
0.865 0.055 0.814 0.119 0.755 0.099 0.848 0.086
MMCI
0.839 0.065 0.813 0.132 0.823 0.082 0.877 0.068
AFNet
0.775 0.068
0.78 0.133 0.735 0.118 0.848 0.075
TANet
0.853 0.046 0.827 0.111 0.835 0.063 0.878 0.060
DMRA
0.906 0.035 0.823 0.111 0.874 0.055 0.867 0.066
S2MA
0.944 0.021 0.862 0.094 0.691 0.138 0.895 0.051
ADF
0.901 0.038 0.863 0.098 0.710 0.120 0.877 0.064
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She is currently an Associate Professor with the School of. Zhengzheng Tu received the M.S. and Ph.D.degrees from the School of Computer Science and Technology. Hefei, ChinaAnhui University ; Computer Science and Technology, Anhui UniversityrespectivelyHer current research interests include computer vision and deep learningZhengzheng Tu received the M.S. and Ph.D.degrees from the School of Computer Science and Technol- ogy, Anhui University, Hefei, China, in 2007 and 2015, respectively. She is currently an Associate Professor with the School of Computer Science and Technology, Anhui University. Her current research interests include computer vision and deep learning.
Yan Ma, 2021. Her current research interests include computer vision and deep learning. Hefei, ChinaYan Ma received the M.S. degree from the School of Computer Science and Technology, Anhui Uni- versity, Hefei, China, in 2021. Her current research interests include computer vision and deep learning.
He is pursuing M.S. degree at the School of Computer Science and Technology. Hefei, ChinaZhun Li received the B.Eng. degree in Anhui University ; Anhui UniversityHis current research interests include computer vision and deep learningZhun Li received the B.Eng. degree in Anhui University, in 2019. He is pursuing M.S. degree at the School of Computer Science and Technology, in Anhui University, Hefei, China. His current research interests include computer vision and deep learning.
Institute of Automation, Chinese Academy of Sciences (CASIA), China. He is currently an Associate Professor at the School of. He was a postdoctoral research fellow at the Center for Research on Intelligent Perception and Computing (CRIPAC), National Laboratory of Pattern Recognition (NLPR). Hefei, China; Guangzhou, ChinaChenglong Li received the M.S. and Ph.D. degrees from the School of Computer Science and Technology, Anhui University ; Sun Yat-sen University ; Computer Science and Technology, Anhui UniversityHis research interests include computer vision and deep learning. He was a recipient of the ACM Hefei Doctoral Dissertation Award in 2016Chenglong Li received the M.S. and Ph.D. degrees from the School of Computer Science and Technol- ogy, Anhui University, Hefei, China, in 2013 and 2016, respectively. From 2014 to 2015, he worked as a Visiting Student with the School of Data and Com- puter Science, Sun Yat-sen University, Guangzhou, China. He was a postdoctoral research fellow at the Center for Research on Intelligent Perception and Computing (CRIPAC), National Laboratory of Pat- tern Recognition (NLPR), Institute of Automation, Chinese Academy of Sciences (CASIA), China. He is currently an Associate Professor at the School of Computer Science and Technology, Anhui University. His research interests include computer vision and deep learning. He was a recipient of the ACM Hefei Doctoral Dissertation Award in 2016.
2021. He is currently pursuing the M.S. degree in College of Computer Science and Technology. Hefei, China; Chengdu, ChinaJieming Xu received the B.Eng. degree from Anhui University ; Sichuan UniversityHis current research interests include bigdata and deep learningJieming Xu received the B.Eng. degree from An- hui University, Hefei, China, in 2021. He is cur- rently pursuing the M.S. degree in College of Com- puter Science and Technology, Sichuan University, Chengdu, China. His current research interests in- clude bigdata and deep learning.
His current research interests include computer vision and deep learning. Hefei, China, in 2021Yongtao Liu received the B.Eng. degree from Anhui UniversityYongtao Liu received the B.Eng. degree from An- hui University, Hefei, China, in 2021. His current research interests include computer vision and deep learning.
| [
"https://github.com/lz118/RGBT-Salient-Object-Detection.",
"https://github.com/lz118/RGBT-Salient-Object-Detection."
]
|
[
"EXISTENCE OF AN EXTREMAL OF SOBOLEV INEQUALITY ASSOCIATED WITH DUNKL GRADIENT AND OF STEIN-WEISS INEQUALITY FOR D-RIESZ POTENTIAL",
"EXISTENCE OF AN EXTREMAL OF SOBOLEV INEQUALITY ASSOCIATED WITH DUNKL GRADIENT AND OF STEIN-WEISS INEQUALITY FOR D-RIESZ POTENTIAL"
]
| [
"Saswata Adhikari ",
"ANDV P Anoop ",
"Sanjay Parui "
]
| []
| []
| In this paper, we prove the existence of an extremal for the Dunkltype Sobolev inequality in case of p = 2. Also we prove the existence of an extremal of the Stein-Weiss inequality for the D-Riesz potential in case of r = 2. | null | [
"https://arxiv.org/pdf/1902.08530v3.pdf"
]
| 104,292,117 | 1902.08530 | ec149dfae9436743717d65758113d350dcc2cdfb |
EXISTENCE OF AN EXTREMAL OF SOBOLEV INEQUALITY ASSOCIATED WITH DUNKL GRADIENT AND OF STEIN-WEISS INEQUALITY FOR D-RIESZ POTENTIAL
Saswata Adhikari
ANDV P Anoop
Sanjay Parui
EXISTENCE OF AN EXTREMAL OF SOBOLEV INEQUALITY ASSOCIATED WITH DUNKL GRADIENT AND OF STEIN-WEISS INEQUALITY FOR D-RIESZ POTENTIAL
arXiv:1902.08530v3 [math.FA] 9 Apr 2019
In this paper, we prove the existence of an extremal for the Dunkltype Sobolev inequality in case of p = 2. Also we prove the existence of an extremal of the Stein-Weiss inequality for the D-Riesz potential in case of r = 2.
Introduction
The classical Sobolev inequality states that for all f ∈ C ∞ c (R d ),
f L q (R d ) ≤ C ∇f L p (R d ) ,(1.1)
where 1 ≤ p < d, q = dp d−p and the constant C > 0 only depends on d. This inequality plays an important role in analysis and and as such it has been studied by many for e.g. see ( [10,15,11]). The problem of finding sharp constant to inequality (1.1) was answered in [17].
One can consider inequality (1.1) in the context of Dunkl setting by replacing the Euclidean gradient ∇f by Dunkl gradient ∇ k f . Sobolev inequality (1.1) associated with Dunkl gradient was derived for 1 < p < d k in [1,8].
In the Eucledean space R d , the negative powers of the Laplacian can be defined as an integral representation in terms of the Riesz potential or fractional integral operator as follows:
(−∆) − α 2 f (x) = I α f (x) = (c α ) −1 R d f (y)|x − y| α−d dy,
where 0 < α < d and c α = 2 α− d 2 Γ( α 2 ) Γ( d−α 2 ) . One fundamental result for the Riesz potential operator is the Stein-Weiss inequality which gives the weighted (L r , L s ) boundedness:
Theorem 1.1. [16] Let d ∈ N, 1 < r ≤ s < ∞, γ > − d s , β ≥ γ, 0 < α < d, β < d r ′ , α + γ − β = d( 1 r − 1 s ). Then |x| γ I α f (x) L s (R d ) ≤ C |x| β f (x) L r (R d ) , ∀ f ∈ L r (R d , |x| βr ).
(1.2) S. Thangavelu and Y. Xu in [19] defined the D-Riesz potential operator on Schwartz spaces as follows:
I k α f (x) = (c k α ) −1 R d τ k y f (x)|y| α−d k w k (y)dy,
where 0 < α < d k and c k α = 2 α− d k
2 Γ( α 2 ) Γ( d k −α 2 )
. In [7], D. V. Gorbachev et all proved the following Stein-Weiss inequality for the D-Riesz potential.
Theorem 1.2. Let d ∈ N, 1 < r ≤ s < ∞, γ > − d k s , β ≥ γ, 0 < α < d k , β < d k r ′ , α + γ − β = d k ( 1 r − 1 s ). Then |x| γ I k α f (x) L s (R d ,w k ) ≤ C k |x| β f (x) L r (R d ,w k ) ∀ f ∈ L r (R d , |x| βr w k ). (1.3)
In this paper, first we consider the Sobolev inequality (1.1) for the case p = 2 associated with Dunkl gradient and are interested to find its extremals. Towards this, for u ∈Ḣ 1 (R d , w k ), we consider the function
F (u) = R d |∇ k u| 2 w k (x)dx R d |u| q w k (x)dx 2 q ,(1.4)
where q = 2d k d k −2 andḢ 1 (R d , w k ) =Ẇ 1,2 (R d , w k ). The aim of this paper is to show that infimum is attained for the function F when the infimum is taken over all nonvanishing functions u ∈Ḣ 1 (R d , w k ). Recently, similar problem has been considered in [20] by A. Velicu, wherein he shows that the function F defined in (1.4) attains an infimum and have found the best constant on a Weyl chamber. Velicu uses Nash's inequality to prove the existence of a minimizer whereas we use Dunkl-type refined Sobolev inequality to prove the existence of a minimizer. Towards this we have made explicit use of Plancherel formula of the Dunkl transform so our proof is different. This type of refined Sobolev inequality on R d is proved in more general setting in [9]. Our approach to this problem is mainly based on [5].
Finally, we consider the problem of finding the existence of an extremals for the inequality (1.3) in case of r = 2. By definition the best constant W k in (1.3) is given by
W k = sup |x| γ I k α f L s (R d ,w k ) |x| β f L 2 (R d ,w k )
, (1.5) where the supremum is taken over all non-vanishing functions f ∈ L 2 (R d , w k ). We first have obtained weighted boundedness for the Dunkl-type heat semi group operator and an improved version of Stein-weiss inequality (1.3) in the Dunkl setting. Then we have proved a compact embeddinġ H α,r β,k (R d ) ⊂ L s (K, |x| γs ), whereḢ
α,r β,k (R d ) = {u = I k α f : f ∈ L r (R d , |x| βr w k )} (1.6)
is the homogenous Sobolev space in the Dunkl setting , which is a Banach space with the norm u Ḣ α,r β,k (R d ) = |x| β f L r (R d ,w k ) . Using the compact embedding, we prove that W k defined in (1.5) has a maximizer. Our approach to this problem is based on [12].
We organize the paper as follows. In section 2, we provide a brief introduction to Dunkl theory and some known results. In section 3, we prove a Dunkl-type refined Sobolev inequality. In section 4, we prove the existence of an extremals for the Sobolev inequality associated with Dunkl gradient in case of p = 2. In section 5, we prove an weighted estimate for the operator e t∆ k . In section 6, we prove an improved version of Stein-Weiss inequality for D-Riesz potential. In section 7, we prove the existence of an extremals of Stein-Weiss inequality for the D-Riesz potential in case of r = 2.
Preliminaries
In this section, we shall briefly introduce the theory of Dunkl operators. For more details on Dunkl theory, we refer to [3,4,18].
For ν ∈ R d \ {0} let σ ν denote the reflection of R d in the hyperplane ν ⊥ given by the following formula:
σ ν (x) = x − 2 ν, x |ν| 2 ν. A finite subset R of R d \ {0} is said to be a root system if R ∩ R ν = {ν, −ν} and σ ν (R) = R, ∀ ν ∈ R.
The set of reflections {σ ν : ν ∈ R} generates the subgroup G := G(R) of the orthogonal group O(d, R), which is known as the reflection group associated with R. From now onwards let R be a fixed root system in R d and G be the associated reflection group . For simplicity, we assume R to be normalized in the sense that ν, ν = 2, ∀ ν ∈ R. A function k : R → C is called a multiplicity function on the root system R if it is invariant under the natural action of G on R, that is, if k(σ ν g) = k(g), ν, g ∈ R.
The set of all multiplicity functions forms a C-vector space and it is denoted by K.
Definition 2.1. Associated with G and k, the Dunkl operator
T ξ := T (ξ)(k) is defined by (for f ∈ C 1 (R d )) T ξ f (x) = ∂ ξ f (x) + ν∈R + k(ν) ν, ξ f (x) − f (σ ν (x)) ν, x , ξ ∈ R d ,
where ∂ ξ denotes the directional derivative in the direction of ξ and R + is a fixed positive subsystem of R.
For ξ = e i , we shall write T i for T e i . We denote Dunkl gradient by ∇ k = (T 1 , T 2 , . . . , T d ) and Dunkl Laplacian by ∆ k = d i=1 T 2 i . Throughout the paper we assume that k ≥ 0. Let w k denote the weight function defined by
w k (x) = ν∈R + | ν, x | 2k(ν) , x ∈ R d ,(2.1)
which is a G-invariant homogeneous function of degree 2γ k with γ k = ν∈R + k(ν).
Let d k = d + 2γ k . Further, we define the constants c k = R d e − |x| 2 2 w k (x)dx and a k = S d−1 w k (x ′ )dx ′ .
Then c k and a k are related by the following formula
c k = 2 d k 2 −1 Γ d k 2 a k . (2.2)
There exists a unique linear isomorphism V k on polynomials, which intertwines the associated commutative algebra of Dunkl operators and the algebra of usual partial differential operators. Using the function V k , one can define the Dunkl kernel E k as follows:
E k (x, y) := V k (e .,y )(x), x ∈ R d , y ∈ C d .
For k = 0, the Dunkl kernel E k reduces to the usual expotential function e ix.y . Alternatively, it is the solution of a joint eigen value problem for the associated Dunkl operators. We collect few properties of the Dunkl kernel E k .
Proposition 2.1. Let k ≥ 0, x, y ∈ C d , λ ∈ C, α ∈ Z d + . (i) E k (x, y) = E k (y, x) (ii) E k (λx, y) = E k (x, λy) (iii) E k (x, y) = E k (x, y). (iv) |∂ α y E k (x, y)| ≤ |x| |α| max g∈G e Re gx,y .
In particular, E k (−ix, y) ≤ 1 and |E k (x, y)| ≤ e |x||y| , ∀ x, y ∈ R d .
Using the Dunkl kernel one can define Dunkl transform , which is the generalization of classical Fourier transform. Dunkl transform enjoins similar properties to that of classical Fourier transform.
Definition 2.2.
For a function f ∈ L 1 (R d , w k ), the Dunkl transform associated with G and k ≥ 0, denoted by F k f , is defined as
F K (f )(ξ) = c −1 k R d f (x)E k (−iξ, x)w k (x)dx, ξ ∈ R d .
When k = 0, the Dunkl transform reduces to the classical Fourier transform. The Dunkl transform can be extended to an isometric isomorphism between L 2 (R d , w k ) and L 2 (R d , w k ) i.e., for f ∈ L 2 (R d , w k ), one has
f 2 L 2 (R d ,w k ) = F k (f ) 2 L 2 (R d ,w k ) . (2.3)
The usual translation operator f −→ f (. − y) leaves the Lebesgue measure on R d invariant. However the measure w k (x)dx is no longer invariant under the usual translation and the Leibniz's formula T i (f g) = f T i g + gT i f does not hold in general. So one can introduce the notion of a generalized translation operator defined on the Dunkl transform by the formula
F k (τ k y f )(ξ) = E k (iy, ξ)F k (f )(ξ). (2.4)
In case when k = 0, τ k y f reduces to the usual translation τ 0 y f (x) = f (x + y). In general, the explicit expression for τ k y f is unknown. It is known only when either f is a radial function or G = Z d 2 . The convolution of two functions f, g ∈ L 2 (R d , w k ) is defined as follows:
(f * k g)(x) = R d f (y)τ k y g(x)w k (y)dy.
The convolution operator satisfies the following basic properties:
(i) F k (f * k g) = F k (f ).F k (g) (ii) f * k g = g * k f
. Using the convolution operator, the heat semi-group operator e t∆ k is defined as follows:
e t∆ k u = u * k q k t , where q k t (x) = (2t) −(γ k + d 2 ) e − |x| 2 4t , x ∈ R d . (2.5)
In [13], it has been shown that the function q k t (x) satisfies the Dunkl-type heat equation ∆ k u − ∂ t u = 0 on R d × (0, ∞). A short calculation using the properties of Dunkl transform shows that
F k (q k t )(ξ) = e −t|ξ| 2 ,(2.6)
and
τ k y q k t (x) = (2t) −(γ k + d 2 ) e − |x| 2 +|y| 2 4t E k x √ 2t , y √ 2t . (2.7)
From (2.7), it is observed that τ k y q k t (x) = τ k x q k t (y). We recall few results which will be useful in this paper.
Theorem 2.1. [6] Let 1 ≤ p ≤ ∞ and g is a Schwartz class radial function. Then for any y ∈ R d , For any non-negative integer m and for any multi-indices α, β, there exists constant C m,α,β > 0 such that for any t > 0 and for any x, y ∈ R d , the following estimate holds:
τ k y g L p (R d ,w k ) ≤ g L p (R d ,w k ) .lim j→∞ X ||f j | p − |f j − f | p − |f | p |dx = 0.|∂ m t ∂ α x ∂ β y h t (x, y)| ≤ C m,α,β t −m− |α| 2 − |β| 2 h 2t (x, y),(2.
9)
where h t (x, y) = τ k y q k t (x).
We observe the following properties of the weight function w k . Properties of w k :
(1) For c > 0,
R d e −c|x| 2 w k (x)dx = c − d k 2 R d e −|y| 2 w k (y)dy.
Proof. By substituting √ cx = y and using (2.1) we get
R d e −c|x| 2 w k (x)dx = c − d 2 R d e −|y| 2 w k y √ c dy = c − d 2 R d e −|y| 2 ν∈R + ν, y √ c 2k(ν) dy = c − d 2 R d e −|y| 2 ν∈R + c −k(ν) | ν, y | 2k(ν) dy = c −(γ k + d 2 ) R d e −|y| 2 w k (y)dy = c − d k 2 R d e −|y| 2 w k (y)dy. (2) For c ∈ R, w k (cx) = |c| 2γ k w k (x).
Proof. The proof follows from the definition of the weight function w k defined in (2.1).
(3) R d e −|y| 2 w k (y)dy = a k 2 Γ d k 2 .
Proof. By substituting x = √ 2y in the integral involved in c k and then using (2.2) as well as property (2), we obtain property (3).
(4) If R > 0 and c < d k , then |y|≤R |x| −c w k (x)dx = a k R d k −c d k −c .
Proof. Using property (2),
|y|≤R |x| −c w k (x)dx = R 0 S d−1 r −c w k (rx ′ )r n−1 dx ′ dr = R 0 S d−1 r −c r 2γ k w k (x ′ )r n−1 dx ′ dr = R 0 r d k −c−1 dr S d−1 w k (x ′ )dx ′ = a k R d k −c d k − c ,
since the integrability condition at 0 is c < d k , thus proving property (4).
Dunkl-type refined Sobolev inequality
The goal of this section is to prove Dunkl-type refined Sobolev inequality (3.1). In order to prove this we first prove the following Pseudo-Poincare inequality in the Dunkl setting for p = 2.
Lemma 3.1. For u ∈ L 2 (R d , w k ), one has u − e t∆ k u 2 L 2 (R d ,w k ) ≤ t ∇ k u 2 L 2 (R d ,w k ) .
Proof. In order to prove the above Lemma, we shall make use of the following inequality.
(1 − e −x ) 2 ≤ 1 − e −x ≤ x, ∀ x ≥ 0.
Now, using the Plancherel formula(2.3) and (2.6), we get
u − e t∆ k u 2 L 2 (R d ,w k ) = F k (u − e t∆ k u ) 2 L 2 (R d ,w k ) = F k (u) − F k (u * k q k t ) 2 = F k (u) − F k (u)F k (q k t ) 2 = R d |F k (u)(ξ) − F k (u)(ξ)F k (q k t )(ξ)| 2 w k (ξ)dξ = R d |F k (u)(ξ)| 2 (1 − e −t|ξ| 2 ) 2 w k (ξ)dξ ≤ t R d |F k (u)(ξ)| 2 |ξ| 2 w k (ξ)dξ = t F k (∇ k u) 2 L 2 (R d ,w k ) = t ∇ k u 2 L 2 (R d ,w k ) . Theorem 3.1. For d ≥ 3, there is a constant C d,k > 0 such that for all u ∈ H 1 (R d , w k ), one has R d |u| q (x)w k (x)dx 1 q ≤ C d,k R d |∇ k u| 2 (x)w k (x)dx 1 q sup t>0 t (d k −2) 4 e t∆ k u ∞ 2 d k , (3.1) with q = 2d k d k −2 .
Proof. Consider the function
e t∆ k u(x) = u * k q k t (x) = R d u(y)(τ k y q k t )(x)w k (y)dy.
Applying Holder's inequality to the function e t∆ k u with p = 2d k d k +2 and q = 2d k d k −2 , we get
|(e t∆ k u)(x)| ≤ u q,w k τ k x q k t p,w k . (3.2)
Now, using (2.8),
(τ k x q k t ) p p,w k = R d |(τ k x q k t )(y)| p w k (y)dy ≤ R d |q k t (y)| p w k (y)dy.
By substituting the value of q k t from (2.5) in the last integral and using property (1) as well as property (3), we obtain
(τ k x q k t ) p p,w k ≤ R d |(2t) −(γ k + d 2 ) e − |y| 2 4t | p w k (y)dy = (2t) −(γ k + d 2 )p R d e − p 4t |y| 2 w k (y)dy = (2t) − d k 2 p p 4t − d k 2 R d e −|y| 2 w k (y)dy = A d,k t − d k 2 (p−1) , (3.3) where A d,k = 2 − d k 2 p ( p 4 ) − d k 2 a k 2 Γ d k 2 with p = 2d k d k +2 . This implies that (τ k x q k t ) p,w k ≤ A 1 p d,k t − d k 2 (1− 1 p ) = A 1 p d,k t − d k 2 (1− d k +2 2d k ) = A 1 p d,k t − d k −2 4 . Then from (3.2), we get e t∆ k u ∞ ≤ A 1 p d,k u q,w k t − d k −2 4 = C d,k u q,w k t − d k −2 4 . Let I[u] = sup t>0 t (d k −2) 4 e t∆ k u ∞ . Then I[u] ≤ C d,k u q,w k . Thus by homogeneity, we can assume that I[u] ≤ 1 , that is, t d k −2 4 e t∆ k u(x) ≤ 1, ∀ t > 0, ∀ x ∈ R d ,(3.4)
and hence in order to prove (3.1), it is enough to show that
R d |u| q (x)w k (x)dx ≤ C q d,k R d |∇ k u| 2 (x)w k (x)dx. (3.5)
Now we will be using some basic measure theory results in the proof. Recall that
|u(x)| q = ∞ 0 χ {|u(x)| q >λ} dλ = q ∞ 0 χ {|u(x)|>τ } τ q−1 dτ.
Existence of an extremal of Sobolev inequality associated with Dunkl gradient and of Stein-Weiss inequality for D-Riesz potential 9
From this one can easily write that
R d |u(x)| q w k (x)dx = q ∞ 0 |{|u| > τ }|τ q−1 dτ (3.6) where |{|u| > τ }| is the measure given by |{|u| > τ }| = R d χ {|u(x)|>τ } w k (x)dx. If we write u = (u − e t∆ k u) + e t∆ k u for some t > 0 chosen later, then |{|u| > τ }| ≤ |{|u − e t∆ k u}| > τ /2| + |{|e t∆ k u| > τ /2}|. Let us now choose t = t τ satisfying τ /2 = t − d k −2 4 , then from (3.4), |{|e tτ ∆ k u| > τ /2}| = 0. Hence by (3.6) we have R d |u| q w k (x)dx ≤ q ∞ 0 |{|u − e tτ ∆ k u| > τ /2}|τ q−1 dτ. (3.7)
For a fixed constant b ≥ 1/16 and for any τ > 0, we define a function u τ on R d as follows:
u τ (x) = (b − 1 16 )τ if u(x) > bτ, u(x) − τ 16 if bτ ≥ u(x) ≥ τ 16 , 0 if τ 16 > u(x) > − τ 16 , u(x) + τ 16 if − τ 16 ≥ u(x) ≥ −bτ, −(b − 1 16 )τ if u(x) < −bτ.
Note that u τ is inḢ(R d , w k ) and
R d |∇ k u τ | 2 w k (x)dx = τ /16≤|u|≤bτ |∇ k u| 2 w k (x)dx. The decomposition u − e tτ ∆ k u = (u τ − e tτ ∆ k u τ ) − e tτ ∆ k (u − u τ ) + (u − u τ ) gives |{|u − e tτ ∆ k u| > τ 2 }| ≤ |{u τ − e tτ ∆ k u τ | > τ 4 }| + |{|u − u τ | > τ 8 }| +|{|e tτ ∆ k (u − u τ )| > τ 8 }|. (3.8)
By using Chebyshev inequality with Lemma 3.1, we get the bound for the first term of the right hand side of (3.8)
|{|u − e tτ ∆ k u| > τ 4 }| ≤ (τ /4) −2 u τ − e tτ ∆ k u τ 2 L 2 (R d ,w k ) ≤ (τ /4) −2 t τ ∇ k u τ 2 L 2 (R d ,w k ) ≤ 4(τ /2) −q τ /16≤|u|≤bτ |∇ k u| 2 w k (x)dx, which implies that ∞ 0 |{|u τ − e −tτ ∆ k u τ | > τ 4 }|τ q−1 dτ = 2 q+2 log(16b) R d |∇ k u| 2 w k (x)dx. (3.9)
Now we need to obtain the bound for the second and third term of the right hand side of (3.8). Towards this, first we observe that
|u τ − u| = |u τ − u|χ {|u|≤bτ } + |u τ − u|χ {|u|>bτ } ≤ τ 16 + |u|χ {|u|>bτ } ,(3.10)
which leads to again by Chebyshev inequality
|{|u − u τ | > τ 8 }| ≤ |{|u|χ {|u|>bτ } > τ 16 }| ≤ (τ /16) −1 R d |u|χ {|u|>bτ } w k (x)dx. (3.11)
Using the properties of Dunkl heat kernel and (3.10),
|e t∆ k u τ − e t∆ k u| ≤ e t∆ k |u τ − u| ≤ τ 16 τ k y q k t L 1 (R d ,w k ) + e t∆ k (|u|χ {|u|>bτ } ) = τ 16 c k + e t∆ k (|u|χ {|u|>bτ } ).
Now we assume the c k ≤ 1. Then
|e t∆ k u τ − e t∆ k u| ≤ τ 16 + e t∆ k (|u|χ {|u|>bτ } ). (3.12)
Hence
|{|e t∆ k (u τ − u)| > τ 8 }| ≤ |{e t∆ k (|u|χ {|u|>bτ } ) > τ 16 }| ≤ (τ /16) −1 R d e t∆ k (|u|χ {|u|>bτ } )w k (x)dx = (τ /16) −1 c k R d |u|χ {|u|>bτ } w k (x)dx ≤ (τ /16) −1 R d |u|χ {|u|>bτ } w k (x)dx.
Then using (3.11), we have the estimate
∞ 0 (|{|e t∆ k u τ − e t∆ k u| > τ 8 }| + |{|u − u τ | > τ 8 }|)τ q−1 dτ = 32 q − 1 b −q+1 R d |u| q w k (x)dx.
Now from (3.6), using (3.9) and the above estimate we obtain for sufficiently large b,
R d |u| q w k (x)dx ≤ q2 q+2 log(16b) 1 − 32q q−1 b −q+1 R d |∇ k u| 2 w k (x)dx. (3.13)
thus proving (3.5).
Now let us assume that c k > 1. Choose b > 1 16c k and for any τ > 0, define the function u τ on R d as follows:
u τ (x) = (b − 1 16c k )τ if u(x) > bτ, u(x) − τ 16c k if bτ ≥ u(x) ≥ τ 16c k , 0 if τ 16c k > u(x) > − τ 16c k , u(x) + τ 16c k if − τ 16c k ≥ u(x) ≥ −bτ, −(b − 1 16c k )τ if u(x) < −bτ.
Now proceeding as before we get,
∞ 0 |{|u τ − e −tτ ∆ k u τ | > τ 4 }|τ q−1 dτ = 2 q+2 log(16bc k ) R d |∇ k u| 2 w k (x)dx.
Also, in this case
|u τ − u| ≤ τ 16c k + |u|χ {|u|>bτ } ≤ τ 16 + |u|χ {|u|>bτ } , and |e t∆ k u τ − e t∆ k u| ≤ e t∆ k |u τ − u| ≤ τ 16c k τ k y q k t L 1 (R d ,w k ) + e t∆ k (|u|χ {|u|>bτ } ) = τ 16 + e t∆ k (|u|χ {|u|>bτ } ).
Then proceeding exactly as before, for sufficiently large b, we have
R d |u| q w k (x)dx ≤ q2 q+2 log(16bc k ) 1 − 32qc k q−1 b −q+1 R d |∇ k u| 2 w k (x)dx,
thus proving (3.5). This completes the proof of Theorem 3.1.
Existence of extremals for Dunkl-type Sobolev inequality
The aim of this section is to prove the existence of a minimizer for the function F defined in (1.4). Now we prove the following corollary.
Corollary 4.1. For d ≥ 3, let (u j ) be a bounded sequence inḢ 1 (R d , w k ).
Then either one of the following statements holds.
(i) (u j ) converges to 0 in L q (R d , w k ).
(ii) There exists a subsequence (u jm ) of (u j ) and sequences (a m )
⊂ R d and (b m ) ⊂ (0, ∞) such that v m (x) = b d k −2 2 m (τ k am u jm )(b m x)
converges weekly inḢ 1 (R d , w k ) to a function v ≡ 0. Moreover, (v m ) converges pointwise a.e. to v.
Proof. Assume that (i) does not hold. Then there exists ǫ > 0 and a subsequence (u j,m ) of (u j ) such that u j,m q,w k ≥ ǫ. We shall denote u j,m by u j itself. Since u j is bounded inḢ 1 (R d , w k ), there exists A > 0 such that ∇ k u j L 2 (R d ,w k ) ≤ √ A ∀ j. Now applying the Dunkl-type Sobolev inequality (3.1) for the function u j , we get
sup t>0 t (d k −2) 4 e t∆ k u j ∞ 2 d k ≥ C −1 d,k A − d k −2 2d k ǫ.
Then there exists t j > 0,
x j ∈ R d such that t d k −2 4 j |e t j ∆ k u j (x j )| ≥ 1 2 C − d k 2 d,k A − d k −2 4 ǫ. (4.1) Let G(y) = 2 − d k 2 e − |y| 2 4 and v j (y) = t d k −2 4 j (τ k −x j u j )( √ t j y). Now consider R d G(y)v j (y)w k (y)dy = R d 2 − d k 2 e − |y| 2 4 t d k −2 4 j (τ k −x j u j )( t j y)w k (y)dy = 2 − d k 2 t d k −2 4 j R d e − |y| 2 4t j (τ k −x j u j )(y)w k y √ t j 1 t d 2 j dy = 2 − d k 2 t d k −2 4 j t − d k 2 j R d e − |y| 2 4t j (τ k −x j u j )(y)w k (y)dy ,
using property (2) of section 2. Then using (4.1)
R d G(y)v j (y)w k (y)dy = t d k −2 4 j R d q t j (y)(τ k −x j u j )(y)w k (y)dy = t d k −2 4 j R d u j (y)(τ k x j q t j )(y)w k (y)dy = t d k −2 4 j R d u j (y)(τ k y q t j )(x j )w k (y)dy = t d k −2 4 j e t j ∆ k u j (x j ) ≥ 1 2 C − d k 2 d,k A − d k −2 4 ǫ. (4.2)
Moreover,
F k (v j )(ξ) = t − d k +2 4 j F k (τ k −x j u j ) ξ √ t j . (4.3)
Existence of an extremal of Sobolev inequality associated with Dunkl gradient and of Stein-Weiss inequality for D-Riesz potential 13
Indeed, by inserting the value of v j in Definition 2.2, we get
F k (v j )(ξ) = c −1 k R d v j (y)E k (−iξ, y)w k (y)dy = c −1 k R d t d k −2 4 j (τ k −x j u j )( t j y)E k (−iξ, y)w k (y)dy.
Now by replacing y by y √ t j and proceeding as before, we obtain
F k (v j )(ξ) = t d k −2 4 j t − d k 2 j c −1 k R d (τ k −x j u j )(y)E k − iξ, y √ t j w k (y)dy = t − d k +2 4 j c −1 k R d (τ k −x j u j )(y)E k − iξ √ t j , y w k (y)dy = t − d k +2 4 j F k (τ k −x j u j ) ξ √ t j ,
using Proposition 2.1(ii) and the definition of Dunkl transform, thus proving (4.3). Therefore, using (4.3)
∇ k v j 2 L 2 (R d ,w k ) = F k (∇ k v j ) 2 L 2 (R d ,w k ) = R d |ξ| 2 |F k (v j )(ξ)| 2 w k (ξ)dξ = t − d k +2 2 j R d |ξ| 2 F k (τ k −x j u j ) ξ √ t j 2 w k (ξ)dξ = t − d k +2 2 j t 1+γ k + d 2 j R d |ξ| 2 |F k (τ k −x j u j )(ξ) 2 w k (ξ)dξ.
Consequently, using (2.4) and Proposition 2.1 (iv), we get
∇ k v j 2 L 2 (R d ,w k ) = R d |ξ| 2 |E k (−ix j , ξ)F k (u j )(ξ)| 2 w k (ξ)dξ ≤ R d |ξ| 2 |F k (u j )(ξ)| 2 w k (ξ)dξ = ∇ k u j 2 L 2 (R d ,w k ) . Thus ∇ k v j 2 L 2 (R d ,w k ) ≤ A for all j.
By Banach-Alaoglu theorem, v j has a weekly convergent subsequence inḢ 1 (R d , w k ) and let it converge to w. Since G ∈Ḣ 1 (R d , w k ) * , G(v j ) converges to G(w). It follows from (4.2) that G(v j ) = 0, ∀ j and therefore, G(w) = 0, which in turn imply that w ≡ 0. This completes the proof of the corollary.
S d,k = inf u∈Ḣ 1 (R d ,w k ) R d |∇ k u| 2 w k (x)dx R d |u| q w k (x)dx 2 q is attained.
Proof. Let (u j ) be a minimizing sequence, which we assume to be normalized in
L q (R d , w k ). Then (u j ) is bounded inḢ 1 (R d , w k ), since the sequence ∇ k u j 2 L 2 (R d ,w k )
converges to S d,k . Moreover, because (u j ) has norm 1 in L q (R d , w k ), from Corollary 4.1, we can say that after a generalized translation and a dilation , (u j ) converges weekly to a non-zero function u a.e. inḢ 1 (R d , w k ). This implies that
R d |∇ k u j | 2 w k (x)dx = R d |∇ k (u j − u)| 2 w k (x)dx + R d |∇ k u| 2 w k (x)dx + o(1).
From Lemma 2.1, we have
1 = R d |u j | q w k (x)dx = R d |u j − u| q w k (x)dx + R d |u| q w k (x)dx + o(1).
As a consequence, as 2 q < 1,
1 = R d |u j | q w k (x)dx 2 q ≤ R d |u j − u| q w k (x)dx 2 q + R d |u| q w k (x)dx 2 q + o(1), since (a + b) u ≤ a u + b u for a, b > 0, 0 ≤ u ≤ 1. Hence S d,k + o(1) = R d |∇ k u j | 2 w k (x)dx ≥ R d |∇ k (u j − u)| 2 w k (x)dx + R d |∇ k u| 2 w k (x)dx + o(1) R d |u j − u| q w k (x)dx 2 q + R d |u| q w k (x)dx 2 q + o(1) ≥ S d,k R d |u j − u| q w k (x)dx 2 q + R d |∇ k u| 2 w k (x)dx + o(1) R d |u j − u| q w k (x)dx 2 q + R d |u| q w k (x)dx 2 q + o(1)
, which implies that
S d,k + o(1) ≥ R d |∇ k u| 2 w k (x)dx + o(1) R d |u| q w k (x)dx 2 q + o(1)
, from which it follows that u is a minimizer.
A weighted estimate for the heat semi group e t∆ k
In this section, we prove the following Proposition involving a weighted estimate for the operator e t∆ k , using which we show that e t∆ k is a compact operator.
Proposition 5.1. Let d ≥ 2, 1 < r < ∞. Assume that 0 < β < d k r ′ and fix t > 0. Then (i) e t∆ k f L ∞ (R d ) ≤ C d,r,β,t,k |x| β f r,w k (ii) |x| w e t∆ k f L ∞ (R d ) ≤ D d,r,β,t,k |x| β f r,w k for β ≥ w > 0. (iii) ∂ x i e t∆ k f L ∞ (R d ) ≤ E d,r,β,t,k |x| β f r,w k , i = 1, 2, . . . , n.
Proof.
Consider
|e t∆ k f (x)| ≤ R d |f (y)||τ k y q k t (x)|w k (y)dy ≤ R d |f (y)| r |y| βr w k (y)dy 1 r R d |τ k y q k t (x)| r ′ |y| −βr ′ w k (y)dy 1 r ′ = |y| β f r,w k (I 1 (x) + I 2 (x) 1 r ′ ,(5.1)
where
I 1 (x) = |y|≤ √ t |τ k y q k t (x)| r ′ |y| −βr ′ w k (y)dy and I 2 (x) = |y|> √ t |τ k y q k t (x)| r ′ |y| −βr ′ w k (y)dy.
Substituting the value of τ k y q k t (x) from (2.7) and then using Proposition 2.1,
|τ k y q k t (x)| = (2t) −(γ k + d 2 ) e − |x| 2 +|y| 2 4t E k x √ 2t , y √ 2t ≤ (2t) −(γ k + d 2 ) e − |x| 2 +|y| 2 4t e |x||y| 2t = (2t) − d k 2 e − (|x|−|y|) 2 4t (5.2) ≤ Ct − d k 2 , ∀ x, y ∈ R d , where C = 2 − d k 2 .
Then
I 1 (x) ≤ C r ′ t − d k 2 r ′ |y|≤ √ t |y| −βr ′ w k (y)dy.
By property (4) of section 2, the above integral is finite if β < d k r ′ and equals to
a k ( √ t) d k −βr ′ d k −βr ′ . Thus I 1 (x) ≤ A d,r,β,k t − d k 2 r ′ t 1 2 (d k −βr ′ ) , where A d,r,β,k = a k C r ′ d k −βr ′ .
On the other hand, over the integral I 2 , since |y| > √ t, |y| −βr ′ ≤ t − βr ′ 2 and hence
I 2 (x) ≤ t − βr ′ 2 |y|> √ t |τ k y q k t (x)| r ′ w k (y)dy ≤ t − βr ′ 2 R d |τ k y q k t (x)| r ′ w k (y)dy ≤ t − βr ′ 2 R d |q k t (y)| r ′ w k (y)dy ≤ B d,r,k t − βr ′ 2 t − d k 2 (r ′ −1) , (5.3) by taking p = r ′ in (3.3), where B d,r,k = 2 − d k 2 r ′ r ′ 4 − d k 2 a k 2 Γ d k 2 .
Finally from (5.1) using the estimates of I 1 , I 2 and the fact that (a + b) u ≤ a u + b u for a, b > 0, u < 1, we obtain
|e t∆ k f (x)| ≤ |y| β f L r (R d ,w k ) A d,r,β,k t − d k 2 r ′ t 1 2 (d k −βr ′ ) + B d,r,k t − βr ′ 2 t − d k 2 (r ′ −1) 1 r ′ ≤ C d,r,β,k |y| β f L r (R d ,w k ) t − d k 2 t 1 2r ′ (d k −βr ′ ) + t − β 2 t − d k 2r (5.4) ≤ C d,r,β,t,k |y| β f L r (R d ,w k ) ,
where C d,r,β,k = (max{A d,r,β,k , B d,r,k })
1 r ′ and C d,r,β,t,k = C d,r,β,k t − d k 2 t 1 2r ′ (d k −βr ′ ) + t − β 2 t − d k 2r
. Thus we have proved that
e t∆ k f L ∞ (R d ) ≤ C d,r,β,t,k |x| β f L r (R d ,w k ) , (5.5)
thus proving (i).
In particular, when r = 2, then from (6.3), we get
e t∆ k f L ∞ (R d ) ≤ C d,β,k t − 1 2 ( d k 2 +β) |x| β f L 2 (R d ,w k ) , (5.6)
where C d,β,k = 2C d,2,β,k . Now we shall prove (ii). since w > 0, using (5.5) we observe that, for |x| ≤ 1,
||x| w e t∆ k f (x)| ≤ |e t∆ k f (x)| ≤ C d,r,β,t,k |x| β f L r (R d ,w k ) . (5.7)
So we assume that |x| > 1. Consider
||x| w e t∆ k f (x)| ≤ |x| w R d |f (y)||τ k y q k t (x)|w k (y)dy ≤ R d |f (y)| r |y| βr w k (y)dy 1 r R d |x| wr ′ |y| −βr ′ |τ k y q k t (x)| r ′ w k (y)dy 1 r ′ ≤ |y| β f r,w k R d |x| βr ′ |y| −βr ′ |τ k y q k t (x)| r ′ w k (y)dy 1 r ′ ,
since |x| > 1 and β ≥ w, it implies that |x| (w−β)r ′ ≤ 1. Then
||x| w e t∆ k f (x)| ≤ |y| β f r,w k (I 1 (x) + I 2 (x)) 1 r ′ , (5.8) where I 1 (x) = |y|≤ |x| 2 |x| βr ′ |y| −βr ′ |τ k y q k t (x)| r ′ w k (y)dy and I 2 (x) = |y|≥ |x| 2 |x| βr ′ |y| −βr ′ |τ k y q k t (x)| r ′ w k (y)dy.
Over the first integral I 1 (x), (|x| − |y|) 2 ≥ |x| 2 4 ∀ y, since |y| ≤ |x| 2 . Consequently, from (5.2) we arrive at the bound τ k y q k t (x) ≤ (2t) − d k 2 e − |x| 2 16t . Then
I 1 (x) ≤ (2t) − d k 2 r ′ |x| βr ′ e r ′ 16t |x| 2 |y|< |x| 2 |y| −βr ′ w k (y)dy.
For β < d k r ′ , the above integral finite and equals to
1 d k −βr ′ ( |x| 2 ) d k −βr ′ . Therefore, I 1 (x) ≤ (2t) − d k 2 r ′ (d k −βr ′ )2 d k −βr ′ |x| d k e r ′ 16t |x| 2 . Since |x| d k e r ′
16t |x| 2 −→ 0 as |x| −→ ∞, it follows that I 1 (x) ≤ A ′ d,r,β,t,k for some constant A ′ d,r,β,t,k > 0. Now we consider the integral I 2 . As |y| > |x| 2 , |x| |y| βr ′ < 2 βr ′ for β, r ′ > 0.
I 2 (x) ≤ 2 βr ′ |y|> |x| 2 |τ k y q k t (x)| r ′ w k (y)dy ≤ 2 βr ′ R d |q k t (y)| r ′ w k (y)dy ≤ 2 βr ′ B d,r,k t − d k 2 (r ′ −1) = B ′ d,r,β,t,k , using (5.3). Hence for |x| > 1, from (5.8) , we get a constant C ′ d k ,r ′ ,t > 0 such that ||x| w e t∆ k f (x)| ≤ C ′ d,r,β,t,k |y| β u r,w k . (5.9)
Consequently, combining (5.7) and (5.9), we get a constant D d,r,β,t,k > 0 such that
|x| w e t∆ k f L ∞ (R d ) ≤ D d,r,β,t,k |x| β f r,w k ,
thus proving (ii). Next we shall prove (iii). Consider
∂ ∂x i (e t∆ k f )(x) = ∂ ∂x i (f * k q k t )(x) = ∂ ∂x i R d f (y)τ k y q k t (x)w k (y)dy = R d f (y) ∂ ∂x i τ k y q k t (x)w k (y)dy ≤ R d |f (y)| ∂ ∂x i τ k y q k t (x) w k (y)dy ≤ Ct − 1 2 R d |f (y)||τ k y q k 2t (x)|w k (y)dy,
by taking m = 0, α = e i , β = 0 in (2.9). Now using (i), there exists constant E d,r,β,t,k > 0 such that
∂ x i e t∆ k f L ∞ (R d ) ≤ E d,r,β,t,k |x| β f r,w k ,
thus proving (iii).
Using Proposition 5.1, we prove the following theorem.
Theorem 5.1. Let d ≥ 2, 1 < r < ∞. If 0 < β < d k r ′ , then for any fixed t > 0, the operator e t∆ k is a compact operator from L r (R d , |x| βr w k ) to L ∞ (R d ).
Proof. Let (u j ) j∈N ∈ L r (R d , |x| βr w k ) be a bounded sequence so that |x| β u j r,w k ≤ C 0 for all j ∈ N. We will prove that the sequence (e t∆ k u j ) j∈N has convergent subsequence in L ∞ (R d ) and that will imply the theorem. Let v j = e t∆ k u j . Since (u j ) j∈N is a bounded sequence, using Proposition 2.1 (i), we have
v j L ∞ (R d ) = e t∆ k u j L ∞ (R d ) ≤ C |x| β u j r,w k ≤ C 0 .
This proves that each v j ∈ L ∞ (R d ) and the collection (v j ) j is equibounded in R d . Moreover, Proposition 2.1 (iii) shows that (v j ) j is also equicontinuos in R d . Now for each n ∈ N, we define the compact set A n := {x ∈ R d : |x| ≤ n}. Then by the Arzelá-Ascoli theorem for each n ∈ N, there exists a subsequence of (v j ) j which converges uniformly in A n . Now by applying diagonal argument, we get a subsequence of (v j ) j which converges uniformly in every A n . let us call this subsequence also by (v j ) j and we can write v j → v uniformly in each A n for some v ∈ L ∞ (R d ).
Now let z be such that 0 < z < β. By Proposition 2.1(ii), we can write
|x| z v j L ∞ (R d ) = |x| z e t∆ k u j L ∞ (R d ) ≤ C 1 |x| β u j r,w k . (5.10)
Now by using (5.10) we get
sup |x|>n |v j | ≤ sup |x|>n |x| n z |v j | ≤ n −z |x| z v j L ∞ (R d ) ≤ C 1 n −z .
Thus we can easily see that v j → v strongly in L ∞ (R d ) and this proves the theorem.
Improved Stein-Weiss inequality for the D-Riesz potential
In this section, we will be focusing on deriving an improved version of the Stein-Weiss inequality for the D-Riesz potential i.e., we are interested to generalize The-orem1.2. Towards this, for any δ > 0, first we define the Dunkl Besov space aṡ
B −δ,k ∞,∞ := {f : f is a tempered distribution on R d and f Ḃ −δ,k ∞,∞ < ∞}, where f Ḃ −δ,k ∞,∞ := sup t>0 t δ/2 e t∆ k f L ∞ . (6.1)
Theorem 6.1. Let d ≥ 2 and 0 < α < d k . Also let β, γ, µ, θ, r and s be such
that 1 < r < s < ∞, γ > − d k s , β < d k r ′ , β ≥ γ θ .
Also it satisfy that µ > 0, max{ r s , µ µ+α } < θ ≤ 1 and d k s
+ γ = β + d k r − α θ + µ(1 − θ). (6.2)
Then for all f ∈ L r (R d , |x| βr w k ) ∩Ḃ −µ−α,k ∞,∞ ,the following inequality holds:
|x| γ I k α f L s (R d ,w k ) ≤ C k x| β f θ L r (R d ,w k ) f 1−θ B −µ−α,k ∞,∞ . (6.3)
Proof. The case θ = 1 reduces to Theorem 1.2. So we will prove the theorem for θ < 1. For f ∈ L r (R d , |x| βr w k ), let u = I k α f . Then u has an integral representation of the following form: Our aim is to find a bound for u. To achieve this we will look for the bounds for L k f and H k f separately. Using the definition of Besov norm in (6.1), we have
u = 1 Γ(α/2)|L k f (x)| ≤ C k T −µ/2 f B −µ−α,k ∞,∞
. Now we will find the bound for H k f . Let
Φ k α,T (x) = 1 Γ( α 2 ) T 0 t α 2 −1 q k t (x)dt.
It can be easily verified that
H k f = Φ k α,T * k f . Fix ǫ = µ/θ−µ 2
> 0. We note that since θ > µ µ+α , α − 2ǫ > 0. Since for a given u > 0, there exists a constant C > 0 such that for any non-zero real x, e −|x| ≤ C |x| u holds, we can write by taking u = (d k − α)/2 + ǫ > 0,
Φ k α,T (x) = 1 Γ( α 2 ) T 0 t α 2 −1 (2t) − d k 2 e − |x| 2 4t dt ≤ C T 0 t (α−d k )/2−1 4t |x| 2 (d k −α)/2+ǫ dt ≤ C k 1 |x| d k −α+2ǫ T 0 t −1+ǫ dt = C k T ǫ |x| d k −α+2ǫ .
Since Dunkl translation is linear and positivity-preserving for radial functions we can
write τ k y (Φ k α,T )(x) ≤ T ǫ τ k y (|.| −(d k −α+2ǫ) )(x). So we obtain H k f (x) ≤ C k T ǫ I k α−2ǫ f (x). Choose T such that T −µ/2 f Ḃ −µ−α,k ∞,∞ = T ǫ I k α−2ǫ f (x), which implies that T = f Ḃ −µ−α,k ∞,∞ I k α−2ǫ f (x) 1/(ǫ+µ/2) .
By substituting the value of T, we arrive at the point wise bound
|u(x)| = |I k α f (x)| ≤ C k I k α−2ǫ f (x) θ f 1−θ B −µ−α,k ∞,∞ .
Hence
|x| γ I k α f L s (R d ,w k ) ≤ C k |x| γ/θ I k α−2ǫ f θ L sθ (R d ,w k ) f 1−θ B −µ−α,k ∞,∞ . (6.4)
If we assume that α ′ = α −2ǫ, γ ′ = γ θ , s ′ = sθ, then it is easy to see by the hypothesis of the theorem and the choice of ǫ that
γ ′ > − d k s ′ , β ≥ γ ′ , 0 < α ′ < d k , r ≤ s ′ , β < d k r ′ and α ′ + γ ′ − β = d k ( 1 r − 1 s ′ ).
Hence by using Theorem 1.2, we get
|x| γ/θ I k α−2ǫ f L sθ (R d ,w k ) ≤ C ′ k |x| β f L r (R d ,w k ) . (6.5)
Now from (6.4) and (6.5), we get the desired inequality
|x| γ I k α f L s (R d ,w k ) ≤ D k |x| β f θ L r (R d ,w k ) f 1−θ B −µ−α,k ∞,∞ .
Remark 6.1. One can prove Theorem 6.1 for the case θ = µ µ+α if the weighted L r -boundedness of the maximal function M k is known for 1 < r < ∞. We recall that for f ∈ S(R d ), S. Thangavelu and Y. Xu [18], defined the maximal function M k as follows:
M k f (x) = sup r>0 R d f (y)τ k x χ Br (y)w k (y)dy Br w k (y)dy .
When θ = µ µ+α , following [12] and the fact that
|H k f (x)| ≤ C k T α 2 M k f (x), we have proved |x| γ I k α f L s (R d ,w k ) ≤ C k |x| β M k f θ L r (R d ,w k ) f 1−θ B −µ−α,k ∞,∞ .
Now in order to obtain (6.3), one has to prove
|x| β M k f L r (R d ,w k ) ≤ |x| β f L r (R d ,w k ) , (6.6)
which is not known to be true in general. For β = 0, (6.6) has been proved by S. Thangavelu and Y. Xu in [18].
Existence of an extremals for Stein-Weiss inequality associated with Dunkl Laplacian
Theorem 7.1. Let d ∈ N, 1 < r ≤ s < ∞, γ > − d k s , β ≥ γ, 0 < α < d k , β < d k r ′ . Further we assume that β + d k r > α > d k r − d k s + β − γ > 0. (7.1)
Then if K ⊂ R d is compact, then one has the compact embeddinġ H α,r β,k (R d ) ⊂ L s (K, |x| γs ), (7.2) where the spaceḢ α,r β,k (R d ) is defined in (1.6).
Proof. Let u ∈Ḣ α,r β,k (R d ). Then u = I k α f , for some f ∈ L r (R d , |x| βr w k ). Now we chooses such that 1
s (d k + γs) = d k r + β − α. (7.3)
We define v =s s andγ = γs s . From (7.1), it follows that v > 1 andγ = γ v . Then (7.3) can be rewritten as
d k 1 r − 1 s = α +γ − β.
We replace γ and s in Theorem 1.2 byγ ands respectively. Since v > 1, r ≤ s implies that r ≤ sv =s and β ≥ γ implies β ≥γ. Alsoγ > − d k s since γ > − d k s . Thus all the conditions of Theorem 1.2 are satisfied and hence from (1.3), we have
|x|γI k α f (x) Ls(R d ,w k ) ≤ C k |x| β f (x) L r (R d ,w k ) . (7.4)
Applying Holder's inequality with components v and v ′ and using the fact that (4) of section 2. Then using (7.4),
γ > − d k s , K |u| s |x| γs w k (x)dx = K |u| s |x| γs v |x| γs v ′ w k (x)dx ≤ K |u| sv |x| γs w k (x)dx 1 v K |x| γs w k (x)dx 1 v ′ ≤ C K K |u|s|x|γsw k (x)dx 1 v , where C K = a k d k +γs 1 v ′ , by propertyK |u| s |x| γs w k (x)dx 1 s ≤ C K K |I k α f |s|x|γsw k (x)dx 1 s ≤ C K |x| β f (x) L r (R d ,w k ) = C K u Ḣ α,r β,k (R d )
, proving that the embedding (7.2) is continuous.
Let us define the kernel of the D-Riesz potential as
K α,k (x) = (c k α ) −1 |x| −(d k −α)
and for t > 0, the truncated kernel as
K t α,k (x) = (c k α ) −1 |x| −(d k −α) χ {|x|>t} .δ = α − ( d k r − d k s + β − γ)
.
Then for any f ∈ L r (R d , |x| βr ) and for any t > 0, K t α,k * k f − K α,k * k f |x| γ s,w k ≤ Ct δ |x| β f r,w k . Now we shall show that the embedding (7.2) is compact. Let {u m } be a bounded sequence inḢ α,r β,k (R d ). Then we can write u m = I k α f m , where {f m } is a bounded sequence in L r (R d , |x| βr w k ). Since L r (R d , |x| βr w k ) is a reflexive space, {f m } has a subsequence, denoted by f m itself such that f m converges weakly to a function f in L r (R d , |x| βr w k ). Let u = I k α f . It is easy to see that u = K α,k * k f . Now let us assume that u t m = K t α,k * k f m and
u t = K t α,k * k f . Consider (u m − u)|x| γ L s (K,w k ) ≤ (u m − u t m )|x| γ L s + (u t m − u t )|x| γ L s + (u t − u)|x| γ L s . Using Lemma 7.1, (u m − u t m )|x| γ L s (K,w k ) = (K α,k * k f m − K t α,k * k f m )|x| γ ≤ Ct δ |x| β f m r, w k ≤ Dt δ .
Similarly,
(u t − u)|x| γ L s (K,w k ) ≤ Ct δ x β f r, w k ≤ Dt δ . Choose ǫ > 0.
For very small t > 0, each of the above estimates can be made less that ǫ 3 for all m. We are left to get bound for (u t m − u t )|x| γ L s (K,w k ) . For radial functions f (x) = f 0 (|x|) ∈ S(R d ) (see [14]), one has
τ k y f (x) = R d f 0 ( |x| 2 + |y| 2 − 2 y, η )dµ k x (η),(7.5)
where dµ k x (η) is a probability Borel measure on R d , whose support is contained in co(G.x), the convex hull of the G-orbit of x in R d . Applying (7.5), for the function K t α,k ,
τ k y K t α,k (x) = R d K t α,k ( |x| 2 + |y| 2 − 2 y, η )dµ k x (η) = R d (c k α ) −1 (|x| 2 + |y| 2 − 2 y, η ) d k −α 2 χ {η:|x| 2 +|y| 2 −2 y,η ≥t 2 } (η)dµ k x (η) = η:A(x,y,η)≥t (c k α ) −1 (A(x, y, η)) d k −α dµ k x (η),(7.6)
where A(x, y, η) = |x| 2 + |y| 2 − 2 y, η . Then
τ k y K t α,k (x) ≤ (c k α ) −1 t d k −α dµ k x {η : A(x, y, η) ≥ t} ≤ (c k α ) −1 t d k −α ,(7.7)
since dµ k x is a probability measure. It is proved in [1] that min g∈G |g.x − y| ≤ A(x, y, η) ≤ max g∈G |g.x − y|, ∀ x, y ∈ R d and η ∈ co(G.x). (7.8)
Let y ∈ K, where K is a compact set in R d . Then there exists R > 0 such that K ⊂ B(0, R).
Consider
R d |τ k y K t α,k (x)| r ′ |x| −βr ′ w k (x)dx = |x|≤2R |τ k y K t α,k (x)| r ′ |x| −βr ′ w k (x)dx + |x|>2R |τ k y K t α,k (x)| r ′ |x| −βr ′ w k (x)dx = I 1 (y) + I 2 (y). (7.9)
Substituting the bound for τ k y K t α,k from (7.7) in I 1 (y), we have
I 1 (y) ≤ (c k α ) −r ′ t r ′ (d k −α) |x|≤2R |x| −βr ′ w k (x)dx = (c k α ) −r ′ t r ′ (d k −α) a k (2R) d k −βr ′ d k − βr ′ = M 1 ,
using property (4) of section 2 with the condition βr ′ < d k i.e., β < d k r ′ . On the other hand, for the integral I 2 (y), substituting the value of τ k y K t α,k from (7.6) and then using (7.8), we get I 2 (y) = |x|>2R η:A(x,y,η)≥t (c k α ) −1 (A(x, y, η)
) d k −α dµ k x (η) r ′ |x| −βr ′ w k (x)dx ≤ |x|>2R (c k α ) −1 min g∈G |gx − y| d k −α r ′ |x| −βr ′ w k (x)dx = (c k α ) −r ′ |x|>2R |x| −βr ′ min g∈G |g.x − y| r ′ (d k −α) w k (x)dx. (7.10)
Since g is a reflection, we observe that |g.x − y| ≥ |g.x| − |y| = |x| − |y|, ∀ g ∈ G,
If y ∈ K and x is in the integration region of I 2 (y), then |y| ≤ R ≤ |x| 2 . So |g.x − y| ≥ |x| 2 . Therefore, min g∈G |g.x − y| ≥ |x| 2 for all y ∈ K. From (7.10),
I 2 (y) ≤ (c k α ) −r ′ 2 r ′ (d k −α) |x|>2R |x| −βr ′ |x| r ′ (d k −α) w k (x)dx ≤ M 2 ,
if d k < βr ′ + r ′ (d k − α) i.e., if α < β + d k r . Thus from (7.9), R d |τ k y K t α,k (x)| r ′ |x| −βr ′ w k (x)dx ≤ M 1 + M 2 , ∀ y ∈ R d . (7.11)
In particular, we have proved that τ k y K t α,k ∈ L r ′ (R d , |x| −βr ′ w k ), ∀ y ∈ K. Since f m converges weakly to f in L r (R d , |x| −βr w k ), K t α,k * k f m (y) converges to K t α,k * k f (y) as a sequence of complex numbers. Thus u t m (y) converges to u t (y) for all y ∈ K. Moreover, since u t m (y) = K t α,k * k f m (y) = R d f m (x)τ k x K t α,k (y)w k (x)dx, using (7.11) and the fact that f m is a bounded sequence in L r (R d , |x| βr w k ),
|u t m (y)| ≤ R d |f m (x)| r |x| βr w k (x)dx 1 r R d |τ k x K t α,k (y)| r ′ |x| −βr ′ w k (x)dx 1 r ′ ≤ A.
By Lebesgue Dominated convergence theorem, u t m − u t |x| γ L s (K,w k ) converges to zero (as the weight |x| γs w k is integrable on K under the condition γ > − d k s by property (4) of section 2). Hence we can make it less than ǫ 3 for large m. Thus (u m − u)|x| γ L s (K,w k ) ≤ ǫ for large m, proving that u m converges to u strongly on L s (K, |x| γs w k ). This completes the proof that the embedding (7.2) is compact. Now using Theorem 6.1 and Theorem 7.1, we are ready to prove our main result that (1.5) has a maximizer.
Theorem 7.2. Let d ≥ 2, 2 < s < ∞, − d k s < γ < β, 0 < β < d k 2 and the relation 1
s − 1 2 = β − γ − α d k (7.12)
holds. Then there exists a maximizer for W k .
Proof. Let {f j } j∈ be a minimizing sequence for W k , which we can take to be normalized i.e., |x| β f j L 2 (R d ,w k ) = 1 and |x| γ I k α f j L s (R d ,w k ) → W k . (7.13)
Now, in Theorem 6.1, we set
µ = d k 2 + β − α,(7.14)
and choose θ such that max 2 s , µ µ + α , γ β < θ < 1.
Because of relation (7.12), equation (6.2) holds for this particular choice of µ. It is also easy to see remaining conditions of Theorem 6.1 hold under the hypothesis of the Theorem and the choice of µ, θ. Also since relation (7.14) holds, from (5.6), . Hence we apply Theorem 6.1 and use (7.13) to get,
f j Ḃ −µ−α,k ∞,∞ ≥ C k > 0.
In other words, sup t>0 t µ+α 2
e t∆ k f j L ∞ ≥ C k > 0.
It then follows that for each j ∈ N, there exists t j > 0 such that Then it is easy to see that |x| βf j L 2 (R d ,w k ) = |x| β f j L 2 (R d ,w k ) . Also, |x| γ I k αf j L s (R d ,w k ) = |x| γ I k α f j L s (R d ,w k ) . (7.16) Indeed, we have
I k αf j (x) = (c k α ) −1 R df j (y)τ k y |.| α−d k (x)w k (y)dy = (c k α ) −1 t 1 2 ( d k 2 +β) j
Lemma 2.1. (Brezis Lieb Lemma) Let (X, dx) be a measure space and (f j ) be a bounded sequence in L p (X), 0 < p < ∞, which converges pointwise a.e. to a function f . Then
Theorem 4 . 1 .
41Let d ≥ 3. Then the infimum
t∆ k f dt. Now, we can write u = H k f + L k f , where H k f := 1 Γ(α/2) T 0 t α/2−1 e t∆ k f dt and L k f := 1 Γ(α/2) ∞ T t α/2−1 e t∆ k f dt.
) e t∆ k f j L ∞ ≤ C d,β,k |x| β f j L 2 = C d,β,k ,showing that f j ∈Ḃ −µ−α,k ∞,∞
Then using (1.3), we have the following Lemma. Lemma 7.1. With the same conditions as that of Theorem 7.1, let
Then by substituting the value of γ from (7.12),s,w k , thus proving (7.16). As a consequence, from (7.13), we havewhich shows that {f j } j is also a maximizing sequence sequence for W k . Moreover, using (2.7), we observe thatExistence of an extremal of Sobolev inequality associated with Dunkl gradient and of Stein-Weiss inequality for D-Riesz potential 27 Using (7.15),Since {f j } j is a bounded sequence in L 2 (R d , |x| 2β w k ), by reflexivity, it has a subsequence still denoted byf j such thatf j converges weakly to a function h in L 2 (R d , |x| 2β w k ). Our aim is to show that h is indeed a maximizer for W k , that is,Now we set u j := I k αf j and v := I k α h. Since by Theorem 5.1, e 1.∆ k is a compact operator from L 2 (R d , |x| 2β w k ) into L ∞ (R d ), passing through a subsequence, we have e 1.∆ kf j converges strongly to e 1.∆ k h in L ∞ (R d ). Then from (7.13), e 1.∆ k h L ∞ ≥ C 2 > 0, which implies that h ≡ 0. Again by taking r = 2, β = γ in Theorem 7.1, we observe that all the conditions of Theorem 7.1 are satisfied under the hypothesis of the given Theorem and therefore, if K is a compact set in R d , we have the compact embeddinġ H α,2 β,k (R d ) ⊂ L s (K, |x| βs w k ). By observing that u j is a bounded sequence inḢ α,2 β,k (R d ) and thereafter following the proof of Theorem 7.1, we can show that u j converges strongly to v in L s (K, |x| βs w k ). Therefore, up to a subsequence, u j converges to v a.e. in K and by using diagonal argument, further, up to a subsequence, u j converges to v a.e. in R d . Now proceeding exactly, as in the proof of Theorem 5.1 in[12], we can prove that |x| β h L 2 (R d ,w k ) = 1 andf j → h strongly in L 2 (R d , |x| 2β w k ). Since by Theorem 1.2, the operator I k α is continuous from L 2 (R d , |x| 2β w k ) into L s (R d , |x| γs w k ), hence u j → v strongly in L s (R d , |x| γs w k ).This implies that u j → v in L s (R d , |x| γs w k ). Now (7.19) will follow from (7.17), which completes the proof.
Riesz transforms for Dunkl transform. B Amri, M Sifi, Ann. Math. Blaise Pascal. 19B. Amri and M. Sifi, Riesz transforms for Dunkl transform, Ann. Math. Blaise Pascal, 19 (2012), 247-262.
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Positive L p −bounded Dunkl-type generalized translation operator and its applications. D V Gorbachev, V I Ivanov, S Yu Tikhonov, doi.org/10.1007/s00365-018-9435-5Constr. Approx. D. V. Gorbachev, V. I. Ivanov, and S. YU. Tikhonov, Positive L p −bounded Dunkl-type generalized translation operator and its applications, Constr. Approx.doi.org/10.1007/s00365- 018-9435-5.
Riesz Potential and maximal function for Dunkl transform CRM Preprint Series number 1238. D V Gorbachev, V I Ivanov, S Yu Tikhonov, D. V. Gorbachev, V. I. Ivanov, and S. YU. Tikhonov, Riesz Potential and maximal function for Dunkl transform CRM Preprint Series number 1238, 2018.
Riesz potentials and fractional maximal function for the Dunkl transform. S Hassani, S Mustapha, M Siphi, J. Lie Theory. 19S. Hassani, S. Mustapha nand M. Siphi, Riesz potentials and fractional maximal function for the Dunkl transform, J. Lie Theory. 19 (2009), 725-734.
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Odisha-752050, India. E-mail address: saswata.adhikari@gmail. Saswata Adhikari, ; Niser Bhubaneswar, Anoop, School of Mathematical Sciences, NISER Bhubaneswar, Odisha-752050. School of Mathematical Sciences. India. E-mail address: [email protected] Adhikari, School of Mathematical Sciences, NISER Bhubaneswar, Odisha- 752050, India. E-mail address: [email protected] V. P. Anoop, School of Mathematical Sciences, NISER Bhubaneswar, Odisha- 752050, India. E-mail address: [email protected]
School of Mathematical Sciences, NISER Bhubaneswar, Odisha-752050, India. E-mail address: [email protected]. Sanjay Parui, Sanjay Parui, School of Mathematical Sciences, NISER Bhubaneswar, Odisha- 752050, India. E-mail address: [email protected]
| []
|
[]
| [
"Shivananda R Poojara \nMobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia\n",
"Chinmaya Kumar Dehury \nMobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia\n",
"Pelle Jakovits \nMobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia\n",
"Satish Narayana Srirama \nSchool of Computer and Information Sciences\nUniversity of Hyderabad\n500 046HyderabadIndia\n"
]
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"Mobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia",
"Mobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia",
"Mobile & Cloud Lab\nInstitute of Computer Science\nUniversity of Tartu\n50090TartuEstonia",
"School of Computer and Information Sciences\nUniversity of Hyderabad\n500 046HyderabadIndia"
]
| []
| With the increasing number of Internet of Things (IoT) devices, massive amounts of raw data is being generated. The latency, cost, and other challenges in cloud-based IoT data processing have driven the adoption of Edge and Fog computing models, where some data processing tasks are moved closer to data sources. Properly dealing with the flow of such data requires building data pipelines, to control the complete life cycle of data streams from data acquisition at the data source, edge and fog processing, to Cloud side storage and analytics. Data analytics tasks need to be executed dynamically at different distances from the data sources and often on very heterogeneous hardware devices. This can be streamlined by the use of a Serverless (or FaaS) cloud computing model, where tasks are defined as virtual functions, which can be migrated from edge to cloud (and vice versa) and executed in an eventdriven manner on data streams. In this work, we investigate the benefits of building Serverless data pipelines (SDP) for IoT data analytics and evaluate three different approaches for designing SDPs: 1) Off-the-shelf data flow tool (DFT) based, 2) Object storage service (OSS) based and 3) MQTT based. Further, we applied these strategies on three fog applications (Aeneas, PocketSphinx, and custom Video processing application) and evaluated the performance by comparing their processing time (computation time, network communication and disk access time), and resource utilization. Results show that DFT is unsuitable for compute-intensive applications such as video or image processing, whereas OSS is best suitable for this task. However, DFT is nicely fit for bandwidthintensive applications due to the minimum use of network resources. On the other hand, MQTT-based SDP is observed with increase in CPU and Memory usage as the number of users rose, and experienced a drop in data units in the pipeline for PocketSphinx and custom video processing applications, however it performed well for Aeneas which had low size data units. | 10.1016/j.future.2021.12.012 | [
"https://arxiv.org/pdf/2112.09974v1.pdf"
]
| 245,335,124 | 2112.09974 | 0af54bc79eacaeca6aee4acc3cb796c7cd83868e |
Shivananda R Poojara
Mobile & Cloud Lab
Institute of Computer Science
University of Tartu
50090TartuEstonia
Chinmaya Kumar Dehury
Mobile & Cloud Lab
Institute of Computer Science
University of Tartu
50090TartuEstonia
Pelle Jakovits
Mobile & Cloud Lab
Institute of Computer Science
University of Tartu
50090TartuEstonia
Satish Narayana Srirama
School of Computer and Information Sciences
University of Hyderabad
500 046HyderabadIndia
Serverless data pipeline approaches for IoT data in fog and cloud computingServerless computingdata pipelinescloud computingfog computingedge computinginternet of things * Corresponding author Email address: satishsrirama@uohydacin (Satish Narayana Srirama)
With the increasing number of Internet of Things (IoT) devices, massive amounts of raw data is being generated. The latency, cost, and other challenges in cloud-based IoT data processing have driven the adoption of Edge and Fog computing models, where some data processing tasks are moved closer to data sources. Properly dealing with the flow of such data requires building data pipelines, to control the complete life cycle of data streams from data acquisition at the data source, edge and fog processing, to Cloud side storage and analytics. Data analytics tasks need to be executed dynamically at different distances from the data sources and often on very heterogeneous hardware devices. This can be streamlined by the use of a Serverless (or FaaS) cloud computing model, where tasks are defined as virtual functions, which can be migrated from edge to cloud (and vice versa) and executed in an eventdriven manner on data streams. In this work, we investigate the benefits of building Serverless data pipelines (SDP) for IoT data analytics and evaluate three different approaches for designing SDPs: 1) Off-the-shelf data flow tool (DFT) based, 2) Object storage service (OSS) based and 3) MQTT based. Further, we applied these strategies on three fog applications (Aeneas, PocketSphinx, and custom Video processing application) and evaluated the performance by comparing their processing time (computation time, network communication and disk access time), and resource utilization. Results show that DFT is unsuitable for compute-intensive applications such as video or image processing, whereas OSS is best suitable for this task. However, DFT is nicely fit for bandwidthintensive applications due to the minimum use of network resources. On the other hand, MQTT-based SDP is observed with increase in CPU and Memory usage as the number of users rose, and experienced a drop in data units in the pipeline for PocketSphinx and custom video processing applications, however it performed well for Aeneas which had low size data units.
Introduction
The advancements in internet technologies like 5G and other allied technologies have accelerated the use of the Internet of Things (IoT) [1,2] at a wider scale. With the increasing number of IoT devices, massive amounts of raw data is being generated. To cope with the ever increasing data, today's industries are looking towards management and computation solutions that enable to manage, process, and control the data flows in realtime, such as for Artificial Intelligence (AI) services.
To extract actionable insights from such raw data, it needs to be preprocessed, transported, transformed, and analyzed before useful knowledge can be extracted from it. To simplify designing and deploying such data processing services, pipelines are commonly used to compose a set of individual data processing processes into a single service, where the output of a component is the input of the next component. This allows to reuse and compose of common data handling processes into more complex data pipeline services [3].
The data pipelines would be deployed as software services running seamlessly in the Cloud or inside on-premise servers.
However, many IoT applications are event-driven and require performing actions in real-time [4], which often requires large and expensive data processing clusters (e.g. Apache Spark, Flink, Storm) to be created to handle large-scale IoT data with self-adapting scaleable processing [5].
However, with the emergence of Serverless computing, a novel cloud computing service model that leverages the function level billing and scaling, designing event-based, real-time and scaleable IoT data processing has been significantly simplified [6]. Furthermore, such cloud-centric approach requires all data to be transported into one central data center and has multiple disadvantages, such as high dependency on end-to-end connectivity, higher latency, higher transfer and storage costs, and other typical issues with centralised data collection. This is where combining Serverless model and data pipelines can produce significant benefits to avoid some of the disadvantages of cloud-centric approach and reduce the complexity of designing multi-layer (Cloud, Edge, Fog) IoT applications. In data pipelines, each task is a process that consumes input data, manipulates it and produces output data and such tasks are composed into pipelines. This makes it easy to compose common data processing tasks into more complex data management services and potentially deploy some of the pipeline tasks closer to the data sources (e.g. Edge, Fog laters). In the Serverless model, functions are individually deployed services which are triggered on certain events (e.g. new database record or REST request arrival), receive data and produce output. Serverless cloud model has several benefits including fine grained auto scaling and increased productivity gains due to reusable serverless functions deployed either on on-premise or on the clouds [7]. It is also significantly easier to deploy individual functions in different locations compared to more monolithic applications (e.g. when compared to Apache Spark data analytic applications).
To combine both models, Serverless Data Pipelines (SDP) can be created when serverless functions are used as pipeline tasks and seamlessly invoked while the data moves through the pipeline. Serverless functions can be deployed in Cloud, Edge or Fog environments and data pipeline technologies are used for data transport, routing and function invocation. To provide one example: due to low latency demand and bandwidth constraints of cloud-centric approach, a novel fog computing architecture [8,1,9] was introduced between IoT devices and Cloud servers. Here, few of data analysis (data pipeline) tasks were moved from far-away clouds to remote fog nodes that are very near to data sources, this eventually improves the performance of realtime services.
This approach is also helped by the advances in fog devices, which can now be equipped with enough compute (even with tensor processing units (TPU) and GPUs ) [10] and storage capacities. It is very useful support to accommodate serverless frameworks that could accelerate the execution of data pipelines. However, the disparity of hardware resources available at the Edge, Fog and cloud leads to an interesting challenge of how to diversify and prioritize data flow between functions residing at fog and cloud to meet the expected Quality of Service (QoS). There are also other related issues which need to be considered and evaluated. Serverless functions are stateless and its frameworks only deal with the runtime management of functions, completely separating it from the data management [11]. This separation simplifies serverless computing but has drawbacks for data-intensive and stream processing pipelines [12] and that can lead to issue of dealing intermediate data between the functions in pipeline.
Considering the above mentioned issues and challenges, this work aims to investigate and compare the suitability of modern off-the-shelf Data Pipeline (DP) tools [13] [14] and other frameworks (such as message brokers and object storage services) which are integrated with serverless frameworks and can be used to design dynamic Serverless Data Pipelines. We utilize three bandwidth and compute intensive real-time fog computing workloads: Aeneas [15], Pocketsphnix [15] and a custom Video processing application to extensively measure the performance (w.r.t CPU, memory, disk and network usage) of different SDP architectures in the fog computing environment.
Motivation
To process this data across edge/fog and cloud environments, a huge amount of resources (more than the actual demand) are allocated to process a user's task in traditional systems, which is challenging and inevitable in resource-constrained edge/fog nodes. In this regard, serverless technology plays a significant role in the deployment of IoT applications by composing into stateless independent serverless functions across edge/fog and cloud environments.
However, serverless functions are stateless with high granular scaling which introduces additional complexity and challenges in data management between functions residing at edge, fog, and cloud nodes. Some enterprise solutions such as Azure IoT or AWS Greengrass use serverless edge functions to preprocess and push data to enterprise clouds. Data movement between functions residing at the edge and cloud is often handled by using object storage services like AWS S3. However, it's challenging when a large set of functions are deployed in edge/fog infrastructure and data needs to be transferred on each function invocation. The object storage may yield higher charges when more data and more function invocations occur. Eventhough, object storage attains the purpose of handling intermediate data but cost, latency, etc., are challenging.
There also exist off-the-shelf DP tools like StreamSet and Apache NiFi, which provide some support for edge/fog environments and can also be utilized to solve the issues, but they usually manage the flow of data in a more centralized manner and often require significant computing resources to run effectively.
Alternatively to object storage, its also possible to use data brokers (e.g. Apache Kafka, MQTT) as Message Queues between serverless functions for designing serverless data pipelines. Compared to object-storage, they would require less storage and may be faster due to more extensive memory usage, which is highly desirable in edge/fog environments. However, compared to NiFi, it may be more difficult to control the precise execution flow of pipelines.
The aforementioned challenges have motivated us to investigate the advantages and disadvantages of different mechanisms for integrating serverless platform with data pipeline platforms. The following subsection will list the contributions in the proposed work.
Contributions
In the above context, our contributions in this work can be summarized as follows:
• We demonstrate how SDP can be deployed in three layered IoT architectures.
• We propose three approaches for designing Serverless Data Pipelines with different data handling mechanisms (Apache Nifi, Message Queues and object storage services such as AWS S3).
• We use real time fog computing workloads such as Aeneas, PocketSphinx and custom Video processing applications to compare the performance (such as processing time) and resource utilization of these different SDP approaches.
• We provide insights on the suitability of these SDPs for different types of fog computing workloads.
The rest of the paper is organized as follows. In section 2, we present literature survey on current state of art data pipeline technologies and ecosystem. The proposed SDP architecture and real time IoT usecases are described in the section 3 and 4, respectively. Following this, three novel SDP approaches are designed and articulated and implemented for real time fog computing use cases in section 5 and compared with different performance metrics in section 6. Finally, the concluding remarks and the future works are discussed in section 7.
Related works
A number of SDP architectures and solutions have been proposed in the field of IoT data management in edge, fog and cloud environments. This section briefly summarizes the recent work done in the context of SDP architectures and models.
The public cloud service providers such as AWS greengrass [16], Google Cloud IoT [17] and Microsoft-Azure IoT Edge [18] have typical IoT data pipeline solutions for industrial, healthcare, smart city and other real time use cases. For example, consider AWS IoT Greengrass, where Lambda service will be executed at the edge layer for data acquisition and pre-processing. Later data is forwarded to the cloud by edge devices and then it passes through pipeline of activities for post processing and finally is delivered to the data sink.
Valeria et al. proposed a solution of IoT data stream processing in distributed fog and edge computing environments with decentralized scaleable manner [5] and further extended to how data processing operators were placed in computing nodes considering the efficiency, application topology and resources configurations [19]. These works provide hint that off-the-shelf data stream processing tools such as Apache Storm can be used for the task. However, these stream processing tools require huge computing clusters and in IoT deployments more often devices are heterogeneous with limited computing capacity. More often IoT workloads are event and time driven which motivates us to investigate the serverless based data processing pipelines. Further, SDPs easily been deployed at various levels in the IoT hierarchy (Edge, Fog and Cloud Infrastructure) with efficient granular scaling of the serverless functions.
Das et al. [20] proposed a model for efficient execution of user tasks as serverless functions in edge/cloud environments and designed a set of data pipelines using AWS Greengrass on edge devices along with Lambda capabilities. Our approach looks similar to this model, however it lacks fog based processing pipeline model.
Dehury et al. [21] designed a framework known as CCo-DaMiC, which aims to ensure data accuracy, trustworthiness, and validation in SDP. This work directly relates to our proposed DFT based SDP approach. However, CCoDaMiC mainly focuses on data accuracy and trustworthiness and not on the performance of the applications. Lixiang et al. [22] designed a framework for video processing using serverless lambda functions known as Sprocket. Authors demonstrated the efficiency of serverless functions for faster execution by constructing pipeline of activities for video handling. However, their primary focus is to reduce latency and cost by using the techniques of parallelism. Interestingly, this work motivated us to consider the complex video processing use case in our proposed research.
Several techniques and methods have been proposed illustrating the use of MQTT for data acquisition from different data sources via publish/subscribe model [23,24,25]. MQTT brokers can act as data carriers and can store data until subscribers consume it. This approach is well suited to store temporary or intermediate data between processing elements in SDP. In our work, one of the approaches uses MQTT together with serverless framework to construct data pipelines from data source to sink.
Thus to the best of our knowledge, none of the research works attempted to investigate and compare the different techniques in the construction of SDP with different approaches for intermediate data handling between serverless functions. Based on the literature survey, we designed an overall SDP architecture and the required software services to handle the flow and execution of data in a pipelined manner, as shown in Figure 1. The data are generated by the IoT devices (such as surveillance cameras, SCARA (Selective Compliance Assembly Robot Arm) robot sensors, health monitoring sensors etc.) and are eventually sent to cloud infrastructure for processing and storage.
Proposed System
Instead of sending the data directly from IoT device to the cloud, the data is preprocessed by different computing environments such as edge computing devices or fog servers through the execution of serverless functions. As in Figure 1, the data are sent to Edge infrastructure, followed by the Fog infrastructure. Both the edge gateways and the fog infrastructure are responsible for providing the infrastructure to host data management and serverless frameworks.
Edge Infrastructure
Edge infrastructure is mainly responsible for receiving the data from IoT devices. For this purpose, different data acquisi-tion tools or software solutions can be used such as MiNiFi or SDC Edge, custom services such as Python or other run-time services, as shown in Figure 1. These solutions usually come with limited capabilities to process the data. Upon receiving and processing, edge infrastructure forwards the data to fog infrastructure. The detailed description of the services used in edge infrastructure are described below.
• MiniFi 1 : Apache MiNiFi is a super light-weight version of NiFi made for the edge devices. It can run as a system service, and it is centrally managed using Apache NiFi. Developers can easily design the pipelines using a set of processors in Apache NiFi and push them into the MiniFi service. These pipelines can handle preliminary data operations near to the source, e.g., compressing a video recorded by drone before sending to cloud/fog to reduce bandwidth consumption, etc. We use this MiniFi service in the implementation of DFT tool based SDP as described in 5.1.
• Custom services: Custom services are similar to the MiniFi processors. Such services need to be created from scratch using a specific programming language. E.g., A Python program can be created to collect and compress the video. Python-based custom services are created and used in the implementation of OSS and MQTT-based SDP as described in 5.2 and 5.3, respectively.
Fog Infrastructure
Fog Infrastructure is mainly responsible for processing the data received from Edge infrastructure, for which, Data pipeline engine (Apache NiFi 2 ), FaaS engine (OpenFaaS 3 ), MinIO 4 , and MQTT 5 services are used. The fog infrastructure includes a group of fog servers deployed in a cluster with a set of particular software services using Docker Container Engine. The processed data are then forwarded to the cloud infrastructure to further process, store, and generate alerts and notifications. The use and necessity of software services are described below:
FaaS Engine: FaaS Engine is primarily one of the core components of this proposed work. Several open-source serverless platforms are available such as Apache OpenWhisk, OpenFaas, Kubeless, etc. OpenFaaS serverless platform is used in this work, as it is lightweight and easy-to-configure over other alternative solutions. OpenFaaS can be installed atop of Docker or Kubernetes platform. Docker containers are used to host and execute the serverless functions. The functions can be invoked using HTTP endpoints with the necessary data. The function invocation is performed in the pipeline by Data pipeline engine, MinIO event notification system, and MQTT event notification service.
Data pipeline engine: As discussed in Section 1, we use Apache NiFi, a data pipeline processing platform, which manages the data flow between the systems. This provides a set of independent processors with specific functionalities to process and manage the data. The data flow between processors is managed via scalable queues. Developers can easily design custom data pipelines using a flexible user interface and automatically configure, control, and deploy the pipelines in Edge infrastructure using MiniFi service. This MiniFi-Nifi integration efficiently manages the orchestrated IoT data processing from the edge, fog, and cloud, and vice-versa.
Message Queue: Message Queues are published/subscribe protocol based data carriers between source and sink. We demonstrate the use of Message Queues to build data pipelines integrated with the OpenFaaS serverless platform. A light-weight messaging protocol, MQTT, is ideal for small sensors and mobile devices and is suitable for high-latency or unreliable networks. MQTT uses different data types such as UTF-8 encoded string, bit/byte integer, binary data, and UTF-8 string pair. A serverless function can publish the processed data to MQTT, and in turn, functions are invoked when data need to be subscribed using web-hooks. The flow of data between MQTT and serverless platform builds consistent, reliable SDP.
Object storage service: An open-source cloud-based storage solution, MinIO, compatible with Amazon S3, stores the IoT data. This provides a RESTful API to access/insert/remove buckets and objects. Moreover, triggers are set to bucket when its content is accessed/written/removed, and corresponding event notifications are generated using techniques such as web-hooks, Message Queues. This is advantageous in IoT applications for handling event-driven data. It is configured as high-availability cluster using docker swarm.
Cloud Infrastructure
Cloud infrastructure is mainly responsible for processing heavy computation data received from Fog infrastructure and storing the data. This is also responsible for generating alerts and notifications to activate other business processes whenever required. To perform this, a set of services from cloud providers or user-configured open-source services are used, such as Data pipeline engine (Apache NiFi, AWS data pipeline, Google Cloud pipeline, etc.), Object storage service (MinIO/AWS S3/ Google object storage), Message queues (MQTT/AWS SNS), and Faas engine (OpenFaaS/AWS Lambda/Google Functions).
The setup and configuration of the FaaS engine, data pipeline engine, and object storage service are the same as that of the fog infrastructure. Along with this, the cloud infrastructure is also responsible for Visualization and Reporting. The primary job of the visualization is to display processed data using visualization tools. The Grafana visualization tool is used to measure Edge/Fog and cloud nodes' performance metrics in this work. Additionally, the Prometheus time-series database is used where performance metrics are collected.
Realtime IoT Use Cases
To compare and evaluate the performance of proposed SDP approaches, a set of standard fog computing workloads or applications are considered from the article [15]. The applications are categorized into latency critical (LC), bandwidth intensive (BI), location aware (LA) and computational intensive (CI). The corresponding applications from the article [15] such as Aeneas (BI), PocketSphinx (BI, CI), Yolo object detection (BI, CI) were considered as real time IoT workloads. These applications are redesigned into sequence of data flow pipelines that can spread across edge, fog and cloud infrastructure. The detail implementation of data flow pipelines are explained below.
Custom video processing using Deep learning based object classification using YOLOv3 and ffmpeg tool
The You Only Look Once (YOLO) makes predictions with a single network evaluation unlike systems like Region-based Convolutional Neural Networks (R-CNN) which require thousands of networks for a single image. Hence, its predictions are 1000x faster than R-CNN and 100x faster than Fast R-CNN. This fog application is ideal candidate to consider because of bandwidth sensitiveness and requires high network bandwidth to send video stream from edge node to cloud and aiming for faster processing with immediate response. Most of the operations are compute intensive and demand for more CPU and Memory resources. So, the application is designed to process preliminary object detection in fog and send to cloud for further analysis. To understand precisely, we use a scenario of real time object detection from the article [26]. The abstract flow of data in this use case is shown in Figure 2. Here a drone is used to capture the video footage and to process this, sequences of operations are carried out as follows:
1. Drone captures video footage and forwards to edge gateway using communication protocol such as MQTT or HTTP or other protocols. 2. Edge gateway compress the video and forwards to fog nodes. 3. In a fog node, video is decompressed using set of tools (gzip or zip). 4. The video will be split in to number of frames that can be processed individually. 5. The frames are passed to YOLOv3 framework and objects are identified from respective frame. 6. The raw output generated from YOLOv3 is processed to json document. This will nicely arrange the raw text into json elements consisting of identified objects. 7. The json documents are stored in storage service for further analysis.
To design the above example in traditional computing platforms, the developer needs to specify a necessary input, configuration, and run-time environment to perform required data operation seamlessly by provisioning resources on the fly. However, it adds an issue of over-provisioning than demand. For example, short-running tasks like triggering an alert message to the end-user when a human/animal object is identified don't require heavy computation. Thankfully, the Serverless platform can subsidize this issue by invoking functions whenever events are triggered by consuming less computation resource. In this application, entire video processing application is decoupled into a set of OpenFaaS-based serverless functions as shown in Table 1. Among which, two major functions are (a) Split and (b) Yolo as described below.
• Split: Splits the video into multiple frames using ffmpeg, an image/video editing tool. The number of splits depends on the value given to the fps argument in ffmpeg command
• Yolo: Yolo is a object detection framework that has a darknet library.
The rest of the functions mentioned in Table 1 are used as subsidiary functions in the pipeline. This video processing application is implemented in all the three SDP approaches, as described in below Section 5.
Aeneas: A text-audio synchronisation
The Aeneas tool is specialized for automated synchronization of audio to given text file also known as forced alignment. It automatically generates a synchronization map between a list of given text fragments and an audio file containing the narration of the text. This fog application is ideal candidate to consider because of bandwidth intensive and consumes huge network bandwidth to stream audio files from large set of end user devices to edge node and cloud node. End users aim for faster response times and to achieve this, few of the operations are performed in fog node. We show the scenario of Aeneas based real time application and its abstract flow in Figure 3.
Here, end-user device such as mobile device or other devices are used to stream the .wav or .mp3 audio files. Then the following operations are carried out as follows:
1. End user offloads a .wav file to edge gateway using mobile device. 2. Edge gateway compress the audio and forwards to fog nodes. 3. In a fog node, audio is decompressed using set of tools (gzip or zip). 4. The file (.xhtml) is downloaded from the cloud or other repository used as a input to Aeneas tool. The file contains a text used for alignment in the audio file. 5. The raw .mp3 or .wav is processed using Aeneas tool along with the given file (.xhtml). The Aeneas tool has facility to generate the alignment output in json. 6. The json documents are stored in storage service for further analysis.
The above traditional data pipeline is redesigned to SDP by composing these operations in to serverless functions. It has one major function:
• aeneas: Aeneas is python library for forced alignment of given text in a audio file and is configured with python3 run time. This function accepts two input files text file (.txt, .xhtml), audio file (.wav,.mp3) and generates the aligned output (.json, .mile). It generates .json as a HTTP response to the invocation.
PocketSphinx: A Speech-to-text conversion
Its a software engine specialized for speaker-independent continuous speech recognition [15]. An audio file (.wav) is converted to a defined language in text form using a pre-trained acoustic model to determine the source and destination language for speech-to-text conversion. The sample audio files are taken from large scale speech repository 6 .
In this fog application, we consider a scenario, where end user submits a .wav file via mobile phone and it needs to be processed (either fog/cloud) to find a given text in the audio file. This application is bandwidth and compute intensive, and requires higher network bandwidth to offload the audio files to cloud. Hence it is feasible to process near to the data source in the fog nodes to achieve higher response times. The Figure 4 shows the abstract view of the application with following set of operations:
1. End user offloads a .wav file to edge gateway node 6 http://www.repository.voxforge1.org/downloads/SpeechCorpus 2. It will be compressed and forwarded to fog/cloud infrastructure 3. Decompress the audio file 4. Process an audio file to text format using PocketSphinx tool 5. Get the text to search in output produced from Pocket-Sphinx operation 6. Perform text search operation 7. Convert into proper json document 8. Store into storage service for further analysis or usage and send corresponding alerts.
The overall flow of the above mentioned PocketSphinx application is designed in to set of serverless functions as shown in Table 1. It has one major function:
• pocketsphinx: Pocketsphinx is python library for speechto-text conversion using predefined acoustic model. This function accepts one audio file (.wav,.mp3) and generates the aligned output (.json). It generates .json as a HTTP response to the invocation. This PocketSphinx application is implemented in all the three SDP approaches, as described in the Section 5.
We represent F = { f 1 , f 2 , . . . , f m } as a set of serverless functions for each application, for example Aeneas for DFT based SDP has m = 2. Further in the below section, we describe the specific implementation of SDP approaches and associated three usecases that are implemented according to design of proposed SDPs.
Serverless Data Pipeline (SDP) approaches
Considering the above overall architecture and challenges mentioned in motivation section, the serverless data pipeline can be designed by following different approaches. In this section, we introduce three approaches: (a) Off-the-shelf data flow tool based SDP, (b) Object storage service based SDP, and (c) MQTT based SDP. Further, we implemented the proposed SDPs for real time IoT usecases as described in the Section 4.
Off-the-shelf data flow tool (DFT) based SDP
In DFT based SDP, the developer need not have to maintain the queue, rather we use the queuing capability of Apache Nifi, which manages the centralized management of data and with integration of serverless platform. In this SDP approach, we have used Apache Minifi in the Edge infrastructure to receive and preprocess the data, as shown in Figure 5. As discussed in Subsection 3.1, MiniFi service can run on resource-constrained devices and is autonomously managed using central Apache NiFi from Fog Infrastructure. Here, sensed data from IoT devices is received into MiniFi using processors such as ConsumeMQTT or ListenHTTP, based on the communication protocol used between edge infrastructure and IoT devices. Some data preprocessing processors such as data compression, filtering, and aggregation are also configured but with limited computing resources and capabilities. MiniFi does not have GUI and is developed with java/C++ libraries and can quickly be started as a system service. The data flow with processor groups is designed in Apache NiFi and is automatically pushed into the MiniFi service. MiniFi performs the data operations and pushes the flow file containing the data to the Apache NiFi service configured in the fog.
Apache NiFi is used to handle data flow in fog and cloud infrastructure. Apache NiFi provides a flexible set of processors for data operations and integration between cross platform systems. This gives the capabilities to seamless integration of serverless platform through specific Nifi processors, such as In-vokeHttp, PutLambda, etc. Such multiple NiFi processors can be connected to others, allowing the developer to invoke multiple serverless functions. The key benefit is that Apache NiFi facilitates queued data that can lend back pressure when limits are attained during data flow processing. Another benefit is that it has priority queuing to set single/multiple prioritization schemes that dictate how data is retrieved from a queue. This allows the developer not to pay much attention to maintain or implement the queue between each pair of serverless function invocations.
For the serverless platform, OpenFaaS is used in both fog and cloud infrastructures. However, public cloud serverless services can also be used in the cloud infrastructure such as Amazon Lambda. The serverless function receives the data flow file as input from invokedHTTP request and sends the processed data as an HTTP response body. Every serverless function is in-voked using an HTTP endpoint with respective HTTP methods (POST or GET). We implemented the DFT based SDP using real time IoT use cases as described in the below paragraphs.
In Custom video processing application, the Drone sends the video footage to the edge node. In the edge node, MiniFi service is configured with GetFile MiNiFi processor to read video file from disk and forward the video to Nifi in the fog node. The Nifi in fog infrastructure consists of three main processors to invoke three different OpenFaaS serverless functions (FFmpeg, Yolo, convertTojson). On the other hand, the Nifi in Cloud infrastructure consists of multiple processors to store the data (received from Fog infrastructure) into MinIO bucket.
Similarly, in the Aeneas application the end-user mobile device sends an audio file to the edge node. In the edge node, MiniFi service is configured with GetFile MiniFi processor to read audio file received and compressed using gzip processor and forward it to NiFi configured in the fog node. The NiFi in fog infrastructure consists of three main processors, first is decompress processor, second is to invoke two different Open-FaaS serverless functions (aeneas, getFile(.xhtml)), third is creating JSON from the output. Finally, this data is stored in the MinIO bucket using multiple NiFi processors.
The PocketSphinx has similar implementation as above, but Nifi in fog infrastructure consists of three main processors; first is decompress processor, second is to invoke two different OpenFaaS serverless functions (pocketsphinx, text-processing), third is creating json from the output.
Object Storage service based SDP
In OSS based SDP, the object storage service will resemble a persistent queue where the developer can visualize the data stored in the storage server as a queue. The capability of object storage service is to trigger an event notification using webhooks, making the integration of object storage with serverless platform flexible.
In this approach, we have used Python service in edge infrastructure to receive, preprocess, and transport data from IoT devices to fog infrastructure, as shown in Figure 6. Here, Python requests library is used to invoke serverless functions from edge Python service to functions residing in fog.
MinIO is used to handle data flow in fog and cloud infrastructure. MinIO is an open-source high-performance scalable storage service, as described in section 3. The flexible bucket notifications are a set of events such as inserted, accessed, deleted and copied. The corresponding events are triggered using Web-hooks. This flexibility makes seamless integration of storage service and serverless platform invocations. Apart from this, the MinIO Python client library makes it easy to code the functions to access the object data from a specific bucket.
For the serverless platform, OpenFaas is used in both fog and cloud infrastructure. Here, the serverless functions are invoked from the gateway node with data or from the events triggered in MinIO buckets. Furthermore, the serverless function may store processed data into buckets using the MinIO client library. Again, events may trigger to invoke functions and continue until the fog node forwards data to cloud infrastructure. In the cloud infrastructure, data flows in a similar fashion over MinIO and OpenFaas serverless platform. This constitutes an SDP, where object storage with persistent mode acts as an intermediate data handling mechanism between serverless functions. The following paragraph will describe the use of OSS based SDP in designing the real time use cases.
In Custom video application, we use a Python service as a drone simulator to send the video file to the gateway node. The gateway node is configured with a Python service to read a video file and send it to the fog node for processing. In fog node, MinIO is configured with two buckets: (a) unprocessed to store raw images and (b) processed to store processed video in JSON format. These buckets are set with web-hook event notifications to trigger serverless functions when a new data object is inserted. This implementation uses five serverless functions, as given in Table 1. However, in Aeneas application, in fog node, MinIO is configured with two buckets: (a) raw-audio to store raw images and (b) syncmap to store synchronization map generated to audio file in JSON format. This implementation uses six serverless functions, as given in Table 1. Finally, the PocketSphnix application in the fog node, MinIO is configured with two buckets: (a) raw-pocketsphinx to store raw images and (b) processed-pocketsphinx to store the converted audio file to text, (c) output-pocketsphinx to store text-processed data. This implementation uses 6 serverless functions, as given in Table 1. In the cloud infrastructure, two MinIO buckets are created, (a) success-pocketsphinx-to store success results from text processing, (b) failure-pocketsphinx to store failure results from text processing. These buckets are responsible to store the processed audio files and output is mainly in text format.
MQTT-based SDP
In this proposed SDP approach, MQTT is used as a queue to store the data, different queues are represented by different MQTT topics and serverless functions can be triggered when new data objects are published into a specific topic. The gateway node receives data from IoT device using the custom Python service (Python code is written to perform specific operation).
The received data are preprocessed and published to the MQTT broker with a topic name, as shown in Figure 7.
The topic names need to be subscribed by the OpenFaaS serverless functions, to consume the data in the queue. For this, Serverless frameworks should be built in with connectors between itself and message broker to subscribe the topics and invoke corresponding functions. Apart from this, MQTT doesn't have the capability to directly trigger an HTTP endpoint upon the arrival of new data. Thanks to the OpenFaaS community for developing the openFaas-mqtt connector, which runs as a service to invoke serverless functions by subscribing to MQTT topics. OpenFaas has multiple connectors supported for different Message Queues. The serverless function processes the data and publishes the output again to the Message Queue. This process continues until the data from the source reaches to the data sink, as shown in Figure 7. The following paragraphs describe how the use cases were implemented using the MQTT based SDP.
Similar to OSS based SDP approach, all the three applications are configured with python service in the edge node. In Custom video processing application, the Python service publishes the video to MQTT broker with topic name. The openfaas-mqtt connector running in fog node subscribes to the topic name and invokes the functions. Here, we use three serverless functions, as given in Table 1 and one openfaas-mqtt connector service. Similarly, in the Aeneas application the Python service publishes the audio file to MQTT broker with topic name. The openfaas-mqtt connector running in fog node subscribes to the topic name and invokes the functions. Here, we use three serverless functions, as given in Table 1 and one openfaas-mqtt connector service. In PocketSphinx application, we use three serverless functions, as given in Table 1 and one openfaas-mqtt connector service.
For calculating the evaluation metrics in Section 6, we represent S = {S 1 , S 2 , . . . , S k } as set of storage units. The storage unit could be processors in NiFi, MQTT queues or MinIO buckets. Next section will describe about experiment details.
Experiment and results
All the proposed SDP approaches are implemented on three applications as described in the Section 4. Further, the goal is to measure the performance w.r.t metrics to understand and investigate efficiency of those SDPs on various applications (text, audio, video and image applications). In the following section, we will discuss the metrics used to measure the performance and analysed the results, further outlined the experience on the SDP implementation and provided future directions.
Performance metrics
In this subsection, we will describe various performance metrics along with their mathematical formulae. The resource utilization metrics are measured cumulatively on all the three layers of infrastructure that consists of utilization of edge (Gateway) resources, fog resources and cloud infrastructure resource. Here, concurrent user requests are generated in the sensor nodes • Processing Time: In IoT environments, computation time and latency are very crucial. In this regard, the SDP processing time is directly proportional to response time of the user requests and therefore we note that these metrics as native pipeline performance metrics. Processing time is measured in seconds, which is defined as the total time taken to process a data in a pipeline from source to destination. The source is a sensor node and sink is a storage/other end point in Cloud Infrastructure as described in Section 3. The processing time is addition of both communication and computation time (latency).
In the below paragraphs, we formulate the mathematical equations used to calculate these metrics. These metrics are calculated using logs of MinIO, MQTT, Apache NiFi and OpenFaaS gateway.
Computation time : The computation time is calculated as summation of time required to compute a data unit by individual processing units (serverless functions) in a data pipeline. Let R be the set of n number of concurrent users requests R = {r 1 , r 2 , . . . , r n }. In our experiments the value of n is considered from 10 to 300 and each user request carries a data unit to be processed by serverless function in the pipeline. The computation time of individual i th user request is
P(r i ) = ∀ f j F P(r i , f j )(1)
where
F = { f 1 , f 2 , .
. . , f m } is a set of m number of functions and P(r i , f j ) represents the time taken by the function f j ∈ F to execute or process the request r i ∈ R.
From equation (1) and equation (2), the communication time is calculated as
C T (r i ) = D(r i ) − P(r i )(3)
Further, In the SDP approaches the data units are stored in the intermediate storage units and served to serverless functions for processing. The total time that data unit resides in the storage unit is considered as disk access time denoted as DAT (r i ) is inclusive of time required to store and access the data units. Let DU at (r i , S j ) and DU dt (r i , S j ) be the arrival time and departure time respectively of the i th user requests' data unit in the storage unit S j ∈ S . The total duration of disk access time is calculated as
DAT (r i ) = S j ∈S DU dt (r i , S j ) − DU at (r i , S j )(4)NCT (r i ) = C T (r i ) − DAT (r i )(5)
• Average CPU utilization: The average CPU utilization is measured in percentage (%) and is calculated over time period from pipeline invocation till the data is received in the final destination.
• Average memory utilization: It is measured in percent (%) and is calculated as sum of total free memory, cache memory, memory in buffer and divided by total memory. Similarly, average disk utilization is measured in percentage (%).
• Network received: This is calculated as bytes per second and is calculated as sum of bytes received on the network over a period of time.
• Network transmitted: It is calculated as bytes per second and is calculated as sum of bytes uploaded on the network over a period of time.
• Disk I/O Read: This is measured in kilobytes and is calculated as sum of bytes read from file system over a period of time.
• Disk I/O Write: It is measured as bytes per second and is calculated as sum of bytes written in to the file system over a period of time.
Experimental Setup
The Docker Container Engine v19.03.12 is installed in both fog and cloud infrastructure in swarm mode. The OpenFaaS serverless platform is used as a FaaS engine configured in fog and cloud. OpenFaaS functions are developed using the programming language templates (bash streaming and Python 3.7). OpenFaaS command line interface (CLI) is used to build and deploy the functions into the OpenFaaS gateway. The Apache NiFi v1.3.2 is used in both fog and cloud as container service. The Apache NiFi user interface is used to design the data flow and monitor the flow files. The MinIO is deployed using docker compose service and volumes are mounted in the host machines. The MinIO client is used to create and configure the settings for event notifications on bucket.
A set of hardware devices and cloud resources are used to deploy and setup application services, as shown in Table 3. Three Raspberry Pi 4B models and MiniX NEO Z83-4U Intel Mini PC are used for setting up fog infrastructure. For cloud infrastructure, the virtual machines of size m2.medium with vCPU and 8GB RAM resembling similar capacity as AWS are provisioned from the University's private OpenStack cloud. The Raspberry Pi 3B model is used as a gateway node, and all the edge and fog devices are connected in a LAN with 1000 Mbps network bandwidth using Inteno DG200 router. The fog devices are connected to cloud services via 1000 Mbps network bandwidth. The network setup used for interconnection between edge, fog and cloud environments are dedicated to these experiments.
Upon setting up of necessary hardware and application services, the use cases (Aeneas, PocketSphinx and custom video processing ) are deployed. The corresponding performance metrics are measured, and results are discussed in below subsections.
Results and Discussion
We considered scaling the number of users as a parameter to measure the performance of the approaches because the rate of concurrent arrival of user requests heavily impacts the pipeline performance. To measure the performance of all the metrics, several users are scaled from 1 to 15 for video applications (we used a chunk of video file as one user request) and 10 to 300 for Aeneas, PocketSphinx applications, and the corresponding SDP performances were measured. However, for calculating the processing time we considered 100 users in Aeneas and PocketSphinx due to data units were started dropping in MQTT based SDP.
Performance metrics observed with Aeneas application
The processing time was studied in all the three SDP approaches, as shown in the Figure 8. Here, the y-axis represents processing time in seconds, and the x-axis shows the # of users. The OSS requires a maximum of 540s to complete the 100 users requests, whereas MQTT based SDP and DFT processed in 330s and 324s, respectively. The OSS had more processing time as the number of users increases, because MinIO notification invocations are synchronous. This pipeline had around Table 1 and two functions were extra to facilitate for retrieving the object data from MinIO buckets and this can lead to extra processing time. The computation time in OSS and MQTT based SDP was higher, where as disk access time was more in DFT. The internal queues in DFT manages efficient flow of data that makes to stay the data units in the queue that increase the DAS time. In OSS and MQTT based SDP, events were triggered as the data units arrived in to storage units which makes openfaas gateway to push these asynchronous user requests to NATs queue that increases the overall function execution time. The average computation time was highest in OSS with 351s but DFT had lesser computation with more disk access time of 280s. The average CPU utilization and Memory utilization were measured, as shown in the Figure 9. The primary vertical y-axis shows an average CPU utilization measured in percentage (%) and the secondary y-axis shows the Memory utilization. The DFT consumed highest CPU of 36%, whereas MQTT based SDP consumed a lesser CPU of 21.06% and OSS had moderate The Object Store and MQTT-based SDP used more number of lightweight python-based serverless functions, and DFT had a higher CPU utilization due to the set of Apache NiFi processors used in the pipeline that require extra computation power apart from serverless functions.
The average Memory utilization at the secondary y-axis is measured in percent (%). The MQTT-based SDP approach has the highest memory usage footprint of an average 45.92%, whereas DFT and MQTT based data pipelines used an average of 40.29% and 36.42% of memory, respectively.
In the Figure 10, the primary y-axis represents disk I/O read and the secondary y-axis shows a disk I/O writes measured in Kilo Bytes (KB). In the case of the OSS SDP approach, 18.38KB disk reads which is maximum as compared to DFT and MQTT with 2.9KB and 1.17KB, respectively for 300 users.
Similarly, OSS had a higher disk writes of 155KB as compared with DFT and MQTT with 83KB and 96KB respectively for 300 users. The OSS has more disk read/writes due to the read/write of bucket values based on each trigger. While in Apache NiFi, data flow is through the queue and doesn't had sever disk read/writes. Network performances of the SDPs were measured as Network receive and transmit bytes as shown in Figure 11. The OSS and MQTT has highest Network receive bytes calculated as average overall users 32KB and DFT performed well with 34KB. But for Network transmit bytes, MQTT had the highest reading with 64.3KB for 300 users. While, DFT had less network transmit bytes of 20KB over 300 users. MQTT recorded with highest values in terms of network performance due to In this application, MQTT based SDP had a lowest computation time with minimum processing time as compared with DFT and OSS. Further, MQTT based SDP did not experience any drop of data units in the pipeline as compared with Pocket-Sphinx and Custom video application. Moreover, CPU, Memory consumption and data Read/Writes metrics were also lowest, but there was raise in Network Receive/Transmit but it was negligible since the data unit size in the pipeline was very minimum. Considering the above metrics and associated SDP performances for Aeneas application, it is evident that MQTT SDP worked better over OSS and DFT, as shown in suitability table Table 4.
Performance metrics of the PocketSphinx application
The processing time of the PocketSphix application over all the proposed SDP's were measured as shown in Figure 12. The OSS had the highest processing time of 851s, whereas DFT had 663s with minimum processing time. This application had large a set of functions in the pipeline and OSS event notifications are set to three buckets to store intermediate results. All the events are triggered asynchronously and lead to a larger processing time. As similar to Aeneas, DFT shared highest disk access time, whereas OSS and MQTT had maximum computation time. In MQTT, major challenge was the data unit drop rate in- The CPU utilization and Memory utilization were observed in % as shown in Figure 13. The OSS and MQTT equally consumed the CPU of 28% calculated over 300 users as shown in the primary y-axis, whereas DFT consumed highest CPU 31%. However, MQTT had more CPU usage when user requests increased which is not suitable in terms of compute intensive and heavy compute bounded workloads. Similarly as in Aeneas, DFT uses Apache NiFi and required more CPU to execute processors concurrently. The memory utilization shown in the secondary y-axis, DFT had highest average memory utilization of 46% whereas MQTT used the highest memory of 48% after 300 users.
The disk I/O read and writes are measured in KB as shown in Figure 14. The OSS leads to more disk read 5KB as compared to DFT with 2KB and least with MQTT-based SDP of 1KB over 300 users. Similarly as in Aeneas, OSS utilizes the MinIO (S3) storage and at each event triggers on bucket notification, data would read from the disk leading to higher disk reads. The disk writes were shown in secondary y-axis, as similar OSS had significant disk writes of 70KB because the number of buckets used were more as compared with Aeneas and Video processing application. The least disk writes by MQTT-based SDP and moderately by DFT is due to data pushed and pulled in the queue neither directly written nor read from the disk. The network receives and transmits were measured in KB as shown in Figure 15. Here, the network received bytes performance of OSS had highest as 6KB while DFT had least with 2KB. The secondary y-axis represents the Network transmit bytes, similarly as above OSS had the highest transmit bytes with 7KB and DFT had least with 5KB over 300 users. Push events and notifications events of MinIO make OSS to consume more Network receive and transmit bytes. Evermore, Pocketsphinx uses more buckets as compared with other applications.
Considering the above performance metrics, its observed that DFT and MQTT performed equally better on Pocketsphinx application as compared with OSS. However, Pocketsphinx required more bandwidth to transfer audio files over the fog network and this motivates to consider DFT as suitable SDP shown in Table 4.
Performance metrics observed for custom video application
The video processing application naturally demands huge computation power and more bandwidth to process and offload the video files. The quality of the video is determined by frame rate. A frame per second (fps) is the speed at which individual still images, known as frames, are displayed in the video. The higher value of fps in the video requires more resources to process. So, it is significantly necessary to investigate the performance metrics based on the change of the fps values. In our work, we considered fps values scaling from 1 to 15 and measured the performance. Along with this, its essential to in- vestigate the rate of arrival of such user videos as measured in earlier applications. So, in this section, we will describe the performance metrics collected based on the change in fps values and arrival rate of user videos.
The processing time was measured in seconds (s) as shown in Figure 16. Here, the primary x-axis shows the number of users The primary y-axis represent the Processing time measured for number of users. The DFT worked better with 4500s as compared with OSS and MQTT based SDP with 5520s, 4515s respectively over 10 users. As like other applications MQTT had major issue of dropping the data units intermediate pipeline and it was approximately 28%. The other challenge was, MQTT openfaas-connector invokes the function carrying heavy data input, which makes maximum openfaas NATs queue memory utilization and this raises an exceptions from openfaas gateway. Similar to other applications, computation time was maximum in OSS where as DFT has minimum disk access time. The event triggers in MinIO and topic publish and subscription with huge multi-media (audio) data took quite higher processing time.
The Figure 17 represents the CPU utilization across scaling of number of users and fps values. Based on scale of users, DFT performed better with CPU utilization of 32%, but MQTT consumed more CPU with 35% over 15 users. However, MQTT worked better based on the scale of fps values with 23% and DFT consumed more CPU with 36%. Considering both of the scenarios, OSS worked well. Even though the MQTT-based SDP works well in case of fps based scenario but consumed more CPU in another scenario.
The memory utilization was shown in Figure 18, MQTT based SDP consumed less memory as compared to OSS and DFT with 34%, 46% and 44% respectively over 15 users. Interestingly, OSS consumed less memory with 35% as compared with MQTT and DFT with 46% and 49% respectively. The MQTT-based SDP and OSS were good in terms of memory consumption considering both of the scenarios. Disk Read for both of the scenarios shown in Figure 19, OSS had very few disk reads in both of the scenarios with average values of 1KB, 6KB respectively. MQTT-based SDP had more disk reads and DFT moderately worked better in both of the scenarios. Figure 20 shows the Disk Writes, MQTT has minimum disk write in both scenarios with an average value of 966KB, 33KB respectively. OSS had the highest disk writes based on the number of users while DFT had more based on the fps. The OSS will have obviously higher disk writes due to objects stored on the disks and DFT had more disk writes due to interaction of NiFi processors with file system.
The Figure 21 and Figure 22 represents the network performance measurement for both of the scenarios in Kilo Bytes (KB). In terms of network receive bytes in scaling of users scenario, OSS worked very well with 2KB and DFT moderately better with 4KB, but MQTT had more network receive bytes with 5.3KB considering 15 users. However, the scaling of fps values scenario, DFT worked well with 59KB, but OSS had maximum values over all the SDPs with 215KB. In network transmit bytes, DFT performance was moderately good with 6KB, 47KB but OSS consumed minimum network resources(receive bytes and transmit bytes) in terms of scaling the users whereas MQTT consumed more network resources. The MQTT based SDP consumed more network bandwidth due to publish of the multi-media data over the tcp network and mqttopenfaas client always listen to this topics leading to higher network consumption.
Considering the various performance metrics based on scaling of users and fps values, its observed that Video processing application consumes more CPU due to ffmpeg and YOLOv3 tools and even demand for more bandwidth to offload the multi media files between edge/fog/cloud infrastructure. However, DFT consumes less network resources but more CPU and disk resources, rather OSS uses less CPU but required more network resources. MQTT based SDP performance shows that its not suitable for Video processing due to heavy usage of resources.
Suitability analysis
All the proposed SDPs were implemented for three fog computing applications, the observed performance of all metrics vs all applications are reported in Table 4. The focus of suitability analysis was to extract insights from an observed experimental results in selection of best fit SDP for Aeneas, Pock-etSphinx and Video processing application. The overall results were summarized with suitability index and corresponding suitable SDP for each application was presented in the Table 4.
In this Table 4, average performance metric values were calculated by averaging the recorded values of SDPs performance across individual application and then minimum and maximum of such average values were noted and corresponding SDPs were chosen, as mentioned in the Table 4. Further, suitabil- templates and associated Docker files of serverless functions including other utility source files are available in GitHub 8 . Apart from the design experience, the resource utilization metrics such as CPU, Memory, Disk Reads and Writes are important in serving the demands of IoT applications, because the resource demand from end-user requests vary with respect to the type of the application. For example, video application demands for maximum compute and bandwidth resources, whereas text processing application demands only for bandwidth. So to investigate and analyze the resource utilization metrics and processing time, we calculated the average over all the three applications (on performance metrics) as shown in the Table 6.3.5.
The DFT consumed highest CPU (69%) and Memory (97%), whereas least in network utilization (15KB, 22KB) and processing time (20.97m), this indicates that DFT is best suitable for applications with huge bandwidth demand such as text processing. Further, OSS consumed moderate CPU (61%), Memory (85%) and network utilization (31KB, 43KB), but had large number of disk read and writes (19KB,197KB). These results show that OSS is best fit for video or image processing applications (bandwidth and compute intensive) due to lesser CPU, Memory and network utilization. On the other-side, MQTTbased SDP utilized the highest network resource (63KB, 33KB) and lesser CPU (52%), disk and moderate memory resources (86%). So, MQTT-based SDP is best suitable for compute intensive applications, however not suitable for bandwidth sensitive applications.
Even though the SDPs were designed and investigated significantly based on different performance metrics, certain challenges still exist and the key improvements can be undertaken.
The future directions should focus on two aspects. Firstly, on how to minimize the processing time and resource utilization. In our analysis, synchronous mechanism was used to invoke the serverless functions leading to larger processing time and they should be tested with asynchronous mode. The serverless function scaling mechanism can also substantially reduce the processing time and using intelligent scaling mechanisms could overcome this issue. Secondly, how can SDPs be executed on fog/cloud in terms of dynamic and stochastic workloads? Serverless operations were focused to fog infrastructure in this work, however, to achieve user QoS expectations and to fulfill the dynamism nature, a part of pipeline could be dynamically executed in fog and rest in the cloud. So, several opportunities exist to implement such intelligent decision mechanisms.
Conclusions and future work
In this paper, we proposed three Serverless Data Pipeline (SDP) approaches: DFT, OSS, and MQTT-based SDP using Apache NiFi, MinIO, and MQTT services, respectively. We applied these approaches to three different fog computing applications namely Aeneas, Pocketsphinx and Video processing application. We investigated their performance using the metrics such as processing time (computation time, disk access time and network access time) and resource utilization (CPU, Memory, Network and Disk utilization) and rigorously analyzed the results by calculating a suitability index for each of them. Results show that MQTT based SDP works best for Aeneas, DFT performs better for PocketSphinx and for video processing application, the OSS performance was good as compared with SDPs. However, an opportunity exist to improve the performance of proposed SDPs by scaling the serverless functions or using asynchronous invocations. Furthermore, the use of dynamic and stochastic workloads in future would help to evaluate the reliably, resilience and throughput etc, of the proposed SDP approaches.
Figure 1 :
1Proposed SDP architecture
Figure 2 :
2Abstract view of video processing data pipeline
Figure 3 :
3Abstract view of Aeneas data pipeline
Figure 4 :
4Pocketsphinx application data pipeline
Figure 5 :
5DFT based SDP approach using Apache Nifi and OpenFaas.
Figure 6 :Figure 7 :
67OSS based SDP approach using MinIO. MQTT based SDP Architecture
Communication time :
:The communication time denoted as C T (r i ) of i th user request is the time required to move data unit from source to sink in pipeline excluding the computation time. It's summation of disk access time (for intermediate storage units) and network communication time. Let D at (r i ) be the arrival time at source and D ct (r i ) be the completion time of the i th user request at sink. The total duration of serving the user request D(r i ) is measured as D(r i ) = D ct (r i ) − D at (r i )
where S = {S 1 , S 2 , . . . , S k } is a set of k number of storage units. Finally, network communication time denoted as NCT (r i ) is the summation of time required to move a data unit (of a user's request) in network from user's device to the sequence of processing units (serverless function) and intermediate storage units until the final data sink. Now from the equation (3) and equation (4) network access time is calculated as
Figure 8 :
8Aeneas application -Processing time measured in seconds
Figure 9 :
9Aeneas application -Average CPU and Memory utilization five serverless functions as shown in
Figure 10 :
10Aeneas application -Average Disk Reads and average Disk Writes measured in Kilobytes CPU utilization of 31%. But MQTT based SDP started using more CPU after 300 users request and further the data units in the pipeline started dropping.
Figure 11 :Figure 12 :
1112Aeneas application -Average Network Transmit and Average Receive data and measured in Kilo Bytes PocketSphinx application -Processing time measured in seconds each data flow with topic will be published and subscribed over the network, even in OSS most of the network operations recorded on buckets with event triggers.
Figure 13 :Figure 14 :
1314PocketSphinx application -Average CPU utilization and average Memory utilization and measured in % PocketSphinx application -Average Disk Reads and average Disk Writes and measured in Kilo Bytes creased as the number of user requests increased and here, drop rate was approximately 2%.
Figure 15 :Figure 16 :
1516PocketSphinx application -Average Network receive and transmit and measured in Kilo Bytes Video processing application-Processing time measured in seconds
Figure 17 :Figure 18 :
1718Video processing application -CPU utilization measured in percent (%) Video processing application -Memory utilization measured in percent (%)
Figure 19 :Figure 20 :
1920Video processing application -Disk Reads measured in Kilo Bytes (KB) Video processing application -Disk Writes measured in Kilo Bytes (KB)
Figure 21 :Figure 22 :
2122Video processing application -Network receive bytes measured in Kilo Bytes (KB) Video processing application -Network Transmit bytes measured in Kilo Bytes (KB)
Table 1 :
1Number of serverless functions used SDP/ApplicationAeneas Pocketsphinx Video proc.DFT based SDP
2
3
3
OSS based SDP
5
6
6
MQTT based SDP
2
3
3
Table 2 :
2List of Notation.Notation Description
P
Computation time
C T
Communication time
F
Set of serverless functions
R
Set of n number of concurrent users' requests
R = {r 1 , r 2 , . . . , r n }
D
Duration of the user request from source to des-
tination (data sink)
D at
Timestamp recorded at arrival from the source
D ct
Timestamp recorded at destination
DAT
Disk Access Time
NCT
Network Communication Time
S
Set of intermediate storage units S
=
{S 1 , S 2 , . . . , S k }
DU at
Timestamp recorded when data unit arrived in
the storage unit
DU dt
Timestamp recorded when data unit departed
from the storage unit
in all the three use cases with corresponding data such that it
mimics the real time application. The Prometheus and Node
Exporter software solutions are used to collect such metrics.
PromoQL (Prometheus Query Language) is used to calculate
the metrics for specific time period.
Table 3 :
3Hardware configuration for experimental setupDevice name
Configuration (Processor, RAM)
Quantity Node type
RPi 4B model
Quad-CoreCortex A72, 4GB LPDDR4
2
Fog node
RPi 3B model
Quad-CoreCortex A53, 4GB LPDDR4
1
Gateway node
Virtual machine
4-Core, 8GB DDR4
1
Cloud node
Minix Neo Z64-W10 Quad Core Z3735F (64 bit), 2GB DDR3
1
Fog node
Router
Inteno DG200 model with 1000Mbps full duplex 1
Network layer
Table 4 :
4Average performance metric values and suitability index calculated across each application
Table 5 :
5Average performance metric values across three applicationsMetric/Application
DFT
OSS
MQTT
Processing Time (m)
20.97 23.97
21.77
CPU Utilization (%)
69
61
52
Memory Utilization (%)
97
85
86
Disk Read (KB)
4
19
3
Disk Writes (KB)
102
197
95
Network Receive (KB)
15
31
33
Network Transmit (KB)
22
43
63
https://nifi.apache.org/minifi/ 2 https://nifi.apache.org/ 3 https://www.openfaas.com/ 4 https://min.io/ 5 https://mqtt.org/
https://github.com/shivupoojar/ServerlessDataPipelines
AcknowledgmentThis work is partially supported by the European Social Fund via IT Academy program and European Union's Horizon 2020 research and innovation project RADON (825040). We also thank financial support to UoH-IoE by MHRD, India (F11/9/2019-U3(A)).Table 4.As mentioned earlier, Aeneas is BI application and according to suitability index, MQTT based SDP is well suited for this application with 43%, in-spite it has huge network consumption which is not acceptable for BI applications. However, it had good performance in other metrics. The DFT is not suited due to higher processing time and disk utilization with 48%. In PocketSphinx application, MQTT based SDP has suitability index of 57%, where as it also has a highest not suitability index with 43%. The OSS had poor performance in all the aspects, but DFT has 0% index for not suitability, this motivates to consider the DFT as a best suited SDP for PocketSphinx. Finally, for Video processing application, the performance of OSS was significant with suitability index of 51% and 71%.Experience and future directionsThe set of experiments and associated results in earlier subsections show that SDPs performance varies significantly according to end user application (CI, BI). An IoT has a stochastic and heterogeneity (latency intensive, CI, BI) nature of workloads. To process such data oriented workloads the placement and design of the data pipeline mechanism on fog/cloud is quite necessary where our research fills this gap.However, while designing and implementing these SDP approaches significant portion of time was consumed in designing and developing the serverless functions in various proposed SDPs. The OSS consumed more time to design as it required more than five number of serverless functions as shown in Table 1, whereas DFT is least with maximum of three functions because of a set of built-in processors in Apache NiFi that could handle necessary utility operations such as PUTS3 to store a data in MinIO were used. But in MQTT based SDP and OSS we need to write them as functions. The DFT was best in designing and implementing, because state of art Apache NiFi data pipeline tool was used, which basically reduced efforts and easily integrated with serverless frameworks. The function
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| [
"https://github.com/shivupoojar/ServerlessDataPipelines"
]
|
[
"Atom-field dressed states in slow-light waveguide QED",
"Atom-field dressed states in slow-light waveguide QED"
]
| [
"Giuseppe Calajó \nVienna Center for Quantum Science and Technology\nStadionallee 21020Atominstitut, Wien, ViennaTUAustria\n",
"Francesco Ciccarello \nDipartimento di Fisica e Chimica\nNEST\nIstituto Nanoscienze-CNR\nUniversita' degli studi di Palermo\nItaly\n",
"Darrick Chang \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain\n",
"Peter Rabl \nVienna Center for Quantum Science and Technology\nStadionallee 21020Atominstitut, Wien, ViennaTUAustria\n"
]
| [
"Vienna Center for Quantum Science and Technology\nStadionallee 21020Atominstitut, Wien, ViennaTUAustria",
"Dipartimento di Fisica e Chimica\nNEST\nIstituto Nanoscienze-CNR\nUniversita' degli studi di Palermo\nItaly",
"ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain",
"Vienna Center for Quantum Science and Technology\nStadionallee 21020Atominstitut, Wien, ViennaTUAustria"
]
| []
| We discuss the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel. Such bound states are formed by an atom and a localized photonic excitation and represent the continuum analog of the familiar dressed states in single-mode cavity QED. Here we present a detailed analysis of the linear and nonlinear spectral features associated with single-and multi-photon dressed states and show how the formation of bound states affects the waveguide-mediated dipole-dipole interactions between separated atoms. Our results provide a both qualitative and quantitative description of the essential strong-coupling processes in waveguide QED systems, which are currently being developed in the optical and the microwave regime. PACS numbers: 42.50.Pq, 42.50 Nn, 03.65.GeThe coupling of atoms or other emitters to the quantized radiation field can result in drastically different physical phenomena depending on the detailed structure of the electromagnetic environment. While in free space atom-light interactions are mainly associated with radiative decay, atoms and photons may undergo processes of coherent emission and reabsorption in the case of a single confined mode as studied in the context of cavity QED [1, 2]. Recently, due in part to exciting experimental developments to interface two-level emitters with nanophotonic waveguides[3][4][5][6][7][8]and to couple superconducting qubits to open transmission lines [9-12], a different paradigm for light-matter interactions has emerged. Waveguide QED refers to a scenario, where single or multiple (artificial) atoms are coupled to a one dimensional (1D) optical channel. The 1D confinement of light brings about that individual photons can be efficiently absorbed by even a single atom or mediate long-range interactions between consecutive atoms along the waveguide. This gives rise to many intriguing phenomena and applications, such as single photon switches and mirrors[13][14][15][16], correlated photon scattering[17][18][19], self-organized atomic lattices[20,21], or the dissipative generation of long-distance entanglement[22][23][24][25]and new realizations of quantum gates[26][27][28].The physics of light-matter interactions in 1D becomes even more involved when the waveguide is engineered to have non-trivial dispersion relations, such as band edges and band gaps[6,8,29], near which the group velocity of photons is strongly reduced or free propagation is completely prohibited. In seminal works by Bykov [30], John and Quang [31] and Kofman et al.[32] the decay of an atom coupled to the band edge of a photonic crystal waveguide was shown to exhibit a non-exponential, oscillatory behavior with a finite non-decaying excitation fraction. This behavior can be attributed to the existence of a localized atom-photon bound state with an energy slightly outside the continuum of propagating modes[29,33,34]. With many atoms, it has been proposed that the long-lived nature of such states can facilitate the exploration of coherent quantum spin dynamics[35,36]or be exploited to engineer long-range photon-photon interactions[37,38].Motivated by the discussion above, here we study a system of a few quantum emitters, which are coupled to a common 'slow-light' photonic waveguide realized by a 1D array of coupled cavities. In the absence of any emitters such a system forms a finite propagating band with an effective speed of light that is fully controlled by the tunnel coupling between neighboring cavities, and thus can in principle be made arbitrarily small. Coupled cavity arrays (CCA) received large attention, in particular, as a platform for observing quantum phase transitions[39][40][41][42]and for the analysis of photon scattering processes in a finite-bandwidth scenario[15,[43][44][45][46][47][48][49][50][51]. Here again the appearance of localized photonic states[43,46,50,51]results in unusual two-photon scattering processes, where, e.g., one photon can remain bound to an atom[44][45][46], while the other one escapes. Such processes are absent in free space or infinite-band waveguides.Building upon those previous findings, we focus in this work specifically on the properties of dressed atomphoton states, which emerge as the elementary excitations of slow-light waveguide QED systems in the moderate to strong coupling regime. We find that an elegant feature of the CCA system is that in various parameter regimes one can recover the behavior of other systems previously discussed (such as single-mode cavity QED, infinite-bandwidth waveguides and band edges), as well as new phenomena not present in those limiting cases. In our analysis we introduce the single-photon bound states in waveguide QED as continuum generalizations of the dressed-states familiar from the Jaynes-Cummings model for single cavities. This analogy then allows us to arXiv:1512.04946v4 [quant-ph] | 10.1103/physreva.93.033833 | [
"https://arxiv.org/pdf/1512.04946v4.pdf"
]
| 54,582,306 | 1512.04946 | b500d5badf37e36cc488c1e9b2cddcc9625c60ff |
Atom-field dressed states in slow-light waveguide QED
6 Jul 2016
Giuseppe Calajó
Vienna Center for Quantum Science and Technology
Stadionallee 21020Atominstitut, Wien, ViennaTUAustria
Francesco Ciccarello
Dipartimento di Fisica e Chimica
NEST
Istituto Nanoscienze-CNR
Universita' degli studi di Palermo
Italy
Darrick Chang
ICFO-Institut de Ciencies Fotoniques
The Barcelona Institute of Science and Technology
08860Castelldefels (Barcelona)Spain
Peter Rabl
Vienna Center for Quantum Science and Technology
Stadionallee 21020Atominstitut, Wien, ViennaTUAustria
Atom-field dressed states in slow-light waveguide QED
6 Jul 2016(Dated: July 7, 2016)
We discuss the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel. Such bound states are formed by an atom and a localized photonic excitation and represent the continuum analog of the familiar dressed states in single-mode cavity QED. Here we present a detailed analysis of the linear and nonlinear spectral features associated with single-and multi-photon dressed states and show how the formation of bound states affects the waveguide-mediated dipole-dipole interactions between separated atoms. Our results provide a both qualitative and quantitative description of the essential strong-coupling processes in waveguide QED systems, which are currently being developed in the optical and the microwave regime. PACS numbers: 42.50.Pq, 42.50 Nn, 03.65.GeThe coupling of atoms or other emitters to the quantized radiation field can result in drastically different physical phenomena depending on the detailed structure of the electromagnetic environment. While in free space atom-light interactions are mainly associated with radiative decay, atoms and photons may undergo processes of coherent emission and reabsorption in the case of a single confined mode as studied in the context of cavity QED [1, 2]. Recently, due in part to exciting experimental developments to interface two-level emitters with nanophotonic waveguides[3][4][5][6][7][8]and to couple superconducting qubits to open transmission lines [9-12], a different paradigm for light-matter interactions has emerged. Waveguide QED refers to a scenario, where single or multiple (artificial) atoms are coupled to a one dimensional (1D) optical channel. The 1D confinement of light brings about that individual photons can be efficiently absorbed by even a single atom or mediate long-range interactions between consecutive atoms along the waveguide. This gives rise to many intriguing phenomena and applications, such as single photon switches and mirrors[13][14][15][16], correlated photon scattering[17][18][19], self-organized atomic lattices[20,21], or the dissipative generation of long-distance entanglement[22][23][24][25]and new realizations of quantum gates[26][27][28].The physics of light-matter interactions in 1D becomes even more involved when the waveguide is engineered to have non-trivial dispersion relations, such as band edges and band gaps[6,8,29], near which the group velocity of photons is strongly reduced or free propagation is completely prohibited. In seminal works by Bykov [30], John and Quang [31] and Kofman et al.[32] the decay of an atom coupled to the band edge of a photonic crystal waveguide was shown to exhibit a non-exponential, oscillatory behavior with a finite non-decaying excitation fraction. This behavior can be attributed to the existence of a localized atom-photon bound state with an energy slightly outside the continuum of propagating modes[29,33,34]. With many atoms, it has been proposed that the long-lived nature of such states can facilitate the exploration of coherent quantum spin dynamics[35,36]or be exploited to engineer long-range photon-photon interactions[37,38].Motivated by the discussion above, here we study a system of a few quantum emitters, which are coupled to a common 'slow-light' photonic waveguide realized by a 1D array of coupled cavities. In the absence of any emitters such a system forms a finite propagating band with an effective speed of light that is fully controlled by the tunnel coupling between neighboring cavities, and thus can in principle be made arbitrarily small. Coupled cavity arrays (CCA) received large attention, in particular, as a platform for observing quantum phase transitions[39][40][41][42]and for the analysis of photon scattering processes in a finite-bandwidth scenario[15,[43][44][45][46][47][48][49][50][51]. Here again the appearance of localized photonic states[43,46,50,51]results in unusual two-photon scattering processes, where, e.g., one photon can remain bound to an atom[44][45][46], while the other one escapes. Such processes are absent in free space or infinite-band waveguides.Building upon those previous findings, we focus in this work specifically on the properties of dressed atomphoton states, which emerge as the elementary excitations of slow-light waveguide QED systems in the moderate to strong coupling regime. We find that an elegant feature of the CCA system is that in various parameter regimes one can recover the behavior of other systems previously discussed (such as single-mode cavity QED, infinite-bandwidth waveguides and band edges), as well as new phenomena not present in those limiting cases. In our analysis we introduce the single-photon bound states in waveguide QED as continuum generalizations of the dressed-states familiar from the Jaynes-Cummings model for single cavities. This analogy then allows us to arXiv:1512.04946v4 [quant-ph]
We discuss the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel. Such bound states are formed by an atom and a localized photonic excitation and represent the continuum analog of the familiar dressed states in single-mode cavity QED. Here we present a detailed analysis of the linear and nonlinear spectral features associated with single-and multi-photon dressed states and show how the formation of bound states affects the waveguide-mediated dipole-dipole interactions between separated atoms. Our results provide a both qualitative and quantitative description of the essential strong-coupling processes in waveguide QED systems, which are currently being developed in the optical and the microwave regime. The coupling of atoms or other emitters to the quantized radiation field can result in drastically different physical phenomena depending on the detailed structure of the electromagnetic environment. While in free space atom-light interactions are mainly associated with radiative decay, atoms and photons may undergo processes of coherent emission and reabsorption in the case of a single confined mode as studied in the context of cavity QED [1,2]. Recently, due in part to exciting experimental developments to interface two-level emitters with nanophotonic waveguides [3][4][5][6][7][8] and to couple superconducting qubits to open transmission lines [9][10][11][12], a different paradigm for light-matter interactions has emerged. Waveguide QED refers to a scenario, where single or multiple (artificial) atoms are coupled to a one dimensional (1D) optical channel. The 1D confinement of light brings about that individual photons can be efficiently absorbed by even a single atom or mediate long-range interactions between consecutive atoms along the waveguide. This gives rise to many intriguing phenomena and applications, such as single photon switches and mirrors [13][14][15][16], correlated photon scattering [17][18][19], self-organized atomic lattices [20,21], or the dissipative generation of long-distance entanglement [22][23][24][25] and new realizations of quantum gates [26][27][28].
The physics of light-matter interactions in 1D becomes even more involved when the waveguide is engineered to have non-trivial dispersion relations, such as band edges and band gaps [6,8,29], near which the group velocity of photons is strongly reduced or free propagation is completely prohibited. In seminal works by Bykov [30], John and Quang [31] and Kofman et al. [32] the decay of an atom coupled to the band edge of a photonic crystal waveguide was shown to exhibit a non-exponential, oscillatory behavior with a finite non-decaying excitation fraction. This behavior can be attributed to the existence of a localized atom-photon bound state with an energy slightly outside the continuum of propagating modes [29,33,34]. With many atoms, it has been proposed that the long-lived nature of such states can facilitate the exploration of coherent quantum spin dynamics [35,36] or be exploited to engineer long-range photon-photon interactions [37,38].
Motivated by the discussion above, here we study a system of a few quantum emitters, which are coupled to a common 'slow-light' photonic waveguide realized by a 1D array of coupled cavities. In the absence of any emitters such a system forms a finite propagating band with an effective speed of light that is fully controlled by the tunnel coupling between neighboring cavities, and thus can in principle be made arbitrarily small. Coupled cavity arrays (CCA) received large attention, in particular, as a platform for observing quantum phase transitions [39][40][41][42] and for the analysis of photon scattering processes in a finite-bandwidth scenario [15,[43][44][45][46][47][48][49][50][51]. Here again the appearance of localized photonic states [43,46,50,51] results in unusual two-photon scattering processes, where, e.g., one photon can remain bound to an atom [44][45][46], while the other one escapes. Such processes are absent in free space or infinite-band waveguides.
Building upon those previous findings, we focus in this work specifically on the properties of dressed atomphoton states, which emerge as the elementary excitations of slow-light waveguide QED systems in the moderate to strong coupling regime. We find that an elegant feature of the CCA system is that in various parameter regimes one can recover the behavior of other systems previously discussed (such as single-mode cavity QED, infinite-bandwidth waveguides and band edges), as well as new phenomena not present in those limiting cases. In our analysis we introduce the single-photon bound states in waveguide QED as continuum generalizations of the dressed-states familiar from the Jaynes-Cummings model for single cavities. This analogy then allows us to arXiv:1512.04946v4 [quant-ph] 6 Jul 2016 (c) Single-photon (i.e., single-excitation) spectrum as a function of the atom-photon coupling g in the case of a single atom (with ωa = ωc) coupled to a cavity array according to Hamiltonian (1).
describe also many properties of the more involved multiphoton and multi-atom settings in terms of the properties of the single-photon dressed state. In particular, we discuss the modefunctions and spectral features of multiphoton dressed states, for which we identify the crossover from a linear regime, where the bound state energies are proportional to the number of excitations, N e , to a nonlinear regime where the splitting of the bound-state energies from the photonic band scales like ∼ √ N e . In the last part of the paper, we show how the usual longrange dipole-dipole interactions between multiple atoms coupled to broadband waveguides are modified in the presence of bound photonic states. Here we observe the formation of meta-bandstructures for delocalized dressed states as well as a partial 'melting' of these bands back into the continuum, when specific coupling conditions are met.
The remainder of the paper is structured as follows. In Sec. I we introduce the basic model of waveguide QED and briefly summarize in Sec. II the atomic master equation, which describes the dynamics of this system in the weak coupling regime. In Sec. III we discuss the properties of single photon-bound states in the absence and presence of decay. Finally, in Secs. IV and V we analyze the properties of multi-photon and multi-atom dressed states, respectively.
I. MODEL
We consider a system as illustrated in Fig. 1 (a), where a set of N a two-level atoms, each with ground (excited) state |g (|e ), are coupled to an optical waveguide of finite bandwidth 4J. We model the waveguide as an array of N → ∞ optical resonators with center frequency ω c and a nearest-neighbor tunnel coupling J. For atoms located at sites x i the total Hamiltonian for this system is ( = 1)
H = ω c x a † x a x − J x (a † x a x−1 + a † x−1 a x ) + Na i=1 ω a |e i e| + g Na i=1 x a x σ i + + a † x σ i − δ x,xi ,(1)
where a x (a † x ) are bosonic annihilation (creation) operators for the individual cavity modes, σ i − = (σ i + ) † = |g i e|, ω a is the atomic transition frequency and g is the atom-photon coupling strength. Note that for the validity of Eq. (1) it has been assumed that ω a ≈ ω c and that both frequencies are much larger than the couplings g and J so that counter-rotating terms can be neglected. Under these assumptions we can eliminate the absolute optical frequencies by changing into a rotating frame with respect to ω c , and the resulting system dynamics depends only on the atom-photon detuning δ = ω a − ω c . To account for atomic emission into other radiation modes as well as the absorption of photons in the waveguide, we introduce a bare atomic decay rate γ a and a photon loss rate γ c for each cavity as additional phenomenological parameters.
The first line of Eq. (1) represents the tight-binding Hamiltonian H c of the waveguide. By introducing momentum operators a k = 1 √ N x e ikx a x , where k ∈ ]−π, π], this can be written in a diagonal form H c = k ω k a † k a k , with mode frequencies
ω k = ω c − 2J cos(k),(2)
lying inside a band of total width 4J and centered around the bare cavity frequency ω c [see Fig. 1 (b)]. The propagation of photons inside the waveguide is characterized by the group velocity
v g (ω) = ∂ω k ∂k ω k =ω = 4J 2 − (ω − ω c ) 2 ,(3)
which vanishes for J → 0 or when operating at frequencies close to the band edges, i.e., ω ≈ ω c ± 2J. In the limit J = 0 the cavities are completely decoupled, each site being thereby described by a single-mode Jaynes-Cummings model [1] with coupling constant g and detuning δ. In this sense the present model captures well finite-bandwidth and bandedge features over a wide range of parameters. However, note that the CCA may only crudely approximates the actual dispersion relation in real photonic bandstructures and does not include effects like directional emission, which can occur in certain waveguide implementations [52][53][54]. . In each case, the photon loss rate has been set to γc/(2J) = 0.14.
II. BROADBAND LIMIT
Let us first consider the weak-coupling or broadband limit g/J → 0. In this regime the photonic waveguide modes simply act as a collective reservoir for the atoms and can be eliminated by using a Born-Markov approximation. As a result we obtain a master equation for the reduced density operator of the atoms (see App. A)
ρ = −i[H a , ρ] + i,j Γ ij 2 2σ j − ρσ i + − σ i + σ j − ρ − ρσ i + σ j − ,(4)
where in the rotating frame with respect to ω c ,
H a = i δ|e i e| + 1 2 i,j U ij σ i + σ j − + σ i − σ j + . (5)
In Eqs. (4) and (5) the Γ ij and U ij represent correlated decay rates and coherent dipole-dipole interactions, respectively, which arise from virtual or real photons propagating along the waveguide. By taking into account small atomic and photonic losses we obtain Γ ij = 2Re{A ij }+γ a and U ij = 2Im{A ij }, where
A ij = g 2 v g (δ) e iK|xi−xj ] ,(6)
and
K = π − arccos δ + iγ c /2 2J .(7)
Here we have introduced a generalized (complex) group velocityṽ
g (δ) = 4J 2 − δ + i γ c 2 2 .(8)
For γ c → 0 and for atomic frequencies within the photonic band this quantity reduces to the conventional group velocity given in Eq. (3). In this case ∼ 1/|v g (δ)| determines the density of photonic modes, or equivalently, the correlation time of the waveguide. In a system with losses this correlation time is now replaced by 1/|ṽ g (δ)|, which is well-defined and non-diverging even at or beyond the band edges (for a related study of the group velocity in lossy waveguides see also Ref. [55]). Therefore, the Born-Markov approximation, which requires
g |ṽ g (δ)|,(9)
can be used for all atomic frequencies provided that the coupling g is sufficiently weak and photon propagation times are negligible (see App. A for additional details on the validity of the Born-Markov approximation). Figure 2 illustrates the dependence of Γ ij and U ij on the interatomic distance for different atom-photon detunings, δ, and a non-vanishing photon loss rate γ c . If instead cavity losses are negligible, Eqs. (5)-(7) reproduce the effective spin model for two-level atoms coupled to an infinite-bandwidth waveguide [22,56]. In particular for frequencies within the propagating band, K becomes purely real and the system thus supports both coherent and dissipative dipole-dipole interactions of equal strength,
Γ ij 2g 2 v g (δ) cos (K|x i − x j |) , U ij 2g 2 v g (δ) sin (K|x i − x j |) .(10)
This coupling is infinite in range, with a phase factor e iK|xi−xj | that reflects the propagation phase of photons at the atomic resonance frequency that mediate the interaction. This behavior can be seen in Fig. 2 for δ = 0 (blue curve), with the deviation from infinite-range interaction due to the finite cavity losses γ c . As expected, by going from the center of the band towards the edge, δ ≈ 2J (red curve), both the coherent couplings as well as the correlated decay rates increase due to a reduction of the group velocity. However, slow propagation also means that the photons have more time to decay and for a finite γ c and large atom-atom distances, there is a tradeoff between an enhanced coupling and a larger propagation loss. For atomic frequencies outside the band there are no longer waveguide modes into which the atom can emit. Therefore, for γ c → 0, the real part of A ij vanishes and the atoms interact predominantly in a coherent way via a virtual exchange of photons. The exponential decay of interactions directly reflects the exponential attenuation of fields propagating through a band gap (see green curve of Fig. 2). In summary Eq. (4) shows that for sufficiently weak coupling the dynamics of the waveguide QED system can be described in terms of atomic excitations, which interact via a quasi-instantaneous exchange of photons. In this regime it is preferential to work near the band edge or to reduce the waveguide bandwidth all together in order to enhance waveguide mediated atom-atom interactions (coherent or dissipative) compared to the bare atomic decay. However, eventually the Markov condition given by Eq. (9) breaks down and for larger couplings the photons emitted by an atom can be coherently reabsorbed before they decay or propagate along the fiber. In this strong coupling regime photons and atoms can be bound together and form new hybridized excitations.
III. ATOM-PHOTON DRESSED STATES
In the absence of other decay channels, the atom-light coupling in Eq. (1) conserves the total number of photons and excited atoms, N e = x a †
x a x + i |e i e i |, and the eigenstates of H can be discussed separately within each subspace of given excitation number. For a given value N e , the Schrödinger equation H|φ = E|φ then has two types of solutions. First, there are scattering states, which are spatially extended over the whole waveguide and have an energy E/N e ∈ [−2J, 2J] within the free N e -photon band. Second, there are states with energy |E|/N e > 2J [57]. These states are energetically separated from the N e -photon continuum and represent bound states with an exponentially localized photonic component. While both types of states are atom-photon dressed states, in this work we are primarily interested in the latter type, namely in bound dressed states.
Note that in the waveguide QED literature the term photon bound state is also used to describe correlated propagating multi-photon wavefunctions scattered by a nonlinear emitter. These states are not localized around the atom but they are typically infinite in spatial extent, and bound only with respect to the relative coordinates of the photons [17]. In this paper we do not consider this kind of states and use the term bound state only for wavefunctions spatially localized around the atomic position.
A. Single-photon dressed states
We first consider the simplest setting of a single photon coupled to a single atom located at position x a . In this case, N e =1 and the solutions of the Schrödinger equation H|φ = E|φ are superpositions of an atomic excitation |e, 0 and single photon states |g, 1 x ≡ a † x |g, 0 (|0 is the field vacuum state). Figure 1 (c) shows the resulting energy spectrum which consists of the above mentioned band of scattering states and two bound states with energies E ± , which are the real solutions of (see App. B)
E ± − δ = g 2 E ± 1 − 4J 2 E 2 ± .(11)
The corresponding bound-state wavefunctions can be written in the form
|φ ± = cos θ ± σ + ± sin θ ± a † λ,± (x a ) |g, 0 ≡ D † ± (x a )|g, 0 .(12)
Here we have defined the normalized bosonic creation operator
a † λ,± (x a ) = x (∓1) |x−xa| e − |x−xa | λ ± coth 1 λ± a † x ,(13)
which creates a photon in an exponentially localized wavepacket around the atom's position x a . In Eqs. (12) and (13) the size of the photonic wavepacket, λ ± = λ(E ± ), and the mixing angle θ ± = θ(E ± ) are functions of the corresponding bound state energies. These two parameters determine the nature of the bound state wave-functions and are given by
cos θ = 1 + g 2 E 2 1 − 4J 2 E 2 3 2 − 1 2 ,(14)
and Figure 3 summarizes the dependence of λ + and the atomic excited-state population p + a = cos 2 (θ + ) on the coupling g and the atom-photon detuning δ. The analogous quantities associated with E − can be inferred through the identities λ − (δ) = λ + (−δ) and θ − (δ) = θ + (−δ).
1 λ = arccosh |E| 2J .(15)
Discussion. In their respective limits, Eqs. (11)-(15) reproduce various results that have been previously obtained for photonic bound states near band edges or in coupled cavity arrays [29,33,34,43,45,46,50,51]. The form of the wavefunction given in Eq. (12) provides a unified description of all those cases in terms of the mixing angles θ ± and the wavepacket lengths λ ± . It also establishes a direct connection to the more familiar dressed states of the single mode Jaynes-Cummings model [1] by
taking the limit J → 0, where E ± = δ 2 ± 1 2 δ 2 + 4g 2 , θ + = θ − − π/2 and λ ± ≈ 0.
For a finite J this singlecavity picture is modified in two ways. First, the photonic component now extends over multiple sites and becomes more and more delocalized the weaker the coupling g. Second, the total atomic contribution to both bound states, cos 2 (θ + ) + cos 2 (θ − ) < 1, is always smaller than one and for |δ| < 2J it vanishes as g/J → 0. Although a bound state solution always exists, both dressed states become more photon-like as g/J decreases and eventually become indistinguishable from the propagating waveguide modes. For atomic frequencies outside the band, e.g., δ > 2J, the upper bound state becomes more atomlike as g/J → 0, but the residual photonic cloud remains localized. Overall, these results show that a simplified model where the waveguide is replaced by an effective cavity of size λ would be incomplete. In particular, such a description misses the fact that for δ = 0 photonic wavefunctions associated with the two dressed states can significantly differ, i.e., λ + = λ − and θ + = θ − .
B. Excitation spectrum
An experimentally relevant quantity to probe the properties of atom-photon dressed states is the atomic excitation spectrum S a (ω), which can be obtained by weakly exciting the atom with a laser of frequency ω and recording the total emitted light. In the weak driving limit the excitation spectrum is given by where
S a (ω) = γ 2 a 4 e, 0| 1 H eff − ω1 |e, 0 2 ,(16)H eff = H − i γ a 2 |e e| − i x γ c 2 a † x a x ,(17)
and the normalization has been set such that S(ω = ω a ) = 1 for g = 0. Figure 4 shows the results for S a (ω) for different coupling strengths g and for the two relevant cases δ = 0 (center of the band) and δ = 2J (upper-band edge). For δ = 0 we observe three different regimes. For very weak coupling there is only a single peak at the atomic frequency with a width ∼ γ a + g 2 /J due to the enhanced emission into the waveguide (recall that in the broadband limit the atom emission rate into the waveguide is g 2 /J, see Section II). At intermediate couplings g/(2J) ∼ 1 the spectrum is completely smeared out. The atom is now partially hybridized with all waveguide modes and there is no longer a well defined frequency associated with the atomic excitation. At larger couplings two dominant resonances at the dressed-state energies E ± appear. As the coupling increases the width of the two bound-state resonances approaches
γ = γ a + γ c 2 ,(18)
as expected from an equal superposition of atomic and photonic excitations. For δ = 2J a significant hybridization between atom and photon is already observed at small g, consistent with the atomic population p + a ≈ 0.67 predicted for the dressed state exactly at the band edge [see Fig. 3 (a)] . However, in this case the transition from waveguide-enhanced decay to atom-photon hybridization is not apparent and will be discussed in more detail in the following.
C. Onset of strong coupling
An important regime of operation in cavity QED is the regime of strong coupling, where the coherent interaction between atoms and photons dominates over the relevant decay processes. For a single cavity in resonance with the atom this regime is usually defined by the condition
g > γ a + γ c 4 .(19)
Our goal is now to identify an equivalent condition for the waveguide QED system, by taking a closer look at the spectral features for g J. Note that the atomic excitation spectrum is in general given by [31,48]
S a (ω) = γ 2 a 4 1 ω − δ + i γa 2 −Σ(ω) 2 ,(20)
whereΣ(ω) = −ig 2 /ṽ g (ω) is the self energy in the presence of dissipation. To bring this result into a more useful form we define
∆ ± (ω) = ω − δ + i γ a 2 ṽ g (ω) ± ig 2 .(21)
It can be shown that ∆ + (ω)∆ − (ω) is a forth order polynomial in ω with two roots given by the complex eigenen-ergiesẼ ± of H eff . We can use this property to further rewrite the spectrum as
S a (ω) = γ 2 a 4 |ṽ g (ω)∆ − (ω)| 2 |(ω −Ẽ + )(ω −Ẽ − )L(ω)| 2 .(22)
Here L(ω) is a quadratic polynomial, which for the limits discussed below has two roots with real parts inside the photonic band, and thus describes the atomic emission into the waveguide continuum. Overall the structure of the spectrum then consists of two external poles with a position and a width given by the real and imaginary parts ofẼ ± and a broader emission peak inside the waveguide. Note that for γ c → 0 the generalized group velocity,ṽ g (ω), and therefore also the spectrum vanishes exactly at the bandedge, ω = ±2J. This is due to a destructive interference between the excitation laser and the long-lived band-edge mode and leads to a Fano-like profile for S a (ω). For non-vanishing γ c this interference effect is washed out. We first consider the case δ = 0, where we obtain to lowest order in g which shows that for not too large decay rates, the position of the external peaks essentially follows the bare energy levels E ± and their width is mainly determined by photon loss. For the polynomial determining the internal peaks we obtain
E ± ±2J ± g 4 16J 3 [1 ± i(γ a − γ c )/(2J)] − i γ c 2 ,(23)L(ω) = ω + i γ a 2 2 + g 2 2J 2 ,(24)
which therefore contributes with two purely imaginary poles at ω = −i(γ a ±g 2 /J)/2. Figure 5 (a) shows a zoomin on the resulting spectrum near the band edge and for different values of γ c . First, we observe that for large γ c the external peak is completely buried within the tail of the broad internal peak and a closer inspection shows that a minimal coupling of
g > Jγ c ,(25)
is required to spectrally resolve the existence of an external bound state. This condition is equivalent to the requirement that the atomic emission rate into the waveguide exceeds the cavity loss rate. Once this condition is fulfilled we can define strong coupling by the requirement that the separation of the external peak from the band edge, Re{Ẽ + − 2J}, exceeds its half-width given by Im{Ẽ + }. Again for γ c , γ a 2J we obtain
g > 4 8J 3 γ c ,(26)
as the strong coupling condition for a resonantly coupled waveguide QED system. Note that since in the present regime the bound states are mainly photonic in nature the atomic decay is relevant only for higher-order corrections.
The second important limit is δ = 2J, which for g/J 1 also corresponds to the quadratic dispersion relation assumed in studies of photonic bound states near the band edge of a photonic crystal waveguide [29,[31][32][33][34].
Note that in this regime the initial scaling of the bound state energy in the absence of losses (γ a = γ c = 0) is given by
E + 2J + g 4 4J 1 3 ,(27)
where the splitting β = 3 g 4 /(4J) can be directly identified with the frequency of coherent atom-photon oscillations at the band edge [34]. In the presence of decay and for g < |γ c − γ a | we obtain instead the modified result
E + 2J − i γ a 2 + g 2 2 J|γ c − γ a | (1 ∓ i),(28)
where the minus (plus) sign is for the case γ c > γ a (γ c < γ a ). This result shows that not only does the presence of loss modify the initial scaling of the bound state energy, Eq. (28) also predicts that at the band edge and for small g the atom is critically damped, i.e., the coupling induced losses are exactly of the same magnitude as the coherent shift of the bound state energy. By increasing the coupling further the imaginary part of the eigenvalueẼ + will eventually saturate at a valueγ/2 [cf. Eq. (18)] corresponding to a fully hybridized state. This hybridized regime is reached for coupling strengths
g > 4 J|γ c − γ a | 3 4 .(29)
Under this condition the separation of the bound-state from the bandedge is then given by β from which we obtain the strong coupling condition β >γ/2, or Figure 5 (b) shows a zoom-in of the atomic spectrum S a (ω) for δ = 2J and for three different values of the photon decay, which correspond to the critically damped, intermediate and strong coupling regime. Note that for δ = 2J the internal poles associated with L(ω), i.e., ω 1 = 2J − i γc 2 and ω 2 = 2J − i γa 2 − g 2 (1 ∓ i)/(2 J|γ c − γ a |) provide an additional background, but do not play a significant role.
g > 4 Jγ 3 /2 .(30)
D. Localization
For the remainder of this work we are mainly interested in coherent effects and for the sake of clarity we will only present results for idealized systems where γ a = γ c = 0. Therefore, the validity of these results in particular requires that the strong-coupling conditions identified in Eqs. (25), (26), (29) and (30) are fulfilled in the respective limits. In addition, it is important to emphasize that all the results discussed in this work are based on the model of a perfectly regular cavity array. In real systems disorder in the cavity frequencies or tunnel couplings introduces an additional localization mechanism, even in the absence of the emitters. To estimate this effect we can consider a simple impurity model, where we add an energy offset to one of the lattice sites, H c → H c + a †
x d a x d . This model is well known in literature [58] and it exhibits a purely photonic bound state with a localization length
1 λ = arcsinh | | 2J .(31)
This means that random energy offsets of typical strength will create bound states that are localized over λ ∼ 2J/| | lattices sites. While atom-photon bound states will also exist in such disordered waveguides, all the predictions in this work are based on the assumption that λ is large compared to the size of the atom-induced bound states, λ ± . For a more accurate treatment of localization in waveguides, see, for example Ref. [59] and the supplementary material of [36].
IV. MULTI-PHOTON DRESSED STATES
While in cavity QED the appearance of a normal-mode splitting (corresponding to the well-known vacuum Rabi frequency) signifies the onset of strong light-matter interactions, the hallmark of a fully quantized radiation coupling lies in the non-linear scaling of this splitting with the number of excitations, ∼ g √ N e . In this section we will address the properties of multi-photon dressed states to see to what extent this quantum signature prevails in the context of waveguide QED. In contrast to the single excitation case, the Schrödinger equation H|ψ = E|ψ for N e > 1 no longer permits simple analytic solutions and for exact results one is restricted to numerical methods in real or momentum space [44][45][46]48]. In this work we perform such calculations by an approximate variational approach, which provides additional intuition on the nature of the multi-photon dressed states, and allows us to evaluate the corresponding bound-state energies for excitation numbers that are no longer trackable by standard numerical methods.
A. Two-photon dressed states
Let us first consider the two-excitation subspace, where a general eigenfunction of Hamiltonian (1) can be written in the form to the set of coupled equations
|φ = x b(x)a † x |e, 0 + 1 √ 2 x,y u(x, y)a † x a †− J [u(x + 1, y) + u(x − 1, y) + u(x, y + 1) +u(x, y − 1)] + g √ 2 [b(x)δ 0,y + b(y)δ 0,x ] = Eu(x, y),(33)
and
− J [b(x + 1) + b(x − 1)] + g √ 2 [u(0, x) + u(x, 0)] = Eb(x) .(34)
These equations, which extend the continuous waveguide [17] case to a discrete model, can be solved numerically and the resulting eigenvalues spectrum is shown in Fig. 6 together with the single excitation energy band discussed in Sec. III A. For our numerical calculations an array of N = 120 coupled resonators with periodic boundary conditions has been assumed. In line with the singleexcitation case, we observe a band of two-photon scattering states with energies E ∈ [−4J, 4J]. In addition, there are two bands with energies E ∈ [E ± −2J, E ± +2J]. These bands can be simply interpreted as the combination of a single-atom bound state with energy E ± and an additional free photon with energy ω k . Finally, we observe two individual lines at energies E (Ne=2) ± above and below all other states, which represent the true twophoton bound states in the N e = 2 sector.
Before proceeding with a more detailed discussion on the two-photon bound states, let us briefly point out another interesting feature in Fig. 6 in the two-excitation manifold, namely the overlap region between the continuum of states with a single bound photon (shaded in green) and the two-photon continuum (shaded in purple). In this region, which extends up to a coupling strength of about g/(2J) 3 scattering processes of the form
|2 in ↔ |1 out |1 bound ,(35)
are energetically allowed, meaning in particular that scattering processes where two incoming photons evolve into a bound photon and an outgoing one are allowed. Such processes have previously been observed in numerical studies [44][45][46] and further investigated in Refs. [48,49].
The energy level diagram shown in Fig. 6 provides simple energetic arguments to determine under which conditions such processes can occur. Note that all the qualitative considerations so far can be extended to the N eexcitation subspace. For example the N e = 3 band structure consists of three-photon continuum of width 12J, two bands of one bound and two free photons of width 8J, two bands with two bound and one free photons of width 4J and two true three-photon bound states, and so on. Therefore, the complete energy spectrum of a single atom waveguide QED system can be constructed from the knowledge of the N e -photon bound state energies E (Ne) ± .
B. Variational wavefunction
While the exact eigenstates of the N e = 2 subspace can be still found numerically, we now consider a variational approach through which additional intuition about the nature of two-photon bound states can be obtained. In particular, within the two excitation subspace, the lower energy two-photon bound state corresponds to the ground state and can be generically written as
|Ψ (2) − = cos(θ)σ + A † 1 − sin(θ)B † 2 |g, 0 ,(36)
where A 1 and B 2 are single-and two-photon operators, respectively. Based on the discussion in Sec. IV A a suitable ansatz for the two-photon state is
B † 2 = 1 N uã † λ1ã † λ2 ,(37)
whereã λ = x e − |x| λ a † x and the normalization constant N u is chosen such that 0|B 2 B † 2 |0 = 1. This two-photon wavepacket is an exact solution of the Schrödinger equation for x, y = 0 with an energy
E (2) − = −2J cosh(1/λ 1 ) − 2J cosh(1/λ 2 ).(38)
For the single-photon operator we demand that the wavefunction also satisfies the first boundary condition, Eq. (33), at x = 0 and y = 0. This leads to where N b is again a normalization constant. By using this ansatz we can now find an upper bound for the twophoton bound state by minimizing E var = Ψ with respect to θ and λ 1,2 . To further reduce the parameter space, it is reasonable to assume that the wavepacket size of the first photon, λ 1 is approximately given by the value of λ − , which we determined for the single-photon bound state in Sec. III A. The variational ansatz is then based on the physical picture of a two-photon dressed state consisting of the single-photon dressed state plus an additional photon, which is more weakly bound and thus less localized, λ 2 > λ 1 . As we will show in more detail in a moment, this ansatz provides very accurate values for the bound-state energies.
A † 1 = 1 N b sinh 1 λ 2 ã † λ1 + sinh 1 λ 1 ã † λ2 ,(39)
C. Multi-photon dressed states
An important aspect of our variational wavefunction approach is that it can be extended to higher excitation numbers N e in a systematic way. To do so we write the wavefunction for the lowest energy state within the N eexcitation subspace as Based on analogous arguments as above, we make the ansatz
B † Ne = 1 N uã † λ1ã † λ2 . . .ã † λ Ne ,(41)
and
A † Ne−1 = 1 N b sinh 1 λ Ne ã † λ1 ...ã † λ Ne −1 ... + sinh 1 λ 1 ã † λ2 ...ã † λ Ne ,(42)
where N u and N b are chosen to normalize each photonic component of the state. To reduce the variational parameter space, the problem can be solved in an iterative manner, i.e., by using the values of λ 1 , . . . λ Ne−1 as input for minimizing the energy E (Ne) − with respect to θ and λ Ne .
Discussion. Figure 7 shows the bound-state energies E (Ne) − obtained from our variational approach for up to N e = 8 photons. For N e = 2, 3 these results are compared in the insets with the energies obtained from exact numerical diagonalization in the crossover regime g/(2J) ∼ 1. The excellent agreement within ∼ 1% (for smaller or larger values of g the agreement is even better) demonstrates that our variational ansatz captures the essential features of the exact wavefunction.
For N e = 1, 2, 3 the shape of the individual photonic wavepackets associated with the operatorsã † λi in Eq. (41) are sketched in Fig. 8 (a) and (b). We see that in particular near the band edge there is a significant difference between λ 1 and λ 2 , while the differences between the λ Ne are less pronounced for higher excitation numbers. It should be noted though that the variational approach, which is constructed to minimize the energy, is not very sensitive to the exponential decay of the wavefunction 0, . . . , 0, x Ne |Ψ (Ne) − ∼ e −|x Ne |/λ Ne . For physical effects that rely on more accurate predictions for the exponential decay we can, instead of simply settingλ Ne = λ Ne , make use of the exact energy relation [see Eq. (38) for
N e = 2] E (Ne) − 2J = Ne n=1 cosh 1 λ n ,(43)
valid at distances far away from the atom. Therefore, from the exact result for λ 1 ≡λ 1 and the set of bound state energies E (Ne) − obtained from our variational calculations, one can iteratively apply Eq. (43) to also calculate values for the asymptotic decay lengthsλ Ne . For N e = 2 the results of this procedure are shown in Fig. 8 (c) and (d) and compared with the asymptotic decay length extracted from the numerical solution of the twophoton wavefunction u(x, y). We observe the same general trend as already mentioned above, but at the same time the use of Eq. (43) provides more accurate quantitative results. Finally, in Fig. 8 (e) and (f) we plot the atomic population of the N e -photon bound states, showing the expected increase of hybridization for higher excitation numbers.
From Fig. 7 we see that for large couplings, g/(2J) 1, the bound-state energies exhibit a splitting from the bare energy by an amount ∼ √ N e , characteristic of the scaling in conventional cavity QED [1]. In this limit all bound photons are essentially localized on the atom site and the single-mode physics is recovered. To characterize the nonlinearity of the spectrum also in the weak and moderate coupling regime we define the nonlinearity parameter
∆ nl (N e ) = |N e E (1) − − E (Ne) − | g|N e − √ N e | .(44)
With this definition ∆ nl (N e ) 1 implies that the excitation spectrum is as nonlinear as cavity QED under resonance conditions, δ = 0, while the opposite limit ∆ nl (N e ) 0 indicates a harmonic spectrum. In Fig. 9 we plot ∆ nl (N e = 2) for different values of g and different atomic detunings. We see that, as expected, in the strong coupling limit, g {J, |δ|}, the waveguide QED system approaches asymptotically the nonlinear behavior of the single-mode Jaynes-Cummings model. It can be seen that although for δ = −2J the nonlinearity (compared to the Jaynes-Cummings nonlinearity) vanishes at small g, it is still much stronger than for the resonant case δ = 0. This is consistent with the observation that for δ = −2J the wavelength of the second photon, λ 2 , can be much larger than the wavelength of the first bound photon, λ 1 . In contrast, for δ = 0 one finds λ 1 ≈ λ 2 . Note that the approximate scaling of the nonlinearity parameter for g → 0 can be understood from the simplified assumption E
(2) − ≈ E (1)
− − 2J, which would correspond to a single photon bound state plus an additional very loosely bound photon at the bandedge. By recalling that E
(1) − −2J − [g 4 /(4J)] 1/3 (see Eq.(27)) we obtain ∆ nl (2) ∼ 3 √ g. For δ = −3J, which
for g → 0 corresponds to an atom-like state inside the bandgap, the nonlinearity parameter diverges. Note that this divergence is a consequence of the chosen normalization for ∆ nl (N e ) and can again be understood from the approximation E
(2) − δ − 2J for small g.
V. DIPOLE-DIPOLE INTERACTIONS BETWEEN DRESSED STATES
Our analysis so far has focused on the bound states forming around a single atom. However, a key element of waveguide QED are the photon-mediated interactions between two or multiple separated emitters. In the weak-coupling regime discussed in Sec. II, we have identified effective dipole-dipole interactions between individual atoms, which can be long-range and scale like U ij ∼ g 2 /J. In the following section we are interested in the corresponding interactions between dressed states, which represent the elementary waveguide excitations in the strong coupling regime. For previous work on bare atom-atom interactions near band-structures see, for example, Refs. [60][61][62][63].
A. Two-atom dressed states
We first consider the case of two atoms located at positions x 1 and x 2 and focus on the single excitation subspace, N e = 1. In this case the Schrödinger equation can still be solved exactly and details are summarized in App. C. The resulting energy spectrum has up to four solutions with energies E ±,s=e,o outside the waveguide continuum given by the real solutions of
E ±,e − δ = g 2 e − |x 1 −x 2 | 2λ cosh |x1−x2| 2λ E ±,e 1 − 4J 2 E 2 ±,e(45)
for the even parity states and
E ±,o − δ = g 2 e − |x 1 −x 2 | 2λ sinh |x1−x2| 2λ E ±,o 1 − 4J 2 E 2 ±,o(46)
for the odd parity states, where λ ≡ λ(E ±,s ) has the same energy dependence as for the single atom case in Eq. (15).
For concreteness and notational simplicity, we restrict the following discussion to the two lower bound states with energies E −,s < −2J below the continuum and even (s = e) or odd (s = o) symmetry of the atom-field system. The corresponding eigenstates can be written as
|φ s=e,o = 1 √ 2 D † s (x 1 ) ± D † s (x 2 ) |g 1 , g 2 , 0 ,(47)
where the + (-) sign holds for the state with even (odd) symmetry. The dressed-state creation operators D † e,o (x i ) are defined as
D † s=e,o (x i ) = cos(θ s )σ i + + sin(θ s )ã † λ,s (x i ) N s ,(48)whereã † λ,s (x i ) = x e − |x−x i | λ a †
x is an unnormalized photonic creation operator and
N e,o = coth 1 λ 1 ± e − |x 1 −x 2 | λ ± |x 1 − x 2 |e − |x 1 −x 2 | λ ,(49)
is the corresponding normalization constant [again, the + (−) sign holds for the even (odd) case]. The mixing angle θ is given by
cos θ s = 1 + g 2 N 2 s 4J 2 sinh 2 1 λ − 1 2 ,(50)
which depends on both the bound-state energy and the distance between the atoms. Discussion. Figure 10 shows the dependence of the two-atom dressed state energies E −,s on the atomic separation |x 1 −x 2 |. For distances which are large compared to λ both energies are approximately equal to the singleatom bound state, E −,e E −,o E − , i.e., there are no long-range interactions. At a large but finite separation |x 1 − x 2 | λ(E − ) the photonic wavefunctions associated with the single-atom bound states start to overlap so as to induce a splitting of the energies such that E −,e < E − < E −,o . As long as this splitting is still small the dressed-states dynamics can be described by the Hamiltonian
H ≈ i=1,2 E − D † i D i + U dd 2 D † 1 D 2 + D 1 D † 2 .(51)
Here the D i ≡ D(x i ) are the single-atom dressed state operators introduced in Eq. (12), which in the approximated model in Eq. (51) are treated as independent, i.e., mutually commuting degrees of freedom. Therefore, Hamiltonian (51) describes a dipole-dipole like coupling between distant dressed states with strength (assuming δ = 0)
U dd J cosh 1 λ 1 + coth 2 1 λ e − |x 1 −x 2 | λ .(52)
This shows that the long-range interactions occurring in the weak-coupling regime become exponentially localized when g/(2J) 1, even in the absence of losses. As the atom-atom separation decreases further, the mutual distortion of the wavepackets must be taken into account. As illustrated in Fig. 11, the even bound state -corresponding to the lower level E −,e -is a 'bonding' state such that the photon becomes more and more localized between the atoms. In contrast, the odd statecorresponding to the upper level E −,o -behaves as an 'anti-bonding' state such that the photon becomes more and more delocalized as the atomic spacing decreases. As a result, two regimes must be distinguished. As shown in more detail in Appendices C and D, for g > g m and δ > −2J, where In the opposite case, g < g m , we find that there is a finite distance x m = (g m /g) 2 > 1 below which the upper bound state E −,o reaches the band edge and disappears (see Fig. 10). This 'melting' of one of the bound states into the waveguide continuum is related to a progressive delocalization of the photonic wavepacket that eventually becomes completely delocalized along the array [see for instance the dashed green line in Fig. 11(c)]. This effect is most relevant for resonantly coupled atoms, δ ≈ 0, and for moderate coupling strengths, while for δ ≤ −2J both the two-atom bound states always exist. Note that the current discussion has been restricted to the two lower dressed states E −,s < − 2J, but analogous results are obtained for the two-atom bound states above the photonic band, E +,s > 2J with the sign of δ reversed. See App. D for more details.
g m = 2J 1 + δ 2J ,(53)
B. Dressed-state bandstructure
The above analysis can be extended to multiple atoms, where for N N a 1 and equidistant spacings, x i+1 − x i = ∆x, the coupling between neighboring atoms leads to the formation of a meta-bandstructure for propagating dressed-state excitations below and above the bare photonic band. This is illustrated in Fig. 12 where in the single-photon bound-state energies for N a = 40 atoms are shown as a function of ∆x. For large ∆x we see that the bound states form a narrow band around the singleatom energies E + and E − with a width of ∆E ≈ U dd . For smaller atomic spacings, the bandwidth grows and -depending on the parameters -it can either partially melt into the waveguide continuum or remain energetically separated.
As shown in App. C 2, the meta-band is bounded by an upper and lower energy E u and E l , which obey the equations
E u − δ = g 2 coth ∆x 2λ E u 1 − 4J 2 E 2 u(54)
and
E l − δ = g 2 tanh ∆x 2λ E l 1 − 4J 2 E 2 l ,(55)
respectively. Similarly to the previous section, it is possible to define a critical coupling
g (Na 1) m = √ 2g m ,(56)
for the multi-atom band, which only differs by a factor √ 2 from the two-atom case g m given in Eq. (53). For g > g (Na) m and |δ|/(2J) < 1, the meta-band is separated from the phonic continuum regardless of ∆x. In the opposite case, g < g (Na) m , a fraction of the dressed-state band disappears in the waveguide continuum, i.e., unlike in a usual band-structure only a fraction of the k-modes are available.
VI. CONCLUSIONS
In summary, we have analyzed the most essential properties of single-photon, multi-photon and multi-atom dressed-state excitations in a slow-light waveguide QED setup. Our results provide a both qualitative and quantitative description of the basic linear and nonlinear optical processes in this system and intuitively explain and connect various effects that have been previously described in different limiting cases. We have derived the necessary requirements that are needed to observe atomphoton bound states under realistic experimental conditions, which can be achieved, for example, with state of the art superconducting circuits [9][10][11][12]. More importantly, our analysis of non-linear and multi-atom effects can serve as a starting point to further explored the complexity of waveguide QED systems, when the regime beyond a few excitation is considered. Note added. After the initial submission of this work a closely related work on multi-photon bound states in waveguide QED systems by T. Shi et al. [71] appeared.
Appendix A: Master equation
In this Appendix we outline the derivation of the master equation (4) in the weak coupling limit g/J → 0. Starting from Hamiltonian (1) we change into an interaction picture with respect to H 0 = i ω a |e i e| + H c and we obtain the atom-field interaction Hamiltonian
H int (t) = g Na i=1 σ i + E(x i , t)e iωat + σ i − E † (x i , t)e −iωat , (A1) where E(x, t) = 1 √ N k e −iω k t e ikx a k ,(A2)
is the field operator at site x and k = 2πm/N with m = −N/2, −N/2 + 1, ..., N/2 − 1. The field operators obey the commutation relations
[E(x, t), E † (x , t )] = Φ(x − x , t − t ),(A3)
where
Φ(z, τ ) = 1 N k e −ikz e −iω k τ = e −iωcτ N N −1 n=0 e −i2πzn/N e i2J cos(2πn/N )τ = e −iωcτ N N −1 n=0 e i2πzn/N ∞ m=−∞ i m J m (2Jτ )e i2πnm/N =e −iωcτ i |z| J |z| (2Jτ ).(A4)
Up to second order in g and by performing the usual Born-Markov approximation [64], we end up with a timelocal master equation governing the time evolution of the atom's reduced density operatoṙ
ρ(t) = − ∞ 0 dτ Tr c {[H int (t), [H int (t − τ ), ρ c ⊗ ρ(t)]]},(A5)
where in the absence of any driving fields ρ c = |0 0| is the vacuum state of the waveguide modes. The master equation can be expressed in the forṁ
ρ = ij A ij σ j − ρσ i + − σ i + σ j − ρ +A * ij σ i − ρσ j + − ρσ j + σ i − , (A6) where A ij =g 2 ∞ 0 dτ E(x i , t)E † (x j , t − τ ) e iωaτ =g 2 ∞ 0 dτ Φ(x i − x j , τ )e iωaτ e −γcτ /2 =g 2 i |xi−xj | ∞ 0 dτ J |xi−xj | (2Jτ )e −( γc 2 −iδ)τ (A7)
and the cavity decay rate γ c appears through the replacement ω c → ω c − iγ c /2. The final integral can now be evaluated with the help of
∞ 0 dτ J m (aτ )e −bτ = 1 √ a 2 + b 2 a b + √ a 2 + b 2 m ,(A8)
and we obtain
A ij = g 2 e iK|xi−xj ] 4J 2 − δ + i γc 2 2 ,(A9)
where K is given in Eq. (7). Finally, since A ij = A ji we can regroup the individual terms into the form given in Eq. (4), where we identify Γ ij = 2Re{A ij } and U ij = 2Im{A ij }. The derivation of the master equations relies on the validity of the Born-Markov approximation, which requires that the kernel in Eq. (A7) either decays faster or oscillates faster than the system evolution time set by the coupling ∼ g. For a single atom this condition is satisfied as long as g |ṽ g (δ)| and by assuming in addition that γ a |ṽ g (δ)|, we can also add to Γ ii the bare atomic decay, without influencing the coupling to the waveguide.
For multiple atoms the Bessel function J |xi−xj | (2Jτ ) reaches its maximum at a finite time
τ ≈ |x i − x j | 2J ,(A10)
which reflects the minimal time it takes a photon to propagate between the atoms. More generally, for the validity of a time-local master equation for N a -atoms with spacing ∆x we must ensure that the maximal retardation time τ R ∼ (N a − 1)∆x/|ṽ g (δ)| is short compared to the system evolution determined by the single-atom spontaneous-emission time Γ −1 with Γ = 2g 2 /|ṽ g (δ)| [see Eqs. (10) and (A9)]. This yields
g |ṽ g (δ)| (N a −1)∆x (A11)
as a slightly more stringent condition for large systems. See also Ref. [23,65].
Appendix B: Single-photon bound states with a single atom
In this Appendix we review the derivation of the eigenvalue equation (11) for the bound states in the case N e = 1 and a single atom located at position x a . In particular, this will provide the basis to derive the analogous results in the multi-atom case. In a frame rotating with frequency ω c , Hamiltonian (1) can be expressed in the momentum space as
H= − 2J k cos(k)a † k a k + Na n=1 δ|e n e| + g √ N Na n=1 k a † k σ n − e ikxn +a k σ n + e −ikxn .(B1)
A state in the single-excitation sector has the form (we set σ ± ≡ σ 1 ± )
|φ = b σ + + k c k a † k |g, 0 .(B2)
Plugging this ansatz into the Schrödinger equation H|φ = E|φ yields the coupled equations
b(E − δ) = g √ N k c k e −ikxa , c k (E + 2J cos k) = g √ N b e ikxa .(B3)
Using the second equation to eliminate c k in the first one, we end up with
E − δ = Σ 1 (E) ,(B4)
where the self-energy Σ 1 (E) (in the continuous limit) is given by
Σ 1 (E) = 1 2π π −π dk g 2 E + 2J cos k = g 2 E 1 − 4J 2 E 2 ,(B5)
where in the last identity we calculated the integral explicitly using that |E| > 2J [58]. Replacing the selfenergy in Eq. (B4) we end up with Eq. (11) in the main text. This equation has two real solutions E ± , where E + (E − ) lies above (below) the continuum E ∈ [−2J, 2J].
The corresponding bound states can be worked out with the help of Eq. (B3) as
|φ ± = b(E ± ) σ + + 1 √ N k ge ikxa E ± + 2J cos k a † k |g, 0 ,
(B6) where, using that the state must be normalized,
b(E) = 1 + g 2 E 2 1− 4J 2 E 2 3 2 − 1 2 .
(B7)
In the real space, the bound state reads
|φ ± = b(E ± ) σ + + g x (∓1) |x−xa| e − |x−xa | λ a † x E ± 1− 4J 2 E 2 ± |g, 0 .
(B8) Exploiting again the normalization of |φ ± , one eventually ends up with Eq. (12) defined in terms of the photonic operators a † λ and the mixing angle θ, defined in Eqs. (13) and (14), respectively.
Appendix C: Single-photon bound states with many atoms
For N e = 1, but considering multiple atoms the bound states can be derived by exploiting the mirror symmetry of the system. For the sake of argument, here we focus on bound states below the continuum, i.e., such that E < −2J [66]. In accordance with the mirror symmetry, we define the pair of collective atomic operators
S s=e,o = Na n=1 (±1) |n+1| σ n − ,(C1)
where the + (-) sign holds for s = e, o. In the case N a = 2, the operators (C1) reduce to the (unnormalized) symmetric and antisymmetric combinations of σ 1 − and σ 2 − . Based on this definition, here we look for bound states of the form
|φ (Na) s = b S † s + k c k a † k |g, . . ., g, 0 .(C2)
If N a > 2, the bound states defined in Eq. (C2) are those whose energies form the boarders of the dressed-state metabands (see Fig. 12). Imposing the ansatz (C2) to be an eigenstate of Hamiltonian (B1) with eigenvalue E yields an eigenvalue equation analogous to Eq. (B4) with the self-energy now given by
Σ s (E) = n (±1) |n+1| 1 2π π −π dk g 2 e ik(xn−xa) E + 2J cos k = Σ 1 (E) f Na,s (E) ,(C3)
where Σ 1 (E) is the single-atom self-energy in Eq. (B5) and
f Na,s (E) = n (±1) |n+1| e − |xn −xa| λ ,(C4)
with λ = λ(E) being the same energy function as in Eq. (15). We introduced the atomic position x a that set the choice of placing the atomic ensemble in the array. As in the one-atom case, in deriving the last identity of Eq. (C3) we used E < −2J to calculate the integral over k through standard methods [58]. The self-energy, hence the eigenvalue equation, is thus determined by the function f Na,s (E) in Eq. (C4). We will analyze this function now in more detail for the paradigmatic cases N a = 2 and N a 1, which are the cases considered in Sec. V.
Two atoms
For N a = 2 and choosing x a = x 1 , Eq. (C4) simply yields
f 2,e = e − ∆x 2λ cosh ∆x 2λ , f 2,o = e − ∆x 2λ
sinh ∆x 2λ , (C5) for the even-and odd-parity states, respectively (recall that ∆x = |x 1 −x 2 |). This provides the self-energy and thus the eigenvalue equation for the energies E −,s [see Eq. (C3)]. The corresponding bound states can be derived in terms of E −,s in a way essentially analogous to that in Appendix B. For bound states below the continuum, this gives
|φ −,s = b(E −,s ) σ 1 + ± σ 2 + + 1 √ N k g(e ikx1 ± e ikx2 ) E −,s + 2J cos k a † k |g 1 , g 2 , 0 ,(C6)
where function b(E) follows from the normalization constraint and reads
b(E) = 2 + g 2 N 2 s 2J 2 sinh 2 1 λ − 1 2 ,(C7)
with N s defined by Eq. (49). In position space, state (C6) reads
|φ −,s = b(E −,s ) σ 1 + ± σ 2 + + g E −,s 1 − 4J 2 E 2 −,s × x e − |x−x 1 | λ ± e − |x−x 2 | λ a † x |g 1 , g 2 , 0 . (C8)
In analogy with the single-atom case, one can arrange such bound states in the form (47) in terms of the polaritonic operators (48) and the mixing angle (50). Regarding bound states above the band, one can follow an analogous reasoning by taking into account the different definition of operators S s [66]. While this affects the expression of the bound states, namely the counterparts of Eqs. (C6) and (C8), Eq. (C3) for the self-energy turns out to be unaffected. The self-energies (C3) thereby hold both above and below the continuum.
At this time, we also mention that -while our approach based on the collective atomic operators (C1) is devised so as to easily tackle the N a 1 limit -in the N a = 2 case an equivalent method would be to block-diagonalize H with the blocks corresponding to even-and odd-parity sectors of the entire single-excitation Hilbert space (including the field). In the even (odd) subspace, the problem is reduced to an effective single atom coupled to the cosine-shaped (sine-shaped) field modes. This approach was followed in Ref. [67], where however the authors focused on bound states in the continuum (BIC) [68] only. The effective Hamiltonian in each parity-definite subspace differs from the Fano-Anderson model in Eq. (B1) (case N a = 1) in that the atom-mode couplings are kdependent. Such "coloured" Fano-Anderson model was first investigated in Ref. [69] in the case of sine-shaped couplings.
2.
Na 1 atoms
In the limiting case of a very large number of equispaced atoms, N a 1, function (C4) can be written in a compact form, by setting x a = x Na/2 [70], since it reduces to a geometric series. By expressing in Eq. (C4) each atomic position as x n = x a + (n − N a /2)∆x, we end up with . We confirmed numerically that the metaband-edge levels (see Fig. 12) for growing N a converge to the numerical solutions of the eigenvalue equation E − δ = Σ 1 (E)f Na 1,s (E). Specifically, above the continuum (E > 2J) the solution for s = e (s = o) gives the upper (lower) metaband edge, while below the continuum s = e (s = o) corresponds to the lower (upper) metaband edge.
Appendix D: Multi-atom bound-states
Here we address a number of properties of the multiatom bound-state levels in the case N a = 2 and N a 1 with the goal of proving the salient features of the energy spectra in Fig. 10 and 12 discussed in the main text.
Na = 2
As discussed in App. C, the bound-state levels are the solutions of the equation E − δ = ∆ s (E) in the domain |E| > 2J. Using Eqs. (C3), (C5) and (15), the self-energy function explicitly reads
Σ s (E)= g 2 E 1− 4J 2 E 2 1± |E| 2J − |E| 2J 1− 4J 2 E 2 |x1−x2| ,
(D1) where as usual the + (−) sign holds for s = e (s = o). The corresponding expression for E < −2J follows straightforwardly from the fact that Σ s (E) is an odd function of E.
Below the continuum, i.e., for E < −2J, both Σ e (E) and Σ o (E) monotonically decrease with E [cf. Eq. (D1)]. Thereby, if the value taken by the linear function y = E − δ at E = −2J lies above Σ s (−2J) then a single bound state (for fixed s) of energy E −,s < −2J certainly occurs. This condition thus explicitly reads −2J − δ > Σ s (−2J). This is always fulfilled for s = e given that Σ e (−2J) = −∞. Instead, for s=o, by calculating Σ o (−2J) = −g 2 |x 1 − x 2 |/(2J) [see Eq. (D1) for E→(−2J) + ], the above condition results in
g > 2J 1 + δ 2J |x 1 − x 2 | = g m |x 1 − x 2 | ,(D2)
where g m is the same as in Eq. (53). Hence, as discussed in Sec. V, both E −,e and E −,o solutions exist for any interatomic distance when g > g m . If instead g < g m , at the critical distance |x 1 −x 2 | = (g m /g) 2 As for bound states above the continuum, a similar reasoning can be carried out. Recalling that Σ s (−E) = −Σ s (E), we have Σ o (2J) = g 2 |x 1 − x 2 |/(2J) and Σ e (2J) = +∞ with both functions Σ s (E) mononically decreasing with E for E > 2J. The condition for the existence of a bound state will now read 2J − δ < Σ s (2J). Again, it is always fulfilled when s = e since Σ e (2J) diverges to +∞. Instead, for s = o the threshold condition for δ < 2J reads
g > 2J 1 − δ 2J |x 1 − x 2 | ,(D3)
which is analogous to Eq. (D2) but the replacement δ → −δ in the expression of g m . For δ > 2J both levels E +,s exist. Moreover, since now Σ o (E) < Σ e (E) we have E +,o < E +,e .
To summarize, outside the continuum, a pair of bound states of even symmetry and energies E ±,e always exist, one above and one below the photonic band. At most two further odd-symmetry bound states of energies E ±,o may be present as well, depending on the values of g, |x 1 − x 2 | and δ. Note that, for |δ| < 2J, the critical coupling strengths appearing in Eqs. (D2) The analysis for N a 1 proceeds similarly to the N a = 2 case. The explicit self-energy functions Σ s=e,o (E) are obtained from Eqs. (C3), (C9) and (15). Like in the 2atom case, Σ e (E) > Σ o (E) [Σ e (E) < Σ o (E)] for E > 2J (E < −2J) with Σ e (E) diverging to +∞ and −∞ for E → (2J) + and E → (−2J) − , respectively. Instead, Σ o (±2J) = ±g 2 ∆x/(4J). Accordingly, the same geometrical criterion as in the previous subsection entails that the conditions for the existence of
PACS numbers: 42.50.Pq, 42.50 Nn, 03.65.Ge
. 1. (a) Sketch of a strongly coupled waveguide QED setup with bound atom-photon dressed states around the atomic locations. The slow-light waveguide can be modelled as a large array of coupled optical resonators with nearest-neighbor coupling J. (b) Band structure of the waveguide without atoms.
FIG. 2 .
2(a) Correlated decay rates Γij against the (discrete) interatomic distance |xi − xj| and (b) coherent dipole-dipole interactions Uij versus |xi − xj| for different detunings δ = ωa − ωc. The solid lines are a guide to the eye obtained from a continuous interpolation of Eq. (6)
FIG. 3 .
3(a) Atomic population p + a = cos 2 (θ+) in the upper bound state as function of the coupling constant g and the atom-field detuning δ. (b) The width of the photonic wavepacket in the upper bound state, λ+, is plotted as a function of g and for three different detunings δ.
FIG. 4 .
4Atomic excitation spectrum Sa(ω) (in logarithmic scale) as function of g and for an atom-cavity detuning (a) δ = 0 and (b) δ = 2J. The dotted lines show the bound-state energies E± in the absence of loss, while the dashed lines correspond to the waveguide band edges. In either case, we have set γa/(2J) = 0.1 and γc/(2J) = 0.2.
FIG. 5 .
5Dependence of the atomic excitation spectrum Sa(ω) near the band edge and for (a) δ = 0 and (b) δ = 2J. In (a) the values g/(2J) = 0.3 and γa/(4J) = 0.02 and in (b) the values g/(2J) = 0.2 and γa/(4J) = 0.05 have been assumed and in both cases the spectrum is plotted for different cavity decay rates γc.
FIG. 6 .
6y |g, 0 .(32) By assuming that the atom is located at x a = 0 the inversion symmetry of the Hamiltonian and the bosonic symmetry of the wavefunction require u(x, y) = u(y, x), u(−x, y) = u(x, y) and b(−x) = b(x). This ansatz leads Sketch of the single-and two-excitation spectrum in a finite-bandwidth waveguide coupled to an atom for δ = 0. See main text for more details.
FIG. 7 .
7The Ne-photon bound-state energies E (Ne) − obtained from a variational approach are plotted for Ne = 1, . . . , 8 in descending order and for (a) δ = 0 and (b) δ = −2J. The dashed lines in the insets show the exact numerical results for Ne = 2 and Ne = 3.
|Ψ(FIG. 8 .
8Ne) − = cos(θ)σ + A † Ne−1 − sin(θ)B † Ne |g, 0 . (40) Sketch of the first three photonic wavefunctions that appear in the variational ansatz, Eq. (41), for the multiphoton bound states. Here, we have set g = 0.6, (a) δ = 0 and (b) δ = −2J − 0 + . Figures (c) and (d) show the exponential decay lengthλN e as a function of g, for Ne = 1, 2, and 3 photons and for δ = 0 and δ = −2J − 0 + , respectively. The dotted line shows the result forλ obtained numerically for the case Ne = 2. In Figs. (e) and (f) the atomic population pa = cos 2 (θ(E Ne − )) is plotted against g for δ = 0 and δ = −2J − 0 + , respectively.
FIG. 9 .
9The nonlinearity parameter ∆ nl (Ne) as defined in Eq. (44) is plotted for Ne = 2 and different atom-photon detunings δ.
FIG. 10 .
10The bound-state energy levels E−,s (left-column panels) and the corresponding atomic populations pa = cos 2 (θ(E−,s)) (right-column panels) are plotted as a function of the interatomic distance for the case of two atoms and for three representative values of g/(2J). For all plots δ = 0 is assumed. For comparison, in each panel the dashed line indicates the corresponding bound-state energy or atomic population for a single atom.
FIG. 11 .
11Spatial profile of the photonic wave function us(x) = x|Φs corresponding to the even (red solid line) and odd (green dashed line) lower band bound states in the case of two atoms for different coupling strengths and interatomic distances. For all plots δ = 0 is assumed. both E −,e and E −,o solutions exists for all |x 1 − x 2 | ≥ 1.
FIG. 12 .
12Single-excitation energy spectrum in the case of Na = 40 equally spaced atoms as a function of the atomic nearest-neighbour distance ∆x. Note the appearance of upper and lower metabands of bound states. For this plot δ/(2J) = 0.6 and g/(2J) = 1 have been assumed.
ACKNOWLEDGMENTS
The authors thank Y. Minoguchi, F. Fratini, H. Pichler, F. Lombardo and G. M. Palma for stimulating discussions. This work was supported by the European Project SIQS (600645), the COST Action NQO (MP1403), and the Austrian Science Fund (FWF) through SFB FOQUS F40, DK CoQuS W 1210 and the START grant Y 591-N16. F.C. acknowledges support from Italian PRIN-MIUR 2010/2011MIUR (PRIN 2010-2011). Work at ICFO has been supported by the ERC Starting Grant FOQAL, the MINECO Plan Nacional grant CANS, and the MINECO Severo Ochoa Grant SEV-2015-0522.
the solution E −,o merges with the continuum, i.e., E −,o = −2J, and it no longer exists for |x 1 − x 2 | < x m (see Fig. 10). Moreover, note that in the light of the geometrical criterion given above if E −,o exists then E −,o > E −,e since Σ o (E) > Σ e (E) [cf. Eq. (D1)]. Eq. (D2) holds for δ > −2J. For δ ≤ −2J, E −,o always exists since −2J − δ is positive while Σ o (2J) is negative anyway.
E +,o and E −,o are the same as in Eqs. (D2) and (D3), respectively, apart from the factor √ 2 on either right-hand side. The same factor thereby appears in Eqs. (D4)-(D6), which are now interpreted as the conditions for establishing whether none [Eq. (D4)], only one [Eq. (D5)] or both [Eq. (D6)] of the metabands merge with the photonic band.
and (D3) are different, which entails that three cases are possible: E +,o exists while E −,o does not (or vicecersa), E ±,o both exist, E ±,o both do not exist. Combining together Eqs. (D2) and (D3), the conditions for these three cases to occur, for |δ| ≤ 2J, read ⇔ neither E +,o nor E −,o exist . (D6)g >
2J 1+
|δ|
2J
√
|x1−x2|
⇔ both E +,o and E −,o exist ,
(D4)
2J 1−
|δ|
2J
√
|x1−x2|
<g<
2J 1+
|δ|
2J
√
|x1−x2|
⇔ only E sgn (δ),o exists, (D5)
g <
2J 1−
|δ|
2J
√
|x1−x2|
2. Na
1
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| []
|
[
"The Reach of Threshold-Corrected Dark QCD",
"The Reach of Threshold-Corrected Dark QCD"
]
| [
"Jayden L Newstead \nDepartment of Physics\nArizona State University\n85287-1504TempeAZ\n",
"Russell H Terbeek \nDepartment of Physics\nArizona State University\n85287-1504TempeAZ\n"
]
| [
"Department of Physics\nArizona State University\n85287-1504TempeAZ",
"Department of Physics\nArizona State University\n85287-1504TempeAZ"
]
| []
| We consider a recently-proposed model which posits the existence of composite dark matter, wherein dark "quarks" transforming as fundamentals under an SU (3) d gauge group undergo a confining phase and form dark baryons. The model attempts to explain both the O(1) relic density ratio, Ω dark /Ω baryon ∼ 5.4, as well as the asymmetric production of both dark and baryonic matter via leptogenesis. Though the solution of β functions for SU (3)c and SU (3) d constitutes the main drive of the model, no threshold corrections were taken into account as the renormalization scale crosses the mass threshold of the heavy new fields in the model. We extend this work by explicitly calculating the threshold-corrected renormalization-group flow for the theory using an effective-field matching technique. We find that the theory has a much wider range of applicability than previously thought, and that a significant fraction of models (defined by the number of fields contained therein) is able to account for the observed relic density. | 10.1103/physrevd.90.074008 | [
"https://arxiv.org/pdf/1405.7427v1.pdf"
]
| 118,510,360 | 1405.7427 | d2995d6a9ba3ac1f70d732d14cd235e5382b77bc |
The Reach of Threshold-Corrected Dark QCD
Jayden L Newstead
Department of Physics
Arizona State University
85287-1504TempeAZ
Russell H Terbeek
Department of Physics
Arizona State University
85287-1504TempeAZ
The Reach of Threshold-Corrected Dark QCD
(Dated: May 2014)
We consider a recently-proposed model which posits the existence of composite dark matter, wherein dark "quarks" transforming as fundamentals under an SU (3) d gauge group undergo a confining phase and form dark baryons. The model attempts to explain both the O(1) relic density ratio, Ω dark /Ω baryon ∼ 5.4, as well as the asymmetric production of both dark and baryonic matter via leptogenesis. Though the solution of β functions for SU (3)c and SU (3) d constitutes the main drive of the model, no threshold corrections were taken into account as the renormalization scale crosses the mass threshold of the heavy new fields in the model. We extend this work by explicitly calculating the threshold-corrected renormalization-group flow for the theory using an effective-field matching technique. We find that the theory has a much wider range of applicability than previously thought, and that a significant fraction of models (defined by the number of fields contained therein) is able to account for the observed relic density.
I. INTRODUCTION
While the basic makeup of the Universe is well known, only 5% is composed of known particles. The remaining portion is 27% dark matter (DM) and 68% dark energy [1]; however, the labeling of these latter components elucidates the extent of our understanding of them. We know the DM exists because of its gravitational phenomena, observed across a wide range of length scales, yet it cannot be observed directly. The success of the standard model (SM) of particle physics behooves us to attempt to fit the DM into this paradigm. By necessity, the particles which make up the DM must (at most) interact weakly with the SM particles, or else experimental efforts to detect them would have already been successful.
There are two core ideas for how the DM was produced in the early universe. The dominant paradigm is that of a thermally-produced weakly-interacting massive particle (WIMP). The WIMP miracle supposes that a particle of weak scale mass and interaction strength would freeze out of the thermal bath and annihilate sufficiently to leave behind the correct relic density. Since ordinary matter has an observed baryon/anti-baryon asymmetry, a more nuanced production mechanism is thus required to account for this fact. Given the distinct origins of the dark and ordinary matter, one has no reason to expect their densities to be of the same order of magnitude. Herein lies the second paradigm: given the similarity of the densities, it is not unreasonable to suppose that the DM may have a common origin with ordinary matter. A broad class of such models, known as asymmetric dark matter (ADM) models, attempt to simultaneously solve the baryon asymmetry problem while tying the abundance of DM to that of ordinary matter (for a recent review of ADM see [13,15]). In doing so the dark sector also becomes asymmetric; in some implementations, this is achieved by hiding the B − L charge in. In recent years a plethora of mechanisms for generating a baryon and dark matter asymmetry have been developed, e.g., cogenesis [8,11], darkogenesis [14], and hylogenesis [10]. More precisely, each mechanism ties the number-density of DM and ordinary particles together, generally up to an O(1) factor (this is not necessarily true in co-genesis models). These models therefore require the DM particle masses to be ∼ 1-10 GeV, but, they often offer no explanation for why this scale should also be the same order as the proton mass.
The scale of the proton mass is effectively set by the QCD confinement scale; thus, in order to justify a commensurate mass of the ADM particle, Bai and Schwaller posited that there are related strong dynamics in the dark sector [3]. In this scenario, the DM particle is the lightest stable hadron of a new 'dark' SU(3) group which is coupled to QCD above some scale M . This model can explain the baryon asymmetry through a leptogensis mechanism; to obtain the correct ratio of energy densities, one must then have the dark confinement scale Λ dQCD ∼ Λ QCD . This can be naturally achieved by decoupling the heavy bi-fundamental fields at some scale M (of the terascale or beyond) and requiring this scale to be at the infrared fixed point (IRFP) of both decoupled beta functions. Below M only low-mass quarks transforming fundamentally under either SU (3) c or SU (3) d influence the running of the beta function. We do not attempt here to re-derive the formalism employed in [3]; we refer the interested reader to that paper for a detailed examination of the leptogenesis-induced baryogenesis mechanism and further implications for LHC phenomenology.
In this paper we aim to extend the analysis of [3] by including threshold corrections to the running of the beta function. Utilizing the space of all possible models (characterized by their field content and mass scale), we constrain the available parameter space by requiring the running of α c to fit the latest experimental data. We then show that the conclusions of [3] are in fact one limiting case of the dark-QCD concept, with a much broader range of applicability.
II. TWO-LOOP β-FUNCTIONS
To the second order in the loop expansion, the βfunction for SU (3) c is given by
β c = g 3 c 16π 2 4 3 T (R f )(n fc + N d n fj ) + 1 3 T (R s )(n sc + N d n sj ) − 11 3 C 2 (G c ) + g 5 c (16π 2 ) 2 10 3 C 2 (G c ) + 2C 2 (R f ) T (R f )2(n fc + N d n fj ) + 2 3 C 2 (G c ) + 4C 2 (R s ) T (R s )(n sc + N d n sj ) − 34 3 C 2 2 (G c )) + g 3 c g 2 d (16π 2 ) [2C 2 (R f )T (R f )2N d n fj + 4C 2 (R s )T (R s )N d n sj ],(1)
with the analogous β-function for the dark coupling g d obtained by swapping the indices c ↔ d. As in [3], the group theory factors are included for complete generality, but we ultimately want the matter fields to transform as fundamentals under SU (N ) c × SU (N ) d for N = 3, and the gauge fields to transform as adjoints. The trace over generators in the fundamental representation is given by
T (R f ) = T (R s ) = 1/2; the quadratic Casimirs of the adjoint representation are C 2 (G c,d ) = N c,d , and those of the fundamental representation are C 2 (R f ) = C 2 (R s ) = (N 2 c,d − 1)/(2N c,d ).
III. METHODS
A given model is completely characterized by its field content:
{n fc , n f c,h , n f d , n fj , n sc , n s d , n sj },(2)
where the indices respectively enumerate the light colored quarks (always 6, as in the Standard Model), heavy colored quarks (transforming under SU (3) c just as conventional quarks do, but with significantly higher masses), dark quarks, bi-fundamental quarks, colored scalars, dark scalars, and bi-fundamental scalars. Bifundamental fields transform fundamentally under the product group SU (3) c × SU (3) d , as the name suggests. For clarity, we use the phrasing e.g. "the field index n f d " rather than "the number of dark-fundamental quarks n f d ." In order to populate all possible models for analysis, we begin by solving the coupled β functions of Eqn. (1) while setting all elements of Eqn.
(2) to zero, save n f c,h . We then determine the maximal field index n max f c,h that would keep the one-loop term of Eqn. (1) negative; if this upper bound were exceeded, the resulting theory could not reasonably be called QCD. Next, we set n f c,h = 0, and vary n fj in likewise fashion. After iterating through the possible index values of all fields transforming non trivially under SU (3) c , we then switch to β d and repeat the process with fields fundamental under SU (3) d . This procedure creates approximately 32 million models for further study. Although they meet the most minimal definition of an acceptable QCD-like theory, further cuts are necessary in order to produce models capable of a GeV-scale DM candidate. Next, we require that the models satisfy:
1. β c,d ≤ 0 for all energies below the IRFP;
2. The value of the color fine structure constant (α c ) may not exceed the experimentally-observed value measured at the highest energy scale currently available; at the time of writing, the CMS collaboration has measured α c (896 GeV) = 0.0889 ± 0.0034 [6];
3. The value of the dark fine structure constant (α d ) may not exceed the Cornwall-Jackiw-Tomboulis bound [9] of π/4 when evaluated at the IRFP.
Upon applying this set of constraints, the number of candidate models reduces to 87,286. Given a set of models with this minimal consistency, it is now possible to begin applying matching conditions, as discussed in [7], from the low-energy effective field theory (EFT) (in this case, just the SM) to the full theory.
In MS-like renormalization schemes, the Appelquist-Carazzone decoupling theorem [2] does not generally apply to quantities that do not represent physical observables; examples include coupling constants and β functions in the absence of ultraviolet completions. This means that heavy fields circulating in loops do not necessarily decouple from such quantities at energies below their mass scale. The solution is to use the EFT formalism, as in [7], and elaborated in [4,12]. In their original studies, Chetyrkin et al. [7] considered, among other things, the decoupling relations that result from integrating out a single quark flavor from an n f -flavor theory, and requiring consistency with the resulting (n f − 1)-flavor EFT that accurately describes the dynamics at lower energies. This consistency condition is
α EFT c (µ 0 ) = ζ 2 c α c (µ 0 ),(3)
where µ 0 is the IRFP scale to be solved for, and the decoupling function for a single heavy quark flavor at oneloop order is
ζ 2 c = 1 − α c (µ) 6π ln µ 2 M 2 .(4)
The work of [7] extends to O(α 3 c ); and at this order in perturbation theory, one must take into account the scheme-dependence of the decoupling function, for instance in MS scheme vs. on-shell scheme. However, since the β functions we use extend to two-loop order, it is only consistent to consider threshold corrections at first order in perturbation theory, and at this order all schemes agree. Furthermore, since we are considering all heavy fields to possess the same mass scale M , it follows that Eqn. (4) ought to be replaced by
ζ 2 c = 1− α c (µ) 6π n f c,h +N d n fj + 1 4 (n sc +N d n s,j ) ln µ 2 M 2 .(5)
When Eqn. (3) is satisfied, one obtains information about the ratio µ 0 /M , rather than M itself. This fact stands in contrast to [3], where, in the absence of threshold effects, the low-energy α c could be evolved directly up to the fixed point determined by solving β c = β d = 0, and the resulting energy scale could then be extracted. This approximation uniquley determines M for a given model. However, when computing the full evolution of the strong couplings to two-loop order and incorporating the necessary threshold/decoupling effects, we find that M is not specified from first principles, and therefore one must consider a continuum of M values. When sampling the range m top ≤ M ≤ 100 TeV, we find that the solutions obtained in [3] are limiting cases of composite DM models with a much broader range of applicability.
In this work, once we specify a model that meets our previously-stated constraints, we randomly select an M uniformly distributed along a logarithmic scale running from m top (taken from [5]) to 100 TeV. This lower limit is set by the non-detection of any novel, fundamental QCD states at energies less than or equal to the top quark mass. Although the upper limit is arbitrary, the distribution of models tends to peak around M ∼ a few TeV (as seen in Fig. 2), and we find that pushing the upper limit much further does not significantly alter the proportion of models with GeV-scale DM candidates.
Once M is fixed, µ 0 may be uniquely determined. One may then independently evolve g c , g d down from the decoupling scale to lower values. Whereas a detailed calculation of the dynamics of chiral symmetry breaking and the resulting hadronization would require a comprehensive lattice study, we follow [3] in adopting the Cornwall-Jackiw-Tomboulis condition for chiral symme-try breaking [9]: when α d (µ)C 2 (R f ) = π/3, or equivalently α d (µ) = π/4, the µ at which this occurs is identified with the chiral symmetry-breaking scale Λ. Working with the low-energy QCD of the Standard Model, one finds the relationship m p ≈ 1.5 Λ. We therefore apply the same relation to learn the approximate dark proton mass, m d , from the chiral symmetry-breaking scale Λ dQCD of dark QCD.
Given that the decoupling scale µ 0 may now be somewhat lower than the scale M , one might inadvertently modify the running of α c at observable scales. We thus employ a χ 2 test to restrict ourselves to models in which measured values of α c are not greatly affected. Using data from CMS (the 31 highest energy data points from [6]) we calculate the figure of merit
t = Σ i (x i − α c,i ) 2 σ 2 i ,(6)
where the x i represent the values of α c we obtain at the energy scale of the ith measurement. The quantity t follows a χ 2 distribution; using this, we calculate a p-value and reject unsatisfactory models at the 95% confidence level.
IV. RESULTS
After performing 20 iterations of the above test per model, and indexing through all models that pass our three criteria, we eventually narrow down the list of candidate solutions from 87,286 to 16,859. It must be noted that this refinement follows from the artificial upper bound of 100 TeV which we set on the universal mass M of all heavy fields. If the upper limit were to be increased further, a model which did not make this most recent cut would have a broader range of M values from which to sample, and one could then find a µ 0 satisfying 0 < µ 0 < M .
When we solve for the fixed-point couplings, α * c,d , we find that there is no correlation whatsoever between the magnitude of the coupling and the precise distribution of field content across the available indices. In fact, for a given number of degrees of freedom N spread across different possible field indices, α * c,d takes on all values in the allowed interval. Even though N and α * are unrelated, there is an argument for the ability of 100 TeVscale particles to influence GeV-scale physics. There exist some models for which α * c is very small, which necessarily means that the coupling of the EFT must be evolved to very large µ 0 in order to meet it. So, to first order in Eqn. (5), one sets α EFT c (µ 0 ) = α * c for very large µ 0 . To next order, one must then satisfy
α EFT c (µ 0 ) = − (α * c ) 2 6π N ln(µ 2 0 /M 2 ),(7)
where N sums over all degrees of freedom that couple to QCD, and still subject to the requirement µ 0 < M .
Taking α EFT c (µ 0 ) = α * c from the previous approximation results in
µ 0 = M exp − 3π N α * c .(8)
Provided that α * c 3πN −1 , Eqn. (8) predicts a natural hierarchy of scales between µ 0 and M . Once SM-and dark-QCD decouple at the scale µ 0 , the evolution of their couplings depends solely on the number of light degrees of freedom transforming fundamentally under one group only. This means that, for 100 TeV M µ 0 , a light baryon-like state can still be achieved even if just a few (n f d 3) matter fields contribute to the beta function, thereby causing α d to run more slowly and trigger chiral symmetry breaking at a lower energy scale. To recapitulate, 1. Low α * c implies large µ 0 ;
2. For α * c 3π N −1 , one finds µ 0 M , allowing for the relevance of very massive fields;
3. Sufficient n f d (from 1 to 3) can evolve α * d slowly from the IRFP value at the high scale to the Cornwall-Jackiw-Tomboulis bound of π/4 at the comparatively low GeV scale.
Below we show the plots for the allowed field content of dark QCD. The blue bars of each histogram indicate the field indices for which the models passed the three criteria outlined in the Section III, whereas the violet bars indicate the field indices corresponding to models for which 1 3
Ω d Ω b obs < Ω d Ω b < 3 Ω d Ω b obs ,(9)
where (Ω d /Ω b ) obs indicates the observed value quoted from the PLANCK collaboration. This factor of three encapsulates in an approximate sense the spread of models that are "near" to the actual relic density ratio observed in the universe today, and how general the reach of dark QCD is in producing candidates that could faithfully replicate the matter density observed in the universe. Though the criterion of Eqn. (9) is not the most precise experimental constraint that could be devised, a more rigorous standard would neglect the fact that the onset of chiral symmetry breaking is a truly non-perturbative phenomenon -whereas we have used a perturbative approximation to derive Λ dQCD -and hence introduces a larger source of error. When we apply Eqn. (9) to the remaining models, we find that 2,578 make the cut, or approximately 15.3% -an O(1) fraction of the total. This suggests an interesting and novel result: for a very diverse range of M , µ 0 , and field indices, the dark QCD theory is capable of producing a DM critical density commensurate with the observed value. Since this result is independent of such idiosyncracies as a precise combination of colored, dark, and bi-fundamental fields, or a narrow range of the M parameter, we are left to conclude that the mechanism itself -a confining SU (3) d gauge group that shares a fixed point with SU (3) c -has some real explanatory reach. The generality of our results bolsters the idea that composite, strongly-interacting DM can provide a natural explanation for the O(1) ratio of dark and baryonic matter densities observed in the universe.
V. CONCLUSIONS
The composition and interactions of dark matter remain a mystery to physics, as does the origin of the matter-antimatter asymmetry. ADM models provide a compelling and elegant way to simultaneously create a matter-antimatter asymmetry and a dark matter component with roughly equal number densities. We have examined in detail one such model, the SU (3) c × SU (3) d theory of [3], in which the relic dark matter candidate is a composite state of "quarks" transforming fundamentally under SU (3) d , and trivially under SU (3) c . Our work has extended the scope of this ADM model in order to include decoupling effects due to heavy fields that influence β c , β d at very high energies. By taking decoupling effects into account at one-loop order, we can relax the restriction M = µ 0 implicit in [3], thereby opening up a much broader range of possible mass values for the candidate fields to sample. We find the remarkable result that thousands of models within the dark-QCD framework are able to produce relic density ratios close to that observed in our own universe. We are therefore led to conclude that ADM models of this type -SU (3) c ×SU (3) d gauge group with an infrared fixed point -can account for the matter density quite naturally, with a broad diversity of different field types involved.
Figure 1 :
1Number of models which pass our three criteria (blue), as well as the subset that comes within a factor of 3 of (Ω d /Ω b ) obs (violet), organized by field index.
Figure 2 :
2Number of models which pass our three criteria (blue), as well as the subset that comes within a factor of 3 of (Ω d /Ω b ) obs (violet), organized by the mass of the heavy fields M and the decoupling scale µ 0 .
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[
"Infinite convolutions of probability measures on Polish semigroups",
"Infinite convolutions of probability measures on Polish semigroups"
]
| [
"Kouji Yano "
]
| []
| []
| This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis-Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.Theorem 4.6, and the study of probability measures with convolution invariance, which will be stated as Proposition 4.5. Theorems 4.6 and 4.9 are based on the product decomposition theorem for completely simple semigroups, which will be called the Rees decomposition and stated as Theorem 2.8. To show that the algebraic decomposition is compatible with a Polish topology, we need Ellis-Żelazko theorem, which will be stated as Theorem 3.2.The Ellis theorem [11](1957) asserts that an algebraic group where the product mapping is separately continuous is a topological group, where the topology is locally compact Hausdorff second countable. It was extended byŻelazko [51](1960) for a completely metrizable topology. The study of infinite convolutions on compact groups dates back to Kawada-Itô [19](1940), which was generalized by Urbanik [49](1957), Kloss [22](1959), and Stromberg [45](1960) and for locally compact groups by Tortrat [48](1964) and Csiszár [7](1966). The convolution invariance Proposition 4.5 is due to Mukherjea [26](1972), which originates from the Choquet-Deny equation [2](1960); for later studies, see [50, 38, 9, 8, 37, 23, 47]. Theorem 4.6 for convolution idempotents is due to Mukherjea-Tserpes [31] (1971); for ealier studies, see Collins [6](1962), Pym [36](1962), Heble-Rosenblatt [14](1963), Schwarz [44](1964), Choy [3](1970), Duncan [10](1970), and Sun-Tserpes [46](1970); see also [12]. Theorem 4.9 for cluster points of infinite convolutions is due to Rosenblatt [39](1960) in the compact case and to Mukherjea [28](1979) in the locally compact case; for studies earlier than [28], see Glicksberg [13](1959), Collins [5](1962), Schwarz [43](1964), Rosenblatt [40](1965), Lin [24](1966), Mukherjea [27](1977), and Mukherjea-Sun [30](1978); for related papers, see [33, 42, 25, 1].This paper is organized as follows. In Section 2 we review the theory of algebraic semigroups. In Section 3 we study the theory of Polish semigroups, where the Ellis-Żelazko theorem is proved and utilized. Section 4 is devoted to the convolution factorization theorems of convolution idempotents and of cluster points of infinite convolutions.Algebraic semigroupWe say that a non-empty set S is a semigroup if it is endowed with multiplication S × S ∋ (a, b) → ab ∈ S (2.1) | 10.1214/22-ps6 | [
"https://arxiv.org/pdf/2108.12588v1.pdf"
]
| 237,353,095 | 2108.12588 | 15c6099256d6ad70d2318985f1a84c66479ba423 |
Infinite convolutions of probability measures on Polish semigroups
Aug 2021
Kouji Yano
Infinite convolutions of probability measures on Polish semigroups
Aug 2021and phrases: Polish semigroupRees decompositionEllis-Żelazko theoremconvolution idem- potentinfinite convolution AMS 2010 subject classifications: 60B15 (60F0560G50)
This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis-Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.Theorem 4.6, and the study of probability measures with convolution invariance, which will be stated as Proposition 4.5. Theorems 4.6 and 4.9 are based on the product decomposition theorem for completely simple semigroups, which will be called the Rees decomposition and stated as Theorem 2.8. To show that the algebraic decomposition is compatible with a Polish topology, we need Ellis-Żelazko theorem, which will be stated as Theorem 3.2.The Ellis theorem [11](1957) asserts that an algebraic group where the product mapping is separately continuous is a topological group, where the topology is locally compact Hausdorff second countable. It was extended byŻelazko [51](1960) for a completely metrizable topology. The study of infinite convolutions on compact groups dates back to Kawada-Itô [19](1940), which was generalized by Urbanik [49](1957), Kloss [22](1959), and Stromberg [45](1960) and for locally compact groups by Tortrat [48](1964) and Csiszár [7](1966). The convolution invariance Proposition 4.5 is due to Mukherjea [26](1972), which originates from the Choquet-Deny equation [2](1960); for later studies, see [50, 38, 9, 8, 37, 23, 47]. Theorem 4.6 for convolution idempotents is due to Mukherjea-Tserpes [31] (1971); for ealier studies, see Collins [6](1962), Pym [36](1962), Heble-Rosenblatt [14](1963), Schwarz [44](1964), Choy [3](1970), Duncan [10](1970), and Sun-Tserpes [46](1970); see also [12]. Theorem 4.9 for cluster points of infinite convolutions is due to Rosenblatt [39](1960) in the compact case and to Mukherjea [28](1979) in the locally compact case; for studies earlier than [28], see Glicksberg [13](1959), Collins [5](1962), Schwarz [43](1964), Rosenblatt [40](1965), Lin [24](1966), Mukherjea [27](1977), and Mukherjea-Sun [30](1978); for related papers, see [33, 42, 25, 1].This paper is organized as follows. In Section 2 we review the theory of algebraic semigroups. In Section 3 we study the theory of Polish semigroups, where the Ellis-Żelazko theorem is proved and utilized. Section 4 is devoted to the convolution factorization theorems of convolution idempotents and of cluster points of infinite convolutions.Algebraic semigroupWe say that a non-empty set S is a semigroup if it is endowed with multiplication S × S ∋ (a, b) → ab ∈ S (2.1)
Introduction
As a natural generalization of random walks on an integer lattice, the theory of infinite convolutions of probability measures on topological semigroups has been extensively studied and widely applied to various problems. For this theory, there are celebrated textbooks Rosenblatt [41], Mukherjea-Tserpes [32] and Högnäs-Mukherjea [16], which include a lot of applications of the theory; see also Mukherjea's lecture notes [29] for applications to random matrices, and Ito-Sera-Yano [17] for applications to the problem of resolution of σ-fields.
The aim of this paper is to help the reader to gain the basic knowledge of this thoery conveniently. We mainly follow [16] and we make some modifications on the proofs. For a potential application, we develop the theory for topological semigroups with a Polish topology, while the textbooks [41,32,16] deal with those with a locally compact Hausdorff second countable topology.
The goal of this paper is the convolution factorization theorem of cluster points of infinite convolutions, which will be stated as Theorem 4.9. The key to the proof is the convolution factorization theorem of convolution idempotents, which will be stated as (1) Graduate School of Science, Kyoto University. which is associative, i.e., (ab)c = a(bc), a, b, c ∈ S.
(2.2)
For two subsets A and B of S, we denote their product by
AB = {ab : a ∈ A, b ∈ B}. (2.3)
We write A 1 = A and A n = A n−1 A for n ≥ 2. We sometimes identify an element a ∈ S with the singleton {a}; for instance, aS = {a}S = {ab : b ∈ S}. An element e ∈ S is called identity if
xe = ex = x, x ∈ S. (2.4)
It is obvious that identity is unique if it exists. For a semigroup S with identity e, we say that y ∈ S is called the inverse of x ∈ S if xy = yx = e. It is obvious that the inverse of an element x ∈ S is unique if it exists. A group is a semigroup S with identity such that all elements have their inverses.
Left and right simplicity
Let S be a semigroup. (ii) S = S 0 a for all a ∈ S.
Proof. Suppose S is a minimal left ideal. Since S 0 a for a ∈ S is a left ideal of S 0 contained in S, we have S = S 0 a by minimality.
Suppose S = S 0 a for all a ∈ S. Let I be a left ideal of S 0 such that I ⊂ S. For any a ∈ I, we have S = S 0 a ⊂ S 0 I ⊂ I, which shows that S is a minimal left ideal of S 0 . The proof of Lemma 2.2 is almost the same as that of Lemma 2.1, and so we omit it. (iv) There exists a semigroup S 0 such that S is a minimal left ideal of S 0 .
(v) For any a, b ∈ S, the equation xa = b has at least one solution x ∈ S.
Proof. [(i) ⇒ (ii)] Suppose that S is a left ideal of a semigroup S 0 and let I be a left iedal of S 0 such that I ⊂ S. Then SI ⊂ S 0 I ⊂ I, and so I is a left ideal of S. Since S is left simple, we have I = S, which shows that S is a minimal left ideal of S 0 .
[(ii) ⇒ (iii) ⇒ (iv)] These are obvious.
[(iv) ⇒ (i)] Suppose that S is a minimal left ideal of S 0 and let I be a left ideal of S. Since S 0 SI ⊂ SI ⊂ I ⊂ S, we see that SI is a left ideal of S 0 with SI ⊂ S. Hence SI = S by minimality. Since I ⊂ S = SI ⊂ I, we have I = S, which implies that S is left simple.
[(iii) ⇒ (v)] This is obvious by S ⊂ Sa. [(v) ⇒ (iii)] Let a ∈ S. Then we have S ⊂ Sa by (v). Since S is a semigroup, we have Sa ⊂ S. Hence we have S = Sa.
For the simplicity, we have the following. (iv) There exists a semigroup S 0 such that S is a minimal ideal of S 0 .
(v) For any a, b ∈ S, the equation xay = b has at least one solution (x, y) ∈ S × S.
The proof of Lemma 2.4 is almost the same as that of Lemma 2.3, and so we omit it.
Proposition 2.5. A semigroup S which is both left and right simple is a group.
Proof. Let a ∈ S. By Lemma 2.3, we have ea = a for some e ∈ S. For any x ∈ S, we have x = ay for some y ∈ S, and so we have ex = eay = ay = x. Similarly, there exists e ′ ∈ S such that xe ′ = x for all x ∈ S. Then we obtain e ′ = ee ′ = e, and thus e is identity of S.
Let x ∈ S. By Lemma 2.3, we have xy = e and y ′ x = e for some y, y ′ ∈ S. Since y ′ = y ′ e = y ′ xy = ey = y, we see that y is the inverse of x.
Left and right groups
Let S be a semigroup. An element e ∈ S is called an idempotent if e 2 = e. We denote the set of all idempotents of S by Proof. Suppose S is right cancellative and let e ∈ E(S). Then xee = xe implies xe = x.
E(S) = {e ∈ S : e 2 = e}
Suppose S is left simple and let e ∈ E(S). By Lemma 2.3, we have S = Se, which yields that xe = x for all x ∈ S. Proof. [(i) ⇒ (ii)] Let e ∈ E(S) be fixed. By Lemma 2.6, we see that e is a right identity.
Suppose xa = ya. By Lemma 2.3, we have ba = e for some b ∈ S. We then have abab = aeb = ab, so that ab ∈ E(S) and ab is a right identity. We then obtain x = xab = yab = y.
[(ii) ⇒ (iii)] Existence follows from left simplicity and Lemma 2.3. Uniqueness follows from right cancellativity.
[(iii) ⇒ (i)] By (iii), we have S = Sa for all a ∈ S, which shows by Lemma 2.3 that S is left simple.
Let a ∈ S and take e ∈ S such that ea = a by (iii). Then we have e 2 a = ea = a, which leads to e 2 = e by right cancellativity.
Rees decomposition
Let S be a semigroup. An idempotent e ∈ E(S) is called primitive if
ex = xe = x ∈ E(S) implies x = e.
(2.6)
We say that S is completely simple if S is simple and contains a primitive idempotent. Then the following assertions hold:
(i) LG = Se is a left group and GR = eS is a right group.
(ii) RL ⊂ G and eL = Re = {e}.
(iii) G = Se ∩ eS is a group where e is its identity.
(iv) S = LGR (This factorization will be called the Rees decomposition of S at e, and G will be called the group factor at e).
(v) The product mapping
ψ : L × G × R ∋ (x, g, y) → (xgy) ∈ LGR (2.8)
is bijective with its inverse given as
ψ −1 : LGR ∋ z → (ze(eze) −1 , eze, (eze) −1 ez) ∈ L × G × R. (2.9)
Proof. (i) It is obvious that Se is a left ideal of S. Let I be a left ideal of S such that I ⊂ Se. Let a ∈ I. Note that ae = a since a ∈ Se. By simplicity of S and Lemma 2.4, we have uav = e for some u, v ∈ S. Set r = eu and s = eve. We then have ras = eu(ae)ve = euave = e, er = r, es = se = s.
(2.10)
If we set t = sra, then et = te = t and t 2 = s(ras)ra = sera = sra = t, (2.11) which yields t = e by primitivity. Since e = t = sra ∈ srI ⊂ I, we have Se ⊂ SI ⊂ I, which shows I = Se and that Se is a minimal left ideal of S. By Lemma 2.3, we see that Se is left simple. Since Se contains an idempotent e, we see that Se is a left group. By a similar argument we see that eS is a right group.
Let us show LG = Se. It is obvious that LG ⊂ Se. Let a ∈ Se. Set g := ea ∈ eSe = G and set b = ag −1 ∈ Se. Since g −1 = g −1 e, we have
b 2 = ag −1 ag −1 = ag −1 (ea)g −1 = ag −1 = b.
(2.12)
Hence we have b ∈ E(Se) = L and a = ae = ag −1 g = bg ∈ LG. We now have LG = Se.
We also have GR = eS similarly.
(ii) RL ⊂ (eS)(Se) ⊂ eSe = G.
Let x ∈ L = E(Se). Since (ex) 2 = e(xe)x = exx = ex and e(ex) = (ex)e = ex, we have ex = e by primitivity. We thus see that eL = {e}. We have Re = {e} similarly.
(iii) It is obvious that G = eSe = eS ∩Se, since x ∈ eS ∩Se implies x = ex = xe = exe. It is also obvious that e is identity of G. Let g ∈ G. Since G ⊂ eSe, we have g = ea for some a ∈ Se. By the left simplicity of Se and by Lemma 2.3, we have ba = e for some b ∈ Se. Since (ab) 2 = a(ba)b = aeb = ab, we see by Lemma 2.6 that ab is right identity. Hence ab = abe = e, which shows that b is the inverse of a. (v) Let z = xgy with (x, g, y) ∈ L × G × R. Since x = xx = xex and since exgye ∈ eSe = G, we have
x = xe = x(exgye)(exgye) −1 = ze(eze) −1 .
(2.13)
We have y = (eze) −1 ez similarly. Since ex = ye = e by (ii), we obtain g = ege = (ex)g(ye) = eze.
(2.14)
The proof is now complete.
Corollary 2.9. Under the same assumptions and notation as Theorem 2.8, it holds that {Sy = LGy : y ∈ R} is the family of all minimal left ideals of S.
Proof. Any minimal left ideal of S is of the form Sz for some z ∈ S. We represent z = xgy and then we obtain Sz = LG(Rx)gy = LGy, since RL ⊂ G.
Conversely, for any z ∈ LGy, we have z = xgy for some (x, g) ∈ L × G, so that we have LGyz = LG(yx)gy = LGy, which shows by Lemma 2.1 that LGy is a minimal left ideal.
Corollary 2.10. Under the same assumptions and notation as Theorem 2.8, the following assertions hold:
(i) For z = xgy with (x, g, y) ∈ L × G × R, z is idemptent if and only if g = (yx) −1 .
(ii) All idempotents of S are primitive.
(iii) Let e ′ be another idempotent of S and represent it as e ′ = a(ba
) −1 b for (a, b) ∈ L×R. Let S = L ′ G ′ R ′ denote the Rees decomposition of S at e ′ .
Then
L ′ G ′ = LGb, G ′ = aGb, G ′ R ′ = aGR. (2.15) Proof. (i) Suppose z 2 = z. Then xgyxgy = xgy. Since eL = Re = {e}, we have gyxg = g, which shows g = (yx) −1 .
Conversely, suppose g = (yx) −1 . Then z 2 = x(gyxg)y = xgy = z.
(ii) Let e 1 , e 2 ∈ S be two idempotents of S and represent them as
e i = a i (b i a i ) −1 b i for (a i , b i ) ∈ L × R, i = 1, 2. Suppose e 1 e 2 = e 2 e 1 = e 2 . Then a 1 ((b 1 a 1 ) −1 (b 1 a 2 )(b 2 a 2 ) −1 )b 2 = a 2 ((b 2 a 2 ) −1 (b 2 a 1 )(b 1 a 1 ) −1 )b 1 = a 2 (b 2 a 2 ) −1 b 2 ,
which shows a 1 = a 2 and b 1 = b 2 by the injectivity of the product mapping ψ. Hence we have e 1 = e 2 , which shows that e 1 is a primitive idempotent.
(iii) We have L ′ G ′ = Se ′ = LG(Ra)(ba) −1 b = LGb and G ′ R ′ = aGR similarly. We also have G ′ = e ′ Se ′ = a(ba) −1 (bL)G(Ra)(ba) −1 b = aGb. Corollary 2.11. A left group S is completely simple. The Rees decomposition of S at e ∈ E(S) is given as S = LG with R = {e}.
Proof. Suppose ex = xe = x ∈ E(S). By Lemma 2.3, we have yx = e for some y ∈ S. Hence x = ex = yxx = yx = e, which shows that e is an primitive idempotent. Hence S is completely simple. Let S = LGR denote the Rees decomposition of S at e. Since S = Se by Lemma 2.3 and since Re = {e}, we obtain S = Se = LGRe = LG.
For later use we prove the following proposition.
Proposition 2.12. Suppose that a semigroup S contains a minimal left ideal A and a minimal right ideal B as well. Then BA is a group and its identity is a primitive idempotent of S. If, in addition, S is simple, then S is completely simple.
Proof. Since (BA)(BA) = (BAB)A ⊂ BA, we see that BA is a subsemigroup of S. To prove right simplicity of BA, let I be a right ideal of BA. Since IB is a right ideal of S and IB ⊂ BAB ⊂ B, we see that IB = B by minimality. Hence BA = IBA ⊂ I, which shows right simplicity of BA. By a similar argument we obtain left simplicity of BA. We thus conclude by Proposition 2.5 that BA is a group.
Let e be the identity of BA and suppose ex = xe = x ∈ E(S). Then x = xx = exxe ∈ (BAS)(SBA) ⊂ BA. Since BA is a group and since x 2 = x, we have x = xx −1 = e, which shows that e is a primitive idempotent of S.
Kernel
A minimal ideal of a semigroup S will be called a kernel of S. Theorem 2.13. Let S be a semigroup. Then the following assertions hold:
(i) If S contains a minimal left ideal, then S contains a unique kernel K, and SzS = K for all z ∈ K.
(ii) If S contains a minimal left ideal and a minimal right ideal as well, then the unique kernel of S is completely simple.
(iii) If S contains a completely simple kernel K, then it is the unique kernel of S. Let K = LGR denote the Rees decomposition at e. Then Sz = Kz = LGz for all z ∈ K.
Proof. (i) Let A denote the family of all minimal left ideals of S and suppose A is not empty. We shall prove that K := A is a unique kernel of S.
Let z ∈ K and take A ∈ A such that z ∈ A. Then Sz = A by Lemma 2.1. For x ∈ S, we see that Ax ∈ A; in fact, for any left ideal I of S such that I ⊂ Ax, we see that
J = {a ∈ A : ax ∈ I} ⊂ A is a left ideal of S, so that J = A by minimality and thus I = Ax. Hence SzS = AS = x∈S Ax ⊂ A = K, which shows by Lemma 2.2 that K is a kernel of S.
Let K ′ be another kernel of S. Since K ∩ K ′ contains KK ′ which is not empty, we see that K ∩ K ′ is an ideal contained both in K and in K ′ . Thus K ∩ K ′ = K = K ′ by minimality.
(ii) By (i) and Lemma 2.2, we see that the unique kernel K of S is both a minimal left ideal of K and a minimal right ideal of K. By Proposition 2.12, we see that K is completely simple.
(iii) Suppose K is a completely simple kernel of S with a primitive idempotent e. By Theorem 2.8, K contains a left group Ke. By Lemma 2.3, we see that Ke is a minimal left idal of S. Hence by (i) the kernel of S is unique.
For z ∈ K, we represent z = xgy ∈ LGR. Then by (i) Sz is a minimal left ideal of K containing y. By Corollary 2.9, we see that Sz = LGy = LGz = Kz.
Topological semigroup
A semigroup S is called topological if S is endowed with a topology such that the product mapping S × S ∋ (x, y) → xy ∈ S is jointly continuous. A semigroup S is called Polish if S is a topological semigroup with respect to a Polish topology, i.e. a separable and completely metrizable topology.
It is well-known (see, e.g. [20,Theorem 1.5.3]) that locally compact Polish is equivalent to locally compact Hausdorff with a countable base. It is elementary that compact Polish is equivalent to compact metrizable.
For a ∈ S and A ⊂ S, we write
a −1 A = {x ∈ S : ax ∈ A}, Aa −1 = {x ∈ S : xa ∈ A}. (3.1)
If S contains identity e and a ∈ S has its inverse a −1 ∈ S, then (a −1 )A = a −1 A; in fact, (iv) For two compact subsets K and K ′ , the product KK ′ is also compact.
(a −1 )A = {a −1 x ∈ S : x ∈ A} = {y ∈ S : ay ∈ A} = a −1 A. (3.2) Lemma 3.1. Let SProof. (i) If we write ψ a : S → S for the translation ψ a (x) = ax, then a −1 A = ψ −1 a (A).
Since ψ a is continuous, we obtain the desired results.
(ii) Let a, b ∈ A and take {a n }, {b n } ⊂ A such that a n → a and b n → b. Then we have ab = lim a n b n ∈ A.
(iii) Let {e n } ⊂ E(A) such that e n → e ∈ S. Since A is closed, we have e ∈ A. Since e 2 n = e n for all n, we have e 2 = e, which shows e ∈ E(A). Let {x n } ⊂ eA such that x n → x ∈ S. Since eA ⊂ A and since A is closed, we have x ∈ A. Then ex = lim ex n = lim x n = x, which shows x = ex ∈ eA.
(iv) Let ψ : S × S → S denote the jointly continuous product mapping: ψ(x, y) = xy. Since KK ′ = ψ(K × K ′ ) and K × K ′ is compact, we see that KK ′ is compact.
Topological group
A group S is called topological if G is a topological semigroup and the inverse mapping G ∋ g → g −1 ∈ G is continuous.
Theorem 3.2 (Ellis [11] andŻelazko [51]). If a group G is a topological semigroup with respect to a completely metrizable topology, then it is a topological group.
Proof. We borrow the proof from Pfister [35]. Let e denote the identity of G and let d be a complete metric of G.
Let U 0 be a open neighborhood of e. By the joint continuity of the product mapping, we can construct a sequence {U n } ∞ n=1 of open balls of e such that the radius of U n decreases to 0 and U n U n ⊂ U n−1 for n = 1, 2, . . ., where U n stands for the closure of U n .
Let {x n } ∞ n=1 be a subsequence of an arbitrary sequence of G which converges to e. It then suffices to construct a subsequence {n(k)} ∞ k=1 of {1, 2, . . .} such that x −1 n(k) → e. We write y k := x n(1) · · · x n(k) .
Set n(0) = 0 and y 0 = x 0 = e. If we have n(0), n(1), . . . , n(k − 1), then we can take n(k) > n(k − 1) such that x n(k) ∈ U k and d(y k , y k−1 ) < 2 −k , since y k−1 x n → y k−1 as n → ∞. By completeness of d, we see that y k converges to a limit y ∈ G. Let n be fixed for a while. Since yU n+1 is a neighborhood of y, we see that y k−1 ∈ yU n+1 for large k. For j > k, we have U j−1 U j ⊂ U j U j ⊂ U j−1 , and hence
y −1 k y j = x n(k+1) · · · x n(j−1) x n(j) ∈ U k+1 · · · U j ⊂ U k ,(3.3)
which implies y −1 k y ∈ U k ⊂ U k−1 . We now obtain
x −1 n(k) = (y −1 k−1 y k ) −1 = y −1 k y k−1 ∈ y −1 k yU n+1 ⊂ U k−1 U n+1 ⊂ U n+1 U n+1 ⊂ U n (3.4)
for large k. Thus we obtain x −1 n(k) → e. Corollary 3.3. Suppose that a Polish semigroup S contains a completely simple kernel K. Let K = LGR denote the Rees decomposition of K at e ∈ E(K). Then it holds that L, G, R and K are closed subsets, and that the product mapping
ψ : L × G × R ∋ (x, g, y) → xgy ∈ LGR (3.5)
is homeomorphic.
Proof. By Corollary 2.13, we have Ke = Se, eK = eS and eKe = eSe. By Lemma 3.1, we see that L = E(Ke), G = eKe and R = E(eK) are all closed. By Theorem 3.2, we see that G is a Polish group. We now see that the inverse
ψ −1 : LGR ∋ z → (ze(eze) −1 , eze, (eze) −1 ez) ∈ L × G × R (3.6)
is continuous. Consequently, we see that K is closed. Proof. Let I denote the family of all closed left ideals of S. The family I contains S and is endowed with a partial order by the usual inclusion. For any linearly ordered subfamily J of I has a lower bound in I; in fact, the intersection J is not empty by compactness of S and is a closed left ideal of S such that J ⊂ J for all J ∈ J . Hence, by Zorn's lemma, we see that I contains a minimal element, say A.
Compact semigroup
Let us prove that A is a minimal left ideal of S. Let I be a left ideal of S such that I ⊂ A. For a ∈ I, we have Sa ∈ I and Sa ⊂ SI ⊂ I ⊂ A, which yields Sa = I = A by the minimality of A in I. This shows that A is a minimal left ideal of S.
Similarly we see that S contains a minimal right ideal. By Theorem 2.13, we see that S contains a completely simple kernel K. By Corollary 3.3, we see that K is a closed subset of S, and hence K is compact. Proof. Let C denote the set of all cluster points of {a n } ∞ n=1 . By the assumption, we see that C is a compact abelian semigroup. By Theorem 3.4, we see that C contains a compact completely simple kernel K. Since the Rees decomposition of K is LGR = GRL = G by commutativity, we see that K is a compact abelian group. Let e denote the identity of K. Then, for any x ∈ C, we can find a subsequence {n(k)} of {1, 2, . . .} such that x = e lim k→∞ a n(k) ∈ eC ⊂ KC ⊂ K ⊂ C, which shows K = eC = C. It is now easy to see that C = {e, ae, a 2 e, . . .}. with e = a rp , where r is the unique integer such that q ≤ rp ≤ q + p − 1. For a ∈ S, we write δ a for the Dirac mass at a: δ a (B) = 1 B (a). It is obvious that
µ * δ x (B) = µ(Bx −1 ), δ x * µ(B) = µ(x −1 B), B ∈ B(S),(4.3)
which will be called translations of µ.
b such that U 1 U 2 ⊂ U, so that µ * ν(U) ≥ 1 U 1 U 2 (xy)µ(dx)ν(dy) ≥ µ(U 1 )ν(U 2 ) > 0,(4.6)
which yields ab ∈ S(µ * ν) and hence S(µ)S(ν) ⊂ S(µ * ν). which shows a ∈ S(µ * ν) c and hence S(µ * ν) ⊂ S(µ)S(ν).
L × G × R → LGR, we denote (z L , z G , z R ) := ψ −1 (z) = (ze(eze) −1 , eze, (eze) −1 ez) ∈ L × G × R, z ∈ LGR. (4.8)
For µ ∈ P(S), we define
µ L (B) = µ(z : z L ∈ B), µ G (B) = µ(z : z G ∈ B), µ R (B) = µ(z : z R ∈ B) (4.9)
for B ∈ B(S). Then, for µ, ν ∈ P(S), it holds that
(µ * ν) L = µ L , (µ * ν) R = ν R . (4.10)
Proof. This is obvious by noting that (z 1 z 2 ) L = z L 1 and (z 1 z 2 ) R = z R 2 .
We equip P(S) with the topology of weak convergence: µ n → µ if and only if f dµ n → f dµ for all f ∈ C b (S), the class of all bounded continuous functions on S. It is wellknown (see, e.g. [34, Theorems 6.2 and 6.5 of Chapter 2]) that P(S) is a Polish space. Proof. Note that, if we take independent random variables X and Y taking values in S such that X d = µ and Y d = ν, then µ * ν coincides with the law of the product XY . The desired result now follows from the Skorokhod coupling thoerem (see, e.g. [18,Theorem 4.30]), which asserts that µ n → µ implies that we can take random variables {X n }, X taking values in S such that X n d = µ n , X d = µ and X n → X a.s. Proof. Note that S(µ) = S(δ x * µ) = xS(µ), x ∈ S, (4.11) which implies that S(µ) is a left ideal of S. Similarly S(µ) is a right ideal of S, and hence S(µ) is an ideal of S.
Translation invariance
Let S be a Polish semigroup. A probability measure µ ∈ P(S) is called ℓ * -invariant [r * -invariant] if δ x * µ = µ [µ * δ x = µ] for all x ∈ S.
Let us prove that, for any x ∈ S(µ), the subsemigroup xS is left-cancellative. Let y, a, b ∈ S be such that (xy)(xa) = (xy)(xb). Since S(µ) = S(µ * δ xyx ) = S(µ)xyx, we can take {z n } ⊂ S(µ) such that z n xyx → x, and hence xa = lim z n xyxa = lim z n xyxb = xb, (4.12) which shows that xS is left-cancellative. Similarly Sx is right-cancellative.
Let a, b ∈ S(µ) be fixed. We shall prove that the subsemigroup D := aS(µ)b contains an idempotent. Note that µ(D) = (δ a * µ * δ b )(D) = µ(a −1 Db −1 ) ≥ µ(S(µ)) = µ(S) = 1, (4.13) which shows µ(D) = 1. For x ∈ D, we have µ(D) ≤ µ(x −1 (xD)) = (µ * δ x )(xD) = µ(xD) ≤ µ(D), (4.14)
which shows µ(xD) = µ(D) = 1. We define two mappings θ, β : S × S → S × S by θ(x, y) = (x, xy), β(x, y) = (y, x). Since (x, y) ∈ θ(D × D) if and only if x ∈ D and y ∈ xD, we have
(µ ⊗ µ)(β • θ(D × D)) = (µ ⊗ µ)(θ(D × D)) = D µ(xD)µ(dx) = µ(D) 2 = 1. (4.16) This shows that β • θ(D × D) ∩ θ(D × D)
is not empty, so that (vw, v) = (x, xy) for some v, w, x, y ∈ D. We now have x(yw) = vwyw = x(yw) 2 , which implies yw = (yw) 2 by left-cancellativity of D.
Let e := yw ∈ E(D) = E(aS(µ)b). By the left-and right-cancellativity of aS(µ)b and by Lemma 2.6, we see that e is identity of aS(µ)b. By Lemma 3.1, we see that By the ℓ * -invariance, we have µ * µ = µ. We now apply [34, Theorem 3.1 of Chapter 3] to obtain the desired result.
S(µ) = eaS(µ)b = e (aS(µ)b) = eS(µ) ⊂ aS(µ)bS(µ) ⊂ aS(µ) ⊂ S(µ),
Convolution invariance
Proposition 4.5 (Mukherjea [26]). Let S be a Polish semigroup and let µ, ν ∈ P(S). Suppose ν = µ * ν = ν * µ.
(4.18)
Then, for any x ∈ S(µ) and any a ∈ S(ν), it holds that ν * δ xa = ν * δ a , δ ax * ν = δ a * ν. Proof. Let a ∈ S(ν), f ∈ C b (S) and ε > 0 be fixed for a while, and set
g(x) = max f d(ν * δ x ) − f d(ν * δ a ) − ε, 0 , x ∈ S. (4.20)
It is obvious that g ∈ C b (S), g is non-negative and g(a) = 0. By ν = ν * µ, we have
f d(ν * δ x ) − f d(ν * δ a ) − ε (4.21) = f d(ν * δ yx ) − f d(ν * δ a ) − ε µ(dy) ≤ g(yx)µ(dy),(4.22)
so that we have g(x) ≤ g(yx)µ(dy), x ∈ S. (4.23)
In addition, by ν = µ * ν, we have
g(x) − g(yx)µ(dy) ν(dx) = gdν − gd(µ * ν) = 0,(4.24)
which shows that the equality in (4.23) holds for ν-a.e. x ∈ S. Since g is continuous, we see that the equality in (4.23) holds for all x ∈ S(ν). Since a ∈ S(ν) and g(a) = 0, we see, again by continuity of g, that g(ya) = 0, y ∈ S(µ). Since ε > 0 is arbitrary, we obtain
f d(ν * δ ya ) ≤ f d(ν * δ a ), a ∈ S(ν), y ∈ S(µ). (4.26) Since ν = ν * µ, we have f d(ν * δ ya ) − f d(ν * δ a ) µ(dy) = 0, which implies f d(ν * δ ya ) = f d(ν * δ a ), a ∈ S(ν), y ∈ S(µ), f ∈ C b (S). (4.27)
Since f ∈ C b (S) is arbitrary, we obtain ν * δ ya = ν * δ a for all a ∈ S(ν) and y ∈ S(µ). We obtain δ ay * ν = δ a * ν similarly.
Convolution idempotent
We denote the n-fold convolution by µ n , i.e. µ 1 = µ and µ n = µ n−1 * µ for n = 2, 3, . . .. Theorem 4.6 (Mukherjea-Tserpes [31]). Let S be a Polish semigroup and let µ ∈ P(S). Suppose that µ 2 = µ. Then S(µ) is completely simple and its group factor is compact. Let S(µ) = LGR denote the Rees decomposition at e ∈ E(S(µ)). Then µ admits the convolution factorization
µ = µ L * ω G * µ R ,(4.28)
where µ L and µ R have been introduced in (4.9) and ω G stands for the normalized unimodular Haar measure on the compact Polish group G.
Remark 4.7. The convolution factorization (4.28) is equivalent to the following assertion: If we let Z be a random variable whose law is µ, then Z L , Z G and Z R are independent and the law of Z G is ω G . Here (Z L , Z G , Z R ) = ψ −1 (Z) with ψ : L × G × R → LGR denoting the product mapping; see Proposition 4.2.
Proof of Theorem 4.6. Since S(µ) = S(µ)S(µ), we see that S(µ) is a closed subsemigroup of S. By Proposition 4.5, we see that, for any a ∈ S(µ), µ * δ xa = µ * δ a , δ ax * µ = δ a * µ, x ∈ S(µ). (4.30)
Then, for a ∈ S(µ), we have µ * δ ay = µ * δ a (y ∈ S(µ * δ a ))., δ za * µ = δ a * µ (z ∈ S(δ a * µ)) (4.31)
In fact, for y ∈ S(µ * δ a ) = S(µ)a, we may take {x n } ⊂ S(µ) such that x n a → y, so that µ * δ a = µ * δ axna → µ * δ ay .
Let a ∈ S(µ) be fixed and set ν = δ a * µ * δ a . Then S(ν) = aS(µ)a is a closed subsemigroup of S. For any y ∈ S(ν) = aS(µ)a, we may take {x n } ⊂ S(µ) such that ax n a → y, so that, using (4.30), we have
ν = δ a * µ * δ a = δ axna 2 * µ * δ a = δ axna * ν → δ y * ν,(4.32)
which shows that ν| S(ν) is ℓ * -invariant. We see similarly that ν| S(ν) is r * -invariant. We may now apply Theorem 4.4 to see that S(ν) = aS(µ)a is a compact Polish group. Its identity is an idempotent of S(µ).
Let e ∈ E(S(µ)). By the above argument with a = e, we see that G := eS(µ)e is a compact Polish group (note that eS(µ)e is closed by Lemma 3.1). Set A := S(µ)e. For y ∈ A, using (4.31), we have Ay = S(µ)ey = S(µ * δ ey ) = S(µ * δ e ) = S(µ)e = A.
(4.33)
Since Ay ∩ eS(µ)e is a left ideal of the group eS(µ)e, we see that Ay ∩ eS(µ)e = eS(µ)e, i.e. eS(µ)e ⊂ Ay, which shows e ∈ Ay. Hence
A = Ae ⊂ AAy ⊂ Ay ⊂ Ay = A,(4.34)
which yields Ay = A for all y ∈ A. By Lemma 2.3, we see that A is a left group. We see similarly that B := eS(µ) is a right group. By Theorem 2.13, we see that S(µ) contains a completely simple kernel K, which is closed by Corollary 3.3.
By (4.30), we have µ * δ e * µ = (µ * δ e * δ a )µ(da) = (µ * δ a )µ(da) = µ * µ = µ. which shows that S(µ) is completely simple.
By (4.31), we see that µ * δ e is r * -invariant on A = S(µ)e = LG, so that µ * δ e = µ * δ e * ω G . Hence, for any B ∈ B(S(µ)),
µ(B) =(µ * δ e * µ)(B) = (µ * δ e * ω G * µ)(B) (4.37) = µ(dz 1 ) µ(dz 2 ) ω G (dg)1 B (z 1 egz 2 ) (4.38) = µ(dz 1 ) µ(dz 2 ) ω G (dg)1 B (z L 1 gz R 2 ) = (µ L * ω G * µ R )(B),(4.39)
which completes the proof.
The following proposition is a converse to Theorem 4.6.
Proposition 4.8. Let S be a Polish semigroup and let µ 1 , µ 2 ∈ P(S). Let G be a compact Polish subgroup of S and suppose that S(µ 2 * µ 1 ) ⊂ G. Then µ := µ 1 * ω G * µ 2 satisfies µ 2 = µ.
Proof. For any B ∈ B(S), we have
µ 2 (B) = (µ 1 * ω G * µ 2 * µ 1 * ω G * µ 2 )(B) (4.40) = µ 1 (dz 1 ) ω G (dg 1 ) (µ 2 * µ 1 )(dg 2 ) ω G (dg 3 ) µ 2 (dz 2 )1 B (z 1 g 1 g 2 g 3 z 2 ) (4.41) = µ 1 (dz 1 ) ω G (dg 1 ) µ 2 (dz 2 )1 B (z 1 g 1 z 2 ) = (µ 1 * ω G * µ 2 )(B) = µ(B),(4.42)
which completes the proof.
Infinite convolutions
Theorem 4.9 (Rosenblatt [39] and Mukherjea [28]). Let S 0 be a Polish semigroup and let µ ∈ P(S 0 ). Suppose that the sequence {µ n } ∞ n=1 is tight. Let S denote the closure of the semigroup generated by S(µ), i.e.
S := ∞ n=1 S(µ) n .
(4.43)
Then the following assertions hold:
(i) There exists ν ∈ P(S) such that ν 2 = ν, µ * ν = ν * µ = ν and
ν = η L * ω G * η R . (4.47)
(v) For g ∈ G, we write ω gH := δ g * ω H . It holds that H is a closed normal subgroup of G and that there exists a Polish group isomorphism F : K → G/H such that
λ = η L * ω F (λ) * η R ,(4.
48)
Consequently, there exists γ ∈ G such that µ k * η admits the convolution factorization µ k * η = η L * ω γ k H * η R , k = 1, 2, . . . , Proof of Theorem 4.9. (i) Let · denote the total variation norm. For j = 1, 2, . . ., we have
µ n − µ j * µ n ≤ 1 n n k=1 µ k − n k=1 µ k+j = 1 n j k=1 µ k − n+j k=n+1 µ k ≤ 2j n −→ n→∞ 0. (4.52)
Since {µ n } is tight, we see that {µ n } is also tight. Let ν 1 , ν 2 be cluster points of {µ n }. For i = 1, 2, we see by (4.52) that µ j * ν i = ν i * µ j = ν i for j = 1, 2, . . ., so that µ n * ν i = ν i * µ n = ν i for n = 1, 2, . . ., which implies ν 1 = ν 1 * ν 2 = ν 2 * ν 1 = ν 2 . Hence we see that {µ n } converges to some ν ∈ P(S 0 ) and we have ν 2 = ν and µ * ν = ν * µ = ν. We may apply Theorem 4.6 to see that S(ν) is a completely simple semigroup and its group factor is compact.
(ii) Let us prove that S(ν) and S(K) := λ∈K S(λ) are ideals of S. Let a ∈ S, x ∈ S(ν) and y ∈ S(K). Then we may take {a n } ⊂ S(µ) m(n) ⊂ S(µ m(n) ) and {y n } ⊂ S(λ n ) such that a n → a and y n → y. Since a n x ∈S(µ m(n) )S(ν) ⊂ S(µ m(n) * ν) = S(ν), (4.53) a n y n ∈S(µ m(n) )S(λ n ) ⊂ S(µ m(n) * λ n ) ⊂ S(K), (4.54)
we obtain ax = lim a n x ∈ S(ν) and ay = lim a n y n ∈ S(K), which shows that S(ν) and S(K) are both left ideals of S. Similarly we see that they are also right ideals of S.
Let U be an open subset containing S(ν). We shall prove that µ n (U) → 1. Let ε > 0. By tightness, we may take a compact subset K 1 such that inf n µ n (K 1 ) > 1 − ε. We may take a compact subset K 2 ⊂ S(ν) such that ν(
K 2 ) > 1 − ε. Since K 1 K 2 ⊂ SS(ν) ⊂ S(ν) ⊂ U, we have K 1 × K 2 ⊂ U := {(x, y) ∈ S 0 × S 0 : xy ∈ U}.such that K 1 ⊂ V 1 , K 2 ⊂ V 2 and V 1 × V 2 ⊂ U , which implies V 1 V 2 ⊂ U. Since µ n → ν, we have lim inf n µ n (V 2 ) ≥ ν(V 2 ) ≥ ν(K 2 ) > 1 − ε.
We may then take some n 0 such that µ n 0 (V ) > 1 − ε. We now have
µ n+n 0 (U) = 1 U (xy)µ n (dx)µ n 0 (dy) ≥ µ n (V 1 )µ n 0 (V 2 ) > (1 − ε) 2 ,(4.55)
which leads to µ n (U) → 1.
By the tightness assumption, we may apply Proposition 3.5 to see that K is a compact abelian group. Let λ ∈ K and let x ∈ S(λ). Suppose that x / ∈ S(ν). We could then take disjoint open sets U and V such that S(ν) ⊂ U and x ∈ V . If we let δ := λ(V )/2 > 0, then µ n (V ) > δ for infinitely many n, and then lim inf n µ n (U) ≤ lim inf n µ n (V c ) ≤ 1 − δ, which would contradict µ n (U) → 1. Hence we obtain S(K) ⊂ S(ν). Since S(ν) is a minimal ideal of S by Lemma 2.4 and since S(K) is an ideal of S, we see that S(K) = S(ν).
(iii) By Theorem 4.6, we see that S(η) is a completely simple semigroup. Let S(η) = LHR denote the Rees decomposition at e ∈ E(S(η)) (hence RL ⊂ H). Then the group factor H is compact and η admits the convolution factorization (4.46).
(iv) We have already seen in (i) that S(ν) is a completely simple kernel of S. Since S(η) ⊂ S(K) = S(ν), we have e ∈ E(S(ν)). Let S(ν) = L ′ GR ′ denote the Rees decomposition at e. As a consequence of Theorem 4.6, we see that ν admits the convolution factorization ν = η L ′ * ω G * η R ′ . Since S(η) ⊂ S(ν) and L = E(S(η)e)) etc., we see that L ⊂ L ′ , H ⊂ G and R ⊂ R ′ .
Let us prove that L ′ = L and R ′ = R. Let z = xgy ∈ L ′ GR ′ . Since S(ν) = S(K), we may take z n ∈ S(λ n ) such that z n → z. Since K is abelian, we have λ n * λ −1 n = λ −1 n * λ n = η, and by Proposition 4.2 we have λ L ′ n = η L ′ = η L and λ R ′ n = η R ′ = η R . Hence we obtain x n := z L ′ n ∈ S(λ L ′ n ) = S(η L ) = L and y n := z R ′ n ∈ S(λ R ′ n ) = S(η R ) = R, and thus x = lim x n ∈ L and y = lim y n ∈ R, which shows L ′ = L and R ′ = R.
(v) Let λ ∈ K. For z = xgy ∈ S(λ) ⊂ S(ν) = LGR, since RL ⊂ H, we have xgy ∈ LgHR ⊂ LHRxgyLHR ⊂ S(η)S(λ)S(η) ⊂ S(η * λ * η) = S(λ).
(4.56)
Hence we have S(λ) = LG λ R for G λ := {gH : z = xgy ∈ S(λ)} ⊂ G, and we also have G λ = {Hg : z = xgy ∈ S(λ)} similarly. Note that G λ H = HG λ = G λ . Take g λ ∈ G such that Hg −1 λ ⊂ G λ −1 . Then we obtain
LHg −1 λ G λ R ⊂ LG λ −1 RLG λ R ⊂ S(λ −1 )S(λ) ⊂ S(λ −1 * λ) ⊂ S(η) = LHR,(4.57)
which yields that Hg −1 λ G λ ⊂ H and hence G λ = g λ H. Similarly, we obtain G λ = Hg λ . For any h ∈ H and g ∈ G ⊂ S(ν) = S(K), we may take z n = x n g n y n ∈ S(λ n ) such that z n → g and consequently g n → g. In a similar way to (4.57), we have g n hg −1 n ∈ (g n H)(Hg −1 n ) = G λn G λ −1 n ⊂ S(η) = LHR, (4.58) which shows g n hg −1 n ∈ eLHRe = H. Letting n → ∞, we obtain ghg −1 ∈ H, which shows that H is a normal subgroup of G. Since G and H are both compact, we see by [15,Theorem 5.22] that the quotient group G/H = {gH : g ∈ G} is also compact. Let π : G → G/H denote the natural projection.
Since S(η R * λ * η L ) = S(η R )S(λ)S(η L ) = RLG λ RL ⊂ Hg λ HH = g λ H, (4.59)
we obtain the convolution factorization λ = ηλη = η L * ω H * (η R * λ * η L ) * ω H * η R = η L * ω g λ H * η R . (4.60)
We now define the mapping F : K → G/H by F (λ) := g λ H. For λ 1 , λ 2 ∈ K, then λ 1 * λ 2 = η L * ω g λ 1 H * (η R * η L ) * ω g λ 2 H * η R = η L * ω (g λ 1 g λ 2 H) * η R , (4.61)
since RL ⊂ H, which shows that F is a group homomorphism. Injectivity of F is obvious by (4.60). Let g ∈ G. As we have seen it above, we may take z n = x n g n y n ∈ S(λ n ) such that g n → g and g n H = g λn H. Then, by (4.60), we have λ n = η L * ω g λn H * η R → η L * ω gH * η R =: λ. This shows that λ ∈ K and F (λ) = gH, which yields surjectivity of F . Suppose K ∋ λ n → λ ∈ K. By (4.60), we have ω F (λn) = δ e * λ n * δ e → δ e * λ * δ e = ω F (λ) in P(G), (4.63) which shows by the continuity of the natural projection π that δ F (λn) = ω F (λn) • π −1 → ω F (λ) • π −1 = δ F (λ) in P(G/H), (4.64) which implies F (λ n ) → F (λ) and we have seen continuity of F . Since K is compact and G/H is Hausdorff, we see by [21, Theorem 9 of Chapter 5] that F is homeomorphic. Since F (µ * η) ∈ G/H, we may take γ ∈ G such that F (µ * η) = γH, and then we obtain (4.49) since (µ * η) k = µ k = µ k * η and F is a group homomorphism.
Finally, let us prove the representations (4.50). Since any λ ∈ K can be represented as λ = λ * η = lim µ n(k) * η, we see that K = {η, µ * η, µ 2 * η, . . .}. Since for any g ∈ G we have F (λ) = gH for some λ = lim µ n(k) * η ∈ K, we obtain gH = F (λ) = lim F (µ n(k) * η) = lim γ n(k) H in G/H, which yields G/H = {H, γH, γ 2 H, . . .}.
( 2 )
2The research of Kouji Yano was supported by JSPS KAKENHI grant no.'s JP19H01791 and JP19K21834 and by JSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921. This research was supported by RIMS and by ISM.
A non-empty subset I is called a left ideal [right ideal ] (of S) if SI ⊂ I [IS ⊂ I]. If S contains no proper left ideal [right ideal], then it is called left simple [right simple]. A non-empty subset I is called a ideal if it is both a left and a right ideal, i.e., SI ∪ IS ⊂ I. If S contains no proper ideal, then it is called simple. Note that both left and right simple implies simple, but the converse is not necessarily true. Lemma 2.1. For a subsemigroup S of a semigroup S 0 , the following are equivalent: (i) S is a minimal left ideal of S 0 .
Lemma 2. 2 .
2For a subsemigroup S of a semigroup S 0 , the following are equivalent: (i) S is a minimal ideal of S 0 . (ii) S = S 0 aS 0 for all a ∈ S.
Lemma 2 . 3 .
23For a semigroup S, the following are equivalent: (i) S is left simple. (ii) For any semigroup S 0 of which S is a left ideal, S is a minimal left ideal of S 0 . (iii) S is a minimal left ideal of S itself (if and only if S = Sa for all a ∈ S by Lemma 2.1).
Lemma 2 . 4 .
24For a semigroup S, the following are equivalent: (i) S is simple. (ii) For any semigroup S 0 of which S is an ideal, S is a minimal ideal of S 0 . (iii) S is a minimal ideal of S itself (if and only if S = SaS for all a ∈ S by Lemma 2.2).
Proposition 2 . 7 .
27For a semigroup S, the following are equivalent: (i) S is a left group. (ii) S is left simple and right cancellative. (iii) For any a, b ∈ S, the equation xa = b has a unique solution x ∈ S.
Theorem 2. 8 (
8Rees decomposition). Let S be a completely simple semigroup and let e be a primitive idempotent of S. Set L := E(Se), G := eSe, R := E(eS).(2.7)
(
iv) LGR = LGGR = SeeS = SeS = S by Lemma 2.4.
(
iii) Let A be a closed subsemigroup of S. Then E(A), eA, Ae and eAe are closed for all e ∈ E(A).
Theorem 3. 4 .
4A compact Polish semigroup S contains a compact completely simple kernel.
Proposition 3 . 5 .
35Let S be a Polish semigroup and let a ∈ S. Suppose that any subsequence of {a n } ∞ n=1 has a convergent further subsequence. Then the set C of all cluster points of {a n } ∞ n=1 is a compact abelian group. If we denote the identity of C by e, then C = {e, ae, a 2 e, . . .}.
Remark 3 . 6 .
36In the settings of Proposition 3.5, suppose that the sequence {a n } ∞ n=1 has multiple points. Let p and q be the smallest positive integers such that a q+p = a q . Then we have {a n : n = 1, 2, . . .} = {a, a 2 , . . . , a q+p−1 } and K = {a q , a q+1 , . . . , a q+p−1 } = {e, ae, . . . , a p−1 e} (3.7)
S be a Polish semigroup. Let B(S) denote the family of all Borel sets of S and P(S) the family of all probability measures on (S, B(S)).
For
µ, ν ∈ P(S), we define the convolution µ * ν ∈ P(S) of µ and ν by µ * ν(B) = 1 B (xy)µ(dx)ν(dy), B ∈ B(S).
For
µ ∈ P(S), we denote its topological support byS(µ) = {x ∈ S : µ(U) > 0 for all open neighborhood U of x}. (4.4)It is obvious that S(µ) is closed and µ(S(µ) c ) = 0.
Lemma 4. 1 .
1For µ, ν ∈ P(S), it holds that S(µ * ν) = S(µ)S(ν).
.
Let a ∈ S(µ) and b ∈ S(ν). For any open neighborhood U of ab, the joint continuity of the product mapping allows us to take open neighborhoods U 1 of a and U 2 of
a ∈ S(µ)S(ν) c . Then we can take an open neighborhood U of a such that U ⊂ {S(µ)S(ν)} c , so that µ * ν(U) ≤ 1 {S(µ)S(ν)} c (xy)µ(dx)ν(dy) ≤ µ(S(µ) c ) + ν(S(ν) c ) = 0, (4.7)
Proposition 4. 2 .
2Let S be a completely simple Polish semigroup. Let S = LGR denote the Rees decomposotion at e ∈ E(S). For the inverse of the product mapping ψ :
Proposition 4. 3 .
3Let S be a Polish semigroup. Then the convolution mapping P(S) × P(S) ∋ (µ, ν) → µ * ν ∈ P(S) is jointly continuous. Consequently, P(S) is a Polish semigroup.
Theorem 4 . 4 .
44Let S be a Polish semigroup and let µ ∈ P(S). Suppose that µ is both ℓ * -invariant and r * -invariant. Then S(µ) is a compact Polish group, and µ coincides with the normalized unimodular Haar measure on S(µ) (see e.g.[4, Chapter 9] for the Haar measure).
aS(µ) = S(µ). Similarly we have S(µ)b = S(µ). By Lemma 2.3, Proposition 2.5 and Theorem 3.2, we see that S(µ) is a Polish group.
2.4, we have K = S(µ)eS(µ), and hence we obtain K = K = S(µ)eS(µ) = S(µ * δ e * µ) = S(µ),(4.36)
((
ii) The family K of cluster points of {µ n : n = 1, 2, . . .} is a compact abelian group such that iii) Let η denote the identity of K. Then S(η) is a completely simple semigroup. Let S(η) = LHR denote the Rees decomposition at e ∈ E(S(η)). Then H is a compact group and η admits the convolution factorizationη = η L * ω H * η R .(4.46) (iv) S(ν) is a completely simple kernel of S containing the idempotent e. The Rees decomposition of S(ν) at e is of the form S(ν) = LGR, where G is a compact group containing H, and ν admits the convolution factorization
( 4 .
449) and furthermore, K and G/H may be represented asK = {η, µ * η,µ 2 * η, . . .}, G/H = {H, γH, γ 2 H, . . .}. (4.50) Remark 4.10. If the order of the group K or G/H is finite, say p, then K = {η, µ * η, . . . , µ p−1 * η}, G/H = {H, γH, . . . , γ p−1 H} (4.51) with γ p ∈ H. It is now obvious that lim n→∞ µ n converges if and only if p = 1.
Note that, if e is an idempotent, then any element of Se is invariant under right multiplication by e, i.e., x ∈ Se implies xe = x. A semigroup S is called left group [right group] if S is left simple [right simple] and contains at least one idempotent. A semigroup S is called left cancellative [right cancellative] if, for any a, x, y ∈ S with ax = ay [xa = ya], we have x = y. An element e ∈ S is called a left identity [right identity] if ex = x [xe = x] for all x ∈ S. Lemma 2.6. Let S be a semigroup. If S is either right cancellative or left simple, then any idempotent of S is a right identity..
(2.5)
be a Polish semigroup. Then the following assertions hold: (i) For a ∈ S and for a closed [open, Borel] subset A, both a −1 A and Aa −1 are also closed [open, Borel].(ii) If A is a subsemigroup of S, then so is its closure A.
By the Wallace theorem (see, e.g., [21, Theorem 12 of Chapter 5]), we may take open subsets V 1 and V 2
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| []
|
[
"Non-destructive photon detection using a single rare earth ion coupled to a photonic cavity",
"Non-destructive photon detection using a single rare earth ion coupled to a photonic cavity"
]
| [
"Chris O'brien \nInstitute for Quantum Science and Technology\nDepartment of Physics and Astronomy\nUniversity of Calgary\nT2N 1N4CalgaryABCanada\n",
"Tian Zhong \nInstitute for Quantum Science and Technology and Department of Physics and Astronomy\nT.J. Watson Laboratory of Applied Physics\nCalifornia Insitute of Technology\n1200 E California Blvd91125PasadenaCAUSA\n",
"Andrei Faraon \nInstitute for Quantum Science and Technology and Department of Physics and Astronomy\nT.J. Watson Laboratory of Applied Physics\nCalifornia Insitute of Technology\n1200 E California Blvd91125PasadenaCAUSA\n",
"Christoph Simon \nUniversity of Calgary\nT2N 1N4CalgaryABCanada\n"
]
| [
"Institute for Quantum Science and Technology\nDepartment of Physics and Astronomy\nUniversity of Calgary\nT2N 1N4CalgaryABCanada",
"Institute for Quantum Science and Technology and Department of Physics and Astronomy\nT.J. Watson Laboratory of Applied Physics\nCalifornia Insitute of Technology\n1200 E California Blvd91125PasadenaCAUSA",
"Institute for Quantum Science and Technology and Department of Physics and Astronomy\nT.J. Watson Laboratory of Applied Physics\nCalifornia Insitute of Technology\n1200 E California Blvd91125PasadenaCAUSA",
"University of Calgary\nT2N 1N4CalgaryABCanada"
]
| []
| We study the possibility of using single rare-earth ions coupled to a photonic cavity with high cooperativity for performing non-destructive measurements of photons, which would be useful for global quantum networks and photonic quantum computing. We calculate the achievable fidelity as a function of the parameters of the rare-earth ion and photonic cavity, which include the ion's optical and spin dephasing rates, the cavity linewidth, the single photon coupling to the cavity, and the detection efficiency. We suggest a promising experimental realization using current state of the art technology in Nd:YVO4. | 10.1103/physreva.94.043807 | [
"https://arxiv.org/pdf/1608.06679v1.pdf"
]
| 40,268,515 | 1608.06679 | 229305bf768f56acd39ff74c340e0462162a53ec |
Non-destructive photon detection using a single rare earth ion coupled to a photonic cavity
24 Aug 2016
Chris O'brien
Institute for Quantum Science and Technology
Department of Physics and Astronomy
University of Calgary
T2N 1N4CalgaryABCanada
Tian Zhong
Institute for Quantum Science and Technology and Department of Physics and Astronomy
T.J. Watson Laboratory of Applied Physics
California Insitute of Technology
1200 E California Blvd91125PasadenaCAUSA
Andrei Faraon
Institute for Quantum Science and Technology and Department of Physics and Astronomy
T.J. Watson Laboratory of Applied Physics
California Insitute of Technology
1200 E California Blvd91125PasadenaCAUSA
Christoph Simon
University of Calgary
T2N 1N4CalgaryABCanada
Non-destructive photon detection using a single rare earth ion coupled to a photonic cavity
24 Aug 2016
We study the possibility of using single rare-earth ions coupled to a photonic cavity with high cooperativity for performing non-destructive measurements of photons, which would be useful for global quantum networks and photonic quantum computing. We calculate the achievable fidelity as a function of the parameters of the rare-earth ion and photonic cavity, which include the ion's optical and spin dephasing rates, the cavity linewidth, the single photon coupling to the cavity, and the detection efficiency. We suggest a promising experimental realization using current state of the art technology in Nd:YVO4.
I. INTRODUCTION
The ability to detect photonic qubits non-destructively would be very useful for many quantum information applications, including long-distance quantum communication [1,2] and photonic quantum computing [3][4][5]. One approach for non-destructive measurement is to use a single atom or ion that is coupled with high cooperativity to an optical cavity [6]. In particular, Ref. [7] suggested realizing a quantum controlled phase-flip (CPHASE) gate between the photon and the ion based on the fact that, depending on the state of the ion, the photon would either be reflected unchanged or with a π-phase shift. The resulting entanglement between the photon and the ion can be used to detect the photon through readout of the ion's state. These ideas have recently been realized in a series of impressive experiments with single trapped atoms inside free-space high-finesse cavities [8][9][10][11].
For more robust and scalable technologies, it would be useful to be able to implement similar protocols in the solid state. A single rare-earth ion (REI) doped into a crystal is very similar to an optically trapped single atom, when the crystal is cooled to cryogenic temperatures in order to avoid dephasing via coupling to phonons. Rare-earth doped crystals have been successfully used for optical quantum memories [12][13][14], and have been suggested for scalable quantum computing [15]. A scheme for performing non-destructive measurements utilizing an ensemble of rare-earth ions coupled to a bulk crystalline waveguide has recently been suggested [16]. It is also now possible to observe single rare-earth ions in bulk crystal * [email protected] [17][18][19][20][21], and to map between ion-spins and a photon's polarization [22]. There has recently been success in coupling Nd ions doped into yttrium orthosilicate (YSO) [23] and yttrium orthovanadate (YVO) [24] crystals with photonic crystal cavities that were fabricated out of bulk crystal. The advantage of using rare-earth ions compared to other solid state emitters like nitrogen-vacancy centers in diamond or semiconductor quantum dots is that they combine narrow inhomogeneous broadening, low spectral diffusion, close to transform-limited optical line-widths and spin states with a long coherence time.
Using rare-earth ions doped into photonic cavities to realize CPHASE gates was first suggested by [25]. Here we perform an in-depth analysis of this idea with a focus on the implementation of non-destructive photon detection, including a detailed scheme and an accounting for the likely fidelity. Rare-earth ions coupled to nano-photonic resonators will enable an on-chip platform where single ions act like optically addressable single quantum bits that can be interfaced via photons, with the possibility for on-chip photon storage into optical quantum memories made from the same atomic species.
The paper is organized as follows. In section II we introduce the protocol for creating the conditional phase shift. In section III we explain how this setup can be used for quantum non-destructive measurements. In section IV we discuss how the state of the ion can be read out. In section V we calculate the fidelity of the non-destructive measurement as a function of the rare-earth ion and photonic cavity parameters, including the dephasing of the ion's optical and spin transitions, the linewidth of the cavity, the single photon coupling between the ion and the cavity, as well as the probability of successful read out. In section VI we discuss a specific implementation of the protocol in Nd:YVO 4 crystals. In section VII we give some concluding remarks.
II. CONDITIONAL PHASE SHIFT
Consider a single rare-earth ion doped directly into a photonic crystal cavity, where one side is partially transparent and the other end is perfectly reflecting. The incoming photon will interact with the ion-cavity system. If the ion is strongly coupled to the cavity, the photon will reflect off the cavity with a phase that depends on the state of the ion. If the ion is not coupled to the cavity, the photon will enter the cavity and receive a π-phase shift. If the ion is coupled to the cavity, the cavity will not be impedance matched with the photon, so the photon will reflect off the cavity without entering and will not have a phase shift. Thus the reflected photon gains a phase that is dependent on whether the ion is in a state that interacts with the cavity, which creates a CPHASE gate.
If we assume the input field is weak enough to have a low probability to excite the ion, we can write down a set of quantum Langevin equations [25]:
a(t) = (−κ − iδ)a(t) + gs(t) − √ 2κa in (t),(1)s(t) = −ga(t) + (−γ/2 − iδ − i∆)s(t).(2)
Here κ is the decay rate of the cavity, δ is the detuning of the incoming photon from the cavity, γ is the decoherence rate of the ion, g is the single photon coupling between the REI and the cavity, a(t) is the photon excitation amplitude, s(t) is the atomic excitation amplitude, and a in is the amplitude of the photon incident on the cavity. We will see that the probability of a single photon entering the cavity and exciting the atom is inversely proportional to the single ion cooperativity C = g 2 /(κγ) which we take as large and thus justify our assumption that the atom remains in its ground state. These equations can be solved under the assumption that the input field has a narrow frequency range with respect to the dynamics of the atom-cavity system such that we can perform adiabatic elimination, i.e. a(t) =ṡ(t) = 0. Assuming there is no initial excitation of the atom then the output photon can be expressed as a function of the input photon.
a out (δ) = g 2 + (iδ + i∆ + γ/2)(iδ − κ) g 2 + (iδ + i∆ + γ/2)(iδ + κ) a in(3)
This expression covers two cases, when the ion is in resonance with the cavity, we can take ∆ = 0, which will give us the case where the photon does not enter the cavity. Then when we do not want the ion interacting with the cavity, we put the ion into a metastable state that is far detuned from the cavity with ∆ ≫ 0, which will allow the photon to enter the cavity and receive a π-phase shift. In general, for a REI, both of the ground states will interact with an upper transition, for the same polarization of applied light, which is not necessarily the case for a trapped ion, which is why we need to consider the case of the non-resonant transition rather than just assuming that one state of the ion does not interact with the cavity at all as [7,25] assumes. Both transitions being allowed is one difference between the protocol in trapped atoms and REI. Another difference is that REI tend to have weaker dipole moments than those of trapped atoms. For trapped atoms that are strongly coupled to cavities it is typical to be in the "good cavity" regime where g ≫ κ ≫ γ, but for the REI-cavity system this is unlikely. Since the ioncavity coupling is usually weaker due to smaller dipole moments. The REI-cavity system is instead in what is called the "bad cavity" regime [25] where κ ≫ g ≫ γ and yet the single photon cooperativity is still high with C ≫ 1.
In order to understand the details of the CPHASE protocol, we develop a clear analytic picture, as shown in Fig.1. Using the decaying-dressed state analysis from [26,27], we can analyze Eq.(3) to get simple analytic expression for the amount of the phase shifts and the bandwidth over which they occur. In the bad cavity limit, we can expand Eq.(3) into partial fractions. If the ion is in resonance such that ∆ = 0, then
a out a in (δ) = 1+ 2iκ 1 − g 2 /κ 2 δ − iκ + ig 2 /κ − 2ig 2 /κ δ − ig 2 /κ − iγ/2 .(4)
The coupling between the atom and cavity creates a broad region with a HWHM of κ−g 2 /κ where the photon enters the cavity and gets a π-phase shift with a narrow central feature with HWHM of g 2 /κ + γ/2 where the interaction with the atom stops the photon from entering the cavity as shown in Fig. 1(a). For near resonance δ ≈ 0, the ratio of output and input is close to unity when the single photon cooperativity is high
a out a in (0, ∆ = 0) = 1 − κγ g 2(5)
such that an incoming photon is reflected with no phase change. The reflectivity is not exactly unity because a small amount of the photon enters the cavity and is scattered by the atom. Now we consider when the ion is in the far-detuned state such that κ,
∆ ≫ g ≫ γ a out a in (δ) = 1 + 2iκ 1 −g 2 (∆+iκ) 2 δ − ∆g 2 ∆ 2 +κ 2 − iκ 1 −g 2 ∆ 2 +κ 2 + 2iκg 2 (∆+iκ) 2 δ + ∆ 1 +g 2 (∆ 2 +κ 2 ) − iγ/2 − iκg 2 (∆ 2 +κ 2 ) .(6)
Since the detuned transition may be weaker due to the partial selection rules of the REI we label the cavityion coupling for this transition asg to distinguish it. If the atom is in its far-detuned state, then the atomic resonance is far-detuned from the photon frequency with ∆ ≫ γ, then there is the cavity interaction centered at δ = ∆g 2 /(∆ 2 + κ 2 ) and a Fano resonance δ = −∆(1 +g 2 /(∆ 2 + κ 2 )) as shown in Fig. 1(b). The first term has a HWHM of κ(1 −g 2 /(∆ 2 + κ 2 )) and is due to interaction with the cavity, where the photon enters the bad cavity and then leaves with a π-phase shift. The second feature is too far detuned to interact directly with the photon. The ratio for a photon with frequency near the cavity resonance δ ≈ 0 is then
a out a in (0) = −1 − 2ig 2 κ∆ .(7)
The imaginary term is due to residual far-detuned interaction with the ion, which causes a small phase shift of the photon. Eq.(3) was derived in the adiabatic limit, dropping the time derivatives of the field and atom. Now consider the case where the photon that reflects off the cavity has a finite bandwidth. Taking a Gaussian pulse with pulse duration HWHM is T p centered at δ = 0, Eq.(3) can be averaged over the bandwidth of the pulse under the assumption that 1/T p > g 2 /κ. This updates Eq.(5) to
a out a in (∆ = 0) = 1 − κγ g 2 e − κ √ log 2 πTpg 2 ,(8)
such that it now applied to a finite pulse.
III. NON-DESTRUCTIVE PHOTON MEASUREMENT
This setup can be used as a method of entangling photons and single rare-earth ions, for use in quantum computing or for non-destructive photon detection. The basic idea is similar to [9]. To generate entanglement we first initialize the ion in a superposition of the ground states, then an incoming photon will be put in a superposition state with a π-phase shift entangled with the ion. Then by performing a rotation and measurement on the ion state, we can read out whether there was an incoming photon.
First, prepare the ion in a superposition of the two ground states
|φ a = 1 √ 2 ( |0 a + |1 a )(9)
where |1 a is the ground state of the far-detuned transition between |1 a and |e a cavity, and |0 a is the ground state of the resonant transition between |0 a and |e a . with the cavity. For a REI, this superposition can be created by using a pair of externally applied far-detuned Raman pulses to drive the system into this state. For a single REI this is more straightforward then for a crystal with a high density of ions, because with a high density of ions, it is required to use extensive hole-burning [28], to isolate those spins with a particular frequency. The prepared superposition state can live only as long as the coherence remains, so once we turn off the external pulses we will only have a limited time to perform the non-destructive photon measurement depending on the decoherence rate of the spin transition. The photon that reflects off of the cavity can be in a superposition state of a single photon |1 p and the state with no photon is |0 p with an arbitrary phase such that our photon state is
|φ p = c 0 |0 p + c 1 |1 p(10)
which as we showed in Sect.II, will give a π-phase shift to the state where there is both a photon present and the atom is in |0 a . This leads to a combined entangled state:
|φ = 1 √ 2 c 0 |0 p ( |0 a + |1 a ) + 1 √ 2 c 1 |1 p ( |1 a − |0 a )(11)
Then performing a π/2 rotation of the ion state which once again can be performed with external pulses such that
R a (π/2) |0 a = 1 √ 2 |1 a − 1 √ 2 |0 a ,(12)R a (π/2) |1 a = 1 √ 2 |1 a + 1 √ 2 |0 a .(13)
Then the ideal entangled state is
|φ ideal =c 0 |0 p |1 a − c 1 |1 p |0 a(14)
Finally a measurement is made on the atom in the population basis. This completes our non-destructive measurement, since now by detecting the photons emitted into the cavity, we know if a photon reflected off of the cavity. The photon reflected off the cavity may have a phase shift, but is otherwise unchanged by the process. Normally, detecting a photon necessitates its destruction. With a unheralded time bin qubit, a non-destructive photon measurement can be made on both bins in order to know there is a photon in one without destroying the qubit [16]. With a heralded time bin photon, we could entangle the REI with the time bin qubit by limiting our QND measurement to one of the time bins.
IV. READ OUT
In order to identify that a photon has reflected off of the cavity it is necessary to detect that the ion was in the state |0 a . This can be accomplished by optically pumping this level to the excited state with a narrow band laser and then detecting the fluorescence. For a REI the probability of detection is limited by the branching ratios for fluorescence from the excited state to lower energy levels. This is greatly improved by interaction with the cavity, due to the Purcell effect. The Purcell effect is due to the density of states for the cavity being much larger than the density of states for free space. The rate of emission into the cavity is enhanced by the Purcell factor defined as:
F P = 3 4π 2 λ n 3 Q V ,(15)
where λ is the wavelength of the cavity, n is the refractive index of the crystal, Q is the quality factor of the cavity, and V is the mode volume of the cavity. For a two level atom, the Purcell factor is related to the single ion cooperativity through the ratio of the radiative line width γ r to the total linewidth γ through F P = (γ/γ r )C, and thus F P is always larger than C. Thus, for C ≫ 1 there is a much higher chance that fluorescence will be into the cavity mode, rather than free space. If a single rare earth ion is strongly coupled to a high quality photonic crystal cavity, it is possible to reach Purcell factors greater than F P = 1, 000. Now in a multi-level atom, there will be multiple channels for fluorescence, the probability in the bare ion to fluoresce in the desired channel is given by the branching ratio β for that transition. This probability is enhanced by interaction with the cavity such that
p cav = F P β 1 − β + F P β(16)
Then for example if we had a branching ratio of β = 15% and F P = 1000, this would give the probability of fluorescence into the cavity as p cav = 0.994. Utilizing preferential emission into the cavity, means the photons emitted into the cavity must be detected. So sometime after the time-bin photon reflects off the cavity, the optical path should be switched such that any future photons emitted from the cavity can be detected by a single photon detector. Then the detection is limited by the efficiency of the single photon detector, which we will assume is p det = 0.9.
When p cav > p det , the best way to read out the atomic state is to drive the cavity transition itself (σ-polarized) and rely on the Purcell effect to preferentially fluoresce into the cavity mode in order to have a cycling transition such that if the ion is in the proper state, many photons are emitted into the cavity. If the ion does not fluoresce into the cavity, the photon is lost and the population cycling ends, which means this method has a maximum efficiency of p cav .
One issue with this method is there is a possibility that photons from the pump will be scattered into the cavity mode. This means that, besides detector dark-counts, scattering will also lead to false-positives. In this case, it is necessary to detect some minimum number of photons n M , in order to discriminate against false positives. Then the detection efficiency can be written as a sum over the number of photons created in the cavity, with a factor giving the probability of n photons being emitted and a factor determining the probability of detecting at least n M photons when n photons are in the cavity
η det = ∞ n=1 p n cav (1 − p cav ) n k=nM n k p k det (1 − p det ) n−k .(17)
For example, if we choose to detect n M = 2 photons, to try to reduce dark counts, with p cav = 0.994 and p det = 0.9, then the detection efficiency will be η det = 0.987. If it is determined that more photons are needed to get a signal above the background of detector dark-counts and scattered light, then this efficiency does not decrease much, for n M = 4, the efficiency is only a little lower η det = 0.974. Thus for the rest of the paper, we will assume this read-out method. The entire detection process must be completed before the state |0 , relaxes to the ground state |1 , but the spin relaxation time T 1 , is quite long for rare-earth ions kept below 7K, on the order of 100ms. This gives plenty of time to complete the read-out process. Another concern is the possibility of false positives due to accidentally driving state |1 to emit a photon into the cavity. Since the Purcell effect guarantees a high probability that a photon will be emitted into the cavity if the off-transition is driven to the alternate excited state labeled |e ′ in Fig.2, we just need to calculate the probability to excite the far-detuned transition. The pump laser needs to have a Rabi frequency Ω =μE p /h that is [29]. In the presence of a 300mT magnetic field, applied at a 45% degree angle with respect to the crystal axis, the detuning for the non-resonant transition is ∆ = 2π ·9GHz.
large enough to achieve Rabi flopping with a significant amount of the population reaching the excited state in order to have fast read-out. Hereμ is the dipole moment of the driven transition, E p is the pump electric field, and h is Planck's constant divided by 2π. At the same time a larger Ω leads to quicker read-out but also leads to a higher chance of driving the off-transition which may lead to a false positive. A good compromise is to take Ω ≥ γ, but of similar magnitude. Any lower Ω will lead to lower excited state population, which slows down the emission while the probability of exciting the off transition is equal to:
p of f = |Ω| 2 ∆ 2 n cycg 2 g 2 > γ 2 ∆ 2 n cycg 2 g 2(18)
where n cyc is the average number of cycles that the driven transition goes through which for p det = 0.994 is n cyc = 116, and there is a factorg 2 /g 2 to account for the possibility that the off-resonant transition is weaker than the resonant transition. We will estimate p of f for our different schemes in Sect.VI, but in general it can be kept small enough to not hamper the fidelity. At the same time, the selection rules for driving transitions in the REI are not perfect, such that sigma polarized light may still drive a predominately π-polarized transition, therefore read out may also lead to a false positive due to population in |1 being driven to |e , then emitting into the cavity. This probability is similar to that given by Eq. (18), now withg being the reduced interaction due to the polarization mismatch and ∆ is just the energy difference between |0 and |1 . Therefore, this probability is also low and can be safely neglected.
There are a few other ways to detect the atomic state.
One approach to spin selective detection is to pump the ground state into a higher level that has a fast nonradiative decay to our excited state, which then will preferentially fluoresce back to the ground state, as demonstrated in Pr:YSO [20]. This method is too slow and does not strongly discriminate between the spin states. Another approach, which is ideal when p det > p cav , is to drive a π-polarization transition of the REI to cycle population back and forth between state |1 and the excited state |e as shown in Fig.2. Since an ion in |1 will cycle until it emits a single photon into the cavity, the detection efficiency is only limited by the detector efficiency η det = p det . Another approach is to use the phase shift present in the CPHASE gate, to detect if the ion is excited by reflecting a weak coherent beam off of the cavity and then measuring the phase shift. The last approach is to utilize the change in reflectivity of the cavity when an ion is coupled to it, by reflecting a weak coherent pulse off the cavity and measuring the transmission as analyzed in [30]. These last two techniques can have close to perfect read-out in a single-pass with the use of many photons. But for the current scheme, the number of photons must be limited to prevent exciting the ion.
V. FIDELITY
In order to calculate the fidelity of the non-destructive photon measurement, we will work through the entire process. In order to consider decoherence, this analysis is performed on a mixed state, using the density matrix formalism. To simplify this analysis we will consider only the case when a single photon is present, which leads to a 2 × 2 atomic density matrix. The presence of a photon is the worse case for the fidelity as in the vacuum case the photon does not interact directly with the cavity, so this assumption is justified. The process starts with the ion in the ground state |1 a . The next step is to rotate the ion into a superposition state ( |0 a + |1 a )/ √ 2 by applying a π/2 rotation through the application of external fields. If this rotation is not perfect, then the rotation would be at an angle π/2 + φ P where φ P is some small angle deviation. Then the ρ a becomes R π/2+φP ρ a R † π/2+φP . Now the ion will undergo dephasing, if we assume this is pure dephasing and not spin flipping, then this is handled by introducing a dephasing rate γ gs . This dephasing continues for the entire time that the atom remains in the superposition state which we will assume is a time period T sp . This period is at least as long as the time-bin photon, but in practice may need to be longer. Then the density matrix is
ρ a = 1 2 − φP 2 ( 1 2 − φ 2 P 4 )e −γgsTsp ( 1 2 − φ 2 P 4 )e −γgsTsp 1 2 + φP 2 .(19)
Now consider the case that a photon reflects off the cavity then from Eq.(8) and Eq. (7), the new density matrix is
ρ a = 1 2 (1 − φ P ) 1 − κγ g 2 2 e −2 κ √ log 2 πTpg 2 − 1 2 (1 − φ 2 P 2 )(1 − κγ g 2 )(1 − 2ig 2 κ∆ )e −γgsTsp e − κ √ log 2 πTpg 2 − 1 2 (1 − φ 2 P 2 )(1 − κγ g 2 )(1 + 2ig 2 κ∆ )e −γgsTsp e − κ √ log 2 πTp g 2 1 2 (1 + φ P )|1 + 2ig 2 κ∆ | 2 . (20)
The state must be rotated again by π/2 for read out. Assuming a small error in creating the phase shift φ R such that we rotate through π/2 + φ R . If we assume each correction is small and keep only the first order terms then the density matrix becomes
ρ a = 1 − κγ g 2 − κ √ log 2 πTpg 2 − 1 2 γ gs T sp − 1 4 φ 2 R + φ 2 p − 1 2 κγ g 2 − 1 2 κ √ log 2 πTpg 2 − i g 2 κ∆ + 1 2 (φ R − φ p ) − 1 2 κγ g 2 − 1 2 κ √ log 2 πTpg 2 + i g 2 κ∆ + 1 2 (φ R − φ p ) 1 2 γ gs T sp + 1 4 φ 2 R + φ 2 p .(21)
Defining the fidelity as
F = min φ ideal | ρ a |φ ideal(22)
where the ideal output state is given by Eq. (14).
|φ ideal = |0 a |1 p .(23)
Then expanding the square root of Eq. (22) and keeping the lowest order term in each correction, the fidelity is approximately
F = η det (1 − κγ 2g 2 − κ √ log 2 2πT p g 2 − 1 4 γ gs T sp − 1 8 (φ 2 R + φ 2 P )),(24)
where η det is the efficiency of detecting the ideal state as discussed in Sect.IV. The fidelity is reduced by κγ 2g 2 due to imperfect reflection of the photon, by κ √ log 2 2πTpg 2 due to the finite bandwidth of the reflected photon, by 1 4 γ gs T sp due to dephasing while the atom is in the superposition state, and by 1 8 (φ 2 R +φ 2 P ) due to imperfect rotations when realizing the CPHASE gate.
For high fidelity we need high cooperativity C = g 2 /(κγ) ≫ 1 which implies we need high quality cavities. But the main limitation on the fidelity is the combination of needing the factor γ gs T sp to be small while the factor κ √ log 2/(2πT p g 2 ) puts a lower limit on the pulse duration, such that the photon spectrum fits into the narrow bandwidth of the resonant feature. The combination of these two factors will limit the overall fidelity, leading to one ideal pulse time, since T p is bound on both sides. The last term due to imperfect rotations is actually quite small, if we make the cautious assumption that the area of the pulses is off by as much as 1%, then the fidelity is only reduced by 0.4%, and likely the pulse areas can be made more accurate than that, so we can safely neglect this term.
VI. IMPLEMENTATION
We need a single ion, strongly coupled to a cavity. Faraon et al. [23,24] are building photonic cavities which strongly couple to a number of rare-earth ions. In order to have a single ion coupled to the photonic cavity, the ion density can be lowered until only a single ion couples to the cavity, but then the single ion may not be near the peak of the cavity mode, and also may not have the right frequency. There has also been work on using an ion beam to implant single REI into a pure crystal with Cerium ions implanted in YAG [31] and Erbium ions implanted in YSO [32], which currently makes small spots of 1,000's of REI, but could be scaled down to implanting a single REI.
In order to implement this protocol in a single rare earth ion, a long lived shelving state is needed, ideally a split ground state. This ground state splitting must be large enough such that one of the states is far-detuned such that it can not interact with the photon and cavity while the other transition is in resonance. Neodymium has a 9GHz separation in the presence of a 300mT magnetic field. Such a large magnetic field is not necessary, but is routinely used. We consider Neodymium because we have reliable data for it in a variety of crystals and coupling to a photonic crystal was already demonstrated, but it could be that other ions will work just as well or better.
Nd:YVO 4 is an attractive implementation, since the Nd has a higher dipole moment in YVO 4 , compared with YSO. The energy diagram is shown Fig.2. High quality resonators which are capable of coupling to a single rareearth ion, have recently been developed [24]. The cavity has a mode volume of V = (λ/n Y V O ) 3 = 0.064µm 3 (where λ = 879.7nm is the Nd linewidth and n Y V O = 2.2 is the refractive index of the YVO 4 crystal) and a quality factor of Q = 20, 000. The electric field for a single photon in the cavity is given by:
E = hω c 2ǫ 0 V(25)
Where ω c is the frequency of the cavity and V is the cavity mode volume. be derived from the cavity frequency and quality factor.
κ = 1 2 ω c Q .(26)
Then the cavity width is κ = 2π · 8.5GHz. The optical T 2 time for the Nd ion doped into YVO was measured to be 27µs with a 1.5T magnetic field [33], which gives a decoherence rate of γ = 2π · 5.9kHz. With this field, the detuning can be as high as ∆ = 2π · 30GHz. The transition in Nd:YVO 4 has a wavelength of 880nm, and according to [25] has a dipole moment is µ = 9.1 · 10 −32 Cm. Then the single photon Rabi frequency or the cavity-photon coupling is:
g = µE 2h ,(27)
such that g = 2π · 30.6MHz. Then the single ion cooperativity C = g 2 /(κγ) = 246. We can also calculate the Purcell factor using Eq.(15), which is F P = 1520. By comparing the radiative decay rate for the transition and the total lifetime of the excited state we can find that the branching ratio from the excited state to the ground state is β = 10.4%, then the probability of emitting into the cavity is p cav = 0.9985, thus the detection efficiency to detect a minimum of two photons, assuming the probability of detecting a single photon is p det = 0.9, given by Eq. (17) is η det = 98.8%. The Nd electron spin lifetime is quite long in Nd:YSO it was measured at T 1 = 100ms [34], giving ample time for the read-out process before the excitation in |0 a decays. The Nd electron spin coherence time is shorter at T 2 = 471µs at 5K [34], which gives γ gs = 2π · 0.34kHz. At low temperatures, similar values should be possible for Nd:YVO 4 . At the same time our pulses need to be long enough to fit into the limited bandwidth of the resonant response from Eq.(8), 1/T p ≪ δω = 2π · 1.3MHz, so we have to make a compromise in pulse duration. We plot the fidelity as a function of T p in Fig.3, here showing that pulses of length T p = 13µs are ideal. Then Eq. (24) gives a maximum fidelity of F = 93.4%. At sub-Kelvin temperatures the spin coherence time can be an order of magnitude higher γ gs = 2π · 34Hz, which improves the fidelity to F = 95.3% as shown in Fig.3.
We also need to make sure the chance of false positives given by Eq. (18) is very low. For Nd:YVO 4 the detuning is large, the width of the ion is small, but both transitions are equally allowed sog = g. These combine to give a probability that is quite low at p of f = 0.01% and can safely be neglected.
There is no reason that cavities can not be improved to reach higher quality factors, in [24] the theoretically possible quality factor is Q = 300, 000 with the same mode volume V = (λ/n Y V O ) 3 . Then κ = 2π · 565MHz and the cavity-photon coupling is still g = 2π · 30.6MHz. Then the single ion cooperativity is C = 7392. The Purcell factor would be F P = 22, 797, giving a detection efficiency of η = 99.1%. Then with γ gs = 2π · 34Hz ideal pulse length is 11µs and the fidelity would be F = 99.5%, as shown in Fig.3.
VII. CONCLUSION
We have demonstrated that a CPHASE gate between a single photon and a rare-earth doped ion coupled to a photonic cavity is possible in the bad cavity regime. We have then shown that this gate can be used to make non-destructive measurements of a single photon. We suggested implementing a non-destructive photon measurement in Nd:YVO 4 and calculated the expected fidelity, concluding that high fidelities are within reach of current technology. A fidelity of 95.3% is currently possible, and a theoretical maximum fidelity of 99.5% could be achieved. Our results show that photonic crystal cavities coupled to individual rare-earth ions are a promising platform for implementing non-destructive photon detection in solid-state systems.
VIII. ACKNOWLEDGMENTS
This work was supported by NSERC (Canada). AF and TZ acknowledge support from National Science Foundation CAREER award 1454607. We thank Dr. John Bartolomew for useful discussion.
FIG. 1 .
1(color online) Plot of the real (solid, red) and imaginary (dashed, blue) parts of aout/ain given by Eq.(3), in the bad cavity regime. The plot lists the frequency widths and amplitudes that are analytically derived in Sect.II. The parameters are normalized to g = 1, with κ = 10g, and γ = 0.01g. (a) When the atom is in the state that is resonant with the cavity, plotted in units of δ/κ. Inset: zooming in on near resonance plotting in units of δ/g. This is the case that limits the bandwidth of an input photon. (b) We plot the ratio aout/ain given by Eq.(3) in units of δ/κ, for when the atom is in the state that is far-detuned from the cavity, with the previous values and a detuning of ∆ = 20g, we also assume thatg = g.
FIG. 2 .
2(color online) Energy level diagram of 879.7nm transition in Nd:YVO4. Showing the qubit states, and how the ion interacts with σ-polarized light, i.e. electric field polarized parallel to the crystal axis. Here the |0 → |e transition is resonant with the cavity. Both optical transitions have the similar g
Then for Nd:YVO 4 we have E = 446, 229V/m. The HWHM linewidth of the cavity canFIG. 3. (color online) Predicted fidelity for Nd:YVO4 as a function of Tp the time width of the pulse to be detected, for the parameters listed in the text. The blue (solid) line is the fidelity for the experimentally demonstrated numbers listed in the text, the green (dot-dashed) line is for experimentally demonstrated numbers except with the lowest observed spin decoherence rate γgs = 2π ·0.34kHz, and the red (dashed) line is for a more optimistic theoretic quality factor of Q=300,000.0
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"Combinatorial Approach to Modeling Quantum Systems",
"Combinatorial Approach to Modeling Quantum Systems"
]
| [
"Vladimir V Kornyak \nLaboratory of Information Technologies\nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n"
]
| [
"Laboratory of Information Technologies\nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia"
]
| []
| Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers and unitarity in the formalism of the quantum mechanics. In our approach, the quantum behavior can be explained by the fundamental impossibility to trace the identity of the indistinguishable objects in their evolution. Any observation only provides information about the invariant relations between such objects. The trajectory of a quantum system is a sequence of unitary evolutions interspersed with observations -non-unitary projections. We suggest a scheme to construct combinatorial models of quantum evolution. The principle of selection of the most likely trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with the appropriate Lagrangians and deterministic evolution equations. 7KLV LV DQ 2SHQ $FFHVV DUWLFOH GLVWULEXWHG XQGHU WKH WHUPV RI WKH &UHDWLYH &RPPRQV $WWULEXWLRQ /LFHQVH ZKLFK SHUPLWV XQUHVWULFWHG XVH GLVWULEXWLRQ DQG UHSURGXFWLRQ LQ DQ\ PHGLXP SURYLGHG WKH RULJLQDO ZRUN LV SURSHUO\ FLWHG 4 7 7Article available at | 10.1051/epjconf/201610801007 | [
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| 119,115,081 | 1507.08896 | 95749081525b6b2bdf20d66ed23649d8b9016a38 |
Combinatorial Approach to Modeling Quantum Systems
Vladimir V Kornyak
Laboratory of Information Technologies
Joint Institute for Nuclear Research
141980Dubna, Moscow RegionRussia
Combinatorial Approach to Modeling Quantum Systems
10.1051/epjconf/201610801007
Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers and unitarity in the formalism of the quantum mechanics. In our approach, the quantum behavior can be explained by the fundamental impossibility to trace the identity of the indistinguishable objects in their evolution. Any observation only provides information about the invariant relations between such objects. The trajectory of a quantum system is a sequence of unitary evolutions interspersed with observations -non-unitary projections. We suggest a scheme to construct combinatorial models of quantum evolution. The principle of selection of the most likely trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with the appropriate Lagrangians and deterministic evolution equations. 7KLV LV DQ 2SHQ $FFHVV DUWLFOH GLVWULEXWHG XQGHU WKH WHUPV RI WKH &UHDWLYH &RPPRQV $WWULEXWLRQ /LFHQVH ZKLFK SHUPLWV XQUHVWULFWHG XVH GLVWULEXWLRQ DQG UHSURGXFWLRQ LQ DQ\ PHGLXP SURYLGHG WKH RULJLQDO ZRUN LV SURSHUO\ FLWHG 4 7 7Article available at
Introduction
Any continuous physical model is empirically equivalent to a certain finite model. This is widely used in practice: solutions of differential equations by the finite difference method or by using truncated series are typical examples. It is often believed that continuous models are "more fundamental" than discrete or finite models. However, there are many indications that the nature is fundamentally discrete at small (Planck) scales, and is possibly finite. 1 Moreover, the description of the physical systems by, e.g., differential equations can not be fundamental in principle, since it is based on approximations of the form f (x) ≈ f (x 0 ) + ∇ f (x 0 )Δx. In this paper we consider some approaches to constructing discrete combinatorial models of the quantum evolution.
The classical description of a reversible dynamical system looks schematically as follows. There are a set W of states 2 and a group G cl ≤ Sym(W) of transformations (bijections) of W. Evolutions of W are described by sequences of group elements g t ∈ G cl parameterized by the continuous time t ∈ T = [t a , t b ] ⊆ R. The observables are functions h : W → R.
An arbitrary set W can be "quantized" by assigning numbers from a number system F to the elements w ∈ W, i.e., by interpreting W as a basis of the module F ⊗W . The quantum description of a a e-mail: [email protected] 1 The total number of binary degrees of freedom in the Universe is about 10 122 as estimated via the holographic principle and the Bekenstein-Hawking formula. 2 The set W often has the structure of a set of functions: W = Σ X , where X is a space, and Σ is a set of local states. dynamical system assumes that the module spanned by the set of classical states W is a Hilbert space H W over the field of complex numbers, i.e., F = C. The transformations g t and the observables h are replaced by unitary, U t ∈ Aut (H W ), and Hermitian, H, operators on H W , respectively. A constructive version of the quantum description is reduced to the following:
• time is discrete and can be represented as a sequence of integers, typically T = [0, 1, . . . , T ];
• the set W is finite and, respectively, the space H W is finite-dimensional;
• the general unitary group Aut (H W ) is replaced by a finite group G;
• the field C is replaced by Kth cyclotomic field Q K , where K depends on the structure of G;
• the evolution operators U t belong to a unitary representation of G in the Hilbert space H W over Q K .
It is clear that a single unitary evolution is not sufficient for describing the physical reality. Such evolution is nothing more than a physically trivial change of coordinates (a symmetry transformation). This means that observable values or relations, being invariant functions of states, do not change with time. As an example, consider a unitary evolution of a pair of state vectors: |ϕ 1 = U |ϕ 0 , |ψ 1 = U |ψ 0 . For the scalar product we have
ϕ 1 | ψ 1 = ϕ 0 | U −1 U |ψ 0 ≡ ϕ 0 | ψ 0 .
There are two ways to obtain observable effects in the scenario of unitary evolution: (a) in quantum mechanics measurements are described by non-unitary operators -projections into subspaces of the Hilbert space; (b) in gauge theories collections of evolutions are considered, and comparing results of different evolutions can lead to observable effects (in the case of a non-trivial gauge holonomy).
The role of observations in quantum mechanics is very important -it is sometimes said that "observation creates reality". 3 We pay special attention to the explicit inclusion of observations in the models of evolution. While the states of a system are fixed in the moments of observation, there is no objective way to trace the identity of the states between observations. In fact, all identificationsi.e., parallel transports provided by the gauge group which describes symmetries of the states -are possible. This leads to a kind of fundamental indeterminism. To handle this indeterminism we need a way to describe statistically collections of parallel transports. Then we can formulate the problem of finding trajectories with maximum probability that pass through a given sequence of states fixed by observations. In a properly formulated model, the principle of selection of the most probable trajectories should reproduce in the continuum limit the principle of the least action.
Constructive description of quantum behavior
The transition from a continuous quantum problem to its constructive counterpart can be done by replacing a unitary group of evolution operators with some finite group. To justify such a replacement [1] one can use the fact from the theory of quantum computing that any unitary group contains a dense finitely generated subgroup. This residually finite [2] group has infinitely many finite homomorphic images. The infinite set of non-trivial homomorphisms allows to find a finite group that is empirically equivalent to the original unitary group in any particular problem.
Permutations and natural quantum amplitudes
As it is well known, any representation of a finite group is a subrepresentation of some permutation representation. Namely, a representation U of G in a K-dimensional Hilbert space H K can be embedded into a permutation representation P of G in an N-dimensional Hilbert space H N , where N ≥ K. The representation P is equivalent to an action of G on a set of things Ω = {ω 1 , . . . , ω N } by permutations. If K = N then U P. Otherwise, if K < N, the embedding has the structure
T −1 PT = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 V H N−K U} H K ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , H N = H N−K ⊕ H K .
Here 1 is the trivial one-dimensional representation. It is a mandatory subrepresentation of any permutation representation. V is an optional subrepresentation. We can treat the unitary evolutions of data in the spaces H K and H N−K independently, since both spaces are invariant subspaces of H N . The embedding into permutations provides a simple explanation of the presence of complex numbers and complex amplitudes in the formalism of the quantum mechanics. We interpret complex quantum amplitudes as projections onto invariant subspaces of vectors with natural components for a suitable permutation representation [1,3,4]. It is natural to assign natural numbers -multiplicities -to elements of the set Ω on which the group G acts by permutations. The vector of multiplicities,
|n = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ n 1 . . . n N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
is an element of the module H N = N N , where N = {0, 1, 2, . . .} is the semiring of natural numbers. The permutation action defines the permutation representation of G in the module H N . Using the fact that all eigenvalues of any linear representation of a finite group are roots of unity, we can turn the module H N into a Hilbert space H N . We denote by N K the semiring formed by linear combinations of Kth roots of unity with natural coefficients. The so-called conductor K is a divisor of the exponent 4 of G.
In the case K > 1 the semiring N K becomes a ring of cyclotomic integers. The introduction of the cyclotomic field Q K as the field of fractions of the ring N K completes the conversion of the module H N into the Hilbert space H N . If K > 2, then Q K is empirically equivalent to the field of complex numbers C in the sense that Q K is a dense subfield of C.
Measurements and the Born rule
A quantum measurement is, in fact, a selection among all the possible state vectors that belong to a given subspace of a Hilbert space. This subspace is specified by the experimental setup. The probability to find a state vector in the subspace is described by the Born rule. There have been many attempts to derive the Born rule from other physical assumptions -the Schrödinger equation, Bohmian mechanics, many-worlds interpretation, etc. However, the Gleason theorem [5] shows that the Born rule is a logical consequence of the very definition of a Hilbert space and has nothing to do with the laws of evolution of the physical systems. The Born rule expresses the probability to register a particle described by the amplitude |ψ by an apparatus configured to select the amplitude |φ by the formula (in the case of pure states):
P(φ, ψ) = | φ | ψ | 2 φ | φ ψ | ψ ≡ ψ| Π φ |ψ ψ | ψ ≡ tr Π φ Π ψ ,
where Π a = |a a| a | a is the projector onto the subspace spanned by |a .
Remark. In the "finite" background the only reasonable interpretation of probability is the frequency interpretation: probability is the ratio of the number of "favorable" combinations to the total number of combinations. So we expect that P(φ, ψ) must be a rational number if everything is arranged correctly. Thus, in our approach the usual non-constructive contrapositioncomplex numbers as intermediate values vs. real numbers as observable values -is replaced by the constructive oneirrationalities vs. rationals. From the constructive point of view, there is no fundamental difference between irrationalities and constructive complex numbers: both are elements of algebraic extensions.
Illustration: constructive view of the Mach-Zehnder interferometer
The Mach-Zehnder interferometer is a simple but important example of a two-level quantum system. The device consists of a single-photon light source, beam splitters, mirrors and photon detectors (see Figure 1). Consider a two-dimensional Hilbert space spanned by the two orthonormal basis vectors | -"right upward beams", and | -"right downward beams". Then the 50/50 beam splitter (i.e., a photon has equal probability of being reflected and transmitted) is described by the matrix
S = 1 √ 2 1 i i 1 .(1)
The mirror matrix is M = 0 i i 0 . Notice that M = S 2 , and, on the other hand, S can be expressed via M as an element of the group algebra: S = 1 √ 2 (I +M), where I is the identity matrix. The scheme in the figure implements the unitary evolution SMS | = S 4 | = − | , which means that only the upper detector will register photons, the lower detector will always be inactive.
This device is able to demonstrate many interesting features of the quantum behavior. Consider, for example, the scheme of quantum interaction-free measurement proposed by Elitzur and Vaidman [6]. The Penrose's version of this example is called the bomb-testing problem. Suppose we have a collection of bombs, of which some are defective. The detonator of a good bomb causes explosion after absorbing a single photon. The detonators of defective bombs reflect photons without any consequences. Classically, the only way to verify that a bomb is good is to touch the detonator. However, as shown in Figure 2, the quantum interference makes it possible to select 25% of good bombs without exploding them: the signal of the lower detector ensures that the unexploded bomb is good.
A slight modification of the scheme shown in Figure 1 allows us to implement any unitary operator U ∈ U(2) by the Mach-Zehnder interferometer. This is easily verified by direct calculation. Since dim U(2) = 4, we should add four parameters in a proper way. For example, we can change the transparency of the beam splitter. Mathematically this means replacing the matrix (1) by another one of the form α I +β M, where |α| 2 + |β| 2 = 1. Another possibility is to introduce phase shifters. The phase shifter matrix related, e.g., to a "right upward beam" has the form e i ω 0 0 1 . Moreover, combining many Mach-Zehnder interferometers [7], one can realize elements of any unitary group U(n).
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01007-p.4 | SMS − −− → − | P = 1 testing defective bomb | Π S − −−→ i √ 2 | P = 1 2 good bomb exploded | Π S M Π S − −−−−−−−− → − 1 2 | P = 1 4 bomb remains untested | Π S M Π S − −−−−−−−− → i 2 | P = 1 4
bomb is good and intact Figure 2. Penrose bomb tester. P is the probability of a branch of evolution. Π a denotes the projector onto |a .
Since a "mirror" is the square of a "beam splitter", any unitary evolution in a sequence of balanced Mach-Zehnder interferometers can be described by degrees of S . The operator S generates the cyclic group Z 8 . The smallest degree faithful action of Z 8 is realized by permutations of 8 objects. Any of the four permutations, that generate Z 8 as a group of permutations, can be put in correspondence with the beam splitter, e.g., S ←→ g = (1, 2, 3, 4, 5, 6, 7, 8). The generator g can be represented by a matrix P g acting in the module N 8 that consists of the vectors with natural components: N = (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 , n 7 , n 8 ) T ∈ N 8 .
To "extract" the beam splitter from the matrix P g we should extend the natural numbers by 8th roots of unity -the conductor K = 8 in this case. Any 8th root of unity can be represented as a power of any of the four primitive roots defined by the cyclotomic polynomial Φ 8 (r) = r 4 +1. Let us denote by N 8 the set of linear combinations of 8th roots of unity with natural coefficients. This is a ring since K = 8 > 1. The ring N 8 is isomorphic to the ring of 8th cyclotomic integers. In principle, due to the projective nature of the quantum states, we could perform all calculations using only natural numbers and roots of unity. But it is convenient to use also the 8th cyclotomic field, which we will denote by Q 8 . The field Q 8 is the fraction field of the ring N 8 .
The matrix P g by a transformation T over the field Q 8 can be reduced to the form
S g = T −1 P g T = A 0 0 S r ,
where A = diag 1, −1, r 2 , − r 2 , r 3 , − r , r is a primitive 8th root of unity, and S r = 1 2 r − r 3 r + r 3 r + r 3 r − r 3
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is the beam splitter matrix S expressed in terms of cyclotomic numbers. The quantum amplitude of the Mach-Zehnder interferometer can be approximated by the projection of the natural vector N into the "splitter" subspace:
|ψ = ψ 1 ψ 2 = 1 8 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ − r 3 (n 1 + n 3 − n 5 − n 7 ) + 1 − r 2 (n 2 − n 6 ) r (n 1 − n 3 − n 5 + n 7 ) + 1 + r 2 (−n 4 + n 8 ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .(3)
It can be shown that the expression (3) can approximate with arbitrary precision any point on the Bloch sphere -a standard representation of the complex projective line CP 1 .
Combinatorial models of evolution
Let us begin with some general considerations concerning the evolution of the probabilistic systems subject to observations. The evolution of such a system can be described as follows. We have a fundamental ("Planck") time which is the sequence of integers:
T = [0, 1, . . . , T ] .(4)
There is also a sequence of "times of observations". For simplicity, we assume that the observation time is a subsequence of the fundamental time
T = [t 0 = 0, . . . , t i−1 , t i , . . . , t N = T ](5)
(otherwise we could assume that the times of the observations are not determined exactly, e.g., they could be random variables with probability distributions localized within subintervals of the fundamental time). Let W t i denote the state of a system observed at the time t i , and
W t 0 → · · · →W t i−1 →W t i → · · · →W t N(6)
denote a trajectory of the system. Whereas the states W t i−1 and W t i are fixed by observation, the transition between them can be described only probabilistically. The selection of the most probable trajectories is the main problem in the study of the evolution. If we can specify P W t i−1 →W t i -the one-step transition probability -then the probability of trajectory (6) can be calculated as the product
P W t 0 →···→W t N = N i=1 P W t i−1 →W t i .(7)
The inconvenience of dealing with the product of large number of multipliers can be eliminated by introducing the entropy, which is defined as the logarithm of probability. The transition to logarithms allows us to replace the products by sums. On the other hand, taking the logarithm does not change the positions of the extrema of a function due to the monotonicity of the logarithm. Thus, for searching the most likely trajectories we introduce the one-step entropy
S W t i−1 →W t i = log P W t i−1 →W t i(8)
and use instead of (7) the entropy of trajectory:
S W t 0 →···→W t N = N i=1 S W t i−1 →W t i .(9)
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The formulation of any dynamical model usually begins with postulating a Lagrangian. However, it would be desirable to derive Lagrangians from more fundamental principles. One can see that continuum approximations of (8) and (9) lead to the concepts of Lagrangian and action, respectively. The reasoning is schematically the following. The states W t i are specified by sets of numerical parameters (coordinates) X t i = X 1,t i , X 2,t i , . . . , X K,t i . For a specific model one-step entropy (8) can be calculated as a function of the coordinates:
S W t i−1 →W t i = S X t i , ΔX t i , where ΔX t i = X t i − X t i−1 .
Assuming that N → ∞, t i −t i−1 → 0 and embedding the sequence X t i into the continuous function X(t), we can represent the one-step entropy in the form S (X(t i ) , ΔX(t i )) . The second order Taylor approximation of this function has the form
S ≈ A + b kk ΔX k (t i ) − ΔX * k (t i ) ΔX k (t i ) − ΔX * k (t i ) , where ΔX * (t i )
is the solution of the system of equations ∂S ∂ΔX(t i ) = 0. Since the discrete time is a dimensionless counter, the differences can be approximated in the continuum limit by introducing derivatives, and we come to the Lagrangian
L = A + B kk dX k dt − a k dX k dt − a k ,
where B kk is a negative definite quadratic form; B kk , A and a k depend on X 1 (t) , X 2 (t) , . . . , X K (t) .
The action S = Ldt is a continuum approximation of the entropy of trajectory (9), so the principle of least action can be treated as a continuous remnant of the principle of selection of the most likely trajectories.
Example: extracting Lagrangian from combinatorics
As an illustration of the above let us consider the one-dimensional random walk. This model studies the statistics of sequences of positive ( + 1) and negative ( − 1) unit steps on the integer line Z. Any statistical description is based on the concepts of microstates and macrostates -the last can naturally be treated as equivalence classes of microstates [8]. In this model, microstates are individual sequences of steps. The probability of a microstate consisting of k + positive and k − negative steps is equal to α k + + α k − − , where α + and α − denote probabilities of single steps (α + + α − = 1). The macrostates are defined by the equivalence relation: two sequences u and v are equivalent if k u
+ + k u − = k v + + k v − = t and k u + − k u − = k v + − k v − = x,
i.e., both sequences have the same length t and define the same point x on Z. The probability of an arbitrary microstate to belong to a given macrostate is described by the binomial distribution, which in terms of the variables x and t takes the form
P (x, t) = t! t+x 2 ! t−x 2 ! 1 + v 2 t+x 2 1 − v 2 t−x 2 ,(10)
where v = α + − α − is the "drift velocity". 5 Obviously, − 1 ≤ v ≤ 1.
Let [x 0 , . . . , x i−1 , x i , . . . , x N ] be a sequence of points (observed values) corresponding to the sequence of times of observations (5). We assume that the time differences Δt i = t i − t i−1 are much larger than the unit of fundamental time (4) but much less than the total time: 1 Δt i T . Applying formula (10) to ith time interval we can write the one-step entropy:
S x i−1 →x i = ln Δt i ! − ln Δt i + Δx i 2 ! − ln Δt i − Δx i 2 ! + Δt i + Δx i 2 ln 1 + v i 2 + Δt i − Δx i 2 ln 1 − v i 2 ,
where Δx i = x i − x i−1 , and v i denotes the drift velocity in the ith interval. Applying the Stirling approximation, ln n! ≈ n ln n − n, we have
S x i−1 →x i ≈ S i = Δt i ln Δt i − Δt i + Δx i 2 ln Δt i + Δx i 1 + v i − Δt i − Δx i 2 ln Δt i − Δx i 1 − v i .(11)
Solving the equation ∂S i /∂Δx i = 0 we obtain the stationary point:
Δx * i = v i Δt i .
Replacing the sequences x i , v i by continuous functions x (t) , v (t) and introducing the approximation Δx i ≈ẋ (t) Δt i in the second order Taylor expansion of (11) around the point Δx * i we have finally
S x i−1 →x i ≈ − 1 2 ẋ (t) − v √ 1 − v 2 2
Δt i .
Thus we come to the Lagrangian
L = ẋ (t) − v √ 1 − v 2 2
with the corresponding Euler-Lagrange equation
d dt ∂L ∂ẋ − ∂L ∂x = 0 =⇒ẍ 1 − v 2 + 2ẋv ∂v ∂t − 1 + v 2 ∂v ∂t = 0 .
Scheme for constructing models of quantum evolution
The trajectory of a quantum system is a sequence of observations with unitary evolutions between them. We propose a scheme to construct quantum models that combine unitary evolutions with observations. The scheme assumes that transitions between observations are described by bunches of properly weighted unitary parallel transports. The standard scheme of quantum mechanics with single unitary evolutions can be reproduced in our scheme by a special choice of weights. But in our scheme such unique evolutions are assumed to be obtained as statistically dominant elements of the bunches. We use the following notations
• H: a Hilbert space;
• Π ψ t 0 , . . . , Π ψ t i , . . . , Π ψ t N : a sequence of observations, where Π ψ t i = ψ t i ψ t i is the projector that fixes ψ t i ∈ H as the result of observation at the time t i ;
• Δt i = t i − t i−1 : the length of ith time interval;
• G = {g 1 , . . . , g M }: a finite gauge group;
• U: a unitary representation of G in the space H;
• γ = g 1 , . . . , g Δt i : a sequence of the length Δt i of elements from G;
• val(γ) = Δt i j=1 g j ∈ G: the (group) value of the sequence γ -the parallel transport; • Γ i = γ 1 , . . . , γ k , . . . , γ K i : an (arbitrary) enumeration of the set of all sequences γ, where K i ≡ |Γ i | = M Δt i is the total number of the sequences;
• w ki : a non-negative weight of kth sequence (in ith time interval).
With these notations we come to the scheme shown in Figure 3. The probability of transition from ψ t i−1 to ψ t i is given by the formula The case of standard quantum mechanics with a single unitary evolution between observations is obtained in our scheme by selecting a sequence γ formed by an element g ∈ G repeated Δt i times.
P ψ t i−1 →ψ t i = K i k=1 w ki ϕ ki | Π ψ t i |ϕ ki , where ϕ ki = U(val(γ k )) ψ t i−1 .
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01007-p.8 Π ψ t 0 Π ψ t i−1 γ 1 , w 1i . . . γ k , w ki . . . γ K i , w K i i Π ψ t i Π ψ t N
The weight of the sequence γ is set to 1, and the weights of all other sequences are equated to 0. In other words, the set of weights is the Kronecker delta on the set of sequences: w ki = δ γ,γ k , γ k ∈ Γ i . Introducing the Hamiltonian H = i ln U(g), we can write the evolution in the usual form
U ≡ U g Δt i = e − i H(t i −t i−1 ) .
Since the notion of Hamiltonian stems from the principle of least action, it is natural to assume the existence of some mechanism of selecting sequences of the form g, g, . . . , g as dominant elements in the set of all sequences. This requires a detailed analysis of the combinatorics of steps in fundamental time (4) for particular models.
Dynamics of observed quantum system. Quantum Zeno effect and finite groups
Consider the issue concerning the connection between the quantum dynamics and the group properties of unitary evolution operators. Namely, we consider the quantum Zeno effect for operators that belong to representations of finite groups. The "quantum Zeno effect" 6 (see the review [10]) is a feature of the quantum dynamics, which is manifested in the fact that frequent measurements can stop (or slow down) the evolution of a system -for example, inhibit decay of an unstable particle -or force it to evolve in a prescribed way. In the latter case, the phenomenon is called the "anti-Zeno effect". Consider a quantum system that evolves from the initial (at t = 0) normalized pure state |ψ 0 under the action of the unitary operator U = e − i Ht , where H is the Hamiltonian. The probability to find the system in the initial state at the time t is the following
p H (t) = ψ 0 e − i Ht ψ 0 2 .(12)
The most important characteristics of any dynamical process are its temporal parameters. For the quantum Zeno effect such a parameter is called the "Zeno time", denoted τ Z . It is determined from the short-time expansion of (12):
p H (t) = 1 − t 2 /τ 2 Z + O t 4 .(13)
Calculation of (13) shows that τ −2
Z = ψ 0 H 2 ψ 0 − ψ 0 H ψ 0 2 .
Let us present the so-called Zeno dynamics in the framework of scheme proposed in Section 3.2. We have here the sequence of observations Π ψ t 0 , Π ψ t 1 , . . . , Π ψ t N , each of which selects the same state ψ 0 , i.e., ψ t 0 = ψ t 1 = · · · = ψ t N ≡ ψ 0 . Assuming that t 0 = 0, t N = T and the times of observations are equidistant: t i − t i−1 = T/N, we can write, using (13), the approximation for the one-step transition probability
P ψ t i−1 →ψ t i ≈ 1 − 1 N 2 T τ Z 2
with the corresponding approximation for the one-step entropy
S ψ t i−1 →ψ t i ≈ − 1 N 2 T τ Z 2 .
For the entropy of the trajectory we have
S ψ t 0 →···→ψ t N = N i=1 S ψ t i−1 →ψ t i ≈ − 1 N T τ Z 2 N → ∞ − −−−−→ 0
and, respectively, for the probability of trajectory:
P ψ t 0 →···→ψ t N N → ∞ − −−−−→ e 0 = 1
. This is precisely the essence of the Zeno effect. Now assume that the evolution operator U belongs to a representation of a finite group G, i.e., U = U (g) , g ∈ G, and the time is the sequence of natural numbers: t = 0, 1, 2, . . . . A natural way to define the Zeno time in this case follows from the observation that the leading part of the expansion (13) vanishes at t = τ Z . By analogy we can define the natural Zeno time τ Z as the first t ∈ [0, 1, 2, . . .] that provides minimum of the expression
p U (t) = ψ 0 U t ψ 0 2 .(14)
Obviously, the expression (14) is either constant (namely, p U (t) = 1) or periodic. In the latter case its period is a divisor of the order of U. The order of an element a of a group is the smallest natural number n > 0 such that a n = e, where e denotes the identity element of the group. The order of a will be denoted ord (a). For the faithful representation, ord (U) ≡ ord (U (g)) = ord (g). Consider, for example, the "Max-Zehnder" representation U MZ of the group Z 8 , i.e., the "beam splitter" matrix (1) is taken as a generator of Z 8 . Table 1 presents the Zeno times for all operators from the representation U MZ . We adopt the convention (motivated by formula (13)) that τ Z = ∞ if the probability (14) is constant.
U = U MZ (g) ord(g) Period(p U (t)) τ Z S 0 = I 1 p U (t) = 1 ∞ S 4 2 p U (t) = 1 ∞ S 2 = M, S 6 4 2 1 S , S 3 , S 5 , S 7 8 4 2
The two-dimensional "Max-Zehnder" representation U MZ can be generalized to the arbitrary cyclic group Z N by replacing the "beam splitter" matrix of the form (2) where r is an Nth primitive root of unity. Figure 4 shows the evolution of the probability to observe the initial state for the evolution operator S 100 in the time interval 0 ≤ t ≤ 100. The quadratic shorttime behavior, described by the formula (13), is clearly visible in the figure. The Zeno time in this example is τ Z = 25. As a non-commutative example, consider the icosahedral group A 5 -the smallest ( |A 5 | = 60) non-commutative simple group. It has applications for model building in the particle physics, especially in issues beyond the standard model, such as the flavor physics [11]. The non-trivial elements of A 5 have orders 2, 3 and 5. The irreducible representations of A 5 are: one trivial singlet, 1, two triplets, 3 and 3 , one quartet, 4, and one quintet, 5. Figure 5 shows the evolution of the "Zeno probabilities" for the following matrices of orders 2, 3 and 5, respectively,
U = 1 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −φ 1/φ 1 1/φ −1 φ 1 φ 1/φ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , V = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 1 1 0 0 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , W = 1 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −φ −1/φ 1 1/φ 1 φ −1 φ −1/φ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(15)
where φ = 1+ √ 5 2 is the "golden ratio". To write these matrices, we added an element of order 3 (the simplest among randomly selected) to the generators of orders 2 and 5 proposed in [12] for the representation 3 .
Figure 1 .
1Mach-Zehnder interferometer. Balanced setup: both beam splitters are 50/50 and there is no phase shift between upper and lower paths.
Figure 3 .
3Scheme of quantum evolution with observations
with the unitary matrixS N = 1 2 r + r N−1 r − r N−1 r − r N−1 r + r N−1 ,
Figure 4 .
4Probability p U (t) vs. time t for the operator U = S 100 ∈ U MZ (Z 100 ).
Figure 5 .
5Zeno dynamics in the representation 3 of the group A 5 for unitary operators (15)
Table 1 .
1Zeno times for all operators from U MZ (Z 8 )
The phrase is often attributed to John Archibald Wheeler.EPJ Web of Conferences01007-p.2
The exponent of a group is defined as the least common multiple of the orders of its elements.Mathematical Modeling and Computational Physics 201501007-p.3
It has been shown[9] that the velocity, defined in a similar way, i.e., as the difference of probabilities of steps in opposite directions, satisfies the relativistic velocity addition rule: w = (u + v) / (1 + uv).
This effect is also known under the name "the Turing paradox".
AcknowledgementsThe work is supported in part by the Ministry of Education and Science of the Russian Federation (grant 3003.2014.2) and the Russian Foundation for Basic Research (grant 13-01-00668).
We adhere to the idea of empirical universality of discrete, more specifically, finite models for describing physical reality. other words, any continuous model can be replaced by a finite. We adhere to the idea of empirical universality of discrete, more specifically, finite models for describing physical reality. In other words, any continuous model can be replaced by a finite
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"Boundary Green's function approach for spinful single-channel and multichannel Majorana nanowires",
"Boundary Green's function approach for spinful single-channel and multichannel Majorana nanowires"
]
| [
"M Alvarado \nDepartamento de Física\nCondensed Matter Physics Center (IFIMAC)\nTeórica de la Materia Condensada C-V\nInstituto Nicolás Cabrera\nUniversidad Autónoma de Madrid\nE-28049MadridSpain\n",
"A Iks \nInstitut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany\n",
"A Zazunov \nInstitut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany\n",
"R Egger \nInstitut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany\n",
"A Levy Yeyati \nDepartamento de Física\nCondensed Matter Physics Center (IFIMAC)\nTeórica de la Materia Condensada C-V\nInstituto Nicolás Cabrera\nUniversidad Autónoma de Madrid\nE-28049MadridSpain\n"
]
| [
"Departamento de Física\nCondensed Matter Physics Center (IFIMAC)\nTeórica de la Materia Condensada C-V\nInstituto Nicolás Cabrera\nUniversidad Autónoma de Madrid\nE-28049MadridSpain",
"Institut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany",
"Institut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany",
"Institut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany",
"Departamento de Física\nCondensed Matter Physics Center (IFIMAC)\nTeórica de la Materia Condensada C-V\nInstituto Nicolás Cabrera\nUniversidad Autónoma de Madrid\nE-28049MadridSpain"
]
| []
| The boundary-Green's function (bGF) approach has been established as a powerful theoretical technique for computing the transport properties of tunnel-coupled hybrid nanowire devices. Such nanowires may exhibit topologically nontrivial superconducting phases with Majorana bound states at their boundaries. We introduce a general method for computing the bGF of spinful multichannel lattice models for such Majorana nanowires, where the bGF is expressed in terms of the roots of a secular polynomial evaluated in complex momentum space. In many cases, those roots, and thus the bGF, can be accurately described by simple analytical expressions, while otherwise our approach allows for the numerically efficient evaluation of bGFs. We show that from the behavior of the roots, many physical quantities of key interest can be inferred, e.g., the value of bulk topological invariants, the energy dependence of the local density of states, or the spatial decay of subgap excitations. We apply the method to single-and two-channel nanowires of symmetry class D or DIII. In addition, we study the spectral properties of multiterminal Josephson junctions made out of such Majorana nanowires. arXiv:1912.07003v2 [cond-mat.mes-hall] | 10.1103/physrevb.101.094511 | [
"https://arxiv.org/pdf/1912.07003v2.pdf"
]
| 209,376,791 | 1912.07003 | c174958f5ac9673889b3c3d0bc866e5807e7fd5b |
Boundary Green's function approach for spinful single-channel and multichannel Majorana nanowires
M Alvarado
Departamento de Física
Condensed Matter Physics Center (IFIMAC)
Teórica de la Materia Condensada C-V
Instituto Nicolás Cabrera
Universidad Autónoma de Madrid
E-28049MadridSpain
A Iks
Institut für Theoretische Physik
Heinrich-Heine-Universität
D-40225DüsseldorfGermany
A Zazunov
Institut für Theoretische Physik
Heinrich-Heine-Universität
D-40225DüsseldorfGermany
R Egger
Institut für Theoretische Physik
Heinrich-Heine-Universität
D-40225DüsseldorfGermany
A Levy Yeyati
Departamento de Física
Condensed Matter Physics Center (IFIMAC)
Teórica de la Materia Condensada C-V
Instituto Nicolás Cabrera
Universidad Autónoma de Madrid
E-28049MadridSpain
Boundary Green's function approach for spinful single-channel and multichannel Majorana nanowires
(Dated: March 13, 2020)
The boundary-Green's function (bGF) approach has been established as a powerful theoretical technique for computing the transport properties of tunnel-coupled hybrid nanowire devices. Such nanowires may exhibit topologically nontrivial superconducting phases with Majorana bound states at their boundaries. We introduce a general method for computing the bGF of spinful multichannel lattice models for such Majorana nanowires, where the bGF is expressed in terms of the roots of a secular polynomial evaluated in complex momentum space. In many cases, those roots, and thus the bGF, can be accurately described by simple analytical expressions, while otherwise our approach allows for the numerically efficient evaluation of bGFs. We show that from the behavior of the roots, many physical quantities of key interest can be inferred, e.g., the value of bulk topological invariants, the energy dependence of the local density of states, or the spatial decay of subgap excitations. We apply the method to single-and two-channel nanowires of symmetry class D or DIII. In addition, we study the spectral properties of multiterminal Josephson junctions made out of such Majorana nanowires. arXiv:1912.07003v2 [cond-mat.mes-hall]
I. INTRODUCTION
The interest in proximitized nanostructures where topological superconductor phases could be engineered is continuing to grow [1][2][3][4][5][6][7][8]. In particular, the case of one-dimensional (1D) semiconducting hybrid nanowires with strong Rashba spin-orbit interaction has been intensely studied as a potential route towards the generation of Majorana bound states (MBSs) [9][10][11][12][13][14][15][16][17]. Such states are of high interest for topological quantum information processing applications [4]. While a phase with broken time-reversal symmetry (class D) can be expected for the cited nanowire experiments because of the presence of a magnetic Zeeman field (we use the abbreviation 'TS' for such topological superconductors below), a time-reversal invariant topological superconductor (TRI-TOPS) phase has been predicted from related wire constructions [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The TRITOPS phase has symmetry class DIII and is still awaiting experimental tests. The interest in hybrid nanowires goes well beyond the generation of topological phases. For instance, recent microwave spectroscopy experiments have investigated the role of spin-orbit coupling effects on the formation of Andreev bound states [36].
The physics of devices made from different types of nanowires coupled by tunneling contacts has been explored by a variety of theoretical models and techniques [1][2][3]5]. On one hand, minimal models restrict the Hilbert space to include only subgap bound states. This key simplification then allows for analytical progress (see, e.g., Refs. [37,38] for early contributions). On the other hand, microscopic models aim for a more detailed understanding of how material properties can influence transport observables (see, e.g., Refs. [39][40][41][42][43][44][45][46]). Recent works along this line have studied the electrostatic potential profile along the nanowire [47][48][49] and the effects of disorder on the phase diagram [50,51]. However, the solution of such microscopic models requires information about many model parameter values and generally can be obtained only by performing a detailed numerical analysis. In this context, theoretical approaches of intermediate complexity are of high interest. Such a framework allows one to describe transport properties by taking into account both subgap and continuum states while keeping the algebra sufficiently simple so as to permit analytical progress. The scattering matrix formalism is a widely known representative for this type of approach (see, e.g., Refs. [52][53][54][55][56][57][58]). The present paper will employ the complementary boundary Green's-function (bGF) method [59][60][61][62][63][64], which is particularly useful for analyzing nonequilibrium transport properties in different types of hybrid nanojunctions. The bGF approach also allows one to examine other electronic properties such as the tunneling density of states (DoSs) or the bulkboundary correspondence expected for topological phases [65][66][67]. Furthermore, electron-phonon and/or electronelectron interaction effects can in principle also be taken into account.
In the present paper, we extend and generalize the bGF approach for 1D or quasi-1D proximitized nanowires, which has been introduced in Refs. [59][60][61][62][63], along several directions. First, we demonstrate that a bGF construction in terms of the roots of a secular equation extended to complex momenta (as discussed in Ref. [59] for the Kitaev chain model) can be generalized to arbitrary spinful multichannel (i.e., quasi-1D) nanowires with topologically nontrivial superconducting phases. In particular, by studying the evolution of the roots in the complex momentum plane under the variation of model parameters, one can readily detect topological transitions, determine bulk topological invariants, or compute the local density of states as a function of energy for translationally invariant cases. In addition, the same roots determine the bGF and thereby give access to the transport properties of devices made from tunnel-coupled (semi-infinite or finite-length) nanowires. In particular, their knowledge also gives access to the spatial decay profile of Majorana states.
Below we investigate the roots and the corresponding bGFs for two widely used spinful single-channel nanowire models harboring topologically nontrivial phases. First, we study TS wires with broken time-reversal invariance using the model by Lutchyn et al. [68] and by Oreg et al. [69]. Second, we consider TRITOPS wires using the model of Zhang et al. [21]. Quasi-1D multichannel models in class D or class DIII are then constructed by coupling several wires of the respective symmetry class by tunnel couplings. We show that also such multichannel models can be efficiently tackled by our bGF method. As application, we will discuss the Josephson current-phase relation both for a multiterminal junction composed of three tunnel-coupled TS wires and for a TRITOPS-TS Josephson junction.
The remainder of this article is organized as follows. In Sec. II, we describe a general formalism for analyzing 1D or quasi-1D lattice models of proximitized nanowires, where we only assume that the hopping amplitudes in the corresponding tight-binding model are of finite range. We show that the real-space bulk Green's function (GF) adopts a compact expression in terms of the roots of the secular polynomial of the bulk Hamiltonian extended into complex momentum space. We also show how the boundary GF can be obtained from the bulk GF by solving a Dyson equation, and we discuss general properties of the corresponding roots. In Sec. III, we consider a discretized version of the single-channel class D model of Refs. [68,69]. We introduce a simple ansatz for the respective roots in the trivial and in the topological phase. This ansatz allows us to obtain analytical insights about the bulk spectral density and the spatial variation of MBSs. In Sec. IV, we extend the analysis to a two-channel model describing two coupled class D wires, where we can study spin-orbit interaction effects in multichannel nanowires [70]. The phase diagram and the spectral density of this model show a richer behavior than in the single-channel case. In Sec. V, we apply our methods to single-and multichannel models for TRITOPS wires. Finally, in Sec. VI, we study the Josephson effect and the formation of Andreev bound states in phase-biased multiterminal TS junctions and for TRITOPS-TS junctions. We finally offer some conclusions in Sec. VII. Technical details have been delegated to two appendices. We often use units with = 1 and focus on the zero-temperature limit throughout.
II. BOUNDARY GREEN'S FUNCTION
A central aim of the present work is to construct the bGF for different hybrid nanowire models which are described by a bulk Hamiltonian of the form
H bulk = 1 2 kΨ † kĤ (k)Ψ k ,(1)
corresponding to an infinitely long and translationally invariant (quasi-)1D chain with lattice spacing a. Here, H(k) is an N ×N Bogoliubov-de Gennes (BdG) Hamiltonian in reciprocal space, and theΨ k are fermionic Nambu spinor fields. Specific examples for these spinor fields will be given in the subsequent sections. The number N may include the Nambu index, the spin degree of freedom, and channel indices for multichannel models. UsingĤ(k + 2π/a) =Ĥ(k), the BdG Hamiltonian can be expanded in a Fourier series,Ĥ(k) = nV n e inka , where Hermiticity impliesV −n =V † n . For simplicity, we here consider only models with nearest-neighbor hopping, V n = 0 for |n| > 1, but the generalization to arbitrary finite-range hopping amplitudes is straightforward.
The retarded bulk GF of the infinite chain is defined asĜ
R (k, ω) = ω + i0 + −Ĥ(k) −1 ,(2)
where the N × N matrix structure is indicated by the hat notation. In real space representation, the GF has the components (j and j are lattice site indices)
G R jj (ω) = a 2πˆπ /a −π/a dk e i(j−j )kaĜR (k, ω).(3)
By the identification z = e ika , this integral is converted into a complex contour integral,
G R jj (ω) = 1 2πi˛| z|=1 dz z z j−j Ĝ R (z, ω).(4)
Introducing the roots z n (ω) of the secular polynomial in the complex-z plane,
P (z, ω) = det ω −Ĥ(z) = 1 z N 2N n=1 [z − z n (ω)] ,(5)
the contour integral (4) can be written as a sum over the residues of all roots inside the unit circle:
G R jj (ω) = |zn|<1 z j−j n (z n , ω) m =n (z n − z m ) ,(6)
whereÂ(z, ω) is the cofactor matrix of [ω − H(z)] z. For notational simplicity, we omit the superscript 'R' in retarded GFs from now on.
Given the real-space components of the bulk GF in Eq. (6), we next employ Ref. [59] (see also Ref. [71]) to derive the bGF characterizing a semi-infinite nanowire. To that effect, we add an impurity potential localized at lattice site j = 0. Taking the limit → ∞, the infinite chain is cut into disconnected semi-infinite chains with j < −1 (left side, L) and j > 1 (right side, R). Using the Dyson equation, the local GF components of the cut nanowire follow as [59]
G jj (ω) =Ĝ jj (ω) −Ĝ j0 (ω) Ĝ 00 (ω) −1Ĝ 0j (ω).(7)
The bGF for the left and right semi-infinite chain, respectively, are with Eq. (7) given bŷ
G L (ω) =Ĝ −1,−1 (ω),Ĝ R (ω) =Ĝ 11 (ω).(8)
We note that by proceeding along the lines of Refs. [60,72], one can also compute reflection matrices from the corresponding bGF,
r L/R = lim ω→0 1 − iV † ±1ĜL/R (ω)V ±1 1 + iV † ±1ĜL/R (ω)V ±1 .(9)
This relation allows one to express topological invariants of the bulk Hamiltonian [66,67] in terms of bGFs. The roots z n (ω) play an important role in what follows. In particular, their knowledge allows us to construct both the bulk and the boundary GFs. In simple cases, this can be done analytically, and otherwise this route offers an efficient numerical scheme. The roots can also provide detailed information about the decay of subgap states localized at the boundaries of semi-infinite wires, and they allow one to compute topological invariants of the bulk system. Let us therefore summarize some general properties of these roots:
(i) Hermiticity of the BdG Hamiltonian implies that every root z n (ω) is accompanied by a root 1/z * n (ω), where ' * ' denotes complex conjugation.
(ii) Electron-hole symmetry of the BdG Hamiltonian implies that z n (ω) = z * n (−ω). In the presence of an additional symmetryĤ(k) =ÛĤ(−k)Û † with a unitary matrixÛ , for every root z n (ω), also z * n (ω) must be a root.
(iii) As a consequence of (i) and (ii), 2N n=1 z n (ω) = 1.
(iv) Topological phase transitions can occur once a pair of zero-energy roots hits the unit circle, |z n (0)| = 1, which corresponds to the closing and reopening of a gap in the bulk spectrum.
(v) Equations (6) and (7) imply that subgap bound states (with energy E) localized near the boundary of a semi-infinite wire decay into the bulk in a manner controlled by max(|z n (E)| < 1).
We illustrate the usefulness of these properties in the following sections for different models of proximitized (quasi-)1D nanowires.
III. SPINFUL SINGLE-CHANNEL HYBRID NANOWIRES
As first example, we consider the spinful single-channel model of Refs. [68,69] for a proximitized semiconductor nanowire. This model has been extensively studied as prototype for 1D wires harboring a TS phase with broken time-reversal invariance. We use the Nambu bispinor Ψ T k = c k↑ , c k↓ , c † −k↓ , −c † −k↑ , i.e., N = 4 in Eq. (1). Here, c kσ is a fermionic annihilation operator for momentum k and spin σ =↑, ↓, and the bulk BdG Hamiltonian in Eq. (1) takes the form
H(k) = k σ 0 τ z + V x σ x τ 0 + α k σ z τ z + ∆σ 0 τ x ,(10)
where σ x,y,z and τ x,y,z are Pauli matrices in spin and Nambu (electron-hole) space, respectively, with the identity matrices σ 0 and τ 0 . Regularizing the continuum model of Refs. [68,69] by imposing a finite lattice spacing a, the kinetic energy k = 2t[1 − cos(ka)] − µ includes the chemical potential µ and the nearest-neighbor hopping amplitude t. Furthermore, V x encapsulates a magnetic Zeeman field oriented along the wire axis, α k = α sin(ka) describes the spin-orbit interaction, and ∆ refers to the proximity-induced on-site pairing amplitude. The bulk dispersion relation, E = E k,± ≥ 0, then follows from [68,69]
E 2 k,± = ∆ 2 + α 2 k + V 2 x + 2 k ± 2 ∆ 2 V 2 x + (α 2 k + V 2 x ) 2 k . (11) This model exhibits a topological transition at V x = V c = ∆ 2 + µ 2 , where the TS phase is realized for V x > V c .
Although it is not essential for the subsequent discussion, the parameters t and α can be assigned values appropriate for InAs nanowires [59]. To that end, we put t = 2 /(2m * a 2 ), where m * is the effective mass, and α = u/a, where u is the spin-orbit parameter [69]. This parameter depends on material properties and can be tuned by an external electric field. Putting a = 10 nm and using typical InAs material parameters, we estimate t ≈ 10 meV and α ≈ 4 meV [59]. On the other hand, a proximity gap of order ∆ ≈ 0.2 meV represents the case of a nanowire in good contact with a superconducting Al layer. (We will use this value in the figures below unless noted otherwise.) The only remaining free variables are then given by V x and µ.
Using Eq. (5) and z = e ika , the roots z n (ω) for this model satisfy the condition
2∆ 2 α 2 (z) + 2 (z) − V 2 x − ω 2 +2α 2 (z) V 2 x − ω 2 − 2 (z) +α 4 (z) + V 4 x + ∆ 4 + ω 2 − 2 (z) 2 − 2V 2 x ω 2 + 2 (z) = 0,(12)
with the functions Equation (12) can be written as
α(z) = −iα(z − z −1 )/2, (z) = −t(z + z −1 − 2) − µ.(13)4 n=1 C n (ω) z n + 1 z n + C 0 (ω) = 0,(14)
where the real coefficients C n (ω) are given in Appendix A. Clearly, Eq. (14) is consistent with the general properties (i) and (ii) listed in Sec. II. Alternatively, Eq. (14) can be expressed as an eighth-order polynomial equation:
8 m=0 a m (ω)z m = 0,(15)
where the coefficients a m are trivially related to the C n and we can impose the normalization conditions a 0 = a 8 = 1.
The resulting roots z n can be grouped into two different classes associated with the two pairing gaps ∆ 1 and ∆ 2 in the bulk spectrum [68,69] [see Fig. 1(a)]. In the limit ∆ → 0, these gaps ∆ 1 and ∆ 2 will also vanish. For ∆ = 0, we find from Eq. (12) that the zero-frequency roots z n (ω = 0) simplify to e ±ik1a and e ±ik2a , with
k 1,2 cos −1 2t(2t − µ) α 2 + 4t 2 ± V 2 x (α 2 + 4t 2 ) + α 4 + 4tµα 2 − α 2 µ 2 α 2 + 4t 2 .(16)
At these momenta, the dispersion relation becomes gapless for ∆ = 0 [see Fig. 1(a)]. We observe from Eq. (16) that k 1 (corresponding to the + sign) becomes purely imaginary for V x > V c . We will then first discuss the topologically trivial regime V x < V c . Figure 1(a) shows that the low-energy physics will be dominated by the regions with |k| ≈ k 1 and |k| ≈ k 2 . The pairing gaps ∆ 1,2 = |E k1,2,− | then follow by substituting k 1,2 into the bulk dispersion relation (11). In particular, we find that ∆ 1 closes and reopens when ramping V x through the topological transition at V x = V c . An approximate expression for the roots is obtained by linearizing the ∆ = 0 dispersion relation in Eq. (11) for electrons and holes near k = k 1 and k = k 2 . Defining the respective velocities as v ν=1,2 = |∂ k E k=kν ,− | ∆=0 , the effective low-energy Nambu Hamiltonian valid near the respective momentum k ν can be written as
H eff,ν=1,2 (k) v ν (k − k ν ) ∆ ν ∆ ν −v ν (k − k ν ) ,(17)
and similarly for k ≈ −k ν . Using ika = ln z, the condition det[ω − H eff,ν (z)] = 0 can readily be solved. In effect, the roots are given by
z ν (ω) 1 ± a v ν ∆ 2 ν − ω 2 e ikν a ,(18)
plus the complex conjugate values. Inspired by Eq. (18), we propose the following ansatz for the roots z n (ω) located inside the unit circle:
z ν (ω) = 1 − τ ν ∆ 2 ν − ω 2 e iδν ,(19)
where τ 1,2 and δ 1,2 are phenomenological coefficients. In addition, the complex conjugate root z * ν (ω) is a solution. This ansatz is expected to work well in the topologically trivial regime V x < V c . For small ∆ and |ω|, Eq. (18) implies the limiting behavior τ ν = a/v ν and δ ν = k ν a. In addition, we also impose the condition
τ 1 ∆ 1 = τ 2 ∆ 2 = η 1,(20)
where η is a small parameter. In the small-∆ case with τ ν ≈ a/v ν , Eq. (20) implies that the effective pairing gap ∆ ν is inversely proportional to the corresponding density of states ∝ 1/v ν . Figure 1(b) shows that this condition is accurately fulfilled as long as V x stays well below V c . However, Eq. (20) becomes less precise for V x → V c .
In Appendix A, we provide more refined analytical expressions that determine the parameters η and δ ν in our ansatz for the roots [see Eqs. (19) and (20)].
Next we turn to the topologically nontrivial regime V x > V c , where the momentum k 1 in Eq. (16) becomes purely imaginary. We should then replace δ 1 → iδ 1 in the above ansatz for the roots. As a consequence, the z ν=1 (ω) roots become real-valued, and the ansatz for V x > V c Figure 2: Behavior of the roots zn(ω) for the spinful singlechannel Majorana wire model [68,69]. We use Vx = 0.5Vc and Vx = 1.2Vc as representatives for topologically trivial and nontrivial cases, respectively, with µ = 5 meV and other parameters as specified in the main text. Upper panels: Roots zn(ω = 0) (black dots) inside the unit circle (red) for (a) Vx < Vc and (b) Vx > Vc. For illustrative purposes, we use ∆ = 1 meV in panels (a) and (b). For additional information, see Supplemental Material [73]. Middle panels: Modulus of the roots inside the unit circle vs ω/∆ for (c) Vx < Vc and (d) Vx > Vc. Solid curves represent numerically exact results and dashed curves follow from Eqs. (19) and (21) takes the form
z 1,± (ω) = 1 ± τ 1 ∆ 2 1 − ω 2 e −δ1 ,z 2,± (ω) = 1 − τ 2 ∆ 2 2 − ω 2 e ±iδ2 ,(21)
where both δ 1 and δ 2 are real positive. We thus have only a single pair of complex conjugate roots (z 2 ) near the unit circle for V x > V c . Accurate analytical results for the δ ν and τ ν parameters can be obtained by solving a cubic equation (see Appendix A). As illustrated in Fig. 2
Q = sgn PfĤ(k = 0) sgn PfĤ(k = π/a) = ±1.(22)
Interestingly, the number N p of complex conjugate root pairs near (but inside) the unit circle is in correspondence with the topological invariant, Q = (−1) Np . These roots can be unambiguously identified as the ones approaching the unit circle from inside in the limit ∆ → 0, corresponding to the Fermi points in the normal phase. For an odd (even) number of pairs, the phase is thus topogically nontrivial with Q = −1 (trivial with Q = 1). The upper panels in Fig. 2 illustrate the distribution of the roots inside the unit circle for the cases V x < V c and V x > V c . We observe that upon entering the topologically nontrivial regime, the complex conjugate z 1 roots coalesce to form an almost degenerate root pair z 1,± [see Eq. (21)] located on the real axis inside the unit circle. The roots on the real axis correspond to additional bands at high energies above ∆. At the same time, a single pair of complex conjugate roots (z 2 ) remains near (but inside) the unit circle, as one expects for a topologically nontrivial phase. As remarked above, this change in the structure of the roots across the transition is consistent with the corresponding change in the topological invariant. The transition between both regions happens when the Pfaffian, or equivalently the Hamiltonian determinant, at k = 0 vanishes. Using the relation detĤ(k = 0) = 8 n=1 (1 − z n (0)), we thus reproduce property (iv) in Sec. II which signals the phase transition. It is also worth mentioning that the bulk invariant (22) can be directly expressed in terms of bGFs for the semiinfinite wire: Using Q = detr L = detr R (see Ref. [72]) the reflection matricesr L/R and therefore also Q can be obtained from the bGFs [see Eq. (9)].
The knowledge of the roots also gives access to other electronic properties of interest. For instance, we can obtain a compact expression for the energy-dependent local DoS at, say, lattice site j = 0 of the translationinvariant chain:
ρ(ω) = − 1 π Im Tr Ĝ 00 (ω) .(23)
We focus on the low-energy limit, where one can expand the cofactor matrixÂ(z, ω) in Eq. (6) to linear order in ω. The local GF then follows aŝ
G 00 (ω) |zn|<1 (z n , ω) + ω (z n , ω) m =n (z n − z m ) ,(24)
where (z n , ω) = d dω (z n (ω), ω). From our ansatz in Eqs. (19) and (21), the sum in Eq. (24) can be reconstructed. A simple approximate expression follows for small ∆ in the low-energy limit, where one needs to keep just the first-order terms ∝ τ ν ∆ 2 ν − ω 2 in the denominator. We then obtain
G 00 (ω) ≈ ν=1,2 ν + ω ν ∆ 2 ν − ω 2 ,(25)
where ν and ν are specified in Appendix A. We note that for V x > V c , the main contribution to Eq. (25) stems from the residues associated to z 2 . The results for ρ(ω) depicted in Figs. 2(e) and 2(f) demonstrate that Eq. (25) accurately reproduces numerically exact calculations, both below and above the topological transition.
Next we turn to the case of a semi-infinite chain in the topological phase, V x > V c . Using the Dyson equation in Eq. (7) and taking into account the behavior of the roots of the infinite chain discussed above, we can deduce the spatial decay profile of the zero-energy Majorana end state into the bulk. Noting that the GF componentsĜ j,0 andĜ 0,j in Eq. (7) are ∝ |z| j , we observe that for V x > V c , the decay is dominated by the z 2 roots since |z 2 | > |z 1 |. Moreover, the decay profile exhibits fast oscillations due to the complex phase δ 2 in Eq. (21), which for µ ∆ can be approximated as δ 2 k F a with k F ≡ k 2 . In this approximation, the local DoS of the ω = 0 MBS thus has the spatial profile
ρ j (ω = 0) ∝ |z 2 (0)| 2j cos 2 (jk F a + χ 0 ) ,(26)
where χ 0 describes a phase shift in the 2k F oscillations. Equation (26) reproduces the numerically exact results obtained from Eq. (7) rather well, as illustrated in Fig. 3. The dashed curve shows that the envelope function is accurately described by |z 2 (0)| 2j , corresponding to an exponential decay into the bulk of the chain.
IV. TWO-CHANNEL CLASS-D NANOWIRE
We next examine the case of spinful multichannel hybrid nanowires with broken time reversal symmetry. The bGF approach could in principle be applied to nanowire models with an arbitrary number of channels. In prac-tice, however, the techniques in Sec. II are less efficient once the degree 2N of the secular polynomial (5) becomes very large. We here restrict ourselves to the two-channel case with N = 8, which can be realized for two singlechannel nanowires coupled by tunneling terms. The resulting model already exhibits many of the features expected for generic multichannel nanowires [74,75].
Our model Hamiltonian is given bŷ
H 2ch (k) = Ĥ (k)T T †Ĥ (k) ,(27)
where the 2 × 2 structure refers to wire space. We consider two identical spinful single-channel Majorana wires described by the model of Refs. [68,69] withĤ(k) in Eq. (10). The interwire tunnel couplings are modeled bŷ
T = −t y σ 0 τ z + iα y σ x τ z + ∆ y σ 0 τ x ,(28)
where t y and α y are spin-conserving and spin-flipping hopping amplitudes, respectively. The coupling α y may arise due the presence of a Rashba spin-orbit coupling produced by an electric field along the z-direction. As in Sec. III, we write t y = 2 /(2m * a 2 y ) and α y = u/a y , with the minimal distance a y between the wires. In the concrete examples shown below, we assume a y = 3a, which corresponds to a subband separation of ≈ 3 meV. The interwire coupling (28) also includes a non-local interwire pairing amplitude ∆ y . For the present class D case, however, we find that allowing for a small ∆ y = 0 does not lead to significant changes in the phase diagram. We thus put ∆ y = 0 in this section.
One can characterize the phase diagram of a translationally invariant two-channel wire by using the bulk topological invariant in Eq. (22) with the replacement H(k) →Ĥ 2ch (k). The Pfaffian at k = 0 is here given by
PfĤ 2ch (0) = α 4 y + (µ − 3t y ) 2 − V 2 x + ∆ 2 × (µ − t y ) 2 − V 2 x + ∆ 2 + 2α 2 y −(µ − 3t y )(µ − t y ) − V 2 x + ∆ 2 .(29)
The boundaries of the topological phase correspond to a vanishing Pfaffian at k = 0, where Eq. (29) implies the two critical Zeeman fields
V c,± = α 2 y + µ 2 − 4µt y + 5t 2 y + ∆ 2 ± 2|µ − 2t y | t 2 y + α 2 y 1/2 .(30)
The resulting phase diagram in the µ-V x plane is illustrated in Fig. 4(a). We observe that the two-channel model (27) exhibits a richer phase diagram than in the single-channel case (see also Refs. [74,75]). We next construct the bGF of a semi-infinite wire by determining the roots of the secular polynomial in Eq. (5), which here is a 16th-order polynomial equation that we solve numerically. Figure 4(b) and 4(c) illustrates the evolution of the energy-dependent local DoS, [73]. Panel (g) shows the evolution of the roots within the topologically trivial regime as Vx increases from 3 to 8 meV at constant chemical potential µ = 2 meV. In panels (d)-(g), we use ∆ = 1 meV. ρ 1 (ω), at the boundary, i.e., taken at site j = 1 of a semi-infinite two-channel wire. We consider two different trajectories in the µ-V x plane as indicated by the arrows in Fig. 4(a). For constant V x [panel b)], there are both topologically nontrivial and trivial regions as µ is varied. In the topologically nontrivial regions, we observe a zeroenergy peak in the local DoS, signaling the presence of MBSs. This ω = 0 peak is absent in the trivial regime. For fixed µ [panel (c)], the topologically nontrivial phase is reached for intermediate values of V x . For larger V x , even though the system is in a trivial phase, we find lowenergy Andreev bound states that approach zero energy as V x increases. This effect has also been described in Ref. [75].
Additional insights follow by analyzing the evolution of the roots z n (ω = 0) inside the unit circle in the complex momentum plane. In Fig. 4(d)-(f), we illustrate their distribution for three different points in the phase diagram. For panels (d) and (f), the system is in a topological phase and, as expected, one finds an odd number of pairs of complex conjugate roots close to the unit circle. As in Sec. III, the roots on the real axis correspond to additional bands at higher energies well above ∆. Panel (e) instead corresponds to a topologically trivial phase with an even number of conjugate root pairs near the unit circle. Finally, Fig. 4(g) illustrates the evolution of the roots in the topologically trivial regime as the Zeeman parameter V x increases. We find that the both roots near the unit circle in the first quadrant become almost degenerate for large V x . Such a behavior effectively amounts to having two replicas of a single-channel TS wire, which in turn helps to explain why Andreev bound states approach the zero energy limit for strong Zeeman field [see Fig. 4(c) and Ref. [75]].
V. TRITOPS NANOWIRES
Next we turn to models for hybrid nanowires of symmetry class DIII. Such TRITOPS wires constitute another interesting system with topologically nontrivial phases. Below we first study single-channel wires and subsequently turn to the two-channel case.
A. Single-channel case
Many different proposals for physical realizations of single-channel TRITOPS wires have been put forward in the recent past [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. For concreteness, we will here focus on the model introduced by Zhang et al. [21]. Using the spin-Nambu basis with N = 4 in Sec. II, the Hamiltonian is given bŷ where in this section, we use
H DIII (k) = k σ 0 τ z + α k σ z τ z + ∆ k σ 0 τ x ,(31)k = −2t cos(ka) − µ, α k = 2α sin(ka), ∆ k = 2∆ cos(ka).(32)
Again t corresponds to a nearest-neighbor hopping amplitude, µ is the chemical potential, a the lattice spacing, and α the spin-orbit coupling strength. The parameter ∆ corresponds to a nearest-neighbor pairing interaction.
In the examples below, we use a = 10 nm, t = 10 meV and α = 4 meV as in Secs. III and IV. By a simple rearrangement of the spin-Nambu spinor Ψ k , one can block-diagonalize the Hamiltonian in Eq. (31),Ĥ DIII = diag(Ĥ − ,Ĥ + ). To that end, upon replacingΨ T k → c k↑ , c † −k↓ , c k↓ , −c † −k↑ , we arrive at the 2 × 2 block Hamiltonianŝ
H ± (k) = ( k ∓ α k )σ z + ∆ kσx = β ± (k) ·σ,(33)
whereσ is the vector ofσ x,y,z Pauli matrices in the respective 2 × 2 space obtained after block diagonalization. Each HamiltonianĤ ± (k) corresponds to a Dirac-type model where
β ± (k) = 2∆ cos(ka) 0 −µ − 2t cos(ka) ± 2α sin(ka) (34)
is a vector field mapping the first Brillouin zone onto a closed curve. At this stage, we can apply the formalism of Ref. [76] for analyzing the roots of the secular polynomial of Diraclike Hamiltonians. By projectingĤ ± to theσ x -σ z plane, we obtain an elliptic curve as illustrated in Fig. 5. According to the arguments in Ref. [76], if the ellipse encloses the origin of theσ x -σ z plane, we know that for a semi-infinite wire,Ĥ ± (k) will generate an edge state with energy equal to the modulus of the component of β ± (k) perpendicular to this plane. In our case, [β ± (k)] y = 0 implies that we have a pair of zero-energy boundary states in the topological phase. In addition, this argument also shows that there are no finite-energy Andreev bound states in the trivial phase (where the ellipse does not contain the origin). For the case in Fig. 5, where the origin is displaced along theσ z -axis, the topological transition occurs at ka = ±π/2 and |µ| = 2α [see Eq. (34)]. This conclusion is consistent with the fact that at the topological transition, one finds roots at z = e ika = ±i (see also Ref. [77]), in agreement with property (v) in Sec. II.
More generally, by determining the roots z n (ω), we can again construct the bGF of a semi-infinite wire. In particular, we thereby obtain the class DIII bulk topological invariant via the reflection matrices in Eq. (9). In the present case, the invariant is given by Q = Pf (ir L,R ) [72]. Furthermore, using the results of Refs. [76,77], an analytical expression for the largest-modulus zero-frequency root, z max , inside the unit circle can be computed from purely geometrical considerations for the ellipse in
B. Two-channel case
As in Sec. IV, we can also extend the TRITOPS model to the two-channel case by coupling two single-channel wires. More general multichannel wire constructions are also possible but will not be pursued here. The corresponding Hamiltonian is with Eq. (31) given bŷ
H DIII,2ch (k) = Ĥ DIII (k)T DIIÎ T † DIIIĤ DIII (k) ,(35)
where the interwire tunneling couplings are modeled in a similar manner as in Eq. (28):
T DIII = −t y σ 0 τ z + iα y σ y τ z + ∆ y σ 0 τ z .(36)
We here allow for spin-conserving (t y ) and spin-flipping (α y ) hopping processes, as well as for nonlocal pairing terms (∆ y ). Below, t y and α y are parametrized as specified in Sec. IV. The resulting phase diagram is illustrated in Fig. 7(a). To make analytical progress, from now on we consider the case ∆ y = 0 and determine the conditions for gap closings, and thus for phase transition curves in the twochannel TRITOPS case. The gap closes again for ka = ±π/2 as in Sec. V A but now for the chemical potential set to one of the critical values
|µ ± | = α 2 y + (t y ± 2α) 2 .(37)
where the topological invariant is related to the product of the signs of the effective pairing amplitude at different Fermi points [66]. As the critical momenta are as in Sec. V A, the pairing function is directly determined by ∆ cos(ka) [see Eq. (31)]. For this reason, the topologically nontrivial (trivial) phase has an odd (even) number of Fermi points between ka = 0 and ka = π/2. The bGF can again be computed from the roots of the secular polynomial. The latter also determine the behavior of the edge modes of a semi-infinite two-channel TRITOPS wire in different regions of the phase diagram. By continuity, the condition of having an odd number of Fermi points with 0 < k F < π/2a corresponds to an odd number N p of roots near the unit circle in the first quadrant. Our results for the roots are illustrated in Fig. 7(c)-7(f). As expected, N p is odd for panels (d) and (f), where panel (b) shows that Majorana end states are present and thus a topological phase is realized. By contrast, panels (c) and (e) show topologically trivial cases with even N p .
VI. PHASE-BIASED TOPOLOGICAL JOSEPHSON JUNCTIONS
In this section, we consider different examples for the equilibrium supercurrent-phase relation in two-and three-terminal Josephson junctions made of nanowires in topologically nontrivial superconducting phases. These wires are coupled together by tunnel junctions. We start in Sec. VI A with the case of a trijunction of TS nanowires (see also Ref. [61]), and then turn to TRITOPS-TS Josephson junctions in Sec. VI B.
A. Three-terminal TS junctions
We first consider a three-terminal junction formed by spinful single-channel nanowires in the TS phase. For a schematic layout, see Fig. 8. Such devices have been suggested, e.g., for Majorana braiding implementations [78][79][80], for the engineering of artificial topological Weyl semimetal phases [81,82], and for the observation of giant shot noise features induced by the single zero-energy MBS localized at the trijunction [83]. While most previous studies have been based on minimal models or on spinless Kitaev chain models, a more realistic description using the spinful nanowire model of Refs. [68,69] discussed in Sec. III is desirable. In particular, one can then assess the role of the spin degree of freedom and the effects of various microscopic parameters such as the angle θ in Fig. 8. We assume that each wire is sufficiently long such that the overlap between MBSs located at different ends of the same wire is negligibly small. The red dots indicate MBSs with Majorana operators γL,R,C near the junction, with tunnel couplings λL,R connecting the L, R wires to the C wire. We assume that no direct tunnel coupling between the L and R wires is present. A Zeeman field Vz is applied perpendicular to the plane containing the three wires. Blue arrows show the positive momentum direction in each wire.
We model each nanowire in the setup of Fig. 8 in terms of the spinful single-channel Hamiltonian of Eq. (10). All three wires lie in a plane, with two of them aligned (L and R in Fig. 8) and the third (the central wire, C, in Fig. 8) at an arbitrary angle θ to the other two. We here assume that the Zeeman field V z is oriented perpendicular to the plane (see Ref. [1]). For simplicity, we consider identical material parameters for the three wires which are chosen such that the TS phase is realized.
Let us next discuss the unitary rotations necessary to adapt the bGFs of Sec. III to a common reference frame for all three wires in Fig. 8. We first perform a π/2 rotation of the spin axis around the y axis, which connects the intrinsic coordinate system of the L and R wires to the common reference frame. Defining
R(ϑ) = [σ 0 cos(ϑ/2) − iσ y sin(ϑ/2)] τ 0 ,(38)
the corresponding rotation matrix, R y = R(ϑ = π/2), transforms a Zeeman field along the x-direction (see Sec. III) into a Zeeman field along the negative zdirection (as in Fig. 8). The bGFs for the L and R wires in Fig. 8 are thus given bŷ
G L/R = R yĜL,R R −1 y ,(39)
withĜ L,R as described in Sec. III. For the C lead, we additionally have to rotate by the angle θ around the global z-axis. The corresponding rotation matrix, R z (θ), follows from Eq. (38) with the replacements σ y → σ z and ϑ → θ. We thereby obtain
G C = R z (θ)R yĜL R −1 y R −1 z (θ).(40)
In what follows, we rewriteĜ L/R →Ĝ L/R to keep the notation simple. The coupling between the L, R wires and the C wire is modeled by a spin-conserving tunneling term,
H T = 1 2 ν=L,RΨ † νλνΨC + H.c.,λ ν = λ ν σ 0 τ z e iτzφν /2 ,(41)
whereΨ L,R,C are boundary spinor fields and φ ν is the phase of the superconducting order parameter in the respective wire. We choose a gauge with φ C = 0 and realvalued tunnel couplings λ ν . The physical properties of the trijunction are then determined by the full bGF,
G 3TS = Ĝ −1 Lλ L 0 λ † LĜ −1 Cλ R 0λ † RĜ −1 R −1 ,(42)
where the 3 × 3 structure refers to wire space. From Eq. (42), the energy dependence of the local DoS at the junction will be given by Figure 9 shows the phase dependence of ρ 3TS (ω) obtained by numerical evaluation of Eqs. (42) and (43) for a trijunction with φ L = −φ R = φ and φ C = 0. (This is the series configuration in the parlance of Ref. [61].)
ρ 3TS (ω) = − 1 π Im Tr Ĝ 3TS (ω) .(43)
Deep in the topological regime, the low-energy properties of the trijunction are well described by a minimal model keeping only the MBSs at the junction. To show this from the above bGFs, we first derive an effective Hamiltonian for each wire that only keeps track of the respective MBS:
H eff,ν = lim ω→0Ĝ −1 ν (ω).(44)
Using Eq. (44) and recalling that the z 2 roots dominate for V x > V c , we can read off the boundary spinors for each of the wires (ν = L, R, C; see Ref. [61]):
Ψ L ∆ 2 t 0 1 −i 0 γ L ,Ψ R ∆ 2 t 0 −i 1 0 γ R , Ψ C ∆ 2 t R z (θ) 0 1 −i 0 γ C ,(45)
where the Majorana operators γ ν satisfy the anticommutation relations {γ ν , γ ν } = δ νν . The pairing gap ∆ 2 has been defined in Sec. III [see also Fig. 1 and Eq. (20)]. Next, we project the tunneling Hamiltonian (41) to the Majorana sector by means of Eq. (45). We thereby arrive at a minimal model Hamiltonian,
H mm = −iΩ L (φ)γ L γ C − iΩ R (φ)γ R γ C ,(46)
with the energies
Ω L (φ) = 2∆ 2 λ L t sin φ + θ 2 , Ω R (φ) = − 2∆ 2 λ R t cos φ − θ 2 .(47)
Equation (46) is easily diagonalized by rotating the γ L,R operators to new Majorana operatorsγ L,R ,
γ L γ R = sin κ − cos κ cos κ sin κ γ L γ R ,(48)
with sin κ = Ω L /Ω and
Ω(φ) = Ω 2 L (φ) + Ω 2 R (φ).(49)
We thereby arrive at
H mm = −iΩ(φ)γ L γ C ,(50)
where the decoupled Majorana operatorγ R describes the remaining zero-energy state [83]. The eigenstates of Eq. (50) correspond to Andreev bound states with the phase-dependent subgap energy [see Eq. (47)],
E ± (φ) = ± 1 2 Ω 2 L (φ) + Ω 2 R (φ).(51)
The phase derivative ∂ φ E − (φ) then yields the Josephson current-phase relation. As illustrated in Fig. 9, Eq. (51) reproduces our numerically exact bGF calculations for small tunnel couplings λ L,R . However, for intermediate-to-large values of the tunnel couplings, the Andreev bound state dispersion may deviate from Eq. (51) [see, e.g., the 'bump'-like features in Fig. 9(c)]. Such deviations are due to the fact that the Majorana operators γ L and γ R will become connected through the virtual excitation of continuum quasiparticle states with above-gap energy E > ∆. Within our minimal model, this physics can be taken into account by adding an effective coupling λ LR between the L and R wires. For λ ν ∆, we estimate λ LR λ L λ R /∆. The corresponding tunneling term is given by
H T,LR = 1 2 λ LRΨ † L σ 0 e iτzφ τ zΨR + H.c.(52)
Using the Majorana spinors in Eq. (45) together with Eq. (46), we arrive at an improved version of the minimal model Hamiltonian:
H mm = −iΩ L (φ)γ L γ C − iΩ R (φ)γ R γ C −iΩ LR (φ)γ L γ R , Ω LR (φ) = 2∆ 2 λ LR t cos φ.(53)
One can easily show that Eq. (53) still predicts a decoupled zero-energy MBS at the trijunction. The hybridization between the remaining two Majorana states yields Andreev bound states with the dispersion relation
E ± (φ) = ± 1 2 Ω 2 L (φ) + Ω 2 R (φ) + Ω 2 LR (φ).(54)
Of course, for λ LR → 0, we recover Eq. (51). Only by including the Ω LR term in Eq. (54), however, the bumps found in the numerically exact dispersion in Fig. 9(c) can be accurately reproduced. We conclude that the minimal model in Eq. (53), which has been derived from the bGF approach, captures the basic physics of the Josephson effect in the three-terminal TS junction shown in Fig. 8. In particular, the dependence of the current-phase relation on the angle θ between the wires resulting from the subgap spectrum in Fig. 8 will be correctly reproduced.
B. TRITOPS-TS junction
We next consider the two-terminal Josephson junction in Fig. 10 Denoting the respective boundary spin-Nambu spinors byΨ L andΨ R , respectively, the tunneling Hamiltonian is given by where φ is the superconducting phase difference across the junction and we assume a real-valued tunnel coupling λ L . Below we assume for simplicity that the pairing gap ∆ is identical for both nanowires. We will allow for a relative angle θ between the directions of the spin-orbit field in each wire, see the schematic device layout in Fig. 10. One could vary θ by changing the orientation of a local electric field applied to the TS wire only, which in turn will affect the corresponding Rashba spin-orbit field. In addition, we need a Zeeman field to induce the topological phase in the TS nanowire (see Sec. III), while no Zeeman field should be present on the time-reversal invariant TRITOPS side. To achieve this goal, one may use mesoscopic ferromagnets for inducing a Zeeman field only locally [84].
H T = 1 2 λ LΨ † L σ 0 e iτzφ/2 τ zΨR + H.c.,(55)
To account for the angle θ, we then apply the unitary transformation R y (θ) to the bGF describing the TS nanowire. This rotation simultaneously affects the spinorbit and the Zeeman field directions in the TS wire such that both directions can never be parallel to each other. The junction spectral properties then follow again from a Dyson equation as in Eq. (41). Assuming that both wires have model parameters putting them deeply into the respective topological regime, we can compare our numerically exact results for the subgap spectral properties to the corresponding predictions of a minimal model Hamiltonian. The latter is obtained by retaining only the MBS degrees of freedom indicated in Fig. 10. To that end, the approximate expression for the boundary spinors can again be derived from the respective bGFs as in Sec. VI A. Those spinors involve the Majorana operators γ L1,L2,R in Fig. 10 and are given bŷ
Ψ L ∆ t 1 0 i 0 γ L1 + ∆ t 0 i 0 1 γ L2 , Ψ R ∆ 2 t R y (θ) i −i 1 1 γ R .(56)
The resulting minimal model Hamiltonian is
H min = −i [w 1 (φ)γ L1 + w 2 (φ)γ L2 ] γ R(57)
with the energies
w 1 (φ) = 2λ L √ ∆∆ 2 t cos φ 2 cos θ 2 , w 2 (φ) = − 2λ L √ ∆∆ 2 t sin φ 2 sin θ 2 .(58)
The structure of H mm in Eq. (57) is similar to the minimal model (46) for the TS trijunction in Sec. VI A without any coupling between the γ L1,L2 operators. The subgap spectrum is therefore characterized by a decoupled zero-energy Majorana state, and the hybridization of the two other Majorana operators yields the Andreev bound state dispersion:
E ± (φ) = ± 1 2 w 2 1 (φ) + w 2 2 (φ).(59)
We compare Eq. (59) to numerically exact results for the subgap spectral properties of the TRITOPS-TS junction in Fig. 11. Clearly, the general subgap spectrum is rather well described by the minimal model (57). In contrast to the case of a tri-terminal TS junction, for TRITOPS-TS junctions it is not necessary to take into account higherorder tunneling processes for obtaining accurate agreement with numerically exact bGF calculations (but see Ref. [85]).
VII. CONCLUDING REMARKS
In the present work, we have generalized the boundary Green's function approach of Refs. [59,61] to quasi-1D spinful models of Majorana nanowires. For singlechannel class D and class DIII wire models, we have obtained an analytical understanding of the behavior of the roots of the corresponding secular polynomial in complex momentum space. This advance helps physical intuition and allows for a practical and numerically efficient method for computing the bGF, and thereby also physical observables. The method has also been extended to spinful multichannel models, where it appears to allow for more efficient numerical bGF calculations than the alternative recursive technique [60,77]. Let us remark that the computational complexity of the method is only limited by the ability to evaluate the roots of a polynomial. Typically, the numerical demands are therefore much smaller than those for a recursive calculation of the bGF.
Given the efficient construction of the bGF put forward in this work, one can now apply the general bGF approach [59] to study the transport properties of many different hybrid devices composed of Majorana nanowires and/or conventional metals or superconductor electrodes. In Sec. VI, we have provided two examples for such devices, namely phase-biased trijunctions of TS wires and TRITOPS-TS junctions. In both cases, we have carried out an analysis of the subgap Andreev (or Majorana) state dispersion at zero temperature.
We believe that this approach offers many interesting perspectives for future research. In particular, one can study nonequilibrium transport properties away from the linear-response regime, and one can also include electronelectron or electron-phonon effects, at least on a perturbative level. We are confident that the results of our work can also be helpful for the interpretation of transport experiments carried out on hybrid devices containing nanowires with topologically nontrivial superconducting phases.
Acknowledgments
We thank L. Arrachea In this appendix, we provide technical details pertaining to our discussion of the spinful single-channel Majorana wire model [68,69] in Sec. II. First, the explicit form of the coefficients C n (ω) in Eq. (14) is given by
C 0 = 3α 4 8 + ∆ 4 + µ 4 − 8µ 3 t + 36µ 2 t 2 − 80µt 3 +70t 4 − 2µ 2 V 2 x + 8µtV 2 x − 12t 2 V 2 x +V 4 x − 2(µ 2 − 4µt + 6t 2 + V 2 x )ω 2 + ω 4 +2∆ 2 (µ 2 − 4µt + 6t 2 − V 2 x − ω 2 ) +α 2 ∆ 2 − µ 2 + 4µt − 5t 2 + V 2 x − ω 2 , C 1 = −(µ − 2t)t(α 2 − 4(∆ 2 + µ 2 − 4µt + 7t 2 −V 2 x − ω 2 )), C 2 = −α 4 + 8t 2 ∆ 2 + 3µ 2 − 12µt + 14t 2 − V 2 x −ω 2 + 2α 2 −∆ 2 + (µ − 2t) 2 − V 2 x + ω 2 /4, C 3 = (tµ − 2t 2 )(α 2 + 4t 2 ), C 4 = t 2 + (α/2) 2 2 (A1)
It is convenient to renormalize these coefficients such that C 4 appears as common factor of the polynomial. The C n coefficients in turn determine the coefficients a m (ω) appearing in the eighth-order polynomial equation (15). The roots z n (ω) therefore have satisfy the Vieta relations S k (z 1 , . . . , z 8 ) = i1<i2<···<i k z i1 z i2 · · · z i k = (−1) 8−k a k a 8 .
Using the condition (20) and the ansatz (19), the first three invariants are given by S 1 = 2AB, S 2 = 2(A 2 − 2)(1 + C) + 4B 2 , and S 3 = 2AB(A 2 − 1) + 4ABC with A = 1 − η + 1 1 − η , B = cos(δ 1 ) + cos(δ 2 ), C = 2 cos(δ 1 ) cos(δ 2 ).
As a consequence, the parameter C obeys a cubic equation that can be solved analytically,
w 3 + w 2 C + w 1 C 2 + C 3 = 0,(A4)
with the coefficients
w 1 = 1 − S 3 S 1 , w 2 = S 2 4 − S 3 S 1 − 1 4 + S 3 2S 1 2 , w 3 = − S 2 S 3 8S 1 − S 2 8 + S 2 1 − 1 4 + S 3 2S 1 2 .(A5)
For V x < V c , the physical solution of Eq. (A4) is given by
C = −2 −Q cos(θ 0 /3) − w 1 /3,(A6)
with
θ 0 = cos −1 − R −Q 3 , Q = 3a 2 − w 2 1 9 , R = 9w 1 w 2 − 27w 3 − 2w 3 1 54 .(A7)
For V x > V c , the solution is given by C = P 1 − Q/P 1 − w 1 /3 (assuming P 1 = 0), with
P 1 = sgn(R) |R| + R 2 + Q 3 1/3 .(A8)
The coefficients A and B then follow from
A 2 = S 3 S 1 + 1 − 2C, B = S 1 2A . (A9)
Finally, the parameters in our ansatz [see Eqs. (19) and (21)], can be determined from the relations
cos δ 1 = B + √ B 2 − 2C 2 , cos δ 2 = B − √ B 2 − 2C 2 , η = 1 − A 2 + A 2 4 − 1.(A10)
We proceed by providing the detailed form of the ma-trices ν and ν in Eq. (25). Using the definition in the main text, for V x < V c , they are with z ν (ω) in Eq. (19) given bŷ
A ν = (z ν ) b ν + (z * ν ) b * ν , ν = (z ν ) b ν + (z * ν ) b * ν ,(A11)
where an expansion of zν =zm (z ν − z m ) to first order in τ ν ∆ 2 ν − ω 2 yields b ν = 32e 3iδν τ ν sin 2 (δ ν ) [cos(δ 2 ) − cos(δ 1 )] 2 .
Explicitly, the components of the symmetric 4 × 4 matrix A in Eq. (A11), ij = ji , follow from
A 11 (z) = −Â 33 (z) = z 3 V 2 x [ (z) −α(z)] + z 3 [ (z) +α(z)] −∆ 2 − ( (z) −α(z)) 2 , A 22 (z) = −Â 44 (z) = z 3 V 2
x [ (z) +α(z)] + z 3 [ (z) −α(z)] −∆ 2 − ( (z) +α(z)) 2 , A 12 (z) =Â 34 (z) = z 3 V x ∆ 2 + 2 (z) −α 2 (z) − V 2
x , A 13 (z) = z 3 ∆ V 2
x − ∆ 2 − [ (z) −α(z)] 2 , A 14 (z) = −Â 23 (z) = 2z 3 V x ∆α(z),
A 24 (z) = z 3 ∆ V 2 x − ∆ 2 − [ (z) +α(z)] 2 .(A13)
Similarly, by taking a derivative with respect to ω, the nonvanishing matrix elements of the symmetric matrix A ij =Â ji follow aŝ A 11 (z) =Â 33 (z) = −z 3 ∆ 2 + [ (z) − α(z)] 2 + V 2 x , A 22 (z) =Â 44 (z) = −z 3 ∆ 2 + [ (z) + α(z)] 2 + V 2
x , A 12 (z) = −Â 34 (z) = 2z 3 V x (z),
A 14 (z) =Â 23 (z) = 2z 3 V x ∆.(A14)
In the topologically nontrivial phase, V x > V c , trigonometric functions associated with the roots z 1,± in Eq. (21) turn into hyperbolic functions. The matrices with ν = 1 in Eq. (A11) are then replaced bŷ
A 1 =Â (z 1,+ ) b 1 −Â (z 1,− ) b 1 , A 1 =Â (z 1,+ ) b 1 −Â (z 1,− ) b 1 ,(A15)
with the quantities b 1 = 32e −3δ1 τ 1 sinh 2 (δ 1 ) [cos(δ 2 ) − cosh(δ 1 )] 2 , b 2 = 32e 3iδ2 τ 2 sin 2 (δ 2 ) [cos(δ 2 ) − cosh(δ 1 )] 2 . (A16)
The ν = 2 matrices follow from Eq. (A11) with the replacement b 2 →b 2 . Finally, we note that for very large V x , one approaches the Kitaev limit of the nanowire, and the relevant residues come from the z 2 roots only.
in Fig. 5), we first compute the rotation angle θ of the ellipse using the eigenvectors of the conic section matrix, cos θ = 1 1 + X 2 /(2t∆) 2 , X = ∆ 2 − t 2 − α 2 − (t 2 + ∆ 2 ) 2 + 2(t 2 − ∆ 2 )α 2 + α 4 . (B4)
As a consequence, l follows from the relation |OF 1,2 | = (f /2) 2 + µ 2 ± µf sin θ.
The largest-modulus root inside the unit circle is then given by (see Refs. [76,77])
|z max | = l + l 2 − f 2 M + m . (B6)
The same result follows for the other block,Ĥ + (k). As discussed in Sec. V A, Eq. (B6) determines the decay length of Majorana end states into the bulk of a TRI-TOPS wire.
Figure 1 :
1Bulk dispersion relation of the spinful singlechannel Majorana wire model[68,69]. (a) E k,− vs k [see Eq.(11)] for the topologically trivial regime Vx < Vc (solid red curve), indicating the two pairing gaps ∆1 and ∆2 at k = k1 and k = k2, respectively [see Eq.(16)]. We use µ = 5 meV, ∆ = 2 meV, and Vx = 0.5Vc. All other parameters are specified in the main text. The dashed yellow curve is for ∆ = 0. (b) Evolution of the two gaps (normalized to the velocities v1,2) vs Zeeman parameter Vx for ∆ = 0.2 meV and µ = 2 meV. We note that ∆1 and v1 simultaneously vanish as Vx → Vc.
, respectively. Bottom panels: Energy dependence of the local bulk DoS, ρ(ω) (in meV −1 ), for (e) Vx < Vc and (f) Vx > Vc. The solid red curves depict numerically exact results using Eq. (3) and the dashed green curves show approximate results obtained from Eq.(25).
Figure 3 :
3Spatial variation of the local DoS, ρj(0) (in meV −1 ), vs distance from the boundary, x = ja (in µm), for the ω = 0 Majorana state in a semi-infinite TS wire with µ = 1 meV and Vx = 2Vc. The solid blue curve gives numerically exact results obtained from Eq. (7). Red-dotted and green-dashed curves show Eq. (26) with and without 2kF oscillations, respectively.
Figure 4 :
4Two-channel spinful Majorana wire model of class D [see Eq. (27)] with parameters as explained in the main text. Panel (a) shows the bulk phase diagram in the µ-Vx plane. Topological nontrivial (trivial) phases are shown in red (blue). Panels (b) and (c) show the energy dependence of the local DoS, ρj=1(ω) (in meV −1 ), at the boundary of a semi-infinite two-channel wire along the trajectories marked by arrows in panel (a). Panels (d)-(f) illustrate the roots zn(0) inside the unit circle at the three points indicated in panel (a) by a triangle (d), a square (e), and a circle (f), respectively. For additional insights, see Supplemental Material
Figure 5 :
5Curve traced out by β−(k) in theσx-σz plane for a single-channel TRITOPS wire in a topologically nontrivial phase [see Eqs. (33) and (34)] with t = 0.5, α = 0.8, ∆ = 1, and µ = 1.04 (all in meV). The evolution of the bulk Hamilto-nianĤ−(k) upon traversal of the Brillouin zone is described by an ellipse containing the origin (O). For details, see main text and Appendix B.
Figure 6 :
6Spatial variation of the local DoS at zero energy (in meV −1 ), corresponding to Majorana end states of a semiinfinite TRITOPS wire in its topological phase [see Eq. (31)] for µ = 0 (blue solid curve). The green dashed curve shows an exponential decay on the length scale λe = − a 2 ln |zmax| [see Eq. (B6)].
Fig. 5 (
5see Appendix B for details). The length scale governing the spatial decay profile of the pair of Majorana states localized near the boundary of a semi-infinite TRITOPS wire then follows as λ e = − a 2 ln |z max | [see Eq. (B6) in Appendix B]. The validity of this expression is confirmed inFig. 6, where we show numerically exact results for the spatial variation of the local DoS at ω = 0 together with the prediction obtained from Eq. (B6).
Figure 7 :
7Two-channel TRITOPS nanowire [see Eq. (35)], with parameters as explained in the main text. Panel (a) shows the phase diagram in the µ-∆y plane, with the topologically nontrivial (trivial) phase in red (blue). (b) Local DoS, ρj=1(ω) (in meV −1 ), at the boundary of a semi-infinite wire in the µ-ω plane for ∆y = 0. Panels (c) to (f) depict the roots zn(ω = 0) inside the unit circle for different µ as indicated by the respective symbol in panel (b). We use ∆ = 1 meV in panels (c)-(f).
Figure 8 :
8Three-terminal junction of spinful TS nanowires (see Sec. III), with two parallel wires (L, R) and a central (C) wire at angle θ.
Figure 9 :
9Phase dependence of the subgap spectrum of the trijunction of TS wires in Fig. 8, with the superconducting phases φL = −φR = φ and φC = 0. The TS wires are modeled as spinful nanowires with µ = 2 meV, Vz = 3Vc, and symmetric couplings, λL = λR = λ. For other parameters, see Sec. III. Panel (a) [(b)] is for λ = 2 meV and θ = π/2 [θ = π/10]. Panel (c) [(d)] is for λ = 5 meV and θ = π/2 [θ = π/10]. From blue to yellow, ρ3TS(ω) (in meV −1 ) gradually increases, where Eq. (43) has been evaluated in a numerically exact manner. White dotted [dashed] curves show the approximate Andreev bound state dispersion relation in Eq. (54) [Eq. (51)].
Figure 10 :
10Sketch of a TRITOPS-TS Josephson junction. Colored dots indicate MBSs corresponding to the Majorana operators γL1,L2,R. The tunnel coupling λL connects both wires, where blue arrows shows the positive momentum direction in each wire. The spin-orbit axes on both sides are tilted by the relative angle θ.
Figure 11 :
11Phase-dependent subgap spectrum of a TRITOPS-TS Josephson junction for different values of the tilt angle θ in Fig. 10. The spinful single-channel model parameters are as described in Secs. III and V, with µ = 1 meV, λL = 2 meV, and Vx = 1.5Vc on the TS side. The tilt angle is θ = 0 in panel (a), θ = 0.3π in panel (b), θ = π/2 in panel (c), and θ = 0.7π in panel (d). From blue to yellow, the color code indicates increasing DoS values at the junction, ρ(ω) (in meV −1 ). White dashed curves show the Andreev bound states (59).
Eq.(21) captures the low-energy behavior of the roots rather well, especially in cases where electronhole symmetry is approximately realized.For this model of symmetry class D, the Z 2 bulk topological invariant takes the form[1] (c)
and 2(d),
between a TRITOPS wire [see Eq. (33) in Sec. V A], and a TS nanowire [see Eq. (10) in Sec. III].
and T. Martin for discussions. This project has been funded by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy, Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 -390534769; by the Spanish Ministry of Economy, Industry, and Competitiveness through Grant No. FIS2017-84860-R; and by the María de Maeztu Programme for Units of Excellence in n Research and Development Grant No. MDM-2014-0377. Appendix A: On the spinful single-channel model
Appendix B: On TRITOPS wiresAccording to Theorem 1 of Ref.[76], the largestmodulus root z max inside the unit circle can be determined from the relative position of the origin inside the ellipse discussed in Sec. V A. For that purpose, we first determine the major (M ) and minor (m) axes of the ellipse inFig. 5. Using Eq.(34)and focusing on the case ofĤ − (k), the defining equation of the ellipse is given by. From the eigenvalues of the 2 × 2 matrix in Eq. (B1),The distance between the foci of the ellipse then follows as f = √ M 2 − m 2 . To obtain the distance l = |OF 1 | + |OF 2 | between the foci and the origin (corresponding to the red dashed line
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"F Cajiao -Vélez ",
"A Jaron "
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| High-order harmonic responses from three C20 isomers: fullerene, ring, and bowl, are calculated within the modified Lewenstein model for molecular systems. Spectra for all three structures exhibit intense modulations of the harmonic spectrum along the plateau and some of them can be interpreted as a consequence of multi-center interference effects. Each structure shows characteristic modulation patterns in peak harmonic intensities, which are directly related to zeroes in the recombination matrix element as a function of the three components of momentum. Different C20 isomers lead to different harmonic polarizations depending on the geometric configuration of carbon atoms and molecular orientation.PACS numbers: | null | [
"https://arxiv.org/pdf/2203.15933v1.pdf"
]
| 247,793,007 | 2203.15933 | a821b22ca5e6336d30c4e9645c350dbc15c79441 |
29 Mar 2022
F Cajiao -Vélez
A Jaron 29 Mar 2022High-order harmonic generation from C 20 isomers
High-order harmonic responses from three C20 isomers: fullerene, ring, and bowl, are calculated within the modified Lewenstein model for molecular systems. Spectra for all three structures exhibit intense modulations of the harmonic spectrum along the plateau and some of them can be interpreted as a consequence of multi-center interference effects. Each structure shows characteristic modulation patterns in peak harmonic intensities, which are directly related to zeroes in the recombination matrix element as a function of the three components of momentum. Different C20 isomers lead to different harmonic polarizations depending on the geometric configuration of carbon atoms and molecular orientation.PACS numbers:
I. INTRODUCTION
In recent years, high-order harmonic generation (HHG) has been a very active research topic in experimental and theoretical fields due to the new and interesting properties of the non-linear interaction between laser radiation and molecules or atoms. This process involves the transformation of multiple low-energy photons coming from the laser field into a single high frequency photon [1]. When the intense laser pulse interacts with a molecule or atom, coherent radiation of frequencies that are integer multiples of the original driving one are emitted [2]. HHG spectra exhibit particular characteristics as a consequence of the non-linear interaction. The first harmonics decrease in intensity rapidly before a plateau, where the strength of the peaks is fairly constant. A sharp cutoff determines a new region characterized by a fast decrease of the harmonic signal. The maximum energy at the end of the plateau can be approximated by the formula [2][3][4]
E cutoff = I p + 3.17U p ,(1)
where I p is the ionization potential and U p = e 2 E 2 0 /(4ω 2 0 ) corresponds to the ponderomotive energy of a free electron driven by a monochromatic plane wave. Here, E 0 represents the amplitude of the electric field and ω 0 is the laser carrier frequency. Atomic units are used along this paper while the electron charge (e < 0) is explicitly written in the formulas. For numerical calculations we set |e| = 1, unless stated otherwise.
As the HHG process is of purely quantum mechanical origin, the most suitable treatment comes from the complete solution of the time-dependent Schrödinger equation (TDSE) which is a prohibitive task for multielectron systems interacting with intense laser fields. Approximate methods such as time-dependent density functional theory (TDDFT) (see, e.g., [5][6][7][8][9]) and time dependent Hartree-Fock theory (see e.g., [? ]) have been applied to intermediate size systems. Note that to solve the TDSE or the TDDFT/TDHF is a challenging task already for two-electron systems. While it is time consuming and requires a big amount of computational effort, those methods can be used to model the interaction of atoms and simple molecules with strong laser fields. Their solution presents, with good agreement with experiments, a plateau before the sharp cutoff [10]. With increasing the laser field intensity or the complexity of the molecule, solving the TDSE or the TDDFT/TDHF becomes prohibitive [11,12].
Many features of the HHG process can be understood by means of the semiclassical three-step model. In the first step, the electron is ionized by tunneling effects due to the distortion of the atomic potential by the oscillating electric field. In the second step, the electron travels in the continuum as a classical particle subject to the Lorentz forces in the laser field. Finally, the electron recombines with the parent ion with the consequent emission of a highly energetic photon [4]. Due to the classical treatment of the propagation step, many quantum-mechanical effects, including the electron wave packet spreading in the continuum and its acquired phase, are ignored.
The Lewenstein model offers a complete quantum-mechanical treatment of HHG by considering the electron wave function, between ionization and recombination events, as a dressed plane wave propagating in the laser field. The Coulomb interaction between parent ion and ejected electron is neglected, so the total evolution in the continuum is governed by a Volkov type evolution operator.
When applied to atoms, the Lewenstein model has proven to give a very good quantitative and qualitative agreement with ab-initio calculations [12] with the advantages of being considerably faster and requiring less computational effort. Such model has been extended and applied to polar and non-polar molecules [12][13][14][15].
An important feature of the Lewenstein model is its absence of gauge invariance and many problems related to the gauge choice may arise, specially when the HHG spectrum from molecules is calculated. For example, if the harmonic emission from an atom displaced from the origin of coordinates is considered, new effects are observed in the response calculated in the length and velocity gauges. Both of them show the presence of even and odd peaks, but in the length gauge the odd harmonics are not invariant under spacial translation [12]. All those drawbacks seem to be partially avoided by the insertion of new terms into the semiclassical action (see, Sec. II). Those terms account for the multi-center features of a molecule and predict interference modulations along the plateau [12]. In Ref. [12] it has been shown that the notorious effects related to the length gauge choice (e.g., nonphysical enhancement of the plateau and increasing harmonic intensities with the internuclear distances) are less important for small molecular sizes. If the nuclear separation is shorter than the quiver radius of the electron [α 0 = |e|E 0 /ω 2 0 ], the artifact features seem to be less intense. This has been proven for diatomic molecules and extended to bigger systems [1,12]. When a large number of atoms is taking into consideration, it is useful to define a parameter Q,
Q = R max 2α 0 .(2)
Here, R max is the maximum distance between two nuclei in the molecule and α 0 (the quiver radius) is the amplitude of classical oscillations of an electron in the laser field. This parameter may help to determine whether or not the length gauge can be applied without introducing nonphysical effects (see, Ref. [1]). The Lewenstein model predicts important contributions to the harmonic response from two different electron trajectories (long trajectory and short trajectory [16]). Such paths are determined by the time spent by the electron in the continuum. It has been proven that different trajectories can interfere constructively or destructively, generating strong modulations on the harmonic peak intensities along the plateau [17,18].
The present paper is devoted to the analysis of the harmonic response from three different C 20 isomers: the fullerene (or cage), the ring, and the bowl (see, Fig. 1). The cage is the smallest possible fullerene, with atomic centers located at the corners of twelve pentagons. The bowl geometry can be considered as a fragment of the buckminsterfullerene C 60 . The planar monocyclic ring is another configuration adopted by the C 20 family. The symmetry and planar structure of such molecule, together with its particular electronic configuration, make the C 20 -ring an interesting harmonic target. Due to the highly symmetric nature of the fullerene and ring, no permanent dipole moment is observed in such structures. As a consequence of the lack of an inversion center, the bowl is characterized by a permanent electric dipole moment pointing along the z-direction. In Table I, we present the values of the mentioned static dipole moments together with the calculated ionization potentials of the Highest Occupied Molecular Orbital (HOMO), (Ip H ) and HOMO-1 (Ip H−1 ). In order to calculate the molecular orbitals for each isomer, Hartree-Fock methods included in the standard quantum chemistry package GAMESS [19] are used after a proper convergence is guaranteed. In Pople's notation, the basis set used in the calculations is written as 6-311G (see, e.g., [20,21]).
In order to illustrate the effects of multi-center and quantum path interferences, a relatively large intensity of I = 5 × 10 14 W/cm 2 and a wavelength of 800nm have been chosen for the calculations. This guarantees that the location of the cutoff is beyond the 60th harmonic order for all structures. The same analysis can be performed for lower intensities and larger frequencies, so the plateau is long enough to observe the interference effects on the harmonic response. In Table II we show the average radius of each C 20 isomer together with the corresponding parameter Q. As it can be seen, the small values of Q suggest that the length gauge formalism can be applied to analyze the harmonic spectrum from the three C 20 structures, given the aforementioned laser parameters, without introducing nonphysical effects [1,22].
Along this paper, different harmonic spectra are studied and related to the geometric and electronic structure of each molecule. Multi-center interference can give important information about the localization of the atoms, as have been shown in Ref. [1]. Certain modulations along the harmonic plateau are direct consequence of multi-center interference effects and can be localized around the points where the recombination matrix element vanishes [1,12]. This relation can be explored in order to analyze and compare the interference effects for different molecular geometries. Other interference patterns are not directly related to the multi-center nature of the molecule and can be used in order to understand the electron dynamics during HHG. II: Average radius and parameter Q for the three C20 isomers. The laser parameters correspond to an intensity of 5 × 10 14 W/cm 2 and a wavelength of 800nm. For the bowl the radius is taken as the maximum distance from the symmetry axis.
II. THEORY
Along this Section we present the main formulas used in the time-dependent dipole moment in the Lewenstein model. The importance of the molecular recombination matrix element and its calculation is shown in Sec. II B.
A. Time-dependent dipole moment in the Lewenstein model Let us consider an electromagnetic wave interacting with a molecule or atom. We assume that the single-activeelectron approximation (SAE) holds, and just one electron is ionized. The Hamiltonian in the length gauge iŝ
H = 1 2p 2 + V (r) Ĥ 0 − eE(t) ·r Ĥ I ,(3)
whereĤ 0 represents the ground state Hamiltonian,Ĥ I is the interaction with the electromagnetic wave, V (r) represents the Coulomb potential energy, and E(t) is the electric field that describes the laser interaction. The total time-dependent dipole moment can be approximated as [23]
d L (t) ≈ −i t −∞ dt ′ ψ 0 (t)|(er)Û (t, t ′ )Ĥ I (t ′ )|ψ 0 (t ′ ) + c.c. ,(4)
where |ψ 0 (t ′ ) = |ψ 0 e −iE0t ′ , is the molecular ground state of energy E 0 at a time t ′ . According to the Lewenstein model, the total evolution operatorÛ (t, t ′ ) is substituted by a Volkov type evolution operator U V (t, t ′ ) which, in the length gauge, reads
U V (t, t ′ ) = dp|p − eA(t) p − eA(t ′ )| × exp − i 2 t t ′ (p − eA(σ)) 2 dσ .(5)
Here, A(t) is the vector potential associated to the oscillating electric field, E(t) = − ∂A(t) ∂t . By replacing the total evolution operator by U V (t, t ′ ) in Eq. 4, the time-dependent dipole moment can be expressed in terms of the recombination and ionization matrix elements (RME and IME, respectively) as
d(t) = ie 2 dp t −∞ dt ′ exp [−iS(p, t, t ′ )] (6) × d * rec (p − eA(t)) d ion (p − eA(t ′ )) · E(t ′ ) + c.c.,
where the semiclassical action S(p, t, t ′ ) is given by
S(p, t, t ′ ) = t t ′ dσ (p − eA(σ)) 2 2m + I p .(7)
Here, I p = −E 0 is the atomic or molecular ionization potential. The recombination and ionization matrix elements in Eq. 6 are written as (see, e.g., Refs. [1,12,23])
d ion (p − eA(t ′ )) = p − eA(t ′ )|r|ψ 0 , (8) d * rec (p − eA(t)) = ψ 0 |r|p − eA(t) .(9)
The molecular ground state |ψ 0 cannot be obtained analytically for large molecules and it is necessary to approximate it by means of a linear combination of atomic orbitals (LCAO). In the position representation, the electron wave function can be considered as a linear superposition of functions centered at the nuclear locations (which are considered static compared to the fast dynamics of the electrons),
r|ψ 0 ≡ ψ 0 (r) = N j=1 n l=1 C l φ l (r − R j ) ,(10)
where R j is the position of the j-th nucleus, N is the total number of atoms constituting the molecule, and n is the number of atomic orbitals considered in the superposition. In Eq. 10, the parameter C l depends on the molecular structure and 'weight' of each orbital contribution. The function φ l (r) represents a superposition of contracted Gaussian functions, which are constructed as a sum over Gaussian primitives (GPs). In a simplified notation and using Cartesian coordinates, we write that
φ l (r) = N l kmax k=1 η k x a y b z c e −α k r 2 ,(11)
where N l is a normalization constant, η k is the superposition coefficient, whereas a, b, and c are integers such that a + b + c = l. The parameter l is related to the angular momentum quantum number of the specific atomic orbital. In addition, k max is the number of functions necessary to model the orbital. α k is the so-called exponent, which is closely related to the 'spreading' of the Gaussian orbital in the molecule. For larger α k values, the atomic orbital shows a larger probability that the electron will be localized near to the nucleus. All constants in Eqs. 10 and 11 are obtained from Hartree-Fock methods for the optimized geometry of the molecules in their ground states. The quantum chemistry computational package GAMESS was used for that effect [19]. Since within LCAO atomic orbitals are centered at each nuclear position, Eqs. 8 and 9 involve new oscillatory factors of the type e ±[i(p−eA)·R] . Such terms have to be included into the semiclassical action, and reflect the molecular multicenter nature of the wave function. The addition of those terms makes the action explicitly dependent on the nuclear coordinates and has important consequences in the overall HHG process [12]. The total time-dependent dipole moment for molecular systems reads
d(t) = ie 2 N i=1 N j=1 dp t −∞ dt ′ e −iS(p,t,t ′ ,Ri,Rj ) (12) ×d * rec2 (p−eA(t), R i )[d ion2 (p−eA(t ′ ), R j )·E(t ′ )]
+c.c., where he modified semiclassical action reads,
S (p, t, t ′ , R i , R j ) = t t ′ dσ (p−eA(σ)) 2 2 + I p (13) +p · (R j − R i ) + eA(t) · R i − eA(t ′ ) · R j .
Recombination and ionization matrix elements in Eq. 12 differ from the ones presented in Eqs. 8 and 9 due to the fact that the fast oscillatory terms were incorporated into the action.
Eq. 12 includes a double sum over the total number of atoms N . Such expression takes into account two different contributions: ionization and recombination at the same atomic center (i = j) and ionization and recombination at two different centers (i = j). The former case gives rise the so-called direct harmonics contribution and the latter case to the transfer harmonics contribution to HHG [12].
In order to calculate the time dependent dipole moment, it is necessary to perform the multidimensional integral in Eq. 12. Integration over momentum can be approximated by means of the saddle-point method due to the highly oscillatory nature of the factor exp [−iS]. For this technique to be applicable, the remaining parts of the integral have to be slow oscillating (or non-oscillatory) functions of momentum. This is only achieved by the introduction of the other oscillatory terms related to recombination and ionization matrix elements into the modified action [12], as it was done in Eq. 13.
The saddle point in momentum, denoted as p s , is obtained from the relation
∇ p S(p, t, t ′ , R i , R j ) = 0 .(14)
From Eqs. 13 and 14 we obtain that
p s = 1 t − t ′ e t t ′ dσA(σ) + (R i − R j ) .(15)
Under the saddle-point approximation, the time-dependent dipole moment becomes [2,12]
d(t) = ie 2 ∞ 0 dτ 2π ǫ+iτ 3/2 N i=1 N j=1 e −iS(ps,t,t−τ,Ri,Rj ) ×d * rec2 (p s − eA(t), R i ) × d ion2 (p s − eA(t − τ ), R j ) · E(t − τ ) + c.c. ,(16)
where ǫ is an infinitesimal regularization constant and τ = t − t ′ is the so-called return time [2]. The integration over τ is done numerically. Let us note that Eq. 15 has important consequences. In the first place, if the semiclassical action was not modified, the saddle point in momentum would always be parallel to the laser polarization direction. The inclusion of the term 1 τ (R i − R j ), when i = j, guarantees that the electron has a momentum component perpendicular to the driving laser field. This new term contributes, in a very important way, to the ellipticity of the resulting harmonics, even if they are originated from linearly polarized radiation [14]. The polarization of the resulting harmonics depends strongly on the molecular orientation, as it was experimentally observed in diatomic molecules (see, e.g., [24]).
B. Recombination matrix element
As it was mentioned before, harmonic responses from molecules can exhibit strong modulations of peaks intensity along the plateau. In general, when the atomic case is considered, the returning electron wave packet collides with a unique center whereas, in the molecular case, many atomic centers are present. This results in pronounced interference modulations of the spectral response. The position and intensity of the suppressed harmonics depend strongly on the molecular orientation [25] and on the orbital symmetry. As the interference effects are directly related to the geometric distribution of the atomic centers, a detailed analysis of the minima can provide valuable information about the molecule. It has been proven that, for other fullerenes and multi-atomic systems, some of the modulations along the plateau are strongly related to the recombination matrix element values, presenting minima in the region when the RME vanishes [1,22]. Other type of local spectral modulations is traditionally related to interferences between different electron trajectories [2,26].
The molecular RME, as a function of the kinetic momentum of the electron, Π(p, t) = p − eA(t) ≡ Π, is given by
d * rec (Π) = 1 (2π) 3/2 N i=1 n l=0 C l N l k η k a,b,c dr ×(x − x i ) a (y − y i ) b (z − z i ) c e −α k (r−Ri) 2 re iΠ·r .(17)
It is evident that the RME values depend on the nuclear positions R i and the set of parameters C l , N l , η k , and α k . As the RME is a function of the kinetic momentum of the electron during recombination and the harmonic response depends on the frequency of the emitted photon, it is necessary to relate those two quantities by means of the energy conservation equation [1,22],
ω = 1 2 Π 2 + I p .(18)
where ω is the frequency of the harmonic photon. Eq. 18 allows us to find the harmonic frequencies for which the RME vanishes.
As it has been pointed out before, when a multi-center system is considered, the saddle point in momentum can have non-zero components in all directions [see, Eq. 15] depending on the molecular geometry and orientation. If the internuclear distance is small enough (more precisely, if the parameter Q [Eq. 2] is small enough), it is expected that the main component of the saddle point in momentum is parallel to the laser polarization direction. The other components can be fairly large as well, especially for small τ values. As the first approximation, the RME is calculated as a function of the harmonic order by increasing the kinetic momentum Π in just one direction and setting the other components as constant. This gives an idea of the approximated behavior of the RME along the plateau, accounting for the three momentum components separately.
III. RESULTS
The goal of this Section is to analyze the relation between the geometric distribution of atomic centers, molecular symmetry and orientation of different C 20 isomers with the spectral properties of the harmonic response. Modulations of peak intensities along the plateau, together with the polarization properties of the emitted radiation, are going to be explored in order to gain a better understanding of the overall HHG process.
In our calculations, the time-dependent dipole moment is calculated by performing the numerical integration over τ in Eq. 16. The dipole acceleration in frequency domain is obtained from the Fourier transform of each one of the time-dependent dipole moment vector components . The laser field is described as a semi-infinite and monochromatic plane wave, with an oscillating electric field given by the relation It is considered that the laser field can be polarized along the x-, y-, or z-directions, whereas the wavelength and intensity are the same as described in Sec. II. In order to obtain an expression for the molecular ground state, the LCAO, with coefficients generated by standard quantum chemistry software, was used.
A linearly-polarized laser field interacting with large, non-linear molecules can generate important harmonic responses in directions perpendicular to the driving field polarization. Thus, the complete study of the HHG should involve three different dipole components [d x (t), d y (t), and d z (t)] for each linear polarization. In total, nine responses are expected from each isomer. Every harmonic response is going to be denoted as d (i) j (i, j = x, y, z), where the bottom index denotes the harmonic polarization whereas the top index relates to the driving field polarization direction with respect to the coordinates and orientation of the systems shown in Fig. 1.
A. Harmonic spectra from the HOMO In this Section, the harmonic responses from the HOMO of the ring, bowl, and cage are going to be analyzed. In Sec. III B, the influence of the HOMO-1 and inner molecular orbitals on the harmonic signal and RME values is going to be explored. from the same structure. The electric field is described by Eq. ?? and it is considered to be polarized along the x-direction. The wavelength corresponds to 800nm and the intensity is I = 5 × 10 14 W/cm 2 .
Results for C20 fullerene
When the laser field polarized along the x-axis interacts with the symmetric cage [ Fig. 1(c)], a strong harmonic response is observed in the x-direction, with a plateau characterized by multiple modulations in peak intensities. The spectral response is presented in panel (b) of Fig. 2. The most visible minima are located in the regions between the 23th and 27th, 31st and 39th, and 55th and 65th harmonic orders (HO).
In panel (a), we present the modulus squared of the recombination matrix element's x-component, calculated according to Eq. 17. As the momentum of the electron is not necessarily parallel to the laser field polarization, |d rec,x (Π)| 2 is shown as a function of Π x , Π y , and Π z , separately. As it was mentioned before, it is expected that, for molecules with small Q values, the major contribution from the saddle points in momentum [Eq. 15] should be parallel to the laser polarization (solid blue line). It can be seen that the spectral minima from the 23th to 27th and from the 55th to 65th harmonic orders (the most prominent modulations) match very well with the zeroes of the RME being the function of Π x . The region from the 19th to 21st (less pronounced) and from the 31st to 39th harmonics are located near the points where the RME as a function of Π y vanishes (dashed red line).
The most pronounced minima agree with the zeroes of the RME and, therefore, one can attribute those modulations to interference effects related to the multi-center nature of the molecule. Similar minima have been observed in other harmonic responses obtained from larger icosahedral fullerenes, and have been attributed to the same cause [1].
It is worth noting that, when the electric field is polarized along the x-direction for the given molecular orientation, the only harmonic response is obtained parallel to it (i.e., the responses d are suppressed). If the driving field is polarized along the y-or z-directions, the harmonic signal shows a different behavior (Fig. 3). In the first place, for each laser polarization we obtain two responses: one parallel to the electric field (d It can be seen from Fig. 3 that, in the plateau region, all spectra present smoother variations of the peak intensities as compared to the d (x)
x case (i.e., the envelope of the peaks shows less modulations). In those cases, the total RME y-and z-components vanish independently of the momentum direction. This can be considered as a consequence of symmetry of the molecular orbitals. As we have checked, if any of the atoms is artificially displaced from its original position the two RME components present strong oscillations, similar to the d (x)
x case. The very smooth modulations along the plateau are, as it will be shown later, consequences of interferences between quantum trajectories.
Up to now, the analysis of modulations of the peak intensities along the HHG plateau from three different C 20 structures has been based on the importance of the RME. It is clear that this quantity contains important information about the molecular configuration and it is related to multi-center interference effects. Another factor which provides substantial information about the process is the modified semiclassical action, as it will become clear along this Section.
The relation between multi-center interference effects in diatomic molecules and its harmonic response was originally proposed by Lein et al. in Refs. [25,26], but other interference features of different nature were ignored. The direct observation of interference between quantum trajectories (or the quantum path interference (QPI) phenomenon) was originally proposed for a single atom [28], where multi-center effects play no role. Intuitively, the QPI can be understood by analyzing the Lewenstein model: the electrons which contribute to HHG can follow two paths, one short and one long. During the excursion to the continuum the electron wave packet spreads and acquires a phase, given by the semiclassical action, which depends on the ionization and recombination times. When the recollision takes place, electrons with different phases interfere. This has direct consequences on the harmonic spectrum. The inclusion of QPI to the analysis of HHG from diatomic molecules has been recently studied by Yang et al. (see, Ref. [29]) and has proven to modify in a very important way the general behavior of the harmonic plateau. New minima were observed with no relation to the multi-center interference but they were unequivocally related to QPI processes, which follows from the time-frequency analysis. Such effects have shown to be important even for ultrashort driving pulses [29].
The QPI analysis in the atomic case is straightforward as just the long and short electron paths are considered. If a multi-center system is taken into account, the complexity of the problem increases with the number of atoms. It is evident from Eq. 13 that the modified semiclassical action does not depend only on the ionization and recombination times, but depends as well on each of the nuclear coordinates. In this sense, short and long trajectories are composed of direct and transfer trajectories which may interfere and generate very rich harmonic responses.
In Sec. III A 1, the HHG spectra from C 20 fullerene have been extensively studied by interpreting the modulations of the peak intensities along the plateau as multi-center interferences, but some of such modulations cannot be matched with minima of the RME. It has been proven that the harmonic responses related to oscillating RMEs exhibit heavily modulated plateaus and many of the minima match fairly good with the zeroes of the RME. The harmonic responses related to non-oscillating RMEs present smooth plateaus with soft modulations of the peak intensities, e.g., the z response shows QPI effects starting from roughly the 30th harmonic, when the intensity of the peaks starts to decrease [see, panels (b) and (d) of Fig. 4]. The most pronounced destructive interference effect is clearly located between the 45th and 55th harmonic orders, which again coincides with the position of the minimum. The time-frequency analysis shows much less pronounced (short)(long?) trajectories.
When the laser field is polarized along the z-axis, fewer interference effects are observed. Particularly interesting is the d Fig. 5 panel (b), right panel] presents relatively small QPIs, being the most intense at around the 25th-35th,53rd and 59th harmonics, when the two trajectories seem to interfere the most and the minimum of the plateau is present. There are other destructive interactions near to the cutoff.
In closing this Section, it is worth noting that the smooth variations of the spectral response for the cases where the multi-center effects are less important can be directly related to QPI effects (i.e., to the acquired phase of the electron and to the modified semiclassical action).
Results for C20 ring
The ring [ Fig. 1(a)] is an interesting C 20 isomer due to its planar configuration and its highly symmetric structure with respect to the z-axis. When the d Fig. 6, the peaks in the range between the 17th and 23rd harmonics present strong modulations. In this region the RME components reach zero at several points.
When the laser field is polarized along the x-or y-directions, two other harmonic responses are observed (d (x) y and d (y)
x ), which are presented in Fig. 7. Due to the highly symmetric structure (wave function and geometry of the molecule) in the xy-plane, one can expect to obtain similar results for both laser polarization along x-and y-directions. This is confirmed by our results, as one can see by comparing panel (a) with panel (b) in Fig. 7, and panels (c) and (d) in Fig. 6. There are relatively small differences in the intensity of some peaks, which are related to imperfect symmetry of the wave function introduced by the numerical error in the ab-initio calculations using the GAMESS code. Nevertheless, the overall spectral trend shows similar characteristics.
When the driving field is polarized along the z-axis, just one harmonic response, d Fig. 16 (c)]. In this case, the z-component of the RME vanishes for all components of momentum, and the resulting harmonic spectrum presents the characteristic smooth plateau.
(z) z , is observed [see,
It is worth noting that, for the ring, all the RME components as a function of Π z vanish. It is not expected that this particular momentum direction would contribute to the harmonic response when the laser field is polarized along the x-or y-directions. This is due to the planar configuration of the isomer and the fact that its symmetry axis is aligned with the z-direction. By inspecting Eq. 15 one can clearly see that the z-component of the saddle point in momentum is zero.
y , (b) d (x) z , (c) d (y) x , (d) d (y) z , (e) d (z) x , and (f) d (z) y .
Results for the C20 bowl
The bowl [ Fig. 1(b)] is the least symmetric of the studied C 20 isomers, which directly influences the harmonic spectrum. In the first place, all different combinations of laser field and harmonics polarizations are nonzero (Figs. 8 and 9). As in the previous cases, the position of the cutoff coincides with the 3.17U p + I p rule and no nonphysical extension of the plateau is observed. The atomic distribution of this molecule generates a static dipole moment pointing along the z-direction, as reported in Table I. It has been shown that the harmonic responses from polar molecules show the presence of even and odd harmonics, which is related to the lack of inversion symmetry (see, e.g. , [27] and references therein). All responses obtained from the bowl present strong even and odd harmonics.
All spectra in Figs. 8 and 9 exhibit several modulations along the plateau, being the less prominent for the d (z) z case. In Fig. 9, the d (x)
x , d z spectra are plotted together with the modulus squared of the three components of the RME as a function of Π x , Π y , and Π z . For this particular structure, due to the generally complicated shape of the plateau, a comparison between RME's zeroes and expected minima of the spectral envelope is not as clear as in the two previous cases. When analyzing the d (x)
x response [ Fig. 9 (d)], a minimum can be found between the 21st and 35th harmonics, which coincides with minima of the modulus squared of the RME's x-component as a function of Π x and Π y (solid blue and dashed red lines in Fig. 9 (a), respectively). Other modulations can be attributed to the combined oscillations of the RME for different momenta. Interference effects between electron quantum trajectories (short and long) starting from different atomic positions are expected to have another important role in the peak intensities modulation. What is certain is that the Π z component does not contribute to destructive interference due to the absence of zeroes in the RME [dotted-dashed green line in Fig. 9 (a)]. A similar analysis can be applied to the spectral response d (y) y shown in Fig. 9 (e). A general decrease of the peaks strength is present between the 25th and 35th harmonics, which corresponds to a minimum in the modulus squared of the RME y-component associated to Π y (dashed red curve in 9 (b)). The following peaks show lower intensities, which can be attributed to the small RME absolute values.
The d (z) z spectrum (see Fig. 9 (f)), shows a different behavior. Even though there are present small modulations in the peak intensities along the plateau, the general trend is more uniform than in the previous two cases. It can be explained by the rapid decrease of the RMEs associated with Π x and Π y (solid blue and dashed red curves in Fig. 9 (c)). The modulus squared of the matrix element as a function of Π z (dotted-dashed green curve) is expected to have a strong influence on the spectrum and it vanishes at the considered frequencies. In the absence of oscillations, the harmonic response presents a relatively smooth plateau.
As it can be seen in all spectra obtained from the bowl, peaks corresponding to even multiples of the driving frequency, in addition to the well defined odd harmonics, are present. This feature is not observed for other two structures, as it is a direct consequence of symmetry of the molecule. Even though the bowl has an axial symmetry, the structure presents a symmetry breakdown with respect to the xy-plane. The appearance of even harmonics in the spectrum has been observed when the Lewenstein model is applied, in both length and velocity gauges, to atoms or molecules displaced from the origin of coordinates [12]. The crucial point to observe the appearance of even harmonics is the absence of an inversion point of symmetry in the molecule.
It is worth noting that all the obtained harmonic responses from C 20 isomers present a sharp cutoff at the position described by the relation (1) and no unphysical extensions are observed. This is expected due to the small Q values and it agrees with the observations presented in Refs. [1,22].
B. Influence of the HOMO-1 orbital
One of the challenges related to studying HHG from large molecules and nanostructures is associated to the fact that the considerable number of electrons can cause that it is relatively easily to ionize the system. Note that the ionization potentials for several valence orbitals can differ by less than one electronovolt (for HOMO and HOMO-1, see, Table I). For this reason, it is often necessary to consider multiple molecular orbitals in the LCAO expansion. The contributions from different orbitals can add new interference features. In this Section, we present how, for the C 20 isomers, the addition of HOMO-1 modifies the overall structure of the HHG spectra. The RMEs behavior as a function of momentum is expected to change as well. Table I. The laser field is approximated as a semi-infinite sinusoidal plane wave with intensity I = 5 × 10 14 W/cm 2 and wavelength λ = 800 nm. (Fig. 9), it is evident that the RME components as a function of Π x and Π y present a different behavior. This is in contrast to the Π z case which does not show important changes. The d x and d (y) y cases, the major differences are located between the 10th and 40th harmonic orders, where the RMEs differ the most. Nevertheless, zeroes and minima of the modulus squared of the RME components are generally located at the same positions. Fig. 11 shows the other harmonic responses obtained from the bowl after the contribution of HOMO-1 is included. Even though the general trend of the plateau in panels (a) through (d) is very similar, some of the peaks present a higher intensity after the last orbital is added. This is particularly true for the region between the 40th harmonic and the cutoff. As expected, the minima seem to be maintained at roughly same locations. The most important changes are present in panels (e) and (f) in Fig. 11. While the responses show a higher intensity, the plateau modulations become less pronounced. ] of the RME and harmonic responses obtained from the C20 fullerene, taking into account the contributions from both HOMO and HOMO-1. The laser field is polarized either along the y-direction (left column) or along the z-direction (right column). The laser field parameters are the same as in Fig. 3. The harmonic responses correspond to d In contrast to the case when just the HOMO is considered, the contribution from HOMO-1 produces a non-vanishing y-and z-components of the RME as a function of Πx. The remaining RMEs vanish along the spectrum, as in the HOMO case.
Considering now the C 20 fullerene with the laser field polarized along the x-direction, one can clearly see that no important modifications of the RME x-component are observed after the addition of the contribution from HOMO-1 (see, Fig. 12 (a)). When the Π z momentum component is taken into account, a general increase of the RME values is observed, but the RME zeroes are located at the same positions. The resulting harmonic response d (x)
x is almost identical as compared to the HOMO case. The difference is a small increase in the peak intensity at the beginning of the spectrum for the combined contributions of HOMO and HOMO-1.
Other harmonic responses from the fullerene are shown in Fig. 13. When the HOMO-1 is included in the calculations, the y-and z-components of the RME as a function of Π x acquire non-zero values (see, panels (a) and (b) of the aforementioned figure). Even though the two RME components which contribute more to the spectral response for the present configurations [d rec,y (Π y ) for a driving field polarized along the y-direction and d rec,z (Π z ) for a driving field polarized along the z-direction] vanish, the contribution from d rec,y (Π x ) or d rec,z (Π x ) can generate minor changes in the modulations along the plateau, as it can be seen by comparing Figs. 13 and 3. The most important of them is for the d yz harmonic response (see panel (b) in Fig. 3 and panel (d) in Fig. 13), where a more pronounced minimum is present at around the 45th harmonic, where the RME y-and z-components decrease strongly.
Finally, the inclusion of the HOMO-1 in calculations of the HHG spectra from the ring seems to leave unchanged the RME x-and y-components x (see, Fig. 15). When the d (z) z harmonic response from the ring includes the contribution of both HOMO and HOMO-1 orbitals, the z-component of the RME as a function of Π x and Π y acquire nonvanishing values, contrary to the case when just the HOMO is considered (see, Fig. 16). For the latter, the z-component of the RME vanishes for Π x , Π y , and Π z . Even though, the RME which contributes the most to the present configuration [d rec,z (Π z )] is zero, it is expected that the two other components change in a minor way the shape of the plateau. As it can be seen in Fig. 16, the harmonic response obtained by accounting for HOMO and HOMO-1 [panel (b)] presents a very similar trend to the spectral response obtained from HOMO [panel (c)] with a general increase in intensity. The differences of the modulations between those two cases are minor and appear in the lower energy part of the plateau, when the z-component of the RME acquires non-zero values. . In panel (c), we present the harmonic response from the ring restricted to the contribution of the HOMO, for which the corresponding RME component vanishes.
IV. CONCLUSIONS
The Lewenstein model is a very useful tool to analyze the harmonic response from atoms and molecules interacting with strong laser fields. We corroborate that the length gauge formalism applied to molecules with small Q values does not involve an unphysical extension of the plateau, as it was pointed out in other works [1,12,22].
Our present calculations, in the length gauge, predict harmonic responses with different polarization directions, which depend on the particular isomer and driving field polarization. We have shown that different molecular arrangements, as in the case of the three C 20 isomers, lead to different spectral responses. Multi-slit interference patterns, which produce intensive modulations of the harmonic responses along the plateau, are related to the nuclear distribution in the molecule and its molecular orbital configuration. The zeroes of the recombination matrix elements as a function of momentum in the three coordinates are closely related to the interference effects evidenced as minima in the plateaus.
We have shown that some of the harmonic responses from the C 20 fullerene, for which the RME is not oscillating, present modulations in peak intensity along the plateau. Such modulations are related to quantum path interferences, which happen when electron wave packets following different trajectories interfere.
We believe that the observation of the harmonic polarization direction, together with the analysis of multi-center interference minima can help in the differentiation between different aligned harmonic targets. Those properties can be used in the development of a simple spectroscopic technique.
FIG. 1 :
1Structure of three C20 isomers. The carbon atoms distributions were obtained from the standard computational chemistry package GAMESS using the 6-311G basis set. The structures correspond to (a) the monocyclic ring, (b) the bowl, and (c) the fullerene (also known as the cage).Isomer Rav(a.u.
FIG. 2 :
2Panel (a) presents the modulus squared of the recombination matrix element x-component, calculated according to Eq. 17, as a function of Πx, Πy, and Πz for the C20 fullerene. Panel (b) shows the harmonic response d(x) x
z
) and one perpendicular to it (d(z) y and d (y) z ). Harmonics in the x-direction are completely suppressed.
FIG. 3 :
3The left column presents the harmonic response from the C20 fullerene when the laser field is polarized along the y-direction [the d(y) y harmonic response is presented in panel (a) and d (y) z response in panel (c)]. The right column corresponds to a polarization of the driving field along the z-direction [d )]. The remaining laser field parameters are the same as in Fig. 2.
FIG. 4 :
4Left panel shows the time-frequency analysis (top) for the d (y) y harmonic response from the cage and the corresponding HHG spectrum is presented in lower figure. The laser parameters are the same as in Fig. 3. The first vertical red line shows the starting point of the region where the interference between quantum trajectories begins to be evident. The second vertical line points at the region where the destructive interference is maximal. Right panel presents the time-frequency analysis of the response d (y) z from the cage (top), which is shown in lower pane. The vertical line points at the minimum of the harmonic spectrum envelope, which can be directly related to QPI effects. FIG. 5: The same as in Fig. 4 but for the time-frequency analysis of the d (z) y [panel (a)] and d (z) z [panel (b)] harmonic responses [panels (c) and (d), respectively]. The red vertical lines point at the regions where the modulations in the spectral response coincide with quantum trajectories interference.
from the fullerene. In order to identify the nature of such soft variations, the time-frequency analysis was performed for the four aforementioned responses and it is presented together with the respective spectra in Figs. 4 and 5. In general, all plots show the well defined long and short trajectories with interference effects at certain frequencies. Those interferences can be observed as bifurcations or interactions between the otherwise clear, well defined, and independent paths. The vertical red lines identify the position of the spectral minima or regions which present modulations that can be related to QPI in the time-frequency analysis. Starting withFig. 4, in panel (a)(top) one can see that the d (y) y response exhibits interference features from around the 20th harmonic order, where a decrease of the peak intensities can be observed. The most evident destructive interference effects are present between the 40th and 50th harmonic orders, where the minimum of the plateau is present [lower figure]. The d (y)
-frequency analysis [Fig. 5, panel (a), left panel top figure]. It exhibits very well defined short and long trajectories except for the region of the 25th-40th harmonics, where the long trajectories are clearly distorted and less pronounced as a consequence of strong interference. This path interference coincides with exact position of the broad spectral minimum. Finally, the d (z) z case [
considered, strongly modulated harmonic plateaus are observed [panels (c) and (d) of Fig. 6]. Similarly to the case of the C 20 fullerene, the modulations of the HHG spectra are related to oscillating components of the RME [with their modulus squared presented in panels (a) and (b) of the same figure]. Some of the most pronounced minima (located from the 37th to 47th and from the 51st to 65th HOs) can be directly related to the zeroes of the RME components with momentum parallel to the polarization axis [solid blue line in panel (a) for d (x) x and red dashed line in panel (b) for d (y) y ]. In panels (c) and (d) of
FIG. 6 :
6Panel (a) presents the modulus squared of the recombination matrix element x-component as a function of either Πx (solid blue line) or Πy (dashed red line) for the ring. The RME as a function of Πz vanishes. Panel (b) shows the same but for the y-component of the RME. In panels (c) and (d) the harmonic responses d , respectively. The remaining laser parameters are the same as in Fig. 2. FIG. 7: HHG spectra generated from the ring. The harmonic responses correspond to (a) d (x) y and (b) d (y) x . The laser parameters are the same as in Fig. 2. FIG. 8: HHG spectra generated from the bowl. The electric field is polarized along the x-axis (top row), along the y-axis (middle row), and along the z-axis (bottom row). Harmonic responses are (a) d (x)
FIG. 9 :
9(a) Modulus squared of the RME x-component as a function of Πx, Πy, and Πz calculated for the bowl. (b) The same as in panel (a) but for the y-component of the RME. (c) The same as in panel (a) and but for the z-component of the RME. (d) Harmonic response d
FIG. 10 :
10(a) Modulus squared of the recombination matrix element x-component as a function of Πx, Πy, and Πz calculated for the bowl. HOMO and HOMO-1 contributions have been included. (b) The same as in panel (a) but for the y-component of the RME. (c) The same as in panel (a) but for the z-component of the RME. (d) Harmonic response d
Fig
. 10 shows the modulus squared of the three RME components, calculated according to Eq. 17, as a function of momentum [panels (a), (b), and (c)] and the corresponding harmonic responses [panels (d), (e), and (f)] for the C 20 bowl. Comparing those results with the calculations including just the HOMO
responses [panel (d) in Figs. 9 and 10] are almost identical, meaning that in this case the contribution of HOMO-1 does not play an important role. In the d (x)
FIG. 13 :
13Presents the modulus squared of the y-[panel (a)] and the z-component [panel (b)
(c), (d), (e), and (f), respectively).
(c) and (d), respectively] are identical to the pure HOMO case. The same can be said about the crossed terms d
FIG. 14 :
14Modulus squared of the x-[panel (a)] and the y-component [panel (b)] of the RME as a function of Πx and Πy calculated for the ring including both HOMO and HOMO-1 contributions. The RME as a function of Πz vanishes. 15: HHG spectra from the ring calculated taking into account contributions from both HOMO and HOMO-1
FIG. 16 :
16Panel (a) presents the modulus squared of the RME z-component for the ring when the contributions from HOMO and HOMO-1 are accounted for. The corresponding spectral response d (z) z is shown in panel (b)
TABLE I :
IIonization potentials and static dipole moments for three stable C20 structures. The data was obtained by Hartree-Fock methods according to the standard quantum chemistry package GAMESS with the 6-311G basis set. The asterisk (*) denotes a two-fold degeneracy of the molecular orbital. In this Table we only present ionization potentials of HOMO and HOMO-1, which are the most important for the present analysis.
TABLE
TABLE III :
IIICutoff position for HOMO and HOMO-1 calculated according to Eq. 1 using the ionization potential values reported in
x , and (f) d x .
acknowledges support from U.S. National Science Foundation (Grant No. PHY-1734006 and Grant No. PHY-2110628). A J , A. J. acknowledges support from U.S. National Science Foundation (Grant No. PHY-1734006 and Grant No. PHY-2110628).
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| []
|
[
"Ground and excited states Gamow-Teller strength distributions of iron isotopes and associated capture rates for core-collapse simulations",
"Ground and excited states Gamow-Teller strength distributions of iron isotopes and associated capture rates for core-collapse simulations"
]
| [
"Jameel-Un Nabi \nThe Abdus Salam ICTP\nStrada Costiera 1134014TriesteItaly\n",
"Jameel-Un Nabi ",
"\nFaculty of Engineering Sciences\nGIK Institute of Engineer-ing Sciences and Technology\nKhyber Pakhtunkhwa23640Topi, SwabiPakistan\n"
]
| [
"The Abdus Salam ICTP\nStrada Costiera 1134014TriesteItaly",
"Faculty of Engineering Sciences\nGIK Institute of Engineer-ing Sciences and Technology\nKhyber Pakhtunkhwa23640Topi, SwabiPakistan"
]
| []
| This paper reports on the microscopic calculation of ground and excited states Gamow-Teller (GT) strength distributions, both in the electron capture and electron decay direction, for 54,55,56 Fe. The associated electron and positron capture rates for these isotopes of iron are also calculated in stellar matter. These calculations were recently introduced and this paper is a follow-up which discusses in detail the GT strength distributions and stellar capture rates of key iron isotopes. The calculations are performed within the framework of the proton-neutron quasiparticle random phase approximation (pn-QRPA) theory. The pn-QRPA theory allows a microscopic state-by-state calculation of GT strength functions and stellar capture rates which greatly increases the reliability of the results. For the first time experimental deformation of nuclei are taken into account. In the core of massive stars isotopes of iron, 54,55,56 Fe, are considered to be key players in decreasing the electron-to-baryon ratio (Y e ) mainly via electron capture on these nuclide. The structure of the presupernova star is altered both by the changes in Y e and the entropy of the core material. Results are encouraging and are compared against measurements (where possible) and other calculations. The calculated electron capture rates are in overall good agreement with the shell model results. During the presupernova evolution of massive stars, from oxygen shell burning stages till around end of convective core silicon burning, the calculated electron capture rates on 54 Fe are around three times bigger than the corresponding shell model rates. The calculated positron capture rates, however, are suppressed by two to five orders of magnitude.Keywords pn-QRPA theory; GT strength distributions; weak interaction; electron and positron capture rates; presupernova evolution of massive stars | 10.1007/s10509-010-0477-9 | null | 118,743,386 | 1203.4675 | 47c5278a87c500fd943831f9336419d6e277531f |
Ground and excited states Gamow-Teller strength distributions of iron isotopes and associated capture rates for core-collapse simulations
21 Mar 2012
Jameel-Un Nabi
The Abdus Salam ICTP
Strada Costiera 1134014TriesteItaly
Jameel-Un Nabi
Faculty of Engineering Sciences
GIK Institute of Engineer-ing Sciences and Technology
Khyber Pakhtunkhwa23640Topi, SwabiPakistan
Ground and excited states Gamow-Teller strength distributions of iron isotopes and associated capture rates for core-collapse simulations
21 Mar 2012
This paper reports on the microscopic calculation of ground and excited states Gamow-Teller (GT) strength distributions, both in the electron capture and electron decay direction, for 54,55,56 Fe. The associated electron and positron capture rates for these isotopes of iron are also calculated in stellar matter. These calculations were recently introduced and this paper is a follow-up which discusses in detail the GT strength distributions and stellar capture rates of key iron isotopes. The calculations are performed within the framework of the proton-neutron quasiparticle random phase approximation (pn-QRPA) theory. The pn-QRPA theory allows a microscopic state-by-state calculation of GT strength functions and stellar capture rates which greatly increases the reliability of the results. For the first time experimental deformation of nuclei are taken into account. In the core of massive stars isotopes of iron, 54,55,56 Fe, are considered to be key players in decreasing the electron-to-baryon ratio (Y e ) mainly via electron capture on these nuclide. The structure of the presupernova star is altered both by the changes in Y e and the entropy of the core material. Results are encouraging and are compared against measurements (where possible) and other calculations. The calculated electron capture rates are in overall good agreement with the shell model results. During the presupernova evolution of massive stars, from oxygen shell burning stages till around end of convective core silicon burning, the calculated electron capture rates on 54 Fe are around three times bigger than the corresponding shell model rates. The calculated positron capture rates, however, are suppressed by two to five orders of magnitude.Keywords pn-QRPA theory; GT strength distributions; weak interaction; electron and positron capture rates; presupernova evolution of massive stars
Introduction
Supernovae of Type Ia and Type II are considered to be the two major contributors to the production of elements in the universe (see for example, (1)). Stars in mass range M ≤ 8M ⊙ (M ⊙ denotes the solar mass) end their life as white dwarfs. The ultimate destiny of more massive stars is even more interesting. Once the mass of the iron core exceeds the Chandrasekhar limit the Pauli principle applied to electrons cannot prevent further gravitational collapse to an even more exotic and denser form of matter than in a white dwarf star. A violent collapse of the core is initiated which ultimately leads to a spectacular supernova explosion (where the luminosity of the star becomes comparable to that of an entire galaxy containing around 10 11 stars!). The supernova leaves behind a compressed ball of hot neutrons -a neutron star. As the hot neutron star cools, any further collapse is prevented by the Pauli principle applied to the neutrons, unless the mass is so great that the star can become a black hole. The structure of the progenitor star has a vital role to play in the mechanism of the explosion. Core-collapse simulators, to-date, find it challenging to convert the collapse into a successful explosion. A lot many physical inputs are required at the beginning of each stage of the entire simulation process including but not limited to collapse of the core, formation, stalling and revival of the shock wave and shock propagation. It is highly desirable to calculate the presupernova stellar structure with the most reliable physical data and inputs.
A smaller precollapse iron core mass and a lower entropy should favor an explosion. A smaller iron core size implies less energy loss by the shock in photodisintegrating the iron nuclei in the overlying onion-like structure whereas a lower entropy environment can assist to achieve higher densities for the ensuing collapse generating a stronger bounce and in turn forming a more energetic shock wave (2). A smaller entropy also helps in reducing the abundance of free protons thereby lowering the electron capture rates on these free protons and resulting in a much higher value of Y e at the time of bounce (at this instant of time the iron core collapses to supranuclear density and rebounds creating a shock wave). As a result, the shock energy
E S ≃ Y 10/3 ef (Y ef − Y ei ),(1)
is larger for larger final lepton fraction Y ef (prior to collapse) and larger difference of initial and final lepton fraction. Electron/positron captures and β ± -decays are amongst the most important nuclear physics inputs that determine both the Y ef and the entropy at the presupernova stage. These nuclear weak-interaction mediated reactions directly affect the lepton-to-baryon ratio. Further the neutrinos and antineutrinos produced as a result of these nuclear reactions are transparent to the stellar matter at presupernova densities and therefore assist in cooling the core to a lower entropy state. These weakinteraction rates are required not only in the accurate determination of the structure of the stellar core but also bear significance in (explosive) nucleosynthesis and element abundance calculations. As such it is imperative to follow the evolution of the presupernova collapse stage with a sufficiently detailed and reliable (microscopically calculated taking into account the nuclear structure details of the individual nuclei) nuclear reaction network that include these weak-interaction mediated rates. Weak interactions in presupernova stars are known to be dominated by allowed Fermi and Gamow-Teller (GT) transitions (2). The exact r-process yields from Type II SNe are not known and are related to the complete understanding of the mechanism of these supernova explosions. Electron capture process is one of the essential ingredients involved in the complex dynamics of core-collapse supernovae and a reliable estimate of these rates can certainly contribute in a better understanding of the explosion mechanism.
SNe Ia are thought to be the explosions of white dwarfs that accrete matter from binary companions and are intensively investigated for their accurate calibration as the cosmological standard candles (one needs to predict the SN Ia peak luminosity at least with a 10% precision). These explosions are also important for stellar nucleosynthesis and are strongly contested to be the main producers of Fe peak elements in the Galaxy (3).
The abundance of the Fe group, in particular of neutron rich species, is highly sensitive to the electron captures taking place in the central layers of SNe Ia explosions (3). These captures drive the matter to larger neutron excesses and thus strongly influence the composition of ejected matter and the dynamics of the explosion. It is highly probable that electron capture occurs in the burning front driving the matter to large neutron excess. In particular, GT properties of nuclei in the region of medium masses around A=56 are of special importance because they are the main constituents of the stellar core in presupernova conditions.
The GT transition is one of the most important nuclear weak processes of the spin-isospin (στ ) type. This charge exchange transition is not only subject of interest in nuclear physics but also of great importance in astrophysics. The GT transitions are involved in many weak processes occurring in nuclei, e.g. nucleosynthesis and stellar core collapse of massive stars preceding the supernova explosion. In the early stages of the collapse, the electron capture and β-decay are important processes. These reactions are dominated by GT (and also by Fermi) transitions. Electron (positron) capture rates are very sensitive to the distribution of the GT + (GT − ) strength. In the GT + strength, a proton is changed into a neutron (the plus sign is for the isospin raising operator (t + ), present in the GT matrix elements, which converts a proton into a neutron). On the other hand the GT − strength is responsible for transforming a neutron into a proton (the minus sign is for the isospin lowering operator (t − ) which converts a neutron into a proton). The total GT + strength is proportional to the electron capture strength (4).
GT + strength distributions on nuclei in the mass range A = 50 -65 have been studied experimentally mainly via (n,p) charge exchange reactions at forward angles (e.g. (4; 5; 6)). Similarly GT − strength distributions were studied keenly using the (p,n) reactions (e.g. (5; 7; 8)). It has been shown that for (p,n) and (n,p) reactions the 0 0 cross sections for such transitions are proportional to the squares of the matrix elements for the GT β decay between the same states (e.g. (9)). Results of these measurements show that, in contrast to the independent particle model, the total GT strength is quenched and fragmented over many final states in the daughter nucleus. Both these effects are caused by the residual interaction among the valence nucleons and an accurate description of these correlations is essential for a reliable evaluation of the stellar weak interaction rates due to the strong phase space energy dependence, particularly of the stellar electron capture rates.
Fuller, Fowler and Newman (FFN) (10) performed the first-ever extensive calculation of stellar weak rates including the capture rates, neutrino energy loss rates and decay rates for a wide density and temperature domain. They performed this detailed calculations for 226 nuclei in the mass range 21 ≤ A ≤ 60. They stressed on the importance of the Gamow-Teller (GT) giant resonance strength in the capture of the electron and estimated the GT centroids using zeroth-order (0 ω ) shell model. Both the decay and capture rates are exponentially sensitive to the location of the GT centroid (11). The location of the GT resonance affects the stellar rates exponentially, while the total strength affects them linearly (11). Few years later Aufderheide et al. (12) extended the FFN work for heavier nuclei with A > 60 and took into consideration the quenching of the GT strength neglected by FFN. They tabulated the 90 top electron capture nuclei averaged throughout the stellar trajectory for 0.40 ≤ Y e ≤ 0.5 (see Table. 25 therein). Later the experimental results of Refs. (4; 5; 6; 7; 8) revealed the misplacement of the GT centroid adopted in the parameterizations of Ref. (10). Since then theoretical efforts were concentrated on the microscopic calculations of GT strength distributions and associated weakinteraction mediated rates specially for iron-regime nuclide. Large-scale shell model (LSSM)(e.g. (13)) and the proton-neutron quasiparticle random phase approximation theory (pn-QRPA) (e.g. (14)) were used extensively and with relative success for the microscopic calculation of stellar weak rates. Monte Carlo shellmodel is an alternative to the diagonalization method and allows calculation of nuclear properties as thermal averages (e.g. (15)). However it does not allow for detailed nuclear spectroscopy and has some restrictions in its applications for odd-odd and odd-A nuclei.
The pn-QRPA theory is an efficient way to generate GT strength distributions. These strength distributions constitute a primary and nontrivial contribution to the capture rates among iron-regime nuclide. The usual RPA was formulated for excitations in the same nucleus. Halbleib and Sorenson (16) generalized this model to describe charge-changing transitions of the type (Z, N ) → (Z ± 1, N ∓ 1) and pn-QRPA first came into existence more than 40 years ago. The model was extended to deformed nuclei (using Nilsson-model wave functions) by Krumlinde and Möller (17). Extension of the model to treat odd-odd nuclei and transitions from nuclear excited states was done by Muto and collaborators (18).
Nabi and Klapdor-Kleingrothaus used the pn-QRPA theory to calculate the stellar weak interaction rates over a wide range of temperature and density scale for sd-(19) and fp/fpg-shell nuclei (14) in stellar matter (see also Ref. (20)). These included the weak interaction rates for nuclei with A = 40 to 44 (not yet calculated by shell model). Since then these calculations were further refined with use of more efficient algorithms, computing power, incorporation of latest data from mass compilations and experimental values, and fine-tuning of model parameters (21; 22; 23; 24; 25; 26; 27; 28; 29; 30). There is a considerable amount of uncertainty involved in all types of calculations of stellar weak interactions. The uncertainty associated with the microscopic calculation of the pn-QRPA model was discussed in detail in Ref. (27). The reliability of the pn-QRPA calculations was discussed in length by . There the authors compared the measured data (half lives and B(GT ± ) strength) of thousands of nuclide with the pn-QRPA calculations and got fairly good comparison.
Three key isotopes of iron, 54,55, 56 Fe, were selected for the calculation of GT ± strength distributions and associated stellar electron and positron capture rates in this phase of the project. Whereas sufficient experimental data are available for the even-even 54, 56 Fe isotopes to test the model, 55 Fe has low-lying excited states which have a finite probability of occupation in stellar conditions and a microscopic calculation of GT ± strengths from these excited states is desirable. Aufderheide and collaborators (12) ranked 54,55, 56 Fe amongst the most influential nuclei with respect to their importance for the electron capture process for the early presupernova collapse. Later Heger et al. (31) studied the presupernova evolution of massive stars (of masses 15M ⊙ , 25M ⊙ , and 40M ⊙ ) and rated 54,55, 56 Fe amongst top seven nuclei considered to be most important for decreasing Y e in 15M ⊙ and 40M ⊙ stars. In 25M ⊙ stars these isotopes of iron were ranked as the top three key nuclei that play the biggest role in decreasing Y e . These isotopes of iron are mainly responsible for decreasing the electron-to-baryon ratio during the oxygen and silicon burning phases. Besides, 55 Fe was also found to be in the top five list of nuclei that increase Y e via positron capture and electron decay during the silicon burning phases. This paper presents the detailed analysis of the improved microscopic calculation of GT ± strength distributions. Details of calculation of stellar electron and positron capture rates for these three isotopes of iron using the pn-QRPA model are also furnished in this manuscript. Comparisons against previous calculations are also presented. These improved calculations were introduced recently in Ref. (28) where it was reported that the betterment resulted mainly from the incorporation of measured deformation values for these nuclei. The idea is to present an alternate microscopic and accurate estimate of weak interaction mediated rates to the collapse simulators which may be used as a reliable source of nuclear physics input to the simulation codes.
The next section briefly describes the theoretical formalism used to calculate the GT strength distributions and the associated electron & positron capture rates. The calculated GT ± strength distributions are presented and compared with measurements and against other calculations in Sec. 3. The pn-QRPA calculated electron and positron capture rates for iron isotopes ( 54,55,56 Fe) are presented and explored in Sec. 4. The main conclusions of this work are finally presented in Sec. 5.
Model Description
The Hamiltonian for the pn-QRPA calculations was taken to be of the form
H pn−QRP A = H sp + V pair + V ph GT + V pp GT ,(2)
where H sp is the single-particle Hamiltonian (single particle energies and wave functions were calculated in the Nilsson model, which takes into account nuclear deformations), V pair is the pairing force (pairing was treated in the BCS approximation), V ph GT is the particlehole (ph) GT force, and V pp GT is the particle-particle (pp) GT force. The proton-neutron residual interactions occurred as particle-hole and particle-particle interaction. The interactions were given separable form and were characterized by two interaction constants χ and κ, respectively. In this work, the values of χ and κ was taken as 0.15 MeV and 0.07 MeV, respectively. Details of choice of these GT strength parameters in the pn-QRPA model can be found in Refs. (32; 33). Other parameters required for the calculation of weak rates are the Nilsson potential parameters, the pairing gaps, the deformations, and the Q-values of the reactions. Nilsson-potential parameters were taken from Ref. (34) and the Nilsson oscillator constant was chosen as ω = 41A −1/3 (M eV ) (the same for protons and neutrons). The calculated half-lives depend only weakly on the values of the pairing gaps (35). Thus, the traditional choice of ∆ p = ∆ n = 12/ √ A(M eV ) was applied in the present work. The deformation parameter is recently argued to be one of the most important parameters in pn-QRPA calculations (36) and as such rather than using deformations calculated from some theoretical mass model (as used in earlier calculations of pn-QRPA capture rates) the experimentally adopted value of the deformation parameters for 54, 56 Fe, extracted by relating the measured energy of the first 2 + excited state with the quadrupole deformation, was taken from Raman et al. (37). The incorporation of experimental deformations lead to an overall improvement in the calculation as discussed earlier in Ref. (28). For the case of 55 Fe (where measurement lacks) the deformation of the nucleus was calculated as
δ = 125(Q 2 ) 1.44(Z)(A) 2/3 ,(3)
where Z and A are the atomic and mass numbers, respectively and Q 2 is the electric quadrupole moment taken from Ref. (38). Q-values were taken from the recent mass compilation of Audi et al. (39). Capture rates on 54,55,56 F e for the following two processes mediated by charge weak interaction were calculated:
1. Electron capture A Z X + e − → A Z−1 X + ν.
Positron capture
A Z X + e + → A Z+1 X +ν.
The electron capture (ec) and positron capture (pc) rates of a transition from the i th state of the parent to the j th state of the daughter nucleus are given by
λ ec(pc) ij = ln 2 D [f ij (T, ρ, E f )] B(F ) ij + g A / g V 2 ef f B(GT ) ij .(4)
The value of D was taken to be 6295s (40). B ′ ij s are the sum of reduced transition probabilities of the Fermi B(F) and GT transitions B(GT). Whereas for 54, 56 Fe phonon transitions contribute, in the case of 55 Fe (odd-A case) two kinds of transitions are possible. One are the phonon transitions, where the odd quasiparticle acts as spectator and the other is the transitions of the odd quasiparticle itself. In the later case phonon correlations were introduced to one-quasiparticle states in first-order perturbation (41). The f ′ ij s are the phase space integrals. Details of the calculations of phase space integrals and reduced transition probabilities can be found in Ref. (19). In Eq. (4) (g A /g V ) ef f is the effective ratio of axial and vector coupling constants and takes into account the observed quenching of the GT strength (42). For this project (g A /g V ) ef f was taken (from Ref. (43)) as:
g A g V 2 ef f = 0.60 g A g V 2 bare ,(5)
with (g A /g V ) bare = -1.254 (44). Interestingly, Vetterli and collaborators (5) and Rönnqvist et al. (4) predicted the same quenching factor of 0.6 for the RPA calculation in the case of 54 Fe when comparing their measured strengths to RPA calculations.
The total electron (positron) capture rate per unit time per nucleus was then calculated using
λ ec(pc) = ij P i λ ec(pc) ij .
( 6) The summation over all initial and final states was carried out until satisfactory convergence in the rate calculations was achieved. Here P i is the probability of occupation of parent excited states and follows the normal Boltzmann distribution. The pn-QRPA theory allows a microscopic state-by-state calculation of both sums present in Eq. (6). This feature of the pn-QRPA model greatly increases the reliability of the calculated rates in stellar matter where there exists a finite probability of occupation of excited states. In order to further increase the reliability of the calculated capture rates experimental data were incorporated in the calculation wherever possible. In addition to the incorporation of the experimentally adopted value of the deformation parameter, the calculated excitation energies (along with their log f t values) were replaced with an experimental one when they were within 0.5 MeV of each other. Missing measured states were inserted and inverse and mirror transitions (if available) were also taken into account. No theoretical levels were replaced with the experimental ones beyond the excitation energy for which experimental compilations had no definite spin and/or parity. A state-by-state calculation of GT ± strength was performed for a total of 246 parent excited states in 54 Fe, 297 states for 55 Fe and 266 states for 56 Fe. For each parent excited state, transitions were calculated for 150 daughter excited states using the pn-QRPA model. The band widths of energy states were chosen according to the density of states to cover an excitation energy of (15-20) MeV in parent and daughter nuclei. The summation in Eq. (6) was done to ensure satisfactory convergence. The use of a separable interaction assisted in the incorporation of a luxurious model space of up to 7 major oscillator shells which in turn made possible to consider these many excited states both in parent and daughter nuclei.
GT ± strength distributions
The isovector response of nuclei may be studied using the nucleon charge-exchange reactions (p, n) or (n, p); by other reactions such as ( 3 He,t), (d, 2 He) or through heavy ion reactions. The 0 0 GT cross sections (∆T = 1, ∆S = 1, ∆L = 0, 0 ω excitations) are proportional to the analogous beta-decay strengths. Charge-exchange reactions at small momentum transfer can therefore be used to study beta-decay strength distributions when beta-decay is not energetically possible. The (p, n) reactions probes the GT − strength (corresponding to beta-minus decay) and the (n, p) reactions gives the strength for β + -decay, i.e. GT + strength.
The study of (p, n) reactions has the advantage over β-decay measurements in that the GT − strength can be investigated over a large region of excitation energy in the residual nucleus. On the other hand the (n, p) reactions populates only T = T 0 + 1 states in all nuclei heavier than 3 He. This means that other final states (including the isobaric analog resonance) are forbidden and GT + transitions can be observed relatively free of background. The study of these reactions suggest that a reduction in the amount of GT strength is observed relative to theoretical calculations. The GT quenching is on the order of 30-40 % (5).
In a sense both β-decay and capture rates are very sensitive to the location of the GT + centroid. An (n, p) experiment on a nucleus (Z, A) shows where in (Z − 1, A) the GT + centroid corresponding to the ground state of (Z, A) resides. Each excited state of (Z, A) has its own GT + centroid in (Z − 1, A) and all of these resonances must be included in the stellar rates. We do not have the ability to measure these resonances. Turning to theory we see that the pioneer calculation done by FFN (10) had to revert to approximations in the form of Brink's hypothesis and "back resonances" to include all resonances in their calculation. Brink's hypothesis states that GT strength distribution on excited states is identical to that from ground state, shifted only by the excitation energy of the state. GT back resonances are the states reached by the strong GT transitions in the inverse process (electron capture) built on ground and excited states. Even the microscopic largescale shell model calculations (13) had to use the Brink assumption to include all states and resonances. On the other hand the pn-QRPA model provides a microscopic way of calculating the GT + centroid and the total GT + strength for all parent excited states and can lead to a fairly reliable estimate of the total stellar rates.
As a starting point, the model was tested for the case of 54, 56 Fe where measurements of total GT ± strength are available. As mentioned earlier an overall quenching factor of 0.6 (43) was adopted in the calculation of GT strength in both directions for all isotopes of iron. There is a considerable amount of uncertainties present in all calculations of stellar rates. The associated uncertainties in the pn-QRPA model was discussed in Ref. (27). Keeping in mind the uncertainties present also in measurements where various energy cutoffs are used as a reasonable upper limit on the energy at which GT strength could be reliably related to measured ∆L = 0 cross-sections, 56 Fe, the measured values of the total GT + was taken from the latest measurement by El-Kateb and collaborators (6). The total measured GT − was taken from Ref. (7). In this case the calculated strengths are relatively more enhanced as compared to LSSM calculated strengths. As mentioned earlier the calculated rates are sensitive to the location of the GT + centroid. For the case of 54 Fe, the calculated centroid of 4.06 MeV (28) is much higher than the LSSM centroid of 3.78 MeV. The numbers are to be compared to the experimental value of 3.7 ± 0.2 MeV which was calculated from the measured data presented in Ref. (4). For 56 Fe, the pn-QRPA model calculated the centroid at an excitation energy of 3.13 MeV (28) in the daughter nucleus 56 Mn whereas LSSM calculated it at an excitation energy of 2.60 MeV. The centroid extracted from the experimental data of El-Kateb et al. (6) comes out to be 2.9 ± 0.2 MeV. Experimental (p,n) data are also available for 54, 56 Fe. According to Anderson and collaborators (8) a large uncertainty exists in the (p,n) measurements and the authors were able to perform measurements on 54 Fe at 135 MeV with significantly better energy resolution than the earlier measurements. Further whereas the GT strength in the discrete peaks were extracted accurately, it was reported that the amount of GT strength in the background and continuum remained highly uncertain. For the sake of completeness a comparison of GT − centroids is also discussed below. The centroid of the data of discrete peaks (presented in Table I of Ref. (8)) was calculated to be 7.63 for 54 Fe which is to be compared with the much lower pn-QRPA calculated value of 5.08. For the case of 56 Fe, the calculated centroid from the data reported by Rapaport et al. (7) comes out to be 8.27. The corresponding pn-QRPA calculated value is 5.61. Despite greater experimental certainties in the (p,n) data one notes that the pn-QRPA calculates the centroid at much lower energies in daughter nuclei as compared to measurements. However it is to be noted that the total strengths (in both direction) compares very well with the measured values. Also the location of GT + centroid is in very good agreement with the measured data which is mainly responsible for the calculation of electron capture rates on iron isotopes. For reasons not known to the author, LSSM did not present the location of GT − centroids in their paper.
The GT + strength distribution for 54 Fe and 56 Fe are depicted in Fig. 1 and Fig. 2 excitation energies up to around 10 MeV. The LSSM strength peaks at much lower excitation energy. The pn-QRPA distribution spreads over higher excitation energies akin to measured strength with a peak around 7.1 MeV for the case of 54 Fe. For the case of 56 Fe the pn-QRPA calculated spread is not as good as in case of 54 Fe. Nonetheless the pn-QRPA calculates its peak around 4 MeV much higher in energy than the LSSM peak accumulating all the measured higher lying strength in a narrow resonance region. Fig. 3 shows the calculated GT + strength distribution for 55 Fe. Here no experimental GT distribution was available for comparison. Instead the calculated strength for ground and first two excited states of 55 Fe (at 0.41 MeV and 0.93 MeV, respectively) is displayed. It may be seen from Fig.3 that the GT strength is fragmented over many daughter excited states and peaks at relatively high excitation energies (around 8 MeV) in the daughter 55 Mn.
Figs. 4, 5 and 6 display the GT − strength distributions for the ground and first two calculated excited states of 54 Fe, 55 Fe and 56 Fe, respectively. One notes the enhancement in the total strength in the GT − direction. For the case of 54 Fe the peaks shift to higher excitation energy in daughter for parent excited states. Correspondingly the energy centroids shift to higher energy for these excited states. For the case of 55 Fe one notes that bulk of the strength resides in a narrow resonance region around 10 MeV in daughter. In the case of 56 Fe the low-lying peaks become considerably small in magnitude for the parent excited states. Figs. 1, 2, 3, 4, 5, and 6 confirm that the calculated strength is fragmented and well spread. At low temperatures and densities these low-lying discrete strengths may very well dominate the rates and play an important role in the state-by-state evaluation of both sums (see Eq. (6)).
The ground and excited states strength functions calculated within the framework of the pn-QRPA model are required in the microscopic calculation of stellar capture rates. The ASCII files of the GT ± strength distributions for all parent excited states are available and can be requested from the author.
Electron and positron capture rates
This section presents and explores the results of the calculated electron and positron capture rates on 54,55, 56 Fe in stellar environment and also compare the pn-QRPA capture rates with the LSSM calculation and against the pioneering calculation of FFN (10). As discussed earlier ground and excited states GT strengths calculated earlier contribute in the state-by-state calculation of electron and positron capture rates (via Eq. 4). Figs. 7, 8 and 9 show the calculated electron capture rates on 54,55, 56 Fe. These figures show four panels depicting the calculated electron capture rates at selected temperature and density domain. The upper left panel shows the electron capture rates in low-density region (ρ[gcm −3 ] = 10 0.5 , 10 1.5 and 10 2.5 ), the upper right in medium-low density region (ρ[gcm −3 ] = 10 3.5 , 10 4.5 and 10 5.5 ), the lower left in medium-high density region (ρ[gcm −3 ] = 10 6.5 , 10 7.5 and 10 8.5 ) and finally the lower right panel depicts the calculated electron capture rates in high density region (ρ[gcm −3 ] = 10 9.5 , 10 10.5 and 10 11 ). The electron capture rates are given in loga- rithmic scales (to base 10) in units of s −1 . T 9 gives the stellar temperature in units of 10 9 K. Figs. 7 and 9 are similar in nature depicting the electron capture rates on even-even isotopes of iron. The rates are relatively enhanced for the case of 55 Fe by orders of magnitude (for the first three panels). It can be seen from these figures that in the low density regions the capture rates, as a function of temperature, are more or less superimposed on one another. This means that there is no appreciable change in the rates when increasing the density by an order of magnitude. However as one goes from the medium-low density region to high density region these rates start to 'peel off' from one another. Orders of magnitude difference in rates are observed (as a function of density) in high density region. For a given density the rates increase monotonically with increasing temperatures. For all three isotopes of iron, the calculated electron capture rates are noticeably bigger than the competing β + decay rates and dominate the weak rates for charge-decreasing nuclear transitions in stellar matter. The effects of positron capture are estimated to be smaller for stars in the mass range (10 ≤ M ⊙ ≤ 40) and could be greater for more massive stars (31). The positron capture rates are normally bigger than the competing β − rates for all isotopes of iron specially at high temperatures. Figs. 10, 11 and 12 again show four panels depicting the calculated positron capture rates at selected temperature and density domain. The upper left panel shows the positron capture rates in low-density region (ρ[gcm −3 ] = 10 0.5 , 10 1.5 and 10 2.5 ), the upper right in medium-low density region (ρ[gcm −3 ] = 10 3.5 , 10 4.5 and 10 5.5 ), the lower left in medium-high density region (ρ[gcm −3 ] = 10 6.5 , 10 7.5 and 10 8.5 ) and finally the lower right panel depicts the calculated positron capture rates in high density region (ρ[gcm −3 ] = 10 9.5 , 10 10.5 and 10 11 ). The positron capture rates are given in logarithmic scales in units of s −1 . T 9 gives the stellar temperature in units of 10 9 K. One should note the order of magnitude enhancement in positron capture rates as the stellar temperature increases. Around presupernova temperatures the positron capture rates are very slow as compared to the corresponding electron capture rates. When the temperature of the stellar core increases further the positron capture rates shoot up. The positron capture rates decrease with increasing densities, in contrast to the electron capture rates which increase as density increases. As temperature rises or density lowers (the degeneracy parameter is negative for positrons), more and more high-energy positrons are created leading in turn to higher capture rates. It is worth mentioning that the positron capture rates are very small numbers and can change by orders of magnitude by a mere change of 0.5 MeV, or less, in parent or daughter excitation energies and are more reflective of the uncertainties in the calculation of energies. In order to present an estimate of the contribution of excited states to the total capture rate, the ground state capture rate and the ratio of this rate to the total rate were computed at selected points of stellar temperature and density. These contributions vary for different isotopes and lead to interesting consequences as presented below.
Tables 2, 3, and 4 show the contribution of excited states in the calculation of total capture rates on 54,55, 56 Fe. In all tables the capture rates and the ratios are calculated at selected points of density and temperature shown in first column. Within the parenthesis the first number gives the density in units of gcm −3 and the second number denotes the stellar temperature in units of 10 9 K. In the second column λ ec (G) gives the ground state electron capture rates whereas R ec (G/T ) denotes the ratios of the ground-state electron capture rate to the total rate. The ground state positron capture rates, λ pc (G), and the ratios of the ground-state positron capture rate to the total rate, R pc (G/T ), are given in the fourth and fifth column, respectively. All capture rates are given in units of sec −1 .
The results for 54 Fe are very interesting and are depicted in Table 2. The total electron capture rate increases with increasing temperatures and densities. At low temperatures and densities (T 9 = 1 and ρY e ≤ 10 3 gcm −3 ) one notes that almost all of the contribution to the total electron capture rate comes from the excited states. At a first glance this might look odd. To explain this result one has to go back to Eq. (4) which tells that the partial capture rate is a product of three factors: a constant, phase space integrals (which are functions of temperature and density) and reduced transition probabilities. For a particular parent state (i) all such transitions are summed over daughter states (j) and then multiplied by the probability of occupa- tion of that parent excited state. These partial capture rates are finally summed over all parent excited states to calculate the total capture rate as a function of stellar temperature and density (Eq. (6)). For the given physical conditions, the ground state electron capture rate ∼ 10 −18 s −1 . Probability of occupation of ground state is essentially 1 at such low temperatures. However for the first excited state (at 1.41 MeV) the partial rate increases roughly by 10 orders of magnitude to around 10 −8 s −1 . This increase is attributed to a simultaneous increase in phase space (39)) as well as an increase in the total GT + strength from 4.26 units to 5.12 units (refer to Table 2 of Ref. (28)). Even after multiplying by the much smaller probability of occupation of parent excited state (∼ 10 −8 ) the partial capture rate from the first excited state is around 3 orders of magnitude bigger than that from ground state at low temperatures. As density increases the phase space from excited states decreases resulting in a larger contribution to total rate from ground state. In the limiting case of densities around 10 11 gcm −3 all the contribution to the total rate comes from ground state. At higher temperatures (3 ≤ T 9 ≤ 10) and densities (≤ 10 7 gcm −3 ) the contribution of ground state to total rate is roughly 50%. At still higher temperatures (T 9 = 30) the ground state contributes around 6-7% at all densities. At T 9 = 30, the partial capture rates are significant also for higher lying excited states (as probability of occupation of these high-lying states then becomes appreciably high) and contribute effectively to the total capture rate. For the corresponding contribution of excited states to total positron capture rate, one sees that these contributions are essentially dependent on stellar temperatures and independent of stellar densities. Further one notes that the contribution of excited states dominate in total positron capture rates. These features are reflective of the fact that the positron capture rates are dominated by phase space integrals. For a given temperature the total positron capture rate increases by a constant factor of the corresponding ground state rate independent of the densities. At low temperatures the ground state contribution is almost negligible for reasons mentioned above. As the stellar temperature increases so does the ground state contribution until it reaches a maximum around T 9 = 10 of the order of 27%. At T 9 = 30 roughly 95% of the contribution comes from excited states.
(= Q + E i − E j where Q = -0.697 MeV
For the case of 55 Fe the story is much different (Table 3). Even though the first excited state is at 0.41 MeV (as against 1.41 MeV for the case of 54 Fe) resulting in a much bigger probability of occupation, the partial electron capture rate from first excited state is around 7 orders of magnitude smaller than the corresponding ground state electron capture rate ∼ 10 −10 s −1 (the total GT + strength decreases for the first excited state from 4.68 units to 4.43 units (28)). As a result the total electron capture rate is determined almost entirely from ground state rate at low temperatures. This is in sharp contrast to the previous case. The low-lying excited states do play their role at higher temperatures and densities. The behavior of ground state contribution to the total electron capture rate is as per expectation and decreases appreciably as the stellar temperature increases. Again at T 9 = 30 almost 97% of the contribution is from the parent excited states at all densities akin to the case of 54 Fe. The excited states contribution to the total positron capture rate follows a similar trend as in previous case with a maximum contribution of around 25% from ground state at T 9 = 10. At T 9 = 3 the ground state contributes ∼ 2% (as against 17% in the case of 54 Fe). At much higher temperature (T 9 ∼ 30) this contribution is again around 2%. The excited states contribution to total electron capture rate on 56 Fe at low temperatures and densities falls roughly in between the two extreme cases of 54 Fe and 55 Fe (Table 4). Here roughly 50% contribution comes from the excited states at low temper- atures and densities. Example giving at T 9 = 1 and ρY e ≤ 10gcm −3 , the total ground state electron capture rate is around 10 −27 s −1 . The contribution from first excited state (0.85 MeV) is around 10 −22 s −1 . This is again attributed partly to the increased total GT + strength of 5.15 units as against 3.71 units (for ground state) (28). This rate when multiplied with the occupation probability of around 10 −5 gives a partial electron capture rate of roughly equal magnitude as that from ground state. The excited states contribution increases at higher temperatures with roughly 95% of the contribution coming from parent excited states at T 9 = 30. One also notes that the ground state contribution to the total electron capture rates increases at ρY e ∼ 10 11 gcm −3 . Specially at low temperatures it is almost totally dictated by ground state rates. This is because ground and excited state rates are of comparable magnitudes at this density. However the probability of occupation of excited states is smaller by orders of magnitude at lower temperatures. The excited states contribution to total positron capture rate follows a similar trend as in previous cases with a maximum contribution of around 30% from ground state at T 9 = 10. Tables 2, 3, and 4 highlight the contribution of partial capture rates from parent excited states to the total capture rate and show disparate behavior of these Table 2 The ground state electron and positron capture rates, λec(G), λpc(G) for 54 F e in units of sec −1 . Given also are the ratios of the ground state capture rates to total capture rates, Rec(G/T ), Rpc(G/T ). The first column gives the corresponding values of stellar density, ρYe (gcm −3 ), and temperature, T9 (in units of 10 9 K), respectively.
contributions for all three cases. This analysis further stresses on the fact that a microscopic calculation of GT strength function from excited states is indeed required for a reliable estimate of the total capture rates. One relevant question would then be to ask how the calculated rates compare with other prominent calculations. For the comparison two important calculations of capture rates were taken into consideration: the pioneer calculation of FFN (10) which is still used in many simulation codes and the microscopic calculation of LSSM.
The comparisons are presented in a tabular form. Tables 5, 6, and 7 show the comparison of calculated electron capture rates with those of FFN and LSSM for 54 Fe, 55 Fe and 56 Fe, respectively. Here the ratios of the calculated electron capture rates to those of FFN and LSSM are presented at selected temperature and density points. For the case of 54 Fe, the calculated electron capture rates are in good agreement with the LSSM results specially at high temperatures and densities.
At low temperatures and densities the reported electron capture rates are bigger than the LSSM capture rates by around a factor of four (Table 5). During the oxygen shell burning and silicon core burning of massive stars (as per the simulation results of (31)) the pn-QRPA calculated rates are enhanced roughly by a factor of three as compared to LSSM rates. Whereas the individual discrete transitions between initial and final states matter at low temperatures and densities, it is the total GT strength that counts at high temperatures and densities. It is again reminded that Brink's hypothesis is not assumed in this calculation (which was adopted in LSSM calculation of weak rates). (Also see the discussion on contribution of excited state partial rates above.) The ground state pn-QRPA centroid is placed at much higher energy in 54 Mn as compared to LSSM centroid (28). However, pn-QRPA calculates a much bigger total strength. FFN rates are up to around an order of magnitude enhanced compared to pn-QRPA rates at low temperatures and densities. FFN did not take into effect the process of particle emission from excited states and their parent excitation energies extended well beyond the particle decay channel. These high lying excited states begin to show their effect specially at high temperatures and densities. Also one notes that FFN neglected the quenching of the total GT strength. The LSSM electron capture rates were smaller than the FFN rates by, on average, an order of magnitude (31). The electron capture rates on 55 Fe are most effective during the oxygen shell burning till around the ignition of the first stage of convective silicon shell burning of massive stars (31). For the corresponding temper-atures and densities, the pn-QRPA rates are in very good agreement with the LSSM rates at all temperatures and densities (Table 6). This excellent agreement is attributed to the fact that the total rates for 55 Fe are commanded by ground state contribution (see Table 3) and both pn-QRPA and LSSM perform a microscopic calculation of the ground-state GT strength function. The comparison with FFN is good at low temperatures only (T 9 ∼ 1). At higher temperatures FFN rates are bigger than the LSSM and pn-QRPA rates by an order of magnitude for reasons mentioned before.
The average comparison of calculated electron capture rates is again very good against the LSSM rates for the case of 56 Fe (Table 7). However at low temperatures and densities the LSSM electron capture rates is around 10% enhanced as compared to pn-QRPA numbers. As mentioned above the comparison of the reported and LSSM electron capture rates may be traced back to the calculation of excited state partial capture rates (Tables 2, 3, and 4). One notes that at T 9 = 1 and ρY e ≤ 10gcm −3 the electron capture rates were dominated by excited state contributions for the case of 54 Fe. For the case of 55 Fe it was entirely dominated by ground state contribution and the case of 56 Fe had a 50-50 contribution. Correspondingly the comparison with LSSM rates is excellent for the case of 55 Fe. The pn-QRPA rates are enhanced by a factor of 4 for 54 Fe and some fluctuations in rate comparison can be seen for the case of 56 Fe. The electron capture rates on 56 Fe are very important for the pre-supernova phase of massive stars. FFN rates are again enhanced by up to an order of magnitude as compared to pn-QRPA and LSSM rates.
Tables 8, 9, and 10 show the corresponding comparison for the calculated positron capture rates. Again the pn-QRPA calculated positron capture rates are compared with the LSSM and FFN rates. It is reminded that the positron capture rates are smaller than the corresponding electron capture rates by orders of magnitude and these small numbers can change appreciably by a mere change of 0.5 MeV in phase space and are actually more reflective of the uncertainties in calculation of the energy eigenvalues (for both parent and daughter states). Further it is also evident from Tables 8, 9, and 10 that positron capture rates have a dominant contribution from parent excited states (the ground state contributes at the maximum by one third). Looking at Table 8 for the case of 54 Fe one notes that the pn-QRPA calculated positron capture rates are suppressed by up to 5 orders of magnitude as compared to LSSM and FFN rates at low temperatures and densities. The comparison improves as the temperature and density increases. At T 9 = 30 the reported rates are in fact enhanced by a factor of 8 as compared to LSSM calculated rates and are in reasonable agreement with FFN rates. The comparison follows a similar trend for the case of 55 Fe (Table 9). Here the pn-QRPA calculated rates are suppressed by as much as 3 orders of magnitude compared with the other calculations at low temperatures. The comparison improves as the density of stellar core stiffens. The reported rates are enhanced at higher temperatures by as much as an order of magnitude as compared to LSSM numbers. FFN rates are again in reasonable agreement with reported rates at T 9 = 30.
The comparison with LSSM rates improves as one matches the results from 54 Fe to 55 Fe and finally to 56 Fe. In the case of 56 Fe the reported positron capture rates are suppressed by around 2 orders of magnitude at T 9 = 1 (Table 10). At higher densities and temperatures the rates are in very good comparison. The FFN rates are enhanced by roughly 4 orders of magnitude at low temperatures and densities. The comparison improves as the density and temperature of stellar core increases.
Summary and conclusions
The microscopic calculation of Gamow-Teller strength distributions (GT ± ) of three key isotopes of iron of astrophysical importance was presented. The calculated strengths were in good comparison to the measured strengths (for the case of 54, 56 Fe). The pn-QRPA calculated total GT strengths were greater in magnitude as compared to those using LSSM. The results also highlighted the fact that the Brink's hypothesis and back resonances may not be a good approximation to use in stellar calculation of weak rates.
The pn-QRPA model was also used to calculate the associated electron and positron capture rates of these isotopes of iron with astrophysical importance. Deformations of nuclei were taken into account and for the first time the reported calculation took into consideration the experimental deformations. The rates are calculated on an extensive temperature-density grid point suitable for interpolation purposes that might be required in collapse simulations. The electronic versions of these files may be requested from the author.
During the oxygen and silicon core burning phase of massive stars the pn-QRPA electron capture rates on 54 Fe are around three times bigger than those calculated by LSSM. The comparison with LSSM gets better for proceeding pre-supernova and supernova phases of stars. The pn-QRPA calculated electron capture rates on 55, 56 Fe are in overall excellent agreement with the LSSM rates. The calculated positron capture rates are generally smaller (by as much as five orders of magnitude) at low temperatures and densities. The calculation further discourages any noticeable contribution of positron capture rates on iron isotopes for the physics of core-collapse.
Due to the weak interaction processes (capture and decay rates) the value of lepton-to-baryon (Y e ) ratio for a massive star changes from 1 (during hydrogen burning) to roughly 0.5 (at the beginning of carbon burning) and finally to around 0.42 just before the collapse to a supernova explosion. The temporal variation of Y e within the core of a massive star has a pivotal role to play in the stellar evolution and a fine-tuning of this parameter at various stages of presupernova evolution is the key to generate an explosion. The electron capture tends to reduce this ratio whereas the positron capture increases it. This paper reported on the microscopic 10gcm −3 10 3 gcm −3 10 3 gcm −3 10 7 gcm −3 10 7 gcm −3 10 11 gcm −3 10 11 gcm −3 1 1.
Fig. 1
1Gamow-Teller (GT+ ) strength distributions for 54 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the measured strength distribution (4) and large-scale shell model calculations (13), respectively. Ej represents daughter states in 54 Mn
Fig. 2
2, respectively. The upper panel displays the calculated GT + distribution using the pn-QRPA model. The middle panel shows the measured data and the bottom panel gives the corresponding LSSM strength distributions. One notes that an appreciable measured strength lies in the daughter Gamow-Teller (GT+ ) strength distributions for56 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the measured values (6) and large-scale shell model calculations (13), respectively. Ej represents daughter states in 56 Mn.
Fig. 3
3Gamow-Teller (GT+ ) strength distributions for 55 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the calculated first and second excited states of 55 Fe. Ej represents daughter states in 55 Mn.
Fig. 4
4Gamow-Teller (GT−) strength distributions for 54 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the calculated first and second excited states of 54 Fe. Ej represents daughter states in 54 Co.
Fig. 5
5Gamow-Teller (GT−) strength distributions for 55 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the calculated first and second excited states of 55 Fe. Ej represents daughter states in 55 Co.
Fig. 6
6Gamow-Teller (GT−) strength distributions for56 Fe. The upper panel shows the pn-QRPA results of GT strength for the ground state. The middle and lower panels show the results for the calculated first and second excited states of56 Fe. Ej represents daughter states in 56 Co.
Fig. 7 (
7Color online) Electron capture rates on 54 Fe, as a function of stellar temperatures, for different selected densities. Temperatures are given in 10 9 K. Densities are given in units of gcm −3 and log10λec represents the log of electron capture rates in units of sec −1 .
Fig. 8 (
8Color online) Same asFig. 7but for electron capture rates on 55 Fe.
Fig. 9 (
9Color online) Same asFig. 7but for electron capture rates on56 Fe.
Fig. 10 (
10Color online) Same asFig. 7but for positron capture rates on 54 Fe.
Fig. 11 (
11Color online) Same asFig. 7but for positron capture rates on 55 Fe.
Fig. 12 (
12Color online) Same asFig. 7but for positron capture rates on56 Fe.
Table 1
1Comparison of measured total GT± strengths with microscopic calculations of pn-QRPA and large scale shell model in 54,56 F e. For references see text.54 Fe
56 Fe
Models
ΣGT−
ΣGT+
ΣGT−
ΣGT+
pn-QRPA
7.56
4.26
10.74
3.71
Experiment 7.5±0.7 3.5±0.7 9.9±2.4 2.9±0.3
LSSM
7.11
3.56
9.80
2.70
Table 1presents the comparison of the total GT ± strengths against measurements and largescale shell model calculations (13) (referred to as LSSM throughout this and proceeding sections) for the case of 54,56 Fe. Throughout this paper all energies are given in units of MeV. For the case of 54 Fe the value of the total GT + was taken from Ref. (4) whereas the total measured GT − was taken from Ref. (8). The total GT − strength calculated using the pn-QRPA model matches very well with the measured strength (Vetterli and collaborators (5) also reported a value of 7.5 ± 1.2 as the total measured GT − strength for 54 Fe). The calculated total GT + strength lies close to the upper bound of the measured value and are higher than the corresponding LSSM calculated strength. For the case of
Table 5
5Ratios of calculations of electron capture rates on 54 F e at different selected densities and temperatures. QRPA implies the reported rates whereas SM and FFN denote rates calculated by Ref. (13) and Ref. (10), respectively.T9
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
10gcm −3
10gcm −3
10 3 gcm −3
10 3 gcm −3
10 7 gcm −3
10 7 gcm −3
10 11 gcm −3
10 11 gcm −3
1
4.49E+00
1.04E-01
4.49E+00
1.05E-01
3.05E+00
1.31E-01
8.59E-01
3.00E-01
3
2.50E+00
1.93E-01
2.50E+00
1.93E-01
2.38E+00
2.36E-01
8.59E-01
2.99E-01
10
9.95E-01
4.36E-01
9.93E-01
4.36E-01
9.95E-01
4.37E-01
8.45E-01
2.99E-01
30
1.14E+00
4.00E-01
1.14E+00
4.00E-01
1.14E+00
4.00E-01
1.05E+00
3.62E-01
Table 4
4Same as Table 2but for56 Fe.
Table 6
6Same as Table 5, but for electron capture on 55 F e.T9
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
10gcm −3
10gcm −3
10 3 gcm −3
10 3 gcm −3
10 7 gcm −3
10 7 gcm −3
10 11 gcm −3
10 11 gcm −3
1
1.01E+00
1.00E+00
1.01E+00
9.98E-01
1.02E+00
1.01E+00
9.04E-01
2.04E-01
3
1.45E+00
3.72E-01
1.44E+00
3.72E-01
1.44E+00
3.58E-01
8.91E-01
2.04E-01
10
1.12E+00
1.12E-01
1.12E+00
1.12E-01
1.12E+00
1.12E-01
9.23E-01
2.19E-01
30
1.72E+00
3.50E-01
1.72E+00
3.51E-01
1.72E+00
3.50E-01
1.44E+00
3.53E-01
Table 7
7Same asTable 5, but for electron capture on 56 F e.T9
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
10gcm −3
10gcm −3
10 3 gcm −3
10 3 gcm −3
10 7 gcm −3
10 7 gcm −3
10 11 gcm −3
10 11 gcm −3
1
8.87E-01
2.77E-01
8.87E-01
2.78E-01
1.01E+00
3.37E-01
1.13E+00
3.37E-01
3
1.12E+00
2.63E-01
1.12E+00
2.63E-01
1.17E+00
2.77E-01
1.10E+00
3.33E-01
10
1.02E+00
4.80E-01
1.02E+00
4.80E-01
1.02E+00
4.80E-01
1.07E+00
3.25E-01
30
1.29E+00
5.04E-01
1.29E+00
5.04E-01
1.29E+00
5.02E-01
1.37E+00
4.14E-01
Table 9
9Same asTable 5, but for positron capture on 55 F e.T9
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
10gcm −3
calculation of electron and positron capture rates on iron isotopes and also highlighted the differences with previous key calculations. What affect the calculated rates may have on the simulation result is difficult to predict at this stage. Core-collapse simulators are urged to check the affect of incorporating pn-QRPA rates in their simulation codes for possible interesting outcome.85E-03
1.57E-03
1.85E-03
1.57E-03
1.85E-03
1.57E-03
9.98E-01
9.98E-01
3
4.73E-01
4.12E-02
4.73E-01
4.12E-02
4.72E-01
4.06E-02
4.69E-01
4.06E-02
10
1.45E+00
1.34E-01
1.45E+00
1.34E-01
1.44E+00
1.34E-01
1.40E+00
1.31E-01
30
1.15E+01
8.39E-01
1.15E+01
8.41E-01
1.15E+01
8.39E-01
1.12E+01
8.24E-01
T9
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
QRPA/SM
QRPA/FFN
10gcm −3
10gcm −3
10 3 gcm −3
10 3 gcm −3
10 7 gcm −3
10 7 gcm −3
10 11 gcm −3
10 11 gcm −3
1
4.12E-02
5.32E-04
4.12E-02
5.32E-04
4.11E-02
5.32E-04
9.98E-01
9.98E-01
3
2.36E+00
7.05E-02
2.36E+00
7.06E-02
2.35E+00
7.03E-02
2.33E+00
7.03E-02
10
1.84E+00
1.07E-01
1.84E+00
1.07E-01
1.84E+00
1.06E-01
1.81E+00
1.03E-01
30
8.15E+00
4.79E-01
8.17E+00
4.80E-01
8.15E+00
4.79E-01
7.96E+00
4.67E-01
This manuscript was prepared with the AAS L A T E X macros v5.2.
Acknowledgements The author would like to acknowledge the kind hospitality provided by the Abdus Salam ICTP, Trieste, where part of this project was completed. The author wishes to acknowledge the support of research grant provided by the Higher Education Commission, Pakistan through the HEC Project No. 20-1283.
. Fe , G) Rec(G/T) λpc(G) Rpc(G/T9Fe (ρYe, T 9 ) λec(G) Rec(G/T) λpc(G) Rpc(G/T)
. Fe , G) Rec(G/T) λpc(G) Rpc(G/T9Fe (ρYe, T 9 ) λec(G) Rec(G/T) λpc(G) Rpc(G/T)
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[
"Space-time: Commutative or noncommutative ?",
"Space-time: Commutative or noncommutative ?"
]
| [
"R Vilela Mendes \nIPFN\nInstituto Superior Técnico\nUniversity of Lisboa\n\n"
]
| [
"IPFN\nInstituto Superior Técnico\nUniversity of Lisboa\n"
]
| []
| Noncommutativity of the spacetime coordinates has been explored in several contexts, mostly associated to phenomena at the Planck length scale. However, approaching this question through deformation theory and the principle of stability of physical theories, one concludes that the scales of noncommutativity of the coordinates and noncommutativity of the generators of translations are independent. This suggests that the scale of the spacetime coordinates noncommutativity could be larger than the Planck length. This paper attempts to explore the experimental perspectives to settle this question, either on the lab or by measurements of phenomena of cosmological origin. | 10.1103/physrevd.99.123006 | [
"https://arxiv.org/pdf/1901.01613v2.pdf"
]
| 119,434,412 | 1901.01613 | 8fe3b975a84e8cf4c29397b87c20e0c60220ff06 |
Space-time: Commutative or noncommutative ?
19 Jan 2019
R Vilela Mendes
IPFN
Instituto Superior Técnico
University of Lisboa
Space-time: Commutative or noncommutative ?
19 Jan 2019
Noncommutativity of the spacetime coordinates has been explored in several contexts, mostly associated to phenomena at the Planck length scale. However, approaching this question through deformation theory and the principle of stability of physical theories, one concludes that the scales of noncommutativity of the coordinates and noncommutativity of the generators of translations are independent. This suggests that the scale of the spacetime coordinates noncommutativity could be larger than the Planck length. This paper attempts to explore the experimental perspectives to settle this question, either on the lab or by measurements of phenomena of cosmological origin.
Introduction
In this paper I will address the following questions:
"Is spacetime a commutative or a noncommutative manifold ? " "Can this question be decided in our time ? ", that is, are there already sufficient experimental results and (or) experimental instruments to decide ?
To sharpen these questions I will borrow from past results and from a few new ideas. The emphasis will be on the experimental perspectives.
To my knowledge the first motivation to explore alternatives to the continuous commutative spacetime manifold, was to cure the divergences arising in the perturbative treatment of quantum fields. In this context several discrete time and (or) discrete space models were proposed. However these proposals violated Lorentz invariance and it was Snyder [1] who made the first Lorentz invariant proposal
[x µ , x ν ] = ia 2 M µν(1)
M µν being the Lorentz group generators. However the full Snyder algebra lacked translation invariance and it was Yang [2] who pointed out that translation invariance would be recovered by interpreting the coordinate operators as generators of linear transformations in 5-dimensional de Sitter space.
In recent years the noncommutativity of the spacetime coordinates, in the sense [x µ , x ν ] = iθ µν (2) where θ µν is either a c-number or an operator, has been explored in many contexts (see for example [3], [4], [5], [6] and references therein). Noncommutative spacetime manifolds and noncommutative geometry techniques appear naturally in the context of string and M-theory but, so far, they lack a solid experimental or compelling physical motivation. An exception might be the work in Refs. [7] [8]. There it is argued that attempts to localize events with extreme precision cause gravitational collapse, so that spacetime below the Planck scale has no operational meaning, leading to spacetime uncertainty relations. However, this compelling reasoning would imply that the noncommutativity and the associated fundamental length would be of the order of Planck's length λ P = G c 3 ≃ 1.6 × 10 −33 cm, far removed from current experimental reach. However, nothing forbids that the noncommutativity of spacetime might occur at a bigger scale. Nevertheless most recent discussions of noncommutativity of spacetime take place in the framework of quantum gravity, therefore at the Planck length scale (see for example the review [9] and references therein).
An alternative approach to the question of noncommutativity of spacetime is based on deformation theory and the stability of physical theories.
Noncommutative spacetime by deformation theory
In the construction of models for the natural world, only those model properties that are robust have a chance to be observed. It is unlikely that properties that are too sensitive to small changes of the parameters will be well described in the model. If a fine tuning of the parameters is needed to reproduce some natural phenomenon, then the model is basically unsound and its other predictions expected to be unreliable. For this reason a good methodological point of view consists in focusing on the robust properties of the models or, equivalently, to consider only models which are stable, in the sense that they do not change, in a qualitative manner, when some parameter changes. This is what has been called the stability of physical theories principle (SPTP) [10]. The stable-model point of view led in the field of non-linear dynamics to the rigorous notion of structural stability [11] [12]. As pointed out by Flato [13] and Faddeev [14] the same pattern seems to occur in the fundamental theories of Nature. In particular the passage from non-relativistic to relativistic and from classical to quantum mechanics, may be interpreted as transitions from two unstable theories to two stable ones. The stabilization of nonrelativistic mechanics corresponds to the deformation of the unstable Galileo algebra to the stable Lorentz algebra and quantum mechanics arises as the stabilization of the Poisson algebra to the stable Moyal algebra. However, when the generators of the Lorentz and the quantum mechanics Heisenberg algebra {M µν , x µ , p µ } are joined together, one finds out that the resulting Poincaré-Heisenberg algebra is also not stable.
The Poincaré-Heisenberg algebra is deformed [15] [16] to the stable algebra ℜ ℓ,φ = {M µν , p µ , x µ , ℑ} defined by the commutators
[M µν , M ρσ ] = i(M µσ η νρ + M νρ η µσ − M νσ η µρ − M µρ η νσ ) [M µν , p λ ] = i(p µ η νλ − p ν η µλ ) [M µν , x λ ] = i(x µ η νλ − x ν η µλ ) [p µ , x ν ] = iη µν ℑ [x µ , x ν ] = −iǫℓ 2 M µν [p µ , p ν ] = −iǫ ′ φ 2 M µν [x µ , ℑ] = iǫℓ 2 p µ [p µ , ℑ] = −iǫ ′ φ 2 x µ [M µν , ℑ] = 0(3)
which, according to the SPTP paradigm, one would expect to be a more accurate model. The stabilizing deformation introduces two new parameters ℓ 2 , φ 2 and two signs ǫ, ǫ ′ . The signs have physical relevance. For example, in the ℓ 2 = 0 case if ǫ = +1 time is discretely quantized and if ǫ = −1 it will be a space coordinate that has discrete spectrum. An important point that this deformation [15] of the Poincaré-Heisenberg algebra puts in evidence is the independence of the deformation parameters ℓ (associated to the noncommutativity of the spacetime coordinates) and φ (associated to the noncommutativity of momenta).
The stable algebra ℜ ℓ,φ = {x µ , M µν , p µ , ℑ}, to which the Poincaré-Heisenberg algebra has been deformed, is isomorphic to the algebra of the 6−dimensional pseudo-orthogonal group with metric
η aa = (1, −1, −1, −1, ǫ, ǫ ′ ), ǫ, ǫ ′ = ±1(4)
Both ℓ and φ −1 have dimensions of length. However they might have different physical status and interpretation. Whereas ℓ might be considered as a fundamental length and a new constant of Nature, φ −1 , being associated to the noncommutativity of the generators of translation of the Poincaré group, is associated to the local curvature of the spacetime manifold 1 and therefore is a dynamical quantity related to the local intensity of the gravitational field. The two deformations, the one in the right-hand side of [p µ , p ν ] and the one in the right-hand side of [x µ , x ν ] are independent from each other. Being associated to the local gravitational field, it is natural that the scale of the deformation in the right-hand side of the [p µ , p ν ] commutator be the Planck length scale (10 −33 cm). However there is no reason for the other one to have the same length scale. A basic conjecture that will be explored in this paper is that ℓ is much larger than φ −1 . In particular, a deformed tangent space would correspond to take the limit φ −1 → ∞ obtaining
[p µ , p ν ]| φ −1 →∞ → 0 and [x µ , ℑ]| φ −1 →∞ → 0 (5)
all the other commutators being the same as in (3), leading to the tangent space algebra ℜ ℓ,∞ = x µ , M µν , p µ , ℑ 2 . The stable Poincaré-Heisenberg algebra in (3), obtained in [15], corresponds to a minimal deformation of the classical Poincaré-Heisenberg algebra. In [15] it is also pointed out that this deformation, not being unique, is the one that seems to be the most reasonable one from a physical point of view. Chryssomalakos and Okon [17] (see also [18] [19]) later careful analysis has then found the structure of the most general deformations of the Heisenberg-Poincaré algebra. This is summarized in the Appendix with a critical analysis of the physical reasoning behind the choice of the deformation in (3).
A first question of interest on the deformed algebras is the form of the dispersion relations. For the deformed tangent space algebra ℜ ℓ,
∞ = x µ , M µν , p µ , ℑ it is p 0 2 − p 1 2 − p 2 2 − p 3 2 = Q 2
the same as in the Poincaré algebra, because this algebra is unchanged in ℜ ℓ,∞ , Q 2 = m 2 being the quadratic Casimir operator.
For ℜ ℓ,φ = {x µ , M µν , p µ , ℑ} it is P 2 + ǫ ′ φ 2 J 2 − K 2 = Q 2 with P 2 = p µ p µ , J i = 1 2 ε ijk M jk , K i = M i0 and Q 2 is the quadratic Casimir operator for SO (3, 2) (ǫ ′ = +1) or SO (4, 1) (ǫ ′ = −1).
The fact that the right-hand-side of the commutator [x µ , x ν ] is a tensor operator rather than a c-number implies that most spacetime global symmetries are preserved (see for example [20]).
The deformed algebra (3) and its tangent space limit (5) have far reaching consequences both for the geometry of spacetime [21] [22], the dimension of the associated differentiable algebra, the interactions of connection related quantum fields [23] and the Dirac equation [24]. Here however I will concentrate mostly on possible experimental tests and estimates of the value of the deformation parameters.
In the past, noncommutativity of the spacetime coordinates has been mostly associated to quantum gravity effects and the Planck length scale. Although, as pointed out in [9], some particular physical situations might greatly amplify the effects, the emphasis on the Planck length scale nature of the noncommutativity has precluded the search for laboratory scale effects. The point of view proposed in this paper is that the formal independence of the deformation parameters ℓ and φ −1 suggests that these two length scales are naturally independent and therefore it makes sense to look in the lab for the possibility of noncommutative effects at a scale larger than the Planck length.
Noncommutative spacetime: experimental perspectives
From the commutator [x µ , x ν ] = −iǫℓ 2 M µν or from a more general one , [x µ , x ν ] = iθ µν , one concludes that in the noncommutative case, the spacetime coordinates cannot be treated in isolation and that at least an extra operator is involved in all calculations in the spacetime manifold. In the ǫ = +1 case the spacetime manifold is locally isomorphic to SO (3,2) and in the ǫ = −1 case to SO(4, 1). Convenient tools for calculations are the representations of these algebras as operators on the corresponding cones (see [21] and the appendixes in [25] and [22]), irreducible representations of these algebras playing the role of "points" in their noncommutative geometry.
Here one analyses a few situations were the noncommutativity of spacetime might be tested and measured as well as some of the instances where such tests seem at present to be unfeasible. When the nature of the noncommutativity is left essentially unspecified, as in [x µ , x ν ] = iθ µν , it is difficult to obtain clearly testable predictions. Therefore here, as a working principle, use will always be made of the commutation relations in (3), in particular in the tangent space limit (5).
Measuring speed
In the noncommutative context, space and time being noncommutative coordinates, they cannot be simultaneously diagonalized and speed can only be defined in terms of expectation values, that is
v i ψ = 1 ψ t , ψ t d dt ψ t , x i ψ t(6)
where ψ is a state with a small dispersion of momentum around a central value p. At time zero
ψ 0 = k 0 −→ k α f p (k) d 3 k(7)
with k 0 = −→ k 2 + m 2 , α standing for the quantum numbers associated to the little group of k and f p (k) is a normalized function peaked at k = p.
In [26] a first order (in ℓ 2 ) derivation of the speed corrections was obtained. Here a more complete treatment will be done. To obtain ψ t one applies to ψ 0 the time-shift operator, which is not e −iap 0 because
e −iap 0 te iap 0 = t + aℑ (8) follows from p 0 , t = iℑ(9)
whereas a time-shift generator Υ should satisfy
[Υ, t] = i1(10)
Here the calculations are carried out in the ℜ ℓ,∞ algebra. To implement the commutation relations of the deformed tangent space algebra ℜ ℓ,∞ = x µ , M µν , p µ , ℑ , use a basis where the 5-variables set p µ , ℑ is diagonalized 3 . In this basis the commutation relations are realized by
x µ = i ǫℓ 2 p µ ∂ ∂ℑ − ℑ ∂ ∂p µ M µν = i p µ ∂ ∂p ν − p ν ∂ ∂p µ(11)
Then, one obtains the following time shift operator Υ in (10), to all ℓ 2 orders
Υ = p 0 ℑ k=0 (−ǫ) k ℓ 2k 2k + 1 p 0 ℑ 2k(12)
To obtain this result, use may be made of t,
ℑ −1 = −iǫℓ 2 p 0 ℑ −2 , which follows from t, ℑℑ −1 = 0 and [t, ℑ] = iǫℓ 2 p 0 .
Alternatively one may check that (12) satisfies (10) using the representation (11) to obtain
Υ, x 0 = i k=0 (−ǫ) k ℓ 2k p 0 ℑ 2k − (−ǫ) k+1 ℓ 2k+2 p 0 ℑ 2k+2
More compact forms of the time-shift operator are
Υ = 1 ℓ tan −1 ℓ p 0 ℑ ǫ = +1 1 ℓ tanh −1 ℓ p 0 ℑ ǫ = −1(13)
Now one computes the time derivative of the expectation value of x i on the time-shifted state
ψ t = exp (−itΥ) k 0 k i α f p k d 3 k(14)
From (11) and (13) one has
x i e −itΥ = e −itΥ t p i p 0 1 − ǫℓ 2 p 0 ℑ 2 1 + ǫℓ 2 p 0 ℑ 2 3
Notice that it is only in the tangent space algebra ℜ ℓ,∞ that the operators p µ , ℑ may be simultaneously diagonalized, not in the full algebra ℜ ℓ,φ .
Therefore the wave packet velocity is
v ψ = p p 0 1 − ǫℓ 2 p 0 ℑ 2 1 + ǫℓ 2 p 0 ℑ 2(15)
a result that holds to all ℓ 2 orders in ℜ ℓ,∞ . In leading order it is v ψ ≃
p p 0 1 − 2ǫℓ 2 p 0 ℑ 2 .
Notice that the correction is negative or positive depending on the sign of ǫ. For example, a massless particles wave packet would be found to travel slower or faster than c according to whether ǫ = +1 (quantized time) or ǫ = −1 (quantized space). Also notice that this deviation from c, for the velocity of the massless particle wave packet, implies no violation of relativity. Both the Lorentz and the Poincaré groups are still exact symmetries in ℜ ℓ,∞ and the velocity corrections do not arise from modifications of the dispersion relation for elementary states, which still is
p 0 2 = − → p 2 + m 2 ,(16)
but from the noncommutativity of time and space. Now some of the existing experimental results will be analyzed to find bounds on the value of ℓ (a fundamental time or a fundamental length).
In the corrected 2012 OPERA data [27] for 17 GeV neutrinos, the reported
result is v − c c = 2.7 ± 3.1 (stat) +3.4 −3.3 (sys) × 10 −6(17)
From
2ǫℓ 2 p 0 ℑ 2 ≤ 3 × 10 −6(18)
with p 0 = 17 GeV and the eigenvalue of the operator ℑ, in the right hand side of the Heisenberg algebra, set to ℑ = 1 4 , it follows 5 ℓ ≤ 1.4 × 10 −18 cm (19) or, equivalently, for the elementary time
τ ≤ 0.5 × 10 −28 sec(20)
From the MINOS [29] data, with neutrino spectrum peaked at
p 0 = 3 GeV v − c c = (5.1 ± 2.9) × 10 −5 (21) 4
In the framework of the representations of some subalgebras [28] of (3), an explicit representation of ℑ as ℑ = 1 + ℓ 2 p 2 1/2 is possible. However this does not change the O ℓ 2 wave packet speed correction. 5 Notice that the correction due to a neutrino mass ∼ 2 eV is smaller, of order 10 −19
2ǫℓ 2 p 0 ℑ 2 ≤ 5 × 10 −5 (22) one obtains ℓ ≤ 3.3 × 10 −17 cm; τ ≤ 10 −27 sec(23)
Assuming a delay of at most a couple of hours between the neutrino and the visible light outbursts from the SN1987A supernova several authors [30] [31]
[32] have estimated v − c c < 2 × 10 −9(24)
which with p 0 ≈ 10 MeV would lead to
ℓ < 6 × 10 −17 cm; τ < 2 × 10 −27 sec(25)
One sees that all this data is compatible with a value ℓ 10 −18 cm or τ 0.3 × 10 −28 sec. Using this value one also sees that the effect is extremely small for visible light. For example with p 0 = 3 eV and ℓ = 10 −18 cm one
obtains v − c c < 4.6 × 10 −26
These are results for elementary states. For slow macroscopic matter instead of (7) the state is
ψ 0 (P ) = |k 1 , k 2 , · · · , k N f P (k 1 , k 2 , · · · , k N ) dk 1 dk 2 · · · dk N(26)
Whenever the coupling energy of the elementary constituents of the macroscopic body is much smaller than their rest masses one may factorize the time shift operator
e −itΥ |k 1 , k 2 , · · · , k N = e −itΥ1 k 1 , e −itΥ2 k 2 , · · · , e −itΥN k N(27)
Therefore for a nonrelativistic body p 0 ≃ m p (the proton mass m p = 938 MeV) leads, with ℓ = 10 −18 cm, to a speed correction
v − p p 0 p p 0 = 0.452 × 10 −8(28)
It does not sound like much, however, for a nominal velocity p p 0 = 10 Km/sec it would lead after one year to a deviation of 1.4 Km.
All the above bounds are much larger than the Planck's time scale and improving them seems in reach of present experimental techniques. In particular, it would be interesting to refine the neutrino wave packet speed measurements, preferably with a larger baseline.
Presumably the best way to test the speed corrections arriving from noncommutativity would be to consider phenomena involving cosmological distances. This is also the point of view of many authors when looking for light velocity modifications as a probe of Lorentz invariance violation (LIV) ( [33] [34] [35] and references therein). In particular special attention has been devoted to gamma ray bursts (GRB). Notice however that in the present paper no LIV is implied, it is the noncommutativity that impacts the group velocity of massless particle wave packets. In any case the LIV-estimates of these authors may in some cases be carried over to the noncommutativity framework and I will comment on that later.
As will be seen, the calculation of cosmological distances (angular diameter and luminosity distance) is affected by the energy-dependent wave packet speed corrections.
One uses the Robertson-Walker metric
(ds) 2 = (dt) 2 − a 2 (t) (dr) 2 1 − Kr 2 + r 2 (dθ) 2 + sin 2 θ (dφ) 2 (29) (c = ℏ = 1).
For a massless wave packet with central energy E moving radially at speed
v (E) v (E (t)) dt a (t) = dr √ 1 − Kr 2 (30) with, in leading ℓ 2 order v (E (t)) = 1 − 2ǫℓ 2 E 2 (t) = 1 − 8π 2 ǫℓ 2 1 λ 2 (t)(31)
λ being the wavelength. Considering now two crests in the central frequency of the packet, using (30) v
(E 0 ) a (t 0 ) λ 0 = v (E e ) a (t e ) λ e
which in leading ℓ 2 order is
λ e λ 0 1 − 8π 2 ǫℓ 2 1 λ 2 e − 1 λ 2 0 = a (t e ) a (t 0 )(32)
λ e being the emitted wavelength at time t e and λ 0 the received one at time t 0 .
Defining 1 + z = λ0 λe a (t 0 ) a (t e ) = 1 + z Γ (λ 0 , z) (33) with Γ (λ 0 , z) = 1 − 8π 2 ǫ ℓ 2 λ 2 0 z (z + 2)(34)
Therefore the relation between the ratio a(t0) a(te) and the redshift z depends on the frequency that is being observed, that is, when using integration over redshift, to obtain the propagation time, one should take into account the wavelength for which the redshift is being measured. From (33) one obtains
dt = 1 H (t) d log Γ (λ 0 , z) dz − 1 1 + z dz (35)
H (t) being the Hubble parameter,
H (t) = • a (t) a (t)(36)
The Friedmann equation becomes
H (t) H 0 = i Ω i,0 1 + z Γ (λ 0 , z) 3 + Ω rad,0 1 + z Γ (λ 0 , z) 4 + Ω K,0 1 + z Γ (λ 0 , z) 2 + Ω Λ,0 = E (λ 0 , z)(37)
the Ω constants related, respectively, to matter, radiation, curvature and vacuum energy. The dependence on λ 0 means that the redshift z is computed from the received light at λ 0 wavelength. The dependence on λ 0 would also have an impact on estimates of the age of the universe
t 0 = 1 H 0 ∞ 0 dz E (λ 0 , z) 1 1 + z − d log Γ (λ 0 , z) dz(38)
For the angular diameter d A and luminosity d L distances one has
d A = Γ (λ 0 , z) 1 + z F K 1 H 0 z 0 (1 + z) 1 1+z − d log Γ(λ0,z) dz 1 − 8π 2 ǫ ℓ 2 (1+z) 2 λ 2 0 Γ (λ 0 , z) E (λ 0 , z) dz (39) d L = (1 + z) 2 Γ 2 (λ 0 , z) d A(40)
with
F K = sin 1 sinh for K = 1 0 −1 .
With these results some experimental information might be obtained from cosmological data. As an example consider the spectral lags [36] [37] [38] [39] [40] in gamma ray bursts (GRB). The spectral lag is defined as the difference in time of arrival of high and low energy photons. It is considered positive when the high energy photons arrive earlier than the low energy ones. The spectral lags being associated to the spectral evolution during the prompt GBR phase, one expects different source types to have different intrinsic lags at the source. In addition, due to the complex nature of the gamma-ray peak structure, the spectral lags, obtained from delayed correlation measurements, have large error bars. Nevertheless they allow access to time scales not achievable in the labs and it might be worthwhile to test whether the lags are also affected by energydependent propagation effects. A few simple hypothesis will be made about the relation between the lag in the production of gamma rays at the source and their observation at earth. Let us consider two gamma pulses at different energies E 1 and E 2 (E 2 > E 1 ) produced with an intrinsic lag α (a) at the source a.
If T
(1) a and T
(2) a are their propagation times from the source a to earth, the spectral lag would be
∆t a = T (1) a − T (2) a + α (a)(41)
From (35) and (37)
T (i) λ (i) 0 , z = 1 H 0 z 0 dz ′ 1 E (z ′ ) 1 1 + z ′ − d dz ′ log Γ λ (i) 0 , z(42)
with λ (i) 0 the wavelengths as observed at earth and E (z) defined in (37). Adopting the nowadays consensus cosmology Ω m,0 = 0.3, Ω Λ,0 = 0.7, Ω k,0 = Ω rad,0 = 0, K = 0, T (i) (λ 0 , z) becomes in leading ℓ 2
λ 2 0 order T (i) λ (i) 0 , z ≃ I 1 (z) H 0 + 4π 2 ǫℓ 2 H 0 λ (i)2 0 I 2 (z)(43)
the integrals I 1 and I 2 being
I 1 (z) = z 0 dz ′ 1 + z ′ 1 Ω m,0 (1 + z ′ ) 3 + Ω Λ,0 1 2 I 2 (z) = z 0 dz ′ Ω m,0 (1 + z ′ ) 2 z ′2 + 2z ′ + 4 + 4 (1 + z ′ ) Ω Λ,0 Ω m,0 (1 + z ′ ) 3 + Ω Λ,0 3 2(44)
From (41) and (43) one sees that the lags are linear on I 2 (z),
∆t a = 4π 2 ǫℓ 2 H 0 I 2 (z) 1 λ (1)2 0 − 1 λ (2)2 0 + α (0)
a for wavelengths at earth or on I2(z) (1+z) 2 for energies at the source
∆t a = 4π 2 ǫℓ 2 H 0 I 2 (z) (1 + z) 2 1 λ (1)2 e − 1 λ (2)2 e + α (0) a = ǫℓ 2 H 0 I 2 (z) (1 + z) 2 E (1)2 e − E (2)2 e + α (0) a(45)
Thus, for fixed E
(1)2 e − E (2)2 e
one may expect the data to be fitted by a few parallel lines, each one corresponding to a particular type of lag mechanism at the source. This analysis is similar to what has been done by other authors (see for example [34] [41]) in the context of searches for LIV.
Let H 0 = 70 Km s −1 , Ω m,0 = 0.3, Ω Λ,0 = 0.7 and to test the hypothesis, use the Swift BAT data on reference [39] for spectral lags of the source-frame bands 100 − 150 KeV and 200 − 250 Kev (E decreases. The fitting accuracy improves appreciably until dimension of − → α (0) equal to 3, but not much afterwards.
(1) 0 = 125 1+z , E(2)
The figure (1) shows the data points and the fitting lines for dim − → α (0) = 3. The error is er = 0.05. The slope β is ≃ −360 corresponding, with E 2 1 − E 2 2 = 35000 KeV 2 to ℓ ≃ 0.95 × 10 −19 cm and ǫ = +1 ( τ = 3 × 10 −30 s). Notice that I 2 (z) grows with z but not I2(z) (1+z) 2 . As shown in Fig.(2) the result is quite similar when one restricts to the GBR's with significance 2σ or greater.
Notice that ǫ = +1 corresponds to higher energy pulses travelling slower than lower energy ones.
A larger set of GRB data with known redshifts is studied in [42]. The main difference from the analysis in [39] is the use of an asymmetric Gaussian model for the cross-correlation function to compute the spectral lags. Otherwise the source frame energy bands (100−150 and 200−250 KeV) are the same as in [39]. The same fitting technique as before was here applied to the 57 GRB's in [42] with the result shown in Fig.3. For 3 intersects (dimension of − → α (0) = 3) the slope that is obtained is β ≃ −330 (er = 0.18) corresponding to ℓ ≃ 0.9 × 10 −19 cm, ǫ = +1, a result consistent with the one obtained before. Notice however that if instead of dimension of − → α (0) = 3 one assumes dimension of − → α (0) = 1 one obtains a worse fit (er = 0.87) and a quite different result, that is ℓ ≃ 0, the dash-dotted green line in Fig.3. This is essentially what has been done in [44] with these authors concluding that there is no evidence for LIV. However that hypothesis (dimension of − → α (0) = 1) assumes that all the intrinsic lags at the source are the same.
In Eq.(45) the α (1+z) 2 for the 57 GRB's in [42] and fitting lines with 3 (red) or one intersect (green) that is replace α
1 + z = ǫℓ 2 H 0 I 2 (z) (1 + z) 3 E (1)2 e − E (2)2 e + α (e) a(46)
With this equation and the data in [42] one obtains the results shown in Fig.4 For short GRB pulses intrinsic lags are in general considered smaller that those of long GRB pulses. Looking for eventual Lorentz invariance violation (LIV), the authors in [43] have analyzed 15 short pulses (on the energy bands 50-100 and 150-200 KeV) concluding that there is no evidence 6 for energy dependence of the light propagation speed. Here the same analyzing technique as described above has been applied to the same data with a different conclusion, as shown in ℓ ≃ 0.9 × 10 −19 cm. Notice that in this case the slope obtained with one or three intersects is essentially the same, suggesting that for this set of pulses the intrinsic lags are identical. The difference to the conclusions of the authors in [43] are not, of course, due to any mistake of these authors but to the fact that they plot the data with respect to a K (z) function, whereas here, according to the calculations above, the z−dependence is coded by the I 2 (z) function (Eq.44).
Of course, all these results, as well as the searches for LIV (see for example [35] and references therein), can only be taken as indicative or as establishing an upper bound on τ because of the large uncertainties on the calculation of the spectral lags, on the statistics of the GRB pulses and even more on the intrinsic spectral lags α a . However, if correct, they have some implications concerning the observation of neutrino emissions from the GRB sources and also on the SN1987A observations. From the SN1987A supernova, neutrinos were observed in the range from 7.5 to 40 Mev[30] [31]. Using Eq.(15) to obtain the propagation time difference over 168000 light years, between visible light and neutrino packets of 10 and 40 Mev, with ℓ = 10 −19 cm, one obtains respectively 1.1 × 10 −3 and 1.8 × 10 −2 seconds. Clearly this does not change the estimate in (24). However for GRB's at cosmological distances the situation is different.
From (35) ℓ = 10 −19 cm and ǫ = +1, neutrinos of energy 40 Mev, would take 27 hours more than visible light to reach earth from a source at z = 2 redshift and 1.7 more hours from a source at redshift z = 1. For 10 MeV neutrinos the result would be 1.7 and 1 hour. For ℓ = 10 −18 cm these numbers would be multiplied by 100 and also grow quadratically with the energy.
Recently a very high energy neutrino was observed from the direction of active galactic nuclei at cosmological distance [45] [46]. If ℓ is in the range discussed above, the conclusion is that it could only have originated from a much earlier event, not a recent flare of gamma activity. Alternatively if by some means its origin is proved to be coincident with recently observed gamma flares, that would mean that ℓ is much smaller than suggested here (that is, ℓ 10 −24 cm). Notice however that dedicated searches [47] [48] [49] for neutrinos in close coincidence with GRB bursts found no or scarce evidence for them.
Wei et al. [50] [51] analyzed a burst GRB160625B with unusually high photon statistics and a steep decline from positive lags to smaller ones with increasing photon energy in the range 8-20 MeV. They have fitted the spectral lag data using a power law for the intrinsic lag and a linear or quadratic term corresponding to the LIV correction. Here the same data has been analyzed using also a power law for the intrinsic lag together with the noncommutativity correction, namely
lag = αE β − ǫℓ 2 H 0 I 2 (1.41) E 2(47)
The least squares result is shown in Fig.(6). One sees that the fitting accuracy is rather poor, what is even more apparent using a linear E 2 axis than in the log-log plot used in [50]. Actually the small statistical significance of the fitting using an equation of the type of Eq.(47) had already been pointed out in [52]. In fact given the probable multiple shock mechanism of the GRB's generation is not likely that a continuous power dependence of the intrinsic lag be a good hypothesis. It seems better to concentrate on the high energy tail of the data and try the equation
lag = αE 0.18+βE 2 − ǫℓ 2 H 0 I 2 (1.41) E 2(48)
This is used to fit the data between 5−20 GeV the result being shown in Fig.(7). s. This is two orders of magnitude smaller than obtained before, but there is small significance of a result obtained with a single burst, compounded with the small quantitative knowledge that still exists about the intrinsic lags at the source. Finally, from (40) one may also estimate the impact of an energy dependent propagation speed on the calculation of the Hubble constant from observations at cosmological distances. Given the luminosity L and the observed flux F o from a standard candle, the luminosity distance d L is
d 2 L = L 4πF o(50)
On the other hand from (39) and (40)
d L = 1 + z Γ (λ 0 , z) I (z)(51)
with
I (z) = 1 H 0 z 0 (1 + z) 1 1+z − d log Γ(λ0,z) dz 1 − 8π 2 ℓ 2 λ 2 0 (1+z) 2 Γ 2 (λ0,z) Γ (λ 0 , z) E (λ 0 , z) dz(52)
Therefore given d L from (50), H 0 is obtained from r = r |r| . One obtains a positive or negative correction (depending on ǫ) of the coupling constant. For classical bound quasi-circular orbits r · p ℑ is very small, therefore any detectable corrections to the classical motion could only be expected for flyby orbits.
With the estimate collision ( r · p) 2 = π 0 |p| 2 cos θdθ = |p| 2 π 2 the approximate correction to the coupling constant would be
G → G 1 + ǫℓ 2 |p| 2 π 2
which for a macroscopic speed 15 Km/sec, the proton mass and ℓ = 10 −19 cm leads to ℓ 2 |p| 2 π 2 = 8.88 × 10 −20 much too small to be observable. Next one computes the modifications to the quantum Coulomb spectrum arising from (54). Because However for muon atoms and large Z, this value is multiplied by a factor ≈ (200 × Z) 2 . For other noncommutative corrections to the Coulomb problem refer to [25] where, in particular, angular momentum effects were taken into consideration.
1 2 ( r · p + p · r) = p r = ℏ i
Phase-space volume effects
The phase space contraction for ǫ = +1 and the phase space expansion for ǫ = −1 have already been described in [25] and [23]. Here I simply rederive this result in the context of the general representation (11) and update the experimental perspectives.
Consider a particular space coordinate x i ⊜ x, p i ⊜ p. Then
x = i ǫℓ 2 p ∂ ∂ℑ + ℑ ∂ ∂p (55)
The eigenstates of this operator are
|x = exp −i x ℓ tanh −1 ℓp ℑ(56)
for ǫ = +1 and
|x = exp −i x ℓ tan −1 ℓp ℑ(57)
for ǫ = −1.
To obtain the wave function of a momentum wave function on the |x basis (ǫ = +1) one projects by integration on the p, ℑ variables
x | k = J p, ℑ dpdℑe i x ℓ tanh −1 ( ℓp ℑ ) δ (p − k)(58)
J p, ℑ being an integration density. To proceed it is convenient to change variables to
x = iℓ ∂ ∂µ p = R ℓ sinh µ ℑ = R cosh µ(59)
and convert (58) into
x | k = dRdµe i x ℓ µ δ µ − sinh −1 ℓk R δ (R − 1) = e i x ℓ sinh −1 (ℓk)(60)
The choice R = 1 corresponds in (59) to the choice of a particular representation of the pseudo Euclidean algebra in two dimensions. It corresponds to the choice of a density J p, ℑ
J p, ℑ = δ ℑ − 1 + ℓ 2 p 2
For ǫ = −1 a similar calculation leads to
x | k = e i x ℓ sin −1 (ℓk)(61)
The density of states is obtained from
x + L ℓ sinh −1 (ℓk n ) = x ℓ sinh −1 (ℓk) + 2πn x + L ℓ sin −1 (ℓk n ) = x ℓ sin −1 (ℓk) + 2πn leading to dn = L 2π dk √ 1+ℓ 2 p 2 for ǫ = +1 dn = L 2π dk √ 1−ℓ 2 p 2 for ǫ = −1 (62) For 3 dimensions [23] dn = V 2π 2 1 ℓ 2 (sinh −1 (ℓ|p|)) 2 dk √ 1+ℓ 2 |p| 2 for ǫ = +1 dn = V 2π 2 1 ℓ 2 (sin −1 (ℓ|p|)) 2 dk √ 1−ℓ 2 |p| 2 for ǫ = −1(63)
As discussed in [25] [23] the contraction or expansion of the phase space has an impact on the cross sections of elementary processes. For example for the cross section of the reaction γ + p → π + N of high energy proton cosmic rays, the contraction of phase space in the ǫ = +1 case would allow cosmic ray protons of higher energies and from further distances to reach the earth. From the calculations performed in [25], one knows that the phase space suppression factor for the photon pion production is a function of α = ω ′2 γ ℓ 2 , ω ′ γ being the photon energy in the proton rest frame, the suppression being only appreciable if α 1. In this case ω ′ γ = 1.49 × 10 −12 p 0 P , p 0 P being the proton energy. Therefore for this reaction the effect would be very small for ℓ = O 10 −19 cm . In any case even for larger values of ℓ the GZK cutoff would not be much changed, the main difference being a bigger size for the GZK sphere, meaning that more cosmic ray protons from further distances would be able to reach the earth.
A better place to look for the effects of this phase space suppression might be an increase (ǫ = +1) or decrease (ǫ = −1) in particle multiplicity in high energy reactions [23]. This effect would be important when ℓk ∼ O (1), k being the typical reaction momentum. For ℓ = 10 −19 cm this would occur for k ≈ 100−200 TeV (in the range of the future FCC).
Diffraction, interference and uncertainty relations
Massless or massive wave equations in the noncommutative context are solutions of [21] [
p µ , [p µ , ψ]] = 0 (64) or [p µ , [p µ , ψ]] − m 2 ψ = 0 (65)
where ψ may either be a scalar or a tensor element of the enveloping algebra U (ℜ ℓ,∞ ) of the algebra ℜ ℓ,∞ = x µ , M µν , p µ , ℑ . They have a general solution
ψ k (x) = exp ik · 1 2 x, ℑ −1 +(66)
from which quantum fields may be constructed [21] with k 2 = 0 or m 2 . Notice that in (66) the x µ 's are simply algebra elements, not the coordinates of the wave. Physical results are obtained from the application of a state to the algebra. From the commutator [p µ , x ν ] = iη µν ℑ it also follows that the wave equations also have factorized solutions
ψ k (x) = 3 µ=0 ψ k µ (x µ ) (67) with ψ k µ (x µ ) = e i 1 2 k µ xµ,ℑ −1 + (fixed µ)(68)
The factorized solutions may be used to study the diffraction problem. A geometry is chosen with one or two long slits along the x 2 coordinate and an incident wave along the third coordinate − → k = k − → e 3 . The wave in the slit(s) will be a superposition of localized states on the first space coordinate x 1 , namely (for a single slit) of width 2∆ (in the p 1 , ℑ representation)
|χ 1 + = ∆ −∆ dx 1 e −i x 1 ℓ tanh −1 ℓp 1 ℑ(69)
for ǫ = +1 and
|χ 1 − = ∆ −∆ dx 1 e −i x 1 ℓ tan −1 ℓp 1 ℑ(70)
for ǫ = −1. Therefore after passing the slit the wave is
|Ψ k = ψ k 0 x 0 dξ ψ ξ x 1 χ ψ ξ x 1 ψ √ k 2 −ξ 2 x 3(71)
The projections ψ ξ x 1 χ of the slit state on the wave equation solution ψ ξ x 1 will be computed in order ℓ 2 . In addition, because of the factorized nature of the solutions, one may use for the operators x 1 , p 1 and ℑ, a subalgebra representation instead of the general representation (11), namely
x 1 = x p 1 = 1 ℓ sinh ℓ i d dx ℑ = cosh ℓ i d dx ǫ = +1 (72) x 1 = x p 1 = 1 ℓ sin ℓ i d dx ℑ = cos ℓ i d dx ǫ = −1 (73) Then in O ℓ 2 ℑ −1 = 1 − ǫ 1 2 ℓ i 2 d 2 dx 2 + 5 4 ℓ i 4 d 4 dx 4 − · · ·(74)
and in the representation (72)-(73) the generalized localized states are simply |χ 1 = δ x 1 − η . The projection ψ ξ x 1 χ becomes
ψ ξ x 1 χ = 1 2π ∆ −∆ e i 1 2 ξ x 1 ,ℑ −1 δ x 1 − η dη = 1 π e −ǫ 1 4 ξ 2 ℓ 2 sin ∆ξ 1 − ǫ 1 6 ξ 2 ℓ 2 ξ − ǫ 1 6 ξ 3 ℓ 2 + O ℓ 4
Therefore the intensity of the diffracted wave at angle θ = sin −1 ξ k is proportional to sin 2 ∆ξ 1 − ǫ 1 6 ξ 2 ℓ 2 ∆ξ 1 − ǫ 1 6 ξ 2 ℓ 2 2 For two slits of width 2∆ at a distance 2Σ the slit states would be
−Σ+∆ −Σ−∆ + Σ+∆ Σ−∆ dx 1 |χ 1
with normalized diffracted intensity sin 2 ∆ξ 1 − ǫ 1 6 ξ 2 ℓ 2 sin 2 Σξ 1 − ǫ 1 6 ξ 2 ℓ 2 ∆ 2 Σ 2 ξ 1 − ǫ 1 6 ξ 2 ℓ 2 4 One sees that the effects of noncommutativity become important for ξℓ ∼ O (1). For ℓ ∼ 10 −19 cm this would be ξ 100 TeV.
On the other hand, the noncommutativity of the spacetime coordinates implies uncertainty relations on the simultaneous measurement of two space coordinates or one space and one time coordinate. From
[x µ , x ν ] = −iǫℓ 2 M µν one obtains for ∆x µ = ψ (x µ ) 2 ψ − ψ |x µ | ψ 2 1 2 ∆x µ ∆x ν ≥ 1 2 ℓ 2 ψ |M µν | ψ
In particular one notices that there is no space-space uncertainty if |ψ is spinless, but time-space uncertainty leads to observable effects.
Remarks and conclusions
1. Approaching the question of noncommutative spacetime from the point of view of deformation theory and the principle of stability of physical theories, the first important observation is the independence of the length scales of noncommutativity of the coordinates (ℓ) and of the momenta (φ −1 ). The scale of φ −1 being associated to the noncommutativity of translations is naturally associated to gravity and the Planck length. However the scale of ℓ might be larger and it makes sense to launch an experimental effort to find upper bounds or even the value of this length scale. At the present time, in addition to a precise analysis of phenomena of cosmological origin and a refinement of the neutrino speed measurements, another possibility lies in phase space modification effects on high energy colliders. 2. The estimates, performed here based on GRB data, point to values of ℓ in the range 10 −19 − 10 −21 cm (or τ ∈ 0.3 × 10 −29 − 0.3 × 10 31 s ) favouring the higher part of this range. However these estimates can only be taken as indicative or as establishing upper bounds because of the large uncertainties on the calculation of the spectral lags, on the statistics of the GRB pulses and on the nature of the intrinsic spectral lags.
2. The deformed ℜ ℓ,∞ = x µ , M µν , p µ , ℑ algebra has also some consequences concerning the structure of the fundamental interactions, in particular those that are associated to connection-valued fields. In particular the additional dimension in the differential algebra may imply the existence of new interactions and states as well as a new extended structure for the Dirac equation. These questions, not dealt with here, because they have a less direct experimental verification, are described elsewhere [21] [23] [21].
3. In the context of deformation theory, the transition from classical to quantum mechanics appears as the stabilization of the unstable Poisson algebra to the stable Moyal algebra. At the level of general nonlinear functions of position and momentum the corresponding Hilbert space algebra of operators is also stable, but the Heisenberg algebra itself and generalizing to x µ and p µ together with compatibility with the Lorentz group would lead to the tangent space deformed algebra ℜ ℓ,∞ = x µ , M µν , p µ , ℑ .
In conclusion, in the framework of stable theories this algebra is already implicit in the transition to quantum mechanics. 4. All calculations in the previous sections were carried out for the algebra of the (noncommutative) tangent space limit φ −1 → ∞. When the full deformed algebra in (3) is used, the noncommutativity of momenta in [p µ , p ν ] = −iǫ ′ φ 2 M µν corresponds to the noncommutativity of spacetime translations. A similar noncommutativity is what occur in a gravitational field. In this sense, gravitation might also be considered an emergent property arising from deformation theory and the principle of stability of physical theories. Considering φ rather than the metric as defining gravitational field, gravitation would be formulated as a SO (3, 3) gauge theory [22]. An interesting consequence is that the gravitational field might be a function of the Casimir invariants of SO (3, 3) and not only of the energy-momentum tensor. is not stable (rigid). Its 2-cohomology group has three nontrivial generators, which lead to the following modified commutators 7 [17] [p µ ,
x ν ] = iη µν ℑ + iβ 3 M µν [x µ , x ν ] = iβ 2 M µν [p µ , p ν ] = iβ 1 M µν [x µ , ℑ] = −iβ 2 p µ + iβ 3 x µ [p µ , ℑ] = iβ 1 x µ − iβ 3 p µ [M µν , ℑ] = 0
There is an instability cone at β 2 3 = β 1 β 2 , but for generic β 1 , β 2, β 3 all these algebras are rigid and are isomorphic to either SO (1,5) or SO (2, 4) or SO (3, 3) (depending on the signs of β 1 and β 2 ). For all these classes there is a representative with β 3 = 0, which is exactly the deformation (3) obtained in [15]. The β 3 = 0 situation may always be obtained by a linear change of coordinates in the algebra. The converse situation β 1 = β 2 = 0 and β 3 = 0 also mentioned in [15] leads to [x µ , x ν ] = [p µ , p ν ] = 0 which does not seem to be physically relevant, because at least the second commutator is expected to be different from zero in the presence of gravity.
Here and elsewhere, I will be interpreting x µ and p ν as the physical coordinates and momenta. In this sense I do not agree with the criticism in [17] about this choice, because not all observables have to be extensive, only those that correspond to symmetry transformations, in this case M µν and p µ .
Comparison of the data with Eq.(45) is performed by minimizing in β and − → α the functionf (β, − → α ) = i min − → α {y i − (βx i + − → α )} 2for several dimensions of the vector − → α (the vector of intrinsic lags). Here the variables y i and x i are respectively the observed lags ∆t a and I2(z) (1+z) 2 . As the dimension of − → α (the number of different lag types at the sources) increases, the fitting error, defined as er = f (β, − → α )
Figure 1 :Figure 2 :
12Lags versus I2(z) (1+z) 2 for 24 GRB's with 1σ significance or greater (data from[39]) Lags versus I2(z) (1+z) 2 for 15 GRB's with significance 2σ or greater (data from[39])
intersects represent several classes of intrinsic lags as seen at earth. It might be better to use the intrinsic lags α
Figure 3 :
3Lags versus I2(z)
Figure 4 :
4Lags/(1+z) versus I2(z) (1+z) 3 for the 57 GRB's in[42] and fitting lines with 3 (red) or one intersect (green) For dimension of − → α (e) = 3 the slope is β = −448 (er = 0.KeV 2 to ℓ ≃ 1.06 × 10 −19 cm. And, as before, a very different result is obtained for dimension of − → α (e) = 1 (er = 0.9), the dash-dotted line inFig.4.
Figure 5 :
5Lags/(1+z) versus I2(z) (1+z) 3 for the 15 short GRB's in[43] and fitting lines with 3 (red) or one intersect (green)The figure shows the fitting of the data assuming either dimension of − → α (e) = 3 (red continuous lines) of dimension of − → α (e) = 1 (dash-dotted green line). The slope is β ≃ −230 corresponding with E
Figure 6 :
6Fitting the GRB160625B data to Eq.(47)
Figure 7 :
7Fitting the GRB160625B data between 5 and 20 GeV by Eq.(48) The minimizing parameters are α = 0.29, β = 1.538 × 10 −10 and ǫℓ 2 H 0 I 2 (1.41) = 7.794 × 10 −8 (49) 0.18 in the exponent being the value obtained in the fitting to Eq.(47). With I 2 (1.41) = 4.3445 one obtains from (49) ℓ = 1.79×10 −21 cm or τ = 0.597×10 −31
2 r acting on ψ (r) as a perturbation. Because of the 1 r factor in ∆ one expects the largest effects to occur for s states. One obtains for the first and second s statesψ 1s , ∆ψ 1s = −3ǫℓ 2 Z 4 m 3 e 8 ℏ 4 ψ 2s , ∆ψ 2s = − 7 16 ǫℓ 2 Z 4 m 3 e 8ℏ 4 and denoting by H 0 the unperturbed Hamiltonian, with ψ ns , H 0 ψ ns = − Z 2 me 4 2ℏ 2 n 2 ψ 1s , ∆ψ 1s ψ 1s , H 0 ψ 1s = 6ǫℓ 2 Z 2 m 2 e 4 ℏ 2 ψ 2s , ∆ψ 2s ψ 2s , H Of more significance is perhaps the mixing matrix element ψ 2s , ∆ψ 1s = − 44 27 ǫℓ 2 Z 4 m 3 e 8 ℏ 4If m = m e , the electron mass, and ℓ = 10
[p, x] = −i1 = 1is not, because the c-number 1 commutes with both p and x. Stabilization would then suggest a deformation to[x, 1] = iǫℓ 2 p; [p, 1] = −iǫ ′ φ 2 x
µν , M ρσ ] = i(M µσ η νρ + M νρ η µσ − M νσ η µρ − M µρ η νσ ) [M µν , p λ ] = i(p µ η νλ − p ν η µλ ) [M µν , x λ ] = i(x µ η νλ − x ν η µλ ) [p µ , x ν ] = iη µν [x µ , x ν
bursts for which the lags were computed with significance 1σ or greater. The following table lists the correspondence of the numbers in the plot with the0
= 225
1+z ), selecting
the 24 burst code.
1
GRB050401
13 GRB080413B
2
GRB050922C 14 GRB080605
3
GRB051111
15 GRB080916A
4
GRB060210
16 GRB081222
5
GRB061007
17 GRB090618
6
GRB061121
18 GRB090715B
7
GRB071010B 19 GRB090926B
8
GRB071020
20 GRB091024
9
GRB080319B 21 GRB091208B
10 GRB080319C 22 GRB100621A
11 GRB080411
23 GRB100814A
12 GRB080413A 24 GRB100906A
In a de Sitter context φ would be the inverse of the (local) curvature radius.
p µ , ℑ denote the tangent space (φ −1 → ∞) limits of the operators, not be confused with the physical p µ , ℑ operators. According to the deformation-stability principle they are stable physical operators only when φ −1 is finite, that is, when gravity is turned on.
Actually the authors conclusion is that the quantum gravity scale E QG 1.5 × 10 16 GeV, which would correspond to a scale ℓ 1.3 × 10 −30 cm.
In the notation of Ref.[17], β i = qα i and ℑ = qM
Then the correction to the H 0 calculation isHowever, for example for z = 0.5 this would be 1 + 17.3 πǫℓ 2, which for visible light (λ 0 = 3.9 − 7 × 10 −9 cm) would be too small to be of any importance. Hence this correction should not be relevant to the present H 0 tension problem.Corrections to the Kepler problemBy the Kepler problem one means motion of a body under the central 1 r potential. In reality an additional inverse cubic term should be added to account for the general relativity corrections. Here only the modifications to the 1 r term arising from noncommutativity will be considered (in first ℓ 2 order). First one considers the corrections to the classical HamiltonianUsing the representation (11) and taking expectation values in a basis e ir· p ℑ (r here is a c-number, not an operator),one sees that the O ℓ 2 corrections to the classical Hamiltonian (53) amount to the replacement
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| []
|
[
"POSITIVITY VS NEGATIVITY OF CANONICAL BASES",
"POSITIVITY VS NEGATIVITY OF CANONICAL BASES"
]
| [
"Yiqiang Li \nDepartment of Mathematics\nSUNY at Buffalo\n14260BuffaloNYUSA\n",
"Weiqiang Wang \nDepartment of Mathematics\nUniversity of Virginia\n22904CharlottesvilleVAUSA\n"
]
| [
"Department of Mathematics\nSUNY at Buffalo\n14260BuffaloNYUSA",
"Department of Mathematics\nUniversity of Virginia\n22904CharlottesvilleVAUSA"
]
| [
"Bulletin of the Institute of Mathematics Academia Sinica (New Series)"
]
| We provide examples for negativity of structure constants of the stably canonical basis of modified quantum gl n and an analogous basis of modified quantum coideal algebra of gl n . In contrast, we construct the canonical basis of the modified quantum coideal algebra of sln, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.Key words and phrases: Canonical basis, quantum coideal subalgebras of gl n and sln, positivity.2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 145 175Thanks to the intersection cohomology construction of the canonical basis for S (n, d) [2], the structure constants P C. This proves the theorem.Proposition 5.7. The bilinear form ·, · onU (sl n ) is non-degenerate.Moreover, the almost orthonormality for the canonical basis holds:Proof. This almost orthonormality follows by an argument entirely similar to [18, Theorem 8.1], and it implies the non-degeneracy of the bilinear form.We have the following positivity for the canonical bases with respect to the bilinear form.Proof. The proof follows very closely McGerty's geometric argument [18, Proposition 6.5, Theorem 8.1], with [18, Corollary 3.3] replaced by [2, Corollary 3.15]. We only sketch the proof with an emphasis on the difference and refer to loc. cit. for further details.By the definition of ·, · , it is reduced to show thatThe positivity of the form ·, · d in the theorem will follow by its identification with another geometrically defined bilinear form ·, · g,d on S (n, d) which manifests the positivity. The latter is defined exactly the same as [18, (6-1)] with the flag variety F a therein replaced by the n-step isotropic flag variety of a (2d+1)-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form.Now arguing similar to [18, Lemma 6.3], we have, for all A minimal with respect to the partial order ≤,where A t is the transpose of A. This implies the analog of [18, Lemma 6.4], which gives the formulas for the adjoints of the Chevalley generators of S (n, d) for the bilinear form ·, · g,d , and we observe that they coincide with 176 YIQIANG LI AND WEIQIANG WANG [June the ones for ·, · d given in [2, Corollary 3.15]. Hence, the identification of the forms ·, · d and ·, · g,d is reduced to show that, then both sides of the above equation are equal to P A,D λ if ro(A) = co(A) = λ, or zero otherwise. The theorem follows. | 10.21915/bimas.2018201 | [
"https://web.math.sinica.edu.tw/bulletin_ns/20182/2018201.pdf"
]
| 116,900,434 | 1501.00688 | f73f8e49a4a962b46baee621798aae65820e6fd5 |
POSITIVITY VS NEGATIVITY OF CANONICAL BASES
2018
Yiqiang Li
Department of Mathematics
SUNY at Buffalo
14260BuffaloNYUSA
Weiqiang Wang
Department of Mathematics
University of Virginia
22904CharlottesvilleVAUSA
POSITIVITY VS NEGATIVITY OF CANONICAL BASES
Bulletin of the Institute of Mathematics Academia Sinica (New Series)
132201810.21915/BIMAS.2018201Dedicated to George Lusztig for his 70th birthday with admiration
We provide examples for negativity of structure constants of the stably canonical basis of modified quantum gl n and an analogous basis of modified quantum coideal algebra of gl n . In contrast, we construct the canonical basis of the modified quantum coideal algebra of sln, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.Key words and phrases: Canonical basis, quantum coideal subalgebras of gl n and sln, positivity.2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 145 175Thanks to the intersection cohomology construction of the canonical basis for S (n, d) [2], the structure constants P C. This proves the theorem.Proposition 5.7. The bilinear form ·, · onU (sl n ) is non-degenerate.Moreover, the almost orthonormality for the canonical basis holds:Proof. This almost orthonormality follows by an argument entirely similar to [18, Theorem 8.1], and it implies the non-degeneracy of the bilinear form.We have the following positivity for the canonical bases with respect to the bilinear form.Proof. The proof follows very closely McGerty's geometric argument [18, Proposition 6.5, Theorem 8.1], with [18, Corollary 3.3] replaced by [2, Corollary 3.15]. We only sketch the proof with an emphasis on the difference and refer to loc. cit. for further details.By the definition of ·, · , it is reduced to show thatThe positivity of the form ·, · d in the theorem will follow by its identification with another geometrically defined bilinear form ·, · g,d on S (n, d) which manifests the positivity. The latter is defined exactly the same as [18, (6-1)] with the flag variety F a therein replaced by the n-step isotropic flag variety of a (2d+1)-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form.Now arguing similar to [18, Lemma 6.3], we have, for all A minimal with respect to the partial order ≤,where A t is the transpose of A. This implies the analog of [18, Lemma 6.4], which gives the formulas for the adjoints of the Chevalley generators of S (n, d) for the bilinear form ·, · g,d , and we observe that they coincide with 176 YIQIANG LI AND WEIQIANG WANG [June the ones for ·, · d given in [2, Corollary 3.15]. Hence, the identification of the forms ·, · d and ·, · g,d is reduced to show that, then both sides of the above equation are equal to P A,D λ if ro(A) = co(A) = λ, or zero otherwise. The theorem follows.
1. Introduction 1.1. In [1], Beilinson, Lusztig and MacPherson realized the quantum Schur algebra S(n, d) geometrically in terms of pairs of partial flags of type A. Furthermore, they construct the modified quantum groupU (gl n ) via a stabilization procedure from the family of algebras S(n, d) as d varies. The IC construction provides a canonical basis for S(n, d) whose structure constants are positive (i.e., in N[v, v −1 ]), which in turn via stabilization leads to a distinguished bar-invariant basis (which we shall refer to as BLM or stably canonical basis) forU (gl n ).
YIQIANG LI AND WEIQIANG WANG [June
Recently the constructions of [1] have been generalized to partial flag varieties of type B and C in [2] (also see [7] for type D). A family of iSchur algebras iS(n, d) was realized geometrically together with canonical (=IC) bases whose structure constants lie in N[v, v −1 ]. Via a stabilization procedure these algebras give rise to a limit algebra which was shown to be isomorphic to the modified quantum coideal algebra iU (gl n ) of gl n , and which also admits a stably canonical basis. The appearance of the quantum coideal algebra was inspired by [3] where a new approach to Kazhdan-Lusztig theory of type B/C via a new theory of canonical bases arising from quantum coideal algebras was developed. Even though the constructions for n odd and even are quite different with the case of even n being more challenging [3], one can carry out the construction in the even n case by relating to the odd n case via a more subtle two-step stabilization [2].
1.2. The original motivation of this paper is to understand the positivity of the stably canonical basis of the modified quantum coideal algebra iU (gl n ). To that end, we have to understand first the same positivity issue forU (gl n ), asU (gl ⌊ n 2 ⌋ ) is simpler and also it appears essentially as a subalgebra of iU (gl n ) with compatible stably canonical bases. The canonical bases arising from quantum groups of ADE type are widely expected to enjoy all kinds of positivity (see [14,15]), and there is no indication in the literature that anything onU (gl n ) (or gl n ) differs substantially from its counterpart oṅ U (sl n ) (or sl n ).
To our surprise, the behavior of the BLM/stably canonical basis oḟ U (gl n ) turns out to be dramatically different, already for n = 2, from the canonical basis ofU (sl n ). In particular, we provide examples that the structure constants of the stably canonical basis are negative, and that the stably canonical basis ofU (gl n ) fails to descent to the canonical basis of the finitedimensional simpleU (gl n )-modules. These examples, though not difficult, are unexpected among the experts whom we have a chance to communicate with, so we write them down hoping to clarify some confusion or false expectation. The fundamental reason behind the failure of positivity of the BLM basis and beyond is that the stabilization process is not entirely geometric (when the involved matrices contain negative diagonal entries).
The structure constants of the canonical basis ofU (sl n ) are positive; this follows easily from combining the positivity of the canonical (=IC) basis of the Schur algebras [1] with a result of McGerty [18,Proposition 7.8] (or with a stronger result of [20], which confirmed Lusztig's conjectures [16,Conjectures 9.2,9.3]). For the reader's convenience, we make explicit this positivity in Proposition 3.1 and supply a short proof, as it could not be explicitly found in these earlier papers.
1.3. Now we focus on the modified quantum coideal algebra iU (sl n ), for n ≥ 2. We construct a canonical basis for the modified quantum coideal algebra iU (sl n ) which shares many remarkable properties of the canonical basis forU (sl n ). In particular, it has positive structure constants, and it is characterized up to sign by the three properties: bar-invariance, integrality, and almost orthonormality with respect to a bilinear form of geometric origin. Moreover, it admits positivity with respect to the geometric bilinear form. In addition, this canonical basis is compatible with Lusztig's under a natural inclusionU (sl ⌊ n 2 ⌋ ) ⊆ iU (sl n ). Our argument largely follows the line in McGerty's work [18] for n odd (the case for n even needs substantial new work), though we have avoided using the non-degeneracy of the geometric bilinear form of iU (sl n ), which was not available at the outset. Instead, the non-degeneracy of the bilinear form is replaced by arguments involving the stably canonical basis of iU (gl n ) from [2] and the non-degeneracy eventually follows from the almost orthonormality of the canonical basis which we establish.
We further show that the transfer map on the iSchur algebras sends every canonical basis element to a positive sum of canonical basis elements or zero. Some basic properties on the transfer map established in [8] are used here. Moreover, the matrix coefficients (with respect to canonical basis) for the action of any canonical basis element in iU (sl n ) on V ⊗d are shown to be positive, where V is the n-dimensional natural representation of iU (sl n ). We remark that the transfer maps on the type A Schur algebras were earlier studied in [16,17,20,18].
As in [3,2], the different behaviors in the cases for n odd and even force us to carry out the studies of the two cases separately in this paper. The case of odd n, indicated by the superscript , is easier and done first, while the remaining case is indicated by the superscript ı. Let us set up some notations used in the main text. For n odd and hence n = n − 1 even, we shall denoteU (gl n ) = iU (gl n ), S (n, d) = iS(n, d),U ı (gl n ) = iU (gl n ), and S ı (n, d) = iS(n, d).
YIQIANG LI AND WEIQIANG WANG [June
There is another purely representation theoretic approach in [4] toward the bilinear forms and canonical bases for general quantum coideal algebras including iU (sl n ), which nevertheless cannot address the positivity of canonical bases. Note that the papers [16,17,20,18] are mostly concerned about the quantum Schur algebras and quantum groups of affine type A. A geometric setting for the quantum coideal algebras of affine type will be pursued elsewhere.
1.4. The paper is organized as follows. In Section 2, we construct examples that a natural shift map (which is an algebra isomorphism) onU (gl n ) does not preserve the BLM basis, that the structure constants of BLM basis foṙ U (gl n ) are negative, and that the BLM basis ofU (gl n ) does not descend to the canonical basis of a finite-dimensional simple module.
In Section 3, we show that the positivity of structure constants for the canonical basis ofU (sl n ) is an easy consequence of McGerty's results. Then we construct a positive basis forU (gl n ) with positive structure constants by transporting the canonical basis ofU (sl n ). We explain several positivity results on the transfer map for Schur algebras.
In Sections 4, 5, and 6, we study the quantum coideal algebras and the associated Schur algebras. In Section 4, we show the stably canonical basis constructed in [2] for the modified quantum coideal algebraU (gl n ) for n odd does not have positive structure constants.
In Section 5, we set n to be odd, and study the behavior of the canonical bases of the Schur algebras S (n, d) and varying d ≫ 0 under the transfer maps. This allows us to construct a canonical basis for the modified quantum coideal algebraU (sl n ). We show that the structure constants of the canonical basis ofU (sl n ) are positive. We further show that the transfer map sends every canonical basis element to a positive sum of canonical basis elements or zero.
In Section 6, we treat S ı (n, d) andU ı (sl n ) for n even, which is more subtle. We show that the main results in Section 5 can be obtained in this case as well though extra technical work is required.
In Section 7, we present explicit formulas of the canonical basis of the rank one iSchur algebra in terms of the standard basis elements. Some interesting combinatorial identities which seem new are obtained along the way. In this section, we construct several examples which show that a natural shift map onU (gl n ) does not preserve the BLM basis, that the structure constants of BLM basis forU (gl n ) are negative, and that the BLM basis oḟ U (gl n ) does not descend to the canonical basis of a finite-dimensional simple modules.
The BLM preliminaries
We recall some basics from [1] (also see [6]). Let v be a formal parameter, [1]) be the quantum Schur algebra over A, which specializes at v = √ q to the convolution algebra of pairs of n-step partial flags in F d q . The algebra A S(n, d) admits a bar involution, a standard basis [A], and a canonical (= IC) basis {A} parameterized by
and A = Z[v, v −1 ]. Let F q be a finite field of order q. Let N = {0, 1, 2, . . .}. Let A S(n, d) (denoted by K d inΘ d = A = (a ij ) ∈ Mat n×n (N)| |A| = d , where |A| = 1≤i,j≤n a ij . Set Θ := ∪ d≥0 Θ d .
The multiplication formulas of the A-algebras A S(n, d) exhibit some remarkable stability as d varies, which leads to a "limit" A-algebra K. The bar involution on A S(n, d) induces a bar involution on K. The algebra K has a standard basis [A] and a BLM (or stably canonical) basis {A}, parameterized bỹ
Θ = {A = (a ij ) ∈ Mat n×n (Z) | a ij ≥ 0 (i = j)}.
Denote by ǫ i the i-th standard basis element in Z n . For 1 ≤ h ≤ n − 1, a ≥ 1 and λ ∈ Z n , we denote by E (a) h,h+1 (λ) the matrix whose (h, h + 1)th entry is a, whose diagonal coincides with λ − aǫ h+1 , and all other entries are zero. Similarly, denote by E (a) h+1,h (λ) the matrix whose (h + 1, h)th entry is a, whose diagonal coincides with λ − aǫ h , and all other entries are zero.
Recall the A-form of the modified quantum gl n , denoted by AU (gl n ), is generated by the idempotents 1 λ (for λ ∈ Z n ) and the divided powers E (a) h 1 λ , for all admissible λ, h and a. We shall always make such an identification K ≡ AU (gl n ) and use only AU (gl n ) in the remainder of the paper.
i 1 λ , F(a
We denote
S(n, d) = Q(v) ⊗ A A S(n, d),U (gl n ) = Q(v) ⊗ A AU (gl n ).
The algebraU (gl n ) is a direct sum of subalgebras:
U (gl n ) = d∈ZU (gl n ) d ,(2.1)
whereU (gl n ) d is spanned by elements of the form 1 λ u1 µ with |µ| = |λ| = d and u ∈U (gl n ); here as usual we denote |λ| = λ 1 + . . . + λ n , for λ = (λ 1 , . . . , λ n ) ∈ Z n .
The elements [E (a) h,h+1 (λ)] for E (a) h,h+1 (λ) ∈ Θ d and [E (a) h+1,h (λ)] for E (a) h+1,h (λ) ∈ Θ d (for all admissible h, a, λ) generate the A-algebra A S(n, d).
Let 0 i,j be the i × j zero matrix. Fix two positive integers m, n such that m < n. Let k ∈ Z. By using the multiplication formulas in [1, 4.6], we note that the assignment The following lemma, which basically follows from the definition of the BLM basis, will be used later on.
Incompatibility of BLM bases under the shift map
Given p ∈ Z, it follows from the multiplication formulas [1, 4.6] that there exists an algebra isomorphism (called a shift map)
ξ p :U (gl n ) −→U (gl n ), ξ p ([A]) = [A + pI],(2.2)
for all A such that A is either diagonal, E h,h+1 (λ) or E h+1,h (λ) for some 1 ≤ h ≤ n − 1 and I denotes the identity matrix. Note that ξ p commutes with the bar involution and ξ p preserves the A-form AU (gl n ). Note also that
ξ −1 p = ξ −p .
Introduce the (not bar-invariant) quantum integers and quantum binomials, for m ∈ Z and b ∈ N,
m b = m b v = 1≤i≤b v 2(m−i+1) − 1 v 2i − 1 , and [m] = m 1 = v 2m − 1 v 2 − 1 . (2.3) Lemma 2.2. Let n = 2. If a 21 ≥ 1, a 22 ≤ −2 and p ≤ 0, then p 1 a 21 a 22 + p = p 1 a 21 a 22 + p − v a 22 +1 [p + 1] p + 1 0 a 21 − 1 a 22 + p + 1 .
Proof. We denote the multiplication inU (gl 2 ) by * to avoid confusion with the usual matrix multiplication. We will repeatedly use the fact that [A] is bar-invariant (divided powers) for A upper-or lower-triangular. The formula in Lemma 2.2 specializes at p = 0 to be
0 1 a 21 a 22 = 0 1 a 21 a 22 − v a 22 +1 1 0 a 21 − 1 a 22 + 1 .
Hence, using (2.6) we have
ξ p 0 1 a 21 a 22 = p 1 a 21 a 22 +p +(v −a 22 −3 [p]−v a 22 +1 ) p + 1 0 a 21 − 1 a 22 +p+1 ,(2.) d − pn with p ∈ Z, we let Φ ′ d (u) = Φ d (ξ p (u)); also let Φ ′ d |U (gl n ) d ′ = 0 unless d ′ ≡ d
Negativity of BLM structure constants
Proposition 2.6. The structure constants for the algebraU (gl n ) with respect to the BLM basis are not always positive, for n ≥ 2. More explicitly, for n = 2, we have
0 1 1 −3 * 0 1 1 −3 = (v + v −1 ) 2 −1 2 2 −4 − (2v −2 + 1 + 2v 2 ) 0 1 1 −3 − (v −4 + v −2 + 2 + v 2 + v 4 ) 1 0 0 −2 .
Proof. It suffices to check the example for n = 2 in view of Lemma 2.1. We will repeatedly use the fact that [A] is bar-invariant (divided powers) for A upper-or lower-triangular.
We claim the following identities hold:
0 1 1 −3 = 0 1 1 −3 − v −2 1 0 0 −2 , (2.8) −1 2 2 −4 = −1 2 2 −4 , 1 0 0 −2 = 1 0 0 −2 . (2.9)
Indeed, (2.8) follows by Lemma 2.2, and the second identity of (2.9) is clear. Moreover, by [1, 4.6(b)] and (2.8), we have
−1 2 2 −4 = 1 0 2 −4 * 1 2 0 −4 + (v −2 + 1 + v 2 ) 0 1 1 −3 −(v −4 + v −2 + 1) 1 0 0 −2 = 1 0 2 −4 * 1 2 0 −4 + (v −2 + 1 + v 2 ) 0 1 1 −3 ,
which is bar invariant. Hence it must be a BLM basis element, whence (2.9).
By [1, 4.6(a),(b)] (also see (2.4)), we have
0 1 1 −3 = 0 1 0 −2 * 0 0 1 −2 − v 2 1 0 0 −2 ,(2.0 0 1 −2 * 0 1 1 −3 = (v + v −1 ) −1 1 2 −3 − (1 + v 2 ) 0 0 1 −2 , (2.11) 0 1 0 −2 * −1 1 2 −3 = (v + v −1 ) −1 2 2 −4 , (2.12) 0 1 0 −2 * 0 0 1 −2 = 0 1 1 −3 + v 2 1 0 0 −2 . (2.13)
Therefore we have
0 1 1 −3 * 0 1 1 −3 = 0 1 0 −2 * 0 0 1 −2 * 0 1 1 −3 − v −2 0 1 0 −2 * 0 0 1 −2 − (v 2 + v −2 ) 0 1 1 −3 + v −2 (v 2 + v −2 ) 1 0 0 −2 = (v+v −1 ) 2 −1 2 2 −4 − (2v −2 + 1 + 2v 2 ) 0 1 1 −3 +(v −4 −v 2 −v 4 ) 1 0 0 −2 ,
(2.14)
where the first identity above uses (2.8) and (2.10), while the second identity above uses (2.11), (2.12) and (2.13).
With the help of (2.8) and (2.9), a direct computation shows the righthand side of the desired identity in the proposition is also equal to (2.14).
The proposition is proved.
Incompatibility of BLM bases forU and L(λ)
Denote by L(λ) theU (gl n )-module of highest weight λ with a highest weight vector u + λ .
Proposition 2.7. There exists a dominant integral weight λ and some BLM basis element C ∈U (gl n ) (for n ≥ 2) such that Cu + λ is not a canonical basis element of L(λ). More explicitly, for n = 2, if a 21 ≥ 1, a 22 ≤ −2 and p ≤ 0,
YIQIANG LI AND WEIQIANG WANG
[June λ = (p + a 21 , a 22 + p + 1), then
p 1 a 21 a 22 + p u + λ = v a 22 +2p+3 [−a 22 − 2p − 3]F (a 21 −1) u + λ .
Proof. It suffices to verify such an example for n = 2 by using Lemma 2.1 where k is chosen such that k ≤ a 22 + p + 1.
By [1, 4.6], we have
p + 1 0 a 21 a 22 + p * p + a 21 1 0 a 22 + p = p 1 a 21 a 22 + p + v a 22 −1 [a 22 + p + 1] p + 1 0 a 21 − 1 a 22 + p + 1 .
By plugging the above equation into the formula in Lemma 2.2 (the assumption of which is satisfied), we obtain that
p 1 a 21 a 22 + p = p + 1 0 a 21 a 22 + p * p + a 21 1 0 a 22 + p + v a 22 +2p+3 [−a 22 − 2p − 3] p + 1 0 a 21 − 1 a 22 + p + 1 ,
where we have used the identity
−v a 22 +1 [p + 1] − v a 22 −1 [a 22 + p + 1] = v a 22 +2p+3 [−a 22 − 2p − 3]
(note this is a bar-invariant quantum integer).
Consider the dominant integral weight λ = (p + a 21 , a 22 + p + 1). We have to the canonical basis of L(λ) when the dominant highest weight λ is assumed to be in Z n ≥0 .
p 1 a 21 a 22 + p u + λ = v a 22 +2p+3 [−a 22 − 2p − 3] p + 1 0 a 21 − 1 a 22 + p + 1 u + λ = v a 22 +2p+3 [−a 22 − 2p − 3]F (a 21 −1) u + λ , which is not a canonical basis element in L(λ) if −a 22 − 2p − 3 > 1.
3. Positivity of Canonical Basis ofU (sl n ) and a Basis ofU (gl n )
In this section we exhibit various kinds of positivity of the canonical basis ofU (sl n ) and Schur algebras in relation to the transfer maps, most of which were known by experts though probably in some other ways. We also construct a positive basis forU (gl n ) by transporting the canonical basis oḟ
U (sl n ) toU (gl n ). 3.1. The algebrasU (gl n ) vsU (sl n )
We identify the weight lattice for gl n as Z n (regarded as the set of integral diagonal n × n matrices inΘ if we think in the setting of K), and we define an equivalence ∼ on Z n by letting µ ∼ ν if and only if µ − ν = k(1, . . . , 1) for some k ∈ Z. Denote by µ the equivalence class of µ ∈ Z n , and we identify the set of these equivalence classesZ n as the weight lattice of sl n . We denote by |µ| ∈ Z/nZ the congruence class of |µ| modulo n. For later use we also extend this definition to define an equivalence relation ∼ onΘ:
A ∼ A ′ if and only if A − A ′ = kI for some k ∈ Z. We set Θ n =Θ/ ∼ . (3.1)
As a variant ofU (gl n ), the modified quantum groupU (sl n ) admits a family of idempotents 1 µ , for µ ∈Z n . The algebraU (sl n ) is naturally a direct sum of n subalgebras:
U (sl n ) = d ∈Z/nZU (sl n ) d , (3.2) whereU (sl n ) d is spanned by 1 µU (sl n )1 λ , where |µ| ≡ |λ| ≡ d mod n. It follows that AU (sl n ) = ⊕d ∈Z/nZ AU (sl n ) d . We denote by πd :U (sl n ) → U (sl n ) d the projection to thed-th summand.
There exists a natural algebra isomorphism
℘ d :U (gl n ) d ∼ =U (sl n ) d (∀d ∈ Z),(3.
3)
156 YIQIANG LI AND WEIQIANG WANG [June which sends 1 λ , E i 1 λ and F i 1 λ to 1λ, E i 1 λ and F i 1 λ respectively, for all r, i, and all λ with |λ| = d. This induces an isomorphism ℘ λ :U (gl n )1 λ ∼ = U (sl n )1 λ , for each λ ∈ Z n , and also an isomorphism µ ℘ λ : 1 µU (gl n )1 λ ∼ = 1 µU (sl n )1 λ , for all λ, µ ∈ Z n with |λ| = |µ|. (These isomorphisms further induce similar isomorphisms for the corresponding A-forms, which match the divided powers.) Combining ℘ d for all d ∈ Z gives us a homomorphism ℘ :U (gl n ) →U (sl n ). It follows by definitions that
℘ • ξ p = ℘, for all p ∈ Z. (3.4)
Recall from Remark 2.5 the surjective algebra homomorphism Φ d :
U (gl n ) → S(n, d). The algebra homomorphism φ d :U (sl n ) → S(n, d) is defined as the composition φ d :U (sl n ) πd −→U (sl n ) d ℘ d −→U (gl n ) d Φ d −→ S(n, d). (3.5)
It follows that φ d |U (sl n ) d′ = 0 ifd ′ =d, and we have a surjective homo-
morphism φ d :U (sl n ) d → S(n, d).
Clearly φ d preserves the A-forms.
Positivity of canonical basis forU (sl n )
The canonical basis of AU (sl n ) (and hence ofU (sl n )) is defined by Lusztig [15], and it is further studied from a geometric viewpoint by McGerty [18]. The following positivity for canonical basis could (and probably should)
have been formulated explicitly in [18], as there is no difficulty to establish it therein. Given an n × n matrix A, we shall denote
p A = A + pI,
where I is the identity matrix.
Proposition 3.1. The structure constants of the canonical basis for the
algebraU (sl n ) lie in N[v, v −1 ], for n ≥ 2.
Proof. LetḂ(sl n ) = ∪d ∈Z/nZḂ (sl n ) d be the canonical basis forU (sl n ), whereḂ(sl n ) d is a canonical basis forU (sl n ) d .
a * b = z∈Ω P z a,b z. (3.6)
It is shown [18] that there exists a positive integer d in the congruence classd and A, B,
C z ∈ Θ d such that φ d+pn (a) = { p A}, φ d+pn (b) = { p B}, φ d+pn (z) = { p C z }, for all p ≫ 0. Hence applying φ d+pn to (3.6) we have { p A} * { p B} = z∈Ω P z a,b { p C z }.
The structure constants for the canonical basis of the Schur algebra S(n, d + pn) are well known to be in N[v, v −1 ] thanks to the intersection cohomology construction [1], and hence P z
a,b ∈ N[v, v −1 ]. Since the algebraU (sl n ) is a direct sum of the algebrasU (sl n ) d for d ∈ Z/nZ, the proposition is proved. Remark 3.2.
The positivity as in Proposition 3.1 was conjectured by Lusztig [15] for modified quantum group of symmetric type. There is a completely different proof of such a positivity in ADE type via categorification technique by Webster [22]. The argument here also shows the positivity of the canonical basis of modified quantum affine sl n , based again on McGerty's work.
Transfer map and positivity
The transfer map for the v-Schur algebras
φ d+n,d : A S(n, d + n) −→ A S(n, d),
or φ d+n,d : S(n, d + n) → S(n, d) by a base change, was defined geometrically by Lusztig [17] and can also be described algebraically as follows. Set . . . , a n ).) Then S(n, d) is generated by these elements (see [1]), and the transfer map φ d+n,d is characterized by
E i;d = λ [E i,i+1 (λ)] summed over all E i,i+1 (λ) ∈ Θ d , F i;d = λ [E i+1,i (λ)] summed over all E i+1,i (λ) ∈ Θ d , and K a;d = b∈N n ,|b|=d v a·b 1 b . (Here a · b = i a i b i for a = (a 1 ,φ d+n,d (E i;d+n ) = E i;d , φ d+n,d (F i;d+n ) = F i;d , φ d+n,d (K a;d+n ) = v |a| K a;d .
YIQIANG LI AND WEIQIANG WANG [June
Recall the homomorphism φ d :U (sl n ) → S(n, d) from (3.5). We have the following commutative diagram by matching the Chevalley generators (see [16,17]): So it suffices to show the positivity of the homomorphism χ : S(n, n) −→
(3.7)Q(v). Recall that the function χ is defined by χ([A]) = v −d A det(A) where d A = i≥k,j<l a ij a kl . (Note that χ([A]) = 0 unless A is a permutation matrix.) We claim that χ({A}) = 1, if A = I, 0, if A = I (3.8)
(recall I is the identity n × n matrix). It suffices to show that the claim holds for all permutation matrices (which form the symmetric group S n ), and we prove this by induction on the length ℓ(w) for w ∈ S n . Recall [1] that the canonical basis {w} for w ∈ S n is simply the Kazhdan-Lusztig with ℓ(w) > 1. We can find an s i such that w = s i w ′ with ℓ(w ′ ) + 1 = ℓ(w
{s i } * {w ′ } = {w} + x:ℓ(x)<ℓ(w ′ ),ℓ(s i x)<ℓ(x) µ(x, w ′ ){x}, µ(x, w ′ ) ∈ A.
(Note the x in the summation satisfies x = I.) Now applying the algebra homomorphism χ to the above identity and using the induction hypothesis, we see that χ({w}) = 0. This finishes the proof of the claim and hence of the theorem. Proof. Let b ∈Ḃ(sl n ). We can assume that b ∈Ḃ(sl n ) d as otherwise we have φ d (b) = 0. By [18, Corollary 7.6, Proposition 7.8], φ d+pn (b) is a canonical basis element in S(n, d + pn), for some p ≫ 0. Using the commutative diagram (3.7) repeatedly, we have
φ d (b) = φ d+n,d φ d+2n,d+n · · · φ d+pn,d+pn−n φ d+pn (b) .
It follows by repeatedly applying Proposition 3.3 that the term on the righthand side above is a sum of canonical basis elements in S(n, d) with coeffi-
cients in N[v, v −1 ].
Remark 3.5. Proposition 3.3 is partly inspired by [18,Remark 7.10], and probably it can also be proved by a possible functor realization of the transfer map, whose existence was hinted at loc. cit. Note that stronger versions of Propositions 3.3 and 3.4 hold (which state that the canonical bases are preserved by φ d+n,d and φ d ), according to the main results of [20] (which proved Lusztig's conjectures [16]). Our short yet transparent proofs of the weaker statements above might be of interest to the reader, and they will be adapted in later sections to the modified quantum coideal algebras and their associated Schur algebras.
Recall [11] that the Schur-Jimbo (S(n, d), H S d )-duality on V ⊗d can be realized geometrically, where V is n-dimensional and H S d is the Iwahori-Hecke algebra associated to the symmetric group S d . Denote by B(n d ) the canonical basis of V ⊗d . The canonical bases on V ⊗d as well as on S(n, d)
YIQIANG LI AND WEIQIANG WANG
[June are realized as simple perverse sheaves, and the action of S(n, d) on V ⊗d is realized in terms of a convolution product. Hence we have the following positivity.
Proposition 3.6. [11] The action of S(n, d) on V ⊗d with respect to the corresponding canonical bases is positive in the following sense: for any canonical basis element a of S(n, d) and any b ∈ B(n d ), we have
a * b = b ′ ∈B(n d ) C b ′ a,b b ′ , where C b ′ a,b ∈ N[v, v −1 ].
We shall take the liberty of saying some action is positive in different contexts similar to the above proposition. Now thatU (sl n ) acts on V ⊗d naturally by composing the action of S(n, d) on V ⊗d with the map φ d : U (sl n ) → S(n, d). We have the following corollary of Propositions 3.4 and 3.6.
Corollary 3.7. The action ofU (sl n ) on V ⊗d with respect to the corresponding canonical bases is positive.
Note by [15, 27.1.7] that the d-th symmetric power S d V (i.e., the simple module of highest weight being d times the first fundamental weight) is a based submodule of V ⊗d in the sense of [15,Chap. 27], and hence S d 1 V ⊗ · · · ⊗ S ds V is also a based submodule of V ⊗d , where the positive integers d i satisfy d 1 + . . . + d s = d. The following is now a consequence (and also a generalization) of Corollary 3.7.
Corollary 3.8. The action ofU (sl n ) on S d 1 V ⊗ · · · ⊗ S ds V with respect to the corresponding canonical bases is positive.
A positive basis forU (gl n )
Note that the BLM basis ofU (gl n ) restricts to a basis ofU (gl n ) d , which does not have positive structure constants in general by Proposition 2.6. However, in light of the positivity in Proposition 3.1, one can transport the canonical basis onU (sl n ) d toU (gl n ) d via the isomorphism ℘ d in (3.3), which has positive structure constants. Let us denote the resulting positive basis (or canonical basis) onU (gl n ) = ⊕ d∈ZU (gl n ) d by B pos (gl n ). By definition, the basis B pos (gl n ) is invariant under the shift maps ξ p for p ∈ Z. Summarizing we have the following. Proposition 3.9. There exists a positive basis B pos (gl n ) for AU (gl n ) (and also forU (gl n )), which is induced from the canonical basis for AU (sl n ).
Recall a 2-categoryU (gl n ) which categorifiesU (gl n ) in [19] is obtained by simply relabeling the objects for the Khovanov-Lauda 2-category which categorifiesU (sl n ) in [13]. We expect that the projective indecomposable 1-morphisms inU (gl n ) categorify the positive basis B pos (gl n ) (instead of the BLM basis which has no positivity).
Modified Quantum Coideal AlgebrasU
(gl n ) andU (sl n ),
for n Odd
In this section and next section, we fix 2 odd positive integers n, D such that n = 2r + 1, D = 2d + 1.
We will almost exclusively use the notation n and d (instead of r and D).
We study the canonical bases for the modified quantum coideal algebraṡ U (gl n ) andU (sl n ) as well as the Schur algebras S (n, d). We will again use the notation {A}, [A], {A} d etc for the bases of these algebras, as these sections are independent from the earlier ones to a large extent. When we occasionally need to refer to similar bases in type A from earlier sections, we shall add a superscript a.
In this section, we show that the stably canonical basis constructed in [2] for the modified quantum coideal algebraU (gl n ) does not have positive structure constants. We also formulate some basic connections betweeṅ U (gl n ) andU (sl n ).
Schur algebras and quantum coideal algebra
We first recall some basics from [2].
Let F q be a finite field of odd order q. Let A S (n, d) (denoted by S in [2]) be the Schur algebra over A, which specializes at v = √ q to the convolution algebra of pairs of n-step partial isotropic flags in F 2d+1 q (with respect to some fixed non-degenerate symmetric bilinear form
Ξ d = A = (a ij ) ∈ Θ 2d+1 a ij = a n+1−i,n+1−j , ∀i, j ∈ [1, n] . (4.1) Set Ξ := ∪ d≥0 Ξ d .
The multiplication formulas of the A-algebras A S (n, d) exhibits some remarkable stability as d varies, which leads to a "limit" A-algebra K .
The bar involution on A S (n, d) induces a bar involution on K [2, §4.1].
The algebra K has a standard basis [A] and a stably canonical basis {A},
parameterized bỹ
Ξ = A =(a ij ) ∈ Mat n×n (Z) | a ij ≥ 0 (i = j), a r+1,r+1 ∈ 2Z + 1, a ij = a n+1−i,n+1−j (∀i, j) . (4.2)
Recall (cf. [3,2] and the references therein) there is a quantum coideal algebra U (gl n ) which can be embedded in U (gl n ), and (U (gl n ), U (gl n )) form a quantum symmetric pair in the sense of Letzter. For our purpose here, its modified versionU (gl n ) is more directly relevant; we recall its presentation below from [2, §4.4] to fix some notation. Let Z n = µ ∈ Z n |µ i = µ n+1−i (∀i) and µ (n+1)/2 is odd .
Let E θ ij be the n × n matrix whose (k, l)-entry is equal to δ k,i δ l,j + δ k,n+1−i δ l,n+1−j . Given λ ∈ Z n , we introduce the short-hand notation λ ± α i whose jth entry is equal to λ j ∓ (δ i,j + δ n+1−i,j ) ± (δ i+1,j + δ n−i,j ). Recall n = 2r + 1. The algebraU (gl n ) is the Q(v)-algebra generated by 1 λ , e i 1 λ , 1 λ e i , f i 1 λ and 1 λ f i , for i = 1, . . . , r and λ ∈ Z n , subject to the following relations, for i, j = 1, . . . , (n − 1)/2 and λ, λ ′ ∈ Z n :
2018]
POSITIVITY VS NEGATIVITY OF CANONICAL BASES 163
x1 λ 1 λ ′ x ′ = δ λ,λ ′ x1 λ x ′ , for x, x ′ ∈ {1, e i , e j , f i , f j }, e i 1 λ = 1 λ−α i e i , f i 1 λ = 1 λ+α i f i , e i 1 λ f j = f j 1 λ−α i −α j e i , if i = j, e i 1 λ f i = f i 1 λ−2α i e i + v λ i+1 −λ i −v λ i −λ i+1 v−v −1 1 λ−α i , if i = n−1 2 , (e 2 i e j + e j e 2 i )1 λ = (v + v −1 )e i e j e i 1 λ , if |i − j| = 1, (f 2 i f j + f j f 2 i )1 λ = (v + v −1 )f i f j f i 1 λ , if |i − j| = 1, e i e j 1 λ = e j e i 1 λ , if |i − j| > 1, f i f j 1 λ = f j f i 1 λ , if |i − j| > 1, (f 2 r e r −(v+v −1 )f r e r f r +e r f 2 r )1 λ = −(v+v −1 ) v λ r+1 −λr−2 +v λr−λ r+1 +2 f r 1 λ , (e 2 r f r −(v+v −1 )e r f r e r +f r e 2 r )1 λ = −(v+v −1 ) v λ r+1 −λr+1 +v λr−λ r+1 −1 e r 1 λ .
It was shown in [2, §4.5] that there is an A-algebra isomorphism K ∼ = AU (gl n ), which matches the Chevalley generators. we shall always make such an identification K ≡ AU (gl n ) and use only AU (gl n ) in the remainder of the paper.
Given m ∈ Z with 0 ≤ 2m ≤ n, let J m be an m × m matrix whose (i, j)-th entry is δ i,m+1−j . Recalling the definition ofΘ depends on n from Section 2.1, we shall writeΘ n forΘ in this paragraph and allow n vary, and so in particularΘ m makes sense. To a matrix A ∈Θ m and k ∈ Z, we define a matrix
τ k m,n (A) = A 0 0 0 2kI + ε 0 0 0 J m AJ m
where ε is the (n − 2m) × (n − 2m) matrix whose only nonzero entry is the very central one, which equals 1. Thus, we have an embedding (We recall here our convention of using the superscript a to denote the corresponding basis in the type A setting from earlier sections.) Note that the homomorphism τ k m,n commutes with the bar involutions on AU (gl m ) and AU (gl n ). The following lemma is immediate from the definitions. We denote
S (n, d) = Q(v) ⊗ A A S (n, d),U (gl n ) = Q(v) ⊗ A AU (gl n ).
The quantum coideal algebra U (sl n ) can be embedded into (and hence identified with a subalgebra of) U (sl n ); cf. [3]. We define an equivalence
relation ∼ on Z n : µ ∼ µ ′ if µ − µ ′ = m n i=1 ǫ i for some m ∈ 2Z. Letμ denote the equivalence class of µ. Put ∧ Z n = Z n / ∼ .
We define the Q(v)-algebraU (sl n ) formally in the same way asU (gl n ) above except now that the weights λ, λ ′ run over ∧ Z n (instead of Z n ). There exists a bar involution onU (sl n ) (as well as onU (gl n )) which fixes all the generators. The A-form AU (sl n ) of the Q(v)-algebraU (sl n ) (as well as the A-form AU (gl n ) ofU (gl n )) is generated by the divided powers
2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 165
The following example arises from discussions with Huanchen Bao.
Proposition 4.2. The structure constants for the stably canonical basis oḟ U (gl n ) are not always positive, for n ≥ 3. More explicitly, for n = 3 and for a, b ∈ Z with a < b ≤ −2, the following identity holds inU (gl 3 ):
{B} * {A} = {C} + (v b+a + v b−a )[b + 1]{D} where [b + 1] ∈ Z ≤0 [v, v −1 ].
Proof. It suffices to check the identity for n = 3, since the general case for n ≥ 4 follows easily from Lemmas 4.1 and 2.6. By using [2, (4.7)] we compute that Also note that v b [b + 1] is a bar-invariant quantum integer. Applying the bar involution to (4.5) and comparing with (4.5) again, we have
[B] * [A] = [C] + v −a v b [b + 1][D].[C] − [C] = (v −a − v a )v b [b + 1][D]. (4.6) By assumption that a < b ≤ −2, we have v a+b [b + 1] ∈ v −1 Z <0 [v −1 ]
, and hence from (4.6) we obtain that
{C} = [C] − v a+b [b + 1][D].
Now the equation (4.5) can be rewritten as
{B} * {A} = {C} + (v a + v −a )v b [b + 1][D]. It is clear that v b [b + 1] = −(v −b + v −b−2 + . . . + v b+2 + v b ) ∈ Z ≤0 [v, v −1 ] for b ≤ −2.
This finishes the proof for n = 3.
RelatingU
(gl n ) toU (sl n )
This subsection, in which we are making a transition fromU (gl n ) tȯ U (sl n ), is a preparation for the next section.
Recall that there is a Schur (S(n, d), H S d )-duality on V ⊗d , where V is an n-dimensional vector space over Q(v). It is shown [10,3] (see also [2]) that there is a Schur-type (S (n, d), H B d )-duality on V ⊗d where H B d is the Iwahori-Hecke algebra associated to the hyperoctahedral group B d . In particular we have algebra homomorphisms
S(n, d) ∼ = −→ End H S d (V ⊗d ), S (n, d) ∼ = −→ End H B d (V ⊗d ).
Recall the sign homomorphism
χ n : S(n, n) −→ Q(v) (4.7)
from the proof of Proposition 3.3 (cf. [17, 1.8]). We have a natural inclusion of algebras
H B d × H Sn ⊆ H B d+n . The transfer map φ d+n,d : S (n, d + n) −→ S (n, d)
is defined as the composition of the homomorphisms
S (n, d + n) ∼ = −→ End H B d+n (V ⊗(d+n) ) ∆ −→ End H B d ×H Sn (V ⊗(d+n) ) ∼ = −→ End H B d (V ⊗d ) ⊗ End H Sn (V ⊗n ) 1⊗χn −→ End H B d (V ⊗d ) ∼ = −→ S (n, d).
(4.8)
This transfer map will be studied in depth from a geometric viewpoint in [8], where the proof of the following lemma can be found.
Lemma 4.3. We have φ d+n,d ([A] d+n ) = [A − 2I] d , if A − 2I ∈ Ξ d , 0, otherwise.
for all A ∈ Ξ d+n such that one of the following matrices is diagonal: A, A − aE θ i+1,i or A − aE θ i,i+1 for some a ∈ N and 1 ≤ i ≤ (n − 1)/2.
Similar to the decomposition (2.1) forU (gl n ), we can decomposeU (gl n ) as a direct sum of subalgebraṡ
U (gl n ) = d∈ZU (gl n ) d ,
2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 167
whereU (gl n ) d is spanned by elements of the form 1 λ u1 µ with |µ| = |λ| = 2d + 1 and u ∈U (gl n ). Also similar to the decomposition (3.2) forU (sl n ), we can decomposeU (sl n ) as a direct sum of n subalgebraṡ
U (sl n ) = d ∈Z/nZU (sl n ) d , whereU (sl n ) d is spanned by 1 µU (sl n )1 λ , where |µ| ≡ |λ| ≡ 2d+1 mod 2n.
Denote by πd :U (sl n ) →U (sl n ) d the natural projection. There exists a natural algebra isomorphism similar to (3.3)
℘ d, :U (gl n ) d ∼ =U (sl n ) d (∀d ∈ Z), (4.9)
which induces a homomorphism ℘ :U (gl n ) →U (sl n ). In the same way as forU (gl n ) defined in (2.2), for each p ∈ 2Z we define a shift map
ξ p :U (gl n ) −→U (gl n ), ξ p ([A]) = [A + pI],(4.φ d :U (sl n ) −→ S (n, d)
to be the compositioṅ
U (sl n ) πd −→U (sl n ) d ℘ −1 d, −→U (gl n ) d Φ d −→ S (n, d). (4.12)
We introduce another homomorphism
Ψ d :U (gl n ) −→ S (n, d) 168
YIQIANG LI AND WEIQIANG WANG [June to be the composition of the following homomorphismṡ
U (gl n ) ℘ −→U (sl n ) φ d −→ S (n, d). Note that Ψ d = Φ d , but Ψ d coincides with Φ d when restricted toU (gl n ) d .
Canonical Basis for Modified Quantum Coideal AlgebraU
(sl n ),
for n Odd
In this section we continue (as in Section 4) to let n = 2r + 1 and D = 2d + 1 be odd positive integers.
We establish some asymptotical behavior for the canonical bases of Schur algebras under the transfer map. This is used to define the canonical basis forU (sl n ) and to show that structure constants of the canonical basis ofU (sl n ) are positive. We further show that the transfer map on the Schur algebras sends every canonical basis element to a positive sum of canonical basis elements or zero, and provide some corollaries.
Asymptotic identification of canonical bases for S (n, d)
Recall a bilinear form ·, · d on S(n, d) is defined in [2, §3.7] (and denoted by (·, ·) D therein with D = 2d + 1). The same argument as for [18,2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 169
Proposition 4.3] shows that
x, y := lim
p→∞ n−1 d=0 φ d+pn (x), φ d+pn (y) d+pn , for x, y ∈U (sl n ), (5.1)
exists as an element in Q(v). Thus we have constructed a bilinear form −, − onU (sl n ).
Recall there is a partial order onΞ [2, (3.22)] by declaring A B if and only if r≤i;s≥j a rs ≤ r≤i;s≥j b rs for all i < j. For an n × n matrix A = (a ij ), let
ro(A) = j a 1j , j a 2j , . . . , j a nj , co(A) = i a i1 , i a i2 , . . . , i a in .
There is a partial order ⊑ onΞ [2, (3.24)], which refines , so that A ′ ⊑ A if and only if A ′ A, ro(A ′ ) = ro(A) and co(A ′ ) = co(A). The following lemma is preparatory.
Lemma 5.1. Fix A = (a ij ) ∈Ξ.
Suppose that p is an even integer such that a ll + p ≥ i =j a ij for all 1 ≤ l ≤ n. If B ∈Ξ satisfies B ⊑ p A, then B ∈ Ξ |pA| , i.e., b ii ≥ 0 for all 1 ≤ i ≤ n.
Proof. We prove by contradiction. Suppose that b i 0 ,i 0 < 0 for some i 0 . We have
j =i 0 b i 0 j > ro(B) i 0 = ro( p A) i 0 ≥ a i 0 i 0 + p ≥ i =j a ij .
This implies that
r≤i 0 ,s≥i 0 +1 b rs + r≥i 0 ,s≤i 0 −1 b rs ≥ j =i 0 b i 0 j > i =j a ij ≥ r≤i 0 ,s≥i 0 +1 a rs + r≥i 0 ,s≤i 0 −1 a rs ,
which contradicts with the condition B ⊑ p A.
Proposition 5.2. Given A ∈Ξ with |A| = 2d 0 + 1, we have, for even integers p ≫ 0,
φ d,d−n ({ p A} d ) = { (p−2) A} d−n ,
YIQIANG LI AND WEIQIANG WANG [June
where we denote d = d 0 + pn/2 so that | p A| = 2d + 1.
Proof. The proof is essentially adapted from that of [18,Proposition 7.8] with minor modifications. Let us go over it for the sake of completeness. [2, (3.25)], (which is denoted by m A therein). By Lemma 4.3 we have
Recall the monomial basis
{ d M A |A ∈ Ξ d } of S (n, d) fromφ d,d−n ( d M A ) = d−n M A−2I , ∀d. (It is understood that d−n M A−2I = 0 if A − 2I ∈ Ξ d−n .)
The proposition is equivalent to the following.
Claim (⋆). Let A ∈Ξ. For all even integer p ≫ 0, we have
{ p A} d = d M pA + A ′ ≺A c A ′ ,A,p d M p A ′ , where c A ′ ,A,p ∈ A is independent of p ≫ 0. Recall [2] that the basis { d M pA } satisfies d M pA = d M pA , d M pA ∈ A S (n, d), and d M p A = { p A} d + B≺A w p A, pB { p B} d , for some w p A,pB ∈ A. (5.2)
We shall argue similarly as for a claim in the proof of [18, Proposition 7.8], with D b A used in loc. cit. replaced by d M p A ; that is, we shall prove Claim (⋆) by induction on A with respect to the partial order . When A is minimal, it follows by (5.2) that d M pA = { p A} d for all p, and hence Claim (⋆) holds. Now assume that Claim (⋆) holds for all B such that B ≺ A. Set • I d is a finite set, and it is independent of p ≫ 0 (recall d = d 0 + pn/2 depends on p).
I d = B ∈Ξ B A, p B ∈ Ξ d ,For u ∈ A = Z[v, v −1 ], let deg(u) be its degree. For x ∈ Span A {{ p B} d |B ∈ I d }, we set n(x) = max deg x, { p B} d d B ∈ I d , B = A , and n p = n( d M pA ).
Suppose that n p ≥ 0. We set
J d = B ∈ I d deg d M pA , { p B} d d = n p .
Then we can write, for each B ∈ I d ,
d M p A , { p B} d d = i≤np c B,p,i v i ∈ Z[v, v −1 ], where c B,p,i ∈ Z (∀i), and c B,p,np = 0, if B ∈ J d , = 0, if B ∈ I d \J d . (5.3)
We define a new bar-invariant element in A S (n, d):
d M ′ pA = d M pA − B∈J d c B,p,np (v np + v −np ){ p B} d , if n p > 0, d M p A − B∈J d c B,p,np { p B} d , if n p = 0.
We now show that n( d M ′ pA ) < n p = n( d M p A ). We give the details for n p > 0, while the case for n p = 0 is entirely similar. By the almost orthonormality of the canonical basis of S (n, d)
[2], we have { p B} d , { p B ′ } d d ∈ δ B,B ′ + v −1 Z[v −1 ]. For B ∈ I d , we have by (5.3) that d M ′ pA , { p B} d d = d M p A , { p B} d d − B ′ ∈J d c B ′ ,p,np (v np + v −np ) { p B} d , { p B ′ } d d ≡ i≤np−1 c B,p,i v i − B =B ′ ∈J d c B ′ ,p,np v np { p B} d , { p B ′ } d d mod v −1 Z[v −1 ],
which implies that n( d M ′ pA ) < n p . By repeating the above procedure with
d M ′ p A in place of d M p A , we produce a bar-invariant element d M ′′ pA in A S (n, d) with degree n( d M ′′ pA ) < n( d M ′ pA )
, and then repeat again and so on. So under the assumption that n p ≥ 0, after finitely many steps we obtain a bar-invariant element in A S (n, d), denoted by b p A , with n(b p A ) < 0.
YIQIANG LI AND WEIQIANG WANG [June
On the other hand, if n p = n( d M pA ) < 0, then we simply set b pA = d M pA .
We now show that b pA = { p A} d . By the above construction and (5.2),
we have b p A = { p A} d + B∈I d f B { p B} d , for some f B ∈ A and f B = f B . If f B = 0 for some B, then n(b pA ) ≥ 0,
which is a contradiction. Hence we have b pA = { p A} d .
In the finite process above of constructing { p A} d (in the form of b pA ) from the monomial basis, we only need the first n p coefficients of d M pA ,
{ p B} d d as well as of { p B ′ } d , { p B} d d for B ∈ I d , B ′ ∈ J d . Recall that the monomial basis {M A |A ∈Ξ} of K from [2, 4.8] satisfies that φ d (M A ) = d M p A if p A ∈ Ξ d .ξ −2 ({ p A}) = { (p−2) A}, ℘ ({ p A}) = ℘ ({ (p−2) A})
for all even integers p ≫ 0, where ξ −2 is defined in (4.10).
Proof. Denote |A| = 2d 0 + 1, and d = d 0 + pn/2. We have the following commutative diagramU
(gl n ) ξ −2 − −−− →U (gl n ) Φ d Φ d−n S (n, d) φ d,d−n − −−− → S (n, d − n) (5.4) i.e., Φ d−n • ξ −2 = φ d,d−n • Φ d . By [2, Appendix A, Theorem A.21], we have Φ d ({ p A}) = { p A} d , Φ d−n ({ (p−2) A}) = { (p−2) A} d−n , ∀p ≫ 0. (5.5)
Moreover, by [2, (4.8)], we have
ξ −2 ({ p A}) = { (p−2) A} + B∈Ξ d−n f B {B}, (for f B ∈ A),(5.6)
where the summation can be taken over B ∈ Ξ d−n is ensured by Lemma 5.1.
Using Proposition 5.2, (5.5), (5.4), and (5.6) one by one, we conclude that
{ (p−2) A} d−n = φ d,d−n • Φ d ({ p A}) = Φ d−n • ξ −2 ({ p A}) = { (p−2) A} d−n + B∈Ξ d−n B⊏ (p−2) A f B {B} d−n .
Hence all f B must be zero, and the first identity in the proposition follows from (5.6). The second identity is immediate from the first one and (4.11). is independent of p and thus a well-defined element inU (sl n ). It follows by definition that ℘ :U (gl n ) →U (sl n ) preserves the A-forms, so we have b A ∈ AU (sl n ).
Canonical basis forU
Proposition 5.4. For A ∈Ξ with |A| = 2d 0 + 1, let d = d 0 + pn/2. Then φ d (b A ) = { p A} d for even integers p ≫ 0.
Proof. We have, for p ≫ 0,
φ d (b A ) = φ d (℘ ({ p A})) = Ψ d ({ p A}) = Φ d ({ p A}) = { p A} d ,
where the first equality follows by definition, the second one is due to (4.13), the third one follows by definition (4.12), and the last one follows from [2, Theorem 6.10]. The proposition is proved.
Theorem 5.5. The setḂ (sl n ) = {b A | A ∈ Ξ} forms a basis ofU (sl n ), and it also forms an A-basis for AU (sl n ).
YIQIANG LI AND WEIQIANG WANG [June
Proof. Observe that ξ p ({A}) = {A + pI} + lower terms. Hence it follows by the surjectivity of ℘ thatḂ (sl n ) is a spanning set for the A-module AU (sl n ). To show thatḂ (sl n ) is linearly independent, it suffices to check thatḂ (sl n ) ∩U (sl n ) d is linearly independent for eachd ∈ Z/nZ. This is then reduced to the Schur algebra level by Proposition 5.4, which is clear.
HenceḂ (sl n ) = {b A | A ∈ Ξ} is an A-basis of AU (sl n )
, and thus it is also a basis ofU (sl n ).
Positivity of the canonical basisḂ
(sl n )
The basisḂ (sl n ) is called the canonical basis (or -canonical basis) oḟ U (sl n ), as we shall show that the canonical basisḂ (sl n ) admits several remarkable properties such as positivity and almost orthonormality just like Lusztig's canonical basis forU (sl n ) (see Proposition 3.1 and [15]).
Given A, B ∈ Ξ, we write b A * b B = C∈ Ξ P C A, B b C , where P C A, B ∈ Z[v, v −1 ]
is zero for all but finitely many C. By Proposition 5.4, we can find some large p (and recall d = d 0 + pn/2) such that p A, p B, p C ∈ Ξ and
φ d (b A ) = { p A} d , φ d (b B ) = { p B} d , φ d (b C ) = { p C} d ,
for all C with C ∈ Ω. So we have the following multiplication of canonical basis in S (n, d):
{ p A} d * { p B} d = C∈Ω P C A, B { p C} d .
Furthermore, we have the following characterization of the signed canonical basis.
Proposition 5.9. The signed canonical basis −Ḃ (sl n ) ∪Ḃ (sl n ) is characterized by the following three properties:
(i) b = b, (ii) b ∈ AU (sl n ), and (iii) b, b ′ ∈ δ b,b ′ + v −1 Z[[v −1 ]].
Proof. It follows by definition and Proposition 5.7 that −Ḃ (sl n ) ∪Ḃ (sl n ) satisfies the three properties above. The characterization claim is then proved in the same way as [15, 14.2.3] for the usual canonical bases. Proof. The strategy of the proof is identical to the one for Proposition 3.3, which is reduced to the positivity of ∆ defined in (4.8) with respect to the canonical bases and the positivity of χ which was already established in (3.8). The proof of the positivity of ∆ is similar to that of ∆ in the proof of Proposition 3.3 (the details are provided in [8] together with other applications in a geometric setting). Proof. This follows by applying (4.13), Proposition 5.4 and Theorem 5.10. The detail is completely analogous to the proof of Proposition 3.4 and hence skipped.
Remark 5.12. Theorem 5.10 provides a strong evidence for a possible functor realization of the transfer map φ d+n,d (cf. [18,Remark 7.10]). In light of [16,20], it is interesting to see if φ d+n,d (and hence φ d ) sends each canonical basis element to a canonical basis element or zero, improving Theorem 5.10 and Proposition 5.11; compare with Remark 3.5.
Recall there is a Schur-type (S (n, d), H B d )-duality on V ⊗d [10,3], where V is n-dimensional, and this duality can be completely realized geometrically [2]. Denote by B (n d ) the -canonical basis of V ⊗d constructed in [3]. These canonical bases on V ⊗d as well as on S (n, d) are realized in [2] as simple perverse sheaves, and the action of S (n, d) on V ⊗d is realized in terms of a convolution product. Hence we have the following positivity.
Proposition 5.13. The action of S (n, d) on V ⊗d with respect to the corresponding -canonical bases is positive in the following sense: for any canonical basis element a of S (n, d) and any b ∈ B (n d ), we have
a * b = b ′ ∈B (n d ) D b ′ a,b b ′ , where D b ′ a,b ∈ N[v, v −1 ].
We obtain a natural action ofU (sl n ) on V ⊗d by composing the action of S (n, d) on V ⊗d with the map φ d :U (sl n ) → S (n, d). As a corollary of Propositions 5.11 and 5.13 we have the following positivity (which is a special case of a conjectural positivity property of the canonical basis for general tensor product modules [3]).
Corollary 5.14. The action ofU (sl n ) on V ⊗d with respect to the corresponding -canonical bases is positive. Given integers k, m with 0 ≤ 2m ≤ n, we recall τ k m,n from (4.3). Fix an m-tuple of integers k = (k 0 , k 1 , . . . , k m−1 ). We define an imbedding τ k d m,n :U (sl m ) d →U (sl n ) d + k d (n − 2m) , for 0 ≤ d < m, to be the composition (or canonical basis) onU (gl n ) = ⊕ d∈ZU (gl n ) d by B pos (gl n ). By definition, the basis B pos (gl n ) is invariant under the shift maps ξ p for p ∈ 2Z. Summarizing we have the following.
Proposition 5.16. There exists a positive basis B pos (gl n ) for AU (gl n ) (and also forU (gl n )), which is induced from the canonical basis for AU (sl n ).
It is clear that the transition matrix between the positive basis and the stably canonical basis of AU (gl n ) is unitriangular.
6. Canonical Basis forU ı (sl n ) for n Even
In this section, we shall construct the canonical basis ofU ı (sl n ) for n even with positivity properties. This is achieved by relating to the case oḟ U (sl n ) for n odd studied in the previous two sections with n = n − 1 ≥ 2 (n even).
6.1. ıSchur algebra S ı (n, d) and the transfer map φ ı d+n,d
Recall A S (n, d) from Section 4.1. We define A S ı (n, d) to be the Asubmodule of A S (n, d) spanned by the standard basis element [A] d , where A runs over the following subset of Ξ d in (4.1).
Ξ ı d = {A ∈ Ξ d |a n 2 +1,j = δ n 2 +1,j , a i, n 2 +1 = δ i, n 2 +1 }. (6.1)
Clearly, this is a subalgebra of A S (n, d) over A. Note that when the parameter v is specialized at √ q, the algebra A S ı (n, d) coincides with the convolution algebra of pairs of n-step partial isotropic flag in F 2d+1 q equipped with a fixed non-degenerate symmetric bilinear form. Moreover, the subset {{A} d |A ∈ Ξ ı d } of the canonical basis of A S (n, d) is an A-basis of A S ı (n, d). Let S ı (n, d) = Q(v) ⊗ A A S ı (n, d).
Recall from (4.8), we have an algebra homomorphism S (n + d, d) → S (n, d) ⊗S(n, n). By restricting to S ı (n, d), we obtain an algebra homomorphism ∆ ı : S ı (n, n + d) → S ı (n, d) ⊗ S(n, n),
YIQIANG LI AND WEIQIANG WANG
[June where we identify S(n, n) with the subalgebra in S(n, n) spanned by the elements [A] whose entries in the ( n 2 + 1)st rows and columns are zero. We refer to [8, Lemma 5.1.1] for a more explicit construction of ∆ ı , which is denoted ∆ ı therein. Recall the sign homomorphism χ n from (4.7), and we define the transfer map φ ı d+n,d : S ı (n, d + n) → S ı (n, d) to be the composition φ ı d+n,d : S ı (n, d + n) We set for all matrices X ∈ Ξ ı d such that either one of the following matrices is diagonal: X, X − E θ n 2 , n 2 +2 , X − aE θ i+1,i or X − aE θ i,i+1 where a ∈ N, 1 ≤ i ≤ n 2 − 1.
I = I − E n+1,
Remark 6.1. As we will show that if X is chosen such that X − aE θ n 2 , n 2 +2
is diagonal for a ≥ 2, the formula (6.2) fails to be true. This makes the construction of canonical basis forU ı (sl n ) more subtle than that ofU (sl n ).
This subtlety boils down to the detailed analysis of the rank-one transfer map, which is the main topic of the following subsection.
6.2. The transfer map on S ı (2, d)
In this subsection, we set n = 2 (hence r = 1) and consider the rank-one transfer map φ ı d,d−2 : S ı (2, d) −→ S ı (2, d − 2). For convenience, we set
A a,b = a 0 b 0 1 0 b 0 a .(6.φ ı d,d−2 ([A a,b ]) = [A a−2,b ] + (v −a+1 − v −a−1 )[A a−1,b−1 ] − v −2a−1 [A a,b−2 ].
Proof. We shall prove the lemma by induction on b. When b = 0, the statement follows from the definition of φ ı d,d−2 . Let b ∈ N, and we assume the formula in the lemma is proved for By induction and using (6.4), we have
φ ı d,d−2 ([A a,b ′ ]),φ ı d,d−2 (t d * [A a,b ]) = φ ı d,d−2 (v −a+b [A a,b ] + v b [b + 1][A a−1,b+1 ] + v b−1 [a + 1][A a+1,b−1 ]) = (v −a+b+2 + v b−1 [b](v −a+1 − v −a−1 ))[A a−2,b ] + v b [b + 1][A a−3,b+1 ] + v b−1 [a − 1] + (v −a+1 − v −a−1 )v −a+b − v −2a+b−3 [b − 1] [A a−1,b−1 ] + v b−2 [a](v −a+1 −v −a−1 )−v −3a+b−3 [A a,b−2 ]−v −2a+b−4 [a+1][A a+1,b−3 ].v b [b + 1]φ ı d,d−2 ([A a−1,b+1 ]) = φ ı d,d−2 (t d * [A a,b ]) − v −a+b φ ı d,d−2 ([A a,b ]) − v b−1 [a + 1]φ ı d,d−2 ([A a+1,b−1 ]) = v b [b + 1][A a−3,b+1 ]+ v −a+b+2 −v −a+b +v b−1 [b](v −a+1 −v −a−1 ) [A a−2,b ] + v b−1 [a − 1] − v −2a b −3 [b − 1] − v b−1 [a + 1] [A a−1,b−1 ] + v b−2 [a](v −a+1 − v −a−1 ) − v −3a+b−3 + v −3a+b−1 − v b−1 [a + 1](v −a − v −a−2 ) [A a,b−2 ] = v b [b + 1] [A a−3,b+1 ] + (v −a+2 − v −a )[A a−2,b ] − v −2a+1 [A a−1,b−1 ] .φ ı d,d−2 ({A a,b }) ∈ [A a−2,b ] + b i=1 v −1 Z[v −1 ][A a−2+i,b−i ]. (6.7) Since φ ı d,d−2 ({A a,b }) is bar invariant, we conclude that φ ı d,d−2 ({A a,b }) = {A a−2,b } if a ≥ 2.
For a = 1, we write In Section 7.3, we will give an explicit formula of the canonical basis in S ı (2, d) in terms of standard basis.
{A 1,b } = [A 1,b ] + b i=1 Q i [A 1+i,b−i ], for some Q i ∈ v −1 Z[v −1 ]. Thus φ ı d,d−2 ({A 1,b }) = (1 − v −2 )[A 0,b−1 ] − v −3 [A 1,b−2 ] + b i=1 Q i φ ı d,d−2 ([A 1+i,b−i ]).
n : AU (gl m ) −→ AU (gl n ).
Lemma 2 . 1 .
21Let m, n, k ∈ Z with 0 < m < n. Then
Proposition 3. 3 .
3The transfer map φ d+n,d : S(n, d + n) −→ S(n, d) sends each canonical basis element to a sum of canonical basis elements with (barinvariant) coefficients in N[v, v −1 ] or zero. Proof. Recall that φ d+n,d is the composition (ξ ⊗ χ)∆, where ξ and ∆ are defined in [17, 2.2, 2.3].The positivity of ξ with respect to the canonical bases is clear from the definition (as it is just a rescaling operator by some v-powers depending on the weights). The positivity of ∆ with respect to the canonical bases follows by its well-known identification with (the function version of) a hyperbolic localization functor and then appealing to the main theorem of Braden[5].
basis for S n . When w = I, the claim holds trivially. Let s i be the ith elementary permutation matrix (corresponding to the ith simple reflection), for 1 ≤ i ≤ n − 1. It is straightforward to check by [1, Lemma 3.8] that {s i } = [s i ] + v −1 [I]. Hence χ({s i }) = v −1 det s i + v −1 det I = 0. Let w ∈ S n 2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 159
Proposition 3. 4 .
4The map φ d :U (sl n ) → S(n, d) sends each canonical basis element to a sum of canonical basis elements with (bar invariant) coefficients in N[v, v −1 ] or zero.
n :Θ m −→Ξ, A → τ k m,n (A).By comparing the multiplication formulas[1, 4.6] in AU (gl m ) and those in AU (gl n ) [2, (4.5)-(4.7)], we have an algebra embedding, also denoted by
Lemma 4. 1 .
1Suppose that 0 ≤ m ≤ (n − 1)/2 and k ∈ Z. Then τ k m,n ( a {A}) = {τ k m,n (A)}, for all A ∈Θ m .
λ for all admissible i, a, λ. For later use we define an equivalence relation ∼ onΞ: A ∼ A ′ if and only if A − A ′ = mI, for some m ∈ 2Z. We set Ξ =Ξ/ ∼ .
{D} = [D], {A} = [A], {B} = [B] since D is diagonal, [A] and [B] are the Chevalley generators ofU (gl 3 ).
Proposition 4. 4 .
4We have the following commutative diagram:
The commutativity of the left upper triangle and the right upper triangle is clear from definition. The commutativity of the bottom triangle follows from a description of the homomorphisms φ d and φ d+n,d in terms of matching the generators by Lemma 4.3.
ro(B) = ro(A), co(B) = co(A) . Then for p ≫ 0, we have by Lemma 5.1 that • I d = {B ∈Ξ|B A, ro(B) = ro(A), co(B) = co(A)};
So by the inductive assumption that any element B ≺ A satisfies Claim (⋆) and the convergence of the bilinear form ·, · d (with d = d 0 +pn/2) in Q((v −1 )) as p → ∞, we conclude that I d , n p and c B,p,i (0 ≤ i ≤ n p ) are all independent of p ≫ 0. Now Claim (⋆) follows by the construction of { p A} d as b pA in terms of the monomial basis above. Proposition 5.3. Given A ∈Ξ, we have
(sl n ) By Proposition 5.3, for A ∈ Ξ (recall Ξ from (4.4)), the elementb A := ℘ ({ p A}),for p ≫ 0
Theorem 5. 6 (
6Positivity). We have P C A, B ∈ N[v, v −1 ], for any A, B, C ∈ Ξ. Proof. Let us write b A * b B = C∈Ω P C A, B b C ,where Ω is the finite set which consists of C ∈ Ξ such that P C A, B = 0. Let us pick representatives A, B, C ∈Ξ such that |A| = |B| = |C| = 2d 0 + 1 for all C ∈ Ω.
5. 4 .
4Positivity of transfer map φ d+n,d We have the following positivity on the transfer map φ d+n,d , generalizing Proposition 3.3 on the positivity of the transfer map φ d+n,d . Theorem 5.10. The transfer map φ d+n,d : S (n, d + n) → S (n, d) sends each canonical basis element to a sum of canonical basis elements with (bar invariant) coefficients in N[v, v −1 ].
Proposition 5 . 11 .
511The map φ d :U (sl n ) → S (n, d) sends each canonical basis element to a sum of canonical basis elements with (bar invariant) coefficients in N[v, v −1 ].
5. 5 .
5Compatibility of canonical basesḂ(sl m ) andḂ (sl n )
→ S ı (n, d) ⊗ S(n, n) 1⊗χn − −−− → S ı (n, d).
d−2 ([A a−1,b+1 ]) = [A a−3,b+1 ] + (v −(a−1)+1 − v −(a−1)−1 )[A a−2,b ] − v −2(a−1)−1 [A a−1,b−1 ]. The coefficients of [A a−1,b−1 ] and [A a,b−2 ] in the expansion of φ ı d,d−2 ([A a,b ]) are in v −1 Z[v −1 ] for a ≥ 2, by Lemma 6.2. Meanwhile, A a ′ ,b ′ A a,bif and only if a ′ ≥ a. So we have
6.2, the coefficient of [A 0,b−1 ] on the RHS of (6.8) is in 1+ v −1 Z[v −1 ] and the coefficients of [A 1+i,b−i ] on the RHS of (6.8) for i ≥ 0 are in v −1 Z[v −1 ]. Now since φ ı d,d−2 ({A a,b }) is bar invariant, the coefficient of [A 0,b−1 ] must be 1, and we have φ ı d,d−2 ({A 1,b }) = {A 0,b−1 }. Now Lemma 6.2 for a = 0 gives us φ ı d,d−2 ([A 0,b ]) = −v −1 [A 0,b−2 ]. Asimilar analysis as for a = 1 shows that the expansion of φ ı d,d−2 ({A 0,b }) with respect to the standard basis [A a,b ] have all coefficients in v −1 Z[v −1 ]. This yields φ ı d,d−2 ({A 0,b }) = 0 due to its bar-invariance property. The proposition is proved.
2018 ]
2018POSITIVITY VS NEGATIVITY OF CANONICAL BASES 1472. Negativity of the Stably Canonical Basis ofU (gl n )
) i 1 λ (for a ≥ 1 and 1 ≤ i ≤ n − 1). It was shown in[1] that there is an A-algebra isomorphism K ∼ = AU (gl n ), which sends [E(a)
h,h+1 (λ)]
148
YIQIANG LI AND WEIQIANG WANG
[June
to E
(a)
h 1 λ and [E
(a)
h+1,h (λ)] to F
(a)
If ξ p preserved the BLM basis, then we would have ξ p ({A}) = {A + pI}The proposition for general n ≥ 2 follows from Lemma 2.1.7)
which can be readily turned into the formula in the proposition by Lemma
2.2.
by definitions, for all A. Hence the formula for ξ p
0 1
a 21 a 22
(with p < 0)
together with the fact ξ −1
p = ξ −p shows that ξ p (for p = 0) does not preserve
the BLM basis.
Remark 2.4. It can be shown similarly that
ξ p
0 1
a 21 a 22
=
p
1
a 21 a 22 + p
,
if a 21 ≥ 1, a 22 ≤ −3 and p > 0.
Indeed precise formulas for both sides of this inequality can be obtained by
(2.5) and (2.7).
Remark 2.5. There exists a surjective algebra homomorphism Φ d :U (gl n ) →
S(n, d) which sends [A] to [A] for A ∈ Θ d or to 0 otherwise. It was shown
in [9] that Φ d preserves the canonical bases, sending {A} to {A} for A ∈ Θ d
or to 0 otherwise. Making a gl n analogy with [16, 9.3], one might modify
the map Φ d to define a new algebra homomorphism Φ ′
d :U (gl n ) → S(n, d)
as follows: for u ∈U (gl n
Let a, b ∈Ḃ(sl n ) d , for POSITIVITY VS NEGATIVITY OF CANONICAL BASES 157 somed. We have, for some suitable finite subset Ω ⊂Ḃ(sl n ) d ,2018]
). The algebra162
YIQIANG LI AND WEIQIANG WANG
[June
A S (n, d) admits a bar involution, a standard basis [A] d , and a canonical (=
IC) basis {A} d parameterized by
(and denoted by φ d therein) which sends [A] to [A] d for A ∈ Ξ d and to zero otherwise. We define10)
where either A, A − E θ
h,h+1 or A − E θ
h+1,h for some 1 ≤ h ≤ n − 1 is diagonal.
It follows by definitions that
℘ • ξ
p = ℘ ,
for all p ∈ 2Z.
(4.11)
Recall a homomorphism Φ
d :U
(gl n ) → S (n, d) was defined in [2, §4.6]
n+1 .
n+1By [8, Corollary 5.1.4], we have φ ı d+n,d ({X} d+n ) = {X − 2I} d , if X − 2I ∈ Ξ ıd ,
0,
otherwise.
(6.2)
3 )
3Thus ifA a,b ∈ Ξ d , we have a + b = d. In this subsection we drop the index d to write [A a,b ] and {A a,b } for [A a,b ] d and {A a,b } d , respectively. We set [A a,b ] = 0, if a < 0 or b < 0.Lemma 6.2. For all a, b ∈ N such that a + b = d, we have
for all b ′ ≤ b and all a. We set t d = {A d−1,1 }. Recall from [2, Lemma A.13] that we have t d * [A a,b ] = v −a+b [A a,b ] + v b [b + 1][A a−1,b+1 ] + v b−1 [a + 1][A a+1,b−1 ].(6.4)
YIQIANG LI AND WEIQIANG WANG[June
AcknowledgmentsWe are grateful to Huanchen Bao and Zhaobing Fan for related collaborations and many stimulating discussions. We thank Olivier Schiffmann and Ben Webster for very helpful comments. The second author is partially supported by NSF DMS-1405131; he thanks the Institute of Mathematics, Academia Sinica (Taipei) and Institut Mittag-Leffler for an ideal working environment and support.These τ k d m,n for all d can be combined into a homomorphism τ k m,n :U (sl m ) → U (sl n ). We recall Θ m from (3.1), which is understood in this subsection to consist of m × m matrices.Proof. We have the following commutative diagram:Let A ∈ Θ m . Pick the preimage (an m×m matrix) A of A with 0 ≤ |A| < m, and set d = |A|. Recall from(3.4)and(4.11) that ℘•ξ 2l = ℘ and ℘ •ξ 2l = ℘ , for l ∈ Z. It follows from these identities, (5.7), and the above commutative diagram that τ k d m,n = ℘ • τ k d +l m,n • ℘ −1 d+2lm . Hence applying[18,Proposition 7.8], Lemma 4.1, and Proposition 5.3 in a row give us (for l ≫ 0)where the last identity uses the fact that A ′ = τ k d m,n (A) and τ k d +l m,n ( 2l A) have the same image in Ξ. The proposition is proved. Now we consider S ı (n, d) for a general even integer n. Recall the monomial basis { d M A |A ∈ Ξ ı d } of S ı (n, d) from [2, Proposition 5.6]; for notation Ξ ı d see(6.1). This is a subset of the monomial basisand Y 1 − aE θ n 2 +1, n 2 for some a ∈ N are diagonal and co(X 1 ) = ro(Y 1 ). Recall that the subset {{A} d |A ∈ Ξ ı d } of the canonical basis of S (n, d) forms a basis for S ı (n, d) and for the twin pair [The following properties of the hybrid monomials d M ı A are the main reasons to introduce them. Proposition 6.5. The following properties hold for a hybrid monomial d M ı A (where A ∈ Ξ ı d ):Proof Recall the algebra K from Section 4.1. This algebra has a standard basis [A] parameterized by the setΞ in (4.2). Let K 1 be the subalgebra of K spanned by the standard basis [A] inΞ such that ro(A) n 2 +1 = co(A) n 2 +1 = 1. Let J 1 be the ideal of K 1 spanned by[A]for all A ∈Ξ ı such that a n 2 +1, n 2 +1 < 0. LetΞ ı be the subset ofΞ consisting of matrices A defined by a n 2 +1,j = δ n 2 +1,j and a i, n 2 +1 = δ i, n 2 +1 for all i, j. We set K ı be the quotient of K 1 by J 1 . It is shown in [2, Appendix A.3] that K ı admits a monomial basis M A + J 1 , a standard basis [A] + J 1 , and a canonical basis {A} + J 1 , for all A ∈Ξ ı . Furthermore, it is shown in[2,Proposition A.11] that K ı is isomorphic to the modified quantum coideal subalgebraU ı (gl n ) of the quantum algebra U(gl n ). We shall identify K ı witḣ U ı (gl n ). Recall that the algebraU ı (gl n ) is an associative Q(v)-algebra generated by the symbols 1 λ , e i 1 λ , 1 λ e i , f i 1 λ , 1 λ f i , t1 λ , and 1 λ t, for i = 1, . . . , n 2 −1 and λ ∈ Z ı n := {λ ∈ Z n |λ n 2 +1 = 1}, subject to the following relations (6.10):(6.10) Here λ ± α i are the short hand notations introduced in Section 4.1. To simplify the notation, we shall writePOSITIVITY VS NEGATIVITY OF CANONICAL BASES 185if the product is not zero.We define an equivalence relation ≈ on Z ı n by setting λ ≈ λ ′ if and only if λ − λ ′ = aI for some a ∈ 2Z. LetẐ ı n be the set Z ı n / ≈ of equivalence classes. LetU ı (sl n ) be the algebra defined in the same fashion asU ı (gl n ) with the parameter set Z ı n replaced byẐ ı n . Similar toU (gl n ), the algebraṡ U ı (gl n ) andU ı (sl n ) admit the following decompositions.µ, λ ∈Ẑ ı n , |µ| ≡ |λ| ≡ 2d + 1 mod 2n. We have the following commutative diagram similar to (4.13):Here the homomorphisms φ ı d and ℘ ı are defined in a similar way as φ d and ℘ in (4.13) respectively, but with I replaced by I.Inner product onULet −, − ı,d be the bilinear form on the ıSchur algebra S ı (n, d) obtained from the bilinear form −, − d on S (n, d) by restriction, thanks to S ı (n, d) ⊂ S (n, d). We define a family of bilinear forms −, − ı,d onU ı (sl n ) by pulling back the one on the Schur algebra level via φ ı d in (6.11), i.e., x,We shall study the behavior of these bilinear forms as d tends to infinity. We need the following analogue of[18,Lemma 4.2].YIQIANG LI AND WEIQIANG WANG [Juneand an integer p 0 ∈ Z such thatfor all even integer p ≥ p 0 .Proof. The proof follows the arguments of[18,Lemma 4.2]and [2, Lemma A.1], except that we need to take care of the new case when k = 1 and A 1 − E θ n 2 , n 2 +2 is diagonal. In this case, we need the following multiplica-The lemma now follows by induction.Remark 6.7. Note that in the multiplication formula in Lemma 6.6, the canonical basis elements are used instead of the standard basis elements, which are the same for all generators except t d .We are ready to state the asymptotic behavior of the form ·, · ı,d .Proposition 6.8. As p goes to infinity, the limit lim p→∞ x, x ′ ı,d+pn , for all x, x ′ ∈U ı (sl n ), converges in Q((v −1 )) to an element in Q(v).2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 187Proof. The proof is similar to[18]. We need the adjointness of the bilinear form ·, · ı,d , which is inherited from that of ·, · d , and in particular we haveThe only difference from[18]is that we work in a larger ringSimilar to the form ·, · , we define a bilinear form −, − ı onU ı (sl n ) (independent of d) by lettingthe set {M ı A |A ∈Ξ ı } forms a basis of AU ı (gl n ),YIQIANG LI AND WEIQIANG WANG [JuneProof. All properties follow readily from the constructions except the last one. As the hybrid monomial bases forU ı (sl n ) and S ı (n, d) are defined multiplicatively by the same procedure, we only need to show that Property(4)in the rank one case. We remind that by construction the hybrid monomial basis in the rank one case is exactly the canonical basis. Hence Property (4) at rank one is exactly the statement of [2, Proposition A.21]. Now since we have Proposition 6.5, the commutative diagram (6.11), Proposition 6.9 at hand, the constructions and results in Sections 5 in the -setting can be rerun for the ı-setting. Let us outline them.for all even integer p ≫ 0, where pI A = A + pI.Proof. The same type arguments of the proof of Proposition 5.3 work here.We define an equivalence onΞ ı by A ≈ B if and only if A − B = pI for some even integer p. We setΞ ı =Ξ ı / ≈ . By Proposition 6.10, the following definition is well defined. Definition 6.11. We define bǍ = ℘ ı ({ pI A}) ∈U ı (sl n ), ∀p ≫ 0,Ǎ ∈Ξ ı . Now as we have the key properties established in Propositions 6.9-6.10, we are in a position to establish the ı-counterparts of results on canonical bases in Sections 5.2−5.3, whose similar proofs will be skipped. Below is a summary of the ı-counterparts of Theorem 5.5, Theorem 5.6, Proposition 5.7, and Theorem 5.8. Theorem 6.12.(1) The set B ı (sl n ) = {bǍ|Ǎ ∈Ξ ı } forms a basis forU ı (sl n ) and for AU ı (sl n ).(2) The structure constants for the algebraU ı (sl n ) with respect to the basis(3) The form −, − ı onU ı (sl n ) is non-degenerate. Moreover, the basis B ı (sl n ) is almost orthonormal and positive with respect to this form, i.e., bǍ, bB ı ∈ δǍ ,2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 189Again, similar to Proposition 5.9, the signed canonical basis −B ı (sl n ) ∪ B ı (sl n ) is characterized by the bar invariance, integrality and almost orthonormality.Remark 6.13. The main results (Theorem 5.10, Proposition 5.11, Proposition 5.13, Corollary 5.14, Propositions 5.15-5.16) in Sections 5.4−5.6 admit ı-analogues here with n replaced by n and by ı, respectively. Remark 6.14. Shigechi[21]has established by combinatorial methods certain positivity of the ı-canonical bases (introduced in [3]) on general tensor products of modules of the quantum coideal algebra of U(sl 2 ), and this supports our general positivity conjectures. See Remark 6.13 for a closely related result.Formulas of Canonical Basis of S ı (2, d)Combinatorial identitiesRecall the quantum v-binomial coefficients were defined in (2.3) for m ∈ Z and b ∈ N. We introduce the following additional notationWe first establish two combinatorial identities which are needed in later computations and could be of some independent interest as well.Lemma 7.1. For any a ∈ Z and p ∈ N, we haveProof. Recall the quantum binomial identityWe prove the lemma by induction on p, with the base case for p = 0 being trivial.190YIQIANG LI AND WEIQIANG WANG [June By (7.1), we can rewrite the sum as a sum of two summands:Setting p ′ = p − 1, s ′ = s − 1 and a ′ = a + 2, we have a + 2s = a ′ + 2s ′ , and thus by the inductive assumption (with p ′ < p) we obtainSetting p ′ = p − 1, by the inductive assumption (with p ′ < p) again we haveSumming up S 1 and S 2 above we have proved the lemma.Proof. Set m = 2n if m is even or m = 2n + 1 otherwise. We first sum up the two summands with j = 2d and j = 2d + 1, for fixed d with 0 ≤ d ≤ n:where the sign '−' is always taken for m = 2n and '+' for m = 2n + 1 on the right-hand side above and similar places below. Note that the above is actually valid for d = n in case m = 2n as well, where the second summand on the left-hand side is simply zero.Hence, notingwhere the last equation uses Lemma 7.1 (where we set a = ∓1 and p = n).The lemma is proved.The bar conjugate of the standard basisLetbe the Q(v)-vector space with the standard basis {[A a,r ]|a, r ∈ N}. As before we set [A a,r ] = 0 if a < 0 or r < 0. We introduce a shorthand notation to denote the monomial basis element M a,r = d M Aa,r . By[2, (5.4)], we haveThen {M a,r |a, r ∈ N, a + b = d} forms a monomial basis for S ı(2, d), and so {M a,r |a, r ∈ N} forms a monomial basis for T. There is a Q-linear bar involution on S ı (2, d) for all d and hence on T, denoted by , which fixes each M a,r . Note thatand M a,r = M a,r .The following theorem is obtained with help from a UVA undergraduate Tahseen Rabbani (supported by NSF), whose computer computation for small values of r was crucial in formulating the precise statement.Proof. The two expressions in the statement are clearly equal. We shall proceed by induction on r. The base case for r = 0 is clear.Assume the formula is verified for [A a,r ′ ] for all a, r ′ ∈ N such that r ′ < r. By (7.2) and M a,r = M a,r , it suffices to verify the formula for [A a,r ] as given in the theorem satisfies thatEquating the coefficients of [A a+m,r−m ] on both sides of the above identity, we are reduced to verifying the following identity for 0 ≤ m ≤ r:POSITIVITY VS NEGATIVITY OF CANONICAL BASES 193= v −am− 1 2 m(m+1) a + m m .(We have used (7.4) on deriving the right-hand side above.)After further simplification using[a]![i]! and i = m − j, the above identity is reduced to the following identity for m ≥ 0:Thanks to [a + m − j]! j u=1 [a + m + 1 − u] = [a + m]!, the above identity is equivalent to the identity in Lemma 7.2. The theorem is proved.Denote the coefficient of A a+i,r−i in Theorem 7.3 above, which is inde-Formulas for canonical basis of S ı (2, d)The canonical basis is the Q(v)-basis {{A a,r }|a, r ∈ N} for T, which is completely determined by the bar invariance together with the following 194 YIQIANG LI AND WEIQIANG WANG [June property:We denote γ a,r (0) = 1.Lemma 7.5. The polynomials γ a,r (i) are independent of r; we shall write γ a (i) = γ a,r (i).Proof. We shall show by induction on i ≥ 0. The case for i = 0 is clear.It follows from this and an easy induction on i that γ a,r (i) is independent of r.The next theorem establishes formulas for the canonical basis {A a,r } for S ı (2, d) for all d, or equivalently by (7.7), determines γ a (i) for all a, i ∈ N. Theorem 7.6.(1) For a, s ∈ N with a even, we have(2) For a, s ∈ N with a odd, we have2018] POSITIVITY VS NEGATIVITY OF CANONICAL BASES 195In other words, these polynomials γ a (r) are all essentially v 2 -binomial coefficients.Proof. Let us rewrite (7.8) asThis formula uniquely determines the polynomials γ a (r) for all a, r ∈ N (by induction on r), which satisfy γ a (0) = 1 and γ a (r) ∈ v −1 Z[v −1 ] for r ≥ 1. It suffices to verify that the formulas for γ a (r) given in the theorem do satisfy (7.9). The verification is divided into 4 very similar cases, depending on the parity of a and the parity of r.Assume first that both a and r are odd. Set r = 2p + 1. Let 0 ≤ s ≤ p.We have, γ a (2s + 1) b r−2s−1 a+2s+1 = v 2s(a+2s)−ar−( r+1 2 )+2a+4s+2The above two formulas have almost identical factors except that γ a (2s)b r−2s a+2s has an extra factor (1− v 2a+4s+2 ) while γ a (2s + 1)b r−2s−1 a+2s+1 has an extra factor v 2a+4s+2 . Hence, On the other hand, we haveTherefore, the verification of the identity (7.9) follows from (7.10)-(7.11) and the identity in Lemma 7.1, and the theorem is proved in the case when both a and r are odd.In the remaining three cases when not both a and r are odd, we have analogous reductions of verification of (7.9) to the same identity in Lemma 7.1, and we shall skip the details.
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[
"Proof Reduction of Fair Stuttering Refinement of Asynchronous Systems and Applications",
"Proof Reduction of Fair Stuttering Refinement of Asynchronous Systems and Applications"
]
| [
"Rob Sumners [email protected] \nCentaur Technology\n\n"
]
| [
"Centaur Technology\n"
]
| [
"ACL2 Theorem Prover and its Applications EPTCS 249"
]
| We present a series of definitions and theorems demonstrating how to reduce the requirements for proving system refinements ensuring containment of fair stuttering runs. A primary result of the work is the ability to reduce the requisite proofs on runs of a system of interacting state machines to a set of definitions and checks on single steps of a small number of state machines corresponding to the intuitive notions of freedom from starvation and deadlock. We further refine the definitions to afford an efficient explicit-state checking procedure in certain finite state cases. We demonstrate the proof reduction on versions of the Bakery Algorithm. | 10.4204/eptcs.249.6 | [
"https://arxiv.org/pdf/1705.01230v1.pdf"
]
| 23,517,141 | 1705.01230 | 9147adf7e365eac28b733af3b84d5873270139b2 |
Proof Reduction of Fair Stuttering Refinement of Asynchronous Systems and Applications
2017
Rob Sumners [email protected]
Centaur Technology
Proof Reduction of Fair Stuttering Refinement of Asynchronous Systems and Applications
ACL2 Theorem Prover and its Applications EPTCS 249
201710.4204/EPTCS.249.6
We present a series of definitions and theorems demonstrating how to reduce the requirements for proving system refinements ensuring containment of fair stuttering runs. A primary result of the work is the ability to reduce the requisite proofs on runs of a system of interacting state machines to a set of definitions and checks on single steps of a small number of state machines corresponding to the intuitive notions of freedom from starvation and deadlock. We further refine the definitions to afford an efficient explicit-state checking procedure in certain finite state cases. We demonstrate the proof reduction on versions of the Bakery Algorithm.
Introduction
Much of hardware and software system design focuses on how to optimize the execution of tasks by dividing the tasks into smaller computations and then scheduling and distributing these computations on the available resources. The natural specification for these systems is an assurance that the systems eventually complete the supplied tasks with results consistent with an atomic (or as atomic as feasible) execution of the task. We refresh the notion of fair stuttering refinements [10] as a means of codifying these specifications -a fair stuttering refinement between two systems ensures that every infinite run of a lower-level system with fair selection and finite stuttering maps to a similarly restricted infinite run of a higher-level system. This notion of refinement can allow sequences of smaller steps in the implementation to be mapped to single steps in the specification while additionally requiring that every task makes progress to completion.
Many previous efforts [10] have attempted to improve the capability of theorem provers in reasoning about refinements for distributed and concurrent systems. Previous efforts in regards to the ACL2 theorem prover [4] focused on trying to reduce the proofs of stuttering refinements with additional structures added to define fair selection and ensuring progress. These efforts generally boiled down to showing that a specification could match the step of an implementation or the implementation stuttered and some rank function decreased. The primary difficulty in these proofs was defining and proving an inductive invariant (either through ACL2 or trying to prove the invariant through some form of state exploration). In addition, the inclusion of additional structures to track fairness and progress as well as the resulting definition of rank functions proved complex. Further, the additional structures at times obfuscated whether the specification was complete and accurate.
In this paper, we take a different tack. We assume certain characteristics of the system we are trying to verify and leverage these characteristics in reducing the proof obligations. In particular, we first assume that the systems we are trying to verify are asynchronous in terms of how tasks make progress to completion. Further, we require the system definition to split the normal next-state transition relation into a next-state relation which only takes forward steps and a blocking relation which defines precisely when a task is blocked from making progress. From these assumed characteristics, we define proof reductions which reduce the goal of proving fair stuttering refinement to proving properties of a few task steps in relation to each other. These proof reductions have been formally defined and mechanically proven in ACL2 and are included in the supporting materials for this paper. In the remainder of this paper, we will cover two stages of proof reductions, review the application of the reductions to a version of the Bakery Algorithm. We conclude the paper with further reductions targeting efficient automatic checks in the finite state case.
Preliminaries
Commonly, systems are defined by an initial state predicate: (init x) and a next-state relation: (next x y). A run of the system is then simply a sequence of states where the first state satisfies (init x) and each pair of states in the sequence satisfies (next x y). We extend this basic construction in a couple of ways.
First, our goal is to reason about fair executions of a system (either as an assumption of fair selection for which task will update next or as a guarantee that every task makes progress). Thus, we assume that there is some set of task identifiers recognized by a predicate (id-p k) and add a task id parameter to the next-state relation: (next x y k) where this now relates state x to state y for an update to the task with id k. We also assume only one task updates at each step of the system without any prescribed order of task updates -essentially, the system is asynchronous at the level of task updates.
Second, we will find it useful to require the definition of an additional relation (blok x k) which returns true when the task identified by k is currently blocked from making progress in state x. Further, with this required definition of (blok x k), we will also require the theorem: (not (next x x k)) be proven and use inequality of next-states as a marker that a task is making progress to completion.
A system is then defined by three functions: (init x), (next x y k), and (blok x k). Our final goal is to prove that the fair runs of an implementation system map to fair runs of a specification system with an allotment for finite stuttering and some guarantee of progress. A run of a system is a function (run i) which takes a natural i and returns a state of the system. Runs will naturally need to satisfy some constraints as detailed in Figure 1. For a given system named sys, the macro (def-inf-run sys) assumes the definition of (sys-init x), (sys-next x y k), (sys-blok x k), (sys-pick i), (sys-run i) and generates the definitions and theorems defining the properties for the run as in Figure 1.
Of particular note, the function (step x y k) relates states x and y via (next x y k) only if k is not blocked in x and we are not stuttering (denoted by the input k being nil) -(as a note, the only requirement we place on id-p is that (not (id-p nil))). So, an infinite run is defined by two functions (run i) which defines the sequence of states and (pick i) which defines the sequence of task identifiers selected. We constrain (pick i) to only return an id-p or nil. We can now naturally define fair selection of (pick i) by positing the existence of a function (fair k i) which returns natural numbers and for each task id k will strictly decrease when k is not selected -see Figure 2. The macro (def-fair-pick sys id-p) assumes the definitions of (sys-pick i), (sys-fair k i), and (id-p k) and produces the theorems in Figure 2. We use the term fair run for an infinite run with a fair picker.
Fair selection of task identifiers ensures that each run only has finite stuttering and that each task gets a chance to make progress, but it does not guarantee that tasks actually make progress. We introduce the term valid run for a run which is not only fair but ensures progress for each task. In order to ensure progress, we define a function (prog k i) similar to (fair k i) but in addition to ensuring pick (encapsulate ((run (i) t) (pick (i) t)) (local (defun run (i) ....)) (local (defun pick (i) ....)) (defun step (x y k) (if (or (null k) ;; finite stutter (blok x k)) ;; or k is blocked in x (equal x y) (next x y k))) (defthm run-init-thm (implies (zp i) (init (run i)))) (defthm run-step-thm (implies (posp i) (step (run (1-i)) (run i) (pick i)))) )
Figure 1: Definition of an infinite run in ACL2
(defthm fair-nat-thm (natp (fair k i))) (defthm pick-fair-thm (implies (and (posp i) (id-p k) (not (equal (pick i) k))) (< (fair k i) (fair k (1-i))))) Figure 2: Fair Runs: fair task selection during a run (defthm prog-is-nat (natp (prog k i))) (defthm run-prog-thm (implies (and (posp i) (id-p k) (or (not (equal (pick i) k)) (equal (run i) (run (1-i))))) (< (prog k i) (prog k (1-i))))) Figure 3: Valid Runs: ensuring task progress during a run eventually equals k, we also need to ensure that a state change actually occurs. The properties in Figure 3 ensure a valid run and the macro (def-valid-run sys id-p) produces these theorems for id-p, sys-run, sys-pick, and sys-prog. We note that a valid run is also a fair run and thus our notion of refinement is compositional -but it is better to prove that all fair runs of the implementation are valid runs and then restrict the refinement to valid runs mapping to valid runs and reduce the proof requirements accordingly at each step. This is straightforward from what we present in this paper but we do not focus on it in this paper.
Proof Reduction to Single System Steps
The principle objective of fair stuttering refinement is to prove that the fair runs of an implementation map to valid runs of a specification. The first set of proof reductions we present refresh similar attempts in past work [10,8] in transferring these proof requirements on infinite runs to properties about single steps of two systems impl and spec. The difference between these past efforts and the work presented is that we directly specify properties related to guaranteeing progress for each task in the system and we leverage the definition of the blocking relation. In addition, while the proof reduction to single step presented in this section could be used as is, the design of the reduction is influenced by the needs of subsequent proof reductions over tasks presented in Section 4. The book "general-theory.lisp" in the supporting materials covers the work in this section.
The goal is to show that if one were to prove certain properties about steps of an implementation system impl and a specification system spec, then one could infer a fair stuttering refinement -every fair run of impl maps to a valid run of spec. We wish to prove this for any specification and implementation system, so specifically, for any impl and spec and any fair run impl-run of the implementation, if we have proven the required properties then we can map impl-run to a valid run spec-run of spec. An overview of the structure of the book "general-theory.lisp" is provided in Figure 4 and attempts to codify this goal. The definitions of the impl and spec systems and the fair run impl-run of impl are constrained within an encapsulate to only have the properties: (def-inf-run impl), (def-fair-pick impl id-p), (def-system-props impl id-p), (def-valid-system impl id-p), and (def-match-systems impl spec id-p). From this fair run impl-run and the properties proven on spec and impl, we can build a valid run spec-run. While it is not possible to make this a closed-form statement of correctness in ACL2, we believe the structure of the book is sufficient to establish the claim.
The function (spec-run i) in Figure 4 defines the spec state at each time to simply be (impl-map (impl-run i)) and the function (spec-pick i) is simply (impl-pick i) except that we introduce finite stutter (i.e. return nil) if the mapped state doesn't change. It is customary to define some notion of observation or labeling of states that must be preserved to ensure correlation of behavior between spec and impl -we assume human review has ensured that the mapping from impl states to spec states preserves any observations relevant to the specification. In this regard, it is relevant that the mapped run on the spec is relatively simple in definition as it avoids errors or oversights in specification due to an obfuscation of how the implementation and specification are correlated.
The properties we need to prove for impl and spec are defined by the macros def-system-props, def-valid-system, and def-match-systems. Along with the functions defining the impl and spec systems, additional definitions are required for each of these macros. We will shortly go into greater detail on the properties we will assume as constraints for these functions, but first, we refer to the listing provided in Figure 5. ;; ASSUMPTIONS: ;; assume relevant properties of given systems impl and spec: (def-system-props impl id-p) (def-valid-system impl id-p) (def-match-systems impl spec id-p)
;; assume an infinite run of the impl system:
(def-inf-run impl) (def-fair-pick impl id-p) )
.... ;; def.s and theorems to establish results.
;; Define the corresponding (assumed to preserve "observations") spec run: (defun spec-run (i) (impl-map (impl-run i))) ;; spec-pick will introduce stutter into spec-run when the mapped state doesn't change: (defun spec-pick (i) (and (not (equal (impl-map (impl-run (1-i))) (impl-map (impl-run i)))) (impl-pick i))) .... ;; additional def.s and theorems to establish results.
;; CONCLUSIONS:
;; and prove that the corresponding spec-run is indeed a valid run of spec: (def-inf-run spec) (def-valid-run spec id-p) Figure 4: Structure of the book ''general-theory.lisp''
• IMPL system definition:
-(impl-init x) -initial predicate on states x for impl system -(impl-next x y k) -state x transitions to state y on selector k -(impl-blok x k) -state x blocked for transitions for selector k
• SPEC system definition:
-(spec-init x) -initial predicate on states x for spec system -(spec-next x y k) -state x transitions to state y on selector k -(spec-blok x k) -state x blocked for transitions for selector k • Definitions needed for (def-system-props impl id-p) macro:
-(impl-iinv x) -inductive invariant for states in impl
• Definitions needed for (def-match-systems impl spec id-p) macro:
-(impl-map x) -maps impl states to corresponding spec states -(impl-rank k x) -ordinal decreases until spec matches transition for k • Definitions needed for (def-valid-system impl id-p) macro: The macro (def-system-props impl id-p) expands into simple theorems ensuring (not (id-p nil)), ensuring (impl-next x x k) is not valid, and ensuring the state predicate (impl-iinv x) is an inductive invariant for impl -namely that (impl-iinv x) holds in the initial state and persists across (impl-next x y k) transitions.
-(impl-noblk k x) -is task id k invariantly unblocked in state x -(impl-nstrv k x) -ordinal decreases until k is in a noblk state -(impl-starver k x) -potential starver of k in x which is not blocked
The (def-match-systems impl spec id-p) macro requires defining (impl-map x), a mapping from impl states to spec states and a ranking function (impl-rank k x) which returns an ordinal for each task id k. The main properties generated by def-match-systems are the following:
(defthm map-matches-next (implies (and (impl-iinv x) (id-p k) (!= (impl-map x) (impl-map y)) (impl-next x y k) (not (impl-blok x k))) (and (spec-next (impl-map x) (impl-map y) k) (not (spec-blok (impl-map x) k))))) (defthm map-finite-stutter (implies (and (impl-iinv x) (id-p k) (= (impl-map x) (impl-map y)) (impl-next x y k)) (o< (impl-rank k y) (impl-rank k x)))) (defthm map-rank-stable (implies (and (impl-iinv x) (id-p k) (id-p l) (!= k l) (impl-next x y l)) (o<= (impl-rank k y) (impl-rank k x))))
The theorem map-matches-next ensures that on any step (impl-next x y k) for task k which is not blocked in x and where the mapped specification state changes (i.e. (!= (impl-map x) (impl-map y))) then the spec must be able to match the transition and the spec state cannot be blocked in the spec for task k. The theorem map-finite-stutter ensures that when the mapped implementation state does not change on an update for task k in impl, then the ordinal returned by impl-rank must strictly decrease and the theorem map-rank-stable ensures that this ordinal does not increase when task k is not selected. The clear intent of these properties is to ensure that as long as a task k is not indefinitely blocked when it is selected for update in impl, then eventually a matching spec transition must be generated. The question is then naturally how to ensure that a task is not indefinitely blocked. This concept of being indefinitely blocked is commonly called "starvation" in the literature and the def-valid-system macro will generate properties intended to ensure that no task is starved.
The (def-valid-system impl id-p) macro requires the definition of a predicate (impl-noblk k x) which is true when the task k can no longer be blocked in state x and a function (impl-nstrv k x) which nominally returns an ordinal that decreases until (impl-noblk k x) is true. Once a task k reaches an impl-noblk state, it can no longer be blocked until it transitions and thus the fair selection of k will ensure a transition of k occurs. Unfortunately, a task's progress to an impl-noblk state may be dependent on any number of other tasks or components in the impl state. At this general level of system definition, we only have system states x and task ids k, so we imagine that for any k and x, we could define a set of task ids called the starve-set which need to make progress before k can reach a noblk state. Updates to ids which are not in this starve-set should simply have no effect on this progress and so we will assume that (impl-nstrv k x) will strictly decrease on transitions for ids in the starve-set and remain unchanged otherwise. Unfortunately, it might be possible that all of the tasks in the starve-set are blocked and so we need the additional definition of an (impl-starver k x) which returns an id in this starve-set which is currently not blocked in state x. Additionally, we need to ensure that when an element outside of the starve-set is chosen, that the (impl-starver k x) remains unchanged. The encoding of these properties as ACL2 theorems are generated from the def-valid-system macro and are listed here:
(defthm noblk-blk-thm (implies (and (iinv x) (id-p k) (noblk k x)) (not (blok x k)))) (defthm noblk-inv-thm (implies (and (iinv x) (id-p k) (id-p l) (!= k l) (next x y l) (noblk k x)) (noblk k y))) (defthm starver-thm (implies (and (iinv x) (id-p k) (not (noblk k x))) (not (blok x (starver k x))))) (defthm nstrv-decreases (implies (and (iinv x) (id-p k) (!= k (starver k x)) (next x y (starver k x)) (not (noblk k x))) (o< (nstrv k y) (nstrv k x)))) (defthm nstrv-holds (implies (and (iinv x) (id-p k) (id-p l) (!= k l) (next x y l) (not (noblk k x))) (o<= (nstrv k y) (nstrv k x)))) (defthm starver-persists (implies (and (iinv x) (id-p k) (id-p l) (!= k l) (!= l (starver k x)) (next x y l) (not (noblk k x)) (= (nstrv k y) (nstrv k x))) (= (starver k y) (starver k x))))
And with these properties assumed as constraints, we return to the goal of proving that the infinite run defined by (spec-run i) and (spec-pick i) from Figure 4 is indeed a valid run of spec. In order to do that we need to define a function spec-prog which satisfies the requirements set out in Figure 3. First, it is useful to define an (impl-prog k i) and show that the impl-run is indeed a valid run.
The definition of (impl-prog k i) is in Figure 6 and essentially looks forward into impl-run until we reach an i where k is picked and the state changes. The key point is obviously the question of what is the measure for demonstrating that this function terminates and this follows from our earlier discussion about the (impl-noblk k x), (impl-nstrv k x), and (impl-starver k x) functions. If we have (impl-noblk k ..) at the current state, then the task with id k cannot be blocked and we can simply countdown the (impl-fair k i) measure until task k is selected -the state will change at that time since k will still be unblocked and impl-next must change the state. If (impl-noblk k ..) does not currently hold then we know there is a task id (impl-starver k ..) which cannot be blocked in the current state and either (impl-nstrv k ..) strictly decreases or (impl-starver k ..) will not change. Thus, at each step, either the impl-nstrv measure strictly decreases or the fair measure for impl-starver counts down and will eventually expire and impl-nstrv will strictly decrease.
This (impl-prog k i) thus ensures that task k is picked and changes state in (impl-run i) but we now must guarantee that the mapped state changes in spec. In the case that the mapped state doesn't change, we know that the (impl-rank k ..) must decrease and that the impl-rank remains unchanged ;; First prove that the implementation run is a valid run... (defun impl-prog (k i) (declare (xargs :measure (if (impl-noblk k (impl-run i)) (impl-fair k i) (ord-nat-pair (impl-nstrv k (impl-run i)) (impl-fair (impl-starver k (impl-run i)) i))))) (cond ((or (not (and (natp i) (id-p k))) ;; ill-formed inputs.. or (and (= (impl-pick (1+ i)) k)
;; impl-pick matches k (!= (impl-run (1+ i)) (impl-run i)))) ;; ..and k makes progress 0) (t (1+ (impl-prog k (1+ i)))))) ;; ...And use that to show that the mapped spec run is also valid (defun spec-prog (k i) (declare (xargs :measure (ord-nat-pair (impl-rank k (impl-run i)) (impl-prog k i)))) (cond ((or (not (and (natp i) (id-p k)))
;; ill-formed inputs.. or (and (= (spec-pick (1+ i)) k)
;; spec-pick matches k (!= (spec-run (1+ i)) (spec-run i)))) ;; ..and k makes progress 0) (t (1+ (spec-prog k (1+ i))))))
Proof Reduction to a Small Bounded Number of Tasks
In the previous section, we presented a proof reduction of the requirements for fair stuttering refinement from reasoning about infinite runs of systems to reasoning about single steps of systems. We did not make any assumption about the state structure of the systems other than that updates occurred asynchronously at some prescribed task level. In this section, we will assume a structure on the states of a system and show how to reduce the requisite properties from across the large state structure to the properties on components of the state. Throughout this section and the next, we will use the set (s k v r) and get (g k r) operations from the records book [5]. In particular, (g k r) takes a record r and returns either the value previously set for key k in record r or nil as default.
The book "trans-theory.lisp" in the supporting materials for this paper includes the definitions and proofs relating to this section. The structure of this book is similar to that shown for "general-theory.lisp" in Figure 4 in that there is an encapsulation which entails the system definitions and properties we want to assume and then outside of the encapsulation, we prove the derived results. For the previous section, in "general-theory.lisp", we proved the property in Figure 8 (in an abuse of notation pretending ACL2 were higher-order for a moment), For this section, our goal is to define systems at a task level and derive the system-level results. In the same higher-level-abuse format as above, we have the property from "trans-theory.lisp" also in Figure 8.
We take the state of the system to be a record associating keys to task states.. what we call t-states. The task id selected on input is now simply one of these keys and the update of the state will only update • TR-IMPL system definition:
-(tr-impl-t-init a k) -initial state predicate for t-state a and key k -(tr-impl-t-next a b x) -t-state a transitions to t-state b in state x -(tr-impl-t-blok a b) -t-state a is blocked from stepping by t-state b
• TR-SPEC system definition:
-(tr-spec-t-init a k) -initial state predicate for t-state a and key k -(tr-spec-t-next a b x) -t-state a transitions to t-state b in state x -(tr-spec-t-blok a b) -t-state a is blocked from stepping by t-state b
• Definitions needed for (def-tr-system-props tr-impl) macro:
-(tr-impl-iinv x) -inductive invariant as previously.. no change at task-level • Definitions needed for (def-match-tr-systems tr-impl tr-spec) macro:
-(tr-impl-t-map a) -maps tr-impl t-states to corresponding tr-spec t-states -(tr-impl-t-rank a) -ordinal decreases until mapped t-state must change • Definitions needed for (def-valid-tr-system tr-impl) macro:
-(tr-impl-t-noblk a b) -is t-state a invariantly not-blocked by t-state b -(tr-impl-t-nstrv a b) -positive natural which strictly decreases until (t-noblk a b) -(tr-impl-t-nlock k x) -ordinal strictly decreases on from k to blocker of k in x (implies (and (def-system-props impl id-p) (def-valid-system impl id-p) (def-match-systems impl spec id-p)) (implies (and (def-inf-run impl) (def-fair-pick impl id-p)) (and (def-inf-run spec) (def-valid-run spec id-p)))) "trans-theory.lisp":
(implies (and (def-tr-system-props tr-impl) (def-valid-tr-system tr-impl) (def-match-tr-systems tr-impl tr-spec)) (and (def-system-props tr-impl key-p) (def-valid-system tr-impl key-p) (def-match-systems tr-impl tr-spec key-p))) Figure 8: High-Level properties in for theory files definitions the corresponding entry of the record. We presume and constrain a fixed finite set of keys -(keys) -of arbitrary size and composition and membership in this set will define the id-p test for task id selection. The state of the system is then a record mapping members of this finite set (keys) to t-states and the system will be defined on the task level. We define task-based systems by assuming the pertinent definitions on task states in the system and derive the system-level definitions across the state. We name these systems derived from the task-level definitions as tr-impl and tr-spec. In Figure 5 from the previous section, we listed the function definitions required for the single-step system-level propertieswe do the same for the single-step task-level properties in Figure 7. Many of the system-level derived functions follow simply from the task-level. The system-level (tr-impl-init x) predicate checks that (tr-impl-t-init (g k x) k) holds for all keys k. The system-level (tr-impl-next x y k) only updates (g k x) as (tr-impl-t-next (g k x) (g k y) x) and leaves all other keys untouched in x. The system-level block function (tr-impl-blok x k) checks if there is any key l such that (tr-impl-t-blok (g k x) (g l x)). The systemlevel mapping function simply goes through all keys and calls tr-impl-t-map for the corresponding t-state and the system level rank just calls (tr-impl-t-rank (g k x)) directly. The inductive invariant does not change; there is just one inductive invariant defined on the entire record defining the system state. Additionally, the system-level proofs for (def-system-props tr-impl key-p) and (def-match-systems tr-impl tr-spec key-p) are straightforward and follow from these systemlevel definitions and properties of task-level definitions.
The functions and properties for proving progress and valid impl runs are more involved. For the sake of brevity and readability, we will drop the tr-impl-prefix from the system-level and task-level defintions for the remainder of this section. In addition to ensuring that t-nlock returns an ordinal and t-nstrv returns a positive natural number 1 , the macro (def-valid-tr-system tr-impl) introduces the following properties:
(defthm t-noblk-blk-thm (implies (and (iinv x) (key-p k) (key-p l) (t-noblk (g k x) (g l x))) (not (t-blok (g k x) (g l x))))) (defthm t-noblk-inv-thm (implies (and (iinv x) (key-p k) (key-p l) (t-noblk (g k x) (g l x)) (t-next (g l x) c x)) (t-noblk (g k x) c))) (defthm t-nlock-decreases (implies (and (iinv x) (key-p k) (key-p l) (t-blok (g k x) (g l x))) (o< (t-nlock l x) (t-nlock k x)))) (defthm t-nstrv-decreases (implies (and (iinv x) (key-p k) (key-p l) (not (t-noblk (g k x) (g l x))) (not (t-noblk (g k x) c)) (t-next (g l x) c x)) (< (t-nstrv (g k x) c) (t-nstrv (g k x) (g l x)))))
The system-level (noblk k x) definition simply checks that (t-noblk (g k x) (g l x)) holds for every key l and as such, the task-level t-noblk-blk-thm and t-noblk-inv-thm are task-level projections of their system-level counterparts and the system-level properties follow fairly easily. The more interesting case comes up in defining the system-level (nstrv k x) and (starver k x). For the task-level, the property t-nlock-decreases ensures that we don't have any "deadlocks" or simply that for any set of keys, there is always some key in that set which is not blocked in x by some other key in that set. The combination of t-nstrv-decreases and the properties of t-noblk ensure that no task can be starved by another task.
The intuition behind defining the system-level (nstrv k x) begins by recognizing that if (not (noblk k x)) then there is some set of keys l such that (not (t-noblk (g k x) (g l x))). We will call this set of keys the may-block set. But since t-noblk persists once we reach it, then we could sum up the (t-nstrv (g k x) (g l x)) for this may-block set and the resulting ordinal would decrease until we reached a state where k was t-noblk for all l and thus noblk. Assume for the moment that k were not blocked (i.e. we could set (starver k x) to be k), then consider an update for some key l. If that key were in the may-block set of k then the ordinal would decrease. If l is not in the mayblock set of k then (t-noblk (g k x) (g l x)) and the transition of l cannot change the blocked status of k and it cannot change the may-block set for k and so progress is made. Unfortunately there is no guarantee that k is not blocked and thus we cannot pick a suitable starver which ensures progress when selected.
But from the property t-nlock-decreases, starting with k in x, we can find a key which is not blocked by checking if the key is blocked and recurring on the first blocking key we find if we are blocked. This is the definition of the function (starver k x) and is included here:
(defun starver (k x) (declare (xargs :measure (t-nlock (g k x)))) (if (and (iinv x) (key-p k) (blok x k)) (starver (pikblk k x) x) k))
The function (pikblk k x) returns the first key we find such that (t-blok (g k x) (g (pikblk k x))). So, from k, we can find a key which is unblocked, but the question is then how to build a measure from the starve-set including k and (starver k x). The answer is to build a natural list where each element is the sum of t-nstrv for the may-block set (as we described before) in each step along the path from k to (starver k x) and define our ordinal as the lexicographic product of the naturals in this list. The first observation is that at the end of this list we will have the summation of t-nstrvs for the may-block set of (starver k x) and since (starver k x) is not blocked, it will make progress as we discussed before. The other key observation is that at each step, the (pikblk k x) key will be in the may-block set of k and thus even though a transition of (pikblk k x) may modify its may-block set and potentially increase the measure from that point, the measure for the may-block set of k will decrease and the ordinal over all will decrease. This list of naturals is defined by the function (nstrvs* k x) as follows where the function (scar s) and (scdr s) return the first element and remainder of a set respectively and (card s) returns the cardinality of the set.
(defun sum-nsts* (k x s) (declare (xargs :measure (card s))) (if (null s) 1 (+ (if (t-noblk (g k x) (g (scar s) x)) 0 (t-nstrv (g k x) (g (scar s) x))) (sum-nsts* k x (scdr s))))) (defun sum-nsts (k x) (sum-nsts* k x (keys))) (defun nstrvs* (k x) (declare (xargs :measure (t-nlock (g k x)))) (if (and (iinv x) (key-p k) (blok x k)) (cons (sum-nsts k x) (nstrvs* (pikblk k x) x)) (list (sum-nsts k x)))) (defun nats->o (n l) (cond ((zp n) 0) ((atom l) (make-ord n 1 (nats->o (1-n) ()))) (t (make-ord n (1+ (car l)) (nats->o (1-n) (cdr l)))))) (defun tr-impl-nstrv (k x) (nats->o (card (keys)) (nstrvs* k x)))
As we mentioned, the function nstrvs* returns a natural list and we build a suitable ordinal from this list using the function nats-o. But because the length of the path to (starver k x) from k could change and thus the length of the nstrvs* list could change, we need to make the defined ordinal "firstaligned" -where the first element in the list is mapped to a coefficient of the same exponent no matter the length of the rest of the list. We use (card keys) as the starting exponent and prove separately that the length of the list returned by nstrvs* can never exceed (card keys).
This construction also shows one of the reasons we assume an arbitrary fixed finite set of (keys) (in order to put a bound on (len (nstrv* k x))), but this restriction makes sense for other reasons as well. If the set of keys were not finite, then we would need some additional requirement to ensure that a task were not persistently blocked by an infinite sequence of newly instantiated tasks. Other options exist to avoid this (such as requiring that all new tasks cannot block existing tasks) but these alternatives end up imposing constraints we believe are too restrictive.
Example -A Bakery Algorithm
We use the Bakery algorithm as an example application of the proof reductions we present in this paper. The Bakery algorithm was developed by Lamport [7] as a solution to mutual exclusion with the additional assurance that every task would eventually gain access to its exclusive section. The Bakery algorithm has also been a focus of previous ACL2 proof efforts [9].
The essential idea of the algorithm is that each task first goes through a phase where it chooses a number (much like choosing a number in a bakery) and then later compares the number against the numbers chosen by the other tasks to determine who should have access to the exclusive section. The version of the Bakery algorithm we will use is defined in Figure 9 (the (upd r .. updates ..) simply expands into a nest of record sets).
Each task will start in program location 0 and start its :choosing phase. During the :choosing phase, the task will grab the current shared max (via the function (curr-sh-max x)) and then set its (defun bake-impl-t-init (a k) (= a (upd nil :loc 0 :key k :pos 1 :old-pos 0 :temp 0 :sh-max 1))) (defun bake-impl-t-next (a b x) (case (g :loc a) (0 (= b (upd a :loc 1 :choosing t))) (1 (= b (upd a :loc 2 :temp (curr-sh-max x)))) (2 (= b (upd a :loc 3 :pos (1+ (g :temp a)) :old-pos (g :pos a) :pos-valid t))) (3 (= b (upd a :loc 4 :sh-max (if (> (curr-sh-max x) (g :temp a)) (curr-sh-max x) (g :pos a))))) (4 (= b (upd a :loc 5 :choosing nil))) (5 (= b (upd a :loc 6))) ;; we are potentially blocked here (6 (= b (upd a :loc 7))) ;; we are potentially blocked here (t (= b (upd a :loc 0 :pos-valid nil))))) (defun bake-impl-t-blok (a b) (or (and (= (g :loc a) 5) (g :choosing b)) (and (= (g :loc a) 6) (and (g :pos-valid b) (lex< (g :pos b) (ndx (g :key b)) (g :pos a) (ndx (g :key a))))))) Figure 9: Bakery Implementation System Definition own position :pos to be 1 more than the shared max. In program :loc 3, a compare-and-swap is implemented and the shared-max is potentially updated. The task then ends its :choosing phase. After the :choosing phase, the task will enter program locations 5 and 6. In these locations, the t-blok predicate ensures that the task wait until other tasks are not :choosing and then wait until it has the least position (where potential ties are broken by comparing the ndx of the :key in the set (keys).
In order to prove (def-valid-tr-system bake-impl), we need to define the t-nlock, t-noblk, and t-nstrv functions. The definition of (t-nlock x k) needs to return an ordinal that is strictly decreasing from the blocked task to the blocking task. From the bake-impl-t-blok relation, we note that :choosing states cannot be blocked and that lex< is already well-founded, so we can devise a suitable bake-impl-t-nlock:
(defun bake-impl-t-nlock (k x) (let ((a (g k x))) (make-ord 2 (if (g :choosing a) 1 2) (make-ord 1 (1+ (nfix (g :pos a))) (ndx (g :key a))))))
For the t-noblk and t-nstrv definitions, we need to analyze where one task can no longer block another task. The simple answer is that (t-noblk a b) is reached once task b has chosen a :pos greater than the one in a, but we also have to make sure that task b is not choosing either. In addition, we note that if a cannot currently be blocked by any task, then we can set t-noblk and task a cannot be blocked if it is not in program locations 5 or 6. With that, we define bake-impl-t-noblk:
(defun bake-impl-t-noblk (a b) (or (and (!= (g :loc a) 5) (!= (g :loc a) 6)) (and (not (g :choosing b)) (> (g :pos b) (g :pos a))))) Finally, we need to define t-nstrv which counts down until we reach the t-noblk state. The simple answer would be to count from the exit of :choosing phase until the next exit from the :choosing phase. Thus, we would return 8 if (g :loc b) was 5 and then proceed down to 6 for 7, then 5 for 0 (wrapping back), then down to 1 for 4 (end of next :choosing). This almost works.. except that it is possible for b to be in :loc 2, 3, or 4 with a :pos lower than a but a has proceeded further. Thus, we need to add a few steps for the case of being in 2,3,4 with a potentially lower :pos but when we come back around for the next :choosing, we will reach noblk:
(defun bake-impl-t-nstrv (a b) (pos-fix (cond ((or (and (= (g :loc b) 2) (< (g :temp b) (g :pos a))) (and (> (g :loc b) 2) (<= (g :pos b) (g :pos a)))) (+ 8 (-8 (g :loc b)))) ((>= (g :loc b) 5) (+ 5 (-8 (g :loc b)))) (t (+ 0 (-5 (g :loc b)))))))
With these definitions and a suitable invariant bake-impl-iinv, we can prove the theorems for (def-valid-tr-system bake-impl) -each of which just blasts into a big case split which pushes through. For the specification of the bakery algorithm, we have a simple system bake-spec defined in Figure 10. Each task in this system goes through the following steps: first, load up a new provisional :pos in the :load variable, then proceed to set the :pos variable and begin to arbitrate in the 'interested state. Tasks are blocked if some other task is in the 'go state or is in the 'interested state and has a lower :pos. The definitions and proof of (def-match-tr-systems bake-impl bake-spec) are fairly straightforward and included in Figure 10. We note that it is feasible (although not required) to define the supporting functions and prove (def-valid-tr-system bake-spec) -this proves that all fair runs of bake-spec are valid while the earlier proofs only ensured that the runs mapped from bake-impl runs were valid.
In previous work [10], a similar proof effort was conducted in proving a fair stuttering refinement for the definition of the Bakery Algorithm. In that effort, the proof was complicated by the need to add additional structures to track fair scheduling and to ensure correlation to a specification which had additional structures to ensure progress for each task. These complications were avoided in the proof here and as such, much less definition and details were required. The reduced proof we present here is primarily the definition and proof of a sufficient inductive invariant but much additional definition and proof was required in the earlier work [10].
(defun bake-impl-t-rank (a) (case (g :loc a) (0 1) (1 0) (2 1) (3 0) (4 2) (5 1) (6 0) (t 0))) Figure 10: Bakery Specification System and Definitions for Proving Matching from Impl This paper focused on mechanized proof reductions for general system definitions, but the work also supports improvements in more efficient automatic verification (in particular when the underlying task state space is finite). For example, take a somewhat draconian restriction that (t-next a b x) can be defined as (t-next a b) and similarly, the initial state predicate ignored the input k -a few things develop in this case. First, we note (somewhat trivially) that for every reachable system state composed of (say) n task states, that every "substate" of n − 1 task states can also be reached. Additionally, if the task state space were finite, then we could compute all of the potential cycles in the blocking relation and for each cycle of size n, we could determine if it was reachable by searching through the system states with only n keys. A similar check could be implemented for the other properties with no more than 2 keys needed.
Of additional interest in this case, is that reachable states of these systems have a particular characterization. Consider any run of a system.. any steps in the run can be permuted as long as the permutation does not change the blocking relationship between the tasks involved. This means that for every reachable state, one can define a set of canonical runs which involves only stepping tasks until the blocking relationship is changed with respect to another task and then switching to the blocking task or stepping back and switching to the blockee task. This property limits the structure of potential invariants and suggests procedures for proving invariants over pairs of states. The inductive invariant iinv over the system state can be defined by invariant definitions on single task states, pairs of states, triples, etc. and in most cases (potentially with additional auxiliary variables), sufficiently defined on single t-states and pairs of t-states. In this case, the requisite properties of the defined t-nlock, t-nstrv, t-noblk, t-map and t-rank definitions could be proven via GL on the specified finite t-state domain using a SAT solver with a sufficient conditions on the t-states assumed. An inductive invariant (defined on single t-states and pairs of t-states) could be defined that proved each of these sufficient condition assumptions as invariant of the system. A model checker could be used to reduce the definitional requirements further by checking invariants (not requiring inductive invariants) and by checking for bad cycles to show that one could infer the existence of suitable t-nlock, t-nstrv, and t-rank. The model checking problems could be limited to a small number of tasks and possibly only single task stepping depending on the conditions of the defintion. The work presented in this paper is a step into many potential future directions.
...) ;; constrained functions defining impl and spec. .... ;; local def.s and prop.s to show constraints.
Figure 5 :
5Function Definitions for Single-Step System-Level Properties
Figure 6 :
6Defined Measure Functions on Infinite Runs when other ids are selected. This is the basis for the definition (spec-prog k i) inFigure 6.
Figure 7 :
7Function Definitions for Single-Step Task-Level Properties"general-theory.lisp":
In the supporting materials for this paper, t-nstrv is generalized to be a list of natural numbers which is then summed and combined into a list of lists of naturals, but for the sake of clarity and brevity in this paper, we keep a simpler definition for t-nstrv. We could not use a generic ACL2 ordinal for t-nstrv since we needed to form lexicographic products of sums of these ordinals and that is not possible for arbitrary ordinals in ACL2.
Further Reductions and ConsiderationsWe conclude this paper with a discussion of further reductions and considerations for search procedures. We first acknowledge that some of the task-based definitions may seem overly restrictive. For example, the (blok a b) relation being defined simply on task states. In essence, this restricts us from supporting systems where a task may be blocked when only some combination of tasks exist. It is possible to extend the notion of blocking to be more general but it comes at the cost of the complexity of other definitions and checks and we have generally found that by adding auxiliary variables to the task state, we can fit any appropriate system under these restrictions.
defun bake-spec-t-init (a k) (declare (ignore k)) (and (= (g :loc a) 'idle) (= (g :pos a) 0) (=. defun bake-spec-t-init (a k) (declare (ignore k)) (and (= (g :loc a) 'idle) (= (g :pos a) 0) (= (g :load a) 0)))
= (g :loc b) 'loaded) (= (g :pos b) (g :pos a)) (natp (g :load b)) (> (g :load b) (max-pos x)) (>= (g :load b) (max-load x)))) (loaded (= b (upd a :loc 'interested :pos (g :load a)))) (interested (= b (upd a :loc 'go))) (go (= b. defun bake-spec-t-next (a b x) (case (g :loc a) (idle (and(defun bake-spec-t-next (a b x) (case (g :loc a) (idle (and (= (g :loc b) 'loaded) (= (g :pos b) (g :pos a)) (natp (g :load b)) (> (g :load b) (max-pos x)) (>= (g :load b) (max-load x)))) (loaded (= b (upd a :loc 'interested :pos (g :load a)))) (interested (= b (upd a :loc 'go))) (go (= b (upd a :loc 'idle)))))
(defun bake-spec-t-blok (a b) (and (= (g :loc a) 'interested) (or (= (g :loc b) 'go) (and (= (g :loc b) 'interested) (< (g :pos b. (defun bake-spec-t-blok (a b) (and (= (g :loc a) 'interested) (or (= (g :loc b) 'go) (and (= (g :loc b) 'interested) (< (g :pos b) (g :pos a))
3) 'loaded) ((4 5 6) 'interested) (t 'go)) :pos (case (g :loc a) (3 (g :old-pos a)) (t (g :pos a))) :load (case (g :loc a. defun bake-impl-t-map (a) (upd nil :loc (case (g :loc a) ((0 1) 'idle. 1t(defun bake-impl-t-map (a) (upd nil :loc (case (g :loc a) ((0 1) 'idle) ((2 3) 'loaded) ((4 5 6) 'interested) (t 'go)) :pos (case (g :loc a) (3 (g :old-pos a)) (t (g :pos a))) :load (case (g :loc a) (2 (1+ (g :temp a))) (t (g :pos a)))))
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[
"I-FAVORABLE SPACES: REVISITED",
"I-FAVORABLE SPACES: REVISITED"
]
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"Vesko Valov "
]
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| The aim of this paper is to extend the external characterization of I-favorable spaces obtained in[13]. This allows us to obtain a characterization of compact I-favorable spaces in terms of quasi κ-metrics. We also provide proofs of some author's results announced in[14]. | null | [
"https://arxiv.org/pdf/1801.07162v1.pdf"
]
| 119,680,381 | 1801.07162 | d21aa3f629564366d61ab1a4a57faaa5d5b22db3 |
I-FAVORABLE SPACES: REVISITED
18 Jan 2018
Vesko Valov
I-FAVORABLE SPACES: REVISITED
18 Jan 2018arXiv:1801.07162v1 [math.GN]
The aim of this paper is to extend the external characterization of I-favorable spaces obtained in[13]. This allows us to obtain a characterization of compact I-favorable spaces in terms of quasi κ-metrics. We also provide proofs of some author's results announced in[14].
separable metric spaces X α and skeletal surjective bounding maps p β α satisfying the following conditions: (1) the index set A is σ-complete (every countable chain in A has a supremum in A); (2) for every countable chain {α n } n≥1 ⊂ A with β = sup{α n } n≥1 the space X β is a (dense) subset of lim K3) ρ(x, C) is continuous function of x for every C; K4) ρ(x, C α ) = inf α ρ(x, C α ) for every increasing transfinite family {C α } of canonically closed sets in X. We say that a function ρ(x, C) is an quasi κ-metric on X if it satisfies the axioms K2) − K4) and the following one: K1 * ) For any C there is a dense open subset V of X \ C such that ρ(x, C) = 0 iff x ∈ X \ V . Our second result provides a characterization of compact I-favorable spaces, which is similar to Shchepin's characterization ( [9], [10]) of openly generated compacta as compact spaces admitting a κ-metric.
Theorem 1.4. A compact space X is I-favorable iff X is quasi κ- metrizable.
Corollary 1.5. Every I-favorable space is quasi κ-metrizable.
The paper is organized as follows: Section 2 contains the proof of Theorem 1.1 and Corollaries 1.2-1.3. The proofs of Theorem 1.4 and Corollary 1.5 are contained in section 3. In section 4 we provide the proof of some results concerning almost continuous inverse systems with nearly open bounding maps, which were announced in [14].
Proof of Theorem 1.1
If follows from the definition of I-favorability that a given space is I-favorable if and only if there are a π-base B and a function σ : n≥0 B n → B such that the union n≥0 U n is dense in X for any sequence σ(∅), U 0 , σ(U 0 ), U 1 , σ(U 0 , U 1 ), U 2 , ..., U n , σ(U 0 , U 1 , .., U n ), U n+1 , , , ,
where U k and σ(∅) belong to B, U 0 ⊂ σ(∅) and U k+1 ⊂ σ(U 0 , U 1 , .., U k ) for every k ≥ 0. Such a function will be also called a winning strategy. Recall that B is a π-base for X if every open set in X contains an element from B. Proposition 2.1. [3] Let B and P be two π-bases for X. Then there is a winning strategy σ : n≥0 B n → B if and only if there is a winning strategy µ : n≥0 P n → P.
Proof. Suppose σ : n≥0 B n → B is a winning strategy. We define a winning strategy µ : n≥0 P n → P by induction. We choose any open non-empty set µ(∅) ∈ P such that µ(∅) ⊂ σ(∅). If V 0 ∈ P is the answer of player II in the game played on P (i.e., V 0 ⊂ µ(∅)), then we choose U 0 ∈ B with U 0 ⊂ V 0 (U 0 can be considered as the answer of player II in the game played on B). Assume we already defined V 0 , .., V n ∈ P and U 0 , .., U n ∈ B such that U k+1 ⊂ V k+1 ⊂ µ(V 0 , .., V k ) ⊂ σ(U 0 , .., U k ) for all k ≤ n − 1. Then, we choose µ(V 0 , .., V n ) ∈ P such that µ(V 0 , .., V n ) ⊂ σ(U 0 , .., U n ). If V n+1 ∈ P is the choice of player II in the game played on P such that V n+1 ⊂ µ(V 0 , .., V n ), we choose U n+1 ∈ B with U n+1 ⊂ V n+1 . This complete the induction. Since σ is a winning strategy and U k ⊂ V k for each k, the union n≥0 V n is dense in X. So, µ is also a winning strategy.
In [13] we considered I-favorable spaces X with respect to the co-zero sets meaning that there is a winning strategy σ : n≥0 Σ n → Σ, where Σ is the family of all co-zero subsets of X. Proposition 2.1 shows that this is equivalent to X being I-favorable. So, all results from [13] are valid for I-favorable spaces.
According to [2,Corollary 1.4], if Y is a dense subset of X, then X is I-favorable if and only Y is I-favorable. So, every compactification of a space X is I-favorable provided X is I-favorable. And conversely, if a compactification of X is I-favorable, then so is X. Because of that, very often when dealing with I-favorable spaces, we can suppose that they are compact.
Let us introduced few more notations. Suppose X ⊂ I A is a compact space and B ⊂ A, where I = [0, 1]. Let π B : I A → I B be the natural projection and p B be restriction map π B |X. Let also X B = p B (X). If U ⊂ X we write B ∈ k(U) to denote that p −1 B p B (U) = U. A base A for the topology of X ⊂ I A consisting of open sets is called special if for every finite B ⊂ A the family {p B (U) : U ∈ A, B ∈ k(U)} is a base for p B (X) and for each U ∈ A there is a finite set B ⊂ A with B ∈ k(U). Proposition 2.2. Let X be a compact I-favorable space and w(X) = τ is uncountable. Then there exists a continuous inverse system S = {X δ , p δ γ , γ < δ < λ}, where λ = cf(τ ), of compact I-favorable spaces X δ and skeletal bonding maps p δ γ such that w(X δ ) < τ for each δ < λ and X = lim ← − S.
Proof. We embed X in a Tychonoff cube I A with |A| = τ and fix a special open base A = {U α : α ∈ A} for X of cardinality τ which consists of open sets such that for each α there exists a finite set H α ⊂ A with H α ∈ k(U α ). Let σ : n≥0 A n → A be a winning strategy. We represent A as the union of an increasing transfinite family {A δ : δ < λ} with |A δ | < τ , and let A δ = {U α : α ∈ A δ } for each δ < λ.
For any finite set C ⊂ A let γ C be a fixed countable base for X C . Observe that for every U ∈ A there exists a finite set B(U) ⊂ A such that B(U) ∈ k(U) and p B(U ) (U) is open in X B(U ) . We are going to construct by transfinite induction increasing families {B δ : δ < λ} and {B δ : δ < λ} ⊂ A satisfying the following conditions for every δ < λ:
(1) A δ ⊂ B δ ⊂ A, A δ ⊂ B δ , |B δ | = |B δ | < τ ; (2) B δ ∈ k(U) for all U ∈ B δ ; (3) p −1 C (γ C ) ⊂ B δ for each finite C ⊂ B δ ; (4) σ(U 1 , .
., U n ) ∈ B δ for every finite family {U 1 , .., U n } ⊂ B δ ; (5) B δ = γ<δ B γ and B δ = γ<δ B γ for all limit cardinals δ.
Suppose all B γ and B γ , γ < δ, have already been constructed for some δ < λ. If δ is a limit cardinal, we put B δ = γ<δ B γ and B δ = γ<δ B γ . If δ = γ + 1, we construct by induction a sequence {C(m)} m≥0 of subsets of A, and a sequence {V m } m≥0 of subfamilies of A such that:
• C 0 = B γ and V 0 = B γ ; • C(m + 1) = C(m) {B(U) : U ∈ V m }; • V 2m+1 = V 2m {σ(U 1 , .., U s ) : U 1 , .., U s ∈ V 2m , s ≥ 1}; • V 2m+2 = V 2m+1 {p −1 C (γ C ) : C ⊂ C(2m + 1) is finite}. Now, we define B δ = m≥0 C(m) and B δ = m≥0 V m .
It is easily seen that B δ and B δ satisfy conditions (1)- (5).
For every δ < λ let X δ = X B δ and p δ = p B δ . Moreover, if γ < δ, we have B γ ⊂ B δ , and let p δ γ = p B δ Bγ . Since A = δ<λ B δ , we obtain a continuous inverse system S = {X δ , p δ γ , γ < δ < λ} whose limit is X. Observe also that each X δ is of weight < τ because p δ (B δ ) is a base for X δ (see condition (3)).
Claim 1. All bonding maps p δ γ are skeletal. It suffices to show that all p δ are skeletal. And this is really true because each family B δ is stable with respect to σ, see (4). Hence, by [6,Lemma 9], for every open set V ⊂ X there exists W ∈ B δ such that whenever U ⊂ W and U ∈ B δ we have V ∩ U = ∅. The last statement yields that p δ is skeletal. Indeed, let V ⊂ X be open, and W ∈ B δ be as above. Then p δ (W ) is open in X δ because of condition (2). We claim that p δ (W ) ⊂ p δ (V ). Indeed, otherwise p δ (W )\p δ (V ) would be a nonempty open subset of X δ . So, p δ (U) ⊂ p δ (W )\p δ (V ) for some U ∈ B δ (recall that p δ (B δ ) is a base for X δ ). Since, by (2), p −1 δ (p δ (U)) = U and p −1 δ (p δ (W )) = W , we obtain U ⊂ W and U ∩ V = ∅ which is a contradiction.
Finally, since the class of I-favorable spaces is closed with respect to skeletal images [5,Lemma 1], all X δ are I-favorable.
An inverse system S = {X α , p β α , α < β < τ }, where τ is a given cardinal, is said to be almost continuous provided for every limit cardinal γ the space X γ is the almost limit of the inverse system S γ = {X α , p β α , α < β < γ}. If X = a − lim ← − S of an almost continuous inverse system S and H ⊂ X, the set
q(H) = {α : Int (p α+1 α ) −1 (p α (H)) \p α+1 (H) = ∅} is called a rank of H. Lemma 2.3. [13, Lemma 3.1] Let X = a − lim ← − S and U ⊂ X be open, where S = {X α , p β α , α < β < τ }
is almost continuous inverse system with skeletal bonding maps. Then we have:
(1) α ∈ q(U) if and only if (p α+1 α ) −1 Intp α (U) ⊂ p α+1 (U); (2) q(U) ∩ [α, τ ) = ∅ provided U = p −1 α (V ) for some open V ⊂ X α . Lemma 2.4. Let S = {X α , p β α , 1 ≤ α < β < τ }
be an almost continuous inverse system with skeletal bonding maps and X = a − lim ← − S. The the following hold for any open U ⊂ X:
(1) If (p α 1 ) −1 Intp 1 (U) ⊂ Intp α (U) for all α < τ , then p −1 1 Intp 1 (U) ⊂ U ; (2) If λ < τ and q(U) ∩ [λ, τ ) = ∅, then p −1 λ Intp λ (U) ⊂ IntU . Proof.
The first item was proved in [13, Lemma 3.2] under the assumption that X = lim ← − S, but the same arguments work in our situ-
ation. Item (2) is equivalent to the inclusion (p λ ) −1 Intp λ (U) ⊂ U. Let A be the set of all α ∈ (λ, τ ) with (p α λ ) −1 Intp λ (U) \ p α (U) = ∅. Suppose A is non-empty and let γ = min A. Observe that γ is a limit cardinal. Indeed, otherwise γ = β + 1 with β ≥ λ, so (p β λ ) −1 Intp λ (U) ⊂ Intp β (U). Since β ∈ q(U), according to Lemma 2.3(1), we have (p γ β ) −1 Intp β (U) ⊂ p γ (U). Hence, (p γ λ ) −1 Intp λ (U) ⊂ p γ (U), a contradiction.
Since S is almost continuous and γ is a limit cardinal, we have
X γ = a − lim ← − S γ , where S γ is the inverse system {X α , p β α , λ ≤ α < β < γ}. Because p γ is skeletal, U γ = Intp γ (U) = ∅. So, we can apply item (1) to X γ , the inverse system S γ and the open set U γ ⊂ X γ , to conclude that (p γ λ ) −1 Intp λ (U) ⊂ p γ (U). So, we obtain again a contradiction, which shows that (p α λ ) −1 Intp λ (U) ⊂ p α (U) for all α ∈ [λ, τ ). Finally, because the system S λ = {X α , p β α , λ ≤ α < β < τ } is almost continuous and X = a − lim ← − S λ , by item (1) we have p −1 λ Intp λ (U) ⊂ IntU .
Next lemma was established in [13] for continuous inverse systems. We present here a simplified proof concerning almost continuous systems.
Lemma 2.5. [13, Lemma 3.3] Let S = {X α , p β
α , α < β < τ } be an almost continuous inverse system with skeletal bonding maps and X = a − lim ← − S. Assume U, V ⊂ X are open with q(U) and q(V ) finite and
U ∩ V = ∅. If q(U) ∩ q(V ) ∩ [γ, τ ) = ∅ for some γ < τ , then Intp γ (U) and Intp γ (V ) are disjoint. Proof. Suppose Intp γ (U) ∩ Intp γ (V ) = ∅. We are going to show by transfinite induction that Intp β (U) ∩ Intp β (V ) = ∅ for all β ≥ γ.
Assume this is done for all β ∈ (γ, α) with α < τ . If α is not a limit cardinal, then α − 1 belongs to at most one of the sets q(U) and q(V ). (1)). Due to our assumption,
Suppose α − 1 ∈ q(V ). Hence, (p α α−1 ) −1 Intp α−1 (V ) ⊂ Intp α (V ) (see Lemma 2.3Intp α−1 (U) ∩ Intp α−1 (V ) = ∅. Moreover, p α α−1 p α (U) is dense in p α−1 (U). Hence, Intp α−1 (V ) meets p α α−1 p α (U) . This yields Intp α (V ) ∩ p α (U) = ∅. Finally, since p α (U) is the closure of its interior, Intp α (V ) ∩ Intp α (U) = ∅. Suppose α > γ is a limit cardinal. Since q(U) ∪ q(V ) is a finite set, there exists λ ∈ (γ, α) such that β ∈ q(U) ∪ q(V ) for all β ∈ [λ, α). Now, we consider the almost continuous inverse system S α = {X δ , p β δ , λ ≤ δ < β < α} with X α = a − lim ← − S α . Let U α = Intp α (U)
and V α = Intp α (V ) and denote by q α (U α ) and q α (V α ) the ranks of U α and V α with respect to the system S α . The, according to Lemma 2.
3(1), β ∈ [λ, α) does not belong to q α (U α ) if and only if (p β+1 β ) −1 Intp α β (U α ) ⊂ p α β+1 (U α ). Since p α β (U α ) = p β (U) and p α β+1 (U α ) = p β+1 (U), we obtain that β ∈ q α (U α ) is equivalent to β ∈ q(U). Similarly, β ∈ q α (V α ) iff β ∈ q(V ). Consequently, β ∈ q α (U α ) ∪ q α (V α ) for all β ∈ [λ, α). Then, according to Lemma 2.4(2), (p α λ ) −1 Intpλ(U) ⊂ Intp α (U) and (p α λ ) −1 Intpλ(V ) ⊂ Intp α (V ). Because Intp λ (U) ∩ Intp λ (V ) = ∅, we finally have Intp α (U) ∩ Intp α (V ) = ∅. This completes the transfinite induction. Therefore, Intp β (U) ∩ Intp β (V ) = ∅ for all β ∈ [γ, τ ). To finish the proof of this lemma, take λ(0) ∈ (γ, τ ) such that q(U) ∪ q(V ) ∩ [λ(0), τ ) = ∅.
Then, according to Lemma 2.4(2) we have the following inclusions:
• p −1 λ(0) Intp λ(0) (U) ⊂ IntU ; • p −1 λ(0) Intp λ(0) (V ) ⊂ IntV .
Since Intp λ(0) (U)∩Intp λ(0) (V ) = ∅, the above inclusions imply U ∩V = ∅, a contradiction. Hence, Intp γ (U) ∩ Intp γ (V ) = ∅.
Next proposition was announced in [14, Proposition 3.2] and a proof was presented in [13,Proposition 3.4] (see Proposition 3.2 below for a similar statement concerning inverse systems with nearly open projections).
Proposition 2.6. [14] Let S = {X α , p β α , α < β < τ } be an almost continuous inverse system with skeletal bonding maps such that X = a − lim ← − S. Then the family of all open subsets of X having a finite rank is a π-base for X. Proposition 2.7. Let X be a compact I-favorable space. Then every embedding of X in another space is π-regular.
Proof. We are going to prove this proposition by transfinite induction with respect to the weight w(X). This is true if X is metrizable, see for example [8, §21, XI, Theorem 2]. Assume the proposition is true for any compact I-favorable space Y of weight < τ , where τ is an uncountable cardinal. Suppose X is compact I-favorable with w(X) = τ . Then, by Proposition 2.2, X is the limit space of a continuous inverse system S = {X α , p β α , α < β < λ}, where λ = cf(τ ), such that all X α are compact I-favorable spaces of weight < τ and all bonding maps are surjective and skeletal. If suffices to show that there exists a π-regular embedding of X in a Tychonoff cube I A for some set A.
By Proposition 2.6, X has a π-base B consisting of open sets U ⊂ X with finite rank. For every U ∈ B let Ω(U) = {α 0 , α, α + 1 : α ∈ q(U)}, where α 0 < λ is fixed. Obviously, X is a subset of {X α : α < λ}. For every U ∈ B we consider the open set Γ(U) ⊂ {X α : α < λ} defined by Γ
(U) = {Intp α (U) : α ∈ Ω(U)} × {X α : α ∈ Ω(U)}. Claim 2. Γ(U 1 ) ∩ Γ(U 2 ) = ∅ whenever U 1 ∩ U 2 = ∅. Moreover, there exists β ∈ Ω(U 1 ) ∩ Ω(U 2 ) with p β (U 1 ) ∩ p β (U 2 ) = ∅. Let β = max{Ω(U 1 ) ∩ Ω(U 2 )}. Then β is either α 0 or max{q(U 1 ) ∩ q(U 2 )} + 1. In both cases q(U 1 ) ∩ q(U 2 ) ∩ [β, λ) = ∅. According to Lemma 2.5, Intp β (U 1 ) ∩ Intp β (U 2 ) = ∅. Since β ∈ Ω(U 1 ) ∩ Ω(U 2 ), Γ(U 1 ) ∩ Γ(U 2 ) = ∅.
For every U ∈ B and α let U α = Intp α (U). Obviously, this is true if |∆| = 1. Suppose it is true for all ∆ with |∆| ≤ n for some n, and let {α 1 , .., α n , α n+1 } be a finite set of n + 1 cardinals < τ .
Then V = i≤n p −1 α i (V α i ) ∩ U = ∅.
Since p α n+1 is a closed and skeletal map, W = Intp α n+1 (V ) is a non-empty subset of X α n+1 and
W ⊂ U α n+1 . Consequently V α n+1 ∩ W = ∅. So, V α n+1 ∩ p α n+1 (V ) = ∅ and i≤n+1 p −1 α i (V α i ) ∩ U = ∅. Claim 4. Γ(U) ∩ X is a non-empty subset of U for all U ∈ B.
We are going to show first that Γ(U) ∩ X = ∅ for all U ∈ B. Indeed, we fix such U and let Ω(
U) = {α i : i ≤ k} with α i ≤ α j for i ≤ j. By Claim 3, there exists x ∈ i≤k p −1 α i (U α i ) ∩ U. So, p α i (x) ∈ U α i for all i ≤ k. This implies Γ(U) ∩ X = ∅. To show that Γ(U) ∩ X ⊂ U, let y ∈ Γ(U) ∩ X and β(U) = max q(U) + 1. Then p β(U ) (y) ∈ Intp β(U ) (U).
Since α ∈ q(U) for all α ≥ β(U), according to Lemma 2.4(2), we have y ∈ p −1 β(U ) Intp β(U ) (U) ⊂ U . This completes the proof of Claim 4. According to our assumption, each X α is π-regularly embedded in I A(α) for some A(α). So, there exists a π-regular operator e α : T Xα → T I A(α) . For every U ∈ B define the open set θ 1 (U) ⊂ α<λ I A(α) , Let show that θ is π-regular. It follows from Claim 2 that θ(G 1 ) ∩ θ(G 2 ) = ∅ provided G 1 ∩ G 2 = ∅. On the other hand, for every open G ⊂ X we have θ(G) ∩ X ⊂ {Γ(U) ∩ X : U ∈ B and U ⊂ G}. Hence, by Claim 4, θ(G) ∩ X ⊂ {U : U ∈ B and U ⊂ G} ⊂ G. To prove that θ(G) ∩ X a dense subset of G it suffices to show that θ 1 (U) ∩ X = ∅ for all U ∈ B with U ⊂ G. To this end, we fix such U and let V α = e α (U α ) ∩ X α for every α ∈ Ω(U). Then V α is a dense open subset of U α , and by Claim 3, V = α∈Ω(U ) p −1 α (V α ) ∩ U is a nonempty subset of θ 1 (U) ∩ X. Therefore, X is π-regularly embedded in I A = α<λ I A(α) .
θ 1 (U) = α∈Ω(U ) e α Intp α (U) × α ∈Ω(U ) I A(α) .
Next proposition was established in [13] (Proposition 3.7) assuming that X is a π-regularly C * -embedded subset of the limit space of a σ-complete inverse system with open bounding maps and second countable spaces. The arguments there work if X is just a π-regularly embedded subset of a product of second countable spaces.
Proposition 2.8. Let X be a π-regularly embedded subspace of a product of second countable spaces. Then X is skeletally generated.
Proof of Theorem 1.1. To prove implication (1) ⇒ (2), suppose X is I-favorable subspace of a space Y . Then X = X βY is a compactification of X. Since X is also I-favorable, according to Proposition 2.7, X is πregularly embedded in βY . This yields that X is π-regularly embedded in Y .
(2) ⇒ (3) Let X be a subset of a Tychonoff cube I A . Then X is π-regularly embedded in I A , and by Proposition 2.8, X is skeletally generated.
The implication (3) ⇒ (1) follows as follows. If X is skeletally generated, then X = a − lim ← S, where S is an almost σ-continuous inverse system of second countable spaces X α , α ∈ A, and skeletal bounding maps p α β . Because each X α is I-favorable, it follows from [4, Theorem 3.3] (see also [6,Theorem 13]) that X is I-favorable too. ✷ Proof of Corollary 1.2. Suppose X is an I-favorable subspace of an extremally disconnected space Y . Then there exists a π-regular operator e : T X → T Y . We need to show that the closure (in X) of every open subset of X is also open. Since Y is extremally disconnected,
e(U) Y is open in Y . So, the proof will be done if we prove that e(U) Y ∩ X = U X for all U ∈ T X . Because e(U) ∩ X is a dense subset of U, we have U X ⊂ e(U) Y ∩X. Assume e(U) Y ∩X\U X = ∅ and choose V ∈ T X with V ⊂ e(U) Y \U X . Then e(V ) ∩ e(U) Y = ∅, so e(V ) ∩ e(U) = ∅.
The last one contradicts U ∩ V = ∅. ✷ Proof of Corollary 1.3. Suppose X is I-favorable and W ⊂ X is open. Then there is a π-regular embedding of X into a product Π of lines. Obviously, W is also π-regularly embedded in Π, and by Proposition 2.8, W is I-favorable. ✷
Quasi κ-metrizable spaces
Proof of Theorem 1.4. Suppose X is a compact I-favorable. We embed X in R τ for some cardinal τ , and let ρ(z, C) be a κ-metric on R τ , see [9]. According to Theorem 1.1, there exists a π-regular function e : T X → T R τ . We define a new function e 1 : T X → T R τ ,
e 1 (U) = {e(V ) : V ∈ T X and V ⊂ U}.
Obviously e 1 is π-regular and it is also monotone, i.e. U ⊂ V implies e 1 (U) ⊂ e 1 (V ). Moreover, for every increasing transfinite family γ = {U α } of open sets in Y we have e 1 ( α U α ) = α e 1 (U α ). Indeed, if z ∈ e 1 ( α U α ), then there is an open set V ∈ T X with V ⊂ α U α and z ∈ e(V ). Since V is compact and the family is increasing, V is contained in some U α 0 . Hence, z ∈ e(V ) ⊂ e 1 (U α 0 ). Consequently, e 1 ( α U α ) ⊂ α e 1 (U α ). The other inclusion follows from monotonicity of e 1 . Now, for every open U ⊂ X and x ∈ X we can define the function d(x, U) = ρ(x, e 1 (U)), where e 1 (U) is the closure of e 1 (U) in R τ . It is easily seen that d(x, U ) satisfies axioms K2) − K3). Let show that it also satisfies K4) and K1 * ). Indeed, assume {C α } is an increasing transfinite family of regularly closet sets in X. We put U α = IntC α for every α and U = α U α . Thus, e 1 (U) = α e 1 (U α ). Since {e 1 (U α )} is an increasing transfinite family of regularly closed sets in R τ ,
d(x, α C α ) = ρ(x, α e 1 (U α )) = inf α ρ(x, e 1 (U α )) = inf α d(x, C α ).
To show that K1 * ) also holds, observe that d(x, U) = 0 if and only if x ∈ X ∩ e 1 (U). Thus, we need to show that there is an open dense
subset V of X \ U such that X ∩ e 1 (U) = X \ V . Because e 1 (U) ∩ X is dense in U, U ⊂ e 1 (U). Hence, V = X \ e 1 (U) is contained in X \ U.
To prove V is dense in X \U , let x ∈ X \U and W x ⊂ X \U be an open neighborhood of x. Then W ∩ U is empty, so e 1 (W ) ∩ e 1 (U) = ∅. This yields e 1 (W ) ∩ X ⊂ V . On the other hand, e 1 (W ) ∩ X is a non-empty subset of W , hence W ∩ V = ∅. Therefore, d is an quasi κ-metric on X.
Suppose X is a compact space and let d(x, U ) be a quasi κ-metric on X. We are going to show that X is skeletally generated. To this end we embed X in I A for some A. Following the notations from the proof of Proposition 2.2, for any countable set B ⊂ A let A B be the countable base for X B = p B (X) consisting of all open sets in X B of the form X B ∩ α∈B V α , where each V α is an open subinterval of I = [0, 1] with rational end-points and V α = I for finitely many α. For any open U ⊂ X denote by f U the function d(x, U ). We also write p B ≺ g, where g is a map defined on X, if there is a map h : p B (X) → g(X) such that g = h • p B . Since X is compact this is equivalent to the following: Claim 5. For every countable set C ⊂ A there is B ∈ D with C ⊂ B. We are going to construct a sequence of countable sets B n ⊂ A such that for every n ≥ 1 we have:
if p B (x 1 ) = p B (x 2 ) for some x 1 , x 2 ∈ X, then g(x 1 ) = g(x 2 ). We say that a countable set B ⊂ A is d-admissible if p B ≺ f p −1 B (V ) for every V ∈ A B .• C ⊂ B n ⊂ B n+1 ; • p B n+1 ≺ f p −1 Bn (V ) for all V ∈ A Bn .
We show the construction of B 1 , the other sets B n can be obtained in a similar way. Every function f p C −1 (V ), V ∈ A C , has a continuous extension f p C −1 (V ) on I A . Moreover, every continuous function g on I A depends on countably many coordinates (i.e., there exists a countable set B g ⊂ A with π Bg ≺ g). This fact allows us to find a countable set
B 1 ⊂ A containing C such that p B 1 ≺ f p C −1 (V ) for all V ∈ A C . Next, let B = n=1 B n . Since A B is the union of all families {(p B Bn ) −1 (V ) : V ∈ A Bn }, n ≥ 1, for every W ∈ A B there is m and V ∈ A Bm with p −1 B (W ) = p −1
Bm (V ). Then, according to the construction of the sets B n ,
we have p B m+1 ≺ f p −1 B (W ) . Hence p B ≺ f p −1 B (W ) for all W ∈ A B , which means that B is d-admissible. Claim 6. For every B ∈ D the map p B is skeletal. Suppose there is an open set U ⊂ X such that the interior in X B of the closure p B (U) is empty. Then W = X B \ p B (U) is dense in X B . Let {W m } m≥1 be a countable cover of W with W m ∈ A B for all m. Since A B is finitely additive, we may assume that W m ⊂ W m+1 , m ≥ 1. Because B is d-admissible, p B ≺ f p −1 B (Wm) for all m. Hence, there are continuous functions h m : X B → R with f p −1 B (Wm) = h m • p B , m ≥ 1. Recall that f p −1 B (Wm) (x) = d(x, p −1 B (W m )) and p −1 B (W ) = m≥1 p −1 B (W m ). There- fore, f p −1 B (W ) (x) = d(x, p −1 B (W )) = inf m f p −1 B (Wm) (x) for all x ∈ X. Moreover, f p −1 B (W m+1 ) (x) ≤ f p −1 B (Wm) (x) because W m ⊂ W m+1 . The last inequalities together with p B ≺ f p −1 B (Wm) yields that p B ≺ f p −1 B (W ) . So, there exists a continuous function h on X B with d(x, p −1 B (W )) = h(p B (x)) for all x ∈ X. Since p B (p −1 B (W )) = W = X B , we have that h is the constant function zero. Then d(x, p −1 B (W )) = 0 for all x ∈ X. But p −1 B (W ) ∩ U = ∅. So, according to K1 * ), there is a dense open subset U ′ of U with d(x, p −1 B (W )) > 0 for each x ∈ U ′ , a contradiction.
It is easily seen that the union of any increasing sequence of dadmissible sets is also d-admissible. This fact and Claims 5 yield that the inverse system S = {X B : p B C : C ⊂ B, C, B ∈ D} is σ-continuous and X = lim ← S. Finally, by Claim 6, all maps p B , B ∈ D, are skeletal. So are the bounding maps p B C in S. Therefore, X is skeletally generated, and hence I-favorable by Theorem 1.1.
Proof of Corollary 1.5. Since Y = βX is I-favorable, by Theorem 1.4 there is a quasi κ-metric d on Y . We are going to show that d X (x, U X ) = d(x, U), U ∈ T X , defines a quasi κ-metric on X, where U X and U is the closure of U in X and Y respectively. Since U is regularly closed in Y , this definition is correct. It follows directly from the definition that d X satisfies axioms K2) and K3). Because for any increasing transfinite family {C α } of regularly closed sets in X the family {C α } is also increasing and consists of regularly closed sets in Y ,
d X (x, α C α X ) = d(x, α C α ) = inf α d(x, C α ) = inf α d X (x, C α ), d X satisfies K4). Finally, d X satisfies also K1 * ). Indeed, for any U ∈ T X there exists V ∈ T Y such that V is dense in Y \ U and d(x, U ) > 0 if and only if x ∈ V . This implies that the set V ∩ X is dense in X \ U X and d X (x, U X ) > 0 iff x ∈ V ∩ X. So, d X is a quasi κ-metric on X.
Inverse systems with nearly open bounding maps
In (1) α ∈ q(U) if and only if (p α+1 α ) −1 Intp α (U) ⊂ p α+1 (U);
(2) q(U)∩[α, τ ) = ∅ provided U = p −1 α (W ) for some open W ⊂ X α ; (3) Suppose q(U) and q(V ) are finite and U ∩ V = ∅. If q(U) ∩ q(V ) ∩ [γ, τ ) = ∅ for some γ < τ , then Intp γ (U) and Intp γ (V ) are disjoint.
Next proposition was announced in [14, Proposition 2.2] without a proof. Note that a similar statement was established in [9] for inverse systems with open bounding maps. Proof. We are going to show by transfinite induction that for every α < τ the open subsets U ⊂ X with q(U) ∩ [1, α] being finite form a base for X. Obviously, this is true for finite α, and it holds for α + 1 provided it is true for α. So, it remains to prove this statement for a limit cardinal α if it is true for any β < α. Suppose G ⊂ X is open and x ∈ G. Since p α is nearly open, G α = Intp α (G) contains p α (G) (here both interior and closure are taken in X α ). Let S α = {X γ , p β γ , γ < β < α}, Y α = lim ← − S α and p α γ : Y α → X γ are the limit projections of S α . Obviously, X α is naturally embedded as a dense subset of Y α and each p α γ restricted on X α is p α γ . So, there exists γ < α and an open set
U γ ⊂ X γ containing x γ = p γ (x) such that ( p α γ ) −1 (U γ ) ⊂ Int Yα G α Yα .
Consequently, (p α γ ) −1 (U γ ) ⊂ G α . We can suppose that U γ = IntU γ . Then, according to the inductive assumption, there is an open set W ⊂ X such that q(W ) ∩ [1, γ] is finite and
x ∈ W ⊂ p −1 γ (U γ ) ∩ G. So, x γ ∈ p γ (W ) ⊂ W γ = Intp γ (W ) and W γ ⊂ U γ . Hence, x ∈ p −1 γ (W γ ) ∩ G ⊂ G.
Next claim completes the induction.
Claim 7. q p −1 γ (W γ ) ∩ G ∩ [1, α] = q(W ) ∩ [1, γ]. Indeed, for every β ≤ γ we have p β p −1 γ (W γ ) ∩ G = p β (W )
. This implies
(6) q(W ) ∩ [1, γ] = q p −1 γ (W γ ) ∩ G ∩ [1, γ].
Moreover, since (p α γ ) −1 (W γ ) ⊂ (p α γ ) −1 (U γ ) ⊂ p α (G), we have p β p −1 γ (W γ ) ∩ G = p β p −1 γ (W γ ) for each β ∈ [γ, α]. Hence, (7) q p −1 γ (W γ ) ∩ G ∩ [γ, α] = q p −1 γ (W γ ) ∩ [γ, α]. Note that, by Lemma 4.1(2), q p −1 γ (W γ ) ∩ [γ, α] = ∅. Then the combination of (1) and (2) provides the proof of the claim.
Therefore, for every α < τ the open sets W ⊂ X with q(W ) ∩ [1, α] being finite form a base for X. Now, we can finish the proof of the proposition. If V ⊂ X is open and x ∈ V we find a set G ⊂ V with x ∈ G = p −1 β (G β ), where G β is open in X β for some β < τ . Then there exists an open set W ⊂ G containing x such that q(W ) ∩ [1, β] is finite. Let W β = Intp β (W ) and U = p −1 β (W β ∩ G β ). It is easily seen that x ∈ U and p ν (U) = p ν (W ) for all ν ≤ β. This yields q(U) ∩ [1, β] = q(W ) ∩ [1, β]. On the other hand, by Lemma 4.1(2), q(U) ∩ [β, τ ) = ∅. Hence U is a neighborhood of x which is contained in V and q(U) is finite.
Similar to the previous proposition, next one was also announced in [14,Proposition 2.3] without a proof. (1) X is regularly embedded in α<τ X α ;
(2) If, additionally, each X α is regularly embedded in a space Y α , then X is regularly embedded in α<τ Y α .
Proof.
(1) We consider the embedding of X in X = α<τ X α generated by the maps p α . According to Proposition 4.2, X has a base B consisting of open sets U ⊂ X with finite rank q(U). As in Proposition 2.7, for every U ∈ B let Ω(U) = {α 0 , α, α + 1 : α ∈ q(U)}, where α 0 < τ is fixed. For all U ∈ B and α < τ let U α = Intp α (U) and Γ(U) ⊂ {X α : α < τ } be defined by
Γ(U) = {U α : α ∈ Ω(U)} × {X α : α ∈ Ω(U)}.
Since p α (U) ⊂ U α for each α, U is contained in Γ(U). Using the arguments from the proof of Proposition 2.7, one can show that Γ(U) ∩ X ⊂ U. Finally, we define the required regular operator e : T X → T X by e(V ) = {Γ(U) : U ∈ B, U ⊂ V }.
(2) For each α < τ let e α : T Xα → T Yα be a regular operator. Define a function θ 1 : B → T Y , where Y = α<τ Y α , by
θ 1 (U) = α ∈Ω(U ) e α (U α ) × α ∈Ω(U ) Y α .
Consider θ : T X → T Y , θ(G) = {θ 1 (U) : U ∈ B and U ⊂ G}. Since θ 1 (U) ∩ X = Γ(U) and U ⊂ Γ(U) ⊂ U for any U ∈ B, θ(G) ∩ X = G. Moreover, Claim 4 implies that θ(G 1 ) ∩ θ(G 2 ) = ∅ provided G 1 ∩ G 2 = ∅. Thus, θ is a regular operator.
Claim 3 .
3α∈∆ p −1 α (V α )∩U = ∅ for every finite set ∆ ⊂ {α : α < λ}, where each V α is an open and dense subset of U α .
Now, we define a function θ from T X to the topology of α<λ I A(α) by θ(G) = {θ 1 (U) : U ∈ B and U ⊂ G}.
Denote by D the family of all d-admissible subsets of A. We are going to show that all maps p B : X → X B , B ∈ D, are skeletal and the inverse system S = {X B : p B C : C ⊂ B, C, B ∈ D} is σ-continuous with X = lim ← S.
this section we consider almost continuous inverse systems with nearly open bounding maps. Recall that a map f : X → Y is nearly open [1] if f (U) ⊂ Intf (U) for every open U ⊂ X. Nearly open maps were considered by Tkachenko [12] under the name d-open maps. The following properties of ranks were established in Lemmas 2.3-2.5 when consider almost continuous inverse systems with skeletal bounding maps. The same proofs remain valid and for inverse systems with nearly open bounding maps.
Lemma 4 . 1 .
41Let X = a − lim ← − S, where S = {X α , p β α , α < β < τ }is almost continuous with nearly open bonding maps. Then for every open sets U, V ⊂ X we have:
Proposition 4.2.[14] Let S = {X α , p β α , α < β < τ } be an almost continuous inverse system with nearly open bonding maps such that X = a − lim ← − S. Then the family of all open subsets of X having a finite rank is a base for X.
Proposition 4.3. [14] Let S = {X α , p β α , α < β < τ } bean almost continuous inverse system with nearly open bonding maps such that X = a − lim ← − S. Then:
Acknowledgments. The author would like to express his gratitude to A. Kucharski for several discussions.
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A private communication. A Kucharski, A. Kucharski, A private communication, May 2017.
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Skeletal maps and I-favorable spaces. A Kucharski, S Plewik, Acta Univ. Carolin. Math. Phys. 51A. Kucharski and S. Plewik, Skeletal maps and I-favorable spaces, Acta Univ. Carolin. Math. Phys. 51 (2010), 67-72.
Inverse systems and I-favorable spaces. A Kucharski, S Plewik, Topology Appl. 1561A. Kucharski and S. Plewik, Inverse systems and I-favorable spaces, Topology Appl. 156 (2008), no. 1, 110-116.
Game approach to universally Kuratowski-Ulam spaces. A Kucharski, S Plewik, Topology Appl. 1542A. Kucharski and S. Plewik, Game approach to universally Kuratowski-Ulam spaces, Topology Appl. 154 (2007), no. 2, 421-427.
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Functors and uncountable degrees of compacta, Uspekhi Mat. E Shchepin, Nauk. 363in RussianE. Shchepin, Functors and uncountable degrees of compacta, Uspekhi Mat. Nauk 36 (1981), no. 3, 3-62 (in Russian).
An external characterization of Dugundji spaces and k-metrizable compacta. L Shirokov, Dokl. Akad. Nauk SSSR. 2635in RussianL. Shirokov, An external characterization of Dugundji spaces and k-metrizable compacta, Dokl. Akad. Nauk SSSR 263 (1982), no. 5, 1073-1077 (in Russian).
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External characterization of I-favorable spaces. V Valov, Mathematica Balkanica. 251-2V. Valov, External characterization of I-favorable spaces, Mathematica Balkanica 25 (2011), no. 1-2, 61-78
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Nipissing University, 100 College Drive. [email protected]. Box. 5002Department of Computer Science and MathematicsDepartment of Computer Science and Mathematics, Nipissing Uni- versity, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada E-mail address: [email protected]
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[
"Chirp assisted ion acceleration via relativistic self induced transparency",
"Chirp assisted ion acceleration via relativistic self induced transparency"
]
| [
"Shivani Choudhary \nDepartment of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia\n",
"Amol R Holkundkar \nDepartment of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia\n"
]
| [
"Department of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia",
"Department of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia"
]
| []
| We study the effect of the chirped laser pulse on the transmission and associated ion acceleration by the sub-wavelength target. In the chirped laser pulses, the pulse frequency has a temporal variation about its fundamental frequency, which manifests to the temporal dependence of the critical density (n c ). In this work we used a chirp model which is beyond the linear approximation. For negatively (positively) chirped pulses, the high (low) frequency component of the pulse interacts with the target initially followed by the low (high) frequency component. The threshold plasma density for the transmission of the pulse is found to be higher for the negatively chirped laser pulses as compared to the unchirped or positively chirped pulses. The enhanced transmission of the negatively chirped pulses for higher densities (6n c ) results in very efficient heating of the target electrons, creating a very stable and persistent longitudinal electrostatic field behind the target. The void of the electrons results in expansion of the target ions in either direction, resulting in the broad energy spectrum. We have introduced a very thin, low density (< n c ) secondary layer behind the primary layer. The ions from the secondary layer are then found to be accelerated as a mono-energetic bunch under the influence of the electrostatic field created by the primary layer upon interaction by the negatively chirped pulse. Under the optimum conditions, the maximum energy of the protons are found to be ∼ 100 MeV for 10 fs (intensity fwhm); Circularly Polarized; Gaussian; negatively chirped laser pulse with peak intensity ∼ 8.5 × 10 20 W/cm 2 . | 10.1063/1.5039918 | [
"https://arxiv.org/pdf/1807.07804v1.pdf"
]
| 53,574,865 | 1807.07804 | 9898b86b8fc3e2e952bb56939db8a2652af7f97f |
Chirp assisted ion acceleration via relativistic self induced transparency
Shivani Choudhary
Department of Physics
Birla Institute of Technology and Science -Pilani
333031RajasthanIndia
Amol R Holkundkar
Department of Physics
Birla Institute of Technology and Science -Pilani
333031RajasthanIndia
Chirp assisted ion acceleration via relativistic self induced transparency
(Dated: October 7, 2018)
We study the effect of the chirped laser pulse on the transmission and associated ion acceleration by the sub-wavelength target. In the chirped laser pulses, the pulse frequency has a temporal variation about its fundamental frequency, which manifests to the temporal dependence of the critical density (n c ). In this work we used a chirp model which is beyond the linear approximation. For negatively (positively) chirped pulses, the high (low) frequency component of the pulse interacts with the target initially followed by the low (high) frequency component. The threshold plasma density for the transmission of the pulse is found to be higher for the negatively chirped laser pulses as compared to the unchirped or positively chirped pulses. The enhanced transmission of the negatively chirped pulses for higher densities (6n c ) results in very efficient heating of the target electrons, creating a very stable and persistent longitudinal electrostatic field behind the target. The void of the electrons results in expansion of the target ions in either direction, resulting in the broad energy spectrum. We have introduced a very thin, low density (< n c ) secondary layer behind the primary layer. The ions from the secondary layer are then found to be accelerated as a mono-energetic bunch under the influence of the electrostatic field created by the primary layer upon interaction by the negatively chirped pulse. Under the optimum conditions, the maximum energy of the protons are found to be ∼ 100 MeV for 10 fs (intensity fwhm); Circularly Polarized; Gaussian; negatively chirped laser pulse with peak intensity ∼ 8.5 × 10 20 W/cm 2 .
I. INTRODUCTION
The advent of high power lasers promised the vast number of applications, covering both applied and fundamental aspects of basic sciences. The laser-plasma based acceleration of the ions and electrons paved the possibility of constructing a table-top [1][2][3], high energy, charged particle beams for medical [4,5] and industrial applications [6,7]. In the last couple of decades, we have already witnessed the experimental realizations of the acceleration of ions to multi MeV of energies via various acceleration mechanisms. The Target Normal Sheath Acceleration (TNSA) [8,9], Radiation Pressure Acceleration (RPA) [10,11], Hole Boring (HB) [12], Breakout Afterburner (BOA) [13], are among few very well studied mechanisms both theoretically and experimentally. A wonderful review of this contemporary field of laser-induced ion acceleration is presented by Macchi et al. [14] in which all the ion acceleration mechanisms are covered in depth.
The laser-driven ion acceleration in the relativistic selfinduced transparency (RSIT) regime is also proving to be the fascinating mechanism to have a very efficient high energy ion and neutron beams [15][16][17][18][19]. The ion and neutron beams with energies around ∼ 180 MeV have been observed in experiments by the RSIT mechanism [16,17]. In general, the dispersion relation for the electromagnetic wave propagation in plasmas, ω 2 = k 2 c 2 + ω 2 p , ignores the interaction of the laser fields with the medium, which in principle alters the electron density and so the dispersion relation via the plasma frequency ω p = n e e 2 /ε 0 m e . The critical plasma density, n c = ε 0 m e ω 2 /e 2 , as predicted by the above dispersion relation has to be corrected for the laser-plasma interaction dynamics. If we take into account the fact, that the plasma electrons will * [email protected] † [email protected] respond to the laser electric field, the modified dispersion relation reads, ω 2 = k 2 c 2 + ω 2 p /γ, where γ = 1 + (p/m e c) 2 and p is the electron momentum. The high-intensity laser beams can efficiently heat the plasma electrons, as a consequence, the ion acceleration can be enhanced [15,[20][21][22][23][24][25].
The introduction of the chirp in the laser pulse is also proving to be a promising way to have enhanced ion energy beams [26][27][28][29][30][31]. In the chirped pulses, the pulse frequency has temporal variation about its fundamental frequency, which manifests in the temporal dependence of the critical density n c as well. In this study, we aim to characterize the effect of laser pulse chirp on the ion energies under RSIT regime. We consider a chirp model which is beyond the linear approximation [27], the chirp model used is in close analogy with the experimental technique for the pulse amplification i.e. Chirped Pulse Amplification. In order to understand how the chirp of the laser pulse affects the transmission through the target, we developed a simplified wave propagation model for the laser with a 0 < 1. The model takes into account the chirp of the pulse while calculating the target density as pulse traverses the target. The results of this simplified wave propagation are found to be consistent with the 1D PIC simulation. Furthermore, we consider the dual layer sub-wavelength target to have a very efficient generation of the accelerated ion bunch from the secondary layer [11,30,32]. Recently, the effect of the pulse shape on the enhanced ion acceleration is reported in Ref. [31] under Radiation Pressure regime of ion acceleration. However, in this study we are concerned in the relativistic transparency regime, and how the chirp of the laser pulse can affect the transmission of the pulse through the target.
The paper is organized as follows. In Section-II, we discuss the simplified wave propagation model for low laser amplitudes (a 0 = 0.5) and its utility to compute the transmission coefficients for different chirp values. We compare the results of the wave propagation model with the 1D PIC simulations. In Section-III, we focus on the study of high intense lasers (a 0 = 20) with the target. The results showing the effect of the laser pulse chirp on the longitudinal electric field for different target parameters is discussed. Moreover, we consider the need for the second layer to obtain an energetic ion bunch. The optimization study for different target parameters is then carried out in Section-IV, and finally, we give the summary and concluding remarks in Section-V.
1.4 E x /a 0 Ω(t) ζ = −5 (a) E x /a 0 Ω(t) (b) ζ = 0 E x /a 0 Ω(t) ωt (c) ζ = 5
II. THEORY AND SIMULATION MODEL
The interaction of the intense laser beams with the overdense plasmas are in general modeled by the cold-relativistic fluid model [20,[33][34][35][36]. The use of the cold-relativistic fluid model is justified, as the quiver velocity of the electrons involved in laser-plasma interaction exceeds the electron temperatures, and ions are considered as immobile in the time scales of the interests. The threshold plasma densities for RSIT as obtained by seeking the stationary solutions of the cold relativistic fluid model for a semi-infinite plasma slab is reported in Refs. [20,34,36].
The set of relativistic cold fluid equations are in general challenging to solve for very thin targets, and hence kinetic simulations are routinely used for studying the laser interaction with thin overdense plasma layers [37][38][39][40]. In this paper, rather than obtaining the stationary solutions for a threshold plasma density, we studied the effect of pulse chirp on the transmission coefficient of the target for a given laser and target parameters. We have used the sub-wavelength target as it allows the transmission from the slightly over-dense plasmas as well. The transmission coefficients are calculated by numerically solving the wave propagation equation along with the corrected electron density, taking into account the timedependent frequency and amplitude of the circularly polarized chirped laser pulse. In the following, we present our simplified wave propagation model to study the effect of the pulse chirping on the transmission coefficient, followed by the comparison with the 1D PIC simulation.
A. Wave propagation model Throughout the paper we will be using the dimensionless units. The laser amplitude is normalized as a = eA/m e c, where A is the vector potential associated with laser, e and m e are charge and mass of the electron. The time and space are normalized against the laser frequency (ω) and wave number (ωt → t and kx → x) respectively. The electron density is normalized against the critical density n c = ε 0 ω 2 m e /e 2 . In the dimensionless form the EM wave propagation in plasma can be written as [41],
∂ 2 a ∂ z 2 − ∂ 2 a ∂t 2 = n e γ e a(1)
where, γ e is the relativisitic factor for electron and n e is the electron density. Using the definition of canonical momentum, the γ e can be expressed as,
γ e = 1 + a 2 + (p e z ) 2(2)
here, p e z is the dimensionless (p e z /m e c → p e z ) longitudinal component of the electron momentum. In general the electron density will be a function of both z and t. The spatial dependence of the electron density is because of the finite target geometry, however the temporal dependence comes via the chirp of the laser pulse. In a chirped laser pulse the frequency varies with time and hence the associated critical density [n c (t) = ε 0 Ω(t) 2 m e /e 2 ] for the laser pulse will also vary accordingly. It should be noted that, here we are not solving the full set of dynamical fluid equations, and hence the electron density is not going to evolve with time by continuity equation. In the later part we will see, that this approximation is valid if we intend to calculate the transmission coefficient of the target for the laser pulses with a 0 < 1. Furthermore, for a 0 < 1 one can ignore the longitudinal electron heating and so Eq. 2 reduces to,
γ e ∼ 1 + a 2 .(3)
The electron density profile in space and time is then given by, n e (z,t) ≡ n e (z)
Ω(t) = n 0 Ω(t) exp − 2 24 ln(2) (z − z 0 ) 24 d 24(4)
here, Ω(t) is a time dependent frequency of the chirped laser pulse (at the peak of the laser pulse Ω = 1), n 0 is the peak target density, d is its thickness and z 0 is the location of the target center. The electron density profile (Eq. 4) serves the dual purpose, not only it mimics the thin layer target, rather it is a continuous function of z as well, which is desirable for numerical stability. The simulation domain is considered to be L = 100λ long, and the target (see Eq. 4) of thickness d is placed at 25λ (z 0 = 25λ + d/2). The transmission coefficient (T ) is calculated by taking the ratio of transmitted pulse energy to the incident pulse energy. The boundary condition on the left of the simulation domain is precisely the temporal profile of the laser pulse. In this paper we have used the laser pulse model as proposed by Mackenroth et. al. [27], because it models the laser pulse chirp beyond the linear approximation [26,[28][29][30] and it is also in close analogy to the model of chirped pulse amplification [42]. The boundary conditions on left side of the simulation domain for a = a xx + a yŷ read as,
a x (0,t) = a 0 √ 2 exp − 4 ln(2) t 2 τ 2 cos[t + g(t, ζ )](5)a y (0,t) = a 0 √ 2 exp − 4 ln(2) t 2 τ 2 sin[t + g(t, ζ )](6)g(t, ζ ) = ζ 4 ln(2) t 2 τ 2 + τ 2 16 ln(2)(1 + ζ 2 ) + tan −1 (ζ ) 2(7)
here, a 0 is peak laser amplitude in dimensionless units, τ is dimensionless FWHM of the laser pulse, and ζ is the chirp parameter. The laser pulse profiles of unchirped, positively and negatively chirped are illustrated in Fig. 1. The time dependent frequency of the laser pulse is then given by [27],
Ω(t) = 1 + ζ 8 ln(2) τ 2 t(8)
For, ζ > 0 (ζ < 0) the low (high) frequency part interacts with the target first, followed by the high (low) frequency part.
To understand how the chirp affects the transmission coefficient of the laser pulse, we have calculated the transmission coefficient for different chirp values by numerically solving the Eqs. 1, 3 -8. The transmission coefficient of a positively, negatively and unchirped laser pulse (a 0 = 0.5, τ = 5 cycles) for different target thickness with n 0 = 1n c is compared, and the results are presented in Fig. 2. We observe from Fig. 2, that as we increase the target thickness the transmission coefficient for the unchirped pulse drops by ∼ 30% with ∼ 100% increase in the target thickness. However, for chirped pulses, the decrease in the transmission coefficient with target thickness is marginal (∼ 5%) with the same variation in the target thickness. It can be understood by the nature of the chirped pulse itself. If the variation in the target thickness is smaller than the wavelength of the pulse, then, in that case, the transmission coefficient associated with the longer (smaller) wavelength (frequency) would not be affected. On the other hand for shorter (larger) wavelength (frequency) component, the skin depth is anyway much larger than the thickness of the target. The collective effect of the chirping would manifest in more or less similar transmission coefficients as we vary the target thickness in the sub-wavelength domain. In the following, we compare the results of this simplified wave propagation model with 1D PIC simulation.
B. Comparison with PIC simulations
Next, we turn to a comparison of the simplified wave propagation model (Fig. 2) with PIC-simulations. The 1D Particle-In-Cell simulation (LPIC++) [43] is carried out to study the effect of target thickness on the transmission coefficient for different chirped values. We have modified this open-source 1D-3V PIC code, to include the multilayer targets, chirped Gaussian laser pulses, and associated diagnostics. In this code the electric fields are normalized as we earlier discussed (a 0 = eE/m e ωc). However space and time are taken in units of laser wavelength (λ ) and one laser cycle τ = λ /c respectively, mass and charge are normalized with electron mass and charge respectively. We have used 100 cells per laser wavelength with each cell having 50 electron and ion macroparticles. The spatial grid size and temporal time step for the simulation are considered to be 0.01λ and 0.01τ respectively.
In Fig. 3, we present the transmission coefficient dependence on the target thickness for different chirped values as calculated by the PIC simulations. The agreement with the simplified wave propagation model (Fig. 2) is found to be excellent. It is clear that for a 0 = 0.5, the electron heating is not very pronounced, or we would have observed the effect of the positive (low frequency interacts first) and negatively (high frequency interacts first) chirped pulses. The approximation we made in wave propagation model regarding the p e z , and Ω(t) are found to be consistent with the PIC simulations as well.
III. RESULTS AND DISCUSSIONS
As we have seen, the chirp of the laser pulse can enhance the effective transmission of the laser pulse over unchirped laser pulses. For a 0 = 0.5 we did not observe a very prominent difference between the positively and the negatively chirped pulses. This is so, as the interaction dynamics is mostly governed by the transverse motion of the electrons. The omission of the p e z in our simplified wave propagation model seems to be consistent with the fully relativistic 1D PIC simulation. However, for a 0 1, this might not be true, as the process is too non-linear to be approximated by this simple wave propagation model.
The chirp effect on the transmission coefficient and on the interaction in general would be much pronounced for a 0 1, as the positively chirped pulse tends to compress the target initially, increasing the target density for the high-frequency part to interact. To study the interaction of the high intense (a 0 1) chirped laser beams with thin targets in RSIT regime, the kinetic simulations are essential, as the fluid model can no longer be used for such scenarios. Now we study the interaction of the Gaussian, Circularly polarized laser pulse having peak amplitude a 0 = 20, and FWHM duration of 5 cycles with a target of thickness 0.75λ . The 1D-3V PIC code LPIC++ is used for this purpose [43]. The simulation domain is considered to be 100λ and the target (protons + electrons) of thickness 0.75λ is placed at 25λ .
The laser incidents on the target from the left side. It should be noted that for the cases when a 0 /[π(n e /n c )(d/λ )] < 1, the ion acceleration is mainly dominated by the Ligh Sail mechanism [44,45]. On the other hand, the RSIT mechanism begins to prevail in the regime where the ratio a 0 /[π(n e /n c )(d/λ )] 1. The parameters used in the current study (a 0 = 20, d = 0.75λ and n e 8n c ) clearly indicates that the RSIT regime would prevail. The laser parameters used i.e. a 0 = 20, Circularly Polarized are routinely accessible in ELI laser facility [46].
A. Chirp effect on threshold plasma density
We present the variation of the threshold plasma density with chirp parameter (ζ ) in Fig. 4. Here, "threshold plasma density" is the target density which allows some percentage fraction of the incident pulse to pass through the target for a given chirp parameter. The threshold plasma density for 80% and 1% transmission coefficients are presented for −5 ≤ ζ ≤ 5 (see, Fig. 4). We observe that the threshold plasma density increases for negatively chirped laser pulses in either scenario (80% and 1% transmission). In case of the positively chirped (ζ > 0) pulses, the low-frequency part interacts with the target initially followed by the high-frequency component. The lowfrequency component tends to compress the electron layer, increasing the electron density for the high-frequency component to interact. However, in case of negatively chirped pulses (ζ < 0) the high-frequency component interact with the target initially followed by the low-frequency part. For highfrequency EM wave, corresponding critical density is also high, which enable it to transmit through the target without much of attenuation.
In Fig. 5, the spatial snapshots (as evaluated at 60τ) of the electron density, ion density, longitudinal field and laser field for different chirp values are shown. The snapshots are also compared for two different target densities, viz 3n c (upper panel) and 6n c (lower panel). We observe that for 3n c case all the quantities are showing the similar characteristics for different chirp values. However, for 6n c case we can see the distinctive spike of the electron density [see, Fig. 5(e)] at ∼ 50λ for ζ = −5, this manifests in enhanced flat longitudinal electrostatic field [see, Fig. 5(g)]. This can be understood from the fact that the transmission coefficient of the target with density 3n c is 80% for −5 ≤ ζ ≤ 5 (see, Fig. 4). Moreover, for 6n c case only the negatively chirped pulse will be having the 1% transmission (see, Fig. 4), in fact the pulse with ζ = −5 can have ∼ 1% transmission for the target with density ∼ 8n c . The transmission of the negatively chirped pulse can be observed in Fig. 5(h). Furthermore, for ζ = 0 and ζ = 5, the transmission is < 1%, as a consequence, the associated pondermotive force of the laser pulse tend to push the electrons inside the target, increasing the electron density as a small spike in electron density around ∼ 50λ coincides with the location of the pulse after transmission, Fig. 5(h).
From the above analysis one can deduce that even 1% transmission of intense laser beams is strong enough to heat the electrons to relativistic energies. As the electrons escape the target for ζ = −5 case, it leaves the target positively charged, resulting in the expansion of the target ions in either direction, as seen in Fig. 5(f).
B. Spatio-temporal evolution of electrostatic field
So far we have learned that the transmission of the pulse for ζ = −5 results in efficient heating of the electrons followed by a very persistent electrostatic field, a few electrons are dragged away from the target. The rapid heating and excursion of the electrons from the target leaves the target charged, as a consequence the target ions expand under its own coulomb repulsion. To further elucidate this fact, in Fig. 6 we present the spatial and temporal evolution of the electron density and longitudinal electrostatic field for ζ = −5, 0 and 5 for the case when target density is 6n c , all other laser parameters are same as Fig. 5. The compression of the target can be seen for unchirped and positively chirped laser pulses. However, for the negatively chirped pulse (ζ = −5), some electrons are accelerated by the transmitted pulse and starts co-moving with the laser pulse, this is mainly due to the nature of the circularly polarized pulse. For circularly polarized pulse the suppression of J×B heating of the electrons leads the push along the direction of the pulse propagation. This excursion of the electrons is the reason; we can see the approximately constant longitudianl electrostatic field configuration for z ≥ 25λ in Fig. 6(d). static field can be harnessed to obtain a mono-energetic proton bunch.
C. Need for secondary layer
We have seen so far, that the negatively chirped pulse efficiently create a very stable electrostatic field, as it transmits through the target. However, the excursion of the electrons leaves the target positively charged, as a consequence the target ions expand in either direction because of the Coulomb repulsion of the ions itself. The expansion of the target ions in either direction manifests in very broad energy distribution, on the contrary for any practical applications, a monoenergetic ion bunches are desirable. To have a mono-energetic ion bunches from the current setup, a very thin, low density (< n c ) secondary layer is introduced just behind the primary layer. The low density of the secondary layer ensures that the interaction dynamics and in general the formation of the electrostatic field by the primary layer remains mostly unaffected even by the presence of the secondary layer. As the laser passes through this composite target (primary + secondary), the electrons are dragged away with the laser pulse, and the ions from the secondary layer experience a very persistent electrostatic field, leading to their acceleration as a monoenergetic ion bunch.
Next, we study the interaction of the negatively chirped (ζ = −5), Gaussian, circularly polarized, 5 cycle laser pulse having peak amplitude a 0 = 20 with the composite target. The thickness (density) of the primary layer (PL) is considered to be d = 0.75λ (n e = 6n c ). However, for the secondary layer (SL) the thickness and density are considered to be 0.2λ and 0.1n c respectively. In Fig. 7, the phasespace plots of ions from PL and SL are presented at 40τ, 50τ and 60τ. The expansion of the ions of the PL in either direction is visible in Fig. 7(a,b,c). However, the ions from the SL are found to be accelerated as a bunch, Fig. 7(d,e,f). The velocity spectrum of the ions from the PL and SL are also illustrated for different time instances in Fig. 7(g,h,i). We present the energy spectrum of the ions from the SL in Fig. 8 at 50τ, 70τ and 90τ. We observe that for the negatively chirped pulse with given laser and target parameters, the maximum number of ions from the SL are accelerated to ∼ 30 MeV, however, the maximum energy (E max ) of the bunch is observed to be ∼ 75 MeV, as evaluated at 90τ. For the sake of completeness, the effect of the pulse chirp on the energy spectrum of the SL is also illustrated in Fig. 8. For the positively and unchriped laser pulses the target density 6n c will be in the opaque regime, as a consequence the laser will be reflected from the PL. The heating of the electrons at the rear side of the PL is not very efficient, and hence the ions from the SL are not very efficiently accelerated for the positively and unchirped laser pulses.
IV. OPTIMIZATION
In Fig. 9 , the variation of the maximum energy of the ions from the SL is presented for different chirp parameters. The effect of the pulse chirp is illustrated for fixed laser amplitude a 0 = 20 and different primary target densities (a), and for fixed primary target density and different laser amplitudes (b). As the target density for a 0 = 20 is varied, we observe that for ζ = −5, the maximum energy of ∼ 105 MeV is obtained for ∼ 6n c . The threshold plasma density for a 0 = 20 and ζ = −5 is ∼ 7n c [see, Fig. 4] and hence the transmission of the pulse for 6n c is > 1%, leading towards the stable electrostatic formation as we discussed in the previous sections. However, for ζ = 5 case, the target with density 6n c would be opaque, and hence ions from SL will not be efficiently accelerated. We further observe that for ζ = 5 the maximum ion energy is seen to be for the case when target density is ∼ 4n c . As we discussed earlier, the leading low-frequency component of the positively chirped pulse tends to compress the target density by the radiation pressure, as a consequence the following high-frequency component interacts with the high-density target, resembling a similar scenario as negatively chirped pulse interacting with
Emax (MeV) X Y Z (c) E z z/λ X Y Z (a) @90τ
No. of Particles
E (MeV) X Y Z (b)
FIG. 10. The longitudinal electrostatic field (a) and energy spectrum of the ions from secondary layer (b) are presented for three different primary target conditions. However, the maximum ion energy of the secondary layer is also presented for different thicknesses and densities of the primary layer (c). The laser pulse with peak amplitude a 0 = 20, duration 5 cycles and chirp parameter ζ = −5 is considered. The target conditions (d/λ , n e /n c ) are X (0.55,5.2), Y (0.75,5.8), and Z (0.95,6.8). All the quantities are evaluated at 90τ. the high-density target. We have also studied the effect of the laser intensity on the maximum ion energy from the secondary layer for fixed target density of 6n c . It can be seen from Fig. 9(b) that for a 0 15, the target with density 6n c and thickness 0.75λ would be opaque, and hence the electrostatic field generation is suppressed and so the acceleration of the ions from the secondary layer. However, as the a 0 increases an efficient acceleration is observed for the negatively chirped pulse.
So far we have seen that the negatively chirped pulses are efficient in generating very persistent and stable electrostatic field behind the target. The effect of the thickness and density of the PL on the maximum ion energy by the negatively chirped pulse is presented in Fig. 10. We have varied the thickness of the PL from 0.5λ − 1λ , and density from 5n c − 8n c , and maximum ion energy (as evaluated at 90τ) of the ions from SL (0.2λ , 0.1n c ) are calculated for a 0 = 20, ζ = −5, 5 cycles, circularly polarized, Gaussian laser pulse. In Fig. 10, we have also presented the electrostatic field and energy spectrum of the ions from the secondary layer as evaluated at 90τ for three different target parameters (d/λ , n e /n c ) namely X (0.55,5.2), Y (0.75,5.8), and Z (0.95,6.8). In all of the three cases we observe very stable flat electrostatic field behind the primary layer [see, Fig. 10(a)]. The optimum target parameters for given laser conditions are found to be Y (0.75,5.8) where maximum ion energy is observed to be ∼ 100 MeV. We have fixed the parameters of the SL throughout the simulations. The main purpose of the SL is to have an accelerated proton bunch. The parameters of the SL neither alter the electrostatic field formed by the primary layer nor affects the laser pulse propagation, and hence the ions of SL mere serve as test particles.
V. CONCLUDING REMARKS
In this work, we have studied the effect of the laser chirping on the acceleration of the protons via relativistic self-induced transparency. In the negatively (positively) chirped pulse, the high (low) frequency component of the pulse interacts with the target initially followed by the low (high) frequency component. The temporal variation of the frequency in the chirped EM pulse manifests in the associated time-dependent critical density, as a consequence the threshold plasma density of the negatively chirped laser pulse is found to be comparatively higher than the unchirped and positively chirped laser pulses. The initial low-frequency interaction of the positively chirped pulse with the target tend to compress the target layer by the radiation pressure, as a consequence, the target would be opaque for positively chirped pulses. Furthermore, as the negatively chirped pulse transmits through the target, the suppression of the J × B heating of the circularly polarized laser results in longitudinal push on the electrons; as a result, few electrons get dragged away and start co-moving with the laser pulse. This imbalance leaves the target positively charged followed by the expansion of the target ions under its own coulomb repulsion. However, the removal of the electrons also generate a very stable and persistent electrostatic field behind the primary layer which can be harnessed by the composite target geometry, comprised of the low density, thin secondary layer behind the primary layer. The ions from the secondary layer are found to be accelerated as a bunch under the effect of the longitudinal field created by the primary layer (secondary layer does not affect the field formation in a profound manner) upon interaction by the negatively chirped laser pulse with this composite target.
Under optimum conditions the maximum energy of the ions from the secondary layer is found to be ∼ 100 MeV for circularly polarized, Gaussian, 5 cycles FWHM, negatively chirped (ζ = −5), laser with peak amplitude a 0 = 20. These parameters for 800 nm laser would translates to ∼ 10 fs (intensity FWHM) pulse with peak intensity ∼ 8.5 × 10 20 W/cm 2 . However, similar energies are reported in the past, but with much higher a 0 values. For example, in Ref. [12] the authors have demonstrated the ion acceleration in the HB-RPA regime and reported the proton energies ∼ 150 MeV by irradiating a laser with peak amplitude a 0 ∼ 90. Similarly, under the RPA regime, ion acceleration to ∼ 150 MeV with a 0 = 108 has also been reported in Ref. [47].
In summary, we have studied the effect of the pulse chirp on the transmission of the laser pulse through the sub-wavelength target and associated ion acceleration. The transmission coefficient for a 0 = 0.5 are estimated by a simplified wave propagation model which takes into account the time dependent critical density of the target. The results of this simplified wave propagation model are found to be consistent with the 1D fully relativistic PIC model. In this work we have used the chirp model which is beyond the linear approximation. The chirp model used in this study is in close analogy of the idea of the Chirped Pulse Amplification. Furthermore, we studied the interaction of the intense laser pulse with a 0 = 20 with the target having the thickness 0.75λ and density 6n c for different chirp parameters. It has been observed that the negatively chirped laser pulse is very efficient in creating a stable and persistent electrostatic field behind the target. The electrostatic field created behind the target can be harnessed by a low density, thin secondary layer behind the target. The optimization of the target parameters are finally carried out to have a maximum energy of the accelerated ions of the secondary layer.
The feasibility of the proposed scheme under experimental scenario needs full 3D Particle-in-Cell simulations, which is currently beyond the scope of the current manuscript. Soon, we plan to study the effect of the higher dimensions on the ion acceleration under RSIT regime. For this purpose we would like to explore the 3D PIC codes like EPOCH [48], or PICCANTE [49].
ACKNOWLEDGMENTS
Authors would like to acknowledge the Department of Physics, Birla Institute of Technology and Science, Pilani, Rajasthan, India for the computational support.
FIG. 1 .
1Laser pulse profiles with negative (a), unchirped (b) and positive (c) chirp parameters. Time dependent frequency is also illustrated (dotted line) for each case as well.
FIG. 2 .
2Transmission coefficient of the laser pulse (a 0 = 0.5, τ = 5 cycles) for chirp parameters ζ = −5, 0, 5 for different target thickness of density n 0 = 1n c is compared. The schematic diagram representing the target geometry (Eq. 4), incident pulse and transmitted pulse is illustrated as an inlet.
FIG. 3 .
3PIC simulation for transmission coefficient of the laser pulse (a 0 = 0.5, τ = 5 cycles) for chirp parameters ζ = −5, 0, 5 for different target thickness of density n 0 = 1n c is compared.
FIG. 4 .
4Variation of threshold target density for 80% and 1% transmission with chirp parameter.
FIG. 5 .
5[see, Fig. 5(e)]. The electrostatic field formed by the compressed electron layer tend to pull the target ions, increasing the ion density as well [see, Fig. 5(f)]. Now, on the contrary for ζ = −5, the transmission coefficient is > 1%, and hence as the pulse exits the target it drags the electrons with it as well. This motion of the electrons can be observed in the Fig. 5(e), The effect of pulse chirp is illustrated for two different target densities, 3n c (upper panel) and 6n c (lower panel). The spatial snapshot at 60τ for the electron density (a,e), ion density (b,f), longitudinal electric field (c,g) and transverse laser profile (d,h) is presented for a 0 = 20, τ = 5 cycles and d = 0.75λ (the target is placed at 25λ ) .
FIG. 6 .
6Spatio-temporal profile of the electron density (upper panel) and the longitudinal electrostatic field (lower panel) are presented for chirp parameters ζ = −5 (left column), ζ = 0 (center column) and ζ = 5 (right column). The laser parameters are same asFig. 5with target density n e = 6n c .
FIG. 7 .
7We will see in the following how this kind of constant electro-Phasespace plots for the ions from the primary layer (d = 0.75λ , n e = 6n c ) at different time instances (left column) and ions from secondary layer (0.2λ , 0.1n c ) are presented (center column). The velocity spectrum for the ions from primary and secondary layers are also illustrated (right column).
FIG. 8 .
8The energy spectrum of the ions from the secondary layer are presented at different time instances for ζ = −5 (a). Moreover, the energy spectrum of secondary ions as evaluated at 90τ for different chirp parameters is also compared (b). The laser parameters are same asFig. 7.
FIG. 9 .
9The effect of the pulse chirping on the maximum energy of the ions from the secondary layer is presented for different primary target density (a). The thickness of the primary layer is 0.75λ and peak laser amplitude is a 0 = 20. The variation of the maximum ion energy with a 0 is presented in (b), here we have fixed the density of the primary layer to 6n c .
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[
"Aspirated capacitor measurements of air conductivity and ion mobility spectra",
"Aspirated capacitor measurements of air conductivity and ion mobility spectra"
]
| [
"K L Aplin \nSpace Science and Technology Department\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxonUK\n"
]
| [
"Space Science and Technology Department\nOX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxonUK"
]
| []
| Measurements of ions in atmospheric air are used to investigate atmospheric electricity and particulate pollution. Commonly studied ion parameters are (1) air conductivity, related to the total ion number concentration, and (2) the ion mobility spectrum, which varies with atmospheric composition. The physical principles of air ion instrumentation are long-established. A recent development is the computerised aspirated capacitor, which measures ions from (a) the current of charged particles at a sensing electrode, and (b) the rate of charge exchange with an electrode at a known initial potential, relaxing to a lower potential. As the voltage decays, only ions of higher and higher mobility are collected by the central electrode and contribute to the further decay of the voltage. This enables extension of the classical theory to calculate ion mobility spectra by inverting voltage decay time series. In indoor air, ion mobility spectra determined from both the novel voltage decay inversion, and an established voltage switching technique, were compared and shown to be of similar shape. Air conductivities calculated by integration were: 5.3 ± 2.5 fSm -1 and 2.7 ± 1.1 fSm -1 respectively, with conductivity determined to be 3 fSm -1 by direct measurement at a constant voltage. Applications of the new Relaxation Potential Inversion Method (RPIM) include air ion mobility spectrum retrieval from historical data, and computation of ion mobility spectra in planetary atmospheres. | 10.1063/1.2069744 | [
"https://export.arxiv.org/pdf/physics/0510149v1.pdf"
]
| 56,308,638 | physics/0510149 | b2704d6f683bb488dd71d560c9abcc721b0ead1d |
Aspirated capacitor measurements of air conductivity and ion mobility spectra
Revised July 26, 2005
K L Aplin
Space Science and Technology Department
OX11 0QXRutherford Appleton Laboratory, Chilton, DidcotOxonUK
Aspirated capacitor measurements of air conductivity and ion mobility spectra
Revised July 26, 20051
Measurements of ions in atmospheric air are used to investigate atmospheric electricity and particulate pollution. Commonly studied ion parameters are (1) air conductivity, related to the total ion number concentration, and (2) the ion mobility spectrum, which varies with atmospheric composition. The physical principles of air ion instrumentation are long-established. A recent development is the computerised aspirated capacitor, which measures ions from (a) the current of charged particles at a sensing electrode, and (b) the rate of charge exchange with an electrode at a known initial potential, relaxing to a lower potential. As the voltage decays, only ions of higher and higher mobility are collected by the central electrode and contribute to the further decay of the voltage. This enables extension of the classical theory to calculate ion mobility spectra by inverting voltage decay time series. In indoor air, ion mobility spectra determined from both the novel voltage decay inversion, and an established voltage switching technique, were compared and shown to be of similar shape. Air conductivities calculated by integration were: 5.3 ± 2.5 fSm -1 and 2.7 ± 1.1 fSm -1 respectively, with conductivity determined to be 3 fSm -1 by direct measurement at a constant voltage. Applications of the new Relaxation Potential Inversion Method (RPIM) include air ion mobility spectrum retrieval from historical data, and computation of ion mobility spectra in planetary atmospheres.
Introduction
Atmospheric molecular cluster-ions are formed by natural radioactive isotopes and cosmic rays, and are central to the electrical properties of air. The original "Gerdien condenser" 1 aspirated capacitor is a widely used instrument for terrestrial atmospheric ion measurements. It consists of a cylindrical outer electrode containing a coaxial central electrode, with a fan to draw air between the electrodes. With an appropriate bias voltage applied across the electrodes, a current flows which is proportional to the air conductivity. (The "conductivity measurement régime" requires an adequate ventilation speed to be maintained for the bias voltage selected; the régime's existence can be verified by an approximately Ohmic response in measured current to a changing bias voltage). Early ion measurements inferred the air conductivity from the rate of voltage decay or relaxation across the electrodes, using a gold-leaf or fibre electrometer 2 . As electronics technology developed, this technique was augmented by direct measurements of the current. Sophisticated contemporary instruments under computer control combine the "current measurement" and "voltage decay" measurement modes for self-calibration 3,4 .
Surface measurements with modern instrumentation suggest that, although generally comparable, conductivities from the two measurement modes (i.e. ion current and voltage decay) are not always completely consistent 3 . The study reported in [3] is believed to be the first direct comparison of the two modes, and motivated reconsideration of the theoretical assumptions underlying air conductivity measurement with an aspirated capacitor in the voltage decay mode. This paper describes a new technique, the Relaxation Potential Inversion Method (RPIM), enabling ion mobility spectra to be retrieved from voltage decay measurements.
Improved ion measurements are needed for solar-terrestrial physics, pollution studies and assessment of long-term geophysical changes in atmospheric electrical parameters 5 .
2.
Classical theory of air conductivity measurement with an aspirated cylindrical capacitor
The electrical conductivity of air σ is the product of air ion number concentration n, and ion mobility µ. The ion mobility spectrum n(µ) describes the distribution of ion number concentration with mobility, and µ is inversely related to the size and molecular mass of the cluster. Molecular ions with 0.5 < µ < 3 cm 2 V -1 s -1 are conventionally defined as "small ions", as their size is limited by thermodynamic constraints on their lifetime, which generally inhibit ion growth to µ ~ 0.5 cm 2 V -1 s -1 6 .
Applying these limits, the air conductivity due to positive or negative ions σ ± is given by
µ µ µ σ d n e s V cm s V cm ∫ − − − − ± ± ± = 1 1 2 1 1 2 3 5 . 0 ) ( Equation 1
where e is the charge on the electron, µ ± is the positive or negative ion mobility, and n ± (µ) the number of positive or negative ions with a given mobility. Because of the large differences between the mobility of small ions and aerosol particles, it is usually possible to assume that only small ions contribute to the conductivity, except in very polluted air. In this case, the lower limit of the integral can change, and intermediate and large ions of lower mobility may also be abundant enough to contribute 7 . Mohnen defined the mean mobility µ as the mode of the ion distribution 8 , usually 1.3-1.6 cm 2 V -1 s -1 for positive and 1.3-1.9 cm 2 V -1 s -1 for the chemically different negative ions 6 .
At typical ionisation rates of 10 cm -3 s -1 , surface continental atmospheric ion concentrations are ~100-2000 cm -3 , and typical air conductivity can therefore vary considerably from ~2-100 fSm -1 . The determining influences on surface conductivity are aerosol pollution number concentrations (aerosol reduces the air conductivity, except in highly polluted air) 6,9 and the ion production rate from cosmic rays and geological sources.
The electrical mobility of small ions and aerosol particles differ by several orders of magnitude, so it is usually assumed that the principal contributions to the air conductivity are from ions with the same µ , and total number concentration N. The capacitance term C accounts for radial electric field variations within the electrode system, usually found empirically to allow for connection and end effects 4 , and ε 0 is the permittivity of free space. Full derivations of Equation 3 are given in [9], [10] and [11].
In the Voltage Decay mode, the voltage established across the capacitor electrodes decays due to the current i flowing through the air, of resistance R, Figure 1. If the instantaneous charge stored by the capacitor is Q, elementary circuit analysis gives
V C Q R dt dQ = − = , Equation 4
As described in [11], Gauss's Law relates i to the air conductivity σ by 0 ε σQ i − = .
Equation 5 Substituting Eq 5 into Eq 4, and differentiating with respect to time, gives
V dt dV 0 ε σ − = . Equation 6
If σ is constant, the solution of Equation 6 gives the instantaneous voltage at a time t, V(t), for an initial applied voltage V 0 ,
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = τ t V t V exp ) ( 0 , Equation 7
where τ is a time constant = ε 0 /σ, so Equation 7 can be rewritten as
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 0 0 exp ) ( ε σ t V t V , Equation 8
Equation 8 has been the standard expression for calculating air conductivity from voltage decay measurements throughout the history of air ion instrumentation, using τ determined from a time series of voltage data 2,3,12 .
It is important to emphasise that the Voltage Decay and Current Measurement modes are fundamentally different in the mobility of ions assumed to be selected. For both modes, the ion mobility contributing to the measurement is assumed to be constant, but the longer duration required for Voltage Decay measurements requires the assumption that the mobility spectrum being sampled is constant for longer than the measurement period. If very long decay timescales are considered, this implies that such measurements could be susceptible to temporal or other fluctuations in ion mobility.
Modification to the classical theory of the Voltage Decay mode
In calculations of the conductivity, the critical mobility µ c is assumed to represent the minimum mobility of ion contributing to the measurement 10 [10] where k is a geometrical constant:
( ) L b a b a k 2 ln ) ( 2 2 − = .
Equation 10
Critical mobility is a function of bias voltage, and therefore ion mobility spectra can be found by changing the voltage at the central electrode 13 . It is possible for some ions with mobility lower than the critical mobility to enter the cylindrical capacitor, but this effect is negligible except in polluted air with very high concentrations of larger charged particles 14 . In this paper it is assumed that only ions with mobility greater than the critical mobility contribute to the measurements. The implications of this assumption will be discussed in section 5.
During a Voltage Decay measurement, both the voltage, and therefore the critical mobility (Equation 9) vary continuously. As a consequence, the decaying voltage across the capacitor's electrodes changes the mobility distribution of the ions contributing to charge exchange. This modulation of the ion mobility spectrum, and therefore, from Equation 1, air conductivity, invalidates the assumption used in the derivation of Equation 7, that the ion spectrum selected for measurement remains constant. The behaviour of an aspirated capacitor in voltage decay mode cannot be completely described by Equation 7: in an ideal instrument, differences from Equation 7 will arise from changes in critical mobility during the decay.
Differences from the exponential decays predicted by Equation 7 appear detectable in measurements of voltage decays in atmospheric air, following a series of measurements made over several months which rarely showed the exponential decays expected based on classical assumptions 9 . Additionally, past voltage decays measured in the free troposphere were also non-exponential 12 . Figure 2 shows exponential fits to typical voltage decay time series in surface atmospheric air. Natural variability in the measurement is expected to cause some fluctuations in the time series, particularly in the polluted boundary layer, but the existence of free tropospheric non-exponential decays is more difficult to explain using the theory outlined in section 2. Classical theory can be modified to account for the variation in critical mobility during voltage decay measurements.
Computing ion spectra from voltage decay measurements
As conductivity is effectively the mobility integral over the ion spectrum (Equation 1), every voltage decay time series contains, in principle, ion spectrum information.
The relationship between voltage decay measurements and the ion spectrum information can be determined by substituting Equation 1 for the conductivity term in Equation 6, and evaluating the resulting integral in two parts. The first part is with respect to mobility, from the maximum ion mobility in the air µ m down to the critical mobility evaluated at time t, µ c (t). The second part is evaluated from µ c (t) to the critical mobility at the start of the decay µ c (0), written as
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − = ∫ ∫ m c c c t t c t d t n d n t e V V µ µ µ µ µ µ µ µ µ µ µ ε ) ( ) ( ) 0 ( 0 0 ) ( ) ( ) ( ln . Equation 11
Although Equation 11 is not generally analytically soluble, n(µ) can be found by using a finite difference numerical method, applicable for small changes. This permits calculation of ion spectra from voltage time series, which is the basis of the Relaxation Probe Inversion Method (RPIM). It can also be used for prediction of voltage decays from a given ion spectrum 9 .
Numerical solution procedure
Writing the two integral terms in Equation 11 as I and M gives During a finite, small time difference between t j-1 and t j+1 the voltage will have decayed slightly, and caused a small increase in the instrument's critical mobility.
Thus Equation 12 can be written in finite difference form as with the changes in I and M, ∆I j and ∆M j approximated at each time j by:
[ ] j j c j c j n t t I ) ( ) ( 1 1 − + − = ∆ µ µ Equation 14 ( ) ( ) ⎪ ⎭ ⎪ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = ∆ + − + − − + j j c j c j j c j c j j c j c j n t t t t t t t M ) ( ) ( 2 1 ) ( ) ( ln ) ( ) ( 1 1 1 1 1 1 µ µ µ µ τ µ µ Equation 15
where µ c (t j ) is the critical mobility evaluated from the voltage (Equation 9)
Numerical example with a synthetic ion spectrum
A numerical example demonstrates the inversion of a voltage decay generated from a synthetic ion mobility spectrum, broadly similar to the observed mean small ion spectrum 6 . A voltage decay time series for the aspirated cylindrical capacitor described in [3] and [9] was generated from the ion spectrum, Figure 4. Inversion of this voltage decay using the RPIM algorithm (Section 4.1) results in an identical spectrum to the original, indicating that the mathematical inversion is exact. However the typical shape of small ion spectrum is generated as an average over many measurements; the spectrum inverted numerically here was an average of 8615 hourly averaged ion spectra taken over 14 months 6 . The example demonstrates that the RPIM can correctly determine ion spectra from voltage decay data, but independent, and ideally synchronous, measurements of voltage decays and ion mobility spectra are necessary in practical evaluations. It should also be noted that the second term in Equation 11 is small compared to typical experimental uncertainty, although it has been included in this numerical example for exactness.
Comparison of ion spectra measured in laboratory air
The RPIM was verified by comparing ion mobility spectra calculated from voltage decays, to spectra computed from the established technique of voltage switching to vary the critical mobility 13,15 . The experiments were carried out in ambient indoor air, in a demonstrably stable atmospheric electrical environment (described in detail in [16]), using the computer-controlled Programmable Ion Mobility Spectrometer where µ is the average mobility. Increasing the bias voltage in steps from V i-1 (through V i ) to V i+1 causes a change in critical mobility from µ c(i-1) to µ c(i+1) , which will increase the magnitude of the ion current from i (i-1) to i (i+1) . The ion concentration in the mobility range centred on µ ci , n i (µ ci ), can be written as
( ) ( ) ( ) ) 1 ( ) 1 ( ) 1 ( ) 1 ( 0 + − + − − − = i i i i i ci i i i V Ce n µ µ ε µ .
Equation 17
Experimentally, the rate of change of current with critical mobility can be determined by using linear regression between a set of measured current and critical mobility values (calculated for each bias voltage using Equation 9). Methods for measuring the capacitance term in Equation 17 are described in [4] and [17]).
Both the RPIM and the voltage switching technique assume that no ions with mobility lower than the critical mobility can enter the cylindrical capacitor. However, some of these larger charged particles, which constitute the particulate space charge, can contribute to the measurement by drifting into the instrument and colliding with the sensing electrodes. The magnitude of the error from particulate space charge can be estimated by measuring the current at the central electrode with zero bias voltage applied, i.e. the current arising from particles unaffected by the electric field. This has been referred to previously as a "dynamic zero". 10 . A more rigorous approach to find the dynamic zero is to calculate the intercept of the bipolar current-voltage plots obtained during the voltage switching measurements, which is a time-averaged dynamic zero. This dynamic zero was subtracted to produce the i-V curve shown in Figure 5b, but corresponded to 2.7 pCm -3 of negative space charge, or a maximum of 17 (singly charged) particles cm -3 .
The mean ion spectrum was calculated using the RPIM as in Section 4.1, and ion spectra from the voltage decay and current mode methods are shown in Figure 6. The spectra are similar in shape, and the mobility of the common peak is consistent with positive ion properties in the literature 8,11 . If the positive air conductivity due to small ions is calculated by integration across the mobility spectrum (Equation 1), then σ (current mode) = 2.7 ± 1.1 fSm -1 , with the error determined from the variability in current measurements. Mean σ (voltage decay mode) = 5.3 ± 2.5 fSm -1 , where the error is the standard deviation across the three measured spectra. This is the same as positive air conductivity measured at the same location by an aspirated cylindrical capacitor operating in current mode at a constant bias voltage, ~3 fSm -1 [16]. The co-located peaks, well-correlated spectral shape and consistent integral spectra all give confidence in the RPIM approach, although there is some disagreement between the concentrations calculated at the extremes of the small ion range, 1.5 > µ > 2.3 cm 2 V -1 s -1 , The comparison of the new RPIM spectra with spectra obtained using the well-established technique shows firstly that the inversion generates reproducible and realistic ion mobility spectra, and secondly that the air conductivity computed by integration across the spectrum is comparable with that found in the same environment by a different method.
Discussion
The classical theory of air conductivity measurement for voltage decays from an aspirated capacitor can be modified to correct for the assumption that the critical mobility of ions sampled by the instrument does not vary during the decay. The spectral information extracted from voltage decay data with the RPIM can be used to calculate conductivity directly by integration rather than with erroneous simplifying assumptions, such as exponential decay.
RPIM has practical as well as theoretical advantages. Measurements of atmospheric ion spectra (reviewed in detail in [9]) are often obtained by varying the bias voltage to change the ion mobilities selected e.g. [13], [14]. As in the example described in and is plotted on the same axes as the original artificial spectrum.
aspirated cylindrical capacitor operating in Current Measurement mode, with sufficient ventilation to ensure ions reaching the central electrode are constantly replenished, the ion current at the central electrode is proportional to the unipolar air conductivity. The motion of an ion in the radial electric field between the cylindrical capacitor's electrodes can be used with Equation 2 to derive the conductivity due to positive or negative ions σ ± from the current i at the central electrode arising from bias voltage V
(Figure 1 ) 10 .
110. It is defined from consideration of the motion of ions in the radial electric field at a ventilated capacitor For a cylindrical geometry, µ c is a function of ventilation speed u, length L, central and outer electrode radii a and b and bias voltage V given by:
at each time j, and τ j the instantaneous decay time constant. The incremental changes, ∆I j and ∆M j can be evaluated for each mobility strip of width [µ c (t j+1 ) -µ c (t j-1 )], with mobility calculated using Equation 9 from V(t j+1 ) and V(t j-1 ). The ion concentration in each mobility strip n j can then be estimated. The inversion yields a mobility spectrum with N-1 points if the original voltage decay has N points; the highest voltage (corresponding to the lowest ion mobility) does not have a corresponding spectral point because it provides the initial voltage V 0 . The steps involved in the inversion procedure are summarised inFigure 3
.
( 5b .
5bPIMS), with sensing electrodes 0.25m long and of radii 11mm and 2mm, ventilated at 2.1 ms -1 3,4 . Voltage decay measurements were carried out using a Keithley 236 SourceMeter instrument to supply, and then measure (to a specified accuracy of to ± 0.025 % + 10mV), the voltage across the ventilated PIMS electrodes. Data was logged by a PC at 1Hz via a GPIB/IEEE interface. The maximum voltage supplied by the Keithley 236 is 110 V, corresponding to a critical mobility of 0.16 cm 2 V -1 s -1 ± 0.025 %. Three voltage decay events from nominally 110 V -3 V, each of duration 2-3 hours were measured on 23 and 24 March 2005, Figure 5a. The Current Mode ion spectrum measurements took place on 24-25 March 2005, at the same temperature and relative humidity as the voltage decay measurements.Voltage switching through a predetermined sequence of error checking modes (as in[4]), and 15 bipolar voltages from -20.8 V -21.1 V was implemented in software, and data logged via the RS232 port. Equation9 can be used to calculate critical mobilities; only results from the positive voltage sequence (3.9V -21.1V) are used here, corresponding to positive ion critical mobilities in the range 0critical mobility result principally from the calibration of the digital to analogue converter used to generate the bias voltage, ± ~10%. Ion currents were sampled at 1Hz and averaged over 20s for each voltage step, and 90s of empirically determined recovery time was allowed between each change in bias voltage. The mean currents are plotted as a function of bias voltage inFigure Combining
Figure Captions
Figure Captions
Figure 1
1Schematic of an aspirated cylindrical capacitor showing a plan view of the end of the tube (centre), a section through the tube (left) and the equivalent circuit (right). The motion of a charged particle through the tube is indicated (left). A charging voltage V 0 is applied and released to measure ions in the Voltage Decay mode (right).
Figure 2
2Consecutive voltage decay time series measured in urban atmospheric air on June 12 1998. (The experimental apparatus and other results are described in detail in [3]). Coefficient of determination (R 2 ) values are shown to indicate the fraction of variance in a data set explained by an exponential model. For 39 voltage decays measured over two weeks, the mean R 2 was 0.9 with a standard deviation of ± 0.1 (the range of values was 0.31-1.00).
Figure 3
3Flow chart illustrating the algorithm developed to invert voltage decay data to obtain an ion mobility spectrum.
Figure 4
4Inversion of an artificial ion mobility spectrum, chosen to closely represent an atmospheric ion mobility spectrum. The experimental voltage decay time series the artificial spectrum would have generated is shown (inset), with a best fit line to an exponential. The voltage decay time series was then inverted back to an ion spectrum,
Figure 5 a
5) Average of three voltage decays in indoor air measured with the Programmable Ion Mobility Spectrometer on 23-24 March 2005. b) Currents measured over a range of bias voltages at the same location on 24-25 March 2005.The y-axis error bars are the standard error of the mean.
Figure 6
6Comparison of average indoor small positive ion spectra generated from the Relaxation Probe Inversion Method and by the established bias voltage switching technique. Typical errors in the x-axis mobility values are ±10%. Estimated errors in ion concentrations are ±30% for the switched voltage spectrum, and ±40% for the RPIM spectrum.
Figure 1
inlet caused by bias voltage switching, which may perturb ion ingress and cause transient saturation in the instrument, briefly preventing any measurements17 . Voltage decay measurements need only the simplest single-channel logging equipment and are ideally suited to remote in situ sensing applications, such as balloon-borne measurements18 . Data processing to compute the ion spectrum and integrated conductivity would typically be carried out off-line.The RPIM assumes that no intermediate or large ions can enter the cylindrical capacitor; however, it can be seen from Section 5 that the magnitude of uncertainty introduced by this assumption is much lower than the variability between individual spectra. RPIM can also be used for extraction of spectral information from historical atmospheric electrical data sets. An important new application is the inversion of voltage decay measurements made in the atmospheres of other planets, such as from the European Space Agency Huygens probe which used voltage relaxation techniques during the first in situ measurements of Titan's atmosphere19 .Section 5, this requires dedicated electronics to switch the bias voltage, and sensitive
current sensing. Both time and mobility resolution can be poor, as compromises must
be made between the voltage size step, and the time the instrument rests at each bias
voltage to obtain an averaged current. Adequate time must also be allowed for the
circuitry to recover from switching transients. RPIM exploits continuous voltage
decay, which avoids delays and errors from switching transients. Smooth voltage
decays also minimise the sharp changes in electric field at the cylindrical capacitor
AcknowledgementsI thank Dr C.F. Clement for his assistance with mathematics, and Dr R.
. H Gerdien, Phys. Zeitung. 6H. Gerdien, Phys. Zeitung 6, 800-801 (1905)
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Formation, nature and mobility of ions of atmospheric importance. V A Mohnen, Electrical processes in atmospheres. Dolezalek H.Darmstadt, GermanySpringer VerlagV.A. Mohnen, Formation, nature and mobility of ions of atmospheric importance, In: Dolezalek H. (ed.), Electrical processes in atmospheres, Springer Verlag, Darmstadt, Germany (1974)
. K L Aplin, UKThe University of ReadingPhD thesisK.L. Aplin, PhD thesis, The University of Reading, UK (2000)
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Atmospheric Electricity. J A Chalmers, Pergamon PressOxford, UK2 nd editionJ.A. Chalmers, Atmospheric Electricity, 2 nd edition, Pergamon Press, Oxford, UK (1967)
Measurement of the electrical potential gradient and conductivity by radiosonde at Poona, India. S P Venkiteshwaran, Recent advances in atmospheric electricity. Smith L.G.Oxford, UKPergamon PressS.P. Venkiteshwaran, Measurement of the electrical potential gradient and conductivity by radiosonde at Poona, India, In: Smith L.G. (ed.), Recent advances in atmospheric electricity, Pergamon Press, Oxford, UK (1958)
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. M Borucki, A Coradini, P Coustenis, E Falkner, R Flamini, M Grard, A M Hamelin, G W Harri, J J Leppelmeier, J A M Lopez-Moreno, C P Mcdonnell, F H Mckay, A Neubauer, G Pedersen, V Picardi, R Pirronello, K Rodrigo, Borucki, M. Coradini, A. Coustenis, P. Falkner, E. Flamini, R. Grard, M. Hamelin, A.M. Harri, G.W. Leppelmeier, J.J. Lopez-Moreno, J.A.M. McDonnell, C.P. McKay , F.H. Neubauer, A. Pedersen, G. Picardi, V. Pirronello, R. Rodrigo, K.
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| []
|
[
"arXiv:astro-ph/0007098v1 7 Jul 2000 Extreme Scattering Events: An Observational Summary",
"arXiv:astro-ph/0007098v1 7 Jul 2000 Extreme Scattering Events: An Observational Summary"
]
| [
"T Joseph \nNaval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n\n",
"W Lazio [email protected] \nNaval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n\n",
"Alan L Fey [email protected] \nNaval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n\n",
"R A Gaume [email protected] \nNaval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n\n"
]
| [
"Naval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n",
"Naval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n",
"Naval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n",
"Naval Research Laboratory\nUS Naval Observatory\nUS Naval Observatory\n"
]
| []
| We review observational constraints on the structures responsible for extreme scattering events, focussing on a series of observations of the quasar PKS 1741−038. VLA observations were conducted to search for changes in the rotation measure and H i absorption during the ESE, while VLBI observations sought ESEinduced changes in the source's image. No RM changes were found implying B || < 12 mG, and no H i opacity changes were found implying N (H i) < 6.4×10 17 cm −2 . No multiple imaging was observed, but the diameter of the source increased by 0.7 mas, contrary to what is predicted by simple refractive lens modeling of ESEs. We summarize what these limits imply about the structure responsible for this ESE. | 10.1023/a:1013190208674 | [
"https://export.arxiv.org/pdf/astro-ph/0007098v1.pdf"
]
| 18,146,951 | astro-ph/0007098 | ae3dfd48f695c0bf77c68db942042fa43d4b6680 |
arXiv:astro-ph/0007098v1 7 Jul 2000 Extreme Scattering Events: An Observational Summary
T Joseph
Naval Research Laboratory
US Naval Observatory
US Naval Observatory
W Lazio [email protected]
Naval Research Laboratory
US Naval Observatory
US Naval Observatory
Alan L Fey [email protected]
Naval Research Laboratory
US Naval Observatory
US Naval Observatory
R A Gaume [email protected]
Naval Research Laboratory
US Naval Observatory
US Naval Observatory
arXiv:astro-ph/0007098v1 7 Jul 2000 Extreme Scattering Events: An Observational Summary
We review observational constraints on the structures responsible for extreme scattering events, focussing on a series of observations of the quasar PKS 1741−038. VLA observations were conducted to search for changes in the rotation measure and H i absorption during the ESE, while VLBI observations sought ESEinduced changes in the source's image. No RM changes were found implying B || < 12 mG, and no H i opacity changes were found implying N (H i) < 6.4×10 17 cm −2 . No multiple imaging was observed, but the diameter of the source increased by 0.7 mas, contrary to what is predicted by simple refractive lens modeling of ESEs. We summarize what these limits imply about the structure responsible for this ESE.
Introduction
Extreme scattering events (ESE) are a class of dramatic decreases ( 50%) in the flux density of radio sources near 1 GHz for several weeks bracketed by substantial increases (Fielder et al., 1994;Fig. 1). Because of their simultaneity at different wavelengths and light travel time arguments, ESEs are likely a propagation effect (Fiedler et al., 1987). First identified toward extragalactic sources, ESEs have since been observed toward pulsars (Cognard et al., 1993;Maitia et al., 1998).
To date, the only other observational constraints on the structures responsible for ESEs-besides the light curves-are the lack of pulse broadening and the variation in the pulse times of arrival during the pulsar ESEs. This paper summarizes constraints obtained during the ESE toward the quasar 1741−038. We discuss Faraday rotation measurements by Clegg et al. (1996) in §2, VLBI imaging by Lazio et al. (2000a) in §3, and H i absorption measurements by Lazio et al. (2000b) in §4. We present our conclusions in §5. Figure 1 shows the ESE of 1741−038 with the epochs of the various observations indicated.
Faraday Rotation Measure Observations
At each epoch, the polarization position angle φ was measured at 6 to 10 frequencies and then fit, by minimizing χ 2 and accounting for nπ ambiguities, as a function of the observing wavelength, λ 2 . The same procedure was used for 1741−038 and for the parallactic angle calibrator 1725+044.
Based on the relative invariability of RM toward 1725+044 and 1741−038, Clegg et al. (1996) conclude ∆RM < 10.1 rad m −2 during the ESE. This upper limit implies an upper limit on the mean magnetic field parallel to the line of sight of B || < 12 mG for a typical value of the free electron column density through the structure, N 0 ∼ 10 −4 pc cm −3 (Clegg et al., 1998).
This upper limit is much larger than the typical interstellar field strength but enhancement of the ambient field to B ∼ 1 mG is possible within a shock front (Clegg et al., 1988). Alternately, ∆RM may be small if the field is disordered or if the ionized region is not magnetized.
VLBI Imaging
Comparison of the visibility data during the ESE to those obtained after the ESE (1994 July 8) show the source to be more resolved during the ESE. The excess angular broadening is ∆θ d 0.7 mas, implying that the ESE structure contributed a scattering measure SM ESE = 10 −2.5 kpc m −20/3 . In turn, the pulse broadening of a background pulsar should be τ d ≤ 1.1D kpc µs at 1 GHz, consistent with the observed lack of broadening during pulsar ESEs (Cognard et al., 1993;Maitia et al., 1998). The refractive models commonly used to explain ESEs predict that the source's flux density and angular diameter should be highly correlated. We observe an anti-correlation. Simple stochastic broadening models require much more scattering (2 mas) than is observed. We consider it likely that both refractive defocussing and stochastic broadening are occurring.
We were unable to test a key prediction of refractive modelsangular position wander of the background source-because these observations had no absolute position information. A second prediction is multiple imaging. During this ESE, any secondary image(s) must have been extremely faint; multiple imaging, with the secondary image slightly offset from the primary, is unlikely to explain the increase in angular diameter because no other effects of strong refraction are seen (cf. also Clegg et al., 1998). We also observe little, if any, ESE-induced anisotropy in the VLBI images. If ESE lenses are filamentary structures (Romani et al., 1987), they must be extended along the line of sight, a possibility also suggested by .
H I Absorption
At all epochs the H i opacity spectra show the presence of a strong absorption feature near 5 km s −1 and a typical rms determined outside the H i line of σ τ ≈ 0.015. There is no gross change in the absorption line during the ESE nor do any additional absorption components appear. Between any two epochs ∆τ ≤ 0.049 (< 2.3σ τ ). This upper limit implies a neutral column density change of ∆N H < 6.4 × 10 17 cm −2 (T s /10 K) for a structure with a spin temperature T s = 10 K. Heiles (1997) proposes interstellar tiny (AU)-scale atomic structures (TSAS) in order to explain small (angular) scale changes in H i opacity. TSAS would have N H ∼ 3 × 10 18 cm −2 and T s ∼ 15 K. Our ∆τ limit marginally excludes a connection between TSAS and ESEs. Walker & Wardle (1998) propose AU-scale molecular clouds in the Galactic halo precisely to explain ESEs. The clouds would be cold, T s 3 K, and dense enough to be opaque in the H i line. ESEs would result from the photoionized skins of the clouds. We see no τ ∼ 1 features (Walker 2000, private communication, has since suggested τ ∼ 0.1). However, the clouds could have velocities approaching 500 km s −1 , while the observed velocity range is no more than 250 km s −1 -significant H i absorption could have been present outside of our velocity range.
Summary
Salient aspects of this observational program are − ∆RM < 10.1 rad m −2 implying a magnetic field within the scatterer of B || < 12 mG.
− No change in the VLBI structure, except a 0.7 mas increase in the angular diameter. This increase is not consistent with that expected from a purely refractive model: ESEs must result from both broadening and defocussing within the ionized structures.
− ∆τ H I < 0.05 implying that the H i column density associated with the ESE structure is N H < 6.4 × 10 17 cm −2 . Tiny-scale atomic structures are marginally ruled out; H i-opaque, halo molecular clouds would be excluded, but the observed velocity range covers only 25% of the allowed range.
The major impediment to improved observational constraints is the lack of a monitoring program that could find additional ESEs.
Figure 1 .
1The light curves of 1741−038 at 2 and 8 GHz showing the extreme scattering event and the epochs of the various observations.
Lazio.preprint.tex; 19/03/2022; 18:23; p.1
Lazio.preprint.tex; 19/03/2022; 18:23; p.2 ESEs: An Observational Summary
Lazio.preprint.tex; 19/03/2022; 18:23; p.3
AcknowledgementsMany people helped make these observations possible, most notably M. Claussen, A. Clegg, B. Dennison, R. Fielder, K. Johnston, and E. Waltman. The NRAO is a facility of the National Science Foundation operated by AUI. Radio astronomy research at the NRL is supported by the Office of Naval Research.
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| []
|
[
"Exponential decay of reconstruction error from binary measurements of sparse signals",
"Exponential decay of reconstruction error from binary measurements of sparse signals"
]
| [
"Richard Baraniuk \nDepartment of Electrical and Computer Engineering\nRice University\n6100 Main Street77005HoustonTXUSA\n",
"Simon Foucart [email protected] \nDepartment of Mathematics\nUniversity of Georgia\n321C Boyd Building30602AthensGAUSA\n",
"Deanna Needell [email protected] \nDepartment of Mathematical Sciences\nClaremont McKenna College\n850 Columbia Ave91711ClaremontCAUSA\n",
"Yaniv Plan \nDepartment of Mathematics\nUniversity of British Columbia\n1984 Mathematics RoadV6T 1Z2VancouverB.C. Canada\n",
"Mary Wootters [email protected] \nDepartment of Mathematics\nUniversity of Michigan\n530 Church Street, Ann Arbor48109MIUSA\n"
]
| [
"Department of Electrical and Computer Engineering\nRice University\n6100 Main Street77005HoustonTXUSA",
"Department of Mathematics\nUniversity of Georgia\n321C Boyd Building30602AthensGAUSA",
"Department of Mathematical Sciences\nClaremont McKenna College\n850 Columbia Ave91711ClaremontCAUSA",
"Department of Mathematics\nUniversity of British Columbia\n1984 Mathematics RoadV6T 1Z2VancouverB.C. Canada",
"Department of Mathematics\nUniversity of Michigan\n530 Church Street, Ann Arbor48109MIUSA"
]
| []
| Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem-e.g., in determining the relationship between genetics and the presence or absence of a disease-or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by onebit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, s-sparse signals in R n can be estimated (up to normalization) from Ω(s log(n/s)) one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor λ := m/(s log(n/s)), where m is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by Ω(λ −1 ). Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization may lower the error rate to e −Ω(λ) . This improves upon guarantees for other methods of adaptive thresholding as proposed in Sigma-Delta quantization. We develop * Authors are listed in alphabetical order. a general recursive strategy to achieve this exponential decay and two specific polynomialtime algorithms which fall into this framework, one based on convex programming and one on hard thresholding. This work is inspired by the one-bit compressed sensing model, in which the engineer controls the measurement procedure. Nevertheless, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression. | 10.1109/tit.2017.2688381 | [
"https://arxiv.org/pdf/1407.8246v1.pdf"
]
| 8,165,418 | 1407.8246 | 8884a2895dd9be69a10809eaf904a9ee14300c6e |
Exponential decay of reconstruction error from binary measurements of sparse signals
August 1, 2014 31 Jul 2014
Richard Baraniuk
Department of Electrical and Computer Engineering
Rice University
6100 Main Street77005HoustonTXUSA
Simon Foucart [email protected]
Department of Mathematics
University of Georgia
321C Boyd Building30602AthensGAUSA
Deanna Needell [email protected]
Department of Mathematical Sciences
Claremont McKenna College
850 Columbia Ave91711ClaremontCAUSA
Yaniv Plan
Department of Mathematics
University of British Columbia
1984 Mathematics RoadV6T 1Z2VancouverB.C. Canada
Mary Wootters [email protected]
Department of Mathematics
University of Michigan
530 Church Street, Ann Arbor48109MIUSA
Exponential decay of reconstruction error from binary measurements of sparse signals
August 1, 2014 31 Jul 20141compressed sensingquantizationone-bit compressed sensingconvex optimiza- tioniterative thresholdingbinary regression
Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem-e.g., in determining the relationship between genetics and the presence or absence of a disease-or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by onebit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, s-sparse signals in R n can be estimated (up to normalization) from Ω(s log(n/s)) one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor λ := m/(s log(n/s)), where m is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by Ω(λ −1 ). Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization may lower the error rate to e −Ω(λ) . This improves upon guarantees for other methods of adaptive thresholding as proposed in Sigma-Delta quantization. We develop * Authors are listed in alphabetical order. a general recursive strategy to achieve this exponential decay and two specific polynomialtime algorithms which fall into this framework, one based on convex programming and one on hard thresholding. This work is inspired by the one-bit compressed sensing model, in which the engineer controls the measurement procedure. Nevertheless, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression.
Introduction
Many practical acquisition devices in signal processing and algorithms in machine learning use a small number of linear measurements to represent a high-dimensional signal. Compressed sensing is a technology which takes advantage of the fact that, for some interesting classes of signals, one can use far fewer measurements than dictated by traditional Nyquist sampling paradigm. In this setting, one obtains m linear measurements of a signal x ∈ R n of the form y i = a i , x , i = 1, . . . , m.
Written concisely, one obtains the measurement vector y = Ax, where A ∈ R m×n is the matrix with rows a 1 , . . . , a m . From these (or even from corrupted measurements y = Ax + e), one wishes to recover the signal x. To make this problem well-posed, one must exploit a priori information on the signal x, for example that it is s-sparse, i.e.,
x 0 def = |supp(x)| = s n,
or is well-approximated by an s-sparse signal. After a great deal of research activity in the past decade (see the website [DSP] or the references in the monographs [EK12,FR13]), it is now well known that when A consists of, say, independent standard normal entries, one can, with high probability, recover all s-sparse vectors x from the m ≈ s log(n/s) linear measurements y i = a i , x , i = 1, . . . , m. However, in practice, the compressive measurements a i , x must be quantized: one actually observes y = Q(Ax), where the map Q : R m → A m is a quantizer that acts entrywise by mapping each real-valued measurement to a discrete quantization alphabet A. This type of quantization with an alphabet A consisting of only two elements was introduced in the compressed sensing setting by [BB08] and dubbed one-bit compressed sensing . In this work, we focus on this one-bit approach and seek quantization schemes Q and reconstruction algorithms ∆ so thatx = ∆(Q(Ax)) is a good approximation to x. In particular, we are interested in the trade-off between the error of the approximation and the oversampling factor λ def = m s log(n/s) .
Motivation and previous work
The most natural quantization method is Memoryless Scalar Quantization (MSQ), where each entry of y = Ax is rounded to the nearest element of some quantization alphabet A. If A = δZ for S n−1
x Figure 1: Geometric interpretation of one-bit compressed sensing. Each quantized measurement reveals which side of a hyperplane (or great circle, when restricted to the sphere) the signal x lies on. After several measurements, we know that x lies in one unique region. However, if the measurements are non-adaptive, then as the region of interest becomes smaller, it becomes less and less likely that the next measurement yields any new information about x.
some suitably small δ > 0, then this rounding error can be modeled as an additive measurement error [DPM09], and the recovery algorithm can be fine-tuned to this particular situation [JHF11]. In the one-bit case, however, the quantization alphabet is A = {±1} and the quantized measurements take the form y = sign(Ax), meaning that sign 1 acts entrywise as
y i = Q MSQ ( a i , x ) = sign( a i , x ), i = 1, . . . , m.
One-bit compressed sensing was introduced in [BB08], and it has generated a considerable amount of work since then, see [DSP] for a growing list of literature in this area. Several efficient recovery algorithms have been proposed, based on linear programming [PV13a,PV13b,GNJN13] and on modifications of iterative hard thresholding [JLBB13,JDDV13]. As shown in [JLBB13], with high probability one can perform the reconstruction from one-bit measurements with error
x −x 2 1 λ for all x ∈ Σ s := {v ∈ R n : v 0 ≤ s, v 2 = 1}.
In other words, a uniform 2 -reconstruction error of at most γ > 0 can be achieved with m γ −1 s log(n/s) one-bit measurements. Despite the dimension reduction from n to s log(n/s), MSQ presents substantial limitations [JLBB13,GVT98]. Precisely, according to [GVT98], even if the support of x ∈ Σ s is known, the best recovery algorithm ∆ opt must obey
x − ∆ opt (Q MSQ (Ax)) 2 1 λ (1)
up to a logarithmic factor. An intuition for the limited accuracy of MSQ is given in Figure 1. Alternative quantization schemes have been developed to overcome this drawback. For a specific signal model and reconstruction algorithm, [SG09] obtained the optimal quantization scheme, but more general quantization schemes remain open.
Recently, Sigma-Delta quantization schemes have also been proposed as a more general quantization model [GLP + 10, KSY14]. These works show that, with high probability on measurement matrices with independent subgaussian entries, r-th order Sigma-Delta quantization can be applied to the standard compressed sensing problem to achieve, for any α ∈ (0, 1), the reconstruction error
x −x 2 r λ −α(r−1/2) (2)
with a number of measurements m ≈ s (log(n/s)) 1/(1−α) .
For suitable choices of α and r, the guarantee (2) overcomes the limitation (1), but it is still polynomial in λ. This leads us to ask whether an exponential dependence can be achieved.
Our contributions
In this work, we focus on improving the trade-off between the error x −x 2 and the oversampling factor λ. To the best of our knowledge, all quantized compressed sensing schemes obtain guarantees of the form
x −x 2 λ −c for all x ∈ Σ s(3)
with some constant c > 0. We develop one-bit quantizers Q : R m → {±1}, coupled with two efficient recovery algorithms ∆ : {±1} → R m that yield the reconstruction guarantee
x − ∆(Q(Ax)) 2 ≤ exp(−Ω(λ)) for all x ∈ Σ s .(4)
It is not hard to see that the dependence on λ in (4) is optimal, since any method of quantizing measurements that provides the reconstruction guarantee x −x 2 ≤ γ must use at least log 2 N (Σ s , γ) ≥ s log 2 (1/γ) bits, where N (·) denotes the covering number.
Adaptive measurement model
A key element of our approach is that the quantizers are adaptive to previous measurements of the signal in a manner similar to Sigma-Delta quantization [GLP + 10]. In particular, the measurement matrix A ∈ R m×n is assumed to have independent standard normal entries and the quantized measurements take the form of thresholded signs, i.e.,
y i = sign( a i , x − τ i ) = 1 if a i , x ≥ τ i , −1 if a i , x < τ i .(5)
Such measurements are readily implementable in hardware, and they retain the simplicity and storage benefits of the one-bit compressed sensing model. However, as we will show, this model is much more powerful in the sense that it permits optimal guarantees of the form (4), which are impossible with standard MSQ one-bit quantization. As in the Sigma-Delta quantization approach, we allow the quantizer to be adaptive, meaning that the quantization threshold τ i of the ith entry may depend on the 1st, 2nd, . . ., (i − 1)st quantized measurements. In the context of (5), this means that the thresholds τ i will be chosen adaptively, resulting in a feedback loop as depicted in Figure 2. The thresholds τ i can also be interpreted as an additive dither, which is oft-used in the theory and practice of analog-to-digital conversion.
In contrast to Sigma-Delta quantization, the feedback loop involves the calculation of the quantization threshold. This is the concession made to arrive at exponentially decaying error rates. It is an interesting open problem to determine low-memory quantization methods with such error rates that do not require such a calculation.
A quantize x ∈ R n Ax ∈ R m y ∈ R m τ ∈ R m
Figure 2: Our closed-loop feedback system for binary measurements.
Overview of our main result
Our main result is that there is a recovery algorithm using measurements of the form (5) and providing a guarantee of the form (4). For clarity of exposition, we overview a simplified version of our main result below. The full result is stated in Section 3.
Theorem 1 (Main theorem, simplified version). Let Q and ∆ be the quantization and recovery algorithms given below in Algorithms 1 and 2, respectively. Suppose that A ∈ R m×n and τ ∈ R m have independent standard normal entries. Then, with probability at least Cλ exp(−cs log(n/s)) over the choice of A and τ , for all x ∈ B n 2 with x 0 ≤ s,
x − ∆(Q(Ax, A, τ )) 2 ≤ exp(−Ω(λ)), where λ = m s log(n/s) .
The quantization algorithm works iteratively. First, a small batch of measurements are quantized in a memoryless fashion. From this first batch, one gains a very rough estimate of x (called x 1 ). The next batch of measurements are quantized with a focus on encoding the difference between x and x 1 , and so on. Thus, the trap depicted in Figure 1 is avoided; each hyperplane is translated with an appropriate dither, with the aim of cutting the size of the feasible region. The recovery algorithm also works iteratively and its iterates are in fact intertwined with the iterates of the quantization algorithm. We artificially separate the two algorithms below.
Note that we present Algorithms 1 and 2 at this point mainly because they are the simplest to state. Below we will provide a more general framework for algorithms that satisfy the guarantees of Theorem 1 and develop a second set of algorithms with computational advantages.
Robustness
Our algorithms are robust to two different kinds of measurement corruption. First, they allow for perturbed linear measurements of the form a i , x + e i for an error vector e ∈ R m with bounded ∞ -norm. Second they allow for post-quantization sign flips, recorded as a vector f ∈ {±1} m .
Formally, the measurements take the form
y i = f i sign( a i , x − τ i + e i ), i = 1, . . . , m.(6)
It is known that for inaccurate measurements with pre-quantization noise on the same order of magnitude as the signal, even unquantized compressed sensing algorithms must obey a lower bound of the form (1) [CD13]. Our algorithms respect this reality and exhibit exponentially fast convergence until the estimate hits the "noise floor"-that is, until the error x −x 2 is on the order of e ∞ . Table 1 summarizes the various noise models, adaptive threshold calculations, and algorithms we develop and study below.
Algorithm 1: Adaptive quantization
Input: Linear measurements Ax ∈ R m ; measurement matrix A ∈ R m×n ; sparsity parameter s; thresholds τ ∈ R m ; parameter q ≥ Cs log(n/s) for the size of batches. Output: Quantized measurements y ∈ {±1} m . T ← m q Partition A and τ into T blocks A (1) , . . . , A (T ) ∈ R q×n and τ (1) , . . . .
τ (T ) ∈ R q . x 0 ← 0 for t = 1, . . . , T do σ (t) ← A (t) x t−1 y (t) ← sign(A (t) x − 2 2−t τ (t) − σ (t) ) z t ← argmin z 1 subject to z 2 ≤ 2 2−t , y (t) i a (t) i , z − 2 2−t τ (t) i ≥ 0 for all i // z t is an approximation of x − x t−1 x t ← H s (x t−1 + z t )
// H s keeps s largest (in magnitude) entries and zeroes out the rest return y (t) for t = 1, . . . , T // Notice that we discard σ (t)
Algorithm 2: Recovery Input: Quantized measurements y ∈ {±1} m ; measurement matrix A; sparsity parameter s; thresholds τ ∈ R m ; size of batches q. Output: Approximationx ∈ R n . T ← m q Partition A and τ into T blocks A (1) , . . . , A (T ) ∈ R q×n and τ (1) , . . . .τ (T ) ∈ R q . x 0 ← 0 for t = 1, . . . , T do
z t ← argmin z 1 subject to z 2 ≤ 2 2−t , y (t) i a (t) i , z − 2 2−t τ (t) i ≥ 0 for all i x t = H s (x t−1 + z t ) return x T
Relationship to binary regression
Our one-bit adaptive quantization and reconstruction algorithms are more broadly applicable to a certain kind of statistical classification problem related to sparse binary regression, and in particular sparse logistic and probit regression. These techniques are often used to explain statistical data in which the response variable is binary. In regression, it is common to assume that the data {z i } is generated according to the generalized linear model, where
z i ∈ {0, 1} is a Bernoulli random variable satisfying E [z i ] = f ( a i , x )(7)
for some function f : R → [0, 1]. The generalized linear model is equivalent to the noisy one-bit compressed sensing model when the measurements y i = 2z i − 1 ∈ {±1} and P (y i = 1) =: f ( a i , x ), Table 1: Summary of the noise models, adaptive threshold calculations, and algorithms considered. See Section 2 for further discussion of the trade-offs between the two algorithms.
Noise model Threshold algorithm Recovery algorithm
Additive error e i in (6) Algorithm 7, instantiated by Algorithm 3
Convex programming: Algorithm 8, instantiated by Algorithm 4
Additive error e i and sign flips f i in (6) Algorithm 7, instantiated by Algorithm 5
Iterative hard thresholding: Algorithm 8, instantiated by Algorithm 6 or equivalently, when
y i = sign( a i , x + e i )
with f (t) := P (e i ≥ −t). In summary, one-bit compressed sensing is equivalent to binary regression as long as f is the cumulative distribution function (CDF) of the noise variable e i . The most commonly used CDFs in binary regression are the inverse logistic link function f (t) = 1 1+e t in logistic regression and the inverse probit link function f (t) = t −∞ N (s)ds in probit regression. These cases correspond to the noise variable e i being logistic and Gaussian distributed, respectively.
The new twist here is that the quantization thresholds are selected adaptively; see Section 6.1 for some examples. Specifically, our adaptive threshold measurement model is equivalent to the adaptive binary regression model
y i = sign( a i , x + e i − τ i ) with P (y i = 1) = P (e i − τ i >= −t) = f (t − τ i ).
The effect of τ i in this adaptive binary regression is equivalent to an offset term added to all measurements y i . Standard binary regression corresponds to the special case with τ i = 0.
Organization
In Section 2, we introduce two methods to recover not only the direction, but also the magnitude, of a signal from one-bit compressed sensing measurements of the form (6). These methods may be of independent interest (in one-bit compressed sensing, only the direction can be recovered), but they do not exhibit the exponential decay in the error that we seek. In Section 3, we will show how to use these schemes as building blocks to obtain (4). The proofs of all of our results are given in Section 4. In Section 5, we present some numerical results for the new algorithms. We conclude in Section 6 with a brief summary.
Notation
Throughout the paper, we use the standard notation v 2 = i v 2 i for the 2 -norm of a vector v ∈ R n , v 1 = i |v i | for its 1 -norm, and v 0 for its number of nonzero entries. A vector v is
called s-sparse if v 0 ≤ s and effectively s-sparse if v 1 ≤ √ s v 2 .
We write H s (v) to represent the vector in R n agreeing with v on the index set of largest s entries of v (in magnitude) and with the zero vector elsewhere. We use a prime to indicate 2 -normalization, so that H s (v) is defined as
H s (v) := H s (v)/ H s (v) 2 . The set Σ s := {v ∈ R n : v 0 ≤ s} of s-sparse vectors is accompanied by the set Σ s := {v ∈ R n : v 0 ≤ s, v 2 = 1} of 2 -normalized s-sparse vectors. For R > 0, we write RΣ s to mean the set {v ∈ R n : v 0 ≤ s, v 2 = R}.
We also write B n 2 = {v ∈ R n : v 2 ≤ 1} for the 2 -ball in R n and RB n 2 for the appropriately scaled version. We consider the task of recovering x ∈ Σ s from measurements of the form (5) or (6) for i = 1, . . . , m. These measurements are organized as a matrix A ∈ R m×n with rows a 1 , . . . , a m and a vector τ ∈ R m of thresholds. Matching the Sigma-Delta quantization model, the a i ∈ R n may be random but are non-adaptive, while the τ i ∈ R may be chosen adaptively, in either a random or deterministic fashion. The Hamming distance between sign vectors y,ỹ ∈ {±1} m is defined as d H (y,ỹ) = i 1 {y i =ỹ i } .
Magnitude recovery
Given an s-sparse vector x ∈ R n , several convex programs are provably able to extract an accurate estimate of the direction of x from sign(Ax) or sign(Ax+e) [PV13b,PV13a]. However, recovery of the magnitude of x is challenging in this setting [KSW14]. Indeed, all magnitude information about x is lost in measurements of the form sign(Ax). Fortunately, if random (non-adaptive) dither is added before quantization, then magnitude recovery becomes possible, i.e., noise can actually help with signal reconstruction. This observation has also been made in the concurrently written paper [KSW14] and also in the literature on binary regression in statistics [DPvdBW14].
Our main result will show that both the magnitude and direction of x can be estimated with exponentially small error bounds. In this section, we first lay the groundwork for our main result by developing two methods for one-bit signal acquisition and reconstruction that provide accurate reconstruction of both the magnitude and direction of x with polynomially decaying error bounds.
We propose two different order-one recovery schemes. The first is based on second-order cone programming and is simpler but more computationally intensive. The second is based on hard thresholding, is faster, and is able to handle a more general noise model (in particular, random sign flips of the measurements) but requires an adaptive dither. Recall Table 1.
Second-order cone programming
The size of the appropriate dither/threshold depends on the magnitude of x. Thus, let R > 0 satisfy x 2 ≤ R. We take measurements of the form
y i = sign( a i , x − τ i + e i ), i = 1, . . . , q,(8)
where τ 1 , . . . , τ q ∼ N (0, 4R 2 ) are known independent normally distributed dithers that are also independent of the rows a 1 , . . . , a q of the matrix A and e 1 , . . . , e q are small deterministic errors (possibly adversarial) satisfying |e i | ≤ cR for an absolute constant c. The following second-order cone program
argmin z 1 subject to z 2 ≤ 2R, y i ( a i , z − τ i ) ≥ 0 for all i = 1, . . . , q(9)
provides a good estimate of x, as formally stated below.
Algorithm 3: T 0 : Threshold production for second-order cone programming
Input: Bound R on x 2 Output: Thresholds τ ∈ R q return τ ∼ N (0, R 2 I q )
Algorithm 4: ∆ 0 : Recovery procedure for second-order cone programming Input: Quantized measurements y ∈ {±1} q ; measurement matrix A ∈ R q×n ; bound R on
x 2 ; thresholds τ ∈ R q . Output: Approximationx return argmin z 1 subject to z 2 ≤ 2R, y i ( a i , z − τ i ) ≥ 0 for all i = 1, . . . , q.
Theorem 2. Let 1 ≥ δ > 0, let A ∈ R q×n have independent standard normal entries, and let τ 1 , . . . , τ q ∈ R be independent normal variables with variance 4R 2 . Suppose that n ≥ 2q and q ≥ C δ −4 s log(n/s).
Then, with probability at least 1 − 3 exp(−c 0 δ 4 q) over the choice of A and the dithers τ 1 , . . . , τ q , the following holds for all x ∈ RB n 2 ∩ Σ s and e ∈ R q satisfying e ∞ ≤ cδ 3 R: for y obeying the measurement model (8), the solutionx to (9) satisfies
x −x 2 ≤ δR.
The positive constants C , c and c 0 above are absolute constants.
Remark 1. The choice of the constraint z 2 ≤ 2R and the variance 4R 2 for the τ i 's allows for the above theoretical guarantees in the presence of pre-quantization error e = 0. However, in the ideal case e = 0, the guarantees also hold if we impose z 2 ≤ R and take a variance of R 2 . This more natural choice seems to give better results in practice, even in the presence of pre-quantization error (as R was already an overestimation for x 2 ). This is the route followed in the numerical experiments of Section 5. It only requires changing 2 2−t to 2 1−t in Algorithms 1 and 2.
To fit into our general framework for exponential error decay, it is helpful to think of the program (9) as two separate algorithms: an algorithm T 0 that produces thresholds and an algorithm ∆ 0 that performs the recovery. These are formally described in Algorithms 3 and 4.
Hard thresholding
The convex programming approach is attractive in many respects; in particular, the thresholds/dithers τ i are non-adaptive, which makes them especially easy to apply in hardware. However, the recovery algorithm ∆ 0 in Algorithm 4 can be costly. Further, while the convex programming approach can handle additive pre-quantization error, it cannot necessarily handle post-quantization error (sign flips). In this section, we present an alternative scheme for estimating magnitude, based on iterative hard thresholding that addresses these challenges. The only downside is that the thresholds/dithers τ i become adaptive within the order-one recovery scheme.
Given an s-sparse vector x ∈ R n , one can easily extract from sign(Ax) a good estimate for the direction of x. For example, we will see that H s (A * sign(Ax)) is a good approximation of x/ x 2 .
Algorithm 5: T 0 : Threshold production for hard thresholding Input: Measurements Ax ∈ R q ; measurement matrix A ∈ R q×n ; sparsity parameter s; bound R on x 2 . Output:
Thresholds τ ∈ R q Partition Ax into A 1 x, A 2 x ∈ R q/2 . u ← H s (A * 1 sign(A 1 x)) v ← V (u) w ← 2R · (u + v) return 0 ∈ R q/2 , A 2 w ∈ R q/2
Algorithm 6: ∆ 0 : Recovery procedure for hard thresholding Input: Quantized measurements y ∈ {±1} q ; measurement matrix A ∈ R q×n ; sparsity parameter s; bound R on x 2 . Output: Approximationx Partition y into y 1 ,
y 2 ∈ R q/2 . u ← H s (A * 1 y 1 ) v ← V (u) t ← −H s (A * 2 y 2 ) return 2Rf ( t, v ) · u, where f (ξ) = 1 − √ 1−ξ 2 ξ
However, as mentioned earlier, there is no hope of recovering the magnitude x 2 of the signal from sign(Ax). To get around this, we use a second estimator, this time for the direction of x − z for a well-chosen vector z ∈ R n obtained by computing H s (A * sign(A(x − z))). This allows us to estimate both the direction and the magnitude of x.
As above, we break the measurement/recovery process into two separate algorithms. The first is an algorithm T 0 describing how to generate the thresholds τ i . The second is a recovery algorithm ∆ 0 that describes how to recover an approximationx to x based on measurements of the form (6), using the τ i as thresholds. These are formally described in Algorithms 5 and 6. In the algorithm statements, V denotes any fixed rule associating to a vector u an 2 -normalized vector V (u) that is both orthogonal to u and has the same support.
The analysis for T 0 and ∆ 0 relies on the following theorems.
Theorem 3. Let 1 ≥ δ > 0 and let A ∈ R q×n have independent standard normal entries. Suppose that n ≥ 2q and q ≥ c 1 δ −7 s log(n/s). Then, with probability at least 1 − c 2 exp(−c 3 δ 2 q) over the choice of A, the following holds for all s-sparse x ∈ R n , all e ∈ R q with e 2 ≤ c 6 √ q x 2 , and all
y ∈ {±1} q : x x 2 − H s (A * y) 2 ≤ δ + c 4 e 2 √ q x 2 + c 5 d H (y, sign (Ax + e)) q(10)
The positive constants c 1 , c 2 , c 3 , c 4 , c 5 , and c 6 above are absolute constants.
The proof of Theorem 3 is given in Section 4. Once Theorem 3 is shown, we will be able to establish the following results when the threshold production and recovery procedures T 0 and ∆ 0 are given by Algorithms 5 and 6.
Theorem 4. Let 1 ≥ δ > 0, let A ∈ R q×n have independent standard normal entries, and let T 0 and ∆ 0 be as in Algorithms 5 and 6. Suppose that n ≥ 2q and q ≥ c 1 δ −7 s log(n/s).
Further assume that whenever a signal z is measured, the corruption errors satisfy e ∞ ≤ cδ z 2 and |{i : f i = −1}| ≤ c δq. Then, with probablity at least 1 − c 7 exp(−c 8 δ 2 q) over the choice of A, the following holds for all x ∈ RB n 2 ∩ Σ s : for y obeying the measurement model (6) with τ = T 0 (Ax, A, s, R), the vectorx = ∆ 0 (y, A, s, R) satisfies
x −x 2 ≤ δR.
The positive constants c 1 , c, c , c 7 , and c 8 above are absolute constants.
Having proposed two methods for recovering both the direction and magnitude of a sparse vector from binary measurements, we now turn to our main result.
Exponential decay: General framework
In the previous section, we developed two methods for approximately recovering x from binary measurements. Unfortunately, these methods exhibit polynomial error decay in the oversampling factor, and our goal is to obtain an exponential decay. We can achieve this goal by applying the rough estimation methods iteratively, in batches, with adaptive thresholds/dithers. As we show below, this leads to an extremely accurate recovery scheme. To make this framework precise, we first define an order-one recovery scheme (T 0 , ∆ 0 ).
Definition 5 (Order-one recovery scheme). An order-one recovery scheme with sparsity parameter s, measurement complexity q, and noise resilience (η, b) is a pair of algorithms (T 0 , ∆ 0 ) such that:
• The thresholding algorithm T 0 takes a parameter R and, optionally, a set of linear measurements Ax ∈ R q and the measurement matrix A ∈ R q×n . It outputs a set of thresholds τ ∈ R q .
• The recovery algorithm ∆ 0 takes q corrupted quantized measurements of the form (6), i.e.,
y i = f i sign( a i , x − τ i + e i ),
where e ∈ R q is a pre-quantization error and f ∈ {±1} q is a post-quantization error. It also takes as input the measurement matrix A ∈ R q×n , a parameter R, and, optionally, a sparsity parameter s and the thresholds τ returned by T 0 . It outputs a vectorx ∈ R n .
• With probability at least 1 − C exp(−cq) over the choice of A ∈ R q×n and the randomness of T 0 , the following holds: for all x ∈ RB n 2 ∩ Σ s , all e ∈ R q with e ∞ ≤ η x 2 , and all f ∈ {±1} q with at most b sign flips, the estimatex = ∆ 0 (y, A, R, s, τ ) satisfies
x −x 2 ≤ R 4 .
Algorithm 7: Q: Quantization Input: Linear measurements Ax ∈ R m ; measurement matrix A ∈ R m×n ; sparsity parameter s; bound R on x 2 ; parameter q ≥ Cs log(n/s) for the size of batches. Output: Quantized measurements y ∈ {±1} m and thresholds
τ ∈ R m T ← m q Partition A into T blocks A (1) , . . . , A (T ) ∈ R q×m x 0 ← 0 for t = 1, . . . , T do R t = 2 −t+1 τ (t) ← T 0 (A (t) (x − x t−1 ), A (t) , R t ) σ (t) ← A (t) x t−1 y (t) ← f (t) sign(A (t) x − τ (t) − σ (t) + e (t) ) x t ← H s (x t−1 + ∆ 0 (y (t) , A (t) , R t , τ (t) ))
return y (t) , τ (t) for t = 1, . . . , T Algorithm 8: ∆: Recovery Input: Quantized measurements y ∈ {±1} m ; measurement matrix A ∈ R m×n ; sparsity parameter s; bound R on x 2 ; thresholds τ ∈ R m ; size of batches q.
Output: Approximationx ∈ R n T ← m q Partition A into T blocks A (1) , . . . , A (T ) ∈ R q×m x 0 ← 0 for t = 1, . . . , T do x t ← H s (x t−1 + ∆ 0 (y (t) , A (t) , R2 −t+1 , τ (t) )) (11) return x T
We saw two examples of order-one recovery schemes in Section 2. The scheme based on secondorder cone programming is an order-one recovery scheme with sparsity parameter s, measurement complexity q = C 0 s log(n/s), and noise resilience η = c 0 R and b = 0. The scheme based on iterated hard thresholding is an order-one recovery scheme with sparsity parameter s, measurement complexity q = C 1 s log(n/s), and noise resilience η = c 1 R and b = c 2 q. Above, c 0 , c 1 , c 2 , C 0 , C 1 > 0 are absolute constants.
We use an order-one recovery scheme to build a pair of one-bit quantization and recovery algorithms for sparse vectors that exhibits extremely fast convergence. Our quantization and recovery algorithms Q and ∆ are given in Algorithms 7 and 8, respectively. They are in reality intertwined, but again we separate them for expositional clarity.
The intuition motivating Step (11) is that ∆ 0 (y (t) , A (t) , R t , τ (t) , ) estimates x − x t−1 ; hence x t approximates x better than x t−1 does. Note the similarity to the intuition motivating iterative hard thresholding, with the key difference being that the quantization is also performed iteratively.
Remark 2 (Computational and storage considerations). Let us analyze the storage requirements and computational complexity of Q and ∆, both during and after quantization.
We begin by considering the approach based on convex programming. In this case, the final storage requirements of the quantizer Q are similar to those in standard one-bit compressed sensing. The "algorithm" T 0 is straightforward: it simply draws random thresholds/dithers. In particular, we may treat these thresholds as predetermined independent normal random variables in the same way as we treat A. If A and τ are generated by a short seed, then all that needs to be stored after quantization are the binary measurements y ∈ {±1} q . During quantization, the algorithm Q needs to store x t . However, this requires small memory since x t is s-sparse.
While the convex programming approach is designed to ease storage burdens, the order-one recovery scheme based on hard thresholding is built for speed. In this case, the threshold algorithm T 0 (Algorithm 5) is more complicated, and the adaptive thresholds τ need to be stored. On the other hand, the computation of x t is much faster, and both the quantization and recovery algorithms are very efficient.
Given an order-one recovery scheme (T 0 , ∆ 0 ), the quantizer Q given in Algorithm 7 and the recovery algorithm ∆ given in Algorithm 8 have the desired exponential convergence rate. This is formally stated in the theorem below and proved in Section 4.
Theorem 6. Let (T 0 , ∆ 0 ) be an order-one recovery scheme with sparsity parameter 2s, measurement complexity q, and noise resilience (η, b). Fix R > 0 and recall that T := m/q . With probability at least 1 − CT exp(−cq) over the choice of A and the randomness of T 0 , the following holds for all x ∈ RB n 2 ∩ Σ s , all e ∈ R m with e ∞ ≤ η2 −T x 2 , and all f ∈ {±1} m with |{i : f i = −1}| ≤ b in the measurement model (6):
for y ∈ {±1} m and τ = Q(Ax, A, s, R, q) ∈ R m , the outputx of ∆(y, A, s, R, τ, q) satisfies
x −x 2 ≤ R 2 −T .(12)
The positive constants η, b, c, and C above are absolute constants.
Our two order-one recovery schemes each have measurement complexity q = Cs log(n/s). This implies the announced exponential decay in the error rate.
Corollary 7. Let Q, ∆ be as in Algorithms 7 and 8 with one-bit recovery schemes (T 0 , ∆ 0 ) given either by Algorithms (3,4) or (5,6). Let A ∈ R m×n have independent standard normal entries. Fix R > 0 and recall that λ = m/(slog(n/s)). With probability at least 1 − Cλ exp(−cs log(n/s)) over the choice of A and the randomness of T 0 , the following holds for all x ∈ RB n 2 ∩ Σ s , all e ∈ R m with e ∞ ≤ η2 −T x 2 , and all f ∈ {±1} m with |{i : f i = −1}| ≤ b in the measurement model (6) (b = 0 if (T 0 , ∆ 0 ) is based on convex programming or b = cs log(n/s) if (T 0 , ∆ 0 ) is based on hard thresholding):
for y ∈ {±1} m τ = Q(Ax, A, s, R, q) ∈ R m , the outputx of ∆(y, A, s, R, τ, q) satisfies
x −x 2 ≤ R 2 −cλ .(13)
The positive constants η, c , c, and C above are absolute constants.
Proofs
Exponentially decaying error rate from order-one recovery schemes
First, we prove Theorem 6 which states that, given an appropriate order-one recovery scheme, the recovery algorithm ∆ in Algorithm 8 converges with exponentially small reconstruction error when the measurements are obtained by the quantizer Q of Algorithm 7.
Proof of Theorem 6. For x ∈ RB n 2 ∩ Σ s , we verify by induction on t ∈ {0, 1, . . . , T } that
x − x t 2 ≤ R2 −t .
This induction hypothesis holds for t = 0. Now, suppose that it holds for t − 1, t ∈ {1, . . . , T }. Consider ∆ 0 (y (t) , A (t) , R t , τ (t) ), the estimate returned by the order-one recovery scheme in (11). By definition, the thresholds τ (t) were obtained in step t by running T 0 on A (t) (x−x t−1 ). Similarly, the quantized measurements y (t) are formed by quantizing (with noise) the affine measurements
A (t) x − σ (t) − τ (t) = A (t) (x − x t−1 ) − τ (t) .
Thus, we have effectively run the order-one recovery scheme on the 2s-sparse vector x − x t . By the guarantee of the order-one recovery algorithm, with probability at least 1 − C exp(−cq),
(x − x t−1 ) − ∆ 0 (y (t) , A (t) , R t , τ (t) ) 2 ≤ R t /4 = R2 −t+1 /4.
Suppose that this occurs. Let
z = x t−1 + ∆ 0 (y (t) , A (t) , R t , τ (t) ), so x − z 2 ≤ R2 −t+1 /4. Since x t = H s (z)
is the best s-term approximation to z, it follows that
x − x t 2 = x − H s (z) 2 ≤ x − z 2 + H s (z) − z 2 ≤ 2 x − z 2 ≤ R2 −t .
Thus, the induction hypothesis holds for t. A union bound over the T iterations completes the proof, since the announced result is the inductive hypothesis in the case that t = T .
Hard-thresholding-based order-one recovery scheme
The proof of Theorem 3 relies on three properties of random matrices A ∈ R q×n with independent standard normal entries. In their descriptions below, the positive constants c, C, and d are absolute constants.
• The restricted isometry property of order s ([FR13, Theorems 9.6 and 9.27]): for any δ > 0, with failure probability at most 2 exp(−cδ 2 q), the estimates
(1 − δ) x 2 2 ≤ 1 q Ax 2 2 ≤ (1 + δ) x 2 2(14)
hold for all s-sparse x ∈ R n provided q ≥ Cδ −2 s log(n/s).
• The sign product embedding property of order s ( [JDDV13,PV13b]): for any δ > 0, with failure probability at most 8 exp(−cδ 2 q), the estimates
π/2 q Aw, sign (Ax) − w, x ≤ δ(15)
hold for all effectively s-sparse w, x ∈ R n with w 2 = x 2 = 1 provided q ≥ Cδ −6 s log(n/s).
• The 1 -quotient property ( [Woj09] or [FR13,Theorem 11.19]): if n ≥ 2q, then with failure probability at most exp(−cq), every e ∈ R q can be written as
e = Au with u 1 ≤ d √ s * e 2 / √ q where s * := q log(n/q) .(16)
Combining the 1 -quotient property and the restricted isometry property (of order 2s for a fixed δ ∈ (0, 1/2), say) yields the simultaneous ( 2 , 1 )-quotient property (use, for instance, [FR13, Theorem 6.13 and Lemma 11.16]); that is, there are absolute constants d, d > 0 such that every e ∈ R q can be written as
e = Au with u 2 ≤ d e 2 / √ q, u 1 ≤ d √ s * e 2 / √ q.(17)
Proof of Theorem 3. We target the inequalities
x x 2 − π/2 q H s (A * y) 2 ≤ δ + c 4 e 2 √ q x 2 + c 5 d H (y, sign (Ax + e)) q .(18)
The desired inequalities (10) then follows modulo a change of constants, because H s (A * y) is the best unit-norm approximation to π/2 q −1 H s (A * y), so that
x x 2 − H s (A * y) 2 ≤ x x 2 − π/2 q H s (A * y) 2 + H s (A * y) − π/2 q H s (A * y) 2 ≤ 2 x x 2 − π/2 q H s (A * y) 2 .
With s * = q/ log(n/q) as in (16), we remark that it is enough to consider the case s = cs * , c := c −1 1 δ 7 . Indeed, the inequality q ≥ c 1 δ −7 s log(n/s) yields q ≥ c −1 s log(n/q), i.e., s ≤ cs * . Then (18) for s follows from (18) for cs * modulo a change of constants because H s (A * y) is the best s-term approximation to H cs * (A * y), so that
x x 2 − π/2 q H s (A * y) 2 ≤ x x 2 − π/2 q H cs * (A * y) 2 + π/2 q H s (A * y) − π/2 q H cs * (A * y) 2 ≤ 2 x x 2 − π/2 q H cs * (A * y) 2 .
We now assume that s = cs * . This reads q = c 1 δ −7 s log(n/q) and arguments similar to [FR13, Lemma C.6(c)] lead to q ≥ (c 1 δ −7 / log(ec 1 δ −7 ))s log(n/s). Thus, if c 1 is chosen large enough at the start, we have q ≥ Cδ −6 s log(n/s). This ensures that the sign product embedding property (15) of order 2s with constant δ/2 holds with high probability. Likewise, the restricted isometry property (14) of order 2s with constant 9/16, say, holds with high probability. In turn, the simultaneous ( 2 , 1 )-quotient property (17) holds with high probability.
We place ourselves in the situation where all three properties hold simultaneously, which occurs with failure probability at most c 2 exp(−c 3 δ 2 q) for some absolute constants c 2 , c 3 > 0. Then, writing S = supp (x) and T = supp (H s (A * y)), we remark that H s (A * y) is the best s-term approximation to A * S∪T y, so that
x x 2 − π/2 q H s (A * y) 2 ≤ x x 2 − π/2 q A * S∪T y 2 + π/2 q H s (A * y) − π/2 q A * S∪T y 2 ≤ 2 x x 2 − π/2 q A * S∪T y 2 .(19)
We continue with the fact that
x x 2 − π/2 q A * S∪T y 2 ≤ x x 2 − π/2 q A * S∪T sign (Ax + e) 2 + π/2 q A * S∪T (y − sign (Ax + e)) 2 .(20)
The second term on the right-hand side of (20) can be bounded with the help of the restricted isometry property (14) as
A * S∪T (y − sign (Ax + e)) 2 2 = A S∪T A * S∪T (y − sign (Ax + e)) , y − sign (Ax + e) ≤ A S∪T A * S∪T (y − sign (Ax + e)) 2 y − sign (Ax + e) 2 ≤ 1 + 9 16 √ q A * S∪T (y − sign (Ax + e)) 2 y − sign (Ax + e) 2 .
Simplifying by A * S∪T (y − sign (Ax + e)) 2 , we obtain A * S∪T (y − sign (Ax + e)) 2 ≤ 5 4 √ q y − sign (Ax + e) 2 = 5 2 √ q d H (y, sign(Ax + e)). (21)
The first term on the right-hand side of (20) can be bounded with the help of the simultaneous ( 2 , 1 )-quotient property (17) and of the sign product embedding property (15). We start by writing Ax + e as A (x + u) for some u ∈ R n as in (17). We then notice that
x + u 2 ≥ x 2 − u 2 ≥ x 2 − d e 2 / √ q ≥ (1 − dc 6 ) x 2 , x + u 1 ≤ x 1 + u 1 ≤ √ s x 2 + d √ s * e 2 / √ q ≤ 1 √ 2 + d c 6 √ 2c √ 2s x 2 .
Hence, if c 6 is chosen small enough at the start, then we have x + u 1 ≤ √ 2s x + u 2 , i.e., x + u is effectively (2s)-sparse. The sign product embedding property (15) of order 2s then implies that w,
x + u x + u 2 − π/2 q A * S∪T sign (Ax + e)
= w, x + u x + u 2 − π/2 q Aw, sign (A (x + u)) ≤ δ 2 for all unit-normed w ∈ R n supported on S ∪ T . This gives
x + u x + u 2 − π/2 q A * S∪T sign (Ax + e) 2 ≤ δ 2 ,
and in turn
x x 2 − π/2 q A * S∪T sign (Ax + e) 2 ≤ δ 2 + x x 2 − x + u x + u 2 2 ≤ δ 2 + 1 x 2 − 1 x + u 2 x 2 + u x + u 2 2 ≤ δ 2 + | x + u 2 − x 2 | x + u 2 + u 2 x + u 2 ≤ δ 2 + 2 u 2 x + u 2 .
From u 2 ≤ d e 2 / √ q and x + u 2 ≥ (1 − dc 6 ) x 2 ≥ x 2 /2 for c 6 is small enough, we derive that
x x 2 − π/2 q A * S∪T (sign (Ax + e)) 2 ≤ δ 2 + 4d e 2 √ q x 2 .(22)
Substituting (21) and (22) into (20) enables us to derive the desired result (18) from (19).
The proof of Theorem 4 presented next follows from Theorem 3.
Proof of Theorem 4. For later purposes, we introduce the constant
C := max ξ∈ 1 √ 2 − 1 20 , 2 √ 5 + 1 20 f (ξ) ≥ 2, f (ξ) := 1 − 1 − ξ 2 ξ .
Given x ∈ RB n 2 ∩ Σ s , we acquire a corrupted version y 1 ∈ {±1} q/2 of the quantized measurements sign(A 1 x). Since the number of rows of the matrix A 1 ∈ R (q/2)×n is large enough for Theorem 3 to hold with δ 0 = δ/(4(1 + 2C)) instead of δ, we obtain
x x 2 − u 2 ≤ δ 0 + c 4 cδ + c 5 c δ ≤ 2δ 0 , u := H s (A * 1 y 1 ),
provided that the constants c and c are small enough. With x denoting the orthogonal projection of x onto the line spanned by u, we have
x − x 2 ≤ x − x 2 u 2 ≤ 2δ 0 x 2 .
We now consider a unit-norm vector v ∈ R n supported on supp(u) and orthogonal to u. The situation in the plane spanned by u and v is summarized in Figure 3. We point out that x ≤ x ≤ R gave x 2 ≤ 2R, but that 2R was just an arbitrary choice to ensure that cos(θ) stays away from 1-here, cos(θ) ∈ [1/ √ 2, 2/ √ 5]. Forming the s-sparse vector w := 2R · (u + v), we now acquire a corrupted version y 2 ∈ {±1} q/2 of the quantized measurements sign(A 2 (x − w)) on the 2s-sparse vector x − w. Since the number of rows of the matrix A 2 ∈ R (q/2)×n is large enough for Theorem 3 to hold with δ 0 = δ/(4(1 + 2C)) instead of δ and 2s instead of s, we obtain
w − x w − x 2 − t 2 ≤ δ 0 + c 4 cδ + c 5 c δ ≤ 2δ 0 , t = −H s (A * 2 y 2 ).
We deduce that t also approximates
(w − x )/ w − x 2 with error w − x w − x − t 2 ≤ w − x w − x 2 − w − x w − x 2 2 + 1 w − x 2 − 1 w − x 2 (w − x) 2 + w − x w − x 2 − t 2 ≤ x − x 2 w − x 2 + w − x 2 − w − x 2 w − x 2 + 2δ 0 ≤ 2 x − x 2 w − x 2 + 2δ 0 ≤ 2 2δ 0 x 2 2R + 2δ 0 ≤ 4δ 0 .
It follows that t, v approximates
(w − x )/ w − x , v = cos(θ) with error | cos(θ) − t, v | = w − x w − x 2 − t, v ≤ w − x w − x 2 − t 2 v 2 ≤ 4δ 0 .
We then notice that x 2 = 2R − 2R tan(θ) = 2Rf (cos(θ)), so that 2Rf ( t, v ) approximates x 2 with error
x 2 − 2Rf ( t, v ) = 2R|f (cos(θ)) − f ( t, v )| ≤ 2R C | cos(θ) − t, v | ≤ 2R C 4δ 0 = 8Cδ 0 R.
Here, we used the facts that cos(θ) ∈ [1/ √ 2, 2/ √ 5] and that t, v ∈ [1/ √ 2 − 4δ 0 , 2/ √ 5 + 4δ 0 ] ⊆ [1/ √ 2 − 1/20, 2/ √ 5 + 1/20]. We derive that
x 2 − 2Rf ( t, v ) ≤ x 2 − x 2 + x 2 − 2Rf ( t, v ) ≤ x − x 2 + x 2 − 2Rf ( t, v ) ≤ 2δ 0 x 2 + 8C δ 0 R ≤ 2(1 + 4C)δ 0 R.
Finally, with the estimatex for x being defined aŝ
x := 2Rf ( t, v ) u,
the previous considerations lead to the error estimate
x −x 2 ≤ x − x 2 u 2 + | x 2 − 2Rf ( t, v )| u 2 ≤ 2δ 0 x 2 + 2(1 + 4C)δ 0 R ≤ 4(1 + 2C)δ 0 R.
Our initial choice of δ 0 = δ/(4(1 + 2C)) enables us to conclude that x −x 2 ≤ δR.
Second-order-cone-programming-based order-one recovery scheme
Proof of Theorem 2. Without loss of generality, we assume that R = 1/2. The general argument follows from a rescaling. We begin by considering the exact case in which e = 0. Observe that, by the Cauchy-Schwarz inequality,
x 1 ≤ x 0 · x 2 ≤ √ s.
Since x is feasible for program (9), we also have x 1 ≤ √ s. The result will follow from the following two observations:
• x,x ∈ √ sB n 1 ∩ B n 2 • sign( a i , x − τ i ) = sign( a i ,x − τ i ), i = 1, . . . , q.
Each equation a i , z −τ i = 0 defines a hyperplane perpendicular to a i and translated proportionally to τ i ; further, x andx are on the same side of the hyperplane. To visualize this, imagine √ sB n 1 ∩B n 2 as an oddly shaped apple that we are trying to dice. Each hyperplane randomly slices the apple, eventually cutting it into small sections. The vectorsx and x belong to the same section. Thus, we ask: how many random slices are needed for all sections to have small diameter? Similar questions have been addressed in a broad context in [PV14]. We give a self-contained proof that O(s log(n/s)) slices suffice based on the following result [PV14, Theorem 3.1].
Theorem 8 (Random hyperplane tessellations of √ sB n 1 ∩ S n−1 ). Let a 1 , a 2 , . . . , a q ∈ R n be independent standard normal vectors. If q ≥ Cδ −4 s log(n/s), then, with probability at least 1 − 2 exp(−cδ 4 q), all x, x ∈ √ sB n 1 ∩ S n−1 with
sign a i , x = sign a i , x , i = 1, . . . , q, satisfy x − x 2 ≤ δ 8 .
The positive constants c and C are absolute constants.
We translate the above result into a tessellation of √ sB n 1 ∩ B n 2 in the following corollary.
Corollary 9 (Random hyperplane tessellations of √ sB n 1 ∩ B n 2 ). Let a 1 , a 2 , . . . , a q ∈ R n be independent standard normal vectors and let τ 1 , τ 2 , . . . , τ q be independent standard normal random variables. If q ≥ Cδ −4 s log(n/s), then, with probability at least 1 − 2 exp(−cδ 4 q), all x, x ∈ √ sB n 1 ∩ B n 2 with
sign( a i , x − τ i ) = sign( a i , x − τ i ), i = 1, . . . , q, satisfy x − x 2 ≤ δ 4 .
be represented by a signal with small 1 -norm. In particular, (17) implies that, with probability at least 1 − exp(−cq), there exists a vector u satisfying e = Au with u 2 ≤ δ/4, u 1 ≤ c 1 δ 3 q/ log(n/q)
where c 1 is an absolute constant which we may choose as small as we need. We may now replace x withx = x + u and proceed as in the proof in the noiseless case. Reconstruction ofx to accuracy δ/4 yields reconstruction of x to accuracy δ/2, as desired. By replacing x withx, we have (mildly) increased the bound on the 1 -norm and the 2 -norm. Fortunately, x 2 ≤ x 2 + u 2 ≤ 1 and thusx remains feasible for the program (9). Further,x is approximately sparse in the sense that x 1 ≤ x 1 + u 1 ≤ √ s + c 1 δ 3 q/ log(n/q) =: √s . To conclude the proof, we must show that the requirement of Theorem 2, namely q ≥ C δ −4 s log(n/s), implies that the required condition of Corollary 9, namely q ≥ Cδ −4s log(n/s), is still satisfied. The result follows from massaging the equations, as sketched below.
If s ≥ c 2 1 δ 6 q/ log(n/q), then √s ≤ 2 √ s and the desired result follows quickly. Suppose then that s < c 2 1 δ 6 q/ log(n/q) and thuss ≤ c 2 δ 6 q/ log(n/q). To conclude, note that Cδ −4s log(n/s) ≤ q · C · c 2 δ 2 log(n/q) · (log(n/q) + log(1/c 2 ) + 6 log(1/δ) + log(log(n/q)) ≤ q,
where the first inequality follows since s log(n/s) is increasing in s and thuss may be replaced by its upper bound, c 2 δ 6 q/ log(n/q). The last inequality follows by taking c 2 small enough. This concludes the proof.
Numerical Results
This brief section provides several experimental validations of the theory developed above. The computations, performed in MATLAB, are reproducible and can be downloaded from the second author's webpage. The random measurements a i were always generated as vectors with independent standard normal entries. As for the random sparse vectors x, after a random choice of their supports, their nonzero entries also consisted of independent standard normal variables. Our first experiment (results not displayed here) verified on a single sparse vector that both its direction and magnitude can be accurately estimated via order-one recovery schemes, while only its direction could be accurately estimated using convex programs [PV13a,PV13b], 1 -regularized logistic regression, or binary iterative hard thresholding [JLBB13]. We also noted the reduction of the reconstruction error by several orders of magnitude from the same number m of quantized measurements when Algorithms 7-8 are used instead of the above methods. We remark in passing that this number m is significantly larger than the number of measurements in classical compressed sensing with real-valued measurements, as intuitively expected.
Our second experiment corroborates the exponential decay of the error rate. The results are summarized in Figure 4, whose logarithmic scale on the vertical axis confirms the behavior log( x− x * 2 / x 2 ) ≤ −cλ for the relative reconstruction error as a function of the oversampling factor λ = m/ log(n/s). The tests were conducted on four sparsity levels s at a fixed dimension n for an oversampling ratio λ varying through the increase of the number m of measurements. The number T of iterations in Algorithms 7 and 8 was fixed throughout the experiment based on hard thresholding and throughout the experiment based on second-order cone programming. The values of all these parameters are reported directly in Figure 4. We point out that we could carry out a more exhaustive experiment for the faster hard-thresholding-based version than for the slower second-order-cone-programming-based version, both in terms of problem scale and of number of tests. Figure 5(a), we observe an error decreasing by a constant factor at each iteration when the measurements are totally accurate. Introducing a pre-quantization noise e ∼ N (0, σ 2 I) in y = sign(Ax + e) does not affect this behavior too much until the "noise floor" is reached. Flipping a small fraction of the bits sign a i , x by multiplying them with f i = ±1, most of which being equal to +1, seems to have an even smaller effect on the reconstruction. However, these bit flips prevent the use of the secondorder-cone-programming-based version, as the constraints of the optimization problems become infeasible. But we still remark that the pre-quantization noise is not very damaging in this case either, see Figure 5(b), where the results of an experiment using 1 -regularized logistic regression in Algorithms 7 and 8 are also displayed. Other settings analyze quantized measurements where the number of bits used depends on signal parameters like sparsity level or the dynamic range [ACS09, GLP + 10, GLP + 13]. Boufounos develops hierarchical and scalar quantization with modified quantization regions which aim to balance the rate-distortion trade-off [Bou11,Bou12]. These results motivate our work but do not directly apply to the compressed sensing setting.
Theoretical guarantees more in line with the objectives of this paper began with Jacques et al. [JLBB13] who proved robust recovery from approximately s log n one-bit measurements. However, the program used has constraints which require sparsity estimation, making it NP-Hard in general. Gupta et al. offers a computationally feasible method via a scheme which either depends on the dynamic range of the signal or is adaptive [GNR10]. Plan and Vershynin analyze a tractable non-adaptive convex program which provides accurate recovery without these types of dependencies [PV13a,PV13b,ALPV14]. Other methods have also been proposed, many of which are largely motivated by classical compressed sensing methods (see e.g. [Bou09,MPD12,YYO12,MBN13,JDDV13]).
In order to break the bound (3) and obtain an exponential rather than polynomial dependence on the oversampling factor, one cannot take traditional non-adaptive measurements. Several schemes have employed adaptive samples including the work of Kamilov et. al. which utilizes a generalized approximate message passing algorithm (GAMP) for recovery, and the adaptive thresholds are selected in line with this recovery method. Adaptivity is also considered in [GNR10] which allows for a constant factor improvement in the number of measurements required. However, to our best knowledge our work is the first to break the bound given by (3).
Regarding the link between our methods and sparse binary regression, there is a number of related theoretical results focusing on sparse logistic regression [NRWY12, Bun08, VDG08, Bac10, RWL10, MVDGB08, KSST10], but these are necessarily constrained by the same limited accuracy of the one-bit compressed sensing model discussed in Section 1.
We also point to the closely related threshold group testing literature, see e.g., [Che13]. In many cases, the statistician has some control over the threshold beyond which the measurement maps to a one. For example, the wording of a binary survey may be adjusted to only ask for a positive answer in an extreme case; a study of the relationship of heart attacks to various factors may test whether certain subjects have heart attacks in a short window of time and other subjects have heart attacks in a long window of time. The main message of this paper is that by carefully choosing this threshold the accuracy of reconstruction of the parameter vector x can be greatly increased.
Conclusions
We have proposed a recursive framework for adaptive thresholding quantization in the setting of compressed sensing. We have developed both a second-order-cone-programming-based method and a hard-thresholding-based method for signal recovery from these type of quantized measurements. Both of our methods feature a bound on the recovery error of the form e −Ω(λ) , an exponential dependence on the oversampling factor λ. To our best knowledge, this is the first result of this kind, and it improves upon the best possible dependence of Ω(1/λ) for non-adaptively quantized measurements.
Figure 3 :
3The situation in the plane spanned by u and v.
Figure 4 :
4Averaged relative error for the reconstruction of sparse vectors (n = 100) by the outputs of Algorithms 7-8 based on (a) hard thresholding and (b) second-order cone programming as a function of the oversampling ratio. Our third experiment examines the effect of measurement errors on the reconstruction via Algorithms 7 and 8. Once again, the problem scale was much larger when relying on hard thresholding than on second-order cone programming. The values of the size parameters are reported on Figure 5. This figure shows how the reconstruction error decreases as the iteration count t increases in Algorithms 7 and 8. For the hard-thresholding-based version, see
Figure 5 :
5one-bit compressed sensing framework developed by Boufounos and Baraniuk [BB08] is a relatively new line of work, with theoretical backing only recently being developed. Empirical evidence and convergence analysis of algorithms for quantized measurements appear in the works Averaged relative error for the reconstruction of sparse vectors (n = 100) by the outputs of Algorithms 7-8 based on (a) hard thresholding (s = 15, m = 10 5 ) and second-order cone programming and (b) 1 -regularized logistic regression (s = 10, m = 2 · 10 4 ) as a function of the iteration count when measurement error is present. of Boufounos et al. and others [Bou09, BB08, LWYB11, ZBC10]. Theoretical bounds on recovery error have only recently been studied, outside from results which model the one-bit setting as classical compressed sensing with specialized additive measurement error [DPM09, JHF11, SG09].
We define sign(0) = 1.
AcknowledgementsWe would like to thank the AIM SQuaRE program for hosting our initial collaboration and also Mr. Lan for discussions around the relationship of our work to logistic regression.The positive constants c and C are absolute constants.Proof. For any z ∈ √ sB n 1 ∩ B n 2 , we notice that sign(we may apply Theorem 8 after projecting on S n to derivewith probability at least 1 − 2 exp(cδ 4 q). We now show that the inequality (23) implies thatx − x 2 ≤ δ/4. First note thatsince x 2 ≤ 1. Subtract and add x / [x , 1] 2 inside the norm and apply triangle inequality to obtainSince x 2 ≤ 1, we may remove x 2 from in front of the second term in parenthesis. Next, use the inequality a + b ≤ √ 2 · √ a 2 + b 2 on the two terms in parenthesis. This bounds the right-hand side by preciselywhich is bounded by δ/4 according to (23).This corollary immediately completes the proof of Theorem 2 in the case e = 0. We now turn to the general problem where e ∞ ≤ cδ 3 and thus e 2 ≤ cδ 3 √ q. We reduce to the exact problem using the simultaneous ( 1 , 2 )-quotient property (17), which guarantees that the error can
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| []
|
[
"Regularity conditions for arbitrary Leavitt path algebras",
"Regularity conditions for arbitrary Leavitt path algebras"
]
| [
"Gene Abrams \nDepartment of Mathematics\nUniversity of Colorado at Colorado Springs Colorado Springs\n80933ColoradoU.S.A\n",
"Kulumani M Rangaswamy \nDepartment of Mathematics\nUniversity of Colorado at Colorado Springs Colorado Springs\n80933ColoradoU.S.A\n"
]
| [
"Department of Mathematics\nUniversity of Colorado at Colorado Springs Colorado Springs\n80933ColoradoU.S.A",
"Department of Mathematics\nUniversity of Colorado at Colorado Springs Colorado Springs\n80933ColoradoU.S.A"
]
| []
| We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L K (E) is locally K-matricial; that is, L K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E:is strongly πregular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions. | 10.1007/s10468-008-9125-2 | [
"https://arxiv.org/pdf/0806.3743v2.pdf"
]
| 2,666,019 | 0806.3743 | 83e1e4d1167a745ef339d670a41173d47e6031b1 |
Regularity conditions for arbitrary Leavitt path algebras
5 Oct 2008
Gene Abrams
Department of Mathematics
University of Colorado at Colorado Springs Colorado Springs
80933ColoradoU.S.A
Kulumani M Rangaswamy
Department of Mathematics
University of Colorado at Colorado Springs Colorado Springs
80933ColoradoU.S.A
Regularity conditions for arbitrary Leavitt path algebras
5 Oct 2008Received: date / Accepted: dateAlgebras and Representation Theory manuscript No. (will be inserted by the editor)Leavitt path algebraacyclic graphvon Neumann regular algebra Mathematics Subject Classification (2000) Primary: 16S99, 16E50 Secondary: 16W50, 46L89
We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L K (E) is locally K-matricial; that is, L K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E:is strongly πregular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.
L K (E) is π-regular. (3) E is acyclic. (4) L K (E) is locally K-matricial (i.e., L K (E) is a direct union of subalgebras, each of which is isomorphic to a finite direct sum of finite matrix rings over K). (5) L K (E) is strongly π-regular. We conclude the article by discussing various additional ring-theoretic conditions which in the context of Leavitt path algebras are equivalent to E being acyclic.
We begin by giving a terse reminder of the germane definitions. For a more complete description and discussion, see e.g. [2] or [8]. A (directed) graph E = (E 0 , E 1 , r, s) consists of two sets E 0 , E 1 and functions r, s : E 1 → E 0 . (The sets E 0 and E 1 are allowed to be of arbitrary cardinality.) The elements of E 0 are called vertices and the elements of E 1 edges. A path µ in a graph E is a sequence of edges µ = e 1 . . . en such that r(e i ) = s(e i+1 ) for i = 1, . . . , n − 1. In this case, s(µ) := s(e 1 ) is the source of µ, r(µ) := r(en) is the range of µ, and n is the length of µ. We view the elements of E 0 as paths of length 0. If µ = e 1 ...en is a path in E, and if v = s(µ) = r(µ) and s(e i ) = s(e j ) for every i = j, then µ is called a cycle based at v. If s −1 (v) is a finite set for every v ∈ E 0 , then the graph E is called row-finite.
Definition 1 Let E be any directed graph, and K any field. The Leavitt path Kalgebra L K (E) of E with coefficients in K is the K-algebra generated by a set {v | v ∈ E 0 } of pairwise orthogonal idempotents, together with a set of variables {e, e * | e ∈ E 1 }, which satisfy the following relations:
(1) s(e)e = er(e) = e for all e ∈ E 1 .
(2) r(e)e * = e * s(e) = e * for all e ∈ E 1 .
(3) (CK1) e * e ′ = δ e,e ′ r(e) for all e, e ′ ∈ E 1 . (4) (CK2) v = P {e∈E 1 |s(e)=v} ee * for every vertex v ∈ E 0 having 1 ≤ |s −1 (v)| < ∞.
For any F ⊆ E 1 the set {e * | e ∈ F } will be denoted by F * . We let r(e * ) denote s(e), and we let s(e * ) denote r(e). If µ = e 1 . . . en is a path, then we denote by µ * the element e * n . . . e * 1 of L K (E). Many well-known algebras arise as the Leavitt path algebra of a graph. For instance, the classical Leavitt algebras L K (1, n) for n ≥ 2 arise as the algebras L K (Rn) where Rn is the "rose with n petals" graph described in Example 1 below. (See e.g. [1,Section 3].) Also, for each n ∈ N = {1, 2, ...}, the full matrix ring Mn(K) arises as the Leavitt path algebra of the oriented n-line graph
• v1 e1 G G • v2 e2 G G • v3
• vn−1 en−1 G G • vn while the Laurent polynomial ring K[x, x −1 ] arises as the Leavitt path algebra of the "one vertex, one loop" graph • v x g g A (possibly nonunital) ring R is called a ring with local units in case for each finite subset S ⊆ R there is an idempotent e ∈ R with S ⊆ eRe. If E is a graph for which E 0 is finite then we have P v∈E 0 v is the multiplicative identity in L K (E); otherwise, L K (E) is a ring with a set of local units consisting of sums of distinct vertices. Conversely, if L K (E) is unital, then E 0 is finite. L K (E) is a Z-graded K-algebra, spanned as a Kvector space by {pq * | p, q are paths in E}. (Recall that the elements of E 0 are viewed as paths of length 0, so that this set includes elements of the form v with v ∈ E 0 .) In particular, for each n ∈ Z, the degree n component L K (E)n is spanned by elements of the form {pq * | length(p) − length(q) = n}. The degree of an element x, denoted deg(x), is the lowest number n for which x ∈ L m≤n L K (E)m. The K-linear extension of the assignment pq * → qp * (for p, q paths in E) yields an involution on L K (E), which we denote simply as * .
A subgraph G of a graph E is called complete in case, for each v ∈ G 0 having 1 ≤
|s −1 G (v)| < ∞, we have s −1 G (v) = s −1 E (v).
(In other words, a subgraph G of E is complete if, whenever v ∈ G 0 emits a nonzero, finite number of edges in G, then necessarily the subgraph G contains all of the edges in E emitted by v.) The natural inclusion map L K (G) → L K (E) is a ring homomorphism precisely when G is a complete subgraph of E, so that complete subgraphs of E naturally give rise to subalgebras of L K (E). One of our main objectives in this article is to show how to construct subalgebras of L K (E) which need not arise in this way. This in turn will allow us to describe algebras of the form L K (E) as unions of subalgebras possessing various ring-theoretic properties, even in situations where E lacks complete subgraphs possessing corresponding graphtheoretic properties. We achieve this objective in Proposition 1. The construction is based on an idea presented by Raeburn and Szymański in [13, Definition 1.1]; this work was brought to our attention by E. Pardo.
Definition 2 Let E be a graph, and let F be a finite set of edges in E. We define s(F ) (resp. r(F )) to be the sets of those vertices in E which appear as the source (resp. range) vertex of at least one element of F . We define a graph E F as follows:
E 0 F = F ∪ (r(F ) ∩ s(F ) ∩ s(E 1 \F )) ∪ (r(F )\s(F )), E 1 F = {(e, f ) ∈ F × E 0 F | r(e) = s(f )} ∪ [{(e, r(e)
) | e ∈ F with r(e) ∈ (r(F )\s(F ))}], and where s((x, y)) = x, r((x, y)) = y for any (x, y) ∈ E 1 F . Note that, since F is finite, the graph E F is finite (regardless of the size of E).
Remarks: 1. It is conventional to define s(v) = v for each vertex v in E. Because of that, the expression in rectangular brackets in Definition 2 for E 1 F is redundant. However, we choose to keep this expression in the definition, as it makes the correspondence between E 1 F and the set G 1 in the proof of Proposition 1 more transparent. 2. While the construction presented in Definition 2 is similar to that given in [13,Definition 1.1], there are indeed some significant differences. For instance, the construction of [13, Definition 1.1] requires that the graph E has no sinks, while the construction presented here has no such stipulation. Additionally, even in situations where E is a graph with no sinks and F is a finite subset of E 1 , the two constructions can in fact yield different corresponding graphs E F . However, the underlying goal of each of the two constructions is the same, namely, to produce a subalgebra of a graph algebra which is isomorphic to the graph algebra of a finite graph.
Example 1 For clarity, we provide an example of the graph E F constructed in the previous definition. Let E be the "rose with n-petals" graph
E = • v y1 g g y2 s s y3 Õ Õ yn ... Let F = {y 1 }. Then E 0 F = {y 1 } ∪ {v}, and E 1 F = {(y 1 , y 1 ), (y 1 , v)}.
Pictorially, E F is given by
E F = • y1 (y1,y1) T T (y1,v) G G • v
This example indicates that various properties of the graph E need not pass to the graph E F . For instance, E is cofinal, while E F is not. In particular, L K (E) is a simple algebra, while L K (E F ) is not. (See [2] for a more complete discussion.)
Our interest in the construction given in Definition 2 can be generally described as follows. We seek to place each finite set of elements taken from the Leavitt path algebra L K (E) inside a subalgebra of L K (E) which possesses certain 'finiteness' properties. In case E is row-finite, by [5, Lemma 3.2] we can realize L K (E) as the direct union of subalgebras of the form L K (E i ) where each E i is a finite, complete subgraph of E. In the general case, however, we need not have such a description of L K (E). For instance, if ℵ is an infinite cardinal, and Clock(ℵ) denotes the 'infinite clock' graph
• • • y y c c G G 1 1 @ @ @ @ @ @ @ (ℵ) • •
having ℵ edges, then there are no nontrivial finite complete subgraphs of Clock(ℵ).
Example 2 It will be instructive to consider the E F construction of Definition 2 within the infinite clock graph E = Clock(ℵ). So let v denote the center vertex, let f denote one of the edges, and let w denote
r(f ). Let F = {f }. Then E 0 F = {f } ∪ {w}, while E 1 F = {(f, w)}. Thus E F is the graph E F = • f (f,w) G G • w
with two vertices, and one edge connecting them. In particular, L(E F ) ∼ = M 2 (K).
Although in general E F need not be a subgraph of E (indeed, as seen in Example 1, E F may contain more vertices than does E), there is an important relationship between the Leavitt path algebras L K (E F ) and L K (E), as we now show.
Proposition 1 Let F be a finite set of edges in a graph E. Then there is an algebra homomorphism θ : L K (E F ) → L K (E) having the properties:
(1) F ∪ F * ⊆ Im(θ).
(2) If w ∈ r(F ), then w ∈ Im(θ).
(3) If w ∈ E 0 has s −1 E (w) = ∅ and s −1 E (w) ⊆ F , then w ∈ Im(θ)
. Proof We define subsets G 0 and G 1 of L K (E) as follows.
G 0 = {ee * | e ∈ F } ∪ {v − X f ∈F,s(f )=v f f * | v ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F )} ∪ {v | v ∈ r(F )\s(F )} and G 1 = {ef f * | e, f ∈ F, s(f ) = r(e)} ∪ {e − X f ∈F,s(f )=r(e) ef f * | r(e) ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F )} ∪ {e ∈ F | r(e) ∈ r(F )\s(F )}.
We define θ :
L K (E F ) → L K (E) as follows.
There are three different types of vertices in E F . If w ∈ E 0 F has form w = e ∈ F , then define θ(w) = ee * .
If w ∈ E 0 F has form w = v with v ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ), then define θ(w) = v − X f ∈F,s(f )=v f f * . If w ∈ E 0 F has form w = v with v ∈ r(F )\s(F ), then define θ(w) = w.
Note that in each case we have θ(w) ∈ G 0 .
There are three different types of edges in
E F . If h ∈ E 1 F has form h = (e, f ) with f ∈ F , then define θ(h) = ef f * . If h ∈ E 1 F has form h = (e, r(e)) with r(e) ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ), then define θ(h) = e − X f ∈F,s(f )=r(e) ef f * .
If h ∈ E 1 F has form h = (e, r(e)) with r(e) ∈ r(F )\s(F ), then define
θ(h) = e.
Note that in each case we have θ(h) ∈ G 1 .
For each h ∈ E 1 F we define θ(h * ) = (θ(h)) * in L K (E).
It is now a long, straightforward check to verify that θ is compatible with the four types of relations which define L K (E F ) (refer to Definition 1). As a representative example of the computations required here, we offer the following. Let w ∈ E 0 F have the form w = e ∈ F . Then the (CK2) relation at e in L K (E F ) is the equation
X g∈E 1 F ,s(g)=e gg * = e. But s(g) = e in E 1 F means g = (e, f ) where either f ∈ F has s(f ) = r(e), or g = (e, r(e)) with r(e) ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ), or g = (e, r(e)) with r(e) ∈ r(F )\s(F ). So the (CK2) relation at e in L K (E F ) takes the form e = X f ∈F,s(f )=r(e) (e, f )(e, f ) * + X w∈r(F )∩s(F )∩s(E 1 \F ),w=r(e) (e, w)(e, w) * + X w∈r(F )\s(F ),w=r(e) (e, w)(e, w) * .
Note that empty sums are interpreted as 0. Also, the final two summation expressions are in fact either singletons or empty, depending on whether
r(e) ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ) or r(e) ∈ r(F )\s(F ).
We must show that the corresponding equation under θ holds in L K (E). In other words, we must show
ee * = X f ∈F,s(f )=r(e) (ef f * )(ef f * ) * + X w∈r(F )∩s(F )∩s(E 1 \F ),w=r(e) [e − X f ∈F,s(f )=r(e) ef f * ][e − X f ∈F,s(f )=r(e) ef f * ] * + X w∈r(F )\s(F ),w=r(e) ee * .
There are two cases. If r(e) ∈ r(F )\s(F ), then this equation simply becomes ee * = ee * and we are done. On the other hand, if r(e) ∈ s(F ), then note the second 'sum'
P w∈r(F )∩s(F )∩s(E 1 \F ),w=r(e) [e − P f ∈F,s(f )=r(e) ef f * ][e − P f ∈F,s(f )=r(e) ef f * ] * is in fact simply the single expression [e− P f ∈F,s(f )=r(e) ef f * ][e− P f ∈F,s(f )=r(e) ef f * ] * . So the right hand side is X f ∈F,s(f )=r(e) (ef f * )f f * e * + [e − X f ∈F,s(f )=r(e) ef f * ][e − X f ∈F,s(f )=r(e) ef f * ] * = X f ∈F,s(f )=r(e) ef f * e * + [ee * − X f ∈F,s(f )=r(e) ef f * e * ]
(by (CK1) and (CK2)) = ee * .
In a similar manner one can verify the compatibility of θ with all the remaining relations which define L K (E F ). Thus we conclude that θ extends to an algebra homomorphism θ :
L K (E F ) → L K (E).
By definition we have Im(θ) is the subalgebra of L K (E) generated by G 0 , G 1 , (G 1 ) * . It will be helpful later to note that for each x ∈ G 1 ∪ (G 1 ) * there exist y, y ′ ∈ G 0 for which yxy ′ = x. In particular, if an element z ∈ L K (E) is orthogonal to every element of G 0 , then necessarily z is orthogonal to every element in Im(θ).
We are now in position to verify the three claimed properties of Im(θ). For (1), we show that every f ∈ F is contained in Im(θ). Suppose first that f ∈ F has s −1
E (r(f )) ⊆ F . If s −1 E (r(f )) = ∅, then r(f ) ∈ r(F )\s(F ) vacuously, so by definition f ∈ G 1 ⊆ Im(θ). On the other hand, if s −1 E (r(f )) = ∅,
then we have f gg * ∈ G 1 for all g ∈ F having r(f ) = s(g). But then by hypothesis this is the same as the collection of g ∈ E 1 having
r(f ) = s(g). Thus we have {f gg * | g ∈ E 1 , s(g) = r(f )} ⊆ Im(θ), so that in particular Im(θ) contains P g∈E 1 ,s(g)=r(f ) f gg * = f · P g∈E 1 ,s(g)=r(f ) gg * = f · r(f ) = f .
On the other hand, suppose that f ∈ F has the property that s −1 E (r(f )) F . Then there are two possibilities. In the first case,
s −1 E (r(f )) = ∅ and s −1 E (r(f )) ∩ F = ∅.
(In other words, there are edges in E which are emitted from r(f ), but none of these edges are in F .) But then r(f ) / ∈ s(F ), so that f ∈ G 1 by definition, so that f ∈ Im(θ). In the second case, suppose
s −1 E (r(f )) ∩ F = ∅. Then either we have s −1 E (r(f )) ⊆ F (in which case we are done by the previous paragraph), or we have r(f ) ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ). In this situation we have f gg * ∈ G 1 ⊆ Im(θ) for all g ∈ F having s(g) = r(f ), so we in particular have P g∈F,s(g)=r(f ) f gg * in Im(θ) as well. But by definition we also have the element f − P g∈F,s(g)=r(f ) f gg * in G 1 ⊆ Im(θ). Then f = ( X g∈F,s(g)=r(f ) f gg * ) + (f − X g∈F,s(g)=r(f ) f gg * ) ∈ Im(θ).
Thus we conclude that F ⊆ Im(θ). But for each x ∈ Im(θ) we have x * ∈ Im(θ) by definition. Thus F ∪ F * ⊆ Im(θ), thereby establishing (1).
In particular, if w = r(f ) for f ∈ F , then w = f * f ∈ Im(θ), yielding (2). For (3), suppose s −1 (w) = ∅, and s −1 (w) ⊆ F . Then each f f * for f ∈ E 1 having s(f ) = w is in G 0 , so that P f ∈E 1 ,s(f )=w f f * is in Im(θ)
. But this last sum is precisely w by (CK2).
We remark here that for θ as given in Proposition 1, θ(w) = 0 for all three possible types of w ∈ E 0 F . (That θ(w) = 0 in case w ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ) hinges on the fact that there exists g ∈ E 1 \F having s(g) = w.) This in turn will allow us to conclude, in certain situations (including the situation where E is acyclic), that θ is in fact a monomorphism. (See e.g. [1].) However, we will not utilize this additional property of θ in the sequel.
With Proposition 1 in hand, we now construct the subalgebras of L K (E) which play the central role in our main result, Theorem 1.
The Subalgebra Construction
Let E be any graph, K any field, and {a 1 , a 2 , ..., a ℓ } any finite subset of nonzero elements of L K (E). For each 1 ≤ r ≤ ℓ write
ar = kc 1 vc 1 + kc 2 vc 2 + ... + kc j(r) vc j(r) + t(r) X i=1 kr i pr i q * ri
where each k j is a nonzero element of K, and, for each 1 ≤ i ≤ t(r), at least one of pr i or qr i has length at least 1. (That such a representation for each ar exists follows from properties of L K (E) mentioned previously.) Let F denote the (necessarily finite) set of those edges in E which appear in the representation of some pr i or qr i , 1 ≤ r i ≤ t(r), 1 ≤ r ≤ ℓ. Now consider the set S = {vc 1 , vc 2 , ..., vc j(r) | 1 ≤ r ≤ ℓ} of vertices which appear in the displayed description of ar for some 1 ≤ r ≤ ℓ. We partition S into subsets as follows:
S 1 = S ∩ r(F ),
and, for the remaining vertices T = S\S 1 , we define
S 2 = {v ∈ T | s −1 E (v) ⊆ F and s −1 E (v) = ∅} S 3 = {v ∈ T | s −1 E (v) ∩ F = ∅} S 4 = {v ∈ T | s −1 E (v) ∩ F = ∅ and s −1 E (v) ∩ (E 1 \F ) = ∅}.
Let E F be the graph as constructed in Definition 2 corresponding to this set F , and let θ : L K (E F ) → L K (E) be the homomorphism described in Proposition 1.
Definition 3 Let E be any graph, K any field, and {a 1 , a 2 , ..., a ℓ } any finite subset of nonzero elements of L K (E). Consider the notation presented in The Subalgebra Construction. We define B(a 1 , a 2 , ..., a ℓ ) to be the K-subalgebra of L K (E) generated by the set Im(θ) ∪ S 3 ∪ S 4 . That is, Proof (1) By Proposition 1 we have that F ∪ F * ∪ S 1 ∪ S 2 ⊆ B(a 1 , a 2 , ..., a ℓ ). Since S 3 ∪ S 4 ⊆ B(a 1 , a 2 , ..., a ℓ ) by construction, (1) follows.
B(a 1 , a 2 , ..., a ℓ ) =< Im(θ), S 3 , S 4 > .
(2) Since the {v i } and {uw j } are pairwise orthogonal idempotents we immediately get that P vi∈S3 Kv i = ⊕ vi∈S3 Kv i , and that P wj ∈S4 Kuw j = ⊕ wj∈S4 Kuw j . We now establish that the three indicated summands are mutually orthogonal, which will establish that the sum Im(θ)
+ (⊕ vi∈S3 Kv i ) + (⊕ wj ∈S4 Kuw j ) is direct. Let v ∈ S 3 .
Then by definition v is neither the source vertex nor range vertex for any of the elements in F . In particular, the one dimensional subalgebra Kv of L K (E) clearly annihilates all the elements of G 0 ; as noted previously, this suffices to yield that Kv indeed annihilates Im(θ). But for any w ∈ S 4 we have that uw
= w − P f ∈F,s(f )=w f f * is orthogonal in L K (E) to v, since S 3 ∩ S 4 = ∅.
So we have shown that ⊕ vi∈S3 Kv i ∩ [Im(θ) + ⊕ wj ∈S4 Kuw j ] = {0}. Thus we need only show that Im(θ) ∩ Kuw = {0} for all w ∈ S 4 , which we establish by showing Kuw ·Im(θ) = Im(θ)·Kuw = {0} and using the fact that each uw is idempotent. Choose any such w. Since by definition w / ∈ r(F ) we have that
uw = w − P f ∈F,s(f )=w f f * is orthogonal in L K (E) to elements of G 0 of the form w ′ − P g∈F,s(g)=w ′ gg * for w ′ ∈ r(F ) ∩ s(F ) ∩ s(E 1 \F ). Similarly, uw is orthogonal in L K (E) to r(e) for each e ∈ F . Now suppose ee * ∈ G 0 with e ∈ F . If s(e) = w then w − P f ∈F,s(f )=w f f * is clearly orthogonal to ee * . On the other hand, if s(e) = w then uw · ee * = (w − P f ∈F,s(f )=w f f * ) · ee * = wee * − P f ∈F,s(f )=w f f * ee * = ee * − ee * = 0,
with the final simplification occurring because f f * ee * = ee * for e = f , and f f * ee * = 0 otherwise by (CK1). Similarly, we have ee * · uw = 0. We conclude that the indicated sums are direct.
We now show that the direct sum in fact equals B(a 1 , a 2 , ..., a ℓ ). By construction, it suffices to show that uw ∈ B(a 1 , a 2 , ..., a ℓ ) for all w ∈ S 4 , and that w ∈ Im(θ) ⊕ (⊕ vi∈S3 Kv i ) ⊕ (⊕ wj ∈S4 Kuw j ) for all w ∈ S 4 . But each of these inclusions follow directly by noting that, for each w ∈ S 4 and each f ∈ F having s(f ) = w, we have f f * ∈ Im(θ) by definition.
(3) As noted previously, E F is a finite graph for each finite subset F of E 1 . In particular, L K (E F ) is a finitely generated K-algebra (with generating set E 0
F ∪ E 1 F ∪ (E 1 F ) * )
. This in turn implies that Im(θ), and hence B(S), is a finitely generated Kalgebra for each finite set S of L K (E). In particular, if S 1 and S 2 are finite subsets of L K (E), we let T 1 (resp. T 2 ) denote a finite set of generators of B(S 1 ) (resp. B(S 2 )).
If T = T 1 ∪ T 2 , then it is clear by construction that B(S 1 ) ∪ B(S 2 ) ⊆ B(T ).
(4) now follows immediately from (1) and (3).
As noted previously, various properties of the graph E need not pass to the graph E F . However, Lemma 1 Let E be any acyclic graph, and F any finite subset of E 1 . Then E F is acyclic.
Proof By contradiction, we show that the existence of a closed path in E F necessarily yields a closed path in E. By definition, a closed path in E F is of the form (e 1 , e 2 ), (e 2 , e 3 ), ..., (en, e 1 ) where (e i , e i+1 ) ∈ E 1 F . Now it is straightforward to show that the indicated sequence of edges in E F yields a sequence e 1 , e 2 , ..., en in E 1 having the desired property.
We recall now some ideas which play central roles in our main result. For additional information about these concepts, see for example [9], [10], and [11].
Definition 4 Let R be a (not necessarily unital) ring.
(1) R is called von Neumann regular in case for every x ∈ R there exists y ∈ R such that x = xyx.
(2) R is called π-regular in case for every x ∈ R there exists y ∈ R and n ∈ N for which x n = x n yx n .
(3) R is called left (resp. right) πregular if for each a ∈ R there exists n ∈ N and b ∈ R such that a n = ba n+1 (resp. a n = a n+1 b). (For rings with local units, this is equivalent to saying that the descending chain of left ideals Ra ⊇ Ra 2 ⊇ ... ⊇ Ra k ⊇ ... (resp. right ideals aR ⊇ a 2 R ⊇ ... ⊇ a k R ⊇ ...) becomes stationary after finitely many terms.) (4) R is called strongly π−regular if its both left and right π-regular.
Clearly any von Neumann regular ring is π-regular. Conversely, the ring Z/4Z provides an easy example of a ring which is π-regular but not von Neumann regular (since2 has no von Neumann regular inverse).
By [7,Lemma 6], if R is a unital strongly π-regular ring then for every element a ∈ R there is a positive integer n and an element x ∈ R such that ax = xa and a n+1 x = a n = xa n+1 . (We will show below that this result holds for rings with local units as well.) From this property it is then easy to show that if R is strongly π-regular, then R is π-regular. Conversely, the ring R = End K (V ) of all linear transformations of an infinite dimensional vector space V over a field K provides an example of a ring which is π-regular (in fact, von Neumann regular), but not strongly π-regular. (Indeed, if α : V → V is the shift transformation given by α(K 1 ) = 0 and α(K i+1 ) = K i for all i > 1, then for any n, ker α n = ⊕ n i=1 K i and so α n = βα n+1 for any n.)
Lemma 2 Let R be a ring with local units. Then R is strongly π-regular ring if and only if for every nonzero idempotent v of R, the subring vRv is strongly π-regular.
Proof Assume R is strongly π-regular. Pick a ∈ vRv. By hypothesis there exists b ∈ R with a n = a n+1 b, and there exists c ∈ R with a m = ca m+1 . But vav = v, so va n v = a n and va n+1 v = a n+1 . Thus, multiplying both sides of the equation a n = a n+1 b by v, we get a n = va n+1 vbv = a n+1 vbv. Since vbv ∈ vRv we have shown that a n = a n+1 b ′ for some b ′ ∈ vRv. A similar computation yields that a m = c ′ a m+1 inside vRv.
Conversely, pick a ∈ R. Then a ∈ vRv for some idempotent v by definition of set of local units. So there exist b and c in vRv, and hence in R, with the appropriate properties.
Although the properties von Neumann regular, strongly π-regular, and π-regular are in general not equivalent, as one consequence of Theorem 1 we conclude that these properties are indeed equivalent in the context of Leavitt path algebras.
We are now in position to establish our main result.
Theorem 1 Let E be an arbitrary graph, and let K be any field. The following are equivalent.
(1) L K (E) is von Neumann regular.
(2) L K (E) is π-regular. (3) E is acyclic. (4) L K (E) is locally K-matricial; that is, L K (E)
is the direct union of subrings, each of which is isomorphic to a finite direct sum of finite matrix rings over K.
(5) L K (E) is strongly π-regular.
Proof (1) ⇒ (2) is immediate.
(2) ⇒ (3). By contradiction, suppose c is a cycle in E, and let v = s(c) = r(c). We show that v + c has no π-regular inverse in L K (E).
Let γ denote v + c. Suppose there exist β ∈ L K (E), n ∈ N such that γ n βγ n = γ n . Note that, since γv = γ = vγ, we have γ n vβvγ n = γ n . Then α = vβv satisfies γ n αγ n = γ n and vαv = α. Moreover, αv = vα = α.
(v + n X k=1 n k ! c k )( N X i=M a i )(v + n X k=1 n k ! c k ) = v + n X k=1 n k ! c k .
Equating the lowest degree terms on both sides, we get va M v = v, so that a M = v.
Since deg(v) = 0, we conclude that M = 0, and that a 0 = v.
Thus α = N P i=0 a i .
Let deg(c) = s > 0. Now every term other than the first on the right hand side has degree ks for some positive integer k ≤ n, and so equating the corresponding graded components on both sides, we conclude that a i = 0 if i is not a multiple of s. We establish by induction that a ks = f k (c) for each k ∈ N, where f k (c) is a polynomial in c with integer coefficients. For k = 1, by equating the degree s components on both sides we obtain vasv + n 1
! ca 0 + a 0 n 1 ! c = n 1 ! c.
This implies that as = −`n 1´c , an integral polynomial in c. Now suppose t > 1, and suppose a ks = f k (c), an integral polynomial in c for all 1 ≤ k < t. We expand the previously displayed equation, and equate the degree ts terms of both sides. This yields
a ts + n 1 ! c[a (t−1)s + a (t−2)s n 1 ! c + a (t−3)s n 2 ! c 2 + ... + a 0 n t − 1 ! c t−1 ] + n 2 ! c 2 [a (t−2)s + a (t−3)s n 1 ! c + ... + a 0 n t − 2 ! c t−2 ] + n 3 ! c 3 [a (t−3)s + a (t−4)s n 1 ! c + ... + a 0 n t − 3 ! c t−3 ] + ... + n t ! c t a 0 = n t ! c t
Substituting for as, ..., a (t−1)s as allowed by the induction hypothesis and solving for a ts , we obtain a ts = f t (c), a polynomial in c with integer coefficients.
In particular, we conclude that every homogeneous component a i of α commutes with c in L K (E). This yields that cα = αc. But then the equation
(v + c) n α(v + c) n = (v + c) n becomes α(v + c) 2n = (v + c) n .
But this is impossible, as follows. Since each a i is a polynomial in c with integer coefficients, we have a i c r = 0 for all r ∈ N. Let i be maximal with the property that a i (v + c) 2n = 0. (Such i exists, since a 0 = v has this property.) Then the left hand side contains terms of degree 2sn + i (namely, a i c 2n ), while the maximum degree of terms on the right hand side is ns.
(3) ⇒ (4). We assume E is acyclic. Let {B(S) | S ⊆ L K (E), S finite} be the collection of subalgebras of L K (E) indicated in Proposition 2(3). By Proposition 2(4), it suffices to show that each such B(S) is of the indicated form. But by Proposition 2(2), B(S) = Im(θ) ⊕ (⊕ vi∈S3 Kv i ) ⊕ (⊕ wj∈S4 Kuw j ). Since terms appearing in the second and third summands are clearly isomorphic as algebras to K ∼ = M 1 (K), it suffices to show that Im(θ) is isomorphic to a finite direct sum of finite matrix rings over K.
Since E is acyclic, by Lemma 1 we have that E F is acyclic. But E F is always finite by definition, so we have by [3,Proposition 3.5
] that L(E F ) ∼ = ⊕ ℓ i=1 Mm i (K)
for some m 1 , ..., m ℓ in N. Since each Mm i (K) is a simple ring, we have that any homomorphic image of L K (E F ) must have this same form. So we get that Im(θ) ∼ = ⊕ L i=1 Mm i (K) for some m 1 , ..., m L in N, and we are done. (As remarked previously, since θ is in fact an isomorphism we have L = ℓ.) (4) ⇒ (1). It is well known that any algebra of the form ⊕ ℓ i=1 Mm i (K) is von Neumann regular. But every element of L K (E) is contained in a subalgebra of L K (E) of this form, so that every element of L K (E) thereby has a von Neumann regular inverse.
(4) ⇒ (5). Suppose L K (E) is locally K-matricial. So every element a ∈ L K (E) is contained in a subring S ∼ = ⊕ ℓ i=1 Mm i (K)
. As any such S is a unital left (resp. right) artinian ring, there is a b ∈ S and a positive integer n such a n = ba n+1 (resp. a n = a n+1 b).
(5) ⇒ (2) By Lemma 2 we have that each a ∈ L K (E) is contained in a strongly π-regular unital subring of the form vL K (E)v for some v = v 2 ∈ L K (E). Then by [7,Lemma 6] there is a positive integer n and an element x ∈ vL K (E)v such that ax = xa and a n+1 x = a n = xa n+1 . Now iterating the substitution a n = a n+1 x = aa n x = a(a n+1 x)x = a n+2 x 2 we get a n = a 2n x n , which using ax = xa gives a n = a n x n a n , which yields (2).
We record the following consequence of Theorem 1, in part because it demonstrates the independence of our results from any cardinality restrictions or graph-theoretic restrictions (e.g. row-finiteness) on the graphs.
Example 3 Let ℵ be any cardinal, and let Clock(ℵ) be the infinite clock graph having ℵ edges. Then for any field K, the Leavitt path algebra L K (Clock(ℵ)) is von Neumann regular. In addition, L K (Clock(ℵ)) is locally K-matricial.
It is worth noting that the locally K-matricial nature of L K (Clock(ℵ)) does not stem from a consideration of the finite complete subgraphs of Clock(ℵ), since as noted previously Clock(ℵ) contains no such nontrivial subgraphs.
As a second consequence of Theorem 1, we see that the ring R = End K (V ) of all linear transformations of an infinite dimensional vector space V over a field K cannot be represented as L K (E) for any graph E, since R is von Neumann regular but not strongly π-regular (as noted earlier). Similarly, let V be a vector space of uncountable dimension over a field K and let S be the (nonunital) subring of End K (V ) consisting of those linear transformations whose images are of at most countable dimension. Then S is a von Neumann regular ring with local units. However, S is not strongly π-regular, so again invoking Theorem 1 we have that S cannot be represented as the Leavitt path algebra of any graph E.
We conclude this article by analyzing two additional "regularity" properties of a ring. We recall the definitions of some ring-theoretic terms.
Definition 5 Let R be a unital ring.
(1) R is called clean if each a ∈ R is of the form a = e + u where e is an idempotent and u is a (two-sided) unit. If in addition aR ∩ eR = 0, we say R is a special clean ring. A clean ring R is said to be strongly clean if in the above definition we can choose e and u which commute.
(2) R is called unit regular in case for each a ∈ R there exists a (two-sided) unit u ∈ R such that aua = a. In particular, every unit regular ring is von Neumann regular.
Additional information about clean rings can be found in [11], while additional information about unit regular rings can be found in [9]. The properties "clean" and "unit regular" are exemplified by matrix rings. Indeed if R is the ring of n × n matrices over a field, then R is both unit regular [9, page 38] and strongly clean [11,Theorem 4.1]. By [6, Theorem 1], a unital ring R is unit regular if and only if R is a special clean ring; in particular, any ring of the form Mn(K) for K a field and n ∈ N is a special clean ring.
While the definitions of von Neumann regularity and π-regularity extend verbatim from unital rings to the nonunital case, the notions of clean and unit regularity require additional attention in the nonunital situation (since each definition refers to a unit in the given ring). We now show how to naturally extend these latter two notions to rings with local units.
Definition 6
Let R be a ring with local units.
(1) R is called locally unit regular if for each a ∈ R there is an idempotent v ∈ R for which a ∈ vRv, and elements u, u ′ ∈ vRv such that uu ′ = v = u ′ u, and aua = a.
(2) R is called locally clean if for each a ∈ R there is an idempotent v ∈ R for which a ∈ vRv, and elements e, u, u ′ ∈ vRv such that e is an idempotent, uu ′ = v = u ′ u, and a = e + u.
That the two notions given in the previous definition are natural generalizations of the corresponding notions for unital rings is established in the following.
Lemma 3 Let R be a unital ring.
(1) R is locally unit regular if and only if R is unit regular.
(2) R is locally clean if and only if R is clean.
Proof For (1), suppose R is a ring with 1 and is locally unit regular. Let a ∈ R, and let v, u, u ′ as given in the definition. Then w = u + (1 − v) and w ′ = u ′ + (1 − v) satisfy ww ′ = 1 = w ′ w and a = awa. Hence R is unit regular. The converse is clear with v = 1. Likewise, for (2), suppose R is a ring with 1 and is locally clean. Let a ∈ R, and write a = u + e as given in the definition. Then a = w + e ′ , where e ′ = e + (1 − v) is an idempotent and
w = u − (1 − v) is a two-sided unit in R (since with w ′ = u ′ − (1 − v),
we have ww ′ = 1 = w ′ w). Thus R is clean. As with (1), the converse follows with v = 1.
Our final result shows that for acyclic graphs E, L K (E) possesses the locally unit regular property, as well as a property involving clean unital subrings.
Theorem 2 Let E be an arbitrary graph, and let K be any field. Then the following conditions are equivalent:
(1) E is is acyclic.
(2) L K (E) is locally unit regular.
(3) L K (E) is a direct limit of unital strongly clean rings, each of which is special.
Proof (1) ⇔ (2) Suppose E is acyclic. Then, by Theorem 1, L K (E) is a direct union of direct sums of matrix rings each of which, by [9, page 38], is unit regular. It is then clear that L K (E) is locally unit regular, where for each a ∈ L K (E) we use for v the identity element of the corresponding subring B(S). Conversely, if L K (E) is locally unit regular, then it is, in particular, von Neumann regular. So, by Theorem 1, E is acyclic.
(2) ⇔ (3) Suppose L K (E) is locally unit regular. Since it is von Neumann regular, Theorem 1 implies that L K (E) is a directed union of direct sums of matrix rings L i each of which, as noted above, is a special clean ring which is, in addition, strongly clean. On the other hand, if L K (E) is a directed union of special clean rings L i , then each L i is unit regular by [6, Theorem 1], and so L K (E) is locally unit regular. (Again for each a ∈ L K (E) we use for v the identity element of the corresponding subring B(S).)
A study of L K (E) for arbitrary graphs E is presented by Goodearl in [8]. Included in [8] is a method to write E as a direct union of countable complete subgraphs. We now show how this approach together with the desingularization process yields an alternate proof of the implication (3) ⇒ (1) of Theorem 1. Our aim in doing so is to contrast this approach with that of using Proposition 1, which helps us to establish not only the von Neumann regularity of L K (E) for an acyclic graph E, but uncovers several internal properties of such an L K (E) (e.g., locally matricial and locally unit regular). (Our approach also shows the coincidence of von Neumann regularity with π-regularity and strong π-regularity for Leavitt path algebras.) One may also note that the desingularization approach as shown below does not work for π-regular rings since π-regularity, unlike von Neumann regularity, is not a Morita invariant (see e.g. [12]). Proposition 1 poses no such restrictions, and provides additional structural insight into these rings.
So suppose E is acyclic. By [8, Proposition 2.7] L K (E) = lim −→α∈A L K (Eα), with the limit taken over the set {Eα | α ∈ A} of countable complete subgraphs of E. So in order to show that L K (E) is von Neumann regular, it suffices to show that each L K (Eα) is von Neumann regular, since the direct limit of von Neumann regular rings is von Neumann regular. Since E is acyclic then necessarily so is each Eα.
Since Eα is countable, we may form a desingularization Fα of Eα. (See e.g. [2].) By construction, Fα is row-finite. Also, since desingularization preserves Morita equivalence, and von Neumann regularity is preserved by Morita equivalence for rings with local units by [4,Proposition 3.1], it suffices to show that each L K (Fα) is von Neumann regular. Since each Eα is acyclic, the desingularization construction shows that each Fα is acyclic as well.
But by [5,Lemma 3.2] Fα is the direct union of G β (the union taken over the set of finite complete subgraphs of Fα), and L K (Fα) = lim −→β∈B L K (G β ). Thus it suffices to show that each L K (G β ) is von Neumann regular. Since G β is a subgraph of Fα we have that G β is acyclic.
So in the end, to establish that L K (E) is von Neumann regular, it suffices to show that for any finite acyclic graph G that L K (G) is von Neumann regular. But by [3,Proposition 3.5] the Leavitt path algebra of a finite acyclic graph is isomorphic to a finite direct sum of finite dimensional matrix rings over the ground field K, and such rings are well known to be von Neumann regular (see e.g. [9, Section 1]).
We conclude this article by noting one more consequence of Theorem 1 (we thank the referee for this suggestion). The proof follows directly from the fact that von Neumann regularity is a Morita invariant for rings with local units. We contrast this result with the aforementioned remark that, in general, the π-regularity property is not a Morita invariant.
Corollary 1
The property of π-regularity is a Morita invariant for Leavitt path algebras; that is, if E and F are graphs with L K (E) Morita equivalent to L K (F ), then L K (E) is π-regular if and only if L K (F ) is π-regular, and in this case E and F are both acyclic.
acknowledgments
Proposition 2
2Let E be any graph, K any field, and {a 1 , a 2 , ..., a ℓ } any finite subset of nonzero elements ofL K (E). Let F denote the subset of E 1 presented in The Subalgebra Construction. For w ∈ S 4 let uw denote the element w − P f ∈F,s(f )=w f f * of L K (E).Then(1) {a 1 , a 2 , ..., a ℓ } ⊆ B(a 1 , a 2 , ..., a ℓ ).(2) B(a 1 , a 2 , ..., a ℓ ) = Im(θ) ⊕ (⊕ vi∈S3 Kv i ) ⊕ (⊕ wj ∈S4 Kuw j ). (3) The collection {B(S) | S ⊆ L K (E), S finite} is an upward directed set of subalgebras of L K (E). (4) L K (E) = lim −→{S⊆L K (E),S f inite} B(S).
where a M = 0, a N = 0, deg(a i ) = i for all nonzero a i having M ≤ i ≤ N , and a i = 0 if i > N or i < M . Since deg(v) = 0, the equation αv = vα implies that a i v = va i = a i for all i. Now expanding the equation γ n αγ n = γ n , we obtain
The authors thank E. Pardo and M. Siles Molina for their valuable discussions during the preparation of this paper. The authors also thank the referee for a very careful reading of, and suggested changes to, the initial version of the manuscript.
The classification question for Leavitt path algebras. G Abrams, P N Ánh, A Louly, E Pardo, J. Algebra. to appear. ArXiV: 0706.3874Abrams, G.,Ánh, P.N., Louly, A., Pardo, E.: The classification question for Leavitt path algebras, J. Algebra, to appear. ArXiV: 0706.3874.
The Leavitt path algebra of arbitrary graphs. G Abrams, G Aranda Pino, Houston J. Math. to appearAbrams, G., Aranda Pino, G.: The Leavitt path algebra of arbitrary graphs, Houston J. Math, to appear.
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Kaplansky, I: Topological representation of algebras. Trans. A.M.S. II1Kaplansky, I: Topological representation of algebras. II, Trans. A.M.S. 68(1), 62-75 (1950).
Clean Rings: a survey. W K Nicholson, Y Zhou, Advances in Ring Theory: Proceedings of the 4th China-Japan-Korea International Conference. Hackensack, N.J.Nicholson, W.K, Zhou, Y.: Clean Rings: a survey. In: Advances in Ring Theory: Pro- ceedings of the 4th China-Japan-Korea International Conference, pp. 181-198. World Sci. Publ., Hackensack, N.J. (2005). ISBN: 981-256-425-X.
Examples of semiperfect rings. L Rowen, Israel J. Math. 653Rowen, L.: Examples of semiperfect rings, Israel J. Math. 65(3), 273-283 (1989).
Cuntz-Krieger algebras of infinite graphs and matrices. I Raeburn, W Szymański, Trans. A.M.S. 3561Raeburn, I., Szymański, W.: Cuntz-Krieger algebras of infinite graphs and matrices, Trans. A.M.S. 356(1) , 39-59 (2003).
| []
|
[
"Weak energy shaping for stochastic controlled port-Hamiltonian systems",
"Weak energy shaping for stochastic controlled port-Hamiltonian systems"
]
| [
"F Cordoni ",
"L Di Persio ",
"R Muradore "
]
| []
| []
| The present work address the problem of energy shaping for stochastic port-Hamiltonian system. Energy shaping is a powerful technique that allows to systematically find feedback law to shape the Hamiltonian of a controlled system so that, under a general passivity condition, it converges or tracks a desired configuration. Energy shaping has been recently generalized to consider stochastic port-Hamiltonian system. Nonetheless the resulting theory presents several limitation in the application so that relevant examples, such as the additive noise case, are immediately ruled out from the possible application of energy shaping. The current paper continues the investigation of the properties of a weak notion of passivity for a stochastic system and a consequent weak notion of convergence for the shaped system considered recently by the authors. Such weak notion of passivity is strictly related to the existence and uniqueness of an invariant measure for the system so that the theory developed has a purely probabilistic flavour. We will show how all the relevant results of energy shaping can be recover under the weak setting developed. We will also show how the weak passivity setting considered draw an insightful connection between stochastic port-Hamiltonian systems and infinite-dimensional port-Hamiltonian system. | null | [
"https://arxiv.org/pdf/2202.08689v1.pdf"
]
| 246,904,693 | 2202.08689 | b155491eb3f6852de294c7ceb997cc09f83a8f33 |
Weak energy shaping for stochastic controlled port-Hamiltonian systems
17 Feb 2022
F Cordoni
L Di Persio
R Muradore
Weak energy shaping for stochastic controlled port-Hamiltonian systems
17 Feb 2022
The present work address the problem of energy shaping for stochastic port-Hamiltonian system. Energy shaping is a powerful technique that allows to systematically find feedback law to shape the Hamiltonian of a controlled system so that, under a general passivity condition, it converges or tracks a desired configuration. Energy shaping has been recently generalized to consider stochastic port-Hamiltonian system. Nonetheless the resulting theory presents several limitation in the application so that relevant examples, such as the additive noise case, are immediately ruled out from the possible application of energy shaping. The current paper continues the investigation of the properties of a weak notion of passivity for a stochastic system and a consequent weak notion of convergence for the shaped system considered recently by the authors. Such weak notion of passivity is strictly related to the existence and uniqueness of an invariant measure for the system so that the theory developed has a purely probabilistic flavour. We will show how all the relevant results of energy shaping can be recover under the weak setting developed. We will also show how the weak passivity setting considered draw an insightful connection between stochastic port-Hamiltonian systems and infinite-dimensional port-Hamiltonian system.
Introduction
In the last decades port-Hamiltonian systems (PHS) have seen a constantly growing interest. The theory of PHS merges two different points of view: (i) the theory of the port-based modelling and bond graphs, [Breedveld, 2006, Breedveld, 2004, Duindam et al., 2009, aiming at providing a unified framework for physical systems belonging to different domains and (ii) Hamiltonian and geometric mechanics, Van Der Schaft, 1998, Dalsmo andVan der Schaft, 1997].
Recently PHSs have been extensively used to tackle optimal control theory, [Ortega et al., 2002, Van Der Schaft and Cervera, 2002, Ortega et al., 1999.
The main object in PHS theory is the Dirac structure, that is a geometric object that describes the geometry of the system. Dirac structures have been introduced in [Courant, 1990] as a general geometric tool to treat degenerate symplectic structures in a unified way. Such objects allow to study and characterize the geometry of a wide variety of physical systems, that encompass presymplectic manifolds, Poisson dynamics and constrained systems, [Dalsmo and Van Der Schaft, 1998, van der Schaft and Maschke, 1995, Dalsmo and Van Der Schaft, 1998, Dalsmo and Van der Schaft, 1997]. The Dirac structure defines an implicit Hamiltonian system, leading to a definition of a Hamiltonian systems in terms of a set of algebraic-differential equations. Such a general description of a physical system allows to a systematic investigation of the interconnection, van der Schaft, 1992, Dalsmo andVan der Schaft, 1997], integrability, [Dalsmo and Van Der Schaft, 1998] and symmetries, [Blankenstein and Van Der Schaft, 2001], and also to study physical systems with nonholonomic constraints, [Gay-Balmaz and Yoshimura, 2015]. Recently, PHSs have been extended to the stochastic case, [Cordoni et al., 2019, Cordoni et al., 2021a, Cordoni et al., 2020, Cordoni et al., 2022, Satoh, 2017, Satoh and Fujimoto, 2012, Satoh and Saeki, 2014, Satoh and Fujimoto, 2010.
Among the most relevant application of PHS is the usage of the geometric properties of interconnected systems to design suitable controls to achieve a precise goal, typically with the aim of stabilizing the overall system at a desidered configuration, or to track a desired trajectory, [Ortega et al., 1999, Ortega et al., 2002. In fact, many physical systems rest at a configuration in which their total energy function assumes a minimum. In case dissipation is present, such configuration is asymptotically stable. Rarely the minimum of the potential energy coincides with the desired configuration, so that the idea is to implement proper control actions able to shape the system energy in order to force a minimum in correspondence with the desired configuration. This control technique is called energy shaping, [Ortega et al., 1999, Secchi et al., 2007. The stabilization of the system follows from the passivity property together with the La Salle's invariance principle.
The generalization of energy shaping techniques to the stochastic case is non trivial. Stabilization of stochastic passive systems has been first studied in [Florchinger, 1999, Florchinger, 1994, Florchinger, 2003, Satoh and Fujimoto, 2012. Energy shaping for stochastic PHS (SPHS) has been addressed in [Haddad et al., 2018], where standard results from deterministic energy shaping have been adapted by considering stochastic PHS. In [Satoh, 2017, Satoh andSaeki, 2014] different notions of stochastic stabilization are considered to include a broad range of possible physical examples. A different and yet related approach to stochastic energy shaping via Casimir functionals, that conserved quantities of the system, is studied in [Arnold et al., 1983]. The authors addressed energy shaping of a class of stochastic Hamiltonian systems via the associated infinitesimal generator. As noted by the authors, their approach is only valid for short time, whereas to look at the long-time behaviour the invariant measure of the stochastic system must be considered. In [Fang and Gao, 2016, Cordoni et al., 2021a, Cordoni et al., 2020 a weak notion of stochastic stability is considered, showing that such notion is in turn strictly related to the invariant measure of the SPHS. Broadly speaking, the weak notion of passivity introduced in [Fang and Gao, 2016, Cordoni et al., 2021a, Cordoni et al., 2020 is not defined on the whole state-space but only outside a ball centred at a specified state. This definition has several desirable implications regarding the limiting distribution of the system. It turns out that this weak notion of passivity is tailor-made to deal with stochastic equations with additive noise, allowing to extend previous results to consider also the case of SPHS's with non vanishing noise.
The present work extensively and systematically studies the notion of weak stochastic passivity used in [Cordoni et al., 2021a] and the related convergence, with particular attention to the energy shaping of SPHS. We will show that the proposed approach generalizes the results in [Haddad et al., 2018], including relevant physical systems that do not fall in their assumption. In particular, results proved in [Haddad et al., 2018], although being a very interesting first step in generalizing energy shaping to a stochastic scenario, have a few practical as well as theoretical limitations. On one side, in order to design the control, strong Casimir are considered. By strong Casimir, following the notation of [Cordoni et al., 2019, Lázaro-Camí andOrtega, 2008], we mean P−a.s. conserved quantities. On the other side, given the notion of convergence used in [Haddad et al., 2018], in order to stabilize the system the noise must vanishes at the desired configuration. The usage of strong Casimir, implies that the control must share the same noise as the physical system to be controlled. This is needed since, in order to obtain a Casimir for the system, the noise of the control must compensate P−a.s. the noise of the system. Such condition, as shown in [Cordoni et al., 2019], is hardly satisfied in real applications since it implies that it is possible to separate at any time the state of the system from the noise. Also, it is worth stressing that strong Casimir functional are rare and difficult to obtain.
The assumption on the vanishing noise has also other restrains. In fact, the idea of the energy shaping approach for PHS is that a controller can in principle stabilize the system around any configuration designing a suitable control law in feedback form so that the resulting controlled PHS is again a PHS with a new Hamiltonian function having a minimum in the desired configuration. This fact, together with the passivity property of the PHS, implies that the system stabilizes around the minimum of the Hamiltonian. Since the fact that the control does not affect the noise of a SPHS, the randomness of a SPHS cannot be changed by any law. Therefore, since the notion of stochastic stability used in [Haddad et al., 2018] requires a vanishing noise, it turns out that a system can be stabilized only around configurations for which the noise vanishes. Such assumption strongly limits the possible configurations around which a SPHS can be stabilized. Even more relevant, above assumption immediately rule out additive noise which is the standard case when real sensors are taken into account.
The approach proposed in the current paper solves all of the above problems. Using a weak notion of passivity we are able to consider a weak notion of convergence, which is strictly related to the invariant measure of a SPHS. Such notion, without any requirement on the vanishing noise, allows to include additive noise as well as to stabilize the system around any configuration. In the case of a vanishing noise we recover stabilization as proved in [Haddad et al., 2018]. Further, control design can be done as in the deterministic case where now the stochastic system oscillates around the desired configuration with a magnitude given by the noise that affects the system. Also, using the notion of weak Casimir we are able to include a wider class of possible Casimir. In order to be as general as possible, we will prove the main results also for the relevant class of stochastic systems with degenerate noise.
We will also show how, as typical in stochastic analysis, the problem of finding an invariant measure for a SPHS can be solve looking at stationary solutions for a deterministic PDE, called the Fokker-Planck equation, [Lorenzi andBertoldi, 2006, Borkar, 2006]. We will show that such deterministic PDE can be proved to be an infinite dimensional PHS in a Lebesgue space weighted by the invariant measure. Such result has a major consequence. It provides a deep and interesting connection between weak energy shaping of SPHS and energy shaping for infinite-dimensional PHS. This allows to tackle the problem of energy shaping either from a deterministic or a stochastic point of view. Such connection is only introduced in the current work and it will be further studied in the future.
The main contributions of the present paper are:
(i) to study a weak notion of stochastic passivity and stability for a wide class of SPHS;
(ii) to investigate the problem of energy shaping via the new proposed notion of stochastic passivity;
(iii) to generalize the energy shaping approaches available in the literature;
(iv) to show that the Fokker-Planck equation associated to a SPHS can be seen as an infinitedimensional deterministic PHS.
The structure of the paper is as follow: in Section 2 we introduce the main notions of stochastic passivity and stability. In Section 3 we introduced rigorously the weak notion of stochastic passivity, while in Section 4 we prove the connection to infinite-dimensional PHS. Section 5 is devoted to studying energy shaping under the weak notion of stochastic passivity introduced; Section 6 shows two examples where explicit invariant measures for a SPHSs are calculated.
2 Stability and passivity for stochastic differential equations Throughout the work we will consider a complete filtered probability space Ω, (F t ) t≥0 , P satisfying usual assumptions.
Before entering into details on energy shaping for stochastic port-Hamiltonian systems (SPHS), in order to make the paper as much self-contained as possible, some key results regarding the stability of a general stochastic differential equation (SDE) are briefly recalled. Consider a stochastic process (X(t)) t≥0 ∈ R n satisfying the following SDE
dX(t) = µ(X(t))dt + σ(X(t))dW (t) , X(s) = x ,(1)
where µ : R n → R n and σ : R n → R n×d are suitable regular enough coefficients, W (t) is a d−dimensional standard Brownian motion and dW (t) is the integration in Itô sense. We will use the convention X s,x (t) to denote the solution of equation (1) at time t starting at time s < t with initial value x. If no confusion is possible we will write for short X(t) = X s,x (t).
Next we recall different possible notions of convergence for a stochastic process X, [Florchinger, 2003, Khasminskii, 2011.
Definition 2.1. The equilibrium solution X(t) ≡ 0 is said: (iii) asymptotically stable in probability if it is stable in probability and for any s ≥ 0 and x ∈ R n it holds P lim t→∞ |X s,x (t)| = 0 = 1 .
We will denote by L the infinitesimal generator of the process (1). Recall that, [Karatzas and Shreve, 1998], for f : R n → R regular enough the infinitesimal generator L of the process X satisfying equation (1) is defined as
Lf (x) := lim t→0 E t,x [f (X(t))] − f (x) t ,
being E t,x the conditional expectation w.r.t. t and x. It can be shown, [Karatzas and Shreve, 1998], that the infinitesimal generator of X satisfying equation (1) is explicitly given by
Lf (x) = n i=1 µ i (x)∂ xi f (x) + 1 2 n i,j=1 σ(x)σ T (x) ij ∂ 2 xi xj f (x) = = µ(x) · ∂ x f (x) + 1 2 T r σ(x)σ T (x)∂ 2 x f (x) ,(2)
where ∂ x and ∂ 2 x are the first and second derivative in x, respectively. Stability of a SDE can be inferred assessing certain properties of the infinitesimal generator L. In particular, the following stochastic Lyapunov theorem holds, [Florchinger, 2003, Khasminskii, 2011.
LV (x) ≤ 0 , resp. LV (x) < 0 ,(3)
then the equilibrium solution X(t) ≡ 0 of the SDE (1) is stable in probability, resp. locally asymptotically stable, in probability. If further D = R n , the Lyapunov function V is said to be proper and the stability to be global.
On the passivity for controlled SDE
The present section is devoted to the introduction of the concept of passivity for controlled SDE and on its relation to stochastic stability.
Consider an input-state-output stochastic process (X(t)) t≥0 ∈ R n satisfying the SDE
dX(t) = µ(X(t), u(t))dt + σ(X(t))dW (t) , y(t) = h(X(t), u(t)) ,(4)
for u ∈ U, being the space of all (F t ) t≥0 −adapted process u : [0, T ] → U so that u ∈ L 2 ([0, T ]) P−a.s. is a U −valued progressively measurable process, where U is a closed subset of R m , representing the domain of the control process acting on the state process X. Also, y ∈ Y, being the space of all (F t ) t≥0 −adapted process P−a.s. is a Y −valued progressively measurable process, being Y a closed subset of R m . We will denote by X 0 (t) the solution to the autonomous system (4), that is the solution with constant null control u ≡ 0, while, L 0 is the infinitesimal generator of the autonomous system (4). Further, for a suitable regular enough function f , ∂ x f (x, u) denoted the partial derivative w.r.t. the first argument whereas ∂ u f (x, u) denotes the partial derivative w.r.t. the second argument.
Next is the definition of stochastic passivity, [Florchinger, 2003, Definition 3.1].
Definition 2.2. The input-state-output system (4) is said to be passive if there exists a Lyapunov function V on R n , called storage function, such that
LV (x) ≤ h T (x, u)u ,(5)
for every (x, u) ∈ R n × U .
A straightforward application of Itô formula and Dynkin lemma, [Karatzas and Shreve, 1998], shows that stochastic passivity, as defined in Definition 2.2, implies that
EV (X(t)) ≤ V (x) + E t 0 h T (X(s), u(s))u(s)ds .
Strictly related to stability of a SDE, the following necessary conditions for the input-stateoutput system (4) to be passive can be state, [Florchinger, 2003]. In the following, we will denote by L 0 the infinitesimal generator of the autonomous system (4), i.e. u ≡ 0.
Theorem 2.2. The following conditions are necessary for system (4) to be passive:
(i) L 0 V (x) ≤ 0, for every x ∈ R n ; (ii) for every x ∈ S := {x ∈ R N : L 0 V (x) = 0} it holds n i=1 ∂ u µ i (x, 0)∂ xi V (x) = h T (x, 0) ; (iii) for every x ∈ S := {x ∈ R N : L 0 V (x) = 0} it holds n i=1 ∂ 2 u u µ i (x, 0)∂ xi V (x) ≤ ∂ u h T (x, 0) + ∂ u h(x, 0) .
In the particular case that the stochastic system (4) is affine in the control, that is dX(t) = (µ(X(t)) +μ(X(t))u(t)) dt + σ(X(t))dW (t) ,
y(t) = h(X(t))(6)
the following stochastic version of the Kalman-Yakubovich-Popov (KYP) property can be proven, [Florchinger, 1999].
Definition 2.3. The stochastic system in affine form (6) satisfies the KYP property if there exists a proper Lyapunov function V such that for every x ∈ R n , it holds
L 0 V (x) ≤ 0 , n i=1 ∂ xi V (x)μ(x) = h T (x) .
We thus have the following.
Theorem 2.3. The stochastic system (6) is passive if and only if it satisfies the KYP property.
2.2 On the passivity for controlled stochastic port-Hamiltonian system
Having introduced the main notations and results regarding stability and passivity of general R n −valued SDE, we can thus introduce the notion of stochastic port-Hamiltonian system.
A stochastic PHS (X(t)) t≥0 is the solution to the SDE
dX(t) = [(J − R)∂ x H(X(t)) + g(X(t))u(t)] dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂H(X(t)) ,(7)
being J = J T a given n × n matrix, R 0 a n × n positive semi-definite matrix representing dissipation, H the Hamiltonian of the system, u ∈ U the control and y ∈ Y the output of the system. As above, W is a standard Brownian motion and dW (t) denoted the integration in the sense of Itô.
Remark 2.4. It is worth remarking that in [Cordoni et al., 2019] a stochastic port-Hamiltonian system has been defined in terms of the Stratonovich integral; the present work adopt the stochastic integration on the Itô sense as passivity is typically addressed considering Itô notion of integration. In [Cordoni et al., 2021a], an energy tank approach for a teleoperated system modelled as SPHS has been investigated. Similarly to the present paper, in [Cordoni et al., 2021a], the Itô point of view has been considered, since the main object of investigation was the passivity of the system. Nonetheless, in that paper, it has been shown how the Itô SPHS can be converted into a corresponding Stratonovich SPHS, showing further how passivity is affected. △
Using Dynkin lemma and Itô formula, it can be seen that the SPHS (7) satisfies the energy preserving property
H(X(t 2 )) − H(X(t 1 )) = t2 t1 LH(X(s))ds + t2 t1 ∂ T H(X(s))σ(X(s))dW (s) ,(8)
being L the infinitesimal generator (2) for the Itô SPHS (7) defined as
LH(x) = [(J − R) ∂ x H(x) + g(x)u] · ∂ x H(x) + 1 2 T r σ(x)σ T (x)∂ 2 x H(x) .(9)
We thus have the following result concerning the passivity and convergence of a SPHS, [Haddad et al., 2018, Theorem 4] or also [Florchinger, 2003, Satoh andFujimoto, 2012].
Proposition 2.5. Consider the stochastic PHS (7), if 2∂ T x H(x)R(x)∂ x H(x) ≥ T r ∂ 2 x H(x)σ(x)σ T (x) ,(10)
then, the SPHS (7) is passive. Further if:
(i) the SPHS (7) is passive;
(ii) the noise vanishes at an equilibrium configuration, that is σ(x e ) = 0;
then the equilibrium solution X(t) ≡ x e is Lyapunov stable in probability. If, in addition
{x ∈ R n : L 0 H(x) = 0} = {x e } ,
then the equilibrium solution X(t) ≡ x e is locally asymptotically stable in probability.
Proof. Using the infinitesimal generator (2), it immediately follows that condition (10) yields
LH(x) ≤ y T (t)u(t) ,
which is the Definition 2.2 of passivity for the SPHS (7). The convergence thus follows using [Khasminskii, 2011, Thm. 5.3, Cor. 5.1, Thm. 5.11].
Remark 2.6. Above results highlight how classical notions of stochastic passivity and stability have clear weaknesses. In particular, a vanishing noise need to be required, that is σ(x e ) = 0 at the desired equilibrium state x e . Such assumption is in general strong, but in the context of energy shaping for SPHS this assumption can have even stronger implications. In particular, it is a system property and cannot be modified in any way suitably shaping the energy of the system. The main idea of energy shaping is to derive a feedback control law u = φ(X) so that the dynamics of the stochastic equation under the feedback law preserves the port-Hamiltonian structure. The final goal is to shape the energy of the system so that it can be stabilized at a certain state x e , which was not the minimum of the original Hamiltonian function. Nonetheless, since only the Hamiltonian function can be suitably shaped and in particular the noise cannot be modified in any way, this new equilibrium point should already be a point for which the noise vanishes from the beginning. △ 3 Ultimately stochastic passivity and stability for controlled stochastic port-Hamiltonian system
In the introduction the main limitations behind the classical notion of passivity have been briefly mentioned, explaining how we intent to weaken certain approaches to suitably extend the idea of energy shaping to a broader stochastic context. The present section is devoted to introducing a weak notion of stochastic passivity and a consequently related notion of stability, that appears to be tailor-made for tackling the problem of energy shaping for SPHS. Such a notion has been first introduced in [Fang and Gao, 2016] and already used in the SPHS in [Cordoni et al., 2021a, Cordoni et al., 2020.
The definition of ultimately stochastic passivity for a stochastic port-Hamiltonian system, [Cordoni et al., 2020] is recalled.
Definition 3.1. [Ultimately stochastic passivity] The stochastic PHS (7) is said to be ultimately stochastic passive if for x e ∈ R n and for any x such that x − x e ≥ C, for a given constant C > 0 called passivity radius, it holds LH(x) ≤ y T u .
If further there exists δ C > 0 such that for x − x e ≥ C it holds
LH(x) ≤ y T u − δ C x − x e 2 ,
then system (7) is said to be strictly ultimately stochastic passive.
As mentioned in [Cordoni et al., 2021a, Cordoni et al., 2020, to highlight the connection of weak passivity with the limiting invariant measure and with the notion of deterministic ultimately bounded process, we will use the name ultimately stochastic passivity instead of weak passivity. In particular, our choice is motivated by the fact that the concept of weak passivity is closely related to a more general notion of convergence in the deterministic setting and called ultimately bounded. In fact, in the presence of a non-vanishing term, as in the present context with an additive noise, the process does not converge to an equilibrium but instead it can be proven to be bounded in a suitable domain. In the deterministic setting, the bounded stability can be proven to hold if there exists a Lyapunov function V , so thatV (x) < 0, ∀ x such that x − x e > C, with a given x e . In the stochastic setting we could retrieve a similar result choosing as candidate Lyapunov function the Hamiltonian of the system. If the process X is ultimately stochastic passive, or equivalently in the terminology of [Fang and Gao, 2016] weakly stochastic passive, then for the autonomous process with null control, i.e., u ≡ 0, it holds LV (
x) < 0, ∀ x such that x − x e > C.
The notion of ultimately stochastic passivity can be thought as follows: for X converging to x e = 0 we have that the system is not passive, as the noise keeps injecting energy into the system preventing the system from asymptotically stabilizing at x e = 0. Nonetheless, the system cannot exhibits non stationary behaviours since, as soon as the process exits a suitable ball of radius C, the system becomes passive and the energy injected by the noise into the system is strictly less then the one dissipated, so that the system recovers stability. It follows that, at large time, the system will be forced to stay in a fixed domain and keeps oscillating around the stationary point x e = 0 according to a suitable invariant law. Remark 3.1. As it will be clear in a while, the notion of strictly ultimately stochastic passivity is fundamental in proving the existence and uniqueness of an invariant measure for a stochastic system. In general to ensure stability of a stochastic system it is not enough to require only ultimately stochastic passivity, namely that there exists C > 0, so that for all x ≥ C, it holds
LH(x) < 0 .
As a counterexample consider the system
dX(t) = X(t) X 2 (t) + 1 dt + dW (t) ,(11)
and as Lyapunov candidate the function
H(x) = log 2 (1 + |x|) .
The explicit computation shows that there exists C > 0 so that, for x ≥ C it holds
LH(x) ≤ 0 .(12)
Nonetheless, system (11) diverges, hence becoming unstable. The key point is that, even if condition (12) holds true, it can be seen that
lim x→±∞ LH(x) = 0 .
To avoid such a phenomenon, thus ensuring system stability, the correct requirement is that there exist C > 0 and ǫ > 0, such that LH(x) < −ǫ .
for x ≥ C. △ Although Definition 3.1 might seem similar to the standard definition, it has some key aspects that makes it more suitable to be adapted to study SPHS and in general to address the problem of energy shaping. Some of these key features will be showed in the remaining of the current section, other will clearly emerges in subsequent sections.
A remarkable aspects of ultimately stochastic passivity of SPHS, is that under general and relevant setting, it can be shown that a SPHS cannot be passive but on the contrary it is always ultimately stochastic passive.
Proposition 3.2. Consider the stochastic PHS (7) with additive noise, i.e. σ(X(t)) ≡ σ, being Σ := σ T σ ≻ 0 a n × n positive definite matrix, and H a quadratic Hamiltonian of the form H(X(t)) = 1 2 X T (t)ΛX(t), with Λ a positive definite symmetric n × n matrix. Then the SPHS is never passive but it is always ultimately stochastic passive.
Proof. Using the skew-symmetric property of the matrix J, the infinitesimal generator (2) of the Itô process SPHS (7) L is given by
LH(x) = ∂ T x H(x) [(J(x) − R(x)) ∂ x H(x) + g(x)u] + 1 2 T r ∂ 2 x H(x)Σ = (13) = −∂ T x H(x)R(x)∂ x H(x) + ∂ T x H(x)g(x)u + 1 2 T r ∂ 2 x H(x)Σ .(14)
Using the quadratic form of the Hamiltonian function it follows that
∂ T x H(x)R(x)∂ x H(x) = x T ΛR(x)Λx ,(15)T r ∂ 2 x H(x)Σ = T r [ΛΣ] .(16)
Since T r [ΛΣ] > 0 is constant and strictly positive, it is immediate to see that for x sufficiently small, that is it exists ǫ > 0 such that for x < ǫ, the stochastic passivity is violated as
x T ΛRΛx < T r [ΛΣ] .(17)
On the contrary, there exists a positive constant C > 0 such that, for x ≥ C it holds
x T ΛR(x)Λx ≥ T r [ΛΣ] > 0 ,(18)
implying that, for x ≥ C, there exists δ C > 0 such that
LH(x) ≤ ∂ T x H(x)g(x)u = y T (t)u(t) − δ C x 2 ,
which is the Definition 3.1 of ultimately stochastic passivity.
A further immediate and relevant consequence of Proposition 3.2 is that, for the class of SPHS considered above, that is SPHS with additive noise and quadratic Hamiltonian, no additional conditions have to be assumed to guarantee the ultimately stochastic passivity of the stochastic system. This is in contrast to the classical notion of stochastic passivity, where an additional condition compared to the standard deterministic setting must be imposed. In fact, if ultimately stochastic passivity is considered, the SPHS considered in Proposition 3.2 is passive under the typically condition R ≻ 0.
The notion of ultimately stochastic passivity is strictly related to a convergence in a suitable weak sense of the SPHS. In particular, we will show that if the SPHS is strictly ultimately stochastic passive, then it converges toward the unique invariant measure of the system.
In order to prove the convergence of the SPHS toward an invariant measure, the Feller property and the transition Markov semigroup for the SPHS have to be studied. We will first recall basic definitions and results about ergodicity and Markov property for stochastic differential equations; we refer the reader to [Da Prato et al., 1996, Borkar, 2006, Khasminskii, 2011 for further details.
In the following we will assume without loss of generality that the SPHS (7) is equipped with the initial condition X(s) = x; we will denote for short by X(t) ≡ X s,x (t) the solution of the SPHS (7) at time t with initial time s < t and initial state x ∈ R n . We will say that the SPHS is a Markov process on R n , if P ( X(t) ∈ B| F s ) = P ( X(t) ∈ B| X(s)) , P − a.s. , for all t ≥ s and Borel set B ∈ B(R n ). In the following we will consider time homogeneous Markov process so that the considered process is invariant up to a time rescaling. This means that it is equivalent to consider as initial time s = 0; for this reason in the following we will omit explicitly the dependence upon the initial time s. For any Markov process we can introduce the notion of Markov transition function p(t, x, B), namely P(X x (t) ∈ B) =: p(t, x, B) , B ∈ B(R n ) .
We will say that the transition semigroup p is called Feller semigroup if the Markov semigroup P t defined as
P t f (x) := Ef (X x (t)) ,
is bounded and continuous for any f ∈ C b (R n ), being C b (R n ) the space of bounded and continuous function on R n . If P t f (x) is continuous and bounded for any t > 0 and for any f ∈ C b (R n ), then it is called strongly Feller semigroup. If, for t > 0, all Markov semigroup P t are equivalent, then P t is called t−regular.
The Markov semigroup and the Markov transition function are connected by the following
P t f (x) = R n f (y)p(t, x, dy) ,(19)
that can be also expressed as
P t ½ B (x) = p(t, x, B) , B ∈ B(R n )) .
Under certain regularity condition, [Lorenzi and Bertoldi, 2006], the function v(t, x) defined through Markov semigroup in equation (19) as
v(t, x) := P t f (x) ,
is the solution of the Cauchy problem
∂ t v(t, x) − Lv(t, x) = 0 , (t, x) ∈ R + × R n , v(0, x) = f (x) ,(20)
where L is the infinitesimal generator of the Markov process with transition function p(t, x, B).
For the definitions of strongly Feller Markov semigroup and regular Markov semigroup, that will be used later, we refer to the literature [Da Prato et al., 1996].
We can give the following definition of invariant measure for a SPHS, [Borkar, 2006, Definition 1.5.14].
Definition 3.2. Consider the SPHS (7), a measure ρ is said to be an invariant measure for the SPHS (7)
if it holds R n p(t, x, B)ρ(dx) = ρ(B) .(21)
The Definition 3.2 can be equivalently written in terms of the Markov transition semigroup P t as
R n P t f (x)ρ(dx) = R n f (x)ρ(dx) , f ∈ B b (R n ) ,(22)
being B b (R n ) the set of Borel and bounded function over R n . If the process X is a Feller process, then it can be shown that, [Borkar, 2006, Lemma 2.6.14], a measure ρ is invariant according to the Definition 3.2 if and only if it is infinitesimally invariant, that is, For what concern existence of an invariant measure, exploiting the ultimately stochastic passivity property of the system, we will show that there exists a unique invariant measure. In particular, the existence follows from the next result, [Lasota and Szarek, 2006, Prop. 3.1], that we report in order to make the treatment as much self-contained as possible.
R n Lf (x)ρ(dx) = 0 ,
Proposition 3.3 (Proposition 3.1 [Lasota and Szarek, 2006]). Let (X, · X ) be a complete separable metric space and let (P t ) t≥0 be the semigroup of Markov operators corresponding to a Markov process which satisfies the Feller property. Assume that there exist a compact set B and a point x ∈ X such that
lim sup T →∞ 1 T T 0 P t ½ B (x)dt > 0 .
Then the Markov process has a stationary distribution.
We thus have the following notion of convergence for a SPHS.
and
lim T →∞ 1 T T 0 p(t, x, B)dt = ρ(B) .(24)
then that the SPHS is ultimately stochastic stable.
We will assume throughout the paper that there exists a feedback control law u(t) = u(X(t)) so that the SPHS admits a global solution. Also, without loss of generality we will assume x e = 0. The next proposition states that if a SPHS is strictly ultimately stochastic passive, then it is ultimately stochastic stable.
Proposition 3.4. Consider the SPHS (7) and assume that:
(i) it is strictly ultimately stochastic passive;
(ii) the noise is non-degenerate, that is the matrix Σ(x) := σ(x)σ T (x) ≻ 0 is positive definite.
Then, the SPHS (7) admits a unique invariant measure and is ultimately stochastic stable.
Proof. For the sake of readability we will divide the proof in several steps: in particular, in step 1 we will prove existence of an invariant measure, in step 2 we will prove the long time convergence of the transition density towards one of the invariant measures and at step 3 we will show uniqueness of the invariant measure.
(Step 1 -existence) Under above assumptions the SPHS admits an invariant measure. In fact, since the SPHS is strictly ultimately stochastic passive, there exists ǫ > 0 such that, for x ≥ C,
C > 0, it holds LH(x) < −ǫ < 0 .(25)
Then, using Itô formula we have that
EH(X(t)) ≤ H(x) + t 0 ELH(X(s))ds .(26)
Using equations (25)-(26) we obtain
ELH(X(s)) ≤ C m P ( |X(s)| ≤ C| X(0) = x) − ǫP ( |X(s)| > C| X(0) = x) = = −ǫ + (C m + ǫ)P ( |X(s)| ≤ C| X(0) = x) ,(27)
being C m the maximum value of LH over the set x ≤ C.
Using the fact that H(x) ≥ 0, it follows from equation (26) − 1 t EH(X(0)) ≤ 1 t t 0 ELH(X(s))ds , so that using estimate (27) we obtain
− 1 t EH(X(0)) + ǫ < (C m + ǫ) 1 t t 0 P (|X(s)| ≤ C) ds .(28)
The existence of the invariant measure thus follows using equation (28) together with Proposition 3.3.
(
Step 2 -convergence) Let τ denote the first time the SPHS X reaches the sphere x ≤ C and by t ∧ τ := min{t, τ }; then, Itô formula yields
EH(X(t ∧ τ )) = H(x) + t∧τ 0 ELH(X(s))ds ≤ H(x) − ǫE[t ∧ τ ] .
The fact that H is non-negative implies
E[t ∧ τ ] ≤ H(x) ǫ .
In particular, assumption (B.2) in [Khasminskii, 2011, Assumption B, Chapter 4] holds true. Therefore, using [Khasminskii, 2011, Theorem 4.2] we obtain that, for a function F integrable with respect to an invariant measure ρ, it holds
P 1 T T 0 F (X(t))dt → R n F (y)ρ(dy) = 1 , as T → ∞ .(29)
If the function F is bounded, equation (29) implies using Lebesgue dominated convergence thereon,
lim T →∞ 1 T T 0 EF (X(t))dt = R n F (x)ρ(dx) ,(30)
which in turn implies, for B ∈ B(R n ), that
lim T →∞ 1 T T 0 p(t, x, B)dt = ρ(B) .(31)
At last, [Khasminskii, 2011, Theorem 4.3] yields that lim t→∞ p(t, x, B) = ρ(B) .
We have thus proven that the PSHS (7) is ultimately stochastic stable.
(
Step 3 -uniqueness) To prove uniqueness of the invariant measure ρ, denote by ρ 1 another invariant measure. In particular, by the definition of invariant measure it holds
R n p(t, x, B)ρ 1 (dx) = ρ 1 (B) .(33)
Integrating equation (33) in [0, T ] we obtain from equation (31) that ρ(B) = ρ 1 (B) from which we infer the uniqueness of the invariant measure ρ.
Proposition 3.4 establish a key result concerning the convergence of a strictly ultimately stochastic passive towards the unique invariant measure. Such result is based, besides ultimately passivity of the SPHS, also on an assumption of non-degeneracy of the noise. Although such assumption can be considered to be fairly general in certain contexts, there is a general class of processes of particular interest that fails to satisfy the above non-degeneracy assumption. In particular, stochastic oscillator equations of the form
dq(t) = f q (p, q)dt , dp(t) = f p (p, q)dt + σ(p, q)dW (t) ,(34)
do not satisfy non-degeneracy assumed in Proposition 3.4. Given the relevance of this class of systems, we will provide an alternative version of Proposition 3.4 dropping the non-degeneracy assumption on the noise.
Proposition 3.5. Consider the SPHS (7) and assume that:
(i) it is strictly ultimately stochastic passive;
(ii) the transition kernel p(t, x, B) is equivalent to the Lebesgue measure for any t > 0 and x ∈ R n .
Then, the SPHS (7) admits a unique invariant measure, which is absolutely continuous with respect to the Lebesgue measure, and it is ultimately stochastic stable.
Proof. As in the proof of Proposition 3.4 we will divide the current proof into three steps. Also, for the sake of brevity steps that follow from the same arguments as in Proposition 3.4 will be skipped.
(
Step 1 -existence) Same arguments as in Step 1 of the proof of Proposition 3.4 yield existence of an invariant measure.
(
Step 2 -convergence) Since p(t, x, A) is absolutely continuous with respect to the Lebesgue measure, denoting with a slight abuse of notation again by p its density, we have that
R n B p(t, x, y)dyρ(dx) = ρ(B) ,
so that Fubini theorem yields that also ρ is absolutely continuous with respect to the Lebesgue measure. In the following we will denote again by ρ the density of the invariant measure ρ. Thus, following [Zakai, 1969, Theorem 3] we can infer equations (29)-(30)-(31)-(32), so that the PSHS (7) is ultimately stochastic stable.
(
Step 3 -uniqueness) Uniqueness of the invariant measure follows from equation (31) as in the proof of Proposition 3.4.
Therefore, to consider stochastic oscillator alike equation (34), the transition kernel of the driving process must be studied. The next example shows how Proposition 3.5 can be applied to a simple and yet relevant class of stochastic oscillator systems.
Example 3.1. Consider the system with additive noise
dq(t) = p(t)dt , dp(t) = f p (p, q)dt + σdW (t) ,(35)
with σ > 0. Since the transition probability kernels are equivalent under a change of probability measure, we can apply Girsanov theorem, [Karatzas and Shreve, 1998], and introduceW , a Brownian motion under the probability measureP equivalent to the original probability measure P, defined asW
(t) := W (t) + σ −1 t 0 f p (p, q)ds .
Equation (35) can be thus rewritten as
dq(t) = p(t)dt , dp(t) = σdW (t) ,(36)
or in compact form as
dX(t) = AX(t)dt + ΣdW (t) , X(t) = (q(t), p(t)) T ,(37)
for some suitable constant matrices A and Σ. An integration with respect to time shows that the transition probability kernel is Gaussian distributed with covariance matrix given by
t 0 e A(t−s) ΣΣ T e A T (t−s) ds .(38)
To show that p is equivalent to the Lebesgue measure we must show that equation (38) is positive definite. This is equivalent to the requirement that the pair (A, Σ) is controllable , that is the matrix [Σ, AΣ, . . . , A n−1 Σ] span R n . Therefore, a direct calculation implies that the transition probability kernel associated to (35) is equivalent to the Lebesgue measure and therefore Proposition 3.5 applies.
This example can be generalized to other systems and the equivalence of the probability kernel can be checked via analogous arguments studying the controllability of the pair (A, Σ). △
Energy balance of SPHS
Consider the SPHS (7), then by mean of Itô-formula, we have the following mean energy balance equation
EH(X(t)) − H(x) = E t 0 u T (s)y(s)ds + E t 0 LH(X(s))ds = = E t 0 u T (s)y(s)ds − d(t), ,(39)
where d represents the dissipation of the system. The objective is thus to find a suitable control law u = φ(x) + κ such that the controlled SPHS has Hamiltonian function H d with minimum in a desired configuration x e .
Consider the SPHS of the form
dX(t) = [(J − R)∂ x H(X) + g(X(t))u(t)] dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂ x H(X(t)) .(40)
We want to design a feedback control u(t) = φ(X(t)) so that the controlled port-Hamiltonian system
dX(t) = [J d − R d ]∂ x H d (X(t))dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂ x H d (X(t)) ,(41)
maintains the port-Hamiltonian structure. The next result is the stochastic counterpart of the energy shaping result proved in [Ortega et al., 2002] in the deterministic setting.
Proposition 3.6. Given the SPHS (40) and assume that it is possible to find φ, J a , R a and K such that
([J + J a ] − [R + R a ]) K(X(t)) = g(X(t))φ(X(t)) − [J a − R a ] ∂ x H(X(t)) ,(42)
and such that the following conditions hold (i) structure preservation:
J d := J + J a = − (J + J a ) T ,(43)R d := R + R a = (R + R a ) T 0 ; (44) (ii) integrability: it holds that ∂ x K(X(t)) = ∂ T x K(X(t)) ;(45)
(iii) equilibrium assignment for a given x e ∈ R n , it holds
K(x e ) = −∂ x H(x e ) ;(46)
(iv) stability for x e ∈ R n , it holds
∂ x K(x e ) > −∂ 2 x H(x e ) ;(47)
(v) ultimately stochastic passivity for all x such that x − x e > C,
LH d (x) ≤ −ǫ < 0 ;(48)
(vi) uniqueness of the invariant measure either one of the following holds true:
(vi a) non-degeneracy the noise is non-degenerate, that is the matrix Σ(x) := σ(x)σ T (x) ≻ 0 is positive definite;
(vi b) equivalence the transition kernel p(t, x, B) is equivalent to the Lebesgue measure for any t > 0 and x ∈ R n .
Then the closed-loop system (40) with control feedback law u(t) = φ(X(t)) is a SPHS of the form
dX(t) = [J d − R d ]∂ x H d (X)dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂ x H(X(t)) .(49)
with H d := H + H a , ∂ x H a = K and there exists a unique invariant measure ρ under which (49) is ultimately stochastic stable.
Proof. Equations (42) ensures that its solution K is such that the closed-loop SPHS is of the form (49) with total energy H d = H + H a . Using Propositions 3.4-3.5 it follows that the invariant measure is unique and the process is ultimately stochastic stable.
Remark 3.7. Several comments on Proposition 3.6 are in order:
1. as regard condition (v) on ultimately stochastic passivity in Proposition 3.6, it is worth noticing that the following holds true
LH d (x) = L 0 H(x) + L a H a (x) + ∂ T x H d (x) [J d − R d ] ∂ x H d (x) ,(50)
where L 0 is the infinitesimal generator of the autonomous SPHS (40) with null control u ≡ 0 and L a is the infinitesimal generator of the autonomous SPHS (40) with structure matrices J a and R a . Using conditions (43), equation (50) reduces to
LH d (x) ≤ L 0 H(x) + L a H a (x) ,(51)
so that if the original SPHS and the SPHS with structure matrices J a and R a are ultimately stochastic passives, then ultimately stochastic passivity holds true because
LH d (x) ≤ L 0 H(x) + L a H a (x) < −ǫ .
2. in [Haddad et al., 2018] an alternative energy shaping was proposed. As already mentioned in the introduction, there is one key difference, with a fundamental implication, between Proposition 3.6 and [Haddad et al., 2018, Theorem 4]. In [Haddad et al., 2018, Theorem 4] it is required that the noise vanishes at the equilibrium, that is σ(x e ) = 0. Such condition has two relevant consequences: (i) an additive noise cannot be considered, and (ii) the Hamiltonian can be shaped only around the points in which the noise vanishes. Such conditions limit the range of application of [Haddad et al., 2018, Theorem 4]. Therefore, our proposed setting aims at filling the gap in which results in [Haddad et al., 2018] cannot be applied. It is worth stressing nonetheless that, in the case of a vanishing noise we recover the same results;
More formally, consider a vanishing small noise σ ε := εσ, either additive or multiplicative, and denote by ρ ε the invariant measure of the SPHS (49) with volatility σ ε . Then, under some mild integrability assumptions on the regularity of the coefficients, is can be seen that, [Huang et al., 2018], ρ ε → ρ, ε → 0, in the weak * topology of probability measure, that is
R n f (x)ρ ε (dx) → R n f (x)ρ(dx) , as ε → 0 ,
for any f ∈ C b (R n ). Also, the limiting measure ρ, corresponding to the deterministic PHS (49) with null volatility σ ≡ 0, can be seen to have support in
supp(ρ) ⊂ S = x ∈ R n : ∂ T x H d (x)R d (x)∂ x H d (x) = 0 .
Therefore in the case S = {x e }, the deterministic PHS is asymptotically stable and ρ ε → δ xe as ε → 0, being δ xe the Dirac measure concentrated in x e , [Huang et al., 2018]. This clarifies the structure of Proposition 3.6 in the sense that the shaped SPHS possesses an invariant measure which is shaped around the equilibrium that is typically specified in the deterministic context. In this sense, since in general the noise is an external disturbance that cannot be removed, the propose weak energy shaping aims at stabilizing a system around a desired configuration given a certain environmental noise that cannot be removed or compensated.
△ Proposition 3.6 clarifies the main idea behind the proposed setting. In general, a stochastic system is a dynamic system subject to external random perturbations. Since in the proposed setting the noise cannot be affected by the control law, the main idea is to shape the dynamic system around the equilibrium desired for the deterministic system. This shaping is done considering ergodic properties of the stochastic systems. In particular, to a smaller noise corresponds to an invariant measure concentrated around the equilibrium of the deterministic PHS. A further relevant point opened by the current research is the opposite point of view. That is, given a deterministic PHS, a controller can inject a suitable noise into the system such that the resulting system enjoys better stability properties, such as faster convergence to the equilibrium, [Arnold et al., 1983]. For instance, this can be achieved via a suitable multiplicative noise of increasing magnitude in certain domains. It is worth stressing that results show that, suitably injecting a random perturbation into a stochastic system can stabilize a deterministic dynamic system that would not be stabilizable otherwise. Such stabilization-by-noise is not treated in the current research and will be the topic of future research.
On the connection with infinite dimensional deterministic Port-Hamiltonian systems
The present Section aims at showing a suggestive connection between SPHS and infinitedimensional deterministic PHS. As introduced in previous Sections, the weak energy shaping approach is based on the invariant measure of an SPHS. In order to study the invariant measure of a SDE, a common and powerful approach is to study the Fokker-Planck equation associated to the SDE. We will show that the related Fokker-Planck equation is an infinite-dimensional PHS on a suitable functional space, so that in turn the problem of weak energy shaping of stochastic PHS can be associated to the problem of energy shaping of an infinite-dimensional controlled PHS.
Consider the SPHS dX(t) = [(J − R)∂ x H(X(t)) + g(X(t))u(t)] dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂H(X(t)) .
As introduced in Section 2, the transition semigroup P t defined as
P t f (x) := Ef (X x (t))
,
for any f ∈ C b (R n ).
Under certain regularity condition, [Lorenzi and Bertoldi, 2006], the function v(t, x) defined through Markov semigroup in equation (19) as
v(t, x) := P t f (x) ,
is the solution of the Cauchy problem
∂ t v(t, x) − Lv(t, x) = 0 , (t, x) ∈ R + × R n , v(0, x) = f (x) ,(53)
where L is the infinitesimal generator of the Markov process defined as
Lf (x) = n i=1 µ i (x)∂ xi f (x) + 1 2 n i,j=1 Σ ij (x)∂ 2 xi xj f (x) ,(54)
where we set for short
µ := [(J − R)∂ x H(X(t)) + g(X(t))u(t)] ,
and Σ(x) := σ T (x)σ(x). The formal adjoint L * in distributional sense, [Krylov, 1995], of the infinitesimal generator L is given by
L * f (x) = − n i=1 ∂ xi (µ i (x)f (x)) + 1 2 n i,j=1 ∂ 2 xi xj (Σ ij (x)f (x)) .(55)
In the following we will denote by L p ρ := L p (R n , ρ), p ∈ [1, ∞), the space of p−integrable functions with respect to the measure ρ; · p is the standard norm in the space L p ρ . Definition 3.2 can be restated in terms of the Markov semigroup P t as
R n P t f (x)ρ(dx) = R n f (x)ρ(dx) , f ∈ B b (R n ) .(56)
A key aspect is that, if P t admits a unique invariant measure according to equation (56), then the semigroup can be extended to a semigroup of bounded operators in the space L p , [Lorenzi and Bertoldi, 2006, Chapter 9]. If no confusion is possible, typically such extension is still denoted by P t . Among the most relevant aspects of the extended semigroup is that it is possible to study into details the long-time behaviour of the semigroup. In particular, it can be proved that the function P t f converges tō
f ρ := R n f (x)ρ(dx) in L p as t → ∞ .
Therefore, if the semigroup P t is the solution of an infinite-dimensional PHS, as we will show in a while, the problem of weak energy shaping for the SPHS (52) can be associated to the energy shaping of the associated infinite-dimensional PHS.
We will not enter into details regarding neither the properties of the Markov semigroup P t or its extensions to a semigroup on L p ρ since it is a topic extensive treated in literature. We refer the reader to [Lorenzi and Bertoldi, 2006] for further details.
Assume that Proposition 3.6 holds and that σ is non-degenerate. Therefore, the SPHS (52) can be rewritten as
dX(t) = [J d − R d ]∂ x H d (X)dt + σ(X(t))dW (t) , y(t) = g T (X(t))∂ x H d (X(t)) .(57)
In particular, the autonomous SPHS (57) admits a unique invariant measure ρ. Therefore, following [Lorenzi and Bertoldi, 2006], we can show that the Markov semigroup P t associated to the autonomous SPHS (57) extends to a strongly continuous semigroup on the space L p ρ , p ∈ [1, ∞). In the following, we assume that the unique invariant measure admits a density and it is invertible. We will consider the Hilbert space setting so that we set p = 2. We recall that, being L 2 ρ a Hilbert space, it can be endowed with a natural inner product defined as
f, g ρ := f (x)g(x)ρ(x)dx .
In such a case we have the following.
Proposition 4.1. The infinitesimal generator L in equation (54) can be decomposed into a symmetric and anti-symmetric operator, i.e.
L = L as + L s ,(58)
with L as , resp. L s , an anti-symmetric, resp. symmetric, operator on L 2 (R n , ρ(x)dx).
Proof. Notice first that, integrating by parts, the following holds true,
g, Lf ρ = = n i=1 g(x)µ i (x)∂ xi f (x)ρ(x)dx + 1 2 n i,j=1 g(x)Σ ij (x)∂ 2 xi xj f (x) = = − n i=1 ∂ xi (g(x)µ i (x)ρ(x)) f (x)dx + 1 2 n i,j=1 ∂ 2 xi xj (g(x)Σ ij (x)ρ(x)) f (x)dx = = ρ −1 L * (ρg), f ρ .(59)
We can thus define
L s f := 1 2 Lf + ρ −1 L * (ρf ) , L as f := 1 2 Lf − ρ −1 L * (ρf ) .(60)
It is immediate to see that L = L as + L s .
It further holds that, using equation (59),
g, L s f ρ = 1 2 g, Lf ρ + 1 2 g, ρ −1 L * (ρf ) ρ = = 1 2 ρ −1 L * (ρg), f ρ + 1 2 Lg, f ρ = L s g, f ρ , and g, L as f ρ = 1 2 g, Lf ρ − 1 2 g, ρ −1 L * (ρf ) ρ = = 1 2 ρ −1 L * (ρg), f ρ − 1 2 Lg, f ρ = − L as g, f ρ ,
so that the operator L as , resp. L s , is anti-symmetric, resp. symmetric.
Therefore, we can establish the following formulation of the Fokker-Planck equation (53) in term of a Hilbert-space valued deterministic PHS, [Le Gorrec et al., 2004, Jacob andZwart, 2012].
Proposition 4.2. The Cauchy problem in equation (53) defines an infinite-dimensional PHS on the Hilbert space L 2 ρ with linear Hamiltonian. In particular, the probability density is conserved. Proof. Using Proposition 4.1 it can be seen that the Cauchy problem (53) can be written in terms of a symmetric and anti-symmetric operator as
∂ t v(t, x) − (L as + L s ) v(t, x) = 0 , (t, x) ∈ R + × R n , v(0, x) = f (x) .(61)
Equation (61) defines therefore a linear PHS on the space L 2 ρ with linear Hamiltonian Hv = v. Using [Lorenzi and Bertoldi, 2006, Proposition 9.1.9] we have that the semigroup P t is conservative on the space L 2 ρ , that is
R n P t 1ρ(x)dx = R n 1ρ(x)dx = 1 .
From the fact that ρ is an invariant measure we immediately have that
R n Lf (x)ρ(x)dx = 0 .
Using [Lorenzi and Bertoldi, 2006, Theorem 1.3.4], it follows
d dt P t f (x) = LP t f (x) .
which yields, after integration in R n , that
d dt R n P t f (x)ρ(x)dx = R n LP t f (x)ρ(x)dx = 0 .
At last, using the fact that v(t, x) :
= P t f (x) , we obtain d dt R n v(t, x)ρ(x)dx = 0 ,
which states that the probability density function is conserved.
Finally, we can prove a convergence result.
Proposition 4.3. For any f ∈ L p ρ , p ∈ [1, ∞), we have lim t→∞ P t f =f := R n f (x)ρ(x)dx in L 2 ρ .
Proof. It follows from [Lorenzi and Bertoldi, 2006, Theorem 9.1.16].
The above result shows a key connection between the weak stochastic energy shaping proposed in the present paper and the infinite dimensional shaping of deterministic Hilbert space valued PHS. The current work addresses the implication that a weak stochastic shaping has on an associated infinite dimensional PHS. Nonetheless in several application the opposite direction is of relevant interest, where deterministic infinite dimensional technique can be used to properly shape a stochastic PHS. Particular interest in this direction is played by the the H-theorem used in physical systems, [Barbu and Röckner, 2019].
Energy shaping for stochastic port-Hamiltonian systems
The final goal of energy shaping techniques is to change the shape of the Hamiltonian function so that it has a minimum at a desired point x e . Under the above described point of view, energy shaping for SPHS assumes a purely probabilistic interpretation. In fact, the final goal will be to properly shape the invariant measure of the stochastic system so that the process will evolve al large time according to a desired invariant law, peaked around the desired equilibrium configuration x e .
Before entering into details let state some results regarding Casimir function for SPHS.
On Casimir for stochastic port-Hamiltonian systems
In the present section we are to extend the notion of Casimir to stochastic PHS. As usual in the stochastic contest, different notions of conserved quantities can be introduced, usually referred to as strong or weak conserved quantities, see, e.g. [Lázaro-Camí andOrtega, 2008, Cordoni et al., 2019].
Consider the stochastic PHS (7). We will use the following definitions of strong and weak Casimir function.
Definition 5.1. (i) A function C : R n → R is called a strong Casimir for the SPHS (7) if dC(X(t)) = 0 .
(ii) A function C : R n → R is called a weak Casimir for the SPHS (7) if LC(X(t)) = 0 .
It is worth mentioning that the notion of strong Casimir is the straightforward generalization of deterministic Casimir, in the sense that a Casimir is a quantity that is conserved along the trajectory of the system. Nonetheless, as already emerged in [Cordoni et al., 2019], it is often too strong to require that a quantity is conserved P−a.s. so that in practical applications it is usually preferable to consider a weak conservation. Such choice of weak conserved quantities will have strong implication in the following treatment.
Notice that an application of Dynkin formula yields that, if C is a weak Casimir, then it holds
EC(X(t)) = C(x) + E t 0 LC(X(s))ds = C(x) = c .(62)
We thus have the following.
Theorem 5.1. Given the autonomous stochastic PHS (7) with constant null control u ≡ 0.
(i) If ∂ T x C(x) [J(x) − R(x)] = 0 , 1 2 T r σ T (x)∂ 2 x C(x)σ(x) = 0 , ∂ T x C(x)σ(x) = 0 ,(63)
then C is a strong Casimir for the SPHS (7).
(ii) If ∂ T x C(x) [J(x) − R(x)] = 0 , 1 2 T r σ T (x)∂ 2 x C(x)σ(x) = 0 ,(64)
then C is a weak Casimir for the SPHS (7).
Proof. (i) Itô formula applied to a function C : R n → R, yields
dC(X(t)) = ∂ T x C(X(t))dX(t) + 1 2 T r σ T (X(t))∂ 2 C(X(t))σ(X(t)) dt = (65) = ∂ T x C(X(t)) [J(x) − R(x)] ∂ x H(X(t)) + 1 2
T r σ T (X(t))∂ 2 C(X(t))σ(X(t)) dt+ (66)
+ ∂ T x C(X(t))σ(X(t))dW (t) .(67)
Using therefore conditions (63) it follows that dC(X(t)) = 0 and according to Definition 5.1 the claim follows.
(ii) Considering the infinitesimal generator L, taking the expected values and using the martingale property of the Itô integral we obtain
LC(x) = ∂ T x C(x) [J(x) − R(x)] ∂ x H(x) + 1 2 T r σ T (x)∂ 2 x C(x)σ(x) ,
so that, using conditions (64), it follows LC(X(t)) = 0 and according to Definition 5.1 we have the claim.
Control as interconnection
The present section is devoted to energy-based control of SPHS; the main goal is to design a feedback interconnection of SPHS such that the closed-loop system is stable in a suitable sense.
A R nC −valued controller in SPHS form is given by
dZ(t) = [(J c (Z(t)) − R c (Z(t))) ∂ z H c (z) + g c (Z(t))u c ] + σ c (Z(t))dB(t) , y c (t) = g T c (Z(t))∂ z H c (Z(t)) ,(68)
where B is a standard Brownian motion independent of W . Figure 1: Block diagram of the control by interconnection scheme with external ports (v, y) and external noise (B, W ).
u c Σ c = {Z, H c , J c , R c , g c , σ c } B
We interconnect the controller (68) to the original SPHS (7) through the power preserving interconnection
u = −y c + v , u c = y + v c .
Therefore it can be seen that, see, e.g. [Cordoni et al., 2019], the interconnected system in Figure 1 is still a SPHS of the form
d X(t) Z(t) = J −gg T c g c g T J c − R 0 0 R c ∂ x H(X(t)) ∂ z H c (Z(t)) dt+ + g 0 0 g c v v c dt + σ(X(t))dW (t) σ c (Z(t))dB(t) y(t) y c (t) = g T 0 0 g T c ∂ x H(X(t)) ∂ z H c (Z(t)) .(69)
Remark 5.2. We stress that, if not otherwise specified, in the present work we will always consider weak Casimir. The choice is motivated by many reasons. Firstly, as already mentioned above, existence of strong Casimir is often unrealistic and simple examples can be found where no strong Casimir exists. Secondly, when the attention is turned to energy shaping, the choice of strong Casimir poses even greater problems. In [Haddad et al., 2018] for instance energy shaping via strong Casimir is studied and in [Haddad et al., 2018, Proposition 5] sufficient conditions are derived in order to ensure that a given function is a strong Casimir. In particular, the authors must assumes that the controller in SPHS form is perturbed by the same noise as the original SPHS. This is due to the fact that, since a quantity must be conserved along the trajectories, the controller must compensate P−a.s. the noise due to the system. Such condition require further a complete knowledge of the noise in the sense that in a real application the controller must be able to discern the contribution due to the noise from the real state of the system. Such assumptions are difficult to be satisfied in practice. The choice of a weak Casimir on the contrary overcomes such issues. In fact, since a quantity, as shown in equation (62), is required to be conserved in mean value, it is enough to design a controller that matches the SPHS on average. Therefore a filtering of the state of the system can be used to disentangle the contribution of the noise from the state of the system. These considerations will emerge in later results.
△ We thus look for (weak) Casimir functions C(x, z) of the form
C i (x, z) = F i (x) − S i (z i ) , i = 1, . . . , n c ,(70)
for some regular enough functions F i : R n → R and S i : R nc → R. We thus have the following.
△
Proof. The proof follows straightforward by checking that if conditions in (73) are valid then (71) follows.
The infinitesimal generator for the closed-loop dynamics of X in equation (69) now becomes, for any ϕ,
Lϕ(x) = ∂ T x ϕ(x) (J(x) − R(x))∂ x H(x) − g(x)g T c (z)∂ z H c (z) + 1 2 T r σ T (x)∂ 2 x ϕ(x)σ(x) .(74)
We can thus restrict the dynamics on the set
{(x, z) : F i (x) + c i = S i (z) , i = 1, . . . , n C } ,
where c i := C i (x) is the constant value assumed by the Casimir C i according to Dynkin formula (62). Thus, since C is a Casimir and using second and third conditions in equation (73) we have that
Lϕ(x) = ∂ T x ϕ(x)(J(x) − R(x)) (∂ x H(x) + ∂ z H C (z)∂ x F (x)) +(75)+ 1 2 T r σ T (x)∂ 2 xx ϕ(x)σ(x) .(76)
Thus, on
z i = F i (x) + c i , setting H d (x) := H(x) + H c (c + F (x)) ,
equation (75) can be rewritten as
Lϕ(x) = ∂ T x ϕ(x)(J(x) − R(x))∂ x H d (x) + 1 2 T r σ T (x)∂ 2 xx ϕ(x)σ(x) ,(77)
so that it is the infinitesimal generator of the stochastic PHS
dX(t) = (J(x) − R(x))∂ x H d (X(t))dt + σ(X(t))dW (t) .(78)
Proposition 5.6. Consider the closed-loop SPHS (69) and assume that it is possible to find F and S such that conditions (73) hold. Assume further that H d (x) := H(x) + H c (S −1 (c + F (x))). If (i) equilibrium assignment: for a given x e ∈ R n , it holds ∂ x H c (S −1 (c + F (x e ))) = −∂ x H(x e ) ;
(ii) stability: for x e ∈ R n , it holds
∂ 2 x H c (S −1 (c + F (x e ))) > −∂ 2 x H(x e ) ;(80)
(iii) ultimately stochastic passivity: for all x such that x − x e > C,
LH d (x) ≤ −ǫ < 0 ;(81)
(iv) uniqueness invariant measure: either (iv b) non-degeneracy: the noise is non-degenerate, that is the matrix Σ(x) := σ(x)σ T (x) ≻ 0 is positive definite;
(iv b) equivalence: the transition kernel p(t, x, A) is equivalent to the Lebesgue measure for any t > 0 and x ∈ R n .
Then there exists a unique invariant measure ρ under which (69) is ultimately stochastic stable.
Proof. The proof follows from above reasoning and proceeds similarly to the proof of Proposition 3.6.
The stochastic RLC circuit
Consider the following SPHS dX 1 (t) = α∂ x 2 H(X 1 (t), X 2 (t)) + Eu(t) dt + √ 2σ 1 dW 1 (t) , dX 2 (t) = − α∂ x 1 H(X 1 (t), X 2 (t)) + 1 RL ∂ x 2 H(X 1 (t), X 2 (t)) dt + √ 2σ 2 dW 2 (t) ,
with Hamiltonian H(x 1 , x 2 ) = 1 2L
x
2 1 + 1 2C x 2 2 ,(88)
where above, X 1 is the inductance flux, X 2 is the charge in the capacitor, α ∈ [0, 1] represents the duty ratio of the PWM, R L is the output load resistance and E is the DC voltage source. Such system is the example considered in [Ortega et al., 2002] with additive stochastic perturbation. The control objective is to drive the output capacitor voltage to some constant desired value V d > E, maintaining internal stability. The equilibrium point is thus given by (x 1;e , x 2;e ) = LV 2 d R L E , CV d .
Proceeding as in [Ortega et al., 2002, Section 7.A], we can compute the feedback control as
u = φ(x) = −E 2 RLE c 1 x 1 + c 2 c 1 x 2 + c 3 ,
where c 1 , c 2 and c 3 are some constants that need to satisfy
R 2 L E 2 4LV 3 d c 3 < c 1 < 1 CV d c 3 , c 3 < 0 , − 1 CV d c 3 < c 1 < R 2 L E 2 4LV 3 d c 3 , c 3 > 0 , c 2 = − 2LV 2 d R 2 L V 2 d + C c 1 − 1 V d c 3 .
The resulting shaped Hamiltonian is thus given by
H d (x 1 , x 2 ) = 1 2L x 2 1 + 1 2C x 2 2 + 1 2c 1 (c 1 x 2 + c 3 ) 2 ( 2 RLE c 1 x 1 + c 2 ) − LV 4 d 2R 2 L E 2 + V d c 3 2c 1 .
To apply Proposition 3.6 we must at last check that the shaped SPHS with Hamiltonian H d is ultimately stochastic passive. Computing the infinitesimal generator we thus have
LH d (x 1 , x 2 ) = − 1 R L 1 C x 2 + c 1 x 2 + c 3 2 RLE c 1 x 1 + c 2 2 + + 1 L + 4c 1 (c 1 x 2 + c 3 ) R 2 L E 2 2 RLE c 1 x 1 + c 2 3 (σ 1 ) 2 + + 1 C + c 1 2 RLE c 1 x 1 + c 2 (σ 2 ) 2 .(89)
Since we have that LH d (x 1;e , x 2;e ) = (σ 1 ) 2 1
L − 4c 1 V 3 d E 2 R 2 L (c 1 CV d + c 3 ) 2 + (σ 1 ) 2 c 3 c 3 C + c 1 C 2 V d < ∞ ,
using the fact lim (x1,x2)→±∞ LH d (x 1 , x 2 ) = −∞ , we can infer that there exist C > 0 and δ C > 0 such that for x − (x 1;e , x 2;e ) ≥ C it holds LH d (x 1 , x 2 ) ≤ −ǫ < 0 .
We can thus conclude that the shaped SPHS is ultimately stochastic passive and thus Proposition 3.6 holds.
It is worth stressing that, given the non-linear potential appearing in the third term of the shaped Hamiltonian H d , it is not possible to compute explicitly the invariant measure for the shaped SPHS. Nonetheless, Proposition 3.6 still apply and we can conclude the existence and uniqueness of the invariant measure of the shaped SPHS. Similarly to what happens in the deterministic case, the strength of Proposition 3.6 is that there is no need to explicitly compute the invariant measure to infer the convergence of the shaped SPHS. Also Remark 3.7 implies that the invariant measure of the shaped SPHS is peaked around the equilibrium we would obtain in the deterministic case with null noise.
Conclusions
The present paper continues the investigation of stochastic PHS started in [Cordoni et al., 2021a, Cordoni et al., 2020, Cordoni et al., 2019, Cordoni et al., 2022, Cordoni et al., 2021b, addressing the problem of energy shaping of SPHS. Such topic has been one of the main interest in the study of deterministic PHS and consequently it has been object of deep investigation also in the stochastic case. Nonetheless, existing results lack of a suitable generality to include relevant examples. In particular, stochastic systems with additive noise are often ruled out from the possible applications of energy shaping due to a non-vanishing noise. In the present paper we therefore generalizes the energy shaping techniques for SPHS. In particular, we introduce a weak notion of passivity, related to the ergodicity and the connected invariant measure for the SPHS. Such definition naturally leads to a weak notion of convergence in terms of transition semigroup. Compared to existing approaches to energy shaping, the one proposed in the current work has a purely probabilistic flavour, where the main objects are the invariant measure of the system and the related transition probabilities. At last, reformulating the problem of stochastic weak energy shaping in terms of the associated Fokker-Planck equation, energy shaping of deterministic infinite-dimensional port-controlled PHS is recovered, highlighting an insightful connection between the stochastic and deterministic energy shaping techniques.
0
0(i) stable in probability if for any s ≥ 0 and ǫ > 0 it holdslim x→0 P sup s≤t |X s,x (t)| > ǫ = 0 ;(ii) locally asymptotically stable in probability if it is stable in probability and for any s ≥
Theorem 2 . 1 .
21Assume there exists a Lyapunov function V ∈ C 2 (D; R) positive definite in a bounded open set D of R n containing the origin. If, for any x ∈ D \ {0},
for any f in the domain of the infinitesimal generator L of X.Further, if the transition semigroup P t admits an invariant measure ρ, then the semigroup can be extended to a semigroup of bounded operators in the space L p (R n , ρ),[Lorenzi and Bertoldi, 2006, Corollary 8.1.7, Proposition 8.1.8]. Such characterization allows to investigate the long-time behaviour of the semigroup; in particular it holds that lim t→∞ P t f −f p = 0 ,f := R n f ρ(dx) , which corresponds to the long-time behaviour of equality (20).
Definition 3. 3 .
3Consider the SPHS (7) and assume that it admits an invariant measure ρ. If for any Borel set B it holds lim t→∞ p(t, x, B) = ρ(B) .
Σ = {X, H, J, R, g, σ}y c
u
−
+ +
y
W
v c
v
+
equation (2.7)
equation (5.7)
. invariant measures for general stochastic systems are not easy to explicitly derive. In fact, besides gradient systems, in which case the invariant measure has an explicit exponential form, a general theory is of difficult derivation and each case must be studied ad hoc. Nonetheless, ergodicity of general stochastic systems is among the most studied properties of stochastic processes, so that a vast literature on this topic exists. In particular, sharp estimates on the support of the invariant measure can be obtained,[Huang et al., 2018, Huang et al., 2015. Nonetheless, as already highlighted in[Ortega et al., 2002], the strengths of the proposed method is that it not necessary to explicitly compute the shaped Hamiltonian, whereas the main objective is to design a feedbackcontrol construction procedure to stabilize a system. As it will be shown later with the aid of a specific example, a similar argument translates in the proposed stochastic setting in the sense that the explicit computation of the invariant measure of the system is not required;4. the function φ in equation(42)can be obtained with analogous techniques as in the deterministic systems with null volatility σ ≡ 0. In particular, comments made in[Ortega et al., 2002, 3.2] can be used to design the feedback control φ. The only additional condition compared to a classical deterministic PHS is that the resulting SPHS(49)is ultimately stochastic passive. This is among the main reasons for developing the weak energy shaping theory for SPHS.
Proposition 5.3. If, for all i = 1, . . . , n c , it holdsthen the functions C i (x, z) = F i (x) − S i (z i ) , i = 1, . . . , n c , are weak Casimirs for the closed-loop SPHS (69).Proof. An application of Itô formula yields thatTaking the expectation implies that if equation(71)holds then C i (x, z) is a weak Casimir according to Definition 5.1.As regard Proposition 5.3 we have the following sufficient conditions for the condition (71).Proposition 5.4. If there exist functions F i and S i such thatthen equation (71) holds and the functionsis a weak Casimir for the closed-loop SPHS (69).Remark 5.5. Sufficient conditions (73) highlight some key aspects of the proposed approach:1. as in the deterministic case, the second conditionimplies that the direction in which dissipation happens cannot be shaped. This limitation is known in literature as dissipation obstacle,[Secchi et al., 2007, Ortega et al., 2002. Nonetheless, conditions (73) are only sufficient so that future research will be devoted to understand if the setting developed can be used to overcome the dissipation obstacle;2. the last condition states that the controller's noise must compensate, on average, the noise of the system. This implies two immediate comments: (i) differently from[Haddad et al., 2018]it is only necessary that the controller's noise compensates the noise of the system on average instead that P−a.s.: such condition is easy to be satisfied in practice, and (ii) the controller in equation(68)has been designed with a noise term so that it can be chosen to match the contribution due to the system noise.ExamplesThe stochastic inverted pendulumConsider an inverted pendulum with additive noise of the formwith g denoting the gravitational acceleration and u being the control. Definingand using the Hamiltonian functionsystem(82)can be seen to be a SPHS of the form(40). We wish the system to converge towards a statex e = (x 1;e , 0). Clearly, as the noise does not vanish at the statex e , there is no way convergence in probability can be achieved. We therefore consider a weak energy shaping setting, so that we will shape the limiting distribution around the desired state.With the control law u = −x 2 − (x 1 − x 1;e ) − g sin x 1 , the system (82) is transformed into the systemwithand shaped HamiltonianA direct computation shows that LH d (x) = −x 2 2 + 1 , so that, for x 2 2 < 1 + ǫ, ǫ > 0, the SPHS (84) is ultimately passive. Further, since the Hamiltonian is quadratic, it can be proved that X is a bivariate Gaussian random variable with invariant measure given by ρ(x)dx = N e − 1 2 [x 2 2 +(x1−x1;e) 2 ] dx . An interesting consequence of the weak energy shaping theory is that, in order to obtain the desired convergence for this system, we are not forced to choose a control that compensates for the sinusoidal term in the Hamiltonian (83). In fact, choosingwe obtain a system alike to (84) with Hamiltonian given byHamiltonian(86)is of the form of equation (85) plus a potential given by V (x 1 ) = g cos x 1 . Explicit calculations implies that the ergodic invariant measure is ρ(x)dx = N e − 1 2 [x 2 2 +(x1−x1;e) 2 −g(cos x1−1)] dx .Since we aims at shaping a limiting distribution peaked around x e , we require ∂ x1 ρ(x)| x1=x1;e = −N ρ(x) (x 1 −x e + g(cos x 1 − 1))| x1=x1;e = 0 , so that by settingx e = x 1;e − g sin x 1;e , we obtain the desired property.
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| []
|
[
"SOBOLEV SPACES FOR THE WEIGHTED ∂-NEUMANN OPERATOR",
"SOBOLEV SPACES FOR THE WEIGHTED ∂-NEUMANN OPERATOR"
]
| [
"Friedrich Haslinger "
]
| []
| []
| We discuss compactness of the ∂-Neumann operator in the setting of weighted L 2 -spaces on C n . In addition we describe an approach to obtain the compactness estimates for the ∂-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich -Kondrachov lemma. ∂ −→ 0 , 2010 Mathematics Subject Classification. Primary 32W05; Secondary 35N15, 46E35. | null | [
"https://arxiv.org/pdf/1707.05136v1.pdf"
]
| 119,698,161 | 1707.05136 | 0be2ccabee799a2249f9a97b6cd1f8d3f3d86584 |
SOBOLEV SPACES FOR THE WEIGHTED ∂-NEUMANN OPERATOR
17 Jul 2017
Friedrich Haslinger
SOBOLEV SPACES FOR THE WEIGHTED ∂-NEUMANN OPERATOR
17 Jul 2017
We discuss compactness of the ∂-Neumann operator in the setting of weighted L 2 -spaces on C n . In addition we describe an approach to obtain the compactness estimates for the ∂-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich -Kondrachov lemma. ∂ −→ 0 , 2010 Mathematics Subject Classification. Primary 32W05; Secondary 35N15, 46E35.
Introduction
Let Ω be a bounded open set in R n , and k a nonnegative integer. We denote by W k (Ω) the Sobolev space W k (Ω) is a Hilbert space. If Ω ⊂ R n , n ≥ 2, is a bounded domain with a C 1 boundary, the Rellich-Kondrachov lemma says that for n > 2 one has W 1 (Ω) ⊂ L r (Ω) , r ∈ [1, 2n/(n − 2)) and that the imbedding is also compact; for n = 2 one can take r ∈ [1, ∞) (see for instance [1]), in particular, there exists a constant C r such that where L 2 (0,q) (Ω) denotes the space of (0, q)-forms on Ω with coefficients in L 2 (Ω). The ∂-operator on (0, q)-forms is given by
(1.3) ∂ J ′ a J dz J = n j=1 J ′ ∂a J ∂z j dz j ∧ dz J ,
where ′ means that the sum is only taken over strictly increasing multi-indices J. The derivatives are taken in the sense of distributions, and the domain of ∂ consists of those (0, q)-forms for which the right hand side belongs to L 2 (0,q+1) (Ω). So ∂ is a densely defined closed operator, and therefore has an adjoint operator from L 2 (0,q+1) (Ω) into L 2 (0,q) (Ω) denoted by ∂ * .
We consider the ∂-complex
(1.4) L 2 (0,q−1) (Ω) ∂ −→ ←− ∂ * L 2 (0,q) (Ω) ∂ −→ ←− ∂ * L 2 (0,q+1) (Ω), for 1 ≤ q ≤ n − 1. We remark that a (0, q + 1)-form u = ′ J u J dz J belongs to C ∞ (0,q+1) (Ω) ∩ dom(∂ * ) if and only if (1.5) n k=1 u kK ∂r ∂z k = 0
on bΩ for all K with |K| = q, where r is a defining function of Ω with |∇r(z)| = 1 on the boundary bΩ. (see for instance [9]) The complex Laplacian = ∂ ∂ * + ∂ * ∂, defined on the domain
dom( ) = {u ∈ L 2 (0,q) (Ω) : u ∈ dom(∂) ∩ dom(∂ * ), ∂u ∈ dom(∂ * ), ∂ * u ∈ dom(∂)}
acts as an unbounded, densely defined, closed and self-adjoint operator on L 2 (0,q) (Ω), for 1 ≤ q ≤ n, which means that = * and dom( ) = dom( * ). Note that
(1.6) ( u, u) = (∂ ∂ * u + ∂ * ∂u, u) = ∂u 2 + ∂ * u 2 ,
for u ∈ dom( ).
If Ω is a smoothly bounded pseudoconvex domain in C n , the so-called basic estimate says that
(1.7) ∂u 2 + ∂ * u 2 ≥ c u 2 , for each u ∈ dom(∂) ∩ dom(∂ * ), c > 0.
This estimate implies that : dom( ) −→ L 2 (0,q) (Ω) is bijective and has a bounded inverse
N (0,q) : L 2 (0,q) (Ω) −→ dom( ). N (0,q) is called ∂-Neumann operator. In addition (1.8) N (0,q) u ≤ 1 c u .
Hence the ∂-Neumann operator N (0,q) is continuous from L 2 (0,q) (Ω) into itself. Compactness of the ∂-Neumann operator is relevant for a number of circumstances ( [9]). From the point of view of the L 2 -Sobolev theory of the ∂-Neumann operator, an important application of compactness is that it implies global regularity. Kohn and Nirenberg ([8]) proved that compactness of N (0,q) on L 2 (0,q) (Ω) implies compactness (in particular, continuity) of N (0,q) from the Sobolev spaces W s (0,q) (Ω) into itself for all s ≥ 0, see also [9]. For this result the Rellich -Kondrachov lemma is important, it holds as Ω is a bounded domain.
The aim of this paper is to study similar properties for the weighted ∂-Neumann operator on C n . Let ϕ : C n −→ R be a plurisubharmonic C 2 -function and let
L 2 (C n , e −ϕ ) = {g : C n −→ C measurable : g 2 ϕ = (g, g) ϕ = C n |g| 2 e −ϕ dλ < ∞}. Let 1 ≤ q ≤ n and f = |J|=q ′ f J dz J ,
where the sum is taken only over increasing multiindices J = (j 1 , . . . , j q ) and dz J = dz j 1 ∧ · · · ∧ dz jq and f J ∈ L 2 (C n , e −ϕ ). We write f ∈ L 2 (0,q) (C n , e −ϕ ) and define
∂f = |J|=q ′ n j=1 ∂f J ∂z j dz j ∧ dz J for 1 ≤ q ≤ n − 1 and dom(∂) = {f ∈ L 2 (0,q) (C n , e −ϕ ) : ∂f ∈ L 2 (0,q+1) (C n , e −ϕ )}.
In this way ∂ becomes a densely defined closed operator and its adjoint ∂ * ϕ depends on the weight ϕ. We consider the weighted ∂-complex
L 2 (0,q−1) (C n , e −ϕ ) ∂ −→ ←− ∂ * ϕ L 2 (0,q) (C n , e −ϕ ) ∂ −→ ←− ∂ * ϕ L 2 (0,q+1) (C n , e −ϕ )
and we set
(0,q) ϕ = ∂ ∂ * ϕ + ∂ * ϕ ∂, where dom( (0,q) ϕ ) = {u ∈ dom(∂) ∩ dom(∂ * ϕ ) : ∂u ∈ dom(∂ * ϕ ), ∂ * ϕ u ∈ dom(∂)}.
It turns out that (0,q) ϕ is a densely defined, non-negative self-adjoint operator, which has a uniquely determined self-adjoint square root ( [3]. Next we consider the Levi matrix
(0,q) ϕ ) 1/2 . The domain of ( (0,q) ϕ ) 1/2 ) coincides with dom(∂)∩dom(∂ * ϕ ), which is also the domain of the corresponding quadratic form Q ϕ (u, v) := (∂u, ∂v) ϕ + (∂ * ϕ u, ∂ * ϕ v) ϕ , see for instanceM ϕ = ∂ 2 ϕ ∂z j ∂z k n j,k=1
and suppose that the lowest eigenvalue µ ϕ of M ϕ satisfies
(1.9) lim inf |z|→∞ µ ϕ (z) > 0.
(1.9) implies that (0,1) ϕ is injective and that the bottom of the essential spectrum σ e ( (0,1) ϕ ) is positive (Persson's Theorem), see [5]. Now it follows that (0,1) ϕ has a bounded inverse, which we denote by
N (0,1) ϕ : L 2 (0,1) (C n , e −ϕ ) −→ L 2 (0,1) (C n , e −ϕ ).
Using the square root of N (0,1) ϕ we get the basic estimates
(1.10) u 2 ϕ ≤ C( ∂u 2 ϕ + ∂ * ϕ u 2 ϕ ), for all u ∈ dom(∂) ∩ dom(∂ * ϕ )
. Now we will study compactness of the weighted ∂-Neumann operator N (0,1) ϕ . For this purpose we will use the description of compact subsets in L 2 -spaces, as it is done in [4] Chapter 11, to derive a sufficient condition for compactness in terms of the weight function. It turns out that compactness of the ∂-Neumann operator N (0,1) ϕ is equivalent to compactness of the embedding of a certain complex Sobolev space into L 2 (0,1) (C n , e −ϕ ).
Definition 1.1. Let W Qϕ = {u ∈ L 2 (0,1) (C n , e −ϕ ) : u ∈ dom(∂) ∩ dom(∂ * ϕ )} with norm (1.11) u Qϕ = ( ∂u 2 ϕ + ∂ * ϕ u 2 ϕ ) 1/2 . So W Qϕ is the form domain of Q ϕ .
Theorem 1.2. Suppose that the weight function ϕ is plurisubharmonic and that the lowest eigenvalue µ ϕ of the Levi -matrix M ϕ satisfies
(1.12) lim |z|→∞ µ ϕ (z) = +∞ .
Then the embedding
(1.13) j ϕ : W Qϕ ֒→ L 2 (0,1) (C n , e −ϕ )
is compact. Consequently, the ∂-Neumann operator N
(0,1) ϕ is compact.
This result can be seen as a Rellich Kondrachov lemma for Sobolev spaces defined by complex derivatives. Notice that is compact if and only if j ϕ is compact. We have to show that the unit ball in W Qϕ is relatively compact in L 2 (0,1) (C n , e −ϕ ). For this purpose we use the characterization of compact subsets in L 2 -spaces (see [4] Chapter 11). For u ∈ W Qϕ we have
N (0,1) ϕ : L 2 (0,1) (C n , e −ϕ ) −→ L 2 (0,1) (C n , e −ϕ ) can be written in the form N (0,1) ϕ = j ϕ • j * ϕ , where j * ϕ : L 2 (0,1) (C n , e −ϕ ) −→ W∂u 2 ϕ + ∂ * ϕ u 2 ϕ ≥ (M ϕ u, u) ϕ .
This implies (1.14)
∂u 2 ϕ + ∂ * ϕ u 2 ϕ ≥ C n µ ϕ (z) |u(z)| 2 e −ϕ(z) dλ(z) ≥ C n \B R µ ϕ (z)|u(z)| 2 e −ϕ(z) dλ(z),
where B R is the ball with center 0 and radius R > 0. Consequently, assumption (1.12) implies that for each ǫ > 0 there is R > 0 such that
(1.15) C n \B R |u(z)| 2 e −ϕ(z) dλ(z) < ǫ,
for all u in the unit ball of W Qϕ . Also, the map u → u| B R is compact from W Qϕ to L 2 (0,1) (B R , e −ϕ ), in view of the ellipticity of ∂ ⊕ ∂ * ϕ . Together with (1.15), this latter fact shows that the image of a bounded set in W Qϕ is pre-compact in L 2 (0,1) (C n , e −ϕ ).
In the following we will describe an approach to obtain the so-called compactness estimates for the ∂-Neumann operator N (0,1) ϕ , where we follow [9], Propostion 4.2. For this purpose we have to define appropriate weighted Sobolev spaces and we need an appropriate Rellich -Kondrachov lemma.
Weighted L 2 -Sobolev spaces
Let z = (z 1 , . . . , z n ) = (x 1 + iy 1 , . . . , x n + iy n ) ∈ C n and write for a multiindex γ = (γ 1 , γ 2 , . . . , γ 2n−1 , γ 2n ) and an appropriate function
∂ γ f = ∂ |γ| f ∂x γ 1 1 ∂y γ 2 1 . . . ∂x γ 2n−1 n ∂y γ 2n n .
Definition 2.1. We denote by W k (C n ) the Sobolev space
W k (C n ) = {f ∈ L 2 (C n ) : ∂ γ f ∈ L 2 (C n ), |γ| ≤ k},
where the derivatives are taken in the sense of distributions and endow the space with the norm
f k = |γ|≤k C n |∂ γ f | 2 dλ 1/2 .
W k (C n ) is a Hilbert space. It is well-known that the embedding ι : W 1 (C n ) ֒→ L 2 (C n ) fails to be compact. In sake of completeness we recall the easy proof: let ψ ∈ C ∞ 0 (C n ) be a smooth function with compact support such that Trψ ⊂ B 1/2 (0) and C n |ψ(z)| 2 dλ(z) = 1. For k ∈ N let ψ k (z) = ψ(z − − → k ), where − → k = (k, 0, . . . , 0) ∈ C n . Then Trψ k ⊂ B 1 ( − → k ) and (ψ k ) k is a bounded sequence in W 1 (C n ). Now let k, m ∈ N with k = m. Due to the fact that ψ k and ψ m have non-overlapping supports we have
ψ k − ψ m 2 = ψ k 2 + ψ m 2 = 2,
and the sequence (ψ k ) k has no convergent subsequence in L 2 (C n ). Let U ϕ : L 2 (C n ) −→ L 2 (C n , e −ϕ ) denote the isometry given by U ϕ (f ) = f e ϕ/2 , for f ∈ L 2 (C n ). The inverse is given by U −ϕ (g) = ge −ϕ/2 , for g ∈ L 2 (C n , e −ϕ ). The appropriate weighted Sobolev spaces are determined as the images of W k (C n ) under the isometry U ϕ . In the following we consider only Sobolev spaces of order 1. Let f ∈ W 1 (C n ). Then f e ϕ/2 , (∂ j f )e ϕ/2 ∈ L 2 (C n , e −ϕ ), where ∂ j f denotes all first order derivatives of f with respect to x j and y j for j = 1, . . . , n. Set h = f e ϕ/2 . Then
∂ j h = (∂ j f )e ϕ/2 + 1 2 f (∂ j ϕ)e ϕ/2 = (∂ j f )e ϕ/2 + 1 2 (∂ j ϕ)h, which implies (∂ j f )e ϕ/2 = ∂ j h − 1 2 (∂ j ϕ)h and U ϕ (W 1 (C n )) = {h ∈ L 2 (C n , e −ϕ ) : ∂ j h − 1 2 (∂ j ϕ)h ∈ L 2 (C n , e −ϕ )
, j = 1, . . . , 2n}. For reasons which will become clear later, we denote
W 1 0 (C n , e −ϕ ) := U ϕ (W 1 (C n )), and we endow the space W 1 0 (C n , e −ϕ ) with the norm h → ( h 2 ϕ + j ∂ j h− 1 2 (∂ j ϕ)h 2 ϕ ) 1/2 . in this way U ϕ : W 1 (C n ) −→ W 1 0 (C n , e −ϕ )
is again isometric and we have the following commutative diagram
W 1 (C n ) ι −−− → L 2 (C n ) Uϕ Uϕ W 1 0 (C n , e −ϕ ) −−− → ιϕ L 2 (C n , e −ϕ )
where ι ϕ : W 1 0 (C n , e −ϕ ) ֒→ L 2 (C n , e −ϕ ) is the canonical embeddings. As U ϕ ι = ι ϕ U ϕ and ι fails to be compact, ι ϕ is also not compact.
Definition 2.2. Let η ∈ R. We denote by W 1 η (C n , e −ϕ ) the Sobolev space
W 1 η (C n , e −ϕ ) = {h ∈ L 2 (C n , e −ϕ ) : ∂ j h − 1+η 2 (∂ j ϕ)h ∈ L 2 (C n , e −ϕ ), j = 1, . . . , 2n}, endowed with the norm h → ( h 2 ϕ + j ∂ j h − 1+η 2 (∂ j ϕ)h 2 ϕ ) 1/2 . We use the notation X j = ∂ ∂x j − 1 + η 2 ∂ϕ ∂x j and Y j = ∂ ∂y j − 1 + η 2 ∂ϕ ∂y j ,
for j = 1, . . . , n. Then
W 1 η (C n , e −ϕ ) = {f ∈ L 2 (C n , e −ϕ ) : X j f, Y j f ∈ L 2 (C n , e −ϕ ), j = 1, . . . , n}, with norm f 2 ϕ,η = f 2 ϕ + n j=1 ( X j f 2 ϕ + Y j f 2 ϕ ).
For suitable weight functions ϕ, we can prove an analogous result to the Rellich Kondrachov lemma. Then the canonical embedding ι ϕ,η : W 1 η (C n , e −ϕ ) ֒→ L 2 (C n , e −ϕ ) is compact. Proof. We adapt methods from [2] , [6] and [7] and use the general result that an operator between Hilbert spaces is compact if and only if the image of a weakly convergent sequence is strongly convergent. In addition we remark that C ∞ 0 (C n ) is dense in all spaces which are involved. For the vector fields X j and their adjoints X * j in the weighted space
L 2 (C n , e −ϕ ) we have X * j = − ∂ ∂x j + 1−η 2 ∂ϕ ∂x j and (2.2) (X j + X * j )f = −η ∂ϕ ∂x j f and [X j , X * j ]f = −η ∂ 2 ϕ ∂x 2 j f, for f ∈ C ∞ 0 (C n ), and (2.3) ([X j , X * j ]f, f ) ϕ = X * j f 2 ϕ − X j f 2 ϕ , (2.4) (X j + X * j )f 2 ϕ ≤ (1 + 1/ǫ) X j f 2 ϕ + (1 + ǫ) X * j f 2 ϕ
for each ǫ > 0, where we used the inequality |a + b| 2 ≤ |a| 2 + |b| 2 + 1/ǫ |a| 2 + ǫ |b| 2 .
Similar relations hold for the vector fields Y j . Now we set Ψ(z) = η 2 |∇ϕ(z)| 2 + (1 + ǫ)η△ϕ(z).
By (2.2), (2.3) and (2.4), it follows that
(Ψf, f ) ϕ ≤ (2 + ǫ + 1/ǫ) n j=1 ( X j f 2 ϕ + Y j f 2 ϕ ).
Since C ∞ 0 (C n ) is dense in W 1 η (C n , e −ϕ ) by definition, this inequality holds for all f ∈ W 1 η (C n , e −ϕ ). If (f k ) k is a sequence in W 1 η (C n , e −ϕ ) converging weakly to 0, then (f k ) k is a bounded sequence in W 1 η (C n , e −ϕ ) and our assumption implies that
Ψ(z) = η 2 |∇ϕ(z)| 2 + (1 + ǫ)η△ϕ(z)
is positive in a neighborhood of ∞. So we obtain
C n |f k (z)| 2 e −ϕ(z) dλ(z) ≤ |z|<R |f k (z)| 2 e −ϕ(z) dλ(z) + |z|≥R Ψ(z)|f k (z)| 2 inf{Ψ(z) : |z| ≥ R} e −ϕ(z) dλ(z) ≤ C ϕ,R f k 2 L 2 (B(0,R)) + C ǫ f k 2 ϕ,η inf{Ψ(z) : |z| ≥ R} .
Notice that in the last estimate the expression Ψ(z) plays a similar role as µ ϕ (z) in (1.14).
It is now easily seen that the sequence (f k ) k converges also weakly to zero in W 1 (B(0, R)).
Hence the assumption and the fact that the embedding
W 1 (B(0, R)) ֒→ L 2 (B(0, R))
is compact (classical Rellich Kondrachov Lemma, see for instance [1]) show that (f k ) k tends to 0 in L 2 (C n , e −ϕ ).
Remark 2.4. If η = 0, we get the case corresponding to W 1 (C n ), whereas η = −1 corresponds to the Sobolev space of all functions h ∈ L 2 (C n , e −ϕ ) such that all derivatives of order 1 satisfy ∂ j h ∈ L 2 (C n , e −ϕ ); in this case the higher order Sobolev spaces are defined as the spaces of all functions h ∈ L 2 (C n , e −ϕ ) such that all derivatives of order k ≥ 1 belong to L 2 (C n , e −ϕ ).
From Theorem 2.3 we can also derive compactness for embeddings in Sobolev spaces without weights. For this purpose we define
Definition 2.5. Let η ∈ R. We define W 1 η (C n , ∇ϕ) := {f ∈ L 2 (C n ) : ∂ j f − η 2 (∂ j ϕ)f ∈ L 2 (C n ), j = 1, . . . , 2n}. Then U ϕ : W 1 η (C n , ∇ϕ) −→ W 1 η (C n , e −ϕ )
is an isometry. We consider the canonical embedding ι η : W 1 η (C n , ∇ϕ) ֒→ L 2 (C n ) and we have the following commutative diagram
W 1 η (C n , ∇ϕ) ιη −−− → L 2 (C n ) Uϕ Uϕ W 1 η (C n , e −ϕ ) −−− → ιϕ,η L 2 (C n , e −ϕ )
Hence the condition (2.1) implies that the canonical embedding ι η : W 1 η (C n , ∇ϕ) ֒→ L 2 (C n ) is compact. Now we return to compactness of the ∂-Neumann operator N (0,1) ϕ . We consider the weighted Sobolev space W 1 1 (C n , e −ϕ ) = {h ∈ L 2 (C n , e −ϕ ) : ∂ j h − (∂ j ϕ)h ∈ L 2 (C n , e −ϕ ), j = 1, . . . , 2n}, and use
X j = ∂ ∂x j − ∂ϕ ∂x j and Y j = ∂ ∂y j − ∂ϕ ∂y j ,
for j = 1, . . . , n. Then
W 1 1 (C n , e −ϕ ) = {f ∈ L 2 (C n , e −ϕ ) : X j f, Y j f ∈ L 2 (C n , e −ϕ ), j = 1, . . . , n}, with norm f 2 ϕ,1 = f 2 ϕ + n j=1 ( X j f 2 ϕ + Y j f 2 ϕ ).
We point out that each continuous linear functional L on W 1 1 (C n , e −ϕ ) is represented by
L(f ) = C n f g 0 e −ϕ dλ + n j=1 C n ((X j f )g j + (Y j f )h j )e −ϕ dλ,
for f ∈ W 1 1 (C n , e −ϕ ) and for some g 0 , g j , h j ∈ L 2 (C n , e −ϕ ), j = 1, . . . , n. In particular, each function in L 2 (C n , e −ϕ ) can be identified with an element of the dual space (W 1 1 (C n , e −ϕ )) ′ =: W −1 1 (C n , e −ϕ ). We denote the norm in W −1 1 (C n , e −ϕ ) by . ϕ,−1 . See [4] Chapter 11, for more details. If we suppose that ϕ is a C 2 -function satisfying (2.5) lim |z|→∞ (|∇ϕ(z)| 2 + (1 + ǫ) △ϕ(z)) = +∞, for some ǫ > 0, then the embedding L 2 (0,1) (C n , e −ϕ ) ֒→ W −1 1,(0,1) (C n , e −ϕ ) is compact by Theorem 2.3 and duality. So, as in [9], Proposition 4.2 or [4], Proposition 11.20, we get the compactness estimates Theorem 2.6. Suppose that the weight function ϕ satisfies (1.9) and lim |z|→∞ (|∇ϕ(z)| 2 + (1 + ǫ) △ϕ(z)) = +∞, for some ǫ > 0, then the following statements are equivalent.
(1) The ∂-Neumann operator N (0,1) ϕ is a compact operator from L 2 (0,1) (C n , e −ϕ ) into itself.
(2) The embedding of the space dom (∂) ∩ dom (∂ * ϕ ), provided with the graph norm u → ( u 2 ϕ + ∂u 2 ϕ + ∂ * ϕ u 2 ϕ ) 1/2 , into L 2 (0,1) (C n , e −ϕ ) is compact. (3) For every positive ǫ ′ there exists a constant C ǫ ′ such that We mention that for the weight ϕ(z) = |z| 2 the ∂-Neumann operator fails to be compact (see [4] Chapter 15), but condition (2.5) is satisfied.
u 2 ϕ ≤ ǫ ′ ( ∂u 2 ϕ + ∂ * ϕ u 2 ϕ ) + C ǫ ′ u 2 ϕ,−1 , for all u ∈ dom (∂) ∩ dom (∂ * ϕ ).
In view of Theorem 2.3 it is clear that for any weight satisfying (1.9) and (2.1) for η ∈ R, η = 0, and for some ǫ > 0, the restriction of the ∂-Neumann operator N (0,1) ϕ to W 1 η,(0,1) (C n , e −ϕ ) is compact as an operator from W 1 η,(0,1) (C n , e −ϕ ) to L 2 (0,1) (C n , e −ϕ ).
ACKNOWLEDGMENT: The author wishes to thank the referee for the valuable comments.
W k (Ω) = {f ∈ L 2 (Ω) : ∂ α f ∈ L 2 (Ω), |α| ≤ k},where the derivatives are taken in the sense of distributions and endow the space with α = (α 1 , . . . , α n ) is a multiindex , |α| = n j=1 α j and∂ α f = ∂ |α| f ∂x α 1 1 . . . ∂x αn n .
(1.1) f r ≤ C r f 1 .
1Now let Ω ⊆ C n ( ∼ = R 2n ) be a smoothly bounded pseudoconvex domain. We consider the
( 4 )
4For every positive ǫ ′ there exists R > 0 such thatC n \B R |u(z)| 2 e −ϕ(z) dλ(z) ≤ ǫ ′ ( ∂u 2 ϕ + ∂ * ϕ u 2 ϕ ) for all u ∈ dom (∂) ∩ dom (∂ * (C n , e −ϕ ) ∩ ker(∂) −→ L 2 (C n , e −ϕ ) (C n , e −ϕ ) ∩ ker(∂) −→ L 2 (0,1) (C n , e −ϕ ) are both compact.Remark 2.7. If lim |z|→∞ µ ϕ (z) = +∞, then the condition of the Rellich-Kondrachov lemma (2.5) is satisfied. This follows from the fact that we have for the trace tr(M ϕ ) of the Levifor any invertible (n × n)-matrix T tr(M ϕ ) = tr(T M ϕ T −1 ), it follows that tr(M ϕ ) equals the sum of all eigenvalues of M ϕ .
Qϕ is the adjoint operator to j ϕ , see [4] Section 6.2, or [9] Section 2.8.It is now clear that N(0,1)
ϕ
Sobolev spaces. R A Adams, J J F Fournier, Pure and Applied Math. 140Academic PressR.A. Adams and J.J.F. Fournier, Sobolev spaces, Pure and Applied Math., vol. 140, Academic Press, 2006.
Conditions suffisantes pour l'injection compacte d'espace de Sobolevà poids. P Bolley, M Dauge, B Helffer, Séminaireéquation aux dérivées partielles (France). 1P. Bolley, M. Dauge, and B. Helffer, Conditions suffisantes pour l'injection compacte d'espace de Sobolevà poids , Séminaireéquation aux dérivées partielles (France), Université de Nantes 1 (1989), 1-14.
E B Davies, Spectral theory and differential operators, Cambridge studies in advanced mathematics. CambridgeCambridge University Press42E.B. Davies, Spectral theory and differential operators, Cambridge studies in advanced mathematics, vol. 42, Cambridge University Press, Cambridge, 1995.
The ∂-Neumann problem and Schrödinger operators, de Gruyter Expositions in Mathematics 59. F Haslinger, Walter De GruyterF. Haslinger, The ∂-Neumann problem and Schrödinger operators, de Gruyter Expositions in Mathe- matics 59, Walter De Gruyter, 2014.
Compactness of the solution operator to ∂ in weighted L 2 -spaces. F Haslinger, B Helffer, J. of Functional Analysis. 243F. Haslinger and B. Helffer, Compactness of the solution operator to ∂ in weighted L 2 -spaces, J. of Functional Analysis 243 (2007), 679-697.
On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb's inequality. J Johnsen, Integral Equations Operator Theory. 36J. Johnsen, On the spectral properties of Witten Laplacians, their range projections and Brascamp- Lieb's inequality , Integral Equations Operator Theory 36 (2000), 288-324.
Equation de Schmoluchowski généralisée. J.-M Kneib, F Mignot, Ann. Math. Pura Appl. (IV). 167J.-M. Kneib and F. Mignot, Equation de Schmoluchowski généralisée , Ann. Math. Pura Appl. (IV) 167 (1994), 257-298.
Non-coercive boundary value problems. J Kohn, L Nirenberg, Comm. Pure and Appl. Math. 18J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure and Appl. Math. 18 (1965), 443-492.
The L 2 -Sobolev theory of the ∂-Neumann problem. E Straube, ESI Lectures in Mathematics and Physics. E. Straube, The L 2 -Sobolev theory of the ∂-Neumann problem , ESI Lectures in Mathematics and Physics, EMS, 2010.
Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria E-mail address: [email protected]. F Haslinger, Fakultät für Mathematik, Universität WienatF. Haslinger: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria E-mail address: [email protected]
| []
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[
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition",
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition",
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition",
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition",
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition",
"Inverse Sturm-Liouville problem with analytical functions in the boundary condition"
]
| [
"Natalia P Bondarenko ",
"Natalia P Bondarenko ",
"Natalia P Bondarenko "
]
| []
| []
| The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case. | 10.1002/mma.7375 | [
"https://arxiv.org/pdf/2002.12076v1.pdf"
]
| 211,532,340 | 2002.12076 | 4868a344e149b011fadf22b7e7ecd27b8150c37c |
Inverse Sturm-Liouville problem with analytical functions in the boundary condition
27 Feb 2020
Natalia P Bondarenko
Inverse Sturm-Liouville problem with analytical functions in the boundary condition
27 Feb 2020inverse spectral problemSturm-Liouville operatoranalytical dependence on the spectral parameteruniquenessconstructive solution AMS Mathematics Subject Classification (2010): 34A55 34B07 34B09 34B24 34L40
The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.
Introduction
The paper aims to solve the inverse spectral problem for the following boundary value problem −y ′′ (x) + q(x)y(x) = λy(x), x ∈ (0, π), (1.1) y(0) = 0, f 1 (λ)y ′ (π) + f 2 (λ)y(π) = 0.
(1.2)
Here (1.1) is the Sturm-Liouville equation with the complex-valued potential q ∈ L 2 (0, π). The boundary condition at x = π contains arbitrary functions f j (λ), j = 1, 2, analytical by the spectral parameter λ in the whole complex plane. The Sturm-Liouville equation (1.1) arises in investigation of wave propagation in various media, heating processes, electron motion, etc.
The case of constant coefficients f 1 and f 2 has been studied fairly completely (see the classical monographs [1][2][3][4] and references therein). There is also a number of studies concerning inverse problems for Sturm-Liouville operators with linear [5][6][7][8] and polynomial [10][11][12][13][14] dependence on the spectral parameter in the boundary conditions.
In this paper, we study the Sturm-Liouville problem with arbitrary entire functions in the boundary condition. Let {λ n } ∞ n=1 be a subsequence of the eigenvalues of the problem L(q). This subsequence may coincide with the whole spectrum or not. Note that the behavior of the spectrum depends very much on the functions f j (λ), j = 1, 2. Since no additional restrictions are imposed on these functions, we cannot investigate certain properties of the spectrum. Nevertheless, we can study the following inverse problem under some additional restrictions on the subspectrum {λ n } ∞ n=1 .
Inverse Problem 1.1. Let the entire functions f j (λ), j = 1, 2, be known a priori. Given the eigenvalues {λ n } ∞ n=1 and the number ω := 1 2 π 0 q(x) dx, find the potential q. Investigation of this problem is motivated by several applications. In recent years, the socalled partial inverse problems have attracted much attention of scholars. In such problems, it is assumed that coefficients of differential expressions (e.g., the Sturm-Liouville potential q(x)) are known a priori on a part of an interval. Therefore less spectral data are required to recover the unknown part of coefficients. A significant part of those partial inverse problems can be reduced to Inverse Problem 1.1 for the operator with analytical dependence on the spectral parameter in the boundary conditions. We provide an example of such reduction for the Hochstadt-Lieberman problem [15] in Section 5. Recently partial inverse problems have been intensively studied for Sturm-Liouville operators with discontinuities (see [16][17][18][19][20]). The latter operators arise in geophysics and electronics. Partial inverse problems have also been investigated for differential operators on geometrical graphs (see [21][22][23][24][25]). Such operators model wave propagation through a domain being a thin neighborhood of a graph and have applications in various branches of science and engineering (see [26]). Another popular problem is the inverse transmission eigenvalue problem arising in acoustics (see [27][28][29][30]). The results of the present paper generalize many known results on the mentioned inverse problems. Note that, in certain applications, the constant ω can be obtained from the eigenvalue asymptotics (e.g., see Section 4).
In this paper, we obtain necessary and sufficient conditions for uniqueness of Inverse Problem 1.1 solution and develop a constructive algorithm for solving this inverse problem. This algorithm will be used in our future study [31] for investigation of solvability and stability for Inverse Problem 1.1. Further this theory can be generalized to other types of differential operators and pencils.
Our method is based on completeness and basisness of special vector-functional sequences in appropriate Hilbert spaces. This method allows us to reduce Inverse Problem 1.1 to the classical Sturm-Liouville inverse problem with constant coefficients in the boundary conditions. In contrast to the majority of the studies on inverse Sturm-Liouville problems, our analysis does not require self-adjointness of the operator. We investigate the most general case, when the potential q(x) is complex-valued and the given eigenvalues can be multiple. For solving the inverse Sturm-Liouville problem with boundary conditions independent of the spectral parameter, we rely on the inverse problem theory for non-self-adjoint Sturm-Liouville operators developed in [4,32,33].
The paper is organized as follows. In Section 2, we introduce the notations, and formulate the main results, in particular, necessary and sufficient conditions for uniqueness of solution (Theorems 2.2 and 2.3) and Algorithm 2.4 for constructive solution of the inverse problem. We also provide some simple conditions on the subspectrum {λ n } ∞ n=1 sufficient for uniqueness and for constructive solution (see Theorem 2.5). The main theorems are proved in Section 3. In Section 4, we apply our results to the Hochstadt-Lieberman problem. In Appendix, Theorem 5.1 on local solvability and stability is proved for the inverse Sturm-Liouville problem by Cauchy data. This result plays an auxiliary role in analysis of Inverse Problem 1.1. However, as far as we know, Theorem 5.1 is new for the case of the complex-valued potential q(x) and so can be treated as a separate result.
Main results
Let us start with some preliminaries. Denote by S(x, λ) the solution of equation (1.1), satisfying the initial conditions S(0, λ) = 0, S ′ (0, λ) = 1. Here and below the prime stands for the derivative by x. For derivatives by λ, we use the following notation:
f <j> (λ) = 1 j! d j dλ j f (λ), j ≥ 0.
The spectrum of L(q) consists of eigenvalues, which coincide with the zeros the the characteristic function ∆(λ) := f 1 (λ)S ′ (π, λ) + f 2 (λ)S(π, λ).
(2.1)
Clearly, the function ∆(λ) is entire in λ-plane. Consider a subsequence {λ n } ∞ n=1 of the spectrum. Any multiple eigenvalue can appear in the sequence {λ n } ∞ n=1 a number of times not exceeding its multiplicity. By the eigenvalue multiplicity we mean the multiplicity of the corresponding zero of the analytic function ∆(λ). In other words, if for some µ we have #{n ∈ N : λ n = µ} = k, then ∆ <j> (µ) = 0, j = 0, k − 1. We call such a sequence {λ n } ∞ n=1 a subspectrum of L(q). Let us add to the the given subspectrum the value λ 0 := 0. Define I := {n ≥ 0 : λ n = λ k , ∀k : 0 ≤ k < n}, m n := #{k ≥ 0 : λ k = λ n }, (2.2) i.e. I is the index set of all the distinct values among {λ n } ∞ n=0 and m n is the multiplicity of λ n for n ∈ I. Without loss of generality, we assume that the equal eigenvalues are consecutive: λ n = λ n+1 = · · · = λ n+mn−1 for all n ∈ I.
Define the functions
s(x, λ) = √ λ sin( √ λx), c(x, λ) = cos( √ λx).
Obviously, the functions λ −1 s(x, λ) and c(x, λ) are entire by λ for each fixed x ∈ [0, π]. Define η 1 (λ) := S(π, λ), η 2 (λ) := S ′ (π, λ). Further we need the following standard relations, which can be obtained by using the transformation operator (see [1,4,39]):
η 1 (λ) = s(π, λ) λ − ωc(π, λ) λ + 1 λ π 0 K(t)c(t, λ) dt, (2.3) η 2 (λ) = c(π, λ) + ωs(π, λ) λ + 1 λ π 0 N(t)s(t, λ) dt, (2.4)
where K, N ∈ L 2 (0, π). The pair of functions {K, N} is called the Cauchy data of the potential q. Consider the following auxiliary inverse problem.
Inverse Problem 2.1. Given the Cauchy data {K, N} and the number ω, find the potential q.
Using the Cauchy data {K, N} and ω, one can easily construct the Weyl function M(λ) := η 2 (λ) η 1 (λ) . It is well-known that the potential q can be uniquely recovered from the Weyl function, e.g., by the method of spectral mappings (see [4,32,33]
λ∆(λ) = f 1 (λ) λc(π, λ) + ωs(π, λ) + π 0 N(t)s(t, λ) dt + f 2 (λ) s(π, λ) − ωc(π, λ) + π 0 K(t)c(t, λ) dt . (2.5)
Introduce the complex Hilbert space of vector-functions:
H := L 2 (0, π) ⊕ L 2 (0, π) = {h = [h 1 , h 2 ] : h j ∈ L 2 (0, π), j = 1, 2}
with the following scalar product and the norm:
(g, h) H := π 0 (g 1 (t)h 1 (t) + g 2 (t)h 2 (t)) dt, h H = (h, h) H , g, h ∈ H, g = [g 1 , g 2 ], h = [h 1 , h 2 ].
Define the vector-functions
u(t) := [N(t), K(t)], v(t, λ) := [f 1 (λ)s(t, λ), f 2 (λ)c(t, λ)]. (2.6)
Clearly, u(.) and v <ν> (., λ) for each fixed λ and ν ≥ 0 belong to H. In view of our notations, the relation (2.5) can be rewritten in the form (u(t), v(t, λ)) H = λ∆(λ) + w(λ), w(λ) := −f 1 (λ)(λc(π, λ) + ωs(π, λ)) − f 2 (λ)(s(π, λ) − ωc(π, λ)).
(2.7)
Here t is the variable of integration in the scalar product. Since (λ∆(λ)) <ν> |λ=λn = 0, n ∈ I, ν = 0, m n − 1,
(2.8) we get (u(t), v <ν> (t, λ n )) H = w <ν> (λ n ), n ∈ I, ν = 0, m n − 1. (2.9) Denote v n+ν (t) := v <ν> (t, λ n ), w n+ν := w <ν> (λ n ), n ∈ I, ν = 0, m n − 1, n + ν ≥ 1, (2.10) v 0 (t) := [0, 1], w 0 := ω. (2.11) Finally, we get (u, v n ) H = w n , n ≥ 0. (2.12)
The relation (2.12) for n ≥ 1 follows from (2.9). For n = 0, (2.12) follows from (2.3), since S(π, λ) is analytical at λ = 0. In view of (2.6), (2.7), (2.10) and (2.11), the vector-functions {v n } ∞ n=0 and the numbers {w n } ∞ n=0 can be constructed by the given data of Inverse Problem 1.1. The components of u can help to find the unknown potential q.
Introduce the following conditions.
(Complete) The sequence {v n } ∞ n=0 is complete in H. (Basis) The sequence {v n } ∞
n=0 is an unconditional basis in H. Indeed, (Basis) implies (Complete). Along with the problem L(q), we consider the problem L(q) of the form (1.1)-(1.2) with another potentialq ∈ L 2 (0, π). The functions f j (λ), j = 1, 2, are the same for these two problems. We agree that, if a certain symbol γ denotes an object related to L(q), the symbol γ with tilde denotes the analogous object related to L(q). Now we are ready to formulate the uniqueness theorem for Inverse Problem 1.1.
Theorem 2.2. Let {λ n } ∞ n=1 and {λ n } ∞ n=1
be subspectra of the problems L(q) and L(q), respectively. Suppose that L(q) and {λ n } ∞ n=0 satisfy the condition (Complete), and let λ n =λ n , n ≥ 1, ω =ω. Then q =q in L 2 (0, π).
The following theorem asserts that the condition (Complete) is not only sufficient but also necessary for uniqueness of solution of Inverse Problem 1.1.
Theorem 2.3. Let {λ n } ∞
n=1 be a subspectrum of the problem L(q). Suppose that the sequence {v n } ∞ n=0 is incomplete in H. Then there exists a complex-valued functionq ∈ L 2 (0, π),q = q such that ω =ω and {λ n } ∞ n=1 is a subspectrum of L(q). Suppose that the condition (Basis) holds. Then one can constructively solve Inverse Problem 1.1, by using the following algorithm. Algorithm 2.4. Let eigenvalues {λ n } ∞ n=1 and the number ω be given. We need to find the potential q.
1. Using f j (λ), j = 1, 2, {λ n } ∞ n=0 and ω, construct the vector-functions {v n } ∞ n=0 and the numbers {w n } ∞ n=0 via (2.6), (2.7), (2.10) and (2.11).
For the basis {v
n } ∞ n=0 , find the biorthonormal basis {v * n } ∞ n=0 , i.e. (v n , v * k ) H = δ nk , n, k ≥ 0, where δ nk is the Kronecker delta.
3. Construct the element u ∈ H, satisfying (2.12), by the formula
u = ∞ n=0 w n v * n .
4. Using the components of u(t) = [N(t), K(t)], solve Inverse Problem 2.1 and find q.
In certain applications, it can be difficult to check the conditions (Complete) and (Basis). Therefore we introduce some other conditions, sufficient for uniqueness and for constructive solution of Inverse Problem 1.1.
(Complete2) The sequence {c <ν> (t, λ n )} n∈I, ν=0,mn−1 is complete in L 2 (0, 2π). (Basis2) The sequence {c <ν> (t, λ n )} n∈I, ν=0,mn−1 is a Riesz basis in L 2 (0, 2π). (Separation) For every n ≥ 0, we have f 1 (λ n ) = 0 or f 2 (λ n ) = 0. (Simple)
There exists an integer n 0 such that m n = 1 and λ n = 0 for n ≥ n 0 .
(Asymptotics) Im ρ n = O(1), n → ∞, and {ρ −1 n } n≥n 0 ∈ l 2 , where ρ n := √ λ n , arg ρ n ∈ − π 2 , π 2 .
These conditions are natural for applications, such as the Hochstadt-Lieberman problem (see Section 5), transmission inverse eigenvalue problem, inverse problems for quantum graphs, etc. The condition (Separation) is essential for investigation of Inverse Problem 1.1. If this condition is violated, i.e. f 1 (λ n ) = f 2 (λ n ) = 0 for some n, in view of (2.1), the eigenvalue λ n carries no information on the potential q. It is easy to check, that (Separation) follows from (Complete), so (Separation) is implicitly required in the uniqueness Theorem 2.2 and in Algorithm 2.4.
Proofs
The aim of this section is to prove Theorems 2. and (2.11), we have v n =ṽ n in H and w n =w n for all n ≥ 0. Hence the relation (2.12) forL has the form (ũ, v n ) H = w n , n ≥ 0.
(3.1)
Subtracting (3.1) from (2.12) and using the completeness of the sequence {v n } ∞ n=0 , we get u =ũ in H, i.e. K =K, N =Ñ in L 2 (0, π). Using the uniqueness of Inverse Problem 2.1 solution, we conclude that q =q in L 2 (0, π).
Proof of Theorem 2.3. Let the problem L(q) and the subspectrum {λ n } ∞ n=1 be such that the sequence {v n } ∞ n=0 is incomplete in H. Then there existsû ∈ H,û = 0, such that
(û, v n ) H = 0, n ≥ 0. (3.2)
Since the relations (3.2) are linear byû, one can chooseû satisfying the estimate û H ≤ ε for ε from Theorem 5.1.
Set u := [N(t), K(t)],ũ := u +û = [Ñ (t),K(t)]
,ũ = u. By Theorem 5.1, there existsq ∈ L 2 (0, π) such that ω =ω and {K,Ñ} are the Cauchy data ofq. Define the functionsη
1 (λ) := s(π, λ) λ − ωc(π, λ) λ + 1 λ π 0K (t)c(t, λ) dt, η 2 (λ) := c(π, λ) + ωs(π, λ) λ + 1 λ π 0Ñ (t)s(t, λ) dt, ∆(λ) := f 1 (λ)η 2 (λ) + f 2 (λ)η 1 (λ).
Clearly,∆(λ) is the characteristic function of L(q). The relations (2.12) and (3.2) yield (3.1). Consequently, the function λ∆(λ) has zeros {λ n } n∈I of the corresponding multiplicities {m n } n∈I . Thus, {λ n } ∞ n=1 is a subspectrum of L(q),q = q. In order to prove Theorem 2.5, we need several auxiliary lemmas.
Lemma 3.1. Suppose that (Separation) is fulfilled. Then there exist coefficients {C n,k } such that the following relations hold:
η <ν> j (λ n ) = (−1) j−1 ν k=0 C n,k f <ν−k> j , j = 1, 2, (3.3)
for n ∈ I\{0}, ν = 0, m n − 1 and for n = 0, ν = 0, m 0 − 2.
Proof. Fix n ∈ I\{0}. The relation (2.1) can be rewritten in the form
η 1 (λ)f 2 (λ) + η 2 (λ)f 1 (λ) = ∆(λ). (3.4)
The condition (Separation) and the relation ∆(λ n ) = 0 imply that
[η 1 (λ n ), η 2 (λ n )] = C n,0 [f 1 (λ n ), −f 2 (λ n )],
where C n,0 is a nonzero constant, i.e. the relation Here and below the arguments (λ n ) are omitted for brevity. Using (3.3) for η <k> j , k = 0, ν − 1, we obtain
η <ν> 1 f 2 + ν−1 k=0 k j=0 C n,j f <k−j> 1 f <ν−k> 2 = −η <ν> 2 f 1 + ν−1 k=0 k j=0 C n,j f <k−j> 2 f <ν−k> 1 .
Calculations show that
η <ν> 1 f 2 + η <ν> 2 f 1 = ν−1 j=0 C n,j ν−1 k=j (f <k−j> 2 f <ν−k> 1 − f <k−j> 1 f <ν−k> 2 ) = ν−1 j=0 C n,j ν−j−1 s=0 f <s> 2 f <ν−j−s> 1 − ν−j s=1 f <s> 2 f <ν−j−s> 1 = ν−1 j=0 C n,j (f 2 f <ν−j> 1 − f <ν−j> 2 f 1 ). Hence f 2 η <ν> 1 − ν−1 j=0 C n,j f <ν−j> 1 = −f 1 η <ν> 2 + ν−1 j=0 C n,j f <ν−j> 2 .
In view of (Separation), f 1 = 0 or f 2 = 0. Consequently, there exists the constant C n,ν such that
η <ν> i + (−1) i ν−1 j=0 C n,j f <ν−j> i = (−1) i C n,ν f i , i = 1, 2.
Thus, the relation (3.3) is proved for n ∈ I\{0}, ν = 0, m n − 1. Obviously, the arguments above are also valid for n = 0, ν = 0, m 0 − 2.
Introduce the vector-functions g(t, λ) := [η 1 (λ)s(t, λ), −η 2 (λ)c(t, λ)], g 0 (t) = [0, 1], (3.5) g n+ν (t) := g <ν> (t, λ n ), n ∈ I, ν = 0, m n − 1, n + ν ≥ 1.
Lemma 3.2. The sequence {v n } ∞ n=0 is complete in H if and only if so does {u n } ∞ n=0 . Proof. Let an element h = [h 1 , h 2 ] ∈ H be such that (h, v n ) H = 0, n ≥ 0. (3.6)
The definitions (2.6), (2.10) and (2.11) imply that V <ν> (λ n ) = 0, n ∈ I, ν = 0, m n − 1,
where
V (λ) := π 0 (h 1 (t)f 1 (λ)s(t, λ) + h 2 (t)f 2 (λ)c(t, λ)) dt.
Obviously,
V <ν> (λ n ) = ν k=0 π 0 (h 1 (t)f <k> 1 (λ n )s <ν−k> (t, λ n ) + h 2 (t)f <k> 2 (λ n )c <ν−k> (t, λ n )) dt. (3.8)
Consider the function
G(λ) := π 0 (h 1 (t)η 1 (λ)s(t, λ) − h 2 (t)η 2 (λ)c(t, λ)) dt. (3.9)
Let us show that G <ν> (λ n ) = 0, n ∈ I, ν = 0, m n − 1.
(3.10)
Using Lemma 3.1, (3.7) and (3.8), we derive By using analogous ideas and Lemma 3.1, we derive
G <ν> (λ n ) = ν k=0 π 0 (h 1 (t)η <k> 1 (λ n )s <ν−k> (t, λ n ) − h 2 (t)η <k> 2 (λ n )c <ν−k> (t, λ n )) dt = ν k=0 k j=0 C n,j π 0 (h 1 (t)f <k−j> 1 (λ n )s <ν−k> (t, λ n ) + h 2 (t)f <k−j> 2 (λ n )c <ν−k> (t, λ n )) dt = ν l=0 C n,ν−l l j=0 π 0 (h 1 (t)f <j> 1 (λ n )s <l−j> (t, λ n ) + h 2 (t)f <j> 2 (λ n )c <l−j> (t, λ n )) dt = ν l=0 C n,ν−l V <l> (λ n ) =G <ν> (0) = ν−1 k=0 (h 1 (t)η <k> 1 (0)s <ν−k> (t, 0) − h 2 (t)η <k> 2 (0)c <ν−k> (t, 0)) dt = ν−1 k=0 k j=0 C n,j π 0 (h 1 (t)f <k−j> 1 (0)s <ν−k> (t, 0) + h 2 (t)f <k−j> 2 (0)c <ν−k> (t, 0)) dt = ν−1 j=0 C n,j ν−j−1 l=0 π 0 (h 1 (t)f <l> 1 (0)s <ν−l−j> (t, 0) + h 2 (t)f <l> 2 (0)c <ν−l−j> (t, 0)) dt = ν−1 j=0 C n,j V <ν−j> (0) = 0, ν = 0, m 0 − 1.
In particular, (3.10) holds for n = 0, ν = m 0 − 1.
The relations (3.9) and (3.10) yield (h, g n ) H = 0, n ≥ 0. (3.12)
Thus, we have shown that (3.12) follows from (3.6). It can be proved similarly that (3.6) follows from (3.12). Therefore the completeness of the sequence {v n } ∞ n=0 is equivalent to the completeness of the sequence {g n } ∞ n=0 . Further we need two auxiliary propositions. Proposition 3.3 is proved in Appendix of [34]. Then the sequence {c <ν> (t, θ n )} n∈J, ν=0,µn−1 is a Riesz basis in L 2 (0, a).
Proposition 3.4. Let G(λ) be an entire function, satisfying the conditions:
|G(ρ 2 )| ≤ C exp(|Imρ|a), ∀λ ∈ C, R |G(ρ 2 )| 2 dρ < ∞.
for some positive constants C and a. Let {λ n } ∞ n=0 be arbitrary complex numbers, and let the set I and the multiplicities {m n } ∞ n=0 be defined by (2.2). Suppose that G <ν> (λ n ) = 0, n ∈ I, ν = 0, m n − 1, and the sequence {c <ν> (t, λ n )} n∈I, ν=0,mn−1 is complete in L 2 (0, a). Then G(λ) ≡ 0.
Proof. By Paley-Wiener Theorem, the function G can be represented in the form G(ρ 2 ) = a 0 r(t) cos(ρt) dt, r ∈ L 2 (0, 2π). Since the sequence {c <ν> (t, λ n )} n∈I, ν=0,mn−1 is complete in L 2 (0, a), we have r = 0 in L 2 (0, a), so G(λ) ≡ 0.
Proof of Theorem 2.5(i). Suppose that the conditions (Separation) and (Complete2) are fulfilled. Let us show that these two conditions imply the completeness of the sequence {g n } ∞ n=0 . Let h ∈ H be such that (3.12) is valid. We have to show that h = 0. Consider the function G(λ) defined by (3.9). The relation (3.12) implies (3.10). In view of (2.3), (2.4), (3.9), (3.10) and (Complete2), the conditions of Proposition 3.4 are fulfilled for a = 2π. Therefore G(λ) ≡ 0, i.e. π 0 (h 1 (t)η 1 (λ)s(t, λ) − h 2 (t)η 2 (λ)c(t, λ)) dt ≡ 0.
(3.16) By virtue of Proposition 5.2, η 1 (λ) has a countable set of zeros {θ n } ∞ n=1 , counted with their multiplicities and satisfying the asymptotic formula (3.13) with a = π. Add the value θ 0 = 0. Define the set J and the multiplicities {µ n } n∈J by (3.14). It follows from (3.16) that π 0 h 2 (t)η 2 (λ)c(t, λ) dt <ν> |λ=θn = 0, n ∈ J, ν = 0, µ n − 1.
(3.17)
Note that η 2 (θ n ) = 0, n ≥ 1. (Otherwise we have S(π, θ n ) = S ′ (π, θ n ) = 0. Together with equation (1.1), this yields the relation S(x, λ) ≡ 0, which is wrong). Consequently, using (3.17) and the equality π 0 h 2 (t) dt = 0, we obtain π 0 h 2 (t)c <ν> (t, θ n ) = 0, n ∈ J, ν = 0, µ n − 1.
According to Proposition 3.3, the sequence {c <ν> (t, θ n )} n∈J, ν=0,µn−1 is complete in L 2 (0, π).
Hence h 2 = 0 in L 2 (0, π). Returning to (3.16), we easily conclude that also h 1 = 0 in L 2 (0, π). Thus, we have shown that (3.12) implies h = 0 in H, so the sequence {g n } ∞ n=0 is complete in H. By Lemma 3.2, the sequence {v n } ∞ n=0 is also complete in H under the assumptions of the theorem.
Lemma 3.5. Let {τ n } n≥0 be arbitrary complex numbers such that τ n = τ k and τ n = τ k for all n = k, n, k ≥ 0. Suppose that the sequence {cos(τ n t)} ∞ n=0 is a Riesz basis in L 2 (0, 2π). Then the sequence {g 0 n } ∞ n=0 is a Riesz basis in H, where g 0 n (t) := [sin(τ n π) sin(τ n t), − cos(τ n π) cos(τ n t)].
G 0 (λ) := π 0 (h 1 (t) sin( √ λπ) sin( √ λt) − h 2 (t) cos( √ λπ) cos( √ λt)) dt
has zeros {τ 2 n } n≥0 . Applying Proposition 3.4, we conclude that G 0 (λ) ≡ 0. Then one can easily show that h 1 = h 2 = 0 in L 2 (0, π), so {g 0 n } ∞ n=0 is complete. Second, we prove the two-side inequality (3.19). Calculations show that
(g 0 n , g 0 k ) H = π 0
(sin(τ n π) sin(τ n t) sin(τ k π) sin(τ k t) + cos(τ n π) cos(τ n t) cos(τ k π) cos(τ k t)) dt = sin(2(τ n − τ k )π) 2(τ n − τ k ) = 2π 0 cos(τ n t) cos(τ k t) dt.
Hence N 0 n=0 b n g 0 n H = N 0 n=0 N 0 k=0 b n b k (g 0 n , g 0 k ) H = N 0 n=0 b n cos(τ n t) L 2 (0,2π)
.
Since the sequence {cos(τ n t)} ∞ n=0 is a Riesz basis in L 2 (0, 2π), the two-side inequality similar to (3.19) is valid for this sequence. Consequently, the inequality (3.19) is also valid for {g 0 n } ∞ n=0 , so {g 0 n } ∞ n=0 is a Riesz basis in H. Proof of Theorem 2.5(ii). Suppose that the conditions (Separation), (Simple), (Asymptotics) and (Basis2) are fulfilled. First, let us show that {g n } ∞ n=0 is a Riesz basis in H. Since (Basis2) implies (Complete2), the conditions of Theorem 2.5(i) hold, so the sequence {g n } ∞ n=0 is complete in H according to the previous proof. Substituting (2.3) and (2.4) into (3.5), we get g(t, ρ 2 ) = [sin(ρπ) sin(ρt), − cos(ρπ) cos(ρt)] + O ρ −1 exp(2|Im ρ|π) , |ρ| → ∞.
Substituting ρ = ρ n into the latter relation and taking the conditions (Simple) and (Asymptotics) into account, we conclude that { g n − g 0 n H } n≥0 ∈ l 2 , where g 0 n is defined by (3.18) for n ≥ 0. Here τ n = ρ n for n ≥ n 0 and {τ n } n 0 −1 n=0 are arbitrary complex numbers, such that τ n = τ k and τ n = τ k for all n = k, n, k ≥ 0. Thus, the sequence {τ n } n≥0 satisfies the conditions of Lemma 3.5. The Riesz-basis property of the sequence {cos(τ n t)} ∞ n=0 follows from (Basis2). Applying Lemma 3.5, we conclude that {g 0 n } ∞ n=0 is a Riesz basis in H. Thus, the sequence {g n } ∞ n=0 is complete and l 2 -close to the Riesz basis {g 0 n } ∞ n=0 in H. Hence {g n } ∞ n=0 is also a Riesz basis.
Second, let us show that the Riesz-basis property of {g n } ∞ n=0 implies that {v n } ∞ n=0 is an unconditional basis in H, i.e. the normalized sequence {v n / v n H } ∞ n=0 is a Riesz basis. By Lemma 3.2, the sequence {v n } ∞ n=0 is complete in H. Recall that, by (Simple), the eigenvalues {λ n } are simple for sufficiently large n. Therefore, by Lemma 3.1, we have g n = k n v n , n ≥ n 0 , where {k n } n≥n 0 are nonzero constants. This fact together with the completeness of {v n } ∞ n=0 yield the claim.
Hochstadt-Lieberman problem
In this section, we show one of the applications of our main results. Consider the following eigenvalue problem: −y ′′ (x) + q(x)y(x) = λy(x), x ∈ (0, 2π), (4.1) y(0) = y(2π) = 0, (4.2)
with a complex-valued potential q ∈ L 2 (0, 2π). Denote by {λ n } ∞ n=1 the eigenvalues of the problem (4.1)-(4.2), counted with their multiplicities and numbered according to their asymptotics
λ n = n 2 + Ω πn + o n −1 , n → ∞,(4.3)
where Ω := 1 2 2π 0 q(x) dx. The Hochstadt-Lieberman problem, also called the half-inverse problem, is formulated as follows.
Inverse Problem 4.1. Suppose that the potential q(x) is known a priori for x ∈ (π, 2π). Given the spectrum {λ n } ∞ n=1 , find the potential q(x) for x ∈ (0, π).
Inverse Problem 4.1 and its generalizations were studied in [15,[36][37][38][39][40] and other papers. In this section, we show that this problem can be treated as a special case of Inverse Problem 1.1.
Denote by S(x, λ) and ψ(x, λ) the solutions of equation (4.1), satisfying the initial conditions S(0, λ) = 0, S ′ (0, λ) = 1, ψ(2π, λ) = 0, ψ ′ (2π, λ) = −1. The eigenvalues of the problem (4.1)-(4.2) coincide with the zeros of the characteristic function ∆(λ) = ψ(π, λ)S ′ (π, λ) − ψ ′ (π, λ)S(π, λ).
(4.4)
Comparing (4.4) with (2.1), we conclude that the eigenvalue problem (4.1)-(4.2) is equivalent to (1.1)-(1.
2) with f 1 (λ) := ψ(π, λ), f 2 (λ) := −ψ ′ (π, λ). (4.5) Note that these functions f j (λ), j = 1, 2, are entire in λ-plane, and they can be constructed by the known part of the potential q(x), x ∈ (π, 2π). The constant ω also can be easily determined by the given data of Inverse Problem 4.1. Indeed, we have Proof. The condition (Basis2) follows from the asymptotics (4.3) and Proposition 3.3. (Separation) is fulfilled, because the functions ψ(π, λ) and ψ ′ (π, λ) do not have common zeros. Indeed, if ψ(π, µ) = ψ ′ (π, µ) = 0 for some µ ∈ C, then ψ(x, µ) is the solution of the initial value problem for equation (4.1) with the zero conditions at x = π. Then ψ(x, µ) ≡ 0, which is impossible. The conditions (Simple) and (Asymptotics) easily follow from the asymptotics (4.3).
ω = 1 2 π 0 q(x) dx = Ω − 1 2 2π π q(x) dx,
Thus, our main results can be applied to the Hochstadt-Lieberman problem. In particular, Theorem 2.2 implies the following corollary, which generalizes the Hochstadt-Lieberman uniqueness theorem [15] to the case of complex-valued potentials. Theorem 4.3. Let {λ n } ∞ n=1 and {λ n } ∞ n=1 be the spectra of the boundary value problems in the form (4.1)-(4.2) for potentials q andq, respectively. Suppose that q(x) =q(x) a.e. on (π, 2π) and λ n =λ n for all n ≥ 1. Then q(x) =q(x) a.e. on (0, π). In other words, the solution of Inverse Problem (4.1) is unique.
Algorithm 2.4 can be used for constructive solution of Inverse Problem 1.1. This algorithm generalizes the methods, developed in parallel by Martinyuk and Pivovarchik [39] and by Buterin [40] for solving the Hochstadt-Lieberman problem.
Appendix. Inverse problem by Cauchy data
The goal of this section is to prove the following theorem on local solvability and stability of Inverse Problem 2.1.
Theorem 5.1. Let q be a fixed complex-valued function from L 2 (0, π), and let {K, N} be the corresponding Cauchy data. Then there exists ε > 0 (depending on q) such that, for any functionsK,Ñ from L 2 (0, π) satisfying the estimate Ξ := max{ K −K L 2 (0,π) , N −Ñ L 2 (0,π) } ≤ ε, (5.1) there exists a unique functionq ∈ L 2 (0, π) such that π 0 (q(x) −q(x)) dx = 0 and {K,Ñ} are the Cauchy data forq. In addition, q −q L 2 (0,π) ≤ CΞ, (5.2) where the constant C depends only on q and not on {K,Ñ}.
Below the symbol C is used for various positive constants. In order to prove Theorem 5.1, we need several auxiliary propositions. Applying the standard approach (see, e.g., [4, Theorem 1.1.3]), based on Rouché's Theorem, one can easily obtain the following result.
Proposition 5.2. Let K(t) be an arbitrary complex-valued function from L 2 (0, π). Then the function η 1 (λ) defined by (2.3) has the countable set of zeros {θ n } ∞ n=1 numbered according to their multiplicities so that |θ n | ≤ |θ n+1 |, n ∈ N, and satisfying the asymptotic formula ν n := θ n = n + O n −1 , n ∈ N.
(5.3)
In view of the asymptotic formula (5.3), we can find the smallest integer n 1 ≥ 2 such that the zeros {θ n } are simple for n ≥ n 1 and |θ n 1 | > |θ n 1 −1 |. Define the contour γ 0 := {λ ∈ C : |λ| = (|θ n 1 | + |θ n 1 −1 |)/2}. Clearly, θ n ∈ int γ 0 for n = 1, n 1 − 1 and the eigenvalues {θ n } ∞ Below we agree that, if a certain symbol γ denotes an object constructed by {K, N, ω}, then the symbolγ with tilde denotes the analogous object constructed by {K,Ñ , ω}.
Lemma 5.3. Let K, N be fixed complex-valued functions from L 2 (0, π), and let ω ∈ C. Then there exists ε > 0 (depending on K, N, ω) such that, for anyK,Ñ ∈ L 2 (0, π) satisfying (5.1), the points {θ n } n 1 −1 n=1 lie strictly inside γ 0 and Proof. Step 1. If the functions K, N,K,Ñ and the number ω satisfy the conditions of the lemma for sufficiently small ε > 0, then (2.3) and (5.1) yield the estimates
|η 1 (λ)|, |η 1 (λ)| ≥ c 0 > 0, λ ∈ γ 0 , (5.6) |η 1 (λ) −η 1 (λ)| ≤ CΞ, λ ∈ γ 0 .
Consequently, for sufficiently small ε > 0, we have |η 1 (λ) −η 1 (λ)| |η 1 (λ)| < 1 on γ 0 . By Rouché's Theorem, the functionη 1 (λ) has inside γ 0 the same number of zeros as η 1 (λ). According to our notations, these zeros ofη 1 (λ) are {θ n } n 1 −1 n=1 . Using (2.3), (5.1) and (5.6), we obtain the estimate (5.4):
|M(λ) −M (λ)| = |η 2 (λ)η 1 (λ) −η 2 (λ)η 1 (λ)| |η 1 (λ)||η 1 (λ)| ≤ CΞ, λ ∈ γ 0 .
Step 2. For n ≥ n 1 , consider the contours γ n,r := {ρ ∈ C : |ρ − ν n | = r}, where r > 0 is fixed and so small that r ≤ |νn−ν n+1 | 2 , n ≥ n 1 . The function η 0 (ρ 2 ) has exactly one zero ν n ∈ int γ n,r in ρ-plane for every n ≥ n 1 . The relations (2.3) and (5.1) yield the estimate
|η 1 (ρ 2 )| ≥ c r n , ρ ∈ γ n,r , n ≥ n 1 ,(5.7)
where the constant c r depends on r and not on ρ and n. For sufficiently small ε > 0, we obtain the estimate |η 1 (ρ 2 ) − η 1 (ρ 2 )| ≤ CΞ n 2 , ρ ∈ γ n,r , n ≥ n 1 .
(5.8)
Using (5.7), (5.8) and applying Rouché's Theorem to the contour γ n,r in ρ-plane, we conclude thatη 1 (ρ 2 ) has exactly one zeroν n ∈ int γ n,r for each n ≥ n 1 . Using the Taylor formula η 1 (ν 2 n ) = η 1 (ν 2 n ) + d dρ η 1 (ρ 2 ) |ρ=ζn (ν n − ν n ), ζ n ∈ int γ n,r , and (2.3), we derive the relation η 1 (ν 2 n ) −η 1 (ν 2 n ) = 1 ν 2 n π 0K (t) cos(ν n t) dt = d dρ η 1 (ρ 2 ) |ρ=ζn (ν n − ν n ), (5.9) whereK := K −K. It is easy to check that d dρ η 1 (ρ 2 ) ≥ C n , ρ ∈ int γ n,r , n ≥ n 1 . (t) cos(ν n t) dt ≤ π 0K (t) cos(nt) dt + π 0K (t)(cos(ν n t) − cos(nt)) dt ≤ |K n | + CΞ n , n ≥ n 1 ,K n := π 0K (t) cos(nt) dt. Combining (5.9), (5.10) and (5.11), we obtain |ν n − ν n | ≤ C|K n | n + CΞ n 2 , n ≥ n 1 . Step 3. Note that {θ n } ∞ n=n 1 are simple poles of M(λ), so M n = Res λ=θn M(λ) = η 2 (θ n ) η 1 (θ n ) , n ≥ n 1 , whereḟ (λ) = d dλ f (λ). If ε > 0 is sufficiently small, the analogous relation is valid forM n , n ≥ n 1 . HenceM n − M n = (η 2 − η 2 )η 1 + η 2 (η 1 −η 1 ) η 1η1 |λ=θn , n ≥ n 1 .
(5.15) Using (5.15) and the following estimates Similarly to (5.14), we obtain The relations (5.14) and (5.16) together imply (5.5).
|η 1 (θ n )| ≥ C n 2 , |η 1 (θ n )| ≥ C n 2 , |η 1 (θ n )| ≤ C n 2 , |η 2 (θ n )| ≤ C, |η 2 (θ n ) − η 2 (θ n )| ≤ C|N n | n + CΞ n 2 , |η 1 (θ n ) −η 1 (θ n )| ≤ C|L n | n 3 +
In [33,41] the following inverse problem has been studied.
Inverse Problem 5.4. Given the data {θ n , M n } ∞ n=1 , find q.
Clearly, Inverse Problem 5.4 is equivalent to Inverse Problem 2.1 by the Cauchy data. In addition, one can uniquely construct M(λ) by {θ n , M n } ∞ n=1 and vice versa. In [41], the following proposition on local solvability and stability of Inverse Problem 5.4 has been proved. Proposition 5.5. Let q ∈ L 2 (0, π) be fixed. Then there exists ε > 0 (depending on q) such that, for any complex numbers {θ n ,M n } ∞ n=1 satisfying the estimate Ω := max max
Theorem 2.5. (i) (Separation) and (Complete2) together imply (Complete); (ii) (Separation), (Simple), (Asymptotics) and (Basis2) together imply (Basis). Thus, one can change the condition (Complete) in Theorem 2.2 to (Separation) and (Complete2) and the condition (Basis) in Algorithm 2.4 to (Separation), (Simple), (Asymptotics) and (Basis2). Those results remain valid.
Proof of Theorem 2.2. Suppose that the problems L(q), L(q) and their subspectra {λ n } ∞ n=1 , {λ n } ∞ n=1 satisfy the conditions of Theorem 2.2. By virtue of the definitions (2.6), (2.7), (2.10)
(3.3) holds for ν = 0. Let us prove (3.4) for ν = 1, m n − 1 by induction. Assume that (3.3) is already proved for η <k> j (λ n ), k = 0, ν − 1, j = 1, 2. Using (3.4) and the relation ∆ <ν> (λ n ) = 0, we get (η 1 f 2 ) <ν> (λ n ) = −(η 2 f 1 ) <ν> (λ n ).
n ∈ I, ν = 0, m n − 1, except for (n, ν) = (0, m 0 − 1). Let us consider the case (n, ν) = (0, m 0 − 1) separately.
c <ν−k> (t, 0)) dt = 0, ν = 0, m 0 − 1.
Proposition 3. 3 .
3Let {θ n } ∞ n=0 be a sequence of complex numbers, satisfying the asymptotic formula θ n = πn a + κ n , {κ n } ∈ l 2 , a > 0.(3.13) Define µ n := #{k ≥ 0 : θ k = θ n }, J := {n ≥ 0 : θ n = θ k , ∀k : 0 ≤ k < n}.(3.14)
t)c <ν> (t, λ n ) dt = 0, n ∈ I, ν = 0, m n − 1.
( 3 . 18 )
318Proof. In view of[35, Theorem 3.6.6], for the sequence {g 0 n } ∞ n=0 to be a Riesz basis in H, it is sufficient to be complete in H and to satisfy the twosequence {b n } ∞ n=0 , every integer N 0 ≥ 0 and some fixed positive constants M 1 and M 2 , independent of {b n } and N 0 .First, we show that the sequence {g 0 n } ∞ n=0 is complete in H. Let h = [h 1 , h 2 ] ∈ H be such that (h, g 0 n ) H = 0 for all n ≥ 0. It means that the function
and the constant Ω can be found from the eigenvalue asymptotics (4.3). Thus, Inverse Problem 4.1 is reduced to Inverse Problem 1.1 by the whole spectrum {λ n } ∞ n=1 of (4.1)-(4.2).
Proposition 4. 2 .
2Let f j (λ), j = 1, 2, be entire functions defined by (4.5), and let {λ n } ∞ n=1 be the eigenvalues of the problem (4.1)-(4.2) counted with their multiplicities, λ 0 := 0. Then the conditions (Basis2), (Separation), (Simple) and (Asymptotics) are fulfilled.
outside γ 0 . Without loss of generality, we may assume that equal eigenvalues in the sequence {θ n } ∞ n=1 are consecutive. Introduce the notations S := {1} ∪ {n ≥ 2 : λ n = λ n−1 }, k n := #{k ∈ N : θ k = θ n }, M(λ) := η 2 (λ) η 1 (λ) , M n+ν := Res λ=θn (λ − θ n ) ν M(λ), n ∈ S, ν = 0, k n − 1.
ξ n := |ν n −ν n | + 1 n 2 |M n −M n |. The constant C in the estimates (5.4) and (5.5) depends only on {K, N, ω}.
sin(nt) dt,N := N −Ñ,L n := π 0 tK(t) sin(nt) dt, we arrive at the estimate |M n − M n | ≤ Cn(|N n | + |L n |) + CΞ, n ≥ n 1 .
the unique complex-valued functionq ∈ L 2 (0, π) being the solution of Inverse Problem 5.4 for {θ n ,M n } ∞ n=1 . Moreover, the estimate (5.2) holds with the constant C depending only on q. Lemma 5.3 and Proposition 5.5 together imply Theorem 5.1.
12 )
12Bessel's inequality for the Fourier coefficients {K n } and (5.1) imply that Combining (5.12) and(5.13), we arrive at the estimate∞
n=n 1
|K n | 2
1/2
≤ CΞ.
(5.13)
∞
n=n 1
n 2 |ν n − ν n | 2
1/2
≤ CΞ.
(5.14)
Acknowledgments. This work was supported by Grant 20-31-70005 of the Russian Foundation for Basic Research.
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. Natalia Pavlovna Bondarenko, Natalia Pavlovna Bondarenko
. Department of Mechanics and Mathematics. 34Department of Applied Mathematics and Physics, Samara National Research University ; Saratov State UniversityAstrakhanskaya. e-mail: [email protected] of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia, 2. Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia, e-mail: [email protected]
| []
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"Resilience in urban networked infrastructure: the case of Water Distribution Systems",
"Resilience in urban networked infrastructure: the case of Water Distribution Systems"
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"Antonio Candelieri [email protected] \nDepartment of Economics, Management and Statistics\nUniversity of Milano-Bicocca\n20126MilanItaly\n",
"Ilaria Giordani \nOaks srl. Milano\n\n",
"Andrea Ponti \nDepartment of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly\n",
"Riccardo Perego \nDepartment of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly\n",
"Francesco Archetti \nDepartment of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly\n"
]
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"Department of Economics, Management and Statistics\nUniversity of Milano-Bicocca\n20126MilanItaly",
"Oaks srl. Milano\n",
"Department of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly",
"Department of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly",
"Department of Computer Science\nSystems and Communication\nUniversity of Milano-Bicocca\n20126MilanItaly"
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| Resilience is meant as the capability of a networked infrastructure to provide its service even if some components fail: in this paper we focus on how resilience depends both on net-wide measures of connectivity and the role of a single component. This paper has two objectives: first to show how a set of global measures can be obtained using techniques from network theory, in particular how the spectral analysis of the adjacency and Laplacian matrices and a similarity measure based on Jensen-Shannon divergence allows us to obtain a characterization of global connectivity which is both mathematically sound and operational. Second, how a clustering method in the subspace spanned by the l smallest eigenvectors of the Laplacian matrix allows us to identify the edges of the network whose failure breaks down the network. Even if most of the analysis can be applied to a generic networked infrastructure, specific references will be made to Water Distribution Networks (WDN). | null | [
"https://arxiv.org/pdf/2006.14622v1.pdf"
]
| 220,128,168 | 2006.14622 | 2db4cc415ce5ea72848370e7daa887d99142c6a8 |
Resilience in urban networked infrastructure: the case of Water Distribution Systems
Antonio Candelieri [email protected]
Department of Economics, Management and Statistics
University of Milano-Bicocca
20126MilanItaly
Ilaria Giordani
Oaks srl. Milano
Andrea Ponti
Department of Computer Science
Systems and Communication
University of Milano-Bicocca
20126MilanItaly
Riccardo Perego
Department of Computer Science
Systems and Communication
University of Milano-Bicocca
20126MilanItaly
Francesco Archetti
Department of Computer Science
Systems and Communication
University of Milano-Bicocca
20126MilanItaly
Resilience in urban networked infrastructure: the case of Water Distribution Systems
Network AnalysisResilienceWater Distribution NetworkCluster- ingSpectral Analysis
Resilience is meant as the capability of a networked infrastructure to provide its service even if some components fail: in this paper we focus on how resilience depends both on net-wide measures of connectivity and the role of a single component. This paper has two objectives: first to show how a set of global measures can be obtained using techniques from network theory, in particular how the spectral analysis of the adjacency and Laplacian matrices and a similarity measure based on Jensen-Shannon divergence allows us to obtain a characterization of global connectivity which is both mathematically sound and operational. Second, how a clustering method in the subspace spanned by the l smallest eigenvectors of the Laplacian matrix allows us to identify the edges of the network whose failure breaks down the network. Even if most of the analysis can be applied to a generic networked infrastructure, specific references will be made to Water Distribution Networks (WDN).
Introduction
Networked infrastructures, as water, energy and transport, have developed similar functional and structural features in their evolution over time: spatial, but also financial, constraints have significantly restricted their connectivity, robustness and their capability to deliver their service with failed or damaged components, in short their resilience. These features have also generated systemic risk and cascading effects exacerbated by the complexity of the infrastructure with up to tens of thousands of components (pipes, valves, pumping stations, tanks and consumption points). Resilience of a Water Distribution Network (WDN) is about delivering services regardless of disruptive events that may occur. Resilience, robustness, reliability and vulnerability are terms strictly linked and often confusingly used.
We propose a framework based on network theory to address structural analysis of any WDN: the growing awareness of the interplay between global (system-wide) and local (individual component) resilience have spawned a line of research directly aimed at resilience in WDN (Yazdani and Jeffrey, 2012), .
Complex networks are instances of real-world graphs.
They include examples such as the Internet, social networks, supply networks, metabolic networks, and critical infrastructures, among other engineered systems. Important characterizations are: smallworld networks (Backstrom et al., 2012), scale-free networks (Barabási et al., 2009) and planarity, which typically characterizes road networks, water distribution networks, energy grids and general networked flow systems. Another important characterization, relevant for resilience, is the community structure (Girvan et al., 2002), meaning that connections are dense among nodes within the same subset (i.e., "community") and sparse among nodes between different subsets.
Related Work
Graph theoretic approaches have been proposed in the literature to address the issue of resilience in WDN both in terms of connectivity and service levels (Diao et al. (Diao et al. 2020). In this paper we focus on connectivity, whose analysis in network models offers important cues to the design and management of the network even without the need of running a hydraulic simulation model. Spectral analysis of networks offers a mathematically principled approach yielding both local and global structural information at a computational cost of O(n 3 ), with n the number of nodes of the network, due to the computation of eigenvalue and eigenvectors. An innovative perspective on the structural characterization of networks is offered in (Schieber et al., 2017), based not only on averages but on distributional properties analyzed by information theoretic models. A related approach to the evaluation of resilience, drawn from physics, is provided by percolation analysis which evaluates the impact of removing nodes/links from the network in terms of how the average length of the shortest paths connecting pairs of nodes increases, to the point of bringing to a disconnected network. Monte Carlo (MC) methods are key for percolation in complex networks (Chen, 2017) in which several random global disruption scenarios are analyzed (Torres et al., 2017). A simulation-based model percolation is flexible and can handle several kinds of network failures ranging from a single node to a scenario in which a critical fraction of the network components has failed.
Our Contributions
The main contribution of this paper is a global approach to the characterization of resilience in urban water distribution network (WDN). Specific contribution is articulated in 3 points:
• To show how a set of global measures can be obtained using techniques from network theory, in particular how the spectral analysis of the adjacency and Laplacian matrices allows us to obtain a characterization of global connectivity which is both mathematically sound and operational. • To show that considering the analysis of the node-node distance distribution and the Network Node Dispersion (NND) (Schieber et al., 2017) can yield additional insights on network structural characterization. • To show how a graph-clustering method, in the subspace spanned by the l smallest eigen-vectors of the Laplacian matrix, allows us to identify the edges whose failure breaks the network into unconnected components.
The proposed approach has been evaluated on both benchmark and real world WDNs, considering breakages on pipes as relevant disruptive events. The structure of the paper is as follows: section 2 gives background notions on graph models and network analysis; section 3 describes the measures and tools provided by spectral analysis. Section 4 describes the different WDNs used in this study and the relevant results; finally, in section 5 some conclusions are provided.
2
Mathematical Background
Graph Theory
Let denote a graph with = ( , ), where V is the set of nodes and E is the set of edges. Each edge of G is represented by a pair of nodes ( , ) with ≠ , and , ∈ and with = | |and = | |. If ( , ) ∈ , and are called adjacent nodes. A graph is undirected if ( , ) and ( , ) represent the same edge. A graph is simple if no self-loops are admitted (edges starting from a node and ending on the same node) and only one edge can exist between each pair of nodes ( , ), with ≠ . The adjacency relationship between the nodes of can be represented through a non-negative × matrix (i.e., the adjacency matrix of ). The entry , of the adjacency matrix is 1 if and are adjacent nodes (i.e., ( , ) ∈ ), and 0 otherwise. Furthermore, = if is undirected and (entries on the diagonal) are 0 if is simple. Let denote with the degree of the node , that is the number of edges having as one of the two nodes on the edge = ∑ , =1
. Anyone of the edges having as one of its nodes is called incident on .
When G is directed, meaning that the order of the two nodes of an edge is relevant for its definition, the can be split into out-degree (number of edges having as first node) and in-degree (number of edges having as second node).
A path in a graph is a sequence of nodes connected by edges the length of the path is the number of edges. A connected component is a maximal subgraph when all nodes can be reached from every other.
The shortest path between and is the one related to the smallest number of arcs from to , which is usually named distance ( , ). The largest distance among each possible pair of nodes in is named diameter.
A subgraph ' = ( ', ') of is a graph such that ' ⊆ and ' ⊆ ; a connected component of is a maximal if is the largest possible graph for which you could not find another node in the graph that could be added to the graph with all the nodes be still connected..
A weight ≥ 0 can also be associated with every edge ( , ) ∈ ; in this case the graph is called weighted and the (weighted) adjacency matrix is a × matrix
having = 0, if G is simple, ≥ 0 and = for each ≠ if G is undirected.
In the case of weighted graphs, the previous definitions, related to degree, path and diameter, can be modified to consider weights on the edges rather than their number. In particular, degree of the node is the sum of the weights of the edges incident on (out-degree is the sum of the weights of the edges starting from , while in-degree is the sum of the weights of the edges ending to ); shortest path between and is the list of adjacent nodes from to with minimal sum of the weights on the correspondent connecting edges; the diameter is the longed shortest path computed.
Network Analysis: the basic measures
The number of edges = 1/2 ∑ =1 = 1 2 ∑ , =1 .
If is the mean vertex degree, = 1 ∑ =1 we get = 2 .
Since the max possible number of edges in G is
( 2 ) = ( −1) 2
we can compute the density of the network as the fraction of edges which are present in the specific graph:
= ( 2 ) = 2 ( − 1) = − 1
the density is in the range (0,1). A pair of nodes is usually connected by many paths which typically share some nodes or edges. If they share no edges, they are called edge independent. No shared nodes imply vertex independence. The number of independent paths between 2 nodes is called connectivity of the 2 nodes. A cut-set, specifically a vertex cut-set, is a set of nodes whose removal disconnects and . A minimum cut-set is the smallest cut-set.
An important concept is the Laplacian matrix of a network = − , where A is the adjacency matrix and D is a × diagonal matrix with = (Brouwer et al., 2011). The eigenvalues of L are of paramount importance in assessing the connectivity. We number them as 0 = 1 ≤ 2 ≤ ⋯ ≤ , so they are not negative. Note that L is singular. If we have ℎ different components of size 1 , 2 , … , ℎ , then is block diagonal and the multiplicity of the zero-eigenvalue is exactly equal to the number of components, which in turn implies that 2 is non zero if and only if the network is connected; 2 is also called algebraic connectivity.
Also important in the analysis of resilience are centrality measures, which address the issue of the relative importance of nodes/edges. The most widely used measures are:
Eigenvector centrality of the vertex , that is = 1 −1 ∑ =1, ≠ , where 1 is the largest eigenvalue of the adjacency matrix, , (aka spectral radius). The eigenvector centrality can be large either because the vertex has many neighbors or because has important neighbors. Katz centrality and Page Rank algorithm are just parametrized version of eigenvector centrality. Closeness centrality measures the mean distance from one vertex to the others. Let be the length of a shortest path from to , that is the number of edges along that path. The closeness centrality is: = ∑ .
Betweenness centrality: let be = 1 if vertex lies on the shortest path from to and 0 otherwise. Then, betweenness centrality is given by = 1 2 ∑ , =1 . We can similarly define an edge betweenness that counts the number of shorter paths that run along the edges. Upon these indices we can build a first characterization of resilience removing the vertex/edge with the highest centrality score until the network splits. As a basic measure of connectivity, the average degree can provide an immediate information about the organization of the network. This measure is also linked to the linkper-node ratio (e), that is computed as the number of edges of a graph with respect to the number of its nodes.
Central point dominance ′ , based on betweenness centrality is a measure for characterizing the organization of a network according to its path-related connectivity;
′ = 1 −1 ∑ ( − ) =1,…,
where ( ) s the betweenness centrality of the node i and is the maximum value of betweenness centrality over all the n nodes of the network.
The evaluation of network resilience requires to extend the analysis other structural features: the clustering coefficient (CC) is used to characterize the resilience of a network according to loops of length three and is computed as the number of triangles with respect to the overall number of possible connected triples, where a triple consists of three nodes connected at least by two edges while a triangle consists of three nodes connected exactly by three edges:
= 3
In this paper, the open-source software Cytoscape (http://www.cytoscape.org/) has been adopted as the basic tool of the analytical framework proposed and its plug-in-ClusterMaker2 (Morris et al., 2011).
Dissimilarity analysis &Network Nodes Dispersion (NND)
The measures introduced in Section 2.2 are based on distances and their average values. Another analysis can be performed also in terms of distributions. This kind of analysis has been inspired by the paper (Schieber et al., 2017) which is based on the vertex-vertex distance distribution. The first step is to consider a measure of the graph heterogeneity through connectivity distances. The shortest path distances between all nodes are arranged in the distance matrix = [ , ], , = 1, … , .
The maximum entry of row , max =1,…, , is known as the eccentricity of node .
The maximum eccentricity among the nodes max , , is equal to the diameter of the network.
For each row we compute ( ) as the fraction of nodes which are connected to at a distance and associate to node the probability distribution of the r.v. ( ).
The Network Node Dispersion (NND) is given by
( ) =
( 1 , … , ) log( + 1) The Jensen-Shannon divergence of the probability distributions , = 1, … , is given by
∑ ( ) log ( ( ) ) , where = ∑ ( ) =1
and is normalized by log( + 1) where is the diameter of the network.
Considering the distance distribution over the whole graph we obtain the average node distance distribution ( ) with average . This enables to compute a measure of similarity with another graph ' through the Jensen-Shannon divergence ( ( ), ( ′ ).
Then we measure the distance between and ′ by
( , ′ ) = 1 √ ( ( ), ( ′ )) log 2 + 2 |√ ( ) − √ ( ′ )| with 1 + 2 = 1.
This distance is different than the one used in Scheiter (2016) which has a third term which takes into account the centrality measures of each node and its connectivity span.
We use instead the set of centrality related measures introduced in 2.2.
Spectral Clustering
Given two sets of nodes 1 and 2 , an n-dimensional vector i.e., n is the number of nodes in the graph) is used to represent the association of each node to cluster 1 or 2 = { +1 i ∈ 1 −1 i ∈ 2 The graph clustering problem can be formulated as the minimization of the following function ( ):
( ) = ∑ ( − ) 2 = , ∈
where are the entries of the Laplacian matrix. The important feature of spectral clustering methods is that the produce a set of balanced clusters. An elegant solution, conceptually simple but computationally inefficient, to the problem was proposed in (Fiedler, 1973) which identified the 2nd smallest eigenvector of the Laplacian matrix (usually known as Fiedler vector) as the vector p which provides the optimal bi-partitioning of the graph. Early applications of this result have permitted to implement recursive bi-partitioning spectral clustering approaches (Hagen and Kahng, 1992) to perform partitioning in > 2 groups. However, this approach requires the computation of matrices, eigenvalues and eigenvectors, for each sub-graph until the desired number of clusters is reached. More effective computational schemes are analyzed in (Luxburg, 2007) and use a data representation in the lower dimensional space spanned by the most relevant eigenvectors. Our approach in this paper consists in ranking in descending order the eigenvalues 1 ≥ 2 ≥ ⋯ ≥ of the adjacency matrix. If the user sets the desired number of clusters as k, k-means clustering is performed on the resulting dataset having n rows (nodes of the graph) and k columns (eigenvectors corresponding to the k largest eigenvalues of A). If a suitable value of k is not known the implementation in Cytoscape ClusterMaker2 computes the ratio +1 ⁄ , = 1, … , − 1 and picks as the smallest integer such the ratio than 1+ε (in the computation reported in Section 4, = 1.02).
Experimental Setting
The Network Models
In this section 4 WDNs are analyzed. The first WDN is a benchmark model often used in different studies on WDN management, namely "Anytown"1. The associated graph consists of 22 nodes and 43 links.
Marnate is a small town in Northern Italy, with an associated graph consisting of 384 nodes and 469 edges.
Neptun is the WDN of the Romanian city of Timisoara, with an associated graph of 333 nodes and 339 edges, analyzed in the European project Icewater.
Abbiategrasso refers to a pressure management zone in Milan (namely, Abbiategrasso) with an associated graph consisting of 1212 nodes and 1385, analyzed in the European project Icewater.
In analyzing WDNs one must consider that most of the end-users are supplied by single connections. To avoid a bias in the analysis, a preliminary preprocessing can be performed by cutting the final connections, that are usually the links between the consumption meters of each building and the main distribution pipes.
Computational results
The characteristic path length is the average number of edges along the shortest path for every possible pair of nodes ( , ). Anytown looks rather more like a "No-town" network, with structural properties far from those of the real WDNs. The three real-world WDNs analyzed are very sparse (with density lower or equal to 0.006) with respect to Anytown (density q around 0.2). The central point dominance ′, instead, is quite similar among all the four WDNs taken into account. The clustering coefficient , diameter d and characteristic path length are quite similar among the three real WDNs and different from those computed on Anytown: the three real-world WDNs are effectively planar and "almost" regular. Anytown is different from the others and the difference is captured quite naturally. Marnate and Neptun are quite similar and different from Abbiategrasso. Actually, Neptun and Marnate have grown out of autonomous urban water distribution networks, constrained in their development by technological and physical constraints. Abbiategrasso is a subnetworkspecifically a Pressure Management Zonecarved out of the whole water distribution network of Milano for administrative reasons and management strategies. This structural difference is captured by the dissimilarity measure.
Clustering
Graph clustering approaches, such as Spectral Clustering, can be used to identify the specific links (pipelines) whose removal may induce a disconnection of the network in two or more sub-networks. In this paper, Spectral Clustering has been performed (through Cytoscape's Cluster plug-in named ClusterMaker2) to identify sub-networks connected by a limited (minimal) number of links, that are pipelines whose breakage implies the disconnection of some WDN portion. In the following figures these pipelines are highlighted; it is important to note that breakages must occur, at the same time, on all the different red edges to imply a hydraulic disconnection. Breakages affecting only one pipe may imply a reduction in the supply service or generate a "stress" condition on the hydraulic infrastructure. A software simulation of the damaged network may be used to evaluate the induced scenario. According to results of Spectral Clustering, the disconnection in two sub-networks is reported for the Marnate WDN, while the disconnection in three sub-networks is depicted for both Bresso-Cormano-Cusano and Abbiategrasso WDNs. More in detail, respect to Anytown, Spectral Clustering is not able to provide a bi-partitioning of the WDN in a reasonable time, mainly due to the high connectivity of the water network. Moreover, the disconnection in three different sub-networks may occur only by the simultaneous breakage of many pipelines.
Conclusion
In this paper the use of network analysis for the evaluation of resilience in a urban networked infrastructure has been proposed. The application to three WDNs from two different projects and one of benchmark has permitted to define the main measures and their characteristic values for real world WDNs, taking into account also previous results reported in the literature.
While general measures have been used in order to evaluate and compare connectivity and resilience of the WDNs considered, the application of spectral clustering has permitted to identify the most critical hydraulic pipelines whose breakage imply structural disconnection and consequent failure of the distribution service (vulnerability).
A further layer of analysis that can be added consists in joining the network analysis, in the abstract graph setting, and hydraulic simulation, provided for instance by EPANET. The set of resilience indices based on network analysis, and adopted in this paper, continues to measure how the failure of a single component impacts the connectivity while the simulation of the damaged network provides a measure about how a damaged component impacts the service level still offered by the WDN.
, 2016), (Shuang et al., 2014), (Archetti et al., 2015), (Candelieri et al., 2017), (Gutiérrez-Pérez et al., 2013), (Herrera et al., 2016), (Di Nardo et al., 2018),
Fig. 1 .
1Anytown (k=3): Critical edges (red) whose removal generates a disconnection and (right) resulting disconnected components.
Fig. 2 .
2Marnate WDN (k=3): Critical edges (red) whose removal generates a disconnection.
Fig. 3 .
3Neptun WDN (k=2): Critical edges (red) whose removal generates a disconnection.
Fig. 4 .
4Abbiategrasso WDN (k=3): Critical edges (red) whose removal generates a disconnection.
Table 1 .
1Structural Analysis of four WDNsMeasure
Anytown Marnate Neptun Abbiategrasso
Density (q)
0.186
0.006
0.005
0.001
Link-per-node ratio (e)
1.954
2.443
0.992
1.156
Central point dominance (cb')
0.230
0.189
0.476
0.303
Clustering coefficient (CC)
0.303
0.007
0.000
0.004
Diameter
7
35
82
83
Characteristic Path Length
2.761
21.696
30.226
31.233
Table 2 .
2SpectralAnalysis of four WDNs
Measure
Anytown
Marnate
Neptun Abbiategrasso
Spectral Gap
1.5149
0.0838
0.0149
0.2132
Algebraic Connectivity
0.1708
0.0046
0.0009
0.0002
The spectral gap is the difference between the two largest eigenvalues of the Laplacian
matrix. The algebraic connectivity is the value of 2 .
The spectral analysis shows that the 3 real world WDN have relatively similar values
and again quite different from Anytown.
Table 3 .
3Dissimilarityof four WDN
Measure
Anytown
Marnate
Neptun
Abbiategrasso
Anytown
0.000
0.670
0.700
0.770
Marnate
0.670
0.000
0.335
0.681
Neptun
0.700
0.335
0.000
0.617
Abbiategrasso
0.770
0.681
0.617
0.000
http://emps.exeter.ac.uk/engineering/research/cws/resources/benchmarks/expansion/anytown.html
AcknowledgementsThis study has been partially supported by the Italian project "PERFORM-WATER 2030"programma POR (Programma Operativo Regionale) FESR (Fondo Europeo di Sviluppo Regionale) 2014-2020, innovation call "Accordi per la Ricerca e l'Innova- We greatly acknowledge the DEMS Data Science Lab for supporting this work by providing computational resources (DEMS -Department of Economics, Management and Statistics).
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| []
|
[
"Theoretical analysis of the role of complex transition dipole phase in XUV transient-absorption probing of charge migration",
"Theoretical analysis of the role of complex transition dipole phase in XUV transient-absorption probing of charge migration"
]
| [
"Yuki Kobayashi \nDepartment of Chemistry\nUniversity of California\n94720BerkeleyCAUSA\n",
"Daniel M Neumark \nDepartment of Chemistry\nUniversity of California\n94720BerkeleyCAUSA\n\nChemical Sciences Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA\n",
"Stephen R Leone \nDepartment of Chemistry\nUniversity of California\n94720BerkeleyCAUSA\n\nChemical Sciences Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA\n\nDepartment of Physics\nUniversity of California\n94720BerkeleyCAUSA\n"
]
| [
"Department of Chemistry\nUniversity of California\n94720BerkeleyCAUSA",
"Department of Chemistry\nUniversity of California\n94720BerkeleyCAUSA",
"Chemical Sciences Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA",
"Department of Chemistry\nUniversity of California\n94720BerkeleyCAUSA",
"Chemical Sciences Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA",
"Department of Physics\nUniversity of California\n94720BerkeleyCAUSA"
]
| []
| We theoretically investigate the role of complex dipole phase in the attosecond probing of charge migration. The iodobromoacetylene ion (ICCBr + ) is considered as an example, in which one can probe charge migration by accessing both the iodine and bromine ends of the molecule with different spectral windows of an extreme-ultraviolet (XUV) pulse. The analytical expression for transient absorption shows that the site-specific information of charge migration is encoded in the complex phase of cross dipole products for XUV transitions between the I-4 and Br-3 spectral windows. Ab-initio quantum chemistry calculations on ICCBr + reveal that there is a constant phase difference between the I-4 and Br-3 transient-absorption spectral windows, irrespective of the fine-structure energy splittings. Transient absorption spectra are simulated with a multistate model including the complex dipole phase, and the results correctly reconstruct the charge-migration dynamics via the quantum beats in the two element spectral windows, exhibiting out-of-phase oscillations. | 10.1364/oe.451129 | [
"https://arxiv.org/pdf/2112.10008v1.pdf"
]
| 245,334,423 | 2112.10008 | ea78c0ac252e93986d2de6055fed990649e9efd8 |
Theoretical analysis of the role of complex transition dipole phase in XUV transient-absorption probing of charge migration
Yuki Kobayashi
Department of Chemistry
University of California
94720BerkeleyCAUSA
Daniel M Neumark
Department of Chemistry
University of California
94720BerkeleyCAUSA
Chemical Sciences Division
Lawrence Berkeley National Laboratory
94720BerkeleyCAUSA
Stephen R Leone
Department of Chemistry
University of California
94720BerkeleyCAUSA
Chemical Sciences Division
Lawrence Berkeley National Laboratory
94720BerkeleyCAUSA
Department of Physics
University of California
94720BerkeleyCAUSA
Theoretical analysis of the role of complex transition dipole phase in XUV transient-absorption probing of charge migration
© 2021 Optica Publishing Group under the terms of the Optica Publishing Group Publishing Agreement
We theoretically investigate the role of complex dipole phase in the attosecond probing of charge migration. The iodobromoacetylene ion (ICCBr + ) is considered as an example, in which one can probe charge migration by accessing both the iodine and bromine ends of the molecule with different spectral windows of an extreme-ultraviolet (XUV) pulse. The analytical expression for transient absorption shows that the site-specific information of charge migration is encoded in the complex phase of cross dipole products for XUV transitions between the I-4 and Br-3 spectral windows. Ab-initio quantum chemistry calculations on ICCBr + reveal that there is a constant phase difference between the I-4 and Br-3 transient-absorption spectral windows, irrespective of the fine-structure energy splittings. Transient absorption spectra are simulated with a multistate model including the complex dipole phase, and the results correctly reconstruct the charge-migration dynamics via the quantum beats in the two element spectral windows, exhibiting out-of-phase oscillations.
Introduction
Recent progress in laser technology has enabled real-time tracking of photochemical dynamics from the most fundamental point of view, i.e., electron motion [1]. One such phenomenon is known as charge migration, in which a coherent superposition of electronic states drives oscillatory motion of electrons from one end of a molecule to another [2,3]. The characteristic few-electronvolt energy spacing of molecular valence orbitals dictates that coherent electronic motion can occur even before nuclear motions set in (e.g., 1 eV of energy spacing corresponds to an oscillation period of 4.1 fs). Observation of charge migration has been a central topic in attosecond science [4,5], and a number of experimental [6][7][8][9] and theoretical [10][11][12][13][14][15][16][17][18][19][20][21][22][23] studies have reported results the last few years. There are both exciting perspectives and unanswered questions regarding charge migration. Potential implications of electronic coherence in photochemisry remain a topic of debate, such as efficient charge transfer in light-harvesting antenna [24] and selective bond dissociation in photoionized peptide molecules [25]. The mechanisms of electronic decoherence due to nuclear motions or many-body interactions have been investigated theoretically and will benefit from more experimental results [2]. Laser-control of charge migration is predicted to be attainable [26], and realization of this can be a fundamental step toward attochemistry [27][28][29].
In order to see charge-migration dynamics in real time, one needs a spectroscopic probe that has, in addition to a sub-femtosecond temporal resolution, the capability to resolve sitespecific electron density. One way to achieve this is to utilize the localized nature of inner-shell electrons, which can be accessed by high-energy photons in the x-ray or extreme-ultraviolet (XUV) regime. Among the various spectroscopic methods in this energy regime [30,31], the technique considered here is transient absorption spectroscopy, wherein an optical pulse triggers photochemical reactions and an XUV pulse imprints the dynamics in the core-to-valence absorption spectra [32]. Combined with attosecond XUV pulses produced by high-harmonic generation, the method has been successfully applied to measure electronic coherences in atoms [33][34][35] and molecules [36,37]. The site specificity of XUV absorption can be most efficiently utilized by accessing multiple tagging elements in a molecule with a broadband XUV light source [38], with the potential to achieve a panoramic reconstruction of charge migration.
Simulation of charge migration and XUV absorption therein requires quantum-mechanical treatments, as charge migration is inherently a quantum-mechanical process. One of the possible ways is to solve the time-dependent Schrödinger equation for a finite number of electronic states, and compute the absorption signals from a coherent superposition system as a Fourier transformation of dipole oscillations [39]. What distinguishes this method, compared to an absorption spectrum that is constructed as a simple sum of the absolute squares of the transition dipole moment (or oscillator strength), is that it can account for quantum interference between multiple transition pathways. Indeed, characteristic quantum beats that are induced by a quantum interference from a pair of valence states that end up at a common final state is the key to attosecond transient-absorption probing of electronic coherence [33,40]. A question arises when considering the aforementioned scheme of multielement probing of charge migration: what determines the relative timing of the quantum beats when multiple final states are present, especially when these final states belong to different elemental spectral windows?
Here, we highlight the role of the complex transition dipole phase in understanding XUV absorption probing of charge migration. The complex dipole phase is lost when absorption signals or transition probabilities are simply considered to be proportional to oscillator strengths. However, as recently shown for solid-state high-harmonic generation [41,42] and attosecond probing of light-dressed states [43], or as traditionally well known in two-pathway excitation experiments (e.g., -3 ) [44,45], the complex dipole phase is not negligible when multiple transition pathways interfere. The target molecule of the present study is iodobromoacetylene (ICCBr). The I-4 and Br-3 core-level absorption edges lie in the XUV regime (∼ 50 eV and ∼ 65 eV, respectively), which can be accessed by a typical high-harmonic generation setup [46,47]. A molecule of similar structure, iodoacetylene (ICCH), was already used for charge migration studies via high-harmonic spectroscopy [7,42], and the additional bromine atom in ICCBr enables the possibility of multi-element probing of charge migration in attosecond transient absorption. We first examine analytical expressions for the attosecond transient absorption signals from a coherent system, and show that the complex phase of cross transition dipole products determines the beat phase for each probe state. Ab-initio calculations are performed for the electronic structure of ICCBr + , and the complex dipole phases are revealed to be constant within each of the I-4 and Br-3 windows irrespective of the spin-orbit coupling, ligand-field effects, and probe direction. Attosecond XUV absorption spectra are simulated by solving the time-dependent Schrödinger equation within a multistate model, and the manifestation of the complex dipole phase is analyzed.
Computational methods
Electronic structure of ICCBr +
The electronic structure of ICCBr + is computed by using the spin-orbit general multiconfigurational quasidegenerate perturbation theory (SO-GMC-QDPT) implemented in a developer version of GAMESS-US [48][49][50][51]. Relativistic model-core potentials and basis sets of triple-zeta quality (MCP-TZP) are used in the calculations [52,53]. A fixed linear structure of the molecule with bond lengths of C-I 1.99 Å, C-C 1.18 Å, C-Br 1.79 Å is used [54]. The active space contains 11 valence orbitals and the 10 core-level orbitals (i.e., I-4 and Br-3 ), and a total of 42 single-hole configurations are included in the state-average calculations. The transition dipole moments including their complex phase are obtained for all combinations of the valence and core-excited states. Note that the wavefunctions of the valence and core-excited states are obtained in a single set of calculations, and the transition dipole moments are calculated straightforwardly by using the Slater-Condon rule. In order to accurately compute the spin-orbit couplings of the halogen atoms, effective nuclear charges of 71.37 and 41.40 are used for the iodine and bromine atoms, respectively. In addition, the spin-orbit matrix coupling constants for the Br-3 and I-4 shells are scaled by factors of 0.577 and 0.665, respectively [38].
Multi-state simulation of core-to-valence absorption spectra
Core-to-valence absorption spectra of ICCBr + are simulated by solving the time-dependent Schrödinger equation for a multi-state model,
Ψ( ) = 0 − Γ 2 − d · E( ) Ψ( ).(1)
In Eq. (1), Ψ( ) is a column vector that contains the complex coefficients for the electronic states, 0 is a field-free Hamiltonian whose diagonal elements correspond to the state energies, Γ is a diagonal matrix for the autoionization lifetime of the core-excited states, d is a transition-dipole matrix, and E( ) is the laser electric field. The model consists of two valence states, X 2 Π 3/2 and A 2 Π 3/2 , and the ten I-4 and Br-3 core-excited states. The energy and dipole moments are obtained from the SO-GMC-QDPT calculations, the autoionization lifetimes of all the core-excited states are assumed to be Γ = 100 meV [55,56], which corresponds to a 1/ lifetime of 6.6 fs. The XUV absorption spectra are obtained by calculating the single-atom absorption cross section,
( ) ∝ Im ( ) ( ) ,(2)
where ( ) and ( ) are the dipole moments and applied laser field, respectively, that are Fourier transformed from the time domain to the frequency domain [39].
Results
3.1.
State-resolved core-to-valence absorption spectra Figure 1 shows the calculated XUV absorption of ICCBr + from 4 different electronic states, X 2 Π 3/2 , A 2 Π 3/2 , B 2 Π 3/2 , and C 2 Σ 1/2 [57]. The absorption spectra are constructed as a sum of the oscillator strengths convoluted with a 0.5-eV Gaussian broadening. The neutral molecule has an electronic configuration of 1 2 1 4 2 4 3 4 , and the inset images show the molecular orbitals that yield the main single-hole configuration of each electronic state. The results show that the different electronic states make unique fingerprints in the XUV absorption spectra. Furthermore, the site specificity of core-to-valence absorption is demonstrated; the 3 and 2 orbitals have comparable weights from the I-5 and Br-4 orbitals, and the absorption signals appear both in the I-4 and Br-3 windows for the X 2 Π 3/2 and A 2 Π 3/2 states [Figs. 1(b) and 1(c)]. The 1 orbital, on the other hand, has little contribution from the I-5 orbital, and the absorption signals from the B 2 Π 3/2 state are highly suppressed in the I-4 window but are strong in the Br-3 window. Similarly, the 1 orbital mostly consists of the I-5 orbital, and the absorption signals from the C 2 Σ 1/2 state are more enhanced in the I-4 window. These site and state specificities of core-level absorption are what make this method an appropriate tool to probe charge migration.
Complex-dipole phase in analytical expression of transient absorption
We next consider charge migration dynamics that arise from a coherent superposition of the X 2 Π 3/2 and A 2 Π 3/2 states of the ion. The energy difference between the two states is 1.90 eV [57], and the corresponding beat period is = 2.2 fs. Figure 2(a) shows the hole density motion of the charge migration, which is calculated as a coherent superposition of the 3 and 2 hole states ( Figs. 1(b) and 1(c)). At one timing, defined as 0 , the hole density is localized on the I atom; a half period later, the hole density migrates to the carbon and Br sites.
(c) A 2 Π 3/2 (d) B 2 Π 3/2 (e) C 2 Σ 1/2 2π 1σ 1π 2 Π 3/2 2 Π 1/2 2 Π 3/2 2 Π 1/2 2 Π 3/2 2 Π 1/2 2 Σ 1/2 X A B C Absorbance (arb. units) (a)
In order to investigate how the core-to-valence absorption from the I-4 and Br-3 windows probe such hole-density motion, we start with a general analytical formula for the attosecond transient absorption spectra of a coherent atomic system, whose derivation is given in Ref. [40]. The single atom response at a probe timing is expressed as,
( , ) = 4 Im ∑︁ , ( ) ∑︁ * Δ , − − Γ /2 ,(3)
where , are the labels for the valence states, is the label for the core-excited states, is the density matrix of the targeted valence states, and Δ is the transition energy. The electronic coherence of the system, i.e., the off-diagonal density matrix element ( ≠ ), gives rise to a quantum beat with its amplitude mainly determined by a cross product of the transition dipole moments * . Specifically at the peak center = Δ , , the equation simplifies to
( = Δ , , ) = − 8 Δ , Γ Re ∑︁ , ( ) ∑︁ * .(4)
This equation shows that the choice of the probe state , be it I-4 or Br-3 core-excited states, is manifested in the following three factors: (i) the XUV absorption peak width via Γ , (ii) the XUV absorption peak strength at line center via Δ , , and (iii) the overall XUV absorption peak strength and time dependence via the cross dipole product, * . Factors (i) and (ii) only affect the static information and are not important here, whereas factor (iii) is directly related to the amplitude and phase of time-dependent spectral absorption and modulation depth that arise from charge migration.
The cross transition dipole products for the specific case of ICCBr + , where = X 2 Π 3/2 and = A 2 Π 3/2 , are calculated and shown in Fig. 2(b). The results are shown for the parallel x,y (perp.) (ΔΩ = 0) and perpendicular (ΔΩ = ±1) excitations, and different sizes and colors indicate the amplitude and phase, respectively. Note that there is an arbitrary phase offset in the cross transition dipole products and only the relative difference for different states has physical meaning. The energy structure of the probe states , i.e., the I-4 and Br-3 core-excited states, can be explained as follows [51]. The single-hole configuration of −1 gives rise to a 2 electronic state in the atomic limit. For the case of a linear molecule, the presence of the neighboring atoms, i.e. the molecular ligand fields [58][59][60], causes energy splittings for the different orbital angular momenta on the order of tens of meV, giving rise to the 2 Δ, 2 Π, and 2 Σ states. Spin-orbit coupling is critical for orbitals, causing few-hundred meV additional energy splittings, and the electronic fine structure shows five final states, 2 Δ 5/2,3/2 , 2 Π 3/2,1/2 , and 2 Σ 1/2 . The results in Fig. 2(b) show that within each elemental window the phase of the cross dipole products is constant despite the energy splittings caused by the ligand-field effects and spin-orbit coupling. Furthermore, the phases between the I-4 and Br-3 windows exhibit a phase difference, both for the parallel and perpendicular probes. This result is directly related to an intuitive picture that the multielement probe of the targeted charge migration will show out-of-phase oscillations in the I-4 and Br-3 element windows.
2 Π 3/2 2 Σ 1/2 2 Δ 5/2 2 Π 1/2 2 Δ 3/2 2 Π 3/2 2 Σ 1/2 2 Δ 5/2 2 Π 1/2 2 Δ 3/2 (b) t = t 0 t = t 0 +T/2 π phase difference
It bears mentioning that the phase difference is not always ; for example, a case of = X 2 Π 3/2 and = B 2 Π 3/2 (see Fig. 1) yields a phase difference of zero between the I-4 and Br-3 windows. This can be understood, for lower electronic states of ICCBr + , by counting the number of nodes in the molecular wavefunctions. The number of nodes along the direction of the chemical bonds is 2, 1, and 0 for the 3 , 2 , and 1 orbitals, respectively (see Fig. 1). The wavefunctions at the bromine and iodine sites are in phase for the 3 and 1 orbitals, whereas they are out of phase for the 2 orbital. Due to the localized nature of the core orbitals, this local phase information of the valence orbitals is directly encoded in the core-to-valence transition dipole moments. Returning to the case mentioned above, the transition dipole moments from the I-4 and Br-3 orbitals are both in phase for the X 2 Π 3/2 and B 2 Π 3/2 states, and thus the phase difference in the cross transition dipole products is zero.
Multistate model transient absorption spectra
By using the complex transition dipoles calculated for ICCBr + , attosecond transient absorption spectra of the charge migration in the X 2 Π 3/2 and A 2 Π 3/2 states are simulated by solving the time-dependent Schrödinger equation in a multistate model [ Fig. 3(a)]. The parallel probe is considered in the following discussion, but the same conclusions are obtained for the perpendicular probe. Time zero is defined as when the I-4 absorption is maximized, and the signals in the Br-3 window (>63 eV) are multiplied by a factor of 4 to improve the visibility.
In each elemental window, there are four absorption signals that correspond to the transitions from the X 2 Π 3/2 and A 2 Π 3/2 states to the 2 Π 3/2 and 2 Δ 3/2 core-excited states. All the signals exhibit quantum beats with a 2.2-fs period, as expected from the 1.90 eV energy difference. The beat phase can be analyzed by performing cosine-function fitting to absorption lineouts at each photon energy with a fixed period of 2.2 fs. Figure 3(b) shows the fitted amplitude (gray area, left axis) and phase (purple and red curves, right axis) of the quantum beats in the I-4 and Br-3 windows. The results show that the beat phase is not constant with XUV photon energy around the signal maxima; for example, the signal around 45.4 eV (X 2 Π 3/2 to 2 Π 3/2 ) exhibits a phase reversal versus XUV photon energy from − /2 to /2, and at the signal center the phase is almost exactly zero.
The beat phase is not constant because, in addition to the signal amplitude, the line shape also exhibits periodic modulations. Figure 3(c) shows the absorption signals in the Br-3 window for one beat cycle. It can be seen that the shape of the absorption signals is changing between Lorentz and Fano lineshapes [61,62], which is a direct outcome of the quantum interference in the core-to-valence transitions. The time and energy dependence of the line shape as well as the spectral overlap of the absorption signals make it challenging to determine the beat phase for each individual signal. Experimentally, it would be desirable to have a simple approach to determine the beat phase for each signal, instead of performing two-dimensional global fitting to the entire spectra with Eq. (3). One such approach is to only examine signals that have little to no overlap with neighboring ones. For example, the lowest signals in each element window, i.e., the X 2 Π 3/2 to 2 Π 3/2 transitions at 45.4 eV and 65.3 eV, exhibit beat phases of 0.01 and 3.16 radians, respectively, which are almost equal to 0 and . Another approach is to take a spectral average and determine one representative phase in each element window. Figure 3(c) shows the time evolution of the averaged absorption for the I-4 (43.0-50.5 eV) and Br-3 (64.0-69.5 eV) windows. This simple analysis yields a phase difference of 3.0 radians, which is close to the expected value of . In either approach, it needs to be verified that the complex dipole phases are expected to be constant within the targeted energy window by performing electronic-structure calculations.
Conclusions
We examined the role of the complex transition dipole phase in the attosecond transient absorption probe of charge migration. An example system of ICCBr + is considered, which allows for multielemental probing of electron dynamics at the molecular termini. The analytical expression for the transient absorption signals shows that the complex phase of the cross transition dipole products plays a critical role in determining the time-dependent spectral absorption and modulation timing. The ab-initio electronic-structure calculations show that the complex phases of the cross transition dipole products are constant within each of the I-4 and Br-3 windows for the X-A charge migration despite the energy splittings caused by the ligand-field effect and spin-orbit coupling. More importantly, there is a phase difference between the two element windows, which corresponds to the intuitive picture of electron density migrating between the iodine and Beat phase (rad.) Fig. 3. (a) Attosecond transient absorption spectra simulated within a multistate model including the complex dipole phase. The probe direction is parallel to the molecular axis. The initial state is a coherent superposition of the X 2 Π 3/2 and A 2 Π 3/2 states with equal weights. The signals in the Br-3 window are multiplied by a factor of 4. (b) The amplitude (gray area, left axis) and phase (purple and red curves, right axis) of the quantum beats determined by cosine-function fitting. It can be seen that the beat phase shows an abrupt change around the signal maxima. (c) Snapshots of the absorption signals from = 0 to = . In addition to the signal amplitude, the line shape also exhibits periodic modulations. (d) Average absorption in the I-4 (43.0-50.5 eV) and Br-3 (64.0-69.5 eV) windows. The average signals show a phase difference of 3.0 radians, a value close to . bromine ends of the molecule. The simulated absorption spectra show that the charge migration causes periodic modulations both in the amplitude and line shape of the XUV absorption. Spectral overlap of the absorption signals can add complexity to experimental characterization of the intrinsic beat phases, and it is suggested that using spectrally isolated signals or taking spectral averages within a targeted window can be a practical way to circumvent the problem. Our results are limited to a multistate model with frozen nuclei, and accurate simulations of the transient absorption probe of charge migration require several additional effects such as
X A 2 Π 3/2 X A 2 Δ 3/2 X A 2 Π 3/2 X A 2 Δ 3/2 (b)
Fig. 1 .
1(a) An energy diagram of valence electronic states of ICCBr + . (b-e) Calculated core-to-valence absorption spectra of the (b) X 2 Π 3/2 , (c) A 2 Π 3/2 , (d) B 2 Π 3/2 , and (e) C 2 Σ 1/2 states. The inset images show the main single-hole configuration of the respective electronic states. The site and state specificities of the core-to-valence absorption signals are shown.
Fig. 2 .
2(a) Snapshots of the charge migration that arises from a coherent superposition of the X 2 Π 3/2 and A 2 Π 3/2 states. (b) Cross products of the transition dipole moments, * . The size and color of the circles represent product magnitude and phase, respectively. The valence states are : X 2 Π 3/2 and : A 2 Π 3/2 . The probe states are the I-4 and Br-3 core-excited states. The calculated complex phases are 0.566 radians (green) and -2.575 radians (purple) in the iodine and bromine windows, respectively, thus yielding a phase difference of .
Photon energy (eV)
molecular vibrations and electron-hole correlation[13,21]. Nevertheless, it is demonstrated that a multistate model can reproduce the panoramic probe of charge migration if the complex dipole
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"Head-to-head domain walls in one-dimensional nanostructures: an extended phase diagram ranging from strips to cylindrical wires",
"Head-to-head domain walls in one-dimensional nanostructures: an extended phase diagram ranging from strips to cylindrical wires"
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| So far magnetic domain walls in one-dimensional structures have been described theoretically only in the cases of flat strips, or cylindrical structures with a compact cross-section, either square or disk. Here we describe an extended phase diagram unifying the two pictures, extensively covering the (width,thickness) space. It is derived on the basis of symmetry and phase-transition arguments, and micromagnetic simulations. A simple classification of all domain walls in two varieties is proposed on the basis of their topology: either with a combined transverse/vortex character, or of the Bloch-point type. The exact arrangement of magnetization within each variety results mostly from the need to decrease dipolar energy, giving rise to asymmetric and curling structures. Numerical evaluators are introduced to quantify curling, and scaling laws are derived analytically for some of the iso-energy lines of the phase diagram.Contents 18References 18 arXiv:1412.0679v1 [cond-mat.mes-hall] 1 Dec 2014 | 10.1016/b978-0-08-100164-6.00025-4 | [
"https://arxiv.org/pdf/1412.0679v1.pdf"
]
| 117,819,703 | 1412.0679 | fe7325d26d00ebc53ed819079a25c41c688722c8 |
Head-to-head domain walls in one-dimensional nanostructures: an extended phase diagram ranging from strips to cylindrical wires
S Jamet
Univ. Grenoble Alpes
Inst NEEL
F-38042GrenobleFrance
CNRS
Inst NEEL
F-38042GrenobleFrance
N Rougemaille
Univ. Grenoble Alpes
Inst NEEL
F-38042GrenobleFrance
CNRS
Inst NEEL
F-38042GrenobleFrance
J C Toussaint
Univ. Grenoble Alpes
Inst NEEL
F-38042GrenobleFrance
CNRS
Inst NEEL
F-38042GrenobleFrance
O Fruchart
Univ. Grenoble Alpes
Inst NEEL
F-38042GrenobleFrance
CNRS
Inst NEEL
F-38042GrenobleFrance
Head-to-head domain walls in one-dimensional nanostructures: an extended phase diagram ranging from strips to cylindrical wires
(Dated: December 3, 2014)
So far magnetic domain walls in one-dimensional structures have been described theoretically only in the cases of flat strips, or cylindrical structures with a compact cross-section, either square or disk. Here we describe an extended phase diagram unifying the two pictures, extensively covering the (width,thickness) space. It is derived on the basis of symmetry and phase-transition arguments, and micromagnetic simulations. A simple classification of all domain walls in two varieties is proposed on the basis of their topology: either with a combined transverse/vortex character, or of the Bloch-point type. The exact arrangement of magnetization within each variety results mostly from the need to decrease dipolar energy, giving rise to asymmetric and curling structures. Numerical evaluators are introduced to quantify curling, and scaling laws are derived analytically for some of the iso-energy lines of the phase diagram.Contents 18References 18 arXiv:1412.0679v1 [cond-mat.mes-hall] 1 Dec 2014
The investigation of magnetic domain walls (DWs) in one-dimensional systems is an important topic in nanomagnetism, which has been active over about twenty years. Such DWs are associated with new physics dealing with their magnetization configuration at rest [1,2], their pinning with roughness and modulations of cross-section [3][4][5], their magnetization dynamics driven by an external field [6,7], in-plane spin-polarized currents [8][9][10] or perpendicular spin currents of either tunnel or spin-Hall origin [11][12][13][14]. They have also been proposed as a basis for low-power devices with logic or memory functionalities, based on DW motion [15,16]. Low-dimensional structures also provide an opportunity to use surface-related effects such as perpendicular anisotropy and the Dzyaloshinskii-Moriya interaction etc. Here we will however consider structures with all dimensions larger than a few nanometers, so these effects may be disregarded. We will focus on low-anisotropy, soft magnetic materials, i.e. displaying so called head-to-head or tail-to-tail DWs.
One may consider two different types of one-dimensional systems. Using a top-down approach, thin films and lithography can be combined, producing mostly strips, i.e. with a rectangular and rather flat cross-section. Strips are ideal for physics and devices thanks to their versatility of fabrication and easiness of inspection. DWs in strips have been investigated theoretically and experimentally extensively over the past twenty years. In a bottom-up approach, pores are formed in, e.g., irradiated polycarbonate or anodized alumina, which can be filled by electroplating [17]. This yields wires, by which we mean with a compact cross-section such as disk or square. More complex structures such as tubes and core-shell can also be produced by atomic layer deposition [18,19] or electroplating [20,21]. There exists a recent review covering both the variety of synthesis strategies and magnetic properties of such magnetic structures. The consideration of DWs in such wires started later than for strips, typically about ten years ago, and so far mostly models and simulations are available [22]. These first considered wires and later on tubes (with an empty or non-magnetic core). Experimental results are emerging only now [23][24][25], however a rising interest is expected as wires is the natural geometry for possibly realizing a three-dimensional race-track memory [16].
Strips and wires have mostly been considered so far as different systems. Thus there has not been a global thinking about the names of DWs in these. This sometimes yields conflicting statements or naming such as the multiple names Bloch-point wall (BPW) [7,26], vortex wall (VW) [27][28][29] or pseudo-vortex wall [30] for the same object in a wire. Also, the latter should not be confused with the vortex wall in strips [1,2], which has a different shape and even topology, as we will detail later. This conflicting name of a so-called vortex wall for walls with distinct topologies in a wire versus in a strip is an issue as a wire with, e.g., a square cross-section, may be obtained continuously by increasing the thickness and/or decreasing the width of a strip. Making this connexion is not a purely theoretical consideration, as processes are emerging for producing wires with a well-defined rectangular or square cross-section [31]. Finally, due to the complexity of displaying and understanding three-dimensional configurations such as found in wires and thick strips, the description of these is sometimes not free of misconceptions, so that a simple description and classification would be useful to guide their understanding. For these two reasons it is desirable to sketch a global phase diagram of DWs in 1D systems. This is the purpose of the present work.
In the following we first propose in sec.II a short overview of the current understanding of domain walls in onedimensional systems. In sec.III we then sketch the extended phase diagram based on general considerations and symmetry arguments, and in sec.IV we refine it based on micromagnetic simulations. The simulations also allow us to discuss the key aspects of DWs in this picture, so at to deliver a simple view and means of classification in terms of a small number of features. In sec.V a few analytical scaling laws are derived to highlight the physics at play in the phase diagram. These resulting sections have been written to be readable mostly as stand-alone sections, depending on the interest of the reader.
II. A SHORT OVERVIEW OF EXISTING KNOWLEDGE
As said above, we consider low-anisotropy, soft magnetic materials and disregard surface effects. In micromagnetics these materials may be described by the sole values of their magnetization M s and exchange stiffness A. With these we define the dipolar exchange length ∆ d = 2A/µ 0 M 2 s , which can be used to scale the present results to any soft magnetic material (for Fe 20 Ni 80 , so-called permalloy, ∆ d ≈ 5 nm). Besides, in absence of effects such as exchange bias the time-reversal symmetry applies, to that the physics of head-to-head and tail-to-tail DWs are identical. In the following we will designate these walls by head-to-head, for the sake of concision.
Before addressing one-dimensional structures, let us recall the basics of DWs in three-and two-dimensional systems, i.e. in bulk and thin-film materials, still in rather soft magnetic materials. This is interesting as some phenomena arising in these, have a similarity with those that we will describe in one-dimensional systems. Besides, identical words are used in both cases such as vortex wall or asymmetry. They are used to describe phenomena sharing common features, however with a different geometry. It is important to review their use in 3d and 2d systems to avoid any x and y along transverse directions, and z along the strip length (b) Phase diagram of DWs in strips known to-date [1,2]. Labels indicate the state of lowest energy among TW, ATW and VW [2]. (c-e) Mid-height views of micromagnetic simulations of the vortex (VW), transverse (TW) and asymmetric transverse (ATW) walls. The strip width is 100 nm and thickness 16 nm for c and d, and 28 nm for e. The color codes the magnetization component along z.
confusion with the 1d case. When a sample is subdivided in several domains, be it for the sake of decreasing magnetostatic energy or because of magnetic history, DWs are present at the domain boundaries. As a general rule DWs arrange themselves to avoid the formation of magnetic charges, which would otherwise give rise to magnetostatic energy and thus to a cost of dipolar energy. In the case of 180 • DWs this results in so-called Bloch DWs, in which magnetization lies in the plane of the wall. More generally, the plane of a domain wall tends to bisect the direction of magnetization in the neighboring two domains. The component of magnetization perpendicular to the wall remains uniform from one domain to the other through the wall, while the field of perpendicular component is similar to a 180 • Bloch wall. This way the volume density of magnetic charges −divM is zero. The picture is more complex in thin films, where magnetic charges may arise at surfaces with an areal density M.n with n the outward normal to the surface. For very thin films perpendicular magnetization cannot be sustained as in a Bloch wall, and magnetization rotates in the plane over a length scale equal to the domain wall width. This is the geometry of the Néel wall [32]. In thicker films domain walls remain of Bloch type deep inside the film, while at surfaces magnetization tends to turn parallel to the surface to avoid the formation of surface charges [33,34]. These features were called later Néel caps [35], due to the profile of surface magnetization being very similar to that of Néel walls. A refinement of this picture is the spontaneous shift of Néel caps along the direction perpendicular to the domain wall plane, so as asymptotically avoid any volume charge even in the vicinity of Néel caps. This arrangement was called an asymmetric Bloch wall [34]. When the thickness of the film is reduced the Néel caps at opposite surfaces become close one to another, so that the cross-section of the domain wall looks like a swirl of magnetization. This feature, taking place to close as much as possible the magnetic flux, has been named a vortex wall [36]. It should not be confused with the so-called vortex wall occurring in thin strips. The history of domain and domain wall structure resulting from the principle of pole avoidance was recently reviewed by A. S. Arrott [37].
Domain walls in strips are known to take two forms: the transverse wall (TW) and the VW, as initially described by McMichael and M. Donahue [1]. In both cases magnetization remains mostly in-plane. In the TW magnetization inside the wall is directed along the in-plane direction transverse to the strip (Fig. 1d). Its shape is roughly triangular, reminiscent of the formation of 90 • walls in thin films to avoid having locally net charges, associated with a longrange magnetostatic energy. In a VW the magnetization is curling within the plane around a small area with perpendicular magnetization called the vortex core, of diameter ≈ 3∆ d (Fig. 1c) [38]. Examination of the detailed magnetic microstructure reveals again 90 • sub-walls. Both types of DWs have the same total charge 2twM s , with t and w the strip thickness and width, respectively. Simulations and magnetic force microscopy provide the picture of a rather uniform distribution of these charges over the entire area of the wall [39], an argument which we will use in the scaling laws section. TWs and VWs have been investigated numerically up to w = 500 nm and t = 20 nm. The TW is of lower energy than the VW for tw 61∆ 2 d , and of higher energy for larger width or thickness (Fig. 1b) [2]. Both DWs persist as a metastable state in a large part of the ground-state domain of the other DW. In experiments it is often the metastable TW which is observed in the region of stable VW, following its initial stability during preparation under a transverse applied field [39,40]. A refinement revealed later is that the TW undergoes a transition towards large and wide strips, from a symmetric shape to an asymmetric one (Fig. 1e). The resulting wall was named an asymmetric transverse wall, which we will write ATW [2].
Letting aside early numerical approaches [41,42], DWs have been investigated thoroughly in cylindrical structures only later, pioneered by R. Hertel [43,44] and H. Forster [27,45]. Two types of DWs were predicted. The first type of DW was named TW in analogy with the case of strips, because its core displayed a significant component of magnetization along a direction transverse to the wire ( Fig. 2a;c). It is again the most stable form for low lateral dimension, either diameter for disk section or side for square section, typically below ≈ 7∆ d [7,27,28]. The main feature of the second type of DW is again a curling of magnetization as for a VW in a strip. However in the wire case the curling occurs around the axis, allowing a three-dimensional flux-closure ( Fig. 2b;d). This curling is made possible in a wire, compared to a strip, because exchange energy is not prohibitively costly along the thickness due to the somewhat large dimension. This orthoradial curling makes it impossible to sustain a radial component of magnetization on the wire axis. Besides, as the head-to-head nature of the DW forbids sustaining a longitudinal component either in the core of the DW, there must exist a point where the magnitude of the magnetization vector vanishes. This quite unusual object is called a Bloch point, predicted and described in the early days of micromagnetism ( Fig. 2e;f) [46][47][48][49]. Its reality is accepted although it has not been imaged directly due to its very small size; only observing extended boundary conditions allows to infer its existence and monitor its dynamics [48,50,51]. Due to the curling this domain wall was initially named a VW, a name still in wide use today. The name Bloch-point wall (BPW) was proposed later by A. Thiaville and Y. Nakatani [7] to avoid the confusion with the strip case. Some other authors call it a pseudo-VW, stressing that this name is intended to avoid confusion with a VW in a strip [30]. In this manuscript we will stick to the name BPW for this reason. Finally, DWs were later studied numerically and analytically in tubes [52][53][54]. The physics is very similar to the case of wires, with the essential difference of the removal of a core of magnetization along the axis, so that the equivalent of a BPW consists mostly of an orthoradial curling, with no Bloch point. It was also mentioned that TWs undergo a distortion upon increasing the wire diameter. This effect has been named helical domain wall [55] or pseudo transverse wall [23] depending on the authors.
In the above we introduced the terms circulation and curling. These are interchangeable, the latter bearing the meaning of the curl operator and related to circulation of magnetization along a closed path (Stokes theorem). The name curling has been used for a long time in magnetism to describe structures with such circulation, introduced in nucleation theory as what is now known as the curling model [56]. Notice that curling is not necessarily a transient feature during magnetization switching: curling structures are known to be relevant for magnetization textures at rest [57]. The notion of curling is more general than that of vortex, which often has a meaning related to a particular distribution of magnetization such as for the VW in a flat strip. Also, the name vortex gives more importance to its core structure, such as in fluid swirls. In the course of this manuscript we will encounter weakly curling structures, for which there exists no well-defined core. Going along this line, we could use vortex state to describe the flux-closure in a flat disk, however curling would be more suited for a ring. Similarly, in a nanotube we could say longitudinal curling wall for the equivalent of Bloch-point wall for a wire, as the core is removed.
III. SKETCHING THE PHASE DIAGRAM
A. Preliminary discussion
It is not our idea to deliver a comprehensive view of all details of domain walls in one-dimensional structures. The first reason is that this would be a task formidable and difficult to summarize, due to the large variety of geometries to be considered, e.g., with cross-sections of type rectangular, disk and elliptical, hollow (for tubes) etc. The second reason is that many fine micromagnetic features appear as the size of a system is increased, such as edge and corner effects, configurational anisotropy [58,59]; the task would be even more formidable. In fact, not all these features are crucial to understand the energetics and possibly the dynamics of the walls. Simplifying the picture by selecting the most important features is therefore desirable. For instance, it should be of use to understand and analyze outputs of micromagnetic simulations in simple words, whereas they are often difficult to display and discuss because of their possibly three-dimensional character.
Most of the concepts discussed and simulations reported here, consider one-dimensional structures with a rectangular cross-section. This choice stems from the whish to span continuously the panorama from strips to wires with an experimentally-relevant geometry. Nevertheless, figures and features are also discussed for a circular cross-section, which so far is the most relevant geometry for wires. As considering domain walls in tubes instead of wires was shown to change numbers however not the general texture of the walls [52][53][54], the case of tubes could probably be extrapolated from the present work, missing only the precise position of the various features in the diagram.
We will build a phase diagram consisting of lines defining regions where such or such domain wall is stable, metastable or does not exist. We will name first-order those lines separating two different states, each existing and being stable or metastable on either side of the line. For instance, the well-known iso-energy line between the TW and the VW in strips is of first order. On graphs these will be depicted with a bold line, full for separating the two states of lowest energy, dotted when the ground state is not one of the two states considered. We will name second-order those lines associated with the continuous rise of an internal degree of freedom in a DW (characterized by an ordering parameter), associated with a breaking of symmetry. In this case only one type of DW exists on either side, and there is no metastability. For instance, this is the case in strips for the transition of the TW to the ATW. Their features have been described in detail for magnetic systems [57]. On graphs these will be depicted with thin lines, full or dotted with the same meaning as indicated above. The features expected for a transition depending on its order are summarized in TABLE I.
Due to the symmetry upon exchanging t and w, any line has its counterpart symmetric with respect to the diagonal t = w, be it of first or second order. In particular, if a line is itself not symmetric with respect to the diagonal, and therefore is not perpendicular to it at the intercept, this implies the existence of two distinct lines intercepting at the same point on the diagonal, related to configurations that can be obtained one from another through a rotation of π/2 around the axis. We will call lower triangle the part for t ≤ w, and upper triangle its symmetric counterpart for t ≥ w. x and y are coordinates along the directions transverse to the wire, while z is used along the wire (Fig. 1a).
Finally it has been said that we restrict the discussion to soft magnetic materials. The discussion and calculations are not material-dependant, once scaled with the dipolar exchange length ∆ d . In the following we use general arguments to derive a simplified sketch for the phase diagram, such as symmetry, phase transition of first or second order, topology and scaling laws (the detail of the latter being found in sec.V). In sec.IV we use micromagnetic simulation to give a firmer basis to the sketch, and refine quantitatively its various elements. We also define and calculate estimators for a few quantities characterizing the internal degrees of freedom, and which may serve as ordering parameters when describing second-order transitions.
B. Transverse versus vortex walls
Let us examine in more detail the structure of the TW and VW in strips. In both cases a tube of magnetization goes through the strip: from edge to edge along x for the TW, from top to bottom along y for the VW. Thus, as a view of mind it is possible through a continuous deformation of the magnetic texture to transform a TW into a VW, following a path in the phase space. In other words, these two DWs share the same topology. Considering a TW and a VW for a given rectangular cross-section, let us imagine that we continuously change the geometry of the strip towards a square cross-section. When t = w the two types of walls should be degenerate and be obtained one from another through a rotation of π/2 around the wire axis. This is illustrated on Fig. 3, based on micromagnetic simulations. These show that a wall for a square cross-section indeed displays both a transverse feature (a flux of magnetization from one side to the opposite one, Fig. 3d) and a vortex feature (circulation of magnetization around the transverse component, Fig. 3c), in the common sense used in strips for a TW and a VW. This is analogous to a liquid-gas transition, which determines a line of first-order transition, however where a path may be found to go continuously from one to the other around a critical point (here: through the critical line t = w). We therefore propose to name this family of walls the transverse-vortex wall, or mixed transverse-vortex wall (TVW). The similarity of both types of walls has already been outlined by some authors, stating e.g. the vertical Bloch line is probably the one that represents closest similarity with a transverse domain wall [60]. Of course, when one feature is dominating the other and no confusion is possible, restricting the name to TW or VW is desirable for clarity. We may also add information when necessary to avoid confusion, about the direction of the flux of magnetization: a x-TW when the transverse component is along x as in flat strips, or a y-TW (or y-TVW) when the transverse component is along y, as may be considered for small cross-section ( Fig. 3a-b). In this case there is little variation of the direction of magnetization within a cross-section, which make it a near realization of the 1d model [7].
Let us come back to the known first-order line separating TVWs in strip with transverse component along x and y respectively, i.e. with predominantly transverse (TW) and vortex (VW) feature. This is depicted on Fig. 4a. In the lower triangle usually displayed, x-TW are the most stable in region 1, and y-VW are in region 2. Considering points symmetric when swapping t with w, a mostly x-TW is transformed into a mostly y-TW, and it is straightforward that this line is symmetric with respect to the diagonal. There is one single TW/VW line in the diagram, and it crosses the diagonal at right angle at a point we will name D tw . In region 3 it is the y-TW which is of lowest energy, and in region 4 it is the x-VW. This may be checked rigorously with the following argument. Let us name ∆E(t, w) the difference of energy between the x-TW and the y-VW at a given point in the phase space. ∆E is of class C 2 for (t, w) > 0 so that we may consider its Taylor expansion around D tw . ∆E remains zero along both the diagonal and the iso-energy line, while being non-zero along other directions pointing into stability/metastability regions, which from the expansion requires that both lines are perpendicular. Note that the symmetry with respect to the diagonal is fortunately consistent with the phenomenological law wt ≈ Cte already reported [1,2] (see also the scaling law in sec.V).
C. Asymmetries
The following discussion is illustrated on Fig. 4b. In the preliminary discussion we mentioned the second-order transition from a x-TW (labeled 1) to an x-ATW (labeled 2) upon increase of, e.g., the thickness of a strip. This asymmetry towards a slanted wall is reminiscent for the zigzag domain walls found in extended thin films with uniaxial anisotropy, as already pointed out in [2]. The asymmetry is characterized by the fact that the locus of entry and outlet of the flux of magnetization on either edge of the strip are shifted along the length of the strip (Fig. 1e). Let us assume that this second-order line exists for a broad range of geometries, and we follow it towards the diagonal and beyond, in the upper left triangle. In this region the broad and narrow dimensions of the rectangular section are interchanged with respect to its symmetric lower-right triangular region, so that starting from a DW with an x-through-flux along the long transverse direction and thus with a dominant transverse character, we end up with a DW still with a x-through-flux however now along the short transverse direction and thus with a dominant x-VW character. The tube of flux is then entering from one flat surface, which may be called bottom, and exiting from the other flat surface, say top. Swapped back into the lower triangle, this is a y-VW wall, labeled 3. Introducing an asymmetry between these two entry/outlet locus transforms the vortex into a Bloch wall of finite length terminated by a surface vortex at either of its ends, labeled 4. This feature of topology is well known for finite-length Bloch walls [61][62][63][64]. This occurrence of asymmetry is thus related to the physics of the transition from a Néel wall to a Bloch wall upon, e.g., applying a transverse magnetic field [36,63,65]. In thin films, it has also been recognized to be of second order [66].
As the diagonal t = w plays a priori no role for this line of second-order transition, there is no reason why it should be perpendicular to it. Its symmetric with respect to the diagonal should therefore be a distinct line, which shall
a) b) c) w t T V W y-TW T W x y y x
x-TW be also added to the phase diagram. In this view of mind, on one side of the diagonal it pertains to the TW/ATW transition, while on the other side it pertains to the transition from Néel or vortex, to the Bloch wall. When wrapped, these two lines are found one above the other on each side, and intercepting the diagonal at the same point (Fig. 4b).
If we focus again on the lower triangle (w > t), the evolution with thickness of the magnetic microstructure of lowest energy is now the following: for low thickness we shall consider TWs (label 1), then for increasing thickness: possibly ATWs (label 2), VWs, and finally the transition from vortex to Bloch wall (labels 3 to 4). This latest transition has been described in detail recently [67]. The occurrence of the Bloch wall allows to increase the length of the entire head-to-head wall by giving it an internal structure similar to a tilted Landau flux-closure pattern (Fig. 6e,f). This again is a clear illustration of the similarity of VW and TW. The Landau-type DW has been described above as an evolution of the VW (Fig. 6c,d). Similarly, it can be viewed as the evolution of an ATW (Fig. 6a,b), the in-plane transverse part having initially a Néel type, being converted to a Bloch type due to the large thickness. In this phase diagram, asymmetric walls are expected for t = w beyond D atw . We will see that micromagnetic simulations will refine this picture.
D. Bloch-point walls
The situation of BPWs versus TVWs is depicted on Fig. 4c. So far the BPW has been discussed in the literature only for the geometry of wires, i.e. for t = w. It was predicted for a disk [27,43] as well as a square [7] cross-section. The experimental confirmation of it existence was only provided recently, based on a disk cross-section [25]. The TVW and the BPW have a different topology. They are separated by an energy barrier and are (meta)stable over a large range of diameters. Along the diagonal they are thus associated with a first-order transition at a point that we will call D bp (Fig. 4c), previously determined to be located at w = t ≈ 7∆ d . Moving along the diagonal from D bp towards larger diameters, the energy of the BPW becomes lower than that of the TVW [7]. As the energy of domain walls is continuous as a function of strip dimensions, this implies that there should exist a region on either side of the diagonal where the BPW is the wall of lowest energy, associated with first-order transition lines that we outline in more details below.
Outside the diagonal a x-TVW (towards a dominant TW feature in the lower triangle) and a y-TVW (towards a dominant VW feature in the lower triangle) have a different energy, so that their must exist two distinct firstorder lines: one for the x-TW-BPW transition (could be called TW-BPW in the lower triangle), another one for the y-TVW-BPW transition (could be called VW-BPW in the lower triangle). On the diagonal the TW and VW are degenerate, so both lines should intercept there, at D bp . As regards this intercept, noticing that upon crossing the diagonal a VW becomes a TW and vice versa, the two lines are mirror one to another with respect to the diagonal. As the energy difference between the TVW and the BPW are anywhere differentiable with class C 2 , no kink is expected at D. Therefore, the two curves display complementary angles with the diagonal. Appending these curves on the phase diagram concerning transitions in the family of TVW requires a further discussion. If D bp were found for a larger diameter than D tw , it is probable that none of these curves intercept the TW/VW first-order transition line. Consequently in the lower triangle it would always be the y-VW which is of lower energy under the TVW/BPW lines, so that the one relevant for determining the ground state would be the VW/BPW one (Fig. 5). Conversely if D bp is found for a smaller diameter than D tw , then the relevant line for determining the ground state is a portion of the TW/BPW one for small width, and the VW/BPW for larger width (Fig. 5). Micromagnetic simulations will show that this is the relevant situation. In that case, notice that the two iso-energy lines intercept at a triple point the TW/VW line: when a TW and a VW have the same energy, their energy difference with that of a BPW is the same, in particular when it is vanishing. To the contrary, there is no reason why D bp and D tw should be identical. On Fig. 4c the energy of the various domain walls is ordered the following way: TW<VW<BPW in region 1, TW<BPW<VW in region 2, VW<TW<BPW in region 3, VW<BPW<TW in region 4, BPW<TW<VW in region 5, and BPW<VW<TW in region 6.
IV. MICROMAGNETIC SIMULATIONS
A. Methods
Micromagnetic simulations have been performed with the parameters of Permalloy Ni 80 Fe 20 : µ 0 M s = 1.0053 T for the spontaneous magnetization, A = 10 pJ/m for the exchange stiffness, and zero magnetocrystalline anisotropy. The resulting dipolar exchange length is ∆ d = 2A/µ 0 M 2 s ≈ 5 nm. Although directly applicable to Py, our simulations are relevant to any other magnetic material with no magnetocrystalline anisotropy, scaling all lengths with ∆ d .
We used two micromagnetic codes. The first code is the finite differences OOMMF[68]. It has been used to cover the phase diagram continuously from the case of flat strips with a rectangular cross-section, to wires with a square cross-section. The cell size was 1 × 1 × 2 nm 3 for strips of width smaller than 60 nm, 2 × 2 × 2 nm 3 for strips with higher widths up to 100 nm, and 4 × 4 × 2 nm 3 above. Damping was set to 1 to speed up convergence, with no consequence on the result as we are interested in equilibrium states. Magnetic moments at the extremities of the strips are fixed to avoid the formation of end domains, and the length of the strips is chosen such that the aspect ratio is at least 10. The second code is feellgood, a home-built code based on a finite element scheme, i.e. using tetrahedra to discretize matter. It is thus better suited to describe curved boundaries, and was used to investigate wires. Both circular and square cross-sections were considered. The size of tetrahedra was about 4 nm. The Landau-Lifshitz-Gilbert equations are integrated using an original semi-implicit scheme which is consistent up to order two in time and unconditionally stable. It combines a linear inner iteration with a renormalization stage for the nodal magnetization for which a rigorous proof of convergence was established [69,70]. The computation of the magnetostatic interactions constitutes the major bottleneck in the efficiency of micromagnetic codes. The solution adopted here uses a time-optimized and parallelized version of the Fast Multiple Method running on multi-core conventional processors [71,72]. The computing time devoted to magnetostatic interactions then becomes comparable to the time required for the other steps in the calculation.
B. First-order transitions
Simulations were first used to test the existence and refine the location of the first-order equilibrium lines (Fig. 5). Series of simulations of the two states to be compared were performed for a given strip width (along x) and variable thickness (along y), and the thickness-dependent energy was fitted with a second-order polynomial. For first-order transitions (TW/VW, TW/BPW and VW/BPW) states of either type are (meta)stable on both sides of the equilibrium line, and at the transition their energy curves cross each other with different slopes. Thus, an accurate determination of the crossing point is straightforward.
The VW/TW iso-energy line was already known for w ≥ 80 nm, although computed mostly in a 2D micromagnetic scheme so far [1,2]. Our 3D calculations essentially confirm this data, and extend the line towards the diagonal. We confirm the intercept of the latter at right angle derived from symmetry arguments, and this for a square side ≈ 43 nm, or ≈ 8.6∆ d . This shows that the phenomenological scaling law tw 61∆ 2 d , derived in the range of width 80−500 nm, remains of high accuracy up to the diagonal.
So far BPWs had only been considered for wires. The iso-energy points had been determined to be ≈ 6.2∆ d for disk [27] and ≈ 7.0∆ d for square [7] cross-sections. We confirm the magnitude of ≈ 7.0∆ d for the transition in square cross-section, and slightly lower for the disk cross-section. Fig. 7 shows the energy of the two types of walls for both types of cross-sections, plotting the energy normalized to the section area. This normalization, following Ref. [7], has the advantage of yielding a quantity converging to a finite value towards low diameter. From this graphs it is clear that the stability of BPWs with respect to TVWs is enhanced in wires with a disk cross-sections, compared to those with a square cross section. The reason for this will be discussed in the next section. Besides this, our simulations confirm that for larger dimensions the BPW remains stable in a significant range of anisotropic cross-sections, over both walls with a dominant TW or VW feature, i.e. cores along the long or short transverse dimension. For instance, for w = 100 nm the BPW remains of lower energy than the VW in the range t ∈ [50−100 ]nm. Finally, the existence of two distinct lines for TW/BPW and VW/BPW transitions is confirmed, with a triple point when meeting the VW/TW transition line.
C. Second-order transitions and estimators for their parameter
For first-order transitions the two states considered were separated by an energy barrier of finite height, and differed in orientation or even in topology. To the contrary, across a second-order transition a symmetric state continuously develops a feature breaking its symmetry. There are two possibilities to determine the transition in this case. The first one is based on energies. The difference between the energy of one state with the extrapolation of the other one (extrapolation because the two states do not exist at the same spot in phase space), is expected to scale with (t − t c ) 1/2 in the Landau theory with m 2 and m 4 terms; t c is the thickness for transition and m an order parameter characterizing the magnitude of the breaking of symmetry. This 1/2 exponent was already reported for second -order transition in micromagnetics, despite the complexity of the system considered [57]. One may then fit with parabola both branches. These parabola should intercept and share the same slope at the transition, so that the latter cannot be determined easily. In practice, closely-spaced data points and a high accuracy in the numerics are required. Second, it is possible to monitor the order parameter m. Extrapolating it towards zero provides the locus of the transition. An exponent 1/2 is also expected from the Landau theory with m 2 and m 4 terms. Such an exponent has already been reported [57,73,74], while other exponents such as 1/6 seem to be possible [2]. We followed the procedure based on energy.
The first case of breaking of symmetry is a one-dimensional asymmetry, such as for the TW-ATW transition for rather thin strips. The examination of energy around the transition confirms that it is of second order. Previously, the transition line had been outlined for strip widths 180 nm and higher [2]. The present simulations extended it for smaller widths, and show that it rises faster than the TW-VW first-order line, and even cross it around w = 115 nm. This will be explained with simple arguments with a scaling law (sec.V).
Simulations confirm that a breaking of symmetry also occurs for thick strips, where the vortex at the center of the VW is transformed into a Bloch wall of finite length (Fig. 6e,f). For strip width 100 nm or higher this line is essentially flat, and located at t = 57.5 ± 2.5 nm. In sec.III we argued that this line may be viewed as the symmetric of the prolongation of the TW/ATW, and that they may intercept on the diagonal. We searched along the diagonal when a TVW may become asymmetric, with entry and exit points of the magnetization flux at different longitudinal positions. Such configurations have not been found, neither for square nor disk cross-sections, and this up to side or diameter equal to 140 nm. The reason why no TW asymmetry is present for wires may be because a more efficient way of reducing magnetostatic energy is developed before, as described below.
We now turn to a third type of second-order transition, which had been described previously in the context of near-single-domain particles [57], however not in the context of domain walls. The well-known VWs in flat strips are characterized by the curling of magnetization around the y-transverse component, i.e. the core of the vortex. In a system of very small size this curling cannot occur, because it would have a prohibitive cost in exchange. Accordingly, it is known and we confirm that for wires of small diameter the magnetization profile of a TW is essentially onedimensional (Fig. 3a,b). For larger diameter curling may be achieved through the continuous orthoradial deformation of an initially symmetric volume, either clockwise or anti-clockwise, and as such may give rise to a second-order transition. Circulation may be proposed as an order parameter, for example any component of the quantity curl m. When integrated on a disk cross-section and normalized with d, (curl m).u z should equal π for a perfectly orthoradial vector field, i.e. close to the situation found at the center of a BPW. The three components of the z-resolved estimator for a TVW in a disk-based wire are plotted against its diameter in (Fig. 8)a, where the transverse component y is chosen to be the azimuth of the core of the TVW: this is a y-TVW. Transverse curling adds a vortex character to the domain wall as seen from (curl m).u y . This DW has thus well-developed transverse and vortex features, visible depending on the cross-section examined (Fig. 3c,d and Fig. 8d-f). When integrated in absolute value over the length of the wire to avoid cancellation (see Fig. 8a), and normalized with d 2 , a curling quantity again of the order of π is expected for a wall of length d. The exact value depends on both the strength of curling inside the wall, and the wall width. Plotting it versus wire diameter allows to investigate the transition (Fig. 8b). Starting from zero for small diameter, the integrated circulation grows rapidly beyond d = 36 nm (≈ 7∆ d ) consistent with a second-order transition, and this simultaneously along the longitudinal and y transverse directions. The integrated x curling is non-zero for small diameter and starting from roughly π/2, because the layer-resolved curling is not zero: each side of the wall having the shape of a quarter vortex whose core would be at the wire side (see Fig. 3b,d). Despite this non-zero background the superimposed second-order-like rise is clearly visible. Finally, while longitudinal curling keeps rising due to the increasing wall length, as discussed later, both transverse components tend to a value close to π, consistent with the picture of a vortex wall. Going back to strips, ≈ 7∆ d is a threshold of the x (resp. y) dimension over which a y-TW (resp. x-TW) transforms into a y-VW (resp: x-TW). These lines x = 7∆ d and t = 7∆ d are omitted for clarity on Fig. 5.
The rise of chirality can be understood as a means to lower magnetostatic energy. The transverse curling around the core of an initially TW is similar to curling in a near-single-domain particle. The driving force for transverse curling is to reduce the magnetic charges on the sides of the wire, associated with the dipolar character of the core of the TW. Transverse curling may also be viewed as putting together two ATWs of opposite asymmetry on either side of the transverse component. This is clear when considering maps of magnetization at the surface of square wires (Fig. 9a). Absence of mirror symmetry is clear on these maps, although it is expressed through curling rather than through the entry and outlet of the flux of magnetization. By this process volume and surface charges are driven more apart one from another, further decreasing magnetostatic energy. This is why, whenever possible, transverse curling takes over asymmetry of entry/outlet points of the magnetization. As regards longitudinal curling (curl m).u z , while it is a well-recognized feature of the BPW, its relevance is less obvious for TWs at first sight, although is has been reported and sometimes called a helical domain wall [55]. In fact, the physics is the same for both types of DWs: the longitudinal curling allows a progressive longitudinal variation of m z from one domain to another, in this manner spreading the volume magnetic charges −∂m z /∂z and thus decreasing magnetostatic energy. Longitudinal curling is accompanied by an increase of the DW width L, as found in the simulations and well reproduced by a simple scaling law L ∼ d 2 , see calculations in sec.V and Fig. 7b. On this figure we computed the DW width as L = +∞ −∞ sin 2 θ dz following Jakubovics definition apart from a factor 2 [75]. Notice that in disk wires with large diameter TVWs and BPWs have a very similar width, as well as similar z-resolved longitudinal curling (Fig. 8a, f-g), confirming the common ingredient of monopolar magnetostatics and curling, independent from the DW internal structure. The variety of directions of circulation motivates our use of the word curling, which had been introduced in this context [56], and is less ambiguous than the word vortex, connected to the existence of a core. We finally stress that what we consider here is the geometrical width under static conditions. While the width is often invoked as a proportionality factor determining DW mobility under applied field, it is a dynamic width that shall be considered in that case, which often differs from the static width discussed, and depends very much on the internal structure of the wall.
Finally, fine points about curling and symmetry are the following. First, in TVWs longitudinal and transverse curling compete with each other and lead to frustration, since the direction of magnetization on either side of the wire needs to be opposite for longitudinal curling, and be the same for transverse curling. This is the reason for the distorted feature of the TVW upon the rise of curling (although the entry/exit fluxes remain symmetric) (Fig. 8c).
In practice transverse curling, characterized by antisymmetry with respect to a plane perpendicular to the wire axis, combined with the natural dihedron shape of a TW, imposes that longitudinal curling is of opposite sign on either sign of the core of the TVW (Fig. 8a,d-f). This frustration is accommodated different ways depending on the wire size ( Fig. 8d-e), or whether of square or disk cross-section ( Fig. 8e-f). In the first case transverse curling is maintained and longitudinal curling yields, while it is the opposite for disk-based sections. This remark is connected to the fact that curling in square cross-sections, especially of longitudinal type, induces an extra cost of energy due to the sharp edges. This decreases the efficiency of curling to lower magnetostatic energy, and thus makes both types of DWs more costly at large diameter where the DW length increases like d 2 . This explains why the wall width is smaller for square cross-sections compared to disks, for a given area of cross-section (Fig. 7b). It particularly increases the energy of the BPW where longitudinal curling is the strongest, making it barely favorable against the TVW in square wires (Fig. 7d). This induces the occurrence of an asymmetric BPW (ABPW), which does not exist in wires with a disk cross-section (Fig. 9d). Again, the surface maps have the same shape as ATW in flat strips. For the same reason of the cost associated with curling, ATWs may be formed in wires with square cross-section, while they do not with those of disk cross-section (not shown here). These ATWs then prevail over symmetric TVW until above 45 nm. Yet another fine feature related to magnetostatics is the rise of a slight backward longitudinal component around a BPW for disk cross-section at large diameter (Fig. 9c). These are the consequence of the distribution of head-to-head magnetic charge centered on the wall, and its principle is similar to concertina features is soft planar magnetic elements. These head-to-head charges are also responsible for the outward radial tilt of magnetization in the longitudinal curling [7] of both TVWs and BPW (see m.n maps on Fig. 9b
D. Fine micromagnetic features
The above description covers the main features of domain walls in one-dimensional systems. Nevertheless finer features may occur, mostly due to the existence of flat surfaces and sharp edges. These features occur predominantly for compact cross-sections with t ≈ w. The resulting increased number of possible states makes the phase diagram more complex to describe close to the diagonal, which explains that we deliberately discontinued some lines in the vicinity of the diagonal (Fig. 5). Besides, the transition from one type to another type of subvariety is associated with minute changes of energy for difference widths, which nonetheless are responsible for the somewhat irregular shape of some curves, e.g. the BPW/TW one.
The finer features may be of two types, shortly described below.
In strips of significant thickness the cost of magnetization perpendicular to an edge becomes prohibitive. Thus, for a VW magnetization tends to be perpendicular to the strip at the two such locus at the strip sides, for strip width larger than a few times ∆ d . These locus are therefore sometimes described as half antivortices [76,77], owing to the distribution of in-plane magnetization, and the existence of a vertically-magnetized core. The question arising, is whether the core of the half antivortices shall be parallel or antiparallel to the vortex core. Various varieties may be found as (meta)stable states for the same geometry, and reached depending on the initial computing conditions: antiparallel, parallel or even one anti-vortex in each direction (Fig. 10a-c). For large and thin strips, i.e. when the DW has a predominant VW feature, the state with lowest energy tend to be with anti-vortices antiparallel to the core of the vortex. The driving force for this is minimization of magnetostatic coupling, while exchange is negligible. For narrower or thicker strips, i.e. when the TW feature of the DW is growing, the state of lower energy tends to be with anti-vortices parallel to the core of the domain wall. The driving force is that by gaining features of a TW, the core grows in size and tends to form a continuous sheet of transverse magnetization, in which switching the direction of magnetization would imply a high cost in exchange energy.
At sharp edges of a 3D magnetic element the density of magnetostatic energy diverges logarithmically [78]. Deviations from uniform magnetization lift the degeneracy expected from demagnetization coefficients calculated for a uniformly-magnetized system. Magnetization prefers to point along the bisector or perpendicular to it, both possibilities being separated by an energy barrier. This has been recognized in near single domain bulk [79] and thin film elements [58] to give rise to so-called leaf or flower states. The same physics may occur, applied to the transverse core of a TVW. The flux of magnetization may cross the strip from one side to the opposite side, or from one edge to the opposite edge, or as an intermediate situation from one side to one of the opposite edges (Fig. 10d-f). The trend seems that the state of lowest energy if the side-to-side configuration for smaller lateral dimensions, and the edge-to-edge configuration for larger lateral dimensions. It has not been attempted to provide a full picture of the phenomenon, due to the large number of states involved. Indeed, this issues holds for both VW and TW, giving rise to numerous sub-varieties.
V. ANALYTICAL SCALING LAWS
In this section we derive scaling laws pertaining to a few aspects of the phase diagram. The scaling laws are not intended to be rigorous nor numerically accurate, however to provide trends and physical insights to the micromagnetic simulations. In the following we will write K d = µ 0 M 2 s /2 the dipolar constant.
A. Transverse versus vortex walls in flat strips
Here we consider the TW/VW iso-energy line in a flat strip already reported, for which a phenomenological law was mentioned in previous works [1,2]. First, we notice on the simulation results that the length of the π/2 sub-walls is identical in both cases. Their energy can therefore be disregarded for the difference. As regards magnetostatic energy, MFM shows that the total head-to-head charge Q = 2M s tw is spread over an area S VW = 2w 2 for the VW and S TW = w 2 for the TW [39]. The areal density of charge σ = Q/S amounts to σ VW = M s t/w and σ TW = 2M s t/w. The resulting magnetostatic energy can be estimated as (1/2)µ 0 H 2 d integrated over area S and height w (used as a cut-off) on either side of the strip, with H d ≈ σ/2. This yields E d,VW = K d t 2 w and E d,TW = 2K d t 2 w. This shows that the TW has a higher magnetostatic energy of head-to-head origin, however the VW also holds extra exchange and magnetostatic energy, related to the vortex core and its surroundings. As the core has a diameter around 3∆ d [38], its energy itself is of the order of 2π(3∆ d /2) 2 K d t, taking into account an equipartition of energy of exchange with dipolar with the front factor 2. Finally, exchange energy integrated from around the core to the strip edge may be estimated at 2πtA ln(2w/3∆ d ) based on the cut-off radius 3∆ d /2. Putting everything together and substituting the numerical value 2 to 3 for the logarithm as valid for the range of geometries studied in [2], one finds the following equation for the iso-energy line:
tw ∼ ∆ 2 d (1)
This provides the good scaling law proposed in the previous reports, with a surprisingly good numerical value (≈ 30) compared to the one fitted to the simulation data (≈ 61).
B. Transverse versus asymmetric transverse walls in flat strips
Examination of TWs and ATWs in strips shows that their difference is essentially restricted close to the edge of the strip, on the large side of the mostly triangular transverse component (upper side on Fig. 1d-e). We therefore focus on this area to evaluate the difference of energy between the two walls. In the TW magnetization is perpendicular to the edge over an edge length of the order of w. Considering the edge as a plane with a charge density M s , implies a magnetostatic cost ∆E d ∼ K d wt 2 , based on arguments similar to those derived in the previous paragraphs. The cost associated with the ATW comes from several aspects. The progressive rotation of magnetization from the center of the strip to the edge implies a cost of density of exchange energy ∼ A/w 2 . Considered over part of the width w, over thickness t and edge length ∼ w implies a total cost ∼ At. Another cost is associated with the pinch of magnetization flux at one corner of the triangle. The associated exchange energy scales with At, while the associated magnetostatic energy may be estimated from that of a sphere with an internal charge Q ≈ π∆ d tM s , π∆ d being used as a cutoff for the concentration of flux. This localized charge becomes increasingly clear for larger thickness (Fig. 6a,b), and is also evidenced with MFM [39]. The magnetostatics of the charged sphere implies a cost ∼ t 2 ∆ d K d , with a radius set to ∆ d . Using this cutoff rather that thickness is required to fit the cutoff ∆ d introduced to estimate the charge within the sphere. In the end, ∆E ATW ∼ At + t 2 ∆ d K d , omitting numerical factors. Equating ∆E TW and ∆E ATW provides the following scaling law:
(w − w 0 )t ∼ ∆ 2 d(2)
with w 0 ∼ ∆ d . For large width this law agrees well with the known phase diagram (Fig. 1.) The offset w 0 explains why the ATW/TW curve, although well below the TW/VW one for large diameter, seemed to cross it towards smaller diameters. Our numerical results confirm this crossing and the then rapid increase in thickness (Fig. 5). It is also consistent with the fact that this line is not expected to reach the y axis, based on topology and symmetry arguments (Fig. 4b).
C. Length and energy of domain walls in cylindrical wires
We consider a domain wall in wires with a disk cross-section, and are interested in its energy and geometrical width at rest. We search for scaling laws and not numerical coefficients, so that the exact definition of DW width is not important (Lilley[80], Jakubovics [75], Thiele [81] or other). It was noticed by Nakatani et al. that for very small radius (R A/K d ) the 1d model is a good approximation for the DW energy and width, using as anisotropy energy K d /2, related to the N = 1/2 demagnetizing coefficient across the wire [2]. A refined model was proposed later to derive the diameter dependence of this law, based on a variational model [60]. However this model is valid only for small diameters as it disregards the monopolar charge of the DW. Here we propose a scaling law for large radius. The total head-to-head charge in the DW is Q = 2πR 2 M s . Assuming that this charge is uniformly distributed in a spherical volume of radius R the associated total magnetostatic energy E d is expected to scale like µ 0 Q 2 /R, so:
E d ∼ K d R 3 .
As the integrated exchange energy of a non-uniform state in such a volume scales like R [46,82], this shows that very rapidly magnetostatic energy becomes the dominant term upon increasing R. To decrease E d , the width of the wall (i.e., along the wire axis) will tend to increase. In practice a balance will be found with exchange energy associated with this stretching, which we evaluate below.
We consider that Q is spread over a length L, so that the volume density of charges is ρ = Q/πR 2 L. Leaving aside numerical factors, and computing magnetostatics as for an infinitely-long cylinder, one finds E d ∼ K d R 4 /L (note the consistency with the above case when L is set to R). To reach this the logarithmic divergence is bound with cutoff at a radius L from the axis. We have seen in the simulations that magnetization tends to curl around the axis to avoid the formation of surface charges while progressively changing the component of magnetization along the axis of the wire. This implies a total cost of exchange energy E ex scaling like LR 2 (A/R 2 ), thus: E ex ∼ AL. Minimizing
E = E d + E ex yields E ∼ AR 2 /∆ d (3) L ∼ R 2 /∆ d(4)
The wall length is therefore expected to increase rapidly with the wire radius, confirming existing simulations [83]. Quantitatively, this scaling law fits well our own simulations (Fig. 7b). The cross-over from the low-diameter regime to the R 2 law may be defined as the intercept of the asymptote of the latter, with the x axis. This occurs around 40 nm ≈ 8∆ d . This happens to be close to the onset of curling, confirming it as the key ingredient. The R 2 scaling law for energy is also in fair agreement with simulations (Fig. 7a).
VI. CONCLUSION AND TRENDS
Through micromagnetic simulations and analytical scaling laws we considered head-to-head magnetic domain walls in one dimensional structures, with geometry spanning from thin strips to wires with a square or disk cross-section. The former are experimentally relevant for structures made by lithography, while the latter are more relevant for bottom-up synthesis. All domain walls found fall into only two varieties, based on their topology: transverse vortex walls (TVWs) and Bloch-point walls (BPWs). For wires the former display both a transverse (flux of magnetization going through the wire) and a vortex (also named transverse curling) feature. When the geometry goes towards thin strips the transverse or vortex feature takes over and yield the already known transverse and vortex walls, depending on the direction of the through flux, in-plane or out-of-plane, respectively. Concerning BPWs, it is found that they exist for non-perfectly-disk-or -square-shaped wires, and may even be the ground state over a significant range of geometries. These are relevant for thick strips made by top-down techniques, so the BPW and its predicted peculiarities of magnetization dynamics may be of more general relevance than previously thought. We deliver a phase diagram of the different types of domain walls, based on their energy and discussed in terms of phase transitions of either first or second order. The latter are concerned with textures of magnetization developed to close as much as possible the flux, and spread the head-to-head charges along the axis of the structure, to lower magnetostatic energy. These textures develop when at least one of the transverse dimensions equals seven times the dipolar exchange length 2A/µ 0 M 2 s . They may take the form of either an asymmetry as in the already known case of asymmetric transverse walls in flat strips, or curling (with both transverse and longitudinal components). Curling generally yields a more efficient decrease of energy than asymmetric domain walls, except for some cases of wires with square cross-section, because curling is associated with magnetostatic energy at edges.
The theoretical and experimental consideration of magnetic domain walls in thin strips is now rather extensive, following about twenty years of reports. To the contrary, domain walls in wires (i.e. with compact cross-section such as square or disk) have mostly been described theoretically, either through analytics or simulations. Since pioneering works around 2000 several tens of papers have been devoted to the expected types, energetics and peculiar dynamics of motion under magnetic field or spin-polarized current. The first experimental results on the statics of such domain walls are emerging [23][24][25]. Investigating their control in wires and tubes and their dynamics is a challenging however timely direction of research. These hold both exciting prospects for fundamental findings such as velocities up to 1 km/s [7] or interaction with spin waves [84], and are directly applicable to the proposal for a three-dimensional racetrack memory making use of densely-packed arrays of magnetic wires [16]. Our extended phase diagram shows that the physics of domain walls in wires, especially associated with the exotic Bloch-point domain wall, may also be searched for in thick strips, and thus be achievable with top-down techniques.
VII. FURTHER INFORMATION
Most of the above results are previously unpublished. Nevertheless, below are a few key references and reviews concerning pre-existing knowledge. R. McMichael and M. Donahue [1], and Y. Nakatani et al. [2] provided the theoretical overview of domain walls in flat strips. The theoretical investigation of domain walls in wires was pioneered by R. Hertel [43,44] and H. Forster [27,45]. Excellent reviews (both literature and original research) for their statics and dynamics under magnetic field and spin-polarized current can be found in two book chapters by A. Thiaville and Y. Nakatani [7,10]. Many of the concepts developed also apply to the geometry of tubes, published later by several authors.
Second-order transitions and estimators for their parameter 12 D. Fine micromagnetic features 15 V. Analytical scaling laws 16 A. Transverse versus vortex walls in flat strips 16 B. Transverse versus asymmetric transverse walls in flat strips 16 C. Length and energy of domain walls in cylindrical wires 17 VI. Conclusion and trends 17 VII. Further information 18 I. INTRODUCTION
Fig. 1 :
1Vortex walls (VWs)Symmetric transverse walls (TWs)A s y m m e t r i c t r a n sv er se wa ll s (AT Ws) Domain walls in strips (a) Coordinates used to describe strips:
Fig. 2 :
2Bloch points and domain walls in one-dimensional structures (a) Transverse wall (TW) and (b) Vortex wall (VW) in a strip. (c) TW and (d) Bloch-point wall (BPW) in a cylindrical wire (e-f) Bloch-points, of types hedgehog and curling. (g) Micromagnetic simulations of a BPW. The top view shows surface magnetization. The bottom one highlights streamlines of magnetization, while the central disk is the surface where mz = 0.
Fig. 3 :
3I: Common features for first and second order transitions, which we will use to describe the transitions from one micromagnetic state to another Rise of curling for transverse-vortex walls (TVWs) in wires with square cross-section. Wires of side (a,b) 30 nm and (c,d) 44 nm. (a;c) show mid-height top views, while (b,d) show mid-depth side views. In both cases the color codes magnetization along y, i.e. the transverse component of the DW. At the bottom are shown open 3D views, for which the areas displayed are framed in the views above.
Fig. 4 :
4Expected phase diagram. The sketches are based on existing knowledge extended by symmetry arguments. Bold (resp. thin) lines mark first-(resp. second) order transitions. Full lines are used when the ground state is one of the two states considered, while dotted lines are used when both states are metastable, the ground state being of another variety. The phase diagram is built from a to c, adding one by one different types of transitions lines (a) First-order line separating transverse walls (TWs) and vortex walls (VWs) (b) Second-order lines separating symmetric/asymmetric TWs, and also VWs walls from Bloch-type walls (also called Landau-type, see text).
Fig. 5 :
5Phase diagram refined with micromagnetic simulations. The notations for the various lines is the same as inFig. 4.
Fig. 6 :
6The various domain walls for thick strips. Views of (a,b) TW, (c,d) VW and (e,f) Landau walls in strips of width 200 nm. The thickness is 16 nm from a to d, and 60 nm for e and f. (a,c,e) display mid-height views with color coding mz. (b,d,f) display top, middle and bottom views with color coding mx. In b and d the color contrast is enhanced tenfold, so that |mx| = 0.1 uses full contrast. In f the large black arrows point at the surface vortices where the flux of magnetization from the Bloch wall flows from the strip.
Fig. 7 :
7DW energy and width for square-and disk-based wires (a) DW energy normalized to cross-section. The dot on the y axis stands for the limit 4 AK d /2 ≈ 5.7 mJ/m[7]. (b) DW width versus squared (main graph) or lateral dimension (inset). The dot on the y axis of inset stands for the limit 2 2A/K d ≈ 14.1 nm[7]. Narrow-dotted lines are guides to the eye. Notice that for a given value d the area of the cross-section is larger by the amount 4/π in squares, compared to disks.
Fig. 8 :
8Curling features. (a) the three components of the layer-resolved curling for a y-TVW, and longitudinal component for a BPW, in a wire with disk section of diameter 80 nm (b) integration along the wire of the absolute value of layer-resolved curling for a y-TVW in a wire with disk section (c) Illustration of the competition of transverse with longitudinal curling for TVWs. The dark arrows stand for the mean direction of magnetization, while the light arrows stand for the curling part. Frustrated areas are highlighted with an exclamation mark. Cross-sections highlighting curling through coloring the x component for a y-TVW DW, for (d) 60 × 60 nm and (e) 80 × 80 nm square cross-sections, and (f) 80 nm-diameter disk cross-section of a y-TVW. The sign of transverse curling is the same in all cases. (g) Longitudinal curling around a BPW for a 80 nm-diameter disk cross-section.
Fig. 9 :
9Unrolled surface maps. In all cases the length of the view is 1000 nm, and the ratio of lengths along z and ϕ is exact (a) m.n maps of TVWs for disk cross-sections with diameters 30 nm and 120 nm (b) m.n and mz maps of a TVW for square cross-section of side 120 nm. The circled (dotted-circled) areas indicate the locus of the outgoing (ingoing) flux of magnetization from the core of the TVW. We show here the sub-variety where the flux enters and exits through edges, as inFig. 10f (c)mz maps of a BPW for square and disk cross-sections of side 30 nm and diameter 120 nm, respectively (d) m.n and mz maps of a BPW for square cross-section of side 120 nm.
Fig. 10 :
10,d, showing an imbalance towards positive values). Finer points (a-c) Three sub-varieties of VW in strips with cross-section 80 × 40 nm (d-f) Three sub-varieties of TVW in wires with square cross-section of side 30 nm, 36 nm and 60 nm from top to bottom.
TABLE
AcknowledgementsAcknowledgementsThe research leading to these results has received funding from the European Unions's 7th Framework Programme under grant agreement n°309589 (M3d). We acknowledge helpful discussions with Ch. Thirion and J. Vogel (Institut NEEL-Grenoble), A. Fernandez-Pacheco (Cavendish Laboratory-Cambridge), and critical reading of the manuscript by A. Thiaville (LPS-Orsay) and R. Schäfer (IFW-Dresden).
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"\nDJORDJO MILOVIC\nInstitute for Advanced Study\nEinstein Drive08540PrincetonNJUSA\n"
]
| [
"DJORDJO MILOVIC\nInstitute for Advanced Study\nEinstein Drive08540PrincetonNJUSA"
]
| [
"Mathematics Subject Classification. 11N45, 11R29, 11R44"
]
| We use a variant of Vinogradov's method to show that the density of the set of prime numbers p ≡ −1 mod 4 for which the class group of the imaginary quadratic number field Q( √ −8p) has an element of order 16 is equal to 1/16, as predicted by the Cohen-Lenstra heuristics. | 10.1007/s00039-017-0419-6 | [
"https://arxiv.org/pdf/1511.07127v2.pdf"
]
| 119,613,784 | 1511.07127 | 057cf6393966daca73339f815bf3aec158d606b7 |
2010
DJORDJO MILOVIC
Institute for Advanced Study
Einstein Drive08540PrincetonNJUSA
Mathematics Subject Classification. 11N45, 11R29, 11R44
2010ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8p) FOR p ≡ −1 mod 4 1 2 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8p) FOR p ≡ −1 mod 4 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8p) FOR p ≡ −1 mod 4 5
We use a variant of Vinogradov's method to show that the density of the set of prime numbers p ≡ −1 mod 4 for which the class group of the imaginary quadratic number field Q( √ −8p) has an element of order 16 is equal to 1/16, as predicted by the Cohen-Lenstra heuristics.
Introduction and motivation
Let Cl(D) denote the (narrow) class group of the quadratic number field Q( √ D) of discriminant D, and let h(D) := #Cl(D) denote its class number. Although the class group Cl(D) encodes important arithmetic information about the ring of integers in Q( √ D), very little is known about its average behavior as D varies in some natural family of discriminants. The 2-part of Cl(D) is perhaps the most accessible. In [12], Gauss proved that the 2-rank (in other words, the "width" of the 2-part) is given by the formula where ω(D) is the number of distinct prime divisors of D. In particular, if ω(D) = 1, then the 2-part of Cl(D) is trivial, so class groups with the simplest non-trivial 2parts arise from discriminants that have exactly two distinct prime divisors. We focus on one family of such discriminants, namely the family {−8p} p , where p ranges over prime numbers congruent to −1 modulo 4. The 2-part of Cl(−8p) is cyclic and hence completely determined by the highest power of 2 dividing h(−8p). Therefore a natural problem is to determine, for each integer k ≥ 1, the natural density of the set of prime numbers p ≡ −1 mod 4 such that 2 k divides h(−8p). In that vein, for each integer k ≥ 1 and real number X > 2, we set ρ(X; 2 k ) := #{p ≤ X : p ≡ −1 mod 4, 2 k |h(−8p)} #{p ≤ X} , and we define ρ(2 k ) := lim X→∞ ρ(X; 2 k ), if the limit exists. Rédei's [19] and Reichardt's [20] work from the 1930's implies that 4 divides h(−8p) if and only if p ≡ −1 mod 8 and that 8 divides h(−8p) if and only if p ≡ −1 mod 16. It now follows from theČebotarev Density Theorem that ρ(2 k ) = 2 −k for 1 ≤ k ≤ 3. We prove that ρ(16) = 1 16 . More precisely, we prove the following equidistribution result. where the implied constant is absolute. In particular, the natural density of the set of prime numbers p such that p ≡ −1 mod 4 and such that 16 divides h(−8p) is equal to 1 16 .
In other words, if we define the 2 k -rank of Cl(D) to be rk 2 k Cl(D) = dim F2 (2 k−1 Cl(D)/2 k Cl(D)), Theorem 1 states that the natural density of the set of prime numbers p such that p ≡ −1 mod 4 and such that rk 16 Cl(−8p) = 1 is equal to 1 16 . Aside from giving a numerical density for the 16-rank, the main novelty of Theorem 1 is that the powersaving in X gives strong evidence that the behavior of the 16-rank is different from the behavior of the lower 2-power ranks in a very essential way. In the terminology used by Serre to describe equidistribution phenomena [22], the 8-rank in families of quadratic fields of the type {Cl(dp)} p , where d is a fixed integer and p varies among primes such that dp is a discriminant, is "motivated" -that is, rk 8 Cl(dp) is given by the trace of the Frobenius conjugacy class of p in a Galois representation associated to a motive depending only on d; however, the 16-rank in the family {Cl(−8p)} p≡−1 mod 4 appears not to be "motivated." We now explain the various consequences of Theorem 1 in more detail.
1.1. Cohen-Lenstra heuristics. Cohen and Lenstra [1] proposed a heuristic model to predict the average behavior of class groups. They stipulate that an abelian group G occurs as the class group of an imaginary quadratic field with probability proportional to the inverse of the size of the automorphism group of G. Hence the cyclic group of order 2 k−1 should occur as the 2-part of the class group of an imaginary quadratic number field twice as often as the cyclic group of order 2 k . As noted above, the 2-part of the class group Cl(−8p) is cyclic, and so it is natural to make the following conjecture.
Conjecture 1. For all k ≥ 1, we have ρ(2 k ) = 2 −k .
Conjecture 1 for k ≤ 3 was implicit in the work of Rédei and Reichardt mentioned above (although the case k = 3 first appeared explicitly in literature as a result Hasse [14]). Moreover, Conjecture 1 is supported by strong numerical evidence (for instance, the percentages of primes p that are < 10 6 such that p ≡ −1 mod 4 and such that rk 2 k Cl(−8p) = 1 for k = 1, 2, 3, 4, 5, and 6 are 50.09, 25.06, 12.53, 6.40, 3.16, and 1.62%, respectively). Theorem 1 gives a positive answer to Conjecture 1 for the case k = 4.
1.2.
Methods for proving density results about the 2-part of the class group. Rédei [19] showed that the 4-rank of Cl(D) is essentially determined by the quadratic residue symbols ( p1 p2 ) for distinct prime divisors p 1 and p 2 of D. Fouvry and Klüners [7,8,9] combined this characterization with analytic and combinatorial techniques to obtain many interesting results about the 4-rank of Cl(D) and the negative Pell equation x 2 − Dy 2 = −1.
The 8-rank is already more subtle. The main method to prove density results for the 8-rank has been to construct certain governing fields and apply theČebotarev Density Theorem. More precisely, Stevenhagen [23] proved that if d is a non-zero integer, then there exists a normal extension M d /Q such that the 8-rank of Cl(dp) (when dp is a fundamental discriminant) is determined by the Artin conjugacy class (p, M d /Q) in Gal(M d /Q). Knowing such a governing field M d explicitly makes it easy to study the density of primes p for which rk 8 Cl(dp) = r for any fixed integer r. For instance, a governing field for the 8-rank in the family {Cl(−8p)} p≡−1 (4) is Q(ζ 16 ), where ζ 16 is a primitive 16th root of unity.
Cohn and Lagarias [2] made the bold conjecture that governing fields M d,2 k for the 2 k -rank in the family {Cl(dp)} p as above should exist for every 2-power 2 k (see also [3]). However, a governing field has not been found for the 16-or higher 2-power ranks in any family. This is the main reason that density results for the 16-rank have been out of reach for such a long time (see [24, p. 16-18]).
Instead of exhibiting a governing field for the 16-rank in the family {Cl(−8p)} p , we introduce another method to the study of the 2-part of class groups of quadratic number fields.
α m β n a mn ,
where a n = exp(2πi √ n), d is any positive integer, and {α n } n and {β m } m are very general sequences of complex numbers that do not grow too quickly. We prove Theorem 1 by using a modern version of Vinogradov's method developed by Friedlander, Iwaniec, Mazur, and Rubin [11, Proposition 5.2, p.722].
One important feature of Vinogradov's estimates is that the bound for the sum over primes saves a power of X, i.e., there exists a small real number δ > 0 such that p≤X a p log p X 1−δ .
On the other hand, the best zero-free regions of classical L-functions generally give the much worse error estimates of the type X log(−c √ log X). This suggests that the a p are not "motivated" -they do not arise naturally as coefficients of a finite sum of classical L-functions, and in particular are not naturally related to an Artin symbol of p in a fixed normal extension M/Q. In other words, the proof of Theorem 1 gives strong evidence that a governing field for the 16-rank in the family {Cl(−8p)} p≡−1(4) in fact does not exist. For a more precise discussion of this phenomenon, see Section 7.
1.4.
A few words about the proof. Leonard and Williams [17] found the following criterion for the 16-rank. Since we were unable to verify their proof of this criterion, we give another proof of a slightly more general statement in Section 2. A prime p ≡ −1 mod 16 can be written as
(1.1) p = u 2 − 2v 2
where u and v are integers, u > 0, and
(1.2) u ≡ 1 mod 16.
Given such a representation, [17, Theorem 3, p.205] (or our Proposition 1) states that
(1.3) e p = v u ,
where e p is defined as in Theorem 1 and · · is the Jacobi symbol. The first few primes satisfying the above criterion are 127, 223, 479, 719, . . . . Note that integers u > 0 and v satisfying (1.1) and (1.2) are not unique. Nonetheless, the criterion (1.3) is valid for any choice of integers u > 0 and v satisfying (1.1) and (1.2). Hence Theorem 1 is a corollary of the following theorem, which we will prove using Vinogradov's method. To apply Vinogradov's method to the sum p≤X e p , the most important task is to define a sequence {e n } n in a way that one can prove good estimates for sums of type I and type II. Generalizing the proof in [10] to our setting is made difficult by the fact that an odd ideal in the quadratic ring Z[ √ 2] does not have a canonical generator -the group of units Z[ √ 2] × is infinite. We resort to averaging over four carefully chosen generators to define an analogous spin symbol. Proving that the resulting quantity is well-defined already requires significant new ideas. Proposition 2 in Section 2 is a key result in this direction; it describes the twisting of v u by the fundamental unit 1 + √ 2. Section 2 also contains the class field theoretic construction of the governing (spin) symbol e p = v u χ(u) for the 16-rank in the family {Cl(−8p)} p≡−1(4) (see Proposition 1). In Section 3, we construct spin symbols that both encode behavior of the 16-rank in our family and are conducive to analytic techniques (see Equations (3.4) and (3.5)). We also reduce Theorem 2 to a purely analytic statement (see Theorem 3) that can be attacked by the machinery of Friedlander, Iwaniec, Mazur, and Rubin (see Proposition 4). The goal of Section 4 is to construct convenient fundamental domains for the multiplicative action of a fundamental unit 1 + √ 2 on Z[ √ 2]. In Section 5, we use a Polya-Vinogradov-type estimate to give bounds for sums of type I for the spin symbol. In Section 6, we give bounds for sums of type II of the spin symbol, thus completing the proof of Theorem 2. In the final section, we discuss the implications of the power-saving bound in Theorem 1 on the existence of governing fields for the 16-rank.
1.5. Generalizations. While it would be desirable to prove density results about the 16-rank in any family of the type {Cl(dp)} p with d fixed and p varying, there are serious technical limitations on both algebraic and analytic sides of the problem. On the algebraic side, the 2-part of Cl(dp) might no longer be cyclic, and hence one would have to account for the possible interactions between the spin symbols arising from different prime divisors of d. On the analytic side, one would have to account for the possibility that the class group of Q( √ d) need not be trivial.
Perhaps an even more basic problem is that applying Vinogradov's method in this setting generally requires one to carry out analytic estimates over a number ring instead of Z, and many such estimates require bounds on incomplete character sums that are well beyond anything currently available. For instance, similar proofs of density results about the 16-rank in the families {Cl(−8p)} p≡1(4) and {Cl(−4p)} p would require Burgess-type estimates for short character (modulo q) sums of length q Theorem 1 thus lives on the very edge of unconditional density results about the 2-part of class groups of quadratic number fields. So while the statement of our main theorem is not as general as one might hope for, our work nevertheless demonstrates two important ideas: that yet another classical analytic method is applicable to modern problems concerning class groups; and that the nature of the 16-rank is of a type not seen before in the study of the 2-part of class groups.
Let χ be a character (Z/16Z) × → C × with kernel {±1}. In other words, we have χ(±1 mod 16) = 1 and χ(±7 mod 16) = −1. Then our generalization of [17,Theorem 3,p.205] is as follows:
Proposition 1. Let p ≡ −1 mod 16 be a prime number. Let u and v be integers such that p = u 2 − 2v 2 and such that u > 0 and v ≡ 1 mod 4. Then
(2.1) rk 16 Cl(−8p) = 1 ⇐⇒ v u χ(u) = 1.
The choice of u and v in the proposition above is not unique. Let
ε = 1 + √ 2 be a fundamental unit in Z[ √ 2]
, so that the group of units Z[ √ 2] × is generated by ε and −1. As the norm of ε is −1, the norm of ε 2 = 3 + 2 √ 2 is 1. Let p ≡ −1 mod 16 be a prime number as in Proposition 1. Given one integer solution (u, v) = (u 0 , v 0 ) to the system
(2.2) p = u 2 − 2v 2 u > 0, v ≡ 1 mod 4 ,
then the complete set of integer solutions (u, v) to the system (2.2) is of the form
u + v √ 2 = ε 2k (u 0 + v 0 √ 2)
for some integer k. An interesting consequence of Proposition 1 is that the quantity v u χ(u) is independent of the choice of u and v satisfying (2.2). For a prime p ≡ −1 mod 16, we can thus define the governing symbol for the 16-rank to be
(2.3) p := v u χ(u),
where u and v are integers satisfying (2.2). The quantity p determines the 16-rank of the class group Cl(−8p). It is interesting to note that the 16-rank of Cl(−8p) depends on a "quantitative" aspect of the splitting behavior of p in Z[ √ 2] that appears to allow no description purely in terms of the "qualitative" splitting behavior of p in some normal extension of Q.
Leonard and Williams claim that [17,Theorem 3,p.205] can be proved by numerous manipulations of Jacobi symbols and applications of quadratic reciprocity. We instead prove Proposition 1 by interpreting the Jacobi symbol v u as an Artin symbol of an ideal u defined via the decomposition p = u 2 − 2v 2 in an extension of Q( √ −8p) defined via the same decomposition p = u 2 − 2v 2 . Moreover, a byproduct of our proof is the following proposition, which turns out to be essential for a successful application of the analytic tools we wish to use. Proposition 2. Let u 1 and v 1 be integers such that u 1 is odd and positive and such that u 2 1 − 2v 2 1 > 0. Define integers u 2 and v 2 by the equality
u 2 + v 2 √ 2 = ε 8 (u 1 + v 1 √ 2).
Then
v 1 u 1 = v 2 u 2 .
In other words, we have the equality of Jacobi symbols v 1
u 1 = 408u 1 + 577v 1 577u 1 + 816v 1 .
The rest of this section is devoted to proving Proposition 1 and Proposition 2.
2.1. Preliminaries.
2.1.1. Galois theory. We will make extensive use of the following lemma from Galois theory (see [16, Chapter VI, Exercise 4, p.321]).
Lemma 1. Let F be a field of characteristic different from 2, let E = F ( √ d), where d ∈ F × \ (F × ) 2 , and let L = E( √ x), where x ∈ E × \ (E × ) 2 . Let N = Norm E/F (x)
. Then we have three cases:
(1) If N / ∈ (E × ) 2 ∩ F × = (F × ) 2 ∪ d · (F × ) 2 , then L/F has normal closure L( √ N ) and Gal(L( √ N )/F ) is a dihedral group of order 8.
(2) If N ∈ (F × ) 2 , then L/F is normal and Gal(L/F ) is a Klein four-group. We will use the following lemma several times. Lemma 2. Let E/F be an abelian extension of number fields, let L/F be a finite extension, and let ι : Gal(EL/L) → Gal(E/F ) be the restriction-to-E map. Then for every prime ideal p of L that is coprime to Disc(E/F ), we have Suppose for the moment that rk 2 n Cl(−8p) = 1. Then 2 n Cl is a subgroup of Cl of index 2 n . We define the 2 n -Hilbert class field H 2 n to be the subfield of H fixed by the the image of 2 n Cl under the isomorphism (2.4). Since the 2-primary part of Cl is cyclic, it follows immediately that H 2 n is the unique unramified, cyclic, degree-2 n extension of K. Moreover, (2.4) induces a canonical isomorphism of cyclic groups of order 2 n
(3) If N ∈ d · (F × ) 2 ,ι p EL/L = Norm L/F (p) E/F .(2.5) · H 2 n /K : Cl/2 n Cl −→ Gal(H 2 n /K).
The main idea of the proof of Proposition 1 is to write down explicitly, for p ≡ −1 mod 8, both
• the 4-Hilbert class field H 4 of K, and
• an ideal u generating a class of order 4 in Cl(−8p)
in terms of integers u and v satisfying p = u 2 − 2v 2 , and then to characterize those p such that
(2.6) u H 4 /K = 1.
The isomorphism (2.5) for n = 2 and the equality (2.6) then imply that the class of order 4 in Cl in fact belongs to 4Cl, which proves that Cl has an element of order 16.
2.1.4. Ring class fields. To prove Proposition 2, we will have to work with a generalization of the Hilbert class field. Let D < 0 be any integer ≡ 0, 1 mod 4 that is not a square, and let O D be the quadratic order of discriminant D, i.e.,
O D = Z[(D + √ D)/2]. Let K = Q( √ D)(2.7) · R D /K : Cl(D) −→ Gal(R D /K).
In the case f = 1, so that D = Disc(K), the ring class field R D coincides with the Hilbert class field of K.
The main property of ring class fields of imaginary quadratic orders that we will use is stated in the following lemma.
Lemma 3. Let K be an imaginary quadratic number field of even discriminant, and let L/K be a cyclic extension such that:
• L/Q is a dihedral extension, and
• the conductor of L/K divides (4).
Then L is contained in the ring class field R D of the imaginary quadratic order O D of discriminant D = 16 · Disc(K).
2.2.
A special family of quadratic fields. Let u and v be coprime integers such that u is odd and positive and such that
(2.8) n = u 2 − 2v 2
is positive as well. Let K be the imaginary quadratic number field defined by
K = Q( √ −2n).
Note that n ≡ ±1 mod 8, and moreover n ≡ 1 mod 8 if and only if v is even. Let m and d be the unique positive integers such that m is squarefree and such that n = d 2 m. Then K = Q( √ −2m) and the discriminant of K/Q is equal to −8m. We emphasize that both m and d are odd. As gcd(u, v) = 1, every prime dividing n splits in Q( √ 2). Hence there exist δ and µ in Q( √ 2) of norm d and m, respectively, such that u + v √ 2 = δ 2 µ.
Let G = K( √ 2)
. Note that G coincides with the genus field of K in the case that n is a prime number congruent to −1 modulo 4. Finally, we define a quadratic extension of G as follows.
Define ν ∈ Z[ √ 2] ⊂ G by setting (2.9) ν = u + v √ 2. Then let L = L u,v = G( √ εν), where ε = 1 + √ 2 as before.
If n is a prime number congruent to −1 modulo 8 and u and v are chosen as in the statement of Proposition 1, we will see that L coincides with the 4-Hilbert class field H 4 of K.
Remark. The fields K and G are determined simply by n. In other words, had we started with another choice of integers u and v giving rise to the same n, the definitions of K and G would not change. However, the field L may depend on the specific choice of u and v. Since we fixed u and v in the beginning of the section, this should not cause any confusion. We now introduce some notation and prove some properties of the extensions K ⊂
G ⊂ L. Let ν = u − v √ 2 be the conjugate of ν in Q( √ 2)
. We now state a few consequences of the assumption that gcd(u, v) = 1. It will be useful to consider the following field diagram.
L = G( √ εν) G = K( √ 2) A = Q( √ 2, √ εν) K = Q( √ −2m) Q( √ 2)
Q Lemma 4. The extension L/K is cyclic of degree 4, and the extension L/Q is dihedral of order 8.
Proof. We have
Norm G/K (εν) = Norm Q( √ 2)/Q (εν) = −νν = −n. As −n = 2 · 1 2 √ −2n 2 ∈ 2 · (K × ) 2 , the first claim follows from Lemma 1, part (3). Now let A = Q( √ 2, √ εν). As −n / ∈ (Q × ) 2 ∪ 2 · (Q × ) 2 , part (1) of Lemma 1 implies that L = A( √ −n) is the normal closure of A/Q and Gal(L/Q) ∼ = D 8 .
Let t denote the prime of K lying above 2.
Lemma 5. L/K is unramified at every prime other than possibly at t.
Proof. Recall that ν = δ 2 µ, so L = Q( √ −2m, √ 2, √ εµ).
As the norm of µ is m, every prime that ramifies in L/Q must divide 2m. Let p be a rational prime dividing m. Suppose p factors as ππ in Z[ √ 2], and, without loss of generality, suppose π divides ν. As u and v are coprime, ν and ν are coprime in Z[ √ 2] and hence π does not ramify in A = Q( √ 2, √ εν). Thus, as p splits in Q( √ 2), its ramification index in L/Q is at most 2. But p already ramifies in K/Q, and hence every prime p of K lying above p must be unramified in L/K. By Lemma 5, the only prime that can divide the conductor f of L/K is the prime t. The following lemma gives the precise power of t dividing f. (1) If v ≡ 1 mod 4, then L/K is unramified and f = 1.
(2) If v ≡ −1 mod 4, then f = t 2 = (2).
(3) If v ≡ 0 mod 2, then f = t 4 = (4).
Proof. Since t is the only prime that can divide f, we only need to study the extensions locally at the primes above 2. Let T be a prime of G lying above t and T a prime of L lying above T. Let K t , G T , and L T denote the completions of K, G, and L with respect to the primes t, T, and T , respectively.
If v is odd, then n ≡ −1 mod 8, and
so K t = Q 2 ( √ −2n) = Q 2 ( √ 2) and G T = K t ( √ 2) = K t . Thus the extension G T /K t is trivial and L T = Q 2 ( √ 2, √ εν). The extension Q 2 ( √ 2, √ εν)/Q 2 ( √ 2) is unramified if and only if εν is a square modulo t 4 ; here t = ( √ 2) is the maximal ideal in Z 2 [ √ 2]. If v ≡ 1 mod 4, then εν = (u + 2v) + (u + v) √ 2 ≡ 1 mod t 4 if u ≡ −1 mod 4, ε 2 mod t 4 if u ≡ 1 mod 4, and hence L T /K t is unramified. This proves part (1) of the lemma. Similarly, if v ≡ 1 mod 4, then εν ≡ 3 or 1 + 2 √ 2 mod t 4 .
In this case εν is not a square modulo t 4 , and so L T /K t is ramified. The ramification is wild, and thus f must be divisible by t 2 . As εν ≡ 1 mod t 2 , the extension L T /K t can be generated by a root of the polynomial
X 2 + √ 2X + 1 − εν 2 = 1 2 √ 2X + 1 2 − εν ,
whose discriminant is 2 mod t 4 . Hence f = t 2 and part (2) of the lemma is proved.
Finally, suppose v ≡ 0 mod 2, so that n ≡ 1 mod 8.
Then K t = Q 2 ( √ −2n) = Q 2 ( √ −2) and G T = K t ( √ 2) = Q 2 (ζ 8 ). The quadratic extension G T /K t is ram- ified of conductor t 2 , where t = ( √ −2) is the maximal ideal in Z 2 [ √ −2]. Let s = 1 + ζ 8 be a generator of the maximal ideal s in Z 2 [ζ 8 ]. Note that s 2 = √ 2 · ζ 8 ε, so εν ≡ 1 mod s 2 .
Hence the extension L T /K t can be generated by a root of the polynomial
X 2 + s 3 ζ 6 8 ε −2 X + 1 − εν s 2 = 1 s 2 (sX + 1) 2 − εν ,
whose discriminant is s 6 mod s 7 . Hence the discriminant of L T /G T is s 6 .
To finish, we use the conductor-discriminant formula, i.e.,
Disc(L T /K t ) = Disc(G T /K t )f(L T /K t ) 2 .
The discriminant formula for the tower of fields
K t ⊂ G T ⊂ L T gives Disc(L T /K t ) = Disc(G T /K t ) 2 Norm G T /Kt (Disc(L T /G T )), so that f(L T /K t ) 2 = Disc(G T /K t )Norm G T /Kt (Disc(L T /G T )).
Substituting Disc(G T /K t ) = t 2 and Disc(L T /G T ) = s 6 into the formula above implies that f(L T /K t ) = t 4 , which completes the proof of part (3) of the lemma. The integers u and v appearing in (2.8) are not unique. Given a representation n = u 2 − 2v 2 , another representation can be obtained by multiplying u + v √ 2 by 3 + 2 √ 2. This transforms (u, v) into (3u + 4v, 2u + 3v).
We will show how the quantity v u χ(u), where χ is a Dirichlet character from Proposition 2, naturally arises in the computation of a certain Artin symbol. This computation is somewhat delicate because the Artin symbol will take a value in a cyclic group of order 4, and such a group has a non-trivial automorphism.
Remark. In [13], Halter-Koch, Kaplan, and Williams compute Artin symbols in similar cyclic field extensions L/K of degree 4. Their results, however, involve computations of Artin symbols of ideals of K of order 2 in the class group of K, and hence only give information about the 8-rank in certain quadratic fields.
Let f ∈ {1, 4}. The case f = 1 will be used to prove Proposition 1, while the case f = 4 will be used to prove Proposition 2.
Let τ = f √ −2n, so that Z[τ ] is the order of K of discriminant −8nf 2 . We define a homomorphism ψ u,v : Z[τ ] → Z/uZ by sending τ → 2vf mod u. This homomorphism is well-defined since τ 2 = −2nf 2 = −2(u 2 − 2v 2 )f 2 ≡ (2vf ) 2 mod u. Let (2.10) u = ker ψ u,v .
It is the ideal of Z[τ ] generated by u and 2vf − τ , i.e., u = (u, 2vf − τ ). In case n = p ≡ −1 mod 8 and f = 1, the ideal class of u turns out to have order 4, as we will see later. We remark that (2.11) 2vf ≡ τ mod u and that
(2.12) Norm(u) = u.
Let √ εν be a square root of εν. Then, by Lemma 1, the extension G( √ εν)/K is cyclic of degree 4. We are interested in computing the Artin symbol u
G( √ εν)/K .
The key idea is to relate this Artin symbol to the Artin symbol associated to a different but related cyclic degree-4 extension of K. Let
(2.13) γ = (2 + √ 2)v ∈ Z[ √ 2].
Then again by Lemma 1, the extension G( √ γ)/K is cyclic of degree 4. The element γ was chosen so that (2.14) εν ≡ γ mod u, and at the same time so that the extension Q( √ γ)/Q mimics the cyclic degree-4 subextension of the cyclotomic extension Q(ζ 16 )/Q. Finally, let F be the compositum of G( √ εν) and G( √ γ). We have the following field diagram.
ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8p) FOR p ≡ −1 mod 4 13 K = Q( √ −2n) G = K( √ 2) K( √ β ) K( √ β) G( √ ενγ) G( √ γ) F = G( √ εν, √ γ) G( √ εν)
Here β and β are elements of K that are conjugate over Q. Let ενγ ∈ Q( √ 2) be the conjugate of ενγ over Q. Since
2ενγ ± 2ενγ 2 = 4v((4u + 6v) ± √ −2n) = 4v f ((4u + 6v)f ± τ ) ,
we can take
β = v((4u + 6v)f − τ ) and β = v((4u + 6v)f + τ ).
The inclusion Gal(F/K( √ β)) ⊂ Gal(F/K) and projections Gal(F/K) Gal(G( √ εν)/K) and Gal(F/K) Gal(G( √ γ)/K) induce canonical isomorphisms
ψ 1 : Gal(F/K( β)) ∼ −→ Gal(G( √ εν)/K) and ψ 2 : Gal(F/K( β)) ∼ −→ Gal(G( √ γ)/K).
Using (2.11), we find that if p is a prime ideal dividing u, then
β p = v((4u + 6v)f − τ ) p = 4v 2 f p = 1,
and so p splits in K( √ β). By Lemma 2, for any prime P of K( √ β) lying above a prime ideal p dividing u, we have
ψ 1 P F/K(β) = p G( √ εν)/K and ψ 2 P F/K(β) = p G( √ γ)/K .
Multiplying over all prime ideals p dividing u, we have proved the following key lemma.
Lemma 8. Let u be defined as in (2.10). Then
ψ 2 • ψ −1 1 u G( √ εν)/K = u G( √ γ)/K . 14 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4 Now we apply Lemma 2 with E = Q( √ −2n), F = Q, and L = Q( √ γ). We have ι u G( √ γ)/K = u Q( √ γ)/Q , so that, by Lemma 8, we have ι • ψ 2 • ψ −1 1 u G( √ γ)/K = u Q( √ γ)/Q . Now observe that Q( √ γ) is a subfield of Q(ζ 16 √ v). Indeed, ζ 16 √ v + ζ −1 16 √ v = γ. There is a canonical isomorphism Gal(Q(ζ 16 √ v)/Q) ∼ = (Z/16Z) × ∼ = −1 mod 16 × 3 mod 16
given by sending
ζ 16 √ v → ζ k 16 √ v → (k mod 16). Then Q( √ γ) is the subfield of Q(ζ 16 √ v) fixed by −1. For each prime q coprime to 2v, we have q Q(ζ 16 √ v)/Q = q v q mod 16, so that if we identify ψ 3 : 3 mod 16 ∼ = µ 4 = i ⊂ C × by sending 3 → i = √ −1, we get ψ 3 q Q( √ γ)/Q = v q χ(q).
Multiplying over all primes q dividing u and using Lemma 8, we finally obtain the following result.
Lemma 9. Let ψ : Gal(G( √ εν)/K) ∼ −→ µ 4 be the isomorphism of cyclic groups of order 4 defined by ψ = ψ 3 • ι • ψ 2 • ψ −1 1 . Then ψ u G( √ εν)/K = v u χ(u).
2.4. An ideal identity. We keep the same notation as in Sections 2.2 and 2.3.
Recall that τ = f √ −2n, where f ∈ {1, 4}
. Let t f be the ideal of Z[τ ] defined as the kernel of the homomorphism
τ f : Z[τ ] → Z/2f 2 Z given by sending τ → 2vf . The homomorphism τ f is well-defined because τ 2 = −2nf 2 = 4v 2 f 2 − 2u 2 f 2 ≡ (2vf ) 2 mod 2f 2 .
Then t f = (2vf − τ, 2f 2 ). The following identity of between ideals in Z[τ ] will be useful in proofs of both Proposition 1 and Proposition 2.
Lemma 10. Let u be defined as in (2.10). Then
(2vf − τ ) = t f u 2 .
Proof. The principal ideal 2vf − τ is invertible of norm 2u 2 f 2 . Since u is odd and gcd(u, v) = 1, we deduce that u is coprime to the discriminant −8nf 2 of Z[τ ] and is thus invertible. No rational primes can divide 2vf − τ and u divides (2vf − τ ) by definition, so it must be that u 2 divides (2vf − τ ).
The ideal t f of norm 2f 2 contains (2vf − τ ) and has the same norm as the invertible ideal (2vf − τ )u −2 . Hence we must have (2vf − τ )u −2 = t f . 2.5.1. A class of order 4. We now produce an ideal generating a class of order 4 in the class group Cl(−8p) when p is a prime ≡ −1 mod 8. This is the main ingredient in [17].
When n = p and f = 1, the ideal t = t f defined in Section 2.4 is the prime ideal lying above 2. If t = (x + y √ −2p) for some x, y ∈ Z, then x 2 + 2py 2 = Norm(t) = 2, which is impossible. Hence the class of t in Cl(−8p) has order 2. Now let u be defined as in (2.10) with u and v as above and f = 1. Lemma 10 shows that u 2 and t are in the same ideal class in Cl(−8p). Hence we have proved the following result. Remark. Perhaps an easier, although more old-fashioned, way to prove Lemma 11 is via the theory of binary quadratic forms, as was done in [17]. Let [a, b, c] denote the SL 2 (Z)-equivalence class of the form ax 2 + bxy + cy 2 . The key observation is that [u, −4v, 2u] has discriminant 16v 2 − 8u 2 = −8p. To compose this class with itself, one can use the special case of the composition law for concordant forms, which yields the class [u, −4v, 2u] 2 = [u 2 , −4v, 2] = [2, 0, p]. The classes [u, −4v, 2u] and [2, 0, p] correspond to the ideal classes of u and t, respectively.
2.5.2.
Generating the 4-Hilbert class field. Let p be a prime congruent to −1 mod 8 and let K = Q( √ −8p). The 2-Hilbert class field, also called the genus field of K, is known to be H 2 = K( √ 2). Lemma 11 implies that rk 4 Cl(−8p) = 1, and our aim is to generate the 4-Hilbert class field H 4 over H 2 by adjoining an element that we can write explicitly in terms of u and v.
Define π ∈ Z[ √ 2] by setting π = ν with ν as in (2.9), i.e., π = u + v √ 2. The following proposition achieves our aim.
Proposition 3. Let K = Q( √ −8p)
, and let π be as above. Then the 4-Hilbert class field of K is
H 4 = H 2 ( √ επ). ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4
Proof. Since the 2-part of the class group Cl(−8p) is cyclic, it suffices to show that H 2 ( √ επ) is an unramified, cyclic, degree-4 extension of K.
We apply the lemmas of Sections 2.2 and 2.3 with n = m = p, e = 1, and u and v as above. By Lemma 4, the extension H 2 ( √ επ)/K is cyclic of degree 4. By Lemma 5, H 2 ( √ επ)/K is unramified over the prime ideal p = (p, √ −2p) of K lying over p. Finally, by part (1) of Lemma 6, H 2 ( √ επ)/K is unramified over the prime ideal t = (2, √ −2p) of K lying over 2. 2.6. Proof of Proposition 2. As in the statement of Proposition 2, let u 1 and v 1 be integers such that u 1 is odd and positive and such that u 2 1 − 2v 2 1 > 0. We define u 2 and v 2 by the equality
(2.16) u 2 + v 2 √ 2 = ε 8 (u 1 + v 1 √ 2) = (577u 1 + 816v 1 ) + (408u 1 + 577v 1 ) √ 2,
where, as before, ε = 1 + √ 2. Our goal is to prove the following equality of Jacobi symbols
(2.17) v 1 u 1 = v 2 u 2 .
By the Euclidean algorithm, we have the equality
gcd(u 1 , v 1 ) = gcd(u 2 , v 2 ).
First, if gcd(u 1 , v 1 ) = gcd(u 2 , v 2 ) > 1, then both sides of (2.17) are equal to 0, and hence (2.17) holds true. Now suppose gcd(u 1 , v 1 ) = gcd(u 2 , v 2 ) = 1. Let
n = u 2 1 − 2v 2 1 = u 2 2 − 2v 2 2 , and let K = Q( √ −2n) as in Section 2.2. Set τ = 4 √ −2n.
Let u 1 (resp. u 2 ) be the ideal of the imaginary quadratic order Z[τ ] (of discriminant 16 · −8n) defined by (2.10) with (u, v) = (u 1 , v 1 ) (resp. (u, v) = (u 2 , v 2 )) and f = 4. The ideals u 1 and u 2 satisfy the following key property. Therefore
(2.18) u 2 2 = 8v 2 − τ 8v 1 − τ u 2 1 .
Let α = (17u 1 + 24v 1 ) + 3τ . We claim that
(2.19) α u 1 2 = 8v 2 − τ 8v 1 − τ .
We first note that
(2.20) 8v 2 − τ 8v 1 − τ = 8v 2 − τ 8v 1 − τ · 8v 1 + τ 8v 1 + τ = 64v 1 v 2 + 32n + 8(v 2 − v 1 )τ 64v 2 1 + 32n = 64v 1 (408u 1 + 577v 1 ) + 32n + 8(408u 1 + 576v 1 )τ 32u 2 1 = n + 2v 1 (408u 1 + 577v 1 ) + (102u 1 + 144v 1 )τ u 2 1 .
Expanding α 2 , we get (2.21)
α 2 = 289u 2 1 + 576v 2 1 + 816u 1 v 1 − 288n + (102u 1 + 144v 1 )τ = u 2 1 + 1152v 2 1 + 816u 1 v 1 + (102u 1 + 144v 1 )τ = n + 1154v 2 1 + 816u 1 v 1 + (102u 1 + 144v 1 )τ = n + 2v 1 (408u 1 + 577v 1 ) + (102u 1 + 144v 1 )τ.
Comparing the last line of (2.21) with the numerator in the last line of (2.20), we obtain (2.19).
Now (2.18) and (2.19) imply that (2.22)
u 2 1 u 2 2 = α 2 u 2 1 . By (2.12), Norm(u 2 ) = u 2 . Hence Norm(u 2 ) is odd, and since u 1 is also odd, we find that u 2 1 u 2 2 is coprime to the conductor f = 4 of Z[τ ], and hence factors uniquely into prime ideals. Therefore (2.22) implies that u 1 u 2 = αu 1 , which proves the lemma.
Remark. There is a shorter proof of Lemma 12 via the theory of binary quadratic forms. The SL 2 (Z)-equivalence classes of binary quadratic forms of discriminant 16 · −8n corresponding to the ideals u 1 Since √ 2 is contained in G = K( √ 2), 8 is a square in G. Hence the fields G( √ εν 1 ) and G( √ εν 2 ) are equal, and so we define
L = G( √ εν 1 ) = G( √ εν 2 ).
By Lemma 7, L is contained in the ring class field of Z[τ ]. Hence, by Lemma 12, the images of both u 1 and u 2 under the map (2.7) coincide, i.e.,
u 1 L/K = u 2 L/K . Applying Lemma 9, we get v 1 u 1 χ(u 1 ) = v 2 u 2 χ(u 2 ).
Equation (2.16) implies that
(2.24) u 2 = 577u 1 + 816v 1 ≡ u 1 mod 16.
Hence, as χ is a character modulo 16, we have χ(u 1 ) = χ(u 2 ), and so Proposition 2 is finally proved.
Sums over primes
Above, we defined the governing symbol p for a prime p ≡ −1 mod 16 in terms of particular integer solutions u and v to the equation p = u 2 − 2v 2 . The main lemma that we will use to prove Theorem 2, i.e., that these governing symbols oscillate, is a proposition due to Friedlander, Iwaniec, Mazur and Rubin [11]. We now state this proposition in our context.
3.1.
A result of Friedlander, Iwaniec, Mazur, and Rubin. Recall that an element w = u+v √ 2 ∈ Z[ √ 2] is totally positive if and only if Norm(w) = u 2 −2v 2 > 0 and u > 0. We sometimes write w 0 to say that w is totally positive.
Since Z[ √ 2
] is a principal ideal domain and since the norm of the fundamental unit ε over Q is −1, an ideal n in Z[ √ 2] can always be generated by a totally positive element. For an ideal n of Z[ √ 2], recall that the norm of n is given by
Norm(n) := u 2 − 2v 2 ,
where u + v √ 2 is a totally positive generator of n.
We now define an analogue of the von Mangoldt function Λ for the ring Z[ √ 2]. For a non-zero ideal n of Z[ √ 2], we set Λ(n) = log(Norm(p)) if n = p k for some prime ideal p and integer k ≥ 1 0 otherwise.
Hence Λ is supported on powers of prime ideals.
Given a sequence of complex numbers {a n } n indexed by non-zero ideals in Z[ √ 2], a good estimate for the sum of a n over prime ideals p of norm Norm(p) ≤ X can usually be derived from a good estimate of the "smoother" weighted sum S(X) := Norm(n)≤X a n Λ(n).
The idea in [11] (and even earlier in [10]), is to bound S(X) by combinations of linear and bilinear sums in a n . Given a non-zero ideal d of Z[ √ 2], we define the linear sum We consider bilinear sums where the complex numbers α m and β n satisfy In other words, power-saving estimates for linear and bilinear sums imply powersaving estimates for sums supported on primes. Note that this result is now classical in the context of rational integers, thanks to the pioneering work of Vinogradov [25]. u ≡ 1 mod 16 implies that χ(u) = 1 and also that −v u = v u (note that one of v and −v is congruent to 1 mod 4). It is also convenient that exactly one of the four (17u+24v, 12u+17v)), and hence u 2 ≡ u 0 +8 mod 16; one can now easily check that multiplying u + v √ 2 successively by ε 2 cycles u mod 16 through the set {1, 7, 9, 15}.
elements ε 2k (u + v √ 2) = u k + v k √ 2 (0 ≤ k ≤ 3) satisfies u k ≡ 1 mod 16. Indeed, multiplying u + v √ 2 by ε 2 (resp. ε 4 ) transforms (u, v) into (3u + 4v, 2u + 3v) (resp.
Proposition 2 states that [w] = [ε 8 w] for any odd and totally positive w ∈ Z[ √ 2], so, in light of the preceding discussion, we might naively define a n = 3 k=0 [ε 2k w], where w 0 is any totally positive generator of n. This definition does not quite suffice for our purposes because we want to isolate those p that are congruent to −1 mod 16 and representations p = u 2 − 2v 2 with u ≡ 1 mod 16. Hence we weigh the expression above by Dirichlet characters modulo 16. More precisely, for each pair of Dirichlet characters φ and ψ modulo 16 and totally positive u + v √ 2, we set
(3.4) [u + v √ 2] φ,ψ := [u + v √ 2]φ(−u 2 + 2v 2 )ψ(u).
For a non-zero ideal n in Z[ √ 2] generated by a totally positive element w, we set
(3.5) a φ,ψ,n := 3 k=0 [ε 2k w] φ,ψ .
This is still well-defined, i.e., independent of the choice of w 0, by Proposition 2 and by (2.24). We will apply Proposition 4 to 8 2 sequences {a φ,ψ,n } n , one for each pair of Dirichlet characters φ, ψ, and then add together the corresponding 8 2 sums S φ,ψ (X) to obtain Theorem 2. It is now easy to check Lemma 13. If p is a prime and p is a prime ideal lying above p, then we have
1 8 2 φ ψ a φ,ψ,p = p if p ≡ −1 mod 16 0 otherwise.
Hence, to prove Theorem 2, it now suffices to prove Theorem 3. Let a φ,ψ,n be defined as in (3.5). For every > 0, there is a constant C > 0 depending only on such that for every X ≥ 2, we have
Norm(n)≤X a φ,ψ,n Λ(n) ≤ C X 149 150 + .
Fundamental domains
In order to obtain power-saving cancellation for linear and bilinear sums as in Proposition 4, we will have to choose generators of n in (3.5) carefully. The problem reduces to constructing a convenient fundamental domain for the action of ε 2 = 3 + 2 √ 2 on totally positive elements of Z[ √ 2]. Such constructions are standard (see for instance [18,Chapter 6] or [11,Section 4]). For the sake of completeness and explicitness, we give a simple argument tailored to our specific needs. Let
(4.1) Ω := (u, v) ∈ R 2 : u > 0, −u < √ 2v < u .
Then the lattice points (u, v) ∈ Ω ∩ Z 2 precisely enumerate the totally positive elements w = u + v √ 2. The group ε 2 of totally positive units of Z[ √ 2] acts on the totally positive elements of Z[ √ 2] by multiplication, and this induces an action ε 2 × Ω → Ω given by
a + b √ 2 · (u, v) := (au + 2bv, bu + av).
Let D be the subset of Ω defined by
(4.2) D := (u, v) ∈ R 2 : u > 0, −u < 2v ≤ u
We claim that the region D is a fundamental domain for the action of ε 2 on Ω in the following sense.
Lemma 14.
For each element (u, v) ∈ Ω ∩ Z 2 , there exists exactly one integer k such that ε 2k · (u, v) ∈ D.
Proof. Since ε 2 = 3+2 √ 2 > 1, we have that ε 2k > ε 2j whenever k > j, that ε 2k → 0 as k → −∞, and that ε 2k → ∞ as k → ∞. Moreover, given (u, v) ∈ Ω ∩ Z 2 , we have
ε 2 (u + v √ 2) ε −2 (u − v √ 2) = ε 4 · u + v √ 2 u − v √ 2 .
Hence, given (u, v) ∈ Ω ∩ Z 2 , there exists a unique integer k such that
ε −2 < ε 2k (u + v √ 2) ε −2k (u − v √ 2) ≤ ε 2 .
The lemma follows upon noticing that for (u, v) ∈ Ω, we have (u, v) ∈ D if and only
if ε −2 < (u + v √ 2)/(u − v √ 2) ≤ ε 2 .
An immediate consequence of Lemma 14 is the following proposition.
Proposition 5. Suppose that n is a non-zero ideal of Z[ √ 2]. Then n has a unique generator in D.
4.1.
Geometry of numbers in the fundamental domain: the Lipschitz principle. We now briefly turn to the problem of counting lattice points and boxes inside certain compact subsets of the fundamental domain D. We state a lemma of Davenport (see [5] and [6]).
Let R be a compact, Lebesgue measurable subset of R n . Suppose that R satisfies the following two conditions: Lemma 15 (Davenport). If R satisfies conditions (1) and (2) above, then
|R ∩ Z n − Vol(R)| ≤ n−1 m=0 h n−m V m
where V m is the sum of the m-dimensional volumes of the projections of R on the various coordinate spaces obtained by equating any n − m coordinates to zero, and V 0 = 1 by convention.
We will apply Lemma 15 to the fundamental domain D ⊂ R 2 as well as certain variations thereof.
Let k ≥ 0 be an integer, and define
D k = D ∪ ε 2 · D · · · ∪ ε 2k · D.
Let X > 0. Then the region
(4.3) D k (X) := {(u, v) ∈ D k : u 2 − 2v 2 ≤ X}
is a compact subset of R 2 and satisfies conditions (1) and (2) above with h = 2. Moreover, one can check that there exist positive real numbers a k and k such that (4.4) Vol(D k (X)) = a k X and Vol(∂(D k (X))) = k X (1) and (2) above, also with h = 2. We define the diameter of L to be diam(L) := |a| + |b| + |c| + |d|. Then Vol(L(D k (X))) = |D|Vol(D k (X)) and Vol(∂(L(D k (X)))) = O(diam(L) · X 1 2 ), where the implied constant is absolute.
Linear sums
In this section we prove that the estimate (A) from Proposition 4 holds for the sequence {a φ,ψ,n } n defined in (3.5) with θ 1 = 1/6. Proposition 6. Let a n = a φ,ψ,n , where a φ,ψ,n is defined as in (3.5), and let A d (X) be defined as in (3.1). Then for all > 0 and all X ≥ 2, we have
A d (X) X 5 6 + .
Proof. Recall that
A d (X) = Norm(n)≤X n≡0 mod d a n .
Since the sequence a n is supported on odd ideals n, we see that A d (X) = 0 unless d is odd. Hence we may assume without loss of generality that d is an odd ideal. Let
(5.1) R(X) := D 4 (X) = (u, v) ∈ D ∪ ε 2 D ∪ ε 4 D ∪ ε 6 D : u 2 − 2v 2 ≤ X .
By Proposition 5 and definition (3.5), we have
A d (X) = (u,v)∈R(X) u+v √ 2≡0 mod d [u + v √ 2] φ,ψ , where [u + v √ 2] φ,ψ is defined as in (3.4).
We now reformulate the congruence condition u + v √ 2 ≡ 0 mod d. Proposition 5 implies that there is an element d 1 + d 2 √ 2 ∈ D which generates d. Then the congruence above is equivalent to saying that there exist integers e 1 and e 2 such that
u + v √ 2 = (d 1 + d 2 √ 2)(e 1 + e 2 √
2), i.e., such that u = d 1 e 1 + 2d 2 e 2
and v = d 2 e 1 + d 1 e 2 . In other words, (u, v) is in the image of the linear transformation and we rewrite the sum A d (X) as
L d := d 1 2d 2 d 2 d 1 : Z 2 → Z 2 of determinant D := Norm(d) = d 2 1 − 2d 2 2 . Hence we define R(d, X) := {(u, v) ∈ R(X) : (u, v) ∈ Image(L d )} ,A d (X) = (u,v)∈R(d,X) [u + v √ 2] φ,ψ .
Using the fact that |[u + v √ 2] φ,ψ | ≤ 1, we obtain the trivial bound
(5.2) |A d (X)| ≤ (u,v)∈R(d,X) 1 = L −1 d R(X)∩Z 2 1.
Since
d 1 + d 2 √ 2 ∈ D, we have the inequalities d 2 1 2 ≤ D ≤ d 2 1 , which implies that diam(L −1 d ) D −1/2 . Hence Lemma 15 gives (5.3) |A d (X)| ≤ a 4 XD −1 + O(D − 1 2 X 1 2 + 1) XD −1 + X 1 2 D − 1 2 + 1,
where the implied constant is absolute. This estimate will be useful when D is large compared to X.
Next we split the sum A d (X) into 8 · 16 sums where the congruence classes of u and v modulo 16 are fixed, say u ≡ u 0 mod 16 and v ≡ v 0 mod 16 for some congruence classes u 0 and v 0 modulo 16 with u 0 invertible modulo 16. For u and v satisfying these congruences, we have
[u + v √ 2] φ,ψ = δ(u 0 , v 0 ) v u ,
where δ(u 0 , v 0 ) ∈ {±1} depends only on the congruence classes u 0 and v 0 modulo 16. Hence it remains to give estimates for sums of the type
A d (u 0 , v 0 , X) := (u,v)∈R(u0,v0,d,X) v u , where R(u 0 , v 0 , d, X) := {(u, v) ∈ R(d, X) : (u, v) ≡ (u 0 , v 0 ) mod 16} .
Splitting the sum according to the value of u, we obtain
(5.4) A d (u 0 , v 0 , X) = 0≤u≤R1(X) u≡u0 mod 16 A u,d (v 0 , X), where A u,d (v 0 , X) := v∈Iu (u,v)∈Ld(Z 2 ) v≡v0 mod 16 v u .
Here R 1 (X) = sup{u ∈ R : (u, v) ∈ R(X)} X 1 2 and I u is an interval (or a union of 2 disjoint intervals) of size ≤ 2R 2 (X), where R 2 (X) = sup{|v| ∈ R : (u, v) ∈ R(X)} X We now unwind the condition (u, v) ∈ L d (Z 2 ), i.e., that (u, v) is in the image of L d . Consider the system of equations in x and y:
(5.5) u = d 1 x + 2d 2 y v = d 2 x + d 1 y.
Let d := gcd(d 1 , d 2 ) and write d 1 = dd 1 , d 2 = dd 2 . Recall that d and so also d 1 is odd, so that d = gcd(d 1 , 2d 2 ). If the system (5.5) has a solution over Z, then d must divide u. This means that
A d (u 0 , v 0 , X) = 0≤u≤R1(X) u≡u0 mod 16 u≡0 mod d A u,d (v 0 , X).
Now suppose u ≡ 0 mod d, and let x u , y u ∈ Z be such that
u = d 1 x u + 2d 2 y u .
Then all solutions (x, y) ∈ Z 2 to the first equation in (5.5) are given by
(x, y) = (x u − 2d 2 k, y u + d 1 k), k ∈ Z. Hence v = d 2 (x u − 2d 2 k) + d 1 (y u + d 1 k) = d 2 x u + d 1 y u + Dk/d,
which means that (5.5) has a solution over Z if and only if
v ≡ d 2 x u + d 1 y u mod D/d.
Note that D is odd, so that D/d and 16 are coprime. Let v u be the congruence class modulo 16D/d such that
v u ≡ d 2 x u + d 1 y u mod D/d v u ≡ v 0 mod 16.
Thus we have proved that if u ≡ 0 mod d, then
A u,d (v 0 , X) = v∈Iu v≡vu mod 16D/d v u .
Let e u = gcd(v u , 16D/d), write 16D/d = e u d u , v u = e u v u , and perform a change of variables v = e u v , so that
A u,d (v 0 , X) = e u u v ∈I u v ≡v u mod du v u ,
where I u = I u /e u . Since gcd(v u , d u ) = 1, we can now detect the congruence condition v ≡ v u mod d u via Dirichlet characters modulo d u . In other words,
(5.6) A u,d (v 0 , X) = 1 ϕ(d u ) e u u χ(v u ) χ mod du v ∈I u χ(v ) v u ,
where v u denotes the multiplicative inverse of v u modulo d u . Let χ be a Dirichlet character modulo d u . If the character
v → χ(v ) v u 26 ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4
is trivial, then u = f g 2 for some f dividing d u (and therefore dividing 16D/d) and some integer g. The number of such u ≤ R 1 (X) is
≤ τ (16D/d)R 1 (X) 1 2 D X 1 4
. In this case we use the trivial bound v ∈I u
χ(v ) v u #I u ≤ #I u X 1 2 ,
where the implied constant in is absolute. Hence the contribution of such u to
A d (u 0 , v 0 , X) is (5.7) D X 3 4 .
On the other hand, if the character v → χ(v ) v u is not trivial, its conductor is at most 16Du/d DX 1 2 , and so the Polya-Vinogradov inequality gives the estimate v ∈I u
χ(v ) v u D 1 2 X 1 4 + .
Combining this with (5.4), (5.6), and (5.7), we have proved the bound
(5.8) A d (X) D 1 2 X 3 4 + .
We use (5.8) for D < X 1/6 and (5.3) for D ≥ X 1/6 to obtain A d (X) X 5 6 + .
Bilinear sums
We are left with proving the estimate (B) from Proposition 4, which we do with θ 2 = 1/12 in much the same way as in [10, Sections 19-21, p. 1018-1028].
Proposition 7. Let a n = a φ,ψ,n , where a φ,ψ,n is defined as in Our basic strategy will be to prove a factorization formula of the type [wz] = [w][z]γ(w, z), where γ(w, z) is a quantity which oscillates in both arguments w, z ∈ Z[
√ 2]
. We first develop some background necessary to define γ(w, z) and then prove power-saving cancellation for general bilinear sums of the type w,z α w β z γ(w, z).
6.1. Primitivity. We say that an ideal a in Z[ √ 2] is primitive if whenever p is a prime ideal dividing a, then p is unramified, of residue degree one, and p does not divide a. Here and after, if x is an element or an ideal in Z[ √ 2] we will use x to denote the conjugate of x over Q. The main property of primitive ideals that we will use is that the inclusion Z → Z[ √ 2] induces an isomorphism
(6.1) Z/(Norm(a)) ∼ → Z[ √ 2]/a.
We call an ideal a (resp. element w) in Z[ √ 2] odd if Norm(a) (resp. Norm(w)) is an odd integer. An ideal in Z[ √ 2] is odd if and only if every prime ideal that divides a is unramified. Hence, an ideal a is primitive if and only if a is odd and there is no rational prime p dividing a (i.e., no rational prime p such that (p) divides a).
Remark. For instance, Norm(7) = 49, but Z[
√ 2]/(7) ∼ = Z[ √ 2]/(3+ √ 2)×Z[ √ 2]/(3− √ 2) ∼ = Z/(7) × Z/(7) Z/(49).
For every integer n we have the equality of quadratic residue symbols
(6.2) n Norm(a) = n a ,
where the symbol on the left is the usual Jacobi symbol while the symbol on the right is the quadratic residue symbol in Z[ Proof. If a is not primitive, then there is a rational prime p dividing a. As p is rational, it also divides a, and so gcd(a, a) = (1). Conversely, if gcd(a, a) = (1), then there is a prime ideal p in Z[ √ 2] such that both p and p divide a. If p is a prime of degree 2, then p = (p) for some rational prime p and automatically a is not primitive. Otherwise, as a is odd and the only prime that ramifies in Q( √ 2)/Q is 2, we conclude that p and p are coprime, and hence that pp divides a. Once again, as pp = (p) for a rational prime p, a is not primitive. Proof. Suppose p is a rational prime such that p divides ab. Then (p) cannot be a prime in Z[ √ 2], because otherwise either a or b is not primitive. Hence there exists a prime ideal p ⊂ Z[ √ 2] such that p = pp and p|a. If p k is the exact power of p dividing ab, then the assumption that a and b are primitive implies that p k |a and p k |b, which is true if and only if p k |r. There is another way to obtain a primitive ideal from a product of two odd primitive ideals a and b. We can write
a = p split p ap p ap and b = p split p bp p bp ,
where a p a p = b p b p = 0 for every p. Let r = gcd(a, b) and let r = Norm(r). If a prime p divides r, after possibly interchanging the roles of p and p in the products above, we can assume that p divides r. For every such prime p, define
c a,p = p ap if a p ≤ b p , 1 otherwise, and c b,p = 1 if a p ≤ b p , p bp otherwise, and set c a = p c a,p , c b = p c b,p , and c = c a c b .
Then clearly Norm(c) = Norm(r) = r.
Moreover, by construction gcd a c a , b c b = (1),
so by Lemma 16, we conclude ab/c is primitive. By construction, c is also primitive and coprime to ab/c. Therefore, using the Chinese Remainder Theorem and applying (6.1) twice, we conclude that
(6.4) Z[ √ 2]/ab ∼ = Z[ √ 2]/(ab/c) × Z[ √ 2]/c ∼ = Z/(W/r) × Z/(r),
where W = Norm(ab).
Finally, we say that an element w ∈ Z[ √ 2] is primitive if and only if the principal ideal generated by w is primitive. An equivalent definition is that w = a + b √ d is odd and gcd(a, b) = 1.
6.2.
A quasi-bilinear symbol with a reciprocity law. For w, z ∈ Z[ √ 2] with w odd, we define the generalized Dirichlet symbol γ(w, z) to be
(6.5) γ(w, z) := wz (w) ,
where · · is the quadratic residue symbol in Q( √ 2). Our choice of terminology is inspired by the Dirichlet symbol defined in a slightly different context in [10, Section 19, p. 1018[10, Section 19, p. -1021.
The symbol γ(w, z) factors as
(6.6) γ(w, z) = m(w) z (w) ,
where, for odd w ∈ Z[ √ 2], we define (6.7) m(w) := γ(w, 1) = w (w) .
By Lemma 6.2, if w ∈ Z[ √ 2] is odd, then m(w) = 0 ⇐⇒ gcd((w), (w)) = (1) ⇐⇒ w is primitive.
Hence the factor m(w) restricts the support of γ(w, z) to w that are primitive. Furthermore, if w is primitive, then gcd((w), (wz)) = gcd((w), (z)), and so in this case γ(w, z) = 0 if and only if gcd((w), (z)) = (1).
The factor z (w)
is completely multiplicative in z, so it follows from (6.6) that
(6.8) γ(w, z 1 )γ(w, z 2 ) = γ(w, z 1 z 2 )m(w),
for any w, z 1 , and z 2 in Z[ √ 2] such that w is odd. Hence the symbol γ(w, z) is multiplicative in z except for a twist by m(w).
The symbol γ(w, z) also satisfies a reciprocity law, which is an important ingredient in our proof of Proposition 7. Proof. We have
γ(w, z)γ(z, w) = wz (w) zw (z) = wz (wz) = m(wz).
Finally, we note that γ(w, z) is periodic in the second argument. In fact, γ(w, z 1 ) = γ(w, z 2 ) whenever z 1 ≡ z 2 mod (w). In other words, γ(w, ·) is a function on Z[
√ 2]/(w). This allows us to prove the following analogue of [10, Lemma 21.1, p. 1025], which will provide all of the cancellation that we need for Proposition 7.
Lemma 19. Let w 1 , w 2 ∈ Z[ √ 2] be primitive. Let r = gcd((w 1 ), (w 2 )), r = Norm(r), W = Norm(w 1 w 2 ). Then
z∈Z[ √ 2]/(W )
γ(w 1 , z)γ(w 2 , z) = W ϕ(r)ϕ(W/r) if W and r are squares 0 otherwise.
Proof. By (6.6), we have
γ(w 1 , z)γ(w 2 , z) = m(w 1 )m(w 2 ) z (w 1 w 2 )
, and, as w 1 and w 2 are odd and primitive, m(w 1 )m(w 2 ) = 0. Hence
z∈Z[ √ 2]/(W ) γ(w 1 , z)γ(w 2 , z) = z∈Z[ √ 2]/(W ) z (w 1 w 2 )
. Now, as W is rational, the map z → z is an automorphism of the group Z[ √ 2]/(W ). Thus, we obtain
z∈Z[ √ 2]/(W ) z (w 1 w 2 ) = z∈Z[ √ 2]/(W ) z (w 1 w 2 ) . b r ,
where the symbols on the right-hand side of the equality are the usual Jacobi symbols. For any positive integer n, we have a∈Z/(n) a n = ϕ(n) if n is a square, 0 otherwise.
Combining all of the equations above, we conclude the proof of the proposition.
We conclude this section by expressing γ(w, z) as a Jacobi symbol. Suppose w = a + b √ 2 and z = c + d √ 2, with w primitive and totally positive. Then
wz (w) = wz + wz (w) = 2ac + 4bd a 2 − 2b 2 .
Moreover, as w is primitive, every prime factor of Norm(w) = a 2 − 2b 2 is congruent to ±1 modulo 8, so 2 a 2 −2b 2 = 1. Hence (6.9) γ(w, z) = ac + 2bd a 2 − 2b 2 .
6.3. Double oscillation of γ(w, z). We can now prove some general bilinear sum estimates that we will use to deduce Proposition 7. Let α = {α w } and β = {β z } be two sequences of complex numbers, each indexed by non-zero elements in Z[ √ 2], such that (6.10) |α w | ≤ log(Norm(w))τ (w) and |β z | ≤ log(Norm(z))τ (z)
for all w and z in Z[
√ 2]
. For a positive real number X, let D(X) = D 0 (X) as in (4.3). Set
C := lim sup X→∞ {u : (u, v) ∈ D(X)} · X − 1 2 .
and note that C < ∞. Next, for a positive real number X, we define the "cone"
B(X) := {(u, v) ∈ Ω : 0 < u ≤ CX 1 2 },
where Ω is the region, defined in (4.1), which enumerates the totally positive elements in Z[ √ 2]. Hence the set of elements in Z[ √ 2] enumerated by ∪ X>0 B(X) = Ω is closed under multiplication. Note also that D(X) ⊂ B(X) for every real number X. For a subset S of R 2 and an element u + v √ 2 ∈ Z[ √ 2], we will say that u + v √ 2 ∈ S to mean that (u, v) ∈ S ∩ Z 2 . Finally, for positive real numbers M and N , we define the bilinear sum Proof. Let Q(M, N ) = Q(M, N ; α, β). Applying the Cauchy-Schwarz inequality to the sum over z and expanding the square, we obtain
|Q(M, N )| 2 ≤ z∈B(N ) |β z | 2 * w1∈D(M ) * w2∈D(M ) α w1 α w2 R(N ; w 1 , w 2 ), where R(N ; w 1 , w 2 ) = z∈B(N ) γ(w 1 , z)γ(w 2 , z).
Since β z is bounded in modulus by N , Lemma 15 applied to L = Id gives (6.12)
z∈B(N ) |β z | 2 N Vol(B(N )) + N O(Vol(∂(B(N ))) + 1) N 1+ .
Next, recall that γ(w, z 1 ) = γ(w, z 2 ) whenever z 1 ≡ z 2 mod Norm(w). Hence we can split the inner sum over z into residue classes modulo W . More precisely, if ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4 ζ = ζ 1 + ζ 2 √ 2, we define L to be the linear transformation L = W · Id + (ζ 1 , ζ 2 ) : R 2 → R 2 . Then Lemma 15 gives
R(N ; w 1 , w 2 ) = ζ mod W γ(w 1 , ζ)γ(w 2 , ζ) z∈B(N ) z≡ζ mod W 1 = ζ mod W γ(w 1 , ζ)γ(w 2 , ζ) 2C 2 N W 2 + O N 1 2 W + 1 = 2C 2 N W 2 ζ mod W γ(w 1 , z)γ(w 2 , z) + O W 2 N 1 2 W + 1 .
Now set r = Norm(gcd((w 1 ), (w 2 ))). Note that W ϕ(r)ϕ(W/r) ≤ W 2 . Then using Lemma 19, we obtain the estimate
R(N ; w 1 , w 2 ) N + W N 1 2 + W 2 if W and r are squares W N 1 2 + W 2 otherwise.
By unique factorization in Z[ √ 2], the number of primitive elements w ∈ D such that Norm(w) = n is at most 2 ω(n) ≤ τ (n) n . Hence, using the bound W M 2 and setting m 1 = Norm(w 1 ) and m 2 = Norm(w 2 ), we get The following method, which appears in [10], exploits the multiplicativity of γ(w, z) in z to improve the quality of the estimate when M and N are close to each other. where α w = m(w) 5 and
|Q(M, N )| 2 N m1,m2≤M m1m2= N + M 2 N 1 2 + M 4 + M 2 M 2 N 1 2 + M 4 (M N ) .β z = z1···z6=z z1,...,z6∈B(N ) β z1 β z2 · · · β z5 β z6 .
Note that β z is supported on z ∈ B(27C 6 N 3 ). Now using Lemma 20 to estimate the sum (6.14), and substituting back into (6.13), we obtain the desired result.
The final step is to exploit the symmetry of the symbol γ(w, z) coming from its reciprocity law. Suppose that w = a+b √ 2 ∈ Z[ √ 2] is primitive and totally positive. By (6.9) and the law of quadratic reciprocity, we have and β(z 0 ) z := β z · 1(z ≡ z 0 mod 8).
m(w) = γ(w, 1) = a a 2 − 2b 2 = (−1) a−1 2 · a 2 −2b 2 −1 2 −2 a ,
Here 1(P ) is the indicator function of a property P . We will now prove Proof. It suffices to establish the desired estimate for the sequences α(w 0 ) and β(z 0 ) for each pair of congruence classes w 0 and z 0 modulo 8Z[ As discussed above, by Lemma 18, we have Q(M, N ; α(w 0 ), β(z 0 )) = δ(w 0 , z 0 ) · Q(N, M ; β(z 0 ), α(w 0 )).
Applying Lemma 21 to the right-hand side above, we also get (6.16) Q(M, N ; α(w 0 ), β(z 0 )) N 11 12 M + N Finally, taking the minimum of the terms in (6.15) and (6.16) in the appropriate ranges, we obtain Q(M, N ; α(w 0 ), β(z 0 )) M 11 12 N + N 11 12 (M N ) , and then the inequality gives the desired result. does not hold for all totally positive w and z. Instead, the equation above becomes essentially valid when twisted by γ(w, z). We now state our result more precisely.
We now introduce notation that will simplify the subsequent arguments. Suppose that f 1 and f 2 are functions Z r → C. For x ∈ Z r , we write f 1 ∼ f 2 (or more conveniently f 1 (x) ∼ f 2 (x)) if there exists a function δ : Z r → {±1} such that δ factors though (Z/16Z) r , i.e., the value of δ(x) depends only on the congruence classes of the coordinates of x modulo 16, and such that
f 1 (x) = δ(x)f 2 (x)
for all x ∈ Z r . For instance, [u + v √ 2] φ,ψ ∼ [u + v √ 2] φ ,ψ for any four Dirichlet characters φ, ψ, φ , ψ modulo 16.
The following proposition is analogous to [10, Lemma 20.1, p. 1021]. It is perhaps the most surprising part of the proof of Proposition 7. Proof. When wz is not primitive, then [wz] = 0 and γ(w, z) = 0, and so the result follows. Hence we may assume that wz is primitive.
First note that wz = (ac + 2bd) + (ad + bc) √ 2.
We set ρ = (a, d) and define a 1 and d 1 by the equalities a = ρa 1 and d = ρd 1 , respectively. Then
[wz] = ad + bc ac + 2bd = ad + bc ρ ad + bc a 1 c + 2bd 1 , and since ρ divides ad, the above simplifies to
[wz] = bc ρ ad + bc a 1 c + 2bd 1 .
Now, since w is primitive, a 1 is relatively prime to b and hence also to a 1 c + 2bd 1 .
Hence we may write c ≡ −2bd 1 /a 1 (mod a 1 c + 2bd 1 ), so that the second factor in the expression above becomes ad + bc a 1 c + 2bd 1 = ad − 2b 2 d 1 /a 1 a 1 c + 2bd 1 = a 1 d 1 a 1 c + 2bd 1 ρ 2 − 2b 2 /a 2 1 a 1 c + 2bd 1 .
As a 2 − 2b 2 = a 2 1 (ρ 2 − 2b 2 /a 2 1 ), we deduce that
[wz] ∼ bc ρ a 1 d 1 a 1 c + 2bd 1 a 2 − 2b 2 a 1 c + 2bd 1 .
We write the last factor in the expression above as a 2 − 2b 2 a 1 c + 2bd 1 = a 2 − 2b 2 ρ a 2 − 2b 2 ac + 2bd , and use the fact that a 2 − 2b 2 ρ = −2b 2 ρ = −2 ρ to conclude that
[wz] ∼ −2bc ρ a 1 d 1 a 1 c + 2bd 1 a 2 − 2b 2 ac + 2bd .
The law of quadratic reciprocity implies that a 2 − 2b 2 ac + 2bd ∼ ac + 2bd a 2 − 2b 2 , so that, by (6.9),
[wz] ∼ −2bc ρ a 1 d 1 a 1 c + 2bd 1 γ(w, z)
We again use the law of quadratic reciprocity to treat the middle term above. We get a 1 a 1 c + 2bd 1 = (−1) ν1(a,b,c,d,ρ) 2 a 1 bd 1 a 1 ,
where ν 1 (a, b, c, d, ρ) ≡ a 1 − 1 2 · r 1 − 1 2 mod 2 and r 1 = a 1 c + 2bd 1 . Similarly, we write d 1 as d 1 = 2 e d 2 ,
Now suppose e = 1. Then splitting into cases similarly as above, we get Finally, suppose e = 0. Then Here α w = α (w) and β z = β (z) , i.e., α w (resp. β z ) depends only on the ideal generated by w (resp. z).
ν 1 + ν 2 + ν 3 + ρ − 1 2 ≡ aν 1 + ν 2 + ν 3 + ρ − 1 2 ≡ a
It is enough to estimate (6.17) for each 0 ≤ k ≤ 3. First, suppose u + v √ 2 0 is primitive and odd. Then by Proposition 8, we have
[ε 2k (u + v √ 2)] ∼ [u + v √ 2][ε 2k ]γ(ε 2k , u + v √ 2) ∼ [u + v √ 2].
We write w = a + b √ 2 and z = c + d Unless both w and z are primitive, wz is not primitive, and hence [wz] = 0 . Using Proposition 8 again and replacing α w by α w [w]1(w ≡ w 0 mod 16) and β z by β z [z]1(w ≡ w 0 mod 16), it now suffices to estimate sums of the type * w∈D(M ) * z∈D(N ) α w β z γ(w, z). This is exactly a sum of the type Q(M, N ; α, β) as in Lemma 22, and so Proposition 7 follows. This completes the proof of Theorem 3 and hence also Theorem 2.
rk 2
2Cl(D) := dim F2 (Cl(D)/2Cl(D)) = ω(D) − 1,
Theorem 1 .
1For a prime number p ≡ −1 mod 16, let e p = 1 if 16 divides h(−8p) and let e p = −1 otherwise. Then for all X > 0,
be the field of fractions of O D . Then K is an imaginary quadratic number field of discriminant Disc(K) satisfying the equality D = f 2 Disc(K) for some positive integer f , called the conductor of O D . Let Cl(D) denote the class group of O D . Then there is a unique abelian extension R D /K called the ring class field of O D such that the Artin map induces a canonical isomorphism of groups
Lemma 6 .
6Let f denote the conductor of L/K. Then:
Lemma 7 .
7L is contained in the ring class field R D of the imaginary quadratic order O D of discriminant D = 16 · −8m. Proof. Combine Lemmas 3, 4, and 6. ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4 2.3. A computation of Artin symbols. This section contains the heart of the proof of both Proposition 1 and Proposition 2.
2. 5 .
5Proof of Proposition 1. We apply the results of Sections 2.3 and 2.4 in the case n = p ≡ −1 mod 8 is a prime number and f = 1. In this case there exist integers u and v such that p = u 2 − 2v 2 , and the congruence p ≡ −1 mod 8 immediately implies that both u and v are odd. Without loss of generality, we may assume that u is positive and(2.15) v ≡ 1 mod 4. Since the 2-part of Cl(−8p) is cyclic, rk 16 Cl(−8p) = 1 if and only if Cl(−8p) has an element of order 16. To get started, we first produce an element of order 4 in Cl(−8p) that we can write explicitly in terms of u and v.
Lemma 11 .
11Let u be the ideal of Z[ √ −2p] defined as above. Then the ideal class of u has order 4 in Cl(−8p).
2.5. 3 .
3Conclusion of the proof of Proposition 1. By Lemma 11, rk 16 Cl(−8p) = 1 if and only if the ideal class of u belongs to Cl(−8p) 4 . By Proposition 3, this is true if and only if the Artin symbol of u in H 4 = H 2 (√ επ) is trivial. In the notation of Section 2.3, we have that H 2 = G, so that rk 16 Cl(−8p) 9, this occurs if and only if v u χ(u) = 1, which proves Proposition 1.
Lemma 12 .
12The ideals u 1 and u 2 belong to the same ideal class in the class group Cl(16 · −8n) of the imaginary quadratic order Z[τ ]. Proof. Let k ∈ {1, 2}. By Lemma 10, we have (8v k − τ ) = t 4,k u 2 k where t 4,k = (8v k − τ, 32) is as in Section 2.4. By (2.16), we have 8v 2 = 8(408u 1 + 577v 1 ) = 8v 1 + 32(102u 1 + 144v 1 ), ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8p) FOR p ≡ −1 mod 4 17so that t 4,2 = (8v 2 − τ, 32) = (8v 1 − τ, 32) = t 4,1 .
Moreover, given two sequences of complex numbers {α m } and {β n }, each indexed by non-zero ideals in Z[ √ 2], we define the bilinear sum (3.2) B(M, N ) := Norm(m)≤M Norm(n)≤N α m β n a mn .
(3. 3 )
3|α m | ≤ Λ(m) and |β n | ≤ τ (n), where τ (n) denotes the number of ideals in Z[ √ 2] dividing n. We now state [11, Proposition 5.2, p.722] that we use to prove Theorem 2.
Proposition 4 .
4Let a n be a sequence of complex numbers bounded by 1 in absolute value and indexed by non-zero ideals of Z[ √ 2]. Suppose that there exist two real numbers 0 < θ 1 , θ 2 < 1 such that: for every > 0, we have(A) A d (X) X 1−θ1+ uniformly for all non-zero ideals d of Z[ √ 2] and all X ≥ 2, and (B) B(M, N ) (M + N ) θ2 (M N ) 1−θ2+ uniformly for all M, N ≥ 2 and sequences of complex numbers {α m } and {β n } satisfying (3.3).Then for all X ≥ 2 and all > 0, we have the bound
( 1 )
1Any line parallel to one of the n coordinate axes intersects R in a set of points which, if not empty, consists of at most h intervals, and (2) The same is true (with m in place of n) for any of the m-dimensional regions obtained by projecting R on one of the coordinate spaces defined by equating a selection of n−m of the coordinates to zero; and this condition is satisfied for all m from 1 to n − 1. ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4
Now let L : R 2 → R 2 be an invertible linear transformation of the form D := ad − bc = 0. Then L(D k (X)) is a compact subset of R 2 that also satisfies conditions
Figure 1 .
1The region R(X) and the lattice points R(d, X) ON THE 16-RANK OF CLASS GROUPS OF Q( √ −8P ) FOR P ≡ −1 MOD 4
(3.5), and let B(M, N ) be defined as in(3.2). Then for all > 0 and all M, N ≥ 2, we have
α, p) = 1 and α ≡ mod a −1 if (α, p) = 1 and α ≡ mod aThe following is yet another characterization of primitive ideals.Lemma 16. Suppose a ⊂ Z[ √ 2] is an odd ideal. Then a is primitive if and only if gcd(a, a) = (1).
Suppose a and b are ideals in Z[ √ 2]. If one of a and b is not primitive, then clearly their product ab is not primitive. Even if both a and b are primitive, the product ab need not be primitive. Nonetheless, we have the following lemma.
Lemma 17 .
17Suppose a and b are primitive. Let r = gcd(a, b) and r = Norm(r). Then ab/(r) is primitive. In particular, ab is primitive if and only if gcd(a, b) = (1).
Lemma 18 .
18Let w, z ∈ Z[ √ 2]such that both w and z are odd. Then γ(w, z)γ(z, w) = m(wz).
( 6 .
611) Q(M, N ; α, β) := * w∈D(M ) z∈B(N ) α w β z γ(w, z),where * w restricts the summation to primitive w. The first result we prove is a standard consequence of the Cauchy-Schwartz inequality and Lemma 19.
Lemma 20 .
20For every > 0, there is a constant C > 0 such that for every pair of sequences of complex numbers α = {α w } and β = {β z } satisfying (6.10) and every pair of real numbers M, N > 1, we have |Q(M, N ; α, β)| ≤ C
(
M N ) , and the inequality M ≥ 1 now implies the desired result.
Lemma 21 .(
21For every > 0, there is a constant C > 0 such that for every pair of sequences of complex numbers α = {α w } and β = {β z } satisfying (6.10) and every pair of real numbers M, N > 1, we have |Q(M, N ; α, β)| ≤ M N ) .Proof. Let Q(M, N ) = Q(M, N ; α, β). We apply Hölder's inequality to get (6.13) |Q(M, N )| 6 ≤
and so m(w) ∈ {±1} depends only on the residue class of w modulo 8Z[ √ 2]. Lemma 18 then implies that for every pair of odd and totally positive w, z ∈ Z[√ 2], we have γ(w, z) = δ · γ(z, w),where δ = δ(w mod 8, z mod 8) := m(wz) ∈ {±1} depends only on the congruence classes of w and z modulo 8Z[ √ 2]. We are thus led to decompose the sum Q(M, N ; α, β) as Q(M, N ; α, β) = w0 mod 8 z0 mod 8 Q(M, N ; α(w 0 ), β(z 0 )), where α(w 0 ) and β(z 0 ) are sequences indexed by non-zero elements of Z[ √ 2] defined by α(w 0 ) w := α w · 1(w ≡ w 0 mod 8)
Lemma 22 .
22For every > 0, there is a constant C > 0 such that for every pair of sequences of complex numbers α = {α w } and β = {β z } such that (6.10) holds and such that β is supported on primitive z ∈ D(N ), and for every pair of real numbers M, N > 1, we have |Q(M, N ; α, β)| ≤ C (M + N )
√ 2 ]
2. So fix congruence classes w 0 and z 0 modulo 8Z[ √ 2]. Note that the sum Q(N, M ; β(z 0 ), α(w 0 )) satisfies the assumptions of Lemma 21. Thus, applying Lemma 21 gives the estimate (6.15) Q(M, N ; α(w 0 ),
Proposition 8 .
8Let w = a+b √ 2 and z = c+d √ 2 be two primitive, totally positive, odd elements of Z[ √ 2]. Then
[
wz] ∼ [w][z]γ(w, z).
Proof of Proposition 7. We are now ready to conclude the proof of Proposition 7. The bilinear sum (3.2) can be written as B k (M, N ) = w∈D(M ) z∈D(N ) α w β z [ε 2k wz] φ,ψ .
√ 2 and split (6.17) into 8 2 · 16 2 sums by fixing congruence classes of a, b, c, and d modulo 16 (where the congruence classes of a and c are invertible). Then it suffices to estimate each sum w∈D(M ) w≡w0 mod 16 z∈D(N ) z≡z0 mod 16 α w β z [wz].
1.3. Vinogradov's method.In late 1940's, I.M. Vinogradov[25,26] was able to prove cancellation in the sum over primesp≤X
exp(2πi
√ p) log p
by expanding it into sums of type I
n≤X, n≡0 mod d
a n
and sums of type II
m≤M,n≤N
Theorem 2 .
2For every > 0, there is a constant C > 0 depending only on such that for every X ≥ 2, we have where, for each prime p in the sum above, u and v are taken to be integers satisfying (1.1) and (1.2).Theorem 2 is an equidistribution result reminiscent of [10, Theorem 2, p. 948]. In[10], Friedlander and Iwaniec associate a spin symbol (i.e., a quantity taking values in S 1 ⊂ C × ) to each non-zero ideal in the Gaussian integers Z[i] and show, also using Vinogradov's method, that its value is equidistributed over prime ideals in Z[i] ordered by their norms.p≤X
p≡−1 mod 16
v
u
≤ C X
149
150 + ,
then L/F is normal and Gal(L/F ) is a cyclic group of order 4. 2.1.2. The Artin map and Artin symbols. Let E/F be a finite abelian extension of number fields. Let I F denote the free abelian group generated by prime ideals of F that are unramified in E. The Artin map is the group homomorphism multiplicatively to I F .·
E/F
: I F → Gal(E/F )
defined as follows. Let p be a prime ideal of F which is unramified in E and let
P be any prime ideal of E lying above p. Let Norm(p) be the cardinality of the
residue field at p. Then the Artin symbol
p
E/F
is the unique element of Gal(E/F ) such that
p
E/F
(α) ≡ α Norm(p) mod P
for all α in E. We then extend
·
E/F
Hilbert class field H of K is the maximal unramified abelian extension of K.Proof. See [15, Proposition 3.1, p. 103].
2.1.3. 2 n -Hilbert class fields. Let K = Q(
√
−8p) and Cl = Cl(−8p). Recall that
the The Artin symbol induces a canonical isomorphism of groups
(2.4)
·
H/K
: Cl −→ Gal(H/K).
and u 2 of Z[τ ] are [u 1 , 16v 1 , 32u 1 ] and [u 2 , 16v 2 , 32u 2 ], respectively. The matrix17 96
3 17
∈ SL 2 (Z)
transforms the quadratic form [u 1 , 16v 1 , 32u 1 ] into [u 2 , 16v 2 , 32u 2 ], which proves the
lemma.
Now, for k ∈ {1, 2}, define ν k = u k + v k
√
2 similarly as in Section 2.2. Then
(2.23)
ν 2 = ε 8 ν 1 .
ON THE 16-RANK OF CLASS GROUPS OF Q(
√
−8P ) FOR P ≡ −1 MOD 4
mod 16 is a prime and p is a prime ideal of Z[ √ 2] lying above p, then a p coincides with the governing symbol p defined in (2.3). We first define a spin symbol [·] for all totally positive elements of Z[3.2. Extending governing symbols. In light of Proposition 4, our current goal
is to define a sequence {a n } indexed by non-zero ideals n of Z[
√
2] so that if p ≡
−1 √
2]. We put
[u + v
√
2] :=
v
u
if u is odd
0
otherwise
If u+v
√
2 0 generates a prime ideal p in Z[
√
2] lying above a prime p ≡ −1 mod 16
and if u ≡ 1 mod 16, then [u+v
√
2] = p , by definition (2.3). Indeed, the condition
ON THE 16-RANK OF CLASS GROUPS OF Q(
√
−8P ) FOR P ≡ −1 MOD 4
6.4. Twisted multiplicativity of governing symbols. Recall that if u + v √ 2 is a totally positive odd element of Z[ Thus [u + v √ 2] = 0 whenever u + v √ 2 is not primitive. A key feature of the governing symbol [·] which leads to significant cancellation in (3.2) is that [·] is not multiplicative, i.e., the relation [wz] = [w][z]√
2], we defined the governing symbol [u + v
√
2]
to be
[u + v
√
2] =
v
u
.
−8p) FOR p ≡ −1 mod 4
Acknowledgments. I would like to give special thanks to my PhD advisorsÉtienne Fouvry and Peter Stevenhagen for checking my work and helping me improve some arguments. I would also like to thank Hendrik Lenstra, Peter Sarnak, and Marco Streng for useful discussions. This research was partially supported by an ALGANT Erasmus Mundus Scholarship and by National Science Foundation agreement No. DMS-1128155.is already a function on Z[ √ 2]/(w 1 w 2 ), andwe have.By (6.4), we havewhere (α) and (β) are coprime primitive ideals of norm W/r and r, respectively, satisfying (w 1 w 2 ) = (αβ). Hencewhere z = z 01 · β · β + z 02 · α · α and α and β are some elements of Z[ √ 2] such that αα ≡ 1 mod β and ββ ≡ 1 mod α. With these choices, we haveThen, by (6.3), we haveCounting primesIn this section we give evidence that a governing field for the16Hence given any two subsets S 1 and S 2 of Gal(M/Q) which are stable under conjugation and of the same cardinality,is the best known bound. Note that this bound is weaker than X 1−δ for any δ > 0.However, we have the following result. Proof. We simply let S 1 be the union of Artin classes c p for primes p satisfying rk 16 Cl(−8p) = 1 and S 2 be the union of Artin classes c p for primes p satisfying rk 8 Cl(−8p) = 1 but rk 16 Cl(−8p) = 0. The result now immediately follows from Theorem 1.However, with our current methods of complex analysis applied to L-functions, we are not able to produce an error term of the form O(x 1−δ M ) for any δ M > 0. This leads us to believe that a governing field M for the 16-rank of the family {Q( √ −8p)} p≡3(4) is unlikely to exist.
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| []
|
[
"Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations",
"Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations"
]
| [
"Claude Bardos [email protected] \nUniversité Denis Diderot and Laboratory J. L. Lions Université Pierre et Marie Curie\nParisFrance\n",
"Jasmine S Linshiz [email protected] \nDepartment of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot\n76100Israel\n",
"Edriss S Titi [email protected] \nDepartment of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot\n76100Israel\n\nDepartment of Mathematics and Department of Mechanical and Aerospace Engineering\nUniversity of California Irvine\n92697-3875CAUSA\n"
]
| [
"Université Denis Diderot and Laboratory J. L. Lions Université Pierre et Marie Curie\nParisFrance",
"Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot\n76100Israel",
"Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot\n76100Israel",
"Department of Mathematics and Department of Mechanical and Aerospace Engineering\nUniversity of California Irvine\n92697-3875CAUSA"
]
| []
| We present an α-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-α equation, remains Lipschitz for all times, (ii) an initially Hölder C 1,β , 0 ≤ β < 1, chord arc curve remains in C 1,β for all times, and finally, (iii) an initially Hölder C n,β , n ≥ 1, 0 < β < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-α and the BR-α descriptions of the vortex sheet motion are equivalent. | null | [
"https://arxiv.org/pdf/0902.3356v2.pdf"
]
| 15,113,807 | 0902.3356 | ded1daef79a43f0dcf3776281a11c0fc1387edd4 |
Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations
1 Oct 2009
Claude Bardos [email protected]
Université Denis Diderot and Laboratory J. L. Lions Université Pierre et Marie Curie
ParisFrance
Jasmine S Linshiz [email protected]
Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot
76100Israel
Edriss S Titi [email protected]
Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot
76100Israel
Department of Mathematics and Department of Mechanical and Aerospace Engineering
University of California Irvine
92697-3875CAUSA
Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations
1 Oct 2009inviscid regularization of Euler equationsEuler-αBirkhoff-RottBirkhoff-Rott-αvortex sheet Mathematics Subject Classification: 76B03, 35Q35, 76B47
We present an α-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-α equation, remains Lipschitz for all times, (ii) an initially Hölder C 1,β , 0 ≤ β < 1, chord arc curve remains in C 1,β for all times, and finally, (iii) an initially Hölder C n,β , n ≥ 1, 0 < β < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-α and the BR-α descriptions of the vortex sheet motion are equivalent.
Introduction
The α-regularization of the Navier-Stokes equations (NSE) is one of the novel approaches for subgrid scale modeling of turbulence. The inviscid Euler-α model was originally introduced in the Euler-Poincaré variational framework in [38,39]. In [13-15, 31, 32] the corresponding Navier-Stokes-α (NS-α) [also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-α (LANS-α)] model, was obtained by introducing the appropriate viscous term into the Euler-α equations. The extensive research of the α-models (see, e.g., [2, 7, 10, 11, 13-18, 20, 31, 32, 34, 34-36, 40, 42, 43, 48, 49, 51, 52, 63, 77]) stems, on the one hand, from the successful comparison of their steady state solutions to empirical data, for a large range of huge Reynolds numbers, for turbulent flows in infinite channels and pipes [13][14][15]. On the other hand, the α-models can also be viewed as numerical regularizations of the original, Euler or Navier-Stokes, systems [7,11,44,52]. The main practical question arising is that of the applicability of these regularizations to the correct predictions of the underlying flow phenomena.
In this paper we present some analytical results concerning the α-regularization of the two-dimensional (2D) Euler equations in the context of vortex sheet dynamics. The incompressible Euler equations are ∂v ∂t + (v · ∇)v + ∇p = 0,
∇ · v = 0, v(x, 0) = v in (x), (1.1)
where v the fluid velocity field and p, the pressure are the unknowns, and v in is the given initial velocity.
A vortex sheet is a surface of codimension one (a curve in the plane) in inviscid incompressible flow, across which the tangential component of the velocity has a jump discontinuity, while the normal component is continuous. The flow outside the sheet is irrotational. The evolution of the vortex sheet can be described by the Birkhoff-Rott (BR) equation [8,67,68]. This is a nonlinear singular integro-differential equation, which can be obtained formally from the Euler equations assuming that the evolution of a vortex sheet retains a curve-like structure:
∂z ∂t (Γ, t) = 1 2πi p.v. ∞ −∞ dΓ ′ z (Γ, t) − z (Γ ′ , t) ,
here z = x + iy is the complex position of the sheet and Γ ∈ (−∞, ∞) represents the circulation, that is, γ = 1/|z Γ | is the vorticity density along the sheet. However, the initial data problem for the BR equation is ill-posed due to the Kelvin-Helmholtz instability [8,69]. Numerous results show that an initially real analytic vortex sheet (curve) can develop a finite time singularity in its curvature. This singularity formation was studied with asymptotic techniques in [23,64] and numerically in [23,46,62]. Specific examples of solutions were constructed in [9,29], where the development, in a finite time, of curvature singularity from initially analytic data was rigorously proved. After the appearance of the first singularity the solution becomes very irregular. This is a consequence of the elliptic nature of the Birkhoff-Rott equations: if solutions have a certain minimal regularity, then they are actually analytic [50,79,80]. An open problem is the determination of this threshold of regularity that will imply analyticity. It was shown in [50] that any solution consisting of a closed chord arc vortex sheet that near a point belongs to C 1,β , β > 0 must be analytic. The conclusion is maintained if the vortex sheet is required to be a Lipschitz chord arc curve [79,80]. The problem of the evolution of a vortex sheet can also be approached, in the general framework of weak solutions (in the distributional sense) of the Euler equations, as a problem of evolution of the vorticity, which is concentrated as a measure along a surface of codimension one. This approach was pioneered by DiPerna and Majda in [26][27][28]. The general problem of existence for mixed-sign vortex sheet initial data remains an open question. However, in 1991, Delort [25] proved a global in time existence of weak solutions of the 2D incompressible Euler equations for the vortex sheet initial data with initial vorticity being a Radon measure of a distinguished sign, see also [30,53,58,59,71,72]. This result was later obtained as an inviscid limit of the Navier-Stokes regularizations of the Euler equations [58,71], and as a limit of numerical vortex methods [53,54,72]. The Delort's result [25] was also extended to the case of mirror-symmetric flows with distinguished sign vorticity on each side of the mirror [57]. It is worth mentioning that uniqueness of solutions of the 2D Euler equations was obtained by Yudovich [81] for initially bounded vorticity, see, also, [76] for an improvement with vorticity in a class slightly larger than L ∞ , and [75] for review of relevant two-dimensional results. This does not include vortex sheets, which admit measure-valued vorticity. There is also a nonuniqueness result for velocity in C (0, T ) , L 2 weak [24,70,73]. However, the problem of uniqueness of a weak solution with a fixed sign vortex sheet initial data is still unanswered, numerical evidences of non-uniqueness can be found, e.g., in [55,66]. Furthermore, the structure of weak solutions given by Delort's theorem is not known, while the Birkhoff-Rott equations assume a priori that a vortex sheet remains a curve at a later time. A proposed criterion for the equivalence of a weak solution of the 2D Euler equations with vorticity being a Radon measure supported on a curve, and a weak solution of the Birkhoff-Rott equation can be found in [56]. Also, another definition of weak solutions of Birkhoff-Rott equation has been proposed in [79,80]. For a recent survey of the subject, see [4].
The Euler-α model [15,21,[37][38][39]61] is an inviscid regularization of the Euler equations (1.1) given by ∂v ∂t
+ (u · ∇) v + j v j ∇u j + ∇π = 0, v = 1 − α 2 ∆ u, ∇ · u = ∇ · v = 0,
u(x, 0) = u in (x).
(1.2)
Here u represents the "filtered" fluid velocity vector, π is the "filtered" pressure, and α > 0 is a regularization lengthscale parameter representing the width of the filter. The question of global existence of weak solutions for the three-dimensional Euler-α equations is still an open problem. On the other hand, the 2D Euler-α equations were studied in [65], where it has been shown that there exists a unique global weak solution to the Euler-α equations with initial vorticity in the space of Radon measures on R 2 , with a unique Lagrangian flow map describing the evolution of particles. In particular, it follows that the vorticity, initially supported on a curve, remains supported on a curve for all times.
In this paper we relate the weak solutions of Euler-α equations with distinguished sign vortex sheet initial data to those of the 2D Euler equations, by proving their convergence, as the length scale α → 0. This produces a variant of the result of Delort [25], by obtaining a weak solution of Euler equations as a limit of an inviscid regularization of Euler equations, in addition to approximations obtained by smoothing the initial data, viscous regularization, or numerical vortex methods [25,53,54,58,59,71,72]. Since a weak solution of Euler equations with vortex sheet is unlikely to be unique, a different regularization could produce a different weak solution.
We also present an analytical study of the α-analogue of the Birkhoff-Rott equation, the Birkhoff-Rott-α (BR-α) model, which is induced by the 2D Euler-α equations. The BR-α results that were reported in a short communication [3] are presented here with full details. The BR-α model was implemented computationally in [41], where a numerical comparison between the BR-α regularization and the existing regularizing methods, such as a vortex blob model [1,19,22,45,53], has been performed. In the BR-α case the singular kernel of the Biot-Savart law determining the velocity in terms of the vorticity is smoothed by a convolution with a smoothing function G α (x) = 1 α 2 1 2π K 0 |x| α , which is the Green function associated with the Helmholtz operator I − α 2 ∆ . The function K 0 is a modified Bessel function of the second kind of order zero. This is similar to vortex blob methods, however, unlike the standard vortex blob methods [1,6,19,22,45,47] (and, in particular, the proof of convergence of vortex blobs methods to a weak solution of 2D Euler equations [53]), the BR-alpha smoothing function G α is unbounded at the origin. Also, unlike the vortex blob methods that regularize the singular Biot-Savart kernel, the Euler-α model regularizes the Euler equations themselves to obtain a smoother kernel. Section 2 contains the preliminaries about the 2D Euler-α equations. In Section 3 we investigate the convergence of solutions of the Euler-α equations for vortex sheet initial data to those of the 2D Euler equations, as the regularization length scale α tends to zero. Specifically, we prove that for the vortex sheet initial data with initial vorticity of a distinguished sign Radon measure one can extract subsequences of weak solutions of the Euler-α equations which converge weak- * in L ∞ [0, T ] ; M(R 2 ) , as α → 0, to a weak solution of the 2D Euler equations. The space M(R 2 ) denotes the space of finite Radon measures on R 2 .
In Section 4 we describe the BR-α equation. Section 5 studies the linear stability of a flat vortex sheet with uniform vorticity density for the 2D BR-α model. The linear stability analysis shows that the BR-α regularization controls the growth of high wave number perturbations, which is the reason for the well-posedness. This is unlike the case for the original BR problem for Euler equations that exhibits the Kelvin-Helmholtz instability, the main mechanism for its ill-posedness. In Section 6 we show global wellposedness of the 2D BR-α model in the space of Lipschitz functions and in the Hölder space C n,β , n ≥ 1, which is the space of n-times differentiable functions with Hölder continuous n th derivative. Specifically, we show that (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-α equation, remains Lipschitz for all times, (ii) an initially Hölder C 1,β , 0 ≤ β < 1, chord arc curve remains in C 1,β for all times, and finally, (iii) an initially Hölder C n,β , n ≥ 1, 0 < β < 1, closed chord arc curve remains in C n,β for all times. Notice that for n > 1 we request β to be strictly larger than zero and the curve to be closed. In all these cases the weak Euler-α and the BR-α descriptions of the vortex sheet motion are equivalent. The convergence of BR-α solutions to the solutions of the original BR system on the short interval of existence of solutions will be reported in a forthcoming paper.
Euler-α equations
In two dimensions, the incompressible Euler equations in the vorticity form are obtained by taking the curl of (1.1) and are given by
∂q ∂t + (v · ∇) q = 0, v = K * q, q(x, 0) = q in (x), (2.1)
where K (x) = 1 2π ∇ ⊥ log |x|, v is the fluid velocity field, q = curl v is the vorticity, and q in is the given initial vorticity. Delort [25] proved a global in time existence of weak solutions of the 2D Euler equations for the vortex sheet initial data with fixed sign initial vorticity in M(
R 2 ) ∩ H −1 loc R 2 . The space M(R 2 ) is the space of finite Radon measures on R 2 with the norm µ M = sup R 2 ϕdµ : ϕ ∈ C 0 R 2 , ϕ L ∞ ≤ 1 , C 0 (R 2 )
is the space of continuous functions vanishing at infinity. The space H −s denotes the dual of the Sobolev space H s . The localized Sobolev space H s loc R 2 , s ∈ R is the set of all distributions f such that ρf ∈ H s (R 2 ) for any ρ ∈ C ∞ c (R 2 ), see, e.g., [33].
A vorticity q ∈ L ∞ [0, T ] , M(R 2 ) ∩ H −1 loc R 2 ∩ Lip [0, T ] , H −L loc R 2 , L > 1, is called a weak solution of (2.1), if for every test function ψ ∈ C ∞ c R 2 × (0, T ) W (q; ψ) ≡ T 0 R 2 ∂ t ψ (x, t) dq (x, t) dt + T 0 R 2 R 2 H ψ (x, y, t) dq (y, t) dq (x, t) dt = 0, (2.2) where H ψ (x, y, t) = 1 4π (x − y) ⊥ · (∇ψ (x, t) − ∇ψ (y, t)) |x − y| 2 . (2.3)
The initial value is q(x, 0) = q in (x) and it makes sense since q ∈ Lip [0, T ] , H −L loc R 2 . The kernel H ψ is bounded, continuous outside the diagonal x = y and vanishes at infinity. This weak vorticity formulation is well-defined, since the H −1 vorticity has no discrete part (i.e., q ({x 0 } , t) = 0 for all x 0 ∈ R 2 ), which implies that the diagonal x = y has q (x, t) q (y, t)-measure zero, see [25,71]. Thorough discussions of Delort's theorem, its extension and different proofs of the result can be found in [12,25,30,53,58,59,71,72].
Taking the curl of (1.2) yields the vorticity formulation of the 2D Euler-α model
∂q ∂t + (u · ∇) q = 0, u = K α * q, q(x, 0) = q in (x).
(2.4)
Here u represents the "filtered" fluid velocity, and α > 0 is a regularization length scale parameter, which represents the width of the filter. At the limit α = 0, we formally obtain the Euler equations (2.1). The smoothed kernel is K α = G α * K, where G α is the Green function associated with the Helmholtz operator I − α 2 ∆ , given by
G α (x) = 1 α 2 G x α = 1 α 2 1 2π K 0 |x| α , (2.5)
here x = (x 1 , x 2 ) ∈ R 2 and K 0 is a modified Bessel function of the second kind of order zero [78]. To see this relationship in R 2 one can take a Fourier transform of v = 1 − α 2 ∆ u, and obtain G α as the inverse Fourier transform of 1 (1+α 2 |k| 2 ) . Notice that
K α (x) = ∇ ⊥ Ψ α (|x|) = x ⊥ |x| DΨ α (|x|) , (2.6) where Ψ α (r) = 1 2π K 0 r α + log r , (2.7) DΨ α (r) = dΨ α dr (r) = 1 2π − 1 α K 1 r α + 1 r ,
and K 1 denotes a modified Bessel functions of the second kind of order one. For details on Bessel functions, see, e.g., [78].
A weak solution of (2.4) is q ∈ C [0, T ] ; M(R 2 ) satisfying W α (q; ψ) ≡ T 0 R 2 ∂ t ψ (x, t) dq (x, t) dt + T 0 R 2 R 2 H α ψ (x, y, t) dq (x, t) dq (y, t) dt = 0, (2.8) for all test functions ψ ∈ C ∞ c R 2 × (0, T ) .
The initial value is q(x, 0) = q in (x) and it makes sense since q ∈ C [0, T ] ; M(R 2 ) . The kernel H α ψ is a continuous vanishing at infinity function given by
H α ψ (x, y, t) = 1 2 DΨ α (|x − y|) (x − y) ⊥ · (∇ψ (x, t) − ∇ψ (y, t)) |x − y| . (2.9)
Oliver and Shkoller [65] showed global well-posedness of the Euler-α equations with initial vorticity in M(R 2 ). Theorem 2.1. (Oliver and Shkoller [65]) For initial data q in ∈ M(R 2 ), there exists a unique global weak solution of Euler-α equations (2.4) in the sense of (2.8).
Let G denote the group of all homeomorphism of R 2 , which preserve the Lebesgue measure and let η α = η α (·, t) denote the Lagrangian flow map induced by (2.4), i.e., which obeys the equation ∂ t η α (x, t) = u (η α (x, t), t) := R 2 K α (η α (x, t), η α (y, t)) dq in (y, t), η α (x, 0) = x. Then the unique Lagrangian flow map η α ∈ C 1 ([0, T ] ; G) exists globally and the vorticity q α is transported by the flow, i.e.,
q α (x, t) = q in • η −1 α (x, t).
Notice that the original BR equations assume a priori that a vortex sheet remains a curve at a later time, however, in the 2D Euler-α case, it follows as a consequence of the existence of the unique Lagrangian flow map, that the vorticity that is initially supported on a curve remains supported on a curve for all times.
3
Convergence of a fixed sign Euler-α vortex sheet to an Euler vortex sheet
Let the initial vorticity q in ∈ M(R 2 ) ∩ H −1 loc R 2 be of a fixed sign, q in ≥ 0, and compactly supported. In this section we show that there is a subsequence of the solutions of 2D Euler-α model with initial data q in , guaranteed by Theorem 2.1, that converge to a weak solution of 2D Euler equations in the sense of (2.2). This produces a variant of the result of Delort [25], by obtaining a weak solution of Euler equations as a limit of solutions of inviscid regularization of Euler equations, namely, the Euler-α equations. The above regularization method is different from the various existing regularizations that are obtained, for instance, by smoothing the initial data, viscous regularization or numerical vortex methods [25,53,54,58,59,71,72]. Since a weak solution of Euler equations with vortex sheet is unlikely to be unique, a different regularization could produce a different weak solution of Euler equations.
In order to prove the convergence of the solutions q α of the Euler-α equations (2.4) to a weak solution of Euler equations (2.1) we follow ideas similar to those reported in [25,58,59,71]. However, due to the structure of the Euler-α equations one needs to deal with various technical estimates concerning the "filtered" vorticity ω α = 1 − α 2 ∆ −1 q α and α 2 ∆ω α = q α − ω α . Specifically, we show in Lemma 3.2 and Lemma 3.3, respectively, that ω α have a uniform decay in small disks, sup α>0,0≤t≤T,0<R<1,x0∈ R 2 |x−x0|<R dω α (x, t) ≤ C (T ) |log R| −1/2 , and the contribution of R 2 d α 2 ∆ω α converges to zero, as α → 0.
Theorem 3.1. Let q α be the solutions of the weak vorticity formulation of Euler-α equations (2.8), guaranteed by Theorem 2.1, with initial data q in ∈ M(R 2 ) ∩ H −1 loc R 2 , q in ≥ 0 and compactly supported and let T > 0. Then there exists a subsequence q αj that weak- * converges to q in L ∞ [0, T ] ; M(R 2 ) and in M R 2 for each fixed t, as α j → 0, and q is a weak solution of the Euler equations (2.1) in the sense of (2.2) with initial data q in .
The weak- * convergence in L ∞ [0, T ] ; M(R 2 ) means that lim αj →∞ T 0 R 2 ϕ (x, t) dq αj (x, t) dt = T 0 R 2 ϕ (x, t) dq (x, t) dt, for all ϕ ∈ L 1 [0, T ] ; C 0 (R 2 ) .
We denote the velocity and the "filtered" velocity by v α and u α , respectively, and their corresponding vorticities by q α = curl v α and ω α = curl u α .
Given q α ∈ M(R 2 ), we define a linear continuous functional ω α = 1 − α 2 ∆ −1 q α acting on every
ϕ ∈ C 0 R 2 by ω α , ϕ = R 2 1 − α 2 ∆ −1 ϕ dq α , (3.1) where ψ = 1 − α 2 ∆ −1 ϕ is defined as the unique, vanishing at infinity, solution of ϕ = 1 − α 2 ∆ ψ, given by 1 − α 2 ∆ −1 ϕ = R 2 1 α 2 1 2π K 0 |y| α ϕ (x − y) dy,(3.2)
the function K 0 is a modified Bessel function of the second kind of order zero, K 0 > 0, ∞ 0 K 0 (r) rdr = 1, see, e.g., [78]. From the above its follows that
1 − α 2 ∆ −1 ϕ L ∞ ≤ ϕ L ∞ . We observe that if q α ≥ 0 then ω α is a nonnegative linear functional. Indeed, let ϕ ∈ C 0 R 2 , ϕ ≥ 0, then 1 − α 2 ∆ −1 ϕ = R 2 1 α 2 1 2π K 0 |y| α ϕ (x − y) dy ≥ 0, and hence by (3.1) ω α , ϕ ≥ 0. Also, | ω α , ϕ | ≤ q α M 1 − α 2 ∆ −1 ϕ L ∞ ≤ q α M ϕ L ∞ .
Therefore, by the Riesz representation theorem (see, e.g., [33,Chapter 7] ) the functional ω α can be represented by a unique nonnegative Radon measure, which we also denote by ω α , and
ω α M ≤ q α M . (3.3)
Again, by the Riesz representation theorem, a linear functional α 2 ∆ω α defined by
α 2 ∆ω α , ϕ = R 2 α 2 ∆ 1 − α 2 ∆ −1 ϕ dq α , (3.4)
for every ϕ ∈ C 0 R 2 , can be identified with a Radon measure, which we also denote by α 2 ∆ω α . Observe that, since for every ϕ ∈ C 0 R 2
α 2 ∆ 1 − α 2 ∆ −1 ϕ = 1 − α 2 ∆ −1 ϕ − ϕ, we have α 2 ∆ω α , ϕ ≤ q α M α 2 ∆ 1 − α 2 ∆ −1 ϕ L ∞ ≤ 2 q α M ϕ L ∞ , that is, α 2 ∆ω α M(R 2 ) ≤ 2 q α M(R 2 ) .
We note that by Theorem 2.1 the solution q α of Euler-α equations (2.8) is transported by the flow, that
is, q α (x, t) = q in • η −1 α (x, t), η α ∈ C 1 ([0, T ] ; G), hence for all t q α (·, t) M = q in M . (3.5)
In addition, if q in ≥ 0, then q α ≥ 0 for all times, and therefore also ω α ≥ 0 for all times.
The kernel H ψ appearing in the non-linear term of (2.2) is discontinuous on the diagonal x = y, so, following [26,59], to prove the convergence of the non-linear term we need the following estimate, which shows uniform decay of the "filtered" vorticity ω α in small disks.
Lemma 3.2. Let q α be the solutions of (2.8) with initial data q in ∈ M(R 2 ) ∩ H −1 loc R 2 , q in ≥ 0 and compactly supported. Then for ω α = 1 − α 2 ∆ −1 q α defined by (3.1), there exists a constant C = C (T ), such that for all α > 0, 0 ≤ t ≤ T , 0 < R < 1 and x 0 ∈ R 2 we have |x−x0|<R dω α (x, t) ≤ C (T ) |log R| −1/2 . (3.6)
Proof. Recall that ω α ≥ 0 for all times. The idea of the proof, which is shown in details below, is to convolve the initial data with a standard C ∞ c R 2 mollifier to obtain a sequence of solutions of the Euler-α equations that has a uniform decay of the circulation on small disks
|x−x0|≤R ω α,ε (x, t) dx ≤ C (T ) |log R| −1/2 , 0 < ε ≤ ε 0 , 0 ≤ t ≤ T , R < 1,
and then the weak- * limit in L ∞ [0, T ] , M R 2 of a subsequence ω α,εj when ε j → 0, which is the solution of Euler-α equations with initial data q in , satisfies a similar bound. We observe that, similarly to the Euler equations, any smooth radially symmetric vanishing at infinity vorticityq (|x|) defines a stationary solution of Euler-α equations
(2.4) with the corresponding velocitȳ v (x) = ∇ ⊥ ∆ −1q (|x|) = x ⊥ |x| 2 |x| 0 sq (s) ds.
This could be seen using the vorticity stream function formulation for Euler-α equations, which is
q t + J (ϕ, ∆ψ) = 0, q = ∆ψ, where J (ϕ, χ) = ∂ϕ ∂x1 ∂χ ∂x2 − ∂ϕ ∂x2 ∂χ ∂x1 is the Jacobian, ψ is the velocity stream function, v = ∇ ⊥ ψ, and ϕ = 1 − α 2 ∆ −1 ψ is the "filtered" stream function, u = ∇ ⊥ ϕ. Since ∆ and 1 − α 2 ∆ are rotationally invariant, we have that the correspondingω = 1 − α 2 ∆ −1q ,ψ = ∆ −1q andφ = 1 − α 2 ∆ −1ψ
are also radially symmetric, therefore J φ, ∆ψ = 0 and henceq defines a stationary solution of Euler-α equations.
Let ρ ∈ C ∞ c R 2 be a standard mollifier, for example,
ρ (x) = C exp 1/ |x| 2 − 1 if |x| < 1, 0 if |x| ≥ 1, R 2 ρ = 1, ρ ε (x) = 1 ε 2 ρ x ε .
Smoothing the initial data by a mollification with ρ ε , q in ε = ρ ε * q in , we have that for all 0 < ε < ε 0 the smoothed initial vorticities satisfy q in
ε ≥ 0, supp q in ε ⊆ {x| |x| < R 0 } (since q in is compactly supported), R 2 q in ε (x) dx = R 2 dq in (x)
. Following [26,59] for the 2D Euler case we decompose the velocity into a combination of a stationary bounded velocity plus a time dependent velocity with finite total energy. Letq (|x|) be any smooth radially symmetric function with compact support, such
that R 2q (|x|) dx = R 2 dq in (x). Definev = K * q,q in ε = q in ε −q andṽ in ε = K * q in ε . Notice, that by direct calculation divv = 0 andv, ∇v, ∂ 2v ∈ L ∞ R 2 . Since R 2q
in ε = 0, andq in ε has compact support we have thatṽ in ε ∈ L 2 R 2 . Also, due to the fact that q in ∈ M(R 2 ) ∩ H −1 loc R 2 with compact support, and hence, for ε ≤ ε 0 , the smooth q in ε are uniformly bounded in L 1 with a common compact support and v in ε = K * q in ε are uniformly bounded in L 2 loc , and sinceq is independent of ε, we have thatṽ in ε are uniformly bounded in
L 2 R 2 , for ε ≤ ε 0 .
Observe, that the stationary part
u (x) = 1 − α 2 ∆ −1v (x) = R 2 1 α 2 1 2π K 0 |y| α v (x − y) dy, satisfies ū L ∞ ≤ v L ∞ , (3.7) ∇ū L ∞ ≤ ∇v L ∞ , ∂ 2ū L ∞ ≤ 1 2π 1 α ∇v L ∞ ,
since K 0 and its derivative are smooth functions outside of the origin, satisfying |K 0 (r)| ≤ C log r, |DK 0 (r)| ≤ Cr −1 and rapidly decaying at infinity. Consider the partial differential equation
∂ ∂tṽ α,ε + (ũ α,ε · ∇)ṽ α,ε + j (ṽ α,ε ) j ∇ (ũ α,ε ) j (3.8) + (ũ α,ε · ∇)v + jv j ∇ (ũ α,ε ) j + (ū · ∇)ṽ α,ε + j (ṽ α,ε ) j ∇ū j + ∇π α,ε = 0, v α,ε = 1 − α 2 ∆ ũ α,ε .
This evolution equation is similar to the Euler-α equations.
Moreover, ifṽ α,ε (x, t) is the solution of the equation (3.8) with initial dataṽ in ε , then v α,ε (x, t) =ṽ α,ε (x, t) +v (x) is the solution of the 2D Euler-α equations (1.2) with initial data v in ε = K * q in ε .
Similarly to the Euler case (see, e.g., [59]) this equation has a unique global infinitely smooth solution, since, as in 2D Euler case, we have an a priori uniform control over the L ∞ norm of theq α,ε , which implies the global existence, as in the proof of the Beale-Kato-Majda criterion [5]. The solutionṽ α,ε is in C 1 [0, ∞) , H s R 2 for all s > 2, and hence, by Sobolev embedding theorem, ∂ kṽ α,ε and, consequently,
∂ kũ α,ε (x) = R 2 1 α 2 1 2π K 0 |y| α ṽ α,ε (x − y) dy are also in C 0 R 2 for all k.
Moreover, the solutionũ α,ε is in L ∞ [0, ∞) ; H 1 R 2 due to the following a priori estimate. Taking the inner product of (3.8) withũ α,ε we have (omitting the subindices α and ε)
1 2 d dt |ũ| 2 L 2 + α 2 |∇ũ| 2 L 2 = α 2 ((ū · ∇) ∆ũ,ũ) − j (ṽ j ∇ū j ,ũ) = I 1 − I 2 .
Since divū = 0, for I 1 we have
I 1 = −α 2 i,j,k ū i ∂ 2ũ j ∂x 2 k ∂ ∂x iũ j = α 2 i,j,k ∂ū i ∂x k ∂ũ j ∂x k ∂ũ j ∂x i + α 2 i,j,k ū i ∂ 2 ∂x i ∂x kũ j ∂ ∂x kũ j .
Since the second term on the right is zero, we obtain that
|I 1 | ≤ Cα 2 ∇ū L ∞ ∇ũ 2 L 2 .
Now we estimate I 2
I 2 = i,j,k ũ j ∇ū j ·ũ − α 2 ∆ũ j ∇ū j ·ũ = I 21 + I 22 .
We have
|I 21 | ≤ C ũ 2 L 2 ∇ū L ∞ and I 22 = α 2 i,j,k ∂ũ j ∂x k ∂ū j ∂x i ∂ũ i ∂x k + α 2 i,j,k ∂ũ j ∂x k ∂ 2ū j ∂x k ∂x iũ i , hence |I 22 | ≤ Cα 2 ∇ũ 2 L 2 ∇ū L ∞ + Cα 2 ∇ũ L 2 ũ L 2 ∂ 2ū L ∞ . To conclude, we obtain 1 2 d dt ũ 2 L 2 + α 2 ∇ũ 2 L 2 ≤ C α 2 ∇ū L ∞ ∇ũ 2 L 2 + ũ 2 L 2 ∇ū L ∞ + α ∇ũ L 2 ũ L 2 α ∂ 2ū L ∞ .
Hence, thanks to (3.7),
1 2 d dt ũ 2 L 2 + α 2 ∇ũ 2 L 2 ≤ C ∇v L ∞ α 2 ∇ũ 2 L 2 + ũ 2 L 2 ,
and by Grönwall inequality
ũ (·, t) 2 L 2 + α 2 ∇ũ (·, t) 2 L 2 ≤ e C ∇v L ∞ t ũ (·, 0) 2 L 2 + α 2 ∇ũ (·, 0) 2 L 2 ≤ e C ∇v L ∞ t ṽ (·, 0) 2 L 2 .
Hence we have that for all 0 < ε ≤ ε 0 , 0 ≤ t ≤ T , the solution of Euler-α equations with the smoothed initial data satisfies (we now put back the subindices α and ε)
u α,ε (·, t) L 2 (B(x0,1)) ≤ ũ α,ε (·, t) L 2 (B(x0,1)) + ū L 2 (B(x0,1)) ≤ ũ α,ε (·, t) L 2 (R 2 ) + π ū L ∞ (R 2 ) ≤ C(T ), where C(T ) = C q in M , q in H −1 , q L ∞ , ε 0 , R 0 e C ∇v L ∞ t + π ū L ∞ (R 2 )
. This is enough to show uniform decay of the vorticity ω α,ε in small disks (see [71], we remark that here the fixed sign of the vorticity comes into place 1 ):
for R < 1, ε ≤ ε 0 sup 0≤t≤T |x−x0|≤R ω α,ε (x, t) dx ≤ C (T ) |log R| −1/2 . By (3.3) ω α,ε (·, t) M ≤ q α,ε (·, t) M = q in ε M = q in M , hence there exists a subsequence ω α,εj which converges weak- * in L ∞ [0, T ]
, M R 2 to the limit ω α . This limit has a similar decay
sup 0≤t≤T |x−x0|<R dω α (x, t) ≤ lim inf εj →0 sup 0≤t≤T |x−x0|<R ω α,εj (x, t) dx ≤ C (T ) |log R| −1/2 .
Furthermore, q α = 1 − α 2 ∆ ω α is the solution of the Euler-α equations (2.4), the passing to the limit
in lim εj →0 W α q α,εj ; ψ = W α (q α ; ψ) is straightforward since H α ψ ∈ C [0, T ] , C 0 R 2 2 and q α,εj are equicontinuous in time with values in a negative Sobolev space W −2,1 (which, together with q α,εj * ⇀ q α in L ∞ [0, T ] , M R 2 , implies q α,εj (x, t) q α,εj (y, t) * ⇀ q α (x, t) q α (y, t) in L ∞ [0, T ] , M R 2 , see [71, Lemma 3.2])
. The equicontinuity follows from the fact that |x| DΨ α (|x|) is bounded (in fact, it is bounded independent of α) and hence we have for all
ψ ∈ C ∞ c R 2 × (0, T ) T 0 R 2 ∂ t ψ (x, t) q α,εj (x, t) dxdt = (3.9) = 1 2 T 0 R 2 R 2 DΨ α (|x − y|) (x − y) ⊥ · (∇ψ (x, t) − ∇ψ (y, t)) |x − y| q α,εj (x, t) q α,εj (y, t) dxdydt ≤ 1 2 |x − y| DΨ α (x − y) L ∞ T 0 D 2 ψ (·, t) L ∞ (R 2 ) R 2 q α,εj (x, t) dx R 2 q α,εj (y, t) dydt ≤ C q in 2 M ψ L 1 ([0,T ],W 2,∞ (R 2 )) ≤ C q in 2 M ψ L 1 ([0,T ],H 4 (R 2 )) ,
where in the last inequality we used the Sobolev embedding theorem. Hence ∂ t q α,εj are uniformly bounded in L ∞ [0, T ] , H −4 R 2 , and hence q α,εj are uniformly bounded in
Lip [0, T ] ; H −4 R 2 .
We also need the following result 1 In [71] to prove the uniform decay of the vorticity in small circles one defines for R < 1
δ R (x) = 8 > > < > > : 1 |x| ≤ R, log( √ R/|x|) log(1/ √ R) R ≤ |x| ≤ √ R, 0 |x| ≥ √ R.
Then
|∇δ R | L 2 ≤ C |log R| −1/2 . We have Z |x−x 0 |≤R ωα,ε (x, t) dx ≤ Z R 2 δ R (x − x 0 ) ωα,ε (x, t) dx ≤˛Z R 2 ∇ ⊥ δ R (x − x 0 ) uα,ε (x, t) dx≤ |∇δ R | L 2 |uα,ε (·, t)| L 2 (B(x 0 ,1)) ≤ C (T ) |log R| −1/2 .
Here in the second transaction we used the fact that ωα,ε ≥ 0.
Lemma 3.3. Let q be a finite Radon measure, q = 1 − α 2 ∆ ω, as defined in (3.1)-(3.4), then R 2 d α 2 ∆ω ≤ Cα q M .
Proof. For the theory of Radon measures, see, e.g., [33]. First, we show that for all compact K ⊂ R 2
α 2 ∆ω (K) ≤ Cα q M .
By Riesz representation theorem (see, e.g., [33,Chapter 7] )
α 2 ∆ω (K) = inf R 2 f d α 2 ∆ω : f ∈ C c R 2 , f ≥ χ K . Let R be such that K ⊂ B (0, R), take θ ∈ C ∞ c (R 2 ) with 0 ≤ θ(x) ≤ 1 for all x, with θ(x) = 1 if |x| ≤ R, θ(x) = 0 if |x| ≥ R + 1. For example, θ = χ B(0,R+1/2) * ρ ε=1/4
. Then by (3.1) and using that
α∆ 1 − α 2 ∆ −1 θ L ∞ ≤ C ∇θ L ∞ (see, e.g., (3.7)), we have α 2 ∆ω (K) ≤ R 2 θd α 2 ∆ω ≤ R 2 α 2 ∆ 1 − α 2 ∆ −1 θ d |q| ≤ Cα q M ∇θ L ∞ ≤ Cα q M .
Now, since a Radon measure is inner regular we have
α 2 ∆ω R 2 = sup α 2 ∆ω (K) : K ⊂ R 2 , K compact ≤ Cα q M .
Now we are ready to prove Theorem 3.1. We notice that (3.9) implies that q α ∈ Lip [0, T ] ; H −4 R 2 . Hence due to (3.5) there exists a subsequence, that we relabel as q α , such that q α ⇀ q weak- * in L ∞ [0, T ] , M R 2 and in M R 2 for each fixed t, as α → 0. Also, due to
q α ∈ Lip [0, T ] ; H −4 R 2 and q α ⇀ q weak- * in L ∞ [0, T ] , M R 2 , we have that q α (x, t) q α (y, t) ⇀ q (x, t) q (y, t) weak- * both in L ∞ [0, T ] , M R 2 and in M R 2 for each fixed t ∈ [0, T ], as α → 0 (see [71, Lemma 3.2]).
Since q α is uniformly bounded in M(R 2 ) and Lip
R 2 ψ (t) ϕ (x) dq α (x, t) for every ψ ∈ C c ([0, T ])
, ϕ ∈ C c R 2 shows thatq = q, and hence the limit q belongs to Lip [0, T ] , H −4 loc R 2 as well. We observe that ω α (t, ·) also weak- * converges to q in M R 2 for every t ∈ [0, T ], as α → 0. Indeed, let ϕ ∈ C c R 2 then
R 2 ϕ (x) dq (x, t) − R 2 ϕ (x) dω α (x, t) ≤ R 2 ϕ (x) dq (x, t) − R 2 ϕ (x) dq α (x, t) + R 2 ϕ (x) dq α (x, t) − R 2 ϕ (x) dω α (x, t)
the first term on the right-hand side converges to 0, since q α * ⇀ q in M R 2 , as α → 0, and the second term is equal to R 2 ϕd α 2 ∆ω α ≤ ϕ L ∞ R 2 d α 2 ∆ω → 0, as α → 0, due to Lemma 3.3. Hence also q decays in small disks, that is, for all 0 ≤ t ≤ T , 0 < R < 1 and
x 0 ∈ R 2 |x−x0|<R dq (x, t) ≤ lim inf α→0 |x−x0|<R dω α (x, t) ≤ C (T ) |log R| −1/2 . (3.10)
Next we show that q is a weak solution of the Euler equations (2.2), namely, for every test function
ψ ∈ C ∞ c R 2 × (0, T ) W (q; ψ) = lim α→0 W α (q α ; ψ) = 0.
The convergence of the linear term is obvious from the weak- * convergence q α ⇀ q in L ∞ [0, T ] ; M(R 2 ) , as α → 0. Hence we need to show the convergence for the non-linear term
lim α→0 W α N L (q α ; ψ) = W N L (q; ψ) . We rewrite W N L (q; ψ) − W α N L (q α ; ψ) as W N L (q; ψ) − W α N L (q α ; ψ) = T 0 R 2 R 2 H ψ (x, y, t) [dq (x, t) dq (y, t) − dq α (x, t) dq α (y, t)] dt + T 0 R 2 R 2 H ψ (x, y, t) − H α ψ (x, y, t) dq α (x, t) dq α (y, t) dt = I 1 + I 2 .
We recall that the kernel H ψ is bounded by a constant times D 2 ψ L ∞ , tends to zero at infinity, and it is discontinuous on the diagonal x = y (see [71]). Let θ (|x|) ∈ C ∞ c (R 2 ) be a fixed cutoff function 0 ≤ θ ≤ 1 with θ = 1 for |x| ≤ 1 and θ = 0 for |x| ≥ 2. Let 0 < δ < 1. Write I 1 as
I 1 = T 0 R 2 R 2 1 − θ |x − y| δ H ψ (x, y, t) (dq (x, t) dq (y, t) − dq α (x, t) dq α (y, t)) dt + T 0 R 2 R 2 θ |x − y| δ H ψ (x, y, t) (dq (x, t) dq (y, t) − dq α (x, t) dq α (y, t)) dt = I 11 + I 12 . Since 1 − θ |x−y| δ H ψ ∈ C [0, T ] , C 0 R 2 2 and q α (x, t) q α (y, t) ⇀ q (x, t) q (y, t) weak- * in L ∞ [0, T ]
, M R 2 as α → 0, then lim α→0 I 11 = 0. Now we estimate I 12
|I 12 | ≤ T 0 R 2 |x−y|<2δ |H ψ (x, y, t)| dq (x, t) dq (y, t) dt + T 0 R 2 |x−y|<2δ |H ψ (x, y, t)| dq α (x, t) dq α (y, t) dt = I 121 + I 122 .
For I 121 , due to uniform decay of the vorticity q in small disks (3.10), we have for 2δ < 1
I 121 ≤ |H ψ | L ∞ T 0 R 2 B(y,2δ) dq (x, t) dq (y, t) dt ≤ C (T ) |log 2δ| −1/2 q in M .
To estimate I 122 we use (3.6) (for 2δ < 1) and Lemma 3.3.
I 122 ≤ |H ψ | L ∞ T 0 R 2 |x−y|<2δ d 1 − α 2 ∆ ω α (x, t) d 1 − α 2 ∆ ω α (y, t))dt = |H ψ | L ∞ T 0 R 2 |x−y|<2δ dω α (x, t) dω α (y, t))dt + |H ψ | L ∞ T 0 2 R 2 dω α (x, t) R 2 d α 2 ∆ω α (x, t) + R 2 d α 2 ∆ω α (x, t) 2 dt ≤ C (T ) |log 2δ| −1/2 q in 2 M + α (1 + α) CT q in 2 M .
Thus, I 12 → 0, as δ and α converge to zero.
It remains to estimate I 2 , by (2.3), (2.9) and (2.7)
I 2 = 1 4π T 0 R 2 R 2 1 α K 1 |x − y| α (x − y) ⊥ ∇ (ψ (x, t) − ψ (y, t)) |x − y| dq α (x, t) dq α (y, t) dt. Now, for r α → ∞ r α K 1 r α ≤ C π 2 1/2 √ r α e r α
→ 0 [78]. Hence, for each ε > 0, there is an L large enough, depending on ε, such that r α K 1 r α < ε, whenever r α ≥ L. Write I 2 as
I 2 = 1 4π T 0 R 2 R 2 1 − θ |x − y| αL 1 α K 1 |x − y| α (x − y) ⊥ ∇ (ψ (x, t) − ψ (y, t)) |x − y| dq α (x, t) dq α (y, t) dt + 1 4π T 0 R 2 R 2 θ |x − y| αL 1 α K 1 |x − y| α (x − y) ⊥ ∇ (ψ (x, t) − ψ (y, t)) |x − y| dq α (x, t) dq α (y, t) dt = I 21 + I 22 .
We have
I 21 ≤ 1 4π T 0 R 2 |x−y| α >L 1 − θ |x − y| αL |x − y| α K 1 |x − y| α |∇ (ψ (x, t) − ψ (y, t))| |x − y| dq α (x, t) dq α (y, t) dt ≤ ε D 2 ψ L ∞ 1 4π T 0 R 2 |x−y| α >L dq α (x, t) dq α (y, t) dt ≤ ε D 2 ψ L ∞ 1 4π T 0 R 2 dq α (x, t) 2 ≤ ε D 2 ψ L ∞ 1 4π T q in 2 M .
Since r α K 1 r α ≤ C for all r (independent of α), then similarly to the bound on I 122 , we have that for α < 1 2L
I 22 ≤ C (T ) |log 2αL| −1/2 D 2 ψ L ∞ q in M + α (1 + α) CT D 2 ψ L ∞ q in 2 M .
Hence for each ε > 0, there is an L large enough, depending on ε, such that (for α < 1 2L )
I 2 ≤ C T, ψ, q in M (ε + |log 2αL| −1/2 + α (1 + α)).
For each ε > 0, there is δ * such that |log r| −1/2 < ε, whenever r < δ * . Hence, for α < min δ * 2L , 1 2L , ε I 2 ≤ εC T, ψ, q in M . Therefore, lim α→0 I 2 = 0. This concludes the proof that q is a weak solution of the Euler equations (2.2) with initial data q in .
Birkhoff-Rott-α equation
The Birkhoff-Rott-α equation, based on the Euler-α equations (2.4) is derived similarly to the original Birkhoff-Rott equation. Detailed descriptions of the Birkhoff-Rott equation as a model for the evolution of the vortex sheet can be found, e.g., in [59,60,68]. We remark, however, that while the BR equations assume a priori that a vortex sheet remains a curve at a later time, in the 2D Euler-α case, if the vorticity is initially supported on a curve, then due to the existence of the unique Lagrangian flow map ∂ t η(x, t) = , t), given by Theorem 2.1 of Oliver and Shkoller [65], it remains supported on a curve for all times. Existence of the unique Lagrangian flow map implies that the BR-α equation gives an equivalent description of the vortex sheet evolution, as the weak solution of 2D Euler-α equations. It is described in the following proposition.
R 2 K α (x, y) dq (y, t), η (x, 0) = x, q (x, t) = q in • η −1 (xProposition 4.1. Let q in ∈ M(R 2 ) supported on the sheet (curve) Σ in = x = x(σ) ∈ R 2 |σ in 0 ≤ σ ≤ σ in 1
, with a density γ in (σ), that is, the vorticity q in satisfies
R 2 ϕ(x)dq in (x) = σ in 1 σ in 0 ϕ (x(σ)) γ in (σ)|x σ (σ) |dσ, for every ϕ ∈ C ∞ c R 2 , γ in ∈ L 1 (|x σ | dσ) 2 .
Let q be the solution of (2.4) in the sense of the Theorem 2.1. Then, for as long as the curve Σ(t) = x = x(σ, t) ∈ R 2 | σ 0 (t) ≤ σ ≤ σ 1 (t) remains nice enough so that x σ makes sense a.e., q has a density γ(σ, t) supported on the sheet Σ(t), γ (·, t) ∈ L 1 (|x σ | dσ), γ (σ, t) |x σ (σ, t)| dσ = γ (σ, 0) |x σ (σ, 0)| dσ and the sheet evolves according to the equation
∂ ∂t x (σ, t) = σ1(t) σ0(t) K α (x (σ, t) − x (σ ′ , t)) γ (σ ′ , t) |x σ (σ ′ , t)| dσ ′ , (4.1)
where K α is given by (2.6).
Additionally, if Γ (σ, t) = σ σ * γ (σ ′ , t) |x σ (σ ′ , t)| dσ ′ , where x (σ * , t)
is some fixed reference point on Σ(t), defines a strictly increasing function of σ (e.g., as in the case of positive vorticity), then the evolution equation is given by the Birkhoff-Rott-α (BR-α) equation
∂ ∂t x (Γ, t) = Γ1 Γ0 K α (x (Γ, t) − x (Γ ′ , t)) dΓ ′ (4.2)
with γ = 1/|x Γ | being the vorticity density along the curve and −∞ < Γ 0 < Γ 1 < ∞.
2 Let Σ be a curve parametrized by x(σ) : [σ 0 , σ 1 ] → R 2 , such that xσ ∈ L 1 ([σ 0 , σ 1 ]), and let q ∈ M(R 2 ) be supported on the curve Σ, with a density γ. Then γ ∈ L 1 (|xσ| dσ) (and vice versa).
Proof. First, assume q ≥ 0, and let θn be a truncating sequence, θn ∈ C ∞ c (R 2 ), θn (x) = θ 1`x n´, θ 1 ∈ C ∞ c (R 2 ), 0 ≤ θ 1 ≤ 1 with θ 1 = 1 for |x| ≤ 1 and θ 1 = 0 for |x| ≥ 2. Then, on the one hand,
Z σ 1 σ 0 θn (x(σ)) γ(σ)|xσ (σ) |dσ ≥ Z {σ:|x(σ)|≤n}∩[σ 0 ,σ 1 ] γ(σ)|xσ (σ) |dσ, on the other hand Z σ 1 σ 0 θn (x(σ)) γ(σ)|xσ (σ) |dσ = Z R 2 θn(x)dq in (x) ≤ θn L ∞ q M ≤ q M , hence Z {σ:|x(σ)|≤n}∩[σ 0 ,σ 1 ] γ(σ)|xσ (σ) |dσ ≤ q M .
Since n can be taken arbitrary large this implies that R σ 1 σ 0 γ(σ)|xσ (σ) |dσ < ∞. Now, for a signed measure we apply the previous result to each of the nonnegative measures q + , q − , given by the Jordan Decomposition of q, q = q + − q − , which is defined by
Z R 2 ϕ(x)dq ± (x) = Z σ 1 σ 0 ϕ (x(σ)) γ ± (σ)|xσ (σ) |dσ.
In Section 6 we show the global well-posedness of the Birkhoff-Rott-α (4.2) in the space of Lipschitz functions and in the Hölder space C n,β , n ≥ 1, which is the space of n-times differentiable functions with Hölder continuous n th derivative. Thus the solutions the Birkhoff-Rott-α and of the Euler-α are equivalent for the initial data being a finite positive Radon measure supported on Lipschitz or Hölder C 1,β ((Γ 0 , Γ 1 )), 0 ≤ β < 1, chord arc curve, or supported on C n,β ((Γ 0 , Γ 1 )), n ≥ 1, 0 < β < 1, closed chord arc curve.
Here σ 0 , σ 1 can represent either a finite length curve, or an infinite one. We remark that the smoothed kernel K α (x) is a bounded continuous function, that for |x| α → 0 behaves asymptotically as K α (x) = − 1 4π 1 α 2 x ⊥ log |x| α +O |x| α 2 , i.e., it is non-singular kernel at the origin. For the case where γ (·, t) ∈ L 1 (|x σ | dσ) we can show the integrability of the relevant terms, even though |K α (x)| is decaying like |x| −1 at infinity.
Linear stability of a flat vortex sheet with uniform vorticity density for 2D BR-α model
The initial data problem for the BR equation is highly unstable due to an ill-posed response to small perturbations called Kelvin-Helmholtz instability [8,69]. The linear stability analysis of the BR-α equation shows that the ill-posedness of the original problem is mollified, and the Kelvin-Helmholtz instability of the original system now disappears. We assume that the vortex sheet can be parameterized as a graph x 2 = x 2 (x 1 , t), the proof can be easily adapted to establish the result in general. In this case, following calculations presented in [60, Section 6.1], one can infer from (4.1) the system
∂x 2 ∂t = − ∂x 2 ∂x 1 u 1 + u 2 , (5.1) ∂γ ∂t = − ∂ ∂x 1 (γu 1 ) ,
where the velocity field u = (u 1 , u 2 ) t is given by t)) t and p.v. is taken at infinity. We recall that K α (x) = x ⊥ |x| DΨ α (|x|) and DΨ α (r) = 1 2π − 1 α K 1 r α + 1 r , and K 1 denotes a modified Bessel functions of the second kind of order one. By linearization of (5.1) about the flat sheet x 0 2 ≡ 0 with uniformly concentrated intensity γ 0 , which is a stationary solution of (5.1), we obtain the following linear system
u (x 1 , t) = p.v. R K α (x (x 1 , t) − x (x ′ 1 , t)) γ (x ′ 1 , t) dx ′ 1 , here x (x 1 , t) = (x 1 , x 2 (x 1 ,∂x 2 ∂t =ũ 2 , ∂γ ∂t = −γ 0 ∂ũ 1 ∂x 1 , whereũ 1 (x 1 , t) = −γ 0 (sgn (x 1 ) DΨ α (|x 1 |)) * ∂x 2 ∂x 1 , u 2 (x 1 , t) = (sgn (x 1 ) DΨ α (|x 1 |)) * γ,
and (x 2 ,γ) is a small perturbation about the flat sheet x 2 ≡ 0 with the constant density γ = γ 0 . See [74] for the description of the original Birkhoff-Rott system in such a case. Consequently, the equation for the Fourier modes (transform) of the above system is given by d dt
x 2 γ = 0 i 2 sgn(k)d(k) −i γ 2 0 2 k 2 sgn(k)d(k) 0 x 2 γ , (5.2) where d(k) = 1 + 1 α 2 k 2 −1/2 − 1.
Observe that in order to calculate the Fourier transform
F (sgn (x 1 ) DΨ α (|x 1 |)) (k) = i 2 sgn(k)d(k),
we used here the integral representation of the modified Bessel function of the second kind K 1 (x 1 ) = x 1 ∞ 1 e −x1t t 2 − 1 1/2 dt, (see, e.g., [78]). The eigenvalues of the coefficient matrix, given in (5.2),
are λ(k) = ± 1 2 |γ 0 | |k| 1 − 1 + 1 α 2 k 2 −1/2 . (5.3)
We observe that while the linear system for the original Birkhoff-Rott equation is elliptic (in space and time)
∂ 2 x 2 ∂t 2 − 1 4 γ 2 0 k 2 x 2 = 0,
for a Birkhoff-Rott-α equation it is no longer an elliptic system
∂ 2 x 2 ∂t 2 − 1 4 γ 2 0 d 2 (k)k 2 x 2 = 0,
since d 2 (k)k 2 is bounded and behaves like 1 α 4 k 2 , as k → ∞ (for α fixed). To conclude, the α-regularization mollifies the Kelvin-Helmholtz instability as follows: we have an algebraic decay of the eigenvalues to zero of order 1 α 2 |k| , as k → ∞ (for α fixed). While, for α → 0, for fixed k, we recover the eigenvalues of the original BR equations ± 1 2 |γ 0 | |k| (see, e.g., [74]). For the sake of comparison, we note that for the vortex blob regularization of Krasny [46], where the singular BR kernel, K(x), was replaced with the smoothed kernel
K δ (x) = K (x) |x| 2 |x| 2 + δ 2 = 1 2π x ⊥ |x| 2 + δ 2 ,
the eigenvalues are λ(k) = ± 1 2 e −δk |γ 0 | |k| with an exponential decay to zero, as k → ∞ (δ > 0 is fixed). As δ → 0, for fixed k, one recovers again the eigenvalues of the original BR equations. The behavior of the eigenvalues of the linearized system (5.2) indicates that high wave number perturbations grow exponentially in time with a rate that decays to zero, as k → ∞, which is the reason for well-posedness of the α-regularized model. This is unlike the original BR problem that exhibits the Kelvin-Helmholtz instability. It is worth mentioning that the α-regularization is "closer" to the original system than the vortex-blob method at the high wave numbers, due to the algebraic decay instead of exponential one in the vortex blob method. This result was also evaluated computationally in [41].
Global regularity for BR-α equation
In this section we present the global existence and uniqueness of solutions of the BR-α equation (4.2) in the space of Lipschitz functions and in the Hölder space C n,β , n ≥ 1, which is the space of n-times differentiable functions with Hölder continuous n th derivative.
Let us first describe the Hölder space C n,β J ⊂ R; R 2 , 0 < β ≤ 1, which is the space of functions x : J ⊂ R → R 2 , with a finite norm
x n,β(J) = n k=0 d k dΓ k x C 0 (J) + d n dΓ n x β(J)
, where
x C 0 (J) = sup Γ∈J |x (Γ)|
and |·| β is the Hölder semi-norm
|x| β(J) = sup Γ,Γ ′ ∈J Γ =Γ ′ |x (Γ) − x (Γ ′ )| |Γ − Γ ′ | β .
The Lipschitz space Lip (J) is the C 0,1 space, that is, with the finite norm x Lip(J) = x C 0 (J) + |x| 1(J) . We also use the notation
|x| * = inf |x (Γ) − x (Γ ′ )| |Γ − Γ ′ | ,
where the infimum is taken over all Γ, Γ ′ ∈ J such that Γ = Γ ′ , or, in the case of a closed curve (without loss of generality, over S 1 ), the infimum is taken over all Γ, Γ ′ ∈ S 1 such that Γ = Γ ′ mod 2π. We consider the BR-α equation as an evolution functional equation in the Banach spaces Lip, C 1 or C n,β , n ≥ 1, 0 < β < 1,
∂x ∂t (Γ, t) = Γ1 Γ0 K α (x (Γ, t) − x (Γ ′ , t)) dΓ ′ , x (Γ, 0) = x 0 (Γ) (6.1)
with γ = 1/|x Γ | being the vorticity density along the sheet and −∞ < Γ 0 < Γ 1 < ∞. Notice that the density γ (Γ) is in C n−1,β ((Γ 0 , Γ 1 )) for the subset {|x| * > 0} of C n,β ((Γ 0 , Γ 1 )), and γ (Γ) ∈ L ∞ ((Γ 0 , Γ 1 )) for the subset {|x| * > 0} of Lip ((Γ 0 , Γ 1 )).
Theorem 6.1. Let −∞ < Γ 0 < Γ 1 < ∞, V be either the space C 1,β ((Γ 0 , Γ 1 )), 0 ≤ β < 1 or the space Lip ((Γ 0 , Γ 1 )) and let x 0 ∈ V ∩ {|x| * > 0}. Then for any T > 0 there is a unique solution x ∈ C 1 ([−T, T ]; V ∩ {|x| * > 0}) of (6.1) with initial value x (Γ, 0) = x 0 (Γ). Furthermore, let x 0 be a closed curve and without loss of generality, we assume x 0 (Γ) ∈ C n,β S 1 ∩ {|x| * > 0}, then for all n ≥ 1, 0 < β < 1, T > 0 there is a unique solution x ∈ C 1 [−T, T ]; C n,β S 1 ∩ {|x| * > 0} of (4.2) with initial value x (Γ, 0) = x 0 (Γ). In particular, if
x 0 ∈ C ∞ S 1 ∩ {|x| * > 0} then x ∈ C 1 [−T, T ]; C ∞ S 1 ∩ {|x| * > 0} .
Notice that for n > 1 we request β to be strictly larger than zero and the curve to be closed. We remark that, although the kernel K α is a continuous bounded function, its derivatives are unbounded near the origin, and the chord arc condition |x| * > 0, which implies simple curves, allows us to show the integrability of the relevant terms.
The following are the main steps involved in the proof of Theorem 6.1. In the first step, we apply the contraction mapping principle to the BR-α equation (4.2) to prove the short time existence and uniqueness of solutions in the appropriate space of functions. Specifically, we show that an initially Lipschitz or C 1,β , 0 ≤ β < 1 smooth solutions of (4.2) remain, respectively, Lipschitz or C 1,β smooth for a finite short time. Next, we derive an a priori bound for the controlling quantity for continuing the solution for all time. At step three we extend the C 1,β , 0 < β < 1, result for higher derivatives for the case of a closed curve.
Step 1. Local well-posedness.
First we show the local existence and uniqueness of solutions. To apply the contraction mapping principle to the BR-α equation (6.1), we first prove the following result Proposition 6.2. Let −∞ < Γ 0 < Γ 1 < ∞, 1 < M < ∞, V be either the space C 1,β ((Γ 0 , Γ 1 )), 0 ≤ β < 1 or the space Lip ((Γ 0 , Γ 1 )), and let K M be the set
K M = x ∈ V : |x| 1 < M, |x| * > 1 M .
Then the mapping
x (Γ) → u (x (Γ)) = Γ1 Γ0 K α (x (Γ) − x (Γ ′ )) dΓ ′ (6.2)
defines a locally Lipschitz continuous map from K M , equipped with the topology induced by the · V norm, into V .
Proof. Notice that K M is an open set in V . We recall that K α (x) = ∇ ⊥ Ψ α (|x|) = x ⊥ |x| DΨ α (|x|) , where Ψ α (r) = 1 2π K 0 r α + log r and DΨ α (r) = dΨ α dr (r) = 1 2π − 1 α K 1 r α + 1 r .
The functions K 0 and K 1 denote the modified Bessel functions of the second kind of orders zero and one, respectively. For details on Bessel functions, see, e.g., [78]. We observe that DΨ α is bounded
DΨ α (r) ≤ C α , (6.3)
derivatives of Ψ α decay to zero as r α → ∞, and as r α → 0 satisfy
DΨ α (r) = − 1 4π r α 2 log r α + O r α 2 , (6.4) D 2 Ψ α (r) = − 1 4π 1 α 2 log r α + O 1 α 2 , D 3 Ψ α (r) = − 1 4π 1 rα 2 + O r α 4 log r α .
The constant C will denote a generic constant independent of the parameters, while, C(⋄) denotes a constant which depends on ⋄.
First we show the local existence and uniqueness of solutions in C 1,β , 0 < β < 1.
We start by showing that u (x (Γ)) maps K M into C 1,β . Let x ∈ K M . Using the boundness of DΨ α (6.3) we have
|u (x (Γ))| ≤ Γ1 Γ0 DΨ α (|x (Γ) − x (Γ ′ )|) dΓ ′ ≤ C α (Γ 1 − Γ 0 ) . (6.5) Using that du dΓ (x (Γ)) = Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) dΓ ′ ,
(which can be justified by applying Lebesgue dominated convergence theorem) and the fact that dx dΓ (Γ) < M , we obtain
du dΓ (x (Γ)) ≤ M Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dΓ ′ , = M (Γ0,Γ1)∩ |Γ−Γ ′ | α <ε ff + (Γ0,Γ1)\ |Γ−Γ ′ | α <ε ff , = M (I 1 + I 2 ) ,
where ε is to be defined later. Due to (2.6), (6.4) and
1 M |Γ − Γ ′ | α < |x| * |Γ − Γ ′ | α ≤ |x (Γ) − x (Γ ′ )| α ≤ dx dΓ C 0 |Γ − Γ ′ | α ≤ M ε, (6.6)
we have that for a fixed small ε
I 1 ≤ |Γ−Γ ′ | α <ε 1 4πα 2 log C (M ) |Γ − Γ ′ | α + C (M ) 1 α 2 dΓ ′ = C (M ) 1 α . For I 2 due to the boundness of |∇K α | in (Γ 0 , Γ 1 ) \ |Γ−Γ ′ | α < ε we obtain I 2 ≤ C (M ) α 2 (Γ 1 − Γ 0 ) . Summing up, Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dΓ ′ ≤ C M, Γ 1 , Γ 0 , 1 α (6.7)
and hence du dΓ
(x (Γ)) ≤ C M, Γ 1 , Γ 0 , 1 α . (6.8)
To show the Hölder continuity of du dΓ (x (Γ)) we write
du dΓ (x (Γ)) − du dΓ x Γ ≤ Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dx dΓ (Γ) − dx dΓ Γ dΓ ′ + Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) − ∇K α x Γ − x (Γ ′ ) dx dΓ Γ dΓ ′ ≤ dx dΓ β |Γ − Γ ′ | β Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dΓ ′ + + M Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) − ∇K α x Γ − x (Γ ′ ) dΓ ′ .
The first term on the right-hand side is bounded by C M, Γ 1 , Γ 0 , 1 α dx dΓ β Γ −Γ β due to (6.7), as for the second one, let r = |Γ−Γ| α , and write
I = Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) − ∇K α x Γ − x (Γ ′ ) dΓ ′ = (Γ0,Γ1)∩ |Γ−Γ ′ | α <2r ff + (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff = I 1 + I 2 .
For I 1 we have |Γ−Γ ′ | α < 2r, and hence |Γ−Γ ′ | α < 3r. Due to (2.6), the fact that
D 2 Ψ α (s) ≤ 1 4π 1 α 2 log s α + C α 2 (6.9)
and (6.6) we obtain
I 1 ≤ (Γ0,Γ1)∩ |Γ−Γ ′ | α <2r ff |∇K α (x (Γ) − x (Γ ′ ))| + ∇K α x Γ − x (Γ ′ ) dΓ ′ ≤ C α 2 |Γ−Γ ′ | α <2r log |x (Γ) − x (Γ ′ )| α dΓ ′ + |Γ−Γ ′ | α <3r log x Γ − x (Γ ′ ) α dΓ ′ + rα ≤ C α 2 |Γ−Γ ′ | α <2r log C (M ) |Γ − Γ ′ | α dΓ ′ + |Γ−Γ ′ | α <3r log C (M ) Γ − Γ ′ α dΓ ′ + rα ≤ C M, 1 α r (|log r| + 1)
.
For I 2 we have |Γ−Γ ′ | α ≥ 2r, and hence |Γ−Γ ′ | α ≥ r.
By the mean value theorem (MVT), (2.6) and the fact that
D 3 Ψ α (s) ≤ 1 4πα 2 1 s + C α 3 (6.10) we have that for Γ ′′ ∈ Γ,Γ ∇K α (x (Γ) − x (Γ ′ )) − ∇K α x Γ − x (Γ ′ ) ≤ rα dx dΓ C 0 C α 2 |x (Γ ′′ ) − x (Γ ′ )| + C α 3 ≤ C M, 1 α r 1 |Γ ′′ − Γ ′ | + 1 ,
we also have that
|Γ ′′ −Γ ′ | α ≥ r and Γ 0 ≤ Γ ′′ ≤ Γ 1 . Hence |I 2 | ≤ r C (M ) α 3 (Γ0,Γ1)∩ |Γ ′′ −Γ ′ | α ≥r ff α |Γ ′′ − Γ ′ | + 1 dΓ ′ ≤ C M, 1 α , Γ 1 , Γ 0 r (1 + |log r|) .
Summing up we obtain
|I| ≤ C M, 1 α , Γ 1 , Γ 0 Γ − Γ log Γ − Γ + 1 ,(6.11)
which implies the Hölder continuity of du dΓ (x (Γ)) for 0 < β < 1. It remains to show that u (x) is locally Lipschitz continuous on K M . We will show that for x ∈ K M , y ∈ C 1,β ((Γ 0 , Γ 1 ))
D x u (x) y 1,β ≤ C 1 α , M, x 1,β , Γ 1 , Γ 0 , β y 1,β .
Hence for any x ∈ K M , let δ be such that B (x, δ) ⊂ K M , then for everyx,x ∈ B (x, δ) by the fundamental theorem of calculus
u (x) − u (x) 1,β = 1 0 d dε u (x + ε (x −x)) dε 1,β = 1 0 D x u (x + ε (x −x)) (x −x) dε 1,β ≤ x −x 1,β 1 0 C 1 α , M, x + ε (x −x) 1,β , Γ 1 , Γ 0 , β dε ≤ C 1 α , M, x 1,β , δ, Γ 1 , Γ 0 , β x −x 1,β ,
that is, the map is locally Lipschitz. Let x ∈ K M , y ∈ C 1,β ((Γ 0 , Γ 1 )), we now compute
D x u (x (Γ)) y (Γ) = d dε u (x (Γ) + εy (Γ)) ε=0 = d dε Γ1 Γ0 K α (x (Γ) + εy (Γ) − x (Γ ′ ) − εy (Γ ′ )) dΓ ′ ε=0 = Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) (y (Γ) − y (Γ ′ )) dΓ ′ .
Now, we show that
D x u (x (·)) y (·) 1,β ≤ C 1 α , M, x 1,β , Γ 1 , Γ 0 , β y 1,β .
To estimate D x u (x) y C 0 we use (6.7)
|D x u (x) y| ≤ C M, Γ 1 , Γ 0 , 1 α y C 0 . (6.12) Next we estimate d dΓ D x u (x) y C 0 . For Γ ′ = Γ, ∇K α (x (Γ) − x (Γ ′ )) (y (Γ) − y (Γ ′ ))
is differentiable in Γ, hence (can be justified by using Lebesgue dominated convergence theorem)
d dΓ D x u (x (Γ)) y (Γ) = Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) dy dΓ (Γ) dΓ ′ + Γ1 Γ0 2 i,j=1 ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) (y j (Γ) − y j (Γ ′ )) dΓ ′ .
Notice, that although, for Γ ′ close to Γ,
D 2 K α (x (Γ) − x (Γ ′ )) = O 1 α 2 |x(Γ)−x(Γ ′ )|
(see (2.6) and (6.4)), the term (y (Γ) − y (Γ ′ )) cancels the singularity in 1 x(Γ)−x(Γ ′ ) , so this is not a singular integral. We have
d dΓ D x u (x (Γ)) y (Γ) ≤ dy dΓ C 0 Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dΓ ′ + dx dΓ C 0 Γ1 Γ0 D 2 K α (x (Γ) − x (Γ ′ )) |y (Γ) − y (Γ ′ )| dΓ ′ .
Write the second integral on the right-hand side as
Γ1 Γ0 D 2 K α (x (Γ) − x (Γ ′ )) |y (Γ) − y (Γ ′ )| dΓ ′ = (Γ0,Γ1)∩ |Γ−Γ ′ | α <ε ff + (Γ0,Γ1)\ |Γ−Γ ′ | α <ε ff = I 1 + I 2 ,
for a fixed small ε. Then due to (2.6), (6.4) and (6.6), we obtain
I 1 ≤ C 1 α 2 dy dΓ C 0 |Γ−Γ ′ | α <ε M |Γ − Γ ′ | |Γ − Γ ′ | dΓ ′ ≤ CM 1 α dy dΓ C 0 .
For I 2 we have
I 2 ≤ C M, 1 α dy dΓ C 0 (Γ0,Γ1)∩{|Γ−Γ ′ |≥εα} |Γ − Γ ′ | dΓ ′ ≤ C M, Γ 1 , Γ 0 , 1 α dy dΓ C 0 . Hence d dΓ D x u (x (Γ)) y (Γ) ≤ y C 1 C M, Γ 1 , Γ 0 , 1 α . (6.13) It remains to estimate d dΓ D x u (x) y β . d dΓ D x u (x (Γ)) y (Γ) − d dΓ D x u x Γ y Γ = Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) dy dΓ (Γ) − ∇K α x Γ − x (Γ ′ ) dy dΓ Γ dΓ ′ + Γ1 Γ0 i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) (y j (Γ) − y j (Γ ′ )) − − i,j ∂ xi ∂ xj K α x Γ − x (Γ ′ ) dx i dΓ Γ y j Γ − y j (Γ ′ ) dΓ ′ = I 1 + I 2 .
We write I 1 as
|I 1 | ≤ Γ1 Γ0 |∇K α (x (Γ) − x (Γ ′ ))| dy dΓ (Γ) − dy dΓ Γ dΓ ′ + Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) − ∇K α x Γ − x (Γ ′ ) dy dΓ Γ dΓ ′ ≤ I 11 + I 12 .
Using (6.7) to bound I 11 and (6.11) to bound I 12 we obtain that
|I 1 | ≤ C M, 1 α , Γ 1 , Γ 0 Γ − Γ β y 1,β .
Now we estimate I 2 . Let r = |Γ−Γ| α , write I 2 as
I 2 = (Γ0,Γ1)∩ |Γ−Γ ′ | α <2r ff + (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff = I 21 + I 22 .
Using (2.6) and (6.10) for I 21 we have
|I 21 | ≤ (Γ0,Γ1)∩ |Γ−Γ ′ | α <2r D 2 K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) |y (Γ) − y (Γ ′ )| dΓ ′ + (Γ0,Γ1)∩ |Γ−Γ ′ | α <3r D 2 K α x Γ − x (Γ ′ ) dx dΓ Γ y Γ − y (Γ ′ ) dΓ ′ ≤ C dx dΓ C 0 dy dΓ C 0 |Γ−Γ ′ | α <3r C α 2 M |Γ − Γ ′ | + C α 3 |Γ − Γ ′ | dΓ ′ ≤ C 1 α , M, Γ 1 , Γ 0 dx dΓ C 0 y 1,β Γ −Γ β .
We write I 22 as
I 22 = (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) y j (Γ) − y j Γ dΓ ′ + (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) − dx i dΓ Γ y j Γ − y j (Γ ′ ) dΓ ′ + i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) − ∂ xi ∂ xj K α x Γ − x (Γ ′ ) x i Γ Γ y j Γ − y j (Γ ′ ) dΓ ′ = I 221 + I 222 + I 223 .
For I 221 :
|I 221 | ≤ dx dΓ C 0 dy dΓ C 0 Γ −Γ (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff D 2 K α (x (Γ) − x (Γ ′ )) dΓ ′ ≤ dx dΓ C 0 dy dΓ C 0 Γ −Γ (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff 1 α 2 M |Γ − Γ ′ | + C α 3 dΓ ′ ≤ C 1 α , M, Γ 1 , Γ 0 , dx dΓ C 0 y 1,β Γ −Γ 1 + log Γ −Γ ,
which implies Hölder continuity for 0 < β < 1.
For I 222 :
|I 222 | ≤ (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff D 2 K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) − dx dΓ Γ y Γ − y (Γ ′ ) dΓ ′ ≤ C dx dΓ β Γ −Γ β dy dΓ C 0 (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff 1 α 2 M |Γ − Γ ′ | + C α 3 Γ − Γ ′ dΓ ′ ≤ C 1 α , M, dx dΓ β , Γ 1 , Γ 0 Γ −Γ β y 1,β ,
here we also used that
Γ − Γ ′ ≤ Γ − Γ + |Γ − Γ ′ | .
For I 223 :
|I 223 | ≤ dx dΓ C 0 dy dΓ C 0 (Γ0,Γ1)∩ |Γ−Γ ′ | α ≥2r ff D 2 K α (x (Γ) − x (Γ ′ )) − D 2 K α x Γ − x (Γ ′ ) Γ − Γ ′ dΓ ′ Since |Γ−Γ ′ | α ≥ 2r and |Γ−Γ ′ | α ≥ r, i.e., D 2 K α (x (Γ) − x (Γ ′ )) − D 2 K α x Γ − x (Γ ′ )
is differentiable in Γ,Γ , we can apply the MVT to obtain that for Γ ′′ ∈ Γ,Γ
D 2 K α (x (Γ) − x (Γ ′ )) − D 2 K α x Γ − x (Γ ′ ) = rα C (M ) α 3 α |Γ ′′ − Γ ′ | 2 + 1 .
We also have that
|Γ ′′ −Γ ′ | α ≥ r. Hence |I 223 | ≤ C 1 α , M dx dΓ C 0 dy dΓ C 0 rα (Γ0,Γ1)∩ |Γ ′′ −Γ ′ | α ≥r α |Γ ′′ − Γ ′ | 2 + 1 Γ − Γ ′ dΓ ′ ≤ C 1 α , M, dx dΓ C 0 , Γ 1 , Γ 0 y 1,β Γ − Γ 1 + log Γ − Γ ,
where we have also used that
Γ − Γ ′ ≤ Γ − Γ ′′ + |Γ ′′ − Γ ′ | ≤ Γ − Γ + |Γ ′′ − Γ ′ | .
This implies Hölder continuity for 0 < β < 1. Summing up we have
d dΓ D x u (x) y β ≤ C 1 α , M, x 1,β , Γ 1 , Γ 0 , β y 1,β .
The above proof deals with C 1,β case, for 0 < β < 1. To show the local existence and uniqueness of solutions in C 1 case, one simply applies directly the above proof (estimates (6.5),(6.8),(6.12),(6.13)) without resorting to the Hölder estimates.
The proof of the Lipschitz case is similar to the proof of the C 1,β case, for example, to show Lipschitz continuity of u (x (Γ)) for x ∈ Lip ((Γ 0 , Γ 1 )), denote r = |Γ−Γ| α and write
u (x (Γ)) − u x Γ ≤ Γ1 Γ0 K α (x (Γ) − x (Γ ′ )) − K α x Γ − x (Γ ′ ) dΓ ′ = (Γ0,Γ1)∩Er + (Γ0,Γ1)\Er = I 1 + I 2 , where E r = Γ ′ ∈ (Γ 0 , Γ 1 ) : |x (Γ) − x (Γ ′ )| α < 2rM . For I 1 , due to 1 M |Γ−Γ ′ | α < |x| * |Γ−Γ ′ | α ≤ |x(Γ)−x(Γ ′ )| α , we have that |Γ−Γ ′ | α < 2rM 2 and hence |Γ−Γ ′ | α < r 1 + 2M 2 .
Thus by (2.6) and (6.3) we obtain
I 1 ≤ (Γ0,Γ1)∩ |Γ−Γ ′ | α <2rM 2 ff |K α (x (Γ) − x (Γ ′ ))| + K α x Γ − x (Γ ′ ) dΓ ′ ≤ C α |Γ−Γ ′ | α <2rM 2 dΓ ′ + |Γ−Γ ′ | α <r(1+2M 2 ) dΓ ′ ≤ C (M ) r. For I 2 due to M |Γ−Γ ′ | α ≥ |x(Γ)−x(Γ ′ )| α ≥ 2rM , we have that |Γ−Γ ′ | α ≥ 2r, and hence |Γ−Γ ′ | α ≥ r, which in turn implies that |x(Γ)−x(Γ ′ )| α > 1 M |Γ−Γ ′ | α ≥ r M . Also, due to x (Γ) − x Γ ≤ M Γ −Γ = M rα we have for every x (Γ ′′ ) ∈ B x (Γ) , x (Γ) − x Γ that |x(Γ ′′ )−x(Γ ′ )| α ≥ M r.
Hence by the mean value theorem and (6.9), we have that for
x (Γ ′′ ) ∈ B x (Γ) , x (Γ) − x Γ K α (x (Γ) − x (Γ ′ )) − K α x Γ − x (Γ ′ ) ≤ |∇K α (x (Γ ′′ ) − x (Γ ′ ))| x (Γ) − x Γ ≤ 1 4π 1 α 2 log |x (Γ ′′ ) − x (Γ ′ )| α + C α 2 x (Γ) − x Γ ≤ C M, 1 α Γ −Γ log |Γ ′′ − Γ ′ | α + 1
Hence
|I 2 | ≤ rC M, 1 α (Γ0,Γ1)∩ |Γ ′′ −Γ ′ | α ≥r ff log |Γ ′′ − Γ ′ | α + 1 dΓ ′ ≤ C M, 1 α , Γ 1 , Γ 0 r (1 + r |log r|) .
Hence u (x (Γ)) is Lipschitz continuous. We remark, that in the proof of the C 1,β part we used partitions using the fact that x (Γ) is a differentiable, however, given the fact that differentiable functions are Lipschitz, one could have used the partitioning introduced in the proof of Lipschitz case on subsets of x (Γ) also for C 1,β results. Proposition 6.2 implies the local existence and uniqueness of solutions: To control the quantities 1 |x(·,t)| * and x (·, t) V we need to bound
Proposition 6.3. Let −∞ < Γ 0 < Γ 1 < ∞, let V be either the space C 1,β ((Γ 0 , Γ 1 )), 0 ≤ β < 1 or the space Lip ((Γ 0 , Γ 1 )), let K M = x ∈ V : |x| 1 < M, |x| * > 1 M and let x 0 ∈ V ∩ {|x| * > 0} ,Tmax 0 ∇ x u (x(·, t), t) L ∞ ((Γ0,Γ1)) dt.
The next proposition provides the bound on the gradient of the velocity .
Proposition 6.4. Let x 0 ∈ Lip ((Γ 0 , Γ 1 )) and |x 0 | * > 0. Suppose the solution exists on [0, T max ), then for t ∈ [0, T max ) we have |∇ x u (x (Γ, t) , t)| ≤ 1 α C (|x 0 | * , C 1 ) e tC1 + 1 ,(6.
14)
where
C 1 = C 1 α 2 (Γ 1 − Γ 0 ). Proof. We write ∇ x u (x(Γ, t), t) as ∇ x u (x(Γ, t), t) = Γ1 Γ0 ∇ x K α (x (Γ, t) − x (Γ ′ , t)) dΓ ′ = (Γ0,Γ1)∩Eε + (Γ0,Γ1)\Eε = I 1 + I 2 , where E ε = Γ ′ ∈ (Γ 0 , Γ 1 ) : |x (Γ, t) − x (Γ ′ , t)| α < ε ,
for a fixed small 0 < ε < 1, to be further refined later. Let the vorticity q(x, t) be supported on the curve {x (Γ, t) : Γ 0 ≤ Γ ≤ Γ 1 }, with a density γ (Γ, t) = 1/|x Γ (Γ, t) | (due to the Lipschitz continuity of x (Γ, t) its derivative exists almost everywhere and is essentially bounded, and also due to {|x| * > 0}, the vorticity density γ (Γ, t) ∈ L ∞ ((Γ 0 , Γ 1 ))), that is for every
ϕ ∈ C ∞ c R 2 R 2 ϕ(x)dq(x, t) = Γ1 Γ0 ϕ (x(Γ, t)) dΓ.
Observe that the vorticity q(x, t) is a finite Radon measure which is the unique weak solution of the Euler equations given by Theorem 2.1 of Oliver and Shkoller [65]. Also, q M = Γ 1 − Γ 0 .
Let η denote the unique Lagrangian flow map ∂ t η(y, t) = R 2 K α (y, z) dq (z, t), η (y, 0) = y, q = q in • η −1 , y ∈ R 2 given by Theorem 2.1. We remark that in the formulation of BR-α model, we assumed the positivity of the vorticity q, see Proposition 4.1. Denote the distance between two points η(y, t) and η(y ′ , t) by r (t) = |η (y, t) − η (y ′ , t)|, where r (0) = |y − y ′ |.
Then, using the estimate (2.14) of [65], we have
d dt r (t) ≤ R 2 |K α (y, z) − K α (y ′ , z)| dq (z, t) ≤ C 1 α ϕ r (t) α q M = C 1 α ϕ r (t) α q in M , where ϕ (r) = 0, r = 0, r (1 − log r) , 0 < r < 1, 1, r ≥ 1.
By comparison with the solution of the differential equation 3
d dt ξ (t) = −C 1 α ϕ ξ (t) α q in M , ξ (0) = |x (Γ, 0) − x (Γ ′ , 0)| ,
we can choose ε small enough, ε < e 1−e tC 1 , where
C 1 = C 1 α 2 q in M = Γ 1 − Γ 0 , e.g., ε = e −e tC 1 , such that, for |x(Γ,t)−x(Γ ′ ,t)| α < ε, we have that |x(Γ,0)−x(Γ ′ ,0)| α = r(0) α < 1. 4 Hence |x (Γ, t) − x (Γ ′ , t)| α ≥ r (t) α = r (0) α e tC 1 e 1−e tC 1 (6.15) = |x (Γ, 0) − x (Γ ′ , 0)| α e tC 1 e 1−e tC 1 .
Now, using also that |x 0 | * is bounded away from zero, we can bound |x(Γ,t)−x(Γ ′ ,t)| α from below, using (6.15),
1 > ε > |x (Γ, t) − x (Γ ′ , t)| α ≥ |x (Γ, 0) − x (Γ ′ , 0)| α e tC 1 e 1−e tC 1 ≥ |x 0 | e tC 1 * |Γ − Γ ′ | α e tC 1 e 1−e tC 1 , 3 ξ (t) α = 8 > > > < > > > : " ξ(0) α " e tC 1 e 1−e tC 1 , ξ(0) α < 1, ξ(0) α − C 1 t, ξ(0) α ≥ 1, t < t * * , e 1−e tC 1 − ξ(0) α +1 , ξ(0) α ≥ 1, t ≥ t * * , where C 1 = C 1 α 2 ‚ ‚ q in ‚ ‚ M , t * * = 1 C 1 " ξ (0) α − 1 « . 4 Otherwise, |x(Γ,0)−x(Γ ′ ,0)| α = r(0) α ≥ 1, hence ε > |x (Γ, t) − x (Γ ′ , t)| α ≥ r (t) α ≥ 8 < : r(0) α − C 1 t, t < t * * , e 1−e tC 1 − r(0) α +1 , t ≥ t * * .
Let t ≥ t * * , then we have ε ≥ e 1−e tC 1 , a contradiction. Otherwise t < t * * = 1
C 1 " r(0) α − 1 " , hence C 1 t < " r(0) α − 1 " ε > r (t) α ≥ r (0) α − C 1 t > r (0) α − r (0) α + 1 = 1, a contradiction.
which in turn implies the bound (using also (2.6) and (6.9))
I 1 ≤ (Γ0,Γ1)∩ |x(Γ,t)−x(Γ ′ ,t)| α <ε 1 2π 1 α 2 log |x (Γ, t) − x (Γ ′ , t)| α + C α 2 dΓ ′ ≤ 1 α C (|x 0 | * ) e tC1 + 1 .
While to bound I 2 , we use the boundness of |∇
x K α (x (Γ, t) − x (Γ ′ , t))| in {Γ ′ ∈ (Γ 0 , Γ 1 ) : |x(Γ,t)−x(Γ ′ ,t)| α ≥ ε}. I 2 ≤ sup |x(Γ,t)−x(Γ ′ ,t)| α ≥ε |∇ x K a (x (Γ, t) − x (Γ ′ , t))| Γ1 Γ0 dΓ ′ ≤ C 1 α 2 (|log ε| + 1) (Γ 1 − Γ 0 ) = C 1 e tC1 + 1 . Now, the bound on x (·, t) C 0 on [0, T max ) follows from dx dt (Γ, t) = u (x (Γ, t)
, t) and the fact that
|u (x (Γ, t) , t)| ≤ Γ1 Γ0 |K a (x (Γ, t) − x (Γ ′ , t))| dΓ ′ ≤ C α (Γ 1 − Γ 0 ) ,
due to the boundness of K a (see (2.6),(6.3)). Also, by Grönwall inequality the bound (6.14) provides bounds on 1 |x(·,t)| * and |x (·, t)| 1 on [0, T max ).
Finally, for the initial data in C 1,β ((Γ 0 , Γ 1 )), the bound (6.14) provides bound on dx dΓ (·, t) C 0 on [0, T max ) by Grönwall inequality. While the bound on dx dΓ (·, t) β on [0, T max ) is a consequence of
d dt x Γ (Γ, t) = ∇ x u (x (Γ, t) , t) · x Γ (Γ, t) ,
the bound (which is shown in a local existence part, see (6.11))
|∇ x u (x (·, t) , t)| β ≤ C 1 α , x Γ (·, t) L ∞ , |x (·, t)| * , Γ 1 , Γ 0 ,(6.
14) and the Grönwall inequality. This yields global in time existence and uniqueness of Lip and C 1,β , 0 ≤ β < 1, solutions of (6.1).
6.3
Step 3. Higher regularity for closed curves. Now we show the higher regularity for an initially closed curve x 0 (Γ) ∈ C n,β S 1 ∩ {|x| * > 0}, n ≥ 1, 0 < β < 1. We remark that the high derivatives of the kernel K α (x) are singular at the origin, thus the condition on closedness of the curve.
To provide an a priori bound for higher derivatives in terms of lower ones, we show that for x ∈ C n,β S 1 ∩ |x| 1 < M, |x| * > 1 M , the map u defined by (6.2) satisfies
u (x) n,β ≤ C 1 α , M, x n−1,β , 1 β x n,β ,
hence by Grönwall inequality and the induction argument, it is enough to control |x| * and x 1,β , to guarantee that x (Γ) ∈ C n,β S 1 , for all n ≥ 1, (and consequently in C ∞ S 1 , whenever x 0 ∈ C ∞ S 1 ∩ {|x| * > 0}).
Lemma 6.5. Let V be the space C n,β S 1 , n ≥ 1, 0 < β < 1, and let u and K M be as defined in Proposition 6.2. Then for x ∈ K M u (x) n,β ≤ C 1 α , M, x n−1,β , 1 β x n,β .
Proof. We show the proof for n = 2, the proof for general n is similar. The derivative of u with respect to Γ (in the sense of distributions) satisfies (see Appendix, Lemma A.1)
d 2 dΓ 2 u (x (Γ)) = 2π 0 ∇K α (x (Γ) − x (Γ ′ )) d 2 x dΓ 2 (Γ) dΓ ′ + p.v. 2π 0 2 i,j=1 ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) dx j dΓ (Γ) dΓ ′ = I 1 + I 2 .
I 1 can be bounded using similar arguments as for (6.7),
|I 1 | ≤ 1 α 2 C 1 α , M x 2,β .
We write I 2 as
I 2 = p.v. |Γ−Γ ′ | α <ε + (0,2π)\ |Γ−Γ ′ | α <ε ff 2 i,j=1 ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) dx j dΓ (Γ) dΓ ′ = I 21 + I 22 ,
where, because the curve is closed, we can fix a small ε < π/2 independent of Γ, by taking For I 21 we have that to desingularize the I 211 . We rewrite
I 21 = 1 4πα 2 p.v. |Γ−Γ ′ | α <ε i,j σ ij (x (Γ) − x (Γ ′ )) dxi dΓ (Γ) |x (Γ) − x (Γ ′ )| dx j dΓ (Γ) dΓ ′ + 1 4πα 2 |Γ−Γ ′ | α <ε i,j O |x (Γ) − x (Γ ′ )| α 2 log |x (Γ) − x (Γ ′ )| α dx i dΓ (Γ) dx j dΓ (Γ) dΓ ′ = I 211 + I 212 ,i,j σ ij (x (Γ) − x (Γ ′ )) dxi dΓ (Γ) |x (Γ) − x (Γ ′ )| dx j dΓ (Γ) = i,j σ ij (x (Γ) − x (Γ ′ )) dxi dΓ (Γ) (Γ − Γ ′ ) |x (Γ) − x (Γ ′ )| (Γ − Γ ′ ) dx j dΓ (Γ) = i,j σ ij (x (Γ) − x (Γ ′ )) dxi dΓ (Γ) (Γ − Γ ′ ) − x i (Γ) + x i (Γ ′ ) |x (Γ) − x (Γ ′ )| (Γ − Γ ′ ) dx j dΓ (Γ) + i,j σ ij (x (Γ) − x (Γ ′ )) (x i (Γ) − x i (Γ ′ )) |x (Γ) − x (Γ ′ )| (Γ − Γ ′ ) dx j dΓ (Γ) = J 1 + J 2 .
Observe that J 2 = 1 (Γ ′ −Γ) dx dΓ (Γ) ⊥ and J 1 ≤ C (M ) x 2 1,β |Γ − Γ ′ | −1+β due to |σ ij | ≤ 1 (see (6.16)) and (6.17). Hence
|I 211 | = 1 4πα 2 |Γ−Γ ′ | α <ε J 1 dΓ ′ + 1 4πα 2 dx (Γ) dΓ ⊥ p.v. |Γ−Γ ′ | α <ε 1 (Γ ′ − Γ) dΓ ′ ≤ C (M ) 1 α 2−β x 2 1,β 1 β ε β .
Summing up, we have that Using the same ideas we also bound d 2 dΓ 2 u (x (Γ)) β .
where for a fixed Γ we take ε < min Γ−Γ0 α , Γ1−Γ α . Denote D = (Γ 0 , Γ 1 ) ∩ |Γ−Γ ′ | α > ε, by integration by parts we get
= lim ε→0 Γ1 Γ0 − ϕ (Γ) ∇K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) ∂D + D ϕ (Γ) d dΓ ∇K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) dΓ dΓ ′ = lim ε→0 Γ1 Γ0 (A + B) dΓ ′ For A we have A = −ϕ (Γ ′ − εα) ∇K α (x (Γ ′ − εα) − x (Γ ′ )) dx dΓ (Γ ′ − εα) + ϕ (Γ ′ + εα) ∇K α (x (Γ ′ + εα) − x (Γ ′ )) dx dΓ (Γ ′ + εα) = I 1 + I 2 + I 3 ,
where
I 1 = [ϕ (Γ ′ + εα) − ϕ (Γ ′ − εα)] ∇K α (x (Γ ′ + εα) − x (Γ ′ )) dx dΓ (Γ ′ + εα) , I 2 = ϕ (Γ ′ − εα) [∇K α (x (Γ ′ + εα) − x (Γ ′ )) − ∇K α (x (Γ ′ − εα) − x (Γ ′ ))] dx dΓ (Γ ′ + εα) , I 3 = ϕ (Γ ′ − εα) ∇K α (x (Γ ′ − εα) − x (Γ ′ )) dx dΓ (Γ ′ + εα) − dx dΓ (Γ ′ − εα) . Now, since for y ∈ R 2 , |y| α → 0 : |∇K α (y)| ≤ − 1 2π 1 α 2 log |y| α + O 1 α 2 and dx dΓ ε ≥ |x(Γ ′ +εα)−x(Γ ′ )| α ≥ |x| * ε we have |∇K α (x (Γ ′ + εα) − x (Γ ′ ))| ≤ − 1 2π 1 α 2 log |x (Γ ′ + εα) − x (Γ ′ )| α + C α 2 ≤ C |x| * , 1 α (log ε + 1) ,
Hence |I 1 | ≤ C |x| * , 1 α dϕ dΓ C 0 dx dΓ C 0 ε (log ε + 1) → 0, as ε → 0.
Similarly,
|I 3 | ≤ C |x| * , 1 α ϕ C 0 d 2 x dΓ 2
C 0 ε (log ε + 1) → 0, as ε → 0.
For I 2 we have
|I 2 | ≤ ϕ C 0 dx dΓ C 0 |∇K α (x (Γ ′ + εα) − x (Γ ′ )) − ∇K α (x (Γ ′ − εα) − x (Γ ′ ))| ,
using that for y ∈ R 2 ∇K α (y) = 1 |y| DΨ α (|y|) (σ (y) + J) − σ (y) D 2 Ψ α (|y|) , for |y| α , |y ′ | α → 0, we obtain |∇K α (y) − ∇K α (y ′ )| ≤ 1 4π 1 α 2 |σ (y) − σ (y ′ )| + C |y| 2 α 4 log |y| α + C |y ′ | 2 α 4 log |y ′ | α + 1 4π 1 α 2 |log |y| − log |y ′ || . Now, due to
|log |x (Γ ′ + εα) − x (Γ ′ )| − log |x (Γ ′ − εα) − x (Γ ′ )|| ≤ C x 1,β , 1 |x| * α β ε β , |σ (x (Γ ′ + εα) − x (Γ ′ )) − σ (x (Γ ′ − εα) − x (Γ ′ ))| ≤ C x 1,β , 1 |x| * α β ε β , we obtain |∇ x K α (x (Γ ′ + εα) − x (Γ ′ )) − ∇ x K α (x (Γ ′ − εα) − x (Γ ′ ))| → 0, as ε → 0.
We also have that
|∇ x K α (x (Γ ′ + εα) − x (Γ ′ )) − ∇ x K α (x (Γ ′ − εα) − x (Γ ′ )) | ≤ 1 2π 1 α 2 + C α 2 dx dΓ 2 C 0 ε 2 log dx dΓ C 0 ε + C α 2 α β ε β dx dΓ β |x| * ≤ C 1 α , dx dΓ 1,β , |x| * ,
hence by the Lebesgue's dominated convergence theorem lim ε→0 Γ1 Γ0 AdΓ ′ = 0.
For B we have
d dΓ ∇K α (x (Γ) − x (Γ ′ )) dx dΓ (Γ) = ∇K α (x (Γ) − x (Γ ′ )) d 2 x dΓ 2 (Γ) + i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) dx j dΓ (Γ) .
Hence
d 2 u dΓ 2 (x (Γ, t) , t) , ϕ (Γ) = Γ1 Γ0 dΓϕ (Γ) · Γ1 Γ0 ∇K α (x (Γ) − x (Γ ′ )) d 2 x dΓ 2 (Γ) dΓ ′ + p.v. Γ1 Γ0 i,j ∂ xi ∂ xj K α (x (Γ) − x (Γ ′ )) dx i dΓ (Γ) dx j dΓ (Γ) dΓ ′ ,
which concludes the proof.
[ 0 ,
0T ] ; H −4 R 2 (by (3.9)), M(R 2 ) ֒→ H −s loc R 2 comp ֒→ H −4 loc R 2 for 1 < s < 4, then by Arzela-Ascoli theorem there is a subsequence of q α that converges to somē q in C [0, T ] ; H −4 loc , and henceq is also in Lip [0, T ] ; H −4 loc . Applying both types of convergence of the q α to the integral T 0
then for any M , 1 < M < ∞, such that x 0 ∈ K M , there exists a time T (M ), such that the system (6.1) has a unique local solution x ∈ C 1 ((−T (M ), T (M )); K M ).6.2 Step 2. Global existence.To show the global existence, we assume by contradiction, thatT max < ∞, where [0, T max ) is the maximal interval of existence, and hence the solution leaves in a finite time the open set K M , for all M > 1, that is, lim sup t→T − max x V = ∞ or lim sup t→T − max 1 |x(·,t)| * = ∞. Therefore, if we show global bounds on
x (·, t) V in [0, T max ), we obtain a contradiction to the blow-up and thus the obtained local solutions can be continued for all time. The result extends to negative times as well.
= (0, 2π) if εα < Γ < 2π − εα, D = (−π, π) if 0 ≤ Γ ≤ εα, or D = (π, 3π) if 2π − εα ≤ Γ ≤ 2π.Treating I 22 as in the local existence proof, we have |I 22 | ≤ 1,β |log ε| .
212 is not a singular integral and due to|x (Γ) − x (Γ ′ )| ≤ x Γ C 0 |Γ − Γ ′ | , We use the observation |f (x) − f (y) − (x − y) f ′ (x)| ≤ |x − y| 1+β |f ′ | β (6.17)
AcknowledgementsThe authors would like to thank the anonymous referee for a careful reading of the manuscript and for valuable comments. C.B. would like to thank the Faculty of Mathematics and Computer Science at the Weizmann Institute of Science for the kind hospitality where this work was initiated. This work was supported in part by the BSF grant no. 2004271, the ISF grant no. 120/06, and the NSF grants no. DMS-0504619 and no. DMS-0708832.AppendixLemma A.1. Let x ∈ C 2.β ((Γ 0 , Γ 1 )) ∩ {|x| * > 0} thenProof. By the definition of the distribution derivative, for all
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| []
|
[
"Interplay between core and corona components in high-energy nuclear collisions",
"Interplay between core and corona components in high-energy nuclear collisions"
]
| [
"Yuuka Kanakubo \nDepartment of Physics\nSophia University\n102-8554TokyoJapan\n",
"Yasuki Tachibana \nAkita International University\n010-1292Yuwa, Akita-cityJapan\n",
"Tetsufumi Hirano \nDepartment of Physics\nSophia University\n102-8554TokyoJapan\n"
]
| [
"Department of Physics\nSophia University\n102-8554TokyoJapan",
"Akita International University\n010-1292Yuwa, Akita-cityJapan",
"Department of Physics\nSophia University\n102-8554TokyoJapan"
]
| []
| We establish the updated version of dynamical core-corona initialization framework (DCCI2) as a unified description from small to large colliding systems and from low to high transverse momentum (pT ) regions. Using DCCI2, we investigate effects of interplay between locally equilibrated and non-equilibrated systems, in other words, core and corona components in high-energy nuclear collisions. Given experimental multiplicity distributions and yield ratios of Ω baryons to charged pions as inputs, we extract the fraction of core and corona components in p+p collisions at √ s = 7 TeV and Pb+Pb collisions at √ sNN = 2.76 TeV. We find core contribution overtakes corona contribution as increasing multiplicity above dN ch /dη |η|<0.5 ∼ 18 regardless of the collision system or energy. We also see that the core contribution exceeds the corona contribution only in 0.0-0.95% multiplicity class in p+p collisions. Notably, there is a small enhancement of corona contribution with ∼ 20% below pT ∼ 1 GeV even in minimum bias Pb+Pb collisions. We find that the corona contribution at low pT gives ∼ 15-38 % correction on v2{2} at N ch 370. This raises a problem in conventional hydrodynamic analyses in which low pT soft hadrons originate solely from core components. We finally scrutinize the roles of string fragmentation and the longitudinal expansion in the transverse energy per unit rapidity, which is crucial in initial conditions for hydrodynamics from event generators based on string models. | 10.1103/physrevc.105.024905 | [
"https://arxiv.org/pdf/2108.07943v2.pdf"
]
| 237,195,099 | 2108.07943 | a54b573c700a696a09261cc7c6454baa6867a723 |
Interplay between core and corona components in high-energy nuclear collisions
Yuuka Kanakubo
Department of Physics
Sophia University
102-8554TokyoJapan
Yasuki Tachibana
Akita International University
010-1292Yuwa, Akita-cityJapan
Tetsufumi Hirano
Department of Physics
Sophia University
102-8554TokyoJapan
Interplay between core and corona components in high-energy nuclear collisions
(Dated: August 31, 2021)numbers: 2575-q1238Mh2575Ld2410Nz
We establish the updated version of dynamical core-corona initialization framework (DCCI2) as a unified description from small to large colliding systems and from low to high transverse momentum (pT ) regions. Using DCCI2, we investigate effects of interplay between locally equilibrated and non-equilibrated systems, in other words, core and corona components in high-energy nuclear collisions. Given experimental multiplicity distributions and yield ratios of Ω baryons to charged pions as inputs, we extract the fraction of core and corona components in p+p collisions at √ s = 7 TeV and Pb+Pb collisions at √ sNN = 2.76 TeV. We find core contribution overtakes corona contribution as increasing multiplicity above dN ch /dη |η|<0.5 ∼ 18 regardless of the collision system or energy. We also see that the core contribution exceeds the corona contribution only in 0.0-0.95% multiplicity class in p+p collisions. Notably, there is a small enhancement of corona contribution with ∼ 20% below pT ∼ 1 GeV even in minimum bias Pb+Pb collisions. We find that the corona contribution at low pT gives ∼ 15-38 % correction on v2{2} at N ch 370. This raises a problem in conventional hydrodynamic analyses in which low pT soft hadrons originate solely from core components. We finally scrutinize the roles of string fragmentation and the longitudinal expansion in the transverse energy per unit rapidity, which is crucial in initial conditions for hydrodynamics from event generators based on string models.
We establish the updated version of dynamical core-corona initialization framework (DCCI2) as a unified description from small to large colliding systems and from low to high transverse momentum (pT ) regions. Using DCCI2, we investigate effects of interplay between locally equilibrated and non-equilibrated systems, in other words, core and corona components in high-energy nuclear collisions. Given experimental multiplicity distributions and yield ratios of Ω baryons to charged pions as inputs, we extract the fraction of core and corona components in p+p collisions at √ s = 7 TeV and Pb+Pb collisions at √ sNN = 2.76 TeV. We find core contribution overtakes corona contribution as increasing multiplicity above dN ch /dη |η|<0.5 ∼ 18 regardless of the collision system or energy. We also see that the core contribution exceeds the corona contribution only in 0.0-0.95% multiplicity class in p+p collisions. Notably, there is a small enhancement of corona contribution with ∼ 20% below pT ∼ 1 GeV even in minimum bias Pb+Pb collisions. We find that the corona contribution at low pT gives ∼ 15-38 % correction on v2{2} at N ch 370. This raises a problem in conventional hydrodynamic analyses in which low pT soft hadrons originate solely from core components. We finally scrutinize the roles of string fragmentation and the longitudinal expansion in the transverse energy per unit rapidity, which is crucial in initial conditions for hydrodynamics from event generators based on string models.
I. INTRODUCTION
Quark-gluon plasma (QGP) is a state of thermalized and chemically equilibrated matter consisting of quarks and gluons deconfined from hadrons at extremely high temperature and density. High-energy nuclear collision experiments in the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and at the Large Hadron Collider (LHC) in CERN provide opportunities to explore properties of the extreme state.
Relativistic hydrodynamics is proven to successfully describe experimental data of relativistic heavy-ion collisions since the first discovery of hydrodynamic behavior of the QGP in the early 2000s [1][2][3][4]. Since final observables reflect all the history of the reaction, it is of significant importance to model each stage of the reaction and to integrate these modules as a whole in a consistent way towards a further comprehensive understanding of the QGP [5]. In particular, modeling of initial preequilibrium and final decoupling stages is needed in addition to a relativistic hydrodynamic model as a framework to describe transient states. Notably, there are some attempts to constrain transport coefficients of the QGP by using state-of-the-art dynamical models based on relativistic hydrodynamics [6][7][8]. As one sees from this, the QGP study is in the middle of a transition to precision science.
Despite the great success of dynamical models based on relativistic hydrodynamics in describing a vast body of experimental data, it poses some open issues for a comprehensive description of the whole reaction in highenergy nuclear collisions. One of the major issues is an initial condition of relativistic hydrodynamic equations which does not respect the total energy of the colliding systems. Initial conditions have been parametrized and put to reproduce centrality dependence of multiplicity or pseudorapidity distributions in a conventional hydrodynamic approach. As a result, the total energy of the initial hydrodynamic fields does not exactly match the collision energy of the system. Even when some outputs from event generators with a given collision system and energy are utilized for initial conditions in hydrodynamic models, an additional scale parameter is commonly introduced to adjust the model outputs of multiplicity. One might not think it is necessary for the energy of the initial hydrodynamic fields to be the same as the total energy of the system. This is exactly a starting point of our discussion in a series of papers [9][10][11]: The relativistic hydrodynamics merely describes a part of system, namely, matter in local equilibrium, while other parts of the system such as propagating jets and matter out of equilibrium are to be described at the same time.
First attempts of simultaneous description of both the QGP fluids in equilibrium and the energetic partons out of equilibrium had been made in Refs. [12][13][14][15][16] 1 . Initial conditions in those studies were still either parametrized via an optical Glauber model [18] or taken from a saturation model [19][20][21][22] so as to reproduce yields of low p T hadrons, while hard partons which undergo energy loss during traversing QGP fluids were supplemented to successfully reproduce the hadron spectra from low to high p T regions [12,14,16]. It was found that an intriguing interplay between soft and hard components brought ones to interpretation of the proton yield anomaly in p T spectra [14]. However, the model lacked back reactions from quenching partons to the QGP fluids and a contribution from fragmentation was cut in low p T regions, both of which obviously violate the energy-momentum conservation law.
Medium responses to propagating energetic partons have been modeled within hydrodynamic equations with source terms by assuming the instantaneous equilibration of the deposited energy and momentum from partons [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Within this approach, the sum of the energy and momentum of fluids and those of traversing partons is conserved as a whole. However, it is not clear how to divide the initial system just after the collision into soft (fluids in equilibrium) and hard (partons out of equilibrium) parts.
To remedy this issue, a dynamical initialization model [9] was proposed to describe the dynamics of gradually forming QGP fluids phenomenologically 2 . In contrast to the conventional hydrodynamic models in which initial conditions are put at a fixed initial time, the QGP fluids are generated locally in time and space in the dynamical initialization framework. Under this framework, all the input of energy-momentum of QGP fluids is the one of partons produced just after the nuclear collisions. Starting with vacuum, energy-momentum of the QGP fluids is dynamically generated by solving hydrodynamic equations with source terms. Consequently, fluids in local equilibrium are generated from the initial partons by depositing the energy and the momentum and surviving partons are considered to remain out of equilibrium. When initial partons are taken, e.g., from event generators, the total energy keeps its value of the colliding two nuclei all the way through the dynamical initialization. Although we successfully separated matter in local equilibrium from initially produced partons in the dynamical initialization framework [9], the fluidization scheme was too simple and phenomenological to describe the transverse momentum spectra and the particle ratios. Then, we introduced the core-corona picture into the dynamical initialization.
The conventional core-corona picture was proposed to explain centrality dependence of strange hadron yield rations in quantitative analysis of parton energy loss was done in Ref. [17]. 2 Dynamical initialization is essential in describing the formation of fluids in lower collision energies in which secondary hadrons are gradually produced in finite time duration due to insufficient Lorentz contraction of colliding nuclei [38,39]. tios [40]. As multiplicity increases, the high-density region, in which the matter is mostly thermalized, is supposed to become larger. As a result, the final hadron yields become dominated by the hadrons from thermalized matter rather than non-thermalized matter created in low-density regions. The former component is referred to as core, while the latter one is referred to as corona. A Monte-Carlo event generator, EPOS [41,42] is widely accepted for its implementation of the core-corona picture. In the latest study in Refs. [43,44], string segments produced in a collision are separated sharply into the core and the corona components depending on their density and transverse momentum at a fixed time. Low-p T string segments in dense regions are fully converted into the thermalized medium fluid, while string segments with high momentum or dilute regions are directly hadronized.
On the other hand, we model the dynamical aspects of the core-corona separation introducing the particle density dependence of the dynamical initialization scheme, which is called the dynamical core-corona initialization (DCCI) [10,11] 3 . One of the key features of the DCCI is to deal with dynamics of the core (equilibrium) and the corona (non-equilibrium) at the same time. With the description of the dynamics, gradual formation of core and corona in spatial and momentum space is achieved. In the DCCI framework, the multiplicity dependence of the hadron yield ratios of multi-strange baryons to pions from small to large colliding systems in a wide range of collision energy is attributed to a continuous change of the fractions of the core and the corona components as multiplicity increases [10,11]. It should be noted here that the "corona" is referred not only to an outer layer in the coordinate space but also to the one in the momentum space: The lower p T partons are more likely to deposit their energy and momentum to form the fluids and the higher p T partons are less likely to be equilibrated during the DCCI processes.
In this paper, we update the DCCI framework towards a more comprehensive description of dynamics in full phase space from small to large colliding systems in a unified manner. Hereafter we call this updated DCCI the DCCI2. In comparison with the previous work [10,11], several crucial updates have been made in this new version including a more sophisticated formula for four-momentum deposition of initial partons, particlization of the fluids on the switching hypersurface through a Monte-Carlo sampler iS3D [45], hadronic rescatterings through a hadron cascade model JAM [46], and modification of structure of color strings inside the fluids. With these updates, the DCCI2 is capable of describing highenergy nuclear collisions from low to high p T region with particle identification in various colliding systems.
We generate initial partons from a general-purpose event generator Pythia8 [47,48] switching off hadronization and make all of them sources of both the core and the corona parts. Here a special emphasis is put on to discriminate between the two terms, "soft-hard" and "corecorona". We call the core when it composes the matter in equilibrium. In DCCI2, the fluids generated through the dynamical initialization correspond to the core part and hardons particlized on switching hypersurface are regarded as the core components. On the other hand, we call the corona when it composes matter completely out of equilibrium. In DCCI2, partons in the dilute regions and/or surviving even through the dynamical initialization correspond to the corona part, and hadrons from string decays are regarded as the corona components. Although the core (corona) component is sometimes identified with the soft (hard) component, it is not the case in the DCCI2: The hadrons from string fragmentation are distributed all the way down to very low p T region, which one cannot consider as "hard" components. To the best of our knowledge, it has been believed so far without any strong justification that the corona components would be negligible in low p T region in heavy-ion collisions. In this paper, we scrutinize the size of the contribution from the corona components in soft observables ("soft-from-corona") and how the fraction of the corona components evolves as multiplicity increases.
One of the main interests in this field is to constrain the transport coefficients of the QGP through the hydrodynamic analysis of anisotropic flow data in low p T regions. It is conventionally assumed that the low p T hadrons are completely dominated by the core components. What if the corona components, whose contribution is often considered to be very small, affect the bulk observables in heavy-ion collisions? The non-equilibrium contribution is at most taken through small corrections to thermodynamic quantities such as shear stress and bulk pressure. Since the corona components are far-fromequilibrium distributions, these must be more important than those dissipative corrections in the hydrodynamic analysis if these are smaller than the core components, but of the same order of the magnitude. Thus, dynamical modeling containing the core-corona picture could become the next-generation model inevitably needed for the precision study of the QGP properties.
The present paper is organized as follow: We explain details about the DCCI2 model in Sec. II. Staring from the general idea of dynamical initialization and the dynamical core-corona initialization, we discuss some new features in the DCCI2 model. In Sec. III, we show the results from the DCCI2 model in p+p and Pb+Pb collisions at the LHC energies. We also discuss the effects of string fragmentation and the longitudinal hydrodynamic expansion on the transverse energy per unit rapidity which play a crucial role in modeling hydrodynamic initial conditions. Section IV is devoted to the summary of the present paper.
Throughout this paper, we use the natural unit, = c = k B = 1, and the Minkowski metric, g µν = diag(1, −1, −1, −1).
II. MODEL
The DCCI2 framework as a multi-stage dynamical model describes high-energy nuclear reactions from p+p to A+A collisions. Before going into the details of the modeling of each stage, we briefly summarize the entire model flow of the DCCI2 framework. Figure 1 represents the flowchart of the DCCI2 framework. First, we obtain event-by-event phase-space distributions of initial partons produced just after the first contact of incoming nuclei using Pythia8.244 or its heavyion mode, Angantyr model [47,48]. Hereafter, we call Pythia8 and Pythia8 Angantyr, respectively. Those initial partons are assumed to be generated at a formation time, τ 0 . Under the dynamical initialization framework, the QGP fluids are generated via energymomentum deposition from those initial partons by solving the relativistic hydrodynamic equations with source terms from τ = τ 0 to the end of the hydrodynamic evolution. The energy-momentum deposition rate of partons is formulated based on the dynamical core-corona picture. Partons which experience sufficient secondary interactions with surrounding partons are likely to deposit their energy-momentum and form QGP fluids. In contrast, partons that do not experience such sufficient secondary interactions give less contribution to the medium formation. Hydrodynamic simulations are performed in the (3+1)-dimensional Milne coordinates incorporating the s95p-v1.1 [49] equation of state (EoS). In the original s95p-v1, an EoS of the (2+1)-flavor lattice QCD at high temperature from HotQCD Collaboration [50] is smoothly connected to that from a hadron resonance gas, whose list is taken from Particle Data group as of 2004 [51], at low temperature. The particular version of EoS, s95p-v1.1, which we employ in the present calculations, is tuned to match the EoS of the hadron resonance gas with the resonances implemented in a hadronic cascade model, JAM, below a temperature of 184 MeV. The fluid elements below the switching temperature T sw can be regarded as hadron gases whose evolution is described by the hadronic cascade model to be mentioned later. Once the temperature of the fluid element goes down to T (x) = T sw , we switch the description from hydrodynamics to hadronic transport. For the switch of the description, we use a Monte-Carlo sampler, iS3D [45], to convert hydrodynamic fields at the switching hypersurface to particles, which we call direct hadrons, in the EoS by sampling based on the Cooper-Frye formula [52]. Hadronization of non-equilibrated partons is performed by the string fragmentation in Pythia8. When a color string connecting partons from Pythia8 has a spatial overlap with the medium fluid, we assume that the string is cut and reconnected to partons sampled from the medium due to the screening effect of the medium. The direct hadrons obtained from both Pythia8 and iS3D are handed to the hadronic cascade model, JAM [46], to perform hadronic rescatterings among them and resonance decays. In the following subsections, we explain the details of each stage.
PYTHIA8/PYTHIA8 Angantyr
A. Generating initial partons
The initially produced partons, i.e., all partons we use as an input of dynamical initialization, are obtained with Pythia8 or Pythia8 Angantyr. Here we summarize settings that we use to generate initial partons. We basically use the default settings in Pythia8 and Pythia8 Angantyr to obtain phase-space distributions of partons except the two parameters,
MultipartonInteractions:pT0Ref and SpaceShower:pT0Ref.
These parameters regularize cross sections of multiparton interactions and infrared QCD emissions [53]. The same value of p T0Ref is used for the both parameters just for simplicity. Detailed discussion is given in Sec. III E.
The information of color strings is given by Pythia8 and Pythia8 Angantyr besides phase-space information. In order to respect the configuration of initially produced color strings, we keep this information for dynamical core-corona initialization. Technically speaking, color and anti-color tags are given to each parton so that one is able to see the configuration of the color strings by tracing the tags. Note that if there exist junctions, which are Y-shaped objects that three string pieces are converged, we keep this information as well to trace all strings generated in the event.
Eventually, we obtain a particle list for each event including particle IDs, phase-space information, and color and anti-color tags. Junctions are added to the particle list if they exist in the event. For heavy-ion collisions obtained with Pythia8 Angantyr, weighted events are generated [54]: The impact parameter is distributed in a way that more central collisions and fewer peripheral collisions are generated than in the minimum bias cases. The corresponding weights are stored to be used in statistical analysis.
B. Dynamical initialization
We phenomenologically and dynamically describe the initial stage of high-energy nuclear collisions through dynamical initialization. Just after the first contact of incoming nuclei or nucleons, quarks and gluons are produced through hard scatterings or initial or final state radiations. Subsequently, some of those initially produced partons experience the secondary scatterings and contribute to forming the equilibrated matter. On the other hand, partons that do not experience the interactions and partons surviving even after the secondary interactions contribute as the non-equilibrated matter.
To describe this stage, we start from the continuity equations of the entire system generated in a single collision event
∂ µ T µν tot (x) = 0.(1)
If we assume that the entire system can be decomposed into equilibrated matter (fluids) and non-equilibrated matter (partons), Eq. (1) can be written as
∂ µ T µν tot (x) = ∂ µ T µν fluids (x) + T µν partons (x) = 0,(2)
where T µν fluids and T µν partons are energy-momentum tensors of equilibrated matter (fluids) and non-equilibrated mat-ter (partons), respectively. Then the space-time evolution of fluids can be expressed as a form of hydrodynamic equations with source terms
∂ µ T µν fluids (x) = J ν (x),(3)
where the source terms are written as
J ν = −∂ µ T µν partons .(4)
Here we assume the energy and momentum deposited from non-equilibrated partons instantly reach a state under local thermal and chemical equilibrium. The exact form of the source terms is obtained by defining phase-space distributions and kinematics of initial partons [11]. For the phase-space distributions, we assume
f (x, p; t)d 3 xd 3 p = i G(x−x i (t))δ (3) (p − p i (t))d 3 xd 3 p,(5)
where G(x−x i (t)) is a three-dimensional Gaussian distribution centered at a position of the ith parton, x i (t), generated in one single event. We assume that each parton traverses along a straight trajectory. Under this assumption, the position of a parton at an arbitrary time is obtained as
x i (t) = p i (t) p 0 i (t) (t − t form,i ) + x form,i ,(6)
where t form,i and x form,i are a formation time and a formation position, respectively. The ith parton is assumed to be formed at a common proper time, τ = τ 0 . With these assumptions, the explicit form of the source terms in Eq. (4) is obtained as [11]
J ν = −∂ µ T µν partons = − i d 3 p p µ p ν p 0 ∂ µ f (x, p; t) = − i dp ν i (t) dt G(x−x i (t)).(7)
As one reads from the last line of Eq. (7), the source terms of fluids are described as a summation of deposited energy-momentum of initial partons. Space-time evolution of fluids is described by ideal hydrodynamics. This does not mean we do not describe any non-equilibrium components within DCCI2: Dissipative hydrodynamics deals only with small non-equilibrium corrections to equilibrium components, while the corona components, which are far from equilibrium, are taken into account in DCCI2. By neglecting dissipative terms for simplicity, energy-momentum tensor of fluids is expressed as
T µν fluids = (e + P )u µ u ν − P g µν ,(8)
where e, P , and u µ are energy density, hydrostatic pressure, and four-velocity of fluids, respectively. In this study, we do not solve conserved charges such as baryon number, strangeness, and electric charges. It would be interesting to investigate them considering the initial distribution of the charges [55,56] .
As we mentioned at the beginning of this section, we perform actual hydrodynamic simulations in the (3+1)dimensional Milne coordinates, τ = √ t 2 − z 2 , x ⊥ = (x, y), and η s = (1/2) ln [(t + z)/(t − z)]. In this case, the Gaussian distribution in Eq. (5) is replaced with
G(x ⊥ −x ⊥,i , η s −η s,i )d 2 x ⊥ τ dη s = 1 2πσ 2 ⊥ exp − (x ⊥ − x ⊥,i ) 2 2σ 2 ⊥ × 1 2πσ 2 ηs τ 2 exp − (η s − η s,i ) 2 2σ 2 ηs d 2 x ⊥ τ dη s ,(9)
where σ ⊥ and σ ηs are transverse and longitudinal widths of the Gaussian distribution, respectively. In the longitudinal direction, a straight trajectory implies η s,i =
y p,i , where η s,i = (1/2) ln [(t + z i )/(t − z i )] and y p,i = (1/2) ln [(E i + p z,i )/(E i − p z,i )]
are space-time rapidity and momentum rapidity of the ith parton, respectively. In the following, we show some formulas in the Cartesian coordinates to avoid the complex representation of them. In any case, all the hydrodynamic simulations are performed in the Milne coordinates.
C. Dynamical core-corona initialization
We establish the dynamical aspect of core-corona picture by modeling the four-momentum deposition of partons dp µ i (t)/dt in Eq. (7) as an extension of the conventional core-corona picture. The dynamical aspect of the core-corona picture, which we are going to model, is as follows: Partons which are to experience sufficient secondary scatterings with others are likely to deposit their energy-momentum and form equilibrated matter (QGP fluids), while partons which are to rarely interact with others are likely to be free from depositing their energymomentum. To model the dynamical energy-momentum deposition under the core-corona picture, we invoke the equation of motion with a drag force caused by microscopic interactions with other particles.
We define the four-momentum deposition rate of the ith parton generated initially at a co-moving frame along η s,i = y p,i , space-time rapidity of the ith parton, as
dp µ i dτ = − coll j σ ijρij |ṽ rel,ij |p µ i ,(10)
where σ ij is a cross section of the collision between the ith and jth partons,ρ ij is an effective density of the jth parton seen from the ith parton which is normalized to be unity, |ṽ rel,ij | is relative velocity between the ith and the jth partons. Variables with tilde are defined at each co-moving frame along η s,i . The Lorentz transformation from laboratory frame to a co-moving frame along η s,i is given as,
Λ µ ν (η s,i ) = cosh η s,i 0 0 − sinh η s,i 0 1 0 0 0 0 1 0 − sinh η s,i 0 0 cosh η s,i . (11)
The summation in Eq. (10) is taken for all partons with non-zero energy that the ith parton will collide. The candidate partons include not only initially produced ones but also ones in the thermalized medium. We explain details on the treatment to pick up the thermalized partons in Sec. II D. We employ an algorithm [57] to evaluate the number of partonic scatterings that a parton undergoes along its trajectory. Under the geometrical interpretation of cross sections, two partons, i and j, are supposed to collide when the closest distance of them is smaller than σ ij /π where σ ij is the same variable used in Eq. (10). The cross section σ ij is defined as
σ ij = min σ 0 s ij /GeV 2 , πb 2 cut ,(12)
where σ 0 is a parameter with a dimension of area, s ij is a Mandelstam variable s ij = (p µ i +p µ j ) 2 , and b cut is a parameter to avoid infrared divergence of the cross section when s ij becomes too small. We neglect possible color Casimir factors in the cross section, and this parametrization is applied for all quarks, anti-quarks, and gluons. It should be emphasized that this energy dependence of the cross section captures the core-corona picture in the momentum space: the rare and the high-energy partons are not likely to deposit the four-momentum during the dynamical initialization process.
The effective density of the jth parton that is seen from the position of the ith one is defined as follows. The value of Gaussian distribution centered atx j is obtained atx i ,
ρ ij = G(x ⊥ ,η s ;x ⊥,j ,η s,j )|x ⊥ =x ⊥,i ,ηs=ηs,i = 1 2πσ 2 ⊥ exp − (x ⊥,i −x ⊥,j ) 2 2σ 2 ⊥ × 1 2πσ 2 ηs τ 2 exp − (η s,i −η s,j ) 2 2σ 2 ηs .(13)Note thatη s,i −η s,j = η s,i − η s,j ,x ⊥,i = x ⊥,i ,σ ⊥ = σ ⊥ , andσ ηs = σ ηs .
The relative velocity is calculated as
|ṽ rel,ij | = p ĩ p 0 i −p j p 0 j .(14)
As a consequence of the modeling for the fourmomentum deposition rate, initial partons traversing dense regions with low energy and momentum tend to deposit their energy-momentum and generate QGP fluids. On the other hand, initial partons traversing dilute regions with high energy tend to relatively keep their initial energy and momentum. Here the factor coll j σ ijρij |ṽ rel,ij |dτ can be regarded as the number of scattering that the ith parton experiences during dτ .
During the DCCI processes, we monitor the change of the invariant mass of a string which is composed of the color-singlet combination of initial partons provided by Pythia8. It should be noted that once the invariant mass of a string becomes smaller than a threshold to be hadronized via string fragmentation in Pythia8, energymomentum of all the partons that compose the string is dumped into fluids. In this model, we use m th = m 1 + m 2 + 1.0 in units of GeV for the threshold, where m 1 and m 2 are masses of each parton at both ends of the string.
We emphasize here that the formulation of the fourmomentum deposition rate of a parton is largely sophisticated from the one introduced in the previous work [11] although the basic concept of the core-corona picture is the same in both cases. Under the previous work, there was a problem that high p T partons suffer from unexpected large suppression even in p+p collisions. The reason is that, since partons in parton showers in a high p T jet are collimated and close to each other in both coordinate and momentum spaces, they had to deposit large four-momentum in our previous prescription in which only density and transverse momentum of the ith parton are taken into account [11]. This problem is reconciled in this sophisticated modeling by considering trajectories of partons and relative velocities of parton pairs |ṽ rel,ij |. Since trajectories of shower partons in a jet are supposed not to cross each other, the four-momentum deposition due to collisions among the shower partons is not likely to be counted in the summation in Eq. (10). Even if one consider trajectories of shower partons at the early time of dynamical core-corona initialization, they are close in space-time coordinates and would unreasonably deposit their four-momentum. The relative velocity avoids this issue because shower partons are supposed to have small relative velocities. Thus, because of the two factors, the dynamical core-corona initialization with Eq. (10) does not cause the unreasonable four-momentum deposition for partons in jets in DCCI2.
D. Sampling of thermalized partons
As we mentioned in the previous subsection, the summation in the right hand side of Eq. (10) is taken for all partons in the system including not only initially produced partons but also thermalized partons which are constituents of the QGP fluids. This enables one to consider the four-momentum deposition due to scatterings with thermalized partons while traversing in the medium. In order to consider the scatterings with thermalized partons, which are described by hydrodynamics, we sample partons from all fluid elements and obtain phase-space distributions of them at each time step.
Although we employ the lattice EoS, we make a massless ideal gas approximation on fluid elements for simplicity. In this approximation, the number density of partons in a fluid element can be estimated as
n = 90d ζ(3) 4π 4 d s EoS (T ),(15)
where s EoS (T ) is the entropy density obtained from the EoS via temperature T at the fluid element. The effective degeneracies of the QGP, d and d , are defined as
d = d F × 7 8 + d B ,(16)d = d F × 3 4 + d B .(17)
The factors 7 8 in Eq. (16) and 3 4 in Eq. (17) originate from differences between Fermi-Dirac and Bose-Einstein statistics in the entropy density and the number density. The degrees of freedom of fermion d F and boson d B are obtains as
d F = d c × d f × d s × d qq = 3 × 3 × 2 × 2 = 36, (18) d B = d c × d s = 8 × 2 = 16,(19)
where d c , d f , d s , and d qq represent the degrees of freedom of color, flavor, spin, and particle-antiparticle, respectively. The number of partons in a fluid element is then, ∆N 0 = n∆x∆yτ ∆η s , where n is the number density of partons obtained in Eq. (15), ∆x, ∆y, and ∆η s are the widths of one fluid element in the Milne coordinates. One can interpret ∆N 0 as a mean value of Poisson distribution and sample the number of partons N with
P (N ) = exp (−∆N 0 ) ∆N N 0 N ! .(20)
For sampled N partons, we stochastically assign species of them. We pick up a quark or an anti-quark with a probability,
P q/q = (3/4)d F (3/4)d F + d B ,(21)
which corresponds to a fraction of the degree of freedom of Fermi particles, while a gluon is picked up with a probability,
P g = 1 − P q/q .(22)
The three-dimensional momentum k of (anti-)quarks or gluons in the local rest frame of the fluid element is assigned according to the normalized massless Fermi or Bose distribution,
P (k)d 3 k = 1 N nom 1 exp (k/T ) ∓ B,F 1 d 3 k,(23)
where N nom is a normalization factor, T is temperature of the fluid element, and ∓ B,F is a sign for Bose (−) and Fermi (+) statistics. Then, the energy and momentum in the lab frame is obtained by performing Lorentz transformation on k µ = (|k| , k) with the velocity of the fluid element. Space coordinates are assigned with a uniform distribution within each fluid element. For partons sampled from a fluid element centered at
x = x i = (x i , y i , η s,i ),
where the index i stands for the numbering of fluid elements, we assign their coordinates with
P uni (x)τ ∆x∆y∆η s = 0 (x < x i − ∆x/2, x i + ∆x/2 < x) 1 (x i − ∆x/2 ≤ x ≤ x i + ∆x/2) .(24)
As discussed in this subsection, the four-momentum deposition caused by collisions between a traversing parton as a corona part and thermalized partons as core parts could be regarded as a toy model of jet quenching. We note that, although the implementation of a more sophisticated jet-quenching mechanism is the future work, the energy loss of traversing partons in the medium is phenomenologically introduced via the dynamical corecorona initialization in Eq. (10).
E. Modification of color strings
The Lund string model is based on a linear confinement picture of color degree of freedom [47,58]. Energy stored between a quark and an anti-quark linearly increases with the separation length of the quark and anti-quark in the vacuum. However, this picture should be modified if we put them into the QGP at finite temperature. It is known that, at high temperature, the string tension becomes so small that color strings would disappear [59]. Since our input is initially produced partons connected with color strings and we generate fluids through their energy-momentum depositions, some color strings should overlap with the fluids in the coordinate space. We phenomenologically incorporate the modification of the color string configuration due to the finite temperature effect in DCCI2.
At τ = τ s (> τ 0 ), we assume that the string fragmentation happens when the entire color string is outside the fluids. Here, "a color string" means chained partons as a color singlet object. When a color string is entirely inside the fluids at τ = τ s , we discard the information of its color configuration and let its constituent partons evolve as individual non-equilibrated partons according to Eq. (6). If a color string is partly inside and partly outside the fluids, the color string is subject to be cut off at the boundary of the fluids. The boundary here is identified with a contour of T (x) = T sw . The color string in a vacuum cut off at the boundary is reattached to a thermal parton picked up from the hypersurface and forms a color-singlet object again to be hadronized via string fragmentation. The thermal parton is sampled in the hypersurface of the fluids. The details of the above treatment of color strings at τ = τ s are explained in Sec. II E 1. As for the rest of the color string left inside the fluids, we discard the information of its color configuration and let its constituent partons evolve as individual non-equililbrated partons likewise the above case.
During the evolution of the fluids, the individual nonequilibrated partons come out from the hypersurface at some point. We assume that such a parton forms parton pairs to become a color-singlet object by picking up a thermalized parton from the hypersurface of the fluids. This prescription is based on exactly the concept of the coalescence models [60][61][62][63][64][65][66][67][68]. We explain the partonpairing treatment at τ > τ s in Sec. II E 2.
We perform the modification only for color strings in which the transverse momentum of all partons forming that color string is less than a cut-off parameter, p T < p T,cut , at τ = τ 0 . Notice that this is merely a criterion of whether the modification of color string is performed and that all initially produced partons, including very high p T ones, nonetheless experience the dynamical core-corona initialization regardless of p T,cut . This treatment avoids modification on p T spectra of final hadrons generated from intermediate to high p T partons which would less interact with fluids rather than low p T partons. Since the modification on the structure of color strings sensitively affect the final hadron distribution in momentum space, we should make a more quantitative discussion on the parameter p T,cut as a future work.
It should also be noted that when we sample thermalized partons, energy-momentum is not subtracted from fluids just for simplicity.
String cutting at τ = τs
We find a crossing point between a color string and the hypersurface of the fluids by tracing partons chained as color strings one by one. Since Pythia8 and Pythia8 Angantyr give us information of structure of color strings, we respect the initially produced color flow.
At τ = τ s , all initial partons are classified into four types: "hard" partons, dead partons inside fluids, surviving partons inside fluids, and partons outside fluids. We regard partons which is chained with at least one high p T (> p T,cut ) parton as "hard" partons. We do not modify color strings composed of these hard partons to keep the initial color flow and hadronize them in a usual way discussed in Sec. II F. During dynamical core-corona initialization, some partons lose their initial energy completely inside fluids. We regard those partons as dead ones and remove them from a list of partons. These dead partons are no longer considered to be hadronized through string fragmentation. For the other partons, we check whether they are inside the fluids one by one and regard them as surviving partons if it is the case. These surviving partons are to be hadronized at τ > τ s if they have sufficient energy to come out from the fluids. This be explained later in Sec. II E 2. The rest of the partons are considered to be partons outside fluids. Since partons outside the fluids cannot form color-singlet strings by themselves, we need to cut the original color strings at crossing points between the hypersurfaces and the color strings by sampling thermal partons.
In the following, we explain how to find crossing points and how to sample thermal partons. We first assume that two adjacent partons in a color string, regardless of their status (dead, surviving, or outside fluids), are chained with linearly stretched color strings between the ith and the (i + 1)th partons in coordinate space. As a simple case, suppose that [T (
x i ) − T sw ][T (x i+1 ) − T sw ] < 0, where x i = (x i , y i , η s,i ) and x i+1 = (x i+1 , y i+1 , η s,i+1 )
are the positions of the ith and the (i + 1)th partons, respectively, there exists a hypersurface of fluids between the ith and the (i+1)th partons. We scan temperature at all fluid elements along the linearly stretched color string from the ith to the (i + 1)th parton to find a crossing point.
Once the crossing point is found, a thermalized parton is picked up to form a color-singlet string. The thermalized partons are sampled by using the information of the crossing point on the hypersurface such as velocity v hyp , temperature T hyp , and coordinates x hyp , which are obtained by taking an average of that of two adjacent fluid elements crossing the hypersurface. For instance, if the two adjacent fluid elements (symbolically denoted as the jth and the (j + 1)th fluid element) have temperature T j > T sw and T j+1 < T sw , respectively, the temperature at the crossing point is obtained as T hyp = (T j+1 + T j )/2. When the hypersurface of the fluids and the configuration of the color string are highly complicated, there could exist more than one crossing point between two adjacent partons. In such a case, we pick up a thermal parton from the closest crossing point for each parton in the string.
The species of the picked-up parton, whether if it is a quark, an anti-quark, or a gluon, is fixed by the configuration of color strings. For a color string that has a quark and an anti-quark at its ends, if string cutting removes the quark(anti-quark) side of the color string, an anti-quark(a quark) is picked up from the crossing point to form a color-singlet string in the vacuum. When there is a color string that consists of two gluons while the only one of the gluons is inside of fluids, we pick up a gluon to make a color-singlet object. On the other hand, for color strings with more than two gluons and no quarks or anti-quarks as their components, the so-called gluon loops, we cut the loop to open and pick up two gluons from the crossing points to make this a color-singlet object again.
A momentum of a picked-up parton is sampled with a normalized Fermi or Bose distribution,
P (p; m)d 3 p = 1 N norm (m) 1 exp p 2 + m 2 /T hyp ± B,F 1 d 3 p, (25) where N norm (m) = 1 exp p 2 + m 2 /T hyp ± B,F 1 d 3 p. (26)
The energy of the (anti-)quarks is assigned so that they are mass-on-shell, which we require to perform hadronization via string fragmentation in Pythia8.
Four-momentum of these partons is Lorentz-boosted by using fluid velocity at the crossing point, v hyp .
We stochastically assign flavors f = u, d, or s for each quark or anti-quark with the following probability, (28) where the mass values of these quarks are taken from general settings in Pythia8.
P f = N norm (m f )/N sum ,(27)N sum = N norm (m u ) + N norm (m d ) + N norm (m s ),
As we mentioned in Sec. II C, there is a threshold of invariant mass of a color string to be hadronized via string fragmentation in Pythia8. If the invariant mass of a modified color string is smaller than the threshold, we remove partons forming the color string from a list of partons.
Parton-pairing for surviving partons
At τ > τ s , we hadronize "surviving partons traversing inside of the fluids" when each of them comes out from fluids. To make the parton color-singlet to hadronize via string fragmentation, the parton picks up a thermal parton around the hypersurface.
Whether a parton comes out from medium or not is determined by the temperature of a fluid element at which the parton is currently located. A surviving parton traverses a fluid according to Eq. (6).
At the kth proper time step τ = τ k , suppose that a parton is at x = x(τ k ), where its temperature is T (x(τ k ), τ k ) > T sw , and will move to x = x(τ k+1 ) at the next time step.
Simply assuming that the hypersurface does not change between the kth and the (k + 1)th time step and see if the temperature satisfies T (x(τ k+1 ), τ k ) < T sw by checking hypersurface only at the kth time step. If the above condition is satisfied, the parton is regarded as coming out from the fluids at x hyp = [x(τ k ) + x(τ k+1 )]/2 at the kth time step. For a quark (an anti-quark) coming out from medium, an anti-quark (a quark) is picked up to form a color-singlet string. On the other hand, for a gluon, a gluon is picked up to do so. A momentum is again sampled by using Eq. (25), while its flavor is sampled with Eq. (27).
Note that, if a surviving parton fails to escape from the fluids by losing its initial energy completely, we regard that parton as a dead one and remove it from a list of surviving partons. Here again, if the invariant mass of the pair of partons is smaller than the threshold to be hadronized via string fragmentation in Pythia8, we remove them from the list of partons.
F. Direct hadrons from core-corona and hadronic afterburner
We switch the description of the hadrons in the core parts from hydrodynamics to particle picture at the T (x) = T sw hypersurface. The particlization of fluids is performed with iS3D [45], which is an open-source code to perform conversion of hypersurface information of fluids into phase-space distributions of hadrons based on Monte-Carlo sampling of the Cooper-Frye formula [52]. Since the original iS3D [45] is not intended for eventby-event particlization, we extend the code so that this can be utilized for our event-by-event analysis. We also change the list of hadrons in iS3D to the one from the hadronic cascade model, JAM [46], which is employed for the hadronic afterburner in the DCCI2 framework. The hypersurface information is stored from τ = τ 0 , the beginning of dynamical core-corona initialization, to the end of hydrodynamic evolution at which temperature of all fluid elements goes below T sw . Fluid elements with p · dσ < 0, which is known as the negative contribution in the Cooper-Frye formula, and those with T < 0.1 GeV are ignored in iS3D. Note that ignoring the negative contributions makes it possible to count all flux generated via source terms in dynamical initialization. In other words, if one integrates all flux including negative ones and neglects deposition of energy inside the fluids, total flux becomes zero due to Gauss's theorem 4 . This is because, under the dynamical initialization framework, our simulation starts from the vacuum, and the deposited energy-momentum is regarded as incoming flux into the hypersurface. We admit that energy-momentum conservation should be improved in a better treatment while we checked the conservation is satisfied within a reasonable range. The space-time coordinates of sampled hadrons (x i , y i , η s,i ) are assigned stochastically in the same way as one used for picking up thermalized partons explained in Sec. II D.
Regarding the corona parts, partons out of equilibrium undergo hadronization through string fragmentation. Until τ = τ s , we assume no hadronization occurs. At τ = τ s , partons outside the fluids hadronize via string fragmentations. If a part of the string is inside the fluids, we modify the color string by cutting it at the crossing point as explained in Sec. II E 1 and hadronize the modified color string. Once surviving partons come out from the hypersurface of fluids after τ = τ s , those partons are hadronized by picking up a thermal parton to form a string as discussed in II E 2. The string composed of at least one high p T (> p T,cut ) parton is hadronized when all the partons chained with this high p T parton come out from the fluids.
The string fragmentation is performed by utilizing Pythia8. The flag ProcessLevel:all=off is set to stop generating events and forceHadronLevel() is called to perform hadronization against the partons manually added as an input. The information of input partons handed to Pythia8 is, particle ID, four-momentum, coordinates, color, and anti-color. As for coordinates, only transverse coordinates, x and y, of partons are handed while t and z are set to be zeros. This is because assigning t and z may cause violation of causality and should be treated carefully in Pythia8. We correct the energy of the partons to be mass-on-shell using their momenta and rest masses to perform string fragmentation in Pythia8. This is the same procedure as we did in the previous work [11] since quarks or anti-quarks that lose their energy in dynamical initialization are mass-off-shell due to the four-momentum deposition of Eq. (10). Information of vertices of generated hadrons are obtained with an option Fragmentation:setVertices = on based on the model proposed in Ref. [69]. We use this information for initial conditions in JAM.
In both the particlization by iS3D and the string fragmentation by Pythia8, we turn off decays of unstable hadrons. Instead, JAM handles decays of the unstable hadrons together with rescatterings while describing their space-time evolution in the late stage. The hadrons obtained from both iS3D and Pythia8 are put into JAM all together to perform the hadronic cascade since both components should interact with each other. In JAM, an option to switch on or off hadronic rescatterings is used to see its effect on final hadrons. It should be also noted that we turn off electroweak decays except Σ 0 → Λ + γ to directly compare our results with experimental observable.
G. Parameter set in DCCI2
Here we summarize all the parameters that we use throughout this paper.
Note that we use the same parameters for both p+p and Pb+Pb collisions except p T0Ref . In the conventional hydrodynamic models, several parameters are used to directly parametrize the initial profiles of hydrodynamic fields. In contrast in DCCI2, how many initial partons are generated is determined in Pythia8 or Pythia8 Angantyr and how much the energy-momentum of these initial partons are converted to the hydrodynamic fields is controlled through the parameters, σ 0 , b cut , σ ⊥ , and σ ηs , in Eq. (10). More details on how to fix these parameters are discussed in Sec. III.
III. RESULTS AND DISCUSSIONS
In this paper, we simulate p+p collisions at √ s = 7 TeV and Pb+Pb collisions at √ s N N = 2.76 TeV with DCCI2. The following results are obtained from full simulations of 300K and 12.5K events for p+p and Pb+Pb collisions, respectively. In Sec. III A, we start with fixing some major parameters in DCCI2 to reproduce the experimental data of the charged particle multiplicity as functions of multiplicity (p+p) or centrality (Pb+Pb) classes and the multiplicity dependence in particle yield ratios of omega baryons to charged pions. As a result of the parameter determination, fractions of core and corona components to final hadronic productions as a function of charged particle multiplicity at midrapidity are extracted. Next, we show the transverse momentum spectrum in p+p and Pb+Pb collisions and its breakdown into core and corona components in Sec. III B. In order to see the interplay between core and corona components on observable obtained from final hadrons, we analyze the mean transverse momentum and the second-order anisotropic flow coefficients as functions of the number of produced charged particles in certain kinematic windows in Sec. III C. Finally, we show multiplicity dependence of radial flow effects based on violation of the mean transverse mass scaling and discuss if the effect can be discriminated from the one originating from pure string fragmentation with color reconnection [70] in Sec. III D.
Due to the two competing particle production mechanisms, it is not trivial to reproduce the multiplicity within a two-component model like DCCI2. We discuss details of this issue in Sec. III E. Let us note that the effects of string cutting explained in Sec. II E are not investigated throughout this paper because the modification on color strings adds another complexity which we want to avoid in this discussion.
A. Parameter determination and fractions of core and corona components
Here we focus on two main parameters in DCCI2, p T0Ref used in the generation of initial partons in Pythia8 and Pythia8 Angantyr, and σ 0 to scale the magnitude of cross sections in Eq. (10). We determine these parameters to reasonably describe both the charged particle multiplicity as a function of multiplicity (p+p) or centrality (Pb+Pb) classes at midrapidity and particle yield ratios of omega baryons to charged pions as functions of charged particle multiplicity.
The multi-strange hadron yield ratios tell us fractions of contributions from thermalized (core) and nonthermalized (corona) matter to total final hadron yields [10,11]. On the other hand, the charged particle yields need to be used in the parameter determination together with the particle yield ratios. These two parameters are highly sensitive to both charged particle multiplicity and particle yield ratios and are strongly correlated. Detailed discussion on this issue is made in Sec. III E. The resultant parameter values are summarized in Table I in Sec. II G. Here, the switching temperature T sw , which controls particle yield ratios as the parameters mentioned above do, is fixed to describe the ratios of omega baryons to charged pions in central Pb+Pb collisions. Figure 2 shows particle yield ratios to charged pions produced in |y| < 0.5 as functions of charged particle multiplicity |η| < 0.5 in p+p and Pb+Pb collisions compared with the ALICE experimental data [71][72][73][74][75][76]. Note that the charged particle multiplicity at midrapidity dN ch /dη |η|<0.5 in the horizontal axes is obtained by using V0M (−3.7 < η < −1.7 and 2.8 < η < 5.1) multiplicity (p+p) or centrality (Pb+Pb) class, which is the same procedure as used in the ALICE data [78]. Determining these classes by using the multiplicity in forward and backward rapidity regions is essential even in theoretical analysis to avoid the effect of self-correlation on observables at midrapidity [74]. Throughout this paper, the "charged particles" mean the sum of charged pions, charged kaons, protons, and antiprotons, which do not contain contributions from weak decays. To obtain particle ratios of primary strange hadrons, which are stable against strong decays, we switch off their weak decays in JAM. Note that we take into account a particular electromagnetic decay, Σ 0 → Λ + γ, in the presented results of Λ yields [79]. Results with switching off hadronic rescatterings are shown to reveal the effect of hadronic rescat-terings [80][81][82] on both core and corona components in the late stage. Results from Pythia8 for p+p collisions and Pythia8 Angantyr for Pb+Pb are also plotted as references.
Overall, smooth changes of the particle yield ratios are observed along charged particle multiplicity, which is consistent with our previous studies [10,11]. Due to the implementation of the core-corona picture in the dynamical initialization framework, particle productions from corona components with string fragmentation are dominant in final hadron yields in low-multiplicity events, while those from core components produced from equilibrated matter are dominant in high-multiplicity events. Thus the overall tendency is that the particle yield ratio at low-multiplicity events almost reflects its value obtained from string fragmentation, while the one at highmultiplicity events reflects the value obtained only from hadronic productions from hydrodynamics. Notice that the particle yield ratios of all hadronic species are almost independent of multiplicity from p+p to Pb+Pb collisions with default Pythia8 and Pythia8 Angantyr respectively, which is one of the manifestations of "jet universality", namely, the string fragmentation being independent of how the string is formed from e + +e − to Pb+Pb collisions [83] 5 .
We tune the parameters in the full simulations of DCCI2 to reasonably reproduce the particle yield ratios of omega baryons to pions, Ω/π, reported by the ALICE Collaboration [71,72]. Although we have to admit that our results do not perfectly describe the experimental data as one sees in Fig. 2 (a), fine-tuning of the parameters is beyond the scope in this paper. In Figs. 2 (b)-(e), we also show results of cascades, lambdas, protons, and phi mesons, respectively. For the ratios of cascades to pions, Ξ/π, in Fig. 2 (b), our results underestimate the experimental data except the lowest-and the highestmultiplicity classes in p+p collisions. For the ratios of lambdas to pions, Λ/π, in Fig. 2 (c), our results from full simulations show smaller values than the experimental data in p+p collisions for almost the entire charged particle multiplicity, while it shows good agreement with the data in Pb+Pb collisions. For the ratios of protons to pions, p/π, in Fig. 2 (d), our full results including hadronic rescatterings through JAM qualitatively describe the decreasing behavior along the charged particle multiplicity in the experimental data in Pb+Pb collisions. This is consistent with a perspective of proton-antiproton annihilations [42,75]. The annihilation effect is seen even in p+p collisions, which leads to a better agreement with the experimental data. For the ratios of phi mesons to pions, φ/π, in Fig. 2 (e), the tendency in the experimental data above dN ch /dη |η|<0.5 ∼ 7 is well captured by our full result. In particular, the dissociation of phi mesons , (e) phi mesons (φ) to charged pions (π + and π − ) as functions of charged particle multiplicity at midrapidity in p+p and Pb+Pb collisions. Results from full simulations of DCCI2 in p+p collisions at √ s = 7 TeV (closed red triangles) and Pb+Pb collisions at √ sNN = 2.76 TeV (closed red diamonds) collisions are compared with the ALICE experimental data in p+p (black pluses) and Pb+Pb (black crosses) collisions [71][72][73][74][75][76]. The Λ/π ratio in Pb+Pb collisions at √ sNN = 2.76 TeV reported by the ALICE Collaboration in Ref. [76] is plotted as a function of the number of participants Npart rather than charged particle multiplicity. The corresponding charged particle multiplicity at midrapidity is taken from Ref. [75]. in hadronic rescatterings plays an important role to describe the suppression at high-multiplicity as observed in the experimental data.
Notably, the increasing behavior along charged particle multiplicity in p+p collisions is achieved in our results with the core-corona picture. It is also discussed that canonical suppression models, which are commonly used in the discussion on multiplicity dependence of particle yield ratios in comparison with the core-corona picture, need to incorporate incompleteness of chemical equilibrium for strangeness due to the hidden strangeness of phi mesons [85,86]. Thus the increasing behavior would clearly represent that the matter formed in p+p collisions is under incompleteness of chemical equilibrium for strangeness.
The upper panel of Fig. 3 (a) shows charged particle multiplicity at midrapidity dN ch /dη |η|<0.5 as a function of multiplicity class σ/σ INEL>0 in p+p events. Here, we take into account only INEL > 0 events in which at least one charged particle is produced within a pseudorapidity range |η| < 1.0 defined in the ALICE experimental analysis [74].
The upper panel of Fig. 3 (b) shows the same observable but as a function of centrality class in Pb+Pb collisions. Here we again note that each multiplicity or centrality class is obtained with V0M multiplicity. In both figures, results from simulations with and without hadronic rescatterings are compared with the ALICE experimental data [74,77]. Each contribution from core and corona components is separately shown as stacked bars for the case without hadronic rescatterings. It should be noted that the separation of core and corona components in DCCI2 is attained only by switching off hadronic rescatterings in JAM. This is because hadronic rescatterings mix those two components up by causing parton exchange between hadrons or formation of excited states. Both of our results show the reasonable description of the ALICE experimental data in p+p and Pb+Pb collisions. From the comparison between with and without hadronic rescatterings, (quasi-)elastic scatterings would be dominant and, as a result, the effect of hadronic rescatterings on multiplicity turns out not to be significant.
The lower panels of Figs. 3 (a) and (b) show the yield fractions of core and corona components to the total from results without hadronic rescatterings, R core and R corona , respectively, as functions of multiplicity (p+p) and centrality (Pb+Pb) classes. Smooth changes along multiplicity and centrality classes are observed in both p+p and Pb+Pb collisions. In Fig. 3 (a), the fraction of core components in p+p collisions almost vanishes for 48-68% and 68-100% multiplicity classes, in which dN ch /dη |η|<0.5 is less than ∼ 5. Then, it increases along multiplicity and reaches R core ∼ 0.53 in the highest multiplicity class 0.0-0.95% in which dN ch /dη |η|<0.5 ∼ 21. One also sees that the contribution of core components overtakes that of corona components only in 0.0-0.95% multiplicity class within our calculations with the current parameter set. This supports a perspective that recent observations of collectivity in high-multiplicity small colliding systems at the LHC energies result from the (partial) formation of the QGP fluids. It should also be noted that the fraction of core components shows R core ∼ 0.12 at dN ch /dη |η|<0.5 ∼ 7 which is minimum-bias multiplicity for INEL > 0 events [87] 6 . The lower panel of Fig. 3 (b) shows results in Pb+Pb collisions. The core components highly dominate, R core 0.90, from 0 to 10% centrality classes where their corresponding multiplicities are above dN ch /dη |η|<0.5 ∼ 10 3 . The corona components become dominant around at 80% centrality class towards peripheral events. It should also be mentioned that the contribution of corona components remains R corona ∼ 0.17-0.22 at midrapidity in intermediate centrality classes (∼ 40-60%) where the whole systems is often assumed to be described by hydrodynamics. Figure 4 shows the fractions of core and corona components in p+p and Pb+Pb collisions simultaneously, which are identical to the results in the lower panels in Fig. 3 but as functions of charged particle multiplicity at midrapidity. Smooth crossover from corona dominance to core dominance appears along multiplicity from p+p to Pb+Pb collisions. The dominant contribution flips at dN ch /dη |η|<0.5 ∼ 18. These results clearly demonstrate that the fractions of contribution from core and corona components are scaled with charged particle mul-tiplicity in DCCI2, regardless of differences in the system size or collision energy between p+p and Pb+Pb collisions. Here we emphasize that, interestingly, the fraction of corona still remains ∼ 10% at the most central events in Pb+Pb collisions. This also implies that both core and corona components should be implemented even in dynamical modeling of high-energy heavy-ion collisions towards precision studies on properties of QCD matter.
B. Transverse momentum dependence of core and corona contribution
It is also interesting to compare the sizes of core and corona contributions in transverse momentum p T spectra of the final state particles. According to the implementation of the core-corona picture by Eq. (10), initial low momentum partons are likely to generate QGP fluids and expected to contribute largely in the low-p T region. Meanwhile, high momentum particles are likely to traverse vacuum or fluids as mostly keeping their initial momentum and supposed to dominate the high-p T region.
Upper panels of Fig. 5 show the charged particle p T spectra at midrapidity |η| < 2.5 in (a) p+p and (b) Pb+Pb collisions. The kinematic cuts and event selections are the same as the ones used in the ATLAS experimental results [88]. Event average is made with at least one charged particle having p T > 0.5 GeV and |η| < 2.5 in both p+p and Pb+Pb collisions, which can be regarded as almost minimum-bias events. Again, a comparison between results obtained from full simulations and ones from simulations without hadronic rescatterings is made here. Each contribution from core and corona components to the final hadrons from simulations without hadronic rescatterings is shown as well. In both p+p and Pb+Pb collisions, the p T spectra of final hadrons without hadronic rescatterings are represented as sums of contributions of core and corona components over the whole p T regions. One also sees that the effect of hadronic rescatterings on the p T spectra of charged particles is almost absent in both p+p and Pb+Pb collisions. Since the charged particles are mainly composed of charged pions, their p T spectra are relatively insensitive to hadronic rescatterings.
The corresponding lower panels of Fig. 5 show the fractions of core and corona components for final hadrons without hadronic rescatterings as functions of p T . As an overall tendency, the dominance of the corona components at high p T regions is seen in both p+p and Pb+Pb collisions, which is exactly what we expect from the core-corona picture in the momentum space encoded in Eq. (10). In p+p collisions, the contribution from the core components reaches R core ∼ 0.3 around p T ∼ 1.0-1.5 GeV and the contribution from the corona components is almost dominant over the whole p T range. On the other hand, the contribution from core components is dominant in low p T regions in Pb+Pb collisions, while the domi- nant contribution is flipped to the corona components at p T ∼ 5.5 GeV towards high p T regions. Remarkably, only within 0.7 p T 3.6 GeV, the core components highly dominate, R core 0.9. The existence of corona components should be considered below ∼ 0.7 GeV and above ∼ 3.6 GeV even in minimum-bias events.
In particular, there is a small peak in the fraction of corona components with R corona ∼ 0.2 at most in p T 1 GeV. This contribution originates mainly from a feeddown from fragmentation of strings including surviving partons during the dynamical initialization stage. This is a consequence of the dynamical core-corona initialization against initially generated partons. Thus there should be a kind of "redshift" of the p T spectrum due to energy loss of traversing partons which contribute as corona components in the soft region. As we emphasized in Introduction, in the core-corona picture, this result exactly illustrates "soft-from-corona" that there exists a non-negligible contribution of non-equilibrated corona components in low p T region. Therefore, in order to properly extract transport coefficients of the QGP fluids from, for example, an analysis of flow observables, hydrodynamic results should be corrected with corona components. We demonstrate this correction within DCCI2 in the next subsection.
C. Correction from corona to flow observable
As we discussed in Secs. III A and III B, both core and corona contributions appear over a wide range of multiplicity. Moreover, each component contributes as a function of p T in a nontrivial way. To investigate how the effects of the interplay between core and corona components appear on observable, we first analyze the mean transverse momentum p T of charged particles at midrapidity as a function of the number of charged hadrons generated at midrapidity N ch in p+p and Pb+Pb collisions. Figure 6 shows the mean transverse momentum p T of charged particles as a function of charged particle multiplicity N ch in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ s N N = 2.76 TeV. Charged particles with 0.15 < p T < 10.0 GeV and |η| < 0.3 are used for evaluation of p T , while N ch is obtained by counting charged particles with |η| < 0.3 (without p T cut), which is the same kinematic range used in Ref. [89]. For p+p collisions in Fig. 6 (a), our result from DCCI2 qualitatively describes the steep enhancement of p T along N ch observed in the ALICE experimental data [89]. Almost no significant difference is seen between results from full simulations and the ones without hadronic rescatterings. This means that the effect of hadronic rescatterings on p T of charged particles is almost negligible due to a small number of final hadrons in p+p collisions. One also sees that the core and corona components show small difference of p T below N ch ∼ 20. This is because, as seen in Fig. 5 (a), there is no large difference for the slopes of p T spectrum of the core and corona components in low p T regions while the particle productions in the region would contribute to p T significantly.
For Pb+Pb collisions presented in Fig. 6 (b), our results from full simulations with DCCI2 reasonably describe the experimental data within the range of experimental data. A slight difference is seen between results from full simulations and the ones from simulations without hadronic rescatterings: Mean transverse momentum is slightly enhanced due to hadronic rescatterings and the effect becomes relatively clear as increasing N ch . On the other hand, the large difference is seen between the results from core and corona components. The core components show larger p T while the corona components show smaller values for almost the entire N ch . The larger p T from core components originates from the flatter slope of p T spectrum, while the smaller p T from corona components originates from the steeper slope of p T spectrum in the low p T region seen in Fig. 5 (b). The difference between the results without hadronic rescatterings and the ones from core components exactly exhibits there exists the sizable correction from non-thermalized matter to the results obtained purely from hydrodynamics. The correction is found to be visible for the entire N ch and to be ∼ 5-11% in N ch 200. Therefore the "soft-from-corona" components are the key to precisely reproduce the multiplicity dependence of the mean transverse momentum 7 . Figure 7 shows the second-order anisotropic flow coefficient of charged particles obtained from two-particle cumulants, v 2 {2}, as a function of N ch in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ s N N = 2.76 TeV. Kinematic cuts for both v 2 {2} and N ch are 0.2 < p T < 3.0 GeV and |η| < 0.8 as used in Ref. [93]. It should be mentioned that insufficient statistics with DCCI2 simulations do not allow us to have a pseudorapidity gap of charged hadron pairs |∆η| > 1.4 in the v 2 {2} analysis unlike in the ALICE analysis. This is the reason why we do not compare our results with the experimental data in this paper. We leave quantitative discussion by comparing with experimental data for future work.
For the results in p+p collisions, v 2 {2} obtained from 7 It is discussed that the centrality dependence of p T is well described by hydrodynamic simulations introducing the finite bulk viscosity [90]. While a recent Bayesian analysis supports the zero-consistent bulk viscosity by analyzing p T -differential observables [7]. Both of them, however, still failed to reproduce pion p T spectra below ∼ 0.3 GeV (see, e.g., Fig. 3 in Ref. [90] and Fig. 20 in Ref. [91]. Another related discussion is in Ref. [92]), which would result in overestimation of p T . The discrepancy between hydrodynamic results and experimental data in this low p T region becomes larger as going to peripheral collisions or small colliding systems [91]. Therefore the deviation between the model and the data in the low p T region could be filled with the corona components and would improve the description of p T . both core and corona components is larger than that from simulations without hadronic rescatterings in 10 N ch 50. This suggests that the event plane angle of core components might be different from that of corona components, which dilutes v 2 {2} of core and corona components with each other.
For the results in Pb+Pb collisions, the v 2 {2} from full simulations reaches a maximum value at N ch ∼ 400, which is similar to the tendency observed in experimental data [93]. From a comparison between the results with and without hadronic rescatterings, one can tell that a slight enhancement of v 2 {2} comes from generation of elliptic flow in the late hadronic rescattering stage [80][81][82]. Here again, one can see the correction from corona components in the comparison between the core result and the inclusive result in the case without hadronic rescatterings. The correction from corona components is found to be ∼ 15-38% below N ch ∼ 370, which originates from the small peak seen at very low p T region in the p T spectra in Fig. 5 (b) 8 . This suggests that one would need to 8 The leftmost point of the contribution from core components is slightly shifted to large N ch since there are some events in which one cannot calculate two-particle cumulants due to less than two charged particles from the core parts are measured in a given kinematic window in this N ch bin. Therefore the event average of N ch for core components is biased to larger N ch .
incorporate corona components in hydrodynamic frameworks to extract transport coefficients from comparisons with experimental data.
In both p+p and Pb+Pb results, there are two factors that would give a finite anisotropy in corona components, which are color reconnection and feed-down from surviving partons. The color reconnection effect implemented in default Pythia8 and Pythia8 Angantyr can arise collectivity [94]. With the color reconnection, dense color strings formed due to multiparton interactions interact with each other and eventually induce flow-like behavior of final hadrons. Its effect can be enhanced due to more multiparton interactions in initial parton generation with DCCI2 compared to default Pythia8 and Pythia8 Angantyr. The detailed discussion on multiparton interactions in initial parton generation is made in Sec. III E. Under the dynamical core-corona initialization, partons originating from hard scatterings and emitted in the back-to-back directions tend to survive. In contrast, soft partons, which originate from multiparton interactions and are randomly directed, tend to be converted into fluids. Since the low p T charged hadrons come from such surviving partons through string fragmentation, v 2 {2} of corona components could reflect that of their parents. As a result, the corona components show larger anisotropy compared to results from the default Pythia8 [94] and Pythia8 Angantyr.
D. Multiplicity dependence of mean transverse mass
The fraction of core components to total hadronic productions increases along charged particle multiplicity as shown in Fig. 4. Since the effects of radial flow are expected to be more pronounced as increasing fraction of core components, we analyze the mean transverse mass for various hadrons in high-and low-multiplicity p+p and Pb+Pb events and see its mass dependence. It has been empirically known that m T spectra in small colliding systems exhibit the m T scaling, i.e., the slope of m T spectra being independent of the rest mass of hadrons [95,96]. Here, m T = m 2 + p 2 T is the transverse mass and m is the rest mass of the hadron. In contrast, in heavy-ion collisions, the slope parameter increases with m and, as a result, the m T scaling is violated, which is regarded as a sign of the existence of radial flow generated [97,98]. Thus whether radial expansion exists in small colliding systems due possibly to the QGP formation can be explored through the empirical scaling behavior and its violation in the mean transverse mass.
We take two event classes, high-multiplicity (0-10%) and low-multiplicity (50-100%) events, in p+p and in Pb+Pb collisions 9 . The multiplicity or centrality classification is performed in the same way as the one used in Fig. 3. In the following, we analyze the mean transverse mass of charged pions (π + and π − ), charged kaons (K + and K − ), protons (p andp), phi mesons (φ), lambdas (Λ andΛ), cascade baryons (Ξ − andΞ + ), and omega baryons (Ω − andΩ + ), in |η| < 0.5 without p T cut. Figure 8 (a) shows the mean transverse mass, m T − m, as a function of the rest mass of hadrons, m, in high-multiplicity (0-10%) and low-multiplicity (50-100%) p+p collisions at √ s = 7 TeV. To pin down the effect of hadronic rescatterings, the results from full simulations and simulations without hadronic rescatterings are compared with each other. The result from Pythia8 with the default settings including color reconnection is also plotted as a reference. Overall, m T − m in highmultiplicity events (0-10%) tends to exhibit an almost linear increase with increasing m except for phi mesons. On the other hand, such a clear mass dependence is not seen in low-multiplicity events (50-100%), which is consistent with the m T scaling. It should also be mentioned that the violation of linear increase of phi mesons in core components appears after resonance decays against direct hadrons (not shown). An apparent flow-like linear mass dependence is seen in results from Pythia8 in highmultiplicity events as well, which is due to the color reconnection [99]. The almost linear increasing behavior in DCCI2 is caused by both radial flow for core components from hydrodynamic expansion and color reconnection for corona components obtained with Pythia8. As a result, the results from both DCCI2 and Pythia8 have similar tendencies. Therefore, it is difficult to discriminate each effect by merely seeing the mean transverse mass. The effect of hadronic rescatterings is almost absent for pions. This comes from an interplay between small pdV work in the late hadronic rescattering stage and approximate conservation of pion number [4]. The small effects of hadronic rescatterings are seen for phi mesons and omega baryons because they do not form resonances in scattering with pions unlike other hadrons [81,82]. almost linear increasing behavior of m T − m appears even in 50-100% centrality class as one can expect from the centrality dependence of the fraction of core components in Fig. 3 (b). The larger enhancement of the mean transverse mass due to hadronic rescatterings, in particular, for protons is seen in high-multiplicity events in comparison with the low-multiplicity events. This is a manifestation of the famous "pion wind" in the late rescattering stage [100][101][102][103].
Figure 8 (c) shows each contribution of core and corona components to the final result without hadronic rescatterings in 0-10% multiplicity class in p+p collisions. The inclusive result here is identical to the one shown as the result without hadronic rescatterings in Fig. 8 (a). The difference between results of core and corona components is seen in protons, lambdas, and omega baryons. The linear mass ordering of m T − m from core components is slightly diluted for protons and lambdas in the inclusive result due to the sizable contribution of corona components. In contrast, the core result and the inclusive result are almost on top of each other since the contribution of corona components for omega baryons is smaller in 0-10% multiplicity class compared to other particle species. Fig. 3 (b), the fraction of core components shows R core ∼ 0.8 to ∼ 0.9 in this centrality range. Eventually the result of core components is found to be slightly diluted by corona components. As shown in Fig. 3 in Sec. III A, we reproduced centrality dependence of charged particle multiplicity in p+p and Pb+Pb collisions within DCCI2. Although the default Pythia8 (or Angantyr model in heavy-ion modes) works reasonably well, reproduction of multiplicity within DCCI2 can be attained only after the considerable change of a parameter p T0Ref from its default value as mentioned in Sec. II G. A nontriviality in DCCI2 stems from different competing mechanisms of how the transverse energy changes during the evolution of the system. In this subsection, we discuss the effects of string formation/fragmentation and longitudinal pdV work on the transverse energy and explain why we needed to change this parameter in DCCI2.
The transverse energy per unit rapidity dE T /dη is a basic observable in high-energy nuclear collisions and contains rich information on the dynamics of an entire stage of the reactions. The transverse energy changes mainly in the initial and the expansion stages of the reactions. In the initial stage, the two energetic hadrons and/or nuclei form color flux tubes between them as they pass through each other. The chromo-electric and magnetic fields in the color flux tubes possess the energy originating from the kinetic energy of colliding hadrons or nuclei. The decays of color flux tubes into partons and subsequent rescatterings among them are supposed to lead to the QGP formation [104][105][106]. Thus, how much energy is deposited in the reaction region is a fundamental problem of the QGP formation and depends on the initial dynamics of high-energy nuclear collisions. On the other hand, in the expansion stage, the pdV work associated with the longitudinal expansion after the QGP formation reduces the energy produced in the initial reaction region [107,108]. The amount of reduced energy is sensitive to viscosity and other transport properties of the QGP [109]. Therefore dE T /dη can be a good measure to scrutinize modeling in the initial and the expansion stages of the reactions.
In Pythia8, partons are first generated through hard scatterings and then, together with partons from initial and final state radiations, form hadron strings which eventually fragment into hadrons. The transverse energy per unit rapidity of final hadrons is always larger than that of initially generated partons around midrapidity. To understand this enhancement around midrapidity, suppose a hadronic string formed from a di-quark in the forward beam rapidity region and a quark in the backward beam rapidity region as an extreme case. Although the partons lie only around beam rapidity regions and the transverse energy of them vanishes around midrapidity, that string fragments into hadrons almost uniformly in rapidity space. Thus, the emergence of the transverse energy at midrapidity is a consequence of the formation of a color string between such partons around beam rapidity. Since parameters in Pythia8 are so tuned to reproduce the final hadron spectra, the initial parameters are highly correlated with parameters in the fragmentation as a whole. Therefore the default parameter set should not be used if the subsequent hydrodynamic evolution, which reduces the transverse energy from its initial value of generated partons, is incorporated in DCCI2. Figure 9 shows dE T /dη of the initial partons before the string hadronization or the dynamical initialization, and that of the hadrons in the final state from both Pythia8 and DCCI2 in minimum-bias (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ s N N = 2.76 TeV.
For the results from Pythia8 in Fig. 9 (a), the transverse energy per unit pseudorapidity, dE T /dη, of the final hadrons is always larger than that of the initial partons in the whole rapidity region except around the beam rapidity. Since the final hadron yield is dominated by the corona components in p+p collisions in DCCI2, these results are almost identical with the ones from Pythia8. For the results from Pythia8 Angantyr in Fig. 9 (b), the behavior of the transverse energy in Pb+Pb collisions is again the same as in p+p collisions. In contrast, to obtain the same amount of the transverse energy in the final state in DCCI2, the transverse energy must be deposited initially ∼ 3 times as large as that in Pythia8 Angantyr at midrapidity to reconcile the reduction of transverse energy due to pdV work. This is exactly possible by considerably decreasing the parameter p T0Ref . The parameter p T0Ref regulates infrared divergence of the QCD cross section, can be interpreted as a parameter p ⊥min to separate soft from hard scales 10 , and controls the number of multiparton interactions in Pythia [110,111]. The smaller the separation scale p ⊥min is, the larger the number of multiparton interaction n MPI (p ⊥min ) = σ 2→2 (p ⊥min )/σ nd is. Here σ 2→2 and σ nd are the perturbative QCD 2 → 2 cross section and the inelastic non-diffractive cross section, respectively. By increasing n MPI (p ⊥min ) as decreasing p T0Ref , initial partons are generated more and bring the sufficient amount of transverse energy in the final hadron state as shown in Fig. 9 (b). In this work, we use p T0Ref =1.8 and 0.9 for p+p and Pb+Pb collisions as mentioned in Sec. III A, which are smaller than the default values, 2.28 and 2.0 for MultipartonInteractions:pT0Ref and SpaceShower:pT0Ref in Pythia.
So far, we have found that hydrodynamics and string fragmentation have different evolution of transverse energy. Thus, the multiplicity of final hadrons from such a two-component model is sensitive to a fraction of each component in a system. To, at least, reproduce multiplicity in DCCI2, we need to change the parameter p T0Ref from its default value. However, as we mentioned at the beginning of this section, the other parameter σ 0 in Eq. (10) has a non-trivial correlation with p T0Ref , which means that we need to tune both the parameters at the same time. Suppose that one firstly tries to reproduce multiplicity by tuning p T0Ref . Since a small p T0Ref gives a rise to the number of multiparton interactions and deposited transverse energy is enhanced, it affects to enhance final hadron multiplicity. On the other hand, since the number of initial partons produced in midrapidity increases, this causes more fluidization in dynamical core-corona initialization. Once a fraction of the core is enhanced, the multiplicity of final hadrons can also decrease since the initial transverse energy deposited in midrapidity region is used for pdV work. As a result of competition between these effects, multiplicity cannot linearly enhance or decrease by decreasing or increasing p T0Ref . Secondly, suppose that one tries to reproduce particle yield ratios as functions of multiplicity by tuning σ 0 . Since changing σ 0 means changing a fraction of core and corona, final multiplicity is easily altered, too. This is the reason why we need to fix both parameters by taking into account multiplicity and particle yield ratios at the same time.
Note that, if we made viscous corrections in the hydrodynamic evolution, the resultant change of p T0Ref from its default value could have been modest due to the less reduction of transverse energy [109], which is beyond the scope of the present paper, but which should be investigated in the future work.
A string melting version of A Multi Phase Transport (AMPT) model [112,113] and the hydrodynamic models using it for generating initial conditions [114,115] avoid this issue of the transverse energy in an "ad hoc" way: The hadrons decaying from a string are re-decomposed into their constitutive quarks and antiquarks, and then form high-energy density partonic matter. Although it is possible to count the energy stored along the string contrary to considering the generated partons directly, this prescription lacks gluons from melting strings. Therefore, we do not pursue this idea in the present paper.
IV. SUMMARY
We studied the interplay between core and corona components establishing the DCCI2, which describes the dynamical aspects of core-corona picture under the dynamical initialization scheme. To develop the DCCI2, we put an emphasis on reconciliation of open issues of dynamical models, mainly relativistic hydrodynamic models, toward a comprehensive description of a whole reaction of highenergy nuclear collisions. One of the important achievements is to generate the initial profiles of hydrodynamics by preserving initial total energy and momentum of the collision systems. This is achieved by adopting hydrodynamic equations with source terms on initial partons obtained from Pythia8, one of the widely accepted generalpurpose Monte-Carlo event generators. Consequently, in addition to the equilibrated matter (core) described by relativistic hydrodynamics, we also consider the existence of non-equilibrated matter (corona) through dynamical initialization with the core-corona picture.
We have updated our model from the previous work. The updates include sophistication of four-momentum deposition of initial partons in dynamical core-corona initialization, samplings of hadrons from hypersurface of core parts (fluids) with iS3D, hadronic afterburner for final hadrons from core and corona parts with a hadronic transport model JAM, and modification on color string structures in corona parts due to co-existence with core parts (fluids) in coordinate space.
Discussion on the interplay between core and corona components is made once fixing major parameters so that our model reasonably describes multiplicity as a function of multiplicity or centrality class and omega baryon yield ratios to charged pions as functions of multiplicity. First we extracted the fractions of core and corona components to the final hadron yields as functions of multiplicity and centrality classes. We found that, as increas-ing multiplicity, the core components become dominant at dN ch /dη |η|<0.5 ∼ 18, which corresponds to about 0.95-4.7% multiplicity class in p+p collisions at √ s = 7 TeV and ∼ 80% centrality class in Pb+Pb collisions at √ s N N = 2.76 TeV. Next, we showed the fractions of core and corona components in charged particle p T spectra. In minimum-bias Pb+Pb collisions, the fraction of core components is dominant below p T ∼ 5.5 GeV, while that of corona components is dominant above that. Interestingly, we found that there was an enhancement in the fraction of corona contribution with R corona ∼ 0.2 at most in p T 1 GeV even in minimum-bias Pb+Pb collisions. From this, the fraction of the corona contribution is anticipated to increase in peripheral collisions. This brings up a problem in all conventional hydrodynamic calculations in which low p T soft hadrons are regarded purely as core components. Since the fraction of each component would exist finite for a wide range of multiplicity and, as a result, there should be interplay between them, we suggest that both small colliding systems and heavy-ion collisions should be investigated in a unified theoretical framework by incorporating both core and corona components.
To investigate the effects of co-existence of core and corona components on observables, we showed p T and v 2 {2} as functions of N ch . In particular, in Pb+Pb collisions, we found that the finite contribution of corona components at midrapidity gives a certain correction on the results obtained purely from core components, which is described by hydrodynamics. The correction is ∼ 5-11% for p T below N ch ∼ 200, while it is ∼ 15-38% for v 2 {2} below N ch ∼ 370. The former correction leads to the reasonable agreement of p T with the experimental data. This suggests that one might need to incorporate corona components in hydrodynamic frameworks to extract transport coefficients from comparisons with experimental data. Finally we explored effects of radial flow based on violation of m T scaling with hadron rest mass by classifying events into high-and low-multiplicity ones. Noteworthy, we found that it is difficult to discriminate the radial flow originated from hydrodynamics from collectivity arisen from color reconnection in Pythia8. We also discussed evolution of transverse energy in the DCCI2. In string fragmentation, final transverse energy is larger than initial transverse energy as producing hadrons around midrapidity. While in hydrodynamics, transverse energy just decreases from its initial value during the evolution due to the longitudinal pdV work. To obtain the same amount of transverse energy in the final state in DCCI2 with default Pythia8 Angantyr in minimum-bias Pb+Pb collisions, it is necessary to have three times larger initial transverse energy than the one of default Pythia8 Angantyr.
For more quantitative discussion on transport properties of the QGP fluids, we admit an absence of viscous corrections to fully equilibrium distribution in our analysis. We leave this as one of our future works. Nevertheless, we emphasize that the corrections from corona components mean the ones from "far from" equilibrium components which should exist nonetheless and would more significantly affect the final hadron distributions than the viscous corrections.
It is known that transverse momentum spectra solely from hydrodynamics or hybrid (hydrodynamics followed by hadronic cascade) models do not perfectly reproduce the experimental data below p T ∼ 0.5 GeV [75], although hydrodynamics is believed to provide a better description in the low p T region in general. The deviation between pure hydrodynamic results and the data in the low p T region could be filled with the corona components. Detailed analyses of centrality dependent particle identified p T spectra from DCCI2, which require high statistics, and its comparison with the experimental data will be made in a future publication.
With this model, we anticipate that it would be interesting to explore planned O+O collisions at LHC [116] since the collision system can provide data around "sweet spot" in which the core components are to be dominant and yet corrections from corona components cannot be ignored at all [117]. In addition, investigation on strangeness enhancement in forward or backward regions might give some insights into ultra high-energy cosmic ray measurements [118,119]. Incorporation of a dynamical description of kinematic and chemical pre-equilibrium stage [120,121] and investigation of medium modification of jets [122,123] are in our interests as well. We leave the discussion on those topics as a future work.
PACS numbers: 25.75.-q, 12.38.Mh, 25.75.Ld, 24.10.Nz
(
3+1)-D hydro FIG. 1. (Color Online) Flowchart of the DCCI2 framework.
•
PartonVertex:setVertex=on • HadronLevel:all=off • MultipartonInteractions:pT0Ref • SpaceShower:pT0Ref
FIG
. 2. (Color Online) Particle yield ratios of (a) omegas (Ω − andΩ + ), (b) cascades (Ξ − andΞ + ), (c) lambdas (Λ andΛ), (d) protons (p andp)
Results without hadronic rescatterings are also plotted in p+p (open orange circles) and Pb+Pb (open orange squared) collisions. Results from Pythia8 in p+p collisions (gray pluses) and from Pythia8 Angantyr in Pb+Pb (gray crosses) collisions are plotted as references.
FIG
. 3. (Color Online) (Upper) (a) Charged particle multiplicity at midrapidity as a function of the fraction of the INEL> 0 cross sections in p+p collisions at √ s = 7 TeV. (b) Charged particle multiplicity at midrapidity as a function of centrality from Pb+Pb collisions at √ sNN = 2.76 TeV. Results from full simulations (orange diamonds) and simulations without hadronic rescatterings (open orange circles) are compared with the ALICE experimental data (black crosses) [74, 77]. Core and corona contributions without hadronic rescatterings are shown in red and blue stacked bars, respectively. (Lower) Fractions of core (red) and corona (blue) components without hadronic rescatterings are plotted in (a) p+p and (b) Pb+Pb collisions.
FIG
. 4. (Color Online) Fractions of core and corona components in the final hadron yields as functions of charged particle multiplicity at midrapidity. Smooth behaviors of fractions of core (open red circles) and corona (open blue circles) contributions in p+p collisions at √ s = 7 TeV are taken over by those of core (closed red squares) and corona (closed blue squares) contributions in Pb+Pb collisions at √ sNN = 2.76 TeV, respectively.
FIG
. 5. (Color Online) (Upper) Transverse momentum spectra of charged particles at midrapidity from DCCI2 in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ sNN = 2.76 TeV. An event average is taken with at least one charged particle having pT > 0.5 GeV and |η| < 2.5 in both cases. Results from full simulations (closed orange diamonds) and simulations without hadronic rescatterings (open orange triangles) are plotted and compared with the ATLAS data (black pluses) [88] only in p+p collisions as a reference. Results from core (open red diamonds) and from corona (open blue squares) are also plotted for simulations without hadronic rescatterings. (Lower) Corresponding fractions of core (red circles) and corona (blue squares) components in the final hadron without hadronic rescatterings are shown as functions of transverse momentum.
FIG. 6 .
6(Color Online) Mean transverse momentum of charged particles as a function of the number of charged particles in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ sNN = 2.76 TeV compared with the ALICE experimental data [89] (gray pluses). Results from full simulations (closed orange squares), from simulations without hadronic rescatterings (closed yellow circles), from core components (open red diamonds), and from corona components (open blue triangles) are shown for comparisons.
FIG. 7 .
7(Color Online) Second order of anisotropic flow coefficient obtained from two-particle correlation for charged hadrons as a function of the number of produced charged particles in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ sNN = 2.76 TeV. Results from full simulations (orange squares) and simulations without hadronic rescatterings (yellow diamonds) are shown with closed symbols. While results from core (red diamonds) and corona (blue triangles) components from simulations without hadronic rescatterings are plotted with open symbols.
Figure 8 (
8b) shows the mean transverse mass as a function of hadron rest mass for 0-10% and 50-100% centrality classes in Pb+Pb collisions at √ s N N = 2.76 TeV. The
Figure 8 (
8d) shows each contribution of core and corona components to the final result without hadronic rescatterings in 0-10% centrality class in Pb+Pb collisions. The linear increase except phi meson is seen very clearly for the core component, and the increase rate is more than the one from p+p results shown inFig. 8 (c).
Figure 8 (
8e) shows the same variable withFig. 8(c) but in 50-100% multiplicity class in p+p collisions. Since the fraction of the core components is less than 10% in this range of multiplicity class as shown inFig. 3 (a), the final result and the result from corona components are almost top of each other showing no significant dependence on hadron rest mass.
Figure 8 (
8f) shows the same variables with Fig. 8 (d) but in 50-100% centrality class in Pb+Pb collisions. According to
scat.(50-100% core) w/o scat.(50-100% corona) FIG. 8. (Color Online) Mean transverse mass, mT − m, as a function of rest mass of hadrons, m, from DCCI2 in (a) p+p collisions at √ s = 7 TeV and (b) Pb+Pb collisions at √ sNN = 2.76 TeV. A comparison of the results from full simulations (closed symbols connected with solid lines) and the ones from simulations without hadronic rescatterings (open symbols connected with dashed lines) is made. Results of high-multiplicity events (0-10%, red) and of low-multiplicity events (50-100%, blue) are shown to see the effects of the fraction of core components. The result from Pythia8 and Pythia8 Angantyr (black symbols) with default parameters including color reconnection is plotted in p+p and Pb+Pb collisions, respectively, as references. Corresponding contributions of core and corona components in 0-10% multiplicity and centrality classes in (c) p+p and in (d) Pb+Pb collisions, respectively. Corresponding contribution of core and corona components in 50-100% multiplicity and centrality classes in (e) p+p and in (f) Pb+Pb collisions, respectively.
Online) Pseudo-rapidity distribution of transverse energy in (a) INEL> 0 p+p collisions at √ s = 7 TeV and (b) minimum-bias Pb+Pb collisions at √ sNN = 2.76 TeV from DCCI2 and Pythia8. A comparison of transverse energy distribution between parton and hadron levels is made for results from DCCI2 and Pythia8. Results from the parton level in DCCI2 (green crosses) and Pythia8 (blue pluses) and the ones from the hadron level (red crosses) in DCCI2 and Pythia8 (orange pluses) are shown for comparison. For Pb+Pb collisions, Pythia8 Angantyr is used to obtain the results.
TABLE I .
IParameter set in DCCI2 used throughout this paper.Parameters
values
p T0Ref (p+p)
1.8 GeV
p T0Ref (Pb+Pb)
0.9 GeV
τ0
0.1 fm
τs
0.3 fm
Tsw
0.165 GeV
σ0
0.4 fm 2
bcut
1.0 fm
pT,cut
3.0 GeV
σ ⊥
0.5 fm
ση s
0.5
∆x
0.3 fm
∆y
0.3 fm
∆ηs
0.15
In fact, this idea was first implemented in Ref.[39] to describe excitation functions of particle ratios at lower collision energies. However, it was applied to the secondary produced hadrons rather than partons.
In the conventional hydrodynamic models, initial conditions of hydrodynamic fields are put at a fixed initial time, τ = τ init , which can be regarded as a negative (in-coming) energymomentum flux from the hypersurface T (x, τ = τ init ) = Tsw. Thus, thanks to the Gauss's theorem, the sum of outgoing energy-momentum fluxes from the hypersurface T (x, τ > τ init ) = Tsw is exactly the same as that of in-coming fluxes at τ = τ init when there are no source terms in hydrodynamic equations.
Note that Pythia8 and Pythia8 Angantyr with rope hadronization show enhancement of strange hadron yield ratios as a function of multiplicity[48,84].
The result from EPOS 3.210 shows ∼ 30% at the same multiplicity[44].
Due to the lack of statistics, we simply divide events into these classes regardless of collision system.
In actual simulations in Pythia, a parameter p T0Ref provides a scale to make a smooth turnoff of hard scattering rather than the sharp separation[110].
ACKNOWLEDGEMENTWe gratefully acknowledge valuable comments from C. Bierlich
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| []
|
[
"Supplementary materials for: Evolution of Pairing Orders between Pseudogap and Superconducting Phases of Cuprate Superconductors",
"Supplementary materials for: Evolution of Pairing Orders between Pseudogap and Superconducting Phases of Cuprate Superconductors"
]
| [
"Wei-Lin Tu \nDepartment of Physics\nNational Taiwan University\nDaan Taipei 10617Taiwan\n\nLaboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance\n\nInstitute of Physics\nAcademia Sinica\nNankang Taipei11529Taiwan\n",
"Ting-Kuo Lee *[email protected] \nInstitute of Physics\nAcademia Sinica\nNankang Taipei11529Taiwan\n"
]
| [
"Department of Physics\nNational Taiwan University\nDaan Taipei 10617Taiwan",
"Laboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance",
"Institute of Physics\nAcademia Sinica\nNankang Taipei11529Taiwan",
"Institute of Physics\nAcademia Sinica\nNankang Taipei11529Taiwan"
]
| []
| In this supplementary material we will discuss some details left out from the main article due to the limit of length. The structure is as the following: In section 1, we will revisit the nPDW and show more results of this exotic state. We will also discuss the discommensurate(DC) patterns first proposed by Mesaros et al.[3]. Then, we head to answer some details only briefly mentioned in the main text in section 2. Section 2.1 will explain how we determined k G by the usage of EDCs. Section 2.2 will discuss again the two-gap plots shown inFig. 2 and 5bin the main text, aiming to show readers that these plots are free from parameter(Γ) chosen. Section 2.3 will analyze the effect of Γ in our results. At last, section 2.4 will revisit the Fermi arc and LDOS for our patterns to help distinguish between nPDW and IPDW.3/8 | 10.1038/s41598-018-38288-7 | null | 59,617,661 | 1811.10190 | b554998502b8a166a68b97e0b24e8ebcd32c7b85 |
Supplementary materials for: Evolution of Pairing Orders between Pseudogap and Superconducting Phases of Cuprate Superconductors
Wei-Lin Tu
Department of Physics
National Taiwan University
Daan Taipei 10617Taiwan
Laboratoire de Physique Théorique
IRSAMC
Université de Toulouse
CNRS
UPS
France
Institute of Physics
Academia Sinica
Nankang Taipei11529Taiwan
Ting-Kuo Lee *[email protected]
Institute of Physics
Academia Sinica
Nankang Taipei11529Taiwan
Supplementary materials for: Evolution of Pairing Orders between Pseudogap and Superconducting Phases of Cuprate Superconductors
In this supplementary material we will discuss some details left out from the main article due to the limit of length. The structure is as the following: In section 1, we will revisit the nPDW and show more results of this exotic state. We will also discuss the discommensurate(DC) patterns first proposed by Mesaros et al.[3]. Then, we head to answer some details only briefly mentioned in the main text in section 2. Section 2.1 will explain how we determined k G by the usage of EDCs. Section 2.2 will discuss again the two-gap plots shown inFig. 2 and 5bin the main text, aiming to show readers that these plots are free from parameter(Γ) chosen. Section 2.3 will analyze the effect of Γ in our results. At last, section 2.4 will revisit the Fermi arc and LDOS for our patterns to help distinguish between nPDW and IPDW.3/8
Various PDW states including discommensurate PDW
The nodal pair density wave(nPDW) state first proposed by Tu and Lee [1] comes from the anti-phase charge density wave(AP-CDW) state but with a non-zero uniform pairing order parameter(UPOP), which is generated from its (quasi-)incommesurate nature. This accords with previous experimental data [4] that within the superconducting dome, the modulations of Cu-O surface observed are incommensurate. Fig. S1 shows some basic characteristics of nPDW in a 32 × 32 lattice size. Fig. S1(a) shows that the hole density is maximum at the domain walls near sites 2,7,10 and 15. For the Fourier transform in Fig. S1(b) and S1(c), it is clear that although there are several peaks, the leading one is the one at π/2(π/4) for hole density(pairing order), which corresponds to the modulation of 4a 0 (8a 0 ). Fig. S1(d) demonstrates the local density of state(LDOS) of several chosen sites and the v-shape near zero energy indicates a d-wave pairing gap with a node is opened. Finally, in Fig. S1(e) the comparison of different form factors confirms the dominance of d form factor [2].
McMillan [5] was the first to define a "discommensuration"(DC) as a defect in a commensurate CDW state. In such state, the phase of the CDW jumps between discrete lattice-locked values. Mesaros et al. [3] showed that this kind of CDW could be what was observed by experiments. Hence, let us consider a sinusoidal modulation in one spatial dimension with 4a 0 modulation but a phase jump between each domain, it can be written as:
ψ(x) = Aexp[i(Q 0 x + φ )] (S8)
where A is the amplitude and Q 0 = 4a 0 . The additional phase φ defines the phase shift for each domain. For example, Fig. S2(a) shows the modulation in x-direction for one of the DC patterns we have obtained. We will name it after the discommensurate nPDW state(DCnPDW). It is clear that there are two separate domains, one with pink color(sites 0-3, 12-15, and 24-27) with φ = 0 and the other with Green color(sites 6-9, 18-21, and 30-33) with φ = π. Moreover, in Fig. S2(b)-(d) we can see its FT shows that the averaged modulation is no longer 4a 0 . This might explain for the reason why there are some experiments which ended up observing the modulation period of 4a 0 but the others with incommensurability. They can originate from the same phase with local 4a 0 feature but a global incommensurability. Even with the discommensurability, however, they still possess the same dominant symmetry. As shown in Fig. S2(b), the leading form factor is still d form.
One of the most important points we need to clarify is that despite the nPDW and DCnPDW already mentioned, we can easily obtain a number of different states by changing the initial inputs or lattice size. Each of them has slightly different values of pairing, charge density and bond orders. Fig. S3(a) lists some of the examples and demonstrates their energies. One can see that in fact their energies are nearly degenerate, as claimed previously [1], although the lattice size is different. Even with the same lattice size, it is also possible to have two distinct patterns, as shown in Fig. S3(a). Within the states shown, there are two of them labeled with QIAPCDW, which is the abbreviation of quasi-incommensurate anti-phase CDW. Different from nPDW, this pattern has zero UPOP, just like AP-CDW in ref. [1]. Amazingly, even though QIAPCDW and nPDW seem very different because of the existence of UPOP, these two still share nearly degenerate energies. This suggests what we have claimed in ref. [1] that in fact all the orders(∆, χ, etc.) are, instead of competing, intertwined and influencing each other. That is why such different states can possess nearly the same energy. We have to make it clear that the patterns listed here are only some of the possibilities and in fact there can be many more different states. Moreover, we like to point out that these QIAPCDW states are quite similar with the IPDW states, except the latter is generated by raising the temperature of the nPDW states. Except for their energies, there are also some characteristics which these states all share with. One of them is the d form factor symmetry. Among all these states, surprisingly, they all have leading d form factor over s and s , which is one of the important feature of nPDW. In Fig. S3(b), we have collected the values of magnitude of d form factor for each states. We can see although for different states their magnitudes of d form factor vary from each other, most values are within the range from 0.15 to 0.2. Moreover, the ending points are all within the range of 0.18∼0.22, which is very near the quantum critical point observed by experiment around 0.19.
Except for the usual FFT, one of the central goals of this work is to investigate the properties of patterns found in k space. Among all, the most important one is the quasiparticle spectra. In Fig. S4, we demonstrate the spectra at (π, 0) and (0, π) for three different patterns chosen: nPDW at 30 and 32 lattice size, and DCnPDW at 36 lattice size. It is clear that, as mentioned in the main text, the spectra at antinodes (π, 0) and (0, π) are quite different but the gap values are very similar [6]. More interestingly, although there are three different states, their quasi-particle spectra are also very close to each other. Even we increase the temperature, there is still not qualitative difference for the spectra, but only that the gap values decrease a bit as temperature rises. One need to note that at T = 0.035t and T = 0.05t the states have already evolved to IPDW states. Once again, this result suggests that although there are numerous possibilities of having a commensurate, (quasi-)incommensurate, or discommensurate state w/wo UPOP, the deeper cause is alway the same: strong correlated Mott physics with the Gutzwiller factors.
Other discussions
In this section, we shall discuss several details left out in the main text, such as the method we used to determined k G , the two gaps and Fermi arcs. We will also discuss the effect of using different Γ in calculating the spectra density.
Method to determine k G
We have mentioned in the main text that as the cut of spectra goes away from the node toward near the antinode, the momentum of gap(k G ) will also deviate from k F . The way of determining k G will require the usage of energy distribution curves(EDCs). In Fig. S5(b), we show the spectra at (π, 0) for nPDW in the doping level of 0.15(30 × 30 lattice size). Next to the spectra, we also put in a series of EDC cuts starting from the point (π, 0) toward (π, π). Just like the ARPES experiment, we can easily determine k G when the minimum gap is reached by looking at the EDC cuts. Note that here we use Γ = 0.25|E|. The difference between k G and k F as a function of doping is shown in Fig. S5(a). The difference becomes smaller as doping increases. This is expected since the gap approaches a pure d-wave gap as doping increases and particle-hole symmetry is recovered for the usual BCS superconductors.
Two-gap plots
Here we discuss the method used to determine gap values in Fig. 2 and 5(b) in the main text, as well as their error bars. First we will explain that in fact there is only small difference if we utilize different ways of determining gap. In the main text, all the values are determined by using EDCs and the horizontal error bars come from the finite size effect, which could be reduced if we further apply supercells with larger size, while the vertical error bars come from either the width of peaks(due to the choice of Γ), or the fact that there are actually several peaks coexisting. But in fact there are different ways of determining gap values and they will provide the same outcomes. For example, the quasiparticle spectra can be also used to determine the gap as explained in the main text. The result are all the same no matter which way we decided to exploit. Fig. S6(a) put together two curves of gap values determined by EDC and quasiparticle spectra. One can see that these two lines are very close and even if there are small differences, they are within the error bars.
Choices of Γ
In the main text, we mentioned that the width Γ is chosen as different values for better demonstration in different plots. But in fact we have done a series of analysis showing that there is no qualitative difference in choosing Γ to be a constant as 0.01t or as 0.25 √ E 2 + T 2 . In Fig. S6(b) we plotted the same figure as Fig. 5(c) in the main text. But here we include also the curve using Γ = 0.01t. One can see clearly that there is only small quantitative differences between two blue curves. Our second proof is to investigate the two-gap plots as Fig. 2 and 5(b) in the main text, with different choice of Γ. In Fig. S6(c) we show 2/8 the curves of gap values for nPDW at δ = 0.15, but under different choices of Γ. We can see that those three curves are nearly the same within error bars.
Last but not least, we also need to check the consistency of quasi-particle spectra. In Fig. S6(d) and (e), we plotted the same spectra but with different Γ, one with Γ = 0.01t(d) and another with Γ = 0.25|E|(e). If we discount the broadening of Fig.S6(e), 6(d) and 6(e) have the same k G .
Fermi arcs and LDOS
We have shown in the main text that the UPOP of nPDW is decreasing when temperature rises. The resulting pattern is called IPDW by us, which is also a PDW phase but UPOP is close to zero. Fig. 4(f) in the main text plots the zero energy quasiparticle spectra weight in momentum space and it reveals the feature of the so-called Fermi arc. However, in experiments arcs usually have x and y rotational symmetry. That is because the experimental detection scans over a region of materials that contains domains with modulations in both x and y direction. Therefore the resulting arcs would have the rotational symmetry. In order to compare with their results, we took average of x and y axis of our arcs and replotted it. The resulting figure is as Fig. S7(b), which looks more like the experimental data.
One of the main differences upon having UPOP or not is to look at the LDOS. Since our nPDW possesses d-wave UPOP, its LDOS will have a v-shape feature near the Fermi energy. However, for IPDW there is no UPOP and therefore the DOS at Fermi energy should be non-zero. Consequently, to further confirm the vanishing UPOP, we compare the LDOS of sites near and away from domain walls in Fig. S7(c) and S7(d), respectively. LDOS for five different temperatures are shown and the state remains nPDW for T = 0 and 47K but becomes IPDW at T = 94K, 163K, and 232K(0.1t ∼ 464K) because of the disappearance of UPOP. According to the LDOS plots, it is also clear that the v-shape feature disappears gradually as temperature rises, confirming that the node has changed into an arc in IPDW state. We list several quasi-particle spectra at antinodes((π, 0)/(0, π)) for three different patterns at different temperatures. Although marked as nPDW in the first column, the patterns become IPDW at T = 0.035t and T = 0.05t. However their spectra do not change much and the differences of gap values at (π, 0) and (0, π) are within 10% [6]. (a) A collection of several data points of k G − k F vs doping at k x = π. The way of determining the difference of k G and k F is shown in (b): k G determined by examining EDCs plotted from k y = 0 toward k y = pi, for dopant concentration 0.15. k F is determined by Fermi liquid surface and marked along with k G on the EDC plot. The quasiparticle spectra is also shown with Gaussian width Γ = α|E| (α = 0.25) and marked with positions of k G and k F . Fig. 3 in the main text but obtained from different approaches: red line is determined by the gap values shown by quasiparticle spectra but green line comes from EDCs, the same way as in main text. (b) Relative DOS as a function of hole concentration as in Fig.7(b) in the main text but put together with two different Γ. The two blue lines are very close to each other. (c) Two gap plots determined by different Γ for nPDW at δ = 0.15. One can see that these lines nearly overlap with each other. Figure (d) and (e) again show the quasi-particle spectra for nPDW at δ = 0.125(for the 32 × 32 lattice) at k x = 0.977π but with different Γ: (d) Γ = 0.01t and (e) Γ = 0.25|E|. Note that in fact (d) is identical as Fig. 1(f) in the main text. We can find that although these two figures look quite different due to the choices of Γ, important features such as location of k G are still the same, only that in (e) the spectra bands are broadened due to larger Γ.
Fig. S 1 .Fig. S 2 .
12Properties of nPDW. (a) The real space modulation of nPDW in 32 × 32 lattice sites with δ = 0.125. Since the pattern repeats itself with an inversion symmetry in the middle bond, here we only show the first 16 sites. The red and black numbers on each bond denote the values of pairing order and the number at each site (black dots) is the hole density. (b)(c) The Fourier transform of the value of hole density(b) and pairing order(c). (d) LDOS of the first 4 sites of this 32 × 32 nPDW. (e) Different form factors. Figures showing the properties of discommensurate nPDW. (a) The phase variation of this pattern. Site 0-3, 12-15, and 24-27 are of phase equal to 0(2π) while sites 6-9, 18-21, and 30-33 are of phase π. (b) Form factors for discommensurate nPDW. We also include the Fourier transform of hole density(c) and pairing order(d).
Fig. S 3 .
3(a) Energies of several states chosen by us. Although we have listed ten different states here, their energies seem to be nearly degenerate and follow the same trend line. (b) Magnitude of d form factor of patterns. Given different states we expect their magnitude to change but still all of them seem to have the same trend: the magnitude maintains the same until doping level exceeds 0.18, where it starts to drop drastically and becomes zero in the range of 0.18∼0.22.
Fig. S 5 .
5Fig. S 5. (a) A collection of several data points of k G − k F vs doping at k x = π. The way of determining the difference of k G and k F is shown in (b): k G determined by examining EDCs plotted from k y = 0 toward k y = pi, for dopant concentration 0.15. k F is determined by Fermi liquid surface and marked along with k G on the EDC plot. The quasiparticle spectra is also shown with Gaussian width Γ = α|E| (α = 0.25) and marked with positions of k G and k F .
Fig. S 6 .
6(a) Two-gap plot for nPDW at δ = 0.125 as shown in
Fig. S 7 .
7(a) and (b) Zero energy quasiparticle spectra in k space before(a) and after(b) taking average of x-and y-directions PDW. (a) is the same as Fig. 4f in the main text and we put it here again for the reason of comparison. Clearly, (b) looks more like the observation by experimental groups. (c) and (d) LDOS at sites near(c) and away from(d) domain walls at different temperatures for nPDW(IPDW) at δ = 0.15. Γ used here is equal to α √ E 2 + T 2 (α = 0.25). All figures shown here are of 30 × 30 lattice size. Its T p1 is around 90K.
Genesis of charge orders in high temperature superconductors. W L Tu, T K Lee, Sci. Rep. 618675Tu, W. L. and Lee, T. K.. Genesis of charge orders in high temperature superconductors. Sci. Rep. 6, 18675 (2016).
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Commensurate 4a 0 -period charge density modulations throughout the Bi 2 Sr 2 CaCu 2 O 8+x pseudogap regime. A Mesaros, Proc. Nat. Acad. Sci. 11312661Mesaros, A. et al.. Commensurate 4a 0 -period charge density modulations throughout the Bi 2 Sr 2 CaCu 2 O 8+x pseudogap regime. Proc. Nat. Acad. Sci. 113, 12661 (2016).
Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state. M H Hamidian, Nature Phys. 12150Hamidian, M. H. et al.. Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state. Nature Phys. 12, 150 (2015).
Theory of discommensurations and the commensurate-incommensurate Charge-density-wave phase transition. W L Mcmillan, Phys. Rev. B. 14McMillan, W. L.. Theory of discommensurations and the commensurate-incommensurate Charge-density-wave phase transition. Phys. Rev. B 14, 1469-1502(1976).
Gap values near the two antinodes my differ by 10% for different nPDW states. This is probably the accuracy of the mean-field theoryGap values near the two antinodes my differ by 10% for different nPDW states. This is probably the accuracy of the mean-field theory.
| []
|
[
"Improving Quality of Service for Users of DAG-based Distributed Ledgers",
"Improving Quality of Service for Users of DAG-based Distributed Ledgers"
]
| [
"Andrew Cullen ",
"Lianna Zhao ",
"Luigi Vigneri ",
"Robert Shorten "
]
| []
| []
| An outstanding problem in the design of distributed ledgers concerns policies that govern the manner in which users interact with the network. Network usability is crucial to the mainstream adoption of distributed ledgers, particularly for enterprise applications in which most users do not wish to operate full node. For DAG-based ledgers such as IOTA, we propose a user-node interaction mechanism that is designed to ensure the risk of a user experiencing a poor quality of service is low. Our mechanism involves users selecting nodes to issue their transactions to the ledger based on quality of service indicators advertised by the nodes. Simulation results are presented to illustrate the efficacy of the proposed policies. | 10.48550/arxiv.2203.12076 | [
"https://arxiv.org/pdf/2203.12076v1.pdf"
]
| 247,618,680 | 2203.12076 | 81faf85bdf3ac84defc7c24a4476f6ac7e6bdac9 |
Improving Quality of Service for Users of DAG-based Distributed Ledgers
Andrew Cullen
Lianna Zhao
Luigi Vigneri
Robert Shorten
Improving Quality of Service for Users of DAG-based Distributed Ledgers
1
An outstanding problem in the design of distributed ledgers concerns policies that govern the manner in which users interact with the network. Network usability is crucial to the mainstream adoption of distributed ledgers, particularly for enterprise applications in which most users do not wish to operate full node. For DAG-based ledgers such as IOTA, we propose a user-node interaction mechanism that is designed to ensure the risk of a user experiencing a poor quality of service is low. Our mechanism involves users selecting nodes to issue their transactions to the ledger based on quality of service indicators advertised by the nodes. Simulation results are presented to illustrate the efficacy of the proposed policies.
I. INTRODUCTION
The past decade has witnessed huge interest in the study of distributed ledger technology (DLT), both in the academic community [1]- [3], and in industry [4]- [6]. The main topics of these studies on DLTs concern consensus mechanisms, scalability and security as well as a number of real use cases and applications, for instance in the areas of transportation [7]- [9]. In particular, ledgers based on directed acyclic graphs (DAGs) have arisen as a promising solution for IoT applications because they can facilitate high throughput and low delays, and they present fewer barriers to participation than blockchain alternatives [10]. As DLTs and their applications become better known and accepted in the wider technology community, the related issue of network usability is being recognised as a bottleneck issue hindering applications of this nascent technology. In some sense, this is to be expected as the underlying technology is new and not yet mature. Nevertheless, the fact that in many ledger architectures, users report lengthy and variable times for transactions to complete, as well as unfairness in the service experienced by users of the technology, is a significant impediment to its adoption. Consequently, the issue of network usability represents a significant research challenge for the DLT community.
In many DLT architectures, the variation in the experience amongst users can be attributed to two factors. The first comes down to the design of core components of the ledger architecture and how these affect the efficiency and performance of the system. The second, equally important factor, is connected to the interaction between human decision makers and the DLT network, and the incentive structure that guides this interaction. This second factor is our primary concern in this work. In a typical DLT architecture, two * These authors contributed equally to this work. distinct types of network actors can be readily identified. First, there are individuals or organisations who maintain a copy of all transactions on the ledger and participate fully in network activities such as consensus and validation. These actors, which we refer to as nodes, have the power to modify the ledger to include transactions which they may use to transact themselves or may offer this as a service, perhaps for economic gain. A second type of network actors are basic actors, referred hitherto as users, who are simply interested in utilising the network as part of some enterprise. For users (humans or software agents), the network is a tool to be utilised as part of some application, and their only objective is to send transactions through the network for a specific purpose. Users can only add data to the ledger by sending it to a node, usually via a digital wallet, and in many cases these users are willing to pay a fee to nodes to achieve a good quality of service (QoS). The experience of users is strongly influenced by the manner in which users and nodes interact, which is driven by the different factors that motivate each actor's participation in a DLT network. Typically, users wish to transact as cheaply as possible while receiving some level of QoS from the network. For example, users may wish to transact with guaranteed instantaneous transaction delay, or with guaranteed maximum delay; minimize the financial cost of transactions or their energy consumption; or be simply assured of fairness and value for money they receive with respect to other users using the network. Nodes, on the other hand, are driven by the desire to use of their resources (e.g., computational power or reputation 1 ) for both personal or societal gain. The desires and objectives of users and nodes are not always mutually inclusive. In other infrastructures, the interaction between users and nodes can sometimes lead to unstable network behaviour. For example, parallels can be found in road networks, where toll roads (nodes) are attractive to vehicles (users) precisely because they offer faster and more reliable transit times. However, if they attract too much traffic, precisely the opposite can be the effect, leading to unpredictable and sometimes catastrophic user experience. Problems of this nature are not unique to transportation and arise also in other domains -for example in job scheduling and queuing problems that arise in many applications (we shall have more to say on this in the next section). Here we simply note that similar issues can arise in DLT networks if the interaction between users and nodes is poorly designed [11] and while node-user interaction 1 Reputation is a numeric value associated to a node which affords it ledger access. In principle, reputation should be difficult to gain and easy to lose. arXiv:2203.12076v1 [cs.DC] 22 Mar 2022 mechanisms have been designed for blockchains (based on transaction fee incentives), there is currently no user-node interaction mechanism designed for DAG-based DLTs of the kind considered here. The problem of designing such a mechanism for DAG-based ledgers differs fundamentally from that of blockchains because nodes can add transactions to DAGs in parallel rather than sequentially as in blockchains. This feature of DAGs restricts users of these networks to selecting specific nodes to process each of their transactions and results in an entirely new paradigm for user-node interaction.
Our goal in this paper is to suggest an interaction mechanism between nodes and users which allows both users and nodes to achieve their objectives and leads to networks that are stable, robust and fully utilised. Our idea is simple. Nodes broadcast a QoS indicator for the service they can offer, and users respond probabilistically to this signal. We shall show that this simple algorithm equalises the QoS experienced by users, avoids large deviations in QoS experienced by individual users, whilst allowing nodes to be rewarded for the service they provide to the network. Moreover, the mechanism is inherently robust to the behaviour of dishonest nodes and users. Nodes advertising false QoS information can be rapidly detected and blacklisted by users. The proposed policies are also agnostic with respect to specific implementation issues, such as node discovery, and assume users have full access to QoS signals from nodes. As a final comment on the contribution of this work, it is important to note once again that the DAG-based DLT setting considered here requires a fundamentally different usernode interaction mechanism than those found in traditional blockchain architectures such as Bitcoin. The key difference lies in the fact that users must select a specific node to issue their transactions in our setting rather than broadcasting them to as many nodes as possible. To the best of our knowledge, this work is the first to consider the design of a user-node interaction mechanism for DLTs of this kind.
Our paper is structured as follows. In the remainder of this section, we present related work. In Section II, we give a basic system model for the relevant components of DAG-based DLT networks including some basic concepts, such as users, nodes and QoS indicators. In Section III, we present a usernode interaction mechanism for DAG-based DLT networks and propose a variety of policies for users and nodes to orchestrate their relationship. The proposed policies range from naive approaches to more sophisticated strategies which take into account different indicators of QoS. In Section IV, a set of simulations are performed to validate and contrast the proposed policies.
A. Related work
Before proceeding, it is worth noting that our research builds on a number of seemingly unrelated prior works. First, the issue of user-network interaction arises in several domains. For example, stochastic policies to guide this interaction (of this nature proposed here) have been studied and analysed in the context of transportation networks [12]- [15]. The authors in [16] and further work in [17] propose an algorithm enabling the number of arriving, departing cars and the instantaneous occupancy to be counted by car parks. Based on broadcasts from the car parks, cars then can predict the parking space available possibility at the estimated time that the car will arrive there. In [18] the authors propose a stochastic policy to associate cars with parking spaces and balance cars across a network of car parks and charge points, in a manner that minimizes the probability of a space not being available when cars arrive at a car part. The problem discussed in this paper is also closely related to a host of other load balancing problems that can be found in the networking literature; one example is that of server farms with immediate dispatching of jobs requiring task assignment policies [19]. Well-known policy examples that arise in this context include the random selection policy, the round-robin selection policy and the join the shortest queue policy (the algorithms proposed in this paper are variants of the random selection policy in which a non-uniform distribution is sampled to select a node). Examples of relevant work in this direction can be found in [20]- [26] and the references therein.
It is also worth noting that the user-node interaction problem arises in the design of other DLTs, namely those based on blockchain technology. The best known such network is the Bitcoin network. In Bitcoin each miner (node) maintains a pool of transactions referred to as a mempool [19]. Miners select transactions from their mempool to include in block proposals, and if they are successful in adding a block, they receive fees from all the included transactions. In the blockchain setting, users aim to send their transactions to the mempools of all miners to maximise their chances of having them included in a block. Due to the sequential nature of how blocks are appended to a blockchain, miners can share identical mempools without the risk of adding the same transaction to the ledger twice. As such, users do not need to send their transactions to specific nodes in blockchain networks but should aim to get their transactions to as many nodes as possible. In the DAGbased DLT setting we consider here, however, nodes cannot share mempools because transactions can be added in parallel rather than sequentially and nodes would risk adding the same transactions more than once causing conflicts in the ledger. While the ability to add transactions in parallel is a desirable property for a DLTs, particularly in IoT settings, it renders the design of a user-node interaction mechanism significantly more complex. Some primitive techniques to improve the usability of DAG-based DLTs have been employed in practice, for example, some of the more strenuous tasks required to issue a transaction were offered as a service by a third party provider in a early implementation of the IOTA network 2 . However, to the best of our knowledge, no solutions for improving the interactions between users and nodes have been proposed in this setting.
II. SYSTEM MODEL
We consider the network architecture as depicted in Figure 1 which shows users accessing a DLT network via nodes, and nodes which communicate directly with one another, forming a distributed ledger network. More specifically, users make a selection from a set of nodes according to some criteria, and then send their transactions to these nodes to be processed. Furthermore, in this initial simplified model (neglecting some networking details such as availability of IP addresses) where users are able to query all nodes, users are free to select a node to process their transactions, and nodes are free to accept or reject transactions from individual users. It is important to note that, in contrast to blockchain-based DLT networks, users only send each transaction to a single node, and nodes do not share these transactions with one another prior to issuing them to the ledger.
Nodes may offer ledger access as a public service and can offer different levels of QoS to users (some nodes may be operated by private organisations who reserve resources for their own gain, however, we do not consider such nodes any further in this work). Generally speaking, nodes must consume resources in order to add transactions to the ledger, and providing better QoS consumes more resources. For example, in a Proof-of-Work (PoW) ledger, the consumed resource is computation power, whereas in Proof-of-Stake (PoS) ledgers, the consumed resource is wealth in the native ledger currency. A generalised version of PoS known as delegated PoS allows this resource to be transferred to nominated nodes, serving as a proxy for reputation. In the remainder of the paper, we refer to this resource simply as reputation, and we denote the reputation of a node i as rep i . Although small inconsistencies in reputation calculation across nodes can typically be permitted and reputation can be changed over time, we assume that each node's reputation is a constant quantity that is agreed upon and publicly known. In what follows, we also make the following assumptions.
• Ledger access is limited by scarce resources so providing ledger access with low delay is costly to nodes. This is generally true for all DLT architectures (beyond the DAG-based DLTs considered here); for example, in a conventional PoW-based ledger, a node must consume more power to find PoW solutions more frequently and issue transactions with lower delay.
• Each node in our network is equipped with a Local Transaction Pool (LTP) that is used as a buffer to store pending transactions. Transactions sent from a user to a particular node enter that node's LTP. Note that similar pools of transactions can also be found in other ledger architectures, such as the mempool in the Bitcoin network, although nodes typically share a common mempool which is not permitted in our setting.
• Nodes can advertise their expected QoS that they can offer via some proxy. For example, nodes may advertise the expected delay from receiving a new transaction to them writing to the ledger. Additionally, nodes may require a fee for issuing a transaction and this can also be taken into account as part of a node's overall QoS.
A model for our user-node interaction mechanism is depicted in Figure 1. The sending rate of user i is denoted by µ i , while the service rate of node i's LTP is denoted by λ i . The QoS indicator for each node is represented by the colored bars above their LTP. Here, a red QoS indicator simply means poor QoS is offered by this node (for example, this could be due to high delay and/or high fees), while yellow indicates average QoS and green indicates good QoS. In what follows, we are specifically interested in designing a mechanism to orchestrate the interaction between users and nodes in DAG-based DLT networks. Our goal is to deliver a uniformly good QoS to all users and ensure that the probability of any user experiencing a very bad QoS is low, while allowing nodes to profit by processing a regular stream and (on average) fair share of transactions. While our results generally apply to any DAG-based DLT, we include some details of IOTA's implementation as a concrete example. Specifically, the rate at which nodes can issue transactions in the IOTA network is regulated by an access control algorithm [27], [28]. Each node regulates its own issue rate via a distributed algorithm based on the additive increase multiplicative decrease (AIMD) algorithm [29]- [31] 3 . Fair issue rates are then enforced throughout the network via a scheduling algorithm based on deficit round robin (DRR) algorithm [32]. The result of this mechanism is a network that is attack resistant and where each node achieves a transaction issue rate proportional to its reputation.
While the definition of QoS is quite broad, we will focus our attention mostly on delay and fees as measures of QoS in this work. In order to provide an indicator of QoS to users, nodes must estimate the expected delay for them to issue a newly arriving transaction. This delay depends on two factors: the number of transactions in the node's LTP at the time of the new transaction's arrival; and the service rate of the LTP. As mentioned above, in the IOTA network, the service rate of each node's LTP is determined by the AIMD-based rate setting algorithm employed by the node (see the Appendix for further detail). Given this background, we now propose policies by which users and nodes can interact with each other in order to achieve good network behaviour.
III. USER-NODE INTERACTION MECHANISM
We now propose simple but powerful policies by which users and nodes can interact with each other in order to achieve good network behaviour. Specifically, we propose allocation strategies for users with the objective of minimizing the probability of experiencing bad QoS when connecting to a given node. Additionally, we propose a policy for nodes to adapt their fees and incentivise users to reduce their demand in the presence of congestion. To do this we borrow ideas from transportation networks; see [18]. As depicted in Figure 2, our policies generally operate as follows.
• Nodes measure the expected delay that is associated with processing a transaction and broadcast it to all users.
To be specific, a max-min filter is designed to estimate nodes' expected delays (see Appendix), which is used to provide a QoS indicator. Nodes may additionally require a fee for their service which can be adapted in response to congestion, and this can also be provided as a QoS indicator for users.
• Users gather these QoS indicators 4 provided by the nodes and construct a vector p representing a probability distribution over the nodes. Users then sample this distribution to select a node based on a probability that is proportional to the QoS indicator (inversely proportional to expected delay/fees) of a given node. Stochastic policies of this nature have been studied and analysed in the context of transportation networks [13]- [15]. We shall not repeat this analysis here. Intuitively, when all users operate this policy, nodes are kept busy, and users minimize their probability of experiencing a large delay. While such policies operate well in transportation and mobility networks, a defining characteristic of DLT based environments is that they are adversarial. Our mechanism can be adapted to deal with an adversarial setting by requiring users to complete a small PoW to prevent spamming or by introducing fees as we shall demonstrate in Section III-C.
A. Naive policies
To provide a benchmark for comparison to the policies we will propose, we begin by describing two stochastic policies that may seem reasonable. We refer to these as naive policies as they do not attempt in any serious manner to shape traffic reaching the nodes based on prevailing network conditions. It is important to reiterate that no state of the art exists in the context of DAG-based DLTs, as user-node interaction usually refers to blockchain where a common mempool is used. The first policy we describe is a simple uniform random node selection (URNS) policy, as outlined in Algorithm 1. The second, less naive, policy which we will compare to our proposed policies is referred to as reputation-based node selection (RBNS), as described in Algorithm 2. j ← random sample from p 3:
Send transactions to node j Algorithm 2 RBNS executed by users
1: p j = repj N i=1 repi , ∀j ∈ [1, N ]
Initialise probablities
Repeat each time a transaction is sent: 2: j ←random sample from p 3:
Send transactions to node j
As can be seen from the above algorithms, URNS involves selecting nodes uniformly at random, without taking in to account any information about the node whatsoever. RBNS, on the other hand, makes use of the globally known reputation of each node which gives some information about the rate at which each node can issue new transactions. Both URNS and RBNS are open loop policies in the sense that they do not make use of feedback from nodes about the current QoS indicators such as delay as we do in the policies which we propose next. As we shall see in Section IV, closed loop policies based on measured QoS metrics offer significant improvement over these open loop alternatives.
B. Selection policy based on pure delay
In order to improve upon the naive policies stated above, we would like to introduce QoS metrics from nodes. We first focus our attention on delay as the primary indicator of QoS.
Algorithm 3 DBNS executed by users
Repeat each time a transaction is sent: 1:
p j ← repj /τj N i=1 repi/τi , ∀j ∈ [1, N ]
Update probabilities 2: j ← random sample from p 3:
Send transaction to node j As such, we refer to our solution as delay-based node selection (DBNS) as presented in Algorithm 3. Algorithm 3 is executed by a user each time it wishes to have a transaction issued to the ledger. The probability update of line 1 is the critical step of the algorithm, namely:
p j = rep j /τ j N i=1
rep i /τ i where p j is the probability that a user choose node j, τ i is the expected delay in node j's LTP, and rep i is the reputation of node i. Note that this policy is decentralised, requiring no information about the operation of other users. The only information required by users from each node is their reputation and their expected delay. Moreover, recall that the reputation of each node is a quantity that is known by all users.
The expected delay, on the other hand, can be calculated by nodes at regular intervals and broadcast to all users. The delay signal, τ i , broadcast by nodes acts as a feedback to users about the state of the network, resulting in a closed loop system. The expected delay for a transaction arriving to node i's LTP at time t can be estimated as follows:
τ i (t) = L i (t) λ i (t)(1)
where L i (t) is the number of transactions in node i's LTP at time t (including the newly arrived transaction).λ i (t) is the expected average service rate of the LTP over the time this new transaction spends waiting to be issued. In a PoW ledger, this would be a fixed value,λ i (t) = λ i , based on the computing power of node i (provided the node's computing power remains constant). In this work, however, we focus our attention on the IOTA protocol in which the service rate of the LTP is governed by an AIMD rate setter [27] as part of IOTA's access control algorithm which replaces the need for PoW. We employ a max-min moving average filter to estimate the issue rate of each node as described in the Appendix.
C. Including transaction fees
The above policy can be extended to include transaction fees which offers improved security for nodes against malicious users and demonstrates the ability of our approach to capture broader notions of QoS. This extension requires a policy for nodes as well as for users. Specifically, nodes require a fee σ i to be included in any transaction and they broadcast this along with their expected delay τ i , to users. When congestion is detected by nodes, they increase their fee to reduce the load on them. To achieve this goal, a simple P controller (it also can be adapted to PI/PID equivalent) is adopted here for node i.
σ i (t) = max (0, K pi (τ i (t) − r i ))(2)
where K pi is the proportional gain (a positive constant) and r i is a delay setpoint. If the expected delay of node i's LTP, τ i (t), is less that r i , then node i will set their fee to zero, and as delay increases beyond this setpoint, the fee σ i (t) will increase at a rate proportional to K pi . Both K pi and r i can be tuned by nodes on an individual basis, but we will assume that they are equal for all nodes in this work.
Users, on the other hand, can compute a cost function based on their desired trade-off between fee and delay to get a QoS metric for each node. The cost function for user m can be specified as follows
c mi (t) = a m τ i (t) + (1 − a m )σ i (t)
where c mi (t) is the QoS metric used by user m for node i at time t, a m determines the weights assigned to delay τ i (t) and fees σ i (t) for user m. Larger values for a m indicate that user m cares greatly about having low delay whilst lower values reflect that this user is more concerned about paying low fees. User also define a cost threshold, c max m , above which this user will not continue sending transactions, which means when congestion increases, fees go up and some users stop sending transactions until the QoS metric is lower than their threshold. The algorithm employed by users, which we call DBNS+, including both delay and fees, is described in Algorithm 4. Send transaction to node j
IV. SIMULATIONS
In each of the following experiments, we simulate 100 users, each continuously sending transactions, and 50 nodes available to issue transactions for the users. Each node in the network is peered with 4 randomly chosen neighbours. The nodes we discuss in this paper are obeying restrictions imposed by the access control algorithm in [28] wherein the issue rates of nodes are controlled by an AIMD algorithm. Transactions issued by nodes need to be forwarded to neighbours to achieve a shared view of the ledger. As described in [27], the total rate at which nodes can process new transaction ν. In the below simulations, ν is set to 50 transactions per second. All simulation results are averaged over 10 Monte Carlo simulations, unless otherwise specified. The reputation distribution of the nodes follows a Zipf distribution with exponent 0.9, as depicted in Figure 3. These statistics are chosen to be representative of real values that have been measured from the IOTA network 5 . For each group of simulations (i)-(iv), we consider three different scenarios based on the combined sending rate of users which represents the total traffic which nodes must process. In all scenarios, users generate transactions to send according to a Poisson process.
• In scenario (a), users send transactions at an average rate that is 90% of the total network scheduling rate. This scenario is depicted in Figure 4(a). Observe that the sending rate of nodes occasionally exceeds the scheduling rate of the nodes.
• In scenario (b), we consider an average sending rate at 98% of the scheduling rate, putting even more strain on nodes, as shown in Figure 4(b).
• Finally, in scenario (c), we set the average sending rate to 120%, as shown in Figure 4(c), to simulate how these approaches deal with high-load situations.
Summary of findings
The delays experienced in each user side and each node's LTP as well as other metrics are shown as follows.
(i) URNS policy : When users select nodes uniformly at random according to Algorithm 1, as depicted in the first row of Figure 5, the delay density of all users across all transactions has a high spread distribution between 0 and 60 seconds. From the perspective of users, this means that there is some chance of getting a good QoS but also a high chance of getting a very poor QoS. Meanwhile, the first row of Figure 6 show traces for the delays experienced at each of the 50 nodes, where the thickness of each trace is proportional to the reputation of the node. We observe that lower reputation nodes experience severe delays and high reputation nodes are underutilised and hence experience very low delays. From the perspective of nodes, traffic from users is very poorly balances, resulting in low reputation nodes being overwhelmed and high reputation nodes being underutilised. In summary, URNS performs poorly from the perspective of both users and nodes.
(ii) RBNS policy : In the second row of Figures 5 and 6, users employ the RBNS policy of Algorithm 2 and the improvement over the URNS policy is immediately evident. As depicted in the second row of Figure 5, the delay density across all users has far lower variance than the case of URNS. Scenarios (a) and (b) in this figure show us that, from the perspective of a user, there is small chance of having a slightly longer delay than others when the network is not congested. Scenario (c), on the other hand, shows us that there is still some chance of high delays in the case of high load. Figure 6 shows traces for the delays experienced at each of the 50 nodes, where the thickness of each trace is proportional to the reputation of the node. We can see from these plots that in at 98% capacity, some fluctuations in delay of some nodes become more evident due to the fact that no feedback is used here. In the high load scenario of 120% capacity, we see that higher reputation nodes experience higher delays which is related to the fact that the issue rates of nodes are not perfectly fair with respect to their reputation. It is clear that a closed loop approach is required to achieve better fairness.
(iii) DBNS policy : As depicted in the third row of Figure 5, the delay density of all users here has a mean closer to zero and lower variance for DBNS when compared to RBNS. However, scenario (c) of this row demonstrates that DBNS does not prevent larger delays in the event of congestion (this requires the introduction of fees). From the perspective of users, there is still a lower chance of getting poor QoS than in the case of RBNS. The third row of Figure 6 again illustrates the delays experienced at each of the 50 nodes, where the thickness of each trace is proportional to the reputation of the node. We see that there is less fluctuation in scenario (a) and (b) than was the case for RBNS, and in case (c) we observe that the traffic is balances more fairly among nodes due to the use of feedback in DBNS.
(iv) DBNS+ policy: Finally, we present preliminary results for DBNS+. The proportional gain K pi is set to 0.8, and desired level of delay r i (t)) is set to 15 for all nodes. a m is set to 0.6 for all users. The resulting delay densities for each scenario are depicted in the fourth row of Figure 5 and the corresponding delays for nodes over time are depicted in the fourth row of Figure 6. As can be seen from these plots, in scenarios (a) and (b), DBNS+ performs similarly to DBNS, but when we introduce congestion in scenario (c), there are clear advantages to the DBNS+ policy. In particular, we see in scenario (c) of Figure 6 that the delay of each node is effectively stabilised by DBNS+. This is also reflected in the density function of Figure 5 where we see a spike at around 16 second delay. The reason for the stabilisation of delays after a certain point is that nodes increase their fees in response to congestion and the QoS metric, c mi (t), for users eventually exceeds the threshold c max m . At this point, users stop sending transactions to node i. Note that the oscillations evident in the plots for scenario (c) in Figure 6 are a direct result of the AIMD algorithm governing the service rate of the LTP which in turn determines the estimated delay serving as feedback to users.
V. CONCLUSION
In this paper, we proposed a user-node interaction mechanism for DAG-based DLTs which seeks to improve the usability of such networks for basic users who do not wish to run a fully operation node. The mechanism involves nodes calculating QoS metrics and providing this information to users. Users then use this feedback in a stochastic policy to select a node. Experiments including contrast experiments were carried out to validate the effectiveness of the proposed algorithms. In particular, our experiments show that by combining fees and measurements of expected delays, QoS can be provided in a fair manner to users and the demand from users can be allocated fairly among nodes. There are a number of simplifying assumptions used in this work, for example, we assumed that users have full knowledge of nodes statistics, which is probably unfeasible for large networks. For future work, it would be interesting to study how our policies perform when users query only a subset of nodes for QoS indicators. It would also be interesting to consider heterogenous user behaviour and to introduce more complex node behaviours such as collaboration between nodes and sharing of fees through mechanisms such as Shapley value. Other work to be considered include attack analyses for the policies presented here. APPENDIX In our simulations nodes use a max-min moving averaging filter. This filter is based heavily on knowledge of the IOTA access control protocol; in particular, on the fact that IOTA uses an AIMD algorithm as part of the rate scheduling algorithm. A typical node-issuing-rate, based on the AIMD algorithm, is depicted in Figure 7. As can be seen from the plot, the issuing rate varies over sawtooth shaped cycles in which the issuing rate moves from a minimum to a maximum issuing rate according to the AIMD algorithm. Based on this knowledge the filter operates as follows.
• Each time a minimum of the sawtooth is detected, a node records the corresponding issuing rate. We denote this value λ M ini (k) where the index k refers to the k th cycle.
• When the next maximum of the sawtooth is detected, a node records this value, denoted λ M axi (k).
• The node then averages the minimum and maximum rate over the k th cycle. We denote this quantity byλ i (k).
• Finally, the node averages these values,λ i (k), λ i (k − 1),....,λ i (k − δ), to obtain a filtered estimate of the issuing rate delay which is namedλ i (k).
Finally, an estimate of the delay for each transaction after the k th cycle is calculated as in (1).
Fig. 1 .
1Basic network model
Fig. 2 .
2User-node interaction mechanism.
Algorithm 4
4DBNS+ executed by user m Repeat each time a transaction is sent: 1: N ← [1, N ] 2: for j ∈ N do 3: c mj = a m τ i + (1 − a m
Fig. 3 .
3Reputation distribution of nodes in all simulations. We organise our simulation results as follows. Experiments were conducted by changing the node selection policy employed by users from the portfolio of policies: (i) URNS (see Algorithm 1); (ii) RBNS (see Algorithm 2); (iii) DBNS (see Algorithm 3); (iv) DBNS+ (see Algorithm 4); but keeping all other simulation parameters the same:.
Fig. 4 .Fig. 6 .
46Sending-rate (sum over all users) as measured over a single simulation of each scenario (a)-(c). (a) 90% capacity. (b) 98% capacity. (c) 120% capacity Fig. 5. Delay density for users under URNS, RBNS, DBNS and DBNS+ policies. Measured transaction delay in each node's LTP under URNS, RBNS, DBNS and DBNS+ policies.
Fig. 7 .
7An example of applying max-min moving average filter for one node.
https://ecosystem.iota.org/projects/powsrv-io
The AIMD algorithm is used to control transmission rate: nodes increase their transmission rate additively until congestion happens, at which point nodes multiplicatively decrease their transmission rate.
We do not specify exactly how users access node QoS indicators in this work, but nodes would most likely broadcast these values by writing to a commonly accessible resource for the users to query.
This model is well suited to reputation systems derived from wealth, i.e., PoS[33].
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| []
|
[
"Quantitative Phase Microscopy Spatial Signatures of Cancer Cells",
"Quantitative Phase Microscopy Spatial Signatures of Cancer Cells"
]
| [
"Darina Roitshtain \nDepartment of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael\n",
"Lauren Wolbromsky \nDepartment of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael\n",
"Evgeny Bal \nDepartment of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael\n",
"Hayit Greenspan \nDepartment of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael\n",
"Lisa L Satterwhite \nDepartment of Biomedical Engineering\nDuke University\n27708DurhamNCUSA\n",
"Natan T Shaked *[email protected] \nDepartment of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael\n"
]
| [
"Department of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael",
"Department of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael",
"Department of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael",
"Department of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael",
"Department of Biomedical Engineering\nDuke University\n27708DurhamNCUSA",
"Department of Biomedical Engineering\nFaculty of Engineering\nTel Aviv University\n69978Tel AvivIsrael"
]
| []
| We present cytometric classification of live healthy and cancer cells by using the spatial morphological and textural information found in the label-free quantitative phase images of the cells. We compare both healthy cells to primary tumor cell and primary tumor cells to metastatic cancer cells, where tumor biopsies and normal tissues were isolated from the same individuals. To mimic analysis of liquid biopsies by flow cytometry, the cells were imaged while unattached to the substrate. We used low-coherence off-axis interferometric phase microscopy setup, which allows a single-exposure acquisition mode, and thus is suitable for quantitative imaging of dynamic cells during flow. After acquisition, the optical path delay maps of the cells were extracted, and used to calculate 15 parameters derived from cellular 3-D morphology and texture. Upon analyzing tens of cells in each group, we found high statistical significance in the difference between the groups in most of the parameters calculated, with the same trends for all statistically significant parameters. Furthermore, a specially designed machine learning algorithm, implemented on the phase map extracted features, classified the correct cell type (healthy/cancer/metastatic) with 81%-93% sensitivity and 81%-99% specificity. The quantitative phase imaging approach for liquid biopsies presented in this paper could be the basis for advanced techniques of staging freshly isolated live cancer cells in imaging flow cytometers. | 10.1002/cyto.a.23100 | [
"https://arxiv.org/pdf/1904.00997v1.pdf"
]
| 3,683,750 | 1904.00997 | 9075f0e3ea8bb51adb142d46cfac4a6e62e6022f |
Quantitative Phase Microscopy Spatial Signatures of Cancer Cells
Darina Roitshtain
Department of Biomedical Engineering
Faculty of Engineering
Tel Aviv University
69978Tel AvivIsrael
Lauren Wolbromsky
Department of Biomedical Engineering
Faculty of Engineering
Tel Aviv University
69978Tel AvivIsrael
Evgeny Bal
Department of Biomedical Engineering
Faculty of Engineering
Tel Aviv University
69978Tel AvivIsrael
Hayit Greenspan
Department of Biomedical Engineering
Faculty of Engineering
Tel Aviv University
69978Tel AvivIsrael
Lisa L Satterwhite
Department of Biomedical Engineering
Duke University
27708DurhamNCUSA
Natan T Shaked *[email protected]
Department of Biomedical Engineering
Faculty of Engineering
Tel Aviv University
69978Tel AvivIsrael
Quantitative Phase Microscopy Spatial Signatures of Cancer Cells
10.1002/cyto.a.23100Submitted to Cytometry Part A. Published version is here: https://doi.Key terms Quantitative phase microscopyDigital holographic microscopyInterferometric imagingCytometryMachine learning
We present cytometric classification of live healthy and cancer cells by using the spatial morphological and textural information found in the label-free quantitative phase images of the cells. We compare both healthy cells to primary tumor cell and primary tumor cells to metastatic cancer cells, where tumor biopsies and normal tissues were isolated from the same individuals. To mimic analysis of liquid biopsies by flow cytometry, the cells were imaged while unattached to the substrate. We used low-coherence off-axis interferometric phase microscopy setup, which allows a single-exposure acquisition mode, and thus is suitable for quantitative imaging of dynamic cells during flow. After acquisition, the optical path delay maps of the cells were extracted, and used to calculate 15 parameters derived from cellular 3-D morphology and texture. Upon analyzing tens of cells in each group, we found high statistical significance in the difference between the groups in most of the parameters calculated, with the same trends for all statistically significant parameters. Furthermore, a specially designed machine learning algorithm, implemented on the phase map extracted features, classified the correct cell type (healthy/cancer/metastatic) with 81%-93% sensitivity and 81%-99% specificity. The quantitative phase imaging approach for liquid biopsies presented in this paper could be the basis for advanced techniques of staging freshly isolated live cancer cells in imaging flow cytometers.
Introduction
Finding cancer in its early curable stages is a clear and critical unmet need. Late stage metastatic forms of cancer are almost always uniformly fatal (1,2). Many cancers are found in early stages only due to incidental or proactive screening, especially in high risk groups (3,4). However, some cancers including pancreatic cancer and colon cancer can develop asymptomatically and thus often escape early detection (5,6). The primary basis for diagnosis of cancer is evaluation of morphological changes in a tissue biopsy by a trained pathologist, a process with inherent subjectivity, performed usually after the location of the suspect tissue is known (7).
Flow cytometry of body fluids obtained by routine medical tests can identify circulating tumor cells after their separation from the other fluid contents (8,9). Interest in using circulating tumor cells found in liquid biopsies to diagnose solid tumors has exploded recently, especially to address difficulties in analyzing tumor biopsies due to intractable anatomical placement or to determine if metastatic disease is present (10 13). In 2004, a landmark study found that the number of circulating tumor cells in blood could predict survival in metastatic breast cancer (14). Cancer-specific signatures in liquid biopsies include DNA sequence, composition of cancer cell exosomes, and unique systemic response reflected by components in blood, tears, saliva or urine assayed by genomics, epigenetics, proteomics and metabolomics (4,15). Increasingly, liquid biopsy has been suggested as a viable and affordable mechanism to safeguard good health and to detect cancer in its asymptomatic early curable stages through proactive monitoring, especially in high risk groups.
Detection of circulating tumor cells requires a highly sensitive method to identify a small number of diseased cells in a large cell population. Isolation of these cancer cells is laborious and typically yields uniformly round cells, which are hard to stage without advanced methods. Thus, other internal features of the cells should be used for evaluation (4).
During the progression of a healthy cell to immortal cancerous cell and later to metastatic cell, the biophysical and morphological phenotypes of the cell change (16). Many scientific efforts have been made to perform cytometry of cancer cells with the goal of revealing the unique biophysical properties of these cells for cancer prognosis include measuring the mechanical (17 20) or optical properties of the cells by labelbased (22) or label-free (23 29) techniques. For example, multiphoton laser tomography combined with fluorescence lifetime imaging has been used to generate 30 optical parameters that are able to distinguish normal nevi from melanoma in situ with high sensitivity and specificity (30). Atypical cells, underlying cytoskeletal disorganization in the epidermis, poorly defined keratinocyte cell borders and presence of dendritic cells typified melanoma tissue compared to normal nevi and resulted in the unique diagnostic optical parameters. Additionally, basal cell carcinoma could be distinguished from melanoma using these parameters. Although markers provide biological assays with a high degree of specificity, using fluorescent markers might cause cytotoxicity by perturbing the cell environment, by influencing its behavior over time and its viability, and eventually damaging the accuracy of the test or prohibiting further clinical use of the isolated cells (31). In flow cytometry for cell sorting, one evaluates cellular features through fluorescence markers and purifies the heterogeneous cell suspension into fractions containing a single cell type (32). However, in addition to possible cytotoxicity, suitable markers might be not available for certain cell types, and some markers might be difficult to use (33). Specifically, fluorescent markers tend to photobleach, which damages the image contrast and the prognosis results (34).
The internal morphology and texture of cancer cells changes during oncogenesis (35)(36)(37). Specifically, the intrinsic refractive index of live cells can indicate abnormal cell morphology. The cell refractive index is related to the optical interaction of the light field with cellular organelles and chemical composition, and thus can be potentially used for quantitative monitoring and diagnosis of the cellular phenotypes (28) and indicate abnormal cell morphology. In addition, dry mass of the cancer cells has been recognized as a possible diagnostic and monitoring marker (38,39).
These cellular changes in cancer cells can be potentially detected by label-free imaging techniques. Without staining, however, biological cells are nearly transparent under bright-field microscopy, as their absorption differs only slightly from that of their surroundings, resulting in a low image contrast. An internal contrast mechanism that can be used when imaging cells without staining is their refractive index. The light beam passing through the imaged cells is delayed, since the cells have a slightly higher refractive index compared with their surroundings. Conventional intensity-based detectors are not fast enough to record this delay directly. Phase imaging methods, on the other hand, use optical interference to record the delays of light passing through the sample, and thus they are able to create label-free contrast in the image (40). In contrast to qualitative phase contrast methods, such as Z interference contrast (DIC) microscopies, quantitative phase microscopy yields the optical thickness map or optical path delay (OPD) map of the cell on all spatial (x-y) points. Per each x-y point of this map, OPD is equal to the integral of the refractive-index values across the cell thickness.
Quantitative phase imaging techniques, interferometric based (23,28,41 43) and non-interferometric based (24), (29) have been used to analyze cell features for red blood cells (41), stem cells (42), cancer tissues (23) and cancer cells (24,28,29,43), and showed the ability to differentiate between various conditions of cells and tissues, based on the average OPD values (23,42,43), or other parameters that are based on the OPD map and the cell visible morphology (24,29), as well as their spatial-frequency content (41).
Machine learning of cytometric data from circulating tumor cells enables automatic analysis of a large number of cells with good classification results (44). Specific to interferometry, after extraction of various cellular features from the cellular OPD maps, machine learning techniques can assign weights to these features for classification. Several machine learning techniques have been previously applied on reconstructed digital interferograms starting from 2005 to identify filamentous microorganisms (45); and later to classify stained and unstained cells, and quantify cell viability and concentration (46); and to grade red blood cells infected by the malaria parasite (47).
In our study, we compared the quantitative phase imaging based features of healthy and cancer cells and of primary cancer and metastatic cancer cells. When doing this comparison, we have chosen pairs of cell lines taken from the same individual to avoid differences that are related to changes between people organs and disease expression. In order to obtain data for significant statistics, cell imaging techniques should have high throughput capabilities, but still be affordable (16). Acquiring the cells during fast flow can enable high throughput. In our research, the cells were alive and not attached to the surface, which allowed for analyzing a large number of cells during flow, in contrast to previous studies that used adherent cells (28,43), fixed cells (24,29), or tissue samples (23), which were limited in the amount of recorded data. In addition, we have used off-axis interferometry, which requires only a single camera exposure, and thus is suitable for acquisition of rapid dynamics, such as the one occurring during cell flowing.
After acquiring off-axis interferograms of the cells and extracting their OPD maps, we calculated cellular features based on these quantitative maps, and applied machine learning approaches for cell group classification. Our quantitative phase imaging approach is expected to yield an automated tool to distinguish oncogenic progression and metastasis based on label-free cancer cell cytometry.
Materials and methods
Low-coherence interferometric phase microscopy (IPM) setup
We used low-coherence interferometric phase microscopy (IPM) to capture the OPD map of healthy, primary cancer and metastatic cells in vitro. The (IPM) system used in the experiments is a low-coherence Mach-Zehnder imaging interferometer for creating off-axis image interferograms of the sample. Figure 1 presents a scheme of a system, which is illuminated by a supercontinuum laser source, coupled to an acousto optical tunable filter (AOTF) (Fianium SC-400-4 and AOTF, 650 nm with FWHM spectral bandwidth of 7nm). The beam is split into a reference and a sample beam by BS1. The sample beam passes through the cell sample S, which is located on an XYZ micrometer stage. This beam is then magnified by microscope objective MO1 (Newport, 60×, 0.85 NA). In parallel, the reference beam propagates through identical microscope objective MO2. Both beams are projected through tube lens L (f=150 mm) onto a CCD digital camera (Thorlabs, DCC1545M, max framerate 25 fps, exposure time 40 ms). The lateral resolution was 0.76 µm. Since low coherence illumination is used to minimize spatial noise and parasitic interferences, retroreflectors RRs on micrometers are utilized to adjust the beam paths of the sample and reference beams. These two beams interfere on the camera at a small off-axis angle and induce straight off-axis fringes. The temporal and spatial OPD (axial) stabilities were 0.62 nm and 1.1 nm, respectively. The cell lines were incubated under standard cell culture conditions at 37°C and 5% CO2 in a humidified incubator until 80% confluence was achieved.
Preparation for imaging
Prior to the imaging experiment, the cells were trypsinized for suspension, supplemented with a suitable medium, and inserted into an adhesive chamber (Grace Biodiameter × 0.15 mm thickness, ports diameter 1.5 mm, Sigma Aldrich SN. GBL611101) attached to a cover slip. This chamber induced a constant thickness value on the entire imaged sample, which is important for the flatness of the final phase map. Another adhesive chamber was filled with the suitable medium and placed in the reference beam propagation path. Then, all cells lines were quantitatively imaged without labeling using the low-coherence IPM system shown in Fig. 1.
Data analysis
To extract the quantitative phase map from the acquired off-axis image interferograms, we used the off-axis interferometry Fourier-based algorithm (56), which includes a 2-D Fourier transform, filtering one of the cross-correlation terms, and an inverse 2-D Fourier transform, where the argument of the resulting matrix is the wrapped phase of the sample. To compensate for stationary aberrations and field curvatures, we subtracted from the wrapped phase map of the sample, a phase map which is extracted from an interferogram acquired with no sample. We then applied the unweighted least squares phase unwrapping (57). The resulting unwrapped phase map is multiplied by the defined as follows:
(1) ) , ( ] ) , ( [ ) , ( y x h n y x n y x OPD c m c c ,
where m n is the refractive index of the medium, c h is the thickness profile of the cell, and c n is the cell integral refractive index, which is defined as follows (58):
(2) h o c c c dz z y x n h y x n ) , ,( 1 )
, ( . To separate single cells from the background and be able to process only the OPD related to the cells, we used the Normalized Cut Algorithm as an edge detector (59). This method is based on a graph formulation, wherein the nodes of the graph are the points in the feature space with a similarity function weight connecting them. The goal is to partition the vertices into disjoint sets
m V V V ,..., , 2 1
, whereby the similarity within a set i V is high, and across different sets is low. After implementation of this algorithm, a morphological opening operator was applied in order to connect the gradient lines that were detected. Next, a morphological dilation operator was applied in order to expand the connected lines. At last, a global threshold was applied in order to remove background pixels that erroneously appeared in the cell area.
Using the above described methods, we could create a data set containing the OPD information of the cell areas only, and calculate the following parameters that are based directly on the OPD map defined in Eq. (1), without decoupling the cellular thickness profile from the refractive index as a prior stage: . As can be seen, the phase volume is proportional to the dry mass of the cell. 5. Phase surface area: This parameter can be calculated as the sum of the upper surface area of the phase profile and the projected area, as follows:
(6) c c S c y x S c S dxdy h h S dA SA 2 / 1 2 2 ) 1 ( ,
where dA is the discrete cell surface area as projected over a single camera pixel, x h and y h are the gradients along the x and y directions of the cell OPD map (63,64). 6. Phase surface area to volume ratio: This parameter is a generalized version of the physical surface area to volume parameter (65,66), but again it takes into consideration phase changes in the cell:
(7) V SA SAV . 7.
Phase surface area to dry mass ratio: This parameter is defined as follows (62):
(8) M SA SDM .
The last two parameters can quantify cell metabolism and describe how much material a surface unit transfers to one volume unit or mass unit. . It is important to mention that the OPD values have coupling between the cellular refractive index and the physical thickness (see Eq. (1)), and the entire analysis was performed directly on the OPD value without decoupling these parameters to allow single exposure mode per each instance of the sample, which is suitable for acquiring cells during fast flow. Note also, that all 15 parameters presented above are based on the quantitative OPD map and thus cannot be calculated based on simple bright-field microscopy or fluorescent microcopy (recording the intensity of light), which is the basis of conventional flow cytometry, or based on non-quantitative interference contrast (DIC) microscopy.
Statistical analysis
In total, we acquired 106, 97, 71, 102, 118, and 163 OPD maps for cell lines Hs 895.Sk, Hs 895.T, WM-115, WM-266-4, SW-480, and SW-620, respectively. To evaluate statistical difference amongst the cell groups, for each cell line pair and for each of the calculated parameters, we used the two-sample t-test for the p values of the data shown in Table 1. In addition, we also implemented the Mann-Whitney test, which yielded similar or even greater statistical difference between the groups.
Machine learning
Following the OPD-map-based feature extraction, our objective was to obtain classification decisions. For this goal, we used state-of-the-art machine learning techniques to perform three separate classifications for each pair of cell lines: Hs 895.Sk versus Hs 895.T, WM 115 versus WM 266-4, and SW 480 versus SW 620.
In general, a classification solution comprises of three main stages: (a) selection of features for the specific task, (b) a dimensionality reduction (and noise removal), and (c) the final classification using a selected classifier. In this work, we used the proposed analysis of the descriptors to select the features.
As for stage (a) of feature selection, for each cell OPD map, a set of 13 features which were found to be statistically discriminative according to their p values (mean, median, projected surface area, phase volume, dry mass, dry mass average density, energy, surface area, phase surface area to volume ratio, phase surface area to dry mass ratio, projected area to volume ratio, sphericity and phase variance), were concatenated into a feature vector of size 13. For each classification task, a feature matrix of size (M+N)×13 was built, where M and N are the numbers of cells in the matching cell line pair.
For stage (b) of dimensionality reduction, a standard Principal Component Analysis (PCA) (71) was performed on the feature matrix. The resulting PCA feature matrix was column normalized to a mean of 0 and a standard deviation of 1. A set of the most informative components can be selected following the PCA stage (71). In this work, after trying other options, we eventually selected the first six components to serve as the input representation.
Next, in stage (c) of classification, since we know all of the cells labels, a supervised learning method was selected. A state-of-the-art Support Vector Machine (SVM) classifier was used for this task. The SVM is widely used for pattern classification problems (72). A Leave-One-Out SVM classification was performed using LibSVM (73). The classification performance was evaluated using the area under curve (AUC) of the receiver operating characteristic (ROC) curve.
Results
We used the low-coherence IPM system presented in Fig. 1
to acquire OPD maps of the six cells lines described above, with the goal of comparing healthy to cancer cells (Hs 895.Sk [skin] vs. Hs 895.T [melanoma]), and primary cancer to metastatic cancer cells (WM-115 [melanoma] vs. WM-266-4 [metastatic melanoma], and SW-480 [adenocarcinoma colon] vs. SW-620 [metastatic adenocarcinoma colon])
. Note that the cells were unattached and therefore had mostly round projected areas. The ability to analyze unattached isolated cells is important for flow cytometry via high-throughput quantitative imaging of cells during flow. Visualization 1 (Fig. 2) presents the OPD map of SW 480 cells flowing in a microfluidics channel (IBIDI, SN. 80666, 1 mm width, 17 mm length, 0.1 mm height). In this situation, when the isolated cells are round and unattached, most of the cells look alike and subjective pathological examination cannot be performed. Indeed, Fig. 3 shows one representative OPD map from each of the cell line groups, demonstrating that even when using quantitative phase imaging, a bare eye cannot see significant differences between each pair of cell lines. This is solved by the automatic machine-learning method analyzing the cell topological OPD maps directly. On the other hand, non-quantitative imaging methods, such as label-free bright-field, Zernike's phase contrast or DIC microscopies, which do not have access to these cell topological OPD maps, do not allow for quantitative automatic identification.
Prior to the analysis of these OPD maps, we applied the segmentation image processing procedure, described in the Methods section, to track the cell edges. Next, we applied Eqs. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) to the cell OPD area selected by the segmentation process to calculate the following parameters: mean, median, projected surface area, dry mass, dry mass average density, phase volume, phase surface area, phase surface area to volume ratio, phase surface area to dry mass ratio, projected area to volume ratio, sphericity, phase variance, kurtosis, skewness and energy.
The corresponding average and standard deviation values for each parameter for all cell lines are summarized in Table 1. As can be seen from this table, 13 out of 15 OPD-based parameters were statistically significant. Figures 4-6 show the histograms of each of 12 parameters for which the two groups of cells compared were statistically significant (excluding the projected surface area parameter, to spare space, that was also statistically significant with p-values of <0.0005). Each subfigure shows the histograms of a pair of cell lines, where each cell line is shown in a different color (red or purple). All these parameters are statistically significant based on p-values of <0.05, <0.005 and even <0.0005. These results demonstrate the parameters ability to statistically discriminate between each cell line in each pair, even if the initial OPD map of the cells look similar.
The 13 successful parameters were used as an input for PCA/SVM analysis (see machine learning details in the Method section) to extract the best combination of these parameters, which is useful for classification between each cell line pair. The ROC curves obtained by using PCA followed by an SVM classification for each pair of cells are presented in Fig. 7. The best results of the classification were obtained for six-first principle components and for linear SVM kernel for all cell line pairs. As seen in this figure, the area under the curve (AUC) in all three classification tasks is high and around 0.9. Table 2 summarizes the results obtained using the SVM learning combined with PCA for each pair of cells, with the AUC and the matching working points (sensitivity/specificity) chosen on the ROC curves. Note that for the definition of sensitivity and specificity, the Hs 895.Sk, SW 480 and WM 115 cell lines were defined as , and the Hs 895.T, SW 620 and WM 266-4 cell lines were defined as .
These results demonstrate an automatic algorithm with an ability to classify cells in different cancer stages using the OPD-map-based parameters. We achieved high classification rates and AUC values for unattached cells, whereby no morphological differences can be observed in the phase maps by the naked eye (see Fig. 3). The AUC values correspond with the separation between the groups presented in the histograms (Figs. [4][5][6], whereby better classification with higher percentages corresponds with lower p-values between the groups in the histograms. Thus, for the WM pair (Fig. 5), we can observe better separation than the Hs pair (Fig. 4) and the SW pair (Fig. 6).
Significantly, since the best PCA/SVM parameters were achieved for all cell line pair classification tasks (sixbest principle components and linear SVM kernel were finally chosen for all cases), it can be assumed that a global machine for cancer grading can be developed.
Discussion and conclusions
Early detection of cancer can prevent tedious and painful treatment process, may prevent recurrence, and improve survival. To identify cancer, pathologists typically use fresh samples that are frozen and sectioned, or fixed samples that are dehydrated, embedded in paraffin, sectioned and stained with dyes and/or antibodies to specific tumor antigens. Our work describes the first steps in the development of optical signatures that can distinguish normal cells from cancer in situ and cancer in situ from metastatic forms without fixation, sectioning or any type of labeling. Thus, the scores can be performed rapidly and automatically and are not only based on simple qualitative cell parameters based on 2-D imaging (such as cell size and general shape), but rather use the cellular optical thickness dimension as well. This yields new parameters based on the cells topological map that have not been available to the pathologists before, with which automatic cell identification can be performed.
The challenge with flow cytometry for cancer diagnosis is the number of cells that are needed, which in turn is dependent on the brightness of the marker. In general, 1000 to 1,000,000 cells are needed from a dispersed tumor biopsy depending on the proportion of cancer cells to non-cancer cells in the sample and tumor heterogeneity (74). There is no universal marker that can be used to detect all cancers. Flow cytometry is used to detect leukemia, other bone marrow-associated cancers and to determine success of stem cell transplantations, but certainly cannot be used to detect circulating tumor cells where greater than or equal to five metastatic breast cancer cells are found in 7.5 mL peripheral blood (75). On the other hand, our method can distinguish between normal and cancerous cells and primary cancer from metastatic cancer cells, with as few as tens of cells, without using fixation, embedding and tumor markers.
To address the clinical need for cancer prognosis, we identified OPD-based diagnostic signatures for live and unattached cells in different stages of oncogenesis. We proposed using spatial morphological and texture parameters, which are based on the cell OPD maps, to compare tumor-derived cancer cell lines of different cancer levels and cell lines derived from a healthy tissue of the same individuals. We showed the feasibility to distinguish between different cell conditions in a label free manner and we have quantified differences among microscopically similar looking cells with statistical significance. We demonstrated that diagnostic optical signatures can be derived from comparisons between single cells, normal fibroblasts derived from living biopsy tissue compared to fibroblasts isolated from melanoma or from comparison of fibroblasts derived from melanoma to cells derived from a melanoma lymph metastasis. Thus, we showed that these quantitative phase based parameters are able to distinguish cancerous cells from the healthy cells and metastatic cancer cells from primary cancer cells with high accuracy.
Specifically, to compare healthy and cancerous cells, normal human skin fibroblasts isolated from a 48 year old Caucasian female (Hs 895.Sk) were compared to fibroblasts isolated a melanoma tumor from the skin of the same individual (Hs.895.T), with classification results of 81% sensitivity and 83% specificity. To compare primary cancer cells and metastatic cancer cells, human fibroblasts isolated from a melanoma tumor in situ from a 58 year old female (WM-115) were compared to fibroblasts isolated from a lymph metastasis in the same individual (WM-266-4), with classification results of 93% sensitivity and 99% specificity. Additionally, cells from colorectal adenocarcinoma in situ (SW-480) were compared to cells from colorectal adenocarcinoma lymph metastasis from the same individual (SW-620), with classification results of 82% sensitivity and 81% specificity.
Tumor microenvironment may differ between melanoma in situ and distant sites, contributing to reorganization of cellular morphology and interaction with quantitative phase. Indeed, metastatic forms of different cancers share morphological similarities leading to the idea that many cancers evolve into a more common metastatic state that is stratified by heterogeneity in tumor microenvironment, which differs depending on the site of metastasis, interaction with the immune system, and chemotherapeutic resistance (76).
Cells in our study were released from the substratum in adherent culture by limited digestion with trypsin resulting in spherical cells in solution, to model the geometry of circulating tumor cells. We compared the OPD profiles of trypsinized spherical normal cells to trypsinized spherical cancer cells and of trypsinized spherical primary cancer cells to trypsinized spherical metastatic cancer cells. The likely contributor to the difference in refractive index is that the cytoskeleton is altered dramatically during oncogenesis. Future experiments will determine how trypsinized spherical cells compare to affinity purified circulating tumor cells. We predict that the circulating tumor cells will resemble trypsinized cells analyzed in our experiments because cells inside a tumor are essentially in 3D adherent culture.
The optical signatures that can be built using the OPD parameters are advantageous in that tumor cells can be imaged without time consuming labeling, sectioning, sequencing or other methods in use today to identify tumor cells by genomic changes.
Thirteen of the parameters defined, which include the OPD mean, median, projected surface area, phase volume, dry mass, dry mass average density, surface area, phase surface area to volume ratio, phase surface area to dry mass ratio, projected area to volume ratio, sphericity, phase variance and energy, showed statistical significance with low p-values for all cell line pairs. Significantly, despite the wide distributions in the features of the various groups, as shown in the histograms in Figs. 4-6, which could be a result from differences in cell cycle length and non-synchronization across the population; low p-values were still obtained between the groups, so that these groups are statistically different. No statistical significance was observed for kurtosis and skewness, which might be explained by the fact that the experiments were done on floating, uniformly spherical cells, so the measures of peaks, flatness or symmetry were less applicable.
It can be seen in Figs. 4-6 that the parameter values decreased with cancer progression (healthy > cancer > metastatic) except for the projected area to volume ratio and sphericity (where the mathematical relation is opposite) that increased with cancer progression. In any case, as can be seen from these results, a similar trend of progression is retained in the average parameters values. We suppose that the observed trends will be kept for other similar cell line pairs. However, since our experiments are based on three cell line pairs, further research is needed to prove this hypothesis. The greater differences in the calculated parameters between the WM cell line cells than between isogenic SW cell line cells might be explained by the fact that that the WM fibroblasts isolated from in situ melanoma compared to the WM fibroblasts isolated from metastatic melanoma are likely to be more similar (both flat well-spread) than the SW epithelial cells isolated from colorectal adenocarcinoma, which are more cuboidal in situ, compared to SW metastatic cells, which are more spread and more motile.
IPM of unattached cells in flow can be performed faster than video processing rate on a regular computer, including 2-D phase unwrapping (56,77) and much faster if using the graphics processing unit (GPU) of the computer (78). Our method can be combined in the clinics using compact modules that can be connected in the output of a conventional microscope (79,80). By combining IPM with real-time fast processing algorithms (56,77,78) and automatic cell detection and machine learning algorithm for evaluating cell condition as presented in our work, quantitative phase microscopy has potential to become a powerful clinical screening tool for cancer diagnosis and might allow pathologists to inexpensively grade circulating tumor cells in liquid biopsies in real time. Thus, together with the analysis tools presented here for cancer monitoring, we believe that in the future IPM can be useful for detecting and monitoring cancer from body fluids in flow cytometry, for routine cell analysis or for the investigation and detection of pathological conditions in a semi-automatic way.
In this work, we used state-of-the-art machine-learning algorithms based on SVM to demonstrate the feasibility of our approach for automatic classification of cancer cells based on IPM data. However, note that there is a great variety of other machine leaning algorithms. One of the possible extensions is neural network related algorithms for deep learning, which require a greater number of samples.
To summarize, the proposed quantitative imaging technique shows preliminary clinical potential for automatic cell flow cytometry. This technique does not require cell staining as conventional flow cytometry, and since it uses the quantitative phase profile it has access to parameters which are not available using bright-field microscopy or other non-quantitative phase imaging techniques. Still, future studies are needed to confirm if the unique quantitative phase signatures developed here can distinguish normal from circulating cancer cells isolated from liquid biopsies, and determine the diagnostic and prognostic value of quantitative phase signatures in determining tumor grade and metastatic potential. *** *** ***
three pairs of isogenic cell lines: normal skin cells Hs 895.Sk (ATCC CRL-7636) and melanoma skin cells Hs 895.T (ATCC CRL-7637), melanoma skin cells WM-115 (ATCC CRL-1675) and metastatic melanoma skin cells WM-266-4 (ATCCCRL-1676), colorectal adenocarcinoma colon cells SW-480 (ATCC CCL-228) and metastatic from lymph node of colorectal adenocarcinoma cells SW-620 (ATCC CCL-227). The first pair of cells was chosen to compare healthy cells versus cancer cells, and the other two pairs of cells were chosen to compare primary cancer cells versus metastatic cells It is important to note that each of the cell line comparisons is from the same individual. The complete growth medium used for the Hs cell pair (ATCC, SN. 30-2002) supplemented with 10% Fetal Bovine Serum (FBS) (BI, SN. 04-007-1A)The complete growth medium used for the WM cell pair is DMEM (DMEM) (BI, SN. 01-055-1A) supplemented with 10% FBS (BI, SN. 04-007-1A) and 2 mM L-glutamine (BI, SN. 03-020-1B)(48,49).The complete growth medium used for the SW cell pair is BI Roswell Park Memorial Institute (RPMI) 1640 Medium without L-glutamine (BI, SN. 01-104-1A) supplemented with 10% FBS (BI, SN. 04-007-1A) and BI 2 mM L-glutamine (BI, SN. 03-020-1B)(50 55).
1 .
1Mean and median of c OPD . 2. Dry mass: This parameter quantifies the mass of the non-aqueous material of the cell, yielding information about cell growth (60,61). Dry mass averaged density: This parameter can be calculated using the cell dry mass as follows (Phase volume: This is not the actual cell volume but only the equivalent of the cell volume that is based on c OPD directly and takes into consideration refractive-index variations inside the cell (62).
8 .
8Projected area to volume ratio: This parameter describes the flatness of the cell (62).
Phase sphericity index: This parameter quantifies the degree of cell roundness. The sphericity of an object is the ratio of object volume and the surface area. Round shape is a value that may imply on cell abnormality(62,67 69). It is a dimensionless constant with values ranging from zero for a laminar disk to unity for a sphere(68). Phase statistical parameters: These parameters describe the dry mass or volume distribution in the cell. They are based on changes in phase values and thus react to structural alternations of cell organelles and factors. To use these parameters, the phase values over the projected cell area need to be written as a single vector containing n values. Then, the following statistical parameters can be defined(62): a. Phase variance: This parameter measures how a set of the cell OPD values is spread out. the mean of the OPD of the cell. b. Phase kurtosis: This parameter measures whether the cell OPD distribution is peaked or flat.
.
Phase skewness: This parameter measures the lack of symmetry of the cell OPD values from the mean value. Energy: This parameter characterizes the cell texture (70).
Figure. 1 .Figure. 3 .
13Low-coherence IPM used in the experiments SC, Super continuum laser source. AOTF, Acousto optical tunable filter. BS1, BS2, Beam splitters. M1, M2, mirrors. RR, retroreflector. S, Sample. MO, Microscope objective. L, tube lens. CCD, Digital camera. Quantitative phase microscopy of unattached cancer cells, demonstrating that most cells are round and look alike in this situation. (a) Hs 895.Sk. (b) Hs 895.T. (c) SW480. (d) SW620. (e) WM 115. (f) WM 266-4. Colorbars represent OPD values in nm.
Figure. 4 .Figure. 5 .Figure. 6 .Figure 7 .
4567Histograms of the parameters based the OPD maps for normal skin cells Hs 895.Sk (red) via melanoma skin cells Hs 895.T parameters (blue). (a) Mean. (b) Median. (c) Phase Volume. (d) Phase Variance. (e) Dry Mass. (f) Dry Mass Average Density. (g) Surface Area. (h) Phase Surface Area to Volume Ratio. (i) Phase Surface Area to Dry Mass Ratio. (j) Projected area to volume ratio. (k) Energy. (l) Sphericity. * denotes p-value<0.05, *** denotes p-value<0.0005. Histograms of the parameters based the OPD maps for melanoma skin cells WM-115 (red) via metastatic melanoma skin cells WM-266-4 (blue). (a) Mean. (b) Median. (c) Phase Volume. (d) Phase Variance. (e) Dry Mass. (f) Dry Mass Average Density. (g) Surface Area. (h) Phase Surface Area to Volume Ratio. (i) Phase Surface Area to Dry Mass Ratio. (j) Projected area to volume ratio. (k) Energy. (l) Sphericity. *** denotes p-value<0.0005. Histograms of the parameters based the OPD maps for colorectal adenocarcinoma colon cells SW-480 (red) via metastatic from lymph node of colorectal adenocarcinoma cells SW-620 (blue). (a) Mean. (b) Median. (c) Phase Volume. (d) Phase Variance. (e) Dry Mass. (f) Dry Mass Average Density. (g) Surface Area. (h) Phase Surface Area to Volume Ratio. (i) Phase Surface Area to Dry Mass Ratio. (j) Projected area to volume ratio. (k) Energy. (l) Sphericity. *** denotes p-value<0.0005. ROC curves of true-positive rate (sensitivity) vs. false-positive rate (one minus specificity), as obtained using SVM learning combined with PCA for the: Hs cell lines (blue curve), WM cell lines (red curve), SW cell lines (green curve). Each graph represents another cell line pair (Hs, WM, or SW), with the resulting AUC written after the cell line pair name. The small circles denote the working points of the classification tasks.
Table 1. Averages and standard deviations for all the parameters calculated for all the cell lines. In each cell, the first and second values correspond to the average and to the standard deviation values, respectively. * denotes p-value<0.05, ** denotes p-value<0.005, *** denotes p-value<0.0005. -denotes p-value>0.05. SAV [10 2 1/nm]1-Specificity
Hs, 0.878
WM, 0.985
SW, 0.897
SW 620
SW 480
WM 266-4
WM 115
Hs 895.T
Hs 895.Sk
286.04
27.82
306.45
34.59
319.90
35.70
474.10
86.58
272.40
48.45
333.09
60.99
Mean [nm]
***
***
***
312.34
32.95
335.89
40.07
342.90
39.04
502.78
93.88
282.53
52.90
348.88
72.37
Median [nm]
***
***
***
70.4
13.53
95.60
21.48
153.14
34.62
342.24
106.06
202.99
84.96
251.87
78.94
Projected surface
area [10
6 nm
2
]
***
***
***
20.34
5.35
29.74
9.29
49.75
14.87
165.93
69.02
57.67
33.01
86.34
38.76
Phase volume
[10
9 nm
3
]
***
***
***
10.17
2.68
14.87
4.64
24.87
7.44
82.96
34.51
28.84
16.51
43.17
19.38
Dry mass
[10 -11 gr]
***
***
***
14.30
1.39
15.32
1.73
15.99
1.79
23.70
4.33
13.62
2.42
16.65
3.05
Dry mass average
density [10
-19 gr/nm
2 ]
***
***
***
17.96
6.93
31.36
13.90
71.57
31.83
376.22
225.56
114.37
100.90
172.16
109.01
Surface area
[10
12 nm
2
]
***
***
***
8.59
1.04
10.17
1.34
13.87
2.08
21.46
4.19
17.37
4.92
18.71
3.53
***
***
*
17.19
2.08
20.35
2.68
27.75
4.16
42.92
8.38
34.75
9.84
37.43
7.06
SDM [10
22 nm/gr]
***
***
*
35.29
3.40
33.07
4.01
31.74
4.64
21.80
4.06
37.77
6.17
31.05
5.84
PAV [10
-4 1/nm]
***
***
***
21.39
4.12
16.13
3.77
11.26
15.59
4.50
1.44
8.45
3.16
6.42
1.95
Sphericity [10
-7 #]
***
***
***
12.34
3.29
16.06
5.34
18.85
5.31
41.38
12.84
13.22
5.69
20.58
7.46
Phase variance
[10
3 nm
2
]
***
***
***
11.79
6.50
10.24
11.09
11.41
17.57
4.85
3.10
29.29
18.86
15.02
10.42
Phase kurtosis
[10
-6 1/nm
4 ]
**
***
-28.89
14.19
-25.29
16.67
-21.86
21.33
-15.04
8.60
-26.75
21.40
-21.57
23.91
Phase skewness
[10 -5 1/nm 3 ]
*
6.61
2.38
10.64
4.60
18.72
7.40
94.82
52.08
19.30
15.01
34.97
21.54
Energy [10
7 nm
2
]
Table 2 .
2Machine learning results for classifications between the groups (PCA-SVM analysis).Sensitivity Specificity
AUC
Hs 895.Sk
Hs 895.T
81%
83%
0.878
WM 115
WM 266-4
93%
99%
0.985
SW 480
SW 620
82%
81%
0.897
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)Hs 895.Sk Hs 895.T
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
AcknowledgmentsThis research was supported by Horizon 2020 European Research Council (ERC) Grant No. 678316. We thank Ruth Gottlieb for significant biological support and advising, Dr. Ksawery Kalinowski for useful technical discussions, and Dr. Itay Barnea for useful biological discussions. *** *** *** ***
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One snapshot from a video (see Visualization 1) presenting quantitative phase microscopy of live unlabeled cancer cells (SW 480 cell line) flowing in a microfluidic channel. Colorbar represents OPD values in nm. Figure. 2. One snapshot from a video (see Visualization 1) presenting quantitative phase microscopy of live unlabeled cancer cells (SW 480 cell line) flowing in a microfluidic channel. Colorbar represents OPD values in nm.
| []
|
[
"A conformal block Farey tail",
"A conformal block Farey tail"
]
| [
"Alexander Maloney \nDepartment of Physics\nMcGill University\nMontréalCanada\n",
"Henry Maxfield \nDepartment of Physics\nMcGill University\nMontréalCanada\n",
"Gim Seng Ng \nDepartment of Physics\nMcGill University\nMontréalCanada\n"
]
| [
"Department of Physics\nMcGill University\nMontréalCanada",
"Department of Physics\nMcGill University\nMontréalCanada",
"Department of Physics\nMcGill University\nMontréalCanada"
]
| []
| We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by P SL(2, Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over P SL(2, Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the P SL(2, Z) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over P SL(2, Z) in CFT 2 has a natural AdS 3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator. | 10.1007/jhep06(2017)117 | [
"https://arxiv.org/pdf/1609.02165v2.pdf"
]
| 118,578,848 | 1609.02165 | 9fa0cbc3b3a51bd3b7db460f84c9b85227cf2187 |
A conformal block Farey tail
July 7, 2017
Alexander Maloney
Department of Physics
McGill University
MontréalCanada
Henry Maxfield
Department of Physics
McGill University
MontréalCanada
Gim Seng Ng
Department of Physics
McGill University
MontréalCanada
A conformal block Farey tail
July 7, 2017
We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by P SL(2, Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over P SL(2, Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the P SL(2, Z) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over P SL(2, Z) in CFT 2 has a natural AdS 3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.
Introduction
The conformal bootstrap is a powerful tool to constrain the spectrum and dynamics of strongly coupled field theories. It is especially powerful in two dimensions [1], where it has led to an exact classification of the rational models [2]. Bootstrap techniques have recently been used to place strong constraints on higher dimensional conformal field theories as well (see e.g. [3,4] and citations therein). The goal of this paper is to describe a somewhat different implementation of the conformal bootstrap program which is inspired by the modular properties of conformal blocks. Most of our explicit computations are in two dimensions, although we expect the general strategy to apply in higher dimensions as well.
We begin with the usual starting point of the conformal bootstrap: the expansion of a CFT correlation function as a sum over intermediate states. For example, the four point function of a scalar operator O can be written as a sum over intermediate
operators φ as 1 O(z 1 )O(z 2 )O(z 3 )O(z 4 ) = G({z a }) φ C 2 OOφ |x| ∆ φ F φ (x,x) .(1)= (1 − x)(1 −x) .(2)
The sum in (1) is over all primary operators φ, with dimensions ∆ φ , and is weighted by the square of the three point coefficient C OOφ . The conformal block F φ (x,x) encodes the contribution of the entire family of conformal descendants of φ, and is a function only of the dimension and spin of φ and O. Equation (1) is a general formula, but many simplifications occur in two dimensions. In this case F φ (x,x) is the product of a left-and a right-moving block. Moreover, in D = 2 the states can be organized into representations of Virasoro symmetry, so F φ (x,x) can be taken to be the full Virasoro block rather than just a global conformal block. (1) is not manifestly invariant under crossing symmetry -the conformal blocks transform in a highly non-trivial way -so this is a strong constraint on the operator dimensions and three point coefficients. In the standard implementation of the conformal bootstrap, one attempts to solve this constraint directly.
The problem is that the constraints of crossing are difficult to write down explicitly. For example, in general the dimensions ∆ φ of the intermediate states are not known, so one must solve for these dimensions at the same time that one is solving for the three point coefficients. However, if we are only interested in the limit x → 0, the sum is dominated by the identity operator, so that 2
O(z 1 )O(z 2 )O(z 3 )O(z 4 ) ≈ G({z i })F 1 (x,x) + . . .(3)
More generally, when x is small we can approximate the four point function by
O(z 1 )O(z 2 )O(z 3 )O(z 4 ) ≈ G({z i })F light (x,x)(4)
where
F light (x,x) ≡ ∆ φ <∆ light C 2 OOφ |x| ∆ φ F φ (x,x)(5)
is the contribution from the "light" operators, i.e. the operators with dimension less than some value ∆ light : This approximation has the advantage that it requires only CFT data involving light operators. As we increase ∆ light the approximation (4) becomes more accurate, but requires more detailed information about the CFT. Our strategy is motivated by the following question: Given only the contribution F light (x,x) from a set of light states, can we construct a consistent "candidate" correlation function
O(z 1 )O(z 2 )O(z 3 )O(z 4 ) candidate(6)
which has all of the desired properties of a true CFT four point function? In particular, we will seek a candidate correlation function that
• matches the x → 0 behaviour of the light operators in (4),
• is crossing symmetric, and
• is a single valued function of the cross ratio x.
The first property is easy to satisfy. We can just take our candidate partition function to be the truncated sum (4), which includes only light operators. The second property is, at least naively, just as straightforward: one could simply sum the result over all possible permutations of the external operators. This has the effect of summing over channels in which the intermediate light states could propagate. The real problem is the third condition. The conformal blocks are not single valued functions of x; they have branch cuts with non-trivial monodromy structure around x = 1, which is the radius of convergence of the OPE expansion (1). We propose to resolve this problem by exploiting the modular structure of conformal blocks. In particular, we will use the fact that conformal blocks are naturally viewed as functions not of cross ratio, but rather as functions of a modular parameter τ which lives on the upper half plane H + . This observation has appeared in the literature before (see e.g. [5]), and will be reviewed in detail in the next section. The upper half plane H + is the universal cover of x-space, so the conformal blocks are single valued functions of τ . In this language, crossing symmetry is simple to state: regarded as a function of τ , the four point function (1) must be invariant under the modular transformations
τ → γτ ≡ aτ + b cτ + d , for all γ = a b c d ∈ SL(2, Z) .(7)
Our question can therefore be rephrased as follows: Given a light contribution F light (τ,τ ), how do we turn it into a modular invariant function of τ ?
Our proposal is that candidate correlation functions should be constructed by averaging the light contribution F light (τ,τ ) over the modular group P SL(2, Z):
O(z 1 )O(z 2 )O(z 3 )O(z 4 ) candidate = G({z i }) 1 N γ∈P SL(2,Z)
F light (γτ, γτ ), (8) where N is a normalization constant. This average, provided it converges, satisfies all of our criteria. It can be viewed as an improved sum of the light contribution F light (x,x) over all possible channels, including those obtained by non-trivial monodromies of the cross-ratio. In the mathematics literature, averages over P SL(2, Z) are known as Poincaré series (see e.g. [6,7]). In the physics literature they are often referred to as Farey tail sums, and have appeared primarily in the context of three dimensional gravity (see e.g. [8][9][10][11][12][13]). Unfortunately, in many cases sums of the form (8) will diverge, and must be regulated; regularizations of sums of this type were considered in [8,10,13,14]. In this work we will focus on this sum primarily in the context of minimal models, where the convergence is manifest. One can view the proposal (8) as a construction of an approximate four point function which has the advantage that it depends only on the light data of the theory, i.e. on the dimensions and three point coefficients of operators with ∆ φ < ∆. The two dimensional case is particularly interesting, because in this case the Virasoro vacuum block itself is non-trivial. So one can take F light (x,x) = F 1 (x,x), including in the sum only the contribution of the vacuum block. This gives candidate four point functions which are determined uniquely in terms of the central charge. Even in higher dimensions, one can imagine including only the contributions of the stress tensor or of other universal light operators as a seed contribution from which to construct the candidate correlation functions 3 .
The utility of this approach becomes clear when we imagine taking the candidate four point function (8) and re-expanding around x → 0 as in (1). In this case we can ask the following: does the candidate four point function reproduce the contribution of heavy states as well? In particular, by expanding around x → 0 one can attempt to extract from our candidate four point function the dimensions and three point coefficients of other operators in the theory. In general there is no guarantee that the resulting coefficients C OOφ extracted in this way would be real. In this case one would discover that additional heavy operators need to be added at a particular dimension. This would provide a novel implementation of the conformal bootstrap strategy.
On the other hand, one might hope that in some cases the candidate four point function constructed in (8) might be exactly correct. This would be a truly miraculous occurrence, since by re-expanding around x → 0 and using (1) one could then read off the dimensions and three-point coefficients of all operators of the theory. We will see that, for rational CFTs in two dimensions, miracles do indeed occur. For example, in the 2D Ising model CFT we will see that all of the correlation functions of the theory are given by modular sums (8), where we include only the Virasoro vacuum block in F light (x,x) = F 1 (x,x). 4 In analogy with [16], we will call a CFT correlation function which has the property that it is equal to the modular average of the vacuum block an "extremal correlator." If a CFT correlator is extremal, then all of the three-point coefficients are determined in terms of the central charge. We will show explicitly that the Ising model correlators are extremal, and present numerical evidence that other minimal model correlators are extremal as well.
The sum over P SL(2, Z) described above has, at least in some cases, a natural AdS/CFT interpretation as a sum over semi-classical bulk saddles. One saddle point contribution to a CFT four point function is described by a pair of bulk worldlines which connect the two pairs of boundary points. For two dimensional CFTs (three dimensional bulk) these worldlines can be topologically non-trivial. We will see that the sum over P SL(2, Z) corresponds precisely to the sum over particle worldlines which map out "rational tangles" in the bulk. When the boundary operators are heavy the worldlines will back-react on the geometry, so the sum over bulk saddles is difficult to compute precisely. We will therefore focus on a particular computation -that of external operators of dimension ∆ = c 16 -where the computation can be performed explicitly. In this case we will show that the holographic computation of the correlation function takes precisely the form of a sum over rational tangles. This is holographic evidence that, at least in some cases, our candidate correlation functions constructed by a modular average are correct.
We note that our computation of the correlation function of ∆ = c 16 operators includes new results on the AdS gravity interpretation of conformal blocks. Our semiclassical, first-quantised description of particles in AdS 3 will naturally compute Virasoro conformal blocks in the boundary CFT. In particular, we will compute from gravity an exact (in dimensions, leading order in c) semiclassical block, where all external and internal operators have dimensions of order c.
In section 2 we will review the relationship between modular transformations and crossing symmetry. In section 3 we will describe in detail our proposal for the candidate correlation function as a modular average. In section 4 we will demonstrate that the candidate correlation functions are exactly correct in certain two dimensional rational CFTs, but that they fail to be exact in other cases. In section 5 we describe the interpretation of the Farey tail approximation in AdS/CFT, where the sum over P SL(2, Z) can be regarded as a sum over saddle point contributions to a semi-classical correlation function. We also give gravitational interpretation of the conformal block when ∆ = c 16 . Much of this section can be read independently from the rest of the paper. We emphasize that, although the general modular structure applies in all dimensions D ≥ 2, in this paper we will describe specific computations of the Farey tail sum only in D = 2.
While this paper was in preparation, we learned that related results will be discussed in [17,18]. We thank these authors for their correspondence.
Crossing symmetry as modular invariance
In this section we will describe the relationship between crossing symmetry and modular invariance. This section is largely a review, although in much of the literature this relationship is not discussed explicitly.
Four-point functions and crossing symmetry
We are interested in the 4-point correlation function
O 1 (z 1 )O 2 (z 2 )O 3 (z 3 )O 4 (z 4 )(9)
of a D dimensional conformal field theory in Euclidean signature. This is a function of 4 points {z a } which transforms covariantly under conformal transformations. We can use these conformal transformations to send z 4 to infinity, z 1 to the origin, z 3 to (1, 0, . . . , 0) and z 2 to a point in the x 1 − x 2 plane. Using complex coordinates, we will denote the resulting position of z 2 by x ∈ C − {0, 1}. 5 The parameter x is the cross-ratio, which can be written in terms of the original four points as
u = z 2 12 z 2 34 z 2 13 z 2 24 = xx , v = z 2 14 z 2 23 z 2 13 z 2 24 = (1 − x)(1 −x)(10)
where z ab = z a − z b . In two dimensions we simply have
x = z 12 z 34 z 13 z 24 .(11)
The configuration space of our four distinct marked points {z a }, modulo conformal transformations, is the twice-punctured plane (or thrice punctured sphere): x ∈ C − {0, 1}. In two dimensions this space is usually denoted M 0,4 , the moduli space of four points on S 2 . We will simply refer to this space as x-space, or cross-ratio space. Our four point function can be written as:
O 1 (z 1 )O 2 (z 2 )O 3 (z 3 )O 4 (z 4 ) = G 0 ({z a }; {∆ a , s a }) G 1234 (x,x).(12)
Here G 0 is a function chosen once and for all, containing only kinematic data, which transforms like a correlation function under conformal transformations. It is convention dependent and depends only on the dimensions ∆ a and spins s a of the operators O a . For operators with spin in D > 2, Equation (12) should include a sum over multiple terms, one for each different tensor structure that can appear in the correlator. We will focus on scalar operators where this is not an issue. For D = 2 we will choose G 0 to contain the correct branch structure encoding the statistics of anyonic operators, so 5 In D > 2 we have the additional freedom to choose Im x ≥ 0, but we will not insist on this here.
G 1234 (x,x) is single-valued. For scalar operators of dimension ∆ i we will choose
G 0 ({z a }; {∆ a }) = a<b |z ab | ∆/3−∆a−∆ b(13)
where ∆ = a ∆ a . This convention has the advantage that G 0 treats the operators democratically, in the sense that it is invariant under permutation of the O a (z a ). Crossing symmetry is the invariance of the four point function under permutations of the operators O a (z a ). The cross ratio transforms under this permutation, and as a result the functions G abcd (x,x) are related by
G 1234 (x,x) = G 1243 x x − 1 ,x x − 1 = G 4231 1 x , 1 x = G 4213 x − 1 x ,x − 1 x = G 3241 1 1 − x , 1 1 −x = G 3214 (1 − x, 1 −x) .(14)
The permutation of the indices, and the action on x, come from the application of six Möbius maps that permute z 1 , z 3 , z 4 . These six permutations form the anharmonic group, which is isomorphic to the symmetric group S 3 . The remaining permutations which interchange z 2 with one of z 1 , z 3 , z 4 give 6
G 1234 (x,x) = G 2143 (x,x) = G 3412 (x,x) = G 4321 (x,x) .(15)
The upper half plane as the universal cover
Conformal blocks are not single-valued functions of the cross-ratio x, so it will be convenient to pass to the universal cover of cross-ratio space. In doing so, we wish to keep the local analytic structure intact. The essential point is that cross-ratio space, viewed as the thrice-punctured sphere, is a Riemann surface. So, as with almost all Riemann surfaces, the universal cover is the upper half plane H + . Concretely, we can take the cross ratio x to be the image of a point τ ∈ H + under the modular λ function:
x = λ(τ ) = √ 2 η(τ /2)η 2 (2τ ) η 3 (τ ) 8 = θ 2 (τ ) θ 3 (τ ) 4 .(16)
The limit x → 0, where the identity block dominates, is given by τ → i∞ on H + . Locally, we can write the inverse of our map as where K(x) is the elliptic integral
τ (x) = i K(1 − x) K(x) (17) (a) τ -plane (b) x-planeK(x) = 1 2 1 0 dt t(1 − t)(1 − xt) = π 2 . 2 F 1 1 2 , 1 2 ; 1; x .(18)
Of course, the inverse (17) is not unique; it describes the infinite set of pre-images τ (x) on H + . This is reflected by the fact that τ (x) is a not a single valued function of x. An advantage of this perspective is that cross-ratio space can now be viewed as a quotient of the upper half plane. In particular, the modular λ function is invariant under the action of the congruence group Γ(2):
λ(τ ) = λ(γτ ), for all γ ∈ Γ(2) .(19)
Here Γ(2) is the index 6 normal subgroup of the modular group P SL(2, Z), which is generated by the two Möbius maps
T 2 : τ → τ + 2, ST 2 S : τ → τ −2τ + 1 .(20)
This means that we can view cross-ratio space as the quotient H + /Γ(2). The group Γ(2) is identified with the fundamental group of cross-ratio space (see fig. 1), C−{0, 1}, which is the free group on two elements. We conclude that we can write the functions G abcd (x) of cross-ratio as functions of G abcd (τ ) on H + which are invariant under Γ(2).
Crossing symmetry as modular invariance
We can now ask how the remaining modular transformations γ ∈ P SL(2, Z) act on the cross ratio x = λ(τ ). The important point is that the generators of P SL(2, Z) act as crossing transformations:
T · x = x x − 1 , S · x = 1 − x.(21)
Indeed, the quotient P SL(2, Z)/Γ(2) = S 3 is precisely the anharmonic group described above, which acts as the six nontrivial Möbius maps on x given in equation (14). So crossing symmetry implies that the G abcd (τ ) collectively transform into one another under modular transformations. Before considering the general case, let us first consider the special case when the four external operators O a are identical. In this case, swapping identical points will leave the correlator invariant. So G abcd (x) ≡ G(x) is invariant under the anharmonic group. Thus, as a function on the upper half plane, G(τ ) must be invariant under the full modular group P SL(2, Z). Any such function can be written as a function of the j-invariant
j(τ ) = 256(1 − x(1 − x)) 3 x 2 (1 − x) 2(22)
which assigns a unique complex number to each point on the fundamental domain H + /SL(2, Z). In cross-ratio space, this fundamental domain is {x : |x−1| < 1, Re(x) < 1/2}. 7 When some, but not all, of the operators O a (z a ) are identical, the function G abcd (τ ) will be invariant under a subgroup Γ ⊆ P SL(2, Z) which contains Γ(2). For three identical operators it is invariant under P SL(2, Z). With two identical operators, or two pairs of identical operators, it is invariant under an index 3 subgroup of P SL(2, Z) (the congruence subgroup Γ 1 (2)), which itself contains Γ(2) as an index 2 subgroup.
It is useful to reformulate this slightly, by regarding the G abcd (τ ) collectively as the components of a vector-valued modular function G(τ ). This means that the components of G(τ ) will in general transform into one another under a modular transformation. This has the effect of restricting the domain to the fundamental region of H + /SL(2, Z), at the expense of introducing multiple functions that map into one another under modular transformations. In particular, we have
G(γτ ) = σ(γ) G(τ )(23)
where σ is a representation of the permutation group. The representation is six dimensional in the general case, three dimensional when two operators or two pairs are identical, or one dimensional when three or all four operators are identical. It is useful to write this all out explicitly. Expressed in terms of τ , crossing symmetry is G abcd (τ ) = G bacd (τ + 1) = G adcb (−1/τ ) (24) along with
G abcd (τ ) = G badc (τ ) = G dcba (τ ) = G cdab (τ ).(25)
Arranging the independent functions in a six-dimensional vector
G(τ ) = (G 1234 (τ ), G 2134 (τ ), G 4132 (τ ), G 1432 (τ ), G 2431 (τ ), G 4231 (τ )) t(26)
the crossing relations can be written as
G(τ + 1) = G(τ ) .(27)
This is a reducible representation of the anharmonic group S 3 . It is the sum of the trivial representation, the one-dimensional sign representation and two copies of the 'standard' two-dimensional representation. One basis for this decomposition is
G triv (τ ) = 1 √ 6 (G 1234 (τ ) + G 2134 (τ ) + G 4132 (τ ) + G 1432 (τ ) + G 2431 (τ ) + G 4231 (τ )) G sign (τ ) = 1 √ 6 (G 1234 (τ ) − G 2134 (τ ) + G 4132 (τ ) − G 1432 (τ ) + G 2431 (τ ) − G 4231 (τ )) G std 1 (τ ) = 1 √ 3 G 1234 (τ ) + ωG 4132 (τ ) + ω 2 G 2431 (τ ) ω 2 G 2134 (τ ) + G 1432 (τ ) + ωG 4231 (τ ) G std 2 (τ ) = 1 √ 3 ωG 2134 (τ ) + G 1432 (τ ) + ω 2 G 4231 (τ ) G 1234 (τ ) + ω 2 G 4132 (τ ) + ωG 2431 (τ )
where ω = e 2πi/3 . Under the modular group, the trivial representation is invariant, the sign representation picks up a (−1) from the action of S or T , and the standard representation in the chosen basis is
ρ std (T ) = 0 ω ω 2 0 ; ρ std (S) = 0 1 1 0 .(28)
A correlation function be described as the collection of four vector valued modular functions for P SL(2, Z), transforming in the above representations. When some of the operators are identical, we may not need all these representations. If all four operators are identical, three representations identically vanish, and only the trivial representation remains. If O 1 ≡ O 2 and O 3 ≡ O 4 , the sign representation vanishes and the two copies of the standard representation are proportional, leaving the trivial representation and one two-dimensional representation.
The description of crossing symmetry as modular transformations is not particularly useful when discussing the correlation functions themselves; we have just replaced the anharmonic group of crossing symmetries with the infinite dimensional modular group P SL(2, Z). For the four point functions themselves, this extra structure is not necessary. The advantage of the present approach is that -because H + is the universal cover of cross-ratio space -the conformal blocks are single valued functions of τ , even though they are multiply valued functions of x.
A Poincaré series for correlation functions
We can now describe our construction of a candidate four point function as a sum over the modular group P SL(2, Z). The expansion of the four point function (12) as a sum over intermediate states takes the form
G abcd (x,x) = p C p ab C p cd F ab,cd p (x,x),(29)
where F ab,cd p (x,x) is the conformal block associated with the primary operator O p . We are still working in general D ≥ 2, although we will later specialize to the case D = 2 where F ab,cd p (x,x) will be the product of left-and right-moving Virasoro blocks. We have absorbed into F ab,cd p (x,x) the usual factors of x, so that
F ab,cd p (x,x) ∼ |x| ∆p−∆/3 + · · ·(30)
as x → 0. In the case of a four point function of identical scalar operators, the conformal block defined here is |x| ∆p times the conformal block in eq. (1). If we wish to approximate our four point function at x → 0, it is sufficient to include only the contributions to G abcd from low-lying operators, i.e. to take
G abcd (x,x) = F light ab,cd (x,x) + · · ·(31)
where
F light ab,cd (x,x) = ∆p≤∆ light C p ab C p cd F ab,cd p (x,x)(32)
includes only contributions from operators below some dimension ∆. For D = 2, even the Virasoro vacuum block contribution is non-trivial.
In the notation of the previous section, where we assemble the six G abcd into a vector according to (26), we can write this more succinctly as
G(x,x) = F light (x,x) + · · ·(33)
In our expansion (29) the four point function G abcd (x,x) is a single valued function of x. The individual conformal blocks, however, are not. They have non-trivial mon-odromies as one moves around in cross-ratio space. Of course, this branch structure will disappear when we perform the sum over p in (29) to obtain the single valued G abcd .
This branch structure means that the conformal blocks should be regarded as function of the covering coordinate τ rather than x. From (21) we see that the monodromy around x = 0 is generated by the shift T 2 : τ → τ + 2 and that the monodromy around x = 1 is generated by ST 2 S. This allows us to completely unwrap the branch structure and view F ab,cd p (τ,τ ) as a single valued function of τ . Our approximate four point function (33) should therefore really be written as an equation on H + , as
G(τ,τ ) = F light (τ,τ ) + · · ·(34)
Our goal is then to ask how this can be completed to a crossing symmetric four point function. In particular, we will fix the · · · terms in (34) by demanding that the four point function obeys
G(γτ, γτ ) = σ(γ) · G(τ,τ )(35)
where σ(γ) is the six dimensional representation of the modular group defined in equation (27). Our ansatz is that we simply average over the modular group, by setting
G candidate (τ,τ ) = 1 N γ∈P SL(2,Z) σ −1 (γ) · F light (γτ, γτ ).(36)
Here N is a normalization constant, which is fixed by demanding that the limit x → 0 (τ → i∞) matches with the light limit (34). Provided the sum converges, G candidate automatically obeys (34) and (35). The ansatz (36) should be viewed as a precise version of the statement that heavy states arise through the propagation of light states in a dual channel.
In the next section we will unpack this statement and perform specific computations with this ansatz for two dimensional minimal model CFTs. Before proceeding, however, we should make a few comments. Sums of this sort appear frequently in number theory, both in the holomorphic and non-holomorphic settings. They have also been considered extensively in the context of three dimensional gravity. One important feature is that the convergence of the sum (36) is not guaranteed, and the regularization can be quite subtle (see e.g. [8,10,13,14]). In some cases the sum can only be defined using zeta function regularization, and the normalization constant N is formally infinite. In some of the explicit computations performed below, however, the sum will collapse to a finite sum in an obvious way, so convergence will not be an issue; a similar phenomenon was noted in [15].
Minimal models
We will now construct correlation functions by performing the modular average explicitly in some unitary minimal models in D = 2. We begin by recalling a few facts on the 2D minimal models 8 . For a pair of coprime integers p and p with p > p , the minimal model M (p, p ) has central charge
c = 1 − 6(p − p ) 2 pp .(37)
The allowed holomorphic dimensions of Virasoro primary operators are labelled by integers r, s, as
h (r,s) = (pr − p s) 2 − (p − p ) 2 4pp , 1 ≤ r < p and 1 ≤ s < p(38)
with the redundancy h (r,s) = h (r+p ,s+p) = h (p −r,p−s) . We may denote such a primary by φ (r,s) in a context where only the holomorphic properties are important. The physical spectrum consists of a collection of primary operators with appropriate holomorphic and antiholomorphic dimensions (constrained by modular invariance of the torus partition function), for example the diagonal series, for which each scalar with allowed dimension h =h = h (r,s) appears exactly once. We shall concentrate almost exclusively on the unitary series, for which p = p − 1.
The section will begin with the discussion of some useful mathematical structure of the space of conformal blocks, action of the modular group, crossing symmetric fourpoint functions and the construction of the modular average. This will specialise the discussion in the previous section to the case with a finite number of primary operators. Next, we will move on to examples in minimal models. At the end of the section, we will present an alternative group theoretic perspective on the modular sum, motivated by results for compact groups.
Mathematical structure
In a two dimensional CFT a conformal block can be written as a product of holomorphic and anti-holomorphic factors, as
F ab,cd p (x,x) = F ab,cd p (x)F ab,cd p (x).(39)
The holomorphic and anti-holomorphic blocks F andF depend only on the left-and right-moving dimensions of the operators, respectively. If we fix the external operators abcd, we can think of the holomorphic blocks as elements of a vector space B (additionally labelled by the external operator dimensions, though we will leave this implicit) with basis F ab,cd p labelled by the holomorphic dimension of the exchanged operator. 9 The antiholomorphic blocks live in the complex conjugate vector spaceB. The correlation function is a sum of products of holomorphic and antiholomorphic blocks, which in this language is an element of B ⊗B. The coef-ficients in the given basis are simply the products of OPE coefficients C p ab C p cd (summed over all exchanged operators with the same dimensions). We may give the correlation function by arranging these in a matrix C, with rows and columns labelled by holomorphic and antiholomorphic dimensions respectively. Scalar operators appear on the diagonal, and operators with spin away from the diagonal. Explicitly we have
G(x,x) = h,h F h (x)C hhFh (x), where C hh = Op: hp=h hp=h C p ab C p cd(40)
where the last sum runs over the OPE coefficients of all operators O p with the given holomorphic and antiholomorphic dimensions.
Concentrating firstly on the case of identical external operators, modular transformations act linearly on B, turning B into a representation of the modular group. On the correlation functions, living in B ⊗B, the modular group acts by γ : C → γ · C · γ † , so, in particular, crossing symmetric correlation functions obey γ · C · γ † = C, for γ in the appropriate representation of P SL(2, Z). Now, so far in the discussion, B could include exchange of any dimension h, in which case it is an infinite-dimensional space, in the worst case perhaps even the space of all holomorphic functions on the upper half plane. But if we start with minimal model central charge and dimensions, the action of the modular group only produces other dimensions h r,s : the action is highly reducible, and the finite-dimensional subspace including only the h r,s in the exchange is invariant under the action. In many cases, including examples below, we can further reduce the representation so that some h r,s do not appear, taking B to be the minimal invariant space including our 'seed'. This is the way in which the exchange spectrum and fusion rules appear in our construction: we do not put these data in by hand, but rather they come out as the set of exchanged blocks generated by modular images of the seed.
With the appropriate representation in hand, our proposed solution to crossing is to start with a seed and sum over images:
C candidate ∝ γ∈P SL(2,Z) γ · C seed · γ † .(41)
In the simplest case, the seed C seed contains just the contribution of the vacuum block.
In general the Virasoro vacuum block will be invariant under some subgroup Γ stab ⊆ P SL(2, Z). In this case we need sum only over the coset
C candidate ∝ γ∈P SL(2,Z)/Γ stab γ · C seed · γ † .(42)
In the simple cases considered below, we will see that Γ stab is often a finite index subgroup of P SL(2, Z). Thus the sum has only a finite number of terms and can be computed explicitly.
It is straightforward to generalise this discussion when the external operators are not identical: we must simply add an additional label to the matrix C to identify the permutation of the external operators (or the irreducible representation of S 3 , as in eq. (26)), and allow the modular group to act additionally by permuting these labels. Abstractly, the space B breaks up into the tensor product of a representation of P SL(2, Z) and a representation of the anharmonic group S 3 .
Examples
In this section, we will perform the modular sum in eq. (41) in three minimal model examples. In these cases, the seed C seed will be taken to be the contribution of the Virasoro vacuum block. We will study the p = 4, 5 and 12 diagonal minimal models. The first two, M (4, 3) and M (5, 4), are the Ising and tricritical Ising models. The third one, M (12,11), is a coset model which will be described below. The Ising and tricritical Ising models are the simplest examples in the diagonal series where the modular average of the vacuum block correctly reproduces all of the identical operator four point functions, allowing us to uniquely determine the three point coefficients. The diagonal M (12, 11) is included as an example where the modular average of the vacuum block alone fails to reproduce the three point coefficients.
Ising model
The Ising model is the p = 4 unitary minimal model, with central charge c = 1 2 . The spectrum includes three scalar primary operators, the identity 1 ≡ φ (1,1) , the spin field σ ≡ φ (1,2) and the energy density ≡ φ (2,1) with dimensions
h (1,1) = 0, h (1,2) = 1 16 , h (2,1) = 1 2(43)
andh = h. The fusion rules are
σ × σ = 1 + , σ × = σ, × = 1.(44)
We will first consider the correlation functions of identical operators,
σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 ) , (x 1 ) (x 2 ) (x 3 ) (x 4 ) .(45)
Of these two, the fusion rule × = 1 implies that the (holomorphic times antiholomorphic) vacuum block is already modular invariant, so the sum over P SL(2, Z) is trivial. We will therefore focus on the σ-four-point function.
For the mixed four-point functions, we will perform a similar computation where the following three correlators
σ(x 1 )σ(x 2 ) (x 3 ) (x 4 ) , σ(x 1 ) (x 2 ) (x 3 )σ(x 4 ) , σ(x 1 ) (x 2 )σ(x 3 ) (x 4 )(46)
are assembled into the components of a vector-valued modular function.
γ γ · C seed · γ † 1 1 0 0 0 S 1 8 4 2 2 1 T S 1 8 4 −2i 2i 1 T 2 S 1 8 4 −2 −2 1 T 3 S 1 8 4 2i −2i 1 ST 2 S 1 4 0 0 0 1
Four-point function of σ:
For the σ operator four point function, there are two relevant blocks:
F σσσσ 1 = 1 √ 2 1 + √ 1 − x ((1 − x)x) 1/12 , F σσσσ = √ 2 1 − √ 1 − x ((1 − x)x) 1/12 .(47)
In the basis {F σσσσ 1 , F σσσσ } the generators S and T are represented by
T = e −iπ/12 1 0 0 i , S = 1 √ 2 1 2 1 2 −1 .(48)
This matrix notation 10 means that S acts on
F j (x) by F i (1 − x) = j S ji F j (x).
We refer the reader to the appendix for details on calculation of these blocks and matrices.
The vacuum block has a finite orbit under the action of the modular group, represented as the six matrices listed in table 1, and the vacuum block is invariant under the index 6 subgroup Γ stab = Γ 1 (4) ⊆ P SL(2, Z). 11 Computing the sum of these six terms we find the correlation function
G candidate = |F σσσσ 1 | 2 + 1 4 |F σσσσ | 2 ,(49)
where the overall normalisation is fixed by demanding that the OPE with the identity is unity, for example from the behaviour as x → 0. Equation (49) is precisely the correct answer for the σσσσ correlation function. We note that the modular average has correctly determined that σσ fuses only to the identity and (h,h) = ( 1 2 , 1 2 ) scalars, and given the relative coefficient of the |F σσσσ 1 | 2 and |F σσσσ | 2 terms. This coefficient is the three point function coefficient
C 2 σσ = 1 4 ,(50)
which is the only non-trivial three point coefficient for the Ising model. In this way we see that the modular average has exactly reproduced the Ising model three point coefficients, taking only the central charge and dimensions of σ as inputs.
Mixed four-point functions:
We now consider the σσ four-point function, the only nontrivial example for which the vacuum block is exchanged in some channel. One might expect that we need to consider a nine-dimensional space of blocks, for the three operator dimensions that are exchanged and the three independent permutations of operators, giving the S,T and U channel for the exchange. However, in this case, only three of these blocks are produced as modular images of the identity exchange. These are one block for each channel, with the unique exchange allowed by the Ising fusion rules.
Explicitly, casting the results of [20,[22][23][24][25] in our conventions, the relevant blocks are
F σσ 1 (x) = 1 − x 2 (1 − x) 5/16 x 3/8 F σ σ σ (x) = 1 + x (1 − x) 3/8 x 5/16 (51) F σ σ σ (x) = 1 − 2x ((1 − x)x) 5/16 .
It is easy to see from these expressions that the Γ(2) subgroup leaving the operators in the original order, generated by monodromies around x = 0 (T 2 ) and around x = 1 (ST 2 S), act on the blocks with a phase only. In the basis
F = {F σσ 1 , F σ σ σ F σ σ σ } ,(52)
the generators of the full modular group act as
T = (−1) 13/8 0 0 0 0 (−1) 27/16 0 (−1) 27/16 0 , S = 0 2 0 1 2 0 0 0 0 −1 .(53)
The vacuum block in the S-channel, represented by the matrix with one in the top left and zero elsewhere, is left invariant under the subgroup Γ(2), as well as under T , which together generate the index three congruence subgroup Γ 1 (2). The modular sum therefore has three terms, and each term reproduces the correlation function, expanded in different channels (with only one block appearing in each channel):
σσ = |F σσ 1 | 2 ; σ σ = 1 4 |F σ σ 1 | 2 ; σ σ = 1 4 |F σ σ 1 | 2 .(54)
This again reproduces the correct OPE coefficient C 2 σσ = 1/4, as well as the fact that this is the only nontrivial fusion.
The above computation can be recast in the language of representations of the anharmonic group, as described in section 2. Since we have a pair of identical operators, the computation will involve only the trivial representation and one copy of the standard representation of S 3 .
Tricritical Ising model
Next in the unitary series is the tricritical Ising model, with c = 7/10. The primary operators and dimensions are
1 : h (1,1) = h (3,4) = 0, : h (1,2) = h (3,3) = 1 10 , : h (1,3) = h (3,2) = 3 5 ,(55)
:
h (1,4) = h (3,1) = 3 2 , σ : h (2,1) = h (2,4) = 7 16 , σ : h (2,2) = h (2,3) = 3 80 .
The four-point functions of all five nontrivial scalars of the model are reproduced from a modular sum with the Virasoro vacuum block as a seed. This is trivial for , since the fusion rule × = 1 implies that the vacuum block alone is modular invariant, and the σ case is very similar to the case of σ in the Ising model. For the other three, the sum does not truncate and we need to include an infinite number of terms 12 .
Modular-averaging four-point functions of identical operators gives all OPE coefficients of the form C OaOaO b . The others can be obtained by considering correlation
functions of two pairs of identical operators O a O a O b O b .
In the tricritical Ising model, the modular average of the vacuum block reproduces all OPE coefficients by studying these two types of four-point functions. We note that for other mixed correlators, for example O a O a O a O b , the identity operator does not appear in the decomposition in terms of the blocks in any channel. In these cases, the light "seed" may be taken as the lightest operator appearing in any of the three channels.
Four-point function of σ
For the σ -four-point function, the modular group acts on the subspace spanned by the vacuum block and the exchange block, with the generators acting as
T = e −7iπ/12 1 0 0 −i , S = 1 √ 2 1 8 7 7 8 −1(56)
taking the result from the appendix. The modular images of the vacuum block generate six distinct terms, and after normalisation this gives the correlation function
G candidate = |F 1 | 2 + 49 64 |F | 2 .(57)
This is the correct correlation function, and gives the right value of the OPE coefficient C σ σ .
Four-point function of
For the four-point function, the action of the modular group on the vacuum block generates only one other internal dimension, from the block. The representation acting on this is
T = e −2iπ/15 1 0 0 e 3iπ/5 , S = √ 5−1 2 √ 5−1 2 Γ( 1 5 )Γ( 8 5 ) Γ( 2 5 )Γ( 7 5 ) Γ( 2 5 )Γ( 7 5 ) Γ( 1 5 )Γ( 8 5 ) − √ 5−1 2 .(58)
In this case, the orbit of the vacuum block under the modular group appears to be infinite, so the sum does not truncate. We can, however, compute the sum numerically and check that it converges to the correct OPE 13 . For our numerical checks, we performed the sum over the distinct images produced by products of S and T acting on F 1 , organised by the number of generators in the element of the modular group. Specifically, defining the length of a generator γ as the minimal k such that we may write
γ = S m 1 T n 1 S m 2 T n 2 · · · S m k T n k(59)
for m i = 0, 1 and non-negative integers n i up to the order of T , we sum over all distinct 13 The sum does not converge in the standard sense, since the individual terms do not tend to zero. However, since we are normalising the result by an overall factor in the end, we can proceed as follows: we first choose an order for the terms, normalise the partial sums by an appropriate factor, and take the limit as we add more and more terms in the chosen order. We expect that an unfortuitous choice of order could lead to any answer, as for conditionally convergent sums, but the hope is that any natural choice of ordering gives the same finite answer. images of the seed, taking words of length at most k max :
G candidate = N (k max ) −1 length(γ)≤kmax |F 1 (γτ )| 2 distinct = |F 1 | 2 + (b(k max )F 1 F * + b(k max ) * F * 1 F ) + c(k max ) |F | 2 .(60)
Here the coefficients of the blocks, after normalising the identity contribution to unity, are given by a complex number b(k max ) and a real number c(k max ). The subscript 'distinct' in the sum is to indicate that we are only summing over distinct terms. With this method, taking k max up to 4, which generates approximately 10 6 distinct terms in the sum, the numerical results are consistent with the sum reproducing the correct OPE coefficients. The off-diagonal terms are small, with |b(k max = 4)| ≈ 10 −9 , while for c(k max ) the result is around 2% from the known exact value, and approaching it as more terms are added as shown in fig. 2:
G candidate | kmax=4 ≈ |F 1 | 2 + 0.381 |F | 2 + O(10 −9 ).(61)
The diagonal M (12, 11) minimal model
Unitary minimal models sometimes have Virasoro scalar primaries with even-integer dimension. In this case the T -matrix will have repeated eigenvalues, and hence invariance under T does not require the four-point function to be a diagonal sum of conformal blocks squared, so non-scalar operators may be exchanged. In this case, there may be more than one solution to the crossing equations 14 . On the other hand, the modular sum of the vacuum block yields a unique crossing-symmetric answer. We can then ask whether the modular average of the vacuum block reproduces the correct three point coefficients.
We will answer this question in the M (12, 11) unitary minimal model, which has central charge 21/22. The diagonal model can be realised as the su(2) coset model 15
su(2) k ⊕ su(2) 1 su(2) k+1(62)
at level k = 9, with 55 Virasoro scalar primaries. We will focus on the φ (1,4) operator, which fuses with itself to four primaries
φ (1,4) × φ (1,4) = 1 + φ (1,3) + φ (1,5) + φ (1,7) (63) with dimensions h (1,4) = 31/16, h (1,3) = 5/6, h (1,5) = 7/2, h (1,7) = 8,(64)
so, in particular, h (1,7) is an even integer.
Four-point function of φ (1,4) The T matrix corresponding to the four point function of φ (1,4) is
T = e − 4 3 πih (1,4) 1 0 0 0 0 e πih (1,3) 0 0 0 0 e πih (1,5) 0 0 0 0 e πih (1,7) = e −31πi/12 1 0 0 0 0 e 5πi/6 0 0 0 0 −i 0 0 0 0 1 . (65)
Invariance under T then allows for off-diagonal terms in the corners of the C matrix, restricting the form of the four point function to be
G = |F 1 | 2 + C 1 F (1,3) 2 + C 2 F (1,5) 2 + C 3 F (1,7) 2 + D 1 F 1 F * (1,7) + D 2 F * 1 F (1,7) . (66)
The S matrix is rather complicated analytically, but its numerical value is 16
S ≈ 0.2989 0.1863 0.0922 1.3807 1.3098 0.5176 0. −4.43 3.6153 0. −0.7071 6.1137 0.2414 −0.1102 0.0273 −0.1094 .(67)
14 See [28,29] for discussions related to non-uniqueness of solutions to crossing equations. 15 For detailed discussion, see Chapter 18.3 of [20]. 16 For this fourth-order correlator, we used the Mathematica codes in [30] to obtain the S matrix. Note that our matrix S is the transpose of that in [30], as explained previously in footnote 10.
Imposing S-invariance allows a one-parameter family of crossing invariant solutions: 7) .
G ≈ |F 1 | 2 + (7.031 − 23.7794D 1 ) F (1,3) 2 + (39.2118 + 66.3091D 1 ) F (1,5) 2 + (0.1749 − 0.2957D 1 ) F (1,7) 2 + D 1 F 1 F * (1,7) + F * 1 F (1,
The diagonal model corresponds to D 1 = 0. Similar to the previous section, the modular sum over the vacuum block can be done numerically. We have performed the sum up to k max = 5, which generates approximately 10 4 distinct terms in the sum. The result is that the modular sum yields the modular invariant OPE with D 1 ≈ 0.12 and, in particular, none of the OPE coefficients implied by G candidate converge to zero.
Given this, if we wish for the modular sum to produce the correct correlation function, it must be in a model whose spectrum contains the φ (1,4) scalar, as well as all the operators appearing in the conformal block expansion: the φ (1,3) , φ (1,5) and φ (1,7) scalars, as well as the chiral φ (1,7) spin 8 current.
There are three modular invariant spectra for the M (12, 11) minimal model, corresponding to (A 10 , A 11 ), (A 10 , D 7 ) and (A 10 , E 6 ) in the ADE classification of minimal model spectra [31][32][33] (reviewed in [20]), these being the pairs of simply laced root systems with dual Coxeter numbers (11,12). The A 11 model is the diagonal one, containing only scalars, so in particular does not have the spin 8 current. The current is also absent in the D 7 model, which in addition lacks a φ (1,4) scalar, so this correlation function is not even part of that theory. Finally, the E 6 model has a spin 8 current in the spectrum, but no φ (1,3) scalar. There is therefore no model containing all the required operators for this correlation function to appear in a modular invariant theory.
This example illustrates that if one takes only the modular average of the vacuum block one will not always correctly reproduce all three point coefficients. This may be improved by adding more information to the seed before performing the sum: for example, the correct correlation function for the diagonal model can be obtained in this instance by including the correct OPE coefficient for the h = 8 scalar in the seed, as well as the vacuum contribution.
A group theoretic perspective
We will now describe a somewhat more mathematical reformulation of the above discussion. This will motivate a redefinition of the modular average, allowing it to be computed more rigorously and systematically for infinite sums.
Abstractly, we can formulate our problem as follows. We have the vector space V spanned by conformal blocks, on which a group Γ acts in some representation R (Γ is P SL(2, Z) if it acts faithfully, modulo the kernel of the representation if not). In the two-dimensional case, V = B ⊗B, and R is the tensor product of a representation on B and its conjugate. A crossing symmetric correlation function is a vector v ∈ V which is invariant under the action of Γ. Our strategy is to start with a choice of 'seed' vector v 0 ∈ V (the vacuum block in the above minimal examples) and to sum over all
its images in Γ: v ∝ γ∈Γ R(γ)v 0 .(69)
Note that (assuming for now that all the relevant sums converge) the dependence on the seed v 0 factors out, so we can solve the problem by finding the linear map P R associated to the representation R defined by
P R ∝ γ∈Γ R(γ).(70)
We will show that for finite groups, P R is a projection canonically associated to the representation R. We can characterise this projection more generally, including cases of relevance in our discussion where the convergence is less obvious. This could be regarded as an alternative proposal to construct correlation functions, motivated by the sum over Γ, and equivalent in many cases, but rigorously defined and sometimes more easily calculable. Let us begin by taking Γ to be a finite group 17 so the sum unambiguously makes sense. Now we can make use of the following standard results in the representation theory of finite groups:
• Every finite-dimensional representation is equivalent to a unitary representation.
• Every finite-dimensional unitary representation is completely reducible (i.e. it decomposes as a direct sum of irreducible representations).
• The grand orthogonality theorem, which states that for irreducible unitary representations R 1 , R 2 , the sums over Γ of matrix elements are orthonormal, in the sense that
1 |Γ| γ∈Γ R 1 (γ) * ij R 2 (γ) i j = 0 R 1 , R 2 inequivalent 1 dim(R 1 ) δ ii δ jj R 1 = R 2(71)
As an immediate corollary of this last statement, choosing R 1 to be the trivial representation, we find that the sum over the group of a matrix element of a nontrivial irreducible unitary representation vanishes. From the first two of the quoted results, we may choose a basis in which R is block-diagonal, with each block being an irreducible unitary representation. There are a number of trivial representations appearing in this decomposition, and the subspace spanned by these representations is exactly the subspace of V left invariant under the group action; it follows that the image of P R must be contained in this subspace. The orthogonality theorem then implies that in this basis where the representation is unitary, P R = 1 |Γ| Γ R(γ) is the diagonal matrix with ones on the diagonal where the 17 More generally, we could take Γ to be compact, with Haar measure µ, and define the average over the group as 1 µ(Γ) Γ dµ. Then the following discussion is essentially unaltered.
trivial representations live and zeroes elsewhere. More abstractly, the conclusion can be simply stated: There is an inner product on V which is invariant under the action of Γ. The sum over Γ is equivalent to the orthogonal projection, with respect to this inner product, onto the invariant subspace of V . This inner product is not quite unique (there is a GL(k)/U (k) choice for each inequivalent irreducible representation appearing k times, for example, an overall scale if k = 1), but the projection is independent of which is chosen. In particular, this definition of P R makes sense for any group, as long as R is equivalent to a unitary representation. We can refine this discussion further in the case of 2D CFTs using the additional structure implied by the factorisation of the blocks, so V = B ⊗B. The action is then by conjugation, so R is the tensor product of some representation R 0 on B and its conjugate R * 0 (or equivalently the dual of R 0 if it is a unitary representation). In this product, the identity representations in R appear in a simple way when R 0 is unitary. This is because Schur's lemma implies that for unitary irreps R 1 and R 2 , the identity appears in the decomposition of the tensor product R 1 ⊗ R * 2 into irreps exactly once when R 1 and R 2 are equivalent, and not at all when they are inequivalent. So writing R 0 = ⊕ i k i R i , where R i denote inequivalent irreps and k i their multiplicities, the dimension of the invariant subspace is i k 2 i . When we have this decomposition into irreducible representations, after changing to the basis where R 0 is block diagonal, the projection acts on the matrix C in a simple way. The elements of a block corresponding to inequivalent irreps acting from the left and right gets set to zero, while the blocks with the same representation acting on both sides (appearing on the diagonal in particular, and off the diagonal when there are multiple copies of the same representation) get projected to a multiple of the identity in that block, while preserving the trace.
The crucial requirement is that the representation R 0 is equivalent to a unitary representation. This is always true when Γ is a finite group. Even in the infinite case there may exist a basis in which the representation is unitary. Indeed, we will show that this is always the case when one considers identical operators in unitary minimal models, and restricts to the minimal subspace of exchange operators generated by the action on the vacuum block. We will give examples to show that relaxing either assumption may lead to a representation which is not equivalent to a unitary one.
To see this, note that when the group acts by conjugation, the solutions to crossing satisfy
γ · C · γ † = C ∀γ ∈ Γ(72)
which means that C is a Hermitian form on B, invariant under the action of Γ, with Hermiticity of C guaranteed by reality of the correlation function. The only basisindependent information in such a form is its signature, the number of positive, negative and zero eigenvalues. Thus there is a basis where the form has only 1, −1 and 0 along the diagonal. If there exists a form with definite signature, i.e. with all eigenvalues having the same sign, then in the basis where the form is proportional to the identity, the invariance under Γ is equivalent to unitarity of the representation. Then we may follow the logic of the above, decomposing R 0 into unitary irreps, and projecting onto the invariant subspace. This happens for correlation functions of identical scalars in unitary minimal models, where we include only a subset of the exchange operators. This is because there is always a positive definite crossing-invariant solution, given by the diagonal minimal model, with squares of OPE coefficients of the scalars coupling to the external operator along the diagonal. Unitarity guarantees that the OPE coefficients are real, so there are no negative eigenvalues, and including only the internal operators appearing when the external operator fuses with itself (a subspace guaranteed to be invariant under Γ) ensures that there are no zero eigenvalues. Note that this only guarantees that the representation is unitary, not that the projection of the vacuum block will reproduce the diagonal model OPE coefficients, as can be seen from the M (12, 11) example. Rescaling the basis vectors by absorbing OPE coefficients again makes the representation of the modular group unitary in this case, and in fact makes the S matrix look much simpler:
T = e −31πi/12 1 0 0 0 0 e 5πi/6 0 0 0 0 −i 0 0 0 0 1 , S = √ 3−1 √ 6 1 √ 3 − 1 3 1 √ 3 1 √ 3 1 √ 3 − 1 3 2 − √ 3 0 − 2 3 √ 3 − 1 1 √ 3 0 − 1 √ 2 1 √ 6 1 √ 3 − 2 3 √ 3 − 1 1 √ 6 √ 3−2 √ 6 .
Now we still have the freedom of a unitary change of basis, while keeping the representation unitary, which we will use to show that this representation is reducible. With change of basis matrix
P = − 1 √ 3 0 0 2 3 0 1 0 0 0 0 1 0 2 3 0 0 1 √ 3 ,(75)
writing γ = P γP −1 , the generators become
T = e −7πi/12 1 0 0 0 0 e 5πi/6 0 0 0 0 −i 0 0 0 0 1 , S = − 2 − √ 3 − √ 3 − 1 0 0 − √ 3 − 1 2 − √ 3 0 0 0 0 − 1 √ 2 1 √ 2 0 0 1 √ 2 1 √ 2 (76)
and after this change of basis, the identity block is represented by
C seed = P 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 P † = 1 3 0 0 − √ 2 3 0 0 0 0 0 0 0 0 − √ 2 3 0 0 2 3 .(77)
After the projection, this will turn into a matrix proportional to the identity in the upper left 2 × 2 block, and twice the identity in the lower right 2 × 2 block. After projecting and changing basis back to the original one, we find an exact result for the OPE coefficients coming out of the modular sum (without the work of performing any sum). Translating to the notation used in section 4.2.3, where the crossing-invariant correlation functions are parametrised by D 1 , the OPE coefficient with the spin 8 current, we obtain D 1 = 7499023/63406080 ≈ 0.118, consistent with the truncated numerical sum. This provides good evidence that the sum, the way we have defined it, does indeed give the same result as the group-theoretic method.
Example 3: Yang-Lee model
To illustrate what happens when we relax the unitarity condition, consider the M (5, 2) minimal model, corresponding to the Yang-Lee edge singularity. This model has central charge c = −22/5, and one primary operator apart from the identity, the scalar Φ with h = −1/5. For the four-point function of Φ, we can compute the action of the modular group on the blocks as before, finding
T = e −2πi/5 1 0 0 e −iπ/5 , S = −ϕ −ϕ/α α ϕ ,(78)
where ϕ = 1+ . This representation has a unique invariant hermitian form (up to scale), with the OPE coefficients C 2 ΦΦ1 = 1 and C 2 ΦΦΦ = −α 2 /ϕ on the diagonal. Since the nontrivial OPE coefficient is imaginary in this model, this form has indefinite signature.
Example 4: Ising
We have already commented that the vacuum block alone is modular invariant for the four-point function of the operator in the Ising model, so no sum is required to find a solution to crossing, since × fuses only to the identity. Despite this, we may still consider the action of the modular group on both the identity and blocks to illustrate the general pattern. The representation is given by
T = e −2πi/3 0 0 −1 , S = 1 10Γ( 2 3 ) 2 9Γ( 1 3 ) 0 −1 (79)
which is reducible, but not completely reducible. This means, in particular, that it cannot be equivalent to a unitary representation, and indeed the only invariant Hermitian form is degenerate, with the exchanged block being a zero eigenvector.
Semiclassical limit
We have motivated the main construction of the paper -a candidate correlation function obtained by summing a conformal block over all channels -as an abstract method for solving the constraints of crossing. We will now explain how the same construction follows naturally from considerations of a semiclassical gravity dual. We will focus again on two dimensional CFTs, and consider gravity in three dimensions with some 'heavy' bulk particles quantised via the worldline formalism. Correlation functions are then found by integrating over all possible worldlines of these particles. In the semiclassical limit this is dominated by solutions to the classical equations of motion, including the backreaction of the particles on the geometry. The action of a classical solution (along with perturbative corrections) will compute a conformal block in the dual CFT, and different channels correspond to different classical solutions. The sum over channels is therefore the same as the sum over saddle points in the bulk path integral.
It is not obvious that the crossing images under P SL(2, Z) for a given conformal block should be in one-to-one correspondence with classical bulk solutions. We will focus on one example -the four-point function of a ∆ = c/16 scalar -where it is possible to classify all the classical solutions and make this correspondence explicit. This will also give us a bulk knot theoretic interpretation of the channels of the conformal block, in terms of 'rational tangles', and make close contact with the older idea of the partition function Farey tail.
The conformal block Farey tail as first-quantised gravity
We will consider a CFT in two dimensions with a semiclassical bulk dual. This means, in particular, that the central charge c = 3 AdS 2G N 1 is large so that the Planck length is small in AdS units. The spectrum of primaries in the theory must also be constrained (see e.g. [34][35][36][37][38]). In particular, there are some 'light' primaries, whose dimension does not scale with c, which are described by perturbative bulk fields 18 . There are also heavy states with dimension of order c, but with ∆ < c/12. These are dual to massive particles in the bulk, which backreact on the geometry to form conical defects. Finally we have states with ∆ > c/12, corresponding to bulk black hole microstates, with asymptotic density of states given by the Cardy (or Bekenstein-Hawking) formula. Our strategy will be to quantise the light bulk fields (including the graviton) as well as the heavy bulk particles by computing perturbatively a path integral with an appropriate action. The contribution of the black hole states will then follow from a non-perturbative sum over bulk saddle points. This is most familiar in the black hole Farey tail [8] (see also [10]), where the partition function of the theory is computed by summing over topologically distinct saddle points. We will start by briefly reviewing this construction, before turning to the analogous computation of correlation functions.
We begin with the computation of the partition function of a two dimensional CFT as a sum over all states, weighted by the Boltzmann factor. This may be organised into a sum over only primary operators, with contributions from descendants packaged into the characters χ p of the Virasoro (or perhaps some other extended) algebra:
Z(τ ) = all states q L 0 − c 24qL 0 − c 24 = all primaries χ p (τ )χ p (τ ).(80)
We wish to compute this using a Euclidean bulk path integral. The path integral is over all bulk solutions whose boundary is a torus, the spatial circle times the Euclidean time circle. In semiclassical gravity this sum is dominated by a set of saddle points, which are the classical solutions of Einstein's equations with torus boundary [39]. The leading order contribution of each saddle point is the classical bulk action, with bulk loops around these solutions giving corrections perturbative in 1/c. One solution is thermal AdS (pure Euclidean AdS with periodic identification of Euclidean time). The action and loop corrections around this solution are computed by the characters of light bulk fields. The Virasoro character comes from the graviton loops and other light primaries give loops for the corresponding bulk fields [40]. The other classical solutions of pure gravity are given by modular transformations of thermal AdS. The sum over saddles is therefore a sum over the modular group, with the summand being the total of the characters of light primaries:
Z(τ ) = saddle points e −c S classical +S one-loop +... = γ∈P SL(2,Z)/Z light primaries χ p (γτ )χ p (γτ ).
At leading order the partition function will be dominated by the geometry with least action. This means that the leading order partition function has first order phase transitions (the Hawking-Page transition [41] in this case) as τ varies and different saddle points exchange dominance. This phase transition will be smoothed out at finite c.
Comparing the CFT and gravity results, we see that the contribution from heavy states is accounted for in gravity by the contribution of a different bulk saddle. In other words, the heavy states come from the light states, but propagating in a different channel (i.e. around a different cycle on the boundary torus). The partition function is constructed as a modular sum over the characters of the light spectrum only. By construction, this is modular invariant, though it may not decompose into a sum over characters with positive density of states [10,13].
Our proposal is that essentially the same strategy should be used to study correlation functions, with the characters now being replaced by conformal blocks. For definiteness, let us consider a description of gravity in which heavy particles (that is, with ∆ of order c) are 'first quantised', in the worldline formulation. The perturbative path integral is therefore over configurations of light fields, as well as over heavy particle worldlines (including interactions where worldlines may split and join). The single-particle states of the massive bulk particles, which we will take to be scalars for simplicity, are dual to CFT primaries with energy of order c, but less than c/12 above the vacuum. The correlation functions of the corresponding heavy primary operators are again given by a bulk path integral, but now imposing the boundary conditions that an appropriate particle worldline ends on the boundary at the insertion point of the heavy operator.
In the large c semiclassical limit, the path integral is dominated by classical solutions, including the backreaction from heavy particles. Each heavy particle worldline contributes a factor of mL to the action, where m is the mass and L is the (regularized) proper length of the worldline. The heavy particles also back-react on the geometry, creating a conical singularity with deficit angle 2π(1 − α), where α = 1 − 6m/c = 1 − 24h/c, along its worldline. In many cases, the action of these solutions corresponds to the contribution to the correlation function from an appropriate semiclassical conformal block [42][43][44][45][46][47][48]. Bulk graviton loops around the solution contribute to the perturbative corrections (in 1/c ∼ G N ) to the semiclassical blocks. Loops from other light bulk fields contribute to blocks where the corresponding light primaries are exchanged.
Once again, the full correlation function should be given as a sum over the contributions from all classical solutions. Schematically
O · · · O = classical solutions light primaries F p (81)
where F p denotes the appropriate conformal block, with the light operator p exchanged 19 .
In the case of the four-point function, this is exactly of the form of our proposed correlation function eq. (36), provided we can show that the sum over classical solutions includes the modular sum over channels described in section 3. We will now explain how the family of classical solutions corresponding to the modular sum arises topologically in the sum over classical worldlines. We then give an explicit example where we can show that there is a unique solution for the topological classes associated with the sum over channels, and no others.
Just as in the case of the partition function, in the semiclassical limit the correlation function will be dominated by a particular classical solution. As we vary the moduli (in this case the cross-ratio) this gives rise to first-order phase transitions in correlation functions. An example of this is the well-known exchange in dominance of Ryu-Takayanagi surfaces for the entanglement entropy of two intervals, which can be thought of as a formal limit of correlation functions of twist operators in cyclic orbifolds of the theory [49]. At finite c this phase transition will be smoothed out by the subleading "instanton" corrections to the correlation function.
Rational tangles and modular invariance
We begin by describing in more detail the bulk interpretation of the different P SL(2, Z) channels which appear in our conformal block Farey tail.
Consider the calculation of the Euclidean four-point function
O(z 1 )O(z 2 )O(z 3 )O(z 4 )(82)
of a heavy primary O which is dual to a massive bulk particle that sources a conical 19 We will discuss examples where some classical solutions correspond to the exchange of a heavy particle dual to a bulk conical defect, represented by an internal worldline of this particle in the bulk. The loop corrections due to light bulk fields do not then literally correspond to a sum over blocks of light primaries. defect. One simple classical bulk contribution to this correlator involves two bulk worldlines of this massive particle which join the z i on the boundary sphere in pairs. In fact, there are many such contributions, with different topology. A simple contribution, which we denote t ∞ for reasons that will become clear below, is shown in fig. 3a; it has one worldline joining z 1 and z 2 and another z 3 and z 4 . This contribution is expected to dominate as we take the cross-ratio
x = (z 2 −z 1 )(z 4 −z 3 ) (z 3 −z 1 )(z 4 −z 2 ) → 0. This is the channel where we fuse O(z 1 ) with O(z 2 ) and O(z 3 ) with O(z 4 )
. The corresponding conformal block comes from the exchange of the identity operators and descendants, along with other light operators which would give additional loop corrections.
Suppose now that we begin with this solution as a function of the insertion points {z i }. We may then generate further solutions by analytic continuation. We continuously vary the z i without bringing two insertion points together, obtaining at the end the same configuration of points we started with, albeit with the z i possibly permuted. For example, we may start with t ∞ and rotate the boundary sphere to cyclically permute the insertion points, obtaining the tangle t 0 shown in fig. 3b. This configuration joins z 1 to z 4 and z 2 to z 3 , corresponding to the T-channel, where we fuse O(z 1 ) with O(z 4 ) and O(z 2 ) with O(z 3 ), dominant as x → 1.
In mathematical terms, the generation of further solutions by analytic continuation can be described as braiding of the insertion points: the four-strand braid group on the sphere B 4 (S 2 ), described in fig. 4, acts on the space of solutions.
As we analytically continue the solutions, we expect the worldlines of the particles to stay apart: they do not intersect or pass through one another, and they do not split and join. This means that the worldlines can be usefully categorised by their topological class, defining what is known in knot theory as a '2-tangle'. An n-tangle is, roughly, a configuration of n strings in the ball B 3 which end on 2n fixed boundary points, with configurations considered equivalent if and only if they can be continuously deformed into one another without strings passing through one another while leaving the boundary anchor points fixed. The braid group B 2n (S 2 ) acts on the space of n-tangles in the obvious way.
The set of solutions we describe, obtainable from analytic continuation of t ∞ , gives Figure 4: The spherical braid group generators and relations: B 4 (S 2 ) is generated by σ 1 , σ 2 , σ 3 , with the relations σ 1 σ 3 = σ 3 σ 1 , σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 , σ 2 σ 3 σ 2 = σ 3 σ 2 σ 3 , and σ 1 σ 2 σ 2 3 σ 2 σ 1 = 1. The outer in inner circles represent cross-sections through a twosphere so, for example, the last braid is trivial because the strand can be unwrapped around the front and back of the internal S 2 . only a limited set of topological classes of tangles. We get the orbit of the 2-tangle t ∞ under the braid group, which is known as the set of rational tangles, denoted R. Informally, R is the set of tangles that can be untangled by moving the boundary anchoring points around on the sphere. This excludes, for example, tangles with a strand that is by itself knotted in the bulk.
(a) σ 1 (b) σ 2 (c) σ 3 = (d) σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 (e) σ 1 σ 2 σ 2 3 σ 2 σ 1 = 1
Rational tangles were classified by Conway [50]: they are in one-to-one correspondence with the rational numbers and infinity, R = {t r |r ∈ Q ∪ {∞}}. A non-trivial rational tangle is shown in fig. 3c. Examples of 2-tangles not included in this set are shown in fig. 5.
To explain this classification, we need one small lemma about rational tangles. Drawing them as in fig. 3, with the tangles anchored at diagonal points (traditionally labelled NE,SE,SW,NW with reference to points of the compass) in the plane of the page, the rational tangles are invariant under rotation by π about any of the three axes running vertically, horizontally, and coming out of the page. This is easy to prove by induction: it is clearly true for t ∞ , and if it holds for some tangle, it is straightforward to verify that it continues to hold after acting with any of the three braid group generators. Notice that these rotations can be performed while keeping the cross-ratio of the boundary points constant.
As a consequence of the lemma, we learn that the braid group does not act faithfully: σ 1 σ −1 3 acts trivially on any rational tangle, since that element has the same effect as a π rotation around the horizontal axis. To find the group that acts faithfully, we should therefore quotient the braid group by the normal subgroup generated by σ 1 σ −1 3 . Write T for the coset of σ 1 (or σ 3 ), and S for the coset of σ 1 σ 2 σ 3 , so that, in particular, σ 2 is in the coset T −1 ST −1 . Then the relations for the quotient can be written as
B 4 (S 2 )/N σ 1 σ −1 3 = S, T |S 2 = 1, (ST ) 3 = 1 .(83)
These are the defining relations of the modular group P SL(2, Z). The action of S and T on tangles is simple, S acting as a π/2 rotation of the knot diagram in the axis coming out of the page (in either sense, since a π rotation acts trivially, so that S = S −1 ), and T by twisting the strands on the left, the same as σ 1 in fig. 4. The set of rational tangles R is given by the orbit of t ∞ under the modular group. By the orbit-stabiliser theorem, this is the same as the set of cosets of the stabiliser, the subgroup leaving t ∞ invariant. It is clear that T acts trivially on t ∞ , and in fact the stabiliser is exactly the Z generated by that element (to show that the stabiliser is no larger is the nontrivial part of Conway's classification; a proof can by found in [51]). We therefore have
R P SL(2, Z) Z Q ∪ {∞}.(84)
The last equality follows by considering the action of P SL(2, Z) on Q ∪ {∞} by fractional linear transformations r → ar+b cr+d ; the stabiliser of r = ∞ is precisely the powers of T , which acts here by T (r) = r + 1 (with S(r) = −1/r). This completes the classification. The action of the modular group on tangles allows one to simply describe the rational tangle t r in terms of the continued fraction decomposition of the rational number r, as noted in fig. 3c.
We have learned that there is a natural action of the modular group on the space of rational tangles. This is the same as the action on conformal blocks. To see this, recall that the braids leaving all rational tangles invariant are precisely those that can be done while keeping the cross-ratio x constant. Other braids will cause x to traverse a topologically nontrivial path through cross-ratio space, ending at x or one of its anharmonic images 1/x, 1 − x, 1 1−x , x x−1 , 1 − 1 x depending on how the operators are permuted. This precisely mirrors the discussion of section 2, in which the different nontrivial paths correspond to distinct channels of a conformal block. We may pass to the universal cover of cross-ratio space, which is the upper half-plane, and the braids will there correspond to a path joining the initial τ to one of its images aτ +b cτ +d under the modular group. The braids corresponding to the generators act on τ in the usual way, S · τ = −1/τ and T · τ = τ + 1 . Finally, the tangle t ∞ corresponds to the usual S-channel block, which is invariant under T as required.
If the operators are not identical, it may be useful to distinguish between operator insertion points. From the cross-ratio point of view, this means we allow x to continue only to a subset of its anharmonic images, so we reduce to a subgroup of the full modular group. From the Q classification, the three ways to join the boundary points with tangles in pairs are distinguished by whether the numerator and denominator of the rational number are even or odd. In particular, if we require the tangles to be joined as in the original configuration of t ∞ , for example to compute the vacuum block of O 1 O 1 O 2 O 2 , we must restrict to rational numbers with even denominator (such as 0!) and odd numerator. For example, the tangle t −29/74 in fig. 3c has associated rational number of this form, and pairs the boundary points in the same way as t ∞ . This set is invariant under the congruence subgroup Γ 1 (2) (a, d odd and c even). If we distinguish all boundary points, we should consider not the braid group B 4 (S 2 ) acting on the boundary, but the pure braid group P 4 (S 2 ), the normal subgroup restricting to braids that do not permute the marked points. A similar analysis leads us to consider the congruence subgroup Γ (2) (a, d odd and b, c even), which can be understood as the subgroup of the modular group that leaves all three possible ways of joining boundary points invariant when acting on tangles. This should be compared with the discussion of section 2. The refined notion of vector valued modular functions used there can be realised for tangles by including a label on the endpoints of the strands, which allows us to act with the entire modular group while keeping track of the permutation of operators.
These topological considerations relate the crossing images of a conformal block to a particular set of bulk solutions. It is not clear that these are the only classical solutions, so the proposed sum over modular images of light blocks may miss some saddle points to the path integral. To help to justify this, we now give one example where we can prove that the rational tangle construction exhausts all solutions.
The semiclassical h = c/32 conformal block
We will now consider an example where all classical saddle points in the worldline formulation can be classified. We consider the four-point correlation function of a dimension h =h = c/32 scalar in a 2D CFT in the semiclassical limit. This example will also make a more direct connection to the partition function Farey tail.
We begin by considering the saddle points contributing to a four point function of a scalar of weight h =h in the semiclassical limit. In this limit, we need to compute the action of a pair of particles which propagate through the bulk between the boundary operator insertion points. Each worldline contributions a factor mL, where m is the mass of the bulk particle and L is the (regulated) proper length of the worldline. When the mass of the particle is of order the central charge, we must also include gravitational backreaction. Each particle creates a delta-function source of stress-energy supported along the worldline, and Einstein's equations then imply that the worldline is replaced by a conical singularity. The conical deficit angle is related to the mass of the particle by 2π(1 − α) = 8πG N m.
In terms of the dimension h of the operator we have
α = 1 − 6m c = 1 − 24h c .(86)
Note that, since 0 < α < 1, the operator must have 0 < h < c 24 . We must now find the gravitational action of the backreacted configuration of two worldlines, where no other particles are exchanged in the bulk. This will give the leading semiclassical contribution to the vacuum conformal block in the channel where the pairs of boundary points joined by the particle worldlines fuse to the identity operator. Generically, the interaction between the two particles means that the geometry cannot be explicitly found. Thus it is not possible to find a closed form expression for this semiclassical block.
However, at the special value h = c/32 the deficit angle is exactly π, which allows us to make progress. The trick is to consider not the original geometry, but the twofold cover, branched along the particle worldlines. This solves the equations of motion, but it is smooth everywhere, since we do not have any other massive particle exchanged. The boundary geometry in this example is particularly simple, being a torus. This is exactly the situation one encounters in the computation of four point functions of twist operators in a Z 2 orbifold theory [52]; indeed such operators have precisely dimension h = c/32. This arises also in the computation of the second Rényi entropy for a pair of intervals [53][54][55].
We must begin by finding the smooth solutions to 3D gravity with torus boundary, which are known [10] to be thermal AdS and the Euclidean BTZ black hole, and their 'SL(2, Z) black hole' generalisations [39], described in more detail below. In all of these solutions, the Z 2 covering group of the boundary extends as an isometry into the bulk. Taking the quotient by this Z 2 gives the desired solutions with conical deficit worldlines. It follows that we have the complete classification of all such classical solutions.
As we will describe below, the solutions with torus boundary are labelled by an upper half-plane parameter τ (parameterising the conformal structure of the torus) as usual. Images of τ correspond to different solutions with the same (or anharmonically related) cross-ratio. Moreover, as described in the previous section, the topology of the conical defects is that of a rational tangle.
Gravity solutions
Let us now be more explicit about the classical solutions. The double cover of the boundary can be written as an elliptic curve
y 2 = z(z − x)(z − 1)(87)
where z is the usual coordinate on the sphere, and y picks up a sign after circling the branch points at 0, x, 1, ∞; this sign labels the two sheets of the cover. With the familiar description of the torus as the complex plane modulo a lattice (u ∈ C, with identifications u ∼ u + 1 ∼ u + τ , for some Im τ > 0), the map to the Riemann sphere giving z in terms of u is a doubly periodic function, which is essentially the Weierstrass ℘-function (up to some Möbius map). This map is two-to-one, mapping u and −u to the same z, excepting at the branch points u = 0, 1/2, 1+τ 2 , τ 2 , which may be chosen to map to 0, x, 1, ∞ respectively. A Möbius map fixes three of these, and then x is determined in terms of τ as the modular λ function x = λ(τ ) as in section 2.
To describe the bulk solutions, it is convenient to write the boundary in terms of the coordinate w = exp(2πiu), which implements the identification u ∼ u+1 automatically. The other identification to obtain the torus becomes w ∼ qw, with q = e 2πiτ , and the Z 2 identification giving the plane is w ∼ 1/w, with fixed points at w = ±1 and w = ±q 1/2 . The fundamental domain for these identifications in the u and w coordinates is shown in fig. 6. Now take the upper half-space model of Lobachevsky space, with coordinates w ∈ C 0 1 2 and y > 0, and metric
1 1 2 + τ τ 1 + τ (a) u-plane q −q 1 −1 (b) w-planeds 2 = dwdw + dy 2 y 2 .(88)
We now may quotient the bulk by isometries which restrict on the boundary to the identifications described above. Firstly, identifying by (w, y) ∼ (qw, |q|y) results in a solid torus, with smooth hyperbolic metric since this map acts without fixed points. This construction gives every such metric with torus boundary (and without cusps). Then the Z 2 to return to the sphere on the boundary extends isometrically into the bulk as (w, y) ∼ 1 ww+y 2 (w, y), resulting in the ball with conical defects along two curves. These defects appear along the fixed points of the isometry, which are the semicircles |w| 2 + y 2 = 1 with w real, and |w| 2 + y 2 = |q| with q −1/2 w real. Now, solutions with q related by a modular transformation (modulo the Z subgroup generated by τ → τ + 1 leaving q invariant) have a torus with the same conformal structure on the boundary, but different topology in the bulk: in terms of the rational number used to classify tangles, the cycle described in the u-plane by a line through u = 0 and u = r + τ (or u = 1 if r = ∞) is contractible in the bulk. Here, r is rational so that this line intersects a lattice point, to form a closed cycle. In the quotient, the fact that the conical defects have the topology of rational tangles follows since the bulk is continuously deformed by moving through τ -space, and the conical defects never intersect.
Finding the on-shell action is easy because of a fortuitous cancellation: the conical defect in the geometry means that there is a delta-function in the curvature supported on the particle worldline, contributing a piece to the Einstein-Hilbert action proportional to the length of the worldline L, but this is precisely cancelled by the particle action mL itself. As a consequence, we need only compute the usual Einstein-Hilbert action away from the defect, without taking the singular piece into account. This is particularly useful in the current context, since it means we may compute the action by passing to the smooth double cover, use existing results, and simply halve that action to find our answer.
We therefore need only the solid torus action, regulated according to the boundary metric ds 2 = dzdz (modified at the operator insertions to regulate the conformal factor between the plane and torus, justified by requiring that the two-point functions are canonically normalised). The action in the flat dudū metric on the torus is straightforward to compute, and to convert this to the required dzdz metric we need a factor from the conformal anomaly, much as in the twist operator correlation function calculation [52] or the transformation between flat metric and 'pillow metric' operators described in [56]. The saddle-point contribution e −S in the end factorises into a holomorphic times (conjugate) antiholomorphic piece, the holomorphic half giving the semiclassical block 20 :
F(c, 0, h = c/32; x) ∼ (2 8 x(1 − x)) −c/48 exp 2πc 48 K(1 − x) K(x) (89) = 4 θ 2 (q)θ 4 (q) θ 3 (q) 2 −c/12 q −c/48 .(90)
Here K is the elliptic integral, which has branch cuts; the expression in the second line in terms of the upper half-plane parameter τ = 1 2πi log q does not suffer from this ambiguity (excepting possibly for the overall phase from a fractional power). The first factor in this result comes from the conformal anomaly, and the second factor from the action in the dudū metric, S = − c 12 2π Im τ , with τ = i K(1−x) K(x) . In fact, we can straightforwardly derive a more general solution than this, describing the same external operators, but instead of the internal primary being the identity, we have the exchange of some heavy primary of arbitrary dimension h p =h p < c/24. This means we have two cubic vertices in the bulk, one on each of the original defect worldlines, and, joining the two, the worldline of the intermediate particle. Since this intermediate particle is also heavy, it too sources a conical defect, of arbitrary strength determined by the particle mass. The trick of taking the double cover branched along the worldlines still works, except now the resulting solid torus is not smooth, but has a conical defect determined by the exchanged particle wrapping the nontrivial cycle. To include this, simply alter the hyperbolic bulk metric eq. (88) by including a defect of the appropriate strength along the line w =w = 0
ds 2 = α 2 p y 2 y |w| 2(1−αp) dwdw + dy 2 ,(91)
and take the same identifications as before.
It is straightforward to generalize the classical action calculation to this case. The simplest way to do this is by differentiating the on-shell action with respect to the mass of the internally exchanged particle. When we differentiate, there is a contribution coming from the variation of the metric and other fields themselves, since the classical solution changes as the mass is changed, but this vanishes because the solution is a 20 Note that the convention used in this section for conformal blocks differs from that in section 3 by a factor of (1 −
x) (htot/3)−h2−h3 x (htot/3)−h1−h2 (where h tot = i h i ).
stationary point of the action. This leaves only a contribution coming from the explicit variation of the parameter appearing in the action, in this case giving dS on-shell dm = L, where L is the length of the worldline of the exchanged particle 21 . In particular, this is why the action reduces simply to worldline length in the limit where h c 1, as in [45], for instance.
In the metric (91), this worldline runs along |w| = 0, between y = 1 and y = |q| α/2 (where the lines of fixed points of the quotient meet the defect at w = 0), giving L = πα Im τ . Integrating this to find the action, the result in the end matches the one found from the Zamalodchikov monodromy method for semiclassical conformal blocks eq. (98) (up to the normalisation, discussed in section 5.3.3), which we now briefly describe.
Monodromy method for semiclassical blocks
A commonly used method for computing semiclassical conformal blocks is the Zamalodchikov monodromy method [5], reviewed in [57,58], which can be understood as coming from the semiclassical limit of Liouville theory. This is essentially equivalent to classical gravity, but since the calculations are, to immediate appearances, rather different, it is nonetheless instructive to include both. It is also a novel example where the monodromy problem can be solved exactly, without any approximations (beyond the semiclassical limit required for its applicability).
Consider the conformal block of four external operators of dimension h i = i c/6, exchanging an operator of dimension h p = p c/6, in the limit c → ∞, with the 's fixed. To leading order in this semiclassical limit, the block exponentiates as
F(c, h p , h i , x) ∼ exp − c 6 f ( p , i , x) .(92)
The function f is found by solving the differential equation
ψ (z) + T c (z)ψ(z) = 0(93)
where T c (z) is given in terms of one unknown function of x, the accessory parameter c 2 (x):
T c (z) = 1 z 2 + 2 (z − x) 2 + 3 (z − 1) 2 + 4 − 1 − 2 − 3 z(z − 1) + x(1 − x)c 2 (x) z(z − x)(z − 1) .(94)
As a second order equation, there are two solutions to (94). These solutions mix when we transport the solution around any topologically non-trivial cycle in the z-plane, i.e. we go around any of the singular points of the differential equation. This mixing is described by a monodromy matrix M , which has unit determinant by the constancy of the Wronskian. The basis independent data of this matrix is then encoded in the trace of the monodromy matrix. We then fix the accessory parameter c 2 (x) by choosing a particular cycle in the z-plane and demanding that the associated Monodromy matrix has
Tr M = −2 cos πα p , where h p = c 24 1 − α 2 p (95)
so the eigenvalues of M are −e ±iπαp . The conformal block is determined by c 2 (x) = ∂f ∂x , the constant of integration determined by normalization (which can be fixed by the behavior as operators become coincident). The choice of cycle in the differential equation determines the channel of the block.
In the present case, we have i = 3/16 and the ODE is solved by
ψ ± (z) = 1 t (z) e ±ikt(z) , with t (z) = 1 z(z − x)(z − 1) .(96)
The accessory parameter is c 2 (x) = 1−2x+8k 2 8x(1−x) . This is the WKB solution used to work out the limit of large internal dimension [5], but for the correct accessory parameter, with these values for the external dimensions, it is in fact an exact solution.
If we choose a cycle enclosing 0 and x the monodromy is diagonal in this basis. In particular, the solutions pick up factors of − exp(±2ikt(x)): the sign comes from the square root in the prefactor, since t (z) winds once round the origin as we traverse the cycle, and the phase comes from integrating t (z) in eq. (96) around the cycle from zero to x and back again. Expressing t(x) as the elliptic integral 2K(x) (with appropriate branch choice) we use the monodromy condition to find
c 2 (x) = 1 − 2x 8x(1 − x) + π 2 α 2 16x(1 − x)K(x) 2(97)
where α = 1 − 4 p . This gives the block
F(c, h p , h i = c/32; x) ∼ 2 4hp (2 8 x(1 − x)) −c/48 exp c 24 − h p π K(1 − x) K(x) .(98)
This reduces to eq. (89) in the case h p = 0, with the additional factors coming from the worldline action of the exchanged particle, as discussed above. In terms of q = e 2πiτ the block is
F(c, h p , h i = c/32; x) ∼ 2 4hp 4 θ 2 (q)θ 4 (q) θ 3 (q) 2 −c/12 q hp/2−c/48 .(99)
These semiclassical blocks give the classical contribution to the correlation function coming from individual saddle points. To find the full correlation function, we should sum over all saddle points, which come from taking τ to one of its modular images. Thus, gravity naturally leads us to the conformal block Farey tail. It is natural to ask now what the full CFT operator content and couplings are that give a correlation function of this form, but we leave this question for the future.
Worldline interpretation of heavy exchange, OPE coefficients, and the semiclassical DOZZ formula
As discussed in section 5.3.1, the modification of the block when we include a heavy internal operator exchange can be understood from the worldline quantized gravity point of view. The change in the action from including the additional defect accounts for the factor of exp −h p π K(1−x)
K(x)
in the block. The prefactor 2 4hp , that we fixed by the x → 0 limit, does not appear in the gravitational action. This is because the saddle point action computes not just the (holomorphic times antiholomorphic) block, but the contribution of the block to the correlation function, which includes (the leading semiclassical part of) the OPE coefficients C 2
OOhp . To find these OPE coefficients, we may compute a three-point function, with a gravitational saddle point consisting of three conical defects from the boundary meeting at a trivalent vertex in the bulk, equivalent to a Liouville theory calculation giving the semiclassical limit of the DOZZ formula [57,59,60]. In the case of interest, when O is the h = c/32 scalar, the relevant OPE coefficient is C OOhp = 2 −4hp , cancelling precisely the prefactor in the block. This can be shown directly in this special case by performing the gravity calculation using a double cover trick similarly to before, but also can be obtained from the more general (though much more complicated) results on the semiclassical DOZZ formula, as we now briefly show.
When properly normalized, the OPE coefficients of heavy h < c/24 scalar operators are given by exp P(η 1 , η 2 , η 3 ), where the dimensions of the operators are h i = c 6 η i (1−η i ) with 0 < η < 1/2, and the function P is [47] P(η 1 , η 2 , η 3 ) (100)
= c 6 F (2η 1 ) − F (η 2 + η 3 − η 1 ) + (1 − 2η 1 ) log(1 − 2η 1 ) + (2 permutations) + F (0) − F (η 1 + η 2 + η 3 ) − 2(1 − η 1 − η 2 − η 3 ) log(1 − η 1 − η 2 − η 3 ) where F (η) = η 1/2 log Γ(x) Γ(1 − x)
dx, for 0 < η < 1 is, roughly speaking, the semiclassical limit of Υ b which appears in the general DOZZ formula. Taking the case of interest, for which η 1 is arbitrary and η 2 = η 3 = 1/4, we find
P η, 1 4 , 1 4 = c 6 F (2η) − F 1 2 − η − F 1 2 + η − 2F (η) + F (0) − 2η log 2 .
To simplify this expression, it is easiest to first differentiate, getting
d dη P η, 1 4 , 1 4 = c 3 log Γ( 1 2 − η)Γ(1 − η)Γ(2η) 2Γ( 1 2 + η)Γ(η)Γ(1 − 2η) = c 3 log 2 4η−2
where the last equality uses the duplication identity Γ(z)Γ(z + 1 2 ) = 2 1−2z √ πΓ(2 − z) once on the top and once on the bottom. Integrating, and fixing the constant by noting P 0, 1 4 , 1 4 = 0 as follows from canonical normalization of the operators, we at last find that P η, 1 4 , 1 4 = −4h p log 2, reproducing the OPE coefficient claimed above.
Connection to twist operator correlation functions and the black hole Farey tail
Finally, let us briefly expand on the connection between these calculations and the four-point function of twist operators in a Z 2 orbifold theory, or equivalently the second Rényi entropy of two intervals. Firstly, to be clear, we do not demand that the our theory is a Z 2 orbifold theory, or that the h = c/32 scalar a twist operator; we only want an operator of this dimension so we can use the convenient trick to find classical solutions, and do not necessarily want, for example, the additional light states that must appear in an orbifold CFT 22 . An example of an orbifold theory containing twist operators with a gravitational dual is given by the D1-D5 system at the orbifold point, though this is very 'stringy' and the low-energy physics bears little resemblance to Einstein gravity. Having said all this, since the conformal block is a universal kinematical function, we may derive it using any theory and operator we like with the correct central charge and dimensions, including an orbifold theory and twist operators. The conformal block for a given internal primary operator can be defined as the correlation function, with the insertion of a projector onto the descendant states of that primary on a cycle separating the points 0, x from 1, ∞. Taking the external operators as Z 2 twist operators, when we pass to the covering space this projection is on a nontrivial cycle of the torus, so we project onto a subset of states propagating round a complementary cycle. It is therefore tempting to use this to identify the conformal block with external weights c/32 with a Virasoro character. But this is not quite right: the projection to obtain the Virasoro character leaves more states intact, because it contains not just descendants in the orbifold theory, but also all descendants in the seed theory, which includes states regarded as Virasoro primaries from the Z 2 orbifold theory. One way of saying this is that the untwisted sector of the orbifold theory (relevant since all states exchanged are untwisted) has an extended algebra, the symmetric product of two Virasoros, one from each copy of the theory; the character includes descendants under this entire algebra, but the block only descendants under the diagonal Virasoro. The character and the block do (when the appropriate conformal anomaly is included) match in the semiclassical limit, but not the perturbative corrections. The OPE coefficient 2 −4hp that appears from the gravitational calculation also matches the coefficient between two twist operators and a third primary operator to which they fuse (which must be untwisted, and of the form φ (1) ⊗ φ (2) , where φ is some primary in the seed theory and the superscript indicates which copy it acts on).
Finally, we directly connect to previous work by noting that the black hole Farey tail is a special case of our conformal block Farey tail, where the CFT is taken to be a Z 2 orbifold of a gravity theory and we consider the correlation function of twist operators, since this is (up to an anomaly term) just the partition function in the original theory [52].
Blocks computed perturbatively in h/c.
As mentioned already, there is a convenient limit in which to study semiclassical blocks, where the dimension of some operator is large, but much less than c. Concretely, one may solve the monodromy problem described in section 5.3.2 perturbatively in = 6h/c for the appropriate operator. To leading order, as discussed above this corresponds to a 'probe limit' in gravity, where the worldlines of the operator in question become geodesics in the background created by other operators.
One might try to apply the ideas discussed in this section to this perturbative limit, for example for the four-point function O H O H O h O h , where we compute exactly in the dimension H (of order c) and perturbatively in h/c. In this example, the perturbation theory describes a conical defect background created by O H , and a geodesic associated with O h in this background. However, in this geodesic limit, one runs into trouble when attempting to analytically continue in the cross-ratio. In particular, along some curve (depending on the dimension H) in τ space, the geodesic intersects the defect, and a naïve analytic continuation of the block past that curve gives results that are not reproduced by any geodesic. From the gravity point of view, there is no reason why analytic continuation should be applicable, since the spacetime is not analytic. Nonetheless there is no obvious breakdown in perturbation theory from the point of view of the monodromy method, so it is likely that this tension can be resolved only by going beyond perturbation theory.
From the gravitational point of view, it is natural that the perturbation theory ceases to be applicable when the geodesic intersects the conical defect: once the worldlines are parametrically close, it is not valid to neglect their mutual gravitational interaction. This interaction may prevent the worldlines from crossing, in which case the topological discussion of rational tangles remains applicable, though the nontrivial tangling of the worldlines may be confined to a parametrically small region of the spacetime.
This question would benefit from more quantitative understanding, particularly as it is an important limit for holographic calculations of entanglement entropy. Another example in this spirit, where progress may be easiest, is for the four-point function of identical operators O h O h O h O h , as considered in [58], relevant for the entanglement entropy of two disjoint intervals. The problem there occurs when the cross-ratio hits the line Im(x) = 0, Re(x) > 1, where two geodesics intersect. Naïve perturbation theory suggests that the geodesics pass through one another, so the conformal blocks are single valued in cross-ratio space, but this seems incompatible with our results at finite .
Appendix: conformal blocks and P SL(2, Z) representations in minimal models
In this appendix, we review the Coulomb-gas representation of the conformal blocks for minimal models, from which we obtain the representation of the modular group associated with various conformal blocks. Our discussion is based mainly on [20,[23][24][25], with some slightly different conventions, more convenient for our purposes. For simplicity, we will focus here on the correlation functions of identical second-order scalar operators. The extension to mixed correlators or operators with spin can be found in [20,[23][24][25].
The four-point function of the scalar operator φ (r,s) of dimensions h =h = h (r,s) φ (r,s) (z 1 )φ (r,s) (z 2 )φ (r,s) (z 3 )φ (r,s) (z 4 ) = |z 12 z 34 | −4h (r,s) (1 − x)
with N (r,s) primary fields appearing in the fusion rule φ (r,s) × φ (r,s) =
N (r,s) i=1 φ (r i ,s i ) .(103)
The conformal block F i is associated with the primary φ (p i ,q i ) appearing in the fusion rule. We shall organize the label i such that h (r i ,s i ) is a non-decreasing function of i, with h (r 1 ,s 1 ) = 0 (i.e. φ (r 1 ,s 1 ) = 1). This definition of F i implies the leading order behaviour F i (x) = x h (r i ,s i ) −4h (r,s) /3 + · · · (104) as x → 0.
In the Coulomb-gas formalism (see section 9.2.3 of [20]), the holomorphic four-point function of φ (r,s) is computed by φ (r,s) (z 1 )φ (r,s) (z 2 )φ (r,s) (z 3 )φ (r,s) (z 4 ) = V (r,s) (z 1 )V (r,s) (z 2 )V (r,s) (z 3 )V (−r,−s) (z 4 )Q r−1
+ Q s−1 − .(105)
The screening operator Q ± is defined by
Q ± ≡ C dw V α ± (w)(106)
with α ± ≡ α 0 ± α 2 0 + 1 and α 0 ≡ 1/(2 p(p − 1)), or equivalently α ± ≡ ± p p−1
± 1 2 .
The contour C must be chosen appropriately to get the full correlation function; a different choice of contour will give the contribution from an individual conformal block as we will see. Let us now focus on the case of (r, s) = (2, 1) 23 , with dimension h (2,1) = p + 2 4(p − 1)
.
For example for (p, p ) = (4, 3) we have c = 1/2 with h 2,1 = 1/2, and for (p, p ) = (5, 4), c = 7/10 and h 2,1 = 7/16, giving the scalar operators usually labelled in the Ising model and σ in the tricritical Ising model. We also have
α + = p p − 1 , α 2,1 = − 1 2 p p − 1 .(108)
The fusion rule of two φ (2,1) operators is given by φ (2,1) × φ (2,1) = φ (1,1) + φ (3,1) = 1 + φ (3,1) ,
the dimension of the φ (3,1) operator given by h 3,1 = (1+p)/(−1+p) = −(2a+1), where we have introduced the parameter a = 2α + α 2,1 = − p p−1 . We will find the conformal blocks for the four-point function of φ (2,1) with these two operators exchanged, in terms of hypergeometric functions.
From the Coulomb gas expressions, the four-point function is given by φ (2,1) (z 1 )φ (2,1) (z 2 )φ (2,1) (z 3 )φ (2,1) (z 4 ) = C dw V (2,1) (z 1 )V (2,1) (z 2 )V (2,1) (z 3 )V (−2,−1) (z 4 )V + (w) , (110) 23 The results for (r, s) = (1, 2) can be obtained straightforwardly from this by taking p → 1 − p in the final expressions and using the formula for the k-point function of vertex operators
V α 1 (z 1 ) . . . V α k (z k ) = i<j (z ij ) 2α i α j , with z i,j ≡ z i − z j(111)
gives the integral expression φ (2,1) (0)φ (2,1) (x)φ (2,1) (1)φ (2,1)
(∞) = [(1 − z)z] 2α 2 2,1 C dw [w(w − 1)(w − x)] a .
(112) The conformal blocks can be extracted from this expression simply by changing the contour of integration C, as
F i (x) = 1 N i [x(1 − x)] −(a+ 1 3 ) C i dw [w(w − 1)(w − x)] a(113)
with C i being the line from 0 to x for the vacuum block, and from 1 to ∞ for the φ (3,1) exchange. The normalisation is fixed by the eq. (104), to give
N 1 = Γ 2 (a + 1) Γ(2a + 2) , N 2 = Γ(−3a − 1)Γ(a + 1) Γ(−2a)(114)
and the blocks can then be expressed in terms of hypergeometric functions as
With these expressions in hand, the T matrix and the S matrix can be read off by using standard identities for hypergeometric functions, which my be derived by deforming the contour of integration (as in figure 9.3 of [20]). For example, for the S matrix, we use F 1 (1 − x) = − 1 2 cos(aπ) F 1 (x) + (1 + 2 cos(2πa)) N 2 N 1 F (3,1) (x) ,
F (3,1) (1 − x) = − 1 2 cos(aπ) N 1 N 2 F 1 (x) − F (3,1) (x) .(116)
The resulting representation of P SL(2, Z), in the basis {F 1 , F (3,1) }, is generated by (1 + 2 cos(2πa)) N 2 N 1 −1 .
T = e
In our conventions, the matrices mean for example, that F i (1 − x) = j S ji F j (x). It is easy to see here that rescaling the basis (picking N 1 = 1 + 2 cos(2πa)N 2 instead of the choices above) can make the representation unitary, as long as 1 + 2 cos(2πa) > 0, as is the case for p ≥ 5. The marginal case p = 4 is discussed in section 4.3.4.
The usual solution to crossing is now simple to obtain. Imposing T -invariance on the correlator restricts it to the form
G = |F 1 | 2 + C 2 (3,1) |F (3,1) | 2 ,(118)
and then S-invariance gives the OPE coefficient as
C (3,1) = ± 2 cos 2πp p − 1 + 1 Γ 2p+1 p−1 Γ − 2 p−1 Γ 2p p−1 Γ − 1 p−1 .(119)
The basic observation is that the four point function O(z 1 )O(z 2 )O(z 3 )O(z 4 ) must be invariant under crossing symmetry, i.e. invariant under permutations of the operators O(z a ). The expansion
Figure 1 :
1Fundamental domain for Γ(2) in the upper half τ -plane, bounded by the blue dashed curves. T 2 identifies the left and right vertical lines, and ST 2 S identifies the semicircles. The three cusps at τ = i∞, 0 and ±1 correspond to x = 0, 1, ∞. The black dashed lines show how this domain breaks up further into six fundamental domains for the modular group Γ (four of which are split in two across the blue lines). These six domains correspond to the images in the cross-ratio x-plane under the anharmonic group, shown in the right figure, where the marked points are at x = 0, 1.
Figure 2 :
2The blue dots show c(k max ), plotted agains k max . The orange line on the RHS is approximately 0.372 -the exact OPE coefficient. The last point on the RHS is approximately 0.381 which is about 2% from 0.372.
of the representation relevant to the four-point function of σ in the Ising model in eq. (48), by rescaling to the basis {F 1 , F /2} the representation becomes unitary: of basis essentially amounts to absorbing the OPE coefficient into the block. This representation is irreducible, so there is a unique (up to multiples) invariant correlation function.To show how the action on the two-dimensional space B extends onto the fourdimensional space V = B ⊗B by conjugation, write the matrix on which it acts in the basis consisting of the identity and three Pauli matrices. Then the representation trivial representation appearing in the upper left component (once, as expected), and an irreducible three-dimensional representation in a second block.4.3.2 Example 2:Four-point function of φ(1,4) in M(12,11)
Figure 3 :
3The rational tangles t ∞ , t 0 and t −29/74 . The last diagram should be compared with the continued fraction −1/(3 − 1/(2 − 1/(−4 − 1/3))) = − 29 74 .
Figure 5 :
5Examples of 2-tangles which are not rational, so may not be untangled only by moving the boundary points. No such worldline topologies appear as saddle points to the gravitational path integral.
Figure 6 :
6The fundamental domain for the plane (hatched) and its double cover, the torus, in the u and w planes. For the torus, the top/bottom and left/right edges of the diamond in the u-plane are identified, and the inner and outer circles in the w-plane (with a twist). The Z 2 further identifies the hatched and unhatched regions.
s)(r,s)(r i ,s i ) F i (x)F i (x)
F 1 F
1((3,1) (x) = [x(1 − x)] −(a+ 1 3 ) 2 F 1 (−a, −3a − 1, −2a, x) = x − 1 3 −a + · · · .
Table 1 :
1Generators of the orbit of the vacuum block.
In this equation we are using conventions where the |x| ∆ φ appears explicitly in front of the conformal block, in order to emphasize that low dimension operators will dominate the sum when x → 0. Later we will absorb this factor |x| ∆ φ into the definition of the conformal block, as is standard in much of the literature.2 In D > 2 the identity block is trivial, but in D = 2 we can (and will) use Virasoro blocks where F 1 (x,x) is non-trivial.
If one takes only the vacuum block as the seed contribution, the sum contains three terms, being the product of two-point functions in S,T and U channels. This gives the disconnected piece of the correlation function, the generalised free field result.
This property was previously observed for the Ising model partition function in[15]; our result is an extension of this to correlation functions.
One may notice that these are the same as the symmetries of the Riemann tensor. This is no accident: for exactly marginal operators, an integrated four-point function gives the curvature of moduli space.
In the special case where G(τ ) is a meromorphic function of τ , G(τ ) will be a rational function of j(τ ) which is uniquely determined by its poles and zeros. This can be used to efficiently compute the correlation function of chiral operators in two dimensional CFTs, as in[19].
Our discussions and conventions are based on[20].9 In this section, we restrict our discussion to finite-dimensional spaces of conformal blocks. For work and subtleties related to extension to the infinite-dimensional spaces, see[21].
The order of the indices here is (perhaps despite appearances) natural, because the blocks are the basis vectors of the space B, and so transform with a transpose relative to the components. This requires care, since it is different from conventions in much of the literature.11 This itself is a subgroup of the kernel of the representation(48), which is Γ 0 (4), with index 24.
The representation of P SL(2, Z) for the four-point function of is the same (up to a phase) as that of . Although is a third-order operator, the fusion rule × = 1 + implies that the modular group acts invariantly on the two-dimensional space spanned by the 1 and blocks. Thus, the analysis of four-point function of is the same as that of , consistent with the fact that C 2 = C 2 . This can be understood as a consequence of the model being secretly supersymmetric, with and in the same supermultiplet[26,27].
A single bulk field does not give rise to a single primary, but rather to a tower of primaries coming from multiparticle states. In the language of large N gauge theories, these are multi trace operators.
This is true in general, but particularly useful here, as the worldline in question does not end on the boundary, so we do not need to regulate the length.
This is different, for example, to the discussion of[61], which requires the defect geometry to be dual to be a twisted sector state in an orbifold theory.
AcknowledgementsWe are grateful to V. Balasubramanian
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| []
|
[
"VCExplorer: A Interactive Graph Exploration Framework Based on Hub Vertices with Graph Consolidation",
"VCExplorer: A Interactive Graph Exploration Framework Based on Hub Vertices with Graph Consolidation"
]
| [
"Huiju Wang \nSchool of Information and Safety Engineering\nZhongnan University of Economics and Law\n430073China\n",
"Zhengkui Wang \nInfoComm Technology\nSingapore Institute of Technology\n138683Singapore\n",
"Kian-Lee Tan \nSchool of Computing\nNational University of Singapore\n117417Singapore\n",
"Chee-Yong Chan \nSchool of Computing\nNational University of Singapore\n117417Singapore\n",
"Qi Fan \nSchool of Computing\nNational University of Singapore\n117417Singapore\n",
"Xiao Yue \nSchool of Information and Safety Engineering\nZhongnan University of Economics and Law\n430073China\n"
]
| [
"School of Information and Safety Engineering\nZhongnan University of Economics and Law\n430073China",
"InfoComm Technology\nSingapore Institute of Technology\n138683Singapore",
"School of Computing\nNational University of Singapore\n117417Singapore",
"School of Computing\nNational University of Singapore\n117417Singapore",
"School of Computing\nNational University of Singapore\n117417Singapore",
"School of Information and Safety Engineering\nZhongnan University of Economics and Law\n430073China"
]
| []
| Graphs have been widely used to model different information networks, such as the Web, biological networks and social networks (e.g. Twitter). Due to the size and complexity of these graphs, how to explore and utilize these graphs has become a very challenging problem. In this paper, we propose, VCExplorer, a new interactive graph exploration framework that integrates the strengths of graph visualization and graph summarization. Unlike existing graph visualization tools where vertices of a graph may be clustered into a smaller collection of super/virtual vertices, VCExplorer displays a small number of actual source graph vertices (called hubs) and summaries of the information between these vertices. We refer to such a graph as a HA-graph (Hub-based Aggregation Graph). This allows users to appreciate the relationship between the hubs, rather than super/virtual vertices. Users can navigate through the HAgraph by "drilling down" into the summaries between hubs to display more hubs. We illustrate how the graph aggregation techniques can be integrated into the exploring framework as the consolidated information to users. In addition, we propose efficient graph aggregation algorithms over multiple subgraphs via computation sharing. Extensive experimental evaluations have been conducted using both real and synthetic datasets and the results indicate the effectiveness and efficiency of VCExplorer for exploration. arXiv:1709.06745v1 [cs.DB] 20 Sep 2017• We conduct extensive experimental evaluation based on both real and synthetic data. The experimental results demonstrate that VCExplorer is effective and efficient. | null | [
"https://arxiv.org/pdf/1709.06745v1.pdf"
]
| 23,471,356 | 1709.06745 | 942c7bddf4334711a29b3fa0390efc32424739a1 |
VCExplorer: A Interactive Graph Exploration Framework Based on Hub Vertices with Graph Consolidation
Huiju Wang
School of Information and Safety Engineering
Zhongnan University of Economics and Law
430073China
Zhengkui Wang
InfoComm Technology
Singapore Institute of Technology
138683Singapore
Kian-Lee Tan
School of Computing
National University of Singapore
117417Singapore
Chee-Yong Chan
School of Computing
National University of Singapore
117417Singapore
Qi Fan
School of Computing
National University of Singapore
117417Singapore
Xiao Yue
School of Information and Safety Engineering
Zhongnan University of Economics and Law
430073China
VCExplorer: A Interactive Graph Exploration Framework Based on Hub Vertices with Graph Consolidation
Graphs have been widely used to model different information networks, such as the Web, biological networks and social networks (e.g. Twitter). Due to the size and complexity of these graphs, how to explore and utilize these graphs has become a very challenging problem. In this paper, we propose, VCExplorer, a new interactive graph exploration framework that integrates the strengths of graph visualization and graph summarization. Unlike existing graph visualization tools where vertices of a graph may be clustered into a smaller collection of super/virtual vertices, VCExplorer displays a small number of actual source graph vertices (called hubs) and summaries of the information between these vertices. We refer to such a graph as a HA-graph (Hub-based Aggregation Graph). This allows users to appreciate the relationship between the hubs, rather than super/virtual vertices. Users can navigate through the HAgraph by "drilling down" into the summaries between hubs to display more hubs. We illustrate how the graph aggregation techniques can be integrated into the exploring framework as the consolidated information to users. In addition, we propose efficient graph aggregation algorithms over multiple subgraphs via computation sharing. Extensive experimental evaluations have been conducted using both real and synthetic datasets and the results indicate the effectiveness and efficiency of VCExplorer for exploration. arXiv:1709.06745v1 [cs.DB] 20 Sep 2017• We conduct extensive experimental evaluation based on both real and synthetic data. The experimental results demonstrate that VCExplorer is effective and efficient.
I. INTRODUCTION
Graphs are powerful tools to model a variety of information networks, such as the Web, biological networks and social networks (e.g. Twitter). In a graph, each vertex usually represents one real world object and each edge indicates the link between two objects. Normally, both vertices and edges may be annotated with attributes or labels.
These graphs contain a wealth of valuable information to support a wide variety of queries for information discovery and decision making. To better understand the information encoded in the underlying graphs, different approaches have been used to explore these data.
On one hand, we have summarized-based methods that aim to simplify or summarize the graphs to provide a coarser and higher level graph that is normally referred to as a view. These approaches include graph summarization [1], graph aggregation in graph OLAP [2], graph clustering and so on. The common methodology of these approaches is to aggregate multiple vertices (resp. edges) into one super node (resp. edge) based on certain rules (e.g. through clustering or aggregating the vertices with the same attributes) to a view with much fewer vertices and edges. This makes it easier to visualize a large and complex graph. On the other hand, we have graphbased methods (e.g. [3]) that convey the content of a graph by displaying the whole graph including all the individual vertices and the links on a screen via graph layout. The mainstream approach of these mechanisms is graph visualization which provides the individual vertices and the links among them in the visualization space.
From users' point of view, graph summarization/aggregation methods show summarized view, but hide the original individual vertices; conversely, graph visualization schemes show all individual vertices, but hide the summarized view. Each of the approaches has its own strengths and limitations in exploring a graph. As the size of the graph increases, what to show and what to hide plays an important role on the effectiveness of graph exploration.
A. A Running Example over Social Network.
Typically, a social network is modeled as a graph. Vertices of the graph represent persons, whereas edges represent relationships between the vertices. Both vertices and edges may have attributes. Figure 1(a) shows such a social network. Each vertex is affiliated with an attribute name, and each edge is affiliated with a relationship type (e.g., friend, relative) between two vertices. Given such a social network, an analyst may be interested to find out how user bingfish is connected with user kristy. Now, each path between bingfish and kristy represents one type of connections between them, and there are potentially an exponential number of such paths. Under the graph-based methods, it is not feasible to show the entire graph (or the subgraph containing all paths between them) to users as the display will become too cluttered (as shown in Figure 1(a)). With summarized-based methods, the resultant view resulted in information "loss" -the vertices of bingf ish and kristy are not shown at certain levels. Therefore, for the aforementioned query, both approaches cannot effectively facilitate exploration.
In this work, we advocate an alterative approach that displays a subgraph (called HA-graph) containing a subset of the actual vertices (called hubs) between bingfish and kristy 1 as well as summaries of the relationships and information between these vertices.
Such an approach allows users to be engaged with the original/source vertices (rather than virtual vertices), and the consolidated summary information of the hidden vertices (i.e, vertices that are not hubs in the current graph). Our approach may be viewed as a generalization of the above two approaches: if all vertices are chosen as hubs, it becomes a graph visualization approach; if no hub is selected, it becomes a graph summarization approach. We have developed VCExplorer (Vertex and Consolidation Based Explorer), a novel graph exploration framework that does just precisely what we advocate. VCExplorer starts by accepting a new type of graph exploration query (denoted as GE-query) that is formally defined in Section II. The following is an example GE-query, denoted by SQ1, on the social network graph G in Figure 1:
SELECT TopMaxDegreeVertices(G', 2) FROM Subgraph(G, kristy, bingfish, 4) G' GROUP BY betweenness() SUMMARIZE BY relationshipStrength(), relationshipType(), vertexCount()
Given a GE-query, VCExplorer first derives the target subgraph to be explored. For social network applications, we expect users to explore relationships among people close to each other. In SQ1, the FROM clause specifies the subgraph of interest to be explored by using a user-defined function Subgraph, which extracts the subgraph G ′ of G that consists of all vertices/edges along paths (with a path length of at most 4 hops apart) between a specific pair of vertices, bingf ish and kristy 3 . The SELECT clause identifies a set of hubs using a user-defined function, TopMaxDegreeVertices(G', 2), which returns a set of two vertices in G ′ with the maximum vertex degree; these hubs represent the two most influential people connecting bingf ish and kristy. For SQ1, suppose that David and karlf un are the top 2 vertices selected. Unlike graph visualization methods, only the hubs will be displayed in the resultant graph (as shown in Figure 1(b)). In this way, it is visually more appealing since fewer but more important vertices are being displayed.
Given the hubs (including vertices kristy and bingfish), the GROUP BY clause then induces a subgraph of G ′ between every pair of the hubs using a user-defined function which determines for each induced subgraph G ′ (x, y) (wrt a pair of hubs x and y in G ′ ) and for each vertex v in G ′ , whether v is contained in G ′ (x, y). For SQ1, the betweenness function in the GROUP BY clause includes a vertex v in an induced 1 Note that both vertices bingfish and kristy are also hubs. 2 The network is consisted of bi-directional edges of the input Twitter network. For clearness, we draw bi-directional edges as undirectional ones in Figure 1 3 If bingf ish and kristy are more than 4 hops apart, then we should use that distance to bound the search space. subgraph G ′ (x, y) if v is along some path between x and y in G ′ . One edge belongs to G'(x, y) if its two vertices are in G'(x, y). Note that a vertex/edge could be contained in multiple induced subgraphs.
The SUMMARIZE BY clause specifies a list of userdefined aggregation functions to compute summary information for each of the induced subgraphs. In SQ1, the user is interested in the following three summary information for each induced subgraph G ′ (x, y). The first is the closeness of the two hubs x and y based on the trust propagation among the users in G ′ (x, y) [5] computed by the relationshipStrength function. The second is the most representative relationship between the two hubs, such as "friend's friend" relationship; the relationshipType function returns the concatenation of the relationship types along the shortest path between x and y. The third is a count of the number of vertices in the induced subgraph G ′ (x, y) which is computed by the vertexCount function.
In general, all the information discovered can be visualized as a graph, referred as a Hub-based Aggregation Graph (HAgraph) in this paper. In the resultant HA-graph, the vertices are the hubs and edges are the connections among them which will be associated with the summarized information. For instance, the resultant HA-graph of SQ1 is shown in Figure 1(b). The HA-graph is much clearer than visualizing all the vertices in the underlying graphs. In addition, the HAgraph allows users to navigate and explore by zooming to the next level. To analyze the reason why kristy and karlf un is weakly connected, the analyst may zoom in to the subgraph between kristy and karlf un by issuing another GE-query. The resultant graph is shown in Figure 1(c).
B. Contributions
Our contributions may be summarized as follows:
• We present VCExplorer, a novel graph exploration framework. VCExplorer combines the innovative ideas of graph visualization and graph summarization. On one hand, it shows a subset of vertices each time without cluttering the display; and on the other hand, it summarizes information of "hidden" vertices. Compared to traditional graph visualization approach, VCExplorer is able to provide much clearer and useful information. It also offers an effective mechanism to navigate through the graph.
• We illustrate how VCExplorer framework can be designed by incorporating existing technologies. Each component of VCExplorer actually covers many research problems and most of them have been studied for a long time. We further study how the newly emerged graph aggregation can be well integrated with the VCExplorer as one approach to summarize the relationship between two hub vertices. We propose and study efficient algorithms to share computations to salvage partial work done.
II. VCEXPLORER: THE BIG PICTURE
It is interesting and challenging to develop techniques to support graph exploration in real-time. In this section, we introduce VCExplorer by giving an overview of its features and components.
A. Graph Exploration Query
The exploration starts by accepting a user's query defined as follows.
Definition 1: A graph exploration query (GE-Query) is used to explore a data graph G by identifying a subgraph G ′ of interest, a subset of interested vertices (i.e. hubs) in G', and computing summarized information for each subgraph induced by every pair of hubs in G ′ . A GE-Query is characterized by five components (G, π, σ, γ, {τ 1 , ⋯, τ n }) which can be expressed using the following syntax:
SELECT σ(G ′ ) FROM π(G) G ′ GROUP BY γ(G ′ , x, y) SUMMARIZE BY τ 1 (G ′ (x, y)),⋯,τ n (G ′ (x, y)) where
• G is an input data graph from which a subgraph G ′ of interest is extracted from a user-defined function π().
• σ is a user-defined function to return a set of hubs from the subgraph of interest G ′ . Possible selection criterias for σ include "selecting vertices with a specific attribute value", "selecting the top k vertices with maximum closeness centrality value" and so on. For each selection criterion, the system may build an index to accelerate the computation of the selection.
• γ is a user-defined function to compute an induced subgraph of G ′ , denoted by G ′ (x, y), for each pair of hubs (x, y) from σ(G ′ ). An example of γ is the InBetween function illustrated in SQ1, whose computation can be accelerated using some reachability index.
• Each τ i is an aggregation function to compute some summarized information for each of the induced subgraphs G ′ (x, y). The summarized information could be pathrelated information (e.g., shortest path length), aggregation information (e.g., aggregate graph based on different attributes like in graph OLAP [2]). In [6], we have developed aggregation sharing algorithms by utilizing the overlaps between subgraphs to share computations.
B. Hub-based Aggregation Graph
The output of a GE-query is formally defined as a HAgraph.
Definition 2: Hub-based Aggregation Graph (HA-graph): Given a GE-query (G, π, σ, γ, {τ 1 , ⋯, τ n }), the result is a graph called the HA-graph H = (V , E), where V is the set of hubs extracted from the subgraph of interest π(G); note that the set of hubs also include any vertex argument in π function for computing the subgraph of interest. E = {(x, y) x, y ∈ V, γ(π(G), x, y) is a non-empty graph}. Each vertex v ∈ V is associated with a set of attribute values inherited from the corresponding vertices in G. Each edge (x, y) in E is associated with a set of summarized values {t 1 , ⋯, t n } where each t i = τ i (γ(π(G), x, y)) is an aggregated value computed by the aggregation function τ i on the induced subgraph for the pair of hubs (x, y). Figure 1(b) shows the resultant HA-graph for SQ1, which consists of two most influential users between kristy and bingf ish. The labeled edges between a pair of vertices indicate the summarized information for the induced subgraph betwen the vertices. For instance, the edge (kristy, karlf un) in Figure 1(b) indicates that the number of vertices in the induced subgraph between kristy and karlf un is 19, the shortest path between them in the induced subgraph is 3
consisting of three f riend edge labels along this shortest path, and they have weaker relationship strength comparing with other pairs of hubs.
C. Navigation
It is essential to provide navigation capabilities in graph exploration. This is to allow users to interact and explore large graphs. In general, zooming operations are quite indispensable and useful. Given a HA-graph, users can zoom-in on an induced subgraph G ′ (x, y) by clicking on its corresponding edge (x, y). Another way for users to zoom-in is to select a subset of the vertices in the HA-graph; the collection of induced subgraphs among the selected vertices would form a new subgraph of interest to be further explored.
III. FRAMEWORK DESIGN
After defining VCExplorer framework, we turn to the design of such framework. Specifically, we discuss how to utilize existing techniques to design efficient algorithm for each component. Due to space limit, we will not go too far into the technical details.
A. Hub Vertex Generation
Hub vertices are selected using the Γ function which is based on some measures, such as vertex attribute, importance values, etc. According to the variability of measure value, we classify them into two categories:
Static function: whose measure values does not change during subgraph navigation. Such measures include vertex attributes and derived attributes. Take Twitter network as example. Γ function "Users whose age is above 80" takes age as measure which is an attribute native to vertex and remain static during navigation. Γ function "Top-10 Americans rank with closeness centrality value in ascending order" is built on closeness centrality. As closeness centrality measure is defined in the context of whole graph, during navigation, the closeness centrality value do not change in the context of new subgraphs. In this context, closeness centrality is in fact a derived attribute for vertex.
Since this kind of measures are static, it can be precomputed (for derived attribute) and indexed. For example, we can precompute the Twitter Closeness Centrality for every vertex and index them using B + -tree. When processing Γ, we can directly refer to the B + -tree for a candidate list and thus boost the Γ computation.
Dynamic Function: whose measure values change accordingly during subgraph navigation. For example, given a Γ which computes "Top-10 Americans rank with Closeness Centrality in ascending order", here Closeness Centrality measure implicitly refers to current subgraph that consists of American users and following relationship between these users. During navigation, since subgraph is changing, the Closeness Centrality also changes.
Dynamic measures are often not easy to index, a commonly used technique is to compute the measure at run-time. When online computing is time consuming, we generally have two alternatives: 1) use approximated measure to speed up. For example in the case of Closeness Centrality, we can adopt the approximate scheme as in [7], [8]. 2) precompute some intermediate results. In the case of Closeness Centrality, we can compute all-pair shortest distance first. Since during navigation, subgraphs are extracted based on reachability property, all shortest distances are valid locally. With the knowledge of shortest distance, the Closeness Centrality can be efficiently computed.
B. Subgraph Extraction
Before consolidation, subgraphs between any pair of hub vertices are extracted. By default, a betweenness function is used. That is for a vertex v and two hub vertices sv 1 , sv 2 , v is between (sv 1 , sv 2 ) iff sv 1 ↝ v and v ↝ sv 2 . Given an exploring graph G and a set of hub vertices SV , subgraph extraction can be translated as: ∀v ∈ G, compute two sets:
S(v) = {sv sv ∈ SV ∧sv ↝ v}, R(v) = {sv sv ∈ SV ∧v ↝ sv}.
The Cartesian product of S(v) and R(v) denotes the set of subgraphs v belongs to. Reachability index can be used to boost the extraction process. If the index is built and extraction is based on betweenness measure, we can use the following approach:
Index based extraction: Given a reachability index, the extraction can be performed as follows: ∀v ∈ G ∀sv ∈ SV , conduct two reachability tests v ↝ sv? and sv ↝ v?. And then update its S and R lists accordingly. Many reachability indices are developed in literature, such as transitive closure, 2-hop [9], highway [10], dual-labeling [11] etc. Due to betweeness measure, it is easy to see that reachabiilty relationship holds in any subgraphs. Therefore, the index can be reused in further navigation.If the reachabality index is unavailable, we can adapt graph traversal based approach instead.
Non-Index based extraction: We first preprocess the graph to assign each vertex v with a topological order number. Circles are condensed and vertices in the same circle share the same order number. Then we process the vertices topologically. For every vertex, it will push its S lists to all its immediate children. Each children unions all the S lists it received from its father to form its own S. The procedure to compute R lists is similar but in a reversed manner. By so doing, the subgraph is extracted, but is slower than index based approach.
C. Consolidation
After subgraph extraction, consolidation is performed on each subgraph. According to object type to be consolidated, graph consolidation can be further classified into following categories:
Attribute-based consolidation: consolidation that is only operate on vertices (edges) attributes or derived attributes. Typical operators are SUM, COUNT, AVG, etc. Since all the vertices (edges) are known at this stage, we can retrieve related attributes from the vertex (edge) attribute table. If any index on vertex(edge) ID is present, we can directly retrieve the target attributes, otherwise one scan on vertex (edge) attribute table will be introduced.
Structure-based consolidation: consolidation that is only related to graph structure. Typical operators include shortest distance (path), minimum cut etc. These problems are wellstudied in the literature. Taking shortest distance as example, we have several algorithms to choose from: 1) In uniweighted graph, a BFS from a hub vertes is sufficient to compute all the shortest distance to other hub vertices; 2) In weighted graph, a Dijkstra's algorithm is applicable; 3) If shortest distance indices [12], [13] is available and subgraphs are extracted based on betweenness function, the distance can be efficiently derived.
Attribute and Structure based consolidation: consolidation that is related to both graph structure and attributes on vertex (edge). Prominent example in this category is graph aggregation. Several algorithms has been developed recently [14], [2].Since these schemes focus on single graph computation, one naive solution is to run these schemes for each subgraph. Unlike the above two category where proper indices can boost the consolidation, graph aggregation is more complex and no indexing scheme is available. In the next Section, we will give an efficient algorithm to perform graph aggregation on multiple graphs.
D. Visualization
A HA-graph usually has at most tens of vertices, and hundreds of edges, thus most layout algorithms [15] is able to handle it. In additional to displaying HA-graph structure, we also display the consolidated information for each edges in the HA-graph. A consolidated information can be a single value (i.e., COUNT(.)), a list (i.e., a shortest path, a set of groupvalue pairs) or a attributed graph (i.e., an aggregate graph). Given the diversity of the information, we create two modes for displaying the information on edges.
Data Mode: we display results in raw data format. A single value is a label to an edge; A path is displayed as a list of vertices and is attached to edge. A graph is displayed as 2-D tables with each row representing an edge or a vertex in aggregated graph.
Graph Mode: we display results in graph format. A single value is still a label; A path is displayed as a chain; and A graph is displayed in vertex-edge format.
Users are flexible to toggle between two modes for each edge. We adapted several other interaction designs which are more friendly to users. In HA-graph, user can enable hover features, then all results on edges are hidden and only shown when mouse is moving over. In data mode, user are freely to perform selection, projection, sorting on 2-D table.
IV. SHARING-BASED ONLINE AGGREGATION Efficient online aggregation for multiple subgraphs is the key to provide user better exploration experience. In this section, we introduce how to conduct the graph aggregation for multiple subgraphs online. We first introduce some preliminaries followed by one naive aggregation algorithm -SN (Shared Nothing) algorithm. Then we introduce the proposed aggregation algorithm, AS (Aggregation Sharing) algorithm, that provides an efficient aggregation by sharing the computation. Graph aggregation offers a high level view of the attribute graph [14]. Integrating graph aggregation with VCExplorer is of great help to provide users the summarized information of the subgraphs which are unable to display. In this work, we will focus on the discussion of the distributive and algebraic functions (e.g. SUM, COUNT, Max, Min etc.) which can be applied to the subset of the edges or vertices in one graph. For these functions, the final results can be further calculated based on the result of each subset. For illustration, we take directed graph in Figure 2 as input graph and use betweenness function to find out target subgraphs one vertex belongs to. Other types of aggregation functions and graphs should be addressed similarly.
A. Preprocessing 1) Handling SCC: For directed graph, when betweenness is set chosen as the manner to extract the influential subgraph between two hub vertices, once one of the vertices in a SCC (strongly connected component) is in the subgraph, the entire SCC will be in the subgraph. Therefore, one optimization can be adopted here is to preprocess each SCC in advance.
In graph aggregation, each SCC can be pre-aggregated together and condensed into a super vertex. The super vertex will be associated with the pre-aggregate value of the SCC. In so doing, the original graph becomes an acyclic graph which is our discussion focused on in the later. Note that many existing works have been proposed and can be adopted here to detect the SCC, such as the Tarjan's strongly connected component algorithm that runs in O(V + E).
2) Tags Generation: For illustration, we first definition vertex and edge tags which will be used later. In G, every vertex is associated with a conceptual tag indicating which influential subgraph it belongs to in the HA-graph G s .
Definition 3: Vertex Tag: T (v) is a tag for every v ∈ G. T (v) = S(v) • R(v), where S(v) = {u u ∈ G s ∧ u ↝ v} and R(v) = {u u ∈ G s ∧ v ↝ u}.
Intuitively, S(v) denotes the hub vertices which can reach v in G and R(v) denotes the hub vertices which can be reached by v in G. T (v) is formed by concatenating the two lists. For instance, Figure 2 indicates a simple graph where vertices 1, 2, 3, 4 and 5 are selected as the hub vertices. In this example, vertex A 1 's tag is < 1 >< 2, 3, 4, 5 >, which means vertex 1 can reach A 1 and A 1 can reach < 2, 3, 4, 5 >. We refer to the list of S(v) as T S (v) and R(v) as T R (v).
On the basis of tag definition, given a tag T (v) of v, Cartesian product of T S (v) and T R (v) represents the infulential subgraphs v belongs to. In addition, we also define the the size of the Cartesian product as the cardinality of T (v). For instance, in Figure 2, A 1 is tagged with < 1 >< 2, 3, 4, 5 > indicating that it belongs to subgraphs < 1, 2 >, < 1, 3 >, < 1, 4 > and < 1, 5 > and the cardinality T (A 1 ) is 4.
Fig. 3: SN-Agg Plan Example
Similarly, we assign the similar tag for edge tag as well. In G, a tag for e(s, t) is denoted as < T S (s), T R (t) >. For instance, e(c1, c3) is tagged with < T X (c1), T R (c3) > (¡1,2,3¿¡4,5¿). The cardinality of e(c1, c3) is 6.
To speedup generating the tags, the reachability index can be adopted here, such as transitive closure or 2-hop. For each v ∈ G and u ∈ G S , we test whether v ↝ u or u ↝ v. The total complexity is O( V * k * r), where k stands for the number of hub vertices and r stands for the cost for reachability testing between two vertices. After generating the tags for each vertex, the edge tags can be easily calculated based on the vertex tags.
B. Share-Nothing Aggregate Algorithm
Recall that there are multiple subgraphs need to be aggregated, each of which corresponds to one edge in G s . To conduct the graph aggregation, one naive approach is to aggregate each subgraph individually. Intuitively, this approach aggregate the subgraph independently without any sharing operation. Thus, we refer to this algorithm as SN algorithm -stands for shared nothing.
In SN algorithm, each subgraph extracts its own vertices and edges and further calculates its own aggregate graph independently. Take the vertex aggregation as in example. Figure 3 shows how the vertices will be processed for different subgraphs. In Figure 3, the bottom lists all the vertices and the top lists all the subgraphs. Each link between the vertex and subgraphs indicates one aggregate operation where the vertex should be aggregated to a corresponding subgraph. Thus, in SN , each subgraph (denoted by tags) receives and aggregates the vertices independently. Given a graph with n vertices, assume that S is the number of subgraphs, the complexity of vertex aggregation is O(m S ).
For the edge aggregation, if the graph is stored in the format as shown in Figure 1, there is a need to convert the vertex IDs of two endpoints of one edge to the vertex aggregate attributes. This can be done by performing a join between the edge attribute table and vertex attribute table. After the conversation, the edge aggregation can be conducted in the similar way as the vertex aggregation. Given the a graph with m edges and S subgraphs, the complexity of edge aggregation is O(m S ).
C. Aggregation Sharing Algorithm
SN is a straightforward approach as it computes the graph aggregation for each subgraph independently. However, it may incur high computation overhead as it may involve many redundant computations.
One observation is that some vertices and edges are involved the same set of multiple subgraphs. This provides us the opportunity to share the computation among different subgraphs.
For instance, in Figure 3 C 1 , C 2 , C 2 have common tag of < 1, 2, 3 >< 4, 5 > which means these three vertices are involved into the same 6 subgraphs < 1, 4 >, < 1, 5 >, < 2, 4 > , < 2, 5 >, < 3, 4 > and < 3, 5 >. Therefore, the aggregation computation can be shared among these subgraphs. C 1 , C 2 , C 2 can be aggregated once and then supply to the 6 subgraph directly, instead of aggregating them 6 times. Similarly, B 1 , B 2 can also be aggregated together then supply the result to their shared subgraphs directly. Figure 4 (a) indicates this procedure where B (resp. C) is the aggregate result of B 1 and B 2 (resp. C 1 , C 2 , C 2 ).
Another observation is that even though the tag are not exactly the same, they may still be able to share the computation once they have the shared subgraphs. One simple example is between B (< 1, 2, 3 >< 4, 5 >) and C (< 2, 3 >< 4, 6 >) which are similar but not the same. It is easy to see that they share 3 subgraphs < 2, 3 >< 4 >. We can pre-aggregate B and C where the result can be directly supplied to the 3 shared subgraphs which is able to reduce the computation overhead. Figure 4 (b) indicates such an idea.
Based on these observations, we propose a new algorithm, AS (Aggregation Sharing), on the principle of sharing the aggregation when the vertices or edges are involved into a common set of subgraphs. We refer to a common set of subgraphs as a shared component(SC). Given two tags t1 and t2, the SC can be calculated by t1.S ∧ t2.S concatenated by t1.R ∧ t2.R where ∧ means intersect. For instance, given t1 (< 1, 2, 3 >< 4, 5 >) and t2(< 2, 3 >< 4, 6 >), the SC can be calculated as < 1, 2, 3 > ∧ < 2, 3 > concatenated by < 4, 5, > ∧ < 4, 6 > which will be < 2, 3 >< 4 >. Note that to speed up SC calculation, the vertex ID lists in the tag can be stored as BitSet where the SC can be simply computed via the AND operation between two BitSets.
AS Algorithm: Discovering all the possible SCs among the tags incurs a high computation complexity that is almost 2 n where n is the number of different tags. As a realtime exploration, finding the optimal solution for finding SCs may not be practical. Therefore, in this paper, we propose a heuristic algorithm to discover the SCs by tag clustering. For illustration, as the aggregating the vertices and edges is under the similar procedure, we focus on introducing the vertex aggregation here. The similar algorithm can be easily adopted for the edge aggregation which will be omitted. The pesudo code of proposed AS algorithm is provided in Algorithm 1.
Given a set of vertices, we first generate tags for each vertex (Line 4) then sort all tags and put into a queue based on their size and their values (Line 5). The benefit of this sorting operation is two-fold. First, after sorting, it is easier cluster:=FindBestCluster(g, clusters) 13: ct:=cluster.tag 14: if Saving of combing g and cluster is positive then 15: cluster.add(g) 16: st:= ct ∧ nt 17: and fast to combine and pre-aggregate all the vertices with the same tag. Second, after sorting by size, we can guarantee that the larger tags can be clustered first. This is designed based on the fact that the longer tag it is, the larger possibility it has to provide a benefited sharing.
As the vertices with the same tags are definitely able to share their computation, for each popped tag in queue, vertices with same tags will be combined together into groups first(Line 9). This same tag combing is conducted until it reaches a different tag. Note that this coming is also a pre-aggregating procedure where the corresponding vertices information is preaggregated.
After the first step of combing vertices with the same tags, we get a list of distinct tags each of which is associated with on group and the pre-aggregated value in the group. For instance, like in Figure 3 (B), after the first combining step, B1 and B2 are combining into one group B with the tag < 1, 2 >< 3, 4, 5 > and C1, C2 and C3 are into another group C with the tag < 1, 2, 3 >< 4, 5, >.
In the second step, we discover more sharing opportunities among these distinct tags by clustering them into clusters according to their similarity. The general idea of this clustering procedure is as follows: Given a new tag, it compares all the existing clusters to find the best cluster which obtains the biggest saving value after adding the new tag into the cluster based on one saving function. The saving function will be provided in Equation 1. If the biggest saving value is negative which means adding the new tag into any of the cluster does not increase the sharing opportunity, this new tag becomes a new cluster itself. This heuristic approach guarantees that the best cluster that increases the computation sharing is chosen in each clustering step. Since the tags are in sorted order, the clustering can stop while the new tag size becomes smaller than a threshold value, like 3. This is because most likely, when the tag size is smaller, the sharing opportunity is slightly small. There is no need to cluster them. Now, we provide the saving equation used during the clustering. Assume for each cluster C i , CT i is the common tag that is the intersection among all the tags in C i . SZ i is the number of tags already in the cluster. Then the saving cost after adding a new tag nt can be calculated as follows:
saving(C i , nt) = CT i ∧ nt × (SZ i + 1) − CT i × SZ i (1)
where CT i ∧ nt is the new common tag of the cluster after adding nt to C i , CT i ∧nt ×(SZ i +1) indicates the total saving of the new cluster after adding nt, CT i × SZ i indicates the aggregation saving of C i before adding the nt. Therefore the difference between these two costs are the benefit of adding a new tag to the cluster.
After the clustering, the aggregation can be conducted for each cluster. Each tag t in one cluster C i is actually split into two parts: one is the common tag CT i and another is the differential tag DT t . Note that DT t of t is the tag that is not covered by the common tag CT i of C i . The DT t can be obtained by
T (t)−CT i . For instance, if t is < 1 >< 3, 4, 5 > and CT i is < 1 >< 3, 4 >, DT t is < 1 >< 5 >.
For the common tag in each cluster, one further aggregation based on all the groups in one cluster can be conducted. The aggregation results can be directly used to all the subgraphs indicating by the common tag. This saves the repeated aggregation among these group for each subgraph. For each member in the cluster, its preaggregate value from the first step needs to send to all the subgraphs representing in its differential tag.
V. EXPERIMENTAL EVALUATION
Environment. We conduct all the experimental evaluations on a platform with an Intel Xeon E5607 4-core CPU C SV 5 10 20 30 40 SN AS SN AS SN AS SN AS SN AS 10 1877 1383 2302 1233 3154 1009 4894 972 6175 990 100 1888 1563 2433 1234 3432 969 5375 961 6862 1001 1000 2110 1904 2823 1308 3828 1071 7281 1099 9264 1274 10000 2656 2148 3084 1506 4454 1364 7696 1522 10367 1877 100000 2927 2382 3192 1869 4370 2588 8403 2977 11452 3141 200000 3028 2620 3215 2115 4608 2469 9174 3116 13100 4049 400000 3039 3070 3294 3404 4664 3296 8587 6146 16380 5678 600000 2933 (2.33GHz), 32GB of memory with running Linux 2.6.32 64bit OS. Implementation. All algorithms are implemented using java. Transitive closure are used as reachability index to support the extraction of subgraphs.
Datasets. We perform our experimental studies on two kinds of datasets including one real Twitter dataset (provided by UIUC [4]) and a set of synthetic datasets. The Twitter dataset contains 284 million following relationships, 3 million user profiles and 50 million tweets. Each user profile has information about account age, location, etc, and Re-tweets contains information about origin, time, content, etc.
The synthetic datasets are generated using the GRAIL graph data generator. Each generated synthetic dataset is a directed attributed graph. Each vertex in the graph is associated with three attributes (vid, v grp, v mr) where v grp and v mr are the group and measure information with integer data type. Each edge is associated with four attributes (src vid, tgt vid, e grp, e mr), where e grp and e mr are edge group and measure information with integer data type as well.
A. Effectiveness
We first show the effectiveness of VCExplorer as a powerful tool to explore the Twitter graph. Given the Twitter graph, we are interested in discovering who are the most active users and what are the distributions of contact frequency among users in the influence subnetwork between them. We use count of tweets between two users to compute their contact frequency. Bigger f requency is, stronger relationship they are. Further, we classify f requency into three categories(Closeness): High, Middle, and Low.
The GE-query may be expressed as follows:
SELECT TopMaxDegreeVertices(twitter,3) FROM twitter GROUP BY betweeness() SUMMARIZE BY COUNT(.) e. Closeness() Resulting graph is shown in Figure 5 (a). Distribution of different closeness categories of each subgraph are annotated on edge. From Figure 5(a) we may see that there is one circle between u1 and u2 which causes other edges (u1, u3) and (u2, u3) have the same distributions. So we change to another betweenness function to eliminate the circle affection: replace the betweenness function with betweenness(h) which check whether one vertex may reach another vertex within h hops. Figure 5(c) is the resulting HA-graph while h = 4. One remarkable change is, high closeness relationships between u1 and u2 has been reduced from 7 to 2. Such remarkable change leads us to analyze the subnetwork between u1 and u2 deeply. We may issue a zoom query over subgraph between u1 and u2 with k = 2 and h = 4, zoom operation output a new HA-graph as shown in Figure 5 (b). From the aggregate values on edge, it is easy to see that most strong relationships between u1 and u2 are between u4 and u5, which indicates that middle users between u4 and u5 have stronger relationships.
B. Performance Evaluation
In this section, we evaluate the performance of our proposed graph aggregation algorithm. Two algorithms are implemented and compared including the baseline algorithm SN -shared nothing algorithm as discussed in section IV-B and the AS algorithm -Aggregation Sharing algorithm as proposed in section IV-C. Note that all the following experiments are conducted three times and the average performance is reported.
The GE-query used is provided as follows:
SELECT TopMaxDegreeVertices(k) FROM G GROUP BY betweeness() SUMMARIZE BY SumVMrByVGrpEGrp(), SumEMrByVGrpEGrp()
For simplicity, the GE-query used during the following experiments is to identify the top k hub vertices with the maximum degree and summarize the relationship between two hub vertices by calculating the aggregate graph based on dimension v grp and e grp using SumVMrByVGrpEGrp() and SumEMrByVGrpEGrp() function which summarize v mr and e mr measures respectively by v grp and e grp.
Towards a comprehensive study, we study the impact of the number of hub vertices, graph dimension cardinality, graph degree and graph size accordingly. It is worthy of noting that the aggregation performance is affected by the cardinality of vertex group-by dimension and edge dimension together. These cardinalities will affect the final total different number of group-by values. Therefore, for simplicity, in the following experiments, we refer to the final total different number of group-by values of both vertices and edges as the cardinality.
Impact of the number of hub vertices. In this experiment, we first study the benefit of graph aggregation sharing over multiple sub-graphs when we vary the number of hub vertices (SV) from 5 to 40. We conduct the experiments over two different types of graphs: one with graph degree 8 representing a relative spare graph and another with degree 40 representing a relative dense graph. All the graphs used in these set of experiments consist of 30K vertices. Table I and Table II show detailed results for the graphs with degree 8 and 40 respectively. Note that each row indicates the execution time of different algorithms while selecting different number of hub vertices on the same graph with a specific cardinality showing the most left column.
From the result, we have the following findings: First, SN and AS have different reactions when change SV. While SV increases, the execution time of shared nothing SN algorithm increases as well. The reason is that, as more hub vertices generate more influential subgraphs which leads more vertices and edges are involved into recomputation. Differently, AS does held this pattern. As shown in the result, while SV increases, the execution time does not increase as much. For some cases, it is even decreasing. For instance, in Table 1, the execution time of AS with SV=10 is always smaller than the one with SV=5 for smaller cardinality. This, however, is reasonable, as more hub vertices and smaller cardinality mean more sharing opportunities.
Second, as SV increases, AS outperforms SV more. As shown in Table 1 and 2, the execution time of SN increases dramatically while SV becomes larger. However, AS is more stable which leads AS outperforms SN more.
Impact of cardinality. Table 1 and 2 also indicate how the performance changes when we vary the cardinality from 10 to 1,000,000. As expected, SN outperforms weakly AS only when the cardinality is large enough and SV is small. For instance, in Table 1, when SV=5, SN becomes faster than AS when the cardinality reaches 400,000(Italic numbers). This is because a larger cardinality reduces the opportunity of sharing operation.
Impact of graph degree. In this set of experiments, we study the performance comparison among different algorithms while we change the graph degree from 2 to 80. These experiments are conducted with SV=20 and C=10K based on the graphs with 10,000 vertices. Figure 6 (a) and (b) show the execution time(solid line) for relative sparse graph and dense graph respectively. From the result, we can see that as the degree increases, the query execution time of all the algorithms increases as well. It also indicates that AS is more stable than SN.
To better understand, how many add operations are saved by the sharing algorithm. We collect the total number of add operations and show them as dash line in Figure 6. From Figure 6, we can see that the reason the AS can outperform SN dramatically is because it saves many add operations by sharing. In average, AS saved 74% and 60% add operations in dense graph and sparse graph respectively compared to SN.
Impact of the number of vertices. In this set of experiments, we study how the performance changes while we fix the graph degree but vary the number of vertices from 10K to 40K. Figure 7 (a) and (b) provide the execution time (solid line) based on the graphs with degree 8 and 40 each of which represents relative sparse or dense graphs. The results indicate that the execution time of SN algorithm increases faster than the execution time of AS algorithm when the number of vertices is increased. We further calculate the number of add operations incur in each experiment as shown as dash line in Figure 7. It is easy to see that the number of add operations in SN algorithm becomes much larger than the ones in AS algorithm. These experiments also show that both SN and AS scale linearly when vertex number increases.
Time Distribution Analysis. To better understand the proposed AS algorithm, we run a set of experiments and count the running time of each part, including tagging generation (referred as Tag), subgraph extraction (referred as SGExt), planning time (referred as Plan) and aggregation time (referred as Agg). The experiments are conducted over a set of graphs with C=10,000 by running the query with different number of hub vertices (5, 20 and 40). Note that the x axis indicates the graph size used. For instance, 10K-80K means the graph consists of 10K vertices and 80K edges. 40K-1600K means the graph consists of 40K vertices and 1600K edges and so on. The results indicate that the planning is very fast compared with other operations. The tagging time and subgraph extraction time occupy about 13% and 38% of total query time in average respectively. In whatever cases, the aggregation time takes the big portion of the total execution time.
VI. RELATED WORK
A great challenge in graph analytic is to deal with the presence of large attributed graphs. Related work on graph analytic can be summarized as follows:
Graph Layout Drawing aims to display whole graph in a user friendly way. Classic graph drawing algorithms are surveyed in [15]. Those algorithms can structurally display small graph on the screen. In order to enable user discerning on interesting vertices and edges, some discriminating methods are proposed in the literature. Position discriminating methods [16], [17] place vertices with high centrality [18], [19], [20] near the center of graph. Some other works [21] use Size discriminating methods by displaying vertices with high importance value in larger circles or using prominent colors [22]. All these algorithms suffer from the volume of graphs. When graph size is up to tens of thousands vertices and edges, the screen will be filled up with dots, and the link information among vertices is barely seen. In contrast, our SVExpolorer displays sketch graph which contains less vertices and consolidated information between vertices. By so doing, user will not get overwhelming points in the display.
Graph Simplification aims to reduce graph size prior to above layout algorithms. Several approaches are developed for this purpose. [23], [24] group strongly connected vertices and edges into metanodes. [25] merges edges in the same simple path or routes, [26], [27] condense non-planar graph into planar graphs, and [28], [29], [30] form edge bundles by some metrics. [31], [32] uses clustering based approach to form hierarchical view of the graph, which supports navigation. [33] reduces graph size by displaying only nodes and neighborhoods that are most subjectively interesting to users. However, all these methods cannot handle attributed graphs as in our case. First, since vertices and edges to be retained in the simplification algorithm is selected automatically, users are not feasible to choose particular points and view the relationships among them. Second, most of these methods only concern the structure of graphs, the attributes of vertices and edges are not preserved. On the contrary, VCExplorer enables users arbitrarily picking of the interesting vertices, and further provides consolidated information among these vertices.
Graph Summarization aims to provide a succinct highlevel graph by consolidating vertex's attributes and edge information. Vertices and edges belonging to the same metric are viewed as metanodes and edges. Aggregated information from detail vertices and edges are attached to the metanodes and edges. [1] develops k-SNAP method for cluster graph into k groups. [2], [14] has proposed graph aggregation methods which group the graph based on vertex and edge attributes. These methods offer good overview of graph attributes in a succinct way, but they do not position important vertices and their relationships. Although [34] summarizes graph according to the importance and relatedness of vertices, it focus mainly detailed vertices. Differently, our VCExplorer displays important vertices as hub vertices and reveals the relationships between them using consolidation techniques.
Fig. 1 :
1A running example of VCExplorer. (a) A derived Twitter network dataset 2 with 5k vertices and 18k edges visualized by Cytoscape [4] (b) output HA-graph of SQ1. (c) HA-graph after zooming in edge (kristy, karlf un) in (b). In (b) and (c), the width of an edge represents the relationship strength of the induced subgraph represented by the edge; and each edge is labeled with its representative relationship type as well as a count of the number of vertices in the associated induced subgraph.
Fig. 4 :
4Sharing Plan Example Algorithm 1 Aggregation Sharing Alogrithm 1: INPUT: vertices 2: aggP lan:=null; 3: clusters:=null; 4: tags:= genTags(vertices) 5: queue:= sort(tags) 6: while !queue.isEmpty do
Fig. 5 :
5HA-Graphs over Twitter Network.
Figure 8 (
8a), (b) and (c) present the overview of query execution time distribution with SV=5, 20 and 40 respectively.
Fig. 6 :
6Scalability vs graph degree.
Fig. 7 :
7Scalability vs graph size.
Fig. 8 :
8Time distribution for AS.
nt s :=nt − s 18: ct s :=ct − s19:
ct:=st
20:
aggP lan.add(ct s , cluster)
21:
aggP lan.add(nt s , aggGraph)
22:
else
23:
clusters.newCluster(tag, g)
24:
end if
25:
else
26:
aggP lan.add(tag, g)
27:
end if
28:
end if
29: end while
30: aggregate(aggP lan)
TABLE I :
IAggregate performance over dense graph (ms)
P
P
P
P
P P
TABLE II :
IIAggregate performance over sparse graph(ms)
P
P
P
P
P P
C
SV
5
10
20
30
40
SN
AS
SN
AS
SN
AS
SN
AS
SN
AS
10
502 418
625 375 664
229
963
227
1136
226
100
513 428
665 377 708
231 1019
234
1357
244
1000
548 464
682 401 776
276 1126
385
1544
345
10000
573 516
713 446 843
360 1240
558
1604
540
50000
587 584
537 389 923
581 1328
721
1677 1012
100000
593 633
760 681 898
841 1284 1013
1775 1219
150000
602 650
584 568 884
863 1340 1155
2221 1581
200000
645 674
763 795 951
996 1305 1286
2508 1699
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| []
|
[
"Edge Profile Super Resolution",
"Edge Profile Super Resolution"
]
| [
"\nSungkyunkwan University\n16419SuwonRepublic of Korea\n",
"\nBig Data & AI Lab\nHana Institute of Technology\nHana TI06133SeoulRepublic of Korea\n"
]
| [
"Sungkyunkwan University\n16419SuwonRepublic of Korea",
"Big Data & AI Lab\nHana Institute of Technology\nHana TI06133SeoulRepublic of Korea"
]
| []
| Jiun LEE 1[0000−0002−7338−4565] , Inyong YUN 2[0000−0001−8082−033X] , and Jaekwang KIM 1[0000−0001−5174−0074] Abstract. In this paper, we propose Edge Profile Super Resolution(EPSR) method to preserve structure information and to restore texture. We make EPSR by stacking modified Fractal Residual Network(mFRN) structures hierarchically and repeatedly. mFRN is made up of lots of Residual Edge Profile Blocks(REPBs) consisting of three different modules such as Residual Efficient Channel Attention Block(RECAB) module, Edge Profile(EP) module, and Context Network(CN) module. RECAB produces more informative features with high frequency components. From the feature, EP module produce structure informed features by generating edge profile itself. Finally, CN module captures details by exploiting high frequency information such as texture and structure with proper sharpness. As repeating the procedure in mFRN structure, our EPSR could extract highfidelity features and thus it prevents texture loss and preserves structure with appropriate sharpness. Experimental results present that our EPSR achieves competitive performance against state-of-the-art methods in PSNR and SSIM evaluation metrics as well as visual results. arXiv:2011.05308v3 [eess.IV] 12 May 2021 | null | [
"https://arxiv.org/pdf/2011.05308v3.pdf"
]
| 226,289,817 | 2011.05308 | 30e5421aad2157ca6925b52e6bf6e299a736564e |
Edge Profile Super Resolution
Sungkyunkwan University
16419SuwonRepublic of Korea
Big Data & AI Lab
Hana Institute of Technology
Hana TI06133SeoulRepublic of Korea
Edge Profile Super Resolution
Jiun LEE 1[0000−0002−7338−4565] , Inyong YUN 2[0000−0001−8082−033X] , and Jaekwang KIM 1[0000−0001−5174−0074] Abstract. In this paper, we propose Edge Profile Super Resolution(EPSR) method to preserve structure information and to restore texture. We make EPSR by stacking modified Fractal Residual Network(mFRN) structures hierarchically and repeatedly. mFRN is made up of lots of Residual Edge Profile Blocks(REPBs) consisting of three different modules such as Residual Efficient Channel Attention Block(RECAB) module, Edge Profile(EP) module, and Context Network(CN) module. RECAB produces more informative features with high frequency components. From the feature, EP module produce structure informed features by generating edge profile itself. Finally, CN module captures details by exploiting high frequency information such as texture and structure with proper sharpness. As repeating the procedure in mFRN structure, our EPSR could extract highfidelity features and thus it prevents texture loss and preserves structure with appropriate sharpness. Experimental results present that our EPSR achieves competitive performance against state-of-the-art methods in PSNR and SSIM evaluation metrics as well as visual results. arXiv:2011.05308v3 [eess.IV] 12 May 2021
Introduction
Single Image Super-Resolution(SISR) [10] has been focused on recently. Generally, SISR targets to reconstruct an accurate high resolution(HR) image from its degraded low resolution(LR) image. Image super-resolution(SR) is usually applied to diverse computer vision tasks (e.g. security and surveillance imaging [55], object recognition [33], image generation [16], and medical imaging [34]). Since there are plenty of solutions for reconstructing any LR inputs, image SR has an ill-posed inverse [8] problem. For highfidelity image, it is necessary to represent details including high frequency components such as texture and structural information. To address this issue, numerous SR methods have been proposed, such as conventional methods [50,7,9,36,44] and deep learning methods [52,23,22,20,42,35].
In conventional methods, edge-based models [9,36,44] enhance sharpness of super resolved image by utilizing edge statistics. They model edge statistical dependencies by estimating structural connectivity between HR and LR. However, edge distribution tends to be heavily dependent on the similarity between training and test datasets. Therefore, the performance lacks consistency. Furthermore, since they focus on sharpness of SR image, they have weakness on improvement of texture restoration. The modeling is proceeded point by point. Hence, the process of edge generation is complex and inflexible.
On the other hand, deep learning methods are more flexible and remarkable in handling probability transformation including pixel distribution. They acquire outstanding results compared with previous methods [11,3] recently. Normally, deep learning methods approach SISR problem by utilizing influential feature representation and deep end-to-end structure. These models [23,20,52] achieve notable improvement in visual quality. In this case, most of them are optimized as measuring pixel distance between SR and its corresponding HR by MSE or L 1 . This optimizing methods tend to make the networks generate an image based on statistical information of possible HR solutions. Even though they reach high numerical value evaluation on peak signal-to-noise ratio(PSNR), general deep learning models show blurry with texture loss and structural trouble results.
To represent texture and preserve image structure, Yang et al. [45] applies simply edge information in deep learning model. However, they utilize edge information as assistant device and design their model to reach higher PSNR evaluation metric and thus using structure information is inadequate. For perceptual improvement, some methods such as [22,42,35] utilize the generative adversarial network(GAN) with perceptual loss to generate photo realistic image. Although these perceptual-driven models bring perceptual enhancement by restoring texture information related to blurry problems, they can not avoid structural distortions in details with definite edges. To overcome the image structural limitation, some models [24,27] utilize structural information by designing additional module for preserving structure. The models feed explicit guidance to an established perceptual-driven model for solving structure problems in SR. Even though they compensate the structural defects of GAN-based model, they do not still reach the visual quality of HR images. Furthermore, since the discriminators may bring unstable factors during optimization procedure, GAN-based models have difficulties in stability of learning process and keeping structural consistency.
In this paper, we propose an Edge Profile Super Resolution(EPSR) method to alleviate the issues that we have mentioned above. In SISR problem, to generate high quality SR image, it is important to represent high frequency details such as structure and texture information. Since these components have frequent pixel variations, they have contextual properties and thus displaying them is the crucial point for high quality results. To achieve the goal, we modify Fractal Residual Network(FRN) as network structure to utilize various information in learning process. we call it modified Fractal Residual Network(mFRN) structure. To draw high frequency components from diverse information, we construct Residual Edge Profile Blocks(REPBs) as basic blocks. REPB consists of Residual Efficient Channel Attention Block(RECAB) module, Edge Profile(EP) module and Context Network(CN) module. For extracting high-fidelity features, it is necessary to utilize informative features which contain detail information. Hence, by referring to previous methods [23,51] and recent research [31], we apply ECA on feature extraction. This systemically organized feature provide abundant information to EP module. EP module feeds thus structural information on the features by generating edge profile itself from the informative features. This module is based on principle of conventional edge extraction. Even though EP module contributes to preserving image structure, exploiting high frequency components such as sharpness and textures should be consid-ered for more high-fidelity results. However, these contextual details contain complex variations in specific regions(i.e. high frequency regions such as edge and texture) and thus there are difficulties to maintain the detail information in process. To exploit high frequency components, we construct Context Network(CN) module. By exposing contextual information, this module captures pixel variation and thus sharpness of results could be enhanced properly and texture loss also be restored. By proceeding repeatedly this process in network, SR results shows structural stability and representing details with reducing texture loss and structure distortions. Experimental results on benchmark datasets demonstrate that our EPSR achieves in improving SR quality.
Related Work
In the computer vision community, various SISR methods have been proposed for several years. To be related with our proposed method, we review on SISR methods into three categories: Edge-related methods [9,36,44,39,54,45,24,27], General deep learning method [5,17,18,37,38,23,52,51,4] and Perceptual-driven method [15,22,33,42,41,32] General deep learning method Recently, general deep learning methods have been mainly studied in single image super resolution. SRCNN proposed by Dong et al. [5] achieves noteworthy performance using three-layer convolutional network. Later, VDSR [17] and DRCN [18] improve accuracy with stacking convolutional networks deeply through residual learning. Tai et al. [37] introduce DRRN, which is a recursive learning model based on parameters sharing and they propose MemNet [38], which consists of memory block for a deep network. EDSR and MDSR by Lim et al. [23] improve significantly the performance by stacking residual blocks very deeply and widely. From the results, the depth of network is a key point in image SR. Since the achievement of deep networks, RDN by Zhang et al. [52] is designed as a deep network based on the dense block for utilizing all of the hierarchical features from all the convolutional layers. Zhang et al. [51] and Dai et al. [4] consider not only increasing the depth of network, but also applying feature correlations in spatial and channel dimension. From the investigations, general deep learning methods target to achieve high PSNR performance by utilizing feature information efficiently. Perceptual-driven method As aforementioned, all general deep learning methods concentrate on achieving high PSNR. However, their results display blurry and unstable structural SR images. For recovering SR image more toward realistic direction, Johnson et al. [15] propose perceptual loss to enhance the visual quality of SR images. Ledig et al. [22] design SRGAN based on adversarial loss and it is the first model that can generate photo-realistic HR images. EnhanceNet by Sajadi et al. [33] shows high-fidelity textures SR images by applying texture loss. Wang et al. [42] propose ESRGAN which enhances the previous frameworks by constructing Residual-in Residual Dense Block(RRDB). On the other hand, Wang et al. [41] generate more natural textures for specific categories by exploiting semantic segmentation maps as priors. In addition, SROBB by Rad et al. [32] is proposed to a objective perceptual loss based on the labels of object, background and boundary. These perceptual-driven methods shows enhancement in overall visual quality. However, they leave problems of structure distortions and fails recovering details such as texture. Edge-related method Edge and gradient information has been utilized in previous SISR works. Fattal [9] proposes a method learning the prior dependencies among edge statistics of image gradients. Sun et al. [36] propose a gradient field transformation to control HR gradient fields and enhance sharpness. Yan et al. [44] propose a method based on gradient profile sharpness extracted from gradient description models. Tai et al. [39] propose an approach to combine edge-directed SR with detail from an image and texture examples. These models are dependent on connectivity and relation HR and LR. Thus the results are decided by similarities between train and test datasets. Furthermore, since the processes are modeled point by point, they are complicated and less flexible. Zhu et al. [54] propose a SISR method based on the gradient reconstruction by collecting a dictionary of gradient patterns. Yang et al. [45] propose a recurrent residual network which applies edge information from off-the shelf edge detector. However, this method targets to restore high-frequency components related to PSNR evaluation. Ma et al. [24] utilizes edge information in perceptual-driven methods as explicit guidance and Nazari et al. [27] propose an edge-informed SR method based on image inpainting task. They contribute to preserving structural information. However, they still have weakness in recovering high frequency components such as sharpness and texture. For high-fidelity image, it is important to represent high frequency components. To draw high quality results, our proposed method aims to exploit high frequency information that are related to structure with proper sharpness and texture by utilizing visual and their contextual properties.
Methodology
In this section, we present the overview of the EPSR. Then we introduce the details of REPB which forms informative features by utilizing structural information and exploiting high frequency components. At the end, we describe objective functions.
Overview
The overall structure is described in Fig.1. As researched in [21,52], we apply one convolution layer to extract the shallow feature from the LR input. To utilize various information in process, we modify the skip connection structure of FRN by Kwak et al. [19] and modify as in Fig.1. we call it mFRN. mFRN consists of REPBs. Since the self-similarity property of mFRN structure gains deep depth and provide very large receptive field size, REPBs can obtain diverse information and generate informative features effectively, which include high frequency components containing details such as structure with sharpness and texture. Then the deep features from the mFRN structure is upscaled by upscale module. We apply this upscaling module such as previous work [6,52]. According to the process, the upscaled feature is then converted into SR image via one convolution layer.
Residual edge profile block(REPB)
Due to self-similarity of mFRN structure, the abundant diverse frequency information can be bypassed. From various information, Our proposed REPBs can focus on exploiting high frequency components by utilizing influential features with structural information and exposing contextual information. Our REPB consists of three parts: RECAB module, EP module and CN module. Residual Efficient Channel Attention Block(RECAB): As proposed in EDSR, MDSR [23], by removing batch normalization layers, we extract the feature. Thus range flexibility of our EPSR can be maintained. So we can formulate feature extraction as
F F E = H F E (F input ),(1)
where the output F F E and H F E (·) stand for the feature and function from feature extraction of REPB block respectively. F input is the input feature of REPB block. In SISR problem, RCAN by Zhang et al. [51] consider feature interdependencies and utilizing mutual independence by applying channel attention process from SENet [12]. However, this process has been shown that dimensionality reduction brings side effects on channel prediction. By messing up the direct correspondence between its channel and weight, it captures unnecessary dependencies across all channels empirically. To avoid this problem, we use efficient channel attention(ECA) by [31]. ECA captures local cross-channel interaction by using 1D convolution of size k, where kernel size k implies the coverage of local cross-channel interaction and the number of neighbors involved in attention prediction of one channel. To embody this process in equation:
w eca = σ(C1D k (g(F F E ))),(2)
where C1D k denotes 1D convolution and w eca is the scale statistics of channel and g(·) stands for global average pooling. Then F F E is rescaled aŝ
F F E = w eca · F F E ,(3)
whereF F E stands for rescaled feature.
To utilize informative features from ECA, we apply residual block on the network. We transform residual block by applying weighted summation on it
F RECAB = w 1 + 2 i=1 w i F input + w 2 + 2 i=1 w iF F E ,(4)
where F RECAB is the final extracted feature and w i is a learnable weight which is a scalar per feature. As applying ReLU each w i , we ensure w i ≥ 0, and fix value as 0.00001 to avoid numerical instability. Similar to interpolation, the values of each weight are ranged from 0 to 1. Since these two weight values are learnable parameters, they find more proper values for producing well-balanced features in every training process. From the process, the informative feature is generated by considering the interdependencies among feature channels and thus it brings connectivity among channels and discriminative ability in network.
Edge Profile(EP) module: In SISR, it is significant point to maintain structure for high quality SR image. For considering structural information, we construct an EP module based on conventional image processing principle. This module extracts edge profile itself from the systemically organized feature by RECAB. Intuitively, edge area has rapid variance of pixel as Fig.2. This means that there are large pixel gradient values in edge area. Next, the onset and end of discontinuities (e.g. step and ramp discontinuities) in image are also described as edge areas. To extract edge profile of image, we consider utilizing discontinuous property of edge. As described in Fig.2, to get edge mask(or profile), we subtract the blurred image from the original. So this process can be formulated as:
g(x, y) = f (x, y) −f (x, y),(5)
where g(x, y), f (x, y) andf (x, y) are edge mask, original image and blurred image respectively. We convert this process to deep learning method. First, we generate an image from feature F RECAB , which comes from feature extraction, using one convolution layer:
I blockSR = H blockSR (F RECAB ),(6)
where I blockSR is a produced image from feature F RECAB , and H blockSR (·) can be denoted as image reconstruction in EP module, which generates RGB-channel image from the 64-channel feature. To form a blurred image, we transfer arithmetic mean filter concept using average pooling. Let's denote S xy as the set of coordinates in a rectangular sub-image window of size m × n where center point is (x, y). Then this filter computes the average values of the original image i(x, y) in the area defined by S xy . In other words,î
(x, y) = 1 mn (s,t)∈Sxy i(s, t),(7)
whereî(x, y) is a blurred image of i(x, y). From this operation, if we define window size as 3 × 3, it can also be average pooling operation. So we form the blurred image by using it.
I blockblur = H blur (I blockSR ),(8)
where I blockblur is the blurred image from I blockSR and H blur (·) denotes average pooling whose kernel size is 3 × 3 and padding margin is 1. From operation in Eq.5, to get a edge profile(or mask), I blockSR is subtracted by I blockblur . Then we apply ReLU operation on edge profile(or mask) for getting outer line.
M = ReLU (I blockSR I blockblur ),(9)
where M denotes edge profile(or mask) in REPB and is element-wise subtraction.
To guide edge in training process, we concatenate I blockSR with M
I guided = Concat(I blockSR , M ),(10)
where I guided and Concat(·) denote a guided image and concatenation operation respectively. In the end, to generate the feature of EP module, we apply one convolution layer, and then give the feature F F E information by using residual structure.
F EP = F F E + H EP (I guided ),(11)
where F EP stands for the feature from EP module, which channel size is 64, and H EP (·) denotes edge profile module of REPB. By extracting structure information itself, we can obtain structure preserving effects. Context Network(CN) module: From EP module, we can obtain informative feature with structural information. This features could be beneficial to preserve structure. However, this module has limitation in handling high frequency components such as texture and sharpness of structure. Since the details have frequent pixel variations, it could be hard to capture. To reveal those contextual components, we construct CN module. Inspired by [46], we design a CN module that is based on dilated convolutions. We apply CN module following EP module. As described in CN part of Fig.1, our CN module consists of four 3 × 3 dilated convolution network, whose dilated factors are 1, To prevent loss of resolution or coverage, we consider expansion of the receptive field to set up dilated factors exponentially. Intuitively, CN module can improve learning the feature maps by passing them through multiple layers that expose contextual information. After that, the output feature is added by the input feature as residual block.
F CN = F EP + H f =1 • H f =4 • H f =2 • H f =1 (F EP ),(12)
where F CN is the output feature of CN module and H f =n (·) denotes dilated convolution whose dilated factor f is n. As this operation captures contextual information from the feature of EP module F EP , our EPSR can minimize loss of texture and recover sharpness. In other words, recovering high frequency components can be ensured with minimizing side effects and damages.
Objective Functions
Our EPSR is optimized with set-up loss functions. Normally, L 1 [20,21,23,52], L 2 [5,17,37,38], adversarial and perceptual losses [15,33] have been used in SR method. To establish the effect of EPSR, we choose two loss functions L 1 and L gradient . As proposed in previous works, we choose L 1 for guaranteeing stable convergence. Let's denote a given training set with N LR images and their HR counterparts as {I i LR , I i HR } N i=1 , and then we can formulate L 1 loss as:
L 1 = 1 N N n=1 ||H EP SR (I i LR ) − (I i HR )|| 1(13)
Since our EPSR utilizes diverse and different features each, REPBs generate edge profiles depending on feature information from feature input of them. To give consistent standard for EP modules in learning process, we consider loss function to guide them. By using Sobel filter [28], we can extract gradient maps of HR and SR and formulate gradient loss function as:
L gradient = 1 N N n=1 ||S(H EP SR (I i LR )) − S((I i HR ))|| 1 ,(14)
where S(·) is gradient function based on Sobel filter [28]. By adding L gradient to L 1 , we can achieve end-to-end network without additional module training. So the goal of training EPSR is to optimize the total loss function:
L T otal (θ) = L 1 + 10 −1 L gradient ,(15)
where θ is the parameter set of EPSR. We set the coefficient as 10 −1 empirically. The loss function is optimized by ADAM gradient descent algorithm.
Experiment Results
Settings
We state the settings of experiment about datasets, degradation models, evaluation, and training settings. Datasets. Following [23,51,52], we set up 800 high resolution images from DIV2K dataset [40] as a training set. For testing, we use 5 standard benchmark datasets: Set5 [2], Set14 [47], B100 [25], Urban100 [13], and Manga109 [26]. Degradation Models. In order to prove the effectiveness of our EPSR, we use 3 degradation models to generate LR images. First, we generate LR images with scaling factor ×2, ×3, ×4 by using Bicubic Interpolation(BI) operation. Second, by using Gaussian kernel of size 7 × 7 with standard deviation 1.6, we blur HR image and downsample it with scaling factor ×3. We denote this process as BD [49]. At last, we downsample HR image with scaling factor ×3 using bicubic interpolation and then add Gaussian noise with level 30. This process is denoted as DN for short.
Ablation Study
As we discussed above, our EPSR concentrates on structure preserving and representing details. To demonstrate effectiveness of our EPSR, we focus on showing influence of EP and CN modules, which could affect quality of SR results. Therefore, we set three comparisons by decomposing REPB, and two comparisons by feature extractions based on RECAB or RCAB by [51]. First, to establish a criterion, we construct basic block without EP module and CN module. That is, by only RECAB, we generate SR images directly. As Fig.4, only using RECAB is quite well in representing texture information. However, it has difficulty to recover image detail and edge components. Continually, we proceed with an experiment by connecting EP module to feature extraction for checking effect of edge profile. Even though edge profile is just provided on network, we can check enhancement of image reconstruction in aspect of structure preserving. Subsequently, adding CN at last, we build full REPB. As we explain details in section 3, CN helps to capture hidden information that include image details. In Fig.4 (e), which is generated by our EPSR, edge and texture information are reconstructed more stable than two images. In terms of PSNR and SSIM evaluations(See Table.1) on all datasets, we can check that utilizing edge properties brings overall significant benefits in each evaluations. It implies that EP module is helpful to preserve structure image in reconstruction process as we can see in SSIM evaluations. Furthermore, by exploiting contextual information as image details such as texture and edge, CN module shows synergy effect with EP module. As a result, the efficacy of EP and CN modules is verified in images and numerical value evaluations both. Additionally, when we remove EP module in our EPSR, we can find some problems in recovering texture and edge information like as in Fig.4 (c). It shows that even if CN module gives benefits to capture contextual information, it could have weakness in exploiting overall features. we can also check this in numerical value evaluation. This indicates the rationality that CN module is plugged into the combination of RECAB and EP modules due to concentrating on capturing contextual information that contain image details not tendency of features.
On top of that, we proceed extra experiments to investigate relationship between EP module and feature extraction. In our EPSR, we choose to use ECA for extracting features in RECAB. To verify the effect of it, we conduct experiment by substituting ECA for feature extraction to Channel Attention(CA) from [51]. we call the substitution as RCAB. As in Fig.4 (d), we can check that EPSR based on RCAB generates well SR image. However, we can see that the direction of edge lines are wrong way. Aforementioned in section 3, since CA has problem about channel predictions, it generates unclear features and it seems that EP module has some difficulties to find right edge lines. On the contrary, EPSR based on ECA feature extraction reconstructs edge and texture successfully. It is revealed visually that is generated by EPSR based on ECA feature extraction and in numerical value evaluations on PSNR and SSIM. This indi- cates that forming proper features is important key for extracting right edge profile to preserve structure in SR. To compare the effectiveness of our network with other methods, we investigate 14 state-of-the-art SR methods including general deep learning models, perceptual-driven models and edge-related models: SRCNN [5], DEGREE [45], VDSR [17], LapSRN [20], EDSR [23], MemNet [38], IDN [14], SRMDNF [49], CARN [1], RDN [52], RCAN [51], SRGAN [22], NatSR [35], SPSR [24]. All of the quantitative comparisons for ×2, ×3 and ×4 SR are shown in Table.2. With rich texture information datasets, such as Set5, Set14, and BSD100, our EPSR obtains better results in SSIM compared to other networks. NatSR gets very high results, it shows weakness in BSD100 dataset specifically. However, our EPSR shows very well balanced results compared to NatSR and acquires high performance on all datasets. Furthermore, in PSNR, it obtains comparable results with RCAN and RDN whose main target is PSNR evaluation metric. In Urban100 and Manga109 datasets that contain rich repeated edge information, our EPSR achieves competitive results in PSNR and SSIM both. Subsequently, we compare our EPSR with SPSR and DEGREE which utilize structure information in super resolution method. They are dependent on artificial edge extracting work presents quite good improvement in structure preserving. However, the results do not reach on our EPSR. Overall, our EPSR shows high and competitive performance on PSNR and SSIM evaluation metrics. Qualitative Comparison. We present visual comparison on scale ×4. From Fig.5, we see that our results are stronger in preserving structure and recovering texture both than other methods. In " img 076 " and " img 093", we observe that most of the compared models cannot reconstruct the lattices and would have trouble in blurring effects. Other methods generate twisted lines and squashed the lattices. On the other hand, EPSR shows strength in recovering structural properties. We can see the capabilities of capturing structural characteristics of objects in image and it contributes to preserving structure information in image and our EPSR captures image details well, which are including high frequency components. In " img 030 " our EPSR shows clear structure in images without damage and distortion, while most of other methods fail to reconstruct fine appearance of the objects. The qualitative comparison verifies that our EPSR generate geometrically more stable image for perceptions by utilizing structural information extracted autonomously and exploiting contextual components.
Result with BD and DN
Quantitative Comparison. model. We compare our EPSR with 8-state-of-the-art SR methods with ×3 scaling factors: SRMSR [30], SRCNN [5], FSRCNN [6], VDSR [17], IRCNN [48], SRMDNF [49], RDN [52], and RCAN [51]. In Table.3 and Table.4, all of the results are stated explicitly. We can observe that our EPSR shows higher performance compared to other methods. These results imply that our EPSR is very effective method for various types of degradation models.
Qualitative Comparison. We also show visual comparisons for challenging problem of blurring(BD) and noising(DN) degradation. First of all, in BD, there are difficulties in restoring definite texture and structural information. In Fig.6, we can see that our results are clearer and more natural than other methods. Even though most methods suffer from heavy blurring problem, our EPSR recovers texture clearer than other methods. Especially, we can check the structure in our results are well-preserved without serious distortions. In succession, in DN, since there is heavy loss of information in LR, it is hard to reconstruct image ordinary. In Fig.7, because of heavy damages of input, other methods have difficulties to overcome a lack of information and restore distortions. However, our EPSR is capable of restoring edge information with preventing texture loss.
This indicates that our EPSR can cope with damage and distortion of texture and structure by utilizing various information effectively. EPSR alleviates these troubles significantly and can reconstruct more details compared to other methods. B100
Conclusion
In this paper, we propose Edge Profile Super Resolution (EPSR) method to preserve structure information and to restore texture in SISR. We construct EPSR by building modified-Fractal Residual Network (mFRN) structures hierarchically and repeatedly. mFRN is composed of residual Edge Profile Blocks (REPBs) consisting of three different modules such as Residual Efficient Channel Attention Block (RECAB) module, Edge Profile (EP) module, and Context Network (CN) module. RECAB generates more informative features with high frequency components. From the feature, EP module produce structure informed features by generating edge profile itself. Finally, CN module captures details by exploiting high frequency information such as texture and structure with proper sharpness. As repeating the procedure in mFRN structure, our EPSR could extract high-fidelity features and thus it prevents texture loss and preserves structure with appropriate sharpness. As our EPSR consider texture loss and structure information by applying conventional principle to deep learning method, high-quality results are obtained. Extension experiments on SR with BI, BD, and DN degradation models show the effectiveness of our EPSR.
Fig. 1 .
1The architecture of the proposed EPSR which follows mFRN structure consisting of REPBs as basic blocks. REPB consists of RECAB module, EP module and CN module. we also include the explanation of components of all blocks.
Fig. 2 .
2Extracting edge profile process
Fig. 3 .
3Extracted edges of " img 048" by edge profile modules 2, 4, and 1 in order.
Fig. 4 .
4Ablation study with Bicubic(BI) degradation(×4) on " img 033" from Urban100.
Evaluation Metrics. The SR results are evaluated with PSNR and SSIM[43] on Y channel(i.e. luminance) of YCbCr space. Training Settings. In training process, the training images are augmented by randomly rotating 90 • ,180 • ,270 • , and horizontally flipping. In each training batch, 8 LR color patches with size 48 × 48 are extracted as input. Our model is trained by ADAM optimizer with β 1 = 0.9, β 2 = 0.99, and = 1e − 8. We set learning rate as 10 −4 initially and then it is reduced to half every 200 epochs. We implement our proposed EPSR using Pytorch[29] on a Tesla V100 GPU.
Table 1 .
1Comparisons of models with different components. The best results are highlighted.Set5
Set14
BSD100
Urban100
Manga109
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
RECAB
32.26/0.8937 28.46/0.7802 27.26/0.7327 26.29/0.7934 30.55/0.9017
RECAB+EP 32.20/0.8932 28.51/0.7823 27.34/0.7356 26.31/0.7943 30.66/0.9062
RECAB+CN 31.44/0.8786 22.75/0.5959 21.33/0.5206 19.40/0.5848 27.05/0.8274
RECAB+EP+CN 32.28/0.8945 28.55/0.7828 27.34/0.7362 26.43/0.7983 30.82/0.9084
RCAB+EP+CN 32.25/0.8939 28.53/0.7831 27.32/0.7349 26.39/0.7975 30.73/0.9076
Table 2 .
2Quantitative results with BI degradation model. Highlight stands for the best performance, red indicates the second, and blue is the third.Method
Set5
Set14
BSD100
Urban100 Manga109
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
Bicubic 2 33.66/0.9229 30.24/0.8688 29.56/0.8431 26.88/0.8403 30.80/0.9339
SRCNN 2 36.66/0.9542 32.45/0.9067 31.36/0.8879 29.50/0.8946 35.60/0.9663
DEGREE 2 37.40/0.9580 32.96/0.9115 31.73/0.8937
-/ -
-/ -
VDSR
2 37.53/0.9587 33.05/0.9127 31.90/0.8960 30.77/0.9141 37.16/0.9740
LapSRN 2 37.52/0.9591 32.99/0.9124 31.80/0.8949 30.41/0.9101 37.53/0.9740
EDSR
2 37.99/0.9587 33.57/0.9175 32.16/0.8994 31.98/0.9272 39.10/0.9773
MemNet 2 37.78/0.9597 33.28/0.9142 32.08/0.8978 31.31/0.9195 37.72/0.9740
IDN
2 37.83/0.9600 33.30/0.9148 32.08/0.8985 31.27/0.9196 38.02/0.9749
SRMDNF 2 37.79/0.9601 33.32/0.9159 32.05/0.8985 31.33/0.9204 38.07/0.9761
CARN
2 37.76/0.9590 33.52/0.9166 32.09/0.8978 31.92/0.9256 38.36/0.9764
RDN
2 38.24/0.9614 34.01/0.9212 32.34/0.9017 32.89/0.9353 39.18/0.9780
RCAN
2 38.27/0.9614 34.12/0.9216 32.41/0.9027 33.34/0.9384 39.44/0.9786
EPSR
2 38.29/0.9618 34.13/0.9227 32.38/0.9046 33.36/0.9401 39.57/0.9788
Bicubic 3 30.40/0.8686 27.54/0.7741 27.21/0.7389 24.46/0.7349 26.95/0.8556
SRCNN 3 32.75/0.9090 29.29/0.8215 28.41/0.7863 26.24/0.7991 30.48/0.9117
DEGREE 3 33.39/0.9182 29.61/0.8275 28.63/0.7921
-/ -
-/ -
VDSR
3 33.66/0.9213 29.78/0.8318 28.83/0.7976 27.14/0.8279 32.01/0.9340
LapSRN 3 33.82/0.9227 29.79/0.8320 28.82/0.7973 27.07/0.8271 32.21/0.9350
EDSR
3 34.37/0.9270 30.28/0.8418 29.09/0.8052 28.15/0.8527 34.17/0.9476
MemNet 3 34.09/0.9248 30.00/0.8350 28.96/0.8001 27.56/0.8376 32.51/0.9369
IDN
3 34.11/0.9253 29.99/0.8354 28.95/0.8013 27.42/0.8359 32.69/0.9378
SRMDNF 3 34.12/0.9254 30.04/0.8382 28.97/0.8025 27.57/0.8398 33.00/0.9403
CARN
3 34.29/0.9255 30.29/0.8407 29.06/0.8034 28.06/0.8493 33.49/0.9440
RDN
3 34.71/0.9296 30.57/0.8468 29.26/0.8093 28.80/0.8653 34.13/0.9484
RCAN
3 34.74/0.9299 30.65/0.8482 29.32/0.8111 29.09/0.8702 34.44/0.9499
EPSR
3 34.73/0.9297 30.52/0.8491 29.15/0.8139 28.96/0.8702 34.46/0.9486
Bicubic 4 28.43/0.8109 26.00/0.7023 25.96/0.6678 23.14/0.6574 25.15/0.7890
SRCNN 4 30.48/0.8628 27.50/0.7513 26.90/0.7103 24.52/0.7226 27.66/0.8580
DEGREE 4 31.03/0.8761 27.73/0.7597 27.07/0.7177
-/ -
-/ -
VDSR
4 31.35/0.8838 28.02/0.7678 27.29/0.7252 25.18/0.7525 28.82/0.8860
LapSRN 4 31.54/0.8866 28.09/0.7694 27.32/0.7264 25.21/0.7553 29.09/0.8900
EDSR
4 32.09/0.8938 28.58/0.7813 27.57/0.7357 26.04/0.7849 31.02/0.9148
MemNet 4 31.74/0.8893 28.26/0.7723 27.40/0.7281 25.50/0.7630 29.42/0.8942
IDN
4 31.82/0.8903 28.25/0.7730 27.41/0.7297 25.41/0.7632 29.40/0.8936
SRMDNF 4 31.96/0.8925 28.35/0.7787 27.49/0.7337 25.68/0.7731 30.09/0.9024
SRGAN 4 32.05/0.8910 28.53/0.7804 27.57/0.7354 26.07/0.7839
-/ -
NatSR
4 32.20/0.8939 28.54/0.7808 27.60/0.7366 26.21/0.7904
-/ -
SPSR
4 31.52/0.8827 27.74/0.7828 27.21/0.7276 24.80/0.8021 30.12/0.9037
CARN
4 32.13/0.8937 28.60/0.7806 27.58/0.7349 26.07/0.7837 30.40/0.9082
RDN
4 32.47/0.8990 28.81/0.7871 27.72/0.7419 26.61/0.8028 31.00/0.9151
RCAN
4 32.63/0.9002 28.87/0.7889 27.77/0.7436 26.82/0.8087 31.22/0.9173
EPSR
4 32.42/0.8969 28.65/0.7867 27.45/0.7403 26.64/0.8038 31.16/0.9127
4.3 Result with BI
Quantitative Comparison.
Table 3 .Table 4 .
34Quantitative results with BD degradation model PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM Bicubic 3 28.78/0.8308 26.38/0.7271 26.33/0.6918 26.88/0.8403 25.46/0.8149 SRMSR 3 32.21/0.9001 28.89/0.8105 28.13/0.7740 25.84/0.7856 29.64/0.9003 SRCNN 3 32.05/0.8944 28.80/0.8074 28.13/0.7736 25.70/0.7770 29.47/0.8924 FSRCNN 3 26.23/0.8124 24.44/0.7106 24.86/0.6832 22.04/0.6745 23.04/0.33.38/0.9182 29.63/0.8281 28.65/0.7922 26.77/0.8154 31.15/0.9245 IRCNN C 3 33.17/0.9157 29.55/0.8271 28.49/0.7886 26.47/0.8081 31.13/0.9236 SRMDNF 3 34.01/0.9242 30.11/0.8364 28.98/0.8009 27.50/0.8370 32.97/0.Quantitative results with DN degradation model SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM Bicubic 3 24.01/0.5369 22.87/0.4724 22.92/0.4449 21.63/0.4687 23.01/0.5381 SRCNN 3 25.01/0.6950 23.78/0.5898 23.76/0.5538 21.90/0.5737 23.75/0.7148 FSRCNN 3 24.18/0.6932 23.02/0.5856 23.41/0.5556 21.15/0.5682 22.39/0.25.70/0.7379 24.45/0.6305 24.28/0.5900 22.90/0.6429 24.88/0.7765 IRCNN C 3 27.48/0.7925 25.92/0.6932 25.55/0.6481 23.93/0.6950 26.07/0.We apply our EPSR with BD degradation model, which is used recently in[51], and following[52], we further compare various SR methods on image with DN degradationFig. 6. Visual Comparison for SR(×3) with BD model on Urban100. The best results are highlighted.Method
Set5
Set14
BSD100
Urban100 Manga109
7927
VDSR
3 33.25/0.9150 29.46/0.8244 28.57/0.7893 26.61/0.8136 31.06/0.9234
IRCNN G 3 9391
RDN
3 34.58/0.9280 30.53/0.8447 29.23/0.8079 28.46/0.8582 33.97/0.9465
RCAN
3 34.70/0.9288 30.63/0.8462 29.32/0.8093 28.81/0.8645 34.38/0.9483
EPSR
3 34.68/0.9288 30.56/0.8484 29.14/0.8130 28.83/0.8667 34.51/0.9476
Method
Set5
Set14
BSD100
Urban100 Manga109
PSNR/7111
VDSR
3 25.20/0.7183 24.00/0.6112 24.00/0.5749 22.22/0.6096 24.20/0.7525
IRCNN G 3 8253
RDN
3 28.47/0.8151 26.60/0.7101 25.93/0.6573 24.92/0.7364 28.00/0.8591
EPSR
3 28.53/0.8142 26.57/0.7105 25.86/0.6588 25.16/0.7477 28.20/0.8634
Fig. 7. Visual Comparison for SR(×3) with DN model on Urban100. The best results are highlighted.(x3):
HR
Bicubic
SRCNN [5]
VDSR[17] IRCNN C[48]
RDN[52]
Ours
302008
PSNR/SSIM 24.58/0.5737 25.60/0.8187 25.77/0.8448 28.45/0.8901 30.84/0.9167 30.90/0.9158
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| []
|
[
"Ab-initio study of the thermopower of biphenyl-based single-molecule junctions",
"Ab-initio study of the thermopower of biphenyl-based single-molecule junctions"
]
| [
"M Bürkle \nInstitute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nDFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n",
"L A Zotti \nDepartamento de Física Teórica de la Materia Condensada\nUniversidad Autónoma de Madrid\nE-28049MadridSpain\n",
"J K Viljas \nLow Temperature Laboratory\nAalto University\nP.O. Box 15100FIN-00076AaltoFinland\n\nDepartment of Physics\nUniversity of Oulu\nP.O. Box 3000FIN-90014Finland\n",
"D Vonlanthen \nDepartment of Chemistry\nUniversity of Basel\nCH-4056BaselSwitzerland\n",
"A Mishchenko \nDepartment of Chemistry and Biochemistry\nUniversity of Bern\nCH-3012BernSwitzerland\n",
"T Wandlowski \nDepartment of Chemistry and Biochemistry\nUniversity of Bern\nCH-3012BernSwitzerland\n",
"M Mayor \nDFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nDepartment of Chemistry\nUniversity of Basel\nCH-4056BaselSwitzerland\n\nInstitute for Nanotechnology\nKarlsruhe Institute of Technology\nD-76344Eggenstein-LeopoldshafenGermany\n",
"G Schön \nInstitute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nDFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nInstitute for Nanotechnology\nKarlsruhe Institute of Technology\nD-76344Eggenstein-LeopoldshafenGermany\n",
"F Pauly \nInstitute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nDFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n\nMolecular Foundry\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA\n"
]
| [
"Institute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"DFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"Departamento de Física Teórica de la Materia Condensada\nUniversidad Autónoma de Madrid\nE-28049MadridSpain",
"Low Temperature Laboratory\nAalto University\nP.O. Box 15100FIN-00076AaltoFinland",
"Department of Physics\nUniversity of Oulu\nP.O. Box 3000FIN-90014Finland",
"Department of Chemistry\nUniversity of Basel\nCH-4056BaselSwitzerland",
"Department of Chemistry and Biochemistry\nUniversity of Bern\nCH-3012BernSwitzerland",
"Department of Chemistry and Biochemistry\nUniversity of Bern\nCH-3012BernSwitzerland",
"DFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"Department of Chemistry\nUniversity of Basel\nCH-4056BaselSwitzerland",
"Institute for Nanotechnology\nKarlsruhe Institute of Technology\nD-76344Eggenstein-LeopoldshafenGermany",
"Institute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"DFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"Institute for Nanotechnology\nKarlsruhe Institute of Technology\nD-76344Eggenstein-LeopoldshafenGermany",
"Institute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"DFG Center for Functional Nanostructures\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany",
"Molecular Foundry\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA"
]
| []
| Employing ab-initio electronic structure calculations combined with the non-equilibrium Green's function technique we study the dependence of the thermopower Q on the conformation in biphenylbased single-molecule junctions. For the series of experimentally available biphenyl molecules, alkyl side chains allow us to gradually adjust the torsion angle ϕ between the two phenyl rings from 0 • to 90 • and to control in this way the degree of π-electron conjugation. Studying different anchoring groups and binding positions, our theory predicts that the absolute values of the thermopower decrease slightly towards larger torsion angles, following an a + b cos 2 ϕ dependence. The anchoring group determines the sign of Q and a, b, simultaneously. Sulfur and amine groups give rise to Q, a, b > 0, while for cyano Q, a, b < 0. The different binding positions can lead to substantial variations of the thermopower mostly due to changes in the alignment of the frontier molecular orbital levels and the Fermi energy. We explain our ab-initio results in terms of a π-orbital tightbinding model and a minimal two-level model, which describes the pair of hybridizing frontier orbital states on the two phenyl rings. The variations of the thermopower with ϕ seem to be within experimental resolution. | 10.1103/physrevb.86.115304 | [
"https://arxiv.org/pdf/1202.5709v1.pdf"
]
| 17,174,218 | 1202.5709 | f86bfc471cd2e814d6a75ca38df08a11e38e27f4 |
Ab-initio study of the thermopower of biphenyl-based single-molecule junctions
M Bürkle
Institute of Theoretical Solid State Physics
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
DFG Center for Functional Nanostructures
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
L A Zotti
Departamento de Física Teórica de la Materia Condensada
Universidad Autónoma de Madrid
E-28049MadridSpain
J K Viljas
Low Temperature Laboratory
Aalto University
P.O. Box 15100FIN-00076AaltoFinland
Department of Physics
University of Oulu
P.O. Box 3000FIN-90014Finland
D Vonlanthen
Department of Chemistry
University of Basel
CH-4056BaselSwitzerland
A Mishchenko
Department of Chemistry and Biochemistry
University of Bern
CH-3012BernSwitzerland
T Wandlowski
Department of Chemistry and Biochemistry
University of Bern
CH-3012BernSwitzerland
M Mayor
DFG Center for Functional Nanostructures
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
Department of Chemistry
University of Basel
CH-4056BaselSwitzerland
Institute for Nanotechnology
Karlsruhe Institute of Technology
D-76344Eggenstein-LeopoldshafenGermany
G Schön
Institute of Theoretical Solid State Physics
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
DFG Center for Functional Nanostructures
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
Institute for Nanotechnology
Karlsruhe Institute of Technology
D-76344Eggenstein-LeopoldshafenGermany
F Pauly
Institute of Theoretical Solid State Physics
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
DFG Center for Functional Nanostructures
Karlsruhe Institute of Technology
D-76131KarlsruheGermany
Molecular Foundry
Lawrence Berkeley National Laboratory
94720BerkeleyCaliforniaUSA
Ab-initio study of the thermopower of biphenyl-based single-molecule junctions
numbers: 8565+h8580Fi7363Rt8107Pr
Employing ab-initio electronic structure calculations combined with the non-equilibrium Green's function technique we study the dependence of the thermopower Q on the conformation in biphenylbased single-molecule junctions. For the series of experimentally available biphenyl molecules, alkyl side chains allow us to gradually adjust the torsion angle ϕ between the two phenyl rings from 0 • to 90 • and to control in this way the degree of π-electron conjugation. Studying different anchoring groups and binding positions, our theory predicts that the absolute values of the thermopower decrease slightly towards larger torsion angles, following an a + b cos 2 ϕ dependence. The anchoring group determines the sign of Q and a, b, simultaneously. Sulfur and amine groups give rise to Q, a, b > 0, while for cyano Q, a, b < 0. The different binding positions can lead to substantial variations of the thermopower mostly due to changes in the alignment of the frontier molecular orbital levels and the Fermi energy. We explain our ab-initio results in terms of a π-orbital tightbinding model and a minimal two-level model, which describes the pair of hybridizing frontier orbital states on the two phenyl rings. The variations of the thermopower with ϕ seem to be within experimental resolution.
Employing ab-initio electronic structure calculations combined with the non-equilibrium Green's function technique we study the dependence of the thermopower Q on the conformation in biphenylbased single-molecule junctions. For the series of experimentally available biphenyl molecules, alkyl side chains allow us to gradually adjust the torsion angle ϕ between the two phenyl rings from 0 • to 90 • and to control in this way the degree of π-electron conjugation. Studying different anchoring groups and binding positions, our theory predicts that the absolute values of the thermopower decrease slightly towards larger torsion angles, following an a + b cos 2 ϕ dependence. The anchoring group determines the sign of Q and a, b, simultaneously. Sulfur and amine groups give rise to Q, a, b > 0, while for cyano Q, a, b < 0. The different binding positions can lead to substantial variations of the thermopower mostly due to changes in the alignment of the frontier molecular orbital levels and the Fermi energy. We explain our ab-initio results in terms of a π-orbital tightbinding model and a minimal two-level model, which describes the pair of hybridizing frontier orbital states on the two phenyl rings. The variations of the thermopower with ϕ seem to be within experimental resolution.
I. INTRODUCTION
Tailored nanostructures hold promise for improved efficiencies of thermoelectric materials. [1][2][3] For this reason there is a growing interest to gain a better understanding of the role of interfaces on thermoelectric properties at the atomic scale. Controlled metal-organic interfaces can be studied using single-molecule junctions, and recently the thermopower of these systems was determined in first experiments. 4 While the thermopower (or Seebeck coefficient) of metallic atomic contacts was measured already several years ago, 5 molecular junctions offer fascinating possibilities to adjust thermoelectric properties due to the control over chemical synthesis and interface structure. Ref. 4 and subsequent experimental studies thus explored the influence of different parameters on the thermopower, such as molecule length, 4,6,7 substituents, 8 anchoring groups, [7][8][9] or electrode metal. 10 On the theory side, the electronic contribution to the thermopower explains important experimental observations. 11 We have shown recently that the thermopower of metallic atomic contacts, which serve as reference systems in molecular electronics, can be understood by considering the electronic structure of disordered junction geometries. 12 Using molecular dynamics simulations of many junction stretching processes combined with tight-binding-based electronic structure and transport calculations, we found thermopower-conductance scatter plots similar to the low-temperature experiment. 5 Such a statistical analysis, although highly desirable for molecular junctions, is complicated by the time-consuming electronic structure calculations needed to describe these heteroatomic systems. Still, early studies of the thermopower based on density functional theory (DFT) for selected geometries explained crucial trends, such as the dependence of the thermopower on molecule length 13 or the influence of substituents and anchoring groups. 13,14 Since the experiments on the thermopower of molecular junctions were all performed at room temperature until now, finite temperature effects may play a role. They can impact the thermopower by fluctuations of the junction geometry and electron-vibration couplings. 12,15,16 While the quantification constitutes an interesting challenge for future work, we will focus here on the purely electronic effects in static ground-state contact structures.
An interesting aspect, not yet addressed in the experiments is the influence of conjugation on the thermopower Q. For the conductance, such studies were carried out by different groups with biphenyl molecules. [17][18][19] The torsion angle ϕ between the phenyl rings was adjusted stepwise by use of appropriate side groups. While such substituents may have a parasitic shifting effect on energies of current-carrying molecular orbitals, the changes in conformation, which control the degree of π-electron conjugation, turned out to domi-arXiv:1202.5709v1 [cond-mat.mes-hall] 26 Feb 2012 nate the behavior of the conductance. 17 The systematic series of biphenyl molecules of Refs. 18-21 uses alkyl chains of various lengths and methyl groups to adjust ϕ and avoids strongly electron-donating and electronwithdrawing substituents. Hence it seems ideal for determining the influence of conjugation on thermopower.
Theoretical work has considered the behavior of Q when ϕ is changed continuously for the thiolated biphenyl molecule contacted to gold (Au) electrodes. 13,22 Both studies agree in the fact that Q is positive for all ϕ. However, while we predicted Q to decrease with increasing ϕ based on DFT calculations and a π-orbital tight-binding model (TBM), 13 work of Finch et al. 22 suggested the opposite for this idealized system. In this study we clarify this contradiction and demonstrate with the help of a two-level model (2LM) that for the off-resonant transport situation the absolute value of Q is expected to decrease when the molecule changes from planar to perpendicular ring orientation. This confirms our previous conclusions. More importantly, this work explores the possibility to measure the dependence of Q on ϕ for the experimentally relevant family of molecules presented in Refs. 18-21. Using DFT calculations of the electronic structure combined with the Landauer-Büttiker scattering formalism expressed with Green's function techniques, we determine the thermopower of biphenyl-derived molecules connected to gold electrodes. The molecules investigated are displayed in Fig. 1. Alkyl chains, one to four CH 2 units long, allow to change ϕ gradually from 0 • to 60 • . To achieve ϕ ≈ 90 • , we included in addition M7, and as a reference also M0, the "standard" biphenyl molecule. For each of the molecules in Fig. 1 we will explore the three different anchoring groups sulfur (S), amine (NH 2 ), and cyano (CN) in various binding geometries.
This work is organized as follows. In Sec. II we introduce the theoretical procedures used in this work. Sec. III presents the main results. We start by discussing models to describe the ϕ dependence of the thermopower, show the DFT-based results for Q, and provide further insights by discussing their relation to the predictions of the TBM and the 2LM. The paper ends with the conclusions in Sec. IV.
II. THEORETICAL METHODS
A. Electronic structure and contact geometries
We determine the electronic structure and contact geometries in the framework of DFT. All our calculations are performed with the quantum chemistry package TURBOMOLE 6.3, 23 and we use the gradient-corrected BP86 exchange-correlation functional. 24,25 For the basis set, we employ def2-SV(P) which is of split-valence quality with polarization functions on all non-hydrogen atoms. 26 For Au an effective core potential efficiently deals with the innermost 60 electrons, 27 while the basis set provides an all-electron description for the rest of the atoms in this work.
The contact geometries for the S-terminated molecules are those of Ref. 28. For the NH 2 and CN anchors we proceed as described in Refs. 19,28 and use for consistency the electrode geometry from Ref. 28.
B. Charge transport
We determine charge transport properties within the Landauer-Büttiker formalism. The transmission function τ (E), describing the energy-dependent transmission probability of electrons through the nanostructure, is calculated with non-equilibrium Green's function techniques. The Green's functions are constructed by use of the DFT electronic structure as obtained for the groundstate molecular junction geometries. A detailed description of our quantum transport method is given in Ref. 29.
The thermopower at the average temperature T is defined as the ratio of the induced voltage difference ∆V in the steady state and the applied temperature difference ∆T between the ends of a sample, Q(T ) = − (∆V /∆T )| I=0 . In the Landauer-Büttiker formalism the electronic contribution to the thermopower can be expressed as 30
Q(T ) = − K 1 (T ) eT K 0 (T ) (1) with K n (T ) =´dEτ (E)(E − µ) n [−∂f (E, T )/∂E], the absolute value of the electron charge e = |e|, the Fermi function f (E, T ) = {exp[(E −µ)/k B T ]+1} −1 ,Q(T ) = −q(T ) ∂ E τ (E) τ (E) E F .(2)
with the prefactor q(T ) = π 2 k 2 B T /(3e) depending linearly on temperature. Table I: Torsion angle ϕ in units of degrees and the thermopower Q at T = 10 K in units of µV/K for all junction geometries.
S-HH
S-BB S-TT1 NH2-TT1 NH2-TT2 CN-TT1 CN-TT2 ϕ Q ϕ Q ϕ Q ϕ Q ϕ Q ϕ Q ϕ QM0
In the following the thermopower is calculated, if not otherwise indicated, for a low temperature of T = 10 K, where our theory is expected to apply best. For the DFT-based results presented below we determine Q by means of Eq. (1), i.e. by taking into account the thermal broadening of the electrodes. For the molecular junctions studied here, the differences to the values obtained via Eq. (2) often turn out to be small even at room temperature (T = 300 K). Hence thermopower values for higher T can be estimated using the values at 10 K through Q(T ) ≈ (T /10 K) × Q(10 K). Since we are not primarily interested in the temperature dependence of Q in this work, we suppress from here on the temperature argument.
III. RESULTS AND DISCUSSION
A. Models for the angle-dependent thermopower
In Ref. 13 we argued that the thermopower should depend on the torsion angle as
Q ϕ ≈ a + b cos 2 ϕ.(3)
Our argument was based on the observation that for a πorbital TBM in the off-resonant transport situation, the transmission of the biphenyl molecule can be expanded in powers of cos 2 ϕ as 13,31,32
τ ϕ (E) = α 2 (E) cos 2 ϕ + α 4 (E) cos 4 ϕ + O(cos 6 ϕ). (4)
Conductance measurements [17][18][19] and corresponding DFT calculations, 31 which both determine the transmission at the Fermi energy, show that α 2 is the dominant term. The leading term in the ϕ dependence of Q is obtained from Eq. (2) by taking into account the energy dependence of the expansion coefficients α j (E) and considering the terms up to j = 4. Then, we obtain Eq. (3) with
a = −q ∂ E α 2 (E) α 2 (E) E=E F ,(5)b = −q α 2 (E)∂ E α 4 (E) − α 4 (E)∂ E α 2 (E) α 2 (E) 2 E=E F .(6)
While this model uses minimal information about the biphenyl molecular junction, a disadvantage is that the magnitude and energy dependence of the coefficients α 2 and α 4 are a priori unknown.
An alternative strategy is to use the 2LM of Ref.
18. This minimal model explains the cos 2 ϕ law of the conductance by considering the pair of hybridizing frontier orbital resonances of the phenyl rings which are closest to E F . Within this model, the transmission is given by 18
τ ϕ (E) = Γt cos ϕ (E −ε s (ϕ) − iΓ/2)(E −ε a (ϕ) − iΓ/2) 2(7)
withε s,a (ϕ) =ε 0 ±t cos ϕ. Here,ε 0 is the relevant frontier molecular orbital energy of the individual phenyl ring. For the biphenyl molecule, it can be determined as the highest occupied molecular orbital (HOMO) or lowest unoccupied molecular orbital (LUMO) energy for vanishing inter-ring couplingt cos ϕ at ϕ = 90 • . The angle-dependent inter-ring coupling leads to a splitting of the pair of degenerate levelsε 0 at energiesε s,a (ϕ) = ε 0 ±t cos ϕ with symmetric and antisymmetric wavefunctions, respectively. In addition, we have made the wide-band approximation with a symmetric and energyindependent couplingΓ to the left and right phenyl rings. The 2LM is hence characterized by the parametersε 0 ,t,Γ.
We setε =ε 0 − E F ,x =t cos ϕ/ ε 2 +Γ 2 /4, and assume |x| 1. Performing a Taylor expansion inx, we obtain Eq. (3) with
a = −q 4ε ε 2 +Γ 2 /4 ,(8)b = −q 4t 2ε ε 2 − 3Γ 2 /4 ε 2 +Γ 2 /4 3 .(9)
These expressions predict that the sign of a, b is determined byε. Thus, whenε changes sign, a, b change sign at the same time. In the typical off-resonant transport situation |ε| t ,Γ, the sign of a ≈ −q4/ε and b ≈ −q4t 2 /ε 3 is identical. However, b may be of a different sign than a in a more on-resonant case when the broadeningΓ is of a similar size asε, i.e., whenε 2 −3Γ 2 /4 changes sign.
B. Thermopower based on density functional theory
For each of the biphenyl molecules in Fig. 1, we study the three different anchoring groups X = S, NH 2 , CN and select a total of seven contact geometries, as displayed in Fig. 2. For S anchors we choose three representative binding sites, 28,33 where S binds covalently either to three Au atoms in the hollow position (S-HH), to two of them in the bridge position (S-BB), or to a single one in the top position (S-TT1). NH 2 -and CN-terminated molecules bind selectively to a single Au electrode atom at each side via the nitrogen lone pair. 19,34 Thus, we consider In Table I we summarize the torsion angle ϕ, which is defined as the dihedral angle between the two phenyl rings (see Fig. 3), and the thermopower for all 42 molecular junctions studied. The data is presented graphically in Fig. 3 by plotting Q as a function of ϕ for each of the seven types of junctions in Fig. 2. We notice that the sign of the thermopower is determined by the anchoring group. For the electron-donating S and NH 2 linkers 35 the energy of the π-electron system of the molecules is increased compared to the hydrogenterminated case (X = H in Fig. 1). The HOMO energy is therefore close to E F , as visible also from the transmission curves in Fig. 4. The hole conduction through the HOMO yields Q > 0, in agreement with previous experimental 4, 6 and theoretical results. 13,14 In contrast to this, for the electron-withdrawing CN anchoring group 35 we have electron transport through the LUMO 8,14,19,36 (see also Fig. 4), and consequently Q < 0.
Considering the absolute values of the thermopower, Fig. 3 shows that Q can differ markedly for the types of contact geometries. Given the off-resonant transport situation suggested by the transmission curves in Fig. 4 and using Eqs. (8) and (9), we can understand the results by changes in the level alignmentε. As we will discuss in more detail below in Sec. III C, level broadeningsΓ and couplingst play no important role in that respect. The level alignment is determined by the charge transfer between the molecule and the electrodes, which is sensitive to the binding site of the anchoring group at the molecule-metal interface. For the thiolated molecules we find that the thermopower for S-BB and S-TT1 is comparable, but the values are significantly larger than those for S-HH. This behavior is related to our recent findings for the conductance of the thiolated molecules, where top and bridge geometries yield similar but much larger conductances than those with hollow sites. 28 Both observations are due to a HOMO level which is more distant from E F for S-HH as compared to S-BB and S-TT1. We explain this by the leakage of electrons from the molecule, including the S atoms, to the Au electrodes, when going from the S-TT1 over the S-BB to the S-HH geometry. 28 For the amines NH 2 -TT1 gives a larger thermopower than NH 2 -TT2. We have checked that this is a result of the larger negative charge on the molecule when bonded in NH 2 -TT1 position as compared to NH 2 -TT2, which moves the HOMO closer to E F . With respect to the thiols we see that both NH 2 -linked geometries give rise to a thermopower well below those of S-BB and S-TT1 but still larger than for S-HH. The CN-linked molecules show the largest |Q|. The more positive charge on the molecules in CN-TT1 as compared to CN-TT2 leads to their smaller, i.e., more negative Q.
Regarding M0 with X = S, NH 2 we can compare to experimental and theoretical results for Q in the literature. has not yet been reported. A trend by the DFT calculations to overestimate the thermopower can nevertheless be recognized. 37 It is expected from the typical overestimation of experimental conductance values, 28 attributed mostly to the interpretation of Kohn-Sham eigenvalues as approximate quasi-particle energies. 38,39 According to Eqs. (8) and (9) an underestimation of |ε| leads to an overestimation of |Q|. However, finite temperature effects due to vibrations, not accounted for in our calculations, may also play a role in the room-temperature experiments.
The transport through the well-conjugated molecules a (µV/K) b (µV/K) r (%) Fig. 3 to fit the DFT results of M1-M4 for each type of junction, and the relative change r of Q between M1 and M4.
M0-M4 is dominated by the π electrons, and we have shown in Refs. 18,19,28 that for these molecules the conductance arises from one transmission eigenchannel of π character. 19,28 Hence we would expect their thermopower to follow Eq. (3). Despite the variations of Q with anchoring groups and binding positions, we find a weak cos 2 ϕ-like decrease of the absolute values for M1-M4 for all types of geometries. M0, however, deviates from this trend. Although the electron-donating effect of the alkyl chains is expected to be small, it increases Q for M1-M4 as compared to M0. To clarify this we calculated by means of electrostatic potential fitting and a Löwdin population analysis the charge transferred from the alkyl side chains to the two phenyl rings for the hydrogen-terminated (X = H in Fig. 1), isolated gasphase molecules. Both methods yield an overall negative charge on the phenyl rings which is practically independent of the alkyl chain length. Therefore the substituentrelated energy shift of frontier orbital levels is similar for M1-M4, and the a + b cos 2 ϕ dependence is observed. Focusing on the thermopower of M1-M4, we extract a and b by fitting their Q with Eq. (3). The precise values are given in Table II, and the corresponding fits are shown as continuous lines in Fig. 3. Additionally, we list in Table II the ratio r = |Q M1 −Q M4 |/(Q M1 +Q M4 ), quantifying the maximal decrease of Q in that subset of molecules. We find it to vary between 4% and 31%. In detail, we observe the largest relative change for S-HH followed by CN-TT2. CN-TT1, S-BB, S-TT1, and NH 2 -TT2 all show a similar r, while it is smallest for NH 2 -TT1.
For M7, Eq. (3) is not expected to hold, because the transport at ϕ 90 • is not π-like but proceeds through transmission eigenchannels of π-σ character. 13,28 Furthermore, M7 shows the largest substituent-related shifting effect on the biphenyl backbone in our family of molecules due to the electron-donating nature of the four attached methyl side-groups. 13,35,40 Its thermopower hence arises from a detailed interplay between the substituent-related shifting and the large torsion angle, as explained in Ref. 13. We find that the absolute values of Q for M7 are generally lower than predicted by the fits with Eq. (3). Only for S-TT1 and NH 2 -TT2 the thermopower seems to follow the a + b cos 2 ϕ dependence but this is likely coincidental.
C. Transport analysis using the π-orbital tight-binding model and the two-level model In order to better understand the differences in the thermopower for the various anchoring groups and binding positions, we need to examine the parametersε,t,Γ of the 2LM which determine the thermopower according to Eqs. (8) and (9). We note that the dominant, angleindependent term a is a function ofε,Γ only. Thus, to discuss main anchor-group-and binding-site-related variations of Q for the seven different junction types of Fig. 2, it is sufficient to concentrate on these two parameters. The ϕ dependence of Q, however, results from the interference of the hybridizing pair of phenyl-ring frontier orbital levels, and b hence depends also ont.
We obtain the parameters of the 2LM from the TBM introduced in Ref. 32. The TBM is sketched in Fig. 4(a). Similar to the 2LM, the Hückel-like TBM is characterized by three parameters which are the on-site energy ε 0 of each carbon atom, the nearest-neighbor hopping t between atoms on each of the phenyl rings, and the electrode-related broadening Γ. The inter-ring hopping is given as t = t cos ϕ. Using the wide-band approximation, we assume all components of the lead self-energy matrices to vanish except for (Σ r L ) αα = (Σ r R ) ωω = −iΓ/2, with α and ω indicating the terminal carbon atoms of the biphenyl molecule as shown in Fig. 4(a).
The parameters ε 0 , t, Γ of the TBM are extracted by fitting τ (E) curves calculated with DFT. We focus on the molecules M1-M4 and set ϕ to the torsion angle realized in the specific junction geometry (see Table I).
Concentrating particularly on the HOMO-LUMO gap and frontier orbital peaks, we find that the fitted TBM generally reproduces well the transmission in that range and that the parameters extracted for M1-M4 are very similar in each of the seven types of junctions. Finally, the parameters of the 2LM are derived from those of the TBM.ε andt are obtained by evaluating appropriate eigenvalues of the angle-dependent Hückel-like Hamiltonian of the TBM. ForΓ we identify imaginary parts of complex eigenvalues of the non-Hermitian matrices (H + Σ r L + Σ r R ) jk for the TBM and the 2LM, respectively. 28 Here, H jk and (Σ r L ) jk , (Σ r R ) jk represent the matrix elements of the Hamiltonian and of the electrode self-energies in the corresponding model. All the parameters determined in this way are listed in Table III. For M2, transmission curves calculated with the DFT, the TBM, and the 2LM are shown in Fig. 4(b) for each of the three anchoring groups.
Using the parameters of Table III we compare in Fig. 4(c) Q as a function of ϕ for the TBM and 2LM fits with the DFT results. We find that the TBM agrees well with the DFT-based values for the illustrated junction geometries S-TT1, NH 2 -TT1, and CN-TT2. The 2LM, instead, overestimates |Q| somewhat. Considering Table III: Parameters of the TBM ε0, t, Γ obtained by fitting the DFT-based transmission curves for M1-M4 for each type of junction. The parametersε,t,Γ of the 2LM are derived from those of the TBM as described in the text. All values are given in units of eV.
Eq. (2) and the transmission curves in Fig. 4(b), we attribute this to an underestimation of τ (E F ) and an overestimation of |∂ E τ (E F )|. All results exhibit a consistent weak dependence of Q on ϕ.
The data in Table III shows that transport through the biphenyl molecules is off-resonant with the relatioñ Γ |ε| being well fulfilled. As argued in Sec. III A, a, b should thus take the same sign and change it together with Q when the transport for S-and NH 2 -linked molecules changes from HOMO-to LUMO-dominated for CN anchors. This is consistent with our findings in Figs. 3 and 4(c), and explains the decrease of |Q| with increasing ϕ. 41 Coming back to our discussion of the differences of the thermopower for the various anchoring groups and binding positions in Fig. 3, we observe thatε is around 0.6 to 0.8 eV closer to E F for S-BB and S-TT1 as compared to S-HH, NH 2 -TT1, and NH 2 -TT2, which explains their larger Q. For the CN-terminated molecules |ε| is comparable to those for S-BB and S-TT1. Slightly larger values of |Q| for CN result from the very small broaden-ingsΓ. Furthermore, for both NH 2 and CN,t andΓ are essentially independent of the binding position, and the difference in Q between TT1 and TT2 hence stems from the changes in the alignment of the HOMO and LUMO levels.
IV. CONCLUSIONS
We have analyzed theoretically the thermopower of single-molecule junctions consisting of biphenyl derivatives contacted to gold electrodes. Or DFT-based study with the three anchors S, NH 2 , and CN shows a positive thermopower for S or NH 2 and a negative one for CN. For the junction geometries considered, different binding sites did not affect the sign of Q but led to variations of absolute value. For thiolated molecules in bridge and top binding sites Q can be up to two orders of magnitude larger than for molecules bonded in hollow position, while the variations for the two considered top binding sites were around a factor of two for NH 2 and CN anchors. We have explained these observations by the changes in the level alignment of current-carrying frontier molecular orbitals. They are caused by the binding-site-dependent charge transfer at the metal-molecule interface.
The main purpose of this work was the study of the dependence of the thermopower on conjugation for an experimentally relevant system. In our set of six biphenyl derivatives, the conjugation was controlled by the torsion angle ϕ between the phenyl ring planes, and it was varied stepwise between 0 and 90 • by means of alkyl side chains attached to the molecules. Despite the sensitivity of the thermopower to the precise geometry at the moleculemetal interface, we observed for all investigated types of junction configurations a decrease in |Q| with increasing ϕ, following a characteristic a + b cos 2 ϕ law. We explained this behavior in terms of a two-level model, which considers the pair of hybridizing frontier orbitals on the phenyl rings. Predictions by this model of a simultaneous change in sign of Q, a, b for a change from HOMOto LUMO-dominated transport in the off-resonant situation are consistent with our DFT results. Overall, the influence of conjugation on the thermopower is much less pronounced than on the conductance.
We propose to measure the a + b cos 2 ϕ dependence of the thermopower for the set of biphenyl molecules studied here. Using alkyl chains of different lengths, parasitic substituent-related shifts in Q, superimposed on the weak a + b cos 2 ϕ dependence, are largely avoided. Depending on binding site and employed anchoring group, relative variations of Q of around 15% are expected between M1 and M4. Since frontier molecular orbital energies are likely positioned closer to E F in our calculations than in the experiment, the relative changes of Q with ϕ are expected to be somewhat smaller than in our theoretical predictions. Nevertheless, we suggest that the variations of the thermopower with torsion angle are experimentally detectable.
PACS numbers: 85.65.+h, 85.80.Fi, 73.63.Rt, 81.07.Pr
Figure 1 :
1Chemical structure of the studied biphenyl molecules with X standing either for the S, NH2, or CN anchoring group.
Figure 2 :
2(Color online) Analyzed types of junctions, shown for M3. For S anchors we consider hollow, bridge, and top binding sites to Au with the corresponding contact geometries called S-HH, S-BB, and S-TT1, respectively. For NH2 and CN we consider binding to single Au atoms in two different top positions with the contacts named NH2-TT1, NH2-TT2 and CN-TT1, CN-TT2.
Figure 3 :
3(Color online) (a) Evolution of Q with increasing ϕ for all contact geometries. The symbols represent the thermopower values calculated with DFT at T = 10 K, and the lines are obtained by fitting Eq. (3) to M1-M4 for each type of junction. (b) Zoom in on the Q values for S-HH, NH2-TT1, and NH2-TT2. (c) Schematic of the studied biphenyl derivatives and definition of the torsion angle ϕ. two different top sites for NH 2 (NH 2 -TT1, NH 2 -TT2) and CN (CN-TT1, CN-TT2), respectively.
For biphenyl-diamine a thermopower of Q NH2-EXP M0 = 4.9±1.9 µV/K was found at T = 300 K, 6 which compares reasonably well to our calculated values of Q NH2-TT1 M0 = 10.52 µV/K and Q NH2-TT2 M0 = 4.6 µV/K for the same T . Furthermore, recent calculations within a DFT approach with an approximate self-interaction correction for comparable geometries showed similar results to ours. 37 For biphenyl-dithiol the comparison is complicated by the fact that our calculated values vary by two orders of magnitude for the different geometries, i.e., Q S-HH M0 = 0.11 µV/K, Q S-BB M0 = 39.14 µV/K, Q S-TT M0 = 28.08 µV/K at T = 300 K. They scatter indeed around the experimental result of Q S-EXP M0 = 12.9 ± 2.2 µV/K. 4 To our knowledge, the thermopower of cyano-terminated biphenyls
Figure 4 :
4(Color online) (a) Schematic of the TBM used to fit DFT-based transmission curves. (b) Transmission of M2 as a function of energy calculated with DFT, and the fits using the TBM and the 2LM. (c) Q as a function of ϕ, comparing values obtained with the TBM and the 2LM to the DFTbased results. In panels (b) and (c) S-TT1, NH2-TT1, and CN-TT2 junction geometries were selected.
AcknowledgmentsWe acknowledge fruitful discussions with A. Bagrets, F. Evers, and V. Meded. R. Ahlrichs and M. Sierka are thanked for providing us with TURBOMOLE. M.B. and G.S. were supported through the DFG Center for Functional Nanostructures (Project C3.6), the DFG priority program 1243, and the Initial Training Network "NanoCTM"
FUNMOLS, the DFG priority program 1243, and the University of Bern. FUNMOLS, the DFG priority program 1243, and the University of Bern.
* Electronic address: marius. [email protected]* Electronic address: [email protected]
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We note that Fig. 2(a) of Ref. 22 suggests that transport is strongly off-resonant also in their calculations, and the positive Q should hence decrease. with ϕ in their Fig. 3(bWe note that Fig. 2(a) of Ref. 22 suggests that transport is strongly off-resonant also in their calculations, and the positive Q should hence decrease with ϕ in their Fig. 3(b).
| []
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[
"Toroidal moments of Schrödinger eigenstates",
"Toroidal moments of Schrödinger eigenstates"
]
| [
"M Encinosa \nDepartment of Physics\nFlorida A&M University\n32307TallahasseeFL\n",
"J Williamson \nDepartment of Physics\nFlorida A&M University\n32307TallahasseeFL\n"
]
| [
"Department of Physics\nFlorida A&M University\n32307TallahasseeFL",
"Department of Physics\nFlorida A&M University\n32307TallahasseeFL"
]
| []
| The Hamiltonian for a particle constrained to motion near a toroidal helix with loops of arbitrary eccentricity is developed. The resulting three dimensional Schrödinger equation is reduced to a one dimensional effective equation inclusive of curvature effects. A basis set is employed to find low-lying eigenfunctions of the helix. Toroidal moments corresponding to the individual eigenfunctions are calculated. The dependence of the toroidal moments on the eccentricity of the loops is reported. Unlike the classical case, the moments strongly depend on the details of loop eccentricity. | null | [
"https://arxiv.org/pdf/1106.4248v2.pdf"
]
| 119,253,889 | 1106.4248 | a18323f19b0779eda0858e8e7cf8dd9f7bfe36bf |
Toroidal moments of Schrödinger eigenstates
25 Aug 2011
M Encinosa
Department of Physics
Florida A&M University
32307TallahasseeFL
J Williamson
Department of Physics
Florida A&M University
32307TallahasseeFL
Toroidal moments of Schrödinger eigenstates
25 Aug 2011arXiv:1106.4248v2 [quant-ph]toroidal helixtoroidal momentcurvature potential
The Hamiltonian for a particle constrained to motion near a toroidal helix with loops of arbitrary eccentricity is developed. The resulting three dimensional Schrödinger equation is reduced to a one dimensional effective equation inclusive of curvature effects. A basis set is employed to find low-lying eigenfunctions of the helix. Toroidal moments corresponding to the individual eigenfunctions are calculated. The dependence of the toroidal moments on the eccentricity of the loops is reported. Unlike the classical case, the moments strongly depend on the details of loop eccentricity.
Introduction
The majority of work directed towards modeling the metaparticle constituents of metamaterials has been performed using classical physics [1]. The characteristic length scales of most currently fabricated metaparticles allow for that approach to be appropriate and productive. However, it is nearly certain that metaparticles will eventually be fabricated on scales at which quantum mechanical methods will prove necessary to capture their physics with good fidelity [2,3,4]. This paper focuses upon two interesting properties common to many metaparticles: they can be approximated as reduced dimensionality systems and they can possess nontrivial topologies. The advent of quasi one and two dimensional curved nanostructures has led to situations wherein formalism developed for particles constrained to curved manifolds has become of practical importance.
Specifically, there exits a prescription that allows for degrees of freedom extraneous to the particle's 'motion' on a curve or surface to be shuttled into effective curvature potentials in the Schrödinger equation [5,6,7,8,9,10,11,12,13,14].
Recently, it was suggested that quantum methods be employed in an effort towards understanding toroidal moments induced by currents supported on nanoscale metaparticles and the interactions of those moments with timedependent electromagnetic fields [15]. Because of the theoretical and practical interest in toroidal moments [16,17,18,19,20,21,22], a toroidal helix (TH) of adjustable eccentricity has been chosen here to investigate the role of quantum effects. Being closed, a TH can support current carrying solutions allowing for the existence of a toroidal moment [23]. Furthermore, the TH has the advantage of having sufficient symmetry to allow for a clean reduction of the full Hamiltonian to a one dimensional effective Hamiltonian.
The goals of this work are threefold. The first is to derive the Hamiltonian for a particle in a coordinate system adapted to include points near the coils of a TH of arbitrary eccentricity. The next deals with reducing the full three dimensional Hamiltonian via a well known procedure [6,24,25] to arrive at an effective one-dimensional Schrödinger equation. The reduction of dimensionality impels the introduction of a curvature potential well known to workers in the field of curved manifold quantum mechanics. A basis set consistent with the periodicity and symmetry of the system is introduced thereafter. Achieving the first two goals and with the basis functions in hand, the spectrum and wave functions of the system (which can be used for applications in the external field and/or time-dependent case) are found. Finally, toroidal moments corresponding to particular eigenstates are determined and their sensitivity to the eccentricity of the loops comprising the TH is investigated.
The remainder of this paper is organized into four sections. Section 2 introduces a parameterization for an ω turn TH in terms of an azimuthal coordinate φ. A three dimensional Hamiltonian H 3 ω appropriate to motion near the TH follows by attaching a Frenet system to the helix and assigning two coordinates q N , q B to describe degrees of freedom away from the coil. Section 3 details the reduction of H 3 ω to a one-dimensional H 1 ω by standard methods, although perhaps unfamiliar to workers in the metamaterial community. As a consequence of the reduction, curvature potentials appear. Their presence has been shown to be essential in properly describing one dimensional systems that exist in an ambient higher dimensional space [26]. Section 4 presents the basis set used to calculate the spectrum, eigenstates and toroidal moments per a given quantum state. Those quantities, along with results showing the dependence of TMs on eccentricity are given. Section 5 is dedicated to conclusions and some remarks concerning future work.
The TH Schrödinger equation
To arrive at the time independent Schrödinger equation H 3 ω (r)Ψ = EΨ = − 1 2 ∇ 2 + V Ψ, the Laplacian must be derived from a suitable parameterization of the TH geometry. Consider a TH with ω equally spaced circular coils. Let R be the distance from the z-axis to a loop center and a the radius of a loop.
First define
W (φ) = R + a cos(ωφ)(1)
with φ the usual cylindrical coordinate azimuthal angle. The circular TH is traced out by the Monge form [27] r(φ) = W (φ)ρ + a sin(ωφ)k.
Generalizing Eq.(2) to coils of arbitrary eccentricity requires only the modifica-
tion r(φ) = W (φ)ρ + b sin(ωφ)k(3)
where a, b may be adjusted to yield the coil shape desired (Fig. 1). To avoid cluttering the narrative with blocks of equations, the expressions that follow will apply to the circular case only. The expressions for arbitrary a and b are given in the appendix.
A three dimensional neighborhood in the vicinity of the TH is built by assigning two coordinates to points near the curve along unit vectors orthogonal to the curve's tangent and to each other. The Frenet-Serret equations [27] provide such an orthonormal coordinate system known as a Frenet trihedron.
The unit tangent to any point on a curve traced by r(φ) iŝ
T = dr(φ) dφ dr(φ) dφ −1(4)
from which the Frenet trihedron can be constructed via the relations
dT dφ = dr(φ) dφ κ(φ)N (5) dN dφ = dr(φ) dφ − κ(φ)T + τ (φ)B (6) dB dφ = − dr(φ) dφ τ (φ)N(7)
where the curvature and torsion of the space curve r(φ) are indicated by κ(φ) and τ (φ) respectively (where again, detailed forms for the expressions in Eqs. (4)(5)(6)(7) appear in the appendix). Points near the TH are located via two perpendicular displacements q NN and q BB . The TH position vector may now be written
x(φ, q N , q B ) = r(φ) + q NN + q BB .(8)
It should be noted that Eq.(8) defines a Cartesian region about a curve traced by r(φ). While it is certainly possible to construct a finite tubular neighborhood about r(φ), the coordinate ambiguity of the azimuthal angle as the radial distance approaches zero causes the limiting procedure to become complicated.
Additionally, the separability of the Schrödinger equation into tangential and normal variables is lost, and with it any real advantage in using the reduced Hamiltonian.
The covariant metric tensor elements g ij can be read off of the quadratic form [28] dx · dx = g ij dq i dq j (9) where in what follows the ordering convention is (q 1 , q 2 , q 3 ) ≡ (φ, q N , q B ). The
Laplacian is
∇ 2 = 1 √ g ∂ ∂q i √ g g ij ∂ ∂q j(10)
with g = det(g ij ) and g ij the contravariant components of the metric tensor.
Before presenting explicit forms for g ij and g ij , it is useful to define
f (φ) = dr(φ) dφ = [(aω) 2 + W (φ) 2 ] 1/2(11)
and
G(φ, q N ) = 1 − q N κ(φ)(12)
after which the covariant metric may be written
g ij = f (φ) 2 [G(φ, q N ) 2 + τ (φ) 2 (q 2 N + q 2 B )] −τ (φ)q B f (φ) τ (φ)q N f (φ) −τ (φ)q B f (φ) 1 0 τ (φ)q N f (φ) 0 1 .(13)
The contravariant form of the metric is obtained straightforwardly;
g ij = 1 f (φ) 2 G(φ, q N ) 2 1 τ (φ)q B f (φ) −τ (φ)q N f (φ) τ (φ)q B f (φ) f (φ) 2 [G(φ, q N ) 2 + τ (φ) 2 q 2 B ] −τ (φ) 2 q N q B f (φ) 2 −τ (φ)q N f (φ) −τ (φ) 2 q N q B f (φ) 2 f (φ) 2 [G(φ, q N ) 2 + τ 2 (φ)q 2 N ] .(14)
It is easy to show that
√ g = f (φ) 1 − q N κ(φ) .
The Laplacian found by directly evaluating Eq.(10) is complicated by the existence of cross terms arising from ∂ 2 /∂q i ∂q j , (i = j), operations. However, all of those terms are multiplied by the distance parameters q N and q B such that in the limit q N , q B → 0 they vanish independently of the derivative operators that follow them. Taking this limit now (it will be taken again later post operation of the q N,B derivatives) leads to a more convenient starting point for developing the reduced Hamiltonian in the ensuing section. Physically, the limiting procedure is effected by external mechanical or electrical constraints; mathematically, they are added by hand into the Schrödinger equation as potentials V n (q) normal to the lower dimensionality base manifold. Their detailed forms are not important. Previous work has shown that even for finite thicknesses, degrees of freedom extraneous to those of the base manifold do not mix with the latter in the sense that their wave functions decouple [26]. Here, for the sake of definiteness, hard wall potentials are assumed for V n (q N ) and V n (q B ) in this and the next section. With this discussion in mind, H 3 ω may be written as (withh = m = 1)
H 3 ω = − 1 2 1 f (φ) 2 ∂ 2 ∂φ 2 − f ′ (φ) f (φ) 3 ∂ ∂φ −κ(φ) ∂ ∂q N + ∂ 2 ∂q 2 N + ∂ 2 ∂q 2 B +V n (q N )+V n (q B ).(15)
Note that the H 3 ω at this stage is still not separable. The procedure for rendering H 3 ω separable and arriving at a simpler effective Hamiltonian is given in the following section.
Constructing the effective Hamiltonian
As the particle is constrained to the toroidal helix, its wave function will decouple into tangent and normal functions (the subscripts t and n denote tangent and normal respectively)
Ψ(φ, q N , q B ) → χ t (φ)χ n (q N )χ n (q B )(16)
and G(φ, q N ) will approach unity. The normalization condition
2π 0 |Ψ(φ, q N , q B )| 2 G(φ, q N )f (φ) dφ dq N dq B = 1 (17) becomes 2π 0 |χ t (φ)| 2 |χ n (q N )| 2 |χ n (q B )| 2 f (φ) dφ dq N dq B = 1.(18)
The norm must be conserved in the decoupled limit [6], which implies
|Ψ(φ, q N , q B )| 2 G(φ, q N ) = |χ t (φ)| 2 |χ n (q N )| 2 |χ n (q B )| 2 .(19)
The wave function
Ψ(φ, q N , q B ) is now related to χ t (φ)χ n (q N )χ n (q B ) by Ψ(φ, q N , q B ) = χ t (φ)χ n (q N )χ n (q B )G −1/2 (φ, q N ).(20)
Applying H 3 ω to Ψ(φ, q N , q B ) and taking the limit as q N , q B → 0 post all derivative operations yields the result
H 3 ω = − 1 2 1 f (φ) 2 ∂ 2 ∂φ 2 − f ′ (φ) f (φ) 3 ∂ ∂φ + 1 4 κ 2 (φ) + ∂ 2 ∂q 2 N + ∂ 2 ∂q 2 B + V n (q N ) + V n (q B ).(21)
Distributing the energy between the (φ, q N , q B ) degrees of freedom by allowing
E = E φ + E N + E B , leads to the decoupled system − 1 2 1 f (φ) 2 ∂ 2 ∂φ 2 − f ′ (φ) f (φ) 3 ∂ ∂φ + 1 4 κ 2 (φ) χ t (φ) = E φ χ t (φ) (22) − 1 2 ∂ 2 χ n (q N ) ∂q 2 N + V n (q N )χ n (q N ) = E N χ n (q N ) (23) − 1 2 ∂ 2 χ n (q B ) ∂q 2 B + V n (q B )χ n (q B ) = E B χ n (q B ).(24)
Since V (q N ) and V (q B ) are the confining potentials effecting the q N , q B → 0 constraint, q N and q B can be considered spectator variables and only the φ-dependent part of the Hamiltonian indicated in Eq. (21) is nontrivial. The Hamiltonian in one dimension H 1 ω is written
H 1 ω = − 1 2 1 f (φ) 2 ∂ 2 ∂φ 2 − f ′ (φ) f (φ) 3 ∂ ∂φ + V c (φ)(25)
with
V c (φ) = − 1 8 κ 2 (φ)(26)
the curvature potential. The curvature potential V c (φ) emerges as an artifact of embedding the particle's one dimensional path of motion in the ambient three dimensional space. The explicit form of the curvature potential in Eq.(26) can be determined from
κ(φ) = [P 1 (φ) 2 + P 2 (φ) 2 ] 1/2(27)
where
P 1 (φ) = − aω 2 + W (φ)cos(ωφ) f (φ) 2(28)
and
P 2 (φ) = sin(ωφ) f (φ) 1 + aω f (φ) 2 .(29)
Explicit forms of the tangent, normal, and binormal vectors, along with other vectors and functions for the circular and elliptic helices are given in the appendix.
A plot of V c (φ) for some representative values of a, b with ω = 4 appears in It is worth stating that instead of parameterizing the TH with φ, it would also be possible to employ an arc length scheme where an arc length parameter λ
is determined from λ = φ 0 f (φ ′ ) dφ ′ .
However, to include the curvature potential as a function of λ, it would be necessary to find φ(λ) along the curve. While this could be accomplished numerically, using the azimuthal angle is somewhat better suited to incorporating external fields [29,30].
Computational methods and results
If the TH is small enough to require a quantum mechanical description, the φ-dependent part of its wave function must obey Bloch's theorem (the tsubscript will be dropped hereafter)
χ k φ + 2π ω = exp ik 2π ω χ k (φ).(30)
A standard choice is [31]
χ k (φ) = exp(ikφ)u k (φ)(31)
where u k (φ + 2π/ω) = u k (φ) is satisfied. Single valuedness requires the Bloch index k ≡ p = integer. A convenient choice for u p (φ) basis elements is
u p (β, φ) = exp[iβφ].(32)
The requirement indicated in Eq.(30) yields β = ωn, n ≡ integer.
From the above considerations, a suitable basis set for the TH is
χ pα (φ) = exp[ipφ] n C pα n exp(inωφ).(34)
The Bloch form introduces sub-states (sub-bands in the case of a continuous rather than discrete index) for each p value which would not be present if the TH were treated as a ring of length L. The C pα m are the expansion coefficients for α-th sub-state of a given p value. In this work, it was found that a five-state expansion proved sufficient to yield basis size independent results for the lower
χ pα (φ) = exp[ipφ] f (φ) 1/2 n C pα n exp(inωφ).(35)
With basis function orthogonality preserved on the right hand side of the Hamiltonian, eigenvalues and eigenvectors are calculated by diagonalizing the matrix comprising the elements
H mn = 1 2π 2π 0 e iω(n−m)φ − 2V c (φ) − (p + ωn) 2 f (φ) 2 + 5 4 f ′ (φ) 2 f (φ) 4 − f ′′ (φ) 2f (φ) 3 − 2i(p + ωn) f ′ (φ) f (φ) 3 dφ.(36)
Once the eigenstates are found, the current in general is calculated with (now with units)
j (φ, q B , q N ) = q eh m e Im Ψ * (φ, q B , q N )∇Ψ(φ, q B , q N ) .(37)
The current density given by Eq.(37) is inclusive of cross-sectional degrees of freedom and yields a current passing through a rectangular area with unit nor-malT. However, in keeping with the intent of this work, the limit of infinitesimal thickness is assumed (or equivalently, the q B , q N degrees of freedom are integrated out) leading to the current expression for the pα-th state
j pα (φ, 0, 0) = q eh m e Im (χ pα (φ)) * f (φ) ∂χ pα (φ) ∂φ T (38)
where the form of the reduced gradient operator is obvious. The quantum mechanical current that stems from Eqs. (35) and (38) becomes
j pα (φ, 0, 0) = q eh m e 1 2π m,n C pα m C pα n (p + ωn) f (φ) 2 cos[ω(n−m)φ]− f ′ (φ) 2f (φ) 3 sin[ω(n−m)φ] T .(39)
When V c (φ) is included in the Hamiltonian the C pα m are modified, causing the current to become inclusive of curvature effects. This current is then used to calculate the toroidal moments according to [32]
T pα M = 1 10 2π 0 [(j pα (φ, 0, 0) · r)r − 2r 2 j pα (φ, 0, 0) ] f (φ)dφ.(40)( dr dφ · r )r − 2r 2 dr dφ f (φ) dφ.(41)
For circular TH, Eq.(41) yields
T p M = − πωIa 2 R 2k(42)
and for the elliptic TH,
T p M = − πωIabR 2k .(43)
As a means of comparison, the current for the p state without curvature effects (i.e. a free particle on a given ω turn helix) is easily determined to be
I = 2πq eh p m e L 2(44)
where the total length of the TH, L, is calculated using
L = 2π 0 f (φ) dφ.
The formalism described in this section was employed to calculate the eigenvalues and eigenstates expressed in terms of the C pα m for several ω and p values. To get a sense of the modifications arising from V c (φ), the eigenvalues and amplitudes for a six-turn eccentric helix in a p = 1 state are listed without (Table 1) and with ( Table 2) the curvature potential being present. The eigenvalue shifts reflect that V c (φ) is always attractive as shown in Fig. (2), and capable of causing amplitude shifts. The reader will note there is no table indicating the shifts for the circular case; the effects are negligible and essentially independent of the coil radius a.
With the C pα m amplitudes in hand, Eq.(39) can be used to find the j pα (φ, 0, 0) necessary for computing TMs. To set a baseline for understanding the effect of including V c (φ), the curvature potential was shut off and j pα (φ, 0, 0) determined for many cases. In Fig. (3), representative results are given for ω = 4, p = 1.
As anticipated, the lowest energy states yield very steady currents; oscillations begin to manifest in the higher sub-band energy states. Turning the potential on produces very little change in the currents; in Fig. (4), it becomes clear that V c (φ) does little, which is consistent with its small amplitude indicated in Fig. (2). Toroidal moment results for ω = 4 are shown in Table 3 for several p-states and their corresponding sub-states. For a given p value, the lowest energy state in Table 3 agrees very well with the value obtained if the current given by Eq. (44) were used, although V c (φ) can shift the ordering of states in a way to reorder the ground state moment as seen for the p = 3 states. In isolation, this is relatively unimportant. However, in a broader context where the natural temperature scale of a 1000Å helix is a few µK, thermodynamic averages of the type
T M = T M (E n )exp[−E n /τ ](45)
will necessitate accounting for proper ordering. Table 3 show a much stronger variation in TM values, consistent with the much larger strength of V c (φ) for a > b relative to the converse. A sense of the dependence of TMs on ω can be gleaned from Table 4 where now ω = 8.
Increased variation is seen for both eccentricities, but the flattened coil case demonstrates appreciable deviation from the classical expression.
Conclusions
In this work a prescription to include curvature at the nanoscale for particles constrained to toroidal helices was presented, which the authors applied toward a quantum mechanical calculation of toroidal moments. It is worth emphasizing that the curvature inclusive reduced dimensionality Schrödinger equation developed here is driven by an interest in having more tractable, effective models for nanomaterials, and is done with the aim of eventually confronting experimental data rather than as a purely theoretical exercise. In that context, the choice to Tailoring the response of toroidal helices to electromagnetic radiation by fabricating objects with curvature as a free parameter is still well outside the reach of current fabrication methods. However, the formalism and basis set established here may serve as means for further investigation of the T M · ∂E/∂t interactions relevant to the coupling of toroidal moments to electromagnetic fields. The extension of the methods here to cases where external vector potentials are present may be naturally developed from work already done for tori immersed in arbitrary magnetic fields [29] and is ongoing with an aim to understanding persistent current effects.
Finally, debate as to whether curvature effects are relevant to, and how they are manifested in, topologically novel nanostructures may eventually be settled by examining systems akin to toroidal helices. By opting to either include or exclude curvature potentials in modeling routines, it may prove true that sensitive quantities like toroidal moments will provide a clear signature as to the influence of V c . Work such as that done in this paper may hopefully contribute to a resolution to the question of how to properly incorporate twists and turns in the quantum mechanical description of bent nanostructures.
This appendix presents a more complete set of formulae for the circular TH as well as a corresponding set for the elliptical case.
A.1 The circular caseθ
= −sin(ωφ)ρ + cos(ωφ)k (A-1) n = cos(ωφ)ρ + sin(ωφ)k (A-2) f (φ) = (a 2 ω 2 + W (φ) 2 ) 1/2 (A-3) e 2 = W (φ)θ − aωφ f (φ) (A-4) P 1 (φ) = − aω 2 + W (φ)cos(ωφ) f (φ) 2 (A-5) P 2 (φ) = sin(ωφ) f (φ) 1 + aω f (φ) 2 (A-6) κ(φ) = (P 2 1 (φ) + P 2 2 (φ)) 1/2 (A-7) T = aωθ + W (φ)φ f (φ) (A-8) N = 1 κ(φ) (P 2 (φ)ê 2 + P 1 (φ)n) (A-9) B = 1 κ(φ) (−P 1 (φ)ê 2 + P 2 (φ)n) (A-10)
A.2 The elliptic case
With W (φ) = R + a cos(ωφ), the equation of the elliptic toroidal helix is
r(φ) = W (φ)ρ + b sin(ωφ)k.
The extension of the results of Sec. A.1 to the elliptic case is straightforward:
P (φ) = [a 2 sin 2 (ωφ) + b 2 cos 2 (ωφ)] 1/2 (A-11) θ E = 1 P (φ)
[−a sin(ωφ)ρ + b cos(ωφ)k] (A-12)
n E = 1 P (φ)
[b cos(ωφ)ρ + a sin(ωφ)k] (A-13) Curvature effects on the current are small in the circular case.
e 2 = W (φ)θ E + P (φ)ωφ f (φ) (A-14) f (φ) = (P (φ) 2 ω 2 + W (φ) 2 ) 1/2 (A-15) P 1 (φ) = − b P (φ) aω 2 + W (φ)cos(ωφ) f (φ) 2 (A-16) P 2 (φ) = sin(ωφ) f (φ) a P (φ) + ω 2 W (φ)(a 2 − b 2 )cos(ωφ) + P (φ) 2 aω 2 f (φ) 2 P (φ) (A-17) κ(φ) = (P 1 (φ) 2 + P 2 (φ) 2 ) 1/2 (A-18) T = P (φ)ωθ E + W (φ)φ f (φ) (A-19) N = 1 κ(φ) (P 2 (φ)ê 2 + P 1 (φ)n E ) (A-20) B = 1 κ(φ) (−P 1 (φ)ê 2 + P 2 (φ)n E ) (A-21)
Fig. 2 .
2Note that the circular case values are negligible in magnitude compared to the eccentric cases, and when a > b, V c (φ) is substantially larger than for the converse. For larger ratios of a to b, V c (φ) can be orders of magnitude larger than indicated in the figure.
p sub-states. For ω turns, values of p consistent with the Bloch theorem, p < ω, are used. For clarity, only p ≥ 0 are discussed. A disadvantage of directly adopting the expression given by Eq.(34) is that the basis functions are not orthogonal over the integration measure f (φ)dφ. A more natural basis set is given by a re-scaled form of Eq.(34)
Equation ( 40 )
40allows calculation of quantum mechanical toroidal moments of ground and excited states for each Block index p. For a macroscopic thin wire where j dτ → Idr is applicable, the toroidal moment for each p reduces to
When eccentricity is introduced by setting a = .75 and b = .25, the results are less trivial. The results displayed in Figs. (5) and (6) are representative of a general trend observed throughout values of ω, p. The curvature potential suppresses the current in every sub-band by a discernable fraction. Similar behavior is observed when a = .25 and b = .75 as shown in Fig. (7) and (8), but note that the magnitudes are substantially different from the converse values of a and b. The Bloch form of the wave function, independent of the presence of V c (φ) (which again lessens the magnitude of the currents), the Bloch form of the wave function and the ab dependence of the Laplacian are sufficient to cause asymmetries in the currents.
The modifications to the TMs for the upright (b > a) coil situation are generally minimal with exceptions only for p = 2. The flattened coil (a > b) results in
consider helices was driven by their capability of producing toroidal moments, which are currently of both theoretical and practical interest. The curvature potential for the helix was derived and shown to be the dominant part of the Hamiltonian for lower energy eigenstates of eccentric helices. An intriguing result that arose here was a demonstrated ab asymmetry in several states of the quantum mechanically calculated T pα M , an asymmetry not exhibited in the classical expression of Eq.(43). The array of results given in this work was limited to relatively small values of ω and to less severe eccentricity because of numerical limitations on evaluating integrals of the type shown in Eq.(36). The extension to larger values of ω were considered (at least currently) outside the scope of what the authors were attempting to accomplish. However, preliminary work gives some indication that Mathematica is capable of performing the necessary integrals, albeit with increased time expense. It would be of interest to investigate more extreme cases of eccentricity and loop number given the enhancement of moments already evidenced by larger ω.
Fig. 1 :
1A toroidal helix where R is the distance from the center of the TH to a center of a loop of the TH. The parameter a is the TH's maximum perpendicular horizontal distance from a concentric cylinder of radius R. The parameter b is the TH's maximum vertical distance from the x-y plane.
Fig. 2 :
2The curvature potential V c (φ) in units ofh 2 /(m e R 2 ) for the case of the circular TH with R = 1, ω = 4, a = b = 0.5 and two elliptic TH cases: R = 1, ω = 4, a = 0.25, b = 0.75 and R = 1, ω = 4, a = 0.25, b = 0.75.
Fig. 3 :
3j pα (φ, 0, 0) ≡ I(φ) in units of q eh /(m e R 2 ) for five eigenstates of the circular TH configuration ω = 4, a = 0.5, b = 0.5, R = 1, p = 1 where V c (φ) is neglected.
Fig. 4 :
4j pα (φ, 0, 0) ≡ I(φ) in units of q eh /(m e R 2 ) for five eigenstates of the circular TH configuration ω = 4, a = 0.5, b = 0.5, R = 1, p = 1 with V c (φ).
Fig. 5 :
5j pα (φ, 0, 0) ≡ I(φ) corresponding to the ground state, and second and fourth excited states in units of q eh /(m e R 2 ), for the elliptic TH configuration ω = 4, a = 0.75, b = 0.25, R = 1, p = 1, without the curvature potential V c (φ).
Fig. 6 :
6j pα (φ, 0, 0) ≡ I(φ) corresponding to the ground state, and second and fourth excited states in units of q eh /(m e R 2 ), for the elliptic TH configuration ω = 4, a = 0.75, b = 0.25, R = 1, p = 1 inclusive of the curvature potential V c (φ). Inclusion of the curvature potential causes a reduction in amplitude for each j.
Fig. 7 :
7j pα (φ, 0, 0) ≡ I(φ) corresponding to the ground state, and second and fourth excited states in units of q eh /(m e R 2 ) for the elliptic TH configuration ω = 4, a = 0.25, b = 0.75, R = 1, p = 1 without curvature potential.
Fig. 8 :
8j pα (φ, 0, 0) ≡ I(φ) corresponding to the ground state, and second and fourth excited states in units of q eh /(m e R 2 ) for the configuration ω = 4, a = 0.25, b = 0.75, R = 1, p = 1 inclusive of the curvature potential. When the curvature potential is included in the calculation of I(φ), the current is reduced in amplitude.
Table 1 :
1Eigenvalues and amplitudes C (α) m for an ω=6, a=0.75, b=0.25, R=1, p=1 elliptic TH neglecting curvature effects.E 0
E 1
E 2
E 3
E 4
-1.4739 -0.0442 2.1393
5.7258
9.2713
m
C
Table 2 :
2Eigenvalues and amplitudes C (α) m for an ω=6, a=0.75, b=0.25, R=1, p=1 elliptic TH including curvature effects.
Table 3 :
3Toroidal moments for two configurations with ω=4. TM's with and without curvature effects, and classical calculation for each case.ω
a
b
4
0.25
0.75
p
TM
TM w/Vc(φ)
ratio
Classical TM
Table 4 :
4Toroidal moments for two configurations with ω=8. TM's with and without curvature effects, and classical calculation for each case.ω
a
b
8
0.25
0.75
p
TM
TM w/Vc(φ)
ratio
Classical TM
-0.3893 -0.3413 1.1407 3 -0.1784 -0.2131 0.8372
-0.4509 -0.4640 0.9718
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| []
|
[
"ON SOME SUMMABILITY METHODS FOR A q-ANALOGUE OF AN INTEGRAL TYPE OPERATOR BASED ON MULTIVARIATE q-LAGRANGE POLYNOMIALS",
"ON SOME SUMMABILITY METHODS FOR A q-ANALOGUE OF AN INTEGRAL TYPE OPERATOR BASED ON MULTIVARIATE q-LAGRANGE POLYNOMIALS"
]
| [
"Purshottam Narain Agrawal ",
"Rahul Shukla \nDepartment of Mathematics\nIndian Institute of Technology Roorkee Roorkee-247667\nIndia\n",
"Behar Baxhaku 3email:[email protected] \nDepartment of Mathematics\nUniversity of Prishtina\nPrishtina, Kosovo\n"
]
| [
"Department of Mathematics\nIndian Institute of Technology Roorkee Roorkee-247667\nIndia",
"Department of Mathematics\nUniversity of Prishtina\nPrishtina, Kosovo"
]
| []
| The present paper considers a q-analogue of an operator defined by 45(1) (2008), 53-67) involving q-Lagrange polynomials in several variables. The Korovkin type theorems in the settings of deferred weighted A-statistical convergence and the power series method are investigated. Keywords: Multivariate Lagrange polynomials, q-multivariate Lagrange polynomials, natural density, A-statistical convergence, deferred weighted A-statistical convergence, convergence by power series method. MSC(2020): 41A25, 41A36, 33C45, 26A15. | 10.1080/01630563.2022.2056199 | [
"https://arxiv.org/pdf/2111.02899v1.pdf"
]
| 242,756,898 | 2111.02899 | 5f0f89b05c5116329f25d861e128208cff004a69 |
ON SOME SUMMABILITY METHODS FOR A q-ANALOGUE OF AN INTEGRAL TYPE OPERATOR BASED ON MULTIVARIATE q-LAGRANGE POLYNOMIALS
2 Nov 2021
Purshottam Narain Agrawal
Rahul Shukla
Department of Mathematics
Indian Institute of Technology Roorkee Roorkee-247667
India
Behar Baxhaku 3email:[email protected]
Department of Mathematics
University of Prishtina
Prishtina, Kosovo
ON SOME SUMMABILITY METHODS FOR A q-ANALOGUE OF AN INTEGRAL TYPE OPERATOR BASED ON MULTIVARIATE q-LAGRANGE POLYNOMIALS
2 Nov 2021Dedicated to Prof. R. P. Agarwal on his 74 th birthday
The present paper considers a q-analogue of an operator defined by 45(1) (2008), 53-67) involving q-Lagrange polynomials in several variables. The Korovkin type theorems in the settings of deferred weighted A-statistical convergence and the power series method are investigated. Keywords: Multivariate Lagrange polynomials, q-multivariate Lagrange polynomials, natural density, A-statistical convergence, deferred weighted A-statistical convergence, convergence by power series method. MSC(2020): 41A25, 41A36, 33C45, 26A15.
Introduction
The past two decades have witnessed to the constructions of linear positive operators by means of multivariate-Lagrange polynomials (and their q-analgoue) and to their approximation behaviour. Let C(I), . be the Banach space of all continuous functions on I = [0, 1] with the sup-norm . . After the introduction of celebrated multivariate Lagrange polynomials (widely known as Chan-Chyan-Srivastava polynomials) [5], for f ∈ C(I), Erkus et al. [8] defined the following sequence of linear positive operators as; where x ∈ I, β (j) = β (j) n are sequences in (0, 1) for each j = 1, 2, · · · r, and (ρ) s denotes the standard Pochhammer symbol. The authors studied the approximation properties of the above operator in the A-statistical settings. For every q ∈ R such that |q| < 1 and n ∈ N 0 = {0, 1, 2, ...}, the q-Pochhammer symbol (ρ; q) n is given by (ρ; q) n = 1, if n = 0, (1 − ρ)(1 − ρq)...(1 − ρq n−1 ), if n ∈ N, and the q-analogue of a natural number (q-integers) is defined by [n] q = 1 − q n 1 − q = 1 + q + q 2 + .... + q n−1 .
Altin et al. [1] proposed a q-multivariable Lagrange polynomials h (η1,··· ηr ) n,q (z 1 , z 2 , · · · z r ) as follows, h (η1,··· ηr ) n,q (z 1 , z 2 , · · · z r ) = l1+l1+···+lr=n r k=1 (q η k , q) l k (z k ) l k (q, q) l k , (1.2) whereas the above multivariate polynomials has the generating function of the following form r k=1 1 (tz k ; q) η k = ∞ n=0 h (η1,··· ηr ) n,q (z 1 , z 2 , · · · z r )t n , (1.3) where |t| < min{|z 1 | −1 , · · · |z r | −1 }. Erkuş-Duman et al. [7] proposed an integral type generalizations of the operator (1.1) in the following manner: and studied its statistical approximation properties by means of modulus of continuity and Peetre's K-functional. Using the generating function given by (1.3), Erkuş-Duman [6] studied the following q-analogue of the operator L β (1) ,··· β (r)
n S β (1) ,··· ,β (r) n,q (f (s); x) = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+l2+···+lr=p (q n ; q) l1 (q n ; q) l2 ...(q n ; q) lr (β (1) n ) l1 (β (2) n ) l2 ...(β (r) n ) lr (q; q) l1 (q; q) l2 ...(q; q) lr f [l r ] q [n + l r − 1] q x p . (1.5)
It has been observe that the above operator (1.5) was also indepedently tackled by Mursaleen et al. [14] but unfortunately the proposed definition was incorrect while Behar et al. [2] extended the study of Erkuş-Duman to the bi-variate and GBS (Generalized Boolean Sum) cases.
The prime objective of this paper is to define a q-analogue of the operators (1.4) by means of Riemann type q-integral, and to study the convergence of such operators via summability methods. In Section 2., we construct the operator and establish some important lemmas to prove the main results. In Section 3., the Korovkin type theorems in the deferred weighted A-statistical approximation are studied for these operators. In the last Section, we establish the basic convergence theorem and an estimate of error in the approximation by using power series summability method.
Construction of the operators and Important Lemmas
Marinković et al. [20] introduced the following Riemann type q-integral
β α f (s)d R q s = (1 − q)(α − β) ∞ j=0 f α + (β − α)q j q j ,(2.1)
where α, β, q ∈ R such that 0 < α < β and q ∈ (0, 1). This definition of q-integral is appropriate to derive some q-analogues of well-known integral inequalities [12]. Using the q-Riemann type integral, for f ∈ C(I), we propose a q-analogue of the operators (1.4) as follows:
K β (1) ,··· ,β (r) n,q (f (s); x) = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+l2+···+lr=p
(q n ; q) l1 (q n ; q) l2 ...(q n ; q) lr
×[n + l r − 1] q q −lr (β (1) n ) l1 (β (2) n ) l2 ...(β (r) n ) lr (q; q) l1 (q; q) l2 ...(q; q) lr [lr +1]q [n+lr−1]q [lr ]q [n+lr−1]q f (s)d R q s x p . (2.2)
In order to discuss our main results, we first give the following lemmas.
Lemma 1. The operators K β (1) ,··· ,β (r) n,q (.; x) verify the assertions:
(i) K β (1) ,··· ,β (r) n,q (1; x) = 1; (ii) K β (1) ,··· ,β (r) n,q (s; x) ≤ xβ (r) n + 1 [2]q[n]q . Moreover, K β (1) ,··· ,β (r) n,q (s; x) − x ≤ x(1 − β (r) n ) + 1 [2] q [n] q . (iii) S β (1) ,··· ,β (r) n,q (s 2 ; x) ≤ 1 [3]q[n] 2 q + xβ (r) n [n]q 1 + 2 [2]q + q(xβ (r) n ) 2 . Also, K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 ≤ 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q + 2x 2 (1 − β (r) n ).
Proof.
K β (1) ,··· ,β (r) n,q (s; x) = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p (q n ; q) l1 · · · (q n ; q) lr [n + l r − 1] q q −lr × (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr [lr +1]q [n+lr−1]q [lr ]q [n+lr −1]q s d R q s x p = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p (q n ; q) l1 · · · (q n ; q) lr [n + l r − 1] q q −lr × (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr 1 [n + l r − 1] 2 q q lr [l r ] q + q 2lr [2] q x p = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr =p (q n ; q) l1 · · · (q n ; q) lr [l r ] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p + r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr =p (q n ; q) l1 · · · (q n ; q) lr q lr [2] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p .
Since,
[lr]q (q;q) lr = 1 (1−q)(q;q) lr −1 , (q n ;q) lr [n+lr −1]q = (1 − q)(q n ; q) lr −1 and q lr [n+lr−1]q ≤ 1 [n]q , we have K β (1) ,··· ,β (r) n,q (s; x) ≤ xβ (r) n r k=1 (xβ (k) n ; q) n ∞ p=1 l1+···+lr−1=p−1 lr ≥1 (q n ; q) l1 · · · (q n ; q) lr −1 (β (1) n ) l1 · · · (β (r) n ) lr −1 (q; q) l1 · · · (q; q) lr −1 x p−1 + 1 [2] q [n] q r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p
(q n ; q) l1 · · · (q n ; q) lr (β
(1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p ≤ xβ (r) n r k=1 (xβ (k) n ; q) n ∞ p=1 h (n,··· ,n) p−1,q (β (1) n , β (2) n , · · · , β (r) n )x p−1 + 1 [2] q [n] q r k=1 (xβ (k) n ; q) n ∞ p=0 h (n,··· ,n) p,q (β (1) n , β (2) n , · · · , β (r) n )x p , from (1.2) ≤ xβ (r) n + 1 [2] q [n] q
, in view of (i).
Using above inequality we can get
K β (1) ,··· ,β (r) n,q (s; x) − x ≤ x β (r) n − 1 + 1 [2] q [n] q . (2.3)
On the other hand, we have
K β (1) ,··· ,β (r) n,q (s; x) ≥ r k=1 (xβ (k) n ; q) n ∞ p=1 l1+···+lr =p lr ≥1 (q n ; q) l1 · · · (q n ; q) lr [l r ] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p ≥ xβ (r) n . (2.4)
Thus by equation (2.3) and (2.4), we get
K β (1) ,··· ,β (r) n,q (s; x) − x ≤ x(1 − β (r) n ) + 1 [2] q [n] q .
(iii) Since from (1.5), we have
K β (1) ,··· ,β (r) n,q (s 2 ; x) = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p (q n ; q) l1 · · · (q n ; q) lr ×[n + l r − 1] q q −lr (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr [lr +1]q [n+lr−1]q [lr ]q [n+lr−1]q s 2 d R q s x p K β (1) ,··· ,β (r) n,q (s 2 ; x) = r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr =p (q n ; q) l1 · · · (q n ; q) lr [n + l r − 1] q q −lr × 1 [n + l r − 1] 3 q q lr [l r ] 2 q + 2q 2lr [l r ] q [2] q + q 3lr [3] q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p = xβ (r) n r k=1 (xβ (k) n ; q) n ∞ p=1 l1+···+lr −1=p−1 lr≥1 (q n ; q) l1 · · · (q n ; q) lr −1 1 + q[l r − 1] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr−1 (q; q) l1 · · · (q; q) lr −1 x p−1 + 2 [2] q r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p (q n ; q) l1 · · · (q n ; q) lr q lr [l r ] q [n + l r − 1] 2 q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p + 1 [3] q r k=1 (xβ (k) n ; q) n ∞ p=0 l1+···+lr=p (q n ; q) l1 · · · (q n ; q) lr q 2lr [n + l r − 1] 2 q (β (1) n ) l1 · · · (β (r) n ) lr (q; q) l1 · · · (q; q) lr x p = 1 + 2 + 3 + 4 say. Now, 1 = xβ (r) n r k=1 (xβ (k) n ; q) n ∞ p=1 l1+···+lr−1=p−1 lr≥1 (q n ; q) l1 · · · (q n ; q) lr−1 1 [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr −1 (q; q) l1 · · · (q; q) lr −1 x p−1 ≤ xβ (r) n [n] q , using 1 [n + l r − 1] q ≤ 1 [n] q and (i). (2.5) Also, 2 = qxβ (r) n r k=1 (xβ (k) n ; q) n ∞ p=1 l1+···+lr −1=p−1 lr≥1 (q n ; q) l1 · · · (q n ; q) lr −1 [l r − 1] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr−1 (q; q) l1 · · · (q; q) lr −1 x p−1 = q(xβ (r) n ) 2 r k=1 (xβ (k) n ; q) n ∞ p=2 l1+···+lr−2=p−2 lr≥2 (q n ; q) l1 · · · (q n ; q) lr −2 1 − q lr−1 (1 − q)[n + l r − 1] q (1 − q n+lr −2 ) 1 − q lr −1 (β (1) n ) l1 · · · (β (r) n ) lr −2 (q; q) l1 · · · (q; q) lr −2 x p−2 = q(xβ (r) n ) 2 r k=1 (xβ (k) n ; q) n ∞ p=2 l1+···+lr−2=p−2 lr≥2 (q n ; q) l1 · · · (q n ; q) lr −2 [n + l r − 2] q [n + l r − 1] q (β (1) n ) l1 · · · (β (r) n ) lr−2 (q; q) l1 · · · (q; q) lr −2 x p−2 . Since [n+lr−2]q [n+lr−1]q < 1, using (i), we get 2 ≤ q(xβ (r) n ) 2 . (2.6)
Similarly, using
1 [n+lr −1]q ≤ 1 [n]q and 1 [n+lr−1] 2 q ≤ 1 [n] 2 q , we obtain 3 = xβ (r) n 2 [2] q [n] q ,(2.7)
and
4 = 1 [3] q [n] 2 q , respectively. (2.8)
Finally, combining the equations (2.5) − (2.8), we have
K β (1) ,··· ,β (r) n,q (s 2 ; x) ≤ 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q + q(xβ (r) n ) 2 . (2.9)
Now, from (2.9)
K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 ≤ x 2 (q(β (r) n ) 2 − 1) + 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q = −x 2 (1 − q(β (r) n ) 2 ) + 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q .
Since q, β (r) n ∈ (0, 1), we get
K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 ≤ 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q .
(2.10)
Using the positivity and linearity of the operators, and equation (2.4), we have
0 ≤ K β (1) ,··· ,β (r) n,q ((s − x) 2 ; x) = K β (1) ,β (2) n,q (s 2 ; x) − 2xK β (1) ,··· ,β (r) n,q (s; x) + x 2 or, − 2x 2 (1 − β (r) n ) ≤ K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 or, − 2x 2 (1 − β (r) n ) − 1 [3] q [n] 2 q − xβ (r) n [n] q 1 + 2 [2] q ≤ K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 .
Hence, in view of (2.10)
−2x 2 (1 − β (r) n ) − 1 [3] q [n] 2 q − xβ (r) n [n] q 1 + 2 [2] q ≤ K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 < 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q + 2x 2 (1 − β (r) n ), thus |K β (1) ,··· ,β (r) n,q (s 2 ; x) − x 2 | ≤ 1 [3] q [n] 2 q + xβ (r) n [n] q 1 + 2 [2] q + 2x 2 (1 − β (r) n ).
Lemma 2. For the operators K β (1) ,··· ,β (r) n,q (.; x), we have the inequality
K β (1) ,··· ,β (r) n,q (s − x) 2 ; x ≤ 2x(1 + x)(1 − β (r) n ) + x [n] q β (r) n 1 + 2 [2] q + 2 [2] q + 1 [3] q [n] 2 q ≤ 4(1 − β (r) n ) + 1 [n] q β (r) n 1 + 2 [2] q + 2 [2] q + 1 [3] q [n] 2 q = γ n,q (β (r)
n ), say. Proof. We can write
K β (1) ,··· ,β (r) n,q (s − x) 2 ; x ≤ K β (1) ,··· ,β (r) n,q s 2 − x 2 ; x + 2x K β (1) ,··· ,β (r) n,q s − x; x .
Now, using Lemma 1, we obtain the required inequality.
From now onwards in Sections 3 and 4, we assume that q = q n ∈ (0, 1) such that q n → 1 and q n n → a ∈ [0, 1) as n → ∞. There are many ways to define the density of the subsets of natural numbers and these definitions are playing a pivotal role in the areas of Number Theory and Graph Theory (see [16,17]). Any sequence x n is called statistically convergent to l if, for each ǫ > 0, we have the following lim n→∞ |{k ∈ N : k ≤ n and |x k − l| ≥ ǫ}| n = 0.
Deferred weighted
In this case, we write stat lim n→∞ x n = l. The definition shows that every convergent sequence is always statistically convergent, while the converse need not to be true in general. Karakaya et al. [10] derived the concept of weighted statistical convergence and the idea later modified by Mursaleen et al. in [13]. Let us assume that s k be a sequence such that s k ≥ 0 and S n = n k=1 s k , s 1 > 0, denotes its partial sum. Now, set
u n = 1 S n n k=1 s k x k , n ∈ N.
Then, the sequence x n is called weighted statistically convergent to a number l if, for any given ǫ > 0, the following holds lim n→∞ |{k ∈ N : k ≤ S n and s k |x k − l| ≥ ǫ}| S n = 0, and we write stat w lim n→∞ x n = l. If X 1 and X 2 are sequence spaces such that for every infinite matrix A = (a n,k ) : X → Y , we have (Ax) n = ∞ k=1 a n,k x k . Then, the matrix A is called regular if lim n→∞ (Ax) n = l whenever lim k→∞ (x) k = l. For a non-negative regular matrix A = (a n,k ), Freedman et al. [9] defined the idea of A-statistical convergence. The sequence (x) n is called A-statistically convergent to a number l, denoted by stat A lim Let c DW S be the space of all deferred weighted summable sequences and (b n ), (c n ) are the sequences of non-negative integers. Then an infinite matrix A = (a n,k ), is called deferred weighted regular matrix if (Ax) n = ∞ k=1 a n,k x k ∈ c DW S for every convergent sequence x = (x n ),
with c DW S − lim n→∞ (Ax) n = stat A lim n→∞ x n .
For a non-negative deferred weighted regular matrix A = (a n,k ) and K ǫ ⊂ N = {k ∈ N : |x k − l| ≥ ǫ}, a sequence x n is said to be deferred weighted A-statistically convergent to l (denoted by
stat DW A − lim n→∞ x n = l) if, for each ǫ > 0, the deferred weighted A-density of K ǫ denoted by d A DW (K ǫ ) is zero. That is d A DW (K ǫ ) = lim n→∞ 1 S n cn m=bn+1 k∈Kǫ s m a m,k = 0.
In our further consideration, in this section we assume A = (a n,k ) to be a non-negative deferred weighted regular matrix. The following theorem shows the deferred weighted A-statistical convergence of the operators K β (1) ,··· ,β (r) n,qn (.; x) defined by (2.2).
Theorem 1. For f ∈ C(I), stat DW A − lim n→∞ K β (1) ,··· ,β (r) n,qn (f ) − f = 0, if and only if stat DW A − lim n→∞ K β (1) ,··· ,β (r)
n,qn
(f i ) − f i = 0 for i = 1, 2 where f i (s) = s i .
Proof. The necessary part is trivial. For the converse, let us assume that stat DW
A − lim n→∞ K β (1) ,··· ,β (r)
n,qn
(f i ) − f i = 0 is true. For f ∈ C(I) there exists a positive constant M f such that f (s) − f (x) ≤ 2M f , for all s, x ∈ I.
In view of the uniform continuity of f on I, for any ǫ > 0 ∃ δ > 0, such that |f (s) − f (x)| < ǫ whenever |s − x| < δ. Hence, for all s, x ∈ I, we can write
f (s) − f (x) ≤ ǫ + 2M f δ 2 (s − x) 2 . (3.1)
Thus, applying the operators K β (1) ,··· ,β (r) n,q (.; x) on the above equation, we obtain
K β (1) ,··· ,β (r)
n,qn
(f ; x) − f (x) ≤ ǫ + 2M f δ 2 K β (1) ,··· ,β (r) n,qn (s 2 ; x) − x 2 + 2|x| K β (1) ,··· ,β (r) n,qn (s; x) − x .
Using Lemma 1 and considering sup norm, we have the following inequality
K β (1) ,··· ,β (r)
n,qn
(f ) − f ≤ ǫ + 2M f δ 2 K β (1) ,··· ,β (r) n,qn (s 2 ) − x 2 + 2 K β (1) ,··· ,β (r) n,qn (s) − x .
Now, for any ǫ ′ > 0, we consider the following sets;
K ǫ ′ := k ∈ N : K β (1) ,··· ,β (r)
n,qn
(f ) − f ≥ ǫ ′ ; K ǫ ′ 2 := k ∈ N : ǫ + 2M f δ 2 K β (1) ,··· ,β (r) n,qn (s 2 ) − x 2 ≥ ǫ ′ 2 ; K ′ ǫ ′ 2 := k ∈ N : 4M f δ 2 K β (1) ,··· ,β (r) n,qn (s) − x ≥ ǫ ′ 2 , thus K ǫ ′ ⊂ K ǫ ′ 2 ∪ K ′ ǫ ′ 8
and therefore
1 S n cn m=bn+1 k∈K ǫ ′ s m a m,k ≤ 1 S n cn m=bn+1 k∈K ǫ ′ 2 s m a m,k + 1 S n cn m=bn+1 k∈K ′ ǫ ′ 2 s m a m,k . (3.2)
Now, using the hypothesis and from Lemma 1, it is obvious that Hence, from equation (3.2), we have
d A DW (K ǫ ′ ) = lim n→∞ 1 S n cn m=bn+1 k∈K ǫ ′ s m a m,k = stat D A − lim n→∞ K β (1) ,··· ,β (r)
n,qn
(f ) − f = 0.
We recall the definition of modulus of continuity. For any continuous function f : I → R and a given δ > 0, the modulus of continuity ω f (δ) is defined as
w f (δ) := sup |s−x|≤δ {|f (s) − f (x)| : s, x ∈ I}.
From the above definition, we have
|f (s) − f (x)| ≤ 1 + (s − x) 2 δ 2 ω f (δ). (3.3)
Let α n be a positive non-increasing sequence of real numbers and δ n be any sequence of positive real numbers. Then, we say the sequence
ω f (δ n ) is deferred weighted A-statistically convergent with o(α n ) if stat DW A − lim n→∞ ω f (δ n ) α n = 0. Theorem 2. For the operator K β (1) ,··· ,β (r) n,qn , if ω f γ n,qn (β (r) n ) = stat D A − o(α n ), then K β (1) ,··· ,β (r) n,qn (f ) − f = stat D A − o(α n ).
Proof. From the inequality given in (3.3) and using Lemma 1 , for any δ > 0, we obtain
K β (1) ,··· ,β (r)
n,qn
(f ; x) − f (x) ≤ {1 + 1 δ 2 K β (1) ,··· ,β (r) n,qn ((s − x) 2 ;
x)}ω f (δ), for all x ∈ I.
Hence in view of Lemma 2, we can write
K β (1) ,··· ,β (r)
n,qn
(f ) − f ≤ ω f (δ) 1 + γ n,qn (β (r) n ) δ 2 .
Now, we choose δ = γ n,qn (β Let p j be a sequence of real numbers such that p 1 > 0 and p j ≥ 0, ∀ j = 2, 3, · · · . Also, suppose that the power series
p(u) = ∞ j=1 p j u j−1 ,(4.1)
has a radius of convergence R ∈ (0, ∞]. Now, a sequence η j is said to be convergent to l in the sense of power series method ( please see [4,11,19]
) if lim u→R− 1 p(u) ∞ j=1 η j p j u j−1 = l, ∀ x ∈ (0, R).
Further, the power series method is called regular [3] if and only if
lim u→R− p j u j−1 p(u) = 0, ∀ j ∈ N. (4.2)
Recently, the power series summability method of convergence has attracted the researchers due to its nature of generality over the classical convergence [23]. For the interested reader, we refer to [15,21,22]. The following result shows the convergence of our operators K β (1) ,··· ,β (r) n,qn (.; x) by means of power series method.
Theorem 3. For f ∈ C(I), the operators K β (1) ,··· ,β (r) n,qn
(.; x) satisfy lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f ) − f p n u n−1 = 0, (4.3)
if and only if
lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f i ) − f i p n u n−1 = 0, (4.4) for i = 1, 2 where f i (s) = s i .
Proof. First assume that (4.3) is true. Then (4.4) is obvious. Conversely, let the condition (4.4) is true. Now, using the inequality given in (3.1), we can write
1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f (s); x) − f (x) p n u n−1 ≤ 1 p(u) ∞ n=1 ǫ + 1 δ 2 K β (1) ,··· ,β (r) n,qn ((s − x) 2 ; x) p n u n−1 ≤ 1 p(u) ∞ n=1 ǫ + 1 δ 2 K β (1) ,··· ,β (r) n,qn (s 2 ; x) − x 2 + 2 K β (1) ,··· ,β (r) n,qn (s; x) − x p n u n−1 ,
for all x ∈ I. Considering sup norm and taking the required limit, in view of (4.1), we obtain
lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f ) − f p n u n−1 ≤ ǫ + 1 δ 2 lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (s 2 ) − x 2 p n u n−1 + lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r)
n,qn (s) − x p n u n−1 . Now, the assertion follows easily on using the hypothesis (4.4) and arbitrariness of ǫ. Remark 1. Let us assume that lim n→∞ β (r) n = 1. In order to show that uniform convergence of
K β (1) ,··· ,β (r)
n,qn (f ) to f on I by power series method, it is sufficient to establish the following;
lim u→R− 1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f i ) − f i p n u n−1 = 0, for i = 1, 2.
Using Lemma 1,
1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f 1 ; x) − x p n u n−1 ≤ 1 p(u) ∞ n=1 (1 − β (r) n ) + 1 [2] qn [n] qn p n u n−1 = J 1 + J 2 , say. (4.5)
Now, let us estimate
J 1 = 1 p(u) ∞ n=1 (1 − β (r) n )p n u n−1 .
Let ǫ > 0 be an arbitrary. Then from the hypothesis ∃ n 0 (ǫ) such that |1−β (r) n | ≤ ǫ 4 for all n > n 0 (ǫ). Then
J 1 ≤ 1 p(u) n0 n=1 (1 − β (r) n )p n u n−1 + ǫ 4p(u) ∞ n=n0+1 p n u n−1 < 1 p(u) n0 n=1 (1 − β (r) n )p n u n−1 + ǫ 4p(u) ∞ n=1 p n u n−1 . Since 1 − β (r) n is a bounded sequence then ∃ M 1 > 0 such that M 1 = max 1≤n≤n0 (1 − β (r)
n ) and using (4.1),
J 1 < M 1 p(u) n0 n=1 p n u n−1 + ǫ 4 .
In view of regularity condition given by (4.2) there exists δ j (ǫ) > 0 such that pj u j−1 p(u) < ǫ 4M1n0 for all R − δ j (ǫ) < u < R, and j = 1, 2, ...n 0 (ǫ). Let us consider δ(ǫ) = min δ 1 (ǫ), δ 2 (ǫ), ..., δ n0 (ǫ) then for every R − δ(ǫ) < u < R and for all n = 1, 2, ...n 0 , we have
J 1 < ǫ 4M 1 n 0 M 1 n 0 + ǫ 4 = ǫ 2 .
Now, we estimate
J 2 = 1 p(u) ∞J 2 < 1 p(u) n1 n=1 1 [2] qn [n] qn p n u n−1 + ǫ 4 < ǫ 4M 2 n 1 M 2 n 1 + ǫ 4 = ǫ 2 ,
for some δ ′ (ǫ) > 0 such that u ∈ (R − δ ′ , R), in view of the regularity condition (4.2). Finally on choosing δ 0 = min δ, δ ′ and using the estimates of I 1 ; I 2 in (4.5), we obtain
1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f 1 ) − f 1 p n u n−1 < ǫ, ∀ u ∈ (R − δ 0 , R).
Next, using Lemma 1, we consider converges to 0 as n → ∞, it will also converge to 0 in the sense of power series method. Further β (r) n + 1 [2]q n [n]q n ≤ 2 for each n ∈ N, hence in view of (4.8), we have lim u→1− 1 p(u) ∞ n=1 p n u n−1 P β (1) ,··· ,β (r) n,qn (f 1 ) − f 1 = 0.
1 p(u) ∞ n=1 K β (1) ,··· ,β (r) n,qn (f 2 ; x) − f 2 p n u n−1 ≤ 1 p(u) ∞n=1
Again, using the definition (4.7) and Lemma 1, we obtain P β (1) ,··· ,β (r) n,qn However, x n is not convergent to 0 as n → ∞ (in the usual sense). Thus, the Korovkin theorem for linear positive operators does not work for the auxiliary operator defined in (4.7). Hence, our Theorem 3 is a non-trivial generalization of the classical Korovkin theorem. Proof. For f ∈ C(I) and δ > 0, using Lemma 2 1 p(u) Hence in view of our hypothesis, the required result follows.
(f 2 ) − f 2 ≤ 1 [3]
(i) Using the definition of q-Riemann integral given by (2.1) and combining (1.2)-(1.3), we have the result. (ii) From (1.5), we have
A-statistical Approximation process via K β(1) ,··· ,β (r) n,q Let M be a subset of the set of natural numbers N and for each n ∈ N, we defineM n = {m ∈ M : m ≤ n}.The density ( or natural density) of the set M , denoted by d(M ), is defined by the limit(if exists) of the sequence |Mn| n
, Srivastava et al.[18] derived a more general concept of A-statistical convergence and called it deferred weighted A-statistical convergence. Suppose (b n ) and (c n ) are the sequences of non-negative integers satisfying the regularity conditions b n < c n ; lim n→∞ c n = ∞. Now, we set S n = cn m=bn+1 s m , for any given sequence (s n ) of non-negative real numbers and its respective deferred weighted mean by ρ n = 1 Sn cn m=bn+1 s m x m . Then, the sequence (x n ) is called deferred weighted summable (denoted by c DW S − lim n→∞ x n = l) to l if lim n→∞ ρ n = l. Also, we call (x n ) to be deferred weighted A-summable to (denoted by c a m,k x k = l.
s
m a m,k = 0.
n
) and consider the hypothesis ω f γ n,qn (β(r) n ) = stat DW A −o(α n ), to reach the assertion. 4. Power Series Summability Approximation Process via K β (1) ,··· ,β (r) n,qn (.; x)
Theorem 4 .
4For f ∈ C(I), ) − f p n u n−1 = O(Ω(u)), as u → R−,where Ω(u) is some positive function on (0, R).
(s − x) 2 ω f (δ)p n u nn
2 γ n,qn (β (r) n ) ω f (δ)p n u n−1for every u ∈ (0, R). Taking δ = γ n,qn (β ) p n u n−1 .
n=1 1 [
12] qn [n] qn p n u n−1 . ]q n [n]q n → 0, as n → ∞ there exists n 1 (ǫ) ∈ N such that 1[2]q n [n]q n < ǫ 4 , for all n > n 1 (ǫ). Let us set M 2 = max 1≤n≤n1(ǫ) 1[2]q n [n]q nSince
1
[2. Then
]q n [n]q n1
[3] qn [n] 2
qn
+
β
(r)
n
[n] qn
1 +
2
[2] qn
+ 2(1 − β (r)
n ) p n u n−1
= K 1 + K 2 + K 3 , say.
(4.6)
Since (1 − β
(r)
n ) +
1
[2
qn [n] 2qn
+
β
(r)
n
[n] qn
1 +
2
[2] qn
+ 2(1 − β (r)
n ) + x n 1 +
1
[n] qn
1 +
2
[2] qn
+
1
[3] qn [n] 2
qn
.
We see that the sequence
1
[3]q n [n] 2
qn
+
β (r)
n
[n]q n
1 + 2
[2]q n
AcknowledgmentsThe authors are grateful to Dr. E. Erkuş-Duman and Dr. O. Duman for their constructive and invaluable suggestions. We believe these suggestions have enrich the presentation and quality of the paper. The second author also gratefully acknowledges the financial support given to him by the Ministry of Education, Govt. of India to carry out the above work.Let the ǫ > 0 be given. Now, first we estimatefor all u ∈ (R − θ, R) and some θ(ǫ) > 0. Similarly, we can show that there exist some θ ′ > 0 and θ ′′ > 0 such thatConsidering θ 0 (ǫ) = min{θ, θ ′ , θ ′′ }, and using the estimates where x n = 1, if n = m 2 , m ∈ N, 0, otherwise.. Now if we take p n = 1 for all n ∈ N, then we obtain p(u) = ∞ n=1 p n u n−1 = 1 1−u , |u| < 1 which implies that R = 1. Further, we note thatSince by Cauchy's root test, the series ∞ m=1 u m 2 is absolutely convergent in the interval |u| < 1, it follows thatHence, the sequence x n converges to zero in the sense of power series method. Using the definition of auxiliary operators and (4.8), we conclude thatn,qnMoreover, from equation (4.7) and Lemma 1, we have P β(1),··· ,β(r)n,qn . Authors' contributions:-All the authors have equally contributed to the conceptualization, framing and writing of the manuscript.
Data availability:-We assert that no data sets were generated or analyzed during the preparation of the manuscript. Data availability:-We assert that no data sets were generated or analyzed during the preparation of the manuscript;
Code availability:-Not applicable. Code availability:-Not applicable;
The q-Lagrange polynomials in several variables. A Altin, E Erkuş, F Taşdelen, Taiwan. J. Math. 5A. Altin, E. Erkuş, F. Taşdelen, The q-Lagrange polynomials in several variables, Taiwan. J. Math. 5 (2006), 1131-1137.
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"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems",
"ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems"
]
| [
"Shreyas Pai [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Gopal Pandurangan [email protected]. \nDepartment of Computer Science\nUniversity of Houston\n77204HoustonTXUSA\n",
"Sriram V Pemmaraju [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Talal Riaz [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Peter Robinson [email protected]. \nDepartment of Computer Science\nRoyal Holloway\nUniversity of London\nUK\n",
"Shreyas Pai [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Gopal Pandurangan [email protected]. \nDepartment of Computer Science\nUniversity of Houston\n77204HoustonTXUSA\n",
"Sriram V Pemmaraju [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Talal Riaz [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Peter Robinson [email protected]. \nDepartment of Computer Science\nRoyal Holloway\nUniversity of London\nUK\n",
"Shreyas Pai [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Gopal Pandurangan [email protected]. \nDepartment of Computer Science\nUniversity of Houston\n77204HoustonTXUSA\n",
"Sriram V Pemmaraju [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Talal Riaz [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Peter Robinson [email protected]. \nDepartment of Computer Science\nRoyal Holloway\nUniversity of London\nUK\n",
"Shreyas Pai [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Gopal Pandurangan [email protected]. \nDepartment of Computer Science\nUniversity of Houston\n77204HoustonTXUSA\n",
"Sriram V Pemmaraju [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Talal Riaz [email protected]. \nDepartment of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA\n",
"Peter Robinson [email protected]. \nDepartment of Computer Science\nRoyal Holloway\nUniversity of London\nUK\n"
]
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"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nUniversity of Houston\n77204HoustonTXUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nRoyal Holloway\nUniversity of London\nUK",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nUniversity of Houston\n77204HoustonTXUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nRoyal Holloway\nUniversity of London\nUK",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nUniversity of Houston\n77204HoustonTXUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nRoyal Holloway\nUniversity of London\nUK",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nUniversity of Houston\n77204HoustonTXUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nThe University of Iowa\n52242Iowa CityIAUSA",
"Department of Computer Science\nRoyal Holloway\nUniversity of London\nUK"
]
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| We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016 for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree ∆ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for somewhat small ∆. Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results:Time Complexity: We show that we can break the O(log n) "barrier" for 2-and 3-ruling sets.We compute 3-ruling sets in O log n log log n rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O log ∆ · (log n) 1/2+ε + log n log log n rounds for any ε > 0, which is o(log n) for a wide range of ∆ values (e.g., ∆ = 2 (log n) 1/2−ε ). These are the first 2-and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. Message Complexity: We show an Ω(n 2 ) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log 2 n) messages and runs in O(∆ log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor). Our results are a step toward understanding the time and message complexity of symmetry breaking problems in the Congest model.Symmetry Breaking in the Congest Model | 10.1145/3087801.3087865 | [
"https://arxiv.org/pdf/1705.07861v1.pdf"
]
| 10,097,747 | 1705.07861 | 505cd0b88913908827a50c5c6041a6ec147b19c7 |
ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems
1998
Shreyas Pai [email protected].
Department of Computer Science
The University of Iowa
52242Iowa CityIAUSA
Gopal Pandurangan [email protected].
Department of Computer Science
University of Houston
77204HoustonTXUSA
Sriram V Pemmaraju [email protected].
Department of Computer Science
The University of Iowa
52242Iowa CityIAUSA
Talal Riaz [email protected].
Department of Computer Science
The University of Iowa
52242Iowa CityIAUSA
Peter Robinson [email protected].
Department of Computer Science
Royal Holloway
University of London
UK
ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems
21998.2 Graph Theory © Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, Peter Robinson; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016 for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree ∆ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for somewhat small ∆. Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results:Time Complexity: We show that we can break the O(log n) "barrier" for 2-and 3-ruling sets.We compute 3-ruling sets in O log n log log n rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O log ∆ · (log n) 1/2+ε + log n log log n rounds for any ε > 0, which is o(log n) for a wide range of ∆ values (e.g., ∆ = 2 (log n) 1/2−ε ). These are the first 2-and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. Message Complexity: We show an Ω(n 2 ) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log 2 n) messages and runs in O(∆ log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor). Our results are a step toward understanding the time and message complexity of symmetry breaking problems in the Congest model.Symmetry Breaking in the Congest Model
Introduction
The maximal independent set (MIS) problem is one of the fundamental problems in distributed computing because it is a simple and elegant abstraction of "local symmetry breaking," an issue that arises repeatedly in many distributed computing problems. About 30 years ago Alon, Babai, and Itai [1] and Luby [28] presented a randomized algorithm for MIS, running on n-node graphs in O(log n) rounds with high probability (whp) 1 . Since then the MIS problem has been studied extensively and recently, there has been some exciting progress in designing faster MIS algorithms. For n-node graphs with maximum degree ∆, Ghaffari [16] presented an MIS algorithm running in O(log ∆) + 2 O( √ log log n) rounds, improving over the algorithm of Barenboim et al. [6] that runs in O(log 2 ∆) + 2 O( √ log log n) rounds. Ghaffari's MIS algorithm is the first MIS algorithm to improve over the round complexity of Luby's algorithm when ∆ = 2 o(log n) and ∆ is bounded below by Ω(log n). 2 While the results of Ghaffari and Barenboim et al. constitute a significant improvement in our understanding of the round complexity of the MIS problem, it should be noted that both of these results are in the Local model. The Local model [34] is a synchronous, message-passing model of distributed computing in which messages can be arbitrarily large. Luby's algorithm, on the other hand, is in the Congest model [34] and uses small messages, i.e., messages that are O(log n) bits or O(1) words in size. In fact, to date, Luby's algorithm is the fastest known MIS algorithm in the Congest model; this is the case even when ∆ is between Ω(log n) and 2 o(log n) . For example, for the class of graphs with ∆ = 2 O( √ log n) , Ghaffari's MIS algorithm runs in O( √ log n) rounds whp in the Local model, but we don't know how to compute an MIS for this class of graphs in o(log n) rounds in the Congest model. It should be further noted that the MIS algorithms of Ghaffari and Barenboim et al. use messages of size O(poly(∆) log n) (see Theorem 3.5 in [6]), which can be much larger than the O(log n)-sized messages allowed in the Congest model; in fact these algorithms do not run within the claimed number of rounds even if messages of size O(poly(log n)) were allowed. Furthermore, large messages arise in these algorithms from a topology-gathering step in which cluster-leaders gather the entire topology of their clusters in order to compute an MIS of their cluster -this step seems fundamental to these algorithms and there does not seem to be an efficient way to simulate this step in the Congest model.
Ruling sets are a natural generalization of MIS and have also been well-studied in the Local model. An (α, β)-ruling set [17] is a node-subset T such that (i) any two distinct nodes in T are at least α hops apart in G and (ii) every node in the graph is at most β hops from some node in T . A (2, β)-ruling set is an independent set and since such ruling sets are the main focus of this paper, we use the shorthand β-ruling sets to refer to (2, β)-ruling sets. (Using this terminology an MIS is just a 1-ruling set.) The above mentioned MIS results due to Barenboim et al. and Ghaffari have also led to the sublogarithmic-round algorithms for β-ruling sets for β 2. The earliest instance of such a result was the algorithm of Kothapalli and Pemmaraju [22] that computed a 2-ruling set in O( log ∆ · (log n) 1/4 ) rounds by using an earlier version of the Barenboim et al. [5] MIS algorithm. There have been several further improvements in the running time of ruling set algorithms culminating in the O(β log 1/β ∆) + 2 O( √ log log n) round β-ruling set algorithm of Ghaffari [16]. This result is based on a recursive sparsification procedure of Bisht et al. [8] that reduces the β-ruling set problem on graphs with maximum degree ∆ to an MIS problem on graphs with degree much smaller. Ghaffari's β-ruling set result is also interesting because it identifies a separation between 2-ruling sets and MIS (1-ruling sets). This follows from the lower bound of Ω min log n log log n , log ∆ log log ∆ for MIS due to Kuhn et al. [23]. Again, we emphasize here that all of these improvements for ruling set algorithms are only in the Local model because these ruling set algorithms rely on Local-model MIS algorithms to "finish off" the processing of small degree subgraphs. As far as we know, prior to the current work there has been no o(log n)-round, β-ruling set algorithm in the Congest model for any β = O (1).
The focus of all the above results has been on the time (round) complexity. Message complexity, on the other hand, has been largely ignored in the context of local symmetry breaking problems such as MIS and ruling sets. For a graph with m edges, Luby's algorithm uses O(m) messages in the Congest model and until now there has been no MIS or ruling set algorithm that uses o(m) messages. We note that the ruling set algorithm of Goldberg et al. [17] which can be implemented in the Congest model [19] also takes at least Ω(m) messages.
The focus of this paper is symmetry breaking problems in the Congest model and the specific question that motivates our work is whether we can go beyond Luby's algorithm in the Congest model for MIS or any closely-related symmetry breaking problems such as ruling sets. In particular, can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier, in the Congest model for MIS and ruling sets? In many applications, especially in resource-constrained communication networks and in distributed processing of large-scale data it is important to design distributed algorithms that have low time complexity as well as message complexity. In particular, optimizing messages as well as time has direct applications to the performance of distributed algorithms in other models such as the k-machine model [21].
We present two sets of results, one set focusing on time (round) complexity and the other on message complexity. 1. Time complexity: (cf. Section 2) We first show that 2-ruling sets can be computed in the Congest model in O log ∆ · (log n) 1/2+ε + log n log log n rounds whp for n-node graphs with maximum degree ∆ and for any ε > 0. This is the first algorithm to improve over Luby's algorithm, by running in o(log n) rounds in the Congest model, for a wide range of values of ∆. Specifically our algorithm runs in o(log n) rounds for ∆ bounded above by 2 (log n) 1/2−ε for any value of ε > 0. Using this 2-ruling set algorithm as a subroutine, we show how to compute 3-ruling sets (for any graph) in O log n log log n rounds whp in the Congest model. We also present a simple 5-ruling set algorithm based on Ghaffari's MIS algorithm that runs in O( √ log n) rounds in the Congest model.
2.
Message complexity: (cf. Sections 3 and 4) We show that Ω(n 2 ) is a fundamental lower bound for computing an MIS (i.e., 1-ruling set) by showing that there exists graphs (with m = Θ(n 2 ) edges) where any distributed MIS algorithm needs Ω(n 2 ) messages. In contrast, we show that 2-ruling sets can be computed using significantly smaller message complexity. In particular, we present a randomized 2-ruling set algorithm that, whp, uses O(n log 2 n) messages and runs in O(∆ log n) rounds. This is the first o(m)-message algorithm known for ruling sets, which takes near-linear (in n) message complexity. This message bound is tight up to a polylogarithmic factor, since we show that any O(1)-ruling set (randomized) algorithm that succeeds with probability 1 − o (1) requires Ω(n) messages in the worst case. We also present a simple 2-ruling set algorithm that uses O(n 1.5 log n) messages, but runs faster -in O(log n) rounds.
Our results make progress towards understanding the complexity of symmetry breaking, in particular with respect to ruling sets, in the Congest model. With regards to time complexity, our results, for the first time, show that one can obtain o(log n) round algorithms for ruling sets in the Congest model. With regards to message complexity, our results are (essentially) tight: while MIS needs quadratic (in n) messages in the worst case, 2-ruling sets can be computed using near-linear (in n) messages. We discuss key problems left open by our work in Section 5.
Distributed Computing Model
We consider the standard synchronous Congest model [34] described as follows.
We are given a distributed network of n nodes, modeled as an undirected graph G. Each node hosts a processor with limited initial knowledge. We assume that nodes have unique IDs (this is not essential, but simplifies presentation), and at the beginning of the computation each node is provided its ID as input. Thus, a node has only local knowledge 3 . Specifically we assume that each node has ports (each port having a unique port number); each incident edge is connected to one distinct port. This model is referred to as the clean network model in [34] and is also sometimes referred to as the KT 0 model, i.e., the initial (K)nowledge of all nodes is restricted (T)ill radius 0 (i.e., just the local knowledge) [3].
Nodes are allowed to communicate through the edges of the graph G and it is assumed that communication is synchronous and occurs in discrete rounds (time steps). In each round, each node can perform some local computation including accessing a private source of randomness, and can exchange (possibly distinct) O(log n)-bit messages with each of its neighboring nodes. This model of distributed computation is called the Congest(log n) model or simply the Congest model [34].
Related Work
As one would expect, Congest model symmetry breaking algorithms are easier for sparse graphs. There is a deterministic (∆+1)-coloring algorithm due to Barenboim,Elkin,and Kuhn [4] that runs in the Congest model in O(∆) + 1 2 log * n rounds. This can be used to obtain an o(log n)-round Congest model MIS algorithm, when ∆ = o(log n). For trees, Lenzen and Wattenhofer [27] presented an MIS algorithm that runs in O( √ log n log log n) rounds whp in the Congest model. More generally, for graphs with arboricity bounded above by α, Pemmaraju and Riaz [35] present an MIS algorithm that runs in O(poly(α) · √ log n log log n) rounds in the Congest model. Other research that is relevant to ruling sets, but is only in the Local model, includes the multi-trials technique of Schneider and Wattenhofer [37] and the deterministic ruling set algorithms of Schneider et al. [36].
As mentioned earlier, in the context of local symmetry breaking problems such as MIS or ruling sets, message complexity has not received much attention. However, in the context of global problems (i.e., problems where one needs to traverse the entire network and, hence, take at least Ω(D) time) such as leader election (which can be thought as a "global" symmetry breaking) and minimum spanning tree (MST), message complexity has been very well studied. Kutten et al. [24] showed that Ω(m) is a message lower bound for leader election and this applies to randomized Monte-Carlo algorithms as well. This lower bound also applies to the Broadcast and MST problems. In a similar spirit, in this paper, we show that in general, Ω(m) is a message lower bound for the MIS problem as well. In contrast, we show that this lower bound does not hold for ruling sets which admit a near-linear (in n) message complexity.
It is important to point out that the current paper as well as most prior work on leader election and MST [2,9,14,15,26,13,12,33,24]) assume the KT 0 model. However, one can also consider a stronger model where nodes have initial knowledge of the identity of their neighbors. This model is called the KT 1 model. Awerbuch et al. [3] show that Ω(m) is a message lower bound for MST for the KT 1 model, if one allows only comparison-based algorithms (i.e., algorithms that can operate on IDs only by comparing them); this lower bound for comparison-based algorithms applies to randomized algorithms as well. Awerbuch et al. [3] also show that the Ω(m) message lower bound applies even to non-comparison based (in particular, algorithms that can perform arbitrary local computations) deterministic algorithms in the Congest model that terminate in a time bound that depends only on the graph topology (e.g., a function of n). On the other hand, for randomized non-comparisonbased algorithms, it turns out that the message lower bound of Ω(m) does not apply in the KT 1 model. King et al. [20] showed a surprising and elegant result (also see [29]): in the KT 1 model one can give a randomized Monte Carlo algorithm to construct a MST or a spanning tree inÕ(n) messages (Ω(n) is a message lower bound) and inÕ(n) time (this algorithm uses randomness and is not comparison-based). While this algorithm shows that one can get o(m) message complexity (when m = ω(n polylog n)), it is not time-optimal (it can take up toÕ(n) rounds).
One can also use the King et al. algorithm to build a spanning tree usingÕ(n) messages and then use time encoding (see e.g., [18,32]) to collect the entire graph topology at the root of the spanning tree. Hence, using this approach any problem (including, MST, MIS, ruling sets, etc.) can be solved usingÕ(n) messages in the KT 1 model. However, this is highly time inefficient as it takes exponential (in n) rounds.
Technical Overview
Time Bounds
The MIS algorithms of Barenboim et al. [6] and Ghaffari [16] use a 2-phase strategy, attributed to Beck [7], who used it in his algorithmic version of the Lovász Local Lemma. In the first phase, some number of iterations of a Luby-type "base algorithm" are run (in the Congest model). During this phase, some nodes join the MIS and these nodes and their neighbors become inactive. The first phase is run until the graph is "shattered", i.e., the nodes that remain active induce a number of "small" connected components. Once the graph is "shattered", the algorithm switches to the second, deterministic phase to "finish off" the problem in the remaining small components. It is this second phase that relies critically on the use of the Local model in order to run fast.
In general, in the Congest model it is not clear how to take advantage of low degree or low diameter or small size of a connected component to solve symmetry-breaking problems (MIS or ruling sets) faster than the O(log n)-round bound provided by Luby's algorithm. In both Barenboim et al. [6] and Ghaffari [16], a key ingredient of the second "finish-off" phase is the deterministic network decomposition algorithm of Panconesi and Srinivasan [31] that can be used to compute an MIS in O(2 √ log s ) rounds on a graph with s nodes in the Local model. If one can get connected components of size O(poly(log n)) then it is possible to finish the rest of the algorithm in 2 O( √ log log n) rounds and this is indeed the source of the "2 O( √ log log n) " term in the round complexity of these MIS algorithms. In fact, the Panconesi-Srinivasan network decomposition algorithm itself runs in the Congest model, but once the network has been decomposed into small diameter clusters then algorithms simply resort to gathering the entire topology of a cluster at a cluster-leader and this requires large messages. Currently, there seem to be no techniques for symmetry breaking problems in the Congest model that are able to take advantage of the diameter of a network being small. As far as we know, there is no o(log n)-round O(1)-ruling set algorithm in the Congest model even for constant-diameter graphs, for any constant larger than 1. To obtain our sublogarithmic β-ruling set algorithms (for β = 2, 3, 5), we use simple greedy MIS and 2-ruling set algorithms to process "small" subgraphs in the final stages of algorithm. These greedy algorithms just exchange O(log n)-bit IDs with neighbors and run in the Congest model, but they can take Θ(s) rounds in the worst case, where s is the length of the longest path in the subgraph. So our main technical contribution is to show that it is possible to do a randomized shattering of the graph so that none of the fragments have any long paths.
Message Bounds
As mentioned earlier, our message complexity lower bound for MIS and the contrasting the upper bound for 2-ruling set show a clear separation between these two problems. At a high-level, our lower bound argument exploits the idea of "bridge crossing" (similar to [24]) whose intuition is as follows. We consider two types of related graphs: (1) a complete bipartite graph and (2) a random bridge graph which consists of a two (almost-)complete bipartite graphs connected by two "bridge" edges chosen randomly (see Figure 1 and Section 4 for a detailed description of the construction). Note that the MIS in a complete bipartite graph is exactly the set of all nodes belonging to one part of the partition. The crucial observation is that if no messages are sent over bridge edges, then the bipartite graphs on either side of the bridge edges behave identically which can result in choosing adjacent nodes in MIS, a violation. In particular, we show that if an algorithm sends o(n 2 ) messages, then with probability at least 1 − o(1) that there will be no message sent over the bridge edges and by symmetry, with probability at least 1/2, two nodes that are connected by the bridge edge will be chosen to be in the MIS. Our 2-ruling set algorithm with low-message-complexity crucially uses the fact that, unlike in an MIS, in a 2-ruling set there are 3 categories of nodes: category-1 (nodes that are in the independent set), category-2 (nodes that are neighbors of category-1) and category-3 nodes (nodes that are neighbors of category-2, but not neighbors of category-1). Our algorithm, inspired by Luby's MIS algorithm, uses three main ideas. First, category-2 and category-3 nodes don't initiate messages; only undecided nodes (i.e., nodes whose category are not yet decided) initiate messages. Second, an undecided node does "checking sampling" (cf. Algorithm 4) first before it does local broadcast, i.e., it samples a few of its neighbors to see if they are any category-2 nodes; if so it becomes a category-3 node immediately. Third, an undecided node tries to enter the ruling set with probability that is always inversely proportional to its original degree, i.e., Θ(1/d(v)), where d(v) is the degree of v. This is unlike in Luby's algorithm, where the marking probability is inversely proportional to its current degree. These ideas along with an amortized charging argument [10] yield our result: an algorithm using O(n log 2 n) messages and running in O(∆ log n) rounds.
Time-Efficient Ruling Set Algorithms in the Congest model
The main result of this section is a 2-ruling set algorithm in the Congest model that runs in O log ∆ · (log n) 1/2+ε + log n log log n rounds whp, for any constant ε > 0, on n-node graphs with maximum degree ∆. An implication of this result is that for graphs with ∆ = 2 O((log n) 1/2−ε ) for any ε > 0, we can compute a 2-ruling set in O log n log log n rounds in the Congest model. A second implication is that using this 2-ruling set algorithm as a subroutine, we can compute a 3-ruling set for any graph in O log n log log n rounds whp in the Congest model. These are the first sublogarithmic-round Congest model algorithms for 2-ruling sets (for a wide range of ∆) and 3-ruling sets. Combining some of the techniques used in these algorithms with the first phase of Ghaffari's MIS algorithm [16], we also show that a 5-ruling set can be computed in O( √ log n) rounds whp in the Congest model.
The 2-ruling Set Algorithm
Our 2-ruling set algorithm (described in pseudocode below) takes as input an n-node graph with maximum degree ∆ 2 √ log n , along with a parameter ε > 0. For ∆ > 2 √ log n , we simply execute Luby's MIS algorithm to solve the problem. The algorithm consists of log ∆ scales and in scale t, 1 t log ∆ , nodes with degrees at most ∆ t := ∆/2 t−1 are processed. Each scale consists of Θ(log 1/2+ε n) iterations. In an iteration i, in scale t, each undecided node independently joins a set M i,t with probability 1/(∆ t · log ε n) (Line 5). Neighbors of nodes in M i,t , that are themselves not in M i,t , are set aside and placed in a set W i,t (Lines 6-8). The nodes in M i,t ∪ W i,t have decided their fate and we continue to process the undecided nodes. At the end of all the iterations in a scale t, any undecided node that still has ∆ t /2 or more undecided neighbors is placed in a "bad" set B t for that scale (Line 11), thus effectively deciding the fate of all nodes with degree at least ∆ t /2. We now process the set of scale-t "bad" nodes, B t , by simply running a greedy 2-ruling set algorithm on B t (Line 13). We also need to process the sets M i,t (Line 15) and for that we rely on a greedy 1-ruling set algorithm (i.e., a greedy MIS algorithm). Note that the M i,t 's are all disconnected from each other since the W i,t 's act as "buffers" around the M i,t 's. Thus after all the scales are completed, we can compute an MIS on all of the M i,t 's in parallel. Since each node in W i,t has a neighbor in M i,t , this will guarantee that every node in W i,t has an independent set node at most 2 hops away. In the following algorithm we use deg S (v) to denote the degree of a vertex v in the G[S], the graph induced by S.
Algorithm 1: 2-ruling Set(Graph G = (V, E), ε > 0): 1 I ← ∅; S ← V ; 2 for each scale t = 1, 2, . . . , log ∆ do 3 Let ∆ t = ∆ 2 t−1 ; S t ← S; 4 for iteration i = 1, 2, . . . , c · log 1/2+ε n do 5
Each v ∈ S marks itself and joins M i,t with probability
1 ∆t·log ε n ; 6
if v ∈ S is unmarked and a neighbor in S is marked then
7 v joins W i,t ; 8 end 9 S ← S \ (M i,t ∪ W i,t ); 10 end 11 B t ← {v ∈ S | deg S (v) ∆ t /2}; 12 S ← S \ B t ; 13 I ← I ∪ GreedyRulingSet(G[S t ], B t , 2); 14 end 15 I ← I ∪ (∪ t ∪ i GreedyRulingSet(G[S t ], M i,t , 1)); 16 return I;
The overall round complexity of this algorithm critically depends on the greedy 2-ruling set algorithm terminating quickly on each B t (Line 13) and the greedy 1-ruling set algorithm terminating quickly on each M i,t (Line 15). To be concrete, we present below a specific β-ruling set algorithm that greedily picks nodes by their IDs from a given node subset R. To show that the calls to this greedy ruling set algorithm terminate quickly, we introduce the notion of witness paths. If GreedyRulingSet(G, R, β) runs for p iterations (of the while-loop), then R must contain a sequence of nodes (v 1 , v 2 , . . . , v p ) such that v i , 1 i p, joins the independent set I in iteration i and node v i , 1 < i p, must contain an undecided node with higher ID in its 1-neighborhood in G, which was removed when v i−1 and its β-neighborhood in G were removed in iteration i − 1. We call such a sequence a witness path for the execution of GreedyRulingSet. Three simple properties of witness paths are needed in our analysis: (i) any two nodes v i and v j in the witness path are at least β + 1 hops away in G, (ii) any two consecutive nodes v i and v i+1 in the witness path are at most β + 1 hops away in G, and (iii) G[R] contains a simple path with (p − 1)(β + 1) + 1 nodes, starting at node v 1 , passing through nodes v 2 , v 3 , . . . , v p−1 and ending at node v p . To show that each M i,t can be processed quickly by the greedy 1-ruling set algorithm we show (in Lemma 1) that whp every witness path for the execution of the greedy 1-ruling set algorithm is short. Similarly, to show that each B t can be processed quickly by the greedy 2-ruling set algorithm we prove (in Lemma 2) that whp a "bad" set B t cannot contain a witness path of length √ log n or longer to the execution of the greedy 2-ruling set algorithm. At the start of our analysis we observe that the set S t , which is the set of undecided nodes at the start of scale t, induces a subgraph with maximum degree ∆ t = ∆/2 t−1 .
Algorithm 2: GreedyRulingSet(Graph G = (V, E), R ⊆ V , integer β > 0): 1 I ← ∅; U ← R; // U is the initial set of undecided nodes 2 while U = ∅ do 3 for each node v ∈ U in parallel do 4 if (v
Lemma 1. For all scales t and iterations
i, GreedyRulingSet(G[S t ], M i,t , 1) runs in O log n ε log log n rounds, whp.
Proof. Consider an arbitrary scale t and iteration i. By Property (iii) of witness paths, there is a simple path P with (2p − 1) nodes in G[S t ], all of whose nodes have joined M i,t . Due to independence of the marking step (Line 5) the probability that all nodes in P join
M i,t is at most (1/∆ t · log ε n) 2p−1 . Since ∆(G[S t ]) ∆ t , the number of simple paths with 2p − 1 nodes in G[S t ] are at most n · ∆ 2p−1 t
. Using a union bound over all candidate simple paths with 2p − 1 nodes in G[S t ], we see that the probability that there exists a simple path in −1) . Picking p to be the smallest integer such that 2p − 1 4 log n ε log log n , we get Pr(∃ a simple path with 2p − 1 nodes that joins M i,t ) n· 1
G[M i,t ] of length 2p − 1 is at most: n · ∆ 2p−1 t · 1 ∆t log ε n 2p−1 = n · 1 (log n) ε(2p(2 log log n ) ε 4 log n ε log log n = n· 1 n 4 = 1 n 3 . We have O(log ∆ · (log n) 1/2+ε ) different M i,t '
s. Using a union bound over these M i,t 's, we see that the probability that there exists an M i,t containing a simple path with 2p − 1 nodes is at most n −2 . Thus with probability at least 1 − 1/n 2 , all of the calls to
GreedyRulingSet(G[S t ], M i,t , 1)) (in Line 15) complete in O log n log log n rounds. Lemma 2. For all scales t, the call to GreedyRulingSet(G[S t ], B t , 2) takes O( √ log n) rounds whp.
Proof. Consider a length-p witness path P for the execution of GreedyRulingSet(G[S t ], B t , 2) (Line 13). By Property (i) of witness paths, all pairs of nodes in P are at distance at least 3 from each other. Fix a scale t. We now calculate the probability that all nodes in P belong to B t . Consider some node v ∈ P . For v to belong to B t , it must have not marked itself in all iterations of scale t and moreover at least ∆ t /2 neighbors of v in S t must not have marked themselves in any iteration of scale t. Since the neighborhoods of any two nodes in P are disjoint, the event that v joins B t is independent of any other node in P joining B t . Therefore,
Pr(P is in B t ) v∈P
Pr(v and at least ∆ t /2 neighbors do not mark themselves in scale t).
This can be bounded above by v∈P
1 − 1 ∆t(log n) ε ∆ t 2 ·c(log n) 1/2+ε exp − c 2 · (log n) 1/2 · p . Plugging in p = √
log n we see that this probability is bounded above by n −c/2 . By Property (ii) of witness paths and the fact that ∆(G[S t ]) ∆ t , we know that there are at most n·(∆ t ) 3p length-p candidate witness paths. Using a union bound over all of these, we get that the probability that there exists a a witness path that joins B t is at most n∆ 3p ·n −c/2 . Plugging in ∆ 2 √ log n and p = √ log n we get that this probability is at most n · n 3 · n −c/2 = n −c/2+4 . Picking a large enough constant c guarantees that this probability is at most 1/n 2 and taking a final union bound over all log ∆ scales gives us the result that all calls to Since all nodes in W i,t are at distance 1 from some node in M i,t and we find an MIS of
the graph G[M i,t ], nodes in ∪ i ∪ t (M i,t ∪ W i,t )
are all at distance at most 2 from some node in I. Nodes that are not in any M i,t ∪ W i,t are in B t for some scale t. We compute 2-ruling sets for nodes in B t and therefore every node is at most 2 hops from some node in I.
The 3-ruling Set Algorithm
5-ruling sets in O( √ log n) rounds
It turns out that for slightly larger but constant β, it is possible to compute β-ruling sets in the Congest model in O( √ log n) rounds. We show this in this section for β = 5. The 5-ruling set algorithm (described in pseudocode below) starts by calling the Sparsify [8,22] subroutine. Recall that the first phase of Ghaffari's algorithm is a "Luby-like" algorithm that runs in the Congest model. Finally, we consider the set of nodes that are still undecided, i.e., nodes in S that are not in I ∪ N (I) and we call Algorithm GreedyRulingSet algorithm with β = 4 in order to compute a 4-ruling set of the as-yet-undecided nodes. The fact that we compute a 4-ruling set in this step, rather than a β-ruling set for some β < 4, is because of independence properties of Ghaffari's MIS algorithm. Steps (1)-(3) of 5-RulingSet run in O( √ log n) rounds either by design or due to properties of Sparsify. The fact that the greedy 4-ruling set algorithm (Step (4)) also terminates in O( √ log n) rounds remains to be shown and this partly depends on the following property of the first phase of Ghaffari's MIS algorithm. [16]) For any constant c > 0, for any set S of nodes that are at pairwise distance at least 5 from each other, the probability that all nodes in S remain undecided after Θ(c log ∆) rounds of the first phase of the MIS algorithm is at most ∆ −c|S| .
Lemma 6. (Lemma 4.1 in
The sparsification step performed by the call to Sparsify in Step (1) along with Lemma 6 and Properties (i) and (ii) of witness paths are used in the following theorem to prove that
Step (4) of 5-RulingSet also completes in O( √ log n) rounds whp. The fact that we are greedily computing a 4-ruling set in Step (4) and any two nodes selected to be in the ruling set are at least 5 hops away from each other provides the independence that is needed to apply Lemma 6.
A Message-Efficient Algorithm for 2-Ruling Set
In this section, we present a randomized distributed algorithm for computing a 2-ruling set in the Congest model that takes O(n log 2 n) messages and O(∆ log n) rounds whp, where n is the number of nodes and ∆ is the maximum node degree. The algorithm does not require any global knowledge, including knowledge of n or ∆. We show in Theorem 14 that the algorithm is essentially message-optimal (up to a polylog(n) factor). This is the first message-efficient algorithm known for 2-ruling set, i.e., it takes o(m) messages, where m is the number of edges in the graph. In contrast, we show in Theorem 15 that computing a MIS requires Ω(n 2 ) messages (regardless of the number of rounds). Thus there is a fundamental separation of message complexity between 1-ruling set (MIS) and 2-ruling set computation.
The Algorithm
Algorithm 4 is inspired by Luby's algorithm for MIS [28]; however, there are crucial differences.
(Note that Luby's algorithm sends Θ(m) messages.) Given a ruling set R, we classify nodes in V into three categories:
category-1: nodes that belong to the ruling set R; category-2: nodes that have a neighbor in R; and category-3: the rest of the nodes, i.e., nodes that have a neighbor in category-2. At the beginning of the algorithm, each node is undecided, i.e., its category is not set and upon termination, each node knows its category.
Let us describe one iteration of the algorithm (Steps 3-19) from the perspective of an arbitrary node v. Each undecided node v marks itself with probably 1/2d(v). If v is marked it samples a set of Θ(log(d(v)) random neighbours and checks whether any of them belong to category-2 -we call this the checking sampling step. If so, then v becomes a category-3 node and is done (i.e., it will never broadcast again, but will continue to answer checking sampling queries, if any, from its neighbors). Otherwise, v performs the broadcast step, i.e., it communicates with all its neighbors and checks if there is a marked neighbor that is of equal or higher degree, and if so, it unmarks itself; else it enters the ruling set and becomes a category-1 node. 4 Then node v informs all its neighbors about its category-1 status causing them to become category-2 nodes (if they are not already) and they are done.
A node that does not hear from any of its neighbors knows that it is not a neighbor of any category-1 node. Note that category-2 and category-3 nodes do not initiate messages, which is important for keeping the message complexity low. Another main idea in reducing messages is the random sampling check of a few neighbors to see whether any of them are category-2. Although some nodes might send O(d(v)) messages, we show in Section 3.2 that most nodes send (and receive) only O(log n) messages in an amortized sense. Nodes that remain undecided at the end of one iteration continue to the next iteration. It is easy to implement each iteration in a constant number of rounds.
Analysis of Algorithm 2-rulingset-msg
One phase of the algorithm consists of Steps 3-19, which can be implemented in a constant number of rounds. We say that a node is decided if it is in category-1, category-2, or category-3. The first lemma, which is easy to establish, shows that if a node is marked, it has a good chance to get decided.
Lemma 8.
A node that marks itself in any phase gets decided with probability at least 1/2 in that phase. Furthermore, the probability that a node remains undecided after 2 log n marked phases is at most 1/n 2 .
Proof. Consider a marked node v. We only consider the probability that v becomes a category-1 node, i.e., part of the independent set. (It can also become decided and become a category-3 or category-2 node .) A marked node becomes unmarked if an equal or higher degree neighbor is marked. The probability of this "bad" event happening is at most
u∈N (v):d(u) d(v) 1 2d(u) u∈N (v):d(u) d(v) 1 2d(v) u∈N (v) 1 2d(v) d(v) 2d(v) = 1 2 .
The probability that a node remains undecided after 2 log n marked phases is at most
1 2 2 log n 1/n 2 .
The next lemma bounds the round complexity of the algorithm and establishes its correctness. The round complexity bound is essentially a consequence of the previous lemma and the correctness of the algorithm is easy to check. If v hears from an equal or higher degree (marked) neighbor then v unmarks itself; 16 If v remains marked, set status v = category-1;
17
Announce status to all neighbors; Proof. Consider a node v with degree d(v). In 16d(v) log n phases, it marks itself 8 log n times in expectation, assuming that it is still undecided. Using a Chernoff bound -lower tail -(cf. Section A), it follows that the node is marked at least 2 log n times with probability at least 1 − 1/n 2 . By Lemma 8, the probability that v is still undecided if it gets marked 2 log n times is at most 1/n 2 . Hence, unconditionally, the probability that a node is still undecided after 16d(v) log n phases is at most 2/n 2 . Applying a union bound over all nodes, the probability that any node is undecided after 16∆ log n phases is at most 2/n. Hence the algorithm finishes in O(∆ log n) rounds with high probability.
From the description of the algorithm it is clear that when the algorithm ends, every node has entered into either category-1, category-2, or category-3. By the symmetry breaking step (Step 15), category-1 nodes form an independent set. They also form a ruling set because, category-2 nodes are neighbors of category-1 nodes (Step 4) and a node becomes category-3 if it is not a neighbor of a category-1, but is a neighbor of a category-2 node (and hence is at distance 2 from a category-1 node).
We now show a technical lemma that is crucially used in proving the message complexity bounds of the algorithm in Lemma 11. It gives a high probability bound on the total number of messages sent by all nodes during the Broadcast step in any particular phase (i.e., Step 14) of the algorithm in terms of a quantity that depends on the number of undecided nodes and their neighbors. While bounding the expectation is easy, showing concentration is more involved. (We note that we really use only part (b) of the Lemma for our subsequent analysis, but showing part (a) first, helps understand the proof of part (b)).
We call a node's checking sampling step a "success", if it results in finding a category-2 node (in this case the node will get decided in Step 11), otherwise, it is called a "failure". N (v)). The total number of messages sent by all nodes in U during the Broadcast step in this phase (i.e., Step 14) of the algorithm is O(|Z (U )| log n) with probability at least 1 − 1/n 3 .
(v)| d(v)/2 (where d(v) is the degree of v), for each v ∈ U . Let Z (U ) = U ∪ (∪ v∈U
Proof.
A node v enters the broadcast step only if it marks itself and if it does not find any category-2 neighbor in its checking sampling step. The marking probability is 1/2d(v). Hence (even) ignoring the checking sampling step, the probability that a node broadcasts is at most 1/2d(v) (note that in the very first phase, checking sampling will result in failure for all nodes). Let random variable (r.v.) X v denote the number of messages broadcast by node v in this phase. Hence,
E[X v ] = 1 2d(v) d(v) = 1/2.
Let random variable X denote the total number of messages broadcast in this phase: X = v∈U X v . By linearity of expectation, the expected number of messages broadcast in one phase is
E[X] = v∈U E[X v ] = k/2, where k = |U |.
We next show concentration of X. We note that X v s are all independent and, for the variance of X v , we get
Var[X v ] = E[X 2 v ] − (E[X v ]) 2 = 1 2d(v) (d(v)) 2 − 1 2 2 = d(v) 2 − 1 4 d(v) 2 .
It follows that
Var[X] = v∈U Var[X v ] v∈U d(v)/2 (|Z(U )| 2 )/4.
We have v∈U d(v) (|Z(U )| 2 )/2, since the latter counts all possible edges in the subgraph induced by U and its neighbors. Furthermore,
X v − E[X v ] d(v) |Z(U )|.
Thus, we can apply Bernstein's inequality (cf. Section A) to obtain
Pr(X k + 4|Z(U )| log n) = exp − 16|Z(U )| 2 log 2 n 2V ar(X) + (8/3)|Z(U )| 2 log n exp − 16|Z(U )| 2 log 2 n (|Z(U )| 2 )/2 + (8/3)|Z(U )| 2 log n ,
which is at most 1/n 2 , completing part (a). To show part (b), let N 2 (v) be the set of category-2 neighbors of a node v ∈ U . Then
N 2 (v) = N (v) − N (v) and v∈U d(v) v∈U |N (v)| + v∈U |N 2 (v)|.(1)
We will now bound from above the two sums on the right-hand side. Note that v∈U |N (v)| |Z (U )| 2 /2 since the latter counts all possible edges in the subgraph induced by U and its category-3 and undecided neighbors. By assumption,
|N 2 (v)| d(v)/2, which means that v∈U |N 2 (v)| v∈U d(v)/2 k|Z (U )| |Z (U )| 2 .
Plugging these bounds into (1), we get
v∈U d(v) |Z (U )| 2 /2 + |Z (U )| 2 (3/2)|Z (U )| 2 .
Hence Var[X] (3/4)|Z (U )| 2 . Let X v and X be defined as above. We have,
X v − E[X v ] d(v) 2|Z (U )|. Now,
applying Bernstein's inequality shows a similar concentration bound for X as in part (a).
Lemma 11. The algorithm 2-rulingset-msg uses O(n log 2 n) messages whp.
Proof. We will argue separately about two kinds of messages that any node can initiate. Consider any node v. 1. type 1 messages: In the checking sampling step in some phase, v samples 4 log d(v) random neighbours which costs O(log d(v)) messages in that phase.
type 2 messages:
In the broadcast step in some phase, v sends to all its neighbors which costs d(v) messages. This happens when all the sampled neighbors in set A v (found in Step 9) are not category-2 nodes. Note that v initiates any message at all, i.e., both type 1 and 2 messages happen, only when v marks itself, which happens with probability 1/2d(v).
We first bound the type 1 messages sent overall by all nodes. By the above statement, a node does checking sampling when it marks itself which happens with probability 1/2d(v). By Lemma 8, with probability at least 1 − 1/n 2 , a node is marked (before it gets decided) at most 2 log n times. Hence, with probability at least 1 − 1/n 2 , the number of type 1 messages sent by node v is at most O(log d(v) log n); this implies, by union bound, that with probability at least 1 − 1/n every node v sends at most O(log d(v) log n) type 1 messages. Thus, whp, the total number of type 1 messages sent is v∈V O(log d(v) log n) = O(n log 2 n).
We next bound the type 2 messages, i.e., messages sent during the broadcast step. There are two cases to consider in any phase.
Case 1.
In this case we focus (only) on the broadcast messages of the set U of undecided nodes v that (each) have at least d(v)/2 neighbors that are in category-3 or undecided (in that phase). We show by a charging argument that any node receives amortized O(log n) messages (whp) in this case. When a node u (in this case) broadcasts, its d(u) messages are charged equally to itself and its category-3 and undecided neighbors (which number at least d(u)/2).
We first show that any category-3 or undecided node v is charged by amortized O(log n) messages in any phase. Consider the set U (v) which is the set of undecided nodes (each of which satisfy Case 1 property of having at least half of its neighbors that are in category-3 or undecided in this phase) in the closed neighborhood of v (i.e., {v} ∪ N (v)). As in
Lemma 10.(b), we define Z (U (v)) = U (v) ∪ (∪ w∈U (v) N (w)), where N (w)
is the set of all undecided or category-3 neighbors of w. Since, by assumption of Case 1, every undecided node u ∈ U (v) has at least d(u)/2 neighbors that are in category-3 or undecided in the current phase, applying Lemma 10 (part (b)) to the set Z (U (v)) tells us that, with probability at least 1 − 1/n 2 , the total number of messages broadcast by undecided nodes in U (v) is O(|Z (U (v))| log n). Hence, amortizing over the total number of (undecided and category-3) nodes in Z (U (v)), we have shown each node in Z (U (v)), in particular v, is charged (amortized) O(log n) in a phase. Taking a union bound, gives a high probability result for all nodes v.
To show that the same node v is not charged too many times across phases, we use the fact that category-2 nodes are never charged (and they do not broadcast). We note that if a node enters the ruling set (i.e., becomes category-1) in some phase, then all its neighbors become category-2 nodes and will never be charged again (in any subsequent phase). Furthermore, since a marked node enters the ruling set with probability at least 1/2, a neighbor of v (or v itself) gets charged at most O(log n) times whp. Hence overall a node is charged at most O(log 2 n) times whp and by union bound, every node gets charged at most O(log 2 n) times whp.
Case 2.
In this case, we focus on the messages broadcast by those undecided nodes v that have at most d(v)/2 − 1 neighbors that are in category-3 or undecided, i.e., at least d(v)/2 + 1 neighbors are in category-2. By the description of our algorithm, a node enters the broadcast step, only if checking sampling step (Step 8) fails to find a category-2 node. The probability of this "bad" event happening is at most 1 d(v) 4 , which is the probability that a category-2 neighbor (of which there are at least d(v)/2 many) is not among any of the 4 log(d(v)) randomly sampled neighbors. We next bound the total number of broadcast messages generated by all undecided nodes in Case 2 during the entire course of the algorithm. By Lemma 8, for any node v, Case 2 can potentially happen only 2 log n times with probability at least 1 − 1/n 2 , since that is the number of times v can get marked. Let r.v. Y v denote the number of Case 2 broadcast messages sent by v during the course of the algorithm. Conditional on the fact that it gets marked at most 2 log n times,
we have E[Y v ] = 2 log n 1 d(v) 4 d(v) = 2 log n 1 d(v) 3 . Let Y = v∈V Y v .
Hence, conditional on the fact that each node gets marked at most 2 log n times (which happens with probability 1 − 1/n) the total expected number of Case 2 broadcast messages sent by all nodes is
E[Y ] = v∈V E[Y v ] = v∈V 2 log n 1 d(v) 3 = O(n log n).
We next show concentration of Y (conditionally as mentioned above). We know that Since the conditioning with respect to the fact that all nodes get marked at most 2 log n times happens with probability at least 1 − 1/n, unconditionally, Pr(Y Θ(n log 2 n)) O(1/n 2 ) + 1/n. Hence, the overall broadcast messages sent by nodes in Case 2 is bounded by O(n log 2 n) whp.
Var[Y v ] = 4 log 2 n( 1 d(v) 2 − 1 d(v) 6 ) 4 log 2 n. Since the random variables Y v are independent, we have Var[Y ] = v∈V V ar(Y v ) = 4n log 2 n. Noting that Y v − E[Y v ]
Combining type 1 and type 2 messages, the overall number of messages is bounded by O(n log 2 n) whp.
Thus we obtain the following theorem. A tight example. We show that the above analysis of the Algorithm 2-rulingset-msg is tight up to a polylogarithmic factor. The tightness of the message complexity follows from Theorem 14 which shows that any O(1)-ruling set algorithm needs Ω(n) messages.
For the time complexity, we show that the analysis is essentially tight by giving a n-node graph where the algorithm takes Ω(n 1− ) rounds in a graph where ∆ = O(n), for any fixed constant > 0. The graph is constructed as follows. The graph consists of a distinguished node s and three sets of nodes -sets A, B, and C. A has n 1− nodes, B has n 1− nodes, where > > 0 and C has n − 1 − n 1− − n 1− nodes. There is a complete bipartite graph between sets A and B and between sets B and C and s is connected to all nodes in A. Hence s has degree n 1− , every node in A has degree Θ(n 1− ), every node in B has degree Θ(n) and every node in C has degree Θ(n 1− ). If we run the algorithm 2-rulingset-msg on this graph, then with high probability at least one node in C will enter the ruling set in the first phase itself; further, with probability at least 1 − o(1), neither s nor any node in sets A and B mark themselves in the first phase. Once a node from C enters the ruling set, all nodes in B become category-2 nodes. On the other hand, nodes in sets A and C (conditioned on not entering the ruling set in the current phase) will become category-3 nodes in the next phase by executing the checking sampling step (which will succeed with high probability). However, since all of the neighbors of s are in category-3, node s is bound to execute Θ(n 1− ) phases until it marks itself and enter the ruling set in expectation. Hence the expected round complexity is Ω(n 1− ). Note that even though this graph has Θ(n 2− ) edges, the algorithm sends only O(n log 2 n) messages (whp).
Message Complexity Lower Bounds
We first point out that the bound of Theorem 12 is tight up to logarithmic factors.
Theorem 14.
Any O(1)-ruling set algorithm that succeeds with probability 1 − o(1) sends Ω(n) messages in the worst case. This is true even if nodes have prior knowledge of the network size n.
Proof. Consider a cycle G of n nodes and, for any t = O(1), suppose that A is a t-ruling set algorithm that has a message complexity of o(n).
We first condition on the assumption that nodes do not have IDs and subsequently remove this restriction. By assumption, there are (1 − 1 2t+1 )n nodes in any run of A that are quiescent, i.e., neither send nor receive any messages. We define a segment to be a sequence of consecutive nodes in the cycle. Let E u be the indicator random variable that is 1 if and only if node u enters the ruling set. It follows that there exists a segment S of 2t + 1 quiescent nodes and we can observe that the random variables in the set
{E v | v ∈ S} are independent.
Consider a sub-segment of 3 nodes in S. Since the network is anonymous, we know that Pr[E u = 1] = Pr[E v = 1] = p, for any u, v ∈ S. Recalling that A succeeds with probability 1−o(1) tells us that the event where two neighbors join the ruling set happens with probability at most 2p 2 (1 − p) = o(1) and hence it must be that p = o (1). On the other hand, since S has length 2t + 1, at least 1 node must enter the ruling set with probability 1 − o(1) and
conversely (1 − p) 2t+1 = o(1), which contradicts t = O(1) and p = o(1).
Finally, suppose that an algorithm B computes a t-ruling set correctly with probability 1 − o(1) requiring o(n) messages in networks where nodes have unique IDs. Similarly to [25], we construct an algorithm A that works in anonymous networks by instructing every node to first uniformly at random choose a unique ID from [1, n c ], where c 4 is any constant, and then run algorithm B with the random ID as input. With high probability, all randomly chosen IDs are unique and hence A also succeeds with probability 1 − o(1), contradicting the lower bound above.
Next, we show a clear separation between the message complexity of computing an t-ruling set (t > 1) and a maximal independent set (i.e., 1-ruling set) by proving an unconditional Ω(n 2 ) lower bound for the latter.
Theorem 15. Any maximal independent set algorithm that succeeds with probability 1 − on connected networks, where 0 < 1 2 is a constant, must send Ω(n 2 ) messages in the worst case. This is true even if nodes have prior knowledge of the network size n.
Proof of Theorem 15. For the sake of a contradiction, assume that there is an algorithm A that sends at most µ = o(n 2 ) messages in the worst case and succeeds with probability 1 − , for some < 1 2 . Consider two copies G and G of the complete bipartite graph on n/2 nodes. 5 For now, we consider the anonymous case where nodes do not have access to unique IDs; we will later show how to remove this restriction. Recall that in our model (cf. Section 1.1), we assume that nodes do no have any prior knowledge of their neighbors in the graph. Instead, each node u has a list ports 1, . . . , deg u , whose destination are wired in advance by an adversary.
We consider two concrete instances of our lower bound network depending on the wiring of the edges. First, let D = (G, G ) be the disconnected graph consisting of G and G and their induced edge sets. It is easy to see that there are exactly 4 possible choices for an MIS on D, as any valid MIS must contain the entire left (resp. right) half of the nodes in G and G and no other nodes. We denote the events of obtaining one of the four possible MISs by LL , LR , RL , RR , where, e.g., RL is the event that the right half of G (i.e. nodes in R) and the left half of G (i.e. nodes in L ) are chosen. Let "on D" be the event that A is executed on graph D. Of course, we cannot assume that algorithm A does anything useful on this graph as we require A only to succeed on connected networks. However, we will make use of the symmetry of the components of D later on in the proof. We now define the second instance of our lower bound graph. Consider any pair of edges e = (u, v) ∈ G = (L, R) and e = (u , v ) ∈ G = (L , R ). We define the bridge graph by removing e and e from G respectively G and, instead, adding the bridge edges b = (u, u ) and b = (v, v ) by connecting the same ports that were used for e and e ; see Figure 1. We use B to denote a graph that is chosen uniformly at random from all possible bridge graphs, i.e., the edges replaced by bridge edges are chosen uniformly at random according to the above construction. Let "G ↔ G " be the event that A sends at least 1 message over a bridge edge and, similarly, we use "G ↔ G " to denote the event that this does not happen. Proof. We start out by observing that nodes do not have any prior knowledge regarding the choice of the bridge edges when A is executed on graph B. Moreover, as long as no message was sent across a bridge edge, every unused port (i.e., over which the algorithm has not sent a message yet) has the same probability of being connected to a bridge edge. It follows that discovering a bridge edge corresponds to sampling without replacement, which we can model using the hypergeometric distribution, where we have exactly 4 bridge ports among the edge set of B and one draw per message sent by the algorithm. Observe that the total number of ports is 2|E(B)| = 4 n 4 2 = n 2 4 . Hence, by the properties of the hypergeometric distribution, the expected number of bridge edges discovered when the algorithm sends at most µ messages is Θ( µ n 2 ). Using Markov's Inequality and the fact that µ = o(n 2 ), it follows that the probability of discovering at least 1 bridge edge is o(1).
A crucial property of our construction is that, as long as no bridge edge is discovered, the algorithm behaves the same on B as it does on D. The following lemma can be shown by induction over the number of rounds. We are now ready to prove Theorem 15. Consider a run of algorithm A on a uniformly at random chosen bridge graph B. Let "A succ." denote the event that A correctly outputs an MIS. Observe that A succeeds when executed on B if and only if we arrive at an output configuration corresponding LR or RL . It follows that which we can plug into (2) to obtain 1 2 − o(1), yielding a contradiction. Finally, we can remove the restriction of not having unique IDs by arguing that the algorithm can generate unique IDs with high probability, since we assume that nodes know n; see the proof of Theorem 14 for a similar argument. This completes the proof of Theorem 15.
Conclusion
We studied symmetry breaking problems in the Congest model and presented time-and message-efficient algorithms for ruling sets. Several key open questions remain. First, can the MIS lower bounds in the Local model shown by Kuhn et al. [23] be extended to 2-ruling sets? In an orthogonal direction, can we derive time lower bounds for MIS in the Congest model, that are stronger than their Local-model counterparts? And on the algorithms side, can we build on the techniques presented here to improve the time bounds in the Congest model? For example, can we solve the 2-ruling set problem in O(log α n) rounds for some constant α < 1. Second, although we have presented near-tight message bounds for 2-ruling sets, we don't have a good understanding of the message-time tradeoffs. In particular, a key question is whether we can design a 2-ruling set algorithm that uses O(n polylog n) messages, while running in O(polylog n) rounds? Such an algorithm will also lead to better algorithms for other distributed computing models such as the k-machine model [21]. More generally, can we obtain tradeoff relationships that characterizes the dependence of one measure on the other or obtain lower bounds on the complexity of one measure while fixing the other measure.
A Concentration Bounds
Theorem 18 (Chernoff Bound [30]). Let the random variables X 1 , X 2 , . . . , X n be independently distributed in [0, 1] and let X = i X i . Then, for 0 < ε < 1,
P r(X < (1 − ε)E[X]) exp − ε 2 2 E[X] .
Theorem 19 (Bernstein's Inequality [11]). Let the random variables X 1 , X 2 , . . . , X n be independent with X i − E[X i ] b for each i, 1 i n. Let X := i X i and let σ 2 := i σ 2 i be the variance of X. Then for any t > 0,
P r(X > E[X] + t) exp t 2 2σ 2 (1 + bt/3σ 2 )
.
has higher ID than all neighbors in U ) then 5 I ← I ∪ {v}; 6 v and nodes within distance β in G are removed from U
GreedyRulingSet(G[S t ], B t , 2) take O( √ log n) rounds whp. Theorem 3. Algorithm 2-RulingSet computes a 2-ruling set in the Congest model in O log ∆ · (log n) 1/2+ε + log n ε log log n rounds, whp. Proof. Note that there are O(log ∆) scales and each scale contains (i) (log n) 1/2+ε iterations, each of which takes O(1) rounds and (ii) one call to GreedyRulingSet which requires O( √ log n) rounds whp by lemma 2. Thus Lines 2-14 take O(log ∆ · (log n) 1/2+ε ) rounds. From Lemma 1, the call to the greedy MIS algorithm in Line 15, takes O( log n log log n ) rounds to compute an MIS of each of the G[M i,t ]'s in parallel. This yields the claimed running time.
This 2 -
2ruling set algorithm can be used to obtain a 3-ruling set algorithm running in O log n log log n rounds for any graph in the Congest model. The 3-ruling set algorithm starts by using the simple, randomized subroutine called Sparsify[8,22] to construct in, say O((log n) 2/3 ) rounds, a set S such that (i) ∆(G[S]) = 2 O((log n) 1/3 ) whp and (ii) every node is in S or has a neighbor in S. The properties of Sparsify are more precisely described in the following lemma.
Lemma 4 .
4(Theorem 1 in[8]) Let G be an n-node graph with maximum degree ∆. Algorithm Sparsify with input G and f runs in O(log f ∆) rounds in the Congest model and produces a set S ⊆ V (G) such that ∆(G[S]) = O(f · log n) whp, and every vertex in V is either in S or has a neighbor in S.Using Algorithm 2-Ruling Set on G[S] yields the following corollary.Corollary 5. It is possible to compute a 3-ruling set in O log n log log n rounds whp in the Congest model. Proof. First call Sparsify with input graph G and parameter f = 2 (log n) 1/3 . Sparsify runs in O((log n) 2/3 ) rounds and returns a set S such that ∆(G[S]) = 2 O((log n) 1/3 ) . We then run 2-RulingSet on G[S] with ε < 1/6, which completes in O log n log log n rounds, returning a 3-ruling set of G.
Algorithm 3: 5 -
5RulingSet(Graph G = (V, E)):1 S ← Sparsify(G, 2 √ log n ); 2 I ← GhaffariMISPhase1(G[S]) for Θ(c · log ∆(G[S])) rounds; 3 R ← S \ (I ∪ N (I)); 4 I ← I ∪ GreedyRulingSet(G[S],R, 4); 5 return I; We use f = 2 √ log n in our call to Sparsify, which implies that Sparsify runs in O( √ log n) rounds. We then run O(log ∆(G[S])) iterations of the first phase of Ghaffari's MIS algorithm on G[S] and this returns an independent set I [16]. Since ∆(G[S]) = 2 O( √ log n) , this is equivalent to running O( √ logn) iterations of the first phase of Ghaffari's MIS algorithm.
Theorem 7 ...
7Algorithm 5-RulingSet computes a 5-ruling set of G in O( √ log n) rounds whp. Proof. Sparsify with paramter 2 √ log n takes O(log 2 √ log n n) = O( √ log n) time. Also, whp the maximum degree of the graph G[S], ∆(G[S]) = O(2 √ log n · log n) and this is bounded above by 2 c √ log n for some constant c . Next, we run the GhaffariMISPhase1 algorithm for Θ(c log ∆(G[S])) = O( √ log n) rounds. We now argue that Step (4) also runs in O( √ log n) rounds whp. Suppose that the call to GreedyRulingSet(G[S], R, 4) takes p iterations, Then, there is a witness path P = (v 1 , v 2 , . . . , v p ) in G[S] to the execution of this algorithm. Then, by Lemma 6 and Property (i) of witness paths: Pr(All nodes in P remain undecided after Step (Now we use Property (ii) of witness paths to upper bound the total number of possible length-p witness paths in G[S] by |S| · ∆(G[S]) 5p . Let E P denote the event that all nodes of a possibe length-p witness path P in G[S] have remained undecided after Step (2). Taking a union bound over all possible length-p witness paths in G[S] we see that Pr(∃ a length-p witness path P in G[S]: E P ) Plugging in p = √ log n and choosing c 5 + 2/c we get an upper bound of 1/n on the probability that a length-p witness path exists after Step (2). This implies that whp Step (4) takes O( √ log n) rounds. Step (4) computes a 4-ruling set of G[R] and this along with the set I form a 4-ruling set of G[S]. Since every node is at most 1 hop away from some node in S, we have computed a 5-ruling set of G.
Algorithm 4 :
4Algorithm 2-rulingset-msg: code for a node v. d(v) is the degree of v. 1 status v = undecided; 2 while status v = undecided do 3 if v receives a message from a category-1 node then 4 Set status v = category-2; 5 end 6 if v is undecided then it marks itself with probability 1 2d(v) ; 7 if v is marked then 8 (Checking Sampling Step:) Sample a set A v of 4 log(d(v)) random neighbors independently and uniformly at random (with replacement) ; 9 Find the categories of all nodes in A v by communicating with them; 10 if any node in A v is a category-2 node then 11 Set status v = category-Broadcast Step:) Send the marked status and d(v) value to all neighbors; 15
Lemma 9 .
9The algorithm 2-rulingset-msg runs in O(∆ log n) rounds whp. In particular, with probability at least 1 − 2/n 2 , a node v becomes decided after O(d(v) log n) rounds. When the algorithm terminates, i.e., when all nodes are decided, the category-1-nodes form a 2-ruling set of the graph. Moreover, each node is correctly classified according to its category.
Lemma 10 .
10Let U ⊆ V be a (sub-)set of undecided nodes at the beginning of a phase. Let N (v) be the set of neighbors of v. Then the following statements hold: (a) Let Z(U ) = U ∪ (∪ v∈U N (v)). The total number of messages sent by all nodes in U during the Broadcast step in this phase (i.e., Step 14) of the algorithm is O(|Z(U )| log n) with probability at least 1 − 1/n 3 . (b) Let N (v) be the set of undecided and category 3 neighbors of v and suppose |N
2n log n, we apply Bernstein's inequality to obtain Pr(Y E[Y ] + 4n log 2 n) exp − 16n 2 log 4 n 8n log 2 n + (2/3)2n log n(4n log 2 n) O(1/n 2 ).
Theorem 12 .
12The algorithm 2-rulingset-msg computes a 2-ruling set using O(n log 2 n) messages and terminates in O(∆ log n) rounds with high probability.
Observation 1 .
1Pr[LL | on D] = Pr[LR | on D] = Pr[RL | on D] = Pr[RR | on D].
Lemma 16 .
16Consider an execution of algorithm A on a uniformly at random chosen bridge graph B. The probability that a message is sent across a bridge is o(1), i.e., Pr[G ↔ G ] = 1 − o(1).
Lemma 17 .
17Let Y be any event that is a function of the communication and computation performed by algorithm A. Then, Pr[Y | G ↔ G ] = Pr[Y | on D].
WFigure 1
1∈{LR ,RL } Pr[W | G ↔ G ] · Pr[G ↔ G ] + Pr[A succ. | G ↔ G ] · Pr[G ↔ G ] 1 − .Theorem 16 tells us that Pr[G ↔ G ] = o(1) and, using Pr[G ↔ G ] 1, allows us to rewrite the above inequality as W ∈{LR ,RL } Pr[W | G ↔ G ] 1 − − o(1). The lower bound graph B(G, G ) for Theorem 14 with bridge edges (u2, u 2 ) and (v n/4 , v 2 ). The disconnected graph D is given by replacing the bridge edges with the dashed edges. Applying Theorem 17 to the terms in the sum, we get W ∈{LR ,RL } Pr[W | on D] 1 − − o(1). LR | on D] + Pr[RL | on D] 1 2 ,
Throughout, we use "with high probability (whp)" to mean with probability at least 1 − 1/n c , for some c 1. 2 For ∆ = o(log n), the deterministic MIS algorithm ofBarenboim, Elkin, and Kuhn [4] that runs O(∆ + log * n) rounds is faster than Luby's algorithm.
Our near-linear message-efficient algorithm (Section 3) does not require knowledge of n or ∆, whereas our time-efficient algorithms (Section 2) assume knowledge of n and ∆ (otherwise it will work up to a given ∆).
Alternately, if v finds any category-2 neighbor (that was missed by checking sampling) during broadcast step it becomes a category-3 node and is done. However, this does not give an asymptotic improvement in the message complexity analysis compared to the stated algorithm.
To simplify our analysis, we assume that n/2 and n/4 are integers.
An O(log n)-round, O(n 1.5 log n) message complexity 2-ruling set algorithmWe show that when m is large, one can design a simple algorithm with o(m) message complexity algorithm that runs in time O(log n). This algorithm requires knowledge of n. The algorithm is as follows.1.Initially all nodes are inactive. 2. Every node with degree less than √ n becomes active and nodes with degree higher than √ n become active independently with probability 2 log n/ √ n. Let S denote the set of active nodes. 3. Nodes in S broadcast their status to all nodes (thus each active node knows its active neighbors, if any). 4. Compute, using Luby's algorithm, an MIS of S and return it. Proof. We show that the set returned, i.e., the MIS of S, is a 2-ruling set. Every node in S (the active set) is either in the MIS or a neighbor of a node in the MIS. We next show that every node in v ∈ V − S has a neighbor in S. If v has a neighbor u of degree less than √ n, then u belongs to S. Otherwise, since v's degree is at least √ n, the probability that at least one of its neighbors (all of which must have a degree at least √ n) belonging to S is at leastHence, by a union bound, every node in V − S has a neighbor in S whp. It follows that an MIS of S is a 2-ruling set of the graph.We next analyze the time and message complexity. The time complexity follows immediately from the run time of Luby's algorithm. We can show that, whp, the number of messages in Step(3)is O(n 1.5 log n) as follows. Nodes with degree less than √ n contribute O(n 1.5 ) messages. For nodes that have a degree higher than √ n, the expected number of neighbors in S is O( √ n log n) and this holds whp (by applying a standard Chernoff bound). Hence the number of messages is O(n 1.5 log n) whp. From the above, it follows that the sum of the degrees of the nodes in S is bounded by O(n 1.5 log n). Hence, we observe that Step (4) -Luby's algorithm on S -requires O(n 1.5 ) messages. Hence, overall the total message complexity is O(n 1.5 log n).
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| []
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"A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket",
"A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket"
]
| [
"Kumiko Hattori [email protected] \nDepartment of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan\n",
"Noriaki Ogo \nDepartment of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan\n",
"Takafumi Otsuka \nDepartment of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan\n"
]
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"Department of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan",
"Department of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan",
"Department of Mathematics and Information Sciences\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan"
]
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| We show that the 'erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the 'standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent ν governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension 1/ν, which is strictly greater than 1.The self-avoiding walk (SAW) and the loop-erased random walk (LERW) are two typical examples of non-Markov random walks on graphs. The self-avoiding walk is defined by the uniform measure on self-avoiding paths of a given length. In this paper we call this model the 'standard' selfavoiding walk ('standard' SAW), for we shall deal with a family of different walks whose paths are self-avoiding. The loop-erased random walk is a random walk obtained by erasing loops from the simple random walk in chronological order (as soon as each loop is made). Although the LERW has self-avoiding paths, it has a different distribution from that of the 'standard' SAW. | 10.3934/dcdss.2017014 | [
"https://arxiv.org/pdf/1511.04840v2.pdf"
]
| 56,574,676 | 1511.04840 | 0046d57819596e9ed4e5e8d652ada14ae4e1bc08 |
A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket
May 2016 May 3, 2016
Kumiko Hattori [email protected]
Department of Mathematics and Information Sciences
Tokyo Metropolitan University
192-0397HachiojiTokyoJapan
Noriaki Ogo
Department of Mathematics and Information Sciences
Tokyo Metropolitan University
192-0397HachiojiTokyoJapan
Takafumi Otsuka
Department of Mathematics and Information Sciences
Tokyo Metropolitan University
192-0397HachiojiTokyoJapan
A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket
May 2016 May 3, 2016loop-erased random walkself-avoiding walkself-repelling walkscaling limitdisplacement exponentfractal dimensionSierpinski gasketfractal MSC2010 Subject Classifications: 60F99, 60G17, 28A80, 37F25, 37F35
We show that the 'erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the 'standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent ν governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension 1/ν, which is strictly greater than 1.The self-avoiding walk (SAW) and the loop-erased random walk (LERW) are two typical examples of non-Markov random walks on graphs. The self-avoiding walk is defined by the uniform measure on self-avoiding paths of a given length. In this paper we call this model the 'standard' selfavoiding walk ('standard' SAW), for we shall deal with a family of different walks whose paths are self-avoiding. The loop-erased random walk is a random walk obtained by erasing loops from the simple random walk in chronological order (as soon as each loop is made). Although the LERW has self-avoiding paths, it has a different distribution from that of the 'standard' SAW.
Two of the basic questions concerning random walks are: (1) What is the asymptotic behavior of the walk as the number of steps tends to infinity? To be more specific, if X(n) denotes the location of the walker starting at the origin after n steps, does the mean square displacement show a power behavior? In other words, does the following hold in some sense?
E[|X(n)| 2 ] ∼ n 2ν ,
where |X(n)| denotes the Euclidean distance from the starting point and ν is a positive constant.
If it is the case, what is the value of the displacement exponent ν?
(2) Does the walk have a scaling limit? A scaling limit is the limit as the edge length of the graph tends to 0. To give some examples, Brownian motion on Z d and Brownian motion on the Sierpiński gasket are obtained as the scaling limit of the simple random walk on their respective graph approximations. The displacement exponent ν governs also the short-time behavior of the scaling limit. Question (1) originated from the problem of the end-to-end distance of long polymers. Since no two monomers can occupy the same place, a self-avoiding walk is expected to model polymers. There have been many works, not only mathematical works, but also computer simulations and heuristics aimed at answering the question, however, for 'standard' self-avoiding walk on Z d with d = 2, 3, 4, it is not solved rigorously yet. Question (2) for Z d , d = 2, 3, 4 has not been given a rigorous answer yet, either, while for Z d with d > 4 the answers are given; the scaling limit is the d-dimensional Browinan motion and ν = 1/2. The difficulties for d = 2, 3, 4 lie in the strong self-avoiding effect in low dimensions. For what is known about 'standard' self-avoiding walks on Z d , see [19].
The situation is quite different for the LERW on Z d . The existence of the scaling limit has been proved for all d, and the asymptotic behavior has been studied in terms of the growth exponent (the reciprocal of the displacement exponent). For d = 2 Schramm-Loewner evolution (SLE) has played an essential role. For some further discussion of the LERW on Z d , see [18], [20], [14] and [17].
The Sierpiński gasket provides a space which is 'low-dimensional', but permits rigorous analysis. For this fractal space, the displacement exponent ν of the 'standard' SAW is obtained in [10]. The scaling limit is studied in [6] and it is proved that the same ν governs the short-time behavior of the limit process X t , that is, there exist positive constants C 1 and C 2 such that
C 1 ≦ E[|X t |] t ν ≦ C 2
holds for small enough t ( [3]). As for the LERW, the scaling limit was obtained by two groups independently, using different methods ( [21], [11]). SLE mentioned above is a profound theory, which goes far beyond the investigation of the scaling limit of the LERW on Z 2 . It is a unified theory for a variety of random curves in R 2 that involves a parameter κ, and different values of κ correspond to different models. κ = 2 corresponds to the scaling limit of the LERW and κ = 8/3 is conjectured to be the scaling limit of the SAW. Thus, SLE is expected to connect the SAW and the LERW on R 2 .
There arises a natural question: Is it possible to construct a model that connects the SAW and the LERW on the Sierpiński gasket continuously in some parameter? In this case we cannot use SLE, for which the conformal invariance of models in R 2 plays an essential role.
In this paper, we construct a one-parameter family of self-avoiding random walks on the Sierpiński gasket continuously connecting the LERW and a SAW which has the same asymptotic behavior as the 'standard' SAW. We prove the existence of the scaling limit and show some path properties: The exponent ν governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension 1/ν, which is strictly greater than 1.
The main ingredients for the model are the one-parameter family of self-repelling walks on the Sierpinski gasket studied in [3] and [7], and the 'erasing-larger-loops-first' (ELLF) method employed in the study of the LERW [11]. A self-repelling walk is a walk that is discouraged, if not prohibited, to return to points it has visited before. There have been a variety of models on Z. See, for example, the survey paper [12] and the references therein. The model we use here is unique in the way of discouraging returns; penalties are given for backtracks and sharp turns, rather than for revisits to the same points or the same edges.
For the 'standard' LERW on graphs, the uniform spanning tree proves to be a powerful tool ( [21]). By 'standard', we mean the loops are erased chronologically as first introduced by G. Lawler ([16]). On the other hand, [11] constructed a LERW on the Sierpiński gasket by ELLF, that is, by erasing loops in descending order of size of loops and proved that the resulting LERW has the same distribution as that of the 'standard' LERW. The uniform spanning tree is powerful in the sense that it can be used on any graph, however, this tool is valid only for loop erasure from simple random walks. We prove that ELLF does work also for other kinds of random walks on some fractals, in particular, for self-repelling walks on the Sierpiński gasket, for the method is based on self-similarity. Thus, our construction is performed by erasing loops from the family of self-repelling walks by the ELLF method.
In Section 2, we describe the set-up and recall the family of self-repelling walks introduced in [3] and [7] in a more concise manner. In Section 3, we describe the ELLF method of loop-erasing in a more organized manner than [11], and apply it to the self-repelling walks to obtain a new family of self-avoiding walks interpolating LERW and SAW. In Section 4 we study the scaling limit. In Section 5 we prove some properties of the limit process concerning the short-time behavior. In Section 6, we give the conclusion and some remarks.
2 Self-repelling walk on the pre-Sierpiński gaskets Let us first recall the definition of the pre-Sierpiński gaskets, that is, graph approximations of the Sierpiński gasket which is a fractal with Hausdorff dimension log 3/ log 2. Let O = (0, 0),
a = ( 1 2 , √3 2
), b = (1, 0) and define F ′ 0 to be the graph that consists of the three vertices and the three edges of △Oab. Define similarity maps f i : R 2 → R 2 , i = 1, 2, 3 by
f 1 (x) = 1 2 x, f 2 (x) = 1 2 (x + a), f 3 (x) = 1 2 (x + b),
and a recursive sequence of graphs
{F ′ N } ∞ N =0 by F ′ N +1 = f 1 (F ′ N ) ∪ f 2 (F ′ N ) ∪ f 3 (F ′ N ).
Let F N be the union of F ′ N and its reflection with respect to the y-axis, and let G N and E N be the sets of the vertices and of the edges of F N , respectively. F 3 is shown in Fig. 1.
Let T M be the set of all upward (closed and filled) triangles which are translations of 2 −M △Oab and whose vertices are in G M ; an element of T M is called a 2 −M -triangle.
For each N ∈ Z + = {0, 1, 2, . . .}, denote the set of finite paths on F N starting from O, not hitting any vertices in G 0 other than O on the way, and stopped at the first hitting time of a by
W N = { w = (w(0), w(1), · · · , w(n)) : w(0) = O, w(n) = a, w(i) ∈ G N , {w(i), w(i + 1)} ∈ E N , w(i) ∈ G 0 \ {O}, 0 ≦ i ≦ n − 1, n ∈ N }.
For a path w = (w(0), w(1), · · · , w(n)) ∈ W N , denote the number of steps by ℓ(w) := n.
If we assign probability (1/4) ℓ(w)−1 to each w ∈ W N , then this determines a probability measure on paths which is the same as that induced by the simple random walk on F N starting We shall assign probabilities such that they give random walks whose revisits to the same points are discouraged. First let us start with paths in W 1 . The idea is that we give a penalty to w ∈ W 1 every time it makes a sharp turn or a backtrack at G 1 \ G 0 , or revisits O. We realize it by using N (w), the total number of sharp turns and backtracks, and M (w), the total number of revisits to O, and by assigning probability u N (w)+M (w) x ℓ(w)−1 u , where u is a parameter taking values in [0, 1] and x u is a positive constant determined so that the sum of the probabilities over W 1 equals to 1. This is a natural way to define a self-repelling walk on F 1 : If u = 1, then we have x 1 = 1/4 and the simple random walk given above, and if u = 0, then the probability is supported on a set of self-avoiding paths. On a general W N , we define the probability recursively.
To give a precise definition, we shall make some preparations. For a path w ∈ ∞ N =1 W N and A ⊂ R 2 , we define the hitting time of A by
T A (w) = inf{j ≧ 0 : w(j) ∈ A},
where we set inf ∅ = ∞. For w ∈ W N and 0 ≦ M ≦ N , we shall define a recursive sequence {T M i (w)} m i=0 of hitting times of G M as follows: Let T M 0 (w) = 0, and for i ≧ 1, let
T M i (w) = inf{j > T M i−1 (w) : w(j) ∈ G M \ {w(T M i−1 (w))}},
here we take m to be the smallest integer such that T M m+1 (w) = ∞. Then T M i (w) is the time (steps) taken for the path w to hit vertices in G M for the (i + 1)-th time, under the condition that if w hits the same vertex in G M more than once in a row, we count it only 'once'.
For each M ∈ Z + , we define a coarse-graining map Q M :
∞ N =M W N → W M by setting (Q M w)(i) = w(T M i (w)) for i = 0, 1, 2, . . . , m,
where m is as above. Note that
Q K • Q M = Q K , if K ≦ M
holds and that if w ∈ W N and M ≦ N , then Q M w ∈ W M . For w ∈ W 1 , define the reversing number N (w) and the revisiting number M (w) by
N (w) = ♯{1 ≦ i ≦ ℓ(w) − 1 : − −−−−−−−− → w(i − 1)w(i) · − −−−−−−−− → w(i)w(i + 1) < 0, w(i) / ∈ G 0 }, (2.1) M (w) = ♯{1 ≦ i ≦ ℓ(w) : w(i) = O}, (2.2) where − → a · − → b denotes the inner product of − → a and − → b in R 2 . For x > 0 and 0 ≦ u ≦ 1, define Φ(x, u) = w∈W 1 u N (w)+M (w) x ℓ(w) . (2.3)
For each u, within the radius of convergence r u > 0 as a power series in x, we have the following explicit form of Φ given in [3]:
Φ(x, u) = x 2 {1 + (1 + u)x − u(1 − u 2 )x 2 + 2(1 − u) 2 u 2 x 3 } (1 + ux)(1 − 2ux) − 4u 2 x 2 {1 + 2(1 − u 2 )x 2 − 2u(1 − u) 2 x 3 } .
.
Proposition 1 (Proposition 2.3 in [3]) (1) For each u ∈ [0, 1], there is a unique fixed point x u of the mapping Φ(·, u) : (0, r u ) → (0, ∞), that is, Φ(x u , u) = x u , x u > 0.
As a function in u, x u is continuous and strictly decreasing on [0, 1].
(2) Letλ u = ∂Φ ∂x (x u , u). Thenλ u > 2 andλ u is continuous in u.
In the two extreme cases, we know that
x 0 = √ 5 − 1 2 ,λ 0 = 7 − √ 5 2
, and x 1 = 1 4 ,λ 1 = 5.
To define a family of probability measures {P u N , u ∈ [0, 1]} on each W N , we consider decompositions of a path based on the self-similarity and the symmetries of the pre-Sierpiński gaskets. Assume w ∈ W N and 0 ≦ M < N and denotew = Q M w. Since the pair of adjacent 2 −M -triangles includingw(i − 1),w(i) andw(i + 1) is similar to F N −M , there is a unique decomposition (w; w 1 , · · · , w ℓ(w) ),w ∈ W M , w i ∈ W N −M , i = 1, · · · , ℓ(w) (2.4) such that the path segment (w(T M i−1 (w)), w(T M i−1 (w) + 1)), · · · , w(T M i (w))) of w is identified with w i ∈ W N −M by appropriate similarity, rotation, translation and reflection so that w(T M i−1 (w)) is identified with O and w(T M i (w)) with a. We shall use this kind of identification throughout this paper. We illustrate a simple example of the decomposition in Fig. 2
. b a O O b a b a O b a O b a O ww w 1 w 2 w 3 Fig 2: w,w, w 1 , w 2 , w 3
First, for each w ∈ W 1 , let
P u 1 (w) = u N (w)+M (w) x ℓ(w)−1 u , (2.5)
and define P u N on W N recursively by
P u N (w) = P u N −1 (w) ℓ(w) i=1 P u 1 (w i ), (2.6) where (w; w 1 , · · · , w ℓ(w) ) is the decomposition of w ∈ W N with M = N − 1 given in (2.4). Denote the image measure of P u N induced by the mapping Q M by P u N • Q −1 M . P u N is self-similar in the sense that P u N • Q −1 M = P u M . (W N , {P u N } u∈[0,1] ) defines a family of self-repelling walks Z u N on F N such that Z u N (w)(i) = w(i), i = 0, · · · , ℓ(w), w ∈ W N . (2.7)
In [3], it is proved that for each u, the sequence {Z u N (λ N u · )} ∞ N =1 of time-scaled self-repelling walks converges to a continuous process as N → ∞. The one-parameter family of the limit processes {Z u ( · ), u ∈ [0, 1]} continuously interpolates a self-avoiding process (u = 0) and Brownian motion (u = 1) on the Sierpiński gasket.
In the next section, we erase loops from this family of self-repelling walks to obtain a oneparameter family of self-avoiding walks. For this purpose, we introduce an auxiliary family of self-repelling walks. Let
F V N = F N ∪ (F ′ N + b)
, which consists of three adjoining copies of F ′ N , and let G V N and E V N be the sets of the vertices and of the edges of F V N , respectively. Denote the set of finite paths on F V N starting from O and stopped at the first hitting time of a by
V 0 N = { w = (w(0), w(1), · · · , w(n)) : w(0) = O, w(n) = a, w(i) ∈ G V N , {w(i), w(i + 1)} ∈ E V N , w(i) = a, 0 ≦ i ≦ n − 1, n ∈ N }. and define ℓ(w), T M i (w)
and Q M in the same way as for W N . Let
V N = { w ∈ V 0 N : Q 0 w = (O, b, a) }.
Paths in V N are allowed to leak into the 'interior' of the third copy of F ′ N . A path w ∈ V N defined in this way consists of two parts, (w(0), w(1), · · · , w(T 0 1 (w))) and (w(T 0 1 (w)), w(T 0 1 (w) + 1), · · · , w(T 0 2 (w))), and they can be identified with some w ′ , w ′′ ∈ W N , respectively. Define a probability measure P ′u
N on V N by P ′u N [w] = P u N [w ′ ] · P u N [w ′′ ], where P u N is defined in (2.5) and (2.6). (V N , {P ′u N } u∈[0,1] ) defines a family of self-repelling random walks Z ′u N on F V N such that Z ′u N (w)(i) = w(i), i = 0, · · · , ℓ(w), w ∈ V N . (2.8)
This is a family of self-repelling walks that hit b 'once' in the sense that Q 0 w = (O, b, a).
For (w(0), w(1), · · · , w(n)) ∈ W N ∪ V N , if there are c ∈ G N , i and j, 0 ≦ i < j ≦ n such that w(i) = w(j) = c and w(k) = c for any i < k < j, we call the path segment [w(i), w(i+1), . . . , w(j)]
a loop formed at c and define its diameter by d = sup i≦k 1 <k 2 ≦j |w(k 1 ) − w(k 2 )|, where | · | denotes the Euclidean distance. Note that a loop can be a part of another larger loop formed at some other vertex. By definition the paths in W N ∪ V N do not have any loops with diameter greater than or equal to 1. Let Γ N be the set of loopless paths on F N from O to a:
Γ 0 = {(O, a), (O, b, a)}, Γ N = { (w(0), w(1), · · · , w(n)) ∈ W N ∪ V N : w(i) = w(j), 0 ≦ i < j ≦ n, n ∈ N }.
Note that any loopless path from O to a is confined in △Oab.
Loop erasure on F V 1
We shall now describe the loop-erasing procedure for paths in W 1 ∪ V 1 :
(i) Erase all the loops formed at O;
(ii) Progress one step forward along the path, and erase all the loops at the new position;
(iii) Iterate this process, taking another step forward along the path and erasing the loops there, until reaching a.
To be precise, for w ∈ W 1 ∪ V 1 , define the recursive sequence {s i } n i=0 ,
s 0 = sup{j : w(j) = O}, s i = sup{j : w(j) = w(s i−1 + 1)}. If s i > s i−1 + 1, then [w(s i−1 + 1), w(s i−1 + 2), . . . , w(s i − 1), w(s i )]
forms a loop or multiple loops at w(s i−1 + 1) = w(s i ), so we erase this part by removing w(s i−1 + 1), w(s i−1 + 2), . . . , w(s i − 2), and w(s i − 1). If w(s n ) = a, then we have obtained a loop-erased path, Note that w ∈ W 1 implies Lw ∈ W 1 ∩ Γ 1 , but that w ∈ V 1 can result in Lw ∈ W 1 ∩ Γ 1 , with b being erased together with a loop. So far, our loop-erasing procedure is the same as the chronological method defined for paths on Z d in [16].
Lw = [w(s 0 ), w(s 1 ), . . . , w(s n )] ∈ Γ 1 , where L : W 1 ∪ V 1 → Γ 1 is the loop-erasing operator. ✷
For a general N , we erase loops from the largest scale loops down, repeatedly applying the loop-erasing procedure on F V 1 .
First step of the induction -erasing largest scale loops
We shall illustrate the first step of loop erasure. Decompose a path w ∈ W N ∪ V N into (Q 1 w; w 1 , · · · , w ℓ(Q 1 w) ), w i ∈ W N −1 ∪ V N −1 i = 1, · · · , ℓ(Q 1 w) as in (2.4). Fig. 4(a) shows w ∈ W N ∪V N and Fig. 4(b) shows Q 1 w. Erase all the loops in chronological order from Q 1 w ∈ W 1 ∪V 1 to obtain LQ 1 w as in Fig. 4(c), then restore the original fine structures to the remaining parts as shown in Fig. 4(d). That is, if we write
LQ 1 w = [w(T 1 0 ), w(T 1 s 1 ), . . . , w(T 1 sn )], n = ℓ(LQ 1 w), a b O w * 1 w * 2 w * 3 w * 4 w * 5 w * 6 w * 7 w * 8 w * 9 w * 10 Fig 3: Loopless paths from O to a on F 1 for each i, fit the path segment w s i +1 = (w(T 1 s i ), w(T 1 s i + 1), · · · , w(T 1 s i +1 )) between w(T 1 s i ) and w(T 1 s i+1 ) of LQ 1 w.
We call the path obtained at this stageLw. Notice that in this stage all the loops with diameter greater than 1/2 have been erased. LetQ 1 w = LQ 1 w. This completes the first induction step. ✷
The idea is to repeat a similar procedure within each 2 −1 -triangle to erase all loops with diameter greater than 1/4, and then within each 4 −1 -triangle, and so on, until there remain no loops. To describe next induction steps more precisely, we make some preparations. For w ∈ W N and M ≦ N , we shall define the sequence (∆ 1 , . . . , ∆ k ) of the 2 −M -triangles w 'passes through', and their exit times
{T ex,M i (w)} k i=1 as a subsequence of {T M i (w)} m i=1 as follows: Let T ex,M 0 (w) = 0.
There is a unique element of T M that contains w(T M 0 ) and w(T M 1 ), which we denote by ∆ 1 . For i ≧ 1, define
J(i) = min{j ≧ 0 : j < m, T M j (w) > T ex,M i−1 (w), w(T M j+1 (w)) ∈ ∆ i }, if the minimum exists, otherwise J(i) = m. Then define T ex,M i (w) = T M J(i) (w)
, and let ∆ i+1 be the unique 2 −M -triangle that contains both w(T ex,M i ) and w(T M J(i)+1 ). By definition, we see that
∆ i ∩ ∆ i+1 is a one-point set {w(T ex,M i )}, for i = 1, . . . , k − 1.
We denote the sequence of these triangles by σ M (w) = (∆ 1 , . . . , ∆ k ), and call it the 2 −M -skeleton of w. We call the sequence
{T ex,M i (w)} i=0,1,...,k exit times from the triangles in the skeleton. For each i, there is an n = n(i) such that T ex,M i−1 (w) = T M n (w). If T ex,M i (w) = T M n+1 (w), we say that ∆ i ∈ σ M (w) is Type 1, and if T ex,M i (w) = T M n+2 (w), Type 2.
If w ∈ Γ N and M ≦ N , then its 2 −M -skeleton is a collection of distinct 2 −M -triangles and each of them is either Type 1 or Type 2. Assume w ∈ W N ∪ V N and M ≦ N . For each ∆ in σ M (w), the path segment of w in ∆ is defined by
w| ∆ = [w(n), T ex,M i−1 (w) ≦ n ≦ T ex,M i (w)]. (3.1)
Note that the definition of T M i allows a path segment w| ∆ to leak into the neighboring 2 −Mtriangles.
If Q M w ∈ Γ M , then w| ∆ ∈ W N −M or w| ∆ ∈ V N −M (identification implied), according to the type of ∆ ∈ σ M (w), where the entrance to ∆ is identified with O and the exit with a. This means that each w satisfying Q M w ∈ Γ M can be decomposed uniquely to
(σ M (w); w| ∆ 1 , · · · , w| ∆ k ), w| ∆ i ∈ W N −M ∪ V N −M , i = 1, · · · , k. (3.2) a O (a) b (b) O a b (c) O a b a O (d) bb) Q 1 w, (c) LQ 1 w =Q 1 w, (d)Lw
Conversely, given a collection of distinct 2 −M -triangles {∆ i } k i=1 such that O ∈ ∆ 1 , a ∈ ∆ k , ∆ i and ∆ i+1 are neighbors, and w ′ i ∈ W N −M ∪ V N −M , i = 1, · · · , k, then we can assemble them to obtain a unique element w of W N ∪ V N .
We call a loop [w(i), w(i + 1), · · · , w(i + i 0 )] a 2 −M -scale loop whenever there exists an M ∈ Z + such that
min{N ′ : w(i) = w(i + i 0 ) ∈ G N ′ } = M, d ≧ 2 −M ,
where d is the diameter of the loop. Using above as a base step, we shall now describe the induction step of our operation:
Induction step Let w ∈ W N ∪ V N . For 1 ≦ M ≦ N ,
assume that all of the 2 −1 to 2 −M -scale loops have been erased from w, and denote by w ′ ∈ W N ∪ V N the path obtained at this stage.
Then Q M w ′ ∈ Γ M . 1) Since Q M w ′ ∈ Γ M , we have the decomposition of w ′ : (σ M (w ′ ); w ′ 1 , · · · w ′ k ), w ′ i ∈ W N −M ∪ V N −M as given in (3.2).
2) From each w ′ i , erase 2 −1 -scale loops (largest scale loops) according to the base step procedure above to obtainLw
′ i ∈ W N −M ∪ V N −M andQ 1 w ′ i ∈ Γ 1.
3) Assemble (σ M (w ′ );Lw ′ 1 , · · · ,Lw ′ k ) and (σ M (w ′ );Q 1 w ′ 1 , · · · ,Q 1 w ′ k ) to obtain w ′′ ∈ W N ∪ V N andQ M +1 w ∈ Γ M +1 , respectively. w ′′ has no 2 −1 to 2 −(M +1) -scale loops.
✷
We then continue this operation until we have erased all of the loops to have Lw =Q N w ∈ Γ N . In this way, the loop erasing operator L defined for W 1 ∪ V 1 has been extended to L :
∞ N =1 (W N ∪ V N ) → ∞ N =1 Γ N with L(W N ∪V N ) = Γ N .
Notice that the operation described above is essentially a repetition of loop-erasing for
W 1 ∪ V 1 .Q M is a map from ∞ N =M (W N ∪ V N ) to Γ M .
In the induction step, we observe thatQ M +1 w = Q M +1 w ′′ . Although it may occur that σ M +1 (w ′′ ) = σ M +1 (w ′ ) because of the erasure of 2 −(M +1) -scale loops, it holds that σ M (w ′′ ) = σ M (w ′ ), which can be extended to σ K (w ′ ) = σ K (w ′′ ) for any K ≦ M . We remark that the procedure implies that for any w ∈ W N ∪ V N ,
σ K (Q M w) = σ K (Q K w) for any N ≧ M ≧ K. (3.3)
In particular,
σ K (Lw) = σ K (Q K w) for K ≦ N. (3.4)
i.e., in the process of loop-erasing, once loops of 2 −K -scale and greater have been erased, the 2 −Kskeleton does not change any more. However, it should be noted that the types of the triangles can change from Type 2 to Type 1.
We induce measuresP u N = P u N • L −1 andP ′u N = P ′u N • L −1 , which satisfyP u N [Γ N ] = 1 and P ′u N [Γ N ] = 1.
For w * 1 , · · · , w * 10 shown in Fig. 3, denote
p i =P u 1 [w * i ] = P u 1 [w : Lw = w * i ], q i =P ′u 1 [w * i ] = P ′u 1 [w : Lw = w * i ].
(3.5)
p i and q i can be obtained as explicit functions of u and x u by direct, but lengthy calculations, which are shown in Appendix. In the case that u = 1 (the ordinary loop-erased random walk), we have x 1 = 1/4, p 1 = 1/2, p 2 = p 3 = p 7 = 2/15, p 4 = p 5 = p 6 = 1/30, p 8 = p 9 = p 10 = 0, q 1 = 1/9, q 2 = q 3 = 11/90, q 4 = q 5 = q 6 = 2/45, q 7 = 8/45, q 8 = 2/9 and q 9 = q 10 = 1/18 as in [11]. For u = 0, we have p 1 = x 0 , p 7 = x 2 0 and p i = 0 otherwise, and q 1 = x 4 0 , q 2 = q 3 = x 3 0 , q 8 = x 2 0 and q i = 0 otherwise, with x 0 = ( √ 5 − 1)/2 as in [8]. P u N andP ′u N define two families of walks on F N obtained by erasing loops from Z u N and Z ′u N , respectively. We remark that that 2 3P
1 N +
3P
′1 N equals to the 'standard' LERW studied in [21]. An important observation is that in the process of erasing loops from Z u N +1 , if we stop at the point where we have obtainedQ N Z u N +1 , it is nothing but the procedure for obtaining LZ u N from Z u N . The same holds also for Z ′u N +1 . This can be expressed as:
P u N +1 [{v ′ :Q N v ′ = v}] =P u N [v], P ′u N +1 [{v ′ :Q N v ′ = v}] =P ′u N [v]. (3.6)
In this stage what is left to do for obtaining LZ u N +1 fromQ N Z u N +1 is a sequence of loop-erasing from Z u 1 or Z ′u 1 . This combined with (3.6) leads to a 'decomposition' of LERW measures. For
w ∈ Γ N +1 ,P u N +1 [ w ] = v∈Γ N P u N +1 [{v ′ : Lv ′ = w}|Q N v ′ = v ] P u N +1 [{v ′ :Q N v ′ = v}] = v∈Γ N ( k i=1P * u 1 [ w i ])P u N [ v ],
where σ N (v) = (∆ 1 , · · · , ∆ k ), w i = v| ∆ i (identification implied),P * u 1 =P u 1 if ∆ i is Type 1, and P * u 1 =P ′u 1 if ∆ i is Type 2. A similar decomposition holds also forP ′u N +1 . This is the key to the recursion relations of generating functions defined below.
For w ∈ Γ N , let us denote the number of 2 −N -triangles of Type 1, (the path passes two of the vertices) and those of Type 2 (the path passes all three vertices) in σ N (w) by s 1 (w) and s 2 (w), respectively. Note that ℓ(w) = s 1 (w) + 2s 2 (w). Define two sequences, {Φ N } N ∈N and {Θ N } N ∈N , of generating functions by:
Φ N (x, y) = w∈Γ NP u N (w)x s 1 (w) y s 2 (w) , Θ N (x, y) = w∈Γ NP ′u N (w)x s 1 (w) y s 2 (w) , x, y ≧ 0.
For simplicity, we shall denoteΦ 1 (x, y) andΘ 1 (x, y) byΦ(x, y) andΘ(x, y) and omit writing u-dependence explicitly. Similar to Proposition 3 in [11], we have Proposition 2 The above generating functions satisfy the following recursion relations for all N ∈ N :Φ (x, y) = p 1 x 2 + (p 2 + p 3 )xy + p 4 y 2 + (p 5 + p 6 )x 2 y + p 7 x 3 ,
Θ(x, y) = q 1 x 2 + (q 2 + q 3 )xy + q 4 y 2 + (q 5 + q 6 )x 2 y + q 7 x 3 + q 8 x 2 y + (q 9 + q 10 )xy 2 ,
Φ N +1 (x, y) =Φ N (Φ(x, y),Θ(x, y)), Θ N +1 (x, y) =Θ N (Φ(x, y),Θ(x, y)), where p i =P u 1 [w * i ] and q i =P ′u 1 [w * i ], i = 1, 2, · · · , 10.
Define the mean matrix by
M = ∂ ∂x Φ(1, 1) ∂ ∂y Φ(1, 1) ∂ ∂x Θ(1, 1) ∂ ∂y Θ(1, 1)
.
(3.7)
It is a strictly positive matrix, and the larger eigenvalue λ = λ(u) is a continuous function of u, satisfying 2 < λ < 3. Let Z u N and Z ′u N be as in (2.7)
The scaling limit
In this section, we investigate the limit of the loop-erased self-repelling walks constructed in Section 3 as the edge length tends to 0. Since it is easier to deal with continuous functions from the beginning, we regard F N 's and F V N 's as closed subsets of R 2 made up of all the points on their edges. We define the Sierpiński gasket by F = cl(∪ ∞ N =0 F N ), where cl denotes closure. We start with a larger space F V = cl(∪ ∞ N =0 F V N ) and let
C = {w ∈ C([0, ∞) → F V ) : w(0) = O, lim t→∞ w(t) = a} .
C is a complete separable metric space with the metric
d(u, v) = sup t∈[0,∞) |u(t) − v(t)| , u, v ∈ C,
where |x − y|, x, y ∈ R 2 , denotes the Euclidean distance. Hereafter, for w ∈ ∞ N =1 (W N ∪ V N ), we define w(t) = a, t ≧ ℓ(w), and interpolate the path linearly,
w(t) = (i + 1 − t)w(i) + (t − i)w(i + 1), i ≦ t < i + 1, i ∈ Z +
so that we can regard w as a continuous function on [0, ∞). We shall regard W N , V N and Γ N as subsets of C. Hitting times, {T M i (w)} m i=1 are defined for w ∈ C as in the previous sections, although the infimum is taken over continuous time:
T M 0 (w) = 0, T M i (w) = inf{t > T M i−1 (w) : w(t) ∈ G M \ {w(T M i−1 (w))}}.
Notice that the condition lim t→∞ w(t) = a makes {T M i (w)} m i=0 a finite sequence. For N ∈ Z + , we define a coarse-graining map Q N : C → C by (Q N w)(i) = w(T N i (w)) for i = 0, 1, 2, . . . , m, and by using linear interpolation
(Q N w)(t) = (i + 1 − t) (Q N w)(i) +(t − i) (Q N w)(i + 1), i ≦ t < i + 1, i = 0, 1, 2, . . . , m − 1, a, t ≧ m.
We define also the 2 −M -skeleton, σ M (w) (a sequence of 2 −M -triangles w passes through), the exit times {T ex,M i } k i=1 and types of triangles in a similar way to their counterparts in Section 3. The loop-erasing operator is regarded as L :
∞ N =1 (W N ∪ V N ) → ∞ N =1 Γ N .Q N '
s are as in Section 3 with resulting paths in Γ N . P u N , P ′u N ,P u N andP ′u N are regarded as probability measures on C.
In order to consider an almost sure limit, we shall couple walks on different pre-Sierpiński gaskets. Let
f a (t) = ta 0 ≦ t ≦ 1, a t > 1, where a = ( 1 2 , √ 3 2 ),
and
Ω ′ = {ω = (ω 0 , ω 1 , ω 2 , · · ·) : ω 0 = f a , ω N ∈ Γ N , ω N −1 ⊲ ω N , N ∈ N}, where ω N ⊲ ω N +1 means that there exists a v ∈ W N +1 ∪ V N +1 such thatQ N v = ω N and Lv = Q N +1 v = ω N +1 .
Namely, v is a path obtained by adding a finer, 2 −(N +1) -scale structure (not loopless yet) to ω N , and erasing 2 −(N +1) -scale loops from v gives ω N +1 . We assumed ω 0 = f a here, for we can deal with the case
ω 0 = f b with f b (t) = tb 0 ≦ t ≦ 1, b + (t − 1)(a − b) 1 < t ≦ 2 a t > 2,
where b = (1, 0), in a similar way. Define the projection onto the first N + 1 elements by π N ω = (ω 0 , ω 1 , . . . , ω N ).
For each u ∈ [0, 1], define a probability measureP N on π N Ω ′ bỹ
P N [(ω 0 , ω 1 , . . . , ω N )] = P u N [ v :Q i v = ω i , i = 0, . . . , N ],
where P u N is defined in Section 2. AlthoughP N depends on u, we shall not write the u-dependence explicitly for simplicity. The following consistency condition is a direct consequence of the looperasing procedure:P
N [(ω 0 , ω 1 , . . . , ω N )] = ω ′P N +1 [(ω 0 , ω 1 , . . . , ω N , ω ′ )], (4.1)
where the sum is taken over all possible ω ′ ∈ Γ N +1 such that ω N ⊲ ω ′ . By virtue of (4.1) and Kolmogorov's extension theorem for a projective limit, there is a probability measure P on Ω 0 = C N = C × C × · · · such that P [ Ω ′ ] = 1,
P • π −1 N =P N , N ∈ Z + ,
where π N denotes the projection onto the first (N + 1) elements also here.
Define The offspring distributions born from a Type 1 triangle and from a Type 2 triangle are equal to those of S and S ′ , respectively. If ∆ i is Type 1, the process starts in state (1, 0), and if ∆ i is Type 2, in state (0, 1).
Y N : Ω 0 → C by Y N ((ω 0 , ω 1 , . . .)) = ω N if (ω 0 , ω 1 , . . .) ∈ Ω ′ , f a otherwise . Then Y N is an F -valued process Y N (ω, t) on (Ω 0 , B, P ),
(1) The generating functions for the offspring distributions are
E[ x S 1 y S 2 ] =Φ(x, y), E[ x S ′ 1 y S ′ 2 ] =Θ(x, y),
where E is the expectation with regard to P .
(2) Let M be the mean matrix given by (3.7).
Then E[ S M +N (Y M +N | ∆ i ) | Y M = v ] = S M (v| ∆ i )M N .
(3) P [S 1 + S 2 ≧ 2] = P [S ′ 1 + S ′ 2 ≧ 2] = 1 (non-singularity).
(4) E[ S i log S i ] < ∞, E[ S ′ i log S ′ i ] < ∞, i = 1, 2.
Proposition 4 suggests that we should consider the time-scaled processes:
X N ( ω, · ) = Y N ( ω, λ N · ), N ∈ Z + ,
where λ is the larger eigenvalue of the mean matrix.
Proposition 5 For M ≦ N , the following holds:
σ M (X N ) = σ M (X M ) = σ M (Y M ), a.s. and X N (T ex,M i (X N )) = X M (T ex,M i (X M )) = Y M (T ex,M i (Y M )), a.s. (4.2)
Note that if σ M (X N ) = (∆ 1 , · · · , ∆ k ), then
T ex,M j (X N ) = λ −N j i=1 (S N 1 (X N |∆ i ) + 2S N 2 (X N |∆ i )), 1 ≦ j ≦ k.
Let u = t (u 1 , u 2 ) and v = (v 1 , v 2 ) be the right and left positive eigenvectors associated with λ such that (u, v) = 1 and (u, 1) = 1. (2) {S * M,i , i = 1, · · · , k} are independent.
(3) There are random variables B 1 and B 2 such that S * M,i is equal in distribution to
λ −M B 1 v if ∆ i is of Type 1, and equal in distribution to λ −M B 2 v if ∆ i is of Type 2.g i (t) = E[exp(−tB i )], t ∈ C
are entire functions on C and are the solution to g 1 (λt) =Φ(g 1 (t), g 2 (t)), g 2 (λt) =Θ(g 1 (t), g 2 (t)), g 1 (0) = g 2 (0) = 1.
(1)-(4) in Proposition 6 are the straightforward consequences of general limit theorems for supercritical multi-type branching processes (Theorem 1 and Theorem 2 in V.6 of [1]). P [B i > 0] = 1 is a consequence ofΦ andΘ having no terms with degree smaller than 2. For the existence of the Laplace transform on the entire C, we need careful study of the recursions. We omit the details here, since they are lengthy and similar to the proof of Proposition 4.5 in [9]. Theorem 7 X N converges uniformly in t a.s. as N → ∞ to a continuous process X.
Proof. Choose ω ∈ Ω ′ such that the following holds for all M ∈ Z + : lim
N 1 = N 1 (ω) ∈ N such that max 1≦i≦k |T ex,M i (X N ) − T * M i | ≦ min 1≦i≦k (T * M i − T * M i−1 ), |T ex,M k (X N ) − T * M k | < ε, (4.4) for N ≧ N 1 . If 0 ≦ t < T * M k , then choose j ∈ {1, · · · , k} such that T * M j−1 ≦ t < T * M j . Then (4.4) implies that T ex,M j−2 (X N ) ≦ t ≦ T ex,M j+1 (X N ), for N ≧ N 1 . Since Proposition 5 shows X N (T ex,M j (X N )) = X M (T ex,M j (X M )), (4.5) for all N with N ≧ M , we have |X N (T ex,M j (X N )) − X N (t)| ≦ 3 · 2 −M . Otherwise, if T * M k ≦ t ≦ T * M k + ε = R, then let j = k. Since T ex,M k−1 (X N ) ≦ t, |X N (T ex,M j (X N )) − X N (t)| ≦ 2 −M .
Therefore, if N, N ′ ≧ N 1 , then for any t ∈ [0, R],
|X N (t) − X N ′ (t)| ≦ |X N (T ex,M j (X N )) − X N (t)| + |X N ′ (T ex,M j (X N ′ )) − X N ′ (t)| +|X N (T ex,M j (X N )) − X N ′ (T ex,M j (X N ′ ))| ≦ 6 · 2 −M ,
where the third term in the middle part is 0 by (4.5). Since M is arbitrary, we have the uniform convergence. ✷ Proposition 6 (5) implies that E[exp tB i ] < ∞ for t > 0, which leads to:
Proposition 8 P [There exist t 0 < t 1 such that X(t) = X(t 0 ) = a for all t ∈ [t 0 , t 1 ]] = 0.
The proof is similar to that in [6].
Proposition 9
The following holds for all M ∈ Z + almost surely:
(1) σ M (X) = σ M (X M ),(2)(3) Let σ M (X M ) = (∆ 1 , · · · , ∆ k M ). If T * M i−1 < t < T * M i , then X(t) ∈ ∆ i \G M , for all 1 ≦ i ≦ k M . In particular, T * M i = T ex,M i (X) = T M i (X).
Proof. (1) and (2) are direct consequences of Proposition 5, (4.3) and Theorem 7. To prove (3), let v i = X(T * M i ), i = 1, · · · , k M and we first prove that if T * M i−1 < t < T * M i , then X(t) ∈ {v i−1 , v i }, by showing none of the following events A j , j = 1, 2, 3, 4 has positive probability.
A 1 : There exists t 1 , T * M i−1 < t 1 < T * M i such that X(t) = v i for all t 1 < t ≦ T * M i holds for some i ∈ {1, · · · , k M }. A 2 : There exists t 1 , T * M i−1 < t 1 < T * M i such that X(t) = v i−1 for all T * M i−1 ≦ t < t 1 holds for some i ∈ {1, · · · , k M }. A 3 : There exist t 1 and t 2 , T * M i−1 < t 1 < t 2 < T * M i
such that X(t 1 ) = v i and X(t 2 ) = v i holds for some i ∈ {1, · · · , k M }. A 4 : There exist t 1 and t 2 ,
T * M i−1 < t 1 < t 2 < T * M i such that X(t 1 ) = v i−1 and X(t 2 ) = v i−1 holds for some i ∈ {1, · · · , k M }.
Proposition 8 guarantees that P [A 1 ] = P [A 2 ] = 0. Since X is the uniform limit of a sequences of self-avoiding walks, we have P [
A 3 ] = P [A 4 ] = 0.
Let σ = (∆ 1 , · · · , ∆ k M ) be a sequence such that P [σ M (X) = σ] > 0. Let ∆ i be one of the triangles in σ, and denote the third vertex of ∆ i (neither the exit or entrance) by v * i . We prove that the probability that X hits v * i at some T * M i−1 < t < T * M i is zero. We can take a decreasing sequence of triangles {∆
(K) i } ∞ K=M such that ∆ (M ) i = ∆ i , ∆ (K) i ∈ T K (a 2 −K -triangle), ∆ (K) i ⊃ ∆ (K+1) i , ∞ K=M ∆ (K) i = {v * i }. Denotep = max{ 10 i=5 p i , 10 i=5 q i } < 1,
where p i and q i are defined by (3.5). For any K, with K ≧ M , (1) implies
P [ ∆ (K) i ∈ σ K (X) | σ M (X) = σ ] = P [ ∆ (K) i ∈ σ K (X K ) | σ M (X K ) = σ ] ≦p K−M .
Thus it follows that
P [ ∆ (K) i ∈ σ K (X) for all K ≧ M | σ M (X) = σ ] = 0 and P [ ∆ (K) i ∈ σ K (X) for all K ≧ M for some 1 ≦ i ≦ k M | σ M (X) = σ ] = 0, therefore, P [ ∆ (K) i ∈ σ K (X) for all K ≧ M for some i ∈ {1, · · · , k M }] = 0.
This implies that the probability that X hits any 'third' vertex of the triangles in its skeleton is zero. This completes the proof of (3). ✷ This proposition further leads to ;
Theorem 10 (1) X is almost surely self-avoiding in the sense that
P [ X(t 1 ) = X(t 2 ), 0 ≦ t 1 ≦ t 2 ≦ T a (X) ] = 0,
where T a (X) = inf{t > 0 : X(t) = a} = T * 0 1 . (2) The Hausdorff dimension of the path X([0, T a (X)]) is almost surely equal to log λ/ log 2, which is a continuous function of u.
(1) is a consequence of Proposition 8 and Proposition 9. To calculate the Hausdorff dimension, we use the fact that if a path w is self-avoiding, then it holds that σ 1 (w) ⊃σ 2 (w) ⊃σ 3 (w) ⊃ · · · → w, in the Hausdorff metric, whereσ M (w) is the union of all the closed 2 −M -triangles in σ M (w). We could call the sample path a 'random graph' directed recursive construction, for the numbers of similarity maps are random variables. We obtain the Hausdorff dimension by applying Thoerem 4.3 in [4] ✷
Path properties of the limit process
In this section we study some more sample path properties of the limit process. We assume 0 < u ≦ 1, for the case of u = 0 is considered in [7]. We shall not explicitly write u-dependence as in the previous section. Let ν = ν(u) = log 2 log λ .
Recall, from Proposition 6 (5) that
g i (t) = E[ exp(−tB i ) ], i = 1, 2
satisfy the functional equations:
g 1 (λt) =Φ(g 1 (t), g 2 (t)), g 2 (λt) =Θ(g 1 (t), g 2 (t)).
Let
h i (t) = −t −ν log g i (t).
The proof of the following proposition uses the explicit forms ofΦ andΘ, but it basically follows those of [2] and [15].
Proposition 11 There exist positive constants C 1 , C 2 and t 0 such that
C 2 ≦ h i (t) ≦ C 1 i = 1, 2 hold for all t ≧ t 0 .
Proof. We prove the upper bound for i = 1. Combining g 1 (λt) =Φ(g 1 (t), g 2 (t)) and the fact that Φ(x, y) contains the term p 1 x 2 , we have g 1 (λt) ≧ p 1 g 1 (t) 2 , which implies h 1 (λt) ≦ a 2 2 t −ν + h 1 (t), where a 2 = − log p 1 > 0. By induction, we have h 1 (λ n t) ≦ a 2 t −ν + h 1 (t), for any t > 0 and n ∈ N. Fix t 1 > 0 arbitrarily. Since h(t) is continuous for t > 0, b 1 := max t∈[t 1 ,λt 1 ] h 1 (t) exists. For t > λt 1 , there is a positive integer m and s ∈ (t 1 , λt 1 ] such that t = λ m s. Then h 1 (t) = h 1 (λ m s) ≦ a 2 s −ν + h(s) ≦ a 2 t −ν 1 + b 1 =: C 1 . Thus we have h 1 (t) ≦ C 1 for any t ≧ t 1 . The proof for i = 2 is similar, with the use of the term q 4 y 2 inΘ(x, y). Note that q 4 > 0 for u > 0. Take the larger C 1 .
To show the lower bound, first note that for x, y ∈ [0, 1], max{Φ(x, y),Θ(x, y)} ≦ max{x 2 , y 2 }, which leads toΦ(x, y) +Θ(x, y) ≦ 2(x + y) 2 . Let g(t) := g 1 (t) + g 2 (t), then g(λt) ≦ 2g(t) 2 . h(t) := −t −ν log g(t) satisfiesh(λt) ≧ (−(1/2) log 2 − log g(t))t −ν = −t −ν (1/2) log 2 +h(t). By induction, we haveh(λ n t) ≧ t −ν (− log 2 − log g(t)). Since − log g(t) → ∞ as t → ∞, we can take t 2 > 0 such that − log 2 − log g(t) > 1 for all t ≧ t 2 , which impliesh(λ n t) ≧ t −ν for all t ≧ t 2 . In a similar way to the proof above, we can show that for any
t ≧ t 2 ,h(t) ≧ (1/2)t −ν 2 =: C 2 , thus h i (t) = −t −ν log g i (t) ≧ −t −ν log g(t) =h(t) ≧ C 2 holds for both i = 1, 2. Let t 0 = max{t 1 , t 2 }. ✷
We now use a Tauberian theorem of exponential type. The following theorem, Corollary A.17 from [5] has a most suitable form for our purpose. If there are constants C 1 > 0, C 2 > 0 and 0 < ν < 1 such that
−C 1 ≦ lim s→∞ s −ν log g(s) ≦ lim s→∞ s −ν log g(s) ≦ −C 2 ,
then there exist C 3 > 0 and C 4 > 0 such that
−C 3 ≦ lim x→0 x ν/(1−ν) log P [[0, x]] ≦ lim x→0 x ν/(1−ν) log P [[0, x]] ≦ −C 4 , x > 0.
LetB i = (v 1 +2v 2 )B i , i = 1, 2, where v = (v 1 , v 2 ) is the positive left eigenvector corresponding to λ introduced just before Proposition 6 in Section 4. Then Proposition 11 and Theorem 12 lead to Corollary 13 There exist positive constants C 5 , C 6 , and x 0 such that
e −C 5 x − ν 1−ν ≦ P [B i ≦ x ] ≦ e −C 6 x − ν 1−ν , i = 1, 2
hold for any x ≦ x 0 .
Remark In [13], supercritical multi-type branching processes are studied and detailed results on the tail behavior of the limit processes are given, but our case does not satisfy the conditions for his results.
Proposition 14 There exist positive constants C 7 , C 8 and K such that
e −C 7 (δt −ν ) 1/(1−ν) ≦ P [ |X(t)| ≧ δ ] ≦ P [ sup 0≦s≦t |X(s)| ≧ δ ] ≦ e −C 8 (δt −ν ) 1/(1−ν) , i = 1, 2 hold for δt −ν ≧ K.
Proof. For an arbitrarily given 0 < δ < 1, take N ∈ N such that 2 −N < δ ≦ 2 −N +1 holds. Recall that if ∆ 1 , the first element of σ N (X), is of Type 1, T ex,N 1 (X) has the same distribution as that of λ −NB 1 , and if of Type 2, the same distribution as that of λ −NB 2 . For i = 1, 2 denote by A i the event that ∆ 1 is of Type i.
For the upper bound, since sup 0≦s≦t |X(s)| ≧ δ implies T ex,N ≦ e −C 6 (λ N t) − ν
1−ν ≦ e −C 8 (δt −ν ) 1/(1−ν) ,
where we assumed that λ N t ≦ x 0 in the second inequality and set C 8 = 2 −1/(1−ν) C 6 . For the lower bound, since T ex,N −1 1 < t implies |X(t)| ≧ δ, we can show that there exists a C 7 > 0 such that P [ |X(t)| ≧ δ ] ≧ e −C 7 (δt −ν ) 1/(1−ν) holds for λ N −1 t ≦ x 0 . Take K = 2x −ν 0 . ✷ Theorem 15 For any p > 0, there are positive constants C 9 and C 10 such that
C 9 ≦ lim t→0 E[|X(t)| p ] t pν ≦ lim t→0 E[|X(t)| p ] t pν ≦ C 10 .
Proof. Proposition 14 implies that the following holds for small enough t: ✷ Corollary 13 and Proposition 14 lead to a law of the iterated logarithm. Since the argument is similar to that in [3], we just give the statement below:
Theorem 16 There are positive constants C 11 and C 12 such that
C 11 ≦ lim t→0 |X(t)| ψ(t) ≦ C 12 , a.s.,
where ψ(t) = t ν (log log(1/t)) 1−ν .
Conclusion and remarks
We constructed a one-parameter family of self-avoiding walks that interpolates the SAW and the LERW on the Sierpiński gasket, and proved that the scaling limit exists. The exponent that governs the short-time behavior and equals to the reciprocal of the path Hausdorff dimension is a continuous function of the parameter. Our construction has proved that the ELLF method does work for non-Markov random walks as well as the simple random walk. Although we restricted ourselves to u ∈ [0, 1] above, all the results hold also for u > 1, that is, for self-attracting walks. By numerical calculations we observe that λ is a decreasing function of u and conjecture that as u → ∞, x * = lim u→∞ ux u , p * i = lim u→∞ p i (x u , u) and q * i = lim u→∞ q i (x u , u) exist with x * ∼ 0.351, p * 1 ∼ 0.206 , p * 2 ∼ 0.124, p * 3 ∼ 0.206, p * 4 ∼ 0.352, p * 5 ∼ 0.083, p * 6 ∼ 0, p * 7 ∼ 0.029, q * 1 ∼ 0.345, q * 2 ∼ 0.034, q * 3 ∼ 0.242, q * 4 ∼ 0.097, q * 5 ∼ 0.208, q * 7 ∼ 0.073 and q * i ∼ 0 otherwise.
Fig 1: F 3
F 3 from
3O and stopped at the first hitting time of a conditioned that the walk does not hit any vertices in G 0 \ {O} on the way. The factor (1/4) −1 comes from this conditioning.
Fig. 3
3shows all the possible loopless paths from O to a on F 1 . Here only the parts in △Oab are shown, for any path cannot go into the other triangles without making a loop.
Fig 4 :
4The loop-erasing procedure: (a) w, (
1 (
1and (2.8). The loop-erasing procedure together with the structure of the Sierpiński gasket leads to (Proposition 4 in [11]) Proposition 3 Let M ≦ N . Conditioned on σ M (LZ u N ) = (∆ 1 , . . . , ∆ k ) and the type of each element of the skeleton, the traverse times of the triangles T ex,M i (LZ u N ) − T ex,M i−1 (LZ u N ), i = 1, 2, . . . , k are independent. Each of them has the same distribution as either T ex,N −M LZ u N −M ) or T ex,N −M 1 (LZ ′u N −M ), according to whether ∆ i is of Type 1 or Type 2.
where B is the Borel algebra on Ω 0 generated by the cylinder sets. Then we have P• Y −1 N =P u N . For N ≧ M and ∆ ∈ T M , denote the the path segment of Y N in ∆ by Y N | ∆ as the continuous version of (3.1). For w ∈ ∞ N =1 Γ N and j = 1, 2, denote by S M j (w) the number of 2 −M -triangles of Type j in σ M (w), and S M (w) = (S M 1 (w), S M 2 (w)). Note that if w ∈ Γ N , then ℓ(w) = S N 1 (w) + 2S N 2 (w). Let S = (S 1 , S 2 ) and S ′ = (S ′ 1 , S ′ 2 ) be (Z + ) 2 -valued random variables on (Ω 0 , B, P ) with the same distributions as those of (S 1 1 , S 1 2 ) underP u 1 and underP ′u 1 , respectively. Proposition 4 Fix arbitrarily v ∈ Γ M , and let σ M (v) = (∆ 1 , . . . , ∆ k ). For each i, 1 ≦ i ≦ k, under the conditional probability P [ · |Y M = v], {S M +N (Y M +N | ∆ i ), N = 0, 1, 2, · · ·} is a two-type supercritical branching process, with the types of children corresponding to the types of triangles.
Proposition 6
6Fix arbitrarily v ∈ Γ M , and let σ M (v) = (∆ 1 , . . . , ∆ k ). For each i, 1 ≦ i ≦ k, under the conditional probability P [ · |Y M = v], we have the following: (1) {λ −(M +N ) S M +N (X M +N | ∆ i ), N = 0, 1, 2, . . .} converges a.s. as N → ∞ to a R 2 -valued random variable S * M,i = (S
P
[B i > 0] = 1, E[B i ] = u i , i = 1, 2.
B
1 and B 2 have strictly positive probability density functions. (5) The Laplace transform of B i , i = 1, 2
(
X N ) = T * M j . (4.3)By virtue of Proposition 5 and Proposition 6, we can prove the almost sure uniform convergence for X N . The proof here closely follows the argument of[2].
all 1 ≦ i ≦ k, where k = k M denotes the number of triangles in σ M (Y M ). Let R = T * 0 1 + ε, where ε > 0 is arbitrary. It suffices to show that X N (ω, t) converges uniformly in t ∈ [0, R]. In fact, if t > R, X N (t) =a for a large enough N . Fix M ∈ Z + arbitrarily. By expressing the arrival time at a as the sum of traversing times of 2 −M -triangles, we have T ex,M k (X N ) = T ex,0 1 (X N ) a.s.. Letting N → ∞, we have T * M k = T * 0 1 a.s.. The choice of ω implies that there exists an
X(T * M i ) = X M (T ex,M i (X M )),
Theorem 12 Assume P is a Borel probability measure supported on [0, ∞), and denote its Laplace transform by g(s) = ∞ 0 e −sξ P [dξ], s > 0.
s)| ≧ δ ] ≦ P [ T ex,N 1 (X) < t ] = P [B 1 < λ N t ] P [ A 1 ] + P [B 2 < λ N t ] P [ A 2 ]
p−1 P i [|X(t)| ≧ δ] dδ ≧ 1 Kt ν δ p−1 P i [|X(t)| ≧ δ] dδ ≧ 1 Kt ν δ p−1 e −C 7 (δt −ν ) 1/(1−ν) dδ = t e −C 7 y 1/(1−ν) dy = C 9 t pν .
dy = C 10 t pν .
AcknowledgmentsOne of the authors (K. Hattori) would like to thank Ben Hambly for suggesting the problem of an interpolation between the SAW and the LERW, and Tetsuya Hattori for helpful discussion. The authors would like to thank Wataru Asada for valuable discussion.AppendixWe show p i , q i introduced in Section 3 as explicit functions of x u and u. Define for i = 1, 2, · · · , 10where the sum i is taken over all w ∈ W 1 such that Lw = w * i and ′ i over all w ∈ V 1 such that Lw = w * i . Substituting x = x u , we haveLet U 1 be a set of single loops formed at O on F 1 :and define N (w) by (2.1). DefineWe obtain the explicit form (Θ in[3]) as follows:We show the explicit forms of p i (x, u) and q i (x, u) below. Each factor in these expressions represents a particular part of paths. The common factor 1/(1 − 2uΞ) comes from the sum over all the possible loops formed at O. In the lengthy expression of q ′ 2 (x, u), the first term is related to those paths with loops that are formed at (1/2, 0) and include b. The factor{2(u 2 − u + 1)x + 3} represents the part from the last hit at O followed immediately by a step to (1/2, 0) then to the first hit of b. The factor 1/(1 − 2uΞ) stands for the sum over all the possible loops formed at b.corresponds to the trip back from b to (1/2, 0) followed immediately by a step to (1/4, √ 3/4), andconcerns the loops formed at (1/4, √ 3/4). The second term is related to paths whose first hit to b occurs in a loop formed at (1/4, √ 3/4)....Thenq ′ 6 (x, u) = 2u(1 + u)x 8 1 +.Using MATHEMATICA, we have confirmed that as functions of x and u, the following holds:as required by the definitions ofP u 1 andP ′u 1 , where Φ(x, u) is defined in (2.3).
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| []
|
[
"Signalling entropy: a novel network-theoretical framework for systems analysis and interpretation of functional omic data",
"Signalling entropy: a novel network-theoretical framework for systems analysis and interpretation of functional omic data"
]
| [
"Andrew E Teschendorff *[email protected] \nCAS-MPG Partner Institute for Computational Biology\nChinese Academy of Sciences\nShanghai Institute for Biological Sciences\n320 Yue Yang Road200031ShanghaiChina\n\nStatistical Cancer Genomics\nUCL Cancer Institute\nPaul O'Gorman Building\n\nUniversity College London\nWC1E 6BTLondonUK\n",
"Peter Sollich \nDepartment of Mathematics\nKing's College London\nWC2R 2LSLondonUK\n",
"Reimer Kuehn \nDepartment of Mathematics\nKing's College London\nWC2R 2LSLondonUK\n"
]
| [
"CAS-MPG Partner Institute for Computational Biology\nChinese Academy of Sciences\nShanghai Institute for Biological Sciences\n320 Yue Yang Road200031ShanghaiChina",
"Statistical Cancer Genomics\nUCL Cancer Institute\nPaul O'Gorman Building",
"University College London\nWC1E 6BTLondonUK",
"Department of Mathematics\nKing's College London\nWC2R 2LSLondonUK",
"Department of Mathematics\nKing's College London\nWC2R 2LSLondonUK"
]
| []
| A key challenge in systems biology is the elucidation of the underlying principles, or fundamental laws, which determine the cellular phenotype. Understanding how these fundamental principles are altered in diseases like cancer is important for translating basic scientific knowledge into clinical advances. While significant progress is being made, with the identification of novel drug targets and treatments by means of systems biological methods, our fundamental systems level understanding of why certain treatments succeed and others fail is still lacking. We here advocate a novel methodological framework for systems analysis and interpretation of molecular omic data, which is based on statistical mechanical principles. Specifically, we propose the notion of cellular signalling entropy (or uncertainty), as a novel means of analysing and interpreting omic data, and more fundamentally, as a means of elucidating systems-level principles underlying basic biology and disease. We describe the power of signalling entropy to discriminate cells according to differentiation potential and cancer status. We further argue the case for an empirical cellular entropy-robustness correlation theorem and demonstrate its existence in cancer cell line drug sensitivity data. Specifically, we find that high signalling entropy correlates with drug resistance and further describe how entropy could be used to identify the achilles heels of cancer cells. In summary, signalling entropy is a deep and powerful concept, based on rigorous statistical mechanical principles, which, with improved data quality and coverage, will allow a much deeper understanding of the systems biological principles underlying normal and disease physiology. | 10.1016/j.ymeth.2014.03.013 | [
"https://arxiv.org/pdf/1408.1002v1.pdf"
]
| 15,084,292 | 1408.1002 | 2c53eacc0761db43fe88404bdbecd554960983f7 |
Signalling entropy: a novel network-theoretical framework for systems analysis and interpretation of functional omic data
5 Aug 2014
Andrew E Teschendorff *[email protected]
CAS-MPG Partner Institute for Computational Biology
Chinese Academy of Sciences
Shanghai Institute for Biological Sciences
320 Yue Yang Road200031ShanghaiChina
Statistical Cancer Genomics
UCL Cancer Institute
Paul O'Gorman Building
University College London
WC1E 6BTLondonUK
Peter Sollich
Department of Mathematics
King's College London
WC2R 2LSLondonUK
Reimer Kuehn
Department of Mathematics
King's College London
WC2R 2LSLondonUK
Signalling entropy: a novel network-theoretical framework for systems analysis and interpretation of functional omic data
5 Aug 2014Preprint submitted to Methods August 6, 2014entropynetworksignallinggenomicsdrug resistancecancerdifferentiationstem cell
A key challenge in systems biology is the elucidation of the underlying principles, or fundamental laws, which determine the cellular phenotype. Understanding how these fundamental principles are altered in diseases like cancer is important for translating basic scientific knowledge into clinical advances. While significant progress is being made, with the identification of novel drug targets and treatments by means of systems biological methods, our fundamental systems level understanding of why certain treatments succeed and others fail is still lacking. We here advocate a novel methodological framework for systems analysis and interpretation of molecular omic data, which is based on statistical mechanical principles. Specifically, we propose the notion of cellular signalling entropy (or uncertainty), as a novel means of analysing and interpreting omic data, and more fundamentally, as a means of elucidating systems-level principles underlying basic biology and disease. We describe the power of signalling entropy to discriminate cells according to differentiation potential and cancer status. We further argue the case for an empirical cellular entropy-robustness correlation theorem and demonstrate its existence in cancer cell line drug sensitivity data. Specifically, we find that high signalling entropy correlates with drug resistance and further describe how entropy could be used to identify the achilles heels of cancer cells. In summary, signalling entropy is a deep and powerful concept, based on rigorous statistical mechanical principles, which, with improved data quality and coverage, will allow a much deeper understanding of the systems biological principles underlying normal and disease physiology.
Introduction
Recent advances in biotechnology are allowing us to measure cellular properties at an unprecedented level of detail [1]. For instance, it is now possible to routinely measure various molecular entities (e.g. DNA methylation, mRNA and protein expression, SNPs) genome-wide in hundreds if not thousands of cellular specimens [2]. In addition, other molecular data detailing interactions between proteins or between transcription factors and regulatory DNA elements are growing at a rapid pace [1]. All these types of data are now widely referred to collectively as "omic" data. The complexity and high-dimensional nature of this omic data presents a daunting challenge to those wishing to analyse and interpret the data [1]. The difficulty of analysing omic data is further compounded by the inherent complexity of cellular systems. Cells are prime examples of organized complex systems, capable of highly stable and predictable behaviour, yet an understanding of how this deterministic behaviour emerges from what is a highly complex and probabilistic pattern of dynamic interactions between numerous intra and extracellular components, still eludes us [1]. Thus, elucidating the systemsbiological laws or principles dictating cellular phenotypes is also key for an improved analysis and interpretation of omic data. Furthermore, important biological phenomena such as cellular differentiation are fundamentally altered in diseases like cancer [3]. Hence, an attempt to understand how cellular properties emerge at a systems level from the properties seen at the individual gene level is not only a key endeavour for the systems biology community, but also for those wanting to translate basic insights into effective medical advances [4,5]. It is now well accepted that at a fundamental level most biological systems are best modeled in terms of spatial interactions between specific entities (e.g. neurons in the case of the brain), which may or may not be dynamically changing in time [6,7]. It therefore seems natural to also use the mathematical and physical framework of networks to help us analyse and in-terpret omic data at a systems level [8,9]. Indeed, the cellular phenotype is determined to a large extent by the precise pattern of molecular interactions taking place in the cell, i.e. a molecular interaction network [10]. Although this network has spatial and dynamic dimensions which at present remain largely unexplored due to technological or logistical limitations, there is already a growing number of examples where network-based analysis strategies have been instrumental [11,12,13]. For instance, a deeper understanding of why sustained treatment with EGFR inhibitors can lead to dramatic sensitization of cancer cell lines to cytotoxic agents was possible thanks to a systems approach [13]. Another study used reverse engineering network approaches to identify and validate drug targets in glioblastoma multiforme, to be further tested in clinical trials [14]. What is key to appreciate here is that these successes have been achieved in spite of noisy and incomplete data, suggesting that there are simple yet deep systems biological principles underlying cellular biology that we can already probe and exploit with current technology and data. Thus, with future improvements in data quality and coverage, network-based analysis frameworks will play an ever increasing and important role in systems biology, specially at the level of systems analysis and interpretation [4]. Therefore, it is also imperative to develop novel, more powerful, network-theoretical methods for the systems analysis of omic data. In adopting a network's perspective for the analysis and interpretation of omic data, there are in principle two different (but not mutually exclusive) approaches one can take. One possibility is to infer (i.e. reverse engineer) the networks from genome-wide data [15]. Most of these applications have done this in the context of gene expression data, with the earliest approaches using clustering or co-expression to postulate gene interdependencies [16]. Partial correlations and Graphical Gaussian Models have proved useful as a means of further refining correlation networks by allowing one to infer the more likely direct interactions while simultaneously also filtering out those which are more likely to be indirect [17]. These methods remain popular and continue to be studied and improved upon [18,19]. Other methods have drawn on advanced concepts from information theory, for instance ARACNe ("Algorithm for the Reconstruction of Accurate Cellular Networks") has been shown to be successful in infering regulatory networks in B-cells [15]. In stark contrast to reverse engineering methods, another class of algorithms have used structural biological networks from the outset, using these as scaffolds to integrate with omic data. Specifically, by using a structural network one can sparsify the correlation networks inferred from reverse-engineering approaches, thus providing another means of filtering out correlations that are more likely to be indirect [9]. Besides, integration with a structural network automatically provides an improved framework for biological interpretation [10,20,21,22]. The structural networks themselves are typically derived from large databases, which detail literature curated experimentally verified interactions, including interactions derived from Yeast 2 Hybrid screens (Y2H) [23]. The main example is that of protein protein interaction (PPI) maps, which have been generated using a number of different complementary experimental and in-silico approaches, and merging these maps from these different sources together has been shown to be an effective approach in generating more comprehensive high-confidence interaction networks [24,25]. PPI networks have been used mainly as a means of integrating and analysing gene expression data (see e.g. [20,26,27]). More recently, this approach has also been successfully applied in the DNA methylation context, for instance it has been shown that epigenetic changes associated with age often target specific gene modules and signalling pathways [28]. Another class of methods that have used structural networks, PPIs in particular, have integrated them with gene expression data to define an approximation to the underlying signaling dynamics on the network, thus allowing more in-depth exploration of the interplay between network topology and gene expression. Typically, these studies have used the notion of random walks on weighted graphs where the weights are constructed from differential expression statistics, and where the aim is to identify nodes (genes) in the network which may be important in dictating the signaling flows within the pathological state. For instance, among these random walk methods is Ne-tRank, a modification of the Google PageRank algorithm, which was able to identify novel, robust, network based biomarkers for survival time in various cancers [29,30]. Other random walk based approaches, aimed at identifying causal drivers of specific phenotypes (e.g. expression or cancer), have modeled signal transduction between genes in the causal and phenotypic layers as flows in an electric circuit diagram, an elegant formulation capable of not only identifying the likely causal genes but also of tracing the key pathways of information flow or dysregulation [31,32]. Random walk theory has also been employed in the development of differential network methodologies. An example is NetWalk [33], which is similar to NetRank but allows differential signaling fluxes to be inferred. This approach was successful in identifying and validating the glucose metabolic pathway as a key determinant of lapatinib resistance in ERBB2 positive breast cancer patients [34].
Another important concept to have emerged recently is that of network rewiring [35,36,37]. This refers to the changes in interaction patterns that accompany changes in the cellular phenotype. Network rewiring embodies the concept that it is the changes in the interaction patterns, and not just the changes in absolute gene expression or protein activity, that are the main determinants of the cellular phenotype. That network rewiring may be key to understanding cellular phenotypes was most convincingly demonstrated in a differential epistasis mapping study conducted in yeast cells exposed to a DNA damaging agent [35]. Specifically, what this study demonstrated is that responses to perturbations or cellular stresses are best understood in terms of the specific rewiring of protein complexes and functional modules. Thus, this conceptual framework of network rewiring may apply equally well to the genetic perturbations and cellular stresses underlying disease pathologies like cancer. In this article we advocate a network-theoretical framework based on statistical mechanical principles and more specifically on the notion of signalling entropy [8,9]. This theoretical framework integrates gene expression (but in principle also other functional data) with a PPI network, merging existing concepts such as signaling dynamics (i.e. random walks) and network rewiring with that of signalling entropy. In previous work we have shown how signalling entropy (i) provides a proxy to the elevation in Waddington's epigenetic landscape, correlating with a cell's differentiation potential [38], (ii) how it can be used to identify signaling pathways and nodes important in differentiation and cancer [38,8,9], and (iii) how it predicts two cancer system-omic hallmarks: (a) cancer is characterised by an increase in signalling entropy and (b) local signaling entropy changes anti-correlate with differential gene expression [8,9]. Here, we present and unify the different signaling entropy measures used previously and further explore a novel application of signalling entropy to understanding drug sensitivity profiles in cancer cell lines. Specifically, we first use simulated data to justify the existence of an entropy-robustness theorem, and subsequently provide empirical evidence for this theorem by demonstrating that increases in local signalling entropy correlate with drug resistance (robustness). We further show the importance of network topology in dictating the signalling entropy changes underlying drug response. In addition, we provide R-functions implementing the entropy rate calculation and ranking of genes according to differential entropy, all freely available from sourceforge.net/projects/signalentropy/files/ .
Materials and Methods
2.1. Basic rationale and motivation for Signalling Entropy: understanding systems biology through uncertainty Loosely defined, entropy of a system, refers to a measure of the disorder, randomness or uncertainty of processes underlying the system's state. In the context of a single cell, signalling entropy will reflect the amount of overall disorder, randomness or uncertainty in how information, i.e. signaling, is passed on in the molecular interaction network. At a fundamental level, all signaling in a cell is probabilistic, determined in part by the relative cellular concentrations of the interacting molecules. Hence, in discussing cellular signalling entropy, it is useful to picture a molecular interaction network in which edges represent possible interactions and with the edge weights reflecting the relative probabilities of interaction (Fig.1A). Thus, an interaction network in a high-entropy state is characterised by signaling interaction probabilities that are all fairly similar in value, whereas a low-entropy state will be characterised by specific signalling interactions possessing much higher weights (Fig.1A).
Why would this type of signalling entropy, loosely defined as the amount of uncertainty in the signaling interaction patterns, be useful to systems biology? One way to think of signalling entropy is as representing signaling promiscuity, which has been proposed as a key systems feature underlying the pluripotent or multipotent capacity of cells (Fig.1B) [39,40,41,42]. Indeed, it has been suggested that pluripotency is an intrinsic statistical mechanical property, best defined at the cellular population level [40]. Specifically, it has been demonstrated that pluripotent stem cells exhibit remarkable heterogeneity in gene expression levels, including well known stem cell markers such as NANOG [40]. It is also well known that a large number of genes, many encoding transcription factors, exhibit low-levels of expression in stem cells, yet simultaneously are being kept in a poised chromatin state, allowing immediate activation if this were required [43]. Thus, in a pluripotent stem cell like state, signal transduction is in a highly egalitarian and, thus, promiscuous state, i.e. a state of high signalling entropy. Conversely, differentiation leads, by necessity, to activation of specific transcription factors and pathways and thus to a lowering in the uncertainty of signaling patterns, and thus to a lowering of entropy. We recently demonstrated, using gene expression profiles of over 800 samples, comprising cells of all major stages of differentiation, including human embryonic stem cells (hESCs), induced pluripotent stem cells (iPSCs), multipotent cell types (e.g. hematopoietic stem cells (HSCs)), and terminally differentiated cells within these respective lineages, that signalling entropy not only correlates with differentiation potential but that it provides a highly quantitative measure of potency [38]. Indeed, we showed that signalling entropy provides a reasonably good approximation to the energy potential in Waddington's epigenetic landscape [38].
Here we decided to explore the concept of signalling entropy in relation to cellular robustness and specifically to drug resistance in cancer. That signalling entropy may be informative of a cell's robustness is a proposal that stems from a general (but unproven) theorem, first proposed by Manke and Demetrius [44,45,46]: namely, that a system's entropy and robustness are correlated. Mathematically, this can be expressed as ∆S∆R > 0, which states that a positive change in a system's entropy (i.e. ∆S > 0) must lead to an increase in robustness (∆R > 0). Now, cells are fairly robust entities, having evolved the capacity to buffer the intra-and-extracellular stresses and noise which they are constantly exposed to [47,39]. Much of this overall stability and robustness likely stems from the topological features of the underlying signaling and regulatory networks, for instance features such as scale-freeness and hierarchical modularity, which are thought to have emerged through natural evolutionary processes such as gene duplication [10,48,49]. However, another key feature which contributes to cellular robustness is cross-talk and signalling pathway redundancy [47]. Pathway redundancy refers to a situation where a cell has the choice of transmitting signals via two or more possible routes. In the language of statistical mechanics, this corresponds to a state of high uncertainty (or entropy) in signaling. High signalling entropy could thus underpin a cell's robustness to perturbations, suggesting that a cell's entropy and robustness may indeed be correlated ( Fig.2A). Consistent with this, pathway redundancy is also well recognized to be a key feature underlying drug resistance of cancer cells [50].
Further supporting the notion that entropy and robustness may be correlated, we previously showed that (i) cancer is characterised by a global increase in signalling entropy compared to its respective normal tissue [9], in line with the observation that cancer cells are specially robust to generic perturbations, and (ii) that a gene's differential entropy between normal and cancer tissue is generally speaking anticorrelated with its differential expression [9], consistent with the view that cancer cells are specially sensitive to drug interventions that target overexpressed oncogenes, a phenomenon known as oncogene addiction (Fig.2B) [3]. Interestingly, the observations that differential entropy and differential expression are anti-correlated and that cancer is characterised globally by an increase in entropy [9], are also consistent with the prevalent view that most driver mutations are indeed inactivating, targeting tumor suppressor genes [51,52,53]. Hence, based on all of these observations and insights, we posited that signalling entropy could also prove useful as a means of predicting drug resistance in cancer cells.
The Signalling Entropy Method: definitions and construction
Briefly, we review the definitions and construction of signalling entropy as used in our previous studies [8,9,38]. The construction relies on a comprehensive and high-confidence PPI network which is integrated with gene expression data [24,9] (see Appendix A). Briefly, the PPI is used as a scaffold, and edge weights are constructed from the gene expression data to approximate the interaction or signaling probabilities between the corresponding proteins in the PPI. Thus, the gene expression data provides the biological context in which to modulate the PPI interaction probabilities. To compute signalling entropy requires the estimation of a stochastic matrix, reflecting these interaction probablities over the network. The construction of the stochastic matrix can proceed in two different ways. In the earliest studies we used a construction which was done at the level of phenotypes [8,9]. Under this model, edge weights w ij between proteins i and j were constructed from the correlation between the expression levels of the corresponding genes i and j, as assessed over independent samples all representing the same phenotype. Estimating the correlations over independent samples, all within the same phenotype, can be viewed as representing a multifactorial perturbation experiment, with e.g. genetic differences between individuals mimicking specific perturbations, and thus allowing influences to be inferred. Thus, this approach hinges on the assumption that highly correlated genes, whose coding proteins interact, are more likely to be interacting in the given phenotype than two genes which do not correlate. The use of a PPI as a scaffold is important to filter out significant correlations which only result from indirect influences. The correlations themselves can be defined in many different ways, for instance, using simple Pearson correlations or nonlinear measures such as Mutual Information. For example, one way to define the weights is as w ij = 1 2 (1 + c ij ) with c ij describing the Pearson correlation coefficient between genes i and j. This definition guarantees positivity, but also treats positive and negative correlations differently, which makes biological sense because activating and inhibitory interactions normally have completely distinct consequences on downstream signalling. Thus, the above weight scheme treats zero or insignificant correlations as intermediate, consistent with the view that an absent interaction is neither inhibitory nor activating. However, other choices of weights are possible: e.g. w ij = |c ij |, which treats negative and positive correlations on an equal footing. Once edge weights are defined as above, these are then normalised to define the stochastic matrix p ij over the network,
p ij = w ij k∈N i w ik ,
with N i denoting the PPI neighbors of gene i. Thus, p ij is the probability of interaction between genes i and j, and as required, j p ij = 1. However, there is no requirement for p ij to be doubly stochastic, i.e. P is in general not a symmetric matrix. Hence, edges are bi-directional with different weights labeling the probability of signal transduction from i to j, and that from j to i (p ij = p ji ). An alternative to the above construction of the stochastic matrix is to invoke the mass action principle, i.e. one now assumes that the probability of interaction in a given sample is proportional to the product of expression values of the corresponding genes in that sample [38]. Thus, the PPI is again used as a scaffold to only allow interactions supported by the PPI network, but the weights are defined using the mass action principle, as
w ij ∝ E i E j
where E i denotes the normalised expression intensity value (or normalised RNA-Seq read count) of gene i. An important advantage of this construction is that the stochastic matrix is now sample specific, as the expression values are unique to each sample. Given a stochastic matrix, p ij , constructed using one of the two methods above, one can now define a local Shannon entropy for each gene i as
S i = − 1 log k i k∈N i p ik log p ik ,
where k i denotes the degree of gene i in the PPI network. The normalisation is optional but ensures that this local Shannon entropy is normalised between 0 and 1. Clearly, if only one weight is non-zero, then the entropy attains its minimal value (0), representing a state of determinism or lowest uncertainty. Conversely, if all edges emanating from i carry the same weight, the entropy is maximal (1), representing the case of highly promiscuous signaling. In principle, local Shannon entropies can thus be compared between phenotypes to identify genes where there are changes in the uncertainty of signaling. In the case where entropies are estimated at the phenotype level, jackknife approaches can be used to derive corresponding differential entropy statistics [9]. Deriving statistics is important because node degree has a dramatic influence on the entropy variance, with high degree nodes exhibiting significantly lower variability in absolute terms, which nevertheless could be highly significant [9]. In the case where entropies are estimated at the level of individual samples, ordinary statistical tests (e.g. rank sum tests) can be used to derive sensible P-values, assuming of course that enough samples exist within the phenotypes being compared.
In addition to the local entropy, it is also of interest to consider statistical properties of the distribution of local entropies, for instance their average. Comparing the average of local entropies between two phenotypes would correspond to a comparison of non-equilibrium entropy rates. To see this, consider the formal definition of the entropy rate SR [54,55], i.e.
SR = n i=1 π i S i ,
where π i is the stationary distribution (equivalently the left eigenvector with unit eigenvalue) of P (i.e. πP = π), and where now
S i = − k∈N i p ik log p ik .
Note that the entropy rate SR is an equilibrium entropy since it involves the stationary distribution of the random walker. As such, the entropy rate also depends on the global topology of the network. Thus, the entropy rate is a weighted average of the local unnormalized entropies, with the weights specified by the stationary distribution. It follows that comparing the unweighted averages of local entropies reflects a comparison of a non-equilibrium entropy rate since the stationary distribution is never used.
2.3. The importance of the integrated weighted network in estimating signalling entropy The entropy rate constructed using the mass action principle is sample specific. We previously demonstrated that this entropy rate was highly discriminative of the differentiation potential of samples within a developmental lineage, as well as being highly discriminative of normal and cancer tissue [38] (Fig.3A). Since the entropy rate takes as input a PPI network and a sample's genome-wide expression profile, the significance of the resulting entropy values, as well as that of the difference between entropy rates, also needs to be assessed relative to a null distribution in which the putative information content between network and gene expression values is non-existent. In fact, since the weights in the network determine the entropy rate and these weights are dependent on both the specific network nodes and their respective gene expression profiles, it is natural to assess the significance of the entropy rate by "destroying" the mutual information between the network nodes and their gene expression profiles, for instance by randomising (i.e. permuting) the gene expression profiles over the network. Thus, under this randomisation, the topological properties of the network remain fixed, but the weights are redefined. Application of this randomisation procedure to the normal/cancer expression set considered previously [38] (Fig.3A) shows that the discriminatory potential is significantly reduced upon permuting the gene expression values over the network (Fig.3B-C). Importantly, we observe that the entropy rate is much higher in the normal and cancer states compared to the rates obtained upon randomisation of the gene expression profiles (Fig.3B), indicating that both normal and cancer states are characterised by a higher level of signaling promiscuity compared to a network with random weights. That the discrimination between normal and cancer is significantly reduced in the randomly weighted network further demonstrates that there is substantial mutual information between the PPI network and the gene expression profiles, thus justifying the signalling entropy approach.
Signalling entropy R-package: input, output and code availability
A vignette/manual and user-friendly R-scripts that allow computation of the entropy rate is available at the following url: sourceforge.net/projects/signalentropy/files/.
Here we briefly describe the salient aspects of this package: The input: The main R-script provided (CompSR) takes as input a userspecified PPI network, and a genome-wide expression vector representing the gene expression profile of a sample. It is assumed that this has been generated using either Affy, Illumina or RNA-Sequencing. In principle one ought to use as gene expression value the normalised unlogged intensity (Affy/Illu) or RNA-seq count, since this is what should be proportional to the number of RNA transcripts in the sample. However, in practice we advise taking the log-transformed normalised value since the log-transformation provides a better compromise between proportionality and regularisation, i.e. some regularisation is advisable since the underlying kinetic reactions are also regular. The output: The R-functions provided in the package then allow the user to estimate the global entropy rate for a given sample, as well as the local normalised entropies for each gene in the integrated network. If a phenotype is specified then genes can be ranked according to the correlation strength of their local entropy to the phenotype. Thus, the signalling entropy method allows us to assess (i) if the overall levels of signalling promiscuity is different between phenotypes, which could be important, for instance, to compare the pluripotent capacity of iPSCs generated via different protocols or to assess the stem cell nature of putative cancer stem cells [38], and (ii) to rank genes according to differential entropy between phenotypes, allowing key signalling genes associated with differentiation, metastasis or cancer to be identified [8,9,38].
Results
Signalling entropy and cellular robustness
Our previous observation that signalling entropy is increased in cancer [9], and that one of the key characteristics of cancer cells is their robustness to intervention and environmental stresses, suggested to us that high cellular signalling entropy may be a defining feature of a cell that is robust to general perturbations. In addition, cancer cells often exhibit the phenomenon of oncogene addiction, whereby they become overly reliant on the activation of a specific oncogenic pathway, rendering them less robust to targeted intervention [3]. Since oncogenes are normally overexpressed in cancer, it follows by the second cancer system-omic hallmark [9], that their lower signalling entropy may underpin their increased sensitivity to targeted drugs (Fig.2). Based on these insights, we posited that cellular signalling entropy may be correlated to the overall cellular system's robustness to perturbations. In order to explore the relation between signalling entropy and robustness in a general context, it is useful to consider another network property of the stochastic matrix, namely the global mixing rate. This mixing rate is defined formally as the inverse of the number of timesteps for which one has to evolve a random walk on the graph so that its probability distribution is close to the stationary distribution, independently of the starting node (Appendix B). This is a notion that is closer to that of robustness or resilience as defined by Demetrius and Manke [44,45,46], allowing the mixing rate to be viewed as a crude proxy of a system's overall robustness to generic perturbations. Furthermore, the global mixing rate can be easily estimated as
µ R = − log SLEM(1)
where SLEM is the second largest (right) eigenvalue modulus of the stochastic matrix (Appendix B). Thus, we first asked how this global mixing rate is related to the signalling entropy rate.
For a regular network of degree d it can be easily shown that the entropy rate SR = log d, whilst results on graph theory also show that for sufficiently large regular graphs, µ R ∝ log d [56]. Hence, at least for regular networks a direct correlation exists. It follows that for non-regular graphs with tight degree distributions, e.g. Erdös-Renyi (ER) graphs, the entropy and mixing rates should also be approximately correlated. Indeed, using an empirical computational approach to evaluate the entropy and mixing rates for ER graphs with variable entropy rates, we were able to confirm this correlation (Appendix C, Fig.4A). Next, we wanted to investigate if this relationship also holds for networks with more realistic topologies than ER graphs. Hence, we generated connected networks on the order of 500 nodes by random subsampling 1000 nodes from our large protein interaction network (∼ 8000 nodes) followed by extraction of the maximally connected component (Appendix D). We verified that these networks possessed approximate scale-free topologies with clustering coefficients which were also significantly higher than for ER graphs. As before, for each generated network, stochastic matrices of variable entropy rates were defined. Signalling entropy and mixing rates were then estimated for each of these networks, and subsequently averaged over an ensemble of such graphs. As with the random Poisson (ER) graphs, average mixing and entropy rates were highly correlated (Fig.4B). Having demonstrated a direct correlation between these two measures on simulated data, we next asked if such a correlation could also be present in the full protein interaction networks and with the stochastic matrices derived from real expression data. Thus, we computed the global entropy and mixing rates for 488 cancer cell lines from the Cancer Cell Line Encylopedia (CCLE) (Appendix E) [57]. Remarkably, despite the dynamic ranges of both entropy and mixing rates being much smaller (Fig.4C) compared to those of the artificially induced stochastic matrices (c.f Fig.4A-B), we were still able to detect a statistically significant correlation between entropy and mixing rates, as estimated across the 488 cell lines (Fig.4C). Thus, all these results support the view that global entropy and mixing rates are correlated, albeit only in an average/ensemble sense.
Local signalling entropy predicts drug sensitivity
In the case of realistic expression and PPI data, the observed correlation between entropy and mixing rates was statistically significant but weak (Fig.4C). This could be easily attributed to the fact that in real biological networks, the global mixing rate is a very poor measure of cellular robustness. In fact, it is well known that cellular robustness is highly sensitive to which genes undergo the perturbation. For instance, in mice some genes are embryonically lethal, whereas others are not [10]. Robustness itself also admits many different definitions. Because of this, we decided to investigate signalling entropy in relation to other more objective measures of cellular robustness. One such measure is drug sensitivity, for instance, IC50 values, which measure the amount of drug dose required to inhibit cell proliferation by 50%. According to this measure, a cell that is insensitive to drug treatment is highly robust to that particular treatment. Since most drugs target specific genes, we decided to explore the relation, if any, between the local signalling entropy of drug targets and their associated drug sensitivity measures. Specifically, we hypothesized that since local entropy provides a proxy for local pathway redundancy, that it would correlate with drug resistance. To test this, we first computed for each of the 8038 genes in the PPI network its local signalling entropy in each of the 488 CCLE cancer cell-lines. To identify associations between the 24 drug sensitivity profiles and the 8038 local network entropies, we computed non-linear rank (Spearman) correlation coefficients across the 488 cancer cell-lines, resulting in 24 × 8038 correlations and associated P-values. We observed that there were many more significant associations than could be accounted for by random chance (Fig.5A), with the overall strength of association very similar to that seen between gene expression and drug sensitivity (Fig.5B).
One would expect the targets of specific drugs to be highly informative of the sensitivity of the corresponding drugs. We used the CancerResource [58] to identify targets of the 24 drugs considered here, and found a total of 154 drug-target pairs. For 134 of these pairs we could compute a P-value of association between local entropy and drug sensitivity with 76 pairs (i.e. 57%) exhibiting a significant association (Fig.5C). This was similar to the proportion (54%) of observed significant associations between gene expression and drug sensitivity (Fig.5D). However, interestingly, only 42 of these significant drug-target pairs were in common between the 76 obtained using signalling entropy and the 72 obtained using gene expression. Importantly, while the significant associations between gene expression and drug sensitivity involved preferentially positive correlations, in the case of signalling entropy most of the significant correlations were negative (Fig.5C-D), exactly in line with our hypothesis that high entropy implies drug resistance. Thus, as expected, cell-lines with highly expressed drug targets were more sensitive to treatment by the corresponding drug, but equally, drug targets exhibiting a relatively low signalling entropy were also predictive of drug sensitivity. To formally demonstrate that local signalling entropy adds predictive power over gene expression, we considered bi-variate regression models including both the target's gene expression as well as the target's signalling entropy. Using such bivariate models and likelihood ratio tests we found that in the majority of cases where signalling entropy was significantly associated with drug sensitivity that it did so independently of gene expression, adding predictive value (Fig.5E). Top ranked drug-target pairs where signalling entropy added most predictive value over gene expression included Topotecan/TP53 and Paclitaxel/MYC (Fig.5F). To further demonstrate that the observed associations between local signalling entropy and drug sensitivity are statistically meaningful, we conducted a control test, in which we replaced in the multivariate model the signalling entropy of the target with a non-local entropy computed by randomly replacing the PPI neighbours of the target with other "far-away" genes in the network. For a considerable fraction (41%) of drug-target pairs, the original multivariate models including the local entropy constituted better predictive models than those with the non-local entropy (false discovery rate < 0.25), indicating that the observed associations are driven, at least partly, by the network structure.
High signalling entropy of intra-cellular signaling hubs is a hallmark of
drug resistance Among the drug-target pairs for which signalling entropy was a significant predictor of drug sensitivity, we observed a striking non-linear association with the topological degree of the targets in the network (Fig.6A). In fact, for hubs in the network, most of which encode nodes located in the intracellular signaling hierarchy, high signalling entropy was exclusively associated with drug resistance (negative SCC). Examples included well-known intracellular signalling hubs like HDAC1, HDAC2, AKT1, TP53, STAT3, MYC). Some intracellular non-hubs (e.g. CASP9, BCL2L1, BIRC3) also exhibited negative correlations between signalling entropy and drug sensitivity. Among targets for which high signalling entropy predicted drug sensitivity, we observed several membrane receptors (e.g ERBB2, ERBB3, EGFR, MET) and growth factors (e.g HBEGF, EGF, TGFA). Given that the correlation coefficients were estimated across samples (cell-lines) and that the underlying network topology is unchanged between samples, the observed non-linear relation between the directionality of the correlation and node degree is a highly non-trivial finding. We also observed a clear dependence on the main signaling domain of the target, with intracellular hubs showing preferential anticorrelations, in contrast to growth factors and membrane receptors which exhibited positive and negative correlations in equal proportion (Fig.6B). Thus, we can conclude from these analyses that cancer associated changes to the interaction patterns of intra-cellular hubs are key determinants of drug resistance. In particular, in cancers where the local entropy at these hubs is increased, as a result of increased promiscuous signaling, drugs targeting these hubs are less likely to be effective.
Discussion and Conclusions
Here we have advocated a fairly novel methodological framework, based on the notion of signalling entropy, to help analyze and interpret functional omic data sets. The method uses a structural network, i.e. a PPI network, from the outset, and integrates this with gene expression data, using local and global signalling entropy measures to estimate the amount of uncertainty in the network signaling patterns. We made the case as to why uncertainty or entropy might be a key concept to consider when analysing biological data. In previous work [9,38], we showed how signalling entropy can be used to estimate the differentiation potential of a cellular sample in the context of normal differentiation processes, as well as demonstrating that signalling entropy also provides a highly accurate discriminator of cancer phenotypes. In this study we focused on a novel application of signalling entropy to understanding cellular robustness in the context of cancer drug sensitivity screens. Our main findings are that (i) local signalling entropy measures add predictive value over models that only use gene expression, (ii) that the local entropy of drug targets generally correlates positively with drug resistance, and (iii) that increased local entropy of intra-cellular hubs in cancer cells is a key hallmark of drug resistance. These results are consistent and suggestive of an underlying entropy-robustness correlation theorem, as envisaged by previous authors [44]. Here, we provided additional empirical justification for such a theorem, using both simulated as well as real data, and using drug sensitivity measures as proxies for local robustness measures. A more detailed theoretical analysis of local mixing and entropy rates and incorporation of additional information (e.g. phosphorylation states of kinases, protein expression,..etc) when estimating entropies on real data, will undoubtedly lead to further improvements in our systemslevel understanding of how entropy/uncertainty dictates cellular phenotypes. From a practical perspective, we have already shown in previous work [38] how local network entropies could be used to identify key signaling pathways in differentiation. It will therefore be interesting in future to apply the signalling entropy framework in more detail to identify specific signaling nodes/pathways underlying drug sensitivity/resistance. Our results have also confirmed the importance of network topology (e.g. hubness and therefore scale-freeness) in dictating drug resistance patterns. Thus, it will be interesting to continue to explore the deep relation between topological features such as scale-freeness and hierarchical modularity in relation to the gene expression patterns seen in normal and disease physiology. It is entirely plausible that, although our current data and network models are only mere caricatures of the real biological networks, that underlying fundamental systems biology principles characterising cellular phenotypes can still be gleaned from this data. Indeed, our discovery that increased signalling entropy correlates with drug resistance demonstrates that such fundamental principles can already be inferred from existing data resources. It will also be of interest to consider other potential applications of the signaling entropy method. For instance, one application could be to the identification of functional driver aberrations in cancer. This would first use epigenomic (e.g. DNA methylation and histone modification profiles) and genomic information (SNPs, CNVs) together with matched gene or protein expression data to identify functional epigenetic/genetic aberrations in cancer. Signalling entropy would subsequently be used as a means of identifying those aberrations which also cause a fundamental rewiring of the network. With multi-dimensional matched omic data readily available from the TCGA/ICGC, this represents another potentially important application of the signalling entropy method. Another important future application of the signaling entropy method would be to single-cell data, which is poised to become ever more important [59]. So far, all signaling entropy analyses have been performed on cell populations, yet single-cell analysis will be necessary to disentangle the entropies at the single-cell and population-cell levels.
In summary, we envisage that signalling entropy will become a key concept in future analyses and interpretation of biological data.
Appendix A. The protein protein interaction (PPI) network
We downloaded the complete human protein interaction network from Pathway Commons (www.pathwaycommons.org) (Jun.2012) [24], which brings together protein interactions from several distinct sources. We built a protein protein interaction (PPI) network from integrating the following sources: the Human Protein Reference Database (HPRD) [23], the National Cancer Institute Nature Pathway Interaction Database (NCI-PID) (pid.nci.nih.gov), the Interactome (Intact) http://www.ebi.ac.uk/intact/ and the Molecular Interaction Database (MINT) http://mint.bio.uniroma2.it/mint/. Protein interactions in this network include physical stable interactions such as those defining protein complexes, as well as transient interactions such as posttranslational modifications and enzymatic reactions found in signal transduction pathways, including 20 highly curated immune and cancer signaling pathways from NetPath (www.netpath.org) [60]. We focused on nonredundant interactions, only included nodes with an Entrez gene ID annotation and focused on the maximally conntected component, resulting in a connected network of 10,720 nodes (unique Entrez IDs) and 152,889 documented interactions.
Appendix B. Mixing rate results for a general random walk on a connected graph
Suppose that we have a connected graph with an ergodic Markov chain defined on it, given by a stochastic matrix P with stationary distribution π (i.e. πP = π). We further assume that the detailed balance equation, π i p ij = π j p ji holds. Defining the diagonal matrix Π ≡ diag(π 1 , ..., π N ), the detailed balance equation can be rewritten as ΠP = P T Π. That the matrix P is stochastic means that each row of P sums to 1. Equivalently, the vector with all unit entries, 1, is a right eigenvector of P , i.e. P 1 = 1. Note also that the stationary distribution π is a left eigenvector of P with eigenvalue 1. The Perron-Frobenius theorem further implies that all other eigenvalues are less than 1 in magnitude. If detailed balance holds, all eigenvalues are also real. The global mixing rate can be defined by considering the rate at which the node visitation probabilities of a random walker approaches that of the stationary distribution, independently of the starting position. Formally, if we let Q i (t) denote the probability that at time t we find the walker at node i, then the mixing rate, µ R , is defined by [61]
µ R = lim t→∞ sup max i |Q i (t) − π i | 1/t .
Denoting by Q(t) the column vector with elements Q i (t), one can write
Q(t) = (P t ) T Q(0).
To determine the elements, Q i (t), of this vector, it is convenient to first introduce the matrix M = Π 1 2 P Π − 1 2 . This is because M t = Π 1 2 P t Π − 1 2 , and so P t can be rewritten in terms of M , but also because M satisfies the following lemma: Proof of Lemma 1. Suppose that M u a = λ a u a . It then follows that
P (Π − 1 2 u a ) = λ a (Π − 1 2 u a ).
In the case of the left-eigenvector, multiply M u a = λ a u a from the left with Π Detailed balance implies that ΠP = P T Π, so P T l a = λ a l a . Taking the transpose of this implies that l a is indeed a left-eigenvector of P .
The significance of the above lemma becomes clear in light of the detailed balance equation, which implies that M = M T , and so M and M t can be orthogonally diagonalized. In particular, we can express Q i (t) in terms of the eigenvalue decomposition of M t , as
Q i (t) = a q a |λ a | tũ ai π 1/2 i , where q a = jũ aj π −1/2 j Q j (0)
and whereũ a is the a'th unit norm eigenvector of M . Since Π 1 2 1 is the top unit norm eigenvector of M (using previous lemma), which has an eigenvalue of 1, it follows that q 1 = 1 and hence that
Q i (t) = π i + a≥2 q a |λ a | tũ ai π 1/2 i . It follows that |Q i (t) − π i | = a≥2 q a |λ a | tũ ai π 1/2 i .
Since 1 = |λ 1 | ≥ |λ 2 | ≥ |λ 3 | . . ., we can conclude that as t → ∞, the rate at which Q i approaches the stationary value π i is determined by the modulus of the second largest eigenvalue (the Second Largest Eigenvalue Modulus-SLEM). The global mixing rate µ R can thus be estimated as
µ R ≈ − log |λ 2 | = − log SLEM .
Appendix C. Entropy and mixing rates in simulated weighted Erdös-Renyi graphs
For large regular graphs of degree d, the mixing rate is proportional to log d [56] and thus directly proportional to the entropy rate (SR = log d for a regular graph of degree d). By extrapolation, we can thus reasonably assume that for any sufficiently large graph with a tight degree distribution, such as random Erdos-Renyi (ER) graphs, that the entropy and mixing rates will also be correlated, albeit perhaps only in an average ensemble sense. The analytical demonstration of this is beyond the scope of this work. Hence, we adopt a computational empirical approach to see if the entropy and mixing rates may indeed be correlated. In detail (and without loss of generality concerning the end result) we considered ER graphs of size 100 nodes and average degree 10 (other values for these parameters gave similar answers). We built an ensemble of 100 distinct ER graphs, and for each of these we constructed a family of weighted networks, parameterised by a parameter which controls the level of signalling entropy. Specifically, we gradually shifted the weight distribution around each node i (with degree d i ), in such a way that p ij = /d i for j = k and p ij = 1 − d i (d i − 1) for j = k, with 0 ≤ ≤ 1 and with k labeling a randomly chosen neighbor of node i. Thus, is a parameter that directly controls the uncertainty in the information flow, i.e. the entropy rate, since for = 1 we have that p ij = A ij /d i (A ij is the symmetric adjacency matrix of the graph), whilst for = 0, p ij = δ ik , i.e. the information flow from node i can only proceed along one node (node k). Thus, one would expect the entropy rate to decrease as is decreased to zero. For each value of we thus have an ensemble of 100 ER-graphs, for each of which the entropy and mixing rates can be computed. Finally, at each value of the entropy and mixing rates are averaged over the ensemble.
Appendix D. Entropy and mixing rates in simulated weighted subgraphs of a PPI network
The analysis described above was performed also for maximally connected subnetworks generated from the underlying realistic PPI network described earlier. Specifically, we randomly subsampled 1000 nodes from the 8038 node PPI network, and extracted the maximally connected subnetwork, which resulted (on average) in a subnetwork of approximately 500 nodes. A family of stochastic matrices of variable entropy rates were constructed as explained above and for each resulting weighted network we estimated the entropy and mixing rates. Finally, ensemble averages over 100 different realisations were computed.
Appendix E. The Cancer Cell Line Encyclopedia (CCLE) data
We used the gene expression data and drug sensitivity profiles as provided in the previous publication [57]. Briefly, integration of the gene expression data with our PPI network resulted in a maximally connected component consisting of 8038 genes/proteins. There were 488 cell-lines with gene expression and drug sensitivity profiles for 24 drugs. As a measure of drug response we used the Activity Area [57] since this measure gave optimal results when correlating drug response to gene expression levels of well established drug targets. Figure 1: Signalling Entropy: understanding systems biology through uncertainty. A) A caricature model of a cellular interaction network with edge widths/color indicating the relative probabilities of interaction. On the left and right, we depict states of high and low signalling entropy, respectively. At the cellular population level, this translates into samples of high and low intra-sample heterogeneity, respectively. B) Signalling entropy correlates with pluripotency as demonstrated in our previous work (Banerji et al 2013). The pluripotent state is a highly promiscuous signaling state, generating high intra-sample heterogeneity, and allowing the stem cell population to differentiate into any cell type. In contrast, in a terminally differentiated state, signaling is almost deterministic, reflecting activation of very specific pathways in the majority of cells, leading to a highly homogeneous and differentiated cell population. Thus, signalling entropy defines the height in Waddington's differentiation landscape. Figure 2: Signalling entropy and cellular robustness: A) Signalling entropy ought to correlate with cellular robustness. The inequality encapsulates this information by stating that a decrease in signalling entropy (i.e. if ∆S < 0), then the system's robustness R must also decrease, i.e. ∆R < 0, so that the product ∆S∆R > 0. Observe how in the low entropy state, random removal of edges through e.g. inactivating mutations, can lead to deactivation of a key signaling pathway connecting a given start and end nodes (shown in orange). In the high entropy state, the same perturbations do not prevent signal transduction between the two orange nodes. B) Depicted are the effects of two major forms of cancer perturbation. In the upper panel, inactivation (typically of tumour suppressors), leads to underexpression and a corresponding loss of correlations/interactions with neighbors in the PPI network. This is tantamount to a state of increased entropy and drug intervention is unlikely to be effective. In the lower panel, we depict the case of an oncogene, which is overexpressed in cancer. This overexpression leads to activation of a specific oncogenic pathway which results in oncogene addiction and increased sensitivity to targeted drug intervention. Thus, local signalling entropy and robustness (as determined by response to a drug), may also correlate locally.
Inactivation of tumour-suppressor low/absent expression high expression
Activation of oncogene
C)
AUC=0.92 AUC=0.67(permuted case) Figure 3: Entropy rate in normal and cancer tissue: A) Boxplots of sample specific entropy rates comparing normal liver and liver cancer samples. Expression data set is the one used in [38]. B) As A), but also shown are the sample specific entropy rates obtained by randomly permuting the gene expression values over the network. Note how the entropy rates for the normal and cancer states are significantly reduced upon permutation and are no longer highly discriminative between normal and cancer. C) ROC curves and associated AUC normal-cancer discriminatory values for the unpermuted and permutated cases depicted in A) and B). Figure 5: Anti-correlation between local signalling entropy and drug sensitivity. A) Histogram of Spearman rank correlation P-values between drug sensitivities (n=24) and local signalling entropies (n=8038 genes), as computed over the 488 CCLE cell-lines. B) As A) but for gene expression instead of signalling entropy. C) Scatterplot of Spearman rank Correlation Coefficient (SCC) between local signalling entropy (sigS) and drug sensitivity (DS) against −log 10 P-value for each of 134 drug gene target pairs. D) As C) but for gene expression instead of local entropy. In C) & D), we give the distribution of significant positive and negative correlations and the associated Binomial test P-value. E) Drug target gene pairs ranked according to negative SCC (cyan color) between signalling entropy and drug sensitivity. Only pairs where at least one of entropy or gene expression were significantly associated are shown. Upper panels show the SCC (cyan=strong negative SCC, white=zero or non-significant SCC, magenta=strong positive SCC), while lower panels show the corresponding P-values with the last row showing the P-value from the Likelihood Ratio Test (LRT) assessing the added predictive value of signalling entropy over gene expression. The darker the tones of red the more significant the P-values are, whilst white indicates non-significance. F) A subset of E), with pairs now ranked according to the LRT P-value. Figure 6: High signalling entropy of intra-cellular hubs confers drug resistance: Upper panel plots the topological degree of drug targets (x-axis) against the Spearman rank Correlation Coefficient (SCC) between its local signalling entropy and drug sensitivity, as assessed over the CCLE samples. EC=target annotated as extra-cellular, MR=target annotated as membrane receptor, IC=target annotated as intra-cellular. Left lower panel shows the difference in SCC values between the IC and EC+MR targets. Wilcoxon rank sum test P-value given. Right lower panel shows the difference in SCC values between the IC and EC+MR targets, where now the SCC value were computed between gene expression and drug sensitivity. Wilcoxon rank sum test P-value given.
Acknowledgements AET is supported by the Chinese Academy of Sciences, Shanghai Institute for Biological Sciences and the Max-Planck Gesellshaft. RK and PS acknowledge funding under FP7/2007-2013/grant agreement nr. 290038.
Lemma 1 .
1M has the same eigenvalues as P and if u a is an eigenvector of M , then r a = Π − 1 2 u a and l a = Π 1 2 u a are right and left eigenvectors of P .
a ) = λ a (Π 1 2 u a ).
Figure 4 :
4Correlation between global entropy and mixing rates. A) Plotted is the entropy rate against the mixing rate for Erdos-Renyi graphs of 100 nodes and average degree 10. The light blue lines show the results over 100 different network realisations, with the dark blue line representing the ensemble average. B) As A) but for connected subnetworks of average size 500 nodes, generated by random subsampling of 1000 nodes from the full PPI network of 8038 nodes. As in A), a range of edge weight distributions were considered reflecting variable entropy rates. The light blue lines show the results over 100 different realisations, with the dark blue line representing the ensemble average. C) Scatterplot of the global entropy versus mixing rates for the 488 Cancer Cell-Line Encyclopedia (CCLE) samples. Spearman rank correlation coefficient (SCC) and associated P-value are given (left panel). Wilcoxon rank sum test P-value between high and low mixing rate groups (as defined by tertiles) (right panel).
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[
"Entanglement fidelity and measurement of entanglement preserving in quantum processes",
"Entanglement fidelity and measurement of entanglement preserving in quantum processes"
]
| [
"Yang Xiang \nNational Laboratory of Solid State Microstructures and Department of Physics\nNanjing University\n210093NanjingChina\n",
"Shi-Jie Xiong \nNational Laboratory of Solid State Microstructures and Department of Physics\nNanjing University\n210093NanjingChina\n"
]
| [
"National Laboratory of Solid State Microstructures and Department of Physics\nNanjing University\n210093NanjingChina",
"National Laboratory of Solid State Microstructures and Department of Physics\nNanjing University\n210093NanjingChina"
]
| []
| The entanglement fidelity provides a measure of how well the entanglement between two subsystems is preserved in a quantum process. By using a simple model we show that in some cases this quantity in its original definition fails in the measurement of the entanglement preserving. On the contrary, the modified entanglement fidelity, obtained by using a proper local unitary transformation on a subsystem, is shown to exhibit the behavior similar to that of the concurrence in the quantum evolution.PACS numbers: 03.67.Mn, 03.65.Ud Quantum entanglement is a key element for applications of quantum communications and quantum information. A complete discussion of this has been given in Ref.[1]. Characterizing and quantifying the entanglement is a fundamental issue in quantum information theory. For pure and mixed states of two qubits this problem about the description of the entanglement has been well elucidated[2,3,4,5,6,7]. Recently, Jordan et al.[8] considered two entangled qubits, one of which interacts with a third qubit named as a control one that is never entangled with either of the two entangled qubits. They found that the entanglement of these two qubits can be both increased and decreased by the interaction with the control qubit on just one of them. If we regard the control qubit as an environment and the state of the qubit interacting with the control qubit as the information source, this example is just a model for the time evolution of quantum information via a noisy quantum channel originating from the interaction with the control qubit. Schumacher [9] and Barnum et al.[10] have investigated a general situation where R and Q are two quantum systems and the joint system RQ is initially prepared in a pure entangled state |Ψ RQ . The system R is dynamically isolated and has a zero internal Hamiltonian, while the system Q undergoes some evolution that possibly involves interaction with the environment. The evolution of Q might represent a transmission process via some quantum channel for the quantum information in Q. They introduced a fidelity F e = Ψ RQ |ρ RQ ′ |Ψ RQ , which is the probability that the final state ρ RQ ′ would pass a test checking whether it agrees with the initial state |Ψ RQ . This quantity is called as entanglement fidelity (referred hereafter as EF). The EF can be defined entirely in terms of the initial state ρ Q and the evolution of system Q, so EF is related to a process, specified by a quantum operation ε Q , which we shall discuss later in more details, acting on some initial state ρ Q . Thus, the EF can be denoted by a function form F e (ρ Q , ε Q ). The EF is usually used to measure how well the state * Electronic address: [email protected] ρ Q is preserved by the operation ε Q and to identify how well the entanglement of ρ Q with other systems is preserved by the operation of ε Q . The complete discussion of EF can be seen in[9,11]. In the present work we will investigate the following question: Is EF a good measurement of the entanglement preserving? Using the example of Jordan et al., we find that in some cases EF defined above completely fails for measuring the entanglement preserving though it may be a good measurement of the entanglement preserving in the case of slight noise. We also find that in order to make the EF indeed equivalent to an entanglement measure the modified entanglement fidelity (MEF) should be used. Some detailed discussions about the MEF have been given in[9,12,13]. Recently, Surmacz et al. [14] have investigated the evolution of the entanglement in a quantum memory and showed that the MEF can be used to measure how well a quantum memory setup can preserve the entanglement between a qubit undergoing the memory process and an auxiliary qubit. For the example of Jordan et al., we derive an analytic expression of the MEF and the comparison of it with the concurrence is given.Quantum operation ε Q is a map for the state of QHere ρ Q is the initial state of system Q, and after the dynamical process the final state of the system becomes ρ Q ′ . Then the dynamical process is described by ε Q .In the most general case, the map ε Q must be a tracepreserving and positive linear map[15,16], so it includes all unitary evolutions. They also include unitary evolving interactions with an environment E. Suppose that the environment is initially in state ρ E . The operator can be written as | 10.1103/physreva.76.014301 | [
"https://arxiv.org/pdf/0704.2973v3.pdf"
]
| 119,630,026 | 0704.2973 | 0f64cfe248633e6ba5b1aa38e79aeed37038d027 |
Entanglement fidelity and measurement of entanglement preserving in quantum processes
13 May 2007 (Dated: February 1, 2008)
Yang Xiang
National Laboratory of Solid State Microstructures and Department of Physics
Nanjing University
210093NanjingChina
Shi-Jie Xiong
National Laboratory of Solid State Microstructures and Department of Physics
Nanjing University
210093NanjingChina
Entanglement fidelity and measurement of entanglement preserving in quantum processes
13 May 2007 (Dated: February 1, 2008)
The entanglement fidelity provides a measure of how well the entanglement between two subsystems is preserved in a quantum process. By using a simple model we show that in some cases this quantity in its original definition fails in the measurement of the entanglement preserving. On the contrary, the modified entanglement fidelity, obtained by using a proper local unitary transformation on a subsystem, is shown to exhibit the behavior similar to that of the concurrence in the quantum evolution.PACS numbers: 03.67.Mn, 03.65.Ud Quantum entanglement is a key element for applications of quantum communications and quantum information. A complete discussion of this has been given in Ref.[1]. Characterizing and quantifying the entanglement is a fundamental issue in quantum information theory. For pure and mixed states of two qubits this problem about the description of the entanglement has been well elucidated[2,3,4,5,6,7]. Recently, Jordan et al.[8] considered two entangled qubits, one of which interacts with a third qubit named as a control one that is never entangled with either of the two entangled qubits. They found that the entanglement of these two qubits can be both increased and decreased by the interaction with the control qubit on just one of them. If we regard the control qubit as an environment and the state of the qubit interacting with the control qubit as the information source, this example is just a model for the time evolution of quantum information via a noisy quantum channel originating from the interaction with the control qubit. Schumacher [9] and Barnum et al.[10] have investigated a general situation where R and Q are two quantum systems and the joint system RQ is initially prepared in a pure entangled state |Ψ RQ . The system R is dynamically isolated and has a zero internal Hamiltonian, while the system Q undergoes some evolution that possibly involves interaction with the environment. The evolution of Q might represent a transmission process via some quantum channel for the quantum information in Q. They introduced a fidelity F e = Ψ RQ |ρ RQ ′ |Ψ RQ , which is the probability that the final state ρ RQ ′ would pass a test checking whether it agrees with the initial state |Ψ RQ . This quantity is called as entanglement fidelity (referred hereafter as EF). The EF can be defined entirely in terms of the initial state ρ Q and the evolution of system Q, so EF is related to a process, specified by a quantum operation ε Q , which we shall discuss later in more details, acting on some initial state ρ Q . Thus, the EF can be denoted by a function form F e (ρ Q , ε Q ). The EF is usually used to measure how well the state * Electronic address: [email protected] ρ Q is preserved by the operation ε Q and to identify how well the entanglement of ρ Q with other systems is preserved by the operation of ε Q . The complete discussion of EF can be seen in[9,11]. In the present work we will investigate the following question: Is EF a good measurement of the entanglement preserving? Using the example of Jordan et al., we find that in some cases EF defined above completely fails for measuring the entanglement preserving though it may be a good measurement of the entanglement preserving in the case of slight noise. We also find that in order to make the EF indeed equivalent to an entanglement measure the modified entanglement fidelity (MEF) should be used. Some detailed discussions about the MEF have been given in[9,12,13]. Recently, Surmacz et al. [14] have investigated the evolution of the entanglement in a quantum memory and showed that the MEF can be used to measure how well a quantum memory setup can preserve the entanglement between a qubit undergoing the memory process and an auxiliary qubit. For the example of Jordan et al., we derive an analytic expression of the MEF and the comparison of it with the concurrence is given.Quantum operation ε Q is a map for the state of QHere ρ Q is the initial state of system Q, and after the dynamical process the final state of the system becomes ρ Q ′ . Then the dynamical process is described by ε Q .In the most general case, the map ε Q must be a tracepreserving and positive linear map[15,16], so it includes all unitary evolutions. They also include unitary evolving interactions with an environment E. Suppose that the environment is initially in state ρ E . The operator can be written as
The entanglement fidelity provides a measure of how well the entanglement between two subsystems is preserved in a quantum process. By using a simple model we show that in some cases this quantity in its original definition fails in the measurement of the entanglement preserving. On the contrary, the modified entanglement fidelity, obtained by using a proper local unitary transformation on a subsystem, is shown to exhibit the behavior similar to that of the concurrence in the quantum evolution. Quantum entanglement is a key element for applications of quantum communications and quantum information. A complete discussion of this has been given in Ref. [1]. Characterizing and quantifying the entanglement is a fundamental issue in quantum information theory. For pure and mixed states of two qubits this problem about the description of the entanglement has been well elucidated [2,3,4,5,6,7]. Recently, Jordan et al. [8] considered two entangled qubits, one of which interacts with a third qubit named as a control one that is never entangled with either of the two entangled qubits. They found that the entanglement of these two qubits can be both increased and decreased by the interaction with the control qubit on just one of them. If we regard the control qubit as an environment and the state of the qubit interacting with the control qubit as the information source, this example is just a model for the time evolution of quantum information via a noisy quantum channel originating from the interaction with the control qubit. Schumacher [9] and Barnum et al. [10] have investigated a general situation where R and Q are two quantum systems and the joint system RQ is initially prepared in a pure entangled state |Ψ RQ . The system R is dynamically isolated and has a zero internal Hamiltonian, while the system Q undergoes some evolution that possibly involves interaction with the environment. The evolution of Q might represent a transmission process via some quantum channel for the quantum information in Q. They introduced a fidelity F e = Ψ RQ |ρ RQ ′ |Ψ RQ , which is the probability that the final state ρ RQ ′ would pass a test checking whether it agrees with the initial state |Ψ RQ . This quantity is called as entanglement fidelity (referred hereafter as EF). The EF can be defined entirely in terms of the initial state ρ Q and the evolution of system Q, so EF is related to a process, specified by a quantum operation ε Q , which we shall discuss later in more details, acting on some initial state ρ Q . Thus, the EF can be denoted by a function form F e (ρ Q , ε Q ). The EF is usually used to measure how well the state * Electronic address: [email protected] ρ Q is preserved by the operation ε Q and to identify how well the entanglement of ρ Q with other systems is preserved by the operation of ε Q . The complete discussion of EF can be seen in [9,11]. In the present work we will investigate the following question: Is EF a good measurement of the entanglement preserving? Using the example of Jordan et al., we find that in some cases EF defined above completely fails for measuring the entanglement preserving though it may be a good measurement of the entanglement preserving in the case of slight noise. We also find that in order to make the EF indeed equivalent to an entanglement measure the modified entanglement fidelity (MEF) should be used. Some detailed discussions about the MEF have been given in [9,12,13]. Recently, Surmacz et al. [14] have investigated the evolution of the entanglement in a quantum memory and showed that the MEF can be used to measure how well a quantum memory setup can preserve the entanglement between a qubit undergoing the memory process and an auxiliary qubit. For the example of Jordan et al., we derive an analytic expression of the MEF and the comparison of it with the concurrence is given.
Quantum operation ε Q is a map for the state of Q
ρ Q ′ = ε Q (ρ Q ).(1)
Here ρ Q is the initial state of system Q, and after the dynamical process the final state of the system becomes ρ Q ′ . Then the dynamical process is described by ε Q .
In the most general case, the map ε Q must be a tracepreserving and positive linear map [15,16], so it includes all unitary evolutions. They also include unitary evolving interactions with an environment E. Suppose that the environment is initially in state ρ E . The operator can be written as
ε Q (ρ Q ) = Tr E U (ρ Q ⊗ ρ E )U † = Tr E U (ρ Q ⊗ i p i |i i|)U † = j E Q j ρ Q E Q † j ,(2)
where i p i |i i| is the spectral decomposition of ρ E , with {|i } being a base in the Hilbert space H E of the environment E, and E Q j = i √ p i j|U |i . Now we can use Eq. (2) to get the intrinsic expression of Ψ RQ |ρ RQ ′ |Ψ RQ , i.e., F e (ρ Q , ε Q ). Because
ρ RQ ′ = I R ⊗ ε Q (ρ RQ ) = j (1 R ⊗ E Q j )ρ RQ (1 R ⊗ E Q j ) † ,(3)
one has
F e = Ψ RQ |ρ RQ ′ |Ψ RQ = j Ψ RQ |(1 R ⊗ E Q j )|Ψ RQ × Ψ RQ |(1 R ⊗ E Q j ) † |Ψ RQ = j (Trρ Q E Q j )(Trρ Q E Q † j ).(4)
If systems R and Q both have zero internal Hamiltonian and there is no interaction between R and Q, the operation ε Q entirely originates from the interaction between Q and the environment. In this sense the example of Jordan et al. is a special case of this situation. We consider two entangled qubits, A and B, and suppose that qubit A interacts with a control qubit C. Then A, B and C respectively correspond to systems Q, R and environment E that we have just referred. We suppose that the initial states of the three qubits are
W = ρ AB ± ⊗ 1 2 1 c ,(5)
where
ρ AB ± = 1 4 (1 ± σ A 1 σ B 1 ± σ A 2 σ B 2 − σ A 3 σ B 3 ),(6)
with σ A(B) i , i = 1, 2, 3, being Pauli matrices for qubit A(B). ρ AB + and ρ AB − are two Bell states, representing the maximally entangled pure states for the combined system of qubits A and B. The total spins of states ρ AB − and ρ AB + are 0 and 1, respectively. We suggest an interaction between qubit A and C described by the unitary transformation
U = e −itH ,(7)
where
H = λσ A 3 2 (|α α| − |β β|),(8)
λ is the strength of the interaction, and |α and |β are two orthonormal vectors for system C. Then the changing density matrix for the combined system of qubits A and B can be calculated as
ρ AB ′ ± = Tr c (U ⊗ 1 B )W (U ⊗ 1 B ) † = 1 4 [1 ± (σ A 1 σ B 1 + σ A 2 σ B 2 ) cos (λt) − σ A 3 σ B 3 ] = ρ AB ± cos 2 ( λt 2 ) + ρ AB ∓ sin 2 (λt 2
).
The changing density matrix ρ AB ′ ± usually represents a mixed state. In order to quantify the entanglement of it we use the Wootters concurrence [5] defined as
C(ρ) ≡ max[0, λ 1 − λ 2 − λ 3 − λ 4 ],(10)
where ρ is the density matrix representing the investigated state of the combined system of A and B, λ 1 , λ 2 , λ 3 , and λ 4 are the eigenvalues of ρσ A 2 σ B 2 ρ * σ A 2 σ B 2 in the decreasing order, and ρ * is the complex conjugation of ρ. From Eq. (9) we can obtain
C(ρ AB ′ ± ) = | cos λt|.(11)
It is found that at time λt = π 2 , the state ρ AB ′ ± is changed from a maximally entangled state at t = 0 to a separable state and at time λt = π the state ρ AB ′ ± returns to the maximally entangled state. The explicit calculation about ρ AB ′ and C(ρ AB ′ ± ) can be seen in [8]. Now we adopt the EF to investigate this example. Using Eqs. (2), (5), (7), and (8), we obtain the quantum operation on qubit A,
ε A (ρ A ) = Tr C U (ρ A ⊗ ρ C )U † = Tr C U ρ A ⊗ ( 1 2 (|α α| + |β β|)) U † = 1 2 e −iσ A 3 ( λt 2 ) ρ A e +iσ A 3 ( λt 2 ) + 1 2 e +iσ A 3 ( λt 2 ) ρ A e −iσ A 3 ( λt 2 ) .(12)So E A α = 1 √ 2 e −iσ A 3 ( λt 2 ) and E A β = 1 √ 2 e +iσ A 3 ( λt 2 )
. Substituting them into Eq. (4) and noting that ρ A ≡ Tr B (ρ AB ± ) = 1 2 1, we can get the EF as
F e = j (Trρ A E A j )(Trρ A E A † j ) = 1 √ 2 Tr e −i λt
We can easily find the disagreement between the evolutions of F e and C(ρ AB ′ ± ). At λt = π, state ρ AB ′ ± returns to the maximally entangled state as can be seen from the concurrence, but its entanglement fidelity is zero (F e = 0). On the contrary, the initial maximally entangled state have been changed to a separable state at λt = π 2 , but the EF at this time is not zero. The evolutions of EF F e and concurrence C(ρ AB ′ ± ) are depicted in Fig. 1.
In fact, F e (ρ Q , ε Q ) = F 2 s (ρ RQ , ρ RQ ′ ), where F s (ρ RQ , ρ RQ ′ ) is the static fidelity [11]. The static fidelity satisfies 0 ≤ F s (ρ RQ , ρ RQ ′ ) ≤ 1, where the first
F e (ρ A , ε Q ) = F 2 s (ρ AB , ρ AB ′ ) = 0 at λt = π.
The concept of the EF arises from the mathematical description for the purification of mixed states. Any mixed state can be represented as a subsystem of a pure state in a larger Hilbert space. The entanglement of a pure state may cause the states of subsystems to be mixed. The EF is usually used to measure how faithfully a channel maintains the purification, or, equivalently, how well the channel preserves the entanglement. In the above simple example, however, we have found that, except for some special cases, only in the case of slight noise, i.e., λt −→ 0, the EF approximately agrees with the concurrence. This means that this quantity may not be a good measurement for the evolution of the entanglement in the processes of interaction with the environment.
In fact, Schumacher [9] has noted that the EF can be lowered by a local unitary operation but the entanglement cannot be so. From this consideration he defined the MEF
F ′ e = max U Q Ψ RQ (1 R ⊗ U Q )ρ RQ ′ (1 R ⊗ U Q ) † Ψ RQ ,(14)
where U Q is any unitary transformation acting on Q. It is clear that F ′ e ≥ F e . Since by using a proper local unitary operation we can make the Bell state ρ AB ± become the Bell state ρ AB ∓ , we can find that in the above example F ′ e = 1 at time λt = π whereas F e = 0 at this time. So at λt = π, the MEF equals the concurrence. By using the quantum operation which we discussed above, we can get the intrinsic expression of the MEF
F ′ e = max U Q j Ψ RQ (1 R ⊗ U Q E Q j ) Ψ RQ × Ψ RQ (1 R ⊗ U Q E Q j ) † Ψ RQ = max U Q j (Trρ Q U Q E Q j )(Trρ Q (U Q E Q j ) † )
. (15) For this example we can derive an analytic expression of F ′ e . Suppose U is an arbitrary unitary operation on a single qubit. Then it can be written as [11]
U = e −iα R z (β)R y (γ)R z (δ) = e −iα e i(−β/2−δ/2) cos γ 2 − e i(−β/2+δ/2) sin γ 2 e i(+β/2−δ/2) sin γ 2 e i(+β/2+δ/2) cos γ 2 ,
where α, β, γ and δ are real numbers, and R y(z) is the rotation operator about the y(z) axis. We have
j (Trρ A U E A j )(Trρ A (U E A j ) † ) = 1 2 1 2 Tr e i(−β/2−δ/2− λt 2 ) cos γ 2 0 0 e i(β/2+δ/2+ λt 2 ) cos γ 2 2 + 1 2 1 2 Tr e i(−β/2−δ/2+ λt 2 ) cos γ 2 0 0 e i(β/2+δ/2− λt 2 ) cos γ 2 2 = 1 2 cos 2 ( γ 2
) cos 2 (β/2 + δ/2 + λt/2)
+ 1 2 cos 2 ( γ 2 ) cos 2 (β/2 + δ/2 − λt/2).(16)
We should find a unitary operator U which make j (Trρ A U E A j )(Trρ A (U E A j ) † ) take its maximum value. Since cos 2 (β/2 + δ/2 + λt/2) ≥ 0 and cos 2 (β/2 + δ/2 − λt/2) ≥ 0, we can take γ = 0. So one obtains
j (Trρ A U E A j )(Trρ A (U E A j ) † )
= 1 + cos 2 (β/2 + δ/2)(2 cos 2 (λt/2) − 1) − cos 2 (λt/2).
When 2 cos 2 (λt/2) − 1 ≥ 0 we take cos 2 (β/2 + δ/2) = 1 and get F ′ e = cos 2 (λt/2); when 2 cos 2 (λt/2) − 1 < 0 we take cos 2 (β/2 + δ/2) = 0 and get F ′ e = 1 − cos 2 (λt/2). The evolutions of the MEF F ′ e and the concurrence C(ρ AB ′ ± ) are depicted in Fig. 2. We can find that the MEF and the concurrence exhibit a similar behavior, although their values do not exactly agree with each other at all moments. When the state ρ AB ′ ± returns to the maximally entangled state, the MEF is equal to 1. The maximal difference between them comes at the separable states where the MEF is equal to 1/2 while the concurrence is zero.
We have mentioned that the EF equals 1 if and only if ρ RQ = ρ RQ ′ . This means that the EF can be use to measure the difference between a quantum channel and the identity channel. If the concern is on the entanglement preserving in an evolution process, however, one has to use the MEF because the EF can be lowered by a local unitary operation in this process but the entanglement cannot be so. If a quantum channel is just a unitary operator, the entanglement is certainly invariant and the MEF always equals to 1 in the quantum process. In this sense the MEF can be used to measure the difference between a quantum channel and an arbitrary unitary operator.
In summary, for the example of Jordan et al., we have derived the analytic expressions of both the EF and the MEF, and show the comparisons of them with the concurrence. From these we find that the MEF may admirably reflects the entanglement preserving in a quantum process.
PACS numbers: 03.67.Mn, 03.65.Ud
FIG. 1 :
1The evolutions of the EF Fe (solid line) and the concurrence C (dashed line). We takeh = 1 so λt is dimensionless.symbol of "≤" becomes equality if and only if ρ RQ and ρ RQ ′ have orthogonal support, and the second symbol becomes equality if and only if ρ RQ = ρ RQ ′ . When λt = π, from Eq. (9) we can see that ρ AB ′ ± = ρ AB ∓ . The ρ AB ± are two different Bell states and correspond respectively to eigenstates of total spin one and total spin zero of the combined system of qubits A and B. So they have orthogonal support in the Hilbert space H A ⊗ H B . This is the reason for the fact that
FIG. 2 :
2The evolutions of the modified entanglement fidelity F ′ e (solid line) and the concurrence C (dashed line).
Acknowledgments We wish to thank K. Surmacz for his stimulating discussion which leads us to note the MEF. This work was supported by National
. R Horodecki, P Horodecki, M Horodecki, K Horodecki, arXiv:quant-ph/0702225R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, arXiv:quant-ph/0702225(2007).
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. K Surmacz, private communicationK. Surmacz, private communication.
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"Majorization bounds for Ritz values of Rayleigh quotients of self-adjoint matrices",
"Majorization bounds for Ritz values of Rayleigh quotients of self-adjoint matrices"
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"Demetrio Stojanoff \nCentro de Matemática\nFCE-UNLP\nIAM-CONICET\nLa PlataArgentina\n",
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"Centro de Matemática\nFCE-UNLP\nIAM-CONICET\nLa PlataArgentina"
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| In this work we obtain a priori, a posteriori and mixed type upper bounds for the absolute change in Ritz values of Rayleigh quotients of self-adjoint matrices in terms of submajorization relations. Some of our results solve recent conjectures by Knyazev, Argentati and Zhu, that extend several known results for one dimensional subspaces to arbitrary subspaces. In particular, we revisit Nakatsukasa's version of the tan Θ theorem of Davies and Kahan and obtain an improved version of this result. As a consequence, we obtain improved quadratic a posteriori bounds for the absolute change in Ritz values of Rayleigh quotients. | null | [
"https://arxiv.org/pdf/1905.06998v1.pdf"
]
| 158,046,606 | 1905.06998 | 86c08697589195585ec225302f6c19b11623480d |
Majorization bounds for Ritz values of Rayleigh quotients of self-adjoint matrices
May 2019
Pedro Massey
Centro de Matemática
FCE-UNLP
IAM-CONICET
La PlataArgentina
Demetrio Stojanoff
Centro de Matemática
FCE-UNLP
IAM-CONICET
La PlataArgentina
Sebastián Zárate
Centro de Matemática
FCE-UNLP
IAM-CONICET
La PlataArgentina
Majorization bounds for Ritz values of Rayleigh quotients of self-adjoint matrices
16May 2019AMS subject classification: 42C1515A60 Keywords: principal anglesRitz valuesRayleigh quotientsmajorization
In this work we obtain a priori, a posteriori and mixed type upper bounds for the absolute change in Ritz values of Rayleigh quotients of self-adjoint matrices in terms of submajorization relations. Some of our results solve recent conjectures by Knyazev, Argentati and Zhu, that extend several known results for one dimensional subspaces to arbitrary subspaces. In particular, we revisit Nakatsukasa's version of the tan Θ theorem of Davies and Kahan and obtain an improved version of this result. As a consequence, we obtain improved quadratic a posteriori bounds for the absolute change in Ritz values of Rayleigh quotients.
Introduction
The study of sensitivity of Ritz values of Rayleigh quotients of self-adjoint matrices (i.e. the changes in the eigenvalues of compressions of a self-adjoint matrix) is a well established and active research field in applied mathematics [1,3,8,9,10,11,13,15,18,19,20,21]. Explicitly, given a d × d complex self-adjoint matrix A and isometries X, Y of size d × k, with ranges X and Y respectively, we are interested in computing upper and lower bounds for |λ(ρ(X)) − λ(ρ(Y ))| = ( |λ i (ρ(X)) − λ i (ρ(Y ))| ) i∈I k ∈ R k ≥0 where ρ(X) = X * A X, ρ(Y ) = Y * A Y are k × k complex self-adjoint matrices known as Rayleigh quotients (RQ) of A, and λ(ρ(X)), λ(ρ(Y )) ∈ R k are the eigenvalues (counting multiplicities and arranged in non-increasing order) also known as Ritz values of the corresponding RQ.
Typically, the bounds for the absolute change in the Ritz values of RQ are obtained in terms of the residuals R X = AX − X ρ(X) and R Y = AY − Y ρ(Y ) or in terms of the principal angles between subspaces (PABS) denoted by Θ(X , Y) ∈ [0, π/2] k . Upper bounds are classified according to which parameters are used to bound the change in Ritz values (see [19]). Indeed, the a priori bounds are those obtained in terms of PABS; the a posteriori bounds are those obtained in terms of (singular values of) residuals while the mixed type bounds are obtained in terms of both PABS and residuals. It is worth pointing out that the PABS appearing in a priori bounds may not be readily available in practice. On the other hand, a posteriori bounds are based on computable singular values of residual matrices. Moreover, bounds based on residuals (i.e. both a posteriori and mixed type) are particularly convenient in case one of the spaces, say X , is A-invariant (as in this case R X = 0), as opposed to (autonomous) a priori bounds.
The abstract matrix analysis formulation of the sensitivity problem stated above makes it possible to apply this theory in a variety of similarly different research areas such as: graph matching [9] in terms of spectral analysis of the graphs; signal distinction in signal processing, where Ritz values serve as harmonic signature to differentiate subspaces; finite element methods (FEM) [8], for approximation of subspaces corresponding to fundamental modes; of course, matrix analysis, e.g. for bounds for eigenvalues after matrix additive perturbations. Also, bounds for changes in Ritz values of RQ play a central role in the analysis of algorithms for simultaneous approximation of eigenvalues based on Rayleigh-Ritz methods (see [16,17] and the references therein). By now, the role of submajorization in obtaining bounds for the change of Ritz values of RQ (recognized in the seminal paper [9]) is well known; this partial pre-order relation is a powerful tool in this context, as bounds in terms of submajorization imply a whole family of inequalities with respect to unitarily invariant norms and with respect to the class of non-decreasing convex functions ( [12]).
In this work we obtain a priori, a posteriori and mixed type upper bounds for the absolute change in Ritz values of RQ of self-adjoint matrices in terms of submajorization. Some of our results solve recent conjectures from [8,19,20] that extend several known results for one dimensional subspaces to arbitrary subspaces. In particular, we revisit Nakatsukasa's version of the tan Θ theorem [14] of Davies and Kahan [4] and obtain an improved version of this result. We have included some (rather simple) examples to establish comparisons with previous work (for a detailed exposition of the context, previous work, our results and some applications, see Section 3). We will consider further applications of the results herein elsewhere.
The paper is organized as follows. In Section 2 we introduce preliminary results in majorization theory and principal angles between subspaces. In Section 3 we develop our main results; our approach to obtain these results is based on methods from abstract matrix analysis, so we delay the proofs of some technical results until an appendix section. Section 3 is divided in three subsections: in Section 3.1 we prove a mixed type upper bound for the change of the Ritz values of RQ that is conjectured in [20] and show that this bound is sharp. We have also included some comments with a comparison of our results with previous works and with future applications of the results of this subsection. In Section 3.2 we establish a link between the results from Section 3.1 and an a priori upper bound for Ritz values of RQ conjectured from [8]. Although the results in this section are not sharp, they can be applied in quite general situations and they capture the order of approximation conjectured in [8]. In Section 3.3 we revisit Nakatsukasa's version of the tan Θ theorem of Davies and Kahan and obtain an improved version of this result; we include an example that shows that this new version of the tan Θ theorem is sharp in cases in which the classical result is not. As an application, we obtain improved quadratic a posteriori error bounds for Ritz values of RQ. The paper ends with an Appendix (Section 4) in which we include a detailed background on majorization theory and present the proofs of some technical results needed in Section 3.
Preliminaries
Throughout our work we use the following Notation and terminology. We let M k,d (C) be the space of complex k × d matrices and write M d,d (C) = M d (C) for the algebra of d × d complex matrices. We denote by H(d) ⊂ M d (C) the real subspace of self-adjoint matrices and by M d (C) + , the cone of positive semi-definite matrices. Also, Gl(d) ⊂ M d (C) and U (d) denote the groups of invertible and unitary matrices respectively, and Gl(d)
+ = Gl(d) ∩ M d (C) + . For d ∈ N, let I d = {1, . . . , d}. Given a vector x ∈ C d we denote by D x the diagonal matrix in M d (C) whose main diagonal is x. Given x = (x i ) i∈I d ∈ R d we denote by x ↓ = (x ↓
i ) i∈I d the vector obtained by rearranging the entries of x in non-increasing order. We also use the notation ( Arithmetic operations with vectors are performed entry-wise i.e., in case
R d ) ↓ = {x ∈ R d : x = x ↓ } and (R d ≥0 ) ↓ = {x ∈ R d ≥0 : x = x ↓ }. For r ∈ N, we let 1 r = (1, . . . , 1) ∈ R r . Given a matrix A ∈ H(d) we denote by λ(A) = (λ i (A)) i∈I d ∈ (R d ) ↓x = (x i ) i∈I k , y = (y i ) i∈I k ∈ C k then x + y = (x i + y i ) i , x y = (x i y i ) i and (assuming that y i = 0, for i ∈ I k ) x/y = (x i /y i ) i ,
where these vectors all lie in C k . Moreover, if we assume further that x, y ∈ R k then we write x ≤ y whenever x i ≤ y i , for i ∈ I k . △
Next we recall the notion of majorization between vectors, that will play a central role throughout our work.
Definition 2.1. Let x, y ∈ R k . We say that x is submajorized by y, and write x ≺ w y, if j i=1 x ↓ i ≤ j i=1 y ↓ i for j ∈ I k .
If x ≺ w y and tr x def = k i=1
x i = tr y, then we say that x is majorized by y, and write x ≺ y. △
There are many fundamental results in matrix theory that are stated in terms of submajorization relations (see for example [2,6,12]). In what follows, we mention some elementary properties of submajorization that we will need in Section 3. We will consider some further properties and results on majorization theory in Section 4. Given f : I → R, where I ⊂ R is an interval, and
z = (z i ) i∈I k ∈ I k we denote f (z) = (f (z i )) i∈I k ∈ R k .
Remark 2.2. Let I ⊂ R be an interval and let f : I → R be a convex function. Then,
1. if x, y ∈ I n satisfy x ≺ y then f (x) ≺ w f (y).
2. If x, y ∈ I n only satisfy x ≺ w y but f is further non-decreasing in I, then f (x) ≺ w f (y). △
Definition 2.3. A norm N in M d (C) is unitarily invariant (briefly u.i.n.) if N (U AV ) = N (A), for every A ∈ M d (C) and U, V ∈ U (d). △
Well known examples of u.i.n. are the spectral norm · sp and the p-norms · p , for p ≥ 1.
s(A) ≺ w s(B). △
Principal Angles Between Subspaces. Let X , Y ⊂ C d denote subspaces, with dim X = h and dim Y = k. Let X ∈ M d,h and Y ∈ M d,k be such that their columns form orthonormal bases of X and Y respectively. Then, the principal angles between X and Y, denoted
π/2 ≥ Θ 1 (X , Y) ≥ . . . ≥ Θ m (X , Y) ≥ 0 where m = min{h, k} -are determined by cos(Θ m−i+1 (X , Y)) = s i (X * Y ) for i ∈ I m .
We further write Θ(X , Y) = (Θ i (X , Y)) i∈Im ∈ (R m ) ↓ for the vector of principal angles between X and Y. Principal angles are a useful tool in describing the relative position and several geometric and metric aspects related with the subspaces X and Y in C d (see [4,5] and the references therein).
Main results
In this section we develop our main results. The section is divided in three parts; first we prove [20, Conjecture 2.1] which establishes a mixed type bound for the error in the (absolute) change of the Ritz values of Rayleigh quotients (RQ). In the second part, we establish connections between the mixed type bounds of the first section and some a priori bounds for the change of Ritz values conjectured in [8,10]. Finally we take a closer look at Nakatsukasa's tan Θ theorem under relaxed conditions from [14] and obtain an improved version of this result. As a consequence we obtain quadratic a posteriori error bounds for the change of the Ritz values of RQ that improve several known bounds. Our approach to obtain these results is based on methods from abstract matrix analysis, so we delay the proofs of some technical results until Section 4, where we have also included several classical results of this area that we will refer to in this section.
We begin by introducing the following Notation 3.1. Throughout this section we consider the following notation and terminology:
1. X , Y ⊂ C d denote two subspaces of dimension k. We fix X, Y ∈ M d,k (C) such that their columns form orthonormal bases of X and Y, respectively.
2. Θ(X , Y) ∈ (R k ≥0 ) ↓ denotes the vector of principal angles between the subspaces X and Y; in this case,
cos(Θ ↑ (X , Y)) = s(X * Y ) = (s 1 (X * Y ), . . . , s k (X * Y )) ∈ (R k ≥0 ) ↓ .
For a (fixed) self-adjoint
A ∈ H(d) we set ρ(X) = X * AX ∈ M k (C), R X = AX − Xρ(X) ∈ M d,k (C) and similarly ρ(Y ) and R Y for Y . Notice that R X = AX − XX * AX = AX − P X AX = (I − P X ) AX = P X ⊥ AX ∈ M d,k (C) ,
where P X ∈ M d (C) denotes the orthogonal projection onto X and X ⊥ denotes the orthogonal complement of X . We consider similar notation and identities for Y.
4. Let X ⊥ ∈ M d , d−k (C) such that their columns form an o.n.b. of X ⊥ . Then the matrix representation of A in the o.n.b. given by the colummns of X and X ⊥ has the form
A = ρ(X) R * X X ⊥ X * ⊥ R X ρ(X ⊥ ) X X ⊥ .
Note that, since R X = (I − P X ) R X , then s(R X ) = s(X * ⊥ R X ), so that we can think of R X as the (2, 1)-block in the block matrix representation of A as above. △
Rayleigh-Ritz majorization error bounds of the mixed type
We consider Notations 3.1; moreover, in this subsection we further assume that X and Y are such that Θ 1 (X , Y) < π 2 that is, that X * Y ∈ Gl(k) is invertible. Our first result concerns a submajorization error bound for the distance of eigenvalue lists of selfadjoint matrices, within the context of matrix analysis theory.
Theorem 3.2. Let C, D ∈ H(k) and let T ∈ Gl(k). Then,
|λ(C) − λ(D)| ≺ w s(T −1 ) s(CT − T D).
(1)
Proof. See the Appendix (Section 4).
The following result is [20, Conjecture 2.1] (see also Corollary 3.4 below).
Theorem 3.3. Consider Notations 3.1 and assume that
Θ 1 (X , Y) < π 2 .
We have that
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w s(P Y R X ) + s(P X R Y ) cos(Θ(X , Y)) and (2) |λ(ρ(X)) − λ(ρ(Y ))| ≺ w [s(P X +Y R X ) + s(P X +Y R Y )] tan(Θ(X , Y)) . (3) Proof. Set T = X * Y and notice that, since Θ 1 (X , Y) < π 2 , T ∈ M k (C) is invertible. Using Theorem 3.2 we get that |λ(ρ(X)) − λ(ρ(Y ))| ≺ w s(T −1 ) s(ρ(X)T − T ρ(Y )) ,(4)
where ρ(X) = X * AX, ρ(Y ) = Y * AY ∈ H(k). By construction we have that
s(T −1 ) = 1 cos(Θ(X , Y)) ∈ (R k >0 ) ↓ .(5)
Arguing as in [20, Thm 4.1] we notice that
ρ(X)T − T ρ(Y ) = X * A XX * Y − X * Y Y * A Y = X * A P X Y − X * P Y A Y = X * A (I − P X ⊥ )Y − X * (I − P Y ⊥ )A Y = X * A Y − X * A P X ⊥ Y − X * A Y + X * P Y ⊥ A Y = −X * A P X ⊥ Y + X * P Y ⊥ A Y .
Using that s(C) = s(C * ) for C ∈ M k (C), we see that
s(X * A P X ⊥ Y ) = s(Y * P X ⊥ A X) = s(P Y P X ⊥ A X) = s(P Y R X ) ∈ (R k ≥0 ) ↓ . Analogously s(X * P Y ⊥ A Y ) = s(P X R Y ).
The previous facts together with the sub-additivity property of taking singular values (item 1. in Theorem 4.1) imply that
s(ρ(X)T − T ρ(Y )) = s(−X * A P X ⊥ Y + X * P Y ⊥ A Y ) ≺ w s(P X R Y ) + s(P Y R X ) .(6)
Now, if we apply Eqs. (5) and (6)
s(P X R Y ) ≺ w s(P X +Y R Y ) sin(Θ(X , Y)) .(7)
Since the entries of these vectors are ordered downwards, by Lemma 4.3 we can deduce that
s(P X R Y ) + s(P Y R X ) ≺ w s(P X +Y R Y ) sin(Θ(X , Y)) + s(P X +Y R X ) sin(Θ(X , Y)) .(8)
Hence, using Eqs. (2) and (8) The fact that Eq. (2) implies Eq. (3) was already observed in [20]; we have included the proof of this fact for the benefit of the reader.
Corollary 3.4. Consider Notations 3.1 and assume that
Θ 1 (X , Y) < π 2 . If we further assume that X is A-invariant then |λ(ρ(X)) − λ(ρ(Y ))| ≺ w s(P X R Y ) cos(Θ(X , Y))
and
(9) |λ(ρ(X)) − λ(ρ(Y ))| ≺ w s(P X +Y R Y ) tan(Θ(X , Y)) .(10)
Proof. In case X is A-invariant notice that R X = 0. The result now follows from Theorem 3.3.
It is natural to wonder whether we can improve the bounds in the previous results. As shown in the following example, the submajorization bounds in Theorem 3.3 and Corollary 3.4 are sharp.
Example 3.5. Let λ = (a, b, c, d) ∈ R 4 , where a < b < c < d, and consider A ∈ H(4) given by A = D λ , i.e. A is the diagonal matrix with main diagonal λ.
Let X be the A-invariant subspace X = span{e 1 , e 2 } spanned by the first two elements of the canonical basis of C 4 . For θ ∈ (0, π/2) let f θ = cos θ e 2 + sin θ e 3 and set Y θ = span{e 1 , f θ }. Then, the principal angles are given by Θ(X , Y θ ) = (θ, 0). Let
X = 1 0 0 1 0 0 0 0 , X ⊥ = 0 0 0 0 1 0 0 1 and Y θ = 1 0 0 cos θ 0 sin θ 0 0 . It is straightforward to check that λ(X * AX) = (b, a) and that λ(Y * θ AY θ ) = (b cos 2 θ + c sin 2 (θ), a). Again, simple computations show that R Y θ = 0 0 0 (b − c) cos θ sin 2 θ 0 (c − b) cos 2 θ sin θ 0 0 , P X R Y θ = 0 0 0 (b − c) cos θ sin 2 θ 0 0 0 0 . Hence, s(P X R Y θ ) = ((c − b) cos θ sin 2 θ, 0). Now, |λ(X * A X) − λ((Y θ ) * A Y θ )| = ((c − b) sin 2 θ, 0) and s(P X R Y θ ) cos(Θ(X , Y θ )) = ((c − b) sin 2 θ, 0) . (11)
That is, Eq. (9) in Corollary 3.4 becomes an equality in this case. This also shows that Eq. (2) is sharp, since Eq. (9) above is a particular case (when X is A-invariant).
Notice that X + Y θ = span{e 1 , e 2 , e 3 }. Since P X +Y θ R Y θ = R Y θ and s(R Y θ ) = ((c − b) cos θ sin θ, 0) then s(P X +Y θ R Y θ ) tan(Θ(X , Y θ )) = ((c − b) sin 2 θ, 0) .(12)
By Eqs. (11) and (12)
|λ(ρ(X)) − λ(ρ(Y ))| 2 ≺ w {s(P Y R X ) + s(P X R Y )} 2 cos 2 (Θ(X , Y)) and (13) |λ(ρ(X)) − λ(ρ(Y ))| 2 ≺ w {s(P X +Y R X ) + s(P X +Y R Y )} 2 tan 2 (Θ(X , Y)) .(14)
Using the fact that f : R ≥0 → R ≥0 given by f (x) = x 2 is an increasing and convex function, then Remark 2.2 shows that Eqs. (13) and (14) follow from Eqs. (2) and (3) from Theorem 3.3. Similarly, using that cos In [20] the authors show that their results can be applied in several situations such as: first order and quadratic a posteriori majorization bounds; bounds for eigenvalues after matrix additive perturbations. The previous remarks show that our bounds can also be applied in these settings. Moreover, Theorem 3.3 allows to formalize the arguments related with bounds for eigenvalues after matrix additive perturbations, and in particular with bounds for eigenvalues after discarding off-diagonal blocks from [20, Section 5] (see the detailed discussion there). △
Θ 1 (X , Y) = cos Θ max (X , Y) ≤ cos Θ i (X , Y), for i ∈ I k ,
The bounds in Theorem 3.3 can be used to perform a detailed analysis and obtain better convergence rates for iterative algorithms related with the Rayleigh-Ritz method (see [16,17,21]). We will consider such applications elsewhere.
Applications: a priori majorization error bounds for Ritz values
In this section we establish a link between the majorization error bounds of the mixed type obtained in the previous section and some a priori majorization error bounds considered in [8,10].
Definition 3.7. Let A ∈ H(d) and let Z ⊂ C d be a subspace with dim Z = p. We consider the (spectral) spread of A relative to Z, denoted Spr(A , Z), given by
Spr(A, Z) = λ(A Z ) − λ ↑ (A Z ) = (λ i (A Z ) − λ p−i+1 (A Z )) i∈Ip ∈ (R p ) ↓ , where A Z = P Z A| Z ∈ L(Z) is a self-adjoint operator. In case Z = C d then we write Spr(A, C d ) = Spr(A). △ Remark 3.8. Let A ∈ H(d) and let X , Y ⊂ C d with dim(X ) = dim(Y) = k. Denote by p = dim X + Y.
In what follows we consider the vector
Spr(A, X + Y) sin(Θ(X , Y)) = ( (λ i (A X +Y ) − λ p−i+1 (A X +Y )) sin(Θ i (X , Y) ) ) i∈I k .
Notice that, by construction, [20]). △ Remark 3.9 (A priori error bounds for changes of Ritz values: conjectures and previous work).
Spr(A, X + Y) sin(Θ(X , Y)) ∈ (R k ≥0 ) ↓ (seeLet A ∈ H(d) and let X , Y ⊂ C d with dim(X ) = dim(Y) = k.
In [8] the authors conjectured that, in general, the following submajorization bound for the Ritz values of Rayleigh quotients holds:
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w Spr(A, X + Y) sin(Θ(X , Y)) .(15)
Moreover, in case X is A-invariant, the authors conjectured that
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w Spr(A, X + Y) sin(Θ(X , Y)) 2 .(16)
These conjectures are natural extensions of results from [10] (that were obtained for k = 1). Although [8, Conjecture 2.1.] claims the validity of Eqs. (15) and (16) for arbitrary subspaces X and Y such that dim X = dim Y, such bounds would become relevant in the particular case when the subspace Y is a (small) perturbation of the subspace X . In this case, the validity of Eqs. (15) and (16) would reveal the different orders of approximation of ρ(X) by ρ(Y ) in terms of PABS as well as in terms of the spectral spread of A (i.e. when considering A as well as X and Y as variables). Notice that these results would have immediate applications in the study of numerical stability and convergence of iterative methods related with the Rayleigh-Ritz type algorithms.
In [8, Theorem 2.1.] the authors showed that, in general,
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w (λ max (A X +Y ) − λ min (A X +Y )) sin(Θ(X , Y)) ,(17)
while, in case X is A-invariant,
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w (λ max (A X +Y ) − λ min (A X +Y )) sin(Θ(X , Y)) 2 ,(18)
where A X +Y = P X +Y A| X +Y ∈ L(X + Y); moreover, in [8, Theorem 2.2.] they showed that in the particular case in which X is the A-invariant subspace corresponding to the k largest eigenvalues of A, then
0 ≤ λ(ρ(X)) − λ(ρ(Y )) ≺ w (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin(Θ(X , Y)) 2 .(19)
Notice that, Eq. (19) is a stronger bound than that in Eq. (18); yet, it is weaker than the bound conjectured in Eq. (16)
, since Spr i (A, X + Y) ≤ λ i (A X +Y ) − λ min (A X +Y ), for i ∈ I k . △
In what follows we apply Theorem 3.3 and obtain some results related with the conjectures from [8] described in Eqs. (15) and (16). In order to obtain these results, we take a closer look at the quantity s(P X R Y ) for arbitrary X and Y, as well as in the case where X is A-invariant.
s(P X R Y ) ≺ w Spr(A, X + Y) sin(Θ(X , Y)) .(20)
Proof. See the Appendix (Section 4).
Theorem 3.11. Let A ∈ H(d), X , Y ⊂ C d subspaces, dim(X ) = dim(Y) = p. If Θ 1 (X , Y) < π 2 , then |λ(ρ(X)) − λ(ρ(Y ))| ≺ w 2 Spr(A, X , Y) sin(Θ(X , Y)) cos(Θ(X , Y)) = 2 Spr(A, X , Y) tan(Θ(X , Y)) .(21)
Proof. Theorem 3.3 establishes that
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w s(P X R Y ) + s(P Y R X ) cos(Θ(X , Y)) .
Proposition 3.10 together with Lemma 4.3 imply that Y)) .
s(P X R Y ) + s(P Y R X ) cos(Θ(X , Y)) ≺ w 2 Spr(A, X , Y) sin(Θ(X , Y)) cos(Θ(X ,
The result follows from combining these two last inequalities.
The next result illustrates the quadratic dependance of s(P X R Y ) from sin(Θ(X , Y)) in case X is A-invariant.
Proposition 3.12. Let A ∈ H(d), X , Y ⊂ C d subspaces with dim(X ) = dim(Y) = k. Assume that X is A-invariant. Then, s(P X R Y ) ≺ w 2 (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin 2 (Θ(X , Y)) .(22)
Proof. See the Appendix (Section 4).
Theorem 3.13. Let A ∈ H(d), X , Y ⊂ C d subspaces, dim(X ) = dim(Y) = k, and assume that X is A-invariant. If Θ 1 (X , Y) < π 2 , then |λ(ρ(X)) − λ(ρ(Y ))| ≺ w 2 (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin 2 (Θ(X , Y)) cos(Θ(X , Y)) .(23)
Proof. The result follows from Corollary 3.4 and Proposition 3.12 with an argument similar to that in the proof of Theorem 3.11 above.
Corollary 3.14. Let A ∈ H(d), X , Y ⊂ C d subspaces, dim(X ) = dim(Y) = k. If Θ 1 (X , Y) < π 2 , then |λ(ρ(X)) − λ(ρ(Y ))| ≺ w 2 cos(Θ 1 (X , Y)) Spr(A, X , Y) sin(Θ(X , Y)) .(24)
If we assume further that X is A-invariant, then
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w 2 cos(Θ 1 (X , Y)) (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin 2 (Θ(X , Y)) .(25)
We end this section with some remarks concerning the relation between Theorems 3.11 and 3.13, Corollary 3.14 and the conjectured bounds in Eqs. (15) and (16). As already mentioned in Remark 3.9, the bounds in Eqs. (15) and (16) would be particularly relevant in case Y is a (small) perturbation of X or, in other terms, in case that X and Y are close subspaces (e.g. Θ 1 (X , Y) is small). In order to simplify the discussion, let us assume that Θ 1 (X , Y) ≤ π/4. We point out that this assumption holds in a number of significant situations (see for example [20, Section 5.2.]). In this case, if A ∈ H(d) then Corollary 3.14 implies that
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w (2 √ 2) Spr(A, X , Y) sin(Θ(X , Y)) .(26)
Hence, under the present assumptions (Θ 1 (X , Y) ≤ π/4), the upper bound in Eq. (26) has the conjectured order of approximation (when considering A as well as the subspaces X and Y as variables), up to the constant factor 2 √ 2.
If we further assume that X is A-invariant then by the same result we get that
|λ(ρ(X)) − λ(ρ(Y ))| ≺ w (2 √ 2) (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin 2 (Θ(X , Y)) .(27)
Again, the upper bound in Eq. (27) has the conjectured order of approximation (when considering A as well as the subspaces X and Y as variables), up to the constant factor 2 √ 2. Moreover, notice that this bound holds for an arbitrary A-invariant subspace X (as opposed the bound in Eq. (19) from [8] that is shown to hold for special choices of A-invariant subspaces X ).
The tan Θ theorem revisited: improved quadratic a posteriori error bounds
In this section we revisit Nakatsukasa's extension of Davies-Kahan's tan(Θ) theorem. Our motivation is the study of an improved version of this result conjectured in [20] (see Corollary 3.22 below). We first recall the separation hypothesis for Nakatsukasa's result. As before, in this section we adopt Notation 3.1.
Definition 3.15. Let A ∈ H(d) and let X , Y ⊂ C d be subspaces with dim X = dim Y = k, such that X is A-invariant. Let [X, X ⊥ ], [Y, Y ⊥ ] ∈ U (d)
be unitary matrices such that the columns of (the d×k matrices) X and Y form ONB's of X and Y respectively. Given δ > 0 we say that (A , X , Y , δ) satisfies the Davies-Kahan-Nakatsukasa (DKN) separation property if there exist a ≤ b such that
1. λ i (X * ⊥ AX ⊥ ) = λ i (P X ⊥ A P X ⊥ ) ∈ [a, b], for i ∈ I d−k ; 2. λ i (Y * AY ) = λ i (P Y A P Y ) ∈ (∞, a − δ] ∪ [b + δ, ∞), for i ∈ I k . △
Next we state Nakatsukasa's tan Θ theorem under relaxed conditions.
Theorem 3.16 ([14]
). Let A ∈ H(d), let X , Y ⊂ C d and δ > 0 be such that (A , X , Y , δ) satisfies the DKN separation property. Then, Θ 1 (X , Y) < π/2 and we have that
δ tan(Θ(X , Y)) ≤ R Y ,
for every unitarily invariant norm · . Equivalently, δ tan(Θ(X , Y)) ≺ w s(R Y ).
Remark 3.17. Let A ∈ H(d), let X , Y ⊂ C d and δ > 0 be such that (A , X , Y , δ) satisfies the DKN separation property. Then, Theorem 3.16 requires the knowledge of the full matrix A in order to bound the (norm of the) vector tan(Θ(X , Y)) from above. Instead, it would be interesting to bound the vector tan(Θ(X , Y)) from above (only) in terms of the self-adjoint operator A X +Y = P X +Y A| X +Y ∈ L(X + Y). In the next result we show that the tan Θ theorem mentioned above allow to obtain such a result. Moreover, we will also see that it is possible to describe separation hypothesis for (A X +Y , X , Y), that are more general than the DKN separation hypothesis for (A, X , Y), for which the tan Θ theorem holds; arguing in terms of interlacing inequalities, we can show that these separation hypothesis on A X +Y provide better separation constants than the DKN conditions on the complete matrix A. △
We formalize the content of the previous remark -with a small variation on the notation -in the following result. First, we recall some facts related with the relative position of two subspaces.
Remark 3.18. Let X , Y ⊂ C d be two subspaces with dim X = dim Y = k. Consider the mutually orthogonal subspaces
H 00 = X ⊥ ∩ Y ⊥ , H 10 = X ∩ Y ⊥ , H 01 = X ⊥ ∩ Y , H 11 = X ∩ Y ,
and H g = C d ⊖ (H 00 ⊕ H 10 ⊕ H 01 ⊕ H 11 ) which is called the generic part of the pair (X , Y). Each of these five (possible zero) subspaces reduces each projection P X and P Y . Moreover, the subspaces X g = X ∩ H g and Y g = Y ∩ H g are in generic position so that H g = X g + Y g . For details of this well known construction and several fundamental results see [5]. △
Theorem 3.19. Let A ∈ H(d), let X , Y ⊂ C d be such that dim X = dim Y = k. Let A X +Y = S * AS ∈ H(p), where S ∈ M d,p (C)
is such that its columns form and ONB for X + Y. Then, 1. If δ > 0 is such that (A , X , Y , δ) satisfies the DKN separation property then there exists δ ′ ≥ δ such that (A X +Y , S * X , S * Y , δ ′ ) satisfies the DKN separation property.
2. If δ ′ > 0 is such that (A X +Y , S * X , S * Y , δ ′ ) satisfies the DKN separation property, then
δ ′ tan(Θ(X , Y)) ≤ A X +Y Y S − Y S (Y * S A X +Y Y S ) = P X +Y R Y ,(28)
for every unitarily invariant norm · , where Y S = S * Y ∈ M p,k (C).
Proof. We first show item 1. Let X, Y ∈ M d,k (C) be such that their columns form orthonormal bases of X and Y, respectively. By hypothesis, there exist a ≤ b such that
λ i (X * ⊥ AX ⊥ ) ∈ [a, b] for i ∈ I d−k and λ i (Y * AY ) ∈ (∞, a − δ] ∪ [b + δ, ∞) for i ∈ I k ,
where X ⊥ ∈ M d,d−k is such that its columns for an ONB for X ⊥ . Let Z = X + Y and notice that S ∈ M d,p (C) is an isometry from C p onto Z. Moreover, the matrix S * A S ∈ H(p). Similarly, X S = S * X, Y S = S * Y ∈ M p,k are isometries from C k onto S * X , S * Y ⊆ C p , respectively. Consider the mutually orthogonal subspaces
H 11 = X ∩ Y , X g = H g ∩ X and X g ⊥ = H g ⊖ X g ,
where H g is the subspace of C d corresponding to the generic part of the pair (X , Y) (see Remark 3.18). By Theorem 3.16 we have that Θ 1 (X , Y) < π/2 so then,
X ⊥ ∩ Y = {0} = X ∩ Y ⊥ . Thus, X = H 11 ⊕ X g , Z = H 11 ⊕ X g ⊕ X g ⊥ and X g ⊥ = Z ⊖ X .
Let X ′ ∈ M d,(p−k) (C) be such that its columns form an orthonormal basis of X g ⊥ ⊂ X ⊥ . Then,
X ′ S = S * X ′ ∈ M p,(p−k) (C) is an isometry from C p−k onto S * X g ⊥ = (S * X ) ⊥ ⊆ C p .
In order to check the DKN separation property for (A X +Y , S * X , S * Y) we consider the eigenvalues of
(X ′ S ) * (S * A S) X ′ S = (X ′ ) * S S * A S S * X ′ = (X ′ ) * A X ′ ∈ H(p − k) ,
since SS * = P Z ∈ M d (C), P Z X ′ = X ′ and (X ′ ) * P Z = (X ′ ) * . Hence, we now see that
λ i ((X ′ S ) * (S * A S) X ′ S ) = λ i (P X g ⊥ A P X g ⊥ ) for i ∈ I p−k .
Since X g ⊥ ⊂ X ⊥ we have that P X g ⊥ A P X g ⊥ is a compression of P X ⊥ A P X ⊥ . Using the interlacing inequalities for compressions of self-adjoint matrices (see [2]), we get that
λ i ((P X ⊥ A P X ⊥ )) ∈ [a, b] for i ∈ I d−k =⇒ λ i (P X g ⊥ A P X g ⊥ ) ∈ [a, b] for i ∈ I p−k . (29)
On the other hand, notice that
Y * S (S * A S) Y S = Y * P Z A P Z Y = Y * A Y
since, as before, SS * = P Z , P Z Y = Y and Y * P Z = Y * . Therefore, we get that
λ i (Y * S (S * A S) Y S ) = λ i (Y * A Y ) ∈ (∞, a − δ] ∪ [b + δ, ∞) for i ∈ I k .(30)
Item 1. now follows from Eqs. (29) and (30) and the fact that S * X ⊆ C p is, by construction, an A X +Y -invariant subspace.
We now show item 2. Fix an unitarily invariant norm · . Using that X , Y ⊂ Z and the fact that S * is an isometry from Z onto C p , we see that Θ(X , Y) = Θ(S * X , S * Y). Then, an application of Nakatsukasa's tan Θ theorem (Theorem 3.16) to the self-adjoint matrix S * AS ∈ H(p) and subspaces S * X , S * Y ⊆ C p shows that
δ ′ tan(Θ(X , Y)) ≤ A X +Y Y S − Y S (Y * S A X +Y Y S ) ,
where Y S = S * Y ∈ M p,k is an isometry from C k onto S * Y. We notice that
A X +Y Y S − Y S (Y * S A X +Y Y S ) = S * A S S * Y − S * Y (Y * S(S * A S)S * Y ) = S * (A Y − Y (Y * A Y )) ,
where we have used that SS * = P Z , P Z Y = Y and Y * P Z = Y * . Hence, it follows that
A X +Y Y S − Y S (Y * S A X +Y Y S ) = P Z (A Y − Y (Y * A Y )) = P X +Y R Y .
Remark 3.20. With the notation of Theorem 3.19 and using Remark 2.4, then Eq. (28) is equivalent to the majorization relation
δ ′ tan(Θ(X , Y) ≺ w s(A X +Y Y S − Y S (Y * S A X +Y Y S )) = s(P X +Y R Y )
in terms of the separation constant δ ′ for A X +Y = S * A S, S * X and S * Y. △
Consider the notation in Theorem 3.19. Let δ > 0 be such that (A , X , Y , δ) satisfies the DKN. Given a unitarily invariant norm · then, Theorem 3.16 allows to bound tan Θ(X , Y) from above by
tan Θ(X , Y) ≤ R Y δ .(31)
On the other hand, by item 2 in Theorem 3.19 there exists δ ′ ≥ δ > 0 such that (A X +Y , S * X , S * Y , δ ′ ) satisfies the DKN separation property, so that we get the upper bound
tan Θ(X , Y) ≤ P X +Y R Y δ ′ .(32)
Since P X +Y R Y ≤ R Y and δ ≤ δ ′ , we immediately see that the upper bound in Eq. (32) improves the classical bound in Eq. (31). In order to compare these two bounds in some more detail, let us consider the following Let X , Y θ ⊂ C 4 be as in Example 3.5 i.e. X = span{e 1 , e 2 } and Y θ = span{e 1 , f θ }. Recall that Θ(X , Y θ ) = (θ, 0). In particular, tan Θ(X , Y θ ) = (tan θ, 0) in this case.
It is clear that X + Y θ = span{e 1 , e 2 , e 3 }. Let
X = 1 0 0 1 0 0 0 0 , X ⊥ = 0 0 0 0 1 0 0 1 and Y θ = 1 0 0 cos θ 0 sin θ 0 0 .
Then, we have that λ(Y * θà Y θ ) = (b cos 2 θ + d sin 2 (θ), a), while λ(X * ⊥à X ⊥ ) = (d, c). Therefore, if we let θ 0 (c) = θ 0 = arcsin c−b d−b and we set δ θ = c − (b cos 2 θ + d sin 2 θ) > 0 for 0 < θ < θ 0 , then (Ã, X , Y θ , δ θ ) satisfies the DKN-separation property, and δ θ is the optimal (largest) separation constant and the separation property holds only for 0 < θ < θ 0 in this case. Again, simple computations show that s(R Y θ ) = ((d − b) cos θ sin θ, 0). Now, Eq. (31) obtained from Theorem 3.16 becomes
tan θ ≤ (d − b) cos θ sin θ c − (b cos 2 θ + d sin 2 θ) for 0 < θ < θ 0 .(33)
Notice that lim c→b + θ 0 = 0 i.e., the range of θ for which we can apply the bound in Eq. (33) tend to become small. In the limit case in which b = c (i.e. multiple eigenvalues) then we can not apply the bound (33) (the separation constant in this case is δ 0 = 0). Finally, if we consider the limit case in which θ becomes small, then the upper bound is comparable with the upper bound
( d−b c−b ) tan θ (> tan θ).
On the other hand, X + Y θ ⊖ X = C e 3 , the subspace spanned by e 3 . In this case, if we let
X ′ = (0, 0, 1, 0) t , it is clear that λ((X ′ S ) * Ã X ′ S ) = d. Therefore, if we let δ ′ θ = d − (b cos 2 θ + d sin 2 θ) = (d − b)
cos 2 θ > 0, for θ ∈ (0, π/2), we get that (Ã X +Y θ , S * X , S * Y θ , δ ′ θ ) satisfies the DKN-separation property, where S ∈ M 4,3 (C) is the matrix whose columns are the first three elements in the canonical basis. In this case we have that
s 1 (P X +Y θ R Y θ ) δ ′ θ = (d − b) cos θ sin θ (d − b) cos 2 θ = tan θ ,
and hence, the upper bound in Eq. (32) coincides with tan θ (where tan Θ(X , Y θ ) = (tan θ, 0)) i.e. the upper bound is sharp. Notice that the bound is applicable for every θ ∈ (0, π/2). △
The following result was conjectured in [20].
Corollary 3.22. Let A ∈ H(d), let X , Y ⊂ C d and δ > 0 be such that (A , X , Y , δ) satisfies the DKN separation property. Then, δ tan(Θ(X , Y)) ≤ P X +Y R Y .
for every unitarily invariant norm · .
Proof. Let S ∈ M d,p (C) be such that its columns form and ONB for X + Y. By item 1. in Theorem 3.19, there exists δ ′ ≥ δ such that (S * A S , S * X , S * Y , δ ′ ) satisfies the DKN separation property. By item 2. of the same result, we have that
δ tan(Θ(X , Y)) ≤ δ ′ tan(Θ(X , Y)) ≤ P X +Y R Y .
Finally, we get the following quadratic a posteriori error bound for the simultaneous approximation of eigenvalues of A by the Ritz values corresponding to Rayleigh quotients for which a DKN separation property holds.
Theorem 3.23. Let A ∈ H(d), let X , Y ⊂ C d and δ > 0 be such that (A , X , Y , δ) satisfies the DKN separation property. Then, for every unitarily invariant norm · we have that
λ(ρ(X)) − λ(ρ(Y )) ≤ P X +Y R Y 2 δ .
Proof. This is a consequence of Corollary 3.4 and Theorem 3.19.
Theorem 3.23 allows to obtain the following extension of [19,Theorem 5.3] (see Remark 3.25 below) which is a quadratic a posteriori majorization error bound for simultaneous approximation of consecutive eigenvalues.
Corollary 3.24. Let A ∈ H(d) and let Y ⊂ C d be such that:
1. λ 1 (Y * AY ) < λ j (A), where j ∈ I d−k is the smallest such index; 2. λ i (Y * AY ) ≥ λ i+j (A), for i ∈ I k .
Let U be the A-invariant space spanned by the eigenvectors associated with λ i (A), for 1 ≤ i ≤ j,
and set X = (I − P U )Y. If we let η = λ j (A) − λ 1 (Y * AY ) > 0 then, we have that (λ i+j (A)) i∈I k − λ(ρ(Y )) ≤ P X +Y R Y 2 η ,
for every unitarily invariant norm · .
Proof. Let V = U + Y and notice that U ∩ Y = {0}; hence, p = dim V = dim U + k i.e. j = dim U = p − k. Moreover, V ⊖ U = (I − P U )Y = X ; then, in particular, dim X = dim Y and V ⊖ X = U . Also notice that Θ 1 (X , Y) < π/2 or otherwise, we would have that U ∩ Y = {0}, since V ⊖ X = U .
Let V ∈ M d,p (C) be such that its columns form a ONB of V and set A V = V * AV ∈ H(p).
Similarly, let X, Y ∈ M d,k (C), U ∈ M d,p−k (C) be such that their columns form ONB's of X , Y and U respectively; set
X V = V * X, Y V = V * Y ∈ M p,k (C) and U V = V * U ∈ M p,p−k (C). Then, the columns of U V span U V ⊂ C p an A-invariant space of A V . In particular, the columns of X V span X V ⊂ C p which is also an A-invariant space of A V . In this case X ⊥ V = U V and Θ 1 (X V , Y V ) = Θ 1 (X , Y) < π/2, where Y V ⊂ C p is the space spanned by the columns of Y V . Notice that, by construction λ i (Y * V A V Y V ) = λ i (Y * A Y ), for i ∈ I k .
Since X ⊂ U ⊥ by the interlacing inequalities for compressions of self-adjoint matrices and Item 2 above, we get that
λ i (X * V A V X V ) = λ i (X * A X) ≤ λ i (A U ⊥ ) = λ j+i (A) ≤ λ i (Y * A Y ) = λ i (Y * V A V Y V )(34)
for i ∈ I k , where U ⊥ ∈ M d,d−j (C) is such that its columns for an ONB for U ⊥ . On the other hand, by hypothesis (A V , X V , Y V , η) satisfies the DKN separation property (recall that X ⊥ V = U V ). Hence, by Theorem 3.23 we conclude that
λ(X * V A V X V ) − λ(Y * V A V Y V ) ≤ P X V +Y V (A V Y V − Y V (Y * V A V Y V )) 2 η .(35)
By Eq. (34) we get that
|(λ i+j (A)) i∈I k − λ(Y * V A V Y V )| ≺ w |λ(X * V A V X V ) − λ(Y * V A V Y V )| .
On the other hand, arguing as in the proof of Theorem 3.19 we see that
P X V +Y V (A V Y V − Y V (Y * V A V Y V )) = P X +Y R Y .
The result follows from these last facts together with Eq. (35) and Remark 2.4.
Remark 3.25. We mention that the hypothesis in item 1. in Corollary 3.24 is that there exists an eigenvalue β of A such that λ 1 (Y * AY ) < β. Indeed, in this case we can apply the interlacing inequalities and get that λ i (Y * AY ) ≥ λ d−k+i (A), for i ∈ I k . Therefore, β = λ j (A) for some
1 ≤ j ≤ d − k.
The hypothesis in item 2. is rather restrictive and difficult to check in general. Nevertheless, we mention two cases in which the hypotheses in Corollary 3.24 can be easily checked:
1. In case the hypothesis in item 1 holds for j = d − k then, by the interlacing inequalities
λ i (Y * AY ) ≥ λ i+d−k (A) for i ∈ I k ,
so the hypothesis in item 2 automatically hold.
2. In case k = 1 that is, if Y = C y for a unit norm vector y ∈ C d , the hypotheses become the existence of j ∈ I d−1 such that λ j+1 (A) ≤ A y, y < λ j (A); then, Corollary 3.24 implies that 0 ≤ Ay, y − λ j+1 (A) ≤ P X +Y (Ay − Ay, y y) λ j (A) − Ay, y ,
where X = C x, for x = (I − P U )y ∈ C d ; this is [19,Theorem 5.3]. As explained in [19], Corollary 3.24 encodes several known bounds related with eigenvalue estimation even when k = 1. △
Appendix
Here we collect several and well known results about majorization, used throughout our work. The first result deals with submajorization relations between singular values of arbitrary matrices in M d (C). For detailed proofs of these results and general references in majorization theory see [2,6,12]. 4. If we assume that CD ∈ H(d) then s(CD) ≺ w s(re(DC)).
For hermitian matrices we have the following majorization relations 3. Let P = {P j } r j=1 be a system of projections (i.e. they are mutually orthogonal projections on C d such that r i=1 P i = I). If C P (C) = r i=1 P i CP i , then λ(C P (C)) ≺ λ(C).
In the next result we describe several elementary but useful properties of (sub)majorization between real vectors. Lemma 4.3. Let x, y, z ∈ R k . Then,
1. x ↓ + y ↑ ≺ x + y ≺ x ↓ + y ↓ ;
2. If x ≺ w y and y, z ∈ (R k ) ↓ then x + z ≺ w y + z;
If we assume further that x, y, z ∈ R k ≥0 then,
3. x ↓ y ↑ ≺ w x y ≺ w x ↓ y ↓ ;
4. If x ≺ w y and y, z ∈ (R k ≥0 ) ↓ then x z ≺ w y z. [Theorem 4.6, [8]] Let X , Y ⊂ C d be such that dim(X ) = dim(Y) = k. Then λ(P X P Y ⊥ P X ) = s(P X P Y ⊥ P X ) = s 2 (P Y P X ⊥ ) = s 2 (P X ⊥ P Y ) = (sin 2 (Θ(X , Y)), 0 d−k ).
Notice that item 2. below is Theorem 3.2 from Section 3.
Theorem 4.6. Let C, D ∈ H(k). Then,
1. if T ∈ Gl(k) + , then s(C − D) ≺ w s(T −1 ) s(CT − T D) . 2. if T ∈ Gl(k), then |λ(C) − λ(D)| ≺ w s(T −1 ) s(CT − T D).
Proof. We first show item 1. Since T is positive and invertible, using Theorem 4.2 (item 3.) we get that
s(C − D) = s(CT 1 2 T − 1 2 − T − 1 2 T 1 2 D)) = s(T − 1 2 (T 1 2 CT 1 2 − T 1 2 DT 1 2 )T − 1 2 ) ≺ w s(T − 1 2 ) 2 s(T 1 2 CT 1 2 − T 1 2 DT 1 2 ) = s(T −1 ) s(T 1 2 (C − D) T 1 2 ) .
By Theorem 4.1 (items 2. and 4.) and the fact that re(DT ) = re(T D) we obtain that
s(T 1 2 (C − D)T 1 2 ) ≺ w s(re[(C − D)T ]) = s(re[CT − T D]) ≺ w s(CT − T D),(36)
By the previous inequalities and Lemma 4.3 we see that
s(C − D) ≺ w s(T −1 ) s(CT − T D) .(37)
In order to show item 2, consider a representation of T given by T = U ΣV * , where U, V ∈ U (k) are unitary matrices and Σ ∈ M k (C) is the diagonal matrix with main diagonal s(T ) ∈ R k ≥0 (notice that such representation follows from the SVD decomposition of T ); note that Σ is definite positive and invertible. Using item 2 in Theorem 4.2 and (the already proved) item 1. of the statement we get
|λ(C) − λ(D)| = |λ(U * CU ) − λ(V * DV )| ≺ w s(U * CU − V * DV ) ≺ w s(Σ −1 ) s(U * CU Σ − ΣV * DV ) = s(T −1 ) s(U * (CT − T D)V ) = s(T −1 ) s(CT − T D) .
In what follows we re-state and prove two propositions of Section 3.2.
Proof. Consider A ∈ H(d) and X , Y ⊂ C d with dim(X ) = dim(Y) = k. In what follows we show that s(P X R Y ) ≺ w Spr(A, X + Y) sin(Θ(X , Y)).
We begin with a simple reduction argument. A simple calculation show that (s(P X R Y ), 0 d−k ) = s(P X (A P Y − P Y A P Y )) ∈ (R d ≥0 ) ↓ . Let Z = X + Y with dim Z = p, and consider the matrix representations with respect to the decomposition C d = Z ⊕ Z ⊥ : P X = P X 0 0 0 , P Y = P Y 0 0 0 and A = A Z * * * ,
where P X , P Y , A Z = P Z A| Z ∈ L(Z) are self-adjoint operators. In this case we have
P X (A P Y − P Y A P Y ) = P X (A Z P Y − P Y A Z P Y ) 0 0 0 .
Hence, (s(P X R Y ), 0 p−k ) = s(P X (A Z P Y − P Y A Z P Y )) = s(P X (I Z − P Y ) A Z P Y ). Thus, we can assume further that C d = Z = X + Y and show that (s(P X R Y ), 0 d−k ) = s(P X (P Y ⊥ A P Y )) ≺ w (Spr(A) sin(Θ(X , Y)), 0 d−k ) .
Now using multiplicative Lidskii's
s(P X P Y ⊥ AP Y ) = s(P X P Y ⊥ P Y ⊥ AP Y ) ≺ w s(P X P Y ⊥ ) s(P Y ⊥ AP Y ).(40)
First notice that by Theorem 4.5, we have that s(P X P Y ⊥ ) = (sin(Θ(X , Y)), 0 d−k ). On the other hand, consider the matrix representation induced by the decomposition C d = Y ⊕ Y ⊥ :
A = A 11 A * 21 A 21 A 22 and set A 1 := A 11 0 0 A 22 , A 2 := 0 A * 21 A 21 0 .(41)
Then, we have that A = A 1 + A 2 . Now, A 1 is a pinching of A (associated with the system of projections
{P Y , P Y ⊥ }) so λ(A 1 ) ≺ λ(A) so then − λ ↑ (A 1 ) ≺ −λ ↑ (A) .(42)
Using Lidskii's additive inequality for A 2 = A − A 1 (see item 1 in Theorem 4.2)
λ(A 2 ) ≺ λ(A) − λ ↑ (A 1 ) .(43)
Combining (42)
where Spr(A) = (Spr i (A)) i∈I d . Using Eqs. (40) and (45) together with Lemma 4.3 we finally get that s(P X P Y ⊥ AP Y ) ≺ w (Spr(A) sin(Θ(X , Y)), 0 d−k ) ∈ (R d ≥0 ) ↓ . Now the result follows from the last submajorization relation, by considering the first k entries of both vectors. Proposition 3.12. Let A ∈ H(d), X , Y ⊂ C d subspaces with dim(X ) = dim(Y) = k. Assume that X is A-invariant. Then, s(P X R Y ) ≺ w 2 (λ i (A X +Y ) − λ min (A X +Y )) i∈I k sin 2 (Θ(X , Y)).
(46)
Proof. Arguing as in the proof of Proposition 3.10, we can assume further that C d = X + Y. With this assumption, we consider first the case where A ∈ M d (C) + and show that s(P X R Y ) ≺ w 2 (λ i (A)) i∈I k sin 2 (Θ(X , Y)) .
Indeed, the A-invariance of X , allows us to write A = P X AP X +P X ⊥ AP X ⊥ . With this decomposition in mind and using the fact that (s(P X R Y ), 0 d−k ) = s(P X P Y ⊥ AP Y ), we have that s(P X P Y ⊥ A P Y ) = s(P X P Y ⊥ P X A P X P Y + P X P Y ⊥ P X ⊥ A P X ⊥ P Y ) ≺ w s(P X P Y ⊥ P X A P X P Y ) + s(P X P Y ⊥ A P X ⊥ P Y ) def = M .
Using Theorem 4.1 (multiplicative Lidskii's), the fact that 0 d ≤ s(P X P Y ) ≤ 1 d and Theorem 4.5, we get M ≺ w s(P X P Y ⊥ P X ) s(A) + s(P X P Y ⊥ ) s(A) s(P X ⊥ P Y )
≺ w 2 λ(A) (sin 2 (Θ(X , Y)), 0 d−k ) ∈ (R d ≥0 ) ↓ ,
since A ∈ M d (C) + is positive semi-definite. The result now follows from the previous facts.
In general, for A ∈ H(d) consider the auxiliary matrixà = A − λ min(A) I ∈ M d (C) + . Notice that
R Y (Ã) =Ã Y − Y (Y * Ã Y ) = A Y − Y (Y * A Y ) = R Y and λ(Ã) = λ(A) − λ min(A) 1 d .
The result now follows from these facts and from Eq. (47) applied toÃ.
the eigenvalues of A counting multiplicities and arranged in non-increasing order. For B ∈ M d (C) we let s(B) = λ(|B|) denote the singular values of B, i.e. the eigenvalues of |B| = (B * B) 1/2 ∈ M d (C) + .
Remark 2. 4 .
4It is well known that (sub)majorization relations between singular values of matrices are intimately related with inequalities with respect to u.i.n's. Indeed, given A, B ∈ M d (C) the following statements are equivalent: 1. For every u.i.n. N in M d (C) we have that N (A) ≤ N (B).
together with Lemma 4.3 we see that Eq. (3) holds.
Proposition 3. 10 .
10Let A ∈ H(d) and let X , Y ⊂ C d with dim(X ) = dim(Y) = k. Then
Example 3 . 21 .
321Letλ = (a, b, d, c) ∈ R 4 , where a < b < c < d, and letà ∈ H(4) be given bỹ A = Dλ. For the purposes of this example, we consider the real parameter c ∈ (b, d) as variable (while a, b, d are fixed).
Theorem 4. 1 .
1Let C, D ∈ M d (C). Then, 1. s(C + D) ≺ w s(C) + s(D); 2. s(re(C)) ≺ w s(C) ; 3. s(CD) ≺ w s(C) s(D);
Theorem 4. 2 .
2Let C, D ∈ H(d). Then, 1. λ(C) − λ(D) ≺ λ(C − D) ≺ λ(C) − λ ↑ (D); 2. |λ(C) − λ(D)| ≺ w s(C − D);
Proposition 4 . 4 .
44Let 1 ≤ k < d and let E ∈ M k,(d−k) (C). Then E = 0 E E * 0 ∈ H(d) and λ(Ê) = (s(E), −s(E * )) ↓ ∈ (R d ≥0 ) ↓ .
Proposition 3. 10 .
10Let A ∈ H(d) and let X , Y ⊂ C d with dim(X ) = dim(Y) = k. Then s(P X R Y ) ≺ w Spr(A, X + Y) sin(Θ(X , Y)) .
and (43), we obtainλ(A 2 ) ≺ λ(A) − λ ↑ (A) = Spr(A) ∈ R d .(44)By Proposition 4.4, we get that λ(A 2 ) = (s(A 21 ), −s(A * 21 )) ↓ ; in particular, s(A 21 ) = (λ i (A 2 )) i∈I k . Now, s(P Y ⊥ AP Y ) = (s(A 21 ), 0 d−k ); thus, we see thats(P Y ⊥ AP Y ) = (s(A 21 ), 0 d−k ) = ((λ i (A 2 )) i∈I k , 0 d−k ) ≺ w ((Spr i (A)) i∈I k , 0 d−k ) ,
we now see that Eq. (10) in Corollary 3.4 becomes an equality in this case. This also shows that Eq. (3) is sharp, since Eq. (10) above is a particular case (when X is A-invariant). △ Remark 3.6 (Relations between our work and previous results). In the vector case, that is when X and Y are one dimensional spaces, Theorem 3.3 implies the upper bounds in [19, Theorem 3.7], which is one the main results of that work (see also Corollary 3.24 and Remark 3.25). In [20] Knyazev and Zhu obtain several bounds for the absolute change of the Ritz values of Rayleigh quotients. Using Notations 3.1, the authors show (see [20, Theorem 4.2 and Corollary 4.4]) that
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| []
|
[
"Equivariant bifurcations in 4-dimensional fixed point spaces",
"Equivariant bifurcations in 4-dimensional fixed point spaces"
]
| [
"Reiner Lauterbach ",
"Sören Schwenker "
]
| []
| []
| In honor of Marty Golubitsky on the occasion of his seventieth birthday.AbstractIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach[14]and Lauterbach and Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples. | 10.1080/14689367.2016.1219696 | [
"https://arxiv.org/pdf/1511.00545v2.pdf"
]
| 119,676,295 | 1511.00545 | ec9ef7f4371fb47768e354303a166204f65eac32 |
Equivariant bifurcations in 4-dimensional fixed point spaces
Reiner Lauterbach
Sören Schwenker
Equivariant bifurcations in 4-dimensional fixed point spaces
In honor of Marty Golubitsky on the occasion of his seventieth birthday.AbstractIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach[14]and Lauterbach and Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.
Introduction
Lauterbach and Matthews [15] have looked at the Ize conjecture:
Conjecture (J. Ize). Let V be a real, linear and absolutely irreducible representation of a finite group or a compact Lie group G. Then there exists an isotropy subgroup H ≤ G with odd-dimensional fixed point space.
They proved that this conjecture is not true by presenting three infinite families of finite groups acting on R 4 , such that any of these groups has only nontrivial isotropy subgroups whose corresponding fixed point spaces are two-dimensional. They also show that for equivariant bifurcations with any group in the first two families at least one of the nontrivial isotropy types is generically symmetry breaking (in the sense of Field and Richardson [7]). In their construction each of these families relates to a compact Lie group, which contains all the groups in the family, however these Lie groups do not a play a substantial role in the analysis. Concerning dimensions of representation spaces which are small multiples of 4 they provide tables presenting computational results on counterexamples to the Ize conjecture including the three mentioned families. It turns out that there are, besides the three families, many more potential counterexamples to the Ize conjecture (however there are no proofs yet). The bifurcation question for all of these groups is completely open. In Lauterbach [14] the third family is analysed including the question concerning the generic bifurcations. Based on this information a new family of infinitely many finite groups acting on R 8 is constructed. For both cases in dimension 4 and in dimension 8 it is shown that generically the (only) nontrivial isotropy type is symmetry breaking in the sense of Field and Richardson. Again there is a compact Lie group which plays no visible role in this context. In part 4 of Theorem B in [14], Lauterbach stated that this Lie group is a counter example to the Ize conjecture. However no proof for this statement is provided and in fact it is not correct as one can easily see. In this paper we investigate infinite families of finite groups whose orders do not form an arithmetic progression as in the previous examples. Moreover we construct a new family acting on R 4 and based on this family a second family acting on R 8 which has a single nontrivial isotropy type and the dimension of its fixed point space is four dimensional. We prove that this isotropy type is generically symmetry breaking. The proofs are substantially different from the previous ones, here we make essential use of the Lie groups containing the groups in the family. The general question whether counterexamples to Ize's conjecture possess isotropy types which are generically symmetry breaking is open, but our technique might provide a tool to either construct counterexamples or to provide proofs.
Main results
Lauterbach and Matthews [15] have constructed three families of groups of orders 16 with ∈ 2N + 1, acting absolutely irreducibly on R 4 and leading to counterexamples to the Ize conjecture. In [14], Lauterbach continues this work and constructs a family of groups of order 64 with ∈ 2N + 1 acting absolutely irreducibly on R 8 with only even-dimensional fixed point subspaces. In this paper we construct groups of order 8m where m is odd and of the form m = a · b with a, b ∈ 2N + 1 and gcd(a, b) = 1 (1.1) (This sequence is listed in the On-Line Encyclopedia of Integer Sequences as sequence A061346 (http://www.oeis.org)). These groups act absolutely irreducibly on R 4 and we use them to construct groups twice their size acting absolutely irreducibly on R 8 . For this step a needs to be of a special form guaranteeing the existence of square roots of −1 modulo a: Proposition 1.1. Let a i = 1 mod 4 be prime and s i ∈ N for i = 1, . . . , r. Furthermore let
a = r i=1 a s i i .
Then there exists ρ ∈ N such that
ρ 2 = −1 mod a.
For further use we denote the set of such a by A: On-Line Encyclopedia of Integer Sequences (http://www.oeis.org)). The groups acting on R 8 have precisely one nontrivial isotropy type which has a 4-dimensional fixed point space and therefore lead to counterexamples to the Ize conjecture in dimension 8.
A = r i=1 a s i i | r ∈ N; a i
The construction in Lauterbach and Matthews [15] relies heavily on the biquaternionic characterization of elements in SO (4) presented in Conway and Smith [3]. We briefly recall the necessary notations. Denote the space of quaternions with the standard basis {1, i, j, k} where i 2 = j 2 = k 2 = −1 by H and let Q ⊂ H be the set of unitary quaternions. The group of ordered pairs of such quaternions Q × Q is isomorphic to the 4-dimensional spin group Spin 4 . Identifying H with R 4 in the obvious manner,
x = (x 1 , x 2 , x 3 , x 4 ) T ↔ x 1 + ix 2 + jx 3 + kx 4 ,
these pairs correspond to elements in SO(4) via [l, r] : x →lxr.
Conway and Smith [3] show that this is a two-to-one map on SO (4) where the needed identification is [1,1] (4) in terms of the biquaternionic notation.
It is a subtle yet very important observation that this map is -when taking the identification into account -a bijection but it is not a group homomorphism. Following Chillingworth, Lauterbach, with the map given in Conway and Smith [3] asl = l −1 for unitary quaternions. The application of [l, r] therefore yields an antirepresentation. The tilde notation is obsolete from now on.
In a similar manner we can construct a map Q × Q → O(4) \ SO(4) using * [l, r] : x → lxr.
We do not need the explicit definition of this map but the fact that it is two-to-one as well turns out to be helpful. We may now define the groups, we want to study in more detail. Let e p = e πi p be one of the primitive p-th root of −1 in C and denote a group that is generated by the elements g 1 , g 2 , . . . by g 1 , g 2 , . . . . Choose a, b ∈ 2N + 1 such that they are relatively prime and define
H a,b = [e a , 1], [1, e b ], [1, j], [j, 1] . (1.3)
We summarize results on the structure and the 4-dimensional representation of these groups in the following theorems. (2) Let b < b where b is odd and relatively prime to a.
If b divides b , H a,b is a subgroup of H a,b , i.e. H a,b ≤ H a,b .
(3) The action of H a,b on H as defined in (1.2) is absolutely irreducible. It has precisely two nontrivial isotropy types. The corresponding fixed point spaces are 2dimensional.
We use this construction to define the family H a for each a ∈ 2N + 1:
H a = {H a,b | b ∈ 2N + 1 and gcd(a, b) = 1} .
The last result of Theorem 1.2 allows us to generate a one-dimensional Lie group for each suitable a as follows: forms a compact Lie group of dimension 1. Its action on H is absolutely irreducible and it possesses isotropy subgroups with one-dimensional fixed point space.
Theorem 1.3. (1) Let a ∈ 2N
(2) Let a, a ∈ 2N + 1 odd with a < a . If a divides a , H a is a subgroup of H a , i.e.
H a ≤ H a .
In the same manner this gives rise to a Lie group of dimension 2:
Theorem 1.4. The set H = a∈2N+1 H a = [1, j], [j, 1], [e iφ , 1], [1, e iψ ] | φ, ψ ∈ S 1
forms a compact Lie group of dimension 2.
To perform the final step in the construction of the groups acting on R 8 we need the matrix representatives of the generating elements of the groups H a,b and denote them as follows:
[e a , 1] ↔ c, [1, e b ] ↔ d, [1, j] ↔ q, [j, 1] ↔ s.
We then look at 8-dimensional representations of the groups constructed so far and extend them so that the representation becomes absolutely irreducible. Let a ∈ A and b ∈ 2N + 1 such that a and b are relatively prime as before. Choose ρ as in Proposition 1.1. Without loss of generality we may assume ρ to be odd. If ρ 2 = −1 mod a then the same holds for −ρ. Since a is odd, either ρ or −ρ is odd. We define a group as follows: We are interested in the direct sum of these two representations. It defines a group action of H a,b on R 8 which is obviously reducible. To guarantee absolute irreducibility we need to supplement the set of generators of H a,b with an element v which exchanges the blocks of the two representations. We define its action on R 8 as follows: let x, y ∈ R 4 then v x y = 1]x .
H = [e a , 1] ρ , [1, e b ],1 4 y [j,
Since v cannot be displayed properly in terms of pairs of unitary quaternions we focus on the matrix representation from now on. To meet the assumptions we have made on the 8-dimensional representation we define the matrix generators as block matrices as follows (see (1.4) and (3.2)):
C(a) = C = c 0 0 c ρ D(b) = D = d 0 0 d Q = q 0 0 −q S = s 0 0 s V = 0 1 4 s 0 .
With these we define the groups acting on R 8 in terms of matrix generators
G a,b = C, D, Q, S, V
where the dependence on a and b lies in the matrices C and D.
We obtain similar results on the structure and the 8-dimensional representations as in the 4-dimensional case.
Theorem 1.5. (1) For each a ∈ A and b ∈ 2N+1 with gcd(a, b) = 1 as above the group G a,b forms a subgroup of O(8) of order 16m where m = a · b.
(2) Let b < b where b is odd and relatively prime to a.
If b divides b , G a,b is a subgroup of G a,b , i.e. G a,b ≤ G a,b .
(3) The natural 8-dimensional representation of G a,b is absolutely irreducible. It has precisely one nontrivial isotropy type. The corresponding fixed point space is 4dimensional.
We define families of these groups for every suitable a as well
G a = {G a,b | b = 1 odd and gcd(a, b) = 1} .
In a similar manner as in the 4-dimensional case we can generate compact Lie groups of dimension 1 from the groups G a,b for every a ∈ A. To do so we adapt notation of the generating matrices to characterize arbitrary rotations. Denote the 2-dimensional rotation matrix by an angle ψ by r(ψ) and write
d(ψ) = r(−ψ) 0 0 r(ψ) ,
where ψ ∈ S 1 (compare this to (3.2)) and
D(ψ) = d(ψ) 0 0 d(ψ) .
Theorem 1.6. Let a ∈ A. Then the set
G a = G∈Ga H = C, Q, S, V, D(ψ) | ψ ∈ S 1
forms a compact Lie group of dimension 1. Its natural 8-dimensional representation is absolutely irreducible and it possesses isotropy subgroups with one-dimensional fixed point spaces.
Furthermore we investigate the equivariant structure and the bifurcation behaviour of G a,b -symmetric systems and obtain the final result. Theorem 1.7. Let a ∈ A with a > 5 and G ∈ G a . The 8-dimensional representation of G has no quadratic equivariants. The space of cubic equivariants P 3 G (R 8 , R 8 ) is 5dimensional. A basis is given by the maps E 1 , . . . , E 5 (see Table 3). Furthermore these are equivariant with respect to the Lie groups G a .
Main Theorem. For the natural 8-dimensional representation of G ∈ G a with 5 < a the only nontrivial isotropy type is generically symmetry breaking. Systems that are symmetric with respect to this representation generically have nontrivial symmetry breaking branches of steady states that are hyperbolic within the fixed point spaces.
Remark. The bifurcation result holds true for the groups H a,b as well. The two nontrivial isotropy types of the 4-dimensional representation are generically symmetry breaking. The proof uses exactly the same techniques as the proof for the main theorem. But as this is no new result on counterexamples to the Ize conjecture in 4-dimensional representations, we omit the details and only present the proof for the 8-dimensional case.
Prime numbers of the form 1 mod 4
To construct the groups acting absolutely irreducibly on R 8 from the ones acting on R 4 it is crucial that we restrict ourselves to numbers a which are products of prime numbers of the form 1 mod 4. We quote some number theoretic results first that eventually deliver square roots of −1 in suitable congruences. The first and easiest result, which is proved using Wilson's theorem, can be found for example in Hardy and Wright [13]. For a more thorough historical discussion of this question see Gauss [12].
Proposition 2.1. Let a be a prime number of the form a = 1 mod 4. Then there exists ρ ∈ N such that
ρ 2 = −1 mod a.
The next step is to apply a method based on Hensel's lemma (see Eisenbud [4] or Milne [17] for the formulation that is used here) that provides the same result for prime powers.
Proposition 2.2. Letã be a prime number of the formã = 1 mod 4 and a =ã s for some s ∈ N. Then there exists ρ ∈ N such that
ρ 2 = −1 mod a.
Proof. We use Hensel's lemma in the formulation given in Milne [17] with the polynomial f (X) = X 2 + 1.
Performing an induction we obtain zeros in congruences of arbitrary powers of a, since we have at least the zero modulo a from Proposition 2.1.
Now we can apply the Chinese remainder theorem (see for example Eisenbud [4]) to obtain the result for arbitrary products of prime powers. Hence this completes the proof for Proposition 1.1.
Families of groups
Representation on R 4
In a first step towards the proof of the results on the 4-dimensional representation we investigate the structure of the groups H a,b as defined before (see (1.3)). Note that the generators are subject to several relations which we summarize in the following lemma.
[e a , 1] a = [1, j] 2 , [1, e b ] b = [1, j] 2 , [j, 1] 2 = [1, j] 2 .
These relations allow us to write every group element h ∈ H a,b in the form
h = [e a , 1] k 1 [1, e b ] k 2 [1, j] l 1 [j, 1] l 2 (3.1)
where k 1 ∈ Z/aZ, k 2 ∈ Z/bZ, l 1 ∈ Z/4Z and l 2 ∈ Z/2Z. We present the proof for Theorem 1.2 in the following lemmas. Proof. Comparing H a,b with Table 4.2 in Conway and Smith [3] and using their notation we find
H a,b = ± [D 2a , D 2b ]
where D 2n is the dihedral group of order 2n. This group is of order 2 · 2a · 2b = 8m. In the notation of Conway and Smith [3], the ±[. . .] is reflected in a factor 2 in the group orders. The order of H a,b can also be derived directly from the representation of group elements in terms of the generators (3.1).
Remark. The definition of H a,b is symmetric in a and b:
H a,b ∼ = H b,a .
However choosing different decompositions for a value of m leads to groups of the same order which are not necessarily isomorphic. At the end of this section we have listed some concrete examples (see Table 1).
Lemma 3.3. Let b < b where b is odd and relatively prime to a. If b divides b , H a,b is a subgroup of H a,b , i.e. H a,b ≤ H a,b .
Proof. Let b and b be as assumed above. There exists q ∈ N with b = bq. Then
e q b = e πiq b = e πi b = e b .
So we obtain [1, e b ] ∈ H a,b and therefore
H a,b ≤ H a,b .
Remark. The same result holds for the parameter a. The proof is completely analog to the one of the previous lemma.
In the next step we consider the action of H a,b on H (see (1.2)). To prove absolute irreducibility of the representation we follow the strategy of Lauterbach and Matthews [15] from where we use Lemma 6.2 (for the necessary background on representation theory see Chapter 4 in Chossat and Lauterbach [2]).
Lemma 3.4. The action of H a,b on H is absolutely irreducible.
Proof. We want to use the two two-to-one maps from the ordered pairs of unitary quaternions Q × Q to SO(4) and O(4) \ SO(4) respectively to find linear maps that commute with the group action. Let [l, r] ∈ Q × Q commute with H a,b . Consider the group element [1, j]. If [l, r] commutes with [1, j], then r commutes with j. This yields r = r 1 + r 2 j with r 1 , r 2 ∈ R. Furthermore [1, e b ] = [1, cos (π/b) + sin (π/b) i] ∈ H a,b and the relation r · e b = e b · r yields r 2 = 0, since sin (π/b) = 0. Therefore r ∈ R and as r is a unitary quaternion this gives r = ±1. Performing the same calculations for l using the elements [j, 1] and [e a , 1], we obtain l = ±1 as well. All pairs of unitary quaternions that commute with H a,b are [±1, ±1]. Application of the two-to-one maps from Q × Q to SO (4) and O(4) \ SO(4) respectively yields that the only elements of O(4) commuting with the group action are ±1. Lemma 6.2 from Lauterbach and Matthews [15] implies absolute irreducibility.
In the following lemmas we investigate the isotropy of the action of H a,b on H. Using Lemmas 6.3 and 6.4 from Lauterbach and Matthews [15] as well as the relations of generating elements (Lemma 3.1) we may prove: (2) h is of order 2;
Lemma 3.5. Let h = [e a , 1] k 1 [1, e b ] k 2 [1, j] l 1 [j, 1] l 2 ∈ H a,b as in (3.1) with k 1 ∈ Z/aZ, k 2 ∈ Z/bZ
(3) For
l 1 = 1 the fixed point space of h is cos 1 2 k 1 a − k 2 b π sin 1 2 k 1 a − k 2 b π 0 0 , 0 0 cos 1 2 k 1 a + k 2 b π sin 1 2 k 1 a + k 2 b π . For l 1 = 3 the fixed point space of h is cos 1 2 k 1 a − k 2 b + 1 π sin 1 2 k 1 a − k 2 b + 1 π 0 0 , 0 0 cos 1 2 k 1 a + k 2 b + 1 π sin 1 2 k 1 a + k 2 b + 1 π .
The previous lemma describes restrictions on the exponents in the representation (3.1) that guarantee nontrivial isotropy. These are in fact all the elements with nontrivial fixed point spaces. Using Lemmas 6.5 and 6.6 in Lauterbach and Matthews [15], we see that elements which do not meet these restrictions fix the origin only.
Lemma 3.6. Let h = [e a , 1] k 1 [1, e b ] k 2 [1, j] l 1 [j, 1] l 2 ∈ H a,b \ {[1, 1]} as in (3.1) with k 1 ∈ Z/aZ, k 2 ∈ Z/bZ as well as l 1 ∈ {1, 3} or l 2 = 1. Then h fixes only 0 ∈ H.
The form of group elements that have nontrivial fixed points from Lemma 3.5 guarantees that nontrivial isotropy subgroups can only contain one such element. The product of two different elements with fixed point space can not fix a point besides 0.
Lemma 3.7. The nontrivial isotropy subgroups of H a,b are generated by precisely one group element.
To shorten notation we want to name the two types of isotropy subgroups as follows for the rest of this subsection
K = h = [e a , 1] k 1 [1, e b ] k 2 [1, j][j, 1] K = h = [e a , 1] k 1 [1, e b ] k 2 [1, j] 3 [j, 1] .
Lemma 3.8. The isotropy groups K and K are conjugate either to [1, j]
[j, 1] = [j, j] or to [1, j] 3 [j, 1] = −[j, j] .
Proof. This can be calculated directly using the relations on the generating elements of K = h and H = h and the fact that a and b are odd. For K we obtain
h = [1, j] 2 [e a , 1] a+1 2 2 k 1 [1, j] 2 [1, e b ] b+1 2 2 k 2 [1, j][j, 1] =h [1, j] 2 k 1 +k 2 [1, j][j, 1]h −1 withh = [e a , 1] k 1 a+1 2 [1, e b ] k 2 b+1 2 . Since [1, j] 2 k 1 +k 2 = [1, 1] for k 1 + k 2 even, [1, j] 2 for k 1 + k 2 odd,
this yields the claim for K. The proof for K is completely alike.
The previous lemma completes the proof for Theorem 1.
, j], [j, 1], [1, e ψi ] | φ ∈ S 1 (note that [1, e (π+ψ)i ] = [−1, e ψi ] for ψ ∈ [0, π)). Write it as follows H a = [e a , 1], [1, j], [j, 1], [1, e ψi ] | ψ ∈ S 1 .
This is a compact 1-dimensional Lie group. It contains an element [e a , e a ] which fixes 1, i as a real subspace of H. Furthermore [j, j] ∈ H a,b fixes the real subspace 1, j . Thus the subgroup generated by these two elements [e a , e a ], [j, j] fixes the one-dimensional real subspace 1 ⊂ H.
(2) Write H a and H a in terms of generators. Then for a, a ∈ 2N + 1 with a < a and a divides a we obtain [e a , 1] ∈ H a . The claim follows as in the proof of Theorem 1.2.
For the construction of the groups acting on R 8 we need the matrix representation of the groups H a,b with respect to the standard basis of R 4 . It can be calculated directly via applying the generators to the basis elements. One obtains
c = r π a 0 0 r π a d = r − π b 0 0 r π b q = 0 0 1 0 0 0 0 1 −1 0 0 0 0 −1 0 0 s = 0 0 −1 0 0 0 0 1 1 0 0 0 0 −1 0 0 (3.2)
where r(ψ) is again the 2-dimensional rotation matrix by an angle ψ. One can see that the first two elements correspond to blockwise rotations in two coordinates respectively.
1 4 , cs = sc −1 = sc 2a−1 , dq = qd −1 = qd 2b−1 , c a = q 2 , d b = q 2 , s 2 = q 2 .
The group algebra software GAP [8] allows to check some of the stated results for low group orders. Among other things, GAP provides a classification scheme for small groups. The GAP-identifiers are composed of two integers. The first one is the group order and the second one enumerates the isomorphism classes of groups of the given order. Here we present the identifiers of the first few groups within our classification which are relevant for our subsequent analysis. In cases like this we have multiple choices for a. Bear in mind that there are more complicated cases in which we have more than two choices for a and b for the same value of m. In these cases, a change of the parameters does not necessarily lead to the same groups. However, the smallest groups for which this occurs are already far beyond the reach of the SmallGroup library from GAP.
Representation on R 8
We want to investigate the groups G a,b in a similar manner as the groups H a,b before. First of all we can calculate relations on the generators. The relations on C, D, Q and S are the same as for c, d, q and s because of the blockdiagonal structure. The relations containing V can be calculated using the ones for the small matrices (see Lemma 3.9).
SC 2a−1 , DQ = QD −1 = QD 2b−1 , C a = Q 2 , D b = Q 2 , S 2 = Q 2 . Adding V yields V C = C ρ V CV = V C −ρ = V C 2a−ρ V D = DV V Q = Q 3 V V S = SV V 8 = 1 8 V 2 = S.
Remark. This is the point where the fact that ρ is odd becomes important. If it were even C a would not be equal to Q 2 = −1 8 but
C a = −1 4 0 0 1 4 .
Just as before this allows us to write every element g ∈ G a,b in the form
g = C k 1 D k 2 Q l 1 S l 2 V m (3.3)
with k 1 ∈ Z/aZ, k 2 ∈ Z/bZ, l 1 ∈ Z/4Z, l 2 ∈ Z/2Z and m ∈ Z/2Z. Using the calculations for the groups G a,b we may state similar results on the structure and isotropy of the groups H a,b .
Lemma 3.11. For each a ∈ A and b ∈ 2N + 1 with gcd(a, b) = 1 the group G a,b forms a subgroup of O(8) of order 16m where m = a · b.
Proof. The elements C, D, Q and S generate a group that is isomorphic to H a,b . Addition of V to the set of generators gives two copies of this group. Therefore |G a,b | = 2 |H a,b | = 16m. Once again this can be calculated directly from the form of the group elements (3.3).
Lemma 3.12. Let b < b where b is odd and relatively prime to a. If b divides b , G a,b is a subgroup of G a,b , i.e. G a,b ≤ G a,b .
Proof. This follows directly from the second statement in Theorem 1.2.
Remark. In contrast to the 4-dimensional case, we do not obtain the same result for the parameter a, which is due to the power ρ in the definition of the generating element C. This power is not necessarily the same for a and a when a divides a . However in this case G a,b is isomorphic to a subgroup of G a ,b .
Lemma 3.13. The natural 8-dimensional representation of G a,b is absolutely irreducible.
Proof. Let L : R 8 → R 8 be a linear map in matrix representation that commutes with the group action of G a,b . We write L in form of a block matrix
L = L 1,1 L 1,2 L 2,1 L 2,2 where L i,j : R 4 → R 4 for i, j ∈ {1, 2}.
In a first step we want to show that L 1,2 = L 2,1 = 0. Then we can use absolute irreducibility of the 4-dimensional representation of H a,b to prove the claim. Using the commutativity assumption and the structure of the generating matrices we obtain
L 1,2 qs = − qsL 1,2 from LQ =QL and LS = SL, L 1,2 c ρ =cL 1,2 from LC =CL, L 1,2 s =L 2,1 from LV =V L.
The first relation yields that L 1,2 is of the form
L 1,2 = 0 * 0 * * 0 * 0 0 * 0 * * 0 * 0 .
We want to apply the second relation and remember that c is the representation matrix of [e a , 1] on H. Thus we calculate the power of c to be
c ρ = r ρπ a 0 0 r ρπ a .
Note that the entries of this matrix contain the real and imaginary part of e ρ a :
cos ρπ a = (e ρ a ) , sin ρπ a = (e ρ a ) .
Now we make use of the special choice of the power ρ to prove that these can not match the real and imaginary part of e a . Since ρ ∈ {0, . . . , a − 1}, we obtain that the only chance for (e ρ a ) = (e a ) is for ρ = 1 or ρ = 2a − 1 which both contradict the fact that ρ 2 = −1 mod a. Considering the imaginary parts, we obtain that the only possibility for (e ρ a ) = (e a ) is if ρ = 1 or ρ = a − 1. Once again this contradicts the choice of ρ. Therefore cos ρπ a = cos π a and sin ρπ a = sin π a .
Omitting the details, this allows us to compute that the remaining entries of L 1,2 are zero as well. Together with the last relation this yields L 1,2 = L 2,1 = 0. Therefore we obtain two linear maps L i,i : R 4 → R 4 for i = 1, 2 that commute with the action of H a,b . From the absolute irreducibility of this action we know that L is of the form
L = γ1 4 0 0 δ1 4 with γ, δ ∈ R. Commutation with V yields γ = δ.
In the next step we investigate the isotropy of the 8-dimensional representation of G a,b . Note that the corresponding results on H a,b mostly rely on the relations of the generating elements. Hence they can be adapted almost directly. Proof. For m = 0 the claim follows directly from Lemmas 3.5 and 3.6 since the other elements keep the two H a,b -blocks intact. Therefore we consider elements of the form (3.3) with m = 1:
g = C k 1 D k 2 Q l 1 S l 2 V.
Suppose x = (ζ, η) with ζ, η ∈ R 4 such that gx = x. Using the structure of the generating matrices, this yields
gx = C k 1 D k 2 Q l 1 S l 2 (η, sζ) = (ζ, η).
Since C, D, S and Q keep the block structure intact, we may split this into two equations:
c k 1 d k 2 q l 1 s l 2 η = ζ c ρk 1 d k 2 q 3l 1 s l 2 +1 ζ = η.
Inserting the second equation in the first one, we obtain
c k 1 d k 2 q l 1 s l 2 c ρk 1 d k 2 q 3l 1 s l 2 +1 ζ = ζ.
Using the relations on the matrix representation of the generators of H a,b (Lemma 3.9), we may then calculate
c k 1 d k 2 q l 1 s l 2 c ρk 1 d k 2 q 3l 1 s l 2 +1 = c k 1 +(−1) l 2 ρk 1 d k 2 +(−1) l 1 k 2 q 2l 2 s.
Since the power of q is even, Lemma 3.6 yields ζ = 0. Inserting this in the second equation gives η = 0 which completes the proof.
Remark. Note that we can use the formulas to compute basis elements of the fixed point spaces (Lemma 3.5) in the case m = 0 as well. We only have to take the powers of c and q in the second block of the matrices C and Q into account. The fixed point spaces are obviously 4-dimensional.
Concerning the isotropy subgroups of G a,b we obtain the same result as in Lemma 3.7 from the fact that C, D, Q, S is isomorphic to H a,b :
Lemma 3.15. The nontrivial isotropy subgroups of G a,b are generated by precisely one group element.
Once more we want to make use of shorter notations. We therefore write
K = g = C k 1 D k 2 QS K = g = C k 1 D k 2 Q 3 S
for the two types of nontrivial isotropy subgroups for the rest of this subsection. Using the element V , we may now show, that in the 8-dimensional case we obtain only one isotropy type:
Lemma 3.16. All nontrivial isotropy subgroups of G a,b are conjugate to QS .
Proof. All nontrivial isotropy subgroups are generated by either g or g which both do not contain a factor V . We may therefore use Lemma 3.8 and the fact that C, D, Q and S are subject to the same relations as [e a , 1], [1, e b ], [1, j] and [j, 1]. This yields that every nontrivial isotropy subgroup is conjugate to either QS or Q 3 S . These two subgroups are conjugate by V :
V QSV −1 = Q 3 SV V −1 = Q 3 S.
Thus all nontrivial isotropy subgroups are conjugate to QS .
This completes the proof for Theorem 1.5. Similarly to the 4-dimensional case these considerations leave the results on the Lie group structure straightforward and we may state the proof of Theorem 1.6:
Proof of Theorem 1.6. The claim follows in the same way as in the proof for the groups H a . Let {ξ 1 , . . . , ξ 8 } be the standard basis of R 8 . The Lie group G a contains the elements C and D (π/a) and their product CD (π/a) fixes the subspace ξ 1 , ξ 2 . Furthermore QS fixes the subspace ξ 1 , ξ 3 , ξ 6 , ξ 8 . Thus the subgroup CD (π/a) , QS generated by these two elements fixes the subspace ξ 1 .
Remark. As mentioned before the subgroup relation for the one-dimensional Lie groups as in Theorem 1.3 does not hold because of the exponent ρ in the construction of the matrix C. Furthermore we do not obtain Lie groups of dimension 2 when considering the closure of the union of the one-dimensional Lie groups G a over all a ∈ A. The reason for this structural difference lies in the power ρ as well. It is a nonconstant natural number depending on the angle which is a rational multiple of π. As such it has no smoothmore precisely, not even a continuous -continuation on all angles φ ∈ S 1 and therefore prevents a smooth structure on the matrices C for all angles. It is unknown whether there exist 2-dimensional Lie groups containing all the groups G a .
We provide the GAP-identifiers for the first groups G a,b (compare to the symmetry in a, b is broken in the case of both factors being in A. This is due to the different construction of the matrix C from c.
Equivariant structure for G a
Since we are interested in bifurcation problems on R 8 with G a,b -symmetry for suitable a and b, we have to investigate smooth G a,b -equivariant maps on R 8 . Using methods from character and invariant theory, we are able to compute dimensions of spaces of homogeneous equivariant polynomial maps for up to third degree. Then we determine the generating functions for the corresponding spaces. This allows us to gain insight in the general bifurcation behaviour of equations with G a,b -symmetry which we will investigate further in the next section.
For a group Γ acting on the real space W define its character as follows
χ : Γ → R, g → tr (g)
and denote the space of smooth Γ-equivariant maps by C ∞ Γ (W, W ). It is well known that the symmetric functions form a module which contains the equivariant polynomials as a dense subset (see for example Chossat and Lauterbach [2] or Field [6]). The space of homogeneous equivariant polynomial maps of degree d shall be denoted by P d Γ (W, W ). To gather information about the equivariant structure of a given representation one often looks at the so called Molien series, a formal power series that carries information about dimensions of these spaces. It is defined as follows
∞ d=0 R d z d where R d = dim P d Γ (W, W )
is the number of linearly independent equivariant polynomial maps of degree d to which we refer as Molien coefficients. In a similar way we consider invariant polynomials from the represention space into the real numbers. These are in a close relationship to the equivariant polynomial maps. We denote the space of invariant homogeneous polynomials of degree d by Π d Γ (W ). Then we obtain for example that for every p ∈ Π d Γ (W ) the gradient ∇p is an equivariant polynomial map: ∇p ∈ P d−1 Γ (W, W ). For more details on this matter and the connection between invariant and equivariant polynomials see Chossat and Lauterbach [2]. The corresponding formal power series
∞ d=0 r d z d with r d = dim Π d Γ (W )
is called Molien series as well. Molien's theorem states a way to calculate these formal power series but it is often difficult to do so. That is why we use a slightly different approach.
Computation of Molien coefficients
We are especially interested in the equivariant structure for low degree polynomial maps. Sattinger [19] proves a formula by which we can calculate the R d for a single d without having to deal with the Molien series. This formula also follows from the results in Zhilinskií [20]. Although it is impractical for large values of d it is very helpful in the cases which we consider. For g ∈ Γ Sattinger defines the quantity
χ (d) (g) = d k=1 k·i k =d χ i 1 (g) · · · χ i d g d 1 i 1 i 1 !2 i 2 i 2 ! · · · d i d i d ! and obtains R d = Γ χ (d) (g)χ(g)dg. (4.1)
Note that in the case of a finite group the integral becomes a normed sum. To compute single Molien coefficients r d for the invariant polynomials there exists a similar formula that can easily be derived from the calculations in Zhilinskií [20]:
r d = Γ χ (d) (g)dg.
For the bifurcation analysis we are only interested in the equivariant structure and we will see later that we only need the data for degrees up to d = 3 (see Section 5). For these cases we can apply formula (4.1) with reasonable effort. In the case of an absolutely irreducible representation, the only linear maps commuting with the group action are multiples of the identity, therefore we immediately obtain R 1 = 1. Furthermore χ (2) reads
χ (2) (g) = 1 2 χ(g 2 ) + χ 2 (g) .
To calculate χ (3) using i 1 + 2i 2 + 3i 3 = 3 we have the choices (3, 0, 0), (1, 1, 0) and (0, 0, 1) for (i 1 , i 2 , i 3 ). Therefore we get
χ (3) (g) = 1 3! χ 3 (g) + 1 2 χ(g)χ(g 2 ) + 1 3 χ(g 3 ).
We want to use formula (4.1) to calculate R 2 and R 3 for the groups G a,b . Let a ∈ A and b ∈ 2N + 1 with gcd(a, b) = 1 be natural numbers as before. In a first step we investigate the character χ : G a,b → R for an arbitrary element g ∈ G a,b . It is very useful to notice that the character is a class function, i.e. it is invariant under conjugation. We have seen that g can be written in the form
g = C k 1 D k 2 Q l 1 S l 2 V m
with k 1 ∈ Z/aZ, k 2 ∈ Z/bZ, l 1 ∈ Z/4Z, l 2 ∈ Z/2Z and m ∈ Z/2Z (see (3.3)). Recall how G a,b is constructed from H a,b and note that C, D, Q and S keep the two H a,b -blocks intact. This yields that every element g with m = 1 is of the form g = 0 * * 0 and therefore χ(g) = 0. Hence we may restrict to the case m = 0:
g = C k 1 D k 2 Q l 1 S l 2 .
These elements are of the form
g = h 0 0 h (4.2)
with h, h ∈ H a,b (in matrix representation) and for their character we obtain
χ(g) = χ 4 (h) + χ 4 (h )
where χ 4 : H a,b → R denotes the character of the 4-dimensional representation of H a,b . Investigating this character provides us with the needed result. To do so we make use of both the biquaternionic and the matrix representation of H a,b . Similar to the form of the group element g ∈ G a,b we may characterize
h = c k 1 d k 2 q l 1 s l 2
for h ∈ H a,b (see (3.1)). In a similar manner as before we obtain h = 0 * * 0 if l 1 + l 2 is odd and therefore χ 4 (h) = 0 in this case. Consider l 1 = 3 and l 2 = 1. In Lemma 3.8 we have seen that elements of this form are conjugate to either qs = diag ((1, −1, 1, −1)) or −qs and therefore χ 4 (h) = 0. For l 1 = 1 and l 2 = 1 we have
h = c k 1 d k 2 qs = −c k 1 d k 2 q 3 s
and by linearity of the character χ 4 (h) = 0 as well. The remaining two cases are l 1 ∈ {0, 2} and l 2 = 0. Once more note that
c k 1 d k 2 q 2 = −c k 1 d k 2 = −h
for h = c k 1 d k 2 and we may make use of the linearity again. The matrix h corresponds to the group element e k 1 a , e k 2 b and we compute it to be
h = r k 1 a − k 2 b π 0 0 r k 1 a + k 2 b π .
From now on let η = k 1 a π and ν = k 2 b π.
Then we obtain χ 4 (h) = 4 cos (η) cos (ν) .
Summarizing this yields the only nonzero cases for l 1 ∈ {0, 2} and l 2 = 0 giving
χ 4 (h) = (−1) l 1 2 4 cos (η) cos (ν) .
Returning back to g ∈ G a,b with m = 0 and using the block structure (4.2) we obtain
h = c k 1 d k 2 q l 1 s l 2 h = c ρk 1 d k 2 q 3l 1 s l 2
for the H a,b -blocks. We investigate the same cases for the powers as before. If l 1 + l 2 is odd, then so is 3l 1 + l 2 and therefore χ 4 (h ) = 0 giving χ(g) = 0. If l 2 = 1 and l 1 is odd then so is 3l 1 and in the same manner we obtain χ(g) = 0. For l 2 = 0 and l 1 even we obtain 3l 1 = 0 for l 1 = 0, 6 = 2 mod 4 for l 1 = 2.
This yields
χ(g) = (−1) l 1 2 4
(cos (η) + cos (ρη)) cos (ν)
if l 1 ∈ {0, 2} and l 2 = 0 and χ(g) = 0 in all other cases. Knowing the character for every element g ∈ G a,b allows us to calculate the quantities χ (d) . Note that we only need them for group elements with χ(g) = 0 because of the corresponding factor in the dimension formula (4.1). To perform these calculations for d = 2, 3 we still need to consider χ g d . Note that for l 1 ∈ {0, 2} and l 2 = m = 0 we obtain
g 2 = C 2k 1 D 2k 2 Q 2l 1 = C 2k 1 D 2k 2 ,
using the relations on the generating elements, and therefore χ(g 2 ) = 4 (cos (2η) + cos (2ρη)) cos (2ν) .
In an analogue way we obtain
g 3 = C 3k 1 D 3k 2 Q l 1
and therefore χ(g 3 ) = (−1) l 1 2 4 (cos (3η) + cos (3ρη)) cos (3ν) .
We can then put the parts together to obtain χ (d) for d = 2, 3 which we use to calculate R 2 and R 3 . The remaining steps are a subtle computation using calculation rules for cosine and the geometric sum formula. The details shall be omitted at this point but can be found in the Appendix (A.1). Performing the calculations we obtain:
Lemma 4.1. The dimensions R d = dim P d G a,b (R 8 , R 8 ) for d = 1, 2, 3 are R 1 = 1, R 2 = 0, R 3 = 8 for a = 5, 5
else.
Equivariant maps in the case a = 5 and b = 3
We want to determine the equivariant structure up to third degree for the smallest group we can construct with the method presented in Sections 1 and 3 which is G 5,3 . The groups in the family G 5 form a special case in our considerations as we have seen from the calculations of the Molien coefficients. We point out when this is important in a remark at the end of the section. By irreducibility we already know that the only linear equivariants are scalar multiples of the identity. Furthermore we have no quadratic G 5,3 -symmetric maps on R 8 , since R 2 = 0 and the space of cubic equivariants is 8-dimensional. There are several ways to find equivariant maps of a given degree. Sattinger [19] investigates some simple examples. Lauterbach and Matthews [15] describe methods for groups that are constructed in a similar way as the ones we consider using complex polynomials. More general results and computer algebra systems can be found in Gatermann [9,10] and Gatermann and Guyard [11]. We have chosen an elementary method to calculate a basis using general homogeneous polynomials and having Maple [16] solve for the coefficients under the assumption of equivariance with respect to the generating matrices. We obtain eight linearly independent polynomial maps E 1 , . . . , E 8 that prove to meet the symmetry condition. They can be found in the Appendix in Table 3.
The general case
We want to use the results for the case a = 5, b = 3 to obtain the full picture for all groups. By construction of the groups G a,b it follows that the only dependence on the parameters a and b is in the matrices C(a) and D(b). A short calculation shows that the vector fields E 1 , . . . , E 5 remain equivariant with respect to the matrices C(a) and D(b) with arbitrary a ∈ A and b ∈ 2N + 1 such that gcd(a, b) = 1. We may even prove that E 1 , . . . , E 5 are equivariant with respect to a matrix D(ψ) that describes an arbitrary angle of rotation ψ ∈ S 1 . This gives us the final result on the equivariant structure and hence completes the proof of Theorem 1.7.
Remark. (1) The dimension of P d Ga (R 8 , R 8 ) is at most the dimension of P d G a,b (R 8 , R 8 ) with G a,b ≤ G a . As a consequence we obtain P 3 Ga (R 8 , R 8 ) = P 3 G a,b (R 8 , R 8 ).
(2) A similar statement holds true for the matrices C. We define the matrix c(φ) to be the representing matrix of [e φi , 1] andC(φ, φ ) as the diagonal blockmatrix of c(φ), c(φ ) for arbitrary distinct angles φ, φ ∈ S 1 . This leads to a compact 3dimensional Lie group
G = Q, S, V,C(φ, φ ), D(ψ) | φ, φ , ψ ∈ S 1 .
It is easy to see that E 1 , . . . , E 5 are equivariant with respect toG. Therefore the space P 3 G (R 8 , R 8 ) is also generated by these vectorfields. But as mentioned beforẽ G is not obtained from the closure of the union of all G a . (4) At first it may appear odd that the number of linearly independent cubic equivariant polynomials is different in the case a = 5. But from equivariance with respect to the Lie groupG it follows that the dimension of cubic equivariants has to become stationary for some value of a. This occurs at the first step from a = 5 to a = 13. Therefore, when investigating G a,b -symmetric dynamical systems, we need to take care of the case a = 5 separately.
Generic symmetry breaking bifurcations
In this section we want to investigate bifurcation problems on R 8 which are symmetric with respect to the groups G a,b that we have constructed before. In order to do so, we use methods proposed by Field [5,6] and Field and Richardson [7]. The authors use techniques from equivariant transversality to develop a complete geometric theory on equivariant dynamics. It allows us, similar to the equivariant branching lemma, to obtain results on bifurcations in generic equations that are symmetric with respect to a given representation. The basic principle is that it suffices to investigate Taylor expansions up to some critical degree to gather information on the dynamical behaviour. Since these polynomials are equivariant as well, we can apply methods from invariant theory to calculate possible terms in the expansion, which is what we have done in the previous section using formula (4.1). The authors even prove that we can always find such a critical degree d in which the branching of solutions is fully determined. We say that the equivariant bifurcation problems are d-determined. However we will not go that far here, as we see that the cubic truncation suffices to prove the bifurcation result. For this reason we do not try to establish determinacy statements. In our case we can apply a polar blowing-up technique from the texts mentioned above to find a nontrivial branch of solutions bifurcating off the trivial one. All the methods used in this section are formulated in Chapter 4 of Field [6], where we can also find the technical details that we partly omit here. Furthermore we use a slight modification of this approach which respects the restriction on fixed point spaces of isotropy subgroups. This will be pointed out explicitly when we make use of it. As the equivariant structure forms a special case for a = 5 (compare to the previous section), we restrict ourselves to a ∈ A with a > 5 for the rest of this section.
Normalized families of equivariant vector fields
Following the notation of Field [6] we let V (R 8 , G) = C ∞ G (R 8 × R, R 8 ) be the set of smooth G-equivariant vector fields, for G ∈ G a and 5 < a ∈ A, depending on a real parameter. The action of G on the product space is defined to be only on the first component. We equip the function space with the C ∞ -topology and subsets with the induced topology. For f ∈ V (R 8 , G) we define the 1-parameter family {f λ } λ of smooth G-symmetric vector fields on R 8 by f λ = f (·, λ). By equivariance we get
f (0, λ) = 0 for every λ ∈ R, D 1 f (0, λ) = σ f (λ)1 8 with σ f ∈ C ∞ (R)
. This set of zeros will be called the branch of trivial zeros and we are looking for solution branches bifurcating off this branch as we vary λ. As long as σ f (λ) = 0 we can use the implicit function theorem to obtain a neighbourhood U of (0, λ) such that the only zeros in U are trivial. We are therefore interested in points λ 0 ∈ R with σ f (λ 0 ) = 0 to find nontrivial solutions. Generically in such a point f will satisfy σ f (λ 0 ) = 0 which we will assume from now on. Furthermore we can assume λ 0 = 0 without loss of generality and use the inverse function theorem to reparametrize λ so that σ f (λ) = λ for λ near 0. The extension of σ f to all the real numbers in the same manner does not impose a loss of generality, since we are only interested in branching close to the trivial solution. These considerations motivate the restriction to the closed affine linear subspace
V 0 = V 0 R 8 , G = f ∈ V R 8 , G | σ f (λ) = λ, λ ∈ R of normalized families of smooth G-equivariant vector fields on R 8 . For f ∈ V 0 we may write f λ (x) = f (x, λ) = λx + F λ (x)
using Taylor's theorem, to which we refer as normalized bifurcation problem.
Nonradial equivariant polynomial maps
To follow the methods of Field the next step is to find G-equivariant polynomial maps on R 8 which are nonradial. A polynomial map P is called radial if it is of the form
P = p1 8
where p : R 8 → R is an invariant polynomial. We call d = d(G, R 8 ) the smallest degree in which nonradial equivariant polynomial maps exist. We have seen before from the Molien coefficients (Section 4) and Table 3 in the Appendix that d = 3. For f ∈ V 0 let R be the Taylor polynomial of f 0 of order d at the origin. Using Taylor's theorem we obtain Field [6]). It is a well known fact (that can be recalled from the results of Chapter 5 in Chossat and Lauterbach [2]) that the homogeneous terms in the Taylor expansion of an equivariant vector field are equivariant as well. The linear part λ1 8 of the Taylor polynomial of f λ vanishes for λ = 0. If we look again at the Molien coefficients, we find that there are no quadratic G-equivariant polynomial maps. Therefore R ∈ P 3 G (R 8 , R 8 ) which is generated by E 1 , . . . , E 5 and hence R must be of the form R = αE 1 + βE 2 + γE 3 + δE 4 + E 5 with α, β, γ, δ, ∈ R.
f (x, λ) = λx + R(x) + F 1 (x) + λF 2 (x, λ) with F 1 (x) = f (x, 0)−R(x) = O( x d+1 ) and F 2 (x, λ) = f (x, λ)−R(x)−F 1 (x) = O( x d ) (see
Phase vector fields and fixed point subspaces
As mentioned in the introduction to this section, the major technical tool to find nontrivial solution branches is a polar blowing-up technique. Using general polar coordinates we can decompose the normalized function f ∈ V 0 into a spherical part -a smooth vector field on the unit sphere -and a radial part perpendicular to the sphere. A suitable solution for the spherical part with the radial coordinate 0 can then be generalized to other radial values, using the implicit function theorem, leading to nontrivial solutions. But first, we adapt Field's method in such a way, that it applies in fixed point spaces of isotropy subgroups. In our case this reduces the dimension by four, which is convenient for the following computations.
As we have seen before (Theorem 1.5) each group G ∈ G a has precisely one nontrivial isotropy type [K] which contains the conjugate subgroups of
K = QS = diag (1, −1, 1, −1, −1, 1, −1, 1) .
The subgroup K obviously fixes elements of the subspace
Fix (K) = {x ∈ R 8 | x 2 = x 4 = x 5 = x 7 = 0} ∼ = R 4 .
Utilizing the symmetry property of f , it therefore suffices to consider Fix (K). Denote the coordinates by y = (y 1 , y 2 , y 3 , y 4 ) ∈ Fix (K) .
By equivariance f λ fixes Fix (K) for each f ∈ V 0 :
kf λ (x) = f λ (kx) = f λ (x)
for k ∈ K and x ∈ Fix (K). Therefore f λ (x) ∈ Fix (K) for all x ∈ Fix (K). With this we may now restrict f λ and R to Fix (K), which is the part that does not appear in the texts by Field and Field & Richardson. Since the blow-up method does not interfere with this reduction, we may perform it in the fixed point space just as well.
To investigate bifurcation behaviour in Fix (K) we calculate the so called phase vector field of the cubic equivariant polynomial maps. For R as before restricted to Fix (K) it is defined as the vector field P R (y) = R(y) − R(y), y y with y ∈ S 3 ⊂ Fix (K) which is the tangential component of the restriction of R to the unit sphere S 3 in R 4 . Up to a factor depending on the radial coordinate, it is equal to the spherical part of R. Furthermore the phase vector field P R coincides with the spherical part of f if the radial coordinate is 0, which is the starting point for the blow-up technique. Note that the phase vector field of a radial polynomial vanishes and therefore cannot provide any information on solutions of the original equation. The projection on the phase vector field is a linear map from P 3
G a,b (R 4 , R 4 ) to P 5 G a,b (S 3 , S 3 ) so P R = αP E 1 + βP E 2 + γP E 3 + δP E 4 + P E 5 .
The phase vector fields of E 1 , . . . E 5 can be found in the appendix (see Table 4). As we have seen in Theorem 1.7 the cubic equivariants E 1 , . . . , E 5 are G a -symmetric as well and the same holds for P E 1 , . . . , P E 5 . Hence they leave the fixed point spaces of isotropy subgroups of G a invariant. In Theorem 1.6 we have proved that G a has subgroups with one-dimensional fixed point spaces. These intersect the sphere in two points and therefore directly lead to zeros of the phase vector fields P E 1 , . . . , P E 5 . For example the group CD − π a ρ , QS < G a fixes the one-dimensional subspace (0, . . . , 0, 1) T ⊂ R 8 . This is a subspace of Fix (K) as well and reads (0, 0, 0, 1) T in the corresponding coordinates. Thus y 0 = (0, 0, 0, 1) T and −y 0 are common zeros of P E 1 , . . . , P E 5 and therefore P R (±y 0 ) = 0 for any linear combination. The Jacobian of P R (y 0 ) has the eigenvalues −α + δ, −α + γ, −α + β and −2α. So we see that y 0 is a hyperbolic zero of P R if α = 0, α = β, α = γ and α = δ. These conditions are met for an open and dense subset of R 5 and therefore y 0 is generically a hyperbolic zero for P R . This allows us to start the blow-up technique which, using the implicit function theorem, provides us with nontrivial hyperbolic solutions to f = 0 depending on the value of the radial coordinate. These can be reformulated into a solution curve bifurcating off the trivial solution where the direction of branching is y 0 . By construction this new branch of solutions lies in the fixed point space Fix (K) meaning that the isotropy type [K] is symmetry breaking. For the technical details of the blow-up method see Lemma 4.8.1. and its proof from Field [6] with the fact that R(y 0 ), y 0 = α which is generically not zero. As we have seen, the branching of steady states occurs for a generic bifurcation problem. This completes the proof for the main theorem on bifurcations with G a,b -symmetry. 6 The special case a = 5
To conclude the above considerations we want to briefly discuss the special case a = 5 and point out that it is not so special after all. As we have seen in section 4 the main difference between the groups G 5,b and G a,b (for admissible values of b) lies in the structure of equivariant polynomial maps. This in turn influences the argumentation to prove the bifurcation result. In subsection 4.2 we have computed the space of equivariant cubic polynomial maps P 3 G 5,b (R 8 , R 8 ) to be generated by the maps E 1 , . . . , E 8 (see Table 3). Furthermore, in subsection 4.3, we have seen that the corresponding spaces are generated by E 1 , . . . , E 5 whenever a > 5. A major aspect for the proof of the bifurcation result is the fact that these maps are equivariant not only with respect to the groups G a,b but also with respect to the Lie groups G a . A short calculation shows that this holds true for the maps E 6 , E 7 and E 8 as well. But, on the contrary to the case a > 5, the additional maps are not equivariant with respect to the largest Lie groupG that we have considered.
Nevertheless we may use the same technique to investigate the bifurcation behavior in the presence of G 5,b symmetry as before. We only sketch the proof here since it is completely analog to the one before. We consider the cubic truncation of a normalized bifurcation problem
R = 8 i=1 α i E i with a i ∈ R
and restrict to the fixed point subspace
Fix (K) = {x ∈ R 8 | x 2 = x 4 = x 5 = x 7 = 0} ∼ = R 4 .
Then we consider the corresponding phase vector field P R (y) = R(y) − R(y), y y with y ∈ S 3 ⊂ Fix (K) which by the same argumentation as before -one-dimensional fixed point space of an isotropy subgroup of G 5 -has the zero y 0 = (0, 0, 0, 1) T . This is once again generically hyperbolic and thus we can apply the polar blowing up method to obtain a branch of zeros for the bifurcation equation bifurcating off the trivial solution.
Summing up we see that the case a = 5, even though it has to be treated separately, does not imply significant differences. It provides the same bifurcation result which can be proved using the same techniques. The main point of interest lies in the structure of the equivariant maps as P 3
G a,b (R 8 , R 8 ) ⊂ P 3 G 5,b (R 8 , R 8 )
as a proper subspace.
Further groups and even dimensional representations
In this paper we have constructed groups of order 16m -here m = a · b with a, b = 1, relatively prime, odd and a > 5 being a product of prime powers of the form 1 mod 4with an 8-dimensional absolutely irreducible representation that provide counterexamples to the Ize conjecture. Furthermore Lauterbach [14] describes groups of order 64 + 128 with ∈ N with the same purpose. In both cases there are generically symmetry breaking isotropy types. But comparing these results to the GAP-calculations provided by Table 7 in Lauterbach and Matthews [15] we still expect a vast number of counterexamples to the Ize conjecture in 8 dimensions that have not yet been investigated systematically. It is an open task to find a reasonable ordering for the groups in terms of their orders and to obtain information on the dynamics in their 8-dimensional representations. In order to do so, a first step could be to slightly adapt the construction of G a,b in such a way that we define the 8 × 8 generating matrix Q to be Q = q 0 0 q instead of the second block being −q. We obtain two isotropy types in this case for which it would be interesting to determine whether both of them are generically symmetry breaking. Another task is to determine the role of the 3-dimensional Lie group containing all the G a . This has not yet been sufficiently investigated. Furthermore we see from Tables 5-10 in Lauterbach and Matthews [15] that there are further groups acting absolutely irreducibly in dimensions 4, 8, 12, 16 and 20 that appear to lead to counterexamples to the Ize conjecture but none with the same property in dimensions 2, 6, 10, 14 and 18 (at least for small group orders). The authors formulate the conjecture: "For dimensions N = 0 mod 4, there are infinitely many groups acting absolutely irreducibly on R N that have no isotropy subgroups with odd-dimensional fixed point spaces. But for dimensions N = 2 mod 4, there are no such groups". There is some evidence for this conjecture to be true as the GAP calculations do not provide counterexamples in the second case for groups of order up to 1000. Furthermore Ruan [18] proves the claim for dimension 6 under the mild additional assumption that the groups are solvable.
These intermediate steps and the conjecture of Lauterbach and Matthews [15] would provide some major insight in the question if absolute irreducible group actions lead to generically symmetry breaking isotropy types. This interpretation of the Ize conjecture from the dynamical systems point of view would be a significant contribution to the understanding of bifurcations in the presence of symmetry. However such a general statement is still far from being proved.
Lemma A.2. Let w ∈ N and l ∈ Z. Then w−1 k=0 cos 2l k w π = 0 for w l, w else.
We want to use this lemma to calculate sums of such cosine terms that contain an even factor in front of (k/w)π. We have to distinguish whether this factor is an integer multiple of 2w. The following lemma performs this distinction in the occurring cases.
Lemma A.3. Let a ∈ A and ρ be chosen as in Proposition 1.1 and odd (compare to the construction of G a,b ). Then
(1) ρ − 1 = 0 mod 2a;
(2) 2(ρ − 1) = 0 mod 2a;
(3) ρ + 1 = 0 mod 2a;
(4) 2(ρ + 1) = 0 mod 2a;
(5) 2ρ = 0 mod 2a;
(6) 4ρ = 0 mod 2a; (2) Suppose 2(ρ − 1) = 0 mod 2a. Then ρ − 1 = 0 mod a and ρ = 1 mod a. The contradiction follows as before.
(3) Suppose ρ+1 = 0 mod 2a. Then ρ = −1 mod a and ρ 2 = 1 mod a which is again a contradiction.
(4) Suppose 2(ρ + 1) = 0 mod 2a. Then ρ + 1 = 0 mod a and ρ = −1 mod a. This is a contradiction as before.
(5) Suppose 2ρ = 0 mod 2a. Then ρ = 0 mod a which contradicts the choice of ρ.
(6) Suppose 4ρ = 0 mod 2a. Then 2ρ = 0 mod a and therefore 4ρ 2 = 0 mod a. But ρ was chosen so that 4ρ 2 = −4 mod a. This is impossible, since a is odd.
(7) Suppose ρ − 3 = 0 mod 2a. Then ρ = 3 mod a and ρ 2 = 9 mod a. But ρ 2 = −1 mod a so 9 = −1 mod a. The only choice is a = 5, since a is odd. The corresponding odd ρ is 3.
(8) Suppose 3ρ − 1 = 0 mod 2a. Then 3ρ = 1 mod a and ρ = 3ρ 2 = −3 mod a. This yields ρ 2 = 9 mod a and as before a = 5. Then ρ = −3 = 2 mod a which is not odd and therefore a contradiction.
(9) Suppose ρ + 3 = 0 mod 2a. Then ρ = −3 mod a and the contradiction follows as before.
(10) Suppose 3ρ + 1 = 0 mod 2a. Then 3ρ = −1 mod a and therefore ρ = −3ρ 2 = 3 mod a. As before we obtain a = 5 and ρ = 3.
These two technical lemmas together with multiple applications of the calculation rules for cosine allow us to compute R 3 explicitly and we obtain Proposition A.4. There are eight linearly independent cubic equivariant maps for the 8dimensional representation of G a,b if a = 5 and five linearly independent cubic equivariant polynomial maps for all other a ∈ A:
R 3 = 8
for a = 5, 5 else.
A.2 Cubic equivariant maps −y 2 (2 y 1 2 y 2 2 + 2 y 3 2 y 4 2 − y 1 2 ) −y 3 (2 y 1 2 y 2 2 + 2 y 3 2 y 4 2 − y 4 2 ) −y 4 (2 y 1 2 y 2 2 + 2 y 3 2 y 4 2 − y 3 2 )
E 1 (x) = (x 2 1 + x 2 2 ) x 1 (x 2 1 + x 2 2 ) x 2 (x 2 3 + x 2 4 ) x 3 (x 2 3 + x 2 4 ) x 4 (x 2 5 + x 2 6 ) x 5 (x 2 5 + x 2 6 ) x 6 (x 2 7 + x 2 8 ) x 7 (x 2 7 + x 2 8 ) x 8 E 5 (x) = −x 3 x 5 x 7 − x 3 x 6 x 8 − x 4 x 5 x 8 + x 4 x 6 x 7 x 3 x 5 x 8 − x 3 x 6 x 7 − x 4 x 5 x 7 − x 4 x 6 x 8 −x 1 x 5 x 7 − x 1 x 6 x 8 + x 2 x 5 x 8 − x 2 x 6 x 7 −x 1 x 5 x 8 + x 1 x 6 x 7 − x 2 x 5 x 7 − x 2 x 6 x 8 x 1 x 3 x 7 + x 1 x 4 x 8 − x 2 x 3 x 8 + x 2 x 4 x 7 x 1 x 3 x 8 − x 1 x 4 x 7 + x 2 x 3 x 7 + x 2 x 4 x 8 x 1 x 3 x 5 − x 1 x 4 x 6 + x 2 x 3 x 6 + x 2 x 4 x 5 x 1 x 3 x 6 + x 1 x 4 x 5 − x 2 x 3 x 5 + x 2 x 4 x 6 E 2 (x) = (x 2 3 + x 2 4 ) x 1 (x 2 3 + x 2 4 ) x 2 (x 2 1 + x 2 2 ) x 3 (x 2 1 + x 2 2 ) x 4 (x 2 7 + x 2 8 ) x 5 (x 2 7 + x 2 8 ) x 6 (x 2 5 + x 2 6 ) x 7 (x 2 5 + x 2 6 ) x 8 E 6 (x) = −x 1 x 3 x 6 + x 1 x 4 x 5 + x 2 x 3 x 5 + x 2 x 4 x 6 x 1 x 3 x 5 + x 1 x 4 x 6 + x 2 x 3 x 6 − x 2 x 4 x 5 −x 1 x 3 x 8 + x 1 x 4 x 7 + x 2 x 3 x 7 + x 2 x 4 x 8 x 1 x 3 x 7 + x 1 x 4 x 8 + x 2 x 3 x 8 − x 2 x 4 x 7 x 3 x 5 x 8 + x 3 x 6 x 7 + x 4 x 5 x 7 − x 4 x 6 x 8 x 3 x 5 x 7 − x 3 x 6 x 8 − x 4 x 5 x 8 − x 4 x 6 x 7 −x 1 x 5 x 8 − x 1 x 6 x 7 − x 2 x 5 x 7 + x 2 x 6 x 8 −x 1 x 5 x 7 + x 1 x 6 x 8 + x 2 x 5 x 8 + x 2 x 6 x 7 E 3 (x) = (x 2 5 + x 2 6 ) x 1 (x 2 5 + x 2 6 ) x 2 (x 2 7 + x 2 8 ) x 3 (x 2 7 + x 2 8 ) x 4 (x 2 3 + x 2 4 ) x 5 (x 2 3 + x 2 4 ) x 6 (x 2 1 + x 2 2 ) x 7 (x 2 1 + x 2 2 ) x 8 E 7 (x) = −2x 5 x 7 x 8 − x 6 x 2 7 + x 6 x 2 8 −x 5 x 2 7 + x 5 x 2 8 + 2x 6 x 7 x 8 x 2 5 x 8 + 2x 5 x 6 x 7 − x 2 6 x 8 x 2 5 x 7 − 2x 5 x 6 x 8 − x 2 6 x 7 x 2 1 x 4 + 2x 1 x 2 x 3 − x 2 2 x 4 −x 2 1 x 3 + 2x 1 x 2 x 4 + x 2 2 x 3 2x 1 x 3 x 4 + x 2 x 2 3 − x 2 x 2 4 −x 1 x 2 3 + x 1 x 2 4 + 2x 2 x 3 x 4 E 4 (x) = (x 2 7 + x 2 8 ) x 1 (x 2 7 + x 2 8 ) x 2 (x 2 5 + x 2 6 ) x 3 (x 2 5 + x 2 6 ) x 4 (x 2 1 + x 2 2 ) x 5 (x 2 1 + x 2 2 ) x 6 (x 2 3 + x 2 4 ) x 7 (x 2 3 + x 2 4 ) x 8 E 8 (x) = x 2 3 x 8 − 2x 3 x 4 x 7 − x 2 4 x 8 −x 2 3 x 7 − 2x 3 x 4 x 8 + x 2 4 x 7 x 2 1 x 6 − 2x 1 x 2 x 5 − x 2 2 x 6 −x 2 1 x 5 − 2x 1 x 2 x 6 + x 2 2 x 5 2x 1 x 7 x 8 + x 2 x 2 7 − x 2 x 2 8 x 1 x 2 7 − x 1 x 2 8 − 2x 2 x 7 x 8 −2x 3 x 5 x 6 − x 4 x 2 5 + x 4 x 2 6 −x 3 x 2 5 + x 3 x 2 6 + 2x 4 x 5 x 6 P E 3 (y) =
−y 1 (y 1 2 y 3 2 + y 4 2 y 1 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 3 2 ) −y 2 (y 1 2 y 3 2 + y 4 2 y 1 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 4 2 ) −y 3 (y 1 2 y 3 2 + y 4 2 y 1 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 2 2 ) −y 4 (y 1 2 y 3 2 + y 4 2 y 1 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 1 2 )
P E 4 (y) =
−y 1 (y 1 2 y 3 2 + y 1 2 y 4 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 4 2 )
−y 2 (y 1 2 y 3 2 + y 1 2 y 4 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 3 2 )
−y 3 (y 1 2 y 3 2 + y 1 2 y 4 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y 1 2 ) −y 4 (y 1 2 y 3 2 + y 1 2 y 4 2 + y 3 2 y 2 2 + y 2 2 y 4 2 − y
prime, a i = 1 mod 4, s i ∈ N for i = 1, . . . , r = {5, 13, 17, 25, 29, 37, 41, 53, 61, 65, . . .} (The sequence of these m is a subsequence of the one listed as sequence A257591 in the
= [−1, −1] yielding [l, r] = [−l, −r], which obviously both map a point x to the same image point. The authors have used this characterization to classify the closed subgroups of SO
Theorem 1. 2 .
2(1) For each odd m ∈ N and each decomposition m = a · b as in (1.1) H a,b forms a subgroup of SO(4) of order 8m.
Lemma 3. 2 .
2For each odd m ∈ N and each decomposition m = a · b as in (1.1) the group H a,b forms a subgroup of SO(4) of order 8m.
as well as l 1 ∈ {1, 3} and l 2 = 1. Then (1) h fixes a 2-dimensional subspace of H;
2 on the groups H a,b and their 4-dimensional representations. The proofs for the Lie group structure are straightforward from the corresponding properties of the finite groups, where the 2-dimensional Lie group (Theorem 1.4) arises in the same manner as the one-dimensional Lie group (Theorem 1.3). Proof of Theorem 1.3. (1) The closure of the union of the groups H a,b over all suitable b is the smallest group that contains the elements [e a , 1], [1
Lemma 3 . 9 .
39The matrix generators are subject to the same relations as the corresponding pairs of quaternions (see Lemma 3.1): cd = dc, cq = qc, ds = sd, qs = sq, q 4 = 1 4 , s 4 =
1: GAP-identifiers of H a,b for small values of m. GAP identifies groups by their order in the first position and an enumeration of the isomorphism classes in the second position. Note the symmetry in a and b in the case m = 65 where both factors are in A.
Lemma 3 . 10 .
310The blockdiagonal generators of H a,b are subject to the following relations: CD = DC, CQ = QC, DS = SD, QS = SQ, Q 4 = 1 8 , S 4 = 1 8 , CS = SC −1 =
Lemma 3 . 14 .
314Let g ∈ G a,b \ {1 8 } be written in the form (3.3). Then g fixes a point x ∈ R 8 \ {0} if and only if l 1 ∈ {1, 3}, l 2 = 1 and m = 0.
( 3 )
3The Lie groupG contains the matrices C(a) and D(b) for arbitrary values of a and b. This could have served as proof for the G a,b -equivariance of E 1 , . . . , E 5 .
( 7 )
7ρ − 3 = 0 mod 2a if and only if a = 5 and ρ = 3;(8) 3ρ − 1 = 0 mod 2a; (9) ρ + 3 = 0 mod 2a; (10) 3ρ + 1 = 0 mod 2a if and only if a = 5 and ρ = 3. Proof. (1) Suppose ρ − 1 = 0 mod 2a. Then ρ = 1 mod a and ρ 2 = 1 mod a which is a contradiction to the choice of ρ.
and Turzi [1] we define a similar map via [l, r] : x → lxr. (1.2) This provides an isomorphism Q × Q → SO(4) using the same identification [1, 1] = [−1, −1]. Therefore we may view it as a group representation. Note that this isomorphism corresponds to the application of [l, r]−1
so the other inclusion holds as well and therefore the groups are equal:H = H a,b . Hence we can define a representation of H a,b on R 4 where [e a , 1] and[1, j] act as [e a , 1] ρ and [1, j] 3 respectively.[1, j] 3 , [j, 1] .
(1.4)
This is obviously a subgroup of H a,b . Furthermore
([e a , 1] ρ ) −1 ρ = [e a , 1] and [j, 1] 2 [1, j] 3 = [1, j]
GAP-id. [120, 10] [280, 9] [312, 17] [360, 9] [408, 9] [440, 19] [520,13] [520,13] m
15
35
39
45
51
55
65
65
(a, b)
(5,3)
(5,7)
(13,3)
(5,9)
(17,3)
(5,11)
(5,13)
(13,5)
Table
Table 1 ):
1m
15
35
39
45
51
55
65
65
(a, b)
(5,3)
(5,7)
(13,3)
(5,9)
(17,3)
(5,11)
(5,13)
(13,5)
GAP-id. [240, 101] [560, 94] [624, 130] [720, 98] [816, 97] [880, 130] [1040,105] [1040,112]
Table 2 :
2GAP-identifiers of G a,b for small values of m. Note that for the groups G a,b
Table 3 :
3Cubic quivariant maps E 1 , . . . , E 8 for G 5,3 .A.3 Phase vector fields
P E 1 (y) =
−y 1 (y 1
4 + y 2
4 + y 3
4 + y 4
4 − y 1
2 )
−y 2 (y 1
4 + y 2
4 + y 3
4 + y 4
4 − y 2
2 )
−y 3 (y 1
4 + y 2
4 + y 3
4 + y 4
4 − y 3
2 )
−y 4 (y 1
4 + y 2
4 + y 3
4 + y 4
4 − y 4
2 )
P E 2 (y) =
−y 1 (2 y 1
2 y 2
2 + 2 y 3
2 y 4
2 − y 2
2 )
Table 4 :
4Phase vector fields of E 1 , . . . , E 5 restricted to S 3 ⊂ Fix (K).
Acknowledgements R.L. would like to thank U. Kühn for some helpful discussions on modular congruences.A AppendixA.1 Calculations of Molien coefficientsIn this section we want to fill the gaps that were left in Section 4 in the calculations ofand note that χ (2) (g) = 2 (cos (2η) + cos (2ρη)) cos (2ν) + 8 (cos (η) + cos (ρη)) 2 cos (ν) 2 which only depends on k 1 and k 2 . Remember that the only nonzero terms occur for l 1 ∈ {0, 2} and l 2 = m = 0. This allows us to calculate This depends on l 1 therefore the terms do not cancel out as easily as in the case d = 2.To be able to compute R 3 , we state two technical lemmas first.
Molien series and lowdegree invariants for a natural action of SO(3) Z 2 ". D R J Chillingworth, S S Lauterbach, Turzi, Journal of Physics A: Mathematical and Theoretical. 481D. R. J. Chillingworth, R Lauterbach, and S. S. Turzi. "Molien series and low- degree invariants for a natural action of SO(3) Z 2 ". Journal of Physics A: Math- ematical and Theoretical 48.1 (2015).
. P Chossat, R Lauterbach, Advanced Series in Nonlinear Dynamics. World Scientific Singapore. 15Methods in Equivariant Bifurcations and Dynamical Systems.P. Chossat and R. Lauterbach. Methods in Equivariant Bifurcations and Dynam- ical Systems. Vol. 15. Advanced Series in Nonlinear Dynamics. World Scientific Singapore, 2000.
On Quaternions and Octonions. Their Geometry, Arithmetic, and Symmetry. A K Peters, Ltd. J H Conway, D A Smith, J. H. Conway and D. A. Smith. On Quaternions and Octonions. Their Geometry, Arithmetic, and Symmetry. A K Peters, Ltd., 2003.
Commutative Algebra with a View Toward Algebraic Geometry. D Eisenbud, Springer-Verlag150New YorkGraduate Texts in MathematicsD. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. Vol. 150. Graduate Texts in Mathematics. Springer-Verlag New York, 1995.
M Field, Symmetry Breaking for Compact Lie Groups. American Mathematical Society120M. Field. Symmetry Breaking for Compact Lie Groups. Vol. 120. Memoirs of the American Mathematical Society 574. American Mathematical Society, 1996.
M Field, Advanced Texts in Mathematics. Imperial College Press London3M. Field. Dynamics and Symmetry. Vol. 3. ICP Advanced Texts in Mathematics. Imperial College Press London, 2007.
Symmetry breaking in equivariant bifurcation problems. M Field, R Richardson, Bulletin of the American Mathematical Society. 22M. Field and R. Richardson. "Symmetry breaking in equivariant bifurcation prob- lems". Bulletin of the American Mathematical Society 22.1 (1990), pp. 79-84.
GAP -Groups, Algorithms, and Programming. Version 4.7.8. The GAP GroupGAP -Groups, Algorithms, and Programming, Version 4.7.8. The GAP Group. 2015. url: http://www.gap-system.org.
Developed Software. K Gatermann, K. Gatermann. Developed Software. 1991/2002. url: http://www.orcca.on.ca/ gatermann/webpage/software.html (visited on 07/18/2015).
Computer Algebra Methods for Equivariant Dynamical Systems. K Gatermann, Lecture Notes in Mathematics. 1728SpringerK. Gatermann. Computer Algebra Methods for Equivariant Dynamical Systems. Vol. 1728. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000.
Gröbner bases, invariant theory and equivariant dynamics. K Gatermann, F Guyard, ZIBTech. rep. SC-96-37K. Gatermann and F. Guyard. Gröbner bases, invariant theory and equivariant dynamics. Tech. rep. SC-96-37. ZIB, 1996.
Disquisitiones arithmeticae. 2. Abdr. Carl Friedrich Gauss Werke. Hrsg. von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. C F Gauss, C. F. Gauss. Disquisitiones arithmeticae. 2. Abdr. Carl Friedrich Gauss Werke. Hrsg. von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 1870.
An introduction to the theory of numbers. G H Hardy, E M Wright, Oxford University Press4th edG. H. Hardy and E. M. Wright. An introduction to the theory of numbers. 4th ed. Oxford University Press, 1968.
Equivariant Bifurcation and Absolute Irreducibility in R 8 : A Contribution to Ize Conjecture and Related Bifurcations. R Lauterbach, Journal of Dynamics and Differential Equations. R. Lauterbach. "Equivariant Bifurcation and Absolute Irreducibility in R 8 : A Contribution to Ize Conjecture and Related Bifurcations". Journal of Dynamics and Differential Equations (2014), pp. 1-21.
Do absolutely irreducible group actions have odd dimensional fixed point spaces?. R Lauterbach, P Matthews, arXiv:1011.3986v1R. Lauterbach and P. Matthews. "Do absolutely irreducible group actions have odd dimensional fixed point spaces?" (2010). arXiv:1011 . 3986v1. (Visited on 07/16/2015).
. J Milne, Elliptic Curves. BookSurge PublishersJ. Milne. Elliptic Curves. BookSurge Publishers, 2006.
Fixed points in absolutely irreducible real representations. H Ruan, The Illinois Journal of Mathematics. 55H. Ruan. "Fixed points in absolutely irreducible real representations". The Illi- nois Journal of Mathematics 55.4 (2011), pp. 1551-1567.
D H Sattinger, Group Theoretic Methods in Bifurcation Theory. Berlin HeidelbergSpringer762Lecture Notes in MathematicsD. H. Sattinger. Group Theoretic Methods in Bifurcation Theory. Vol. 762. Lec- ture Notes in Mathematics. Springer Berlin Heidelberg, 1979.
Theory of complex molecular spectra. B I Zhilinskií, Russian. Moscow University PressB. I. Zhilinskií. Theory of complex molecular spectra. Russian. Moscow University Press, 1989.
| []
|
[
"Polynomial complementarity problems",
"Polynomial complementarity problems"
]
| [
"M Seetharama Gowda [email protected] \nDepartment of Mathematics and Statistics\nUniversity of Maryland\nBaltimore County Baltimore\n21250MarylandUSA\n"
]
| [
"Department of Mathematics and Statistics\nUniversity of Maryland\nBaltimore County Baltimore\n21250MarylandUSA"
]
| []
| Given a polynomial map f from R n to itself and a vector q ∈ R n , the polynomial complementarity problem, PCP(f, q), is the nonlinear complementarity problem of finding an x ∈ R n such thatx ≥ 0, y = f (x) + q ≥ 0, and x, y = 0.It is called a tensor complementarity problem if the polynomial map is homogeneous. In this paper, we establish results connecting the polynomial complementarity problem PCP(f, q) and the tensor complementarity problem PCP(f ∞ , 0), where f ∞ is the leading term in the decomposition of f as a sum of homogeneous polynomial maps. We show, for example, that PCP(f, q) has a nonempty compact solution set for every q when zero is the only solution of PCP(f ∞ , 0) and the local (topological) degree of min{x, f ∞ (x)} at the origin is nonzero. As a consequence, we establish Karamardian type results for polynomial complementarity problems. By identifying a tensor A of order m and dimension n with its corresponding homogeneous polynomial F (x) := Ax m−1 , we relate our results to tensor complementarity problems. These results show that under appropriate conditions, PCP(F +P, q) has a nonempty compact solution set for all polynomial maps P of degree less than m−1 and for all vectors q, thereby substantially improving the existing tensor complementarity results where only problems of the type PCP(F, q) are considered. We introduce the concept of degree of an R0-tensor and show that the degree of an R-tensor is one. We illustrate our results by constructing matrix based tensors.Given a (nonlinear) map f : R n → R n and a vector q ∈ R n , the nonlinear complementarity problem, NCP(f, q), is to find a vector x ∈ R n such thatx ≥ 0, y = f (x) + q ≥ 0, and x, y = 0. This reduces to a linear complementarity problem when f is linear and is a special case of a variational inequality problem. With an extensive theory, algorithms, and applications, these problems have been well studied in the optimization literature, see e.g., [3], [4], and [5]. When f is a polynomial map (that is, when each component of f is a real valued polynomial function), we say that the above nonlinear complementarity is a polynomial complementarity problem and denote it by PCP(f, q). While the entire body of knowledge of NCPs could be applied to polynomial complementarity problems, because of the polynomial nature of PCPs, one could expect interesting specialized results and methods for solving them. PCPs appear, for example, in polynomial optimization (where a real valued polynomial function is optimized over a constraint set defined by polynomials). In fact, minimizing a real valued polynomial function over the nonnegative orthant leads (via KKT conditions) to a PCP. Polynomial complementarity problems include tensor complementarity problems which have attracted a lot of attention recently in the optimization community, see e.g., [1], [2], [8], [13], [16], [17], and [18] and the references therein. Consider a tensor A of order m and dimension n given by | null | [
"https://arxiv.org/pdf/1609.05267v1.pdf"
]
| 109,936,700 | 1609.05267 | 41e4c92887187de72de5c5d46d9c9a40075d17ec |
Polynomial complementarity problems
17 Sep 2016 September 20, 2016
M Seetharama Gowda [email protected]
Department of Mathematics and Statistics
University of Maryland
Baltimore County Baltimore
21250MarylandUSA
Polynomial complementarity problems
17 Sep 2016 September 20, 2016Nonlinear complementarity problemvariational inequalitypolynomial complementarity prob- lemtensortensor complementarity problemdegree Mathematics Subject Classification: 90C33 1
Given a polynomial map f from R n to itself and a vector q ∈ R n , the polynomial complementarity problem, PCP(f, q), is the nonlinear complementarity problem of finding an x ∈ R n such thatx ≥ 0, y = f (x) + q ≥ 0, and x, y = 0.It is called a tensor complementarity problem if the polynomial map is homogeneous. In this paper, we establish results connecting the polynomial complementarity problem PCP(f, q) and the tensor complementarity problem PCP(f ∞ , 0), where f ∞ is the leading term in the decomposition of f as a sum of homogeneous polynomial maps. We show, for example, that PCP(f, q) has a nonempty compact solution set for every q when zero is the only solution of PCP(f ∞ , 0) and the local (topological) degree of min{x, f ∞ (x)} at the origin is nonzero. As a consequence, we establish Karamardian type results for polynomial complementarity problems. By identifying a tensor A of order m and dimension n with its corresponding homogeneous polynomial F (x) := Ax m−1 , we relate our results to tensor complementarity problems. These results show that under appropriate conditions, PCP(F +P, q) has a nonempty compact solution set for all polynomial maps P of degree less than m−1 and for all vectors q, thereby substantially improving the existing tensor complementarity results where only problems of the type PCP(F, q) are considered. We introduce the concept of degree of an R0-tensor and show that the degree of an R-tensor is one. We illustrate our results by constructing matrix based tensors.Given a (nonlinear) map f : R n → R n and a vector q ∈ R n , the nonlinear complementarity problem, NCP(f, q), is to find a vector x ∈ R n such thatx ≥ 0, y = f (x) + q ≥ 0, and x, y = 0. This reduces to a linear complementarity problem when f is linear and is a special case of a variational inequality problem. With an extensive theory, algorithms, and applications, these problems have been well studied in the optimization literature, see e.g., [3], [4], and [5]. When f is a polynomial map (that is, when each component of f is a real valued polynomial function), we say that the above nonlinear complementarity is a polynomial complementarity problem and denote it by PCP(f, q). While the entire body of knowledge of NCPs could be applied to polynomial complementarity problems, because of the polynomial nature of PCPs, one could expect interesting specialized results and methods for solving them. PCPs appear, for example, in polynomial optimization (where a real valued polynomial function is optimized over a constraint set defined by polynomials). In fact, minimizing a real valued polynomial function over the nonnegative orthant leads (via KKT conditions) to a PCP. Polynomial complementarity problems include tensor complementarity problems which have attracted a lot of attention recently in the optimization community, see e.g., [1], [2], [8], [13], [16], [17], and [18] and the references therein. Consider a tensor A of order m and dimension n given by
Introduction
A := [a i1 i2 ··· im ],
where a i1 i2 ··· im ∈ R for all i 1 , i 2 , . . . , i m ∈ {1, 2, . . . , n}. Let F (x) := Ax m−1 denote the homogeneous polynomial map whose ith component is given by
(Ax m−1 ) i := n i2,i3,...,i k =1 a i i2 ··· im x i2 x i3 · · · x im .
Then, for any q ∈ R n , PCP(F, q) is called a tensor complementarity problem, denoted by TCP(A, q). Now consider a polynomial map f : R n → R n , which is expressed, after regrouping terms, in the following form:
f (x) = A m x m−1 + A m−1 x m−2 + · · · + A 2 x + A 1 ,(1)
where each term A k x k−1 is a polynomial map, homogeneous of degree k − 1, and hence corresponds to a tensor A k of order k. We assume that A m x m−1 is nonzero and say that f is a polynomial map of degree The main focus of this paper is to exhibit some connections between the complementarity problems corre-sponding to the polynomial f and its leading term f ∞ (or the tensor A m ). Some connections of this type have already been observed in [6] for multifunctions satisfying the so-called 'upper limiting homogeneity property'. A polynomial map, being a sum of homogeneous maps, satisfies this upper limiting homogeneity property (see remarks made after Example 2 in [6]). The results of [6], specialized to a polynomial map f , connect PCP(f, q) and PCP(f ∞ , 0) (which is TCP(A m , 0)) and yield the following.
m − 1.
• Suppose f is copositive, that is, f (x), x ≥ 0 for all x ≥ 0, and let S denote the solution set of PCP(f ∞ , 0). If q is in the interior of the dual of S, then PCP(f, q) has a nonempty compact solution set.
• If PCP(f ∞ , 0) and PCP(f ∞ , d) have (only) zero solutions for some d > 0, then for all q, PCP(f, q) has a nonempty compact solution set.
The first result, valid for an 'individual' q, is a generalization of a copositive LCP result (Theorem 3.8.6 in [3]); it is new even in the setting of tensor complementarity problems. The second result is a 'Karamardian type' result that yields 'global' solvability for all qs. Reformulated in terms of tensors, it says the following: If A is a tensor of order m for which the problems TCP(A, 0) and TCP(A, d) have (only) zero solutions, then for F (x) = Ax m−1 , PCP(F + P, q) has a nonempty compact solution set for all polynomial maps P of degree less than m − 1 and for all vectors q. This is a substantial improvement over the existing results where only problems of the type TCP(A, q) (= PCP(F, q)) are considered.
Our objectives in this paper are to prove similar but refined results, address uniqueness issues, and provide examples. Our contributions are as follows.
• Assuming that zero is the only solution of PCP(f ∞ , 0) and the local (topological) degree of min{x, f ∞ (x)} at the origin is nonzero, we show that for all q, PCP(f, q) has a nonempty compact solution set.
• Assuming that PCP(f ∞ , 0) and PCP(f, d) (or PCP(f ∞ , d)) have (only) zero solutions for some d > 0, we show that for all q, PCP(f, q) has a nonempty compact solution set.
• Analogous to the concept of degree of an R 0 -matrix, we define the degree of an R 0 -tensor. We show that when the degree of an R 0 -tensor A is nonzero, PCP(f, q) has a nonempty compact solution set for all polynomial maps f with f ∞ (x) = Ax m−1 . We further show that the degree of an R-tensor is one.
• We construct matrix based tensors. Given a matrix A ∈ R n×n and an odd (natural) number k, we define a tensor A of order m (= k + 1) by Ax m−1 = (Ax) [k] and show that many solution based complementarity properties of A (such as R 0 , R, Q, and GUS-properties) carry over to A.
These results clearly exhibit some close connections between polynomial complementarity problems and tensor complementarity problems. In particular, they show the usefulness of tensor complementarity problems in the study of polynomial complementarity problems.
Preliminaries
Notation
Here is a list of notation, definitions, and some simple facts that will be used in the paper.
• R n carries the usual inner product and R n + denotes the nonnegative orthant; we write x ≥ 0 when x ∈ R n + and x > 0 when x ∈ int(R n + ). For two vectors x and y in R n , we write min{x, y} for the vector whose ith component is min{x i , y i }. We note that min{x, y} = 0 ⇔ x ≥ 0, y ≥ 0, and x, y = 0.
(
Given a vector y ∈ R n and a natural number k, we write y [k] for the vector whose components are
(y i ) k . When k is odd, we similarly define y [ 1 k ] .
• f denotes a polynomial map from R n to itself.
• A nonconstant polynomial map F from R n to itself is homogeneous of degree k (which is a natural number) if F (λ x) = λ k F (x)
for all x ∈ R n and λ ∈ R. For a tensor A of order m ≥ 2, the polynomial map F (x) := Ax m−1 is homogeneous of degree m − 1.
• Given f represented as in (1), f ∞ (x) denotes the leading term.
• The solution set of PCP(f, q) is denoted by SOL(f, q). • For a tensor A of order m and q ∈ R n , we let TCP(A, q) denote PCP(F, q), where F (x) := Ax m−1 .
• f q (x) := min{x, f (x) + q}, f (x) := min{x, f (x)}, and f ∞ (x) := min{x, f ∞ (x)}.
We write SOL(A, q) for the corresponding solution set.
For a polynomial map f , PCP(f, q) is equivalent to PCP(f − f (0), f (0) + q). Because of this and to avoid trivialities, throughout this paper, we assume that f (0) = 0 and f is a nonconstant polynomial, so that m ≥ 2 in (1).
Analogous to various complementarity properties that are studied in the linear complementarity literature [3], one defines (similar) complementarity properties for polynomial or tensor complementarity problems.
In particular, we say that the polynomial map f has the Q-property if for all q, PCP(f, q) has a solution and f has the GUS-property (that is, globally uniquely solvable property) if PCP(f, q) has a unique solution for all q. Similarly, we say that a tensor A has the Q-property (GUS-property) if F has the Q- Here is a new definition.
We say that a tensor A has the strong Q-property if PCP(f, q) has a nonempty compact solution set for all q ∈ R n and for all polynomial maps f with f ∞ (x) = Ax m−1 or equivalently, PCP(F + P, q) has a nonempty compact solution set for all q ∈ R n and for all polynomial maps P of degree less than m − 1.
We note an important consequence of the Q-property of a polynomial map f : Given any vector q, ifx is a solution of PCP(f, q − e), where e is a vector of ones, then,x ≥ 0 and f (x) + q ≥ e > 0. By perturbingx we get a vector u such that u > 0 and f (u) + q > 0. This shows that when f has the Q-property, for any q ∈ R n , the (semi-algebraic) set {x ∈ R n : x ≥ 0, f (x) + q ≥ 0} has a Slater point.
In this paper, we use degree-theoretic ideas. All necessary results concerning degree theory are given in [4], Prop. 2.1.3; see also, [12], [15]. Here is a short review. Suppose Ω is a bounded open set in R n , g : Ω → R n is continuous and p ∈ g(∂ Ω), where Ω and ∂ Ω denote, respectively, the closure and boundary of Ω. Then the degree of g over Ω with respect to p is defined; it is an integer and will be denoted by deg (g, Ω, p). When this degree is nonzero, the equation g(x) = p has a solution in Ω. Suppose g(x) = p has a unique solution, say, x * in Ω. Then, deg (g, Ω ′ , p) is constant over all bounded open sets Ω ′ containing x * and contained in Ω. This common degree is called the local (topological) degree of g at x * (also called the index of g at x * in some literature); it will be denoted by deg (g,
x * ). In particular, if h : R n → R n is a continuous map such that h(x) = 0 ⇔ x = 0,
Bounded solution sets
Many of our results require (and imply) bounded solution sets. The following is a basic result.
Proposition 2.1.
For a polynomial map f , consider the following statements:
(i) SOL(f ∞ , 0) = {0}. (ii) For any bounded set K in R n , q∈K SOL(f, q) is bounded. Then, (i) ⇒ (ii). The reverse implication holds when f is homogeneous (that is, when f = f ∞ ).
Proof. Assume that (i) holds. We show (ii) by a standard 'normalization argument' as follows. If possible, let K be a bounded set in R n with q∈K SOL(f, q) unbounded. Then, there exist sequences q k in K and
x k ∈ SOL(f, q k ) such that ||x k || → ∞ as k → ∞. Now, from (2), min{x k , f (x k ) + q k } = 0 ⇒ min x k ||x k || , f (x k ) + q k ||x k || m−1 = 0.
Let k → ∞ and assume (without loss of generality) lim x k ||x k || = u. As m ≥ 2, from (1) and the boundedness
of the sequence q k , we get f (x k ) ||x k || m−1 → f ∞ (u) and q k ||x k || m−1 → 0; hence min{u, f ∞ (u)} = 0.
From (i), u = 0. As ||u|| = 1, we reach a a contradiction. Thus, (ii) holds.
Now, if f is homogeneous, that is, if f = f ∞ , (ii) implies that SOL(f ∞ , 0) is bounded.
As this set contains zero and is invariant under multiplication by positive numbers, we see that SOL(f ∞ , 0) = {0}. This concludes the proof.
Remarks 1.
As the solution set of any PCP(f, q) is always closed, we see that
When SOL(f ∞ , 0) = {0}, the solution set SOL(f, q)
is compact for any q (but may be empty).
A degree-theoretic result
The following result and its proof are slight modifications of Theorem 3.1 in [8] and its proof.
(a) f ∞ (x) = 0 ⇒ x = 0 and (b) deg f ∞ , 0 = 0.
Then, for all q ∈ R n , PCP(f, q) has a nonempty compact solution set.
Proof. From the representation (1), we can write
f (x) = f ∞ (x) + p(x), where p(x)
is the sum of the lower order terms in f (x). We fix a q and consider the homotopy So, H(·, 1), that is, min{x, f (x) + q} has a zero in Ω. This proves that PCP(f, q) has a solution. The compactness of the solution set follows from the previous proposition and Remark 1.
H(x, t) := min x, (1 − t)f ∞ (x) + t[f (x) + q] = min x, f ∞ (x) + t[p(x) + q] , where t ∈ [0, 1]. Then, H(x, 0) = min{x, f ∞ (x)} and H(x, 1) = min{x, f (x) + q}. Since min{x, f ∞ (x)} = 0 ⇒ x = 0,
Remarks 2.
We make two important observations. First, note that the conditions (a) and (b) in the above theorem are imposed only on the leading term of f . This means that in the conclusion, the lower order terms of f are quite arbitrary. Second, the above theorem yields a stability result: If g is a polynomial map with g ∞ sufficiently close to f ∞ and q ∈ R n , then PCP(g, q) has a nonempty compact solution set. To make this precise, suppose conditions (a) and (b) are in place and let Ω be any bounded open set in R n containing zero. Let ε be the distance between zero and (the compact set) f ∞ (∂ Ω) in the ∞-norm. Then, for any polynomial map g on R n with sup Ω || f ∞ (x) − g ∞ (x)|| ∞ < ε and any q ∈ R n , PCP(g, q) has a nonempty compact solution set. This follows from the nearness property of degree, see [4], Proposition 2.1.3(c).
To motivate our next concept, consider an R 0 -matrix A on R n so that for Φ(x) := min{x, Ax}, Φ(x) = 0 ⇒ x = 0. Then, the local (topological) degree of Φ at the origin is called the degree of A in the LCP literature [7], [3]. Symbolically,
deg(A) := deg (Φ, 0).
An important result in LCP theory is:
An R 0 -matrix with nonzero degree is a Q-matrix.
We now extend this concept and result to tensors. By definition, A has the strong Q-property.
Matrix based tensors
In order to illustrate our results, we need to construct polynomials or tensors with specified complementarity properties. With this in mind, we now describe matrix based tensors. First, we prove a result that connects complementarity problems corresponding to a homogeneous polynomial and its power. Proof. (a) As k is odd, the univariate function t → t k is strictly increasing on R. Hence, the following statements are equivalent:
• x ≥ 0, G(x) + q ≥ 0, and x i G(x) + q i = 0 for all i. • x ≥ 0, F (x) + q [ 1 k ] ≥ 0, and x i F (x) + q [ 1 k ] i = 0 for all i.H(x, t) := min x, (1 − t)F (x) + tG(x) ,
where t ∈ [0, 1]. We show that H(x, t) = 0 ⇒ x = 0 for all t.
Clearly, this holds for t = 0 and t = 1 as H(x, 0) = F (x) and H(x, 1) = G(x). For 0 < t < 1,
H(x, t) = min x, F (x) [(1 − t) + tF (x) [k−1] ] .
As k is odd, each component in the factor ] ] is always positive and hence,
[(1 − t) + tF (x) [k−1H(x, t) = 0 ⇒ min{x, F (x)} = 0 ⇒ x = 0.
Let Ω be any bounded open set containing 0. Then, by the homotopy invariance of degree,
deg F , 0 = deg F , Ω, 0 = deg G, Ω, 0 = deg G, 0 .
As an illustration, let A be tensor of order m and dimension n with the corresponding homogeneous map only if A has the GUS-property.
As a further illustration, we construct matrix based tensors. Let A be an n × n real matrix. For any odd natural number k, define a tensor A of order k + 1 and dimension n by
Ax (k+1)−1 := (Ax) [k] .
We say that A is a matrix based tensor induced by the matrix A and exponent k. It follows from the above result that
SOL(A, q) = SOL(A, q [ 1 k ] ),(3)
where SOL(A, q) denotes the solution set of the linear complementarity problem LCP(A, q).
We have the following result.
Proposition 4.2. Consider a matrix based tensor A corresponding to a matrix A and odd exponent k. Then the following statements hold:
(1) The set of all q's for which TCP(A, q) has a solution is closed.
(2) If A is an R 0 -matrix, then A has the R 0 -property. In this setting, deg(A) = deg(A).
(3) If A is an R-matrix, then A has the R-property.
(4) If A is a Q-matrix, then A has the Q-property.
Proof.
(1) For any matrix A, the set D := {q ∈ R n : SOL(A, q) = ∅} is closed (as it is the union of complementary cones [3]). As SOL(A, q) = SOL(A, q [ 1 k ] ), we can write D = {p [k] ∈ R n : SOL(A, p) = ∅}. Since k is odd, the map p → p [k] is a homeomorphism of R n ; hence set {p ∈ R n : SOL(A, p) = ∅} is closed.
The statements (2) − (4) follow easily from Theorem 4.1.
Combining this with Theorem 3.2, we get the following.
A Karamardian type result
A well-known result of Karamardian [9] deals with a positively homogeneous continuous map h : R n → R n . It asserts that for such a map, if NCP(h, 0) and NCP(h, d) have trivial/zero solutions for some d > 0, then NCP(h, q) has nonempty solution set for all q. Below, we prove a result of this type for polynomial maps.
Theorem 5.1. Let f : R n → R n be a polynomial map with leading term f ∞ . Suppose there is a vector d > 0 in R n such that one of the following conditions holds:
(a) SOL(f ∞ , 0) = {0} = SOL(f ∞ , d). (b) SOL(f ∞ , 0) = {0} = SOL(f, d).
Then, deg f ∞ , 0 = 1. Hence, for all q ∈ R n , PCP(f, q) has a nonempty compact solution set.
Note: We recall our assumption that f (0) = 0. In the case of (a), the second part of the conclusion has already been noted in Theorem 3 of [6]; here we present a different proof.
Proof. Let g denote either f ∞ or f . Then, for any t ∈ [0, 1], the leading term of (
1 − t)f ∞ (x) + t[g(x) + d] is f ∞ . Now consider the homotopy H(x, t) := min x, (1 − t)f ∞ (x) + t[g(x) + d] ,
where t ∈ [0, 1]. Since the condition SOL(f ∞ , 0) = {0} is equivalent to min{x, f ∞ (x)} = 0 ⇒ x = 0, by a normalization argument (as in the proof of Proposition 2.1), we see that the zero set
x : H(x, t) = 0 for some t ∈ [0, 1] is bounded, hence contained in some bounded open set Ω in R n . Then, with g d (x) = min{x, g(x) + d}, by the homotopy invariance of degree,
deg f ∞ , 0 = deg f ∞ , Ω, 0 = deg g d , Ω, 0 = deg g d , 0 ,
where the last equality holds due to the implication min{x, g(x) + d} = 0 ⇒ x = 0. Now, when x is close to zero, g(x) + d is close to g(0) + d = d > 0 (recall that f (0) = 0). Hence for x close to zero, g d = min{x, g(x) + d} = x. So, the (local) degree of g d at the origin is one. This yields deg f ∞ , 0 = 1. The second part of the conclusion comes from Theorem 3.1.
We now have a useful consequence of the above theorem. (e) Any tensor A induced by an R-matrix A and an odd exponent k.
Note: By Corollary 5.2, all the tensors mentioned above will have the strong Q-property.
Example 1. We now provide an example of an R 0 -tensor with a nonzero degree which is not an R-tensor. Consider the 2 × 2 matrix
A = −1 1 3 −2 .
This is an N-matrix of first category (which means that all principle minors of N are negative and A has some nonnegative entries). Kojima and Saigal [10] have shown that such a matrix is an R 0 -matrix with degree −1. Now, for any odd number k, consider the tensor induced by A, that is, for which Ax m−1 = (Ax) [k] .
Then, A is an R 0 -tensor with degree −1. By Theorem 3.2, this A has the strong Q-property; it cannot be an R-tensor by Corollary 5.2.
The global uniqueness in PCPs
In the NCP theory, a nonlinear map f on R n is said to have the GUS-property if for every q ∈ R n , NCP(f, q) has a unique solution. One sufficient condition for this property is the 'uniform P-property' of f on R n + ([4], Theorem 3.5.10): There exists a positive constant α such that
max 1≤i≤n (x − y) i [f (x) − f (y)] i ≥ α||x − y|| 2 ∀ x, y ∈ R n + .
Another is the 'positively bounded Jacobians' condition of Megiddo and Kojima [14]. The GUS-property in the context of tensor complementarity problems has been addressed recently in [1], [2], and [8]. In this section, we address the global uniqueness property in PCPs.
Theorem 6.1. Suppose f is a polynomial map such that SOL(f ∞ , 0) = {0}.
Then the following are equivalent:
(a) f has the GUS-property.
(b) PCP(f, q) has at most one solution for every q.
Moreover, condition (b) holds when f satisfies the P-property on R n + :
max i (x − y) i f (x) − f (y) i > 0 for all x, y ≥ 0, x = y.(4)
Proof. Clearly, (a) ⇒ (b). Suppose (b) holds. As f (0) = 0, SOL(f, d) = {0} for every d > 0. Since (by assumption) SOL(f ∞ , 0) = {0}, by Theorem 5.1, for every q, PCP(f, q) has a solution, which is unique by (b). Thus f has the GUS-property.
Now suppose f satisfies the additional condition (4). We verify condition (b). If possible, suppose x and y are two solutions of PCP(f, q) for some q. Then, for some i,
0 < (x − y) i [f (x) − f (y)] i = − x i (f (y) + q) i + y i (f (x) + q) i ≤ 0
yields a contradiction. Thus (b) holds and hence (a) holds.
We remark that when f is homogeneous (in which case, f = f ∞ ), the condition SOL(f ∞ , 0) = {0} in the above theorem is superfluous. It is not clear if this is so in the general case. . This shows that when the order is more than 2, one can never get uniqueness in all perturbed problems.
The following result gives us a way of constructing tensors with the GUS-property. Proposition 6.3. Suppose A is an P-matrix and k is an odd natural number. Then, the tensor defined by Ax m−1 = (Ax) [k] has the GUS-property as well as the strong Q-property.
Proof. We have, from (3), SOL(A, q) = SOL(A, q [ 1 k ]). As A is a P-matrix, all related LCPs will have unique solutions. Thus, TCP(A, q) has exactly one solution for all q and so, A has the GUS-property. Since a P-matrix is an R-matrix, the strong Q-property of A comes from Corollary 5.2.
Copositive PCPs
We say that a polynomial map f is copositive if
f (x), x ≥ 0 for all x ≥ 0.
For example, f is copositive in the following situations:
(i) f is monotone, that is, f (x) − f (y), x − y ≥ 0 for all x, y ∈ R n (recall our assumption that f (0) = 0).
(ii) In the polynomial representation (1), each tensor A k is nonnegative.
(iii) In the polynomial representation (1), the leading tensor A m is nonnegative and other (lower order) homogeneous polynomials are sums of squares.
We remark that testing the copositivity of a polynomial map or more generally that of nonnegativity of a real-valued polynomial function on a semi-algebraic set is a hard problem in polynomial optimization. These generally involve SOS polynomials, certificates of positivity (known as positivestellensatz) and are related to some classical problems (example, Hilbert's 17th problem) in algebraic geometry [11].
Our first result in this section gives the solvability for (individual) qs when f is copositive. We let It is easy to see that f ∞ is copositive when f is copositive. This raises the question whether the above result continues to hold if the copositivity of f is replaced by that of f ∞ . The following example (modification of Example 5 in [6]) shows that this cannot be done.
Example 2. Let
A = 0 −1 1 0 , q = 2 −2 , and f (x) = ||x|| 2 Ax − 2 √ 2x. Clearly, f ∞ (x) = ||x|| 2 Ax. Since A is skew-symmetric, x, f ∞ (x) = 0 for all x. Thus, f ∞ is copositive.
An easy calculation shows that S is the nonnegative real-axis in R 2 , so that S * is the closed right halfplane and q ∈ int(S * ). We claim that PCP(f, q) has no solution. Suppose that x ∈ SOL(f, q). Since A is skew-symmetric, the complementarity condition f (x) + q, x = 0 becomes q, x = 2 √ 2||x|| 2 , which, by Cauchy-Schwarz inequality, gives ||x|| ≤ 1. Further, the nonnegativity condition f (x) + q ≥ 0 implies that ||x|| 2 x 1 − 2 √ 2x 2 − 2 ≥ 0 where x 1 and x 2 are the first and the second components of x respectively. But this cannot hold since x 2 ≥ 0 and ||x|| 2 x 1 ≤ ||x|| 3 ≤ 1. Hence the claim.
The following result shows that Theorem 7.1 continues to hold if the copositivity of f is replaced by that of
f ∞ provided we assume S = {0}.
Corollary 7.2. For a polynomial map f , suppose f or f ∞ is copositive, and S = {0}. Then, for all q ∈ R n , PCP(f, q) has a nonempty compact solution set.
Proof. As observed previously, f ∞ is copositive when f is copositive. So we assume that f ∞ is copositive. Then, for any d > 0, we claim that SOL(f ∞ , d) = {0}. To see this, suppose x ∈ SOL(f ∞ , d). Then x ≥ 0 and 0 = x, f ∞ (x) + d = x, f ∞ (x) + x, d ≥ x, d due to the copositity condition. Since d > 0 and x ≥ 0, we see that x = 0. As SOL(f ∞ , 0) = {0} = SOL(f ∞ , d), from Theorem 5.1, we see that PCP(f, q) has a nonempty compact solution set.
We now state Theorem 7.1 for tensors. For a polynomial map f , consider the set D := {q ∈ R n : SOL(f, q) = ∅}. When f is linear, this set is closed as it is a finite union of polyhedral cones. It is also closed in some special situations (see e.g., Proposition 4.2). As the following example shows, this need not be the case for a general (homogeneous) polynomial map.
Example 3. On R 2 , consider the map F (x, y) = x 2 − y 2 − (x − y) 2 , x 2 − y 2 + 2(x − y) 2 .
We show that (i) The image of R n + under F is not closed, and (ii) the set D := {q ∈ R n : SOL(F, q) = ∅} is not closed.
Item (i) follows from the observations 1, 1 + 3 4k 2 = F k + 1 2k , k ∈ F (R n + ) and (1, 1) ∈ F (R n ).
To see Item (ii), let q k := −F k + 1 2k , k = − 1, −1 − 3 4k 2 and q = (−1, −1).
Clearly, (k + 1 2k , k) ∈ SOL(F, q k ) and q k → q as k → ∞. We claim that SOL(F, q) = ∅. Assuming the contrary, let (x, y) ∈ SOL(F, q). Since F (x, y) + q ≥ 0, we must have x 2 − y 2 − (x − y) 2 − 1 ≥ 0. Hence, neither x nor y can be zero. When both x and y are nonzero, by complementarity conditions, we must have x 2 − y 2 − (x − y) 2 − 1 = 0 and x 2 − y 2 + 2(x − y) 2 − 1 = 0. Upon subtraction, we get (x − y) 2 = 0, that is, x = y. But then, −1 = 0 yields a contradiction. Hence, for the given map F , the set of all solvable qs is not closed.
Acknowledgments
Part of this work was carried out while the author was visiting Beijing Jiaotong University in Beijing, China, during May/June 2016. He wishes to thank Professors Naihua Xiu and Lingchen Kong for the invitation and hospitality.
=
A m x m−1 denote the 'leading term' of f . Then, for all q ∈ R n , PCP(f ∞ , q) ≡ TCP(A m , q).
Note that f q (x) = 0 if and only if x ∈ SOL(f, q), etc. Also, as f ∞ is homogeneous, SOL(f ∞ , 0) contains zero and is invariant under multiplication by positive numbers. Moreover, SOL(f ∞ , 0) = {0} if and only if f ∞ (x) = 0 ⇒ x = 0 .
property (respectively, GUS-property), where F (x) := Ax m−1 . A tensor A is said to have the R 0 -property if SOL(A, 0) = {0} and has the R-property if it has the R 0 -property and SOL(A, d) = {0} for some d > 0.
then, for any bounded open set containing 0, we have deg (h, 0) = deg (h, Ω, 0); moreover, when h is the identity map, deg (h, 0) = 1. Let H(x, t) : R n × [0, 1] → R n be continuous (in which case, we say that H is a homotopy) and the zero set {x : H(x, t) = 0 for some t ∈ [0, 1]} be bounded. Then, for any bounded open set Ω in R n that contains this zero set, we have the homotopy invariance of degree: deg H(·, 1), Ω, 0 = deg H(·, 0), Ω, 0 .
Theorem 3. 1 .
1Let f be a polynomial map and f ∞ (x) := min{x, f ∞ (x)}. Suppose the following conditions hold:
a normalization argument (as in the proof of Proposition 2.1) shows that the zero set x : H(x, t) = 0 for some t ∈ [0, 1] is bounded, hence contained in some bounded open set Ω in R n . Then, by the homotopy invariance of degree, we have deg H(·, 1), Ω, 0 = deg H(·, 0), Ω, 0 = deg f ∞ , 0 = 0.
Let
A be an R 0 -tensor. Then, with F (x) = Ax m−1 and F (x) := min{x, F (x)}, we have F (x) = 0 ⇒ x = 0; hence deg ( F , 0) is defined. We call this number, the degree of A. Symbolically, deg(A) := deg ( F , 0). We now state the tensor version of Theorem 3.1. Recall that A has the strong Q-property if PCP(f, q) has a nonempty compact solution set for all polynomial maps f with f ∞ (x) = Ax m−1 and all q ∈ R n .Theorem 3.2. Suppose A is an R 0 -tensor with deg(A) = 0. Then, A has the strong Q-property.Proof. Let f be any polynomial map with f ∞ (x) = Ax m−1 . Then, the assumed conditions on A translate to conditions (i) and (ii) in Theorem 3.1. Thus, PCP(f, q) has a nonempty compact solution set for all q.
Theorem 4. 1 .
1Suppose F : R n → R n is a homogeneous polynomial map and k is an odd natural number. Define the map G by G(x) = F (x)[k] for all x. Then the following statements hold:(a) SOL(G, q) = SOL(F, q [ 1 k ] ) for all q ∈ R n . In particular, SOL(G, 0) = SOL(F, 0). (b) If SOL(F, 0) = {0}, then deg F , 0 = deg G, 0 .
.
Then, SOL(G, 0) = {0} from (a). These are equivalent to the implications F (x) = 0 ⇒ x = 0 and G(x) = 0 ⇒ x = 0. Consider the homotopy
F
(x) := Ax m−1 . Let k be an odd natural number. Define a tensor B of order l := k(m − 1) + 1 by Bx l−1 := (Ax m−1 ) [k] . Then for all q, SOL(B, q) = SOL(A, q [ 1 k ] ). In particular, B has the Q-property if and only if A has the Q-property and B has the GUS-property if and
Corollary 4 . 3 .
43Suppose A is an R 0 -matrix with deg(A) = 0. Then, the corresponding tensor A has the strong Q-property. Remarks 3. Extending the ideas above, we now outline a way of constructing (more) R 0 -tensors with the strong Q-property. Let A be an R 0 -matrix with deg(A) = 0 and k be an odd natural number. Let θ(x) be a homogeneous polynomial function such that θ(x) > 0 for all 0 ≤ x = 0. (For example, θ(x) = ||x|| 2r , where r is a natural number.) Define a tensor B by Bx m−1 = θ(x)(Ax) [k] . Then, as in the proof of Theorem 4.1, we can show that for all t ∈ [0, 1], min x, t(Ax) [k] + (1 − t)θ(x)(Ax) [k] = 0 ⇒ x = 0. This implies that B is an R 0 -tensor and (by homotopy invariance of degree) deg(B) = deg(A) = deg(A) = 0. Hence B has the strong Q-property by Theorem 3.2.
Corollary 5 . 2 .
52The degree of an R-tensor is one. Hence, every R-tensor has the strong Q-property.Proof. LetA be an R-tensor so that for some d > 0, SOL(A, 0) = {0} = SOL(A, d). Written differently, SOL(F, 0) = {0} = SOL(F, d), where F (x) = Ax m−1 . Now, let f be any polynomial map with f ∞ = F. Then, SOL(f ∞ , 0) = {0} = SOL(f ∞ , d). From the above theorem, deg(A) := deg ( F , 0) = deg f ∞ , 0 = 1.The additional statement about the strong Q-property now comes from Theorem 3.2.
Remarks 4 .
4The class of R-tensors is quite broad. It includes the following tensors.(a) Nonnegative tensors with positive 'diagonal'. These are tensors A = [a i1 i2 ··· im ] with a i1 i2 ··· im ≥ 0 for all i 1 , i 2 , . . . , i m and a i i ··· i > 0 for all i.
( b )
bCopositive R 0 -tensors. These are tensors A = [a i1 i2 ··· im ] satisfying the property Ax m−1 , x ≥ 0 for all x ≥ 0 and SOL(A, 0) = {0}.(c) Strictly copositive tensors. These are tensors A = [a i1 i2 ··· im ] satisfying the property Ax m−1 , x > 0 for all 0 = x ≥ 0. (d) Strong M-tensors. A tensor A = [a i1 i2 ··· im ] is said to be a Z-tensor if all the off-diagonal entries of A are nonpositive. It is a strong M-tensor [8] if it is a Z-tensor and there exists d > 0 such that Ad m−1 > 0.
Proposition 6. 2 .
2For a tensor A, the following are equivalent: (a) A has the GUS-property.(b) TCP(A, q) has at most one solution for all q. Moreover, when these conditions hold, A has the strong Q-property. Proof. Obviously, (a) ⇒ (b). When (b) holds, SOL(A, 0) = {0} = SOL(A, d) for any d > 0. Thus, A is an R-tensor. By Corollary 5.2, A has the strong Q-property. In particular, TCP(A, q) has a solution for all q and by (b), the solution is unique. Thus (b) ⇒ (a) and we also have the strong Q-property. Remarks 5. Consider a tensor A with the GUS-property. The above result shows that for every polynomial map f with f ∞ (x) = Ax m−1 and for all q, PCP(f, q) has a solution. Can we demand that all these PCP(f, q)s have unique solution(s)? The following argument shows that this can never be done when the order is more than 2. Let A be any tensor of order m > 2 and F (x) = Ax m−1 . With e denoting the vector of ones in R n , define the vector d := −Ae m−1 − e and let D be the diagonal matrix with d as its diagonal. Let f (x) := Ax m−1 + Dx. Then, it is easy to see that 0 and e are two solutions of PCP(f, e)
S
:= SOL(f ∞ , 0). Theorem 7.1. ([6], Theorem 2) Suppose the polynomial map f is copositive. If q ∈ int(S * ), then, PCP(f, q) has a nonempty compact solution set. Moreover, when the set D := {q ∈ R n : SOL(f, q) = ∅} is closed, PCP(f, q) has a solution for all q ∈ S * .
Corollary 7. 3 .
3Suppose A is a copositive tensor, that is, Ax m−1 , x ≥ 0 for all x ≥ 0. Let S = SOL(A, 0). If q ∈ int(S * ),then, TCP(A, q) has a nonempty compact solution set. Moreover, when the set D := {q ∈ R n : SOL(A, q) = ∅} is closed, TCP(A, q) has a solution for all q ∈ S * . 8 On the closedness of the set of all solvable qs
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[
"THE MUTUAL SINGULARITY OF HARMONIC MEASURE AND HAUSDORFF MEASURE OF CODIMENSION SMALLER THAN ONE",
"THE MUTUAL SINGULARITY OF HARMONIC MEASURE AND HAUSDORFF MEASURE OF CODIMENSION SMALLER THAN ONE"
]
| [
"Xavier Tolsa "
]
| []
| []
| Let Ω ⊂ R n+1 be open and let E ⊂ ∂Ω with 0 < H s (E) < ∞, for some s ∈ (n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with H s on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that Ω is a uniform domain.Partially supported by by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain). | 10.1093/imrn/rnz197 | [
"https://arxiv.org/pdf/1901.07783v2.pdf"
]
| 119,162,793 | 1901.07783 | 5964a972c27ee7f3aa4b55e183a98ed1aa430588 |
THE MUTUAL SINGULARITY OF HARMONIC MEASURE AND HAUSDORFF MEASURE OF CODIMENSION SMALLER THAN ONE
26 Jan 2019
Xavier Tolsa
THE MUTUAL SINGULARITY OF HARMONIC MEASURE AND HAUSDORFF MEASURE OF CODIMENSION SMALLER THAN ONE
26 Jan 2019
Let Ω ⊂ R n+1 be open and let E ⊂ ∂Ω with 0 < H s (E) < ∞, for some s ∈ (n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with H s on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that Ω is a uniform domain.Partially supported by by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain).
INTRODUCTION
In this paper we study the relationship between harmonic measure and Hausdorff measure of codimension smaller than 1 in R n+1 . The importance of harmonic measure is mainly due to its connection with the Dirichlet problem for the Laplacian. Indeed, recall that given a domain Ω ⊂ R n+1 and a point p ∈ Ω, the harmonic measure with pole at p is the measure ω p satisfying the property that, for any function f ∈ C(∂Ω) ∩ L ∞ (∂Ω), the integral f dω p equals the value at p of the harmonic extension of f to Ω.
The study of the metric and geometric properties of harmonic measure has been a classical subject in mathematical analysis since the Riesz brothers theorem [RR] asserting that harmonic measure is absolutely continuous with respect to arc length measure on simply connected planar domains with rectifiable boundary. In the plane, complex analysis plays a very important role in connection with harmonic measure, essentially because of the invariance of harmonic measure by conformal mappings. This fact makes the case of planar domains rather special.
In the plane it is known that the dimension of harmonic measure is at most 1 by a celebrated result of Jones and Wolff [JW]. This means that there exists a set E ⊂ ∂Ω of Hausdorff dimension at most 1 which has full harmonic measure. Furthermore, such set E can be taken so that it has σ-finite length, as shown by Wolff [Wo1]. More precise results for simply connected planar domains had been obtained previously by Makarov [Mak1], [Mak2].
In higher dimensions one has to use real analysis techniques to study harmonic measure. The codimension 1 is still quite special, mainly because of relationship between harmonic measure and rectifiability. For instance, in [AHM 3 TV] it was shown that the mutual absolute continuity between harmonic measure and n-dimensional Hausdorff measure on a subset E ⊂ ∂Ω, Ω ⊂ R n+1 , implies the nrectifiability of E. Also, under the assumption that ∂Ω is AD-regular, that is H n (B(x, r) ∩ ∂Ω) ≈ r n for all x ∈ ∂Ω, 0 < r ≤ diam(∂Ω), many recent works have been devoted to relate quantitative properties of harmonic measure and other analytic or geometric properties of the domain. See for example [Az1], [GMT], [HLMN], [HM1], [HM2], [HMM], [HMU], [MT], etc.
One of the main differences between the planar case and the higher dimensional case is that in R n+1 , with n ≥ 2, there exist domains where the dimension of harmonic measure is larger than n. This was proved by Wolff in [Wo2]. An important open problem consists of finding the sharp value for the upper bound of the dimension of harmonic measure in R n+1 , n ≥ 2. In [Bo] Bourgain showed that this sharp value is strictly smaller than n+1. In [Jo] Jones conjectured that the sharp bound should be n+1−1/n. However, for the moment there have been no significative advances on this open problem. On the other hand, the techniques of Bourgain [Bo] play an important role in more recent results asserting that in some classes of sets (for example, in some self-similar sets) the dimension of harmonic measure is strictly smaller than the dimension of the set. See [Ba1], [Ba2], and [Az2].
As mentioned above, the current paper deals with harmonic measure in the case of codimension less than 1. Although the main result of the paper is not directly related to the above Jones' conjecture, I think that this contributes to a better understanding of the behavior of harmonic measure in this codimension.
To state precisely the main result, we need some additional notation. For n ≥ 2, let Ω ⊂ R n+1 be open and let E ⊂ ∂Ω be a non-empty set. We say that the local capacity density condition (or local CDC) holds in E is there exists constants r E > 0 and c E > 0 such that
(1.1) Cap(B(x, r) ∩ Ω c ) ≥ c E r n−1 for all x ∈ E and 0 < r ≤ r E ,
where Cap stands for the Newtonian capacity (see Section 2.2 for the definition). We denote by ω the harmonic measure in Ω.
The main result of this paper is the following. In other words, harmonic measure cannot be mutually absolutely continuous with Hausdorff measure of codimension less than 1 in any subset of positive harmonic measure, under the local CDC assumption. Recall that the same result was proved in [AM] by Azzam and Mourgoglou under the additional assumption that Ω is a uniform domain. Recall that, roughly speaking, a domain is called uniform if it satifies an interior porosity assumption (the so-called interior corkscrew condition), and a quantitative connectivity condition in terms of Harnack chains.
The methods in the current paper are very different from the ones used in [AM]. The new main tool is an identity obtained by integration by parts (see Section 3.1), whose application requires later some rather delicate stopping time arguments. On the other hand, the arguments in [AM] use blowups and tangent measures, and it seems that the uniformity assumption is important. In fact, in their work, Azzam and Mourgoglou leave as an open question the possibility of eliminating the uniformity assumption. They also ask the same questions about the CDC: can this be avoided? While Theorem 1.1 confirms that uniformity is not necessary, it is still an open problem to know if the CDC is required.
In fact, in [AM] a non-degeneracy condition weaker (at least, a priori) than the CDC is used. I think that, quite likely, in the arguments below one may be able to replace the local CDC assumption by the non-degeneracy condition of Azzam-Mourgoglou. However, I have preferred to state Theorem 1.1 in terms of the local CDC, which is closer to the usual CDC. On the other hand, the techniques in the current paper do not look very useful for codimensions larger than 1, unlike the arguments in [AM], which are applied by the authors to derive other related results.
The aforementioned integration by parts formula (see (3.1)) required for the proof of Theorem 1.1 is a generalization of a formula that has already been applied to some problems involving harmonic or elliptic measure and rectifiability in works such as [HLMN] or [AGMT], and it goes back to some work of Lewis and Vogel [LV], at least.
PRELIMINARIES
In the paper, constants denoted by C or c depend just on the dimension and perhaps other fixed parameters (such as the constant c E involved the local CDC, for example). We will write a b if there is C > 0 such that a ≤ Cb . We write a ≈ b if a b a.
2.1. Measures. The set of (positive) Radon measure in R n+1 is denoted by M + (R n+1 ). The Hausdorff s-dimensional measure and Hausdorff s-dimensional content are denoted ty H s and H s ∞ , respectively. Given µ ∈ M + (R n+1 ), its supper s-dimensional density at x is defined by
Θ s, * (x, µ) = lim sup r→0 µ(B(x, r)) (2r) s .
Recall that, given an
H s -measurable set E ⊂ R n+1 with 0 < H s (E) < ∞, we have (2.1) 2 −s ≤ Θ s, * (x, H s | E ) ≤ 1 for H s -a.e. x ∈ E.
See [Mat,Chapter 6], for example.
2.2. Newtonian capacity and harmonic measure. The fundamental solution of the minus Laplacian in R n+1 , n ≥ 2, equals
E(x) = c n |x| n−1 ,
where c n = (n − 1)H n (S n ), with S n being the unit hypersphere in R n+1 .
The Newtonian potential of a measure µ ∈ M + (R n+1 ) is defined by
U µ(x) = E * µ(x),
and the Newtonian capacity of a compact set F ⊂ R n+1 equals
Cap(F ) = sup µ(F ) : µ ∈ M + (R n+1 ), supp µ ⊂ F, U µ ∞ ≤ 1 .
It is well known that U µ ∞ = U µ ∞,F , and that there exist a unique measure that attains the supremum in the definition of Cap(F ). If µ attains that supremum, then it turns out that U µ(x) = 1 for quasievery x ∈ F (denoted also q.e. in F ), i.e., for all x ∈ F with the possible exception of a set of zero Newtonian capacity. The probability measure
µ F = 1 Cap(F ) µ
is called equilibrium measure (of F ), and so it holds that
U µ F (x) = 1 Cap(F ) for q.e. x ∈ F .
Recall that we denote by ω the harmonic measure on an open set Ω. The associated Green function is denoted by g(·, ·). The following result is quite well known, but we prove it here for the reader's convenience.
Lemma 2.1. Given n ≥ 2, let Ω ⊂ R n+1 be open and let B be a closed ball centered at ∂Ω. Then
ω x (B) ≥ c(n) Cap( 1 4 B ∩ ∂Ω) r(B) n−1 for all x ∈ 1 4 B ∩ Ω,
with c(n) > 0.
Proof. Let µ 1 4 B∩∂Ω be the equilibrium measure for 1 4 B ∩ ∂Ω, and let µ = Cap( 1 4 B ∩ ∂Ω) µ 1 4 B∩∂Ω , so that U µ ∞ ≤ 1 and µ = Cap( 1 4 B ∩ ∂Ω). Notice that, for all x ∈ B c ,
U µ(x) = c n |x − y| n−1 dµ(y) ≤ c n µ ( 3 4 r(B)) n−1 . Consider the function f (x) = U µ(x) − cn µ ( 3 4 (B)) n−1 . Using that f (x) ≤ 0 in B c , f ∞ ≤ 1
, and that f is harmonic in Ω, by the maximum principle we deduce that, for all x ∈ Ω,
ω x (B) ≥ f (x).
In particular, for x ∈ 1 4 B we have
ω x (B) ≥ c n |x − y| n−1 dµ(y) − c n µ ( 3 4 r(B)) n−1 ≥ c n µ ( 1 2 r(B)) n−1 − c n µ ( 3 4 r(B)) n−1 = c n 2 n−1 − ( 4 3 ) n−1 Cap( 1 4 B ∩ ∂Ω) r(B) n−1 ,
which proves the lemma.
We recall also the following lemma, whose prove can be found in [AHM 3 TV].
Lemma 2.2. Let n ≥ 2 and Ω ⊂ R n+1 be open. Let B be a closed ball centered at ∂Ω. Then, for all a > 0,
(2.2) ω x (aB) inf z∈2B∩Ω ω z (aB) r(B) n−1 g(x, y) for all x ∈ Ω\2B and y ∈ B ∩ Ω,
with the implicit constant independent of a.
Combining the two preceding lemmas, choosing a = 8, we obtain:
Lemma 2.3. Let n ≥ 2, s > n − 1, and Ω ⊂ R n+1 be open. Let B be a closed ball centered at ∂Ω. Then,
(2.3) ω x (8B) n Cap(2B ∩ ∂Ω) r(B) n−1 g(x, y)
for all x ∈ Ω\2B and y ∈ B ∩ Ω.
THE KEY IDENTITY AND THE MAIN IDEA
3.1. The key identity.
Lemma 3.1 (Key identity). Let Ω ⊂ R n+1 be open, let ψ ∈ C ∞ c (Ω)
, and let u : Ω → R be harmonic and positive in supp ψ. Then, for any α > 0,
|∇ 2 u| 2 u α ψ dx = 1 2 α(α − 1) |∇u| 4 u α−2 ψ dx (3.1) − 1 2 ∇(|∇u| 2 ) · ∇ψ u α dx + 1 2 |∇u| 2 ∇(u α ) · ∇ψ dx.
In the lemma we denoted
|∇ 2 u| 2 = i,j (∂ i,j u) 2 .
The identity (3.1), in the particular case α = 1, was already used in connection with harmonic measure in [LV] and [HLMN].
Proof. Notice that |∇ 2 u| 2 = i |∇∂ i u| 2 .
So (3.1) follows by summing from i = 1 to n + 1 the following identity:
|∇∂ i u| 2 u α ψ dx = 1 2 α(α − 1) |∂ i u| 2 |∇u| 2 u α−2 ψ dx (3.2) − 1 2 ∇(|∂ i u| 2 ) · ∇ψ u α dx + 1 2 |∂ i u| 2 ∇(u α ) · ∇ψ dx.
To prove this, we integrate by parts:
|∇∂ i u| 2 u α ψ dx = ∇∂ i u · ∇∂ i u u α ψ dx = ∇∂ i u · ∇ ∂ i u u α ψ dx − ∇∂ i u · ∇ u α ψ ∂ i u dx.
The first integral on the right hand side vanishes because u is harmonic:
∇∂ i u · ∇ ∂ i u u α ψ dx = − ∆(∂ i u) ∂ i u u α ψ dx = 0. Using also ∂ i u ∇∂ i u = 1 2 ∇(|∂ i u| 2 ), we get |∇∂ i u| 2 u α ψ dx = − 1 2 ∇(|∂ i u| 2 ) · ∇ u α ψ dx (3.3) = − 1 2 ∇(|∂ i u| 2 ) · ∇ u α ψ dx − 1 2 ∇(|∂ i u| 2 ) · ∇ψ u α dx = − 1 2 ∇(|∂ i u| 2 ψ) · ∇ u α dx + 1 2 |∂ i u| 2 ∇ψ · ∇ u α dx − 1 2 ∇(|∂ i u| 2 ) · ∇ψ u α dx.
Finally, integrating by parts and taking into account that ∆ u α = α(α − 1)|∇u| 2 u α−2 , we deduce that the first term on the right hand side satisfies
− 1 2 ∇(|∂ i u| 2 ψ) · ∇ u α dx = 1 2 |∂ i u| 2 ψ ∆ u α dx = 1 2 α(α − 1) |∂ i u| 2 ψ |∇u| 2 u α−2 dx.
Plugging this into (3.3), we get (3.2).
3.2. The strategy of the proof. Let s > n be as in Theorem 1.1. By Bourgain's theorem [Bo], it is clear that we can assume s ∈ (n, n + 1). Let a ∈ (0, 1) be such that s = n + a, and let
α = 1 − a 1 + a ,
so that α ∈ (0, 1) too. We will apply the identity (3.1) with u equal to the Green function g(·, p) and a suitable function ψ. The choice of the preceding value of α is motivated by the fact that then the integrals that appear in (3.1) scale like
ω(·) ω(·) ℓ s α+1 ,
under the assumption that that u = g(·, p) scales like ω(·)ℓ 1−n .
A key fact in our arguments is that the first term on the right hand side of (3.1) is negative (because α(α − 1) < 0), while the left hand side is positive. These two terms should be considered as the main ones in (3.1), and the last two integrals should be considered as "boundary terms" because of the presence of ∇ψ in their integrands.
Writing
g(x) = g(x, p), from (3.1) we get |α(α − 1)| |∇g| 4 g α−2 ψ dx (3.4) ≤ ∇(|∇g| 2 ) · ∇ψ g α dx + |∇g| 2 ∇(g α ) · ∇ψ dx − 2 |∇ 2 g| 2 g α ψ dx.
Using this inequality and assuming the existence of a set E ⊂ ∂Ω with ω(E) > 0 such that the harmonic measure and the Hausdorff measure H s are absolutely continuous on E, we will get a contradiction. To this end, we will construct an appropriate function ψ by some stopping time arguments involving the set E, and with this choice we will show that the integral on left hand side of (3.4) is much larger than the right hand side.
4. THE BALL B 0 , THE STOPPING CONSTRUCTION, AND THE FUNCTION ψ 4.1. The ball B 0 . From now we assume that we are under the assumptions of Theorem 1.1. We consider a point p ∈ Ω and we denote by ω the harmonic measure for Ω with respect to the pole p. We also denote µ = H s | E and we assume that 0 < µ(E) < ∞ and that µ is absolutely continuous with respect to ω. Our objective is to find a contradiction. By replacing E by a suitable subset if necessary, by standard methods (taking into account the upper bound for the upper density of µ in (2.1)) we may assume that there exists some δ 0 > 0 such that µ(B(x, r)) ≤ 3 s r s for all x ∈ E and 0 < r ≤ δ 0 .
Since µ ≪ ω, there exists some non-negative function h ∈ L 1 (ω) such that µ = h ω. We consider a point x 0 ∈ E satisfying the following: x 0 is a Lebesgue point for h with h(x 0 ) > 0 and a density point of E (both with respect to ω), and there exists a sequence of radii r k → 0 such that
(4.1) ω(B(x 0 , 200r k )) ≤ (200) n+2 ω(B(x 0 , r k )).
For this last property, see, for example, Lemma 2.8 in [To]. Now, given some κ 0 ∈ (0, 1/10), let δ 1 ∈ (0, δ 0 ] be such that
(4.2) 1 ω(B(x, r)) B(x,r) |h − h(x 0 )| dω ≤ κ 0 h(x 0 ) for all r ∈ (0, δ 1 ] and (4.3) ω(B(x, r) \ E) ≤ κ 0 ω(B(x, r)) for all r ∈ (0, δ 1 ].
The parameter κ 0 will be fixed below, and depends only on n. We take now a radius r ∈ 0, 1 300 min(r E , δ 1 , |p − x 0 |) such that (4.1) holds for r = r k (recall that r E is defined by local CDC in (1.1)), and we denote
B 0 = B(x 0 , 2 r). From (4.2) we deduce that, for all r ∈ (0, 100r(B 0 )], µ(B(x 0 , r)) = B(x 0 ,r) h dω ≤ h(x 0 ) ω(B(x 0 , r)) + B(x 0 ,r) |h − h(x 0 )| dω ≤ 2 h(x 0 ) ω(B(x 0 , r)). Analogously, µ(B(x 0 , r)) ≥ h(x 0 ) ω(B(x 0 , r)) − B(x 0 ,r) |h − h(x 0 )| dω ≥ 1 2 h(x 0 ) ω(B(x 0 , r)).
We collect some of the properties about B 0 in the next lemma.
Lemma 4.1. For all r ∈ (0, 100r(B 0 )], we have (4.4) 1 2 h(x 0 ) ω(B(x 0 , r)) ≤ µ(B(x 0 , r)) ≤ 2 h(x 0 ) ω(B(x 0 , r)).
We also have
(4.5) ω(100B 0 ) ≤ (200) n+1 ω( 1 2 B 0 ) and µ(100B 0 ) ≤ 4(200) n+1 µ( 1 2 B 0 ). Proof.
The estimates in (4.4) have been shown above, as well as the first inequality in (4.5). The second inequality follows from the preceding estimates:
µ(100B 0 ) ≤ 2 h(x 0 ) ω(100B 0 ) ≤ 2 h(x 0 ) (200) n+1 ω( 1 2 B 0 ) ≤ 4(200) n+1 µ( 1 2 B 0 ).
4.2.
The bad balls and the function d(·). We consider the constant
(4.6) A = 4 ω(5B 0 ) µ( 1 2 B 0 )
, Notice that, by Lemma 4.1, A ≈ h(x 0 ) −1 . For each x ∈ ∂Ω ∩ 2B 0 and r ∈ (0, r(B 0 )], we say that the ball B(x, r) is bad (and we write B(x, r) ∈ Bad) if ω(B(x, r)) > A µ(B(x, 10r)). Given some fixed parameter ρ 0 ∈ (0, 1 10 r(B 0 )], if there exists some r ∈ (ρ 0 , r(B 0 )] such that B(x, r) is bad, we denote
(4.7) r 0 (x) = sup r ∈ (ρ 0 , r(B 0 )] : B(x, r) is bad .
Otherwise, we set r 0 (x) = ρ 0 . Using the openness of the balls in the definition of r 0 (x), it is easy to check that the supremum in (4.7) is attained and thus the ball B(x, r 0 (x)) is bad if r 0 (x) > ρ 0 .
Next we define the following regularized version of r 0 (·):
d(x) = inf y∈2B 0 ∩∂Ω (r 0 (y) + |x − y|), for x ∈ R n+1 .
It is immediate to check that d(·) is a 1-Lipschitz function. Further, since r 0 (x) ≤ r(B 0 ) for any x ∈ 2B 0 ∩ ∂Ω, we infer that
d(x) ≤ r(B 0 ) for any x ∈ 2B 0 ∩ ∂Ω too.
We need the following auxiliary result. , 32r)).
Lemma 4.2. Let x ∈ 2B 0 ∩ ∂Ω. For all r ∈ [d(x), r(B 0 )], (4.8) ω(B(x, r)) ≤ A µ(B(x
Proof. Suppose first that r ≥ r(B 0 )/3. In this case, using just that B(x, r) ⊂ 3B 0 , B 0 ⊂ B(x, 3r(B 0 )), and the choice of A in (4.6), we infer that , 9r)).
ω(B(x, r)) ≤ ω(3B 0 ) ≤ A µ(B 0 ) ≤ A µ(B(x, 3r(B 0 ))) ≤ A µ(B(x
Assume now that r < r(B 0 )/3. Let y ∈ 2B 0 ∩ ∂Ω be such that
2d(x) ≥ r 0 (y) + |x − y|.
Using that B(x, r) ⊂ B(y, |x − y| + r) ⊂ B(y, 3r) (because |x − y| ≤ 2d(x) ≤ 2r) and that 3r ≥ 3d(x) ≥ 3 2 r 0 (y) and 3r ≤ r(B 0 ), we get ω(B(x, r)) ≤ ω(B(y, 3r)) ≤ A µ(B(y, 30r)). Now we take into account that B(y, 30r) ⊂ B(x, |x − y| + 30r) ⊂ B(x, 32r) (again because |x − y| ≤ 2r), and we derive ω(B(x, r)) ≤ A µ(B(y, 30r)) ≤ A µ (B(x, 32r)).
Now we apply Vitali's 5r-covering theorem to get a finite subfamily of balls (4.9)
{B i } i∈I ⊂ {B(x, 1 2000 d(x))} x∈2B 0 ∩∂Ω such that
• the balls B i , i ∈ I, are pairwise disjoint, and • x∈2B 0 ∩∂Ω B(x, 1 2000 d(x)) ⊂ i∈I 5B i . In the next lemma we show some elementary properties of the family {B i } i∈I . Lemma 4.3. Let {B i } i∈I be the family of balls defined above. The following holds:
(a) For each i ∈ I, r(B i ) ≤ 1 2000 r(B 0 ) and 1000B i ⊂ 3B 0 . (b) For all x ∈ 1000B i , with i ∈ I, 1000 r(B i ) ≤ d(x) ≤ 3000 r(B i ). (c) If 1000B i ∩ 1000B j = ∅, for i, j ∈ I, then 1 3 r(B i ) ≤ r(B j ) ≤ 3r(B i ). (d) The balls 1000B i , i ∈ I, have finite superposition. That is, i∈I χ 1000B i ≤ C 1 , for some absolute constant C 1 . Proof. Denote by x i the center of B i , so that B i = B(x i , 1 2000 d(x i )).
The statement in (a) is due to the fact that, for each i ∈ I, we have
r(B i ) = 1 2000 d(x i ) ≤ 1 2000 r 0 (x i ) ≤ 1 2000 r(B 0 ), with x i ∈ 2B 0 .
On the other hand, notice that, for all x ∈ 1000B i ,
|d(x) − d(x i )| ≤ |x − x i | ≤ 1000 2000 d(x i ), and thus 1 2 d(x i ) ≤ d(x) ≤ 3 2 d(x i ),
which gives (b). Concerning (c), given 1000B i and 1000B j with non-empty intersection, we consider x ∈ 1000B i ∩ 1000B j and we deduce that
1 2 d(x i ) ≤ d(x) ≤ 3 2 d(x j ).
Together with the converse estimate, this shows that
r(B i ) ≤ 3r(B j ) ≤ 9r(B i ).
To prove (d), let B i 1 ,. . . , B im be such that m j=1 1000B i j = ∅.
Suppose that B i 1 has maximal radius among the balls B i 1 ,. . . , B im , so that m j=1 1000B i j ⊂ 3000B i 1 .
Since the balls B i 1 ,. . . , B im are pairwise disjoint, by the properties (c), (b) and the usual volume considerations we deduce that
m 3 n+1 r(B i 1 ) n+1 ≤ m j=1 r(B i j ) n+1 ≤ (3000 r(B i 1 )) n+1 ,
and thus m 1.
4.3.
The function ψ. Let ϕ be a radial C ∞ function such that χ B(0,1.1) ≤ ϕ ≤ χ B(0,1.2) , and let (4.10)
ϕ i (x) = ϕ x − x i 5r i ,
where x i is the center of B i and r i its radius. Notice that ϕ ≡ 1 on 5.5B i and vanishes out of 6B i . Next we need to define some auxiliary functions θ j . First, by applying the 5r-covering theorem, we consider a covering of 3B 0 \ i∈I 1.1B i with balls of the form B(z j , 10 −5 d(z j )), with z j ∈ 3B 0 \ i∈I 1.1B i , so that the balls 1 5 B(z j , 10 −5 d(z j )) are disjoint. This implies that the dilated balls 1.2B(z j , 10 −5 d(z j )) have finite superposition, by arguments analogous to the ones in Lemma 4.3. For each j ∈ J, we define
θ j (x) = ϕ x − z j 10 −5 d(z j )
.
In this way, using the property (b) in the preceding lemma, for any j ∈ J,
(4.11) supp θ j ∩ i∈I 5B i = ∅.
We consider the functions
ϕ i = ϕ i j∈I ϕ j + j∈j θ j , i ∈ I.
Notice that the denominator above is bounded away from 0 in supp ϕ i , and thus ϕ i ∈ C ∞ , with
∇ ϕ i ∞ r −1 i . Further, by construction 0 ≤ i∈I ϕ i ≤ 1 in R n+1 .
Also, taking into account (4.11),
i∈I ϕ i ≡ 1 in i∈I 5B i and, since supp ϕ i ⊂ 6B i , i∈I ϕ i ≡ 0 in R n+1 \ i∈I 6B i .
We also denote
ψ 0 = 1 − i∈I ϕ i ϕ x − x 0 r(B 0 )
(recall that x 0 is the center of B 0 ). Finally, we let ψ = ψ 4 0 . Lemma 4.4. The following holds:
(a) supp ψ 0 ⊂ 2B 0 \ i∈I 5B i and ψ 0 ≡ 1 in B 0 \ i∈I 6B i . (b) supp ∇ψ 0 ⊂ i∈I A(x i , 5r i , 6r i ) ∪ A(x 0 , r(B 0 ), 2r(B 0 )). (c) |∇ψ 0 (x)| 1 r(B i ) for all x ∈ 6B i . (d) |∇ψ 0 (x)| 1 r(B 0 ) for all x ∈ 2B 0 \ i∈I 6B i .
The same properties hold for ψ.
The proof of the lemma follows easily from the construction above, and we leave it for the reader.
4.4. The sets V , V , and F . By Vitali's 5r-covering theorem, there exists a subfamily Bad V ⊂ Bad such that
• the balls from Bad V are pairwise disjoint, and • any ball from Bad is contained in some ball 5B, with B ∈ Bad V .
We denote
V = B∈Bad V 5B, V = B∈Bad V
10B.
Notice that V ⊂ V and that all the bad balls are contained in V (not only the ones with radius larger than ρ 0 ). In the next lemma we show that V is rather small, because of our choice of A above.
Lemma 4.5. We have
µ V ≤ B∈Bad V µ(10B) ≤ 1 4 µ( 1 2 B 0 ).
Proof. By the definition of bad balls and the disjointness of the family Bad V , we get
µ( V ) ≤ B∈Bad V µ(10B) ≤ 1 A B∈Bad V ω(B) ≤ 1 A ω(5B 0 ),
where, in the last inequality, we took into account that the bad balls are centered at 2B 0 and have radius at most r(B 0 ). By the choice of A in (4.6), we are done.
Next we need to consider another kind of bad set. We let F be the subset of the points x ∈ E ∩ 1 2 B 0 for which there exists some r ∈ (0, 1 4 r(B 0 )] such that ω(B(x, r)) ≤ κ 0 h(x 0 ) −1 µ(B(x, r)) (recall that κ 0 ∈ (0, 1/10) will be fixed below).
Lemma 4.6. We have µ(F ) ≤ Cκ 0 µ( 1 2 B 0 ). Proof. By the Besicovitch covering theorem, there exists a covering of F by a family of balls B(z i , s i ),
with z i ∈ F , 0 < s i ≤ r(B 0 )/4, such that ω(B(z i , s i )) ≤ κ 0 h(x 0 ) −1 µ(B(z i , r i ))
, and having finite superposition. That is, i χ B(z i ,s i ) 1. Then we have:
ω(F ) ≤ i ω(B(z i , s i )) ≤ κ 0 h(x 0 ) −1 i µ(B(z i , s i )) ≤ C κ 0 h(x 0 ) −1 µ(B 0 ) ≤ C κ 0 h(x 0 ) −1 µ( 1 2 B 0 )
, taking into account that all the balls B(z i , s i ) are contained in B 0 and the finite superpostion of the balls in the before to last inequality, and the fact that 1 2 B 0 is doubling with respect to µ, by (4.5), in the last inequality. As a consequence,
µ(F ) = F h dω ≤ h(x 0 ) ω(F ) + 1 2 B 0 |h − h(x 0 )| dω ≤ Cκ 0 µ( 1 2 B 0 ) + κ 0 h(x 0 ) ω( 1 2 B 0 ) ≤ C κ 0 µ( 1 2 B 0 ).
PROOF OF THEOREM 1.1
Let s > n be as in Theorem 1.1. Recall that we assume s ∈ (n, n + 1) and we denote a = s − n. Also, we take
α = 1 − a 1 + a ,
so that both a, α ∈ (0, 1). We will apply the identity (3.1) with u equal to the Green function g(·, p), the function ψ constructed in Section 4, and the preceding value of α. Recall that ψ supported in 2B 0 and vanishes in a neighborghood of ∂Ω ∩ 2B 0 . Thus, g = g(·, p) is harmonic in supp ψ. Recall also that we have |α(α − 1)| |∇g| 4 g α−2 ψ dx (5.1) ≤ ∇(|∇g| 2 ) · ∇ψ g α dx + |∇g| 2 ∇(g α ) · ∇ψ dx − 2 |∇ 2 g| 2 g α ψ dx.
To achieve the desired contradiction to prove Theorem 1.1 we will show that the integral on the left hand side tends to ∞ as ρ 0 → 0, while the right hand side is much smaller than the left hand side.
We denote I 0 = |∇g| 4 g α−2 ψ dx,
I 1 = ∇(|∇g| 2 ) · ∇ψ g α dx,
I 2 = |∇g| 2 ∇(g α ) · ∇ψ dx, I 3 = |∇ 2 g| 2 g α ψ dx.
5.1.
Estimate of I 1 . Using the fact that |∇(|∇g| 2 )| |∇ 2 g| |∇g| and Hölder's inequality and recalling that ψ = ψ 4 0 , we get
|I 1 | |∇ 2 g| |∇g| g α ψ 3 0 |∇ψ 0 | dx ≤ |∇ 2 g| 2 g α ψ 4 0 dx 1/2 |∇g| 2 g α ψ 2 0 |∇ψ 0 | 2 dx 1/2 .
Observe that the first integral on the right hand side coincides with I 3 . To deal with the last one, we split it as follows:
(5.2) |∇g| 2 g α ψ 2 0 |∇ψ 0 | 2 dx ≤ i∈I 6B i . . . + 2B 0 \ i∈I 6B i . . . .
By Lemma 4.4 (c) and
Caccioppoli's inequality, for each i ∈ I we obtain
6B i |∇g| 2 g α |∇ψ 0 | 2 dx 1 r 2 i sup 6B i g(x) α 6B i |∇g| 2 dx (5.3) 1 r 4 i sup 6B i g(x) α 12B i |g| 2 dx r n−3 i sup 12B i g(x) α+2 . By (2.3), we have (5.4) sup 12B i g(x) ω(96B i ) r n−1 i .
Therefore, by the choice of α,
(5.5) 6B i |∇g| 2 g α |∇ψ 0 | 2 dx r n−3 i ω(96B i ) r n−1 i α+2 = ω(96B i ) ω(96B i ) r s i α+1 .
By Lemma 4.2,
(5.6) ω(96B i ) ≤ ω(B(x i , d(x i )) ≤ A µ(B(x i , 32d(x i )) Ad(x i ) s ≈ Ar s i . Thus, (5.7) 6B i |∇g| 2 g α |∇ψ 0 | 2 dx A α+1 ω(96B i ).
Finally, Lemma 4.3 (a) and (d),
(5.8) i∈I 6B i |∇g| 2 g α |∇ψ 0 | 2 dx A α+1 i∈I ω(96B i ) A α+1 ω(3B 0 ) A α+2 µ( 1 2 B 0 ).
Next we deal with the last integral on the right hand side of (5.2). We argue as in (5.2)-(5.8), but now we use the fact that |∇ψ 0 | 1/r(B 0 ) in 2B 0 \ i∈I 6B i and we replace 6B i by 2B 0 . Then, as in (5.5), we get (5.9)
2B 0 \ i∈I 6B i |∇g| 2 g α |∇ψ 0 | 2 dx ω(32B 0 ) ω(32B 0 ) r(B 0 ) s α+1 .
Using now (4.5) and (4.6), we derive (5.10)
2B 0 \ i∈I 6B i |∇g| 2 g α |∇ψ 0 | 2 dx A α+2 µ( 1 2 B 0 ).
Altogether, we obtain
(5.11) |I 1 | ≤ CA α+2 µ(B 0 ) 1/2 I 1/2 3 ≤ CA α+2 µ( 1 2 B 0 ) + I 3 .
5.2. Estimate of I 2 . Using that ∇(g α ) = α g α−1 ∇g, ∇ψ = 4ψ 3 0 ∇ψ 0 , and Hölder's inequality, we get
|I 2 | |∇g| 3 g α−1 ψ 3 0 |∇ψ 0 | dx ≤ |∇g| 4 g α−2 ψ 4 0 dx 3/4 g α+2 |∇ψ 0 | 4 dx 1/4 .
Observe that the first integral on the left hand side equals I 0 . To estimate the second one we split it:
(5.12) g α+2 |∇ψ 0 | 4 dx ≤ i∈I 6B i . . . + 2B 0 \ i∈I 6B i . . . .
By Lemma 4.4 (c), for each i ∈ I, we obtain
6B i g α+2 |∇ψ 0 | 4 dx r n−3 i sup 6B i g(x) α+2 .
As in (5.4), we have
sup 6B i g(x) ≤ sup 12B i g(x) ω(96B i ) r n−1 i .
Then, operating exactly as in (5.5)-(5.8), we derive
i∈I 6B i g α+2 |∇ψ 0 | 4 dx A α+2 µ( 1 2 B 0 ).
To estimate the last integral on the right hand side of (5.12) we use the fact that |∇ψ 0 | 1/r(B 0 ) in 2B 0 \ i∈I 6B i and we apply (2.3). Then we get
2B 0 \ i∈I 6B i g α+2 |∇ψ 0 | 4 dx r(B 0 ) n−3 sup 2B 0 g(x) α+2 ω(32B 0 ) ω(32B 0 ) r(B 0 ) s α+1 ,
which is the same estimate as in (5.9). Then, as in (5.10), we deduce
2B 0 \ i∈I 6B i g α+2 |∇ψ 0 | 4 dx A α+2 µ( 1 2 B 0 ).
Therefore,
|I 2 | ≤ I 3/4 0 (A α+2 µ( 1 2 B 0 )) 1/4 ≤ |α(1 − α)| 2 I 0 + C(α)A α+2 µ( 1 2 B 0 ).
5.3.
Lower estimate of I 0 . From the identity (5.1) and the estimates for I 1 and I 2 we derive |α(α − 1)| I 0 = |α(α − 1)| |∇g| 4 g α−2 ψ dx
≤ I 1 + I 2 − 2I 3 ≤ CA α+2 µ( 1 2 B 0 ) + I 3 + |α(1 − α)| 2 I 0 + C(α)A α+2 µ( 1 2 B 0 ) − 2I 3 .
Hence,
(5.13) I 0 ≤ C(α)A α+2 µ( 1 2 B 0 )
. In this section, by estimating I 0 from below, we will contradict this inequality.
To get a lower estimate for I 0 we need to define some reasonably good set contained in 1 2 B 0 ∩∂Ω. To this end, we need first to introduce another type of balls. Let I b ⊂ I be the subfamily of indices i such that r(B i ) > 1 2000 ρ 0 . Recall that I is the set of indices in (4.9) and r(
B i ) = 1 2000 d(x i ) ≤ 1 2000 r 0 (x i ). So if i ∈ I b , then r 0 (x i ) > ρ 0 and thus B(x i , r 0 (x i ))
is a bad ball. We say that a ball B is useless (and we write B ∈ Uss) if it is centered at 1 2 B 0 ∩ ∂Ω \ V and (5.14) µ
i∈I b :6B i ∩B =∅ 960B i > ε 1 µ(B) and µ(B) ≥ 3 −s r( 1 2 B) s ,
where ε 1 ∈ (0, 1/10) is a small parameter to be fixed below that will depend only on n. Recall now that, by Lemma 4.5,
B∈Bad V µ(10B) ≤ 1 4 µ( 1 2 B 0 ).
Hence there exists some ρ 1 > 0 such that
(5.15) B∈Bad V :r(B)≤ρ 1 µ(10B) ≤ ε 2 1 µ( 1 2 B 0 ).
Notice that ρ 1 may depend here on the particular measure µ, not only on n. We define
U (ρ 1 ) = B∈Uss: r(B)≤ρ 1 B.
Lemma 5.1. We have µ(U (ρ 1 )) ε 1 µ( 1 2 B 0 ). Proof. Let B ∈ Uss with r(B) ≤ ρ 1 and let B i , i ∈ I b , be such that 6B i ∩ B = ∅. Notice that 2000B i is contained in some bad ball (because d(x i ) > ρ 0 ), which in turn is contained in some ball 5B ′ , with B ′ ∈ Bad V . Thus, 2000B i ⊂ 5B ′ . Now note that B is centered at ∂Ω \ V ⊂ (10B ′ ) c , and observe that the condition 6B i ∩ B = ∅ implies that B intersects 5B ′ . These two facts ensure that
(5.16) r(B) ≥ r(5B ′ ) ≥ r(2000B i ).
Then we deduce that
960B i ⊂ 2000B i ⊂ 3B.
The first inequality in (5.16) also implies that r(B ′ ) ≤ ρ 1 , which in turn gives that by (5.15). From the first condition in (5.14), we deduce that
960B i ⊂ B ′′ ∈Bad V :r(B ′′ )≤ρ 1 5B ′′ ⊂ B ′′ ∈Bad V :r(B ′′ )≤ρ 1 10B ′′ =: V 0 , with (5.17) µ( V 0 ) ≤ ε 2 1 µ( 1 2 B 0 ),µ(3B ∩ V 0 ) ≥ µ i∈I b :6B i ∩B =∅ 960B i > ε 1 µ(B).
Using also the fact that µ(15B) r(B) s µ(B) (by the second condition in (5.14)), we get
(5.18) µ(3B ∩ V 0 ) ≥ c ε 1 µ(15B).
Now we apply the 5r-covering theorem to get a subfamily I U from the balls in Uss with radius not exceeding ρ 1 such that
• the balls 3B with B ∈ I U are pairwise disjoint, and
• U (ρ 1 ) ⊂ B∈I U 15B.
From these properties and (5.18) and (5.17), we obtain
µ(U (ρ 1 )) ≤ B∈I U µ(15B) 1 ε 1 B∈I U µ(3B ∩ V 0 ) ≤ 1 ε 1 µ( V 0 ) ≤ ε 1 µ( 1 2 B 0 ).
Now we are ready to define the aforementioned reasonably good set contained in 1
2 B 0 ∩ ∂Ω. First denote G 0 = 1 2 B 0 ∩ ∂Ω \ F ∪ V ∪ U (ρ 1 )
, and recall that, by Lemmas 4.6, 4.5, and 5.1,
µ(G 0 ) ≥ µ( 1 2 B 0 ) − Cκ 0 µ( 1 2 B 0 ) − 1 4 µ( 1 2 B 0 ) − Cε 1 µ( 1 2 B 0 ).
We assume κ 0 to be an absolute constant small enough so that Cκ 0 ≤ 1/4 and also ε 1 small enough so that Cε 1 ≤ 1/4, and then we obtain
µ(G 0 ) ≥ 1 4 µ( 1 2 B 0 )
. Next we need to define some families of balls centered at G 0 inductively. Let G be the subset of those x ∈ G 0 such that Θ s, * (x, µ) ≥ 2 −s . Note that, by (2.1), µ(G 0 \ G) = 0. By definition, for each η k ∈ (0, r(B 0 )/10], for µ-a.e. x ∈ G there exists a ball B i x centered at x with radius r(B i x ) ≤ η k such that
(5.19) µ(B i x ) ≥ 3 −s r(B i x ) s .
Hence, by the 5r-covering theorem, we can extract a subfamily F k ⊂ {2B i x } x∈G such that (a) G ⊂ B∈ F k 80B, and (b) the balls 16B, B ∈ F k , are disjoint.
Further, we can still extract a finite subfamily F k ⊂ F k such that
(5.20) µ B∈F k 80B ≥ 1 2 µ(G) ≥ 1 8 µ( 1 2 B 0 ).
Now we fix inductively the parameters η k as follows: first we take η 1 = r(B 0 )/10, and next we set
η k = ε 0 min B∈F k−1 r(B),
where ε 0 ∈ (0, 1/100) is some small constant to be chosen below. Notice that this choice ensures that the balls from the family F k are much smaller than the ones of the preceding families F 1 , . . . , F k−1 .
Remark that the balls B(x, r) centered at G with radius r ∈ (0, r(B 0 )/4] (like the balls from the families F k ) satisfy
(5.21) ω(B(x, r)) ≥ κ 0 h(x 0 ) −1 µ(B(x, r)),
and the ones with radius r ∈ [ρ 0 , r(B 0 )],
(5.22) ω(B(x, r)) ≤ A µ(B(x, 10r)) ≤ CA r s ≈ h(x 0 ) −1 r s ,
by (4.8) and the choice of A.
Lemma 5.2. Let B ∈ F k \ Uss and suppose that ρ 0 ≤ 1 2 η k+1 . Denote
(5.23) B = B \ B ′ ∈F k+1 B ′ ∪ i∈I 6B i .
Then,
(5.24) B |∇g| 4 g α−2 ψ dx A α+2 µ(B).
Proof. Denote by x B the center of B and let
ϕ B (y) = ϕ y − x B 1 2 r(B) ,
where ϕ is the radial C ∞ function appearing in (4.10), so that supp ϕ B ⊂ B, ϕ ≡ 1 in 1 2 B, and ∇ϕ B ∞ 1/r(B). Then we have
1 r(B) B |∇g| dx ∇g · ∇ϕ B dx = ϕ B dω ≥ ω( 1 2 B) ≥ κ 0 h(x 0 ) −1 µ( 1 2 B),
taking into account (5.21) for the last inequality. Next we will show that 1 r(B) B\B |∇g| dx is small if the parameters ε 0 and ε 1 above are small. To this end, given any ball B ′ intersecting B and centered at x B ′ ∈ B 0 ∩ ∂Ω with radius r(B ′ ) ∈ [d(x B ′ ), r(B)], we write:
B∩B ′ |∇g| dx r(B ′ ) n sup 2B ′ g(x),
applying Hölder's inequality and Cacciopoli's inequality. By (2.3), the last supremum can be bounded above by ω(16B ′ ) r(B ′ ) 1−n , and then we get
(5.25) B∩B ′ |∇g| dx r(B ′ ) ω(16B ′ ).
We split
(5.26) B\B |∇g| dx ≤ B∩ B ′ ∈F k+1 B ′ |∇g| dx + B∩ i∈I\I b 6B i |∇g| dx + B∩ i∈I b 6B i |∇g| dx.
To deal with the first integral on the right hand side, recall that the balls 16B ′ , with B ′ ∈ F k+1 , are disjoint and their radius is at most ε 0 r(B), by construction. Thus,
1 r(B) B∩ B ′ ∈F k+1 B ′ |∇g| dx 1 r(B) B ′ ∈F k+1 :16B ′ ⊂2B r(B ′ ) ω(16B ′ ) ε 0 ω(2B).
The second integral on the right hand side of (5.26) is estimated analogously. In this case we use that all the balls B i with i ∈ I \ I b have radius equal to ρ 0 /2000 ≪ r(B), and that the balls 96B i , i ∈ I \ I b , have bounded overlap, by Lemma 4.3 (d). Then, by (5.25), we deduce
1 r(B) B∩ i∈I\I b 6B i |∇g| dx 1 r(B) i∈I\I b :96B i ⊂2B r(B i ) ω(96B i ) ε 0 ω(2B).
Regarding the last integral on the right hand side of (5.26), we will use the fact that
(5.27) µ i∈I b :6B i ∩B =∅ 960B i ≤ ε 1 µ(B),
because B is assumed to be not useless. Note that given i ∈ I b such that B ∩ 6B i = ∅, we have r(6B i ) ≤ r(B). Otherwise, B ⊂ 18B i , which violates the condition (5.27). Then, applying (5.25) again, we deduce
1 r(B) B∩ i∈I b 6B i |∇g| dx 1 r(B) i∈I b :6B i ∩B =∅ r(B i ) ω(96B i ) i∈I b :6B i ∩B =∅ ω(96B i ) A i∈I b :6B i ∩B =∅ µ(960B i ) A µ i∈I b :6B i ∩B =∅ 960B i Aε 1 µ(2B),
by the finite superposition of the balls 960B i and (5.27). We take into account now that, by (5.22), the s-growth of µ, and (5.19),
ω(2B) h(x 0 ) −1 r(2B) s ≈ h(x 0 ) −1 r( 1 2 B) s h(x 0 ) −1 µ( 1 2 B). Therefore, 1 r(B) B∩ B ′ ∈F k+1 B ′ |∇g| dx + 1 r(B) B∩ i∈I\I b 6B i |∇g| dx ε 0 h(x 0 ) −1 µ( 1 2 B).
Then, using also that µ( 1 2 B) r s µ(2B), h(x 0 ) −1 ≈ A, and recalling the splitting (5.26), we deduce 1
r(B) B\ B |∇g| dx ≥ (κ 0 − Cε 0 − Cε 1 ) h(x 0 ) −1 µ( 1 2 B) ≥ κ 0 2 h(x 0 ) −1 µ( 1 2 B) ≈ κ 0 A µ( 1 2 B) ≈ κ 0 A µ(B),
assuming ε 0 and ε 1 small enough.
Next, applying Hölder's inequality, we get To finish the proof of the lemma it just remains to notice that κ 0 is some absolute constant depending just on n and that ψ = 1 on B \ B.
Now we are ready to obtain the lower estimate for I 0 required to complete the proof of Theorem 1.1. Let N > 1 be an arbitrarily large integer, and choose ρ 0 ∈ (0, 1 2 η N +1 ] such that ρ 0 ≪ ρ 1 too. Let k 0 be the minimal integer such that 2η k 0 ≤ ρ 1 . If B ∈ F k with k ∈ [k 0 , N ], then r(B) ≤ ρ 1 by construction and so B ∈ Uss (since B is centered in U (ρ 1 ) c ). Then we can apply Lemma 5.2 to deduce that B |∇g| 4 g α−2 ψ dx A α+2 µ(B), with B defined in (5.23). Then we get
|∇g| 4 g α−2 ψ dx ≥ N k=k 0 B∈F k B |∇g| 4 g α−2 ψ dx N k=k 0 B∈F k A α+2 µ(B),
using the fact that the regions B above do not overlap. Now, by the doubling property of the balls from F k and (5.20), for each k we have
B∈F k µ(B) ≈ B∈F k µ(80B) ≥ 1 2 µ(G) ≥ 1 8 µ( 1 2 B 0 ).
Thus, I 0 = |∇g| 4 g α−2 ψ dx (N − k 0 )A α+2 µ( 1 2 B 0 ).
Taking N big enough, we contradict (5.13), as wished. E-mail address: [email protected]
.
To estimate the last integral on the right hand side we take into account that, by (
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"Fine \"mist\" vs large droplets in phase separated manganites",
"Fine \"mist\" vs large droplets in phase separated manganites"
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| [
"L Khomskii \nLaboratory of Solid State Physics\nSt Edmund's College\nCambridge University\nCB3 0BNCambridgeUK\n",
"D Khomskii \nGroningen University\nNijenborgh 49747 AGGroningenThe Netherlands\n"
]
| [
"Laboratory of Solid State Physics\nSt Edmund's College\nCambridge University\nCB3 0BNCambridgeUK",
"Groningen University\nNijenborgh 49747 AGGroningenThe Netherlands"
]
| []
| The properties of phase-separated systems, e.g. manganites, close to a I order phase transition between charge-ordered insulator and ferromagnetic metal, are usually described by the percolation picture. We argue that the correlated occupation of metallic sites leads to the preferential formation of larger metallic clusters, which explains the often observed inverse, or "overshot" hysteresis in manganites (when the resistivity with increasing temperature is larger than with decreasing T ). It also explains the recently discovered thermal cycling effect in manganites. Thus in treating this and similar systems in percolation picture, not only the total concentration of metallic phase, but also the distribution of metallic clusters by shape and size may significantly influence the properties of such systems. | null | [
"https://arxiv.org/pdf/cond-mat/0210616v1.pdf"
]
| 119,387,969 | cond-mat/0210616 | 6b7ba71e7b51ef0d037dc9f18d3117faee82c584 |
Fine "mist" vs large droplets in phase separated manganites
28 Oct 2002
L Khomskii
Laboratory of Solid State Physics
St Edmund's College
Cambridge University
CB3 0BNCambridgeUK
D Khomskii
Groningen University
Nijenborgh 49747 AGGroningenThe Netherlands
Fine "mist" vs large droplets in phase separated manganites
28 Oct 2002
The properties of phase-separated systems, e.g. manganites, close to a I order phase transition between charge-ordered insulator and ferromagnetic metal, are usually described by the percolation picture. We argue that the correlated occupation of metallic sites leads to the preferential formation of larger metallic clusters, which explains the often observed inverse, or "overshot" hysteresis in manganites (when the resistivity with increasing temperature is larger than with decreasing T ). It also explains the recently discovered thermal cycling effect in manganites. Thus in treating this and similar systems in percolation picture, not only the total concentration of metallic phase, but also the distribution of metallic clusters by shape and size may significantly influence the properties of such systems.
PACS numbers: 71. 10.w, 75.30.Kz
Phase separation seems to be the generic feature of doped strongly correlated systems such as manganites La 1−x M x MnO 3 (M = Ca, Sr). It is observed in many situations experimentally and is obtained in many theoretical models [1,2,3], both at low doping range (x < 1) and close to a half-doped case (x ∼ 0.5). Apparently one can speak of two different types of phase separation: microscopic phase separation, which is most often discussed by theoreticians and which is observed e.g. by the small-angle neutron scattering [4], and the large-scale, macroscopic phase separation. The later type is typically met close to a first-order phase transition and leads to a percolation-like behaviour of the system. Two unusual effects were observed recently in studying the behaviour of certain manganites close to a I order insulator-metal transition -in (PrCa)MnO 3 [5], Pr 0.5 Ca 0.5 (MnCr)O 3 [6] and in some others [7]. In these systems there apparently occurs with decreasing temperature a transition from a charge-ordered (CO) insulator to a ferromagnetic metallic (FM) phase, accompanied by a sharp drop of resistivity. This drop has a large hysteresis. But with increasing temperature from the FM phase an inverse, or "overshot" hysteresis was observed in [5,6,7], schematically shown in fig. 1. The nature of this behaviour was not clarified; there were even suggestions [5] that there exist two different CO phases, one appearing with decreasing temperature, and another -when the temperature increases.
Another unusual phenomenon was found in one of these systems, Pr 0.5 Ca 0.5 (MnCr)O 3 [6]: when after the first decrease of the temperature it was increased and then the cycle was repeated, the resistivity, having the behaviour like the one shown in fig. 1, in each following cycle became higher and higher (if temperature was not increased beyond the shaded region of fig. 1). In some cases after several cycles the resistivity became insulating down to the lowest temperatures. Simultaneous magnetic measurements did not show any significant decrease of the total magnetization, i.e. the total fraction of the FM phase did not decrease strongly with such "training."
We want to suggest here a simple explanation of these effects, introducing the notion generalizing the standard percolation picture. This idea was first put forth in 1999 [8]; closely related ideas were discussed recently in [9].
The main idea is the following. When there occurs a first order phase transition to a metallic phase with decreasing temperature, there appear FM droplets in a CO insulating matrix. With further decrease of temperature they grow and start to coalesce until a percolation limit is reached, after which the system behaves as a metallic one. During this process the metallic droplets are first formed close to some nucleation centres, so that there appear T ρ Figure 1: Schematic behaviour of resistivity in certain manganites close to CO-FM transition [5][6][7]. Shaded is the region of "overshot" hysteresis. many very small droplets -like a fog on a cold evening. With further decrease of temperature these droplets grow, and, as is well known, bigger droplets grow faster, gradually "consuming" smaller ones. This is caused by the larger vapour pressure above droplets with smaller curvature [10], or, in other words, by the tendency to reduce total surface energy. Finally the FM phase occupies (almost) the whole sample, which occurs at low temperatures.
With the following increase of the temperature the FM phase gradually starts to "evaporate", but this process is accompanied by hysteresis -resistivity is initially lower than at the first decrease of temperature. However after a percolation metallic cluster is broken, there may occur an inverse situation -inverse, or "overshot" hysteresis, at which the resistivity is higher than at the first downward run. This is a natural consequence of different distribution of FM clusters by size and shape during decreasing and increasing temperature: whereas in going from the CO insulator phase we create a lot of small FM clusters (fine "fog"), in the opposite process, when we increase temperature starting from the FM phase occupying large part of the sample (big FM "pools") , these big droplets would survive even at high temperature. Thus at a given temperature (in the "shaded" region of fig. 1) the total volume of a FM phase may still be the same, but the distribution, the typical sizes of these FM clusters would be different: a lot of small droplets with decreasing T , and much smaller number of bigger droplets when we increase temperature from the FM phase. This would naturally lead to an increase of resistivity in a reverse run -an overshot hysteresis: not only would FM droplets be bigger, but also insulating regions between them would increase, which would lead to larger resistivity.
One can illustrate this conclusion on a simple picture shown in fig. 2, in which we substituted random distribution of FM and CO phases by a regular stripe-like structure. A fine "fog" (large number of small FM and CO regions), realized with the decreasing T , is modelled by the situation of fig. 2a, and a situation which should be realized with the temperature increase is illustrated in fig. 2b. One immediately sees that the resistivity in the first case is given by the expression
ρ 1a = ρ 0 n e V /kT(1)
where n is the number of insulating barriers (white stripes in fig. 2a), and V is the value of each of these barriers (which for simplicity we take equal).
On the other hand, in the case of fig. 2b, instead of having n small barriers, we have smaller number -in a limiting case only one barrier, but with the width n times bigger. As a result we would get the resistivity
ρ 1b = ρ 0 e nV /kT(2)
-much larger than that given by Eq. (1). Of course this model is strongly oversimplified, and in reality the difference between resistivities would be much smaller due to random distribution of different regions by size, shape and position; but the physics of "overshot hysteresis" may be explained by the picture described above. Thus when considering the percolation conductivity, we have to take into account not only the relative volume, occupied by one or another phase, but also the distribution of these phases by the size. This is usually not done in a standard treatment of percolation; but, as we argued above, this may be a very important factor [11].
We checked this picture by a computer simulation. We modelled the percolation in phase-separated manganites by first randomly putting the "metallic atoms" (black points) on a 200 × 200 square lattice. The resulting distribution of metallic clusters for certain concentration n 0 , smaller than the percolation threshold n c ∼ 0.59 [12] (here for n 0 = 0.125) is shown in fig. 3a. This distribution is on the average the same for increasing and for decreasing n.
To model the physical situation described above (the preferable formation of large clusters) the algorithm was modified in such a way that the probability of adding new metallic atom at a certain cite is larger when there are already occupied sites adjacent to it (i.e. the probability to occupy the site is the larger, the more neighbouring sites are occupied). The resulting structures are shown in fig. 3b, 3c. Fig. 3b shows the distribution of occupied sites at n 0 = 0.125 reached by increasing occupation n from zero with correlated occupation as explained above. Fig. 3c shows the distribution at the same concentration n 0 as in fig. 3b, but reached by first increasing n from the situation of fig. 3b to n ∼ 0.75 (above percolation threshold) and then decreasing n down to 0.125; in reducing n we used the same algorithm as when increasing it, i.e. the probability to remove an atom from a given site is larger when there are fewer occupied sites nearby.
As we see by comparing fig. 3a with 3b and 3c, the resulting distribution of metallic sites at the same total concentration (here 0.125) depends on whether we have random or correlated percolation: the clusters are bigger for correlated occupation. But more interestingly, the resulting distribution also depends on history: in accordance with our general expectations, for correlated occupation we indeed obtain many small clusters with increasing n (or decreasing temperature), fig. 3b, and smaller number of bigger clusters, with bigger insulating barriers between them, with decreasing n (increasing temperature), fig. 3c.
We can also add yet another ingredient in our computer modelling, imitating annealing: after several steps of adding particles, we allowed for their redistribution, removing and adding particles in the same correlated fashion, but keeping their total number fixed. This leads to some "rounding off" of the clusters, whose boundaries become smoother, and this annealing somewhat enhances the tendency described above: that for correlated occupation we obtain, on the average, larger droplets, see fig. 3d, and they become even larger at the reverse process of decreasing n.
(a) (b) (d) (c)
One also sees in these simulations that the percolation limit n c itself does not depend on whether we increase or decrease n, even for correlated occupation; but the value of n c seems to decrease somewhat, from ∼ 0.593 for the usual percolation [12] to about 0.576 for correlated occupation.
Thus our computer modelling confirms our general physical arguments and shows that in a more realistic picture of phase separation, which takes into account correlation in occupation of particles and which leads to the preferential formation of bigger clusters, the resulting picture depends on the history: we have many small clusters (fine "fog") with increasing the fraction of metallic sites (decreasing temperature), and smaller number of bigger clusters with increasing T . As argued above, on the insulating side of the transition (below percolation threshold) this would lead to an increase of resistivity (inverse, or "overshot" hysteresis) which may explain the experimental observations of [5,6,7]. Thus, the correlated occupation of sites makes the system "more insulating" on the insulating side of the percolation transition and "more metallic" on the metallic side (the sharpening of the transition on the metallic side was also seen in the calculations of Ref. [9]). The most interesting feature of this picture is the dependence of it on the thermal history, shown above.
The picture suggested above may also explain the "training" effect observed in [6]. Indeed, one may expect that after the first cycle not all small droplets disappear. But with further cycling the larger and larger droplets will be formed, eventually with larger insulating barriers between them, which can lead to the behaviour observed in [6]. The requirement is that the upper temperature during cycling should fall within the "shaded" region of fig. 1a (and should not exceed it) so that the large droplets, which would serve as condensation centres during the next cycle, should not disappear.
Our computer modelling also confirms these qualitative considerations. In fig. 4a-d we show typical results of the distribution of FM (black) regions obtained after cycling. The procedure was first to increase n from 0 to certain n 0 < n c (here again n 0 = 0.125), and then cycling n several times between n 0 and n ∼ 0.75. All the time we used the same algorithm as before (probability of adding and removing particles depending on the number of occupied neighbours). We see that, indeed, with increasing number of cycles the number of FM droplets for the same n 0 decreases, and their size and distance between them increase, which, according to the arguments leading to Eqs. (1)-(2), gives an increase of resistivity with "training", even for the same concentration of the FM phase. This can explain the experimental results of Mahendiran et al. [6] (similar behaviour is also seen in the data of [5]).
Summarizing, we proposed that the properties of inhomogeneous systems like some manganites close to an insulator-metal transition may be explained if we add to the conventional percolation picture another ingredient -that not only the net concentration of metallic phase, but also the distribution of these metallic inclusions by size and shape may be different, which may strongly influence the properties. We argue that the metallic droplets formed with decreasing temperature take the form of a fine "fog" -a lot of small droplets formed at many different nucleation centres, whereas with increasing temperature, going from the metallic state, predominantly large metallic droplets survive. The picture we propose seems to be quite natural and agrees with what we know from other fields of physics and even from our everyday experience; it is also confirmed by our computer modelling. It can explain the "inverse hysteresis" and the change of properties during thermal cycling, observed in some manganites in the inhomogeneous phase close to an insulator-metal transition.
The general conclusion is that when treating the properties of inhomogeneous systems in percolation picture, not only the total fraction of one or another phase, but also distribution of these phases by size and shape may be crucial. It would be very interesting to verify the proposed picture experimentally, e.g. by small angle neutron scattering or by light scattering in manganites during thermal cycling.
We are grateful to E. Dagotto
Figure 2 :
2Schematic picture illustrating the dependence of resistivity on the distribution of metallic regions (shaded) by size.
Figure 3 :
3Distribution of metallic clusters (black) for the filling n 0 = 0.125 reached at random occupation (3a) and for correlated occupation of metallic sites (3b-3c), see text.Fig. 3bcorresponds to a state reached by increasing n from 0 to n 0 , andfig. 3c-by first increasing n still further to ∼ 0.75 and then reducing it back to n 0 .Fig. 3dshows typical distribution of FM clusters at the same n 0 reached with annealing.
Figure 4 :
4The effect of thermal cycling. The occupied metallic clusters (black) for n 0 = 0.125 after first increasing n from 0 to n 0 (a), further increasing n to 0.75 and then decreasing it back to n 0 (b) and after 2 (c) and 5 (d) cycles.(Figs. 4a and 4b correspond to figs. 3b and 3c.)
and M. Mostovoy for useful discussions and to Th. Lorenz and R. Mahendiran for informing us of their experimental results. This work was supported by the Netherlands Foundation for the Fundamental Study of Matter (FOM).
. E Dagotto, Phys. Reports. 3441E. Dagotto et al., Phys. Reports 344, 1 (2001)
. E L Nagaev, Phys. Usp. 39781E. L. Nagaev, Phys. Usp. 39, 781 (1996)
. D I Khomskii, Physica B. 280325D. I. Khomskii, Physica B 280, 325 (2000)
. M Hennion, Phys. Rev. Lett. 811957M. Hennion et al., Phys. Rev. Lett. 81, 1957 (1998)
report at the OXSEN meeting, Groningen. Th, J.-P Lorenz, Renard, unpublishedTh. Lorenz, J.-P. Renard et al., report at the OXSEN meeting, Gronin- gen, 1999; Th. Lorenz et al., unpublished
. R Mahendiran, J. Appl. Phys. 902422R. Mahendiran et al., J. Appl. Phys. 90, 2422 (2001);
. Phys. Rev. B. 6464424Phys. Rev. B 64, 064424 (2001)
. N A Babushkina, Phys. Rev. B. 596081Phys. Rev. BN. A. Babushkina et al., Phys. Rev. B 59, 6994 (1999), Phys. Rev. B 62, R6081 (2000);
. Y Tomioka, J. Phys. Soc. Jap. 70302Appl. Phys. Lett.Y. Tomioka et al., Appl. Phys. Lett. 70, 3609 (1997), J. Phys. Soc. Jap. 66, 302 (1997);
. V Podzorov, Phys. Rev. B. 64140406V. Podzorov et al., Phys. Rev. B 64, 140406(R) (2001);
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D Khomskii, OXSEN meeting. GroningenD. Khomskii, OXSEN meeting, Groningen, 1999
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. L D Landau, E M Lifshitz, Statistical Physics. L. D. Landau and E. M. Lifshitz, Statistical Physics, Nauka, Moscow 1964
The physics invoked above is actually well known from our everyday experience: in the right weather conditions, when the temperature drops, e.g. in the evening, an overcooled water vapour can often form fog; but after big water droplets are finally formed, they survive when the temperature increases (e.g. in the morning), and the fog rarely forms. we rather have dew-drops insteadThe physics invoked above is actually well known from our everyday ex- perience: in the right weather conditions, when the temperature drops, e.g. in the evening, an overcooled water vapour can often form fog; but after big water droplets are finally formed, they survive when the tem- perature increases (e.g. in the morning), and the fog rarely forms, we rather have dew-drops instead.
Introduction to Percolation Theory. D Stauffer, A Aharoni, Taylor & FrancisD. Stauffer and A. Aharoni, Introduction to Percolation Theory, Taylor & Francis, 1994
| []
|
[
"A General Glivenko-Gödel Theorem for Nuclei",
"A General Glivenko-Gödel Theorem for Nuclei"
]
| [
"Giulio Fellin [email protected] ",
"Peter Schuster [email protected] ",
"\nUniversità di Verona\nStrada le Grazie 1537134VeronaItaly\n",
"\nUniversità di Trento\nItaly University of Helsinki\nFinland\n",
"\nUniversità di Verona\nStrada le Grazie 1537134VeronaItaly\n"
]
| [
"Università di Verona\nStrada le Grazie 1537134VeronaItaly",
"Università di Trento\nItaly University of Helsinki\nFinland",
"Università di Verona\nStrada le Grazie 1537134VeronaItaly"
]
| [
"MFPS 2021 EPTCS 351"
]
| Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an inductively generated abstract consequence relation, and from propositional logic to any set of objects whatsoever. The resulting conservation theorem comes with precise criteria for its validity, which allow us to instantly include Gödel's counterpart for first-order predicate logic of Glivenko's theorem. The open nucleus gives us a form of the deduction theorem for positive logic, and the closed nucleus prompts a variant of the reduction from intuitionistic to minimal logic going back to Johansson. | 10.4204/eptcs.351.4 | [
"https://arxiv.org/pdf/2112.14049v1.pdf"
]
| 237,413,382 | 2112.14049 | dc121637d4aff45dce96217c01c3d743cd1afba3 |
A General Glivenko-Gödel Theorem for Nuclei
2021
Giulio Fellin [email protected]
Peter Schuster [email protected]
Università di Verona
Strada le Grazie 1537134VeronaItaly
Università di Trento
Italy University of Helsinki
Finland
Università di Verona
Strada le Grazie 1537134VeronaItaly
A General Glivenko-Gödel Theorem for Nuclei
MFPS 2021 EPTCS 351
202110.4204/EPTCS.351.4
Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an inductively generated abstract consequence relation, and from propositional logic to any set of objects whatsoever. The resulting conservation theorem comes with precise criteria for its validity, which allow us to instantly include Gödel's counterpart for first-order predicate logic of Glivenko's theorem. The open nucleus gives us a form of the deduction theorem for positive logic, and the closed nucleus prompts a variant of the reduction from intuitionistic to minimal logic going back to Johansson.
Introduction
Double negation over intuitionistic logic is a typical instance of a nucleus [4,37,43,50,65,74,75,8]. Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of the double negation of that formula [35]. This stood right at the beginning of the success story of negative translations, which have been put into the context of nuclei [8] or monads [27]. As compared to recent literature on Glivenko's theorem [36,28,52,51,57,46,56,42,31,34], 1 the purpose of the present paper is to generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an abstract consequence relation, and from propositional logic to any set of objects whatsoever.
To this end we move to a nucleus j over a Hertz-Tarski consequence relation in the form of a (singleconclusion) entailment relation ⊲à la Scott [72,13]. Assuming that ⊲ is inductively generated by axioms and rules, we propose two natural extensions (Section 3): ⊲ j generalises the provability of double negation, and ⊲ j is inductively defined by adding the generalisation of double negation elimination to the inductive definition of ⊲. By their very definitions, ⊲ j satisfies all axioms and rules of ⊲, and ⊲ j satisfies all axioms of ⊲. But when does ⊲ j also satisfy all rules of ⊲? Our main result, Theorem 3.8, says that ⊲ j extends ⊲ j , and that the two relations coincide precisely when ⊲ j is closed under the nonaxiom rules that are used to inductively generate ⊲, which of course is the case whenever there are no such non-axiom rules (Corollary 3.9).
In logic this gives us a multi-purpose conservation criterion (Theorem 5.2), by which propositional and predicate logic can be handled in parallel. The prime instance of course is Glivenko's theorem (Application 5.3(i)) as a syntactical conservation theorem (see also [31,32]):
Γ ⊢ c ϕ ⇐⇒ Γ ⊢ i ¬¬ϕ
where ⊢ c and ⊢ i denote classical and intuitionistic propositional logic. Simultaneously we re-obtain Gödel's theorem (Application 5.3(ii)) which states that
Γ ⊢ Q c ϕ ⇐⇒ Γ ⊲ Q * ¬¬ϕ
where ⊢ Q c denotes classical predicate logic, and ⊲ Q * is any extension (by additional axioms) of intuitionistic predicate logic that satisfies the double negation shift:
∀x¬¬ϕ ⊲ ¬¬∀xϕ
While the double negation nucleus jϕ ≡ ¬¬ϕ is an instance of the continuation monad, it is tantamount to the same case jϕ ≡ ¬ϕ → ϕ of the Peirce monad [27]. What does our main result mean for other nuclei in logic? The Dragalin-Friedman nucleus jϕ ≡ ϕ ∨ ⊥, a case of the closed nucleus, yields a variant of the reduction from intuitionistic to minimal logic going back to Johansson (Application 5.4). Last but not least, the open nucleus jϕ ≡ A → ϕ prompts a form of the deduction theorem for positive logic (Application 5.5).
Preliminaries
We intend to proceed in a constructive and predicative way, keeping the concepts elementary and the proofs direct. If a formal system is desired, our work can be placed in a suitable fragment of Aczel's Constructive Zermelo-Fraenkel Set Theory (CZF) [1,2,3,5,6] based on intuitionistic first-order predicate logic.
By a finite set we understand a set that can be written as {a 1 , . . . , a n } for some n ≥ 0. Given any set S, let Pow(S) (respectively, Fin(S)) consist of the (finite) subsets of S. We refer to [62] for further provisos to carry over to the present note. 2
Entailment relations
Entailment relations are at the heart of this note. We briefly recall the basic notions, closely following [61,62].
Let S be a set and ⊲ ⊆ Pow(S) × S. Once abstracted from the context of logical formulae, all but one of Tarski's axioms of consequence [76] 3 can be put as
U ∋ a U ⊲ a ∀b ∈ U (V ⊲ b) U ⊲ a V ⊲ a U ⊲ a ∃U 0 ∈ Fin(U )(U 0 ⊲ a)
where U,V ⊆ S and a ∈ S. These axioms also characterise a finitary covering or Stone covering in formal topology [66]; 4 see further [16,15,49,50,67,68]. The notion of consequence has presumably been described first by Hertz [38,39,40]; see also [9,45].
Tarski has rather characterised the set of consequences of a set of propositions, which corresponds to the algebraic closure operator U → U ⊲ on Pow(S) of a relation ⊲ as above where
U ⊲ ≡ {a ∈ S : U ⊲ a} .
Rather than with Tarski's notion, we henceforth work with its (tantamount) restriction to finite subsets, i.e. a (single-conclusion) entailment relation. 5 This is a relation ⊲ ⊆ Fin(S) × S such that
U ∋ a (R) U ⊲ a V ⊲ b V ′ , b ⊲ a (T) V,V ′ ⊲ a U ⊲ a (M) U,U ′ ⊲ a for all finite U,U ′ ,V,V ′ ⊆ S and a, b ∈ S, where as usual U,V ≡ U ∪ V and V, b ≡ V ∪ {b}.
Our focus thus is on finite subsets of S, for which we reserve the letters U,V,W, . . .; we sometimes write a 1 , . . . , a n in place of {a 1 , . . . , a n } even if n = 0. Remark 2.1. The rule (R) is equivalent, by (M), to the axiom a ⊲ a.
Redefining
T ⊲ ≡ {a ∈ S : ∃U ∈ Fin(T )(U ⊲ a)}
for arbitrary subsets T of S gives back an algebraic closure operator on Pow(S). By writing T ⊲ a in place of a ∈ T ⊲ , the entailment relations thus correspond exactly to the relations satisfying Tarski's axioms above.
Given an entailment relation ⊲, by setting a ≤ b ≡ a ⊲ b we get a preorder on S; whence the conjunction a ≈ b of a ≤ b and b ≤ a is an equivalence relation.
Quite often an entailment relation is inductively generated from axioms by closing up with respect to the three rules above [64]. Some leeway is required in the present paper by allowing for generating rules other than (R), (M), and (T). If, however, these three rules are the only rules employed for inductively generating an entailment relation, we stress this by saying that this is generated only by axioms. Given an inductively generated entailment relation ⊲ and a set of axioms and rules P, then we call ⊲ plus P the entailment relation inductively generated by all axioms and rules that either are used for generating ⊲ or belong to P.
A main feature of inductive generation is that if ⊲ is an entailment relation generated inductively by certain axioms and rules, then ⊲ ⊆ ⊲ ′ for every entailment relation ⊲ ′ satisfying those axioms and rules. By an extension ⊲ ′ of an entailment relation ⊲ we mean in general an entailment relation ⊲ ′ such that ⊲ ⊆ ⊲ ′ . We say that an extension ⊲ ′ of ⊲ is conservative if also ⊲ ⊇ ⊲ ′ and thus ⊲ = ⊲ ′ altogether [61,62,31,32].
Nuclei over entailment relations
Throughout this section, fix a set S endowed with an entailment relation ⊲. We say that a function j : S → S is a nucleus (over ⊲) if for all a, b ∈ S and U ∈ Fin(S) the following hold:
U, a ⊲ jb L j U, ja ⊲ jb U ⊲ b R j U ⊲ jb
Unlike L j, by (R) and (T) the rule R j can be expressed by an axiom, viz.
b ⊲ jb (1) 5 In the present paper there is no need for abstract multi-conclusion consequence or entailmentà la Scott [71,72,73], Lorenzen's contributions to which are currently under scrutiny [22,23]. The relevance of multi-conclusion entailment to constructive algebra, point-free topology, etc. has been pointed out in [13], and has widely been used, e.g. in [17,19,20,18,21,24,25,53,60,79,63,69,70,80,61,62,53,47].
Remark 3.1. The above notion of a nucleus includes as a special case the notion of a nucleus on a locale [4,43,50,65,74,75], which is well-known as a point-free way to put subspaces. In fact, if S is a locale with partial order ≤, then U ⊲ a ⇐⇒ U ≤ a defines an entailment relation [25] such that any given map j : S → S is a nucleus on ⊲ precisely when j is a nucleus on the locale S. The latter means that j satisfies
ja ∧ jb ≤ j(a ∧ b)(2)
on top of the conditions for j being a closure operator on S, which can be put as a ≤ ja and
a ≤ jb =⇒ ja ≤ jb .(3)
In the presence of a ≤ ja, which is nothing but (1), the conjunction of (2) and (3) is equivalent to
c ∧ a ≤ jb =⇒ c ∧ ja ≤ jb ,
which in turn subsumes L j. So the two notions of a nucleus coincide.
Example 3.2.
1. Every entailment relation ⊲ has the trivial nucleus j ≡ id.
2. Consider an algebraic structure S with a unary self-inverse function j (e.g. take a group as S and the inverse as j). The entailment relation ⊲ of S-substructures is inductively defined by a 1 , ..., a n ⊲ f (a 1 , ..., a n )
for every n-ary function f in the language of S, including j. We want to show that j is a nucleus on ⊲. Axiom (1) is just (4) for f ≡ j, therefore rule R j holds. In particular, j 2 = id implies j(a) ⊲ a, which, together with (T), gives rule L j. In conclusion, j is a nucleus on ⊲.
3. Double negation ¬¬ is a nucleus over intuitionistic logic ⊢ i as an entailment relation (see Subsection 5.1 for further details and Subsections 5.2-5.3 for more nuclei in logic).
Entailment relations induced by a nucleus, and conservation
Consider a nucleus j over an entailment relation ⊲. We define
-the weak j-extension (or Kleisli extension) of ⊲ as the relation ⊲ j ⊆ Fin(S) × S defined by U ⊲ j a ⇐⇒ U ⊲ ja
-the strong j-extension as the entailment relation ⊲ j ⊆ Fin(S) × S inductively generated by the axioms and rules of ⊲ plus the stability axiom for j:
ja ⊲ j a(5)
In the terminology coined before, ⊲ j is nothing but ⊲ plus the stability axiom for j. Remark 3.3. By (R) in the form of a ⊲ a (Remark 2.1), stability holds for ⊲ j too, that is, ja ⊲ j a.
Under appropriate circumstances Remark 3.3 will help to obtain ⊲ j ⊆ ⊲ j ; see Theorem 3.8 and Corollary 3.9.
Lemma 3.4. Let S be a set with an entailment relation ⊲ and let j be a nucleus on ⊲.
1. ⊲ j is an entailment relation that extends ⊲.
⊲ j is an entailment relation that extends ⊲.
Proof. (i) holds by the very definition of ⊲ j . As for (ii): By (1) and Remark 2.1, rule (R) is bestowed from ⊲ to ⊲ j . Rule (M) is inherited from ⊲, and so is rule (T) in view of L j:
U ⊲ ja V, a ⊲ jb L j V, ja ⊲ jb (T) U,V ⊲ jb
Finally, also ⊲ ⊆ ⊲ j is a consequence of (1).
Remark 3.5. The nucleus j on ⊲ is a nucleus also on ⊲ j and ⊲ j . In fact, by Lemma 3.4 both extensions inherit axiom (1) from ⊲, and actually satisfy the following strengthening of L j:
U, a ⊲ b L j + U, ja ⊲ b .
While L j + for ⊲ j is just L j for ⊲, stability ja ⊲ a is tantamount to L j + for any entailment relation ⊲ whatsoever.
To understand better whether and when ⊲ j coincides with ⊲ j , we first consider a concrete example.
Example 3.6. Consider deduction in minimal logic ⊢ m with the nucleus jϕ ≡ ϕ ∨ ⊥ (see Subsection 5.2 below for details). Propositional minimal logic ⊢ m is inductively generated by certain axioms plus the rule Γ, ϕ ⊢ m ψ R→ Γ ⊢ m ϕ → ψ which cannot be expressed as an axiom. By its very definition, ⊢ j m too satisfies R→. Does also ⊢ m j satisfy this rule? If this were the case, then by definition of ⊢ m j we would have
Γ, ϕ ⊢ m ψ ∨ ⊥ Γ ⊢ m (ϕ → ψ) ∨ ⊥ As ⊥ ⊢ m ψ ∨ ⊥, we would obtain ⊢ m (⊥ → ψ) ∨ ⊥.
However, since minimal logic has the disjunction property and neither disjunct is provable in general, this cannot be the case. So ⊲ j does not satisfy rule R→.
The moral of Example 3.6 is that ⊲ may already have non-axiom rules, such as R→, which carry over to ⊲ j by its very definition, and thus need to hold in ⊲ j too for the former to be conservative over the latter. To deal with this issue, we say that a rule r that holds for ⊲ is compatible with j if r also holds for ⊲ j . 2. Every composition r of compatible rules is compatible. In fact, the derivation that gives r in ⊲ can be translated smoothly into ⊲ j , as all applied rules are compatible. This is very useful: if we want to check compatibility for all rules of an entailment relation ⊲, it suffices to check compatibility for any set of rules that generate ⊲.
3. Every axiom a 1 , ..., a n ⊲ b can be viewed as a rule with no premiss, and as such is compatible with every nucleus j, simply by R j. Moreover, rules
U, b ⊲ c U, a 1 , ..., a n ⊲ c U ⊲ a 1 ... U ⊲ a n U ⊲ b
which are known respectively as left and right rule [64,30] 6 are provably equivalent to the axiom a 1 , ..., a n ⊲ b and therefore are compatible with j.
4. If an entailment relation ⊲ is generated only by axioms, then every rule that holds for ⊲ is compatible with any nucleus j over ⊲.
Theorem 3.8 (Conservation for nuclei). Let S be a set with an entailment relation ⊲ inductively generated by axioms and rules, and let j be a nucleus on ⊲.
Then ⊲ j extends ⊲ j , that is ⊲ j ⊆ ⊲ j . Moreover, the following are equivalent:
(a) ⊲ j is conservative over ⊲ j , that is, ⊲ j ⊆ ⊲ j ;
(b) All non-axiom rules that generate ⊲ are compatible with j.
Proof. First recall that, by its very definition, ⊲ j is inductively generated by rules (R), (M), (T), stability (5), and all rules that generate ⊲. In particular, ⊲ ⊆ ⊲ j . Now suppose that U ⊲ j b, i.e. U ⊲ jb. Since ⊲ ⊆ ⊲ j , also U ⊲ j jb. Then apply
U ⊲ j jb jb ⊲ j b (T) U ⊲ j b to show ⊲ j ⊆ ⊲ j .
(a)⇒(b) (b) follows directly from (a) and the fact that ⊲ j satisfies all rules that generate ⊲. -⊲ j satisfies stability (5) by Remark 3.3.
-⊲ j satisfies all rules that generate ⊲ since they are either compatible with j by hypothesis or axioms and thus compatible with j by Remark 3.7.
As ⊲ j is the smallest extension of ⊲ satisfying these axioms and rules, we get ⊲ j ⊆ ⊲ j .
Corollary 3.9. Let S be a set with an entailment relation ⊲ inductively generated only by axioms, and let j be a nucleus on ⊲. Then ⊲ j = ⊲ j , that is, ⊲ j is a conservative extension of ⊲ j .
Let j be a nucleus over an entailment relation ⊲ inductively generated by axioms and rules, and let ⊲ * be an extension of ⊲. We say that ⊲ * is an intermediate j-extension of ⊲ if ⊲ * is ⊲ plus * where * is a collection of axioms that are valid in ⊲ j . In particular, ⊲ ⊆ ⊲ * ⊆ ⊲ j . Remark 3.10. Since ⊲ ⊆ ⊲ * , we have ⊲ j ⊆ ⊲ j * . On the other hand, as all axioms in * already hold for ⊲ j , we also have ⊲ j * ⊆ ⊲ j . Therefore ⊲ j * = ⊲ j .
Corollary 3.11 (Conservation for intermediate j-extensions). Let S be a set with an entailment relation ⊲ inductively generated by axioms and rules, let j be a nucleus on ⊲, and let ⊲ * be an intermediate j-extension of ⊲.
Then ⊲ j extends ⊲ * j , that is ⊲ * j ⊆ ⊲ j . Moreover, the following are equivalent:
(a) ⊲ j is conservative over ⊲ * j , that is, ⊲ j ⊆ ⊲ * j ;
(b) All non-axiom rules that generate ⊲ hold for ⊲ * j .
Proof. Follows from Theorem 3.8 for ⊲ * by noticing that ⊲ j * = ⊲ j (Remark 3.10) and that all additional rules of ⊲ * are axioms and thus already compatible with j (Remark 3.7).
The following characterisation will prove useful in several applications: Lemma 3.12. Let S be a set with an entailment relation ⊲, and let j be a nucleus on ⊲. Let r be a rule holding for ⊲. The following are equivalent:
(a) Rule r is compatible with j.
(b) For every instance U 1 ⊲ b 1 ... U n ⊲ b n U ⊲ b
of rule r, there is β ∈ S such that β ⊲ jb and
U 1 ⊲ jb 1 ... U n ⊲ jb n U ⊲ β (6)
Proof. (a)⇒(b) If we take β ≡ jb, then (b) immediately follows by reflexivity and compatibility. (b)⇒(a) Recall that b ⊲ jb, and that from U ⊲ β and β ⊲ jb follows U ⊲ jb by (T).
Logic as entailment
Throughout this section, the overall assumption is that S is a set of propositional or (first-order) predicate formulae containing ⊤, ⊥, and closed under the connectives ∨, ∧, →, ¬ for propositional logic and also under the quantifiers ∀, ∃ for predicate logic. Following [58,10], by (propositional) positive logic ⊢ p we mean the positive fragment of propositional intuitionistic logic. More precisely, we define ⊢ p as the least entailment relation ⊲ that satisfies the deduction theorem Γ, ϕ ⊲ ψ R→ Γ ⊲ ϕ → ψ and the following axioms: Table 1: Sequent calculus-like rules for positive propositional logic [58] following [10].
ϕ, ψ ⊲ ϕ ∧ ψ ϕ ∧ ψ ⊲ ϕ ϕ ∧ ψ ⊲ ψ ϕ ⊲ ϕ ∨ ψ ψ ⊲ ϕ ∨ ψ ϕ ∨ ψ, ϕ → δ , ψ → δ ⊲ δ ϕ, ϕ → ψ ⊲ ψ Γ, ϕ, ψ ⊲ δ L∧ Γ, ϕ ∧ ψ ⊲ δ Γ ⊲ ϕ Γ ⊲ ψ R∧ Γ ⊲ ϕ ∧ ψ Γ, ϕ ⊲ δ Γ, ψ ⊲ δ L∨ Γ, ϕ ∨ ψ ⊲ δ Γ ⊲ ϕ R∨ 1 Γ ⊲ ϕ ∨ ψ Γ ⊲ ψ R∨ 2 Γ ⊲ ϕ ∨ ψ Γ ⊲ ϕ Γ, ψ ⊲ δ L→ Γ, ϕ → ψ ⊲ δ Γ, ϕ ⊲ ψ R→ Γ ⊲ ϕ → ψ R⊤ Γ ⊲ ⊤
Of course, we understand this as an inductive definition. The above system for positive logic [58] is equivalent to the G3-style calculus in Table 1 taken from [10]; they inductively generate the same entailment relation.
On top of ⊢ p we consider the following additional axioms:
ϕ → ⊥ ≈ ¬ϕ (PC) ⊥ ⊲ ϕ (EFQ) ¬¬ϕ ⊲ ϕ (RAA)
They are known as principium contradictionis, ex falso quodlibet sequitur and reductio ad absurdum.
The two directions of PC can also be expressed via the rules
Γ ⊲ ϕ Γ, ⊥ ⊲ ψ L¬ Γ, ¬ϕ ⊲ ψ Γ, ϕ ⊲ ⊥ R¬ Γ ⊲ ¬ϕ
In the presence of EFQ, the rule L¬ can be simplified as
Γ ⊲ ϕ L¬ Γ, ¬ϕ ⊲ ψ
Axiom EFQ is sometimes considered as a rule without premises:
L⊥ Γ, ⊥ ⊲ ϕ
We define:
minimal logic ⊢ m as ⊢ p plus PC, intuitionistic logic ⊢ i as ⊢ m plus EFQ, classical logic ⊢ c as ⊢ i plus RAA.
Let ⊢ * be ⊢ p plus additional axioms. In particular, ⊢ * satisfies the deduction theorem R→. The (first-order) predicate version ⊢ Q * of ⊢ * , which we also refer to as ⊢ * plus quantifiers, is then obtained by adding quantifiers ∀ and ∃ to the language and the following rules to the inductive definition of ⊢ * :
ϕ[t/x], Γ, ∀xϕ ⊲ δ L∀ Γ, ∀xϕ ⊲ δ Γ ⊲ ϕ[y/x] R∀ Γ ⊲ ∀xϕ Γ, ϕ[y/x] ⊲ δ L∃ Γ, ∃xϕ ⊲ δ Γ ⊲ ϕ[t/x] R∃ Γ ⊲ ∃xϕ
with the condition that y has to be fresh in L∃ and R∀. Rules L∀ and R∃ can be expressed as axioms:
∀xϕ ⊲ ϕ[t/x] ϕ[t/x] ⊲ ∃xϕ
The definition of a nucleus j given in [8] requires j to be compatible with substitution, that is,
j(ϕ[t/x]) ≡ ( jϕ)[t/x]
We prefer not to have this as a general assumption, but to make explicit whenever we need it.
Conservation for nuclei in logic
Among the usual logical rules, R→, R∀ and L∃ are the only ones that cannot be expressed as axioms. Rule L∃ is compatible with j for every nucleus j as it does not affect the right-hand side of the sequent. Therefore, when checking compatibility of rules with j, if we do not add other rules that cannot be expressed as axioms, then the only rules we have to check are R→ and R∀.
Lemma 5.1. Let ⊢ * be ⊢ p plus additional axioms, and let j be a nucleus on ⊢ * . Consider ⊢ * as ⊲.
R→ is compatible with j if and only if
ϕ → jψ ⊢ * j(ϕ → ψ)
If j is compatible with substitution, then R∀ is compatible with j if and only if
∀x jϕ ⊢ Q * j∀xϕ
Proof. We prove (i), the proof of (ii) is analogous. As for "if", by Lemma 3.12, R→ is compatible with j if and only if for every instance
Γ, ϕ ⊢ * ψ Γ ⊢ * ϕ → ψ of R→ there is β ∈ S such that β ⊢ * j(ϕ → ψ) and Γ, ϕ ⊢ * jψ Γ ⊢ * β
By R→, the latter condition is satisfied if we set β ≡ ϕ → jψ, for which the former condition reads as ϕ → jψ ⊢ * j(ϕ → ψ).
As for "only if", compatibility directly entails the desired criterion. In fact, as an instance of modus ponens we have ϕ → jψ, ϕ ⊢ * jψ, which by the very definition of ⊢ j is nothing but ϕ → jψ, ϕ ⊢ * j ψ.
By compatibility, the deduction theorem carries over from ⊢ * to ⊢ * j . Hence we get
ϕ → jψ ⊢ * j ϕ → ψ,
which again by the definition of ⊢ j yields the desired criterion:
ϕ → jψ ⊢ * j(ϕ → ψ).
This gives us the following version of Corollary 3.11:
Theorem 5.2 (Conservation for nuclei in logic). Let ⊢ be ⊢ p plus additional axioms, let j be a nucleus on ⊢, and let ⊢ * be ⊢ plus additional axioms such that ⊢ * ⊆ ⊢ j .
1. The following are equivalent in propositional logic:
(a) Γ ⊢ j ϕ ⇐⇒ Γ ⊢ * jϕ for all Γ, ϕ (b) ⊢ * satisfies the following axiom:
ϕ → jψ ⊢ * j(ϕ → ψ)
2. Let ⊢ Q , ⊢ Q * , ⊢ Q j be ⊢, ⊢ * , ⊢ j plus quantifiers. If j is compatible with substitution, then the following are equivalent in predicate logic:
(a) Γ ⊢ Q j ϕ ⇐⇒ Γ ⊢ Q * jϕ for all Γ, ϕ (b) ⊢ Q
* satisfies the following axioms:
ϕ → jψ ⊢ Q * j(ϕ → ψ) ∀x jϕ ⊢ Q * j∀xϕ
The Glivenko nucleus
Take intuitionistic logic ⊢ i as ⊲, and define
jϕ ≡ ¬¬ϕ .
This j is well-known to be a nucleus over ⊢ i [8,65], which we call the Glivenko nucleus. As stability (5) equals RAA, the strong extension ⊢ j i of intuitionistic logic ⊢ i is nothing but classical logic ⊢ c . Since ϕ → ¬¬ψ ⊢ i ¬¬(ϕ → ψ) follows, e.g., from [26,Lemma 6.2.2], and the Glivenko nucleus is compatible with substitution, we get the following instance of Theorem 5.2 where ⊢ is ⊢ i : [28]. Now let jϕ ≡ ¬ϕ → ϕ. This j is a nucleus [8,65], which we call the Peirce nucleus, as it is a special case of the Peirce monad [27]. Over intuitionistic logic, it is easy to show that the Glivenko nucleus is equivalent to the Peirce nucleus, i.e., ¬¬ϕ ≈ i ¬ϕ → ϕ for every ϕ.
Application 5.3. 1. (Glivenko's Theorem) Γ ⊢ c ϕ ⇐⇒ Γ ⊢ i ¬¬ϕ
The Dragalin-Friedman nucleus
Take minimal logic ⊢ m as ⊲, and define jϕ ≡ ϕ ∨ ⊥ .
This j is a nucleus, in fact a closed nucleus [8,65]. We refer to this j as the Dragalin-Friedman nucleus. As stability (5) is equivalent to EFQ, the strong extension ⊢ j m of minimal logic ⊢ m is nothing but intuitionistic logic ⊢ i .
Since the Dragalin-Friedman nucleus is compatible with substitution, we get the following instance of Theorem 5.2 where ⊢ is ⊢ m : Application 5.4. Let ⊢ * be ⊢ m plus additional axioms such that ⊢ * ⊆ ⊢ i .
1. The following are equivalent in propositional logic:
(a) Γ ⊢ i ϕ ⇐⇒ Γ ⊢ * ϕ ∨ ⊥ for all Γ, ϕ; (b) ϕ → (ψ ∨ ⊥) ⊢ * (ϕ → ψ) ∨ ⊥(a) Γ ⊢ Q i ϕ ⇐⇒ Γ ⊢ Q * ϕ ∨ ⊥ for all Γ, ϕ; (b) ϕ → (ψ ∨ ⊥) ⊢ Q * (ϕ → ψ) ∨ ⊥ and ∀x(ϕ ∨ ⊥) ⊢ Q * (∀xϕ) ∨ ⊥ for all ϕ, ψ.
The deduction nucleus
Let ⊢ be ⊢ p or ⊢ Q p plus additional axioms. We fix a propositional formula A and set
jϕ ≡ A → ϕ .
This j, which we call the deduction nucleus, is an instance of the open nucleus [8,65]. As for this j stability (5) is equivalent to ⊢ A, the strong extension ⊢ j is the smallest extension of ⊢ in which A is derivable. The deduction nucleus is compatible with substitution, and the following axioms are easy to show (see, e.g., [26, Lemma 6.2.1] for the case of intuitionistic logic):
ϕ → (A → ψ) ⊢ A → (ϕ → ψ) ∀x(A → ϕ) ⊢ A → ∀xϕ
Hence we get the following instance of Theorem 5.2 where ⊢ = ⊢ * is ⊢ p or ⊢ Q p plus additional axioms: As Γ ⊢ j ϕ also means that ϕ is derivable from Γ ∪ {A}, Application 5.5 is a variant of the deduction theorem:
Γ, A ⊢ ϕ ⇐⇒ Γ ⊢ A → ϕ 6 Related and future work
In propositional lax logic (PLL) [29] the modality is characterised by axioms and rules corresponding [29, p. 2, (2)] to the ones of a (logical) nucleus. Also the rules L j and R j of the present paper are counterparts of the rules L and R of PLL [29, p. 5]. We expect to gain insight by relating our approach to PLL, its semantics and applications. To start with, in the vein of [29, Lemma 2.1] rule R j is tantamount to the inverse of L j.
It is known that in certain cases, given a nucleus j, it is possible to define a function J : S → S, known as Kuroda-style j-translation, such that U ⊲ j b implies JU ⊲ j Jb, which can be viewed as conservation of ⊲ j over ⊲ j modulo j-translation. Some particular instances of this are discussed in [8], Proposition 4. Is there a general result for arbitrary entailment relations?
It will be a challenge to include also other proof translation methods. For instance, Friedman's Atranslation [33] makes use of the closed nucleus to prove Markov's rule; and Ishihara and Nemoto [41] use the same translation but work with the open nucleus to prove the independence-of-premiss rule.
We will further study nuclei about other forms of negation: weak negation over positive logic [10], co-negation over dual logics [7] and strong negation over extensions of intuitionistic logic [44,78].
Remark 3.7. 1. Rules (R), (M), (T) are compatible with every nucleus j, by Lemma 3.4.
(b)⇒(a) Let us consider one by one the axioms and rules that generate ⊲ j : -⊲ j satisfies (R), (M), (T), since ⊲ j is an entailment relation by Lemma 3.4.
for all Γ, ϕ in propositional logic. 2. (Gödel's Theorem) Let ⊢ * be ⊢ i plus additional axioms such that ⊢ * ⊆ ⊢ c , and let ⊢ Q i , ⊢ Q * and ⊢ Q c be ⊢ i , ⊢ * and ⊢ c plus quantifiers. The following are equivalent in predicate logic: (a) Γ ⊢ Q c ϕ ⇐⇒ Γ ⊢ Q * ¬¬ϕ for all Γ, ϕ; (b) ∀x¬¬ϕ ⊢ Q * ¬¬∀xϕ for all ϕ. Condition (b) in Application 5.3 is called Double Negation Shift (DNS) and is known to define a proper intermediate logic ⊢ Q DNS , that is, ⊢ Qi
⊢ Q
DNS ⊢ Q
c
for all ϕ, ψ. 2. Let ⊢ Q m , ⊢ Q * and ⊢ Q i be ⊢ m , ⊢ * and ⊢ i plus quantifiers. The following are equivalent in predicate logic:
Application 5.5. Let ⊢ be ⊢ p or ⊢ Q p plus additional axioms. ThenΓ ⊢ j ϕ ⇐⇒ Γ ⊢ A → ϕ that is, A → ϕ is derivable from Γ ifand only if ϕ is derivable from Γ when assuming that A is derivable.
This list of references is by no means meant exhaustive.
For example, we deviate from the terminology prevalent in constructive mathematics and set theory[5,6,11,12,47,48]: to reserve the term 'finite' to sets which are in bijection with {1, . . . , n} for a necessarily unique n ≥ 0. Those exactly are the sets which are finite in our sense and are discrete too, i.e. have decidable equality[48].3 Tarski has further required that S be countable.4 This is from where we have taken the symbol ⊲, used also[77,14] to denote a 'consecution'[59].
A reader familiar with structural proof theory may be reminded of the notion of left and right rules in sequent calculus[54,55]. Though they look similar, the two concepts are not to be confused.
AcknowledgementsThe present study was carried out within the projects "A New Dawn of Intuitionism: Mathematical and Philosophical Advances" (ID 60842) funded by the John Templeton Foundation, and "Reducing complexity in algebra, logic, combinatorics -REDCOM" belonging to the programme "Ricerca Scientifica di Eccellenza 2018" of the Fondazione Cariverona. Both authors are members of the "Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni" (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM). Last but not least, the authors are grateful to Daniel Wessel for his ideas, interest and suggestions, and to Tarmo Uustalo for pointing out propositional lax logic.
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[
"Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis",
"Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis"
]
| [
"Wei Bian ",
"Senior Member, IEEEDacheng Tao "
]
| []
| []
| Fisher's linear discriminant analysis (FLDA) is an important dimension reduction method in statistical pattern recognition. It has been shown that FLDA is asymptotically Bayes optimal under the homoscedastic Gaussian assumption. However, this classical result has the following two major limitations: 1) it holds only for a fixed dimensionality D, and thus does not apply when D and the training sample size N are proportionally large; 2) it does not provide a quantitative description on how the generalization ability of FLDA is affected by D and N . In this paper, we present an asymptotic generalization analysis of FLDA based on random matrix theory, in a setting where both D and N increase and D/N −→ γ ∈ [0, 1). The obtained lower bound of the generalization discrimination power overcomes both limitations of the classical result, i.e., it is applicable when D and N are proportionally large and provides a quantitative description of the generalization ability of FLDA in terms of the ratio γ = D/N and the population discrimination power. Besides, the discrimination power bound also leads to an upper bound on the generalization error of binary-classification with FLDA.Index TermsFisher's linear discriminant analysis, asymptotic generalization analysis, random matrix theory May 2, 2014 DRAFT 2 and the robustness of a decision system [3] [4] [5] [6]. During the past decades, FLDA has been applied to a wide range of areas, from speech/music classification [7] [8], face recognition [9] [10] to financial data analysis [11] [12]. An important property of FLDA is its asymptotic Bayes optimality under the homoscedastic Gaussian assumption [13] [14] [15] , which is a corollary of classical results from multivariate statistics [16]. Actually, as training sample size N goes to infinity, both the within-class scatter matrix Σ (sample covariance) and the between-class scatter matrix S converge to their population counterparts Σ and S. Therefore, the empirically optimal projection matrix W of FLDA, obtained by generalized eigendecomposition over Σ and S, also converges to its population counterpart W. Thanks to the asymptotic Bayes optimality, we can expect an acceptable performance of FLDA as long as N is sufficiently large. However, this classical result, i.e., the asymptotic Bayes optimality, suffers from two major limitations:1) It is obtained by fixing the dimensionality D and letting only N increase to infinity.But in practice, D and N can be proportionally large, which makes the classical result inapplicable.2) It does not provide quantitative description on the performance of FLDA, especially, how the generalization ability of FLDA is affected by D and N.A. The Contribution of this PaperTo address aforementioned limitations of the classical result, in this paper, we present an asymptotic generalization analysis of FLDA. Our analysis is superior from two aspects. First, we modify the setting of analysis by allowing both D and N to increase and assuming the dimensionality to training sample size ratio γ = D/N has a limit in [0, 1). This makes our result applicable in the case where D and N are proportionally large. Second, we quantitatively examine the generalization ability of FLDA. Denoting by ∆(Σ, S| W) the generalization discrimination power of FLDA, we intend to bound it from the lower side in terms of D and N, with respect to the population discrimination power ∆(Σ, S|W). Taking a binary-class problem, for example:suppose ∆(Σ, S|W) = λ and γ = D/N, then our asymptotic generalization bound shows that ∆(Σ, S| W) is almost surely larger than cos 2 (arccos( λ/(λ + γ)) + arccos( 1 − γ))λ, May 2, 2014 DRAFT | 10.1109/tpami.2014.2327983 | [
"https://arxiv.org/pdf/1208.3030v2.pdf"
]
| 14,822,342 | 1208.3030 | 1b9a35ad5d2707481cbf1a132acd14a23bb28367 |
Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis
22 Apr 2013
Wei Bian
Senior Member, IEEEDacheng Tao
Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis
22 Apr 2013arXiv:1208.3030v2 [stat.ML] 1
Fisher's linear discriminant analysis (FLDA) is an important dimension reduction method in statistical pattern recognition. It has been shown that FLDA is asymptotically Bayes optimal under the homoscedastic Gaussian assumption. However, this classical result has the following two major limitations: 1) it holds only for a fixed dimensionality D, and thus does not apply when D and the training sample size N are proportionally large; 2) it does not provide a quantitative description on how the generalization ability of FLDA is affected by D and N . In this paper, we present an asymptotic generalization analysis of FLDA based on random matrix theory, in a setting where both D and N increase and D/N −→ γ ∈ [0, 1). The obtained lower bound of the generalization discrimination power overcomes both limitations of the classical result, i.e., it is applicable when D and N are proportionally large and provides a quantitative description of the generalization ability of FLDA in terms of the ratio γ = D/N and the population discrimination power. Besides, the discrimination power bound also leads to an upper bound on the generalization error of binary-classification with FLDA.Index TermsFisher's linear discriminant analysis, asymptotic generalization analysis, random matrix theory May 2, 2014 DRAFT 2 and the robustness of a decision system [3] [4] [5] [6]. During the past decades, FLDA has been applied to a wide range of areas, from speech/music classification [7] [8], face recognition [9] [10] to financial data analysis [11] [12]. An important property of FLDA is its asymptotic Bayes optimality under the homoscedastic Gaussian assumption [13] [14] [15] , which is a corollary of classical results from multivariate statistics [16]. Actually, as training sample size N goes to infinity, both the within-class scatter matrix Σ (sample covariance) and the between-class scatter matrix S converge to their population counterparts Σ and S. Therefore, the empirically optimal projection matrix W of FLDA, obtained by generalized eigendecomposition over Σ and S, also converges to its population counterpart W. Thanks to the asymptotic Bayes optimality, we can expect an acceptable performance of FLDA as long as N is sufficiently large. However, this classical result, i.e., the asymptotic Bayes optimality, suffers from two major limitations:1) It is obtained by fixing the dimensionality D and letting only N increase to infinity.But in practice, D and N can be proportionally large, which makes the classical result inapplicable.2) It does not provide quantitative description on the performance of FLDA, especially, how the generalization ability of FLDA is affected by D and N.A. The Contribution of this PaperTo address aforementioned limitations of the classical result, in this paper, we present an asymptotic generalization analysis of FLDA. Our analysis is superior from two aspects. First, we modify the setting of analysis by allowing both D and N to increase and assuming the dimensionality to training sample size ratio γ = D/N has a limit in [0, 1). This makes our result applicable in the case where D and N are proportionally large. Second, we quantitatively examine the generalization ability of FLDA. Denoting by ∆(Σ, S| W) the generalization discrimination power of FLDA, we intend to bound it from the lower side in terms of D and N, with respect to the population discrimination power ∆(Σ, S|W). Taking a binary-class problem, for example:suppose ∆(Σ, S|W) = λ and γ = D/N, then our asymptotic generalization bound shows that ∆(Σ, S| W) is almost surely larger than cos 2 (arccos( λ/(λ + γ)) + arccos( 1 − γ))λ, May 2, 2014 DRAFT
I. INTRODUCTION
Fisher's linear discriminant analysis (FLDA) [1] [2] is one of the most representative dimension reduction techniques in statistical pattern recognition . By projecting examples into a low dimensional subspace with maximum discrimination power, FLDA helps improve the accuracy under mild conditions. Further, as a corollary of the discrimination power bound, we also obtain an asymptotic generalization error bound for binary classification with FLDA.
Based on the obtained asymptotic generalization bound, we can get better insight of FLDA.
It is commonly known that the performance of covariance estimation has a severe influence to the generalization ability of FLDA. By assuming a sufficient population discrimination power so as to eliminate the influence from between-class matrix estimation, we show that the mere influence from covariance estimation is proportional to the ratio γ = D/N < 1, i.e., due to the imperfection of covariance estimation, ∆(Σ, S| W) is about 1 − γ times of ∆(Σ, S|W). It is worth noticing that such result holds independent of the covariance Σ. Besides, the bound shows that the performance of FLDA is substantially determined by the ratio γ = D/N, given a fixed population discrimination power ∆(Σ, S|W). Therefore, N only needs to scale linearly with respect to D for an acceptable generalization ability of FLDA, although a quadratic number of parameters are to be estimated in the sample covariance.
B. Tools
The technical tools used in our asymptotic generalization analysis are from random matrix theory (RMT) [17] [18] [19] [20], the main goal of which is to provide understanding of the statistics of eigenvalues of matrices with entries drawn randomly from various probability distributions. RMT was originally motivated by applications in nuclear physics in 1950's, and then it was intensively studied in mathematics and statistics. It also found successful applications in engineering fields, e.g., wireless communications [21], recently. In this paper, we make use of two important results from RMT. The first one is the Marčenko-Pastur Law [19], which states that the empirical spectral distribution of a Wishart random matrix converges almost surely to a deterministic distribution F γ (λ) as lim γ = D/N ∈ [0, 1). The second one is the almost sure convergence of the extreme singular values of a large Gaussian random matrix [20]. We formulate these two results in following propositions.
Proposition 1: Given G ∈ R D×N , whose entries are independently sampled from standard Gaussian distribution N (0, 1), then as both D and N −→ ∞ and D/N −→ γ ∈ [0, 1), the empirical distribution of the eigenvalues of 1 N GG T , i.e.,
F N (λ) = 1 D D i=1 1 λ i 1 N GG T ≤ λ , λ ≥ 0,(1)
May 2, 2014 DRAFT converges almost surely to a deterministic limit distribution F γ (λ) with density
dF γ (λ) = (λ + − λ)(λ − λ − ) 2πγλ dλ,(2)
where
λ + = (1 + √ γ) 2 and λ − = (1 − √ γ) 2 .(3)
Proposition 2: Letting G ∈ R D×m with i.i.d. entries sampled from N (0, 1), then as m/D −→ γ ∈ [0, 1),
1 √ D σ max (G)
a.s.
−→ 1 + √ γ,(4)and 1 √ D σ min (G) a.s. −→ 1 − √ γ.(5)
C. Notations
Throughout this paper, we will use the following notations. Bold lower case letter a denotes a vector. Bold upper case letter A denotes a matrix. R D denotes a D-dimensional vector space. [e 1 , ..., e D ] is the canonical basis of R D .
II. MAIN RESULT
A. Bounding Generalization Discrimination Power
Suppose we have c + 1 classes, represented by homoscedastic Gaussian distributions in a high-dimensional space R D , N i (µ i , Σ), i = 1, 2, ..., c + 1, with class means µ i ∈ R D and the May 2, 2014 DRAFT common covariance matrix Σ ∈ S D×D ++ . Assuming the classes have equal prior probability 1 c+1 1 , the following matrix S, which is referred to as the between-class scatter matrix, gives a measure of class separation,
S = 1 c + 1 c+1 i=1 (µ i − µ)(µ i − µ) T , with µ = 1 c + 1 c+1 i=1 µ i .(6)
Suppose the eigendecomposition of Σ −1 S has (at most) c nonzero eigenvalues λ i , i = 1, 2, ..., c, and associated eigenvectors W = [w 1 , .., w c ]. FLDA uses W as a projection matrix to obtain a low-dimensional data representation, and according to Fisher's criterion, the discrimination power in the dimension reduced space is given by [22]
∆(Σ, S|W) = Tr (W T ΣW) −1 W T SW = c i=1 λ i .(7)
In practice, we do not have access to population parameters Σ and S, but their estimates, i.e., the sample covariance Σ and the sample between-class scatter matrix S via sample class means µ i . Denoting by W the empirical projection matrix obtained from generalized eigendecomposition of Σ and S, the generalization discrimination power of FLDA is given by
∆(Σ, S| W) = Tr ( W T Σ W) −1 W T S W ,(8)
which measures how the classes are separated in the dimension reduced space. When data dimensionality D is fixed and training sample size N goes to infinity, the generalization discrimination power (8) will converge to its population counterpart (7), since W converges to W. However, such classical result is invalid when D increases proportionally with N. Regarding this, the following theorem gives a new asymptotic result on FLDA's generalization ability, in a setting where D and N increase to infinity proportionally.
Theorem 1: Suppose the population discrimination power ∆(Σ, S|W) = c i=1 λ i . The generalization discrimination power ∆(Σ, S| W) can be factorized as
∆(Σ, S| W) = c i=1 δ i λ i(9)
where 0 ≤ δ i ≤ 1. Further, as both the dimensionality D and the training sample size N increase (N > D) and D/N −→ γ ∈ [0, 1), it holds asymptotically
δ i λ i ≥ max 2 cos(arccos( λ i /(λ i + γ)) + arccos( 1 − γ)), 0 λ i , a.s.(10)
Theorem 1 gives an asymptotically lower bound on the generalization ability of FLDA, in terms of the population discrimination power λ i and the dimensionality to training sample size ratio γ = D/N. An important feature of the bound is that it is determined by the ratio γ = D/N rather than the dimensionality D. In other words, a good generalization performance of FLDA only requires a training sample size that scales linearly with respect the dimensionality, although there are a quadratic number of parameters to be estimated in the sample covariance. Figure 1 (a) gives an illustration of the bound under different values of the ratio γ = D/N.
Besides, according to (10), the influence of the ratio γ = D/N to the lower bound comes from two aspects, each through the term λ i /(λ i + γ) and the term
√ 1 − γ. Note that λ i /(λ i + γ)
allows a tradeoff between λ i and γ, i.e., when λ i is sufficiently large, arccos( λ i /(λ i + γ)) approaches 0 and thus vanishes from the lower bound (10). The second term √ 1 − γ only depends on γ, and later proofs reveal that it measures how covariance estimation influences the generalization of FLDA. Assuming a sufficient large λ i such that λ i /(λ i + γ) ≈ 1, we have
δ i λ i ≈ (1 − γ)λ i ,(11)
which shows that the loss of discrimination power due to the imperfection of covariance estimation is approximately proportion to γ. To the best of our knowledge, this is the simplest quantitative result on the influence of covariance estimation to FLDA, compared with related studies in the literature [14] [23] [24]. It is worth noticing that, as long as Σ ∈ S D×D ++ , the result is independent of the spectrum of the population covariance Σ, e.g., the extreme eigenvalues λ min (Σ) and λ max (Σ), or the conditional number λ max (Σ)/λ min (Σ).
B. Bounding Generalization Error of Binary Classification
In binary-class case, FLDA can also be regarded as a linear classifier, where the hyperplane of the linear classifier is perpendicular to the one-dimensional projection vector w 1 of dimension reduction. Without loss of generality, suppose w T 1 (µ 1 − µ 2 ) ≥ 0, the generalization error P of May 2, 2014 DRAFT binary classification with FLDA can be calculated analytically by [25]
P = 0.5Φ − w T 1 µ 1 − 0.5 w T 1 ( µ 1 + µ 2 ) w T 1 Σ w 1 + 0.5Φ − 0.5 w T 1 ( µ 1 + µ 2 ) − w T 1 µ 2 w T 1 Σ w 1 ,(12)
where Φ(·) is the cumulative distribution function (CDF) of the standard Gaussian. If we replace w 1 and µ i by its population counterpart w 1 and µ i , then (12) gives the Bayes error P Bayes , i.e.,
P Bayes = Φ − 0.5w T 1 (µ 1 − µ 2 ) w T 1 Σw 1 = Φ − w T 1 Sw 1 w T 1 Σw 1 = Φ − λ 1 .(13)
Below, we present a corollary of Theorem 1, which gives an asymptotic upper bound of P in terms of P Bayes and γ = D/N.
P ≤ Φ −̺ λ 1 , a.s.(14)
where ̺ = max cos(arccos( λ 1 /(λ 1 + γ)) + arccos( 1 − γ)), 0 .
P ≤ Φ ̺Φ −1 (P Bayes ) , a.s.(16)
with ̺ = max cos arccos
(Φ −1 (P Bayes )) 2 ((Φ −1 (P Bayes )) 2 + γ + arccos( 1 − γ) , 0 .(17)
Similar to the discrimination power bound, Corollary 1 shows that, given a binary classification problem with Bayes error P Bayes , the generalization error of FLDA is also determined by the dimensionality to training sample size ratio γ = D/N. Figure 1
C. Related Work
In recent years, asymptotic analysis on FLDA have also been performed in the case where D > N. For example, [14] found that when D increases faster than N the the pseudo-inverse based FLDA approaches to a random guess and therefore suggested a "naive Bayes" approach in this situation. A more detailed analysis on pseudo-inverse FLDA was given in [24] by investigating the estimation error of pseudo-inverse of the sample covariance. Random matrix theory, e.g.,
Marčenko-Pastur Law, was also utilized in [24], so as to bound the expected estimation error in the asymptotic case. The result in this paper provides a complementary theory of FLDA in the setting of D < N, which shows that the generalization ability of FLDA in such situation is mainly determined by the ratio γ = D/N.
III. PROOF OF MAIN RESULT
In this section, we present the proof of Theorem 1, which are mainly based upon the asymptotic results on eigensystems of the sample covariance and the sample between-class scatter matrix.
A. On ∆(Σ, S| W)
We begin the proof by bounding the generalization discrimination power ∆(Σ, S| W) in terms of eigenvalues and/or eigenvectors of a normalized version of the sample covariance and sample between-class scatter matrix.
Lemma 1: Given a problem with population discrimination power ∆(Σ, S|W) = c i=1 λ i , there is a nonsingular matrix X that simultaneously diagonalizes Σ and S, i.e.,
X T ΣX = I and X T SX = Λ 0 ,(18)
where
Λ 0 = diag(λ 1 , ..., λ c , 0, ..., 0).
Lemma 2:
Given the normalized estimates Σ 0 = X T ΣX and S 0 = X T SX, and their eigende-
compositions Σ 0 = UΛ( Σ 0 )U T and S 0 = VΛ( S 0 )V T , the generalization discrimination power ∆(Σ, S| W) can be expressed as ∆(Σ, S| W) = c i=1 δ i λ i ,(19)
where
δ i = R T Λ −1 ( Σ 0 )U T V 1:c U T e i 2 .(20)
Lemma 3: Given Λ( Σ 0 ) and V 1:c from Lemma 2, it holds
δ i ≥ max 2 cos arccos( V T 1:c e i ) + arccos ξ T Λ −1 ( Σ 0 )ξ ξ T Λ −2 ( Σ 0 )ξ , 0 . (21)
where ξ is a unit-length random vector uniformly distributed on the unit sphere S D−1 .
Lemma 2 and Lemma 3 show that the generalization discrimination power of FLDA are determined by the eigensystems of the normalized estimates Σ 0 and S 0 . Since Σ 0 is actually an estimate of the identity covariance matrix I, we have that given the population discrimination power
∆(Σ, S|W) = c i=1 λ i , the generalization ability of FLDA, i.e., ∆(Σ, S| W) = c i=1 δ i λ i , is
independent of the population covariance Σ. Next, we present properties on the eigensymstems of Σ 0 and S 0 , which are necessary for evaluating the lower bound of δ i in (21).
B. Properties of Σ 0
We have the following lemma on the eigensystem of the normalized sample covariance Σ 0 .
Lemma 4: Given the eigendecomposition Σ 0 = UΛ( Σ 0 )U T , it holds 1) U and Λ( Σ 0 ) are independent random variables;
2) U follows the Haar distribution, i.e., it is uniformly distributed on the set of all orthonormal matrices in R D×D ;
3) denoting by F N (λ) the empirical spectral distribution of the eigenvalues of Σ 0 , i.e.,
F N (λ) = 1 D D i=1 1{λ i ( Σ 0 ) ≤ λ}, λ ≥ 0,(22)then, as D/N −→ γ ∈ [0, 1), F N (λ) a.s. −→ F γ (λ),(23)
where the limit distribution F γ (λ) has the density
dF γ (λ) = 1 2πγ (λ + − λ)(λ − λ − ) λ dλ,(24)
with
λ + = (1 + √ γ) 2 and λ − = (1 − √ γ) 2 .(25)
The first and the second statements in Lemma 4 can be understood by the fact that Σ 0 is an empirical estimate of I, whose probability density is invariant to any orthogonal transformation.
The last statement is a corollary of the Marčenko-Pastur law, i.e., Proposition 1, which says that the empirical spectral distribution of the matrix 1 N GG T , wherein G ∈ R D×N has i.i.d entries sampled from N (0, 1), converges almost surely to the deterministic distribution F γ (λ) as
D/N −→ γ ∈ [0, 1).
Further, we need the following lemma on the inverse of the eigenvalues Λ( Σ 0 ), which says that the energy of Λ −1 ( Σ 0 ) and Λ −2 ( Σ 0 ) projected onto a random direction is almost surely deterministic in the limit. It is worth noticing that the results in Lemma 5 generalize the results [24].
on the expectations E[ i λ −1 i ( Σ 0 )] and E[ i λ −2 i ( Σ 0 )] in
Lemma 5: Suppose ξ is a unit-length random vector uniformly distributed on the unit sphere S D−1 and it is independent of Σ 0 , then as D/N −→ γ ∈ [0, 1),
ξ T Λ −1 ( Σ 0 )ξ a.s. −→ λ −1 dF γ (λ) = 1 1 − γ ,(26)
and
ξ T Λ −2 ( Σ 0 )ξ a.s. −→ λ −2 dF γ (λ) = 1 (1 − γ) 3 .(27)
C. Properties of S 0
We have the following lemma on the eigenvectors of S 0 .
Lemma 6: Given the eigendecomposition S 0 = VΛ( S 0 )V T , then as D/N −→ γ ∈ [0, 1), lim D/N −→γ V T 1:c e i 2 ≥ λ i λ i + γ , a.s., i = 1, 2, ..., c,(28)
where λ i is from the population discrimination power ∆(Σ, S|W) = c i=1 λ i . Recalling Lemma 1, the population counterpart of S 0 is actually the diagonal matrix Λ 0 = X T SX. Therefore, we expect the first c eigenvectors V 1:c of S 0 to be close to
I 1:c = [e 1 , ..., e c ].
Lemma 6 shows that the performance of eigenvector estimation is determined by the λ i and γ, and in particular, as γ approaches 0 the estimation becomes consistent.
D. Proof of Theorem 1
Now, we are ready to prove our main result Theorem 1, which is a conclusion out of the combination of Lemmas 2, 3, 5 and 6.
Proof: By Lemma 5, we have
lim D/N −→γ ξ T Λ −1 ( Σ 0 )ξ ξ T Λ −2 ( Σ 0 )ξ = 1 1−γ 1 (1−γ) 1.5 = 1 − γ, a.s.(29)
By Lemma 6, we have
lim D/N −→γ V T 1:c e i ≥ λ i /(λ i + γ), a.s.(30)
Then the proof is completed by substituting (29) and (30) into Lemma 2 and Lemma 3.
May 2, 2014 DRAFT
IV. EMPIRICAL EVALUATIONS
A. On the Bound of Generalization Discrimination Power
According to Theorem 1, the generalization discrimination power of FLDA for dimension reduction can be factorized as
∆(Σ, S| W) = c i=1 δ i λ i ,
where λ i measures the population discrimination power, and each component δ i λ i of the generalization discrimination power can be lower bounded by
δ i λ i ≥ max 2 cos(arccos( λ i /(λ i + γ)) + arccos( 1 − γ)), 0 λ i .
We evaluate this result on both simulated and real datasets by comparing δ i λ i with the lower bound above.
For simulated data, we fix the ratio γ = D/N = 0.5, with D = 50 and N = 100. Note the settings give moderate size problems; however, due to the asymptotic characteristic of the bound, which inherently fits to large size problem, the evaluation on moderate size problems is more critical. We generate 1,000 experiments, each having 5 classes with randomly generated population covariance Σ and class means µ i , i = 1, ..., 5. The population discrimination power λ i , i = 1, ..., 4, are calculated via eigendecomposition of Σ −1 S, where S is the between-class scatter matrix. For the generalization discrimination power δ i λ i , the factor δ i has a close form formulation as shown by Lemma 2, i.e.,
δ i = R T (Λ −1 ( Σ 0 )U T V 1:c )U T e i 2 ,
where Λ( Σ 0 ) and U are the eigensystems of Σ 0 and V 1:c are the first c eigenvectors of S 0 , with Σ 0 = X T ΣX and S 0 = X T SX being the normalized sample covariance and betweenclass scatter matrix and X simultaneously diagonalizing Σ and S. Since a larger discrimination power means a better separation between classes, we expect that on most of the experiments the generalization discrimination power of FLDA can be bounded from the lower side by the generalization bound. Indeed, as shown by Figure 2, the bound holds with an overwhelming probability in the empirical sense (i.e., on more than 990 out of the 1,000 experiments).
We further evaluate the bound of generalization discrimination power on four benchmark datesets from the UCI machine learning repository [28]: 1) the image segmentation (ImageSeg) May 2, 2014 DRAFT or moderate size problems.
B. On the Bound of Generalization Errors
According to Corollary 1, suppose the Bayes error of a binary classification problem is P Bayes , then the generalization error P of FLDA can be boudned by
P ≤ Φ(̺Φ −1 (P Bayes )),
where Φ(·) is the CDF of the standard Gaussian distribution and ̺ = max cos arccos (Φ −1 (P Bayes )) 2 ((Φ −1 (P Bayes )) 2 + γ + arccos( 1 − γ) , 0 .
To evaluate this result, we perform binary classification with FLDA on 1,000 experiments, with randomly generated covariance matrix and class means. The same as in previous simulation, we fix the ratio γ = D/N = 0.5, with D = 50 and N = 100. Figure 4 shows the result, where the generalization error of FLDA is properly bounded by the upper bound.
In addition, we run experiments on the previous four real datasets to evaluate the generalization error bound. We randomly select class pairs from each dataset to perform binary classification.
We hold out 10% data as the evaluation set, which is used to estimate the "Bayes" error and generalization error. The "Bayes" classifier is obtained by training FLDA on the rest 90% data, May 2, 2014 DRAFT
V. PROOFS OF LEMMAS AND COROLLARY
This section provides detailed proofs of Lemmas in Section III and Corollary 1 in Section II.
May 2, 2014 DRAFT
A. Proof of Lemma 1
It is a direct result of the simultaneous diagonalization theorem for a pair of semidefinite matrices [22].
B. Proof of Lemma 2
The proof is divided into two steps.
i) Since X in Lemma 1 is nonsingular, there exists some Q ∈ R D×c such that W = XQ. Then,
∆(Σ, S| W) = Tr(( W T Σ W) −1 W T S W) = Tr((Q T X T ΣXQ) −1 Q T X T SXQ) = Tr((Q T Q) −1 Q T X T ΛQ) = Tr((Q T Q) −1 Q T 1 Λ 1 Q 1 ) = Tr(Q 1 (Q T Q) −1 Q T 1 Λ 1 ) = c i=1 δ i λ i ,(31)
where Q 1 contains the first c rows of Q and Λ 1 is the upper-left c × c submatrix of Λ, and clearly,
δ i = {Q 1 (Q T Q) −1 Q T 1 } ii .(32)
ii) In FLDA, W are the eigenvectors of Σ −1 S, and we can restrict the scale of W such that
W T Σ W = I c and W T S W = Λ 1 ,(33)
where Λ 1 is some c × c diagonal matrix. Substituting W = XQ into (33) and recalling Σ 0 = X T ΣX and S 0 = X T SX, we get
Q T Σ 0 Q = I c and Q T S 0 Q = Λ 1 .(34)O T Λ − 1 2 ( Σ 0 )U T VΛ( S 0 )V T UΛ − 1 2 ( Σ 0 )O = Λ 1 .(36)
In addition, since S 0 has rank c, we can rewrite (36) as
O T Λ − 1 2 ( Σ 0 )U T V 1:c Λ 1 2 1 ( S 0 )Λ 1 2 1 ( S 0 )V T 1:c UΛ − 1 2 ( Σ 0 )O = Λ 1 ,(37)
where
Λ 1 ( Σ 0 ) is the upper-left c × c submatrix of Λ( Σ 0 ). (37) implies the columns of O must be the left singular vectors of Λ − 1 2 ( Σ 0 )U T V 1:c Λ 1 2 1 ( S 0 ). Thus, O spans the range space of Λ − 1 2 ( Σ 0 )U T V 1:c Λ 1 2 1 ( S 0 ) and therefore the range space of Λ − 1 2 ( Σ 0 )U T V 1:c . Then, there must exist some matrix A ∈ R c×c such that Λ − 1 2 ( Σ 0 )U T V 1:c = OA, and thus O = Λ − 1 2 ( Σ 0 )U T V 1:c A −1 ,(38)
where the nonsingularity of A is implied by the nonsingularity of Λ − 1 2 ( Σ 0 )U T .
By (35) and (38), we have
Q = UΛ −1 ( Σ 0 )U T V 1:c A,(39)
and
Q 1 = I T 1:c UΛ −1 ( Σ 0 )U T V 1:c A.(40)
Therefore,
{Q 1 (Q T Q) −1 Q 1 } ii = e T i UΛ −1 ( Σ 0 )U T V 1:c (V T 1:c UΛ −2 ( Σ 0 )U T V 1:c ) −1 V T 1:c UΛ −1 ( Σ 0 )U T e i .(41)Letting R = R(Λ −1 ( Σ 0 )U T V 1:c ), then RR T = Λ −1 ( Σ 0 )U T V 1:c (V T 1:c UΛ −2 ( Σ 0 )U T V 1:c ) −1 V T 1:c UΛ −1 ( Σ 0 ),(42)
which together with (41) gives
{Q 1 (Q T ℓ Q ℓ ) −1 Q 1 } ii = e T i URR T U T e i = R T U T e i 2 = R T (Λ −1 ( Σ 0 )U T V 1:c )U T e i 2 .(43)
This completes the proof.
May 2, 2014 DRAFT
C. Proof of Lemma 3
Recall Lemma 2 that
δ i = R T (Λ −1 ( Σ 0 )U T V 1:c )U T e i 2 . Denote by ∡(U T e i , R(Λ −1 ( Σ 0 )U T V 1:c ))
the angle between vector U T e i and subspace R T (Λ −1 ( Σ 0 )U T V 1:c ), we have
δ i = cos 2 (∡(U T e i , R(Λ −1 ( Σ 0 )U T V 1:c ))).(44)
Two basic facts that hold for arbitrary vector a 1 , a 2 and subspace A are
∡(a 1 , A) ≤ ∡(a 1 , a 2 ) + ∡(a 2 , A).(45)
and
∡(a 1 , A) ≤ ∡(a 1 , a), if a ∈ A.(46)
Then, by using (45) and (46), we get
∡(U T e i , R(Λ −1 ( Σ 0 )U T V i )) ≤∡(U T e i , U T V 1:c V T 1:c e i ) + ∡(U T V 1:c V T 1:c e i , R(Λ −1 ( Σ 0 )U T V 1:c )) ≤∡(U T e i , U T V 1:c V T 1:c e i ) + ∡(U T V 1:c V T 1:c e i , Λ −1 ( Σ 0 )U T V 1:c V T 1:c e i ) =θ 1 + θ 2 .(47)
Denoting θ = θ 1 + θ 2 , since cos(x) is positive and decreasing on [0, π/2], x 2 is increasing on [0, 1], and δ i is nonnegative, we have
δ i ≥ cos 2 (θ), θ ≤ π 2 0, else = max 2 {cos(θ), 0}.(48)
It remains to calculate θ 1 and θ 2 . For θ 1 , We have
cos 2 (θ 1 ) = |e i V T 1:c UU T V 1:c e i | 2 U T V 1:c V T 1:c e i 2 = |e T i V 1:c V T 1:c e i | 2 e T i V 1:c V T 1:c e i = V T 1:c e i 2 ,(49)
which gives
θ 1 = arccos( V T 1:c e i ).(50)
May 2, 2014 DRAFT For θ 2 , as rescaling does not change the direction of a vector, we can rewrite θ 2 as
θ 2 = ∡(U T ζ, Λ −1 ( Σ 0 )U T ζ),(51)
where
ζ = V 1:c V T 1:c e i V 1:c V T 1:c e i .(52)
Note that ζ is a unit-length random vector and is independent of U due to the independency between V 1:c and U. Then, we have
cos 2 (θ 2 ) = |ζ T UΛ −1 ( Σ 0 )U T ζ| 2 Λ −1 ( Σ 0 )U T ζ 2 = (ζ T UΛ −1 ( Σ 0 )U T ζ) 2 ζ T UΛ −2 ( Σ 0 )U T ζ .(53)
We have known, from Lemma 4, U is uniformly distributed on the set of all orthonormal matrices in R D×D , and ζ is a unit-length random vector independent of U. Thus, ξ = U T ζ must be a unit-length random vector uniformly distributed on the unit sphere S D−1 . Finally, (53) gives
θ 2 = arccos ξ T Λ −1 ( Σ 0 )ξ ξ T Λ −2 ( Σ 0 )ξ .(54)
This completes the proof.
D. Proof of Lemma 4
Since Σ 0 = X T ΣX is a normalized sample covariance, wherein X T ΣX = I, we have
Σ 0 = 1 N c+1 i=1 n j=1 (x i j −x i )(x i j −x i ) T ,(55)
where x i j is sampled from some N (µ i , I) andx i is the sample mean. Letting z i j = x i j − µ i , which implies z i j is sampled from the standard Gaussian distribution N (0, I), andz i =x i − µ i , then Σ 0 can be rewritten as
Σ 0 = 1 N c+1 i=1 n j=1 (z i j −z i )(z i j −z i ) T ,(56)
One property of Σ 0 in (56) is that, as a random variable, its distribution is invariant to orthogonal similarity transformation, i.e., Σ 0 and O Σ 0 O T , wherein U T U = I, have the same distribution.
This is due to the fact that O T Σ 0 O corresponds to (56) in the case of replacing z i j by Oz i j while
Oz i j has the same distribution with z i j , i.e., the standard Gaussian distribution N (0, I). Then, according to Theorem 3.2 in [29], the invariant property to orthogonal similarity transformation implies that the distribution of Σ 0 is independent of its eigenvectors U but only depends on its eigenvalues Λ( Σ 0 ), and U is a random matrix uniformly distributed on the set of all possible orthonormal matrices in R D×D . This completes the statements 1) and 2) in Lemma 4.
Further, (56) can be rewritten as
Σ 0 = 1 N c+1 i=1 n j=1 z i j z iT j − 1 c + 1 c+1 i=1z iziT = 1 N c+1 i=1 n j=1 z i j z iT j − 1 (c + 1)n c+1 i=1 √ nz i √ nz iT = 1 N G 1 G T 1 − 1 N G 2 G T 2 = T 1 + T 2 .(57)
where G 1 ∈ R D×N and G 2 ∈ R D×(c+1) . For the first term T 1 = 1 N G 1 G T 1 , by Proposition 1, we know that the empirical distribution of its eigenvalues converges almost surely to F γ (λ) with density,
dF γ (λ) = (λ + − λ)(λ − λ − ) 2πγλ dλ,(58)
where γ = lim D/N and
λ + = (1 + √ γ) 2 and λ − = (1 − √ γ) 2 .(59)
For the second term T 2 = 1 N G 2 G T 2 , clearly it has finite rank c + 1. According to [30], a finite rank perturbation does not effect the convergence of the empirical spectral distribution, i.e., lim F N (λ(T 1 + T 2)) = lim F N (λ(T 1 )) = F γ (λ). This completes the proof.
E. Proof of Lemma 5
The condition that ξ is a unit-length random vector uniformly distributed on the unit sphere S D−1 can be replaced by ξ ∈ R D with entries independently sampled from N (0, 1/D). This is because, in the later case, ξ/ ξ is uniformly distributed on S D−1 , and ξ 2 a.s. −→ 1 due to the Strong Law of Large Numbers.
For (26), we divide the proof into two steps. First, we show that
ξ T Λ −1 ( Σ 0 )ξ a.s. −→ λ −1 dF γ (λ),
and then we calculate the integral.
i) Recall λ − = (1 − √ γ) 2 , and let Λ −1 ( Σ 0 ) = diag(min{λ − , λ −1 i ( Σ 0 )}), i.e., a truncated version of Λ −1 ( Σ 0 ) by clamping λ −1 i ( Σ 0 ) to be λ −1 − if λ −1 i ( Σ 0 ) ≥ λ −1 − .
Then, we divide the left-hand side of (26) into three terms
ξ T Λ −1 ( Σ 0 )ξ − ξ T Λ −1 ( Σ 0 )ξ,(60)ξ T Λ −1 ( Σ 0 )ξ − 1 D Tr(Λ −1 ( Σ 0 )),(61)and 1 D Tr(Λ −1 ( Σ 0 )) − λ −1 dF γ (λ).(62)
We show that all the three terms converge almost surely to zero.
For the first term (60), we have
0 ≤ξ T (Λ −1 ( Σ 0 ) − Λ −1 ( Σ 0 ))ξ ≤ ξ 2 max{0, λ −1 min ( Σ 0 ) − λ −1 − }.(63)
By the same argument in the proof of Lemma 4, we know that
lim λ min ( Σ 0 ) = lim λ min 1 N c+1 i=1 n j=1 z i j z iT j = lim 1 √ N σ min (Z) 2 ,(64)
where Z = [z 1 1 , ..., z c+1 n ] ∈ R D×N , with entries independently sampled from N (0, 1). By Proposition 2, we have lim 1 √ N σ min (Z) = 1− √ γ, and thus λ min ( Σ 0 ) a.s.
−→ (1− √ γ) 2 = λ − . Accordingly, max{0, λ −1 min ( Σ 0 ) − λ −1 − } a.s. −→ 0.(65)
Then, by ξ 2 a.s. −→ 1, (63) and (65), we have
ξ T Λ −1 ( Σ 0 )ξ − ξ T Λ −1 ( Σ 0 )ξ a.s. −→ 0.(66)
For the second term (61), since Λ −1 ( Σ 0 ) ≤ λ − for all D, i.e., it is uniformly bounded, we apply Theorem 3.4 in [21] and get
ξ T Λ −1 α ( Σ 0 )ξ − 1 D Tr(Λ −1 α ( Σ 0 )) a.s. −→ 0.(67)
For the third term (62), since dF γ (λ) is nonzero only on [λ − , λ + ], it is sufficient to examine
1 D Tr(Λ −1 ( Σ 0 )) − λ −1 dF γ (λ) = ∞ 0 min(λ − , λ −1 )dF N (λ) − λ + λ − λ −1 dF γ (λ) = λ + λ − λ −1 d(F N (λ) − F γ (λ)) + λ −1 − λ − 0 dF N (λ) + ∞ λ + λ −1 dF N (λ). (68) Sine F N (λ) a.s. −→ F γ (λ) and λ −1 is bounded on [λ − , λ + ], it holds [31] λ + λ − λ −1 d(F N (λ) − F γ (λ)) a.s. −→ 0.(69)
Further, sine F γ (λ − ) = 0 and F γ (λ + ) = 1, it holds
λ − 0 dF N (λ) = F N (λ − ) a.s. −→ F γ (λ − ) = 0,(70)
and
0 ≤ ∞ λ + λ −1 dF N (λ) ≤ λ −1 + (1 − F N (λ + )) a.s. −→ λ −1 + (1 − F γ (λ + )) = 0.(71)
Thus,
1 D Tr(Λ −1 α ( Σ 0 )) − λ −1 dF γ (λ) a.s. −→ 0.(72)
ii) We now calculate the integral
I = λ −1 dF γ (λ) = λ + λ − (λ + − λ)(λ − λ − ) 2πγλ 2 dλ(73)
where λ + = (1 + √ γ) 2 and λ − = (1 − √ γ) 2 .
Letting λ = 1 + γ − 2 √ γ cos x, x ∈ [0, π] and substituting it into (73), we have
I = 2 π π 0 sin 2 x (1 + γ − 2 √ γ cos x) 2 dx.(74)
Further, letting t = tan x 2 , we have
I = 2 π ∞ 0 2t 1+t 2 2 1 + γ − 2 √ γ 1−t 2 1+t 2 2 2 1 + t 2 dt = 16 π ∞ 0 t 2 (1 + γ)(t 2 + 1) − 2 √ γ(1 − t 2 ) 2 1 1 + t 2 dt = 16 π ∞ 0 t 2 (1 + √ γ) 2 t 2 + (1 − √ γ) 2 2 1 1 + t 2 dt = 16 π(1 + √ γ) 4 ∞ 0 t 2 t 2 + 1− √ γ 1+ √ γ 2 2 1 1 + t 2 dt.(75)∞ 0 t 2 (t 2 + α 2 ) 2 1 1 + t 2 dt = ∞ 0 − 1 (1−α 2 ) 2 t 2 + 1 dt + ∞ 0 1 (1−α 2 ) 2 t 2 + α 2 dt + ∞ 0 − α 2 (1−α 2 ) (t 2 + α 2 ) 2 dt.(76)
Denoting by I 1 , I 2 and I 3 the terms in the righthand side of (76), we have
I 1 = ∞ 0 − 1 (1−α 2 ) 2 t 2 + 1 dt = −1 (1 − α 2 ) 2 ∞ 0 d arctan t = −π 2(1 − α 2 ) 2 ,(77)I 2 = ∞ 0 1 (1−α 2 ) 2 t 2 + α 2 dt = 1 α(1 − α 2 ) 2 ∞ 0 d arctan t α = π 2α(1 − α 2 ) 2 ,(78)I 3 = ∞ 0 − α 2 (1−α 2 ) (t 2 + α 2 ) 2 dt = −1 2(1 − α 2 ) ∞ 0 d t t 2 + α 2 + −1 2(1 − α 2 ) ∞ 0 1 t 2 + α 2 dt = 0 + −π 4α(1 − α 2 ) = −π 4α(1 − α 2 ) .(79)
Combining (75) to (79) and noticing α =
1− √ γ 1+ √ γ , we get I = 16 π(1 + √ γ) 4 −π 2(1 − α 2 ) 2 + π 2α(1 − α 2 ) 2 + −π 4α(1 − α 2 ) = 16 π(1 + √ γ) 4 π 4α(1 + α) 2 = 1 1 − γ .(80)
This completes the proof of (26).
For (27), by the same strategy as used in the proof of (26), we have ξ T Λ −2 ( Σ 0 )ξ a.s.
−→
λ −2 dF γ (λ). Below, we calculate the integral.
I = λ −2 dF γ (λ) = λ + λ − (λ + − λ)(λ − λ − ) 2πγλ 3 dλ,(81)
where
λ + = (1 + √ γ) 2 and λ − = (1 − √ γ) 2 . Letting λ = 1 + γ − 2 √ γ cos x, x ∈ [0, π] and
substituting it into (73), we have
I = 2 π π 0 sin 2 x (1 + γ − 2 √ γ cos x) 3 dx.(82)
Further, letting t = tan x 2 , we have
I = 2 π ∞ 0 2t 1+t 2 2 1 + γ − 2 √ γ 1−t 2 1+t 2 3 2 1 + t 2 dt = 16 π ∞ 0 t 2 (1 + γ)(t 2 + 1) − 2 √ γ(1 − t 2 ) 3 dt = 16 π ∞ 0 t 2 (1 + √ γ) 2 t 2 + (1 − √ γ) 2 3 dt = 16 π(1 + √ γ) 6 ∞ 0 t 2 t 2 + 1− √ γ 1+ √ γ 2 3 dt.(83)Letting α = 1− √ γ 1+ √ γ , we have ∞ 0 t 2 (t 2 + α 2 ) 3 dt = − 1 4 ∞ 0 d t (t 2 + α 2 ) 2 + 1 4 ∞ 0 1 (t 2 + α 2 ) 2 dt = π 16α 3 .(84)Thus, by α = 1− √ γ 1+ √ γ , we get I = 16 π(1+ √ γ) 6 π 16α 3 = 1 (1−γ) 3 .
This completes the proof of (27).
F. Proof of Lemma 6
By Lemmas 1 and 2, S 0 is an estimate of X T SX = Λ 0 = diag(λ 1 , ..., λ c , 0, ..., 0). Suppose the original distributions of the c + 1 classes are N (µ i , Σ) and the between-class scatter matrix is S.
Then, Λ 0 should be the between-class scatter matrix of an equivalent problem with distributions
N (µ ′ i , I), wherein µ ′ i = X T µ i . Therefore, Λ 0 = 1 c+1 c+1 i=1 (µ ′ i − µ ′ )(µ ′ i − µ ′ ) T , with µ ′ = 1 c+1 c+1 i=1 µ ′ i . Letting M = [µ ′ 1 , ..., µ ′ c+1 ] and E ∈ R (
Accordingly,
V T 1:c e i 2 = R T ([ξ 1 , ..., ξ c ])e i 2 ≥ R T (ξ i )e i 2 = 1 ξ i 2 |ξ T i e i | 2 = |e T i (c + 1)λ i e i + e T i X(I − E)Q i | 2 (c + 1)λ i e i + X(I − E)Q i 2 ≥ (c + 1)λ i + |e T i X(I − E)Q i | 2 − 2 (c + 1)λ i |e T i X(I − E)Q i | (c + 1)λ i + X(I − E)Q i 2 + 2 (c + 1)λ i e T i X(I − E)Q i .(89)
It can be verified that as N = (c + 1)n −→ ∞
|e T i X(I − E)Q i | ≤ e T i X = c+1 j=1 X 2 ij a.s. −→ 0,(90)
where the inequality is due to (I − E)Q i ≤ (I − E) Q i ≤ 1 and the limit is because X ij follows the distribution N (0, 1 n ). In addition, by Proposition 2 and letting G = √ nX, we have
X = 1 √ n G a.s. −→ D n = (c + 1)D N −→ (c + 1)γ.(91)
Thus,
X(I − E)Q i ≤ X a.s. −→ (c + 1)γ.(92)
Combining (89), (90) and (92), we obtain
lim D/N −→γ V T 1:c e i 2 ≥ λ i λ i + γ , a.s.(93)
This completes the proof.
P = 0.5Φ − w T 1 µ 1 − 0.5 w T 1 ( µ 1 + µ 2 ) w T 1 Σ w 1 + 0.5Φ − 0.5 w T 1 ( µ 1 + µ 2 ) − w T 1 µ 2 w T 1 Σ w 1 ,(94)assumed w T 1 (µ 1 − µ 2 ) ≥ 0. First, we have − w T 1 µ 1 − 0.5 w T 1 ( µ 1 + µ 2 ) w T 1 Σ w 1 = − 0.5 w T 1 (µ 1 − µ 2 ) w T 1 Σ w 1 + 0.5 w T 1 (( µ 1 + µ 2 ) − (µ 1 + µ 2 )) w T 1 Σ w 1 = − w T 1 S w 1 w T 1 Σ w 1 + 0.5 w T 1 (( µ 1 + µ 2 ) − (µ 1 + µ 2 )) w T 1 Σ w 1 = − δ 1 λ 1 + 0.5T,(95)
and similarly
− 0.5 w T 1 ( µ 1 + µ 2 ) − w T 1 µ 2 w T 1 Σ w 1 = − 0.5 w T 1 (µ 1 − µ 2 ) w T 1 Σ w 1 − 0.5 w T 1 (( µ 1 + µ 2 ) − (µ 1 + µ 2 )) w T 1 Σ w 1 = − δ 1 λ 1 − 0.5T,(96)
As long as T a.s. −→ 0, we have by Theorem 1 that
P = Φ(− δ 1 λ 1 ) ≤ Φ(−̺ λ 1 )(97)
with ̺ = max cos(arccos( λ i /(λ i + γ)) + arccos( 1 − γ)), 0 .
Below, we verify that it indeed holds
T = w T 1 (( µ 1 − µ 1 ) + ( µ 2 − µ 2 )) w T 1 Σ w 1 a.s. −→ 0.(99)
By using similar strategy in the proof of Lemma 2, in particular (39), we have w 1 = Xq, wherein X satisfies X T ΣX = I and q = aU T Λ −1 ( Σ 0 )UX T ( µ 1 − µ 2 ), for some a = 0,
since X( µ 1 − µ 2 ) is the first eigenvector of the normalized sample between-scatter matrix May 2, 2014 DRAFT S 0 = X T SX. Substituting (100) into T , we have
T = ( µ 1 − µ 2 ) T XU T Λ −1 ( Σ 0 )UX T (( µ 1 − µ 1 ) + ( µ 2 − µ 2 )) ( µ 1 − µ 2 ) T XU T Λ −2 ( Σ 0 )UX T ( µ 1 − µ 2 ) .(101)
For the numerator, we have
( µ 1 − µ 2 ) T XU T Λ −1 ( Σ 0 )UX T (( µ 1 − µ 1 ) + ( µ 2 − µ 2 )) =( µ 1 − µ 1 ) T XU T Λ −1 ( Σ 0 )UX T ( µ 1 − µ 1 ) − ( µ 2 − µ 2 ) T XU T Λ −1 ( Σ 0 )UX T ( µ 2 − µ 2 ) + (µ 1 − µ 2 ) T XU T Λ −1 ( Σ 0 )UX T (( µ 1 − µ 1 ) + ( µ 2 − µ 2 )) =T 1 − T 2 + T 3 .(102)
Due to the normalization, we know that ξ 1 = UX T ( µ 1 − µ 1 ) follows the multivariate Gaussian distribution N (0, 1 n I), with n = N/2 being the training data number per class. Then, by Lemma 5 and ξ 1 2 a.s. −→ 2γ, we have
T 1 = ξ T 1 Λ −1 ( Σ 0 )ξ 1 = ξ 1 2 ξ T 1 ξ 1 Λ −1 ( Σ 0 ) ξ 1 ξ 1 a.s. −→ 2γ 1 − γ .(103)
Similarly, letting ξ 2 = UX T ( µ 2 − µ 2 ), the same argument gives T 2 a.s.
−→ 2γ 1−γ . Denoting ξ 3 = Λ −1 ( Σ 0 )UX T (µ 1 − µ 2 ) and recalling Lemma 5, we have
ξ 3 2 = (µ 1 − µ 2 ) T XU T Λ −2 ( Σ 0 )UX T (µ 1 − µ 2 ) a.s. −→ X T (µ 1 − µ 2 ) 2 (1 − γ) 3 < ∞.(104)
Then, since ξ follows N (0, 1 n I) and ξ 3 has bounded entries due to (104), we have
ξ T 3 ξ 1 a.s. −→ 0.(105)
Similarly, ξ T 3 ξ 2 a.s. −→ 0. Thus,
T 3 = ξ T 3 (ξ 1 + ξ 2 ) a.s. −→ 0.(106)
Therefore, we have the numerator T 1 − T 2 + T 3 a.s.
−→ 0.
For the dominator, letting ζ = UX T ( µ 1 − µ 2 ), we have
( µ 1 − µ 2 ) T XU T Λ −2 ( Σ 0 )UX T ( µ 1 − µ 2 ) = ζ ζ T ζ Λ −2 ( Σ 0 ) ζ ζ a.s. −→ lim ζ (1 − γ) 3/2 .(107)
Note that lim ζ > 0, because µ 1 = µ 2 almost surely. Thus, the dominator must be positive.
Therefore, we have T in (99) has limit 0.
VI. CONCLUSION
FLDA is an important statistical model in pattern recognition. The result obtain in this paper enriches the existing theory of FLDA, by showing that the generalization ability of FLDA is mainly determined by the dimensionality to training sample size ratio γ = D/N, given D and N are reasonably large and N > D. Important conclusions from this result include: 1) to ensure FLDA performing well, training sample size only needs to scale linearly with respect to data dimensionality, although a quadratic number of parameters are to be estimated in the sample covariance; and 2) the generalization ability of FLDA (with respect to the Bayes optimum) is independent of the spectral structure of the population covariance, given its nonsingularity and above conditions.
R D 1
1×D 2 denotes the set of all D 1 by D 2 matrices. A ii or {A} ii denotes the i-th diagonal entry of a symmetric matrix A. A i denotes the i-th column of A. A 1:c denotes the matrix composed by the first c columns of A. S D−1 denotes the D-dimensional unit sphere located on the original point. S D×D ++ denotes the set of all D by D positive definite matrices. a denotes the ℓ 2 norm of a. σ max (A) and σ min (A) are the extreme singular values of A. A = σ max (A) denotes the operator norm of A. λ i (A) denotes the i-th eigenvalue of A, sorted in a descent order. Λ(A) denotes the diagonal matrix composed of the eigenvalues of A, with the eigenvalues sorted in a descent order. R(A) denotes an orthogonal basis of the range or the column space of A.
Fig. 1 .
1Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis.
Corollary 1 :
1For binary classification with equal prior probabilities, suppose the population discrimination power ∆(Σ, S|w 1 ) = λ 1 , then if both dimensionality D and training sample size N increase (N > D) and D/N −→ γ ∈ [0, 1), the generalization error P of FLDA can be upper bounded asymptotically by
(b) gives an illustration of the generalization error bound under different values of γ.
In contrast to asymptotic analysis, generalization bounds in finite sample case were derived most recently in both linear and kernel spaces, and by using random projection as regularizationif D > N [23] [26][27]. The advantage of these results is they provide explicit probability bounds for finite N and D, while asymptotic results inherently require sufficient large N and D. However, we would like to emphasize that the bounds obtained in this paper have their own merit, by linking the generalization discrimination power (or generalization error) to the population discrimination power (or Bayes error) directly in terms of the ratio γ = D/N. Besides, as shown by empirical evaluation in later section IV, the bounds hold with high probability (in the empirical sense) for moderate D and N, though they are obtained asymptotically.
Fig. 2 .Fig. 3 .Fig. 4 .
234Evaluation of the Generalization Discrimination Power Bound with Simulated Data. dataset 2 , which contains 7 classes and in total 2,310 examples from R 18 ; 2) the Landsat dataset, which constants 6 classes and in total 6,435 examples from R 36 ; 3) the optical recognition of handwritten digits (Optdigits) dataset, which contains 10 classes and in total 5,620 examples from R 60 ; and 4) the USPS handwritten digits dataset, which contains 10 classes and in total 9,298 examples from R 256 . Note that for real dataset, the population parameters Σ and S are unknown. Thus, we use the entire dataset to get their estimates and treat them as population parameters. Again, we fix the ratio γ = D/N = 0.5, i.e., we randomly select examples twice Evaluation of the Generalization Discrimination Power Bound with Real Data. of the dimensionality as the training data. The generalization discrimination powers over 1,000 random experiments are shown in Figure 3. On the panel for each dataset, the columns of the scatters correspond to different components of the generalization discrimination power δ i λ i , and the horizontal axis location of each column equals the population discrimination power λ i (the column number is class number minus 1). On three out of the four datasets, including LandSat, Optdigits and USPS, the generalization discrimination power is properly bounded by the lower bound, with a high probability in the empirical sense. On the ImageSeg dataset, the bound does not hold with high probability as on the other three datasets. The major reason is that the size of the problem is considerably small, with D = 18 and N = 36, while the bound favors large Evaluation of the Generalization Error Bound with Simulated Data.
Fig. 5 .
5Evaluation of the Generalization Error Bound with Real Data. and the empirical classifier is trained with a subset of the rest data, such that N = 2D, namely fixing the ratio γ = D/N = 0.5. On each dataset, 1,000 random experiments are performed, with the result shown in Figure 5. Similar to the result in Figure 3, on three out of the four datasets, the generalization error can be bounded by the upper bound, while the bound does not dominate all the experiment on the ImageSeg dataset due to the small size of the problem.
Given the eigendecomposition Σ 0 =
0UΛ( Σ 0 )U T , we have from the first equation in (34) that there must exist some orthogonal matrix O ∈ R D×c , O T O = I c , such that Q = UΛ − 1 2 ( Σ 0 )O. given the eigendecomposition S 0 = V T Λ( S 0 )V, we get from the second equation in (34) that
c+1)×(c+1) with all entries equal to 1 c+1 , we have Λ 0 = 1 c+1 M(I −E)(I −E) T M T . Similarly, we have S 0 = 1 c+1 M(I −E)(I −E) T M T , where M = [ µ ′ 1 , ..., µ ′ c+1 ] and µ ′ 1 is an estimate of µ ′ 1 . As there are n training examples per class, we have M = M + X, where the entries of X ∈ R D×(c+1) are i.i.d. samples from N (0, 1/n). Note that the nonzero diagonal entries of Λ 0 are λ i , i = 1, 2, ..., c, which are actually eigenvalues of Λ 0 , associated with eigenvectors e i , i = 1, 2, ..., c. Thus, Λ 0 = 1 c+1 M(I − E)(I − E) T M T implies that M(I − E) has singular values (c + 1)λ i , i = 1, 2, ..., c and left singular vectors
I 1:c = [e 1 , ..., e c ]. Denoting by Q ∈ R (c+1)×c the right singular vectors of M(I − E), Q T Q = I c , we have M(I − E)Q = (c + 1)λ 1 e 1 , ..., (c + 1)λ c e c . (85) Consequently, by M = M + X, we have M(I − E)Q = (c + 1)λ 1 e 1 , ..., (c + 1)λ c e c + X(I − E)Q = [ξ 1 , ..., ξ c ], (86) where ξ i = (c + 1)λ i e i + X(I − E)Q i , i = 1, 2, ..., c. (87) Then, by S 0 = 1 c+1 M(I − E)(I − E) T M T , we have for the first c eigenvectors V 1:c of S 0 that V 1:c = R( M(I − E)) = R( M(I − E)Q) = R([ξ 1 , ..., ξ c ]).
For the convenience of expression, we assume an equal prior probability. This does not substantially change the analysis throughout this paper.
May 2, 2014DRAFT
The original dataset has 19 features; however the 3rd feature is a constant for all examples, and therefore is discarded in the experiments.
The authors are with the Centre for Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology, Sydney, Australia. E-mail: [email protected] and [email protected]
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| []
|
[
"False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time",
"False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time"
]
| [
"Anja Rey \nInstitut für Informatik Heinrich-Heine-Universität Düsseldorf\n40225DüsseldorfGermany\n",
"Jörg Rothe \nInstitut für Informatik Heinrich-Heine-Universität Düsseldorf\n40225DüsseldorfGermany\n"
]
| [
"Institut für Informatik Heinrich-Heine-Universität Düsseldorf\n40225DüsseldorfGermany",
"Institut für Informatik Heinrich-Heine-Universität Düsseldorf\n40225DüsseldorfGermany"
]
| []
| False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al.[ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley-Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely. | 10.1613/jair.4293 | [
"https://arxiv.org/pdf/1303.1691v1.pdf"
]
| 45,059,322 | 1303.1691 | ee130dfef96cd5d922a2ded5320d2e0ef97fa2d0 |
False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time
7 Mar 2013 June 9, 2021
Anja Rey
Institut für Informatik Heinrich-Heine-Universität Düsseldorf
40225DüsseldorfGermany
Jörg Rothe
Institut für Informatik Heinrich-Heine-Universität Düsseldorf
40225DüsseldorfGermany
False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time
7 Mar 2013 June 9, 2021
False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al.[ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley-Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.
Introduction
Weighted voting games are an important class of succinctly representable, simple games. They can be used to model cooperation among players in scenarios where each player is assigned a weight, and a coalition of players wins if and only if their joint weight meets or exceeds a given quota. Typical real-world applications of weighted voting games include decision-making in legislative bodies (e.g., parliamentary voting) and shareholder voting (see the book by Chalkiadakis et al. [CEW11] for further concrete applications and literature pointers). In particular, the algorithmic and complexitytheoretic properties of problems related to weighted voting have been studied in depth, see, e.g., the work of Elkind et al. [ECJ08,EGGW09], Bachrach et al. [BEM + 09], Zuckerman et al. [ZFBE08], and [CEW11] for an overview.
Bachrach and Elkind [BE08] were the first to study false-name manipulation in weighted voting games: Is it possible for a player to increase her power by splitting into several players and distributing her weight among these false identities? Relatedly, is it possible for two or more players to increase their power in a weighted voting game by merging their weights? The most prominent measures of a player's power, or influence, in a weighted voting game are the Shapley-Shubik and Banzhaf power indices. Merging and extending the results of [BE08] and [AP09], Aziz et al. [ABEP11] in particular study the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik index [Sha53,SS54] and the normalized Banzhaf index [Ban65] (see Section 2 for formal definitions). Rey and Rothe [RR10] extend this study for the probabilistic Banzhaf index proposed by Dubey and Shapley [DS79]. All these results, however, provide merely NP-hardness lower bounds. Aziz et al. [ABEP11, Remark 13 on p. 72] note that "it is quite possible that our problems are not in NP" (and thus are not NP-complete). Faliszewski and Hemaspaandra [FH09] provide the best known upper bound for the beneficial merging problem with respect to the Shapley-Shubik index: It is contained in the class PP, "probabilistic polynomial time," which is considered to be by far a larger class than NP, and they conjecture that this problem is PP-complete. Rey and Rothe [RR10] observe that the same arguments give a PP upper bound also for beneficial merging in terms of the probabilistic Banzhaf index, and they conjecture PP-completeness as well. 1 We resolve these conjectures in the affirmative by proving that beneficial merging and splitting (for any fixed number of false identities) are PP-complete problems both for the Shapley-Shubik and the probabilistic Banzhaf index. Beneficial splitting in general (i.e., for an unbounded number of false identities) belongs to NP PP and is PP-hard for the same two indices. Thus, none of these six problems can be in NP, unless the polynomial hierarchy collapses to its first level, which is considered highly unlikely.
Preliminaries
We will need the following concepts from cooperative game theory (see, e.g., the textbook by Chalkiadakis et al. [CEW11]). A coalitional game with transferable utilities, G = (N, v), consists of a set N = {1, . . . , n} of players (or, synonymously, agents) and a coalitional function v :
P(N) → R with v( / 0) = 0, where P(N) denotes the power set of N. G is monotonic if v(B) ≤ v(C) whenever B ⊆ C for coalitions B,C ⊆ N,
and it is simple if it is monotonic and v : P(N) → {0, 1}, that is, v maps each coalition C ⊆ N to a value that indicates whether C wins (i.e., v(C) = 1) or loses (i.e., v(C) = 0), where we require that the grand coalition N is always winning. The probabilistic Banzhaf power index of a player i ∈ N in a simple game G (see [DS79]) is defined by
Banzhaf(G , i) = 1 2 n−1 ∑ C⊆N {i} (v(C ∪ {i}) − v(C)) .(1)
Intuitively, this index measures the power of player i in terms of the probability such that i turns a losing coalition C ⊆ N {i} into a winning coalition by joining it, and therefore is pivotal for the success of C. (For comparison, the normalized Banzhaf index of i in G defined by Banzhaf [Ban65], who rediscovered a notion originally introduced by Penrose [Pen46], is obtained by dividing the raw Banzhaf index of i in G , which is the term (1), not by 2 n−1 , but by the sum of the raw Banzhaf indices of all players in G ; see [DS79,FM05,RR10] for a discussion of the differences between these two power indices.) Unlike the Banzhaf indices, the Shapley-Shubik index of i in G takes the order into account in which players enter coalitions and is defined by
∑ C⊆N {i} (v(C ∪ {i}) − v(C)) inShapleyShubik(G , i) = 1 n! ∑ C⊆N {i} C ! · (n − 1 − C )! · (v(C ∪ {i}) − v(C)) .
Since the number of coalitions is exponential in the number of players, specifying coalitional games by listing all values of their coalitional function would require exponential space. For algorithmic purposes, however, it is important that these games can be represented succinctly. Simple games can be compactly represented by weighted voting games. A weighted voting game (WVG) G = (w 1 , . . . , w n ; q) consists of nonnegative integer weights w i , 1 ≤ i ≤ n, and a quota q, where w i is the ith player's weight. For each coalition C ⊆ N, letting w(C) denote ∑ i∈C w i , C wins if w(C) ≥ q, and it loses otherwise. Requiring the quota to satisfy 0 < q ≤ w(N) ensures that the empty coalition loses and the grand coalition wins. Weighted voting games have been intensely studied from a computational complexity point of view (see, e.g., [ECJ08, EGGW09, BEM + 09, ZFBE08] and [CEW11, Chapter 4] for an overview). Aziz et al. [ABEP11] introduce the merging and splitting operations for WVGs. We use the following notation. Given a WVG G = (w 1 , . . . , w n ; q) and a nonempty 2 coalition S ⊆ {1, . . . , n}, let G &S = (w(S), w j 1 , . . . , w j n− S ; q) with { j 1 , . . . , j n− S } = N S denote the new WVG in which the players in S have been merged into one new player of weight w(S). Similarly, given a WVG G = (w 1 , . . . , w n ; q), a player i, and an integer m ≥ 2, define the set of WVGs G i÷m = (w 1 , . . . , w i−1 , w i 1 , . . . , w i m , w i+1 , . . . , w n ; q) in which i with weight w i is split into m new players i 1 , . . . , i m with weights w i 1 , . . . , w i m such that ∑ m j=1 w i j = w i . (Note that there is a set of such WVGs G i÷m , since there might be several possibilities of distributing i's weight w i to the new players i 1 , . . . , i m satisfying ∑ m j=1 w i j = w i .) For a power index PI, the beneficial merging and splitting problems are defined as follows.
PI-BENEFICIALMERGE
Given:
A WVG G = (w 1 , . . . , w n ; q) and a nonempty coalition S ⊆ {1, . . . , n}. Question: Is it true that PI(G &S , 1) > ∑ i∈S PI(G , i)?
We distinguish between two splitting problems: In the first problem, the number m of false identities some player splits into is not part of the given problem instance (rather, the problem itself is parameterized by m), whereas m is given in the instance for the second problem. (This distinction wouldn't make sense for beneficial merging.)
PI-m-BENEFICIALSPLIT
Given:
A WVG G = (w 1 , . . . , w n ; q) and a player i. Question: Is it possible to split i into m new players i 1 , . . . , i m with weights w i 1 , . . . , w i m satisfying ∑ m j=1 w i j = w i such that in this new WVG G i÷m , it holds that ∑ m j=1 PI(G i÷m , i j ) > PI(G , i)?
PI-BENEFICIALSPLIT
Given:
A WVG G = (w 1 , . . . , w n ; q), a player i, and an integer m ≥ 2. Question: Is it possible to split i into m new players i 1 , . . . , i m with weights w i 1 , . . . ,
w i m satisfying ∑ m j=1 w i j = w i such that in this new WVG G i÷m , it holds that ∑ m j=1 PI(G i÷m , i j ) > PI(G , i)?
The goal of this paper is to classify these problems in terms of their complexity for both the Shapley-Shubik and the probabilistic Banzhaf index. We assume that the reader is familiar with the basic complexity-theoretic concepts such as the complexity classes P and NP, the polynomial-time many-one reducibility, denoted by ≤ p m , and the notions of hardness and completeness with respect to ≤ p m (see, e.g., the textbook by Papadimitriou [Pap95]). Valiant [Val79] introduced #P as the class of functions that give the number of solutions of the instances of NP problems. For a decision problem A ∈ NP, we denote this function by #A. For example, if SAT is the satisfiability problem from propositional logic, then #SAT denotes the function mapping any boolean formula ϕ to the number of truth assignments satisfying ϕ. There are various notions of reducibility between functional problems in #P (see [FH09] for an overview, literature pointers, and discussion). Here, we need only the most restrictive one: We say a function f parsimoniously reduces to a function g if there exists a polynomial-time computable function h such that for each input x, f (x) = g(h(x)). That is, for functional problems f , g ∈ #P, a parsimonious reduction h from f to g transfers each instance x of f into an instance h(x) of g such that f (x) and g(h(x)) have the same number of solutions. We say that g is #P-parsimonious-hard if every f ∈ #P parsimoniously reduces to g. We say that g is #P-parsimonious-complete if g is in #P and #P-parsimonious-hard. It is known that, given a WVG G and a player i, computing the raw Banzhaf index is #P-parsimonious-complete [PK90], whereas computing the raw Shapley-Shubik index is not [FH09], although it, of course, is in #P as well.
Gill [Gil77] introduced the class PP ("probabilistic polynomial time") that contains all decision problems X for which there exist a function f ∈ #P and a polynomial p such that for all instances x, x ∈ X if and only if f (x) ≥ 2 p(|x|)−1 . It is easy to see that NP ⊆ PP; in fact, PP is considered to be by far a larger class than NP, due to Toda's theorem [Tod91]: PP is at least as hard (in terms of polynomial-time Turing reductions) as any problem in the polynomial hierarchy (i.e., PH ⊆ P PP ). NP PP , the second level of Wagner's counting hierarchy [Wag86], is the class of problems solvable by an NP machine with access to a PP oracle; Mundhenk et al. [MGLA00] identified NP PP -complete problems related to finite-horizon Markov decision processes.
Beneficial Merging and Splitting is PP-Hard
In this section we prove that beneficial merging and splitting is PP-hard, and we provide matching upper bounds for beneficial merging and splitting (for any fixed number of false identities) both for the Shapley-Shubik and the probabilistic Banzhaf index. We start with the latter.
The Probabilistic Banzhaf Power Index
We will use the following result due to Faliszewski and Hemaspaandra [FH09, Lemma 2.3].
Lemma 3.1 (Faliszewski and Hemaspaandra [FH09])
Let F be a #P-parsimonious-complete
function. The problem COMPARE-F = {(x, y) | F (x) > F(y)} is PP-complete.
The well-known NP-complete problem SUBSETSUM (which is a special variant of the KNAP-SACK problem) asks, given a sequence (a 1 , . . . , a n ) of positive integers and a positive integer q,
do there exist x 1 , . . . , x n ∈ {0, 1} such that ∑ n i=1 x i a i = q?
It is known that #SUBSETSUM is #Pparsimonious-complete (see, e.g., the textbook by [Pap95] for parsimonious reductions from #3-SAT via #EXACTCOVERBY3-SETS to #SUBSETSUM). Hence, by Lemma 3.1, we have the following.
Corollary 3.2 COMPARE-#SUBSETSUM is PP-complete.
Our goal is to ≤ p m -reduce COMPARE-#SUBSETSUM to Banzhaf-BENE-FICIALMERGE. However, to make this reduction work, it will be useful to consider two restricted variants of COMPARE-#SUBSETSUM, which we denote by COMPARE-#SUBSET-SUM-R and COMPARE-#SUBSETSUM-RR, show their PP-hardness, and then reduce COMPARE-#SUBSETSUM-RR to the problem Banzhaf-BENEFICIALMERGE. This will be done in Lemmas 3.3 and 3.4 and in Theorem 3.5. In all restricted variants of COMPARE-#SUBSETSUM we may assume, without loss of generality, that the target value q in the related #SUBSETSUM instances ((a 1 , . . . , a n ), q) satisfies 1 ≤ q ≤ α − 1, where α = ∑ n i=1 a i , such that #SUBSETSUM remains #P-parsimonious-complete.
COMPARE-#SUBSETSUM-R
Given:
A sequence A = (a 1 , . . . , a n ) of positive integers and two positive integers q 1 and q 2 with 1 ≤ q 1 , q 2 ≤ α − 1, where α = ∑ n i=1 a i . Question: Is the number of subsequences of A summing up to q 1 greater than the number of subsequences of A summing up to q 2 , that is, does it hold that #SUBSETSUM((a 1 , . . . , a n ), q 1 ) > #SUBSETSUM((a 1 , . . . , a n ), q 2 )?
Lemma 3.3 COMPARE-#SUBSETSUM ≤ p m COMPARE-#SUBSETSUM-R. PROOF. Given an instance (X ,Y ) of COMPARE-#SUBSETSUM, X = ((x 1 , . . . , x m ), q x ) and Y = ((y 1 , . . . , y n ), q y ), construct a COMPARE-#SUBSETSUM-R instance (A, q 1 , q 2 ) as follows. Let α = ∑ m i=1 x i and define A = (x 1 , . .
. , x m , 2αy 1 , . . . , 2αy n ), q 1 = q x , and q 2 = 2αq y . This construction can obviously be achieved in polynomial time.
It holds that integers from A can only sum up to q x < α −1 if they do not contain multiples of 2α, thus #SUBSETSUM(A, q 1 ) = #SUBSETSUM((x 1 , . . . , x m ), q x ). On the other hand, q 2 can only be obtained by multiples of 2α, since ∑ m i=1 x i = α is too small. Thus, it holds that #SUBSETSUM(A, q 2 ) = #SUBSETSUM((y 1 , . . . , y n ), q y ). It follows that (X ,Y ) is in COMPARE-#SUBSETSUM if and only if (A, q 1 , q 2 ) is in COMPARE-#SUBSETSUM-R. ❑
In order to perform the next step, we need to ensure that all integers in a COMPARE-#SUB-SETSUM-R instance are divisible by 8. This can easily be achieved, since for a given instance ((a 1 , . . . , a n ), q 1 , q 2 ), we can multiply each integer by 8, obtaining ((8a 1 , . . . , 8a n ), 8q 1 , 8q 2 ) without changing the number of solutions for both related SUBSETSUM instances. Thus, from now on, without loss of generality, we assume that for a given COMPARE-#SUBSET-SUM-R instance ((a 1 , . . . , a n ), q 1 , q 2 ), it holds that a i , q j ≡ 0 mod 8 for 1 ≤ i ≤ n and j ∈ {1, 2}. Now, we consider our even more restricted variant of this problem.
COMPARE-#SUBSETSUM-RR
Given:
A sequence A = (a 1 , . . . , a n ) of positive integers. Question: Is the number of subsequences of A summing up to ( α /2) − 2 greater than the number of subsequences of A summing up to ( α /2) − 1, i.e., #SUBSETSUM((a 1 , . . . , a n ), ( α /2) − 2) > #SUBSETSUM((a 1 , . . . , a n ),
( α /2) − 1), where α = ∑ n i=1 a i ? Lemma 3.4 COMPARE-#SUBSETSUM-R ≤ p m COMPARE-#SUBSETSUM-RR.
PROOF. Given an instance (A, q 1 , q 2 ) of COMPARE-#SUBSETSUM-R, where we assume that A = (a 1 , . . . , a n ), q 1 , and q 2 satisfy a i , q j ≡ 0 mod 8 for 1 ≤ i ≤ n and j ∈ {1, 2}, we construct an instance B of COMPARE-#SUBSETSUM-RR as follows. (This reduction is inspired by the standard reduction from SUBSETSUM to PARTITION due to Karp [Kar72].) Letting α = ∑ n i=1 a i , define B = (a 1 , . . . , a n , 2α − q 1 , 2α + 1 − q 2 , 2α + 3 + q 1 + q 2 , 3α).
This instance can obviously be constructed in polynomial time. Observe that
T = n ∑ i=1
a i + (2α − q 1 ) + (2α + 1 − q 2 ) + (2α + 3 + q 1 + q 2 ) + 3α = 10α + 4, and therefore, ( T /2) − 2 = 5α and ( T /2) − 1 = 5α + 1. We show that (A, q 1 , q 2 ) is in COMPARE-#SUBSETSUM-R if and only if the constructed instance B is in COMPARE-#SUBSETSUM-RR.
First, we examine which subsequences of B sum up to 5α. Consider the following cases: If 3α is added, 2α + 3 + q 1 + q 2 cannot be added, as it would be too large. Also, 2α + 1 − q 2 cannot be added, leading to an odd sum. So, 2α − q 1 has to be added, as the remaining α are too small. Since 3α + 2α − q 1 = 5α − q 1 , 5α can be achieved by adding some a i 's if and only if there exists a subset A ′ ⊆ {1, . . . , n} such that ∑ i∈A ′ a i = q 1 (i.e., A ′ is a solution of the SUBSETSUM instance (A, q 1 )). If 3α is not added, but 2α + 3 + q 1 + q 2 , an even number can only be achieved by adding 2α + 1 − q 1 such that α − 4 − q 1 remain. So, 2α − q 1 is too large, while no subsequence of A sums up to α − 4 − q 1 , because of the assumption of divisibility by 8. If neither 3α nor 2α + 3 + q 1 + q 2 are added, the remaining 5α + 1 − q 1 − q 2 are too small. Thus, the only possibility to obtain 5α is to find a subsequence of A adding up to q 1 . Thus, #SUBSETSUM(A, q 1 ) = #SUBSETSUM(B, 5α).
Second, for similar reasons, a sum of 5α + 1 can only be achieved by adding 3α + (2α + 1 − q 2 ) and a term ∑ i∈A ′ a i , where A ′ is a subset of {1, . . . , n} such that ∑ i∈A ′ a i = q 2 . Hence, #SUBSETSUM(A, q 2 ) = #SUBSETSUM(B, 5α + 1).
Thus, the relation #SUBSETSUM(A, q 1 ) > #SUBSETSUM(A, q 2 ) holds if and only if #SUBtextscsetsum(B, 5α) > #SUBSETSUM(B, 5α + 1), which completes the proof. ❑
We now are ready to prove the main theorem of this section.
i = α − 1 = 1 2 n+3 A ′ ⊆ N ∑ i∈A ′ 2a i = α − 1 + 3 · A ′ ⊆ N 1 + ∑ i∈A ′ 2a i = α − 1 (2) +3 · A ′ ⊆ N 2 + ∑ i∈A ′ 2a i = α − 1 + A ′ ⊆ N 3 + ∑ i∈A ′ 2a i = α − 1 (3) = 1 2 n+3 3 · A ′ ⊆ N ∑ i∈A ′ 2a i = α − 2 + A ′ ⊆ N ∑ i∈A ′ 2a i = α − 4 ,
since the 2a i 's can only add up to an even number. The first of the four sets in (2) and (3) refers to those coalitions that do not contain any of the players n + 1, n + 3, and n + 4; the second, third, and fourth set in (2) and (3) refers to those coalitions containing either one, two, or three of them, respectively. Since they all have the same weight, players n+ 3 and n+ 4 have the same probabilistic Banzhaf index as player n + 2. Furthermore, the new game after merging is G &{n+2,n+3,n+4} = (3, 2a 1 , . . . 2a n , 1; α) and, similarly to above, the Banzhaf index of the first player is calculated as follows:
Banzhaf G &{n+2,n+3,n+4} , 1 = 1 2 n+1 C ⊆ {2, . . . , n + 2} ∑ i∈C w i = α − 1 = 1 2 n+1 A ′ ⊆ N ∑ i∈A ′ 2a i ∈ {α − 3, α − 2, α − 1} + A ′ ⊆ N 1 + ∑ i∈A ′ 2a i ∈ {α − 3, α − 2, α − 1} = 1 2 n+1 2 · A ′ ⊆ N ∑ i∈A ′ 2a i = α − 2 + A ′ ⊆ N ∑ i∈A ′ 2a i = α − 4 .
Altogether, it holds that
Banzhaf G &{n+2,n+3,n+4} , 1 − ∑ i∈{n+2,n+3,n+4} Banzhaf(G , i) = 1 2 n+1 2 · A ′ ⊆ N ∑ i∈A ′ 2a i = α − 2 + A ′ ⊆ N ∑ i∈A ′ 2a i = α − 4 − 3 2 n+3 3 · A ′ ⊆ N ∑ i∈A ′ 2a i = α − 2 + A ′ ⊆ N ∑ i∈A ′ 2a i = α − 4 = 1 2 n+1 · 2 − 3 2 n+3 · 3 A ′ ⊆ N ∑ i∈A ′ 2a i = α − 2 + 1 2 n+1 − 3 2 n+3 A ′ ⊆ N ∑ i∈A ′ 2a i = α − 4 = − 1 2 n+3 · A ′ ⊆ N ∑ i∈A ′ a i = α 2 − 1 + 1 2 n+3 · A ′ ⊆ N ∑ i∈A ′ a i = α 2 − 2 ,
which is greater than zero if and only if
{A ′ ⊆ N | ∑ i∈A ′ a i = ( α /2) − 2} is greater than {A ′ ⊆ N | ∑ i∈A ′ a i = ( α /2) − 1} ,
which in turn is the case if and only if the original instance A is in COMPARE-#SUBSETSUM-RR. ❑
It is known (see [RR10]) that both the beneficial merging problem for a coalition S of size 2 and the beneficial splitting problem for m = 2 false identities can trivially be decided in polynomial time for the probabilistic Banzhaf index, since the sum of power (in terms of this index) of two players is always equal to the power of the player that is obtained by merging these two players. Analogously to the proof of Theorem 3.5, it can be shown that the beneficial splitting problem for a fixed number of at least three false identities is PP-complete.
Since we allow players with zero weight, we need another simple fact for the analysis of the beneficial splitting problem to be used in the proofs of Theorems 3.7 and 3.10.
Lemma 3.6 For both the probabilistic Banzhaf index and the Shapley-Shubik index, given a weighted voting game, adding a player with weight zero does not change the original players' power indices, and the new player's power index is zero.
The proof of Lemma 3.6 is straightforward and therefore omitted. We are now ready to prove Theorem 3.7, which states that Banzhaf-m-BENEFICIALSPLIT is PP-complete for each m ≥ 3.
Theorem 3.7 Banzhaf-m-BENEFICIALSPLIT is PP-complete for each m ≥ 3.
PROOF. As already mentioned in the proof of Theorem 3.5, comparing the values of two #P functions on two (possibly different) inputs reduces to a PP-complete problem and thus is is in PP. In particular, this is true for the problem of comparing (sums of) probabilistic Banzhaf indices in possibly different WVGs, such as testing whether the sum of the new players' raw Banzhaf indices is greater than 2 m−1 times the raw Banzhaf index of the original player i (which is equivalent to
"∑ m j=1 Banzhaf(G i÷m , i j ) > Banzhaf(G , i)" from the definition of Banzhaf-m-BENE-FICIALSPLIT), where i is split into m new players i 1 , . . . , i m .
The main difference between the beneficial merging and splitting problems is that before comparing the two #P functions associated with beneficial splitting, one has to choose a right way of distributing i's weight among the m false identities of i. Since m is fixed, there are only polynomially many (specifically, some number in O(w m i )) ways of doing so, i.e., of finding nonnegative integers w i 1 , . . . , w i m satisfying ∑ m j=1 w i j = w i . Thus, this comparison can be done in PP for each such weight distribution. As PP is closed under union, Banzhaf-m-BENEFICIALSPLIT is in PP.
In order to show PP-hardness for Banzhaf-3-BENEFICIALSPLIT, we use the same techniques as in Theorem 3.5, appropriately modified.
First, we slightly change the definition of COMPARE-#SUBSETSUM-RR by switching ( α /2) − 2 and ( α /2) − 1. The problem (call it COMPARE-#SUBSETSUM-R R ) of whether the number of subsequences of a given sequence A of positive integers summing up to ( α /2) − 1 is greater than the number of subsequences of A summing up to ( α /2) − 2, is PP-hard by the same proof as in Lemma 3.4 with the roles of q 1 and q 2 exchanged. Now, we reduce this problem to Banzhaf-3-BENEFICIALSPLIT by constructing the following instance of the beneficial splitting problem from an instance A = (a 1 , . . . , a n ) of the problem COM-PARE-#SUBSETSUM-R R . Let G = (2a 1 , . . . , 2a n , 1, 3; α), where α = ∑ n j=1 a j , and let i = n + 2 be the player to be split. G is (apart from the order of players) equivalent to the game obtained by merging in the proof of Theorem 3.5. Thus, letting N = {1, . . . , n}, Banzhaf(G , n + 2) equals
1 2 n+1 2 · A ′ ⊆ N ∑ j∈A ′ 2a j = α − 2 + A ′ ⊆ N ∑ j∈A ′ 2a j = α − 4 .
Allowing players with weight zero, there are different possibilities to split player n + 2 into three players. By Lemma 3.6, splitting n + 2 into one player with weight 3 and two others with weight 0 is not beneficial. Likewise, splitting n + 2 into two players with weights 1 and 2 and one player with weight 0 is not beneficial, by Lemma 3.6 and since splitting into two players is not beneficial (by the remark above Theorem 3.7). Thus, the only possibility left is splitting n + 2 into three players of weight 1 each. This corresponds to the original game in the proof of Theorem 3.5, G i÷3 = (2a 1 , . . . , 2a n , 1, 1, 1, 1; α). Therefore,
Banzhaf(G i÷3 , n + 2) = Banzhaf(G i÷3 , n + 3) = Banzhaf(G i÷3 , n + 4) = 1 2 n+3 3 · A ′ ⊆ N ∑ j∈A ′ 2a j = α − 2 + A ′ ⊆ N ∑ j∈A ′ 2a j = α − 4 .
Altogether, as in the proof of Theorem 3.5, the sum of the three new players' probabilistic Banzhaf indices minus the probabilistic Banzhaf index of the original player is greater than zero if and only if
A ′ ⊆ N ∑ j∈A ′ a j = ( α /2) − 1 > A ′ ⊆ N ∑ j∈A ′ a j = ( α /2) − 2 ,
which is true if and only if A is in COMPARE-#SUBSETSUM-R R . This result can be expanded to all m ≥ 3 by splitting into additional players with weight 0. More precisely, if m > 3, we consider the same game G as above and split into three players of weight 1 each and m − 3 players of weight 0 each. By Lemma 3.6, the sum of the m new players' Banzhaf power is equal to the combined Banzhaf power of the three players. Thus, PP-hardness holds by the same arguments as above. ❑
On the other hand, a PP upper bound for the general beneficial splitting problem cannot be shown in any straightforward way. Here, we can only show membership in NP PP , and we conjecture that this problem is even complete for this class. PROOF. With m being part of the input, there are exponentially many possibilities to distribute the split player's weight to her false identities. Nondeterministically guessing such a distribution and then, for each distribution guessed, asking a PP oracle to check in polynomial time whether their combined Banzhaf power in the new game is greater than the original player's Banzhaf power in the original game, shows that Banzhaf-BENEFICIALSPLIT is in NP PP .
Since Banzhaf-3-BENEFICIALSPLIT is a special variant of the general problem Banzhaf-BENE-FICIALSPLIT, PP-hardness is implied immediately by Theorem 3.7. ❑
The Shapley-Shubik Power Index
In order to prove PP-hardness for the merging and splitting problems with respect to the Shapley-Shubik index, we need to take a further step back. EXACTCOVERBY3-SETS (X3C, for short) is another well-known NP-complete decision problem: Given a set B of size 3k and a family S of subsets of B that have size three each, does there exist a subfamily S ′ of S such that B is exactly covered by S ′ ? Theorem 3.9 ShapleyShubik-BENEFICIALMERGE is PP-complete, even if only two players of equal weight merge.
PROOF. The PP upper bound, which has already been observed for two players in [FH09], can be shown analogously to the proof of Theorem 3.5.
For proving the lower bound, observe that the size of a coalition a player is pivotal for is crucial for determining the player's Shapley-Shubik index. Pursuing the techniques of Faliszewski and Hemaspaandra [FH09], we examine the problem COMPARE-#X3C, which is PP-complete by Lemma 3.1. We will apply useful properties of X3C instances shown by Faliszewski and Hemaspaandra [FH09, Lemma 2.7]: Every X3C instance (B ′ , S ′ ) can be transformed into an X3C instance (B, S ), where B = 3k and S = n, that satisfies k /n = 2 /3 without changing the number of solutions, i.e., #X3C(B, S ) = #X3C(B ′ , S ′ ). Now, by the properties of the standard reduction from X3C to SUBSETSUM (which in particular preserves the number of solutions, i.e., #X3C parsimoniously reduces to #SUBSETSUM, as well as the "input size" n and the "solution size" k), we can assume that in a given COMPARE-#SUBSETSUM instance each subsequence summing up to the given integer q is of size 2n /3. Following the track of the reductions from COMPARE-#SUBSETSUM via COMPARE-#SUBSETSUM-R to COMPARE-#SUBSETSUM-RR in Lemmas 3.3 and 3.4, a solution A ′ ⊆ {1, . . . , n} to a given instance A = (a 1 , . . . , a n ) of the latter problem (A ′ satisfying either ∑ i∈A ′ a i = ( α /2) − 2 or ∑ i∈A ′ a i = ( α /2) − 1, where α = ∑ n i=1 a i ) can be assumed to satisfy A ′ = k = (n+2) /3. Under this assumption, we show PP-hardness of ShapleyShubik-BENEFICIALMERGE via a reduction from COMPARE-#SUBSETSUM-RR. Given such an instance, we construct the WVG G = (a 1 , . . . , a n , 1, 1; α /2) and consider coalition S = {n + 1, n + 2}. Define X = #SUBSETSUM(A, ( α /2) − 1) and Y = #SUBSETSUM(A, ( α /2) − 2). Letting N = {1, . . . , n}, it holds that ShapleyShubik(G , n + 1) = ShapleyShubik(G , n + 2)
= 1 (n + 2)! ∑ C⊆N such that ∑ i∈C a i =( α /2)−1 C !(n + 1 − C )! + ∑ C⊆N such that ∑ i∈C a i =( α /2)−2 ( C + 1)!(n − C )! = 1 (n + 2)! (X · k!(n + 1 − k)! +Y · (k + 1)!(n − k)!) .
Merging the players in S, we obtain G &S = (2, a 1 , . . . , a n ; α /2). The Shapley-Shubik index of the new player in G &S is
ShapleyShubik(G &S , 1) = 1 (n + 1)! ∑ C⊆N such that ∑ i∈C a i ∈ {( α /2)−1,( α /2)−2} C !(n − C )! = 1 (n + 1)! (X +Y ) · (k + 1)!(n − k)!.
All in all, ShapleyShubik(G &S , 1) − (ShapleyShubik(G , n + 1) + ShapleyShubik(G , n + 2)) = (X +Y ) · (k + 1)!(n − k)! (n + 1)! − 2 (X · k!(n + 1 − k)! +Y · (k + 1)!(n − k)!) (n + 2)! = k!(n − k)! (n + 2)! (n − 2k)(−X +Y ).
Since we assumed that k = (n+2) /3 and we can also assume that n > 4 (because we added four integers in the construction in the proof of Lemma 3.4), it holds that
n − 2k = n − 4 3 > 0.
Thus the term (4) is greater than zero if and only if Y is greater than X , which is true if and only if A is in COMPARE-#SUBSETSUM-RR. ❑ Theorem 3.10 ShapleyShubik-m-BENEFICIALSPLIT is PP-complete for each m ≥ 2.
PROOF. PP membership can be shown analogously to the PP upper bound in the proof of Theorem 3.7. PP-hardness can also be shown analogously to the proof of Theorem 3.7, appropriately modified to use the arguments from the proof of Theorem 3.9 instead of those from the proof of Theorem 3.5. ❑ Theorem 3.11 ShapleyShubik-BENEFICIALSPLIT is PP-hard and belongs to NP PP .
PROOF. The upper bound of NP PP holds due to analogous arguments as in the proof of Theorem 3.8. Also, analogously to the proof of Theorem 3.8, since ShapleyShubik-2-BENEFICIAL-Split is a special variant of the general ShapleyShubik-BENEFICIAL-SPLIT problem, PP-hardness is implied immediately by Theorem 3.10. ❑
Conclusions and Open Questions
Solving previous conjectures in the affirmative, we have pinpointed the precise complexity of the beneficial merging problem in weighted voting games for the Shapley-Shubik and the probabilistic Banzhaf index by showing that it is PP-complete. We have obtained the same result for beneficial splitting (a.k.a. false-name manipulation) whenever the number of false identities a player splits into is fixed. For an unbounded number of false identities, we raised the known lower bound from NP-hardness to PP-hardness and showed that it is contained in NP PP . For this problem, it remains open whether it can be shown to be complete for NP PP , a huge complexity class that by Toda's theorem [Tod91] contains the entire polynomial hierarchy. NP PP is an interesting class, but somewhat sparse in natural complete problems. The only (natural) NP PP -completeness results we are aware of are due to Littman et al. [LGM98], who analyze a variant of the satisfiability problem and questions related to probabilistic planning, and due to Mundhenk et al. [MGLA00], who study problems related to finite-horizon Markov decision processes. Another interesting open question is whether our results can be transferred also to the beneficial merging and splitting problems for the normalized Banzhaf index. Finally, it would be interesting to know to which classes of simple games, other than weighted voting games, our results can be extended.
from COM-PARE-#SUBSETSUM-RR, which is PP-hard by Corollary 3.2 via Lemmas 3.3 and 3.4. Our construction is inspired by the NP-hardness results by Aziz et al.[AP09] and Rey and Rothe[RR10].Given an instance A = (a 1 , . . . , a n ) of COMPARE-#SUBSETSUM-RR, construct the following instance for Banzhaf-BENEFICIALMERGE. Let α = ∑ n i=1 a i . Define the WVG G =(2a 1 , . . . , 2a n , 1, 1, 1, 1; α), and let the merging coalition be S = {n + 2, n + 3, n + 4}. Letting N = {1, . . . , n}, it holds that Banzhaf(G , n + 2) = 1 2 n+3 C ⊆ {1, . . . , n + 1, n + 3, n + 4} ∑ i∈C w
Theorem 3. 8
8Banzhaf-BENEFICIALSPLIT is PP-hard and belongs to NP PP .
We show PP-hardness of Banzhaf-BENEFICIALMERGE by means of a ≤Theorem 3.5 Banzhaf-BENEFICIALMERGE is PP-complete, even if only three players of equal
weight merge.
PROOF. Membership of Banzhaf-BENEFICIALMERGE in PP has already been observed in [RR10,
Theorem 3]. It follows from the fact that the raw Banzhaf index is in #P and that #P is closed
under addition and multiplication by two, 3 and, furthermore, since comparing the values of two
#P functions on two (possibly different) inputs reduces to a PP-complete problem. This technique
(which was proposed by Faliszewski and Hemaspaandra [FH09] and applies their Lemma 2.10)
works, since PP is closed under ≤
p
m -reducibility.
They also note that the same arguments cannot be transferred immediately to the corresponding problem for the normalized Banzhaf index.
We omit the empty coalition, since this would slightly change the idea of the problem.
Again, note that this idea cannot be transferred straightforwardly to the normalized Banzhaf index, since in different games the indices have possibly different denominators, not only different by a factor of some power of two, as is the case for the probabilistic Banzhaf index.
Acknowledgments:We thank Haris Aziz for interesting discussions. This work has been supported in part by DFG grant RO-1202/14-1.
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| []
|
[
"POSITIVSTELLENSATZË FOR NONCOMMUTATIVE RATIONAL EXPRESSIONS",
"POSITIVSTELLENSATZË FOR NONCOMMUTATIVE RATIONAL EXPRESSIONS"
]
| [
"J E Pascoe "
]
| []
| []
| We derive some Positivstellensatzë for noncommutative rational expressions from the Positivstellensatzë for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-that is, we obtain a so-called "perfect Positivstellensatz" on convex sets. | 10.1090/proc/13773 | [
"https://arxiv.org/pdf/1703.06951v1.pdf"
]
| 119,586,405 | 1703.06951 | f6b9fbe3463048c5906843c4dcb128d37d43ce28 |
POSITIVSTELLENSATZË FOR NONCOMMUTATIVE RATIONAL EXPRESSIONS
20 Mar 2017
J E Pascoe
POSITIVSTELLENSATZË FOR NONCOMMUTATIVE RATIONAL EXPRESSIONS
20 Mar 2017arXiv:1703.06951v1 [math.FA]
We derive some Positivstellensatzë for noncommutative rational expressions from the Positivstellensatzë for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-that is, we obtain a so-called "perfect Positivstellensatz" on convex sets.
Introduction
We consider the positivity of noncommutative rational functions on polynomially convex sets. The theory on positive noncommutative polynomials has been well-studied [3,6,4], essentially inspired by the operator theoretic methods from the theory of positive (commutative) polynomials on polynomially convex sets originating in the work [10,9]. We note that going from the polynomial to the rational case is less clear than in the noncommutative case because we cannot "clear denominators," as it were.
A noncommutative polynomial (over C) in d-variables is an element of the free associative algebra over C in the noncommuting letters x 1 , . . . , x d . For example 1000x 1 x 2 x 1 − x 2 2 and x 2 1 + x 1 x 2 are noncommutative polynomials in two variables. A matricial noncommutative polynomial is a matrix with noncommutative polynomial entries. For example, is a matricial noncommutative polynomial. We define an involution * on matricial noncommutative polynomials to be the involution which treats each x i as a self-adjoint variable. For example,
7i 1000x 1 x 2 x 1 −x 2 2 x 2 1 +x 1 x7i 1000x 1 x 2 x 1 −x 2 2 x 2 1 +x 1 x 2 0 * = −7i x 2 1 +x 2 x 1 1000x 1 x 2 x 1 −x 2 2 0 .
We say a collection P of square matricial noncommutative polynomials is Archimedian if P contains elements of the form C i − x 2 i for some real numbers C i and each element of P is self-adjoint.
Let H be the infinite dimensional separable Hilbert space. For a selfadjoint operator T , we say T ≥ 0 if T is positive semidefinite, we say T > 0 if T is strictly positive definite in the sense that the spectrum of T is contained in (0, ∞). We define Helton and McCullough showed the following Positivstellensatz for matricial noncommutative polynomials. [6]). Let P be an Archimedian collection of matricial noncommutative polynomials. Let q be a square matricial noncommutative polynomial. If q > 0 on D P , then
D P = {X ∈ B(H) d |p(X) ≥ 0, ∀p ∈ P, X i = X * i }. Previously,Theorem 1.1 (Helton, McCulloughq = finite s * i s i + finite r * j p j r j
where s i , r j are all matricial noncommutative polynomials and p j ∈ P.
The rational positivstellensatz
A noncommutative rational expression is a syntactically correct expression involving +, (, ), −1 the letters x 1 , . . . , x d and scalar numbers. We say two nondegenerate expressions are equivalent if they agree on the intersection of their domains. (Nondegeneracy means that the expression is defined for at least one input, or equivalently that the domain is a dense set with interior. That is, examples such as 0 −1 are disallowed.) Examples of noncommutative rational expressions include
1, x 1 x −1 1 , 1 + x 2 (8x 3 1 x 2 x 1 + 8) −1 .
We note that the first two are equivalent.
A matricial noncommutative rational expression is a matrix with noncommutative rational expression entries.
We show the following theorem.
Theorem 2.1. Let P be an Archimedian collection of noncommutative polynomials. Let q be a square matricial noncommutative rational expression defined on all of D P . If the noncommutative rational expression q > 0 on D P , then
q ≡ finite s * i s i + finite r * j p j r j (2.1)
where s i , r j are all matricial noncommutative rational expressions defined on D P and p j ∈ P.
Proof. We let g j (x) be such that the term g j (x) −1 occurs in q. The proof will go by strong induction on the number of such terms. Define
O = P∪{±[1−u j g j (x)] * [1−u j g j (x)], ±[1−g j (x)u j ] * [1−g j (x)u j ]}∪{D j −u * j u j } where D j are positive real scalars chosen to be large enough so that D j − [g j (x) −1 ] * g j (x) −1 is positive on D P .
We now define a self-adjoint noncommutative polynomialq(x, u) so thatq(x, g) = q(x). Nowq is a noncommutative polynomial in terms of x i and u j . Moreover, in terms of the x i and u j , we see that q(x, u) is positive on D O , so by Theorem 1.1,
q = s * i s i + r * j o j r j for some o j ∈ O.
We now analyze each term of the form t j = r * j o j r j . We need to show that t j (x, g) is of the form (2.1). If o j ∈ P, we are fine. If o j = ±[1 − u j g j (x)] * [1 − u j g j (x)], we are also fine, since t j (x, g) = 0, and similarly for the reversed case.
If o j = D j − [u j ] * u j we note that o j (x, g) = D j −[g j (x) −1 ] * g j (x) −1 = [g j (x) −1 ] * [D j g j (x) * g j (x)−1]g j (x) −1 ,
and since D j g j (x) * g j (x) − 1 > 0 on D P , by induction it is of the form (2.1), so we are done.
We note that the same proof can be adapted for the hereditary case in [6]. Moreover, we note that this implies the Agler model theory for rational functions on polynomially convex sets established variously in [2,1].
The convex perfect rational positivstellensatz
It is important to note that in Theorem 1.1 and Theorem 2.1, the complexity of the sum of squares representation is unbounded and we needed strict inequality. Specifically, in (2.1), the number of terms in each sum and the degree of each s i and r j are not bounded in the statement of the theorem. However, Helton, Klep and McCullough [4] showed that bounds do exist when we additionally assume that D P is convex and contains 0 and moreover that P consists of a single monic linear pencil, L, a self-adjoint linear matrix polynomial such that L(0) is the identity. We note that for any finite set P of noncommutative polynomials such that D P is convex and contains 0, there exists such an L [7].
Our goal is to prove the following: Proof. Given an expression r(x), we consider the expressionr(x, u) where each g j (x) −1 occurring in r has been replaced by u j as in the proof of Theorem 2.1. First we consider the minimal set C r of rational expressions such that:
(
1) ab ∈ C r ⇒ b ∈ C r ,(2)(a + b)c ∈ C r ⇒ ac ∈ C r , bc ∈ C r , (3) a + b ∈ C ⇒ a ∈ C r , b ∈ C r , (4) a −1 b ∈ C r ⇒ aa −1 b ∈ C r .
From C r , form a setC r by replacing each occurence of g j (x) −1 in elements of C r with a new symbol u j . We define the set of M r to be
M r = {g j (x)u j b − b|g j (x)u j b ∈C r }. Define Z r = {(X, U, v)|m(X, U)v = 0, m ∈ M r , L(X) ≥ 0}.
We note that for (X, U, v) ∈ Z r andã(x, u) ∈C r , one can show we have thatã(X, U)v =ã(x, g(X) −1 )v via a recursive argument. We see that
r(x, u) satisfies r(X)v, v = r(X, U)v, v ≥ 0,
on Z r sincer(X, U)v = r(X)v on Z r by construction. Now, we apply the Helton-Klep-Nelson convex Positivstellensatz[5, Theorem 1.9], where the variety is given by Z r and the convex set is {(X, U)|L(X) ≥ 0}, to get that:
r(x, u) = s * is i + r * j Lr j + ι * k m k + m * k ι k
where each ι k is in the real radical of the ideal generated by the elements of M r . That is, each ι k (X, U)v vanishes on Z r . So, substituting g j (x) −1 for u j we get that
r(x) ≡ s * i s i + r * j Lr j .
We note that we could have proved a bit more: that on the variety Z r thatr is positive and given by a sum of squares. This would essentially correspond to the so-called Moore-Penrose evaluation in [8]. Moreover, we note that the main result on positive rational functions, the noncommutative analogue of Artin's solution to Hilbert's seventeenth problem, that regular positive rational expressions are sums of squares [8], follows from our present theorem by taking an empty monic linear pencil, in fact, we obtain a slightly better matricial version of that result. Moreover, one has size bounds inherited from the Helton-Klep-Nelson convex Positivstellensatz [5], that is, checking that a noncommutative rational expression is effective using the algorithms given in [5].
Theorem 3. 1 .
1Let L be a monic linear pencil. Suppose D {L} is convex. Let r be a square matricial noncommutative rational expression defined on all of D {L} . The noncommutative rational expression r ≥ 0 on all of D {L} if and only if s i , r j are all matricial noncommutative rational expressions defined on all of D {L} .
2 0
2Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary 13J30, 16K40, 47L07; Secondary 15A22, 26C15, 47A63. Key words and phrases. Noncommutative rational function, positive rational function, Hilberts 17th problem, noncommutative Positivstellensatz. Research supported by NSF Mathematical Science Postdoctoral Research Fellowship DMS 1606260.
Global holomorphic functions in several noncommuting variables. J Agler, J E Carthy, Canad. J. Math. 67J. Agler and J.E. M c Carthy. Global holomorphic functions in several noncom- muting variables. Canad. J. Math., 67:241-285, 2015.
Conservative Structured Noncommutative Multidimensional Linear Systems. Joseph A Ball, Gilbert Groenewald, Tanit Malakorn, Birkhäuser Basel, BaselJoseph A. Ball, Gilbert Groenewald, and Tanit Malakorn. Conservative Structured Noncommutative Multidimensional Linear Systems, pages 179-223. Birkhäuser Basel, Basel, 2006.
Positive noncommuative polynomials are sums of squares. J W Helton, Ann. of Math. 1562J. W. Helton. Positive noncommuative polynomials are sums of squares. Ann. of Math., 156(2):675-694, 2002.
The convex Positivstellensatz in a free algebra. J W Helton, I Kelp, S Mccullough, Adv. Math. 231J.W. Helton, I. Kelp, and S. McCullough. The convex Positivstellensatz in a free algebra. Adv. Math., 231:516-534, 2012.
Noncommutative polynomials nonnegative on a variety intersect a convex set. J W Helton, I Klep, S Mccullough, C Nelson, J. Funct. Anal. 26612J.W. Helton, I. Klep, S. McCullough, and C. Nelson. Noncommutative poly- nomials nonnegative on a variety intersect a convex set. J. Funct. Anal., 266(12):6684-6752, 2014.
A Positivstellensatz for non-commutative polynomials. J W Helton, S Mccullough, Trans. AMS. 356J.W. Helton and S. McCullough. A Positivstellensatz for non-commutative polynomials. Trans. AMS, 356:3721-3737, 2004.
Every convex free basic semi-algebraic set has an LMI representation. J W Helton, S Mccullough, Ann. of Math. 1762J.W. Helton and S. McCullough. Every convex free basic semi-algebraic set has an LMI representation. Ann. of Math., 176(2):979-1013, 2012.
Regular and positive noncommutative rational functions. Igor Klep, James Eldred Pascoe, Jurij Volcic, J. Lond. Math. Soc. to appearIgor Klep, James Eldred Pascoe, and Jurij Volcic. Regular and positive non- commutative rational functions. J. Lond. Math. Soc., 2017. to appear.
Positive polynomials on compact semi-algebraic sets. M Putinar, Indiana Univ. Math. J. 42M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J., 42:969-984, 1993.
The K-moment problem for compact on semi-algebraic sets. K Schmüdgen, Math. Ann. 2892K. Schmüdgen. The K-moment problem for compact on semi-algebraic sets. Math. Ann., 289(2):203-206, 1991.
E-mail address: pascoej@math. Brookings Drive, St. Louis, MO 63130, USACampus Box 1146. wustl.eduBrookings Drive, Campus Box 1146, St. Louis, MO 63130, USA. E-mail address: [email protected]
| []
|
[
"Supplementary Online Material for: Alternative paths to realize Majorana Fermions in Superconductor- Ferromagnet Heterostructures",
"Supplementary Online Material for: Alternative paths to realize Majorana Fermions in Superconductor- Ferromagnet Heterostructures"
]
| [
"G Livanas \nDepartment of Physics\nNational Technical University of Athens\nGR-15780AthensGreece\n",
"G Varelogiannis \nDepartment of Physics\nNational Technical University of Athens\nGR-15780AthensGreece\n",
"M Sigrist \nInstitut für Theoretische Physik\nETH Zürich\n8093ZürichSwitzerland\n",
"\nQUARTETS INVOLVED AND INDUCED FIELDS\n\n"
]
| [
"Department of Physics\nNational Technical University of Athens\nGR-15780AthensGreece",
"Department of Physics\nNational Technical University of Athens\nGR-15780AthensGreece",
"Institut für Theoretische Physik\nETH Zürich\n8093ZürichSwitzerland",
"QUARTETS INVOLVED AND INDUCED FIELDS\n"
]
| []
| In previous work by one of us, it was demonstrated that four fields or order parameters with respective matrix representations for example A, B, C and D that obey the quartet rule: A B C D = ± 1 form a quartet meaning that the presence of any three of them generates the missing fourth one via what was named the quartet rule coupling[1]. Note that the rule holds as well for all cyclic permutations of the matrix representations in the product. The phenomenon of quartet rule coupling has also been proven numerically on dozens of different examples over the last years and there are many cases where quartets extracted from the above rule have been shown to provide significant insight into various puzzling problems of correlated electron systems especially when multiple order parameters or fields are involved[2]. Here we focus on the quartets that are relevant for our Majorana engineering approach illustrating how the phenomenon of quartet rule coupling manifests there.More precisely, in this section we demonstrate how triplet superconducting (SC) p fields and spin-orbit coupling (SOC) are induced when a Zeeman field h and a charge current J couple to a conventional singlet SC field ∆ and a chemical potential µ respectively. For simplicity we consider a 1D system though it is straightforward to extent the same results to systems with arbitrary dimensions and lattice symmetry. The Hamiltonian for the particular fields acquires the following form in coordinate space.where we have introduced the spin dependent Nambu spinor Ψ † i = ψ † i,↑ , ψ † i,↓ , ψ i,↑ , ψ i,↓ and the Pauli matrices τ and σ acting on Nambu and spin space respectively. With d h we denote the vector of the Zeeman field in spin space, i the coordinate along the x-coordinate axis, whileσ = (σ 1 , τ 3 σ 2 , σ 3 ). InTable Iwe demonstrate how the terms of the Hamiltonian transform under the inversion symmetry operation I acting in coordinate space, the time reversal symmetry operation T effected by the anti-unitary operator Θ = iσ 2 K, where K acts as complex-conjugation and the combined symmetry operation R = IT .According to [1], the charge current J and the Zeeman field hd h couple to a chemical potential µ and a conventional SC field ∆ to induce SOC of the form ±iαd h ·σδ j,i±1 and the triplet p SC field ±i∆ p τ 1 d h · iσ 2σ δ j,i±1 respectively. Indeed, matrix representations of those fields A, B, C and D obey the quartet coupling rule A B C D = ± 1 [1] forming the overlapping quartets A and B (see table I) that have common elements the charge current and the Zeeman field. The spin space vector of the induced triplet fields is determined by the Zeeman field while their spatial configuration by the charge current.Regarding the symmetry operations considered above SOC is even under T and odd under I and R. Considering a charge current J, a Zeeman field hd h and a chemical potential µ we observe that their product corresponds to a field with the same behaviour under these three symmetry operations as SOC. Therefore when the above three fields coexist SOC is effectively induced. A triplet SC field has the same transformation properties as SOC. However in addition pairing fields lock the charge conjugation symmetry operation. Therefore triplet SC fields are effectively induced when a charge current J and a Zeeman field hd h coexist with a conventional SC field instead of a chemical potential. Notice that according to [1] every field is equivalent to a product of at least three distinct fields.While the validity of the quartet coupling rule [1] is generic, to illustrate how this coupling emerges in quartets A and B we present analytical results considering a 1D translationally invariant system for the particular case where d h = (1, 0, 0) for the Zeeman field. In momentum space the Hamiltonian acquires the following form | 10.1038/s41598-019-42558-3 | null | 118,648,590 | 1606.05623 | 36fd3b1bafb7b1e3de4c03f3df875f6dd8d9ee56 |
Supplementary Online Material for: Alternative paths to realize Majorana Fermions in Superconductor- Ferromagnet Heterostructures
G Livanas
Department of Physics
National Technical University of Athens
GR-15780AthensGreece
G Varelogiannis
Department of Physics
National Technical University of Athens
GR-15780AthensGreece
M Sigrist
Institut für Theoretische Physik
ETH Zürich
8093ZürichSwitzerland
QUARTETS INVOLVED AND INDUCED FIELDS
Supplementary Online Material for: Alternative paths to realize Majorana Fermions in Superconductor- Ferromagnet Heterostructures
1
In previous work by one of us, it was demonstrated that four fields or order parameters with respective matrix representations for example A, B, C and D that obey the quartet rule: A B C D = ± 1 form a quartet meaning that the presence of any three of them generates the missing fourth one via what was named the quartet rule coupling[1]. Note that the rule holds as well for all cyclic permutations of the matrix representations in the product. The phenomenon of quartet rule coupling has also been proven numerically on dozens of different examples over the last years and there are many cases where quartets extracted from the above rule have been shown to provide significant insight into various puzzling problems of correlated electron systems especially when multiple order parameters or fields are involved[2]. Here we focus on the quartets that are relevant for our Majorana engineering approach illustrating how the phenomenon of quartet rule coupling manifests there.More precisely, in this section we demonstrate how triplet superconducting (SC) p fields and spin-orbit coupling (SOC) are induced when a Zeeman field h and a charge current J couple to a conventional singlet SC field ∆ and a chemical potential µ respectively. For simplicity we consider a 1D system though it is straightforward to extent the same results to systems with arbitrary dimensions and lattice symmetry. The Hamiltonian for the particular fields acquires the following form in coordinate space.where we have introduced the spin dependent Nambu spinor Ψ † i = ψ † i,↑ , ψ † i,↓ , ψ i,↑ , ψ i,↓ and the Pauli matrices τ and σ acting on Nambu and spin space respectively. With d h we denote the vector of the Zeeman field in spin space, i the coordinate along the x-coordinate axis, whileσ = (σ 1 , τ 3 σ 2 , σ 3 ). InTable Iwe demonstrate how the terms of the Hamiltonian transform under the inversion symmetry operation I acting in coordinate space, the time reversal symmetry operation T effected by the anti-unitary operator Θ = iσ 2 K, where K acts as complex-conjugation and the combined symmetry operation R = IT .According to [1], the charge current J and the Zeeman field hd h couple to a chemical potential µ and a conventional SC field ∆ to induce SOC of the form ±iαd h ·σδ j,i±1 and the triplet p SC field ±i∆ p τ 1 d h · iσ 2σ δ j,i±1 respectively. Indeed, matrix representations of those fields A, B, C and D obey the quartet coupling rule A B C D = ± 1 [1] forming the overlapping quartets A and B (see table I) that have common elements the charge current and the Zeeman field. The spin space vector of the induced triplet fields is determined by the Zeeman field while their spatial configuration by the charge current.Regarding the symmetry operations considered above SOC is even under T and odd under I and R. Considering a charge current J, a Zeeman field hd h and a chemical potential µ we observe that their product corresponds to a field with the same behaviour under these three symmetry operations as SOC. Therefore when the above three fields coexist SOC is effectively induced. A triplet SC field has the same transformation properties as SOC. However in addition pairing fields lock the charge conjugation symmetry operation. Therefore triplet SC fields are effectively induced when a charge current J and a Zeeman field hd h coexist with a conventional SC field instead of a chemical potential. Notice that according to [1] every field is equivalent to a product of at least three distinct fields.While the validity of the quartet coupling rule [1] is generic, to illustrate how this coupling emerges in quartets A and B we present analytical results considering a 1D translationally invariant system for the particular case where d h = (1, 0, 0) for the Zeeman field. In momentum space the Hamiltonian acquires the following form
I. QUARTETS INVOLVED AND INDUCED FIELDS
In previous work by one of us, it was demonstrated that four fields or order parameters with respective matrix representations for example A, B, C and D that obey the quartet rule: A B C D = ± 1 form a quartet meaning that the presence of any three of them generates the missing fourth one via what was named the quartet rule coupling [1]. Note that the rule holds as well for all cyclic permutations of the matrix representations in the product. The phenomenon of quartet rule coupling has also been proven numerically on dozens of different examples over the last years and there are many cases where quartets extracted from the above rule have been shown to provide significant insight into various puzzling problems of correlated electron systems especially when multiple order parameters or fields are involved [2]. Here we focus on the quartets that are relevant for our Majorana engineering approach illustrating how the phenomenon of quartet rule coupling manifests there.
More precisely, in this section we demonstrate how triplet superconducting (SC) p fields and spin-orbit coupling (SOC) are induced when a Zeeman field h and a charge current J couple to a conventional singlet SC field ∆ and a chemical potential µ respectively. For simplicity we consider a 1D system though it is straightforward to extent the same results to systems with arbitrary dimensions and lattice symmetry. The Hamiltonian for the particular fields acquires the following form in coordinate space.
H = 1 2 i,j Ψ † i [(µτ 3 + ∆τ 2 σ 2 − hτ 3 d h ·σ)δ i,j + tδ j,i±1 τ 3 ± iJδ j,i±1 ] Ψ j ,(S1)
where we have introduced the spin dependent Nambu spinor Ψ † i = ψ † i,↑ , ψ † i,↓ , ψ i,↑ , ψ i,↓ and the Pauli matrices τ and σ acting on Nambu and spin space respectively. With d h we denote the vector of the Zeeman field in spin space, i the coordinate along the x-coordinate axis, whileσ = (σ 1 , τ 3 σ 2 , σ 3 ). In Table I we demonstrate how the terms of the Hamiltonian transform under the inversion symmetry operation I acting in coordinate space, the time reversal symmetry operation T effected by the anti-unitary operator Θ = iσ 2 K, where K acts as complex-conjugation and the combined symmetry operation R = IT .
According to [1], the charge current J and the Zeeman field hd h couple to a chemical potential µ and a conventional SC field ∆ to induce SOC of the form ±iαd h ·σδ j,i±1 and the triplet p SC field ±i∆ p τ 1 d h · iσ 2σ δ j,i±1 respectively. Indeed, matrix representations of those fields A, B, C and D obey the quartet coupling rule A B C D = ± 1 [1] forming the overlapping quartets A and B (see table I) that have common elements the charge current and the Zeeman field. The spin space vector of the induced triplet fields is determined by the Zeeman field while their spatial configuration by the charge current.
Regarding the symmetry operations considered above SOC is even under T and odd under I and R. Considering a charge current J, a Zeeman field hd h and a chemical potential µ we observe that their product corresponds to a field with the same behaviour under these three symmetry operations as SOC. Therefore when the above three fields coexist SOC is effectively induced. A triplet SC field has the same transformation properties as SOC. However in addition pairing fields lock the charge conjugation symmetry operation. Therefore triplet SC fields are effectively induced when a charge current J and a Zeeman field hd h coexist with a conventional SC field instead of a chemical potential. Notice that according to [1] every field is equivalent to a product of at least three distinct fields.
While the validity of the quartet coupling rule [1] is generic, to illustrate how this coupling emerges in quartets A and B we present analytical results considering a 1D translationally invariant system for the particular case where d h = (1, 0, 0) for the Zeeman field. In momentum space the Hamiltonian acquires the following form Induced spin-orbit coupling by combining a charge current J with a chemical potential µ and a Zeeman field hd h where d h is a vector in spin space. The particular fields transform symmetrically(anitsymmetrically), denoted by +(−), under the symmetry operations of inversion I (i ↔ j), time reversal T (Θ = iσ 2 K) and the combined operation R = IT . In order for spin-orbit coupling to be induced in the presence of a charge current T and R have to break by the application of a chemical potential and a Zeeman field respectively. (right) Quartet B : In the case of induced triplet p SC field the symmetry arguments are the same if instead of the chemical potential we consider a conventional SC field ∆.
Quartet A I T R Charge current ±iJδj,i±1 − − + Zeeman field hτ3d h ·σδj,i + − − Chemical potential µτ3δj,i + + + spin-orbit coupling ±iαd h ·σδj,i±1 − + − Quartet B I T R Charge current ±iJδj,i±1 − − + Zeeman field hτ3d h ·σδj,i + − − Conventional SC ∆τ2σ2δj,i + + + Triplet p SC ±i∆ p τ1d h · iσ2σδj,i±1 − + −H = 1 2 k Ψ † k [(µ + (k))τ 3 + ∆τ 2 σ 2 + J(k) − h x τ 3 σ 1 ] Ψ k ,(S2)
where we have introduced the spinor Ψ † k = c † k,↑ , c † k,↓ , c −k,↑ , c −k,↓ and the Pauli matrices τ and σ acting on Nambu and spin space respectively. We have also added a kinetic term (k) = (−k) present in most realistic models, while for the current term by definition we have J(k) = −J(−k). The equal spin pairing P x and the spin-orbit coupling A x correlations are calculated through the following equations.
P x = 1 2N k,iωn f Px (k)T r{τ 1 σ 3 G(k; iω n )} = 1 2N k,n f Px (k)[U (k) † τ 1 σ 3 U (k) ] nn n F (E n (k)) → P x = 1 N k,m,s f Px (k) [m · s]∆ ∆ 2 + [µ + (k)] 2 n F (E m,s (k)) ,(S3)A x = 1 2N k,iωn f Ax (k)T r{σ 1 G(k; iω n )} = 1 2N k,n f Ax (k)[U (k) † σ 1 U (k) ] nn n F (E n (k)) → A x = k,m,s −[m · s]f Ax (k)(µ + (k)) N ∆ 2 + [µ + (k)] 2 n F (E m,s (k)) (S4)
where N the number of the lattice points of the system or equivalently the number of momenta k within the first Brillouin zone, n=1-4 corresponding to the dimension of the spinor, m, s = ±, G(k; iω n ) the momentum space Matsubara Green functions defined as G(k;
iω n ) = [iω n − H k ] −1 , andÛ (k) † = exp [iπ/4τ 1 σ 2 ] exp [(k) = f Ax (−k), f Px (k) = f Px (−k)
as imposed by the anticommutation rules for fermions. Notice that the induced equal spin pairing correlations and spin-orbit coupling are of the particular representation τ 1 σ 3 and σ 1 respectively and in fact are the only induced correlations, as only for these representationsÔ the matrix [Û (k)ÔÛ (k) † ] acquires non-zero diagonal elements, or equivalently the T r{ÔG(k; iω n )} can be finite. Indeed
we have [Û (k)τ 1 σ 3Û (k) † ] = τ 3 σ 3 , while [Û (k)σ 1Û (k) † ] = −τ 3 σ 3 ,
leading to the final form of the induced correlations for the particular system. From Eq. S3 it is straightforward that P x = 0 when ∆ = 0. However only when also h x = 0 and J = 0, finite P x correlations are induced, since for h x = 0 we get n F (E m,+ (k)) = n F (E m,− (k)) → P x = 0 and for J=0 again we have n F (E m,s (k)) = n F (E m,s (−k)) → P x = 0 due to f Px (k) = −f Px (−k). Based on similar considerations we can show that spin-orbit coupling correlations A(x) become finite only when J = 0, µ = 0 and h x = 0 while the momentum configuration functions f Ax (k) is determined again by the corresponding function f J (k) for the current. In Fig. S1 we demonstrate how P x triplet equal spin pairing correlations and A x spin-orbit coupling are induced for particular values of the singlet pairing field ∆ and the chemical potential µ as the charge current J and the Zeeman field h x increase. Finally we have to remark that the induced correlations and their conjugate fields vanish when the momentum configuration functions f Px (k)(f Ax (k)) for P x (A x ) and f J (k) for the charge current J, belong to different irreducible representations of the corresponding point symmetry group of the system. FIG. S1: a) For singlet pairing field ∆ = 0.5t and chemical potential µ = 0 the induced P x τ 1 σ 3 correlations as the charge current J and the Zeeman field h x increase. The induced P x correlations are normalized with respect to the highest value for this range of parameters. b) For the same parameters and h x = 0.1t the induced P x correlations with respect to charge current J. Notice that the sign of the induced correlations is determined by the direction of the current. c) For chemical potential µ = 0.5t the induced A x σ 1 spin-orbit coupling correlations for increasing values of the charge current J and the Zeeman field h x . d) The induced A x correlations with respect to the chemical potential µ for current J = 0.2t and h x = 0.4t. The sign of the induced A x correlations is determined by the chemical potential.
Using the same symmetry arguments we anticipate triplet odd in time reversal ±i ∆ p τ 2 d p · iσ 2σ δ j,i±x SC fields to emerge in a conventional SC ∆τ 2 σ 2 with SOC ±iαd α ·σδ j,i±x when a Zeeman field hτ 3 d h ·σ is applied. As presented in Table. II SOC α and Zeeman fields h break the inversion I and time reversal T symmetries respectively in order for a triplet p SC field to be induced. In this case the vector d p in spin space of the induced p SC field derives from d p = d h × d α and therefore the three triplet fields must be orthogonal to each other, while the two SC fields acquire a π/2 relative phase. In a 1D translationally invariant system of a conventional SC with SO interactions α x (k) = −α x (−k) under the effect of a Zeeman field h z τ 3 σ 3 , described by the following Hamiltonian
H = 1 2 k Ψ † k [α x (k)σ 1 + ∆τ 2 σ 2 + h z τ 3 σ 3 ] Ψ k ,(S5)
the induced P y τ 2 correlations derive from the following equation
P y = k m,s f py (k)[m · s] α x (k) [∆ + sh z ] 2 + [a x (k)] 2 n F (E m,s (k)) ,(S6)
where E m,s (k) = m [∆ + sh z ] 2 + [α x (k)] 2 the excitation spectrum of the system and f py (k) = −f py (−k) the configuration function of the induced p correlations. From the above equation it is straightforward that P y = 0 when a x = 0, or ∆ = 0 [as n F (E +,s (k)) = n F (E −,s (k))], or h z = 0 [as n F (E m,+ (k)) = n F (E m,− (k))].
In the same way triplet ±i ∆ p τ 2 d p · iσ 2σ δ j,i±x SC fields are anticipated to emerge when a Zeeman field hτ 3 d h ·σ and a chemical potential µ are applied to a triplet ±i∆ p τ 1 d p ·iσ 2σ δ j,i±x SC. The two SC fields acquire a relative phase φ = π/2 and the vector d p in spin space of the induced field derives from d p = d p × d h . Particularly considering a 1D translationally invariant system of a equal spin triplet ∆ p x (k)τ 1 σ 3 SC under the effect of a Zeeman field h z τ 3 σ 3 and a chemical potential µ, described by the following Hamiltonian
H = 1 2 k Ψ † k [∆ p x (k)τ 1 σ 3 + µτ 3 + h z τ 3 σ 3 ] Ψ k ,(S7)
the induced P y τ 1 correlations derive from the following equation
P y = k m,s f py (k)[m · s] ∆ p x (k) [µ + sh z ] 2 + [∆ p x (k)] 2 n F (E m,s (k)) , (S8) where E m,s (k) = m [µ + sh z ] 2 + [p x (k)] 2 the corresponding eigenvalues. Again P y = 0 when ∆ p x = 0, or h z = 0 [as n F (E m,+ (k)) = n F (E m,− (k))], or µ = 0 [as n F (E +,s (k)) = n F (E −,s (k))].
Finally according to the quartet coupling rule of induced fields [1] the singlet pairing field ∆(τ 2 σ 2 ), the potential field µ(τ 3 ), the Zeeman field hτ 3 d h ·σ, spin-orbit interactions ±iαd α ·σδ j,i±x and p SC fields ±i∆ p τ 1 d p · iσ 2σ δ j,i±x , ±i ∆ p τ 2 d p · iσ 2σ δ j,i±x form a closed pattern of coupled fields when
d p = d α × d h = d p × d h and d α = d p .
The pattern is closed in the sense that any subset of these fields will induce a field, if any, which belongs to the above set of six fields. A system with all these fields belongs to the BDI symmetry class as elaborated in Appendix III. II: (left) Quartet C : Triplet p SC field is induced in a conventional SC ∆ with SOC a when a Zeeman field h is applied. The singlet SC field ∆ transforms symmetrically (denoted by +) under the symmetry operations of inversion I (i ↔ j), time reversal T (Θ = iσ 2 K) and the combined operation R = IT . An odd under I and T triplet p field is induced when SOC α and a Zeeman field h break I and T respectively. The induced p SC field acquires a π/2 phase with respect to the ∆ conventional SC field, while its vector in spin space derives from
Quartet C I T R Conventional SC ∆τ2σ2δj,i + + + Zeeman field hτ3d h ·σδj,i + − − SO interactions ±iαdα ·σδj,i±1 − + − Triplet p SC ±i ∆ p τ2d p · iσ2σδj,i±1 − − + Quartet D I T R Triplet p SC ±i∆ p τ1dp · iσ2σδj,i±1 − + − Zeeman field hτ3d h ·σδj,i + − − Chemical potential µτ3δj,i + + + Triplet p SC ±i ∆ p τ2d p · iσ2σδj,i±1 − − +d p = d h × d α . (right) Quartet D :
In the same way a triplet p SC field is induced in a π/2 relative phase with respect to another p SC field when a Zeeman field and a chemical potential are applied. The vector of the induced field is also determined by
d p = d p × d h .
FIG. S2: Left: Ferromagnetic wire embedded in a conventional superconductor. The magnetization of the wire is considered perpendicular to the surface of the superconductor(green arrows). The magnetic field due to the magnetization of the wire is expected to stimulate a supercurrent flow (black arrow) and contribute an in-plane Zeeman field h y (yellow arrows) with opposite polarization at the each side of the wire. Right: The effective 2-D lattice scheme on which our calculations are based (Hamiltonian (Eq. 1) in the Letter) describes a perpendicularly polarized 1D FM wire (red sites) embedded in a SC (blue sites). The magnetic field of the FM wire triggers a super-current flow in the SC (black arrows) and contributes to small in plane Zeeman field components (yellow arrows).
II. INDUCED FIELDS ON FERROMAGNET-SUPERCONDUCTOR HETEROSTRUCTURES
In this section we demonstrate how triplet SC correlations and SOC are induced in FM/SC heterostructures, due to a supercurrent flow J and a transverse to the magnetisation of the FM Zeeman field h.
a. 1D wire on a conventional SC First we consider the case of a 1D FM wire embedded in a conventional SC (Fig. S2). We assume the magnetisation of the FM wire perpendicular to the SC surface and the corresponding magnetic field B z , triggers a supercurrent flow J in the SC region. Moreover the rotation of the magnetic field away from the FM wire contributes to a Zeeman field h y component transverse to the magnetisation of the wire which is not screened by the supercurrent flow. Therefore in lattice points within the SC region and adjacent to the FM wire we consider a finite supercurrent flow and a Zeeman field h y as depicted in Fig. S2. For the SC region we consider a finite on-site SC field ∆ and we choose ∆ = 0 over the FM wire although we allow the SC field to be induced in the FM wire by proximity. For this heterostructure we utilise a 2D lattice model described by the following simple Hamiltonian,
H = i,j,s (tf i,j + J x i g x i,j + J y i g y i,j )ψ † i,s ψ j,s + i µ i · n i − i ψ † i,s (h i · σ) ss ψ i,s + i (∆ i ψ † i,↑ ψ † i,↓ + h.c.) , (S9)
where ψ i,s is the annihilation operator of fermions at lattice site i with spin s, n i is the local charge density operator, while µ and h stands for the chemical potential and Zeeman field respectively. With t we denote the transfer integral associated with the even in inversion connection matrix f i,j = δ j,i±x +δ j,i±y , where x and y the unit vectors along the x-and y-coordinate axes, corresponding to nearest neighbours lattice points. On the other hand for the charge current J x and J y we associate the odd in inversion connection matrices g x i,j = ±iδ j,i±x and g y i,j = ±iδ j,i±y respectively. Hamiltonian Eq. (S9) acquires the following compact form when we introduce the spinor Ψ † i = ψ † i,↑ , ψ † i,↓ , ψ i,↑ , ψ i,↓ and employ the usual Pauli matrices τ , σ acting on the Nambu and spin space respectively.
H = 1 2 i,j Ψ † iĤ i,j Ψ j ,Ĥ i,j = tf i,j τ 3 + J x i g x i,j + J y i g y i,j + (µ i τ 3 − τ 3 h i ·σ + ∆ i τ 2 σ 2 )δ i,j ,(S10)
whereσ = (σ 1 , τ 3 σ 2 , σ 3 ). We diagonalise the above Hamiltonian by solving the Bogoliubov-de Gennes (BdG) equation
jĤ i,j U j,n = E n U i,n , where E n and U i,n = u * i,↑ , u * i,↓ , v i,↑ , v i,↓ T
the energy eigenvalues and the corresponding eigenfunctions respectively. Using the eigenvalues and eigenstates of the system we can calculate any correlations at lattice site i corresponding to the representationÔ in the SU(4) spin-Nambu space based on the following equation
j Ψ † iÔ f r i,j Ψ j = n,j [U † n,iÔ f r i,j U j,n ]n F (E n ) ,(S11)
where n F (E n ) = 1 1+e βEn the Fermi distribution and the connection matrix f r i,j , a basis function of the C 4u point group symmetry which corresponds to the coordinate space configuration of the particular correlations.
Due to the coupling of the J, h y and ∆ fields, an equal spin pairing field of the form ∆ p y g x i,j τ 2 emerges (Quartet B). In the same way J and h y couple to the chemical potential of the SC or to an effective chemical potential present in the boundaries of the FM/SC heterostructure due to a Fermi mismatch of the two systems, to induce SO interactions of the form α y g x i,j τ 3 σ 2 (Quartet A). These fields which emerge on the boundaries of the FM/SC heterostructure mediate by proximity within the FM wire.
In Fig. S3 we present typical results obtained by diagonalising Eq. S10 for a 91 × 11 SC with a 1D FM wire with length L = 83 located at the center of the system, considering ∆ = 1, µ SC = 0 for the SC and h z = 4, µ F M = 3.8 (all in t units) for the FM wire. For these parameters the FM wire is strongly polarised with a single spin band occupied. Notice also that due to their antiparallel with respect to the FM wire configuration the supercurrent and the transverse Zeeman field vanish along the wire. We demonstrate (Fig. S3 a) and b)) how equal spin SC correlations ∆ p y and SO correlations A y emerge within the wire as the supercurrent flow J and the Zeeman field h y acquire finite values. Notice that for J = 0 or h y = 0 there are no induced triplet correlations ∆ p y and A y and therefore a topological non-trivial SC phase cannot be realised. In Fig. S3 c) and d) we present how the energy gap and a single zero-energy excitation emerges with increasing h y and J respectively due to the induced p SC fields. The zero-energy excitation localised at the edges of the FM wire emerges only when both current J and Zeeman field h y are finite Fig. S3 e). Particularly for |h y | = 0.1 and |J| = 0.2 the local density of states of the FM wire (Fig. S3 g)) reveals the energy gap in the quasiparticle excitation spectrum of the FM wire due to the induced p SC field. For the same parameters a pair of zero-energy bound states emerges localised at the edges of the wire (Fig. S3 f)) . The zero energy Majorana bound states derive by expressing the two zero-energy eigenstates of Hamiltonian Eq. The following 1D effective Hamiltonian for the FM wire including the induced fields a y and ∆ p y , gives a better understanding for the emergence of MFs in the particular FM/SC heterostructure.
H 1D,ef f F M = 1 2 i Ψ † i t f x i,j τ 3 + (µ F M τ 3 − h z τ 3 σ 3 + ∆ τ 2 σ 2 )δ i,j + α y τ 3 σ 2 + ∆ p y τ 2 + ∆ p x τ 2 σ 3 g x i,j Ψ j ,(S12)
We have also included the induced by proximity conventional SC field ∆ and the ∆ p x g x i,j τ 2 σ 3 SC field induced by the coupling of chemical potential µ F M and Zeeman field h z with ∆ p y (see Sec. I). Moreover we take into account a possible renormalisation of the dispersion of the FM wire t ≤ t due to its coupling with the SC [3]. As elaborated in Sec. III Hamiltonian Eq. S12 belongs to the chiral BDI symmetry class with a strong integer Z topological invariant. The criteria for realising the topologically non-trivial phase with topological invariant W = 1
are |2t − h 2 z − ∆ 2 | < |µ F M | < |2t + h 2 z − ∆ 2 |
corresponding to a single energy band of the FM wire occupied. The chemical potential of the FM wire for the particular case presented above is within the boundaries set by the topological criteria. Thus, the emergence of the zero energy MFs at the edges of the FM wire is completely justified.
b. Quasi-1D FM wire Next we elaborate the case of a quasi-1D FM wire with finite width. In this case we have also to take into account the induced SOC of the form α x g y i,j σ 1 due to the rotation of the Zeeman field around the x-axis [4]. As presented in Sec. I these spin-orbit interactions couple to the singlet SC field ∆, the Zeeman field h z and the chemical potential µ F M of the wire, to induce triplet p SC fields of the form ∆ p y g y i,j τ 1 and ∆ p x g y i,j τ 1 σ 3 . In Fig. S4 e) and f) we present the induced fields on the W=3 FM wire embedded in a conventional SC derived by diagonalising Eq. S10 considering ∆ = 2 (only within the SC region while ∆ = 0 within the wire), µ SC = 0, h z = 4, |h y | = 1.6 and |J| = 0.2. From Fig. S4 b) we observe that zero-energy modes (which acquire a small finite energy due to finite length of the wire) emerge for 0.9 < µ F M < 2.4 (see Fig. S4 c) and 5.8 < µ F M < 7.2 when one transverse sub-band is occupied and for 3.6 < µ F M < 4.5 when 3 transverse sub-bands are occupied. For 3.6 < µ F M < 4.5 apart from the zero-energy localised modes (red line) two gapped states corresponding to complex fermions also localised at the wire's edges emerge (green,purple lines). Finally in Fig. S4 d) we present the lowest (almost zero)-energy eigenstate for µ = 0.95h z in the Majorana basis. Again we interpret our results according to the following effective Hamiltonian for the quasi-1d FM wire, which includes the induced fields.
H Q1D,ef f F M = 1 2 i,j Ψ † i [t f i,j τ 3 + (µ F M τ 3 − h z τ 3 σ 3 − h y σ 2 + ∆ τ 2 σ 2 )δ i,j + (α y τ 3 σ 2 + ∆ p y τ 2 + ∆ p x τ 2 σ 3 + J x )g x i,j + (α x σ 1 + ∆ p x τ 1 σ 3 + ∆ p y τ 1 + J y )g y i,j Ψ j ,(S13)
We have also included small components of the supercurrent flow J which are induced in the FM wire by proximity. For simplicity we ignore in our analysis the dependence of the fields in the transverse direction. Moreover as explained in the main article the in-plane Zeeman field h y in this case can also emerge from a rotation of the magnetisation of the FM wire (see Fig. S4 a)). Therefore we have also included the corresponding term h y σ 2 in the effective Hamiltonian Eq. S13. As described in Sec. III the SOC α x and the Zeeman field h y as well as the induced triplet SC fields ∆ p x and ∆ p y don 't satisfy the condition d p = d α × d h = d p × d h and therefore break the chiral symmetry. magnetization is perpendicular to SC surface (green arrows). The magnetic field created by wire magnetization stimulates a supercurrent flow around the wire (black arrow). A rotation of the magnetization of the wire contributes a Zeeman field hy (yellow arrows) which is not screened by the super-current and leaks into the SC region. b) Schematic 2D lattice modelization of the three rows wide wire (red sites) embedded in a SC (blue sites) corresponding to the Hamiltonian (Eq. 1) in the Letter. c) For a three rows wide FM wire with length L=83, the low-energy excitation spectrum as the chemical potential of the FM wire increases, taking ∆ = 2, µSC = 0, |hy| = 1.6, |J| = 0.2 and hz = 4. A single zero-energy excitation emerges (red line) when an odd number of bands of the wire cross the Fermi level. d) For µF M = 2t the two Majorana modes localized at the edges of the wire. e) and f) The induced on the FM wire correlations conjugate to the triplet SC field ∆ p y g x i,j τ2 (blue line) and SOC αyg x i,j τ3σ2 (red line) along the x-axis and the same for the triplet SC field ∆ p y g y i,j τ1 (black line) and SOC αxg y i,j σ1 (green line) along the y-axis, for the edge rows and the middle row respectively. The overall induced SOC αyg x i,j + αxg y i,j is of the Rashba form while ∆ p y g x i,j + ∆ p y g y i,j corresponds to a chiral SC field. All values are normalized with respect to the value of the correlations for the ∆ p y g y i,j τ1 field in the middle of the wire.
Since the time reversal symmetry is also broken the system described by the Hamiltonian Eq. S13 belongs to the D symmetry class which accepts a Z 2 topological invariant (1D systems). Quasi-1D systems described by Hamiltonian S13 without the current J and h y terms, were investigated in Ref. [5]. There it was found that when the width W of the system is smaller than the pairing coherence length ξ 0 ∼ t/∆ p where ∆ p the single electron species pairing field, a single pair of Majorana zero energy modes localised at the edges of the wire emerge when odd number of transverse sub-bands are occupied. We obtain the same results in our model where the p-wave pairing field is induced due to the supercurrent flow J and the transverse Zeeman field h y , even though the latter fields acquire a small but finite value within the wire. According to our results in this case 1 ξ ∼ ∆ p /t < 0.1 and the corresponding coherence length ξ > W suffice for Majorana zero-energy modes to emerge localised at the edge of the wire. Moreover from the topological phase diagram (Fig. S4 c)) we conclude that the transverse sub-bands of the quasi-1D wire are slightly renormalised due to the coupling with the SC [3] and particularly 2 sub-bands acquire a bandwidth which corresponds to t 0.9 while for the third sub-band we get t 0.85. Regarding the case for the quasi-1D wire coupled to a SC with ∆ = 1 presented in the main article, from the corresponding topological phase diagram (Fig. 4) we derive the renormalised kinetic terms t 0.875 (for two sub-bands) and t 0.775 (for one sub-band). In general we conclude that the renormalisation is not the same for all the transverse sub-bands, while as anticipated, it increases as the singlet pairing field ∆ of the substrate SC decreases.
(tf i,j + J g i,j )ψ † i,l,s ψ j,l,s − i,l µ l · n i,l − ψ † i,l,s (h l · σ) ss ψ i,l,s + ∆ l ψ † i,l,↑ ψ † i,l,↓ + ∆ l ψ i,l,↓ ψ i,l,↑ + i,l,l ,s t l,l ψ † i,l,s ψ i,l ,s + c.c. , (S14)
where l is the layer index. In addition to Eq. S9 we have considered the simplest coupling term between the layers t l,l ψ † i,l,s ψ i,l ,s , while we have dropped the lattice site index i from the fields J ,µ ,h and ∆ as they are considered to be uniform within each layer. Introducing the spinor Ψ † i,l = ψ † i,l,↑ , ψ † i,l,↓ , ψ i,l,↑ , ψ i,l,↓ for each layer and the Pauli matrices τ , σ, Hamiltonian Eq. S15 simplifies to
H = 1 2 i,j,l Ψ † i,lĤ i,j,l Ψ j,l + 1 2 i,l,l Ψ † i,lĤ i,l,l Ψ i,l , H i,j,l = t l f i,j τ 3 + J l g i,j + (µ l τ 3 − τ 3 h l ·σ + ∆ l τ 2 σ 2 )δ i,j ,Ĥ i,l,l = t l,l τ 3 ,(S15)
Particularly we consider a finite singlet pairing field ∆ = 0 only for the two superconducting layers and a Zeeman field h z τ 3 σ 3 = 0 within the ferromagnetic layer only. Moreover a supercurrent flow J x l g x i,j along the x-axis and a Zeeman field h y σ 2 are externally applied to the two superconducting layers in an antiparallel configuration as depicted in Fig. S5. For the coupling term t l,l τ 3 we consider a finite value only between the ferromagnetic and the superconducting layers, while the two superconducting layers are not coupled with each other. As elaborated Appendix I the current J x g x i,j and the Zeeman field h y σ 2 couple to the singlet pairing field ∆τ 2 σ 2 and the chemical potential µτ 3 to induce a triplet pairing field ∆ p y τ 2 g x i,j and SOC a y τ 3 σ 2 g x i,j respectively. The pairing field ∆ p y and the SOC a y are subsequently induced by proximity to the ferromagnetic layer (see Fig. S5). Finally the pairing field ∆ p y τ 2 g x i,j couples to Zeeman field h z τ 3 σ 3 and the chemical potential µτ 3 of the ferromagnetic layer to induce an additional triplet pairing field ∆ p x τ 2 σ 3 g x i,j . In Fig. S5 we present the results derived from Eq. S15 for a trilayer with ∆ = 2, µ SC = 0, |J x | = 0.2, |h y | = 0.8 for the SC layers and h z = 8, µ F M = 6.4 for the FM layer. Notice the corresponding correlations for the induced fields ∆ p y and a y in Fig. S5 b) and c) respectively. In this case the effective Hamiltonian of the ferromagnetic layer including the induced fields acquires the following form.
H 2D,ef f F M = 1 2 i,j Ψ † i t f i,j τ 3 + (µ F M τ 3 + h z τ 3 σ 3 + ∆ τ 2 σ 2 )δ i,j + (α y τ 3 σ 2 + ∆ p y τ 2 + ∆ p x τ 2 σ 3 )g x i,j Ψ j , (S16)
where we consider t ≤ t as the renormalised kinetic term within the ferromagnetic layer due to coupling with the superconducting layers. Because all the induced fields follow the direction of the applied current flow the only transverse term in Eq.S13 is the kinetic term t f i,j τ 3 . Thus we can rewrite the above Hamiltonian in the following form
H 2D,ef f F M = 1 2 i,j,ν,ν Ψ † i,ν [t f i,j τ 3 + (µ F M τ 3 + h z τ 3 σ 3 + ∆ τ 2 σ 2 )δ i,j + (α y τ 3 σ 2 + ∆ p y τ 2 + ∆ p x τ 2 σ 3 )g x i,j + t ⊥ δ ν,ν±1 τ 3 Ψ j,ν H 2D,ef f F M = 1 2 i,ν Ψ † i,ν H 1D i,j,ν + t ⊥ δ ν,ν±1 τ 3 Ψ j,ν ,(S17)
where ν ∈ (1, 2, ..., N y ) indicates each row of the ferromagnetic layer for which we have introduced the spinor Ψ † i,ν = ψ † i,ν,↑ , ψ † i,ν,↓ , ψ i,ν,↑ , ψ i,ν,↓ . The 1D Hamiltonian H 1D i,j,ν for each row has exactly the same form as Eq. S12. First we assume periodic boundary conditions along the transverse direction. Because the fields in Eq. S15 are uniform we can drop the row index from the 1D effective Hamiltonians, H 1D i,j,ν = H. Since the system acquires the translation symmetry across the transverse direction we can block diagonalise Hamiltonian Eq. S17 by conducting a Fourier transformation from y-coordinate to k y -momenta space. After the transformation the effective Hamiltonian of the ferromagnetic layer acquires the following form H ef f F M = 1 2 i,ky Ψ † i,ky H 1D i,j + t ⊥ τ 3 λ ky Ψ j,ky , where λ ky = 2 cos(k y ) the eigenvalues of the transverse Hamiltonian H ⊥ = δ ν,ν±1 with k y = 2π(n−1) Eq. S17 acquires a block diagonal form H ef f F M = 1 2 i,n Ψ † i,n H 1D i,j,n + t ⊥ τ 3 λ n Ψ j,n [6], with λ n the eigenvalues of the transverse part of the Hamiltonian H ⊥ = δ ν,ν±1 where in this case terms connecting the first and the last row are absent. In both cases notice that the kinetic term along the transverse direction acts as an effective chemical potential for each of the N y independent subsystems described by the Hamiltonian H 1D i,j,n which belongs to the BDI chiral symmetry class. Thus the overall system can support W = n W n pairs of zero-energy Majorana modes, where W n is the integer topological invariant for each subsystem H 1D i,j,n + t ⊥ τ 3 λ n . Due to the effective chemical potential t ⊥ λ n the topological criteria are modified accordingly and particularly for realising a phase with topological invariant |W n | = 1 we get |2t − h 2 z − ∆ 2 | < |µ + t ⊥ λ n | < |2t + h 2 z − ∆ 2 | (see Sec. III). For closed boundary conditions we observe that apart from k y = 0 (and in addition k y = π when N y is even) eigenvalues λ n are doubly degenerate. Moreover when N y is even the eigenvalues are symmetric with respect to 0. Therefore when h 2 z − ∆ 2 > 2t , for N y even the topological invariant W is odd apart from |µ − h 2 z − ∆ 2 | < 2|(t − t ⊥ )|, while for N y odd the topological invariant is odd only when µ > 2(t
⊥ − t ) + h 2 z − ∆ 2 .
For open boundary conditions the eigenvalues λ n are always non-degenerate and therefore transitions among topological phases with odd and even topological invariant are in general observed. The symmetry of the eigenvalues with respect to zero is reflected in the symmetry of the topological phases with respect to h 2 z − ∆ 2 in Fig. 4b of the main article. d. Results for self-consistently determined pairing field ∆. Finally in Fig. S6 we present typical results for the 1D and quasi-1D wire on a conventional SC where the singlet pairing field ∆ of the superconductor is determined self-consistently considering an attractive on-site potential U. Including the on-site interaction term Hamiltonian Eq. S9 modifies to
H = i,j,s (tf i,j + J x i g x i,j + J y i g y i,j )ψ † i,s ψ j,s + i µ i · n i − i ψ † i,s (h i · σ) ss ψ i,s + i U i ψ i,↓ ψ i,↑ ψ † i,↑ ψ † i,↓ . (S18)
From the above Hamiltonian for interacting fermions we retrieve the corresponding non-interacting fermions Hamiltonian Eq. S9 using the mean field approach for the singlet pairing field Particularly Fig. S6 a) is the topological phase diagram with respect to the chemical potential µ F M of the 1D FM wire considering U = 1 only within the SC substrate and µ SC = 0, h z = 4, |h y | = 0.4 and |J| = 0.2. This is the same case presented in the main article where we have considered a fixed singlet pairing field ∆ = 1 instead. Notice that the topological phase diagram is very similar to that presented in Fig. 2 of the main article. In Fig. S6 b) we present the lowest energy eigenstate for µ F M = 4 (left) and the corresponding Majorana modes localised at the edges of the wire. Moreover for the quasi-1D wire with width W=3 and length L=83 we solve Eq. S18 self-consistently for the pairing field ∆ considering an attractive potential U = 2 only within the SC, while |h y | = 0.8, |J| = 0.2, µ SC = 0 and h z = µ F M = 6 for the FM wire (Fig. S6 c) and d)).
∆ i = U i < ψ i,↓ ψ i,↑ >= 1 2 U i j Ψ † i [τ 2 σ 2 δ i,j ]Ψ j ,(S19)
III. TOPOLOGICAL PHASE TRANSITIONS
In this section we review the topological criteria for 1D wires where SC fields and SOC are induced by proximity. For a translationally invariant system we consider only the local and nearest neighbors extended fields included in our discussion in previous sections. Introducing the spin-dependent Nambu spinor Ψ k = (c k,↑ , c k,↓ , c † −k,↑ , c † −k,↓ ) T and the Pauli matrices (τ ) for the particle-hole and (σ) for the spin space, the Hamiltonian of the wire acquires in this case the following form
H = 1 2 k Ψ † k [(2t cos k + µ)τ 3 − hτ 3 d h ·σ + ∆τ 2 σ 2 + sin k(∆ p τ 1 iσ 2 d p + ∆ p τ 2 iσ 2 d p + αd α ) ·σ] Ψ k , (S20) whereσ = (σ 1 , τ 3 σ 2 , σ 3 )
, h a Zeeman field with vector d h in spin space, ∆ a local SC field, µ and t the chemical potential and the nearest neighbors hopping term respectively, ∆ p and ∆ p triplet nearest neighbors extended SC fields with a relative phase φ = π/2 and a SO interactions. For the triplet fields we have considered the simplest odd in inversion function sin(k). The reality conditions in momentum space according to which free electron systems are classified to ten symmetry classes are ΘH(k)Θ −1 = H(−k) and ΞH(k)Ξ −1 = −H(−k) where Θ and Ξ anti-unitary operators [7]. In the particular case with d p = d h × d α = d h × d p and d p = d α the reality conditions are satisfied for Ξ = τ 1 K with Ξ 2 = I and Θ = iτ 3 d P · iσ 2σ K with Θ 2 = I. In addition a unitary chiral symmetry operator S = Θ · Ξ = τ 2 d p · iσ 2σ anticommutes with the Hamiltonian. Thus when the triplet fields satisfy the condition d p = d h × d α = d h × d p , the system belongs to the BDI symmetry class and is characterised by a strong integer Z topological invariant. Next we consider the particular case with d h = (0, 0, 1), d p = (0, 1, 0) and d α = d p = (1, 0, 0) corresponding to the Hamiltonian Eq. S12 relevant to the case we study in the main article. Since in this case the system belongs to the BDI symmetry class with Θ = iτ 3 σ 3 K Eq. S20 acquires a block off-diagonal form H (k) = 0 A(k) A(k) † 0 in the eigenbasis of the chiral operator S = τ 2 σ 3 . Therefore by applying the transformation U s = i[τ 1 + iτ 2 − τ 3 ]σ 2 + [τ 1 − iτ 2 + τ 3 ]σ 1 we obtain A(k) = [−h z + i ∆ p y sin k]σ 3 − [2t cos k + µ + i∆ p x sin k] + [a x sin k + i∆]σ 1 . We define as topological invariant the winding number [8]
W = −i 2π k=2π k=0 dz(k) z(k) = 1 2π k=2π k=0 dθ(k) ,(S21)
where z(k) the unimodular complex number defined as z(k) = exp[iθ(k)] = Det(A(k))/|Det(A(k))| ∈ U (1). Since the one dimensional Hamiltonian Eq. S20, belongs to the BDI class, the topological charge W is in general an integer, W ∈ Z. In order to determine W, for the particular system we get z(k) = Det(A(k)) + i Det(A(k)), θ(k) = tan −1 Det(A(k)) Det(A(k)) Det(A(k)) = ∆ 2 + (2t cos k + µ) 2 − (h z ) 2 + ([ ∆ p y ] 2 − α 2 x − [∆ p x ] 2 ) sin 2 k Det(A(k)) = 2 ∆α x − h z ∆ p y + (2tcosk + µ)∆ p x sin k (S22)
Topological phase transitions occur when the winding number is ill-defined which is the case when |Det(A(k))| = 0 or equivalently the Hamiltonian acquires zero eigenvalues. The quasiparticles eigenenergy spectrum of the particular Hamiltonian derives from the following equation
E(k) = ± h 2 z + ∆ 2 + [2t cos(k) + µ] 2 + [[∆ p x ] 2 + a 2 x + [ ∆ p y ] 2 ] sin 2 (k) ± 2 √ B B = [∆∆ p
x + (2t cos(k) + µ)a x ] 2 sin 2 (k) + [ ∆ p y ∆ p x sin 2 (k) − (2t cos(k) + µ)h z ] 2 + [ ∆ p y a x sin 2 (k) + ∆h z ] 2 (S23) and for k = 0, π acquires the following simple form E s,m,± = −sh z + m ∆ 2 + [2t ± µ] 2 (S24)
where s = ± ,m = ± and E s,m,+ (E s,m,− ) corresponds to k = 0(k = π).
In the following we derive the criteria with respect to the exchange energy h z and the chemical potential µ of the wire, for topological charge W = 1. From Eq. S24 we derive that for
|h z | = (2t ± µ) 2 + ∆ 2 ,(S25)
the eigenenergy spectrum acquires a node at k = 0(+) or k = π(-). For these momenta we get in general Det(A(0)) = Det(A(π)) = 0 and for (2t − µ) 2 + ∆ 2 < |h z | < (2t + µ) 2 + ∆ 2 , µ > 0 (2t + µ) 2 + ∆ 2 < |h z | < (2t − µ) 2 + ∆ 2 , µ < 0 ,
we observe that Det(A(0)) Det(A(π)) < 0 and therefore |θ(0) − θ(π)| = π . Since θ(k) is a continuous function of momenta k and because θ(−k) = θ(k) + π we conclude that for (2t − |µ|) 2 + ∆ 2 < |h z | < (2t + |µ|) 2 + ∆ 2 (S27) W = 1. This is the condition for a topological non-trivial phase which can support a single pair of zero energy Majorana modes. Equivalently for the chemical potential we get the following condition
|2t − h 2 z − ∆ 2 | < |µ| < |2t + h 2 z − ∆ 2 | .(S28)
Finally we remark that when a charge current term J is present or the triplet fields don't satisfy the condition d p = d h × d α = d h × d p , as for example in the case of quasi-1D FM wire embedded in a SC where d h = (1, 0, 1) and d p = d α = d p = (1, 1, 0), reality condition ΘH(k)Θ −1 = H(−k) cannot be satisfied for any anti-unitary operator. Therefore in this case the system breaks the chiral symmetry S and belongs to the D symmetry class with a strong Z 2 topological invariant for one-dimensional systems.
FIG. S3: 1D FM wire with length L=83 lattice sites on a SC with dimensions 91 × 11. For the SC region we consider ∆ = 1 and µSC = 0, while for the FM wire hz = 4 and µF M = 3.8 (all in t units) corresponding to a single spin band occupied. In the middle of the FM wire the induced equal spin SC Py (blue line) and SO Ay correlations (red line) a) for J = 0.08 as the Zeeman field hy increases and b) for hy = 0.025hz for increasing supercurrent flow J. The correlations are normalised with respect to the maximum value of Py correlations. For J = 0.08 c) and hy = 0.025hz d) the energy gap in the FM wire created due to the induced py SC field as the transverse Zeeman field hy and the supercurernt flow J increase respectively. Because for hz = 4 and µF M = 3.8 the FM wire is in a non-trivial topolocigal phase the lowest energy (red line) corresponding to a MF tends to zero. e) The lowest energy |Ψ| 2 wavefunction for hz = 4 and for hy = 0.05hz, |J| = 0 (right), hy = 0, |J| = 0.08 (middle), hy = 0.05hz, |J| = 0.08 (left). Only when both hy and J fields are finite, a localised eigenstate emerges at the edges of the FM wire. f) The MFs wavefunctions for hy = 0.05hz, |J| = 0.08. Experimentally the MFs wavefunctions can by identified by zero-bias conductance maps based on scanning tunneling microscopy measurement. g) The local density of states along the FM wire for hy = 0.4, |J| = 0.2. A peak at E=0 appears only at the edge of the FM wire (x=1-5), while an energy gap ∆E 0.05 emerges over the middle area of the FM wire. We remark that the local density of states presented here, is a possible experimental signature for the emergence of zero energy MFs in scanning tunneling microscopy experiments.
S10 in the Majorana basisΓ = γ i,↑ , γ i,↓ , γ i,↑ , γ i,↓ T , where γ i,s = γ † i,s and γ i,s = γ † i,s the Majorana operators. The Majorana basis relates to the Nambu spin basis in the following way Γ = AΨ where A = e i π 4 τ2 e −i π 4 τ1 e i π 4 .
FIG. S4: a) A quasi-1D FM wire embedded in conventional superconductor. The main component of the wire's
c. Superconductor-Ferromagnet-Superconductor trilayers Finally we present the induced SO interactions and the p pairing field for the SC/FM/SC trilayers case with [001] interfaces. The microscopic Hamiltonian in this case acquires the following form
N
Y and n ∈ (1, 2, ..., N y ). For open boundary conditions the system maintains only the reflection symmetry along the y-axis direction and therefore the 1D Hamiltonians H 1D i,j,ν in general can be related in the following way H 1D i,j,m+1 = H 1D i,j,Ny−m where m ∈ [0, (N y − 2)/2] for N y even and m ∈ [0, (N y − 3)/2] for N y odd. In the eigenbasis of the reflection symmetry matrix R Hamiltonian FIG. S5: a) A superconductor-ferromagnet-superconductor trilayer, where supercurrent flows (black arrows) and Zeeman fields (yellow arrows), perpendicular to the magnetisation of the FM (green arrows), are externally applied in an antiparallel configuration on the SC layers. The induced triplet pairing Pyτ2g x i,j b) and SO Ayτ3σ2g x i,j c) correlations within the FM layer with hz = 8 and µF M = 6.4 coupled to two SC layers with ∆ = 2, |hy| = 0.8, |J x | = 0.4 and tc = 0.8 for the interlayer hopping term. The induced correlations are normalised to their maximum value. d) For the same parameters the low-energy excitation spectrum. Notice with red color the 5 pairs of almost zero energy eigenstates separated by the rest excitations by an energy gap ∆E 0.04. For µ = 6.4 the topological invariant is indeed W = 5 (see Fig. 4b in main article). The left edge localised Majorana wavefunction corresponding to eigenenergy E = 2.5 · 10 −4 within the FM layer e) and the two SC layers f). The Majorana wavefunction corresponds to spin down within the FM layer and to spin up within the SC.
FIG. S6: a) The low-energy excitation spectrum for a 1D FM wire with length L=63 lattice sites on a conventional SC region with dimensions 71 × 11. The SC field was determined self-consistently considering an on-site potential U=1 only within the SC region and µSC = 0, hz = 4, |hy| = 0.4 and |J| = 0.2. The lowest eigenergy (red line) approaches to zero for hz − 1.2 < µF M < hz + 1.2 as anticipated by the topological criterion Eq. S28 corresponding to a remormalised dispersion t = 0.6t. b) left) The lowest energy wavefunction |Ψ| 2 for µF M = 4 and right) one of the corresponding Majorana fermions localised at one edge of the FM wire. c) The excitation energy spectrum for a 3 rows wide FM wire coupled to a conventional SC. The SC field was determined self-consistently for on-site potential U=2 and µSC = 0 |hy| = 0.8, |J| = 0.2 and , hz = µF M = 6. d) The near zero energy eigenstates (within red circle) separated by a ∆E 0.07 energy gap from the rest excitation spectrum, correspond to the two Majorana fermions localised at the edges of the wire.
TABLE I :
I(left) Quartet A :
iπ/4τ 2 ] the unitary transformation which diagonalises the Hamiltonian matrix for each momentumÛ (k) † HÛ (k) = E m,s (k), with E m,s (k) = J(k) − sh x + m ∆ 2 + [µ + (k)] 2 , the excitation spectrum. The momentum configuration functions for both terms are antisymmetric f Ax
TABLE
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| []
|
[
"Single photons from coupled quantum modes",
"Single photons from coupled quantum modes"
]
| [
"T C H Liew \nInstitute of Theoretical Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n",
"V Savona \nInstitute of Theoretical Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n"
]
| [
"Institute of Theoretical Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland",
"Institute of Theoretical Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nCH-1015LausanneSwitzerland"
]
| []
| Single photon emitters often rely on a strong nonlinearity to make the behaviour of a quantum mode susceptible to a change in the number of quanta between one and two. In most systems the strength of nonlinearity is weak, such that changes at the single quantum level have little effect. Here, we consider coupled quantum modes and find that they can be strongly sensitive at the single quantum level, even if nonlinear interactions are modest. As examples, we consider solid-state implementations based on the tunneling of polaritons between quantum boxes or their parametric modes in a microcavity. We find that these systems can act as promising single photon emitters. | 10.1103/physrevlett.104.183601 | [
"https://arxiv.org/pdf/1002.4341v2.pdf"
]
| 2,497,283 | 1002.4341 | e439ab540fde2b6b205b5ad0964ff55a550e4f08 |
Single photons from coupled quantum modes
5 May 2010 (Dated: May 6, 2010)
T C H Liew
Institute of Theoretical Physics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
CH-1015LausanneSwitzerland
V Savona
Institute of Theoretical Physics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
CH-1015LausanneSwitzerland
Single photons from coupled quantum modes
5 May 2010 (Dated: May 6, 2010)
Single photon emitters often rely on a strong nonlinearity to make the behaviour of a quantum mode susceptible to a change in the number of quanta between one and two. In most systems the strength of nonlinearity is weak, such that changes at the single quantum level have little effect. Here, we consider coupled quantum modes and find that they can be strongly sensitive at the single quantum level, even if nonlinear interactions are modest. As examples, we consider solid-state implementations based on the tunneling of polaritons between quantum boxes or their parametric modes in a microcavity. We find that these systems can act as promising single photon emitters.
Introduction.-The construction of single photon sources [1,2] is a current aim of quantum nonlinear optics. Aside contributing to the security of quantum cryptography [3], single photon sources are useful elsewhere, for example in schemes for quantum computation using only linear optics and photodetection [4]. For some applications it is enough to reduce the intensity of a laser source to obtain single photons with a probability limited by the Poisson distribution. To do better than Poisson statistics one requires some form of nonlinearity. However, when one works in the single photon regime a strong nonlinearity is not so easy to find.
In semiconductor microcavities, light is strongly coupled to quantum well excitons resulting in new quasiparticles known as polaritons. Taking the best from both parents, polaritons have attracted particular attention for over a decade due to their strong nonlinearity (inherited from excitons) as well as their fast dynamics, long coherence and ability to couple to external light (features of photons). Polariton-polariton interactions have resulted in micron-sized optical parametric oscillators [5][6][7], optical gates [8], spontaneous coherence [9][10][11], low threshold lasing at room temperature [12][13][14] and superfluidity [15]. Whilst these effects involve many polaritons at once, we wish to focus on the single quantum regime. In planar cavities, quantum effects such as squeezing have been reported and several studies on quantum correlations undertaken [16][17][18][19]. More pronounced effects at the single polariton level are expected in quantum boxes [20][21][22], where polaritons are fully confined in three-dimensions and forced to interact even more strongly. Available recently, such confinement has encouraging prospects for single photon sources.
It has been predicted that for a very strong nonlinearity, the presence of a single polariton can block the resonant injection of another [23], analogous to the photon blockade [24] of nonlinear cavities. However, to obtain a strong enough nonlinearity for a single photon source, an extremely small quantum box is required (with size of the order of 200nm). Although one may anticipate such a system in the future, current systems do not display such a strong nonlinearity -whilst high nonlinearity is present in semiconductor microcavities, the energy shift caused by two interacting polaritons remains small.
We consider theoretically two coupled quantum boxes and show that the coupling can dramatically enhance the characteristics of single photon devices. By solving the quantum master equation for the density matrix, we find strong single photon statistics for values of the polaritonpolariton interaction strength corresponding to today's systems. We show that this is due to correlations between the quantum fluctuations in the two boxes, allowing a much stronger sensitivity of the system to the population compared to the single mode case. We expect that the coupling under study can also be exploited in analogous systems such as coupled nonlinear cavities or coupled photonic crystal cavities [25]. Finally, we show how mode coupling in parametric oscillators can also enhance single photon statistics due to selection rules.
A pair of linearly coupled modes.-Consider a pair of quantum modes described by creation operatorsâ † 1 and a † 2 respectively. As an example, we imagine the lowest energy polariton modes of two spatially separated microcavity quantum boxes. In each box, polariton-polariton interactions are characterized by an interaction strength α. The boxes are spatially separated such that there are no significant nonlinear interactions between boxes. However, the boxes are close enough together such that particles can tunnel from one box to the other, at a rate given by the tunneling constant J. The Hamiltonian is:
H = E 1â † 1â 1 + E 2â † 2â 2 + α â † 1â † 1â 1â1 +â † 2â † 2â 2â2 − J(â † 1â 2 +â † 2â 1 ) + Fâ † 1 + F * â 1(1)
where E 1 and E 2 are the uncoupled energy levels of the two quantum modes and F represents a coherent excitation of the first mode. With quantum boxes, this would be a laser excitation focused onto the first quantum box. We define our energy scale such that the pump energy is zero. The evolution equation of the corresponding density matrix, ρ, is:
i dρ dt = Ĥ , ρ + i Γ 2 2 n=1 2â n ρâ † n −â † nân ρ − ρâ † nân(2)
where the last term represents the standard Lindblad dissipation characterized by decay rate Γ. Equation (2) can be solved by expanding the density matrix over a particle number basis in a similar way to that done in Ref. [23]; one truncates at a given particle number and propagates in time from the vacuum to the steady state [26]. For a pair of quantum boxes of 3µm size, separated by 1µm, a typical value of the tunnel constant is J = 0.5meV. Since J > Γ, strong coupling takes place and the single particle eigenmodes are the symmetric and antisymmetric modes spanning the two wells [27]. We take α = 0.012meV, a value measured in Ref. [28] for condensed polaritons occupying spot sizes of ∼ 3µm. Although the pump acts directly on the first well it effectively pumps the second well due to an interference effect. This can be understood by considering the Heisenberg equations for the symmetric (â + = sin φâ 1 + cos φâ 2 ) and antisymmetric (â − = cos φâ 1 − sin φâ 2 ) field operators, with eigenenergies E + and E − respectively:
dâ + dt = E +â+ + F sin φ; dâ − dt = E −â− + F cos φ For E 1 ≈ E 2 , φ ≈ π/4 and in the steady state one finds thatâ ± = F E± √ 2 . Since E + and E − have different signs, a + andâ − are excited with different signs such thatâ 2 is excited instead ofâ 1 .
A key quantity in quantum optics is the second order correlation function, defined as:
g 2,nm (t − t ′ ) = â † n (t ′ )â † m (t)â m (t)â n (t ′ ) â † n (t ′ )â n (t ′ ) â † m (t)â m (t)(3)
When n and m correspond to the same mode and t = t ′ , this quantity also measures the performance of a single photon source; an ideal source has g 2,nn (0) = 0, whilst a classical source has g 2,nn (0) = 1. For fixed pump intensity, the dependence of g 2,11 (0) on the energy levels of the two wells is shown in Fig. 1a. The optimum (smallest) g 2,11 (0) was attained when E 1 = 0.07meV and E 2 = 0.05meV. The g 2,11 (0) depends mostly on the energy of the second well, E 2 , and this variation is shown again in Fig. 1b along with the variation of g 2,22 (0), and the average well populations, N 1 and N 2 respectively. Unlike in Fig. 1a, the pump energy is also varied to maintain a constant detuning between the pump and the lowest energy (symmetric) single particle eigenstate. This allows a better test of the variation of E 2 since the average populations do not change drastically and reveals that whilst the second well has g 2,22 (0) ≈ 1, varying E 2 has a dramatic effect on g 2,11 (0).
Utilizing the first well as a single photon source, one finds that at a pump amplitude giving N 1 = 0.02 the probability of having more than one photon is 0.18%; this is five times better than the failure rate of devices based on spontaneous parametric down conversion [1]. Spectral filtering could further reduce emission from the n 1 ≥ 2 states, providing extra improvement. Fig. 1c shows the unequal time second order correlation functions [23], which oscillate at half the Rabi oscillation period arising from the J = 0.5meV coupling.
To better understand the low value of g 2,11 (0) we carried out analytical calculations, extending the method of Ref. [29], which applies directly to the single mode case. We use stochastic (Langevin-type) equations for the evolution of quantum fields [30]. Lowest-order fluctuations of the fields around their mean values can be found by solving the linearized (â n →ā n + δa n ) version of the equations [26]. Choosing the convention thatā 1 is real, the 2 nd order correlation can be written [30]:
g 2,11 = 1 + 2 n 1 [ δa * 1 δa 1 + ℜe { δa 1 δa 1 }](4)
which yields [26]:
g 2,11 = 1 + 2 n 1 [ δα * 1 δα 1 ] − 2α (E 1 + 4αn 1 ) (1 + ζ 2 ) + 2Jℜe { δa 1 δa 2 } n 1 (E 1 + 4αn 1 ) (1 + ζ 2 ) − 2ζJℑm { δa 1 δa 2 } n 1 (E 1 + 4αn 1 ) (1 + ζ 2 )(5)
where ζ = Γ/ (2(E 1 + 4αn 1 )). The mean field values, a 1 , can be obtained as in Ref. [27]. To calculate all second order correlations we have extended the method of Ref. [31]. In Fig. 1d we compare the result of Eq. 5 for the two mode (solid line) and single mode (dotted line) cases. The single mode value is obtained by setting J = 0 and matches the result from Ref. [29]. For parameters corresponding to the optimum g 2,11 , the last term in Eq. 5 makes a strong negative contribution [26]. In other words, the correlated noise fluctuations δa 1 δa 2 drive the low value of g 2, 11 . These correlations are a result of the interplay between nonlinearity (in the limit α → 0, δa 1 δa 2 → 0) and tunnelling. This is a very different mechanism from the polariton blockade [29], which originate from the third term in Eq. 5 that vanishes in the present regime where αn 1,2 ≪ Γ.
For pioneering experiments one may also consider replacing the coupled quantum box modes by the circularly polarized spin modes of a single quantum box or a localized state in a planar microcavity. Magnetic fields parallel and perpendicular to the growth direction would allow the tuning of the spin energy levels (as suggested as a control method of the original polariton blockade [32]) and the coupling constant J, respectively.
Parametrically coupled modes.-Given the attention devoted to parametric processes in microcavities at the beginning of the millennium [5][6][7], it is only natural for us to ask whether sub-Poisson statistics can also be derived from pair scattering processes. Again, a variety of systems can be imagined, including the eigenmodes of a localized quantum box [33]. The Hamiltonian now includes three separate modes characterized by creation operatorsâ † 1 ,â † 2 andâ † 3 respectively:
H = 3 n=1 E nâ † nâ n + αâ † nâ † nâ nân + 2α m =nâ † nâ † mâ nâm + 2α â † 1â † 3â 2â2 +â † 2â † 2â 1â3 + Fâ † 2 + F * â 2(6)
The scattering terms represent the same selection rules as when one deals with a pump mode that can scatter in pairs to signal and idler modes, as in intra-branch [5,6] and inter-branch [7] scattering in planar microcavities. The Hamiltonian can be diagonalized on a number state manifold in the absence of the pump terms. The energy levels are shown in Fig. 2. The only state in the n 1 = 2 manifold that can pollute the value of g 2,11 (0) is the lowest lying |200 state. An analysis of the allowed transitions shows that this state can only be reached from decay of an n 1 = 3 state. In the low occupation limit the system is not expected to visit the n 1 = 3 manifold very frequently, leading us to expect a low value of g 2,11 (0). Using a similar evolution equation to Eq.(2), g 2,11 (0) is calculated in Fig. 3a for different values of the energy levels, E 2 and E 3 . A study varying also E 1 , showed that the optimum parameters are for the case when the pumped mode is at an energy slightly higher than the pump energy, E 2 = 0.12meV and the sum of the signal and idler mode energies is resonant with the pump (see Fig. 3b). In fact g 2,11 (0) is found only to vary if E 1 +E 3 is changed, that is, a selection of values of E 1 and E 3 can be used to obtain optimum results. The time dependent second order correlation is shown in Fig. 3c. For zero delay we obtain g 2,11 (0) = 0.28. It is important to remember that we are working with a value of the polariton-polariton interaction strength available in two dimensional planar cavities [28]; much lower values of g 2,11 (0) would appear in more confined systems in which the strength of interactions is higher.
For comparison, the second order coherence function is shown in Fig. 3d for three cases using the same value of the interaction strength: the single mode polariton blockade [23]; the two coupled well case (studied in the previous section); and the case of three parametrically coupled modes. It is clear that the use of schemes involving two or three coupled modes can give statistics closer to that of a single photon device than the single mode case. Whilst in all schemes the value of g 2,11 (0) decreases with the average population of the signal mode, for the single mode and three mode case g 2,11 (0) tends to a constant value as N is decreased.
Finally, we have considered the effect of pure dephasing (associated with exciton-phonon scattering) by adding the term:
Γ P 2 n 2â † nân ρâ † nân −â † nânâ † nân ρ − ρâ † nânâ † nân(7)
to the right-hand side of Eq.(2). Taking Γ P = 0.3µeV (an upper estimate of the dephasing from Ref. [34]) gives the dashed curves shown in Fig. 3d.
Conclusion.-We considered the use of a pair of coupled quantum boxes as a single photon source. We present the general idea that coupling can dramatically improve the single photon statistics compared to the single-mode case, through noise correlations. With competitive characteristics one may choose quantum boxes in a solid-state system, which offer fast (picosecond scale) relaxation rates; compact size; emission into a welldefined spatial mode (a luxury not always present when working with quantum dots); and wavelengths compatible with transmission through silica fibres and photodetection with silicon based photon-counters. Alternatively, one may consider coupling nonlinear cavities or using parametrically coupled modes in planar microcavities. Indeed we anticipate more studies specific to each system and experimental verifications. The nonlinearity is available with our present technology and we have several paths to choose from.
We thank M. Wouters and A. Fiore for useful discus-sions. This work is supported by NCCR Quantum Photonics (NCCR QP), research instrument of the Swiss National Science Foundation (SNSF).
FIG. 1
1: a) Variation of g2,11(0) with E1 and E2. b) Dependence of the equal time correlation function and average populations on E2 for E1 = 0.07meV. c) g2,11(t) for the optimum parameters from (a). d) g2,11(0) from Eq.(5) (solid) and comparison to the single mode case (dotted). In all panels Γ = 0.2meV and in (a-c) F = 0.1meV.
FIG. 2 :
2Eigenstates from the diagonalization of the Hamiltonian Eq.(6) without the pump terms on the particle number manifold. Parameters: E1 = 0meV, E1 = 0.12meV, E2 = 0meV, α = 0.012meV, Γ = 0.1meV. The transitions caused by optical pumping and decay are shown by the solid and dashed arrows respectively. Variation of g2,11(0) with E2 and E3 for E1 = 0, F = 0.1meV. b) The optimum energy levels. c) g2,11 for the optimum parameters. Other parameters: Γ = 0.1meV, α = 0.012meV. d) Comparison of the g2,11(0) available from the single mode polariton blockade (with pump resonant with the bare mode energy), the two coupled mode case (fromFig 1)and the three parametrically coupled modes. g2,11(0) is plotted as a function of the average occupation number of the signal mode. Dashed curves show the effect of dephasing.
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| []
|
[
"A Fully Abstract Symbolic Semantics for Psi-Calculi",
"A Fully Abstract Symbolic Semantics for Psi-Calculi"
]
| [
"Magnus Johansson ",
"Björn Victor ",
"Joachim Parrow "
]
| []
| [
"6th Workshop on Structural Operational Semantics (SOS'09) EPTCS 18"
]
| We present a symbolic transition system and bisimulation equivalence for psi-calculi, and show that it is fully abstract with respect to bisimulation congruence in the non-symbolic semantics.A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Psi-calculi can be more general than other proposed extensions of the pi-calculus such as the applied pi-calculus, the spi-calculus, the fusion calculus, or the concurrent constraint pi-calculus.Symbolic semantics are necessary for an efficient implementation of the calculus in automated tools exploring state spaces, and the full abstraction property means the semantics of a process does not change from the original. | 10.4204/eptcs.18.2 | [
"https://arxiv.org/pdf/1002.2867v1.pdf"
]
| 10,346,763 | 1002.2867 | bbb9e119cbf973e892b5e38c70cfda43a3b22487 |
A Fully Abstract Symbolic Semantics for Psi-Calculi
2010
Magnus Johansson
Björn Victor
Joachim Parrow
A Fully Abstract Symbolic Semantics for Psi-Calculi
6th Workshop on Structural Operational Semantics (SOS'09) EPTCS 18
201010.4204/EPTCS.18.2
We present a symbolic transition system and bisimulation equivalence for psi-calculi, and show that it is fully abstract with respect to bisimulation congruence in the non-symbolic semantics.A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Psi-calculi can be more general than other proposed extensions of the pi-calculus such as the applied pi-calculus, the spi-calculus, the fusion calculus, or the concurrent constraint pi-calculus.Symbolic semantics are necessary for an efficient implementation of the calculus in automated tools exploring state spaces, and the full abstraction property means the semantics of a process does not change from the original.
Introduction
A multitude of extensions of the pi-calculus have been defined, allowing higher-level data structures and operations on them to be used as primitives when modelling applications. Ranging from integers, lists, or booleans to encryption/decryption or hash functions, the extensions increase the applicability of the basic calculus. In order to implement automated tools for analysis and verification using state space exploration (e.g. bisimilarity or model checking), each extended calculus needs a symbolic semantics, where the state space of agents is reduced to a manageable size -the non-symbolic semantics typically generates infinite state spaces even for very simple agents.
The extensions thus require added efforts both in developing the theory of the calculus for each variant, and in constructing specialised symbolic semantics for them. As the complexity of the extensions increases, producing correct results in these areas can be very hard. For example the labelled semantics of applied pi-calculus [2] and of CC-Pi [14] have both turned out to be non-compositional; another example is the rather complex bisimulations which have been developed for the spi-calculus [3] (see [12] for an overview of non-symbolic bisimulations, or [11,13,10] for symbolic ones).
The psi-calculi [5] improve the situation: a single framework allows a range of specialised calculi to be formulated with a lean and compositional labelled semantics: with the parameters appropriately instantiated, the resulting calculus can be used to model applications such as cryptographic protocols and concurrent constraints, but also more advanced scenarios with polyadic synchronization or higher-order data and logics. The expressiveness and modelling convenience of psi-calculi exceeds that of earlier pi-calculus extensions, while the purity of the semantics is on par with the original pi-calculus. Its metatheory has been proved mechanically using the theorem prover Isabelle [6].
In this paper we develop a symbolic semantics for psi-calculi, admitting large parts of this range of calculi to be verified more efficiently. We define a symbolic version of labelled bisimulation equivalence, and show that it is fully abstract with respect to bisimulation congruence in the original semantics. This means that our new symbolic semantics does not change which processes are considered equivalent.
A symbolic semantics abstracts the values received in an input action. Instead of a possibly infinite branching of concrete values, a single name is used to represent them all. When the received values are used in conditional constructions (e.g. if-then-else) or as communication channels, we do not know their precise value, but need to record the constraints which must be satisfied for a resulting transition to be valid.
A (non-symbolic) psi-calculus transition has the form Ψ £ P α −→ P , with the intuition that P can perform α leading to P in an environment that asserts Ψ. For example, if P can do an α to P then case prime(x) : P can make an α-transition to P if we can deduce prime(x) from the environment, e.g.
{x = 3} £ case prime(x) : P α −→ P .
In the symbolic semantics where we may not have the precise value of x, we instead decorate the transition with its requirement, so
Ψ £ case prime(x) : P α −−−−−−−−−→ C∧{|Ψ prime(x)| } P (for any Ψ)
where C is the requirement for P to do an α to P in the environment Ψ. Constraints also arise from communication between parallel agents, where, in the symbolic case, the precise channels may not be known; instead we allow communication over symbolic representations of channels and record the requirement in a transition constraint. As an example consider a(x) . a(y) . (x x . P | y(z) . Q) which after its initial inputs only has symbolic values of x and y. The resulting agent has the symbolic transition
x x . P | y(z) . Q τ − −−−−− → {|Ψ x . ↔y| } P | Q[z := x] where x .
↔ y means that x and y represent the same channel, but might not have a τ transition in the non-symbolic semantics. Communication channels in psi-calculi may be structured data terms, not only names. This leads to a new source of possibly infinite branching: a subject in a prefix may be rewritten to another equivalent term before it is used in a transition. E.g., when first(x, y) and x represent the same channel,
P = first(a, b)c . P ac −→ P , but also P first(a,c)c −−−−−→ P , etc.
The possibility of using structured channels gives significant expressive power (see [5]). Our symbolic semantics abstracts the equivalent forms of channel subject by using a fresh name as subject, and adds a suitable constraint to the transition label (see Section 3).
Comparison to related work
Symbolic bisimulations for process calculi have a long history. Our work is to a large extent based on the pioneering work by Hennessy and Lin [17] for value-passing CCS, later specialised for the pi-calculus by Boreale and De Nicola [9] and independently by Lin [18,19]. While [17] is parameterised by general boolean expressions on an underlying data signature it does not handle names and mobility; on the other hand [9,18,19] handle only names and no other data structures. The number of (direct or indirect) follow-up works to these is huge, with applications ranging from pi-calculus to constraint programming; here we focus on the relation to the ones for applied pi-calculus and spi-calculus.
The existing tools for calculi based on the applied pi-calculus (e.g. [1,7,8]), are not fully abstract wrt bisimulation. A symbolic semantics and bisimulation for applied pi-calculus has been defined in [15], but it is not complete. Additionally, the labelled (non-symbolic) bisimulation of applied pi-calculus is not compositional (see [5]). The situation for the spi-calculus is better: fully abstract symbolic bisimulation for hedged bisimulation has been defined in [10], and for open hedged bisimulation (a finer equivalence) in [13]. According to those authors, neither is directly mechanizable. The only symbolic bisimulation which to our knowledge has been implemented in a tool is not fully abstract [11].
It can be argued [11] that incompleteness is not a problem when verifying authentication and secrecy properties of security protocols, which appears to have been the main application of the applied picalculus so far. When going beyond security analysis we claim (based on experience from the Mobility Workbench [22]) that completeness is very important: when analysing agents with huge state spaces, a positive result (the agents are equivalent) may be more difficult to achieve than a negative result (the agents differ). However, such a negative result can only be trusted if the analysis is fully abstract.
Our symbolic semantics is relatively simple, compared to the ones presented for applied pi-calculus or spi-calculus. In relation to the former, we are helped significantly by the absence of structural equivalence rules, which in applied pi-calculus are rather complex. In [15] an intermediate semantics is used to overcome the complexity. In contrast we can directly relate the original and symbolic semantics. In relation to the symbolic semantics for spi-calculus, our semantics has a straight-forward treatment of scope opening due to the simpler psi-calculi semantics. In addition, the complexities of spi-calculus bisimulations are necessarily inherited by the symbolic semantics, introducing e.g. explicit environment knowledge representations with timestamps on messages and variables. In psi-calculi, bisimulation is much simpler and the symbolic counterpart is not significantly more complex than the one for valuepassing CCS.
In the light of these complications, the relevance of precise encodings of the applied pi-calculus or spi-calculus as psi-calculi, or comparing the resulting bisimulation equivalences is questionable. Our interest is in handling and analysing the same type of applications, and also the more advanced kinds of applications mentioned in the beginning of this section.
Disposition. In the next section we review the basic definitions of syntax, semantics, and bisimulation of psi-calculi. Section 3 presents the symbolic semantics and bisimulation, while Section 4 illustrates the concrete and symbolic transitions and bisimulations by examples. In Section 5 we show our main results: the correspondence between concrete and symbolic transitions and bisimulations. Section 6 concludes, and presents plans and ideas for future work.
Psi-calculi
This section is a brief recapitulation of psi-calculi and nominal data types; for a more extensive treatment including motivations and examples see [5].
Nominal data types
We assume a countably infinite set of atomic names N ranged over by a, b, . . . , x, y, z. Intuitively, names will represent the symbols that can be statically scoped, and also represent symbols acting as variables in the sense that they can be subjected to substitution. A nominal set [20,16] is a set equipped with name swapping functions written (a b), for any names a, b. An intuition is that for any member X it holds that (a b) · X is X with a replaced by b and b replaced by a. One main point of this is that even though we have not defined any particular syntax we can define what it means for a name to "occur" in an element: it is simply that it can be affected by swappings. The names occurring in this way in an element X constitute the support of X, written n(X).
We write a#X, pronounced "a is fresh for X", for a ∈ n(X). If A is a set of names we write A#X to mean ∀a ∈ A . a#X. We require all elements to have finite support, i.e., n(X) is finite for all X.
A function f on nominal sets is equivariant if (a b) · f (X) = f ((a b) · X) holds for all X, a, b, and similarly for functions and relations of any arity. Intuitively, this means that all names are treated equally.
A nominal data type is just a nominal set together with a set of functions on it. In particular we require a substitution function [6], which intuitively substitutes elements for names. If X is an element of a data type,ã is a sequence of names without duplicates andỸ is an equally long sequence of elements, the substitution X[ã :=Ỹ ] is an element of the same data type as X.
Agents
A psi-calculus is defined by instantiating three nominal data types and four operators:
Definition 1 (Psi-calculus parameters). A psi-calculus requires the three (not necessarily disjoint) nominal data types:
T the (data) terms, ranged over by M, N C the conditions, ranged over by ϕ A the assertions, ranged over by Ψ and the four equivariant operators:
.
↔: T × T → C Channel Equivalence ⊗ : A × A → A Composition 1 : A Unit ⊆ A × C Entailment
The binary functions above will be written in infix. Thus, if M and N are terms then M . ↔ N is a condition, pronounced "M and N are channel equivalent" and if Ψ and Ψ are assertions then so is Ψ⊗Ψ . Also we write Ψ ϕ, "Ψ entails ϕ", for (Ψ, ϕ) ∈ .
The data terms are used to represent all kinds of data, including communication channels. Conditions are used as guards in agents, and M . ↔ N is a particular condition saying that M and N represent the same channel. The assertions will be used to declare information necessary to resolve the conditions. Assertions can be contained in agents and thus represent information postulated by that agent; they can contain names and thereby be syntactically scoped and thus represent information known only to the agents within that scope. The intuition of entailment is that Ψ ϕ means that given the information in Ψ, it is possible to infer ϕ. We say that two assertions are equivalent if they entail the same conditions: Definition 2 (Assertion equivalence). Two assertions are equivalent, written Ψ Ψ , if for all ϕ we have that Ψ ϕ ⇔ Ψ ϕ.
A psi-calculus is formed by instantiating the nominal data types and morphisms so that the following requisites are satisfied:
Definition 3 (Requisites on valid psi-calculus parameters).
Channel Symmetry:
Ψ M . ↔ N =⇒ Ψ N . ↔ M Channel Transitivity: Ψ M . ↔ N ∧ Ψ N . ↔ L =⇒ Ψ M . ↔ L Weakening: Ψ ϕ =⇒ Ψ⊗Ψ ϕ Composition: Ψ Ψ =⇒ Ψ⊗Ψ Ψ ⊗Ψ Identity: Ψ⊗1 Ψ Associativity: (Ψ⊗Ψ )⊗Ψ Ψ⊗(Ψ ⊗Ψ ) Commutativity: Ψ⊗Ψ Ψ ⊗Ψ
Our requisites on a psi-calculus are that the channel equivalence is a partial equivalence relation, that ⊗ preserves equivalence, and that the equivalence classes of assertions form an abelian monoid. We do not require that channel equivalence is reflexive. There may be terms M such that M . ↔ M does not hold. By transitivity and symmetry then M . ↔ N holds for no N, which means that M cannot be used as a channel at all. In this way we accommodate data structures which cannot be used as channels. The requisite of weakening (which is not present in [5]) excludes some non-monotonic logics; it simplifies our proofs in the present paper although we do not know if it is absolutely necessary. It is only used in one place in the proof of Theorem 20.
In the followingã means a finite (possibly empty) sequence of names, a 1 , . . . , a n . The empty sequence is written ε and the concatenation ofã andb is writtenãb. When occurring as an operand of a set operator,ã means the corresponding set of names {a 1 , . . . , a n }. We also use sequences of terms, conditions, assertions etc. in the same way.
A frame can intuitively be thought of as an assertion with local names:
Definition 4 (Frame). A frame F is of the form (ν b)Ψ where b
is a sequence of names considered bound in the assertion Ψ. We use F, G to range over frames. 1 Name swapping on a frame F = (ν b)Ψ just distributes to its two components. We identify alpha equivalent frames, so n(F) = n(Ψ) − n( b). We overload 1 to also mean the least informative frame (νε)1 and ⊗ to mean composition on frames defined by
(ν b 1 )Ψ 1 ⊗(ν b 2 )Ψ 2 = (ν b 1 b 2 )Ψ 1 ⊗Ψ 2 where b 1 # b 2 , Ψ 2 and vice versa. We write (νc)((ν b)Ψ) for (νc b)Ψ,
and when there is no risk of confusing a frame with an assertion we write Ψ for (νε)Ψ.
Definition 5 (Equivalence of frames). We define F ϕ to mean that there exists an alpha variant (ν b)Ψ of F such that b#ϕ and Ψ ϕ. We also define F G to mean that for all ϕ it holds that F ϕ iff G ϕ.
Intuitively a condition is entailed by a frame if it is entailed by the assertion and does not contain any names bound by the frame. Two frames are equivalent if they entail the same conditions. Definition 6 (Psi-calculus agents). Given valid psi-calculus parameters as in Definitions 1 and 3, the psi-calculus agents, ranged over by P, Q, . . ., are of the following forms.
M N.P Output M(x).P Input case ϕ 1 : P 1 [] · · · [] ϕ n : P n Case (νa)P Restriction P | Q Parallel !P Replication (|Ψ| ) Assertion
In the Input M(x).P, x binds its occurrences in P. Restriction binds a in P. An assertion is guarded if it is a subterm of an Input or Output. In a replication !P there may be no unguarded assertions in P.
In the Output and Input forms M is called the subject and N and x the objects, respectively. Output and Input are similar to those in the pi-calculus, but arbitrary terms can function as both subjects and objects. Note that differently from [5], for simplicity the input is not pattern matching (see Section 6 for a discussion). The case construct works by performing the action of any P i for which the corresponding ϕ i is true. So it embodies both an if (if there is only one branch) and an internal nondeterministic choice (if the conditions are overlapping).
Some notational conventions: We define the agent 0 as (|1| ). The construct case ϕ 1 : P 1 [] · · · [] ϕ n : P n is sometimes written as case ϕ : P, or if n = 1 as if ϕ 1 then P 1 . The input subject is underlined to facilitate parsing of complicated expressions; in simple cases we often conform to a more traditional notation and omit the underline.
Formally, we define name swapping on agents by distributing it over all constructors, and substitution on agents by distributing it and avoiding captures by binders through alpha-conversion in the usual way. We identify alpha-equivalent agents; in that way we get a nominal data type of agents where the support n(P) of P is the union of the supports of the components of P, removing the names bound by Input and ν, and corresponds to the names with a free occurrence in P. Definition 7 (Frame of an agent). The frame F (P) of an agent P is defined inductively as follows:
F (M(x).P) = F (M N.P) = F (case ϕ : P) = F (!P) = 1 F ((|Ψ| )) = (νε)Ψ F (P | Q) = F (P) ⊗ F (Q) F ((νb)P) = (νb)F (P)
Operational semantics
The presentation of psi-calculi in [5] gives a semantics of an early kind, where input actions are of kind M N. Here we give an operational semantics of the late kind, meaning that the labels of input transitions contain variables for the object to be received. With this kind of semantics it is easier to establish a relation to the symbolic semantics. We also establish precisely how it relates to the original. Definition 9 (Transitions). A transition is of the kind Ψ £ P α −→ P , meaning that when the environment contains the assertion Ψ the agent P can do an α to become P . The transitions are defined inductively in Table 1.
Note that Ψ in Table 1 expresses the effect that the environment has on the agent, by enabling conditions in CASE, by giving rise to action subjects in IN and OUT and by enabling interactions in COM.
Both agents and frames are identified by alpha equivalence. This means that we can choose the bound names fresh in the premise of a rule. In a transition the names in bn(α) count as binding into both the action object and the derivative, and transitions are identified up to alpha equivalence. This means that the bound names can be chosen fresh, substituting each occurrence in both the object and the derivative. This is the reason why bn(α) is in the support of the output action: otherwise it could be alpha-converted in the action alone. Table 2 gives the rules for input and communication of an early kind used in [5]. The following lemma clarifies the relation between the two semantics: In the rule COM we assume that F (P) = (ν b P )Ψ P and F (Q) = (ν b Q )Ψ Q where b P is fresh for all of Ψ, b Q , Q, M and P, and that b Q is correspondingly fresh. In the rule PAR we assume that F (Q) = (ν b Q )Ψ Q where b Q is fresh for Ψ, P and α. In OPEN the expressionã ∪ {b} means the sequenceã with b inserted anywhere. Table 2: Early structured operational semantics. All other rules are as in the late semantics of Fig. 1. The proof is by induction over the transition derivations. In the proof of (2), the case α = τ needs both (1) and the case where α is an output.
IN Ψ M . ↔ K Ψ £ M(x).P K(x) −−→ P OUT Ψ M . ↔ K Ψ £ M N.P K N − − → P CASE Ψ £ P i α −→ P Ψ ϕ i Ψ £ case ϕ : P α −→ P COM Ψ Q ⊗Ψ £ P M (ν a)N − −−−− → P Ψ P ⊗Ψ £ Q K(x) −−→ Q Ψ⊗Ψ P ⊗Ψ Q M . ↔ K Ψ £ P | Q τ −→ (ν a)(P | Q [x := N]) a#Q PAR Ψ Q ⊗Ψ £ P α −→ P Ψ £ P|Q α −→ P |Q bn(α)#Q SCOPE Ψ £ P α −→ P Ψ £ (νb)P α −→ (νb)P b#α, Ψ OPEN Ψ £ P M (ν a)N − −−−− → P Ψ £ (νb)P M (ν a∪{b})N −−−−−−−→ P b# a, Ψ, M b ∈ n(N) REP Ψ £ P | !P α −→ P Ψ £ !P α −→ PIN Ψ M . ↔ K Ψ £ M(x).P K N − − → P[x := N] COM Ψ⊗Ψ P ⊗Ψ Q M . ↔ K Ψ Q ⊗Ψ £ P M (ν a)N − −−−− → P Ψ P ⊗Ψ £ Q K N − − → Q Ψ £ P | Q τ −→ (ν a)(P | Q ) a#Q
Bisimulation
We proceed to define early bisimulation with the late semantics: We define P . ∼ Q to mean that there exists a bisimulation R such that R(1, P, Q). We also define P ∼ Q to mean that P[x := L]
. ∼ Q[x := L] for all x, L. The relation between this definition and the original definition of bisimulation in [5] is clarified by the following: Lemma 12. For the psi-calculi in the present paper, a relation is a bisimulation according to Def. 11 precisely if it is a bisimulation according to [5].
The proof is straightforward using Lemma 10. As a corollary the algebraic properties of ∼ established in [5] hold, notably that it is a congruence.
Symbolic semantics and equivalence
The idea behind a symbolic semantics is to reduce the state space of agents. One standard way is to avoid infinite branching in inputs by using a fresh name to represent whatever was received.
In psi-calculi there is an additional source of infinite branching: a subject in a prefix may get rewritten to many terms. Also here we use a fresh name to represent these terms. This means that the symbolic actions are the same as the concrete actions with the exception that only names are used as subjects.
A symbolic transition is of form
Ψ £ P α − → C P
The intuition is that this represents a set of concrete transitions, namely those that satisfy the constraint C.
Before the formal definitions we here briefly explain the rationale. Consider a psi-calculus with integers and integer equations; for example a condition can be "x = 3". An example agent is P = case x = 3 : P . If P α − − → true P , where true is a constraint that is always true, then there should clearly be a transition P α − → C P for some constraint C that captures that x must be 3. One context that can make this constraint true is an input, as in a(x).P. The input will give rise to a substitution for x, and if the substitution sends x to 3 the constraint is satisfied. In this way the constraints are similar to those for the pi-calculus [9,18]. In psi-calculi there is an additional way that a context can enable the transition: it can contain an assertion as in (|x := 3| ) | P. Concretely this agent has a transition (|x := 3| ) | P α −→ (|x := 3| ) | P since x := 3 x = 3. Therefore a solution of a constraint will contain both a substitution of terms for names (representing the effect of an input) and an assertion (representing the effect of a parallel component). Definition 13. The atomic constraints are of the form (ν a){|Ψ ϕ| } where a are binding occurrences into Ψ and ϕ. A solution of an atomic constraint is a pair (σ , Ψ ) where σ is a substitution of terms for names such that a#σ , Ψ and Ψσ ⊗Ψ ϕσ . We adopt the notation (σ , Ψ) |= C to say that (σ , Ψ) is a solution of C, and write sol(C) for {(σ , Ψ) : (σ , Ψ) |= C}.
The transition constraints are the atomic constraints C and conjunctions of atomic constraints C ∧C , where the solutions are the intersection of the solutions for C and C and we let (ν a)(C ∧ C ) mean (ν a)C ∧ (ν a)C . In the rule COM we assume that F (P) = (ν b P )Ψ P and F (Q) = (ν b Q )Ψ Q where b P is fresh for all of Ψ, b Q , Q and P, and that b Q is correspondingly fresh. We also assume that y, z#Ψ, b P , P, b Q , Q, N, b P , b Q , a.
IN Ψ £ M(x).P y(x) − −−−−− → {|Ψ M . ↔y| } P y#Ψ, M, P, x CASE Ψ £ P i α − → C P Ψ £ case ϕ : P α − −−−−− → C∧{|Ψ ϕ i | } P OUT Ψ £ M N.P y N − −−−−− → {|Ψ M . ↔y| } P y#Ψ, M, N, P COM Ψ⊗Ψ Q £ P y(ν a)N − −−−−−−−−−−−− → (ν b P ){|Ψ M P . ↔y| }∧C P P Ψ⊗Ψ P £ Q z(x) − −−−−−−−−−−−− → (ν b Q ){|Ψ M Q . ↔z| }∧C Q Q Ψ £ P | Q τ − −−−−−−−−−−−−−−−−− → (ν b P , b Q ){|Ψ M P . ↔M Q | }∧C P ∧C Q (ν a)(P | Q [x := N]) a#Q, y#z Ψ = Ψ⊗Ψ P ⊗Ψ Q PAR Ψ⊗Ψ Q £ P α − → C P Ψ £ P | Q α − −−− → (ν b Q )C P | Q bn(α)#Q α = τ ∨ subj(α)#Q SCOPE Ψ £ P α − → C P Ψ £ (νa)P α − −− → (νa)C (νa)P a#α, Ψ OPEN Ψ £ P y(ν a)N − −−− → C P Ψ £ (νa)P y(ν a∪a)N −−−−−→ (νa)C P a ∈ n(N) a# a, Ψ, y REP Ψ £ P | !P α − → C P Ψ £ !P α − → C P
In the rule PAR we assume that F (Q) = (ν b Q )Ψ Q where b Q is fresh for Ψ, P and α. In OPEN the expressionã ∪ {a} means the sequenceã with a inserted anywhere.
A transition constraint C defines a set of solutions sol(C), namely those where the entailment becomes true by applying the substitution and adding the assertion. For example, the transition constraint {|1 x = 3| } has solutions ([x := 3], 1) and (Id, x = 3), where Id is the identity substitution.
The structured operational symbolic semantics is defined in Table 3. First consider the OUT rule:
Ψ £ M N.P y N − −−−−− → {|Ψ M . ↔y| } P.
The symbolic subject y must be chosen fresh and has a constraint associated with it: the transition can be taken in any solution that implies that the subject M of the syntactic prefix is channel equivalent to y.
The rule COM is of particular interest. The intuition is that the symbolic action subjects are placeholders for the values M P and M Q . In the conclusion the constraint is that these are channel equivalent, while y and z will not occur again.
We will often write P
Symbolic bisimulation
In order to define a symbolic bisimulation we need additional kinds of constraints. If a process P does a bound output y (ν a)N that is matched by a bound output y (ν a)N from Q we need constraints that keep track of the fact that N and N should be syntactically the same, and that a is sufficiently fresh. Definition 14. The constraints include the transition constraints, {|M = N| }, and {|a#X| }, where X is any nominal data type. The solutions of the last two are all pairs (Ψ, σ ) such that Mσ = Nσ and a#(Xσ ) respectively. We also include conjunction of constraints C ∧C , where the set of solutions is the intersection of the solutions for C and C .
Note that the assertion part of the solution is irrelevant for constraints of kind {|M = N| } and {|a#X| }, and that the substitution does not affect a in {|a#X| }. The constraint {|M = N| } is used in the bisimulation for matching output objects, and {|a#X| } is used in the bisimulation for recording what an opened name must be fresh for. This corresponds to distinctions in open bisimulation for the pi-calculus [21]. We define true to be {|M = M| }, we write {|a#X,Y | } for {|a#X| } ∧ {|a#Y | }, and we extend the notation to sets of names, e.g. {| a#X| }. Definition 15 (Constraint implication). A constraint C implies another constraint D, written C ⇒ D, iff sol(C) ⊆ sol(D). We write C ⇒ C iff for each (σ , Ψ) ∈ sol(C) there exists a C ∈ C such that (σ , Ψ) ∈ sol(C ).
Before we can give the definition of symbolic bisimulation we need to define a symbolic variant of the concrete static equivalence. Definition 16 (Symbolic static equivalence). Two processes P and Q are statically equivalent for C, written P C Q, if for each (σ , Ψ) ∈ sol(C) we have that Ψ⊗F (P)σ Ψ⊗F (Q)σ .
We now have everything we need to define symbolic bisimulation. This definition follows the definition in [17] closely. Definition 17 ((Early) Symbolic bisimulation). A symbolic bisimulation S is a ternary relation between constraints and pairs of agents such that S (C, P, Q) implies all of 1. P C Q, and 2. S (C, Q, P), and 3. If P α − → C P P , bn(α)#(P, Q,C,C P , subj(α)) and subj(α)#(P, Q,C) then there exists a set of constraints C such that C ∧C P ⇒ C and for all C ∈ C there exists Q , α , and C Q such that We write P ∼ s Q if (true, P, Q) ∈ S for some symbolic bisimulation S , and say that P is symbolically bisimilar to Q. The set C allows a case analysis on the constraint solutions, as examplified in the next section. The output objects need to be equal in a solution to C . Since the solutions of {|N = N | } only depend on the substitutions, this constraint corresponds to the fact that the objects must be identical in the concrete bisimulation. Note that bn(α) may occur in C. Based on [9,18], we conjecture that adding the requirement bn(α)# C would give late symbolic bisimulation.
Examples
We now look at a few examples to illustrate the concrete and symbolic transitions and bisimulations. First consider a simple example from the pi-calculus. This can be expressed as a psi-calculus: let the only data terms be names, the only assertion be 1, the conditions be equality and inequality tests on names, and entailment defined by ∀a.1 a = a, ∀a, b : a = b.1 a = b and ∀a.1 a . ↔ a. For a more thorough discussion, see [5]. In the following examples we drop a trailing .0. Consider the two agents P 1 and Q 1 :
P 1 = a(x) . P 1 where P 1 = a b . a b Q 1 = a(x) . Q 1 where Q 1 = (case x = b : a b . a b [] x = b : a b . a b)
These are bisimilar. A concrete bisimulation between these agents is
{(1, P 1 , Q 1 )} ∪ n∈N {(1, P 1 , Q 1 [x := n]} ∪ {(1, a b, a b)}
The bisimulation needs to be infinite because of the infinite branching in the input. In contrast, a symbolic bisimulation only contains four triples:
(true, P 1 , Q 1 ), (true, P 1 , Q 1 ), ({|1 x = b| }, a b, a b), ({|1 x = b| }, a b, a b)
When checking the second triple (true, P 1 , Q 1 ), the transition of P 1 is matched by a case analysis: C in the definition of symbolic bisimulation (Def. 17) is {{|1 x = b| }, {|1 x = b| }}, and a matching transition for Q 1 can be found for each of these cases, so the agents are bisimilar. In contrast, they are not equivalent in the incomplete symbolic bisimulations in [11] and [15].
Next we look at an example where we have tuples of channels and projection, e.g. the entailment relation gives us that 1 first(M, N)
. For another example, consider the two agents
P 2 = F N . P Q 2 = 0
where F is a term such that for no Ψ, M does it hold that Ψ F . ↔ M, i.e., F is not a channel. Then we have that P 2 and Q 2 are concretely bisimilar since neither one of them has a transition. But symbolically P 2 has the transition P 2
y N −−−−−→ {|1 F .
↔y| } P , while Q 2 has no symbolic transition. Perhaps surprisingly they are still symbolically bisimilar: Def. 17 requires that we find a disjunction C such that C ∧ C P ⇒ C, or in this case such that true ∧ {|1 F . ↔ y| } ⇒ C. Since F is not channel equivalent to anything, the left hand side has no solutions, which means that any set C will do, and in particular the empty one. The condition "for all C ∈ C " in the definition becomes trivially true, so Q 2 does not have to mimic the transition.
A final example shows the use of cryptographic primitives. Here the terms contains enc(M, k) and dec(M, k), assertions are variable assignments, e.g. x := M, the conditions are equality tests between terms, and the entailment relation is parametrised by an equation system which contains the equation dec(enc(M, k), k) = M. Consider
P 3 = (νa, k) ((|x := enc(a, k)| ) | b(z) . b k . (case z = a : c d)) Q 3 = (νa, k) ((|x := enc(a, k)| ) | b(z) . b k)
Here the environment can use x, the result of encrypting a with k, but not the bound a or k. Intuitively these agents are bisimilar since the key k is not revealed until after the agents receive z, which therefore cannot be equal to a. The first symbolic transitions of the agents are
P 3 y(z) − −−−−−−−− → (νa,k){|1 b . ↔y| } (νa, k)((|x := enc(a, k)| ) | b k . (case z = a : c d)) = P 3 Q 3 y(z) − −−−−−−−− → (νa,k){|1 b . ↔y| } (νa, k)((|x := enc(a, k)| ) | b k) = Q 3
and the second transitions are
P 3 y (νk)k − −−−−−−−− → (νa,k){|1 b . ↔y | } (νa)((|x := enc(a, k)| ) | (case z = a : c d)) = P 3 Q 3 y (νk)k − −−−−−−−− → (νa,k){|1 b . ↔y | } (νa)((|x := enc(a, k)| )) = Q 3
A symbolic bisimulation, where we for simplicity ignore the constraints that arise for subjects, is
{(true, P 3 , Q 3 ), (true, P 3 , Q 3 ), ({|k#P 3 , Q 3 | }, P 3 , Q 3 )}
Here the constraint {|k#P 3 , Q 3 | } will among other things imply that k#z. The final transition of P 3 has the constraint (νa){|1 z = a| }, so we must find a disjunction C such that k#P 3 ,
Q 3 ∧ (νa){|1 z = a| } ⇒ C.
Since a is bound, the only way to find a solution to the left hand side is to find a value for z that evaluates to a. One candidate for a solution is ([z := dec(x, k)], 1), but because of the constraint k#z this does not work. In fact, there is no solution to the left hand side because of the freshness constraint on k and the fact that a is bound. This means that, as in the previous example, any disjunction C will do, and in particular the empty disjunction, and trivially Q 3 does not have to mimic the transition.
In contrast, if we swap the order of the inputs and the outputs in P 3 and Q 3 and try to construct the bisimulation relation we will discover that we do not get the constraint k#z. This means that ([z := dec(x, k)], 1) is a solution to C ∧C P in the definition of bisimulation, and that Q 3 must mimic the transition from P 3 . In this case the agents are not bisimilar.
Results
We now turn to showing that the concrete and symbolic equivalences coincide.
We define substitution on symbolic actions by τσ = τ, (y(x))σ = yσ (xσ ), and (y (ν a)N)σ = yσ (ν a)Nσ , where x, a#σ . We define the substitution σ · [y := M] for y#σ by (σ · [y := M])(x) = M if x = y, and σ (x) otherwise.
The following two lemmas show the operational correspondence between the symbolic semantics and the concrete semantics: given a symbolic transition where the transition constraint has a solution, there is always a corresponding concrete transition (Lemma 18) and vice versa (Lemma 19).
Lemma 18 (Correspondence symbolic-concrete). We assume in 1 and 2 that F (P) = b P , Ψ P and b P , b#y, Ψ, σ , P.
The proofs are by induction over the transition derivation (one case for each rule).
Theorem 20 (Soundness). Assume S is a symbolic bisimulation and let R = {(Ψ, Pσ , Qσ ) : ∃C.(σ , Ψ) |= C and (C, P, Q) ∈ S }. Then R is a concrete bisimulation.
The proof idea to show that R is a concrete bisimulation is to assume (Ψ, Pσ , Qσ ) ∈ R and that Pσ has a transition in environment Ψ. We use Lemma 19 to find a symbolic transition from P, then the fact that S is a symbolic bisimulation to find a simulating symbolic transition from Q, and finally Lemma 18 to find the required concrete transitions from Qσ .
Similarly to [17] we need an extra assumption about the expressiveness of constraints: for all R, P, Q such that R is a concrete bisimulation there exists a constraint C such that (Ψ, σ ) |= C ⇐⇒ (Ψ, Pσ , Qσ ) ∈ R. In order to determine symbolic bisimilarity in an efficient way we need to compute this constraint, which is easy for the pi-calculus [9,18,19] and harder (but in many practical cases possible) for cryptographic signatures [10]. These results suggest that our constraints are sufficiently expressive, but for other instances of psi-calculi we may have to extend the constraint language. We leave this as an area of further research.
Theorem 21 (Completeness). Assume that R is a concrete bisimulation and let S = {(C, P, Q) : (σ , Ψ) |= C implies (Ψ, Pσ , Qσ ) ∈ R}. Then S is a symbolic bisimulation.
The proof idea is the converse of the proof for Theorem 20. The expressiveness assumption of constraints mentioned above is needed in order to construct the disjunction of constraints in the symbolic bisimulation. From these two theorems we get:
Corollary 22 (Full abstraction). P ∼ Q if and only if P ∼ s Q.
Conclusion and Future Work
We have defined a symbolic operational semantics for psi-calculi and a symbolic bisimulation which is fully abstract wrt the original semantics. While the developments in [5] give meta-theory for a wide range of calculi of mobile processes with nominal data and logic, the work presented in this paper gives a solid foundation for automated tools for the analysis of such calculi.
As mentioned in the introduction, the purity of the original semantics of psi-calculi has made the symbolic semantics easier to develop. There are no structural equivalence rules (which are a complication in applied pi-calculus), the scope opening rule is because of this straight-forward which makes knowledge representation simpler than in spi-calculi, and the bisimulation less complex. Nevertheless, the technical challenges have not been absent: the precise design of the constraints and their solution has been delicate. Since assertions may occur under a prefix, the environment can change after a transition. Keeping the assertion Ψ in the transition constraints (on the form (ν a){|Ψ ϕ| }) essentially keeps a snapshot of the environment that gives rise to the transition. An alternative would be to use time stamps to keep track of which environment made which condition true, but that approach seems more difficult.
Our symbolic bisimulation is a strong equivalence which does not abstract the internal τ transitions. This is less useful for verification than a weak observational equivalence, but still a significant step towards mechanized verification. We are currently developing a weak bisimulation for psi-calculi, and are studying the correspondence to a barbed bisimulation congruence. Preliminary results indicate that lifting the symbolic bisimulation presented here to weak bisimulation will be unproblematic.
The original psi-calculi admit pattern matching in inputs. In a symbolic semantics this would lead to complications in the COM-rule, which should introduce a substitution for the names bound in the pattern. This means introducing more fresh names and constraints, and it is not clear that the convenience of pattern matching outweighs such an awkward semantic rule. We leave this as an area for further study.
For future work, we need to develop an algorithm for deciding symbolic bisimulation and implement it in a tool. A natural basis for this would be the algorithm given in [17]. Furthermore, the termination of the algorithm will depend on the properties of the parameters of the particular psi-calculus: it is easy to construct a psi-calculus where the entailment relation or static equivalence is not decidable, but in many practical cases it will be [10,4]. We intend to use constraint solvers developed for specific application domains (e.g. security) in a future generic tool. We will also produce mechanized proofs of the adequacy of the symbolic semantics, using the Isabelle theorem prover.
When typing schemes have been developed for psi-calculi, a natural progression would be to take advantage of those also in the symbolic semantics, to further constrain the possible values and thus the size of state spaces.
Definition 8 (
8Actions). The actions ranged over by α, β are of the following three kinds: M (νã)N (Output), M(x) (Input), and τ (Silent). For actions we refer to M as the subject and N and x as the objects. We let subj(M (ν a)N) = subj(M(x)) = M. We define bn(M (νã)N) =ã, bn(M(x)) = {x}, and bn(τ) = / 0. We also define n(τ) = / 0 and n(α) = n(N) ∪ n(M) if α is an output or input. As in the pi-calculus, the output M (νã)N represents an action sending N along M and opening the scopes of the namesã. Note in particular that the support of this action includesã. Thus M (νa)a and M (νb)b are different actions.
→ Q in the early semantics iff there exist Q and x such that Ψ £ P M(x) − −− → Q in the late semantics, where Q = Q [x := N]. 2. For output and τ actions, Ψ £ P α −→ Q in the early semantics iff the same transition can be derived in the early semantics.
Definition 11 ((Early) Bisimulation). A bisimulation R is a ternary relation between assertions and pairs of agents such that R(Ψ, P, Q) implies all of 2. Symmetry: R(Ψ, Q, P) 3. Extension of arbitrary assertion: ∀Ψ . R(Ψ⊗Ψ , Q and R(Ψ, P [x := L], Q [x := L]). (b) otherwise: Ψ £ P α −→ P =⇒ ∃Q . Ψ £ Q α −→ Q and R(Ψ, P , Q ).
) C ⇒ C Q , and (c) if α = y (ν a)N then α = y (ν a)N , C ⇒ {|N = N | }, and (C ∧ {| a#P, Q| }, P , Q ) ∈ S otherwise α = α and (C , P , Q ) ∈ S
↔ M. Consider the agent R = M N . R Concretely this agent has infinitely many transitions even in an empty frame: R M N − − → R , and equivalent actions first(M, K) N for all K, and first(first(M, L), K) N for all L and K, etc. Symbolically, however, it has only one transition:
for all (σ , Ψ) ∈ sol(C) s.t. x#σ we have that Ψ £ Pσ for all (σ , Ψ) ∈ sol(C) s.t. a#σ we have that Ψ £ Pσ for all (σ , Ψ) ∈ sol(C) we have that Ψ £ Pστ −→ P σ .Lemma 19 (Correspondence concrete-symbolic).1. If Ψ £ Pσ M(x) − −− → P σ ,y#P, σ , M, N, x, and x#σ , P then there exists b, M P , and C P (σ · [y := M], Ψ) ∈ sol((ν b){|Ψ P M P . ↔ y| } ∧C P ). 2. If Ψ £ Pσ M (ν a)Nσ −−−−−→ P σ , y#P, σ , M, a, and a#σ , P then there exists b, M P , (σ · [y := M], Ψ) ∈ sol((ν b){|Ψ P M P . ↔ y| } ∧C P ).3. If Ψ £ Pσ τ −→ P σ then there exists C such that P (σ , Ψ) ∈ sol(C).
Table 1 :
1Late operational semantics. Symmetric versions of COM and PAR are elided.
Table 3 :
3Transition rules for the symbolic semantics. Symmetric versions of COM and PAR are elided.
In some presentations frames have been written just as pairs b, Ψ . The notation in this paper better conveys the idea that the names bind into the assertion, at the slight risk of confusing frames with agents. Formally, we establish frames and agents as separate types, although a valid intuition is to regard a frame as a special kind of agent, containg only scoping and assertions. This is the view taken in[2].
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"Alexandre Bouchard-Côté \nDepartment of Statistics\nDepartment of Computer Science\nColumbia Department of Pathology and Laboratory Medicine\nColumbia Department of Molecular Oncology, BC Cancer Agency\nUniversity of British Columbia\nUniversity of British\nUniversity of British\n\n",
"Andrew Roth [email protected] \nDepartment of Statistics\nDepartment of Computer Science\nColumbia Department of Pathology and Laboratory Medicine\nColumbia Department of Molecular Oncology, BC Cancer Agency\nUniversity of British Columbia\nUniversity of British\nUniversity of British\n\n"
]
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"Department of Statistics\nDepartment of Computer Science\nColumbia Department of Pathology and Laboratory Medicine\nColumbia Department of Molecular Oncology, BC Cancer Agency\nUniversity of British Columbia\nUniversity of British\nUniversity of British\n",
"Department of Statistics\nDepartment of Computer Science\nColumbia Department of Pathology and Laboratory Medicine\nColumbia Department of Molecular Oncology, BC Cancer Agency\nUniversity of British Columbia\nUniversity of British\nUniversity of British\n"
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| Bayesian feature allocation models are a popular tool for modelling data with a combinatorial latent structure. Exact inference in these models is generally intractable and so practitioners typically apply Markov Chain Monte Carlo (MCMC) methods for posterior inference. The most widely used MCMC strategies rely on an element wise Gibbs update of the feature allocation matrix. These element wise updates can be inefficient as features are typically strongly correlated. To overcome this problem we have developed a Gibbs sampler that can update an entire row of the feature allocation matrix in a single move. However, this sampler is impractical for models with a large number of features as the computational complexity scales exponentially in the number of features. We develop a Particle Gibbs sampler that targets the same distribution as the row wise Gibbs updates, but has computational complexity that only grows linearly in the number of features. We compare the performance of our proposed methods to the standard Gibbs sampler using synthetic data from a range of feature allocation models. Our results suggest that row wise updates using the PG methodology can significantly improve the performance of samplers for feature allocation models. arXiv:2001.09367v1 [stat.CO] 25 Jan 2020 Arnaud Doucet and Adam M Johansen. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009.Paul Fearnhead and Peter Clifford. On-line inference for hidden markov models via particle filters. | null | [
"https://arxiv.org/pdf/2001.09367v1.pdf"
]
| 210,921,186 | 2001.09367 | 85184c7d71843ed92d362ff1271856bf983e5968 |
Particle-Gibbs Sampling For Bayesian Feature Allocation Models
Alexandre Bouchard-Côté
Department of Statistics
Department of Computer Science
Columbia Department of Pathology and Laboratory Medicine
Columbia Department of Molecular Oncology, BC Cancer Agency
University of British Columbia
University of British
University of British
Andrew Roth [email protected]
Department of Statistics
Department of Computer Science
Columbia Department of Pathology and Laboratory Medicine
Columbia Department of Molecular Oncology, BC Cancer Agency
University of British Columbia
University of British
University of British
Particle-Gibbs Sampling For Bayesian Feature Allocation Models
Corresponding address: 2366 Main Mall, Vancouver, BC, Canada V6T 1Z4 Editor:
Bayesian feature allocation models are a popular tool for modelling data with a combinatorial latent structure. Exact inference in these models is generally intractable and so practitioners typically apply Markov Chain Monte Carlo (MCMC) methods for posterior inference. The most widely used MCMC strategies rely on an element wise Gibbs update of the feature allocation matrix. These element wise updates can be inefficient as features are typically strongly correlated. To overcome this problem we have developed a Gibbs sampler that can update an entire row of the feature allocation matrix in a single move. However, this sampler is impractical for models with a large number of features as the computational complexity scales exponentially in the number of features. We develop a Particle Gibbs sampler that targets the same distribution as the row wise Gibbs updates, but has computational complexity that only grows linearly in the number of features. We compare the performance of our proposed methods to the standard Gibbs sampler using synthetic data from a range of feature allocation models. Our results suggest that row wise updates using the PG methodology can significantly improve the performance of samplers for feature allocation models. arXiv:2001.09367v1 [stat.CO] 25 Jan 2020 Arnaud Doucet and Adam M Johansen. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009.Paul Fearnhead and Peter Clifford. On-line inference for hidden markov models via particle filters.
Introduction
Bayesian feature allocation models posit that observed data is generated by a collection of latent features with the aim of obtaining an interpretable and sparse representation of the data. A concrete way to represent a feature allocation is using a binary matrix, where the rows of this matrix represent data points or observations and the columns represent features. Common prior distributions for these binary matrices include the finite dimensional Beta-Bernoulli (FBB) model and the nonparametric Indian Buffet process (IBP) (Griffiths and Ghahramani, 2011). Exact inference for models which use these prior distributions is generally intractable, so practitioners often appeal to Monte Carlo Markov Chain (MCMC) approaches. A straightforward Gibbs sampler can be derived for such models which proceeds by updating a single entry of the binary matrix conditioned on the values of the remaining entries. While relatively simple to implement, this sampler can be extremely slow to mix due to the correlation among the feature allocation variables. In this work we show that it is possible to derive a simple Gibbs sampler which updates the entire feature usage vector of a data point (row of the binary matrix) jointly. When the number features (columns of the matrix), K, is small this sampler is practical and can significantly improve the mixing of the MCMC chain. However, this sampler is computationally expensive, requiring 2 K evaluations of the likelihood. We show that it is possible to sample efficiently from the distribution targeted by this row Gibbs update using the Particle Gibbs (PG) methodology (Andrieu et al., 2010). Our PG sampling approach has computational complexity that scales linearly with the number of features.
In sequel we will first review Bayesian feature allocation models and the relevant prior distributions on binary matrices. Next we will describe the new row wise Gibbs update and explain how to use the PG methodology to efficiently sample from the target distribution. We then compare these new samplers to existing approaches on a range of synthetic datasets using several previously published models. Finally, we conclude with a discussion and some thoughts on future directions.
Related work
The widely used Gibbs sampler which updates the feature allocation of each data point sequentially was first introduced by Ghahramani and Griffiths (2006). Later Meeds et al. (2007) described the use of Metropolis-Hastings (MH) moves to update multiple components of the feature allocation vector for a data point. They observed that larger moves in the space of feature allocations improved mixing, though this was never formally benchmarked. While the MH move partially addresses the issue of highly correlated features, it becomes impractical as the number of features grows, as large moves proposed at random will increasingly be rejected. An alternative approach to speeding up sampling for feature allocation models was proposed by Doshi-Velez and Ghahramani (2009). The main idea of that work was to partially marginalize elements of the model to improve mixing. This is not a general strategy however, as it requires conjugacy. Wood and Griffiths (2007) proposed the use of particle filters to fit matrix factorization models using IBP priors. In contrast to our approach, they used a single pass particle filter sampling the entire feature allocation matrix. They showed that this approach could significantly outperform single entry Gibbs sampling. However, the single pass particle filter approach does not scale well to models with large numbers of data points or features due to the degeneracy of standard particle filter methods. In contrast PG algorithms are not subject to this degeneracy. Broderick et al. (2013) pointed out the predictive distribution of the feature allocation models could be written as a product of Bernoulli distributions. However, they did not appear to pursue the obvious row wise Gibbs sampler that this implies. Fox et al. (2014) proposed the use of split-merge moves to improve the mixing of features. While the sequential nature of these proposals bear some similarity to our method, they differ in that this previous work updates the columns of the feature allocation matrix as opposed to the rows. As a result they need to be interleaved with element wise Gibbs updates to obtain adequate mixing. The methods we propose in this work can be used in place of the element wise Gibbs update with the moves proposed by Fox et al. (2014) to further improve performance.
Methods
Here we review the basic background about feature allocation priors and introduce the standard Gibbs updating procedure. We then explain how to implement an exact Gibbs sampler for updating an entire row of the feature allocation matrix. Next we show how to construct a PG sampler to target the conditional distribution the row wise Gibbs samples from. We then discuss two strategies for improving the efficiency of the basic PG algorithm.
Notation
We use bold letters for (random) vectors, capital letters for matrices and normal fonts for (random) scalars and sets. For quantities such as an individual observation x n , or a parameter θ, which can be either scalars or vectors without affecting our methodology, we consider them as scalars without loss of generality. Given a vector z = (z 1 , . . . , z K ), and i ≤ j, we use z i:j to denote the sub-vector z i:j = (z i , z i+1 , . . . , z j ). For a permutation, σ, we let y[σ] = (y σ(1) , . . . , y σ(K) ) denote vector obtained by permuting the entries of y by σ. For a permutation σ we define the inverse permutation σ −1 to be the permutation such that (y[σ])[σ −1 ] = y. To simplify notation, we do not distinguish random variables from their realization. We define discrete probability distributions with their probability mass functions, and continuous probability distributions with their density functions with respect to the Lebesgue measure.
Feature allocation
Intuitively a feature allocation model ascribes a set of features that are exhibited by a set of data points x n . At the core of these models is the combinatorial stochastic feature allocation object. To formally define a feature allocation we follow the description in Broderick et al. (2013). Let [N ] = {1, . . . , N }, then a feature allocation f N of [N ] is defined to be a multi-set of non-empty sets of [N ]. Let f N = {A N 1 , . . . , A N K } where we refer to the elements A N k as blocks. Each block represents the assignment of data points to a feature. For example consider the feature allocation f 3 = {{1}, {1, 2}, {2, 3}}. In this feature allocation the first feature is exhibited by data point 1, the second feature by data points 1 and 2, and the third feature by data points 2 and 3. In contrast to partitions which are frequently used in clustering models, feature allocations do not require data points to be in mutually exclusive blocks or in fact to be in any block. If we let z n,k = I(n ∈ A N k ) then we can map the feature allocation f N to a binary matrix Z ∈ {0, 1} N ×K . The rows of Z represent data points and the columns represent features. We use the notation z n = (z n,1 . . . z n,K ) to denote the n th row of Z, that is the vector indicating which features data point n uses. Note that the ordering of features is arbitrary so that the matrix Z is only defined up to a permutation of the columns and strictly speaking the feature allocation prior distribution is defined on the equivalence class of matrices that are identical up to a permutation of their columns. An alternative way to define this equivalence class is as the set of matrices which are equivalent when put into left ordered form (Griffiths and Ghahramani, 2011). In sequel we will abuse notation and not make the distinction between a feature allocation f N and its binary matrix representation Z.
Feature allocation prior distributions
To specify a Bayesian feature allocation model we need to define a prior distribution for the feature allocation. In this work we consider the two most widely used prior distributions for feature allocations, the Finite Beta-Bernoulli (FBB) distribution and Indian Buffet Process (IBP). Below we give: the probability mass function of these distributions; the probability that a data point n exhibits feature k, ρ n,k ; and the predictive distribution when adding a new data point. The predictive distribution is defined as
p(f N +1 |f N ) = p(f N +1 ) p(f N ) . Let K N = |f N | and m k = |A N k | = N n=1
z n,k , then these quantities are as follows:
• FBB with K features p(f N ) = I(K N = K) K k=1 Γ(m k + a)Γ(N − m k + b) Γ(N + a + b) ρ N +1,k = m k + a N + a + b p(f N +1 |f N ) = K k=1
Bernoulli (z N +1,k |ρ N +1,k )
• Indian Buffet Process
p(f N ) = α K N K N ! K N k=1 Γ(m k )Γ(N − m k + 1) Γ(N + 1) ρ N +1,k = m k N + 1 p(f N +1 |f N ) = Poisson K + N +1 α N + 1 K N k=1 Bernoulli (z N +1,k |ρ N +1,k ) where K + N +1
is the number of singletons (unique) features exhibited by data point N + 1. We note that this definition of the IBP prior differs slightly from the original one defined in Ghahramani and Griffiths (2006). This construction is due to Broderick et al. (2013) and results in an exchangeable prior as the probability mass functions only depend on the number of features and size of blocks. As we will see later this is useful for defining a Gibbs sampler for updating the feature allocation variable.
Bayesian feature allocation models
To fully specify a Bayesian feature allocation model we need two additional elements. First, a set of parameters θ = (θ 1 , . . . , θ K ) associated with the features. We will assume that θ k are drawn i.i.d. from a common distribution so that the features are exchangeable. Second, we need to define a likelihood for our data X = (x 1 , . . . , x N ) T which depends on our feature allocation matrix Z and the feature parameters θ. We also assume that the data points are exchangeable so the likelihood takes the form p(X|θ, Z) = N n=1 p(x n |z n , θ). In order for the model to remain exchangeable we require that for any permutation, σ, that p(x n |θ, z n ) = p(x n |θ[σ], z n [σ]). With these assumptions the full joint distribution is given by Equation 1.
p(X, Z, θ) = p(Z) K k=1 p(θ k ) N n=1 p(x n |z n , θ)(1)
In general the component distributions will also depend on additional hyper-parameters which may also have prior distributions. For notational clarity we have suppressed these terms and any dependencies on these hyper-parameters. As a concrete example consider the linear Gaussian feature allocation model.
Z ∼ IBP(α) θ k ∼ N (0, I) x n |σ, θ k , z n ∼ N k z n,k θ k , σ 2 I
This model assumes the feature parameters follow a multivariate normal distribution, and the data follow a multivariate normal distribution with a mean which is the sum of the features that a data point exhibits. Note that the likelihood is invariant to permutations of the feature indexes due to the linear sum construction. This model is thus exchangeable in both data points and features.
Gibbs updates
Since data points are exchangeable we can use p(f N +1 |f N ) to derive a simple Gibbs sampler to update the entries of Z by assuming we are observing the last data point to be assigned. Let
Z (−n) = {z i } ip(z n,k = 1|x n , Z (−n) , θ) ∝ ρ n,k × p(x n |z n , θ)(2)
The update for the IBP prior is slightly more complex and is performed in two parts. We update columns for features which are also exhibited by other data points using the Gibbs update in Algorithm 1 as for the FBB. The columns for features exhibited only by the current data point, singletons, are then updated with another move which leaves the target distribution invariant. The simplest of these is to use a Metropolis-Hastings update where the number of singletons is proposed from the Poisson with parameter α N , and the corresponding feature values from their prior distributions. The methods we describe in this work only applies to the non-singleton updates, and can be used with any update for the singletons.
Algorithm 1 Sample a row of the feature allocation using the element wise Gibbs update.
1: function ElementWiseGibbsUpdate(x n , ρ n , z n , σ) 2:
for k ∈ σ do Iterate over columns in random order.
3:
z n,k ← 0 4: p 0 ← (1 − ρ n,k ) × p(x n |z n , θ) 5: z nk ← 1 6: p 1 ← ρ n,k × p(x n |z n , θ) 7: p 1 ← p 1 p 0 +p 1 8: z nk ∼ Bernoulli(·|p 1 ) 9:
end for 10:
return z n 11: end function 2.6 Row wise Gibbs updates
The element wise Gibbs update has been widely used. It only requires O(K) evaluations of the likelihood function to update a row. However, the resulting sampler can be extremely slow to mix due to correlations between the features. The form of the predictive distributions for the FBB and IBP priors suggests an alternative Gibbs update that could potentially lead to better mixing. Rather than sample a single entry at a time, instead update an entire row, z n , of the feature allocation matrix. This can be done by using the update defined by Equation 3 leading to Algorithm 2 for updating a row. Again, this update only applies to the non-singleton entries when using the IBP prior.
p(z n = z|X, Z (−n) , θ) ∝ p(x n |z n , θ) K k=1 Bernoulli (z k |ρ n,k )(3)
In order to sample from distribution defined by Equation 3 we need to enumerate all possible binary vectors of length K and evaluate the likelihood function. This approach leads to a sampler with computational complexity O(2 K ). For moderate values of K, particularly if we are using the parametric FBB prior, this is a practical sampler. However, the exponential scaling in K will render this approach infeasible for larger numbers of features. This is especially problematic when using the IBP prior, as K varies between iterations.
Algorithm 2 Sample a row of the feature allocation using the row wise Gibbs update. for z ∈ {0, 1} K do Iterate over all possible feature allocation vectors.
5:
j ← j + 1 6:
S ← (S, z) Add z to list of visited vectors 7: We now describe how to sample from p(z n = z|X, Z (−n) , θ) with computational complexity O(K) by using the Particle Gibbs (PG) methodology (Andrieu et al., 2010). PG sampling is a form of Sequential Monte Carlo (SMC) sampling (Doucet and Johansen, 2009). Like all SMC algorithms the PG approach proceeds by approximating a sequence of distribution using a set of interacting particles. Resampling is periodically used to prune particles which, informally, are exploring low probability regions. There are three key quantities that need to be defined when constructing an SMC sampler:
p j ← p(x n |z, θ) k (1 − ρ n,k ) (1−z k ) ρ z k n,
1. The sequence of target distributions {γ t } T t=1 used to weigh the particles at each time step.
2. The sequence of proposal distributions {q t } T t=1 used to extend particles between time steps.
3. The resampling distribution r(·|w t−1 ).
The key difference between PG and other SMC approaches is that we are updating a set of variables which have already been instantiated. We would like to do this in way that leads to a valid kernel targeting the conditional distribution p(z n = z|X, Z (−n) , θ). To accomplish this we need to include a conditional path, that is a particle trajectory which follows the sequence of choices required to generate the initial value before the update. This trajectory will always be included after the resampling steps. Thus the resampling step is conditional on including the particle representing this trajectory. Intuitively this forces the sampler to explore regions of space around the existing value.
To simplify the bookkeeping and algorithm implementation we always assume the first particle is the conditional path. The PG sampler is still valid when this is done as shown by Chopin et al. (2015). SMC algorithms are commonly used for models with a natural sequential structure, such as state space models. This in turn identifies a natural sequence of target distributions defined on an expanding state space. Our setup is non-standard in that no natural sequential structure is defined. To sample from p(z n = z|X, Z (−n) , θ) we will define a sequence of distribution which updates one entry of the feature allocation vector z n at each time step. Thus if we have K features we will define a sequence of T = K target distributions. For the FBB we take T to be the fixed value of K and update all elements. For the IBP T is taken to be the number of elements such m (−n) k > 0 and we only update the corresponding feature assignments.
At time step t of the algorithm the particles will take values ξ t ∈ {0, 1} t , that is we record the sequence of binary decisions up to point t. In order to evaluate the likelihood term we need to set the values of the feature vector which have not been updated at time t. To do this we introduce an auxiliary variable which we call the the test path denotedz. We discuss and compare possible strategies for selectingz later. An illustration of the method is given in Figure 1.
We randomly order the features before each update by a permutation σ so that at time t we sample component σ(t) of the feature allocation vector. To obtain a complete feature vector to evaluate the likelihood we define the function given by Equation 4 which returns a binary vector where the entries σ(1 : t) have been set to the sampled values and the remaining entries are set to the test path. The entries are then reordered by the inverse permutation σ −1 . z(t, σ, ξ t ,z) = (ξ 1 , . . . , ξ t ,z σ(t+1) , . . . ,z σ(T ) )[σ −1 ]
For the IBP the singleton entries are fixed to one and deterministically inserted when evaluating the likelihood. We use the sequence of target distributions defined in Equation 5. We have γ T (ξ T |ρ n , σ,z) ∝ p(z n = z|X, Z (−n) , θ), that it the target density at the final iteration is proportional to the density of the distribution of interest. This is the key constraint required to define a valid sequence of target distributions.
γ t (ξ t |ρ n , σ,z) = p(x n |z(t, σ, ξ t ,z), θ) t s=1 ρ (1−ξs) nσ(s) ρ ξs nσ(s)(5)
The second component we need for our algorithm is a sequence of proposal distributions. Here we exploit the fact that our proposal space is {0, 1} and use the fully adapted proposal kernel defined in Equations 6 and 7. We use (ξ t−1 , ξ t ) to denote the concatenation ξ t to ξ t−1 and ξ t = (ξ t−1 , ξ t ).
q 1 (ξ 1 ) = γ 1 (ξ 1 |ρ n , σ,z) ξ 1 ∈0,1 γ 1 ((ξ 1 )|ρ n , σ,z) (6) q t (ξ t |ξ t−1 ) = γ t (ξ t |ρ n , σ,z) ξt∈0,1 γ t ((ξ t−1 , ξ t )|ρ n , σ,z)(7)
Given our choice of proposal and target distributions the incremental weight functions are defined by Equations 8 and 9. To reduce computational overhead p(x n |z(t, σ, ξ t ,z), θ) can be cached to avoid re-evaluation of the likelihood term in the denominator of Equation 9.
w 1 (ξ 1 ) = γ 1 (ξ 1 |ρ n , σ,z) q 1 (ξ 1 ) = ξ 1 ∈{0,1} γ 1 ((ξ 1 )|ρ n , σ,z) (8) w t (ξ t |ξ t−1 ) = γ t (ξ t |ρ n , σ,z) γ t−1 (ξ t−1 |ρ n , σ,z)q t (ξ t |ξ t−1 ) = ξt∈{0,1} γ t ((ξ t−1 , ξ t )|ρ n , σ,z) γ t−1 (ξ t−1 |ρ n , σ,z)(9)
The final component we need to define our PG algorithm is a resampling distribution. For simplicity we use multinomial resampling, however more sophisticated approaches such as stratified sampling could also be use. Our resampling distribution deterministically includes the conditional path, which we arbitrarily assign to particle index 1. The conditional multinomial resampling distribution is given by Equation 10 where a ∈ {1, . . . , P } P is the vector of ancestor indices, w the vector of normalized particle weights and P is the number of particles.
r(a|w) = I (a 1 = 1) P i=2 P j=1 w I(a i =j) i(10)
Algorithm 3 Sample a row of the feature allocation using the particle Gibbs update.
1: function ParticleGibbsUpdate(x n , z n , σ, ρ n ,z) 2: One potential pitfall of the target distribution is that due to the correlation among features, it is difficult to change a feature from its current values. This will be particularly acute if there is a need to move through a low probability configuration. A simple strategy to mitigate this is to consider an different family of target distributions which anneals the likelihood defined in Equation 11.
ξ 1 T ← z n [σ] Set conditional path 3: for t ∈ {1, . . . , T − 1} do 4: ξ 1 t ← (ξ 1 T ) 1if (P i (w i t−1 ) 2 ) −1 < τξ i t ∼ q t (·|ξ a i t−1 ) 25: ξ i t ← (ξ a i t−1 , ξ i t )w i t ← w a i t−1 w t (ξ i t |ξ a i tz ∼ P i=1 w i T δ ξ i T(γ β,t (ξ t |ρ n , σ,z) = p(x n |ξ t , θ) ( t T ) β t s=1 ρ (1−ξs) nσ(s) ρ ξs nσ(s)(11)
It can easily be checked that γ β,T (z) ∝ p(z n = z|X, Z (−n) , θ) so this sequence of target densities does indeed target the correct distribution. Also note, the original sequence of densities is recovered if β = 0.
2.9 Discrete particle filtering SMC algorithms are known to be inefficient in cases where the target distribution is discrete. This is due to the computation and storage of redundant particles. Fearnhead and Clifford (2003) addressed this problem by designing an SMC approach tailored to discrete state spaces. The key difference is that their approach deterministically expands each particle to test all available extensions, which is possible due to the discrete nature of the space. In order to avoid storing an exponentially expanding system of particles, they introduce an approach to deterministically keep particles with high weights while resampling from those with low weights. Their approach guarantees that no more that |X |M particles will be created, where X is the discrete state space and M is a user specified value. Whiteley et al. (2010) later showed that this approach could be adapted to the Particle Gibbs framework. We refer to this approach as the discrete particle filter (DPF). In practice we use a slightly different version which was proposed by Barembruch et al. (2009). This version sets the expected number of particles to M instead of fixing it at exactly M . We have found this implementation to be more stable numerically. The resampling procedure is outlined in Algorithm 4. There is no proposal densities in the DPF algorithm so we obtain a different set of weight functions from the PG algorithm which are given by Equations 12 and 13. As for the PG algorithm it is useful to cache p(x n |z(t, σ, ξ t ,z), θ) to avoid re-evaluation of the likelihood term in the denominator. When using annealing the corresponding target densities are substituted in the weight functions. The full details of the DPF are given in Algorithm 5. Again to simplify the bookkeeping our proposed algorithm always assigns the conditional path to the first particle, and this is enforced during resampling.
w 1 (ξ 1 ) = γ 1 (ξ 1 |ρ n , σ,z) (12) w t (ξ t |ξ t−1 ) = γ t (ξ t |ρ n , σ,z) γ t−1 (ξ t−1 |ρ n , σ,z)(13)
Algorithm 4 Conditional resampling for DPF. return a, w new , j 28: end function Algorithm 5 Sample a row of the feature allocation using the discrete particle filter update.
1: function ResampleDPF(w, M , P ) 2: c ← findRoot( P i=1 min(1, xw i ) − M ) Find
1: function DiscreteParticleFilter(x n , z n , σ, ρ n ,z, M ) 2:
ξ 1 T ← z[σ] Set conditional path 3: for t ∈ {1, . . . , T − 1} do 4:
ξ 1 t ← (ξ T ) 1:t First particle of each generation matches conditional path 5:
ξ 2 t ← (ξ 1 t−1 , 1 − (ξ 1 t ) t ) Expandfor i ∈ {1, 2} do 9: w i 1 ← w 1 (ξ i 1 ) Initialize incremental importanceξ j t ← (ξ a i t−1 , z) Expandw i t ← w a i t−1 w t (ξ i t |ξ a i t−1 ) Update incremental importancew i t ← w i t j w j tz ∼ P i=1 w i T δ ξ i T (·) Sample updated feature allocation 36: z ← z[σ −1 ]
Reorder sampled feature allocation vector by inverse of σ 37:
return z 38: end function
Results
We first demonstrate the potential slow mixing of the standard Gibbs sampler on a toy dataset and illustrate how the row wise Gibbs updates can alleviate this problem. Next we explore how to tune the parameters of the PG and DPF samplers. We then compare the behaviour of the Gibbs sampler and our proposed methods on a number of synthetic datasets.
We have compared the performance of the Gibbs, Row Gibbs (RG), Particle Gibbs (RG) and Discrete Particle Filter (DPF) using three models. The first model we tested with was the Linear Gaussian (LG) model, which has been widely used in the IBP literature (Griffiths and Ghahramani, 2011). The second model we considered was the Latent Feature Relational Model (LFRM) proposed by Miller et al. (2009). The final model we consider is a modified version of the PyClone model used for inferring population structure from admixed data in cancer genomics (Roth et al., 2014). The original PyClone model clusters sets of mutations which appear in a similar proportion of cells.
We have modified this model to use feature allocations to indicate which cell populations have each mutation. Full details of the models and the updates used for parameters are in the Section 5.1.
When comparing methods we applied the Friedman test to see if there were any significant difference in performance between the methods (p-value < 0.001). If the Friedman test was significant we then applied the post-hoc Nemenyi test with a Bonferroni correction to all pairs of models to determine which models showed significantly different performance from each other (p-value < 0.001) (Demšar, 2006). All statements of significance are with respect to this test. Because the samplers have different computational complexity per iteration, we report the results using wall clock time instead of per iteration. This ensures a fair comparison, as for example, we can perform many more updates using the Gibbs sampler than the PG sampler in a given time interval. We report the relative log density when comparing methods to better represent how far away from convergence the samplers are. Letˆ be the log density of the data under the true parameters used for simulation and the observed log density. The relative log density is given by −ˆ ˆ . Code implementing the samplers and models is available online at https://github.com/ aroth85/pgfa. All experiments were done using version 0.2.2 of the software. Code for performing the experiments is available online at https://github.com/aroth85/pgfa_experiments.
Row updates improve mixing
To illustrate the potential benefits of using row wise updates, we first consider a simple pedagogical example. We simulated N = 100 data points from the linear Gaussian model with D = 1, K = 2, τ v = 0.25, and τ x = 25. We set the value of the feature parameters V to be 100 for both features with half of the data points using the first feature and half using the second feature. For inference we used the FBB prior with K = 2, a = 0.5 and b = 1. This prior distribution for the feature allocation heavily favours the configuration where all data points use one feature and the other is not used. Because both features have identical values, there is no difference is likelihood for a data point to use one feature or the other. Thus, if a sampler is mixing efficiently it should quickly assign all data points to one feature and none to the other. We ran both the element wise Gibbs and row wise Gibbs samplers for 100 seconds recording the value of the log joint probability and number of data points that used each feature at each iteration. We set all model parameters except the feature allocation to their true values, and did not update them in contrast to the remaining experiments where the feature parameters are updated. We show the trace of the log joint probability in Figure 2 a). The element wise Gibbs sampler (blue) is clearly trapped in a local mode from initialization and cannot move away from the initial configuration. This is due to the need for the element wise Gibbs sampler to traverse a region of low probability to use the other feature. Specifically, a data This contrived example clearly illustrates the potential for slow mixing that element wise updates can cause and that row wise updates can solve the problem. We will see that this behaviour is a general phenomenon of the element wise Gibbs sampler, even when the initialization is not constructed to be adversarial as in this case.
Setting tuning parameters
The PG and DPF samplers have a number of tuning parameters which affect performance. We explored the impact these parameters have on performance using synthetic data generated from the LG model. We generated four datasets and four sets of initial parameter values. For all combinations of datasets and initial parameters we performed five random restarts of the sampler. Thus we executed 80 chains for each method considered, all with the same data and parameter initialization. Data was simulated from the LG model using the FBB prior with α = 2, τ v = 0.25, τ x = 25, D = 10, K = 20 and N = 100. These parameters were chosen to generate datasets where we would expect the sampler to converge to the true parameter values used for simulation. We randomly assigned 10% of the data matrix to be missing and used these entries to compute root mean square reconstruction error (RMSE). For each experiment we varied a single tuning parameter, setting the remaining parameters to default values, namely an annealing power of 1.0, a number of particles of 20, a resampling threshold of 0.5, and test paths consisting of a vector of zeros.
Number of particles
The first parameter we explore is the number of particles. For standard SMC algorithms a large number of particles are typically used, as this parameter ultimately controls the quality of the Monte Carlo approximation. In contrast to standard SMC, the Particle Gibbs framework is less sensitive to the number of particles. This is primarily a result of the fact many conditional SMC (cSMC) moves can be used within a sampling run, in contrast to the one shot approach of SMC. For our particular problem the length of the cSMC runs also tends to be short because we rarely expect very large numbers of features to be used in the model. As a result, our updates will also be less sensitive to path degeneracy. We benchmarked the PG and DPF algorithms using varying number of particles ( Figures S5-S7). Both algorithms appear to be relatively insensitive to the number of particles used. Using 50 and 100 particles leads to significantly worse performance for both the PG (Tables S1 and S2) and DPF samplers (Tables S3 and S4) than using fewer particles after the algorithms have run for 10 seconds. However, after 1000 seconds there were no significant differences between the runs with different numbers of particles for either method. One surprising feature is that runs using as few as two particles still work well. We caution this observation may not hold for other models or larger numbers of features, and more particle may be required. We also note that it is possible to parallelize these samplers across particles which would allow for more particles to be used, though we did not investigate this.
Resampling threshold
We use an adaptive resampling scheme for the PG algorithm, whereby resampling only occurs if the relative effective sample size (ESS) falls below a specified threshold. The DPF algorithm does not require this tuning parameter as the resampling mechanism is deterministic. Figures S8 and S9 shows the results of the benchmark experiment. The performance of the PG algorithm was insensitive to the value of this parameter with the exception of using a threshold of 1.0 which corresponds to always resampling. Always resampling performed significantly worse (Tables S5 and S6) than several other thresholds at all time points. Somewhat surprisingly when the resampling threshold is 0.0, that is never resampling, the sampler still performed well.
Annealing power
As discussed in the methods we can use a sequence of target distribution which anneals the data likelihood. In principle this allows the method to defer resampling away particles with low data likelihood at early stages. We explore the impact of the annealing parameter in Figures S10-S12. The PG sampler using no annealing, that is setting the power to zero, performed significantly worse (Tables S7 and S8). The actual value of the annealing power seemed to be less important provided it was larger than zero. The DPF sampler was generally insensitive to this parameter. The only significant difference observed was between using a power of 0.0 and 3.0 after 10 seconds (Tables S9 and S10) and this difference disappears for later times. This is likely due to the fact all possible paths from early time steps are included in by the DPF sampler, and are not resampled away.
Test path
In order to evaluate the data likelihood term in the target distributions, we must instantiate the values of the feature allocation vector that have not been updated yet. We consider several strategies for doing so:
• Conditional: Use the value of the conditional path.
• Ones: Set the value of all features to one.
• Random: Draw the value of the feature vector uniformly at random.
• Two stage: Run an unconditional SMC sampler using the conditional path to draw a test path.
• Unconditional: Similar to two stage but using zeros as the test path for the first pass unconditional SMC.
• Zeros: Set the value of all features to zero.
The Conditional and Two Stage strategies do not lead to valid Gibbs updates due to the dependency on the conditional path. However, we include them in this analysis as they could be used during a burnin phase. After burnin, another strategy which does lead to a valid Gibbs update could be used. The Two Stage and Unconditional strategies both use a pilot run of unconditional SMC. This increases run time, and introduces additional tuning parameters. For the purpose of this experiment, we set the tuning parameters of both the SMC and cSMC passes to the same values. Figures S13-S15 show the results of the experiment. For the PG sampler the Ones and Random test paths performed significantly worse than other approaches. At early time points the Conditional and Zeros strategies were the best, but at later time points the Two Stage and Unconstrained approaches were not significantly worse (Tables S11 and S12). For the DPF algorithm the Conditional, Random, and Zeros methods significantly outperformed other approaches after 10 seconds (Tables S13 and S14). Both the Conditional and Zeros methods significantly outperformed the Random method at this time. For later time points no methods had significantly different performance. This result suggests that the simple approach of using a test path of zeros is effective, though there may still be some benefit of using the Conditional strategy for burnin. This experiment also suggests that the PG sampler is sensitive to this parameter, whereas the DPF sampler is quite robust.
Summary
Based on these results we used the following parameter values for subsequent experiments.
• Annealing power -1.0
• Number of particles -20
• Resample threshold -0.5
• Test path -Zeros
These were not necessarily the optimal parameters based on the experiments, but were reasonably close to optimal. Note we use the Zeros test path strategy to ensure we have a valid Markov Chain kernel targeting the correct distribution.
Method comparison
To compare the performance of our proposed approaches to the standard Gibbs sampler we generated synthetic data from three feature allocation models. For all comparisons we ran 80 chains for each sampler as in the tuning experiments. We simulated data with parameter values which should lead to easily identifiable solutions and thus we would expect the samplers to converge to a distribution concentrated on the parameters used for simulation.
Linear Gaussian model
We generated datasets using two sets of model parameters. The first dataset was simulated using the FBB prior and K = 5 and the second was simulated using the FBB model with K = 20. We fit the second dataset using both the FBB prior with K = 20 and the IBP prior. For both datasets we simulated N = 1000 data points from the linear Gaussian model with α = 2, τ v = 0.25, and τ x = 25.
The results of these experiments are shown in Figure 3 and Figures S16-S18. For the first experiment with K = 5 it was computationally feasible to use the RG sampler. Because we use 20 particles for the DPF algorithm it is equivalent to the RG sampler in this case. The RG sampler serves as the gold standard for the DPF and PG methods in this experiment. For the K = 5 dataset the RG sampler significantly outperforms the Gibbs sampler, supporting the results of our initial toy data experiment (Tables S15 and S16). The DPF and RG samplers do not perform significantly different as expected, and outperform the other two approaches. The PG sampler does not significantly outperform the Gibbs sampler.
For the second dataset, fitting the FBB prior model (K = 20) the DPF sampler does not significantly outperform the Gibbs sampler after 100 seconds (Tables S17 and S18). However, for longer runs the performance advantage of the DPF sampler becomes significant. At the earliest time point the Gibbs sampler significantly outperforms the PG sampler, but the situation reverse at later time points.
The results are somewhat different for the third experiment fitting the IBP model. In this case we see that the Gibbs sampler outperformed the PG sampler at early time points (Tables S19 and S20). As the samplers were run longer the PG sampler began to outperform the Gibbs sampler. The DPF sampler outperforms both approaches. One explanation for the better performance of the Gibbs sampler over the PG sampler is that the Gibbs sampler can propose more moves to alter the dimensionality of the model in the same time period. Thus during the burnin phase the Gibbs sampler can more efficiently move the model to the correct number of features which improves performance. However, the fact the DPF sampler outperforms both, suggests that the ability to perform efficient updates on the non-singleton columns dominates this effect.
Latent Feature Relational Model
We next explored performance using the LFRM model described by Miller et al. (2009). As for the LG experiment, we generated datasets using two sets of model parameters. We fit the second dataset using both the FBB prior with K = 20 and the IBP prior. We again executed 80 runs for all samplers using the same strategy as previous experiments. We simulated N = 100 data points with parameters α = 2 and τ = 0.25 from the non-symmetric LFRM model. We randomly assigned 5% of the data matrix to be missing. In addition to the relative log density we report the reconstruction error of the model for the entire data matrix, both observed and missing values.
The result of these experiments are shown in Figures S19-S22. The RG and DPF methods significantly outperformed the other two methods for the K=5 experiment in terms of relative log density (Tables S21 and S22). However, the difference in reconstruction error was not significant. There was no significant difference between samplers for the other two runs (Tables S23-S26).
PyClone model
The final model we tested with was a modified version of the PyClone model described in Roth et al. (2014). The modifications are described in the supplement. We simulated three datasets using the FBB prior with α = 2, a V = b V = 1 with N = 200 data points. For the first dataset we set D = 4 and K = 4, the second we set D = 10 and K = 8 and the third D = 10 and K = 12. We did not fit the model using the IBP prior as we could not develop an efficient proposal for updating singleton entries. In addition to the relative log density we computed the B-Cubed F-Measure (Amigó et al., 2009). The B-Cubed metric is a measure of feature allocation accuracy analogous to the V-Measure metric (Rosenberg and Hirschberg, 2007) used to evaluate clustering algorithms. We focused on feature allocation accuracy as the features are interpretable quantities that we wish to infer in this application. The results of the experiments are shown in Figure 4 and Figures S23-S25. Note that we exclude the PG method from these figures as the performance was so poor as to obscure the scales of the plots. For the first dataset the RG and DPF sampler both outperformed the other approaches (Tables S27 and S28). The PG approach performed significantly worse than all other approaches including the Gibbs sampler. The two other datasets presented similar trends, with the DPF sampler outperforming both approaches and the PG sampler performing the worst (Tables S29-S32). The performance of the Gibbs sampler did not improve from times 1000 to 10000. Both the PG and DPF samplers show improved performance as sampling is run for longer. This suggests that the Gibbs sampler is potentially trapped in the vicinity of a local optima, which it cannot escape from.
Discussion
In this work we have developed several methods for updating an entire row of a feature allocation matrix. Our results suggest that such samplers can significantly improve performance compared to the widely used single entry Gibbs sampler. Directly implementing row wise Gibbs updates is intractable for more than a small number of features due to the exponential number of feature allocations. We overcome this limitation by using the PG methodology to develop an algorithm which scales linearly in the number of features. When coupled with the DPF framework we obtain significantly better performance than the standard Gibbs sampler. The use of the DPF framework appears to be critical, as the standard PG sampler did not always perform well. In particular, the performance of the PG sampler when applied to the PyClone model was significantly worse than the standard Gibbs sampler. However, the DPF approach significantly outperformed both the Gibbs and PG methods. Furthermore, when applied to models such as the LFRM where the standard Gibbs sampler performs well, our approach does not perform significantly worse. This suggests that despite the increased computational complexity of the PG framework, there is little downside to employing this approach. Taken together our results suggest the DPF algorithm is a computationally efficient and generally applicable approach for performing Bayesian inference for feature allocation models. Our algorithm is applicable to both the parametric FBB and nonparametric IBP model.
We have focused on developing row wise updates for the feature allocation matrix. When applied to the parametric FBB model these updates can significantly improve performance. However, when applied to the non-parametric models using the IBP prior we did not see consistent improvement. We believe a major problem in the non-parametric regime is the updates for the singleton features. The most general approach of using MH updates with proposals from the priors seems to lead to very slow mixing. While this is an issue for the Gibbs sampler as well, the low computational cost of updating non-singleton entries allows this sampler to perform more singleton updates. We believe that the development of efficient schemes to update the columns in a single move will be particularly useful. This has already been explored to some extent in Fox et al. (2014), where splitmerge moves are used as proposals for an MH update. It should be possible to further improve upon these split-merge style moves using the PG framework, in a similar way to what has been done in the Bayesian clustering literature (Bouchard-Côté et al., 2017). Such updates would complement the approach we have developed in this work.
We have not exploited the potential for performing parallel computation that is offered by the PG framework. In particular we could parallelize any loops over particles in Algorithm 3 which could potentially yield significant speed-ups. It has been noted by Whiteley et al. (2010) and Lindsten et al. (2014) that the use of backward or ancestor sampling can significantly reduce the effect of path degeneracy for SMC models. These approaches could naturally be combined with our method, and could allow for the use of fewer particles. Nick Whiteley, Christophe Andrieu, and Arnaud Doucet. Efficient bayesian inference for switching state-space models using discrete particle markov chain monte carlo methods. arXiv preprint arXiv:1011.2437, 2010.
Frank Wood and Thomas L Griffiths. Particle filtering for nonparametric bayesian matrix factorization. In Advances in neural information processing systems, pages 1513-1520, 2007.
Appendix
Models
We describe the three models we used for performance comparisons. We use the notation Z | α ∼ FAM(· | α) to describe sampling from a feature allocation distribution, either the FBB or IBP priors. The number of features K is implicitly determined by the number of columns of Z. We place a Gamma(· | 1, 1) prior on the hyper-parameter α and use a random walk Metropolis-Hastings kernel to update the variable. When referring to the Normal distribution we use the mean/precision parametrization. When referring to the Gamma distribution we use the shape/rate parametrization.
Linear Gaussian
The linear Gaussian model has been widely used, particularly in the IBP literature (see Griffiths and Ghahramani (2011) for example). One reason for the model's popularity is that it is possible to marginalize the feature parameters, so a collapsed sampler can be developed. In this work we do not exploit this, and instead work with uncollapsed model. The hierarchical model is as follows:
Z | α ∼ FAM(· | α) τ v | a v , b v ∼ Gamma(· | a v , b v ) S v = τ v I D τ x | a x , b x ∼ Gamma(· | a x , b x ) S x = τ x I D v k | τ v ∼ Normal(· | 0, S v ) x n | {v k } K k=1 , τ x , z n ∼ Normal(· | K k=1 z nk v k , S x )
We use a Gibbs kernels to update v k , τ v and τ x . When using the IBP prior we use a collapsed Metropolis-Hastings step to update the singletons (Doshi-Velez and Ghahramani, 2009).
Linear Feature Relational Model
The LFRM model was proposed by Miller et al. (2009). The observed data is a binary matrix X ∈ {0, 1} N ×N which encodes interactions between entities. It could for example be used to model relationships on a social network. The model posits that an underlying set of features encoded by Z governs whether the entries in X are on or off. The hierarchical model is as follows:
Z | α ∼ FAM(· | α) τ | a, b ∼ Gamma(· | a, b) v kl | τ ∼ Normal(·|0, τ ) x ij | {v kl }, Z ∼ Bernoulli · | σ K k=1 K l=1 z ik z jl v kl
where σ(x) = 1 1+e −x . Note that the model can be symmetric so that v kl = v lk or non-symmetric by letting these parameters vary independently. We use random walk Metropolis-Hastings kernels to update τ and v kl . When using the IBP prior we use a Metropolis-Hastings kernel with proposals from the prior to update the singletons.
PyClone
The original PyClone model was proposed by Roth et al. (2014). The model assumes we observe data a nm , b nm ∈ N which represent the number of sequencing reads without and with mutation n in sample m. We refer to d nm = a nm + b nm as the sequencing depth. We refer to the proportion of cells with mutation n in sample m, φ nm , as the cellular prevalence. We can model the probability of observing b nm reads with the mutation in the samples by a density g(b nm | d nm , φ nm , * ) where * indicates other quantities which are not relevant to the discussion. In the original PyClone model φ n is assumed to be sampled from a Dirichlet process so that mutations appearing at similar cellular prevalences are clustered. This corresponds to the biological assumption mutations appear within sub-populations of cells, and that the cellular prevalence is the sum of the proportion of cells in the sub-populations containing the mutation. We can alter this model to explicitly identify which sub-populations have the mutation using a feature allocation model. Let f km be the proportion of cells from population k in sample m. We use the feature allocation vector z n for mutation n to encode which sub-populations have the mutation. The cellular prevalence is then given by φ nm = K k=1 z nk f km . Substituting this into the observation density g gives the new model. The hierarchical model is as follows:
Z | α ∼ FAM(· | α) v km | a v , b v ∼ Gamma(· | a v , b v ) f km | v km = v km K l=1 v lm φ nm | {f km } K k=1 , z n = K k=1 z nk f km b nm | d nm , {f km } K k=1 , z n , * ∼ g(· | d nm , φ nm , * )
Updating v km was somewhat difficult for this model so we used a number of MCMC kernels which included random walk Metropolis-Hastings kernels on either individual v km values or blocks. We also used a Metropolis-Hastings kernel where the proposal was a random permutation of the values for a sample. The final kernel was the Multiple-Try-Metropolis kernel (Liu, 2008). Figure S5: Trace plots of PG sampler using different number of particles. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds. Figure S6: Trace plots of DPF sampler using different number of particles. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds. Linear Gaussian D=10,K=20,N=100 Figure S15: Performance of the PG (top) and DPF (bottom) samplers as function of the test path used to evaluate the data likelihood. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and root mean square error reconstruction of missing values (left).
Supplementary figures
Figure 1 :
1Illustration of the PG update procedure. Note we suppress the random ordering of features defined by σ for clarity. (top) Select a row for the update, shown in grey. (middle) Run a conditional particle filter and sample new row. (bottom) Update row with sample for particle filter, shown in grey. The stars (*) indicate values of the test path. We discuss how these values can be selected in Section 3.2.4.
Figure 2 :
2Comparison of element wise Gibbs to row Gibbs sampler. a) Log joint probability of the samplers over time. b) Number of data points assigned to each feature over time. Lines for the Gibbs sampler are jittered away from 50 for visibility. point must use both features or neither feature in one iteration before it can then only use the other feature in the next iteration.Figure 2b) supports this hypothesis as we see that the number of data points using each feature never changes over the course of sampling. In contrast the row wise Gibbs sampler (red) rapidly increases the joint probabilityFigure 2 a) and moves all data points to a single feature Figure 2 b).
Figure 3 :
3Performance of different samplers using synthetic data from the LG model. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (left, higher is better) and root mean square error reconstruction of missing values (right, lower is better).
Figure 4 :
4Performance of different samplers using synthetic data from the PyClone model. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and B-Cubed F-measure (left).
Edward
Meeds, Zoubin Ghahramani, Radford M Neal, and Sam T Roweis. Modeling dyadic data with binary latent factors. In Advances in neural information processing systems, pages 977-984, 2007. Kurt Miller, Michael I Jordan, and Thomas L Griffiths. Nonparametric latent feature models for link prediction. In Advances in neural information processing systems, pages 1276-1284, 2009. Andrew Rosenberg and Julia Hirschberg. V-measure: A conditional entropy-based external cluster evaluation measure. In Proceedings of the 2007 joint conference on empirical methods in natural language processing and computational natural language learning (EMNLP-CoNLL), pages 410-420, 2007. Andrew Roth, Jaswinder Khattra, Damian Yap, Adrian Wan, Emma Laks, Justina Biele, Gavin Ha, Samuel Aparicio, Alexandre Bouchard-Côté, and Sohrab P Shah. Pyclone: statistical inference of clonal population structure in cancer. Nature methods, 11(4):396, 2014.
Figure S7 :
S7Performance of the PG (top) and DPF (bottom) samplers as function of the number of particles. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and root mean square error reconstruction of missing values (left).
Figure S8 :Figure S9 :
S8S9Trace plots of PG sampler using different resampling thresholds. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds. Performance of the PG samplers as function of the resampling threshold. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and root mean square error reconstruction of missing values (left).
Figure S10 :
S10Trace plots of PG sampler using different annealing powers. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S11 :Figure S12 :
S11S12Trace plots of DPF sampler using different annealing powers. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds. Performance of the PG (top) and DPF (bottom) samplers as function of the annealing power of the intermediate target distribution. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and root mean square error reconstruction of missing values (left).
Figure S13 :
S13Trace plots of PG sampler using different test paths. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S16 :
S16Trace plots of sampling algorithms with simulated data from the linear Gaussian model with K=5. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S17 :
S17Trace plots of sampling algorithms with simulated data from the linear Gaussian model with K=20. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S18 :
S18Trace plots of sampling algorithms with simulated data from the linear Gaussian model with K=20 using an IBP prior. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S19 :
S19Trace plots of sampling algorithms with simulated data from the non-symmetric LFRM model with K=5. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S20 :
S20Trace plots of sampling algorithms with simulated data from the non-symmetric LFRM model with K=20. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S21 :Figure S22 :
S21S22Trace plots of sampling algorithms with simulated data from the non-symmetric LFRM model with K=20 using an IBP prior. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds. Performance of different samplers using synthetic data from the LFRM model. The box plots represent the distribution of values from 80 random starts of each parameter setting. We show the values of the relative log density (right) and reconstruction error (left).
Figure S23 :
S23Trace plots of sampling algorithms with simulated data from the PyClone model with D=4 and K=4. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S24 :
S24Trace plots of sampling algorithms with simulated data from the PyClone model with D=10 and K=8. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
Figure S25 :
S25Trace plots of sampling algorithms with simulated data from the PyClone model with D=8 and K=12. See main text for other model parameters. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.
=n indicate the entries of Z minus the n th row. Let m = i =n z i,k and ρ n,k be defined by replacing m k in the definition of ρ N +1,k from the previous section with m Gibbs update takes the form given by Equation 2 for the FBB model leading to Algorithm 1 for updating a row.(−n)
k
(−n)
k
. The
element wise
1 :
1function RowWiseGibbsUpdate(x n , ρ n , K)2:
j ← 0
Counter for number of vectors
3:
S ← ()
List to store vectors
4:
then Resample only if the relative ESS below threshold τ18:
a ∼ r(·|w t−1 )
Conditional resampling
19:
w t−1 ← (1, . . . , 1)
Reset incremental weights to one
20:
else
21:
a ← (1, 2, . . . , P )
Resampling skipped set a to identity map
22:
end if
23:
for i ∈ {2, . . . , P } do
Propose new feature usage for feature σ(t)
24:
Figure S14: Trace plots of DPF sampler using different test paths. Rows are datasets and columns are initial parameter settings. Error bars are averaged over five restarts of the sampler with different random seeds.Conditional
Ones
Random
Two stage
Unconditional
Zeros
3
2
1
0
Relative log density
10.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
100.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
Test path
1000.0
Linear Gaussian D=10,K=20,N=100
Conditional
Ones
Random
Two stage
Unconditional
Zeros
0.5
1
1.5
RMSE
10.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
100.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
Test path
1000.0
Linear Gaussian D=10,K=20,N=100
Conditional
Ones
Random
Two stage
Unconditional
Zeros
3
2
1
0
Relative log density
10.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
100.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
Test path
1000.0
Linear Gaussian D=10,K=20,N=100
Conditional
Ones
Random
Two stage
Unconditional
Zeros
0.5
1
1.5
RMSE
10.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
100.0
Conditional
Ones
Random
Two stage
Unconditional
Zeros
Test path
1000.0
Table S1 :
S1Comparison of the performance of PG algorithm
using different number of particles. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001) are
indicated in bold.
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
10
10
100
2.7480 0.0000
-0.5587 0.0000
2
-0.2048
0.5277
0.0359
0.9929
20
0.3871
0.0245
-0.0349
0.9845
5
-0.1429
0.6517
-0.0092
1.0000
50
1.4779 0.0000
-0.2271 0.0001
100
2
-2.9527 0.0000
0.5946 0.0000
20
-2.3609 0.0000
0.5238 0.0000
5
-2.8909 0.0000
0.5495 0.0000
50
-1.2701
0.0477
0.3316 0.0001
2
20
0.5918 0.0000
-0.0707
0.7447
5
0.0619
1.0000
-0.0450
0.9981
50
1.6826 0.0000
-0.2630 0.0000
20
5
-0.5300 0.0000
0.0257
0.9640
50
1.0908 0.0000
-0.1922
0.0031
5
50
1.6208 0.0000
-0.2179 0.0000
100
10
100
0.1149
0.0477
0.0470
NS
2
-0.0446
0.3834
0.0728
NS
20
0.0268
0.9879
0.0589
NS
5
-0.0005
1.0000
0.0379
NS
50
0.0207
0.9845
0.1246
NS
100
2
-0.1595 0.0000
0.0258
NS
20
-0.0881
0.2976
0.0119
NS
5
-0.1154
0.0218
-0.0092
NS
50
-0.0942
0.3180
0.0776
NS
2
20
0.0713
0.0720
-0.0139
NS
5
0.0441
0.5526
-0.0350
NS
50
0.0653
0.0651
0.0518
NS
20
5
-0.0272
0.9485
-0.0211
NS
50
-0.0061
1.0000
0.0657
NS
5
50
0.0212
0.9393
0.0867
NS
1000
10
100
0.0185
0.5776
-0.0345
NS
2
-0.0104
1.0000
-0.0136
NS
20
0.0113
0.5776
-0.0082
NS
5
0.0154
0.9998
0.0311
NS
50
0.0055
1.0000
-0.0106
NS
100
2
-0.0289
0.6025
0.0209
NS
20
-0.0071
1.0000
0.0263
NS
5
-0.0030
0.8071
0.0656
NS
50
-0.0129
0.6517
0.0239
NS
2
20
0.0218
0.6025
0.0054
NS
5
0.0259
0.9999
0.0447
NS
50
0.0160
1.0000
0.0030
NS
Continued on next page
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
20
5
0.0041
0.8071
0.0393
NS
50
-0.0058
0.6517
-0.0024
NS
5
50
-0.0099
1.0000
-0.0417
NS
Table S2: Comparison of the performance of PG algorithm
using different number of particles. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001) are
indicated in bold.
Metric Relative log density RMSE
Time
10.0
0.0000
0.0000
100.0
0.0000
0.0000
1000.0 0.0000
0.0000
Table S3: Comparison of the performance of DPF algorithm
using different number of particles. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001) are
indicated in bold.
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
10
10
100
2.4108 0.0000
-0.4485 0.0000
2
0.0585
0.4064
-0.0628
0.5028
20
0.0847
1.0000
0.0521
0.9051
5
-0.0195
1.0000
0.0348
0.9290
50
1.1853 0.0000
-0.1525
0.0017
100
2
-2.3523 0.0000
0.3856 0.0000
20
-2.3261 0.0000
0.5006 0.0000
5
-2.4303 0.0000
0.4833 0.0000
50
-1.2254
0.0477
0.2960 0.0001
2
20
0.0262
0.5776
0.1149
0.0385
5
-0.0780
0.2976
0.0977
0.0477
50
1.1269 0.0000
-0.0897
0.3834
20
5
-0.1042
0.9995
-0.0173
1.0000
50
1.1007 0.0000
-0.2046 0.0000
5
50
1.2048 0.0000
-0.1873 0.0000
100
10
100
0.0153
0.7872
0.0301
0.9703
2
0.0090
0.9947
-0.1183
0.1919
20
-0.0077
1.0000
-0.0148
0.9998
5
0.0003
1.0000
-0.0283
1.0000
50
0.0028
1.0000
0.0044
1.0000
100
2
-0.0062
0.9879
-0.1484
0.0152
20
-0.0229
0.7664
-0.0449
0.8609
5
-0.0149
0.8768
-0.0584
0.9758
50
-0.0125
0.8440
-0.0256
0.9879
2
20
-0.0167
0.9929
0.1035
0.3834
5
-0.0087
0.9991
0.0900
0.1772
50
-0.0063
0.9981
0.1227
0.1379
20
5
0.0080
1.0000
-0.0135
0.9997
50
0.0104
1.0000
0.0193
0.9987
5
50
0.0024
1.0000
0.0327
1.0000
1000
10
100
0.0307
0.6025
-0.0039
1.0000
2
-0.0022
0.9995
-0.0688
0.5776
20
0.0036
1.0000
-0.0359
0.9845
5
-0.0074
0.9051
-0.0555
0.4539
50
0.0080
1.0000
-0.0355
0.6025
100
2
-0.0329
0.3180
-0.0649
0.4299
20
-0.0271
0.5776
-0.0319
0.9485
5
-0.0381
0.0588
-0.0516
0.3180
50
-0.0227
0.6757
-0.0316
0.4539
2
20
0.0058
0.9997
0.0330
0.9640
5
-0.0052
0.9906
0.0133
1.0000
50
0.0102
0.9981
0.0333
1.0000
Continued on next page
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
20
5
-0.0110
0.9176
-0.0197
0.9176
50
0.0044
1.0000
0.0003
0.9703
5
50
0.0153
0.8609
0.0200
1.0000
Table S4: Comparison of the performance of DPF algorithm
using different number of particles. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001) are
indicated in bold.
Metric Relative log density RMSE
Time
10.0
0.0000
0.0000
100.0
0.0000
0.0581
1000.0 0.0000
0.0001
Table S5:
Comparison of the performance of PG algo-
rithm using different resampling thresholds. P-Values are
computed using the Friedman test. Significant values at
(p ≤ 0.001) are indicated in bold.
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
10
0.0
0.25
-0.0913
0.9846
-0.0222
0.9740
0.5
-0.0146
1.0000
-0.0581
0.3502
0.75
-0.0197
1.0000
-0.0552
0.7304
1.0
0.5900 0.0000
-0.1732 0.0000
0.25
0.5
0.0767
0.9885
-0.0359
0.8245
0.75
0.0716
0.9671
-0.0330
0.9885
1.0
0.6813 0.0000
-0.1510 0.0006
0.5
0.75
-0.0051
1.0000
0.0029
0.9916
1.0
0.6047 0.0000
-0.1152
0.0467
0.75
1.0
0.6097 0.0000
-0.1180
0.0070
100
0.0
0.25
0.0173
0.7797
-0.0516
NS
0.5
0.0330
0.5100
-0.0577
NS
0.75
0.0300
0.8643
-0.0361
NS
1.0
0.1700 0.0000
-0.1385
NS
0.25
0.5
0.0158
0.9983
-0.0062
NS
0.75
0.0127
1.0000
0.0155
NS
1.0
0.1528 0.0001
-0.0869
NS
0.5
0.75
-0.0030
0.9916
0.0217
NS
1.0
0.1370 0.0008
-0.0807
NS
0.75
1.0
0.1400 0.0001
-0.1024
NS
1000
0.0
0.25
0.0507
0.0414
-0.0787
0.2196
0.5
0.0312
0.4010
-0.0618
0.7797
0.75
0.0466
0.0664
-0.0985
0.0527
1.0
0.1380 0.0000
-0.1780
0.0017
0.25
0.5
-0.0195
0.9134
0.0168
0.9390
0.75
-0.0041
1.0000
-0.0198
0.9916
1.0
0.0873
0.0020
-0.0993
0.5947
0.5
0.75
0.0154
0.9590
-0.0366
0.6505
1.0
0.1067 0.0000
-0.1161
0.1139
0.75
1.0
0.0913
0.0010
-0.0795
0.9134
Table S6:
Comparison of the performance of PG algo-
rithm using different resampling thresholds. P-Values are
computed using the Friedman test. Significant values at
(p ≤ 0.001) are indicated in bold.
Metric Relative log density RMSE
Time
10.0
0.0000
0.0000
100.0
0.0000
0.0000
1000.0 0.0000
0.0000
Table S7: Comparison of the performance of PG algorithm
using different annealing powers. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001)
are indicated in bold.
Table S8 :
S8Table S12: Comparison of the performance of PG algorithm using different test paths. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S13: Comparison of the performance of DPF algorithm using different test paths. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. RMSE Mean difference P-Value Mean difference P-ValueTable S14: Comparison of the performance of DPF algorithm using different test paths. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S15: Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=5. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. RMSE Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Comparison of the performance of PG algorithm
using different annealing powers. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001)
are indicated in bold.
Metric Relative log density RMSE
Time
10.0
0.0000
0.0028
100.0
0.0000
0.0000
1000.0 0.0000
0.0000
Table S9: Comparison of the performance of DPF algorithm
using different annealing powers. P-Values are computed
using the Friedman test. Significant values at (p ≤ 0.001)
are indicated in bold.
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1 Param 2
10
0.0
1.0
-0.1168
0.0227
0.0676
NS
2.0
-0.1317
0.1023
0.0702
NS
3.0
-0.1392 0.0002
0.0842
NS
1.0
2.0
-0.0149
0.9820
0.0026
NS
3.0
-0.0224
0.7219
0.0166
NS
2.0
3.0
-0.0075
0.3735
0.0140
NS
100
0.0
1.0
-0.0108
0.9996
0.0429
0.9667
2.0
-0.0034
0.9946
0.0387
0.7514
3.0
-0.0149
0.9874
0.0622
0.4032
1.0
2.0
0.0074
0.9751
-0.0043
0.9820
3.0
-0.0041
0.9982
0.0193
0.8066
2.0
3.0
-0.0114
0.8970
0.0236
0.9820
1000
0.0
1.0
-0.0076
0.9999
0.0200
0.9999
2.0
-0.0168
0.6912
-0.0073
0.8319
3.0
-0.0191
0.9446
0.0342
0.9915
1.0
2.0
-0.0093
0.7797
-0.0272
0.7514
3.0
-0.0116
0.9751
0.0142
0.9982
2.0
3.0
-0.0023
0.9820
0.0415
0.5624
Table S10:
Comparison of the performance of DPF al-
gorithm using different annealing powers.
P-Values are
computed using the Friedman test. Significant values at
(p ≤ 0.001) are indicated in bold.
Metric Relative log density RMSE
Time
10.0
0.0000
0.0000
100.0
0.0000
0.0000
1000.0 0.0000
0.0000
Table S11: Comparison of the performance of PG algorithm
using different test paths. P-Values are computed using the
Friedman test. Significant values at (p ≤ 0.001) are indicated
in bold.
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1
Param 2
10
Conditional
Ones
1.9441 0.0000
-0.3619 0.0000
Random
1.2298 0.0000
-0.2472 0.0000
Two stage
1.2831 0.0000
-0.1914 0.0000
Unconditional
1.2887 0.0000
-0.2109 0.0000
Zeros
0.2714
0.4539
-0.0870
0.1055
Ones
Random
-0.7143 0.0000
0.1147
0.1502
Two stage
-0.6610 0.0002
0.1705
0.0047
Unconditional
-0.6554 0.0001
0.1510
0.0062
Zeros
-1.6727 0.0000
0.2749 0.0000
Random
Two stage
0.0533
0.9998
0.0558
0.9176
Unconditional
0.0589
1.0000
0.0362
0.9393
Zeros
-0.9584 0.0000
0.1602
0.0011
Two stage
Unconditional
0.0056
1.0000
-0.0195
1.0000
Zeros
-1.0117 0.0000
0.1044
0.0588
Unconditional Zeros
-1.0173 0.0000
0.1239
0.0477
100
Conditional
Ones
0.5421 0.0000
-0.1969 0.0000
Random
0.2535 0.0000
-0.1982 0.0002
Two stage
-0.0286
0.3834
0.0011
1.0000
Unconditional
0.1017
0.1156
-0.1003
0.4064
Zeros
0.0404
0.9393
-0.0214
0.9845
Ones
Random
-0.2886
0.0345
-0.0013
0.9998
Two stage
-0.5707 0.0000
0.1981 0.0000
Unconditional
-0.4403 0.0000
0.0966
0.0794
Zeros
-0.5017 0.0000
0.1755
0.0013
Random
Two stage
-0.2821 0.0000
0.1993 0.0002
Unconditional
-0.1517
0.0062
0.0979
0.1919
Zeros
-0.2131 0.0000
0.1768
0.0054
Two stage
Unconditional
0.1303 0.0001
-0.1015
0.4064
Zeros
0.0690
0.0308
-0.0226
0.9845
Unconditional Zeros
-0.0614
0.6993
0.0789
0.8915
1000
Conditional
Ones
0.1854 0.0000
-0.1978 0.0006
Random
0.1033 0.0000
-0.2113 0.0001
Two stage
-0.0091
0.9981
-0.0177
1.0000
Unconditional
0.0219
0.9929
-0.1663
0.0047
Zeros
0.0045
1.0000
-0.0879
0.5028
Ones
Random
-0.0820
0.5028
-0.0135
0.9997
Two stage
-0.1944 0.0000
0.1801 0.0004
Unconditional
-0.1634 0.0000
0.0315
0.9987
Zeros
-0.1808 0.0000
0.1099
0.2411
Random
Two stage
-0.1124 0.0000
0.1936 0.0001
Unconditional
-0.0814 0.0004
0.0450
0.9703
Zeros
-0.0988 0.0000
0.1234
0.0961
Continued on next page
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1
Param 2
Two stage
Unconditional
0.0310
0.8768
-0.1486
0.0036
Zeros
0.0136
0.9987
-0.0702
0.4539
Unconditional Zeros
-0.0174
0.9906
0.0784
0.5526
Metric Relative log density RMSE
Time
10.0
0.0000
0.0000
100.0
0.0000
0.0000
1000.0 0.0000
0.0000
Time Param 1
Param 2
10
Conditional
Ones
1.0392 0.0000
-0.2413 0.0000
Random
0.6055 0.0000
-0.1644 0.0003
Two stage
1.2704 0.0000
-0.2645 0.0000
Unconditional
1.1626 0.0000
-0.2435 0.0000
Zeros
0.1129
0.9879
-0.0881
0.2239
Ones
Random
-0.4337 0.0005
0.0769
0.2075
Two stage
0.2312
0.8261
-0.0232
1.0000
Unconditional
0.1234
0.9879
-0.0021
0.9987
Zeros
-0.9263 0.0000
0.1533 0.0003
Random
Two stage
0.6649 0.0000
-0.1001
0.1772
Unconditional
0.5571 0.0000
-0.0790
0.5028
Zeros
-0.4926 0.0000
0.0764
0.4299
Two stage
Unconditional
-0.1078
0.9972
0.0211
0.9972
Zeros
-1.1575 0.0000
0.1765 0.0002
Unconditional Zeros
-1.0497 0.0000
0.1554
0.0023
100
Conditional
Ones
0.0369
0.7872
-0.1604 0.0001
Random
0.0150
0.9906
-0.0855
0.7224
Two stage
-0.0170
0.8071
-0.0123
0.9991
Unconditional
0.0134
0.9290
-0.0791
0.3609
Zeros
-0.0023
1.0000
-0.0235
0.8915
Ones
Random
-0.0219
0.9929
0.0749
0.0308
Two stage
-0.0539
0.0720
0.1481 0.0006
Unconditional
-0.0235
0.9999
0.0813
0.1379
Zeros
-0.0391
0.8768
0.1369
0.0104
Random
Two stage
-0.0320
0.3391
0.0732
0.9393
Unconditional
-0.0016
0.9998
0.0064
0.9981
Zeros
-0.0173
0.9981
0.0620
0.9999
Two stage
Unconditional
0.0304
0.1633
-0.0668
0.6757
Zeros
0.0147
0.6993
-0.0112
0.9906
Unconditional Zeros
-0.0157
0.9703
0.0556
0.9758
1000
Conditional
Ones
-0.0017
1.0000
-0.0590
0.6993
Random
0.0009
0.9997
-0.0246
1.0000
Two stage
-0.0103
0.8261
-0.0273
0.9906
Unconditional
0.0185
0.9929
-0.0909
0.3609
Zeros
-0.0075
0.9972
-0.0097
0.9972
Ones
Random
0.0026
1.0000
0.0344
0.8440
Two stage
-0.0086
0.9290
0.0317
0.9805
Unconditional
0.0202
0.9640
-0.0319
0.9987
Zeros
-0.0059
0.9999
0.0493
0.9567
Random
Two stage
-0.0112
0.9640
-0.0027
0.9991
Unconditional
0.0176
0.9290
-0.0663
0.5277
Zeros
-0.0084
1.0000
0.0149
0.9999
Continued on next page
Relative log density
RMSE
Mean difference P-Value Mean difference P-Value
Time Param 1
Param 2
Two stage
Unconditional
0.0288
0.3834
-0.0637
0.8261
Zeros
0.0028
0.9879
0.0176
1.0000
Unconditional Zeros
-0.0261
0.8609
0.0812
0.7447
Metric
Relative log density RMSE
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
10000.0 0.0000
0.0000
Relative log density
100
DPF
Gibbs
8.7843 0.0000
-0.1836 0.0000
PG
7.8931 0.0000
-0.1154 0.0000
RG
-0.2859
0.9999
0.0073
0.9948
PG
Gibbs
0.8912
0.9736
-0.0683
0.3831
RG
-8.1790 0.0000
0.1226 0.0000
RG
Gibbs
9.0702 0.0000
-0.1909 0.0000
1000
DPF
Gibbs
8.7537 0.0000
-0.1851 0.0000
PG
6.0201 0.0000
-0.0833 0.0000
RG
-0.4225
0.9614
0.0055
0.6499
PG
Gibbs
2.7337
0.1589
-0.1018
0.0209
RG
-6.4425 0.0000
0.0888 0.0000
RG
Gibbs
9.1762 0.0000
-0.1906 0.0000
10000
DPF
Gibbs
9.0712 0.0000
-0.1974 0.0000
PG
4.3285 0.0000
-0.0861 0.0000
RG
-0.0766
0.6499
-0.0145
0.9071
PG
Gibbs
4.7427
0.1802
-0.1112
0.0209
RG
-4.4052 0.0000
0.0716 0.0000
RG
Gibbs
9.1479 0.0000
-0.1829 0.0000
Table S16 :
S16Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=5. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S17: Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=20. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. RMSE Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Metric
Relative log density RMSE
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
10000.0 0.0000
0.0000
Relative log density
100
DPF
Gibbs
0.7041
0.0307
-0.0430
0.0246
PG
8.0163 0.0000
-0.1185 0.0000
PG
Gibbs
-7.3122 0.0000
0.0754 0.0004
1000
DPF
Gibbs
4.4600 0.0000
-0.1574 0.0000
PG
1.8011 0.0000
-0.0790 0.0000
PG
Gibbs
2.6589 0.0000
-0.0784
0.0015
10000
DPF
Gibbs
5.2375 0.0000
-0.1793 0.0000
PG
1.8331 0.0000
-0.0657 0.0000
PG
Gibbs
3.4044 0.0000
-0.1136 0.0006
Table S18 :
S18Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=20. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S19: Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=20 and an IBP prior. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Relative log density RMSE Mean difference P-Value Mean difference P-ValueMetric
Relative log density RMSE
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
100000.0 0.0000
0.0000
Time
Sampler 1 Sampler 2
100
DPF
Gibbs
1.7953
0.9695
-0.0275
0.4151
PG
5.0827 0.0000
-0.0856 0.0000
PG
Gibbs
-3.2874 0.0000
0.0582 0.0003
1000
DPF
Gibbs
1.5649
0.1180
-0.0974 0.0000
PG
2.7708 0.0000
-0.0646 0.0000
PG
Gibbs
-1.2059 0.0000
-0.0328
0.7088
100000
DPF
Gibbs
2.4641 0.0000
-0.1124 0.0000
PG
1.3703
0.0246
-0.0334
0.0045
PG
Gibbs
1.0938 0.0000
-0.0790 0.0008
Table S20 :Table S21 :Table S23 :Table S24 :Table S25 :Table S26 :
S20S21S23S24S25S26Comparison of the performance of sampling algorithm with simulated data from a linear Gaussian model with K=20 and an IBP prior. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=5. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Reconstruction error Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Table S22: Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=5. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=20. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Reconstruction error Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2 Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=20. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=20 and an IBP prior. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Reconstruction error Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2 Comparison of the performance of sampling algorithm with simulated data from a non-symmetric LFRM model with K=20 and an IBP prior. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S27: Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=4 and K=4. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Relative log density B-Cubed F-Measure Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Metric
Relative log density Reconstruction error
Time
100.0
0.0000
0.0030
1000.0
0.0000
0.0264
10000.0 0.0000
0.0151
Relative log density
100
DPF
Gibbs
0.0074 0.0002
-40.6125
NS
PG
0.0038
0.3831
-16.1375
NS
RG
-0.0006
0.9900
-2.0500
NS
PG
Gibbs
0.0036
0.0583
-24.4750
NS
RG
-0.0045
0.2287
14.0875
NS
RG
Gibbs
0.0081 0.0001
-38.5625
NS
1000
DPF
Gibbs
0.0081 0.0000
-31.5625
NS
PG
0.0009
0.0681
-5.3125
NS
RG
-0.0009
0.4940
4.0125
NS
PG
Gibbs
0.0072
0.0143
-26.2500
NS
RG
-0.0018
0.7253
9.3250
NS
RG
Gibbs
0.0091 0.0003
-35.5750
NS
10000
DPF
Gibbs
0.0086 0.0000
-39.3000
NS
PG
0.0010
0.3831
-9.1125
NS
RG
-0.0006
0.9071
-9.7125
NS
PG
Gibbs
0.0076
0.0011
-30.1875
NS
RG
-0.0016
0.7949
-0.6000
NS
RG
Gibbs
0.0092 0.0000
-29.5875
NS
Metric
Relative log density Reconstruction error
Time
100.0
0.0004
0.0031
1000.0
0.0031
0.4296
10000.0 0.7886
0.8229
Relative log density
100
DPF
Gibbs
0.0051
0.0011
7.2875
NS
PG
0.0024
0.9695
-57.6125
NS
PG
Gibbs
0.0026
0.0026
64.9000
NS
1000
DPF
Gibbs
0.0048
NS
8.7000
NS
PG
0.0024
NS
-2.8375
NS
PG
Gibbs
0.0024
NS
11.5375
NS
10000
DPF
Gibbs
-0.0014
NS
0.4000
NS
PG
0.0015
NS
-1.8875
NS
PG
Gibbs
-0.0029
NS
2.2875
NS
Metric
Relative log density Reconstruction error
Time
100.0
0.0094
0.0004
1000.0
0.0571
0.4949
100000.0 0.4328
0.6771
Relative log density
100
DPF
Gibbs
0.0028
NS
9.3250
0.5345
PG
0.0007
NS
-16.4625
0.0175
PG
Gibbs
0.0021
NS
25.7875 0.0004
1000
DPF
Gibbs
0.0013
NS
6.3250
NS
PG
-0.0010
NS
-3.2625
NS
PG
Gibbs
0.0023
NS
9.5875
NS
100000
DPF
Gibbs
0.0002
NS
-4.4125
NS
PG
-0.0004
NS
-6.1000
NS
PG
Gibbs
0.0006
NS
1.6875
NS
Metric
Relative log density B-Cubed F-Measure
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
10000.0 0.0000
0.0000
100
DPF
Gibbs
0.9702 0.0000
0.2967 0.0000
PG
92.2997 0.0000
0.6791 0.0000
RG
-0.0144
0.2559
-0.0303
0.4940
PG
Gibbs
-91.3295 0.0000
-0.3824 0.0000
RG
-92.3141 0.0000
-0.7093 0.0000
RG
Gibbs
0.9845 0.0000
0.3270 0.0000
1000
DPF
Gibbs
0.9738 0.0000
0.3128 0.0000
PG
89.3427 0.0000
0.6741 0.0000
RG
-0.0116
1.0000
-0.0270
0.9282
PG
Gibbs
-88.3689 0.0000
-0.3614 0.0000
RG
-89.3544 0.0000
-0.7012 0.0000
RG
Gibbs
0.9854 0.0000
0.3398 0.0000
10000
DPF
Gibbs
0.9813 0.0000
0.3402 0.0000
PG
85.0018 0.0000
0.6797 0.0000
RG
-0.0084
0.9948
-0.0083
0.9900
PG
Gibbs
-84.0205 0.0001
-0.3395 0.0000
RG
-85.0103 0.0000
-0.6879 0.0000
RG
Gibbs
0.9898 0.0000
0.3484 0.0000
Table S28 :
S28Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=4 and K=4. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S29: Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=10 and K=8. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Relative log density B-Cubed F-Measure Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Metric
Relative log density B-Cubed F-Measure
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
10000.0 0.0000
0.0000
100
DPF
Gibbs
0.4032 0.0000
0.4152 0.0000
PG
60.3425 0.0000
0.7977 0.0000
PG
Gibbs
-59.9393 0.0000
-0.3825
0.2207
1000
DPF
Gibbs
0.4452 0.0000
0.5236 0.0000
PG
58.3945 0.0000
0.9094 0.0000
PG
Gibbs
-57.9493 0.0000
-0.3858
0.0568
10000
DPF
Gibbs
0.4391 0.0000
0.5242 0.0000
PG
53.8505 0.0000
0.8962 0.0000
PG
Gibbs
-53.4114 0.0000
-0.3719 0.0000
Table S30 :
S30Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=10 and K=8. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Table S31: Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=10 and K=12. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold. Relative log density B-Cubed F-Measure Mean difference P-Value Mean difference P-Value Time Sampler 1 Sampler 2Table S32: Comparison of the performance of sampling algorithm with simulated data from the PyClone model with D=10 and K=12. See main text for other model parameters. P-Values are computed using the Friedman test. Significant values at (p ≤ 0.001) are indicated in bold.Metric
Relative log density B-Cubed F-Measure
Time
100.0
0.0000
0.0000
1000.0
0.0000
0.0000
10000.0 0.0000
0.0000
100
DPF
Gibbs
0.2250 0.0002
0.1678 0.0000
PG
29.3156 0.0000
0.4260 0.0000
PG
Gibbs
-29.0906 0.0000
-0.2582
0.8022
1000
DPF
Gibbs
0.3318
0.0015
0.3967 0.0000
PG
24.7551 0.0000
0.6541 0.0000
PG
Gibbs
-24.4233 0.0000
-0.2574 0.0002
10000
DPF
Gibbs
0.3240 0.0000
0.4466 0.0000
PG
13.4891 0.0000
0.5367 0.0000
PG
Gibbs
-13.1651 0.0000
-0.0902
0.0379
A comparison of extrinsic clustering evaluation metrics based on formal constraints. Enrique Amigó, Julio Gonzalo, Javier Artiles, Felisa Verdejo, Information retrieval. 124Enrique Amigó, Julio Gonzalo, Javier Artiles, and Felisa Verdejo. A comparison of extrinsic cluster- ing evaluation metrics based on formal constraints. Information retrieval, 12(4):461-486, 2009.
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| [
"https://github.com/aroth85/pgfa_experiments."
]
|
[
"Maxwell's equations with hypersingularities at a conical plasmonic tip",
"Maxwell's equations with hypersingularities at a conical plasmonic tip"
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| [
"Anne-Sophie Bonnet-Ben Dhia s:[email protected] \nLaboratoire Poems\nCNRS/INRIA\nENSTA Paris\nInstitut Polytechnique de Paris\n828 Boulevard des Maréchaux91762PalaiseauFrance\n",
"Lucas Chesnel [email protected] \nINRIA/Centre de mathématiques appliquées\nÉcole Polytechnique\nInstitut Polytechnique de Paris\nRoute de Saclay91128PalaiseauFrance\n",
"Mahran Rihani [email protected] \nLaboratoire Poems\nCNRS/INRIA\nENSTA Paris\nInstitut Polytechnique de Paris\n828 Boulevard des Maréchaux91762PalaiseauFrance\n\nINRIA/Centre de mathématiques appliquées\nÉcole Polytechnique\nInstitut Polytechnique de Paris\nRoute de Saclay91128PalaiseauFrance\n"
]
| [
"Laboratoire Poems\nCNRS/INRIA\nENSTA Paris\nInstitut Polytechnique de Paris\n828 Boulevard des Maréchaux91762PalaiseauFrance",
"INRIA/Centre de mathématiques appliquées\nÉcole Polytechnique\nInstitut Polytechnique de Paris\nRoute de Saclay91128PalaiseauFrance",
"Laboratoire Poems\nCNRS/INRIA\nENSTA Paris\nInstitut Polytechnique de Paris\n828 Boulevard des Maréchaux91762PalaiseauFrance",
"INRIA/Centre de mathématiques appliquées\nÉcole Polytechnique\nInstitut Polytechnique de Paris\nRoute de Saclay91128PalaiseauFrance"
]
| []
| In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical L 2 framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle. Proposition 1.1. Under Assumption 1, the embeddings of X T (1) in L 2 (Ω) and of X N (1) in L 2 (Ω) are compact. And there is a constant C > 0 such that∀u ∈ X T (1) ∪ X N (1).Therefore, in X T (1) and in X N (1), curl · Ω is a norm which is equivalent to · H(curl ) . | 10.1016/j.matpur.2022.03.001 | [
"https://arxiv.org/pdf/2010.08472v1.pdf"
]
| 223,956,796 | 2010.08472 | 8a03e6eff6aace318fcbcba47d6e81ea69236ee9 |
Maxwell's equations with hypersingularities at a conical plasmonic tip
Anne-Sophie Bonnet-Ben Dhia s:[email protected]
Laboratoire Poems
CNRS/INRIA
ENSTA Paris
Institut Polytechnique de Paris
828 Boulevard des Maréchaux91762PalaiseauFrance
Lucas Chesnel [email protected]
INRIA/Centre de mathématiques appliquées
École Polytechnique
Institut Polytechnique de Paris
Route de Saclay91128PalaiseauFrance
Mahran Rihani [email protected]
Laboratoire Poems
CNRS/INRIA
ENSTA Paris
Institut Polytechnique de Paris
828 Boulevard des Maréchaux91762PalaiseauFrance
INRIA/Centre de mathématiques appliquées
École Polytechnique
Institut Polytechnique de Paris
Route de Saclay91128PalaiseauFrance
Maxwell's equations with hypersingularities at a conical plasmonic tip
(October 19, 2020)Time-harmonic Maxwell's equationsnegative metamaterialsKondratiev weighted Sobolev spacesT -coercivitycompact embeddingsscalar and vector potentialslimiting absorp- tion principle
In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical L 2 framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle. Proposition 1.1. Under Assumption 1, the embeddings of X T (1) in L 2 (Ω) and of X N (1) in L 2 (Ω) are compact. And there is a constant C > 0 such that∀u ∈ X T (1) ∪ X N (1).Therefore, in X T (1) and in X N (1), curl · Ω is a norm which is equivalent to · H(curl ) .
Introduction
For the past two decades, the scientific community has been particularly interested in the study of Maxwell's equations in the unusual case where the dielectric permittivity ε is a real-valued sign-changing function. There are several motivations to this which are all related to spectacular progress in physics. Such sign-changing ε appear for example in the field of plasmonics [3,33,6]. The existence of surface plasmonic waves is mainly due to the fact that, at optical frequencies, some metals like silver or gold have an ε with a small imaginary part and a negative real part. Neglecting the imaginary part, at a given frequency, one is led to consider a real-valued ε which is negative in the metal and positive in the air around the metal. A second more prospective motivation concerns the so-called metamaterials, whose micro-structure is designed so that their effective electromagnetic constants may have a negative real part and a small imaginary part in some frequency ranges [46,45,44]. Let us emphasize that for such metamaterials not only the dielectric permittivity ε may become negative but the magnetic permeability µ as well. At the interface between dielectrics and negative-index metamaterials, one can observe a negative refraction phenomenon which opens a lot of exciting prospects. Finally let us mention that negative ε also appear in plasmas, together with strong anisotropic effects. But we want to underline a main difference between plasmas and the previous applications. In the case of plasmonics and metamaterials, ε is sign-changing but does not vanish (and similarly for µ), while in plasmas, ε vanishes on some particular surfaces, leading to the phenomenon of hybrid resonance (see [21,39]). The theory developed in the present paper does no apply to the case where ε vanishes.
The goal of the present work is to study the Maxwell's system in the case where ε, µ change sign but do not vanish. In case of invariance with respect to one variable, the analysis of timeharmonic Maxwell's problem leads to consider the 2D scalar Helmholtz equation
div 1 ε ∇ϕ + ω 2 µϕ = f.
Here f denotes the source term and the unknown ϕ is a component of the magnetic field. For this scalar equation, only the change of sign of ε matters because roughly speaking, the term involving µ is compact (or locally compact in freespace). In the particular case where ε takes constant values ε + > 0 and ε − < 0 in two subdomains separated by a a curve Σ, the results are quite complete [8]. If Σ is smooth (of class C 1 ), the equation has the same properties in the H 1 framework as in the case of positive coefficients, except when the contrast κ ε := ε − /ε + takes the particular value −1. One way to show this consists in finding an appropriate operator T such that the coercivity of the variational formulation is restored when testing with functions of the form Tϕ (instead of ϕ ). This approach is called the T-coercivity technique. When κ ε = −1, Fredholmness is lost in H 1 but some results can be established in some weighted Sobolev spaces where the weight is adapted to the shape of Σ [41,37,42]. The picture is quite different when Σ has corners. For instance, in the case of a polygonal curve Σ, Fredholmness in H 1 is lost not only for κ ε = −1 but for a whole interval of values of κ ε around −1. We name this interval the critical interval. The smaller the angle of the corners, the larger the critical interval is. In fact, we can still find a solution in that case but this solution has a strongly singular behaviour at the corners in r iη where r is the distance to the corner and η is a real coefficient. In particular, this hypersingular solution does not belong to H 1 . It has been shown that Fredholmness can be recovered in an appropriate unusual framework [10] which is obtained by adding a singular function to a Kondratiev weighted Sobolev space of regular functions. The proof requires to adapt Mellin techniques in Kondratiev spaces [30] to an equation which is not elliptic due to the change of sign of ε (see [20] for the first analysis). From a physical point of view, the singular 1 function corresponds to a wave which propagates towards the corner, without never reaching it because its group velocity tends to zero with the distance to the corner [7,25,26]. In the literature, this wave which is trapped by the corner is commonly referred to as a black-hole wave. It leads to a strange phenomenon of leakage of energy while only non-dissipative materials are considered.
The objective of this article is to extend this type of results to 3D Maxwell's equations. The case where the contrasts in ε and µ do not take critical values has been considered in [9]. Using the T-coercivity technique, a Fredholm property has been proved for Maxwell equations in a classical functional framework as soon as two scalar problems (one for ε and one for µ) are well-posed in H 1 . The case where these problems satisfy a Fredholm property in H 1 but with a non trivial kernel has also been treated in [9]. Let us finally mention [38] where different types of results have been established for a smooth inclusion of class C 1 . In the present work, we consider a 3D configuration with an inclusion of material with a negative dielectric permeability ε. We suppose that this inclusion has a tip at which singularities of the electromagnetic field exist. The objective is to combine Mellin analysis in Kondratiev spaces with the T-coercivity technique to derive an appropriate functional framework for Maxwell's equations when the contrast κ ε takes critical values (but not the contrast in µ). We emphasize that due to the non standard singularities we have to deal with, the results we obtain are quite different from the ones existing for classical Maxwell's equations with positive materials in non smooth domains [4,14,5,16,15].
The outline is as follows. In the remaining part of the introduction, we present some general notation. In Section 2, we describe the assumptions made on the dielectric constants ε, µ. Then we propose a new functional framework for the problem for the electric field and show its wellposedness in Section 3. Section 4 is dedicated to the analysis of the problem for the magnetic field. We emphasize that due to the assumptions made on ε, µ (the contrast in ε is critical but the one in µ is not), the studies in sections 3 and 4 are quite different. We give a few words of conclusion in Section 5 before presenting technical results needed in the analysis in two sections of appendix. The main outcomes of this work are Theorem 3.6 (well-posedness for the electric problem) and Theorem 4.9 (well-posedness for the magnetic problem).
All the study will take place in some domain Ω of R 3 . More precisely, Ω is an open, connected and bounded subset of R 3 with a Lipschitz-continuous boundary ∂Ω. Once for all, we make the following assumption:
Assumption 1.
The domain Ω is simply connected and ∂Ω is connected.
When this assumption is not satisfied, the analysis below must be adapted (see the discussion in the conclusion). For some ω = 0 (ω ∈ R), the time-harmonic Maxwell's equations are given by
curl E − iω µ H = 0 and curl H + iω ε E = J in Ω.(1)
Above E and H are respectively the electric and magnetic components of the electromagnetic field. The source term J is the current density. We suppose that the medium Ω is surrounded by a perfect conductor and we impose the boundary conditions
E × ν = 0 and µH · ν = 0 on ∂Ω,(2)
where ν denotes the unit outward normal vector field to ∂Ω. The dielectric permittivity ε and the magnetic permeability µ are real valued functions which belong to L ∞ (Ω), with ε −1 , µ −1 ∈ L ∞ (Ω) (without assumption of sign). Let us introduce some usual spaces in the study of Maxwell's equations:
L 2 (Ω) := (L 2 (Ω)) 3 H 1 0 (Ω) := {ϕ ∈ H 1 (Ω) | ϕ = 0 on ∂Ω} H 1 # (Ω) := {ϕ ∈ H 1 (Ω) |ˆΩ ϕ dx = 0} H(curl ) := {H ∈ L 2 (Ω) | curl H ∈ L 2 (Ω)} H N (curl ) := {E ∈ H(curl ) | E × ν = 0 on ∂Ω}
and for ξ ∈ L ∞ (Ω):
X T (ξ) := {H ∈ H(curl ) | div(ξH) = 0, ξH · ν = 0 on ∂Ω} X N (ξ) := {E ∈ H N (curl ) | div(ξE) = 0} .
We denote indistinctly by (·, ·) Ω the classical inner products of L 2 (Ω) and L 2 (Ω). Moreover, · Ω stands for the corresponding norms. We endow the spaces H(curl ), H N (curl ), X T (ξ), X N (ξ) with the norm · H(curl ) := ( · 2 Ω + curl · 2 Ω ) 1/2 . Let us recall a well-known property for the particular spaces X T (1) and X N (1) (cf. [47,1]).
Assumptions for the dielectric constants ε, µ
In this document, for a Banach space X, X * stands for the topological antidual space of X (the set of continuous anti-linear forms on X). In the analysis of the Maxwell's system (1)-(2), the properties of two scalar operators associated respectively with ε and µ play a key role. Define A ε : H 1 0 (Ω) → (H 1 0 (Ω)) * such that
A ε ϕ, ϕ =ˆΩ ε∇ϕ · ∇ϕ dx, ∀ϕ, ϕ ∈ H 1 0 (Ω)(3)
and
A µ : H 1 # (Ω) → (H 1 # (Ω)) * such that A µ ϕ, ϕ =ˆΩ µ∇ϕ · ∇ϕ dx, ∀ϕ, ϕ ∈ H 1 # (Ω).
Assumption 2.
We assume that µ is such that
A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism.
Assumption 2 is satisfied in particular if µ has a constant sign (by Lax-Milgram theorem). We underline however that we allow µ to change sign (see in particular [17,11,8,9] for examples of sign-changing µ such that Assumption 2 is verified). The assumption on ε, that will be responsible for the presence of (hyper)singularities, requires to consider a more specific configuration as explained below.
Conical tip and scalar (hyper)singularities
We assume that Ω contains an inclusion of a particular material (metal at optical frequency, metamaterial, ...) located in some domain M such that M ⊂ Ω (M like metal or metamaterial). We assume that ∂M is of class C 2 except at the origin O where M coincides locally with a conical tip. More precisely, there are ρ > 0 and some smooth domain of the unit sphere
S 2 := {x ∈ R 3 | |x| = 1} such that B(O, ρ) ⊂ Ω and M ∩ B(O, ρ) = K ∩ B(O, ρ) with K := {r θ | r > 0, θ ∈ }.
Here B(O, ρ) stands for the open ball centered at O and of radius ρ. We assume that ε takes the constant value ε − < 0 (resp. ε
+ > 0) in M ∩ B(O, ρ) (resp. (Ω \ M) ∩ B(O, ρ)
). And we assume that the contrast κ ε := ε − /ε + < 0 and (which characterizes the geometry of the conical tip) are such that there exist singularities of the form
s(x) = r −1/2+iη Φ(θ, φ)(4)
satisfying div(ε∇s) = 0 in K with η ∈ R, η = 0. Here (r, θ, φ) are the spherical coordinates associated with O while Φ is a function which is smooth in and in S 2 \ . We emphasize that since the interface between the metamaterial and the exterior material is not smooth, singularities always exist at the conical tip. However, here we make a particular assumption on the singular exponent which has to be of the form −1/2 + iη with η ∈ R, η = 0. Such singularities play a particular role for the operator A ε introduced in (3) because they are "just" outside H 1 . More precisely, we have s / ∈ H 1 (Ω) but r γ s ∈ H 1 (Ω) for all γ > 0. With them, we can construct a sequence of functions u n ∈ H 1 0 (Ω) such that ∀n ∈ N, u n H 1 (Ω) = 1 and lim n→+∞ div(ε∇u n ) (H 1 0 (Ω)) * + u n Ω = 0.
Then this allows one to prove that the range of A ε : H 1 0 (Ω) → (H 1 0 (Ω)) * is not closed (see [12,8,10] in 2D). Of course, for any given geometry, such singularities do not exist when κ ε > 0 because we know that in this case A ε : H 1 0 (Ω) → (H 1 0 (Ω)) * is an isomorphism. On the other hand, when = {(cos θ cos φ, sin θ cos φ, sin φ) | −π ≤ θ ≤ π, −π/2 ≤ φ < −π/2+α} for some α ∈ (0; π) (5) Figure 1: The domain Ω with the inclusion M exhibiting a conical tip.
(the circular conical tip, see Figure 1), it can be shown that such s exists for κ ε > −1 (resp. κ ε < −1) and |κ ε + 1| small enough (see [28]) when α < π/2 (resp. α > π/2). For a general smooth domain ⊂ S 2 and a given contrast κ ε , in order to know if such s exists, one has to solve the spectral problem
Find (Φ, λ) ∈ H 1 (S 2 ) \ {0} × C such that S 2 ε∇ S Φ · ∇ S Φ ds = λ(λ + 1)ˆS 2 εΦ Φ ds, ∀Φ ∈ H 1 (S 2 ),(6)
and see if among the eigenvalues some of them are of the form λ = −1/2 + iη with η ∈ R, η = 0. Above, ∇ S stands for the surface gradient. With a slight abuse, when ε is involved into integrals over S 2 , we write ε instead of ε(ρ ·). Note that since ε is real-valued, if λ = −1/2 + iη is an eigenvalue, we have λ(λ + 1) = −η 2 − 1/4, so that λ = −1/2 − iη is also an eigenvalue for the same eigenfunction. And since λ(λ + 1) ∈ R, we can find a corresponding eigenfunction which is real-valued. From now on, we assume that Φ in (4) is real-valued. Let us mention that this problem of existence of singularities of the form (4) is directly related to the problem of existence of essential spectrum for the so-called Neumann-Poincaré operator [29,43,13,27]. A noteworthy difference with the 2D case of a corner in the interface is that several singularities of the form (4) with different values of |η| can exist in 3D [28] (this depends on ε and on ). To simplify the presentation, we assume that for the case of interest, singularities of the form (4) exist for only one value of |η|. Moreover we assume that the quantity´S 2 ε|Φ| 2 ds does not vanish. In this case, exchanging η by −η if necessary, we can set η so that ηˆS 2 ε|Φ| 2 ds > 0.
For the 2D problem, it can be proved that the quantity corresponding to´S 2 ε|Φ| 2 ds vanishes if and only if the contrast κ ε coincides with a bound of the critical interval. We conjecture that this also holds in 3D. Note that when´S 2 ε|Φ| 2 ds = 0, the singularities have a different form from (4). To fix notations, we set
s ± (x) = χ(r)r −1/2±iη Φ(θ, φ)(8)
In this definition the smooth cut-off function χ is equal to one in a neighbourhood of 0 and is supported in [−ρ; ρ]. In particular, we emphasize that s ± vanish in a neighbourhood of ∂Ω.
In order to recover Fredholmness for the scalar problem involving ε, an important idea is too add one (and only one) of the singularities (8) to the functional framework. From a mathematical point of view, working with the complex conjugation, it is obvious to see that adding s + or s − does not change the results. However physically one framework is more relevant than the other. More precisely, we will explain in §3.7 with the limiting absorption principle why selecting s + , with η such that (7) holds, together with a certain convention for the time-harmonic dependence, is more natural.
Kondratiev functional framework
In this paragraph, adapting what is done in [10] for the 2D case, we describe in more details how to get a Fredholm operator for the scalar operator associated with ε. For β ∈ R and m ∈ N, let us introduce the weighted Sobolev (Kondratiev) space V m β (Ω) (see [30]) defined as the closure of
C ∞ 0 (Ω \ {O}) for the norm ϕ V m β (Ω) = |α|≤m r |α|−m+β ∂ α x ϕ 2 L 2 (Ω) 1/2 .
Here C ∞ 0 (Ω \ {O}) denotes the space of infinitely differentiable functions which are supported in Ω \ {O}. We also denoteV 1 β (Ω) the closure of C ∞ 0 (Ω \ {O}) for the norm · V 1 β (Ω) . We have the characterisationV
1 β (Ω) = {ϕ ∈ V 1 β (Ω) | ϕ = 0 on ∂Ω}. Note that using Hardy's inequalitŷ 1 0 |u(r)| 2 r 2 r 2 dr ≤ 4ˆ1 0 |u (r)| 2 r 2 dr, ∀u ∈ C 1 0 [0; 1), one can show the estimate r −1 ϕ Ω ≤ C ∇ϕ Ω for all ϕ ∈ C ∞ 0 (Ω \ {O}). This proves that V 1 0 (Ω) = H 1 0 (Ω). Now set β > 0. Observe that we have V 1 −β (Ω) ⊂ H 1 0 (Ω) ⊂V 1 β (Ω) so that (V 1 β (Ω)) * ⊂ (H 1 0 (Ω)) * ⊂ (V 1 −β (Ω)) * . Define the operators A ±β ε :V 1 ±β (Ω) → (V 1 ∓β (Ω)) * such that A ±β ε ϕ, ϕ =ˆΩ ε∇ϕ · ∇ϕ dx, ∀ϕ ∈V 1 ±β (Ω), ϕ ∈V 1 ∓β (Ω).(9)
Working as in [10] for the 2D case of the corner, one can show that there is β 0 > 0 (depending only on κ ε and ) such that for all β ∈ (0; β 0 ), A β ε is Fredholm of index +1 while A −β ε is Fredholm of index −1. We remind the reader that for a bounded linear operator between two Banach spaces T : X → Y whose range is closed, its index is defined as ind T := dim ker T − dim coker T , with dim coker T = dim (Y/range(T )). On the other hand, application of Kondratiev calculus guarantees that if ϕ ∈V 1 β (Ω) is such that A +β ε ϕ ∈ (V 1 β (Ω)) * (the important point here being that (V 1 β (Ω)) * ⊂ (V 1 −β (Ω)) * ), then there holds the following representation
ϕ = c − s − + c + s + +φ with c ± ∈ C andφ ∈V 1 −β (Ω).(10)
Note that s ± , with s ± defined by (8), belongs toV 1 β (Ω), but not to H 1 0 (Ω), and a fortiori not to V 1 −β (Ω). Then introduce the spaceV out := span(s + ) ⊕V 1 −β (Ω), endowed with the norm
ϕ V out = (|c| 2 + φ 2 V 1 −β (Ω)) ) 1/2 , ∀ϕ = c s + +φ ∈V out ,(11)
which is a Banach space. Introduce also the operator A out ε such that for all ϕ = c s + +φ ∈V out and ϕ ∈ C ∞ 0 (Ω \ {O}),
A out ε ϕ, ϕ =ˆΩ ε∇ϕ · ∇ϕ dx = −cˆΩ div(ε∇s + )ϕ dx +ˆΩ ε∇φ · ∇ϕ dx.
Note that due to the features of the cut-off function χ, we have div(ε∇s + ) ∈ L 2 (Ω). And since div(ε∇s + ) = 0 in a neighbourhood of O, we observe that there is a constant C > 0 such that
| A out ε ϕ, ϕ | ≤ C ϕ V out ϕ V 1 β (Ω) .
The density of C ∞ 0 (Ω \ {O}) inV 1 β (Ω) then allows us to extend A out ε as a continuous operator fromV out to (V 1 β (Ω)) * . And we have A out ε ϕ, ϕ = −cˆΩ div(ε∇s + )ϕ dx +ˆΩ ε∇φ · ∇ϕ dx, ∀ϕ = c s + +φ, ϕ ∈V 1 β (Ω).
Working as in [10] (see Proposition 4.4.) for the 2D case of the corner, one can prove that A out ε :V out → (V 1 β (Ω)) * is Fredholm of index zero and that ker A out ε = ker A −β ε . In order to simplify the analysis below, we shall make the following assumption.
Assumption 3. We assume that ε is such that for β ∈ (0; β 0 ), A −β ε is injective, which guarantees that A out ε :V out → (V 1 β (Ω)) * is an isomorphism.
In what follows, we shall also need to work with the usual Laplace operator in weighted Sobolev spaces. For γ ∈ R, define A γ :
V 1 γ (Ω) → (V 1 −γ (Ω)) * such that A γ ϕ, ϕ =ˆΩ ∇ϕ · ∇ϕ dx, ∀ϕ ∈V 1 γ (Ω), ϕ ∈V 1 −γ (Ω)
(observe that there is no ε here). Combining the theory presented in [32] (see also the founding article [30] as well as the monographs [34,36]) together with the result of [31, Corollary 2.2.1], we get the following proposition.
Proposition 2.1. For all γ ∈ (−1/2; 1/2), the operator A γ :V 1 γ (Ω) → (V 1 −γ (Ω)) * is an isomor- phism.
Note in particular that for γ = 0, this proposition simply says that ∆ : H 1 0 (Ω) → (H 1 0 (Ω)) * is an isomorphism. In order to have a result of isomorphism both for A out ε and A β , we shall often make the assumption that the weight β is such that
0 < β < min(1/2, β 0 )(12)
where β 0 is defined after (9). To measure electromagnetic fields in weighted Sobolev norms, in the following we shall work in the spaces
V 0 β (Ω) := (V 0 β (Ω)) 3 V 1 β (Ω) := (V 1 β (Ω)) 3 . Note that we have V 0 −β (Ω) ⊂ L 2 (Ω) ⊂ V 0 β (Ω).
Analysis of the problem for the electric component
In this section, we consider the problem for the electric field associated with (1)- (2). Since the scalar problem involving ε is well-posed in a non standard framework involving the propagating singularity s + (see (11)), we shall add its gradient in the space for the electric field. Then we define a variational problem in this unsual space, and prove its well-posedness. Finally we justify our choice by a limiting absorption principle.
A well-chosen space for the electric field
Define the space of electric fields with the divergence free condition
X out N (ε) := {u = c∇s + +ũ, c ∈ C,ũ ∈ L 2 (Ω) | curl u ∈ L 2 (Ω), div(εu) = 0 in Ω \ {O}, u × ν = 0 on ∂Ω}.(13)
In this definition, for u = c∇s + +ũ, the condition div(εu) = 0 in Ω \ {O} means that there holdŝ
Ω εu · ∇ϕ dx = 0, ∀ϕ ∈ C ∞ 0 (Ω \ {O}),(14)
which after integration by parts and by density of
C ∞ 0 (Ω \ {O}) in H 1 0 (Ω) is equivalent to − cˆΩ div(ε∇s + )ϕ dx +ˆΩ εũ · ∇ϕ dx = 0, ∀ϕ ∈ C ∞ 0 (Ω).(15)
Note that we have X N (ε) ⊂ X out N (ε) and that dim (X out N (ε)/X N (ε)) = 1 (see Lemma D.1 in Appendix). For u = c∇s + +ũ with c ∈ C andũ ∈ L 2 (Ω), we set
u X out N (ε) = (|c| 2 + ũ 2 Ω + curl u 2 Ω ) 1/2 .
Endowed with this norm, X out N (ε) is a Banach space. Lemma 3.1. Pick some β satisfying (12). Under Assumptions 1 and 3, for
any u = c∇s + +ũ ∈ X out N (ε), we haveũ ∈ V 0 −β (Ω) and there is a constant C > 0 independent of u such that |c| + ũ V 0 −β (Ω) ≤ C curl u Ω .(16)
As a consequence, the norm · X out N (ε) is equivalent to the norm curl · Ω in X out N (ε) and X out N (ε) endowed with the inner product (curl ·, curl ·) Ω is a Hilbert space.
Proof. Let u = c∇s + +ũ be an element of X out N (ε). The fieldũ is in L 2 (Ω) and therefore decomposes asũ = ∇ϕ + curl ψ (17) with ϕ ∈ H 1 0 (Ω) and ψ ∈ X T (1) (item iv) of Proposition A.1). Moreover, since u × ν = 0 on ∂Ω and since both s + and ϕ vanish on ∂Ω, we know that curl ψ × ν = 0 on ∂Ω. Then noting that
−∆ψ = curlũ = curl u ∈ L 2 (Ω), we deduce from Proposition A.2 that curl ψ ∈ V 0 −β (Ω) with the estimate curl ψ V 0 −β (Ω) ≤ C curl u Ω .(18)
Using (14), the condition div(εu (10) we know that there are some complex constants c ± and someφ ∈V 1 −β (Ω) such that
) = 0 in Ω \ {O} implieŝ Ω ε∇(c s + + ϕ) · ∇ϕ dx = −ˆΩ εcurl ψ · ∇ϕ dx, ∀ϕ ∈V 1 −β (Ω), which means exactly that A β ε (c s + + ϕ) = −div(ε curl ψ) ∈ (V 1 −β (Ω)) * . Since additionally −div(ε curl ψ) ∈ (V 1 β (Ω)) * , fromc s + + ϕ = c − s − + c + s + +φ.
This implies c − = 0, c + = c (because ϕ ∈ H 1 0 (Ω)) and so ϕ =φ is an element ofV 1 −β (Ω). This shows that c s + +ϕ ∈V out and that A out
ε (c s + +ϕ) = −div(ε curl ψ). Since A out ε :V out → (V 1 β (Ω)) * is an isomorphism, we have the estimate |c| + ϕ V 1 −β (Ω) ≤ C div(ε curl ψ) (V 1 β (Ω)) * ≤ C curl ψ V 0 −β (Ω) .(19)
Finally gathering (17)- (19), we obtain thatũ ∈ V 0 −β (Ω) and that the estimate (16) is valid. Noting that ũ Ω ≤ C ũ V 0 −β (Ω) , this implies that the norms · X out N (ε) and curl · Ω are equivalent in X out N (ε). Thanks to the previous lemma and by density of
C ∞ 0 (Ω \ {O}) inV 1 β (Ω), the condition (15) for u = c∇s + +ũ ∈ X out N (ε) is equivalent to − cˆΩ div(ε∇s + )ϕ dx +ˆΩ εũ · ∇ϕ dx = 0, ∀ϕ ∈V 1 β (Ω)(20)
where all the terms are well-defined as soon as β satisfies (12).
Definition of the problem for the electric field
Our objective is to define the problem for the electric field as a variational formulation set in X out N (ε). For some γ > 0, let J be an element of V 0 −γ (Ω) such that div J = 0 in Ω. Consider the problem
Find u ∈ X out N (ε) such that Ω µ −1 curl u · curl v dx − ω 2 Ω εu · v dx = iωˆΩ J · v dx, ∀v ∈ X out N (ε),(21)
where the term
Ω εu · v dx(22)
has to be carefully defined. The difficulty comes from the fact that X out N (ε) is not a subspace of L 2 (Ω) so that this quantity cannot be considered as a classical integral.
Let u = c u ∇s + +ũ ∈ X out N (ε). First, forṽ ∈ V 0 −β (Ω) with β > 0, it is natural to set Ω εu ·ṽ dx :=ˆΩ εu ·ṽ dx.(23)
To complete the definition, we have to give a sense to (22) when v = ∇s + . Proceeding as for the derivation of (20), we start from the identitŷ
Ω εu · ∇ϕ dx = −c uˆΩ div(ε∇s + )ϕ dx +ˆΩ εũ · ∇ϕ dx, ∀ϕ ∈ C ∞ 0 (Ω \ {O}). By density of C ∞ 0 (Ω \ {O}) inV 1 β (Ω), this leads to set Ω εu · ∇ϕ dx := −c uˆΩ div(ε∇s + )ϕ dx +ˆΩ εũ · ∇ϕ dx, ∀ϕ ∈V 1 β (Ω).(24)
With this definition, condition (20) can be written as
Ω εu · ∇ϕ dx = 0, ∀ϕ ∈V 1 β (Ω).
In particular, since s + ∈V 1 β (Ω), for all u ∈ X out N (ε) we have Ω εu · ∇s + dx = 0 and soˆΩ εũ · ∇s + dx = c uˆΩ div(ε∇s + )s + dx.
Finally for all u = c u ∇s + +ũ and v = c v ∇s + +ṽ in X out N (ε), using (23) and (25), we find
Ω εu · v dx =ˆΩ εu ·ṽ dx = c uˆΩ ε∇s + ·ṽ dx +ˆΩ εũ ·ṽ dx.
But since v ∈ X out N (ε), we deduce from the second identity of (25) that
Ω ε∇s + ·ṽ dx = c vˆΩ div(ε∇s + )s + dx.(26)
Summing up, we get
Ω εu · v dx = c u c vˆΩ div(ε∇s + )s + dx +ˆΩ εũ ·ṽ dx, ∀u, v ∈ X out N (ε).(27)
Remark 3.2. Even if we use an integral symbol to keep the usual aspects of formulas and facilitate the reading, it is important to consider this new quantity as a sesquilinear form
(u, v) → Ω εu · v dx on X out N (ε) × X out N (ε).
In particular, we point out that this sesquilinear form is not hermitian on
X out N (ε) × X out N (ε). Indeed, we have Ω εv · u dx =ˆΩ εũ ·ṽ dx + c u c vˆΩ div(ε∇s + )s + dx so that Ω εu · v dx − Ω εv · u dx = 2ic u c v m ˆΩ div(ε∇s + ) s + dx .(28)
But Lemma C. 1 and assumption (7) show that
m ˆΩ div(ε∇s + ) s + dx = 0.
In the sequel, we denote by a N (·, ·) (resp. N (·)) the sesquilinear form (resp. the antilinear form) appearing in the left-hand side (resp. right-hand side) of (21).
Equivalent formulation
Define the space H out N (curl ) := span(∇s + ) ⊕ H N (curl ) ⊃ X out N (ε) (without the divergence free condition) and consider the problem
Find u ∈ H out N (curl ) such that a N (u, v) = N (v), ∀v ∈ H out N (curl ),(29)
where the definition of
Ω εu · v dx
has to be extended to the space H out N (curl ). Working exactly as in the beginning of the proof of Lemma 3.1, one can show that any u ∈ H out N (curl ) admits the decomposition
u = c u ∇s + + ∇ϕ u + curl ψ u ,(30)
with c u ∈ C, ϕ u ∈ H 1 0 (Ω) and ψ u ∈ X T (1), such that curl ψ u ∈ V 0 −β (Ω), for β satisfying (12).
Then, for all u = c u ∇s + + ∇ϕ u + curl ψ u and v = c v ∇s + + ∇ϕ v + curl ψ v in H out N (curl )
, a natural extension of the previous definitions leads to set
Ω εu · v dx :=ˆΩ ε (∇ϕ u + curl ψ u ) · (∇ϕ v + curl ψ v ) dx +ˆΩ c u ε∇s + · curl ψ v + c v ε curl ψ u · ∇s + dx −ˆΩ c u c v div(ε∇s + )s + + c u div(ε∇s + )ϕ v + c v ϕ u div(ε∇s + ) dx.(31)
Note that (31) is indeed an extension of (27). To show it, first observe that for u = c u ∇s + + ∇ϕ u + curl ψ u , v = c v ∇s + + ∇ϕ v + curl ψ v in X out N (ε), the proof of Lemma 3.1 guarantees that (12). This allows us to integrate by parts in the last two terms of (31) to get
ϕ u , ϕ v ∈V 1 −β (Ω) with β satisfyingΩ εu · v dx :=ˆΩ ε (∇ϕ u + curl ψ u ) · (∇ϕ v + curl ψ v ) dx +ˆΩ c u ε∇s + · (∇ϕ v + curl ψ v ) + c v ε (∇ϕ u + curl ψ u ) · ∇s + dx −c u c vˆΩ div(ε∇s + )s + dx.(32)
Using (25), (26), the second line above can be written aŝ
Ω c u ε∇s + · (∇ϕ v + curl ψ v ) + c v ε (∇ϕ u + curl ψ u ) · ∇s + dx = c u c vˆΩ div(ε∇s + )s + dx + c u c vˆΩ div(ε∇s + )s + dx.(33)
Inserting (33) in (32) yields exactly (27). (21) if and only if it solves the problem (29). (29), and using that div J = 0 in Ω, we get (14), which implies that u ∈ X out N (ε). This shows that u solves (21).
Lemma 3.3. Under Assumptions 1 and 3, the field u is a solution of
Proof. If u ∈ H out N (curl ) satisfies (29), then taking v = ∇ϕ with ϕ ∈ C ∞ 0 (Ω \ {O}) inNow assume that u ∈ X out N (ε) ⊂ H out N (curl ) is a solution of (21). Let v be an element of H out N (curl ). As in (30), we have the decomposition v = c v ∇s + + ∇ϕ v + curl ψ v ,(34)
with c v ∈ C, ϕ v ∈ H 1 0 (Ω) and ψ v ∈ X T (1) such that curl ψ v ∈ V 0 −β (Ω) for all β satisfying (12). By Assumption 3, there is ζ ∈V out such that
A out ε ζ = −div(ε curl ψ v ) ∈ (V 1 β (Ω)) * .(35)
The function ζ decomposes as ζ = αs + +ζ withζ
∈V 1 −β (Ω). Finally, set v = curl ψ v − ∇ζ = v − ∇(c v s + + ϕ v + ζ).
The functionv is in X out N (ε), it satisfies curlv = curl v and from (25), we deduce that
Ω εu ·v dx = Ω εu · v dx.
Using also that J ∈ V 0 −γ (Ω) for some γ > 0 and is such that div J = 0 in Ω, so that
Ω J ·v dx =ˆΩ J · v dx, this shows that a N (u, v) = a N (u,v) = N (v) = N (v)
and ends the proof.
In the following, we shall work with the formulation (21) set in X out N (ε). The reason being that, as usual in the analysis of Maxwell's equations, the divergence free condition will yield a compactness property allowing us to deal with the term involving the frequency ω.
Main analysis for the electric field
Define the continuous operators A out
N : X out N (ε) → (X out N (ε)) * and K out N : X out N (ε) → (X out N (ε)) * such that for all u, v ∈ X out N (ε), A out N u, v =ˆΩ µ −1 curl u · curl v dx, K out N u, v = Ω εu · v dx.
With this notation, we have (A out N (u, v).
N + K out N )u, v = aProposition 3.4. Under Assumptions 1-3, the operator A out N : X out N (ε) → (X out N (ε)) * is an isomorphism.
Proof. Let us construct a continuous operator T :
X out N (ε) → X out N (ε) such that for all u, v ∈ X out N (ε),ˆΩ µ −1 curl u · curl (Tv) dx =ˆΩ curl u · curl v dx.
To proceed, we adapt the method presented in [9]. Assume that v ∈ X out N (ε) is given. We construct Tv in three steps.
1) Since curl v ∈ L 2 (Ω) and A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism, there is a unique ζ ∈ H 1 # (Ω) such that Ω µ∇ζ · ∇ζ dx =ˆΩ µ curl v · ∇ζ dx, ∀ζ ∈ H 1 # (Ω). Then the field µ(curl v − ∇ζ) ∈ L 2 (Ω) is divergence free in Ω and satisfies µ(curl v − ∇ζ) · ν = 0 on ∂Ω. 2) From item ii) of Proposition A.1, we infer that there is ψ ∈ X N (1) such that µ(curl v − ∇ζ) = curl ψ.
Thanks to Lemma A.5, we deduce that ψ ∈ V 0 −β (Ω) for all β ∈ (0; 1/2) and a fortiori for β satisfying (12).
3) Suppose now that β satisfies (12). Then we know from the previous step that div(εψ) ∈ (V 1 β (Ω)) * . On the other hand, by Assumption 3, A out
ε :V out → (V 1 β (Ω)) * is an isomorphism. Consequently we can introduce ϕ ∈V out such that A out ε ϕ = −div(εψ). Finally, we set Tv = ψ − ∇ϕ. Clearly Tv is an element of X out N (ε). Moreover, for all u, v in X out N (ε), we havê Ω µ −1 curl u · curl Tv dx =ˆΩ µ −1 curl u · curl ψ dx =ˆΩ curl u · curl v dx −ˆΩ curl u · ∇ζ dx =ˆΩ curl u · curl v dx.
From Lemma 3.1 and the Lax-Milgram theorem, we deduce that T * A out N : X out N (ε) → (X out N (ε)) * is an isomorphism. And by symmetry, permuting the roles of u and v, it is obvious that T * A out N = A out N T, which allows us to conclude that A out N : X out N (ε) → (X out N (ε)) * is an isomorphism.
Proposition 3.5. Under Assumptions 1 and 3, if
(u n = c n ∇s + +ũ n ) is a sequence which is bounded in X out N (ε), then we can extract a subsequence such that (c n ) and (ũ n ) converge respectively in C and in V 0 −β (Ω) for β satisfying (12). As a consequence, the operator K out N :
X out N (ε) → (X out N (ε)) * is compact.
Proof. Let (u n ) be a bounded sequence of elements of X out N (ε). From the proof of Lemma 3.1, we know that for n ∈ N, we have u n = c n ∇s + + ∇ϕ n + curl ψ n (36) where the sequences (c n ), (ϕ n ), (ψ n ) and (curl ψ n ) are bounded respectively in C,V 1 −β (Ω), X T (1) and V 0 −β (Ω). Observing that curl u n = curl curl ψ n = −∆ψ n is bounded in L 2 (Ω), we deduce from Proposition A.3 that there exists a subsequence such that curl ψ n converges in V 0 −β (Ω). Moreover, by (19), we have
|c n − c m | + ϕ n − ϕ m V 1 −β (Ω) ≤ C curl (ψ n − ψ m ) V 0 −β (Ω) ,
which implies that (c n ) and (ϕ n ) converge respectively in C and inV 1 −β (Ω). From (36), we see that this is enough to conclude to the first part of the proposition. Finally, observing that
K out N u (X out N (ε)) * ≤ C ( ũ V 0 −β (Ω) + |c u |), we deduce that K out N : X out N (ε) → (X out N (ε)) * is a compact operator.
We can now state the main theorem of the analysis of the problem for the electric field.
Theorem 3.6. Under Assumptions 1-3, for all
ω ∈ R the operator A out N − ω 2 K out N : X out N (ε) → (X out N (ε)) * is Fredholm of index zero.
Proof. Since K out N : X out N (ε) → (X out N (ε)) * is compact (Proposition 3.5) and A out N :
X out N (ε) → (X out N (ε)) * is an isomorphism (Proposition 3.4), A out N −ω 2 K out N : X out N (ε) → (X out N (ε)) * is Fredholm of index zero.
The previous theorem guarantees that the problem (21) is well-posed if and only if uniqueness holds, that is if and only if the only solution for J = 0 is u = 0. Since uniqueness holds for ω = 0, one can prove with the analytic Fredholm theorem that (21) is well-posed except for at most a countable set of values of ω with no accumulation points (note that Theorem 3.6 remains true for ω ∈ C). However this result is not really relevant from a physical point of view. Indeed, negative values of ε can occur only if ε is itself a function of ω. For instance, if the inclusion M is metallic, it is commonly admitted that the Drude's law gives a good model for ε. But taking into account the dependence of ε with respect to ω when studying uniqueness of problem (21) leads to a non-linear eigenvalue problem, where the functional space X out N (ε) itself depends on ω. This study is beyond the scope of the present paper (see [24] for such questions in the case of the 2D scalar problem). Nonetheless, there is a result that we can prove concerning the cases of non-uniqueness for problem (21). Proposition 3.7. If u = c∇s + +ũ ∈ X out N (ε) is a solution of (21) for J = 0, then c = 0 and u ∈ X N (ε).
Proof. When ω = 0, the result is a direct consequence of Theorem 3.6 (because zero is the only solution of (21) for J = 0). From now on, we assume that ω ∈ R \ {0}. Suppose that u = c∇s + +ũ ∈ X out N (ε) is such that
Ω µ −1 curl u · curl v dx − ω 2 Ω εu · v dx = 0, ∀v ∈ X out N (ε).
Taking the imaginary part of the previous identity for v = u, we get m Ω εu · u dx = 0.
On the other hand, by (27), we have
Ω εu · u dx =ˆΩ ε|ũ| 2 dx + |c| 2ˆΩ div(ε∇s + ) s + dx, so that |c| 2 m ˆΩ div(ε∇s + ) s + dx = 0.
The result of the proposition is then a consequence of Lemma C.1 in Appendix where it is proved that
m ˆΩ div(ε∇s + ) s + dx = ηˆS 2 ε|Φ| 2 ds,
and of the assumption (7).
Remark 3.8. As a consequence, from Lemma 3.1, we infer that elements of the kernel of
A out N − ω 2 K out N are in V 0 −β (Ω)
for all β satisfying (12).
Problem in the classical framework
In the previous paragraph, we have shown that the Maxwell's problem (21) for the electric field set in the non standard space X out N (ε), and so in H out N (curl ) according to Lemma 3.3, is wellposed. Here, we wish to analyse the properties of the problem for the electric field set in the classical space X N (ε) (which does not contain ∇s + ). Since this space is a closed subspace of X out N (ε), it inherits the main properties of the problem in X out N (ε) proved in the previous section. More precisely, we deduce from Lemma 3.1 and Proposition 3.5 the following result. Proposition 3.9. Under Assumptions 1 and 3, the embedding of X N (ε) in L 2 (Ω) is compact, and curl · Ω is a norm in X N (ε) which is equivalent to the norm · H(curl ) .
Note that we recover classical properties similar to what is known for positive ε, or more generally [9] for ε such that the operator A ε : H 1 0 (Ω) → (H 1 0 (Ω)) * defined by (3) is an isomorphism (which allows for sign-changing ε). But we want to underline the fact that under Assumption 3, these classical results could not be proved by using classical arguments. They require the introduction of the bigger space X out N (ε), with the singular function ∇s + . Let us now consider the problem
Find u ∈ X N (ε) such that Ω µ −1 curl u · curl v dx − ω 2ˆΩ εu · v dx = iωˆΩ J · v dx, ∀v ∈ X N (ε).(37)
An important remark is that one cannot prove that problem (37) is equivalent to a similar problem set in H N (curl ) (the analogue of Lemma 3.3). Again, the difficulty comes from the fact that A ε is not an isomorphism, and the trouble would appear when solving (35). Therefore, a solution of (37) is not in general a distributional solution of the equation
curl µ −1 curl u − ω 2 εu = iωJ .
To go further in the analysis of (37), we recall that X N (ε) is a subspace of codimension one of X out N (ε) (Lemma D.1 in Appendix). Let v 0 be an element of X out N (ε) which does not belong to X N (ε). Then we denote by 0 the continuous linear form on X out N (ε) such that:
∀v ∈ X out N (ε) v − 0 (v)v 0 ∈ X N (ε).(38)
Let us now define the operators A N : X N (ε) → (X N (ε)) * and K N :
X N (ε) → (X N (ε)) * by A N u, v =ˆΩ µ −1 curl u · curl v dx, K N u, v =ˆΩ εu · v dx.
Proposition 3.10. Under Assumptions 1-3, the operator
A N : X N (ε) → (X N (ε)) * is Fredholm of index zero.
Proof. Let u ∈ X N (ε). By Proposition 3.4, for the operator T introduced in the corresponding proof, one has:
u 2 X N (ε) = curl u 2 Ω = A out N u,
Tu . Then, using (38), we get:
u 2 X N (ε) = A N u, Tu − 0 (Tu)v 0 + A out N u, 0 (Tu)v 0 , which implies that u X N (ε) ≤ C A N u (X N (ε)) * + | 0 (Tu)| .
The result of the proposition then follows from a classical adaptation of Peetre's lemma (see for example [48,Theorem 12.12]) together with the fact that A N is bounded and hermitian.
Combining the two previous propositions, we obtain the (37) is well-posed, it does not provide a solution of Maxwell's equations.
Theorem 3.11. Under Assumptions 1-3, for all ω ∈ R, the operator
A N − ω 2 K N : X N (ε) → (X N (ε)) * is Fredholm of index zero.
But as mentioned above, even if uniqueness holds and if Problem
Expression of the singular coefficient
Under Assumptions 1-3, Theorem 3.6 guarantees that for all ω ∈ R the operator A out N − ω 2 K out N : X out N (ε) → (X out N (ε)) * is Fredholm of index zero. Assuming that it is injective, the problem (21) admits a unique solution u = c u ∇s + +ũ. The goal of this paragraph is to derive a formula allowing one to compute c u without knowing u. Such kind of results are classical for scalar operators (see e.g. [22], [32,Theorem 6.4.4], [18,19,2,23,49,40]). They are used in particular for numerical purposes. But curiously they do not seem to exist for Maxwell's equations in 3D, not even for classical situations with positive materials in non smooth domains. We emphasize that the analysis we develop can be adapted to the latter case.
In order to establish the desired expression, for ω ∈ R, first we introduce the field w N ∈ X out N (ε) such that
Ω µ −1 curl v · curl w N dx − ω 2 Ω εv · w N dx =ˆΩ εṽ · ∇s + dx, ∀v ∈ X out N (ε).(39)
Note that Problem (39) is well-posed when A out N − ω 2 K out N is an isomorphism. Indeed, using (28), one can check that it involves the operator (A out N − ω 2 K out N ) * , that is the adjoint of A out N − ω 2 K out N . Moreover v →´Ω εṽ · ∇s + dx is a linear form over X out N (ε).
Theorem 3.12.
Assume that ω ∈ R, Assumptions 1-3 are valid and A out N − ω 2 K out N : X out N (ε) → (X out N (ε)) * is injective. Then the solution u = c u ∇s + +ũ of the electric problem (21) is such that
c u = iωˆΩ J · w N dx ˆΩ div(ε∇s + ) s + dx.(40)
Here w N is the function which solves (39).
Remark 3.13. Note that in practice w N can be computed once for all because it does not depend on J . Then the value of c u can be determined very simply via Formula (40).
Proof. By definition of u, we havê
Ω µ −1 curl u · curl w N dx − ω 2 Ω εu · w N dx = iωˆΩ J · w N dx.
On the other hand, from (39), there holdŝ
Ω µ −1 curl u · curl w N dx − ω 2 Ω εu · w N dx =ˆΩ εũ · ∇s + dx.
From these two relations as well as (25), we get iωˆΩ J · w N dx =ˆΩ εũ · ∇s + dx = c uˆΩ div(ε∇s + ) s + dx.
But Lemma C.1 in Appendix guarantees that m´Ω div(ε∇s + ) s + dx = 0. Therefore we find the desired formula.
Limiting absorption principle
In §3.4, we have proved well-posedness of the problem for the electric field in the space X out N (ε). But up to now, we have not explained why we select this framework. In particular, as mentioned in §2.1, well-posedness also holds in X in N (ε) where X in N (ε) is defined as X out N (ε) with s + replaced by s − (see (8) for the definitions of s ± ). In general, the solution in X in N (ε) differs from the one in X out N (ε). Therefore one can build infinitely many solutions of Maxwell's problem as linear interpolations of these two solutions. Then the question is: which solution is physically relevant?
Classically, the answer can be obtained thanks to the limiting absorption principle. The idea is the following. In practice, the dielectric permittivity takes complex values, the imaginary part being related to the dissipative phenomena in the materials. Set
ε δ := ε + iδ
where ε is defined as previously (see (2)) and δ > 0 (the sign of δ depends on the convention for the time-harmonic dependence (in e −iωt here)). Due to the imaginary part of ε δ which is uniformly positive, one recovers some coercivity properties which allow one to prove well-posedness of the corresponding problem for the electric field in the classical framework. The physically relevant solution for the problem with the real-valued ε then should be the limit of the sequence of solutions for the problems involving ε δ when δ tends to zero. The goal of the present paragraph is to explain how to show that this limit is the solution of the problem set in X out N (ε).
Limiting absorption principle for the scalar case
Our proof relies on a similar result for the 3D scalar problem which is the analogue of what has been done in 2D in [9,Theorem 4.3]. Consider the problem
Find ϕ δ ∈ H 1 0 (Ω) such that − div(ε δ ∇ϕ δ ) = f,(41)
where f ∈ (H 1 0 (Ω)) * . Since δ > 0, by the Lax-Milgram lemma, this problem is well-posed for all f ∈ (H 1 0 (Ω)) * and in particular for all f ∈ (V 1 β (Ω)) * , β > 0. Our objective is to prove that ϕ δ converges when δ tends to zero to the unique solution of the problem
Find ϕ ∈V out such that A out ε ϕ = f.(42)
We expect a convergence in a spaceV 1 β (Ω) with 0 < β < β 0 . We first need a decomposition of ϕ δ as a sum of a singular part and a regular part. Since problem (41) is strongly elliptic, one can directly apply the theory presented in [32]. On the one hand, from the assumptions of Section 2, one can verify that for δ small enough, there exists one and only one singular exponent λ δ ∈ C such that e λ δ ∈ (−1/2; −1/2 + β 0 − √ δ). We denote by s δ the corresponding singular function such that s δ (r, θ, ϕ) = r λ δ Φ δ (θ, φ).
Note that it satisfies div(ε δ ∇s δ ) = 0 in K. As in (8) for s ± , we set
s δ (x) = χ(r) r −1/2+iη δ Φ δ (θ, φ),(43)
where η δ ∈ C is the number such that λ δ = −1/2 + iη δ . By applying [32, Theorem 5.4.1], we get the following result.
Lemma 3.14. Let 0 < β < β 0 and f ∈ (V 1 β (Ω)) * . The solution ϕ δ of (41) decomposes as ϕ δ = c δ s δ +φ δ (44) where c δ ∈ C andφ δ ∈V 1 −β (Ω). Let us first study the limit of the singular function. (8)).
Proof. The pair (Φ δ , λ δ ) solves the spectral problem
Find (Φ δ , λ δ ) ∈ H 1 (S 2 ) \ {0} × C such that S 2 ε δ ∇ S Φ δ · ∇ S Ψ ds = λ δ (λ δ + 1)ˆS 2 ε δ Φ δ Ψ ds, ∀Ψ ∈ H 1 (S 2 ).(45)
Postulating the expansions Φ δ = Φ 0 +δΦ +. . . , λ δ = λ 0 +δλ +. . . in this problem and identifying the terms in δ 0 , we get Φ 0 = Φ and we find that λ 0 = −1/2 + iη 0 where η 0 coincides with η or −η (see an illustration with Figure 2). At order δ, we get the variational equalitŷ
S 2 ε∇ S Φ · ∇ S Ψ ds + iˆS 2 ∇ S Φ · ∇ S Ψ ds = λ 0 (λ 0 + 1) ˆS 2 εΦ Ψ ds + iˆS 2 Φ Ψ ds +λ (2λ 0 + 1)ˆS 2 εΦ Ψ ds, ∀Ψ ∈ H 1 (S 2 ).(46)
Taking Ψ = Φ in (46), using (6) and observing that λ 0 (λ 0 + 1) = −η 2 − 1/4, this implieŝ
S 2 |∇ S Φ| 2 + (η 2 + 1/4)|Φ| 2 ds = λ 2η 0ˆS 2 ε|Φ| 2 ds.
Thus λ is real. Since by definition of λ δ , we have e λ δ > −1/2 for δ > 0, we infer that λ > 0. As a consequence, we have η 0ˆS 2 ε|Φ| 2 ds > 0 which according to the definition of η in (7) ensures that η 0 = η. Therefore the pointwise limit of s δ when δ tends to zero is indeed s + and not s − . This is enough to conclude that s δ converges to s + inV 1 β (Ω) for β > 0. Then proceeding exactly as in the proof of [10,Theorem 4.3], one can establish the following result.
Lemma 3.16. Let 0 < β < β 0 and f ∈ (V 1 β (Ω)) * . If Assumption 3 holds, then (ϕ δ = c δ s δ +φ δ ) converges to ϕ = c s + +φ inV 1 β (Ω) as δ tends to zero. Moreover, (c δ ,φ δ ) converges to (c,φ) in C ×V 1 −β (Ω). In this statement, ϕ δ (resp. ϕ) is the solution of (41) (resp. (42)). Note that the results of Lemma 3.16 still hold if we replace f by a family of source terms (f δ ) ∈ (V 1 β (Ω)) * that converges to f in (V 1 β (Ω)) * when δ tends to zero.
Limiting absorption principle for the electric problem
The problem
Find u δ ∈ X N (ε δ ) such that curl µ −1 curl u δ − ω 2 ε δ u δ = iωJ ,(47)with X N (ε δ ) = {E ∈ H N (curl ) | div(ε δ E) = 0}
, is well-posed for all ω ∈ R and all δ > 0. This result is classical when µ takes positive values while it can be shown by using [9] when µ changes sign. We want to study the convergence of u δ when δ goes to zero. Let (δ n ) be a sequence of positive numbers such that lim n→+∞ δ n = 0. To simplify, we denote the quantities with an index n instead of δ n (for example we write ε n instead of ε δn ).
Lemma 3.17.
Suppose that (u n ) is a sequence of elements of X N (ε n ) such that (curl u n ) is bounded in L 2 (Ω). Then, under Assumption 3, for all β satisfying (12), for all n ∈ N, u n admits the decomposition u n = c n ∇s n +ũ n with c n ∈ C andũ n ∈ V 0 −β (Ω). Moreover, there exists a subsequence such that (c n ) converges to some c in C while (ũ n ) converges to someũ in V 0 −β (Ω). Finally, the field u := c∇s + +ũ belongs to X out N (ε). Proof. For all n ∈ N, we have u n ∈ X N (ε δ ) ⊂ L 2 (Ω). Therefore, there exist ϕ n ∈ H 1 0 (Ω) and ψ n ∈ X T (1), satisfying curl ψ n × ν = 0 on ∂Ω such that u n = ∇ϕ n + curl ψ n . Moreover, we have the estimate ∆ψ n Ω = curl u n Ω ≤ C. As a consequence, Proposition A.2 guarantees that (curl ψ n ) is a bounded sequence of V 0 −β (Ω), and Proposition A.3 ensures that there exists a subsequence such that (curl ψ n ) converges in V 0 −β (Ω). Now from the fact that div(ε n u n ) = 0, we obtain div(ε n ∇ϕ n ) = −div(ε n curl ψ n ) ∈ (V 1 β (Ω)) * . By Lemmas 3.14 and 3.16, this implies that the function ϕ n decomposes as ϕ n = c n s n +φ n with c n ∈ C andφ n ∈V 1 −β (Ω). Moreover, (c n ) converges to c in C while (φ n ) converges toφ in V 1 −β (Ω).
Summing up, we have that u n = c n ∇s n +ũ n whereũ n = ∇φ n + curl ψ n converges toũ in V 0 −β (Ω). In particular, this implies that u n converges to u = c∇s + +ũ in V 0 γ (Ω) for all γ > 0. It remains to prove that u ∈ X out N (ε), which amounts to show that u satisfies (25). To proceed, we take the limit as n → +∞ in the identity −c nˆΩ div(ε n ∇s n )ϕ dx +ˆΩ ε nũn · ∇ϕ dx = 0 which holds for all ϕ ∈V 1 β (Ω) because u n ∈ X N (ε n ).
Theorem 3.18.
Let ω ∈ R. Suppose that Assumptions 1, 2 and 3 hold, and that u = 0 is the only function of X N (ε) satisfying
curl µ −1 curl u − ω 2 εu = 0.(48)
Then the sequence of solutions (u δ = c δ ∇s δ +ũ δ ) of (47) converges, as δ tends to 0, to the unique solution u = c∇s + +ũ ∈ X out N (ε) of (21) in the following sense: (c δ ) converges to c in C, (ũ δ ) converges toũ in V 0 −β (Ω) and (curl u δ ) converges to curl u in L 2 (Ω).
Proof. Let (δ n ) be a sequence of positive numbers such that lim n→+∞ δ n = 0. Denote by u n the unique function of X N (ε n ) such that
curl µ −1 curl u n − ω 2 ε n u n = iωJ .(49)
Note that we set again ε n instead of ε δn . The proof is in two steps. First, we establish the desired property by assuming that ( curl u n Ω ) is bounded. Then we show that this hypothesis is indeed satisfied. First step. Assume that there is a constant C > 0 such that for all n ∈ N curl u n Ω ≤ C.
By lemma 3.17, we can extract a subsequence from (u n = c n ∇s n +ũ n ) such that (c n ) converges to c in C, (ũ n ) converges toũ in V 0 −β (Ω), with u =ũ + c∇s + ∈ X out N (ε). Besides, since for all n ∈ N, curl u n ∈ L 2 (Ω), there exist h n ∈ H 1 # (Ω) and w n ∈ X N (1), such that
µ −1 curl u n = ∇h n + curl w n .(51)
Observing that (w n ) is bounded in X N (1), from Lemma A.5, we deduce that it admits a subsequence which converges in V 0 −β (Ω). Multiplying (49) taken for two indices n and m by (w n − w m ), and integrating by parts, we obtain
Ω |curl w n − curl w m | 2 dx = ω 2ˆΩ (ε n u n − ε m u m ) (w n − w m ) dx.
This implies that (curl w n ) converges in L 2 (Ω). Then, from (51), we deduce that div (µ∇h n ) = −div (µ curl w n ) in Ω.
By Assumption 2, the operator A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism. Therefore (∇h n ) converges in L 2 (Ω). From (51), this shows that (curl u n ) converges to curl u in L 2 (Ω). Finally, we know that u n satisfieŝ
Ω µ −1 curl u n · curl v dx − ω 2ˆΩ ε n u n · v dx = iωˆΩ J · v dx for all v ∈ V 0
−β (Ω). Taking the limit, we get that u satisfieŝ
Ω µ −1 curl u · curl v dx − ω 2 Ω εu · v dx = iωˆΩ J · v dx(52)
for all v ∈ V 0 −β (Ω). Since in addition, u satisfies (25), (52) also holds for v = ∇s + and we get that u is the unique solution u of (21). Second step. Now we prove that the assumption (50) is satisfied. Suppose by contradiction that there exists a subsequence of (u n ) such that curl u n Ω → +∞ and consider the sequence (v n ) with for all n ∈ N, v n := u n / curl u n Ω . We have v n ∈ X N (ε n ) and curl µ −1 curl v n − ω 2 ε n v n = iωJ / curl u n Ω .
Following the first step of the proof, we find that we can extract a subsequence from (v n ) which converges, in the sense given in the theorem, to the unique solution of the homogeneous problem (21) with J = 0. But by Proposition 3.7, this solution also solves (48). As a consequence, it is equal to zero. In particular, it implies that (curl v n ) converges to zero in L 2 (Ω), which is impossible since by construction, for all n ∈ N, we have curl v n Ω = 1.
Analysis of the problem for the magnetic component
In this section, we turn our attention to the analysis of the Maxwell's problem for the magnetic component. Importantly, in the whole section, we suppose that β satisfies (12), that is 0 < β < min(1/2, β 0 ). Contrary to the analysis for the electric component, we define functional spaces which depend on β:
Z out T (µ) := {u ∈ L 2 (Ω) | curl u ∈ span(ε∇s + ) ⊕ V 0 −β (Ω)
, div(µu) = 0 in Ω, µu · ν = 0 on ∂Ω} and for ξ ∈ L ∞ (Ω),
Z ±β T (ξ) := {u ∈ L 2 (Ω) | curl u ∈ V 0 ±β (Ω)
, div (ξu) = 0 in Ω and ξu · ν = 0 on ∂Ω}.
Note that we have Z −β T (µ) ⊂ Z out T (µ) ⊂ Z β T (µ).
The conditions div(µu) = 0 in Ω and µu · ν = 0 on ∂Ω for the elements of these spaces boil down to imposê Ω µu · ∇ϕ dx = 0, ∀ϕ ∈ H 1 # (Ω).
Remark 4.1.
Observe that the elements of Z out T (µ) are in L 2 (Ω) but have a singular curl. On the other hand, the elements of X out N (ε) are singular but have a curl in L 2 (Ω). This is coherent with the fact that for the situations we are considering in this work, the electric field is singular while the magnetic field is not.
The analysis of the problem for the magnetic component leads to consider the formulation
Find u ∈ Z out T (µ) such that Ω ε −1 curl u · curl v dx − ω 2ˆΩ µu · v =ˆΩ ε −1 J · curl v, ∀v ∈ Z β T (µ),(54)
where J ∈ V 0 −β (Ω). Again, the first integral in the left-hand side of (54) is not a classical integral. Similarly to definition (25), we set Ω ∇s + · curl v dx := 0, ∀v ∈ Z β T (µ).
As a consequence, for u ∈ Z out T (µ) such that curl u = c u ε∇s + + ζ u (we shall use this notation throughout the section) and v ∈ Z β T (µ), there holds
Ω ε −1 curl u · curl v dx =ˆΩ ε −1 ζ u · curl v dx.(55)
Note that for u, v in Z out T (µ) such that curl u = c u ε∇s + + ζ u , curl v = c v ε∇s + + ζ v , we have
Ω ε −1 curl u · curl v dx =ˆΩ ε −1 ζ u · (c v ε∇s + + ζ v ) dx =ˆΩ ε −1 ζ u · ζ v dx − c vˆΩ div(ζ u ) s + dx =ˆΩ ε −1 ζ u · ζ v dx + c u c vˆΩ div(ε∇s + ) s + dx.(56)
We denote by a T (·, ·) (resp. T (·)) the sesquilinear form (resp. the antilinear form) appearing in the left-hand side (resp. right-hand side) of (54).
Remark 4.2. Note that in (54), the solution and the test functions do not belong to the same space. This is different from the formulation (21) for the electric field but seems necessary in the analysis below to obtain a well-posed problem (in particular to prove Proposition 4.7)
. Note also that even if the functional framework depends on β, the solution will not if J is regular enough (see the explanations in Remark 4.11).
Equivalent formulation
Define the spaces
H β (curl ) := {u ∈ L 2 (Ω) | curl u ∈ V 0 β (Ω)} H out (curl ) := {u ∈ L 2 (Ω) | curl u ∈ span(ε∇s + ) ⊕ V 0 −β (Ω)}.
Lemma 4.3. Under Assumptions 1-2, the field u is a solution of (54) if and only if it solves the problem
Find u ∈ H out (curl ) such that a T (u, v) = T (v), ∀v ∈ H β (curl ).(57)
Proof. If u satisfies (57), then taking v = ∇ϕ with ϕ ∈ H 1 # (Ω) in (57), we get that u ∈ Z out T (µ). This proves that u solves (54).
Assume now that u is a solution of (54). Let v be an element of H β (curl ). Introduce ϕ ∈ H 1 # (Ω) the function such thatˆΩ µ∇ϕ · ∇ϕ dx =ˆΩ µv · ∇ϕ dx, ∀ϕ ∈ H 1 # (Ω).
The fieldv := v − ∇ϕ belongs to Z β T (µ). Moreover, there holds curlv = curl v and since for u ∈ Z out T (µ), we haveˆΩ µu · ∇ϕ dx = 0, ∀ϕ ∈ H 1 # (Ω),
we deduce that a T (u, v) = a T (u,v) = T (v) = T (v).
Norms in Z ±β T (µ) and Z out T (µ)
We endow the space Z β T (µ) with the norm
u Z β T (µ) = ( u 2 Ω + curl u 2 V 0 β (Ω) ) 1/2 ,
so that it is a Banach space. Proof. Let u be an element of Z β T (µ). Since u belongs to L 2 (Ω), according to the item v) of Proposition A.1, there are ϕ ∈ H 1 # (Ω) and ψ ∈ X N (1) such that
Lemma 4.4. Under Assumptions 1-2, there is a constant
C > 0 such that for all u ∈ Z β T (µ), we have u Ω ≤ C curl u V 0 β (Ω) . As a consequence, the norm · Z β T (µ) is equivalent to the norm curl · V 0 β (Ω) in Z β T (µ).u = ∇ϕ + curl ψ.(58)
Lemma A.5 guarantees that ψ ∈ V 0 −β (Ω) with the estimate
ψ V 0 −β (Ω) ≤ C curl ψ Ω .(59)
Multiplying the equation curl curl ψ = curl u in Ω by ψ and integrating by parts, we get
curl ψ 2 Ω ≤ curl u V 0 β (Ω) ψ V 0 −β (Ω) .(60)
Gathering (59) and (60) leads to
curl ψ Ω ≤ C curl u V 0 β (Ω) .(61)
On the other hand, using that
Ω µu · ∇ϕ dx = 0, ∀ϕ ∈ H 1 # (Ω)
and that A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism, we deduce that ∇ϕ Ω ≤ C curl ψ Ω . Using this estimate and (61) in the decomposition (58), finally we obtain the desired result.
If u ∈ Z out T (µ), we have curl u = c u ε∇s + + ζ u with c u ∈ C and ζ u ∈ V 0 −β (Ω). We endow the space Z out T (µ) with the norm
u Z out T (µ) = ( u 2 Ω + |c u | 2 + ζ u 2 V 0 −β (Ω) ) 1/2 ,
so that it is a Banach space.
Lemma 4.6. Under Assumptions 1-3, there is C > 0 such that for all u ∈ Z out T (µ), we have
u Ω + |c u | ≤ C ζ u V 0 −β (Ω) .(62)
As a consequence, the norm u Z out T (µ) is equivalent to the norm ζ u V 0 −β (Ω) for u ∈ Z out T (µ).
Proof. Let u be an element of Z out T (µ). Since Z out T (µ) ⊂ Z β T (µ), Lemma 4.4 provides the estimate
u Ω ≤ C curl u V 0 β (Ω) ≤ C (|c u | + ζ u V 0 −β (Ω) ).(63)
On the other hand, taking the divergence of curl u = c u ε∇s + + ζ u , we obtain c u div(ε∇s + ) = −div ζ u . Using the fact that A out ε :V out → (V 1 β (Ω)) * is an isomorphism, we get
|c u | ≤ C div ζ u (V 1 β (Ω)) * ≤ C ζ u V 0 −β (Ω) .
Using this inequality in (63) leads to (62).
Main analysis for the magnetic field
Define the continuous operators A out
T : Z out T (µ) → (Z β T (µ)) * and K out T : Z out T (µ) → (Z β T (µ)) * such that for all u ∈ Z out T (µ), v ∈ Z β T (µ), A out T u, v = Ω ε −1 curl u · curl v dx, K out T u, v =ˆΩ µu · v dx.(64)
With this notation, we have (A out u, v). Proposition 4.7. Under Assumptions 1-3, the operator A out
T − ω 2 K out T )u, v = a T (T : Z out T (µ) → (Z β T (µ)) * is an iso- morphism. Proof. We have A out T u, v =ˆΩ ε −1 ζ u · curl v dx, ∀u ∈ Z out T (µ), ∀v ∈ Z β T (µ).
Let us construct a continuous operator T :
Z β T (µ) → Z out T (µ) such that A out T Tu, v =ˆΩ r 2β curl u · curl v dx, ∀u, v ∈ Z β T (µ).(65)
Let u be an element of Z β T (µ). Then the field r 2β ε curl u belongs to V 0 −β (Ω). Since A out ε : V out → (V 1 β (Ω)) * is an isomorphism, there is a unique ϕ = α s + +φ ∈V out such that A out ε ϕ = −div(r 2β ε curl u). Observing that w := r 2β curl u − ∇ϕ ∈ V 0 β (Ω) is such that div w = 0 in Ω, according to the result of Proposition B.2, we know that there is a unique ψ ∈ Z β T (1) such that curl ψ = ε (r 2β curl u − ∇ϕ).
At this stage, we emphasize that in general ∇ϕ ∈ V 0 β (Ω) \ L 2 (Ω). This is the reason why we are obliged to establish Proposition B.2. Since ψ is in L 2 (Ω), when A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism, there is a unique φ ∈ H 1 # (Ω) such that Ω µ∇φ · ∇φ dx =ˆΩ µψ · ∇φ dx, ∀φ ∈ H 1 # (Ω).
Finally, we set Tu = ψ − ∇φ. It can be easily checked that this defines a continuous operator T : Z β T (µ) → Z out T (µ). Moreover we have curl Tu = α ε∇s + + ζ Tu with ζ Tu = ε (r 2β curl u − ∇φ).
As a consequence, indeed we have identity (65). From Lemma 4.4, we deduce that A out T T : Z β T (µ) → (Z β T (µ)) * is an isomorphism, and so that A out T is onto. It remains to show that A out T is injective.
If u ∈ Z out T (µ) is in the kernel of A out T , we have A out T u, v = 0 for all v ∈ Z β T (µ).
In particular from (56), we can write
A out T u, u =ˆΩ ε −1 |ζ u | 2 dx + |c u | 2ˆΩ div(ε∇s + )s + dx = 0.
Taking the imaginary part of the above identity, we obtain c u = 0 (see the details in the proof of Proposition 4.10). We deduce that u belongs to Z −β T (µ) and from (56), we infer that A out T u, Tu = A out T Tu, u . This gives 0 =ˆΩ r 2β |curl u| 2 dx = 0 and shows that u = 0.
Analysis in the classical framework
In the previous paragraph, we proved that the formulation (54) for the magnetic field with a solution in Z out T (µ) and test functions in Z β T (µ) is well-posed. Here, we study the properties of the problem for the magnetic field set in the classical space X T (µ). More precisely, we consider the problem
Find u ∈ X T (µ) such that Ω ε −1 curl u · curl v dx − ω 2ˆΩ µu · v =ˆΩ ε −1 J · curl v, ∀v ∈ X T (µ).(66)
Working as in the proof of Lemma 4.3, one shows that under Assumptions 1, 2, the field u is a solution of (66) if and only if it solves the problem
Find u ∈ H(curl ) such that Ω ε −1 curl u · curl v dx − ω 2ˆΩ µu · v =ˆΩ ε −1 J · curl v, ∀v ∈ H(curl ).(67)
Define the continuous operators A T : X T (µ) → (X T (µ)) * and K T :
X T (µ) → (X T (µ)) * such that for all u ∈ X T (µ), v ∈ X T (µ), A T u, v =ˆΩ ε −1 curl u · curl v dx, K T u, v =ˆΩ µu · v dx.
As for A N and K N , we emphasize that these are the classical operators which appear in the analysis of the magnetic field, for example when ε and µ are positive in Ω.
Proposition 4.12. Under Assumptions 1-3, for all
ω ∈ C the operator A T − ω 2 K T : X T (µ) → (X T (µ)) * is not Fredholm.
Proof. From [9, Theorem 5.3 and Corollary 5.4], we know that under the Assumptions 1, 2, the embedding of X T (µ) in L 2 (Ω) is compact. This allows us to prove that K T : X T (µ) → (X T (µ)) * is a compact operator. Therefore, it suffices to show that A T : X T (µ) → (X T (µ)) * is not Fredholm. Let us work by contraction assuming that A T is Fredholm. Since this operator is self-adjoint (it is symmetric and bounded), necessarily it is of index zero.
If A T is injective, then it is an isomorphism. Let us show that in this case, A ε : H 1 0 (Ω) → (H 1 0 (Ω)) * is an isomorphism (which is not the case by assumption). To proceed, we construct a continuous operator T :
H 1 0 (Ω) → H 1 0 (Ω) such that A ε ϕ, Tϕ =ˆΩ ε∇ϕ · ∇(Tϕ ) dx =ˆΩ ∇ϕ · ∇ϕ dx, ∀ϕ, ϕ ∈ H 1 0 (Ω).(68)
When A T is an isomorphism, for any ϕ ∈ H 1 0 (Ω), there is a unique ψ ∈ X T (µ) such that
Ω ε −1 curl ψ · curl ψ dx =ˆΩ ε −1 ∇ϕ · curl ψ dx, ∀ψ ∈ X T (µ).
Using item iii) of Proposition A.1, one can show that there is a unique Tϕ ∈ H 1 0 (Ω) such that
∇(Tϕ ) = ε −1 (∇ϕ − curl ψ).
This defines our operator T : H 1 0 (Ω) → H 1 0 (Ω) and one can verify that it is continuous. Moreover, integrating by parts, we indeed get (68) which guarantees, according to the Lax-Milgram theorem, that A ε : H 1 0 (Ω) → H 1 0 (Ω) is an isomorphism.
If A T is not injective, it has a kernel of finite dimension N ≥ 1 which coincides with span(λ 1 , . . . , λ N ),
where λ 1 , . . . , λ N ∈ X T (µ) are linearly independent functions such that (curl λ i , curl λ j ) Ω = δ ij (the Kronecker symbol). Introduce the spacẽ
X T (µ) := {u ∈ X T (µ) | (curl u, curl λ i ) Ω = 0, i = 1, . . . N }.
as well as the operatorà T :X T (µ) →X T (µ) such that
à T u, v =ˆΩ ε −1 curl u · curl v dx, ∀u, v ∈X T (µ).
Thenà T is an isomorphism. Let us construct a new operator T : H 1 0 (Ω) → H 1 0 (Ω) to have something looking like (68). For a given ϕ ∈ H 1 0 (Ω), introduce ψ ∈X T (µ) the function such that
Ω ε −1 curl ψ · curl ψ dx =ˆΩ(ε −1 ∇ϕ − N i=1 α i curl λ i ) · curl ψ dx, ∀ψ ∈X T (µ),(69)
where for i = 1, . . . , N , we have set α i :=´Ω ε −1 ∇ϕ · curl λ i dx. Observing that (69) is also valid for ψ = λ i , i = 1, . . . , N , we infer that there holdŝ
Ω ε −1 curl ψ · curl ψ dx =ˆΩ(ε −1 ∇ϕ − N i=1 α i curl λ i ) · curl ψ dx, ∀ψ ∈ X T (µ).
Using again item iii) of Proposition A.1, we deduce that there is a unique Tϕ ∈ H 1 0 (Ω) such that
∇(Tϕ ) = ε −1 (∇ϕ − curl ψ) − N i=1 α i curl λ i .
This defines the new continuous operator T : H 1 0 (Ω) → H 1 0 (Ω). Then one finds
A ε ϕ, Tϕ =ˆΩ ε∇ϕ · ∇(Tϕ ) dx =ˆΩ ∇ϕ · ∇ϕ dx − N i=1 α iˆΩ ε∇ϕ · curl λ i dx, ∀ϕ, ϕ ∈ H 1 0 (Ω).
This shows that T is a left parametrix for the self adjoint operator A ε . Therefore, A ε : H 1 0 (Ω) → H 1 0 (Ω) is Fredholm of index zero. Note that then, one can verify that dim ker A ε = dim ker A T . And more precisely, we have ker A ε = span(γ 1 , . . . , γ N ) where γ i ∈ H 1 0 (Ω) is the function such that ∇γ i = ε −1 curl λ i (existence and uniqueness of γ i is again a consequence of item iii) of Proposition A.1). But by assumption, A ε is not a Fredholm operator. This ends the proof by contradiction.
Remark 4.13.
In the article [9], it is proved that if A ε : H 1 0 (Ω) → H 1 0 (Ω) is an isomorphism (resp. a Fredholm operator of index zero), then A T : X T (1) → (X T (1)) * is an isomorphism (resp. a Fredholm operator of index zero). Here we have established the converse statement. Remark 4.14. We emphasize that the problems (37) for the electric field and (66) for the magnetic in the usual spaces X N (ε) and X T (µ) have different properties. Problem (37) is well-posed but is not equivalent to the corresponding problem in H N (curl ), so that its solution in general is not a distributional solution of Maxwell's equations. On the contrary, problem (66) is equivalent to problem (67) in H(curl ) but it is not well-posed.
Expression of the singular coefficient
Under Assumptions 1-3, Theorem 4.9 guarantees that for all ω ∈ R the operator A out T − ω 2 K out T : Z out T (µ) → (Z β T (µ)) * is Fredholm of index zero. Assuming that it is injective, the problem (54) admits a unique solution u with curl u = c u ε∇s + + ζ u . As in §3.6, the goal of this paragraph is to derive a formula for the coefficient c u which does not require to know u.
For ω ∈ R, introduce the field w T ∈ Z β T (µ) such that
Ω ε −1 ζ v · curl w T dx − ω 2ˆΩ µv · w T dx =ˆΩ ζ v · ∇s + dx, ∀v ∈ Z out T (µ).(70)
Note that w T is well-defined because (A out
T − ω 2 K out T ) * : Z β T (µ) → (Z out T (µ)) * is an isomorphism.
Theorem 4.15. Assume that ω ∈ R, Assumptions 1-3 are valid and A out
T − ω 2 K out T : Z out T (µ) → (Z β T (µ)) * is injective.
Let u denote the solution of the magnetic problem (54). Then the coefficient c u in the decomposition curl u = c u ε∇s + + ζ u is given by the formula
c u = iωˆΩ ε −1 J · curl w T dx ˆΩ div(ε∇s + ) s + dx.(71)
Here w T is the function which solves (70).
Proof. By definition of u, we havê
Ω ε −1 ζ u · curl w T dx − ω 2ˆΩ µu · w T dx = iωˆΩ ε −1 J · curl w T dx.
On the other hand, from (70), we can writê
Ω ε −1 ζ u · curl w T dx − ω 2ˆΩ µu · w T dx =ˆΩ ζ u · ∇s + dx.
From these two relations, using (56), we deduce that iωˆΩ ε −1 J · curl w T dx =ˆΩ ζ u · ∇s + dx = c uˆΩ div(ε∇s + ) s + dx.
This gives (71).
Conclusion
In this work, we studied the Maxwell's equations in presence of hypersingularities for the scalar problem involving ε. We considered both the problem for the electric field and for the magnetic field. Quite naturally, in order to obtain a framework where well-posedness holds, it is necessary to modify the spaces in different ways. More precisely, we changed the condition on the field itself for the electric problem and on the curl of the field for the magnetic problem. A noteworthy difference in the analysis of the two problems is that for the electric field, we are led to work in a Hilbertian framework, whereas for the magnetic field we have not been able to do so.
Of course, we could have assumed that the scalar problem involving ε is well-posed in H 1 0 (Ω) and that hypersingularities exist for the problem in µ. This would have been similar mathematically. Physically, however, this situation seems to be a bit less relevant because it is harder to produce negative µ without dissipation. We assumed that the domain Ω is simply connected and that ∂Ω is connected. When these assumptions are not met, it is necessary to adapt the analysis (see §8.2 of [9] for the study in the case where the scalar problems are well-posed in the usual H 1 framework). This has to be done. Moreover, for the conical tip, at least numerically, one finds that several singularities can exist (see the calculations in [28]). In this case, the analysis should follow the same lines but this has to be written. On the other hand, in this work, we focused our attention on a situation where the interface between the positive and the negative material has a conical tip. It would be interesting to study a setting where there is a wedge instead. In this case, roughly speaking, one should deal with a continuum of singularities. We have to mention that the analysis of the scalar problems for a wedge of negative material in the non standard framework has not been done. Finally, considering a conical tip with both critical ε and µ is a direction that we are investigating.
A Vector potentials, part 1
Proposition A.1. Under Assumption 1, the following assertions hold.
i) According to [1,Theorem 3.12], if u ∈ L 2 (Ω) satisfies div u = 0 in Ω, then there exists a unique ψ ∈ X T (1) such that u = curl ψ.
ii) According to [1,Theorem 3.17]), if u ∈ L 2 (Ω) satisfies div u = 0 in Ω and u · ν = 0 on ∂Ω, then there exists a unique ψ ∈ X N (1) such that u = curl ψ.
iii) If u ∈ L 2 (Ω) satisfies curl u = 0 in Ω and u × ν = 0 on ∂Ω, then there exists (see [35,Theorem 3.41]) a unique p ∈ H 1 0 (Ω) such that u = ∇p. with p ∈ H 1 # (Ω) and ψ ∈ X N (1) which are uniquely defined.
Proposition A.2. Under Assumption 1, if ψ satisfies one of the following conditions i) ψ ∈ X N (1) and ∆ψ ∈ L 2 (Ω), ii) ψ ∈ X T (1), curl ψ × ν = 0 on ∂Ω and ∆ψ ∈ L 2 (Ω), then for all β < 1/2, we have curl ψ ∈ V 0 −β (Ω) and there is a constant C > 0 independent of ψ such that curl ψ V 0 −β (Ω) ≤ C ∆ψ Ω .
Proof. It suffices to prove the result for β ∈ (0; 1/2). Let ψ ∈ X N (1)∪X T (1). Since curl curl ψ = −∆ψ, integrating by parts we get curl ψ 2 Ω = −ˆΩ ∆ψ · ψ dx.
Note that the boundary term vanishes because either ψ × ν = 0 or curl ψ × ν = 0 on ∂Ω. This furnishes the estimate curl ψ Ω ≤ C ∆ψ Ω .
Now working with cut-off functions, we refine the estimate at the origin to get (72). Let us consider a smooth cut-off function χ, compactly supported in Ω, equal to one in a neighbourhood of O. To prove the proposition, it suffices in addition to (73) to prove that curl (χψ) ∈ V 0 −β (Ω) together with the following estimate curl (χψ) V 0 −β (Ω) ≤ C ∆ψ Ω . First of all, since curl (χψ) ∈ L 2 (Ω) and div(χψ) = ∇χ · ψ ∈ L 2 (Ω), we know that χψ i ∈ H 1 0 (Ω) for i = 1, 2, 3 and we have curl (χψ) 2 Ω + div(χψ) 2
Ω = 3 i=1 ∇(χψ i ) 2 Ω .
From the previous identity, (73) and Proposition 1.1, we deduce ψ 2 Ω + The next two lemmas are results of additional regularity for the elements of classical Maxwell's spaces that are direct consequences of Propositions A.2 and A.3.
Lemma A.4. Under Assumption 1, for all β ∈ (0; 1/2), X T (1) is compactly embedded in V 0 −β (Ω). In particular, there is a constant C > 0 such that
u V 0 −β (Ω) ≤ C curl u Ω , ∀u ∈ X T (1).(76)
Proof. Let u be an element of X T (1). From the item ii) of Proposition A.1, we know that there exists ψ ∈ X N (1) such that u = curl ψ. Using that −∆ψ = curl u ∈ L 2 (Ω), from Proposition A.2, we get that u ∈ V 0 −β (Ω) together with the estimate curl ψ V 0 −β (Ω) ≤ C curl u Ω . This gives (76). Now suppose that (u n ) is a bounded sequence of elements of X T (1). Then there exists a bounded sequence (ψ n ) of elements of X N (1) such that u n = curl ψ n . Since (curl u n = −∆ψ n ) is bounded in L 2 (Ω), the first item of Proposition A.3 implies that there is a subsequence such that (u n ) converges in V 0 −β (Ω).
Lemma A.5. Under Assumption 1, for all β ∈ (0; 1/2), X N (1) is compactly embedded in V 0 −β (Ω). In particular, there is a constant C > 0 such that
u V 0 −β (Ω) ≤ C curl u Ω , ∀u ∈ X N (1).
Proof. The proof is similar to the one of Lemma A.4. Proof. Let ψ be an element of Z β T (1). According to Proposition 2.1, there is a unique ϕ ∈V 1 −β (Ω) such thatˆΩ ∇ϕ · ∇ϕ dx =ˆΩ r 2β curl ψ · ∇ϕ dx, ∀ϕ ∈V 1 β (Ω).
B Vector potentials, part 2
Then denote Tψ ∈ Z −β T (1) the function such that curl (Tψ) = r 2β curl ψ − ∇ϕ.
Observe that Tψ is well-defined according to the item i) of Proposition A.1. This defines a continuous operator T : Z β T (1) → Z −β T (1). We have B T ψ, Tψ =ˆΩ curl ψ · curl (Tψ) dx = r β curl ψ 2 Ω = curl ψ 2 V 0 β (Ω) .
Adapting the proof of Lemma 4.4, one can show that curl · V 0 β (Ω) is a norm which is equivalent to the natural norm of Z β T (1). Therefore, from the Lax-Milgram theorem, we infer that T * B T is an isomorphism which shows that B T is injective and that its image is closed in (Z −β T (1)) * . And D Dimension of X out N (ε)/X N (ε) Lemma D.1. Under Assumptions 1-3, we have dim (X out N (ε)/X N (ε)) = 1.
Proof. If u 1 = c 1 ∇s + +ũ 1 , u 2 = c 2 ∇s + +ũ 2 are two elements of X out N (ε), then c 2 u 1 − c 1 u 2 ∈ X N (ε), which shows that dim (X out N (ε)/X N (ε)) ≤ 1. Now let us prove that dim (X out N (ε)/X N (ε)) ≥ 1. Introduces ∈V out the function such that A out εs = div(ε∇s − ). Note that since div(ε∇s − ) vanishes in a neighbourhood of the origin, it belongs to (V 1 γ (Ω)) * for all γ ∈ R. Then set s = s − +s.
Observe that s ∈V 1 γ (Ω) for all γ > 0 and that div(ε∇s) = 0 in Ω \ {O} (s is a non zero element of ker A γ ε for all γ > 0). Letũ ∈ (C ∞ 0 (Ω \ {O})) 3 be a field such that´Ω εũ · ∇s dx = 0. The existence of such aũ can be established thanks to the density of (C ∞ 0 (Ω \ {O})) 3 in L 2 (Ω), considering for example an approximation of 1 B ∇s ∈ L 2 (Ω) where 1 B is the indicator function of a ball included in M. Introduce ζ = c s + +ζ ∈V out , with c ∈ C,ζ ∈V 1 −β (Ω), the function such that A out ε ζ = −div(εũ). This is equivalent to have −cˆΩ div(ε∇s + )ϕ dx +ˆΩ ε∇ζ · ∇ϕ dx =ˆΩ εũ · ∇ϕ dx, ∀ϕ ∈V 1 β (Ω).
Clearly ∇ζ −ũ = c∇s + + (∇ζ −ũ) is an element of X out N (ε). Moreover taking ϕ = s above, we get −cˆΩ div(ε∇s + )s dx =ˆΩ εũ · ∇s dx = 0.
This shows that c = 0 and guarantees that dim (X out N (ε)/X N (ε)) ≥ 1.
Lemma 3 . 15 .
315For all β > 0, when δ tends to zero, the function s δ converges inV 1 β (Ω) to s + and not to s − (see the definitions in
Figure 2 :
2Behaviour of the eigenvalue λ δ close to the line e λ = −1/2 as the dissipation δ tends to zero. Here the values have been obtained solving the problem (45) with a Finite Element Method. We work in the conical tip defined via (5) with α = 2π/3 and κ ε = −1.9.
Remark 4. 5 .
5The result of Lemma 4.4 holds for all β such that 0 ≤ β < 1/2 and not only for 0 < β < min(1/2, β 0 ).
iv) Every u ∈ L 2 (Ω) can be decomposed as follows ([35, Theorem 3.45]) u = ∇p + curl ψ, with p ∈ H 1 0 (Ω) and ψ ∈ X T (1) which are uniquely defined. v) Every u ∈ L 2 (Ω) can be decomposed as follows ([35, Remark 3.46]) u = ∇p + curl ψ,
First
we establish an intermediate lemma which can be seen as a result of well-posedness for Maxwell's equations in weighted spaces with ε = µ = 1 in Ω. Define the continuous operatorB T : Z β T (1) → (Z −β T (1)) * such that for all ψ ∈ Z β T (1), ψ ∈ Z −β T (1), B T ψ, ψ =ˆΩ curl ψ · curl ψ dx.Lemma B.1. Under Assumption 1, for 0 ≤ β < 1/2, the operator B T : Z β T (1) → (Z −β T (1)) * is an isomorphism.
Proof. Set Ω τ := Ω \ B(O, τ ). Noticing that div(ε∇s + ) vanishes in a neighbourhood of the origin, we can writeˆΩ div(ε∇s + ) s + dx = lim τ →0ˆΩ div(ε∇s + ) s + dx ε|∇s + | 2 dx −ˆ∂ Taking the imaginary part and observing thatτ
= lim
τ →0
−ˆΩ
τ
B(O,τ )
ε
∂s +
∂r
s + ds .
∂B(O,τ )
ε
∂s +
∂r
s + ds = −
1
2
+ iη
ˆS
2
ε|Φ| 2 ds,
the result follows.
From now on, we simply write "singular" instead of "hypersingular".
ε|Φ| 2 ds.
Proposition 4.8. Under Assumptions 1-3, the embedding of the space Z outT (µ) in L 2 (Ω) is compact. As a consequence, the operator K out T : Z out T (µ) → (Z β T (µ)) * defined in (64) is compact.Proof. Let (u n ) be a sequence of elements of Z out T (µ) which is bounded. For all n ∈ N, we have curl u n = c un ε∇s + + ζ un . By definition of the norm of Z out T (µ), the sequence (c un ) is bounded in C. Let w be an element of Z out T (µ) such that c w = 1 (if such w did not exist, then we would have Z out T (µ) = Z −β T (µ) ⊂ X T (µ) and the proof would be even simpler). The sequence (u n − c un w) is bounded in X T (µ). Since this space is compactly embedded in L 2 (Ω) when A µ : H 1 # (Ω) → (H 1 # (Ω)) * is an isomorphism (see[9,Theorem 5.3]), we infer we can extract from (u n − c un w) a subsequence which converges in L 2 (Ω). Since clearly we can also extract a subsequence of (c un ) which converges in C, this shows that we can extract from (u n ) a subsequence which converges in L 2 (Ω). This shows that the embedding of Z out T (µ) in L 2 (Ω) is compact. Now observing that for all u ∈ Z out T (µ), we haveWe can now state the main theorem of the analysis of the problem for the magnetic field.Theorem 4.9. Under Assumptions 1-3, for allFinally we establish a result similar to Proposition 3.7 by using the formulation for the magnetic field.Proposition 4.10. Under Assumptions 1 and 3, if u ∈ Z outT (µ) is a solution of (54) for J = 0, then u ∈ Z −γ T (µ) ⊂ X T (µ) for all γ satisfying(12).Taking the imaginary part of this identity for v = u, since ω is real, we getIf curl u = c u ε∇s + + ζ u with c u ∈ C and ζ u ∈ V 0 −β (Ω), according to (56), this can be written asThen one concludes as in the proof of Proposition 3.7 that c u = 0, so that curl u ∈ V 0 −β (Ω).−γ (Ω) for all γ satisfying(12). This shows that u ∈ Z −γ T (µ) for all γ satisfying (12).Remark 4.11. Assume that J ∈ V 0 −γ (Ω) for all γ satisfying(12). Assume also that zero is the only solution of (54) with J = 0 for a certain β 0 satisfying(12). Then Theorem 4.9 and Proposition 4.10 guarantee that (54) is well-posed for all γ satisfying(12). Moreover Proposition 4.10 allows one to show that all the solutions of (54) for γ satisfying(12)Note that,(74)is also valid if we replace χ by any other smooth function with compact support in Ω. Now setting f i = ∆(χψ i ) for i = 1, 2, 3, we haveBy writing that ∇χ · ∇ψ i = div(ψ i ∇χ) − ψ i ∆χ and replacing χ by ∂ j χ in (74) for j = 1, 2, 3, we deduce that for i = 1, 2, 3, f i belongs to L 2 (Ω) and satisfiesNote that since β ∈ (0; 1/2), we haveV 1As a consequence, curl (χψ) ∈ V 0 −β (Ω) andwhich concludes the proof.Proposition A.3. Under Assumption 1, the following assertions hold: i) if (ψ n ) is a bounded sequence of elements of X N (1) such that (∆ψ n ) is bounded in L 2 (Ω), then one can extract a subsequence such that (curl ψ n ) converges in V 0 −β (Ω) for all β ∈ (0; 1/2); ii) if (ψ n ) is a bounded sequence of elements of X T (1) such that curl ψ n × ν = 0 on ∂Ω and such that (∆ψ n ) is bounded in L 2 (Ω), then one can extract a subsequence such that (curl ψ n ) converges in V 0 −β (Ω) for all β ∈ (0; 1/2). Proof. Let us establish the first assertion, the proof of the second one being similar. Let (ψ n ) be a bounded sequence of elements of X N (1) such that (∆ψ n ) is bounded in L 2 (Ω). Observing that curl curl ψ n = −∆ψ n , we deduce that (curl ψ n ) is a bounded sequence of X T (1). Since the spaces X N (1) and X T (1) are compactly embedded in L 2 (Ω) (see Proposition 1.1), one can extract a subsequence such that both (ψ n ) and (curl ψ n ) converge in L 2 (Ω). Then, working as in the proof of Proposition A.2, we can show that for a smooth cut-off function χ compactly supported in Ω and equal to one in a neighbourhood of O, the sequence (χψ n ) is bounded in V 2 γ (Ω) := (V 2 γ (Ω)) 3 for all γ > 1/2. To obtain this result, we use in particular the fact that if O ⊂ R 3 is a smooth bounded domain such that O ∈ O, then ∆ :is an isomorphism for all γ ∈ (1/2; 3/2) (see[34, §1.6.2]). Finally, to conclude to the result of the proposition, we use the fact V 2 γ (O) is compactly embedded in V 1 γ (O) a soon as γ − 1 < γ ([32, Lemma 6.2.1]). This allows us to prove that for all β < 1/2, the subsequence (χψ n ) converges in V 1 −β (Ω), so that (curl ψ n ) converges in V 0 −β (Ω).from that, we deduce that B T is onto if and only if its adjoint is injective. The adjoint of B T is the operator B * T :(77), we obtain curl ψ Ω = 0. Since Z −β T (1) ⊂ X T (1) and curl · Ω is a norm in X T (1) (Proposition 1.1), we deduce that ψ = 0. This shows that B * T is injective and that B T is an isomorphism.Now we use the above lemma to prove the following result which is essential in the analysis of the Problem (54) for the magnetic field. This is somehow an extension of the result of item i) of Proposition A.1 for singular fields which are not in L 2 (Ω).Proposition B.2. Under Assumption 1, for allProof. Let u ∈ V 0 β (Ω) be such that div u = 0 in Ω. According to Lemma B.1, we know that there is a unique ψ ∈ Z β T (1) such thatThen we haveˆΩSince u is divergence free in Ω, we also havêNow if v is an element of V 0 −β (Ω) ⊂ L 2 (Ω), from item iv) of Proposition A.1, we know that there holds the decomposition v = ∇p + curl ψ ,for some p ∈ H 1 0 (Ω) and some ψ ∈ X T (1). Taking the divergence in (80), we getFrom Proposition 2.1, since 0 ≤ β < 1/2, we know that (81) admits a solution inV 1 −β (Ω) ⊂ H 1 0 (Ω). Using uniqueness of the solution of (81) in H 1 0 (Ω), we obtain that p ∈V 1 −β (Ω). This implies that curl ψ = v − ∇p ∈ V 0 −β (Ω) and so ψ ∈ Z −β T (1). From(78)and(79), we infer thatThis shows that u = curl ψ. Finally, if ψ 1 , ψ 2 are two elements of Z β T (1) such that u = curl ψ 1 = curl ψ 2 , then ψ 1 − ψ 2 belongs to X T (1) and satisfies curl (ψ 1 − ψ 2 ) = 0 in Ω. From Proposition 1.1, we deduce that ψ 1 = ψ 2 .C Energy flux of the singular functionLemma C.1. With the notations of (4), we have m ˆΩ div(ε∇s + ) s + dx = ηˆS
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| []
|
[
"Heisenberg antiferromagnets with exchange and cubic anisotropies",
"Heisenberg antiferromagnets with exchange and cubic anisotropies"
]
| [
"G Bannasch \nMPI für Physik komplexer Systeme\n01187DresdenGermany\n",
"W Selke [email protected] \nInstitut für Theoretische Physik\nRWTH Aachen University and JARA-SIM\n52056AachenGermany\n"
]
| [
"MPI für Physik komplexer Systeme\n01187DresdenGermany",
"Institut für Theoretische Physik\nRWTH Aachen University and JARA-SIM\n52056AachenGermany"
]
| []
| We study classical Heisenberg antiferromagnets with uniaxial exchange anisotropy and a cubic anisotropy term on simple cubic lattices in an external magnetic field using ground state considerations and extensive Monte Carlo simulations. In addition to the antiferromagnetic phase field-induced spin-flop and non-collinear, biconical phases may occur. Phase diagrams and critical as well as multicritical phenomena are discussed. Results are compared to previous findings. | 10.1088/1742-6596/200/2/022057 | [
"https://arxiv.org/pdf/0907.4173v1.pdf"
]
| 14,819,580 | 0907.4173 | d10643de523c489102f5478ff83944d8bc5e6e45 |
Heisenberg antiferromagnets with exchange and cubic anisotropies
23 Jul 2009
G Bannasch
MPI für Physik komplexer Systeme
01187DresdenGermany
W Selke [email protected]
Institut für Theoretische Physik
RWTH Aachen University and JARA-SIM
52056AachenGermany
Heisenberg antiferromagnets with exchange and cubic anisotropies
23 Jul 2009
We study classical Heisenberg antiferromagnets with uniaxial exchange anisotropy and a cubic anisotropy term on simple cubic lattices in an external magnetic field using ground state considerations and extensive Monte Carlo simulations. In addition to the antiferromagnetic phase field-induced spin-flop and non-collinear, biconical phases may occur. Phase diagrams and critical as well as multicritical phenomena are discussed. Results are compared to previous findings.
Introduction
Recently, there has been a renewed interest in uniaxially anisotropic Heisenberg antiferromagnets in a field, for many years known to display antiferromagnetic and field-induced spin-flop phases. This interest is due to various reasons, among others, (i) to clarify the phase diagram of the prototypical XXZ model, especially for the square lattice [1,2,3], (ii) to study multicritical, like bi-and tetracritical, behavior [4,5], and (iii) to elucidate ground states as well as thermal properties of low-dimensional quantum magnets exhibiting, possibly, 'supersolid' magnetic (i.e. non-collinear 'biconical' [6]) structures [7,8,9]. Of course, rather recent pertinent experiments also should be mentioned [10,11].
Results
As a starting point of theoretical studies on uniaxially anisotropic Heisenberg antiferromagnets, one often considers the XXZ model, with the Hamiltonian
H XXZ = J i,j ∆(S x i S x j + S y i S y j ) + S z i S z j − H i S z i(1)
where J > 0 is the exchange coupling between spins being located on neighboring lattice sites i and j. ∆ is the exchange anisotropy, 1 > ∆ > 0, and H is the applied magnetic field along the easy axis, the z-axis. The model has been found to display in the (temperature T, field H)-plane antiferromagnetic (AF) and spin-flop (SF) phases on square and simple cubic lattices.
In two dimensions, the transition between the two phases had been argued to be of first order in case of the spin-1/2 quantum case [3,12]. In contrast, in the classical case (with spin vectors of length one) there appears to be a narrow disordered phase in between the two ordered phases. The three phases seem to meet at zero temperature in a 'hidden tetracritical point' [2,3] at the critical field H c . That point is a highly degenerate ground state, where also biconical (BC) spin configurations are stable [3]. The antiferromagnetic, biconical, and spin-flop classical spin configurations are shown in Fig. 1. For the XXZ model on the simple cubic lattice, the phase diagram has been believed to show the same topology as in the mean-field approximation, with a direct transition of first order between the AF and SF phases, ending in a bicritical point at which the two critical phase boundary lines between the paramagnetic (P) phase and the AF as well as the SF phases meet with the AF-SF transition line [13]. Recently, this scenario has been scrutinized using renormalization group methods [4,5]. One of the aims of our study is to shed light upon this issue, applying extensive Monte Carlo simulations. The phase diagram obtained from our simulations, analyzing systems with up to 32 3 spins in runs of, at least, 10 7 Monte Carlo steps per spin, is shown in Fig. 2. Indeed, setting ∆ = 0.8, we can locate the triple point of the AF-P, SF-P, and AF-SP boundary lines accurately, at k B T t /J = 1.025 ± 0.015 and H t /J = 3.90 ± 0.03, differing quite substantially from the old estimate based on appreciably shorter simulations [13]. Moreover, studying the transitions in the vicinity of that triple point, we do not observe deviations from the previously anticipated bicritical scenario [13], with the AF-P boundary line belonging to the Ising, and the SF-P boundary line belonging to the XY universality class. We identify the universality classes from determining critical exponents, e.g., of the (staggered) longitudinal and tranverse susceptibilities, and from determining the critical Binder cumulants of the AF and SF order parameters [14]. Note that in a renormalization group calculation to high loop order, a different scenario had been put forward [4], favouring the existence of a tetracritical point or some kind of critical end point. In that scenario, one expects either a stable BC phase near (T t , H t ) or at least one of the AF-P and SF-P boundaries should become a line of transitions of first order near that point [4]. In our simulations, we observe that biconical spin configurations show up close to the AF-SF phase boundary at low temperatures, reflecting the, again, high degeneracy of the ground state at the critical field separating the AF and SF structures. However, these configurations do not destroy, in contrast to the situation in two dimensions, the direct AF-SF transition of first order [15]. Actually, applying also renormalization group arguments, it has been suggested very recently that the type of the triple point may depend on the strength of the anisotropy, allowing for a bicritical point [5]. Perhaps, Monte Carlo studies on different values of the anisotropy may provide further insights.
θ θ θ A B SF z x y (a) (b) (c)
Adding, especially, single-ion terms due to crystal-field anisotropies to the XXZ model, eq. (1), BC spin configurations may be stabilized over a range of fields in the ground state. This behavior has been observed for quantum and classical spins on chains and square lattices for a quadratic single-ion term [7,8,9,16]. Here we shall add to H XXZ a cubic anisotropy term of the form [17]
H CA = F i (S x i ) 4 + (S y i ) 4 + (S z i ) 4(2)
where F denotes the strength of the cubic anisotropy. The sign of F determines whether the spins tend to align along the cubic axes, for F < 0, case 1, or, for F > 0, case 2, in the diagonal directions of the lattice. Because of these tendencies, the BC, (i.e. BC1 or BC2), structures, as well as the SF structures, show no full rotational invariance in the xy-plane, in contrast to the XXZ case discussed above. Now, obviously, the discretized spin projections in the xy-plane favour four directions [14,15]. The resulting ground state phase diagram of the full Hamiltonian, H XXZ + H CA with fixed exchange anisotropy, ∆ = 0.8, and varying cubic term, F , may be determined numerically without difficulty, as depicted in Fig. 3. Note that, for F < 0, the transitions to the BC1 structures are typically of first order, with a jump in the tilt angles, Θ, with respect to the z-axis, see Fig. 1, characterizing the BC configurations. However, in the reentrance region between the SF and BC1 structures at F/J close to -1, the change in the tilt angles seems to be smooth at the transitions [14].
Obviously, at non-zero temperatures, several interesting scenarios leading, possibly, to multicritical behaviour, where AF, SF, BC, and P phases meet, may exist. So far, we focussed attention on two cases: (a) Positive cubic anisotropy F > 0, at constant field H/J = 1.8 [15], see Fig. 3. At small values of F , there is an AF ordering at low temperatures. Above a critical value, F c = 0.218...J, the low-temperature phase is of BC2 type, followed by the AF and P phases, when increasing the temperature. The transition between the AF and P phases is found to belong to the Ising universality class, while the transition between the BC2 and AF phases seems to belong to the XY universality class, with the cubic term being then an irrelevant perturbation [15]. (b) Negative F , fixing the cubic term, F/J = −2, and varying the field, see Fig. 3. In accordance with the ground state analysis, we observe, at sufficiently low temperature, k B T /J = 0.2, AF, SF, BC1, and P phases, when increasing the field, as depicted in Fig. 4. Interestingly, the BC1 phase seems to become unstable when raising the temperature, with the other phases being still present [14], see Fig. 5. This may suggest that the three boundary lines between the BC1-P, SF-BC1, and SF-P phases meet at a multicritical point. Of course, further clarification and a search for other, possibly multicritical scenarios at different strengths of the cubic term, F , are desirable. In summary, we have studied antiferromagnets with fixed exchange and varying cubic anisotropies in a field on the simple cubic lattice. In the case of the XXZ model, the nature of the triple point and the thermal role of BC structures have been clarified. By adding the cubic anisotropy, discretized BC phases may be stabilized leading to interesting critical and multicritical phenomena.
Figure 1 .
1Spin orientations on neighboring sites showing antferromagnetic (a), biconical (b), and spin-flop (c) ground state structures in the XXZ model.
Figure 2 .
2Phase diagram of the XXZ antiferromagnet, with ∆ = 0.8. Inset: Vicinity of the triple point.
Figure 3 .
3Ground states of the full Hamiltonian with exchange anisotropy, ∆ = 0.8, and the cubic term.
Figure 4 .
4Staggered magnetizations versus field at F/J = −2 and k B T /J = 0.2, indicating the AF, SF, BC1, and P phases, when increasing the field.
Figure 5 .
5Staggered magnetizations versus field, at F/J = −2 and k B T /J = 0.4, with the BC1 phase being squeezed out. Systems with 16 3 spins are simulated.
AcknowledgmentsWe should like to thank A. N. Bogdanov, T.-C. Dinh, R. Folk, M. Holtschneider, D. P. Landau, and D. Peters for useful discussions and information.
. M Holtschneider, W Selke, R Leidl, Phys. Rev. B. 7264443Holtschneider M, Selke W and Leidl R 2005 Phys. Rev. B 72 064443.
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. R Folk, Holovatch Yu, G Moser, Phys. Rev. E. 7841124Folk R, Holovatch Yu and Moser G 2008 Phys. Rev. E 78 041124.
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. R Leidl, R Klingeler, B Büchner, M Holtschneider, W Selke, Phys. Rev. B. 73224415Leidl R, Klingeler R, Büchner B, Holtschneider M and Selke W 2006 Phys. Rev. B 73 224415.
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Diploma thesis. G Bannasch, RWTH AachenBannasch G 2008 Diploma thesis, RWTH Aachen.
. G Bannasch, W Selke, Eur. Phys. J. B. in printBannasch G and Selke W 2009 Eur. Phys. J. B in print.
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. A Aharony, Phys. Rev. E. 84270Aharony A 1973 Phys. Rev. E 8 4270.
| []
|
[
"FACETS OF THE SPINLESS SALPETER EQUATION",
"FACETS OF THE SPINLESS SALPETER EQUATION"
]
| [
"Wolfgang Lucha \nInstitut für Hochenergiephysik\nOsterreichische Akademie der Wissenschaften\nInstitut für Theoretische Physik\nUniversität Wien\nNikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria\n",
"Franz F Schöberl \nInstitut für Hochenergiephysik\nOsterreichische Akademie der Wissenschaften\nInstitut für Theoretische Physik\nUniversität Wien\nNikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria\n"
]
| [
"Institut für Hochenergiephysik\nOsterreichische Akademie der Wissenschaften\nInstitut für Theoretische Physik\nUniversität Wien\nNikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria",
"Institut für Hochenergiephysik\nOsterreichische Akademie der Wissenschaften\nInstitut für Theoretische Physik\nUniversität Wien\nNikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria"
]
| []
| The spinless Salpeter equation represents the simplest, and most straightforward, generalization of the Schrödinger equation of standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics. Moreover, it can be also regarded as a well-defined approximation to the Bethe-Salpeter formalism for descriptions of bound states in relativistic quantum field theories. The corresponding Hamiltonian is, in contrast to all Schrödinger operators, a nonlocal operator. Because of the nonlocality, constructing analytical solutions for such kind of equation of motion proves difficult. In view of this, different sophisticated techniques have been developed in order to extract rigorous analytical information about these solutions. This review introduces some of these methods and compares their significance by application to interactions relevant in physics. | null | [
"https://arxiv.org/pdf/hep-ph/0408184v1.pdf"
]
| 119,179,147 | hep-ph/0408184 | 6ba218de3bdc0344572a160eb0dc9cfd56954ceb |
FACETS OF THE SPINLESS SALPETER EQUATION
17 Aug 2004 July 2004
Wolfgang Lucha
Institut für Hochenergiephysik
Osterreichische Akademie der Wissenschaften
Institut für Theoretische Physik
Universität Wien
Nikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria
Franz F Schöberl
Institut für Hochenergiephysik
Osterreichische Akademie der Wissenschaften
Institut für Theoretische Physik
Universität Wien
Nikolsdorfergasse 18, Boltzmanngasse 5A-1050, A-1090Wien, WienAustria, Austria
FACETS OF THE SPINLESS SALPETER EQUATION
17 Aug 2004 July 2004numbers: 0365Ge0365Pm1110St
The spinless Salpeter equation represents the simplest, and most straightforward, generalization of the Schrödinger equation of standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics. Moreover, it can be also regarded as a well-defined approximation to the Bethe-Salpeter formalism for descriptions of bound states in relativistic quantum field theories. The corresponding Hamiltonian is, in contrast to all Schrödinger operators, a nonlocal operator. Because of the nonlocality, constructing analytical solutions for such kind of equation of motion proves difficult. In view of this, different sophisticated techniques have been developed in order to extract rigorous analytical information about these solutions. This review introduces some of these methods and compares their significance by application to interactions relevant in physics.
Bethe-Salpeter Formalism in the "Instantaneous Approximation"
Within quantum field theory, the appropriate framework for the description of bound states is the Bethe-Salpeter formalism [1]. Therein, all bound states of two particles (in fact, of any two fermionic constituents) are governed by the homogeneous Bethe-Salpeter equation. Here we are interested in a particular well-defined approximation to this formalism, obtained by several simplifying steps:
1. The instantaneous approximation, neglecting any retardation effect, considers all interactions of the (two) bound-state constituents in their static limit.
2. The additional assumption that all the bound-state constituents propagate as free particles with some effective mass m yields the Salpeter equation [2]. 3. A disregard of all of their spin degrees of freedom focuses on the treatment of scalar bound particles. 4. In technical respect, the canonical transformation
x → λ x , p → p λ(1)
of position (x) and momentum (p) variables casts in the case of particles of equal mass m for a scale factor λ = 2 this approach into one-particle form.
(For more details of the derivation, consult, for instance, Refs. [3][4][5] and references therein.) Refraining from the nonrelativistic limit, we get the (nonlocal!) Hamiltonian
H = T + V .(2)
This operator is composed of the "square-root operator" T of the relativistically correct expression for the kinetic or free energy of a particle of mass m and momentum p,
T = T (p) ≡ p 2 + m 2 ,(3)
and a (coordinate-dependent) static interaction potential
V = V (x) ;
frequently, the potential V (x) is assumed to be a central potential that depends merely on the radial coordinate r:
V = V (r) , r ≡ |x| .
The eigenvalue equation for this particular Hamiltonian, H |χ k = E k |χ k , k = 0, 1, 2, . . . ,
defining a complete system of Hilbert-space eigenstates |χ k of H corresponding to its (energy) eigenvalues E k , E k ≡ χ k | H |χ k χ k |χ k , k = 0, 1, 2, . . . , is commonly known as the "spinless Salpeter equation."
The Relativistic Virial Theorem
Useful general statements about the solutions of explicit or implicit eigenvalue equations may be proved with the help of virial theorems obtained by generalization [6] of the well-known result of nonrelativistic quantum theory. (Ref. [7] is a brief review of relativistic virial theorems.) For eigenvalue equations of the form (4), the derivations of such virial theorems can be traced back to the (trivial) observation that expectation values taken with respect to given eigenstates |χ k of H -or matrix elements taken with respect to arbitrary pairs of degenerate eigenstates, |χ i and |χ j , of H, i. e., eigenstates satisfying E i = E j -of the commutators [G, H] of the operator H and any other operator G (the domain of which must be assumed to contain the domain of H) clearly vanish. Suppressing the subscript that enumerates the eigenstates, this means
χ| [G, H] |χ = 0 .(5)
For the symmetrized, self-adjoint generator of dilations,
G ≡ 1 2 (x · p + p · x) ,(6)
and H of the form (2) their commutator [G, H] becomes
[G, H] = i p · ∂ T ∂p (p) − x · ∂ V ∂x (x) .
In this case Eq. (5) yields the master virial theorem [6,7]
χ p · ∂ T ∂p (p) χ = χ x · ∂ V ∂x (x) χ ; (7)
this relation expresses the equality of all the expectation values of the momentum-space radial derivative of T (p) and the (configuration-space) radial derivative of V (x), it produces the specific virial theorem for a particular H. For any nonrelativistic (Schrödinger) Hamiltonian, i. e.,
H = H S = m + p 2 2 m + V (x) ,
Theorem (7) entails, retaining the conventional factor 1 2 ,
χ p 2 2 m χ = 1 2 χ x · ∂ V ∂x (x) χ . (8)
For the semirelativistic "spinless-Salpeter" Hamiltonian (2), involving the square-root operator of the relativistic kinetic energy (3), our master virial theorem (7) leads to
χ p 2 p 2 + m 2 χ = χ x · ∂ V ∂x (x) χ . (9)
In the nonrelativistic limit m → ∞ (i. e., for p 2 ≪ m 2 ), this spinless-Salpeter relativistic virial theorem, Eq. (9), necessarily reduces to its nonrelativistic counterpart (8).
Similarly, the virial theorem for the Dirac equation [8,9] is easily deduced [7] from our master virial theorem (7).
Bounds to Energy Eigenvalues of a Spinless-Salpeter Hamiltonian
The precise determination of eigenvalues of operators is of particular importance for any formulation of quantum theory. Unfortunately, for most cases it is not possible to determine the point spectrum (the set of all eigenvalues) of a given operator analytically. Several powerful tools, however, allow to derive analytic bounds to eigenvalues; applications of these techniques to the spinless-Salpeter operator (2) are reviewed, for instance, in Refs. [10][11][12][13].
Minimum-maximum principle
The theoretical foundation of any derivation of a system of rigorous upper bounds to the (isolated) eigenvalues of some operator H in Hilbert space and hence the primary tool for any localization of the discrete spectrum of H is the well-known minimum-maximum principle [14][15][16].
Its precise formulation is based on several prerequisites:
• Let this operator H be some self-adjoint operator.
• Assume that this operator is bounded from below.
• Define the eigenvalues of H, E k , k = 0, 1, 2, . . . , by the eigenvalue equation, with eigenstates |χ k ,
H |χ k = E k |χ k , k = 0, 1, 2, . . . .
• Let these eigenvalues E k be ordered, according to
E 0 ≤ E 1 ≤ E 2 ≤ · · · .
• Consider only the eigenvalues E k below the onset of the essential spectrum of the above operator H.
• Restrict all considerations to some d-dimensional
subspace D d ⊂ D(H) of the domain D(H) of H.
Then this theorem asserts that every eigenvalue E k of H -counting multiplicity of degenerate levels -satisfies
E k ≤ sup |ψ ∈D k+1 ψ| H |ψ ψ|ψ for all k = 0, 1, 2, . . . .
In the case of one-dimensional subspaces, that is, d = 1, the minimum-maximum theorem reduces to Rayleigh's principle: the ground-state eigenvalue E 0 of an operator H is less than, or equal to, every expectation value of H:
E 0 ≤ ψ| H |ψ ψ|ψ , |ψ ∈ D(H) .
Given some operator inequality satisfied by the operator H, the minimum-maximum principle may be employed to derive, by comparison, upper bounds on the (discrete) eigenvalues of H, provided that a few assumptions hold:
• The operator H, exhibiting all properties required by the minimum-maximum principle, is bounded from above by some other operator called O, i. e., it is subject to an (operator) inequality of the form
H ≤ O .
Applying both the minimum-maximum principle and this operator inequality, any eigenvalue E k of H must be bounded from above by the supremum of the expectation values of the operator O within the (k+1)-dimensional subspace D k+1 of D(H):
E k ≡ χ k | H |χ k χ k |χ k ≤ sup |ψ ∈D k+1 ψ| H |ψ ψ|ψ ≤ sup |ψ ∈D k+1
ψ| O |ψ ψ|ψ for all k = 0, 1, 2, . . . . (10)
• All eigenvalues E k of O are ordered according to
E 0 ≤ E 1 ≤ E 2 ≤ · · · .
• Every k-dimensional subspace D k in the chain of inequalities which constitutes Eq. (10)
sup |ψ ∈D k+1 ψ| O |ψ ψ|ψ = E k .
Consequently, an eigenvalue E k , k = 0, 1, 2, . . . , of the discrete spectrum of H(≤ O) is bounded from above by the corresponding eigenvalue E k , k = 0, 1, 2, . . . , of O:
E k ≤ E k for all k = 0, 1, 2, . . . .
It remains to prove an "appropriate" operator inequality.
(Summaries of the idea to find bounds by combining the minimum-maximum principle with reasonable operator inequalities may be found, e. g., in Refs. [12,13,17,18].)
Analytical upper bounds
The trivial nonrelativistic Schrödinger bound
The inequality (T −m) 2 ≥ 0 expressing nothing but the positivity of the square of the operator T −m may be, for m > 0, written as an inequality for the kinetic energy T :
T ≤ m + p 2 2 m .
(The right-hand side is the tangent line to the square root in the relativistic kinetic energy T in the point of contact p 2 = 0.) This result proves [17] that H is bounded from above by a nonrelativistic Schrödinger Hamiltonian H S :
H ≤ H S = m + p 2 2 m + V .
For a pure Coulomb potential V (r) = −α/r, the energy eigenvalues of the Schrödinger Hamiltonian H S depend only on the principal quantum number n, related to both radial and orbital angular-momentum quantum numbers by n = n r +ℓ+1, with n r = 0, 1, 2, . . . , ℓ = 0, 1, 2, . . .:
E S,n = m 1 − α 2 2 n 2 .
A "squared" bound
A relation between the (semirelativistic) Hamiltonian H and a nonrelativistic Schrödinger operator may be found [17] by considering the square H 2 of H and by realizing that the anticommutator T V +V T of relativistic kinetic energy T and potential V generated by the square fulfils
T V + V T ≤ p 2 + V 2 + 2 m V ,
as may be shown [17] by inspecting some consequences of the positivity of the square of the operator T −m−V :
H 2 = T 2 + V 2 + T V + V T ≤ Q ≡ 2 p 2 + m 2 + 2 V 2 + 2 m V .
With this inequality, the minimum-maximum principle, recalled in Subsect. 3.1, immediately guarantees that the energy eigenvalues E k of H are bounded from above by the square root of the corresponding eigenvalues E Q,k of the Schrödinger operator Q, constructed by squaring H:
E k ≤ E Q,k , k = 0, 1, 2, . . . .
For the case of a pure Coulomb potential V (r) = −α/r, the operator Q has the same structure as the Schrödinger Hamiltonian H S of Subsect. 3.2.1, with ℓ replaced by an effective orbital angular momentum quantum number L involving both the usual ℓ and the Coulomb coupling, α:
L (L + 1) = ℓ (ℓ + 1) + α 2 , ℓ = 0, 1, 2, . . . . (11)
The set of eigenvalues E Q of a "Coulombic" operator Q,
E Q,N = m 2 1 − α 2 2 N 2 ,
is determined by the effective principal quantum number N = n r + L + 1 , n r = 0, 1, 2, . . . .
Unfortunately, in the Coulomb case the squared bounds are above, and thus worse than, the Schrödinger bounds.
Rigorous semianalytical upper bound
We regard an energy bound as semianalytical if it can be derived by an (in general, numerical) optimization of an analytically given expression over a single real variable. Taking advantage, as a straightforward generalization of the (simple) line of argument sketched in Subsect. 3.2.1, of the inequality (T −µ) 2 ≥ 0 requiring an arbitrary real parameter µ of mass dimension 1 and obviously holding for all self-adjoint T [19] implies, for the kinetic energy,
T ≤ p 2 + m 2 + µ 2 2 µ for all µ > 0 .
This translates [17] to a set of inequalities for H, each of these involving a Schrödinger-like Hamiltonian, H S (µ):
H ≤ H S (µ) = p 2 + m 2 + µ 2 2 µ + V for all µ > 0 .
The best "Schrödinger-like" upper bound on any energy eigenvalue E k of H is then provided by the minimum of the µ-dependent energy eigenvalues of H S (µ), E S,k (µ):
E k ≤ min µ>0 E S,k (µ) , k = 0, 1, 2, . . . .
For a pure Coulomb potential V (r) = −α/r, the energy eigenvalues E S,n (µ) of H S (µ) read, with n = n r +ℓ+1,
E S,n (µ) = 1 2 µ m 2 + µ 2 1 − α 2 n 2 .
Here, minimizing E S,n (µ) with respect to µ entails [17] min µ>0
E S,n (µ) = m 1 − α 2 n 2 .
This (exact) upper bound [17] to the energy eigenvalues of the so-called "spinless relativistic Coulomb problem" holds for all those values of the Coulomb coupling α for which the Hamiltonian H with a Coulomb potential can be regarded as a reasonable operator and arbitrary levels of excitation, and for any value of the principal quantum number n it definitely improves the Schrödinger bound:
min µ>0 E S,n (µ) < E S,n for α = 0 .
Clearly, fixing µ = m recovers the Schrödinger bounds.
Exact semianalytical upper and lower bounds from the "envelope technique"
Rigorous semianalytical expressions for both upper and lower bounds to the eigenvalues E nℓ of the Hamiltonian H are found by a geometrical operator comparison in an approach called "envelope theory." The envelope theory constructs bounds on E nℓ by comparing the spectrum of H with the one of a conveniently formulated "tangential Hamiltonian" H involving some "basis potential" h(r),
H = m 2 + p 2 + c h(r) + const. , c > 0 ,
for which sufficient spectral information (i. e., either the exact eigenvalues or suitable bounds on these) is known.
Let V (r) be a smooth transformation V = g(h) of h(r),
with definite convexity of g(h). After optimization with respect to the point of contact of V (r) and the tangential potential, this technique produces bounds on E nℓ : lower bounds for g(h) convex (g ′′ > 0), and upper bounds for g(h) concave (g ′′ < 0). Suppressing for the moment the quantum numbers nℓ, all these bounds on E may be cast into a common generic form [20][21][22][23][24][25] with the individual bounds discriminated by a dimensionless parameter, P :
E ≈ min r>0 m 2 + 1 r 2 + V (P r) .(12)
Here, that cryptic sign of approximate equality indicates that for any definite convexity of g(h) all expressions on the right-hand side represent a lower bound for a convex g(h) and an upper bound for a concave g(h). The value of the parameter P used in Eq. (12) is determined by the algebraic structure of the interaction potential V (r), and by its convexity with respect to the basis potential, h(r):
• The spinless relativistic Coulomb problem posed by V (r) = −α/r is well-defined if its coupling α is constrained to α < 2/π [26]. The bottom of the corresponding spectrum of H (or, its ground-state energy eigenvalue, E 0 ) is bounded from below by
E 0 ≥ m 1 − α 2 P 2 ,
with the lower-bound parameter P given either by P = 2/π for α fulfilling 0 ≤ α < 2/π [26], or by
P ≡ P (α) = 1 2 1 + 1 − 4 α 2 for 0 ≤ α ≤ 1 2 , which obviously covers the range P ( 1 2 ) = 1 √ 2 ≤ P (α) ≤ P (0) = 1
, as derived by weakening [21][22][23]25] an improved lower bound to E 0 valid only for 0 ≤ α ≤ 1 2 [27].
If V (r) is a convex transform V = g(h), g ′′ > 0, of the Coulomb potential h(r) = −1/r, the above envelope approximation generates a lower bound [21][22][23]25] on the ground-state eigenvalue E 0 (on the entire spectrum) of the Hamiltonian H for any choice of the Coulomb lower-bound parameter P.
Clearly, the quoted upper bounds on the Coulomb coupling α apply also to any "effective" Coulomb coupling in H. Consequently, they translate into a constraint on all coupling constants introduced by the interaction potential V (r) under investigation.
(An example for these restrictions enforced by the Coulomb menace will be given in Subsect. 3.7.2.)
• For V (r) a concave transform V = g(h), g ′′ < 0, of the harmonic-oscillator potential h(r) = r 2 , a straightforward application of the above envelope approximation yields upper bounds [21][22][23]25] to all the eigenvalues E nℓ of the Hamiltonian H; the parameter P for a given energy level identified by quantum numbers nℓ is, in this case, related to the explicitly algebraically known eigenvalues E nℓ of the (nonrelativistic) Schrödinger operator p 2 +r 2 :
P ≡ P nℓ (2) = 1 2 E nℓ = 2 n + ℓ − 1 2 .
• For V (r) a concave transform V = g(h), g ′′ < 0, of the linear potential h(r) = r, the application of a "generalized" envelope approximation provides upper bounds [23,25] to all eigenvalues E nℓ of H if the parameters P which characterize the energy levels are given, in terms of the eigenvalues E nℓ of the nonrelativistic Schrödinger operator p 2 +r, by
P ≡ P nℓ (1) = 2 1 3 E nℓ 3/2 ;(13)
the parameter values P nℓ (1) corresponding to the lowest-lying energy levels can be found in Table 1 (for more details see, for instance, Refs. [20][21][22][23]).
If the potential V (r) is the sum of several distinct terms,
V (r) = i V i (r) , V i (r) = c i h i (r) ,
where every component problem defined by the operator
m 2 + p 2 + c i h i (r)
supports, for a sufficiently large c i , a discrete eigenvalue E i,0 at the bottom of its spectrum and information about the lowest energy eigenvalue, E i,0 , is available, all these pieces of information can be combined to a lower bound to E 0 [24]; for sums of pure power-law terms sgn(q) r q , where the coefficients a(q) of the pure power-law terms, sgn(q) r q , in the potential are positive, that is, a(q) ≥ 0, and do not vanish all, this yields the "sum lower bound"
V PL (r) = q a(q) sgn(q) r q ,(14)E 0 ≥ min r>0 m 2 + 1 r 2 + q a(q) sgn(q) (P (q) r) q
provided that some set of lower-bound parameters P (q) can be derived such that, whenever V (r) consists of just one single component, the above inequality yields either the corresponding exact ground-state energy eigenvalue or, at least, a rigorous lower bound to this latter quantity:
• For Coulomb components, that is, h i (r) = −1/r, P (−1) is the Coulomb lower-bound parameter P.
• For linear components, that is, h i (r) = r, P (1) is derived from the lowest eigenvalue E 0 of p 2 +r,
P (1) = 1 4 E 2 0 = 1.2457 .
It is straightforward to (try to) generalize these envelope techniques from the simpler one-body case summarized in this review to systems composed of arbitrary numbers of relativistically moving interacting particles described by a semirelativistic spinless Salpeter equation [28][29][30].
At least for the particular case of all harmonic-oscillator potentials V (r) = c r 2 with c > 0 the generalized upper bounds presented in Subsect. 3.3 and the envelope upper bounds can be shown to be equivalent to each other [23].
Rayleigh-Ritz (variational) technique
An immediate consequence of the minimum-maximum principle is the "Rayleigh-Ritz (variational) technique:"
• Introduce the restriction H of some operator H to a subspace D d by orthogonal projection P to D d :
H ≡ H D d := P H P .
• Identify all d eigenvalues E k , k = 0, 1, . . . , d−1, of the restricted operator H as the solutions of the eigenvalue equation of H for the eigenstates | χ k :
H | χ k = E k | χ k , k = 0, 1, . . . , d − 1 .
• Let these eigenvalues E k be ordered, according to
E 0 ≤ E 1 ≤ · · · ≤ E d−1 .
Then every (discrete) eigenvalue E k of H -if counting the multiplicity of degenerate levels -is bounded from above by the eigenvalue E k of the restricted operator H:
E k ≤ E k for all k = 0, 1, . . . , d − 1 .
If that d-dimensional subspace D d is spanned by any set of d (of course, linearly independent) basis vectors |ψ k , k = 0, 1, . . . , d−1, the eigenvalues E k can immediately be determined, by the diagonalization of the d×d matrix
ψ i | H |ψ j , i, j = 0, 1, . . . , d − 1 ,
that is, as the d roots of the characteristic equation of H,
det ψ i | H |ψ j − E ψ i |ψ j = 0 , i, j = 0, 1, . . . , d − 1 .
To establish this, expand any eigenvector | χ k of H over the basis {|ψ i , i = 0, 1, . . . , d−1} of the subspace D d . For spherically symmetric (central) potentials V (r), that is, for all potentials which depend only on the radial coordinate r ≡ |x|, a convenient and thus rather popular choice for the basis vectors {|ψ i , i = 0, 1, . . . , d−1} is that one the configuration-space representation of which involves the complete orthogonal system of generalized Laguerre polynomials [12,13,31,32] -cf. Appendix A.
Variational upper bounds
In the one-dimensional case [33] realized in the notation of Appendix A if all quantum numbers k = ℓ = m = 0, the Laguerre basis collapses to just a single basis vector:
ψ(x) ≡ ψ 0,00 (x) = µ 3 π exp(−µ r) , µ > 0 .
With a trial state |ψ represented by this exponential and the trivial (nevertheless fundamental) general inequality
| ψ| O |ψ | ψ|ψ ≤ ψ| O 2 |ψ ψ|ψ ,
which holds for any self-adjoint, but otherwise arbitrary, operator O (O † = O), Rayleigh's principle entails, after optimization with respect to the variational parameter µ, for a Coulomb potential V (r) = −α/r the upper bound
E 0 ≤ m 1 − α 2 ;
this is identical to the "generalized" upper energy bound on the ground-state or n = 1 eigenvalue of the Coulomb operator H found by different reasoning in Subsect. 3.3.
Application to illustrative interactions
Let us appreciate the above bounds' beauty at examples.
Trivial "testing ground:" Coulomb potential
Our first example clearly must be the Coulomb potential
V (r) = − α r , α > 0 ;
this potential arises from the exchange of some massless boson between the interacting objects. Therefore it is of particular interest in many areas of physics. Its effective interaction strength is given by a coupling α, identical to the fine structure constant in electrodynamics. We study
•E 0 (P ) ≈ m 1 − α 2 P 2 ,(15)
where for the ground state characterized by the quantum numbers n = 1, ℓ = 0 the (single) parameter P is given,
• for the "Coulomb lower bound" (Subsect. 3.4), by
P ≡ P C = P (α) = 1 2 1 + 1 − 4 α 2 ,
• for the generalized upper bound (Subsect. 3.3), by
P ≡ P G = n = 1 ,
• for the "linear upper bound" (Subsect. 3.4), as can be simply read off from the first row in Table 1, by P ≡ P L = P 10 (1) = 1.37608 ,
• for the "squared upper bound" (Subsect. 3.2.2), in accordance with the solution of Eq. (11) for L, by
P ≡ P Q = √ 2 N = 1 + √ 1 + 4 α 2 √ 2 ,
• and, in the case of the "harmonic-oscillator upper bound" (Subsect. 3.4), from the P nℓ (2) results, by
P ≡ P H = P 10 (2) = 3 2 .
It is a very trivial observation that, for fixed values of the Coulomb coupling, the ground-state energy bounds (15) are (monotone) increasing with increasing parameter P :
∂ E 0 (P ) ∂P ≥ 0 .
Thus it is straightforward to convince oneself that all the Coulomb (E C ), generalized (E G ), nonrelativistic (E N ), linear (E L ), squared (E Q ) and harmonic-oscillator (E H ) bounds on the ground-state energy eigenvalue E 0 of the semirelativistic Coulomb Hamiltonian H have to satisfy
E C ≤ E 0 ≤ E G ≤ E N ≤ E L ≤ E Q ≤ E H for α ≤ α 0 ≡ 3 8 3 − 2 √ 2 , E C ≤ E 0 ≤ E G ≤ E N ≤ E L ≤ E H ≤ E Q for α ≥ α 0 ≡ 3 8 3 − 2 √ 2 ,
taking into account the crossing of the upper bounds E H and E Q at
α 2 0 = 3 8 (3−2 √ 2), i. e., E Q (α 0 ) = E H (α 0 ).
For Coulomb-like interactions the only dimensional quantity among the parameters of this theory is the mass m of the interacting particles. Consequently, in this case all energy eigenvalues are proportional to m: the energy scale is set by m. The ratio E k /m is a universal function of the coupling α; w. l. o. g. it thus suffices to fix m = 1. Figure 1 compares for the ground state (n r = ℓ = 0) of the spinless relativistic Coulomb problem the various bounds to the lowest energy eigenvalue, E 0 , listed at the beginning of this subsection. Inspecting Fig. 1, we note:
• the squared, harmonic-oscillator, and linear upper bounds are numerically comparable to each other;
• likewise the nonrelativistic and generalized upper bounds are close to each other for all couplings α;
• using a Laguerre trial space of dimension d = 25, the Rayleigh-Ritz variational upper bound can be expected to come already pretty close to the exact eigenvalue E 0 -which, in turn, clearly indicates that it is highly desirable to find improvements for the lower bounds, in particular for large couplings α (this stimulated, e. g., the analysis of Ref. [34]).
Coulomb-plus-linear (or "funnel") potential
Within the field of elementary particle physics, quantum chromodynamics (QCD) is generally accepted to be that relativistic quantum field theory that describes all strong interactions between quarks and gluons by assigning the so-called "colour" degrees of freedom to these particles. In the instantaneous approximation inherent to all of the QCD-inspired quark potential models developed for the purely phenomenological description of experimentally observed hadrons, as bound states of constituent quarks, the strong forces are assumed to derive from an effective potential generating the bound states (this description of hadrons within the framework of quark potential models involving either nonrelativistic or relativistic kinematics is reviewed, for instance, in Refs. [3,35].) The prototype of all "realistic," that is, phenomenologically acceptable (static) interquark potentials V (r) consists of the sum of • a Coulomb contribution generated by a one-gluon exchange between quark bound-state constituents (dominating the potential at short distances r) and
• a linear term including all nonperturbative effects (that dominates the potential at large distances r).
The resulting interaction potential V (r) is characterized by a "funnel-type" Coulomb-plus-linear form; therefore it is called the Coulomb-plus-linear, or funnel, potential. Upon factorizing off a constant v, which spans the range 0 < v ≤ 1 in order to parametrize an overall interaction strength, we (prefer to) analyze this potential in the form
V (r) = − c 1 r + c 2 r = v − a r + b r .(16)
Clearly, given the overall coupling strength v, the actual shape of this potential is fixed by the ratio of the positive parameters a > 0 and b > 0; the coupling constants that enter, on the one hand, in the general expression (14) for sums of pure power-law terms and, on the other hand, in our funnel potential (16) must be identified according to
a(−1) ≡ c 1 ≡ a v > 0 , a(1) ≡ c 2 ≡ b v > 0 .
In view of the lack of fully analytical bounds we explore
• the three "basic" envelope bounds of Subsect. 3.4, distinguished by the adopted basis potential, viz.,
-the upper bound from a harmonic oscillator, -the upper bound involving a linear potential, -the lower bound due to a Coulomb potential,
• the envelope sum lower bound, derived in the sum approximation recalled by Subsect. 3.4, as well as
• the "Rayleigh-Ritz" upper bound of Subsect. 3.6.
For definiteness, let us fix the potential parameters a and b to a = 0.2, b = 0.5. As done in the Coulomb-potential example (in Subsect. 3.7.1) in order to take advantage of upper and lower bounds, we investigate the ground-state energy E 0 . The basic envelope bounds are computed by application of Eq. (12), for the appropriate parameter P :
• for the "harmonic-oscillator upper bound" we use P ≡ P H = P 10 (2) = 3 2 ; • for the "linear upper bound" we find from Table 1 P ≡ P L = P 10 (1) = 1.37608 ;
• for the "Coulomb lower bound," in order to derive the maximum value P consistent with 0 < v ≤ 1, we are forced to evaluate that "Coulomb coupling constant constraint" mentioned in Subsect. 3.4, in its form [22,23] fixed by our funnel potential (16),
c 1 + P 4 1 − P 2 c 2 m 2 ≤ P 1 − P 2 ,
for the maximum values of c 1 and c 2 , which gives P ≡ P C = 0.728112397 for m = 1 .
The "sum lower bound" is extracted from the expression given explicitly in Subsect. 3.4 for power-law potentials by insertion of the lower-bound parameters P (q = ±1):
• the Coulomb lower-bound parameter P (α) leads, for the relevant maximum coupling α = a = 0.2, in the Coulomb term of the sum approximation to P (−1) = P (α) = P (a) = 0.9789063 ;
• the lower-bound parameter required for any linear part of sum potentials is copied from Subsect. 3.4, P (1) = 1.2457 .
As before, the Rayleigh-Ritz or variational upper bound is found in a trial space of dimension d = 25 spanned by the generalized Laguerre basis (summarized in App. A). Figure 2 depicts the bounds to the lowest eigenvalue E 0 of H as function of the overall coupling strength v in the funnel potential (16). Remarkably, variational upper and sum lower bounds now restrict E 0 to a narrow band.
Approximate Solutions: Quality
Having determined -for instance, by application of the Rayleigh-Ritz technique sketched in Subsect. 3.5 -for some k = 0, 1, 2, . . . the state | χ k ∈ D d corresponding to any upper bound E k on the exact eigenvalue E k of H, one question immediately arises: how closely resembles the approximate solution | χ k the exact eigenstate |χ k ? Standard criteria, such as the (relative) distance between E k and true E k , or the overlap of approximate and exact eigenstates, require the knowledge of the exact solution. In contrast to this, the virial theorem (Sect. 2) represents an indicator for the accuracy of approximate eigenstates that merely uses information provided by the variational approach: Since all eigenstates of H satisfy any relation of the form (5), a significant imbalance in Eq. (7) reveals that this approximation is far from optimum [23,36,37]. Of course, because of the involvement (5) of the dilation generator (6) in the derivation of Eq. (7), any variational solution found by minimization of expectation values of H with respect to the scale transformations, or dilations, (1) will necessarily satisfy our master virial theorem (7).
Summary, Concluding Remarks
The various efficient approaches presented here allow to analyze the semirelativistic Hamiltonians of the spinless Salpeter equation analytically; this is crucial for general considerations that aim to answer questions of principle, like operator boundedness. For numerical methods, see, for instance, Refs. [10,38,39] and the references therein.
A The Generalized Laguerre Basis
Assume every basis function of L 2 (R 3 ) to factorize into a function of the radial variable and the angular term. Its configuration-space representation has the general form ψ k,ℓm (x) = Φ k,ℓ (r) Y ℓm (Ω r ) , r ≡ |x| ; the spherical harmonics Y ℓm (Ω) for angular momentum ℓ and projection m depend on the solid angle Ω ≡ (θ, φ) and satisfy a well-known orthonormalization condition:
dΩ Y * ℓm (Ω) Y ℓ ′ m ′ (Ω) = δ ℓℓ ′ δ mm ′ .
The most popular choice [12,13,31,32] for the basis states which span the Hilbert space L 2 (R + ) of [with the weight w(x) = x 2 ] square-integrable functions f (x) on the positive real line R + -which is the Hilbert space of radial trial functions Φ k,ℓ (r) -involves the generalized Laguerre polynomials L The basis states defined by the generalized-Laguerre choice for the radial basis functions Φ k,ℓ (r) involve two parameters, both of which may be subsequently adopted for variational purposes: µ (with the dimension of mass) and β (dimensionless); requirements of normalizability of our basis states constrain the parameters to the ranges 0 < µ < ∞ , −1 < 2 β < ∞ .
Therein, the orthonormality of the generalized Laguerre polynomials, inherent to their definition, is equivalent to the orthonormality of the radial basis functions Φ k,ℓ (r): ∞ 0 dr r 2 Φ k,ℓ (r) Φ k ′ ,ℓ (r) = δ kk ′ , k, k ′ = 0, 1, . . . ; this condition fixes the normalization constant N (µ,β) k,ℓ to N (µ,β) k,ℓ = (2 µ) 2 ℓ+2 β+1 k! Γ(2 ℓ + 2 β + k + 1) .
Fortunately the assumed factorization of every basis function persists in its momentum-space representation: ψ k,ℓm (p) = Φ k,ℓ (p) Y ℓm (Ω p ) , p ≡ |p| .
Analytical statements about Hamiltonians that involve a kinetic-energy operator nonlocal in configuration space, such as a relativistic square root (3), are facilitated by an explicit knowledge of the momentum-space basis states. One of the great advantages of the generalized-Laguerre basis is the availability of its analytic Fourier transform.
For all factorizations into radial and angular parts, as consequence of the Fourier transformation acting on the Hilbert space L 2 (R 3 ) of the square-integrable functions on the three-dimensional space R 3 , the radial parts of all basis functions that represent the chosen basis vectors in configuration space and momentum space, respectively, are related by so-called Fourier-Bessel transformations: Φ k,ℓ (r) = i ℓ 2 π ∞ 0 dp p 2 j ℓ (p r) Φ k,ℓ (p) , Φ k,ℓ (p) = (−i) ℓ 2 π ∞ 0 dr r 2 j ℓ (p r) Φ k,ℓ (r) , for all k = 0, 1, . . . , ℓ = 0, 1, . . . ;
the angular-integration remnants j n (z) (n = 0, ±1, . . .) label the spherical Bessel functions of the first kind [40]. For the generalized-Laguerre basis under consideration, these radial basis functions become in momentum space Φ k,ℓ (p) = N (µ,β) k,ℓ (−i) ℓ p ℓ 2 ℓ+1/2 Γ ℓ + 3 2 × k t=0 (−1) t t! k + 2 ℓ + 2 β k − t × Γ(a t,ℓ;β ) (2 µ) t (p 2 + µ 2 ) a t,ℓ;β /2 × F a t,ℓ;β 2 , − β + t 2 ; ℓ + 3 2 ; p 2 p 2 + µ 2 , with the hypergeometric series F (u, v; w; z), defined, in terms of the gamma function Γ, by the power series [40] F (u, v; w; z)
= Γ(w) Γ(u) Γ(v) ∞ n=0 Γ(u + n) Γ(v + n)
Γ(w + n) z n n! , and the simplifying abbreviation a t,ℓ;β ≡ 2 ℓ+β +t+2. Clearly, the momentum-space radial basis functions Φ k,ℓ (p) have to satisfy the orthonormalization condition ∞ 0 dp p 2 Φ * k,ℓ (p) Φ k ′ ,ℓ (p) = δ kk ′ , k, k ′ = 0, 1, . . . .
quality achieved by the variational solution of some eigenvalue problem depends decisively on the definition of the trial subspace D d employed by the Rayleigh-Ritz technique briefly sketched in Subsect. 3.5: enlarging D d to higher dimensions d or choosing a more sophisticated basis {|ψ i , i = 0, 1, . . . , d−1} which spans D d will, in general, increase the accuracy of the obtained solutions.
Figure 1 :
1Both upper (full lines) and lower (dashed line) bounds on the ground-state energy eigenvalue (E) of the semirelativistic Hamiltonian H with Coulomb potential V (r) = −a/r as a function of the Coulomb coupling, a, for the squared (Q), harmonic-oscillator (H), linear (L), nonrelativistic (N), generalized (G), variational (V) and Coulomb (C) [using the "optimized" P (a)] approaches.
Figure 2 :
2Three upper (full lines) and two lower (dashed lines) bounds on the ground-state energy eigenvalue (E) of the semirelativistic Hamiltonian H with the so-called funnel potential V (r) = v (−a/r+b r), where a = 0.2, b = 0.5, m = 1. These include: the harmonic-oscillator (H), linear (L) and variational (V) upper bounds and the sum-approximation (S) and Coulomb (C) lower bounds.
kk ′ , k, k ′ = 0, 1, . . . .
Table 1 :
1Numerical values of the parameter P nℓ (1) used in the linear-potential-based lower envelope bounds and defined in Eq. (13) for the lowest-lying energy levels nℓ.n ℓ
P nℓ (1)
1 0
1.37608
2 0
3.18131
3 0
4.99255
4 0
6.80514
5 0
8.61823
1 1
2.37192
2 1
4.15501
3 1
5.95300
4 1
7.75701
5 1
9.56408
1 2
3.37018
2 2
5.14135
3 2
6.92911
4 2
8.72515
5 2 10.52596
n ℓ
P nℓ (1)
1 3
4.36923
2 3
6.13298
3 3
7.91304
4 3
9.70236
5 3 11.49748
1 4
5.36863
2 4
7.12732
3 4
8.90148
4 4 10.68521
5 4 12.47532
1 5
6.36822
2 5
8.12324
3 5
9.89276
4 5 11.67183
5 5 13.45756
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| []
|
[
"IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE [PREPRINT 2020] 1 Infinite Feature Selection: A Graph-based Feature Filtering Approach",
"IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE [PREPRINT 2020] 1 Infinite Feature Selection: A Graph-based Feature Filtering Approach"
]
| [
"Giorgio Roffo ",
"Member, IEEESimone Melzi ",
"Umberto Castellani ",
"Member, IEEEAlessandro Vinciarelli ",
"Member, IEEEMarco Cristani "
]
| []
| []
| We propose a filtering feature selection framework that considers subsets of features as paths in a graph, where a node is a feature and an edge indicates pairwise (customizable) relations among features, dealing with relevance and redundancy principles. By two different interpretations (exploiting properties of power series of matrices and relying on Markov chains fundamentals) we can evaluate the values of paths (i.e., feature subsets) of arbitrary lengths, eventually go to infinite, from which we dub our framework Infinite Feature Selection (Inf-FS). Going to infinite allows to constrain the computational complexity of the selection process, and to rank the features in an elegant way, that is, considering the value of any path (subset) containing a particular feature. We also propose a simple unsupervised strategy to cut the ranking, so providing the subset of features to keep. In the experiments, we analyze diverse settings with heterogeneous features, for a total of 11 benchmarks, comparing against 18 widely-know comparative approaches. The results show that Inf-FS behaves better in almost any situation, that is, when the number of features to keep are fixed a priori, or when the decision of the subset cardinality is part of the process. - | 10.1109/tpami.2020.3002843 | [
"https://arxiv.org/pdf/2006.08184v1.pdf"
]
| 219,687,197 | 2006.08184 | a481a1279cb5fd9f79c3b4f20d01cdec287803a3 |
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE [PREPRINT 2020] 1 Infinite Feature Selection: A Graph-based Feature Filtering Approach
Giorgio Roffo
Member, IEEESimone Melzi
Umberto Castellani
Member, IEEEAlessandro Vinciarelli
Member, IEEEMarco Cristani
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE [PREPRINT 2020] 1 Infinite Feature Selection: A Graph-based Feature Filtering Approach
Index Terms-Feature selectionfilter methodsMarkov chains
We propose a filtering feature selection framework that considers subsets of features as paths in a graph, where a node is a feature and an edge indicates pairwise (customizable) relations among features, dealing with relevance and redundancy principles. By two different interpretations (exploiting properties of power series of matrices and relying on Markov chains fundamentals) we can evaluate the values of paths (i.e., feature subsets) of arbitrary lengths, eventually go to infinite, from which we dub our framework Infinite Feature Selection (Inf-FS). Going to infinite allows to constrain the computational complexity of the selection process, and to rank the features in an elegant way, that is, considering the value of any path (subset) containing a particular feature. We also propose a simple unsupervised strategy to cut the ranking, so providing the subset of features to keep. In the experiments, we analyze diverse settings with heterogeneous features, for a total of 11 benchmarks, comparing against 18 widely-know comparative approaches. The results show that Inf-FS behaves better in almost any situation, that is, when the number of features to keep are fixed a priori, or when the decision of the subset cardinality is part of the process. -
INTRODUCTION
O VER the last few decades, successful approaches to machine learning problems have been based initially on hand-crafted features (e.g., SIFT and HOG-like [1], [2], [3], [4], dictionarybased [5]) that evolved into automatically learned ones with the diffusion of deep learning models [6], [7], [8]. Through these advancements, feature selection (FS) still remains an active and growing research area that enables both dimensionality reduction and data interpretability, looking for features which are relevant and not redundant [9], [10], [11].
In this paper we introduce a fast graph-based feature filtering approach that ranks and selects features by considering the possible subsets of features as paths on a graph, and works in an unsupervised or supervised setup.
Our framework is composed by three main steps. In the first step, an undirected fully-connected weighted graph is built, where the node v i , 1 ≤ i ≤ n, corresponds to the feature f i , and each edge connecting v i to v j has associated a weight, or value, modeling the expectation that features f i and f j are relevant and not redundant. The weight comes from customizable pairwise relations among feature distributions, which can be easily crafted by the user, and, as a future perspective, learned directly from data. Here we present two instances of pairwise relations: one exploiting class information (Inf-FS S ), the other one being completely agnostic (Inf-FS U ).
In the second step, the weighted adjacency matrix associated to the graph is employed to assess the value of each feature (i.e., a Manuscript received August 5, 2019. node in the graph) while considering possible subsets of features (i.e., subsets of nodes) as they were paths of variable length. Two interpretations can be exploited: one comes from the properties of power series of matrices, the other one from the concept of absorbing Markov chain. In both the cases, we compute a vector which at the i-th entry expresses the value (or probability) of having a particular feature in a subset of any length, summing for all the possible lengths, until infinite. Going to infinite allows us to reduce the computational complexity from O(n 3 lT ) (n features, l path length, T samples) to O(n 3 T ). For this reason, we dubbed our approach Infinite Feature selection (Inf-FS). Ranking the values of the "infinite" vector gives the ordered importance of the features. In the third step, a threshold over the ranking is automatically selected by clustering over the ranked value. The rationale is to individuate at least two distributions, one which contains the features to keep with higher value, the other the ones to discard.
The proposed framework is compared against 18 comparative approaches of feature selection, with the goal of feeding the selected features into an SVM classifier.
As for the datasets, we selected 11 publicly available benchmarks to deal with diverse FS scenarios and challenges. In particular we consider five DNA microarray datasets for cancer classification (Colon [12], Lymphoma [13], Leukemia [13], Lung [14], Prostate [15]), handwritten character recognition (GINA [16]), general classification tasks from the NIPS feature selection challenge (MADELON, GISETTE [17], DEXTER [18]), and two classic object recognition datasets with convolutional neural networks (CNNs) features (PASCAL VOC 2007 [19] and CalTech 101 [20]).
One of the most interesting aspects shown in the experiments is the flexibility of Inf-FS, both in its unsupervised and supervised version: independently on the scenario (small-sample+high dimensional, unbalanced classes, severe interclass overlap, noise) Inf-FS overcomes the competitors, and if not, it gives the second arXiv:2006.08184v1 [cs.CV] 15 Jun 2020 or third best performance, promoting itself as all-purpose feature selection strategy. Another important achievement is in the automatic thresholding, which is simple yet effective in deciding which features to keep on any dataset. Finally, Inf-FS operates also on neural features, acting over cues that have been the state-of-the-art for long time [21].
The proposed framework generalizes the previously published Infinite Feature Selection (Inf-FS) [22], [23] presented as an unsupervised filtering approach, explained by algebraic motivations. Here we introduce a supervised counterpart and a strategy to select a subset of features, supported by a novel alternative way to explain the Inf-FS thanks to Markov chains fundamentals.
The rest of the paper is organized as follows: Sec. 2 illustrates the related literature, including the comparative approaches we consider in this study. Sec. 3 introduces our approach showing how the fully-connected graph is built for both the unsupervised and supervised variants. Sec. 3.5 connects the proposed approach to the absorbing Markov chain framework, deriving the subset selection strategy. Extensive experiments are reported in Sec. 4, and, finally, in Sec. 5, conclusions are given and future perspectives are envisaged.
STATE OF THE ART
Feature selection algorithms are partitioned into three main classes [24], [25]: filters, wrappers and embedded methods. Filter methods make use of the intrinsic properties of the data (e.g., correlation, variance, locality, information gain, etc.) to evaluate the value of a feature. In contrast, wrapper methods assign an importance score to each feature based on the performance of a predictor, which is considered as a black box.Finally, embedded methods include the feature selection process as part of an internal regression model which estimates the relationships among variables. Inf-FS belongs to the filter approaches, since it deals with the sole properties of the data, without relying on a specific predictor.
Within each of the above families of algorithms, FS techniques can be further classified into two sub-categories, unsupervised and supervised, depending on the use of class-label information in the selection process. In this paper we present an example of Inf-FS for both the scenarios.
Most of the feature selection algorithms evaluates an initial feature set, providing a ranking on them as a final output. Subsequently, the ranking is cut by subset selection strategies, commonly performed by cross-validation strategies on validation data [25].
The section overviews the three families of FS methods, separating their unsupervised and supervised versions.
Filter methods
Unsupervised approaches
In unsupervised scenarios, filter methods are mainly based on locality preserving principia found by clustering. The Laplacian Score (LS) [26] evaluates the value of a feature as its tendency to preserve spatial relationships which ensure intra-cluster proximity. Technically, LS constructs a nearest neighbor graph and ranks high those features that are consistent with Gaussian Laplacian matrix [26]. Similarly, in the multi-cluster feature selection approach (MCFS) [27], features are selected based on spectral analysis and solving a sparse regression problem, encouraging the formation of compact clusters. Local learning clustering (LLCFS) method [9] is a kernel learning method that weights features and exploits the weights to regularize the clustering. Uninformative features are left out before the clustering. These solutions, included in the experiments, are computationally expensive since rely on clustering. In contrast, our approach is faster since it only uses intrinsic properties of the data.
Supervised approaches
A standard two-class filter method is Relief and its multi-class extension Relief-F [28]. In general, the strategy evaluates feature value differences between nearest neighbor pairs and scores features according to how well they contribute to the overall class separation. A common criticism of Relief is that it selects redundant subsets, since it is not controlling feature correlation. A solution is given by the minimum Redundancy and Maximum Relevancy (mRMR) algorithm [29], minimizing the redundancy and maximizing the relevance of the set of features.This is obtained by maximizing the joint mutual information (using Parzen Gaussian windows [30]) between the values of a given feature and the membership to a particular class. The mRMR suffers from an expensive computational cost (i.e., O(n 2 T 3 ) where n is the number of features and T the number of samples [25], [29], [31]). Another weakness of mRMR comes with the approximation of the mutual information, which is inaccurate when the number of training samples is small [30]. A faster filter approach is the Fisher score [32], which scores the features individually, according to the ratio of inter-class separation and intra-class variance.
Several algorithms employ mutual information to select the features. The simple method proposed in [33] estimates the mutual information between feature distributions and class labels. All the features are evaluated independently, one by one, obtaining a score used to do ranking. The recent Max-Relevance and Max-Independence (MRI) [34] introduces a relevancy additional constraint, by maximizing the classification accuracy while minimizing the redundancy between features. Other approaches such as CIFE [35], MIFS [36] and ICAP [37] quantify the redundancy (or dependency) among the set of feature distributions by proposing slightly different variations of the objective function, i.e., the conditional likelihood of the training labels. Similarly, the joint mutual information (JMI) [38] and conditional mutual information (CMIM) [39] may be included in this group. The common assumption behind all these methods is that independency among features can positively affect the classification performance.
The Inf-FS framework is attractive because, when computing the weighted adjacency matrix, allows to include inter/intra class reasoning without relying on a specific strategy: in fact, the supervised Inf-FS S proposed here makes use of a fast computation of the mutual information and the Fisher criterion, but other alternatives are possible. Especially in the case of large number of samples, mutual information may be dropped in favor of other relations faster to be computed. Another difference with Inf-FS is that the MI-based approaches take into account pairwise (featureclass label) dependencies, while our approach extends the 2-nd order to n-th order by considering subsets of features as paths on a graph.
Recently, other graph-based approaches have been proposed such as the eigenvector centrality (ECFS) [40], [41], [42] and the infinite latent feature selection (ILFS) [22], which is an extension of the unsupervised Inf-FS U . The ECFS ranks features according to a centrality measure over the graph of features (eigenvector centrality), and should be considered a lighter version of Inf-FS U , see Sec. 3 for further details. In ILFS, the features are grouped into token by probabilistic latent semantic analysis (PLSA), which in practice learns the weights of the adjacency graph of Inf-FS as to provide better class separability. Instead, our framework requires to explicitly craft the weights; despite the experiments show that our approach overcomes ILFS, we think learning the weights is a convenient direction, which we are interested at the present moment.
Summarizing, some advantages of using filter methods are:
• faster than wrapper and embedded methods; • scalable; • classifier independent (better generalization). On the other hand, disadvantages are related to a generic lower performance if compared to supervised approaches, since filters are independent on the specific classifier.
Wrapper approaches
Unsupervised approaches
In the dependence-guided unsupervised feature selection (DGUFS) [43], feature selection is performed by graph-based clustering through the optimization of two terms: one term individuates dependence among samples by clustering, while the other term votes for features which minimize the intra-cluster variance. This approach is showed to be prone to local minima. The feature selection with adaptive structure learning (FSASL) [44] is an iterative approach that captures the global structure of data within a sparse representation framework, where the reconstruction coefficient is learned from the selected features. Its main drawback is the high computational complexity (see Table 1). Finally, the unsupervised feature selection with ordinal locality (UFSOL) [45] is a clustering-based approach that preserves the relative neighborhood proximities of the samples through distance-based clustering.
Similarly to our approach, these last three methods estimate inter-relationships among features, but in an iterative fashion which makes the entire process expensive and prone to local minima. Conversely, Inf-FS is one-shot.
Supervised approaches
The support vector machine with recursive feature elimination (RFE) [46] is a popular wrapper method that eliminates useless features in a sequential, backward fashion, ranking high a feature if it separates (by a linear SVM) samples of different classes. However, the performance of the RFE becomes unstable at some values of the filter-out factor (i.e., the number of features eliminated in each iteration) [47]. To overcome this weakness many different variants of RFE have been proposed, where the initial feature subset is selected using several SVM models with different filter-out factors, and in the second stage, features are selected by eliminating one feature at each iteration. For example, the sample weighting SW SVM-RFE [48] gives more weight to those samples that are close to the separating hyperplane. The Ensemble SVM-RFE [48] aggregates the results of several SVM-RFE selectors which are applied to randomized training data. Finally, the recursive cluster elimination (RCE) [49] has been introduced to overcome the RFE instability; it is a backward elimination algorithm that combines K-means to identify correlated clusters of features.
Some advantages of wrapper methods are:
• exploit the advantages of specific classifiers; • in general, higher classification accuracy than filters. The advantages of wrappers are also disadvantages, since they are suitable for some data only if their associated classifiers are, limiting the overall portability; additionally, wrappers tend to be computational expensive. On the contrary, Inf-FS is classifier agnostic, focusing only on intrinsic properties of data and their labels. We omit the RFE-X approaches in the experiments since they have been already shown to be inferior to Inf-FS in [23].
Embedded methods
Embedded methods include the selection process as part of an internal regression model (e.g., L1, LASSO regularization, decision tree), and the overall ranking process is less prone to overfitting than wrappers.
Unsupervised approaches
An example of unsupervised embedded method is the L 2,1norm regularized discriminative feature selection for unsupervised learning (UDFS) [50]. UDFS optimizes an objective function representing a L 2,1 -norm regularized minimization problem with orthogonal and locality preserving constraints [51] so that it simultaneously exploits discriminative information and feature correlations. However, such optimization problems are difficult to solve due to the non-smooth objective function and non-convex constraints [51].
Supervised approaches
In supervised learning scenarios, support vector machines play a role in many embedded approaches. The Feature Selection concaVe method (FSV) [11] generates a separating plane by maximizing the usual margin, minimizing at the same time the number of dimensions (= features) where the plane is defined. Another SVM-based feature selection approach minimizes the 0norm with (L0) SVMs [52], encouraging sparsity. The least square regression (LSR) has also been frequently employed for feature selection. The classical embedding approach is the regression by LASSO [53], where feature selection takes place by selecting the variables that have non-zero weighting coefficients. For classification, LASSO is modified by exploiting a hinge loss (LASSO h ), which penalizes linearly with respect to the correct classification labels [54]. More recently, unhinged losses have shown to be more robust against biased estimates [55], which are a known issue of LASSO (LASSO u ). In the experiments we consider as comparative approaches both LASSO h and LASSO u .
Another way to avoid bias comes with non-convex optimization strategies, for example with hard-thresholding approaches, which work under the hypotheses of strong restricted convexity/smoothness of the function to be minimized. Recent hard thresholding approaches are GraHTP [56], [57] and NHTP [58], the latter included as comparative approach.
Advantages and disadvantages of using embedded methods are similar to those of wrappers (they depend on external techniques), however, they are less prone to over-fitting. Inf-FS is conceptually different, being a filter which prepares the data to a subsequent, independent classification step.
OUR APPROACH
We propose two different versions of Inf-FS: the unsupervised Inf-FS U and the supervised Inf-FS S . In both the cases, we build upon a weighted undirected fully-connected graph G = (V, E) with node set V = { v 1 , ..., v n } representing a set of n feature distributions F = {f 1 , ..., f n }, and edge set E modeling relations among pairs of nodes (i.e., relations among distributions). In the following, the terms feature and feature distribution will be used interchangeably.
Let us represent G with its adjacency matrix A, where each of its elements A(i, j), 1 ≤ i, j ≤ n, models the confidence that features f i and f j (the nodes v i and v j ) are both good candidates to be selected, thanks to an associated weight function ϕ(·, ·):
A(i, j) = ϕ( v i , v j ),(1)
where ϕ(·, ·) is a positive, real-valued function defining the value of each edge. In the unsupervised version of our approach, referred as Inf-FS U , the function ϕ U (·, ·) is modeled as a function of both the variance and correlation of the features, while in its supervised form (Inf-FS S ), the function ϕ S (·, ·) adds the class information using the Fisher criterion and the mutual information. It is worth noting that other types of functions can be built, with the only constraint that the higher the value of the function, the stronger the preference of selecting both the features.
Graph Building for Inf-FS U
For the unsupervised scenario, ϕ U (·, ·) is a weighted linear combination of two pairwise measures relating the features f i and f j , defined as:
ϕ U ( v i , v j ) = αE ij + (1 − α)corr ij ,(2)
with E ij indicating the maximal normalized standard deviation over the two distributions, i.e., E ij = max (σ i , σ j ), where σ i is the standard deviation over the samples {f i }, normalized to the range [0, 1] by the maximum standard deviation over the set F . The second term is the opposite of the correlation corr ij = 1−|Spearman(f i , f j )|, with Spearman indicating Spearman's rank correlation coefficient. The α is a loading coefficient ∈ [0, 1], with its value being estimated during the experiments by cross validating on the training set for the classification tasks. In practice, ϕ U (·, ·) ∈ [0, 1] analyzes two feature distributions, accounting for the maximal feature dispersion (the standard deviation) and how much they are uncorrelated (the Spearman rank correlation coefficient).
Graph Building for Inf-FS S
The Inf-FS S introduces measures which consider class membership information, where we assume to have G classes into play.
The function ϕ S ( v i , v j ) is formed by three factors: the first is the Fisher criterion [59]:
h i = |µ i,1 − µ i,2 | 2 σ 2 i,1 + σ 2 i,2 ,(3)
where µ i,g and σ i,g are the mean and standard deviation, respectively, assumed by the i-th feature when considering the samples of the g-th class, 1 ≤ g ≤ G. The multi-class generalization is given by:
h i = G g=1 (µ i,g −μ i ) 2 E 2 i(4)
whereμ i and E i denote the mean and standard deviation of the whole data set corresponding to the f i feature (i.e.,
E 2 i = G g=1 (σ i,g ) 2 )
. This is considering intra-class compactness and inter-class separation induced by different features. The final scores are normalized to have maximum 1 and minimum 0. The closer h i to 1, the less redundant is the i-th feature, since its domain does not overlap with the other ones.
The second factor is the normalized mutual information m i between the features samples of the i-th class and the class label [60]:
m i = y∈Y z∈fi p(z, y)log p(z, y) p(z)p(y) ,(5)
where Y is the set of class labels and p(·, ·) stands for the joint probability distribution. Its normalized version is obtained by normalizing over all the n computed values (one for each feature into play). In practice, m i measures the amount by which the knowledge provided by the feature vector decreases the uncertainty about a class, summed over all the classes. The third factor is the normalized standard deviation σ i as computed for the unsupervised case.
The three factors are weighted linearly:
s i = h i α 1 + m i α 2 + σ i α 3 (6) with 1 ≤ i, j ≤ n.
The parameters α k are mixing coefficients, 0 ≤ α k ≤ 1, k α k = 1, and their values have been estimated during the experiments by cross validating on the training set for the classification tasks. Summarizing, the score s i indicates how much a feature is not redundant (Fisher criterion) and relevant (mutual information, standard deviation) w.r.t. the other classes. Finally, the weights of the adjacency matrix A are obtained by coupling the correspondent s as follows:
ϕ S ( v i , v j ) = A(i, j) = s i s j .(7)
It is worth noting that the formulation above is just one among the many possible alternatives that computes the value of features i and j taken together. Studying how to estimate this value in an end-to-end fashion would be probably more effective, and is subject of current work.
Feature Ranking Procedure
The Inf-FS procedure can be explained in two ways: with the properties of power series of matrices, or borrowing from the concept of absorbing Markov chain. Next, the analysis with the power series of matrices is presented, while the Markov chain view is given at Sec. 3.5.
Let γ = { v 0 = i, v 1 , ..., v l−1 , v l =
j} denote a path of length l between nodes i and j, that is, features f i and f j , passing through generic nodes v 1 , ..., v l−1 . Let us suppose that the length l of the path is less than the total number of nodes n in the graph. In this case, a path is simply a subset of the features.
We define the overall weight associated to γ as
π γ = l−1 k=0 A( v k , v k+1 ),(8)
where π γ is actually the value of the path accounting for all the features pairs that belong to it. There can be more than one path of length l connecting nodes i and j. Therefore, we define the set P l i,j as containing all the paths of length l between two nodes i and j. To estimate the overall contribution of all these paths, we calculate the following sum:
R l (i, j) = γ∈P l i,j π γ ,(9)
which, following standard matrix algebra, gives:
R l = A l ,(10)
that is, the power iteration of the adjacency matrix A. R l contains now cycles, and in our feature selection view, this is equivalent to evaluate each feature several times, possibly associated to itself in a self-cycle. This is a side effect that arises with this kind of network, but this possibility holds for all the features, and is taken into account by R l . We can evaluate the single feature score for the feature x (i) at a given path length l as
c l (i) = j∈V R l (i, j) = j∈V A l (i, j).(11)
In practice, Eq.11 models the value of the feature x (i) when considered in whatever selection of l features; the higher c l (i), the better. Therefore, a possible strategy could be that of ordering the features decreasingly by c l , taking the first m obtain a relevant set. Unfortunately, the computation of c l is expensive, bounded by (O((l − 1) · n 3 )): in fact, l is of the same order of n, so the computation turns out to be O(n 4 ) and becomes impractical for large sets of features to select (> 10K); our approach addresses this issue 1) by expanding the path length to infinity l → ∞ and 2) using notions from algebra to analytically solve the ranking problem in a computationally convenient way. Eq.11 estimates the score for feature f i when injected in whatever subset of l features. Taking into account all the possible path lengths can be elegantly modeled by letting l → ∞.
c(i) = ∞ l=1 c l (i) = ∞ l=1 j∈V R l (i, j) .(12)
Let C be the geometric series of adjacency matrix A:
C = ∞ l=1 A l ,(13)
It is worth noting that C can be used to obtain c(i) as
c(i) = ∞ l=1 c l (i) = [( ∞ l=1 A l )e] i = [Ce] i ,(14)
where e indicates a 1D vector of ones, and the square bracket indicates the extraction of an entry of the vector, specified by the index i. The problem is, summing infinite A l terms could lead to divergence; in which case, regularization is needed, in the form of generating functions [61], usually employed to assign a consistent value for the sum of a possibly divergent series. There are different forms of generating functions [62]. We define the generating function for the l-path aš
c(i) = ∞ l=1 r l c l (i) = ∞ l=1 j∈V r l R l (i, j),(15)
where r is a real-valued regularization factor, and r l can be interpreted as the weight for paths of length l. The parameter r has been defined as r = 0.9/ρ(A), with ρ(A) spectral radius of A (more on this at Sec. 3.4), ensuring that the infinite sum converges.
From an algebraic point of view,č(i) can be efficiently computed by using the convergence property of the geometric power series of a matrix (for a proof, see Sec. 3.4):
C = (I − rA) −1 − I,(16)
MatrixČ encodes the partial scores of our set of features. The goodness of this measure is strongly related to the choice of parameters that define the adjacency matrix A.
We can obtain final relevancy scores for each feature by marginalizing this quantity:
c(i) = [Če] i .(17)
Ranking in decreasing order theč vector gives the output of the algorithm: a ranked list of features where the most discriminative and relevant features are positioned at the top of the list. The gist of the Inf-FS is to provide a score of importance for each feature as a function of the importance of its neighbors. See Algorithms 1 (unsupervised) and 2 (supervised) for a sketch of our approaches.
Choice of the regularization parameter r
In this section, we want to justify the correctness of the method in terms of convergence. The value of r (used in the generating function, and introduced in the previous section, Eq. 15) can be determined by relying on linear algebra [63]. Let us define {λ 0 , ..., λ n−1 } as the eigenvalues of the matrix A; drawing from linear algebra, we can define the spectral radius ρ(A) as:
ρ(A) = max λi∈{λ0,...,λn−1} |λ i | .
For the theory of convergence of the geometric series of matrices, we also have::
lim l→∞ A l = 0 ⇐⇒ ρ(A) < 1 ⇐⇒ ∞ l=1 A l = (I − A) −1 − I.
Furthermore, Gelfand's formula [64] states that for every matrix norm, we have:
ρ(A) = lim k−→∞ ||A k || 1 k .
This formula leads directly to an upper bound for the spectral radius of the product of two matrices that commutes, given by the product of the individual spectral radii of the two matrices, that is, for each pair of matrices A and B, we have:
ρ(AB) ≤ ρ(A)ρ(B).
Starting from the definition ofš(i) and from the following trivial consideration:
r l A l = r l I A l = [(rI) A] l ,
we can use Gelfand's formula on the matrices rI and A and thus obtain:
ρ (rI) A ≤ ρ(rI)ρ(A) = rρ(A).(18)
Algorithm 1 Unsupervised Infinite Feature Selection
Input: F = { f 1 , .
.., f n } , α Output:č final scores for each feature + Building the graph for i = 1 : n do for j = 1 : n do
σ ij = max(std(f i ), std(f j )) corr ij = 1 − |Spearman(f i , f j )| A(i, j) = ασ ij + (1 − α)i = 1 : n do h i K k=1 (µ i,k −μi) 2 E 2 i m(i) = y∈Y z∈fi p(z, y)log p(z,y) p(z)p(y) Compute σ i s i = h i α 1 + m i α 2 + σ i α 3 end for for i = 1 : n do
for j = 1 : n do A(i, j) = s i s j end for end for + Letting paths tend to infinite r = 0.9
ρ(A) C = (I − rA) −1 − Ǐ c =Č e returnč
For the property of the spectral radius: lim l→∞ (rA) l = 0 ⇐⇒ ρ(rA) < 1. Thus, we can choose r, such as 0 < r < 1 ρ(A)
; in this way we have:
0 < ρ(rA) = ρ (rI) A ≤ ρ(rI)ρ(A) = rρ(A) < 1 ρ(A) ρ(A) = 1(19)
that implies ρ(rA) < 1, and so:
C = ∞ l=1(
An alternative view of Inf-FS as absorbing random walks
This section provides a different perspective of the proposed framework in terms of absorbing Markov chains and random walks. Following standard theory on stochastic processes [65], any m × m transition matrix T of a discrete time, first-order Markov chain with m states can be written in the canonical form, which separates absorbing states (having probability of self-transition = 1) from transient ones by re-ordering rows and columns as follows:
T = I 0 RÃ(20)
whereà is the square submatrix of size n × n giving the transition probabilities from non-absorbing to non-absorbing states (n ≤ m), R is the non-null rectangular submatrix of size n × k giving transition probabilities from non-absorbing to absorbing states (k = m − n), I is the identity matrix of size k × k, and 0 is a rectangular matrix of zeros of size k × n.
When k > 0, it means we have non-null probability of ending in a absorbing state, with R andà that are both substochastic, meaning that summing (separately) over their rows gives at least one row less than 1; in the case of k = 0 we have that the matrices R, I, 0 vanish, and the transition matrix T =à is stochastic and has no absorbing state. In the following, we assume that all of the rows ofà are substochastic, so that necessarily there is at least one absorbing state, so that k > 0.
With the canonical form, it becomes easy to compute different quantities, all related to the probability of having a particular random walk associated to T . In particular, the probability of having a walk of l steps 1 from state i to state j, 1 ≤ i, j ≤ m is given by
T l = I 0 (I +Ã +Ã 2 + ... +Ã l−1 )RÃ l(21)
The fact thatà is substochastic in all its rows is a sufficient condition which tells us that its spectral radius is ρ(Ã) < 1 [66], which is the same condition that we required for the convergence of the infinite sum at Sec. 3.4, this implyingà = rA. Therefore, let us suppose thatà = rA, for a specific r which will be discussed next, and A built as described in Sec. 3.2 and Sec. 3.1, so thatÃ(i, j) indicates the probability of choosing feature j after having selected i. Under this probabilistic view, the higherÃ(i, j), the higher the complimentarity between j and i. Going from a (transient) state ofà into an absorbing state b, 1 ≤ b ≤ k, driven by probabilityÃ(i, b), would mean to end the feature selection process. Intuitively, a highÃ(i, b) would mean that no other transient state (feature) j, k + 1 ≤ j ≤ k + n is complimentary w.r.t. i. Following this perspective, we may compute T ∞ as containing the probability of going from two states in an infinite number of steps by rewriting Eq. 21 withà → ∞ = 0 and
T ∞ = I 0 CR 0(22)
where the matrix
C = I +Ã +Ã 2 + ... +Ã ∞ = (I −Ã) −1(23)
1. Here step means a single iteration of the stochastic process modeled by the Markov chain At this point, interesting facts do emerge:
• the matrix C of Eq. 23 resembles the matrix of Eq. 16Č = (I − rA) −1 − I, withà = rA and a difference given by the identity matrix I. • In the Markov chain hypothesis, matrix C expresses with C(i, j) the expected number of visits to transient state j starting from transient state i, before to go into an absorbing state. In our feature selection case, C(i, j) could be seen as the length of the path enabled by feature i before to end the process of selection: a long path means that there is a pool of features, including necessarily i and j, which are strongly complimentary among each other (that have high probability to have transitions among themselves). In the same way, considering c = Ce, c i indicates how much, in general, feature i enable long paths, irrespective of the arrival feature j. The longer the path, the more complimentary is the feature i with respect to all the other features. • Unfortunately, the matrix A that we build with the procedures in Sec. 3.1 and Sec.3.2, in general, could be not substochastic, neither could be their regularized versions rA of Sec. 3.4. In fact, Sec. 3.4 indicates a necessary and sufficient condition for making rA convergent to 0 at infinity, which is not sufficient for being substochastic. The three observations above suggest a different, stronger regularization than the one expressed by Sec. 3.4 (r = 0.9/ρ(A)), in order to be compatible with the Markov chain paradigm; in practice, we need to have rA with r = 0.9/r max , where r max = max i n j=1 A(i, j) is the max summation over the rows of the original matrix adjacency A. This makes rA both convergent to 0 at infinity, and substochastic, unlocking an alternative, more interpretable view of our selection process.
At the same time, with the above regularization, theČ of matrix Eq. 16 measuring the value of a couple of features at infinity can be computed as the C matrix at Eq. 23, and, consequently, the vectors to be ordered becomeč =Če and c = Ce.
It is worth noting thatč and c give rise to the same ranking, so choosing one regularization r = 0.9/ρ(A) or the other r = 0.9/r max , in practice, makes absolutely no difference: the two regularizations give just two different interpretations of the same process.
Selection of the number of features
The vectorč obtained by Eq.17 contains at the i-th entry, in term of power series of matrices, the cumulative cost of having a particular feature in any (possibly infinite) subset of features. Equivalently, in terms of Markov chain, c i of Sec. 3.5 represents the expected number of selections of features which are complimentary to i that have been chosen before to finish the process of feature selection.
Ranking the c vector for feature filtering under the former perspective amounts to rank features which ensures paths of higher costs, where the cost, by construction, is higher for features which are relevant and redundant. Choosing the high-ranked features ensures to consider features of high value. In the Markov chain assumption, ranking the c vector amounts to promote features which are highly complimentary to each other.
Looking at how the values ofč (or, equivalently, c ) are distributed will give a global view of the features into play. Experimentally, we have found that the features are bipartite (especially in the supervised case), expressing features which are useful for the classification process and features that carry few or Acronym Type Class Comp. complexity LLCFS [9] f u N/A LS [26] f u N/A MCFS [27] f u N/A Relief-F [28] f s O(iT nG) MI [33] f s O(T 2 n 2 ) Fisher [32] f s O(T n) ECFS [40], [41] f s O(T n + n 2 )
ILFS [22] f s O(n 2.37 +in+T +G) CFS [46] f u O( n 2 2 T ) UDFS [50] f u N/A DGUFS [43] w u N/A FSASL [44] w u O(n 3 + T n 2 ) UFSOL [45] w u O(iT Gn 3 ) RFE [46] w s O(T 2 nlog 2 n) FSV [11] e s O(T 2 n 2 ) LASSO (hinged) [54] (unhinged) [55] e s O(T 2 n 2 ) The methods follow the taxonomy of Sec. 2, and are characterized by type (f=filter, w=wrapper, e=embedded), class (u = unsupervised, s = supervised) and computational complexity. As for the complexity, T is the number of samples, n is the number of initial features, i is the number of iterations in the case of iterative algorithms, and G is the number of classes.
NHTP [58] e s N/A Inf-FS U f u O(n 3 (1 + T )) Inf-FS S f s O(T 2 + n 3 (1 + T ))
no value. In other words, it is easy to spot a structure in this data, which can be extracted by a clustering procedure.
In this paper we propose to select a particular number of features, by considering the distribution of the {c i } values, and select by a clustering method the features which include the first ranked feature. Different clustering strategies can be taken into account: in our case, we consider 1D Mean-shift with automatic bandwidth selection [67], which showed to be highly effective in the experiments.
Future work will be devoted in looking for alternative ways to cluster the data: in particular, we spot few cases in which the Mean-shift was not working, due to Pareto-like distributions.
EXPERIMENTS AND RESULTS
In this section, we compare our framework with several feature selection methods considering both recent approaches [22], [40], [41], [43], [45], [55], [58], as so as some established algorithms [11], [27], [28], [32], [33], [46], [50], [54]. Methods are selected to cover the three different families presented in Sec. 2, i.e., filter, wrapper and embedded approaches. Tab. 1 lists the methods included in the experiments, reporting their type (f = filters, w = wrappers, e = embedded methods), and their class ( s = supervised or u = unsupervised). Additionally, the table shows the computational complexity whereas it has been provided.
The experiments are performed on 11 different publicly available benchmarks, whose characteristics are summarized in small-sample, high-dimensional scenarios, studying the strengths and weaknesses of the unsupervised and supervised Inf-FS on heterogeneous datasets, dealing then with features produced by deep learning algorithms. All of these experiments evaluate the feature selection approaches when they are constrained to provide a definite number b of features; different b's are considered (see in the following sections). In addition, we evaluate the automatic subset selection capability, where the optimal number of features has also to be decided. A conclusive statistics shows the Inf-FS framework as the most versatile and effective generalpurpose algorithm among the considered competitors. All of the (MATLAB) code is available at https://github.com/giorgioroffo/ Infinite-Feature-Selection.
Challenge 1: Small-sample, high-dimensional
Treating few samples described by many features is a traditional feature selection challenge. For example, in the medical field [68] observations are often difficult to collect (e.g., in the case of rare diseases), while the number of measurements performed on each sample can easily reach the order of thousands (e.g., set of DNA sequences). The small-sample, high-dimensional scenario holds in many other fields like business intelligence [69], geoscience [70] and the automatic analysis of behavioural cues and social signals [71], [72]).
Here we consider five widely used small-sample, highdimensional 2-class microarray datasets: Colon [12], Lymphoma [13], Leukemia [13], Lung [14], and Prostate [15]. They have been chosen for their variability in terms of number of features (from 2000 to 12533, see Tab. 2) which characterize 45 to 181 samples, because they deal with balanced and unbalanced classes, and because they are widely used in the literature. An exhaustive list of microarray small-sample, high-dimensional datasets can be found in https://bit.ly/2OSlOfv, while an essay on generic microarray datasets can be found in [73].
The experimental protocol consists in splitting the samples of the dataset in 70% for training and 30% for testing. The training procedure consists in building the matrix A as described in Sections 3.1 and 3.2. In the case of Inf-FS S , the class labels are taken into account, while in the unsupervised case they are ignored. After the training, a selection of the ranked features is considered, by keeping the top-b features, with b variable. The selected features are used to train a linear SVM, where a 5-fold cross-validation on training data is used to set the best C regularization parameter. The same experimental protocol has been applied to all the comparative feature selection approaches.
The number b of selected features varies (i.e., b =10, 50, 100, 150, and 200) in order to show the performance at different regimes. The performance is specified in terms of classification accuracy. In order to avoid any bias induced by a particularly favourable split, this procedure is repeated 20 times by shuffling the data (keeping training and testing separated) and the results are averaged over the trials. A cross-validation is carried out on each training partition of the datasets to select the {α} parameters introduced in Sec. 3.1 and 3.2. Fig. 1 depicts the results: on the left, the average performance obtained over all of the datasets by the unsupervised approaches are reported; on the right, supervised approaches are shown.
On Fig. 1 (left and right), it can be seen that in both the unsupervised and supervised case, the performance improves substantially with the number of the selected features up to a knee around 50 features; after 150 features, in general, the performance tends to saturate. On the left, it can be seen that Inf-FS U outperforms the existing methods with a mild but consistent average gap. On the right, Inf-FS S achieves definitely the best performance, in particular when the number of selected features is fixed to be small (from 10 to 100).
Comparing Inf-FS U and Inf-FS S (Fig. 1, left and right) one can see that, in general, Inf-FS S works better than Inf-FS U , since it uses class-label information to guide the FS process. Nonetheless, it is worth knowing (no curves are reported here) that on some datasets (COLON, LEUKEMIA and LUNG) the performance of the two approaches is comparable. This interesting aspect will be further discussed in Sec. 4.4 and Sec. 4.3.
Challenge 2: Inf-FS U VS Inf-FS S
This section compares the supervised and unsupervised versions of Inf-FS. Essentially, the difference between the two approaches consists of the type of functions used for weighting the graph. In fact, Inf-FS U does not employ any class-label information according to Eq. 2, while Inf-FS S is a combination of three different terms, two of them making use of the class labels (Fisher criterion and mutual information, see Eq. 6). When the difficulty of a classification problem depends on classes that overlap, Inf-FS S can naturally favour those features that best represent the explanatory factors of the dissimilarity among the classes. On the other side, Inf-FS S suffers when features are severely correlated, even if they are representative for a specific class. In this case, variance and correlation computed by Inf-FS U do represent a very convenient option.
To validate these considerations, we consider four additional datasets from the NIPS feature selection challenge, namely: DEXTER [18], GISETTE, MADELON [17] and GINA [16]. GISETTE and GINA present severely overlapped classes. Indeed, the GISETTE dataset [74] has instances of "4" and "9", two confusable handwritten digits (i.e., two overlapped classes) extracted from the MNIST data [75]. Features consist of normalized pixels and quantities derived from their combination.
The task of GINA is again handwritten digit recognition, but in this case, the two classes are even and odd 2-digit numbers. Obviously, only the unit digit is informative. In addition to the overlapping issues among the single digits (which are taken again from the MNIST data), a further consistent overlap is caused by the digits indicating the tens.
As for a dataset with non-descriptive features, we selected the DEXTER dataset [18], composed by sparse continuous bagof-words histograms, extracted from the Reuters text categorization benchmark [74]. Noise is coming from 10, 053 distractors (features having no discriminative power) put voluntarily in the dataset.
A benchmark where Inf-FS U should perform comparably if not superior to Inf-FS S is MADELON [17]. In fact, MADELON is an artificial dataset containing data points grouped in 32 clusters placed on the vertices of a five-dimensional hypercube and randomly labelled +1 or -1. The five dimensions constitute 5 informative features. 15 linear combinations of those features were added to form a set of 20 (redundant) informative features. Based on those 20 features one must separate the examples into the 2 classes (corresponding to the +1, -1 labels). A number of distractor features (480) called "probes" have no predictive power. Other than this, correlated features are present.
The results are shown in Fig. 2. In general, Inf-FS S outperforms Inf-FS U on DEXTER, GINA and GISETTE and achieves a absolute top performance in most of the cases. On the other hand, Inf-FS U achieves a better performance on MADELON at 10 features w.r.t. the supervised counterpart, by discarding the several correlated features in the set, and behaves comparably with Inf-FS S at the other regimes.
Considering each dataset separately, on GISETTE (Fig. 2 topleft) Inf-FS S betters all the comparative approaches when using 10 features, having NHTP close to its performance, while in the other supervised cases the gap is substantial. Unsupervised approaches do comparably to supervised ones when it comes to 10 features, but this is probably due to the fact that 10 features are definitely too few over the 5K which are originally available, and where many of them are probably equally useful. In fact, when the number of allowed features is growing (150, 200), it is visible that most supervised approaches better the unsupervised ones. Among the unsupervised approaches, our Inf-FS U ranks approximately third after LLCFS [9] and LS [26], since the former is driven by variance and correlation, and this does not allow to unveil features which are overlapped among classes. Notably, LLCFS [9] and LS [26] select features which are locality preserving, i.e., which agree on a clustering over the data. We may think that this clustering is capable to naturally separating the digits data, providing a more powerful solution than Inf-FS U .
On GINA instead (Fig. 2 top-right), supervised approaches show at just 10 features a consistent advantage over the unsupervised methods. Here, Inf-FS S is on pair with the mutual information MI [33] and the Fisher approach [32]. In fact, Inf-FS S contains both of them in the adjacency matrix A (see Sec. 3.2), and they are useful to highlight features that do not overlap across classes, i.e., which are non linearly correlated with the class information. Inf-FS U gives here the worst performances, ranking approximately fourth with respect to slower and more complex approaches (MCFS [27], LLCFS [9], DGUFS [43]) which once again exploit the hypothesis that data is organized in multiple clusters which we are ignoring with Inf-FS U . DEXTER (Fig. 2 bottom-left) has the highest number of features (20K) so that restricting to only 10-200 features opens to many equivalent selections, which anyway are better individuated by INf-FS S (among the supervised approaches, except the 10 features case where LASSO shows to be better) and by INf-FS U (among the unsupervised approaches, on pair with LS [26] which is better at 100-200 features).
On MADELON we already have discussed above the results of Fig. 2 (bottom-right) .
Challenge 3: Feature selection on CNN Features
Applying feature selection on deep learning-based cues is a recent trend in image recognition [76], [77]. In fact, recent studies show that feature learning and deep learning are not immune to produce redundant or introduce useless information in the learned representations. For example, [77] proposed a generic framework for network compression and acceleration where CNNs are pruned
Avg. Rank
RANKING (UNSUPERVISED)
CFS [47] DGUFS [44] FSASL [45] LLCFS [9] LS [28] MCFS [29] UDFS [52] UFSOL [46] Inf
Avg. Rank
RANKING (SUPERVISED)
NHTP [59] ECFS [42] Fisher [33] FSV [11] ILFS [22] LASSOU [56] LASSOH [55] MI [34] ReliefF [30] RFE [47] Inf-FSS by removing neurons with least importance, resulting in more robust networks. Neuron importance scores (usually associated to the last layer of the network, before classification) are computed by Inf-FS U as a function of the importance of all the other neurons in the layer.
In this subsection, we evaluate the performance of the proposed approach on features learned by the very deep ConvNet [21] framework, where the pre-trained model used for the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC) is adopted. We use the 4, 096-dimension activations of the last layer as image descriptors (L2-normalized afterwards), and we focus on the CALTECH 101 and PASCAL VOC-2007 datasets. These datasets allow for a systematic testing of the feature selection approaches taken into account in this paper, in a reasonable amount of time. We omit to choose other benchmarks (Imagenet for example) since for some of the comparative methods (LASSO and MCFS) the running time for a single trial is exceeding the week. Indeed, for each comparative approach, we perform a total of 200 runs.
According to the experimental protocol provided by the VOC challenge, a one-vs-rest SVM classifier is trained for each class (where cross-validation is used to find the best parameter C) and evaluated independently. Fig. 3 reports the performance curves obtained with the 18 feature selection approaches (solid lines for supervised approaches, dotted lines for unsupervised ones). In this case, the goal was to investigate the classification while keeping the first 5%, 10%, 15%, 20%, 25% of the features, corresponding to 205, 410, 614, 819 and 1024 characteristics. From Fig. 3 (Left), it can be seen that the supervised Inf-F S S reaches good performance in general, with a slightly superior performance w.r.t. the eigenvector centrality-based approach (ECFS). In general, the supervised approaches are organized into two groups, the most performing ones are the INFFS, ECFS, that, together with MI and ILFS gives an increase in the classification performance when adding more features. The other supervised approaches (RFE, FSV and RELIEF) seems to have a lower trend. Viceversa, all of the unsupervised approaches are more consistent among themselves, with Inf-FS positioning in the top 3 positions after LS and LLCFS. In the case of CALTECH 101 it is easy to see that the task is easier, with all of the approaches positioning in a narrow band of performance. Notably, Inf-FS S and Inf-FS U are on pair at the top position.
On the PASCAL 2007, we performed an additional experi- ment, aimed at exploring the performances when spanning the number of features retained from 5% to 100% (Fig. 5). The idea is to check how much difference holds when keeping a small number of features with respect to the whole set. In fact, feature selection approaches often represent a compromise between admitting a lower classification performance at the price of a faster time of task execution [25]. We apply both Inf-FS S and Inf-FS U . Noteworthy, both of the approaches provide features subsets leading to a performance (mAP) superior to the one obtained with the entire pool. In particular, with 25% of features, Inf-FS S raises the classification performances of barely 1 percentage point (83.8% against 83.1% fo the full set). Better performances are obtained in the range of 25%-45%. The Inf-FS S shows that there is a 10% of features ranked last which cause a slight bending of the performances (see the 90%-100% range). Inf-FS U has a similar behaviour, but lower in mAP score: The peak is at 45% of features (83.6%).
To further explore the behavior of the approach in the range of best performance (25%-45% for Inf-FS S and 35%-45% for Inf-FS U ) we perform a fine-grained cardinality analysis reaching the absolute best of Inf-FS S at 31.5% features (84.18% mAP) and 36.5% for Inf-FS U (83.91% mAP).
The versatility of Inf-FS U and Inf-FS S
In this section we want to summarize the diverse experiments carried out so far, demonstrating that one of the most valuable merit of the Inf-FS framework is that it applies favorably on every genre of feature selection scenario. To this sake, we set up in Fig. 4 two bubble-plots showing the average ranking (the lower, the better) for each compared approach (y-axis), considering all of the used datasets (except CALTECH 101 and PASCAL VOC where LASSO did not apply, and where we evaluated different numbers of features), separating the unsupervised and supervised approaches that we have considered in the experiments. In practice, the ranking represents the position of an approach (as classification accuracy) with respect to all the others. In the case a given approach has the best accuracy for a given benchmark, its rank on that benchmark is 1, in the case it gives the second-best accuracy the rank is 2, and so on. The average ranking shows how an approach, independently on the accuracy score, is generically better than the others, exhibiting a relative ordering.
The average ranking is computed with respect to different subsets of features (x-axis), and is enriched by the standard deviation in the ranking (how consistently an approach had a particular rank), depicted by the size of the blob (the larger the size, the higher the ranking variance).
The figures convey a clear message, since both Inf-FS unsupervised and supervised have the best rank, with a variance of 0.23 which indicates a stable behavior of both the approaches. Notably, Inf-FS S is definitely the most effective choice when it comes to few features selected; the mutual information-based MI [33] and the Fisher criterion for feature selection [32] follow. In the case of unsupervised approaches, Inf-FS U is first, followed by the clustering based approaches LS [26] and MCFS [27].
Challenge 4: Automatic Subset Selection
In this section, we test the process of selecting a subset of relevant features from the ranking provided by Inf-FS, explained in Sec. 3.6.
To this sake, we repeat all of the experiments with Inf-FS S and Inf-FS U on the 11 datasets examined so far, selecting as relevant features the ones indicated by the cluster which includes the firstranked feature, and using them for the classification tasks. As comparative approach, we consider LASSO learned with hinge loss [54] and unhinged loss [55], since it is the only which allows to automatically select a precise number of features, that is, the ones which survive the shrinking process during the training stage. In particular, we individuate the best-performing LASSO by 5-fold cross-validating the regularization parameter over the training set of each benchmark, for both the hinged and unhinged versions. The results are reported in Table 3 For each pair < dataset, method >, we report four different quantities: in the Subset column we show in round brackets the number of selected features, and alongside the classification accuracy obtained with that number of features. In the Best Prev. Perf. column, we report in round brackets the number of features that provided the best performance obtained in the previous experiments (following on the right). In the table, bold scores indicate the highest classification performance among the scores obtained by the automatic selection of feature subset, not the highest absolute.
From the results, several observations can be drawn:
• The automatic selection of the number of features allow Inf-FS U and Inf-FS S to provide higher performances than LASSO on 9 out of 11 cases, with LASSO unhinged beating the Inf-FS framework on GISETTE and GINA; • Tightly connected with the previous point, and worth noting, the Inf-FS framework selects definitely less features than the LASSO approaches (apart from the microarray datasets, where anyway LASSO unhinged is giving scarce performance). LASSO unhinged tends to keep features in a number which is highly variable; for example, it suggests a very large amount of features (2126 for GISETTE) or very few (the five microarray datasets); this seems to be correlated with the number of samples in the dataset, that, for the microarray datasets, is quite small. LASSO hinge appears to be more stable (but it gives the highest number of features). • Inf-FS S requires for all of the datasets less features than Inf-FS U (operating with the automatic selection), showing that the class information enriches the discriminative power of the cues. • Inf-FS performance with the automatic selection remain competitive in every scenario, while LASSO unhinged performs very poor on the small-sample, high dimensional case.
CONCLUSIONS
In this work we considered the feature selection problem under a brand-new perspective, i.e., as a regularization problem, where features are nodes in a weighted fully-connected graph, and a selection of l features is a path of length l through the nodes of the graph. Under this view, the proposed Inf-FS framework associates each feature to a score originating from pairwise functions (the weights of the edges) that measure relevance and non redundancy. This score has different explanations: under a power series of matrices view indicates the value that a feature can bring in a possibly infinite selection of features. Alternatively, under an absorbing Markov chain perspective, the score indicates how many times a feature would be associated to the other cues as complementary, before to end the process of selection. A precise subset of features can be provided, by examining the distribution of these scores. Inf-FS can be customized by hand-crafting the pairwise functions, and here we presented two customizations, for unsupervised and supervised scenarios, respectively. Future work will be spent in designing an end-to-end system capable to infer the optimal pairwise functions.
Supervised Infinite Feature Selection Input: F = {f 1 , ..., f n } , Y = {1, ..., G} , α 1 , α 2 , α 3 Output:č final scores for each feature + Building the graph for
rA) l = (I − rA) −1 − I This choice of r allows us to have convergence in the sum that definesč(i). Particularly, in the experiments, we use r = 0.9 ρ(A) , leaving it fixed for all the experiments.
Fig. 4 .
4Bubble plot showing the average ranking performance (y-axis) overall the datasets while increasing the number of selected features for the unsupervised approaches (left) and supervised ones (right). The area of each circle is proportional to the variance of the ranking.
Fig. 5 .
5Varying the cardinality of the selected features on VOC 2007. Mean average precision instead of classification accuracy is provided here.
TABLE 1
1Feature selection approaches considered in the experiments of Sec. 4.
Table 2 .
2The benchmarks allow to evaluate the proposed approach on supervised classification problems, focusing first onDataset
Ref. #Samples #Classes #Feat. few train unbal. (+/-) overlap noise sparse
COLON
[12]
62
2
2K
X
(40/22)
n.s.
X
LEUKEMIA
[13]
72
2
7129
X
(47/25)
n.s.
X
LUNG
[14]
181
2
12533
X
(31/150)
n.s.
X
LYMPHOMA
[13]
45
2
4026
X
(23/22)
n.s.
PROSTATE
[15]
102
2
6033
X
(50/52)
n.s.
DEXTER
[18]
2600
2
20K
(1,3K/1,3K)
X
X
X
GISETTE
[17]
6000
2
5K
(3K/3K)
X
X
GINA
[16]
3153
2
970
(1,5K/1,6K)
X
MADELON
[17]
2000
2
500
(1K/1K)
X
X
VOC 2007
[19]
10K
20
4096
X
X
X
CalTech 101
[20]
10K
102
4096
X
X
X
TABLE 2
2Datasets and the challenges for the feature selection scenario. The abbreviation n.s. stands for not specified (for example, in the DNA microarray datasets, no information on class overlap is given in advance).Fig. 1. Classification results on the small-sample, high-dimensional challenge. On the left, the average performance curves for unsupervised approaches, and on the right, supervised methods are shown. In all of the cases, the performance is measured at different numbers of selected features (on the x-axis).10
50
100
150
200
Subset Size
65
70
75
80
85
90
95
Accuracy
MICROARRAY -PERFORMANCE (UNSUPERVISED)
CFS [48]
DGUFS [51]
FSASL [52]
LLCFS [37]
LS [35]
MCFS [36]
UDFS [57]
UFSOL [53]
Inf-FSU [8]
10
50
100
150
200
Subset Size
71
76
81
86
91
96
Accuracy
MICROARRAY -PERFORMANCE (SUPERVISED)
NHTP [59]
ECFS [42]
Fisher [33]
FSV [11]
ILFS [22]
LASSOU [56]
LASSOH [55]
MI [34]
ReliefF [30]
RFE [47]
Inf-FSS
Fig. 2. Comparison between Inf-FS U and Inf-FS S . All the supervised approaches are reported by solid lines and the unsupervised ones by dotted lines. Results are expressed in terms of classification accuracy (%).Accuracy
Performance on GISETTE
NHTP
CFS
DGUFS
ECFS
Fisher
FSASL
FSV
ILFS
LASSOU
LASSOH
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
10
50
100
150
200
Subset Size
52
55
58
61
64
67
70
73
76
79
82
85
88
Accuracy
Performance on GINA
NHTP
CFS
DGUFS
ECFS
Fisher
FSASL
FSV
ILFS
LASSOU
LASSOH
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
10
50
100
150
200
Subset Size
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
Accuracy
Performance on DEXTER
NHTP
CFS
DGUFS
ECFS
Fisher
FSASL
FSV
ILFS
LASSOU
LASSOH
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
10
50
100
150
200
Subset Size
49
52
55
58
61
64
Accuracy
Performance on MADELON
NHTP
CFS
DGUFS
ECFS
Fisher
FSASL
FSV
ILFS
LASSOU
LASSOH
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
5%
10%
15%
20%
25%
Subset Size
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
mAP
Performance on PASCAL VOC 2007
NHTP
CFS
DGUFS
ECFS
Fisher
FSV
ILFS
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
5%
10%
15%
20%
25%
Subset Size
87
88
89
90
91
92
Accuracy
Performance on CALTECH 101
NHTP
CFS
DGUFS
ECFS
Fisher
FSASL
FSV
ILFS
LLCFS
LS
MCFS
MI
ReliefF
RFE
UDFS
UFSOL
Inf-FSU
Inf-FSS
Fig. 3. Performance achieved for the image classification task reported in terms of mAP (VOC 2007) and classification accuracy (Caltech-101) while
selecting the first 5%, 10%, 15%, 20%, and 25% features. Solid lines individuate supervised feature selection approaches, dotted lines indicate
unsupervised approaches.
TABLE 3
3The feature subset selection results reported in terms of accuracy (%). The values enclosed in round brackets show the number of the features kept. In bold the best performance for the Subset selection problem.
ACKNOWLEDGMENTSGiorgio Roffo is with the University of Glasgow where he is a Research Associate at the School of Computing Science. He received the European PhD degree in computer science from the University of Verona, Italy. Previously, he was with the Italian Institute of Technology (IIT), Genoa, Italy. His primary research interests are in the areas of machine learning, deep learning and computer vision. He contributed to the research field by publishing more than 10 articles in prestigious journals and conferences. He is in the technical program committee of leading conferences in computer vision and pattern recognition.Simone Melzi is a Post Doctoral researcher at Universit degli Studi di Verona (Italy). He received his PhD in Computer Science at Universit degli Studi di Verona (2018) and graduated in math summa cum laude from the University of Milan "La Statale" (2013). He received the EG-Italy PhD thesis award (2018). His main research interests are geometry processing, shape matching and 3D shape analysis. He has authored over 10 publications in leading journals and conferences. He is or has been Principal Investigator of several national and international projects, including a Centre for Doctoral training (texttthttp://social-cdt.org), a European Network of Excellence (the SSP-Net, www.sspnet.eu), and more than 10 projects funded by the Swiss National Science Foundation end the UK Engineering and Physical Sciences Research Council. Last, but not least, Alessandro is co-founder of Klewel (www.klewel.com), a knowledge management company recognized with national and international awards, and scientific advisor of Neurodata Lab (http://neurodatalab.com).Marco Cristani is Associate Professor (Professore Associato) at the Computer Science Department, University of Verona, Associate Member at the National Research Council (CNR), External Collaborator at the Italian Institute of Technology (IIT). His main research interests are in statistical pattern recognition and computer vision, mainly in deep learning and generative modeling, with application to social signal processing and fashion modeling. On these topics he has published more than 170 papers, including two edited volumes, 6 book chapters, 40 journal articles and 129 conference papers. He has organized 11 international workshops, cofounded a spin-off company, Humatics, dealing with e-commerce for fashion. He is or has been Principal Investigator of several national and international projects, including PRIN and H2020 projects. He is member of the editorial board of the Pattern Recognition and Pattern Recognition Letters journals. He is Managing Director of the Computer Science Park, a technology transfer centre at the University of Verona. Finally, he is ACM, IEEE and IAPR member.
. Lasso Dataset, DEXTER (10) 80.3% (50) 82.9% (2343) 79.9% (10) 81.1% (466) 83.8% (200) 84.8% (339) 92.8% (150) 92.9%Inf-FSU Inf-FSS Subset Best Prev. Perf. Subset Best Prev. Perf. Subset Best Prev. Perf. Subset Best Prev. Perf. 54unhinged) [55] LASSO (hingeDataset LASSO (unhinged) [55] LASSO (hinge) [54] Inf-FSU Inf-FSS Subset Best Prev. Perf. Subset Best Prev. Perf. Subset Best Prev. Perf. Subset Best Prev. Perf. DEXTER (10) 80.3% (50) 82.9% (2343) 79.9% (10) 81.1% (466) 83.8% (200) 84.8% (339) 92.8% (150) 92.9%
. Gisette, 2126) 95.3% (200) 90.3% (2482) 85.9% (200) 83.5% (707) 87.7% (200) 90.2% (638) 94.1% (200) 93.3GISETTE (2126) 95.3% (200) 90.3% (2482) 85.9% (200) 83.5% (707) 87.7% (200) 90.2% (638) 94.1% (200) 93.3
. Madelon, 233) 55.9% (150) 56.9% (396) 54.3% (150) 53.9% (48) 58.7% (10) 61.0% (32) 57.1% (200) 60.0 COLON (22) 66.7% (50) 85.5% (1131) 84.4% (200) 80.0% (326) 91.1% (150) 92.7% (174) 91.1% (100) 92.7%MADELON (233) 55.9% (150) 56.9% (396) 54.3% (150) 53.9% (48) 58.7% (10) 61.0% (32) 57.1% (200) 60.0 COLON (22) 66.7% (50) 85.5% (1131) 84.4% (200) 80.0% (326) 91.1% (150) 92.7% (174) 91.1% (100) 92.7%
. 18) 79.5% (200) 97.1% (1810) 93.8% (150) 93.3% (618) 94.7% (10) 94.7% (242) 95.2% (10) 94.8%LEUKEMIA. LEUKEMIA (18) 79.5% (200) 97.1% (1810) 93.8% (150) 93.3% (618) 94.7% (10) 94.7% (242) 95.2% (10) 94.8%
. 43) 87.0% (100) 95.3% (3168) 90.7% (150) 93.7% (1014) 93.0% (100) 93.3% (563) 94.7.6% (150) 96.6%PROSTATE. PROSTATE (43) 87.0% (100) 95.3% (3168) 90.7% (150) 93.7% (1014) 93.0% (100) 93.3% (563) 94.7.6% (150) 96.6%
3% LUNG (49) 89.8% (50). 96.6% (5297) 96.2% (150) 97.6% (400) 99.9% (200) 99.8% (361) 99.9%LYMPHOMA. 98% VOC 2007 N/A N/A N/A N/A (1,883) 83.6% (1024) 83.1% (696) 83.5% (819) 83.8% CalTech 101 N/A N/A N/A N/A (2250) 92.0% (1024) 92.1% (942) 91.8% (1024) 91LYMPHOMA (13) 56.7% (150) 91.6% (2105) 86.7% (150) 75.8% (674) 93.3% (150) 93.3% (395) 95.8% (200) 98.3% LUNG (49) 89.8% (50) 96.6% (5297) 96.2% (150) 97.6% (400) 99.9% (200) 99.8% (361) 99.9% (200) 99.8% VOC 2007 N/A N/A N/A N/A (1,883) 83.6% (1024) 83.1% (696) 83.5% (819) 83.8% CalTech 101 N/A N/A N/A N/A (2250) 92.0% (1024) 92.1% (942) 91.8% (1024) 91.9%
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Design of experiments of the nips 2003 variable selection benchmark. I Guyon, 9I. Guyon, "Design of experiments of the nips 2003 variable selection benchmark," 2003. 9
Y Lecun, C Cortes, MNIST handwritten digit database. 9Y. LeCun and C. Cortes, "MNIST handwritten digit database," http://yann.lecun.com/exdb/mnist/, 2010. 9
Predicting parameters in deep learning. M Denil, B Shakibi, L Dinh, N De Freitas, Advances in neural information processing systems. M. Denil, B. Shakibi, L. Dinh, N. De Freitas et al., "Predicting param- eters in deep learning," in Advances in neural information processing systems, 2013, pp. 2148-2156. 9
. R Yu, A Li, C.-F Chen, J.-H Lai, V I Morariu, X Han, M Gao, C.-Y , R. Yu, A. Li, C.-F. Chen, J.-H. Lai, V. I. Morariu, X. Han, M. Gao, C.-Y.
Nisp: Pruning networks using neuron importance score propagation. L S Lin, Davis, IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Lin, and L. S. Davis, "Nisp: Pruning networks using neuron importance score propagation," IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. 9
| [
"https://github.com/giorgioroffo/"
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|
[
"Deep Learning with Nonparametric Clustering",
"Deep Learning with Nonparametric Clustering"
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| [
"Gang Chen "
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| Clustering is an essential problem in machine learning and data mining. One vital factor that impacts clustering performance is how to learn or design the data representation (or features). Fortunately, recent advances in deep learning can learn unsupervised features effectively, and have yielded state of the art performance in many classification problems, such as character recognition, object recognition and document categorization. However, little attention has been paid to the potential of deep learning for unsupervised clustering problems. In this paper, we propose a deep belief network with nonparametric clustering. As an unsupervised method, our model first leverages the advantages of deep learning for feature representation and dimension reduction. Then, it performs nonparametric clustering under a maximum margin frameworka discriminative clustering model and can be trained online efficiently in the code space. Lastly model parameters are refined in the deep belief network. Thus, this model can learn features for clustering and infer model complexity in an unified framework. The experimental results show the advantage of our approach over competitive baselines.Recent advances in deep learning[10,26,3]have attracted great attention in dimension reduction[9,27]and classification problems[10,15,23]. The advantages of deep learning are that they give mappings which can capture meaningful structure information in the code space and introduce bias towards configurations of the parameter space that are helpful for unsupervised learning[6]. More specifically, it learns the composition of multiple non-linear transformations (such as stacked 1 arXiv:1501.03084v1 [cs.LG] | null | [
"https://arxiv.org/pdf/1501.03084v1.pdf"
]
| 18,134,251 | 1501.03084 | 8eb427352a2c51779f3a5158ed42ae2152adcb58 |
Deep Learning with Nonparametric Clustering
January 14, 2015
Gang Chen
Deep Learning with Nonparametric Clustering
January 14, 2015
Clustering is an essential problem in machine learning and data mining. One vital factor that impacts clustering performance is how to learn or design the data representation (or features). Fortunately, recent advances in deep learning can learn unsupervised features effectively, and have yielded state of the art performance in many classification problems, such as character recognition, object recognition and document categorization. However, little attention has been paid to the potential of deep learning for unsupervised clustering problems. In this paper, we propose a deep belief network with nonparametric clustering. As an unsupervised method, our model first leverages the advantages of deep learning for feature representation and dimension reduction. Then, it performs nonparametric clustering under a maximum margin frameworka discriminative clustering model and can be trained online efficiently in the code space. Lastly model parameters are refined in the deep belief network. Thus, this model can learn features for clustering and infer model complexity in an unified framework. The experimental results show the advantage of our approach over competitive baselines.Recent advances in deep learning[10,26,3]have attracted great attention in dimension reduction[9,27]and classification problems[10,15,23]. The advantages of deep learning are that they give mappings which can capture meaningful structure information in the code space and introduce bias towards configurations of the parameter space that are helpful for unsupervised learning[6]. More specifically, it learns the composition of multiple non-linear transformations (such as stacked 1 arXiv:1501.03084v1 [cs.LG]
Introduction
Clustering methods, such as k-means, Gaussian mixture model (GMM), spectral clustering and non-parametrical Bayesian methods, have been widely used in machine learning and data mining. Among various clustering methods, nonparametric Bayesian model is one of promising approaches for data clustering, because of its ability to infer the model complexity from the data automatically. To mine clusters or patterns from data, we can group them based on some notion of similarity. In general, calculating the clustering similarity is dependent on the features describing data. Thus, feature representation is vital for successful clustering. Just as common for other clustering methods, the presence of noisy and irrelevant features can degrade clustering performance, making feature representation an important factor in cluster analysis. Moreover, different features may be relevant or irrelevant in the high dimensional data, suggesting the need for feature learning.
restricted Boltzmann machines), with the purpose to yield more abstract and ultimately more useful representations [3]. In addition, deep learning with gradient descent scales linearly in time and space with the number of train cases, which makes it possible to apply to large scale data sets [9].
Unfortunately, little work has been done to leverage the advantages of deep learning for unsupervised clustering problems. Moreover, unsupervised clustering also presents a challenge in the deep learning framework, compared to supervised methods in the final fine-tuning process. Another important research topic in clustering analysis is how to adapt model complexity for increasing volumes in the era of big data [21,4,24]. However, most approaches are generative models and have restrictions on the prior base measures.
In this paper, we are interested in clustering problems and propose a deep belief network (DBN) with nonparametric clustering. This approach is an unsupervised clustering method, inspired by the advances in unsupervised feature learning with DBN, as well as nonparametric Bayesian models [1,7,4]. On the one hand, clustering performance depends heavily on data representation, which implies the need for feature learning in clustering. On the other hand, while the nonparametric Bayesian model can perform model selection and data clustering, it is intractable for non-conjugate prior; furthermore, it may not perform well on high-dimensional data, especially in terms of space and time complexity. Thus, we propose the deep learning with nonparametric maximum margin model for clustering analysis. Essentially, we first pre-train DBN for feature learning and dimension reduction. Then, we will learn the clustering weights discriminatively with nonparametric maximum margin clustering (NMMC), which can be updated online efficiently. Finally, we fine-tune the model parameters in the deep belief network. Refer to Fig. (1) for visual understanding to our model. Hence, our framework can handle high-dimensional input features with nonlinear mapping, and cluster large scale data sets with model selection using the online nonparametric clustering method.
Our contributions can be mainly summarized as: (1) leveraging unsupervised feature learning with DBN for clustering analysis; (2) a discriminative approach for nonparametric clustering under maximum margin framework. The experimental results show advantages of our model over competitive baselines.
Related work
Clustering has been an interesting research topic for decades, including a wide range of techniques, such as generative/discriminative and parametric/nonparametric approaches. As an discriminative method, maximum margin clustering (MMC) treats the label of each instance as a latent variable and uses SVM for clustering with large margins. However, they [2,28] either cannot learn parameters online efficiently or need to define the number of clusters like other clustering approaches, such as k-means, Gaussian mixture model (GMM) and spectral clustering. Considering the weakness of parametric models mentioned above, many nonparametric methods [4,14,8,12] have been proposed to handle the model complexity problems. One of the widely used nonparametric models for clustering is Dirichlet process mixture (DPM) [1,7]. DPM can learn the number of mixture components without specified in advance, which can grow as new data come in. However, the behavior of the model is sensitive to the choice of prior base measure G 0 . In addition, DPM of Gaussians need to calculate mean and covariance for each component, and update covariance with Cholesky decomposition, which may lead to high space and time complexity in high-dimensional data. Unsupervised feature learning with deep structures was first proposed in [9] for dimension reduction. Later, this unsupervised approach was developed into semi-supervised embedding [27] and supervised mapping [16] scenarios. Many other supervised approaches also exploit deep learning for feature extraction and then learn a discriminative classifier with objectives, e.g., square loss [9], logistic regression [15] or support vector machine (SVM) [13,23] for classification in the code space. The success behind deep learning is that it can learn useful information for data visualization and classification [6,3]. Thus, it is desirable to leverage deep learning for clustering analysis, because the performance for clustering depends heavily on data representation. Unfortunately, little attention has been paid to leveraging deep learning for unsupervised clustering problems.
A recent interesting approach is the implicit mixture of RBMs [17]. Instead of modeling each component with Gaussian distribution, it models each component with RBM. It is formulated as a third-order Boltzmann machine with cluster label as the hidden variable for each instance. However, it also requires the number of clusters specified as input.
In this paper, we are interested in deep learning for unsupervised clustering problems. In our framework, we take advantage of deep learning for representation learning, which is helpful for clustering analysis. Moreover, we take an discriminative approach, namely nonparametric maximum margin clustering to infer model complexity online, without the prior measure assumption as DPM.
Deep learning with nonparametric maximum margin clustering
In this section, we will first review RBM and DBN for feature learning. Then, we will introduce nonparametric maximum margin clustering (NMMC) method given the feature learned from DBN.
Finally, we will fine-tune our model given the clustering labels for the data.
Feature learning with deep belief network
Assume that we have a training set
D = {v i } N i=1 , where v i ∈ R d .
An RBM with n hidden units is a parametric model of the joint distribution between a layer of hidden variables h = (h 1 , ..., h n ) and the observations v = (v 1 , ..., v d ). The RBM joint likelihood takes the form:
p(v, h) ∝ e −E(v,h)(1)
where the energy function is
E(v, h) = −h T W 1 v − b T v − c T h(2)
And we can compute the following conditional likelihood:
p(v|h) = i p(v i |h) (3a) p(v i = 1|h) = logistic(b i + j W 1 (i, j)h j ) (3b) p(h i = 1|v) = logistic(c i + j W 1 (j, i)v j ) (3c)
where logistic(x) = 1/(1 + e −x ). To learn RBM parameters, we need to optimize the negative log likelihood −logp(v) on training data D, the parameters updating can be calculated with a efficient stochastic descent method, namely contrastive divergence (CD) [10].
A Deep Belief Network (DBN) is composed of stacked RBMs [9] learned layer by layer greedily, where the top layer is an RBM and the lower layers can be interpreted as a directed sigmoid belief network [3], shown in Fig. (1). Suppose the DBN used here has L layers, and the weight for each layer is indicated as W i for i = {1, .., L}. Specifically, we think RBM is a 1-layer DBN, with weight W 1 . Thus, DBN can learn parametric nonlinear mapping from input v to output x, f : v → x. For example, for 1-layer DBN, we have x = logistic(W 1 T v + c). After we learn the representation for the data, we use NMCC for clustering analysis to model the data distribution.
Nonparametric maximum margin clustering
Nonparametric maximum margin clustering (NMMC) is a discriminative clustering model for clustering analysis. Given the nonlinear mapping with DBN, we can first map the original training data
D = {v i } N i=1 into codes X = {x i } N i=1 in the embedding space. Then, with X = {x i } N i=1 and its the cluster indicators z = {z i } N i=1
, we propose the following conditional probability for nonparametric clustering:
P (z, {θ k } K k=1 |X ) ∝ p(z) N i=1 p(x i |θ z i ) K k=1 p(θ k )(4)
where K is the number of clusters, p(x i |θ z i ) is the likelihood term defined in Sec. 3.2.1 and p(θ k ) can be thought as the Gaussian prior for k = [1, ..., K]. Note that the prior p(θ k ) will be used in the maximum margin learning in Eq. (12).
p(z) = Γ(α) K k=1 Γ(n k +α/K) Γ(n+α)Γ(α/K) K
is the symmetric Dirichlet prior, where n k is the number of element in the cluster k, and α is the concentration parameter.
Recall that Dirichlet process mixture (DPM) [1,7] is the widely used nonparametric Bayesian approach for clustering analysis and model learning, specified with DP prior measure G 0 and α. As a joint likelihood model, it has to model p(X ), which is intractable for non-conjugate prior. The essential difference between our model and DPM is that we maximize a conditional probability, instead of joint probability as in DPM [14]. Moreover, our approach is a discriminative clustering model with component parameters learned under maximum margin framework.
To maximizing the objective function in Eq. (4), we hope the higher within-cluster correlation and lower correlation between different clusters. Given z, we will need to learn {θ k } K k=1 to keep each cluster as compact as possible, which in turn will help infer better K. In other words, to keep the objective climbing, we need higher likelihood p(x i |θ z i ) with higher correlation within-cluster, which can be addressed with discriminative clustering. Given the component parameters, {θ k } K k=1 , we need to decide the label for each element for better K. For each round (on the instance level), we use Gibbs sampling to infer z i for each instance x i , which in turn can be used to estimate {θ k } K k=1 with online maximum margin learning. For each iteration (on the whole dataset), we also update α with adaptive rejection sampling [18].
Gibbs sampling
Given the data points X = {x i } N i=1 and its the cluster indicators z = {z i } N i=1 , the Gibbs sampling involves iterations that alternately draw samples from conditional probability while keeping other variables fixed. For each indicator variable z i , we can derive its conditional posterior as follows:
p(z i = k|z −i , x i , {θ k } K k=1 , α, λ) (5) = p(z i = k|x i , z −i , {θ k } K k=1 ) (6) ∝ p(z i = k|z −i , {θ k } K k=1 )p(x i |z i = k, {θ k } K k=1 ) (7) = p(z i = k|z −i , α)p(x i |θ k )(8)
where the subscript −i indicates all indices except for i, p(z i = k|z −i , α) is determined by Chinese restaurant process, and p(x i |θ k ) is the likelihood for the current observation x i . For DPM, we need to maximize the conditional posterior to compute θ k , which depends on observations belonging to this cluster and prior G 0 .
In our conditional likelihood model, we define the following likelihood for instance x i
p(x i |θ k ) ∝ exp(x T i θ k − λ||θ k || 2 )(9)
where λ is a regularization constant to control weights between the two terms above. By default, the prediction function should be proportional to arg max k (x T i θ k ), for k ∈ [1, K]. In other words, higher correlation between x i and θ k indicates higher probability that x i belongs to cluster k, which further leads to higher objective in Eq. (4). In our likelihood definition, we also subtract λ||θ k || 2 in Eq. (9), which can keep the maximum margin beneficial properties in the model to separate clusters as far away as possible. Another understanding for the above likelihood is that Eq. (9) satisfies the general form of exponential families, which are functions solely of the chosen sufficient statistics [22]. Thus, such probability assumption in Eq. (9) make it general to real applications.
Plug Eq. (9) into Eq. (8), we get the final Gibbs sampling strategy for our model
p(z i = k|z −i , x i , {θ k } K k=1 , α, λ) ∝ p(z i = k|z −i , α)exp(x T i θ k − λ||θ k || 2 )(10)
We will introduce online maximum margin learning for component parameters {θ k } K k=1 in Sec 3.2.2. For the newly created cluster, we assume θ K+1 is sampled from multivariate t-distribution.
Online maximum margin learning
We follow the passive aggressive algorithm (PA) [5] below in order to learn component parameters in our discriminative model with maximum margins [25].
We denote the instance presented to the algorithm on round t by x t ∈ R n , which is associated with a unique label z t ∈ [1, K]. Note that the label z t is determined by the above Gibbs sampling algorithm in Eq. (10). We shall define Θ = [θ 1 , ..., θ K ] a parameter vector by concatenating all the parameters {θ k } K k=1 (that means Θ zt is z t -th block in Θ, or says Θ zt = θ zt ), and Φ(x t , z t ) is a feature vector relating input x t and output z t , which is composed of K blocks, and all blocks but the z t -th are set to be the zero vector while the z t -th block is set to be x t . We denote by Θ t the weight vector used by the algorithm on round t, and refer to the term γ
(Θ t ; (x t , z t )) = Θ t · Φ(x t , z t ) − Θ t · Φ(x t ,ẑ t )
as the (signed) margin attained on round t. In this paper, we use the hinge-loss function, which is defined by the following,
(Θ; (x t , z t )) = 0 if γ(Θ t ; (x t , z t )) ≥ 1 1 − γ(Θ t ; (x t , z t )) otherwise(11)
Following the passive aggressive (PA) algorithm [5], we optimize the objective function:
Θ t+1 = arg min Θ 1 2 ||Θ − Θ t || 2 + Cξ s.t. (Θ; (x t , z t )) ≤ ξ(12)
where the l 2 norm of Θ on the right hand size can be thought as Gaussian prior in Eq. (4). If there's loss, then the updates of PA-1 has the following closed form
Θ zt t+1 = Θ zt t + τ t x t , Θẑ t t+1 = Θẑ t t − τ t x t ,(13)
whereẑ t is the label prediction for x t , and τ t = min{C, (Θt;(xt,zt)) ||xt|| 2 }. Note that the Gibbs sampling step can decide the indicator variable z t for x t . Given the cluster label (the ground truth assignment) for x t , we update our parameter Θ using the above Eq. (13). For convergence analysis and time complexity, refer to [5].
Fine-tuning the model
Having determined the number of clusters and labels for all training data, we can take the fine-tuning process to refine the DBN parameters. Note that the objective function in Eq. (12) takes the l 1 hinge loss as in [23]. Thus, one possible way is that we can take the sub-gradient and backpropagate the error to update DBN parameters. In our approach, we employ another method and only update the top layer weights W L and Θ in the deep structures. This fine-tuning process is inspired by the classification RBM [15] for model refining. Basically, we assume the top DBN layer weight W L and SVM weight Θ can be combined into a classification RBM as in [15] by maximizing the joint likelihood p(x, z) after we infer the cluster labels for all instances with NMMC. Note that there is mapping from SVM's scores to probabilistic outputs with logistic function [19], which can maintain label consistency between the SVM classifier and the softmax function. Thus, the SVM weight Θ can be used to initialize the weight of the softmax function in the classification RBM. After the fine-tuning process, we can max z p(z|v) for z ∈ [1, K] to label the unknown data v. For 1-layer DBN, we can get the following classification probability:
p(z|v) = e dz n j=1 1 + e c j +Θ jz + i W 1 (i,j)v i z * e d z * n j=1 1 + e c j +Θ jz * + i W 1 (i,j)v i(14)
where d z for z ∈ [1, K] is the bias of clustering labels, and c j for j ∈ [1, n] are biases of the hidden units. Note that Θ has been reshaped into n × K matrix before updating in the fine-tuning process. For the deep neural network with more than one layer, we first project v into the coding space x, then use the above equation for classification.
In our algorithm, we only fine-tune in the top layer because of the following reasons: (1)
Experimental Results
In order to analyze our model, we performed clustering analysis on two types of data: images and documents, and compared our results to competitive baselines. For all experiments, including pretraining and fine-tuning, we set the learning rate as 0.1, the maximum epoch to be 100, and used CD-1 to learn the weights and biases in the deep belief network. We used the adjusted Rand Index [11,20] to evaluate all the clustering results.
Clustering on MNIST dataset: The MNIST dataset 1 consists of 28 × 28-size images of handwriting digits from 0 through 9 with a training set of 60,000 examples and a test set of 10,000 examples, and has been widely used to test character recognition methods. In the experiment, we randomly sample 5000 images from the training sets for parameter learning and 1000 examples from the testing sets to test our model. After learning the features with DBN in the pre-training stage, we used NMMC for clustering, with setting α = 4, λ = 15 and C = 0.001. In the experiment, λ The clustering performance of our method (DBN+NMMC) is shown in Table (1), where "pretrain" and "fine-tune" indicate how the accuracy changes before and after the fine-tuning process for the same parameter setting on the same dataset. The results with 2-layer DBN in Table (1) demonstrate that our method significantly outperforms baselines. It also shows that fine-tuning process can greatly improve accuracy, especially on the testing data. In Table (1), we think the largest train/test difference for the least complex model is caused by biases between before and after finetuning. In other words, the fine-tuning step can learn better biases via classification RBM and improve testing performance. We also visualize how the weights change before and after the fine-tuning process in Fig. (2).
We also evaluate how the depth and dimensionality of deep structures influence clustering accuracy. Fig. 3(a) shows how adjusted Rand Index changes with the number of dimensions for 1-layer DBN (or RBM), and it demonstrates that higher dimensionality does not mean higher performance. In Fig. 3(a), we can see fine-tuning severely hurt performance on the training set on higher dimension coding space, we guess it is caused by overfitting problem in the complex model. In other words, the wrong clustering prediction will deteriorate the clustering performance even further through fine-tuning. That makes sense because we treat the wrong labeling as the correct one in the finetuning stage. It also verifies that it is reasonable by just fine-tuning the model in the top layer, instead of the whole network, with the purpose to reduce the overfitting problem. Fig. 3(b) shows that given the 100 hidden nodes in the top layer, how the performance changes with the depth of DBN structure. It seems that the deeper complex model cannot guarantee better performance.
To verify whether our NMMC is effective for data clustering and model selection, we also compare our NMMC to DPM given the same DBN for feature learning. The results in Fig. (4) demonstrates that NMMC outperforms DPM significantly and also shows that our NMMC can always converge after 100 iterations. The time complexity comparison between our method and DPM is shown in Fig. 5 in the DBN projection space. It shows that our method is significantly efficient, compared to DPM. To manifest how effective our method is, we also show the upper bound DBN+GMM, with 2 layers n = [400, 100] in Table ( 1). It shows that features learned with DBN are helpful for clustering, compared to raw data. It also shows that our method yields better clustering results than the upper bound.
Clustering on 20 newsgroup: We also evaluated our model on 20 newsgroup datasets for document categorization. This document dataset has 20 categories, which has been widely used in text categorization and document classification. In the experiment, we tested our model on the binary version of the 20 newsgroup dataset 2 . We used the training set for training and tested the model on the testing dataset. After we learned features in the DBN, we used NMMC for clustering, with setting α = 4, λ = 30 and C = 0.001. To make an fair comparison, we basically took a similar setting as in the MNIST dataset, for both NMMC and DPM in order to generate the number of clusters which is comparable for both methods. Baselines such as k-means and GMM should be thought of as upper bound because they need to specify the number of clusters K = 20.
The clustering performance of our method (DBN+NMMC) on 20 newsgroups is shown in Table. (2). It also demonstrates that the fine-tuning process can greatly improve accuracy, especially on the testing data. Although our model cannot beat baselines on the training set, our model can achieve better evaluation performance on the testing set (better than GMM and k-means on the raw data clustering). To verify whether our NMMC is effective for data clustering and model selection, we also compare our NMMC to DPM given the same DBN for feature learning. The results in Fig Table 2: The experimental comparison on the 20 newsgroup dataset, where "train" means for training data, "test" indicates testing data. It demonstrates that the fine-tuning process in our model can improve clustering performance. We compare the performances between our method and other baselines. It demonstrates that our method (DBN+NMMC) yields clustering accuracy comparable to baselines, and performs better on the testing sets with 1-layer DBN.
method is more efficient in practice.
To sum up, our model can converge well after 100 iterations from the experiments above. Moreover, the fine-tuning process in our model can greatly improve the performance on the test sets. Thus, it also shows that the parameters learned with NMMC can be embedded well in the deep structures.
Conclusion
Clustering is an important problem in machine learning and its performance highly depends on data representation. And, how to adapt the model complexity with data also pose a challenge.
In this paper, we propose a deep belief network with nonparametric maximum margin clustering. This approach is inspired by recent advances of deep learning for representation learning. As an unsupervised method, our model leverages deep learning for feature learning and dimension reduction. Moreover, our approach with nonparametric maximum margin clustering (NMMC) is a discriminative clustering method, which can adapt model size automatically when data grows. In addition, the fine-tuning process can incorporate NMMC well in the deep structures. Thus, our approach can learn features for clustering and infer model complexity in an unified framework. We currently use DBN [10] instead of deep autoencoders [9] for fast feature learning because the latter is time-consuming for dimension reduction. In future work, we will explore deep autoencoders to learn better feature representation for clustering analysis. Another interesting topic to be explored is how to optimize the depth of deep learning structures in order to improve clustering performance.
the objective function in Eq. (4) with deep feature learning is non-convex, which can be easily trapped into local minimum with L-BFGS [9]; (2) if there was clustering error in the top layer, it could be easily propagated in the backpropagation stage; (3) To only update the top layer can effectively handle the overfitting problem.
Figure 2 :Figure 3 :
23The visualization of learned weights in the pre-training and fine-tuning stages respectively with 1-layer DBN for n = 100 on the MNIST dataset. How the dimensionality and structural depth influence performance on MNIST dataset. (a) how the Rand Index changes with the encoded data dimension; (b) how the Rand Index changes with the depth of deep structures. It demonstrates the fine-tuning process is helpful to improve clustering performance. It also shows that complex deep structures cannot improve clustering accuracy.plays a vital role on the final number of clusters. Higher λ, larger number of clusters generated. To make an fair comparison, we basically tuned parameters to keep the number of generated clusters close to the groundtruth in the training stage. For example, in the MNIST experiment, we keep it around 5 to 20 in the training set for both NMMC and DPM. The results from baselines such as k-means and GMM should be conceived as upper bound (specify the number of clusters K = 10).
Figure 4 :Figure 5 :
45The performance comparison between DPM and NMMC on the MNIST dataset with the same DBN structure for feature learning. (a) it is a 1-layer DBN (or RBM) with the number of hidden nodes n = 100; (b) it is a 2-layers DBN, with n = [400, 100] for each layer. It demonstrates that with the same DBN for feature learning, NMMC outperforms DPM remarkably.(6) demonstrate that NMMC outperforms DPM remarkably. To test how time complexity changes with respect to the number of dimensions in the projected space, we tried different coding spaces and compared our method with DPM, with results shown in Fig. 5. Again, it demonstrates our The complexity comparison between DPM and NMMC in the data projection space. (a) shows how the time complexity varies with the number of training data on the MNIST data set, under the 1-layer DBN with 100 hidden nodes; (b) shows how the time complexity changes with the number of hidden nodes on the 20 newsgroup dataset, under the 1-layer DBN. It shows that our method is more efficient than DPM on the data clustering. Model rand Index F-value train test train test DBN+NMMC (pre-train, n = 200) 0.059 ± 0.02 0.034 ± 0.016 0.131 ± 0.017 0.11 ± 0.012 DBN+NMMC (fine-tune, n = 200) 0.069 ± 0.023 0.065 ± 0.025 0.142 ± 0.019 0.141 ± 0.02 DBN+NMMC (pre-train, n = [1000, 200]) 0.048 ± 0.014 0.028 ± 0.007 0.109 ± 0.005 0.098 ± 0.007 DBN+NMMC (fine-tune, n = [1000, 200]) 0.047 ± 0.015 0.043 ± 0.013 0.108 ± 0.006 0.104 ± 0.004 PCA+NMMC (n = 200) 0.036 ± 0.005 0.016 ± 0.012 0.11 ± 0.005 0.087 ± 0.010 IMRBM [17] (n = 200, K = 20) 0.015 ± 0.005 0.013 ± 0.002 0.096 ± 0.004 0.093 ± 0.004 k-means (K = 20) 0.075 ± 0.02 0.032 ± 0.004 0.140 ± 0.019 0.109 ± 0.016 GMM (K = 20) 0.075 ± 0.021 0.051 ± 0.006 0.140 ± 0.019 0.114 ± 0.016 Spectral Clustering (K = 20) 0.058 ± 0.02 0.061 ± 0.017 0.126 ± 0.013 0.129 ± 0.006 DBN + Kmeans (K = 20) 0.237 ± 0.007 0.06 ± 0.036 0.279 ± 0.008 0.119 ± 0.026 DBN + GMM (K = 20) 0.239 ± 0.009 0.125 ± 0.056 0.281 ± 0.006 0.185 ± 0.045
Figure 6 :
6The performance comparison between DPM and NMMC with the same DBN structure for feature learning on 20 newsgroups. (a) it is a 1-layer DBN (or RBM) with the number of hidden nodes n = 200; (b) it is a 2-layers DBN, with n = [1000, 200] for each layer. It demonstrates that with the same DBN for feature learning, NMMC outperforms DPM remarkably.
Figure 1: In this DBN, L indicates the total number of hidden layers, W i is the weight between adjacent layers, for i = {1, .., L} and Θ is the weight for clustering learned with NMMC.h 1
W 1
V
h L-1
W L-1
W L
Θ
h L
Feature Learning with DBN
NMMC
Fine-tuning Process
This graph
demonstrates 3 steps in our model: (1) Feature learning with deep belief network (DBN), with
weights learned layer by layer as described above; (2) Perform clustering analysis with NMMC,
which can assign a cluster label for each element in the data; (3) Update the model parameters with
fine-tuning process (only for W L and Θ).
.Model
rand Index
F-value
train
test
train
test
DBN+NMMC (pre-train, n = 100)
0.363 ± 0.038
0.181 ± 0.07
0.442 ± 0.032 0.285 ± 0.063
DBN+NMMC (fine-tune, n = 100)
0.371 ± 0.039 0.392 ± 0.043 0.447 ± 0.033 0.467 ± 0.036
DBN+NMMC (pre-train, n = [400, 100])
0.419 ± 0.022
0.232 ± 0.09
0.483 ± 0.02
0.319 ± 0.07
DBN+NMMC (fine-tune, n = [400, 100])
0.428 ± 0.021 0.453 ± 0.02
0.492 ± 0.02 0.513 ± 0.016
DBN+NMMC (pre-train, n = [400, 400, 100])
0.302 ± 0.017 0.218 ± 0.055 0.394 ± 0.014 0.317 ± 0.046
DBN+NMMC (fine-tune, n = [400, 400, 100])
0.309 ± 0.015 0.326 ± 0.015
0.40 ± 0.012
0.415 ± 0.02
DBN+NMMC (pre-train, n = [400, 300, 200, 100]) 0.334 ± 0.05
0.31 ± 0.08
0.423 ± 0.04
0.40 ± 0.07
DBN+NMMC (fine-tune, n = [400, 300, 200, 100])
0.34 ± 0.051
0.364 ± 0.054
0.433 ± 0.04
0.45 ± 0.045
PCA+NMMC (n = 100)
0.381 ± 0.02
0.251 ± 0.022
0.452 ± 0.02
0.353 ± 0.02
IMRBM [17] (n = 100, K = 10)
0.13 ± 0.04
0.10 ± 0.03
0.23 ± 0.02
0.22 ± 0.02
k-means (K = 10)
0.356 ± 0.029
0.367 ± 0.03
0.446 ± 0.026 0.451 ± 0.026
GMM (K = 10)
0.356 ± 0.029
0.394 ± 0.04
0.446 ± 0.025 0.465 ± 0.026
Spectral Clustering (K = 10)
0.354 ± 0.057 0.359 + 0.035 0.423 ± 0.045
0.423 ± 0.03
DBN + kmeans (K = 10)
0.411 ± 0.016 0.316 ± 0.027 0.473 ± 0.015 0.401 ± 0.019
DBN + GMM (K = 10)
0.411 ± 0.016 0.406 ± 0.022 0.473 ± 0.015 0.467 ± 0.024
Table 1: The experimental comparison on MNIST dataset, where "train" means the training data,
"test" indicates the testing data, n specifies the number of hidden variables for each layer (for
example, n = [400, 100] indicates DBN has two layers, the first layer has 400 hidden nodes, and the
second layer has 100 hidden nodes). For PCA+NMMC, we first use PCA project the data into 100
dimensions, then perform NMMC for clustering. It demonstrates that the fine-tuning process in
our model can improve clustering performance greatly, and our method (DBN+NMMC) beats the
baselines remarkably when n = [400, 100].
http://yann.lecun.com/exdb/mnist/
http://www.cs.toronto.edu/~larocheh/public/datasets/20newsgroups/20newsgroups_{train,valid, test}_binary_5000_voc.txt
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| []
|
[]
| [
"M Temmer \nIGAM/Kanzelhöhe Observatory\nInstitute of Physics\nDepartment of Physics and Astronomy\nUniversität Graz\nUniversitätsplatz 5A-8010GrazAustria\n",
"A M Veronig \nIGAM/Kanzelhöhe Observatory\nInstitute of Physics\nDepartment of Physics and Astronomy\nUniversität Graz\nUniversitätsplatz 5A-8010GrazAustria\n",
"Kontar E P \nSpace Sciences Laboratory\nUniversity of Glasgow\nG12 8QQUK\n",
"Krucker S [email protected] \nFaculty of Geodesy\nUniversity of California\nCA94720-7450Berkeley\n",
"B Vršnak [email protected] \nUniversity of Zagreb\nKačićeva 26HR-10000ZagrebCroatia\n"
]
| [
"IGAM/Kanzelhöhe Observatory\nInstitute of Physics\nDepartment of Physics and Astronomy\nUniversität Graz\nUniversitätsplatz 5A-8010GrazAustria",
"IGAM/Kanzelhöhe Observatory\nInstitute of Physics\nDepartment of Physics and Astronomy\nUniversität Graz\nUniversitätsplatz 5A-8010GrazAustria",
"Space Sciences Laboratory\nUniversity of Glasgow\nG12 8QQUK",
"Faculty of Geodesy\nUniversity of California\nCA94720-7450Berkeley",
"University of Zagreb\nKačićeva 26HR-10000ZagrebCroatia"
]
| []
| Using the potential of two unprecedented missions, STEREO and RHESSI, we study three well observed fast CMEs that occurred close to the limb together with their associated high energy flare emissions in terms of RHESSI HXR spectra and flux evolution. From STEREO/EUVI and STEREO/COR1 data the full CME kinematics of the impulsive acceleration phase up to ∼4 R ⊙ is measured with a high time cadence of ≤2.5 min. For deriving CME velocity and acceleration we apply and test a new algorithm based on regularization methods. The CME maximum acceleration is achieved at heights h ≤ 0.4 R ⊙ , the peak velocity at h ≤ 2.1 R ⊙ (in one case as small as 0.5 R ⊙ ). We find that the CME acceleration profile and the flare energy release as evidenced in the RHESSI hard X-ray flux evolve in a synchronized manner. These results support the "standard" flare/CME model which is characterized by a feed-back relationship between the large-scale CME acceleration process and the energy release in the associated flare. | 10.1088/0004-637x/712/2/1410 | [
"https://arxiv.org/pdf/1002.3080v1.pdf"
]
| 119,233,328 | 1002.3080 | 0a41b86a2aa585c36ce4d11cf6441ea012cdf7c2 |
16 Feb 2010
M Temmer
IGAM/Kanzelhöhe Observatory
Institute of Physics
Department of Physics and Astronomy
Universität Graz
Universitätsplatz 5A-8010GrazAustria
A M Veronig
IGAM/Kanzelhöhe Observatory
Institute of Physics
Department of Physics and Astronomy
Universität Graz
Universitätsplatz 5A-8010GrazAustria
Kontar E P
Space Sciences Laboratory
University of Glasgow
G12 8QQUK
Krucker S [email protected]
Faculty of Geodesy
University of California
CA94720-7450Berkeley
B Vršnak [email protected]
University of Zagreb
Kačićeva 26HR-10000ZagrebCroatia
16 Feb 2010Received ; accepted -2 -Combined STEREO/RHESSI study of CME acceleration and particle acceleration in solar flares Subject headings: Sun: coronal mass ejections (CMEs) -Sun: flares -Sun: hard X-rays
Using the potential of two unprecedented missions, STEREO and RHESSI, we study three well observed fast CMEs that occurred close to the limb together with their associated high energy flare emissions in terms of RHESSI HXR spectra and flux evolution. From STEREO/EUVI and STEREO/COR1 data the full CME kinematics of the impulsive acceleration phase up to ∼4 R ⊙ is measured with a high time cadence of ≤2.5 min. For deriving CME velocity and acceleration we apply and test a new algorithm based on regularization methods. The CME maximum acceleration is achieved at heights h ≤ 0.4 R ⊙ , the peak velocity at h ≤ 2.1 R ⊙ (in one case as small as 0.5 R ⊙ ). We find that the CME acceleration profile and the flare energy release as evidenced in the RHESSI hard X-ray flux evolve in a synchronized manner. These results support the "standard" flare/CME model which is characterized by a feed-back relationship between the large-scale CME acceleration process and the energy release in the associated flare.
Introduction
Solar flares and coronal mass ejections (CMEs) are the most violent phenomena in our solar system. Many aspects of the basic physics of these events are still not well understood.
In the "standard" model it is envisaged that the erupting filament or CME stretches the coronal magnetic field lines to build up a vertical current sheet, where magnetic reconnection sets in to explosively release vast amounts of free magnetic energy, previously stored in the corona in non-potential magnetic fields (for a review see e.g. Forbes 2000). The released energy goes into plasma heating, acceleration of particles to suprathermal velocities as well as into kinetic energy of the eruption. In this model, a close relation between the kinematics of the CME and the energy release in the associated flare is expected. A significant fraction of the primary flare energy goes directly into acceleration of fast electrons (e.g. Hudson et al. 1992). As the accelerated electrons precipitate downward along the newly closed magnetic field lines to the lower lying denser atmospheric layers, they are collisionally stopped and heat and ionize the chromosphere and lower transition region.
This can then be observed as enhanced UV and Hα radiation. If the beam flux is high enough, they will also emit detectable hard X-rays (HXRs) via nonthermal bremsstrahlung when the electrons scatter off ions of the ambient thermal plasma. Thus, HXR emission provides the most direct indicator of the evolution of the energy release in a flare (e.g.
Fletcher & Hudson 2001).
Several authors found a correlation between the CME acceleration and the flare soft X-ray emission (Zhang et al. 2001(Zhang et al. , 2004Zhang & Dere 2006;Vršnak et al. 2004;Maričić et al. 2007;Vršnak et al. 2007). In a recent case study by Temmer et al. (2008) an almost synchronized behavior between CME acceleration and flare HXR emission was obtained for two on-disk events. Such results reveal that, for some associated flare-CME events, the reconnection process during the flare is closely related to the CME kinematical -4evolution. The question remains if it is also possible to relate the flare HXR emission (count rate, spectral parameters) to the CME acceleration magnitude. The Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI; Lin et al. 2002) delivers HXR spectra and images at high temporal and spectral resolution, which enables us to study in detail the flare energy release process.
To study the flare energy release in relation to the CME kinematics, the impulsive or main acceleration phase of a CME has to be covered. Since the impulsive acceleration phase of a CME takes place at distances of R 3 R ⊙ (MacQueen & Fisher 1983;St. Cyr et al. 1999;Vršnak 2001;Zhang et al. 2001;Temmer et al. 2008) observations at low coronal heights and of high temporal cadence are required but are only limited available from white light coronagraphs. The Extreme Ultraviolet Imager instruments aboard the twin spacecraft of the Solar Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008) mission have a large field of view and a high time resolution in the 171Å passband, perfectly suiting such studies. Together with COR1 observations, the inner coronagraph aboard STEREO, the impulsive acceleration phase of a CME is fully covered from its launch in the low corona up to 4.0 R ⊙ .
In this paper we study and analyze the CME dynamics and HXR emission of the associated flare for three well observed CME/flare events, using the potential of two unprecedented missions, STEREO and RHESSI. The events are selected to be located close to the limb in order to minimize the effect of projection in the CME kinematics.
In addition, we apply a new algorithm (based on regularization methods) to derive the CME velocity and acceleration from the measured height-time data. A systematic test of the regularization method as well as a least-squares spline algorithm to a variety of synthetic CME acceleration profiles is performed, in order to evaluate the limitations and uncertainties of both methods.
-5 -
Data and Methods
The Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) instrument package is part of each of the STEREO twin spacecraft, STEREO-A(head) and STEREO-B(ehind). It includes among others the Extreme Ultraviolet Imager (EUVI; Wuelser et al. 2004) and the white-light inner coronagraph COR1 (Thompson et al. 2003). EUVI has a field of view (FoV) of 1.7 R ⊙ which enables us to follow the erupting CME structure during its initiation and early propagation phase.
Combining EUVI with COR1 observations, which have a FoV of 1.4-4.0 R ⊙ , we can derive a complete velocity and acceleration profile for the early CME evolution. The partially overlapping FoVs between EUVI and COR1 allow us to check if the same features are followed in both instruments.
Further advantages are the high time cadence of EUVI and COR1 observations which allow us to study the evolution of impulsive CME events low in the corona. EUVI observes in the EUV 171Å passband (Fe x,xi: T ∼ 1 × 10 6 K) with a nominal cadence of 2.5 min;
COR1 has a cadence of 10 min. Each day from 18:00 to 22:00 UT, EUVI and COR1 gather observations with increased cadence, namely 75 sec for EUVI and 5 min for COR1 in order to coordinate with Mauna Loa Solar Observatory as well as during observing campaigns.
For the study of the CME kinematics, we combine EUVI 171Å filtergrams and total brightness images calculated from COR1 polarization sequence triplets. The level-0 data are properly reduced using SolarSoft routines, all images are background subtracted (for COR1 monthly background is subtracted from each polarization component) and rotated to solar north up.
RHESSI performs imaging spectroscopy of solar flares in the energy range from 3 keV to 17 MeV with high temporal, spatial and spectral resolution . The flares under study are fully covered by RHESSI observations during their peak phase which -6enables us to study in detail their HXR spectra and flux evolution in relation to the kinematics of the associated CMEs. RHESSI spectra were integrated over 20 sec during the flare peak with a spectral resolution of 1 keV and fitted with a thermal plus non-thermal power-law component using OSPEX (Schwartz et al. 2002). From the power-law component we derive the amplitude of the HXR photon spectrum at 50 keV, the spectral index γ of the photon spectrum (slope of the power law), and the power in electrons above a cutoff energy of 25 keV for thick-target emission (Brown 1971). RHESSI images were reconstructed during the flare peak using the CLEAN algorithm including front detectors 2 to 8 (except 5 and 7).
Derivation and test of CME acceleration profiles
For our study the reliable derivation of CME velocity and acceleration profiles from the measured height-time (HT) data is crucial. The central problem is to properly smooth the noisy data and to estimate the impact of measurement errors on the derived quantities, especially on the acceleration profile. This issue received a lot of attention for deriving CME kinematical curves from SOHO/LASCO data (see Wen et al. 2007). Here, we apply a new type of technique to derive velocity and acceleration profiles from HT measurements, namely an enhanced regularization algorithm originally developed by Kontar et al. (2004) to invert solar X-ray spectra measured by RHESSI.
The main problem of time derivatives of data with measurement errors is that its finite difference estimate always leads to an amplification of the error. This can be expressed as the sum of discretization (finite time cadence) and propagation errors (Groetsch 1984).
The former error is proportional to the time cadence of the measurements, while the latter is inversely proportional to it. Therefore, the total derivative error always has a minimum (Hanke & Scherzer 2001). Following Hanke & Scherzer (2001), and Kontar & MacKinnon -7 -(2005) the regularized method searches for a model-independent velocity and acceleration estimate (regularized solution) with a minimum error, which produces smooth derivatives and avoids additional errors typical of finite differences. In addition, the regularized method of Kontar & MacKinnon (2005) provides the confidence interval for the first and second derivative, i.e. for CME velocity and acceleration profiles. The outcome of the regularization algorithm is controlled by two parameters. The number of time bins (which has a smoothing effect) and the regularization "tweak", an important parameter that regulates how reliable are the uncertainties of the input data (e.g. "tweak" of one means that the errors follow a normal distribution without systematics or correlation).
To test the reliability of the results we compare the CME acceleration profiles 1 derived by the regularization algorithm with those derived from a least-squares spline fit method which is an advanced method already used in previous CME studies (Maričić et al. 2004;Vršnak et al. 2007). (Note that we did not compare less sophisticated methods, like direct derivation of the HT data or 3-point Lagrangian interpolation since these would give worse results.) Based on the formulas given in Gallagher et al. (2003), we generate synthetic kinematical curves of CME propagation representing different scenarios of CME evolution (impulsive and gradual acceleration profiles).
We then reduce the sampling rate of the generated curve and add some noise. The noise consists of random noise as well as of manually shifting individual data points. With this we aim to take into account the decrease in the dynamic range of CME observations from the inner to the outer corona as well as inconsistencies of tracking features across different wavelength regimes (EUV and white light). From these "imperfect" data points 1 Deriving the acceleration profile, i.e. calculating the second derivative, highly intensifies noise on the data. The outcome of the acceleration is critical in order to reliably compare with HXR emission.
-8the acceleration profile is derived and compared to the acceleration profile from the original "perfect" synthetic curve using the regularization and the spline method. To test the stability of the two methods, we simulate HT data with different errors and sampling rates.
For both methods the normalized residuals are derived, i.e. the differences between the measured HT values and the values resulting from the fitting procedure and the regularized solution, respectively, divided by the error.
In addition to the regularization method we use the spline fit routine which computes least squares splines with equally spaced nodes to the HT data points (SPLFIT in IDL).
From the spline fit curve, we numerically derive the second order derivative to determine the acceleration. The smoothing effect of the spline fit is controlled by the number of nodes n (cf. Vršnak et al. 2007). To illustrate the way in which this affects the shape of the derived acceleration curve, we present the results of the spline fit for three different numbers of nodes (n − 1, n, n + 1). The nodes are chosen in such a way that the residuals for each of the curves would be equally small.
We first test the reliability of the methods and derive the acceleration from the synthetical HT data without noise ( Fig. 1 left panels). For comparison, the right panels of Fig. 1 show the results derived from HT data with noise added. We note that the second part of the synthetic curve (decreasing height) is unrealistic for a CME evolution, but the profile was used to provide a challenge to both methods. As can be seen from the left panels, without noise the outcome from the regularization tool represents the acceleration curve better than that from the spline fit with respect to the timing of the acceleration peak and the acceleration duration, though the spline fit method reproduces the HT curve quite well. The peak amplitude of the CME acceleration is well represented by the regularization tool, whereas it is underestimated (by about 20%) by the spline fit method. In addition, the regularization tool reveals horizontal error bars (±1.0 min) to account for the uncertainty in -9timing due to the reduced sampling rate. From the right panels of Fig. 1 it can be seen that the acceleration curve derived from the spline fit applied on noisy data has only slightly changed, whereas those from the regularization tool did since it considers the information provided by the error bars to yield an adequate error estimation which we do not get from the spline fit method. Strictly speaking, as soon as we add noise to the true HT data, the "true" velocity or acceleration cannot be reproduced. So the prime objective of any method is to provide the range of velocities/accelerations where the "true" solution should be. Figure 2 shows the outcome for a HT curve of an impulsively accelerated CME evolution with large noise added (average error of 0.33 R ⊙ , i.e. larger than typically derived from real observations). We see that the acceleration peak is neither well represented by the regularization tool nor by the spline fits. However, the intrinsic errors in time which we get from the regularization tool (±2.0 min) include the true solution. The overall shape, i.e. the duration of acceleration, is well represented by both methods.
Figure 3 (left) shows a CME evolution curve with small errors (±0.07 R ⊙ ) and a sampling rate increasing from 2 to 10 minutes, simulating a cadence of measurement points comparable to the actual observations in the present study. The spline fit acceleration varies only little when applying different numbers of nodes (6,7,8) and matches well the peak but not the duration of acceleration. The outcome of the regularization tool for the most part matches the true solution within its error bars (uncertainty in the timing of the acceleration peak about ±3.0 minutes). The right panel of Fig. 3 shows the same kinematical curve, however, with a higher sampling rate of 1 and 2 min, respectively. The acceleration peak derived from the spline fit using different nodes shows a larger discrepancy. The timing of the acceleration peak calculated from the regularization tool is consistent with the true curve. However, its uncertainty is estimated to about ±4.0 min. To summarize, a sampling rate of ∼2 min covering the impulsive acceleration phase reproduces the true curve in a -10reasonable way. Higher time cadence data produce a better estimate of the acceleration duration. Somewhat surprisingly, the horizontal uncertainties on the acceleration peak are larger when a higher sampling rate is used (<2 min). This is due to the fact that small time steps between the HT data points with errors cause larger uncertainties in the subsequent derivative. Thus, the enhancement of the time cadence can only improve the acceleration profiles when also the errors in the CME HT measurements are reduced. In EUVI 171Å (T ∼ 1 × 10 6 K) typical CME features as the frontal rim and the cavity are observed (see also Aschwanden et al. 2009). However, the embedded cooler prominence material (T ∼ 1 × 10 4 K), visible in the subsequent coronagraph images, is missing. This is reasonable, since the contrast of prominences against the coronal background drops sharply for lines formed at T ≥ 3 × 10 5 K (Noyes et al. 1972). The obvious similarity of the morphology and the temporal evolution of the developing/propagating CME structure justifies the link between EUV and coronagraph observations (for a discussion see Maričić et al. 2004). By combining measurements of the leading edge of the erupting structure from EUVI and COR1 images we derive the kinematics of the CME with typical uncertainties of ±0.02-0.05 R ⊙ for EUVI and ±0.12 R ⊙ for COR1. We stress that for each event under study the location of the CME source (estimated by the flare position) is close to the limb (±10-25 • ). Thus, we expect that the influence of projection effects on the derived kinematics is small. We also note that for all events large-scale coronal waves were together with its spectrum (bottom) integrated over 20 sec around the peak of the HXR emission. The spectrum was fitted with a thermal plus non-thermal model in the energy range 6-200 keV. The HXR source, located close to the eastern limb, is compact with some indication of two footpoints (but not clearly resolved). From the RHESSI spectral fit we derive a photon spectral index γ = 2.4 which indicates a very flat (i.e. hard) spectrum and a photon flux density at 50 keV during the event peak, F 50 = 0.93 photons s −1 cm −2 keV −1 . Figure 6 shows the CME distance, velocity, and acceleration time-profiles for the distance range 1.0-3.3 R ⊙ (left panels from top to bottom). To enlarge the details on the impulsive acceleration phase, the right panels of Fig. 6 show the same curves but for the distance range up to 1.7 R ⊙ . For comparison, the RHESSI HXR flux of the associated flare in the energy range 30-50 keV is overplotted in the bottom panels of Fig. 6. RHESSI reveals impulsive and powerful HXR emission with a distinct burst of short duration (less than 2 min). The evolution of the acceleration profile of the CME and the evolution of the flare HXR flux are highly synchronized peaking at 09:27 UT with an uncertainty of ±1.25 min -12 -(CME) and 09:27:14 UT (RHESSI 30-50 keV HXR flux). The CME reveals a large peak velocity (∼1170 km s −1 ) and peak acceleration (∼5.1 km s −2 ), which occur within the EUVI FoV, i.e. below 1.7 R ⊙ . The CME accelerates within 8 min up to ∼1170 km s −1 , then drops within the next 2 min down to a velocity of ∼700 km s −1 and further decelerates over the following 10 min until reaching a constant velocity of ∼500 km s −1 . If one would determine the CME speed solely from coronagraph observations, the velocity peak would be missed. (Masuda et al. 1994). For that day, MESSENGER (Solomon et al. 2001) is at a heliographic longitude of about E165 and observed the non-occulted flare in soft-X rays (similar to GOES). The flux measured by MESSENGER is about 2.8 times higher than the flux observed by GOES which makes the flare actually of M2 class (Krucker et al. 2010). Figure 8 shows the RHESSI HXR image reconstructed in the energy band 20-50 keV (top) together with its spectrum (bottom) integrated over 20 sec around the peak of the HXR emission and fitted in the energy range 6-90 keV. The image reveals that the strongest HXR emission comes from above the eastern limb, hence the footpoints of the flare are occulted. This means that the HXR flux entirely originates from the coronal source (e.g. Krucker et al. 2008). For this event we derive the RHESSI spectral fit parameters as γ = 4.5, which means a softer HXR spectrum compared to the previous event, and -13 -F 50 = 0.17 photons s −1 cm −2 keV −1 . The considerably lower photon flux density and the softer spectrum as compared to the 2007 June 3 event are not surprising since loop-top sources are generally less bright than the footpoints (Krucker & Lin 2008).
At that day, STEREO-B had a position angle of 23 • with respect to Earth, hence the CME was observed 11 • off the STEREO-B plane-of-sky. Figure 9 shows the CME distance-time, velocity, and acceleration profile for the measured distance range 1-5 R ⊙ (left panels). The closeup view (right panels) shows the distance range up to 2.5 R ⊙ . We stress that for this event one data point (COR1) in the overlapping FoVs of EUVI and COR1
(1.4-1.7 R ⊙ ) reveals a shift between the both instruments, but which lies within the applied error bars. The CME reaches a maximum velocity of ∼790 km s −1 within <10 min, with a maximum acceleration of ∼1.3 km s −2 . The impulsive acceleration phase of the CME is finished within the EUVI FoV and shows a peak at 00:50 UT (±2.5 min) which (within the uncertainties) is in accordance with the HXR evolution and peak time at 00:47:48 UT ( Fig. 9, bottom panels). This is in particular interesting, as in this event we do not observe the HXR emission from the footpoints (which are occulted) but from a coronal source.
2008 March 25: M1.7 flare/CME event
The 2008 March 25 CME event is associated with a M1.7 GOES class flare. The source region lies at S10E87 (EUVI event catalog; Aschwanden et al. 2009) and the position angle of STEREO-B with respect to Earth is 24 • , i.e. the CME propagates about 27 • off the STEREO-B plane-of-sky. The event is observed with high cadence by EUVI and COR1 (75 sec and 5 min, respectively). RHESSI observed the peak phase of the flare HXR emission, but missed part of the rising phase due to spacecraft night. Figure 11 shows the RHESSI HXR image in the energy band 18-30 keV (top) together -14with its spectrum (bottom) integrated over 20 sec around the peak of the HXR emission.
The spectrum was fitted in the energy range 6-100 keV, giving γ = 3.5 and F 50 = 0.23 photons s −1 cm −2 keV −1 . The image shows a rather compact source of HXR emission revealing a weaker component in the corona in addition to the on-disk footpoint emission, suggesting that the second footpoint lies occulted behind the eastern limb (the energy range 18-30 keV is dominated by non-thermal emission, see spectrum in Fig. 11). In EUVI 304Å images a bright ejection is observed which may coincide with the coronal RHESSI source. Figure 10 shows the sequence of STEREO-B EUVI running difference and COR1
images. The images reveal a very thin and distinct spherically shaped front of the CME, reminding of the cross-section of a flux-rope torus. This shape is maintained in COR1 white-light images (see also Aschwanden et al. 2009). The distance-time, velocity, and acceleration profiles are shown in Fig. 12, overplotted with the RHESSI HXR flux in the energy range 18-30 keV. The COR1 data point lying in the overlapping FoVs of EUVI and COR1 fits well to the CME HT data measured by EUVI. Due to RHESSI night, the initial flare phase is missed, but the peak is clearly observed. RHESSI comes out of spacecraft night at ∼18:44 UT revealing enhanced HXR emission with a maximum at 18:51:34 UT.
The CME velocity starts to increase at 18:35 UT, the acceleration reaching a peak at 18:50 UT (±3.5 min). We note that the acceleration time-profile shows two acceleration steps within the EUVI FoV. The CME reaches a peak velocity of 970 km s −1 within ∼20 min with an acceleration maximum of ∼1 km s −2 .
Summary & Discussion
Both the spline fit and the regularization tool determine the acceleration phase of a CME (peak and duration) reasonably well. The main plus of the regularization tool over -15the spline fit method is the provision of errors in time and amplitude. In all test cases, the true solution was included within this error estimation. We come to the somewhat surprising result that an increased image cadence (<2 min) does not reduce the uncertainty in determining the peak time of CME acceleration. This is due to the measurement errors of the CME leading edge. The inconsistencies of the CME leading edge as measured in different wavelength regimes (EUV and white light) as well as the change of intensity over the FoV of a single instrument do not allow us to narrow the measurement errors. The same holds for the acceleration amplitude for which the uncertainties lie in the range between 10% and 50%. As a check for the goodness of the obtained velocity and acceleration profiles from the regularization tool, the normalized residuals between the regularized solutions and the distance-time measurements are investigated and found to be small (≤1.5). Table 1 summarizes the relevant parameters derived from the CME and flare. From the three events under study we obtain that the peak acceleration of the CME is reached within a few minutes after its launch and at low distances of ≤0.4 R ⊙ from the flare site, which we use as an estimate of the CME source region location. We would like to note that all events occurred close to the limb. Thus, projection effects are small, and the derived values should be close to the "true" ones. In these events, the impulsive acceleration phase is already finished before the CME is observed in the coronagraph COR1 FoV. The peak velocity of the CME is reached within heights of <2.2 R ⊙ above the source region. The CME acceleration profile and the flare HXR emission are highly synchronized and peak almost simultaneously. The differences between the flare HXR peak and CME acceleration peak are in the range ∆t ≤ 2 min. Such differences lie within the limitations for deriving the CME acceleration peak time, which are caused by the finite cadence and measurement errors in the HT measurements. The uncertainties in the peak time of the CME acceleration estimated from the inversion method lie in the range 2-3.5 min (see Section 3).
-16 -We note that there is no clear relation between CME velocity and GOES class of the associated flare. For the events under study these parameters vary only by a factor of ∼2. Statistical studies comparing the flare SXR peak flux and CME peak velocity find a linear correlation coefficient of about r ∼0.47 which is not very strong (Moon et al. 2002).
However, in our three events there seems to be a correlation indicated between the CME acceleration peak and the flare HXR peak flux, which both vary over a factor of 5 in the three events under study. Also some relation between the spectral slope of the HXR spectra and the CME acceleration is suggested (harder flare spectrum seems to be related to larger CME acceleration). Of course, three events are inconclusive for such a relationship but our results suggest that this should be tested on a larger event sample.
Especially for the 2007 June 3 event intriguing results are derived. The single HXR burst of only ∼2 min duration goes along with a rather high HXR flux density, CME peak acceleration (acceleration phase of ∼8 min), and a very hard HXR spectrum. In contrast to that, a relatively low total energy in flare-accelerated electrons and low GOES classification is revealed. Further, the 2007 June 3 CME shows quite unusual behavior with respect to its very strong deceleration profile with values of −3 km s −2 and the velocity decreases from 1100 km s −1 to 500 km s −1 very low in the corona (<0.5 R ⊙ ). Typical CME deceleration values are in the order of −0.01 to −0.1 km s −2 (e.g. Vršnak et al. 2004a) and are due to the interaction with the solar wind flow. After the magnetic driving force of the eruption ceases, the CME is slowed down by the drag force until its velocity adjusts to the speed of the solar wind (e.g. Cargill 2004;Manoharan 2006;, and references therein). Reeves (2006) studied the relationship between the CME acceleration and the thermal energy release rate in a loss-of-equilibrium flux rope model (cf. Lin & Forbes 2000;Lin 2002). They found that for high background magnetic fields and fast reconnection rates the evolution of the CME acceleration and flare energy release rate are well correlated. Reeves (2006) also points out that due to rapid reconnection the formation of a long current -17sheet in the wake of a CME is prevented and consequently the energy release is inhibited.
For the 2007 June 3 event, we therefore assume a localized strong magnetic field from which the CME erupts as well as strong overlying fields. On the one side this yields high reconnection rates over a short time range, on the other side a strong overlying magnetic field would drastically decelerate the CME at low coronal heights.
Well observed flare/CME events close to the limb, such as presented in this study, are less affected by projection effects when analyzing the CME parameters and enable us to reliably determine its kinematics and acceleration values. However, close to the limb the associated flare is observed either as "classical" on-disk HXR emitting source (footpoints of flare loops at chromospheric heights) or as coronal loop-top emission only, when the bright footpoint emission is occulted by the solar limb. In partially disk-occulted events, loop-top sources could be studied on a statistical basis (cf. Krucker & Lin 2008) and it is most probable that they are related to the initial location of particle acceleration (e.g. Krucker et al. 2008). Battaglia & Benz (2006) studied five RHESSI flares where HXR emission from both footpoints and the loop-top was observed. They found that the coronal and footpoint sources are well correlated with respect to their temporal and spectral evolution. Our results show in fact that the CME acceleration peak phase is well correlated with both types of HXR sources, i.e. coronal and footpoint HXR emission of the associated flare.
In the present study we found for all three events under study, that the impulsive acceleration of the CME is finished at distances of ≤0.4 R ⊙ above the solar surface, i.e. within the EUVI FoV and before reaching the COR1 FoV. Hence, from coronagraphic observations only, the early CME evolution may be considerably misinterpreted. Vršnak (1990) and Chen & Krall (2003) obtained from analytical models, in which CMEs are treated as toroidal field structures, that the peak of the acceleration should be reached -18when the half-separation of the CME footpoints is comparable to the CME height above the flare site. For the 2008 March 25 event the CME footpoint separation can be derived with adequate accuracy (see Fig. 7). We obtained a CME half-separation of ∼210±30 Mm, which is comparable to its peak height of ∼280 Mm.
Our results point to a possible relation between the flare HXR spectra and the CME acceleration (the event with the highest CME acceleration has the highest HXR flux density and flattest spectrum) whereas the relation to the peak power in electrons and the GOES soft X-ray peak flux seems weak. Maričić et al. (2007) pointed out that the reconnection rate is more relevant for the CME acceleration than a strong heating and non-thermal particle acceleration. From our results it seems that the efficiency of accelerating particles to high energies is better correlated to the CME acceleration than the total number and energy in electrons (though we note that more statistics on this is needed). Such results may provide new constraints on electron acceleration models and its magnetic geometry as the particle acceleration mechanism has to go along with a rapid closing of magnetic field lines which is needed to drive the CME.
Conclusions
The present study based on high cadence data from STEREO-EUVI, STEREO-COR1, and RHESSI evidences a very close relation between flares and CMEs. From three well observed impulsive events that occurred close to the limb, we derive that the CME acceleration profile and energy release of the associated flare evolve in a synchronized manner. This supports the "standard" flare/CME model which predicts a feed-back relationship between the large-scale CME acceleration and the energy release process in the associated flare (e.g. Lin 2004;Maričić et al. 2007;Temmer et al. 2008;Vršnak & Cliver 2008): After the magnetic structure looses equilibrium and starts rising, a current sheet is -19formed below the rising structure (presumably a flux-rope), becoming a site of magnetic field reconnection. The reconnection has two important consequences for the CME acceleration:
it reduces the downward-acting tension of the overlying field (Lin 2004), and it supplies additional poloidal flux to the flux rope , which is considered to be the main driver of the eruption (e.g. Chen 1989;Vršnak 1990;Chen 1996;Kliem & Török 2006;Subramanian & Vourlidas 2007). On the other hand, the upward moving CME drives mass inflow into the current sheet, driving further magnetic reconnection and particle acceleration (see the cartoon in Fig. 13). Thus, reconnection directly relates the CME acceleration and the energy release in the associated flare, leading to the close synchronization of these two phenomena.
The results from the presented study provide strong evidence for the feed-back mechanism between the flare energy release and the CME acceleration. In our study, all events were fast eruptions, hence, we can not draw conclusions on other types of eruptions (failed eruptions or gradual CMEs without flares). According to model calculations by Reeves (2006) a good correlation (within 2 min) between the flux rope acceleration and thermal energy release rate is expected for fast reconnection events with high background magnetic fields. This should relate to impulsive and strong events, whereas no synchronization is expected for weak and gradual events. Finally, we note that high cadence observations of CMEs in non-coronagraphic images enable us to study in detail the CME impulsive acceleration phase revealing peak acceleration as high as ∼5 km s −2 even for C-class flare events. We also note that the height of the CME peak acceleration in such impulsive events is much lower (≤0.4 R ⊙ ) than previously assumed. shows the regularized solution returned from the inversion technique. The grey shaded area in the CME velocity and acceleration curves indicates the 95% confidence level. quence of EUVI 171Å running difference images and COR1 images. Crosses indicate the measured leading edge from which the CME kinematics is derived (Fig. 9). leading edge from which the CME kinematics is derived (Fig. 12).
-36 - -38 - Fig. 13.-Cartoon illustrating the flare-CME feedback mechanism between the large-scale CME dynamics and small-scale flare processes. The upward moving CME evacuates the area in its wake, boosting mass inflow from aside into the reconnection region. The more mass and frozen-in magnetic field is transported into the region, the higher is the magnetic reconnection rate leading to larger flare energy release and to more efficient acceleration of particles. The successive closing of magnetic field lines due to reconnection increases the poloidal flux B φ in the eruption, which leads to a stronger upward oriented magnetic driving force (Lorentz force).
4. CME acceleration compared to flare energy release The flare/CME events under study are: 2007 June 3 (C5.3), 2007 December 31 (C8.3), 2008 March 25 (M1.7). Figures 4, 7, and 10 show for the three events the evolution of the erupting CME in the low corona in EUVI 171Å and white light images from COR1. In both instruments we follow the leading edge of the CME as indicated by crosses in the figures.
3 CME event is associated with a C5.3 GOES class flare at heliographic position S08E67 (EUVI event catalog;Aschwanden et al. 2009). The position angle of STEREO-A with respect to Earth is 7 • , hence for STEREO-A the CME was observed ∼16 • off the spacecraft plane-of-sky. The event was observed by EUVI with a high cadence of 75 sec.
Figure 5
5shows the RHESSI HXR image (top) reconstructed in the energy band 30-50 keV with the CLEAN algorithm
31 CME event is associated with a C8.3 GOES class flare. The source region lies behind the limb at roughly S09E102(Krucker et al. 2010;Dai et al. 2010) which makes this event special with respect to observations from Earth-view (GOES, RHESSI). From the RHESSI perspective the footpoints of the flare are occulted and we actually observe a coronal HXR source at non-thermal energies, with a good count statistics in the range 20-50 keV. This might be interpreted as Masuda type source
Fig. 2 .Fig. 5 .
25-Top to bottom: same as in Fig. 1. Left: Impulsively accelerated CME evolution with rather high noise and errors. Right: Close up view for the time range during the peak acceleration phase.-28 -Fig. 3.-Top to bottom: same as in Fig. 1. Realistic scenario of CME evolution applying low noise to the data and error bars comparable to those from the observations. Left: Simulated is a time cadence of data points comparable to the actual observations. Right: same curve but with higher time cadence. -29 -Fig. 4.-2007 June 3, C5.3 flare/CME event observed with STEREO-A. Sequence of EUVI 171Å running difference images and COR1 total brightness images. Crosses indicate the measured leading edge from which the CME kinematics (shown in Fig. 6) is derived. -2007 June 3 C5.3 flare. Top: RHESSI 30-50 keV HXR image integrated over 20 sec around the flare peak using front detectors 2 to 8 (except 5 and 7) reconstructed with the CLEAN algorithm. Bottom: RHESSI spectrum and fit components for the energy range 6-200 keV. The isothermal fit is indicated as dashed-dotted line, the non-thermal power-law fit as dashed line, and the sum of both components as thick gray line. The derived parameters, electron flux density at 50 keV F 50 [photons s −1 cm −2 keV −1 ], photon spectral index γ, and power P 25 [erg s −1 ] in electrons above a cutoff energy of 25 keV are given in the legend.
Fig. 6 .
6-CME kinematics for the 2007 June 3 C5.3 flare/CME event. Top to bottom: CME distance, velocity, and acceleration against time together with the background subtracted flare HXR flux (red). Left: full height range, right: closeup view onto the early evolution phase. The pink dashed line in the right top panel (CME distance-time measurements)
Fig. 7 .
7-2007 December 31, C8.3 (M2) flare/CME event observed with STEREO-B. Se-
Fig. 8 .Fig. 9 .
89-Same as in Fig. 5 but for the 2007 December 31 C8.3 (M2) flare. The HXR image is integrated over the energy band 20-50 keV. -Same as Fig. 6 but for the 2007 December 31 C8.3 (M2) flare/CME event. The gray bar indicates the time range of RHESSI night (N). -35 -Fig. 10.-2008 March 25, M1.7 flare/CME event observed with STEREO-B. Sequence of EUVI 171Å running difference images and COR1 images. Crosses indicate the measured
Fig. 11 .
11-Same as in Fig. 5 but for the 2008 March 25 M1.7 flare. The HXR image is integrated over the energy band 18-30 keV. Contours show levels of 50%, 70%, and 90% of maximum emission for the energy band 30-50 keV. -37 -Fig. 12.-Same as Fig. 6 but for the 2008 March 25 M1.7 flare/CME event. The gray bar indicates the time range of RHESSI night (N).
Table 1 :
1Summary of the CME and flare characteristics for each event. We give the date, the GOES flare class, the difference ∆t [min] between CME peak acceleration and flare HXR peak, the derived CME peak velocity v max [km s −1 ] and peak acceleration a max [km s −2 ] with errors (95% confidence level), the CME height h [R ⊙ ] from the source region, the electron flux density at 50 keV, F 50 [photons s −1 cm −2 keV −1 ], the flare HXR spectral slope γ, and the total power in electrons P 25 [10 27 erg s −1 ].Fig. 1.-Top: Synthetic curve of CME kinematics (black) together with the data (asterisks), results derived from the regularization tool (dashed blue) and the spline fit (pink). Middle:derived acceleration profile (black: true solution, blue: regularization tool result, pink: spline fit result). Bottom: normalized residuals (we note that in the left panel case the residuals make no sense due to the division by errors of the order of zero). Left: sampling rate is smaller than for the "true" curve, but the data are without noise. Right: same sampling rate as left but with noise added to the data.Date
Class
∆t v max (h)
a max (h)
F 50 γ
P 25
03-Jun-07 C5.3
0.1 1171±359 (0.48) 5.1±2.4 (0.26) 0.93 2.4 1.75
31-Dec-07 C8/M2 2.0 785±125 (2.10) 1.3±0.4 (0.25) 0.17 4.5 4.09
25-Mar-08 M1.7
1.5 967±173 (1.97) 1.0±0.2 (0.40) 0.23 3.5 1.88
The authors appreciate and thank the anonymous referee for his/her constructive
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| []
|
[
"High-Temperature Decomposition of Diisopropyl Methylphosphonate (DIMP) on Alumina: Mechanistic Predictions from Ab Initio Molecular Dynamics",
"High-Temperature Decomposition of Diisopropyl Methylphosphonate (DIMP) on Alumina: Mechanistic Predictions from Ab Initio Molecular Dynamics"
]
| [
"Sohag Biswas \nDepartment of Chemical & Environmental Engineering\nMaterials Science & Engineering Program\nDepartment of Physics & Astronomy\nDepartment of Chemistry\nUniversity of California-Riverside\n92521RiversideCaliforniaUSA\n",
"Bryan M Wong [email protected] \nDepartment of Chemical & Environmental Engineering\nMaterials Science & Engineering Program\nDepartment of Physics & Astronomy\nDepartment of Chemistry\nUniversity of California-Riverside\n92521RiversideCaliforniaUSA\n"
]
| [
"Department of Chemical & Environmental Engineering\nMaterials Science & Engineering Program\nDepartment of Physics & Astronomy\nDepartment of Chemistry\nUniversity of California-Riverside\n92521RiversideCaliforniaUSA",
"Department of Chemical & Environmental Engineering\nMaterials Science & Engineering Program\nDepartment of Physics & Astronomy\nDepartment of Chemistry\nUniversity of California-Riverside\n92521RiversideCaliforniaUSA"
]
| []
| The enhanced degradation of organophosphorous-based chemical warfare agents (CWAs) on metal-oxide surfaces holds immense promise for neutralization efforts; however, the underlying mechanisms in this process remain poorly understood. We utilize large-scale quantum calculations for the first time to probe the high-temperature degradation of diisopropyl methylphosphonate (DIMP), a nerve agent simulant. Our Born-Oppenheimer molecular dynamics (BOMD) calculations show that the γ-Al 2 O 3 surface shows immense promise for quickly adsorbing and destroying CWAs. We find that the alumina surface quickly adsorbs DIMP at all temperatures, and subsequent decomposition of DIMP proceeds via a propene elimination. Our BOMD calculations are complemented with metadynamics simulations to produce free energy paths, which 1 arXiv:2203.08035v1 [physics.chem-ph] 15 Mar 2022show that the activation barrier decreases with temperature and DIMP readily decomposes on γ-Al 2 O 3 . Our first-principle BOMD and metadynamics simulations provide crucial diagnostics for sarin decomposition models and mechanistic information for examining CWA decomposition reactions on other candidate metal oxide surfaces. | 10.1021/acs.jpcc.1c05632 | [
"https://arxiv.org/pdf/2203.08035v1.pdf"
]
| 244,582,092 | 2203.08035 | 255c9391e1e1436f49020a3917154b818e0523ab |
High-Temperature Decomposition of Diisopropyl Methylphosphonate (DIMP) on Alumina: Mechanistic Predictions from Ab Initio Molecular Dynamics
Sohag Biswas
Department of Chemical & Environmental Engineering
Materials Science & Engineering Program
Department of Physics & Astronomy
Department of Chemistry
University of California-Riverside
92521RiversideCaliforniaUSA
Bryan M Wong [email protected]
Department of Chemical & Environmental Engineering
Materials Science & Engineering Program
Department of Physics & Astronomy
Department of Chemistry
University of California-Riverside
92521RiversideCaliforniaUSA
High-Temperature Decomposition of Diisopropyl Methylphosphonate (DIMP) on Alumina: Mechanistic Predictions from Ab Initio Molecular Dynamics
The enhanced degradation of organophosphorous-based chemical warfare agents (CWAs) on metal-oxide surfaces holds immense promise for neutralization efforts; however, the underlying mechanisms in this process remain poorly understood. We utilize large-scale quantum calculations for the first time to probe the high-temperature degradation of diisopropyl methylphosphonate (DIMP), a nerve agent simulant. Our Born-Oppenheimer molecular dynamics (BOMD) calculations show that the γ-Al 2 O 3 surface shows immense promise for quickly adsorbing and destroying CWAs. We find that the alumina surface quickly adsorbs DIMP at all temperatures, and subsequent decomposition of DIMP proceeds via a propene elimination. Our BOMD calculations are complemented with metadynamics simulations to produce free energy paths, which 1 arXiv:2203.08035v1 [physics.chem-ph] 15 Mar 2022show that the activation barrier decreases with temperature and DIMP readily decomposes on γ-Al 2 O 3 . Our first-principle BOMD and metadynamics simulations provide crucial diagnostics for sarin decomposition models and mechanistic information for examining CWA decomposition reactions on other candidate metal oxide surfaces.
Introduction
The neutralization of chemical warfare agents (CWAs) continues to be a pressing area of interest for the safe and effective removal of these hazardous compounds. Among the various CWAs, the most nefarious are organophosphate nerve agents (such as sarin), which contain P=O, P-O-C, and P-C bonds that enable lethal phosphorylating mechanisms. 1 Over the past few decades, a variety of destruction and neutralization methods have been used to safely eliminate these hazardous compounds. For example, destruction-based methods (typically pyrolysis) allow a one-step approach for the complete disposal of CWAs at the expense of using specialized equipment under extreme conditions. 1,2 On the other hand, neutralization methods offer potentially reversible chemical treatments, leading to possible CWA precursors under less severe conditions. 1 Recent studies have shown that metal oxides can effectively destroy CWAs due to their high surface area, a large number of highly reactive edges, corner defect sites, unusual lattice planes, and high surface-to-volume ratios. In particular, metal oxides, such as CaO, 3,4 MgO, 5,6 ZnO, 7-9 TiO 2 , 10-16 Al 2 O 3 , 17-20 Fe 3 O 4 , 21 and CuO 22 are candidates as adsorbents for enhancing the decomposition of CWAs. In γ-Al 2 O 3 , Al atoms in the bulk exhibit either a tetrahedral or octahedral coordination. However, depending on the exposed crystallographic surface, Al atoms on the surface can display penta-, tetra, and tri-coordination and exhibit Lewis acidity. [23][24][25][26] Because of its high degree of surface heterogeneity, γ-Al 2 O 3 offers a high catalytic activity and is a promising candidate for the decomposition of various CWAs. 27 In this work, we present the first ab initio molecular dynamics study for probing hightemperature decomposition dynamics of diisopropyl methylphosphonate (DIMP) on the γ-Al 2 O 3 surface. Due to its structural similarity with sarin, DIMP has been used in experiments to mimic the decomposition reaction mechanism of CWAs. DIMP is the only surrogate with the same isopropyl (-O-C 3 H 7 ) group found in sarin gas (the only structural difference between sarin and DIMP is the substitution of a fluorine for an isopropyl group in the former). Several techniques have also been used to probe DIMP decomposition, including microwave discharge approaches, 28 pyrolysis on porous substrates, 29 laser heating, 30 and thermal decomposition at 700 -800 K. 31 Despite these experimental studies, theoretical analyses of DIMP decomposition on metal oxides are scarce, except for a previous study on mechanism and rates of thermal decomposition of DIMP. 32 In this work, we utilize largescale ab initio molecular dynamics simulations to probe the adsorption dynamics and time scales of DIMP decomposition at various temperatures. In addition, we present new ab initio-based metadynamics simulations to calculate free-energy barriers for various bond breaking decomposition reactions of DIMP. These computational techniques allow us to (1) predict activation energies and detailed mechanistic pathways at various temperatures and (2) establish accurate sarin decomposition models on metal-oxide surfaces to guide experimental efforts for neutralizing DIMP.
Simulation Details
Molecular Dynamics Simulations
Density functional theory (DFT) based Born-Oppenheimer molecular dynamics (BOMD) simulations were carried out using the CP2K 33 software suite. We have specifically chosen to use this software package since the implementation of linear-scaling Kohn-Sham approaches in CP2K allows robust and efficient electronic structure calculations for large systems. The Perdew-Burke-Ernzerhof (PBE) 34 was used for the DFT calculations with Grimmes's D3 method to account for dispersion forces. 35 To obtain reasonable accuracy, we utilized a DZVP (double zeta valence polarized) basis set for Al in the DFT calculations and the TZV2P basis set for C, O, H, and P atoms with the Goedecker, Teter, and Hutter (GTH) pseudopotentials 36,37 for atomic core electrons. A similar basis set for Al atoms was also used to accurately calculate methane activation 38 and alkane dehydrogenation 39 on the γ-
Al 2 O 3 surface.
The orbital transformation method with an electronic gradient tolerance value of 1 × 10 −5 atomic units was adopted as the convergence criteria for the SCF cycle. 40 A kinetic energy cut-off of 400 Ry for the auxiliary plane-wave basis with a 0.5 fs timestep was employed to integrate the equations of motion. The initial guess was furnished by the stable predictor-corrector extrapolation method at each molecular dynamics step. 41,42 We have carried out several molecular dynamics simulations in the 200 -1000 • C temperature range in increments of 100 • C (i.e., a total of nine independent trajectories were considered).
All simulations were carried out using a Nose-Hoover chain thermostat with the canonical ensemble (NVT). 43,44 The nonspinel model of the bulk γ-alumina unit cell (shown in Figure 1a) was built using the crystallographic model by Dinge et al., 23,45 which has been shown to match well with experimental structural parameters. 46 Our calculated cell parameters (a = 7.90Å, b = 7.93 A, and c = 8.07Å) 23,45 for bulk γ-alumina are in excellent agreement and within 2% of the experimental cell parameters (a = b = 7.96Å and c = 7.81Å). 46 In our simulations, we used the (100) facet of this model, which is the lowest-energy facet reported for this material. 23 It is worth mentioning that our calculations represent a simplified model of the metal-oxide surface, and impurities (such as H 2 O, OH, SOx, etc.) may be present and play a crucial role in its reactivity. Nevertheless, our large-scale BOMD calculations still provide critical atomistic insight into the reactivity of the original pristine material, which serves as a baseline for comparing its catalytic activity against other mixed metal oxide surfaces (and their associated impurities), which we save for future studies.
For our NVT simulations, a 3 × 1 × 2 supercell of γ-alumina and a single DIMP molecule ( Figure 1b) containing a total of 268 atoms were used. We introduced a single DIMP molecule 5 -6Å above the center of the alumina slab along the y-direction, as shown in Figure 1c.
Periodic boundary conditions were applied in the x and z directions. Thus, the xz plane of the slab is parallel to the surface, and the y-axis forms the surface normal where DIMP interacts with the alumina surface. We introduced a vacuum layer of 15Å on top of the surface to avoid any significant overlap between the electronic density of periodically translated cells.
Our BOMD simulations show that the DIMP molecule moves extremely fast and explores a large space of configurations. To limit this exploration to regions where DIMP dissociation on alumina might occur, it is essential to restrict the movement of the DIMP molecule.
Specifically, an external spherical potential, which only acts on the DIMP molecule, was placed at the center of the system. Due to the computationally expensive nature of these simulations, we performed 12 ps NVT simulations for each trajectory. We also calculated adsorption energies using conventional geometry optimizations using the Broyden-Fletcher-Gold-farb-Shanno (BFGS) minimization algorithm until the forces converged to 4.0 × 10 −4
Bohr with an SCF convergence criteria of 1 × 10 −5 au. For the geometry optimization, we used a relatively small supercell (1 × 2 × 2) compared to our BOMD simulations.
Metadynamics Simulations
Throughout our BOMD simulations, we did not observe any decomposition of DIMP on alumina within the 200 -600 • C temperature range (decomposition did occur at higher temperatures, which is discussed later in this paper). At lower temperatures, the high com-
CV or CN = 1 − d AB d 0 p 1 − d AB d 0 p+q ,(1)
where d AB is the distance between atoms A and B, and d 0 is the reference distance or fixed cut-off parameter (this parameter characterizes the standard bond distance between atoms A and B). The variables p and q in Equation 1 are constants, which were fixed to p = q = 6. simulations. In this work, well-tempered MetaD (wt-MetaD) simulations were used 48,49 with the deposition rate of the Gaussian hills set to 20 steps. The well-tempering was implemented using a Gaussian height damping factor of ∆T such that the ratio ∆T+T T was equal to 6. We performed two sets of wt-MetaD simulations for each temperature from different initial conditions, which were extracted from the pre-equilibrated BOMD simulations. We confirmed a representative sampling and convergence of the reactant, product, transition state, and free energy differences, particularly for the dissociation of the C-O bond in DIMP on the alumina surface.
Results and Discussions
Adsorption of DIMP on the Alumina Surface To complement our MetaD simulations, we also calculated adsorption energies, E ads , using the following expression:
E ads = E DIMP+surface − E surface − E DIMP ,(2)
where E surface is the energy of the alumina surface, E DIMP is the energy of an isolated gasphase DIMP molecule, and E DIMP + surface represents the energy of the adsorbed molecule on the surface. A negative value of E ads corresponds to an exothermic process and a stable adsorption configuration. The tri-coordinated Al surface atoms are known to be strong Lewis Figure 6b shows that when the temperature is raised to 700 -1000 • C, the C-O bond lengthens significantly, signifying decomposition. It is worth noting that the timescales for DIMP decomposition varies with temperature, and we observe a very rapid decomposition at 1000 • C.
Free energy profiles
The propene end-products predicted by our BOMD simulations corroborates previous experiments on DIMP, which include thermal decomposition studies, 55 that when the C-O bond is broken, subsequent reforming of this bond is prohibited ( Figure S19), which indicates that the reaction is irreversible. The Supporting Information ( Figures S20 and S21) provides further details on the convergence of our free energy profiles between the transition state and reactant basin as a function of time, which demonstrates that our metadynamics simulations are fully converged. Figure 9: Snapshots from the well-tempered metadynamics simulations within the 700 -1000 • C temperature range. The R, TS, and P labels illustrate the reactant, transition state, and product, respectively.
The free energy surface at 1000 • C is shown in Figure 8 (a). The two CVs that capture the energetics of DIMP decomposition are presented in Figure S1. In this mechanism, the reactant (R) goes to a product (P) via a transition state (TS) and a small tiny intermediate in Figure 9(a). In the Supplementary Material, we provide an additional free energy profile ( Figure S2) at 1000 • C that utilizes different initial conditions. For this separate case, the reactant proceeds to the product state via a stable transition state with a tiny minimum, and the activation barrier is ∼6.36 kJ/mol. Collectively, the average free energy value from these independent trajectories is 5.24 kJ/mol.
The reaction mechanism at 900 • C is very similar to that of 1000 • C, which also shows a tiny intermediate between the product and transition state, as shown in Figure 8(b). The free energy barrier value is ∼11.85 kJ/mol, and Figure S3 in the Supplementary Material shows the corresponding evolution of CV1 and CV2. The geometries of the reactant, transition state, and product along the metadynamics trajectory are reported in Figure 9(b). We have also performed additional metadynamics calculations using different initial conditions to test the reproducibility of these results. In these additional calculations (depicted in Figure S4 in the Supplementary propene is formed. The net free energy barrier for this activation process is ∼22.53 kJ/mol.
The additional free energy profile at 800 • C from the different initial conditions is shown in Figure S7. A very similar reaction mechanism is also observed at 700 • C with a slightly higher activation barrier of ∼26.65 kJ/mol. Figure Figure 11(a), and the corresponding free energy surface is depicted in Figure 11(b). The reaction mechanism takes place in two substeps, as shown in Figure 11(c). Figure S22 in the Supporting Information. A Gaussian with a height of 0.0001 Hartree was used for these simulations. Figure S23 in the Supporting Information illustrates free energy profiles for the C-H activation within the 200 -500 • C temperature range. In Figure S24, we show free energy profiles in the 600 -900 • C temperature range. The free energy profile at 1000 • C is shown in Figure S25a, and Figure S25b summarizes the free energy activation barrier as a function of temperature. The free energy activation barriers range from 3.91 -1.29 kJ/mol for the 200 -1000 • C temperature range. We find that the free energy activation barrier value decreases with increasing the temperature. The small free energy barrier value in the 600 -1000 • C temperature range lies between 5 -28 kJ/mol, which can be easily accessible by experiment. Our theoretical findings are also consistent with experimental studies, which also identified propene as the main gas-phase product of the DIMP decomposition. In summary, our AIMD simulations show that the γ-Al 2 O 3 surface can trap and subsequently decompose DIMP due to strong electrostatic attractions between the phosphoryl oxygen and surface Al atoms.
Conclusions
In this work, we have harnessed large-scale ab initio molecular dynamics calculations to investigate the adsorption and decomposition of DIMP on the γ-alumina surface over a wide range of temperatures. Our DFT-based molecular dynamics calculations predict a spontaneous decomposition of DIMP with propene as the main by-product within the 700 -1000 • C temperature range (the decomposition reaction leads to propene and an Al-OCHCH 3 CH 3 adsorbate at 700 • C). Due to the short-time scales inherent to BOMD simulations, it is likely that a similar decomposition would also occur at lower temperatures but would take longer to happen. Well-tempered metadynamics AIMD simulations were performed for temperatures ranging from 200 -1000 • C to provide atomistic-level details of the reaction path and associated energetics of DIMP decomposition. Our metadynamics calculations also reveal that the free energy barrier value decreases with temperature. The low free energy barrier values at higher temperatures suggest that that reaction is extremely fast at higher temperatures and is likely to occur at a lower rate at lower temperatures.
In our study, we obtained free energy values of 62.75 to 5.24 kJ/mol within the 200 -1000 • C temperature range. Recent experiments on CWAs have also been carried out under similar temperatures, including vapor phase decomposition of DMMP (a sarin surrogate) at 500 -800 K, 58 sarin decomposition on TiO 2 nanoparticles at 1000 K, 53 and decomposition of CWAs at 2000 K. 31,59,60 Collectively, all of these experiments showed that the temperature ranges studied in this work can also be achieved under operational conditions, and DIMP would decompose on the γ-Al 2 O 3 surface. Due to the structural similarity between DIMP and sarin, our calculations provide additional insight into decomposition mechanisms of both these molecules and elucidate atomic details of sarin decomposition on candidate metal-oxide surfaces. As a final remark, this study serves as a convincing demonstration of the use of DFT-based molecular dynamics simulations for investigating the interactions of CWAs with existing metal-oxides, which can be used to guide experimental efforts on these hazardous compounds.
Figure 1 :
1(a) Bulk structure of γ-Al 2 O 3 . (b) Molecular structure of diisopropylmethylphosphonate (DIMP). (c) Top view of DIMP adsorbed on the γ-Al 2 O 3 surface. (d) Side view of DIMP adsorbed on the γ-Al 2 O 3 surface. The blue, red, orange, brown, and light pink spheres represent Al, O, P, C, and H atoms, respectively.
putational cost of these simulations only permits explorations of short time intervals; therefore, BOMD simulations alone do not permit a routine exploration of the complete reaction dynamics. One can sidestep this limitation by using advanced computational techniques such as metadynamics simulations (MetaD), 47 which we describe further below. In short, MetaD bypasses the sampling limitations of traditional molecular dynamics by applying a history-dependent biasing potential as a function of time to enable efficient sampling of the free energy surface. The free energy surface itself is defined over a set of collective variables (CVs), which are carefully chosen to provide a complete description of the system's slow degrees of freedom. The decomposition process is characterized by the breaking of various bonds within the DIMP molecule on the alumina surface. For this reason, we utilized two CVs, shown in Figure 2, for our accelerated sampling simulations. The CV specifies the coordination number (CN) during the MetaD simulations, which is expressed as a function of the distance between two atoms:
Figure 2 :
2Collective variables (CVs) used to describe the adsorption dynamics of DIMP on the alumina surface.In this work, we chose CV1 to be the distance between the C and O atoms within DIMP, and CV2 as the distance between the O atom in DIMP and an Al atom on the alumina surface, as shown inFigure 2. MetaD simulations were carried out by depositing Gaussians with heights of 0.02 and 0.001 Hartree for the 200 -500 • C and 600 -1000 • C temperature ranges, respectively. The widths of the Gaussians were set to 0.1 for both the CV1 and CV2
Figure 3 :
3Time-dependent fluctuations of various Al-O distances, which confirm DIMP adsorption throughout the BOMD simulations for all temperatures. Metal oxide surfaces are typically catalytic and can adsorb and subsequently decompose CWAs into benign products. Prior experiments 50 and theoretical 27 calculations have shown that sarin simulants can adsorb on metal oxides by forming a bond with metal atoms via a phosphoryl oxygen. Initially, we observed the DIMP molecule interacting with an Al atom via the O atom of the P-OC 3 H 7 moiety. To further explore these dynamical effects with our BOMD simulations, Figure 3 plots the distance between one of the surface Al atoms and the oxygen of the P-OC 3 H 7 moiety in DIMP at various temperatures. These calculations show that DIMP adsorption occurs within two picoseconds for all temperatures, and the oxygen of the P-OC 3 H 7 moiety of DIMP interacts with the tetra-coordinated Al center. We did not observe any desorption of DIMP on the alumina surface, as indicated by the small Al48-O3 bond distances in Figure 3 (as a side note, the 600 • C plot in the center of Figure 3 does show dissociation of the Al48-O3 bond, but a new Al11-O2 bond quickly forms thereafter, and the DIMP molecule still remains on the alumina surface).Figure 4shows snapshots from our simulations depicting various DIMP adsorption configurations on the alumina surface.As suggested by prior theoretical calculations, 51-53 the adsorption via the O atom of the P-OC 3 H 7 moiety is crucial for propene elimination. As the MD simulations progresses, we also observe interactions between the O (in the -P=O group) and Al atoms. In short, the DIMP molecule remained adsorbed on the alumina surface in our simulations for all temperatures.
Figure 4 :
4Snapshots illustrating the adsorption of DIMP on the alumina surface. From 200 -500 • C, adsorption occurs via the formation of the Al48-O3 bond. At 600 • C, adsorption proceeds via the formation of the Al48-O3 and Al11-O2 bonds. At 700 and 1000 • C, the Al48-O3 bond is formed during the adsorption process, and from 800 -900 • C, an Al43-O2 bond is formed.
Figure 5 :
5Various adsorption configurations of DIMP on the γ-Al 2 O 3 (100) surface. Bond distances are shown in angstroms. acid-type catalytic sites 54 and give large adsorption energies when the DIMP molecule binds to them. Figure 5 illustrates selected optimized structures of DIMP on the alumina surface. In panel 5(a), the DIMP molecule is bonded to two tetra-coordinated Al atoms via the phosphoryl oxygen, resulting in a bridging adsorption with E ads = −48.63 kcal/mol. In panel 5(b), the alkoxy oxygen is bonded to a tri-coordinated Al atom, giving an adsorption energy of -101.40 kcal/mol. The largest adsorption energy (-153.55 kcal/mol) was obtained for the configuration shown in panel 5(c) where the alkoxy and phosphoryl oxygen atoms are bonded to two different tri-coordinated Al centers. In panel 5(d), the phosphoryl oxygen is bonded to a tri-coordinated Al center, giving an adsorption energy of -93.12 kcal/mol. The adsorption energy for configuration in panel 5(e) is -117.97 kcal/mol in which the DIMP molecule binds with two tetra-coordinated Al atoms via two alkoxy oxygen atoms. Our calculated adsorption energy for the configuration shown in panel 5(a) is in good agreement with a previously studied similar configuration of sarin adsorption on γ-Al 2 O 3 (-49.2 kcal/mol). 27 Decomposition of DIMP on the Alumina Surface We next examine DIMP decomposition mechanisms on the alumina surface. Previous studies have shown that one of the decomposition pathways of DIMP proceeds through the breaking of a C-O bond. 28 Figure 6a displays time-dependent variations of C-O bond distances from 200 -600 • C, which fluctuate around 1.50Å (the covalent C-O bond distance), indicating that the C-O bond remains intact. On the other hand,
Figure 7
7depicts snapshots at crucial points for the decomposition of DIMP on the alumina surface for a few representative MD trajectories. As mentioned in the previous section, adsorption initially occurs via the oxygen atom of the P-OC 3 H 7 moiety in DIMP and an Al atom on the alumina surface (which occurs at all temperatures). At 700 • C, the 3 panels
Figure 6 :
6Evolution of C-O distances as a function of time on the alumina surface at various temperatures, which describes DIMP decomposition. The corresponding atom labeling is shown in the inset panel of (a).
Figure 7 :
7Snapshots of DIMP decomposition from representative MD trajectories at various temperatures. at the top of Figure 7 depict a possible decomposition pathway in which the C3-O3 bond breaks after 2 ps and an Al-O bond forms with the alumina surface. During this process, a surface-bound n-alkyl (-C 3 H 7 ) species is formed. At ∼3 ps, a single terminal C-H bond from the n-alkyl molecule is broken when a proton is abstracted by an oxygen atom on the alumina surface. This forms propene, an unsaturated compound, as a by-product. As the simulation further proceeds, the P-O bond in DIMP dissociates at 7 ps and forms a new Al-OCHCH 3 CH 3 adsorbate on the alumina surface. In addition, once the P-O bond in DIMP is cleaved, a new P-O bond is formed between the phosphorus atom and a di-coordinated surface oxygen atom (see the 3 panels at the top of Figure 7). At 800 -1000 • C (depicted in the bottom 9 panels of Figure 7), DIMP decomposition involves only the propene elimination via a two-step process. The first step of the propene elimination is associated with the dissociation of the C2-O2 (800 -900 • C) or C3-O3 (1000 • C) bond. The second step comprises the migration of the hydrogen atom from the methyl group of the -C 3 H 7 fragmentto one of the surface oxygen atoms. Within these simulated timescales, we did not observe any further decomposition of DIMP within the 800 -1000 • C temperature range.
Figure 8 :
8,56 pyrolysis and combustion in nitrogen/oxygen-rich environments, 57 as well as microwave 28 and laser-induced 30 decomposition under inert environments. Collectively, all of these prior experimental studies detected propene as one of the main by-products of DIMP decomposition, which further supports our BOMD predictions. To further investigate the decomposition of DIMP on alumina, we utilized well-tempered metadynamics simulations by adopting two collective variables. We performed two sets of metadynamics simulations (for each temperature within the 200 -1000 • C range) using different initial conditions. Obtaining accurate free energy profiles from metadynamics simulations requires (1) longer simulation times until all acces-Reconstructed free energy surface for the decomposition of DIMP on the alumina surface at (a) 1000 • C, (b) 900 • C, (c) 800 • C, and (d) 700 • C. The R, TS, I, and P labels in each free energy surface correspond to the reactant, transition state, intermediate, and product, respectively. sible regions of the potential are explored (with trajectories spanning forward and backward many times between reactant and product states) and (2) a careful selection of collective variables. Our well-tempered metadynamics simulations at various temperatures suggests
(
I) in between the transition state and product. The reactant state is defined by CV1 = 0.75 (C-O = 1.66Å) and CV2 = 0.11 (Al-O = 2.17Å ), where DIMP is bonded to the surface Al atom. The decomposed DIMP product, P, is characterized by CV1 = 0.03 (C-O = 3.12Å) and CV2 = 0.40 (Al-O = 1.77Å), in which the C-O bond of DIMP is cleaved. At this stage, the H atom from the -C 3 H 7 moiety is transferred to a surface oxygen atom, and propene is formed. The C-O bond length stretches to 1.87Å at the transition state (which has an activation barrier of only 4.11 kJ/mol) and subsequently dissociates. The reactant, transition state, and product from our well-tempered metadynamics simulations at 1000 • C are shown
Information), the reactant proceeds to the product through a single transition state with a free energy barrier value of ∼15.73 kJ/mol. The average free energy barrier value obtained from these different initial conditions is 13.79 kJ/mol. The reconstructed free energy surfaces at 800 and 700 • C are shown in Figures 8(c) and 8(d), and fluctuations of the corresponding CV values are presented in Figures S5 and S6, respectively. In contrast to the higher temperatures discussed previously, the decomposition of DIMP at 800 and 700 • C proceeds via a single transition state. At 800 • C, the collective variables CV1 (0.71) and CV2 (0.12) define the reactant state R, at which DIMP forms a bond with the surface Al atom (typical C-O and Al-O bond distances at this geometry are 1.53 and 2.25Å, respectively). The C-O bond in DIMP stretches to 1.91Å to form a transition state, and the reaction progresses to the product where the C-O bond subsequently dissociates, and
Figure 10 :
10S8 in the Supplementary Material represents the topology of another free energy surface at 700 • C from different initial conditions. The reactant, transition state, and product geometries from these metadynamics simulations are shown in Figures 9(c) and 9(d) for 800 and 700 • C, respectively. A detailed analysis of Free energy activation barrier as a function of temperature for the decomposition of DIMP on the alumina surface. The free energy activation barrier value is calculated by averaging over two metadynamics simulations at each temperature. free energy profiles for DIMP decomposition on alumina at various temperatures (200 -600 • C) with different initial conditions is given in the Supplementary Material (Figures S9-S18). Figure 10 summarizes the free energy activation barrier as a function of temperature, which shows that the free energy activation barrier decreases with temperature. Static DFT calculations have obtained free energy activation barrier values of 113 kJ/mol (ZnO surface), 51 108.0 kJ/mol (rutile surface), 53 122.6 kJ/mol (anatase surface), 53 and 53.7 kJ/mol (MoO 2 surface).52 However, the free energy activation barriers for C-O bond breaking from our simulations are much lower than these previously reported values. We also note that the free energy activation barrier value at 600 • C is very similar to that of 700 • C.
Figure 11 :
11Panel (a) depicts collective variables used in studying the formation of an Al-OCHCH 3 CH 3 adsorbate at 700 • C, and panel (b) shows the 3D reconstructed free energy surface, where R, TS1, I, TS2, and P correspond to the reactant, transition state 1, intermediate, transition state 2, and product, respectively. Panel (c) shows snapshots of R, TS1, I, TS2, and P along the metadynamics trajectory.At 700 • C, we observed formation of an Al-OCHCH 3 CH 3 adsorbate in addition to propene. To further analyze these energetics with metadynamics simulations, we utilized two collective variables: CV3 [P1-O2] and CV4 [P1-064], which denote the coordination number of the P1 phosphorus with respect to the O2 and O64 oxygen atoms, respectively, in the DIMP molecule. The CVs are shown in
Collectively, our AIMD calculations show that the decomposition of DIMP most likely progresses via a propene elimination on the alumina surface. The final step of the propene elimination occurs via the abstraction of a hydrogen atom by the surface oxygen atom of gamma-alumina. We also performed metadynamics simulations to evaluate the free energy activation of this process. The C-H coordination (CV5) and H-O coordination (CV6) were selected as collective variables, as shown in
The reactant is described by CV3 = 0.80 and CV4 = 0.05, in which the P1-O2 and P1-O64 bond distances are 1.69 and 3.02Å, respectively. The P1-O2 bond first stretches to 1.95Å and forms the first transition state (TS1). The P1-O2 bond then dissociates to 2.46 and creates a stable intermediate, I (CV3 = 0.15, CV4 = 0.04), giving a free energy activation of 18.51 kJ/mol. The reaction proceeds over a second transition state (TS2) in which the P1-O2 bond is further stretched and the resulting -OCHCH 3 CH 3 fragment binds to the surface Al atom. In the product state, the distance from the P1 atom to the surface oxygen (O64) shortens (1.74Å), indicating the formation of a P1-O64 covalent bond. In contrast, the P1-O2 bond length further increases, showing the complete decomposition of the DIMP molecule.
AcknowledgementThe project or effort depicted was or is sponsored by the Department of the Defense, Defense
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"Massive Type IIA Supergravity and E 10",
"Massive Type IIA Supergravity and E 10"
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| [
"Marc Henneaux \nPhysique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium\n\nCentro de Estudios Científicos (CECS)\nCasilla 1469ValdiviaChile\n",
"Ella Jamsin \nPhysique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium\n",
"Axel Kleinschmidt \nPhysique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium\n",
"Daniel Persson \nPhysique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium\n\nFundamental Physics\nChalmers University of Technology\nSE-412 96GöteborgSweden\n"
]
| [
"Physique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium",
"Centro de Estudios Científicos (CECS)\nCasilla 1469ValdiviaChile",
"Physique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium",
"Physique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium",
"Physique Théorique et Mathématique\nUniversité Libre de Bruxelles & International Solvay Institutes\nULB-Campus Plaine\nC.P. 231B-1050BruxellesBelgium",
"Fundamental Physics\nChalmers University of Technology\nSE-412 96GöteborgSweden"
]
| []
| In this talk we investigate the symmetry under E10 of Romans' massive type IIA supergravity. We show that the dynamics of a spinning particle in a non-linear sigma model on the coset space E10/K(E10) reproduces the bosonic and fermionic dynamics of massive IIA supergravity, in the standard truncation. In particular, we identify Romans' mass with a generator of E10 that is beyond the realm of the generators of E10 considered in the eleven-dimensional analysis, but using the same, underformed sigma model. As a consequence, this work provides a dynamical unification of the massless and massive versions of type IIA supergravitiy inside E10. | 10.1002/prop.200900040 | [
"https://arxiv.org/pdf/0901.4848v1.pdf"
]
| 14,361,193 | 0901.4848 | 007b6dd19e4b588b374c02a5ed6428ddde319fda |
Massive Type IIA Supergravity and E 10
30 Jan 2009 Fortschritte der Physik, 30 January 2009
Marc Henneaux
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
ULB-Campus Plaine
C.P. 231B-1050BruxellesBelgium
Centro de Estudios Científicos (CECS)
Casilla 1469ValdiviaChile
Ella Jamsin
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
ULB-Campus Plaine
C.P. 231B-1050BruxellesBelgium
Axel Kleinschmidt
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
ULB-Campus Plaine
C.P. 231B-1050BruxellesBelgium
Daniel Persson
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
ULB-Campus Plaine
C.P. 231B-1050BruxellesBelgium
Fundamental Physics
Chalmers University of Technology
SE-412 96GöteborgSweden
Massive Type IIA Supergravity and E 10
30 Jan 2009 Fortschritte der Physik, 30 January 2009Copyright line will be provided by the publisher
In this talk we investigate the symmetry under E10 of Romans' massive type IIA supergravity. We show that the dynamics of a spinning particle in a non-linear sigma model on the coset space E10/K(E10) reproduces the bosonic and fermionic dynamics of massive IIA supergravity, in the standard truncation. In particular, we identify Romans' mass with a generator of E10 that is beyond the realm of the generators of E10 considered in the eleven-dimensional analysis, but using the same, underformed sigma model. As a consequence, this work provides a dynamical unification of the massless and massive versions of type IIA supergravitiy inside E10.
Introduction
An important class of supergravity theories is provided by deformed maximal supergravities, that are theories that cannot be obtained directly by standard toroidal Kaluza-Klein reduction of eleven-dimensional supergravity. They have as an important feature that they admit domain walls solutions, without which the duality symmetry of the underlying string theory cannot be verified. In particular, massive type IIA supergravity, unlike its massless sister, supports a D8-brane solution, that in IIA string theory can be reached from lower-dimensional branes by sequences of T-dualities [1]. Therefore, any decription of M-theory should include deformed supergravities.
A possible approach to M-theory is via Kac-Moody symmetries, notably E 10 [2,3,4,5] and E 11 [2,3,4,6,7,8]. The E 10 proposal, on which we focus in this talk, has two main motivations. First the E 11−D symmetry appearing in the reduction of eleven-dimensional supergravity to dimensions D ≥ 2 naturally leads to the conjecture that the reduction to one dimension should be invariant under E 10 [2,3]. Second, it is remarkable that the same intuition comes from cosmological billiards: close to a spacelike singularity (the BKL limit), eleven-dimensional supergravity becomes explicitly symmetric under the Weyl group of E 10 [5].
Moreover, recent development observed the relevance of E 10 and E 11 in the framework of deformed supergravities, where the deformation parameters are identified with forms of high rank, specifically (D − 1)-forms for deformations in D dimensions [9,10,11].
The purpose of this talk is to explain how the deformation parameter of massive IIA supergravity enters the dynamics of the geodesic model of E 10 , and our analysis includes the fermions. We show that the mass enters as the dual to a generator that is outside the realm of the generators considered usually. Importantly, all terms associated with the mass coincide perfectly. Here we focus on the general features and refer the reader to [12], upon which this talk is based, for the technical details.
Massless IIA
Massive IIA Massive deformation F (2) → F (2) + mA (2) No known D = 11 origin Fig. 1 Massive IIA supergravity from D = 11 supergravity: Massive type IIA supergravity is obtained as a deformation of the standard type IIA supergravity, but unlike the latter, it does not possess any known eleven-dimensional origin. See Figure 2 for a pictorial description of how D = 11 supergravity and massive IIA supergravity are unified inside E10.
E 10 ℓ 1 L SUGRA11 ⊂ L E10/K(E10) L mIIA ⊂ L E10/K(E10) (ℓ 1 , ℓ 2 ) 'Dimensional Reduction' ℓ 2
Fig. 2
This picture describes the common E10 origin of eleven-dimensional supergravity and massive type IIA supergravity. First, if one considers a level ℓ1 decomposition of E10 with respect to A9 (cf. Figure 3), one sees that the first levels (ℓ1 = 0 to ℓ1 = 3) of an E10/K(E10) sigma-model correspond to a truncated version of eleven-dimensional supergravity, with Lagrangian LSUGRA 11 [5]. Taking this as a starting point, we can perform an additional level ℓ2 decomposition on the sigma model. On the lower ℓ1 levels (ℓ1 = 0 to ℓ1 = 3) this is equivalent to a dimensional reduction of eleven-dimensional supergravity, which gives massless IIA supergravity (cf. Figure 1). However, if one includes one of the generators appearing at ℓ1 = 4, this leads to a theory that coincides with a truncated version of massive IIA supergravity, with Lagrangian LmIIA. This procedure is equivalent to a multi-level (ℓ1, ℓ2) decomposition of E10 with respect to A8.
Massive IIA supergravity
The first construction of massive type IIA supergravity is due to Romans [13] and its main step is to give a mass to the two-form potential of standard IIA supergravity [14,15,16] through the replacement (1) . The one-form potential A (1) is then gauged away, which leads to terms depending on m −1 in the supersymmetry variations and thus obscures the massless limit. This is remedied by a field redefinition presented in [17,18], that we employ in the analysis. A more democratic version of massive IIA supergravity is given in [18,19]. It includes a nine-form dual to the mass m ∝ * 10 dA (9) , and it is precisely that dual nine-form that we will be able to identify with a nine-form appearing in a certain decomposition of E 10 . Moreover, massive IIA supergravity has in common with many other deformed maximal supergravities that it does not possess any known higher-dimensional origin, as illustrated in Figure 1. A consequence of the present work is to show that, although they are not related by dimensional reduction, eleven dimensional supergravity and massive IIA supergravity have the same E 10 origin as displayed in Figure 2, see also [20,21,22].
F (2) → F (2) + mA (2) , where F (2) = dA
In the form we consider in this work, the bosonic sector of massive IIA supergravity contains a metric, a dilaton, a one-form, a two-form, a three-form, and a real mass parameter m. On the fermionic side, we have two gravitini, combined in a single 10 × 32 component vector-spinor, and two dilatini, combined in a single 32 component Dirac-spinor, which decompose into two fields of opposite chirality under SO (1,9). The full expression of the Lagrangian in our conventions is given in [12].
3 E 10 and the geodesic sigma model for E 10 /K(E 10 )
3.1 Generalities on E 10 and K(E 10 ).
Here we summarize important features about the Kac-Moody algebras e 10 and k(e 10 ), the groups of which we shall denote by E 10 and K(E 10 ). More details can be found in [5,23,12].
The split real form of e 10 is generated by ten triples (e i , f i , h i ), i = 1, . . . , 10, of Chevalley generators, each triple making up a distinguished sl(2, R) subalgebra, These subalgebras are intertwined inside e 10 according to the stucture of the Dynkin diagram in Figure 3.
The maximal compact subalgebra k(e 10 ) ⊂ e 10 is defined as the subalgebra which is invariant under the Chevalley involution ω, which is defined through its action on each triple (e i , f i , h i ):
ω(e i ) = −f i , ω(f i ) = −e i , ω(h i ) = −h i .(1)
The subalgebra k(e 10 ) enters the so-called Iwasawa decomposition of e 10 ,
e 10 = k(e 10 ) ⊕ h ⊕ n + ,(2)
where h is the Cartan subalgebra, generated by the h i and n + is the infinite-dimensional positive nilpotent subalgebra, generated by the positive step operators e i .
The A 8 level decomposition of E 10
The correspondence between e 10 and eleven-dimensional supergravity is made by introducing an A 9 ∼ = sl(10, R) level decomposition of E 10 , where the level ℓ 1 of a root α of e 10 is its integer coordinate in the direction of the simple root α 10 (associated to node 10 in the Dynkin diagram in Figure 3) [5]. For each value of the level ℓ 1 , one has a finite number of representations of A 9 . The correspondence was established up to level ℓ 1 = 3 (with some minor exceptions [24]).
In the case we are interested in here, one needs to perform a further decomposition ℓ 2 associated to the root 9. Hence, we write any root α of e 10 in terms of the ten simple roots as
α = ℓ 1 α 10 + ℓ 2 α 9 + 8 i=1 m i α i .(3)
The level ℓ := (ℓ 1 , ℓ 2 ) is now two-folded and corresponds to a decomposition under the A 8 ∼ = sl(9, R) subalgebra of e 10 , defined by nodes 1, . . . , 8 in the Dynkin diagram in Figure 3. At level (0, 0), there is a copy of gl(9, R), K a b , and a scalar generator, T , associated with the dilaton. The generators of e 10 at higher levels are sl(9, R)-tensors of higher and higher rank E a1···a k ∈ e 10 , where k = 2ℓ 1 + ℓ 2 and a i = 1, . . . , 9. The full table up to ℓ = (4, 1) can be found in [12].
In particular, at ℓ = (4, 1), one has a nine-form generator E a1···a9 whose accompanying nine-form field will be identified with the dual to the mass of massive IIA supergravity [20,25]. This is intriguing since in D = 11 the matching between supergravity and e 10 has only been successful up to ℓ 1 = 3. Hence, the mass term in D = 10 is outside this and provides a non-trivial check of e 10 beyond its 'sl(10, R)-covariantized e 8 ' subset, i.e. the generators of e 8 and their images under (the Weyl group of) sl(10, R).
Construction of the non-linear sigma model
We here describe how to build the non-linear sigma model with rigid E 10 invariance and local K(E 10 ) invariance. Thanks to the Iwasawa decomposition (2), one can choose a representative of the coset space E 10 /K(E 10 ) in the so-called partial 'Borel gauge' by taking only exponentials of h and n + :
V(t) = e h a b (t)K b a e φT e A(t)⋆E ∈ E 10 /K(E 10 ),(4)
where A(t) ⋆ E is a sum over the positive level generators E a1···a k of E 10 with coefficients A a1···a k (t).
The coset representative V transforms under global g ∈ E 10 -transformations from the right and local k ∈ K(E 10 )-transformations from the left V −→ kVg. From V(t), one can construct the Lie-algebra element in Maurer-Cartan form
v(t) = ∂ t VV −1 = P(t) + Q(t),(5)
that decomposes, under the Chevalley involution, into an invariant part (Q ∈ k(e 10 )) and an anti-invariant part (P ∈ e 10 ⊖ k(e 10 )).
In the next section, we will identify the fields of massive IIA supergravity with the components of P(t) and Q(t) in the A 8 level decomposition of e 10 ⊖ ke 10 or ke 10 respectively, that we will note P (ℓ) and Q (ℓ) at level ℓ. Because of the choice of the partial Borel gauge for V, P (ℓ) = Q (ℓ) , ∀ℓ = (0, 0).
The bosonic part
A manifestly E 10 × K(E 10 ) local -invariant Lagrangian is constructed as follows [5,24]
L [B] E10/K(E10) = 1 4 n(t) −1 P(t) P(t) ,(6)
where the bracket represents an invariant inner product over E 10 and the lapse function n(t) ensures invariance under reparametrizations of the geodesic parameter t. The equations of motion for P (in the gauge n = 1) read
DP := ∂ t P − [Q, P] = 0,(7)
where we defined the K(E 10 )-covariant derivative D.
The fermionic part
In order to build the fermionic part of the E 10 /K(E 10 ) sigma model, one needs to introduce spinorial representations of k(e 10 ). In the case of eleven-dimensional supergravity, a good correspondence is obtained using two finite-dimensional (unfaithful) representations. The first one transforms as a 32dimensional Dirac-spinor representation ǫ of so(10) ⊂ k(e 10 ) and corresponds to the supersymmetry parameter. The second one transforms as a 320-dimensional vector-spinor representation Ψȧ,ȧ = (10, a), of so(10) ⊂ k(e 10 ) and is identified with the gravitino [26,27,28,29]. Upon reduction to the IIA theory (through the additional level decomposition with respect to ℓ 2 ), while the supersymmetry parameter stays unchanged, the gravitino decomposes into a 32-dimensional spinor Ψ 10 (to be associated with the ten-dimensional dilatino) and a 288-dimensional vector spinor Ψ a of so(9) (related to the gravitino) that will mix under k(e 10 ) [30]. Supergravity E 10 /K(E 10 ) Bianchi identities and bosonic equations of motion ∂ t P − [Q, P] = 0
Fermionic equations of motion ∂ t Ψ − Q · Ψ = 0
Supersymmetry variation of ψ t δ ǫ Ψ t = Dǫ Table 1 Each line of this table contains the equations to be compared to each other in order to make the correspondence between massive IIA supergravity and E10 explicit.
The fermionic degrees of freedom are included in the Lagrangian through the spinor representation Ψ as follows [27,28,29]
L [F ] E10/K(E10) = − i 2 Ψ DΨ ,(8)
where the bracket now denotes an invariant inner product on the representation space. The associated 'Dirac equation' reads
DΨ := ∂ t Ψ − Q · Ψ = 0.(9)
The bosonic equations of motion (7) were written for the gauge choice n = 1. The lapse function n has a superpartner Ψ t , which is a Dirac spinor under k(e 10 ), as is the supersymmetry parameter, and the associated supersymmetry transformations are
δ ǫ n = iǫ T Ψ t , δ ǫ Ψ t = Dǫ.(10)
The fermionic equations of motion are then valid in the 'supersymmetric gauge' Ψ t = 0.
The correspondence
In order to compare the equations (of motion and of supersymmetry) of supergravity to the equations of our sigma model, we need to rewrite the former. First, as is customary in the correspondence between E 10 and supergravity we split the indices into temporal and spatial indices and adopt a pseudo-Gaussian gauge for the ten-dimensional vielbein. In addition we demand that the spatial trace of the spin connection vanishes. We also choose temporal gauges for all supergravity gauge potentials. Moreover, we can only expect that a truncated version of the supergravity equation corresponds to the coset model equations. This truncation was originally devised in the context of eleven-dimensional supergravity, where it was strongly motivated by the billiard analysis of the theory close to a spacelike singularity (the 'BKL-limit') [5,24]. In this limit, spatial points decouple and the dynamics becomes effectively time-dependent, ensuring that the truncation is a valid one in this regime. In this paper, we analyse the same question in the context of massive IIA supergravity, and an identical procedure requires the truncation of a set of spatial gradients. These can be obtained from a BKL-type analysis of massive IIA.
One can now proceed to the comparison between the two theories. In practice, we compare the equations as prescribed in Table 1.
As a result, we obtain a dictionary between the bosonic and fermionic fields of massive IIA supergravity and the representations of e 10 and k(e 10 ) that we defined in the previous section. The schematic correspondence is presented in Table 2.
This correspondence works perfectly up to level (4, 1) for all equations but one: the Einstein equation does not fit perfectly in this picture. More precisely, two terms do not match completely with the corresponding sigma model equation. These discrepancies can however be traced back to D = 11 supergravity where both mismatches were part of the D = 11 Ricci tensor [24]. In this sense this is not a new discrepancy but a known one. It is to be noted that all the terms involved in the mismatch are related to Supergravity E 10 /K(E 10 ) Bosonic fields P (ℓ) (ℓ ≥ 0), Q (0)
Fermionic fields Ψ t , Ψ a , Ψ 10 Supersymmetry parameter ǫ Table 2 This table shows schematically which fields of the two theories are identified in the correspondence.
contributions to the Lagrangian which would give rise to walls corresponding to imaginary roots in the cosmological billiards picture [5]. Moreover, in particular, and most importantly, one notices that the mass enters all equations correctly when identified with the nine-form P a1···a9 of E 10 at level (4, 1) in the following way:
P a1···a9 = 1 2 N e 5φ/2 ǫ a1···a9 m,(11)
where N is the lapse and φ the dilaton of massive IIA supergravity. Further aspects of gauge fixing and the consistency of the gauge algebra and supersymmetry with the correspondence can be found in [12].
Fig. 3
3The Dynkin diagram of e10 with the nodes associated with the level decomposition indicated in white.
Copyright line will be provided by the publisher
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| []
|
[
"Thimble regularisation of YM fields: crunching a hard problem",
"Thimble regularisation of YM fields: crunching a hard problem"
]
| [
"Francesco Di Renzo [email protected] \nDipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly\n",
"Simran Singh [email protected] \nDipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly\n",
"Kevin Zambello [email protected] \nDipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly\n"
]
| [
"Dipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly",
"Dipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly",
"Dipartimento di Scienze Matematiche\nFisiche e Informatiche\nUniversità di Parma\nINFN\nGruppo Collegato di ParmaI-43100ParmaItaly"
]
| [
"The 38th International Symposium on Lattice Field Theory"
]
| Thimble regularisation of Yang Mills theories is still to a very large extent terra incognita. We discuss a couple of topics related to this big issue. 2d YM theories are in principle good candidates as a working ground. An analytic solution is known, for which one can switch from a solution in terms of a sum over characters to a form which is a sum over critical points. We would be interested in an explicit realisation of this mechanism in the lattice regularisation, which is actually quite hard to work out. A second topic is the inclusion of a topological term in the lattice theory, which is the prototype of a genuine sign problem for pure YM fields. For both these challenging problems we do not have final answers. We present the current status of our study. | 10.22323/1.396.0233 | [
"https://arxiv.org/pdf/2112.00062v1.pdf"
]
| 244,773,052 | 2112.00062 | a9524fc9afd568fab8cdf8dd8ff0a8e30203f7f3 |
Thimble regularisation of YM fields: crunching a hard problem
LATTICE2021 26th-30th July, 2021
Francesco Di Renzo [email protected]
Dipartimento di Scienze Matematiche
Fisiche e Informatiche
Università di Parma
INFN
Gruppo Collegato di ParmaI-43100ParmaItaly
Simran Singh [email protected]
Dipartimento di Scienze Matematiche
Fisiche e Informatiche
Università di Parma
INFN
Gruppo Collegato di ParmaI-43100ParmaItaly
Kevin Zambello [email protected]
Dipartimento di Scienze Matematiche
Fisiche e Informatiche
Università di Parma
INFN
Gruppo Collegato di ParmaI-43100ParmaItaly
Thimble regularisation of YM fields: crunching a hard problem
The 38th International Symposium on Lattice Field Theory
LATTICE2021 26th-30th July, 2021Zoom/Gather@Massachusetts Institute of Technology * Speaker
Thimble regularisation of Yang Mills theories is still to a very large extent terra incognita. We discuss a couple of topics related to this big issue. 2d YM theories are in principle good candidates as a working ground. An analytic solution is known, for which one can switch from a solution in terms of a sum over characters to a form which is a sum over critical points. We would be interested in an explicit realisation of this mechanism in the lattice regularisation, which is actually quite hard to work out. A second topic is the inclusion of a topological term in the lattice theory, which is the prototype of a genuine sign problem for pure YM fields. For both these challenging problems we do not have final answers. We present the current status of our study.
A thimble primer
QCD at finite baryon density is still to a large extent terra incognita, due to the infamous sign problem. The latter is in fact more general (and fundamental, in a sense): we have to tackle it every time the action of a quantum field theory is complex valued. A number of possible solutions have been put forward, among which thimble regularisation [1,2]. In a light notation in which a field theory looks like an ordinary integral, the thimble approach to field theories is quite easy to describe in terms of the following recipes:
1. We first need to complexify the degrees of freedom, i.e. → = + and ( ) = ( ) + ( ) → ( ).
2. We then need to find the critical points, i.e. those points where = 0.
3. We define the thimble J attached to each critical point as the union of all the Steepest Ascent paths (SA), the latter being the solutions of =¯stemming from the critical point. 4. If the Hessian of the action has no zero eigenvalue, one can immediately prove that the thimble is a manifold of the same real dimension as the original manifold we started from.
5. Due to the holomorphic nature of , is increasing along the ascent and thus on the thimble the original integral is convergent, while stays constant. 6. Sadly, the sign problem is not completely killed, since the integration measure (encoding the orientation of the thimble with respect to the embedding manifold) reintroduces a residual sign problem due the so-called residual phase.
The last point would deserve much more attention than we can pay here; the interested reader can look at [3] for our (basic, but stemming from first principles) solution to the computation of the residual phase. While going through all the steps needed to carry out a computation on thimbles can be a non trivial task, our efforts are fully rewarded by Lefschetz/Picard theory, which states that a thimble decomposition for the original path integral holds
< > = − ( ) ∫ J − − ( ) ∫ J −(1)
In (1) is no longer such a big problem, while the residual sign problem is due to the residual phases . Notice that both the numerator and the denominator (i.e. the partition function) receive contributions in principle by all the critical points. This is not really the case, since the intersection numbers can be zero for possibly many critical points. It can be shown that = 0 for a critical point when the associated unstable thimble does not intersect the original integration manifold .
It is important to remind that the action is complex since the very beginning, even for real degrees of freedom. The unstable thimble is defined as the union of the Steepest Descent (SD) paths stemming from a critical point.
Thimble regularisation of gauge theories
We started our discussion on motivations for thimbles putting forward the big issue of QCD at finite density. How far are we from actually tackling that? Honestly, quite a lot. We have in recent years taken some steps in that direction, but that has been done in the context of two theories (0 + 1 QCD [4] and the so called Heavy Dense QCD [5]) for which gauge invariance in practice does not show up in its full glory. Other groups have perhaps moved a bit further than we have done till now [6,7]. In the following we will try to pin down a sort of status report on our attempts at a thimble regularisation of gauge theories. This will make us discuss 2 YM theories and the inclusion of a -term.
Construction of the thimble
Mimicking thimble construction for gauge theories is not that difficult. The first step (complexification) amounts to
SU ( ) = → = ( + ) ∈ SL ( , C) .(2)
The main thing we should notice is that
SU ( ) † = − → − = − ( + ) = −1 ∈ SL ( , C) .
With this caveat in mind, we can proceed to defining the SA d dˆ(
; ) = ∇ ,ˆ[ ( )] ˆ( ; )(3)
which are written in terms of the Lie derivative
∇ ( ) = lim →0 1 − ( ) = =0 .
Notice that, since d d =∇ ,ˆ¯∇ ,ˆ+ ∇ ,ˆ∇ ,ˆw e have that
d d = 1 2 d d +¯ = 1 2 ∇ ,ˆ¯∇ ,ˆ+ ∇ ,ˆ∇ ,ˆ¯ = ∇ 2 ≥ 0 and d d = 1 2 d d −¯ = 1 2 ∇
,ˆ¯∇ ,ˆ− ∇ ,ˆ∇ ,ˆ¯ = 0, that is, the main properties we expect from the SA are satisfied. Among all the solutions of Eq. (3), we have to look for the ones whose union defines the thimble. Naively, we could think we need to consider those stemming from a critical point . The fact is critical points in gauge theories come along with an entire orbit, which is made by all the gauge replicas of the given critical point. This is not the end of the story, since complexification has left us with two possible candidates. The action is now invariant under the gauge group G = ( , C), so that one orbit we could think of is
M = { ∈ ( , C) | ∃ ∈ ( , C) : = }.
This is not the right choice, and we need to consider instead
N = { ∈ ( , C) | ∃ ∈ ( ) : = } ⊂ M .(4)
In general the relevant gauge group is H = ( ), and as a matter of fact the thimble itself is invariant under ( ) (as it should be), and not under ( , C). All in all, the thimble e.g. associated to = 0 for the (3) Yang-Mills action is defined by
J 0 := ∈ ( (3, C)) 4 | ∃ ( ) solution of Eq. (3) | (0) = & lim →−∞ ( ) ∈ N (0) .
(5) One could think that what we have gone through till now can be summarised as "going from critical points to critical submanifold". We have to admit we have been cheating a little bit: what we actually did (and this is actually the right thing to do) was to go from non-degenerate critical points to non-degenerate critical submanifolds [8]. In Sec. 1 we said that in order for the thimble construction to work we need a non-degenerate critical point, with no zero eigenvalue in the Hessian. Due to gauge modes, this is not the case for gauge theories. The good news is that this is not such a big problem: gauge invariance is realised in a very neat way on the thimble. Indeed the main gauge invariant property of the construction is summarised in the cartoon of the total number of degrees of freedom and nG = V (N 2 1) the number of gauge degrees of freedom, which means that n+ = V (d 1)(N 2 1). We can easily compute the Takagi vectors {v G(i) } spanning T U G 0 J0 given the Takagi vectors {v (i) } spanning TU 0 J0. Consider a couple of configurations U (t0) and U G (t0) with |ci| ⌧ 1, so that they are close to M0, that is 63
Uμ(n; t0) = e i P i civ (i) nμ,a T a U0μ(n) U Ĝ µ (n; t0) = e i P i civ G(i) nμ,a T a U G 0μ (n)
Let us set G(n) = e i gn,aT a
The previous considerations lead to setting U Ĝ µ (n; t0) = G(n)Uμ(n; t0)G † (n +μ), which imply 62 Directions tangent to M0 at U0 represent infinitesimal gauge transformations around U0. 63 We generically take |ci| ⌧ 1 in order not to leave TU J0 while leaving the critical point U . This condition is automatically ensured for directions corresponding to i > 0: for these directions ci = nie i t0 with t0 ! 1, so that we can safely take ni = O(1). For directions corresponding to i = 0, however, the coefficients ci have to be taken small explicitly. so that we get
Morse theory and gauge symmetry
In the last section we have introduced the thimble formalism for gauge theories. We now address the problem of gauge symmetry within the framework of Morse theory. We try to be as general as possible, deferring a more detailed discussion to the study of Yang-Mills theory in Section 10. We consider a set of fields {Uk} on a manifold Y over which we wish to integrate, with dimRY = n and a suitable complexification X , with dimRX = 2n. Let S : X ! C be a holomorphic function that is invariant under transformations of a gauge group G, which is the complexification of a compact gauge group H. We call g the Lie algebra of G and h the Lie algebra of H, with dimRh = nG and dimRg = 2nG. We assume that a critical point U 2 X of S changes non-trivially under transformations of G, that is U ! U G 6 = U . As a consequence, U belongs to a manifold of critical points continuously connected by transformations G 2 G. We call such manifold M M ⌘ U 2 X 9 G 2 G : U G = U ⇢ X which has dimRM = 2nG. On M the action S takes on the same value S(U ); thus, considering SR = <(S), the Hessian H(SR; U ) for U 2 M is degenerate. 27 In particular, the Hessian of SR is a 2n ⇥ 2n real, symmetric matrix with 2nG zero eigenvalues (corresponding to directions of gauge invariance of SR), n nG positive eigenvalues and n nG eigenvalues which are opposite in sign. We say that M is a non degenerate critical submanifold of X for SR : X ! R if dSR = 0 along M and the Hessian H(SR; U ) (for . Moreover, we will see that factors other than symmetry may lead to a degenerate Hessian (for example, torons in pure Yang-Mills theory); this case will be worth of a detailed discussion in Section 10.
28 Actually, things are much more involved, but, for the cases of our interest, our brief discussion suffices. See the point ( ) which is obtained by ascending from the critical point 0 : this point is uniquely defined by selecting a given direction on the tangent space to the thimble at 0 and a given ascend time . Ascent paths of this type are defined selecting directions associated to positive eigenvalues of the Hessian. As for (zero) gauge modes, their role is well understood by looking at the point ( ). All in all, if we take a SA from 0 (i.e. = 0), at any stage (i.e. from any ( )) we can perform a gauge transformation and this will take us to a point ( ( )) starting from which the SD (Steepest Descent) path will make us eventually land on another point on the gauge orbit attached to 0 ; this point ( 0 ) is obtained from 0 by the gauge transformation we choose.
We denote N (0) the orbit N associated to = 0. We provide a quick-and-dirty argument for what a non-degenerate critical submanifold is; all this can be much better understood from [8].
We are still thinking of the critical point = 0, which in the Wilson action is associated to its exponential 0 . This is always known by solving a convenient Takagi problem; see [3].
All that we have said till now is not the end of the story, and we have been once again cheating a little bit. In order to preserve the right number of zero modes, in YM theories we have to kill torons, which thing can be done by going for twisted boundary conditions. Because of this, our preferred critical point is the so-called twist-eater. (A good reference for all that has to do with this is [9].)
What we have, what we can do and what we lack
We have a (working) code (some results in Fig. 2) by which we can simulate 2 YM theories; to be definite, we have been mainly focusing on (2). A sign problem (an artificial one, in a sense) is generated by computing for complex values of the coupling. Notice that the theory is fully solved, so that we can check the results we get. One thing we do is hunting for critical points, the main goal being to compute on different thimbles. This is in the logic of (dis)proving whether one single thimble is enough to reproduce the known result (and this indeed appears not to be the case: see Fig. 2). In all that we have just mentioned there is indeed something we (dramatically) miss: there is no general proof of something like a thimble decomposition for gauge theories. This is the genuine motivation of ours for probing 2 YM theories: the solution is known in a way that is intriguing, sort of alluding to thimbles. (2) YM theory.
The closest to thimble decomposition for gauge theories we can currently think of getting
Indeed we think 2 YM theories provides us with an understanding which is the closest to thimble decomposition for gauge theories we can currently think of getting. What we mean is displayed (once again) in graphical form in Fig. 3. For a full account of the results we are going to quote a first reference is [10]. First of all look at the first row of Fig. 3. All the critical points that are classical solutions of the YM action have been classified by Atiyah and Bott in [8]. In [10] Witten first obtains the partition function as a sum over representations (this is a type of result which is well known to lattice practitioners; see later) on a generic Riemann surface of genus , which for = 1 reduces to the expression in the up-left corner of Fig. 3. Via a tool as simple as the Poisson resummation, he then turns this sum into a sum over critical points (up-right corner). Needless to say, this is the intriguing result we are mostly interested in: this really sounds like thimble decomposition. Now look at the second row. Down-left corner of Fig. 3 is the sum over 1 M partition function on the lattice is given by(See Giovanni's thesis for the
Z (pbc) ( ) = e V 1 X n=1 2 I n ( ) V
e re-written as: representation which is well known to the lattice community, i.e. the Migdal solution [11]. It is well known that one can take the continuum limit and go from down-left to up-left [12]. Finally, what we regard as the big issue: can we fill the down-right corner, i.e. provide a realisation of the mechanism in the first row in the lattice regularisation? The complete task would entail taking the continuum limit and go from down-right to up-right.
Z (pbc) ( ) = 1 X n=1 e V +ln (2 V V I n ( ) V ) Z (pbc) ( ) = 1 X n=1 e V +V ln 2+V ln I n ( ) V ln
Looking into -term
While we have been working on 2 YM theories, we have adapted our code to also include 4 YM in the presence of a -term. Once again we do not have yet definite results, but it is easy to list a few reasons for being interested in this.
• This is a prototype of a genuine sign problem in Euclidean Yang Mills.
• Being the equations of motion unaffected by a -term, the topological charge is conserved while you ascent on the thimble; in a way, this is a genuine way of computing at frozen topological charge.
• We have already made the point that the gauge invariance of the thimble is that of ( ); this holds for the topology as well.
• All in all, the really intruiging (super-hard!) goal would be that of finding the weights of the various topological sectors in the functional integral.
Needless to say, one should be well aware of the effects of lattice artifacts (they must show up, as in any lattice computations of topology). And of course, all this is indeed interesting, but it is not at all guaranteed that we can get positive results in a short time.
Figure 10 . 2 :
102Integration of SA curves (in red) starting from U0 as well as from a gauge-transformed configuration U G 0 , both belonging to the critical manifold M0 (in black). The thimble is pictorially represented with a bowl emanating from M0. The gauge transformation G connecting U (t) and U G (t) is shown in green. The same gauge transformation connects U0 and U G 0 in the critical manifold M0.the nG Takagi vectors of H(S; U0) with zero Takagi value 62 and N + U0 M0 spanned by the n+ Takagi vectors of H(S; U0) with positive Takagi value. The number of such vectors is n+ = n nG, with n = V d(N 2 1)
the very same as(4.4), with zk,c being the components of some tangent space basis vector.
U 2 M
2) is non degenerate on the normal bundle ⌫(M ). The normal bundle of M is subject to the decomposition ⌫(M ) = ⌫ + (M ) dimRN ± M = n nG. N + M is the (normal) space at U 2 M spanned by eigenvectors of H(SR; U ) with positive eigenvalues, while N M is the normal space at U spanned by eigenvectors of H(SR; U ) with negative eigenvalues. We now construct an n-cycle J attached to U . Morse theory [66, 34, 26] tells us 28 that such an n-cycle is constructed by considering all the SA curves (that is, those making up a stable thimble) attached to a middle dimensional manifold N ⇢ M (dimRN = nG, hence the name). The most natural choice for N is N ⌘ U 2 X 9 H 2 H : U H = U ⇢ M 27 We are generically referring to "gauge symmetry", but the reader should keep in mind that symmetries can also arise in scalar field theories; see [39] for a discussion of O(N ) symmetry and [36]
[34] for a detailed treating of symmetries and Morse theory. 34 real thing (a cartoon…) le is no longer attached to a oint, but to a critical submanifold e is gauge invariant under SU(N), ) usual) you pave your way for every point by ascenting as we do
Figure 1 :
1The cartoon for the gauge invariance property of the construction of Eq. 5.
Figure 2 :
2Average Plaquette vs real for the numerical (only dominant thimble) vs analytical results for 2
Figure 3 :
3The flow chart of a possible argument in favor of thimble decomposition in 2 (2) YM theory.
ConclusionsWe have provided a status report of our attempts at formulating gauge theories on thimbles. We do not have positive results to present, but we have a couple of lines of research that are hard to crunch, but fascinating. We would like to find a lattice realisation of the mechanism that in 2 gauge theories enables to go from a solution in terms of a sum over characters to a form which is a sum over critical points. 4 YM in the presence of a -term is a second subject which is rich of interesting features to investigate.AcknowledgmentsThe REALLY BIG ISSUE is the Fourier transform of f . ed to do is compute the fourier transform of exp ( ✏⇡ 2 n 2 ). This is easily seen ( ⇡n 2 ) ✏⇡ . But ✏ 0 = 4pi 2 ✏, giving us:Itamar's discussion wheref is the Fourier transform of f .All we need to do is compute the fourier transform of exp ( ✏⇡ 2 n 2 ). This is easi to be exp ( ⇡n 2 ) ✏⇡ . But ✏ 0 = 4pi 2 ✏, giving us:Notes from Itamar's discussionWe need solutions for the equation : Df = 0, with f 6 = 0.• We know that flat connections are defined as F = dA + [A, A] = 0. Those tr satisfy Df = 0. We are looking for other non-trivial solutions.uum limit of the 2D SU(2) YM lattice partinction the continuum limit are shown. All the formulas for first method are taken mf.nist.gov/10
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"arXiv:astro-ph/0504628v3 15 Sep 2005 GRAVOTURBULENT FORMATION OF PLANETESIMALS"
]
| [
"A Johansen \nMax-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany\n",
"H Klahr \nMax-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany\n",
"Th Henning \nMax-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany\n"
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| We explore the effect of magnetorotational turbulence on the dynamics and concentrations of boulders in local box simulations of a sub-Keplerian protoplanetary disc. The solids are treated as particles each with an independent space coordinate and velocity. We find that the turbulence has two effects on the solids. 1) Meter and decameter bodies are strongly concentrated, locally up to a factor 100 times the average dust density, whereas decimeter bodies only experience a moderate density increase. The concentrations are located in large scale radial gas density enhancements that arise from a combination of turbulence and shear. 2) For meter-sized boulders, the concentrations cause the average radial drift speed to be reduced by 40%. We find that the densest clumps of solids are gravitationally unstable under physically reasonable values for the gas column density and for the dust-to-gas ratio due to sedimentation. We speculate that planetesimals can form in a dust layer that is not in itself dense enough to undergo gravitational fragmentation, and that fragmentation happens in turbulent density fluctuations in this sublayer. Subject headings: instabilities -MHD -planetary systems: formation -planetary systems: protoplanetary disks -turbulence 1 The code is available at | 10.1086/498078 | [
"https://arxiv.org/pdf/astro-ph/0504628v3.pdf"
]
| 6,422,719 | astro-ph/0504628 | 180f6de10b1d962a50bb6daaa35a89792978b518 |
arXiv:astro-ph/0504628v3 15 Sep 2005 GRAVOTURBULENT FORMATION OF PLANETESIMALS
Draft version March 1, 2018 Draft version March 1, 2018
A Johansen
Max-Planck-Institut für Astronomie
Königstuhl 1769117HeidelbergGermany
H Klahr
Max-Planck-Institut für Astronomie
Königstuhl 1769117HeidelbergGermany
Th Henning
Max-Planck-Institut für Astronomie
Königstuhl 1769117HeidelbergGermany
arXiv:astro-ph/0504628v3 15 Sep 2005 GRAVOTURBULENT FORMATION OF PLANETESIMALS
Draft version March 1, 2018 Draft version March 1, 2018Preprint typeset using L A T E X style emulateapj v. 11/26/04Subject headings: instabilities -MHD -planetary systems: formation -planetary systems: proto- planetary disks -turbulence
We explore the effect of magnetorotational turbulence on the dynamics and concentrations of boulders in local box simulations of a sub-Keplerian protoplanetary disc. The solids are treated as particles each with an independent space coordinate and velocity. We find that the turbulence has two effects on the solids. 1) Meter and decameter bodies are strongly concentrated, locally up to a factor 100 times the average dust density, whereas decimeter bodies only experience a moderate density increase. The concentrations are located in large scale radial gas density enhancements that arise from a combination of turbulence and shear. 2) For meter-sized boulders, the concentrations cause the average radial drift speed to be reduced by 40%. We find that the densest clumps of solids are gravitationally unstable under physically reasonable values for the gas column density and for the dust-to-gas ratio due to sedimentation. We speculate that planetesimals can form in a dust layer that is not in itself dense enough to undergo gravitational fragmentation, and that fragmentation happens in turbulent density fluctuations in this sublayer. Subject headings: instabilities -MHD -planetary systems: formation -planetary systems: protoplanetary disks -turbulence 1 The code is available at
INTRODUCTION
Planets are believed to form from micrometer-sized dust grains that grow by collisional sticking in protoplanetary gas discs (Safronov 1969, see reviews by Lissauer 1993 andBeckwith et al. 2000). Once the bodies reach a size of around one kilometer, the growth to Moon-sized protoplanets and later real planets is achieved by gravitationally induced collisions (Thommes et al. 2003). Although significant progress has been made in the understanding of the initial conditions of grain growth (Henning et al. 2005), we nevertheless do not yet have a complete picture of how the solids grow 27 orders of magnitude in mass to form kilometer-sized planetesimals.
Growth by coagulation can take place when there is a relative speed between the solids. Various physical effects induce relative speeds at different grain size scales. This allows for a definition of distinct steps in the growth from micrometer dust grains to meter-sized boulders in a turbulent protoplanetary disc. Microscopic dust grains gain their relative speed due to Brownian motion. This process forms relatively compact cluster-cluster aggregates (Dominik & Tielens 1997). The speed of the Brownian motion falls rapidly with increasing grain mass, and so the time-scale for building up larger compact bodies this way becomes prohibitively large, compared to the lifetime of a protoplanetary disc.
When Brownian motion is no longer important, the relative speed is dominated by the differential vertical settling in the disc. The vertical component of the central star's gravity causes the gas to be stratified. Dust grains do not feel the pressure gradient of the gas and thus continue to fall towards the mid-plane with a velocity given by the balance between vertical gravity and the drag force. Larger grains fall faster than smaller grains due to the size-dependent coupling to the gas (actually bodies that are so massive that they are starting to decouple from the gas will rather move on inclined orbits relative to the disc, i.e. perform damped oscillations around the mid-plane). As they fall, they are thus able to sweep up smaller grains in a process that is qualitatively similar to rainfall in the Earth's atmosphere. Upon arrival at the mid-plane, the largest solids can reach sizes of a few centimeters (Safronov 1969). These bodies have grown as compact particle-cluster aggregates with a high porosity.
Turbulent gas motions cause the sedimented solids to diffuse away from the mid-plane Dubrulle et al. 1995), where they can meet and collide with a reservoir of microscopic grains. These tiny grains still hover above the mid-plane because their sedimentation time-scale is so long that turbulent diffusion can keep them well-mixed with the gas over a large vertical extent. Turbulence also plays a role for equal-sized macroscopic bodies by inducing a relative collision speed that is much larger than the Brownian motion contribution (Völk et al. 1980;Weidenschilling 1984).
When estimating the outcome of an interaction between macroscopic bodies, the issues of collision physics must be taken into account. For relative speeds above a certain threshold, the bodies are likely to break up when they collide rather than to stick (Chokshi et al. 1993;Blum & Wurm 2000). This is a problem for macroscopic bodies where the sticking threshold is a few meters per second. Fragmentation caused by high-speed encounters continuously replenish the reservoir of microscopic dust grains. These can then be swept up by the boulders that are lucky enough to avoid critical encounters. However, the sweeping up of smaller dust grains by a macroscopic body has its limitations when the relative speed exceeds some 10 meter per second (Wurm et al. 2001). At larger relative velocities of up to a hundred meters per second, which are likely to occur due to the high speed of larger bodies, the small grains will erode the boulder.
The time evolution of the size-distribution of solids can be calculated by solving the coagulation equation numerically (e.g. Wetherill 1990;Weidenschilling 1997;Suttner & Yorke 2001). Recently, performed numerical simulations of the coagulation for realistic disc environments. Starting with micrometer-sized grains only, they find that a narrow peak of 0.1-10 meter-sized boulders can form in 10 4 -10 5 years, when fragmentation is ignored. On the other hand, in a more realistic situation high speed impacts lead to fragmentation. Here find that once the size distribution reaches the meter regime, still around 75% of the mass is maintained in microscopic bodies, which are the fragments of larger bodies that have been destroyed in collisions. This picture is given some credit by the fact that microscopic dust grains are observed in protoplanetary discs of millions of years of age, whereas the timescale for depleting grains of those sizes is only around 1,000 years in the absence of fragmentation.
Besides the problem of getting macroscopic bodies to stick, meter-sized boulders quickly drift radially inward toward the central star due to their aerodynamic friction with the gas in a typical sub-Keplerian disc (Weidenschilling 1977). The drift time-scale can be as short as 100 years. To avoid evaporation in the inner disc or in the central star, the bodies must grow by least an order of magnitude in size (three orders of magnitude in mass) in a time shorter than this! A possibility to overcome the growth obstacles was suggested independently by Safronov (1969) and by Goldreich & Ward (1973). The general idea is that boulders sediment towards the mid-plane and form a particle sublayer that undergoes a gravitational instability, forming the planetesimals in a spontaneous event (gelation) rather than by continuous growth (coagulation). The weakest point in this model is that it requires a laminar disc in order to work. Even a tiny amount of turbulence in the disc will prevent the boulders from an efficient sedimentation towards the mid-plane, and the instability will never occur (Weidenschilling & Cuzzi 1993). Thus disc turbulence had always to be avoided in order to allow for self-gravity assisted planetesimal formation. However, even in a completely laminar disc, the settled dust induces a vertical shear in the gas rotation profile (Weidenschilling 1980;Nakagawa et al. 1986). This can be unstable to a Kelvin-Helmholtz instability. The subsequent Kelvin-Helmholtz turbulence puffs up the dust layer so that the densities needed for a gravitational instability are usually not achieved, unless a dust-to-gas ratio many times higher than the solar composition is adopted (Youdin & Shu 2002).
Nevertheless, solids can reach sizes of around one meter without the help of self-gravity. In this size regime the gradual decoupling from the gas motion enables the bodies to move independently from the gas. This can cause them to be trapped in turbulent features of the gas flow. An important theoretical discovery is that meter-sized boulders are concentrated in gaseous anticyclonic vortices (Barge & Sommeria 1995;Chavanis 2000;Johansen et al. 2004). Inside such vortices the dust density can locally be enhanced to values sufficient either for enhanced coagulation or even for gravitational fragmentation. Also the radial drift of particles trapped in the vortices is significantly reduced (de la Fuente Marcos & Barge 2001). Theoretical attention has furthermore been given to the trap-ping of dust grains in high pressure regions. Since dust grains do not feel pressure forces, any pressure-supported gas structure must cause dust grains to move in the direction of the pressure gradient (Klahr & Lin 2001;Haghighipour & Boss 2003;Klahr & Lin 2005). Recently, Rice et al. (2004) demonstrated that this can lead to large concentrations (a density increase of up to a factor 50) of meter-sized boulders in the high density spiral arms of self-gravitating discs. The same mechanism can drain millimeter-sized dust grains from the underdense regions around a protoplanet that is not massive enough to open a gap in the gaseous component of the disc (Paardekooper & Mellema 2004).
Giant long-lived vortices may form in protoplanetary disc due to a baroclinic instability (Klahr & Bodenheimer 2003), but the conditions for the baroclinic instability in protoplanetary discs are still not clear (Klahr 2004). Magnetorotational turbulence (MRI) on the other hand is expected to occur in all discs where the ionization fraction is sufficiently high (Gammie 1996;Fromang et al. 2002;Semenov et al. 2004). A search for dust concentrations in magnetorotational turbulence was done by Hodgson & Brandenburg (1998) who found no apparent concentrations. On the other hand, recently Johansen & Klahr (2005, hereafter referred to as JK05) found evidence for centimeter-sized dust grains being trapped in short-lived turbulent eddies present in magnetorotational turbulence. That work was, however, limited by the fluid description of dust grains, i.e. the friction time must be much shorter than the orbital period, and could not handle grains larger than a few centimeters.
In this paper, we expand the work done in JK05 by putting meter-sized dust particles, represented by real particles rather than by a fluid, into magnetorotational turbulence. We show that magnetorotational turbulence (Balbus & Hawley 1991) is not actually an obstacle to the self gravity-aided formation of planetesimals, but rather can be a vital agent to produce locally gravitational unstable regions in the solid component of the disc when the average density in solids would not allow for fragmentation. This process is very similar to the gravoturbulent fragmentation of molecular clouds into protostellar cores (Klessen et al. 2000;Padoan & Nordlund 2004).
DYNAMICAL EQUATIONS
For the purpose of treating meter-sized dust boulders we have adapted the Pencil Code 1 (see also Brandenburg 2003) to include the treatment of solid bodies as particles with a freely evolving (x, y, z)-coordinate on top of the grid. This is necessary because the mean free path of the boulders, with respect to collisions with the gas molecules, is comparable to the scale height of the disc. Thus the dust component can no longer be treated as a fluid, but must be treated as particles each with a freely evolving spatial coordinate x i and velocity vector v i . In other words, it is no longer possible to define a unique velocity field at a given point in space for the particles, because they keep a memory of their previous motion. Friction only erases this memory for small grains.
Drag Force
The particles are coupled to the gas motion by a drag force that is proportional to the velocity difference between the particles and the gas,
f drag = − 1 τ f (v i − u) .(1)
Here u is the gas velocity at the location of particle i and τ f is the friction time. The friction time depends on the solid radius a • and the solid density ρ • as
τ f = a 2 • ρ • min(a • c s , 9 2 ν)ρ ,(2)
where ν is the molecular viscosity of the gas, c s is the sound speed and ρ is the gas density. This expression is valid when the particle speed is much lower than the sound speed (Weidenschilling 1977). Using the kinetic theory expression for viscosity ν = c s λ/2, where λ is the mean free path of the gas molecules, the friction time can be divided into two regimes: the Epstein regime is valid when a • < 9/4λ. Here the mean free path of the gas molecules is longer than the size of the dust grain, so the gas can not form any flow structure around the object. The friction time is proportional to the solid radius in this regime. In the Stokes regime, where a • > 9/4λ, a flow field forms around the object. Now the friction time is proportional to solid radius squared, so the object decouples faster from the gas with increasing size. For an isothermal and unstratified disc, one can treat the friction time τ f as a constant. The distinction between the Epstein and the Stokes regime is then only important for translating the friction time into a solid radius (see end of this section).
To determine the gas velocity in equation (1) at the positions of the particles, we use a three-dimensional firstorder interpolation scheme, using the eight grid corner points surrounding a given particle. For multiprocessor runs the particles can move freely between the spatial intervals assigned to each processor using MPI (Message Passing Interface) communication.
Disc Model
We consider a protoplanetary disc in the shearing sheet approximation, but for a disc with a radial pressure gradient ∂ ln P/∂ ln r = α (or P ∝ r α ). In the shearing sheet approximation this gradient produces a constant additional force that points radially outwards (because the pressure falls outwards). Making the variable transformation ln ρ → ln ρ + (1/r 0 )αx, the standard isothermal shearing sheet equation of motion (e.g. Goldreich & Tremaine 1978) gets an extra term,
∂u ∂t + (u · ∇)u = −2Ω 0 × u + 3Ω 2 0 x − c 2 s ∇ ln ρ − c 2 s 1 r 0 αx .
(3) The terms on the right-hand-side of equation (3) are the Coriolis force, the centrifugal force plus the radial gravity expanded to first order, and the two terms representing local and global pressure gradient. The coordinate vector (x, y, z) is measured from the comoving radial position r 0 from the central source of gravity, with x pointing radially outwards and y along the Keplerian flow. At r = r 0 the Keplerian frequency is Ω 0 . The shearing sheet approximation is valid when all distances are much shorter than r 0 . The balance between pressure gradient, centrifugal force and gravity is given for a sub-Keplerian rotation of the disc,
u (0) y = − 3 2 Ω 0 x + c 2 s 2Ω 0 1 r 0 α ,(4)
where the first term on the right-hand-side is the purely Keplerian rotation profile, while the second (constant) term is the adjustment due to the global pressure gradient. We now measure all velocities relative to the sub-Keplerian flow using the variable transformation u → u + u 0 . This changes equation (3) into
∂u ∂t + (u · ∇)u + u (0) y ∂u ∂y = f (u) − c 2 s ∇ ln ρ .(5)
Here the last term on the left-hand-side represents the advection due to the rotation of the disc relative to the center of the box (which moves on a purely Keplerian orbit). The function f is defined as
f (u) = 2Ω 0 u y − 1 2 Ω 0 u x 0 .(6)
When making the same variable transformation in the equation of motion of the dust particles, there is however Run Note. -First column: name of run; second column: friction time; third column: global pressure gradient parameter; fourth column: maximum particle density in units of the average density; fifth column: radial velocity averaged over space and time; sixth column: predicted radial drift in a non-turbulent disc; seventh to tenth columns: velocity dispersion averaged over space and time. Averages are taken from 5 orbits and beyond. Grid cells with 0 or 1 particles have been excluded for the calculations of velocity dispersions.
Ω 0 τ f β max(n) vx v(
no global pressure gradient term to balance the extra Coriolis force imposed by the sub-Keplerian part of the motion, so the result is
∂v i ∂t = f (v i ) − 1 τ f (v i − u) + c 2 s 1 r 0 αx .(7)
The modified Coriolis force f appears again because of the presence of x i (t) in u 0 . The last term on the righthand-side reflects the head wind that the dust feels when it moves through the slightly sub-Keplerian gas. The reason that the term appears in the radial component of the equation of motion is that all velocities are measured relative to the rotational velocity of the gas. A dust particle moving at zero velocity with respect to the gas thus experiences an acceleration in the radial direction.
The explicit presence of r 0 in equation (7) is nonstandard in the shearing sheet. It may seem that the term vanishes for r 0 → ∞. But this is actually not the case, since the natural timescale of the disc, Ω −1 0 , also depends on r 0 , so that at large radii there is an immense amount of time at hand to let the tiny global pressure gradient force work. One can quantify this statement by dividing and multiplying by the scale-height H in the last term of equation (7) to obtain the result
∂v (i) x ∂t = . . . + c s Ω 0 H r 0 α .(8)
Here H/r 0 ≡ ξ is the ratio of the scale height to the orbital radius, a quantity that is below unity for thin discs. Depending on the temperature profile of a disc, the typical value of ξ is between 0.001 and 0.1. We define the pressure gradient parameter β as β ≡ αξ.
For the simulations, we adopt the following dynamical equations for gas velocity u, magnetic vector potential A, gas density ρ, particle velocities v i and particle coordinates x i :
∂u ∂t + (u · ∇)u + u (0) y ∂u ∂y = f (u) − c 2 s ∇ ln ρ + 1 ρ J × B + f ν (u, ρ)(9) ∂A ∂t + u (0) y ∂A ∂y = u × B + 3 2 Ω 0 A yx + f η (A) (10) ∂ρ ∂t + u · ∇ρ + u (0) y ∂ρ ∂y = −ρ∇ · u + f D (ρ) (11) ∂v i ∂t = f (v i ) + c s Ω 0 βx − 1 τ f (v i − u) (12) ∂x i ∂t = v i + u (0) yŷ(13)
The functions f ν , f η and f D are hyperdiffusivity terms present to stabilize the finite difference numerical scheme of the Pencil Code. This is explained in more detail in JK05. We shall ignore the effect of the global pressure gradient on the dynamics of the gas, since for ξ ≪ 1 the increase in density due to the global gradient is much smaller than the average density in the box. Thus we set simply u (0) y = −3/2Ω 0 x. We also ignore the contribution from the global density on the Lorentz force term in equation (9) and the advection of global density in equation (11). Furthermore we do not include vertical gravity in the simulations. This means that we solve exactly the same equations for the gas as in JK05, i.e. without radial pressure stratification. The radial drift of solids then originates exclusively from the dynamical equations of the particles.
We solve the dynamical equations (9)-(13) for various values of the friction time and of the box size. The typical resolution is 64 3 for a box size of 1.32H on all sides. A similar setup was used in JK05 to calculate the turbulent diffusion coefficient of dust grains in magnetorotational turbulence. In the present work we expand the model by letting 2,000,000 particles represent the dust grains. Thus the dust component is typically represented by approximately 8 particles per grid cell. We set the strength of the radial pressure gradient by the parameter β = −0.04. This would represent e.g. a disc with a global pressure gradient given by α = −1 and a scale-height-to-radius ratio of ξ = 0.04, which is typical for a solar nebula model (Weidenschilling & Cuzzi 1993). We consider friction times of Ω 0 τ f = 0.1, 1, 10. The translation from friction time into grain size depends on whether the friction force is in the Epstein or in the Stokes regime, but the two drag laws yield quite similar grain sizes in the transition regime. Thus, at the radial location of Jupiter in a typical protoplanetary disc, the friction time corresponds to grains of approximately 0.1,
Ω 0 τ f =0.1 Ω 0 τ f =10.0 Fig. 2.-
The number of particles in the densest grid cell as a function of time, here for runs B (decimeter-sized boulders) and C (decameter-sized boulders). The first shows only very moderate overdensities, whereas the latter is similar in magnitude to run A (metersized boulders), but with broader peaks.
1 and 10 meters in size.
The simulation parameters are given in Table 1. We let the boulders have random initial positions from the beginning and let them start with zero velocity.
PARTICLE CONCENTRATIONS
In Fig. 1 we plot the number of particles in the densest grid cell as a function of time for run A (meter-sized boulders, see Table 1). The average number of particles per grid cell is 7.6. Evidently there is more than 100 particles in the densest grid cell at most of the times, and at some times the number is even above 600. This is more than 80 times the average dust number density. In Fig. 2 we plot the maximum particle density for runs with Ω 0 τ f = 0.1 (run B, gray curve) and Ω 0 τ f = 10 (run C, black curve). The decimeter-sized boulders are obviously not as strongly concentrated as the meter-sized boulders, whereas the decameter-sized boulders have concentrations that are similar in magnitude to run A. The measured values of the maximum particle density for all the
−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 x/(c s Ω 0 −1 ) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 y/(c s Ω 0 −1 ) 0.97 1.03 Σ/Σ 0 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 x/(c s Ω 0 −1 ) −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 y/(c s Ω 0 −1 ) 0.0 5.0 Σ d /(ε 0 Σ 0 ) Fig. 3.
-Gas column density Σ (left panel) and dust column density Σ d (right panel). The gas column density only varies by a few percent over the box, but still a slightly overdense region is seen near the center of the box. The dust column density in the same region is up to 5 times the average dust column density. runs can be found in Table 2.
To examine whether some structures in the gas density are the source of the high particle densities, we plot in Fig. 3 the column densities of gas Σ and of dust particles Σ d at a time of 50.9 orbits for run A. The gas column density varies only by a few percent over the box, since the turbulence is highly subsonic, but a region of moderate overdensity is seen around the middle of the box. The dust column density is very high in about the same region as the gas overdensity, around a factor of five higher than the average dust column density in the box, so dust particles have moved from the regions that are now underdense into the overdensity structure near the center of the box.
We explore the radial density structure of the gas and the dust in the box in more detail in Fig. 4. Here the azimuthally averaged gas and dust column densities are shown as a function of radial position x and time t measured in orbits. Apparently large scale gas density fluctuations live for a few orbits at a constant radial position before decaying and reappearing at another radial position. The fluctuation strength is less than 1% of the average density. The dust density shows strong peaks at the locations of the gas density maxima. The explanation for this correlation is as follows. Locations of maximal gas density are also local pressure maxima. Such pressure maxima can trap dust grains (Klahr & Lin 2001;Haghighipour & Boss 2003) as they are locations of Keplerian gas motion. The inner edge of a pressure maximum must move faster than the Keplerian speed because the pressure gradient mimics an additional radial gravity. At the outer edge of a radial pressure enhancement the outwards-directed pressure gradient mimics a decreased gravity, and the gas must move slower than the Keplerian speed. Dust grains do not feel the pressure gradient and are thus forced to move into the pressure bump. In our simulations the radial gas overdensities have a typical lifetime at a given radial position on the order of a few orbits. When the gas overdensity eventually disappears, the particle overdensity is only slowly getting dissolved, and the particles drift and concentrate towards the location of the next gas overdensity. The gas density structure in the azimuthal and vertical directions does not show a similar density increase, and as expected there is also no significant concentration of particles with respect to these two directions. The density fluctuations thus have the form of two-dimensional sheets.
In Fig. 5 we plot the maximum density experienced by a 200 particle subset of the 2,000,000 particles during the 100 orbits. The distribution function ξ(n) is defined as the fraction of particles that have been the center of a number density of at least n over the size of a grid cell. The curves clearly show how large the concentrations are. For decimeter-sized boulders, 95% of them have experienced a 5 times increase in dust density, whereas only around 2% have been part of a 10 times increase. For meter-sized particles, 70% have been part of a 10 times increase in dust density, and 1% even took part in a 20 times increase. Particles of decameter-size had more than 10% taking part in a 30 times increase of dust density. This is very similar to the concentrations that Rice et al. (2004) find in the spiral arms of self-gravitating discs.
In Fig. 6 correlations between gas flow and particle density are shown for run D (without global pressure gradient). Here we have taken data at every full orbit, starting at 5 orbits when the turbulence has saturated, and calculated the average particle density in bins of various gas parameters. We also plot the spread in the particle density in each bin. The top two panels show the correlation with two components of the vorticity ω = ∇ × u. There is some correlation between vertical vorticity component and the particle density, but the spread in each vorticity bin is larger than the average value. The correlation indicates that some trap- show the fraction ξ(n) of particles that have been part of a given particle density during the 100 orbits.
For Ω 0 τ f = 0.1, only concentrations up to 10 are common, whereas for Ω 0 τ f = 1, 70% of the particles have experienced at least a 10 times increase in density and 2% even a 20 times increase. For massive boulders with Ω 0 τ f = 10, more than 10% were part of a 30 times increase in density.
ping of particles is happening in anticyclonic regions, and that regions of cyclonic flow are expelling particles (Barge & Sommeria 1995). Meter-sized particles should be optimally concentrated by vorticity, so the weak correlation between n and ω z is surprising, considering that for centimeter-sized particles, JK05 find an almost linear relation between n and ω z with very small spread. The explanation may be that the friction time is so high that particle concentrations stay together even after the gas feature which created them has decayed or moved to another location. The limited life-time of the concentrating features weakens the measured correlation with the gas flow.
The lower two panels of Fig. 6 show the correlation with divergence of pressure gradient flux and with gas density. In a steady flow, particles accelerate towards an equilibrium velocity where the drag force is in balance with the other forces working on the particles. The equilibrium velocity is
v = τ f ρ −1 (∇P − J × B) ≡ τ f F .(14)
This is the mechanism for pressure gradient-trapping. Places with a negative value of ∇ · F should produce a high particle density (see JK05). The correlation between ∇ · F and n is existent, but is very weak. The last panel, however, shows that there is a clear correlation between gas density and particle density, as is also evident from Fig. 4. All in all, the correlations, even though some of the are quite weak, give the necessary information about the source of the dust concentrations. The concentrations are primarily due to pressure gradienttrapping in the gas flow. There is also evidence of some vorticity-trapping happening on top of that.
Increases in density of up to two orders of magnitude will make a difference in the coagulation process, because at places of larger concentration more collisions (both destructive and constructive) are possible. Also there is a chance of increasing the density to such high values that a gravitational instability can occur in the densest places. We will consider this last point in more detail in Sect. 5. In the following section we show that the turbulence not only causes concentrations, but also changes the radial drift velocity of the boulders. -Correlations between particle number density and various gas parameters. The first row considers two components of the vorticity. There is some correlation between n and ωz, indicating that particles are trapped in regions of anticyclonic flow. In the second row we consider the correlation between particle density and pressure gradient flux (explained in the text) and gas density, respectively. The first correlation is very weak, whereas there is evidently a correlation between gas density and particle density, although the fluctuation bars are significant.
The global pressure gradient on the gas forces solids to fall radially inwards. If the gas motion in the disc was completely non-turbulent, then the equilibrium radial drift velocity arising from the head wind term present in equation (12)
would be v x = β Ω 0 τ f + (Ω 0 τ f ) −1 c s .(15)
We derived this expression by solving for ∂v/∂t = 0 in equation (12). The highest drift speed occurs for particles with Ω 0 τ f = 1 with a laminar drift velocity of v x /c s = β/2. We have checked by putting particles of different friction times into a non-turbulent disc that the measured drift velocities are in complete agreement with equation (15). The effect of a real turbulent disc on the average drift velocity is seen in Fig. 7. Here the average radial velocity of all the particles is shown as a function of time for run A. For reference we overplot the laminar drift velocity (v x = −0.02 c s ) from equation (15) and the timeaveraged drift velocity (v x = −0.012 c s ). The mean drift velocity is noticeably affected by the turbulence and its absolute value is reduced by 40% compared to the laminar value. The influence that turbulence can have on the mean drift velocity of the particles can be quantified with some simple analytical considerations. Considering the particles for a moment as a fluid with a number density scalar field n and a velocity vector field w, the average radial velocity can be calculated with the expression
w x = x1 x0 nw x dx n L x .(16)
Here we have weighted the drift velocity with the number density so that we are effectively measuring the average momentum. We consider now for simplicity particles that have been accelerated by the gas to their terminal velocity (eq.
[15] including the fluctuation pressure gradient),
w x = ǫc s β + ∂ ln ρ ∂x ,(17)
where ǫ is defined as ǫ = 1/[Ω 0 τ f + (Ω 0 τ f ) −1 ]. Inserting now equation (17) into equation (16), the resulting drift velocity is found to consist of two terms,
w x = ǫβc s + ǫc s x1 x0 n ∂ ln ρ ∂x dx n L x .(18)
The first term on the right-hand-side of equation (18) represents the contribution to the average drift velocity from the global pressure gradient (eq.[15]). The other term is an extra contribution due to any non-zero correlation between number density n and radial pressure gradient ∂ ln ρ/∂x. This situation is sketched in Fig. 8. Here we sketch the global density gradient β (full line) and a sinusoidal density fluctuation ln ρ(x) (dotted line). Particles concentrate in regions where the gas density fluctuation is positive, because there the divergence of the particle velocity is negative. Due to the total pressure gradient, the newly produced particle clumps drift inwards until the point where the outwards drift towards the fluctuation density maximum balances the inwards drift from the global pressure gradient. This is exactly around the location of the box in Fig. 8. Here the correlation between n and ∂ ln ρ/∂x leads to a positive value of the integral in equation (18). A closer inspection of Fig. 4 reveals that the dust overdensities are situated slightly downstream of the gas density fluctuation peaks, which is in good agreement with the the prediction in Fig. 8. If a significant fraction of the particles end up in such regions, the average drift speed is reduced 2 . For runs B and C, there is no significant reduction of the drift speed (see Table 2), but there the predicted drift speed is also ten times lower than for meter-sized objects. Thus the measurement is not as reliable because the random velocity fluctuations of the particles dominate over the radial drift.
Due to the periodic boundary conditions in the y-directions, density structures quickly pass the yboundaries, by shear advection, and thus possibly have some interference with themselves. To see the effect of the toroidal box size on the radial drift, we have run simulations with a box size of 1.32 × 5.28 × 1.32 (run E) and 1.32 × 10.56 × 1.32 (run F), keeping the resolution constant by adding the appropriate number of grid points in the y-direction. The time evolution of the mean radial drift velocity is shown in Fig. 9. It is evidently very similar to Fig. 7, so the toroidal size of the box does not influence the radial drift reduction noticeably. As seen in Table 2, the maximum particle density for runs E and F is quite high at 50 times the average density in the box, but not as high as in run A. However, simulations E and F only ran for 24 and 16 orbits, respectively, because of computational requirements due to the many grid points.
In simulations of the interaction between a planet and a magnetorotationally turbulent disc, Nelson & Papaloizou (2004) find that the average migration velocity of the planets is not changed by the presence of MRI turbulence (whereas the spread in drift velocity causes some planets to even drift outwards). On the other hand, recent simulations by indicate that the mean migration of planets can indeed change because of turbulence. The fluctuations in 2 A more graphic explanation of the speed reduction is to consider a car race over a distance of 100 km. Half of the distance is sand, where the cars can run 50 kilometers per hour, and the other half asphalt, where the cars go 150 kilometers an hour. The average speed of a single car reaching the finish line is less than 100 kilometers per hour, simply because that car spent more time on sandy terrain than on asphalt. migration speed are however much stronger than the average (so that hundreds of orbits are needed for a reliable estimate of the average). This is a very different kind of drift behavior than for the boulders in the current work, where the fluctuations in the drift speed are actually much smaller than the average. The presence of long-lived attracting regions in the gas may be the reason why boulders react on turbulence in a completely different way than planets do.
Diminishing the radial drift for meter-sized objects by roughly one half may not be saving the boulders from their fate of decaying into the star. One will have to investigate this process by additionally looking at the growth behavior of the boulders which are sweeping up small grains on their way inwards. This sweeping up is determined by the actual drift speed with respect to the local gas motion. Even if the mean drift speed is above the threshold for effective sticking, there will be phases of much lower radial drift, where growth can occur. The overdense regions would also greatly increase the rate of destructive encounters between larger bodies, and thus the reservoir of small bodies would be stronger replenished there. This would not only influence growth of larger bodies, but also possibly have observational consequences.
The present simulations are done in the gentle situation of turbulence in a local box. Global disc simulations have stronger turbulence and larger density fluctuations. One can predict that it would thus also lead to a larger decrease in radial drift speed. This would possibly give the meter-sized boulders enough time to grow to a size safe for radial drift. However, this yet has to be demonstrated in global simulations 3 .
GRAVITATIONAL INSTABILITY
We already showed that turbulence can strongly influence the growth of boulders by slowing them down and by concentrating them locally. These results can be incorporated into standard evolution codes for the solid ma-terial (e.g. Weidenschilling 1997;, which try to grow planetesimals from dust grains via coagulation. On the other hand the high local concentration can also lead to a different way of planetesimal formation, i.e. gelation. In the gelation case a cloud of boulders is so dense that gravitational attraction becomes important. While we will not study self-gravity by an N -body approach in this work (as one should), we want at least demonstrate by simple estimations under what conditions the concentration of boulders could clump into planetesimals.
The gravity constant G enters in self-gravity calculations, and thus the equations are no longer scale-free, but depend on the adopted disc model. We characterize a disc model by a column density Σ 0 , an average dust-togas mass density ratio ǫ 0 (for boulders of the considered size range) and a scale-height-to-radius ratio of ξ. Of course, ǫ 0 will be smaller than the global dust-to-gas ratio ∼ 0.02, because only a part of the mass will be present in boulders of the considered size range. We choose for simplicity the value ǫ 0 = 0.01, assuming that 50% of the total dust mass is in bodies of the considered size, and we shall later discuss in how far this value is reasonable for a protoplanetary disc.
The apparently large number of particles in our numerical simulations is still orders of magnitude away from any real number of boulders in the volume of the protoplanetary disc considered in our simulations. Thus it is necessary and validated to let one superparticle represent an entire swarm of many particles of similar location and velocity in the disc. Superparticle means in this context that one particle has the aerodynamic behavior of a single boulder, but represents a mass of trillions of such bodies as it mimics an entire swarm of protoplanetesimals. Similar assumptions are common in simulations of giant planet core formation from colliding planetesimals (Kokubo & Ida 2002;Thommes et al. 2003) as well as in cosmological N -body simulations (Sommer-Larsen et al. 2003). We let the simulation box represent the protoplanetary disc in the mid-plane. Each superparticle then contains the mass m = ǫ 1 ρ 1 V /N , where V is the volume of the box, N is the number of superparticles, and ǫ 1 and ρ 1 are the dust-to-gas ratio and the gas density in the mid-plane of the disc. We shall use the isothermal disc expression ρ 1 = Σ 0 /( √ 2πH) to calculate the mass density in the mid-plane.
To calculate the dust-to-gas ratio in the mid-plane, ǫ 1 , one needs to take into account the effect of vertical settling of solid material. Solids move in the direction of higher gas pressure. In the case of vertical stratification, that means that the boulders must sediment towards the mid-plane. An equilibrium is reached when the sedimentation is balanced by the turbulent diffusion, with diffusion coefficient D t (Schräpler & Henning 2004), away from the mid-plane. This leads to a Gaussian profile of the dust-to-gas ratio (Dubrulle et al. 1995),
ǫ = ǫ 1 exp[−z 2 /(2H 2 ǫ )] ,(19)
with the dust-to-gas ratio scale height given by the expression H 2 ǫ = D t /(τ f Ω 2 0 ). The dust-to-gas ratio at z = 0 is where H = c s Ω −1 0 is the scale height of the gas. We now proceed by writing the turbulent diffusion coefficient as D t = δ t c 2 s Ω −1 0 , where δ t is the turbulent diffusion equivalent of α t of Shakura & Sunyaev (1973). Then the mid-plane dust-to-gas ratio ǫ 1 can be written as
ǫ 1 = ǫ 0 H H ǫ 2 + 1 ,(20)ǫ 1 ǫ 0 = Ω 0 τ f δ t + 1 ≈ Ω 0 τ f δ t ,(21)
where the approximate expression is valid for Ω 0 τ f ≫ δ t . For δ t = α t = 0.002 and Ω 0 τ f = 1, this gives ǫ 1 ≈ 22.4ǫ 0 , so starting from a dust-to-gas ratio of ǫ 0 = 0.01, the mid-plane dust-to-gas ratio can be expected to rise to ǫ 1 = 0.22 due to vertical settling. Such a low dustto-gas ratio alone will not for any physically reasonable column density cause gravitational fragmentation (Goldreich & Ward 1973) or be subject to vertical stirring by the Kelvin-Helmholtz instability (the Richardson number Ri is around unity, see e.g. Sekiya 1998, and stratification with Ri > 0.25 should be stable). Even at such a high dust-to-gas ratio we are still in the gasdominated regime where the back-reaction from the dust on the gas can be neglected. The turbulent dust concentrations are assumed to occur in such a vertically settled dust layer. Now the most overdense regions will have a dust-to-gas ratio of unity and beyond. But we have measured that only about 3% of the grid cells have a dust-to-gas ratio of above unity at any given time, and thus it is still reasonable as a first approximation to ignore the back-reaction of the dust on the gas, although a more advanced study should include this effect as well.
To find out if a given overdense clump is gravitationally unstable, we shall compare the different time-scales and length-scales involved in fragmentation by self-gravity in a Jeans-type stability analysis. First we investigate if the clump is gravitationally bound. We consider a clump of radius R, mass M and velocity dispersion σ. The velocity dispersion must include the dispersion due to the background shear. For such a clump with a given mass to be gravitationally unstable, it must have a radius that is smaller than the Jeans radius given by
R J = 2GM σ 2 .(22)
If this first criterion is fulfilled, then it is also important that the collapse time-scale of the structure is shorter than the life-time of the overdense clump t cl . Only then we can be sure that the changing gas flow will not dissolve the concentration before it has had time to contract significantly. The fragmentational collapse happens on the free-fall time-scale
t ff = R 3 GM .(23)
The condition for gravitational instability is now that R < R J and that at the same time t ff < t cl . We do not have to check separately that the collapse happens faster than a shear time t sh = Ω −1 0 , since the effect of the background shear is already included in the velocity dispersion.
We now try to find out the smallest value of Σ 0 that gives rise to a gravitational instability. Then we can see whether this is a value that occurs in nature or not.
For Ω 0 τ f = 1 (run A) the minimum value of the column density turns out to be around Σ 0 = 900 g cm −2 (6 times the minimum mass solar nebula value at 5 AU), whereas Fig. 10, but for run C (decameter-sized bodies) at a time of 53 orbits. Here the minimum mass solar nebula column density is sufficient to have a gravitational instability. This is mainly because the velocity dispersion is smaller than for run A. Also the high density region has a larger extent.
for Ω 0 τ f = 10 (run C), a gravitationally unstable cluster of protoplanetesimals is achieved already at the minimum mass solar nebula value Σ 0 = 150 g cm −2 .
First run A is considered. Here we can calculate the mass in each superparticle. With Σ 0 = 900 g cm −2 , ξ = 0.04, r = 5 AU and ǫ 1 = 0.22, we get ρ 1 = 1.2 × 10 −10 g cm −3 and m = 8 × 10 20 g. Thus each superparticle represents about 3 × 10 14 meter-sized protoplanetesimals. This is five orders of magnitude more mass than in a kilometer-sized planetesimal, but since we are interested in identifying gravitationally bound regions with the mass of thousands and thousands of planetesimals, this is not a problem. Actually resolving the mass of even one single planetesimal with meter-sized objects would require on the order of a billion particles, which is way beyond current computational resources.
We examine the region around the densest grid point of run A at a time of t = 50.9 orbits in more detail. This time is chosen because there occurs a large concentration of particles, see Fig. 1. We consider the j nearest particles to the densest point and calculate for j between 1 and 200,000 the particle number density n, the velocity dispersion σ and its directional components, the free-fall time t ff , and the radius of the clump together with the Jeans radius R J . The results are shown in Fig. 10. It is reasonable to require at least j = 100 for a measurement to be statistically significant (for j ≥ 100 the relative counting error falls below 10%, see e.g. Casertano & Hut 1985). It is also reasonable to require that the size of the clump be larger than the size of a grid cell, since any structure in the concentration within a single grid is not well-resolved. The same is true for the velocity dispersion. At j = 100 the dust number density is more than 130 times the average, but the radius of the j = 100 clump is only around 0.007, which is smaller than the grid cell radius of δx/2 = 0.01. At j ≃ 500 the clump has the size of a grid cell, and here the number density is more like 100 times the average. This must be multiplied by the enhancement by sedimentation, which is around 20, to give a dust-to-gas ratio increase by a factor of 2000 compared to the original value in the disc. The velocity dispersion is around σ ∼ 0.02 . . . 0.03 c s . That includes the velocity dispersion due to the background shear, but this is not a very important effect anyway because the size of the overdense clump is very small. At small scales the velocity dispersion is completely dominated by the radial component, according to Fig. 10, whereas the shear only takes over at larger scales.
The free-fall time is a bit below the clump life-time, which is typically one shear time (see insert in Fig. 1; note that the time unit is in orbits). For calculating the Jeans radius we have had to adopt a column density as high as Σ 0 = 900 g cm −2 in order to have the clump to be gravitationally unstable. This is mainly due to the high velocity dispersion. The radius of the clump is around one Jeans radius at j = 1000, so the clump is gravitationally bound at this scale and would be subject to further contraction by self-gravity. The gravitationally unstable region is around three grid cells in diameter, but even though this is well within the dissipative scales of the turbulence, the effect of the unresolved turbulence on the motion of the particles should be very little, as such small scale turbulence has short life-times and low amplitudes compared to the large scales. Extrapolating the resolved large scale turbulence to the grid-scale with a Kolmogorov law gives lower turbulent velocities than the particle velocity dispersion that we already measure at the grid scale. Thus we conclude that the unresolved turbulence has little or no influence on the particle dynamics. The concentrations and velocity dispersions are exclusively driven by the large resolved scales of the gas motion.
The solid size of the forming object would be roughly 400 km if all the 1000 superparticles end up in just one large body. On the other hand, the outcome of such a collapse may also favor the further fragmentation of the clump. This all depends on how the velocity dispersion behaves with increasing density. In the N -body simulations of Tanga et al. (2004) gas drag works as an efficient way to dissipate the gravitational energy that is released in the contraction of protoplanetesimal clusters. Only such simulations, that include self-gravity and gas drag, could show the further evolution of the overdense boulder clumps that we see in the present work.
For decameter-sized bodies (run C), we plot in Fig. 11 the same quantities as in Fig. 10 around the densest point at a time of 53 orbits. This time we adopt the minimum mass solar nebula column density of Σ 0 = 150 g cm −2 , which gives a mid-plane density of ρ 1 = 2×10 −11 g cm −3 .
Because of the high friction time, the dust-to-gas ratio in the mid-plane (eq.[21]) is now 0.71. The Richardson number is correspondingly lower at around Ri = 0.4, so it is still stable to Kelvin-Helmholtz instability. The mass of the individual superparticles is here m = 4 × 10 20 g. The density is slightly smaller than for meter-sized bodies, at statistically significant counts around 50 times the average, but the velocity dispersion is lower, and also the overdense region is much larger than it was for metersized boulders. Thus already a minimum mass solar nebula can produce a gravitational instability. The unstable region is as large as 10 grid cells in diameter, and contains around 10 5 particles. The size of a solid object consisting of this number of superparticles is roughly 1,400 kilometers. Again there is also the possibility that millions of 10-kilometer objects form instead.
The preceding calculations are of course only an estimation of the potential importance of self-gravity. In a real protoplanetary disc there will be a distribution of dust grain sizes present at any time. If e.g. fragmentation is important, as discussed in the introduction, then the greater part (80%) of the mass may still be present in bodies that are well below one meter in radius . With only 20% of the mass in the size range between one and ten meters, the critical column density could be as much as a factor of two higher than stated above. However, this is still in the range of the masses derived for circumstellar discs. So the qualitative picture that the clumps are gravitationally unstable for physically reasonable gas column densities is robust.
To quantify the velocity dispersion in the entire box, we have calculated the average values over all the grid cells. The results are shown in the last four columns of Table 2. Grid cells with 0 or 1 particles have been excluded from the average because the velocity dispersion is per definition zero in these underresolved cells.
The meter-sized bodies have the highest velocity dispersion, around σ 1 ≈ 0.02 c s , whereas decimeter bodies have σ 0.1 ≈ 0.014 c s and decameter bodies have a value of σ 10 ≈ 0.017 c s . These values are similar to the turbulent velocities of the gas at the largest scales of the box (see Fig. 2 in JK05), which again shows that these large scales are the drivers of the particle dynamics. Interestingly run D, which is similar to run A only without the radial pressure gradient, has the same velocity dispersion as run A, so the radial pressure gradient does not add extra velocity dispersion to the boulders. The toroidal component of the velocity dispersion is similar for all the runs because it is dominated by the shear over a grid cell. Run C has a twice as large radial velocity dispersion as run B. This can be explained because the large particles in run C react much slower to the local behavior of the gas, and thus particles of different velocities and histories are mixed in together.
The behavior of the velocity dispersion with increasing dust density is relevant for gravitational instability calculations. The average velocity dispersion, and the fluctuation width, as a function of the number of particles in a grid cell is shown in Fig. 12. Again it is evident that the velocity dispersion for Ω 0 τ f of unity is largest. For all runs the velocity dispersion typically rises until there are around 50 particles in the cell. Then the dispersions stay constant all the way to 200 particles. Thus the equation of state of the particles is isothermal, at least up to 30 times the average dust density.
SUMMARY AND DISCUSSION
We have considered the effect of magnetorotational turbulence on the motion of dust particles with a freely evolving space coordinate. The particle treatment was necessary over the fluid treatment, because the mean free path of the macroscopic dust boulders is so long that they can no longer be treated as a fluid. The use of magnetorotational turbulence may not be completely justified in the mid-plane of the disc where the ionization fraction due to radiation and cosmic particles is low. But due to its Kolmogorov-like properties, where energy is injected at the unstable large scales and then cascades down to smaller and smaller scales, magnetorotational turbulence can be seen as a sort of "generic disc turbulence".
We find that the turbulence acts on the particles by concentrating meter-sized boulders locally by up to a factor of 100 and by reducing their radial drift by 40%. Both the concentrations and the reduced radial drift happen because the dust particles are temporarily trapped in radial density enhancements. One would not expect such structures to be long-lived in a general turbulent flow, but magnetorotational turbulence in accretion discs is subject to a strong shear that favors elongated toroidal structures. In the presented simulations the typical lifetime of the structures is on the order of a few orbits, corresponding to tens or even hundreds of years in the outer parts of a protoplanetary disc. When the density structures eventually dissolve, new structures appear at other locations. We find a strong correlation between a gas column density of a few percent above the average and a several times increase in the dust column density. We have also seen some evidence for increased dust density in regions of anticyclonicity, but the long friction time of the dust particles makes it difficult to identify the gas flow that caused a given concentration, because the concentration may drift away from the creation site.
The large concentrations naturally occur near the grid scale. In finite resolution computer simulations the dissipative length scale must necessarily be moved from the extremely small dissipative scales of nature to the smallest scales of the simulation box. Thus the turbulence is not well-resolved near the grid scale. On the other hand, the concentrations are driven by the largest scales of the turbulence, because there are the largest velocities and the longest lived features (Völk et al. 1980). Already the other well-resolved but slightly smaller scales fluctuate too quick and at too low speeds to influence the path of an object that is one meter in size or larger. This argument is given support by the fact that we measure particle velocity dispersions in the grid cells that are comparable to the velocity amplitude of the gas at the largest scales of the simulation. Thus, one should not expect higher resolution to change the concentrations or the velocity dispersions significantly.
Our estimation of the minimum gas column density that would make the densest protoplanetesimal clumps gravitationally unstable is necessarily based on many assumptions. We assumed that half of the dust mass in the disc was present in bodies of the considered size, whereas in real discs an even larger part of the dust mass may be bound in small fragments that result from catastrophic collisions. We also ignored the back-reaction from the dust on the gas. The background state has, both for meter and decameter bodies, a dust-to-gas ratio just below unity (where the back-reaction becomes important). The effect of dust drag on the magnetorotational instability has to our knowledge never been considered. One can speculate that the drag force will mimic a strong viscosity and thus disable the source of turbulence where the dust density is high. For the treatment of Kelvin-Helmholtz instability we based it simply on a criterion on the Richardson number Ri. There is some indication that this may be too simplistic and that in protoplanetary discs much higher Richardson numbers are also unstable (Gómez & Ostriker 2005), but one can also speculate that the full inclusion of dust particles in simulations of Kelvin-Helmholtz turbulence would show strong local concentrations like we see here for magnetorotational turbulence. Thus the exact values of six times the minimum solar nebula for meter-sized boulders and just the minimum mass solar nebula for decameter-sized boulders should only be considered as rough estimates. Still, the result that the clumps are gravitationally unstable for reasonable gas column densities is robust enough to warrant further investigations that include treatment of self-gravity between the boulders.
Thus we find that the gravoturbulent formation of planetesimals from the fragmentation of an overdense swarm of meter-sized rocks is possible. Turbulence is in this picture not an obstacle, but rather the ignition spark, as it is responsible for generating the local gravitationally bound overdensities in the vertically sedimented layer of boulders.
Computer simulations were performed at the Danish Center for Scientific Computing in Odense and at the RIO cluster at the Rechenzentrum Garching. Our re-search is partly supported by the European Community's Human Potential Programme under contract HPRN-CT-2002-00308, PLANETS. We would like to thank the anonymous referee for a number of useful comments that helped to greatly improve the original manuscript.
Fig. 1 .
1-The number of particles in the densest grid cell as a function of time for run A (meter-sized boulders). The maximum density is generally around 20 times the average, but peaks at above 80 times the average particle density. The insert shows a magnification of the time between 50 and 51 orbits.
Fig. 4 .Fig. 5 .
45-Azimuthally averaged gas and dust column densities as a function of radial position relative to the center of the box x and time t. Black contour lines are shown at gas density fluctuations of 0.5% from the average value. Large scale density fluctuations are seen to have lifetimes on the order of a few orbits before moving to other radial positions. The dust column density peaks strongly at the locations of the maximal gas column density. -Distribution of maximum particle densities. The curves
Fig. 6.-Correlations between particle number density and various gas parameters. The first row considers two components of the vorticity. There is some correlation between n and ωz, indicating that particles are trapped in regions of anticyclonic flow. In the second row we consider the correlation between particle density and pressure gradient flux (explained in the text) and gas density, respectively. The first correlation is very weak, whereas there is evidently a correlation between gas density and particle density, although the fluctuation bars are significant.
Average radial particle velocity as a function of time for meter-sized boulders. The non-turbulent drift velocity is v (lam) x = −0.02 cs (indicated with a dashed line), while the average drift velocity in the turbulent case is only around vx = −0.012 cs, a reduction by around 40% in speed.
Fig. 8 .Fig. 9 .
89-Sketch of how turbulent density fluctuations can cause the average drift velocity to change. The full line shows the global density as a function of radial distance from the center of the box. On top of this we sketch a large-scale sinusoidal density fluctuation (dotted line) and the total density (dashed line). Dust particles are concentrated in the positive part of the fluctuation. At the same time the concentration drifts towards the location of the box where the total drift speed is zero. If a significant fraction of the particles end up in such regions, then the average drift speed can decrease. -Drift velocity for simulations E and F with larger ydomains. The expected drift velocity in a laminar disc is indicated with a dashed line. The measured drift velocity is approximately the same as for the cube simulations, so the periodic y-boundary is not the reason for the reduced drift speed. Rather it is a side-effect of trapping the particles in radial density enhancements.
Fig
. 10.-Particle number density n in units of average density n 0 , velocity dispersion σ in units of sound speed cs, free-fall time t ff relative to the clump life-time t cl , and clump radius R together with Jeans radius R J , all as a function of the number of included particles around the densest grid point in the box at a time of 50.9 orbits of run A. The vertical and horizontal dot-dot-dot-dashed lines indicate the regions of gravitational instability for the choice of disc model parameters.
Fig. 11.-Same as Fig. 10, but for run C (decameter-sized bodies) at a time of 53 orbits. Here the minimum mass solar nebula column density is sufficient to have a gravitational instability. This is mainly because the velocity dispersion is smaller than for run A. Also the high density region has a larger extent.
Fig
. 12.-Average velocity dispersion and fluctuation interval as a function of the number of particles in a grid cell. The dispersion rises until there are around 50 particles in a grid cell, and is then constant up to 200 particles, or around 30 times the average dust density. This corresponds to an isothermal equation of state for the boulders.
TABLE 1
1size of the box measured in scale heights; fourth column: grid dimension; fifth column: number of particles per grid cell; sixth column: friction time; seventh column: global pressure gradient parameter; eighth column: number of orbits that the simulation has run.Simulation parameters
Run
N
Lx × Ly × Lz
nx × ny × nz n 0 Ω 0 τ f
β
∆t
A
2 × 10 6
1.32 × 1.32 × 1.32
64 × 64 × 64 7.6
1.0 −0.04 100
B
2 × 10 6
1.32 × 1.32 × 1.32
64 × 64 × 64 7.6
0.1 −0.04 100
C
2 × 10 6
1.32 × 1.32 × 1.32
64 × 64 × 64 7.6
10.0 −0.04 100
D
2 × 10 6
1.32 × 1.32 × 1.32
64 × 64 × 64 7.6
1.0
0.00 100
E
2 × 10 6
1.32 × 5.28 × 1.32 64 × 256 × 64 1.9
1.0 −0.04
24
F
2 × 10 6 1.32 × 10.56 × 1.32 64 × 512 × 64 1.0
1.0 −0.04
16
Note. -First column: name of run; second column: number of particles; third
column:
TABLE 2 Results
2
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[
"Spectral analysis of the Neumann-Poincaré operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance *",
"Spectral analysis of the Neumann-Poincaré operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance *"
]
| [
"Younghoon Jung ",
"Mikyoung Lim "
]
| []
| []
| We consider the Neumann-Poincaré operator on a planar domain enclosed by two touching circular boundaries. This domain, which is a crescent-shaped domain or touching disks, has a cusp at the touching point of two circles. We analyze the operator via the Fourier transform on the boundary circles of the domain. In particular, we define a Hilbert space on which the operator is bounded, self-adjoint. We then obtain the complete spectral resolution of the Neumann-Poincaré operator. On both the crescent-shaped domain and touching disks, the Neumann-Poincaré operator has only absolutely continuous spectrum on the closed interval [−1/2, 1/2]. As an application, we analyze the plasmon resonance on the crescent-shaped domain and touching disks. | null | [
"https://arxiv.org/pdf/1810.12486v5.pdf"
]
| 246,822,911 | 1810.12486 | 976dac62170cd7755dd61d62239dd22101b5105d |
Spectral analysis of the Neumann-Poincaré operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance *
13 Feb 2022 February 15, 2022
Younghoon Jung
Mikyoung Lim
Spectral analysis of the Neumann-Poincaré operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance *
13 Feb 2022 February 15, 2022AMS subject classifications 35P05, 35J05, 31A10 Key words Neumann-Poincaré operatorTouching disksSpectral resolutionResonance
We consider the Neumann-Poincaré operator on a planar domain enclosed by two touching circular boundaries. This domain, which is a crescent-shaped domain or touching disks, has a cusp at the touching point of two circles. We analyze the operator via the Fourier transform on the boundary circles of the domain. In particular, we define a Hilbert space on which the operator is bounded, self-adjoint. We then obtain the complete spectral resolution of the Neumann-Poincaré operator. On both the crescent-shaped domain and touching disks, the Neumann-Poincaré operator has only absolutely continuous spectrum on the closed interval [−1/2, 1/2]. As an application, we analyze the plasmon resonance on the crescent-shaped domain and touching disks.
Introduction
Spectral analysis of the Neumann-Poincaré (NP) operator has received much attention in recent years due to its applications to electromagnetic problems in metamaterials, such as localized surface plasmon resonance of nanoparticles and invisibility cloaking [1,5,12,13,19,20,32,33]. The NP operator is a singular integral operator which naturally appears when one solves interface problems for the Laplacian by using the layer potentials. More precisely, given a simply connected bounded Lipschitz domain Ω ⊂ R 2 , the NP operator K * ∂Ω is defined by
K * ∂Ω [ψ](x) = p.v. 1 2π ∂D x − y, ν x |x − y| 2 ψ(y) dσ(y), x ∈ ∂Ω,(1.1)
for a density function ψ ∈ L 2 (∂Ω), where p.v. denotes the Cauchy principal value, and ν x denotes the outward unit normal vector to ∂Ω at x ∈ ∂Ω. The NP operator is similarly defined in three dimensions [25,39]. In this paper, we investigate the spectrum of the Neumann-Poincaré operator on a planar domain that is enclosed by two touching circular boundaries: a crescentshaped domain and touching disks (see Figure 1.1). This domain has a cusp at the touching point of the two boundary circles. To the best knowledge of the authors, this is the first article providing the spectral resolution of the NP operator on a planar domain with a cusp. For the case of a Lipschitz domain (without a cusp point on its boundary), the spectral property of the NP operator has been studied in many literatures. Let us review some essential results. We refer to a review article [7] and the references therein for more results. The NP operator is not symmetric on L 2 (∂Ω) unless Ω is a disk or a ball [30]. However, it can be realized as a self-adjoint operator on H −1/2 0 (∂Ω) with a new inner-product which is differently defined but equivalent to the original inner-product, based on Plemelj's symmetrization principle [4,22,26]. Here, H −1/2 0 (∂Ω) is the Sobolev space H −1/2 (∂Ω) with the mean-zero condition. We denote by H * the space H −1/2 0 (∂Ω) equipped with the new inner product. Since K * ∂Ω is self-adjoint on H * , its spectrum σ(K * ∂Ω ) is a closed set contained in the real line. In fact, the spectrum of the NP operator on L 2 0 (∂Ω) (or H * ) is contained in (−1/2, 1/2) [16,17,25] (see also [24,27] for the permanence of the spectrum for the NP operator with different norms).
For a C 1,α domain, K * ∂Ω on H * is compact as well as symmetric so that it admits only discrete real eigenvalues, namely λ j , as its spectrum, where zero is the only possible accumulation point. The NP operator admits the spectral decomposition
K * ∂Ω = ∞ j=1 λ j ψ j ⊗ ψ j ,(1.2)
where ψ j are eigenvectors corresponding to eigenvalues λ j . We refer to [6,21,34] for the decay estimates for the eigenvalues of the NP operator on a planar domain (see also [9] for the symmetricity of the spectrum on a planar domain). For simple shapes such as disks or ellipses, the complete sets of eigenvalues are known [2]. We refer to [2,18] for the eigenvalues of K * ∂Ω of an ellipse or an ellipsoid and to [3] for the eigenvalue property of the NP operator of tori.
The NP operator can also be defined for domains with separated components. For example, the eigenvalues of a domain consisting of two separated unit disks were explicitly expressed in terms of the distance between the disks [12] (see also [13,31]). For a Lipschitz domain with corners, the NP operator admits a continuous spectrum as well as eigenvalues, and it can be decomposed into three parts: the absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum (namely, σ ac (K * ∂Ω ), σ sc (K * ∂Ω ) and σ pp (K * ∂Ω ), respectively). It holds the following integral expression by the spectral theorem on a bounded, self-adjoint operator on a Hilbert space (see, for instance, [38,40]):
K * ∂Ω = σ(K * ∂Ω ) t dE(t),(1.3)
where E(t) is a resolution of identity for K * ∂Ω . Various studies have investigated the spectral properties of the NP operator for cornered domains [20,23,29,36,37]. It was shown that the essential spectrum on a bounded planar domain with corners is an interval determined by the corner angles [20,23,35,36,37]. For intersecting disks, the complete spectral resolution of the NP operator was obtained [23], where σ(K * ∂Ω ) consists of only the absolutely continuous spectrum. In [20], a numerical method to determine the spectrum was developed by using its relation to the plasmon resonance rate. We refer to see [4,7,10,11,14,29] for more results on the spectral properties of the NP operator and plasmon resonance. Numerical examples obtained in [20] show that all three kinds of spectrums (that is, σ ac (K * ∂Ω ), σ sc (K * ∂Ω ) and σ pp (K * ∂Ω )) can appear depending on the domains. The existence of embedded eigenvalues within the essential spectrum was verified numerically [20] and analytically [29]. Also, it was shown that infinitely many embedded eigenvalues appear for a perturbed sphere by smoothly attaching a conical singularity [28].
Touching disks can be considered as the limiting geometry of two different types of shapes. One is the limiting geometry of two separated disks as the distance between them tends to zero, and the other is that of intersecting disks whose external corner angle tends to zero. The spectrum σ(K * ∂Ω ) for the domain consisting of two separated unit disks with the distance δ > 0 is a sequence of eigenvalues given by (see [12])
λ ± n = ± (1 + δ − δ(2 + δ) ) 2n 2 , n = 1, 2, . . . . (1.4)
As δ tends to zero, λ ± n become densely located in [−1/2, 1/2]. For intersecting disks with external angle θ ∈ (0, π) at the corner points, it holds that (see [23])
σ(K * ∂Ω ) = σ ac (K * ∂Ω ) = [−b, b], b = |1/2 − θ/π|. (1.5)
As the external angle θ tends to zero, the spectral bound b tends to 1/2. The extension of the results on the spectrum of the separated disks and intersecting disks, (1.4) and (1.5), to touching disks is not straightforward as it involves the layer potential technique on a domain with a cusp point, which, to the best knowledge of the authors, has not been established. We need to generalize the NP operator defined on a Lipschitz domain to the considered domains with a cusp point in a suitable function space.
In the present paper, for Ω a crescent-shaped domain and touching disks, we first define the NP operator on L 2 0 (∂Ω) as a boundary integral operator similarly to the Lipschitz domain case. We then further investigate the NP operator via the Fourier transform on the boundary circles of the domain. More precisely, two touching circular boundaries of the domain are mapped to two parallel lines with a Möbius transformation, and the Fourier transform is applied on the two parallel lines. By using the expression of K * ∂Ω on L 2 0 (∂Ω) in terms of the Fourier transform, we define a Hilbert space (denoted by K −1/2 0 ), which is analogous to H * of the Lipschitz domain case, and generalize K * ∂Ω on L 2 0 (∂Ω) to K −1/2 0 such that K * ∂Ω is bounded, self-adjoint. As a main result, we derive the complete spectral resolution of the NP operator on the space K −1/2 0 . It turns out that the spectrums of the NP operators on a crescent-shaped domain and touching disks are both σ(K * ∂Ω ) = σ ac (K * ∂Ω ) = [−1/2, 1/2]. Note that this is identical to the limit of the spectrum of the separated disks in (1.4) and that of intersecting disks in (1.5) as δ and θ tend to zero, respectively. The analyses for the crescent-shaped domain and touching disks go similarly. We provide the full details for the crescent-shaped domain and briefly state the results for touching disks without a detailed proof.
As an application of the spectral resolution of the NP operator, we compute the order of plasmon resonance on a crescent-shaped domain. It is worth remarking that the plasmon resonance for the crescent-shaped domain was studied in [8] but without mathematical rigor.
The remainder of this paper is organized as follows. In Section 2, we briefly explain the properties of the layer potential operators on a Lipschitz domain and plasmon resonance. Section 3, Section 4, and Section 5 address the crescent-shaped domain case. Section 3 is devoted to deriving expressions of the layer potential operators by using a Möbius transformation and the Fourier transform. In Section 4, we define the Hilbert space K −1/2 0 and derive the spectral resolution of the NP operator. We then analyze the plasmon resonance in Section 5. In Section 6, we derive the spectral resolution of the NP operator for touching disks.
Preliminary: layer potential operators on a Lipschitz domain
Let D be a simply connected bounded Lipschitz domain in R 2 . For a density function ψ ∈ L 2 (∂D), the single-layer potential S ∂D [ψ] is defined by
S ∂D [ψ](x) = ∂D Γ(x − y)ψ(y) dσ(y), x ∈ R 2 ,
where Γ(x) is the fundamental solution to the Laplacian, that is Γ(x) = 1 2π ln |x|. The singlelayer potential is harmonic in R 2 \ ∂D and satisfies the jump relations [39]:
S ∂D [ψ] + (x) = S ∂D [ψ] − (x) a.e. x ∈ ∂D, ∂ ∂ν S ∂D [ψ] ± (x) = ± 1 2 I + K * ∂D ψ(x) a.e. x ∈ ∂D, (2.1)
where the NP operator K * ∂D is given by (1.1), the symbol + and − stand for the limit to ∂D from the outside and inside from ∂Ω, respectively, and p.v denotes the Cauchy principal value.
The NP operator K * ∂D is in general not self-adjoint on L 2 (∂D); however, it can be symmetrized using Plemelj's symmetrization principle (see [26]):
S ∂D K * ∂D = K ∂D S ∂D , (2.2) where K ∂D is the L 2 adjoint of K * ∂D . We denote by H * the space H −1/2 0 (∂D) equipped with the inner product ϕ, ψ H * := − ∂D ϕ S ∂Ω [ψ] dσ for ϕ, ψ ∈ H −1/2 0 (∂D) (2.3)
and let · H * be the corresponding norm, which is equivalent to the H −1/2 norm, i.e.,
C 1 ϕ H −1/2 ≤ ϕ H * ≤ C 2 ϕ H −1/2
for all ϕ ∈ H −1/2 0 (∂D) with some positive constants C 1 and C 2 . From (2.2), K * ∂D is self-adjoint on H * . As a result, the spectrum of K * ∂D lies on the real axis. In fact, the spectrum of K * ∂D on H * lies in (−1/2, 1/2) [15,39].
When the domain D is the disk of radius r 0 > 0 centered at c, it holds that
K * ∂D [ϕ] = 1 4πr 0 ∂D ϕ dσ on ∂D (2.4)
and that the spectrum of K * ∂D consists of eigenvalues 0 and 1 2 . The eigenspace that corresponds to the eigenvalue 0 is H * (∂D), and a constant function is an eigenfunction of K * ∂D corresponding to the eigenvalue 1 2 . The single-layer potential for the constant function 1 r 0 is
S ∂D 1 r 0 (x) = ln r 0 if x ∈ D, ln |x − c| if x ∈ R 2 \ D.
(2.5)
Suppose that the domain D is occupied by a homogeneous material with the dielectric constant ǫ c + iδ (δ is the dissipation factor) and that the matrix R 2 \ D has the dielectric constant ǫ m . We assume that ǫ m = 1. We express the dielectric constant of the entire space as
ǫ = (ǫ c + iδ)χ D + χ R 2 \D ,(2.6)
where χ A means the characteristic function of a set A. We now consider the potential problem
∇ · ǫ∇u δ = f in R 2 , u δ (z) = O(|z| −1 ) as |z| → ∞, (2.7)
where f is a source function that is compactly supported in R 2 \ D satisfying R 2 f dz = 0. An example of the source function is a polarized dipole f (z) = p · ∇δ q (z), where δ q is the Dirac mass at q, and p, q are constant vectors. The solution u δ can be expressed as
u δ = F + S ∂D [ϕ δ ] in R 2 ,
where F denotes the Newtonian potential of f , i.e., F (z) = R 2 Γ(z − y)f (y) dy, and the density function ϕ δ satisfies
(λ δ I − K * ∂D )[ϕ δ ] = ∂ ν F on ∂D (2.8)
with λ δ given by
λ δ = ǫ c + 1 + iδ 2(ǫ c − 1) + 2iδ = ǫ c + 1 2(ǫ c − 1) + O(δ). (2.9)
The plasmon resonance ∇u δ L 2 (D) → ∞ as δ → 0, (2.10) may occur depending on ǫ c and the spectrum of K * ∂D . The blow-up rate (or, the resonance rate) in (2.10) is essentially related to the blow-up feature of the norm of the density function ϕ δ (see, for instance, [23, Section 5] and (5.9) in Section 5).
One can classify the spectrum of the NP operator on a Lipschitz domain by the resonance rate [20]. For g ∈ H * (∂D), let ϕ t,δ be the solution to 20]). Let g ∈ H * . For t ∈ (−1/2, 1/2), the following holds.
((t + iδ)I − K * ∂D )[ϕ t,
(a) If α g (t) > 0, then t ∈ σ(K * ∂D ). (b) If α g (t) = 1 and t is isolated, then t ∈ σ pp (K * ∂D ).
(c) If 1 2 ≤ α g (t) < 1, then t ∈ σ(K * ∂D ).
Layer potential operators on a crescent-shaped domain
We consider a crescent-shaped domain Ω that is the region enclosed by the boundaries of two touching disks B R and B r such that B r ⊂ B R . In other words, Ω = B R \ B r . The boundary of Ω is composed of two circles ∂B R and ∂B r that are tangent at the origin point; see the left figure in Figure 3.1. As Ω has a cusp on its boundary, one cannot apply the results of the layer potential operators of Lipschitz domains. Instead, we will generalize the concepts of the single-layer potential and the NP operator to the crescent-shaped domain by using a Möbius transformation. We will then derive the integral expressions of the layer potential operators via the Fourier transform on R.
Möbius transformation
We identify z = (z 1 , z 2 ) ∈ R 2 with z = z 1 + iz 2 ∈ C. Let Ψ : C \ {0} → C \ {0} be the Möbius transformation, that is,
w = Ψ(z) = 1 z . (3.1)
Obviously, Ψ is a conformal mapping. We set w = x + iy, (x, y) ∈ R 2 . Note that Ψ −1 (w) = Ψ(w) = 1 w . The scale factors of the mapping w −→ Ψ −1 (w) with respect to x and y coincide. We denote the scale factor by
h(x, y) = |Ψ ′ (w)| = 1 x 2 + y 2 .
The Möbius transformation Ψ maps the left half-plane to the left half-plane, and maps the right half-plane to the right half-plane. In particular, Ψ maps a disk of radius |a| centered at (a, 0) to the half-plane determined by x > 1 2a if a > 0, and to the half plane determined by
x < 1 2a if a < 0. We set for a = 0 ∈ R that B a = (x, y) ∈ R 2 | |x + iy − a| ≤ |a| , h a (y) = h 1 2a , y = 1 1 2a 2 + y 2 . (3.2)
The disks B R and B r are defined in this sense. The crescent-shaped domain Ω = B R \ B r is mapped via the Möbius transformation onto the vertical strip (see Figure 3.1)
S := Ψ(Ω) = 1 2R , 1 2r × (−∞, ∞),
and vice versa. For later use, we denote by q the width of S, i.e.,
q = 1 2r − 1 2R > 0. (3.3)
We denote by ν the outward unit normal vector to ∂Ω, except at the touching point of ∂B R and ∂B r . Note that ν is directed toward the exterior of B R on ∂B R , but toward the interior of B r on ∂B r ; see Figure 3.1. Following our normal vector convention, we then define the normal derivative of a function v: at z ∈ ∂Ω with Ψ(z) = x + iy,
(a) z-plane, z = z1 + iz2 (b) w-plane, w = x + iy∂v ∂ν = − 1 h R (y) ∂(u • Ψ) ∂x for z ∈ ∂B R \ {0}, ∂v ∂ν = 1 h r (y) ∂(u • Ψ) ∂x for z ∈ ∂R r \ {0}. (3.4)
Recall that the value of an integrand function at a set of measure zero doesn't affect the integral value. We disregard the origin point, where two normal vectors are defined, in the layer potential formulation on the crescent-shaped domain Ω in the following subsection.
Generalization of the layer potential operators to the crescent-shaped domain
A density function ϕ ∈ L 2 (∂Ω) can be decomposed as
ϕ = ϕ χ ∂B R + ϕ χ ∂Br =: ϕ R + ϕ r . (3.5)
The normal vector ν of ∂Ω points toward the exterior of B R on ∂B R and toward the interior of B r on ∂B r , as mentioned before. We define K * ∂Br and K * ∂B R by the integral expression (1.1) with this normal vector convention. We now define the single-layer potential and the NP operator on the crescent-shaped domain as follows.
Definition 1. For ϕ = ϕ R + ϕ r ∈ L 2 (∂Ω), we define S ∂Ω [ϕ] := S ∂B R [ϕ R ] + S ∂Br [ϕ r ] on R 2 (3.6)
and
K * ∂Ω [ϕ] := K * ∂B R [ϕ R ] + ∂ ∂ν S ∂Br [ϕ r ] ∂B R χ ∂B R + ∂ ∂ν S ∂B R [ϕ R ] ∂Br + K * ∂Br [ϕ r ] χ ∂Br on ∂Ω.
(3.7)
Lemma 3.1. We have
S ∂Ω [ψ] + (x) = S ∂Ω [ψ] − (x) a.e. x ∈ ∂Ω, ∂ ∂ν S ∂Ω [ϕ] ± ∂Ω (x) = ± 1 2 I + K * ∂Ω [ϕ](x) a.e. x ∈ ∂Ω.
Proof. The continuity of the single-layer potential S ∂Ω [ϕ] across ∂Ω directly follows from the
continuity of S ∂B R [ϕ R ] and S ∂Br [ϕ r ].
On the boundary circles ∂B R and ∂B r , we can apply the results of the layer potential operators on Lipschitz domains that are described in Subsection 2. Applying the jump relation (2.1), we have
∂ ∂ν S ∂B R [ϕ R ] ± ∂B R = ± 1 2 I + K * ∂B R [ϕ R ] on ∂B R , ∂ ∂ν S ∂Br [ϕ r ] ± ∂Br = ± 1 2 I + K * ∂Br [ϕ r ] on ∂B r ,
where ν is the outward normal vector to ∂Ω, the interior and exterior limits are defined corresponding to the direction of ν, and K * ∂Br and K * ∂B R are also defined with this normal vector convention. Also, we have
∂ ∂ν S ∂Br [ϕ r ] ± ∂B R = ∂ ∂ν S ∂Br [ϕ r ] ∂B R on ∂B R \ {0}, ± 1 2 I + K * ∂Br [ϕ r ] at 0 and ∂ ∂ν S ∂B R [ϕ R ] ± ∂Br = ∂ ∂ν S ∂B R [ϕ R ] ∂Br on ∂B r \ {0}, ± 1 2 I + K * ∂B R [ϕ R ] at 0.
Hence, we prove the lemma.
We emphasize that it is necessary to analyze the mapping properties of ∂ ∂ν S ∂Br [ϕ r ] ∂B R and ∂ ∂ν S ∂B R [ϕ R ] ∂Br to understand the spectral structure of the NP operator on the crescent-shaped domain.
As in the previous subsection, we set
z = 1 x + iy for z = 0. (3.8) Let z t = Ψ −1 ( 1 2r + it) on ∂B r and z t = Ψ −1 ( 1 2R +it) on ∂B R ; then we have |z−z t | = |x− 1 2r +i(y−t)| |x+iy|| 1
2r +it| on ∂B r and a similar relation on ∂B R , respectively. We identify ϕ R , ϕ r in (3.5) with the functions on R given by
ϕ R (y) = (ϕ R • Ψ −1 ) 1 2R + iy , ϕ r (y) = (ϕ r • Ψ −1 ) 1 2r + iy for y ∈ R.
(3.9)
Then, the single-layer potential (3.6) satisfies
S ∂Ω [ϕ](z) = 1 2π ∂Br ln |z − z t |ϕ r (z t ) dσ(z t ) + 1 2π ∂B R ln |z − z t |ϕ R (z t ) dσ(z t ) = 1 4π ∞ −∞ ln x − 1 2r 2 + (y − t) 2 − ln 1 2r 2 + t 2 ϕ r (t)h r (t) dt + 1 4π ∞ −∞ ln x − 1 2R 2 + (y − t) 2 − ln 1 2R 2 + t 2 ϕ R (t)h R (t) dt − 1 4π ln(x 2 + y 2 ) ∂Ω ϕ(z) dσ(z) (3.10) and ∂ ∂ν S ∂Br [ϕ r ] ∂B R (z) = 1 2π ∂Br ∂ ∂ν z ln |z − z t |ϕ r (z t ) dσ(z t ) = 1 2π 1 h R (y) ∞ −∞ q q 2 + (y − t) 2 + 1 2R ( 1 2R ) 2 + y 2 ϕ r (t)h r (t) dt = 1 2π 1 h R (y) ∞ −∞ q q 2 + (y − t) 2 ϕ r (t)h r (t) dt + 1 4πR ∞ −∞ ϕ r (t)h r (t) dt, ∂ ∂ν S ∂B R [ϕ R ] ∂Br (z) = 1 2π 1 h r (y) ∞ −∞ q q 2 + (y − t) 2 ϕ R (t)h R (t) dt − 1 4πr ∞ −∞ ϕ R (t)h R (t) dt, where q denotes the width of the strip Ψ(Ω) (that is, q = 1 2r − 1 2R ). Furthermore, it holds from (2.4) that K * ∂B R [ϕ R ] = 1 4πR ∂B R ϕ R dσ on ∂B R , K * ∂Br [ϕ r ] = − 1 4πr ∂Br ϕ r dσ on ∂B r . Hence, for ϕ = ϕ R + ϕ r ∈ L 2 (∂Ω), it holds that K * ∂Ω [ϕ](z) = 1 2πh R (y) ∞ −∞ q q 2 + (y − t) 2 ϕ r (t) h r (t) dt + 1 4πR ∂Ω ϕ dσ for x = 1 2R , 1 2πh r (y) ∞ −∞ q q 2 + (y − t) 2 ϕ R (t) h R (t) dt − 1 4πr ∂Ω ϕ dσ for x = 1 2r
(3.11) with ϕ R and ϕ r given by (3.9).
Layer potential operators in terms of the Fourier transform
The Fourier transform and its inversion in R are defined as
F[f ](k) = 1 √ 2π ∞ −∞ f (y)e −iky dy, F −1 [f ](k) = 1 √ 2π ∞ −∞ f (y)e iky dy.
Recall that we identify ϕ ∈ L 2 (∂Ω) with two functions ϕ R , ϕ r given by (3.9). We can further identify ϕ, via the Fourier transform, with
U [ϕ] := F[h R ϕ R ] F[h r ϕ r ] . (3.12)
The inversion of the operator U is
U −1 f R f r (z) = 1 h R (y) F −1 [f R ](y) for z ∈ ∂B R , 1 h r (y) F −1 [f r ](y) for z ∈ ∂B r ,
where z and y satisfy the relation (3.8).
We now express the single-layer potential and the NP operator for ϕ in terms of U [ϕ] as follows.
Lemma 3.2. Let ϕ = ϕ R + ϕ r ∈ L 2 0 (∂Ω). For z = Ψ −1 (x + iy) ∈ ∂Ω, we have S ∂Ω [ϕ](z) = − 1 √ 2π ∞ −∞ 1 2|k| e −|x− 1 2R ||k| F[h R ϕ R ](k) + e −|x− 1 2r ||k| F[h r ϕ r ](k) e iky dk + C,
where C is the constant given by
C = 1 √ 2π ∞ −∞ 1 2|k| e − 1 2R |k| F[h R ϕ R ](k) + e − 1 2r |k| F[h r ϕ r ](k) dk. Proof. The assumption ϕ ∈ L 2 0 (∂Ω) implies that h R | ϕ R | 2 , h r | ϕ r | 2 ∈ L 1 (R).
Since h R and h r are bounded and integrable on R, we have
h R ϕ R , h r ϕ r ∈ L 1 (R) ∩ L 2 (R).
(3.13)
Note that for fixed a, b, A, B, the function ln(
A 2 + (a − t) 2 ) − ln(B 2 + (b − t) 2 ) is square integrable on any bounded interval of t. Furthermore, we have ln A 2 + (a − t) 2 − ln B 2 + (b − t) 2 = (−2a + 2b) 1 t + (−a 2 + A 2 + b 2 − B 2 ) 1 t 2 + O 1 t 3 as t → ∞, (3.14)
where O( 1 t 3 ) is uniformly bounded with respect to small |b| ≥ 0 (with fixed a, A, B). Applying the dominated convergence theorem to (3.10), we obtain
S ∂Ω [ϕ](Ψ −1 (x + iy)) = lim c→y 1 4π ∞ −∞ ln x − 1 2R 2 + (y − t) 2 − ln 1 2R 2 + (t − y + c) 2 ϕ R (t)h R (t) dt + 1 4π ∞ −∞ ln x − 1 2r 2 + (y − t) 2 − ln 1 2r 2 + (t − y + c) 2 ϕ r (t)h r (t) dt .
The last term in (3.10) vanishes assuming the mean-zero condition on ϕ. The Fourier transform of ln(y 2 + a 2 ) is
F[ln(y 2 + a 2 )](k) = − √ 2π e −|a||k| |k| + 2γ E δ(k) ,(3.15)
where 1 |k| is defined in the sense of principal value and γ E denotes Euler's constant. The convolution theorem of the Fourier transform, i.e., From the mean-zero assumption on ϕ, we have
F[f 1 * f 2 ] = √ 2πF[f 1 ] F[f 2 ], leads to the relation S ∂Ω [ϕ](z) = lim c→y −1 √ 2π ∞ −∞ 1 2|k| e −|x− 1 2R ||k| − e − |k| 2R e −ikc F[h R ϕ R ](k)e iky dk + ∞ −∞ 1 2|k| e −|x− 1 2r ||k| − e − |k| 2r e −ikc F[h r ϕ r ](k)e iky dk .∞ −∞ ϕ R h R dy + ∞ −∞ ϕ r h r dy = 0 (3.17)
and, hence,
F[h R ϕ R ](k) + F[h r ϕ r ](k) = 1 √ 2π ∞ −∞ ( ϕ R h R )(y) e −iky − 1 dy + 1 √ 2π ∞ −∞ ( ϕ r h r )(y) e −iky − 1 dy.
From (3.13) and the fact that h R is uniformly bounded, we have
∞ −∞ ( ϕ R h R )(y) e −iky − 1 dy ≤ ϕ R (h R ) 1 2 L 2 (R) (h R ) 1 2 e −iky − 1 L 2 (R) ≤ C( |y|< 1 √ |k| e −iky − 1 2 1 4R 2 + y 2 dy + |y|≥ 1 √ |k| e −iky − 1 2 1 4R 2 + y 2 dy) 1 2 ≤ C( |y|< 1 √ |k| |ky| 2 1 4R 2 dy + |y|≥ 1 √ |k| 1 y 2 dy) 1 2 ≤ C|k| 1 4
for some positive constant C, and a similar relation holds for ϕ r h r . Thus, for any constants a and b, it holds that
e −|a||k| F[h R ϕ R ](k) + e −|b||k| F[h r ϕ r ](k) = O(|k| 1 4 ) as |k| → 0. (3.18)
Then, we can apply the dominated convergence theorem to (3.16) and, as a result, change the order of the limit and integration. Hence, we prove the lemma.
Recall that q = 1 2r − 1 2R > 0. We set
P := 1 √ 2 −1 1 1 1 . (3.19)
This matrix satisfies P = P −1 = P T and, for any d 1 , d 2 ,
1 2 d 1 + d 2 −d 1 + d 2 −d 1 + d 2 d 1 + d 2 = P −1 d 1 0 0 d 2 P. (3.20) Lemma 3.3. For ϕ = ϕ R + ϕ r ∈ L 2 0 (∂Ω), it holds that U [K * ∂Ω [ϕ]] = P −1 1 2 e −q|k| −1 0 0 1 P U [ϕ]. (3.21)
Proof. It holds that
F q q 2 + y 2 (k) = π 2 e −q|k| .
From the assumption ϕ = ϕ R + ϕ r ∈ L 2 0 (∂Ω), we have ∂Ω ϕ dσ = 0. It then follows by applying the convolution theorem of the Fourier transform to (3.11) that
U [K * ∂Ω [ϕ]] = 1 2 e −q|k| 0 1 1 0 F[h R ϕ R ] F[h r ϕ r ] = 1 2 e −q|k| 0 1 1 0 U [ϕ].
From (3.20), we obtain (3.21).
Lemma 3.4. For ψ, ϕ ∈ L 2 0 (∂Ω), we have ∂Ω ψ(z) S ∂Ω [ϕ](z) dσ(z) = ∞ −∞ U [ψ](k) T P −1 − 1 2|k| 1 − e −q|k| 0 0 1 + e −q|k| P U [ϕ](k) dk.
Proof. Set ψ = ψ R + ψ r and ϕ = ϕ R + ϕ r . From Lemma 3.2, we have
∂Ω ψ S ∂Ω [ϕ] dσ = ∞ −∞ ψ R (y) S ∂Ω [ϕ] 1 1/(2R) + iy h R (y) dy + ∞ −∞ ψ r (y) S ∂Ω [ϕ] 1 1/(2r) + iy h r (y) dy = − 1 √ 2π ∞ −∞ ∞ −∞ ( ψ R h R )(y) 1 2|k| F[h R ϕ R ](k) + e −q|k| F[h r ϕ r ](k) e −iky dk dy − 1 √ 2π ∞ −∞ ∞ −∞ ( ψ r h r )(y) 1 2|k| e −q|k| F[h R ϕ R ](k) + F[h r ϕ r ](k) e −iky dk dy.
By changing the order of integrations, one obtains
− ∂Ω ψ S ∂Ω [ϕ] dσ = ∞ −∞ 1 2|k| U [ψ](k) T 1 e −q|k| e −q|k| 1 U [ϕ](k) dk.
In view of (3.20), this completes the proof.
Spectral resolution of the NP operator on a crescent-shaped domain
We denote the two matrix-valued functions in Lemma 3.3 and Lemma 3.4 as follows:
S = − 1 2|k| 1 − e −q|k| 0 0 1 + e −q|k| , k ∈ R \ {0}, K = 1 2 e −q|k| −1 0 0 1 , k ∈ R.
(4.1)
In terms of these matrix-valued functions, we define a Hilbert space K −1/2 0 that extends L 2 0 (∂Ω). We then generalize the layer potential operators to be defined on K
Hilbert space
K −1/2 0 For ϕ ∈ L 2 0 (∂Ω), it holds that ϕ = U −1 P ϕ with ϕ = 1 √ 2 −1 1 1 1 F[h R ϕ R ] F[h r ϕ r ] . From (3.18), we have ∞ −∞ ϕ T (−S) ϕ dk < ∞.
Based on these relations, we define a Hilbert space:
Definition 2. We define K −1/2 0 := ϕ = U −1 P ϕ ϕ = ϕ 1 ϕ 2 satisfying ∞ −∞ ϕ T (−S) ϕ dk < ∞ , (4.2)
where ϕ 1 and ϕ 2 are measurable functions on R, and P is given by (3.19).
The inverse Fourier transform U −1 in (4.2) is defined in the tempered distribution sense. Indeed, for any function f on R satisfying
∞ −∞ 1 1 + |k| |f (k)| 2 dk < ∞,(4.3)
it holds that, for any ψ in the Schwartz class S,
∞ −∞ f ψ dk ≤ ∞ −∞ 1 1 + |k| |f | 2 dk 1/2 ∞ −∞ (1 + |k|)|ψ| 2 dk 1/2 ≤ C ∞ −∞ (1 + |k|) 4 |ψ| 2 1 (1 + |k|) 2 dk 1/2 ≤ C (1 + |k|) 2 ψ(k) ∞ ≤ C α≤2 ψ α,0 < ∞,
where φ α,β := sup x∈R |x α ∂ β φ(x)|. Therefore, we have f ∈ S ′ (R), where S ′ (R) denotes the class of tempered distributions. The Fourier transform and its inversion on L 2 (R) can be extended to S ′ (R), where the inversion formula still holds. Since ϕ 1 and ϕ 2 satisfy the decay condition (4.3), we can define the inverse transform for P ϕ.
As the (2, 2)-component of S blows up as |k| −1 near k = 0, the condition ∞ −∞ ϕ T (−S) ϕ dk < ∞ implies decay of ϕ 2 near k = 0; we highlight this property by adding the subscript 0 in K
−1/2 0 .
On the other hand, we add the superscript −1/2 in K −1/2 0 since we define an inner product in a similar way to (2.3) (see (4.5) below).
It is straightforward to obtain the following. Note that K −1/2 0 is defined in the weighted L 2 sense. We accordingly define the inner product so that K −1/2 0 is complete. In other words, we equip this space with the inner product
ψ, ϕ −1/2 := − ∞ −∞ ψ 1 ψ 2 T S ϕ 1 ϕ 2 dk (4.5) = 1 2 ∞ −∞ ψ 1 (k) ϕ 1 (k) 1 − e −q|k| |k| dk + 1 2 ∞ −∞ ψ 2 (k) ϕ 2 (k) 1 + e −q|k| |k| dk,
where ψ, ϕ ∈ K −1/2 0 are given by
ϕ = U −1 P ϕ 1 ϕ 2 , ψ = U −1 P ψ 1 ψ 2 . (4.6)
We denote the associated norm by
ϕ −1/2 := ( ϕ, ϕ −1/2 ) 1 2 = − ∞ −∞ ϕ 1 ϕ 2 T S ϕ 1 ϕ 2 dkS ∂Ω [ϕ](z) := − 1 √ 2π ∞ −∞ 1 2|k| e −|x− 1 2R ||k| ϕ R (k) + e −|x− 1 2r ||k| ϕ r (k) e iky dk + 1 √ 2π ∞ −∞ 1 2|k| e − |k| 2R ϕ R (k) + e − |k| 2r ϕ r (k) dk,(4.9)
where ϕ R and ϕ r is given by
ϕ R ϕ r = P ϕ 1 ϕ 2 = 1 √ 2 − ϕ 1 + ϕ 2 ϕ 1 + ϕ 2 . (4.10)
It is worth emphasizing that (4.8) and (4.9) hold for ϕ ∈ L 2 0 (∂Ω). In other words, (4.8) and (4.9) are natural extensions of the NP operator and the single-layer potential on L 2 0 (∂Ω) to K −1/2 0 . We observe that
K * ∂Ω [ϕ] 2 −1/2 = − ∞ −∞ (K ϕ) T S(K ϕ) dk ≤ 1 4 ϕ 2 −1/2 for any ϕ ∈ K −1/2 0 .
Hence, K * ∂Ω is a bounded linear operator on K −1/2 0 and its operator norm is bounded by 1 2 . Since S and K are real diagonal matrices, we have SK = K T S. This induces that
ψ, K * ∂Ω [ϕ] −1/2 = K * ∂Ω [ψ], ϕ −1/2 for all ϕ, ψ ∈ K −1/2 0 .
In other words, K * ∂Ω is self-adjoint on K −1/2 0 . In view of (4.8), K * ∂Ω is identical to the matrix K via the transformation U −1 P . From this, one can infer that the spectrum of K * ∂Ω on K −1/2 0 lies in the interval [−1/2, 1/2], that is the spectrum of K. In Subsection 4.3, we will prove it by deriving the spectral resolution of the NP operator on K
−1/2 0 .
In the remainder of this subsection, we obtain properties of the layer potential operators by assuming a decay condition on the density function as k tends to zero.
∂ ∂z 1 S ∂Ω [ϕ](z), ∂ ∂z 2 S ∂Ω [ϕ](z) = (x 2 + y 2 ) O |x| − 5 8 as |x| → ∞,
where O(|x| − 5 8 ) is uniform with respect to y.
(c) The single-layer potential is harmonic, i.e., ∆ (z 1 ,z 2 ) S ∂Ω [ϕ](z) = 0 in C \ ∂Ω.
Proof. From the assumption that ϕ 2 (k) = O(|k| 1 4 ), the integral (4.9) is finite for any z ∈ C. One can also show that S ∂Ω [ϕ](z) is continuous in the whole complex plane by applying the dominated convergence theorem. We can rewrite (4.9) as
S ∂Ω [ϕ](z) = − 1 √ 2π ∞ −∞ 1 2|k| (e −|x− 1 2R ||k| − e − |k| 2R )e iky + e − |k| 2R (e iky − 1) ϕ R (k) dk − 1 √ 2π ∞ −∞ 1 2|k|
(e −|x− 1 2r ||k| − e − |k| 2r )e iky + e − |k| 2r (e iky − 1) ϕ r (k) dk.
(4.11)
It then follows that
S ∂Ω [ϕ](z) = ∞ −∞ O(|x| + |y|)| ϕ R (k)| dk + ∞ −∞ O(|x| + |y|)| ϕ r (k)| dk as |x + iy| → 0,
where O(|x| + |y|) terms are uniform with respect to k. This proves (a). From (4.9), we have and
∂ ∂y S ∂Ω [ϕ](z) = − 1 √ 2π ∞ −∞ ik 2|k| √ 2e −|x− 1 2R ||k| ϕ 2 (k)e iky dk − 1 √ 2π ∞ −∞ ik 2|k| e −|x− 1 2r ||k| − e −|x− 1 2R ||k| ϕ r (k)e iky dk∂ ∂x S ∂Ω [ϕ](z) = 1 2 √ 2π ∞ −∞ e −(x− 1 2R )|k| ϕ R (k) + e −(x− 1 2r )|k| ϕ r (k) e iky dk for x > 1 2r , 1 2 √ 2π ∞ −∞ e −(x− 1 2R )|k| ϕ R (k) − e (x− 1 2r )|k| ϕ r (k) e iky dk for 1 2R < x < 1 2r , 1 2 √ 2π ∞ −∞ − e (x− 1 2R )|k| ϕ R (k) − e (x− 1 2r )|k| ϕ r (k) e iky dk for x < 1 2R .
Because of ϕ 1 , ϕ 2 ∈ L 1 (R), ∂ ∂x S ∂Ω [ϕ](z) and ∂ ∂y S ∂Ω [ϕ](z) are uniformly bounded in C. We have
∂ ∂x S ∂Ω [ϕ](z), ∂ ∂y S ∂Ω [ϕ](z) = ∞ −∞ e −|x− 1 2R ||k| O |k| 1 4 1 + ϕ r (k) dk = O |x| − 5 8 as |x| → ∞,
where the second equality can be derived by splitting the integral into |k| < |x − 1 2R | − 1 2 and |k| > |x − 1 2R | − 1 2 . This proves (b). Recall that Ψ is a conformal mapping. By taking the Laplacian for the right-hand side of (4.9) (switching the order of differentiation and integration), we observe (c).
Lemma 4.3. Let ϕ ∈ K −1/2 0 be given by ϕ = U −1 P ϕ satisfying ϕ 1 , ϕ 2 ∈ L 1 (R) and ϕ 2 (k) = O(|k| 1 4 ) as |k| → 0. We have ∇S ∂Ω [ϕ] 2 L 2 (Ω) = ϕ, 1 2 I − K * ∂Ω [ϕ] −1/2 < ∞, (4.13) ∇S ∂Ω [ϕ] 2 L 2 (R 2 \Ω) = ϕ, 1 2 I + K * ∂Ω [ϕ] −1/2 < ∞.
(4.14)
Proof. We define ϕ R and ϕ r as in (4.10). Note that
ϕ R (k) + ϕ r (k) = √ 2 ϕ 2 (k) = O(|k| 1 4 ) as |k| → 0. (4.15)
For fixed s > 0, we set
Ω s := z = Ψ(w) w = x + iy with 1 2R < |x| < 1 2r , |y| < s .
Then, Ω s is a Lipschitz domain. We identify z = z 1 + iz 2 with z = (z 1 , z 2 ) ∈ R 2 . Applying the divergence theorem, we obtain (4.17) where the first equality holds similarly to (3.4) and the second one is from (4.9). Applying the Riemann-Lebesgue lemma to (4.17), we obtain
Ωs |∇S ∂Ω [ϕ]| 2 dz = ∂Ωs S ∂Ω [ϕ] ∂ ∂ν S ∂Ω [ϕ] − dσ = x= 1 2R ,∂ ∂ν S ∂Ω [ϕ] − h(x, y) = ± ∂ ∂y S ∂Ω [ϕ] y=±s = ∓ 1 √ 2π ∞ −∞ ik 2|k| e −|x− 1 2R ||k| ϕ R (k)e ±iks dk + ∞ −∞ ik 2|k| e −|x− 1 2r ||k| ϕ r (k)e ±iks dk ,∂ ∂ν S ∂Ω [ϕ] − h(x, y) → 0 as s → ∞.
Note that S ∂Ω [ϕ](z) and h(x, y)∇ (z 1 ,z 2 ) S ∂Ω [ϕ](z) are uniformly bounded independent of s for (x, y) satisfying 1 2R < x < 1 2r , |y| ≥ s. It then holds by applying the dominated convergence theorem that III → 0 as s → ∞.
S ∂Ω [ϕ](z) = − F −1 1 2|k| ϕ R (k) + e −|k|q ϕ r (k) (y) + C in I, S ∂Ω [ϕ](z) = − F −1 1 2|k| e −|k|q ϕ R (k) + ϕ r (k) (y) + C in II,
where C is the constant given by
C = 1 √ 2π ∞ −∞ 1 2|k| e − |k| 2R ϕ R (k) + e − |k| 2r ϕ r (k) dk. We also have ∂ ∂ν S ∂Ω [ϕ] − h R (y) = − ∂ ∂x S ∂Ω [ϕ] + x= 1 2R = 1 2 F −1 − ϕ R (k) + e −|k|q ϕ r (k) (y) in I, ∂ ∂ν S ∂Ω [ϕ] − h r (y) = ∂ ∂x S ∂Ω [ϕ] − x= 1 2r = − 1 2 F −1 − e −|k|q ϕ R (k) + ϕ r (k) (y) in II.
In other words,
F S ∂Ω [ϕ] x= 1 2R − C F S ∂Ω [ϕ] x= 1 2r − C T = ϕ R ϕ r T P −1 S P (4.19) and F ∂ ∂ν S ∂Ω [ϕ] − h R F ∂ ∂ν S ∂Ω [ϕ] − h r = −P −1 ( 1 2 I − K)P ϕ R ϕ r . (4.20)
Applying the Plancherel theorem, we derive that as s → ∞,
I + II → ∞ −∞ ϕ 1 ϕ 2 T (−S)( 1 2 I − K) ϕ 1 ϕ 2 dk + C ′ with C ′ = C 2 lim s→∞ s −s F −1 ( ϕ R ) + F −1 ( ϕ r ) − F −1 e −|k|q ϕ R − F −1 e −|k|q ϕ r dy.
We claim that C ′ = 0. Let g ∈ L 1 (R) and g(k) = O(|k| 1 4 ) as |k| → 0. It then holds by Fubini's theorem and the dominated convergence theorem that s −s
F −1 ( g )dy = s −s ∞ −∞ g(k)e iky dk dy = ∞ −∞ g(k) s −s e iky dy dk = 2 ∞ −∞ g(k)
k sin(ks) dk.
By the Riemann-Lebesgue lemma, the last term converges to 0 as s → ∞ and, thus,
lim s→∞ s −s F −1 ( g )dy = 0.
This implies that C ′ = 0. From (4.5), (4.8), (4.16) and (4.18), we prove (4.13). Note that
Ω e = z = Ψ(w) w = x + iy with x ∈ ( 1 2r , ∞) ∪ (−∞, 1 2R ) . From Lemma 4.2 (b), we obtain Ω e |∇S ∂Ω [ϕ]| 2 ≤ C x∈(−∞, 1 2R )∪( 1 2r ,∞) ∞ −∞ |x| − 5 8 1 x 2 + y 2 dy dx < ∞ and Ω e |∇S ∂Ω [ϕ]| 2 dz = − ∂Ω e S ∂Ω [ϕ] ∂ ∂ν S ∂Ω [ϕ] + dσ = − x= 1 2R , y∈R S ∂Ω [ϕ] ∂ ∂ν S ∂Ω [ϕ] + h R (y)dy − x= 1 2r , y∈R S ∂Ω [ϕ] ∂ ∂ν S ∂Ω [ϕ] + h r (y)dy = : I + II.
Note that
∂ ∂ν S ∂Ω [ϕ] + h R (y) = − ∂ ∂x S ∂Ω [ϕ] − x= 1 2R = 1 2 F −1 ϕ R (k) + e −|k|q ϕ r (k) (y) in I, ∂ ∂ν S ∂Ω [ϕ] + h r (y) = ∂ ∂x S ∂Ω [ϕ] + x= 1 2r = − 1 2 F −1 − e −|k|q ϕ R (k) − ϕ r (k) (y) in II, which implies that F ∂ ∂ν S ∂Ω [ϕ] + h R F ∂ ∂ν S ∂Ω [ϕ] + h r = P −1 ( 1 2 I + K)P ϕ R ϕ r . (4.21)
Applying also (4.19), one can derive that
I + II = const. + ∞ −∞ ϕ 1 ϕ 2 T (−S)( 1 2 I + K) ϕ 1 ϕ 2 dk.
One can show that the constant term is zero similarly to the proof of C ′ = 0. Hence, we prove (4.14).
Spectral resolution of
K * ∂Ω on K −1/2 0
To derive the spectral resolution of the NP operator on K −1/2 0 , we define a pair of orthogonal projection operators P 1 (s) and P 2 (s) on K −1/2 0 for each s ∈ R ∪ {∞}. Let ϕ ∈ K −1/2 0 be given by ϕ = U −1 P ϕ and ϕ = [ ϕ 1 , ϕ 2 ] T . We define
P 1 (s)ϕ = U −1 P χ (−∞, s ] ϕ 1 0 , P 2 (s)ϕ = U −1 P 0 χ (−∞, s ] ϕ 2 for s ∈ R and P 1 (∞)ϕ = U −1 P ϕ 1 0 , P 2 (∞)ϕ = U −1 P 0 ϕ 2 .
Note that P 1 (∞) + P 2 (∞) = I, where I is the identity operator on K The limit in (4.23) is in the sense of strong convergence, i.e., lim t→s + E(t)ψ = E(s)ψ for all ψ ∈ K −1/2 0 , which holds from the dominated convergence theorem and the fact that the integral in (4.7) is finite. In short, we have the following lemma. in the sense of strong convergence.
We have
ψ, P 1 (s)ϕ −1/2 = s −∞ ψ 1 ψ 2 T (−S) ϕ 1 0 dk,
which is almost everywhere differentiable in s from the Lebesgue differentiation theorem. Similarly, ψ, E(t)ϕ −1/2 is almost everywhere differentiable in t.
Lemma 4.5. For all ϕ, ψ ∈ K −1/2 0 , it holds that
ψ, ϕ −1/2 = 1 2 − 1 2 d ψ, E(t)ϕ −1/2 . (4.25)
Proof. Let ϕ, ψ be given by (4.6). It then holds from (4.5) that
ψ, P 1 (±k)ϕ −1/2 = 1 2 ±k −∞ ψ 1 (s) ϕ 1 (s) 1 − e −q|s| |s| ds for k > 0. (4.26)
From (4.26) and a similar derivation, we obtain
d dk ψ, P 1 (k) − P 1 (−k) ϕ −1/2 = 1 2 ψ 1 (k) ϕ 1 (k) 1 − e −q|k| |k| + 1 2 ψ 1 (−k) ϕ 1 (−k) 1 − e −q|k| |k| , d dk ψ, P 2 (k) − P 2 (−k) ϕ −1/2 = 1 2 ψ 2 (k) ϕ 2 (k) 1 + e −q|k| |k| + 1 2 ψ 2 (−k) ϕ 2 (−k) 1 + e −q|k| |k| .
We then obtain that, letting
t = − 1 2 e −qk ∈ [−1/2, 0), 1 2 ∞ −∞ ψ 1 (k) ϕ 1 (k) 1 − e −q|k| |k| dk = ∞ 0 d dk ψ, P 1 (k) − P 1 (−k) ϕ −1/2 dk = 0 − 1 2 d ψ, P 1 − ln(−2t) q − P 1 ln(−2t) q ϕ −1/2
and that, letting t = 1 2 e −qk ∈ (0, 1/2], 1 2
∞ −∞ ψ 2 (k) ϕ 2 (k) 1 + e −q|k| |k| dk = ∞ 0 d dk ψ, P 2 (k) − P 2 (−k) ϕ −1/2 dk = 0 1 2 d ψ, P 2 − ln(2t) q − P 2 ln(2t) q ϕ −1/2 .
Since ψ, Iϕ −1/2 is constant, we have d ψ, Iϕ −1/2 = 0. Hence, we complete the proof. We have the spectral resolution of K *
∂Ω on K −1/2 0 as K * ∂Ω = 1 2 − 1 2 t d E(t).
In other words, it holds that
ψ, K * ∂Ω [ϕ] −1/2 = 1 2 − 1 2 t d ψ, E(t)ϕ −1/2 for all ψ, ϕ ∈ K −1/2 0 . (4.27)
Proof. Let ϕ, ψ be given by (4.6). From the definition of the NP operator and the inner product on K −1/2 0 , we have
ψ, K * ∂Ω [ϕ] −1/2 = − 1 4 ∞ −∞ e −q|k| ψ 1 ϕ 1 1 − e −q|k| |k| dk + 1 4 ∞ −∞ e −q|k| ψ 2 ϕ 2 1 + e −q|k| |k| dk = : I + II.
Following the derivation in the proof of Lemma 4.5, we obtain
I = ∞ 0 − 1 2 e −q|k| d dk ψ, P 1 (k) − P 1 (−k) ϕ −1/2 dk = 0 − 1 2 t d ψ, P 1 − ln(−2t) q − P 1 ln(−2t) q ϕ −1/2 and II = ∞ 0 1 2 e −q|k| d dk ψ, P 2 (k) − P 2 (−k) ϕ −1/2 dk = 0 1 2 t d ψ, P 2 − ln(2t) q − P 2 ln(2t) q ϕ −1/2 .
This proves the theorem.
The condition lim t→s − E(t) = E(s) characterizes the point spectrum of K * ∂Ω . By Lemma 4.4, we conclude that K * ∂Ω has only a continuous spectrum. We need the following lemma to prove K * ∂Ω has only an absolutely continuous spectrum. Lemma 4.7. We have
d dt ψ, E(t)ϕ −1/2 = 1 + 2t 2|t ln(2|t|)| ψ 1 (k) ϕ 1 (k) + ψ 1 (−k) ϕ 1 (−k) , t ∈ [−1/2, 0), 1 + 2t 2|t ln(2|t|)| ψ 2 (k) ϕ 2 (k) + ψ 2 (−k) ϕ 2 (−k) , t ∈ (0, 1/2]
with k = − 1 q ln(2|t|). Proof. First, we consider the case t ∈ [−1/2, 0). From the derivation in the proof of Lemma 4.5, we have E(t) = P 1 (k) − P 1 (−k) with t = − 1 2 e −qk (k > 0) and
ψ, E(t)ϕ −1/2 = 1 2 k −k ψ 1 (s) ϕ 1 (s) 1 − e −q|s| |s| ds, d dt ψ, E(t)ϕ −1/2 = 1 2 ψ 1 (k) ϕ 1 (k) + ψ 1 (−k) ϕ 1 (−k) 1 − e −q|k| |k| dk dt . (4.28)
Now, let t ∈ (0, 1/2]. We have E(t) = −P 2 (k) + P 2 (−k) + I with t = 1 2 e −qk (k > 0) and
ψ, E(t)ϕ −1/2 = − 1 2 k −k ψ 2 (s) ϕ 2 (s) 1 + e −q|s| |s| ds + const., d dt ψ, E(t)ϕ −1/2 = − 1 2 ψ 2 (k) ϕ 2 (k) + ψ 2 (−k) ϕ 2 (−k) 1 + e −q|k| |k| dk dt . (4.29)
From (4.28) and (4.29), we prove the lemma.
For any ϕ ∈ K −1/2 0 , it holds from (4.28) and (4.29) that
1 2 − 1 2 d dt ϕ, E(t)ϕ −1/2 dt = 1 2 ∞ −∞ 1 − e −q|k| |k| | ϕ 1 (k)| 2 + 1 + e −q|k| |k| | ϕ 2 (k)| 2 dk < ∞. (4.30)
In other words, d dt ϕ, E(t)ϕ −1/2 is integrable for any ψ ∈ K −1/2 0 , which implies that d dt ψ, E(t)ϕ −1/2 is integrable for any ψ, ϕ ∈ K −1/2 0 . In view of (4.27), we obtain the following theorem.
Plasmon resonance on a crescent-shaped domain
In this section, we analyze the plasmon resonance on a crescent-shaped domain Ω = B R \ B r . We consider the transmission problem (2.7) with ǫ given by
ǫ = (ǫ c + iδ)χ Ω + χ R 2 \Ω . (5.1)
Let f be compactly supported away from Ω so that the Newtonian potential F of f is smooth in a neighborhood of Ω. It then holds from Lemma 4.
1 that ∂ ν F ∈ K −1/2 0 . Set λ δ as in (2.8). Because of λ δ ∈ R 2 \ [− 1 2 , 1 2 ], the problem (λ δ I − K * ∂Ω ) ϕ δ = ∂ ν F (5.2) is solvable in K −1/2 0 . The solution ϕ δ ∈ K −1/2 0 is of the form (see (4.2)) ϕ δ = U −1 P ϕ δ with ϕ δ = ϕ δ 1 ϕ δ 2 ,
which, by Theorem 4.6, admits the following integral expression:
ϕ δ = 1 2 − 1 2 1 λ δ − t d E(t)[∂ ν F ]. (5.3)
Recall the definition of U in (3.12). For any fixed positive integer n, if g ∈ L 1 (R) satisfies ∂ α g ∈ L 1 (R) and ∂ α g(x) → 0 as |x| → ∞ for all |α| ≤ n, then |F[g](k)| ≤ C (1+|k|) n with some positive constant C. From the smoothness of ∂ ν F on ∂B R and ∂B r , each component of U [∂ ν F ] then belongs to L 1 (R) ∩ L 2 (R) so that
P U [∂ ν F ] ∈ L 1 (R) ∩ L 2 (R). (5.4)
From (4.1), (4.8) and (5.2), ϕ δ satisfies
(λ δ I − K)[ ϕ δ ] = P U [∂ ν F ],
which leads to
ϕ δ = (λ δ − 1 2 e −|k|q ) −1 0 0 (λ δ + 1 2 e −|k|q ) −1 P U [∂ ν F ]. (5.5)
Because of ∂ ν F ∈ L 2 0 (∂Ω), from (3.18), the second entry of P U [∂ ν F ] satisfies O(|k| 1 4 ) as k → 0. From the assumption that λ δ ∈ R 2 \ [− 1 2 , 1 2 ], it then holds that
ϕ δ 1 , ϕ δ 2 ∈ L 1 (R) ∩ L 2 (R) and ϕ δ 2 ∈ O(|k| 1 4 ).
Then, from Lemma 4.2 (a), S ∂Ω [ϕ δ ] is continuous in R 2 , harmonic in R 2 \ ∂Ω, and harmonic at infinity. Furthermore, from Lemma 4.3, |∇S ∂Ω [ϕ]| ∈ L 2 (R 2 ). As ϕ δ satisfies (5.2), the solution to (2.7) with ǫ given by (5.1) satisfies
u δ (z) = F (z) + S ∂Ω ϕ δ (z).
Since F is smooth in a region containing Ω, (2.10) is equivalent to ∇S ∂Ω [ϕ δ ] L 2 (Ω) → ∞ as δ → 0. From (5.3) and (2.9), we have
ϕ δ 2 −1/2 = 1 2 − 1 2 1 |λ δ − t| 2 d ∂ ν F, E(t)∂ ν F −1/2 (5.6) with λ δ − t = (λ 0 − t) + 1−2t 2(ǫc−1) iδ 1 + 1 ǫc−1 iδ with λ 0 = ǫ c + 1 2(ǫ c − 1)
.
(5.7)
Note that λ converges to λ 0 as δ → 0. We see from Lemma 4.3 and Theorem 4.6 that
∇S ∂Ω [ϕ δ ] 2 L 2 (Ω) = 1 2 − 1 2 1 2 − t d ϕ δ , E(t) ϕ δ −1/2 = 1 2 − 1 2 1 2 − t 1 |λ δ − t| 2 d ∂ ν F, E(t)∂ ν F −1/2 .
For any ϕ ∈ K −1/2 0 , we have d dt ϕ, E(t)ϕ −1/2 ≥ 0 from Lemma 4.7. Then, (5.6) leads us to
∇S ∂Ω [ϕ δ ] L 2 (Ω) ≤ ϕ δ −1/2 . (5.8)
We now assume that λ 0 < 1 2 . Then, we can take t 1 , t 2 independent of δ such that λ 0 < t 1 < t 2 < 1 2 . Then, we have
∇S ∂Ω [ϕ δ ] 2 L 2 (Ω) = t 2 − 1 2 + 1 2 t 2 1 2 − t 1 |λ δ − t| 2 d ∂ ν F, E(t)∂ ν F −1/2 ≥ 1 2 − t 2 t 2 − 1 2 1 |λ δ − t| 2 d ∂ ν F, E(t)∂ ν F −1/2 = 1 2 − t 2 ϕ δ 2 −1/2 − 1 2 t 2 1 |λ δ − t| 2 d ∂ ν F, E(t)∂ ν F −1/2 ≥ 1 2 − t 2 ϕ δ 2 −1/2 − 1 |λ δ − t 2 | 2 ∂ ν F 2 −1/2 .
Hence, we have
C 1 ϕ δ −1/2 − C 2 ≤ ∇S ∂Ω [ϕ δ ] L 2 (Ω) ≤ ϕ δ −1/2 for λ 0 < 1 2 , (5.9)
where C 1 , C 2 are some positive constants independent of δ. Hence, for λ 0 < 1 2 , the resonance condition ∇S ∂Ω [ϕ δ ] L 2 (Ω) → ∞ as δ → 0 is equivalent to that ϕ δ −1/2 → ∞ as δ → 0. In the following, we characterize the resonance depending on λ 0 . Lemma 4.7 leads us to
d ∂ ν F, E(t)∂ ν F −1/2 = Q(t) dt (5.10) with Q(t) = 1 + 2t 2|t ln(2|t|)| [ f 1 (k) 2 + f 1 (−k) 2 ] for t ∈ [−1/2, 0), 1 + 2t 2|t ln(2|t|)| [ f 2 (k) 2 + f 2 (−k) 2 ]
for t ∈ (0, 1/2], (5.11) where k = − 1 q ln(2|t|) and [ f 1 , f 2 ] T = P U (∂ ν F ). From (4.30), Q belongs to L 1 (R). The two functions f 1 , f 2 are bounded, continuous, and vanishing at infinity since ∂ ν F is smooth on ∂B R and ∂B r . Since ∂ ν F has a zero mean on each ∂B r and ∂B R , the two components of U (∂ ν F ) are both zero at k = 0. Hence,
Q(t) is continuous for t ∈ (− 1 2 , 0) ∪ (0, 1 2 ). For t = ± 1 2 , we have lim t→(− 1 2 ) + Q(t) = 2 f 1 (0) 2 = 0, lim t→ 1 2 − Q(t) = 2 f 2 (0) 2 = 0. (5.12)
In view of (5.9), we can determine the order of resonance from the following propositions:
Proposition 5.1. For λ 0 ∈ [− 1 2 , 0) ∪ (0, 1 2 )
, let ϕ δ be the solution to (5.2) and [ f 1 , f 2 ] T = P U (∂ ν F ). Then, it holds that
lim δ→0 δ ϕ δ 2 −1/2 = |ǫ c − 1| 1 2 − λ 0 Q(λ 0 ) if λ 0 ∈ (− 1 2 , 0) ∪ (0, 1 2 ), 0 if λ 0 = − 1 2 ,
where Q is given by (5.11).
Proof. Let λ 0 ∈ [− 1 2 , 0) ∪ (0, 1 2 ). Then, we can assume that ǫ c is bounded. From (5.6) and (5.7), we have
1 |λ δ − t| 2 = 1 + ( 1 ǫc−1 δ) 2 (λ 0 − t) 2 + ( 1 2 − t) 2 ( 1 ǫc−1 δ) 2 = 1 +δ 2 (λ 0 − t) 2 + ( 1 2 − t) 2δ2 = 1 t − 2λ 0 +δ 2 2(1+δ 2 ) 2 +δ 2 (1−2λ 0 ) 2 4(1+δ 2 ) 2 ,δ = δ ǫ c − 1 .
We then have from (5.6) and (5.10) that
ϕ δ 2 −1/2 = π 1 +δ 2 δ 1 2 − λ 0 ∞ −∞ 1 πδ (1−2λ 0 ) 2(1+δ 2 ) t − 2λ 0 +δ 2 2(1+δ 2 ) 2 +δ 2 (1−2λ 0 ) 2 4(1+δ 2 ) 2 Q(t) χ (− 1 2 , 1 2 ) dt. (5.13)
Now, we simplify (5.13) using the property for the Poisson kernel of the upper half plane: for g ∈ L 1 (R) and x ∈ R,
lim y→0 + 1 π ∞ −∞ y (x − t) 2 + y 2 g(t)dt = g(x − ) + g(x + ) 2
if g(x − ) and g(x + ) exist. Here, g(x − ) and g(x + ) indicate the limit from the left and the right, respectively. From (5.12) and the continuity of Q(t) on t ∈ (− 1 2 , 0) ∪ (0, 1 2 ), we complete the proof of the lemma.
Proposition 5.2. Suppose λ 0 = 0 (or, equivalently, ǫ c = −1). Let ϕ δ be the solution to (5.2). Then, it holds that lim δ→0 δ 2 ϕ δ 2 −1/2 = 0.
Proof. One can derive that lim δ→0 δ 2 ϕ δ 2 −1/2 = 0 by following the proof of Proposition 4 in [23] with the fact that Q ∈ L 1 (R).
Analysis of the NP operator on touching disks
In this section, we provide the spectral resolution of the NP operator on the touching disks Ω = B R ∪ B −r , following the notation in (3.2). The two disks B R and B −r are tangent to each other at the origin (see the left figure in Figure 6.1). We follow the derivations for the result on a crescent-shaped domain in the previous sections. We omit most proofs since the analysis is almost the same as for the crescent-shaped domain case.
The touching disks Ω is mapped onto the region
S := Ψ(Ω) = − ∞, − 1 2r × (−∞, ∞) ∪ 1 2R , ∞ × (−∞, ∞)
via the Möbius mapping Ψ defined in (3.1), and vice versa. The outward normal vector convention is described in Figure 6.1. For a function v, the normal derivative at z ∈ ∂Ω with (a) z-plane, z = z1 + iz2 (b) w-plane, w = x + iy ). Arrows indicate the outward normal vectors to ∂Ω or to ∂S.
Ψ(z) = x + iy is ∂u ∂ν = − 1 h R (y) ∂(u • Ψ) ∂x for z ∈ ∂B R \ {0}, ∂u ∂ν = 1 h −r (y) ∂(u • Ψ) ∂x for z ∈ ∂B −r \ {0},
where h R (y) and h −r (y) are defined as in (3.2). Similar to the crescent-shaped domain case, we denote byq the distance between the two boundary lines of S, i.e., q := 1 2R + 1 2r .
(6.1)
Generalization of the layer potential operators on the crescent-shaped domain
A density function ϕ ∈ L 2 (∂Ω) can be decomposed as
ϕ = ϕ χ ∂B R + ϕ χ ∂B −r =: ϕ R + ϕ −r . (6.2)
We identify ϕ R , ϕ −r with the functions on R given by
ϕ R (y) = (ϕ R • Ψ −1 ) 1 2R + iy , ϕ −r (y) = (ϕ −r • Ψ −1 ) − 1 2r + iy for y ∈ R. (6.3)
We now define the single-layer potential and the NP operator on touching disks. One can easily find that the jump relations in Lemma 3.1 holds for touching disks.
Definition 4. For ϕ = ϕ R + ϕ −r ∈ L 2 (∂Ω), we define
S ∂Ω [ϕ] := S ∂B R [ϕ R ] + S ∂B −r [ϕ −r ] on R 2 (6.4)
and
K * ∂Ω [ϕ] := K * ∂B R [ϕ R ] + ∂ ∂ν S ∂B −r [ϕ −r ] ∂B R χ ∂B R + ∂ ∂ν S ∂B R [ϕ R ] ∂B −r + K * ∂B −r [ϕ −r ] χ ∂B −r on ∂Ω.
We set z = 1 x+iy for z = 0. Let z t = Ψ −1 (− 1 2r + it) on ∂B −r and z t = Ψ −1 ( 1 2R + it) on ∂B R ; then the single-layer potential (6.4) satisfies
S ∂Ω [ϕ](z) = 1 2π ∂B −r ln |z − z t |ϕ −r (z t ) dσ(z t ) + 1 2π ∂B R ln |z − z t |ϕ R (z t ) dσ(z t ) = 1 4π ∞ −∞ ln x + 1 2r 2 + (y − t) 2 − ln 1 2r 2 + t 2 ϕ −r (t)h −r (t) dt + 1 4π ∞ −∞ ln x − 1 2R 2 + (y − t) 2 − ln 1 2R 2 + t 2 ϕ R (t)h R (t) dt − 1 4π
ln(x 2 + y 2 ) ∂Ω ϕ(z) dσ(z). Thus, we have
∂ ∂ν S ∂B −r [ϕ −r ] ∂B R (z) = − 1 2πh R (y) ∞ −∞q q 2 + (y − t) 2 ϕ −r (t)h −r (t) dt + 1 4πR ∞ −∞ ϕ −r (t)h −r (t) dt, ∂ ∂ν S ∂B R [ϕ R ] ∂Br (z) = − 1 2πh −r (y) ∞ −∞q q 2 + (y − t) 2 ϕ R (t)h R (t) dt + 1 4πr ∞ −∞ ϕ R (t)h R (t) dt
withq given by (6.1). The remaining terms of K * ∂Ω are
K * ∂B R [ϕ R ] = 1 4πR ∂B R ϕ R dσ on ∂B R , K * ∂B −r [ϕ −r ] = 1 4πr ∂B −r ϕ −r dσ on ∂B −r .
As a result, we obtain that
K * ∂Ω [ϕ](z) = − 1 2πh R (y) ∞ −∞q q 2 + (y − t) 2 ϕ −r (t)h −r (t)dt + 1 4πR ∂Ω ϕ dσ for x = 1 2R , − 1 2πh −r (y) ∞ −∞q q 2 + (y − t) 2 ϕ R (t)h R (t)dt + 1 4πr ∂Ω ϕ dσ for x = − 1 2r .
(6.6) Similar to the crescent-shaped domain case, we define
U [ϕ] = F[h R ϕ R ] F[h −r ϕ −r ] ,(6.7)
where F is the Fourier transform (see Subsection 3.3 for the definition). By applying the Fourier transform to (6.5) and (6.6), we express the layer potential operators as follows.
Lemma 6.1. Let ϕ = ϕ R + ϕ −r ∈ L 2 0 (∂Ω). For z = Ψ −1 (x + iy) ∈ ∂Ω, we have
S ∂Ω [ϕ](z) = − 1 √ 2π ∞ −∞ 1 2|k| e −|x− 1 2R ||k| F[h R ϕ R ](k) + e −|x+ 1 2r ||k| F[h −r ϕ −r ](k) e iky dk + C,
where C is the constant given by withq given by (6.1).
C = 1 √ 2π ∞ −∞ 1 2|k| e − 1 2R |k| F[h R ϕ R ](k) + e − 1 2r |k| F[h −r ϕ −r ](k) dk.
Spectral resolution of the NP operator on touching disks
We define two matrix-valued functions in Lemma 6.1 and Lemma 6.2 as S = − 1 2|k|
1 − e −q|k| 0 0 1 + e −q|k| , k ∈ R \ {0}, K = 1 2 e −q|k| 1 0 0 −1 , k ∈ R.
(6.8)
Note that the definition of the matrix S is identical to S of the crescent-shaped domain case in (4.1), except that q is now replaced byq. Note also that the signs of the diagonal entries of K are changed from the diagonal entries of K of the crescent-shaped domain case (see (4.1)).
Definition 5. We define K −1/2 0 as the same as Definition 2 with S replaced by S. In other words,
K −1/2 0 := ϕ = U −1 P ϕ ϕ = ϕ 1 ϕ 2 satisfying ∞ −∞ ϕ T (− S) ϕ dk < ∞ ,
where ϕ 1 and ϕ 2 are measurable functions on R, and P is given by (3.19). Also, we define the inner product ·, · −1/2 and the norm · −1/2 on K −1/2 0 as the same as in Section 4.1 with S replaced by S.
We now naturally extend the single-layer potential and the NP operator on L 2 0 (∂Ω) to K −1/2 0 by generalizing the formulas in Lemma 6.1 and Lemma 6.2, respectively. e −|x− 1 2R ||k| ϕ R (k) + e −|x+ 1 2r ||k| ϕ r (k) e iky dk
+ 1 √ 2π ∞ −∞ 1 2|k| e − |k| 2R ϕ R (k) + e − |k| 2r ϕ r (k) dk,(6.10)
where ϕ R and ϕ r is given by
ϕ R ϕ r = P ϕ 1 ϕ 2 = 1 √ 2 − ϕ 1 + ϕ 2 ϕ 1 + ϕ 2 . (6.11)
From arguments similar to those for the crescent-shaped domain case, it can be shown that K * ∂Ω is a bounded linear operator on K −1/2 0 whose operator norm is bounded by 1 2 . In addition, K * ∂Ω is self-adjoint on K −1/2 0 , and the spectrum of K * ∂Ω on K −1/2 0 lies in the interval [−1/2, 1/2]. In the remainder of this subsection, we derive the spectral resolution of the operator.
To derive the spectral resolution of the NP operator on K −1/2 0 , we define a pair of orthogonal projection operators P 1 (s) and P 2 (s) on K −1/2 0 for each s ∈ R ∪ {∞}. Let ϕ ∈ K −1/2 0 be given by ϕ = U −1 P ϕ and ϕ = [ ϕ 1 , ϕ 2 ] T . We define P 1 (s)ϕ = U −1 P χ (−∞, s ] ϕ 1 0 , P 2 (s)ϕ = U −1 P 0 χ (−∞, s ] ϕ 2 for s ∈ R and P 1 (∞)ϕ = U −1 P ϕ 1 0 , P 2 (∞)ϕ = U −1 P 0 ϕ 2 .
Note that I = P 2 (∞) + P 1 (∞), where I is the identity operator on K The right-hand side of (6.15) has different sign of from that of (4.27) because the signs of the diagonal entries of K are changed from the diagonal entries of K in (4.1). In view of (6.12), we obtain the following.
Figure 1 .
11: A crescent-shaped domain (left) and touching disks (right).
δ ] = g on ∂D and define an indicator function α g (t) := sup{α | lim sup δ→0 δ α ϕ t,δ H * = ∞}, t ∈ (−1/2, 1/2).Then, 0 ≤ α g (t) ≤ 1 for all t.
Figure 3 .
31: A crescent-shaped domain Ω = B R \ B r (gray region in the left figure) and the vertical strip S = Ψ(Ω) (gray region in the right figure). Arrows indicate the outward normal vectors to ∂Ω or ∂S.
We define the NP operator and the single-layer potential on K
Definition 3 .
3Let by ϕ = U −1 P ϕ.
∂Ω [ϕ] := U −1 P K ϕ.(4.8) (b) We define the single-layer potential of ϕ: for z = Ψ −1 (x + iy) ∈ C,
by ϕ = U −1 P ϕ satisfying ϕ 1 , ϕ 2 ∈ L 1 (R) and ϕ 2 (k) = O(|k| 1 4 ) as |k| → 0. Set z = z 1 + iz 2 = Ψ(x + iy) ∈ C. Then, we have the following.(a) The single-layer potential S ∂Ω [ϕ](z) is continuous and uniformly bounded in C and S ∂Ω [ϕ](z) = O(|z| −1 ) as |z| → ∞. (b) The partial derivatives ∂ x S ∂Ω [ϕ](z), ∂ y S ∂Ω [ϕ](z) are uniformly bounded for x = 1 2R , 1 2r and
we estimate I + II. From (3.4) and (4.9), we have
t) E(s) = E(min(t, s)), lim t→s + E(t) = E(
Theorem 4. 6 .
6Let E(t) t∈[−1/2,1/2] be the resolution of the identity on K
Figure 6 . 1 :
61Touching disks Ω = B R ∪ B −r (gray region in the left figure) and S = Ψ(Ω) (gray region in the right figure
Lemma 6. 2 .
2For ϕ ∈ L 2 0 (∂Ω), it holds that U [K * ∂Ω [ϕ]] = P
Definition 6 .
6Let by ϕ = U −1 P ϕ.
∂Ω [ϕ] := U −1 P K ϕ. (6.9) (b) We define the single-layer potential of ϕ: for z = Ψ −1 (x + iy) ∈ C, S ∂Ω [ϕ](z)
sense of strong convergence. Following the derivations of (4.25) and (4.27) for the case of the crescent-shaped domain, it can be shown that, for all ψ, ϕ ∈ K
|y|<sWe first estimate III as s → ∞. For (x, y) in the domain of the integral III, we haveS ∂Ω [ϕ]
∂
∂ν
S ∂Ω [ϕ]
−
h R (y)dy +
x= 1
2r , |y|<s
S ∂Ω [ϕ]
∂
∂ν
S ∂Ω [ϕ]
−
h r (y)dy
+
1
2R <x< 1
2r , |y|=s
S ∂Ω [ϕ]
∂
∂ν
S ∂Ω [ϕ]
−
h(x, y)dx
= : I + II + III.
(4.16)
Theorem 6.4. Let E(t) t∈[−1/2,1/2] be the resolution of the identity on K −1/2 0 given by(6.13).Then we have the following spectral resolution of K *In other words, it holds thatThe condition lim t→s − E(t) = E(s) characterizes the point spectrum of K * ∂Ω . From (6.14), we conclude that K * ∂Ω has only a continuous spectrum. In view of (6.12) and Lemma 4.7, we havewith k = − 1 q ln(2|t|). Similar to(4.29), it holds thatFinally, we obtain the following theorem.
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| []
|
[
"Secular resonance sweeping of the main asteroid belt during planet migration",
"Secular resonance sweeping of the main asteroid belt during planet migration"
]
| [
"David A Minton [email protected] \nDepartment of Planetary Sciences\nSouthwest Research Institute\nNASA Lunar Science Institute\n1050 Walnut St., Suite. 30080302BoulderCO\n",
"Renu Malhotra \nUniversity of Arizona\n1629 East University Boulevard85721TucsonAZ\n"
]
| [
"Department of Planetary Sciences\nSouthwest Research Institute\nNASA Lunar Science Institute\n1050 Walnut St., Suite. 30080302BoulderCO",
"University of Arizona\n1629 East University Boulevard85721TucsonAZ"
]
| []
| We calculate the eccentricity excitation of asteroids produced by the sweeping ν 6 secular resonance during the epoch of planetesimal-driven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the ν 6 sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid's initial orbit. Based on the slowest rate of ν 6 sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn's migration speed of ∼ 0.15 AU My −1 during the era that the ν 6 resonance swept through the inner asteroid belt (semimajor axis range 2.1-2.8 AU). This rate limit is for Saturn's current eccentricity, and scales with the square of Saturn's eccentricity; the limit on Saturn's migration rate could be lower if Saturn's eccentricity were lower during its migration. Applied to an ensemble of fictitious asteroids, our calculations show that a prior single-peaked distribution of asteroid eccentricities would be transformed into a double-peaked distribution due to the sweeping of the ν 6 . Examination of the orbital data of main belt asteroids reveals that the proper eccentricities of the known bright (H ≤ 10.8) asteroids may be consistent with a double-peaked distribution. If so, our theoretical analysis then yields two possible solutions for the migration rate of Saturn and for the dynamical states of the pre-migration asteroid belt: a dynamically cold state (single-peaked eccentricity distribution with mean of ∼ 0.05) linked with Saturn's migration speed ∼ 4 AU My −1 , or a dynamically hot state (single-peaked eccentricity distribution with mean of ∼ 0.3) linked with Saturn's migration speed ∼ 0.8 AU My −1 . | 10.1088/0004-637x/732/1/53 | [
"https://arxiv.org/pdf/1102.3131v2.pdf"
]
| 38,040,202 | 1102.3131 | debc2ef9891d46df4391b942c4232f78fdec5067 |
Secular resonance sweeping of the main asteroid belt during planet migration
4 Mar 2011
David A Minton [email protected]
Department of Planetary Sciences
Southwest Research Institute
NASA Lunar Science Institute
1050 Walnut St., Suite. 30080302BoulderCO
Renu Malhotra
University of Arizona
1629 East University Boulevard85721TucsonAZ
Secular resonance sweeping of the main asteroid belt during planet migration
4 Mar 201132 pages, 7 figures. Accepted for publication in ApJ on March 1, 2011
We calculate the eccentricity excitation of asteroids produced by the sweeping ν 6 secular resonance during the epoch of planetesimal-driven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the ν 6 sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid's initial orbit. Based on the slowest rate of ν 6 sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn's migration speed of ∼ 0.15 AU My −1 during the era that the ν 6 resonance swept through the inner asteroid belt (semimajor axis range 2.1-2.8 AU). This rate limit is for Saturn's current eccentricity, and scales with the square of Saturn's eccentricity; the limit on Saturn's migration rate could be lower if Saturn's eccentricity were lower during its migration. Applied to an ensemble of fictitious asteroids, our calculations show that a prior single-peaked distribution of asteroid eccentricities would be transformed into a double-peaked distribution due to the sweeping of the ν 6 . Examination of the orbital data of main belt asteroids reveals that the proper eccentricities of the known bright (H ≤ 10.8) asteroids may be consistent with a double-peaked distribution. If so, our theoretical analysis then yields two possible solutions for the migration rate of Saturn and for the dynamical states of the pre-migration asteroid belt: a dynamically cold state (single-peaked eccentricity distribution with mean of ∼ 0.05) linked with Saturn's migration speed ∼ 4 AU My −1 , or a dynamically hot state (single-peaked eccentricity distribution with mean of ∼ 0.3) linked with Saturn's migration speed ∼ 0.8 AU My −1 .
Introduction
The dynamical structure of the Kuiper Belt suggests that the outer solar system experienced a phase of planetesimal-driven migration in its early history (Fernandez & Ip 1984;Malhotra 1993Malhotra , 1995Hahn & Malhotra 1999;Levison et al. 2008). Pluto and other Kuiper belt objects that are trapped in mean motion resonances (MMRs) with Neptune are explained by the outward migration of Neptune due to interactions with a more massive primordial planetesimal disk in the outer regions of the solar system (Malhotra 1993(Malhotra , 1995. In addition, the so-called the scattered disk of the Kuiper belt can also be explained by the outward migration of Neptune (Hahn & Malhotra 2005), or by the effects of a high eccentricity phase of ice giant planet evolution during the outward migration of Neptune (Levison et al. 2008). The basic premise of planetesimal-driven migration is that the giant planets formed in a more compact configuration than we find them today, and that they were surrounded by a massive (∼ 50 M ⊕ ) disk of unaccreted icy planetesimals that was the progenitor of the currently observed Kuiper belt (Hahn & Malhotra 1999). When planetesimals are preferentially scattered either inward (toward the Sun) or outward (away from the Sun), net orbital angular momentum is transferred between the disk and the large body, causing a drift in the large body's semimajor axis (Fernandez & Ip 1984;Kirsh et al. 2009). In many simulations of giant planet migration, icy planetesimals are preferentially scattered inward by each of the three outer giant planets (Saturn, Uranus, and Neptune) causing these planets to migrate outward. Due to Jupiter's large mass, planetesimals that encounter Jupiter are preferentially ejected out of the solar system, leading to a net loss of mass from the solar system and an inward migration of Jupiter.
Planetesimal-driven giant planet migration has been suggested as a cause of the Late Heavy Bombardment (LHB) Strom et al. 2005), however the link be-tween these two events has yet to be definitively established (Chapman et al. 2007;Ćuk et al. 2010;Malhotra & Strom 2010). Such migration would have enhanced the impact flux of both asteroids and comets onto the terrestrial planets in two ways. First, many of the icy planetesimals scattered by the giant planets would have crossed the orbits of the terrestrial planets. Second, as the giant planets migrated, locations of mean motion and secular resonances would have swept across the asteroid belt, raising the eccentricities of asteroids to planet-crossing values.
Recently, Minton & Malhotra (2009) showed that the patterns of depletion observed in the asteroid belt are consistent with the effects of sweeping of resonances during the migration of the giant planets. The Jupiter-facing sides of some of the Kirkwood gaps (regions of the asteroid belt that are nearly empty due to strong jovian mean motion resonances) are depleted relative to the Sun-facing sides, as would be expected due to the inward migration of Jupiter and the associated inward sweeping of the jovian mean motion resonances. The region within the inner asteroid belt between semimajor axis range 2.1-2.5 AU also has excess depletion relative to a model asteroid belt that was uniformly populated and then subsequently sculpted by the gravitational perturbations of the planets over 4 Gy, as would be expected due to the outward migration of Saturn and the associated inward sweeping of a strong secular resonance, the so-called ν 6 resonance, as explained below. In our 2009 study, we concluded that the semimajor axis distribution of asteroids in the main belt is consistent with the inward migration of Jupiter and outward migration of Saturn by amounts proposed in previous studies based on the Kuiper belt resonance structure(e.g., Malhotra 1995). However, in that study the migration timescale was not strongly constrained, because only the relative depletion of asteroids in nearby semimajor axis bins could be determined, not their overall level of depletion. In the present paper, we explore in more detail the effect that planet migration would have had on the asteroid belt due to asteroid eccentricity excitation by the sweeping of the ν 6 secular resonance. From the observed eccentricity distribution of main belt asteroids, we find that it is possible to derive constraints on the secular resonance sweeping timescale, and hence on the migration timescale. Secular resonances play an important role in the evolution of the main asteroid belt. The inner edge of the belt nearly coincides with the ν 6 secular resonance which is defined by g ≈ g 6 , where g is the rate of precession of the longitude of pericenter, ̟, of an asteroid and g 6 is the sixth eigenfrequency of the solar system planets (approximately the rate of precession of Saturn's longitude of pericenter). The ν 6 resonance, is important for the delivery of Near Earth Asteroids (NEAs) to the inner solar system (Scholl & Froeschle 1991). Williams & Faulkner (1981) showed that the location of the ν 6 resonance actually forms surfaces in a − e − sin i space, and Milani & Knezevic (1990) showed that those surfaces approximately define the "inner edge" of the main asteroid belt. However, as mentioned above, Minton & Malhotra (2009) found that, with the planets in their present configuration, planetary perturbations over the age of the solar system cannot fully account for the detailed orbital distribution of the asteroids in the inner asteroid belt. The pattern of excess depletion in inner asteroid belt noted by Minton & Malhotra (2009) is consistent with the effect of the inward sweeping of the ν 6 secular resonance. In general, the direction of motion of the ν 6 is anticorrelated with that of Saturn, so an inward sweeping of the ν 6 would be produced by an outwardly migrating Saturn.
Sweeping, or scanning, secular resonances have been analyzed in a number of previous works. Sweeping secular resonances due to the changing quadrupole moment of the Sun during solar spin-down have been explored as a possible mechanism for explaining the eccentricity and inclination of Mercury (Ward et al. 1976). Secular resonance sweeping due to the effects of the dissipating solar nebula just after planet formation has also been investigated as a possible mechanism for exciting the orbital eccentricities of Mars and of the asteroid belt (Heppenheimer 1980;Ward 1981). The dissipating massive gaseous solar nebula would have altered the secular frequencies of the solar system planets in a time-dependent way, causing locations of secular resonances to possibly sweep across the inner solar system, thereby exciting asteroids into the eccentric and inclined orbits that are observed today. This mechanism was revisited by Nagasawa et al. (2000), who incorporated a more sophisticated treatment of the nebular dispersal. However, O'Brien et al. (2007) have argued that the excitation (and clearing) of the primordial asteroid belt was unlikely due to secular resonance sweeping due to the dispersion of the solar nebula.
The special case of asteroids on initially circular orbits being swept by the ν 6 and ν 16 resonances has been investigated by Gomes (1997). In this paper, we consider the more general case of non-zero initial eccentricities; our analysis yields qualitatively new results and provides new insights into the dynamical history of the asteroid belt. This extends the work of Ward et al. (1976) and Gomes (1997) in developing analytical treatments of the effects of sweeping secular resonances on asteroid orbits. In doing so, we have developed an explicit relationship between the migration rate of the giant planets, the initial eccentricity of the asteroid and its initial longitude of perihelion, and the final eccentricity of the asteroid after the passage of the resonance. We show that for initially non-zero asteroid eccentricity, the sweeping of the ν 6 resonances can either increase or decrease asteroid eccentricities. Examining the orbits of observed main belt asteroids we find evidence for a double-peaked eccentricity distribution; this supports the case for a history of ν 6 sweeping. Quantitative comparison of our analytical theory with the semimajor axis and eccentricity distribution of asteroids yields new constraints on the timescale of planet migration.
We note that although our analysis is carried out in the specific context of the sweep-ing ν 6 resonance during the phase of planetesimal-driven migration of Jupiter and Saturn, the techniques developed here may be extended to other similar problems, for example, the sweeping of the inclination-node ν 16 resonance in the main asteroid belt, the ν 8 secular resonance in the Kuiper belt, and farther afield, the sweeping of secular resonances in circumstellar or other astrophysical disks.
Analytical theory of a sweeping secular resonance
We adopt a simplified model in which a test particle (asteroid) is perturbed only by a single resonance, the ν 6 resonance. We use a system of units where the mass is in solar masses, the semimajor axis is in units of AU, and the unit of time is (2π) −1 y. With this choice, the gravitational constant, G, is unity. An asteroid's secular perturbations close to a secular resonance can be described by the following Hamiltonian function (Malhotra 1998):
H sec = −g 0 J + ε √ 2J cos(w p − ̟),(1)
where w p = g p t + β p describes the phase of the p-th eigenmode of the linearized eccentricitypericenter secular theory for the Solar system planets (Murray & Dermott 1999), g p is the associated eigenfrequency, ̟ is the asteroid's longitude of perihelion, J = √ a 1 − √ 1 − e 2 is the canonical generalized momentum which is related to the asteroid's orbital semimajor axis a and eccentricity e; −̟ and J are the canonically conjugate pair of variables in this 1-degree-of-freedom Hamiltonian system. The coefficients g 0 and ε are given by:
g 0 = 1 4a 3/2 j α 2 j b (1) 3/2 (α j )m j ,(2)ε = 1 4a 5/4 j α 2 j b (2) 3/2 (α j )m j E (p) j ,(3)
where the subscript j refers to a planet, E (p) j is the amplitude of the g p mode in the j th planet's orbit, α j = min{a/a j , a j /a}, m j is the ratio of the mass of planet j to the Sun, and b
(1) 3/2 (α j ) and b
(2) 3/2 (α j ) are Laplace coefficients; the sum is over all major planets. The summations in equations (2)-(3) are over the 8 major planets, for the greatest accuracy; however, we will adopt the simpler two-planet model of the Sun-Jupiter-Saturn in §3, in which case we sum over only the indices referring to Jupiter and Saturn; then g p is an eigenfrequency of the secular equations of the two planet system.
With fixed values of the planetary masses and semimajor axes, g 0 , g p and ε are constant parameters in the Hamiltonian given in equation (1). However, during the epoch of giant planet migration, the planets' semimajor axes change secularly with time, so that g 0 , g p and ε become time-dependent parameters. In the analysis below, we neglect the time-dependence of g 0 and ε, and adopt a simple prescription for the time-dependence of g p (see equation 8 below). This approximation is physically motivated: the fractional variation of g 0 and ε for an individual asteroid is small compared to the effects of the "small divisor" g 0 − g p during the ν 6 resonance sweeping event.
It is useful to make a canonical transformation to new variables (φ, P ) defined by the following generating function,
F (−̟, P, t) = (w p (t) − ̟)P(4)
Thus, φ = ∂F /∂P = (w p (t) − ̟) and J = −∂F /∂̟ = P . The new Hamiltonian function is
H sec = H sec + ∂F /∂t,H sec = (ẇ p (t) − g 0 )J + ε √ 2J cos φ,(5)
where we have retained J to denote the canonical momentum, since P = J. It is useful to make a second canonical transformation to canonical eccentric variables,
x = √ 2J cos φ, y = − √ 2J sin φ,(6)
where x is the canonical coordinate and y is the canonically conjugate momentum. The Hamiltonian expressed in these variables is
H sec = (ẇ p (t) − g 0 ) x 2 + y 2 2 + εx.(7)
As discussed above, during planetary migration, the secular frequency g p is a slowly varying function of time. We approximate its rate of change,ġ p = 2λ, as a constant, so thaṫ w p (t) = g p,0 + 2λt.
We define t = 0 as the epoch of exact resonance crossing, so that g p,0 = g 0 (cf. Ward et al. 1976). Then,ẇ p (t) − g 0 = 2λt, and the equations of motion from the Hamiltonian of equation (7) can be written as:ẋ = 2λty,
y = −2λtx − ε.(9)
These equations of motion form a system of linear, nonhomogenous differential equations, whose solution is a linear combination of a homogeneous and a particular solution. The homogeneous solution can be found by inspection, giving:
x h (t) = c 1 cos λt 2 + c 2 sin λt 2 ,
y h (t) = −c 1 sin λt 2 + c 2 cos λt 2 ,
where c 1 and c 2 are constant coefficients. We use the method of variation of parameters to find the particular solution. Accordingly, we replace the constants c 1 and c 2 in the homogeneous solution with functions A(t) and B(t), to seek the particular solution of the form
x p (t) = A(t) cos λt 2 + B(t) sin λt 2 (13) y p (t) = −A(t) sin λt 2 + B(t) cos λt 2 .(14)
Substituting this into the equations of motion we now have:
A cos λt 2 +Ḃ sin λt 2 = 0,
−Ȧ sin λt 2 +Ḃ cos λt 2 = −ε;
thereforeȦ = ε sin λt 2 ,(16)B = −ε cos λt 2 .(17)
Equations (17) and (18) do not have a simple closed-form solution, but their solution can be expressed in terms of Fresnel integrals (Zwillinger 1996). The Fresnel integrals are defined as follows:
S(t) = t 0 sin t ′2 dt ′ ,(19)C(t) = t 0 cos t ′2 dt ′ .(20)
and have the following properties:
S(−t) = −S(t),(21)C(−t) = −C(t),(22)S(∞) = C(∞) = π 8 .(23)
Therefore
A(t) = ε |λ| S t |λ| ,(24)B(t) = − ε |λ| C t |λ| .(25)
We denote initial conditions with a subscript i, and write the solution to equations (9) and (10) as
x(t) = x i cos λ t 2 − t 2 i + y i sin λ t 2 − t 2 i + ε |λ| (S − S i ) cos λt 2 − (C − C i ) sin λt 2 ,(26)y(t) = −x i sin λ t 2 − t 2 i + y i cos λ t 2 − t 2 i − ε |λ| (C − C i ) cos λt 2 + (S − S i ) sin λt 2 .(27)
Because the asteroid is swept over by the secular resonance at time t = 0, we can calculate the changes in x, y by letting t i = −t f and evaluating the coefficients C i , C f , S i , S f far from resonance passage, i.e., for t f |λ| ≫ 1, by use of equation (23). Thus we find
x f = x i + ε π 2|λ| cos λt 2 i − sin λt 2 i ,(28)y f = y i − ε π 2|λ| cos λt 2 i + sin λt 2 i .(29)
The final value of J long after resonance passage is therefore given by
J f = 1 2 x 2 f + y 2 f = 1 2 x 2 i + y 2 i + πε 2 2|λ| + ε π 2|λ| [x i (cos λt 2 i − sin λt 2 i ) − y i (cos λt 2 i + sin λt 2 i )] = J i + πε 2 2|λ| + ε 2πJ i |λ| cos(φ i − λt 2 i − π 4 ).(30)
With a judicious choice of the initial time, t i , and without loss of generality, the cosine in the last term becomes cos ̟ i , and therefore
J f = J i + πε 2 2|λ| + ε 2πJ i |λ| cos ̟ i .(31)
The asteroid's semimajor axis a is unchanged by the secular perturbations; thus, the changes in J reflect changes in the asteroid's eccentricity e. For asteroids with non-zero initial eccentricity, the phase dependence in equation (31) means that secular resonance sweeping can potentially both excite and damp orbital eccentricities. We also note that the magnitude of eccentricity change is inversely related to the speed of planet migration. In linear secular theory (equation (1)), eccentricity and inclination are decoupled, and therefore the effect of the sweeping ν 6 does not depend on the inclination. However, as Williams & Faulkner (1981) showed, the location of the ν 6 does depend on inclination, but the dependence is weak for typical inclinations of main belt objects. Nevertheless, there are populations of main belt asteroids at high inclination (such as the Hungaria and Phocaea families), and an analysis of secular resonance sweeping that incorporates coupling between eccentricity and inclination would be valuable for understanding the effects of planet migration on these populations; we leave this to a future investigation.
For small e, we can use the approximation J ≃ 1 2 √ ae 2 . Considering all possible values of cos ̟ i ∈ {−1, +1}, an asteroid with initial eccentricity e i that is swept by the ν 6 resonance will have a final eccentricity in the range e min to e max , where
e min,max ≃ |e i ± δ e | ,(32)
and
δ e ≡ ε π |λ| √ a .(33)
Equations (31)-(33) have the following implications:
1. Initially circular orbits become eccentric, with a final eccentricity δ e .
2. An ensemble of orbits with the same a and initial non-zero e but uniform random orientations of pericenter are transformed into an ensemble that has eccentricities in the range e min to e max ; this range is not uniformly distributed because of the cos ̟ i dependence in equation (31), rather the distribution peaks at the extreme values (see Fig. 1
below).
3. An ensemble of asteroids having an initial distribution of eccentricities which is a singlepeaked Gaussian (and random orientations of pericenter) would be transformed into one with a double-peaked eccentricity distribution.
Application to the Main Asteroid Belt
In light of the above calculations, it is possible to conclude that if the asteroid belt were initially dynamically cold, that is asteroids were on nearly circular orbits prior to secular resonance sweeping, then the asteroids would be nearly uniformly excited to a narrow range of final eccentricities, the value of which would be determined by the rate of resonance sweeping.
Because asteroids having eccentricities above planet-crossing values would be unlikely to survive to the present day, it follows that an initially cold asteroid belt which is uniformly excited by the ν 6 sweeping will either lose all its asteroids or none. On the other hand, an initially excited asteroid belt, that is a belt with asteroids that had non-zero eccentricities prior to the ν 6 sweeping, would have asteroids' final eccentricities bounded by equation (32), allowing for partial depletion and also broadening of its eccentricity distribution. In this section, we apply our theoretical analysis to the problem of the ν 6 resonance sweeping through the asteroid belt, and compare the theoretical predictions with the observed eccentricity distribution of asteroids.
Parameters
In order to apply the theory, we must find the location of the ν 6 resonance as a function of the semimajor axes of the giant planets orbits, and also obtain values for the parameter ε (equation 3), for asteroids with semimajor axis values in the main asteroid belt. The location of the ν 6 resonance is defined as the semimajor axis, a ν 6 , where the rate, g 0 (equation 2), of pericenter precession of a massless particle (or asteroid) is equal to the g 6 eigenfrequency of the solar system. In the current solar system, the g 6 frequency is associated with the secular mode with the most power in Saturn's eccentricity-pericenter variations. During the epoch of planetesimal-driven planet migration, Jupiter migrated by only a small amount but Saturn likely migrated significantly more (Fernandez & Ip 1984;Malhotra 1995;Tsiganis et al. 2005), so we expect that the variation in location of the ν 6 secular resonance is most sensitive to Saturn's semimajor axis. We therefore adopt a simple model of planet migration in which Jupiter is fixed at 5.2 AU and only Saturn migrates. We neglect the effects of the ice giants Uranus and Neptune, as well as secular effects due to the previously more massive Kuiper belt and asteroid belt. With these simplifications, the g 6 frequency varies with time as Saturn migrates, so we parametrize g 6 as a function of Saturn's semimajor axis. In contrast with the variation of g 6 , there is little variation of the asteroid's apsidal precession rate, g 0 , as Saturn migrates. Thus, finding a ν 6 is reduced to calculating the dependence of g 6 on Saturn's semimajor axis.
To calculate g 6 as a function of Saturn's semimajor axis, we proceed as follows. For fixed planetary semimajor axes, the Laplace-Lagrange secular theory provides the secular frequencies and orbital element variations of the planets. This is a linear perturbation theory, in which the disturbing function is truncated to secular terms of second order in eccentricity and first order in mass (Murray & Dermott 1999). In the planar two-planet case, the secular perturbations of planet j, where j = 5 is Jupiter and j = 6 is Saturn, are described by the following disturbing function:
R j = n j a 2 j 1 2 A jj e 2 j + A jk e 5 e 6 cos(̟ 5 − ̟ 6 ) ,(34)
where n is the mean motion, and A is a matrix with elements
A jj = +n j 1 4 m k M ⊙ +m j α jkᾱjk b (1) 3/2 (α jk ),(35)A jk = −n j 1 4 m k M ⊙ +m j α jkᾱjk b (2) 3/2 (α jk ),(36)
for j = 5, 6, k = 6, 5, and j = k; α jk = min{a j /a k , a k /a j }, and
α jk = 1 : a j > a k a j /a k : a j < a k .(37)
The secular motion of the planets is then described by a set of linear differential equations for the eccentricity vectors,
e j (sin ̟ j , cos ̟ j ) ≡ (h j , k j ), h j = + 6 p=5 A pj k j ,k j = − 6 p=5 A pj h j .(38)
For fixed planetary semimajor axes, the coefficients are constants, and the solution is given by a linear superposition of eigenmodes:
{h j , k j } = p E (p) j {cos(g p t + β p ), sin(g p t + β p )},(39)
where g p are the eigenfrequencies of the matrix A and E (p) j are the corresponding eigenvectors; the amplitudes of the eigenvectors and the phases β p are determined by initial conditions. In our 2-planet model, the secular frequencies g 5 and g 6 depend on the masses of Jupiter, Saturn, and the Sun and on the semimajor axes of Jupiter and Saturn.
For the current semimajor axes and eccentricities of Jupiter and Saturn the Laplace-Lagrance theory gives frequency values g 5 = 3.7 ′′ y −1 and g 6 = 22.3 ′′ y −1 , which are lower than the more accurate values given by Brouwer & van Woerkom (1950) by 14% and 20%, respectively (Laskar 1988). Brouwer & van Woerkom (1950) achieved their more accurate solution by incorporating higher order terms in the disturbing function involving 2λ 5 − 5λ 6 , which arise due to Jupiter and Saturn's proximity to the 5:2 resonance (the so-called "Great Inequality"). By doing an accurate numerical analysis (described below), we found that the effect of the 5:2 resonance is only important over a very narrow range in Saturn's semimajor axis. More significant is the perturbation owing to the 2:1 near-resonance of Jupiter and Saturn. Malhotra et al. (1989) developed corrections to the Laplace-Lagrange theory to account for the perturbations from n+1 : n resonances in the context of the Uranian satellite system. Applying that approach to our problem, we find that the 2:1 near-resonance between Jupiter and Saturn leads to zeroth order corrections to the elements of the A matrix 1 . Including these corrections, we determined the secular frequencies for a range of values of Saturn's semimajor axis; the result for g 6 is shown in Fig. 2a (dashed line).
We also calculated values for the eccentricity-pericenter eigenfrequencies by direct numerical integration of the full equations of motion for the two-planet, planar solar system. In these integrations, Jupiter's initial semimajor axis was 5.2 AU, Saturn's semimajor axis, a 6 , was one of 233 values in the range 7.3-10.45 AU, initial eccentricities of Jupiter and Saturn were 0.05, and initial inclinations were zero. The initial longitude of pericenter and mean anomalies of Jupiter were ̟ 5,i = 15 • and λ 5,i = 92 • , and Saturn were ̟ 6,i = 338 • and λ 6,i = 62.5 • . In each case, the planets' orbits were integrated for 100 myr, and a Fourier transform of the time series of the {h j , k j } yields their spectrum of secular frequencies. For regular (non-chaotic) orbits, the spectral frequencies are well defined and are readily identified with the frequencies of the secular solution. The g 6 frequency as a function of Saturn's semimajor axis was obtained by this numerical analysis; the result is shown by the solid line in Fig. 2a.
The comparison between the numerical analysis and the Laplace-Lagrange secular theory indicates that the linear secular theory, including the corrections due to the 2:1 nearresonance, is an adequate model for the variation in g 6 as a function of a 6 . We adopted the latter for its convenience in the needed computations. The value of a ν 6 as a function of Saturn's semimajor axis was thus found by solving for the value of asteroid semimajor axis where g 0 = g 6 ; g 0 was calculated using equation (2) and g 6 is the eigenfrequency associated with the p = 6 eigenmode (at each value of Saturn's semimajor axis). The result is shown in Fig. 2b.
We also used the analytical secular theory to calculate the eigenvector components E (6) j in the secular solution of the 2-planet system, for each value of Saturn's semimajor axis. We adopted the same values for the initial conditions of Jupiter and Saturn as in the direct numerical integrations discussed above. Finally, we computed the values of the parameter ε at each location a ν 6 of the secular resonance. The result is plotted in Fig. 3. Despite the 1 Corrections due to near-resonances are of order e (n−1) , where e is eccentricity and n is the order of the resonance. The 5:2 is a third order resonance, so its effect is O(e 2 ). The 2:1 is a first order resonance, so that its effect does not depend on e. Therefore the discrepancy between linear theory and numerical analysis (or the higher order theory of Brouwer & van Woerkom) arising from the Great Inequality would be much less if Jupiter and Saturn were on more circular orbits, but the effect due to the 2:1 resonance would remain. complexity of the computation, the result shown in Fig. 3 is approximated well by a simple exponential curve, ε ≈ 3.5×10 −9 exp(2a ν 6 /AU), in the semimajor axis range 2 < a ν 6 /AU < 4.
Four test cases
We checked the results of our analytical model against four full numerical simulations of the restricted four-body problem (the Sun-Jupiter-Saturn system with test particle asteroids) in which the test particles in the asteroid belt are subjected to the effects of a migrating Saturn. The numerical integration was performed with an implementation of a symplectic mapping (Wisdom & Holman 1991;Saha & Tremaine 1992), and the integration stepsize was 0.01 y. Jupiter and Saturn were the only massive planets that were integrated, and their mutual gravitational influence was included. The asteroids were approximated as massless test particles. The current solar system values of the eccentricity of Jupiter and Saturn were adopted and all inclinations (planets and test particles) were set to zero. An external acceleration was applied to Saturn to cause it to migrate outward linearly and smoothly starting at 8.5 AU at the desired rate. As shown in Fig. 2b, the ν 6 resonance was located at 3 AU at the beginning of the simulation, and swept inward past the current location of the inner asteroid belt in all simulations. In each of the four simulations, 30 test particles were placed at 2.3 AU and given different initial longitudes of pericenter spaced 12 • apart. The semimajor axis value of 2.3 AU was chosen because it is far away from the complications arising due to strong mean motion resonances. The only parameters varied between each of the four simulations were the initial osculating eccentricities of the test particles, e i , and the migration speed of Saturn,ȧ 6 . The parameters explored were: These values were chosen to illustrate the most relevant qualitative features. The migration rates of 0.5 AU My −1 and 1.0 AU My −1 are slow enough so that the change in eccentricity is substantial, but not so slow that the non-linear effects at high eccentricity swamp the results. These test cases illustrate both how well the analytical model matches the numerical results, and where the analytical model breaks down.
Two aspects of the analytical model were checked. First, the perturbative equations of motion, equations (9) and (10), were numerically integrated, and their numerical solution compared with the numerical solution from the direct numerical integration of the full equations of motion. For the perturbative solution, we adopted values for λ that were approximately equivalent to the values ofȧ 6 in the full numerical integrations. Second, the eccentricity bounds predicted by the analytical theory, equation (32), were compared with both numerical solutions. The results of these comparisons for the four test cases are shown in Fig. 4. We find that the analytically predicted values of the maximum and minimum final eccentricities (shown as horizontal dashed lines) are in excellent agreement with the final values of the eccentricities found in the numerical solution of the perturbative equations, and in fairly good agreement with those found in the full numerical solution. Not surprisingly, we find that the test particles in the full numerical integrations exhibit somewhat more complicated behavior than the perturbative approximation, and equation (32) somewhat underpredicts the maximum final eccentricity: this may be due to higher order terms in the disturbing function that have been neglected in the perturbative analysis and which become more important at high eccentricity; effects due to close encounters with Jupiter also become important at the high eccentricities.
Comparison with observed asteroid eccentricities
Does the eccentricity distribution of main belt asteroids retain features corresponding to the effects of the ν 6 resonance sweeping? To answer this question, we need to know the main belt eccentricity distribution free of observational bias, and also relatively free of the effects of ∼ 4 Gy of collisional evolution subsequent to the effects of planetary migration. We therefore obtained the proper elements of the observationally complete sample of asteroids with absolute magnitude H ≤ 10.8 from the AstDys online data service (Knežević & Milani 2003); we excluded from this set the members of collisional families as identified by Nesvorný et al. (2006). These same criteria were adopted in Minton & Malhotra (2010) in a study of the long term dynamical evolution of large asteroids. This sample of 931 main belt asteroids is a good approximation to a complete set of large asteroids that have been least perturbed by either dynamical evolution or collisional evolution since the epoch of the last major dynamical event that occurred in this region of the solar system; therefore this sample likely preserves best the post-migration orbital distribution of the asteroid belt. The proper eccentricity distribution of these asteroids is shown in Fig. 5. This distribution has usually been described in the literature by simply quoting its mean value (and sometimes a dispersion) (Murray & Dermott 1999;O'Brien et al. 2007). Our best fit single Gaussian distribution to this data has a mean, µ e and standard deviation, σ e , given by µ e = 0.135 ± 0.00013 and σ e = 0.0716 ± 0.00022, and is plotted in Fig. 5. However, we also note (by eye) a possible indication of a double-peak feature in the observed population. Our best fit double Gaussian distribution (modelled as two symmetrical Gaussians with the same standard deviation, but different mean values) to the same data has the following parameters:
µ ′ e,1 = 0.0846 ± 0.00011, µ ′ e,2 = 0.185 ± 0.00012, σ ′ e = 0.0411 ± 0.00020.
More details of how these fits were obtained are described in Appendix A, where we also discuss goodness-of-fit of the single and double gaussian distributions. A Kolmogorov-Sinai test shows that there is only a 4.5% probability that the observed eccentricities actually are consistent with the best-fit single Gaussian distribution, and a 73% probability that they are consistent with the best-fit double Gaussian distribution; while the double gaussian is apparently a better fit to the data compared to the single gaussian, a K-S probability of only 73% is quite far from a statistically significant level of confidence. A dip test for multi-modality in the observed eccentricity distribution (Hartigan & Hartigan 1985) is also inconclusive; the likely reason for this is that our data sample is not very large, and the separation between the two putative peaks is too small in relation to the dispersion. Nevertheless, we bravely proceed in the next section with the implications of a double-Gaussian eccentricity distribution, with the caveat that some of the conclusions we reach are based on this statistically marginal result.
A constraint on Saturn's migration rate
By relating the g 6 secular frequency to the semimajor axis of Saturn,ġ 6 can be related to the migration rate of Saturn,ȧ 6 . In this section only the effects of the sweeping ν 6 secular resonance will be considered, and effects due to sweeping jovian mean motion resonances will be ignored. Because Jupiter is thought to have migrated inward a much smaller distance than Saturn migrated outward during planetesimal-driven migration, the effects due to migrating jovian MMRs were likely confined to narrow regions near strong resonances (Minton & Malhotra 2009). In the inner asteroid belt between 2.1-2.8 AU, these would include the 3:1 and 5:2 resonances, currently located at approximately 2.5 AU and 2.7 AU, respectively. As shown in Fig. 2b, plausible parameters for the outward migration of Saturn would have allowed the ν 6 to sweep across the entire inner asteroid belt. Therefore the ν 6 resonance would have been the major excitation mechanism across the 2.1-2.8 AU region of the main belt (and possibly across the entire main asteroid belt, depending on Saturn's pre-migration semimajor axis) during giant planet migration.
We used the results of our analytical model to set limits on the rate of migration of Saturn assuming a linear migration profile, with the caveat that many important effects are ignored, such as asteroid-Jupiter mean motion resonances, and Jupiter-Saturn mean motion resonances (with the exception of the 2:1 resonance). We have confined our analysis to only the region of the main belt spanning 2.1-2.8 AU. Beyond 2.8 AU strong jovian mean motion resonance become more numerous. Due to the high probability that the icy planetesimals driving planet migration would be ejected from the solar system by Jupiter, Jupiter likely migrated inward. The migration of Jupiter would have caused strong jovian mean motion resonances to sweep the asteroid belt, causing additional depletion beyond that of the sweeping ν 6 resonance (Minton & Malhotra 2009). A further complication is that sweeping jovian mean motion resonances may have also trapped icy planetesimals that entered the asteroid belt region from their source region beyond Neptune ). The effects of these complications are reduced when we consider only the inner asteroid belt. From Fig. 2b, we find that the ν 6 would have swept the inner asteroid belt region between 2.1-2.8 AU when Saturn was between ∼ 8.5-9.2 AU. Therefore the limits onȧ 6 that we set using the inner asteroid belt as a constraint are only applicable for this particular portion of Saturn's migration history.
Our theoretically estimated final eccentricity as a function of initial asteroid semimajor axis and eccentricity is shown in Fig. 6 for three different adopted migration rates of Saturn. The larger the initial asteroid eccentricities, the wider the bounds in their final eccentricities. If we adopt the reasonable criterion that an asteroid is lost from the main belt when it achieves a planet-crossing orbit (that is, crossing the orbits of either Jupiter or Mars) and that initial asteroid eccentricities were therefore confined to 0.4, then from Fig. 6 Saturn's migration rate must have beenȧ 6 0.15 AU My −1 when the ν 6 resonance was sweeping through the inner asteroid belt. Our results indicate that if Saturn's migration rate had been slower than 0.15 AU My −1 when it was migrating across ∼ 8.5-9.2 AU, then the inner asteroid belt would have been completely swept clear of asteroids by the ν 6 resonance.
In light of our analysis and the observed dispersion of eccentricities in the asteroid belt (Fig. 5), we can also immediately conclude that the pre-migration asteroid belt between 2.1 and 2.8 AU had significantly non-zero eccentricities. This is because, as discussed at the top of §3, an initially cold asteroid belt swept by the ν 6 resonance would either lose all its asteroids or none, and very low initial eccentricities would result in final asteroid eccentricities in a very narrow range of values (cf. equation 32), in contradiction with the fairly wide eccentricity dispersion that is observed. This conclusion supports recent results from studies of planetesimal accretion and asteroid and planet formation that the asteroids were modestly excited at the end of their formation (e.g., Petit et al. 2002).
In Appendix A we show that the double-gaussian distribution is a slightly better fit to the main belt asteroid eccentricity distribution, but the statistical tests do not rule out a single-peaked distribution. We boldly proceed with considering the implications of the double-peaked eccentricity distribution to further constrain the migration rate of Saturn, with the caveat that these results can only be said to be consistent with the observations, rather than uniquely constrained by them.
If the pre-sweeping asteroid belt had a Gaussian eccentricity distribution, then the lower peak of the post-sweeping asteroid belt should be equal to the lower bound of equation (32). We use the analytical theory to make a rough estimate of the parameter λ (and henceȧ 6 ) that would yield a final distribution with lower peak near 0.09 and upper peak near 0.19 (which is similar to the best-fit double Gaussian in Fig. 5).
Applying equation (32), we see that there are two possible solutions: e i = 0.14,δ e = 0.05 and e i = 0.05,δ e = 0.14. A corresponding migration rate of Saturn can be estimated from the value of δ e using equation (33), and the parameter relationships plotted in Figs. 2 and 3. The former solution (δ e = 0.05) requires a migration rate for Saturn oḟ a 6 = 30 AU My −1 . We mention this implausible solution here for completeness, but we will not discuss it any further. The latter solution (δ e = 0.14) requires a migration rate for Saturn ofȧ 6 = 4 AU My −1 . We dub this solution the "cold belt" solution. This rate is comparable to the rates of planet migration found in the "Jumping Jupiter" scenario proposed by Brasser et al. (2009). A third solution exists if we consider that eccentricities in the main belt are restricted by the orbits of Mars and Jupiter on either side, such that stable asteroid orbits do not cross the planetary orbits. This limits asteroid eccentricities to values such that neither the aphelion of the asteroid crosses the perihelion distance of Jupiter, nor the perihelion of the asteroid crosses the aphelion distance of Mars. Maximum asteroid eccentricity is therefore a function of semimajor axis, where e max = min(1 − Q M ars /a − 1, q Jupiter /a − 1), where Q and q are planet aphelion and perihelion, respectively, and a is the semimajor axis of the asteroid. In this case, an initial single Gaussian eccentricity distribution with a mean greater than ∼ 0.3 would be severely truncated, therefore we need only fit the lower peak of the double Gaussian distribution at e = 0.09. Applying equation (32), we find that δ e = 0.21 provides a good fit. The corresponding migration rate of Saturn isȧ 6 = 0.8 AU My −1 . We dub this solution the "hot belt" solution.
We illustrate the two possible solutions for an ensemble of hypothetical asteroids having semimajor axes uniformly distributed randomly in the range 2.1 AU to 2.8 AU. In Fig. 7a the initial eccentricity distribution is modeled as a Gaussian distribution with a mean e i = 0.05 and a standard deviation of 0.01. This initial standard deviation was chosen so that the final standard deviation would be the same as that of the observed main belt. Fig. 7b shows the eccentricity distribution after ν 6 resonance sweeping has occurred due to the migration of Saturn at a rate of 4 AU My −1 . The final distribution was calculated with equation (31); we used values of ε shown in Fig. 3, and the value of λ was calculated with the aid of Fig. 2 which relates the value of g p to the semimajor axis of Saturn. As expected, when an ensemble of asteroids with a single-peaked eccentricity distribution is subjected to the sweeping secular resonance, the result is a double-peaked eccentricity distribution. Because of the slight bias towards the upper limit of the eccentricity excitation band, proportionally more asteroids are found in the upper peak.
In Fig. 7c the initial eccentricity distribution is modeled as a truncated Gaussian: a Gaussian with mean e i = 0.4 and standard deviation 0.1, but truncated at the semimajor axis-dependent Mars-crossing value. We used equation (31) to calculate the eccentricity distribution of this hypothetical ensemble after ν 6 resonance sweeping withȧ 6 = 0.8 AU My −1 . Again, allowing that only those asteroids whose final eccentricities are below the Marscrossing value will remain, the resulting post-migration eccentricity distribution is shown in Fig. 7d. In this case, we find the lower peak at the same eccentricity value as the lower peak in the observed main belt distribution (see Fig. 5).
In both cases of possible solutions (initially cold main belt with e i = 0.05 andȧ 6 = 4 AU My −1 ; and initially hot main belt with e i = 0.4 andȧ 6 = 0.8 AU My −1 ), the theoretical models yield an excess of asteroids with eccentricities greater than 0.2 than in the observed main belt. However, as shown by Minton & Malhotra (2010), on gigayear timescales, the e 0.2 population of the asteroid belt is dynamically more unstable than the e 0.2 population. Thus, both solutions may be consistent with the observations, as post-sweeping dynamical erosion could result in a final eccentricity distribution resembling more closely the observed distribution.
The estimates of Saturn's migration rate quoted above depend strongly on the eccentricities of the giant planets during their migration. In deriving the above estimates, we adopted the present values of the giant planets' orbital eccentricities. The ν 6 resonance strength coefficient ε (equation 3) is proportional to the amplitude of the p = 6 mode, which is related to the eccentricities of the giant planets (namely Saturn and Jupiter). From equation (39), and the definition e j (sin ̟ j , cos ̟ j ) ≡ (h j , k j ), the value of E (p) j is a linear combination of the eccentricities of the giant planets. Because Saturn is the planet with the largest amplitude of the p = 6 mode, from equation (31) the relationship between the sweep rate and the value of Saturn's eccentricity is approximately λ min ∝ e 2 6 . Therefore, to increase the limiting timescale by a factor of ten would only require that the giant planets' eccentricities were ∼ 0.3× their current value (i.e., e 5,6 ≈ 0.015).
Conclusion and Discussion
Based on the existence of the inner asteroid belt, we conclude that Saturn's migration rate must have been 0.15 AU My −1 as Saturn migrated from 8.5 to 9.2 AU (as the ν 6 resonance migrated from 2.8 AU to 2.1 AU). Migration rates lower than ∼ 0.15 AU My −1 would be inconsistent with the survival of any asteroids in the inner main belt, as the ν 6 secular resonance would have excited asteroid eccentricities to planet-crossing values. This lower limit for the migration rate of Saturn assumes that Jupiter and Saturn had their current orbital eccentricities; the migration rate limit is inversely proportional to the square of the amplitude of the g 6 secular mode; if Jupiter and Saturn's eccentricities were ∼ 0.3× their current value (i.e., e 5,6 ≈ 0.015) during the planet migration epoch, the limit on the migration rate decreases by a factor of ∼ 10. (This caveat also applies to the migration rate limits quoted below.)
Our analysis of secular resonance sweeping predicts that a single-peaked eccentricity distribution will be transformed into a double-peaked eccentricity by secular resonance sweeping. We find that the observed eccentricity distribution of asteroids may be consistent with a double-peaked distribution function, although we acknowledge that the statistics are poor. We used a double-Gaussian function to model the observed asteroid eccentricity distribution to set even tighter, albeit model-dependent, constraints on the migration rate of Saturn.
We identified two possible migration rates that depend on the pre-migration dynamical state of the main asteroid belt. The first, the "cold primordial belt" solution, has an asteroid belt with an initial eccentricity distribution modeled as a Gaussian with e i = 0.05; Saturn's migration rate of 4 AU My −1 yields a final eccentricity distribution consistent with the observed asteroid belt. The second, the "hot primordial belt" solution, has an asteroid belt with an initial eccentricity distribution modeled as a Gaussian with e i = 0.4, σ e = 0.1, but truncated above the Mars-crossing value of eccentricity; in this case, Saturn's migration rate of 0.8 AU My −1 is generally consistent with the observed asteroid belt.
Each of these solutions has very different implications for the primordial excitation and depletion of the main asteroid belt. The cold belt solution, withȧ 6 = 4 AU My −1 , would lead to little depletion of the asteroid belt during giant planet migration, as the ν 6 resonance would be unable to raise eccentricities to Mars-crossing values. This implies an initial dynamically quite cold asteroid belt with not more than ∼ 2 times the mass of the current main belt; the latter estimate comes from accounting for dynamical erosion over the age of the solar system (Minton & Malhotra 2010).
The hot belt solution, withȧ 6 = 0.8 AU My −1 , would lead to loss of asteroids directly due to excitation of eccentricities above planet-crossing values, by about a factor of ∼ 2. In this case, the main asteroid belt was more dynamically excited prior to resonance sweeping than we find it today. This implies much greater loss of asteroids prior to ν 6 -sweeping, as the peak of the eccentricity distribution would be near the Mars-crossing value, and subject to strong dynamical erosion (Minton & Malhotra 2010).
Each of these solutions has different implications for the model that the Late Heavy Bombardment of the inner solar system is linked to the epoch of planetesimal-driven giant planet migration Strom et al. 2005). These implications will be explored in a future paper.
We remind the reader that in order to elucidate the effects of ν 6 resonance sweeping, we have made a number of simplifying assumptions to arrive at an analytically tractable model. These simplifications include neglecting the effects of planets other than Jupiter and Saturn, the effects of sweeping jovian mean motion resonances on asteroids, the effects of a presumed massive Kuiper belt during the epoch of planet migration, and the self-gravity and collisional interactions of a previously more massive asteroid belt. In addition, our analysis was carried out in the planar approximation, thereby neglecting any eccentricity-inclination coupling effects. These neglected effects can be expected to reduce somewhat the lower limit on Saturn's migration speed that we have derived, because in general they would reduce the effectiveness of the ν 6 in exciting asteroid eccentricities. Perhaps more importantly, giant planet migration would also lead to the sweeping of the main asteroid belt by the ν 16 inclination secular resonance (Williams & Faulkner 1981) whose effects could be used to infer additional constraints.
Recently, Morbidelli et al. (2010) found through numerical simulations with slow rates of planet migration (e-folding timescales exceeding 5 Myr) that the surviving asteroids in the main belt tend to be clumped around mean motion resonances. The semimajor axis distribution of survivors is found very different from the observed distribution for asteroids, and also that a large proportion of survivors have inclinations above 20 • , both inconsistent with the current main asteroid belt; comparisons made with the unbiased main belt asteroid distributions as described in Minton & Malhotra (2009). This is likely due to the effects of the sweeping ν 16 inclination-longitude of ascending node secular resonance, analogous to the sweeping ν 6 eccentricity-pericenter secular resonance that we analyzed in the present study. While the effects of the sweeping ν 16 resonance are analogous to the ν 6 , only affecting inclinations instead of eccentricities, a full analysis of the asteroid belt inclinations is beyond the scope of the present work, but will be explored in a future study.
A number of other studies have derived limits on the speed of planetesimal-driven giant planet migration. Murray-Clay & Chiang (2005) exclude an e−folding migration timescale τ ≤ 1 My to 99.65% confidence based on the lack of a large observed asymmetry in the population of Kuiper belt objects in the two libration centers of the 2:1 Neptune mean motion resonance. Boué et al. (2009) exclude τ ≤ 7 My based on the observed obliquity of Saturn. The latter lower limit on the migration timescale is slightly incompatible with the lower limit on the rate of Saturn's migration ofȧ 6 > 0.15 AU My −1 we derive based on the existence of the inner asteroid belt. One way these can be reconciled is if Saturn's orbital eccentricity were a factor ∼ 2 smaller than its present value as it migrated from 8.5 AU to 9.2 AU; then, some mechanism would need to have increased Saturn's eccentricity up to its present value by the time Saturn reached its present semimajor axis of ∼ 9.6 AU.
The authors would like to thank the anonymous reviewer and the editor Eric Feigelson for useful comments. This research was supported in part by NSF grant no. AST-0806828 and NASA:NESSF grant no. NNX08AW25H. The work of David Minton was additionally partially supported by NASA NLSI/CLOE research grant no. NNA09DB32A
A. The Main belt eccentricity distribution
The binned eccentricity distribution may be modeled as a Gaussian probability distribution function, given by:
p(x) = 1 σ √ 2π exp − (x − µ) 2 2σ 2 ,(A1)
where σ is the standard deviation, µ is the mean, and x is the random variable; in our case x is the eccentricity. With an appropriate scaling factor, equation (A1) can be used to model the number of asteroids per eccentricity bin. However, rather than fit the binned distribution, we instead perform a least squares fit of the unbinned sample to the Gaussian cumulative distribution function given by:
P (x) = 1 2 + 1 2 erf − x − µ σ √ 2 .(A2)
For the eccentricities of our sample of 931 main belt asteroids, the best fit parameters are: µ e = 0.135 ± 0.00013, σ e = 0.0716 ± 0.00022.
We also fit the data to a double-Gaussian distribution,
p 2 (x) = A ′ σ ′ √ 2π exp − (x − µ ′ 1 ) 2 2σ ′2 + exp − (x − µ ′ 2 ) 2 2σ ′2 .(A3)
The cumulative distribution function for equation (A3) is
P 2 (x) = 1 2 + 1 4 erf − x − µ ′ 1 σ √ 2 + erf − x − µ ′ 2 σ √ 2 .(A4)
For the eccentricities of our sample of 931 main belt asteroids, we performed a least squares fit to equation (A4) and obtained the following best-fit parameters:
µ ′ e,1 = 0.0846 ± 0.00011, µ ′ e,2 = 0.185 ± 0.00012, σ ′ e = 0.0411 ± 0.00020.
We evaluated the goodness of fit using the Kolmogorov-Smirnov (K-S) test. The K-S test determines the probability that two distributions are the same, or in our case how well our model distributions fit the observed data (Press et al. 1992). The K-S test compares the cumulative distribution of the data against the model cumulative distribution function. We found that our asteroid sample has a probability of 4.5 × 10 −2 that it comes from the best fit single Gaussian (equation (A2)), but a probability of 0.73 that it comes from the double-Gaussian (equation (A4)). Therefore, the K-S tests indicate that the double-Gaussian is a better fit to the data than the single-Gaussian.
We performed Hartigan's dip test (Hartigan & Hartigan 1985) to test whether the observational data is consistent with a multi-peaked distribution. Hartigan's dip test calculates the probability that the distribution being tested has a single peak. Applying Hartigan's dip test to a given distribution yields in a test statistic; together with the sample size, the test statistic is matched to a p-value range in a precomputed table provided by Hartigan & Hartigan (1985). The p-value is a measure of the probability that the distribution actually has only one peak (the null-hypothesis, for this problem). The smaller the calculated p-value, the less likely is the distribution to have a single peak and the more likely it is to have at least two significant peaks. A p-value of < 0.05 indicates that the null-hypothesis is very unlikely, and that the given distribution has more than one peak. We applied the dip test to the eccentricity distribution of our sample of 931 main belt asteroids; we determined that the test statistic is 0.0107. From Hartigan & Hartigan (1985), this corresponds to a p-value range of 0.6 < p < 0.7 (based on a sample size of 1000). This indicates that the Hartigan dip test does not rule out the null hypothesis, i.e., a single-peaked distribution cannot be ruled out.
To further aid the interpretation of this test, we compare this result to that obtained by applying Hartigan's dip test to synthetic distributions that were explicitly double-peaked by construction. Ten model distributions (of 1000 eccentricity values) were generated from the double-gaussian function of equation (A4), with parameter values µ 1 = 0.0846, µ 2 = 0.185, and σ = 0.0411 (same parameter values as the best-fit for our sample of asteroids). Only the random seed was varied between each model distribution. Applying the Hartigan dip test, we find that the test statistic ranged between 0.00852 and 0.0126, corresponding to p-values between 0.98 and 0.3. This means that, according to the dip test, the null hypothesisi.e., a single-peaked distribution -could not be ruled out for any of the model distributions (since none had p < 0.05), despite the fact that they were each drawn from an explicitly double-peaked distribution by construction. This illustrates that even with a sample size of nearly 1000, the dispersion in eccentricity around the two peaks is too large in comparison to the distance between the peaks, that the dip test is not sufficiently sensitive to detect the double-peaked underlying distribution. We interpret this to mean that the results of both the K-S test and Hartigan's dip test indicate that the main asteroid belt eccentricity distribution is consistent with being drawn from a double-peaked distribution, but that this cannot be definitely shown. (Malhotra et al. 1989). The solid line shows the result from numerical spectral analysis of 233 solar system integrations (see text for explanation). The locations of Jupiter-Saturn MMRs which have an effect on the value of g 6 are indicated by vertical dotted lines. b) The location of the ν 6 resonance (at zero inclination) as a function of Saturn's orbit. The frequencies g 6 and g 0 were calculated for each value of Saturn's semimajor axis, a 6 , and then the location a ν 6 was determined by finding where g 0 − g 6 = 0. The dashed line shows the result from linear secular theory, with a correction for the effect of the 2:1 near-MMR between Jupiter and Saturn (Malhotra et al. 1989). The solid line was obtained by using the g 6 eigenfrequencies obtained from spectral analysis of the 233 numerical integrations, as shown in (a). (Knežević & Milani 2003). Family members identified by Nesvorný et al. (2006) were excluded. The solid lines are the best fit Gaussian distribution to the observational data. The dashed line is the best fit double-Gaussian distribution. Fig. 6.-The final eccentricity of asteroids as a function of asteroid semimajor axis and eccentricity for three different migration rates of Saturn, estimated from equation (32). Asteroids swept by the ν 6 resonance can have a range of final eccentricities depending on their apsidal phase, ̟ i . The outermost shaded region demarcates the range of final eccentricities for asteroids with an initial eccentricity e i = 0.4. The innermost shaded region demarcates the range of final eccentricities for asteroids with an initial eccentricity e i = 0.2. The solid line at the center of the shaded regions is the final eccentricity for an asteroid with an initial eccentricity e i = 0.
10
a) e i = 0.2,ȧ 6 = 1.0 AU My −1 ; b) e i = 0.2,ȧ 6 = 0.5 AU My −1 ; c) e i = 0.1,ȧ 6 = 1.0 AU My −1 ; d) e i = 0.3,ȧ 6 = 1.0 AU My −1 .
Fig. 1 .
1-The final eccentricity distribution of an ensemble of particles, all having initial eccentricity e i = 0.1 but uniformly distributed values of the phase angle ̟ i . The effect due to the sweeping ν 6 resonance was modeled using equation (31), with parameters chosen to simulate asteroids at a = 2.3 AU, and withȧ 6 = 1 AU My −1 .
Fig. 2 .
2a) The g 6 eigenfrequency as a function of Saturn's semimajor axis, for Jupiter fixed at 5.2 AU. The dashed line shows the result from linear secular theory, with a correction for the effect of the near 2:1 mean motion resonance between Jupiter and Saturn
Fig. 3 .Fig. 4 .
34-The value of the coefficient ε defined by equation (3) as a function of the zero inclination location of the ν 6 resonance. The values of E (i) j were calculated using first order Laplace-Lagrange secular theory with corrections arising from the 2:1 Jupiter-Saturn mean motion resonance. -Comparison between the numerical solution of the averaged equations (equations 9-10) and full numerical integrations of test particles at a = 2.3 AU. The dashed lines represent the envelope of the predicted final eccentricity, equation(32). The values of λ given are in the canonical unit system described in §2. Each panel labeled a-d plots both an analytical theory result and a numerical integration result for each of the four test cases labeled ad in section 3.2. The integrations were performed with Saturn starting at 8.5 AU and migrating outward linearly, while Jupiter remained fixed at 5.2 AU. Jupiter and Saturn had their current eccentricities but zero inclination. The thirty test particles in each numerical simulation were placed at 2.3 AU with zero inclination, but with longitudes of perihelion spaced 12 • apart. Time zero is defined as the time when the ν 6 resonance reached 2.
Fig. 5 .
5-Proper eccentricity distribution of the 931 observed asteroids with absolute magnitude H ≤ 10.8, excluding members of collisional families. The proper elements were taken from the AstDys online data service
Fig. 7 .
7-The effects of the sweeping ν 6 resonance on an ensemble of fictitious asteroids with semimajor axes 2.1-2.8 AU and a uniform distribution of pericenter longitudes. (a) Initial distribution of eccentricities, with mean 0.05 and standard deviation 0.01 (the "cold belt" solution). (b) The final distribution of eccentricities after ν 6 sweeping, estimated with the analytical theory (equation(31)), forȧ 6 = 4.0 AU My −1 . The two peaks in the final eccentricity distribution are at approximately the same values as the observed peaks in the main asteroid belt eccentricity distribution shown inFig. 5. (c) Initial distribution of eccentricities, with mean 0.40 and standard deviation 0.1 (the "hot belt" solution). Asteroids with eccentricities greater than the Mars-crossing value were discarded. (d) The final distribution of eccentricities after ν 6 sweeping forȧ 6 = 0.8 AU My −1 . The final distribution in d is depleted by a factor of 2.3 from the initial distribution in c. The ordinates in the four panels are not to the same scale.
Saturn migration rate: 0.15 AU/My0.0
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| []
|
[
"A route to room temperature ferromagnetic ultrathin SrRuO 3 films",
"A route to room temperature ferromagnetic ultrathin SrRuO 3 films"
]
| [
"Liang Si \nInstitute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria\n",
"Zhicheng Zhong \nInstitute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria\n",
"Jan M Tomczak \nInstitute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria\n",
"Karsten Held \nInstitute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria\n"
]
| [
"Institute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria",
"Institute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria",
"Institute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria",
"Institute of Solid State Physics\nVienna University of Technology\nA-1040ViennaAustria"
]
| []
| Experimental efforts to stabilize ferromagnetism in ultrathin films of transition metal oxides have so far failed, despite expectations based on density functional theory (DFT) and DFT+U. Here, we investigate one of the most promising materials, SrRuO3, and include correlation effects beyond DFT by means of dynamical mean field theory. In agreement with experiment we find an intrinsic thickness limitation for metallic ferromagnetism in SrRuO3 thin films. Indeed, we demonstrate that the realization of ultrathin ferromagnetic films is out of reach of standard thin-film techniques. Proposing charge carrier doping as a new route to manipulate thin films, we predict room temperature ferromagnetism in electron-doped SrRuO3 ultra thin films. | 10.1103/physrevb.92.041108 | [
"https://arxiv.org/pdf/1503.00640v1.pdf"
]
| 117,896,286 | 1503.00640 | 63484dc71d5453a6ac1ff2eb4c942afcdbd1cdff |
A route to room temperature ferromagnetic ultrathin SrRuO 3 films
2 Mar 2015
Liang Si
Institute of Solid State Physics
Vienna University of Technology
A-1040ViennaAustria
Zhicheng Zhong
Institute of Solid State Physics
Vienna University of Technology
A-1040ViennaAustria
Jan M Tomczak
Institute of Solid State Physics
Vienna University of Technology
A-1040ViennaAustria
Karsten Held
Institute of Solid State Physics
Vienna University of Technology
A-1040ViennaAustria
A route to room temperature ferromagnetic ultrathin SrRuO 3 films
2 Mar 2015
Experimental efforts to stabilize ferromagnetism in ultrathin films of transition metal oxides have so far failed, despite expectations based on density functional theory (DFT) and DFT+U. Here, we investigate one of the most promising materials, SrRuO3, and include correlation effects beyond DFT by means of dynamical mean field theory. In agreement with experiment we find an intrinsic thickness limitation for metallic ferromagnetism in SrRuO3 thin films. Indeed, we demonstrate that the realization of ultrathin ferromagnetic films is out of reach of standard thin-film techniques. Proposing charge carrier doping as a new route to manipulate thin films, we predict room temperature ferromagnetism in electron-doped SrRuO3 ultra thin films.
Thin films and heterostructures of the 4d perovskite SrRuO 3 (SRO) are intensively studied and used, in particular, as gate electrodes for novel oxide-based electronic devices [1,2]. The reason for this is that SRO is a conductor with good thermal properties [3] and high chemical stability, allowing for epitaxial growth on various substrates, as well as the combining with other perovskitebased materials to form complex heterostructures [4,5]. In the bulk, SRO is a ferromagnetic (FM) metal with a, for a 4d oxide, remarkably high Curie temperature, T C = 160K, and an experimental magnetic moment in the range of 0.8 to 1.6 µ B [6][7][8][9]. SRO further attracted fundamental research interests regarding, among others, magnetic monopoles [10], non-Fermi Liquid [11], spin freezing [12], and the debate of itinerant [13] versus localized magnetism [14].
However, the FM moment and Curie temperature get dramatically suppressed below a sample thickness of 4 unit cells [15][16][17][18], and eventually single unit cell SRO films turn antiferromagnetic (AF) and insulating [17,18]. This led to the pertinent question whether there is a fundamental thickness limit for ferromagnetism [18], and concerted efforts to stabilize ferromagnetism in ultrathin SRO films by compressive and tensile strain or capping layers [19,20]. However, hitherto ferromagnetism in single unit cell films remains unattainable for SRO or any other oxide material, even in a heterostructured setup.
On the theoretical side, previous attempts to understand the electronic structure and the transition to an AF insulator resorted to density functional theory (DFT) and the static mean-field DFT+U approach. The former failed to reproduce the transition [21], while the latter found a transition to an AF insulating state below four layers when assuming an artificial RuO 2 terminated surface [22], while experimentally samples are found to have a SrO termination [5]. DFT+U further predicted a spin-polarized highly confined half-metallic state for an SRO mono-layer when either sandwiched with SrTiO 3 (STO) [19] or grown on a strained STO substrate [20]. However, such a state could not be confirmed in experiment [23]. The apparent discrepancy between experi-ments and results from standard band-structure methods calls for a more sophisticated treatment of electronic correlation effects. Indeed already in the bulk, SRO displays signatures of electronic correlations, such as many-body satellites in photoemission or the violation of the Ioffe-Regel limit in the resistivity [8,24]. Hence, SRO is to be considered an -at least-moderately correlated system. Note that a dimensional reduction/geometric constraints in thin films can be expected to further enhance electronic correlations.
For a better and unbiased treatment of these correlations effects in various SRO films and heterostructure setups, we employ realistic DFT + dynamical mean-field theory (DMFT) [25][26][27][28][29] calculations. Our main findings are: (1) Both the SRO mono-layer and bi-layer are AF insulators. (2) We demonstrate that standard thin film manipulation techniques such as strain and surface capping can neither restore ferromagnetism nor metallicity to a SRO mono-layer; interestingly, we find that surface capping pushes the AF insulator towards a paramagnetic (PM) insulator. (3) With new insight regarding the microscopic origin for the transition, we identify carrier doping as the best option to generate FM properties that are on a par with those of the bulk. We find the FM moments of doped SRO films to be stable even at room temperature, heralding a great potential for technological applications.
Method. We use the experimental orthorhombic crystal structure of SRO [30] for the various setups of bulk, films and heterostructures. In the films and heterostructure both the internal positions and lattice constant are relaxed; the in-plane lattice constants in films are fixed to the experimental ones of STO. Fig. 1 exemplary shows the SRO mono-layer grown on 4 layers of STO substrate. The atomic relaxations are carried out with the VASP program package [31,32] using the PBE functional [33]. For the optimized atomic positions, we subsequently perform WIEN2K [34] electronic structure calculations with the mBJ exchange [35] and PBE correlation functional [36], and a Wannier function projection onto maximally localized [37] t 2g Wannier orbitals [38] using the Wien2Wannier program package [39]. This t 2g Hamiltonian is supplemented by a local Kanamori interaction and solved within DMFT using Wien2Dynamics [40], employing a hybridization expansion [41] continuous-time quantum Monte Carlo (CTQMC) algorithm. For the Coulomb interaction strengths, we adopt a Hund's exchange (J = 0.3 eV), intra-(U = 3.0 eV) and interorbital Coulomb repulsion (U ′ = 2.4 eV). These values are chosen not only because they are in between the constrained random phase approximation (cRPA) values for (i) free standing cubic SRO mono-layer (0.3eV, 3.5eV, and 2.9eV) and (ii) orthorhombic bulk and (0.3eV, 2.3eV, and 1.7eV), but also because they reproduce the FM metallic state for orthorhombic bulk SRO.
Bulk SrRuO 3 . The moderately correlated electronic structure of bulk SRO was successfully captured in both many-body perturbation theory [42] and realistic DMFT calculations [43,44]. Also DFT calculations correctly predict that SRO is an itinerant ferromagnet with a moment ranging from 1.5 to 1.6µ B [8,45]; similar moments have also been obtained within DFT+DMFT [44,46]. Using DFT+DMFT, we indeed find orthorhombic bulk SRO to be a FM metal with orbital occupations of 0.867 (0.466) for the majority (minority) spin of all three t 2g orbitals at T = 100K. This corresponds to a FM magnetic moment of ∼ 1.2µ B . A GdFeO 3 -type distortion in which the corner-sharing octahedral tilt around the y-axis and rotate around the z-axis lifts, in principle, the t 2g degeneracy. The effect on the crystal field and orbital occupations of orthorhombic SRO is however minute. Our DFT+DMFT finds SRO to be a PM metal above T c ∼ 150K which is close to the experimental Curie temperature of 160K (cf. Fig. 3 below).
Thin films. We now consider SRO grown on STO, and study the evolution of the electronic structure when reducing the number of SRO layers. We find that FM is suppressed: the mono-and bi-layer SRO on STO are AF insulators, in congruence with experiments [18,23] that show a dramatic drop in the FM moment and an insulating behavior for 4 layers or lower [15][16][17][18]23]. Indicative of an itinerant origin of ferromagnetism, the critical thickness of the magnetic and electronic transition coincide. Fig. 2 (a) shows the spectral function of the mono-layer. The system is gapped by ∼1.0eV and displays a large orbital polarization: the xy orbital is fully filled, and the xy and xz orbitals are half-filled and fully spin-polarized, resulting in an AF moment of ∼ 2µ B . This finding is supported by recent exchange bias measurements [17].
We note that for the particular case of the SRO monolayer, also LDA+U [20] can seemingly give a qualitative correct picture, as the system is orbitally and spinpolarized. However, the underlying physics is very different: When heating the mono-layer above its Néel temperature within DMFT, the system remains insulating at non-integer filling [0.88 (0.88), 0.56 (0.56), and 0.56 (0.56) for the spin up (down) xy, yz and xz orbitals at 1000K]. This complex Mott physics [48] reveals that the AF insulating phase is beyond a simple Slater description, and thus not describable by LDA+U.
Physical origin of transition. Let us now investigate the microscopic origin of the FM-metal to AF-insulator transition. Whereas the crystal field splitting is minute for the bulk, in the SRO mono-layer the xy-orbital is energetically lower than the yz and xz orbitals, because the (cubic) crystal symmetry is strongly broken at the surface. This is already the case for the DFT Wannier Hamiltonian, but correlation effects boost the crystal field splitting [48][49][50] of the SRO mono-layer (see Supplemental materials Table I). Therewith, the xy-orbital become essentially fully occupied, and the two remaining electrons occupy the yz and xz orbitals: The singlelayer SRO is an effective half-filled two-band system, favourable to AF order.
Tuning the properties of the SRO mono-layer. The prime motivation for SRO-based thin films are the advantageous properties of the FM metallic bulk. However, the desired features such as the magnetic moment strongly decrease for thinner films and eventually ultrathin films become non-FM, in agreement with our calculations. A natural question is whether the bulk properties can be restored, at least partially, by tuning the geometry of the films.
First we discuss the influence of straining/tensioning the mono-layer.
This can be realized experimentally by choosing an appropriate substrate. Indeed, previous DFT+U calculations [20] predicted a straininduced FM half-metallic state for the SRO mono-layer. DFT+DMFT, however, does not show any tendency towards a FM half-metallic state at least for realistic U values, see Table II in the Supplemental Materials. Also, the effective crystal-field splitting ∆ ef f , shown in Table I of Supplemental Materials, can only be tuned slightly through straining/tensioning.
Another way to influence the crystal-field splitting is through the deployment of capping layers. Here, we study the effect of capping the SRO mono-layer with additional layers of STO. Specifically, we consider a (STO) 5 :(SRO) 1 superlattice [19] consisting of 5 layers of STO alternating with a mono-layer of SRO. This restores, at least partially, the hopping amplitudes in the out-ofplane direction. Compared to the SRO mono-layer, the DFT crystal field splitting, shown in Supplemental materials Table I, is now much smaller (-0.05eV) approaching the negligible value of the bulk. As a result, the t 2g orbital occupations are more balanced. However, this causes only a slight reduction of the AF magnetic moment (1.92µ B at 150K and 1.48µ B at 300K, cf. Fig. 3 (b)) with respect to the un-capped mono-layer. Our finding of a non-FM insulating state with a gap of ∼1.0eV for the capped mono-layer is consistent with experiments [23,51], where it was concluded that SRO capped by STO leads to a insulator without a net moment. We note that previous DFT+U calculations [19] instead predicted a FM half-metal, at variance with experiment.
The above calculations reveal that the non-FM insulating state is a robust feature of the SRO mono-layer. Only for an unrealistically small U -to-bandwidth ratio a FM phase can be stabilized (see Supplemental materials). In consequence, none of the standard manipulation strategies available to the production of thin films can tune this ratio sufficiently to induce ferromagnetism to the SRO mono-layer. This stability explains why all experimental efforts to create ultrathin ferromagnetic films have so far been unsuccessful.
Doping. Here we propose an alternative route to achieve ultrathin FM films: doping. This strategy may seem counter-intuitive at first glance, as carrier doping causes a deterioration of the desired properties in the bulk. As we shall see, the situation for ultra thin films is different: Using the virtual crystal approximation to simulate carrier doping within DFT+DMFT, we obtain for the SRO mono-layer and the (STO) 5 :(SRO) 1 superlattice the magnetic moments shown in Fig. 3. For the SRO mono-layer a significant doping corresponding to 3.5 and 4.5 electrons/site is needed to turn the AF state into a PM at both low (150K) and high temperature (300K), see Fig. 3 (a). The AF magnetic moment is essentially symmetric around the filling with four electrons. The reason for this is the previously mentioned crystal field effect that results in an almost fully occupied xy orbital and a half-filled (particle-hole symmetric) yz/xz orbital doublet. We note that in all cases, the doping away from four electrons induces a metallic state.
Let us now turn to the more important case, the (STO) 5 :(SRO) 1 superlattice: At low temperatures, e.g. at 150K as shown in Fig. 3 (a), both hole and electron doping can induce strong FM states. In the case of electron doping (filling > 4) the FM state is accompanied by an alternating orbital ordering of the xz/yz minority spin. The spectral function corresponding to 4.3 electrons/site is shown in Fig. 2 For hole doping, e.g. at 3.7 electrons, our DMFT results indicate that the system is a FM half-metal with a moment of 2.20 µ B /Ru at 150K, see Fig. 3 (b) and the Supplemental Materials for the corresponding spectral functions. The half-metallic behavior makes this setup a prospective candidate for spintronics applications. To put this finding into perspective, we recall that bulk SRO has an FM moment of 2µ B /Ru or less, and an experimental (theoretical) Curie temperature of "only" 160K (150K). One might thus wonder whether the ferromagnetism of the doped supercell is actually superior to the hailed properties of stoichiometric bulk SRO, which we were striving to restore. To investigate this, we perform calculations as a function of temperature, see Fig. 3 (b). We find that for both hole and electron doping, magnetic moments and Curie temperatures of the supercell are remarkably higher than for bulk SRO. In particular, for 4.2 electrons/Ru site, a sizable magnetic moments survives up to room temperature 300K, see Fig. 3 (b). The magnetization curve has a similar shape as for the bulk [46], despite the much higher T c and the orbitalordering. In the Supplemental Material we also go beyond the virtual crystal approximation and show that a (STO) 5 :(La 0.25 Sr 0.75 RuO 3 ) 1 superlattice is indeed FM. Our findings pave the road for realizing FM oxide devices that can be operated at room temperature.
Conclusion. Including many-body effects by means of DFT+DMFT, we show that the SRO mono-layer is an AF Mott insulator owing to a correlation enhanced crystal-field splitting at the surface/interface. While the bare (one-particle) crystal-field splitting can be tuned to almost zero by STO capping layers, electronic correlations are still strong enough to boost the orbital separation so that also the capped SRO layer remains an AF insulator. A FM metallic state is only realized for an interaction-to-bandwidth ratio that cannot be realized by experiment. This explains why no ultrathin FM films could be stabilized in experiment to date. Given the robustness of the AF state of SRO mono-layer setups to standard thin film manipulation techniques, we propose an alternative route to realize a FM state: Our study suggests that carrier doping drives ultrathin SRO films capped with STO into a strong FM state, whose ordered moment and Curie temperature even exceed the values realized in stoichiometric bulk SRO. To achieve the longstanding quest for a FM ultrathin film in practice, we consider inducing oxygen or Sr vacancies [52] or doping potassium into STO:SRO superlattices [23] as the most promising means.
Our study also opens a new, general, perspective: Producing heterostructures based on materials with optimized bulk properties (e.g. stoichiometric SRO) is actually not always the optimal way for achieving those properties in a film geometry. Indeed the electronic structure of the thin film is so different from the bulk that it can be viewed as a completely different material. A manipulation (in our case doping) that decreases the quality of the bulk, can in fact enhance the sought-after property (FM magnetic moment) for the film setup. This suggests in turn that rather inconspicuous bulk materials might actually be good candidates for specific functionalities when deployed in a film or heterostructure. With this observation the repertoire of materials to be evaluated for oxide-electronics applications is significantly enlarged.
FIG. 1 .
1Right: Structure of a SRO mono-layer grown on 4 layers of STO. Upper left: Top view of the same structure. The indicated √ 2 × √ 2 supercell was adopted for allowing AF-ordering in each RuO2 layer. Lower left: atomic labels and coordinate system (Figures drawn with the Vesta code [47]).
FIG. 2 .
2DFT+DMFT spectral functions of (a) the AF SRO mono-layer on a STO substrate and (b) the FM doped (4.3 electrons/Ru) superlattice (STO)5:(SRO)1 at 150K. Insets: electronic occupations.
mono-layer), as obtained from spin-polarized DFT and DFT+DMFT calculations in comparison with experiment. FM-M: ferromagnetic metal, AF-I: antiferromagnetic insulator; NM-I: non-ferromagnetic insulator (the magnetic nature of the experimental non-ferromagnetic state has not been fully determined yet; the exchange bias behavior hints at antiferromagnetism[17]
FIG
. 3. (a) Magnetic moment of SRO mono-layer (ML) and (STO)5:(SRO)1 superlattice (SL) vs. doping at 150K and 300K (positive [negative] moments denote FM [AF] ordering). (b) Magnetic moments vs. temperature for orthorhombic bulk SRO, mono-layer, undoped and doped (STO)5:(SRO)1 superlattice.
(b). There, one Ru site has the orbital occupations for up (down) spin: xy 1.00 (0.34), yz 0.97 (0.83), xz 0.97 (0.19); while for the second Ru site: xy 1.00 (0.34), yz 0.97 (0.19), xz 0.97 (0.83).
TABLE
ZZ acknowledges financial support by the Austrian Science Fund through the SFB ViCoM F4103-N13. LS also thanks the support from Doctoral School Solids4Fun (Building Solids for Function). Calculations have been done on the Vienna Scientific Cluster (VSC). 306447gram (FP/2007-2013)/ERC through grant agreement n. 306447, ZZ acknowledges financial support by the Aus- trian Science Fund through the SFB ViCoM F4103- N13. LS also thanks the support from Doctoral School Solids4Fun (Building Solids for Function). Calculations have been done on the Vienna Scientific Cluster (VSC).
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| []
|
[
"Constellation Design for Multi-color Visible Light Communications",
"Constellation Design for Multi-color Visible Light Communications"
]
| [
"Qian Gao [email protected] ",
"Chen Gong ",
"Rui Wang \nTongji University\nShanghaiChina\n",
"Zhengyuan Xu ",
"Yingbo Hua [email protected] \nUniversity of California at Riverside\nCaliforniaUSA\n",
"\nUniversity of Science and Technology of China\nHefeiAnhuiChina\n"
]
| [
"Tongji University\nShanghaiChina",
"University of California at Riverside\nCaliforniaUSA",
"University of Science and Technology of China\nHefeiAnhuiChina"
]
| []
| In this paper, we propose a novel high dimensional constellation design scheme for visible light communication (VLC) systems employing red/green/blue light emitting diodes (RGB LEDs). It is in fact a generalized color shift keying (CSK) scheme which does not suffer efficiency loss due to a constrained sum intensity for all constellation symbols. Crucial lighting requirements are included as optimization constraints. To control non-linear distortion, the optical peak-to-averagepower ratio (PAPR) of LEDs is individually constrained. Fixing the average optical power, our scheme is able to achieve much lower bit-error rate (BER) than conventional schems especially when illumination color is more "unbalanced". When cross-talks exist among the multiple optical channels, we apply a singular value decomposition (SVD)-based pre-equalizer and redesign the constellations, and such scheme is shown to outperform postequalized schemes based on zero-forcing or linear minimummean-squared-error (LMMSE) principles. To further reduce system BER, a binary switching algorithm (BSA) is employed the first time for labeling high dimensional constellation. We thus obtains the optimal bits-to-symbols mapping. | null | [
"https://arxiv.org/pdf/1410.5932v1.pdf"
]
| 3,258,290 | 1410.5932 | 05fbf3a0a353fa7456336b6740523eab14bcc341 |
Constellation Design for Multi-color Visible Light Communications
Qian Gao [email protected]
Chen Gong
Rui Wang
Tongji University
ShanghaiChina
Zhengyuan Xu
Yingbo Hua [email protected]
University of California at Riverside
CaliforniaUSA
University of Science and Technology of China
HefeiAnhuiChina
Constellation Design for Multi-color Visible Light Communications
Index Terms-Optical wireless communicationconstellation designconstellation labelingmulti-color opticalCSKIM/DD
In this paper, we propose a novel high dimensional constellation design scheme for visible light communication (VLC) systems employing red/green/blue light emitting diodes (RGB LEDs). It is in fact a generalized color shift keying (CSK) scheme which does not suffer efficiency loss due to a constrained sum intensity for all constellation symbols. Crucial lighting requirements are included as optimization constraints. To control non-linear distortion, the optical peak-to-averagepower ratio (PAPR) of LEDs is individually constrained. Fixing the average optical power, our scheme is able to achieve much lower bit-error rate (BER) than conventional schems especially when illumination color is more "unbalanced". When cross-talks exist among the multiple optical channels, we apply a singular value decomposition (SVD)-based pre-equalizer and redesign the constellations, and such scheme is shown to outperform postequalized schemes based on zero-forcing or linear minimummean-squared-error (LMMSE) principles. To further reduce system BER, a binary switching algorithm (BSA) is employed the first time for labeling high dimensional constellation. We thus obtains the optimal bits-to-symbols mapping.
I. INTRODUCTION
In recent years, indoor visible light communication by light emitting diodes (LEDs) has attracted extensive academic attention [1], [2] (and references therein), driven by advancements in designing and manufacturing of LEDs [3]. Adoption of LEDs as lighting source can significantly reduce energy consumption and at the same time offering high speed wireless communication, which is the primary focus of visible light communication (VLC) research [4]- [6]. Most of the existing schemes employ blue LEDs with a yellow phosphor coating, while with red/green/blue (RGB) LEDs higher data rate is possible because of wavelength division multiplexing.
With RGB LEDs, color-shift keying (CSK) was recommended by the IEEE 802.15.7 Visible Light Communication Task Group [7]. A few authors have promoted this idea by designing constellations using signal processing tools. Drost et al. proposed an efficient constellation designed for CSK based on billiard algorithm [8]. Monteiro et al. designed the CSK constellation using an Interior Point Method, operating with peak and color cross-talk constraints [9]. Bai et al. considered the constellation design for CSK to minimize the bit error rate (BER) subject to some lighting constraints [10].
Despite the fact that the three-dimensional constellation design problems have been formulated in [8]- [10], a few important questions have not been addressed. They include how to compare a system with CSK employed and a conventional decoupled system, the constellation design, and the peak-to-average power ratio (PAPR) reduction [11]. In this paper, we propose a novel constellation design scheme in high dimensional space, termed CSK-Advanced. In our design, arbitrary number of red, blue, and green LEDs can be selected. With any average optical intensity and average color selected, we formulate an optimization problem to minimize the system symbol error rate (SER) by maximizing the minimum Euclidean distance (MED) among designed symbol vectors. Further, other important lighting factors such as color rendering index (CRI) and luminous efficacy rate (LER) are also considered. Further, optical PAPR is included as an additional constraint. The remainder of this paper is organized as follows. In Section II, we consider the constellation design problem assuming ideal channel. In Section III, we consider the constellation design for channel with cross-talks (CwC). An SVD-based pre-equalizer is applied and the constellations are redesigned subject to a transformed set of constraints. In Section IV, we discuss the optimization of constellations under arbitrary color illuminations. In Section V, we compare our scheme with a decoupled scheme and provide performance evaluation. Finally, Section VI provides conclusions. The system diagram is shown in Fig. 1, with N r red LEDs, N g green LEDs, and N b blue LEDs 1 . In one symbol interval of length T s , a random bit sequence b of size N B × 1 is first mapped by a BSA mapper f (·) to a symbol vector c of size
II. CONSTELLATION DESIGN WITH IDEAL CHANNEL
Imaging Detector
N T × 1, where N T = N r + N g + N b . The symbol c is chosen from a constellation C = (c 1 , c 2 , . . . , c Nc ),(1)
where N c = 2 N B denotes the constellation size. Each component c i,j is applied to the corresponding LED as intensity to transmit, such that c i ≥ 0. The intensity vector c i is then multiplied with the optical channel H of size N T × N T . 2 The output of the color filters can be written as follows,
y = γηHc + n,(2)
where η is the electro-optical conversion factor, γ is the photodetector responsivity. Without loss of generality (w.l.o.g.), assume γη = 1. The noise n is the combination of shot noise and thermal noise [12], assuming n ∼ N (0, I · N 0 /2). It should be noted that the imaging detector is followed by imperfect color filters such that cross-talks may exist. The received intensity vector y is passed through a symbol detector to obtain an estimate of the transmitter symbol, which is then de-mapped by f −1 (·) to recover the bit sequence. We assume line-of-sight (LOS) links without inter-symbol interference. We first consider ideal channel, i.e. H = I N T . Define a joint constellation vector c T = [c T 1 c T 2 . . . c T Nc ] T , and the i-th symbol is written as
c i = [c r1 i . . . c r Nr i c g1 i . . . c g Ng i c b1 i . . . c b N b i ] T = J i c T ,(3)
where
J i = [O N T . . . I N T . . . O N T ]
is a selection matrix with all zeros except for an identity matrix at the i-th block.
A. The objective function
Our objective is to minimize the system SER subject to several visible lighting constraints. We aim to max the minimum MED d min , i.e., maximize t such that the following holds for all l [15] c
T T F l c T ≥ t.(4)
where the parameter t will be optimized and we obtain d min through this optimization. F l(p,q) = E pq , E p = e T p ⊗ I Nc (Kronecker product), e p of size N T × 1 has all zeros except the p-th element being one,
E pq = E T p E p − E T p E q − E T q E p + E T q E q , and l ∼ = (p−1)N c − p(p + 1) 2 +q, p, q ∈ 1, 2, . . . , N c , p < q. (5)
The distance constraints (4) are nonconvex in c T . We approximate (4) by a first order Taylor series approximation around 1 Commercialized products as such are available to our knowledge. 2 We assume perfect channel knowledge in this paper and do not account for channel estimation errors.
c T (0) , i.e. c T T F l c T ∼ = 2c T (0)T F l c T − c T (0)T F l c T (0) h (0) l (c T ) ≥ t,(6)
where c T (0) is either a random initialization point or a previously attained estimate.
B. The average color and average power constraint
A designer may wish to constrain the average color, as nonwhite illumination could be useful in many places. The average of all LEDs' intensities can be written as the following N T ×1 vectorc
= 1 N c Nc i=1 J i c T =Jc T = [c r1 . . .c r Nrc g1 . . .c g Ngc b1 . . .c b N b ] T .(7)
We consider the average power of each color, i.e., a 3 × 1 vector c 3 given as follows,
c 3 = P o · [c rcgcb ] T = Kc = KJc T ,(8)
where K is a selection matrix summing up r/g/b intensities accordingly, P o is the average optical power, and
c r +c g +c b = 1,(9)P oc x =c x1 + . . . +c x Nx ,(10)
where x ∈ {r, g, b}. By properly selecting c 3 , the CRI and LER constraints can be met [13].
C. The optical PAPR constraint
For each LED, the optical PAPR is defined as the ratio of the highest power over the average power. Mathematically, the PAPR of the j-th LED can be written as follows,
Φ j = max(K j c T ) 1/N c · (K j c T ) , ∀j ∈ [1, N T ],(11)
where (a) denotes the summation of all elements of vector,
K j is a selection matrix of size N c × N c N T , max(K j c T ) denotes the largest element of vector K j c T .
The PAPR of an individual LED can be constrained as follows
Φ j ≤ α j , ∀j ∈ [1, N T ].(12)
D. CRI and LER constraints
CRI stands for a quantitative measure of ability of light sources to reproduce the colors of objects faithfully, comparing with an ideal lighting source [18]. LER measures how well light sources creates visible light. It is the ratio of luminous flux to power. Depending on context, the power can be either the radiant flux of the source's output, or it can be the total power (electric power, chemical energy, or others) consumed by the source [19]. The CRI and LER are important practical lighting constraints. By properly selecting c 3 , specific CRI and LER constraints can be met.
E. The optimization problem
When H = I N T the problem can be formulated as follows,
max cT,t t s.t. KJc T = c 3 , c T ≥ 0, h (0) l (c T ) ≥ t ∀l, N c max(K j c T ) − α j (K j c T ) ≤ 0, ∀j,(13)
which can be straightforwardly proven as a convex optimization problem. With the first three constraints, it is termed as a regular optimization problem and with all constraints a PAPR-constrained problem. By iteratively solving (13), a local optimal constellation c T 1 can be obtained 3 . With multiple runs starting from different initial point c T (0) , the best of solutions, c T * is selected.
III. CONSTELLATION DESIGN WITH CWC The channel cross-talks exist when the transmitting LED's emission spectral does not match the receiver filter's transmission spectral. It can be described by the following structure assuming single RGB LED is employed based on [8], [9] and experiments,
H c = 1 − 0 1 − 2 0 1 − ,
where the parameter ∈ [0, 0.5) characterizes both attenuation and interference effects. By singular value decomposition (SVD), H c = USV H , where U and V are unitary matrices of size N T × r, S is a diagonal matrix of size rank(H c ) × rank(H c ). In this case, r is the dimension of space for constellation design instead of N T . We apply a pre-equalizer P = VS −1 at the transmitterside and a post-equalizer U H at the receiver-side to equalize the channel 4 . Define P T = I Nc ⊗ P, and the optimization in (13) can be transformed as
max cT,t t s.t. KJP T c T = c 3 , P T c T ≥ 0, h (0) l (c T ) ≥ t ∀l, N c max(K j P T c T ) − α j sum(K j P T c T ) ≤ 0, ∀j.(14)
It should be noted that c T now is of dimension N c r × 1, i.e., the constellation is designed in a r-dimensional space.
To further minimize the system BER with a fixed SER, a good bit-to-symbol mapping function f (·) as shown in Fig. 1 need to be designed. In this paper, we apply the binary switching (BSA) algorithm to optimize the mapping. Since it is not the main focus of this paper, the details of BSA are omitted (we refer the readers interested to [16]).
IV. OPTIMIZED CONSTELLATIONS WITH TARGETED
COLOR ILLUMINATION We provide numerical illustration of advantages of the CSK-Advanced with one RGB LED, i.e. N r = N g = N b = 1. Both the CSK-Advanced and the conventional decoupled scheme can work with arbitrary color illumination. With one RGB LED, c i (i ∈ [1, 3]) for the decoupled scheme takes value from OOK (2-PAM) constellations. To make a fair comparison, an 8-CSK-Advanced constellation is designed, with equal spectrum efficiency (bits/sec/Hz), equal average optical powers, and equal average color.
A. Constellation design with ideal channel 1) Balanced lighting system: If the average intensity of each color is similar, we call the corresponding system "Bal- The MED for each branch is d min 6.67. For our scheme, the optimized constellation is as follows (column 1 to 4 and column 5 to 8 are separated due to space limit.) The MED equals 7.27, such that we could expect a lower SER with sufficient SNR. The asymptotic power gain is approximately 0.86dB (=10 × log(7.27/6.67)). With our scheme, the optimized constellation is as follows The MED is approximately 7.26, which is smaller than MED of one branch but larger than MEDs of two branches of the conventional scheme. The MED is approximately 6.31, which is smaller than MED of one branch but larger than MEDs of two branches of the conventional scheme. Table I. It can be observed that there is a tradeoff between minimum distance and PAPR for the three cases. With extremely low PAPR, e.g. α = 1.5, the system suffers from severe power loss. With a PAPR increase of 3dB, e.g. from α = 2 to α = 4 which is typically tolerable, the power gain of unbalance systems are larger than balanced system.
V. PERFORMANCE EVALUATION
We simulate using bit sequence of length N = 9 × 10 6 for selected cases above to compare the BER performance among different systems versus different optical SNR, defined as [17, Eq.27] as follows,
γ o = 10 log 10 P o √ N b N 0 .(15)
Selected BER curves versus optical SNR are included in Fig. 2 and Fig. 3. In Fig. 2, CSK-Advanced system applies constellation C 8 (c B
3 ) and in Fig. 3 constellation C 8 (c E 3 ) is used. It can be observed that with CSK-Advanced scheme non-trivial power gain is obtained over the conventional system, especially when the average color is not balanced. The optimized mapping by BSA offers additional power gain for all OSNR range. Histogram of MEDs of 1000 local optimal constellations for the balanced system is shown by Fig. 4. It can be seen that approximately 1/4 of the runs will converge to satisfactory MEDs. Therefore, we would suggest only 20−30 runs in practice to reduce complexity.
A. Constellation labeling
The optimized bit-to-symbol mapping obtained by the BSA when γ o = 5dB is included in Table II. With optimized mapping, only 1.33 out of 3 bits on average are in error when a symbol error occurs. Without BSA based mapping how average, 1.73 out of 3 bits on average (over results observed from 100 random labelings) are mis-interpreted instead. The optimized mapping tables can be computed offline. B. Constellation design with CwC 1) SVD-based pre-equalizer: Consider the following 3 × 3 channel with moderate cross-talks, e.g. 1 = 0.1,
H 1 = 0.9 0.1 0 0.1 0.8 0.1 0 0.1 0.9 . By SVD we have H 1 = U 1 S 1 V H 1 .
The pre-equalizer P 1,pr = V 1 S −1 1 and post-equalizer P 1 ,po = U H 1 . The corresponding optimized constellation for balanced system is The MED with varying area of overlap for balanced, unbalanced, and extremely unbalanced systems are summarized in Table III. In practice, the mismatch between the emission spectra of the transmitter LEDs and the transmission spectra of the receiver filters is restricted, and cases with ≥ 0.2 are very rare. 2) Comparison with post-equalized systems: Instead of redesign the constellations subject to a transformed set of constraints due to employment of a pre-equalizer P, zero-forcing (ZF) G Z or linear minimum-mean-squared-error (LMMSE) based post-equalizer G L can be employed at the receiver [20] to mitigate the cross-talks. Fig. 5 shows the corresponding BERs against increased crosstalks for a balanced system employing different schemes when OSNR is fixed to 5dB. It is seen that our SVD-based scheme significantly outperforms systems employing either ZF or LMMSE post-equalizers. Fig. 6 shows the BERs against OSNR for a balanced system when is fixed to 0.1. With this particular parameters chosen, there is no significant difference between ZF and LMMSE based system performance and therefore we only included the LMMSE based results.
VI. CONCLUSION
A novel constellation design scheme, named CSK-Advanced, for VLC with arbitrary number of RGB LEDs, is proposed in this paper. With both optimized constellation and bits-to-symbols mapping, significant power gains are observed compared with conventional decoupled systems. For more unbalanced color illumination, the larger power gains can be expected. To avoid excessive nonlinear distortion, optical PAPR constraints is included into the optimization. Furthermore, to deal with CwC, an SVD-based pre-equalizer is introduced. It is shown by simulations that the proposed scheme significantly outperforms various benchmarks employing ZF or LMMSEbased post-equalizers.
Fig. 1 :
1System diagram of the proposed CSK-Advanced System.
anced lighting system". For example, we choose average color as c B 3 = 10·[1/3, 1/3, 1/3] T and the average power P o = 10. For the conventional scheme, each LED can simply take value independently from the following binary constellations C B,r = C B,g = C B,b = [0, 6.67].
2 )
2Unbalanced lighting system: We choose the average color as c U 3 = 10 · [0.44, 0.33, 0.22] T . With the conventional scheme, the LEDs take value from constellations C U,r = [0, 8.88] C U,g = [0, 6.66] C U,b = [0, 4.44].
PAPR-constrained system: If identical individual PAPR constraints are added, i.e. α j = α into the optimization. The corresponding MEDs with varying PAPR are summarized in
Fig. 3 :
3Extremely Unbalanced conventional system vs CSK-Advanced systems.
Fig. 4 :
4Histogram of MEDs of 1000 local optimal constellations.
Fig. 6 :
6BER against OSNR with = 0.1 for a balanced system.
TABLE I :
IMED with varying PAPR and average color.
TABLE II :
IIOptimized Bit-to-symbol mapping with OSNR=5dB.Constellation Point Optimized Labeling
(0, 0, 7.27)
000
(0, 0, 0)
001
(0, 14.55, 0)
010
(0, 7.27, 0)
011
(0, 0, 14.55)
100
(7.27, 0, 0)
101
(14.55, 0, 0)
110
(4.85, 4.85, 4.85)
111
TABLE III :
IIIMED with varying area of overlap.d min
C 8
B,D
C 8
U,D
C 8
E,D
= 0
7.2727 7.2590 6.3139
= 0.05 6.7621 6.6748 5.9275
= 0.1
6.3275 6.1464 5.5657
= 0.15 5.9462 5.7769 5.1635
= 0.2
5.5670 5.3692 4.7727
One can refer to a similar problem in[15] for convergence, complexity and performance analysis.4 n and U H n have the same distribution, since U H is unitary.
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| []
|
[
"On the K-theory of the coordinate axes in the plane",
"On the K-theory of the coordinate axes in the plane"
]
| [
"Lars Hesselholt "
]
| []
| []
| Let k be a regular Fp-algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I = (x, y) be the ideal that defines the intersection point. We evaluate the relative K-groups Kq(A, I) in terms of the groups of big de Rham-Witt forms of k. The result generalizes previous results for K 1 and K 2 by Dennis and Krusemeyer. | 10.1017/s0027763000025757 | [
"https://arxiv.org/pdf/math/0508251v2.pdf"
]
| 14,544,994 | math/0508251 | 955bfab7be5e163dfa30a83c6d16c2e4b3eacf40 |
On the K-theory of the coordinate axes in the plane
19 Oct 2005
Lars Hesselholt
On the K-theory of the coordinate axes in the plane
19 Oct 2005
Let k be a regular Fp-algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I = (x, y) be the ideal that defines the intersection point. We evaluate the relative K-groups Kq(A, I) in terms of the groups of big de Rham-Witt forms of k. The result generalizes previous results for K 1 and K 2 by Dennis and Krusemeyer.
Introduction
Let k be a ring, and let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane. The K-groups of A decompose as the direct sum K q (A) = K q (k) ⊕ K q (A, I) of the K-groups of the ground ring k and the relative K-groups of A with respect to the ideal I = (x, y). In this paper we evaluate the groups K q (A, I) completely in the case where k is a regular F p -algebra. The result is stated in terms of the groups of big de Rham-Witt forms of k as follows.
Theorem A. Let k be a regular F p -algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I ⊂ A be the ideal generated by x and y. Then for all integers q, there is a canonical isomorphism
K q (A, I) ∼ ← − m 1 W m Ω q−2m k
where W m Ω j k is the group of big de Rham-Witt j-forms of k. The group K 2 (A, I) was evaluated by Dennis and Krusemeyer [3] twenty-five years ago. But it was previously known only that the higher relative K-groups are p-primary torsion groups [17]. The group of big de Rham-Witt j-forms W n Ω j k was introduced in [12, Def. 1. 1.6]. It decomposes as a product of the more familiar ptypical de Rham-Witt j-forms W s Ω j k defined by Bloch-Deligne-Illusie [14]. Indeed, by [12,Cor. 1.2.6] there is a canonical isomorphism
W m Ω j k ∼ − → d W s Ω j k ,
where the product ranges over all integers 1 d m that are not divisible by p, and where s = s(m, d) is the unique positive integer with p s−1 d m < p s d. The
The author was partially supported by COE (Japan) and the National Science Foundation. and it is the bi-relative topological cyclic homology groups on the right-hand side that we evaluate here. The method is similar to the calculation of the topological cyclic homology of the ring of dual numbers by the author and Madsen [11,12]. We first prove a general formula that expresses the bi-relative topological cyclic homology groups above in terms of the RO(T)-graded equivariant homotopy groups TR n α (k; p) = [S α ∧ (T/C p n−1 ) + , T (k)] T of the topological Hochschild T-spectrum T (k). Here T is the circle group and C r ⊂ T is the subgroup of order r. To state the formula, which is valid for any ring k, we let λ i be the complex T-representation C(1) ⊕ · · · ⊕ C(i).
Theorem B. Let k be an F p -algebra, let A = k[x, y]/(xy) and B = k[x] × k[y], and let I be the common ideal of A and B generated by x and y. Then for all integers q, there is a canonical isomorphism
TC q (A, B, I; p) ∼ − → lim R TR r q−λ p r−1 d (k; p)
where the product ranges over the positive integers d that are not divisible by p. The analogous statement for the groups with Z/p v -coefficients is valid for any ring k.
The limit system on the right-hand side of the statement of Thm. B stabilizes in the sense that the structure map
R : TR r q−λ p r−1 d (k; p) → TR r−1 q−λ p r−2 d (k; p) is an isomorphism for q < dim R (λ p r−1 d ). See Lemma 2.3 below.
If k is a regular F p -algebra, the structure of the groups on the right-hand side of the statement of Thm. B was determined in [12,Thm. 2.2.2]; see also [9,Thm. 16]. We recall the result in Sect. 3 below and complete the proof of Thm. A.
Finally, we mention that the analog of Thm. A for k a regular Q-algebra is known. Indeed, by a recent theorem of Cortiñas [2, Thm. 0.1] (which inspired us to prove [4, Thm. A]), the trace map induces an isomorphism
K q (A, B, I) ⊗ Q ∼ − → HC − q (A ⊗ Q, B ⊗ Q, I ⊗ Q)
and the bi-relative negative cyclic homology groups on the right-hand side were evaluated long ago by Geller, Reid, and Weibel [5,6]. The result is that, if k is a regular Q-algebra, then there is a canonical isomorphism
K q (A, I) ∼ ← − m 1 Ω q−2m k
where Ω j k is the group of absolute Kähler j-forms of k. This formula differs from the formula of Thm. A in degrees q 4. Indeed, the group
K 4 (A, I) is isomorphic to Ω 2 k ⊕ k, if k is a regular Q-algebra, to Ω 2 k ⊕ W 2 (k), if k is a regular F 2 -algebra, and to Ω 2 k ⊕ k ⊕ k, if k is a regular F p -algebra and p > 2.
We also remark that, if k is a regular F p -algebra, and if p s−1 n < p s , then the group K 2n (A, I) has exponent exactly p s .
The result of Thm. A was announced in [4, Thm. C]. All rings considered in this paper are assumed to be commutative. We write N and I p for the sets of positive integers and positive integers prime to p, respectively. We say that a map of T-spectra is an F -equivalence if the induced map of C-fixed point spectra is a weak equivalence, for all finite subgroups C ⊂ T.
Finally, the author would like to thank an anonymous referee for a very careful reading of an earlier version of this paper and for a number of helpful suggestions on improving the exposition.
Topological Hochschild homology
The proof of Thm. B of the introduction is based on a description of the birelative topological Hochschild T-spectrum T (A, B, I) defined to be the iterated mapping fiber of the following diagram of topological Hochschild T-spectra. We refer [9] for an introduction to topological Hochschild and cyclic homology and for further references. In this section, we shall prove the following result. , and let I be the common ideal generated by x and y. Then there is a canonical
F -equivalence of T-spectra i∈N T (k) ∧ S λi ∧ (T/C i ) + [1] ∼ − → T (A, B, I)
where, on the left-hand side, [1] indicates desuspension.
The rings B = k[x] × k[y] and B/I = k × k are both product rings. Moreover, topological Hochschild homology preserves products of rings in the sense that for every pair of rings R and S, the canonical map of T-spectra
T (R × S) → T (R) × T (S)
is an F -equivalence [1,Prop. 4.20]. Hence, the canonical map from T (A, B, I) to the iterated mapping fiber of the following diagram of T-spectra is an F -equivalence.
(1.3) T (k[x, y]/(xy)) ǫ / / (φ,φ ′ ) T (k) ∆ T (k[x]) × T (k[y]) ǫ×ǫ / / T (k) × T (k).
The rings that occur in this diagram are all pointed monoid algebras. By definition, a pointed monoid Π is a monoid in the symmetric monoidal category of pointed sets and smash product, and the pointed monoid algebra k(Π) is the quotient of the monoid algebra k[Π] by the ideal generated by the base-point of Π. The diagram of rings (1.3) is then induced from the diagram of pointed monoids
Π 2 ǫ / / (φ,φ ′ ) Π 0 ∆ Π 1 × Π 1 ǫ×ǫ / / Π 0 × Π 0 where Π 0 = {0, 1} with base-point 0, where Π 1 = {0, 1, z, z 2 , . . . } with base-point 0, and where Π 2 = {0, 1, x, x 2 , .
. . , y, y 2 , . . . } with base-point 0 and with multiplication given by xy = 0. The map φ (resp. φ ′ ) takes the variables x and y to z and 0 (resp. to 0 and z), and the maps labeled ǫ take the variables x, y, and z to 1.
The topological Hochschild T-spectrum of a pointed monoid algebra k(Π) decomposes, up to F -equivalence, as the smash product
(1.4) T (k) ∧ N cy (Π) ∼ − → T (k(Π))
of the topological Hochschild T-spectrum of the coefficient ring k and the cyclic bar-construction of the pointed monoid Π. This is proved in [11, Thm. 7.1] but see also [9,Prop. 4]. The cyclic bar-construction is the geometric realization of the pointed cyclic set with m-simplices
N cy (Π)[m] = Π ∧ · · · ∧ Π (m + 1 factors)
and with the Hochschild-type cyclic structure maps
d i (π 0 ∧ · · · ∧ π m ) = π 0 ∧ · · · ∧ π i π i+1 ∧ · · · ∧ π m , 0 i < m, = π m π 0 ∧ π 1 ∧ · · · ∧ π m−1 , i = m, s i (π 0 ∧ · · · ∧ π m ) = π 0 ∧ · · · ∧ π i ∧ 1 ∧ π i+1 ∧ · · · ∧ π m , 0 i m, t m (π 0 ∧ · · · ∧ π m ) = π m ∧ π 0 ∧ π 1 ∧ · · · ∧ π m−1 .
It is a pointed T-space by the theory of cyclic sets [16, 7.1.9]. It follows that the T-spectrum T (A, B, I) is canonically F -equivalent to the iterated mapping fiber of the following diagram of T-spectra.
T (k) ∧ N cy (Π 2 ) ǫ / / (φ,φ ′ ) T (k) ∧ N cy (Π 0 ) ∆ (T (k) ∧ N cy (Π 1 )) × (T (k) ∧ N cy (Π 1 )) ǫ×ǫ / / (T (k) ∧ N cy (Π 0 )) × (T (k) ∧ N cy (Π 0 )).
The cyclic bar-constructions of Π 1 and Π 2 have natural wedge-decompositions which we now explain.
We define N cy (Π 1 , i)[m] to be the subset of N cy (Π 1 )[m] that consists of the base-point and of the simplices z i0 ∧ · · · ∧ z im with i 0 + · · · + i m = i. It is clear that the pointed set N cy (Π 1 )[m] decomposes as the wedge-sum of the pointed subsets N cy (Π 1 , i)[m] where i ranges over the non-negative integers. The cyclic structure maps preserve this decomposition, and hence, the geometric realization decomposes accordingly as a wedge-sum of pointed T-spaces
N cy (Π 1 ) = N cy (Π 1 , i)
indexed by the non-negative integers. To state the analogous wedge-decomposition of N cy (Π 2 ), we first recall the notion of a cyclical word.
A word of length m with letters in a set S is a function
ω : {1, 2, . . . , m} → S.
The action by the cyclic group C m of order m on the set {1, 2, . . . , m} by cyclic permutation induces an action on the set of words of length m in S. A cyclical word of length m with letters in S is an orbit for the action of C m on the set of words of length m in S. We writeω for the orbit through ω. By the period ofω, we mean the length of the orbitω. In particular, the set that consists of the empty word is a cyclical word of length 0 and period 1.
We associate a word ω(π) with letters in x and y to every non-zero element π ∈ Π 2 . A non-zero element π ∈ Π 2 is either of the form π = x i or π = y i . In the former case, we define ω(π) to be the unique word of length i all of whose letters are x, and in the latter case, we define ω(π) to be the unique word of length i all of whose letters are y. More generally, we associate to every (m+1)-tuple (π 0 , . . . , π m ) of non-zero elements of Π the word ω(π 0 , . . . , π m ) = ω(π 0 ) * · · · * ω(π m ) defined to be the concatenation of the words ω(π 0 ), . . . , ω(π m ). Now, for every cyclical wordω with letters x and y, we define
N cy (Π 2 ,ω)[m] ⊂ N cy (Π 2 )[m]
to be the subset that consists of the base-point and the elements π 0 ∧ · · · ∧ π m , where (π 0 , . . . , π m ) ranges over all (m + 1)-tuples of non-zero elements of Π 2 such that ω(π 0 , . . . , π m ) ∈ω. As m 0 varies, these subsets form a cyclic subset
N cy (Π 2 ,ω)[−] ⊂ N cy (Π 2 )[−],
and we define N cy (Π 2 ,ω) ⊂ N cy (Π 2 ) to be the geometric realization. It is clear that the cyclic set N cy (Π 2 )[−] decomposes as the wedge-sum of the cyclic subsets N cy (Π 2 ,ω)[−], whereω ranges over all cyclical words with letters x and y. Hence, the geometric realization decomposes as a wedge-sum of pointed T-spaces
N cy (Π 2 ) = N cy (Π 2 ,ω)
indexed by all cyclical words with letters x and y.
Lemma 1.5. There is a canonical F -equivalence of T-spectra T (k) ∧ N cy (Π 2 ,ω) ∼ − → T (A, B, I),
where the wedge-sum on the left-hand side ranges over all cyclical words of period s 2 with letters x and y.
Proof. Letω be a cyclical word of period s ≥ 2. Then every word ω ∈ω has both of the letters x and y. Therefore, the compositions of the canonical map Lemma 1.6. Letω be a cyclical word of period s 2 with letters x and y. The homotopy type of the pointed T-space N cy (Π 2 ,ω) is given as follows.
T (k) ∧ N cy (Π 2 ,ω) → T
(i) Ifω has period s = 2 and length m = 2i, then a choice of representative word ω ∈ω determines a T-equivariant homeomorphism
S R[Cm]−1 ∧ Ci T + ∼ − → N cy (Π 2 ,ω),
where R[C m ] − 1 is the reduced regular representation of C m .
(ii) Ifω has period s > 2, then N cy (Π 2 ,ω) is T-equivariantly contractible.
Proof. We refer the reader to [16] for the basic theory of cyclic sets and their geometric realization. Letω be a cyclical word of period s ≥ 2 and length m = si with letters x and y, and let ω ∈ω. We let (π 0 , . . . , π m−1 ) be the unique m-tuple of non-zero elements in Π 2 such that ω(π 0 , . . . , π m−1 ) = ω. Then the pointed cyclic set N cy (Π 2 ,ω)[−] is generated by the (m − 1)-simplex π 0 ∧ · · · ∧ π m−1 . Hence, there is a unique surjective map of pointed cyclic sets
f ω : Λ m−1 [−] + → N cy (Π 2 ,ω)[−]
that maps the canonical generator of the cyclic standard (m − 1)-simplex to the generator π 0 ∧ · · · ∧ π m−1 . We recall that the automorphism group of the pointed cyclic set Λ m−1 [−] + is a cyclic group order m generated by the dual of the cyclic operator t m . Since the cyclic operator t s m fixes the generator π 0 ∧ · · · ∧ π m−1 , we obtain a factorization of the map f ω over the quotient by the subgroup of the automorphism group or order i = m/s,
f ω : (Λ m−1 [−]/C i ) + → N cy (Π 2 ,ω)[−].
We next recall that the geometric realization of the cyclic standard (m − 1)-simplex is T-equivariantly homeomorphic to ∆ m−1 ×T, where T acts by multiplication in the second factor. Moreover, the homeomorphism may be chosen in such that the dual of the cyclic operator t m acts on ∆ m−1 by the affine map that cyclically permutes the vertices and on T by rotation through 2π/m; see [11,Sect. 7.2]. It follows that the map f ω gives rise to a continuous T-equivariant surjection
f ω : (∆ m−1 × Ci T) + → N cy (Π 2 ,ω).
There is a canonical C i -equivariant homeomorphism
∆ s−1 * · · · * ∆ s−1 ∼ − → ∆ m−1 ,
where
∆ s−1 * · · · * ∆ s−1 × [0, 1] → ∆ s−1 * · · · * ∆ s−1 ,
and since the face F is not collapsed to the base-point by the map f ω , this induces a T-equivariant null-homotopy
N cy (Π 2 ,ω) ∧ [0, 1] + → N cy (Π 2 ,ω).
This completes the proof of the statement for s > 2.
Remark 1.7. The statement of Lemma 1.6 may be viewed as a topological refinement of the calculation in [7] of the Hochschild homology of the pointed monoid ring Z(Π 2 ) = Z[x, y]/(xy). Indeed, for any pointed monoid Π, the reduced singular homology groupsH * (N cy (Π); Z) and the Hochschild homology groups HH * (Z(Π)) are canonically isomorphic.
Proof of Prop. 1.2. It follows from Lemmas 1.5 and 1.6 that we have an F -equivalence of T-spectra
T (k) ∧ S R[C2i]−1 ∧ Ci T + ∼ − → T (A, B, I)
where the wedge sum ranges over all positive integers i. The equivalence depends on a choice, for every cyclical wordω with letters x and y of period 2, of a representative word ω ∈ω. We choose the representative ω = xy . . . xy. Now, as a representation of the subgroup C i ⊂ C 2i , the regular representation R[C 2i ] is isomorphic to the complex representation λ i = C(1)⊕· · ·⊕C(i), where C(t) denotes the representation of T on C through the t-fold power map. Hence, a choice of such an isomorphism determines a T-equivariant homeomorphism
S λi ∧ Ci T + ∼ − → S R[C2i] ∧ Ci T + .
Moreover, since the C i -action on λ i extends to a T-action, we further have the canonical T-equivariant homeomorphism
S λi ∧ (T/C i ) + ∼ − → S λi ∧ Ci T +
that takes (w, zC i ) to the class of (z −1 w, z). The completes the proof.
Topological cyclic homology
In this section, we prove the formula for the bi-relative topological cyclic homology groups TC q (A, B, I; p) that was stated in Thm. B of the introduction. We derive this formula from the corresponding formula for topological Hochschild homology that we proved in Prop. 1.2 above. The argument is very similar to the analogous argument in the case of truncated polynomial algebras [10,12,9]. We refer the reader to [9, 3.7] for the definition of topological cyclic homology.
We have from Prop.
1.2 an F -equivalence of T-spectra i∈N T (k) ∧ S λi ∧ (T/C i ) + [1] ∼ − → T (A, B, I),
and we wish to evaluate the homotopy groups of the C p n−1 -fixed point spectra. To this end, we reindex the wedge-sum on the left-hand side after the p-adic valuation of i ∈ N. The left-hand side is then rewritten as
d∈N T (k) ∧ S λ p n−1 d ∧ (T/C p n−1 d ) + [1] ∨ n−1 r=1 d∈Ip T (k) ∧ S λ p r−1 d ∧ (T/C p r−1 d ) + [1],
where N and I p are the sets of positive integers and positive integers that are not divisible by p, respectively. Hence, the T-spectrum ρ * p n−1 T (A, B, I) C p n−1 is equivalent to the wedge-sum
d∈N ρ * p n−1 (T (k) ∧ S λ p n−1 d ∧ (T/C p n−1 d ) + ) C p n−1 [1] ∨ n−1 r=1 d∈Ip ρ * p n−r (ρ * p r−1 (T (k) ∧ S λ p r−1 d ∧ (T/C p r−1 d ) + ) C p r−1 ) C p n−r [1].
Now, for every T-spectrum T , there is a natural equivalence of T-spectra
ρ * p m T C p m ∧ ρ * p m ((T/C p m d ) + ) C p m ∼ − → ρ * p m (T ∧ (T/C p m d ) + ) C p m
and the p m th root defines a T-equivariant homeomorphism
(T/C d ) + ∼ − → ρ * p m ((T/C p m d ) + ) C p m .
Hence, we can rewrite the wedge-sum above as follows.
d∈N ρ * p n−1 (T (k) ∧ S λ p n−1 d ) C p n−1 ∧ (T/C d ) + [1] ∨ n−1 r=1 d∈Ip ρ * p n−r (ρ * p r−1 (T (k) ∧ S λ p r−1 d ) C p r−1 ∧ (T/C d ) + ) C p n−r [1].
We recall from [13,Lemma 3.4.1] that if T is a T-spectrum, if d ∈ I p , and if ι : {C d } → T/C d is the canonical inclusion, then the map
V m ι * + dV m ι * : π q (T ) ⊕ π q−1 (T ) → π q (ρ * p m (T ∧ (T/C d ) + ) C p m )
is an isomorphism. It follows that the group TR n q (A, B, I; p) is canonically isomorphic to the sum
(2.1) d∈N TR n q+1−λ p n−1 d (k; p) ⊕ TR n q−λ p n−1 d (k; p) ⊕ n−1 r=1 d∈Ip TR r q+1−λ p r−1 d (k; p) ⊕ TR r q−λ p r−1 d (k; p) .
We consider the groups TR n q (A, B, I; p) for varying n 1 as a pro-abelian group whose structure map is the Frobenius map The Frobenius map takes the summand with index d ∈ N in the top line of (2.1) for n to the summand with index pd ∈ N in the top line of (2.1) for n − 1. It takes the summand with indices d ∈ I p and 1 r < n − 1 in the bottom line of (2.1) for n to the summand with the same indices in the bottom line of (2.1) for n − 1. Finally, it takes the summand with indices d ∈ I p and r = n − 1 in the bottom line of (2.1) for n to the the summand with index d ∈ N in the top line of (2.1) for n − 1. It follows that the sub-pro-abelian group of TR n q (A, B, I; p) given by the top line of (2.1) is Mittag-Leffler zero, since the sum in (2.1) is finite. Hence, the projection onto the quotient pro-abelian group of TR n q (A, B, I; p) given by the bottom line of (2.1) is an isomorphism of pro-abelian groups. The value of the Frobenius map on the bottom line of (2.1) follows immediately from the relations F V = p and F dV = d. Indeed, the Frobenius preserves the direct sum decomposition and restricts to the maps F = p : TR r q+1−λ (k; p) → TR r q+1−λ (k; p), F = id : TR r q−λ (k; p) → TR r q−λ (k; p), respectively, on the first and second summand of the bottom line of (2.1).
We now assume that the group TR r q−λ (k; p) is annihilated by p m , for some m. If k is an F p -algebra, then this group is annihilated by p r . For a general ring k, we must instead consider the group TR r q−λ (k; p, Z/p v ) which is annihilated by p v . It follows that the iterated Frobenius F m induces the zero map from the first term in the bottom line of (2.1) for m + n to the first term in the bottom line of (2.1) for n. Hence, the canonical projection onto the second term on bottom line of (2.1),
TR n q (A, B, I; p) → n−1 r=1 d∈Ip TR r q−λ p r−1 d (k; p),
is an isomorphism of pro-abelian groups. Here, we recall, the structure map in the limit system on the left-hand side is the Frobenius map and in the limit system on the right-hand side is the canonical projection. The group TR r q−λ p r−1 d (k; p) is zero, if q < dim R (λ d ) = 2d, and hence, the limit group is the product
(2.2) TF q (A, B, I; p) ∼ − → r∈N d∈Ip TR r q−λ p r−1 d (k; p).
We can now evaluate the bi-relative topological cyclic homology groups that are given by the long-exact following long-exact sequence. Indeed, under the isomorphism (2.2), the map R corresponds to the endomorphism of the product on the right-hand side of (2.2) that is induced from the map
R : TR r q−λ p r−1 d (k; p) → TR r−1 q−λ p r−2 d (k; p).
Hence, the kernel of the map R − id in (2.2) is identified with the limit
d∈Ip lim R TR r q−λ p r−1 d (k; p)
and the cokernel is identified with the corresponding derived limit. The following Lemma 2.3 shows, in particular, that the limit system satisfies the Mittag-Leffler condition. Hence, the derived limit vanishes and we obtain an isomorphism
TC q (A, B, I; p) ∼ − → d∈Ip lim R TR r q−λ p r−1 d (k; p)
as stated in Thm. B.
Lemma 2.3. The restriction map
R : TR r q−λ p r−1 d (k; p) → TR r−1 q−λ p r−2 d (k; p)
is an isomorphism, for q < 2p r−1 d.
Proof. We recall from [11, Thm. 2.2] that there is a long-exact sequence
· · · → H q (C p r−1 , T (k) ∧ S λ p r−1 d ) → TR r q−λ p r−1 d (k; p) R − → TR r−1 q−λ p r−2 d (k; p) → · · ·
and that the left-hand groups are given by a spectral sequence E 2 s,t = H s (C p r−1 , TR 1 q−λ p r−1 d (k; p)) ⇒ H q (C p r−1 , T (k) ∧ S λ p r−1 d ). The groups in the E 2 -term do not depend on the representation λ p r−1 d beyond its dimension, and they are zero if t < dim R (λ p r−1 d ) = 2p r−1 d. It follows that the map R is an isomorphism, if q < 2 p r−1 d as stated.
Regular F p -algebras
Let k be a regular F p -algebra. The structure of the groups TR n q−λ (k; p) that occur on the right-hand side of the statement of Thm. B of the introduction is given by [12,Thm. 2.2.2], but see also [9,Thm. 11]. If λ is a finite dimensional complex T-representation, we define ℓ r = ℓ r (λ) = dim C (λ C p r ) and ℓ −1 = ∞ such that we have a descending sequence
∞ = ℓ −1 ℓ 0 ℓ 1 · · · ℓ r ℓ r+1 · · · ℓ ∞ = dim C (λ T ).
Then the following result is [12, Thm. 2.2.2].
Theorem 3.1. Let k be a regular F p -algebra, and let λ be a finite dimensional complex T-representation. There is a canonical isomorphism of abelian groups W s Ω q−2m k ∼ − → TR n q−λ (k; p) where the sum runs over all integers m ℓ ∞ , and where s = s(n, m, λ) is the unique integer such that ℓ n−s ≤ m < ℓ n−1−s . The group W s Ω j k is understood to be zero for non-positive integers s. Remark 3.2. It appears to be an important problem to determine the structure of the RO(T)-graded equivariant homotopy groups TR n α (k; p) = [S α ∧ (T/C p n−1 ) + , T (k)] T for a general virtual T-representation α. The precise definition of RO(G)-graded equivariant homotopy groups in given in [15,Appendix]. One might well hope that the RO(T)-graded equivariant homotopy groups TR n α (k; p) admit an algebraic description similar to that of the Z-graded equivariant homotopy groups TR n q (k; p) given in [8,Thm. B].
We can now complete the proof of Thm. A of the introduction. is an isomorphism for q < dim R (λ p n d ) = 2p n d. But for λ = λ p n d we have ℓ −1 = ∞ ℓ 0 = p n d ℓ 1 = p n−1 d ≥ · · · ℓ n = d ℓ n+1 = ℓ ∞ = 0.
Hence, Thm. 3.1 gives a canonical isomorphism of abelian groups where s = s(n, m, λ) is the minimum of n and the unique positive integer t that satisfies p t−1 d m < p t d. The statement follows.
Remark 3.3. We conclude this paper with a conjecture on the relationship of the K-groups of the rings k[x, y]/(xy) and k[t]/(t 2 ). The element f = x − y of the ring A = k[x, y]/(xy) is a non-zero-divisor with quotient ring A/f A = k[t]/(t 2 ). It follows that as an A-module A/f A has projective dimension 1, and hence we have a push-forward map on the associated K-groups i * : K q (k[t]/(t 2 )) → K q (k[x, y]/(xy)).
The additivity theorem implies that the image of the map i * is contained in the subgroup K q (A, I). In particular, we obtain an induced push-forward map i * : K q (k[t]/(t 2 ), (t)) → K q (k[x, y]/(xy), (x, y)).
For k a regular F p -algebra, the relative K-groups on the right and left-hand sides were evaluated in Thm. A and [12, Thm. A], respectively. On the one hand, there is a natural long-exact sequence of abelian groups · · · → K q (k
−→
TC q (A, B, I; p, Z/p v )
T
(A/I) T (B) / / T (B/I).
Proposition 1. 2 .
2Let k be any ring, let A = k[x, y]/(xy) and B = k[x] × k[y]
maps φ : T (A) → T (B), φ ′ : T (A) → T (B), and ǫ : T (A) → T (A/I) are all equal to the constant map. Hence, we obtain a canonical map of T-spectra T (k) ∧ N cy (Π 2 ,ω) → T (A, B, I) and the wedge-sum of these maps constitute the map of the statement. The diagram (1.3) and the F -equivalence (1.4) shows that this map is an F -equivalence.
the group C i cyclically permutes the i factors in the join on the left-hand side, and the map f ω collapses the join of a number of codimension 1 faces of ∆ s−1 to the base-point. If the period ofω is s = 2, then the map f ω collapses the whole boundary ∂∆ m−1 ⊂ ∆ m−1 to the base-point. Hence, in this case, we have a T-equivariant homeomorphismf ω : (∆ m−1 /∂∆ m−1 ) ∧ Ci T + ∼ − → N cy (Π 2 ,ω).The simplex ∆ m−1 embeds as the convex hull of the group elements in the regular representation R[C m ]. This identifies the C m -space ∆ m−1 /∂∆ m−1 with the onepoint compactification of the reduced regular representation R[C m ] − 1 as stated. This completes the proof of the statement for s = 2. If the period s > 2, then there exists a codimension 1 face F ⊂ ∆ s−1 that is not collapsed to the base-point. unreduced cone on the face F onto the simplex ∆ s−1 . The canonical nullhomotopy of the unreduced cone induces a C i -equivariant null-homotopy
·
· · → TC q (A, B, I; p) → TF q (A, B, I; p) R−id −−−→ TF q (A, B, I; p) → . . .
the other hand, there is a canonical isomorphism of abelian groups I : m 1 W m Ω q−2m k ∼ − → K q (k[x, y]/(xy), (x, y)).
Proof of Thm. A. It suffices by Thm. B to show that for positive integers d prime to p, there is a canonical isomorphism of abelian groups m 0 W s Ω q−2m k ∼ − → lim TR r q−λ p r−1 d (k; p) where s = s(m, d) is the unique integer that satisfies p s−1 d ≤ m < p s d. It follows from [11, Thm. 1.2] that the canonical projection lim TR r q−λ p r−1 d (k; p) → TR n q−λ p n−1 d (k; p)R
R
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| []
|
[
"Improving the estimation of the odds ratio using auxiliary information",
"Improving the estimation of the odds ratio using auxiliary information"
]
| [
"Camelia Goga :[email protected] \nInstitut de Mathématiques de Bourgogne\nUniversité de Bourgogne\nDijonFrance\n",
"Anne Ruiz-Gazen [email protected] \nToulouse School of Economics\nUniversité Toulouse 1 Capitole\nToulouseFrance\n"
]
| [
"Institut de Mathématiques de Bourgogne\nUniversité de Bourgogne\nDijonFrance",
"Toulouse School of Economics\nUniversité Toulouse 1 Capitole\nToulouseFrance"
]
| []
| The odds ratio measure is used in health and social surveys where the odds of a certain event is to be compared between two populations. It is defined using logistic regression, and requires that data from surveys are accompanied by their weights. A nonparametric estimation method that incorporates survey weights and auxiliary information may improve the precision of the odds ratio estimator. It consists in B-spline calibration which can handle the nonlinear structure of the parameter. The variance is estimated through linearization. Implementation is possible through standard survey softwares. The gain in precision depends on the data as shown on two examples. | null | [
"https://arxiv.org/pdf/1406.7691v1.pdf"
]
| 88,507,703 | 1406.7691 | 25e82ae6e325876b4ee1f20cf3f2f899b6cecd07 |
Improving the estimation of the odds ratio using auxiliary information
July 1, 2014
Camelia Goga :[email protected]
Institut de Mathématiques de Bourgogne
Université de Bourgogne
DijonFrance
Anne Ruiz-Gazen [email protected]
Toulouse School of Economics
Université Toulouse 1 Capitole
ToulouseFrance
Improving the estimation of the odds ratio using auxiliary information
July 1, 2014B-spline functionscalibrationestimating equationinfluence functionlin- earizationlogistic regression
The odds ratio measure is used in health and social surveys where the odds of a certain event is to be compared between two populations. It is defined using logistic regression, and requires that data from surveys are accompanied by their weights. A nonparametric estimation method that incorporates survey weights and auxiliary information may improve the precision of the odds ratio estimator. It consists in B-spline calibration which can handle the nonlinear structure of the parameter. The variance is estimated through linearization. Implementation is possible through standard survey softwares. The gain in precision depends on the data as shown on two examples.
Introduction
We study the use of nonparametric weights for estimating the odds ratio when the risk variable, which is the explanatory variable in the logistic regression, is either a continuous or a binary variable. The odds ratio is used to describe the strength of association or nonindependence between two binary variables defining two groups experiencing a particular event. One binary variable defines a group at risk and a group not at risk; the second binary variable defines the presence or absence of an event related to health. The odds ratio is the ratio of the odds of the event occurring in one group to the odds of the same event occurring in the other group. An odds ratio equal to 1 means that the event has the same odds in both groups; an odds ratio greater than 1 means that the event has a larger odds in the first group; an odds ratio under 1 means that the event has a smaller odds in the first group.
When both variables are categorical, the odds ratio estimator is obtained from a contingency table, as the ratio of the estimated row ratios, then, as a function of four numbers. As suggested by a reviewer, this definition leads to an estimator which takes survey weights into account and yields confidence intervals after linearization. However, this simple definition is not adapted to a continuous risk variable. In this case, the odds ratio measures the change in the odds for an increase of one unit in the risk variable. It is defined through the logistic regression.
For a binary risk variable, the odds ratio is the exponential of the difference of two logits, the logit function being the link function in the logistic regression. So the logistic regression coefficient for a binary risk variable corresponds to the logarithm of the odds ratio associated with this risk variable, net the effect of the other variables. When the risk variable is continuous, the regression coefficient represents the logarithm of the odds ratio associated with a change in the risk variable of one unit, net the effect of the other variables.
The regression coefficient is a solution of a population estimating equation using the theory developed in Binder (1983) for making inference. The sampling design must not be neglected especially for cluster sampling (Lohr, 2010). Korn and Graubard (1999) and Heeringa et al. (2010) give details and examples of estimating an odds ratio but ignore auxiliary information. Korn and Graubard (1999: 169-170) advocate the use of weighted odds ratios contrary to Eideh and Nathan (2006). Rao et al. (2002) suggest using post-stratification information to estimate parameters of interest obtained as solution of an estimating equation. The vector of parameters in the logistic regression is an example. Deville (1999) suggested "allocating a weight w k to any point in the sample and zero to any other point, regardless of the origin of the weights (Horvitz-Thompson or calibration)." Goga and Ruiz-Gazen (2014) use auxiliary information to estimate nonlinear parameters through nonparametric weights. The solutions of estimating equations are nonlinear but Goga and Ruiz-Gazen (2014) give no detail. Our project is the estimation of the odds ratio with auxiliary information.
In Section 2, we recall the definition of the odds ratio and express the B-spline calibration estimator. In Section 3, we use linearization to derive the asymptotic variance of the estimator under broad assumptions. We infer a variance estimator together with asymptotic normal confidence intervals. In Section 4, we draw guidelines for practical implementation and show the properties of our estimator on two case studies.
2 Estimation of the odds ratio with survey data 2.1 Definition of the parameter The odds ratio, denoted by OR, is used to quantify the association between the levels of a response variable Y and a risk variable X. The value taken by Y is y i and the value taken by X is x i for the i-th individual in a population U = {1, . . . , N }.
The logistic regression
logit(p i ) = log p i 1 − p i = β 0 + β 1 x i ,(1)where p i = P (Y = 1|X = x i ) implies that p i = exp(β 0 + β 1 x i )(1 + exp(β 0 + β 1 x i )) −1 .(2)
The odds ratio is (Agresti, 2002):
OR = odds(Y = 1|X = x i + 1) odds(Y = 1|X = x i ) = exp β 1 .(3)
With a binary variable X, the OR has a simpler form and can be derived from a contingency
OR = N 00 N 11 N 01 N 10 ,(4)
where N 00 , N 01 , N 10 , and N 11 are the population counts associated with the contingency
β = (β 0 , β 1 ) byβ = (β 0 ,β 1 ),
The regression parameters β 0 and β 1 are obtained by maximization of the population likelihood:
L(y 1 , . . . , y N ; β) = i∈U p y i i (1 − p i ) 1−y i .(6)
The maximum likelihood estimator of β satisfies:
i∈U (y i − p i ) = 0 ,(7)i∈U (y i − p i )x i = 0.(8)Let x i = (1 x i ) and µ(x i β) = exp(x i β)(1 + exp(x i β)) −1 .
We write Eq. (7) and (8) in the equivalent form
i∈U x i (y i − µ(x i β)) = 0 (9) or, with t i (β) = x i (y i − µ(x i β)), i∈U t i (β) = 0.(10)
The regression estimator of β is defined as an implicit solution of the estimating Eq. (9). We use iterative methods to compute it.
The B-spline nonparametric calibration
For s a sample selected from the population U according to a sample design p(·) , we denote by π i > 0 the probability of unit i to be selected in the sample and π ij > 0 the joint probability of units i and j to be selected in the sample with π ii = π i . We look for an estimator of β and of OR taking the auxiliary variable Z, with values z 1 , . . . , z N , into account. Deville and Särndal (1992) suggest deriving the calibration weights w ks as close as possible to the Horvitz-Thompson sampling weights d i = 1/π i while satisfying the calibration constraints on known totals Z:
i∈s w is z i = i∈U z i .(11)
This method works well for a linear relationship between the main and the auxiliary variables.
When this relationship is no longer linear, the calibration constraints must be changed while keeping the property that the obtained weights do not depend on the main variable. Basis functions that are more general than the ones defined by constants and z i , include B-splines, which are simple to use (Goga and Ruiz-Gazen, 2013), truncated polynomial basis functions, and wavelets.
B-spline functions
Spline functions are used to model nonlinear trends. A spline function of degree m with K interior knots is a piecewise polynomial of degree m − 1 on the intervals between knots, smoothly connected at knots.
The B-spline functions B 1 , . . . , B q of degree m with K interior knots, q = m + K are among the possible basis functions (Dierckx, 1993). Other basis functions exist such as the truncated power basis (Ruppert et al., 2003). For m = 1, the B-spline basis functions are step functions with jumps at the knots; for m = 2, they are piecewise linear polynomials, and so on. Figure 1 shows the six B-spline basis functions obtained for m = 3 and K = 3. Figure 2 gives the approximation of the curve f (x) = x + sin(4πx) taking the noisy data points into account and using the B-spline basis. Even if the function f is nonlinear, the B-spline approximation almost coincides with it. The user chooses the spline degree m and the total number K of knots. There is no general rule giving the total number of knots but Ruppert al. (2003) recommend m = 3 or m = 4 and no more than 30 to 40 knots. They also give a simple rule for choosing K (Ruppert et al., 2003: 126). Usually, the knots are located at the quantiles of the explanatory variable (Goga and Ruiz-Gazen, 2013).
Nonparametric calibration with B-spline functions
The B-splines calibration weights w b is are solution of the optimization problem:
(w b is ) i∈s = argmin w i∈s (w i − d i ) 2 q i d i (12) subject to i∈s w b is b(z i ) = i∈U b(z i ),(13)
where b(z i ) = (B 1 (z i ), . . . , B q (z i )) and q i is a positive constant. They are given by
w b is = d i 1 − q i b (z i )( i∈s d i q i b(z i )b (z i )) −1 (t b,d − t b )(14)witht b,d = i∈s d i b(z i ), t b = i∈U b(z i ).
The weights w b is depend only on the auxiliary variable and are similar to Deville and Särndal's weights. The calibration equation implies (Goga, 2005):
0.0 0.2 0.4 0.6 0.8 1.0 x B 1 (x) 0 K1 K2 K3 1 0.0 0.2 0.4 0.6 0.8 1.0 x B 2 (x) 0 K1 K2 K3 1 0.0 0.2 0.4 0.6 0.8 1.0 x B 3 (x) 0 K1 K2 K3 1 0.0 0.2 0.4 0.6 0.8 1.0 x B 4 (x) 0 K1 K2 K3 1 0.0 0.2 0.4 0.6 0.8 1.0 x B 5 (x) 0 K1 K2 K3 1 0.0 0.2 0.4 0.6 0.8 1.0 x B 6 (x) 0 K1 K2 K3 1i∈s w b is = N and i∈s w b is z i = i∈U z i . If q i = 1 for all i ∈ U, we obtainw b is = d i t b k∈s d k b(z k )b (z k ) −1 b(z i ).(15)
Goga and Ruiz-Gazen (2014) use these weights to estimate totals for variables, which are related nonlinearly to the auxiliary information and to estimate nonlinear parameters such as a Gini index. We use w b is to estimate the logistic regression coefficient and the odds ratio efficiently.
Estimation of OR using B-spline nonparametric calibration
The regression coefficient β is a nonlinear finite population function defined by the implicit Eq. (9). The functional method by Deville (1999), specified for the nonparametric case by Goga and Ruiz-Gazen (2014), is used to build a nonparametric estimator of β defined through the weights of Eq. (15). M is the finite measure assigning the unit mass to each y i , i ∈ U , and zero elsewhere:
M = i∈U δ y i(16)
where δ y i is the Dirac function at y i , δ y i (y) = 1 for y = y i and zero elsewhere. The functional
T defined with respect to the measure M and depending on the parameter β defined by
T (M ; β) = i∈U x i (y i − µ(x i β)).(17)
The regression coefficient β is the solution of the implicit equation
T (M ; β) = 0.(18)
Eq. (18) is called the score equation. The measure M may be estimated using the Horvitz-Thompson weights d k = 1/π k or the linear calibration weights (Deville, 1999). We suggest using the nonparametric weights derived in Eq. (15):
w b is = d i k∈U b(z k ) k∈s d k b(z k )b (z k ) −1 b(z i )(19)
and estimate M by
M = i∈s w b is δ y i .(20)
Plugging M into the functional expression of β given by Eq. (18) yields the B-spline calibrated estimator β of β:
T ( M ; β) = 0,(21)
which means that β is the solution of the implicit equation:
i∈s w b is x i (y i − µ(x i β)) = 0.(22)
The functional method allows us to incorporate auxiliary information for estimating the logistic regression coefficient and any parameter β defined as a solution of estimating equations.
The functional T is differentiable with respect to β and
∂T ∂β = − i∈U ν(x i β)x i x i = X Λ(β)X := J(β),(23)with X = (x i ) i∈U and Λ(β) = −diag(ν(x i β)) with ν(x i β) = µ(x i β)(1 − µ(x i β)) the derivative of µ.
The 2 × 2 matrix X Λ(β)X is invertible and J(β) is definite negative. From Eq. (23), the matrix J(β) is a total estimated using the nonparametric weights w b is by:
J w (β) = − i∈s w b is ν(x i β)x i x i = X s Λ(β)X s ,(24)
where Λ(β) = −diag(w b is ν(x i β)) i∈s and X s = (x i ) i∈s .
An iterative Newton-Raphson method is used to compute β. The r-th step of the Newton-Raphson algorithm is:
β r = β r−1 − J w ( β r−1 )T ( M ; β r−1 ),(25)
where β r−1 is the value of β obtained at the (r−1)-th step. J w ( β r−1 ) is the value of J w (β) and
T ( M ; β r−1 ) the value of T ( M ; β) evaluated at β = β r−1 . Iterating to convergence produces the nonparametric estimator β and the estimated Jacobian matrix J w ( β). The odds ratio is estimated by OR = exp(β 1 ) and J w ( β) is used in section 3 to estimate the variance ofβ.
3 Variance estimation and confidence intervals
Variance estimation
The coefficient β of the logistic regression is nonlinear and nonparametric weights w b is to estimate β add more nonlinearity. We approximate β in Eq. (21) by a linear estimator in two steps: we first treat the nonlinearity due to β, and second the nonlinearity due to the nonparametric estimation. This procedure is different from Deville (1999).
From the implicit function theorem, there exists a unique functional T such that
T (M ) = β and T ( M ) = β.(26)
Moreover, the functional T is also Fréchet differentiable with respect to M . The derivative of T with respect to M , called the influence function, is defined by
I T (M, ξ) = lim λ→0 T (M + λδ ξ ) − T (M ) λ ,(27)
where δ ξ is the Dirac function at ξ. We give a first-order expansion ofT in M /N around
M/N, T M N = T M N + +∞ −∞ I T M N , ξ d M N − M N (ξ) + o p (n −1/2 ),(28)
which is also:
T ( M ) = T (M ) + +∞ −∞ I T (M, ξ) d( M − M )(ξ) + o p (n −1/2 ),(29)= − (X Λ(β)X) −1 x i (y i − µ(x i β)) = −J −1 (β) · x i (y i − µ(x i β)).(30)
The linearized variable u i = (u i,0 , u i,1 ) is a two-dimensional vector depending on the unknown parameter β and on totals contained in the matrix J(β). Eq. (29) becomes:
β − β i∈s w b is u i − i∈U u i .(31)
The second component u i,1 of u i , is the linearized variable of β 1 . With binary data, the odds ratio is given by Eq. (4), which implies that ln(OR) = ln(N 00 ) + ln(N 11 ) − ln(N 01 ) − ln(N 10 ).
In this case, the linearized variable of β 1 has the expression:
u i,1 = 1 {x i =0,y i =0} N 00 + 1 {x i =1,y i =1} N 11 − 1 {x i =1,y i =0} N 10 − 1 {x i =0,y i =1} N 01(33)
and the same expression is obtained from Eq. (30) after some algebra. When the weights w b is are equal to the sampling weights, namely w b is = 1/π i , Eq. (31) implies that the asymptotic variance ofβ is:
AV(β) = Var i∈s d i u i = J −1 (β) V ht (t d (β)) J −1 (β),(34)
where V ht (t d (β)) is the Horvitz-Thompson variance oft d (β) = i∈s t i (β)/π i with t i (β) =
x i (y i − µ(x i β)): V ht (t d (β)) = Var i∈s t i (β) π i = i∈U i∈U (π ij − π i π j ) t i (β) π i t j (β) π j .(35)
Binder (1983) gives the same asymptotic expression for the variance.
For B-spline basis functions formed by step functions on intervals between knots (m = 1), the weights w b is yield the post-stratified estimator of β (Rao et al., 2002). Linear calibration weights lead to the case treated by Deville (1999).
For the general case of nonparametric calibration weights w b is , a supplementary linearization step is necessary. The right hand side of Eq. (31) is a nonparametric calibration estimator for the total of the linearized variable u i . It can be written as a generalized regression estimator (GREG):
i∈s w b is u i − i∈U u i = i∈s u i − θ u b(z i ) π i + i∈U θ u b(z i ) − i∈U u i ,(36)where θ u = ( i∈s d i b(z i )b (z i )) −1 ( i∈s d i b(z i )u i ).
We explain the linearized variable by means of a piecewise polynomial function. This fitting allows more flexibility and implies that the residuals u i − θ u b(z i ) have a smaller dispersion than with a linear fitting regression.
In order to derive the asymptotic variance of the nonparametric calibrated estimator, we assume that ||x i || < C for all i ∈ U with C a positive constant independent of i and N . The Euclidian norm is denoted || · ||. The matrix norm || · || 2 is defined by ||A|| 2 2 = tr(A A). The
linearized variable verifies N ||u i || = O(1) uniformly in i, because N ||u i || ≤ ||N J −1 (β)|| 2 ||x i || |y i − µ(x i β))| = O(1),(37)
where the Jacobian matrix J(β) contains totals
J(β) = − i∈U ν(x i β) i∈U x i ν(x i β) i∈U x i ν(x i β) i∈U x 2 i ν(x i β)(38)
and
1 N i∈U ν(x i β) 2 ≤ 1 N i∈U (ν(x i β)) 2 = O(1)(39)
because ν(x i β) < 1. Under the assumptions of theorem 7 in Goga and Ruiz-Gazen (2014),
the nonparametric calibrated estimator i∈s w b is u i is asymptotically equivalent to i∈s w b is u i − i∈U u i i∈s u i − θ u b(z i ) π i + i∈U θ u b(z i ) − i∈U u i ,(40)where θ u = ( i∈U b(z i )b (z i )) −1 i∈U b(z i )u i .
The variance ofβ is approximated by the Horvitz-Thompson variance of the residuals u i − θ u b(z i ),
AV(β) = Var i∈s u i − θ u b(z i ) π i = i∈U i∈U (π ij − π i π j ) u i − θ u b(z i ) π i u j − θ u b(z j ) π j .(41)
Eq. (40) states that the B-spline nonparametric calibration estimator of i∈U u i is asymptotically equivalent to the generalized difference estimator. We interpret this result as fitting a nonparametric model on the linearized variable u i taking into account the auxiliary information z i . Nonparametric models are a good choice when the linearized variable obtained from the first linearization step does not depend linearly on z i , as it is the case in the logistic regression, which implies a second linearization step.
We write the asymptotic variance in Eq. (41) in a matrix form similar to Eq. (34).
Consider again t i (β) = x i (y i − µ(x i β)) and let θ t = ( i∈s b(z i )b (z i )) −1 i∈s b(z i )t i (β). We have
u i − θ u b(z i ) = −J −1 (β) t i (β) − θ t b(z i ) ,(42)
and the asymptotic variance ofβ is:
AV(β) = J −1 (β) V ht (ê d (β)) J −1 (β) (43) whereê d (β) = i∈s e i (β) π i is the Horvitz-Thompson estimator of the residual e i (β) = t i (β) − θ t b(z i ) of t i (β) using B-spline calibration.
Eq. (43) shows that improving the estimation of β is equivalent to improving the estimation of the score equation
t i = x i (y i − µ(x i β)).
The quantity of interest is the asymptotic variance ofβ 1 . It is the (2, 2) element of the matrix AV(β) given by Eq. (41). We have u i = (u i,0 , u i,1 ) and
u i − θ u b(z i ) = u i,0 − θ u 0 b(z i ) u i,1 − θ u 1 b(z i )(44)
where
θ u 0 = ( i∈U b(z i )b (z i )) −1 i∈U b(z i )u i,0 and θ u 1 = ( i∈U b(z i )b (z i )) −1 i∈U b(z i )u i,1 . We obtain AV(β 1 ) = Var i∈s u i,1 − θ u 1 b(z i ) π i .(45)
The linearized variable u i is unknown and is estimated by:
u i = − J −1 w ( β) x i (y i − µ(x i β))(46)= − J −1 w ( β)t i(47)
where the matrix J w is computed according to Eq. (24) andt i is the estimation of t i (β)
for β = β. The asymptotic variance AV( β) given in Eq. (41) is estimated by the Horvitz-Thompson variance estimator with u i replaced byû i given in Eq. (46):
V ( β) = V ht i∈sû i − θ u b(z i ) π i (48) where θ u = ( i∈s d i b(z i )b (z i )) −1 i∈s d i b(z i )û i .
The variance estimator ofβ 1 is given bŷ
V (β 1 ) = Var i∈sû i,1 − θ û 1 b(z i ) π i .(49)( i∈s d i b(z i )b (z i )) −1 i∈s d i b(z i )t i and V ( β)
is written as:
V ( β) = J −1 w ( β) V ht (ê d ( β)) J −1 w ( β)(50)
where V ht (ê d ) is the Horvitz-Thompson variance estimator ofê d (β) obtained by replacing
e i (β) withê i (β) =t i − θ t b(z i ), V ht (ê(β)) = i∈s i∈s π ij − π i π j π ijê i (β) π iê j (β) π j .(51)
Confidence interval for the odds ratio
The variance estimator ofβ 1 is obtained from Eq. (50) as:
V (β 1 ) = J −1 w ( β) V ht (ê 2 ( β)) J −1 w ( β),(52)
whereê 2 ( β) is the second component ofê( β) so that, under regularity conditions, the (1−α)% normal interval for β 1 is:
CI 1−α (β 1 ) = β 1 − z α/2 V (β 1 ) 1/2 ,β 1 + z α/2 V (β 1 ) 1/2 ,(53)
where z α/2 is the upper α/2-quantile of a N (0, 1) variable. Then the confidence interval for OR is:
CI 1−α (OR) = exp β 1 − z α/2 V (β 1 ) 1/2 , exp β 1 + z α/2 V (β 1 ) 1/2 ,(54)
which is not symmetric around the estimated odds ratio but provides more accurate coverage rates of the true population value for a specified α (Heeringa et al., 2010).
4 Implementation and case studies 4.1 Implementation 1. Compute the B-spline basis functions B j , for j = 1, . . . , q. The B-spline basis functions are obtained using SAS or R. The user has only to specify the degree m and the total number of knots.
2. Use the sampling weights d i = 1/π i and the B-spline functions to derive the nonparametric weights w b is and the estimated parameter β.
3. Compute the linearized variable u i estimated byû i .
Compute the estimated predictions
θ u b(z i ) with θ u = i∈s d i b(z i )b (z i ) −1 i∈s d i b(z i ) u i(55)
and the associated residualsû i − θ u b(z i ).
5. Use a standard computer software able to compute variance estimators and apply it to the previously computed residuals.
Case studies
We compare the asymptotic variance of different estimators of the odds ratio in the simple case of one binary risk variable for two data sets. In this case, the odds ratio is a simple function of four counts given by Eq. (4). We focus on the simple random sampling without replacement and compare three estimators. The first one is the Horvitz-Thompson estimator which does not use the auxiliary variable and whose asymptotic variance is given by Eq. (34).
The second estimator is the generalized regression estimator which takes the auxiliary variable into account through a linear model fitting the linearized variable against the auxiliary variable. The third estimator is the B-spline calibration estimator with an asymptotic variance given by Eq. (43). In order to gain efficiency, the auxiliary variable is related to the linearized variable. In the context of one binary factor, the linearized variable is given by Eq. (33) and takes four different values, which depend on the values of the variables X and Y . In order to be related to the linearized variable, the auxiliary variable is related to the product of the two variables X and Y , which is a strong property. Moreover, because u i,1 , X, and Y are discrete, using auxiliary information does not necessarily lead to an important gain in efficiency as the first health survey example will show. The gain in efficiency however is significant in some cases. In the second example using labor survey data, the gain in using the B-splines calibration estimator compared to the Horvitz-Thompson estimator is significant because the auxiliary variable is related to the variable Y but also to the factor X; X and Y being related to one another, too.
Example from the California Health Interview Survey
The data set comes from the Center for Health Policy Research at the University of California. It was extracted from the adult survey data file of the California Health Interview Survey in 2009 and consists of 11074 adults. The response dummy variable equals one if the person is currently insured; the binary factor equals one if the person is currently a smoker.
The auxiliary variable is age and we consider people who are less than 60 years old. The data are presented in detail in Lumley (2010).
We compare the Horvitz-Thompson, the generalized regression, and the B-splines calibration estimators in terms of asympotic variance. In order to calculate the B-splines functions, we use the SAS procedure transreg and take K = 15 knots and B-splines of degree m = 3.
The gain in using the generalized regression estimator compared to the Horvitz-Thompson estimator is only 0.01%. It is 1.5% when using B-splines instead of the generalized regression. When changing the number of knots and the degree of the B-spline functions, the results remain similar and the gain remains under 2%. In this example, there is no gain in using auxiliary information even with flexible B-splines, because the auxiliary variable is not related enough to the linearized variable. The linearized variable takes negative values for smokers without insurance and non smokers with insurance, positive values for smokers with insurance and non smokers without insurance. Age is not a good predictor for this variable, because we expect to find sufficient people of any age in each of the four categories (smokers/non smokers × insurance/no insurance). Incorporating this auxiliary information brings no gain.
Example from the French Labor Survey
We consider 14621 wage-earners under 50 years of age, from the French labour force survey. The initial data set consists of monthly wages in 2000 and 1999. A dummy variable W00 equals one if the monthly wage in 2000 exceeds 1500 euros and zero otherwise. The same for W99 in 1999. The population is divided in lower and upper education groups. The value of the categorical factor DIP equals one for people with a university degree and zero otherwise. W00 corresponds to the binary response variable Y while the diploma variable DIP corresponds to the risk variable X. The variable W99 is the auxiliary variable Z.
To compare the Horvitz-Thompson estimator with the generalized regression estimator and the B-splines calibration estimator, we calculate the gain in terms of asympotic variance.
We consider K = 15 knots and the degree m = 3. The gain in using the generalized estimator compared to the Horvitz-Thompson estimator is now 20%. It is 33% when using B-splines.
The result is independent of the number of knots and, of the degree of B-spline functions.
When the total number of knots varies from 5 to 50 and the degree varies from 1 to 5, the gain is between 32% and 34%. The nonlinear link between the linearized variable of a complex parameter with the auxiliary variable explains the gain in using a nonparametric estimator compared to an estimator based on a linear model (Goga and Ruiz-Gazen, 2013).
For the odds ratio with one binary factor, the linearized variable is discrete and the linear model does not fit the data.
Conclusion
Estimating the variance of parameter estimators in a logistic regression is not straightforward especially if auxiliary information is available. We applied the method of Goga and
where x denotes the transpose of x. Eq. (3) yields the estimator of OR: OR = expβ 1 .
Figure 1 :Figure 2 :
12B-spline basis functions with K = 3 interior knots and m = 3. B-spline approximation of f (x) = x + sin(4πx) with K = 3 interior knots and m = 3. The crosses correspond to the noisy data. The solid line is the true function f ; the dashed line is the B-spline approximation.
because T is a functional of degree zero, namely T (M/N ) = T (M ) and I T (M/N, ξ) = N I T (M, ξ) (Deville, 1999). For all i ∈ U , the linearized variable u i of T (M ) = β is defined as the value of the influence function I T at ξ = y i : u i = I T (M, y i ) = − ∂T ∂β −1 IT (M, y i ; β)
table . The
.OR is equal to
table .
.In order to estimate the OR of Eq. (3), we estimate first the regression coefficient
The asymptotic variance of the estimator incorporates residuals of the model that we assume between the linearized variable and the auxiliary variable. The gain in using auxiliary information is thus based on the fitting quality of the model for the linearized variable. Because of the complexity of linearized variables, linear models that incorporate auxiliary information seldom fit linearized variables and we use nonparametric B-spline estimators. A particular case is post-stratification. Ruiz-Gazen, The method relies on a linearization principle. Using the influence function defined by Eq. (27), we derive the asymptotic variance of the estimatorsRuiz-Gazen (2014) to the case of parameters defined through estimating equations. The method relies on a linearization principle. The asymptotic variance of the estimator in- corporates residuals of the model that we assume between the linearized variable and the auxiliary variable. The gain in using auxiliary information is thus based on the fitting quality of the model for the linearized variable. Because of the complexity of linearized variables, linear models that incorporate auxiliary information seldom fit linearized variables and we use nonparametric B-spline estimators. A particular case is post-stratification. Using the influence function defined by Eq. (27), we derive the asymptotic variance of the estimators
Asymptotic integrated mean square error using least squares and bias minimizing splines. G G Agarwal, W J Studden, The Annals of Statistics. 8Agarwal, G. G. and Studden, W. J. (1980), Asymptotic integrated mean square error using least squares and bias minimizing splines. The Annals of Statistics, 8: 1307-1325.
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Variance estimation for complex statistics and estimators: linearization and residual techniques. J.-C Deville, Survey Methodology. 25Deville, J.-C. (1999). Variance estimation for complex statistics and estimators: lineariza- tion and residual techniques. Survey Methodology, 25: 193-203.
Calibration estimation in survey sampling. J.-C Deville, C.-E Särndal, Journal of the American Statistical Association. 418Deville, J.-C. and Särndal, C.-E. (1992). Calibration estimation in survey sampling. Jour- nal of the American Statistical Association, 418: 376-382.
Curves and Surfaces Fitting with Splines. P Dierckx, Clarendon PressUnited KingdomDierckx, P. (1993). Curves and Surfaces Fitting with Splines. United Kingdom: Clarendon Press.
The analysis of data from sample surveys under informative sampling. A A H Eideh, G Nathan, Acta et Commentationes Universitatis Tartuensis de Mathematica. 10Eideh, A. A. H. and Nathan, G. (2006). The analysis of data from sample surveys under informative sampling. Acta et Commentationes Universitatis Tartuensis de Mathe- matica, 10: 1-11.
Réduction de la variance dans les sondages en présence d'information auxiliaire : une approche nonparamétrique par splines de régression. C Goga, The Canadian Journal of Statistics/Revue Canadienne de Statistique. 332Goga, C. (2005). Réduction de la variance dans les sondages en présence d'information auxiliaire : une approche nonparamétrique par splines de régression. The Canadian Journal of Statistics/Revue Canadienne de Statistique, 33(2): 1-18.
Efficient estimation of nonlinear finite population parameters using nonparametrics. C Goga, A Ruiz-Gazen, Journal of the Royal Statistical Society series B. 76Goga, C. and Ruiz-Gazen, A. (2014). Efficient estimation of nonlinear finite population parameters using nonparametrics. Journal of the Royal Statistical Society series B, 76, 113-140.
Applied Survey Data Analysis. S G Heeringa, B T West, P A Berglund, Chapman and Hall/CRCHeeringa, S. G., West, B. T., and Berglund, P. A. (2010). Applied Survey Data Analysis. Chapman and Hall/CRC.
A generalization of sampling without replacement from a finite universe. D G Horvitz, D J Thompson, Journal of the American Statistical Association. 47Horvitz, D .G. and Thompson, D. J. (1952). A generalization of sampling without re- placement from a finite universe. Journal of the American Statistical Association, 47: 663-685.
Analysis of Health Survey. E L Korn, B I Graubard, John WileyNew YorkKorn, E. L. and Graubard, B. I. (1999). Analysis of Health Survey. New York: John Wiley.
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Estimating equations for the analysis of survey data using post-stratification information. J N K Rao, W Yung, M A Hidiroglou, Sankhya: The Indian Journal of Statistics. 64Rao, J. N. K., Yung, W., and Hidiroglou, M. A. (2002). Estimating equations for the analysis of survey data using post-stratification information. Sankhya: The Indian Journal of Statistics, 64: 364-378.
D Ruppert, M P Wand, Caroll , R J , Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics. New YorkCambridge University PressRuppert, D., Wand, M. P., and Caroll, R.J. (2003). Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics. New York: Cambridge University Press.
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